Nonnegative Reals Research Articles

The essential spectrum of Neumann Laplacians on some bounded singular domains

In the present paper we consider Neumann Laplacians on singular domains of the type “rooms and passages” or “combs” and we show that, in typical situations, the essential spectrum can be determined from the geometric data. Moreover, given an arbitrary closed subset S of the non-negative reals, we construct domains Ω = Ω(S) such that the essential spectrum of the Neumann Laplacian on Ω is just this set S.

Continuous isometric semigroups and reflexivity

We consider the reflexivity of a WOT-closed algebra generated by continuous isometric semigroups parametrized by the semigroup of non-negative reals or the semigroup of finite sequences of non-negative reals. It is also proved that semigroups o

Fast parallel complex discrete fourier transforms using a multichannel optical correlator

Based on a multichannel incoherent optical correlator, a new simple scheme is proposed for performing a complex discrete Fourier transform. A complex value is represented by using three nonnegative reals, and every real is encoded with the area of a rectangular aperture. A 3 N×3 N-element kernel mask together with a 3 N-element 1D-pin-hole-array illuminated with an extended source is used to calculate an 1D-DFT of an N-point input. Some properties of DFT are demonstrated in experiments.

Three problems concerning ideals of differentiable functions

In this paper, we study the validity of the following two statements in the internal logic of the toposes of Synthetic Differential Geometry: 1. (1) The integral of f is non-negative if f is non-negative; 2. (2) If f=0 in the set of non-negative reals, and f=0 in the set of non-negative reals, then f=0. We find statements (1) and (2) to be true in the toposes considered. We also prove that 3. (3) For n greater than two, the arrow t n from the line to itself is not a stable effective epic. This answers a question raised by Quê-Moerdijk-Reyes.

Finding the vertices nearest to a point in a hypercube

An algorithm is presented and analyzed for the following problem: given a point in the unit hypercube, iteratively output its nearest vertex, its second nearest vertex, and so forth. A more general version of this problem is also addressed, namely: given a d-long vector w of nonnegative reals, iteratively output nonnegative integer-valued vectors v 1, v 2,… where v n gives the n th smallest value of w T· v.

Note on the Ahlswede-Daykin inequality

Given a finite boolean lattice L and four functions α, β, γ, δ of L to the nonnegative reals with α( x) β( gg)⩽ γ( x ∨ y) δ( x ∧ y) for all x, y ∈ L. We show that ∑ xϵX α(x)β(x′)⩽ ∑ rϵXvX′ γ(r)σ(r′) forx⊆L . Here x′ denotes the complement of x and X ∨ X′ stands for { x 1 ∨ x′ 2 | x 1, x 2 ∈ X}. This inequality turns out to be equivalent in a certain sense to the Ahlswede-Daykin inequality. We also apply our basic ideas to a special case of a conjecture of Fürstenberg.

Likelihood estimation for generalized mixed exponential distributions

Abstract : The class of probability functions expressed as linear (not necessarily convex) combinations of negative exponential densities is dense in the set of all distribution functions on the nonnegative reals. Because of this and resultant mathematical properties, such forms would appear to have excellent potential for wide application in stochastic modeling. This work documents the development and testing of a practical procedure for maximum-likelihood estimation for these generalized exponential mixtures. The algorithm offered for the problem is of the Jacobi type and guarantees that the result will provide a legitimate probability function of the prescribed type. Extensive testing has been performed and results are very favorable: convergence is rapid and the use of computer resources rather limited. (Author)

Entropies of Automorphisms of a Topological Markov Shift

Let $\sigma$ be a mixing topological Markov shift, $\lambda$ a weak Perron number, $q\left ( t \right )$ a polynomial with nonnegative integer coefficients, and $r$ a non-negative rational. We construct a homeomorphism commuting with $\sigma$ whose topological entropy is $\log {\left [ {q\left ( \lambda \right )q\left ( {1/\lambda } \right )} \right ]^r}$. These values are shown to include the logarithms of all weak Perron numbers, and are dense in the nonnegative reals.

Entropies of automorphisms of a topological Markov shift

Let σ \sigma be a mixing topological Markov shift, λ \lambda a weak Perron number, q ( t ) q\left ( t \right ) a polynomial with nonnegative integer coefficients, and r r a non-negative rational. We construct a homeomorphism commuting with σ \sigma whose topological entropy is log ⁡ [ q ( λ ) q ( 1 / λ ) ] r \log {\left [ {q\left ( \lambda \right )q\left ( {1/\lambda } \right )} \right ]^r} . These values are shown to include the logarithms of all weak Perron numbers, and are dense in the nonnegative reals.

A strengthening of Leth’s uniqueness condition for sequences

A series ∑ a i \sum {a_i} of nonnegative reals summing to 1 such that a i ≤ ∑ j > i a j {a_i} \leq {\sum _{j > i}}{a_j} for eacn i i is uniquely characterized by the equalities of the form ∑ J a i = ∑ K a k {\sum _J}{a_i} = {\sum _K}{a_k} . This characterization is an improvement of one given by Leth.

Distances between fuzzy sets

Given two fuzzy subsets μ and ν of a metric space S (e.g., the Euclidean plane), we define the ‘shortest distance’ between μ and ν as a density function on the non-negative reals; our definition is applicable both when μ and ν are discrete-valued and when they are ‘smooth’ (i.e., differentiable), and it generalizes the definition of shortest distance for crisp sets in a natural way. We also define the mean distance between μ and ν, and show how it relates to the shortest distance. the relationship to earlier definitions of distance between fuzzy sets [1,3] is also discussed.

