### Abstracts

On the representation theory of affine vertex algebras and W-algebras
Department of Mathematics, Faculty of Science, University of Zagreb

In the first part of the talk we shall briefly review development and main results in the representation theory of affine vertex algebras and $$W$$–algebras obtained in the last few years. We shall discuss problems related with rationality, $$C_2$$–cofiniteness, classification of irreducible modules and relations with quantum groups.

Next we shall present our main results in this direction. A particular emphasis will be put on the following topics:

• Representation theory of vertex algebras appearing in logarithmic conformal field theory. We shall review some results obtained on triplet vertex algebras from joint papers with A. Milas. These results gave one of the first examples of non-rational vertex algebras having finitely many irreducible modules.

• Classification of irreducible Wakimoto and Whittaker modules for affine vertex algebras.

• Explicit realizations of superconformal vertex algebras and $$W$$–algebras and their applications in the theory of conformal embeddings.

References:

D. Adamović, A. Milas, On the triplet vertex algebra $$W(p)$$, Advances in Mathematics 217 (2008), 6; 2664–2699

D. Adamović, R. Lu, K. Zhao, Whittaker modules for the affine Lie algebra $$A_1 ^{(1)}$$, Advances in Mathematics 289 (2016) 438–479

D. Adamović, A realization of certain modules for the $$N=4$$ superconformal algebra and the affine Lie algebra $$A_2 ^{(1) }$$, to appear in Transformation Groups (2016)

D. Adamović, V. G. Kac, P. Möseneder Frajria, P. Papi, O. Perše, Conformal embeddings of affine vertex algebras in minimal $$W$$-algebras I, II, arXiv:1602.04687, arXiv:1604.00893.

The Core Ingram Conjecture in non-recurrent critical orbit case
Ana Anušić
Faculty of Electrical Engineering and Computing, University of Zagreb

In the early 90’s Tom Ingram posed the problem of classifying inverse limit spaces on intervals with the single tent bonding map. These spaces revealed very rich structures and seemed notoriously difficult to classify. Finally, Barge, Bruin and Štimac in 2012. showed that nondegenerate inverse limits of tent maps with different slopes are non-homeomorphic. However, the proof crucially depends on the ray compactifying on the core of the inverse limit space leaving the core version of the conjecture open. We will give a quick overview of the topological properties of these spaces and discuss the recent partial result showing that all cores of inverse limits of tent maps with non-recurrent critical orbits are non-homeomorphic.

This is a joint work with H. Bruin and J. Činč from the University of Vienna.

Frames and outer frames in Hilbert $$C^*$$-modules
Ljiljana Arambašić
Department of Mathematics, Faculty of Science, University of Zagreb

In this talk we present some new results in the theory of frames for countably generated Hilbert $$C^*$$-modules over arbitrary $$C^*$$-algebras. In investigating the non-unital case we introduce the concept of an outer frame as a sequence in the multiplier module $$M(X)$$ that has the standard frame property when applied to elements of the ambient module $$X$$. Given a Hilbert $$\mathcal{A}$$-module $$X$$, we show that there is a bijective correspondence of the set of all adjointable surjections from the generalized Hilbert space $$\ell^2(\mathcal{A})$$ to $$X$$ and the set consisting of all both frames and outer frames for $$X$$.

The presentation is based on a joint work with Damir Bakić.

Generalization of Jensen’s inequality by Euler’s identity and related results
Gorana Aras-Gazić
Faculty of Arhitecture, University of Zagreb

In this paper we consider $$n$$-convex functions. Using Euler’s identity, the result concerning for Jensen’s inequality and converses of Jensen’s inequality for signed measure are presented. As a consequence, also the results concering to the Hermite-Hadamard inequalities are presented. Using these inequalities, we produce new exponentially convex functions. Finally, we give several examples of the families of functions for which the obtained results can be applied.

This is joint work with J. Pečarić and A. Vukelić.

References:

M. Abramowitz and I. A. Stegun (Eds), Handbook of mathematical functions with formulae,graphs and mathematical tables, National Bureau of Standards, Applied Math.Series 55, 4th printing, Washington, 1965.

K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, 1989.

I. S. Berezin and N. P. Zhidkov, Computing methods, Vol. I, Pergamon Press, Oxford, 1965.

W. Ehm, M. G. Genton, T. Gneiting, Stationary covariance associated with exponentially convex functions, Bernoulli 9(4) (2003), 607-615.

J. Jakšetić, J. Pečarić, Exponential Convexity Method, Journal of Convex Analysis {\bf 20}(1) (2013), 181-197.

S. Karlin, Total Positivity, Stanford Univ. Press, Stanford, 1968.

V. I. Krylov, Approximate calculation of integrals, Macmillan, New York-London, 1962.

P. Lah, M. Ribarić, Converse of Jensen's inequality for convex functions. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. {\bf 412-460} (1973), 201-205.

J. E. Pečarić, F. Proschan and Y. L. Tong, Convex functions, partial orderings, and statistical applications, Mathematics in science and engineering 187, Academic Press, 1992.

T. Popoviciu, Sur l'approximation des fonctions convexes d'ordre superieur, Mathematica 10, (1934), 49-54.

$$D(4)$$-pairs $$\{F_{2k},F_{2k+6}\}$$ and $$\{P_{2k},P_{2k+4}\}$$
Ljubica Baćić
Primary school Nikola Andrić

We call the set of m positive distinct integers a $$D(4)$$-$$m$$-tuple if the product of any of its two elements increased by 4 is a perfect square. Let $$k\geq1$$ be an integer and let $$F_k$$ be the $$k$$-th Fibonacci number and $$P_k$$ $$k$$-th Pell number. In this paper we prove that the pairs $$\{F_{2k},F_{2k+6}\}$$ and $$\{P_{2k},P_{2k+4}\}$$ cannot be extended to a $$D(4)$$-quintuple. This is a joint work with A. Filipin.

Expansions from frame coefficients with erasures
Damir Bakić
Department of Mathematics, Faculty of Science, University of Zagreb

We propose a new approach to the problem of recovering signal from frame coefficients with erasures. Such problems arise naturally from applications where some of the coefficients could be corrupted or erased during the data transmission. Provided that the erasure set satisfies the minimal redundancy condition, we construct a suitable synthesizing dual frame which enables us to perfectly reconstruct the original signal without recovering the lost coefficients. Such dual frames which compensate for erasures will be described from various viewpoints. In addition, frames robust with respect to finitely many erasures will be discussed. We characterize all full spark frames for finite-dimensional Hilbert spaces. In particular, we show that each full spark frame is generated by a totally non-singular matrix. Finally, we give a method, applicable to a large class of frames, for transforming general frames into Parseval ones.

The presentation is based on a joint work with Ljiljana Arambašić.

Structure-preserving low multilinear rank approximation of antisymmetric tensors
Erna Begović Kovač
Faculty of Chemical Engineering and Technology, University of Zagreb

A tensor $$\mathcal{A} \in \mathbb{R}^{n\times \cdots \times n}$$ of order $$d\geq 2$$ is called antisymmetric if its entries $$\mathcal{A}(i_1,i_2,\ldots, i_d)$$ change sign when permuting pairs of indices. Antisymmetric tensors play a major role in quantum chemistry, where the Pauli exclusion principle implies that wave functions of fermions are antisymmetric under permutations of variables.

Here we are concerned with finding an approximation $$\mathcal{B}$$ to a given antisymmetric tensor $$\mathcal{A}$$ such that $$\mathcal{B}$$ has a data-sparse representation and is again antisymmetric. More specifically, we consider an approximation of multilinear rank $$r$$ in the structure-preserving Tucker decomposition $$\mathcal{B} = \mathcal{S} \times_1 U \times_2 U \cdots \times_d U$$, where $$\mathcal{S} \in \mathbb{R}^{r\times \cdots \times r}$$ for some $$r \leq n$$ is again antisymmetric and $$U \in \mathbb{R}^{n\times r}$$ has orthonormal columns. We show which ranks can be attained by an antisymmetric tensor. In order to preserve antisymmetry we discuss the adaption of existing approximation algorithms, most notably a Jacobi-type algorithm.

Particular attention is paid to the important special case when choosing the rank equal to the order of the tensor. We show that a best unstructured rank-$$1$$ approximation can always be turned into a best antisymmetric multilinear rank-$$d$$ approximation. This allows the straightforward application of the higher-order power method, for which we consider effective initialization strategies.

This is a joint work with Daniel Kressner (EPF Lausanne).

Model Order Reduction of Linear Stochastic Systems Driven by Lévy Noise
Peter Benner
Max Planck Institute, Magdeburg

Model order reduction is a technique to reduce the simulation and optimization time for mathematical model of physical systems. It is employed in many areas of the sciences and engineering to accelerate numerical computations in a multi-query context, i.e., when the same model is repeatedly evaluated with varying forcing functions, boundary conditions, etc. The increased need for stochastic modeling to capture model uncertainties in real life applications leads to the demand for such techniques also in the context of models described by stochastic ordinary or partial differential equations.

Here, we discuss model order reduction for linear dynamical systems driven by Lévy processes. In particular, we investigate balanced truncation and singular perturbation approximation for these systems. These methods are standard tools in systems and control engineering. Their popularity is based on the fact that they come with computable error bounds and that they preserve stability in the deterministic setting. We will discuss in how far these properties carry over to the stochastic case, and we will derive error bounds for the methods discussed. Numerical experiments illustrate our findings.

Iterations of the generalized Gram–Schmidt procedure for generating Parseval frames
Tomislav Berić
Department of Mathematics, Faculty of Science, University of Zagreb

A sequence $$(f_i)_{i \in I}$$ in a Hilbert space $$H$$ is called a frame for $$H$$ if there exist constants $$0 < A \le B < \infty$$ such that $A \| f \|^2 \le \sum_{i \in I} | \left< f, f_i \right> |^2 \le B \| f \|^2, \quad \text{for all } f \in H.$ Among all frames, those for which $$A = B = 1$$, called Parseval frames, have proved to be most useful in applications since they provide a simple reconstruction formula $f = \sum_{i \in I} \left< f, f_i \right> f_i, \quad \text{for all } f \in H.$ In this paper we investigate some properties of the generalized Gram–Schmidt procedure (GGSP) for generating Parseval frames which was first introduced by Casazza and Kutyniok (2007). Motivated by iterative algorithms such as the frame algorithm for vector reconstruction and the gradient descent of the frame potential used for construction of approximate unit-norm tight frames, we investigate iterations of the GGSP and its limit. We show that regardless of the starting frame, in the limit case we always get an orthonormal basis with added zeros. Moreover, the position of zero vectors is known in advance.

Upper bound on number of $$D(4)$$-quintuples
Marija Bliznac
Faculty of Civil Engineering, University of Zagreb

We call the set of $$m$$ positive distinct integers $$D(n)$$-$$m$$-tuple if the product of any of its two elements increased by $$n$$ is a perfect square. It is conjectured that there does not exist a $$D(4)$$-quintuple and we will present some results that support that conjecture. We will show how we improved previous results on the upper bound of $$D(4)$$-quintuples, more precisely, we prove there is at most $$6.8587\cdot 10^{29}$$ $$D(4)$$-quintuples. In the proof we use the standard methods used in solving similar problems, solving the system of simultaneous Pellian equations, and combine them with the most recent methods used for bounding the number of $$D(1)$$-quintuples. This is a joint work with A. Filipin.

Parallel CPU/GPU m–Hessenberg–Triangular–Triangular Reduction
Nela Bosner
Department of Mathematics, Faculty of Science, University of Zagreb

The $$m$$–Hessenberg–Triangular-Triangular (mHTT) reduction is a simultaneous orthogonal reduction of three matrices to condensed form. It has applications, for example, in solving shifted linear systems arising in various control theory problems. This talk presents a new heterogeneous CPU/GPU implementation of the mHTT reduction, and compares it with an existing CPU implementation. The algorithm offloads the compute-intensive matrix–matrix multiplications to the GPU and keeps the inner loop, which is memory-intensive and has a complicated control flow, on the CPU. Computations on the CPU are also parallelized, and load is balanced among the threads. Experiments demonstrate that the heterogeneous implementation can be superior to the existing CPU implementation on a system with $$2 \times 8$$ CPU cores and one GPU.

Application of Chebyshev splines on image resampling
Tina Bosner
Department of Mathematics, Faculty of Science, University of Zagreb

Digital raster images often need to be represented in higher and lower resolutions. Resampling of digital images is an essential part of image processing. The most efficient and sufficiently accurate image resampling techniques can produce spurious oscillations near sharp transitions of color. To improve that, we use several techniques based on Chebyshev splines applied dimension by dimension. The main idea is to introduce smooth, shape preserving approximations, which produce sharp, non-oscillating image edges. To demonstrate the advantages of our algorithms, we compare them with other algorithms on real digital images.

On codes, designs and their $$q$$-analogs
Michael Braun
Fachbereich Informatik, Hochschule Darmstadt

An $$(n,k,\delta,q)$$-constant dimension code $$C$$ is a set of $$k$$-dimensional subspaces of an $$n$$-dimensional vector space over the finite field with $$q$$ elements such that the subspace distance $$d(X,Y)=\dim(X+Y)-\dim(X\cap Y)$$ is at least $$\delta$$ for all $$X,Y\in C$$. In 2008 Koetter and Kschischang showed that subspace codes define exactly the class of codes that is required for error and erasure detection and correction in random network coding. In this talk l introduce subspace codes from a lattice point of view, show the analogy to ordinary binary codes and the equivalence to $$q-$$analogs of packing designs in projective geometry. Finally, one major challenge of subspace codes will be discussed—the determination of the maximum cardinality of an $$(n,k,\delta,q)$$-constant dimension code.

Two divisors of $$(n^2+1)/2$$ summing up to $$\delta n + \delta \pm 2$$, $$\delta$$ even
Sanda Bujačić
Department of Mathematics, University of Rijeka

Ayad and Luca \cite{ayad} have proved that there does not exist an odd integer $$n>1$$ and two positive divisors $$d_1, d_2$$ of $$(n^2+1)/2$$ such that $d_1+d_2=n+1.$ Dujella and Luca \cite{dujella} have dealt with more general issue, where $$n+1$$ was replaced with an arbitrary linear polynomial $$\delta n+\varepsilon$$, when $$\delta>0$$ and $$\varepsilon$$ are given integers and they have focused on the case when $$\delta\equiv\varepsilon\equiv1\pmod{2}$$. Bujačić \cite{bujacic} deals with the case when coefficients $$\delta, \ \varepsilon$$ are even, or more precisely when $\delta\equiv\varepsilon+2\equiv0 \enspace \textrm{or}\enspace 2\pmod{4},$ for some fixed $$\delta$$ and $$\varepsilon$$. In this talk, we deal with one–parametric families of even coefficients $$\delta, \varepsilon$$, when $\varepsilon = \delta \pm 2$ and we prove that there exist infinitely many odd integers $$n$$ with the property that there exists a pair of positive divisors $$d_1, d_2$$ of $$(n^2+1)/2$$ such that $d_1+d_2=\delta n+(\delta+2).$ We also prove that there exist infinitely many odd integers $$n$$ with the property that there exists a pair of positive divisors $$d_1, d_2$$ of $$(n^2+1)/2$$ such that $d_1+d_2=\delta n+(\delta-2),\enspace \textrm{for} \enspace \delta\equiv4, 6\pmod{8}.$ We use Diophantine equations and their properties, with a special accent on Pellian equations and their properties.

References:

M. Ayad and F. Luca, Two divisors of $$(n^2+1)/2$$ summing up to $$n+1$$, J. Théor. Nombres Bordeaux 19 (2007), 561--566.

S. Bujačić, Two divisors of $$(n^2+1)/2$$ summing up to $$\delta n+\varepsilon$$, for $$\delta$$ and $$\varepsilon$$ even, Miskolc Math. Notes, 15 (2) (2014), 333--344.

A. Dujella and F. Luca, On the sum of two divisors of $$(n^2+1)/2$$, Period. Math. Hungar. 65 (2012), 83--96.

A hybrid algorithm for solving shifted linear systems
Zvonimir Bujanović
Department of Mathematics, Faculty of Science, University of Zagreb

In this talk, we propose a new algorithm for solving dense shifted linear systems with multiple right-hand sides and a large number of shifts. Such problems arise e.g. in control theory when computing the frequency response of a LTI system, and in many other applications as well. The new algorithm is designed for hybrid computer architectures that use classical multicore processors in combination with GPU accelerators with Nvidia CUDA technology.

The algorithm consists of two phases: first, a hybrid CPU-GPU routine is used in order to transform the linear system to a so-called controller–Hessenberg form. This reduction is done only once, regardless of the number of shifts, and allows us to later solve the systems with far less computational effort. In the second phase, the transformed systems are solved by means of a pure GPU algorithm, simultaneously for a large batch of shifts. The solver combines custom made highly parallel kernels with the efficient cuBLAS routines.

Such distribution of computational load shows significant performance benefits compared to classical CPU-bound algorithms, which we demonstrate by numerical experiments.

