Croatian Mathematical Congress is a central mathematical event in Croatia organized every four years. The scientific program features plenary and invited talks by distinguished senior scientists, Croatian Mathematical Society Award Lecture, as well as contributed section talks and poster presentations. A primary aim of the congress is to present a broadest possible overview of the activities of the mathematical community as well as to host panel discussions on topics of current interest to mathematical community.
Supported by Grad Osijek.
In applied mathematics, for instance through the work of Papanicolaou,
it has been known that convection may lead to a substantial increase of
the effective diffusivity, here of a passive tracer. We consider a diffusion process with a random time-independent and spatially stationary drift that de-correlates on large scales. The two-dimensional case is scaling-wise critical; we focus on a divergence-free drift, which can be written as the curl of what is known as the Gaussian free field. In the presence of a small-scale cut-off, we prove that the process is borderline super-diffusive: Its annealed second moments grow like $t\sqrt{\mathrm{ln}\,t}$ for $t \gg 1$. This refines older results of Tóth and Valkó and recent result of Cannizzaro, Haunschmid-Sibitz and Toninelli; the method however is completely different and appeals to quantitative stochastic homogenization of the generator that can be reformulated as a divergence-form second-order elliptic operator.
This is joint work with Chatzigeorgiou, Morfe, and Wang.
Over the last few decades, shape optimization techniques have undergone significant development and have become reliable tools in many applications. However, in some applications the available data are not fully known, and the optimal shapes may show a strong dependence on these parameters. The effects of such variabilities can be analyzed by introducing multiple state variants of the problem.
We will address these questions within the framework of conductivity optimal design problems featuring two isotropic phases, since multiple state problems have been well-studied in this framework, both theoretically and numerically.
We consider principal subspaces and Feigin-Stoyanovsky's type subspaces associated with integrable highest weight modules of affine Kac-Moody Lie algebras. By using the quasi-particle bases of principal subspaces, we construct combinatorial bases of the standard modules of rectangular highest weights and their parafermionic spaces for twisted affine Lie algebras. From quasi-particle bases, we obtain characters of parafermionic spaces and standard modules. We also discuss how vertex algebraic methods can be applied to the construction of combinatorial bases of Feigin-Stoyanovsky's type subspaces associated to level one standard modules of twisted affine Lie algebra of type $D_{l+1}^{(2)}$.
Superpositions of Ornstein-Uhlenbeck type processes (supOU) provide a rich class of stationary stochastic processes for which the marginal distribution and the dependence structure may be modeled independently. Limit theorems will be presented both for the finite and infinite variance integrated supOU processes. Moreover, it will be shown that integrated supOU processes may exhibit a phenomenon of intermittency meaning that their higher order moments grow faster than it would be expected from the limit theorems. We will then discuss how intermittency affects the limiting behavior of the process through large deviations theory. We also present the results on the almost sure rate of growth and the law of iterated logarithm type results for certain cases. Even though the growth of moments may suggest differently, the almost sure growth is of the same order as in the weak limit theorems.
We will survey a number of recent results dealing with statistical properties of dynamical systems exhibiting some hyperbolicity. These include various limit theorems (central limit theorem, large deviation principle, almost sure invariance principle etc.) as well as statistical stability and linear response.
References:
[1] D. Dragičević, G. Froyland, C. Gonzalez-Tokman and S. Vaienti, A spectral approach for quenched limit theorems for random expanding dynamical systems, Comm. Math. Phys. 360 (2018), 1121-1187.
[2] D. Dragičević, G. Froyland, C. Gonzalez-Tokman and S. Vaienti, Almost sure invariance principle for random piecewise expanding maps, Nonlinearity 31 (2018), 2252-2280.
[3] D. Dragičević, G. Froyland, C. Gonzalez-Tokman and S. Vaienti, A spectral approach for quenched limit theorems for random hyperbolic dynamical systems, Trans. Amer. Math. Soc. 373 (2020), 629-664.
[4] D. Dragičević and Y. Hafouta, Almost sure invariance principle for random dynamical systems via Gouëzel's approach, Nonlinearity 34 (2021), 6773-6798.
[5] D. Dragičević and J. Sedro, Statistical stability and linear response for random hyperbolic dynamics, Ergodic Theory Dynam. Systems 43 (2023), 515-544.
[6] D. Dragičević, Y. Hafouta and J. Sedro, A vector-valued almost sure invariance principle for random expanding on average cocycles, J. Stat. Phys. 190 (2023), 38pp.
[7] D. Dragičević, P. Giulietti and J. Sedro, Quenched linear response for smooth expanding on average cocycles, Comm. Math. Phys. 399 (2023), 423-452.
[8] D. Dragičević and Y. Hafouta, Effective quenched linear response for random dynamical systems, in preparation.
In the talk I will provide an overview of how elementary properties known for the weak operator topology can be used to obtain new results in homogenisation. In particular, we shall demonstrate that nonlocal problems can be treated right away and that the gathered insights help to obtain continuous dependence results for time-dependent partial differential equations under mild conditions on the convergences involved and the regularity of the underlying spatial domains.
A unital $C^*$-algebra $A$ is said to satisfy the Dixmier property if for each element $x\in A$ the closed convex hull of the unitary orbit of $x$ intersects the centre $Z(A)$ of $A$. It is well-known that all von Neumann algebras satisfy the Dixmier property and that any unital $C^*$-algebra $A$ that satisfies the Dixmier property is necessarily weakly central, that is, for any pair of maximal ideals $M_1$ and $M_2$ of $A$, $M_1 \cap Z(A) =M_2 \cap Z(A)$ implies $M_1=M_2$. However, weak centrality is not sufficient to guarantee the Dixmier property, as even simple $C^*$-algebras can fail to satisfy it. In fact, a famous result of Haagerup and Zsido from 1984 states that a unital simple $C^*$-algebra satisfies the Dixmier property if and only if it admits at most one tracial state.
In this talk we shall present the overview of the Dixmier property and weak centrality for $C^*$-algebras with an emphasis on more recent results.
M.Primc and T.Šikić have described a combinatorial spanning set for a standard module for the affine Lie algebra of the type $C_\ell^{(1)}$ and have conjectured that this set is linearly independent. We will prove linear independence for certain classes of these modules by establishing a connection with combinatorial bases of Feigin-Stoyanovsky's type subspace of standard modules for the affine Lie algebra of the type $C_{2\ell}^{(1)}$.
We present some results on the double Yangian associated with the Lie superalgebra $\mathfrak{gl}_{m|n}$. We establish the Poincaré-Birkhoff-Witt Theorem for the double Yangian. Then, we construct the dual counterpart of the quantum contraction for the dual Yangian and we show that its coefficients are central elements. As an application, we introduce reflection algebras, certain left coideal subalgebras of the level 0 double Yangian, and we find their presentations by generators and relations. Finally we generalize the notion of quantum Berezinian to the double Yangian associated with $\mathfrak{gl}_{m|n}$ and we show that its coefficients form a family of algebraically independent topological generators of the center of $DY(\mathfrak{gl}_{m|n})$.
We consider W-algebras, a class of vertex algebras that non-linearly generalize the affine Kac-Moody Lie algebras and the Virasoro Lie algebra. They are obtained from the affine Kac-Moody Lie algebras through quantum Hamiltonian reductions parameterized nilpotent orbits. In this talk, we discuss how W-algebras are related to each other, which is partially anticipated from their connections to four-dimensional physics and the Whittaker models of $p$-adic groups.
In this talk, we will discuss a concept of deformation of quantum vertex algebra module by braiding. Furthermore, we will present its applications to Yangians, their generalizations and reflection algebras. This is a joint work with Lucia Bagnoli.
In this talk we give general Opial type inequality. We
consider two functions, convex and concave and prove a new general inequality on a measure space $(\Omega,\Sigma,\mu)$. The obtained inequalities are not direct generalizations of the Opial inequality but are of Opial type because the integrals contain function and its integral representation. We apply our result to new Green functions and obtain new results.
The Hanin inequality is a kind of a reverse of the non-weighted arithmetic-quadratic mean inequality defined for non-negative real $n$-tuples, involving maxima of the corresponding $n$-tuple.
In this talk, our goal is to extend the Hanin inequality in several directions. We first give two-parametric extension of the basic inequality, as well as its refinement. Then, we discuss the corresponding weighted forms of the established results. Finally, we derive several complex extensions of the Hanin inequality.
The extension of the weighted Montgomery identity is established by using the general integral formula. Further, by using this extended weighted Montgomery identity for functions whose derivatives of order $n-1$ are absolutely continunous functions, new inequalities of the weighted Hermite-Hadamard type are obtained. Also, applications of these results are given for various types of weight function.
Extensions of Stolarsky's and Pinelis' inequalities are considered. New results of that type for q-integrals are obtained.
Regular directed graph $\Gamma$ of degree $k$ with $n$ vertices is directed strongly regular graph, $DSRG(n,k,\lambda,\mu,t)$, if number of directed paths of length two from every vertex $v$ to every vertex $w$ is $\lambda$ if there exists directed edge $v\to w$, $t$ if $v=w$ and $\mu$ if there is no edge $v\to w$. Directed strongly regular graphs were introduced by Art Duval in 1988.
One can construct 1-design by defining a basic block as union of $G_{\alpha}$-orbits of transitive permutation group. Using that, we construct directed regular and strongly regular graphs from transitive groups.
We study the $Kf- $ Šoltés problem, which is related to the resistance distance in a graph. While the original Šoltés problem deals with the identification of all graphs for which the removal of an arbitrary vertex preserves the Wiener index, the $Kf- $ Šoltés problem deals with graphs for which the removal of any vertex preserves the Kirchhoff index.
Currently, the only known solution to the $Kf- $Šoltés problem is the cycle $C_5$. We consider the relaxed version of the problem, which is called the $Kf_\beta\, - $ Šoltés problem: find graphs whose proportion of vertices that preserve the Kirchhoff index is equal to $\beta$. We show that for $0< \beta < 2/3$ the $Kf_{\beta}\, - $ Šoltés problem is rich with solutions. Namely, we construct an inifinite family of $Kf_{1/2}-$Šoltés graphs and build a family of graphs for which $\beta$ tends to $2/3$. We also study $Kf_{\beta}\,-$ Šoltés problem on unicyclic and bicyclic graphs.
Fibonacci numbers are one of the most famous and investigated sequences. They can be found almost everywhere. For example, the number of ways to tile a $1\times n$ rectangular strip using squares and dominoes is counted by Fibonacci numbers, as is the number of subsets of the set $\left\lbrace 1,2,\dots,n\right\rbrace$ without consecutive elements. Here we consider $2\times n$ hexagonal strips and count the number of ways to divide such strips into a given number of parts. We prove that such divisions are enumerated by the odd-indexed Fibonacci numbers. In this talk, we present three different proofs of this result.
We find the number of homogeneous polynomials of degree $d$ such that they vanish on cuspidal modular forms of even weight $m\geq 2$ that form a basis for $S_m(\Gamma_0(N))$. We use these cuspidal forms to embedd $X_0(N)$ to projective space and we find the Hilbert polynomial of the graded ideal of the projective curve that is the image of this embedding.
Let $ \Gamma $ be a congruence subgroup of $ \mathrm{Sp}_{2n}(\mathbb Z) $. Using Poincaré series of $ K $-finite matrix coefficients of integrable discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_\rho(\Gamma) $ of Siegel cusp forms for $ \Gamma $ of weight $ \rho $, where $ \rho $ is an irreducible polynomial representation of $ \mathrm{GL}_n(\mathbb C) $ of highest weight $ \omega=(\omega_1,\ldots,\omega_n)\in\mathbb Z^n $ with $ \omega_1\geq\ldots\geq\omega_n>2n $. We study the non-vanishing of constructed Siegel cusp forms and their role in the theory of Siegel modular forms.
