Plenary lectures

Invited lectures

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.

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

Hybrid CUR-type decomposition of tensors in the Tucker format

Erna Begović Kovač
University of Zagreb, Croatia

Low rank approximation obtained by SVD is often hard to interpret in applications. Besides, it does not keep useful matrix properties like sparsity and non-negativity. Therefore, in recent years attention is given to low-rank approximations obtained by interpolatory factorizations. These approximations keep the properties mentioned above and are more suitable in some applications where the columns and/or rows should keep their original meaning. The best known example of the interpolatory factorizations is CUR factorization.
A matrix CUR factorization of \(A\in\mathbb{C}^{m\times n}\) is decomposition of the form \(A\approx CUR\), where \(C\in\mathbb{C}^{m\times k}\) contains \(k\) columns of \(A\), \(R\in\mathbb{C}^{k\times n}\) contains \(k\) rows of \(A\) and \(U\in\mathbb{C}^{k\times k}\). A CUR-type decomposition of a tensor \(\mathcal{A}\in\mathbb{C}^{n_1\times n_2\times\cdots\times n_d}\) is given by \[\mathcal{A}\approx\mathcal{S}\times_1C_1\times_2C_2\times_3\cdots\times_dC_d,\] where \(\mathcal{S}\in\mathbb{C}^{r_1\times r_2\times\cdots\times r_d}\) is a core tensor, matrices \(C_j\in\mathbb{C}^{n_j\times r_j}\), \(1\leq j\leq d\), contain \(r_j\) mode-\(j\) fibers of \(\mathcal{A}\), and \(\times_j\) stands for mode-\(j\) product. The error of such approximation depends on the tensor dimension \(d\). This can be a problem for high-dimensional tensors. At the same time in the applications it is not always important to keep the original entries in all modes.
In this talk we introduce a hybrid approach to the CUR-type decomposition of tensors in the Tucker format. The main idea of the hybrid algorithm is to write a tensor \(\mathcal{A}\) as a product of a core tensor \(\mathcal{S}\), a matrix \(C\) obtained by extracting mode-\(k\) fibers of \(\mathcal{A}\), and matrices \(U_j\), \(j=1,\ldots,k-1,k+1,\ldots,d\), chosen to minimize the approximation error \(\mathcal{E}\), \[\mathcal{A}=\mathcal{S}\times_1U_1\times_2\cdots\times_{m-1}U_{m-1}\times_mC_m\times_{m+1}U_{m+1}\cdots\times_dU_d+\mathcal{E}.\] The approximation error obtained this way is smaller than the one from the standard tensor CUR-type method. This difference gets more important as the tensor dimension increases. We give the error bound for the new method and compare it to the error resulting from the standard CUR approach.

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:
  1. I. Matić, Composition factors of a class of induced representations of classical p-adic groups, Nagoya Mathematical Journal 227 (2017), 16-48.
  2. G. Muić, Composition series of generalized principal series; the case of strongly positive discrete series, Israel J. Math. 140 (2004), 157–202.
  3. 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.

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
University of Rijeka, Department of Mathematics, 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 \(x\leq y\in \frac{1}{2}\mathbb{Z}+1\).

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.

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
Department 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.

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:
  1. 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.
  2. S. L. Druţă-Romaniuc, J. Inoguchi, M. I. Munteanu, A. I. Nistor, Magnetic curves in cosymplectic manifolds, Report Math. Phys. 78 (2016), 33-47.
  3. Z. Erjavec, J. Inoguchi, Magnetic curves in \(Sol_3\), J. Nonlinear Math. Phys. 25 (2), (2018) 198-210.
  4. 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.
  5. J. Inoguchi, M. I. Munteanu, A. I. Nistor, Magnetic curves in quasi-Sasakian 3-manifolds, Anal. Math. Phys. 9, (2019) 43-61.
  6. M. I. Munteanu, Magnetic curves in a Euclidean space: One example, several approaches, Publ. de L'Institut Math. 94 (108) (2013), 141-150.
  7. 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:
  1. Gotovac V (2019) Similarity between random sets consisting of many components. Image Anal Stereol. 38(2):185-99


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.

