Positive Weight Function Research Articles

Photon-added Barut$ndash$Girardello coherent states of the pseudoharmonic oscillator

In a previous paper (Popov D 2001 J. Phys. A: Math. Gen. 34 5283) we have constructed the coherent states (CSs) of the pseudoharmonic oscillator (PHO). These states are of the Barut–Girardello type. In the present paper, we have constructed and investigated some properties of the photon-added Barut–Girardello coherent states (PA-BGCSs) of the PHO. These states are not orthogonal but are normalized and satisfy a unity resolution relation (the completeness relation). We have found the analytical form for the positive weight function in the resolution of unity. By using these states, we have calculated some expectation values (the first two powers of the number operator N), which have evinced some non-classical properties. In order to examine the statistical properties of a PHO gas, which obeys the quantum canonical distribution, the diagonal P-representation of the density operator ρ(m) is constructed and the thermal expectation values are calculated. All these quantities are expressed in terms of Meijer G-functions, and so the PA-BGCSs are a new field of application for these functions. Also, the time dependence of the PA-BGCSs is examined. If in the obtained results we put m = 0, we recover all the results from our previous paper.

A biharmonic maximum principle for hyperbolic surfaces

Let W be a simply connected two-dimensional Riemannian manifold with a Cysmooth metric ds. We can always introduce local isothermal coordinates near any point of W, that is, such coordinates ðx; yÞ that the metric is represented in the form dsðx; yÞ 1⁄4 oðx; yÞðdx þ dyÞ, for some positive weight function o. And since W is orientable as a two-dimensional simply-connected manifold, these local isothermal coordinates can serve as conformal charts for a complex structure on W (see [1], pp. 124–126). The Kœbe uniformization theorem then says that W is conformally equivalent to one of the three sets: the Riemann sphere S 1⁄4 CW fyg, the complex plane C, or the open unit disk D. This equivalence, together with the choice of isothermal coordinates, allows us to identify W with one of the above three sets W supplied with the isothermal Riemannian metric

Weighted domination in triangle-free graphs

A weighted graph (G,w) is a graph G together with a positive weight-function on its vertex set w : V(G)→ R >0 . The weighted domination number γ w(G) of (G,w) is the minimum weight w(D)=∑ v∈Dw(v) of a set D⊆V(G) such that every vertex x∈V(D)−D has a neighbor in D. If ∑ v∈V(G)w(v)=|V(G)| , then we speak of a normed weighted graph. Recently, we proved that γ w(G)γ w( G ̄ )⩽ n 2 8 1+ 2 n−2 and γ w(G)+γ w( G ̄ )⩽ 3n 4 + n 2(n−2) for normed weighted bipartite graphs (G,w) of order n such that neither G nor the complement G ̄ has isolated vertices. In this paper we will extend these Nordhaus–Gaddum-type results to triangle-free graphs.

Designer polynomials, discrete variable representations, and the Schrödinger equation

AbstractThe general procedure for constructing a set of orthonormal polynomials is given for an arbitrary positive definite weight function, w(x), in the interval [a, b]. The Lanczos method is used to generate the three‐term recursion relation, which is then used to produce the polynomial coefficients. A discrete variable representation (DVR) is constructed from Gaussian nodes and weights that result from the three‐term recursion relation. These are termed “designer polynomials” and the associated “designer DVRs.” It will be shown by construction that every such set of “synthetic polynomials” carries an associated DVR. The term “designer” derives from the fact that the interval [a, b] and the weight function w(x) are arbitrary (except that w(x) must be positive definite on [a, b] and must have continuous derivatives except at a finite number of isolated discontinuities) and may be adapted to the physical problem of interest. The difficulties of applying a DVR to a “bare” Coulomb problem will be illustrated on a “toy” model in one dimension (1‐D hydrogen atom). A solution for the 1‐D Coulomb problem will be given, thereby motivating the need for designer DVRs. In doing so, a new set of polynomials is defined with a weight function w(x) = |x|kexp(−α|x|), (such that k = −1, 0, +1, +2, …) between the symmetrical limits [−∞, +∞]. These are called “synthetic Cartesian exponential polynomials (SCEP).” These polynomials are then used in a spectral and pseudospectral (DVR) representation to solve the 1‐D hydrogen atom problem. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

