Decomposing Euclidean space with a small number of smooth sets

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.

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.

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.

In this paper, we study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve. We obtain the Frenet apparatus of these spherical indicatrices and the characterizations of being general helix and slant helix. Moreover we give some properties between the spherical indicatrices and their associated curves.

{\it Separoids\/} capture the combinatorial structure which arises from the separations by hyperplanes of a family of convex sets in some Euclidian space. Furthermore, as we prove in this note, every abstract separoid ]]> ]]> S$ can be represented by a family of convex sets in the $(|S|-1)$-dimensional Euclidian space. The {\it geometric dimension\/} of the separoid is the minimum dimension where it can be represented and the upper bound given here is tight. Separoids have also the notions of {\it combinatorial dimension\/} and {\it general position\/} which are purely combinatorial in nature. In this note we also prove that: {\it a separoid in general position can be represented by a family of points if and only if its geometric and combinatorial dimensions coincide\/}.

Abstract. In this paper, we consider and study a new class of general-ized vector quasi-variational type inequalities and obtain some existencetheorems for both under compact and noncompact assumptions in topo-logical vector spaces without using monotonicity. For the noncompactcase, we use the concept of escaping sequences. 1. IntroductionThe vector variational inequality problem was initiated by Giannessi [11] in nite dimensional Euclidean spaces with applications. Later on many authors[6, 7, 12, 15, 24] generalized vector variational inequalities in abstract spacesin several ways. The vector variational-like inequality is one of the generalizedforms of vector variational inequalities [11]. In 1989, Parida et al [19] stud-ied the existence of solutions for variational-like inequalities in R n space andhave shown a relationship between variational like inequalities problems andconvex programming as well as with complementarity problems. The vectorvariational like inequality and generalized vector variational like inequality arepowerful tools to study non-convex vector optimization problems and convexand nondi erentiable vector optimization problems respectively, see [10, 18, 21].In 1973, Bensoussan and Lions [3] introduced the quasi-variational inequalityproblem (QVIP). Since then, many generalizations of the QVIP have beenappeared in the literature (see, for example, [1, 2]). In the last decade, becauseof applications in the optimization problems, mathematical programming andequilibrium problems, the QVIP has been intensively studied by many authors[5, 8, 16, 20, 22, 23, 25].In this paper, we consider the generalized vector quasivariational type in-equality problem (GVQVTIP) and obtain some existence theorems for solutions

Let M be an n-dimensional orientable compact hypersurface in an (n + 1)-dimensional Euclidean space R n+1 , n ≥ 2. If the gradient ∇α of the mean curvature α and the scalar curvature S of the hypersurface M satisfy the inequality Sk0 ≤ n(n − 1)k0α 2 − (n − 1) �∇ α� 2 where k0 is the infimum of the sectional curvatures of the hypersurface, then it is shown that α is a constant and M is the sphere in S n (α).

Let X be a subset of n points of the Euclidean space, and let 0 < e < 1. A classical result of Johnson and Lindenstrauss [JL] states that there is a projection of X onto a subspace of dimension O(e-2 log n) with distortion ≤ 1+ e. We show a natural extension of the above result to a stronger preservation of the geometry of finite spaces. By a k-fold increase of the number of dimensions used compared with [JL], a good preservation of volumes and of distances between points and affine spaces is achieved. Specifically, we show how to embed a subset of size n of the Euclidean space into a O(e-2 log n)-dimensional Euclidean space, so that no set of size s ≤ k changes its volume by more than (1 + es-1. Moreover, distances of points from affine hulls of sets of at most k - 1 points in the space do not change by more than a factor of 1 + e. A consequence of the above with k = 3 is that angles can be preserved using asymptotically the same number of dimensions as the one used in [JL]. Our method can be applied to many problems with high-dimensional nature such as Projective Clustering and Approximated Nearest Affine Neighbour Search. In particular, it shows a first polylogarithmic query time approximation algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced by Feige [F], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph.