Congruences Concerning Values of the Modular Functions <Emphasis Type="Italic">j</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The j-function j(z) = q⁻¹+ 744 + 196884q + ⋅s plays an important role in many problems. In [7], Zagier, presented an interesting series of functions obtained from the j-function: j_m(ζ) = (j(ζ) – 744)∨T₀(m), where T₀(m) is the usual m′th normalized weight 0 Hecke operator. In [3], Bruinier et al. show how this series of functions can be used to describe all meromorphic modular forms on SL₂(ℤ). In this note we use these functions and basic notions about modular forms to determine previously unidentified congruence relations between the coefficients of Eisenstein series and the j-function. 2000 Mathematics Subject Classification: Primary–11B50, 11F03, 11F30 The author thanks the National Science Foundation for their generous support.     

Zinbiel algebras under <Emphasis Type="Italic">q</Emphasis>-commutators

An algebra with the identity t ₁(t ₂ t ₃) = (t ₁ t ₂+t ₂ t ₁)t ₃ is called Zinbiel. For example, ℂ[x] under the multiplication is Zinbiel. Let a ○ _q b = a ○ b + q b ○ a be a q-commutator, where q ∈ ℂ. We prove that for any Zinbiel algebra A the corresponding algebra under the commutator A ⁽⁻¹⁾ = (A, ○₋₁) satisfies the identities t ₁ t ₂ = −t ₂ t ₁ and (t ₁ t ₂)(t ₃ t ₄) + (t ₁ t ₄)(t ₃ t ₂) = jac(t ₁, t ₂, t ₃)t ₄ + jac(t ₁, t ₄, t ₃)t ₂, where jac(t ₁, t ₂, t ₃) = (t ₁ t ₂)t ₃ + (t ₂ t ₃)t ₁ + (t ₃ t ₁)t ₂. We find basic identities for q-Zinbiel algebras and prove that they form varieties equivalent to the variety of Zinbiel algebras if q ² ≠ 1. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 57–78, 2005.     

Extrapolation results for <Emphasis Type="Italic">q</Emphasis> < 1

Given a sublinear operator T such that is bounded, it can be shown that is bounded, with constant C/(1−q), for every 0 < q < 1. In this paper, we study the converse result, not only for sequence spaces, but for general measure spaces proving that, if T : L ^q (μ) → X is bounded, with constant C/(1−q), for every and X is Banach, then T : L log (1/L)(μ) → X is bounded. Moreover, this result is optimal. We also show that things are quite different if the Banach condition on X is dropped. This work has been partially supported by MTM2004-02299 and by 2005SGR00556.     

Tight sets,weighted <Emphasis Type="Italic">m</Emphasis>-covers,weighted <Emphasis Type="Italic">m</Emphasis>-ovoids,and minihypers

Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342–349, 2008; Govaerts and Storme 4:279–286, 2004; Govaerts et al. 28:659–672, 2002). In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q⁻(5,3), to 56-caps contained in this elliptic quadric Q⁻(5,3).      

Degenerate principal series representations of Sp(<Emphasis Type="Italic">p,q</Emphasis>)

Letp>q and letG=Sp(p, q). LetP=LN be the maximal parabolic subgroup ofG with Levi subgroupL≅GL _q (ℍ)×Sp(p−q). Forsεℂ andμ a highest weight of Sp(p−q), let п_s,μ be the representation ofP such that its restriction toN is trivial and ⊠T _p-q ^μ , where det _q is the determinant character of GL _q (ℍ) andT _p-q ^μ is the irreducible representation of Sp(p−q) with highest weightμ. LetI _p,q(s, μ) be the Harish-Chandra module of the induced representation Ind _P ^G . In this paper, we shall determine the module structure and unitarity ofI _{p, q}(s, μ). Partially supported by NUS grant R-146-000-026-112.     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where

Multiplicities of simple modules in the Sp(4, <Emphasis Type="Italic">q</Emphasis>)-permutation module on <Emphasis Type="Italic">P</Emphasis>(3, <Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">q</Emphasis> even

In this paper we shall determine the multiplicities of simple modules in characteristic 2 in the Sp(4, q)-permutation module on projective 3-space P(3, q), q = 2 ⁿ .     

Charakterisierung der <Emphasis Type="Italic">q</Emphasis>-Orthogonalpolynome in <Emphasis Type="Italic">x</Emphasis>

Characterization of q-Orthogonal Polynomials. Im Anschluß an die Arbeit Orthogonalpolynome in x und q^–x als Lösungen von reellen q-Operatorgleichungen zweiter Ordnung (Monatsh. Math. 132, 123–140 (2001); im folgenden als [4] zitiert) werden alle Möglichkeiten für q-Orthogonalpolynome in x als Lösungen von q-Operatorgleichungen zweiter Ordnung angegeben (Orthogonalität im positiv definiten Sinne). Dabei erfolgt die Numerierung der Abschnitte und die Angabe der Formel-nummern unter Einbeziehung von [4].     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

(1, 1)-<Emphasis Type="Italic">q</Emphasis>-coherent pairs

In this paper, we introduce the concept of (1, 1)-q-coherent pair of linear functionals (U,V)(\mathcal{U},\mathcal{V}) as the q-analogue to the generalized coherent pair studied by Delgado and Marcellán in (Methods Appl Anal 11(2):273–266, 2004). This means that their corresponding sequences of monic orthogonal polynomials {P _n (x)} _n ≥ 0 and {R _n (x)} _n ≥ 0 satisfy

Codeterminants and <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

We study codeterminants in the q-Schur algebra S _q (n,r) and prove that the standard ones form a basis of S _q (n,r), using a quantized version of the Désarménien matrix. We find elements of the form F _S 1_λ E _T in Lusztig’s modified enveloping algebra of gl(n), which, up to powers of q, map to the basis of standard codeterminants, where F _S ∈U ^– and E _T ∈U ⁺ are explicitly given products of root vectors, depending on Young tableaux S and T.     