In this talk we give an explicit characterization of all bases of $\varepsilon-$canonical number systems ($\varepsilon-$CNS) with finiteness property in quadratic number fields for all values $\varepsilon\in\lbrack0,1)$. This result is a consequence of the recent result of Jadrijević and Miletić on the characterization of quadratic $\varepsilon-$CNS polynomials. Our result includes the well-known characterization of all bases of classical CNS ($\varepsilon=0$) with finiteness property in quadratic number fields. It also fits into the general framework of generalized number systems (GNS) introduced by A. Peth\H{o} and J. Thuswaldner.
Since their introduction the Bent functions had an important role in designing S-boxes for block and stream chipers. While the theory behind Bent functions has substantially been developed, their count is still unknown for number of variables $n>8,$ and is probably the most important question left. Recently, some improvements on the upper bound of number of the Bent functions are presented based on the dimensionality of bent rectangles. In this work a novel approach has been established based on the number of minterms of an characteristic function $\chi\colon\mathbb{F}_2^{2^n}\mapsto\mathbb{F}_2$ for the Bent functions. The sets of minterms of the $\chi$ function are carefully built by means of Binary Decision Diagrams and their cardinality is counted from them. A two-dimensional recurrence relation between cardinalities leads to much improved upper bounds on number of Bent functions.
Structural optimization involves strategically arranging certain materials within a structure to enhance its properties with respect to some optimality criteria. This optimization typically entails minimizing or maximizing an integral functional, which depends on the rearrangement of materials within the domain, and the solution of a partial differential equation that models the underlying physics.
We present a novel optimality criteria method for optimal design problems in the setting of linearized elasticity. More precisely, we derive a method for maximizing the first eigenvalue in a mixture of two isotropic elastic materials in a bounded domain, with a volume constraint for the most rigid material. The algorithm is based on necessary conditions of optimality for problem which was obtained by relaxing the original one via the homogenization method. Since the method relies on explicit expressions for the lower Hashin–Shtrikman bound on the complementary energy and information on the microstructure that saturates the bound, implementing the algorithm in three space dimensions was not feasible until a recent explicit calculation was done by Burazin, Crnjac and Vrdoljak (2024). We demonstrate the method on a number of examples of two- and three-dimensional eigenfrequency maximization problems.
We focus on addressing optimal design challenges involving second-order elliptic partial differential equations. Our objective is to determine the optimal outer shape of the domain and the distribution of two isotropic materials within the domain, considering predetermined amounts, to minimize a given functional. The optimization algorithm employed in this study integrates the homogenization method and the shape optimization method. We use the level set function to propagate the movement of the outer boundary through a calculated shape derivative. In the interior optimization, an optimality criteria method is employed to address multiple-state optimal design problems. We propose and test a numerical scheme on various examples.
We consider the solution of sequences of parametrized Lyapunov equations. Solutions of such equations can be encountered in many application settings, and they are often intermediate steps of an overall procedure whose main goal is the computation of quantities of the form $f(X)$ where $X$ denotes the solution of a Lyapunov equation.
We are interested in addressing problems where the parameter dependency of the coefficient matrix is encoded as a low-rank modification to a seed, fixed matrix. We propose two novel numerical procedures that fully exploit such a standard structure. The first one builds upon recycling Krylov techniques, and it is well-suited for small dimensional problems as it uses dense numerical linear algebra tools. The second algorithm can instead address large-scale problems by relying on state-of-the-art projection techniques based on the extended Krylov subspace.
We test the new algorithms on several problems arising in the study of damped vibrational systems and the analyses of output synchronization problems for multi-agent systems.
We consider damping optimization for vibrating systems described by
a second-order differential equation. The goal is to determine position and viscosity
values of the dampers in the system such that the system produces the lowest total
average energy. We propose new framework using relaxed weighted $l_1$ minimization
and pruning techniques to determine the number and positions of the dampers. The
efficiency and performance of our new approach are verified and illustrated on several
numerical examples.
We present a novel numerical method for optimal design problems in the setting of the Kirchhoff-Love model, where the mechanical behaviour of the domain is modeled with fourth order elliptic
equation, and we restrict ourselves to domains ﬁlled with two isotropic elastic materials.
Since the classical solution usually does not exist, we use homogenization theory to prove general existence theorems and to provide efficient numerical algorithm for their
computation.
Moreover, we give an explicit calculation of the Hashin-Shtrikman bounds on the complementary energy. They are of a great significance in optimal design problems, since the necessary conditions of optimality are easily derived and expressed via lower Hashin-Shtrinkman bound on the complementary energy. This enables a development of the optimality criteria method for ﬁnding an approximate solution.
In this talk, we propose mathematical tools for studying oscillation and concentration effects at infinity in sequences of absolutely continuous functions. These tools can be applied beyond the classical setting, especially in cases where the fundamental theorem of Young measures or the fundamental theorem of H-measures cannot be used. Examples of aforementioned cases have already been studied in papers
A. Raguž, Some results in asymptotic analysis of finite-energy sequences of Cahn-Hilliard functional with non-standard two-well potential, Glas. Mat. (2024) (to appear),
A. Raguž, A priori estimates for finite-energy sequences of Cahn-Hilliard functional with non-standard multi-well potential, Math. Commun. (2024) (to appear),
which deal with asymptotic properties of finite-energy sequences of one-dimensional Cahn-Hilliard functional as a typical example of a problem of minimization of integral functionals with singularly perturbed non-convex integrands. Herein we expand our consideration by presenting further results along the same lines.
In this work we consider a planar periodic network of elastic rods. As the model for a structure made of elastic rods we use a one-dimensional model of Naghdi/Timošenko type allowing for membrane, stretching and bending deformations with the Kirchhoff type junction conditions. Using a mesh two-scale convergence, a variant of the two-scale convergence adapted for models given on lower-dimensional objects, we show that the equilibrium solutions of this elastic network with periodicity size $\delta$ converge when $\delta$ tends to zero to the solution of the plate equation. The elasticity tensor in the effective plate equation is obtained as the solution of the network problem on the unit cell,
as usual in homogenization.
We consider long-time behavior of solutions to the thin-film equation $ \partial_th = -\partial_x(h \partial_x^3h)$ on the real line with initial datum of finite second moment. The equation describes the interface dynamics of a thin fluid neck of thickness $2h(x,t)$ in the Hele-Shaw cell. Upon rescaling the equation in such a way that the second moment is constant in time, precise rate of convergence to the steady state is given in terms of corresponding relative Rényi entropy. For intermediate times, this improves the rate of convergence given in terms of the usual relative entropy and thus, improves the rate of convergence in the $L^1$-norm. The result is based on ideas developed by Carrillo and Toscani (Nonlinearity 27 (2014), 3159-3177) for second-order nonlinear diffusion equations and utilizes the concavity property of the Rényi entropy power.
In 1971., Steven J. Takiff, while studying invariant polynomial rings, defined a specific extension of the finite-dimensional Lie algebra. In his honor, mathematicians call the corresponding extensions of finite-dimensional simple Lie algebras, but also of some infinite-dimensional Lie algebras, Takiff algebras. Takiff algebras are also known as truncated current Lie algebras or polynomial Lie algebras. They are strongly related to the Galilean algebras known in (mathematical) physics.
In this talk, we will define the Takiff algebra obtained from the affine Lie algebra of type $A_1^{(1)}$ and explain construction of its associated vertex algebra and the vertex operator algebra. In addition, we will present some important properties of the previous structures.
We investigate the representation theory of simple affine vertex algebra $L_k(\mathfrak{g})$ at special non-admissible levels $k_n=-\frac{2n+1}{2}$ for $\mathfrak{g} = \mathfrak{sl}_{2n}$. We classify irreducible $L_{k_n}(\mathfrak{sl}_{2n})$-modules in category $KL_{k_n}(\mathfrak{sl}_{2n})$ and prove that $KL_{k_n}(\mathfrak{sl}_{2n})$ is a semi-simple, rigid braided tensor category.
In addition, we present a new method for proving simplicity of quotients of universal affine vertex algebras $V^{k_n}(\mathfrak{sl}_{2n})$. We use this result to prove that in the case $n=3$ a maximal ideal is generated by one singular vector of conformal weight 4. As a byproduct, we classify irreducible modules in the category $\mathcal{O}$ for $L_{-7/2}(\mathfrak{sl}_6)$.
The talk is based on joint papers with D. Adamović, T. Creutzig and O. Perše.
The first story concerns the real integral expression for the normalization constant $Z(\lambda, \nu)$ which occurs in the Conway-Maxwell distribution which is a generalization of the Poisson law. It turns out that $Z(\lambda, \nu)$ can be connected to the Le Roy-type hypergeometric function.
The second part of the talk resolves the representation of the Pólya constant in terms of Lauriella generalized hypergeometric function $F_C^{(d)}$ of $d$ variables. The Pólya constant describes the probability $p(d)$ that a simple symmetric random walk on the $d$-dimensional lattice $\mathbb Z^d$ returns to origin, for $d \in \mathbb N$. A famous 103 years old result of Pólya states that $p(1) = p(2) = 1$ but $p(d) < 1$ for $d \geq 3$.
Applications of modified Bessel functions frequently occur in statistics, for instance, it is a constituting term of the probability density function of the non-central $\chi^2$ distribution, having $n$ degrees of freedom and non-centrality parameter $\lambda>0$. The random variable with such distribution is usually denoted by $\chi_n'^{\;2}(\lambda)$.
Bearing in mind a great application of the non-central $\chi^2$ distribution, for example in finance, estimation and decision theory, in mathematical physics and, among others, in communication theory in which case the appropriate cumulative distribution function (CDF) is given in terms of the so-called generalized Marcum Q-function, the appropriate CDF has been widely considered in mathematical literature.
The aim of this talk is to present several new formulae for such CDF in the case of even number of the degrees of freedom. The main advantages of these expressions are that they are given in terms of some familiar special functions as the modified Bessel functions and generalized incomplete gamma function, which have computational advantages and a wide range of applications having numerous in build routines. Also, numerical simulations shows the quality of newly derived formulae in comparison with certain earlier results.
Spaces invariant under unitary group representations have been extensively studied in the recent decades due to their importance in various areas, such as the theory of Gabor systems, wavelets and approximation theory. Dual integrable representations form a large and important class among the unitary representations, for which the properties of the cyclic subspaces and their generating orbits can be described in terms of the associated bracket map; the existence of an operative bracket map is, thefore, crucial for the utility of the definition. The concept was first introduced and studied for the abelian groups, and later for some specific classes in the non-commutative setting. We have recently introduced the definition for the entire class of locally compact groups. Being based on the non-commutative integration theory, the methods in the non-abelian case are quite different from the ones in the abelian setting; however, most of the main properties and fundamental results remain valid or have the appropriate analogues. In this talk, we present the main results concerned with the study of the dual integrability.
The talk is based on joint work with Hrvoje Šikić.
