Two geometric representation theorems for separoids

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.

The main task of classical multidimensional differential geometry is to study the properties of various n-surfaces. Often these studies use torsion coefficients that are defined for any n-surface with codimension k > 1 in (n + k)-dimensional Euclidean space. For hypersurfaces, the torsion coefficients are not defined.Another important concept used to study the properties of n-surfaces is the spherical Gaussian mapping. The Gaussian mapping defined on submanifolds of Euclidean and pseudo-Euclidean spaces allows one to study the external properties of a submanifold immersed in a Euclidean or pseudo-Euclidean space. In a number of papers, the properties of the Gaussian mapping are studied, as well as the geometric characteristics of the images of submanifolds under a spherical mapping, which are submanifolds of a hypersphere or a Grassmannian.In this article, we study the local properties of the spherical image of a regular n-surface of arbitrary codimension. The spherical mapping is defined for n-surfaces with codimension greater than one in Euclidean space by means of a regular vector field. Each vector of this field at a point of the submanifold is orthogonal to the tangent space of the submanifold at the chosen point.The article uses the methods of differential and Riemannian geometry, as well as tensor analysis to study n-surfaces with a codimension greater than one. Under some additional conditions, a connection is established between the curvature tensor of a given surface and the curvature tensor of its spherical image. Under the same additional conditions, some geometric characteristics of the points of the spherical image of the original n-surface are studied.

Consider an n-point metric space M = (V,δ), and a transmission range assignment r: V → ℝ + that maps each point v ∈ V to the disk of radius r(v) around it. The symmetric disk graph (henceforth, SDG) that corresponds to M and r is the undirected graph over V whose edge set includes an edge (u,v) if both r(u) and r(v) are no smaller than δ(u,v). SDGs are often used to model wireless communication networks.Abu-Affash, Aschner, Carmi and Katz (SWAT 2010, [1]) showed that for any n-point 2-dimensional Euclidean space M, the weight of the MST of every connected SDG for M is O(logn) ·w(MST(M)), and that this bound is tight. However, the upper bound proof of [1] relies heavily on basic geometric properties of constant-dimensional Euclidean spaces, and does not extend to Euclidean spaces of super-constant dimension. A natural question that arises is whether this surprising upper bound of [1] can be generalized for wider families of metric spaces, such as high-dimensional Euclidean spaces.In this paper we generalize the upper bound of Abu-Affash et al. [1] for Euclidean spaces of any dimension. Furthermore, our upper bound extends to arbitrary metric spaces and, in particular, it applies to any of the normed spaces ℓ p . Specifically, we demonstrate that for any n-point metric space M, the weight of the MST of every connected SDG for M is O(logn) ·w(MST(M)).KeywordsSpan TreeTransmission RangeMinimum Span TreeHamiltonian PathUnit Disk GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

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.

We study the basic properties of the $d$-plane transform on the Euclidean space as a Fourier integral operator and its application to the microlocal analysis of streaking artifacts in its filtered back-projection. The $d$-plane transform is defined by integrals of functions on the $n$-dimensional Euclidean space over all the $d$-dimensional planes, where $0<d<n$. This maps functions on the Euclidean space to those on the affine Grassmannian $G(d,n)$. This is said to be X-ray transform if $d=1$ and Radon transform if $d=n-1$. When $n=2$ the X-ray transform is thought to be measurements of computed tomography (CT) scanners. In this paper we obtain a concrete expression of the canonical relation between the $d$-plane transform and the quantitative properties of the filtered back-projection of the product of the images of the $d$-plane transform. The FBP of the product is related to the metal streaking artifacts of CT images. Our result is a generalization of recent results of Park, Choi, and Seo [Comm. Pure Appl. Math., 70 (2017), pp. 2191--2217] and Palacios, Uhlmann, and Wang [SIAM J. Math. Anal., 50 (2018), pp. 4914--4936] for the X-ray transform on the plane.

