Bipartite Matching and Van der Waerden Conjecture

In this paper I show that the free energy F and the cost C associated to a bipartite matching problem can be explicitly estimated in term of the solution of a suitable system of equations (cavity equations in the following). The proof of these results relies on a well known result in combinatorics: the Van der Waerden conjecture (Egorychev–Falikman Theorem). Cavity equations, derived by a mean field argument by Mèzard and Parisi, can be considered as a smoothed form of the dual formulation for the bipartite matching problem. Moreover cavity equation are the Euler–Lagrange equations of a convex functional G parameterized by the temperature T. In term of their unique solution it is possible to define a free-energy-like function of the temperature g(T). g is a strictly decreasing concave function of T and C=g(0). The convexity of G allows to define an explicit algorithm to find the solution of the cavity equations at a given temperature T. Moreover, once the solution of the cavity equations at a given temperature T is known, the properties of g allow to find exact estimates from below and from above of the cost C.     

A very strong difference property for semisimple compact connected lie groups

Let G be a topological group. For a function f: G → ℝ and h ∈ G, the difference function Δ _h f is defined by the rule Δ _h f(x) = f(xh) − f(x) (x ∈ G). A function H: G → ℝ is said to be additive if it satisfies the Cauchy functional equation H(x + y) = H(x) + H(y) for every x, y ∈ G. A class F of real-valued functions defined on G is said to have the difference property if, for every function f: G → ℝ satisfying Δ _h f ∈ F for each h ∈ G, there is an additive function H such that f − H ∈ F. Erdős’ conjecture claiming that the class of continuous functions on ℝ has the difference property was proved by N. G. de Bruijn; later on, F. W. Carroll and F. S. Koehl obtained a similar result for compact Abelian groups and, under the additional assumption that the other one-sided difference function ∇ _h f defined by ∇ _h f(x) = f(xh) − f(x) (x ∈ G, h ∈ G) is measurable for any h ∈ G, also for noncommutative compact metric groups. In the present paper, we consider a narrower class of groups, namely, the family of semisimple compact connected Lie groups. It turns out that these groups admit a significantly stronger difference property. Namely, if a function f: G → ℝ on a semisimple compact connected Lie group has continuous difference functions Δ _h f for any h ∈ G (without the additional assumption concerning the measurability of the functions of the form ∇ _h f), then f is automatically continuous, and no nontrivial additive function of the form H is needed. Some applications are indicated, including difference theorems for homogeneous spaces of compact connected Lie groups.     

Completely integrable classical systems connected with semisimple lie algebras

The complete integrability of a class of dynamical systems, more general than those considered recently by Moser and Calogero, is proved. It is shown that these systems are connected with semisimple Lie algebras.Some of the results of this paper were announced in [1].     

The Gell-Mann formula for representations of semisimple groups

A method for constructing representations of non-compact semisimple groups from representations of semidirect product groups is presented. Necessary and sufficient algebraic conditions for the method to work are given, and these are applied to cases of possible interest for the classification of elementary particles.This work was supported by the Office of Naval Research, # NONR 3656 (09).     

Embedding of the Galilei algebra into complex semisimple lie algebras

The following result is proven: Complex simple Lie algebras L? have subalgebras ε(?) isomorphic to the complex Galilei algebra ? if and only if L? ∈ {A_r (r ? 5); B_r (r ? 3); C_r, D_r (r ? 4); F₄; E₆; E₇; E₈}.The set of subalgebras ε(?) in A₅ decomposes intoconjugacy classes labelled by a complex number; there is only one conjugacy class in B?₃ and in C₄.     

Spherically symmetric equations in gauge theories for an arbitrary semisimple compact lie group

An explicit construction of spherically symmetric equations (not only static and/or self-dual) in gauge theories for the minimal embedding of SU(2) in an arbitrary semisimple compact Lie group G is given. The final equations are written in a form containing only gauge invariant quantities in R₂. The whole group structure is concentrated in the only matrix, which is directly related to the Cartan matrix of G. In particular, the developed technique allows to generalize the Witten duality equation [1] and to obtain the spectrum of pointlike solutions in G.     

The Van Cittert-Zernike theorem for optical fibres

The Van Cittert-Zernike theorem determines the degree of coherence γ of light from a source radiating in a uniform medium in terms of the angular size α of the source. For the fields within an optical fibre far from the source, γ is found by replacing α by the critical angle for total internal reflection.     

Compact involutions of semisimple quantum groups

It is proved that a complex cosemisimple Hopf algebra has at most one compact involution modulo automorphisms.     

On the Integrability Conditions for Infeld-Van der Waerden Spin-Affine Connexions

The integrability condition that must be formally fulfilled by any Infeld-Van der Waerden spin-affine connexion is derived explicitly. Some geometric properties of the action of torsionless covariant-derivative commutators on arbitrary spin tensors and densities are then brought out. The relevant calculations supply a set of new differential expressions which will presumably ensure the consistency of any formulation that involves utilizing normally the traditional two-component spinor methods for classical general relativity.     

Bilocal lie groups and solitons

A group theoretical scheme where solition equations are associated with the integrability conditions for differential system is proposed. These conditions ensure the existence of a bilocal Lie group structure which naturally generates a set of conserved currents for arbitrary space-time dimensions.     

The Poincare-Dulac theorem for nonlinear representations of nilpotent lie algebras

The aim of this Letter is to show that the Poincare-Dulac theorem for holomorphic finite-dimensional representation, is valid for any nilpotent Lie algebrag. We reduce the classification problem of representations with a semisimple linear part satisfying the Poincaré condition to an algebraic problem. We develop a complete computation in a particular case.     

Cayley parametrization of semisimple lie groups and its application to physical laws in a (3+1)-dimensional cubic lattice

The assumption of a discrete space-time is expressed mathematically by restricting the space-time variables to the field of integer numbers, and by restricting to the field of rational numbers the functions describing the laws of motion. This rational character must be preserved under the transformations connecting different systems of reference. The Cayley parametrization of semisimple Lie groups, and in particular of the Lorentz group, satisfies this condition if we require these parameters to take only integer values. The rational points of the most frequently used transcendental functions are obtained with the help of the integer complex and hypercomplex numbers. Some applications are made concerning the laws of motion in special relativity defined over a (3+1)-dimensional cubic lattice.     

Factorization of the ground-state wavefunctions of quantum systems related to semisimple lie algebras

Conditions are considered under which the ground-state wavefunctions of quantum systems connected with a semisimple Lie algebra are factorizable and may be found explicitly.     

Null Infeld-van der Waerden Electromagnetic Fields from Geometric Sources

One of the formalisms of Infeld and van der Waerden for general relativity suggests defining geometric sources for electromagnetic fields that are inextricably involved in underlying spin curvatures. It is shown explicitly that such fields must bear nullity whenever a formal continuity equation is effectively implemented.     

Functorial geometric quantization and Van Hove's theorem

A classical theorem of Van Hove in conjunction with a formalism developed by Weinstein is used to prove that a quantization functor does not exist. In the proof a category of exact transverse Lagrangian submanifolds is introduced which provides a functorial link between Schrödinger quantization and the prequantization/polarization theory of Kostant and Souriau.     

Differential calculus on quantized simple lie groups

Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU _q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q are also discussed.     

Geometrization of magnetohydrodynamic equations via lie groups

The equations of magnetohydrodynamics of a perfect fluid are classified with respect to the Coriolis parameter, and all essentially different solutions of rank one are indicated. The geometry of streamlines is discussed.