We consider germs $f$ with asymptotic logarithmic bounds, i.e. $f(z)=\lambda z+o(zL(z))$, $0<|\lambda |<1$, uniformly as $|z|\to 0$, on Riemann surface of the logarithm, where $L(z)$ is, roughly speaking, some product of powers of iterated logarithms . Instead of the standard $z$-chart, we consider the logarithmic chart $\zeta :=-\log z$ which is a global chart for Riemann surface of the logarithm. In this chart germ $f$ can be written in the form $\widehat{f}(\zeta )=\zeta -\log \lambda + o(\zeta^{-1}\widehat{L}(\zeta ))$, uniformly as $\mathrm{Re} \, \zeta \to + \infty $. In order to linearize such germs, we are motivated by famous Koenigs' Theorem about linearization of analytic hyperbolic diffeomorphisms at zero. But instead of open balls around zero, we consider open domains on Riemann surface of the logarithm which spiral around the origin. The spiraling bound of the domain can be of any kind, but we choose special domains which we call the admissible ones. The importance of these domains is their $\widehat{f}$-invariance. Therefore, we consider the Koenigs' sequence $(\widehat{f}^{\circ n})$ and prove its uniform convergence towards the unique linearization of the germ $\widehat{f}$ on some admissible domain. This result is simultaneously a generalization of Koenigs' Theorem and the recent Dewsnapp-Fischer's result about the linearization of real germs with logarithmic asymptotic bounds. We apply our result to the problem of linearization of Dulac germs which are specal kinds of such germs that appear naturally in solutions of the famous Dulac problem of nonaccumulation of limit cycles on a hyperbolic polycycle of an analytic planar vector field. This is joint work with M. Resman, J.P. Rolin and T. Servi.
Whenever we are given a selfmap f of a compact metric space
X, we can associate with it the induced mapping C(f) on the
hyperspace C(X) of continua in X, defined in a natural way.
In this talk we discuss and provide the answer to the following question:
Let f be a selfmap of a topological tree T, and let x be a periodic point of f of given period p. What are the possible periods of periodic points of C(T) containing x?
We will argue the significant importance of this result when studying some other features of the system (C(T),C(f)).
The talk is based on a joint work with Piotr Oprocha.
It is already well known that the multiplicity/cyclicity of limit cycles and weak foci of analytic planar vector fields can be determined from the Minkowski dimension of spiral trajectories near such limit periodic sets. In these configurations the intersection of a spiral trjectory with any transversal to the limit cycle/weak focus has the same Minkowski dimension. On the other hand, non-regular limit periodic sets such as saddle-loops and polycycles don't necessarily satisfy this property. The lack of such a property is one of the obstacles in finding the link between the Minkowski dimension of spiral trajectories and that of the intersections of the trajectory with transversals to the limit periodic set. The main difficulty arises near singular points on limit periodic set. The dimension of the intersection of the spiral trajectory with a transversal changes as we move the transversal through a singular point. We present two new results that deal with these difficulties and allow us to express the dimension of the parts of spiral trajectories near two types of singularities: hyperbolic saddles and semi-hyperbolic singularities with saddle-like behaviour. We also present the application of these results to two examples of non-regular limit periodic sets: the saddle-loop and the hyperbolic $2$-cycle. Based on previous well established theory, we provide a way to obtain upper bounds on the cyclicity of such objects using the Minkowski dimension of (any) spiral trajectories(y) near them.
This talk is based on joint work with Renato Huzak and Maja Resman.
Inquisitive logic is a generalization of classical logic that can express questions. The language of inquisitive logic is obtained by extending the language of classical propositional logic with a new connective, the inquisitive disjunction. Similarly, inquisitive modal logic, InqML, is a generalization of standard modal logic with $\boxplus$ as the basic modal operator. In this talk, we will introduce the selection method for inquisitive modal logic. For a given satisfiable formula $\varphi$, we will show that $\varphi$ is also satisfiable in a finite tree-like model. From this we conclude that InqML has the finite model property, i.e., every satisfiable formula is satisfiable in a finite model.
The basic semantics for the interpretability logics are Veltman models.
R. Verbrugge defined a new relational semantics for interpretability logics, which today is called Verbrugge semantics in her honor.
It has turned out that this semantics has various good properties.
Bisimulations are the basic equivalence relations between Veltman models.
M. Vuković and T. Perkov used them to prove the van Benthem's characterization theorem with respect to Veltman semantics.
Van Benthem's characterization theorems belong to the field of correspondence theory which systematically investigates the relationship between modal and classical logic.
They show that modal languages correspond to the bisimulation invariant fragment of first–order languages.
M. Vuković defined bisimulations for Verbrugge semantics. However, in [2] we proved that with such a definition of bisimulation for Verbrugge semantics, the analogues of some of the results for the Veltman semantics do not hold in case of Verbrugge semantics.
Therefore, in [2] we gave a new version of bisimulations, which we called weak bisimulations.
In this talk, we will make an overview of the results that can be obtained using weak bisimulations.
Among these results, we will especially highlight van Benthem's theorem with respect to Verbrugge semantics, which we proved in [1].
Finally, we will give an overview of the main steps required to prove the van Benthem - Rosen theorem with respect to Verbrugge semantics,
which is a version of the van Benthem theorem for finite models.
References:
[1] S. Horvat, T. Perkov, Correspondence theorem for interpretability logic with respect to Verbrugge semantics, preprint, 2024.
[2] S. Horvat, T. Perkov, M. Vuković, Bisimulations and bisimulation games for Verbrugge semantics, Mathematical Logic Quarterly 69(2023), 231-243
Let $\mathcal R$ be a commutative ring with unity and $n\in \mathcal R$, $n\not=0$. A $D(n)$-quadruple in $\mathcal R$ is a set of four elements in $\mathcal{R}\backslash\{0\}$ with the property that the product of any two of its distinct elements increased by $n$ is a square in $\mathcal{R}$. It is interesting that in some rings $D(n)$-quadruples are related to the representations of $n$ by the binary quadratic form
$x^2 - y^2$. Moreover, there are many examples of rings of integers of number fields in which a $D(n)$-quadruple exists if and only if $n$ can be written as a difference of two squares in $\mathcal R$. Here we investigate the connection between “D(n)-quadruples and
differences of squares” in the ring of polynomials with integer coefficients, $\mathbb Z[X]$ and show that there is no polynomial $D(n)$-quadruple in $\mathbb Z[X]$ for certain $n\in\mathbb Z[X]$ that are not representable as a difference of squares of two polynomials in
$\mathbb Z[X]$.
For a non-zero element $n\in \mathbb{Z}[X]$, a set of $m$ distinct non-zero elements from $\mathbb{Z}[X]$, such that the product of any two of them increased by $n$ is a square of an element of $\mathbb{Z}[X]$, is called a Diophantine $m$-tuple with the property $D(n)$ or simply a $D(n)$-$m$-tuple in $\mathbb{Z}[X]$. We prove that there does not exist a $D(2X+1)$-quadruple in $\mathbb{Z}[X]$, which is one counterexample for the thesis that $n\in\mathbb{Z}[X]$ is representable as a difference of squares of polynomials if and only if there exists $D(n)$-quadruple in $\mathbb{Z}[X]$.
A set $\{a, b, c, d\}$ of four distinct non-zero polynomials in $\mathbb{R}[X]$, which are not all constants, is called a polynomial $D(4)$-quadruple in $\mathbb{R}[X]$ if the product of any two of its distinct elements, increased by 4, is a square of a polynomial in $\mathbb{R}[X]$.
We prove some properties of these sets, and to tackle the problem of regularity of polynomial $D(4)$-quadruples in $\mathbb{R}[X]$, we investigate whether the equation $$(a+b-c-d)^2=(ab+4)(cd+4)$$ is satisfied by each such polynomial $D(4)$-quadruple in $\mathbb{R}[X]$. Our earlier research focused on the regularity of the polynomial $D(4)$-quadruple in $\mathbb{Z}[i][X]$, and we now compare these results with the recent findings from $\mathbb{R}[X]$.
The Jacobi method is a well known iterative method for solving the symmetric eigenvalue problem. Efficiency of the Jacobi method can be improved if the algorithm works on the matrix blocks instead of the elements.
In this talk we consider the convergence of the complex block Jacobi diagonalization methods under the large set of the generalized serial pivot strategies. We present the convergence results of the block methods for Hermitian, normal and $J$-Hermitian matrices. Moreover, we consider the convergence of a general block iterative process in order to obtain the convergence results for the block methods that solve other eigenvalue problems, such as the generalized eigenvalue problem.
We present algorithms for solving the eigenvalue problem for the arrowhead and diagonal-plus-rank-$k$ matrices of quaternions. The algorithms use the Rayleigh Quotient Iteration with double shift combined with Wielandt's deflation technique. Since each eigenvector can be computed in $O(n)$ operations, the algorithms require $O(n^2)$ floating-point operations, $n$ being the order of the matrix. The algorithms are backward stable in the standard sense. The algorithms are elegantly implemented in the programming language Julia.
A plunge into the world of large Erd\H{o}s-R\'enyi graphs, either regular or with non-trivial irregularities. The scaling limits of connected component masses are called the multiplicative coalescents. While a number of their properties have been known for 25 or more years, some interesting questions were answered only recently, and many others remain unsolved. The talk will attempt to give an overview of a few recent directions of research, which all stem from an excursion representation via ``simultaneous breadth-first walks''.
We study maxima of linear processes with heavy-tailed innovations and random coefficients. Using the point process approach we derive functional convergence of the partial maxima stochastic process in the space of cadlag functions on $[0,1]$ endowed with the Skorohod $M_{1}$ topology.
By utilizing integral arithmetic mean $F$ defined as
$$
F(x,y)=\left\{\begin{array}{cc}
\frac{1}{y-x}\int_{x}^{y}f(t)dt, & x,y\in I,~x\neq y,\\
f(x), & x=y\in I
\end{array}\right.
$$
and applying a generalized form of Niezgoda's inequalty, we present new proofs concerning the $(m,n)$-convexity of the integral arithmetic mean function. This paper aims to contribute novel proofs to D. E. Wulbert's findings, which demonstrate that the integral arithmetic mean exhibits convexity on $I^{2}$ provided $f$ is convex on $I$.
Additionally, we establish an inequality for divided differences by employing the extended form of Niezgoda's inequality. Moreover, we demonstrate the convexity of the function defined by these divided differences.
The bicentric $n$-gon is a polygon with $n$ sides that are tangential for incircle and chordal for circumcircle. The connection between the radius ($R$) of circumcirle, the radius ($r$) of incircle and the distance ($d$) of their centers represents the Fuss' relation for the certain bicentric $n$-gon. According to Poncelet's porism, there exist infinitely many bicentric $n$-gons with these characteristics.
In this presentation, we derive relations for the largest and smallest area as a function of the parameters $R$, $r$ and $d$ for the bicentric quadrilateral, hexagon and octagon. We used the works [1]-[2] by Josefson to find out which $n$-gons have extreme areas for bicentric quadrilaterals, and work [3] by Radić for the bicentric hexagon and octagon. So far, only the relation for calculating the area of a bicentric quadrilateral is known. The corresponding relations are derived in [1], [2], [4].
References:
Digital game-based learning (DGBL) is regarded as a engaging teaching approach that has attracted the interest of researchers and has become prominent as a research topic. However, the potential benefits of gaming on students' academic achievement, motivation, and skills are still a topic of debate.
Given the relatively limited adoption of digital games in upper secondary education, particularly in mathematics education, it is important to understand secondary students' perspectives regarding their learning experience following their engagement with DGBL.
GAMe-based learning in MAthematics (GAMMA) was an Erasmus+ project primarily focused on implementing DGBL in upper secondary school mathematics education. Over the project's duration, seven digital games and eight teaching scenarios were created. The educators involved in the project incorporated these scenarios into the classroom, while students gave feedback on their experience with DGBL through a brief survey. This study presents a qualitative and quantitative analysis of student responses to post-piloting surveys.
