Plenary lectures
On the evolution of the theory of regular polyhedra and polytopes
Asia Ivić Weiss
York University, Toronto, Canada
We give an overview of the theory of regular polytopes and introduce the concept of a hypertope (a generalization of a polytope which in turn generalizes the concept of a map and hypermap to higher rank objects). Of particular interest are locally spherical hypertopes which can be of spherical, euclidean, or hyperbolic type. We investigate finite hypertopes arising from them, giving some examples and summarizing known results.
From average sampling to Whittaker sampling
Tibor K. Pogány
Sveučilište u Rijeci, Hrvatska
Alkalmazott Matematika Intézet, Óbudai Egyetem, Budapest, Magyarország
Alkalmazott Matematika Intézet, Óbudai Egyetem, Budapest, Magyarország
(joint work with Andriy Ya. Olenko and Zurab A. Piranashvili)
In the first part of the talk the harmonizable Piranashvili-type stochastic processes are approximated by finite
time shifted average sampling sums. Explicit truncation error upper bounds are established.
The Whittaker-type derivative sampling reconstruction formula was established by J. R. Higgins for deterministic signals. In [4] Higgins' result is established by another method and it is extended for the stochastic process class \(L^\alpha(\Omega);\; \alpha \in [0, 2]\) in the \(\alpha\)-mean and in the almost sure sense, when the input processes possess spectral representation. The \((p, q)\)-order weighted differential operator's Whittaker-Higgins sampling reconstruction formula is also given for entire functions coming from Leont'ev functions space \(\mathfrak S_{[2, \pi \psi/2]}, \psi>0\) and the Piranashvili stochastic processes, applying the circular truncation error's upper bound, which vanishes with exponential rate.
References:
The Whittaker-type derivative sampling reconstruction formula was established by J. R. Higgins for deterministic signals. In [4] Higgins' result is established by another method and it is extended for the stochastic process class \(L^\alpha(\Omega);\; \alpha \in [0, 2]\) in the \(\alpha\)-mean and in the almost sure sense, when the input processes possess spectral representation. The \((p, q)\)-order weighted differential operator's Whittaker-Higgins sampling reconstruction formula is also given for entire functions coming from Leont'ev functions space \(\mathfrak S_{[2, \pi \psi/2]}, \psi>0\) and the Piranashvili stochastic processes, applying the circular truncation error's upper bound, which vanishes with exponential rate.
References:
- A. Ya. Olenko, T. K. Pogány. Time shifted aliasing error upper bounds for truncated sampling cardinal series. J. Math. Anal. Appl. 324(2006), No. 1, 262-280.
- A. Ya. Olenko, T. K. Pogány. Average sampling restoration of harmonizable processes. Commun. Stat., Theory Methods 40 (2011), No. 19-20, 3587-3598.
- Z. A. Piranashvili Z.A., T. K. Pogány. Some new generalizations of the Kotel'nikov-Shannon formula for stochastic signals. Probability Theory and Mathematical Statistics Conference Dedicated to 100th Annyversary of A.N.Kolmogorov, held at Tbilisi, Georgia, September 21-27, 2003.
- T. K. Pogány. Whittaker-type derivative sampling reconstruction of stochastic \(L^\alpha(\Omega)\)-processes. Appl. Math. Comput. 187 (2007), No. 1, 384-394.
- Z. A. Piranashvili, T. K. Pogány. {\it On generalized derivative sampling series expansion}. In Dutta H., Kočinac Lj., Srivastava H.M. (Eds.) Current Trends in Mathematical Analysis and its Interdisciplinary Applications, Birkhäuser Verlag, Springer Basel AC, 2019, 491-519.
- T. K. Pogány. Whittaker-type derivative sampling and \((p, q)\)-order weighted differential operator. In S. D. Casey, M. M. Dodson, P. J. S. G. Ferreira, A. Zayed (Eds.) Sampling, Approximation, and Signal Analysis (Harmonic Analysis in the Spirit of J. Rowland Higgins), 2022, (to appear).
Restricting representations of general linear groups
Gordan Savin
University of Utah, Salt Lake City, USA
A fundamental problem in representation theory is decomposing a representation of a group into
indecomposable summands. One class of such problems arises when we restrict irreducible representations of a larger group to
a smaller group, for example, we can take the larger group to be the general linear group \(GL_{n+1}(F)\) and the smaller \(GL_n(F)\), where \(F\) is a field.
In this lecture I will explain a homological variant of this problem due to Dipendra Prasad, and progress on it from a work with Kei Yuen Chan, in the case
when \(F\) is a \(p\)-adic field.
Time permitting I may touch a similar problem for orthogonal groups addressed in a work with Petar Bakić.
Wavelets and MRA Filters
Hrvoje Šikić
Department of Mathematics, Faculty of Science, University of Zagreb
We will present some results from the newly developed theory of wavelets; based on the joint
work with the following authors:
\(\left[ 1 \right] \) P.M.Luthy, H.Šikić, F.Soria, G.L.Weiss, E.N.Wilson. One-Dimensional Dyadic Wavelets. To appear in the Memoir of the AMS, vii + 152 pages.
For the last three decades the research of our group has been highly influenced by the connection between wavelets and Fourier analysis as outlined in the influential book by E.Hernandez and G.Weiss. The approach was to study wavelets and other reproducing function systems via the four basic equations that characterized various properties of wavelet systems, like frame and basis properties, completeness, orthogonality, etc. Despite hundreds of research papers and the impressive development of the theory of such systems, some questions remain open even in the basic case of dyadic wavelets on the real line. In \(\left[ 1 \right]\) we address several of these questions. Among the most thourough treatments that we provide in this lengthy paper is the issue of the understanding of the low-pass filters associated with the MRA structure. In this talk we will focus on some of these results. As it turned out, a more general and abstract approach to the problem, using the study of dyadic orbits and newly introduced Tauberian function, reveals several interesting properties and opens an interesting context for some older results. We shall compare our reults with older results achieved by Cohen, Lawton, Gundy, and many others, including some members of our group.
Guido Weiss, who recently passed away, was the plenary lecturer for the Third Croatian Math Congress, also in Split; it was the first congress held outside of Zagreb, and as such of a particular importance for the future of Croatian mathematical congresses.
\(\left[ 1 \right] \) P.M.Luthy, H.Šikić, F.Soria, G.L.Weiss, E.N.Wilson. One-Dimensional Dyadic Wavelets. To appear in the Memoir of the AMS, vii + 152 pages.
For the last three decades the research of our group has been highly influenced by the connection between wavelets and Fourier analysis as outlined in the influential book by E.Hernandez and G.Weiss. The approach was to study wavelets and other reproducing function systems via the four basic equations that characterized various properties of wavelet systems, like frame and basis properties, completeness, orthogonality, etc. Despite hundreds of research papers and the impressive development of the theory of such systems, some questions remain open even in the basic case of dyadic wavelets on the real line. In \(\left[ 1 \right]\) we address several of these questions. Among the most thourough treatments that we provide in this lengthy paper is the issue of the understanding of the low-pass filters associated with the MRA structure. In this talk we will focus on some of these results. As it turned out, a more general and abstract approach to the problem, using the study of dyadic orbits and newly introduced Tauberian function, reveals several interesting properties and opens an interesting context for some older results. We shall compare our reults with older results achieved by Cohen, Lawton, Gundy, and many others, including some members of our group.
Guido Weiss, who recently passed away, was the plenary lecturer for the Third Croatian Math Congress, also in Split; it was the first congress held outside of Zagreb, and as such of a particular importance for the future of Croatian mathematical congresses.
The unitary dual problem
David Vogan
Cambridge, MA, USA
(joint work with Jeffrey Adams, Marc van Leeuwen, and Stephen Miller)
A fundamental problem in abstract harmonic analysis is the unitary dual problem: given a group \(G\), what are the possible ways that \(G\) can act by unitary operators on a Hilbert space?
Understanding this as a fundamental problem goes back to work of Gelfand almost a hundred years ago; I will recall a little about that background.
I will talk about work over the past twenty years by a collaboration Atlas of Lie Groups and Representations, aimed at creating computer software able to answer this question.
Sophus Lie's classification of simple Lie groups divides them into two kinds: the classical groups, related to linear algebra, to which all kinds of combinatorial and algebraic tools may be applied; and the exceptional groups, a finite set which can if necessary be studied using case-by-case calculations. Work on the unitary dual problem has generally followed this division.
Understanding this as a fundamental problem goes back to work of Gelfand almost a hundred years ago; I will recall a little about that background.
I will talk about work over the past twenty years by a collaboration Atlas of Lie Groups and Representations, aimed at creating computer software able to answer this question.
Sophus Lie's classification of simple Lie groups divides them into two kinds: the classical groups, related to linear algebra, to which all kinds of combinatorial and algebraic tools may be applied; and the exceptional groups, a finite set which can if necessary be studied using case-by-case calculations. Work on the unitary dual problem has generally followed this division.
Control and Machine Learning
Enrique Zuazua
Friedrich Alexander Universität Erlangen Nürnberg - Alexander von Humboldt Professorship, Germany
Fundación Deusto, Bilbao
Universidad Autónoma de Madrid
Fundación Deusto, Bilbao
Universidad Autónoma de Madrid
In this lecture we shall present some recent results on the interplay between control and Machine Learning, and more precisely, Supervised Learning and Universal Approximation.
We adopt the perspective of the simultaneous or ensemble control of systems of Residual Neural Networks (ResNets). Roughly, each item to be classified corresponds to a different initial datum for the Cauchy problem of the ResNets, leading to an ensemble of solutions to be driven to the corresponding targets, associated to the labels, by means of the same control.
We present a genuinely nonlinear and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies.
This property is rarely fulfilled by the classical dynamical systems in Mechanics and the very nonlinear nature of the activation function governing the ResNet dynamics plays a determinant role. It allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill.
The turnpike property is also analyzed in this context, showing that a suitable choice of the cost functional used to train the ResNet leads to more stable and robust dynamics.
This lecture is inspired in joint work, among others, with Borjan Geshkovski (MIT), Carlos Esteve (Cambridge), Domènec Ruiz-Balet (IC, London) and Dario Pighin (Sherpa.ai).
We adopt the perspective of the simultaneous or ensemble control of systems of Residual Neural Networks (ResNets). Roughly, each item to be classified corresponds to a different initial datum for the Cauchy problem of the ResNets, leading to an ensemble of solutions to be driven to the corresponding targets, associated to the labels, by means of the same control.
We present a genuinely nonlinear and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies.
This property is rarely fulfilled by the classical dynamical systems in Mechanics and the very nonlinear nature of the activation function governing the ResNet dynamics plays a determinant role. It allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill.
The turnpike property is also analyzed in this context, showing that a suitable choice of the cost functional used to train the ResNet leads to more stable and robust dynamics.
This lecture is inspired in joint work, among others, with Borjan Geshkovski (MIT), Carlos Esteve (Cambridge), Domènec Ruiz-Balet (IC, London) and Dario Pighin (Sherpa.ai).
Invited lectures
Minimal dynamical systems with closed relations
Iztok Banič
University of Maribor, Slovenia
(joint work with Goran Erceg, Rene Gril Rogina and Judy Kennedy)
We introduce the dynamical systems \((X,G)\) with closed relations \(G\) on compact metric spaces \(X\) and discuss different types of minimality of such dynamical systems, all of them generalizing minimal dynamical systems \((X,f)\) with continuous function \(f\) on a compact metric space \(X\).
Transcendence and continued fraction expansion of values of Hecke-Mahler series
Yann Bugeaud
Université de Strasbourg, France
(joint work with Michel Laurent)
Let \(\theta\) be an irrational real number in \((0, 1)\).
It is well-known that the characteristic Sturmian word of slope \(\theta\) is the limit
of a sequence of finite words \((M_k)_{k \ge 0}\), with \(M_k\) of length \(q_k\) (the denominator of the \(k\)-th
convergent to \(\theta\)) being a suitable concatenation of copies of \(M_{k-1}\) and one copy of \(M_{k-2}\).
We extend this to any Sturmian word.
Let \(b \ge 2\) be an integer.
As a first application, we give the continued fraction expansion of any real number \(\xi\) whose \(b\)-ary
expansion is a Sturmian word \({\bf x}\) over the alphabet \(\{0, b-1\}\). This
extends a classical result of Böhmer (1927) who considered only the case where \({\bf x}\) is characteristic.
Consequently, we obtain a formula for the irrationality exponent of \(\xi\)
in terms of the slope and the intercept of \({\bf x}\).
As a second application, we show that, for any real number \(\rho\) in \([0, 1)\), the Hecke-Mahler series
$$
F_{\theta, \rho} (z_1, z_2)
= \sum_{k_1 \ge 1} \, \sum_{k_2 = 1}^{\lfloor k_1 \theta + \rho \rfloor} \, z_1^{k_1} z_2^{k_2},
$$
where \(\lfloor \cdot \rfloor\) denotes the integer part function,
takes transcendental values at any algebraic point \((\beta, \alpha)\) with
\(0 < |\beta|, |\beta \alpha^\theta | < 1\).
Integer partitions and characters of affine Lie algebras
Jehanne Dousse
CNRS and Université Lyon 1
(joint work with Isaac Konan)
A partition of a positive integer \(n\) is a non-increasing sequence of positive integers whose sum is \(n\). In the 1980's, Lepowsky and Wilson established a connection between the Rogers-Ramanujan partition identities and representation theory of the affine Lie algebra \(A_1^{(1)}\). Other representation theorists have then discovered new identities yet unknown to combinatorialists.
After presenting the history of the interactions between the two fields, we will introduce a new generalisation of partitions which is well-suited to make the connection with representation theory, and show how it can be used to prove refined partition identities and non-specialised character formulas.
Nonautonomous linearization: survey of some recent results
Davor Dragičević
Faculty of Mathematics, University of Rijeka, Croatia
I will survey a number of recent results concerned with both topological and smooth linearization of several classes of nonautonomous dynamical systems. The talk will be based on joint works with Lucas Backes, Ken Palmer, Weinian Zhang and Wenmeng Zhang.
References:
References:
- L. Backes and D. Dragičević, Smooth linearization of nonautonomous coupled systems, preprint, arXiv:2202.12367
- L. Backes, D. Dragičević and K. Palmer, Linearization and Hölder Continuity for Nonautonomous Systems, J. Differential Equations 297 (2021), 536-574.
- D. Dragičević, W. Zhang and W. Zhang, Smooth linearization of nonautonomous difference equations with a nonuniform dichotomy, Math. Z. 292 (2019), 1175-1193.
- D. Dragičević, W. Zhang and W. Zhang, Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy, Proc. Lond. Math. Soc. 121 (2020), 32-50.
Monogenity and power integral bases
István Gaál
University of Debrecen,Debrecen, Hungary
Monogenity and power integral bases is a classical topic of algebraic number theory.
Recently there are several results in this area.
In our survey we describe the most important ineffective, effective and
constructive/algorithmic results and tools, the most important methods
that can be applied to prove monogenity or non-monogenity of number fields
and certain classes of number fields.
High Order Approximations of the Operator Lyapunov Equation Have Low Rank
Luka Grubišić
Department of Mathematics, Faculty of Science, University of Zagreb
(joint work with Harri Hakula)
The feature underpinning the modern data driven approximation is the low rank structure of the solution (manifold) caused by the high order regularity of the dependence of the solution on the model parameters.
We present a low-rank greedily adapted hp-finite element algorithm for computing an approximation to the solution of the Lyapunov operator equation \(AX +XA = bb^*\). In the case in which the coefficient \(A\) is self-adjoint and positive definite, the Lyapunov equation has the unique positive and self-adjoint solution \(X\). We interpret the problem of finding the low rank approximation of \(X\) as the problem of approximating the dominant eigenvalue cluster of a bounded self-adjoint operator. We show that there is a hidden regularity in eigenfunctions of the solution of the Lyapunov equation which can be utilized to justify the use of high order finite element spaces.
We test our methods on several benchmark problems which test the influence of the critical restrictions of the theorems. For instance, to study the influence of the lower elliptic regularity we use an example of the Lyapunov equation whose coefficient is a Laplace operator defined on the dumbbell domain (two separate identical squares connected by a small bridge). Our numerical experiments indicate that we achieve eight figures of accuracy for computing the trace of the solution of the Lyapunov equation posed in this dumbbell-domain using a finite element space of dimension of only ten thousand degrees of freedom. Even more surprising is the observation that hp-refinement has an effect of reducing the rank of the approximation of the solution.
We present a low-rank greedily adapted hp-finite element algorithm for computing an approximation to the solution of the Lyapunov operator equation \(AX +XA = bb^*\). In the case in which the coefficient \(A\) is self-adjoint and positive definite, the Lyapunov equation has the unique positive and self-adjoint solution \(X\). We interpret the problem of finding the low rank approximation of \(X\) as the problem of approximating the dominant eigenvalue cluster of a bounded self-adjoint operator. We show that there is a hidden regularity in eigenfunctions of the solution of the Lyapunov equation which can be utilized to justify the use of high order finite element spaces.
We test our methods on several benchmark problems which test the influence of the critical restrictions of the theorems. For instance, to study the influence of the lower elliptic regularity we use an example of the Lyapunov equation whose coefficient is a Laplace operator defined on the dumbbell domain (two separate identical squares connected by a small bridge). Our numerical experiments indicate that we achieve eight figures of accuracy for computing the trace of the solution of the Lyapunov equation posed in this dumbbell-domain using a finite element space of dimension of only ten thousand degrees of freedom. Even more surprising is the observation that hp-refinement has an effect of reducing the rank of the approximation of the solution.
Approximate maximum likelihood estimation of drift parameters in an ergodic diffusion model
Miljenko Huzak
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
A simple approximate maximum likelihood method of estimation of nonlinear drift parameters based on
discrete observations of an ergodic diffusion path is analyzed asymptotically. In case of an
equidistant sampling such that the maximal time of observation tends to infinity and size of time
interval subdivision tends to zero in a way that their product tends to zero too, it is proved that
the difference between approximate maximum likelihood estimator (AMLE) and maximum likelihood
estimator based on continuous observation (MLE) of the same path and over the same time interval, goes
to zero with rate equals to square root of time subdivision size. This property can be used for (1.)
proving that AMLE is consistent and asymptotically efficient
estimator and normally distributed, (2.) proving its local mixed normality, and (3.) for estimation
of its standard error that accounts effect of discretization.
On Banach space isometries with finite spectrum
Dijana Ilišević
University of Zagreb
(joint work with Fernanda Botelho, Chih-Neng Liu and Ngai-Ching Wong)
This talk is related to the following problem: when is a given finite set of modulus one complex numbers the spectrum of a linear isometry on a complex Banach space? Necessary conditions on such a set will be presented. The problem of determining sufficient conditions seems to be much more complicated and related to the structure of specific Banach spaces. A particular emphasis will be given to \(C_0(\Omega)\), the Banach space of continuous complex-valued functions on a connected locally compact Hausdorff space \(\Omega\) vanishing at infinity.
A generalized approach to differentiability
Nikola Koceić-Bilan
Faculty of science, University of Split, Croatia
The differentiability of scalar and vector functions of multiple variables is
defined only at the interior points of the domain of these functions, which
results in the traditional consideration of functions only with an open domain
in \(\mathbb{R}^{n}\). This significantly narrows the possibility of applying
potential techniques and tools of differential calculus to a wider class of
functions. Although there is a strong need for it in various problems of
mathematical analysis and other mathematical branches, so far, the notion of
differentiability of a function has not been considered or successfully
defined at points outside the interior of domain of a function. In this talk,
we will define the differentiability at all points of a domain \(X\subseteq
\mathbb{R}^{n}\) of a function \(f:X\rightarrow\mathbb{R}^{m}\) in which that
notion makes sense. These are the points that admit neighborhood ray in \(X\) which is the minimum condition for the notion of linearization of a function (the essential property of differentiable functions) to make sense. In such a way, the notion of differentiability is significantly expanded, leading to a new theory of differentiable functions that offers completely unexpected phenomena and pathologies (such as the non-uniqueness of differentials, the discontinuity of differentiable functions...), but also reveals some common misconceptions. However, if one reduces this theory only to the points with particularly special properties (points that admit raylike neighborhood and a linearization space with dimensions equal to the dimension of the Euclidean space to which the domain belongs), then all properties and assertions of the extended theory remain the same. Moreover, all known theorems and techniques of the differential calculus can be successfully generalized and support the new theory, whereby the derivatives in the direction of the chosen vectors take over the role of partial derivatives. This is especially important for the functions which are differentiable at the point where there are no partial derivatives of them. If \(P\in X\subseteq\mathbb{R}^{n}\) admits neighbourhood ray in \(X\) in the direction of some \(n\) linear independent vectors in \(\mathbb{R}^{n}\) we will investigate under which conditions the existence of derivatives in the direction of those vectors at the point \(P\) implies the differentiability of a function \(f:X\rightarrow\mathbb{R}^{m}\) at \(P\) .
Several new characterizations of inner product spaces with regard to the \(p\)-angular distance
Mario Krnić
University of Zagreb, Faculty of Electrical Engineering and Computing, Zagreb, CROATIA
(joint work with Nicusor Minculete)
In this talk we derive some new bounds for the \(p\)-angular distance
$$\alpha_p[x,y]=\big\Vert \Vert x\Vert^{p-1}x- \Vert y\Vert^{p-1}y\big\Vert$$ in a normed linear space \(X\)
and consequently, we establish the corresponding new characterizations of inner product spaces. Our first goal is to prove a characterization of an inner space which looks familiar to the Hile inequality, although it shows significantly different behavior since the Hile inequality holds in every normed space. More precisely, we prove that if \(|p|\geq |q|\), \(p\neq q\), then \(X\) is an inner product space if and only if for every \(x,y\in X\setminus \{0\}\),
$${\alpha_p[x,y]}\geq \frac{{\|x\|^{p}+\|y\|^{p} }}{\|x\|^{q}+\|y\|^{q} }\alpha_q[x,y].$$
Next, we establish several new characterizations of an inner space based on the Maligranda bounds for the angular distance. Finally, our results are compared with some previously known results from the literature.
Fluid - Poroelastic Structure Interactions
Boris Muha
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with L. Bociu, M. Bukač, S. Čanić and J. Webster)
We consider the interaction between an incompressible, viscous fluid modeled by the dynamic Stokes equation and a multilayered poroelastic structure which consists of a thin, linear, poroelastic plate layer (in direct contact with the free Stokes flow) and a thick Biot layer. The fluid flow and the elastodynamics of the multilayered poroelastic structure are fully coupled across a fixed interface through physical coupling conditions (including the Beavers-Joseph-Saffman condition).
We prove existence of weak solutions to this fluid-structure interaction problem with either (i) a linear, dynamic Biot model, or (ii) a nonlinear quasi-static Biot component, where the permeability is a nonlinear function of the fluid content (as motivated by biological applications). The proof is based on constructing approximate solutions through Rothe's method, and using energy methods and a version of Aubin-Lions compactness lemma (in the nonlinear case) to recover the weak solution as the limit of approximate subsequences. We also provide uniqueness result for the linear problem and a weak-strong uniqueness type of result for the nonlinear problem.
Finally, we discuss some further direction of research such as diffuse interface methods and moving boundary fluid - poroelastic structure interaction problems.
We prove existence of weak solutions to this fluid-structure interaction problem with either (i) a linear, dynamic Biot model, or (ii) a nonlinear quasi-static Biot component, where the permeability is a nonlinear function of the fluid content (as motivated by biological applications). The proof is based on constructing approximate solutions through Rothe's method, and using energy methods and a version of Aubin-Lions compactness lemma (in the nonlinear case) to recover the weak solution as the limit of approximate subsequences. We also provide uniqueness result for the linear problem and a weak-strong uniqueness type of result for the nonlinear problem.
Finally, we discuss some further direction of research such as diffuse interface methods and moving boundary fluid - poroelastic structure interaction problems.
The effective boundary condition on a porous wall
Igor Pažanin
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with Eduard Marušić-Paloka)
The aim of this talk is to present the derivation of the new effective boundary condition for the fluid flow in a domain with porous boundary. Starting from the Stokes system in a domain with an array of small holes on the boundary and using the homogenization and the boundary layers, we find an effective law in the form of the generalized Darcy law. If the pores geometry is isotropic, then the condition splits in Beavers-Joseph type condition for the tangential flow and the standard Darcy condition for the normal flow. We will also study the roughness-induced effects on the proposed Darcy-type boundary condition.
