Volume 4 (2004), Number 2. Abstracts

G. Carlet, B. Dubrovin, and Y. Zhang. The Extended Toda Hierarchy [PDF]

Using construction of logarithm of a difference operator, we present the Lax pair formalism for certain extension of the continuous version of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the CP¹ topological sigma model and the extended Toda hierarchy. We also establish an equivalence of the latter with certain extension of the nonlinear Schrödinger hierarchy.

Keywords. Toda lattice, Lax representation, bihamiltonian structure, tau function.

2000 Mathematics Subject Classification. Primary: 37K10; Secondary: 53D45.

Yu. Flicker and D. Zinoviev. Twisted character of a small representation of PGL(4) [PDF]

We compute by a purely local method the elliptic θ-twisted character χ_π of the representation π=I_(3,1)(1_3) of PGL(4,F). Here F is a p-adic field; θ is the ``transpose-inverse'' automorphism of G=PGL(4,F); π is the representation of PGL(4,F) normalizedly induced from the trivial representation of the maximal parabolic subgroup of type (3,1). Put C = {(g₁,g₂) ∈ GL(2) × GL(2): det(g₁) = det(g₂)}/G_m (G_m embeds diagonally). It is a θ-twisted elliptic endoscopic group of PGL(4). We deduce from the computation that χ_π is an unstable function: its value at one twisted regular elliptic conjugacy class with norm in C=C(F)$ is minus its value at the other class within the twisted stable conjugacy class, and 0 at the classes without norm in C. Moreover π is the unstable endoscopic lift of the trivial representation of C.

Naturally, this computation plays a role in the theory of lifting from C (= ``SO(4)'') and PGp(2) to G = PGL(4) using the trace formula, to be discussed elsewhere ([F']).

Our work develops a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced with Kazhdan in [FK] in a 3-dimensional case. It uses the classification of twisted stable and unstable regular conjugacy classes in PGL(4,F) of [F], motivated by Weissauer [W]. It extends the local method of computation introduced by us in [FZ]. An extension of our work here to apply to similar representations of GL(4,F) whose central character is not trivial has recently been given in [FZ'].

Keywords. Representations of p-adic groups, explicit character computations, twisted endoscopy, transpose-inverse twisting, instability.

2000 Mathematics Subject Classification. 10D40, 10D30, 12A67, 12A85, 14G10, 22E55, 11F27, 11R42, 11S40.

L. Friedlander. Remarks on the Membrane and Buckling Eigenvalues for Planar Domains [PDF]

I present a counter-example to the conjecture that the first eigenvalue of the clamped buckling problem in a planar domain is not smaller than the third eigenvalue of the fixed membrane in that domain. I also prove that the conjecture holds for domains that are invariant under rotation by angle π/2.

Keywords. Eigenvalue inequalities, small eigenvalues, buckling problem.

2000 Mathematics Subject Classification. 35P15, 47A55, 47F05.

A. Gorodentsev and S. Kuleshov. Helix Theory [PDF]

This is a detailed review of helix theory, which describes exceptional sheaves and exceptional bases for derived categories of coherent sheaves on Fano varieties. We explain systematically all basic ideas and constructions related to exceptional objects. Projective spaces and Del Pezzo surfaces are considered especially extensively. Some arithmetic relationships with the mirror symmetry phenomenon are discussed as well. This paper may be considered as a necessary supplement to the book [HuLe], which completely ignores rich structures beyond the zero-dimensional moduli spaces.

Keywords. Exceptional collections, mutations, semiorthogonal decompositions in triangulated categories.

2000 Mathematics Subject Classification. 14F05, 14J60, 18F30, 32L10.

V. Malyshev, S. Pirogov, and A. Rybko. Random Walks and Chemical Networks [PDF]

We consider continuous-time random walks with bounded jumps but unbounded rates (they depend polynomially on coordinates in the orthant). In applications, the case when the rates are bounded corresponds in applications to queueing theory, more precisely, to Markovian communication networks. The goal of this paper is to discuss the situation for polynomial rates; we show that the boundaries often play no role, but new effects and complicated behaviour may arise due to time scales and nonlinearity.

Keywords. Random walks, chemical kinetics, entropy.

2000 Mathematics Subject Classification. 90B10, 94C99, 37-XX.

P. Mardešić, R. Roussarie, and C. Rousseau. Modulus of Analytic Classification for Unfoldings of Generic Parabolic Diffeomorphisms [PDF]

In this paper we give a complete modulus of analytic classification under weak equivalence for generic analytic 1-parameter unfoldings of diffeomorphisms with a generic parabolic point. The modulus is composed of a canonical parameter associated to the family, together with an unfolding of the Ecalle–Voronin modulus. We then study the fixed points bifurcating from a parabolic point with nontrivial Ecalle–Voronin modulus and show that some of the non-hyperbolic resonant ones are non integrable.

In the Addendum it is shown that weak equivalence can be replaced by conjugacy.

Keywords. Analytic classification, parabolic germ of diffeomorphism, modulus, saddle-node vector field.

2000 Mathematics Subject Classification. 34C, 58F.

F. Pablos Romo. Theta Groups over Extensions of Abelian Varieties by Unipotent Groups [PDF]

Let 0 → K_U →ⁱ Y →^π X → 0 be a sequence of morphisms of algebraic groups over an algebraically closed field k, where X is an abelian variety, K_U is a unipotent, connected and commutative group scheme, and (X, π) is a geometric quotient of Y by K_U.

If L is an invertible sheaf over X, in this paper we generalize to \overline L = π^* L the notion of a theta group associated with an invertible sheaf given by D. Mumford for an abelian variety.

Keywords. Theta group, invertible sheaf, unipotent group, abelian variety.

2000 Mathematics Subject Classification. 14K05, 14K30, 14L15.

D. Ueltschi. Cluster Expansions and Correlation Functions [PDF]

A cluster expansion is proposed, that applies to both continuous and discrete systems. The assumption for its convergence involves an extension of the neat Kotecký Preiss criterion. Expressions and estimates for correlation functions are also presented. The results are applied to systems of interacting classical and quantum particles, and to a lattice polymer model.

Keywords. Cluster expansion, correlation functions.

2000 Mathematics Subject Classification. {82B05, 82B10.

I. Zverovich. Weighted Well-Covered Graphs and Complexity Questions [PDF]

A weighted graph G is called well-covered if all its maximal independent sets have the same weight. Let S be an independent set of G (possibly, S = ∅). The subgraph G - N[S] is called a co-stable subgraph of G. We denote by CSub(G) the set of all co-stable subgraphs of G considered up to isomorphism. A class of weighted graphs P is called co-hereditary if it is closed under taking co-stable subgraphs, i.e., G ∈ P implies CSub(G) ⊆ P.

Note that the class WWELL of all weighted well-covered graphs is co-hereditary. We characterize WWELL in terms of forbidden co-stable subgraphs.

Then we use a reduction from Satisfiability to show that the following decision problems are NP-complete.

Decision Problem 1 (Co-Stable Subgraph).

Instance: A graph G and a set U ⊆ V(G) that induces a subgraph H.

Question: Is H a co-stable subgraph of G?

Decision Problem 2 (Co-Stable Subgraph H).

Instance: A graph G.

Question: Is H a co-stable subgraph of G?

Let Δ(G) be the maximum vertex degree of a graph G. We show that recognizing weighted well-covered graphs with bounded Δ(G) can be done in polynomial time.

Keywords. Weighted well-covered graph, co-stable subgraph.

2000 Mathematics Subject Classification. 05C85.

Moscow Mathematical Journal
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Online ISSN 1609-4514
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