Symplectic semifield spreads of <Emphasis Type="Italic">PG</Emphasis>(5, <Emphasis Type="Italic">q</Emphasis>) and the veronese surface

In this paper we show that starting from a symplectic semifield spread S{\mathcal{S}} of PG(5, q), q odd, another symplectic semifield spread of PG(5, q) can be obtained, called the symplectic dual of S{\mathcal{S}}, and we prove that the symplectic dual of a Desarguesian spread of PG(5, q) is the symplectic semifield spread arising from a generalized twisted field. Also, we construct a new symplectic semifield spread of PG(5, q) (q = s ², s odd), we describe the associated commutative semifield and deal with the isotopy issue for this example. Finally, we determine the nuclei of the commutative pre-semifields constructed by Zha et al. (Finite Fields Appl 15(2):125–133, 2009).     

On the parity of the class number of an imaginary abelian field of conductor 2<Superscript><Emphasis Type="Italic">a</Emphasis></Superscript><Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">b</Emphasis></Superscript>

Let p be an odd prime number, and p^n₀{p^{n_0}} the highest power of p dividing 2 ^p−1 − 1. Let K_n=Q(z_pⁿ⁺¹){K_n={\bf Q}(\zeta_{p^{n+1}})} and L_n,j=K_n⁺(z_2^j+2){L_{n,j}=K_n^+(\zeta_{2^{j+2}})} for j ≥ 0. Let h_n^*{h_n^*} be the relative class number of K _n , and h _n,j the class number of L _n,j , respectively. Let n be an integer with n ≥ n ₀. We prove that if the ratio h_n^*/h_n-1^*{h_n^*/h_{n-1}^*} is odd, then h _n,j /h _n−1,j is odd for any j ≥ 0.     

Uniform bound for asymptotic normality of discounted <Emphasis Type="Italic">m</Emphasis>-dependent random variables using Heinrich's method

We estimate the difference | F_Zv(x) - F(x) | \left| {{F_{{Z_v}}}(x) - \Phi (x)} \right| , where F_Zv(x) {F_{{Z_v}}}(x) is the distribution function of normalized series Z _v = B ⁻¹ _v Σ^∞ _j=0 v ^j X _j with B ² _v = \mathbb E \mathbb {E} (Σ^∞ _j=0 v ^j X _j ) > 0 and the discount factor v, 0 < v < 1; X ₀,X ₁,X ₂,… is a sequence of m-dependent random variables, and Φ(x) is the standard normal distribution function. In a particular case, the obtained upper bound is of order O((1−v)^1/2).     

Prime Ideals of <Emphasis Type="Italic">q</Emphasis>-Commutative Power Series Rings

We study the “q-commutative” power series ring R: = k _q [[x ₁,...,x _n ]], defined by the relations x _i x _j  = q _ij x _j x _i , for mulitiplicatively antisymmetric scalars q _ij in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In particular, we prove that the prime spectrum of R is normally separated and is finitely stratified by commutative noetherian spectra. Combining this normal separation with results of Chan, Wu, Yekutieli, and Zhang, we are able to conclude that R is catenary. Following the approach of Brown and Goodearl, we also show that links between prime ideals are provided by canonical automorphisms. Moreover, for sufficiently generic q _ij , we find that R has only finitely many prime ideals and is a UFD (in the sense of Chatters).     

CONVERGENCE RATE FOR ITERATES OF q-BERNSTEIN POLYNOMIALS

Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q ≠ 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein polynomials and their Boolean sum are estimated. Moreover, the saturation of {Bn（., qn）} when n → ∞ and convergence rate of Bn（f,q;x） when f ∈ C^n-1 [0, 1], q → ∞ are also presented.     

On the Riesz means of coefficients of <Emphasis Type="Italic">m</Emphasis>th symmetric power <Emphasis Type="Italic">L</Emphasis>-functions

Let c _n be the Fourier coefficients of L(sym ^m   f, s), and Δρ(x; sym ^m   f) be the error term in the asymptotic formula for ∑ _n≪x c _n . In this paper, we study the Riesz means of Δρ(x; sym ^m   f) and obtain a truncated Voronoi-type formula under the hypothesis Nice(m, f).     

<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Subalgebras Generated by a Single Operator in <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">H</Emphasis>)

In this paper, we characterize a C ^*-subalgebra C ^*(x) of B(H), generated by a single operator x. We show that if x is polar-decomposed by aq, where a is the partial isometry part and q is the positive operator part of x, then C ^*(x) is *-isomorphic to the groupoid crossed product algebra A_q×_a\mathbbG_a\mathcal{A}_{q}\times_{\alpha }\mathbb{G}_{a} , where A_q=C^*(q)\mathcal{A}_{q}=C^{*}(q) and \mathbbG_a\mathbb{G}_{a} is the graph groupoid induced by a partial isometry part a of x.     

Semiclassical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Estimates of Quasimodes on Curved Hypersurfaces

Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L ² normalized family of functions such that P(h)u(h) is O(h) in L ²(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L ^p norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch ^−δ(n,k,p) where k=dim H (except for a logarithmic divergence in the case k=n−2, p=2). When H is a hypersurface, i.e., k=n−1, we have δ(n,n−1, 2)=1/4, which is sharp when M is the round n-sphere and H is an equator.