A new proof of regularity of weak solutions of the <Emphasis Type="Italic">H</Emphasis>-surface equation

Abstract. We give a new proof of a theorem of Bethuel, asserting that arbitrary weak solutions of the H-surface system are locally H?lder continuous provided that H is a bounded Lipschitz function. Contrary to Bethuel's, our proof completely omits Lorentz spaces. Estimates below natural exponents of integrability are used instead. (The same method yields a new proof of Hélein's theorem on regularity of harmonic maps from surfaces into arbitrary compact Riemannian manifolds.) We also prove that weak solutions with continuous trace are continuous up to the boundary, and give an extension of these results to the equation of hypersurfaces of prescribed mean curvature in , this time assuming in addition that decays at infinity like . Received: 10 May 2001 / Accepted: 7 June 2001 / Published online: 18 January 2002 The author gratefully acknowledges the generous support of Alexander von Humboldt Foundation, and the hospitality of Mathematisches Institut der Universit?t Bonn, where this research has been carried out. In particular, many thanks are due to Professor Stefan Hildebrandt.     

The Visual Separability of Plotting Symbols in Scatterplots

Abstract

Which symbols should be used to represent different groups of data in the same scatterplot? Hypotheses are derived to predict which symbol pairs should lead to good separability, based on the contrast of the symbols' visual properties or “features.” In two experiments, experimental scatterplots were shown to subjects on a computer screen; the dependent variable was the decision time to judge which of the two presented symbols was the more frequent one. Analyses of the within-subject effects yielded the following results: (1) Important feature contrasts are brightness, number of line endings, and curvature. (2) Symbols that differ simultaneously in two feature dimensions may be more separable than symbols that differ only in either one. (3) The contrasts between circular symbols and radial line symbols like the plus sign or the asterisk are excellent. Practical applications of these findings are discussed, as well as their contribution to the theory of visual perception.     

On Spectral Properties of Translationally Invariant Magnetic Schrödinger Operators

We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction. Our goal is to study basic spectral properties of the Schr?dinger operator with such a potential. In particular, we show that the spectrum of is absolutely continuous and we find its location. Then we study the long-time behaviour of solutions exp of the time dependent Schr?dinger equation. It turns out that a quantum particle remains localized in the plane orthogonal to the direction of the potential. Its propagation in this direction is determined by group velocities. It is to a some extent similar to an evolution of a one-dimensional free particle but “exits” to +∞ and −∞ in the direction of the potential might be essentially different. Submitted: June 7, 2007. Accepted: August 20, 2007.     

Solutions of the equation \documentclass{article}\pagestyle{empty}\begin{document}$$ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $$\end{document}. Exact solutions of some partial differential equations in mechanics and physics

The solutions of the equation $ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $, for L? a linear operator are derived. Different forms for L? whether it is time independent or time dependent and self-commutative (or not) at different times are considered separately. By using the results obtained, exact solutions of some partial differential equations are found for the first time.     

On second-order periodic elliptic operators in divergence form

We consider second-order, strongly elliptic, operators with complex coefficients in divergence form on . We assume that the coefficients are all periodic with a common period. If the coefficients are continuous we derive Gaussian bounds, with the correct small and large time asymptotic behaviour, on the heat kernel and all its H?lder derivatives. Moreover, we show that the first-order Riesz transforms are bounded on the -spaces with . Secondly if the coefficients are H?lder continuous we prove that the first-order derivatives of the kernel satisfy good Gaussian bounds. Then we establish that the second-order derivatives exist and satisfy good bounds if, and only if, the coefficients are divergence-free or if, and only if, the second-order Riesz transforms are bounded. Finally if the third-order derivatives exist with good bounds then the coefficients must be constant. Received in final form: 28 February 2000 / Published online: 17 May 2001     

