One of the most common types of functions in mathematics, physics, and engineering is a sum of products, sometimes called a partition function. After “normalization,” a sum of products has a natural graphical representation, called a normal factor graph (NFG), in which vertices represent factors, edges represent internal variables, and half-edges represent the external variables of the partitio...

An efficient technique for the analysis of scattering by uniaxial anisotropic objects is presented. The technique is based on the method of higher order MoM of the surface integral equations. This higher order MoM solution uses the higher order hierarchical basis functions which are based on the modified Legendre polynomials. Numerical results are given to demonstrate that the higher order hier...

We argue that the AdS dual of the three dimensional critical O(N) vector model can be evaluated using the Legendre transform that relates the generating functionals of the free UV and the interacting IR fixed points of the boundary theory. As an example, we use our proposal to evaluate the minimal bulk action of the scalar field that it is dual to the spin-zero “current” of the O(N) vector mode...

The paper introduces a way of re-constructing a loss function from predictive complexity. We show that a loss function and expectations of the corresponding predictive complexity w.r.t. the Bernoulli distribution are related through the Legendre transformation. It is shown that if two loss functions specify the same complexity then they are equivalent in a strong sense. The expectations are als...

in this paper, we develop an efficient legendre wavelets collocation method for well known time-fractional heat equation. inthe proposed method, we apply operational matrix of fractionalintegration to obtain numerical solution of the inhomogeneoustime-fractional heat equation with lateral heat loss and dirichletboundary conditions. the power of this manageable method isconfirmed. moreover, the ...

Abstract. Motivated by questions on the preconditioning of spectral methods, and independently of the extensive literature on the approximation of zeroes of orthogonal polynomials, either by the Sturm method, or by the descent method, we develop a stationary phase-like technique for calculating asymptotics of Legendre polynomials. The difference with the classical stationary phase method is tha...

In the main theorem of this paper we show that any involution on the class of lower semi-continuous convex functions which is order-reversing, must be, up to linear terms, the well known Legendre transform.

The purpose of this work is to carry out a systematic study of a special class of convex functions defined over the space Sn of symmetric matrices of order n×n. The functions under consideration (Φ : Sn → R∪ {+∞}) are spectrally defined in the sense that the value Φ(A) depends only on the spectrum {λ1(A), . . . , λn(A)} of the matrix A ∈ Sn. Fenchel–Legendre conjugation, firstand second-order s...

We extend the definition of the classical Jacobi polynomials withindexes α,β > −1 to allow α and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the...

We describe an application of the Legendre transform to communication networks. The Legendre transform applied to max-plus algebra linear systems corresponds to the Fourier transform applied to conventional linear systems. Hence, it is a powerful tool that can be applied to max-plus linear systems and their identification. Linear max-plus algebra has been already used to describe simple data co...