Exact solutions of multiple state optimal design problems
Krešimir Burazin
Department of Mathematics, University of Osijek

We consider multiple state optimal design problems with elliptic state equations in the case of two isotropic phases, aiming to minimize a weighted sum of compliances. It is well-known that such problems do not have classical solutions, and thus a relaxation is needed by introducing generalized materials. We consider (proper) relaxation by the homogenization method which consists in introducing composite materials, which are mixtures of original materials on the micro-scale.

It is well known that for conductivity problems with one state equation, there exist relaxed solutions which correspond to simple laminates at each point of the domain. As a consequence, one can write down a simpler relaxation, ending by a convex minimization problem.

For multiple state optimal design problems this does not hold in general, but we derive analogous result if the number of states is strictly less than dimension of domain, as well as for arbitrary number od states in the spherically symmetric case. In both cases we prove that solving relaxed problem is equivalent to solving the simpler (relaxed) problem. Since this simpler relaxation is a convex optimization problem, one can easily derive the necessary and sufficient conditions of optimality and use them to calculate the optimal design. In the spherically symmetric case, we also prove that there exists a radial optimal design. We demonstrate this procedure on some examples of optimal design problems, where the presented method enables us to explicitly calculate the unique solution.

This is joint work witk Marko Vrdoljak.

Asymptotic analysis of the iterative means
Tomislav Burić
Faculty of Electrical Engineering and Computing, University of Zagreb

We analyze the asymptotic behaviour of the compound mean obtained by the iterative process of two means. Stationary and convergence properties in this expansions are presented. We study some particular cases and show how this approach can be used in comparison of various means.

A structure-preserving $$QR$$ factorization for centrosymmetric real matrices
University of Zagreb

We construct a $$QR$$ factorization of a given centrosymmetric real matrix $$A$$ into centrosymmetric real matrices $$Q$$ and $$R$$. We describe in detail a Householder-type algorithm based on perplectic orthogonal block-reflectors to obtain such a factorization and demonstrate an application of this result to solving centrosymmetric linear systems of full rank.

Quasi-particle bases of principal subspaces for affine Lie algebra of the type $$G_2^{(1)}$$
Marijana Butorac
Department of Mathematics, University of Rijeka

In this talk I’ll describe a quasi-particle basis of principal subspace of standard module of highest weight $$k\Lambda_0$$ of level $$k\geq 1$$ of affine Lie algebra of type $$G_2^{(1)}$$ by means of which we obtain basis for the principal subspace of generalized Verma module of highest weight $$k\Lambda_0$$, $$k \in \mathbb{N}$$. From quasi-particle bases we established a new Rogers-Ramanujan type identity obtained from the characters of principal subspace of generalized Verma module and a new Rogers-Ramanujan type fermionic formula for principal subspace of standard module.

The Zelevinsky classification of unramified representations of the metaplectic group
Igor Ciganović
Department of Mathematics, Faculty of Science, University of Zagreb

The goal is to present results from the article The Zelevinsky classification of unramified representations of the metaplectic group, recently published in the Journal of algebra, a joint work with Neven Grbac.

The metaplectic group over a $$p$$-adic field, where $$p\neq 2$$, is the unique non-trivial two-fold central extension of the $$p$$-adic symplectic group. It appears, together with a classical group, as a member of a dual pair in the theory of theta correspondence and the Weil representation, while its unramified representations occur as local components of an automorphic representation at all but finitely many places of the number field.

The Zelevinsky classification of irreducible unramified representations of the metaplectic group consists of three steps. First, every unramified representation is a fully parabolically induced representation from unramified characters of general linear groups and a negative unramified representation of a smaller metaplectic group. Then, negative unramified representations are described in terms of parabolic induction from unramified characters of general linear groups and strongly negative unramified representation of a smaller metaplectic group, while strongly negative unramified representations are classified in terms of Jordan blocks.

Complex Friedrichs systems and applications
Ivana Crnjac
Department of Mathematics, University of Osijek

Symmetric positive systems of first-order linear partial differential equations were introduced by Kurt Otto Friedrichs (1958) while treating equations that change their type, like the equations modelling transonic fluid flow. This system, today also known as a Friedrichs system, appeared to be convenient for the numerical treatment of various boundary value problems which also inspired Ern, Guermond and Caplain (2007) to express the theory in terms of operators acting in abstract Hilbert spaces obtaining well-posedness result in this abstract setting. Although some evolution (non-stationary) problems can be treated within this framework, their theory is not suitable for evolution problems like the initial-boundary value problem for the non-stationary Maxwell system, or the Cauchy problem for the symmetric hyperbolic system. Some numerical treatment of such non-stationary problems was done by Burman, Ern and Fernandez (2010), while the existence and uniqueness result was recently provided by Burazin and Erceg.

Most classical papers deal with Friedrichs systems in real space setting. We extend both stationary and non-stationary theory to the complex Hilbert space setting and, more general, to complex Banach spaces. We also prove the existence and uniqueness result taking the pivot space $$H^s(\mathbf{R}^d ;\mathbf{C}^r)$$ instead of $$L^2(\mathbf{R}^d ;\mathbf{C}^r)$$. Besides linear, semilinear problems can also be treated in complex case. We apply these results to some tipical examples of complex systems of partial differential equation, such as Dirac, Dirac-Klein-Gordon and Dirac-Maxwell system.

Joint work with Nenad Antonić, Krešimir Burazin and Marko Erceg

Cooperative multi-agent border patrol model
Bojan Crnković
Department of Mathematics, University of Rijeka

Illegal border crossing is an ongoing international security and defense problem. Agents of security forces need to patrol the border and minimize undetected border crossing using limited resources. We present a 1D border patrol model and a centralized feedback control for a system of mobile security/surveillance agents. The model predicts the probability of undetected border crossings that depends on properties and the number of agents in a system. Control is adapted for a cooperative surveillance using heterogeneous multi-agent system that can adapt to changes in the environment and can be used with various boundary conditions of the border.

On the share of closed IL formulas which are also in GL
Vedran Čačić
Department of Mathematics, Faculty of Science, University of Zagreb

This is a presentation of results in the article at http://arxiv.org/abs/1309.3408. Interpretability logic is a generalization of ordinary provability logic GL, and it’s interesting to note that although it is more expressive, a great share of its closed formulas still belong to one of several classes where they have equivalent formulas in GL. Here we present a framework where shares of many such classes can be explicitly calculated. Although the exact numbers depend on syntactical choices, the main result is surprisingly robust with regard to changes in the syntax.

Machine learning algorithm based on preference multigraphs
Lavoslav Čaklović
Department of Mathematics, Faculty of Science, University of Zagreb

Preference multi-graph is directed multi-graph with non-negative weights which may be interpreted as an aggregated preferences of a group of decision makers or several criteria. The nodes in the graph represents the alternatives in consideration, and the arc-weights represents the intensity of the preference between two nodes. The ranking of the graph nodes is obtained as the solution of the Laplace equation of the graph.

We shall give several applications: Missing data in measurement, Sparse multi-criteria decision problem and Medical diagnosis.

A model for diagnosis prediction is formulated as a self-dual hierarchy with symptoms as a double layer and diseases as the third layer in the hierarchy; let us denote it: $$S \rightarrow S \rightarrow D \rightarrow S$$. The first step $$S \rightarrow S$$ in the sequence represents the preference structure of the synergy among the symptoms, $$S \rightarrow D$$ represents the preference structure of the symptom-to-disease relation, and $$D \rightarrow S$$ represents the preference structure of the disease-to-symptom feed-back.

The weights of the symptoms in the first layer are obtained as a fixed point of the self-dual ranking operator. The proposed model is illustrated on several data given by experts in medical diagnosis.

Fluid-composite structure interaction and blood flow
Sunčica Čanić
Department of Mathematics, University of Houston

Fluid-structure interaction problems with composite structures arise in many applications. One example is the interaction between blood flow and arterial walls. Arterial walls are composed of several layers, each with different mechanical characteristics and thickness. No mathematical results exist so far that analyze existence of solutions to nonlinear, fluid-structure interaction problems in which the structure is composed of several layers. In this talk we summarize the main difficulties in studying this class of problems, and present a computational scheme based on which a proof of the existence of a weak solution was obtained. Our results reveal a new physical regularizing mechanism in FSI problems: inertia of thin fluid-structure interface with mass regularizes evolution of FSI solutions. Implications of our theoretical results on modeling the human cardiovascular system will be discussed.

This is a joint work with Boris Muha (University of Zagreb, Croatia), and with Martina Bukac (U of Notre Dame, US). Numerical results with vascular stents were obtained with S. Deparis and D. Forti (EPFL, Switzerland).

On Candido identity
Zvonko Čerin
Department of Mathematics, Faculty of Science, University of Zagreb, retired

We shall describe several results that are similar to the well-known Candido identity from 1951. which says that the square of the sum of squares of three consecutive Fibonacci numbers is equal twice the sum of their fourth powers.

Convergent finite difference scheme for a compressible micropolar fluid flow with a free boundary
Nelida Črnjarić-Žic
Faculty of Engineering, University of Rijeka

In this work the nonstationary flow of the compressible viscous and heat-conducting micropolar fluid is studied. We focus on the fluid flow between a static impermeable solid wall and a free boundary connected to a vacuum state. In the corresponding initial-boundary value problem we suppose that the homogeneous boundary conditions for velocity, microrotation and heat flux on the solid border are valid and that the normal stress, heat flux and microrotation are equal to zero on the free boundary. The aim of this work is to define the convergent numerical scheme for this type of problem and to analyse some numerical solutions. We use the finite difference approach on the staggered mesh and apply it to the model in Lagrangian description to avoid the difficulties associated with the moving boundary in Eulerian coordinate system. It has been shown that the defined scheme is convergent. The numerical solutions obtained by the proposed scheme on the chosen test examples are presented. This is a joint work with prof. Nermina Mujaković.

Topological coarse shape groups
Zdravko Čuka
Faculty of civil engineering, architecture and geodesy, University of Split

Coarse shape groups are well known algebraic invariants in (coarse) shape theory and homotopy theory, as well. Recently, topology was constructed on classical shape groups and it is proved that multiplication and taking inverses behave nice in respect of that topology, so we get topological shape groups. In this talk we will define topology on coarse shape groups, in some way a generalisation of previously mentioned topology, and we will prove that it also induces topological groups structure, which we naturally call topological coarse shape groups. They are topological-algebraic invariants of coarse shape theory and they have some good properties like completely regularity and independence of basepoints. It can be easily shown that we can consider topological shape groups as closed topological subgroups of topological coarse shape groups. Main result is that topological coarse shape groups can be obtained as the inverse limit of an inverse system of discrete topological groups. At the end we will discuss significance of these new structures.

Hölder continuity of Oseledets splittings for semi–invertible operator cocycles
Davor Dragičević
University of New South Wales

For Hölder continuous cocycles over an invertible, Lipschitz base, we establish the Hölder continuity of Oseledets subspaces on compact sets of arbitrarily large measure. This extends a result of Araújo, Bufetov, and Filip [1] by considering possibly noninvertible cocycles, which in addition may take values in the space of compact operators on a Hilbert space. The talk is based on a joint work with G. Froyland [2].

References:

[1] V. Araujo, A. Bufetov and S. Filip, On Hölder-continuity of Oseledets subspaces. Journal of the London Mathematical Society, 93 (2016), 194-218.

[2] D. Dragičević and G. Froyland, Holder continuity of Oseledets splittings for semi-invertible operator cocycles, Ergodic Theory and Dynamical Systems, to appear.

Global existence theorem for 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry
Ivan Dražić
Faculty of Engineering, University of Rijeka

We consider nonstationary 3-D flow of a compressible viscous and heat-conducting micropolar fluid in the domain to be the subset of $$\mathbf{R}^3$$ bounded with two coaxial cylinders that present the solid thermoinsulated walls. In the thermodynamical sense fluid is perfect and polytropic. We also assume that the initial density and temperature are strictly positive, as well as that initial data are only radially dependent.

We know that corresponding homogeneous initial-boundary problem has a unique generalized solution locally in time. In this work, with the help of derived a priori estimates, we prove that the generalized solution exists globally in time as well.

Accurate computations with ill-conditioned matrices and appplications
Zlatko Drmač
Department of Mathematics, Faculty of Science, University of Zagreb

We discuss the importance of robust and accurate implementation of core numerical linear algebra procedures in computational methods for system and control theory. In particular, we stress the importance of error and perturbation analysis that identifies relevant condition numbers and guides computation with noisy data, and careful software implementation. The theme used as a case study is rational matrix valued least squares fitting (e.g. least squares fit to frequency response measurements of an LTI system), in particular with respect to accurate computations with scaled Cauchy, Vandermonde and Hankel matrices.

On rational Diophantine sextuples
Andrej Dujella
Department of Mathematics, Faculty of Science, University of Zagreb

A rational Diophantine $$m$$-tuple is a set of $$m$$ nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuplets. In 1999, Gibbs found the first example of a rational Diophantine sextuple. In this talk, we describe the construction of infinitely many rational Diophantine sextuples. The construction involves elliptic curves, induced by rational Diophantine triples, with torsion group $$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$$. This is joint work with Matija Kazalicki, Miljen Mikić and Márton Szikszai.

On construction of 2-designs and related self-orthogonal codes
Doris Dumičić Danilović
Department of Mathematics, University of Rijeka

We describe a method for constructing 2-designs admitting a solvable automorphism group using tactical decomposition of design, and construct new symmetric $$(78,22,6)$$ designs. Besides that, we show that up to isomorphism there is exactly one symmetric $$(78,22,6)$$ design admitting an automorphism group isomorphic to $$Frob_{39} \times Z_2$$, namely the design constructed by Zvonimir Janko and Tran van Trung. This result implies the nonexistence of a (78,22,6) difference set in the group $$Frob_{39} \times Z_2$$.

In this talk we show that under certain conditions both fixed and non-fixed part of an orbit matrix span a self-orthogonal code over the finite field $$F_{p^n}$$ or over the ring $$Z_m$$. Further, we study binary linear codes spanned by incidence matrices of the constructed designs.

Topological entropy for set-valued functions
Goran Erceg
Faculty of Science, University of Split

We generalize the definition of topological entropy due to Adler, Konheim, and McAndrew to set-valued functions from a closed subset of the interval to closed subsets of the interval. We view these set-valued functions, via their graphs, as closed subsets of $$[0, 1]^2$$ and define topological entropy using new tool, the Mahavier product, defined by J. Kennedy and S. Greenwood. We show that many of the topological entropy properties of continuous functions of a compact topological space to itself hold in our new setting, but not all. We compute the topological entropy of various examples.

This is joint work with Judy Kennedy.

The propagation principle for fractional H-measures
Marko Erceg
Department of Mathematics, Faculty of Science, University of Zagreb

Classical H-measures introduced by Tartar (1990) and independently by Gérard (1991) are mostly suited for hyperbolic equations while parabolic equations fit in the framework of the parabolic H-measures developed by Antonić and Lazar (2007–2013). Although the majority of important equations of mathematical physics can be treated by one of the mentioned variants (having ratios 1:1 or 1:2 between the order of time and spatial derivatives), recently the study of differential relations with fractional derivatives enhanced requiring the extension of the theory to arbitrary ratios. The first generalisation was pointed by Mitrović and Ivec (2011) and applied on fractional conservation laws.

The propagation principle is the basis for more challenging applications of both classical and parabolic H-measures, and the proof relies on the suitable variant of the Second commutation lemma. We extend the Second commutation lemma and prove the propagation principle for fractional H-measures, giving the unification of the known results for classical and parabolic H-measures. At the end, we illustrate possible applications on one example.

This is a joint work with Ivan Ivec.

Comparing of two models of $$\widetilde{SL(2,\mathbb{R})}$$ space
Zlatko Erjavec
Faculty of organization and informatics, University of Zagreb

The $$\widetilde{SL(2,\mathbb{R})}$$ geometry is one of eight homogeneous Thurston 3-geometries $E^{3}, S^{3}, H^{3}, S^{2}\times \mathbb{R}, H^{2}\times \mathbb{R}, {\tiny \widetilde{SL(2,\mathbb{R})}}, Nil, Sol.$ The Riemannian manifold $$(M,g)$$ is called homogeneous if for any $$x,y \in M$$ an isometry $$\Phi :M \rightarrow M$$ exists such that $$y=\Phi (x).$$ A surface is minimal if its mean curvature vanishes identically.

Minimal surfaces in $$\widetilde{SL(2,\mathbb{R})}$$ geometry are researched for matrix model ([1]) and ([2]) and for hyperboloid model ([3]). We compare these models and exhibit graphics of minimal surfaces obtained in matrix model in hyperboloid model.

Key words: $$\widetilde{SL(2,\mathbb{R})}$$ geometry, minimal surface
MSC 2010: 53C30, 53B25.

References:

[1] M. Kokubu, On minimal surfaces in the Real Special Linear Group $$SL(2,\mathbb{R})$$, Tokyo J. Math., Vol. 20 (2) (1997), 287-297.

[2] J. Inoguchi Invariant minimal surfaces in the real special linear group of degree 2, Ital. J. Pure Appl. Math., Vol. 16 (2004), 61-80.

