Values Of Certain Functions Research Articles

On the distribution of angles between increasingly many short lattice vectors

Following Södergren, we consider a collection of random variables on the space Xn of unimodular lattices in dimension n: normalizations of the angles between the N=N(n) shortest vectors in a random unimodular lattice, and the volumes of spheres with radii equal to the lengths of these vectors. We investigate the expected values of certain functions (whose support depends on a parameter K=K(n)) evaluated at these random variables in the regime where K and N are allowed to tend to infinity with n at the rate KN=o(n1/6). Our main result is that as n⟶∞, these random variables exhibit a joint Poissonian and Gaussian behavior.

Non-dense orbits on homogeneous spaces and applications to geometry and number theory

AbstractLet G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.

A Lower Bound of the Dimension of the Vector Space Spanned by the Special Values of Certain Functions

Let $K$ be a number field. Fix a finite set of analytic functions $\mathbf{f}_{\infty}:=\{f_{1,\infty}(x),\ldots,f_{s,\infty}(x) \}$ defined on $\{x\in \mathbb{C} \mid |x|>1\}$ (resp. $\mathbb{C}_p$-valued functions $\mathbf{f}_{p}:=\{f_{1,p}(x),\ldots,f_{s,p}(x) \}$ defined on $\{x\in \mathbb{C}_p \mid |x|_p>1\}$). For $\beta\in K$, we denote the $K$-vector space spanned by $f_{1,\infty}(\beta),\ldots,f_{s,\infty}(\beta)$ by $V_K(\mathbf{f}_{\infty},\beta)$ (resp. $f_{1,p}(\beta),\ldots,f_{s,p}(\beta)$ by $V_K(\mathbf{f}_{p},\beta)$). In this article, under some assumptions for $\mathbf{f}_{\infty}$ (resp. $\mathbf{f}_{p}$), we give an estimation of a lower bound of the dimension of $V_K(\mathbf{f}_{\infty},\beta)$ (resp. $V_K(\mathbf{f}_{p},\beta)$) (see Theorem~2.4 for Archimedean case and Theorem~8.6 for $p$-adic case). Applying our estimation, we give a lower bound of the dimension of the $K$-vector space spanned by the special values of the Lerch functions over a number field in $\mathbb{C}$ (see Theorem~1.1 and Remark~1.2) and the $p$-adic analog of the above result (see Theorem~1.3 and Remark~1.4). Furthermore, we also give a lower bound of the $K$-vector space spanned by the special values of certain $p$-adic functions related with $p$-adic Hurwitz zeta function (see Theorem~1.5).

Combined approach to an assessment of maintenance of electrical equipment on traction rolling stock

A possible approach to assessment of factors influencing the service-life parameters of electrical equipment on locomotives and the variants of their states is considered. The scheme proposed for the variants of states can be carried out as a system of rolling-stock maintenance. The need to increase the reliability of electronic and electrical equipment is especially acute during the operation of high-speed rolling stock, given that circuitry and structural solutions are extremely complex. In developing the computational systems and their capacity, it becomes possible to calculate large data volumes and update the problem of constructing optimal variants of state development in complex technical systems. The costs for rolling-stock maintenance and repair can be reduced by using both maintenance-free elements of rolling stock and an assembly or a part of rolling stock with a complicated structure. In connection with this, new assemblies consist of several elements with known indices of reliability and service life, but have different characteristics as a whole. An approach to technical indices of assemblies and elements as static values of certain functions is considered. A decomposition approach to an integrated reliability index and variants of planning the extents of repair and maintenance that need to be undertaken is carried out depending on the variants of states of an electromechanical assembly in rolling stock.

Minimal unfolded regions of a convex hull and parallel bodies

The minimal unfolded region (or the heart) of a bounded subset Ω in the Euclidean space is a subset of the convex hull of Ω the definition of which is based on reflections in hyperplanes. It was introduced to restrict the location of the points that give extreme values of certain functions, such as potentials whose kernels are monotone functions of the distance, and solutions of differential equations to which Aleksandrov's reflection principle can be applied. We show that the minimal unfolded regions of the convex hull and parallel bodies of Ω are both included in that of Ω.

