Number theory group

• Lucija Ružman, prof., KTF, Split

Research areas:

Number theory (Diophantine approximations and applications, Thue equations, Index form equations, Diophantine m-tuples)

Research description:

We will try to establish some new methods for improvement of the classical inequalities for convex functions, such as the Hermite-Hadamard inequalities, Jensen’s inequality and the converse Jensen inequality. As special cases of those improvements we can obtain refinements of the converse Holder inequality and the converse Minkowski inequality. The obtained improvements and of the Jensen inequality and its converse can be used to establish new inequalities related to the Shannon entropy and the Zipf-Mandelbrot law both of which have considerable importance in Information theory. We also investigate various classes of generalized convex functions as well as some classes of functions which produce sharper variants of the classical inequalities for convex functions, for instance superquadratic functions and strongly convex functions. We will try to establish certain integral inequalities which will enable us to obtain better bounds for the reminder of some corrected quadrature formulae.

Contacts with academic and other institutions:

– PMF-Matematički odsjek, Sveučilište u Zagrebu
– Građevinski fakultet, Sveučilište u Zagrebu
– Odjel za matematiku, Sveučilište u Rijeci
– Matematički institut, Sveučilište u Debrecenu, Mađarska
– Institut za analizu i teoriju brojeva, TU-Graz, Austrija
– Matematički odjel, Sveučilište u Salzburgu, Austria
– Institut za napredna istraživanja u matematici, Sveučilište u Strasbourgu, Francuska
– Glavni laboratorij za numeričke simulacije, Neijiang Normal University, Sečuan, Kina
– Matematički odjel, Purdue University North Central, SAD
– Pedagoški fakultet, Sveučilište u Bihaću, BIH

Description of the current cooperation:

The head of research group is a member of the Seminar on Number Theory and Algebra on Department of Mathematics, Faculty of Science, University of Zagreb. Also, she was a team member on several Croatian-Austrian, Croatian-Hungarian, Croatian-French bilateral projects and projects supported by The Ministry of Science, Education and Sports of the Republic of Croatia whose leader or co-leader was Academician Andrej Dujella from Department of Mathematics, Zagreb. Currently, she is a team member on the project “Diophantine m-tuples, elliptic curves, Thue and index form equations” (supported by Croatian Science Foundation) whose leader is also Academician Andrej Dujella.
With Professor A. Dujella she worked on problems related to solving parametric families of quartic Thue equations. In the framework of this cooperation, the head of research group prepared her doctoral thesis (supervisor was A. Dujella) and published 4 research papers (two authored and two co-authored with A. Dujella).

Plans for future research:

We plan to investigate the problems related to determining the minimal index and all elements with minimal index for some other algebraic extension of Q, and analog problems for the relative extensions. Also, we intend to apply some tools from Diophantine approximations to study Diophantine m-tuples (sets with the property that the product of its any two distinct elements increased by 1 is a perfect square) and their generalizations.

List of selected research papers:

1. A. Dujella and B. Jadrijević, A parametric family of quartic Thue equations, Acta Arith. 101 (2002), 159-170;
2. A. Dujella and B. Jadrijević, A family of quartic Thue inequalities, Acta Arith., 111 (2004), 61-76;
3. B. Jadrijević, A system of Pellian equations and related two-parametric family of quartic Thue equations, Rocky Mountain J. Math. 35, No. 2 (2005), 547-572;
4. B. Jadrijević and V. Ziegler, A system of relative Pellian equations and a related family of relative Thue equations, Int. J. Number Theory 2, No. 4 (2006), 569-590;
5. C. Fuchs and B. Jadrijević, On a parametric family of Thue inequalities over function fields, Math. Proc. Cambridge Philos. Soc. 143 (2007), 9-23;
6. B. Jadrijević, Establishing the minimal index in a parametric family of bicyclic biquadratic fields, Period. Math. Hungar. 58 (2) (2009), 2; 155-180.
7. B. He, B. Jadrijević and A. Togbé, Solutions of a class of quartic Thue inequalities, Glas. Mat., 44 (2009), 2; 309-321;
8. A. Dujella, B. Ibrahimpašić and B. Jadrijević, Solving a family of quartic Thue inequalities using continued fractions, Rocky Mountain J. Math. 41 (2011), 1173-1182.;
9. B. Jadrijević, On elements with index of the form 2^a3^b in a parametric family of biquadratic fields, Glas. Mat., 50 (2015), 1, 43-63;
10. M. Bliznac, A. Filipin, An upper bound for the number of Diophantine quintuple, Bulletin of the Australian Mathematical Society, 94 (2016), 3; 384-394.;
10. I. Gaal, B. Jadrijević, Determining elements of minimal index in an infinite family of totally real bicyclic biquadratic number fields, JP Journal of Algebra, Number Theory and Applications, 39 (2017), 3; 307-326.;
11. Z. Franušić, B. Jadrijević, Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields, Publicationes Mathematicae Debrecen, 92 (2018), 3-4; 293-315;
12. M . Bliznac Trebješanin, A. Filipin, A. Jurasić, On the polynomial quadruples with the property D(−1 ; 1), Tokyo Journal of Mathematics (prihvaćen za objavljivanje);
13. M . Bliznac Trebješanin, A. Filipin, Nonexistence of D(4)-quintuples, (u postupku recenzije);
14. I. Gaal, B. Jadrijević, L. Remete, Totally real Thue equations over imaginary quadratic fields, (u postupku recenzije);
15. I. Gaal, B. Jadrijević, L. Remete, Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields, (u postupku recenzije)