Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg–Landau equation

In this article, unconditional superconvergent analysis of a linearized fully discrete mixed finite element method is presented for a class of Ginzburg–Landau equation based on the bilinear element and zero‐order Nédélec's element pair (Q₁₁/Q₀₁ × Q₁₀). First, a time‐discrete system is introduced to split the error into temporal error and spatial error, and the corresponding error estimates are deduced rigorously. Second, the unconditional superclose and optimal estimate of order O(h² + τ) for u in H¹‐norm and p = ?u in L²‐norm are derived respectively without the restrictions on the ratio between h and τ, where h is the subdivision parameter and τ, the time step. Third, the global superconvergent results are obtained by interpolated postprocessing technique. Finally, some numerical results are carried out to confirm the theoretical analysis.     

b-Functions associated with quivers of type A

We study the b-functions of relative invariants of the prehomogeneous vector spaces associated with quivers of type A. By applying the decomposition formula for b-functions, we determine explicitly the b-functions of one variable for each irreducible relative invariant. Moreover, we give a graphical algorithm to determine the b-functions of several variables.     

Formal verification of tail distribution bounds in the HOL theorem prover

Tail distribution bounds play a major role in the estimation of failure probabilities in performance and reliability analysis of systems. They are usually estimated using Markov's and Chebyshev's inequalities, which represent tail distribution bounds for a random variable in terms of its mean or variance. This paper presents the formal verification of Markov's and Chebyshev's inequalities for discrete random variables using a higher‐order‐logic theorem prover. The paper also provides the formal verification of mean and variance relations for some of the widely used discrete random variables, such as Uniform(m), Bernoulli(p), Geometric(p) and Binomial(m, p) random variables. This infrastructure allows us to precisely reason about the tail distribution properties and thus turns out to be quite useful for the analysis of systems used in safety‐critical domains, such as space, medicine or transportation. For illustration purposes, we present the performance analysis of the coupon collector's problem, a well‐known commercially used algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

Analyzing linear mergers

Mergers are functions that transform k (possibly dependent) random sources (distributions) into a single random source, in a way that ensures that if one of the input sources has min‐entropy rate δ then the output has min‐entropy rate close to δ. Mergers have proven to be a very useful tool in explicit constructions of extractors and condensers, and are also interesting objects in their own right. In this work we give a refined analysis of the merger constructed by [Raz, STOC'05] (based on [Lu, Reingold, Vadhan, and Wigderson, STOC'03 pp. 602–611, 2003]). Our analysis uses min‐entropy instead of Shannon's entropy to derive tighter results than the ones obtained in [Raz STOC'05]. We show that for every constant r and k it is possible to construct a merger that takes as input k strings of length n bits each, and outputs a string of length n/r bits, such that if one of the input sources has min‐entropy b, the output will be close to having min‐entropy b/(r + 1). This merger uses a constant number of additional uniform bits. One advantage of our analysis is that b (the min‐entropy of the “good” source) can be as small as a constant (this constant depends on r and k), while in the analysis given in [Raz STOC'05], b is required to be linear in n. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Jessen–Wintner type random variables and fractal properties of their distributions

Formulas are given for the Lebesgue measure and the Hausdorff–Besicovitch dimension of the minimal closed set S_ξ supporting the distribution of the random variable ξ = 2^–k τ_k, where τ_k are independent random variables taking the values 0, 1, 2 with probabilities p _0k , p _1k , p _2k , respectively. A classification of the distributions of the r.v. ξ via the metric‐topological properties of S_ξ is given. Necessary and sufficient conditions for superfractality and anomalous fractality of S_ξ are found. It is also proven that for any real number a ₀ ∈ [0, 1] there exists a distribution of the r.v. ξ such that the Hausdorff–Besicovitch dimension of S_ξ is equal to a ₀. The results are applied to the study of the metric‐topological properties of the convolutions of random variables with independent binary digits, i.e., random variables ξⁱ = , where η_k are independent random variables taking the values 0 and 1. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A direct method for a regularized least‐squares problem