[3] Z. Erjavec, Minimal surfaces in $$\widetilde{SL(2,\mathbb{R})}$$ space, Glas. Mat. Ser. III, Vol. 50 (2015), 207-221.

On the extendibility of Diophantine pairs
Alan Filipin
Faculty of Civil Engineering, University of Zagreb

A set of $$m$$ positive integers is called a Diophantine $$m$$-tuple if the product of any two of its distinct elements increased by $$1$$ is a prefect square. There is a folklore conjecture that there does not exist a Diophantine quintuple. Moreover, there is a stronger version of that conjecture, that every Diophantine triple can be extended to a quadruple with a larger element in the unique way. Precisely, if $$\{a,b,c,d\}$$ is Diophantine quadruple such that $$a < b < c < d$$, then $d=d_{+}=a+b+c+2(abc+rst),$ where $$r$$, $$s$$ and $$t$$ are positive integers satisfying $$r^2=ab+1$$, $$s^2=ac+1$$ and $$t^2=bc+1$$.

Let $$\{a,b,c,d\}$$ such that $$a < b < c < d$$ be a Diophantine quadruple. In this talk we give an upper bound for minimal $$c$$ such that $$d\neq d_{+}$$. It helps us to prove the strong version of the conjecture for various families of Diophantine triples. As corollary it furthermore implies the non-extendibitily of parametric families of Diophantine pairs to a quintuple.

It is joint work with Yasutsugu Fujita and Alain Togbé.

Analysis of the nonlinear 3D fluid-stent-shell interaction problem
Marija Galić
Department of Mathematics, Faculty of Science, University of Zagreb

We consider a nonlinear moving boundary fluid-structure interaction problem in which the fluid is modeled with 3D incompressible Navier-Stokes equations. The structure is modeled as a 2D shell coupled with 1D hyperbolic net. The motivation comes from studying blood flow through a compliant vessel treated with vascular stents. We design a numerical scheme based on Lie splitting and prove its stability. Furthermore, we define a weak solution and, using the ideas from the numerical scheme, study the existence of a weak solution. Theoretical results are illustrated by numerical examples.

Testing equality in distribution of random convex compact sets via theory of random hedgehogs and $$\mathfrak{N}$$-distances
Vesna Gotovac
Faculty of Science, Universitiy of Split

Convex compact subsets of $$\mathbb{R}^d$$ have a nice property that they are determined by a type of real continuous functions on unit sphere in $$\mathbb{R}^d$$ called support functions. Space of convex compact subsets is equipped with Minkowski addition and multiplication by non negative real numbers and these operations correspond to operations on their support functions, but it doesn’t form a vector space. Using the vector space of hedgehogs, which are defined as differences of convex compact sets, standard definition of random convex compact set can be expanded to random hedgehogs and their characteristic functions can be defined. Also, using theory of $$\mathfrak{N}$$-distances, the newly defined terms can be used to derive a statistical test for testing equality in distribution of two random hedgehogs.

In this talk we will give an introduction to the aforementioned statistical test and random hedgehogs.

Superpositions of Ornstein-Uhlenbeck type processes and intermittency
Danijel Grahovac
Department of Mathematics, University of Osijek

Superpositions of Ornstein-Uhlenbeck type processes provide a class of stationary processes with flexible dependence structure including long-range dependence. The asymptotic properties of cumulants and moments are considered for the two variants of cumulative process: the integrated process and the partial sum process. Both of these are shown to have a particular rate of growth of moments. This property is characterized as intermittency and implications on the asymptotic behaviour of the process are discussed.

Endoscopic transfer and automorphic $$L$$-functions
Neven Grbac
Department of Mathematics, University of Rijeka

There are two approaches to the spectral decomposition of the discrete part in the space of square-integrable adèlic automorphic forms on a reductive group over a number field. One is the endoscopic classification of Arthur using the trace formula, the other is the Langlands spectral theory using analytic properties of Eisenstein series. Comparing the two approaches yields information about automorphic representations and their $$L$$-functions. We explain in the talk how this strategy can be applied to prove the holomorphy in the critical strip of the symmetric square, exterior square and Asai $$L$$-functions associated to cuspidal automorphic representations of the general linear group.

Degenerate Eisenstein series with some applications
Marcela Hanzer
Department of Mathematics, Faculty of Science, University of Zagreb

In this talk we will discuss Eisenstein series for general linear and for classical groups. We shall illustrate how the global representation theory in question is directly applicable to give results for a classical topic in number theory. Most of the talk is related to the author’s joint work with Goran Muić.

On the Convergence of the Block Jacobi Method for Hermitian Matrices
Vjeran Hari
Department of Mathematics, Faculty of Science, University of Zagreb

Jacobi-type eigenvalue methods are generally known for their relative accuracy and inherent parallelism. To enhance their efficiency they have been lately modified and implemented as BLAS 3 algorithms. The common name for such algorithms is block Jacobi-type methods. Here we study the global convergence of the block Jacobi method for Hermitian matrices.

Given a Hermitian matrix $$A$$ of order $$n$$, the method generates a sequence of matrices by the rule $$A^{(k+1)}=U_k^*A^{(k)}U_k$$, $$k\geq0$$, $$A^{(0)}=A$$. The matrices $$U_k$$ are unitary elementary block matrices, which differ from the identity matrix $$I_n$$ in four blocks, two diagonal and the two corresponding off-diagonal blocks. So far it is known that the method globally converges under the serial pivot strategies and the strategies that are equivalent to them.

Our aim is to essentially enlarge the classes of usable cyclic and quasi-cyclic strategies for which the method globally converges. To this end we first provide a convergence proof for the basic class of generalized serial strategies. Then permutation equivalent strategies are investigated. By making link of several equivalence relations on the set of cyclic strategies, we essentially enlarge the basic class of “convergent strategies”.

The results are cast in the stronger form: $$\mbox{off}(A')\leq c_n\,\mbox{off}(A)$$, where $$A'$$ is the matrix obtained from $$A$$ after one full cycle, $$c_n < 1$$ a constant depending just on $$n$$, and $$\mbox{off}(A)$$ is the off-norm of $$A$$. This enables using different techniques, including the theory of block Jacobi operators. That theory can be used to extend the convergence analysis to more general block Jacobi-type methods like those for the generalized eigenvalue problem.

As an immediate consequence, all obtained results hold for the standard (element-wise) Jacobi method for Hermitian or symmetric matrices. In the case $$n=4$$, it is shown that the standard Jacobi method converges globally under any cyclic strategy.

Vjeran Hari, University of Zagreb, Department of Mathematics, Bijenička cesta 30, 10000 Zagreb, Croatia

Erna Begović Kovač, University of Zagreb, Faculty of Chemical Engineering and Technology, Marulićev trg 19, 10000 Zagreb, Croatia

Box dimension as specific property of hyperbolic and nonhyperbolic fixed points and singularities of dynamical systems in $$\mathbb{R}^{n}$$
Lana Horvat Dmitrović
Faculty of Electrical Engineering and Computing, University of Zagreb

This work gives the specification of hiperbolic and nonhyperbolic fixed points or singularities by using the box dimension. We study the box dimension of an orbit around the hyperbolic and nonhyperbolic fixed point in the discrete dynamical systems. It is already known that the orbits around the hyperbolic fixed point in one-dimensional discrete dynamical system has the box dimension equal to zero. In this paper we generalise that result to n dimensions, that is, we show that the box dimension of an orbit equals 0 is property of hyperbolic fixed points of discrete dynamical systems in $$\mathbb{R}^{n}$$. On the other hand, near nonhyperbolic fixed points orbits have positive box dimension. In the process of determining values of box dimension we use the stable, unstable and center manifolds of systems. We also apply this results to the hyperbolic and nonhyperbolic singularities of continuous dynamical systems in $$\mathbb{R}^{n}$$ by using the unit-time map. Moreover, we will connect box dimension as projection property to some specific types of nonhyperbolic points such as fold-flip.

Separable computability structures
Zvonko Iljazović
Department of Mathematics, Faculty of Science, University of Zagreb

We examine separable computability structures on a metric space. In particular, we examine conditions under which a maximal computability structure is separable.

Exponential convexity induced by Bellman-Steffensen functional
Julije Jakšetić
Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb

We first generalize an approach to Steffensen’s inequality, originally due to Bellman, in the class of convex functions. From deduced inequalities we form a linear functional whose actions on carefully chosen families of convex functions produces exponential convexity. Lower and upper estimation of Bellman-Steffensen functional is also discussed.

Convergence in distribution of random elements in metric and submetric spaces
Nicolaus Copernicus University

We shall motivate and introduce a new definition of the notion of convergence in distribution of random elements with tight laws. This new definition coincides with the usual one on metric spaces and spaces of distributions (like $$\mathcal{S}',\mathcal{D}'$$).

The advantage is that the new definition allows us to preserve the whole power of the metric theory (including the direct and the converse Prohorov theorems and the Skorokhod a.s. representation) in a large class of topological spaces, called submetric spaces.

The developed theory brings a new light even in the case of metric spaces, by showing that the crucial property is rather the shape of compact sets and not the completeness.

The tools of the theory are presently used in the area of stochastic partial differential equations, stochastic analysis and mathematical finance.

On new summations of first and second-type Neumann series
Dragana Jankov Maširević
Department of Mathematics, J. J. Strossmayer University of Osijek

Certain closed form expressions for the first and second-type Neumann series which members contain modified Bessel functions of the first kind $$I_\nu$$ are derived. Also, summation formulas for Neumann series with members containing Bessel functions of the first kind $$J_\nu$$ are derived as a by-product of these results.

Representation theory of hermition quaternionic groups over p-adic fields
Nevena Jurčević Peček
Departement of Mathematics, University of Rijeka

In this talk the reducibility of representations of $$p$$-adic hermitian quaternionic groups that are parabolically induced from cuspidal and (essentially) square-integrable representations of the Levi factors of standard parabolic subgroups is studied. Main results generalize the reducibility criteria of Tadi' c for split symplectic and special orthogonal groups to the case of arbitrary hermitian quaternionic groups. Proofs rely on the Jacquet module techniques and use the structure formula and the theory of $$R$$-groups. It is proved that for hermitian quaternionic groups the structure formula holds and the $$R$$-groups are determined.

Asymptote and asymptotic behavior as bodies of knowledge in the praxeologies of graphing functions and curves
Ana Katalenić
Faculty of Education, Josip Juraj Strossmayer University of Osijek

Asymptote and asymptotic behavior are bodies of knowledge present in various areas of mathematics. Moreover, the notion of an asymptote appears also in the secondary mathematics education - as a theoretical concept, a part of a procedure or as a self-sufficient task in different contexts and different educational cycles. Hence, connections with a wide range of teaching contents and mathematical bodies of knowledge makes it an interesting object of research.

In this presentation we elaborate some results from our survey on the didactic transposition of this body of knowledge in the general secondary education in Croatia. The survey is conducted within the theoretical framework of the anthropological theory of the didactics, developed by a French mathematician Y. Chevallard especially for research in mathematics education. The main idea of this theory is to determine the relation $$R_I(p,O)$$ between a body of knowledge O and a person that occupies position p in a institution I. For this purpose mathematical knowledge and activities are described in terms of a praxeology $$\left[T,\tau,\theta,\Theta\right]$$, where its practical component is represented with task $$T$$ and technique $$\tau$$ and discursive or theoretical component with technology $$\theta$$ and theory $$\Theta$$.

In our setting, we questioned the relations $$R_B(p,O)$$ and $$R_S(p,O)$$, where O consists of the task of graphing elementary functions or curves and corresponding techniques, while considered insitutions are mathematics textbooks B and prospective mathematics teachers S. Our results show that: (1) dominant techniques are drawing a curve through corresponding points and drawing a graph on the account of function properties determined using calculus, (2) chosen techniques are not the most efficient for the task in question, (3) asymptotic behavior is available but not fully utilized in praxeologies relevant to graphing functions or curves.

This is joint work with Aleksandra Čižmešija and Željka Milin Šipuš.

Diophantine quadruples in finite fields and modular forms
Matija Kazalicki
Department of Mathematics, Faculty of Science, University of Zagreb

Let $$p$$ be a prime. A Diophantine quadruple in the finite field $$\mathbb{F}_p$$ is the set of four distinct elements of $$\mathbb{F}_p$$ with the property that the product of any two of its distinct elements plus one is a square in $$\mathbb{F}_p$$.

In this talk we will present the formula for the number of Diophantine quadruples in $$\mathbb{F}_p$$. The Fourier coefficients of certain modular forms appear in the formula. This is a joint work with Andrej Dujella.

$$\eta$$-quotients and models for $$X_0(N)$$
Iva Kodrnja
Faculty of Civil Engineering, University of Zagreb

$$\eta$$-quotients are a class of functions derived from the Dedekind $$\eta$$ function and under certain assumptions they are modular forms on $$\Gamma_0(N)$$. Using a method by G. Muić, we map the modular curve $$X_0(N)$$ to the projective plane with three linearly independent modular forms of an even weight on $$\Gamma_0(N)$$. From the theory of compact Riemann surfaces and divisors, he proved a formula that connects the degree of the map and degree of the image curve which is an irreducible plane curve. If the degree of the map equals one, we have a model for $$X_0(N)$$.

Using $$\eta$$-quotients that are modular forms on $$\Gamma_0(N)$$ we can construct various maps from the modular curve $$X_0(N)$$ to the projective plane and calculate the degrees of the maps.

Geometric concepts in parallelogram spaces
Zdenka Kolar-Begović
Department of Mathematics, University of Osijek; Faculty of Education, University of Osijek

Some properties of parallelogram spaces will be examined. Some concrete examples of the quaternary relation on special sets which satisfy the required properties of a parallelogram space will be mentioned. The concepts of a vector and translation in a parallelogram space will be introduced. The concept of symmetry with respect to the pair of points will also be defined. A geometrical representation of the introduced concepts and relations between them will be given.

This is a joint work with Ružica Kolar-Šuper and Vladimir Volenec.

Ideal submodules of Hilbert C*-modules revisited
Biserka Kolarec
Faculty of Agriculture, University of Zagreb

Theory of extensions of Hilbert C*-modules can be rephrased in a category whose objects are full Hilbert C*-modules and whose morphisms are ternary homomorphisms. It turns out that in this category ideal submodules of Hilbert C*-modules can easily be identified as closed ternary ideals. Therefore, it is possible to work with ideal submodules without bearing in mind underlying C*-algebras and their morphisms.

Arc-length preserving smooth deformations of curves
Mate Kosor
Naval department, University of Zadar

The talk will discuss two connected problems. First, how to find a curve from a given class of curves with fixed length, that satisfy some given constraints on the end points. For example, tehniques to solve the first problem are used in CAD design to fit curves to points. The second connected problem investigates what classes of curves permit smooth arc-length preserving displacements that result in a curve which satisfy constraints on the end points. This occurs, for example in ODE boundary value problems such as curved rod theory. Presented result will provide a quite general theoretical solution to both of the problems. The solution utilizes basic concepts from real analysis, measure theory, differential topology and Lie group theory. For example, mollifier which is standarly used on algebras in theory of differential equations, is here used on Lie groups.

Scattering theory analogues of several classical estimates in Fourier analysis
Vjekoslav Kovač
Department of Mathematics, Faculty of Science, University of Zagreb

The AKNS scattering transforms can be viewed as nonlinear analogues of the Fourier transform. One of the simplest instances for which this analogy is only partially understood is the Dirac scattering transform. It is defined using an ordinary differential equation taking values in the matrix group $$SU(1,1)$$. We will present several open problems in the field, along with the partial progress towards their resolution.

On unitary representations of disconnected real reductive groups
Domagoj Kovačević
Faculty of Electrical Engineering and Computing, University of Zagreb

Let $$G$$ be the real reductive group and let $$G_0$$ be the identity component. Let us assume that the unitary dual $$\widehat{G_0}$$ of $$G_0$$ is known. Our goal is to construct the unitary dual $$\hat{G}$$ of $$G$$. Automorphisms of $$G_0$$ generated by elements of $$G$$ are the main ingredient of our construction. If the automorphism is outer, one has to consider the corresponding intertwining operators $$S$$. Automorphisms of the Lie algegra $$\mathfrak{g}_0$$ are closely related to automorphisms of $$G_0$$ and we have parametrized them. In particular, automorphisms of $${\mathfrak so}(4,4)$$ are analyzed.

