Numerical evaluation of gauge invariants for <Emphasis Type="Italic">a</Emphasis>-gauge solutions in open string field theory

We explore the vacuum structure in the bosonic open string field theory expanded near an identity-based solution parameterized by a (≥ −1/2). Analyzing the expanded theory using the level-truncation approximation up to the level 20, we find that the theory has the tachyon vacuum solution for a ≥ −1/2. We also find that at a = −1/2, there exists an unstable vacuum solution in the expanded theory and the solution is expected to be the perturbative open string vacuum. These results reasonably support the hypothesis that the identity-based solution is a trivial pure gauge configuration for a > −1/2, but it can be regarded as the tachyon vacuum solution at a = −1/2.     

Stefan problem

We prove the existence of a global classical solution of the multidimensional two-phase Stefan problem. The problem is reduced to a quasilinear parabolic equation with discontinuous coefficients in a fixed domain. With the help of a small parameter ε, we smooth coefficients and investigate the resulting approximate solution. An analytical method that enables one to obtain the uniform estimates of an approximate solution in the cross-sections t = const is developed. Given the uniform estimates, we make the limiting transition as ε → 0. The limit of the approximate solution is a classical solution of the Stefan problem, and the free boundary is a surface of the class H ^2+α,1+α/2.     

Existence and stability of a summable generalized solution to a mathematical model of a catalytic fluidized bed process

For a system of first-order partial differential equations describing a catalytic process in a fluidized bed, we consider a mixed problem in the half-strip 0 ≤ x ≤ h, t ≥ 0. We prove the existence and uniqueness of a bounded summable generalized solution and study its stability. We prove the stabilization as t → ∞ of the values of some physically meaningful functionals of the solution.     

The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its H?lder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.     

Asymptotic behavior of the solution to the two-dimensional stationary problem of flow past a body far from it

In the exterior domain Ω⊂ℝ² we consider the two-dimensional Navier-stokes system Δu-▽p=(u,▽)u, div u=0 whose solution possesses a finite Dirichlet integral and satisfies the condition lim_|x|→∞ u(x)=(1, 0). For this solution, we establish the estimate |u(x)−(1, 0)|≤c|x| ^−α, where α>1/4. This estimate implies an asymptotic expression for the solution indicating the presence of a track behind the body. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 246–253, February, 1999.     

On the covergence of solutions of parabolic equations with rapidly oscillating coefficients in perforated domains

The behavior of the solution of a boundary value problem for a parabolic equation with rapidly oscillating coefficientsɛ ⁻¹ x,ɛ ^−2k t), (k⋝0) in a perforated domain for ε→0 is studied. Some estimates of the deviation of the solution and energy for the original boundary value problem from the solution and energy of the corresponding homogenized problem are found. In this investigation methods developed by Oleinik, Zhikov, Kozlov, Bensoussan, Lions, Papanikolaou, Cioranescu, and Paulin are used. Bibliography: 15 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 27–50, 1994.     

Disagreement point axioms and the egalitarian bargaining solution

We provide new characterizations of the egalitarian bargaining solution on the class of strictly comprehensive n-person bargaining problems. The main axioms used in all of our results are Nash’s IIA and disagreement point monotonicity—an axiom which requires a player’s payoff to strictly increase in his disagreement payoff. For n = 2 these axioms, together with other standard requirements, uniquely characterize the egalitarian solution. For n > 2 we provide two extensions of our 2-person result, each of which is obtained by imposing an additional axiom on the solution. Dropping the axiom of anonymity, strengthening disagreement point monotonicity by requiring player i’s payoff to be a strictly decreasing function of the disagreement payoff of every other player j ≠ i, and adding a “weak convexity” axiom regarding changes of the disagreement point, we obtain a characterization of the class of weighted egalitarian solutions. This “weak convexity” axiom requires that a movement of the disagreement point in the direction of the solution point should not change the solution point. We also discuss the so-called “transfer paradox” and relate it to this axiom.     

Analytic study on angular fields of HRR solution

By using the constructing function method, a systematic and strict analysis is carried out on the angular distribution field near a crack tip in a power-law hardening material and the analytic solution is provided for HRR problem. In addition, the equivalence of H equation and RR equation is proved. The present analytic solutions for HRR problem can reduce to (he well-known Williams solution in the limit case ofN→1 (orn→1) and Prandtl solution in the limit case ofN→0 (orn→∞). It is particularly interesting that from the deformation theory of plasticity one obtains the Prandtl solution based on the increatment theory of plasticity. Project supported by the National Natural Science Foundation of China (Grant No. 19132022).     

Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation

We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied earlier by the author and present two classes of special functions, namely, ultraexponential and infralogarithm f -type functions. As a result of this investigation, we obtain a general solution of the Abel equation α(f(x)) = α (x) + 1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that an infralogarithm f -type function is its unique solution. We also show that an infralogarithm f -type function is an essentially unique solution of the Abel equation. Similar theorems are proved for ultraexponential f -type functions and their functional equation β(x) = f(β(x − 1)), which can be considered as dual to the Abel equation. We also solve a certain problem unsolved before and study some properties of two considered functional equations and some relations between them.     

New approach to the numerical solution of forward-backward equations

This paper is concerned with the approximate solution of functional differential equations having the form: x′(t) = αx(t) + βx(t - 1) + γx(t + 1). We search for a solution x, defined for t ∈ [−1, k], k ∈ ℕ, which takes given values on intervals [−1, 0] and (k-1, k]. We introduce and analyse some new computational methods for the solution of this problem. Numerical results are presented and compared with the results obtained by other methods.      

