Actions of arithmetic groups on homology spheres and acyclic homology manifolds

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary $p$ -groups in the groups concerned. In some cases, including ${\mathrm{Sp}}(2n,\mathbb Z )$ , the bounds we obtain are sharp: if $X$ is a generalized $\mathbb Z /3$ -homology sphere of dimension less than $2n-1$ or a $\mathbb Z /3$ -acyclic $\mathbb Z /3$ -homology manifold of dimension less than $2n$ , and if $n\ge 3$ , then any action of ${\mathrm{Sp}}(2n,\mathbb Z )$ by homeomorphisms on $X$ is trivial; if $n=2$ , then every action of ${\mathrm{Sp}}(2n,\mathbb Z )$ on $X$ factors through the abelianization of ${\mathrm{Sp}}(4,\mathbb Z )$ , which is $\mathbb Z /2$ .     

Lax embeddings of the Hermitian unital

In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital ${\mathcal{U}}$ of ${\mathsf{PG}(2,\mathbb{L}), \mathbb{L}}$ a quadratic extension of the field ${\mathbb{K}}$ and ${|\mathbb{K}| \geq 3}$ , in a ${\mathsf{PG}(d,\mathbb{F})}$ , with ${\mathbb{F}}$ any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ (and d = 7) or it consists of the projection from a point ${p \in \mathcal{U}}$ of ${\mathcal{U}{\setminus} \{p\}}$ from a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ into a hyperplane ${\mathsf{PG}(6,\mathbb{K}^{\prime})}$ . In order to do so, when ${|\mathbb{K}| >3 }$ we strongly use the linear representation of the affine part of ${\mathcal{U}}$ (the line at infinity being secant) as the affine part of the generalized quadrangle ${\mathsf{Q}(4,\mathbb{K})}$ (the solid at infinity being non-singular); when ${|\mathbb{K}| =3}$ , we use the connection of ${\mathcal{U}}$ with the generalized hexagon of order 2.     

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$ , where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$ -rational points on ${\mathcal X}$ and E is a divisor on ${\mathcal X}$ . If ${\deg(E)\geq 2g+1}$ where g is the genus of ${\mathcal X}$ , then there is an embedding of ${\mathcal X}$ onto ${\mathcal Y}$ in the projective space of the linear series of the divisor E. Moreover, if ${\deg(E)\geq 2g+2}$ , then ${I(\mathcal Y)}$ , the vanishing ideal of ${\mathcal Y}$ , is generated by ${I_2(\mathcal Y)}$ , the homogeneous elements of degree two in ${I(\mathcal Y)}$ . If ${n >2 \deg(E)}$ , then ${I_2(\mathcal Y)=I_2(\mathcal Q)}$ , where ${\mathcal Q}$ is the image of ${\mathcal P}$ under the map from ${\mathcal X}$ to ${\mathcal Y}$ . These three results imply that, if ${2g+2\leq m < \frac{1}{2}n}$ , an AG representation ${(\mathcal Y, \mathcal Q, F)}$ of the code ${\mathcal C}$ can be obtained just using a generator matrix of ${\mathcal C}$ where ${\mathcal Y}$ is a normal curve in ${\mathbb{P}^{m-g}}$ which is the intersection of quadrics. This fact gives us some clues for breaking McEliece cryptosystem based on AG codes provided that we have an efficient procedure for computing and decoding the representation obtained.     

Localization and delocalization of eigenvectors for heavy-tailed random matrices

Consider an $n \times n$ Hermitian random matrix with, above the diagonal, independent entries with $\alpha $ -stable symmetric distribution and $0 < \alpha < 2$ . We establish new bounds on the rate of convergence of the empirical spectral distribution of this random matrix as $n$ goes to infinity. When $1 < \alpha < 2$ and $ p > 2$ , we give vanishing bounds on the $L^p$ -norm of the eigenvectors normalized to have unit $L^2$ -norm. On the contrary, when $0 < \alpha < 2/3$ , we prove that these eigenvectors are localized.     