TRICOMI COHERENT STATES

We construct explicit families of coherent states whose parametrization permits an arbitrarily small deviation from conventional coherent states. They possess the resolution of unity with a positive weight function given as the solution of a Stieltjes power-moment problem, with moments analytically expressible through Tricomi's confluent hypergeometric function. These states are in general squeezed, can be either sub- or super-Poissonian in nature, and induce a deformation of the metric of the complex plane.

Bernstein equiconvergence and Fejér-type theorems for general rational Fourier series

Let w( θ) be a positive weight function on the interval [−π,π] and associate the positive-definite inner product on the unit circle of the complex plane by 〈F,G〉 w= 1 2 π ∫ − π π F( e iθ ) G( e iθ ) w(θ) dθ. For a sequence of points { α k } k=1 ∞ included in a compact subset of the open unit disk, we consider the orthogonal rational functions ( ORF) { φ k } k=0 ∞ that are obtained by orthogonalization of the sequence {1, z/π 1, z 2/π 2,…} where π k(z)= ∏ j=1 k (1− α ̄ jz) , with respect to this inner product. In this paper we prove that s n(z)− S n(z) tends to zero in | z|⩽1 if n tends to ∞, where s n is the nth partial sum of the expansion of a bounded analytic function F in terms of the ORF { φ k } k=0 ∞ and S n is the nth partial sum of the ordinary power series expansion of F. The main condition on the weight is that it satisfies a Dini–Lipschitz condition and that it is bounded away from zero. This generalizes a theorem given by Szegő in the polynomial case, that is when all α k =0. As an important consequence we find that under the above conditions on the weight w and the points { α k } k=1 ∞, the Cesàro means of the series s n converge uniformly to the function F in | z|⩽1 if the boundary function f(θ) ≔ F( e iθ ) is continuous on [0,2π]. This can be seen as a generalization of Fejér's Theorem.

Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems

We construct a large number of coherent states satisfying the resolution of unity with a positive weight function, obtained through analytic solutions of the Stieltjes moment problem (coherent states on a plane) and the Hausdorff moment problem (coherent states on a disk). These solutions are obtained through the method of inverse Mellin transform. In addition, these coherent states induce a deformation of the metric that has been calculated analytically.

On the completeness of photon-added coherent states

We demonstrate explicitly the completeness of photon-added coherent states (PACSs), introduced by Agarwal and Tara (Agarwal G S and Tara K 1991 Phys. Rev. A 43 492) and defined, up to normalization, by (â†)M|z⟩, M = 0,1,2,...,, where ↠is the boson creation operator and |z⟩ are conventional Glauber-Klauder coherent states. We find the analytical form of the positive weight function in their resolution of unity by solving the associated Stieltjes power-moment problem. We furnish an example of generation of another set of PACSs which are complete.

Theory of Boundary Eigensolutions in Engineering Mechanics

A theory of boundary eigensolutions is presented for boundary value problems in engineering mechanics. While the theory is quite general, the presentation here is restricted to potential problems. Contrary to the traditional approach, the eigenproblem is formed by inserting the eigenparameter, along with a positive weight function, into the boundary condition. The resulting spectra are real and the eigenfunctions are mutually orthogonal on the boundary, thus providing a basis for solutions. The weight function permits effective treatment of nonsmooth problems associated with cracks, notches and mixed boundary conditions. Several ideas related to the convergence characteristics are also introduced. Furthermore, the connection is made to integral equation methods and variational methods. This paves the way toward the development of new computational formulations for finite element and boundary element methods. Two numerical examples are included to illustrate the applicability.