Conformal embeddings, collapsing levels, affine vertex algebras and \(\mathcal{W}\)-algebras
Ozren Perše
University of Zagreb, Croatia
(joint work with Dražen Adamović, Victor G. Kac, Pierluigi Möseneder Frajria and Paolo Papi)
In this talk we review recent results on conformal embeddings of affine vertex algebras and \(\mathcal{W}\)-algebras obtained by quantum Hamiltonian reduction. We also study the notion of collapsing level for \(\mathcal{W}\)-algebras, and its application to categories of representations of associated affine vertex algebras
Relative algebraic geometry and difference algebra
Ivan Tomašić
Queen Mary University of London, United Kingdom
Grothendieck believed that topos theory unifies the continuous
and the discrete phenomena perfectly. The old adage of categorical logic stating that any topos can serve as a universe for building Mathematics naturally follows. We extend the techniques from Hakim's 1970's monograph developing algebraic geometry over a base topos to include the theories of the fundamental groupoid and the étale cohomology of relative schemes over a general base topos.
As a proof of concept, we apply these methods to redevelop difference algebra, founded by Ritt in the 1930s as the study of rings and modules with distinguished endomorphisms thought of as 'difference operators’. Our novel perspective is that difference algebra is best viewed as the study of algebraic objects in the topos \(\text{B}\mathbb{N}\) of difference sets, i.e., actions of the additive monoid of natural numbers. We develop difference algebraic geometry as relative algebraic geometry over the base topos \(\text{B}\mathbb{N}\). This approach allows us to overcome traditional obstacles and to make rapid advances in the theory.
As a proof of concept, we apply these methods to redevelop difference algebra, founded by Ritt in the 1930s as the study of rings and modules with distinguished endomorphisms thought of as 'difference operators’. Our novel perspective is that difference algebra is best viewed as the study of algebraic objects in the topos \(\text{B}\mathbb{N}\) of difference sets, i.e., actions of the additive monoid of natural numbers. We develop difference algebraic geometry as relative algebraic geometry over the base topos \(\text{B}\mathbb{N}\). This approach allows us to overcome traditional obstacles and to make rapid advances in the theory.
Stochastic homogenization of high-contrast media
Igor Velčić
University of Zagreb, Faculty of Electrical Engineering and Computing
(joint work with Mikhail Cherdantsev (University of Cardiff) and Kirill Cherednichenko (University of Bath))
We discuss the limiting spectrum of the elliptic equation representing the media with random high-contrast inclusions. We separate the case of finite and infinite domain.
In the case of finite domain we
show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of
equations corresponding to the coupled “macroscopic” and “microscopic” components of the field, giving
rise to an analogue of the “Zhikov function”, which represents the effective dispersion of the medium. We
demonstrate that, under some lenient conditions within the new framework, the spectra of the original
problems converge to the spectrum of their homogenization limit. In the case of infinite domain we show the existence of additional spectrum that is not the part of the spectrum of the limit operator.
Short communications
Class of \((h,g;m)\)-convex functions and certain types of inequality
Maja Andrić
Faculty of Civil Engineering, Architecture and Geodesy, University of Split, Croatia
(joint work with Josip Pečarić)
The recently introduced class of \((h,g;m)\)-convex functions unifies a certain range of convexity, which allows generalizations of known results. For this class, we present several types of inequality such as Hermite-Hadamard, Fejér and Lah-Ribarič, which generalize and extend corresponding inequalities. We also point out some special refined results.
Evaluating topological ordering in directed acyclic graphs
Suzana Antunović
Faculy of Civil Engineering, Architecture and Geodesy, University of Split, Croatia
(joint work with Damir Vukičević)
Directed acyclic graphs are often used to model situations and problems in real life. If we consider topological ordering of the graph as a process of arranging the vertices in best possible way considering the constraints caused by the direction of edges, then it makes sense to try to optimize this process by minimizing the distances between vertices in the ordering. For this purpose, we define measures based on distances between vertices in the topological ordering that allow us to construct a graph with optimal topological ordering regarding a specific measure thus minimizing the complexity of the system represented by the graph. We explore minimal and maximal values of the defined measures and comment on the topology of graphs for which maximal and minimal values are obtained.
Planar embeddings of chainable continua
Ana Anušić
University of São Paulo, Brazil
Continuum is a compact and connected metric space, and it is chainable if it can be covered
by an arbitrary small chain, or equivalently if it can be represented as an inverse limit on intervals.
It is well-known that every chainable continuum can be embedded in the plane (Bing 1951,
Anderson and Choquet 1959), but it is still not known how to describe possibly different (non-equivalent)
planar embeddings of a chainable continuum. For example, one might wonder which points in a planar
representation are accessible, and which are not. A point \(p\) in a planar continuum \(X\) is called accessible
(from the complement of \(X\)) if there is an arc \(A\) in the plane which intersects \(X\) only in \(p\). Nadler and Quinn
asked in 1972 whether for every chainable continuum \(X\) and a point \(p\in X\), there exists a planar
embedding of \(X\) such that \(p\) is accessible. This question is still open. We show that for every interval map \(f\) which is piecewise
monotone, post-critically finite, and locally eventually onto, every point of \(\underleftarrow{\lim}(I,f)\) can be embedded accessible. In particular, every point of Minc's continuum can be embedded accessible. Minc's continuum was
introduced in 2001, and was considered to be a likely counterexample to the Nadler-Quinn question. We will discuss some ideas for further generalization, which are part of current research with L. Hoehn.
Symmetrized strong Birkhoff-James orthogonality in $C^*$-algebras
Ljiljana Arambašić
University of Zagreb, Faculty of Science, Department of Mathematics, Croatia
(joint work with Alexander Guterman, Bojan Kuzma, Rajna Rajić, Svetlana Zhilina)
There are several nonequivalent extensions of orthogonality from inner product spaces to general normed spaces. One of the most well-known is the Birkhoff-James orthogonality: if \(X\) is a normed space and \(x,y\in X\) then \(x\) is Birkhoff-James orthogonal to \(y\) if
\(\|x+\lambda y\|\ge \|x\|\) for all scalars \(\lambda\). In a \(C^*\)-algebra \(A\) we can also discuss the case when \(x\in A\) is
Birkhoff-James orthogonal to all the elements of the form \(ya\), \(a\in A\).
In this case we say that \(x\) is strongly Birkhoff-James orthogonal to \(y\). We discuss this kind of orthogonality in some special classes of \(C^*\)-algebras.
Howe duality for exceptional theta correspondences
Petar Bakić
University of Utah, Salt Lake City, Utah, USA
(joint work with Gordan Savin)
The theory of local theta correspondence is built up from two ingredients: a reductive dual pair inside a symplectic group, and a Weil representation of its metaplectic cover. Exceptional correspondences arise similarly: dual pairs inside exceptional groups can be constructed using so-called Freudenthal Jordan algebras, while the minimal representation provides a suitable replacement for the Weil representation. Focusing on a particular dual pair, we explain how one obtains Howe duality for the correspondence in question. Finally, we discuss how these results can be used to construct certain families of Arthur packets for the exceptional group \(G_2\).
Representations of \(SL_2(\mathbb Q_p)\) on \(p\)-adic Banach spaces
Dubravka Ban
Southern Illinois University, Carbondale, USA
(joint work with Matthias Strauch)
We give a classification of
absolutely irreducible admissible unitary \(E\)-Banach space representations of \(H=SL_2(\mathbb Q_p)\), where \(E\) is a finite extension of \(\mathbb Q_p\). Our approach is based on the \(p\)-adic Langlands correspondence for \(G=GL_2(\mathbb Q_p)\) and restriction
of representations.
Let \(\psi\) be an absolutely irreducible 2-dimensional Galois representation over \(E\) and let \(\Pi = \Pi(\psi)\) be the irreducible unitary \(E\)-Banach space representation of \(G\) associated to \(\psi\) by the \(p\)-adic local Langlands correspondence. We consider the corresponding projective representation \(\overline{\psi}\) and the restriction \(\Pi|_H\). Then \(\Pi|_H\) decomposes as a direct sum of \(r \le 2\) irreducible representations. We prove that \(r\) is equal to the cardinality \(s\) of the centralizer of \(\overline{\psi}\) in \(PGL_2\). The restriction is multiplicity-free, except if \(\psi\) is triply-imprimitive, in which case \(s=4\) and the restriction of \(\Pi\) is a direct sum of two equivalent representations.
Let \(\psi\) be an absolutely irreducible 2-dimensional Galois representation over \(E\) and let \(\Pi = \Pi(\psi)\) be the irreducible unitary \(E\)-Banach space representation of \(G\) associated to \(\psi\) by the \(p\)-adic local Langlands correspondence. We consider the corresponding projective representation \(\overline{\psi}\) and the restriction \(\Pi|_H\). Then \(\Pi|_H\) decomposes as a direct sum of \(r \le 2\) irreducible representations. We prove that \(r\) is equal to the cardinality \(s\) of the centralizer of \(\overline{\psi}\) in \(PGL_2\). The restriction is multiplicity-free, except if \(\psi\) is triply-imprimitive, in which case \(s=4\) and the restriction of \(\Pi\) is a direct sum of two equivalent representations.
Self-orthogonal \(\mathbb{Z}_{2^k}\)-codes constructed from Boolean functions
Sara Ban
University of Rijeka, Faculty of Mathematics, Croatia
(joint work with Sanja Rukavina)
A Boolean function on \(n\) variables is a mapping \(f\colon \mathbb{F}_2^n \rightarrow \mathbb{F}_2\).
A bent function is a Boolean function \(f\) such that \(W_f (v) =\sum_{x\in \mathbb{F}_2^n} (-1)^{f(x)+\left\langle v, x\right\rangle}=\pm 2^{\frac{n}{2}}\), for every \(v\in \mathbb{F}_2^n\).
The subject of this talk is a construction of self-orthogonal codes over \(\mathbb{Z}_{2^k}\) from Boolean functions.
First, we give a construction of a self-orthogonal \(\mathbb{Z}_4\)-code of length \(2^{n+1}\) from a pair of bent functions on \(n\) variables. We prove that for \(n\geq 4\) those codes can be extended to Type IV-II \(\mathbb{Z}_4\)-codes. From that family of Type IV-II \(\mathbb{Z}_4\)-codes, we construct a family of self-dual Type II binary codes by using the Gray map.
We construct a self-orthogonal \(\mathbb{Z}_{2^k}q)-code of length \(2^{n+1}\) with all Euclidean weights divisible by \(2^{k+2}\) from a pair of bent functions on \(n\) variables, for every \(k\geq 3\).
Moreover, we give a construction of a self-orthogonal \(\mathbb{Z}_{2^k}\)-code of length \(2^{n+2}\) with all Euclidean weights divisible by \(2^{k+1}\) from a pair of Boolean functions on \(n\) variables, for every \(3\leq k\leq n\).
The subject of this talk is a construction of self-orthogonal codes over \(\mathbb{Z}_{2^k}\) from Boolean functions.
First, we give a construction of a self-orthogonal \(\mathbb{Z}_4\)-code of length \(2^{n+1}\) from a pair of bent functions on \(n\) variables. We prove that for \(n\geq 4\) those codes can be extended to Type IV-II \(\mathbb{Z}_4\)-codes. From that family of Type IV-II \(\mathbb{Z}_4\)-codes, we construct a family of self-dual Type II binary codes by using the Gray map.
We construct a self-orthogonal \(\mathbb{Z}_{2^k}q)-code of length \(2^{n+1}\) with all Euclidean weights divisible by \(2^{k+2}\) from a pair of bent functions on \(n\) variables, for every \(k\geq 3\).
Moreover, we give a construction of a self-orthogonal \(\mathbb{Z}_{2^k}\)-code of length \(2^{n+2}\) with all Euclidean weights divisible by \(2^{k+1}\) from a pair of Boolean functions on \(n\) variables, for every \(3\leq k\leq n\).
Estimates for some quadrature rules via weighted Hermite-Hadamard inequality
Josipa Barić
University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Split, Croatia
(joint work with Lj. Kvesić, J. Pečarić and M. Ribičić Penava)
In this work, new estimates for some quadrature rules are given using the weighted Hermite-Hadamard inequality
for higher order convex functions and weighted version of the integral identity expressed by w-harmonic sequences
of functions. The obtained results are applied to the weighted one-point formula for numerical integration to derive
new estimates of definite integral values.
One-dimensional model of the flow and thermal explosion of a reactive real micropolar gas
Angela Bašić-Šiško
University of Rijeka, Faculty of Engineering
We consider a model of one-dimensional flow and thermal explosion of a reactive real micropolar gas characterized by a generalized equation of state. This model describes the behavior of gasses in microtubes during combustion.
The use of the generalized equation of state instead of the equation of state of the ideal gas ensures that the behavior of the gasses is better captured by the model. Also, the consideration of micro-effects in a continuum using the micropolar fluid model favors this model over the classical one in describing the flow of a small characteristic dimension of a fluid with a pronounced particle structure.
Here we present existence and uniqueness results for the described model.
Semilinear equations for non-local operators: Beyond the fractional Laplacian
Ivan Biočić
University of Zagreb, Croatia
(joint work with Zoran Vondraček and Vanja Wagner)
In this talk, we study semilinear problems in general bounded open sets for non-local operators
with exterior and boundary conditions. The operators are more general than the
fractional Laplacian. We also give results in case of bounded \(C^{1,1}\) open sets.
Representations induced from essentially Speh and strongly positive discrete series of classical \(p\)-adic groups
Barbara Bošnjak
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
An important role in the classification of the unitary dual of general linear group over \(p\)-adic field, obtained in [3], is played by the so-called Speh and essentially Speh representations. Description of the unitary dual of other classical \(p\)-adic groups is still not known, besides some low-rank cases. One possible approach is to understand composition factors of representations induced from essentially Speh on general linear part and unitary representation on classical part as a natural step toward better understanding and construction of unitary representations of classical \(p\)-adic groups.
In this work, we determine composition factors of representations which represent generalization of so far established results in described direction in [1,2].
References:
References:
- I. Matić, Composition factors of a class of induced representations of classical p-adic groups, Nagoya Mathematical Journal 227 (2017), 16-48.
- G. Muić, Composition series of generalized principal series; the case of strongly positive discrete series, Israel J. Math. 140 (2004), 157–202.
- M. Tadić, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 335–382.
On unitary dual of \(p\)-adic group \(SO(7)\) with support on minimal parabolic subgroup
Darija Brajković Zorić
Department of Mathematics, J. J. Strossmayer University of Osijek
Determination of the unitary dual of a reductive algebraic group \(G\) over a local non-archimedean field \(F\) is a significant aspect of representation theory research in the last few decades. The basic approach for determining the unitary dual consists of two main steps: a complete description of the nonunitary dual and the extraction of the classes of unitarizable representations among the obtained irreducible subquotients. Besides directly studying all the possible cases of induced representations while exploring the induced representations of groups of a small rank, one can also gain a good sense of the structure of a unitary dual in a general case. In this talk we will consider the unitary dual of \(p\)-adic group \(SO(7)\) with support on minimal parabolic subgroup.
This research was supported in part by the Croatian Science Foundation under the project IP-2018-01-3628.
This research was supported in part by the Croatian Science Foundation under the project IP-2018-01-3628.
Polynomial \(D(4)\)-quadruples over Gaussian integers
Sanda Bujačić Babić
Faculty of Mathematics, University of Rijeka, Croatia
(joint work with Marija Bliznac Trebješanin)
A set \(\{a, b, c, d\}\) of four distinct nonzero polynomials in \(\mathbb{Z}[i][X]\) is called a polynomial Diophantine \(D(4)\)-quadruple if the product of any two of its distinct elements increased by 4 is a square of a polynomial in \(\mathbb{Z}[i][X]\).
In this paper we prove that every polynomial \(D(4)\)-quadruple in \(\mathbb{Z}[i][X]\) is regular, or in other words that the equation $$(a+b-c-d)^2=(ab+4)(cd+4)$$ holds for every polynomial \(\{a, b, c, d\}\) \(D(4)\)-quadruple in \(\mathbb{Z}[i][X]\).
Asymptotic behavior of \(L^p\) estimates for a class of multipliers with homogeneous unimodular symbols
Aleksandar Bulj
University of Zagreb
(joint work with Vjekoslav Kovač)
Boundedness of Fourier multiplier operators on \(L^p\) spaces is a well studied problem in harmonic analysis, but since the boundedness of some important multipliers is still unknown, it is still an active area of research.
The aim of this talk is to present the recent joint work with Vjekoslav Kovač regarding the asymptotic behaviour when \(\lambda \to\infty \) of \(L^p\) norms of the Fourier multiplier operators associated to the homogeneous symbols \(e^{i\lambda \phi( \cdot /|\cdot|)}\), where \(\phi\) is a smooth function on \((n-1)\)-dimensional sphere in \(\mathbb{R}^n\). We prove sharp bounds for such operators in even dimensions, giving a negative answer to a question posed by Vladimir Maz'ya regarding the asymptotic growth of the norms of such operators when \(n\ge 2\). Concrete operators that fall into the studied class are the multipliers forming the two-dimensional Riesz group and we give sharp asymptotic bounds for that family of operators, answering the question raised in the work of Dragičević, Petermichl, and Volberg about the sharpness of the asymptotic bounds for the aforementioned family of operators.
The aim of this talk is to present the recent joint work with Vjekoslav Kovač regarding the asymptotic behaviour when \(\lambda \to\infty \) of \(L^p\) norms of the Fourier multiplier operators associated to the homogeneous symbols \(e^{i\lambda \phi( \cdot /|\cdot|)}\), where \(\phi\) is a smooth function on \((n-1)\)-dimensional sphere in \(\mathbb{R}^n\). We prove sharp bounds for such operators in even dimensions, giving a negative answer to a question posed by Vladimir Maz'ya regarding the asymptotic growth of the norms of such operators when \(n\ge 2\). Concrete operators that fall into the studied class are the multipliers forming the two-dimensional Riesz group and we give sharp asymptotic bounds for that family of operators, answering the question raised in the work of Dragičević, Petermichl, and Volberg about the sharpness of the asymptotic bounds for the aforementioned family of operators.
Complete Asymptotic Expansion of the Gaussian Compound Mean
Tomislav Burić
University of Zagreb, Faculty of Electrical Engineering and Computing, Zagreb, Croatia
(joint work with Lenka Mihoković)
We present a complete asymptotic expansion of the Gaussian compound of two arbitrary homogeneous symmetric means \(M\otimes_g N\) and derive an efficient algorithm for computing coefficients in this expansion. This new approach leads to a simple recursive formula for coefficients in the expansion of the arithmetic-geometric mean. We also give application to the Gaussian compounds of some other classical means.
Principal subspaces of standard modules of affine Lie algebras of type \(B_l^{(1)}\), \(C_l^{(1)}\), \(F_4^{(1)}\) and \(G_2^{(1)}\)
Marijana Butorac
Faculty of Mathematics, University of Rijeka, Croatia
We consider principal subspace of standard
module \(L( k_0\Lambda_0+k_j\Lambda_j )\), where \(\Lambda_j\) denote the fundamental weights of level one, for affine Lie algebra of type \(B_l^{(1)}\), \(C_l^{(1)}\), \(F_4^{(1)}\) and \(G_2^{(1)}\). By using the theory of vertex operator algebras, we find combinatorial bases of principal subspaces in terms of quasi-particles, introduced by B. Feigin and A. Stoyanovsky. From quasi-particle bases, we obtain characters of principal subspaces.
Homogenization of elasto-plastic plate equations
Marin Bužančić
University of Zagreb, Faculty of Chemical Engineering and Technology
(joint work with Elisa Davoli and Igor Velčić)
In this talk, we will consider a lower dimensional homogenized thin plate model within the framework of linearized elasto-plasticity. Starting from the energetic formulation of the quasistatic evolution, we analyse the behavior of the elastic energies and dissipation potentials, as well as the displacements and strain tensors, when the period of oscillation of the heterogeneous material \(\varepsilon\) and the thickness of the thin body \(h\) simultaneously tend to zero. In order to derive convergence results for energy functionals and the associated energy minimizers, we base our approach on \(\Gamma\)-convergence techniques and the two-scale convergence method adapted to dimension reduction.
Composition series of a class of induced representations
Igor Ciganović
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
We determine compostion series of a class of parabolically induced representation \(\delta([\nu^{-b}\rho,\nu^c\rho])\times \delta([\nu^{\frac{1}{2}}\rho,\nu^a\rho])\rtimes \sigma\)
of \(p\)-adic symplectic group in terms of Mœglin Tadić clasification.
Here \( \frac{1}{2}\leq a < b < c\in \frac{1}{2}\mathbb{Z}+1\) are half integers,
\(\nu=|det |_{F}\) where \(F\) is a \(p\)-adic field,
\(\rho\) is a cuspidal representation of a general linear group,
\(\sigma\) is a cuspidal representation of a \(p\)-adic symplectic group
such that \(\nu^\frac{1}{2}\rho\rtimes \sigma\) reduces
and \(\delta([\nu^{x}\rho,\nu^y\rho])\hookrightarrow \nu^y\rho\times\cdots \times \nu^{x}\rho\) is a discrete series representation for \(x\leq y\in \frac{1}{2}\mathbb{Z}+1\).
Multi-UAV trajectory planning for 3D visual inspection
Bojan Crnkovic
Faculty of mathematics, University of Rijeka
(joint work with Stefan Ivić, Luka Grbčić, Lea Matleković)
This work presents a new trajectory planning algorithm for three-dimensional autonomous UAV volume coverage and visual inspection. The algorithm is an extension of a HEDAC multi-agent area coverage algorithm for three-dimensional domains. With a given target exploration density field, the algorithm designs a potential field to achieve the minimization of the remaining density and generate trajectories using potential gradients to direct UAVs to the regions of higher potential, i.e., higher values of remaining density.
The proposed algorithm provides collision avoidance, load balancing and successful planning of visual inspection. It offers flexibility in various setup parameters and is applicable in real-world inspection task which is demonstrated with realistic simulations of visual inspection.
Monolithic Characters of Finite Groups
Burcu Çınarcı
Piri Reis University, Istanbul,
Turkey
Let \(G\) be a finite group and let \(\chi\) be an irreducible character
of \(G\). If the factor group \(G/\textrm{ker}\chi\) has a unique
minimal normal subgroup, then the character \(\chi\) is called
monolithic. In fact, the monolithic characters of a group are
abundant and they determine the normal structure of \(G\), that is,
every proper normal subgroup of \(G\) is an intersection of kernels of
monolithic characters. There have been several studies devoted to
studying irreducible characters of finite groups. However, if we
look at the studies published on the monolithic characters of finite
groups, we can see that there is not much work on this subject. To
the best of our knowledge, the first studies on monolithic
characters were considered by Berkovich and others. Before
discussing some results about monolithic characters of a finite
solvable group, we will give an overview of the background and
recent related works in the literature.
References:
References:
- Y.G. Berkovich, On Isaacs' three character degrees theorem, Proc. Am. Math. Soc. 125(3), 669-677 (1997).
- Y. Berkovich and E.M. Zhmud, Characters of Finite Groups Part 2, American Mathemetical Society, (1999).
- Y. Berkovich, I. M. Isaacs and L. Kazarin, Groups with distinct monolithic character degrees, J. Algebra 216 (1999) 448-480.
Computable type of certain quotient spaces
Matea Čelar
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with Zvonko Iljazović)
Topology plays an important role in determining the relationship between different levels of computability of sets in computable topological spaces. In particular, semicomputable sets with certain topological properties are necessarily fully computable. This is expressed in the notion of computable type: a space \(\Sigma\) is said to have computable type if every semicomputable set homeomorphic to \(\Sigma\) must be computable. Examples of sets with computable type include topological manifolds and Hausdorff continua.
In this talk, we consider the effect of quotients on preserving computable type, focusing primarily on locally Euclidean spaces. We provide some sufficient conditions under which space \(A\) having computable type implies the quotient space \(A/B\) has computable type and vice versa, as well as some interesting counterexamples.
This work has been fully supported by Croatian Science Foundation under the project 7459 CompStruct.
In this talk, we consider the effect of quotients on preserving computable type, focusing primarily on locally Euclidean spaces. We provide some sufficient conditions under which space \(A\) having computable type implies the quotient space \(A/B\) has computable type and vice versa, as well as some interesting counterexamples.
