Minimal full polarized embeddings of dual polar spaces

Let Δ be a thick dual polar space of rank n ≥ 2 admitting a full polarized embedding e in a finite-dimensional projective space Σ, i.e., for every point x of Δ, e maps the set of points of Δ at non-maximal distance from x into a hyperplane e∗(x) of Σ. Using a result of Kasikova and Shult [11], we are able the show that there exists up to isomorphisms a unique full polarized embedding of Δ of minimal dimension. We also show that e∗ realizes a full polarized embedding of Δ into a subspace of the dual of Σ, and that e∗ is isomorphic to the minimal full polarized embedding of Δ. In the final section, we will determine the minimal full polarized embeddings of the finite dual polar spaces DQ(2n,q), DQ ⁻(2n+1,q), DH(2n−1,q ²) and DW(2n−1,q) (q odd), but the latter only for n≤ 5. We shall prove that the minimal full polarized embeddings of DQ(2n,q), DQ ⁻(2n+1,q) and DH(2n−1,q ²) are the `natural' ones, whereas this is not always the case for DW(2n−1, q).B. De Bruyn: Postdoctoral Fellow of the Research Foundation - Flanders.     

A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}

Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2ⁿ - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2ⁿ - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.     

The combinatorial properties of the hyperplanes of DW(5, q) arising from embedding

In De Bruyn Discrete math(to appear), one of the authors proved that there are six isomorphism classes of hyperplanes in the dual polar space DW(5, q), q even, which arise from its Grassmann-embedding. In the present paper, we determine the combinatorial properties of these hyperplanes. Specifically, for each such hyperplane H we calculate the number of quads Q for which is a certain configuration of points in Q and the number of points for which is a certain configuration of points in . By purely combinatorial techniques, we are also able to show that the set of hyperplanes of DW(5, q), q odd, which arise from its Grassmann-embedding can be divided into six subclasses if one takes only into account the above-mentioned combinatorial properties. A complete classification of all hyperplanes of DW(5, q), q odd, which arise from its Grassmann-embedding, i.e. the division of the above-mentioned six classes into isomorphism classes, will unlike in De Bruyn (to appear) most likely need a group-theoretical approach. Postdoctoral Fellow of the Research Foundation—Flanders (Belgium).     

On an Isomorphism Problem of Some Dual Hyperovals in PG(2d + 1, q) with q even

Let q = 2^l with l≥ 1 and d ≥ 2. We prove that any automorphism of the d-dimensional dual hyperoval over GF(q), constructed in [3] for any (d + 1)-dimensional GF(q)-vector subspace V in GF(qⁿ) with n≥ d + 1 and for any generator σ of the Galois group of GF(qⁿ) over GF(q), always fixes the special member X(∞). Moreover, we prove that, in case V = GF(q^d+1), two dual hyperovals and in PG(2d + 1,q), where σ and τ are generators of the Galois group of GF(q^d+1) over GF(q), are isomorphic if and only if (1) σ = τ or (2) σ τ = id. Therefore, we have proved that, even in the case q > 2, there exist non isomorphic d-dimensional dual hyperovals in PG(2d + 1,q) for d ≥ 3.     

Locally singular hyperplanes in thick dual polar spaces of rank 4

We study (i-)locally singular hyperplanes in a thick dual polar space Δ of rank n. If Δ is not of type DQ(2n,K), then we will show that every locally singular hyperplane of Δ is singular. We will describe a new type of hyperplane in DQ(8,K) and show that every locally singular hyperplane of DQ(8,K) is either singular, the extension of a hexagonal hyperplane in a hex or of the new type.     

Dual partially harmonic tensors and Brauer-Schur-Weyl duality

Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let $ \mathfrak{B}_n^{(f)} Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let \mathfrakB_n^(f) \mathfrak{B}_n^{(f)} be the two-sided ideal of the Brauer algebra \mathfrakB_n( - 2m ) {\mathfrak{B}_n}\left( { - 2m} \right) over K generated by e ₁ e _3⋯ e _2f-1 where 0 ≤ f ≤ [n/2]. Let HT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} be the subspace of partial-harmonic tensors of valence f in V ^⊗n . In this paper we prove that dimHT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} and dim \textEn\textd_K\textSp(V)( V ^?n \mathord

Approaching pooling design with smaller efficient ratio

In this paper, we employ affine symplectic space ASG(2n,\mathbbF_q){ASG(2\nu,\mathbb{F}_q)} as a tool to construct two new classes of d ^e -disjunct matrices. The efficiency ratio of new d ^e -disjunct matrices is smaller than that of D’yachkov et al. (J Comput Biol 12:1129–1136, 2005).     

Weak infinite powers of Blaschke products

Letb be a Blaschke product with zeros {z _n } in the open unit disk Δ. Let be the set of sequences of non-negative integersp=(p ₁,p ₂,…) such that ∑ _n=1 ^∞ p _n (1 − |z _n |) < ∞ andp _n →∞ asn→∞. We study the class of weak infinite powers ofb, Properties of these classes depend on the setS(b) of the cluster points in ∂Δ of {z _n }. It is proved thatS(b)=∂Δ if and only if , the Douglas algebra generated by . Also, it is proved thatdθ(S(b))=0 if and only if there exists an interpolating Blaschke productB such that .     

