Distribution of Distances between Elements in a Compact Set

Distance distributions in random networks

Let $\mathbb{F}_q$ be an arbitrary finite field, and $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Let $Δ(\mathcal{E})$ be the set of distances determined by pairs of points in $\mathcal{E}$. By using the Kloosterman sums, Iosevich and Rudnev proved that if $|\mathcal{E}|\ge 4q^{\frac{d+1}{2}}$, then $Δ(\mathcal{E})=\mathbb{F}_q$. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. In this paper, we use the recent point-plane incidence bound due to Rudnev to prove that if $\mathcal{E}$ has Cartesian product structure in vector spaces over prime fields, then we can break the exponent $(d+1)/2$, and still cover all distances. We also show that the number of pairs of points in $\mathcal{E}$ of any given distance is close to its expected value.

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painleve–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution– like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of the Euclidean space evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.

Vector space models of word meaning typically represent the meaning of a word as a vector computed by summing over all its corpus occurrences. Words close to this point in space can be assumed to be similar to it in meaning. But how far around this point does the region of similar meaning extend? In this paper we discuss two models that represent word meaning as regions in vector space. Both representations can be computed from traditional point representations in vector space. We find that both models perform at over 95% F-score on a token classification task.

Modeling the process of information relay through inter-vehicle communication

We demonstrate the robustness of BosonSampling to imperfections in the linear\noptical network that cause a small deviation in the matrix it implements. We\nshow that applying a noisy matrix $\\tilde{U}$ that is within $\\epsilon$ of the\ndesired matrix $U$ in operator norm leads to an output distribution that is\nwithin $\\epsilon n$ of the desired distribution in variation distance, where\n$n$ is the number of photons. This lets us derive a sufficient tolerance each\nbeamsplitters and phaseshifters in the network.\n This result considers only errors that result from the network encoding a\ndifferent unitary than desired, and not other sources of noise such as photon\nloss and partial distinguishability.\n

This paper describes the effects of a complex scalar scaling field on quantum mechanics. The field origin is an extension of the gauge freedom for basis choice in gauge theories to the underlying scalar field. The extension is based on the idea that the value of a number at one space time point does not determine the value at another point. This, combined with the description of mathematical systems as structures of different types, results in the presence of separate number fields and vector spaces, both as structures, at different space time locations. Complex number structures and vector spaces at each location, are scaled by a complex space time dependent scaling factor. The effect of this scaling factor on several physical and geometric quantities has been described in other work. Here the emphasis is on quantum mechanics of one and two particles, their states and properties. Multiparticle states are also briefly described. The effect shows as a complex, nonunitary, scalar field connection on a fiber bundle description of nonrelativistic quantum mechanics. The lack of physical evidence for the presence of this field so far means that the coupling constant of this field to fermions is very small. It also means that the gradient of the field must be very small in a local region of cosmological space and time. Outside this region there are no restrictions on the field gradient.

In this project the genetic diversity of Arabis holboellii and the closely related A. drummondii should be analysed on the molecular level with the help of DNA sequences (plastid and nuclear DNA) and microsatellites for several purposes. First, the degree of apomixis within a population has a great impact on the distribution of its total genetic variation. Therefore, populational genetic diversity and the intra-populational degree of apomixis should be analysed using molecular markers and cytological investigations. Second, geneticvariation on the populational level should be used to estimate interpopulational genetic distances and to infer a phylogenetic network of Arabis populations. As a third step, correlations between geograph ic distribution of molecular variation, apomixis and genetic distances between populations should help to elucidate the degree of apomictic variation within and between populations.

Existence of quasiequilibria in metric vector spaces

The distribution of the zeros of the Hermite-Padé polynomials of the first kind for a pair of functions with an arbitrary even number of common branch points lying on the real axis is investigated under the assumption that this pair of functions forms a generalized complex Nikishin system. It is proved (Theorem ) that the zeros have a limiting distribution, which coincides with the equilibrium measure of a certain compact set having the -property in a harmonic external field. The existence problem for -compact sets is solved in Theorem .The main idea of the proof of Theorem consists in replacing a vector equilibrium problem in potential theory by a scalar problem with an external field and then using the general Gonchar-Rakhmanov method, which was worked out in the solution of the ‘’-conjecture.The relation of the result obtained here to some results and conjectures due to Nuttall is discussed.Bibliography: 51 titles.

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese 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.