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.

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.

A Szemerédi-type theorem for subsets of the unit cube

Vjekoslav Kovač
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with Polona Durcik)

We are interested in arithmetic progressions in positive measure subsets of \([0,1]^d\) . After a counterexample by Bourgain, it seemed as if nothing could be said about the longest interval formed by sizes of their gaps. However, Cook, Magyar, and Pramanik gave a positive result for \(3\) -term progressions if their gaps are measured in the \(\ell^p\) -norm for \(p\) other than \(1\) , \(2\) , and \(\infty\), and the dimension \(d\) is large enough. We establish an appropriate generalization of their result to longer progressions. The main difficulty lies in handling a class of multilinear singular integrals associated with arithmetic progressions that includes the well-known multilinear Hilbert transforms, bounds for which still constitute an open problem. As a substitute, we use the previous work of Durcik, Thiele, and the author on power-type cancellation of those transforms, which was, in turn, motivated by a desire to quantify the results of Tao and Zorin-Kranich.

Parafermionic bases of standard modules for affine Lie algebras

Slaven Kožić
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
(joint work with Marijana Butorac and Mirko Primc)

The parafermionic currents present a remarkable class of nonlocal vertex operators with variables in fractional powers which go back to A. B. Zamolodchikov and V. A. Fateev. In this talk, we present a construction of combinatorial bases of parafermionic spaces associated with the standard modules of the rectangular highest weights for the non-simply-laced untwisted affine Lie algebras. The construction relies on the quasi-particle bases of the corresponding principal subspaces found by M. Butorac and the author. As an application, we obtain a proof of the character formulae of A. Kuniba, T. Nakanishi and J. Suzuki.
This research has been supported in part by Croatian Science Foundation under the project 8488.

On joint weak convergence of partial sum and maxima processes

Danijel Krizmanić
Department 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.

A glance at \(p\)-adic integration techniques: Deriving combinatorial sums and identities containing generalized factorials

Irem Kucukoglu
Alanya Alaaddin Keykubat University Faculty of Engineering, Department of Engineering Fundamental Sciences, Antalya, Turkey

