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.     

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.     

On estimates for the Fourier-Bessel integral transform in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(ℝ<Subscript>+</Subscript>)

Two estimates useful in applications are proved for the Fourier-Bessel integral transform in L ₂(?₊) as applied to some classes of functions characterized by a generalized modulus of continuity.     

On the measurability of the conic sections’ family in the projective space<Emphasis FontCategory="NonProportional">P</Emphasis><Subscript>3</Subscript><Subscript>3</Subscript>

In this paper we prove that the non degenerate conic sections’ family in the projective spaceP ₃ is measurable.     

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:

Density of the Medvedev lattice of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

The partial ordering of Medvedev reducibility restricted to the family of ⁰₁ classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a ⁰₁ class, which we call a ``c.e. separating class'. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes. Mathematics Subject Classification (2000): 03D30, 03D25     

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.     

Approximation of the classes <Emphasis Type="Italic">C</Emphasis><Stack><Subscript>β</Subscript><Superscript>ψ</Superscript></Stack><Emphasis Type="Italic">H</Emphasis><Subscript>ω</Subscript> by generalized Zygmund sums

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set H _ω in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.     

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.      

The James constant for the <Emphasis Type="Italic">l</Emphasis><Subscript>3</Subscript> − <Emphasis Type="Italic">l</Emphasis><Subscript>1</Subscript> space

In this note, the exact value of the James constant for the l₃-l₁ space is obtained, J(l₃-l₁)=1.5573…. This result improves the known inequality, J(l₃-l₁)≤4/³√10, which was given by Dhompongsa, Piraisangjun and Saejung.     

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.     

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.     

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.     

A smoothing Newton-type method for solving the <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> spectral estimation problem with lower and upper bounds

This paper discusses the L ₂ spectral estimation problem with lower and upper bounds. To the best of our knowledge, it is unknown if the existing methods for this problem have superlinear convergence property or not. In this paper we propose a nonsmooth equation reformulation for this problem. Then we present a smoothing Newton-type method for solving the resulting system of nonsmooth equations. Global and local superlinear convergence of the proposed method are proved under some mild conditions. Numerical tests show that this method is promising.     

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 .     

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.     

On ℤ<Subscript>2</Subscript>-graded identities of the super tensor product of <Emphasis Type="Italic">UT</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">F</Emphasis>) by the Grassmann algebra

Let F be a field of characteristic zero and E be the unitary Grassmann algebra generated over an infinite-dimensional F-vector space L. Denote by \(\mathcal{E} = \mathcal{E}^{(0)} \oplus \mathcal{E}^{(1)}\) an arbitrary ?₂-grading of E such that the subspace L is homogeneous. Given a superalgebra A = A ⁽⁰⁾ ⊕ A ⁽¹⁾, define the superalgebra \(A\hat \otimes \mathcal{E}\) by \(A\hat \otimes \mathcal{E} = (A^{(0)} \otimes \mathcal{E}^{(0)} ) \oplus (A^{(1)} \otimes \mathcal{E}^{(1)} )\). Note that when E is the canonical grading of E then \(A\hat \otimes \mathcal{E}\) is the Grassmann envelope of A. In this work we find bases of ?₂-graded identities and we describe the ?₂-graded codimension and cocharacter sequences for the superalgebras \(UT_2 (F)\hat \otimes \mathcal{E}\), when the algebra UT ₂(F) of 2 ×2 upper triangular matrices over F is endowed with its canonical grading.     

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     

On the <Emphasis Type="Italic">ω</Emphasis><Subscript>1</Subscript> limits of subsets of the real line

We point out that it is consistent with ZFC that 2 ^ω > ℵ₁ and every subset of ℝ is the ω ₁ limit of a sequence of G _δ sets in ℝ. We prove also that assuming cov ( ) > ℵ₁, not every set in ℝ is the ω ₁ limit of a sequence of measurable sets. This solves two problems of T. Natkaniec and J. Wesołowska.