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Stochastic model-based minimization of weakly convex functionsMar 17 2018Aug 26 2018We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural ... More

Inexact alternating projections on nonconvex setsNov 03 2018Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex sets is typically ... More

Stochastic model-based minimization under high-order growthJul 01 2018Given a nonsmooth, nonconvex minimization problem, we consider algorithms that iteratively sample and minimize stochastic convex models of the objective function. Assuming that the one-sided approximation quality and the variation of the models is controlled ... More

The nonsmooth landscape of phase retrievalNov 09 2017Jan 07 2018We consider a popular nonsmooth formulation of the real phase retrieval problem. We show that under standard statistical assumptions, a simple subgradient method converges linearly when initialized within a constant relative distance of an optimal solution. ... More

An optimal first order method based on optimal quadratic averagingApr 22 2016Feb 28 2017In a recent paper, Bubeck, Lee, and Singh introduced a new first order method for minimizing smooth strongly convex functions. Their geometric descent algorithm, largely inspired by the ellipsoid method, enjoys the optimal linear rate of convergence. ... More

The proximal point method revisitedDec 17 2017In this short survey, I revisit the role of the proximal point method in large scale optimization. I focus on three recent examples: a proximally guided subgradient method for weakly convex stochastic approximation, the prox-linear algorithm for minimizing ... More

Subgradient methods for sharp weakly convex functionsMar 06 2018Subgradient methods converge linearly on a convex function that grows sharply away from its solution set. In this work, we show that the same is true for sharp functions that are only weakly convex, provided that the subgradient methods are initialized ... More

Nonsmooth optimization using Taylor-like models: error bounds, convergence, and termination criteriaOct 11 2016We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our main result ... More

Efficient quadratic penalization through the partial minimization techniqueJun 08 2016Sep 17 2017Common computational problems, such as parameter estimation in dynamic models and PDE constrained optimization, require data fitting over a set of auxiliary parameters subject to physical constraints over an underlying state. Naive quadratically penalized ... More

The Euclidean Distance Degree of Orthogonally Invariant Matrix VarietiesJan 26 2016We show that the Euclidean distance degree of a real orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances. ... More

Counting real critical points of the distance to orthogonally invariant matrix setsFeb 06 2015Jun 16 2015Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to compute and ... More

Noisy Euclidean distance realization: robust facial reduction and the Pareto frontierOct 24 2014Aug 26 2015We present two algorithms for large-scale low-rank Euclidean distance matrix completion problems, based on semidefinite optimization. Our first method works by relating cliques in the graph of the known distances to faces of the positive semidefinite ... More

Graphical Convergence of Subgradients in Nonconvex Optimization and LearningOct 17 2018Dec 17 2018We investigate the stochastic optimization problem of minimizing population risk, where the loss defining the risk is assumed to be weakly convex. Compositions of Lipschitz convex functions with smooth maps are the primary examples of such losses. We ... More

Stochastic subgradient method converges at the rate $O(k^{-1/4})$ on weakly convex functionsFeb 08 2018Feb 19 2018We prove that the proximal stochastic subgradient method, applied to a weakly convex problem, drives the gradient of the Moreau envelope to zero at the rate $O(k^{-1/4})$. As a consequence, we resolve an open question on the convergence rate of the proximal ... More

Sweeping by a tame processDec 30 2014Aug 24 2015We show that any semi-algebraic sweeping process admits piecewise absolutely continuous solutions, and any such bounded trajectory must have finite length. Analogous results hold more generally for sweeping processes definable in o-minimal structures. ... More

Stochastic subgradient method converges at the rate $O(k^{-1/4})$ on weakly convex functionsFeb 08 2018We prove that the projected stochastic subgradient method, applied to a weakly convex problem, drives the gradient of the Moreau envelope to zero at the rate $O(k^{-1/4})$.

