A construction for infinite families of semisymmetric graphs revealing their full automorphism group

We give a general construction leading to different non-isomorphic families $\varGamma_{n,q}(\mathcal{K})$ of connected q-regular semisymmetric graphs of order 2q ⁿ⁺¹ embedded in $\operatorname{PG}(n+1,q)$ , for a prime power q=p ^h , using the linear representation of a particular point set $\mathcal{K}$ of size q contained in a hyperplane of $\operatorname{PG}(n+1,q)$ . We show that, when $\mathcal{K}$ is a normal rational curve with one point removed, the graphs $\varGamma_{n,q}(\mathcal{K})$ are isomorphic to the graphs constructed for q=p ^h in Lazebnik and Viglione (J. Graph Theory 41, 249–258, 2002) and to the graphs constructed for q prime in Du et al. (Eur. J. Comb. 24, 897–902, 2003). These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For q≥n+3 or q=p=n+2, n≥2, we obtain their full automorphism group from our construction by showing that, for an arc $\mathcal{K}$ , every automorphism of $\varGamma_{n,q}(\mathcal{K})$ is induced by a collineation of the ambient space $\operatorname{PG}(n+1,q)$ . We also give some other examples of semisymmetric graphs $\varGamma _{n,q}(\mathcal{K})$ for which not every automorphism is induced by a collineation of their ambient space.     

On mappings preserving orthogonality of non-singular vectors

Letq be a regular quadratic form on a vector space (V,F) and letf be the bilinear form associated withq. Then, \(\dot V: = \{ z \in V|q(z) \ne 0\} \) is the set of non-singular vectors ofV, and forx, y ∈ \(\dot V\) , ?(x, y) ?f(x, y) ²/(q(x) · q(y)) is theq-measure of (x, y), where ?(x,y)=0 means thatx, y are orthogonal. For an arbitrary mapping \(\sigma :\dot V \to \dot V\) we consider the functional equations $$\begin{gathered} (I)\sphericalangle (x,y) = 0 \Leftrightarrow \sphericalangle (x^\sigma ,y^\sigma ) = 0\forall x,y \in \dot V, \hfill \\ (II)\sphericalangle (x,y) = \sphericalangle (x^\sigma ,y^\sigma )\forall x,y \in \dot V, \hfill \\ (III)f(x,y)^2 = f(x^\sigma ,y^\sigma )^2 \forall x,y \in \dot V, \hfill \\ \end{gathered} $$ and we state conditions on (V,F,q) such thatσ is induced by a mapping of a well-known type. In case of dimV ∈N?{0, 1, 2} ∧ ∣F∣ > 3, each of the assumptions (I), (II), (III) implies that there exist aρ-linear injectionξ :V →V and a fixed λ ∈F?{0} such thatF x ^σ =F x ^ξ ?x ∈ \(\dot V\) andf(x ^ξ,y ^ξ)=λ · (f(x, y))^ρ ?x, y ∈V. Moreover, (II) implies ρ =id _F ∧q(x ^ξ) = λ ·q(x) ?x ∈ \(\dot V\) , and (III) implies ρ=id _F ∧ λ ∈ {1,?1} ∧x ^σ ∈ {x ^ξ, ?x ^ξ} ?x ∈ \(\dot V\) . Other results obtained in this paper include the cases dimV = 2 resp. dimV ?N resp. ∣F∣ = 3.     

Blocking Sets in PG(r, q n )

