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t-wise Berge and t-heavy hypergraphsFeb 08 2019In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A hypergraph $\mathcal{H}$ ... More

Jones representations of Thompson's group $F$ arising from Temperley-Lieb-Jones algebrasJan 29 2019Following a procedure due to V. Jones, using suitably normalized elements in a Temperley-Lieb-Jones (planar) algebra we introduce a 3-parametric family of unitary representations of the Thompson's group $F$ equipped with canonical (vacuum) vectors and ... More

Polynomial to exponential transition in Ramsey theoryJan 17 2019Given $s \ge k\ge 3$, let $h^{(k)}(s)$ be the minimum $t$ such that there exist arbitrarily large $k$-uniform hypergraphs $H$ whose independence number is at most polylogarithmic in the number of vertices and in which every $s$ vertices span at most $t$ ... More

Algorithms for $\ell_p$-based semi-supervised learning on graphsJan 15 2019We develop fast algorithms for solving the variational and game-theoretic $p$-Laplace equations on weighted graphs for $p>2$. The graph $p$-Laplacian for $p>2$ has been proposed recently as a replacement for the standard ($p=2$) graph Laplacian in semi-supervised ... More

Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learningDec 31 2018Jan 29 2019The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group-equivariant non-expansive ... More

Isogeometric Mortar Coupling for Electromagnetic ProblemsDec 27 2018This paper discusses and analyses two domain decomposition approaches for electromagnetic problems that allow the combination of domains discretised by either N\'ed\'elec-type polynomial finite elements or spline-based isogeometric analysis. The first ... More

On ultrafilter extensions of first-order models and ultrafilter interpretationsDec 15 2018There exist two known canonical concepts of ultrafilter extensions of first-order models; one comes from modal logic and universal algebra, another one from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups as its main ... More

Sensitivity-driven adaptive sparse stochastic approximations in plasma microinstability analysisNov 30 2018Quantifying uncertainty in predictive simulations for real-world problems is of paramount importance - and far from trivial, mainly due to the large number of stochastic parameters and significant computational requirements. Adaptive sparse grid approximations ... More

Maximal $L^p$-regularity for perturbed evolution equations in Banach spacesOct 21 2018The main purpose of this paper is to investigate the concept of maximal $L^p$-regularity for perturbed evolution equations in Banach spaces. We mainly consider three classes of perturbations: Miyadera-Voigt perturbations, Desch-Schappacher perturbations, ... More

A Discontinuous Galerkin Fast Spectral Method for the Full Boltzmann Equation with General Collision KernelsSep 26 2018Nov 02 2018The Boltzmann equation, an integro-differential equation for the molecular distribution function in the physical and velocity phase space, governs the fluid flow behavior at a wide range of physical conditions, including compressible, turbulent, as well ... More

A Physical Perspective on Control Points and Polar Forms: Bézier Curves, Angular Momentum and Harmonic OscillatorsSep 19 2018Bernstein polynomials and B\'ezier curves play an important role in computer-aided geometric design and numerical analysis, and their study relates to mathematical fields such as abstract algebra, algebraic geometry and probability theory. We describe ... More

SNS: A Solution-based Nonlinear Subspace method for time-dependent model order reductionSep 11 2018Feb 05 2019Several reduced order models have been successfully developed for nonlinear dynamical systems. To achieve a considerable speed-up, a hyper-reduction step is needed to reduce the computational complexity due to nonlinear terms. Many hyper-reduction techniques ... More

Data Assimilation: The Schrödinger PerspectiveJul 22 2018Dec 02 2018Data assimilation addresses the general problem of how to combine model-based predictions with partial and noisy observations of the process in an optimal manner. This survey focuses on sequential data assimilation techniques using probabilistic particle-based ... More

Geometric Shape Features Extraction Using a Steady State Partial Differential Equation SystemJun 13 2018Jan 25 2019A unified method for extracting geometric shape features from binary image data using a steady state partial differential equation (PDE) system as a boundary value problem is presented in this paper. The PDE and functions are formulated to extract the ... More