We consider a linear system of the form A₁x₁+ A₂x₂+η=b₁. The vector η consists of identically distributed random variables all with mean zero. The unknowns are split into two groups x₁ and x₂. In the model usually there are more unknowns than observations and the resulting linear system is most often consistent having an infinite number of solutions. Hence some constraint on the parameter vector x is needed. One possibility is to avoid rapid variation in, e.g. the parameters x₂. We formulate the problem as a partially regularized least‐squares problem, and propose a direct solution method based on the QR decomposition of matrix blocks. Further we consider regularizing using one and two regularization parameters, respectively. We also discuss the choice of regularization parameters, and extend Reinsch's method to the case with two parameters. Also the cross‐validation technique is treated. We present test examples taken from an application in modelling of the substance transport in rivers. Copyright © 2009 John Wiley & Sons, Ltd.     

Long‐time dynamics of KdV solitary waves over a variable bottom

We study the variable‐bottom, generalized Korteweg—de Vries (bKdV) equation ?_tu = ??_x(?u + f(u) ? b(t,x)u), where f is a nonlinearity and b is a small, bounded, and slowly varying function related to the varying depth of a channel of water. Many variable‐coefficient KdV‐type equations, including the variable‐coefficient, variable‐bottom KdV equation, can be rescaled into the bKdV. We study the long‐time behavior of solutions with initial conditions close to a stable, b = 0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave whose center and scale evolve according to a certain dynamical law involving the function b(t,x) plus an H¹(?)‐small fluctuation. © 2005 Wiley Periodicals, Inc.     

On the decay rates of Timoshenko system with second sound

In this work, we study the well‐posedness and the asymptotic stability of a one‐dimensional linear thermoelastic Timoshenko system, where the heat conduction is given by Cattaneo's law and the coupling is via the displacement equation. We prove that the system is exponentially stable provided that the stability number χ_τ=0. Otherwise, we show that the system lacks exponential stability. Furthermore, in the latter case, we show that the solution decays polynomially. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundary value problems with eigenvalue depending boundary conditions

We investigate some classes of eigenvalue dependent boundary value problems of the form where A ? A⁺ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ₀, Γ₁ are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent P_K (à – λ)^–1|_K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Herbrandizing search problems in Bounded Arithmetic

We study search problems and reducibilities between them with known or potential relevance to bounded arithmetic theories. Our primary objective is to understand the sets of low complexity consequences (esp. Σ^b₁ or Σ^b₂) of theories Sⁱ₂ and Tⁱ₂ for a small i, ideally in a rather strong sense of characterization; or, at least, in the standard sense of axiomatization. We also strive for maximum combinatorial simplicity of the characterizations and axiomatizations, eventually sufficient to prove conjectured separation results. To this end two techniques based on the Herbrand's theorem are developed. They characterize/axiomatize Σ^b₁‐consequences of Σ^b₂‐definable search problems, while the method based on the more involved concept of characterization is easier and gives more transparent results. This method yields new proofs of Buss' witnessing theorem and of the relation between PLS and Σ^b₁(T¹₂), and also an axiomatization of Σ^b₁(T²₂). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bivariate spline method for numerical solution of time evolution Navier‐Stokes equations over polygons in stream function formulation

We use a bivariate spline method to solve the time evolution Navier‐Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier‐Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth‐order equation, Crank‐Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L₂(0, T; H²(Ω)) ∩ L_∞(0, T; H¹(Ω)) of the 2D nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C¹ cubic splines are implemented in MATLAB for solving the Navier‐Stokes equations numerically. Our numerical experiments show that the method is effective and efficient. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 776–827, 2003.     

A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function ₂F₁(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given.     

