Trace operators and theta series for Γ<Subscript>0,n</Subscript>[q] and Γ<Subscript>n</Subscript>[q]

We consider the action of suitable trace operators on non homogeneous theta series that are Siegel modular forms for the principal congruence subgroups of the symplectic group of level q, _n[q]. Then, we prove that modular forms for the Hecke subgroup of level q, _0,n[q], that are linear combination of such theta series, can also be expressed as combination of (homogeneous) theta series that are modular forms with respect to _0,n[q].     

Inverting the Satake map for <Emphasis Type="Italic">Sp</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> and applications to Hecke operators

By compatibly grading the p-part of the Hecke algebra associated to Sp _n (ℤ) and the subring of ℚ[x ₀^±1,…,x _n ^±1] invariant under the associated Weyl group, we produce a matrix representation of the Satake isomorphism restricted to the corresponding finite dimensional components. In particular, using the elementary divisor theory of integral matrices, we show how to determine the entries of this matrix representation restricted to double cosets of a fixed similitude. The matrix representation is upper-triangular, and can be explicitly inverted. To address the specific question of characterizing families of Hecke operators whose generating series have “Euler” products, we define (n+1) families of polynomial Hecke operators t _k ⁿ (p ^ℓ ) (in ℚ[x ₀^±1,…,x _n ^±1]) for Sp _n whose generating series ∑t _k ⁿ (p ^ℓ )v ^ℓ are rational functions of the form q _k (v)⁻¹, where q _k is a polynomial in ℚ[x ₀^±1,…,x _n ^±1][v] of degree (2 ⁿ if k=0). For k=0 and k=1 the form of the polynomial is essentially that of the local factors in the spinor and standard zeta functions. For k>1, these appear to be new expressions. Taking advantage of the generating series and our ability to explicitly invert the Satake isomorphism, we explicitly compute the classical operators with the analogous properties in the case of genus 2. It is of interest to note that these operators lie in the full, but not generally the integral, Hecke algebra.      

Moduli spaces ℳ<Subscript>2,1</Subscript> and ℳ<Subscript>3,1</Subscript>

The cell structure of the spaces ℳ_2,1 and ℳ_3,1 is considered. These are the spaces of complex curves of genus 2 and 3 with one marked point. For the space ℳ_2,1, nine cells of the highest dimension 8 are described and their adjacency is studied. For the space ℳ_3,1, a list of all 1726 cells of the highest dimension 14 (with orientation) is obtained. The list of adjacent couples of cells is also obtained. These lists can be found on the web.     

Entropy and escape of mass for SL<Subscript>3</Subscript>(ℤ) SL<Subscript>3</Subscript>(ℝ)

We study the relation between measure theoretic entropy and escape of mass for the case of a singular diagonal flow on the moduli space of three-dimensional unimodular lattices.     

Approximation Representations for Δ<Subscript>2</Subscript> Reals

We study ₂ reals x in terms of how they can be approximated symmetrically by a computable sequence of rationals. We deal with a natural notion of approximation representation and study how these are related computationally for a fixed x. This is a continuation of earlier work; it aims at a classification of ₂ reals based on approximation and it turns out to be quite different than the existing ones (based on information content etc.)Mathematics Subject Classification (2000): 03F60, 03D30     

Eisenstein series associated with Γ<Subscript>0</Subscript>(2)

In this paper, we define the normalized Eisenstein series ℘, e, and associated with Γ₀(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ₀(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.      

Local structure of Rogers semilattices of Σ<Subscript>n</Subscript><Superscript>0</Superscript>-computable numberings

We deal in specific features of the algebraic structure of Rogers semilattices of _n ⁰-computable numberings, for n 2. It is proved that any Lachlan semilattice is embeddable (as an ideal) in such every semilattice, and that over an arbitrary non 0-principal element of such a lattice, any Lachlan semilattice is embeddable (as an interval) in it.Supported by INTAS grant No. 00-499, by FP Universities of Russia grant UR.04.01.013, and by the Grant Center for Fundamental Research (GCFR), project PD02-1.1-475.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 148–172, March–April, 2005.     

<Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>-Singular and <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript>-singular operators between vector-valued Banach lattices

Given an operator T : X → Y between Banach spaces, and a Banach lattice E consisting of measurable functions, we consider the point-wise extension of the operator to the vector-valued Banach lattices T _E : E(X) → E(Y) given by T _E (f)(ω) = T(f(ω)). It is proved that for any Banach lattice E which does not contain c ₀, the operator T is an isomorphism on a subspace isomorphic to c ₀ if and only if so is T _E . An analogous result for invertible operators on subspaces isomorphic to ℓ ₁ is also given.     

Multi-self-similar Markov processes on ℝ<Subscript>+</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> and their Lamperti representations

 A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably time-changing the exponential of the Lévy process. In this paper we generalise Lamperti's result to processes in n dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher dimensions, referred to as multi-self-similarity in the paper. The special case where the Lévy process ξ is standard Brownian motion in n dimensions is studied in detail. There are also specific comments on the case where ξ is an n-dimensional compound Poisson process with drift. Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond to n-dimensional subordinators. Received: 22 August 2002 / Revised version: 10 February 2003 Published online: 15 April 2003 RID="*" ID="*" MaPhySto – Centre for Mathematical Physics and Stochastics, funded by a grant from the Danish National Research Foundation Mathematics Subject Classification (2000): 60G18, 60G51, 60J25, 60J60, 60J75 Key words or phrases: Lévy process – Self-similarity – Time-change – Exponential functional – Brownian motion – Bessel process – Piecewise deterministic Markov process – Moment sequence     

A Ring Theoretic Construction of Hadamard Difference Sets in ℤ<Subscript>8</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℤ<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let be the Galois ring of characteristic 2³ and rank n and let . We give an explicit construction of Hadamard difference sets in .}Research supported by NSA grant MDA 904-02-1-0080.     

Combined ℓ<Subscript>2</Subscript> data and gradient fitting in conjunction with ℓ<Subscript>1</Subscript> regularization

We are interested in minimizing functionals with ℓ₂ data and gradient fitting term and ℓ₁ regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ₁ norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ₂ gradient fitting term can be used to avoid both edge blurring and staircasing effects.      

Containment of c<Subscript>0</Subscript> and ℓ<Subscript>1</Subscript> in π<Subscript>1</Subscript>(<Emphasis Type="Italic">E,F</Emphasis>)

Suppose π₁(E, F) is the space of all absolutely 1-summing operators between two Banach spacesE andF. We show that ifF has a copy of c₀, then π₁ (E, F) will have a copy of c₀, and under some conditions ifE has a copy of ℓ₁ then π₁ (E, F) would have a complemented copy of ℓ₁.     

Equitable total coloring of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> ⋁ <Emphasis Type="Italic">W</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χ _t (G). If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χ _et (G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order. Supported by the National Natural Science Foundation of China (No. 10771091).     

New multiple <Subscript>6</Subscript><Emphasis Type="Italic">ψ</Emphasis><Subscript>6</Subscript> summation formulas and related conjectures

Three new summation formulas for ₆ ψ ₆ bilateral basic hypergeometric series attached to root systems are presented. Remarkably, two of these formulae, labelled by the A_2n−1 and A_2n root systems, can be reduced to multiple ₆ φ ₅ sums generalising the well-known van Diejen sum. This latter sum serves as the weight-function normalisation for the BC _n q-Racah polynomials of van Diejen and Stokman. Two ₈ φ ₇-level extensions of the multiple ₆ φ ₅ sums, as well as their elliptic analogues, are conjectured. This opens up the prospect of constructing novel A-type extensions of the Koornwinder–Macdonald theory.     

On ℑ<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-normal subgroups of finite groups

Given a class ℑ of finite groups, a subgroup H of a group G is called ℑ _n -normal in G if there exists a normal subgroup T of G such that HT is a normal subgroup of G and (H ∩ T)H _G /H _G is contained in the ℑ-hypercenter Z _∞^ℑ (G/H _G ) of G/H _G . We obtain some results about the ℑ _n -normal subgroups and use them to study the structure of some groups.     

Stabilization of Linear Switched Delay Systems: ℋ<Subscript>2</Subscript> and ℋ<Subscript>∞</Subscript> Methods

This paper presents new results pertaining to the delay-dependent stability and control synthesis of a class of linear switched continuous-time systems with time-varying delays. A new state transformation is introduced to exhibit the delay-dependent dynamics in the slow-time scale. For stability, we construct an appropriate selective Lyapunov functional to derive delay-dependent LMI-based sufficient conditions under arbitrary switching and without relying to overbounding. For the control synthesis, we design switched feedback schemes based on quadratic ℋ₂, ℋ_∞ and simultaneous ℋ₂/ℋ_∞ performance criteria. Under the developed transformation, it is established that both the instantaneous and delayed feedback control yield identical results. Numerical examples are presented to illustrate the analytical development.     

BMO Solvability and the <Emphasis Type=Italic>A</Emphasis><Subscript>∞</Subscript> Condition for Elliptic Operators

We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form operator with bounded measurable coefficients with the solvability of an end-point BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO Dirichlet problem implies L ^p solvability for all p>p ₀.     

Parabolic twists for the linear algebras <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>−1</Subscript>

New solutions of twist equations for the universal enveloping algebras U (A_n−1) are found. These solutions can be represented as products of full chains of extended Jordanian twists Abelian factors (“rotations”) , and sets of quasi-Jordanian twists . The latter are generalizations of Jordanian twists (with carrier b²) for special deformed extensions of the Hopf algebra U (b²). The carrier subalgebra for the composition is a nonminimal parabolic subalgebra in A _n−1 such that . The parabolic twisting elements are obtained in an explicit form. Details of the construction are illustrated by considering the examples n = 4 and n = 11. Bibliography: 21 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 187–213.