Complexity of finding near-stationary points of convex functions stochasticallyFeb 21 2018In a recent paper, we showed that the stochastic subgradient method applied to a weakly convex problem, drives the gradient of the Moreau envelope to zero at the rate $O(k^{-1/4})$. In this supplementary note, we present a stochastic subgradient method ... More

Efficiency of minimizing compositions of convex functions and smooth mapsApr 30 2016Aug 14 2017We consider global efficiency of algorithms for minimizing a sum of a convex function and a composition of a Lipschitz convex function with a smooth map. The basic algorithm we rely on is the prox-linear method, which in each iteration solves a regularized ... More

Foundations of gauge and perspective dualityFeb 28 2017Jun 18 2018We revisit the foundations of gauge duality and demonstrate that it can be explained using a modern approach to duality based on a perturbation framework. We therefore put gauge duality and Fenchel-Rockafellar duality on equal footing, including explaining ... More

Complexity of a Single Face in an Arrangement of s-Intersecting CurvesAug 22 2011Consider a face F in an arrangement of n Jordan curves in the plane, no two of which intersect more than s times. We prove that the combinatorial complexity of F is O(\lambda_s(n)), O(\lambda_{s+1}(n)), and O(\lambda_{s+2}(n)), when the curves are bi-infinite, ... More

An accelerated algorithm for minimizing convex compositionsApr 30 2016We describe a new proximal algorithm for minimizing compositions of finite-valued convex functions with smooth mappings. When applied to convex optimization problems having an additive composite form, the algorithm reduces to FISTA. The method both realizes ... More

Optimality, identifiability, and sensitivityJul 27 2012Around a solution of an optimization problem, an "identifiable" subset of the feasible region is one containing all nearby solutions after small perturbations to the problem. A quest for only the most essential ingredients of sensitivity analysis leads ... More

Error bounds, quadratic growth, and linear convergence of proximal methodsFeb 22 2016Jun 27 2016The proximal gradient algorithm for minimizing the sum of a smooth and a nonsmooth convex function often converges linearly even without strong convexity. One common reason is that a multiple of the step length at each iteration may linearly bound the ... More

Generic nondegeneracy in convex optimizationMay 06 2010We show that minimizers of convex functions subject to almost all linear perturbations are nondegenerate. An analogous result holds more generally, for lower-C^2 functions.

Semi-algebraic functions have small subdifferentialsApr 02 2010Oct 29 2011We prove that the subdifferential of any semi-algebraic extended-real-valued function on $\R^n$ has $n$-dimensional graph. We discuss consequences for generic semi-algebraic optimization problems.

Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferentialApr 25 2012We prove that uniform second order growth, tilt stability, and strong metric regularity of the limiting subdifferential --- three notions that have appeared in entirely different settings --- are all essentially equivalent for any lower-semicontinuous, ... More

Variable projection without smoothnessJan 19 2016Variable projection is a powerful technique in optimization. Over the last 30 years, it has been applied broadly, with empirical and theoretical results demonstrating both greater efficacy and greater stability than competing approaches. In this paper, ... More

Orthogonal Invariance and IdentifiabilityApr 03 2013Apr 11 2013Orthogonally invariant functions of symmetric matrices often inherit properties from their diagonal restrictions: von Neumann's theorem on matrix norms is an early example. We discuss the example of "identifiability", a common property of nonsmooth functions ... More

Orbits of geometric descentDec 03 2013Dec 05 2013We prove that quasiconvex functions always admit descent trajectories bypassing all non-minimizing critical points.

An optimal first order method based on optimal quadratic averagingApr 22 2016Apr 25 2016In a recent paper, Bubeck, Lee, and Singh introduced a new first order method for minimizing smooth strongly convex functions. Their geometric descent algorithm, largely inspired by the ellipsoid method, enjoys the optimal linear rate of convergence. ... More

Variable projection for nonsmooth modelsJan 19 2016Dec 02 2018Variable projection solves structured optimization problems by completely minimizing over a subset of the variables while iterating over the remaining variables. Over the last 30 years, the technique has been widely used, with empirical and theoretical ... More

Quadratic Penalization Through the Variable Projection TechniqueJun 08 2016Numerous problems in control require data fitting over auxiliary parameters subject to affine physical constraints over an underlying state. Naive quadratically penalized formulations have some conceptual advantages, but suffer from ill-conditioning, ... More

Composite optimization for robust blind deconvolutionJan 06 2019Jan 18 2019The blind deconvolution problem seeks to recover a pair of vectors from a set of rank one bilinear measurements. We consider a natural nonsmooth formulation of the problem and show that under standard statistical assumptions, its moduli of weak convexity, ... More

Clarke subgradients for directionally Lipschitzian stratifiable functionsNov 15 2012Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain and ... More

Stochastic subgradient method converges on tame functionsApr 20 2018May 26 2018This work considers the question: what convergence guarantees does the stochastic subgradient method have in the absence of smoothness and convexity? We prove that the stochastic subgradient method, on any semialgebraic locally Lipschitz function, produces ... More