This work has been fully supported by Croatian Science Foundation under the project 7459 CompStruct.
Solving polynomial and rational eigenvalue problems through linearizations
Ranjan Kumar Das
Department of Mathematics, J. J. Strossmayer University of Osijek, Croatia
(joint work with Rafikul Alam (IIT Guwahati, India))
In numerous applications, such as vibration analysis of machinery, buildings, and vehicles, control theory and linear systems theory, and approximation of various nonlinear eigenvalue issues, polynomial and rational matrices are used. We will explore how to develop a new family of Fiedler-like linearizations (named EGFPR) from the given data of the eigenvalue problems in order to provide a direct solution for polynomial and rational eigenvalue problems. EGFPR broadens the class of linearizations by incorporating all Fiedler-like pencils in the literature, resulting in a huge number of structure-preserving linearizations for structured (Symmetric, Hermitian, Palindromic, and other) eigenvalue problems. Finally, we discuss operation-free eigenvector and minimal base recovery, as well as minimal indices of polynomial and rational matrices.
This work is partially supported by the Croatian Science Foundation project: Vibration Reduction in Mechanical Systems (IP-2019-04-6774, VIMS).
This work is partially supported by the Croatian Science Foundation project: Vibration Reduction in Mechanical Systems (IP-2019-04-6774, VIMS).
Reformulation of Multifluid Plasma Equations in terms of Hyperbolic Octonions
Süleyman Demir
Department of Physics, Eskişehir Technical University, Turkey
(joint work with Erdinç Zeren)
In this work, using the theoretical analogy between basic equations of electrodynamics and fluid dynamics, the Maxwell type equations of multifluid plasma are expressed in terms of hyperbolic octonions. The presented model allows us to generalize the species generalized Lamb vector and species generalized vorticity of plasma by a hyperbolic octonion. Thus, the basic equations of multifluid plasma are derived in a form similar to electromagnetic and gravitational counterparts previously given using this formalism.
Application of Karamata's theory in the study of asymptotic properties of the second-order quasilinear \(q\)-difference equation
Katarina Djordjević
Faculty of Sciences and Mathematics, University of Niš, Serbia
(joint work with Jelena Manojlović)
The Karamata's theory of regularly varying functions has been recognized as a powerful tool in the study of asymptotic properties of differential, difference and \(q\)-difference equations.
Subsequently, the concept of \(q\)-regularly varying functions was also introduced on the \(q\)-lattice \(\displaystyle q^{\mathbb{N}_0}=\{q^n \, : \, n \in \mathbb{N}_0\}\) with \(q>1,\)
A substantial pool of research contains results about application of the theory of \(q\)-regularly varying functions to the asymptotic analysis of \(q\)-difference linear or half-linear equations. This theory has not been used in the asymptotic analysis of any other type of second-order nonlinear \(q\)-difference equation.
In this talk we will consider the second-order quasilinear \(q\)-difference equation $$ D_q(a(t)\Phi_{\alpha}(D_q(x(t))))=b(t)\Phi_{\beta}(x(qt)), $$ where \(\alpha>\beta>0,\) in the framework of \(q\)-regular variation. Under the assumption that coefficients \(a\,,b\) are \(q\)-regularly varying functions necessary and sufficient conditions for the existence of strongly decaying and strongly increasing solutions of this equation will be established. Moreover, it will be shown that the existing strongly decaying and strongly increasing solutions are \(q\)-regularly varying with a certain regularity index and their precise asymptotic formulae will be determined.
The obtained results will enable the complete structure of the set of q-regularly varying solutions to be presented.
In this talk we will consider the second-order quasilinear \(q\)-difference equation $$ D_q(a(t)\Phi_{\alpha}(D_q(x(t))))=b(t)\Phi_{\beta}(x(qt)), $$ where \(\alpha>\beta>0,\) in the framework of \(q\)-regular variation. Under the assumption that coefficients \(a\,,b\) are \(q\)-regularly varying functions necessary and sufficient conditions for the existence of strongly decaying and strongly increasing solutions of this equation will be established. Moreover, it will be shown that the existing strongly decaying and strongly increasing solutions are \(q\)-regularly varying with a certain regularity index and their precise asymptotic formulae will be determined.
The obtained results will enable the complete structure of the set of q-regularly varying solutions to be presented.
Doubly regular Diophantine quadruples
Andrej Dujella
University of Zagreb, Croatia
(joint work with Vinko Petričević)
For a nonzero integer \(n\), a set of distinct nonzero integers
\(\{a_1, a_2, \ldots , a_m\}\) such that \(a_i a_j + n\) is a perfect square
for all \(1 \leq i < j \leq m\), is called a \(D(n)\)-\(m\)-tuple.
By using properties of so-called regular Diophantine \(m\)-tuples
and certain family of elliptic curves, we show that
there are infinitely many essentially different sets consisting of perfect squares
which are simultaneously \(D(n_1)\)-quadruples and \(D(n_2)\)-quadruples
with distinct non-zero squares \(n_1\) and \(n_2\).
Construction of periodic Golay pairs using PBDs
Doris Dumičić Danilović
Faculty of Mathematics, University of Rijeka, Croatia
(joint work with D. Crnković, R. Egan and A. Švob)
Let \(a = [a_{0},\ldots,a_{v-1}]\) be a \(\{\pm 1 \}\)-sequence of length \(v\). The periodic autocorrelation function of \(a\) with shift \(s\) is given by
\(\displaystyle\mathrm{PAF}_{s}(a) = {\textstyle{\sum}}_{i=0}^{n-1}a_{i}a_{i+s}\), where the sequence indices are read modulo \(v\). A pair \((a,b)\) of \(\{\pm 1\} \)-sequences is a
periodic Golay pair (PGP) if \(\mathrm{PAF}_{s}(a) + \mathrm{PAF}_{s}(b) = 0\) for all \(1 \leq s \leq v-1\). PGPs generalize the better known Golay pairs which are known to have applications in multislit spectroscopy, signal processing, digital communications, etc.
In this talk we will describe a relationship between pairwise balanced designs with \(v\) points and PGPs of length \(v\), in order to classify periodic Golay pairs of length less than 40. Using the theory of orbit matrices and using isomorph rejection which is compatible with equivalence of corresponding PGPs, all pairwise balanced designs with \(v\) points under specific block conditions having an assumed cyclic automorphism group are constructed.
Similar tools to construct new periodic Golay pairs of lengths greater than 40 are used, but classifications remain incomplete. Under some extra conditions on its automorphism group, a PGP of length 90 does not exist, but length 90 still remains the smallest length for which existence of a PGP is undecided. Also, we demonstrate how orbit matrices can be used to construct some quasi-cyclic self-orthogonal linear codes.
In this talk we will describe a relationship between pairwise balanced designs with \(v\) points and PGPs of length \(v\), in order to classify periodic Golay pairs of length less than 40. Using the theory of orbit matrices and using isomorph rejection which is compatible with equivalence of corresponding PGPs, all pairwise balanced designs with \(v\) points under specific block conditions having an assumed cyclic automorphism group are constructed.
Similar tools to construct new periodic Golay pairs of lengths greater than 40 are used, but classifications remain incomplete. Under some extra conditions on its automorphism group, a PGP of length 90 does not exist, but length 90 still remains the smallest length for which existence of a PGP is undecided. Also, we demonstrate how orbit matrices can be used to construct some quasi-cyclic self-orthogonal linear codes.
One analytical method for the approximation of solutions of neutral stochastic differential equations
Dušan Đorđević
Faculty of Sciences and Mathematics, University of Niš, Serbia
(joint work with Miljana Jovanović and Marija Milošević)
A class of \(d\)-dimensional neutral stochastic differential equations with constant delay
$$\quad\quad
d\Big[x(t)-D\big(t,x(t-\tau)\big)\Big] = f\big(t,x(t),x(t-\tau)\big)dt + g\big(t,x(t),x(t-\tau)\big)dw_t,\quad\quad \text{(1)}
$$
with initial data \(\eta=\{\eta(\theta), \, \theta\in[-\tau,0]\}\) on a finite time interval \(t\in[t_0,T]\), is considered. Under certain assumptions for the coefficients of Eq. (1), one of which is the polynomial condition, which are more general than the widely used linear growth or Lipschitz conditions, the approximate solution to the one of Eq. (1) is presented. The approximate solution is constructed via the Taylor expansions of the coefficients of Eq. (1) and the closeness is estimated almost surely and in the \(L^p\) sense. Finally, an example which illustrates the theoretical results and shows that the class of functions which satisfy our conditions is nonempty, is displayed.
RADIAL SYMMETRY FOR AN ELLIPTIC PDE WITH A FREE BOUNDARY
Layan El Hajj
American university in Dubai
(joint work with Henrik Shahgholian)
In this paper we prove symmetry for solutions to the semi-linear elliptic equation
$$
\Delta u = f(u) \quad \hbox{ in } B_1,
\qquad 0 \leq u < M, \quad \hbox{ in } B_1, \qquad u = M, \quad \hbox{ on } \partial B_1,
$$
where \(M>0\) is a constant, and \(B_1\) is the unit ball.
Under certain assumptions on the r.h.s. \(f (u)\), the \(C^1\)-regularity of the free boundary \(\partial \{u>0\}\)
and a second order asymptotic expansion for \(u\) at free boundary points,
we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a
boundary Harnack principle (with r.h.s.) that replaces Serrin's celebrated boundary point lemma, which is not available in our case due to lack of \(C^2\)-regularity of solutions.
A sufficient condition for non-zero entropy of closed relations on \([0,1]\)
Goran Erceg
University of Split, Faculty of Science, Croatia
(joint work with Iztok Banič and Judy Kennedy)
We introduce the topological entropy of a closed relation \(G\) on any compact metric space \(X\). We show that periodic points, finitely generated Cantor sets, Mahavier products and the entropy of closed relations are preserved by topological conjugations. This generalizes the well-known results about the topological conjugacy of continuous mappings. We prove a theorem, giving sufficient conditions for a closed relation \(G\) on \([0,1]\) to have a non-zero entropy. Then we present various examples of closed relations \(G\) on \([0,1]\) with non-zero entropy.
Homogenisation of nonlocal linear elliptic operators
Marko Erceg
University of Zagreb
(joint work with Krešimir Burazin and Miłosz Krupski)
The theory of homogenisation can be seen as a way to simplify complicated
heterogeneous models (i.e. to average heterogeneous media in order to derive effective properties) or to pass from the micro-scale to the macro-scale (e.g. to derive physical laws on a macro-scale starting from the atomic scale).
In the mathematical theory of homogenisation this is achieved by studying differential equations
with parameter-dependent coefficients for which we want to pass to the limit
(with respect to that parameter) and get the effective equation. In this talk we are interested in nonlocal linear elliptic operators with non-symmetric kernels.
On magnetic curves in almost cosymplectic Sol space
Zlatko Erjavec
University of Zagreb, Faculty of Organization and Informatics, Croatia
(joint work with Jun-ichi Inoguchi)
Magnetic curves represent trajectories of charged particles moving on a Riemannian manifold under the action of a magnetic field.
The study of magnetic curves in arbitrary Riemannian manifolds was developed in early 1990's, even though related works can be found earlier (see [4, 7]). Recently there are interesting results on magnetic curves in Euclidean space [6], Sasakian manifolds [1], cosymplectic manifolds [2], Sol space [3] and quasi Sasakian manifolds [5].
In this talk we consider magnetic curves with respect to the almost cosymplectic structure of the \(\mathrm{Sol}_3\) space.
References:
The study of magnetic curves in arbitrary Riemannian manifolds was developed in early 1990's, even though related works can be found earlier (see [4, 7]). Recently there are interesting results on magnetic curves in Euclidean space [6], Sasakian manifolds [1], cosymplectic manifolds [2], Sol space [3] and quasi Sasakian manifolds [5].
In this talk we consider magnetic curves with respect to the almost cosymplectic structure of the \(\mathrm{Sol}_3\) space.
References:
- S. L. Druţă-Romaniuc, J. Inoguchi, M. I. Munteanu, A. I. Nistor, Magnetic curves in Sasakian manifolds, J. Nonlinear Math. Phys. 22 (2015) 3, 428-447.
- S. L. Druţă-Romaniuc, J. Inoguchi, M. I. Munteanu, A. I. Nistor, Magnetic curves in cosymplectic manifolds, Report Math. Phys. 78 (2016), 33-47.
- Z. Erjavec, J. Inoguchi, Magnetic curves in \(Sol_3\), J. Nonlinear Math. Phys. 25 (2), (2018) 198-210.
- V. L. Ginzburg, A charge in a magnetic field: Arnold's problems 1981-9, 1982-24, 1984-4, 1994-14, 1994-35, 1996-17,1996-18, in Arnold's problems (V. I. Arnold ed.) (Springer-Verlag and Phasis, 2004) 395-401.
- J. Inoguchi, M. I. Munteanu, A. I. Nistor, Magnetic curves in quasi-Sasakian 3-manifolds, Anal. Math. Phys. 9, (2019) 43-61.
- M. I. Munteanu, Magnetic curves in a Euclidean space: One example, several approaches, Publ. de L'Institut Math. 94 (108) (2013), 141-150.
- T. Sunada, Magnetic flows on a Riemann surface, in Proc. KAIST Mathematics Workshop: Analysis and Geometry, (KAIST, Taejeon, Korea, 1993), pp. 93-108.
Similarity between random sets based on their realisations consisting of many components
Vesna Gotovac Đogaš
Faculty of Science, University of Split, Split, Croatia
In recent years random sets were recognised as a valuable tool in modelling different processes from fields like biology, biomedicine or material sciences. These random sets could be of very ragged shaped and therefore difficult to compare or describe using simple models.
In this talk, we propose a statistical procedure for comparing two random sets whose realisations consist of many components. It involves a construction of a similarity measure between two realisations of random sets that takes into account the shapes and the positions of the components. The similarity measure is defined as the \(p\)-value of tests of equality in distribution based on \(\mathfrak{N}\)-distances.
The methodology is justified by a simulation study and applied to real biomedical data of histological images of mammary tissue.
References:
In this talk, we propose a statistical procedure for comparing two random sets whose realisations consist of many components. It involves a construction of a similarity measure between two realisations of random sets that takes into account the shapes and the positions of the components. The similarity measure is defined as the \(p\)-value of tests of equality in distribution based on \(\mathfrak{N}\)-distances.
The methodology is justified by a simulation study and applied to real biomedical data of histological images of mammary tissue.
References:
- Gotovac V (2019) Similarity between random sets consisting of many components. Image Anal Stereol. 38(2):185-99
On some results about LCD codes based on two-class association schemes
Ana Grbac
Faculty of Mathematics, University of Rijeka, Croatia
(joint work with Dean Crnković and Andrea Švob)
Linear codes with complementary duals, shortly named LCD codes, are linear codes whose intersection with their duals is trivial. In this talk, we present a construction for LCD codes over finite fields from the adjacency matrices of two-class association schemes. These schemes consist of either strongly regular graphs (SRGs) or doubly regular tournaments (DRTs). Further, we show conditions under which the introduced construction method gives LCD codes over the fields \(\mathbb{F}_2\), \(\mathbb{F}_3\) and \(\mathbb{F}_4\). Lastly, we analyse some LCD codes obtained using this method of construction from some families of strongly regular graphs and some doubly regular tournaments.
Arcs, circles, finite graphs and inverse limits of set-valued functions on intervals
Sina Greenwood
University of Auckland
We investigate conditions for an inverse limit of set-valued functions on intervals to be finite a graph, and in particular an arc or circle.
We strengthen a result by Nall and Vidal-Escobar who showed that if an inverse limit of set-valued functions on intervals is a finite graph, then it is homeomorphic to the Mahavier product of the first \(n\) functions of the sequence for some \(n\in\mathbb N\), and we extend the notion of a splitting sequence introduced by G and Štimac in their characterisation of inverse limits of continuous functions on intervals that are arcs.
Torsion growth of rational elliptic curves
Tomislav Gužvić
University of Zagreb, Faculty of Science, Department of Mathematics, Croatia
We discuss torsion growth of rational elliptic curves over number fields of small degree. If \(E/\mathbb{Q}\) is an elliptic curve, a famous result of Mazur shows that there are only 15 possibilities for \(E(\mathbb{Q})_{tors}\). It is natural to ask what are the possibilities for \(E(F)_{tors}\), where \(F\) is a number field of degree \(d\) over \(\mathbb{Q}\). We give a partial result in the case \(d=6\) and discuss what is needed to complete the result.
On Diagonalization Methods for PGEP
Vjeran Hari
University of Zagreb, Croatia
In this brief communication we describe our latest results on diagonalization methods
for the positive definite generalized eigenvalue problem (PGEP) \(Ax=\lambda Bx\). Here \(A\) and \(B\)
are symmetric or complex Hermitian matrices such that \(B\) is positive definite. We have
considered both real and complex diagonalization methods. The methods include: Hari-Zimmermann (HZ),
Cholesky-Jacobi (CJ) method, and the real block Jacobi method for PGEP.
In particular, we proved the asymptotic quadratic convergence of the real and complex HZ method. We proved the global convergence of block Jacobi method under a large class of generalized serial pivot strategies. Together with Josip Matejaš we proved the high relative accuracy of the real HZ and CJ method.
My latest research also includes the global convergence of the real and complex HZ and CJ method under deRijk pivot strategy. Recent numerical tests have shown that the deRijk pivot strategy significantly decreases number of sweeps needed for a diagonalization and thus reduces the overall CPU time. The convergence proof also includes the case of the standard Jacobi method for the symmetric and Hermitian matrices.
This work has been partially supported by Croatian Science Foundation under the project IP-2020-02-2240.
In particular, we proved the asymptotic quadratic convergence of the real and complex HZ method. We proved the global convergence of block Jacobi method under a large class of generalized serial pivot strategies. Together with Josip Matejaš we proved the high relative accuracy of the real HZ and CJ method.
My latest research also includes the global convergence of the real and complex HZ and CJ method under deRijk pivot strategy. Recent numerical tests have shown that the deRijk pivot strategy significantly decreases number of sweeps needed for a diagonalization and thus reduces the overall CPU time. The convergence proof also includes the case of the standard Jacobi method for the symmetric and Hermitian matrices.
This work has been partially supported by Croatian Science Foundation under the project IP-2020-02-2240.
Non-existence of block-transitive subspace designs
Daniel Hawtin
Faculty of Mathematics, University of Rijeka, Croatia
(joint work with Jesse Lansdown)
A subspace design is the \(q\)-analogue of a balanced incomplete block design. Subspace designs have generated a great deal of interest lately due to their application in network coding. We prove that there are no non-trivial subspace designs admitting a group of automorphisms that acts transitively on the set of blocks of the design.
Poroelastic plate model obtained by simultaneous homogenization and dimension reduction
Pedro Hernández-Llanos
Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb, Croatia
(joint work with Igor Velčić and Josip Žubrinić)
In this talk, the starting point of our analysis is coupled system of elasticity and weakly compressible fluid. We consider two small parameters: the thickness \(h\) of the thin plate and the pore scale \(\varepsilon_h\) which depend on \(h\). We will focus specifically on the case when the pore size is small relative to the thickness of the plate. The main goal here is derive a model for a poroelastic plate from the \(3D\) problem as \(h\) goes to zero using simultaneous homogenization and dimension reduction techniques. The obtained model generalizes the poroelastic plate model derived in [1] by dimension reduction techniques from \(3D\) Biot's equations.
References:
- A. Marcianiak-Czochra, A. Mikelić, A rigurous derivation of the equations for the clamped Biot-Kirchhoff-Love poroelastic plate, Arch. Rational Mech. Anal. 215 (2015), 1035-1062.
A new notion of bisimulations of Verbrugge semantics
Sebastijan Horvat
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with Tin Perkov and Mladen Vuković)
The basic semantics for the interpretability logics are Veltman models
defined by D. de Jongh and F. Veltman in 1988.
They were used by V. Švejdar in 1991.
to prove some independence results.
De Jongh tried to generalize Švejdar's arguments and came up
with the notion of generalized Veltman semantics.
Since R. Verbrugge worked this out in an unpublished note,
this semantics has recently been named after her.
The basic equivalence between Veltman models are bisimulations.
M. Vuković defined bisimulations
(and their finite approximations called \(n\)-bisimulations)
for Verbrugge semantics.
It was proved by M. Vuković and D. Vrgoč that \(n\)-bisimilar worlds are modally
\(n\)-equivalent, i.e. they satisfy the same IL-formulas
of modal depth up to \(n\).
We have shown that the converse is generally not true.
So we have defined in [1] a new notion of bisimulations for Verbrugge semantics
called w-bisimulations.
In this talk we will present them and show the desired converse:
\(n\)-equivalent worlds are \(n\)-w-bisimilar.
In order to do that we define Verbrugge model comparison games called w-games and show
that w-bisimulation relations may be understood as descriptions of winning
strategies for one player in a w-game.
References:
References:
- S. Horvat, T. Perkov, M. Vuković, Bisimulations and bisimulations games for Vebrugge semantics, preprint, 2022.
Approximation of CDF of non-central \(\chi^2\) distribution by second mean-value theorem for integrals
Dragana Jankov Maširević
Department of Mathematics, J. J. Strossmayer University of Osijek
(joint work with Tibor K. Pogány)
Applications of modified Bessel function of the first kind occur in statistics; it is, for instance, a
constituting term of the probability density function of the non-central Chi-squared distribution
\(\chi_n'^{\;2}(\lambda)\) with \(n\) degrees of freedom and non-centrality parameter \(\lambda>0\). The
related cumulative distribution function (CDF) \(F_{n,\lambda}\) has an integral representation
with the modified Bessel function of the first kind \(I_{n/2-1},\, n \in \mathbb N\)
in the integrand. Accordingly, the main aim of this talk is to present certain approximation formulae
for \(F_{n,\lambda}\) established by the Bonnet type, and the Okamura's variant of du Bois-Reymond type
second mean value theorem for definite integrals. Several related results are exposed also in terms of
the Marcum \(Q\)-function. Numerical simulations show the quality of approximations of the CDF
in comparison with certain earlier results.
The finite coarse shape - inverse systems approach and intrinsic approach
Ivan Jelić
Faculty of science, University of Split, Croatia
(joint work with Nikola Koceić-Bilan)
Given an arbitrary category \(C\), a category \(pro^{*^f}\)-\(C\) is constructed using an inverse system approach. Known \(pro\)-\(C\) category can be considered as a subcategory of \(pro^{*^f}\)-\(C\) and \(pro^{*^f}\)-\(C\) is a subcategory of \(pro^*\)-\(C\). Analogously to the construction of the shape category \(Sh_{\left(C,D\right)}\) and the coarse shape category \(Sh^*_{\left(C,D\right)}\), an abstract finite coarse shape category \(Sh^{*^f}_{\left(C,D\right)}\) is obtained. In a special case \(C=HTop\) and \(D=HPol\), one speaks of a topological finite coarse shape category \(Sh^{*^f}\). The finite coarse shape category is also defined using an intrinsic approach in a compact metric case. The categorical isomophism of the finite coarse shape categories obtained by these two approaches is constructed.
On the existence of \(D(-3)\)-quadruples in \(\mathbb{Z}[X]\)
Ana Jurasić
Department of Mathematics, University of
Rijeka
(joint work with Alan Filipin)
Definition Let \(m\geq 2\) and let \(R\) be a commutative ring with unity. Let \(n\in R\) be a non-zero element and let \(\{a_{1}, \ldots ,a_{m}\}\) be a set
of \(m\) distinct non-zero elements from \(R\) such that \(a_{i}a_{j}+n\) is a square of an element of \(R\) for \(1\leq i < j\leq m\). The set \(\{a_{1}, \ldots ,a_{m}\}\) is called a Diophantine \(m\)-tuple with the property \(D(n)\) or simply a \(D(n)\)-\(m\)-tuple in \(R\).
Brown [1] proved that if \(n\) is an integer and \(n\equiv 2\ ({\rm mod}\ {4})\), then there is no \(D(n)\)-quadruple of integers. Dujella [3] proved that if \(n\) is an integer \(n\not\equiv 2\ ({\rm mod}\ {4})\), and \(n\notin S=\{-4,-3,-1,3,5,8,12,20\}\), then there exists at least one \(D(n)\)-quadruple of integers. There is also Dujella's conjecture [4]:
For \(n\in S=\{-4,-3,-1,3,5,8,12,20\}\) there does not exist a \(D(n)\)-quadruple of integers.