The main aim of this presentation is to investigate and survey the \(p\)-adic integration techniques to show how these techniques allow us to derive combinatorial sums and identities, in the case of their blending with generating function techniques. As the main results, by \(p\)-adic integral representations of some special functions associated with generalized factorials, some combinatorial sums and identities are derived in this presentation. Finally, this presentation is ended by providing some remarks and observations on the main results.
Let \(\mathbb{Z}_{p}\) be the set of \(p\)-adic integers. If we assume that \(f\left(x\right)\) is a uniformly differentiable function on \(\mathbb{Z}_{p}\), then its bosonic \(p\)-adic integral (also called Volkenborn integral) is defined by \[ \int\limits_{\mathbb{Z}_{p}}f\left( x\right) d\mu _{1}\left( x\right) =\lim\limits_{N\rightarrow \infty }\frac{1}{p^{N}}\sum_{x=0}^{p^{N}-1}f% \left( x\right) \label{bosonic} \] where \(\mu _{1}\left( x\right)\) denotes the Haar distribution defined as \[ \ \mu _{1}\left( x\right) =\mu _{1}\left( x+p^{N}\mathbb{Z}_{p}\right) = \frac{1}{p^{N}} \] on the set of \(p\)-adic integers \(\mathbb{Z}_{p}\) cf. [1, 2, 3, 4, 5, 6].
The fermionic \(p\)-adic integral of \(f\left(x\right)\) on \(\mathbb{Z}_{p}\) is defined by \[ \int\limits_{\mathbb{Z}_{p}}f\left( x\right) d\mu _{-1}\left( x\right) =\lim\limits_{N\rightarrow \infty }\sum_{x=0}^{p^{N}-1}\left( -1\right) ^{x}f\left( x\right) \label{fermionic} \] where \[ \mu _{-1}\left(x\right)=\mu _{-1}\left( x+p^{N}\mathbb{Z}_{p}\right) =\frac{% (-1)^{x}}{p^{N}} \] on the set of \(p\)-adic integers \(\mathbb{Z}_{p}\) cf. [1, 2].
It is well-known that the \(p\)-adic integrals have connections with the concepts of bosons and fermions in quantum physics. Researchers interested in the \(p\)-adic integration techniques and their applications may especially consult the works of Schikhof [4], Kim [1], and also Simsek [5] and [6] for further details and examples.
In the light of the above preliminary information about \(p\)-adic integration techniques, by applying the aforementioned \(p\)-adic integrals to some special functions associated with generalized factorials, we derive some combinatorial sums and identities in this presentation. Finally, we end this presentation by providing some remarks and observations on the main results.
As conclusion, the results of this presentation have potential to attract the attention of researchers who work on not only mathematics, mathematical physics and quantum physics, but also other relevant areas concerned with \(p\)-adic integration techniques.
References:
  1. T. Kim. \(q\)-Volkenborn integration. Russ. J. Math. Phys. 19 (2002), 288-299.
  2. T. Kim. \(q\)-Euler numbers and polynomials associated with \(p\)-adic \(q\)-integral and basic \(q\)-zeta function. Trends Int. Math. Sci. Study 9 (2006), 7-12.
  3. T. Kim. An invariant \(p-adic \(q\)-integral on \(\mathbb{Z}_p\). Appl. Math. Lett. 21 (2008), 105-108.
  4. W. H. Schikhof. Ultrametric Calculus: An Introduction to \(p\)-adic Analysis. Cambridge Stud. Adv. Math. 4, Cambridge University Press, Cambridge, 1984.
  5. Y. Simsek. Formulas for \(p\)-adic \(q\)-integrals including falling-rising factorials, combinatorial sums and special numbers. arXiv:1702.06999v1, 2017.
  6. Y. Simsek, Explicit formulas for \(p\)-adic integrals: Approach to \(p\)-adic distributions and some families of special numbers and polynomials. Montes Taurus J. Pure Appl. Math. 1 (2019), No. 1, 1-76.


Construction of strongly regular graphs having an automorphism group of composite order

Marija Maksimović
Department of Mathematics, University of Rijeka
(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 \(\mathcal{H}_{n}\).

Asymptotic expansions and 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 mean \(M_0\) can be defined in several ways. One is related to the group structure on \(\mathcal M\) and other is the functional symmetric mean with respect to fixed mean \(M_0\). We discuss some properties of such symmetries trough connection with asymptotic expansion of a mean.

On Ideals Defining Irreducible Representations of Reductive \(p\)-adic Groups

Goran Muić
Department of Mathematics, Faculty of Science, University of Zagreb, Croatia

Let \(G\) be a reductive \(p\)-adic group. Assume that \(L\subset G\) is an open-compact subgroup, and \(\mathcal H_L\) is the Hecke algebra of \(L\)-biinivariant complex functions on \(G\). It is a well-known and standard result on how to prove existence of a complex smooth irreducible \(G\)-module out of a maximal left ideal \(I\subset \mathcal H_L\). Using theory on Bernstein center we make this construction explicit. This leads us to some very interesting questions.
The author acknowledges Croatian Science Foundation grant no. 3628.

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.