New recursive approximations for variable-order fractional operators with applicationsApr 04 2018To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving variable-order fractional ... More

PDE-constrained optimization in medical image analysisFeb 28 2018PDE-constrained optimization problems find many applications in medical image analysis, for example, neuroimaging, cardiovascular imaging, and oncological imaging. We review related literature and give examples on the formulation, discretization, and ... More

A Probabilistic Subspace Bound with Application to Active SubspacesJan 02 2018Given a real symmetric positive semi-definite matrix E, and an approximation S that is a sum of n independent matrix-valued random variables, we present bounds on the relative error in S due to randomization. The bounds do not depend on the matrix dimensions ... More

Hypergraphs not containing a tight tree with a bounded trunkDec 12 2017An $r$-uniform hypergraph is a tight $r$-tree if its edges can be ordered so that every edge $e$ contains a vertex $v$ that does not belong to any preceding edge and the set $e-v$ lies in some preceding edge. A conjecture of Kalai [Kalai], generalizing ... More

On Mubayi's Conjecture and conditionally intersecting setsNov 15 2017Oct 31 2018Mubayi's Conjecture states that if $\mathcal{F}$ is a family of $k$-sized subsets of $[n] = \{1,\ldots,n\}$ which, for $k \geq d \geq 2$, satisfies $A_1 \cap\cdots\cap A_d \neq \emptyset$ whenever $|A_1 \cup\cdots\cup A_d| \leq 2k$ for all distinct sets ... More

Rigorous numerics of finite-time singularities in dynamical systems - methodology and applicationsNov 06 2017This paper aims at providing rigorous numerical computation procedure for finite-time singularities in dynamical systems. Combination of time-scale desingularization as well as Lyapunov functions validation on stable manifolds of invariant sets for desingularized ... More

Circulant embedding with QMC -- analysis for elliptic PDE with lognormal coefficientsOct 25 2017Apr 02 2018In a previous paper (J. Comp. Phys. 230 (2011), 3668--3694), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. ... More

Efficient Statistically Accurate Algorithms for the Fokker-Planck Equation in Large DimensionsSep 16 2017Solving the Fokker-Planck equation for high-dimensional complex turbulent dynamical systems is an important and practical issue. However, most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed ... More

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified SettingsSep 01 2017Oct 30 2018This paper presents a convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than ... More

A Koksma-Hlawka-Potential Identity on the $d$ Dimensional Sphere and its Applications to DiscrepancyJul 27 2017Let $d\geq 2$ be an integer, $S^d\subset {\mathbb R}^{d+1}$ the unit sphere and $\sigma$ a finite signed measure whose positive and negative parts are supported on $S^d$ with finite energy. In this paper, we derive an error estimate for the quantity $\left|\int_{S^d}fd\sigma\right|$, ... More

Asymptotic expansion for solutions of the Navier-Stokes equations with non-potential body forcesJul 25 2017Nov 21 2017We study the long-time behavior of spatially periodic solutions of the Navier-Stokes equations in the three-dimensional space. The body force is assumed to possess an asymptotic expansion or, resp., finite asymptotic approximation, in either Sobolev or ... More

Paths in hypergraphs: a rescaling phenomenonJun 26 2017Let $P^k_\ell$ denote the loose $k$-path of length $\ell$ and let define $f^k_\ell(n,m)$ as the minimum value of $\Delta(H)$ over all $P^k_\ell$-free $k$-graphs $H$ with $n$ vertices and $m$ edges. In the paper we study the behavior of $f^4_2(n,m)$ and ... More

Warps and grids for double and triple vector bundlesMay 02 2017May 22 2017A triple vector bundle is a cube of vector bundle structures which commute in the (strict) categorical sense. A grid in a triple vector bundle is a collection of sections of each bundle structure with certain linearity properties. A grid provides two ... More

A hybridizable discontinuous Galerkin method for the Navier--Stokes equations with pointwise divergence-free velocity fieldApr 25 2017Jan 31 2018We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier--Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells [SIAM J. Sci. ... More

A Numerical Algorithm for C2-splines on Symmetric SpacesMar 28 2017Mar 07 2018Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example computer vision and quantum control, there is a growing need for spline ... More