In this article we present a comparison of the performance between a metaheuristic optimization method, Game of Patterns (GofP), so-called by the author, and the well-known genetic algorithms (GAs), through two implementations, namely: the GA of Scilab (SGA); and the GA of the R Project for Statistical Computing (RGA). For this purpose, we have selected a set of multimodal objective functions in the n-dimensional Euclidean space \(\mathbb {R}^{n}\) with a unique global minimum. For comparing both metaheuristic optimization approaches, a performance indicator of quality, denoted Q(p, n, s), was defined, which allows us to measure the quality of the obtained global optimal solution for each pth problem, in the n-dimensional space, when it is solved by each metaheuristic optimization method \(s\in \{\texttt {GofP},\texttt {SGA},\texttt {RGA}\}\). The indicator Q(p, n, s) then depends on: the number of evaluations of the pth optimization problem in the Euclidean space \(\mathbb {R}^{n}\), which has required the s metaheuristic optimization method for identifying the global minimum; and the distance between the location of its respective unique global minimum and the location of the minimum that has been identified by the s metaheuristic optimization method. The paper also offers a brief explanation of the GofP method, which has been developed for solving unconstrained mixed integer problems in the \(n\times m\)-dimensional Euclidean space \(\mathbb {R}^{n}\times \mathbb {Z}^{m}\).

The purpose of this paper is to provide necessary and sufficient conditions on a continuous and matrix valued radial kernel on a Euclidean space in order that it be conditionally positive definite of a fixed order. Except for the one dimensional Euclidean space, the strict conditional positive definiteness of the kernel is fully characterized.

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 the classical (“smooth”) mathematical analysis, a differentiable function is studied by means of the derivative (gradient in the multidimensional space). In the case of nondifferentiable functions, the tools of nonsmooth analysis are to be employed. In convex analysis and minimax theory, the corresponding classes of functions are investigated by means of the subdifferential (it is a convex set in the dual space), quasidifferentiable functions are treated via the notion of quasidifferential (which is a pair of sets). To study an arbitrary directionally differentiable function, the notions of upper and lower exhausters (each of them being a family of convex sets) are used. It turns out that conditions for a minimum are described by an upper exhauster, while conditions for a maximum are stated in terms of a lower exhauster. This is why an upper exhauster is called a proper one for the minimization problem (and an adjoint exhauster for the maximization problem) while a lower exhauster will be referred to as a proper one for the maximization problem (and an adjoint exhauster for the minimization problem). The directional derivatives (and hence, exhausters) provide first-order approximations of the increment of the function under study. These approximations are positively homogeneous as functions of direction. They allow one to formulate optimality conditions, to find steepest ascent and descent directions, to construct numerical methods. However, if, for example, the maximizer of the function is to be found, but one has an upper exhauster (which is not proper for the maximization problem), it is required to use a lower exhauster. Instead, one can try to express conditions for a maximum in terms of upper exhauster (which is an adjoint one for the maximization problem). The first to get such conditions was Roshchina. New optimality conditions in terms of adjoint exhausters were recently obtained by Abbasov. The exhauster mappings are, in general, discontinuous in the Hausdorff metric, therefore, computational problems arise. To overcome these difficulties, the notions of upper and lower coexhausters are used. They provide first-order approximations of the increment of the function which are not positively homogeneous any more. These approximations also allow one to formulate optimality conditions, to find ascent and descent directions (but not the steepest ones), to construct numerical methods possessing good convergence properties. Conditions for a minimum are described in terms of an upper coexhauster (which is, therefore, called a proper coexhauster for the minimization problem) while conditions for a maximum are described in terms of a lower coexhauster (which is called a proper one for the maximization problem). In the present paper, we derive optimality conditions in terms of adjoint coexhausters.

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.

Introduction Convex sets play an important role in integral geometry. For this reason we will review here their principal properties, especially those which will be needed in the following sections. In this chapter we consider convex sets in the plane. For convex sets in n -dimensional euclidean space, see Chapter 13. For a more complete treatment, refer to the classical books of Blaschke [50] and Bonnesen and Fenchel [63], or to the more modern texts of Benson [27], Eggleston [162], Grunbaum [247], Jaglom and Boltjanski [320], Hadwiger [270], Hadwiger and coauthors [282], and Valentine [683]. A set of points K in the plane is called convex if for each pair of points A ∈ K , B ∈ K it is true that AB ⊂ K , where AB is the line segment joining A and B . For convenience we shall assume throughout that the convex sets are bounded and closed. A curve with end points P, Q , is called convex if its point set, together with the segment PQ , bounds a convex set. If the convex set K is bounded and has interior points, then the boundary of K is called a closed convex curve . Throughout, we will denote by ∂ K the boundary of the set K . If all the points of K belong to ∂ K , then K is a line segment. We can prove that (a) All convex curves are piecewise differentiable (i.e., they are the union of a countable set of arcs with continuously turning tangent); in other words, convex curves have at most a countable set of corners; (b) All bounded convex curves are rectifiable.