Let l be a compact convex subset of a Hausdorfftopological vector space $(\mathcal{E},\tau)$ and $\sigma$ anotherHausdorff vector topology in $\mathcal{E}$. We establish an approximatefixed point result for sequentially continuous maps f:(l,$\sigma$)$\to$ (l,$\tau$). As application, weobtain the weak-approximate fixed point property for demicontinuousself-mapping weakly compact convex sets in general Banach spaces and use this toprove new results in asymptotic fixed point theory. These resultsare also applied to study the existence of limiting-weak solutions for differentialequations in reflexive Banach spaces.

We consider the expressive power of the first-order structure 〈Ω,C 〉 where Ω is either of two of different domains of extended regions in Euclidean space, and C(x,y) is the topological relation ‘Region x is in contact with region y.’ We prove two main theorems: Let P[ℚ] be the domain of bounded, non-empty, rational polyhedra in two- or three-dimensional Euclidean space. A relation Γ over P[ℚ] is definable in the structure 〈P[ℚ], C〉 if and only if Γ is arithmetic and invariant under rational PL-homeomorphisms of the space to itself. We also extend this result to a number of other domains, including the domain of all polyhedra and the domain of semi-algebraic regions. Let R be the space of bounded, non-empty, closed regular regions in n-dimensional Euclidean space. Any analytical relation over lower dimensional (i.e. empty interior) compact point sets that is invariant under homeomorphism is implicitly definable in the structure 〈R⁠,C〉.

Runs statistics have found many applications in various fields and have attracted attentions of many researchers. Traditional methods used to investigate run-related distributions are ad hoc and not easy to generalize. The problems often become tedious because of the combinatorial complexity involved. Several unified methods have been devised to overcome the combinatorial difficulties. One of them is the finite Markov chain imbedding approach. Recently we used a different systematic approach that is inspired by methods in statistical physics. In this approach the study of runs distributions is decoupled into two easy independent steps. In the first step, elements of each of the same object are considered in isolation without regards of elements of the other objects. By considering only one kind of object each time the complexity involved is reduced considerably. In the second step, general formulas in matrix or explicit forms combine the results from the first step into a whole multi-object system with potential nearest neighbor interactions. The method reduces the complicated combinatorics to simple mechanical manipulations of binomial and multinomial coefficients, which can be carried out automatically by the Gosper-Zeilberger method. Many commonly studied run and pattern distributions, such as exact length runs, runs of Type I, II, III, the longest runs, rises and falls, among others, can be handled by the method. Novel run-related distributions have been derived using the method. We’ll discuss the general framework of the method and use some examples to illustrate its application to distributions of runs, including the derivation of the m th longest runs distributions of multivariate random sequences.

The aim is to analyze the development of the industry of accommodation of tourists in Rivne region in space and time. Method. The method of research of the industry of means of accommodation of tourists is used. The analysis of hotels by their number, capacity and number of beds; the number of specialized accommodation facilities for tourists and persons staying in institutions and the regional provision of farmsteads were revealed. The system-structural approach, method of generalization and abstraction, economic-static and graphic methods are applied. Regional statistics and reports of relevant agencies were processed. Results. The regional aspects of hotel development as a means of accommodation of tourists in Rivne region are studied, the number and capacity of establishments are estimated. As of 2017, there are 43 units in the region. Most hotels are concentrated within the city of Rivne and Rivne district. In the dynamics we observe an increase in the number of hotel establishments. In 2017, the capacity of hotels in the region decreased by 1.5 times, compared to 2010. We analyzed the health facilities of the region, which are undergoing significant transformations. The number of sanatoriums has decreased to 9 by 2015 and remains unchanged until now. This testifies to the crisis-stagnation phenomena in the sanatorium-resort sphere of the region. The number of people using the services of health resorts is decreasing. In the structure of the tourist accommodation industry an important place belongs to agro-villages. There were 60 rural estates in the region, which are considered as private accommodation for tourists. Most within the territories of Bereznivsky, Demydivsky and Zarichnensky districts, which are characterized by high natural resource potential. To increase the demand for accommodation for tourists in the region will help create a highly profitable tourism sector, able to produce and sell quality, competitive in the tourist market. Scientific novelty. The geographical and chronological aspects of the tourist accommodation industry of the studied region are revealed. Hotels and similar means of accommodation of tourists, specialized establishments as a part of sanatoria and camps of rest and a farmstead are systematically analyzed on an example of the Rivne area. Practical significance. The results of the work reveal the level and state of providing the Rivne region with accommodation facilities for tourists. Can be used in the preparation of regional socio-economic development programs. To be used in the development of possible recommendations for improving and justifying management decisions in the tourism industry on the example of regions.