On the quantifier complexity of Δ<Subscript> n+1</Subscript> (T)– induction

In this paper we continue the study of the theories I _n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that I _n+1 (T) is _n+2 –axiomatizable. In particular, I _n+1 (I _n+1 ) gives an axiomatization of Th _n+2 (I _n+1 ) and is not finitely axiomatizable. This fact relates the fragment I _n+1 (I _n+1 ) to induction rule for _n+1 –formulas. Our arguments, involving a construction due to R. Kaye (see [9]), provide proofs of Parsons conservativeness theorem (see [16]) and (a weak version) of a result of L.D. Beklemishev on unnested applications of induction rules for _n+2 and _n+1 formulas (see [2]).Research partially supported by grant PB96–1345 (Spanish Goverment)Mathematics Subject Classification (2000): 03F30, 03H15     

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.     

Consistency method for measurements of the support function of a convex body in the metric of L<Subscript>∞</Subscript>

A new algorithm is proposed for estimation of convex body support function measurements in L _∞ metric, which allows us to obtain the solution in quadratic time (with respect to the number of measurements) not using linear programming. The rate of convergence is proved to be stable for quite weak conditions on input data. This fact makes the algorithm robust for a wider class of problems than it was previously. The implemented algorithm is stable and predictable unlike other existing support function estimation algorithms. Implementation details and testing results are presented.     

On the iwasawa algebra associated to a normal element of ℂ<Subscript><Emphasis Type=Italic>p</Emphasis></Subscript>

Given a prime number p and the Galois orbit O(x) of a normal element x of ℂ _p , the topological completion of the algebraic closure of the field of p-adic numbers, we study the Iwasawa algebra of O(x) with scalars drawn from ℚ _p and relate it with ℚ _p -distributions and functionals.     

New Lower Bounds for the Number of (≤ k)-Edges and the Rectilinear Crossing Number of K<Subscript>n</Subscript>

We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for the number of (≤ k)-edges is at least

Cofinality of normal ideals on<Emphasis Type="Italic">P</Emphasis><Subscript>κ</Subscript>(λ) II

For an idealJ on an infinite setX with add(J)=κ, let be the smallest size of any subfamilyY ofJ with the property that any member ofJ can be covered by less than κ members ofY. We study the value of forA in , where denotes the smallest [δ]^<θ ideal onP _κ(λ). We also discuss the problem of whether there exists a setA such that , or even . Some of the material in this paper originally appeared as part of the author's doctoral dissertation completed at the Université de Caen, 1998. Partially supported by the Israel Science Foundation. Publication 813.     

On the power of standard information for <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> approximation in the randomized setting

We study approximation of multivariate functions from a general separable reproducing kernel Hilbert space in the randomized setting with the error measured in the L _∞ norm. We consider algorithms that use standard information consisting of function values or general linear information consisting of arbitrary linear functionals. The power of standard or linear information is defined as, roughly speaking, the optimal rate of convergence of algorithms using n function values or linear functionals. We prove under certain assumptions that the power of standard information in the randomized setting is at least equal to the power of linear information in the worst case setting, and that the powers of linear and standard information in the randomized setting differ at most by 1/2. These assumptions are satisfied for spaces with weighted Korobov and Wiener reproducing kernels. For the Wiener case, the parameters in these assumptions are prohibitively large, and therefore we also present less restrictive assumptions and obtain other bounds on the power of standard information. Finally, we study tractability, which means that we want to guarantee that the errors depend at most polynomially on the number of variables and tend to zero polynomially in n ⁻¹ when n function values are used.     

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.     

On linear isometries of Banach lattices in <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>(Ω)-spaces

Consider the space C0(Ω) endowed with a Banach lattice-norm ‖ · ‖ that is not assumed to be the usual spectral norm ‖ · ‖_∞ of the supremum over Ω. A recent extension of the classical Banach-Stone theorem establishes that each surjective linear isometry U of the Banach lattice (C ₀(Ω), ‖ · ‖) induces a partition Π of Ω into a family of finite subsets S ⊂ Ω along with a bijection T: Π → Π which preserves cardinality, and a family [u(S): S ∈ Π] of surjective linear maps u(S): C(T(S)) → C(S) of the finite-dimensional C*-algebras C(S) such that

Asymptotic results for partitions (II) and a conjecture of Bateman and Erdös

Asymptotic results are obtained for p_A^(k)(n), the kth difference of the function p_A(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that

$\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k+1)}}\text{(n)}\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k)}}\text{(n)}\text{= O(n}{}^{\mathrm{?1/2}}\text{) (n→ ∫)}$

thereby verifying for these A a conjecture of Bateman and Erdös.     