Nonnegative rank-preserving operators

Analogues of characterizations of rank-preserving operators on field-valued matrices are determined for matrices witheentries in certain structures S contained in the nonnegative reals. For example, if S is the set of nonnegative members of a real unique factorization domain (e.g. the nonnegative reals or the nonnegative integers), M is the set of m× n matrices with entries in S , and min( m, n)⩾4, then a “linear” operator on M preserves the “rank” of each matrix in M if and only if it preserves the ranks of those matrices in M of ranks 1, 2, and 4. Notions of rank and linearity are defined analogously to the field-valued concepts. Other characterizations of rank-preserving operators for matrices over these and other structures S are also given.

An Embedding Characterization of Almost Compact Spaces

We characterize almost compact and almost realcompact spaces in terms of their situation in the product ${(J,u)^C}$. In the characterization of almost compactness $J$ is the two point set or the unit interval; in the characterization of almost realcompactness $J$ is the set of nonnegative integers or the nonnegative reals. $u$ is the upper topology on the real line restricted to $J$.

An embedding characterization of almost compact spaces

We characterize almost compact and almost realcompact spaces in terms of their situation in the product ( J , u ) C {(J,u)^C} . In the characterization of almost compactness J J is the two point set or the unit interval; in the characterization of almost realcompactness J J is the set of nonnegative integers or the nonnegative reals. u u is the upper topology on the real line restricted to J J .

Abstract Cauchy problems that involve a product of two Euler-Poisson-Darboux operators

Let X be a Banach space, let B be the generator of a continuous group in X, and let A = B 2. Assume that D( A r ) is dense in X for r an arbitrarily large positive integer and that a and b are non-negative reals. Solution representations are developed for the abstract differential equation (D 2 t + b t D t − A) · (D 2 t + a t D t − A) u(t) = 0, t > 0 corresponding to initial conditions of the form: (i) u(0+) = φ, u ( j) (0+) = 0, j = 1, 2, 3 and (ii) u 2(0+) = φ, u j (0+) = 0, j = 0, 1, 3 (with φ∈ D( A r )) for all choices of a and b.

Nonnegative integral generalized inverses

Necessary and sufficient conditions are given for a nonnegative integral matrix to have nonnegative integral generalized inverses of various types, and the possible ranks of these inverses are determined. More generally, conditions are also given for matrices to have generalized inverses over certain subsets of the nonnegative reals forming monoids under addition and multiplication. Many of our results are adapted from results of Berman and Plemmons [3–6] on real matrices.

A stochastic difference equation

where a(.) and t(.) are random processes with values, respectively, in the non-negative reals, and 5%““. That is, a(n, w) and c(n, w) depend on LL, in a common probability space 0, and (0.1) is assumed to hold for every n ENi for almost every o E 8. Equation (0.1) represents the motion of a particle which, at time i’ = n, moves in the randomly perturbed direction -f( Y(n j) + t(n) by a proportionate step a(n). For example, f(y(n)) might describe the performance (output) of a complex physical system with input value u(n). The performance function f(.) should be assumed to be unknown; however, “noise” disturbed observations of the performance, f(u(n)) &z), are available with each trial run (observation) at t = M. Then, (0.1) can be thought of as a recursive adjustment process under uncertainty, in which the “controllers” of the system seek an input level which yields a desired performance. Henceforth, let 0 E 9’” be the desired performance value. The example is typical of stochastic approximation (SA) or adaptive adjustment problems, and is elaborated in Example (1.2), below. The intuition source, and touchstone for comparison of our assumptions will be SA (see, e.g., [ 1 ] ). We study (0.1) by looking at the continuous-parameter version, the “random” differential equation

A general theory of measures for nonnegative matrices

By a measure μ on the set N of m × n nonnegative matrices we mean that μ is a function from N to the nonnegative reals such that (i) μ( λA)= λμ( A) for all nonnegative λ and all A ∈ N, and (ii) μ( A + B) ⩾ μ( A) for all A, B ϵ N. This paper develops a theory of such measures and shows how this theory can be applied to particular problems.

Remarks on a paper by F. Browder about contraction

THROUGHOUT this note, X denotes a complete metric space with distance function d, and T is a map from X into itself. Powers of T are defined by Tax = x and T”+ lx = T(T”x), n 2 0. Occasionally, we use the notation xk = Tkx, in particular x0 = x, x1 = TX, for the sake of brevity. The set {x, TX, T'x, . . . , T”x,. . .} is called an orbit (starting at x) and denoted by O(x). The set 0(x, y) is the union of the two orbits starting at x and y, 0(x, y) = {xk, yk: k 3 O}. The diameter diam A of a nonempty subset A of X is the supremum of the numbers d(x, y), where x and y vary in A. The letter 4 denotes a contractive gauge function, i.e. a continuous, increasing function from R,, the non-negative reals, into R, which satisfies 4(s) 0.

A characterization of threshold matroids

We show that for a matroid M=(E, F) , where E={1, 2, …, n}, there exist nonnegative reals a 1, a 2, …, a n and b such that F={S⊆E: Σ iϵSa i⩽b} if and only if for each pair of subsets R and S of E, such that | R∪ S|⩽3, either 1. ∀ T⊆ E−( R∪ S), (R∪T)ϵ F⇒(S ∪ T) ϵ F or 2. (ii)|∀ T⊆ E−( R∪ S), (S∪T)ϵ F⇒(R ∪ Y) ϵ F .