On Hankel Nonnegative Definite Sequences, the Canonical Hankel Parametrization, and Orthogonal Matrix Polynomials

This paper continues recent investigations started in Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) into the structure of the set H_q,2n ³ {\mathcal{H}_{q,2n}^{\ge}} of all Hankel nonnegative definite sequences, (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}}, of complex q × q matrices and its important subclasses H_q,2n ^{3 ,e}{\mathcal{H}_{q,2n}^{\ge,{\rm e}}} and ${\mathcal{H}_{q,2n}^>}${\mathcal{H}_{q,2n}^>} of all Hankel nonnegative definite extendable sequences and of all Hankel positive definite sequences, respectively. These classes of sequences arise quite naturally in the framework of matrix versions of the truncated Hamburger moment problem. In Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) a canonical Hankel parametrization [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} consisting of two sequences of complex q × q matrices was associated with an arbitrary sequence (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} of complex q × q matrices. The sequences belonging to each of the classes H_q,2n ³ , H_q,2n ^{3 ,e}{\mathcal{H}_{q,2n}^{\ge}, \mathcal{H}_{q,2n}^{\ge,{\rm e}}}, and ${\mathcal{H}_{q,2n}^>}${\mathcal{H}_{q,2n}^>} were characterized in terms of their canonical Hankel parametrization (see, Dyukarev et al. in Complex anal oper theory 3(4):759–834, 2009; Proposition 2.30). In this paper, we will study further aspects of the canonical Hankel parametrization. Using the canonical Hankel parametrization [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} of a sequence (s_j)_j=0²ⁿ ? H_q,2n ³ {(s_{j})_{j=0}^{2n} \in \mathcal{H}_{q,2n}^{\ge}}, we give a recursive construction of a monic right (resp. left) orthogonal system of matrix polynomials with respect to (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} (see Theorem 5.5). The matrices [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} will be expressed in terms of an arbitrary monic right (resp. left) orthogonal system with respect to (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} (see Theorem 5.11). This result will be reformulated in terms of nonnegative Hermitian Borel measures on \mathbbR{\mathbb{R}}. In this way, integral representations for the matrices [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} will be obtained (see Theorem 6.9). Starting from the monic orthogonal polynomials with respect to some classical probability distributions on \mathbbR{\mathbb{R}}, Theorem 6.9 is used to compute the canonical Hankel parametrization of their moment sequences. Moreover, we discuss important number sequences from enumerative combinatorics using the canonical Hankel parametrization.     

Abelian coverings of finite general linear groups and an application to their non-commuting graphs

In this paper we introduce and study a family A_n(q)\mathcal{A}_{n}(q) of abelian subgroups of GL_n(q){\rm GL}_{n}(q) covering every element of GL_n(q){\rm GL}_{n}(q). We show that A_n(q)\mathcal{A}_{n}(q) contains all the centralizers of cyclic matrices and equality holds if q>n. For q>2, we obtain an infinite product expression for a probabilistic generating function for |A_n(q)||\mathcal{A}_{n}(q)|. This leads to upper and lower bounds which show in particular that

Hereditarily indecomposable Banach algebras of diagonal operators

We provide a characterization of the Banach spaces X with a Schauder basis (e _n ) _n∈ℕ which have the property that the dual space X* is naturally isomorphic to the space L _diag(X) of diagonal operators with respect to (e _n ) _n∈ℕ. We also construct a Hereditarily Indecomposable Banach space $ \mathfrak{X} $ \mathfrak{X} _D with a Schauder basis (e _n ) _n∈ℕ such that $ \mathfrak{X} $ \mathfrak{X} *_D is isometric to L _diag($ \mathfrak{X} $ \mathfrak{X} _D) with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every T ∈ L _diag($ \mathfrak{X} $ \mathfrak{X} _D) is of the form T = λI + K, where K is a compact operator.     

When is the Uvarov transformation positive definite?

Let L\cal{L} be a positive definite bilinear functional, then the Uvarov transformation of L\cal{L} is given by  U(p,q) = L(p,q) + m p(a)[`(q)](a<sup>-1</sup>) +[`(m)] p([`(a)]<sup>-1</sup>)\,\mathcal{U}(p,q) = \mathcal{L}(p,q) + m\,p(\alpha)\overline{q}(\alpha^{-1}) + \overline{m}\,p(\overline{\alpha}^{-1}) [`(q)]([`(a)])\overline{q}(\overline{\alpha}) where $|\alpha| > 1, m \in \mathbb{C}$|\alpha| > 1, m \in \mathbb{C}. In this paper we analyze conditions on m for U\cal{U} to be positive definite in the linear space of polynomials of degree less than or equal to n. In particular, we show that m has to lie inside a circle in the complex plane defined by α, n and the moments associated with L\cal{L}. We also give an upper bound for the radius of this circle that depends only on α and n. This and other conditions on m are visualized for some examples.     

