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...
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).
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...
There are many methods used to study the asymptotic behavior of nonautonomous systems, but one of the most famous is admissibility method. For an arbitrary noninvertible evolution family on the half-line and for $\rho:\left[ 0,\infty \right) \mapsto \left[ 0,\infty \right)$
in a large class of rate functions, we consider the notion of a $\rho$
-dichotomy with respect to a family of norms and...
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...
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,...
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...
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...
We derive asymptotic expansions of the Gaussian compound mean $M \otimes_g N$ and the Archimedean compound mean $M \otimes_a N$, where $M$ and $N$ are arbitrary non-symmetric means which possess asymptotic expansions in terms of negative powers. We present applications to some classical means, such as neo-Pythagorean means, and analyze the existence of means with the same Gaussian and...
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...
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...
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...
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....
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...
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...
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...
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,...
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....
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...
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...
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...
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...
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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,...
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.
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.
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...
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....
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...
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...
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.
We study the spectral problems for integral operators with $L^1$ integrabile kernel with degenerate coefficients. The kernel, besides being degenerate, is becoming local as small parameter $\varepsilon$ tends to zero. We discuss the limit operator in the sense of two-scale resolvent convergence and its spectrum as well as the limiting spectra of the problems as $\varepsilon$ tends to zero,...
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...
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....
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...
Extensions of Stolarsky's and Pinelis' inequalities are considered. New results of that type for q-integrals are obtained.
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...
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 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...
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.
In their recent work, I. Banič and A. Taranenko introduced a new notion, the span of a graph. Their goal was to solve the problem of keeping the safety distance while two players are moving through a graph. They presented three different types of graph spans, depending on the movement rules and relating them to the strong, direct and Cartesian graph products. We observed the same goal, but...
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...
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...
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...
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...
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...
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.
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...
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...
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...
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...
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.
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...
Molecular descriptor is a graph-theoretical invariant (value assigned to a graph that is invariant to isomorphism). There are thousands of molecular descriptors that are of interest to mathematics and chemistry. They have been used to predict the properties of different chemical compounds even before such compounds are synthesized (in so-called “in silico” experiments). In order to do so, it...
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...
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...
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 $|f^{(n)}|^q$.
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...
Let $\widehat{\mathfrak g}$ be an affine Lie algebra of type $C_\ell^{(1)}$, $\ell\geq2$. In [CMPP] it is conjectured that for every integrable highest weight $\widehat{\mathfrak g}$-module $L(k_0,\dots,k_\ell)$ there is a Rogers-Ramanujan type combinatorial identity, that is, the infinite product $\prod_{k_0,\dots,k_\ell}(q)$ obtained from the Weyl-Kac character formula is the generating...
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...
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...
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...
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 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...
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...
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...
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...
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...
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...
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...
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 $\mathbb{Z}[X]$,...
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...
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...
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...
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...
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...
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...
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,...
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...
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...
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...
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.
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...
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...
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...
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...
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...
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{\bf\Large READING MULTIPLICITY IN UNFOLDINGS
FROM $\varepsilon$-NEIGHBORHOODS OF ORBITS
...
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.
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...
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...
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...
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...
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 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...
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...
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 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...
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...
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...
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,...
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...
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,...
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...
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...
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...
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, $C(\Omega_1)$...
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...
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...
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...
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 ...
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...
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.
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...
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...
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...
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...
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 $...
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...
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...