The growth of solutions of <Emphasis Type="Italic">f</Emphasis>″ + e<Superscript>-z</Superscript><Emphasis Type="Italic">f</Emphasis>′+ Q(z)<Emphasis Type="Italic">f</Emphasis> = 0 where the order (Q) = 1

This paper investigates the growth of solutions of the equation f″ + e^-z f′ + Q(z)f = 0 where the order (Q) = 1. When Q(z) = h(z)e^bz, h(z) is nonzero polynomial, b ≠ - 1 is a complex constant, every solution of the above equation has infinite order and the hyper-order 1. We improve the results of M. Frei, M. Ozawa, G. Gundersen and J. K. Langley.     

On exact estimates of the convergence rate of fourier series for functions of one variable in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript>[–π, π]

The work is devoted to exact estimates of the convergence rate of Fourier series in the trigonometric system in the space of square summable 2π-periodic functions with the Euclidean norm on certain classes of functions characterized by the generalized modulus of continuity. Some N-widths of these classes are calculated, and the residual term of one quadrature formula over equally spaced nodes for a definite integral connected with the issues under consideration is found.     

On the basis property of eigenelements of ordinary linear differential operators in a space of the type <Emphasis Type=Italic>L</Emphasis><Subscript><Emphasis Type=Italic>p</Emphasis></Subscript>, <Emphasis Type=Italic>p</Emphasis> ≥ 2

For a linear differential expression with matrix coefficients in the class L _p , p ≥ 2, and with a parameter λ, we consider a boundary value problem with boundary conditions at the endpoints of the interval [a, b]. Under the condition that the problem is regular, we obtain a formula for the Fourier series expansion of an arbitrary vector function of the class L _p in the root functions of the problem.     

An Improved Bound for the Crossing Number of C<Subscript><Emphasis Type="Italic">m</Emphasis></Subscript>C<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>: a Self–Contained Proof Using Mostly Combinatorial Arguments

We present some surprisingly elementary arguments to prove that for every >0, if m is sufficiently large, then the crossing number of the Cartesian product C_m×C_n is at least (0.8–)mn, for every nm. The self–contained proof we give involves only one (rather elementary) geometrical result. The rest of the proof involves purely combinatorial arguments.Supported by CONA-CYT (Grant J32168E) and by FAI-UASLPFinal version received: July 14, 2003     

Discrete series of representations for the modular double of the quantum group <Emphasis Type="Italic">U</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(sl(2, ℝ))

The modular double of the quantum group U _q (sl(2)) with deformation parameter q = e ^iπτ is a natural object explicitly taking into account the duality τ ? 1/τ. The use of the modular double in conformal field theory allows one to consider the region 1 < c < 25 for the central charge of the Virasoro algebra when |τ| = 1. In this paper, a new discrete series of representations for the modular double of U _q (sl(2, ?)) is found for such τ.     

A characterization of the algebra <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">β</Emphasis></Subscript>(Ω)

We study some properties of algebras of continuous functions on a locally compact space, these algebras being equipped with the topology defined by a family of multiplication operators (β-uniform algebras). We prove an analog of a theorem due to Sheinberg for β-uniform algebras [see Uspekhi Mat. Nauk, 32:5 (197) (1977), 203–204].     

On the solvability of a singular boundary-value problem for the equation <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">t,x, x</Emphasis>′, <Emphasis Type="Italic">x</Emphasis>″) = 0

In this work, we consider boundary-value problems of the form

<Emphasis Type="Italic">OD</Emphasis>-Characterization of the automorphism groups of <Emphasis Type="Italic">O</Emphasis><Stack><Subscript>10</Subscript><Superscript>±</Superscript></Stack>(2)

The prime graph of a finite group was introduced by Gruenberg and Kegel. The degree pattern of a finite group G associated to its prime graph was introduced in [1] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions (1) |G| = |H| and (2) D(G) = D(H). Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Till now a lot of finite simple groups were shown to be OD-characterizable, and also some finite groups especially the automorphism groups of some finite simple groups were shown not being OD-characterizable but k-fold OD-characterizable for some k > 1. In the present paper, the authors continue this topic and show that the automorphism groups of orthogonal groups O ₁₀⁺(2) and O ₁₀⁻(2) are OD-characterizable.