A new optimal quaternary sequence family of length 2(2 n − 1) obtained from the orthogonal transformation of Families {\mathcal{B}} and {\mathcal{C}}

In this paper, a general orthogonal transformation on the optimal quaternary sequence Families ${\mathcal{B}}$ and ${\mathcal{C}}$ is presented. Consequently, the known optimal Family ${\mathcal{D}}$ and a new optimal Family ${\mathcal{E}}$ are produced in a uniform method. In contrast to the known optimal Family ${\mathcal{D}}$ , the new Family ${\mathcal{E}}$ has the same parameters such as the sequence length 2(2 ⁿ ? 1), the family size 2 ⁿ , and the maximal nontrivial correlation value ${2^{\frac{n+1}{2}}+2}$ , where n is a positive integer, but with a different correlation function.     

Stability of contact discontinuity for steady Euler system in infinite duct

In this paper, we prove stability of contact discontinuities for full Euler system. We fix a flat duct ${\mathcal{N}_0}$ of infinite length in ${\mathbb{R}^2}$ with width W ₀ and consider two uniform subsonic flow ${{U_l}^{\pm}=(u_l^{\pm}, 0, pl,\rho_l^{\pm})}$ with different horizontal velocity in ${\mathcal{N}_0}$ divided by a flat contact discontinuity ${\Gamma_{cd}}$ . And, we slightly perturb the boundary of ${\mathcal{N}_0}$ so that the width of the perturbed duct converges to ${W_0+\omega}$ for ${|\omega| < \delta}$ at ${x=\infty}$ for some ${\delta >0 }$ . Then, we prove that if the asymptotic state at left far field is given by ${{U_l}^{\pm}}$ , and if the perturbation of boundary of ${\mathcal{N}_0}$ and ${\delta}$ is sufficiently small, then there exists unique asymptotic state ${{U_r}^{\pm}}$ with a flat contact discontinuity ${\Gamma_{cd}^*}$ at right far field( ${x=\infty}$ ) and unique weak solution ${U}$ of the Euler system so that U consists of two subsonic flow with a contact discontinuity in between, and that U converges to ${{U_l}^{\pm}}$ and ${{U_r}^{\pm}}$ at ${x=-\infty}$ and ${x=\infty}$ respectively. For that purpose, we establish piecewise C ¹ estimate across a contact discontinuity of a weak solution to Euler system depending on the perturbation of ${\partial\mathcal{N}_0}$ and ${\delta}$ .     

On Suborbital Graphs for the Extended Modular Group {\hat{\Gamma}}

In this paper, we show that the extended modular group ${\hat{\Gamma}}$ acts on ${\hat{\mathbb{Q}}}$ transitively and imprimitively. Then the number of orbits of ${\hat{\Gamma} _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ is calculated and compared with the number of orbits of ${\Gamma _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ . Especially, we obtain the graphs ${\hat{G}_{u, N}}$ of ${\hat{\Gamma}_{0}(N)}$ on ${\hat{\mathbb{Q}}}$ , for each ${N\in\mathbb{N}}$ and each unit ${u \in U_{N} }$ , then we determine the suborbital graph ${\hat{F}_{u,N}}$ . We also give the edge conditions in ${\hat{G}_{u, N}}$ and the necessary and sufficient conditions for a circuit to be triangle in ${\hat{F}_{u, N}.}$      

SIP: critical value functions have finite modulus of non-convexity

We consider semi-infinite programming problems ${{\rm SIP}(z)}$ depending on a finite dimensional parameter ${z \in \mathbb{R}^p}$ . Provided that ${\bar{x}}$ is a strongly stable stationary point of ${{\rm SIP}(\bar{z})}$ , there exists a locally unique and continuous stationary point mapping ${z \mapsto x(z)}$ . This defines the local critical value function ${\varphi(z) := f(x(z); z)}$ , where ${x \mapsto f(x; z)}$ denotes the objective function of ${{\rm SIP}(z)}$ for a given parameter vector ${z\in \mathbb{R}^p}$ . We show that ${\varphi}$ is the sum of a convex function and a smooth function. In particular, this excludes the appearance of negative kinks in the graph of ${\varphi}$ .     