On the problem of resilience in flow networks

Jelena Sedlar
University of Split, Faculty of Civil Engineering, Architecture and Geodesy, Croatia

We consider the problem of flow in directed networks with one source and one sink with respect to its resilience to the network disruptions. The upper and the lower bound of the capacity are given for each edge in the network, while the cost of each edge is given as the function of edge capacity. The problem of network design consists of selecting a subset of edges in the given network, which induces an optimal subnetwork to be resilient after disruptive event. The restoration behaviour of each edge in a network after the disruptive event is described by using a non-linear function that enables the modelling of three components affecting resilience: the remaining capacity of the edge after the disruption, the degree to which capability can be recovered and the recovery speed. Three different models for designing a resilient network are proposed and then formulated as problems of non-linear optimisation. A simple genetic algorithm using stochastic ranking, which can be used to approach all three proposed network design problems, is proposed. One numerical example is used to illustrate the proposed procedure and the effectiveness of the proposed algorithm.

An approach for deriving identities and relations involving Bezier curves and exponential Euler splines

Yilmaz Simsek
Akdeniz University Faculty of Science, Department of Mathematics, Antalya, Turkey

In this presentation, we investigate and survey some properties of the Bezier curves and exponential Euler splines. Schoenberg [2, 3] gave many properties of the cardinal splines of degree \(n\), \(S_{n}(x;u)\), which satisfies \[S_{n}(x+1;u)=uS_{n}(x;u),\] where \(u \neq 0\) cf. [2, 3]. Schoenberg prove that there is a unique cardinal spline of degree \(n\) related to the exponential Euler splines. These splines are associated with the Euler-Frobenius polynomials and numbers. These numbers and polynomials are defined by means of the following generating functions, respectively: \[\tag{1} F_{P}(z,x;u)=\frac{(1-u)e^{zx}}{e^{t}-u}=\sum_{n=0}^{\infty }H_{n}(x;u)\frac{z^{n}}{n!}, \label{A0} \] and \[ F_{N}(t,u)=\frac{1-u}{e^{t}-u}=\sum_{n=0}^{\infty }H_{n}(u)\frac{t^{n}}{n!}, \] \(u\neq 0, 1\).
The well-known exponential Euler splines are defined by \[\tag{2} S_{n}(x;u)=\frac{H_{n}(x;u)}{H_{n}(u)}, \label{A} \] where \(H_{n}(u)\neq 0\) and \(0\leq x <1\) cf. [2, 3, 5].
By using generating functions and zeta functions, we derive some identities and relations including the exponential Euler splines, the Euler-Frobenius polynomials and numbers. Moreover, we give relations among the the exponential Euler splines, the Bezier curves, and the Bernstein polynomials. These well-known famous Bezier curves are given as follows: \[ B(x;n)=\sum\limits_{j=0}^{n }P_{k}B_{k}^{n}(x), \] where \(P_{0}, P_{1},\dots, P_{n}\) denote the control points and \(B_{k}^{n}(x)\) denote the Bernstein basis functions, which are defined by means of the following generating function: \[\tag{3} \frac{(tx)^{k}}{k!}e^{(1-x)t}=\sum\limits_{n=0}^{\infty }B_{k}^{n}(x), \label{B} \] where \[\tag{4} B_{k}^{n}(x)=\binom{n}{k}x^{k}(1-x)^{n-k}, \label{A.Berns.} \] \(n\) and \(k\) are nonnegative integers and also \(0\leq k\leq n\) cf. [1, 4, 6].
By using equations (1), (2), (3), and (4), some formulas and combinatorial sums involving the aforementioned numbers, polynomials and splines are given.
The results of this presentation have potentially used in mathematics, in physics, and in engineering and also in other areas.
The present investigation was supported by the Scientific Research Project Administration of Akdeniz University: Project ID: FDK-5276 and Project Code: 5276.
References:
  1. R. Goldman. Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling. Morgan Kaufmann Publishers, R. Academic Press, San Diego, CA, 2002.
  2. I. J. Schoenberg. Cardinal interpolation and spline functions. J. Approx. Theory 2 (1969), 167-206.
  3. I. J. Schoenberg. Cardinal Interpolation and Spline Functions IV. The Exponential Euler Splines. In: P. L. Butzer, J. P. Kahane, B. Szökefalvi--Nagy (eds.) Linear Operators and Approximation. International Series of Numerical Mathematics vol 20. Birkhäuser, Basel, 1972, 382-404.
  4. Y. Simsek. Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions. Fixed Point Theory Appl. 2013, 2013: 80, 1-13.
  5. Y. Simsek. Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl. 2013, 2013: 87, 1-28.
  6. Y. Simsek. Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers. Math. Methods Appl. Sci. 38 (2015), No. 14, 3007-3021.