Similarity measurement using term negative weight and its application to word similarity

A “term weighting” is a useful technique for keyword extraction and document classification. The traditional approach depends on high frequency terms, called positive weight ( PW) function. This paper presents a new weighting method that depends on low frequency terms, called negative weight ( NW) function. In this paper word similarity for typical verbs and objects is focused as an example for the application field. Negative weighted inverse verb frequency ( NWIVF) function is well defined in this study and new similarity measurement is presented by combining the NWIVF and PWIVF ( positive weighted inverse verb frequency) functions. The proposed method is applied to 11,000 relationships between verbs and nouns extracted from a large tagged corpus. By using this new method both recall and precision have improved by 33% and 18% respectively, over the positive weight method.

On the completeness of coherent states generated by binomial distribution

We demonstrate the completeness of boson- and spin-coherent states generated by binomial distribution by furnishing the exact analytical form of their respective positive weight functions in resolution of unity. These weight functions are solutions of the associated truncated Stieltjes power-moment problem. We elaborate on the non-unique character of these solutions.

Boundary asymptotics for orthogonal rational functions on the unit circle

Let w(θ) be a positive weight function on the unit circle of the complex plane. For a sequence of points { αk }k = 1∞ included in a compact subset of the unit disk, we consider the orthogonal rational functions φn that are obtained by orthogonalization of the sequence { 1, z / π1, z2 / π2, ... } where \( {\pi }_k \left( {\rm Z} \right) = \prod\nolimits_{j^{ = 1} }^k {\left( {1 - \overline {\alpha } j{\rm Z}} \right)} \), with respect to the inner product \(\left\langle {f,g} \right\rangle = \frac{1}{{2{\pi }}}\int_{{ - \pi }}^{\pi } {f\left( {{e}^{i\theta } } \right)} \overline {g\left( {{e}^{i\theta } } \right)} w\left( {\theta } \right){d\theta }\) In this paper we discuss the behaviour of φn(t) for ∣ t ∣ = 1 and n → ∞ under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szegő in the polynomial case, that is when all αk = 0.

Mittag-Leffler coherent states

We create a family of boson coherent states using the functions of Mittag-Leffler (ML) E(z), (>0) and their generalizations E,(z), (,>0) instead of exponentials. These states are shown to satisfy the usual requirements of normalizability, continuity in the label and the resolution of unity with a positive weight function. This last quantity is found for arbitrary ,>0 as a solution of an associated Stieltjes moment problem. In addition, for = m = 1,2,3 ... and = 1 (corresponding to Em(z)) we propose and analyse special q-deformations (0<q1) of the functions Em(z) which serve as a tool to define q-deformed coherent states of ML type. We provide the expressions for expectation values of physical quantities for all the above states. We discuss physical properties of these states, noting that they are squeezed. The ML coherent states are sub-Poissonian in nature, whereas the q-deformed ML states can be sub- and super-Poissonian depending on q. All these states are shown to be eigenstates of deformed boson operators whose commutation relations are given.

On the Polynomials Orthogonal on Regular Polygons

The two-parameter Pastro–Al-Salam–Ismail (PASI) polynomials are known to be bi-orthogonal on the unit circle with continuous weight function when 0<q<1. We study the case ofqa root of unity. It is shown that corresponding PASI polynomials are orthogonal on the unit circle with discrete measure located on the vertices of the regularN-gon. Cases leading to a positive weight function are analyzed. In particular, we obtain trigonometric analogs of the Askey–Szegő polynomials which are orthogonal on regularN-gons with positive weight function.

Tau-Function Constructions of the Recurrence Coefficients of Orthogonal Polynomials

In this paper we compute the recurrence coefficients of orthogonal polynomials using τ-function techniques. It is shown that for polynomials orthogonal with respect to positive weight functions on a noncompact interval, the recurrence coefficient can be expressed as the change in the chemical potential which, for sufficiently largeNis the second derivative of the free energy with respect toN, the particle number. We give three examples using this technique: Freud weights, Erdős weights, and weak exponential weights.