Considering the Gaussian mixture model the restriction-based iterative method is developed. The proposed method applied the conditional expectations for parameter re-estimation in an iterative process, where the rejection of the model component's tails of low probabilities is considered. In order to study the proposed method, continuous and discrete random variable cases are observed, where the results and properties are conducted by considering the Banach, and the derivative of Brouwer's fixed point theorem, respectively. Finally, the various numerical examples and the application of the proposed method in digital image processing for image segmentation and pattern recognition are presented.
In proportional electoral systems, party vote counts must be converted to seat allocations within a parliament of fixed size. Divisor methods are the most common approach to this problem, but different divisor methods often give different seat allocations. To highlight these differences, the effects of various divisor methods on a party’s seat allocation are expressed as intervals of the party’s vote count within which the seat allocation is unchanged, assuming other parties’ votes are fixed. Knowing these intervals of vote counts could help in the analysis of elections, planning of an electoral strategy and understanding of different divisor methods. Numerical examples are included.
The stochastic version of the SIRV (susceptible-infected-recovered-vaccinated) epidemic model in the population of non-constant size and finite period of immunity is considered. Among many parameters influencing the dynamics of this model, the most important parameter is the contact rate, i.e. the average number of adequate contacts of an infective person, where an adequate contact is one which is sufficient for the transmission of an infection if it is between a susceptible and an infected individual. It is expected that this parameter exhibits time-space clusters which reflect interchanging of periods of low and steady transmission and periods of high and volatile transmission of the disease. The stochastics in the considered SIRV model comes from the noise represented as the sum of the conditional Brownian motion and Poisson random field, closely related to the corresponding time-changed Brownian motion and the time-changed Poisson random measure. The existence and uniqueness of positive global solution of the corresponding system of stochastic differential equations is proven by classical techniques. Furthermore, persistence and extinction of infection in population in long-run scenario are analyzed. In particular, conditions depending on parameters of the model and the underlying measure, under which the persistence and the extinction of the disease appear, are derived. The theoretical results are illustrated via simulated examples. In particular, transmission coefficient is simulated as the mean-reverting diffusion with jumps with different propositions for the absolutely continuous time-change process.
Randomized algorithms have gained significant attention in numerical linear algebra during the last decade. In particular, randomized sketching is used as a simple but effective technique in which a random matrix acts as a dimension reduction map: a problem that features a potentially large input matrix $A \in \mathbb{R}^{n \times n}$ is reduced to a smaller one by replacing $A$ with $A \Omega$, where $\Omega \in \mathbb{R}^{n \times \ell}$ is a random matrix with $\ell \ll n$. This has been used with great success in, e.g., randomized SVD and subspace projection methods for large-scale eigenvalue problems, such as FEAST.
Algorithms based on sketching typically draw random matrices $\Omega$ from standard distributions, such as Gaussian. However, in certain applications it may be advantageous to run computations with $\Omega$ that is compatible with the underlying structure of the problem. In this talk we discuss algorithms that use random Khatri-Rao product matrices: each column of $\Omega$ is generated as the Kronecker product of two Gaussian random vectors. This will allow for faster operations when the matrix $A$ is represented as a short sum of Kronecker products, which arises frequently, e.g., from the discretization of PDEs on tensor product domains. We focus on applications in large-scale eigenvalue computation, and provide theoretical and numerical evidence that the use of random Khatri-Rao product matrices $\Omega$ instead of unstructured Gaussian random matrices leads to good estimates.
This is joint work with Luka Grubišić, Daniel Kressner and Hei Yin Lam.
Determining the rank of an elliptic curve $E/\mathbb{Q}$ is a difficult problem. In applications such as the search for curves of high rank, one often relies on heuristics to estimate the analytic rank (which is equal to the rank under the Birch and Swinnerton-Dyer conjecture).
This talk discusses a novel rank classification method based on deep convolutional neural networks (CNNs). The method takes as input the conductor of $E$ and a sequence of normalized Frobenius traces $a_p$ for primes $p$ in a certain range ($p<10^k$ for $k=3,4,5$), and aims to predict the rank or detect curves of ``high'' rank. We compare our method with eight simple neural network models of the Mestre-Nagao sums, which are widely used heuristics for estimating the rank of elliptic curves.
We evaluate our method on the LMFDB dataset and a custom dataset consisting of elliptic curves with trivial torsion, conductor up to $10^{30}$, and rank up to $10$. Our experiments demonstrate that the CNNs outperform the Mestre-Nagao sums on the LMFDB dataset. The performance of the CNNs and the Mestre-Nagao sums is comparable on the custom dataset.
This is joint work with Domagoj Vlah.
Additionally, we will elaborate on an ongoing project with Zvonimir Bujanović, focusing on a detailed analysis of some aspects of Mestre-Nagao sums through the use of neural networks.
In this talk, we discuss pure-jump Markov processes on smooth open sets whose jumping kernels vanishing at the boundary and part processes obtained by killing at the boundary or (and) by killing via the killing potential. The killing potential may be subcritical or critical.
This work can be viewed as developing a general theory for non-local singular operators whose kernel vanishing at the boundary. Due to the possible degeneracy at the boundary, such operators are, in a certain sense, not uniformly elliptic. These operators cover the restricted, censored and spectral Laplacians in smooth open sets and much more.
The main results are the boundary Harnack principle and its possible failure, and sharp two-sided Green function estimates.
We present a fundamentally new proof of the dimensionless L^p boundedness of the Bakry Riesz vector on manifolds with bounded geometry. The key ingredients of the proof are sparse domination and probabilistic representation of the Riesz vector. This type of proof has the significant advantage that it allows for a much stronger conclusion, giving us a new range of weighted estimates. The talk is based on joint work with K. Domelevo and S. Petermichl.
On the one hand, the classical Banach-Stone theorem shows that the topological structure of a compact Hausdorff space $\Omega$ is determined by the geometry of $C(\Omega)$, the Banach space of continuous scalar-valued functions on $\Omega$, while on the other hand, it gives an explicit description of surjective linear isometries between two Banach spaces of continuous functions, $C(\Omega_1)$ and $C(\Omega_2)$. This theorem has been generalized in various ways. In this talk, following a long line of work on analogues of this classical theorem in the framework of $C^*$-algebras, we will arrive at a recent result of this type, without requiring that the isometries be linear or that the $C^*$-algebras be unital. This is a result from a joint work with C. Bénéteau, F. Botelho, M. Cueto Avellaneda, J. E. Guerra, S. Kazemi and S. Oi.
In this talk we establish new upper bounds for the norm of the sum of two Hilbert space operators and their Kronecker product. The obtained results extend some previously known results from the set of positive operators to arbitrary ones and refine several existing bounds. In particular, applications of the established bounds include refining celebrated numerical radius inequalities and the celebrated operator Cauchy-Schwarz norm inequality.
Sea transport handles more than 90% of global trade, with over 15% relying on container shipment. Consequently, container transport plays a crucial role in global trade. Containers await loading onto ships in container yards, where limited capacity often leads to stacking containers on top of each other. The sequence for loading these containers onto ships is usually unknown, making it impossible to prearrange them to avoid relocations when loading them. The Container Relocation Problem (CRP) is used to find the sequence for relocating containers to fulfill a defined order for loading while optimizing various criteria. Because CRP belongs to NP-hard problems, exact solutions to CRP are typically unattainable, which is why heuristic approaches are mainly used. Relocation Rules (RRs) are straightforward heuristic methods for CRP, offering speed and simplicity. However, creating RRs is usually a trial-and-error process for which domain expertise is needed. In this presentation, Genetic Programming (GP) will be used to automate this process and mitigate this challenge. Achieved results showed that GP-evolved RRs outperform manually designed rules and have strong generalization ability, making this approach a good choice for solving CRP.
The Art Gallery problem is a well-known combinatorial optimization problem in computational geometry that seeks to minimize the number of guards required to ensure visibility within a simple polygon. Specifically, the goal is to identify a minimal set of points (guards) within the polygon such that every internal point remains visible to at least one guard. This visibility criterion is established by line segments connecting each point to a guard, all contained within the polygon. In our study, we focus on the variant where guards are selected from the set of polygon vertices. However, finding the optimal guard configuration is NP-hard, necessitating the use of approximation algorithms. To address this problem, we propose a novel approach leveraging deep learning techniques. Specifically, we employ pointer networks, a specialized type of neural network designed for subset selection based on specific criteria.
Our empirical findings demonstrate that our approach achieves near-optimal guard counts for Art Gallery instances consistent with the model’s training data sizes. However, performance deteriorates when dealing with instance sizes larger than those encountered during training.
A computable metric space $(X, d, \alpha)$ is computably categorical if every two effective separating sequences in $(X, d)$ are equivalent up to isometry. It is known that computable metric space which is not effectively compact is not necessarily computably categorical.
We examine conditions under which an effectively compact metric space is computably categorical.
We show that every effectively compact metric space with locally Euclidean isometry group is computably categorical.
In this talk, we will present some previously unknown information on the life and work of prominent Croatian mathematician Stjepan Bohniček (Vinkovci, 1872 – Zagreb, 1956), corresponding member of the Croatian Academy of Sciences and Arts and the first Croatian expert in the field of number theory. Bohniček studied reciprocity laws, Diophantine equations and quadratic forms. His results on reciprocity laws are cited in several modern monographies on the history of number theory.
A translation curve in a Thurston space is a curve such that for given unit vector at the origin, translation of this vector is tangent to the curve in every point of the curve. In most Thurston spaces translation curves coincide with geodesic lines. However, this does not hold for Thurston spaces equipped with twisted product. In these spaces translation curves seem more intuitive and simpler than geodesics.
We present geodesics and translation curves in $\mathrm{Sol}_0^4$ space and explain the curvature properties of translation curves.
In 2001 Sir M. F. Atiyah formulated a conjecture $C1$ and later with P.Sutcliffe two stronger conjectures $C2$ and $C3$. These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for any configuration of $n$ points in the Euclidean three space. The conjecture $C1$ is proved for $n = 3$ in [1] and for $n=4$ in [2], and $C1-C3$ in [3]. After two decades we succeeded in verifying $C1$ for arbitrary five points in the Euclidean plane. The computer symbolic certificate produces a new remarkable universal ('hundred pages long') positive polynomial invariant (for any five planar points), in terms of newly discovered shear coordinates. This refines the original Atiyah’s conjecture and we are optimistic for its verification for $n$ greater than five (less optimistic variant $\ldots$ 'It remains a conjecture for $300$ years (like Fermat)', see Atiyah: Edinburgh Lectures..2010). In 2013.\ Atiyah's conjectures were put on the new list of nine open problems [4] (hopefully easier than remaining nine millennium problems!).
[1] M. Atiyah. The geometry of classical particles. Surveys in Differential Geometry (International Press) 7 (2001).
[2] M. Eastwood and P. Norbury, A proof of Atiyah’s conjecture on configurations of four points in Euclidean three-space. Geometry \& Topology 5 (2001), 885–893.
[3] D. Svrtan, A proof of All three Euclidean Four Point Atiyah-Sutcliffe Conjectures, https://emis.de/journals/SLC/wpapers/s73vortrag/svrtan.pdf
[4] Open problems in Honor of Wilfried Schmied\\
https://legacy-www.math.harvard.edu/conferences/schmid\_2013/problems/index.html\\
For a given irreducible and monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree $4$, we consider the quadratic twists by square-free integers $q$ of the genus one quartic ${H\, :\, y^2=f(x)}$
$H_q \, :\, qy^2=f(x)$.
Let $L$ denotes the set of positive square-free integers $q$ for which $H_q$ is everywhere locally solvable. For a real number $x$, let ${L(x)= \#\{q\in L:\, q \leq x\}}$ be the number of elements in $L$ that are less then or equal to $x$.
