On transformations of constant depth propositional proofs

This paper studies the complexity of constant depth propositional proofs in the cedent and sequent calculus. We discuss the relationships between the size of tree-like proofs, the size of dag-like proofs, and the heights of proofs. The main result is to correct a proof construction in an earlier paper about transformations from proofs with polylogarithmic height and constantly many formulas per cedent.     

Tusnady's Lemma, 24 years later

The optimal coupling between a variable with the Bin(n,1/2) distribution and a normal random variable lies at the heart of the proof of the KMT Theorem for the empirical distribution function. Tusnády's Lemma (published in 1977 in his dissertation and in Hungarian) provides an inequality with explicit absolute constants which says that for this coupling, the distance between the random variables remains bounded in probability. In the appendix of a joint work with Jean Bretagnolle (1989), we have proposed a proof of Tusnády's Lemma which though elementary is highly technical and considered as rather obscure, at least this is what we have understood from several conversations with motivated readers. The purpose of this paper is to provide an alternative proof which is still based on elementary computations but which we hope to be simpler and more illuminating. This new proof also leads to a slight improvement on the original result in terms of constants.     

Exact Boundary Controllability of Electromagnetic Fields in a General Region

We prove exact controllability for Maxwell's system with variable coefficients in a bounded domain by a current flux in the boundary. The proof relies on a duality argument which reduces the proof of exact controllability to the proof of continuous observability for the homogeneous adjoint system. There is no geometric restriction imposed on the domain. Accepted 25 June 2001. Online publication 19 October 2001.     

Corrections to the paper “The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces”

In this paper the author proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with variable exponent. As an application he proved the boundedness of certain sublinear operators on the weighted variable Lebesgue space. The proof of the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent does not contain any mistakes. But in the proof of the boundedness of certain sublinear operators on the weighted variable Lebesgue space Georgian colleagues discovered a small but significant error in my paper, which was published as R.A.Bandaliev, The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces, Czech. Math. J. 60 (2010), 327–337.     

The polynomial bounds of proof complexity in Frege systems

The concept of a determinative set of variables for a propositional formula was introduced by one of the authors, which made it possible to distinguish the set of hard-determinable formulas. The proof complexity of a formula of this sort has exponential lower bounds in some proof systems of classical propositional calculus (cut-free sequent system, resolution system, analytic tableaux, cutting planes, and bounded Frege systems). In this paper we prove that the property of hard-determinability is insufficient for obtaining a superpolynomial lower bound of proof lines (sizes) in Frege systems: an example of a sequence of hard-determinable formulas is given whose proof complexities are polynomially bounded in every Frege system.     

Boundedness of solutions to a time-dependent semiconductor model

In this paper we obtain the boundedness of solutions to a time-dependent semiconductor model with variable electron mobility. The proof is based upon an interpolation inequality which is of interest on its own right.     

The basis suppression method

The Basis Suppression algorithm is a simplex-based procedure which allows the efficient extension of current special structure algorithms to problems of special structure except for a single complicating side variable. A basis free of the complicating variable is maintained in this algorithm. Various properties of the algorithm are presented, including a proof of convergence. Computational effectiveness is discussed and has been verified by using the procedure to solve the maximal concurrent flow problem.     

Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension

In this paper, we consider the semilinear wave equation with a power nonlinearity in one space dimension. We exhibit a universal one-parameter family of functions which stand for the blow-up profile in self-similar variables at a non-characteristic point, for general initial data. The proof is done in self-similar variables. We first characterize all the solutions of the associated stationary problem, as a one parameter family. Then, we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm, in the non-characteristic case, and to a finite decoupled sum of such a solution in the characteristic case. Finally, in the case where this sum is reduced to one element, which is the case for non-characteristic points, we use modulation theory coupled with a nonlinear argument to show the exponential convergence (in the self-similar time variable) of the various parameters and conclude the proof. This step provides us with a result of independent interest: the trapping of the solution in self-similar variables near the set of stationary solutions, valid also for non-characteristic points. The proof of these results is based on a new analysis in the self-similar variable.     

对复合函数求导定理证明的评论与研究

本文指出了一些广泛使用的教科书对复合函数求导定理的证明是不严格，并创造性地举例阐明存在这样的复合函数f(u)，它对自变量u可导，而对中间变量u=g(x)不可导。本文也给出了该定理的一个严格证明，同时揭示了复合函数求导法则的表达形式与其内容并不完全一致。     

Semidefinite resolution and exactness of semidefinite relaxations for satisfiability

This paper explores new connections between the satisfiability problem and semidefinite programming. We show how the process of resolution in satisfiability is equivalent to a linear transformation between the feasible sets of the relevant semidefinite programming problems. We call this transformation semidefinite programming resolution, and we demonstrate the potential of this novel concept by using it to obtain a direct proof of the exactness of the semidefinite formulation of satisfiability without applying Lasserre’s general theory for semidefinite relaxations of 0/1 problems. In particular, our proof explicitly shows how the exactness of the semidefinite formulation for any satisfiability formula can be interpreted as the implicit application of a finite sequence of resolution steps to verify whether the empty clause can be derived from the given formula.     

