Brief Announcement

In metric Diophantine approximation, one frequently encounters the problem of showing that a limsup set has positive or full measure. Often it is a set of points in m m -dimensional Euclidean space, or a set of n n -by- m m systems of linear forms, satisfying some approximation condition infinitely often. The main results of this paper are bootstraps: if one can establish positive measure for such a limsup set in m m -dimensional Euclidean space, then one can establish positive or full measure for an associated limsup set in the setting of n n -by- m m systems of linear forms. Consequently, a class of m m -dimensional results in Diophantine approximation can be bootstrapped to corresponding n n -by- m m -dimensional results. This leads to short proofs of existing, new, and hypothetical theorems for limsup sets that arise in the theory of systems of linear forms. We present several of these.

Helices and constant procession curves are special examples of slant curves. However, there is no example of a $k$-slant curve for a positive integer $k\geq 2$ in three dimensional Euclidean spaces. Furthermore, the position vector of a $k$-slant curve for a positive integer $k\geq 2$ has not been known thus far. In this paper, we propose a method for constructing $k$-slant curves in three dimensional Euclidean spaces. We then show that spherical $k$-slant curves and $N_{k}$-constant procession curves can be derived from circles, for $k \in \mathbb{N}$, the set of all nonnegative integers. In addition, we provide a new proof of the spherical curve characterization and define a curve in the sphere called a spherical prime curve. Afterward, we apply $k$-slant curves to magnetic curves. Finally, we discuss the need for further research.

The best constant of Sobolev inequality in an $n$ dimensional Euclidean space is found by means of the theory of reproducing kernel and Green function. The concrete form of the best constant is also found in the case of Sobolev space $W^2(\mathbf{R}^n)$ ($n=2,3$).

This work develops the notion of a kernel function for the heat equation in certain regions of n +1 -dimensional Euclidean space and applies that notion to the study of the boundary behavior of nonnegative temperatures.The regions in question are bounded between spacelike hyperplanes and satisfy a parabolic Lipschitz condition at points on the lateral boundary.Kernel functions (normalized, nonnegative temperatures which vanish on the parabolic boundary except at a single point) are shown to exist uniquely.A representation theorem for nonnegative temperatures is obtained and used to establish the existence of finite parabolic limits at the boundary (except for a set of heat-related measure zero).0. Introduction.The notion of a kernel function has been developed in the case of the Laplace operator by Hunt and Wheeden [4], who prove existence and uniqueness for such functions in Lipschitz domains of -dimensional Euclidean space Rn and use these functions to study the nontangential boundary behavior of harmonic functions which have a one-sided bound in the domain.In [7] we reported analogous results for the heat equation in certain regions of the plane.These results had been obtained in [6].There, difficulties in the application of the techniques of [4] to the heat equation were overcome and some simplification of those techniques was achieved, notably in the proof of uniqueness of kernel functions.However, technical problems prevented treatment of the heat equation in more than one space dimension.The present work extends those results to the case of several space variables.In 1 existence and uniqueness of kernel functions is established and a representation theorem is obtained for temperatures with a one-sided bound in a region satisfying a certain mixed-Lipschitz condition (conditions LI and L2 below).The main result of 2 is the existence almost everywhere (with respect to caloric measure) on the parabolic boundary of finite parabolic limits for temperatures with a one-sided bound in the region.This generalizes the work of Jones and Tu [5] and Hattemer [2], who considered regions less general than those dealt with here.

In this paper we define Weyl semiparallel ( ·h= 0) and Weyl 2-semiparallel ( ·∇̅h= 0) submanifolds. We considern-dimensional normally flat submanifolds satisfying these curvature conditions in (n+d)-dimensional Euclidean space n+d. We also consider normally flat submanifolds in (n+d)-dimensional Euclidean space n+dsatisfying the condition  ·h=LQ(g,h) and  ·∇̅h=LQ(g,∇̅h).

Stability analysis and motion planning strategies with fuzzy coordinator and learning for mobile base robotic part macro/micro-assembly tasks

This paper concerns random walks on periodic graphs embedded in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional Euclidian space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and obtains asymptotic expansions of the Green functions of them up to the second order term, which, expressed fairly explicitly, are easily computable for many examples. The result is used to derive an asymptotic form of the hitting distribution of a hyperplane of co-dimension one, which involves not only the first but also second order terms of the expansion of the Green function. We also give similar expansions of the transition probabilities of the walks.

We prove two results about the Hadwiger problem of finding the Helly number for line transversals of disjoint unit disks in the plane, and about its higher-dimensional generalization to hyperplane transversals of unit balls in d -dimensional Euclidean space. These consist of (a) a proof of the fact that the Helly number remains 5 even for arbitrarily large sets of disjoint unit disks—thus correcting a 40-year-old error; and (b) a lower bound of d+3 on the Helly number for hyperplane transversals to suitably separated families of unit balls in Rd .

Curve motion inducing modified Korteweg-de Vries systems

Let Xt be a symmetric stable process on d‐dimensional Euclidean space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {R}}^d$\end{document}. Let F(x, y) be a symmetric positive bounded function on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {R}}^d\times {\mathbb {R}}^d$\end{document} vanishing on the diagonal set and define a discontinuous additive functional by At(F) = ∑0 &lt; s ⩽ tF(Xs −, Xs). We establish the large deviation principle of At(F)/t by employing the Gärtner‐Ellis theorem. © 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim

In this paper we study strictly positive definite functions on the unit sphere of the m m -dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are actually strictly positive definite. The study of this problem was initiated recently by Xu and Cheney (Proc. Amer. Math. Soc. 116 (1992), 977–981), where certain sufficient conditions were derived. A new approach, which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres, is presented here. The relevant interpolation problem is subsequently analyzed by three different complementary methods. The first is based on the de Boor-Ron general “least solution for the multivariate polynomial interpolation problem”. The second, which is suitable only for m = 2 m=2 , is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials, and reduces the problem to the structure of the integer zeros of bounded univariate exponentials. Finally, the last method invokes the realization of harmonic polynomials as the polynomial kernel of the Laplacian, thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels.

Combinatorics of lopsided sets

Asymptotically free quantum field theories with planar Feynman diagrams [such as SU(∞) gauge theory] are considered in 4 dimensional Euclidean space. It is shown that if all particles involved have non-vanishing masses and if the coupling constant(s) γ (org2) are small enough (λ≦λcrit), then an absolutely convergent procedure exists to obtain Green functions that uniquely solve the Dyson-Schwinger equations.

Rational interpolation on a hypersphere

The solution of many applied problems is to find a solution of nonlinear equations systems in finite- dimensional Euclidean spaces. The problem of finding the solution of a nonlinear system is divided into two problems: 1. The existence of a solution of a nonlinear equations system; in the case of nonunique of the solution, it is necessary to find the number of these solutions and their surroundings. 2. Finding the solution of a system of nonlinear equations with a given accuracy. Many publications are devoted to solving problem 2, namely the construction of iterative methods, their convergence and estimates of the solution accuracy. In contrast to problem 2, for problem 1 there is no general algorithm for solving this task, there are no constructive conditions for the existence of a solution of a nonlinear equations system in Euclidean spaces. In this article, in finite-dimensional Euclidean spaces, the constructive conditions for the existence of a solution of nonlinear systems of polynomial form are found. The connection of these conditions with the linear polynomial interpolant of the minimum norm, generated by a scalar product with Gaussian measure and the conditions of its existence, is given.