Constructible Sandwich Cut

Here RN is N-dimensional Euclidean space. We use Cech homology with integer coefficient group [1]. Plane will be used for N-1 dimensional hyperplane. A support plane, P, is a plane meeting K where K is in just one of the two closed half spaces determined by P. KflP is the support set for P. A weaker theorem, requiring the added hypothesis of local contractibility for K was proved by Kuhn [2 ] who pointed out this hypothesis was essential for any proof based on his methods. Liberman [3] has established a slightly weaker theorem requiring contractibility for the support sets as the culmination of a long chain of subsidiary results. PROOF. We need Rn, the linear extension of K, so n 1. Let Sn-' and Y refer to the n-1 topological sphere and to the metric unit n -1 sphere respectively. Each point y in Y determines a unique supporting hyperplane P2, whose normals pointing away from K are parallel translates in Rn of the vector oy. We write X for the set of boundary points of K and Xv for the support set P/nX. Let

The vector space model of information retrieval is one of the classical and widely applied retrieval models. Paradoxically, it has been characterized by a discrepancy between its formal framework and implementable form. The underlying concepts of the vector space model are mathematical terms: linear space, vector, and inner product. However, in the vector space model, the mathematical meaning of these concepts is not preserved. They are used as mere computational constructs or metaphors. Thus, the vector space model actually does not follow formally from the mathematical concepts on which it has been claimed to rest. This problem has been recognized for more than two decades, but no proper solution has emerged so far. The present article proposes a solution to this problem. First, the concept of retrieval is defined based on the mathematical measure theory. Then, retrieval is particularized using fuzzy set theory. As a result, the retrieval function is conceived as the cardinality of the intersection of two fuzzy sets. This view makes it possible to build a connection to linear spaces. It is shown that the classical and the generalized vector space models, as well as the latent semantic indexing model, gain a correct formal background with which they are consistent. At the same time it becomes clear that the inner product is not a necessary ingredient of the vector space model, and hence of Information Retrieval (IR). The Principle of Object Invariance is introduced to handle this situation. Moreover, this view makes it possible to consistently formulate new retrieval methods: in linear space with general basis, entropy‐based, and probability‐based. It is also shown that Information Retrieval may be viewed as integral calculus, and thus it gains a very compact and elegant mathematical way of writing. Also, Information Retrieval may thus be conceived as an application of mathematical measure theory.

Carnap's Normative Theory of Inductive Probability

Background. The area formulae are well known for surfaces embedded into a finite-dimensional Euclidean space. However, in the case of an infinite-dimensional Banach manifold, such formulae cannot be used. Thus, a problem of finding an alternative approach to the surface measures construction appears, that, on the one hand, leads to classical results in finite-dimensional case, and on the other hand, can be used for infinite-dimensional Banach manifolds. Objective. The aim of the paper is to get a construction of surface measure induced by the Lebesgue measure and the associated form for a parametrically defined surface embedded into finite-dimensional Euclidean space. Show the consistency of surface area calculation by this construction with an area calculated by using well-known classical formulae. Methods. Basic results of mathematical analyses, measure theory and differential geometry are used. Results. An alternative construction of surface measures induced by the Lebesgue measure on surfaces in finite-dimensional space is obtained. It is shown that such approach is consistent with the classical definition of the surface area. Conclusions. The construction of surface measures suggested for infinite-dimensional spaces is a generalization of the classical approach in finite-dimensional spaces. Therefore further investigation of the described approach seems to be reasonable.

The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications.

This paper deals with Ramsey-type theorems for metric spaces. Such a theorem states that every n point metric space contains a large subspace which can be embedded with some fixed distortion in a metric space from some special class.Our main theorem states that for any e>0, every n point metric space contains a subspace of size at least n1-e which is embeddable in an ultrametric with O(log(1/e)/e distortion. This in particular provides a bound for embedding in Euclidean spaces. The bound on the distortion is tight up to the log(1/e) factor even for embedding in arbitrary Euclidean spaces. This result can be viewed as a non-linear analog of Dvoretzky's theorem, a cornerstone of modern Banach space theory and convex geometry.Our main Ramsey-type theorem and techniques naturally extend to give theorems for classes of hierarchically well-separated trees which have algorithmic implications, and can be viewed as the solution of a natural clustering problem.We further include a comprehensive study of various other aspects of the metric Ramsey problem.

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.