Quantum vertex algebras and double Yangians
Slaven Kožić
Department of Mathematics, Faculty of Science, University of Zagreb & School of Mathematics and Statistics, University of Sydney

In this talk, I will present a joint work with N. Jing, A. Molev and F. Yang. We consider the vacuum module $$\mathcal{V}_{c}(\mathfrak{gl}_N)$$ for the double Yangian $$DY(\mathfrak{gl}_N)$$. By the results of P. Etingof and D. Kazhdan the vacuum module $$\mathcal{V}_{c}(\mathfrak{gl}_N)$$ has a quantum vertex algebra structure. We recall some general properties of quantum vertex algebras and introduce the definition of their center. Finally, we present the explicit description of the center of $$\mathcal{V}_{c}(\mathfrak{gl}_N)$$ at the critical level.

$$K-$$structure of the $$U(\mathfrak{g})^K-$$module $$U(\mathfrak{g})$$ for $$\mathfrak{g}=\mathfrak{s}\mathfrak{u}(n,1)$$ and $$\mathfrak{g}=\mathfrak{s}\mathfrak{o}(n,1)$$
Hrvoje Kraljević
Croatian Mathematical Society

Let $$\mathfrak{g}$$ be a simple real Lie algebra, $$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$$ its Cartan decomposition, $$G$$ the adjoint group of $$\mathfrak{g},$$ $$K$$ its maximal compact subgroup with Lie algebra $$\mathfrak{k}.$$ Further, denote by $$U(\mathfrak{g})$$ and $$U(\mathfrak{k})\subseteq U(\mathfrak{g})$$ the complexified universal enveloping algebras of $$\mathfrak{g}$$ and $$\mathfrak{k}$$ and let $$Z(\mathfrak{g})$$ and $$Z(\mathfrak{k})$$ be its centers. Let $$U(\mathfrak{g})^K$$ be the subalgebra of $$K-$$invariants in $$U(\mathfrak{g}).$$ Then obviously we have a morphism of algebras $$Z(\mathfrak{g})\otimes Z(\mathfrak{k})\longrightarrow U(\mathfrak{g})^K$$ defined by the multiplication. Knop has proved that for noncompact $$\mathfrak{g}$$ this morphism is always injective and that its image is exactly the center of the algebra $$U(\mathfrak{g})^K.$$ Furthermore, the algebra $$U(\mathfrak{g})^K$$ is commutative, i.e. isomorphic to $$Z(\mathfrak{g})\otimes Z(\mathfrak{k}),$$ if and only if $$\mathfrak{g}$$ is either $$\mathfrak{s}\mathfrak{u}(n,1)$$ or $$\mathfrak{s}\mathfrak{o}(n,1).$$ In these cases $$U(\mathfrak{g})$$ is free as a $$U(\mathfrak{g})^K-$$module. We show that in these cases the multiplication defines an isomorphism of $$K-$$modules and $$U(\mathfrak{g})^K-$$modules $$U(\mathfrak{g})^K\otimes H\longrightarrow U(\mathfrak{g}),$$ where $$H$$ is the subspace of $$U(\mathfrak{g})$$ spanned by all powers $$x^k,$$ $$k\in\mathbb{Z}_+,$$ $$x\in{\cal N}_K,$$ and $${\cal N}_K$$ is the variety of all nilpotent elements in $$\mathfrak{g}^{\mathbb{C}}$$ whose projection to $$\mathfrak{k}^{\mathbb{C}}$$ along $$\mathfrak{p}^{\mathbb{C}}$$ is nilpotent in the reductive Lie algebra $$\mathfrak{k}^{\mathbb{C}}.$$ Furthermore, we study the structure of the $$K-$$module $$H$$ and show that the multiplicity of every irreducible representation $$\delta$$ of $$K$$ in it equals its dimension $$d(\delta).$$ In other words, as a $$K-$$module $$H$$ is equivalent to the regular representation of $$K.$$ A simple consequence of this result is that for any complex finitedimensional $$K-$$module $$V$$ the space of $$K-$$invariants $$(U(\mathfrak{g})\otimes V)^K$$ is free as a $$U(\mathfrak{g})^K-$$module of rank $$\dim\,V.$$

Positive solutions of quasilinear elliptic equations with strong dependence on the gradient and their qualitative properties
Catholic University of Croatia

We study the existence and nonexistence of positive, spherically symmetric solutions of the next quasilinear elliptic equation with an arbitrary positive growth rate $$e_0$$ on the gradient on the right-hand side:

$\label{pde1}\tag{1} \left\{ \begin{array}{ll} -\Delta_p u=\tilde g_0|x|^m+\tilde f_0|\nabla u|^{e_0}\quad \mbox{in B\setminus\{0\},}&\\ \mbox{\phantom{-\Delta_p}u=0 on \partial B,}&\\ \mbox{u(x) spherically symmetric and decreasing.}& \end{array} \right.$

Here $$B$$ is an open ball of radius $$R$$ centered at the origin in $$\mathbb{R}^N$$, $$1<p<\infty$$, $$\Delta_pu=\text{div}(|\nabla u|^{p-2}\nabla u)$$ is $$p$$-Laplacian. The Lebesgue measure (volume) of $$B$$ in $$\mathbb{R}^N$$ is denoted by $$|B|$$, and the volume of the unit ball is denoted by $$C_N$$. The dual exponent of $$p>1$$ is defined by $$p'=\frac p{p-1}$$. We assume that $$\tilde g_0$$, $$\tilde f_0$$ and $$e_0$$ are positive real numbers. We show that $$e_0=p-1$$ is the critical exponent: for $$e_0<p-1$$ there exists a strong solution for any choice of the coefficients which is a known result, while for $$e_0>p-1$$ we have existence-nonexistence splitting of the coefficients $$\tilde f_0$$ and $$\tilde g_0$$. The elliptic problem is studied by relating it to the corresponding singular ODE of the first order. We provide a sufficient conditions for a strong radial solution to be the weak solution. We also examined when $$\omega$$-solutions of ([pde1]) are weak solutions. We found conditions under which the strong solutions are the weak solutions in the critical case of $$e_0=p-1$$.

We also study regularity of strong and weak solutions and we prove a result that gives the conditions for the strong solution of the quasilinear elliptic problem to be a classical solution.

Non–crossing partitions for reflection groups and their cyclic sieving
Christian Krattenthaler
Fakultät für Mathematik, Universität Wien

Kreweras defined non-crossing partitions in 1972. Thirty years later, Bessis, and independently Brady and Watt discovered that there is a natural generalisation of Kreweras’ non-crossing partitions for all finite reflection groups. Kreweras’ original objects arise as the special case where the reflection group is the symmetric group. This definition has been further extended to “$$m$$-divisible” non-crossing partitions by Armstrong. I shall recall and illustrate the corresponding definitions, not assuming any prior knowledge of the theory of reflection groups. Next I shall explain the so-called “cyclic sieving phenomenon”, a phenomenon originally formulated by Reiner, Stanton, and White, which has since then been found to be stunningly widespread. Finally, I shall connect the two by showing that the “$$m$$-divisible” non-crossing partitions satisfy two instances of the cyclic sieving phenomenon.

On cyclic unitals
Departement of Mathematics, Faculty of Science, University of Zagreb

A unital of order $$q$$ is a Steiner $$2$$-design with parameters $$2$$-$$(q^3+1,q+1,1)$$. We shall survey known results about unitals with cyclic automorphisms of order $$q^3+1$$ and present some new examples.

More accurate Heinz operator inequalities
Mario Krnić
Faculty of Electrical Engineering and Computing, University of Zagreb

Motivated by the well-known Heinz norm inequalities, in this talk we study the corresponding Heinz operator inequalities. We derive several refinements of these operator inequalities, first with the help of the well-known Hermite–Hadamard inequality, and then, utilizing the parametrized family of the so-called Heron means. In such a way, we obtain improvements of some recent results, known from the literature.

Damping optimization over the arbitrary time of the excited mechanical systems
Ivana Kuzmanović
Department of Mathematics, University of Osijek

In this presentation we will consider damping optimization in parameterized mechanical systems excited by an external force, described by the system of ordinary differential equations $M \ddot x(t) + D(v) \dot x(t) + K x(t) = g(t),$ where mass and stiffness matrices $$M$$ and $$K$$ are positive definite real matrices of order $$n$$ and the vector $$g(t)\in\mathbb{R}^n$$ corresponds to an external force. The damping matrix $$D(v)\in \mathbb{R}^{n \times n}$$ is a positive semidefinite matrix which depends on damping positions and viscosity parameter $$v\in (0, \infty)$$.
We considered the problem of damping optimization, that is, for the given matrices $$M$$, $$K$$ and external force, we want to find an optimal damping positions and viscosities of dampers in order to avoid unwanted oscillations. We will use two optimization criteria based on considered the problem of damping optimization, that is, for the given matrices $$M$$, $$K$$ and external force, we want to find an optimal damping positions and viscosities of dampers in order to avoid unwanted oscillations. We will use two optimization criteria based

• minimizing average energy amplitude over the arbitrary time

• average displacement over the arbitrary time.

As the main result we will derive explicit formulas for objective functions. These formulas can be implemented efficiently and allow efficient damping optimization, which will be illustrated by example.
Presentation which will be given is based on the results from the paper I.Kuzmanović, Z.Tomljanović, N.Truhar, Damping optimization over the arbitrary time of the excited mechanical system, Journal of Computational and Applied Mathematics, accepted, 2016.

Some aspects of representation theory of the general linear group over a local non-archimedean field
Erez Lapid
The Weizmann Institute of Science, Rehovot

Orthonormal and Parseval wavelets with integer dilations
Ana Laštre
Faculty of Science, University of Split

We describe the class of Parseval multiwavelets associated with a given admissible GMRA. Moreover, we give a method of construction of multiwavelets associated with a GMRA with arbitrarily many generators of the core space. Dilations are induced by an arbitrary expanding matrix with integer coefficients. The method described includes a description and a construction of the so-called characteristic matrix which is a generalization of the high pass filters from the one-dimensional diadic case.

References:

L.W. Baggett, P.E.T. Jorgensen, K.D. Merrill, J.A. Packer, Construction of Parseval wavelets from redundant filter systems, J. Math. Phys. 46(2005)083502.

D. Bakić, Semi-orthogonal Parseval frame wavelets and generalized multiresolution analyses, Appl. & Comp. Harmonic Analysis 21(2006), 281-304.

D. Bakić, On admissible generalized multiresolution analysis, Grazer Mathematische Berichte 348 (2006), 15-30.

D. Bakić, I. Krishtal, E.N. Wilson, Parseval frame wavelets with $$E_n^{(2)}$$-dilations, Appl. & Comp. Harmonic Analysis 19(2005), 386-431.

M. Bownik, Z. Rzeszotnik, Construction and reconstruction of tight framelets and wavelets via matrix mask functions, J. Functional Analysis, 256(2009), 1065-1105.

M. Paluszynski, H. Šikić, G. Weiss, S. Xiao, Generalized low pass filters and MRA frame wavelets, J. Geom. Anal., 11(2001), 311-342.

M. Paluszynski, H. Šikić, G. Weiss, S. Xiao, Tight frame wavelets, their dimension functions, MRA tight frame wavelets and connectivity properties, Adv. Comput. Math., 18(2003), 297-327.

G. Weiss, E. Wilson, The mathematical theory of wavelets, in: Proc. of the NATO-ASI Meeting "Harmonic Analysis - A celebration", Kluwer, Dordrecht, 2001.

Greedy control
Martin Lazar
University of Dubrovnik

Greedy control represents a new notion in the control of parameter dependent systems. It is based on adaptation of (weak) greedy algorithms, developed and explored so far for constructing an approximative solutions to parametric PDEs. The idea is to identify the most distinguished parameter values describing the whole range of admissible controls. The algorithm consists of the (possible expensive) offline part devoted to the selection of parameter representatives and the online one enabling a fast computation of an approximative control for a given value of the parameter within a prescribed accuracy. Our results lead to optimal approximation rates expressed in terms of Kolmogorov widths. These results are applied to the approximate control of finite-difference approximations of the heat and the wave equation. The numerical experiments confirm the efficiency of the methods and show that the number of weak-greedy samplings that are required is particularly low when dealing with heat-like equations, because of the intrinsic dissipativity that the model introduces for high frequencies.

Ubiquitous Doubling Algorithms, General Theory, and Applications
Ren-Cang Li
University of Texas at Arlington

Iterative methods are widely and indispensably used in numerical approximations. Basically, any iterative method is a rule that produces a sequence of approximations and with a reasonable expectation that newer approximations in the sequence are better. The goal of a doubling algorithm is to significantly speed up the approximation process by seeking ways to skip computing most of the approximations in the sequence but sporadically few, in fact, extremely very few: only the $$2^i$$-th approximations in the sequence, kind of like computing $$\alpha^{2^i}$$ via repeatedly squaring. However, this idea is only worthwhile if there is a much cheaper way to directly obtain the $$2^i$$-th approximation from the $$2^{i-1}$$-th one than simply following the rule to generates every approximations between the $$2^{i-1}$$-th and $$2^i$$-th approximations in order to obtain the $$2^i$$-th approximation. Anderson (1978) had sought the idea to speed up the simple fixed point iteration for solving the discrete-time algebraic Riccati equation via repeatedly compositions of the fixed point iterative function. As can be imagined, under repeatedly compositions, even a simple function can usually and quickly turn into nonetheless a complicated and unworkable one, which is the case in Anderson’s doubling iterations. In the last 20 years or so in large part due to an extremely elegant way of formulation and analysis, the researches in doubling algorithms thrived and continues to be very active, leading to numerical effective and robust algorithms not only for the continuous-time and discrete-time algebraic Riccati equations from optimal control that motivated the researches in the first place but also for $$M$$-matrix algebraic Riccati equations (MARE), structured eigenvalue problems, and other nonlinear matrix equations. But the resulting theory is somewhat fragmented and sometimes ad hoc. In this talk, we will seek to provide a general and coherent theory, discuss new highly accurate doubling algorithm for MARE, and look at several important applications.

Tits’ construction of exceptional Lie algebras and Moy-Prasad filtrations
Hung Loke
National University of Singapore

In this talk, I will first quickly review Tits’ construction of exceptional Lie algebras using a Jordan algebra and an octonion algebra. Next I will explain how this will lead to embeddings of buildings of exceptional groups. Finally we recall Gan and Yu results in which they identify the building of split $$F_4$$ (resp. building of $$G_2$$) with certain lattice functions on the Jordan algebra (resp. octonion algebra). We will use these lattice functions and Tits’ construction to obtain Moy-Prasad filtrations of the split Lie algebra of type $$E_8$$.

Local asymptotic mixed normality of approximate maximum likelihood estimator of drift parameters in diffusion model
Snježana Lubura
Department of Matematics, Faculty of Science, University of Zagreb

Let $$X$$ be a diffusion which satisfies a stochastic differential equation of the form: $$dX_t=\mu(X_t,\theta)dt+\sigma_0\nu(X_t)dW_t$$, Let $$X$$ be a diffusion which satisfies a stochastic differential equation of the form: where drift parameter $$\theta$$ is unknown and diffusion coefficient parameter $$\sigma_0$$ is known. We have discrete observations $$(X_{t_i},0\leq i\leq n)$$ along fixed time interval $$[0,T]$$. Let $$\bar{\theta}_n$$ be approximate maximum likelihood estimator of drift parameter obtained from discrete observations and let $$\hat{\theta}$$ be maximum likelihood estimator obtained from continuous observations $$(X_t,0\leq t\leq T)$$ along fixed time interval $$[0,T]$$. We proved that $$\bar{\theta}_n$$, when $$\Delta_n =\max_{1\leq i\leq n}(t_i-t_{i-1})$$ tends to zero, is locally asymptotic mixed normal, with covariance matrix which depends on MLE $$\hat{\theta}$$ and on path $$(X_t,0\leq t\leq T)$$.

Form orbit matrices to strongly regular graphs and codes
Marija Maksimović
Department of Mathematics, University of Rijeka

In this talk we will introduce the definition of orbit matrices with parameters $$(v,k, \lambda, \mu)$$ and orbit lengths distribution $$(n_1, \ldots ,n_b)$$.

We will talk about construction of orbit matrices with parameters $$(v,k, \lambda, \mu)$$ and orbit lengths distribution $$(n_1, \ldots ,n_b)$$ and construction of strongly regular graphs and codes from these matrices.

The construction of strongly regular graphs is a generalization of work of Behbahani and Lam that in 2011 introduced an algorithm for construction strongly regular graphs from their orbit matrices under the action of the automorphism group of prime order.

On a problem concerning the shape of Cartesian products
Sibe Mardešić
Department of Mathematics, Faculty of Science, University of Zagreb

Let $$\mathbb{H}$$ denote the Hawaiian earring and $$\mathbb{P}$$ the wedge of a sequence of 1-spheres. Let $$\mathbb{H}\times \mathbb{P}$$ be their Cartesian product with canonical projections $$\pi_{\mathbb{H}}$$ and $$\pi_{\mathbb{P}}$$. The natural basepoints of these spaces are denoted by $$*$$. The shape-theoretic problem asks if there exist a polyhedron $$Z$$ and a shape morphism $$H\colon Z\to\mathbb{H}\times\mathbb{P}$$, different from the constant morphism $$*$$, such that $$\pi_{\mathbb{H}}H=*$$ and $$\pi_{\mathbb{P}}H=*$$. The main result of the paper establishes equivalence between the shape-theoretic problem and the following problem concerning the standard resolution $$\textbf{Y}=(Y_{\mu},q_{\mu\mu'},M)$$ of $$\mathbb{H}\times\mathbb{P}$$. Is there a polyhedron $$Z$$ and are there phantom mappings $$h_{\mu}\colon Z\to Y_{\mu}$$, $$\mu\in M$$, such that $$h_{\mu}\simeq q_{\mu\mu'}h_{\mu'}$$, for $$\mu\leq\mu'$$, and for a cofinal set of indices $$\mu\in M$$, the mappings $$h_{\mu}$$ are not homotopic to $$*$$. Answering these problems in the affirmative would prove that $$\mathbb{H}\times\mathbb{P}$$ is not a product in the shape category of topological spaces. The latter assertion would give a partial answer to a problem raised by Y. Kodama in 1977.