Computation Models for Parameterized Complexity

A parameterized computational problem is a set of pairs (x, k), where k is a distinguished item called “parameter”. FPT is the class of fixed-parameter tractable problems: for any fixed value of k, they are solvable in time bounded by a polynomial of degree α, where α is a constant not dependent on the parameter. In order to deal with parameterized intractability, Downey and Fellows have introduced a hierarchy of classes W[l] ? W[2] ? ? containing likely intractable parameterized problems, and they have shown that such classes have many natural, complete languages. In this paper we analyze several variations of the halting problem for nondeterministic Turing machines with parameterized time, and we show that its parameterized complexity strongly depends on some resources like the number of tapes, head and internal states, and on the size of the alphabet. Notice that classical polynomial-time complexity fails in distinguishing such features. As byproducts, we show that parameterized complexity is a useful tool for the study of the intrinsic power of some computational models, and we underline the different “computational powers” of some levels of the parameterized hierarchy.     

Linear time isotonic and unimodal regression in the and norms

We consider L₁-isotonic regression and L_∞ isotonic and unimodal regression. For L₁-isotonic regression, we present a linear time algorithm when the number of outputs are bounded. We extend the algorithm to construct an approximate isotonic regression in linear time when the output range is bounded. We present linear time algorithms for L_∞ isotonic and unimodal regression.     

Solution of axisymmetric Maxwell equations

In this article, we study the static and time‐dependent Maxwell equations in axisymmetric geometry. Using the mathematical tools introduced in (Math. Meth. Appl. Sci. 2002; 25 : 49), we investigate the decoupled problems induced in a meridian half‐plane, and the splitting of the solution in a regular part and a singular part, the former being in the Sobolev space H¹ component‐wise. It is proven that the singular parts are related to singularities of Laplace‐like or wave‐like operators. We infer from these characterizations: (i) the finite dimension of the space of singular fields; (ii) global space and space–time regularity results for the electromagnetic field. This paper is the continuation of (Modél. Math. Anal. Numér. 1998; 32 : 359, Math. Meth. Appl. Sci. 2002; 25 : 49). Copyright © 2003 John Wiley & Sons, Ltd.     

Finite‐time stability and boundedness of switched nonlinear time‐delay systems under state‐dependent switching

This article investigates the finite‐time stability, stabilization, and boundedness problems for switched nonlinear systems with time‐delay. Unlike the existing average dwell‐time technique based on time‐dependent switching strategy, largest region function strategy, that is, state‐dependent switching control strategy is adopted to design the switching signal, which does not require the switching instants to be given in advance. Some sufficient conditions which guarantee finite‐time stable, stabilization, and boundedness of switched nonlinear systems with time‐delay are presented in terms of linear matrix inequalities. Detail proofs are given using multiple Lyapunov‐like functions. A numerical example is given to illustrate the effectiveness of the proposed methods. © 2014 Wiley Periodicals, Inc. Complexity 21: 267–275, 2015     

On random cartesian trees

Cartesian trees are binary search trees in which the nodes exhibit the heap property according to a second (priority) key. If the search key and the priority key are independent, and the trees is built based on n independent copies, Cartesian trees basically behave like ordinary random binary search trees. In this article, we analyze the expected behavior when the keys are dependent: in most cases, the expected search, insertion, and deletion times are Φ(√n). We indicate how these results can be used in the analysis of divide-and-conguer algorithms for maximal vectors and convex hulls. Finally, we look at distributions for which the expected time per operation grows like n^a for a ?[1/2, 1]. © 1994 John Wiley & Sons, Inc.     

On distinct unit fractions whose sum equals 1

Questions, partial and complete answers about the diophantine equation in distinct positive integers are given when additional requirements are asked on the x_i's such as: being large, odd, even or x_ix_j for i≠j. Various combinations of the above conditions are also considered.     