In this paper, we obtain that
$L(x) = c_f \frac{x}{(\ln{x})^{m}}+O\left(\frac{x}{(\ln{x})^\alpha}\right)$
for some constants $c_f>0$, $m$ and $\alpha$ only depending on $f$ such that $m<\alpha \leq 1+m$.
We also express the Dirichlet series $F(s)=\sum_{n \in L} n^{-s}$ associated to the set $L$ in terms of Dedekind zeta functions of certain number fields.
We determine all possible degrees of cyclic isogenies of elliptic curves with rational $j$-invariant defined over degree $d$ extensions of $\mathbb Q$ for $d=3,5,7$.
The same question for $d=1$ has been answered by Mazur and Kenku in 1978-1982, and Vukorepa answered the question for $d=2$. All possible prime degrees of isogenies were previously found by Najman.
We use well known results about images of Galois representations of elliptic curves, as the images of those representations are closely related to the existence of isogenies. The possible images of $p$-adic representations have mostly been determined for primes of our interest, and we use these results to reduce to a finite number of cases. The remaining cases are solved by finding all rational points on some modular curves, which is done with the assistance of the computer algebra system Magma.
Gonality of an algebraic curve is defined as the minimal degree of a nonconstant morphism from that curve to the projective space $\mathbb{P}^1$. In this talk, I will present the methods used to determine the $\mathbb{Q}$-gonality of the modular curve $X_0(N)$ and its quotients.
The Dynamic Mode Decomposition (DMD) is a tool of the trade in computational data driven analysis of complex dynamical systems, e.g. fluid flows, where it can be used to reveal coherent structures by decomposing the flow field into component fluid structures, called DMD modes, that describe the evolution of the flow. The theoretical underpinning of the DMD is the Koopman composition operator that can be used for a spectral analysis of nonlinear dynamical system.
One of the computational/numerical challenges in the Koopman/DMD framework is the case of non-normal operator, when the eigenvectors of the discretized problem become severely ill-conditioned. To alleviate the potential problem of ill-conditioned eigenvectors in the existing implementations of the Dynamic Mode Decomposition (DMD) and the Extended Dynamic Mode Decomposition (EDMD), we introduce a new theoretical and computational framework for a data driven Koopman mode analysis of nonlinear dynamics.
The new approach introduces a Koopman-Schur decomposition that is entirely based on unitary transformations. The analysis in terms of the eigenvectors as modes of a Koopman operator compression is replaced with a modal decomposition in terms of a flag of invariant subspaces that correspond to selected eigenvalues.
The main computational tool from the numerical linear algebra is the partial ordered Schur decomposition that provides convenient orthonormal bases for these subspaces. The new computational scheme is presented in the framework of the Extended DMD, with the same functionalities (snapshot reconstruction and forecasting) and the kernel trick is used.
We propose an effective numerical scheme involving deep learning to approximate solution to bilevel optimization problems of size that is considered computationally intractable using known approaches. The lower level is bypassed completely by training a deep neural network to approximate the relevant lower-level effect on the upper level. We illustrate this method on solving bilevel power system optimization problems, commonly arising after the deregulation of the power industry.
This is a joint work with Karlo Šepetanc and Hrvoje Pandžić.
Single-cell RNA-seq (scRNA-seq) produces a plethora of data from which one can derive information about gene expression levels for individual cells. In order to efficiently classify cells based on the tissues they originated from, it is crucial to identify and select informative genes is preserve the differences occurring between distinct cell types while excluding as much redundant information as possible. Finding such a subset is a computationally challenging combinatorial optimization problem in scRNA-seq data analysis. Several state-of-the-art methods tackle this issue in different ways. The aim of this study is to evaluate state-of-the-art marker gene selection methods, comparing their classification accuracy, running time, and memory consumption using real-world datasets. Additionally, we will modify one of the methods under consideration, scGeneFit, allowing it to achieve higher accuracy while having significantly lower running times. We will compare it to the original implementation and the remaining state-of-the-art methods.
Modular curves are moduli spaces of elliptic curves with prescribed images of their Galois representations. They are a key tool in studying torsion groups, isogenies and, more generally, Galois representations of elliptic curves. In recent years great progress, in many directions, has been made in our understanding of points on modular curves of low degree. In this talk I will describe some of these recent results, their consequences, as well as open problems in the field.
A Hadamard matrix of order $n$ is a $n \times n$ $(-1, 1)$-matrix $H$ such that $HH^{\top} = n I_{n}$. In this talk, we are concerned with constructing doubly even self-dual binary codes from Hadamard matrices. More precisely, to a Hadamard matrix of order $8t$ we relate a doubly even self-dual binary code of length $8t$, and give explicit constructions of doubly even self-dual binary codes from skew-type Hadamard matrices and conference graphs. It is known that a doubly even self-dual binary code yields an even unimodular lattice. Consequently, this construction of skew-type Hadamard matrices gives us a series of even unimodular lattices of rank $2^{i+2}t$, $i$ a positive integer.
References
We first outline the application of energy methods to analyzing dynamics of semilinear partial differential equations, including exciting connections to geometry, optimisation, theory of inequalities, and others.
We then focus on developing the theory for equations on unbounded domains, by addressing the challenge that the classical energy function is not well defined. We consider a family of 'stacked' dissipative structures and establish their abstract properties. We then prove new convergence and in some cases global existence of solutions results for a general class of reaction diffusion equations, and new convergence results for a Navier Stokes equation in 2d.
This is a joint work with Thierry Gallay.
In this talk, relying on Foster-Lyapunov drift conditions, we will discuss subexponential upper and lower bounds on the rate of convergence in the Lp-Wasserstein distance for a class of irreducible and aperiodic Markov processes. We will further discuss these results in the context of Markov Lévy-type processes. In the lack of irreducibility and/or aperiodicity properties, we will comment on exponential ergodicity in the Lp-Wasserstein distance for a class of Ito processes under an asymptotic flatness (uniform dissipativity) assumption. Lastly, applications of these results to specific processes will be presented, including Langevin tempered diffusion processes, piecewise Ornstein–Uhlenbeck processes with jumps under constant and stationary Markov controls, and backward recurrence time chains, for which we will provide a sharp characterization of the rate of convergence via matching upper and lower bounds.
By using the interpolation of Jensen's inequality and the integral representation of multivariate B-splines, an estimate of various moments of multivariate B-splines in the class of convex functions has been made.
The two-point Abel-Gontscharoff interpolation problem is a special case of the Abel-Gontscharoff interpolation problem introduced by Whittaker, Gontscharoff and Davis. Over the years, many generalizations of Steffensen's inequality and related identities connected to these generalizations have been established.
For the purpose of this talk, we will use the identities related to generalizations of Steffensen’s inequality via two-point Abel-Gontscharoff interpolation polynomials. By using these identities in the weighted Hermite-Hadamard inequality for (n+2)-convex function, we will prove new Steffensen-type inequalities.
The concept of strong F-convexity is a natural generalization of strong convexity. Although strongly concave functions are rarely mentioned and used, we show that in more effective and specific analysis this concept is very useful, and especially its generalization, namely strong F-concavity. Using this concept refinements of the Young inequality are given as a model case. A general form of self improving property for Jensen's type inequalities is presented. We show that a careful choice of control functions for convex or concave functions can give a control over these refinements and produce refinements of the power mean inequalities.
Genetic algorithms are search methods used in computing whose objective is to find exact or approximate solutions to optimization and search problems. A genetic algorithm mimics natural evolution, that is, it is based on optimizing a population (a subset of the entire search space). As in nature, the population consists of individuals that can reproduce and that can be affected by certain mutations, thus creating new individuals with better or worse properties than the previous ones. The goal of the algorithm is to direct the population towards creating better individuals, which can result in finding optimal solutions to a given problem.
In this talk, we will describe the use of a genetic algorithm for the construction of strongly regular graphs and directed strongly regular graphs from equitable partitions (i.e. orbit matrices) with a prescribed automorphism group.
So far, there are only four known Steiner 2-designs $S(2,6,91)$ which have been found by C.J.Colbourn, M.J.Colbourn and W.H.Mills. Each of them is cyclic, i.e. having a cyclic automorphism group acting transitively on points. For more than 30 years no results about that designs have been published, and the last one is from 1991, when Z.Janko and V.D.Tonchev showed that any point-transitive $2$--$(91,6,1)$ design with an automorphism group of order larger than $91$ is one of the four known designs.
In this talk, we show that there are exactly two Steiner $2$--designs $S(2,6,91)$ with a non-abelian automorphism group of order 26 (i.e. the Frobenius group $\textrm{Frob}_{26}$), and they are isomorphic to the already known designs. Still remains an open question whether there exists a $2$--$(91,6,1)$ design which is not cyclic.
The concept of higher-dimensional combinatorial designs was introduced by Warwick de Launey in 1990. Recently we have studied higher-dimensional incidence structures. Since symmetric designs have the same number of points and blocks, their incidence matrices can easily be superposed to get a 3- or higher-dimensional binary cube. We have focused on 3-dimensional cubes of symmetric $(v,k,\lambda)$ designs, hence 3-dimensional binary matrices of order $v$. They show to be equivalent to $(v,k,\lambda)$ difference sets by an easy-to-follow construction. Therefore, we tried and succeeded to manage to construct examples where the cube does not come from a difference set in this straightforward way. In addition, we have been successful in constructing cubes for which the 2-dimensional slices are incidence matrices of non-isomorphic designs. In this talk we will provide constructive examples and open questions on this topic.
In this talk, we study the Brocard triangles of a triangle in the isotropic plane. We present some statements about the first and the second Brocard triangle in the isotropic plane and consider the relationships between Brocard triangles and some other objects related to a triangle in the isotropic plane. We also investigate some interesting properties of these triangles and consider relationships between the Euclidean and the isotropic case.
Any triangle in an isotropic plane has a circumcircle $u$ and an incircle $i$. It turns out that there are infinitely many triangles with the same circumcircle $u$ and incircle $i$. This one-parameter family of triangles is called a poristic system of triangles.
We prove that all triangles in a poristic system share the centroid and the Feuerbach point. The symmedian point and the Gergonne point of the triangle $P_1P_2P_3$ move on the lines while the triangle traverses the poristic family. The Steiner point of $P_1P_2P_3$ traces a circle, and the Brocard points of $P_1P_2P_3$ trace a quartic curve.
We also study the traces of some further points associated with the triangles of the poristic family. The vertices of the tangential triangle move on a circle while the initial triangle traverses the poristic family, and the centroid of the tangential triangle is fixed.
In this talk, we observe a one-parameter triangle family,
where two vertices are fixed and the third vertex lies on
a given line. For this family of triangles, we observe the
loci of centroids, orthocenters, circumcenters, incenters,
excenters and some triangle elements associated to these
triangle points.
We describe refinements of Euler's inequality that the circumradius is at least n times larger than the inradius of an n-simplex T, provide explicit refinements for n = 2,3,4 and a recursive procedure for higher dimensions. We describe probability application in astrophysics. The final remarks are on Grace-Danielsson-Drozdev theorem (2024) on the upper bound of the distance between the circumcenter and incenter of spheres of the simplex (Egan's conjecture) and corresponding conjectures.
A set of $m$ distinct nonzero rationals $\{a_1, a_2, ... , a_m\}$ such that $a_ia_j + 1$ is a perfect square for all $1 \leq i < j \leq m$, is called a rational Diophantine $m$-tuple. If, in addition, $a_i^2 + 1$ is a perfect square for $1 \leq i \leq m$, then we say the $m$-tuple is strong. In this talk, we will describe a construction of infinite families of rational Diophantine sextuples containing a strong Diophantine pair. In particular, we will show that infinitely many rational Diophantine sextuples contain a strong Diophantine pair $\{30464/2223, 22815/5168\}$.
