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.     

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.     

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.      

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.     

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.     

The Algorithm Z and Ramanujan’s <Subscript>1</Subscript><Emphasis Type="Italic">ψ</Emphasis><Subscript>1</Subscript> summation

We use the Algorithm Z on partitions due to Zeilberger, in a variant form, to give a combinatorial proof of Ramanujan’s ₁ ψ ₁ summation formula.     

The Szlenk index and local ℓ<Subscript>1</Subscript>-indices

We introduce two new local ₁-indices of the same type as the Bourgain ₁-index; the ⁺₁-index and the ⁺₁-weakly null index. We show that the ⁺₁-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain ₁. The ⁺₁-weakly null index has the same form as the Bourgain ₁-index: if it is countable it must take values for some <₁. The different ₁-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its ₁-index, via the ⁺₁-weakly null index. We show that I(C(^{))=^1++1}.     

Patchworking Singularities <Emphasis Type="Italic">A</Emphasis><Subscript>μ</Subscript> and <Emphasis Type="Italic">D</Emphasis><Subscript>μ</Subscript> and Meanders of Their Smoothings

Let an algebraic curve f have a singular point of type A_μ or D_μ. Let be the curve obtained by smoothing the singular point of f. In this paper, local maximal meanders appearing under an M-smoothing in a neighborhood of the singular point are studied. A local maximal meander means that the number of real points of the intersection of with a coordinate axis in the neighborhood is maximal and the points belong to one of the components of . An M-smoothing means that the number of components of which appear in the neighborhood under the smoothing is also maximal. Bibliography: 9 titles. __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 193–217.     

The narrow class groups of the ℤ<Subscript>17</Subscript>- and ℤ<Subscript>19</Subscript>-extensions over the rational field

Let p be either 17 or 19, let ℤ _p denote the ring of p-adic integers, and let l be a prime number which is a primitive root modulo p ². We shall prove, with the help of a computer, that the l-class group of the ℤ _p -extension over the rational field is trivial. We shall also prove the triviality of the narrow 2-class group of the same ℤ _p -extension.     

Distance-Regular Graph with <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript> > 1 and <Emphasis Type="Italic">a</Emphasis><Subscript>1</Subscript> = 0 < <Emphasis Type="Italic">a</Emphasis><Subscript>2</Subscript>

Let Γ be a distance-regular graph of diameter d ≥ 3 with c ₂ > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:

Borel equivalence relations between <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript> and <Emphasis Type="Italic">ℓ</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we show that, for each p 〉 1, there are continuum many Borel equivalence relations between Rω/l1 and Rω/p ordered by ≤B which are pairwise Borel incomparable.     

<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.     

Embeddings into the Medvedev and Muchnik lattices of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

Let _w and _M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty ₁⁰ subsets of 2, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of _w. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of _M.Simpsons research was partially supported by NSF Grant DMS-0070718. We thank the anonymous referee for a careful reading of this paper and helpful comments.     

Fractional Modified Dyadic Integral and Derivative on ℝ<Subscript>+</Subscript>

For functions in the Lebesgue space L(ℝ₊), a modified strong dyadic integral J _α and a modified strong dyadic derivative D ^(α) of fractional order α > 0 are introduced. For a given function f ∈ L(ℝ₊), criteria for the existence of these integrals and derivatives are obtained. A countable set of eigenfunctions for the operators J _α and D ^(α) is indicated. The formulas D ^(α)(J ^α(f)) = f and J _α(D ^(α)(f)) = f are proved for each α > 0 under the condition that . We prove that the linear operator is unbounded, where is the natural domain of J _α. A similar statement for the operator is proved. A modified dyadic derivative d ^(α)(f)(x) and a modified dyadic integral j _α(f)(x) are also defined for a function f ∈ L(ℝ₊) and a given point x ∈ ℝ₊. The formulas d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) are shown to be valid at each dyadic Lebesgue point x ∈ ℝ₊ of f.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 64–70, 2005Original Russian Text Copyright © by B. I. GolubovSupported by the Russian Foundation for Basic Research (grant no. 05-01-00206).     

