The Stablility of a Loop Formed the Separatrix of a Saddle Point and the Condition to Produce a Limit Cycle

In this paper,we discuss a loop by the separatrix of a saddle point of two dimensional autonmoussystem.It contains two sections.In section 1,a short proof of the criterion theorem of stability ofthe loop of saparatrix is given wi~h ~he help o~th~ ~h~orjr 0~ i'otatin~ vector field, and some in~ormatioafor the study of stability of that loop in critical case is supplied.In section 2,we give a sufficientcondition under which a limit cycle is produced from the loop of separatrix.     

Controllability for a parabolic equation with a nonlinear term involving the state and the gradient

In this paper, we prove the controllability of a quasi-linear heat equation involving gradient terms with Fourier boundary conditions in a bounded domain of ? ^N . The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.     

On the Truth of a Conjecture Concerning the q-Derangement Numbers

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On a transformation of the wiener process in ℝ m by a functional of the local time type on a surface

A transformation of the Wiener process _t in ^m is considered. This transformation is realized by a multiplicative functional _l=u(_l/u(₀), where the functionu is constructed in a certain way by using a functional of the local time type on a surface. It is proved that this transformation is equivalent to the successive application of an absolutely continuous change of a measure and killing on the surface.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 863–866, June, 1993.     

When is the commutant of a Bol loop a subloop?

A left Bol loop is a loop satisfying . The commutant of a loop is the set of elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order , odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to , the commutant generates an abelian group of order dividing the order of the loop. This generalizes a well-known result for Moufang loops. After describing all extensions of a loop such that is in the left and middle nuclei of the resulting loop, we show how to construct classes of Bol loops with a non-subloop commutant. In particular, we obtain all Bol loops of order with a non-subloop commutant.

     

Can a ground-based vehicle hear the shape of a room?

Assume that a ground-based vehicle moves in a room with walls or other planar surfaces. Can the vehicle reconstruct the positions of the walls from the echoes of a single sound event? We assume that the vehicle carries some microphones and that a loudspeaker is either also mounted on the vehicle or placed at a fixed location in the room. We prove that the reconstruction is almost always possible if (1) no echoes are received from floors, ceilings, or sloping walls and the vehicle carries at least three noncollinear microphones, or if (2) walls of any inclination may occur, the loudspeaker is fixed in the room and there are four noncoplanar microphones. The difficulty lies in the echo-matching problem: How to determine which echoes come from the same wall. We solve this by using a Cayley–Menger determinant. Our proofs use methods from computational commutative algebra.     

Inequality for the Moment of a Function of a Random Variable

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On the Covering of the Vertices of a Graph by Cliques

In conversation I was told by Professor R.Brigham the following conjecture [1].Let G(n) be a graph of n vertices.Denote by f(G(n))=t the smallest integer for which the vertices of G(n) can be covered by t cliques. Denote further by h(G(n)) =l the largest integer for which there are l edges of our G(n) no two of Which are in the same clique.Clearly h(G(n)) can be much larger than f(G(n))e.g.if n=2m and G(n) is the complete bipartite graph of m white and m black vertices.Then l(G(n))=m and l(G(n))=m~2. It was conjectured that if G(n).has no isolated vertices then     

Correction to the paper “on the curvature of a generalization of a contact metric manifolds”

We considered in Example 3.1 of the paper [1] an S-structure on R^2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.     

Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Maz'ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

What Is the Object of the Encapsulation of a Process?

Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this “object” produced by the “encapsulation” of a process? Here, we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray, and Tall) and consider the nature of the mental objects (apparently) produced through encapsulation and their role in the wider development of mathematical thinking. Does the same developmental route occur in geometry as in arithmetic and algebra? Is the same development used in axiomatic mathematics? What is the role played by imagery?     

Expansion in the eigenfunctions of the schrödinger equation with a potential having a periodic asymptotic behavior

A theorem is proved regarding the expansion in the eigenfunctions of the one-dimensional Schrödinger equationL = ?d ^z/dx ²+q(x)(?∞<x<∞)with a potential q(x), satisfying the condition $$\int\limits_0^{ + \infty } {(1 + x^2 )|q(x) - q_ \pm (x)|dx< \infty ,} $$ where q±(x) are periodic functions.     

On the traces of Sobolev functions on the boundary of a cusp with a Hölder singularity

For the Sobolev classes W _p ¹ on a “zero” cusp with a Hölder singularity at the vertex, we consider the question of compactness of the embedding of the traces of Sobolev functions into the Lebesgue classes on the boundary of the cusp.     

Estimating the Volumes of the Cones in a Gröbner Fan

We present a stochastic method for estimating the relative volumes of the Gröbner cones of a Gröbner fan without computing the actual fan. The method is particularly useful when the dimension of the Gröbner fan is large and/or the volumes of several or all cones need to be estimated. A Macaulay 2 implementation for uniform sampling from the Gröbner fan is provided by the author.     

Rate-of-convergence bounds of a finite-difference approximation of the Dirichlet problem for the Helmholtz equation with a coefficient from the “negative” space

Exact difference scheme operators are used to construct a difference scheme that approximates the Dirichlet problem for the equation

On the Properties of the Solution of a Strongly Degenerate Parabolic Equation

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