Irreducible Wakimoto-like Modules for the Lie Superalgebra D(2,1;α)

By using the idea of Wakimoto's free field,we construct a class of representations for the Lie superalgebra D(2,1;α) on the tensor product of a polynomial algebra and an exterior algebra involving one parameter λ.Then we obtain the necessary and sufficient condition for the representations to be irreducible.In fact,the representation is irreducible if and only if the parameter λ satisfies(λ + m)(λ-(1+α)/αm) = 0 for any m∈Z_+.     

On blocks with <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)−<Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)=1

We use the method of local representation and original method of Brauer to study the block with K(B)−L(B)=1, and get some properties on the defect group and the structure of this kind of blocks. Then, we show that K(B) conjecture holds for this kind of blocks.     

Derivatives for smooth representations of <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℝ) and <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℂ)

The notion of derivatives for smooth representations of GL(n, ? _p ) was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the “adduced” representation. In this paper we define derivatives of all orders for smooth admissible Fréchet representations of moderate growth. The real case is more problematic than the p-adic case; for example, arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation.In the companion paper [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations.We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS13a].     

Deforming the lie superalgebra <Emphasis Type="Italic">D</Emphasis>(2, 1; α) inside the superconformal algebra <Emphasis Type="Italic">K</Emphasis>′(4)

We describe nontrivial deformations of the embedding of the Lie superalgebra D(2, 1; α) into the centrally extended derived contact superconformal algebra obtained in [16], and realize them as (4 × 4)-matrices over the Weyl algebra. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

Period of the adelic Ikeda lift for <Emphasis Type="Italic">U</Emphasis>(<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">m</Emphasis>)

We give a period formula for the adelic Ikeda lift of an elliptic modular form f for U(m, m) in terms of special values of the adjoint L-functions of f. This is an adelic version of Ikeda’s conjecture on the period of the classical Ikeda lift for U(m, m).     

On the equation Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)<Emphasis Type="Italic">u</Emphasis> = 0

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δu + q(x)u = 0, where either q(x) ≠ const or q(x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.     

On Analytic Campanato and <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">s</Emphasis>) Spaces

We investigate the relation between analytic Campanato spaces \(\mathcal {AL}_{p,s}\) and the spaces F(p, q, s), characterize the bounded and compact Riemann–Stieltjes operators from \(\mathcal {AL}_{p,s}\) to \(F(p,p-s-1,s)\). We also describe the corona theorem and the interpolating sequences for the class \(F(p,p-2,s)\), which is the Möbius invariant subspace of the analytic Besov type spaces \(B_p(s)\).     

On common values of <Emphasis Type="Italic">φ</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) and <Emphasis Type="Italic">σ</Emphasis>(<Emphasis Type="Italic">m</Emphasis>). I

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x/(log x)^1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sum-of-divisors function.     

Congruences for Ramanujan’s <Emphasis Type="Italic">ω</Emphasis>(<Emphasis Type="Italic">q</Emphasis>)

Recently, Bruinier and Ono investigated the arithmetic of the coefficients of Ramanujan’s mock theta function ω(q). In Ramanujan J. (submitted) they obtained congruences with respect to the modulus 512. Here we show that ω(q) modulo 5 is dictated by an elliptic curve.     

Characterization of <Emphasis Type="Italic">α</Emphasis>1 and <Emphasis Type="Italic">α</Emphasis>2-matrices

This paper deals with some properties of α1-matrices and α2-matrices which are subclasses of nonsingular H-matrices. In particular, new characterizations of these two subclasses are given, and then used for proving algebraic properties related to subdirect sums and Hadamard products.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">μ</Emphasis>) for vector-valued measure <Emphasis Type="Italic">μ</Emphasis>

Let μ be a measure from a σ-algebra of subsets of a set T into a sequentially complete Hausdorff topological vector space X. Assume that μ is convexly bounded, i.e., the convex hull of its range is bounded in X, and denote by L ¹(μ) the space of scalar valued functions on T which are integrable with respect to the vector measure μ. We study the inheritance of some properties from X to L ¹(μ). We show that the bounded multiplier property passes from X to L ¹(μ). Answering a 1972 problem of Erik Thomas, we show that for a rather large class of F-spaces X the non-containment of c ₀ passes from X to L ¹(μ).     

Dirichlet expression for <Emphasis Type="Italic">L</Emphasis>(1, <Emphasis Type="Italic">χ</Emphasis>) with general Dirichlet character

In the famous work of Dirichlet on class number formula, L(s, χ) at s = 1 has been expressed as a finite sum, where L(s, χ) is the Dirichlet L-series of a real Dirichlet character. We show that this expression with obvious modification is valid for the general primitive Dirichlet character χ.     

On Lie’s problem and differential invariants of ODEs <Emphasis Type="Italic">y</Emphasis>″ = <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">x</Emphasis>, <Emphasis Type="Italic">y</Emphasis>)

A point classification of ordinary differential equations of the form y″ = F(x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F(x, y) is calculated, and Lie’s problem on the point equivalence of such equations is solved.     