The dimension of semialgebraic subdifferential graphsFeb 20 2011Examples exist of extended-real-valued closed functions on ${\bf R}^n$ whose subdifferentials (in the standard, limiting sense) have large graphs. By contrast, if such a function is semi-algebraic, then its subdifferential graph must have everywhere constant ... More

Variational analysis of spectral functions simplifiedJun 16 2015Jul 22 2015Spectral functions of symmetric matrices -- those depending on matrices only through their eigenvalues -- appear often in optimization. A cornerstone variational analytic tool for studying such functions is a formula relating their subdifferentials to ... More

The coloring problem for classes with two small obstructionsJul 01 2013The coloring problem is studied in the paper for graph classes defined by two small forbidden induced subgraphs. We prove some sufficient conditions for effective solvability of the problem in such classes. As their corollary we determine the computational ... More

Formal Languages, Formally and CoinductivelyNov 29 2016Traditionally, formal languages are defined as sets of words. More recently, the alternative coalgebraic or coinductive representation as infinite tries, i.e., prefix trees branching over the alphabet, has been used to obtain compact and elegant proofs ... More

Formal Languages, Formally and CoinductivelyNov 29 2016Sep 18 2017Traditionally, formal languages are defined as sets of words. More recently, the alternative coalgebraic or coinductive representation as infinite tries, i.e., prefix trees branching over the alphabet, has been used to obtain compact and elegant proofs ... More

A necessary flexibility condition of a nondegenerate suspension in Lobachevsky 3-spaceAug 14 2012We show that some combination of the lengths of all edges of the equator of a flexible suspension in Lobachevsky 3-space is equal to zero (each length is taken either positive or negative in this combination).

Hopf-Galois extensions with central invariantsJul 16 1997Sep 22 1997We study Hopf-Galois extensions with central invariants for a finite dimensional Hopf algebra. We collect general facts about them and discuss some examples arising in the study of restricted Lie algebras and quantum groups at roots of unity. Our focus ... More

Cyclic shifts of the van der Corput setNov 12 2008In [13], K. Roth showed that the expected value of the $L^2$ discrepancy of the cyclic shifts of the $N$ point van der Corput set is bounded by a constant multiple of $\sqrt{\log N}$, thus guaranteeing the existence of a shift with asymptotically minimal ... More

The Proof of CSP Dichotomy ConjectureApr 06 2017Dec 18 2017Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The standard way to ... More

The existence of a near-unanimity function is decidableAug 08 2011We prove that the following problem is decidable: given a finite set of relations, decide whether this set admits a near-unanimity function.

Numerically explicit version of the Pólya--Vinogradov inequalityJul 02 2011In this paper we proved a new numerically explicit version of the P\'{o}lya--Vinogradov inequality. Our proof is based on the new ideas of V.A. Bykovskii and improves a recent inequality obtained by C. Pomerance.

On a problem of Dobrowolski--WilliamsJul 02 2011In this paper we prove new upper bounds for the sum $\sum_{n=a+1}^{a+N}f(n)$, for a certain class of arithmetic functions $f$. Our results improve the previous results of G. Bachman and L. Rachakonda.

Lie algebras in symmetric monoidal categoriesMay 16 2012Jun 18 2013We study algebras defined by identities in symmetric monoidal categories. Our focus is on Lie algebras. Besides usual Lie algebras, there are examples appearing in the study of knot invariants and Rozansky-Witten invariants. Our main result is a proof ... More

Algebraic geometry of Hopf-Galois extensionsJul 16 1997Sep 25 1997We continue the study of Hopf-Galois extensions with central invariants for a finite dimensional Hopf algebra. We concentrate on the geometrical side on the subject. We understand how to localize Hopf-Galois extensions and to paste them from local datum. ... More

Miscroscopic origin of de Sitter entropyJan 09 2018It has been argued recently that the entropy of black holes might be associated with soft scalar, graviton and photon states at the event horizon, as number of such possible soft states is proportional to the horizon area. However, the coefficient of ... More

An infinitesimally nonrigid polyhedron with nonstationary volume in the Lobachevsky 3-spaceFeb 20 2010Jan 02 2011We give an example of an infinitesimally nonrigid polyhedron in the Lobachevsky 3-space and construct an infinitesimal flex of that polyhedron such that the volume of the polyhedron isn't stationary under the flex.

