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1.
Temporal logic methods and algorithms are used for verifying programs. A method employing temporal semantic tables is proposed for studying the properties of dynamic processes. 相似文献
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We investigate read-once branching programs for the
following search problem: given a Boolean
m × n matrix with m > n, nd either an all-zero row, or two
1s in some column. Our primary motivation is that this models
regular resolution proofs of the pigeonhole principle
, and that for
m >
n
2
no lower bounds are known for the length of such proofs. We
prove exponential lower bounds (for arbitrarily large
m!) if we further restrict
this model by requiring the branching program
either to finish one row of
queries before asking queries about another row (the
row model)
or put the dual column
restriction (the column
model).Then we investigate a special class of resolution proofs
for
that operate with
positive clauses of rectangular shape; we call this fragment the
rectangular calculus. We show
that all known upper bounds
on the size of resolution proofs of
actually give rise to
proofs in this calculus and, inspired by this fact, also give a
remarkably simple rectangular reformulation of the
Haken–Buss–Turán lower bound for the case m
n
2. Finally
we show that the rectangular calculus is equivalent to the
column model on the one hand, and to transversal calculus on the other hand,
where the latter is a natural proof system for estimating from
below the transversal size of set families. In particular, our
exponential lower bound for the column model translates both to
the rectangular and transversal calculi.* Part of the work was done while this author was
visiting Special Year on Logic and Algorithms at DIMACS,
Princeton. Also supported by Russian Basic Research Foundation
grant 96-01-01222. Part of this work was done while on sabbatical
leave at the Institute for Advanced Study and Princeton
University, Princeton. This work was supported by USA-Israel BSF
grant 92-00106 and by a Wolfson research award administered by
the Israeli Academy of Sciences, as well as a Sloan Foundation
grant. This work was supported in part by National
Science Foundation and DARPA under grant CCR-9627819, and by
USA-Israel BSF grant 92-00106. 相似文献
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We show how B-series may be used to derive in a systematic way the analytical expressions of the high-order stroboscopic averaged equations that approximate the slow dynamics of highly oscillatory systems. For first-order systems we give explicitly the form of the averaged systems with O(ej)mathcal{O}(epsilon^{j}) errors, j=1,2,3 (2π ε denotes the period of the fast oscillations). For second-order systems with large O(e-1)mathcal{O}(epsilon^{-1}) forces, we give the explicit form of the averaged systems with O(ej)mathcal{O}(epsilon^{j}) errors, j=1,2. A variant of the Fermi–Pasta–Ulam model and the inverted Kapitsa pendulum are used as illustrations. For the former it is shown that our approach establishes the adiabatic invariance of the oscillatory energy. Finally we use B-series to analyze multiscale numerical integrators that implement the method of averaging. We construct integrators that are able to approximate not only the simplest, lowest-order averaged equation but also its high-order counterparts. 相似文献
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A wide class of Hamiltonian systems exhibit a mixture of slow motion with superimposed fast oscillations. Under the assumption of scale separation, these systems can be investigated using the principle of adiabatic invariance. In this paper, we start with a review of some of the main theoretical and numerical findings. We then briefly summarize a few important implications for molecular dynamics (MD) before we provide a more extensive discussion of numerical weather prediction (NWP). In particular, the conservative Hamiltonian particle-mesh (HPM) method is extended to Euler's equation and the fundamental concepts of geostrophic and hydrostatic balance are illustrated on the level of fluid blobs. We also demonstrate numerically that symplectic time-stepping methods are able to maintain hydrostatic balance to high accuracy. 相似文献
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The paper considers non-autonomous oscillatory systems of ordinary differential equations with d≥1 non-resonant constant frequencies ω 1,…,ω d . Formal series like those used nowadays to analyze the properties of numerical integrators are employed to construct higher-order averaged systems and the required changes of variables. With the new approach, the averaged system and the change of variables consist of vector-valued functions that may be written down immediately and scalar coefficients that are universal in the sense that they do not depend on the specific system being averaged and may therefore be computed once and for all given ω 1,…,ω d . The new method may be applied to obtain a variety of averaged systems. In particular, we study the quasi-stroboscopic averaged system characterized by the property that the true oscillatory solution and the averaged solution coincide at the initial time. We show that quasi-stroboscopic averaging is a geometric procedure, because it is independent of the particular choice of co-ordinates used to write the given system. As a consequence, quasi-stroboscopic averaging of a canonical Hamiltonian (respectively, of a divergence-free) system results in a canonical (respectively, in a divergence-free) averaged system. We also study the averaging of a family of near-integrable systems where our approach may be used to construct explicitly d formal first integrals for both the given system and its quasi-stroboscopic averaged version. As an application we construct three first integrals of a system that arises as a nonlinear perturbation of five coupled harmonic oscillators with one slow frequency and four resonant fast frequencies. 相似文献
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Nobito Yamamoto Mitsuhiro T. Nakao Yoshitaka Watanabe 《Numerical Functional Analysis & Optimization》2013,34(11):1190-1204
We give a theoretical result with respect to numerical verification of existence and local uniqueness of solutions to fixed-point equations which are supposed to have Fréchet differentiable operators. The theorem is based on Banach's fixed-point theorem and gives sufficient conditions in order that a given set of functions includes a unique solution to the fixed-point equation. The conditions are formulated to apply readily to numerical verification methods. We already derived such a theorem in [11], which is suitable to Nakao's methods on numerical verification for PDEs. The present theorem has a more general form and one may apply it to many kinds of differential equations and integral equations which can be transformed into fixed-point equations. 相似文献
11.