Relative <Emphasis Type="Italic">t</Emphasis>-designs in binary Hamming association scheme <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, 2)

A relative t-design in the binary Hamming association schemes H(n, 2) is equivalent to a weighted regular t-wise balanced design, i.e., certain combinatorial t-design which allows different sizes of blocks and a weight function on blocks. In this paper, we study relative t-designs in H(n, 2), putting emphasis on Fisher type inequalities and the existence of tight relative t-designs. We mostly consider relative t-designs on two shells. We prove that if the weight function is constant on each shell of a relative t-design on two shells then the subset in each shell must be a combinatorial \((t-1)\)-design. This is a generalization of the result of Kageyama who proved this under the stronger assumption that the weight function is constant on the whole block set. Using this, we define tight relative t-designs for odd t, and a strong restriction on the possible parameters of tight relative t-designs in H(n, 2). We obtain a new family of such tight relative t-designs, which were unnoticed before. We will give a list of feasible parameters of such relative 3-designs with \(n \le 100\), and then we discuss the existence and/or the non-existence of such tight relative 3-designs. We also discuss feasible parameters of tight relative 4-designs on two shells in H(n, 2) with \(n \le 50\). In this study we come up with the connection on the topics of classical design theory, such as symmetric 2-designs (in particular 2-\((4u-1,2u-1,u-1)\) Hadamard designs) and Driessen’s result on the non-existence of certain 3-designs. We believe Problems 1 and 2 presented in Sect. 5.2 open a new way to study relative t-designs in H(n, 2). We conclude our paper listing several open problems.     

On Lagrange interpolation for |<Emphasis Type="Italic">X</Emphasis>|<Superscript><Emphasis Type="Italic">α</Emphasis></Superscript> (0 < <Emphasis Type="Italic">α</Emphasis> < 1)

We study the optimal order of approximation for |x| ^α (0 < α < 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained. Supported by the NSFC, 10601065.     

A nondegeneracy criterion for the <Emphasis Type="Italic">sn</Emphasis>(<Emphasis Type="Italic">n</Emphasis> + 1)/2-parameter Lie group of transformations of the space ℝ<Superscript><Emphasis Type="Italic">sn</Emphasis></Superscript>

We prove a nondegeneracy criterion for the sn(n + 1)/2-parameter local Lie group of local transformations of ℝ ^sn .     

On the Spectrum of the Sizes of Maximal Partial Line Spreads in <Emphasis Type="Italic">PG</Emphasis>(2<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">n</Emphasis> ≥ 3

A lot of research has been done on the spectrum of the sizes of maximal partial spreads in PG(3,q) [P. Govaerts and L. Storme, Designs Codes and Cryptography, Vol. 28 (2003) pp. 51–63; O. Heden, Discrete Mathematics, Vol. 120 (1993) pp. 75–91; O. Heden, Discrete Mathematics, Vol. 142 (1995) pp. 97–106; O. Heden, Discrete Mathematics, Vol. 243 (2002) pp. 135–150]. In [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129], results on the spectrum of the sizes of maximal partial line spreads in PG(N,q), N 5, are given. In PG(2n,q), n 3, the largest possible size for a partial line spread is q^2n-1+q^2n-3+...+q³+1. The largest size for the maximal partial line spreads constructed in [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129] is (q²ⁿ⁺¹–q)/(q²–1)–q³+q²–2q+2. This shows that there is a non-empty interval of values of k for which it is still not known whether there exists a maximal partial line spread of size k in PG(2n,q). We now show that there indeed exists a maximal partial line spread of size k for every value of k in that interval when q 9.J. Eisfeld: Supported by the FWO Research Network WO.011.96NP. Sziklai: The research of this author was partially supported by OTKA D32817, F030737, F043772, FKFP 0063/2001 and Magyary Zoltan grants. The third author is grateful for the hospitality of Ghent University.     

Vector-valued interval functions and the Dedekind completion of <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>, <Emphasis Type="Italic">E</Emphasis>)

We consider interval valued functions with values in a Banach lattice E. Certain notions of continuity introduced earlier for real interval valued functions are generalised to the more general case considered here. As an application, we characterise the Dedekind completion of the space of continuous, E-valued functions on a paracompact \(T_{1}\)-space, extending a result of Anguelov.     

A theorem of the alternatives for the equation |<Emphasis Type="Italic">Ax</Emphasis>| − |<Emphasis Type="Italic">B</Emphasis>||<Emphasis Type="Italic">x</Emphasis>| = <Emphasis Type="Italic">b</Emphasis>

A theorem of the alternatives for the equation \({|Ax|-|B||x|=b\ (A,B\in{\mathbb{R}}^{n\times n},\, b\in{\mathbb{R}}^n)}\) is proved and several consequences are drawn. In particular, a class of matrices A, B is identified for which the equation has exactly 2 ⁿ solutions for each positive right-hand side b.     

Restriction of Representations of GL (<Emphasis Type="Italic">n</Emphasis> + 1, <Emphasis Type="Italic">ℂ</Emphasis>) to GL (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">ℂ</Emphasis>) and Action of the Lie Overalgebra

Consider a restriction of an irreducible finite dimensional holomorphic representation of \(\text {GL}(n + 1,\mathbb {C})\) to the subgroup \(\text {GL}(n,\mathbb {C})\). We write explicitly formulas for generators of the Lie algebra \(\mathfrak {g}\mathfrak {l}(n + 1)\) in the direct sum of representations of \(\text {GL}(n,\mathbb {C})\). Nontrivial generators act as differential-difference operators, the differential part has order n ??1, the difference part acts on the space of parameters (highest weights) of representations. We also formulate a conjecture about unitary principal series of \(\text {GL}(n,\mathbb {C})\).