Quasirandom permutations are characterized by 4-point densities

For permutations ${\pi}$ and ${\tau}$ of lengths ${|\pi|\le|\tau|}$ , let ${t(\pi,\tau)}$ be the probability that the restriction of ${\tau}$ to a random ${|\pi|}$ -point set is (order) isomorphic to ${\pi}$ . We show that every sequence ${\{\tau_j\}}$ of permutations such that ${|\tau_j|\to\infty}$ and ${t(\pi,\tau_j)\to 1/4!}$ for every 4-point permutation ${\pi}$ is quasirandom (that is, ${t(\pi,\tau_j)\to 1/|\pi|!}$ for every ${\pi}$ ). This answers a question posed by Graham.     

Characterization of some simple groups by the multiplicity pattern

Let $G$ be a finite group and let ${\mathrm{Irr}}(G)$ denote the set of all complex irreducible characters of $G.$ Let ${\mathrm{cd}}(G)$ be the set of all character degrees of $G.$ For each positive integer $d,$ the multiplicity of $d$ in $G$ is defined to be the number of irreducible characters of $G$ having the same degree $d.$ The multiplicity pattern ${\mathrm{mp}}(G)$ is the vector whose first coordinate is $|G:G^{\prime }|$ and for $i\ge 1,$ the $(i+1)$ th-coordinate of ${\mathrm{mp}}(G)$ is the multiplicity of the $i$ th-smallest nontrivial character degree of $G.$ In this paper, we show that every nonabelian simple group with at most $7$ distinct character degrees is uniquely determined by the multiplicity pattern.     

A characterization of {\square(\kappa^{+})} in extender models

We prove that, in any fine structural extender model with Jensen’s λ-indexing, there is a ${\square(\kappa^{+})}$ -sequence if and only if there is a pair of stationary subsets of ${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$ without common reflection point of cofinality ${ < \kappa}$ which, in turn, is equivalent to the existence of a family of size ${ < \kappa}$ of stationary subsets of ${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$ without common reflection point of cofinality ${ < \kappa}$ . By a result of Burke/Jensen, ${\square_\kappa}$ fails whenever ${\kappa}$ is a subcompact cardinal. Our result shows that in extender models, it is still possible to construct a canonical ${\square(\kappa^{+})}$ -sequence where ${\kappa}$ is the first subcompact.     

On Extending {\mathcal{A}}-Modules Through the Coefficients

In classical linear algebra, extending the ring of scalars of a free module gives rise to a new free module containing an isomorphic copy of the former and satisfying a certain universal property. Also, given two free modules on the same ring of scalars and a morphism between them, enlarging the ring of scalars results in obtaining a new morphism having the nice property that it coincides with the initial map on the isomorphic copy of the initial free module in the new one. We investigate these problems in the category of free ${\mathcal{A}}$ -modules, where ${\mathcal{A}}$ is an ${\mathbb{R}}$ -algebra sheaf. Complexification of free ${\mathcal{A}}$ -modules, which is defined to be the process of obtaining new free ${\mathcal{A}}$ -modules by enlarging the ${\mathbb{R}}$ -algebra sheaf ${\mathcal{A}}$ to a ${\mathbb{C}}$ -algebra sheaf, denoted ${\mathcal{A}_\mathbb{C}}$ , is an important particular case (see Proposition 2.1, Proposition 3.1). Attention, on the one hand, is drawn on the sub- ${_{\mathbb{R}}\mathcal{A}}$ -sheaf of almost complex structures on the sheaf ${{_\mathbb{R}}\mathcal{A}^{2n}}$ , the underlying ${\mathbb{R}}$ -algebra sheaf of a ${\mathbb{C}}$ -algebra sheaf ${\mathcal{A}}$ , and on the other hand, on the complexification of the functor ${\mathcal{H}om_\mathcal {A}}$ , with ${\mathcal{A}}$ an ${\mathbb{R}}$ -algebra sheaf.     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

Admissible closures of polynomial time computable arithmetic

We propose two admissible closures ${\mathbb{A}({\sf PTCA})}$ and ${\mathbb{A}({\sf PHCA})}$ of Ferreira??s system PTCA of polynomial time computable arithmetic and of full bounded arithmetic (or polynomial hierarchy computable arithmetic) PHCA. The main results obtained are: (i) ${\mathbb{A}({\sf PTCA})}$ is conservative over PTCA with respect to ${\forall\exists\Sigma^b_1}$ sentences, and (ii) ${\mathbb{A}({\sf PHCA})}$ is conservative over full bounded arithmetic PHCA for ${\forall\exists\Sigma^b_{\infty}}$ sentences. This yields that (i) the ${\Sigma^b_1}$ definable functions of ${\mathbb{A}({\sf PTCA})}$ are the polytime functions, and (ii) the ${\Sigma^b_{\infty}}$ definable functions of ${\mathbb{A}({\sf PHCA})}$ are the functions in the polynomial time hierarchy.     

