Some remarks on Smale's “Algorithms for solving equations”

In his report at ICM, Smale posed a problem whether the inequalityα(t, ψ _d )≤1 holds for allt∈(0,1), where . This paper gives a negative answer. In addition, elementary proofs are given for the theorems on the tractability of random algorithm and on the average area of approximate zeros. Meanwhile, discussion about the algorithm of the global Newton's method is devoted in this paper. Supported by the National Natural Science Foundation of China, and of Zhejiang Province Dedicated to the 70th birthday of Prof. Feng Kang     

A modification of the method of linearization for solving a mathematical programming problem on a simple set of “parallelepiped” type

We propose a modification of B. N. Pshenichnyi's method of linearization for solving mathematical programming problems on a simple set of parallelepiped type. We study in detail the computational aspects and the efficiency of the approach in question. Two tables. Bibliography: 3 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 24–28.     

A note on “A new method for solving fully fuzzy linear programming problems”

This note shows that solving fully fuzzy linear programming (FFLP) model presented by Kumar et al. [A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzy linear programming problems, Appl. Math. Model. 35 (2011) 817–823] needs some corrections to make the model well in general. A new version is provided in this note. A simple example is also presented to demonstrate the new form.     

Effect of the CYBER 205 on the choice of method for solving the eigenvalue problem (A − λM)x = 0

We consider the generalized eigenvalue problem. (A − λM)x = 0, where A and M are large, sparse, symmetric matrices. For large problems finding only a few eigenpairs involves a major computational task. In a typical example from structural dynamic analysis with matrices of order 8000, O(10⁹) operations are required to compute 50 eigenpairs. It is therefore interesting to examine the advantage that vector computers such as CYBER 205 can offer.We adopted our best versions of the Subspace Iteration Method and the simple Lanczos Method in order to take advantage of the special vector processor of the CYBER 205. Both techniques lend themselves to vectorization. Our extensive comparisons support the following general statements. Both methods require the triangular factorization of the same large n by n matrix. This factorization dominates the total computation as n → ∞ provided that the number of wanted eigenpairs, p, remains fixed (independent of n). However, simple Lanczos is at least an order of magnitude more efficient (in CPU-time) for the remainder of the computation. For p = 40, n = 500 the factorization time is not important and the full order of magnitude difference is seen in the total CPU-time. When p = 40, n = 8000 simple Lanczos is only 4 times faster than Subspace Iteration on the CYBER 205. This confirms experience on serial computers.For problems that cannot fit into primary storage, input/output becomes increasingly important. We found that the cost of input/output dominated over the CPU-cost for a problem that required twice the available primary storage on our CYBER 205. However, this will depend on the billing algorithm of the computer center. We conclude that problems which have a substantial overhead in reading and writing the matrices, should not be solved by the simple Lanczos Method, but by a Block Lanczos Method.     

Session of the seminar “algebra i logika”

Session of the seminar algebra i logika     

Sessions of the seminar “algebra i logika”

Sessions of the seminar algebra i logika     

Characters of representations of the quantum toroidal algebra : Plane partitions with “stands”

An expression for the generating function of plane partitions a _i,j subject to the constraints a _m,n = 0 and a _i,j ? k _j , 1 ? j ? n, which is the character of an irreducible representation of the quantum toroidal algebra , is obtained.     

Statistical algorithms for solving the Cauchy problem for second-order parabolic equations: The “dual” scheme

This paper is a continuation of [A. S. Sipin, “Statistical Algorithms for Solving the Cauchy Problem for Second-Order Parabolic Equations,” Vestn. S.-Peterburg. Univ., Mat. Mekh. Astron., No. 3, 65–74 (2011)]. A new algorithm of the Monte Carlo method for solving the Cauchy problem for a second-order parabolic equation with smooth coefficients is considered. Unbiased estimators for functionals of the solutions of this problem are constructed. Unlike in the paper cited above, the “dual” scheme of constructing unbiased estimators for functionals of the solutions of an integral equation equivalent to the Cauchy problem is considered. This simplifies the modeling procedure, because the boundaries of the spectrum for the matrix of the leading coefficients in the equation are not required to be known.     

Methods for solving systems of linear and convex inequalities based on the Fejér principle

The paper contains a brief review of the author’s results concerning the technique of constructing Fejér contraction mappings, which are used in iterative processes of solving linear and convex systems of inequalities as well as accompanying optimization problems. The general approach is based on the notion of M-Fejér step “p → q” defined by the property

$$\left| {q - y} \right| < \left| {p - y} \right|,\forall y \in M$$

. This property (postulate) assumes the existence of a point p ∈ \(\overline {convM} \) and of a proper sufficiently arbitrary point q. Some of the problems considered in the paper are illustrated by schemes reflecting the analytics of these problems.

     

Bounds on the degree of APN polynomials: the case of x ?1?+?g(x)

In this paper we consider APN functions ${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$ of the form f(x) = x ^?1 + g(x) where g is any non ${\mathcal{F}_{2}}$ -affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields ${\mathcal{F}_{2^m}}$ . Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ?? 3 where these functions are equivalent to x ³.     

Study on convergence and error of a numerical method for solving systems of nonlinear Fredholm–Volterra integral equations of Hammerstein type

The aim of this paper is to present an efficient analytical and numerical procedure for solving systems of nonlinear Fredholm–Volterra integral equations of the Hammerstein type with the aid of fixed point techniques and the usual Schauder basis in an adequate Banach space.     

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : I. Asymptotic Symmetry for the Cauchy Problem

We consider quasilinear parabolic equations on ? ^N satisfying certain symmetry conditions. We prove that bounded positive solutions decaying to zero at spatial infinity are asymptotically radially symmetric about a center. The asymptotic center of symmetry is not fixed a priori (and depends on the solution) but it is independent of time. We also prove a similar theorem on reflectional symmetry.     

“Compactly” supported frames for spaces of distributions on the ball

Frames are constructed on the unit ball B ^d in ${\mathbb{R}^d}$ consisting of smooth functions with small shrinking supports. The new frames are designed so that they can be used for decomposition of weighted Triebel–Lizorkin and Besov spaces on B ^d with weight ${w_\mu(x):=(1-|x|^2)^{\mu-1/2}, \mu}$ half integer,?μ?≥ 0.     

On the Periodic Structure of Delayed Difference Equations of the Form x n = f ( x n − k ) on I and S 1

The aim of this paper is to give an account of some results recently obtained in Combinatorial Dynamics and apply them to get for k S 2 the periodic structure of delayed difference equations of the form x n = f ( x n m k ) on I and S 1 .