On factorization of elements in Pimenov algebrasMar 20 2013We consider the operation of division in Pimenov algebras. We obtain necessary and sufficient conditions for prime elements in Pimenov algebras with a number of generators less than five. We adduce examples of the factorization of elements in these algebras. ... More

The Size of Generating Sets of PowersApr 08 2015In the paper we prove for every finite algebra A that either it has the polynomially generated powers (PGP) property, or it has the exponentially generated powers (EGP) property. For idempotent algebras we give a simple criteria for the algebra to satisfy ... More

Key (critical) relations preserved by a weak near-unanimity functionJan 19 2015Nov 26 2016In the paper we introduce a notion of a key relation, which is similar to the notion of a critical relation introduced by Keith A.Kearnes and \'Agnes Szendrei. All clones on finite sets can be defined by only key relations. In addition there is a nice ... More

Dorronsoro's theorem and a small generalizationJun 21 2015We give a simple proof of Dorronsoro's theorem and use similar ideas to establish an equivalence for embeddings of vector fields.

Quadratic growth and critical point stability of semi-algebraic functionsSep 05 2013Dec 20 2014We show that quadratic growth of a semi-algebraic function is equivalent to strong metric subregularity of the subdifferential --- a kind of stability of generalized critical points. In contrast, this equivalence can easily fail outside of the semi-algebraic ... More

Level-set methods for convex optimizationFeb 03 2016Convex optimization problems arising in applications often have favorable objective functions and complicated constraints, thereby precluding first-order methods from being immediately applicable. We describe an approach that exchanges the roles of the ... More

Projection methods in quantum information scienceJul 24 2014We consider the problem of constructing quantum operations or channels, if they exist, that transform a given set of quantum states $\{\rho_1, \dots, \rho_k\}$ to another such set $\{\hat\rho_1, \dots, \hat\rho_k\}$. In other words, we must find a {\em ... More

The two-dimensional small ball inequality and binary netsNov 23 2015In the current paper we present a new proof of the small ball inequality in two dimensions. More importantly, this new argument, based on an approach inspired by lacunary Fourier series, reveals the first formal connection between this inequality and ... More

Stability of Skorokhod problem is undecidableJul 10 2010Skorokhod problem arises in studying Reflected Brownian Motion (RBM) on an non-negative orthant, specifically in the context of queueing networks in the heavy traffic regime. One of the key problems is identifying conditions for stability of a Skorokhod ... More

Correlation Functions in N=3 Superconformal TheoryJun 17 2010Jan 31 2011Using a superspace representation of the N=3 Neveau-Schwarz super Virasoro algebra, we find solutions of N=3 super Ward identities. Global transformations generated by the non-abelian supercurrent require not only superfields, but also functions of Grassmann ... More

Ergodic Volterra Quadratic Transformations of SymplexMay 17 2012In the paper a Volterra quadratic stochastic operators of three dimensional simplex into itself is considered.The full description of ergodic properties such operators is given.

Relative velocity of dark matter and baryonic fluids and the formation of the first structuresMay 13 2010Oct 18 2010At the time of recombination, baryons and photons decoupled and the sound speed in the baryonic fluid dropped from relativistic to the thermal velocities of the hydrogen atoms. This is less than the relative velocities of baryons and dark matter computed ... More

Switchability and collapsibility of Gap AlgebrasOct 21 2015Let A be an idempotent algebra on a 3-element domain D that omits a G-set for a factor. Suppose A is not \alpha\beta-projective (for some alpha, beta subsets of D) and is not collapsible. It follows that A is switchable. We prove that, for every finite ... More

Multiparametric Passive Linear Stationary Dynamical Scattering Systems: Discrete Case, II: Existence of Conservative DilationsOct 19 1998Mar 28 1999In the present paper we introduce the notion of dilation of a multiparametric linear stationary dynamical system (systems of this type, in particular passive, and conservative scattering ones were first introduced in func-an/9804130). We establish the ... More

On decoherence in quantum gravityAug 21 2015Feb 26 2017It was previously argued that the phenomenon of quantum gravitational decoherence described by the Wheeler-DeWitt equation is responsible for the emergence of the arrow of time. Here we show that the characteristic spatio-temporal scales of quantum gravitational ... More

Regular bipartite graphs and intersecting familiesNov 09 2016Oct 18 2017In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erd\H os--Ko--Rado theorem, the Hilton--Milner theorem, a theorem due to Frankl concerning the size of intersecting ... More