This is a joint work with Matija Kazalicki and Vinko Petričević.
A set of $m$ non-zero elements of a commutative ring $R$ with unity $1$ is called a $D(−1)$-$m$-tuple if the product of any two of its distinct elements decreased by $1$ is a square in $R$.
We investigate $D(−1)$-tuples in rings $\mathbb{Z}[\sqrt{-k}]$, k ≥ 2. We prove that, under certain technical conditions, there does not exist a $D(−1)$-quadruple of the form $\{a, 2^i
p^j,c,d\}$ in $\mathbb{Z}[\sqrt{-k}]$, with an odd prime $p$, positive integers $a,c,d,j$ and $i\in\{0,1\}$. The main tools in the proof are properties of the related (generalized) Pellian equations.
The set $\{a_1, a_2, \ldots , a_m\}$ in a commutative ring $R$ such that $a_i\ne 0$, $i=1,\ldots,m$ and $a_ia_j+n$ is a square in $R$ for all $1\le i< j\le m$ is called a Diophantine $D(n)$-$m$-tuple in the ring $R$.
Let $N$ be a positive integer such that $4N^2+1=q^j$, $q$ is a prime and $j$ is a positive integer. In this talk, we will discuss the extendibility of the Diophantine $D(-1)$-triple of the form $S_N=\{1,4N^2+1,1-N\}$. More precisely, we will show that the set $S_N$ cannot be extended to a $D(-1)$-quadruple in the ring $\mathbb{Z}[\sqrt{-N}]$, with a non-square integer $N$. If $N>1$ is a square, then the set $\{1,4N^2+1,1-N,1+N\}$ is a $D(-1)$-quadruple in the ring $\mathbb{Z}[\sqrt{-N}]$, so in the ring of the Gaussian integers as well.
Let $1< a < b < c$ be multiplicatively dependent integers (i.e., there exist nontrivial integer exponents $x, y, z$, such that $a^x b^y c^z = 1$). Is it possible that $a+1, b+1, c+1$ are multiplicatively dependent as well? It turns out that this is easy to answer. We will discuss related, more difficult questions, which will lead to Diophantine equations. We will solve some of them using lower bounds for linear forms in logarithms. Joint work with Volker Ziegler and work in progress.
In this talk, the starting point of our analysis is coupled system of elasticity and weakly compressible fluid. We consider two small parameters: the thickness $h$ of the thin plate and the pore scale $\varepsilon_h$ which depend on $h$. We will focus specifically on the case when the pore size is small relative to the thickness of the plate. The main goal here is derive a model for a poroelastic plate from the $3D$ problem as $h$ goes to zero using simultaneous homogenization and dimension reduction techniques. The obtained model generalizes the poroelastic plate model derived in ``A. Marcianiak-Czochra, A. Mikelić, A rigurous derivation of the equations for the clamped Biot-Kirchhoff-Love poroelastic plate, Arch. Rational Mech. Anal. 215 (2015), 1035-1062´´ by dimension reduction techniques from $3D$ Biot's equations.
Solving parameter-dependent Partial Differential Equations (PDE) for multiple parameters is often needed in physical, biomedical and engineering applications.Usually,PDEs are solved by transforming PDE to weak formulation. Usually, approximation of the solutions is found in finite-dimensional function space using Finite Element Method. This results in large number of equations and solving them form multiple parameters often becomes too prohibitive.
In order to mitigate these solutions, Reduced Order Models (ROM) is constructed. Classically, it is done by projecting solutions from Finite Element Method to lower-dimensional space. When PDE is linear, this results in smaller number of equations. However, classical ROM may be difficult to adapt to nonlinear methods.
To mitigate these difficulties, multiple techniques using Deep Learning for constructing ROMs are developed. This talk discusses our extension of encoder-decoder type Deep ROM for different domains. Our approach works when domains are given from measurements only, when we have it's exact parametrization and even when domain has varying number of components, or varying number of holes. We demonstrate proposed method on 2D problems. Furthermore, we will elaborate on possible extensions of our work.
In this talk, we present regularity results for entropy solutions to a family of second order degenerate (the diffusion matrix is only positive semi-definite) parabolic partial differential equations under a quantitative variant of the non-degeneracy condition. The proof is based on the kinetic reformulation, which allows for estimating the solution on the Littlewood-Paley dyadic blocks of the dual space. The results primarily refer to the homogeneous case. Additionally, we will provide some comments on the heterogeneous case as well. This is joint work with Darko Mitrović.
The theory of abstract Friedrichs operators, introduced by
Ern, Guermond and Caplain (2007), proved to be a successful setting
for studying positive symmetric systems of first order partial
differential equations (Friedrichs, 1958),
nowadays better known as Friedrichs systems.
Recently, a characterisation of abstract Friedrichs operators in terms of skew-symmetric operators and bounded selfadjoint operators has been established.
In this presentation we shall see the non-stationary theory of abstract Friedrichs operators along with the theory of skew-symmetric operators. We use the von Neumann extension theory for the connection between the theories of these two types of operators. A boundary quadruple/triplet approach has been used to study the semigroup theory.
This is a joint work with Marko Erceg funded by Croatian Science Foundation.
Random Projections have been widely used to generate embeddings for various large graph tasks due to their computational efficiency in estimating relevance between vertices. The majority of applications have been justified through the Johnson-Lindenstrauss Lemma. We take a step further and investigate how well dot product and cosine similarity are preserved by Random Projections. Our analysis provides new theoretical results, identifies pathological cases, and tests them with numerical experiments. We find that, for nodes of lower or higher degrees, the method produces especially unreliable embeddings for the dot product, regardless of whether the adjacency or the transition (normalized version) is used. With respect to the noise introduced by Random Projections, we show that cosine similarity produces remarkably more precise approximations. This work builds on many experiments the Graph Intelligence Sciences team at Microsoft did to compute relevance between entities (email, documents, people, events, ...) in Office 365. Joint work with Cassiano Becker and Jennifer Neville.
In case of an equidistant sampling of an ergodic diffusion path such that
the maximal time of observation tends to infinity and size of time interval subdivision tends to zero, we have investigated an asymptotic normality of the difference between approximate maximum likelihood estimator (AMLE) and maximum likelihood estimator based on continuous observation (MLE).
We use this property for estimating AMLE’s standard errors that accounts
effect of discretization.
The concept of regular variation plays pivotal role in understanding the extreme behavior of stochastic processes. It also finds applications in various areas ranging from random networks to stochastic geometry. We discuss some developments in this field and show how one can generalize regular variation to rather abstract settings, provided compatible notions of scaling and boundedness can be introduced. Based on the joint work with Nikolina Milinčević, Ilya Molchanov, and Hrvoje Planinić.
In this talk I will discuss the potential theory of Dirichlet forms on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-\alpha}\mathcal{B}(x,y)$ and the killing potential $\kappa x_d^{-\alpha}$, where $\alpha\in (0, 2)$ and $\mathcal{B}(x,y)$ can blow up to infinity at the boundary. The jump kernel and the killing potential depend on several parameters. For all admissible values of the parameters involved and all $d \ge 1$, we prove that the boundary Harnack principle holds, and establish sharp two-sided estimates on the Green functions of these processes.
In this talk we shall review the definition and use of Eisenstein series in various cases in the theory of automorphic forms and in the local representation theory. We shall give examples from the seminal classical results for the general linear and classical groups and also some of the authors' results in the case of general linear, classical and exceptional groups. Some of these results are obtained in the joint works with G. Muić and G. Savin.
A clique is a fully connected subset of an undirected graph. Finding the largest clique is NP complete problem (it takes exponential time to solve the problem).
However, if we know that the largest clique is not too large, then it is a polynomial problem. If the graph is also very sparse, then it does not have to be a polynomial of high degree, and it can be solved up to some predetermined limits.
Let's look at an ancient problem: If in the set of numbers $\{1,3,8\}$ we multiply each of its two different elements and add the number $1$ to the product, the result will be the square of a number: $$1\cdot3+1=2^2,\qquad1\cdot8+1=3^2,\qquad3\cdot8+1=5^2.$$
The curiosity of mathematicians has been tickled by similar sets for millennia, so solving such problems, the coming centuries gave birth to many new mathematical disciplines. With the advent of computers and supercomputers, many new interesting sets have been found (and are still being found).
We use the one-point integral formula to obtain identities that are related to the classical Steffensen inequality. We give some new weight inequalities of the Hermite-Hadamard type for these identities. At the end, we present Hermite-Hadamard type bounds for the obtained identities by applying convex/concave functions of the form $|f^{(n)}|^q$.
Let $n\neq0$ be an integer. A set of $m$ distinct positive integers is called a $D(n)$-$m$-tuple if the product of any two of its distinct elements increased by $n$ is a perfect square. We present known results and mention open problems and conjectures related to $D(4)$-$m$-tuples. Also, the latest results from $[1]$ are shown.
References:
$[1]$ Bliznac Trebješanin, Marija; Radić, Pavao,
On extensions of D(4)-triples by adjoining smaller elements, Publicationes mathematicae, (2025)
The classical Ostrowski inequality states:
$
\left\vert f(x)-\frac{1}{b-a}\int_{a}^{b}f(t)dt\right\vert \leq\left[
\frac{1}{4}+\frac{\left( x-\frac{a+b}{2}\right) ^{2}}{\left( b-a\right)
^{2}}\right] \left( b-a\right) \left\Vert f^{\prime}\right\Vert _{\infty
},
$
for all $x \in \left[a,b\right]$, where $f:\left[a,b\right]\to \mathbb{R}$ is continuous on $\left[a,b\right]$ and differentiable on $\left( a,b \right)$ with bounded derivative. Ostrowski type inequalities have been largely investigated in the literature since they are very useful in numerical analysis and probability theory.
The main purpose of this talk is to present new Ostrowski type inequalities for $3$-convex functions and for functions whose modulus of derivatives are convex, using the weighted Montgomery identity. We also derive certain Hermite-Hadamard inequalities for $3$-convex functions by applying those results.
Gyroscopic systems are mechanical systems described by the equation:
$$M \ddot x(t) + G\dot x(t) + K x(t) = 0,$$
where the mass matrix $M\in\mathbb{R}^{n\times n}$ is symmetric positive definite, the gyroscopic matrix $G \in\mathbb{R}^{n\times n}$ is skew-symmetric, the stiffness matrix $K\in\mathbb{R}^{n\times n}$ is symmetric, and $x=x(t)$ is a time-dependent displacement vector. The properties of above system are determined by the algebraic properties of the quadratic matrix polynomial:
$$
\mathcal{G}(\lambda ) = \lambda^2 M + \lambda G + K,$$
and the corresponding quadratic eigenvalue problem:
$$
\mathcal{G}(\lambda)x=(\lambda^2 M+\lambda G+K)x=0, \quad x\in\mathbb{C}^{n}, x\ne 0. $$
Perturbation bounds for mechanical systems are crucial for understanding the stability and behaviour of these systems under external disturbances or variations in parameter values. Stability in this context implies that all eigenvalues of the system are purely imaginary and semi-simple. We present an upper bound for the relative change in eigenvalues, as well as a $\sin\Theta$ type bound for the corresponding eigenvectors for stable gyroscopic systems under perturbations of the system matrices. The case when $K$ is positive definite is treated separately from the case when it is negative definite. To demonstrate the effectiveness of the obtained bounds, we illustrate their performance through numerical experiments.