Existence and Uniqueness of Entropy Solution of Scalar Conservation Laws with a Flux Function Involving Discontinuous Coefficients

ABSTRACT

In this paper, the question of existence and uniqueness for entropy solutions of scalar conservation laws with a flux function which is discontinuous with respect to the space variable is investigated. We show that no extra assumption of convexity or genuine non-linearity with respect to the state variable of the flux function is required for the problem to be well-posed and prove it. The proof uses a kinetic formulation of the conservation law.     

Elementary Exponential Error Estimates for the Adiabatic Approximation

We present an elementary proof that the quantum adiabatic approximation is correct up to exponentially small errors for Hamiltonians that depend analytically on the time variable. Our proof uses optimal truncation of a straightforward asymptotic expansion. We estimate the terms of the expansion with standard Cauchy estimates.     

Proving programs correct: Some techniques and examples

Proving the correctness of computer programs is justified as both advantageous and feasible. The discipline of proof provides a systematic search for errors, and a completed proof gives sufficient reasons why the program must be correct. Feasibility is demonstrated by exhibiting proofs of five pieces of code. Each proof uses one or more of the illustrated proof techniques of case analysis, assertions, mathematical induction, standard prose proof, sectioning and a table of variable value changes. Proofs of other programs, some quite lengthy, are cited to support the claim that the techniques work on programs much larger than the examples of the paper. Hopefully, more programmers will be encouraged to prove programs correct.     

Fredholm property of non-smooth pseudodifferential operators

In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator P which symbol is in a Hölder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.     

《概率论与数理统计》中有关独立性一反例的深入分析

独立性是《概率论与数理统计》是的一个非常重要的概念.教学中在说明随机变量函数独立性时会涉及许多反例.本文就有关随机变量函数独立性的一个反例作了进一步的推广分析.     

A stochastic representation for backward incompressible Navier-Stokes equations

By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer’s forward formulations in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero.     

Uniqueness in One Inverse Problem for the Elasticity System

We consider an inverse problem for the stationary elasticity system with constant Lame coefficients and a variable matrix coefficient depending on the spatial variables and frequency. The right-hand side contains a delta-function whose support (source) varies in some domain disjoint from the support of the variable coefficient. The inverse problem is to find the coefficient from the scattered wave measured at the same point at which the perturbation originates. A uniqueness theorem is proven. The proof bases on reduction of the inverse problem to a family of equations with the M. Riesz potential.     

Resolutions of monomial ideals of projective dimension 1

We show that a monomial ideal I in a polynomial ring S has projective dimension ≤ 1 if and only if the minimal free resolution of S∕I is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the S∕I. We also provide a new characterization of quasi-trees, which we use to give a new proof to a result by Herzog, Hibi, and Zheng which characterizes monomial ideals of projective dimension 1 in terms of quasi-trees.     

On the Trotter-Kato theorem in a variable space

In this paper, the proof of a Trotter-Kato type theorem in a variable Banach space is given and some special cases and examples are considered.     

Backward Uniqueness for Thermoelastic Plates with Rotational Forces

This paper provides a backward uniqueness theorem for thermoelastic plate models which account for rotational forces, under all sets of canonical boundary conditions, including the most challenging case of so-called free boundary conditions. The proof is abstract and accomodates space variable coefficients in the model. This result is derived in two steps. First, in Section 3, a new backward uniqueness theorem for strongly continuous semigroups is given, which is of interest in itself. It is based on the assumption that the resolvent operator of the generator be bounded on suitable rays of the complex plane. Its proof uses the Phragmen-Lindelof Theorem. Next, the paper verifies a fortiori the required resolvent conditions, under all sets of canonical boundary conditions. The explicit proof (in Section 4) considers the most demanding case of free boundary conditions. An abstract version of this proof, and a corresponding backward uniqueness result, are then noted in Section 5, which gives the most general result of this paper. It covers thermoelastic wave equations as well. The results here presented were motivated by, and hence have important implications in, continuous observability/exact controllability problems for thermoelastic plates, and boundary observations/controls, see [8]. March 12, 1999