On the code generated by the incidence matrix of points and hyperplanes in <Emphasis Type="Italic">PG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>) and its dual

In this paper, we study the p-ary linear code C(PG(n,q)), q = p ^h , p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar’s lower bound (Geom Dedicata 8:407–415, 1979). G. Van de Voorde’s research was supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).     

Upper bound theorems for homology manifolds

In this paper we prove the Upper Bound Conjecture (UBC) for some classes of (simplicial) homology manifolds: we show that the UBC holds for all odd-dimensional homology manifolds and for all 2k-dimensional homology manifolds Δ such that β _k (Δ)⩽Σ{β _i (Δ):i ≠k-2,k,k+2 and 1 ⩽i⩽2k-1}, where β _i (Δ) are reduced Betti numbers of Δ. (This condition is satisfied by 2k-dimensional homology manifolds with Euler characteristic χ≤2 whenk is even or χ≥2 whenk is odd, and for those having vanishing middle homology.) We prove an analog of the UBC for all other even-dimensional homology manifolds. Kuhnel conjectured that for every 2k-dimensional combinatorial manifold withn vertices, . We prove this conjecture for all 2k-dimensional homology manifolds withn vertices, wheren≥4k+3 orn≤3k+3. We also obtain upper bounds on the (weighted) sum of the Betti numbers of odd-dimensional homology manifolds.     

Minimax goodness-of-fit testing in multivariate nonparametric regression

We consider an unknown response function f defined on Δ = [0, 1] ^d , 1 ≤ d ≤ ∞, taken at n random uniform design points and observed with Gaussian noise of known variance. Given a positive sequence r _n → 0 as n → ∞ and a known function f ₀ ∈ L ₂(Δ), we propose, under general conditions, a unified framework for goodness-of-fit testing the null hypothesis H ₀: f = f ₀ against the alternative H ₁: f ∈ $ \mathcal{F} $ \mathcal{F} , ∥f − f ₀∥ ≥ r _n , where $ \mathcal{F} $ \mathcal{F} is an ellipsoid in the Hilbert space L ₂(Δ) with respect to the tensor product Fourier basis and ∥ · ∥ is the norm in L ₂(Δ). We obtain both rate and sharp asymptotics for the error probabilities in the minimax setup. The derived tests are inherently non-adaptive. Several illustrative examples are presented. In particular, we consider functions belonging to ellipsoids arising from the well-known multidimensional Sobolev and tensor product Sobolev norms as well as from the less-known Sloan-Woźniakowski norm and a norm constructed from multivariable analytic functions on the complex strip.     

Derivation modules of orthogonal duals of hyperplane arrangements

Let A be an n × d matrix having full rank n. An orthogonal dual A^⊥ of A is a (d-n) × d matrix of rank (d−n) such that every row of A^⊥ is orthogonal (under the usual dot product) to every row of A. We define the orthogonal dual for arrangements by identifying an essential (central) arrangement of d hyperplanes in n-dimensional space with the n × d matrix of coefficients of the homogeneous linear forms for which the hyperplanes are kernels. When n ≥ 5, we show that if the matroid (or the lattice of intersection) of an n-dimensional essential arrangement contains a modular copoint whose complement spans, then the derivation module of the orthogonally dual arrangement ^⊥ has projective dimension at least ⌈ n(n+2)/4 ⌉ - 3.Hal Schenck partially supported by NSF DMS 03-11142, NSA MDA 904-03-1-0006, and ATP 010366-0103.     

A new characterization of L 2(q) by its noncommuting graph

Let G be a non-abelian group and associate a non-commuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this short paper we prove that if G is a finite group with ∇(G) ≅ ∇(M), where M = L ₂(q) (q = p ⁿ , p is a prime), then G ≅ M.      

Lattice construction of logarithmic modules for certain vertex algebras

A general method for constructing logarithmic modules in vertex operator algebra theory is presented. By utilizing this approach, we give explicit vertex operator construction of certain indecomposable and logarithmic modules for the triplet vertex algebra W(p){\mathcal{W}(p)} and for other subalgebras of lattice vertex algebras and their N = 1 super extensions. We analyze in detail indecomposable modules obtained in this way, giving further evidence for the conjectural equivalence between the category of W(p){\mathcal{W}(p)}-modules and the category of modules for the restricted quantum group [`(U)]_q(sl₂){\overline{\mathcal{U}}_q(sl_2)} , q = e ^{π i/p} . We also construct logarithmic representations for a certain affine vertex operator algebra at admissible level realized in Adamović (J. Pure Appl. Algebra 196:119–134, 2005). In this way we prove the existence of the logarithmic representations predicted in Gaberdiel (Int. J. Modern Phys. A 18, 4593–4638, 2003). Our approach enlightens related logarithmic intertwining operators among indecomposable modules, which we also construct in the paper.     

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets of the natural numbers such that the Robin inequality holds for all but finitely many . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.