If \(R\) is a polynomial ring and \(n\) is a non-zero integer, we assume that not all polynomials in a \(D(n)\)-tuple are constant. In integer case, the set \(S\) can be reduced to the set \(S'=\{-3, -1, 3, 5, 8, 20\}\), but not for polynomials. Filipin and Jurasić [5] proved that there does not exist a \(D(n)\)-quadruple in \(\mathbb{Z}[X]\) for a positive integer \(n\) which is not a perfect square. Dujella and Fuchs [4] proved that there is no such \(D(-1)\)-quadruple. Bonciocat, Cipu and Mignotte [2] proved that there are no \(D(-1)\)-quadruple and \(D(-4)\)-quadruple of integers, so there does not exist \(D(-4)\)-quadruple in \(\mathbb{Z}[X]\). We consider the case where \(R=\mathbb{Z}[X]\) and \(n=-3\) and, by the following theorem, we complete the proof of a polynomial variant of Dujella's conjecture:
Theorem There does not exist a \(D(-3)\)-quadruple in \(\mathbb{Z}[X]\).
Let \(\mathbb{Z}^{+}[X]\) be the set of all polynomials with integer coefficients with positive leading coefficient. For \(a,b\in\mathbb{Z}[X]\), \(a means that \(b-a\in\mathbb{Z}^{+}[X]\). Let \(\{a,b,c\}\), such that \(0< a< b< c\), be a \(D(-3)\)-triple in \(\mathbb{Z}[X]\) and \begin{eqnarray*} ab-3=r^{2},\ ac-3=s^{2},\ bc-3=t^{2}, \end{eqnarray*} where \(r,s,t\in\mathbb{Z}^{+}[X]\). The proof that a \(D(-3)\)-triple in \(\mathbb{Z}[X]\) can not be extended to a \(D(-3)\)-quadruple in \(\mathbb{Z}[X]\) starts with transforming the problem of extending a \(D(-3)\)-triple \(\{a,b,c\}\) to a \(D(-3)\)-quadruple \(\{a,b,c,d\}\) in \(\mathbb{Z}[X]\) into solving a system of simultaneous pellian equations. Solutions of that system appear from the intersections of the obtained binary recurrent sequences of polynomials, for which we completely determine possible initial terms. Since, if \(\{a,b,c,d\}\) is a \(D(-3)\)-quadruple in \(\mathbb{Z}[X]\), then \(\big\{\frac{a}{\sqrt{3}},\frac{b}{\sqrt{3}},\frac{c}{\sqrt{3}},\frac{d}{\sqrt{3}}\big\}\) is a \(D(-1)\)-quadruple in \(\mathbb{R}[X]\) and also \(\big\{\frac{a}{\sqrt{3}}i,\frac{b}{\sqrt{3}}i,\frac{c}{\sqrt{3}}i,\frac{d}{\sqrt{3}}i\big\}\) is a \(D(1)\)-quadruple in \(\mathbb{C}[X]\), to complete our proof we use the parallelism of the original problem with the problems in \(\mathbb{R}[X]\) and \(\mathbb{C}[X]\).
References:
Brown [1] proved that if \(n\) is an integer and \(n\equiv 2\ ({\rm mod}\ {4})\), then there is no \(D(n)\)-quadruple of integers. Dujella [3] proved that if \(n\) is an integer \(n\not\equiv 2\ ({\rm mod}\ {4})\), and \(n\notin S=\{-4,-3,-1,3,5,8,12,20\}\), then there exists at least one \(D(n)\)-quadruple of integers. There is also Dujella's conjecture [4]:
For \(n\in S=\{-4,-3,-1,3,5,8,12,20\}\) there does not exist a \(D(n)\)-quadruple of integers.
If \(R\) is a polynomial ring and \(n\) is a non-zero integer, we assume that not all polynomials in a \(D(n)\)-tuple are constant. In integer case, the set \(S\) can be reduced to the set \(S'=\{-3, -1, 3, 5, 8, 20\}\), but not for polynomials. Filipin and Jurasić [5] proved that there does not exist a \(D(n)\)-quadruple in \(\mathbb{Z}[X]\) for a positive integer \(n\) which is not a perfect square. Dujella and Fuchs [4] proved that there is no such \(D(-1)\)-quadruple. Bonciocat, Cipu and Mignotte [2] proved that there are no \(D(-1)\)-quadruple and \(D(-4)\)-quadruple of integers, so there does not exist \(D(-4)\)-quadruple in \(\mathbb{Z}[X]\). We consider the case where \(R=\mathbb{Z}[X]\) and \(n=-3\) and, by the following theorem, we complete the proof of a polynomial variant of Dujella's conjecture:
Theorem There does not exist a \(D(-3)\)-quadruple in \(\mathbb{Z}[X]\).
Let \(\mathbb{Z}^{+}[X]\) be the set of all polynomials with integer coefficients with positive leading coefficient. For \(a,b\in\mathbb{Z}[X]\), \(a means that \(b-a\in\mathbb{Z}^{+}[X]\). Let \(\{a,b,c\}\), such that \(0< a< b< c\), be a \(D(-3)\)-triple in \(\mathbb{Z}[X]\) and \begin{eqnarray*} ab-3=r^{2},\ ac-3=s^{2},\ bc-3=t^{2}, \end{eqnarray*} where \(r,s,t\in\mathbb{Z}^{+}[X]\). The proof that a \(D(-3)\)-triple in \(\mathbb{Z}[X]\) can not be extended to a \(D(-3)\)-quadruple in \(\mathbb{Z}[X]\) starts with transforming the problem of extending a \(D(-3)\)-triple \(\{a,b,c\}\) to a \(D(-3)\)-quadruple \(\{a,b,c,d\}\) in \(\mathbb{Z}[X]\) into solving a system of simultaneous pellian equations. Solutions of that system appear from the intersections of the obtained binary recurrent sequences of polynomials, for which we completely determine possible initial terms. Since, if \(\{a,b,c,d\}\) is a \(D(-3)\)-quadruple in \(\mathbb{Z}[X]\), then \(\big\{\frac{a}{\sqrt{3}},\frac{b}{\sqrt{3}},\frac{c}{\sqrt{3}},\frac{d}{\sqrt{3}}\big\}\) is a \(D(-1)\)-quadruple in \(\mathbb{R}[X]\) and also \(\big\{\frac{a}{\sqrt{3}}i,\frac{b}{\sqrt{3}}i,\frac{c}{\sqrt{3}}i,\frac{d}{\sqrt{3}}i\big\}\) is a \(D(1)\)-quadruple in \(\mathbb{C}[X]\), to complete our proof we use the parallelism of the original problem with the problems in \(\mathbb{R}[X]\) and \(\mathbb{C}[X]\).
References:
- E. Brown, Sets in which \(xy + k\) is always a square, Math. Comp. 45 (1985), 613-620.
- N. Ciprian Bonciocat, M. Cipu and M. Mignotte, There is no Diophantine \(D(-1)\)-quadruple, J. London Math. Soc. 105 (2022), 63-99.
- A. Dujella, Generalization of a problem of Diophantus, Acta Arith. 65 (1993), 15-27.
- A. Dujella and C. Fuchs, A polynomial variant of a problem of Diophantus and Euler, Rocky Mountain J. Math. 33 (2003), 797-811. A. Filipin and A. Jurasić, A polynomial variant of a problem of Diophantus and its consequences, Glas. Mat. Ser. III, 54 (2019), 21-52.
Two lines and a Lelek fan
Judy Kennedy
Lamar University, USA
(joint work with Iztok Banič and Goran Erceg)
We give an example of an inverse limit with with bonding map one set-valued function such that the resulting space is a Lelek fan. Also, the induced shift map on the inverse limit has one fixed point, no other periodic points, and has positive topological entropy.
The Zero Entropy Locus for the Lozi Maps
Kristijan Kilassa Kvaternik
University of Zagreb, Croatia
(joint work with M. Misiurewicz and S. Štimac)
The Lozi map family is a 2-parameter family of piecewise affine homeomorphisms of the Euclidean plane given by
$$L_{a,b}\colon\mathbb{R}^2\rightarrow\mathbb{R}^2,\ L_{a,b}(x,y)=(1+y-a|x|,bx),$$
where \(a,b\in\mathbb{R}\). In this talk we will present an expansion of the known results about the topological entropy of the Lozi map, \(h_{top}(L_{a,b})\), by proving that \(h_{top}(L_{a,b})=0\) in a specific region in the parameter space for which the period-two orbit is attracting and there are no homoclinic points for the fixed point \(X\) in the first quadrant. This is joint work with Michał Misiurewicz (IUPUI, Indianapolis) and Sonja Štimac (University of Zagreb).
Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary
Panki Kim
Department of Mathematical Sciences,
Seoul National University, Seoul, Republic of Korea
(joint work with Renming Song and Zoran Vondraček)
In this talk we discuss the potential theory of Markov processes with
jump kernels degenerate at the boundary.
To be more precise, we consider processes in \({\mathbb R}^d_+\) with jump kernels of the form \({\mathcal B}(x,y) |x-y|^{-d-\alpha}\) and
killing potentials \(\kappa(x)=cx_d^{-\alpha}\), \(0<\alpha<2\).
The boundary part \({\mathcal B}(x,y)\) is comparable to
the product of four terms with parameters \(\beta_1, \beta_2\), \(\beta_3\) and
\(\beta_4\) appearing as exponents in these terms,
and \({\mathcal B}(x,y)\) is allowed to decay at the boundary.
The constant \(c\) in the killing term can be written as a function of \(\alpha\), \({\mathcal B}\) and
a parameter \(p\in ((\alpha-1)_+, \alpha+\beta_1)\),
which is strictly increasing in \(p\),
decreasing to 0 as \(p\downarrow (\alpha-1)_+\) and increasing to \(\infty\) as \(p\uparrow\alpha+\beta_1\).
We establish sharp two-sided estimates on the Green
functions of these processes for
all \(p\in ((\alpha-1)_+, \alpha+\beta_1)\) and all admissible values of
\(\beta_1, \beta_2\), \(\beta_3\) and \(\beta_4\).
Depending on the regions where \(\beta_1\), \(\beta_2\) and \(p\) belong,
the estimates on the Green functions are different. In fact, the estimates have
three different forms depending on the regions the parameters belong to.
As applications,
we completely determine the region of the parameters where the boundary Harnack principle holds.
Chebyshev-Steffensen inequality involving the inner product
Milica Klaričić Bakula
Faculty of Science, University of Split, Croatia
(joint work with Josip Pečarić)
The classical Chebyshev inequality states that
$$
\sum_{j=1}^{n}p_{j}\sum_{i=1}^{n}p_{i}a_{i}b_{i}\geq
\sum_{j=1}^{n}p_{j}a_{j}\sum_{i=1}^{n}p_{i}b_{i}
$$
whenever \(\boldsymbol{a}=\left( a_{1},\ldots ,a_{n}\right)\), \(\boldsymbol{b}
=\left( a_{1},\ldots ,b_{n}\right)\) are real \(n\)-tuples monotonic in the
same direction, and \(\boldsymbol{p}=\left( p_{1},\ldots ,p_{n}\right)\) a
positive \(n\)-tuple. J. Pečarić proved that the assumptions on
\(\boldsymbol{p}\) can be relaxed to Steffensen's conditions: namely, if
\(\boldsymbol{p}\) is a real \(n\)-tuple such that
$$
0\leq P_{i}=p_{1}+\cdots +p_{i}\leq P_{n},\ i\in \left\{ 1,\ldots
,n-1\right\} ,
$$
and \(P_{n}>0\), the Chebyshev inequality, now called the
Chebyshev-Steffensen inequality, still holds.
We present the Chebyshev-Steffensen inequality involving the inner product on the real \(m\)-space, and some new upper bounds for the weighted Chebyshev-Steffensen functional as well as the Jensen-Steffensen functional involving the inner product under various conditions. We show how these results can be used to establish new upper bounds for the Jensen-Steffensen functional for certain generalized convex functions such as \(P\)-convex functions and functions with nondecreasing increments.
We present the Chebyshev-Steffensen inequality involving the inner product on the real \(m\)-space, and some new upper bounds for the weighted Chebyshev-Steffensen functional as well as the Jensen-Steffensen functional involving the inner product under various conditions. We show how these results can be used to establish new upper bounds for the Jensen-Steffensen functional for certain generalized convex functions such as \(P\)-convex functions and functions with nondecreasing increments.
What does an orbit tell about a parabolic diffeomorphism?
Martin Klimeš
Sveučilište u Zagrebu, Fakultet elektrotehnike i računarstva
(joint work with Pavao Mardešić, Goran Radunović and Maja Resman)
Let \(f(x)=x+\ldots\) be a germ of an analytic diffeomorphism that is tangent to the identity,
and let \(\mathbb A=\{x_0,\ x_1=f(x_0),\ x_2=f(f(x_0)),\ \ldots\}\) be an orbit under the positive iteration by \(f\) of some point \(x_0\in\mathbb C\setminus\{0\}\), such that \(x_n\to 0\) as \(n\to+\infty\).
The questions are: What can such single orbit \(\mathbb A\) say about the germ \(f\)? and: What information about \(f\) can be obtained through fractal analysis of \(\mathbb A\)?
In particular, we are interested in the analytic invariants of \(f\) which express the intrinsic properties of \(f\) that are preserved under analytic conjugation
\(f\sim \phi^{-1}\circ f\circ\phi\) associated with analytic changes of coordinate \(z'=\phi(z)\).
These invariants, first described in the works of Birkhoff (1939), Écalle (1975, 1981) and Voronin (1981), rely on existence of sectorial realizations of
a divergent formal Fatou coordinate, and express the general mismatch of these realizations on different sectors.
In our work we introduce a dynamical theta function \(\Theta_{\mathbb A}\) of the set \(\mathbb A\),
and show that one can obtain the sectorial Fatou coordinates as its Laplace type integral transforms.
As a consequence, the analytic invariants of \(f\) can be read from the function \(\Theta_{\mathbb A}\).
Robust Variants of the Maximum WeightedIndependent Set Problem on Trees
Ana Klobučar Barišić
Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Croatia
(joint work with Robert Manger)
An independent set is a set of vertices of a graph G that are not adjacent to each
other. Suppose that vertices of G are given non-negative weights. Then, the weight
of an independent set is defined as the sum of its vertex weights. The maximum
weighted independent set (MWIS) problem consists of finding an independent set
whose weight is as large as possible. The MWIS problem is important because it
models different applications such as facility location, selection of non-overlapping
time slots, labelling of digital maps, etc. To model mentioned applications even
more accurate, we need to take into account uncertainty of input parameters.
For such reason, this work studies robust variants of the MWIS problem. Uncertainty of vertex weights is given by discrete set of scenarios. The study is restricted to cases where the involved graph is a tree.
Since the conventional MWIS problem is NP-hard in general, all its robust variants must also be NP-hard in general. But MWIS problem on trees can be solved in polynomial time. This gives rise to a hope that robust variants of the same problem on trees can also be solved more efficiently than in the general case.
In this work we first prove that most robust variants of MWIS problem on trees are unfortunately NP-hard. Next, we give a heuristic for solving the considered robust MWIS variants, which is customized for trees. Finally, we demonstrate by experiments that our algorithm produces high-quality solutions and runs much faster than a general-purpose optimization software.
For such reason, this work studies robust variants of the MWIS problem. Uncertainty of vertex weights is given by discrete set of scenarios. The study is restricted to cases where the involved graph is a tree.
Since the conventional MWIS problem is NP-hard in general, all its robust variants must also be NP-hard in general. But MWIS problem on trees can be solved in polynomial time. This gives rise to a hope that robust variants of the same problem on trees can also be solved more efficiently than in the general case.
In this work we first prove that most robust variants of MWIS problem on trees are unfortunately NP-hard. Next, we give a heuristic for solving the considered robust MWIS variants, which is customized for trees. Finally, we demonstrate by experiments that our algorithm produces high-quality solutions and runs much faster than a general-purpose optimization software.
Primitive Element Theory in Algebraic Function Fields Associated with Modular Curves
Iva Kodrnja
University of Zagreb, Croatia
(joint work with Goran Muić)
When mapping modular curves to the projective plane via modular forms, the image curve's function field is an algebraic function field generated by quotients of defining modular forms. Using the theory of primitive elements in this finite separable extension, we can find generators of this extension and prove birationality of the map, thus obtaining equations for these planar models of modular curves.
In the case of function fields, finding primitive elements is a complex task and we offer two methods; one is the trial method and the other is the use of estimates for primitive elements.
In the case of function fields, finding primitive elements is a complex task and we offer two methods; one is the trial method and the other is the use of estimates for primitive elements.
Bershadsky-Polyakov vertex algebras at positive integer levels and duality
Ana Kontrec
Max Planck Institute for Mathematics, Bonn, Germany and University of Zagreb, Croatia
(joint work with Dražen Adamović)
One of the simplest examples of \(\mathcal{W}\)-algebras is the Bershadsky-Polyakov vertex algebra \(\mathcal{W}^k(\mathfrak{g}, f_{min})\), associated to \(\mathfrak{g} = sl(3)\) and the minimal nilpotent element \(f_{min}\).
We study the simple Bershadsky-Polyakov algebra \(\mathcal W_k\) at positive integer levels and obtain a classification of their irreducible modules.
In the case \(k=1\), we show that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple affine vertex superalgebra \(L_{k'} (osp(1 \vert 2))\) for \(k'=-5/4\).
Schur-convexity property of the general twopoint weighted quadrature formula
Sanja Kovač
University of Zagreb, Faculty of Geotechnical Engineering, Varaždin, Croatia
Recently, the Schur-convexity property of the weighted midpoint and trapezoid quadrature formula has been investigated. Now, we shall generalize those results on the general two-point integral formula and derive the necessary and sufficient assumptions for Schur-convexity property.
On the quantum affine vertex algebras in type \(A\)
Slaven Kožić
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
One interesting problem in the theory of vertex algebras is to associate certain vertex algebra-like objects, the so-called quantum vertex algebras, to
various classes of quantum groups, such as quantum affine algebras.
Roughly speaking, the goal is to establish a correspondence between these structures that goes in
parallel with
the already established connection between affine Lie algebras and vertex
algebras.
In my talk, I will discuss this problem in the context of Etingof-Kazhdan's quantum vertex algebra associated with the trigonometric \(R\)-matrix of type \(A\). More specifically, in this setting, the usual notion of (quantum) vertex algebra module does no longer seem to be suitable, so we use Li's notion of \(\phi\)-coordinated module instead. This allows us to prove that a certain broad class of modules for the (suitably completed) quantum affine algebra of type \(A\) coincides with the class of \(\phi\)-coordinated modules for the aforementioned quantum vertex algebra.
On variants of Pillai's problems with polynomials
Dijana Kreso
TU Graz
(joint work with Robert F. Tichy)
In this talk I will discuss several polynomial variants of some well-known Pillai's Diophantine problems. In particular, in my focus is the equation
\[
a_1p(x)^{n_1}+b_1q(x)^{m_1}=a_2p(x)^{n_2}+b_2q(x)^{m_2}=f(x)
\]
in complex polynomials \(f, p, q\) with \(f\) nonzero and \(p\) and \(q\) nonconstant, nonzero complex numbers \(a_i, b_i\) and positive integers \(n_i, m_i\). Of particular importance is the special case \(a_1=a_2=1\) and \(b_1=b_2=-1\) corresponding to a well-studied Pillai's Diophantine equation \(a^{n_1}-b^{m_1}=a^{n_2}-b^{m_2}\) in positive integers \(n_i, m_i, a, b\) with \(a>1\) and \(b>1\).
Furthermore, I will show that for nonconstant coprime complex polynomials \(p\) and \(q\), the number of solutions \((n, m)\in \mathbb{N}^2\) of \(0\leq \deg \left(p(x)^n-q(x)^m\right)\leq d\) is asymptotically equal to \(d^2/(\deg p\deg q)\) as \(d\to \infty\), as well as consider a generalization of this problem where the powers of polynomials are replaced by the sums of powers of polynomials.
Furthermore, I will show that for nonconstant coprime complex polynomials \(p\) and \(q\), the number of solutions \((n, m)\in \mathbb{N}^2\) of \(0\leq \deg \left(p(x)^n-q(x)^m\right)\leq d\) is asymptotically equal to \(d^2/(\deg p\deg q)\) as \(d\to \infty\), as well as consider a generalization of this problem where the powers of polynomials are replaced by the sums of powers of polynomials.
On joint weak convergence of partial sum and maxima processes
Danijel Krizmanić
Faculty of Mathematics, University of Rijeka, Croatia
For a stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint
regular variation with index \(\alpha \in (0,2)\) and weak dependence conditions. The convergence takes place in the space
of \(\mathbb{R}^{2}\)-valued càdlàg functions on
\([0,1]\) with the Skorohod weak \(M_{1}\) topology, and the limiting process consists of an
\(\alpha\)-stable Lévy process and an extremal process.
We also describe the dependence between these two components of the limit, and show that the weak \(M_{1}\)
topology in general can not be replaced by the standard \(M_{1}\) topology.
Some new inequalities involving the generalized Hardy operator
Kristina Krulić Himmelreich
University of Zagreb Faculty of Textile Technology, Croatia
In this talk we derive new inequalities involving the generalized Hardy operator. The obtained results generalized known inequalities involving the Hardy operator. We also get new inequalities involving the classical Hardy-Hilbert inequality.
REGULARITY CRITERIA FOR THE NAVIER-STOKES EQUATIONS
Petr Kučera
Czech Technical University in Prague, Czech Republic
We deal with the initial-boundary value Navier-Stokes problem for incompressible fluid in a bounded domain. We study the conditional regularity of the weak solutions to the Navier-Stokes equations. We prove an optimal regularity criterion in terms of \( u \cdot \nabla | u|^\lambda\) for bounded domains with Navier's, Navier-type and Dirichlet boundary conditions. In particular choise of \(\lambda=2\) yields that the control of the energy flow in the critical norms \(L_t^pL_x^r\) provides the regularity of solutions. This criterion is proved by two different techniques. Further, we prove the regularity criteria which are based on the directional derivatives of several fundamental quantities along the streamlines, the pressure, the velocity field and the Bernoulli pressure.
Subgeometric ergodicity of regime-switching diffusion processes
Petra Lazić
University of Zagreb, Faculty of Science, Department of Mathematics
(joint work with Nikola Sandrić)
I will discuss subgeometric ergodicity of a class of regime-switching diffusion processes. These are processes that, beside the continuous, diffusive one, have a second, discrete component which changes the behaviour of the process at random times. The theory about them is quite interesting as it shows that in many aspects they exhibit different characteristics than classical diffusion processs. In this talk, I will derive conditions on the drift and diffusion coefficients which result in subgeometric ergodicity of the corresponding semigroup, that is, which allow us to find explicit bounds on the rate of the convergence with respect to two distance functions: the total variation distance and the class of Wasserstein distances.
Asymptotic mixed normality of approximate maximum likelihood estimator of drift parameters in ergodic diffusion model
Snježana Lubura Strunjak
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with Miljenko Huzak)
We assume that the ergodic diffusion \(X\) satisfies a stochastic differential equation of the form:
\(dX_t=\mu(X_t,\theta)dt+\sigma_0\nu(X_t)dW_t\), with unknown drift parameter \(\theta\) and known diffusion coefficient parameter \(\sigma_0\). We prove that the difference between approximate maximum likelihood estimator of drift parameter \(\bar{\theta}_{n,T}\) obtained from discrete
observations \((X_{i\Delta_n},0\leq i\leq \lfloor nT\rfloor)\) along time interval \([0,T]\), and MLE obtained from continuous observations \((X_t,0\leq t\leq T)\) along time interval \([0,T]\) has asymptotic mixed normal distribution with covariance matrix which depends on true parameter value and on path \((X_t, t\geq 0)\) with the rate of convergence \(\sqrt{\Delta_n T}\), when \(n\) and \(T\) goes to infinity in a way that \(T\Delta_n\) goes to zero.