Hausdorff distance between some sets of points
Tomislav Marošević
Department of Mathematics, Josip Juraj Strossmayer University of Osijek

Hausdorff distance (i.e.  Pompeiu-Hausdorff distance) between some sets of points can be used by problems in pattern recognition, in shape matching and comparison, in computer vision, etc. We look at some simple sets of points in the plane, such as segments, circles, ellipses, and obtain the expressions of corresponding Hausdorff distances.
For example, Hausdorff distance between two circles, $$k_1$$ (with center $$C_1$$ and radius $$r_1$$) and $$k_2(C_2,r_2)$$, is given by $$\quad d_H(k_1,k_2)=d(C_1,C_2)+|r_2-r_1|\,$$ (where $$d$$ denotes Euclidean distance).

Families of Identities for the Integer Partition Function
Ivica Martinjak
Faculty of Science, University of Zagreb

We present recursive identities for the number of partitions with exactly $$k$$ parts and with constraints on both the minimal difference among the parts and the minimal part. Using these results we demonstrate that the number of partitions of $$n$$ is equal to the number of partitions of $$2n+d{n \choose 2}$$ of length $$n$$, with $$d$$-distant parts. We also provide a direct proof for this identity. Joint work with D. Svrtan.

MSC: 05A17, 11P84 Keywords: Euler function, partition function, partition identity, Rogers-Ramanujan identities

On the HZ Method for PGEP
Josip Matejaš
Faculty of Economics and Business, University of Zagreb

We study a variant of the Falk-Langemeyer method for solving the positive definite generalized eigenvalue problem (PGEP) $$Ax=\lambda Bx$$. Here $$A$$ and $$B$$ are symmetric matrices and $$B$$ is positive definite. The considered HZ method assumes that $$B$$ has ones along the diagonal and is designed to retain this property during iteration. The method has proved to be very fast when properly implemented as one-sided block Jacobi-type algorithm for the generalized singular value problem. Then it is almost perfectly parallelizable, so parallel shared memory versions of the algorithm are highly scalable, and their speedup almost solely depends on the number of cores used.

We investigate the element-wise method theoretically. Its derivation, high relative accuracy, the global and asymptotic convergence. The first results include the derivation of the method and its high relative accuracy when the both matrices $$A$$ and $$B$$ are positive definite. The asymptotic quadratic convergence problem is challenging because of the special structure in almost diagonal symmetric matrices $$A$$ and $$B$$ when the pair $$(A,B)$$ has multiple eigenvalues. We hope to prove the global convergence of the method under the wide class of generalized serial strategies.

Vjeran Hari, University of Zagreb, Department of Mathematics, Bijenička cesta 30, 10000 Zagreb, Croatia

Josip Matejaš, University of Zagreb, Faculty of Economics, Kennedyjev trg 6, 10000 Zagreb, Croatia

Asymptotic expansions and integral means
Lenka Mihoković
Faculty of Electrical Engineering and Computing, University of Zagreb

Series $$\sum_{n=0}^{\infty} a_n x^{-n}$$ is said to be an asymptotic expansion of a function $$f(x)$$ as $$x\to \infty$$ if for each $$N\in\mathbf{N}$$ $f(x)=\sum_{n=0}^{N}a_nx^{-n} +o(x^{-N}).$

In recent papers asymptotic expansions of many bivariate classical and parameter means were found, using the explicit formulas for observed means. The coefficients of asymptotic expansions are very useful in analysis of considered means.

Integral means are important class of bivariate means. Assuming that the function $$f$$ possesses an asymptotic expansion, we shall derive the asymptotic expansion of its integral mean $I_f(x+s,x+t)= f^{-1}\left(\frac{1}{t-s}\int_s^t f(x+u) du \right) \sim x \sum_{n=0}^{\infty} a_n x^{-n}.$ This is only a part of the general problem of solving the equation $B(A(x))=C(x),$ where asymptotic expansions of $$B$$ and $$C$$ are known, in terms of asymptotic series. The solution is possible under reasonable conditions posed on series $$B$$ and $$C$$. An efficient recursive algorithm for calculation of the coefficients in asymptotic series of $$A(x)$$ are derived.

This method becomes noticable when there is a difficulty with finding inverse formula. The results are illustrated by calculation of some important integral means connected with gamma function.

On some codes invariant under the action of the Held group
Vedrana Mikulić Crnković
Department of Mathematics, University of Rijeka

Lately, some methods for constructing self-orthogonal codes generated by an orbit matrix of a 2-design or a strongly regular graph were developed. We take into consideration linear codes (over binary field) generated by the incidence matrices of 1-designs, and by orbit matrices of those structures. More precisely, one can construct a binary self-orthogonal code of a self-orthogonal and weakly self-orthogonal 1-design. We determine when the code generated by an orbit matrix of a 1-design is self-orthogonal.

We construct self-orthogonal 1-designs from the sporadic simple group He with large number of points. Furthermore, we define binary codes of the constructed designs and their orbit matrices and analyze their properties.

This talk is joint work with Dean Crnković.

Unrolled quantum groups and vertex algebras
Antun Milas
SUNY Albany

The representation theory of quantum groups has proven to be a useful tool in several areas of mathematics. It is known that "small" quantum groups give rise to modular tensor categories (with categorical $$\mathrm{SL}(2,\mathbb{Z})$$-action), which can be used to construct 3-manifold invariants. Moreover, these categories are equivalent (thanks to Kazhdan-Lusztig's correspondence) to module categories for affine Kac-Moody Lie algebras at positive integral levels. "Unrolled" quantum groups are certain Hopf algebras whose representation categories are neither semi-simple nor do they have finitely many simple objects. These quantum groups have been recently used to construct powerful link invariants whose asymptotic behavior is related to the Volume Conjecture. However, they seem not to carry an $$\mathrm{SL}(2,\mathbb{Z})$$-action. In this talk, I will first give an overview of these developments. Then I will discuss certain false/mock modular forms appearing in representation theory of vertex algebras, whose modular-like properties are supposedly shadowing the missing $$\mathrm{SL}(2,\mathbb{Z})$$-action on the quantum group side. In particular, we show that asymptotic dimensions of modules, often also called quantum dimensions, have an interpretation via open Hopf link invariants. This is a joint project with T. Creutzig.

Detecting a definite Hermitian matrix pair using a subspace algorithm
Marija Miloloža Pandur
Department of Mathematics, J.J. Strossmayer University of Osijek

Hermitian matrix pair $$(A,B)$$ is called positive definite if there exist a real number $$\lambda_0$$ such that the matrix $$A-\lambda_0 B$$ is positive definite. We propose a subspace algorithm for detecting definiteness of given large and sparse Hermitian matrix pair. In the positive case, algorithm returns one such number $$\lambda_0.$$ Since our algorithm is based on a trace minimization property, if allowed to continue, it computes few eigenvalues that a closest to the given shift $$\lambda_0$$ (from both sides) and corresponding eigenvectors.

Exponential decay of measures and Tauberian theorems
Ante Mimica
Department of Mathematics, Faculty of Science, University of Zagreb

We study behavior of a measure on $$[0,\infty)$$ by considering its Laplace transform. If it is possible to extend the Laplace transform to a complex half-plane containing the imaginary axis, then the exponential decay of the tail of the measure occurs and under certain assumptions we show that the rate of the decay is given by the so called abscissa of convergence and extend some known results. Under stronger assumptions we give behavior of density of the measure by considering its Laplace transform. In situations when there is no exponential decay we study occurrence of heavy tails and give an application in the theory of non-local equations.

Double Angle Theorems for Definite Matrix Pairs
Suzana Miodragović
Department of Mathematics, J.J. Strossmayer University of Osijek

We present new double angle theorems for relative perturbations of Hermitian matrix pairs $$(A,B)$$, where $$A$$ is a non-singular matrix which can be factorized as $$A=GJG^*$$, $$J=\textrm{diag}(\pm)$$ and $$B$$ is non-singular. The rotation of eigenspaces is measured in the matrix dependent scalar product. We assess the sharpness of the new estimates in terms of the effectivity quotients (the quotient of the measure of the perturbation and the estimator). The known double angle theorems for relative perturbations of the single matrix Hermitian eigenspace problem are included as special cases in our approach. Our bound is dependent on the norm of the block diagonalizing $$J$$-unitary matrix $$F$$ $$(F^*JF=J)$$ which can be efficiently bounded for the case when the matrix $$A$$ is quasi-definite.
This paper is joint work with Luka Grubišić and Ninoslav Truhar.

Number theory, topology and geometry
Ivan Mirković
University of Massachusetts

The goal of the talk is to present some elementary geometric ideas which originated in Number Theory but are now seen to have a more universal nature. All notions will be introduced at the necessary level.

The central result of Number Theory is the so called Class Field Theory. It describes the case when symmetries of a number-theoretic problem are commutative. In 1968 Robert Langlands has proposed an extension to arbitrary symmetries and the verification of these predictions has since become one of central efforts in contemporary mathematics. Its geometric form, the geometric Langlands program, has come to play the role of a bridge between Algebraic Geometry, Representation Theory and Particle Physics (Quantum Field Theory and String Theory). On the road towards understanding the geometric Langlands program, the aim of the talk is to provide a transparent topological formulation of the geometric Class Field Theory in terms of a new kind of Algebraic Topology that is being developed in the setting of Algebraic Geometry.

Clustering-based model order reduction for multi-agent system
Petar Mlinarić
Max Planck Institute, Magdeburg

We present an efficient clustering-based model order reduction method for multi-agent systems with Laplacian-based dynamics. The method combines an established model order reduction method and a clustering algorithm to produce a graph partition used for reduction, thus preserving the network structure and consensus. By the Iterative Rational Krylov Algorithm, a good reduced order model can be found which is not necessarily structure preserving. However, based on this we can efficiently find a partition using the QR decomposition with column pivoting as a clustering algorithm, so that the structure can be restored. We illustrate the effectiveness on an example from the literature.

Singular BGG resolutions over generalized flag manifolds
Rafael Mrđen
Faculty of Civil Engineering, University of Zagreb

It is well known that a finite dimensional representation of a semisimple complex Lie group $$G$$ can be resolved on generalized flag manifold $$G/P$$ by a sequence of invariant differential operators acting between homogeneous sheaves - so called Bernstein-Gelfand-Gelfand (BGG for short) resolution. We explain how to construct analogous resolutions in singular infinitesimal character, in cases when $$G=Sp(2n,\mathbb{C})$$ or $$SO(2n,\mathbb{C})$$, and $$P$$ maximal, $$|1|$$-graded parabolic subgroup. Our main tool is the Penrose transform. Since duals of homogeneous sheaves are generalized Verma modules, one obtains representation-theoretic consequences.

Torsion of elliptic curves over number fields
Filip Najman
Department of Mathematics, Faculty of Science, University of Zagreb

We will give an overview of known results about the torsion of elliptic curves over number fields, focusing on recent developments in the subject.

Continuous forms of classical inequalities
Ludmila Nikolova
Sofia University, Dept. of Mathematics and Informatics

The aim of this talk is to focus on the question of development of classical inequalities in a more general “continuous” form (involving infinite many functions and/or spaces). First we discuss such developments concerning H" older’s and Minkowski’s inequalities. After that we present such new general developments of Popoviciu’s and Bellman’s inequalities. Finally we present some applications, possible extensions and questions for further research.

This is joint work with L.-E.Persson and S.Varošanec.

Van der Corput property of polynomials
Marina Ninčević
Department of Mathematics, Faculty of Science, University of Zagreb

Let $$P$$ be an integer polynomial of degree $$k\geq 3$$ such that $$P(0)=0$$. A non-negative valued, normed trigonometric polynomial with the spectrum in the set of integer values of $$P$$ not greater than $$n$$ and a small free coefficient $$a_0=O((\log\log n)^{-1/k^2})$$ is constructed. Given an odd integer polynomial $$P$$ of degree $$k\geq 3$$ such that $$P(0)=0$$, a much better estimate of $$O((\log n)^{-1/k})$$ for van der Corput property is obtained. It is done by averaging of trigonometric polynomials that approximate the Fejer kernel.

Codes from M11
Ivona Novak
Department of Mathematics, University of Rijeka

$$M_{11}$$ is the smallest of five sporadic simple Mathieu groups and it can be represented as primitive permutation group on 11, 12, 55, 66 and 165 points. Using orbits of a point stabilizer acting on the set of points as a basic block of a design, we can obtain designs, and from them codes, invariant under the action of $$M_{11}.$$ In this talk, I will present 1-designs and codes obtained from $$M_{11}$$ on 55, 66 and 165 points, and some of their properties. This talk is based on joint work with dr. sc. Vedrana Mikulić Crnković.

Investigation an overdetermined system of linear equations
Vedran Novoselac
Faculty of Mechanical Engineering in Slavonski Brod, University of Osijek

This presentation demonstrates the application of convex functionals in solving of an overdetermined system of linear equations. The problems are considered by using convex functionals generated by p-norms, where p is a real number greater than or equal to 1. Relying on the convex properties, we examine the minimization properties involving regression, scale and affine equivariant properties. The problem of finding data weighted mean and median is illustrated as an example of the application of overdetermined systems and convex functionals.

MSC2010: 15A06, 65F20, 58K70

Keywords: overdetermined system, p-norm, convex functional, equivariant property

Dirac cohomolgy
Pavle Pandžić
Department of Mathematics, Faculty of Science, University of Zagreb

Dirac operators were introduced into representation theory of real reductive Lie groups by Parthasarathy in the 1970s. The point was to explicitly construct the discrete series representations, as kernel of Dirac operators on certain spin bundles on the symmetric space $$G/K$$. Parthasarathy also noticed that Dirac operators provide a necessary condition for unitarity of Harish-Chandra modules, via the Dirac inequality $$D^2\geq 0$$.

In the 1990s, Vogan started a program of studying the action of Dirac operators on Harish-Chandra modules. The main new concept was Dirac cohomology, the kernel of the Dirac operator divided by the intersection of the image and the kernel. Vogan conjectured that Dirac cohomology, if nonzero, determines the infinitesimal character of the module, i.e., the scalars by which the elements of the center of the enveloping algebra of the Lie algebra acts on the module. This conjecture was proved by Huang and Pandžić in 2002.

Since then, Dirac cohomology was proven to be an interesting invariant, related to several other important invariants, like n-cohomology, (g,K)-cohomology, characters and associated cycles. The theory was also developed in several other settings: quadratic Lie algebras (Kostant), noncommutative equivariant cohomology (Alekseev, Meinrenken; Kumar), Lie superalgebras (Huang, Pandžić), affine Lie algebras (Kac, Moseneder-Frajria, Papi), graded affine Hecke algebras and p-adic groups (Barbasch, Ciubotaru, Trapa).

In the talk I will first recall the definition of Dirac cohomology and some basic results about it, and then I will present some new results and applications.

Spectral representation of transition densities of fractional Pearson diffusions
Ivan Papić
Department of Mathematics, J.J. Strossmayer University of Osijek

Heavy-tailed fractional Pearson diffusions are a class of sub-diffusions with marginal heavy-tailed Pearson distributions: reciprocal gamma, Fisher-Snedecor and Student distributions. They are governed by the time-fractional diffusion equations with polynomial coefficients depending on the parameters of the corresponding Pearson distribution. We derive the spectral representation of transition densities of fractional Fisher-Snedecor and reciprocal - gamma diffusions, which depend heavily on the structure of the spectrum of the infinitesimal generator of the corresponding non-fractional Pearson diffusion.

Mosaics of Combinatorial Designs
Mario Pavčević
Faculty of Electrical Engineering and Computing, University of Zagreb

Looking at incidence matrices of $$t$$-$$(v,k,\lambda)$$ designs as $$v \times b$$ matrices with $$2$$ possible entries, each of which indicates incidences of a $$t$$-design, we introduce the notion of a $$c$$-mosaic of designs, having the same number of points and blocks, as a matrix with $$c$$ different entries, such that each entry defines incidences of a design. In fact, a $$v \times b$$ matrix is decomposed in $$c$$ incidence matrices of designs, each denoted by a different color, hence this decomposition might be seen as a tiling of a matrix with incidence matrices of designs as well. We give some explicit constructions of mosaics of designs including some infinite series. We prove that resolvable designs with parameters $$t$$-$$(v,k,\lambda)$$ lead immediately to $$\frac{v}{k}$$- mosaics of designs.

$$\left(U,V\right)$$- chains and computable transition points
Bojan Pažek
Faculty of Architecture, University of Zagreb

We observe a computable metric space, a computable continuum $$K$$ and disjoint computably enumerable open sets $$U$$ and $$V$$ in this space such that $$K$$ intersects both $$U$$ and $$V$$. We examine conditions under which the set $$K\cap S$$ contains a computable point, where $$S=X\setminus (U\cup V)$$. We point out that a sufficient condition for this is that $$K$$ is an arc. Moreover, we observe the more general case when $$K$$ is a chainable continuum and present that $$K\cap S$$ contains a computable point under assumption that $$K \cap S$$ is totally disconnected. We also point out that $$K\cap S$$ contains a computable point if $$K$$ is a chainable continuum and $$S$$ any co-computably enumerable closed set such that $$K\cap S$$ has an isolated and decomposable connected component. Related to this, we examine semi-computable chainable continua and we present some results regarding approximations of such continua by computable subcontinua.