We propose a fractional diffusion process based on the (non-fractional) Bessel process with constant negative drift. The model is obtained as stochastically time-changed Bessel process with constant negative drift through inverse stable subordinator of order $0<\alpha <1$. Spectral representation of the transition density of fractional Bessel process is calculated. Based on this representation, we are able to provide explicit fractional representation of the model via time-fractional backward Kolmogorov equation. Moreover, we provide the corresponding stationary distribution, discuss the long-range dependence property and solve fractional Cauchy problems involving the generator of the process.
We use the tools of topological data analysis to detect some features of random sets in order to detect outliers or do goodness of fit testing. Persistence diagram is a key object of our interest and we view it as an empirical measure. Statistical depth can be defined on that (random) measure, for example by using support functions of corresponding lift zonoid and applying methods for assigning depth to functional data. We also use the persistence diagram for testing goodness of fit to Boolean model using so called accumulated persistence functions as test functions for global envelope test and compare the results with other test functions such as capacity functional, support function of the lift zonoid and spherical contact distribution function. It seems like it is very useful tool in recognizing whether clustering or repulsiveness occurs and can be used for detecting outliers.
The topic of finding point configurations in large subsets of the Euclidean space lies at the intersection of combinatorics, geometry, and analysis. Euclidean Ramsey theory tries to identify patterns that are present in every finite coloring of the space, while a part of geometric measure theory studies patterns in sets of positive measure or positive density (as opposed to lower-dimensional sets or fractals). The two mathematical branches share many ideas, and they often successfully apply techniques from Fourier analysis and its generalizations. Consequently, many questions that have been posed in the 1980s, can now be fully or partially resolved using analytical tools that have been developed in the meantime. We will give a brief overview of the topic and then present some recent results and open problems.
A transversal in a $n \times n$ latin square is a set of $n$ entries not repeating any row, column, or symbol. A famous conjecture of Brualdi, Ryser, and Stein predicts that every latin square has at least one transversal provided $n$ is odd. We will discuss an approach motivated by the circle method from the analytic number theory which enables us to count transversals in latin squares which are quasirandom in an appropriate sense.
Understanding the structure of parabolically induced representations is one of the most important tasks in the representation theory of classical p-adic groups. We will discuss recent results on this topic, and provide an overview of methods related to the determination of reducibility and identification of composition factors of representations obtain by the parabolic induction from particular representations of the maximal Levi subgroup.
The concepts of jet scheme and arc space over an algebraic variety were introduced by John Nash in his 1968 preprint on singularities. In the last two decades, many exciting new discoveries have connected arc and jet algebras with the theory of partitions, modular forms, and algebraic geometry. Arc algebras/spaces have recently acquired increased interest within the field of vertex algebra, primarily due to their significance in the context of 4d/2d dualities in physics.
In my talk, we will focus on n-jet algebras and arc algebras that are relevant to representations of vertex algebras. We also explain how approaches rooted in vertex (super)algebras, particularly through the notion of "classical freeness" of vertex algebra and modules, offer effective solutions to the problem of determining Hilbert series of arc algebras in some cases.
Several parts of this talk are based on a joint work with H. Li.
Green functions are very interesting from different aspects and are used
for solving wide variety of problems in many fields, specifically in
quantum field theory, aerodynamics, aeroacoustics, electrodynamics and
seismology. Considering Green functions, we obtain new results on the
Hardy-type inequality in the general context, in terms of measure spaces
with positive $\sigma$-finite measures. We investigate the difference
operator derived from the Hardy-type inequality on the one hand and the
expression containing the interpolating polynomial of Abel-Gontscharoff
and the four Green functions on the other hand and make connections
between them. We discuss the n-convexity of the function and consider
the result depending on the parity of the indexes. Further, we present
results obtained by using the H\" older inequality for conjugate
exponents $p$ and $q$. Finally, we conclude with upper bounds for the
remainder, obtained from the main result, using the Čebyšev
functional.
We study the properties of a complete quadrangle in the Euclidean plane. Many of them are known from earlier, published in different journals and periods and proved each using different methods. Hereby, we use rectangular coordinates symmetrically on four vertices and four parameters $a, b, c, d$ and prove all properties by the same analytical method. We put the complete quadrangle into such a coordinate system that its circumscribed hyperbola is rectangular. This is possible for each quadrangle for which the opposite sides are not perpendicular. We are focused on the properties of a complete quadrangle related to the center and anticenter of the quadrangle where the center of the quadrangle is the center of its circumscribed rectangular hyperbola and the anticenter of the quadrangle is the point symmetric to the center with respect to the centroid of the quadrangle. In this procedure, we obtain some new results as well.
We study 3-convex functions, which are characterized by the third order divided differences, and for them we derive a class of inequalities of the Jensen and Edmundson-Lah-Ribarič type involving positive linear functionals that does not require convexity in the classical sense. A
great number of theoretic divergences, i.e. measures of distance between two probability distributions, are special cases of Csiszár f-divergence for different choices of the generating function f. We apply our results to the generalized f-divergence functional in order to obtain some lower and upper bounds.
A generalized helix is a space curve whose tangent vectors make a constant angle with a fixed straight line, called the axis of a generalized helix. Among such curves, the ones that lay on a sphere show interesting geometric properties.
In Euclidean space, spherical generalized helices have a property that their orthogonal projections onto a plane normal to their axis appear as epicycloids, what contributes to the widespread of epicycloids in physics, as well as in robotics. Therefore, we were motivated to consider spherical generalized helices in 3-dimensional Lorentz-Minkowski space, the ambient space for general relativity theory.
We provide their characterizations in terms of curvature and torsion and analyze their projections onto planes orthogonal to their axes.
These projections appear as Euclidean or Lorentzian cycloidal curves, so we also introduce natural equations and parametrizations of Lorentzian cycloidal curves.
We assume that the one-dimensional diffusion $X$ satisfies a stochastic differential equation of the form:
$dX_t=\mu(X_t)dt+\nu(X_t)dW_t$, $X_0=x_0$, $t\geq 0$.
Let $(X_{i\Delta_n},0\leq i\leq n)$ be discrete observations along fixed time interval $[0,T]$. We prove that the random vectors which $j$-th component is $\frac{1}{\sqrt{\Delta_n}}\sum_{i=1}^n\int_{t_{i-1}}^{t_i}g_j(X_s)(f_j(X_s)-f_j(X_{t_{i-1}}))dW_s$, for $j=1,\dots,d$, converge stably in law to mixed normal random vector with covariance matrix which depends on path $(X_t,0\leq t\leq T)$, when $n\to\infty$. We use this result to prove stable convergence in law for $\frac{1}{\sqrt{\Delta_n}}(\int_0^Tf(X_s)dX_s-\sum_{i=1}^nf(X_{t_{i-1}})(X_{t_i}-X_{t_{i-1}}))$.
The Nuttall function has a wide range of application, mostly in the communication theory and related areas. We obtain new integral and series representation formulas for the Nuttall function via hypergeometric and related special functions and derive the closed form formula in terms of upper incomplete gamma function which simplifies some known results. We also present the computational efficiency of the new derived formulas and compare them to already known results.
For three homogeneous symmetric bivariate means $K$, $M$, $N$, let
$$
\mathcal{R}(K,M,N)(s,t)=K\bigl(M\bigl(s,N(s,t)\bigr),M\bigl(N(s,t),t\bigr) \bigr)
$$ be their resultant mean--map. Mean $M$ is said to be stable (or balanced), if $\mathcal{R}(M,M,M)=M$. Mean $M$ is called $(K,N)$-sub(super)-stabilizable, if $\mathcal{R}(K,M,N)\le(\ge)\, M$, and $M$ is between $K$ and $N$, where $K$ and $N$ are two non-trivial stable comparable means.
We present some results on $(K,N)$-sub/super-stabilizability where $K$ and $N$ belong to the class of power means, denoted by $B_p$, and $M$ is one of the classical or recently studied new means. Assuming that means $K$, $M$ and $N$ have asymptotic expansions, we present the complete asymptotic expansion
$$
R(x-t,x+t)= \mathcal{R}(K,M,N)(x-t,x+t)\sim\sum_{m=0}^\infty a_m^R t^{m} x^{-m+1},\quad x\to\infty.
$$
As an application of the obtained asymptotic expansions and the asymptotic inequality between $M$ and $\mathcal{R}(B_p,M,B_q)$, we show how to find the optimal parameters $p$ and $q$ for which $M$ is $(B_p,B_q)$ sub(super)-stabilizable.
The constructions of self-orthogonal codes from orbit matrices of $2$-designs has been extensively studied. In this talk we present new constructions of self-orthogonal codes from orbit matrices of $2$-designs for the cases not covered previously. We apply this construction on orbit matrices of $2$-$(1024, 496, 249)$ and $2$-$(45, 5, 1)$ designs and obtain some optimal self-orthogonal codes.
The doubling method is a method for constructing Type II $\mathbb{Z}_4$-codes from a given Type II $\mathbb{Z}_4$-code.
Extremal Type II $\mathbb{Z}_4$-codes are a class of self-dual $\mathbb{Z}_4$-codes with Euclidean weights divisible by eight and the largest possible minimum Euclidean weight for a given length. A small number of such codes is known for lengths greater than or equal to $48.$
The subject of this talk is a construction of new extremal Type II $\mathbb{Z}_4$-codes of length $64$ by the doubling method.
We develop a method to construct new extremal Type II $\mathbb{Z}_4$-codes starting from an extremal Type II $\mathbb{Z}_4$-code of type $4^k$ with an extremal residue code and length $48, 56$ or $64$.
Using this method, we construct three new extremal Type II $\mathbb{Z}_4$-codes of length $64$ and type $4^{31}2^2$. Extremal Type II $\mathbb{Z}_4$-codes of length $64$ of this type were not known before.
In this talk we will describe binary LDPC LCD codes spanned by the adjacency matrices of the odd graphs as their parity-check matrices. For the odd graph $O_n$ ($n\geq 3$), the obtained code $C_n$ is an $(n,n)$-regular binary LDPC code of length ${2n-1 \choose n-1}$, dimension ${2n-2 \choose n-2}$, minimum distance $n+1$ and girth equal to $6$, which is also an LCD code (i.e. $C_n\cap C_n^{\bot}=\{0\}$).
Construction and classification of self-orthogonal and self-dual codes is an active field of research. A code for which all codewords have weight divisible by four is called doubly even. Among self-orthogonal, especially self-dual codes, doubly even codes attract special attention. In this talk, we are dealing with some constructions of doubly even self-orthogonal linear codes from incidence matrices and orbit matrices of quasi-symmetric designs. Quasi-symmetric design meaning that this design has exactly two block intersection numbers. In particular, we consider constructions from quasi-symmetric designs of Blokhuis-Haemers type.
Topology can play a surprisingly important role in determining the relationship between different aspects of computability of sets in computable metric spaces. In particular, semicomputable sets with certain topological properties will automatically be fully computable. To express this property, we use the notion of computable type: a space $A$ is said to have computable type if every semicomputable set homeomorphic to $A$ must be computable.
The study of computable type is usually restricted to compact spaces, most notably compact manifolds and simplicial complexes. However, a more general approach can yield similar results for non-compact spaces. Some examples of non-compact spaces with computable type include 1-manifolds and generalized graphs (i.e. graphs with potentialy "infinite edges"), as well as certain (very specific) manifolds of arbitrary dimension.
In this talk, we begin by contrasting two different techniques used to study computable type of non-compact spaces. We focus on the notion of pseudocompactification and show how it can be utilized to obtain more general results than those known so far. In particular, we prove that the infinite cylinder $\mathbb S^1 \times \mathbb R$ (and $\mathbb S^n \times \mathbb R$ in general) has computable type.
