For learning about the Langlands program, knowledge of Lie-group structure theory, algebraic number theory, algebraic geometry, modular forms, and infinite-dimensional representation theory is appropriate. This article describes how one can get a glimpse of the program with less background than this.

This paper deals with two interrelated issues. One is an invariant subspace approach to finding solutions for the algebraic Riccati equation for a class of infinite dimensional systems. The second is approximation of the solution of the algebraic Riccati equation by finite dimensional approximants. The theory of exponentially dichotomous operators and bisemigroups is instrumental in our approach.

The Dirac theory in the Euclidean Taub-NUT space gives rise to a large collection of conserved operators associated to genuine or hidden symmetries. They are involved in interesting algebraic structures as dynamical algebras or even infinite-dimensional algebras or superalgebras. One presents here the infinite-dimensional superalgebra specific to the Dirac theory in manifolds carrying the Gross...

The Dirac theory in the Euclidean Taub-NUT space gives rise to a large collection of conserved operators associated to genuine or hidden symmetries. They are involved in interesting algebraic structures as dynamical algebras or even infinite-dimensional algebras or superalgebras. One presents here the infinite-dimensional superalgebra specific to the Dirac theory in manifolds carrying the Gross...

A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we have found a Poissonian behavior, whereas level repulsion close to ...

We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of Q. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown...

The perfect Golay codes lifted over the finite rings Z2m and Z3m, or over their infinite counterparts, namely the 2-adic and 3-adic rings, are considered, and their algebraic decoding is obtained as a non-trivial extension of the algebraic decoding of the perfect Golay codes. Mathematics Subject Classification: 94B15, 94B35, 11D88, 14G50

In this paper, biochemical reaction problem is given in the form of a system of non-linear differential equations involving Caputo fractional derivative. The aim is to suggest an instrumental scheme to approximate the solution of this problem. To achieve this goal, the fractional derivation terms are expanded as the elements of shifted Legendre scaling functions. Then, applying operational matr...

In this article we propose algebraic universal procedure for deriving ”fusion rules” and Baxter equation for any integrable model with Uq(ŝl2) symmetry of Quantum Inverse Scattering Method. Baxter Qoperator is got from the certain infinite dimensional representation of the Universal Rmatrix for Uq(ŝl2) affine algebra. The algebraic properties of Q-operator are examined.

We consider fragments of first-order logic and as models we allow finite and infinite words simultaneously. The only binary relations apart from equality are order comparison < and the successor predicate +1. We give characterizations of the fragments Σ2 = Σ2[<,+1] and FO2 = FO2[<,+1] in terms of algebraic and topological properties. To this end we introduce the factor topology over infinite wo...