Volume 2 (2002), Number 4. Abstracts
S. Gindikin. An Analytic Separation of Series of Representations for SL(2;R) [PDF]
For the group SL(2;R), holomorphic wave fronts of the projections on different series of representations are contained in some disjoint cones. These cones are convex for holomorphic and antiholomorphic series, which corresponds to the well-known fact that these projections can be extended holomorphically to some Stein tubes in SL(2;C). For the continuous series, the cone is not convex, and the projections are boundary values of 1-dimensional \bar ∂-cohomology in a non-Stein tube.
Keywords. Integral geometry, horospherical transform, series of representations, \bar ∂-cohomology, holomorphic wave front.
2000 Mathematics Subject Classification. 22E46, 32A45, 44A12.
D. Grigoriev, E. Hirsch, and D. Pasechnik. Complexity of Semialgebraic Proofs [PDF]
It is a known approach to translate propositional formulas into systems of polynomial inequalities and consider proof systems for the latter. The well-studied proof systems of this type are the Cutting Plane proof system (CP) utilizing linear inequalities and the Lovász--Schrijver calculi (LS) utilizing quadratic inequalities. We introduce generalizations LSd of LS that operate on polynomial inequalities of degree at most d.
It turns out that the obtained proof systems are very strong. We construct polynomial-size bounded-degree LSd proofs of the clique-coloring tautologies (which have no polynomial-size CP proofs), the symmetric knapsack problem (which has no bounded-degree Positivstellensatz calculus proofs), and Tseitin's tautologies (which are hard for many known proof systems). Extending our systems with a division rule yields a polynomial simulation of CP with polynomially bounded coefficients, while other extra rules further reduce the proof degrees for those examples.
Finally, we prove lower bounds on the Lovász--Schrijver ranks and on the size and the ``Boolean degree'' of Positivstellensatz calculus refutations. We use the latter bound to obtain an exponential lower bound on the size of Positivstellensatz calculus, static LSd, and tree-like LSd refutations.
Keywords. Computational complexity, propositional proof system.
2000 Mathematics Subject Classification. Primary: 03F20; Secondary: 68Q17.
B. Gross. Unramified Reciprocal Polynomials and Coxeter Decompositions [PDF]
We classify certain automorphisms of an even unimodular lattice L with fixed irreducible and unramified characteristic polynomial. The question which automorphisms are conjugate to their inverses in the orthogonal group of L is investigated.
Keywords. Reciprocal polynomial, unimodular lattice, Coxeter decomposition, Salem numbe.
2000 Mathematics Subject Classification. 11H56, 11R06, 20F55.
N. Koblitz. Good and Bad Uses of Elliptic Curves in Cryptography [PDF]
In the first part of this article I describe the construction of cryptosystems using elliptic curves, discuss the Elliptic Curve Discrete Logarithm Problem (upon which the security of all elliptic curve cryptosystems rests), and survey the different types of elliptic curves that can be chosen for cryptographic applications. In the second part I describe three unsuccessful approaches to breaking various cryptosystems by means of liftings to global elliptic curves. I explain how the failure of these attacks is caused by fundamental properties of the global curves.
Keywords. Public key cryptography, elliptic curve cryptography, discrete logarithm, digital signature, global elliptic curve, index calculus, canonical height, torsion group, uniform boundedness.
2000 Mathematics Subject Classification. Primary: 14G50, 11G05, 94A60; Secondary: 11T71, 14H52.
I. Krichever. Isomonodromy Equations on Algebraic Curves, Canonical Transformations and Whitham Equations [PDF]
We construct the Hamiltonian theory of isomonodromy equations for meromorphic connections with irregular singularities on algebraic curves. We obtain an explicit formula for the symplectic structure on the space of monodromy and Stokes matrices. From these we derive Whitham equations for the isomonodromy equations. It is shown that they provide a flat connection on the space of spectral curves of Hitchin systems.
Keywords. Algebaric curves, meromorphic connections, monodromy matrices, Hamiltonian theory, symplectic form.
2000 Mathematics Subject Classification. 14, 79.
I. Penkov and V. Serganova. Generalized Harish-Chandra Modules [PDF]
Let g be a complex reductive Lie algebra and h be a Cartan subalgebra of g. If k is a subalgebra of g, we call a g-module M a strict (g,k)-module if k coincides with the subalgebra of all elements of g which act locally finitely on M. For an intermediate k, i.e., such that h ⊂ k ⊂ g, we construct irreducible strict (g,k)-modules. The method of construction is based on the D-module localization theorem of Beilinson and Bernstein. The existence of irreducible strict (g,k)-modules has been known previously only for very special subalgebras k, for instance when k is the (reductive) subalgebra of fixed points of an involution of g. In this latter case strict irreducible (g,k)-modules are Harish-Chandra modules.
We also give separate necessary and sufficient conditions on k for the existence of an irreducible strict (g,k)-module of finite type, i.e., an irreducible strict (g,k)-module with finite k-multiplicities. In particular, under the assumptions that the intermediate subalgebra k is reductive and g has no simple components of types Bn for n > 2 or F4, we prove a simple explicit criterion on k for the existence of an irreducible strict (g,k)-module of finite type. It implies that, if g is simple of type A or C, for every reductive intermediate k there is an irreducible strict (g,k)-module of finite type.
Keywords. Complex reducive Lie algebra, (g,k)-module, Harish-Chandra module.
2000 Mathematics Subject Classification. Primary 17B10; Secondary 22E46.
M. Wodzicki. Generalized Harish-Chandra Modules [PDF]
Trace functionals on ideals in the algebra B(H) of bounded operators on a separable Hilbert space H are constructed and studied.
Keywords. Operator ideals, exotic traces, renormalization.
2000 Mathematics Subject Classification. Primary: 47L20; Secondary: 46L80, 81R60.
Moscow Mathematical Journal