Fluids, Geometry, and the Onset of Navier-Stokes Turbulence in Three Space DimensionsDec 27 2016Oct 03 2017A theory for the evolution of a metric $g$ driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional ... More

On the homogenization of the Helmholtz problem with thin perforated walls of finite lengthNov 18 2016Jun 22 2017In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in $\mathbb{R}^2$ with a thin and periodic layer of finite length. The layer may consists of a periodic pertubation of the material coefficients ... More

On the homogenization of the Helmholtz problem with thin perforated walls of finite lengthNov 18 2016In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in $\mathbb{R}^2$ with a thin and periodic layer of finite length. The layer may consists of a periodic pertubation of the material coefficients ... More

High codegree subgraphs in weakly quasirandom 3-graphsOct 20 2016We show that we can extract large subgraphs with high minimum codegree from sequences of weakly quasirandom $3$-graphs, for a particular notion of weakly quasirandom studied by Reiher, R\"odl and Schacht. In particular for any family of nonempty $3$-graphs ... More

Besov Regularity for the Stationary Navier-Stokes Equation on Bounded Lipschitz DomainsAug 02 2016We use the scale $B^s_{\tau}(L_\tau(\Omega))$, $1/\tau=s/d+1/2$, $s>0$, to study the regularity of the stationary Stokes equation on bounded Lipschitz domains $\Omega\subset\mathbb{R}^d$, $d\geq 3$, with connected boundary. The regularity in these Besov ... More

Dimensions of multi-fan algebrasJul 13 2016Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. ... More

Dimensions of multi-fan algebrasJul 13 2016Dec 01 2016Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. ... More

Fluids, Elasticity, Geometry, and the Existence of Wrinkled SolutionsMay 10 2016Sep 24 2016We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the interconnected nonlinear partial differential equations. In this paper, we develop such ... More

Fluids, Elasticity, Geometry, and the Existence of Wrinkled SolutionsMay 10 2016Aug 26 2017We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections ... More

A Monte Carlo method for integration of multivariate smooth functions I: Sobolev spacesApr 20 2016We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by $n^{-1/2}$ ... More

A Monte Carlo method for integration of multivariate smooth functionsApr 20 2016Jun 21 2017We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by $n^{-1/2}$ ... More

Scaling invariant Harnack inequalities in a general settingApr 06 2016Jul 13 2016In a setting, where only "exit measures" are given, as they are associated with an arbitrary right continuous strong Markov process on a separable metric space, we provide simple criteria for the validity of Harnack inequalities for positive harmonic ... More

On a generalized uniform zero-two law for positive contractions of non-commutative $L_1$-spaces and its vector-valued extensionMar 25 2016First, Ornstein and Sucheston proved that for a given positive contraction $T:L_1\to L_1$ there exists $m\in N$ such that $\big\|T^{m+1}-T^m\|<2$ then $$ \lim_{n\to\infty}\|T^{n+1}-T^n\|=0. $$ Such a result was labeled as "zero-two" law. In the present ... More

The coherent matching distance in 2D persistent homologyMar 12 2016Comparison between multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions associated with lines ... More

Position paper: Towards an observer-oriented theory of shape comparisonMar 07 2016In this position paper we suggest a possible metric approach to shape comparison that is based on a mathematical formalization of the concept of observer, seen as a collection of suitable operators acting on a metric space of functions. These functions ... More

A sharp interface immersed boundary method for solving flow with arbitrarily irregular and changing geometryFeb 22 2016In this paper, a sharp interface immersed boundary method is developed for efficiently and robustly solving flow with arbitrarily irregular and changing geometries. The proposed method employs a three-step prediction-correction flow reconstruction scheme ... More

Mathematical Modeling of Myosin Induced Bistability of Lamellipodial FragmentsNov 27 2015For various cell types and for lamellipodial fragments on flat surfaces, externally induced and spontaneous transitions between symmetric nonmoving states and polarized migration have been observed. This behavior is indicative of bistability of the cytoskeleton ... More