Let $\mathcal S$ be a Desarguesian (n – 1)-spread of a hyperplane Σ of PG(rn, q). Let Ω and ${\bar B}$ be, respectively, an (n – 2)-dimensional subspace of an element of $\mathcal S $ and a minimal blocking set of an ((r – 1)n + 1)-dimensional subspace of PG(rn, q) skew to Ω. Denote by K the cone with vertex Ω and base ${\bar B}$ , and consider the point set B defined by $$B=\left(K\setminus\Sigma\right)\cup \{X\in \mathcal S\, : \, X\cap K\neq \emptyset\}$$ in the Barlotti–Cofman representation of PG(r, q ⁿ ) in PG(rn, q) associated to the (n – 1)-spread $\mathcal S$ . Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61–81, 2006), under suitable assumptions on ${\bar B}$ , we prove that B is a minimal blocking set in PG(r, q ⁿ ). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking sets of size q ⁿ⁺² + 1 in PG(r, q ⁿ ), 3 ≤ r ≤ 6 and n ≥ 3, and of size q ⁴ + 1 in PG(r, q ²), 4 ≤ r ≤ 6; actually, in the second case, these blocking sets turn out to be the union of q ³ Baer sublines through a point. Moreover, for q an even power of 3, we construct examples of minimal blocking sets of PG(4, q) of size at least q ² + 2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4, q ²) of size q ⁴ + 1, for any q a power of 3.     

On Bipartite Distance-Regular Graphs with Intersection Numbers c i = (q i ? 1)/(q ? 1)

Let q denote an integer at least two. Let ?? denote a bipartite distance-regular graph with diameter D ?? 3 and intersection numbers c _i = (q ⁱ ? 1)/(q ? 1), 1 ?? i ?? D. Let X denote the vertex set of ?? and let ${V = \mathbb{C}^X}$ denote the vector space over ${\mathbb{C}}$ consisting of column vectors whose coordinates are indexed by X and whose entries are in ${\mathbb{C}}$ . For ${z \in X}$ , let ${{\hat z}}$ denote the vector in V with a 1 in the z-coordinate and 0 in all other coordinates. Fix ${x, y \in X}$ such that ?(x, y) = 2, where ? denotes the path-length distance function. For 0 ?? i, j ?? D define ${w_{ij} = \sum {\hat z}}$ , where the sum is over all ${z \in X}$ such that ?(x, z) = i and ?(y, z) = j. We define W?=?span{w _ij | 0 ?? i, j ?? D}. In this paper we consider the space ${MW={\rm span} \{mw \mid m \in M, w \in W\}}$ , where M is the Bose?CMesner algebra of ??. We observe that MW is the minimal A-invariant subspace of V which contains W, where A is the adjacency matrix of ??. We give a basis for MW that is orthogonal with respect to the Hermitean dot product. We compute the square-norm of each basis vector. We compute the action of A on the basis. For the case in which ?? is the dual polar graph D _D (q) we show that the basis consists of the characteristic vectors of the orbits of the stabilizer of x and y in the automorphism group of ??.     

Linear and Algebraic Independence of q-Zeta Values

In the paper, results on linear and algebraic independence of q-series of the form $\varsigma _q (s) = \sum\nolimits_{n = 1}^\infty {\sigma _{s - 1} (n)q^n }$ over the field ?(q) are obtained, where $\sigma _{s - 1} (n) = \sum\nolimits_{d|n} {d^{s - 1} }$ , s = 1, 2,... .     

On the cosets of the 2q-power group in the unit group modulop

Letp be a prime number ≡ 3 mod 4,G ^p the unit group of ?/p?, andg a generator ofG _p. Letq be an odd divisor ofp - 1 andG _p ^2q = {t ^2q;t ∈G _pthe subgroup of index2q inG _p. The groupG _p ² / _p ^2q consists of the classes \(\bar g^{2j} \) ,j = 0,...,q – 1. In this paper we study the ’excesses’ of the classes \(\bar g^{2j} \) in {l,...,(_p–l)/2}, i.e., the numbers \(\Phi _j = \left| {\left\{ {k;1 \leqslant k \leqslant \left( {p - 1} \right)/2,\bar k \in \bar g^{2j} } \right\}} \right| - \left| {\left\{ {k;\left( {p - 1} \right)/2 \leqslant k \leqslant p - 1,\bar k \in \bar g^{2j} } \right\}} \right|\) ,j = 0.....q — 1. First we express therelative class number h _2q of the subfieldK _2q? ?(e^2#x03C0;i/p ) of degree [K _2q: ?] =2q in terms of these excesses. We use this formula to establish certaincongruences for the Ф^j. E.g., ifq ∈ {3,5,11}, each number Фj is congruent modulo 4 to each other iff 2 dividesh _2q ^- . Finally we study thevariance of the excesses, i.e., the number \(\sigma ^2 = ((\Phi _0 - \hat \Phi )^2 + \ldots + (\Phi _{q - 1} - \hat \Phi )^2 )/(q - 1)\) , where \(\hat \Phi \) is the mean value of the numbers Ф_j. We obtain an explicit lower bound for σ² in terms ofh _2q ^- /h ₂ ^- . Moreover, we show that log σ² is asymptotically equal to 21og(h _2q ^- h ₂ ^- )/(q - 1) forp→∞. Three tables illustrate the results.     