Applications of Schneider’s $$p$$-adic continued fractions
Tomislav Pejković
Department of Mathematics, Faculty of Science, University of Zagreb

We give an overview of the properties of Schneider’s version of $$p$$-adic continued fractions with emphasis on their use in Diophantine approximation. Application to a problem on the difference between Mahler’s and Koksma’s exponents measuring the quality of quadratic approximation is explained in more details.

A new iterative method for solving multiobjective programming problem
Tunjo Perić
Faculty of economics and business, University of Zagreb

Multiobjective programming problem (MOPP) is a practical, frequently encountered problem where several decision makers optimize their utilities, or one decision maker optimizes different goals, at the same time and on the same constraint set. They can achieve their aspirations at different optimal points. However, only one point has to be chosen where a compromise solution will be achieved. Such problem is known as MOPP. There were many attempts at solving it (especially the linear one) and a variety of methods have been proposed. The existing methods are burdened by the complicated solving procedure when a larger number of decision makers are involved. The proposed methodology is quite complicated and decision makers can hardly be able to understand it and trust the results. This is the main disadvantage of existing methods. We have developed a new simple method for solving MOPP with an arbitrary number of decision makers. The method is iterative and it is based on the principles of game theory. Each step of the method yields a unique solution which respects the aspirations of decision makers within the frame of given possibilities. Each decision maker is assigned an objective indicator which shows the reality of his aspiration. This can be a guideline in defining the strategy for the next step. The indicators clearly show the moves which need to be made to come closer to the desired equilibrium state. Thus, at each stage of the process decision makers are able to understand why such a solution is obtained and also to see what should be done to drive the solution in the desired direction. The method also allows that a quite different strategy can be applied if necessary. As a result, decision makers can adjust their aspirations until they reach the state of equilibrium which is satisfactory to ev eryone. The method also enables the players to detect if a such state does not exist.We have developed numerical method for linear, fractional and general nonlinear MOPP and simple graphical method for these problems with two variables. As an application we use our method to solve the problem of distribution and we propose an investment model for economic recovery.

Decidability of interpretability logics via filtrations
Tin Perkov
Zagreb University of Applied Sciences

The filtration method is often used to prove the finite model property of modal logics. Finite model property, together with recursive enumerability of the set of theorems and the set of finite models up to isomorphism, yields the decidability. We adapt the filtration technique to the generalized Veltman semantics for interpretability logics. In order to preserve the defining properties of generalized Veltman models, we use bisimulations to define adequate filtrations. We give an alternative proof of the finite model property of the basic interpretability logic IL with respect to Veltman models, and we prove the finite model property of systems ILM and ILM$$_0$$ with respect to generalized Veltman models. As a consequence, we obtain the decidability of ILM$$_0$$.

On conformal embeddings of affine vertex algebras and branching rules
Ozren Perše
Department of Mathematics, Faculty of Science, University of Zagreb

In this talk we study branching rules for conformal embeddings of simple affine vertex algebras. We discuss finiteness properties of these decompositions. An emphasis will be put on the conformal embedding of simple affine vertex algebras $$V_{\mathbf{k}}({\mathfrak g} ^0)\subset V_{k}({\mathfrak g})$$, corresponding to an embedding of a maximal equal rank reductive subalgebra $${\mathfrak g} ^0$$ into a simple Lie algebra $${\mathfrak g}$$.

The talk is based on joint work with Dražen Adamović, Victor G. Kac, Pierluigi Moseneder Frajria and Paolo Papi.

Generalizations of Steffensen’s inequality via the extension of Montgomery identity
Anamarija Perušić Pribanić
Faculty of Civil Engineering, University of Rijeka

In this talk we present some new identities related to the generalizations of the Steffensen’s inequality obtained by using the extension of the Montgomery identity via the Taylor’s formula .

Moreover, by using these new identities we obtain some new generalizations of Steffensen’s inequality for $$n$$-convex functions. Since $$1$$-convex functions are nondecreasing functions, new inequalities generalize Steffensen’s inequality. Further, Ostrowski type inequalities that are related to our new generalizations are provided. Using the Čebyšev and Gruss type inequalities, bounds for the reminders in the new identities are given.
This is joint work with Andrea Aglić Aljinović and Josip Pečarić
MSC2010:26D15, 26A51.
Keywords: Steffensen’s inequality, generalizations, $$n$$-convex functions, Montgomery identity, Ostrowski-type inequality, Grüss-type inequality.

Analytic proof of the result of Doney
Tibor Pogány
Faculty of Maritime Studies, University of Rijeka

Let $$X$$ denote a spectrally positive stable process of index $$\alpha \in (1, 2)$$ whose Lévy measure has density $$c x^{-\alpha - 1}$$, $$x > 0$$ and let $$S = \sup_{0 \leq t \leq 1} X_t$$. Doney proved in his note [R. A. Doney, A note on the supremum of a stable process, Stochastics 80 (2008), 151–155.] that the density of $$S$$ say $$s$$ behaves as $$s(x) \sim c x^{-\alpha - 1}$$ as $$x \to \infty$$. Together with S. Nadarajah [T. K. Pogány, S. Nadarajah, On the result of Doney, Electron. Commun. Probab. 20 (2015), No. 58, 1–4.] we: i) give a shorter and a more general proof of the same result; ii) derive the first known closed form expressions for $$s(x)$$ and the corresponding cumulative distribution function; iii) derive the order of the remainder in the asymptotic expansion for $$s(x)$$.

Something about the Boas inequality
Dora Pokaz
Faculty of Civil Engineering, University of Zagreb

The well-known Hardy inequality was studied and generalized by many mathematicians including R.P.Boas. His inequality was also further generalized. We present some Boas-type inequalities that we have obtained in the last few years. We recall a weighted general Boas-type inequality in a setting with a topological space and $$\sigma$$-finite Borel measures as well as Boas-type inequality for $$3$$-convex functions.

On the representation theory of the vertex algebra $$W_\infty$$
Marijan Polić
Department of Mathematics, Faculty of Science, University of Zagreb

In this talk we shall present certain results on the vertex algebra $$W_\infty$$ and its representation theory. We shall prove that the simple vertex algebra $$W_\infty$$ for central charge $$c=-2$$ is isomorphic to the vertex algebra $$W(2,3)$$.

We shall also present the structure for the vertex algebras $$W_\infty$$ with central charges $$c=-4$$ and $$c=2$$. We shall realise simple $$W_\infty$$ algebra on the vertex superalgebra associated to the symplectic fermions. We shall decompose symplectic fermions and find all corresponding singular vectors.

Minding isometries of ruled surface in Minkowski space
Ljiljana Primorac Gajčić
Department of Mathematics, University J.J.Strossmayer in Osijek

A ruled surface is a surface traced out by a straight line moving along a curve $$c$$ and therefore it admits a local parametrization of the form $f(u,v)=c(u)+ve(u),u\in I\subset \mathbf{R}, v\in \mathbf{R}.$ The curve $$c$$ is the base curve (the generating curve) and the straight lines with directions $$e(u)$$ are the rulings of a surface. In classical Euclidean differential geometry, ruled surfaces are described, up to Euclidean motion, by the moving Sannia frame (G. Sannia, 1925.) which provides a complete system of the Euclidean invariants of a ruled surface – curvature $$\kappa$$, torsion $$\tau$$ and striction $$\sigma$$.
In Euclidean geometry, for a skew ruled surface, i.e. ruled surface on which consecutive rulings do not intersect, mappings that preserve the rulings of ruled surface were studied by F.Minding and therefore called Minding isometries. A differential treatment of that problem goes to E. Kruppa (1951.). Here we study the same problem in a different ambient setting – in 3-dimensional Minkowski space $$\mathbf{R}^3_1$$. The space $$\mathbf{R}^3_1$$ is a real affine space whose underlying vector space is endowed with a indefinite symmetric bilinear form of index 1 (a pseudoscalar product). Ruled surfaces in 3-dimensional Minkowski space are classified with respect to the casual character of their base curve and their rulings (spacelike, timelike or null (lightlike, isotropic)). We are specially interested in the surfaces with null rulings, so-called class $$\mathbf{M}_0$$. Among them, $$B$$-scrolls of null Frenet curves are of further interest. We analyze Minding isometries of ruled surface in terms of curvature, torsion and striction of surfaces, with respect to different types of ruled surfaces in $$R^3_1.$$

$$K$$–invariants in the algebra $$U(\mathfrak{g}) \otimes C(\mathfrak{p})$$ for the group $$SU(2,1)$$
Ana Prlić
Department of Mathematics, Faculty of Science, University of Zagreb

Let $$\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$$ be the Cartan decomposition of the complexified Lie algebra $$\mathfrak{g}=\mathfrak{sl}(3,\mathbb{C})$$ of the group $$G=SU(2,1)$$. Let $$K=S(U(2)\times U(1))$$; so $$K$$ is a maximal compact subgroup of $$G$$. Let $$U(\mathfrak{g})$$ be the universal enveloping algebra of $$\mathfrak{g}$$, and let $$C(\mathfrak{p})$$ be the Clifford algebra with respect to the trace form $$B(X,Y)=\text{tr}(XY)$$ on $$\mathfrak{p}$$. We are going to prove that the algebra of $$K$$–invariants in $$U(\mathfrak{g}) \otimes C(\mathfrak{p})$$ is generated by five explicitly given elements. This is useful for studying algebraic Dirac induction for $$(\mathfrak{g},K)$$-modules. Along the way we will also recover the (well known) structure of the algebra $$U(\mathfrak{g})^K$$.

Heat conduction through dilated pipe
Marija Prša
Faculty of Graphic Arts, University of Zagreb

We study the heat conduction through a pipe filled with incompressible viscous fluid. The goal of this work is to take into account the effects of the pipe’s dilatation due to the heating. In view of that, we assume that the longitudinal dilatation of the pipe is described by a linear heat expansion law. We prove the existence and uniqueness theorems for the corresponding boundary-value problem. The main difficulty comes from the fact that the flow domain changes depending on the solution of the heat equation leading to a non-standard coupled governing problem. This is a joint work with Eduard Marušić-Paloka and Igor Pažanin from University of Zagreb.

On free field realization of Heisenberg-Virasoro and W(2,2) vertex-operator algebras
Faculty of Science, University of Split

We present a free field realization of the Heisenberg-Virasoro vertex-operator algebra $$L^{\mathcal{H}}(c_L,c_{L,I})$$ and its impact on representation theory of the twisted Heisenberg-Virasoro Lie algebra at level 0 (denoted by $$\mathcal{H}$$). Furthermore, we will see that W(2,2) vertex operator algebra $$L^{W(2,2)}(c_L,c_W)$$ embeds in $$L^{\mathcal{H}}(c_L,c_{L,I})$$ and discuss $$W(2,2)$$–structure of irreducible highest weight $$\mathcal{H}$$–modules.
This is a joint work with Dražen Adamović.

A Minkowski measurability criterion for relative fractal drums via complex dimensions
Faculty of Electrical Engineering and Computing, University of Zagreb

We establish a Minkowski measurability criterion for a large class of relative fractal drums or, in short RFDs, in Euclidean spaces of arbitrary dimension in terms of their complex dimensions. The complex dimensions are defined as poles or, more generally, singularities of their associated Lapidus fractal zeta functions. Relative fractal drums represent a far reaching generalization of bounded subsets of Euclidean spaces as well as of fractal strings. In fact, the Minkowski measurability criterion established here is a generalization of the corresponding one obtained for fractal strings by M. L. Lapidus and M. van Frankenhuijsen. We illustrate the obtained criterion on a number of interesting examples.

Various types of orthogonality in Hilbert $$C^*$$-modules
Rajna Rajić
Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb

Orthogonality in Hilbert $$C^*$$-modules can be defined in different ways. In this talk, we discuss and compare the orthogonality with respect to the $$C^*$$-valued inner product, the Birkhoff–James orthogonality and the strong Birkhoff–James orthogonality. In particular, we characterize the classes of Hilbert $$C^*$$-modules in which any two types of these orthogonalities coincide.

This is a joint work with Ljiljana Arambašić.

Normal forms and embeddings for power-log transseries
Maja Resman
Faculty of Electrical Engineering and Computing, University of Zagreb

First return maps in the neighborhood of hyperbolic polycycles have their asymptotic expansion as Dulac series, which are series with power-logarithm monomials. We extend the class of Dulac series to an algebra of power-logarithm transseries. Inside this new algebra, we provide formal normal forms of power-log transseries and a formal embedding theorem. The questions of classifications and of embeddings of germs into flows of vector fields are common problems in dynamical systems. Aside from that, our motivation for this work comes from fractal analysis of orbits of first return maps around hyperbolic polycycles. This is a joint work with Pavao Mardešić, Jean-Philippe Rolin and Vesna Županović.

Integration of oscillatory and subanalytic functions
Jean-Philippe Rolin
Universite de Bourgogne

The theory of oscillatory integrals with analytic phase is a classical topic. We introduce an algebra $$E$$ of functions containing the subanalytic functions and their complex exponentials. We give an explicit system of generators of $$E$$ and prove its stability under integration and under Fourier transform. The methods of proof involve in particular the theory of almost periodic functions, and provides a new example of fruitful interaction between analysis and singularity theory. Joint work with R. Cluckers, G. Comte, D. Miller and T. Servi

Invariants and time-reversibility in polynomial systems of ODE’s
Valery G. Romanovskij
Center for Applied Mathematics and Theoretical Physics and Faculty of Natural Science and Mathematics, University of Maribor, Slovenia

We present some results related to the theory of invariants of ordinary differential equations. Invariants of a group of orthogonal transformations of two-dimensional systems are considered in details. An algorithm to compute a generalizing set of invariants is given and an interconnection of the invariants and time-reversibility is shown. Some generalizations to the case of three-dimensional systems are discussed as well.

Ergodicity and fluctuations of a fluid particle driven by a diffusion process with jumps
Nikola Sandrić
University of Zagreb

Turbulence is one of the most important phenomena in nature and engineering. It is a flow regime characterized by the presence of irregular eddying motions, that is, motions with high level of vorticity. The key problem is to describe the chaotic motion of a turbulent fluid. In practice this is done by tracking the evolution of a specially marked physical entity which is immersed in the fluid, the so-called fluid particle. Clearly, such a particle must be light and small enough (noninertial) so that its presence does not affect the flow pattern. In this way, the motion of the fluid may be visualized through the evolution of this passively advected particle which follows the streamlines of the fluid. In this talk, I will discuss the long-time behavior of a fluid particle immersed in a turbulent fluid flow driven by a diffusion process with jumps, that is, Feller process associated with a non-local operator. I will present the law of large numbers and central limit theorem for the evolution process of the tracked particle in the cases when the driving process: (i) has periodic coefficients, (ii) is ergodic, or (iii) is a class of Levy processes. The presented results generalize the classical and well-known results for fluid flows driven by diffusions.

Bernstein Zelevinsky derivatives via Hecke algebras
Gordan Savin
Department of Mathematics, University of Utah

For the Gelfand-Graev representation of a split reductive group, we give an explicit description of the corresponding module for the affine Hecke algebra. Then, as an application, we develop a theory of Bernstein-Zelevinsky derivatives for the affine Hecke algebra of GL(n). This is a join work with Kei Yuen Chan.

Dislocated cone metric spaces and some fixed point results of contractive functions
Eriola Sila
University of Tirana, Faculty of Natural Science, Department of Mathematics

In 2000, Hitzler and Seda introduced the concept of dislocated metric space where the distance of each point from itself is not zero. They have studied dislocated topologies as generalization of common topology. They also have proved a fixed point theorem in complete dislocated cone metric space, which generalize the contraction of Banach. There are many authors who have studied the existence and the uniqueness of fixed point in these spaces.

In 2007, Huang and Zhang introduced the concept of cone metric space replacing the set of real numbers with an ordered Banach vector space. They studied fixed points in these spaces. There are many authors as Rezapour, Abdeljawad, who have worked with these ideas. Sh. Rezapour and P. D. Proinov have defined the topology in cone metric space.

In these paper we will introduce the concept of dislocated cone metric space We study some properties related with open and closed sets, the convergent and Cauchy sequences. Also we give some theorems related with completeness and compactness of dislocated cone metric space. Our results are generalizations of many theorems in metric space, dislocated metric space and cone metric space.