On the complexity of branch‐and‐bound search for random trees

Let T_n be a b‐ary tree of height n, which has independent, non‐negative, identically distributed random variables associated with each of its edges, a model previously considered by Karp, Pearl, McDiarmid, and Provan. The value of a node is the sum of all the edge values on its path to the root. Consider the problem of finding the minimum leaf value of T_n. Assume that the edge random variable X is nondegenerate, has E {X^θ}<∞ for some θ>2, and satisfies bP{X=c}<1 where c is the leftmost point of the support of X. We analyze the performance of the standard branch‐and‐bound algorithm for this problem and prove that the number of nodes visited is in probability (β+o(1))ⁿ, where β∈(1, b) is a constant depending only on the distribution of the edge random variables. Explicit expressions for β are derived. We also show that any search algorithm must visit (β+o(1))ⁿ nodes with probability tending to 1, so branch‐and‐bound is asymptotically optimal where first‐order asymptotics are concerned. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14: 309–327, 1999     

On the solution of a nonlinear programming problem by decomposition

A decomposition method, used in least-weight plastic design, is extended to solve problems with nonlinearity arising from variable structure geometry. The problem considered is that of finding vectorsx ₁,x ₂, andq that minimize [l max{|x ₁|, |x ₂|}], subject toAx ₁=b ₁ andAx ₂=b ₂, where both the vectorl and the matrixA are nonlinear functions ofq.     

Initial-boundary value problems in linear viscoelasticity on the half-space

After deriving the linear hereditary constitutive laws for viscoelasticity, deducing frequency representation and the correspondence principle to linear elastodynamics the weak form of the equations of motion and their decomposition into pseudo-wave equations are stated. Applying a Laplace transform in the time domain the Green's tensor is constructed by means of a spatial distributional Fourier transform. A detailed discussion of the four main initial-boundary value problems with prescribed displacement and traction components on the plane {x₃ = 0} leads to half-space representations by inverse Fourier integrals. Finally some asymptotic behaviour of the solution in the original time domain is deduced.     

Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

A linearized three‐step backward differential formula (BDF) Galerkin finite element method (FEM) is developed for nonlinear Sobolev equation with bilinear element. Temporal error and spatial error are discussed through introducing a time‐discrete system. Solutions of the time‐discrete system are bounded in H²‐norm by the temporal error. Superconvergence results of order O(h² + τ³) in H¹‐norm for the original variable are deduced based on the spatial error. Some new tricks are utilized to get higher order of the temporal error and the spatial error. At last, two numerical examples are provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.     

Regular orientable embeddings of complete bipartite graphs

In this paper, it will be shown that the isomorphism classes of regular orientable embeddings of the complete bipartite graph K_n,n are in one‐to‐one correspondence with the permutations on n elements satisfying a given criterion, and the isomorphism classes of them are completely classified when n is a product of any two (not necessarily distinct) prime numbers. For other n, a lower bound of the number of those isomorphism classes of K_n,n is obtained. As a result, many new regular orientable embeddings of the complete bipartite graph are constructed giving an answer of Nedela‐?koviera's question raised in 12 . © 2005 Wiley Periodicals, Inc. J Graph Theory     

Local limit theorems for occupancy models

We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with d neighbors in a germ‐grain model, and the number of degree‐d vertices in an Erd?s‐Rényi random graph. In both cases, the error rate is optimal, up to logarithmic factors.     

Forecasting retained earnings of privately held companies with PCA and L1 regression

We use proprietary data collected by SVB Analytics, an affiliate of Silicon Valley Bank, to forecast the retained earnings of privately held companies. Combining methods of principal component analysis (PCA) and L¹/quantile regression, we build multivariate linear models that feature excellent in‐sample fit and strong out‐of‐sample predictive accuracy. The combined PCA and L¹ technique effectively deals with multicollinearity and non‐normality of the data, and also performs favorably when compared against a variety of other models. Additionally, we propose a variable ranking procedure that explains which variables from the current quarter are most predictive of the next quarter's retained earnings. We fit models to the top five variables identified by the ranking procedure and thereby, discover interpretable models with excellent out‐of‐sample performance. Copyright © 2013 John Wiley & Sons, Ltd.