Crushing runtimes in adiabatic quantum computation with Energy Landscape Manipulation (ELM): Application to Quantum FactoringOct 26 2015We introduce two methods for speeding up adiabatic quantum computations by increasing the energy between the ground and first excited states. Our methods are even more general. They can be used to shift a Hamiltonian's density of states away from the ... More

Stability and Turán numbers of a class of hypergraphs via LagrangiansOct 12 2015Given a family of $r$-uniform hypergraphs ${\cal F}$ (or $r$-graphs for brevity), the Tur\'an number $ex(n,{\cal F})$ of ${\cal F}$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain any member of ${\cal F}$. A pair $\{u,v\}$ ... More

A green perspective on capacitated time-dependent vehicle routing problem with time windowsSep 29 2015Sep 17 2017This study presents a novel approach to the vehicle routing problem by focusing on greenhouse gas emissions and fuel consumption aiming to mitigate adverse environmental effects of transportation. A time-dependent model with time windows is developed ... More

A green perspective on capacitated time-dependent vehicle routing problem with time windowsSep 29 2015This study presents a novel approach to the vehicle routing problem by focusing on greenhouse gas emissions and fuel consumption aiming to mitigate adverse environmental effects of transportation. A time-dependent model with time windows is developed ... More

Reducing multi-qubit interactions in adiabatic quantum computation without adding auxiliary qubits. Part 1: The "deduc-reduc" method and its application to quantum factorization of numbersAug 19 2015Oct 01 2015Adiabatic quantum computing has recently been used to factor 56153 [Dattani & Bryans, arXiv:1411.6758] at room temperature, which is orders of magnitude larger than any number attempted yet using Shor's algorithm (circuit-based quantum computation). However, ... More

Exact simulation of the Wright-Fisher diffusionJun 23 2015Jul 22 2016The Wright-Fisher family of diffusion processes is a widely used class of evolutionary models. However, simulation is difficult because there is no known closed-form formula for its transition function. In this article we demonstrate that it is in fact ... More

Turán numbers of hypergraph treesMay 13 2015An $r$-graph is an $r$-uniform hypergraph tree (or $r$-tree) if its edges can be ordered as $E_1,\ldots, E_m$ such that $\forall i>1 \, \exists \alpha(i)<i$ such that $E_i\cap (\bigcup_{j=1}^{i-1} E_j)\subseteq E_{\alpha(i)}$. The Tur\'an number $ex(n,{\cal ... More

Asymptotics of parabolic Green's functions on latticesApr 10 2015Jun 29 2016For parabolic spatially discrete equations, we consider Green's functions, also known as heat kernels on lattices. We obtain their asymptotic expansions with respect to powers of time variable $t$ up to an arbitrary order and estimate the remainders uniformly ... More

Existence results in dislocation based rate-independent isotropic gradient plasticity with kinematical hardening and plastic spin: The case with symmetric local backstressApr 08 2015In this paper we use convex analysis and variational inequality methods to establish an existence result for a model of infinitesimal rate-independent gradient plasticity with kinematic hardening and plastic spin, in which the local backstress tensor ... More

A Survey of Manoeuvring Target Tracking MethodsMar 06 2015A comprehensive review of the literature on manoeuvring target tracking for both uncluttered and cluttered measurements is presented. Various discrete-time dynamical models including non-random input, random-input and switching or hybrid system manoeuvre ... More

On "stability" in the Erdős-Ko-Rado theoremFeb 19 2015Denote by $K_p(n,k)$ the random subgraph of the usual Kneser graph $K(n,k)$ in which edges appear independently, each with probability $p$. Answering a question of Bollob\'as, Narayanan, and Raigorodskii,we show that there is a fixed $p<1$ such that a.s. ... More

Function of Forgetfulness for the Tedium of Oblivion on Liquidity of Ontology MatchingJan 02 2015The shallow and fragile knowledge on the Web does not examine in depth the things: it behaves lightly. The conditions created by the Web makes our attention labile and especially fickle, it's unable to concentrate for long as we are trained to "surf" ... More