The Connes character formula for locally compact spectral triplesMar 05 2018May 04 2018A fundamental tool in noncommutative geometry is Connes' character formula. This formula is used in an essential way in the applications of noncommutative geometry to index theory and to the spectral characterisation of manifolds. A non-compact space ... More

On Boolean Control Networks with Maximal Topological EntropyJul 05 2014Boolean control networks (BCNs) are discrete-time dynamical systems with Boolean state-variables and inputs that are interconnected via Boolean functions. BCNs are recently attracting considerable interest as computational models for genetic and cellular ... More

Galois invariants of K_1-groups of Iwasawa algebrasJun 28 2010We study Galois descent of K_1 of group algebras with coefficients in certain subrings of the ring of integers of C_p, the completion of an algebraic closure of Q_p.

Discrete maximal parabolic regularity for Galerkin finite element methods for non-autonomous parabolic problem sJul 28 2017The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the lowest order time discontinuous Galerkin solution of linear parabolic equations with time-dependent coefficients. Such estimates ... More

Kac-Moody Groups and CompletionsJun 26 2017May 30 2018In this paper we construct a new "pro-p-complete" topological Kac-Moody group and compare it to various known topological Kac-Moody groups. We come across this group by investigating the process of completion of groups with BN-pairs. We would like to ... More

Cascade Connection of Multiparametric Linear Systems and a Conservative Realization of Decomposable Inner Operator Functions in BidiskFeb 25 2000We introduce the notion of cascade connection of multiparametric discrete time-invariant linear dynamical systems with unit delay. This allows us to construct the explicit example of conservative realization of a decomposable operator-valued function ... More

Dark matter in a Simplest Little Higgs with T-parity modelMar 27 2009Little Higgs models may provide a viable alternative to supersymmetry as an extension of the Standard Model. After the introduction of a discrete $Z_2$ symmetry, dubbed T-parity into Little Higgs models they also contain a promising dark matter candidate. ... More

Global and interior pointwise best approximation results for the gradient of Galerkin solutions for parabolic problemsJun 20 2016In this paper we establish best approximation property of fully discrete Galerkin solutions of the second parabolic problems on convex polygonal and polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Om))$ norm. The discretization method consists of ... More

Discrete maximal parabolic regularity for Galerkin finite element methodsMay 18 2015Feb 05 2016The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the time discontinuous Galerkin solution of linear parabolic equations. Such estimates have many applications. They are essential, ... More

Regular bipartite graphs and intersecting familiesNov 09 2016In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erd\H{o}s-Ko-Rado theorem, the Hilton-Milner theorem, a theorem due to Frankl concerning the size of intersecting ... More

Parametric interaction and intensification of nonlinear Kelvin wavesJul 03 2008Observational evidence is presented for nonlinear interaction between mesoscale internal Kelvin waves at the tidal -- $\omega_t$ or the inertial -- $\omega_i$ frequency and oscillations of synoptic -- $\Omega $ frequency of the background coastal current ... More

Subregular representations of $\sl_n$ and simple singularities of type $A_{n-1}$Jan 15 2001Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem ... More

Functions whose Fourier transform vanishes on a surfaceJan 18 2016Jan 24 2016We study the subspaces of $L_p(\mathbb{R}^d)$ that consist of functions whose Fourier transforms vanish on a smooth surface of codimension $1$. We show that a subspace defined in such a manner coincides with the whole $L_p$ space for $p > \frac{2d}{d+1}$. ... More

Bilinear embedding theorems for differential operators in $\mathbb{R}^2$Jun 08 2014We prove bilinear inequalities for differential operators in $\mathbb{R}^2$. Such type inequalities turned out to be useful for anisotropic embedding theorems for overdetermined systems and the limiting order summation exponent. However, here we study ... More

Equivariant epsilon conjecture for 1-dimensional Lubin-Tate groupsSep 18 2013In this paper we formulate a conjecture on the relationship between the equivariant \epsilon-constants (associated to a local p-adic representation V and a finite extension of local fields L/K) and local Galois cohomology groups of a Galois stable \mathbb{Z}_{p}-lattice ... More

Sequential cavity method for computing free energy and surface pressureJul 09 2008We propose a new method for the problems of computing free energy and surface pressure for various statistical mechanics models on a lattice $\Z^d$. Our method is based on representing the free energy and surface pressure in terms of certain marginal ... More