On the existence in Gevrey classes of local solutions to the Cauchy problem for nonlinear hyperbolic systems with Hölder continuous coefficients

Abstract In this paper we shall solve locally in time the solutions to the Cauchy problem for first order quasilinear hyperbolic systems of which coefficients of principal part and of lower order terms are μ- H?lder and - H?lder continuous in time variable respectively and in Gevrey class of index s with respect to space variables under the assumption , where ν denotes the maximal muliplicity of characteristics of systems. Keywords: Nonlinear hyperbolic systems, Cauchy problem, Gevrey classes     

Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides

We consider a domain Ω in ?ⁿ of the form Ω = ?^l × Ω′ with bounded Ω′ ? ?^n?l. In Ω we study the Dirichlet initial and boundary value problem for the equation ? u + [(? ? ?… ? ?)^m + (? ? ?… ? ?)^m]u = fe^?iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like t^α (α rational, 0<α<1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.     

Farthest-point queries with geometric and combinatorial constraints

In this paper we discuss farthest-point problems in which a set or sequence S of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. First, we give a data structure that can be used to compute the point farthest from a query line segment in O(log²n) time. Our data structure needs O(nlogn) space and preprocessing time. To the best of our knowledge no solution to this problem has been suggested yet. Second, we show how to use this data structure to obtain an output-sensitive query-based algorithm for polygonal path simplification. Both results are based on a series of data structures for fundamental farthest-point queries that can be reduced to each other.     

An adaptive finite element method for a time‐dependent Stokes problem

In this article, we conduct an a posteriori error analysis of the two‐dimensional time‐dependent Stokes problem with homogeneous Dirichlet boundary conditions, which can be extended to mixed boundary conditions. We present a full time–space discretization using the discontinuous Galerkin method with polynomials of any degree in time and the ? ₂ ? ?₁ Taylor–Hood finite elements in space, and propose an a posteriori residual‐type error estimator. The upper bounds involve residuals, which are global in space and local in time, and an L ²‐error term evaluated on the left‐end point of time step. From the error estimate, we compute local error indicators to develop an adaptive space/time mesh refinement strategy. Numerical experiments verify our theoretical results and the proposed adaptive strategy.     

The lower tail: Poisson approximation revisited

The well‐known “Janson's inequality” gives Poisson‐like upper bounds for the lower tail probability when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation‐based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case ) of Janson, ?uczak and Ruciński. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 219–246, 2016     

On Fictitious Domain Formulations for Maxwell''s Equations

We consider fictitious domain-Lagrange multiplier formulations for variational problems in the space H(curl: Ω{\bf)} derived from Maxwell's equations. Boundary conditions and the divergence constraint are imposed weakly by using Lagrange multipliers. Both the time dependent and time harmonic formulations of the Maxwell's equations are considered, and we derive well-posed formulations for both cases. The variational problem that arises can be discretized by functions that do not satisfy an a-priori divergence constraint.     

A note on proving the strong NP-hardness of some scheduling problems with start time dependent job processing times

In this paper, we show that the strong NP-hardness proofs of some scheduling problems with start time dependent job processing times presented in Gawiejnowicz (Eur J Oper Res 180:472–478, 2007) and Zhao and Tang (Optim Lett 5:183–190, 2011) are incorrect. Namely, the applied transformations from 4-Product problem to the considered scheduling problems are polynomial not pseudopolynomial. Thus, the related problems are NP-hard, but their complete computational status is still an open issue: ordinary or strongly NP-hard?     

Non-intersecting paths, random tilings and random matrices

 We investigate certain measures induced by families of non-intersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abc-hexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as the eigenvalue measures in random matrix theory like GUE, which can in fact can be obtained from non-intersecting Brownian motions. The derivations of the measures are based on the Karlin-McGregor or Lindstr?m-Gessel-Viennot method. We use the measures to show some asymptotic results for the models. Received: 1 December 2000 / Revised version: 20 May 2001 / Published online: 17 May 2002     

Automorphisms and forms of lie algebras of cartan type

Let L be any of simple Lie algebras of Cartan type L = X⁽²⁾(m : n : Φ), X = 5, H or K, n≠1, over a field F of characteristic p > 3, and R a commutative ring extension of F. Then AutR(R?L)≌ Aut R(R?21 (m : n) : R?L). It follows that, all forms of L are standard and thereby are determined