Hermite–Hadamard-type inequalities for Riemann–Liouville fractional integrals via two kinds of convexity

In this article, two fundamental integral identities including the second-order derivatives of a given function via Riemann–Liouville fractional integrals are established. With the help of these two fractional-type integral identities, all kinds of Hermite–Hadamard-type inequalities involving left-sided and right-sided Riemann–Liouville fractional integrals for m-convex and (s,?m)-convex functions, respectively. Our methods considered here may be a stimulant for further investigations concerning Hermite–Hadamard-type inequalities involving Hadamard fractional integrals.     

On Cayley-Transform Methods for the Discretization of Lie-Group Equations

In this paper we develop in a systematic manner the theory of time-stepping methods based on the Cayley transform. Such methods can be applied to discretize differential equations that evolve in some Lie groups, in particular in the orthogonal group and the symplectic group. Unlike many other Lie-group solvers, they do not require the evaluation of matrix exponentials. Similarly to the theory of Magnus expansions in [13], we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals can be evaluated to high order by using special quadrature formulas similar to the construction in [13]. Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual advantages associated with time symmetry, e.g., even order of approximation. However, time symmetry (with its attendant benefits) is attained when exact integrals are replaced by certain quadrature formulas. March 7, 2000. Final version received: August 10, 2000. Online publication: January 2, 2001.     

Laplace Invariants of Two-Dimensional Open Toda Lattices

We show that Toda lattices with the Cartan matrices A _n , B _n , C _n , and D _n are Liouville-type systems. For these systems of equations, we obtain explicit formulas for the invariants and generalized Laplace invariants. We show how they can be used to construct conservation laws (x and y integrals) and higher symmetries.     

Low-Temperature Localisation for Continuous Spin-Systems

We study the free energy of continuous spin-systems on Z ^d , in the framework of Laplace integrals and transfer operators. Under a weak coupling condition, we show that the free energy in the low-temperature limit is determined, up to an exponentially small error, by the restriction to a neighbourhood of global minima of the energy. We have results for some single- and double-well problems.     

Berezin–Toeplitz quantization for lower energy forms

Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin–Toeplitz quantization of the open set of M where the curvature on L is nondegenerate. In particular, we quantize any manifold admitting a positive line bundle. The quantum spaces are the spectral spaces corresponding to [0,k^?N], where N>1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powers L^k. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclassical limit k→∞ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin–Toeplitz quantization for semi-positive and big line bundles.     

Some theorems on integral transforms

A theorem is proved which establishes a connection among Laplace, Kantorovich-Lebedev, Meier, and the generalized Mehler-Focktransforms. Some improper integrals are calculated by using this theorem and its immediate generalization.     

Characterizations of m-EP elements in rings

Let R be a ring with involution. In this paper, we extend the notions of m-EP matrices and m-EP operators to an arbitrary ring case. A number of new characterizations of m-EP elements in rings are presented. In particular, the existence criteria for 1-EP (i.e. EP) elements are obtained by means of the group inverse, Moore–Penrose inverse, and core inverse. Some properties of 2-EP are also given.     

A Systematic “Saddle Point Near a Pole” Asymptotic Method with Application to the Gauss Hypergeometric Function

In recent works [ 1 ] and [ 2 ], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [ 1 ] and [ 2 ] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions). In particular, when the order of the saddle point is two, the basic approximant is given in terms of the error function (as in the standard method). As an application, we obtain new asymptotic expansions of the Gauss Hypergeometric function ₂F₁(a, b, c; z) for large b and c with c > b + 1 .     

On Certain Fejér and Tchebychev Type Integrals

Let m and n denote a pair of positive integers. In this paper, we call upon the Hadamard product and computer algebra techniques to evaluate the Fejér integral π ^?1 ∫₀ ^π (sin mθ / sin θ) ²ⁿ dθ. Using symmetry arguments, it is proved that the value of this integral is an odd polynomial in m of degree 2n ? 1. This permits using polynomial curve fitting methods and mathematical software packages to obtain evaluation formulas for n relatively small. Some cases of the above integral with 2n replaced by 2n + 1 are also discussed. A familiar identity shows that these yield evaluations of integrals of powers of certain Tchebychev polynomials.     

TRACE EXPANSIONS FOR THE ZAREMBA PROBLEM

ABSTRACT

We consider a Laplace operator for sections of a vector bundle on a manifold M, with mixed boundary conditions, the so-called Zaremba problem. The boundary consists of three disjoint parts, ?M_D , ?M_N , together with Σ, their common boundary relative to ?M. Dirichlet conditions are imposed along ?M_D and Neumann conditions along ?M_N . It turns out that a condition must be imposed along Σ as well. In order to apply earlier work [Bruening and Seeley, Journal of Functional Analysis 1991, 95, 255–290], we impose Dirichlet conditions along Σ, giving the Friedrichs extension of the operator with the given conditions along ?M_D and ?M_N . We obtain a complete asymptotic expansion of the trace of an appropriate power of the resolvent, and hence also the heat trace, with the usual powers of t. The coefficients are given as integrals over M, over ?M, and over Σ. The logarithmic terms which might be expected are absent in this case; this is the main new result. Similar results are suggested for other conditions along Σ, and for the case of mixed absolute and relative conditions on differential forms. The expansion for these cases requires an extension of the paper cited above.     