Liquidity on Web Dynamic NetworkDec 31 2014Nowadays, the exponentially growing of the Web renders the problem of correlation among different topics of paramount importance. The proposed model can be used to study the evolution of network depicted by different topics on the web correlated by a ... More

On 3-uniform hypergraphs without a cycle of a given lengthDec 27 2014Dec 30 2014We study the maximum number of hyperedges in a 3-uniform hypergraph on $n$ vertices that does not contain a Berge cycle of a given length $\ell$. In particular we prove that the upper bound for $C_{2k+1}$-free hypergraphs is of the order $O(k^2n^{1+1/k})$, ... More

Efficient XVA Management: Pricing, Hedging, and Attribution using Trade-Level Regression and Global ConditioningDec 17 2014Dec 22 2014Banks must manage their trading books, not just value them. Pricing includes valuation adjustments collectively known as XVA (at least credit, funding, capital and tax), so management must also include XVA. In trading book management we focus on pricing, ... More

Infinite loop spaces and positive scalar curvatureNov 26 2014Mar 02 2016We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary ... More

Computing minimal interpolants in $C^{1,1}(\mathbb{R}^d)$Nov 20 2014Oct 26 2016We consider the following interpolation problem. Suppose one is given a finite set $E \subset \mathbb{R}^d$, a function $f: E \rightarrow \mathbb{R}$, and possibly the gradients of $f$ at the points of $E$. We want to interpolate the given information ... More

Computing minimal interpolants in $C^{1,1}(\mathbb{R}^d)$Nov 20 2014Jun 17 2016We consider the following interpolation problem. Suppose one is given a finite set $E \subset \mathbb{R}^d$, a function $f: E \rightarrow \mathbb{R}$, and possibly the gradients of $f$ at the points of $E$. We want to interpolate the given information ... More

Representation of group isomorphisms. The compact caseNov 06 2014Dec 18 2014Let $G$ be a discrete group and let $\mathcal A$ and $\mathcal B$ be two subgroups of $G$-valued continuous functions defined on two $0$-dimensional compact spaces $X$ and $Y$. A group isomorphism $H$ defined between $\mathcal A$ and $\mathcal B$ is called ... More

Well-posedness for dislocation based gradient visco-plasticity with isotropic hardeningNov 05 2014In this work we establish the well-posedness for infinitesimal dislocation based gradient viscoplasticity with isotropic hardening for general gradient monotone plastic flows. We assume an additive split of the displacement gradient into non-symmetric ... More

Partition of unity systems and B-splinesOct 20 2014This paper presents the basic principles of partition of unity systems and B-splines. Analysis of these systems is performed using Fourier analysis, multi-resolution analysis, and wavelet analysis.

Fluctuation Analysis of Adaptive Multilevel SplittingAug 27 2014Sep 18 2015Multilevel Splitting is a Sequential Monte Carlo method to simulate realisations of a rare event as well as to estimate its probability. This article is concerned with the convergence and the fluctuation analysis of Adaptive Multilevel Splitting techniques. ... More

Entropy dissipation estimates for the Landau equation in the Coulomb case and applicationsAug 26 2014Oct 28 2014We present in this paper an estimate which bounds from below the entropy dissipation D(f) of the Landau operator with Coulomb interaction by a weighted H^1 norm of the square root of f. As a consequence, we get a weighted L^1_t(L^3_v) estimate for the ... More

A stochastic Gauss-Bonnet-Chern formulaAug 25 2014Apr 28 2015We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem ... More

Min-max representations of viscosity solutions of Hamilton-Jacobi equations and applications in rare-event simulationJun 13 2014In this paper a duality relation between the Ma\~{n}\'e potential and Mather's action functional is derived in the context of convex and state-dependent Hamiltonians. The duality relation is used to obtain min-max representations of viscosity solutions ... More

VAR and ES/CVAR Dependence on data cleaning and Data Models: Analysis and ResolutionMay 29 2014Historical (Stressed-) Value-at-Risk ((S)VAR), and Expected Shortfall (ES), are widely used risk measures in regulatory capital and Initial Margin, i.e. funding, computations. However, whilst the definitions of VAR and ES are unambiguous, they depend ... More

On hypergraph LagrangiansApr 30 2014It is conjectured by Frankl and F\"uredi that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$ edges in \cite{FF}. ... More

On particle Gibbs Markov chain Monte Carlo modelsApr 23 2014Oct 25 2014This article analyses a new class of advanced particle Markov chain Monte Carlo algorithms recently introduced by Andrieu, Doucet, and Holenstein (2010). We present a natural interpretation of these methods in terms of well known unbiasedness properties ... More

The Gauss-Bonnet-Chern theorem: a probabilistic perspectiveApr 21 2014Jan 18 2016We prove that the Euler form of a metric connection on real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section ... More

Measurable bundles of Banach algebrasApr 13 2014In the present paper we investigate Banach--Kantorovich algebras over faithful solid subalgebras of algebras measurable functions. We prove that any Banach--Kantorovich algebra over faithful solid subalgebras of algebra measurable functions represented ... More

Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the planeMar 11 2014In the present article we consider several issues concerning the doubly parabolic Keller-Segel system in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. More specifically, we analyze the global existence of integral ... More

A family of energy stable, skew-symmetric finite difference schemes on collocated gridsFeb 05 2014A simple scheme for incompressible, constant density flows is presented, which avoids odd-even decoupling for the Laplacian on a collocated grids. Energy stability is implied by maintaining strict energy conservation. Momentum is conserved. Arbitrary ... More

Combining persistent homology and invariance groups for shape comparisonDec 27 2013Jan 28 2016In many applications concerning the comparison of data expressed by $\mathbb{R}^m$-valued functions defined on a topological space $X$, the invariance with respect to a given group $G$ of self-homeomorphisms of $X$ is required. While persistent homology ... More

Gevrey Smoothing Effect for Solutions of the Non-Cutoff Boltzmann Equation in Maxwellian Molecules CaseDec 20 2013In this paper we study the Gevrey regularity for the weak solutions to the Cauchy problem of the non-cutoff spatially homogeneous Botlzmann equation for the Maxwellian molecules model with the singularity exponent $s\in (0,1)$. We establish that any weak ... More

Evaluation of layer potentials close to the boundary for Laplace and Helmholtz problems on analytic planar domainsOct 20 2013Boundary integral equations are an efficient and accurate tool for the numerical solution of elliptic boundary value problems. The solution is expressed as a layer potential; however, the error in its evaluation grows large near the boundary if a fixed ... More

Rigorous high-precision computation of the Hurwitz zeta function and its derivativesSep 11 2013We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function $\zeta(s,a)$ for $s, a \in \mathbb{C}$, along with an arbitrary number of derivatives with respect to $s$, to arbitrary precision with rigorous error bounds. ... More

A Lognormal Central Limit Theorem for Particle Approximations of Normalizing ConstantsJun 30 2013This paper deals with the numerical approximation of normalizing constants produced by particle methods, in the general framework of Feynman-Kac sequences of measures. It is well-known that the corresponding estimates satisfy a central limit theorem for ... More

Subgroups of direct products closely approximated by direct sumsJun 17 2013Let $I$ be an infinite set, $\{G_i:i\in I\}$ be a family of (topological) groups and $G=\prod_{i\in I} G_i$ be its direct product. For $J\subseteq I$, $p_{J}: G\to \prod_{j\in J} G_j$ denotes the projection. We say that a subgroup $H$ of $G$ is: (i) \emph{uniformly ... More

Hypergraph Turán numbers of vertex disjoint cyclesMay 23 2013The Tur\'an number of a $k$-uniform hypergraph $H$, denoted by $e{x_k}\left({n;H} \right)$, is the maximum number of edges in any $k$-uniform hypergraph $F$ on $n$ vertices which does not contain $H$ as a subgraph. Let $\mathcal{C}_{\ell}^{\left(k \right)}$ ... More

Turán Numbers for Forests of Paths in HypergraphsMar 20 2013Feb 24 2014The Tur\'an number of an r-uniform hypergraph H is the maximum number of edges in any r-graph on n vertices which does not contain H as a subgraph. Let P_l^(r) denote the family of r-uniform loose paths on l edges, F(k,l) denote the family of hypergraphs ... More