A linear mapping is a technique that stores objects with multi-dimensional properties in data storage in a certain order. The mapping aims at preserving spatial locality, which means that objects in the same cluster are likely to be accessed together. Locality is a crucial property to improve performance of accessing data. The level of efficiency of a linear mapping relies on distance metric. Existing linear mapping techniques such as Hilbert curve may maximize their efficiency in Euclidean space. For linear mapping, objects in indoor spaces should be dealt with in a different manner because distance measurement in indoor space differs from that in Euclidean space. In this paper, we specify requirements of linear mappings in indoor space and suggest a linear mapping based on heuristics for indoor space objects.

This paper puts forward a simple mean-field game that captures some of the key dynamic features of crowd and pedestrian flows in multilevel building evacuations. It considers both microscopic and macroscopic route choice by strategic agents. To achieve this, we use mean-field differential game with local congestion measure based on the location of the agent in the building. Including the local mean-field term and its evolution along the path causes a sort of dispersion of the flow: the agents will try to avoid high density areas in order to reduce their overall walking costs and queuing costs at the stairs and exits. Each agent state is represented by a center of a box that follows a simple first-order dynamical system in an Euclidean space. Each agent will move to one of the closest exits that is safer and with less congested path. First, we formulate the problem and derive optimality equations using maximum principle and dynamic programming with boundary conditions. Second, well posedness and existence results are provided. Numerics and simulations are carried out to illustrate mean-field equilibria of a safer evacuation process. Finally, the methodology is shown to be flexible enough to include movement noises and stochastic structural component of the building.

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 \(\mathbb {R}^{n}\times \mathbb {Z}^{m}\).

An effective algorithm for planning a collision-free path based on linear parametric curve is developed. A collision-free path is viewed as a series of segmented polynomial curves in a space that does not interfere with any object in a given work space. It is assumed that the path connecting start and target points has no width. The algorithm presented here uses a linear parametric curve as a base curve and maps objects in Euclidean Space (ES) into objects in Control Point Space (CPS) through intersection checks between path and obstacles. A path having a single control point is investigated here. The Free Space (FS) of CPS identifies a collision-free path in ES, hence any point in the FS of CPS is a candidate for a collision-free path. A CPS completely occupied by obstacle images indicates no collision-free path is available with a single control point and a search based on a multiple control point is required. The shortest path with minimum search time is obtained by setting CPS in elliptic coordinate. Considerations are given to get a path with the smallest maximum curvature by selecting the control point toward the bisection line of ST and smoothing the path. ©1997 John Wiley & Sons, Inc.

Detecting community structure in networks via consensus dynamics and spatial transformation

A (finite or infinite) word is said to be k-th power-free if it does not contain k consecutive equal blocks. A colouring of the integer lattice points in the n-dimensional Euclidean space is power-free if there exists a positive integer k such that the sequence of colours of consecutive points on any straight line is a k-th power-free word. The Thue threshold of mathbb Z^n is the least number of colours t(n) allowing a power-free colouring of the integer lattice points in the n-dimensional Euclidean space.Answering a question of Grytczuk (2008), we prove that t(2)=t(3)=2. Moreover, we show the existence of a 2-colouring of the integer lattice points in the Euclidean plane such that the sequence of colours of consecutive points on any straight line does not contain squares of length larger than 26.In order to obtain these results, we study repetitions in Toeplitz words. We show that the Toeplitz word generated by any sequence of primitive partial words of maximal length k is k-th power-free. Moreover, adding a suitable hypothesis on the positions of the holes in the generating sequence, we obtain that also the subwords occurring in the considered Toeplitz word according to an arithmetic progression of suitable difference, are k-th power-free words.

In this paper we consider the solution formula for Stokes equation system without surface tension in half-space. More precisely, we deal with the solution of velocity and density for the model problem. This result is the basic step to estimate the solution operator of the model problem. We investigate the solution operator for the model problem in N-Dimensional Euclidean space (N>=2) In this paper we consider the solution formula for Stokes equation system without surface tension in half-space. More precisely, we deal with the solution of velocity and density for the model problem. This result is the basic step to estimate the solution operator of the model problem. We investigate the solution operator for the model problem in N-Dimensional Euclidean space ( N≥ 2 )

We examine a class of geometric theorems on cyclic 2n-gons. We prove that if we take n disjoint pairs of sides, each pair separated by an even number of polygon sides, then there is a linear combination of the angles between those sides which is constant. We present a formula for the linear combination, which provides a theorem statement in terms of those angles. We describe a program which uses this result to generate new geometry proof problems and their solutions.