Wiener Integrals with Respect to the Hermite Process and a Non-Central Limit Theorem

Abstract

We introduce Wiener integrals with respect to the Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein–Uhlenbeck process.     

Approximate formulas for moderately small eikonal amplitudes

We consider the eikonal approximation for moderately small scattering amplitudes. To find numerical estimates of these approximations, we derive formulas that contain no Bessel functions and consequently no rapidly oscillating integrands. To obtain these formulas, we study improper integrals of the first kind containing products of the Bessel functions J₀(z). We generalize the expression with four functions J₀(z) and also find expressions for the integrals with the product of five and six Bessel functions. We generalize a known formula for the improper integral with two functions J_υ (az) to the case with noninteger υ and complex a.     

Transformation of hypersingular integrals and black-box cubature

In this paper, we will consider hypersingular integrals as they arise by transforming elliptic boundary value problems into boundary integral equations. First, local representations of these integrals will be derived. These representations contain so-called finite-part integrals. In the second step, these integrals are reformulated as improper integrals. We will show that these integrals can be treated by cubature methods for weakly singular integrals as they exist in the literature.

     

Integral Representations of Derivatives and Integrals with Respect to the Order of the Bessel Functions Jv(t), Iv(t), the Anger Function Jv(t) and the Integral Bessel Function Jiv(t)

^*To whom Correspondence should be addressed. On sabbatical leave in the University of Alberta, Department of Chemical Engineering, Edmonton, Alberta, Canada, T6G 2G6. Integral representations of integrals and derivatives with respectto the order of the Bessel functions J_v(t) I_v(t), the integralBessel function Ji_v(t) and the Anger function J_v(t) are presented.The Laplace transform technique is applied to derive them. Theintegral representations permit the evaluation of a number oftrigonometric integrals.     

Talbot quadratures and rational approximations

Many computational problems can be solved with the aid of contour integrals containing e ^z in the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly O(3^-N ), where N is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, O(9.28903^-N ), can be achieved by using a different approach: best supremum-norm rational approximants to e ^z for z∈(–∞,0], following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody–Meinardus–Varga approximants by the method of Carathéodory–Fejér approximation. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65D30, 41A20     

Numerical Computation of Multivariate Normal and Multivariate-t Probabilities over Convex Regions

Abstract

A methodology has been developed and Fortran 90 programs have been written to evaluate multivariate normal and multivariate-t integrals over convex regions. The Cholesky transformation is used to transform the integrand into a product of standard normal or spherically symmetric t variables. For any random direction from the origin, an unbiased estimate of the value of the integral is Pr[X ² ≥ r²] (multivariate normal) or Pr[F ≥ r²/k] (multivariate-t), where r is the distance from the origin to the boundary in a randomly chosen direction, and k is the dimension of the integral. Two Fortran 90 programs have been written. MVI uses the average of many estimates. MVIB uses a binning procedure to obtain an empirical distribution of the distance from the origin to the boundary. Gauss-Legendre quadrature is then used to estimate the value of the integral. The running time for MVIB is modestly smaller than that for MVI. However, in solving certain integral equations (e.g., using an iterative procedure to find the percentage point of a statistic), using MVIB usually requires no Monte Carlo sampling after the first iteration, and is considerably more efficient. MVIB and MVI are highly efficient for the evaluation of integrals whose value is large. “Naive” Monte Carlo (MC) may be competitive with MVI or MVIB only if the value of the probability integral is small or the shape of the region is “extreme.”     

Functional Compatibility,Markov Chains,and Gibbs Sampling with Improper Posteriors

Abstract

The members of a set of conditional probability density functions are called compatible if there exists a joint probability density function that generates them. We generalize this concept by calling the conditionals functionally compatible if there exists a non-negative function that behaves like a joint density as far as generating the conditionals according to the probability calculus, but whose integral over the whole space is not necessarily finite. A necessary and sufficient condition for functional compatibility is given that provides a method of calculating this function, if it exists. A Markov transition function is then constructed using a set of functionally compatible conditional densities and it is shown, using the compatibility results, that the associated Markov chain is positive recurrent if and only if the conditionals are compatible. A Gibbs Markov chain, constructed via “Gibbs conditionals” from a hierarchical model with an improper posterior, is a special case. Therefore, the results of this article can be used to evaluate the consequences of applying the Gibbs sampler when the posterior's impropriety is unknown to the user. Our results cannot, however, be used to detect improper posteriors. Monte Carlo approximations based on Gibbs chains are shown to have undesirable limiting behavior when the posterior is improper. The results are applied to a Bayesian hierarchical one-way random effects model with an improper posterior distribution. The model is simple, but also quite similar to some models with improper posteriors that have been used in conjunction with the Gibbs sampler in the literature.     

Some Maximal Operators Related to Families of Singular Integral Operators

In this paper, we shall study L^p-boundedness of two kinds of maximal operators related to some families of singular integrals.