one can ask is whether there is an efficient. way t.o simulate a network of LTEs with big weight.s by a network of LTEs with small weights. We prove the following results: 1) every LTE with big weights can be simulated by a depth-3, polynomial size network of LTEs with small weights, 2) every depth-d, polynomial size network of LTEs with big weights can be simulated by a depth-(2d+ 1), polynomi...

Abstract. We show that in the large matrix limit, the eigenvalues of the normal matrix model for matrices with spectrum inside a compact domain with a special class of potentials homogeneously fill the interior of a polynomial curve uniquely defined by the area of its interior domain and its exterior harmonic moments which are all given as parameters of the potential. Then we consider the ortho...

We consider a polynomial – which we term the “slope polynomial” – that encodes information about slopes of lines defined by a point-set in finite affine planes. When the aforementioned point-set is the graph of a permutation, we show that constancy of the polynomial is equivalent to the permutation being linear. This has immediate consequences for the structure of the 3-uniform hypergraph of co...

The q–deformation Uq(h4) of the harmonic oscillator algebra is defined and proved to be a Ribbon Hopf algebra. Associated with this Hopf algebra we define an infinite dimensional braid group representation on the Hilbert space of the harmonic oscillator, and an extended Yang–Baxter system in the sense of Turaev. The corresponding link invariant is computed in some particular cases and coincides...

We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a DarbouxCrum transformation to...

Brolin-Lyubich measure λR of a rational endomorphism R : Ĉ → Ĉ with degR ≥ 2 is the unique invariant measure of maximal entropy hλR = htop(R) = log d. Its support is the Julia set J(R). We demonstrate that λR is always computable by an algorithm which has access to coefficients of R, even when J(R) is not computable. In the case when R is a polynomial, BrolinLyubich measure coincides with the h...

The idea of using physical systems symmetries to study degenerate energy levels has been adopted since the early days of quantum mechanics. So ladder operators which connect all the eigen-states with a given energy lead a good method to solve this problem. For linear systems, such as Hydrogen atom and isotropic harmonic oscillator, Lie algebra can work out these problems well. Generally, the N ...

It is shown that the higher order supersymmetric partners of the harmonic oscillator Hamiltonian provide the simplest non-trivial realizations of the polynomial Heisenberg algebras. A linearized version of the corresponding annihilation and creation operator leads to a Fock representation which is the same as for the harmonic oscillator Hamiltonian. 1. Introduction. The non-linear algebras have...

In this work, the problem of Rayleigh wave propagation is considered in the context of the theory of thermoelastic diffusion. The formulation is applied to a homogeneous isotropic thermoelastic half space with mass diffusion at the stress free, isothermal, isoconcentrated boundary. Using the potential functions and harmonic wave solution, three coupled dilatational waves and a shear wave is obt...