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.

Inequalities of the Bellman-Steffensen type for positive measures

Ksenija Smoljak Kalamir
University Of Zagreb Faculty of Textile Technology, Zagreb, Croatia

Since Bellman's \(L^p\) generalization of Steffensen's inequality in 1959, which was incorrect as stated, many papers have been devoted to its modifications. Many corrected versions of Bellman's result were obtained in different settings. In this talk we generalize some Bellman-Steffensen type inequalities in a measure theoretic settings. We show that by using Jensen's inequality and a sub-linearity of a class of convex functions we can obtain nonnormalized versions of Steffensen's inequality.

Computing constants in degenerate subspaces of the multiparametric quon algebra \({\mathcal{B}}\)

Milena Sošić
Department of Mathematics, University of Rijeka, 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:
  1. 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.
  2. M. Sošić, Computation of constants in multiparametric quon algebras. A twisted group algebra approach, Math. Commun. 22 No.2, (2017), 177-192.


Convergence of ergodic-martingale paraproducts

Mario Stipčić
University of Zagreb
(joint work with Vjekoslav Kovač)

We discuss paraproduct-type bilinear operators that simultaneously generalize martingales and ergodic averages. We establish their \(L^p\) convergence for a certain range of exponents and pose an open question about their a.e. convergence.

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.

Limit theorems for a stable sausage

Stjepan Šebek
University of Zagreb, Croatia
(joint work with Wojciech Cygan (TU Dresden) and Nikola Sandrić (University of Zagreb))

In this article, we study fluctuations of the volume of a stable sausage defined via a \(d\)-dimensional rotationally invariant \(\alpha\)-stable process with \(d > 3\alpha/2\), and a closed unit ball. As the main results, we establish a functional central limit theorem with a standard one-dimensional Brownian motion in the limit, and an almost sure invariance principle for the process of the volume of a stable sausage. As a consequence, we obtain Khintchine's and Chung's laws of the iterated logarithm for this process.

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 \(k=2\).

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.

Teaching of initial multiplication concepts and skills in Croatia

Anđa Valent
Zagreb University of Applied Sciences, Zagreb, Croatia
(joint work with Tihana Baković and Goran Trupčević)

In our work we study the treatment of the initial learning of multiplication in Croatia. Since teaching is a complex system rooted in a certain cultural script (Stiegler, Hiebert, 1999), in order to notice some of its attributes that from the inside appear to be self-evident, when describing it, it is necessary to step out of this cultural frame.
For this reason, we compared the teaching of initial multiplication concepts and skills, up to the multiplications table, in two series of textbooks from Croatia and Singapore. In the analysis of the textbooks we used an adapted framework from Charalambous, Delaney, Hsu and Mesa (2010) that looks at a textbook as an environment for construction of knowledge of a single mathematical concept.
Preliminary findings (Baković, Trupčević, Valent, 2019) indicate that in Croatia learning of initial multiplication concepts and skills heavily relies on practice, without given support in underlying constructs and representations or different multiplication strategies. Hence it is expected that multiplications table is understood by Croatian students as something that is expected to be memorized, and not something to be understood and be able to develop on their own.

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.

Constructions of block designs from orbit matrices using a modified genetic algorithm

Tin Zrinski
Department 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.

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.

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.

Posters

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.

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