On large maximal partial ovoids of the parabolic quadric Q(4, q)

We use the representation ${T_2(\mathcal{O})}$ for Q(4, q) to show that maximal partial ovoids of Q(4, q) of size q ² ? 1, q = p ^h , p an odd prime, h > 1, do not exist. Although this was known before, we give a slightly alternative proof, also resulting in more combinatorial information of the known examples for q an odd prime.     

A characterization of the Grassmann embedding of H(q), with q even

In this note, we characterize the Grassmann embedding of H(q), q even, as the unique full embedding of H(q) in PG(12, q) for which each ideal line of H(q) is contained in a plane. In particular, we show that no such embedding exists for H(q), with q odd. As a corollary, we can classify all full polarized embeddings of H(q) in PG(12, q) with the property that the lines through any point are contained in a solid; they necessarily are Grassmann embeddings of H(q), with q even.     

Tangent-Point Repulsive Potentials for a Class of Non-smooth m-dimensional Sets in ℝ n . Part I: Smoothing and Self-avoidance Effects

We consider repulsive potential energies $\mathcal {E}_{q}(\Sigma)$ , whose integrand measures tangent-point interactions, on a large class of non-smooth m-dimensional sets Σ in ? ⁿ . Finiteness of the energy $\mathcal {E}_{q}(\Sigma)$ has three sorts of effects for the set Σ: topological effects excluding all kinds of (a priori admissible) self-intersections, geometric and measure-theoretic effects, providing large projections of Σ onto suitable m-planes and therefore large m-dimensional Hausdorff measure of Σ within small balls up to a uniformly controlled scale, and finally, regularizing effects culminating in a geometric variant of the Morrey–Sobolev embedding theorem: Any admissible set Σ with finite $\mathcal {E}_{q}$ -energy, for any exponent q>2m, is, in fact, a C ¹-manifold whose tangent planes vary in a Hölder continuous manner with the optimal Hölder exponent μ=1?(2m)/q. Moreover, the patch size of the local C ^1,μ -graph representations is uniformly controlled from below only in terms of the energy value $\mathcal {E}_{q}(\Sigma)$ .     

Optimal Sobolev Type Inequalities in Lorentz Spaces

It is well known that the classical Sobolev embeddings may be improved within the framework of Lorentz spaces L ^p,q : the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, embeds into $L^{p^*,q}(\mathbb R^n)$ , p?≤?q?≤?∞. However, the value of the best possible embedding constants in the corresponding inequalities is known just in the case $L^{p^*,p}(\mathbb R^n)$ . Here, we determine optimal constants for the embedding of the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, into the whole Lorentz space scale $L^{p^{\ast}, q}(\mathbb R^n)$ , p?≤?q?≤?∞, including the limiting case q?=?p of which we give a new proof. We also exhibit extremal functions for these embedding inequalities by solving related elliptic problems.     

Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order

Let Q₀ be the classical generalized quadrangle of order q = 2ⁿ(n≥2) arising from a non-degenerate quadratic form in a 5-dimensional vector space defined over a finite field of order q. We consider the rank two geometry \(\mathcal {X}\) having as points all the elliptic ovoids of Q₀ and as lines the maximal pencils of elliptic ovoids of Q₀ pairwise tangent at the same point. We first prove that \(\mathcal {X}\) is isomorphic to a 2-fold quotient of the affine generalized quadrangle Q?Q₀, where Q is the classical (q,q²)-generalized quadrangle admitting Q₀ as a hyperplane. Further, we classify the cliques in the collinearity graph Γ of \(\mathcal {X}\). We prove that any maximal clique in Γ is either a line of \(\mathcal {X}\) or it consists of 6 or 4 points of \(\mathcal {X}\) not contained in any line of \(\mathcal {X}\), accordingly as n is odd or even. We count the number of cliques of each type and show that those cliques which are not contained in lines of \(\mathcal {X}\) arise as subgeometries of Q defined over \(\mathbb {F}_{2}\).     

On the functional codes defined by quadrics and Hermitian varieties

In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD Thesis, 2007). This work was continued in (Edoukou et al., Des Codes Cryptogr 56:219–233, 2010; Edoukou et al., J Pure Appl Algebr 214:1729–1739, 2010; Hallez and Storme, Finite Fields Appl 16:27–35, 2010), where the results of the thesis were improved and extended. In particular, Hallez and Storme investigated the functional codes ${C_2(\mathcal{H})}$ , with ${\mathcal{H}}$ a non-singular Hermitian variety in PG(N, q ²). The codewords of this code are defined by evaluating the points of ${\mathcal{H}}$ in the quadratic polynomials defined over ${\mathbb{F}_{q^2}}$ . We now present the similar results for the functional code ${C_{Herm}(\mathcal{Q})}$ . The codewords of this code are defined by evaluating the points of a non-singular quadric ${\mathcal{Q}}$ in PG(N, q ²) in the polynomials defining the Hermitian varieties of PG(N, q ²).     

Square-free values of f(p), f cubic

Let \({f \in \mathbb{Z}[x]}\) , \({\deg f =3}\) . Assume that f does not have repeated roots. Assume as well that, for every prime q, \({f(x)\not\equiv 0}\) mod q ² has at least one solution in \({(\mathbb{Z}/q^2 \mathbb{Z})^*}\) . Then, under these two necessary conditions, there are infinitely many primes p such that f(p) is square-free.     

Problems similar to the additive divisor problem

For multiplicative functions ?(n), let the following conditions be satisfied: ?(n)≥0 ?(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants $A_\varepsilon$ ,α>0 such that $f(n) \leqslant A_\varepsilon n^\varepsilon$ and Σ _p≤x ?(p) lnp≥αx. For such functions, the following relation is proved: $$\sum\limits_{n \leqslant x} {f(n)} \tau (n - 1) = C(f)\sum\limits_{n \leqslant x} {f(n)lnx(1 + 0(1))}$$ . Hereτ(n) is the number of divisors ofn andC(?) is a constant.     

The \bar \partial -problem on q-pseudoconvex domains with applications

For a q-pseudoconvex domain Ω in ? ⁿ , 1 ≤ q ≤ n, with Lipschitz boundary, we solve the $\bar \partial $ -problem with exact support in Ω. Moreover, we solve the $\bar \partial $ -problem with solutions smooth up to the boundary over Ω provided that it has smooth boundary. Applications are given to the solvability of the tangential Cauchy-Riemann equations on the boundary.     

L 2 existence theorems for the\bar \partial _b - Neumann problem on strongly pseudoconvex CR manifoldsproblem on strongly pseudoconvex CR manifolds