Restrictions of higher derivatives of the Fourier transformSep 11 2018Nov 08 2018We consider several problems related to the restriction of $(\nabla^k) \hat{f}$ to a surface $\Sigma \subset \mathbb R^d$ with nonvanishing Gauss curvature. While such restrictions clearly exist if $f$ is a Schwartz function, there are few bounds available ... More

Extreme point inequalities and geometry of the rank sparsity ballJan 20 2014We investigate geometric features of the unit ball corresponding to the sum of the nuclear norm of a matrix and the $l_1$ norm of its entries --- a common penalty function encouraging joint low rank and high sparsity. As a byproduct of this effort, we ... More

Cascade Connections of Linear Systems and Factorizations of Holomorphic Operator Functions Around a Multiple Zero in Several VariablesFeb 04 2000We show that the factorization problem $\theta (z)=\theta_2(z)\theta_1(z)$ is solvable in the class of Hilbert space operator-valued functions holomorphic on some neighbourhood of $z=0$ in $\nspace{C}{N}$ and having a zero at $z=0$ (here $\theta (z)$ ... More

On the von Neumann Inequality for Linear Matrix Functions of Several VariablesNov 13 1997The theorem on the existence of three commuting contractions on a Hilbert space and of a linear homogeneous matrix function of three independent variables for which the generalized von Neumann inequality fails is proved.

Distribution of the first particle in discrete orthogonal polynomial ensemblesApr 01 2002We show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure, if the (difference or q-) log-derivative of the weight function is rational. In a number ... More

Pointwise best approximation results for Galerkin finite element solutions of parabolic problemsAug 05 2015Feb 17 2016In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty$ norm. The discretization method uses of continuous ... More

BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimensionNov 21 2014Jul 09 2015In the current paper we obtain discrepancy estimates in exponential Orlicz and BMO spaces in arbitrary dimension $d \ge 3$. In particular, we use dyadic harmonic analysis to prove that for the so-called digital nets of order $2$ the BMO${}^d$ and $\exp ... More

Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential EquationsDec 30 2018Jan 09 2019The article is devoted to the comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic differential equations in the framework of the method of approximation ... More

Optimal a priori error estimates of parabolic optimal control problems with a moving point controlJan 11 2017Mar 06 2017In this paper we consider a parabolic optimal control problem with a Dirac type control with moving point source in two space dimensions. We discretize the problem with piecewise constant functions in time and continuous piecewise linear finite elements ... More

Quantum OptimizationJun 20 2000We present a quantum algorithm for combinatorial optimization using the cost structure of the search states. Its behavior is illustrated for overconstrained satisfiability and asymmetric traveling salesman problems. Simulations with randomly generated ... More

On decoherence in non-renormalizable field theories and quantum gravityAug 21 2015It was previously argued that the phenomenon of quantum gravitational decoherence described by the Wheeler-DeWitt equation is responsible for the emergence of the arrow of time. Here we show that the characteristic spatio-temporal scales of quantum gravitational ... More

Viscous compressible homogeneous multi-fluids with multiple velocities: barotropic existence theoryOct 18 2016We consider the model of viscous compressible homogeneous multi-fluids with multiple velocities. We review different formulations of the model and the existence results for boundary value problems. We analyze crucial mathematical difficulties which arise ... More

Twisted Crystalline Differential OperatorsDec 15 1999May 14 2002This paper has been withdrawn by the authors.

Geometric Representation Theory of Restricted Lie Algebras of Classical TypeSep 10 1999We modify the Hochschild $\phi$-map to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a group scheme which leads to a geometric construction of unrestricted representations. For a classical semisimple Lie ... More

2-Groups, 2-Characters, and Burnside RingsApr 11 2016We study 2-representations, i.e. actions of 2-groups on 2-vector spaces. Our main focus is character theory for 2-representations. To this end we employ the technique of extended Burnside rings. Our main theorem is that the Ganter-Kapranov 2-character ... More

On the volume of spherical Lambert cubeDec 21 2002The calculation of volumes of polyhedra in the three-dimensional Euclidean, spherical and hyperbolic spaces is very old and difficult problem. In particular, an elementary formula for volume of non-euclidean simplex is still unknown. One of the simplest ... More

Existence of cube terms in finite finitely generated clonesJan 15 2019We study the problem of whether a given finite clone generated by finitely many operations contains a cube term and give both structural and algorithmic results. We show that if such a clone has a cube term then it has a cube term of dimension at most ... More