This work is part of the research programme that is trying to determine
which conditions render semicomputable subsets of computable metric
spaces computable. In particular, we study generalized topological
graphs, which are obtained by gluing arcs and rays together at their
endpoints. We prove that every semicomputable generalized graph in
a computable metric space can be approximated, with arbitrary
precision, by a computable subgraph with computable endpoints.
We examine conditions under which a semicomputable set is computable. It is known that a semicomputable continuum which is chainable from $a$ to $b$ is computable if $a$ and $b$ are computable points. We generalize this result by showing that a semicomputable continuum which is irreducible from $a$ to $b$ is computable if $a$ and $b$ are computable points. We also examine conditions under which a semicomputable irreducible continuum (that is not necessarily computable) contains a computable point.
We examine conditions under which a metric basis of a computable metric space has to be computable. We focus mainly on spaces which have one-point metric bases. We prove that in an effectively compact metric space with finitely many connected components any one-point metric basis is computable.
We study the motion of a rigid body within a compressible, isentropic, and viscous fluid contained in a fixed bounded domain $\Omega \subset \mathbb{R}^3$. The fluid's behavior is described by the Navier-Stokes equations, while the motion of the rigid body is governed by ordinary differential equations representing the conservation of linear and angular momentum. We prescribe a time-independent fluid velocity along the boundary of $\Omega$ and a time-independent fluid density at the inflow boundary of $\Omega$. Additionally, we assume a no-slip boundary condition at the interface between the fluid and the rigid body. Our goal is to establish the existence of a weak solution to the given problem within a time interval where the rigid body does not touch the boundary $\partial\Omega$.
The fluid temperature in a heat conduction problem in a dilated pipe with a small circular cross-section is considered. The fluid flow is governed by the pressure drop. The heat exchange between the fluid and the environment is described by Newton’s cooling law and the temperature is described by the convection-diffusion equation with a stationary Poiseuille velocity. Due to pipe dilation, the fluid domain is not fixed and changes depending on the unknown temperature. By introducing a suitable change of variables, the domain becomes fixed, but the PDE becomes non-linear.
An asymptotic temperature is obtained by asymptotic analysis with respect to the small parameters (coefficient of heat expansion and ratio between pipe thickness and length). The approximation is first defined on a fixed domain (defined by the change of variables) and then on a dilated domain. By proving the error estimate for the approximation on the extended domain, the justification of the effective model is given.
We consider a system of nonlinear partial differential equations that describes two-phase, two-component fluid flow in porous media. The system equations are obtained from the mass conservation law written for each component, to which initial and boundary conditions are added. For main unknows we select persistent variables, capable of modelling the flow in both the two-phase flow and the one-phase flow regions. We use an artificial persistent variable called global pressure and then rewrite this system in terms of global pressure and gas phase pseudo-pressure. As the obtained system contains degeneracy, we regularize it by a small parameter $\eta$. Furthermore, we apply time discretization, and introduce another persistent variable, called capillary pseudo-pressure. Since the global pressure partially decouples equations, we apply Schauder fixed point theorem easily and obtain the existence of solution at discrete time level. By passing to the limit in the time discretization parameter and afterwards in regularization parameter $\eta$, we prove the existence of weak solutions of the introduced initial-boundary value problem.
We study the system of partial differential equations describing a two-phase two-component flow in heterogeneous porous media. The unknowns of the problem are global pressure, gas pseudo-pressure and capillary pseudo-pressure, artificial persistent variables that describe the flow both in the two-phase and single-phase regions and at the same time allow to decouple the starting equations. Under some realistic assumptions on the data, we rigorously obtain a nonlinear homogenized problem by using the two-scale convergence.
There is the conjecture stating that the Aubert involution preserves unitarity. After finding the unitary dual of $p$-adic group $SO(7)$ with support on minimal parabolic subgroup, we found it intriguing to look at all the Aubert duals of all irreducible unitarizable subquotients that form that unitary dual. In doing that we confirmed the aforementioned conjecture for this case. This work is supported (in part) by the Croatian Science Foundation under the project number HRZZ-IP-2022-10-4615.
We consider composition series of representations of a classical group over p-adic field, induced from two irreducible representations of GL, attached to a certain class of segments, and a cuspidal representation of a smaller classical group, where cuspidal reducibility is one half.
We start with the basic notions of a hyperplane arrangement on $\mathbb{R}^n$ and then explain the braid arrangement on $\mathbb{R}^n$, which consists of $\frac{n(n-1)}{2}$ hyperplanes $H_{ij}=\{(x_1,x_2,\dots, x_n)\in \mathbb{R}^n\mid x_i=x_j\}$, $1\le i < j\le n$. Each region $P_\sigma$ of this arrangement is in one-to-one correspondence with a permutation $\sigma$ as follows
\begin{align}
P_{\sigma} =\{(x_1,x_2,\dots, x_n)\in \mathbb{R}^n\mid x_{\sigma_1}<x_{\sigma_2}<\dots < x_{\sigma_n} \}.
\end{align}
If we introduce the orientation of the braid arrangement, we obtain the oriented braid arrangement on $\mathbb{R}^n$ consisting of $n(n-1)$ open half-spaces
\begin{equation}
H_{ij}^+=\{(x_1,x_2,\dots, x_n)\in \mathbb{R}^ n\mid x_i>x_j\},
\end{equation}
\begin{equation}
H_{ij}^-=\{(x_1,x_2,\dots, x_n)\in \mathbb{R}^ n\mid x_i<x_j\}
\end{equation}
for all ${1\le i < j\le n}.$
Then to every open half-space ${H_{ij}^+}$ we associate a weight $q_{ij}$ and similarly to every open half-space ${H_{ij}^-}$ we associate a weight $q_{ji}$ in the polynomial ring in complex variables $q_{ij}$, ${1\le i\ne j\le n}$.
The quantum bilinear form $\mathit{B}^*_n$ is defined on the regions of that arrangement by
$$\mathit{B}^*_n(P_{\sigma},P_{\tau})=\prod_{(a,b)\in I(\tau^{-1}\sigma)} q_{\sigma(a)\sigma(b)}$$
with $I(\tau^{-1}\sigma)={(a,b)\mid a<b, \ \tau^{-1}\sigma(a)>\tau^{-1}\sigma(b) }$.
The matrix $${B}^*_n=\left({B}^*_n(P_{\sigma},P_{\tau})_{\sigma,\tau\in S_n}\right)$$ is known as the Varchenko matrix of the oriented braid arrangement.
To calculate the inverse of the matrix $${B}^*_n,$$ we need to use some special matrices and their factorizations in the form of simpler matrices. To simplify the calculation of the matrices, we first introduce a twisted group algebra $\mathcal{A}(S_{n}) = R_{n}\rtimes {\mathbb{C}}[S_n]$ of the symmetric group $S_{n}$ with coefficients in the polynomial ring $R_{n}$ of all polynomials in $n^2$ variables $X_{a\,b}$ over the set of complex numbers and then use a natural representation of some elements of the algebra ${\mathcal{A}(S_{n})}$ on the generic weight subspaces of the multiparametric quon algebra ${\mathcal{B}}$. In this way, we directly obtain the corresponding matrices of the quantum bilinear form.
We present the exact realization of the extended Snyder model.
Using similarity transformations we construct realizations of the original Snyder and the extended Snyder models.
Finally, we present the exact new realization of the $\kappa$–deformed extended Snyder model.
In topological dimension theory, a well known Hurewicz theorem for dimension-lowering maps states that if $f:X\to Y$ is a closed map of metric spaces, then
$\dim X \leq \dim Y + \dim (f)$, where $\dim(f) := \sup\ \{ \dim(f^{-1}(y))\ | \ y\in Y\}$. This theorem was extended to asymptotic dimension $\mathrm{asdim}$, and in particular to $\mathrm{asdim}$ of groups - in 2006, Dranishnikov and Smith proved a Hurewicz-type formula, which states that if $f:G\to H$ is a group homomorphism, then $\mathrm{asdim} \ G\leq \mathrm{asdim}\ H + \mathrm{asdim}\ \mathrm{ker}(f)$.
We will show that the analogous formula is true for countable approximate groups, i.e., we will present the proof of the following:
Theorem: Let $(\Xi, \Xi^\infty)$, $(\Lambda, \Lambda^\infty)$ be countable approximate groups and let $f: (\Xi, \Xi^\infty) \to (\Lambda, \Lambda^\infty)$ be a global morphism. Then
$ \mathrm{asdim}\ \Xi \leq \mathrm{asdim} \ \Lambda + \mathrm{asdim}\ ([\mathrm{ker}(f)]_c).$
References:
[1] N. Brodskiy, J. Dydak, M. Levin, and A. Mitra, A Hurewicz theorem for the Assouad-Nagata dimension, J. Lond. Math. Soc. (2), 77(3):741--756, 2008.
[2] M. Cordes, T. Hartnick and V. Tonić, Foundations of geometric approximate group theory, preprint, 2024, https://arxiv.org/pdf/2012.15303.pdf
[3] A. Dranishnikov and J. Smith, Asymptotic dimension of discrete groups,
Fund. Math. 189(1):27--34, 2006.
[4] T. Hartnick and V. Tonić, Hurewicz and Dranishnikov-Smith theorems for asymptotic dimension of countable approximate groups, preprint, 2024.
For a given topological dynamical system $(X,f)$, where $X$ is a non-empty compact metric space and $f : X \to X $ a continuous function, we define an equivalence relation on $X$ and study quotients of dynamical systems. Using those results we produce on the Lelek fan and the Cantor fan a chaotic and mixing homeomorphism as well as a chaotic and mixing mapping, which is not a homeomorphism.
This is joined work with Iztok Banič, Judy Kennedy, Chris Mouron and Van Nall.
In this talk we discuss the specification property from a topological point of view. We show that dynamical system $(X,f)$, where $f$ is a surjective mapping, has specification property if and only if dynamical system $(\underset{\leftarrow}{\lim}(X,f),\sigma)$ has specification property. Of particular interest to us is the application of the obtained results to fans (e.g. Lelek fan).
In this talk we show that a topological pair of a chainable graph and the set of its endpoints has computable type.
The notion of a chainable graph is inspired by the notion of a graph - a set which consists of finitely many arcs such that distinct arcs intersect in at most one endpoint. It is known that if $G$ is a graph and $E$ set of all endpoints of $G$, that then $(G,E)$ has computable type.
We define the following: Suppose $A$ is a topological space. Let $V \subseteq A$ be a finite subset of $A$ and let $\mathcal{K}$ be a finite family of pairs $(K, \{a,b\})$ where $a,b \in V$, $a \neq b$ and $K \subseteq A$ is a continuum chainable from $a$ to $b$.
Suppose
$$A = V \cup \bigcup_{(K,\{a,b\}) \in \mathcal{K}}K$$
and that the following holds:
if $(K, {a,b})$, $(L,{c,d}) \in \mathcal{K}$ and $K \neq L$, then $\operatorname{card} (K \cap L) < \aleph_0$.
Then the triple $(A, \mathcal{K}, V)$ is called a **chainable graph.** We say that $a \in V $ is an **endpoint** of $(A, \mathcal{K}, V)$ if there exist only one $K \subseteq A$ and at least one $b \in V$ such that $(K, {a,b}) \in \mathcal{K}$.
For such an object, it holds:
If $(A,\mathcal{K},V)$ is a chainable graph and $B$ is the set of all its endpoints, then $(A,B)$ has computable type.
Local order isomorphisms of matrix and operator domains will be discussed. A connection with Loewner's theorem and the fundamental theorem of chronogeometry will be explained. The first one characterizes operator monotone functions while the second one describes the general form of bijective preservers of light-likeness on the classical Minkowski space.