Hypergraph Turan numbers of linear cyclesFeb 11 2013A k-uniform linear cycle of length s is a cyclic list of k-sets A_1,..., A_s such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k at least 5 and s at least 3 and sufficiently large n we determine ... More

Continuous mappings with null supportFeb 09 2013Jan 23 2014Let $X$ be a (topological) space and let ${\mathscr I}$ be an ideal in $X$, that is, a collection of subsets of $X$ which contains all subsets of its elements and is closed under finite unions. The elements of ${\mathscr I}$ are called null. The space ... More

Representations of certain Banach algebrasFeb 08 2013Dec 24 2013For a space $X$ denote by $C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on $X$ and denote by $C_0(X)$ the set of all elements in $C_b(X)$ which vanish at infinity. We prove that certain Banach subalgebras $H$ of $C_b(X)$ ... More

Blow-up dynamics of self-attracting diffusive particles driven by competing convexitiesJan 29 2013In this paper, we analyze the dynamics of an $N$ particles system evolving according the gradient flow of an energy functional. The particle system is a consistent approximation of the Lagrangian formulation of a one parameter family of non-local drift-diffusion ... More

On the evaluation of prolate spheroidal wave functions and associated quadrature rulesJan 08 2013As demonstrated by Slepian et. al. in a sequence of classical papers, prolate spheroidal wave functions (PSWFs) provide a natural and efficient tool for computing with bandlimited functions defined on an interval. Recently, PSWFs have been becoming increasingly ... More

The dual space of precompact groupsDec 22 2012For any topological group $G$ the dual object $\hat G$ is defined as the set of equivalence classes of irreducible unitary representations of $G$ equipped with the Fell topology. If $G$ is compact, $\hat G$ is discrete. In an earlier paper we proved that ... More

Distribution of Aligned Letter Pairs in Optimal Alignments of Random SequencesNov 23 2012Considering the optimal alignment of two i.i.d. random sequences of length $n$, we show that when the scoring function is chosen randomly, almost surely the empirical distribution of aligned letter pairs in all optimal alignments converges to a unique ... More

How many colors guarantee a rainbow matching?Nov 05 2012Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow $t$-matching in an r-partite r-uniform multi-hypergraph ... More

Detailed analysis of prolate quadratures and interpolation formulasAug 23 2012As demonstrated by Slepian et. al. in a sequence of classical papers, prolate spheroidal wave functions (PSWFs) provide a natural and efficient tool for computing with bandlimited functions defined on an interval. As a result, PSWFs are becoming increasing ... More

Multifractal analysis via scaling zeta functions and recursive structure of lattice stringsJul 28 2012Jan 25 2013The multifractal structure underlying a self-similar measure stems directly from the weighted self-similar system (or weighted iterated function system) which is used to construct the measure. This follows much in the way that the dimension of a self-similar ... More

Intelligent Interface Architectures for Folksonomy Driven Structure NetworkMar 28 2012The folksonomy is the result of free personal information or assignment of tags to an object (determined by the URI) in order to find them. The practice of tagging is done in a collective environment. Folksonomies are self constructed, based on co-occurrence ... More

Generalised Hunter-Saxton equations, optimal information transport, and factorisation of diffeomorphismsMar 20 2012Feb 28 2014We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher's information metric on the space of smooth probability densities. The right reduced geodesic ... More

Regularity properties for general HJB equations. A BSDE methodFeb 07 2012In this work we investigate regularity properties of a large class of Hamilton-Jacobi-Bellman (HJB) equations with or without obstacles, which can be stochastically interpreted in form of a stochastic control system which nonlinear cost functional is ... More

Measurable bundles of $C^*$-dynamical systems and its applicationsJan 20 2012In the present paper we investigate $L_0$-valued states and Markov operators on $ C^*$-algebras over $L_0$. In particular, we give representations for $L_0$-valued state and Markov operators on $ C^*$ algebras over $L_0$, respectively, as measurable bundles ... More