Arithmetic Properties of Solutions of Certain Functional Equations with Transformations Represented by Matrices Including a Negative Entry

Mahler's method gives algebraic independence results for the values of functions of several variables satisfying certain functional equations under the transformations of the variables represented as a kind of the multiplicative action of matrices with integral entries. In the Mahler's method, the entries of those matrices must be nonnegative; however, in the special case stated in this paper, one can admit those matrices to have a negative entry. We show the algebraic independence of the values of certain functions satisfying functional equations under the transformation represented by such matrices, expressing those values as linear combinations of the values of ordinary Mahler functions.

An extension of the Krylov method for calculating the coefficients of the minimal polynomial

The concept of a k-minimal polynomial of an operator is introduced, and a method for approximate calculation of the coefficients of this polynomial is proposed. The method uses the calculated values of certain functionals on iterations of the operator. Special features emerging when the algorithm is used in combination with the Monte Carlo method are discussed, and numerical results are given.

Determination of the right-hand side of the Navier-Stokes system of equations and inverse problems for the thermal convection equations

Inverse problem for an evolution equation with a quadratic nonlinearity in the Hilbert space is considered. The problem is, given the values of certain functionals of the solution, to find at each point in time the right-hand side that is a linear combination of those functionals. Sufficient conditions for the nonlocal (in time) existence of a solution (on the whole time interval) are established. An application to the inverse problems for the three-dimensional thermal convection equations of viscous incompressible fluid is considered. Unique nonlocal (in terms of time) solvability of the problem of determining the density of heat sources under the regularity condition of the initial data and sufficiently large dimension of the observation space is proved.

Four parameter proximal point algorithms

Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm — one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) [7] while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) [9] and later developed by Xu (2006) [8]. The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature.

SEMIDEFINITE PROGRAMMING AND MULTIVARIATE CHEBYSHEV BOUNDS

Chebyshev inequalities provide bounds on the probability of a set based on known expected values of certain functions, for example, known power moments. In some important cases these bounds can be efficiently computed via convex optimization. We discuss one particular type of generalized Chebyshev bound, a lower bound on the probability of a set defined by strict quadratic inequalities, given the mean and the covariance of the distribution. We present a semidefinite programming formulation, give an interpretation of the dual problem, and describe some applications.

Computation of Spatial Covariance Matrices

In an earlier article, Ghosh derived the density for the distance between two points uniformly and independently distributed in a rectangle. This article extends that work to include the case where the two points lie in two different rectangles in a lattice. This density allows one to find the expected value of certain functions of this distance between rectangles analytically or by one-dimensional numerical integration.In the case of isotropic spatial models or spatial models with geometric anisotropy terms for agricultural experiments one can use these theoretical results to compute the covariance between the yields in different rectangular plots. As the numerical integration is one-dimensional these results are computed quickly and accurately. The types of covariance functions used come from the Matérn and power families of processes. Analytic results are derived for the de Wijs process, a member of both families and for the power models also.Software in R is available. Examples of the code are given for fitting spatial models to the Fairfield Smith data. Other methods for the estimation of the covariance matrices are discussed and their pros and cons are outlined.

Wave scattering by surface-breaking cracks and cavities

Here we suggest a new approach to the analysis of two-dimensional wave scattering by arbitrarily shaped irregularities of a boundary of the half-space. This approach employs the probabilistic method of a rigorously justified representation of the scattered waves by explicit formulas involving the computation of mathematical expectations which average the values of certain functionals computed along the trajectories of the random motion governed by specified stochastic differential equations. As typical for the random walk method, the obtained solutions admit numerical implementations by simple algorithms which are practically independent of the particular geometry of the scatterer, which are inexpensive in terms of computer memory requirements, and which have virtually unrestricted capability for parallel processing. Illustrative numerical examples include problems of wave scattering by a tilted straight crack and by a circular cavity.

Correlation Functions of Strict Partitions and Twisted Fock Spaces

Using twisted Fock spaces, we formulate and study two twisted versions of the n-point correlation functions of Bloch–Okounkov, and then identify them with q-expectation values of certain functions on the set of (odd) strict partitions. We find closed formulas for the 1-point functions in both cases in terms of Jacobi θ-functions. These correlation functions afford several distinct interpretations.