An Efficient Approach to the Numerical Verification for Solutions of Elliptic Differential Equations
The authors and their colleagues have developed numerical verification methods for solutions of second-order elliptic boundary value problems based on the infinite-dimensional fixed-point theorem using the Newton-like operator with appropriate approximation and constructive a priori error estimates for Poisson's equations. Many verification results show that the authors' methods are sufficiently useful when the equation has no first-order derivative. However, in the case that the equation includes the term of a first-order derivative, there is a possibility that the verification algorithm does not work even though we adopt a sufficiently accurate approximation subspace. The purpose of this paper is to propose an alternative method to overcome this difficulty. Numerical examples which confirm the effectiveness of the new method are presented. 相似文献
12.
运用Schwarz-Christoffel变换方法,建立多边形区域到带状区域共形映射数学模型.对于模型中的约束条件和奇异积分问题,根据Riemann(黎曼)原理,建立复参数与实参数互逆变换,消除非线性系统的约束条件;经过合理积分路径的确定,模型中的奇异积分转化为Gauss-Jacobi(高斯 雅可比)型积分;采用Levenberg-Marquardt算法对非线性系统模型进行求解.根据第一类椭圆函数性质,建立了矩形区域到带状区域共形映射数学模型,通过复参数椭圆函数的计算,得到矩形边界与带状区域边界的关系.最后,对8点对称多边形区域与27点不规则条带状区域计算,将不规则封闭区域边界映射到矩形区域边界,矩形区域内的正交网格,通过变换之后在多边形区域内依然满足正交性,为研究不规则区域到规则区域映射的数值计算奠定基础. 相似文献
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K. Nagatou N. Yamamoto M. T. Nakao 《Numerical Functional Analysis & Optimization》2013,34(5-6):543-565
We propose a numerical method to verify the existence and local uniqueness of solutions to nonlinear elliptic equations. We numerically construct a set containing solutions which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space. By using the finite element approximation and constructive error estimates, we calculate the eigenvalue bound with smallest absolute value to evaluate the norm of the inverse of the linearized operator. Utilizing this bound we derive a verification condition of the Newton-Kaiitorovich type. Numerical examples are presented. 相似文献
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Shuting Cai Kaori Nagatou Yoshitaka Watanabe 《Numerical Functional Analysis & Optimization》2013,34(10):1195-1220
We propose a numerical method to enclose a solution of the FitzHugh-Nagumo equation with Neumann boundary conditions. We construct, on a computer, a set which satisfies the hypothesis of Schauder's fixed point theorem for a compact map in a certain Sobolev space, which, therefore contains a solution. Several verified results are presented. 相似文献
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Kouji Hashimoto Kenta Kobayashi Mitsuhiro T. Nakao 《Numerical Functional Analysis & Optimization》2013,34(4-5):523-542
We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given. 相似文献
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A technique is presented in this paper to verify the order of accuracy of asymptotic expansion of Van der Pol's equation. The technique is focused on using numerical solutions as an independent means of verifying the validity of asymptotic expansions. 相似文献