Topological recursion relation for <Emphasis Type="Italic">ψ</Emphasis><Subscript>1</Subscript><Emphasis Type="Italic">ψ</Emphasis><Subscript>2</Subscript> in

We give the definition of the full potential with descendants, which is a genus expansion of the Barannikov-Kontsevich solution of the WDVV equation. This potential satisfies the Getzler relation, which comes from the geometry of the moduli space of curves Open image in new window .     

The Tukey Order and Subsets of <Emphasis Type="Italic">ω</Emphasis><Subscript>1</Subscript>

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map ? : P → Q carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients are said to be Tukey equivalent. Let X be a space and denote by \(\mathcal {K}(X)\) the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of \(\mathcal {K}(S)\) corresponding to various subspaces S of ω ₁, their Tukey invariants, and hence the Tukey relations between them. It is shown that ω ^ω is a strict Tukey quotient of \({\Sigma }(\omega ^{\omega _{1}})\) and thus we distinguish between two Tukey classes out of Isbell’s ten partially ordered sets from (Isbell, J. R.: J. London Math Society 4(2), 394–416, 1972). The relationships between Tukey equivalence classes of \(\mathcal {K}(S)\), where S is a subspace of ω ₁, and \(\mathcal {K}(M)\), where M is a separable metrizable space, are revealed. Applications are given to function spaces.     

Cuts,Trees and ℓ<Subscript>1</Subscript>-Embeddings of Graphs*

Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into ₁ space. The main results are:1. Explicit constant-distortion embeddings of all series-parallel graphs, and all graphs with bounded Euler number. These are the first natural families known to have constant distortion (strictly greater than 1). Using the above embeddings, algorithms are obtained which approximate the sparsest cut in such graphs to within a constant factor.2. A constant-distortion embedding of outerplanar graphs into the restricted class of ₁-metrics known as dominating tree metrics. A lower bound of (log n) on the distortion for embeddings of series-parallel graphs into (distributions over) dominating tree metrics is also presented. This shows, surprisingly, that such metrics approximate distances very poorly even for families of graphs with low treewidth, and excludes the possibility of using them to explore the finer structure of ₁-embeddability.* A preliminary version of this work appeared in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, 1999, pp. 399–408. This work was done while the author was at the University of California, Berkeley. Supported in part by NSF grants CCR-9505448 and CCR-9820951.     

Weak-Polynomial Convergence on Spaces ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is concerned with the study of the set P ^-1(0), when P varies over all orthogonally additive polynomials on _p and L _p spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials.     

On Λ<Subscript>m</Subscript>-sets and related topological spaces

Summary We introduce and investigate three topological spaces <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(X,\Lambda_m)$, $(X,\Lambda_{mc}^*)$ and $(X,\Lambda_{g\Lambda_m})$ by using $\Lambda_m$-sets, $(\Lambda, m)$-closed sets and generalized $\Lambda_m$-sets, respectively. Especially, we study properties of weak separation axioms on these topological spaces. The investigation enables us to obtain a unified theory of notions related to $\Lambda$-sets [21], semi-$\Lambda$-sets [5] and pre-$\Lambda$-sets [15] in topological spaces.     

Spin structures and spectra of ℤ<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>-manifolds

We give necessary and sufficient conditions for the existence of pin^± and spin structures on Riemannian manifolds with holonomy group ₂^k. For any n4 (resp. n6) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M₁, M₂, non homeomorphic to each other, that are Laplace isospectral on functions and on p-forms for any p and such that M₁ admits a pin^± (resp. spin) structure whereas M₂ does not.Mathematics Subject Classification (2000):58J53, 57R15, 20H15Partially supported by Conicet and grants from SecytUNC, Foncyt and AgCba.