Construction of strongly regular graphs having an automorphism group of composite order
Marija Maksimović
Faculty of Mathematics, University of Rijeka, Croatia
(joint work with Dean Crnković)
In this talk we will outline a method for constructing strongly regular graphs from orbit matrices admitting an automorphism group of composite order. The construction is a generalization of work of C. Lam and M. Behbahani that in 2011. introduced an algorithm for construction of orbit matrices of strongly regular graphs with a presumed automorphism group of prime order, and
construction of corresponding strongly regular graphs. We applied this method and constructed some new strongly regular graphs.
GENERALIZED WEYL ALGEBRA APPLIED TO REALIZATIONS OF \(SO(n)\), LORENTZ ALGEBRA AND THEIR EXTENSIONS
Tea Martinić Bilać
Department of Mathematics, Faculty of science, Univesity of Split Croatia
(joint work with Saša Krešić-Jurić and Stjepan Meljanac)
We introduce the generalized Heisenberg algebra \(\mathcal{H}_{n}\) and construct the Weyl realization of the orthogonal algebra \(so(n)\) and Lorentz algebra \(so(1, n-1)\) by formal power series in semicompletion of \(\mathcal{H}_{n}\). The obtained realizations are given in terms of the generating function for the Bernoulli numbers. We also introduce an a extension of the orthogonal and Lorentz algebras by quantum angles and study realizations of the extended algebras in \(\mathcal{H}_{n}\).
Symmetric Realization of the Stasheff Polytope
Ivica Martinjak
Zagreb, Croatia
(joint work with Ana Mimica)
One of the fruitfull problems arising from the determinant theory was the Alternating Sign Matrix Conjecture, addressing enumeration of these matrices. An evaluation of determinants called condensation has a natural interpretation. These are matrices that generalize permutation matrices and which entries are -1, 0 and 1, alternating in sign. Moreover, the sum of each row and column of these matrices (ASM) is equal to 1. Although the ASM Conjecture has been solved, many intriguing problems remain.
We study a family of AMS with pattern avoidance and the property that the rightmost 1 in the row \(i+1 \ge 2\) is to the right of the leftmost 1 in the row \(i\), and further families with analogue properties. A recursive nature of the family provides enumeration of the matrices. Stasheff's polytope, also known as asociahedron, first appeared in Jim Staheff's work in the 1960's, and had many realizations latter. A \(d\)-asociahedron is a convex polytope having incidence structure isomorphic to the lattice of convex \((d+3)\)-gon triangulation. We establish a one to one correspondence between the family of AMS of order \(d+2\) and vertices of \(d\)-asociahedron. Further geometrical and combinatorial properties of this realization of the Stasheff polytope are studied, as well as related polytopes arising from the family of ASM.
We study a family of AMS with pattern avoidance and the property that the rightmost 1 in the row \(i+1 \ge 2\) is to the right of the leftmost 1 in the row \(i\), and further families with analogue properties. A recursive nature of the family provides enumeration of the matrices. Stasheff's polytope, also known as asociahedron, first appeared in Jim Staheff's work in the 1960's, and had many realizations latter. A \(d\)-asociahedron is a convex polytope having incidence structure isomorphic to the lattice of convex \((d+3)\)-gon triangulation. We establish a one to one correspondence between the family of AMS of order \(d+2\) and vertices of \(d\)-asociahedron. Further geometrical and combinatorial properties of this realization of the Stasheff polytope are studied, as well as related polytopes arising from the family of ASM.
On the High Relative Accuracy of the HZ Method
Josip Matejaš
Faculty of Economics Business, University of Zagreb, Croatia
(joint work with Vjeran Hari)
We prove the high relative accuracy of the two-sided Hari-Zimmermann (HZ) method that solves the generalized eigenvalue problem (GEP)
$$ Ax=\lambda Bx\,, \quad x\neq 0\,, $$
where \(A\), \(B\) are real symmetric positive definite matrices of order \(n\). The HZ method can be considered as a normalized version of the Falk-Langemeyer (FL) method. Both methods diagonalize the pivot submatrices in each step, so they extend the standard Jacobi eigenvalue method to matrix pairs.
The HZ method first normalizes \(B\) to have the unit diagonal and then maintains that property during the iterative process.
We prove that the HZ method computes the eigenvalues of the GEP with high relative accuracy on the class of well-behaved pairs of real positive definite matrices. These are the pairs of matrices \((A,B)\) for which the spectral conditions \(\kappa_2(A_S)\) and \(\kappa_2(B_S)\) are small where \(A_S=D_A^{-1/2}AD_A^{-1/2}\), \(B_S=D_B^{-1/2}BD_B^{-1/2}\) and \(D_A=\mbox{diag}(A)\), \(D_B=\mbox{diag}(B)\). Numerical tests confirm the obtained theoretical results.
The proof considers one step of the method. It uses sophisticated error analysis that allows finding exact expressions for the relative errors of key variables in the algorithm. In this way, we get a deeper insight into the structure of errors and what they depend on. Such an approach gives very sharp accuracy bounds. If future computers are to be fast at the expense of accuracy, such fine error analysis can be critical to obtaining usable accuracy results.
High relative accuracy qualifies the HZ method to serve as the core algorithm within the block methods for solving the GEP and the generalized singular value decomposition problem (GSVDP). The block methods are the primary choice for solving GEP and GSVDP on modern parallel CPU and GPU computing machines. The HZ method can be used as a stand-alone method to solve the GEP, while its one-sided version is suitable for solving the GSVDP on high-performance computers.
We prove that the HZ method computes the eigenvalues of the GEP with high relative accuracy on the class of well-behaved pairs of real positive definite matrices. These are the pairs of matrices \((A,B)\) for which the spectral conditions \(\kappa_2(A_S)\) and \(\kappa_2(B_S)\) are small where \(A_S=D_A^{-1/2}AD_A^{-1/2}\), \(B_S=D_B^{-1/2}BD_B^{-1/2}\) and \(D_A=\mbox{diag}(A)\), \(D_B=\mbox{diag}(B)\). Numerical tests confirm the obtained theoretical results.
The proof considers one step of the method. It uses sophisticated error analysis that allows finding exact expressions for the relative errors of key variables in the algorithm. In this way, we get a deeper insight into the structure of errors and what they depend on. Such an approach gives very sharp accuracy bounds. If future computers are to be fast at the expense of accuracy, such fine error analysis can be critical to obtaining usable accuracy results.
High relative accuracy qualifies the HZ method to serve as the core algorithm within the block methods for solving the GEP and the generalized singular value decomposition problem (GSVDP). The block methods are the primary choice for solving GEP and GSVDP on modern parallel CPU and GPU computing machines. The HZ method can be used as a stand-alone method to solve the GEP, while its one-sided version is suitable for solving the GSVDP on high-performance computers.
On Čech systems
Vlasta Matijević
Faculty of Science, University of Split, Croatia
(joint work with L. Rubin, University of Oklahoma, USA)
To each space \(X\) is associated its Čech system \((|N(\mathcal{U}
)|,\left[ p_{\mathcal{UV}}\right] ,\operatorname{Cov}(X))\) which is an
inverse system in the homotopy category \(HTop\). We consider a question whether
it is possible to select a map \(q_{\mathcal{UV}}\in\left[ p_{\mathcal{UV}
}\right] \) in each homotopy class \(\left[ p_{\mathcal{UV}}\right] \) such
that the induced system \((|N(\mathcal{U})|,q_{\mathcal{UV}},\operatorname{Cov}(X))\) is an approximate system (in the sense of
Mardešić-Watanabe). It is already known that, generally, the answer
to that question is negative since each chainable continuum \(X\) provides a
counterexample. Now, we give a full answer by proving that a paracompact
Hausdorff space \(X\) admits an approximate system \((|N(\mathcal{U}
)|,q_{\mathcal{UV}},\operatorname{Cov}(X))\) if and only if \(X\) is strongly
\(0\)-dimensional. The proof is based on certain properties of simplicial
systems, i.e systems consisting of simplicial complexes and simplicial bonding maps.
Chebyshev's inequality of the Mercer type
Anita Matković
Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Croatia
(joint work with Josip Pečarić)
We consider the discrete Jensen-Mercer inequality and Chebyshev's inequality of the Mercer type. We present bounds for Chebyshev's functional of the Mercer type and bounds for the Jensen-Mercer functional in terms of the discrete Ostrowski inequality. We show how these results can be used to establish new refinements of the considered inequalities.
Coinciding mean for the two symmetries on the set of mean functions
Lenka Mihoković
Faculty of Electrical Engineering and Computing, Zagreb, Croatia
On the set \(\mathcal M\) of mean functions
the symmetric mean of \(M\) with respect to fixed mean \(M_0\)
can be defined in several ways.
One definition is related to the group structure on \(\mathcal M\) retrieved from the abelian group of antisymmetric functions
and the other one is defined trough Gauss' functional equation also known as the invariance equation.
We discuss some properties of such symmetries trough connection with asymptotic expansions of means involved. As a result of coefficient comparison, new class of means was discovered which interpolates between harmonic, geometric and arithmetic mean.
Generalized Weierstrass Points and Modular Curves
Damir Mikoč
Department of Teacher Studies in Gospić, University of Zadar
(joint work with Goran Muić)
We are interested in modular curves and modular forms for some Fuchsian group of the first kind \(\Gamma\), especially for the group \(\Gamma_0(N)\).
For an even integer \(m\ge 4\), we are studying Weierstrass and \(\frac{m}{2}\)-Weierstrass points, on curve \(\mathfrak{R}_\Gamma\) in the language of modular forms. We describe algorithm implemented in SAGE for checking whether cusp \(\mathfrak{a}_\infty\) is a \(\frac{m}{2}\)-Weierstrass point on \(\mathfrak{R}_\Gamma\). As a consequence of that algorithm, we calculated the equations of all canonical curves of type \(X_0(N) \), genus \(3 \le g \le 5\), which are not hyperelliptic.
Continuous transformation of regularly varying elements on bornological spaces
Nikolina Milinčević
University of Zagreb, Croatia
(joint work with Bojan Basrak and Ilya Molchanov)
We consider regular variation on Polish spaces using the notion of bornologies. We impose conditions on a continuous transformation \(\psi\) between two bornological spaces \((\mathbb{X}, \mathcal{S})\) and \((\mathbb{Y}, \mathcal{S}')\) in order to keep regularly varying property. In other words, we show that under certain conditions, if \(X\) is regularly varying random element in \((\mathbb{X}, \mathcal{S})\), then \(\psi(X)\) is regularly varying in \((\mathbb{Y}, \mathcal{S}')\).
Identifying parameter regions for oscillations in reaction networks models
Maya Mincheva
Northern Illinois University, Dekalb IL, USA
(joint work with Carsten Conradi, HTW-Berlin, Germany)
Parametric differential equations' models of biochemical reaction networks often show oscillations. Finding parameter values for oscillations usually involves an application of a Hopf bifurcation criterion where the signs of the Hurwitz determinants have to be determined. The Hurwitz determinants are large parametric polynomials whose dominant terms can determine their signs. We propose a method for finding dominant terms in the Hurwitz determinants associated with the product of diagonal entries in the Hurwitz matrix. The method is applied to a cyclic dual phosphorylation network showing periodic concentrations. (Preliminary report.)
DRGs and new block designs obtained from the Mathieu groups
Nina Mostarac
Faculty of Mathematics, University of Rijeka, Croatia
(joint work with Dean Crnković and Andrea Švob)
We describe a construction of distance-regular graphs admitting a vertex transitive action of the five Mathieu groups. From the binary code spanned by an adjacency matrix of the strongly regular graph with parameters
\((176,70,18,34)\) we obtain block designs having
the full automorphism groups isomorphic to the Higman-Sims finite simple group. From the same code we also obtain eight 2-designs whose full automorphism group
is isomorphic to the Mathieu group \(M_{22}\), and whose existence cannot be explained neither by the Assmus-Mattson
theorem nor by a transitivity argument.
Modification of the search algorithm for extremal \(\mathbb{Z}_4\)-codes
Matteo Mravić
Faculty of mathematics, University of Rijeka, Croatia
(joint work with Sanja Rukavina)
A \(\mathbb{Z}_4\)-code \(C\) of length \(n\) is a \(\mathbb{Z}_4\) sub-module of \(\mathbb{Z}_4^n\). With respect to the standard inner product modulo 4, the dual code \(C^\bot\) of the \(\mathbb{Z}_4\)-code \(C\) is defined. The code \(C\) is self-dual if \(C=C^\bot\). There are two binary codes associated with a \(\mathbb{Z}_4\)-code \(C\) called a residue code and a torsion code. These two codes are a starting point in the construction of self-dual \(\mathbb{Z}_4\)-codes by the method given in [1]. For \(\mathbb{Z}_4\)-codes, the Euclidean weight of codeword \(x\) is defined by \(n_1\left(x\right)+4n_2\left(x\right)+n_3\left(x\right)\), where \(n_i\left(x\right)\) is the number of components of \(x\) which are equal to \(i\). A \(\mathbb{Z}_4\)-code \(C\) of length \(n\) is said to be extremal if its minimal Euclidean weight is \(8\left\lfloor\frac{n}{24}\right\rfloor\ +\ 8\). In this talk, we will discuss an algorithm that improves the search for extremal self-dual \(\mathbb{Z}_4\)-codes which we used to obtain some new extremal codes of lengths 32 and 40.
References:
References:
- V. Pless, J. S. Leon, J. Fields, All \(\mathbb{Z}_4\)-codes of Type II and length 16 are known, J. Comb. Theory Ser. A. 78 (1997) 32-50.
Extractors in Paley graphs: a random model
Rudi Mrazović
University of Zagreb, Croatia
A well-known conjecture in analytic number theory states that for every pair of sets \(X,Y\subset\mathbf{Z}/p\mathbf{Z}\), each of size at least \(\log ^C p\) (for some constant \(C\)) we have that for \((\frac12+o(1))|X||Y|\) of the pairs \((x,y)\in X\times Y\), \(x+y\) is a quadratic residue modulo \(p\). We address the probabilistic analogue of this question, that is for every fixed \(\delta>0\), given a finite group \(G\) and \(A\subset G\) a random subset of density \(\frac12\), we prove that with high probability for all subsets \(|X|,|Y|\geq \log ^{2+\delta} |G|\) for \((\frac12+o(1))|X||Y|\) of the pairs \((x,y)\in X\times Y\) we have \(xy\in A\).
Hilbert's Irreducibility, Modular Forms, and Computation of Certain Galois Groups
Goran Muić
Department of Mathematics, Faculty of Science,
University of Zagreb,
Croatia
(joint work with I. Kodrnja)
We consider congruence subgroups \(\Gamma_0(N)\), \(N\ge 1\), and the corresponding compact Riemann surface \(X_0(N)\) which we initially consider as a complex irreducible smooth projective curve.
The \(\mathbb Q\)-structure on \(X_0(N)\) is defined in a standard way using \(j\)-function i.e., the field of rational functions over \(\mathbb Q\) on \(X_0(N)\) is given by
\(\mathbb Q\left(X_0(N)\right)= \mathbb Q(j, j(N \cdot))\). If we take that \(f, g, h\) are linearly independent modular forms of same even weight \(m\ge 2\)
with rational \(q\)-expansions for \(\Gamma_0(N)\), we can construct an irreducible over \(\mathbb Z\) homogeneous polynomial with integral coefficients \(P_{f, g, h}\) such that \(P_{f, g, h}(f, g, h)=0\) in
\(\mathbb Q\left(X_0(N)\right)\). Let \(Q_{f, g, h}\) be its dehomogenization with respect to the last variable. Again, we obtain
an irreducible over \(\mathbb Z\) polynomial with integral coefficients. By means of Hilbert's irreducibility theorem
we obtain a family of irreducible over \(\mathbb Z\)
polynomials with integral coefficients \(Q_{f, g, h}(\lambda, \cdot)\), where \(\lambda\) ranges over a certain thin set.
In this way we obtain a family of number fields determined as splitting fields of these polynomials -all of them have the same Galois group \(G_{f, g, h}\) which is the Galois group of the splitting field of
\(Q_{f, g, h}(g/f, \cdot)\) over \(\mathbb Q(g/f)\). The goal of present talk is to explain
these objects as an application of the theory developed in our earlier papers. We elaborate on example \(\Gamma_0(72)\).
The author acknowledges Croatian Science Foundation grant no. 3628.
The author acknowledges Croatian Science Foundation grant no. 3628.
Cores and some other properties of complementary prisms
Marko Orel
University of Primorska, Koper, Slovenia
Let \(\Gamma\) be a finite simple graph. Its complementary prism is the graph~\(\Gamma\bar{\Gamma}\), which obtained from \(\Gamma\) and its complement \(\bar{\Gamma}\) by connecting each vertex in \(\Gamma\) with its copy in \(\bar{\Gamma}\). In particular, \(\Gamma\bar{\Gamma}\) is the Petersen graph if \(\Gamma\) is the pentagon. The Petersen graph is an example of a core, which means that all its endomorphisms are automorphisms. The main question that the talk will address is the following: Which assumptions on \(\Gamma\) guarantee that \(\Gamma\bar{\Gamma}\) is a core? Here, the focus will be on graphs which either admit some degree of symmetry or have nice combinatorial properties. The existence of Hamiltonian paths/cycles of \(\Gamma\bar{\Gamma}\) will be addressed as well. On the other hand, the automorphism group and the isoperimetric number of a complementary prism will be presented for arbitrary finite simple graph \(\Gamma\). The talk will be based on:
M.Orel. A family of non-Cayley cores based on vertex-transitive or strongly regular self-complementary graph, https://arxiv.org/pdf/2110.10416.pdf (2021).
M.Orel. A family of non-Cayley cores based on vertex-transitive or strongly regular self-complementary graph, https://arxiv.org/pdf/2110.10416.pdf (2021).
Dirac index and associated cycles of Harish-Chandra modules
Pavle Pandžić
University of Zagreb, Faculty of Science, Zagreb, Croatia
(joint work with Salah Mehdi, David Vogan and Roger Zierau)
We show how, for certain Harish-Chandra
modules, the polynomial giving the dimension of the Dirac index of the corresponding coherent family can be expressed as an integer linear combination of the multiplicities in the characteristic cycle.
Whittaker modules for \(\widehat{\mathfrak gl}\) and \(\mathcal W_{1+ \infty}\)-modules which are not tensor products
Veronika Pedić Tomić
University of Zagreb, Croatia
(joint work with Dražen Adamović)
We consider the Whittaker modules \(M_{1}({\lambda},{\mu})\) for the Weyl vertex algebra \(M\) (also called \(\beta \gamma\) vertex algebra), constructed in our recent paper, where it was proved that these modules are irreducible for each finite cyclic orbifold \(M^{\mathbb{Z}_n}\). In this paper, we consider the modules \(M_{1}({\lambda},{\mu})\) as modules for the \({\mathbb{Z}}\)-orbifold of \(M\), denoted by \(M^0\). \(M^0\) is isomorphic to the vertex algebra
\(\mathcal W_{1+\infty, c=-1} = \mathcal M(2) \otimes M_1(1)\) which is the tensor product of the Heisenberg vertex algebra \(M_1(1)\) and the singlet algebra \(\mathcal M(2)\). Furthermore, these modules are also modules
of the Lie algebra \(\widehat{\mathfrak gl}\) with central charge \(c=-1\). We prove that they are reducible as \(\widehat{\mathfrak gl}\)-modules (and therefore also as \(M^0\)-modules), and we completely describe their irreducible quotients
\(L(d,{\lambda},{\mu})\).
We show that \(L(d,{\lambda},{\mu})\) in most cases are not tensor product modules for the vertex algebra \(\mathcal M(2) \otimes M_1(1)\). Moreover, we show that all constructed modules are typical in the sense that they are irreducible for the Heisenberg-Virasoro vertex subalgebra of \(\mathcal W_{1+\infty, c=-1}\).
On bounding the diameter of a distance-regular graph
Safet Penjić
University of Primorska,
Andrej Marušič Institute, Koper, Slovenia
(joint work with Arnold Neumaier (Universität Wien))
In this talk we investigate how to use an initial portion of the intersection array of a distance-regular graph to give an upper bound for the diameter of the graph. We prove three new diameter bounds. Our bounds are tight for the Hamming \(d\)-cube, doubled Odd graphs, the Heawood graph, Tutte's 8-cage and 12-cage, the generalized dodecagons of order (1,3) and (1,4), the Biggs--Smith graph, the Pappus graph, the Wells graph, and the dodecahedron.
This is joint work with Arnold Neumaier. The results was recently published in the journal Combinatorica, available at http://dx.doi.org/10.1007/s00493-021-4619-1
This is joint work with Arnold Neumaier. The results was recently published in the journal Combinatorica, available at http://dx.doi.org/10.1007/s00493-021-4619-1
Analytic linearization of hyperbolic (complex) Dulac germs
Dino Peran
Faculty of Science, University of Split, Croatia
(joint work with M. Resman, J.-P. Rolin and T. Servi)
Dulac germs are analytic germs defined on special subdomains of the Riemann surface of the logarithm called standard quadratic domains. Furthermore, they admit certain logarithmic asymptotic expansions at zero called Dulac series. The first return maps of hyperbolic polycycles of analytic planar vector fields turn to be Dulac germs, which relates them to the Dulac problem of non-accumulation of limit cycles on a hyperbolic or semi-hyperbolic polycycle of an analytic planar vector field, solved independently by Ilyashenko [2] and Écalle [1].
First, we consider analytic maps on certain invariant complex domains with logarithmic asymptotic bounds. We present sufficient conditions for such maps to be analytically linearized [3]. Afterwards, we present the formal linearization result for hyperbolic Dulac series with complex coefficients [4] and define complex Dulac germs (a generalization of Dulac germs with complex coefficients in their asymptotic expansions at zero, [3]). We apply these formal and analytic results on the class of all hyperbolic (complex) Dulac germs in order to prove the linearization theorem [3] for hyperbolic (complex) Dulac germs, which is the main goal of this talk.
References:
First, we consider analytic maps on certain invariant complex domains with logarithmic asymptotic bounds. We present sufficient conditions for such maps to be analytically linearized [3]. Afterwards, we present the formal linearization result for hyperbolic Dulac series with complex coefficients [4] and define complex Dulac germs (a generalization of Dulac germs with complex coefficients in their asymptotic expansions at zero, [3]). We apply these formal and analytic results on the class of all hyperbolic (complex) Dulac germs in order to prove the linearization theorem [3] for hyperbolic (complex) Dulac germs, which is the main goal of this talk.
References:
- J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques, Hermann, Paris, 1992.
- Y. Il'yashenko, Finiteness theorems for limit cycles, Translations of Mathematical Monographs, vol.94, American Mathematical Society, Providence, RI, 1991.
- D. Peran, M. Resman, J. P. Rolin and T. Servi, Linearization of complex hyperbolic Dulac germs, Journal of Mathematical Analysis and Applications, 508(1), 1-27, 2022, https://doi.org/10.1016/j.jmaa.2021.125833
- D. Peran, M. Resman, J. P. Rolin and T. Servi, Normal forms of hyperbolic logarithmic transseries, submitted, 2021. https://arxiv.org/pdf/2105.10660.pdf
On characterization of entrywise positivity preservers in fixed dimension
Ivan Perić
University of Zagreb, Croatia
For \(N\in\mathbb{N}\) and an interval \(I\subseteq\mathbb{R}\) let \(\mathbb{P}_N\left(I\right)\) denotes the set of \(N\times N\) positive semidefinite matrices with entries in \(I\). A function \(f:I\to\mathbb{R}\) is said to be entrywise positivity preserving on \(\mathbb{P}_N\left(I\right)\) if \(f\left[A\right]\) is a positive semidefinite matrix for all \(A\in\mathbb{P}_N\left(I\right)\), where \(f\left[A\right]=f\left[\left(a_{i,j}\right)_{i,j=1}^N\right]:=\left[f\left(a_{i,j}\right)_{i,j=1}^N\right]\) denotes entrywise acting of \(f\) on \(A\). The question of characterizing functions which are entrywise positivity preservers for every \(N\in\mathbb{N}\) is answered by Isaac J. Schönberg 1942 (as absolutely monotonic functions). A complete characterization of entrywise positivity preservers for a fixed \(N\geq 3\) (even for polynomials) remains unknown. For \(N=2\) and \(\mathbb{P}_2\left(\left(0,\infty\right)\right)\) the problem was resolved by Harkrishan L. Vasudeva in 1979.