MEDIANS OF GRAPHS AND KINGS OF TOURNAMENTS*

We first prove that for any graph $G$ with a positive vertex weight function $w$, there exists a graph $H$ with a positive weight function $w'$ such that $w(v)=w'(v)$ for all vertices $v$ in $G$ and whose $w'$-median is $G$. This is a generalization of a previous result for the case in which all weights are 1. The second result is that for any $n$-tournament $T$ without transmitters, there exists an integer $m\leq 2n-1$ and an $m$-tournament $T'$ whose kings are exactly the vertices of $T$. This improves upon a previous result for $m\leq 2n$.

Finite-Sample Confidence Envelopes for Shape-Restricted Densities

A conservative finite-sample simultaneous confidence envelope for a density can be found by solving a finite set of finite-dimensional linear programming problems if the density is known to be monotonic or to have at most $k$ modes relative to a positive weight function. The dimension of the problems is at most $(n/\log n)^{1/3}$, where $n$ is the number of observations. The linear programs find densities attaining the largest and smallest values at a point among cumulative distribution functions in a confidence set defined using the assumed shape restriction and differences between the empirical cumulative distribution function evaluated at a subset of the observed points. Bounds at any finite set of points can be extrapolated conservatively using the shape restriction. The optima are attained by densities piecewise proportional to the weight function with discontinuities at a subset of the observations and at most five other points. If the weight function is constant and the density satisfies a local Lipschitz condition with exponent $\varrho$, the width of the bounds converges to zero at the optimal rate $(\log n/n)^{\varrho/(1+2\varrho)}$ outside every neighborhood of the set of modes, if a "bandwidth" parameter is chosen correctly. The integrated width of the bounds converges at the same rate on intervals where the density satisfies a Lipschitz condition if the intervals are strictly within the support of the density. The approach also gives algorithms to compute confidence intervals for the support of monotonic densities and for the mode of unimodal densities, lower confidence intervals on the number of modes of a distribution and conservative tests of the hypothesis of $k$-modality. We use the method to compute confidence bounds for the probability density of aftershocks of the 1984 Morgan Hill, CA, earthquake, assuming aftershock times are an inhomogeneous Poisson point process with decreasing intensity.

Transient effects in diffusion-controlled absorption by a nonuniform sink of arbitrary constitution

The time-dependent rate coefficient for absorption of diffusing particles by a nonuniform sink of arbitrary constitution is studied on the basis of the Smoluchowski equation. It is shown that the amplitude of the long-time tail can be expressed in terms of the asymptotic rate coefficient. The time-dependent rate coefficient is expressed as the sum of the asymptotic steady-state coefficient and an integral of purely decaying exponentials with a positive weight function. It is shown that the rate coefficient can be represented accurately over a wide range of time by an approximation to its Laplace transform, characterized by a small number of poles in the complex square root of frequency plane.

On the maximal possible growth rate for normal eigenfunctions of a class of Sturm?Liouville operators with continuous positive weight function

On the maximal possible growth rate for normal eigenfunctions of a class of Sturm?Liouville operators with continuous positive weight function

Distribution of points for convergent interpolatory integration rules on (??, ?)

Letw be a “nice” positive weight function on (−∞, ∞), such asw(x)=exp(−⋎x⋎α) α>1. Suppose that, forn≥1, $$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} } f(x_{jn} )$$ is aninterpolatory integration rule for the weightw: that is for polynomialsP of degree ≤n-1, $$I_n [P]: = \int\limits_{ - \infty }^\infty {P(x)w(x)dx.} $$ Moreover, suppose that the sequence of rules {I n} t8 isconvergent: $$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - \infty }^\infty {f(x)w(x)dx} $$ for all continuousf:R→R satisfying suitable integrability conditions. What then can we say about thedistribution of the points {x jn} ,n≥1? Roughly speaking, the conclusion of this paper is thathalf the points are distributed like zeros of orthogonal polynomials forw, and half may bearbitrarily distributed. Thus half the points haveNevai-Ullmann distribution of order α, and the rest are arbitrarily distributed. We also describe the possible distributions of the integration points, when the ruleI n has precision other thann-1.