Optimal control of a coefficient in modification Navier-Stokes equations

This paper deals with the optimal control of a coefficient in the modification of Navier-Stokes equations. Namely, the motion of the viscous incompressible fluid for a small gradient of velocity is described by Navier-Stokes equations where the coefficient of the kinematic viscosity ν is the positive constant (ν ₀). For a greater gradient of velocity the coefficient of kinematic viscosity is a positive function of the gradient of velocity, that is ν (|∇u|). In our case ν (|∇u|) = ν ₀ + ν ₁ a (|∇u|) where ν ₀, ν ₁ ∈ ℝ₊. The function a is positive and monotone and it is taken as a control variable. The existence of a solution of the optimal control problem is proved. Further, the approximation of the control problem by the finite-dimensional control problem is performed. The proof of the existence of a solution of that aproximate problem has been brought into light. Finally, the connection between the solution of the control problem and the solution of the approximate control problem is established.     

Two-phase Stefan problem with vanishing specific heat

The unique solvability of the two-phase Stefan problem with a small parameter ε ∈ [0; ε ₀] at the time derivative in the heat equations is proved. The solution is obtained on a certain time interval [0; t ₀] independent of ε. The solution of the Stefan problem is compared with the solution to the Hele–Shaw problem corresponding to the case ε = 0. The solutions of both problems are not assumed to coincide at the initial moment of time. Bibliography: 18 titles. Dedicated to Vsevolod Alekseevich Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 337–363.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

Maximal solutions of semilinear elliptic equations with locally integrable forcing term

We study the existence of a maximal solution of −Δu+g(u)=f(x) in a domain Ω ∈ ℝ ^N with compact boundary, assuming thatf ∈ (L _loc ¹ (Ω))₊ and thatg is nondecreasing,g(0)≥0 andg satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classicalC _1,2 Wiener criterion, then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition, we discuss the question of uniqueness of large solutions. This research was partially supported by an EC Grant through the RTN Program “Front-Singularities”, HPRN-CT-2002-00274.     

Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method

In the case of the Dirichlet problem for a singularly perturbed ordinary differential reaction-diffusion equation, a new approach is used to the construction of finite difference schemes such that their solutions and their normalized first- and second-order derivatives converge in the maximum norm uniformly with respect to a perturbation parameter ɛ ∈(0, 1]; the normalized derivatives are ɛ-uniformly bounded. The key idea of this approach to the construction of ɛ-uniformly convergent finite difference schemes is the use of uniform grids for solving grid subproblems for the regular and singular components of the grid solution. Based on the asymptotic construction technique, a scheme of the solution decomposition method is constructed such that its solution and its normalized first- and second-order derivatives converge ɛ-uniformly at the rate of O(N ⁻²ln² N), where N + 1 is the number of points in the uniform grids. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed such that its solution and its normalized first and second derivatives converge ɛ-uniformly in the maximum norm at the same rate of O(N ⁻⁴ln⁴ N).     

Positive versus free boundary solutions to a singular elliptic equation

The equation-δu = χ uo(-1/u^Β + λf (x, u)) in Ω with Dirichlet boundary condition on ∂Ω has a maximal solution uλ ≥0 for every λ 0. For λ less than a constant λ*, the solution vanishes inside the domain; and for λ λ*, the solution is positive. We obtain optimal regularity ofuλ even in the presence of the free boundary. Supported in part by H. J. Sussmann’s NSF Grant DMS01-03901. Supported by FAPESP. He also thanks Rutgers University.     

An efficient algorithm for the generalized centro-symmetric solution of matrix equation <Emphasis Type="Italic">A X B</Emphasis> = <Emphasis Type="Italic">C</Emphasis>

In this paper, an iterative algorithm is constructed for solving linear matrix equation AXB = C over generalized centro-symmetric matrix X. We show that, by this algorithm, a solution or the least-norm solution of the matrix equation AXB = C can be obtained within finite iteration steps in the absence of roundoff errors; we also obtain the optimal approximation solution to a given matrix X ₀ in the solution set of which. In addition, given numerical examples show that the iterative method is efficient.     

A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local dirichlet problems

In this paper, we propose a new method to compute the numerical flux of a finite volume scheme, used for the approximation of the solution of the nonlinear partial differential equation u_t+div(qf(u))−ΔΦ(u)=0 in a 1D, 2D or 3D domain. The function Φ is supposed to be strictly increasing, but some values s such that Φ′(s)=0 can exist. The method is based on the solution, at each interface between two control volumes, of the nonlinear elliptic two point boundary value problem (qf(υ)+(Φ(υ))′)′=0 with Dirichlet boundary conditions given by the values of the discrete approximation in both control volumes. We prove the existence of a solution to this two point boundary value problem. We show that the expression for the numerical flux can be yielded without referring to this solution. Furthermore, we prove that the so designed finite volume scheme has the expected stability properties and that its solution converges to the weak solution of the continuous problem. Numerical results show the increase of accuracy due to the use of this scheme, compared to some other schemes.     

A two-space dimensional semilinear heat equation perturbed by (Gaussian) white noise

A two-space dimensional heat equation perturbed by a white noise in a bounded volume is considered. The equation is perturbed by a non-linearity of the type λ : f(AU) :, where :: means Wick (re)ordering with respect to the free solution;λ, A are small parameters, U denotes a solution, f is the Fourier transform of a complex measure with compact support. Existence and uniqueness of the solution in a class of Colombeau-Oberguggenberger generalized functions is proven. An explicit construction of the solution is given and it is shown that each term of the expansion in a power series in λ is associated with an L ²-valued measure when A is a small enough. Received: 20 July 1997 / Revised version: 1 February 2001 / Published online: 9 October 2001     

Analysis of search-extension method for finding multiple solutions of nonlinear problem

For numerical computations of multiple solutions of the nonlinear elliptic problemΔu f(u)=0 inΩ, u=0 onΓ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u∈H~(1 α),α>0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.