We propose a method which gives a characterization for entrywise positivity preserving on \(\mathbb{P}_N\left(\left(0,\rho\right)\right)\), \(0<\rho\leq \infty\), for arbitrary fixed \(N\in\mathbb{N}\) and the complete characterization for generalized polynomials (monomials are of the form \(cx^{\alpha},\;\alpha\in\mathbb{R}\)). Comparing to Apoorva Khare and Terence Tao 2017 arXiv paper (Amer. J. Math. 2021), this method gives more simple and complete answer on characterizing generalized polynomials which are entrywise positivity preservers on \(\mathbb{P}_N\left(\left(0,\infty\right)\right)\) but not on \(\mathbb{P}_{N+1}\left(\left(0,\infty\right)\right)\).
The proposed method is based on the transformation \(f\to\bar{f}\), \(\bar{f}(x):=f\left(e^x\right)\), Karlin's theory of totally positive kernels and extensive use of the Wronskian determinants of the form \(W\left(\bar{f},\bar{f}',\ldots ,\bar{f}^{(k)}\right)\).
We propose a method which gives a characterization for entrywise positivity preserving on \(\mathbb{P}_N\left(\left(0,\rho\right)\right)\), \(0<\rho\leq \infty\), for arbitrary fixed \(N\in\mathbb{N}\) and the complete characterization for generalized polynomials (monomials are of the form \(cx^{\alpha},\;\alpha\in\mathbb{R}\)). Comparing to Apoorva Khare and Terence Tao 2017 arXiv paper (Amer. J. Math. 2021), this method gives more simple and complete answer on characterizing generalized polynomials which are entrywise positivity preservers on \(\mathbb{P}_N\left(\left(0,\infty\right)\right)\) but not on \(\mathbb{P}_{N+1}\left(\left(0,\infty\right)\right)\).
The proposed method is based on the transformation \(f\to\bar{f}\), \(\bar{f}(x):=f\left(e^x\right)\), Karlin's theory of totally positive kernels and extensive use of the Wronskian determinants of the form \(W\left(\bar{f},\bar{f}',\ldots ,\bar{f}^{(k)}\right)\).
Refinement of the Jensen and the Lah-Ribarič inequality and applications on Csiszár divergence
Jurica Perić
University of Split, Croatia
In this talk we give refinements of the discrete and integral form of Jensen's inequality and the Lah-Ribarič inequality. Using these results a refinement of the Hölder inequality, and a refinement of some inequalities for power means and quasi-arithmetic means are obtained. We also give applications in information theory, namely, we give some
interesting estimates for the integral Csiszár divergence and for its
important special cases.
Hennessy-Milner theorem for bisimulations between Veltman models and Verbrugge models
Tin Perkov
Faculty of Teacher Education, University of Zagreb
Bisimulation is the basic equivalence between models in modal logic. Interpretability logics are modal logical systems corresponding to the notion of relative interpretability between first-order arithmetical theories. Veltman semantics is the basic Kripke-like semantics for interpretability logics. Verbrugge semantics is a generalized Veltman semantics in the sense that there is an obvious transformation of any Veltman model to an equivalent Verbrugge model. Bisimulations are normally defined between structures of the same kind. In this work, we establish a notion of bisimulation between Verbrugge models and Veltman models and prove an analogue of Hennessy-Milner theorem for this notion.
This work is supported by Croatian Science Foundation (HRZZ) under the projects UIP-2017-05-9219 and IP-2018-01-7459.
This work is supported by Croatian Science Foundation (HRZZ) under the projects UIP-2017-05-9219 and IP-2018-01-7459.
Generalizations of Hardy type inequalities by Abel-Gontscharoff's interpolating polynomial
Dora Pokaz
Faculty of Civil Engineering, University of Zagreb, Croatia
(joint work with K. Krulić Himmelreich, J. Pečarić and M. Praljak)
In this talk we present how to extend Hardy's type inequalities to convex functions of the higher order. The notion of \(n\)-convexity, defined in terms of divided differences, is useful when discussing degree of smoothness. The techniques that we use in this presentation are based on classical real analysis and the application of the Abel-Gontscharoff interpolation.
We obtain upper bounds for generalized Hardy's inequality. In proofs of some estimations we include recent results related to the Chebyshev functional. Finally, we conclude the subject with some applications.
We obtain upper bounds for generalized Hardy's inequality. In proofs of some estimations we include recent results related to the Chebyshev functional. Finally, we conclude the subject with some applications.
New partition identities from \(C^{(1)}_\ell\)-modules
Mirko Primc
University of Zagreb, Croatia
(joint work with Stefano Capparelli, Arne Meurman and Andrej Primc)
In this work we conjecture combinatorial Rogers-Ramanujan type colored partition identities related to standard representations of the affine Lie algebra of type \(C^{(1)}_\ell\), \(\ell\geq2\), and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras. The conjectures are supported by numerical evidence.
Translation surfaces with constant curvatures in 3-dimensional Lorentz-Minkowski space
Ljiljana Primorac Gajčić
Department of Mathematics, J. J. Strossmayer University of Osijek, Croatia
(joint work with Ivana Filipan and Željka Milin Šipuš)
Translation surface is a surface formed by two curves moving along each
other. In Lorentz-Minkowski space, which is the smooth manifold \(\mathbb{R}^3
\) with flat Lorentzian pseudometric, translation
surfaces
are classified with respect to the causal character of their generating
curves (spacelike, timelike or null (lightlike)). We are specially interested in translation surfaces generated by at least one null curve, which we refer to as null-translation
surfaces. Such surfaces have no Euclidean counterparts. In this talk we present all null-translation surfaces of constant
mean curvature and show that only null-translation surfaces of constant Gaussian
curvature are cylindrical surfaces.
On the classification of unitary highest weight modules
Ana Prlić
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with Pavle Pandžić, Vladimír Souček and Vít Tuček)
Let \(G\) be a connected non-compact simple Lie group which has unitary highest weight representations. Let \(K\) be a maximal compact subgroup of \(G\).
Then \(G/K\) has a Hermitian structure, and the center of \(K\) is one-dimensional.
The proof of the classification of unitary highest weight representations of \(G\) given by Enright, Howe, Wallach is based on Jantzen’s formula and Howe’s theory of dual pairs where one group in the pair is compact.
The aim of this talk is to prove the same results using a completely different tool - the Dirac inequality.
Incentive-Compatibility: Binary Case
Sonja Radas
Institute of Economics, Zagreb, Croatia
(joint work with J.Cvitanić (Caltech), D.Prelec (MIT), and H.Šikić (Univ of Zagreb))
Consider a finite set of respondents or players, each receiving a signal, and also having beliefs about the other
player's signal, on a multiple-answer question with a finite number of possible answers (we also call them
{\em types}). We take a position of an ignorant planner, who wants to elicit truthful responses to a survey
question about player's signals and beliefs. This approach leads to the problem of designing an IC-mechanism,
i.e., the one in which, assuming that all other players tell the truth, the highest expected reward for
the player is to tell the truth, as well. We provide a detailed characterization of all such mechanisms in
the binary case and provide the analysis of various special cases which are typically of interest to the
practitioners.
On (strong) Birkhoff-James orthogonality in Hilbert \(C^*\)-modules
Rajna Rajić
Faculty of Mining, Geology and Petroleum Engineering,
University of Zagreb, Croatia
(joint work with Lj. Arambašić)
In this talk, we present some results on the Birkhoff-James orthogonality and its strong version in Hilbert \(C^*\)-modules. We describe the class of full Hilbert \(C^*\)-modules in which the (strong) Birkhoff-James orthogonality is a symmetric relation. We also characterize the class of (surjective) linear mappings \(\Phi \colon \mathbb{B}(H)\rightarrow \mathbb{B}(H)\) that preserve the (strong) Birkhoff-James orthogonality.
This research was supported in part by the Croatian Science Foundation under the project IP-2016-06-1046.
This research was supported in part by the Croatian Science Foundation under the project IP-2016-06-1046.
Nonzero boundary condition for the unsteady micropolar pipe flow: well-posedness and asymptotics
Borja Rukavina
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with Michal Beneš, Igor Pažanin and Marko Radulović)
In this talk, we consider the unsteady flow of a micropolar fluid through a thin pipe with the nonzero boundary condition for microrotation. We consider the well-posedness of the corresponding initial-boundary value problem governing the flow. Then, using asymptotic analysis with respect to the pipe’s thickness, we construct the higher-order approximation of the solution. The proposed approximation is given in explicit form, taking into account the effects of the boundary conditions, the micropolar nature of the fluid as well as the time derivative. A detailed study of the boundary layers in the vicinity of the pipe’s ends is also provided along with a numerical example illustrating the behaviour of the derived asymptotic solution.
Asymptotically sharp nonlinear Hausdorff-Young inequalities for continuous and discrete SU(1,1)-scattering transforms
Jelena Rupčić
Faculty of Transport and Traffic Sciences, University of Zagreb, Croatia
(joint work with Vjekoslav Kovač and Diogo Oliveira e Silva)
In this talk we will discuss two cases of the nonlinear Hausdorff-Young inequality. The first one is the Hausdorff-Young inequality for the continuous SU(1,1)-scattering transform, which is a nonlinear version of the Fourier transform, established by Christ and Kiselev. The second one is the Hausdorff-Young inequality for the SU(1, 1)-valued Fourier products. In both cases the behavior of sharp constants remains largely unresolved. In the continuous case we consider the particular case of functions that are sufficiently small in the \(L^1\) norm. In the second, discrete, case we consider either sufficiently small sequences or sequences that are far from being the extremizers of the linear Hausdorff–Young inequality.
On the counterexample for the Local Irregularity Conjecture
Jelena Sedlar
University of Split, Faculty of Civil Engineering, Architecture and Geodesy, Croatia
(joint work with Riste Škrekovski)
A locally irregular graph is a graph in which the end-vertices of every edge
have distinct degrees. A locally irregular edge coloring of a graph \(G\) is
any edge coloring of \(G\) such that each of the colors induces a locally
irregular subgraph of \(G\). A graph \(G\) is colorable if it admits a locally
irregular edge coloring. The locally irregular chromatic index of a
colorable graph \(G\) is the smallest number of colors used by a locally
irregular edge coloring of \(G\). The Local Irregularity Conjecture claims
that all graphs, except odd length path, odd length cycle and a certain
class of cacti, are colorable by 3 colors. It is already known that the
conjecture holds for trees, and we further consider the locally irregular
edge coloring of trees which enables us to establish that the conjecture
also holds for unicyclic graph and cacti with vertex disjoint cycles. We
provide a bow-tie graph \(B\), which is a colorable cactus graph in which
cycles are not vertex disjoint, for which the conjecture does not hold. The
consideration of the conjecture is concluded with some further remarks on
coloring of cacti.
Linearization for difference equations with infinite delay
Lokesh Singh
University of Rijeka, Croatia
One of the most common technique to study a nonlinear dynamics is to find an equivalent linear dynamics. The process of constructing a map, which transforms a nonlinear dynamics into a linear dynamics is commonly known as Linearization.
Let \((X, \| \cdot \|)\) be an arbitrary Banach space and let \begin{align*} \mathcal{B}:= \{ \phi: \mathbb{Z}^- \to X \, \big| \quad \| \phi \|_{\mathcal{B}} < \infty \} \end{align*} be a Banach space of sequences with appropriate norm \(\|\cdot \|_{\mathcal{B}}\). Given \(m \in \mathbb{Z}^+\) and a sequence \(x: \mathbb{Z} \to X\), define a new sequence \(x_m : \mathbb{Z}^- \to X\) given by \(x_m (j) := x(m+j)\) for all \(j \in \mathbb{Z}^-\).
In this talk, I am going to present a result on the Linearization of Difference equation with infinite delay, \begin{equation}\label{dis semilin eq} \quad\quad\quad\quad x(m+1)=A_m x_m + f_m(x_m) \qquad \text{ for all } m\in \mathbb{Z}^+, \quad\quad\quad\quad\text{(1)} \end{equation} in a Banach Space \(X\). Here we assume that for each \(m \in \mathbb{Z}^+\), \(A_m: \mathcal{B} \to X\) is a bounded linear map and the perturbation \((f_m)_{m \in \mathbb{Z}^+}\) is small and Lipschitz. In this result, a sequence of continuous and one-one maps, \((h^m)_{m \in \mathbb{Z^+}}\), is constructed which gives equivalency between the nonlinear dynamics (1) and its linear counterpart. We also showed that when \((A_m)_{m \in \mathbb{Z^+}}\) admits exponential dichotomy, our result is applicable. Here are some references.
References:
Let \((X, \| \cdot \|)\) be an arbitrary Banach space and let \begin{align*} \mathcal{B}:= \{ \phi: \mathbb{Z}^- \to X \, \big| \quad \| \phi \|_{\mathcal{B}} < \infty \} \end{align*} be a Banach space of sequences with appropriate norm \(\|\cdot \|_{\mathcal{B}}\). Given \(m \in \mathbb{Z}^+\) and a sequence \(x: \mathbb{Z} \to X\), define a new sequence \(x_m : \mathbb{Z}^- \to X\) given by \(x_m (j) := x(m+j)\) for all \(j \in \mathbb{Z}^-\).
In this talk, I am going to present a result on the Linearization of Difference equation with infinite delay, \begin{equation}\label{dis semilin eq} \quad\quad\quad\quad x(m+1)=A_m x_m + f_m(x_m) \qquad \text{ for all } m\in \mathbb{Z}^+, \quad\quad\quad\quad\text{(1)} \end{equation} in a Banach Space \(X\). Here we assume that for each \(m \in \mathbb{Z}^+\), \(A_m: \mathcal{B} \to X\) is a bounded linear map and the perturbation \((f_m)_{m \in \mathbb{Z}^+}\) is small and Lipschitz. In this result, a sequence of continuous and one-one maps, \((h^m)_{m \in \mathbb{Z^+}}\), is constructed which gives equivalency between the nonlinear dynamics (1) and its linear counterpart. We also showed that when \((A_m)_{m \in \mathbb{Z^+}}\) admits exponential dichotomy, our result is applicable. Here are some references.
References:
- Palmer, K., A generalization of Hartmans linearization theorem, J. Math. Anal. Appl. 41, pp. 753–758 (1973).
- Barreira, L., Valls, C., Perturbations of delay equations, J. Differ. Equ. 269,pp. 7015-7041 (2020).
Conditional regularity of weak solutions to the Navier-Stokes equations
Zdenek Skalak
Czech Technical University, Prague, Czech Republic
We study the conditional regularity of the weak solutions to the Navier-Stokes equations in the whole three-dimensional space. For an incompressible fluid with the constant density the Navier-Stokes equations are given by
\begin{eqnarray}
\label{eq:e1} && \frac {\partial {\bf u}} {\partial t} - \Delta {\bf u} + ({\bf u}
\cdot \nabla) {\bf u} + \nabla \mathcal{P} = 0 \quad \textrm{in} \ \mathbb{R}^3
\times (0,\infty), \\
\label{eq:e2} && \nabla \cdot {\bf u} = 0 \quad \textrm{in} \
\mathbb{R}^3 \times (0,\infty), \\ \label{eq:e3} && {\bf u}|_{t=0}={\bf u}_0,
\end{eqnarray}
where \({\bf u}={\bf u}(x,t) = (u_1(x,t), u_2(x,t), u_3(x,t))\) and \(\mathcal{P}=\mathcal{P}(x,t)\)
denote the unknown velocity and pressure, \({\bf u}_0 = {\bf u}_0(x)\) is the initial velocity and the viscosity is taken to be 1.
We present a new class of regularity criteria in which we assume the control of the instantaneous rate of change of some relevant Navier-Stokes quantities along the streamlines, such as the velocity, pressure, velocity magnitude, kinetic energy, Bernoulli pressure, etc. These criteria are scale-critical, they are mostly proved for the whole range of parameters and their proofs are easy and natural. It is all in the striking contrast to the criteria in which the quantities are differentiated in fixed directions.
Mention as an example a criterion in terms of one partial derivative of the pressure. The best (sub-critical) result reached so far says that \({\bf u}\) is regular if \(\partial_3 \mathcal{P} \in L^p(0,T;L^q)\), where \(2/p + 3/q = 2 + 1/q\) and \(q \in (1,\infty)\). If we replace the direction \((0,0,1)\) with the direction \(\hat{{\bf u}} = {\bf u}/|{\bf u}|\), i.e. if we differentiate the pressure along the streamlines, we obtain the following criterion: \({\bf u}\) is regular if \(\nabla \mathcal{P} \cdot \hat{{\bf u}} \in L^p(0,T;L^q)\), where \(2/p + 3/q = 3\) and \(q \in (1,\infty)\).
We present a new class of regularity criteria in which we assume the control of the instantaneous rate of change of some relevant Navier-Stokes quantities along the streamlines, such as the velocity, pressure, velocity magnitude, kinetic energy, Bernoulli pressure, etc. These criteria are scale-critical, they are mostly proved for the whole range of parameters and their proofs are easy and natural. It is all in the striking contrast to the criteria in which the quantities are differentiated in fixed directions.
Mention as an example a criterion in terms of one partial derivative of the pressure. The best (sub-critical) result reached so far says that \({\bf u}\) is regular if \(\partial_3 \mathcal{P} \in L^p(0,T;L^q)\), where \(2/p + 3/q = 2 + 1/q\) and \(q \in (1,\infty)\). If we replace the direction \((0,0,1)\) with the direction \(\hat{{\bf u}} = {\bf u}/|{\bf u}|\), i.e. if we differentiate the pressure along the streamlines, we obtain the following criterion: \({\bf u}\) is regular if \(\nabla \mathcal{P} \cdot \hat{{\bf u}} \in L^p(0,T;L^q)\), where \(2/p + 3/q = 3\) and \(q \in (1,\infty)\).
Maximal cyclic subspaces for dual integrable representations
Ivana Slamić
University of Rijeka, Croatia
(joint work with Hrvoje Šikić)
Consider a unitary representation of a countable discrete group on a separable Hilbert space. The study of closed subspaces invariant under these representations plays an important role in harmonic analysis. It is known that any such subspace can be decomposed into an orthogonal sum of countably many cyclic subspaces, the subspaces generated by an orbit of a single element. If the representation is dual integrable, then the properties of orbits and the invariant subspaces which they generate can be analyzed in terms of the associated bracket map. This setting includes the system of integer translates of a square integrable function, which are studied in terms of the periodization function. We know that such system is \(\ell^2\)-linearly independent precisely when the periodization function is strictly positive a.e., while on the other hand, this condition is equivalent to the corresponding principal shift invariant subspace being maximal. Motivated by this connection, we study several questions concerned with maximal cyclic subspaces for the general group setting.
Matrix Algorithms over Associative Algebras
Ivan Slapničar
University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Split, Croatia
(joint work with Nevena Jakovčević Stor)
Standard linear algebra algorithms for systems of linear equations, matrix inverses, QR factorization, least squares, eigenvalue decomposition, singular value decomposition, etc., naturally extended to matrices
with elements from associative non-commutative algebras. Examples of such algebras include matrices of quaternions and block matrices. We review definitions that hold in this more general setting and provide some algorithms for general and structured matrices.
While many basic algorithms are simple generalizations of their standard counterparts, finer details of modern algorithms over real or complex fields are difficult to implement in the non-commutative setting. The presented efficient and elegant algorithms are derived using the multiple dispatch feature of the Julia programming language. This work has been supported in part by Croatian Science Foundation under the project IP-2020-02-2240.
Bounds of Hermite-Hadamard-type for generalizations of Steffensen's inequality
Ksenija Smoljak Kalamir
University Of Zagreb Faculty of Textile Technology
(joint work with Josip Pečarić and Anamarija Perušić Pribanić)
In this talk, we will give bounds on identities related to generalizations of Steffensen's inequality using weighted Hermite-Hadamard inequalities.
As we will show, these bounds can be obtained utilizing Hölder's inequality and convex or concave functions of the form \(|f^{(n)}|^q\), for \(n\geq 2\) and \(q\geq 1\).
\(D(-1)\)-tuples in the ring \(\mathbb{Z}[\sqrt{-k}]\) with \(k>0\)
Ivan Soldo
University of Osijek, Croatia
(joint work with Yasutsugu Fujita)
Let \(n\) be a non-zero integer and \(R\) a commutative ring. A \(D(n)\)-\(m\)-tuple in \(R\) is a set of \(m\) non-zero elements
in \(R\) such that the product of any two distinct elements plus \(n\) is a perfect square in \(R\). We prove
that there does not exist a \(D(-1)\)-quadruple \(\{a,b,c,d\}\) in the ring \(\mathbb{Z}[\sqrt{-k}]\), \(k\ge 2\) with positive integers
\(a<b\le8a-3\) and negative integers \(c\) and \(d\). By using that result we show that such a \(D(-1)\)-pair
\(\{a,b\}\) cannot be extended to a \(D(-1)\)-quintuple \(\{a,b,c,d,e\}\) in the ring \(\mathbb{Z}[\sqrt{-k}]\) with integers \(c,d\) and \(e\).
Moreover, we apply the obtained result to the \(D(-1)\)-pair \(\{p^i, q^j\}\) with an arbitrary different
primes \(p\), \(q\) and positive integers \(i\), \(j\).
Classical Friedrichs operators in one dimensional scalar case
Sandeep Kumar Soni
University of Zagreb, Faculty of Science, Croatia
(joint work with Marko Erceg)
The theory of abstract Friedrichs operators, introduced by
Ern, Guermond and Caplain (2007), proved to be a successful setting
for studying positive symmetric systems of first order partial
differential equations (Friedrichs, 1958),
nowadays better known as Friedrichs systems.
Recently, Antonić, Michelangeli and Erceg (2017) presented a purely
operator-theoretic description of abstract Friedrichs operators,
allowing for application of the universal operator extension theory
(Grubb, 1968).
In this presentation we shall see a nice decomposition of the graph space (maximal domain) as a direct sum of the minimal domain and the kernels of corresponding adjoints, which might be a significant ingredient in further studies of abstract Friedrichs operators. As an immediate application of the decomposition we shall see a concrete pair of mutually adjoint bijective realisation relative to the given joint pair of abstract Friedrichs operators. We shall also see a complete classification of admissible boundary conditions of classical Friedrichs differential operators in one dimensional scalar case with variable coefficients.
This is a joint work with Marko Erceg funded by Croatian Science Foundation.
In this presentation we shall see a nice decomposition of the graph space (maximal domain) as a direct sum of the minimal domain and the kernels of corresponding adjoints, which might be a significant ingredient in further studies of abstract Friedrichs operators. As an immediate application of the decomposition we shall see a concrete pair of mutually adjoint bijective realisation relative to the given joint pair of abstract Friedrichs operators. We shall also see a complete classification of admissible boundary conditions of classical Friedrichs differential operators in one dimensional scalar case with variable coefficients.
This is a joint work with Marko Erceg funded by Croatian Science Foundation.
Computing constants in degenerate subspaces of the multiparametric quon algebra \({\mathcal{B}}\)
Milena Sošić
Faculty of Mathematics, University of Rijeka, Croatia
One of fundamental problems in multiparametric quon algebra \({\mathcal{B}}\) equipped with a multiparametric \(\mathbf{q}\)-differential structure is a determination of the space of all constants.
According to a direct sum decomposition of the algebra \({\mathcal{B}}\) into the generic subspace (spanned by all multilinear monomials) and the degenerate subspace (spanned by all monomials which are nonlinear in at least one variable), the given fundamental problem can be reduced to the following two problems, first to the determination of constants in all generic subspaces of \({\mathcal{B}}\) and then in all degenerate subspaces of \({\mathcal{B}}\).
The first problem is solved in detail in [2], where the author has proven an explicit formula for calculating constants in any generic subspace of the algebra \({\mathcal{B}}\) in terms of certain iterated \(\mathbf{q}\)-commutators.
It should be emphasized that finding an explicit formula for determining constants in any degenerate subspaces of \({\mathcal{B}}\) is a much more difficult problem because there is no unique formula.
The aim of this presentation is to show that any constant in the degenerate subspace of the algebra \({\mathcal{B}}\) can be constructed from the formula given in [2, Theorem3] by a certain specialization procedure.
Then we get formulas for calculating constants in some degenerate subspaces of the algebra \({\mathcal{B}}\).
References:
References:
- S. Meljanac, D. Svrtan, Study of Gram matrices in Fock representation of multiparametric canonical commutation relations, extended Zagier's conjecture, hyperplane arrangements and quantum groups, Math. Commun. 1 (1996), 1-24.