This chapter investigates the objects and mappings of algebraic geometry from a geometric point of view, making use especially of the algebraic tools of Chapter VII and of Sections 7–10 of Chapter VIII. In Sections 1–12, k denotes a fixed algebraically closed field. Sections 1–6 establish the definitions and elementary properties of varieties, maps between varieties, and dimension, all over k. Sections 1–3 concern varieties and dimension. Affine algebraic sets, affine varieties, and the Zariski topology on affine space are introduced in Section 1, and projective algebraic sets and projective varieties are introduced in Section 3. Section 2 defines the geometric dimension of an affine algebraic set, relating the notion to Krull dimension and transcendence degree. The actual context of Section 2 is a Noetherian topological space, the Zariski topology on affine space being an example. In such a space every closed subset is the finite union of irreducible closed subsets, and the union can be written in a certainway thatmakes the decomposition unique. Every nonempty closed set has a meaningful geometric dimension. In affine space the irreducible closed sets are the varieties, and each variety acquires a geometric dimension. The discussion in Section 2 applies in the context of projective space as well, and thus each projective variety acquires a geometric dimension. Moreover, any nonempty open subset of a Noetherian space is Noetherian. A nonempty open subset of an affine variety is called quasi-affine, and a nonempty open subset of a projective variety is called quasiprojective. Each quasi-affine variety or quasiprojective variety has a dimension equal to that of its closure, which is a variety. Sections 4–6 take up maps between varieties. Section 4 introduces spaces of scalar-valued functions on quasiprojective varieties–rational functions, functions regular at a point, and functions regular on an open set. The section goes on to relate these notions for the different kinds of varieties. Section 5 introduces morphisms, which are a restricted kind of function between varieties. The tools of Sections 4–5 together show that for many purposes all the different kinds of varieties can be treated as quasiprojective varieties. Section 6 introduces rational maps between varieties; these are not everywhere-defined functions, but each can be restricted to an open dense subset on which it is a morphism. Rational maps with dense image correspond to field mappings of the fields of rational functions, with the order of the mappings reversed. Section 7 concerns singularities at points of varieties, still over the field k. Zariski's Theorem was stated in Chapter VII for affine varieties and partly proved at that time. In the current context it has a meaning for any point of any quasiprojective variety. The section proves the full theorem, which characterizes singular points in a way that shows they remain singular under isomorphisms of varieties. Section 8 concerns classification questions over k for irreducible curves, i.e., quasiprojective varieties of dimension 1. From Section 6 it is known that two irreducible curves are equivalent under rational maps if and only if their fields of rational functions are isomorphic. The main theorem of Section 8 is that each such equivalence class of irreducible curves contains an everywhere nonsingular projective curve, and this curve is unique up to isomorphism of varieties. The points of this curve are parametrized by those discrete valuations of the underlying function field that are defined over k. Sections 9–12 relate the general theory of Sections 1–6 to the topic of solutions of simultaneous solutions of polynomial equations, as treated at length in Chapter VIII. Section 9 treats monomial ideals in k[X1, . . . , Xn], identifying their zero loci concretely and computing their dimension. The section goes on to introduce the affineHilbert function of this ideal, which measures the proportion of polynomials of degree≤ s not in the ideal. In the way that this function is defined, it is a polynomial for large s called the affine Hilbert polynomial of the ideal. Its degree equals the dimension of the zero locus of the ideal. Section 10 extends this theory from monomial ideals to all ideals, again concretely computing the dimension of the zero loci, obtaining an affine Hilbert polynomial, and showing that its degree equals the dimension of the zero locus of the ideal. Section 11 adapts the theory to homogeneous ideals and projective algebraic sets by making use of the cone in affine space over the set in projective space. Section 12 applies the theory of Section 11 to address the question how the dimension of a projective algebraic set is cut down when the set is intersected with a projective hypersurface. A consequence of the theory is the result that a homogeneous system of polynomial equations over an algebraically closed field with more unknowns than equations has a nonzero solution. Section 13 is a brief introduction to the theory of schemes, which extends the theory of varieties by replacing the underlying algebraically closed field by an arbitrary commutative ring with identity.

Tiling sets and spectral sets over finite fields

We consider a finite subgroup Θ n of the group O(N) of orthogonal matrices, where N = 2n, n = 1, 2 .... This group was defined in [7]. We use it in this paper to construct spherical designs in 2n-dimensional Euclidean space RN. We prove that representations of the group Θ n on spaces of harmonic polynomials of degrees 1, 2 and 3 are irreducible. This and the earlier results [1–3] imply that the orbit Θ n,2 x t of any initial point x on the sphere S N − 1 is a 7-design in the Euclidean space of dimension 2n.