Keywords:dislocated metric space, topology, completeness, compactness, convergence

3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry: uniqueness of the solution
Loredana Simčić
Faculty of Engineering, University of Rijeka

This is joint work with Nermina Mujaković and Ivan Dražić. In this work we analyse the system of eight partial differential equations which present the model for nonstationary 3-D flow of a compressible viscous and heat-conducting micropolar fluid under the assumptions of cylindrical symmetry. We introduced the homogeneous boundary conditions for velocity, microrotation velocity and heat flux as well as cylindrically symmetric and smooth enough initial data. We also assume that the fluid is in thermodynamical sense perfect and polytropic. If we assume that initial density and temperature are strictly positive, we know that the generalized solution to the described problem exists locally in time. In this work we analyse the problem in Lagrangian setting and prove that the corresponding generalized solution is unique.

A new approach to Arnold’s diffusion
Siniša Slijepčević
Department of Mathematics, Faculty of Science, University of Zagreb

The Arnold’s diffusion is a phenomena in Hamiltonian dynamical systems, where one can often show existence of unstable orbits propagating through the phase space for Hamiltonian systems arbitrarily close to integrable. It remains an open conjecture that this is a generic property of Hamiltonian dynamical systems. The classical approaches to prove it for a given system are either geometric (based on the Melnikov method), or variational. We develop a third approach, by considering formally gradient dynamics of the action functional. For example, in the case of the classical Arnold’s example, we consider dynamics of a pair of coupled reaction-diffusion equations. We obtain new examples and bounds on Arnold’s diffusion for several classes of systems. For the Arnold’s example with the Arnold’s parameter $$\mu$$, we then show that the topological entropy is a power-law in $$\mu$$ (improving known bounds exponential in $$\mu$$).

Weighted Steffensen type inequalities
Ksenija Smoljak Kalamir
Faculty of Textile Technology, University of Zagreb

The well-known Steffensen inequality $\int_{b-\lambda}^b f(t)dt \leq \int_a^b f(t)g(t)dt \leq \int_a^{a+\lambda}f(t)dt,$ where $$f$$ and $$g$$ are integrable functions defined on $$(a,b)$$, $$f$$ is nonincreasing, $$0\leq g\leq 1$$ and $$\lambda=\int_a^b g(t)dt$$, has been the subject of investigation by many mathematicians. Since its appearance Steffensen’s inequality has been generalized, refined and sharpened in many applications.

The object of this talk are weighted Steffensen type inequalities for the class of convex functions. Additionally, we give weaker conditions for obtained weighted Steffensen type inequalities. Moreover, by further generalizations of these inequalities we obtain refined and sharpened versions.

The kernel of multidegree operator on generic subspaces of algebra $$B$$
Milena Sošić
Department of Mathematics, University of Rijeka

In this presentation we consider a free unital associative complex algebra $$B$$ equipped with a multiparametric q-differential structure given by linear operators that act as twisted derivations on $$B$$. The algebra $$B$$ is naturally graded by total degree and more generally it has a finer decomposition into multigraded components $$B_{Q}$$ called weight subspaces. Of particular interest are generic weight subspaces corresponding to a set $$Q$$ of the cardinality $$n$$. We define a multidegree operator $$\partial$$ on $$B$$ and special we consider its restriction $${\partial^Q}$$ to $$B_{Q}$$. Our motivation is to determine the kernel of the operator $${\partial^Q}$$, where we will show that the operator $${\partial^Q}$$ can be factorized in terms of simpler operators. In order to simplify this computation, we have used a twisted group algebra approach. Therefore, first we have introduced a twisted group algebra $$A(S_{n})$$ of the symmetric group $$S_{n}$$ with coefficients in the polynomial ring $$R_{n}$$ in $$n^2$$ commuting variables $$X_{a\,b}$$, and then we have used a natural representation of $$A(S_{n})$$ on the generic weight subspaces $$B_{Q}$$ of the algebra $$B$$. Then we have studied factorization of certain canonical defined elements in the algebra $$A(S_{n})$$. In this way, we have obtained the corresponding factorizations that we have used in the factorization of the operator $${\partial^Q}$$. Here we will also show that the elements in the kernel of the operator $${\partial^Q}$$ can be expressed in terms of certain iterated q-commutators.

Mixed Atiyah Determinants for Graphs in Euclidean or Hyperbolic Space
Dragutin Svrtan
Department of Mathematics, Faculty of Science, University of Zagreb

In 2001 Sir Michael Atiyah, inspired by physics (Berry Robbins problem related to spin statistics theorem of quantum mechanics) associated a remarkable determinant to any $$n$$ distinct points in Euclidean 3-space (or hyperbolic 3-space), via an elementary construction. Although the problem of nonvanishing of the Atiyah determinants is very intricate (Atiyah’s first conjecture), we shall show how one can associate a mixed Atiyah determinant to any graph with the given points as vertices. For the sum of all mixed determinants we can prove an identity ($$n!$$ conjecture) which implies that for any configuration of $$n$$ distinct points in hyperbolic 3-space at least one of the mixed determinants is nonzero. For two stronger Atiyah-Sutcliffe conjectures, in case of four Euclidean points, a direct geometric proof was obtained five years ago by the speaker (and presented first at MATH/CHEM/COMP 2010, Dubrovnik, CROATIA, 7-12, 06.2010). Recently another proof is obtained, via linear programming, by M.J.Khuzam and M.J.Johnson. Both proofs use the famous Eastwood-Norbury formula for $$4$$-pt Atiyah determinant (hyperbolic analogue of which is not yet completely known!).

Second order Krylov Schur algorithm with arbitrary filter
Ivana Šain
Department of Mathematics, Faculty of Science, University of Zagreb

One way of solving quadratic eigenvalue problem (QEP) is to linearize it and use well known techniques to solve linear eigenproblem. However, eigenvalue properties are not preserved due to the fact that linearization does not preserve structural properties of original problem. Hence, we need methods which are applied to the QEP directly. Bai and Su defined second order Krylov subspace, and second order Arnoldi procedure for generating an orthonormal basis for the subspace. By applying Rayleigh - Ritz orthogonal projection they derived an SOAR method for solving large QEPs. Jia and Sun then introduced generalized second order Arnoldi method (GSOAR) which allows implicit restarting just like in IRAM (Implicitly restarted Arnoldi method).

In our talk we propose second order Krylov Schur method which is inspired by Krylov Schur method for linear problem and SOAR method for QEP. As in IRAM, implicitly restarted GSOAR has difficulties with preserving the structure of the decomposition. We define second order Krylov decomposition which has no structure constraints. This makes restarting the method easier. The choice of shifts for QEP is delicate issue, which is why our algorithm can use arbitrary filter while restarting. We also discuss influence of eigenvalue parameter scaling on convergence of our method.

Combinatorial bases of basic modules for $$C_{n}\sp{(1)}$$ and the conjecture for all standard modules
Tomislav Šikić
Faculty of Electrical Engineering and Computing, University of Zagreb

In a joint work, A. Meurman and M. Primc constructed a combinatorial bases of integrable highest weight modules for affine Lie algebra $$A_1^{(1)}$$. In this construction they used vertex operator algebra theory and combinatorial arguments. A ”representation theory part” of that construction has been extended to all affine Lie algebras, whereas the ”combinatorial part” remained to be an open problem for general affine Lie algebras. The similar generalized method for construction combinatorial bases of basic modules for affine Lie algebras of type $$C_{n}^{(1)}$$ will be presented in this talk. Special accent of this talk will be devoted to the combinatorial parametrization of leading terms of defining relations for all standard modules for affine Lie algebra of type $$C_{n}^{(1)}$$. This parametrization is the base of conjecture on the standard modules and the corresponding colored Rogers-Ramanujan type combinatorial identities where $$n\geq 2$$ and $$k\geq 2$$.
This talk is based on joint work with Mirko Primc (arXiv:1506.05026;1603.04399).

On Harmonic Quadrangle in the Isotropic Plane
Marija Šimić Horvath
Faculty of Architecture, University of Zagreb

In this talk we present several results concerning the geometry of the harmonic quadrangle in the isotropic plane. We consider the standard cyclic quadrangle with the circumscribed circle given by $$y=x^2$$ and vertices are chosen to be $$A=(a, a^2), B=(b, b^2), C=(c, c^2)$$, and $$D=(d, d^2),$$ with $$a, b, c, d$$ being mutually different real numbers, where $$a < b < c < d$$. The harmonic quadrangle in the isotropic plane is the standard cyclic quadrangle with a special property: the vertices $$A,B,C$$ and $$D$$ are chosen in a way that tangents $$\mathcal{A}$$ and $$\mathcal{C}$$ at the vertices $$A$$ and $$C$$, respectively, are intersected in the point incident with $$BD$$, and tangents $$\mathcal{B}$$ and $$\mathcal{D}$$ at the vertices $$B$$ and $$D$$, respectively, are intersected in the point incident with $$AC$$. Accordingly, $$2(ac+bd)=(a+c)(b+d)$$ follows.

This is a joint work with E. Jurkin, V. Volenec and J. Beban-Brkić.

Coherent states for quantum groups
Zoran Škoda
Department of education of teachers, University of Zadar

In 1973. Perelomov generalized classical Schroedinger coherent states to the setting of unitary representations of Lie groups. Interpreting his construction via geometric quantization, one can view coherent states as vectors dual to certain evaluation functionals on the Hilbert space of horizontal section of the quantization line bundle over a homogeneous space. Coherent states form an overcomplete family and play role in a resolution of the identity formula in terms of projection operators related to an invariant measure on the space of coherent states. I will show an axiomatics allowing to extend this definition to the quantum homogeneous spaces of real forms of Hopf algebras with Haar measure and satisfying certain axiomatics motivated by noncommutative geometry. I will show how this axiomatics can explicitly be realized for the quantum groups of type A using quantum Gauss decomposition and an interesting yoga of noncommutative localization functors which play role similar to open coordinate neighborhoods in usual geometry. In the SU(2) case, the corresponding resolution of unity formula can explicitly be realized in terms of a q-integral formula related to the beta function of Ramanujan.

Symbolic dynamics for Lozi maps
Sonja Štimac
University of Zagreb & IUPUI

We study the family of the Lozi maps $$L_{a,b} : {\mathbb R}^2 \to {\mathbb R}^2$$, $$L_{a,b}(x,y) = (1 + y - a|x|, bx)$$, and their strange attractors $$\Lambda_{a,b}$$. We introduce the set of kneading sequences for the Lozi map and prove that it determines the symbolic dynamics for that map.

(Coauthor: Michal Misiurewicz)

Statistical analysis of Fisher-Snedecor diffusion in stationary and non-stationary setting
Department of Mathematics, J.J. Strossmayer University of Osijek

Heavy-tailed Pearson diffusions are a class of diffusion processes with heavy-tailed marginal distributions from the Pearson family (Fisher-Snedecor, reciprocal gamma and Student distributions). These diffusions are characterized by the linear drift and at most quadratic diffusion coefficients, allowing explicit representations of transition densities in terms of the spectral characteristics of the corresponding infinitesimal generators. These spectral representations of transition densities are useful in deriving the explicit results in statistical analysis of these processes, e.g. in calculation of the explicit covariance matrices in analysis of asymptotic normality of the moment based estimators of their parameters. However, the methodology of computation in the non-stationary setting significantly differs from the rather simple methodology in the stationary case. This methodology will be illustrated on the example of Fisher-Snedecor diffusion - parameters of this diffusion will be estimated by the generalized method of moments and the brief sketch of the proof of their P-consistency and asymptotic normality, as well as the methodology of calculation of explicit asymptotic covariances, will be presented in both stationary and non-stationary setting.

Can one make a laser out of cardboard?
University of Zagreb, Microsoft Corporation and University of Washington

We consider two dimensional and three dimensional semi-infinite tubes made of “Lambertian” material, so that the distribution of the direction of a reflected light ray has the density proportional to the cosine of the angle with the normal vector. If the light source is far away from the opening of the tube then the exiting rays are (approximately) collimated in two dimensions but are not collimated in three dimensions. An observer looking into the three dimensional tube will see “infinitely bright” spot at the center of vision. In other words, in three dimensions, the light brightness grows to infinity near the center as the light source moves away. Joint work with Krzysztof Burdzy.

Quantitative unique continuation principles and application to control theory
Martin Tautenhahn
Technische Universität Chemnitz

This talk/poster is divided into three parts. In the first part we will recall a recent scale-free and quantitative unique continuation principle for spectral projectors of Schrödinger operators. Let $$\Lambda_L = (-L,L)^d$$, $$H_L = -\Delta + V$$ be a Schrödinger operator on $$L^2 (\Lambda_L)$$ with a bounded potential $$V$$, and $$b > 0$$. The unique continuation principle states that for any $$\phi \in \operatorname{Ran} \chi_{(-\infty , b]} (H_L)$$ we have $\label{quc}\tag{1} \lVert \phi \rVert_{\Lambda_L}^2 \leq C_{\rm sfuc} \lVert \phi \rVert_{W_\delta (L)}^2,$ where $$W_\delta (L) \subset \Lambda_L$$ is a union of equidistributed $$\delta$$-balls in $$\Lambda_L$$ and $$C_{\rm sfuc} = C_{\rm sfuc} (d , b , \delta , \lVert V \rVert)$$ some explicitly given and $$L$$-independent constant.

In the second part of the talk we will discuss an applications thereof to control theory. Here we consider the controlled heat equation $\label{eq:parabolic}\tag{2} \begin{cases} \partial_t u - \Delta u + Vu = f\chi_{W_\delta (L)}, & u \in L^2([0,T] \times \Lambda_L), \\ u = 0, & \text{on}\ (0,T) \times \partial \Lambda_L , \\ u(0,\cdot) =u_0, & u_0\in L^2(\Lambda_L) , \end{cases}$ where the control function $$f$$ acts on the set $$W_\delta (L)$$ only. It is a classical fact that the system is Null-controllabe at time $$T$$, i.e. there is a control function $$f$$ such that $$u(T,\cdot) = 0$$. Our main result is an estimate on the costs of the form $$\lVert f \rVert_{L^2([0,T]\times \Lambda_L )} \leq C \lVert u_0 \rVert_{L^2 (\Lambda_L)}$$. The novelity of our result is that we give explicit bounds on $$C$$ in terms of the radius $$\delta$$ and that $$C$$ is $$L$$-independent.

In the third part we discuss a generalization of to general second order elliptic operators instead of Schrödinger operators with Lipschitz continuous corfficients. This involves the study of Carleman estimates, where a precise knowledge of the weight function depending on the coefficients of the operator is essential.

The harmonic analysis of two commuting transformations
Christoph Thiele
Mathematical Institute of the University of Bonn

A transference principle allows to relate questions in ergodic theory with questions in harmonic analysis. Following this principle, convergence questions for bilinear ergodic averages with respect to two commuting transformations have lead to a particular program of study in harmonic analysis which has seen steady progress in recent years. We discuss several results including some joint work with P. Durcik, V. Kovac, and K. Skreb. on optimal quantitative norm convergence of ergodic averages for two commuting transformations.

Probabilistic limit theorems in Number Theory and Harmonic Analysis
Robert Tichy
Technische Universität Graz

We start from a classical central limit theorem of Kac (1946) for lacunary sequences, i.e. for sequences $$(n_{k}x)$$, where $$x\in \mathbb{R}$$ and $$n_{k}$$ positive integers with $$n_{k+1}/n_{k}\geq q> 1$$. The purpose of this talk is to present recent developments in the lacunary and sublacunary case as well as applications to diophantine number theory. This includes discrepancy estimates and quantitative results for normal numbers. Furthermore it is studied whether such limit theorems remain valid after rearranging the elements of the sequences (”permutation invariance”). The methods come from harmonic analysis, probability theory and the theory of $$S$$-unit equations. It turns out that arithmetic properties of the integer valued sequence $$(n_{k})$$ depend the probabilistic behaviour of $$(n_{k}x)$$. A focus lies on weak and strong laws of the interated logarithm for such sequences. This is joint work C. Aistleitner und I. Berkes.

$$\mathcal{H}_2$$ and $$\mathcal{H}_\infty$$ semi-active damping optimization
Zoran Tomljanović
Department of Mathematics, University J.J. Strossmayer in Osijek

We consider the problem of optimization of semi-active damping of vibrating systems. For this damping optimization we use a minimization criteria based on the $$\mathcal{H}_2$$ and $$\mathcal{H}_\infty$$ system norm.

Since the objective function is a non-convex function, this damping optimization problem usually requires a large number of function evaluations, thus we propose an optimization approach that calculates the reduced system such that we can accelerate optimization process.

Reduced systems within the $$\mathcal{H}_2$$ system norm can be generated using the parametric dominant pole algorithm or algorithm based on interpolatory model reduction. The optimization process is additionally accelerated with a modal approach. The initial parameters for the parametric model reduction can be calculated using greedy or adaptive approaches which we will compare in numerical examples.

In the setting of $$\mathcal{H}_\infty$$ system norm we propose an approach that is based on gradient of objective function where the optimal parameters can be calculated using hybrid solvers for non-differentiable functions.

With our approach we provide a significant acceleration of the optimization process which is also illustrated in numerical experiments.

Some families of Fourier-Mathieu series
Department of Mathematics, University of Rijeka

Various interesting results are proved for the Fourier- Mathieu series (which are introduced in [1]) and for their nth partial sums by applying some known theorems and lemmas for trigonometric series given by (for example) Telyakovskii, Ul’yanov, Sidon-Fomin, Bojanic and Stanojevic, and others.