- M. Sošić, Computation of constants in multiparametric quon algebras. A twisted group algebra approach, Math. Commun. 22 No.2, (2017), 177-192.
Quantitative bounds for products of simplices in subsets of the unit cube
Mario Stipčić
Chapman University, Orange, California, USA
(joint work with Polona Durcik)
Let us consider the following variant of the Euclidean density theorems.
- Let \(n \geq 1\). For each \(1 \leq i \leq n\), let \(k_i \geq 1\) and let \(\Delta_i\) be a set of vertices of a non-degenerate \(k_i\)-dimensional simplex in \(\mathbb{R}^{k_i}\). For every \(\delta \in (0,1/2]\) there exists \(\varepsilon = \varepsilon(\Delta_1,\dots,\Delta_n,\delta) > 0\) such that the following holds. For every \(A \subseteq [ 0,1 ]^{k_1+1} \times \cdots \times [ 0,1 ]^{k_n+1}\) satisfying \(|A| \geq \delta\) there exists an interval \(I=I(\Delta_1,\dots,\Delta_n,A) \subseteq (0,\infty)\) of length at least \(\varepsilon\) such that for each \(\lambda \in I\) we have \(\Delta_1' \times \cdots \times \Delta_n' \subseteq A\), where \(\Delta_i'\) is an isometric copy of \(\lambda \Delta_i\) for each \(1 \leq i \leq n\).
Using symbolic computations to determine largest small polygons
Dragutin Svrtan
Zagreb, Croatia
(joint work with Charles Audet and Pierre Hansen)
A small polygon is a convex polygon (in a plane) of unit diameter. The question of finding the largest area of small \(n\)-gons has been answered for some values of \(n\) by K. Reinhardt in (Extremale Polygone gegebener Durchmessers Jahr. DMV 1922). He showed that regular \(n\)-gons are optimal when \(n\) is odd and kites with unit length diagonals are optimal when \(n=4\). For \(n=6\) the largest area is a root of a degree 10 polynomial with integer coefficients and height 221360. This famous Graham's largest little hexagon (Jour. Comb. Th., 1975) was obtained by factoring (on a supercomputer) of an intermediate 40-degree polynomial with 25-digit height. As a tool R. Graham introduced the diameter graphs by joining the vertices at maximal distance. For \(n=6\) (resp. 8) there are 10 (resp. 31) possible diameter graphs. The case \(n=8\) was attacked by C. Audet, P. Hansen, F. Messine and J. Xiong (in Jour. Comb. Th. and Appl. 2002) via global optimization (with 10 variables and 20 constraints) which produced (an approximate) famous Hansen's little octagon.
In this talk we report, for \(n=6\), on a smaller auxiliary polynomial of degree 14 (instead of 40) obtained by some rational substitutions (a 'missed opportunity' in Graham and Johnson's approach). In case \(n=8\) under the axial symmetry conjecture, we obtained for the first time, an explicit minimal equations for largest small octagons (resp. decagons) of degree 42 with height 23588130061203336356460301369344 (resp. 152 and 146-digit height 18560040113130148551005860074337195069126245618474566073569475945972172527876136797043381755516092115941578415
203761516711300570773982329173966848) via intriguing iterated discriminants computations symbolicaly done by using MAPLE, (for decagon one intermediate polynomial has integer coefficients with almost three thousand digits). We have also discovered a new algorithm that uses Laurent polynomials and computations in complex numbers, instead of rational substitutions and computations in real numbers. This algorithm was presented at several international conferences including International Congress of Mathematicians in 2018 in Rio de Janeiro, Brasil. Axial symmetry conjecture is not yet proven even for \(n=8\). So for general octagons, we may need nowadays supercomputers (with several TB of RAM).This is a nice example that symbolic computations are extremely powerfull but may require huge amount of memory (in contrast with Numerical Linear Algebra). For nonlinear systems we may never have at hand powerfull enough computers or even quantum computers to do the job. Thus we need to explore deeper the geometry of the problem to relax the algebra.
Note. The first of our papers "Using symbolic calculations to determine largest small polygons" is accepted in the well known Journal of Global Optimization and it will appear in print soon.
In this talk we report, for \(n=6\), on a smaller auxiliary polynomial of degree 14 (instead of 40) obtained by some rational substitutions (a 'missed opportunity' in Graham and Johnson's approach). In case \(n=8\) under the axial symmetry conjecture, we obtained for the first time, an explicit minimal equations for largest small octagons (resp. decagons) of degree 42 with height 23588130061203336356460301369344 (resp. 152 and 146-digit height 18560040113130148551005860074337195069126245618474566073569475945972172527876136797043381755516092115941578415
203761516711300570773982329173966848) via intriguing iterated discriminants computations symbolicaly done by using MAPLE, (for decagon one intermediate polynomial has integer coefficients with almost three thousand digits). We have also discovered a new algorithm that uses Laurent polynomials and computations in complex numbers, instead of rational substitutions and computations in real numbers. This algorithm was presented at several international conferences including International Congress of Mathematicians in 2018 in Rio de Janeiro, Brasil. Axial symmetry conjecture is not yet proven even for \(n=8\). So for general octagons, we may need nowadays supercomputers (with several TB of RAM).This is a nice example that symbolic computations are extremely powerfull but may require huge amount of memory (in contrast with Numerical Linear Algebra). For nonlinear systems we may never have at hand powerfull enough computers or even quantum computers to do the job. Thus we need to explore deeper the geometry of the problem to relax the algebra.
Note. The first of our papers "Using symbolic calculations to determine largest small polygons" is accepted in the well known Journal of Global Optimization and it will appear in print soon.
Expected volume of the convex hull of the time-space trajectory of Brownian motion
Stjepan Šebek
University of Zagreb, Croatia
(joint work with Wojciech Cygan (TU Dresden) and Nikola Sandrić (University of Zagreb))
In this talk, we will develop a closed formula for the expected value of the \(d\)-dimensional volume of the convex hull spanned by the time-space trajectory of \((d-1)\)-dimensional Brownian motion run up to time one.
Combinatorial base of standard module \(L(2\Lambda_0)\) for affine Lie algbera \(\widetilde{\mathfrak{sp}}_{2n}(\mathbb{C})\)
Tomislav Šikić
University of Zagreb, Faculty of Electrical Engineering and Computing
(joint work with Mirko Primc)
At the beginning of this talk will be presented the construction of combinatorial bases of basic modules for affine symplectic Lie algebras \(\widetilde{\mathfrak{sp}}_{2n}(\mathbb{C})\) (J. of Math.Physics, http://dx.doi.org/10.1063/1.4962392). Special accent of this talk will be devoted to the combinatorial parametrization of leading terms of defining relations for all standard modules \(L(k\Lambda_0)\) for symplectic affine Lie algebra (The Ramanujan Journal, https://doi.org/10.1007/s11139-018-0052-5). By using above results we conjecture that the PBW spannig set reduced by difference conditions which are paramatrized by leading terms is in fact a basis of standard module \(L(k\Lambda_0)\) for \(\widetilde{\mathfrak{sp}}_{2n}(\mathbb{C})\). At the end of the talk, will be presented some progres in proving of mentioned conjecture for the case \(k=2\).
LDPC codes and cubic graphs
Marina Šimac
Faculty of Mathematics - University of Rijeka, Rijeka, Croatia
(joint work with Dean Crnković and Sanja Rukavina)
Low-density parity-check (LDPC) codes have been the focus of much research interest in the recent years and it has been shown
that they perform close to the Shannon limit.
The aim of this talk is to present the construction of families of LDPC codes using cubic graphs. For the construction, bipartite cubic symmetric and cubic semisymmetric graphs have been used. In the talk we will discuss some of the properties of the constructed codes and present information on the codes. Computational and simulation results will be presented as well.
The Odd Coloring
Riste Škrekovski
Faculty of Mathematics and Physics, University of Ljubljana & Faculty of Information Studies, Novo mesto
Slovenia
A proper vertex coloring \(\varphi\) of graph \(G\) is said to be odd if for each non-isolated vertex \(x\in V(G)\) there exists a color \(c\) such that \(\varphi^{-1}(c)\cap N(x)\) is odd-sized. The minimum number of colors in any odd coloring of \(G\), denoted \(\chi_o(G)\), is the odd chromatic number. Odd colorings were recently introduced in [M. Petruševski, R. Škrekovski: Colorings with neighborhood parity condition].
In the talk we discuss various basic properties of this new graph parameter, establish several upper bounds, several characterizatons, and pose some questions and problems about odd colorings. We will also consider another new and related coloring, so called the proper conflict-free coloring.
In the talk we discuss various basic properties of this new graph parameter, establish several upper bounds, several characterizatons, and pose some questions and problems about odd colorings. We will also consider another new and related coloring, so called the proper conflict-free coloring.
Classification of the Lozi maps
Sonja Štimac
University of Zagreb, Zagreb, Croatia
(joint work with Jan Boronski)
In this talk, I will show how we classified the Lozi maps (up to conjugacy, for Misiurewicz’s set of parameters) by using two powerful tools, the symbolic dynamics, and the inverse limit spaces.
Modeling interaction of continua one of which is thin
Josip Tambača
University of Zagreb, Croatia
(joint work with Matko Ljulj, Eduard Marušić Paloka and Igor Pažanin)
In cases when a thin body is in contact to another body the thin body is often neglected. In this talk it will be shown that
depending on the regime, related to the ratio of the material properties and the small thickness, more than one limit model is possible.
Furthermore it is possible to build a model replacing the thin continua by a lower-dimensional model and that approximates the limit models in all regimes. The models will be considered in context of elasticity and the heat conduction.
GBS complexes, GBS groups, and the solenoids
Mathew Timm
Bradley University, Peoria, United States. University of Split, Split, Croatia.
A generalized Baumslag-Solitar (GBS) complex is formed by gluing a collection of annuli together along their boundaries. The topology of these spaces is closely related to the group theory of a class of groups known as GBS groups. Certain of the GBS complexes are closely related to the classical solenoids. We explore some of the connections between the GBS complexes, the GBS groups, and the solenoids.
Asymptotic dimension of hyperbolic, geodesic, proper, quasi-cobounded spaces
Vera Tonić
University of Rijeka, Croatia
(joint work with Tobias Hartnick)
A well-known theorem from geometric group theory, by S. Buyalo and N. Lebedeva, states that for a hyperbolic, geodesic, proper, cobounded space \(X\), the equality \(\mathrm{asdim} X = \mathrm{dim}\ \partial X +1\) holds. In particular, this equality holds for hyperbolic groups.
We intend to show that Buyalo-Lebedeva's theorem can be generalized to hyperbolic, geodesic, proper and quasi-cobounded spaces. Quasi-coboundedness means that for a metric space \(X\), there is a uniform collection \(A\) of quasi-isometries of \(X\) and a constant \(R>0\) such that for any chosen base point \(o\in X\), and for any \(x\in X\), there is a \(g\in A\) so that \(g(x)\in B(o, R)\). As a consequence, the equality mentioned above is also true for hyperbolic approximate groups.
We intend to show that Buyalo-Lebedeva's theorem can be generalized to hyperbolic, geodesic, proper and quasi-cobounded spaces. Quasi-coboundedness means that for a metric space \(X\), there is a uniform collection \(A\) of quasi-isometries of \(X\) and a constant \(R>0\) such that for any chosen base point \(o\in X\), and for any \(x\in X\), there is a \(g\in A\) so that \(g(x)\in B(o, R)\). As a consequence, the equality mentioned above is also true for hyperbolic approximate groups.
Category Composition and Special Morphisms
Nikica Uglešić
Veli Råt, Croatia
The special properties of an abstract category morphism (for instance, being
an identity, an isomorphism, an epimorphism., a monomorphism...) fully
depend on the category composition. Consequently, an isomorphic category to a
concrete category may be not concrete, i.e., the concreteness is not a
category invariant. Further, every small category is isomorphic to a small
category whose objects are sets and whose morphisms are functions between
those sets.
Letters from William Feller to Vladimir Varićak
Anđa Valent
Zagreb University of Applied Sciences, Zagreb, Croatia
(joint work with Ivica Vuković)
We will present so far unknown and unpublished letters that William (Vilim) Feller sent to Vladimir Varićak. Feller’s biography is well known; he was one of the most distinguished mathematicians in the area of probability theory of the 20th century. After two years of studying in Zagreb, he continued his studies in Göttingen where he received his PhD. During that time, Feller sent six letters to Varićak, describing his life and work in Göttingen. The other six letters we will present were sent in the 1937-39 period. The most intriguing and emotional ones are two letters Feller sent just before leaving Europe and immediately after fleeing to America at the beginning of WW2.
Refined Euler's inequalities in plane geometries and spaces
Darko Veljan
Department of Mathematics, University of Zagreb
Refined famous Euler's inequalities \(R \geq n r\) of an \(n\)-dimensional simplex
for \(n = 2, 3\) and \(4\) as well as of non-Euclidean triangles
in terms of symmetric functions of edge lengths of a triangle
or a simplex in question are shown.
Here \(R\) is the circumradius and \(r\) the inradius of the simplex.
We also provide an application to geometric probabilities of our results
and an example from astrophysics to the position of a planet within the space of four stars. We briefly discuss a recursive algorithm to get similar inequalities in higher dimensions.
An approximate maximum likelihood estimator of drift parameters in a multidimensional diffusion model
Andreja Vlahek
University of Zagreb, Faculty of Chemical Engineering and Technology, Croatia
(joint work with Snježana Lubura Strunjak and Miljenko Huzak)
For fixed \(T\), we analyze a \(k\)-dimensional vector stochastic differential equation over time interval \([0,T]\):
$$dX_t=\mu(X_t, \theta)\,dt+\nu(X_t)\sigma\,dW_t,$$
where \(\mu(X_t, \theta)\) is a \(k\)-dimensional vector and \(\nu(X_t)\) is a \(k \times k\)-dimensional matrix, both consisting of sufficiently smooth functions. \(\left(W_t, \, t \geq 0\right)\) is a \(k\)-dimensional standard Brownian motion whose components \(W_t^1, W_t^2, \dots, W_t^k\) are independent scalar Brownian motions. Matrix of diffusion parameters \(\sigma\) is known, and vector of drift parameters \(\theta\) is unknown. We prove that approximate maximum likelihood estimator of drift parameter \(\hat{\theta}_n\) obtained from discrete observations \((X_{i\Delta_n}, 0 \leq i \leq n)\), when \(\Delta_n=T/n\) tends to zero, is locally asymptotic mixed normal with covariance matrix that depends on maximum likelihood estimator \(\hat{\theta}_T\) obtained from continuous observations \((X_t, 0\leq t\leq T)\), and on path \((X_t, 0 \leq t\leq T)\). To prove the desired result, we emphasize the importance of the so-called uniform ellipticity condition of diffusion matrix \(S(x)=(\nu(x)\sigma)(\nu(x)\sigma)^T\).
A probabilistic approach to a non-local quadratic form and its connection to the Neumann boundary condition problem
Zoran Vondraček
University of Zagreb, Faculty of Science, Croatia
In this talk I will discuss a probabilistic approach to a non-local quadratic form that has lately attracted some interest. This form is related to a recently introduced non-local normal derivative. The goal is to construct two Markov processes: one corresponding to that form and the other which is related to a probabilistic interpretation of the Neumann problem. I will also discuss the Dirichlet-to-Neumann operator for non-local operators.
Explicit expression of second order shape derivative for disks in conductivity problems
Marko Vrdoljak
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with Petar Kunštek)
We consider conductivity optimal design problems for two isotropic phases, possibly with several state equations,
maximizing the energy functional.
The homogenization theory enables one to express the problem as a minimax problem, which can be solved explicitly in spherically symmetric case. The solution is unique and spherically symmetric, except in some singular cases. We shall study a possible appearance of local maxima.
For this purpose, we calculate the second order shape derivative in the boundary representation. It can be expressed in an explicit manner, by using classical Fourier analysis techniques, enabling one to analyze the second order optimality conditions.
The homogenization theory enables one to express the problem as a minimax problem, which can be solved explicitly in spherically symmetric case. The solution is unique and spherically symmetric, except in some singular cases. We shall study a possible appearance of local maxima.
For this purpose, we calculate the second order shape derivative in the boundary representation. It can be expressed in an explicit manner, by using classical Fourier analysis techniques, enabling one to analyze the second order optimality conditions.
Quadratic points on bielliptic modular curves
Borna Vukorepa
University of Zagreb
(joint work with Filip Najman)
Bruin and Najman, Ozman and Siksek, and Box described all the quadratic points on the modular curves of genus \(2\leq g(X_0(n)) \leq 5\). Since all the hyperelliptic curves \(X_0(n)\) are of genus \(\leq 5\) and as a curve can have infinitely many quadratic points only if it is either of genus \(\leq 1\), hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves \(X_0(n)\) naturally arises; this question has recently also been posed by Mazur.
We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves \(X_0(n)\) for which this has not been done already. The values of \(n\) that we deal with are \(n=60,62,69,79,83,89,92,94,95\), \(101,119\) and \(131\); the curves \(X_0(n)\) are of genus up to \(11\). We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Siksek's and Box's relative symmetric Chabauty method and an application of a moduli description of \(\mathbb{Q}\)-curves of degree \(d\) with an independent isogeny of degree \(m\), which reduces the problem to finding the rational points on several quotients of modular curves.
We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves \(X_0(n)\) for which this has not been done already. The values of \(n\) that we deal with are \(n=60,62,69,79,83,89,92,94,95\), \(101,119\) and \(131\); the curves \(X_0(n)\) are of genus up to \(11\). We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Siksek's and Box's relative symmetric Chabauty method and an application of a moduli description of \(\mathbb{Q}\)-curves of degree \(d\) with an independent isogeny of degree \(m\), which reduces the problem to finding the rational points on several quotients of modular curves.
On the representation theory of the vertex algebra \(L_{-5/2}(sl(4))\)
Ivana Vukorepa
University of Zagreb, Croatia
(joint work with Dražen Adamović and Ozren Perše)
In this talk we study the representation theory of non-admissible simple affine vertex algebra \(L_{-5/2}(sl(4))\). This case is of particular interest since it appears in conformal embeddings of affine vertex algebras. We determine an explicit formula for the singular vector of conformal weight four in the universal affine
vertex algebra \(V^{-5/2}(sl(4))\) and show that it generates the maximal ideal in \(V^{-5/2}(sl(4))\). We classify
irreducible \(L_{-5/2}(sl(4))\)–modules in the category \(\mathcal{O}\), and determine the fusion rules between irreducible modules in the category of ordinary modules \(KL_{-5/2}\). We also prove semi-simplicity of \(KL_{-5/2}\). In our proofs we use Zhu's theory, the notion of collapsing level for the affine \(\mathcal{W}\)–algebra, and the properties of
conformal embedding \(gl(4) \hookrightarrow sl(5)\) at level \(k= -5/2\). We show that \(k= -5/2\) is a collapsing
level with respect to the subregular nilpotent element \(f_{subreg}\) and we prove
certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction
functor \(H_{f_{subreg}}\).
Zdenka pl. Makanec Blašković - the first woman in Croatia to receive a doctorate in mathematics
Ivica Vuković
Zagreb University of Applied Sciences
(joint work with Anđa Valent)
The last year was the 100th anniversary of the first woman from Croatia receiving a doctorate in mathematics.
The doctoral degree was earned by Zdenka Blašković born Makanec, at Zagreb University, under the supervision of Vladimir Varićak.
In our talk we will present so far unknown data on Makanec's family background, education, scientific and artistic interests and papers that she had published. The results we present are obtained from archive records that we have explored, as well as from documents kept in private collections.
We will also correct some false data on Makanec’s bibliography that was previously published.
An overview of types of bisimulations for Verbrugge semantics (or generalized Veltman semantics)
Mladen Vuković
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
The language of the interpretability IL contains propositional letters \(p_0 , \ p_1, \ldots ,\) the logical connectives \(\wedge, \ \vee ,
\rightarrow,\) \(\leftrightarrow\) and \(\neg\) , the unary modal operator \(\Box\) and the binary modal operator \(\triangleright \).
We consider Verbrugge semantics for interpretability logic.
The paper [1] provides the necessary definitions and detailed explanation on IL.
There are several kinds of bisimulations for Verbrugge semantics. We define a notion of bisimulation between two Verbrugge models in [3], and prove Hennessy-Milner theorem. We study various kinds of bisimulations of Verbrugge models in [2]. In [4] is defined a new notion of bisimulation.
References:
There are several kinds of bisimulations for Verbrugge semantics. We define a notion of bisimulation between two Verbrugge models in [3], and prove Hennessy-Milner theorem. We study various kinds of bisimulations of Verbrugge models in [2]. In [4] is defined a new notion of bisimulation.
References:
- A. Visser, An overview of interpretability logic, In: K. Marcus (ed.) et al., Advances in modal logic. Vol. 1. Selected papers from the 1st international workshop (AiML'96), Berlin, Germany, October 1996, Stanford, CA: CSLI Publications, CSLI Lect. Notes. 87(1998), 307-359
- D. Vrgoč, M. Vuković, Bisimulations and bisimulation quotients of generalized Veltman models, Logic Journal of the IGPL, 18(2010), 870-880
- M. Vuković, Hennessy-Milner theorem for interpretability logic, Bulletin of the Section of Logic, 34(2005), 195-201
- S. Horvat, T. Perkov, M. Vuković, Bisimulations and bisimulations games for Vebrugge semantics, preprint, 2022.
Nonlocal quadratic forms with visibility constraint
Vanja Wagner
Faculty of Science, University of Zagreb
(joint work with Moritz Kassmann)
Given a subset \(D\) of the Euclidean space, we study nonlocal quadratic forms that
take into account tuples \((x, y) \in D \times D\) if and only if the line segment between \(x\) and \(y\) is contained in \(D\). We discuss regularity of the corresponding Dirichlet form leading to the existence of a jump process with visibility constraint. Our main aim is to investigate corresponding Poincaré inequalities and their scaling properties. For dumbbell shaped domains we show that the forms satisfy a Poincaré inequality with
diffusive scaling.
Constructions of block designs from orbit matrices using a modified genetic algorithm
Tin Zrinski
Faculty of Mathematics, University of Rijeka, Croatia
Genetic algorithms (GA) are search and optimization heuristic population-based methods which are inspired by the natural evolution process. In this talk, we will present a method of constructing incidence matrices of block designs combining the method of construction with orbit matrices and a modified genetic algorithm.
Norm-resolvent estimates for elastic heterogeneous rods
Josip Žubrinić
University of Zagreb, Faculty of Electrical Engineering and Computing
(joint work with Kirill Cherednichenko, Serena D'Onofrio and Igor Velčić)
We provide norm-resolvent estimates for the class of problems in linear elasticity describing infinite heterogeneous rods. The estimates are provided with respect to the period of material oscillations in the setting of simultaneous homogenization and dimension reduction, while assuming that the period and the rod thickness are of the same order.
On the Schwartz space \( \mathcal S(G(k)\backslash G(\mathbb A)) \)
Sonja Žunar
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with Goran Muić)
Let \(G\) be a connected reductive group defined over a number field \( k \), and let \( \mathbb A \) be the adele ring of \( k \). We introduce the Schwartz space \( \mathcal S(G(k)\backslash G(\mathbb A)) \) - an adelic version of Casselman's Schwartz space \( \mathcal S(\Gamma\backslash G_\infty) \), where \( \Gamma \) is a discrete subgroup of \( G_\infty:=\prod_{v\in V_\infty}G(k_v) \). Our main object of interest is the strong dual \( \mathcal S(G(k)\backslash G(\mathbb A))' \) - the space of tempered distributions on \( G(k)\backslash G(\mathbb A) \), carrying the contragredient representation
\( r' \) of the right regular representation
\( (r , \mathcal S(G(k)\backslash G(\mathbb A))) \) of \( G(\mathbb A) \). We prove that the (naturally defined) Gårding subspace of \( \mathcal S(G(k)\backslash G(\mathbb A))' \) may be identified with the space \( C^\infty_{umg}(G(k)\backslash G(\mathbb A))\) of smooth functions of uniform moderate growth.
We describe the closed irreducible admissible \( G(\mathbb A) \)-invariant subspaces of \( \mathcal S(G(k)\backslash G(\mathbb A))' \) and discuss applications to automorphic forms.