LetM be the boundary of a strongly pseudoconvex domain in \(\mathbb{C}^n \) ,n≥4 and ω be an open subset inM such that ?ω is the intersection ofM with a flat hypersurface. We establish theL ² existence theorems of the \(\bar \partial _b - Neumann\) problem on ω. In particular, we prove that the \(\bar \partial _b - Laplacian\) \(\square _b = \bar \partial _b \bar \partial _b^* + \bar \partial _b^* \bar \partial _b \) equipped with a pair of natural boundary conditions, the so-called \(\bar \partial _b - Neumann\) boundary conditions, has closed range when it acts on (0,q) forms, 1≤q≤n?3. Thus there exists a bounded inverse operator for \(\square _b \) , the \(\bar \partial _b - Neumann\) operatorN _b, and we have the following Hodge decomposition theorem on ω for \(\bar \partial _b \bar \partial _b^* N_b \alpha + \bar \partial _b^* \bar \partial _b N_b \alpha \) , for any (0,q) form α withL ²(ω) coefficients. The proof depends on theL ^p regularity of the tangential Cauchy-Riemann operators \(\bar \partial _b u = \alpha \) on ω?M under the compatibility condition \(\bar \partial _b \alpha = 0\) , where α is a (p, q) form on ω, where 1≤q≤n?2. The interior regularity ofN _b follows from the fact that \(\square _b \) is subelliptic in the interior of ω. The operatorN _b induces natural questions on the regularity up to the boundary ?ω. Near the characteristic point of the boundary, certain compatibility conditions will be present. In fact, one can show thatN _b is not a compact operator onL ²(ω).     

Homoclinic Solutions for a Class of Second Order Hamiltonian Systems

In this paper we consider the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system $${\ddot q}-L(t)q+\nabla W(t,q)=0, \quad\quad\quad\quad\quad\quad\quad (\rm HS)$$ where ${L\in C({\mathbb R},{\mathbb R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in {\mathbb R}}$ , W(t, q)?=?a(t)U(q) with ${a\in C({\mathbb R},{\mathbb R}^+)}$ and ${U\in C^1({\mathbb R}^n,{\mathbb R})}$ . The novelty of this paper is that, assuming L is bounded from below in the sense that there is a constant M?>?0 such that (L(t)q, q)?≥ M |q|² for all ${(t,q)\in {\mathbb R}\times {\mathbb R}^n}$ , we establish one new compact embedding theorem. Subsequently, supposing that U satisfies the global Ambrosetti–Rabinowitz condition, we obtain a new criterion to guarantee that (HS) has one nontrivial homoclinic solution using the Mountain Pass Theorem, moreover, if U is even, then (HS) has infinitely many distinct homoclinic solutions. Recent results from the literature are generalized and significantly improved.     

Transitive A 6-invariant k-arcs in PG(2, q)

For q = p ^r with a prime p ≥ 7 such that ${q \equiv 1}$ or 19 (mod 30), the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to the alternating group A ₆ of degree 6. For a projectivity group ${\Gamma \cong A_6}$ of PG(2, q), we investigate the geometric properties of the (unique) Γ-orbit ${\mathcal{O}}$ of size 90 such that the 1-point stabilizer of Γ in its action on ${\mathcal O}$ is a cyclic group of order 4. Here ${\mathcal O}$ lies either in PG(2, q) or in PG(2, q ²) according as 3 is a square or a non-square element in GF(q). We show that if q ≥ 349 and q ≠ 421, then ${\mathcal O}$ is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601,661. Interestingly, ${\mathcal O}$ is the smallest known complete arc in PG(2,601) and in PG(2,661). Computations are carried out by MAGMA.     

On solvability of derivative dependent doubly singular boundary value problems

In this paper we establish existence of solutions of singular boundary value problem ?(p(x)y ^′(x))^′=q(x)f(x,y,py′) for 0<x≤b and $\lim_{x\rightarrow0^{+}}p(x)y^{\prime}(x)=0$ , α ₁ y(b)+β ₁ p(b)y ^′(b)=γ ₁ with p(0)=0 and q(x) is allowed to have integrable discontinuity at x=0. So the problem may be doubly singular. Here we consider $\lim_{x\rightarrow0^{+}}\frac{q(x)}{p'(x)}\neq0$ therefore $\lim_{x\rightarrow0^{+}}p(x)y'(x)=0$ does not imply y′(0)=0 unless $\lim_{x\rightarrow0^{+}}f(x,y(x),p(x)y'(x))=0$ .