On the Dempster-Shafer evidence theory and non-hierarchical aggregation of belief structures

The Dempster's rule of combination is a widely used technique to integrate evidence collected from different sources. In this paper, it is shown that the values of certain functions defined on a family of belief structures decrease (by scale factors depending on the degree of conflict) when the belief structures are combined according to the Dempster's rule. Similar results also hold when an arbitrary belief structure is prioritized while computing the combination. Furthermore, the length of the belief-plausibility interval is decreased during a nonhierarchical aggregation of belief structures. Several types of inheritance networks are also proposed each of which allows considerable flexibility in the choice of prioritization.

Why Equilibrium Statistical Mechanics Works: Universality and the Renormalization Group

Discussions of the foundations of Classical Equilibrium Statistical Mechanics (SM) typically focus on the problem of justifying the use of a certain probability measure (the microcanonical measure) to compute average values of certain functions. One would like to be able to explain why the equilibrium behavior of a wide variety of distinct systems (different sorts of molecules interacting with different potentials) can be described by the same averaging procedure. A standard approach is to appeal to ergodic theory to justify this choice of measure. A different approach, eschewing ergodicity, was initiated by A. I. Khinchin. Both explanatory programs have been subjected to severe criticisms. This paper argues that the Khinchin type program deserves further attention in light of relatively recent results in understanding the physics of universal behavior.

The constraint rule of the maximum entropy principle

The principle of maximum entropy is a method for assigning values to probability distributions on the basis of partial information. In usual formulations of this and related methods of inference one assumes that this partial information takes the form of a constraint on allowed probability distributions. In practical applications, however, the information consists of empirical data. A constraint rule is then employed to construct constraints on probability distributions out of these data. Usually one adopts the rule that equates the expectation values of certain functions with their empirical averages. There are, however, various other ways in which one can construct constraints from empirical data, which makes the maximum entropy principle lead to very different probability assignments. This paper shows that an argument by Jaynes to justify the usual constraint rule is unsatisfactory and investigates several alternative choices. The choice of a constraint rule is also shown to be of crucial importance to the debate on the question whether there is a conflict between the methods of inference based on maximum entropy and Bayesian conditionalization.

Combinatorial constructions for integrals over normally distributed random matrices

Recent results of Hanlon, Stanley, and Stembridge give the expected values of certain functions of matrices of normal variables in the real and complex cases. They point out that in both cases the results are equivalent to combinatonal results and suggest further that these results may have purely combinatonal proofs, in this way avoiding the use of the theory of spherical functions. Such proofs are given in this paper. In the complex case we use the familiar cycle decomposition for permutations. In the real case we introduce an analogous decomposition into cyclically ordered sequences, called chains, which makes the real and complex cases strikingly similar.

Necessary conditions for stochastic dominance: A general approach

AbstractIn this paper a general method for developing necessary conditions for all degrees of stochastic dominance is derived. The method, a minimization of the expected value of certain functions of the random variable, is used to rederive known necessary conditions for dominance and is then used to derive new necessary conditions. Some of the old and new conditions are then compared empirically using a data set of security returns.

A Logical Function Method for Dynamic and Design Sensitivity Analysis of Mechanical Systems With Intermittent Motion

Dynamic and design sensitivity analysis of mechanisms and machines with intermittent motion are accomplished through introduction of “logical functions” to approximate discontinuities and special features of system motion. The Heaviside step function and the delta and unit doublet distributions are introduced to represent discontinuities and to determine the values of certain functions of isolated times. These functions and distributions are approximated by smooth functions, and validity of the approximation is argued both mathematically and physically. A numerical method is then presented for analysis of the approximate problem. An elementary and a complex, realistic example are presented to illustrate applications of the method.

On generalized Boolean functions i

In this paper we introduce the concept of generalized Boolean function. Such a function has its arguments and values in a Boolean algebra and can be written in a manner similar to the canonical disjunctive form, but instead of the product of simple or complemented variables, the product of values of certain functions is used. Every Boolean function is a generalized Boolean one but the converse is not true. The set of all generalized Boolean function “generated” by some fixed function is a Boolean algebra.