References:

[1] H.M. Srivastava, Živorad Tomovski, Delco Leskovski, Some Families Of Mathieu Type Series And Hurwitz-lerch Zeta Functions And Associated Probability Distributions, Appl. Comput. Math., V.14, N.3, Special Issue, 2015, pp.349-380.

New methods for $$G$$-acyclic resolutions in cohomological dimension
Vera Tonić
University of Rijeka

The Edwards-Walsh cell-like resolution theorem (1981) states that for all $$n\in\mathbb{N}$$ and every compact metrizable space $$X$$ with $$\mathrm{dim}_{\mathbb{Z}} X \leq n$$, there exists a compact metrizable space $$Z$$ with $$\mathrm{dim}\, Z\leq n$$ and a cell-like map of $$Z$$ onto $$X$$. A. Dranishnikov proved the $$\mathbb{Z}/p$$-resolution theorem (1988) and M. Levin proved the $$\mathbb{Q}$$-resolution theorem (2005), and these theorems say: if $$G\in\{\mathbb{Z}/p,\mathbb{Q}\}$$, and a compact metrizable space $$X$$ has $$\mathrm{dim}_G X\leq n$$, then there exists a compact metrizable space $$Z$$ with $$\mathrm{dim}\,Z\leq n$$ and a $$G$$-acyclic map of $$Z$$ onto $$X$$ ($$n\geq2$$ in case $$G=\mathbb{Q}$$).

In all three of the proofs, the space $$X$$ was represented as the limit of an inverse sequence of finite triangulated polyhedra, and a significant part of the proofs required the construction of so-called Edwards-Walsh complexes, which are geometric extensions built upon the $$n$$-skeleta of these polyhedra, and which become rather complicated, especially for $$\mathbb{Q}$$-resolutions. Our aim is to present proofs of these resolution theorems by a new method that in the case of the cell-like resolution theorem requires no extensions at all, in the $$\mathbb{Z}/p$$-resolution theorem uses only Moore spaces, and in the $$\mathbb{Q}$$-resolution theorem uses only the $$(n+1)$$-skeleton of the Eilenberg-MacLane complex $$K(\mathbb{Q},n)$$.

(joint work with Leonard Rubin, University of Oklahoma)

Parameter dependent quadratic eigenvalue problem
Ninoslav Truhar
Department of Mathematics, University of Osijek

We consider the quadratic eigenvalue problem (QEP): $(\mu^2 M(v) + \mu C(v) + K(v)) x =0 \,,$ where $$M(v), C(v), K(v) \in \mathbb{C}^{n\times n}$$ are Hermitian matrices, which depends on a real parameter $$v \in \mathbb{R}^n$$.

We will present a several results on eigenvalues as well as on the eigensubspaces behavior for such QEP, dependent on the structure of the the above QEP.

For the case when the matrices linearly depend on a small perturbation (for example $$M(v)=M + v \delta M$$) we will present the relative perturbation bounds for eigenvalues and eigensubspaces which uniformly depend only on unperturbed quantities and on perturbation matrices $$\delta M$$, $$\delta C$$, $$\delta K$$.

On the other hand if $$M(v), C(v), K(v) \in \mathbb{C}^{n\times n}$$ depend smoothly on $$v$$ we will present an efficient way how to track the movements of eigenvalues in the complex plane depending on the parameter $$v$$.

Finally, let $$(\mu^2 M + \mu C + K) x =0$$ be considered QEP, with given Hermitian matrices $$M$$, $$C$$ and $$K$$. We will present a new algorithm to solve the frequency isolation problem, that is we will modifying masses, stiffnesses and damping matrix along directions in parameter space which insures that the eigenvalues of re-designed system $$(\mu^2 M(v_1) + \mu C(v_2) + K(v_3)) x =0$$ lie outside the resonance band.

Particle basis of Feigin-Stoyanovsky’s type subspaces of level one $$\tilde{\mathfrak{sl}}_{\ell+1}(\mathbb{C})$$-modules
Goran Trupčević
Faculty of Teacher Education, University of Zagreb

We construct particle basis for Feigin-Stoyanovsky’s type subspaces of level $$1$$ standard $$\tilde{\mathfrak{sl}}_{\ell+1}(\mathbb{C})$$-modules. From the description of bases, we obtain character formulas for these subspaces.

Structures of Croatian mathematics textbooks: before and after
Anđa Valent
Zagreb University of Applied Sciences

Over the past 40 years, the Croatian education system has gone through two major reforms. We use TIMSS analytical framework to analyze how these changes are reflected in the structures of mathematical textbooks for the last year of high school.

(joint with Goran Trupčević)

Regularity of intrinsically convex $$W^{2,2}$$ Sobolev isometries with application to homogenization of bending shell
Igor Velčić
Faculty of Electrical Engineering and Computing, University of Zagreb

We will prove the additional regularity of $$W^{2,2}$$ Sobolev isometries of intrinsically convex surfaces Namely, under the assumption on the regularity of the metric, it can be proved that these isometries posses $$C^{\infty}$$ regularity. This result can be used to derive the homogenized bending shell model by means of $$\Gamma$$-convergence from $$3d$$ nonlinear elasticity. However, the model is derived under additional boundary regularity of the limit deformation. The work is collaboration with Peter Hornung.

The AM-GM inequality, the AGM and mixed means
Darko Veljan
Department of Mathematics, Faculty of Science, University of Zagreb

The arithmetic-geometric mean (AM-GM) inequality is one of the well known folklore inequalities and basic principles, like Newton’s and Kepler’s law in physics, the second law of thermodynamics or triangle inequality in geometry. We provide some visual (“astronomy” and “satellite”) proofs in two variables and a combinatorial proof as well as a physical interpretation in more variables and recall some important applications: Euler’s inequality and general isoperimetric inequality in geometry and Motzkin’s example of certain polynomial in algebra. Next, we discuss the arithmetic-geometric mean (AGM) of two numbers and some of its properties including the complete elliptic integrals of the first kind investigated by Gauss and Abel. We introduce the higher mixed means of more variables and more parameters by using recurrences and previous knowledge. The evaluations and hierarchies of these higher mixed means are quite elusive, but we present some conjectures, whose very tiny special case is the AM-GM inequality. The better insight into the mixed mean theory can be helpful in statistics, large network theory and other fields where the “average” is considered only as the arithmetic (or sometimes the geometric) mean.

Quantitative uncertainty principles in harmonic analysis and mathematical physics
Ivan Veselić
TU Chemnitz

In harmonic analysis the uncertainty principle asserts that it is impossibe that a function as well as its Fourier transform are simultaneously compactly supported. In quantum mechanics the uncertainty principle asserts that it is impossible to measure two conjugate observables with arbitraty precision simultaneously. We present recent quantitative versions of uncertainty principles as well as their relations and applications in the theory of partial differential equations and random Schroedinger operators.

The Napoleon-Barlotti theorem in hexagonal quasigroups
Stipe Vidak
University of Zagreb

Hexagonal quasigroups are idempotent medial quasigroups in which the additional identity of semisymmetricity, $$ab \cdot a=b$$, holds. The famous Napoleon-Barlotti theorem of Euclidean geometry says: The centres of the regular $$n$$-gons constructed on the sides of an affine regular $$n$$-gon form a regular $$n$$-gon. In this talk the concepts of parallelogram, regular triangle and its centre, regular hexagon and its centre and affine regular hexagon are introduced in hexagonal quasigroups. Some illustrations of these concepts are given in the model $$C(q)$$, where $$q$$ is a solution of the equation $$q^2-q+1=0$$. The Napoleon-Barlotti theorem in the cases $$n=3$$ and $$n=6$$ is stated and proved in a general hexagonal quasigroup.

This is joint work with Mea Bombardelli.

Fractal Properties of Oscillatory Integrals and Singularities of Differentiable Maps
Domagoj Vlah
Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb

It is well known that theory of singularities is closely related to the study of asymptotic of oscillatory integrals. We investigate the fractal properties of a geometrical representation of oscillatory integrals $I(\tau)=\int_{\mathbb{R}^n}e^{i\tau f(x)}\phi(x) dx,$ for large values of a real parameter $$\tau$$, where $$f$$ is the analytic phase and $$\phi$$ is the smooth amplitude with compact support. We are motivated by a geometrical representation of Fresnel oscillatory integrals by a spiral called the clothoid, and the idea to produce a classification of singularities using the fractal dimension.

We measure the oscillatority by the Minkowski dimension of the planar curve parametrized by the real part $$X$$ and imaginary part $$Y$$ of the integral $$I$$. Also, we measure the oscillatory dimension that is defined as the Minkowski dimension of the graph of the function $$x(t) = X(1/t)$$, near $$t=0$$, and analogously for $$Y$$. We provide explicit formulas connecting these Minkowski dimensions and associated Minkowski contents with asymptotics of the integral $$I$$ and the type of the critical point of the phase $$f$$.

The phase and amplitude of oscillatory integrals can depend also on additional parameters. The phase could have either nondegenerate or degenerate critical points, depending on the value of the parameters. The caustic is a hypersurface in the parameter space that is the set of all values of the parameters such that the phase has degenerate critical points. Finally, we show an example of a family of caustics that undergoes a bifurcation, which can be seen using the fractal properties approach.

Used techniques include Newton diagrams and the resolution of singularities. The Newton diagram technique is commonly used in the analysis of vector fields and maps, and also for the bifurcation analysis.

This is a joint work with Jean-Philippe Rolin and Vesna Županović.

Homogenized model of immiscible incompressible two-phase flow in double porosity media
Anja Vrbaški
Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb

We present a new proof of the homogenization result for immiscible incompressible two-phase flow in double porosity media. Our assumptions on the data allow the discontinuous global pressure and saturation functions which results in the more general and physically more realistic problem. The fractured medium is composed of periodically repeating homogeneous matrix blocks and fractures, where the absolute permeability is discontinuous at the boundary between the two media. The microscopic model consists of the mass conservation laws of both fluids along with the standard Darcy-Muskat law. The problem is written in terms of the phase formulation, i.e. where the phase pressures and the phase saturations are primary unknowns. The convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilatation technique.

Construction of flag-transitive block designs using flag-transitive incidence structures
Tanja Vučičić
Faculty of Science, University of Split

From geometrical point of view the most interesting block designs are those admitting a comparatively large automorphism group. One class of such block designs are flag-transitive designs; these are designs with an automorphism group acting transitively on the set of flags (ordered pairs of incident points and blocks).

On flag-transitive designs the methods involving finite permutation groups are applicable. A transitive permutation group belongs to either class: primitive or imprimitive. In this talk we consider imprimitive flag-transitive designs.

Each imprimitive flag-transitive block design can be observed as the substructure of a flag-transitive Cameron-Praeger design. Theoretically, it is possible to obtain all flag-transitive block designs starting from these designs whose full automorphism group is of rather large size. Descending down the lattice of its subgroups and taking into consideration ever smaller automorphism groups we peel back the desired substructures. Very often and relatively quickly in this procedure one encounters a product of two symmetric groups.

We boil down the construction of block designs on which a product of two symmetric groups acts flag-transitively to the construction of flag-transitive incidence structures, which is simpler. In this manner we obtain flag-transitive designs for symmetric groups of degree not greater than 15 and describe theoretically the corresponding incidence structures. Using calculation by computer we prove that, in the given degree range, these are all existing nonisomorphic designs.

Bisimulation between different kinds of models
Department of Mathematics, Faculty of Science, University of Zagreb

There are several kinds of semantics for the interpretability logic. The basic semantics are Veltman models. D. de Jongh and F. Veltman in [2] prove the completeness of IL w.r.t. Veltman models. There are two main reasons for other semantics. First one is a complexity of the proofs of arithmetical completeness of IL. Second one is that the characteristic classes Veltman frames of some principles of interpretability are equal. Generalized Veltman models were defined by D. de Jongh. We use generalized Veltman models in [6] and prove independences between principles of interpretability. A. Visser (see [3]) use Friedman models in the proof of arithmetical completeness of ILP. A. Berarducci in [1] use Visser’s simplified models in the proof of the arithmetical completeness of ILM.

One can naturally pose the question on connection between different kinds of models for interpretabiltiy logics, i.e.  models for nonstandard modal logic.

We define a notion of bisimulation between two generalized Veltman models in [7], and prove Hennessy–Milner theorem for generalized Veltman semantics. We study various kinds of bisimulations of Veltman models in [5]. In [4] bisimulation quotients of generalized Veltman models are considered. We prove in [8] that there is a bisimulation between Veltman model and generalized Veltman model. The existence of a bisimulation in general setting is an open problem.

References:

[1] A. Berarducci, The Interpretability Logic of Peano Arithmetic, Journal of Symbolic Logic, 55(1990), 1059-1089

[2] D. de Jongh, F. Veltman, Provability Logics for Relative Interpretability, In: Mathematical Logic, (P. P. Petkov, Ed.), Proceedings of the 1988 Heyting Conference, Plenum Press, New York, 1990, 31--42

[3] A. Visser, Interpretability logic, In: P. P. Petkov (ed.), Mathematical Logic, Proceedings of the 1988 Heyting Conference, Plenum Press, New York, 1990, 175--210

[4] D. Vrgoč, M. Vuković, Bisimulations and bisimulation quotients of generalized Veltman models, Logic Journal of the IGPL, 18(2010), 870--880

[5] D. Vrgoč, M. Vuković, Bismulation quotients of Veltman models, Reports on Mathematical Logic, 46(2011), 59--73

[6] M. Vuković, The principles of interpretability, Notre Dame Journal of Formal Logic, 40(1999), 227--235

[7] M. Vuković, Hennessy--Milner theorem for interpretability logic, Bulletin of the Section of Logic, 34(2005), 195--201

[8] M. Vuković, Bisimulations between generalized Veltman models and Veltman models, Mathematical Logic Quarterly, 54(2008), 368--373

Censored Lévy processes
Vanja Wagner
Department of Mathematics, Faculty of Science, University of Zagreb

We consider the construction and main properties of a special class of symmetric Lévy processes censored on an open set $$D$$. So far, these processes have been considered only in a special case when the underlying Lévy process is the symmetric $$\alpha$$-stable process, $$\alpha\in (0,2)$$ (e.g. K. Bogdan, K. Burdzy, i Z.-Q. Chen, Censored stable processes, Probab.Theory Relat. Fields (2003), no. 127, 89-152). As a natural generalization of the symmetric stable Lévy process, we focus on the subordinate Brownian motion whose characteristic exponent satisfies certain scaling conditions.

The censored Lévy process on set $$D$$ is obtained by suppressing jumps of the Lévy process outside of the set $$D$$ by restricting the corresponding Lévy measure on $$D$$. There are three equivalent constructions of such a process – via the corresponding Dirichlet form, through the Feynman-Kac transform of the process killed outside of the set $$D$$ and from the same killed process by the Ikeda-Nagasawa-Watanabe piecing together procedure. We address the question of the boundary behaviour of the censored process and the conditions under which the process approaches the boundary of the set $$D$$ in finite time. To that end, we establish a connection between the censored Lévy and the corresponding reflected process through the trace theorem for generalized Bessel spaces. Furthermore, using the 3G inequality we prove the Harnack inequality for harmonic functions of the censored Lévy process.

We establish a connection between the subordinate Brownian motion censored on the positive half-line and the absolute value of the same subordinate Brownian motion killed at zero. Properties of harmonic functions of these processes are also considered.

From singular dimensions and fractal analysis of vector fields to Lapidus zeta functions
Darko Žubrinić
Faculty of Electrical Engineering and Computing, University of Zagreb

The talk will be concentrated on some problems of fractal analysis appearing in several areas of Mathematics. It will consist of three parts, as indicated in the title. The first part will be dedicated to the study of the “wildest” possible functions from a given space of real functions, as well as to the “wildest” weak solutions corresponding to a given class of elliptic boundary value problems. We achieve this by the simultaneous use of the Hausdorff and box (or Minkowski) dimensions. In the second part, we shall provide a brief sketch of the joint work initiated by Vesna Županović, University of Zagreb, dealing with fractal analysis of vector fields, including the Hopf-Takens bifurcation. The third part of the talk will present a short survey of the higher-dimensional theory of complex dimensions, initiated in 2009 by Michel L. Lapidus, University of California, Riverside, developed over the past seven years of our joint work, together with Goran Radunović, University of Zagreb. More details about our current work can be found on the web pages of the Centre for Nonlinear Dynamics, Zagreb: www.math.hr/cnd. We express our gratitude to the Croatian Science Foundation for its support of the project IP-2014-09-2285.

References:

M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal zeta functions and complex dimensions of relative fractal drums, J. Fixed Point Theory and Appl. No. 2, 15 (2014), 321–378. Festschrift issue in honor of Haim Brezis’ 70th birthday. (DOI: 10.1007/s11784-014-0207-y.) (Also: e-print, arXiv:1407.8094v3 [math-ph], 2014.)

M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, research monograph, Springer, New York, 2016, to appear, approx. 660 pages.

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