Fractal analysis of planar nilpotent singularities and numerical applications
Vesna Županović
University of Zagreb,Faculty of Electrical Engineering and Computing,
Department of Applied Mathematics, Zagreb, Croatia
The goal of our work is to give a complete fractal classification of planar analytic nilpotent singularities. For the classification, we use the notion of box dimension of (two-dimensional) orbits on separatrices generated by the unit time map. We also show how the box dimension of the one-dimensional orbit generated by the Poincaré map, defined on the characteristic curve near the nilpotent center/focus, reveals an upper bound for the number of limit cycles near the singularity. We introduce simple formulas for numerical calculation of the box dimension of one- and two-dimensional orbits and apply them to nilpotent singularities.
References:
References:
- Lana Horvat Dmitrović, Renato Huzak, Domagoj Vlah, Vesna Županović, Fractal analysis of planar nilpotent singularities and numerical applications, Journal of Differential Equations, Volume 293, 2021, Pages 1-22, ISSN 0022-0396, https://doi.org/10.1016/j.jde.2021.05.015.
Posters
The regularity of \(D(4)\)-\(m\)-tuples
Marija Bliznac Trebješanin
University of Split, Faculty of Science, Split, Croatia
Let \(n\neq0\) be an integer. A set of \(m\) distinct positive integers is called a \(D(n)\)-\(m\)-tuple if the product of any two of its distinct elements increased by \(n\) is a perfect square.
We give an overview of the current state of the problems related to \(D(4)\)-\(m\)-tuples. More precisely, we present some results supporting the validity of a conjecture about the regularity of \(D(4)\)-quadruples.
Redistribution of geometry for enhanced NURBS model
Andrijana Ćurković
Faculty of Science, University of Split, Croatia
(joint work with Milan Ćurković and Damir Vučina)
Surface meshes of 3D objects have to be converted into parametric models in order to be used for shape optimization. We have developed an enhanced NURBS (Non-uniform rational B-splines) parametrization based on fitting parameter values that is capable of handling dynamically changing shapes. By geometric redistribution the capabilities of NURBS are extended towards to those of hierarchical NURBS and T-splines. Compared to hierarchical NURBS and T-splines, the proposed single-patch parameterization results in lower dimensionality enabling optimizers to operate on geometric parameters. Avoiding subdivision surfaces and continuity problems for piecewise NURBS and reparameterizations for T-splines speeds up optimization. This parameterization can be used as an approximation or an initial solution for piecewise NURBS and T-splines.
Modelling of epidemics with time-change Lèvy process
Jasmina Đorđević
Department of Mathematics, University of Oslo, Norway
(joint work with Giulia Di Nunno
and Nenad Šuvak)
We introduce time-changed Lèvy noises when the time-change is independent of the Lèvy process in compartment SIRV model. It is proposed that transmission rate is described with mean reverting process via time changed diffusion process with jumps.
Existence and uniqueness of positive solution of SIRV system with time changed Lèvy process is proved. Furthermore the problems of persistence and extinction are analyzed.
Formulas for some Diophantine quintuples in quadratic fields
Zrinka Franušić
University of Zagreb, Croatia
(joint work with Andrej Dujella and Vinko Petričević)
We present some constructions of special form Diophantine quintuples in rings \(\mathcal R=
\mathbb{Z}[\sqrt D]\) for certain positive integer \(D\). A Diophantine \(m\)-tuple in \(\mathcal{R}\) is a set of \(m\) elements in \(\mathcal{R}\backslash\{0\}\) with the property that the product of any two of its distinct elements increased by the unity is a square in \(\mathcal{R}\). The term "special form" refers to Diophantine quintuples of the form \(\{e, a+ b\sqrt{D},a- b\sqrt{D},c+ d\sqrt{D},c- d\sqrt{D}\}\), where \(a,b,c,d,e\) are integers, i.e. quintuples containing two pairs of conjugates.
Refined Bijections for Recursive Sequences
Petra Marija Gojun
University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Split, Croatia
We study integer sequences defined by the recurrence \(U_{n+1} = p U_{n+1} + U_n \) and the initial values \(U_0=a\), \(U_1= 1\), for \(n \ge 1\). Using a combinatorial interpretation by means of tiling, we prove families of identities involving these sequences, some of them in full generality. In particular, we present the tiling interpretation of the alternating sign dual of the first Sury's identity.
Majorization inequalities obtained via convexity and superquadraticity with applications
Slavica Ivelić Bradanović
Faculty of Civil Engineering, Architecture and Geodesy, University of Split, Croatia
Different terms such as variability, inequality and dispersion, which occur
in various engineering problems and scientific fields, in mathematics are
most simply described by the concept of majorization, a powerful
mathematical tool which allows one to see the existing connections between
vectors that can be used. In majorization theory, majorization inequalities
play an important role. Here we show how some well known properties of
superquadratic functions can be used to obtain certain extensions and
improvements of majorization inequalities as well as some conversions. For
superquadratic functions, which are not convex, we obtain results analog
ones for convex functions. We also discuss about some applications to \(f\)
-divergences.
Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems
Nevena Jakovčević Stor
University of Split, Faculty of Electrical Engineering, Mechanical Engineering and
Naval Architecture, Split, Croatia
(joint work with Ivan Slapničar and Zoran Tomljanović)
A fast algorithm for computing optimal viscosities of dampers of a linear vibrational system is presented. The vibrational system is first modeled using the second-order structure which is a standard approach. This structure yields a quadratic eigenvalue problem which is then linearized. Optimal viscosities are those for which the trace of the solution of the Lyapunov equation with the linearized matrix is minimal. Here, the free term of the Lyapunov equation is a low-rank matrix that depends on the eigenfrequencies that need to be damped. The optimization process in the standard approach requires \(O(n^3)\) floating-point operations. In our approach, we transform the linearized matrix into
an eigenvalue problem of a diagonal-plus-low-rank matrix whose eigenvectors have a Cauchy-like structure.
Our algorithm is based on a new fast eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast multiplication of linked Cauchy-like matrices, yielding computation of optimal viscosities for each choice of external dampers in \(O(kn^2)\) operations, \(k\) being the number of dampers. The accuracy of our algorithm is compatible with the accuracy of the standard approach.
On a floor identity
Julije Jakšetić
Unversity of Zagreb, Croatia
In this poster presentation, using some combinatorics and involutory property
of the function \(f,\
f(x)=\frac{r}{x},\ r\in \mathbb{R}_+\) we give a geometric interpretation of
the identity
$$\sum\limits_{k=1}^\infty\left\lfloor\frac{r}{k}\right\rfloor^2=\sum
\limits_{k=1}^\infty(2k-1)\left\lfloor\frac{r}{k}\right\rfloor.$$
The presentation is based on the paper [1].
References:
References:
- J. Jakšetić, Proof Without Words: From Floor to Stairs, Math. Intell.,43, 3 (2021),29-29.
Computability of chainable graphs
Matea Jelić
Faculty of Civil Engineering, Architecture and Geodesy, University of Split, Croatia
(joint work with Zvonko Iljazović)
We investigate conditions under which a semicomputable set in a computable topological space is computable. We have examined spaces obtained by gluing chainable continua in certain ways. It is known that in a computable topological space a semicomputable set \(S\) is computable if \(S\) is a continuum chainable from \(a\) to \(b\), where \(a\) and \(b\) are computable points, or \(S\) is a circulary chainable continuum which is not chainable. However, we have proved that if \(S\) and \(T\) are semicomputable sets in a computable topological space and \(S\) is obtained by gluing finitely many (circulary) chainable continua to \(T\) (in a specific way), then \(S\) is computable. Also, it is known that if \(S\) and \(T\) are semicomputable sets such that \(S\) is a certain topological graph and \(T\) is the set of its endpoints, than \(S\) is a computable set. We have considered more general graphs \(S’\) by taking edges of \(S’\) to be chainable continua and we have proved that in this case the same conclusion holds.
Dynamical analysis of a stochastic delayed epidemic model with Lévy jumps and regime switching for SARS-CoV-2 virus
Bojana Jovanović
Faculty of Sciences and Mathematics, University of Niš, Serbia
(joint work with Jasmina Djordjević)
In this paper, a delayed stochastic SLVIQR epidemic model for the new coronavirus COVID-19 is derived. Model is constructed by assuming that transmission rate satisfies the mean-reverting Ornstain-Uhlenbeck process and, besides a standard Brownian motion, another two driving processes are considered: a stationary Poisson point process and a continuous finite-state Markov chain. For the constructed model, the existence and uniqueness of positive global solution is proved. Also, sufficient conditions under which the disease would lead to extinction or be persistent in the mean are established and it is shown that constructed model has a richer dynamic analysis compered to existing models. In addition, numerical simulations are given to illustrate the theoretical results.
Weird \(K-\)actions on \(U(\mathfrak{g})\) for \(\mathfrak{g}_0=\mathfrak{s}\mathfrak{o}(3,1)\)
Hrvoje Kraljević
Department of Mathematics, University of Zagreb, Croatia
Let \(\mathfrak{g}_0\) be either \(\mathfrak{s}\mathfrak{o}(n,1)\) or \(\mathfrak{s}\mathfrak{u}(n,1)\), \(\mathfrak{g}\) its complexification, \(K\) a maximal compact subgroup of the adjoint group of \(\mathfrak{g}_0\), \(U(\mathfrak{g})\) the universal enveloping algebra of \(\mathfrak{g}\) and \(U(\mathfrak{g})^K\) its subalgebra of \(K-\)invariants. By our results in [1], besides the usual adjoint action of \(K\) on \(U(\mathfrak{g})\) there is another action of \(K\) commuting with the adjoint action of \(K\) and leaving \(U(\mathfrak{g})^K\) pointwisely invariant. This weird action of \(K\) is not by automorphisms of the algebra \(U(\mathfrak{g});\) it gives automorphisms only after localizing both \(U(\mathfrak{g})\) and \(U(\mathfrak{g})^K\) over \(U(\mathfrak{g})^K\setminus\{0\}\). In the case \(\mathfrak{g}_0=\mathfrak{s}\mathfrak{o}(2,1)\) (or \(\mathfrak{s}\mathfrak{u}(1,1)\)) everything is trivial since \(K\) is commutative. We investigate the weird action of \(K\) in the simplest nontrivial case \(\mathfrak{g}_0=\mathfrak{s}\mathfrak{o}(3,1)\). In this case we found also a third action of \(K\) on \(U(\mathfrak{g})\) commuting with the adjoint action of \(K\) and leaving \(U(\mathfrak{g})^K\) invariant (not pointwisely).
Reference:
Reference:
- H. Kraljevic, The structure of the algebra \((U(\mathfrak{g})\otimes C(\mathfrak{p}))^K\) for the groups \(\mathrm{SU}(n,1)\) and \(\mathrm{SO}_e(n,1)\), Math. Commun. 27(2022),11-18
Csiszér divergence functional and the concept of superadditivity
Neda Lovričević
Faculty of Civil Engineering, Architecture and Geodesy, University of Split, Croatia
(joint work with Đilda Pečarić and Josip Pečarić)
Superadditivity property of the Jensen functional is put in relation to the Csiszér divergence functional. Monotonicity of the Jensen functional and specific bounds obtained consequently, via interpolating Jensen inequalities, reflect within well known \(f\)-divergences. The analyzed inequalities are presented for the Zipf-Mandelbrot law and accentuated for its special form, the Zipf law with its interdisciplinary role.
Digital games in the mathematics classrooms: teacher perspectives from different countries
Darija Marković
Department of Mathematics, J. J. Strossmayer University of Osijek
(joint work with Ljerka Jukić Matić)
Traditional teaching methods are often criticized for not being able to hold students' attention and not giving them the autonomy to acquire knowledge on their own. For this reason, game-based learning (GBL), especially digital game-based learning (DGBL), seems to be a promising approach for learning and teaching mathematics due to its interactive nature.
As part of the ERASMUS+ project Game-based learning in mathematics (GAMMA), we conducted a survey among secondary mathematics teachers from Croatia, Finland, Greece and the Netherlands. The aim of the survey was to investigate the factors that influence teachers' use of digital games in mathematics classrooms. In this poster, we present some of the results of this study and highlight some similarities and differences in the perceptions and attitudes of teachers from different countries towards the use of digital games in mathematics education.
As part of the ERASMUS+ project Game-based learning in mathematics (GAMMA), we conducted a survey among secondary mathematics teachers from Croatia, Finland, Greece and the Netherlands. The aim of the survey was to investigate the factors that influence teachers' use of digital games in mathematics classrooms. In this poster, we present some of the results of this study and highlight some similarities and differences in the perceptions and attitudes of teachers from different countries towards the use of digital games in mathematics education.
Linear codes constructed from weakly self-orthogonal designs
Vedrana Mikulić Crnković
Faculty of Mathematics, Universita of Rijeka, Rijeka, Croatia
(joint work with Ivona Traunkar)
A 1-design is weakly self-orthogonal if all the block intersection
numbers have the same parity. If both \(k\) and the block intersection
numbers are even then 1-design is called self-orthogonal and its
incidence matrix generates a self-orthogonal code. We analyze
extensions of the incidence matrix and an orbit matrix of a weakly
self-orthogonal 1-design, that generates a binary self-orthogonal or LCD code.
Additionally, we study methods for constructing self-orthogonal and LCD codes over an arbitrary field by extending
the incidence
matrix and an orbit matrix of suitable 1-design.
A Visual Proof for an Infinite Alternating Sign Series
Ana Mimica
University of Dubrovnik, Croatia
We provide a visual proof that an infinite alternating sign series $$
1 - \frac{1}{4} + \frac{1}{16} - \frac{1}{64} + \cdots
$$
converge to \(4/5\). Our basic shape consist of 5 equal squares. Correspondence of white and colored areas completes the proof.
The presentation is based on the paper [1].
References:
References:
- I. Martinjak, A. Mimica, Proofs Without Words: A Visual Proof for an Infinite Alternating Sign Series, The College Mathematics Journal, 52:3, 204-204, 2021.
The reduction of the \(pro^{*}\text{-}Grp\) category to \(pro\text{-}Grp\)
Ivančica Mirošević
University of Split, Croatia
(joint work with Nikola Koceić Bilan)
The intention of the coarse shape theory, as a generalisation of the shape theory, is to give a tool for classifying localy bad topological spaces. Some of essential invariants in this theory are coarse shape groups, and a new characterisation of them will be presented.
A newly constructed functor \(\widetilde R\) from \(pro^{*}\text{-}Grp\) to \( pro\text{-}Grp\) enables us to represent morphisms in \(pro^{*}\)-category as morphisms in \(pro\)-category between more complex objects.
It is proven that, for every pointed topological space \(\left(X,x_0\right) \), the functor \( \widetilde R\) applied on \( pro^{*}\text{-}\pi _k \left(X,x_0\right) \) results with a \(pro\text{-}\)coarse shape group \(pro\text{-} \check\pi _k^*\left(X,x_0\right)\). Therefore, since \(\displaystyle \lim\limits_{\leftarrow}pro\text{-} \check\pi _k^*=\check\pi _k^*\) holds, \(pro\text{-} \check\pi _k^*\) is a full analog of \(pro\text{-}\pi_k\).
On this basis, a coarse shape homology group of a topological space is defined and coarse shape groups and coarse shape homology groups are proven to be related in a manner of the Hurewicz theorem, fundamental result of algebraic topology that relates homotopy and homology groups. In particular, the first nontrivial coarse shape group and coarse shape homology group of a pointed continuum are isomorphic, which is an assertion that does not hold for shape groups.
It is proven that, for every pointed topological space \(\left(X,x_0\right) \), the functor \( \widetilde R\) applied on \( pro^{*}\text{-}\pi _k \left(X,x_0\right) \) results with a \(pro\text{-}\)coarse shape group \(pro\text{-} \check\pi _k^*\left(X,x_0\right)\). Therefore, since \(\displaystyle \lim\limits_{\leftarrow}pro\text{-} \check\pi _k^*=\check\pi _k^*\) holds, \(pro\text{-} \check\pi _k^*\) is a full analog of \(pro\text{-}\pi_k\).
On this basis, a coarse shape homology group of a topological space is defined and coarse shape groups and coarse shape homology groups are proven to be related in a manner of the Hurewicz theorem, fundamental result of algebraic topology that relates homotopy and homology groups. In particular, the first nontrivial coarse shape group and coarse shape homology group of a pointed continuum are isomorphic, which is an assertion that does not hold for shape groups.
Generalizations of Steffensen's inequality by Lidstone's polynomial and related results
Anamarija Perušić Pribanić
Faculty of Civil Engineering, University of Rijeka, Croatia
(joint work with Josip Pečarić and Ana Vukelić)
In this poster we present few valuable identities obtained by Lidstone's polynomial. Using these identities we prove new generalizations of Steffensen's inequality for \((2n)\)-convex and \((2n+1)\)-convex functions. Further, using Čebyšev and
Grüss type inequalities we consider the bounds for the integrals in the perturben versions of the previously described identities.
\(D(n)\)-QUINTUPLES WITH SQUARE ELEMENTS
Vinko Petričević
University of Zagreb, Croatia
(joint work with Andrej Dujella and Matija Kazalicki)
For an integer \(n\), a set of \(m\) distinct nonzero integers \(\{a_1,a_2,\ldots,a_m\}\)
such that \(a_ia_j+n\) is a perfect square for all \(1\leq i < j\leq m\), is called a
\(D(n)\)-\(m\)-tuple. We have shown that there are infinitely many essentially different \(D(n)\)-quintuples with square elements.
In this presentation, I will show computational background of this search.
References:
In this presentation, I will show computational background of this search.
References:
- A. Dujella, M. Kazalicki and V. Petričević, D(n)-quintuples with square elements, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 115 (2021), Article 172, (10pp)
The truncated Euler–Maruyama approximate solutions for neutral stochastic differential equations with time-dependent delay and convergence rate
Aleksandra Petrović
University of Niš, Faculty of Science and Mathematics,
Niš, Serbia
In the existing literature, the sufficient conditions under which the truncated Euler-Maruyama solutions for neutral stochastic differential equations with time dependent delay converge in the \(L^p\)-sense to the exact solution of the equation are successfully determined. However, the convergence was in the asymptotic form without the convergence rate. As a continuation of that, the main aim of
in this talk will be presented
the convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay in a finite time interval.
Also, in previous considerations some restrictions on the truncated functions are imposed and as a consequence, the application of the truncated Euler–Maruyama method was limited. The convergence rate without these restrictions will be the main aim of this talk. In order to overcome the difficulties due to the removal of these restrictions, as well as those predicted by the presence of the neutral term and delay function, some new mathematical techniques have been developed. The main theoretical result, that is, the convergence rate is illustrated by an example and numerical simulation.
References:
References:
- A. Petrović, M. Milošević, The truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay, Filomat 35:7 (2021), 2457–2484.
- L. Hu, X. Li, X. Mao, Convergence rate and stability of the truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 337 (2018) 274-289.
Tilings of a Honeycomb Strip and Higher Order Fibonacci Numbers
Luka Podrug
University of Zagreb, Faculty of Civil Engineering
(joint work with Tomislav Došlić)
In this poster we present two types of tilings of a honeycomb strip and derive some closed form formulas for the number of tilings. Furthermore, we obtain some new identities involving tribonacci numbers, Padovan numbers and Narayana’s cow sequence and present combinatorial proofs for several known identities about those numbers.
On the converse Jensen-type inequality for generalized \(f\)-divergences
Mirna Rodić
University of Zagreb Faculty of Textile Technology, Croatia
Motivated by some recent investigations about the sharpness of the Jensen inequality, this poster presentation deals with the sharpness of the converse of the Jensen inequality.
These results are then used for deriving new inequalities for different types of generalized \(f\)-divergences. As divergences measure
the differences between probability distributions, these new inequalities can also be applied on the Zipf-Mandelbrot law as a special kind of a probability distribution.
Flag-transitive and point-imprimitive symmetric designs with \(\lambda \leq 10\)
Aljoša Šubašić
University of Split, Faculty of Science, Split, Croatia
(joint work with Joško Mandić)
In a paper published in 2005, Praeger and Zhou improved the upper bound on
the number of points of a flag-transitive, point-imprimitive, symmetric
design in terms of the number of blocks containing two points. Here we
present an improvement of their result, devise a list of feasible parameter
sequences for which two points are contained in at most ten blocks, and also
develop new methods for the elimination of all but two such sequences. We
used those methods to list all designs with given properties, with the
exclusion of designs with said two sets of parameters for which we give
limitations for their group of automorphisms.
Stochastic SEIPHAR model for epidemic of the SARS-CoV-2 virus
Nenad Šuvak
J.J. Strossmayer University of Osijek, Department of Mathematics
(joint work with Jasmina Djordjević, Bojana Jovanović, Jelena Manojlović and Ivan Papić)
A refined version of the classical SEIR (susceptible-exposed-infected-recovered) model for the epidemic of the SARS-CoV-2 virus is proposed. The compartment of infected individuals is divided into four disjoint classes: symptomatic infected individuals (I), superspreaders (P), hospitalized infected individuals (H) and asymptomatic infected individuals (A). Among many parameters influencing the dynamics of epidemic within this model, the most important ones are the transmission coefficients: \(\beta\) (due to class \(I\)), \(l \beta\) with \(l>0\) (due to class \(H\)) and \(\beta^{\prime} > \beta\) (due to class \(P\)).
The model is constructed from the system of ordinary differential equations describing the dynamics of epidemic, by perturbing transmission coefficients \(\beta\) and \(\beta^{\prime}\) by two independent Brownian motions with different intensities. The resulting system of stochastic differential equations (SDEs) is called the stochastic SEIPHAR model.
The main results to be presented are the proof of existence and uniqueness of the positive global solution of the corresponding system of SDEs and analysis of the basic reproduction number \(R_{0}^{d}\) (related to deterministic model) and its counterpart \(R_{0}^{s}\) (related to stochastic model). It is shown that the threshold between the extinction of the virus and its persistence in mean can be described by the value of \(R_{0}^{d}\) and \(R_{0}^{s}\). Also, sensitivity of \(R_{0}^{d}\) and \(R_{0}^{s}\) to the change in parameter values is analyzed, highlighting the importance of superspreaders and noises in stochastic model.
The theoretical results are illustrated via simulations based on the data from the early phase of the epidemic in Wuhan (January 4 - March 9, 2020).
References:
The model is constructed from the system of ordinary differential equations describing the dynamics of epidemic, by perturbing transmission coefficients \(\beta\) and \(\beta^{\prime}\) by two independent Brownian motions with different intensities. The resulting system of stochastic differential equations (SDEs) is called the stochastic SEIPHAR model.
The main results to be presented are the proof of existence and uniqueness of the positive global solution of the corresponding system of SDEs and analysis of the basic reproduction number \(R_{0}^{d}\) (related to deterministic model) and its counterpart \(R_{0}^{s}\) (related to stochastic model). It is shown that the threshold between the extinction of the virus and its persistence in mean can be described by the value of \(R_{0}^{d}\) and \(R_{0}^{s}\). Also, sensitivity of \(R_{0}^{d}\) and \(R_{0}^{s}\) to the change in parameter values is analyzed, highlighting the importance of superspreaders and noises in stochastic model.
The theoretical results are illustrated via simulations based on the data from the early phase of the epidemic in Wuhan (January 4 - March 9, 2020).
References:
- J. Djordjević, I. Papić, N. Šuvak (2021) A two diffusions stochastic model for the spread of the new corona virus SARS-CoV-2, Chaos, Solitons and Fractals, 148: 110991,
- J. Djordjević, B. Jovanović, J. Manojlović, N. Šuvak (2022) Analysis of stability and sensitivity of deterministic and stochastic models for the spread of the new corona virus SARS-CoV-2, Filomat (to appear)
- F. Ndaïrou, I. Area, J.J. Nieto, D.F.M. Torres (2020) Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos, Solitons and Fractals 135: 109846
- G. Pang, E. Pardoux (2022) Functional Limit Theorems for Non-Markovian Epidemic Models, arXiv:2003.03249v3
NEW INEQUALITIES FOR THE DISCRET ČEBYSEV FUNCTIONAL
Sanja Tipurić-Spužević
Faculty of Chemistry and Technology, University of Split
(joint work with Saad Ihsan Butt, Josip Pečarić)
New upper and lower bounds for the discret Čebyšev functional
involving two n-tuples of real numbers in which the bounding constants are
improved with bounding n-tuples of real numbers by employing the Sonin's
identity.