On additive decompositions of the set of primitive roots modulo p

It is conjectured that the set ${\mathcal {G}}$ of the primitive roots modulo p has no decomposition (modulo p) of the form ${\mathcal {G}= \mathcal {A} +\mathcal {B}}$ with ${|\mathcal {A}|\ge 2}$ , ${|\mathcal {B} |\ge 2}$ . This conjecture seems to be beyond reach but it is shown that if such a decomposition of ${\mathcal {G}}$ exists at all, then ${|\mathcal {A} |}$ , ${|\mathcal {B} |}$ must be around p ^1/2, and then this result is applied to show that ${\mathcal {G}}$ has no decomposition of the form ${\mathcal {G} =\mathcal {A} + \mathcal {B} + \mathcal {C}}$ with ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ .     

Bent functions embedded into the recursive framework of {\mathbb{Z}} -bent functions

Suppose that n is even. Let ${\mathbb{F}_2}$ denote the two-element field and ${\mathbb{Z}}$ the set of integers. Bent functions can be defined as ± 1-valued functions on ${\mathbb{F}_2^n}$ with ± 1-valued Fourier transform. More generally we call a mapping f on ${\mathbb{F}_2^n}$ a ${\mathbb{Z}}$ -bent function if both f and its Fourier transform ${\widehat{f}}$ are integer-valued. ${\mathbb{Z}}$ -bent functions f are separated into different levels, depending on the size of the maximal absolute value attained by f and ${\widehat{f}}$ . It is shown how ${\mathbb{Z}}$ -bent functions of lower level can be built up recursively by gluing together ${\mathbb{Z}}$ -bent functions of higher level. This recursion comes down at level zero, containing the usual bent functions. In the present paper we start to study bent functions in the framework of ${\mathbb{Z}}$ -bent functions and give some guidelines for further research.     

Helly-type theorems for intersections of sets starshaped via orthogonally convex paths

Let ${\mathcal{K}}$ be a family of simply connected sets in the plane. If every countable subfamily of ${\mathcal{K}}$ has an intersection that is starshaped via orthogonally convex paths, then ${\mathcal{K}}$ itself has such an intersection. For the d-dimensional case, let ${\mathcal{K}}$ be a family of compact sets in ${\mathbb{R}^d}$ . If every finite subfamily of ${\mathcal{K}}$ has an intersection that is starshaped via orthogonally convex paths, again ${\mathcal{K}}$ itself has such an intersection.     

Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions

Let (M,g) be an n-dimensional, compact Riemannian manifold and ${P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}$ be a semiclassical Schrödinger operator with ${\hbar \in (0,\hbar_0]}$ . Let ${E(\hbar) \in [E-o(1),E+o(1)]}$ and ${(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}$ be a family of L ²-normalized eigenfunctions of ${P_0(\hbar)}$ with ${P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}$ . We consider magnetic deformations of ${P_0(\hbar)}$ of the form ${P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}$ , where ${\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}$ . Here, u is a k-dimensional parameter running over ${B^k(\epsilon)}$ (the ball of radius ${\epsilon}$ ), and the family of the magnetic potentials ${(w_u)_{u\in B^k(\epsilon)}}$ satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of ${(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}$ , given by ${(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}$ , where $$\phi_{\hbar}^{(u)}:= {\rm e}^{-it_0 P_u(\hbar)/\hbar}\phi_{\hbar}$$ for ${|t_0|\in (0,\epsilon)}$ ; the latter functions are themselves eigenfunctions of the ${\hbar}$ -elliptic operators ${Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}$ with eigenvalue ${E(\hbar)}$ and ${Q_0(\hbar) = P_{0}(\hbar)}$ . Our main result, Theorem1.2, states that for ${\epsilon >0 }$ small, there are constants ${C_j=C_j(M,V,\omega,\epsilon) > 0}$ with j = 1,2 such that $$C_{1}\leq \int\limits_{\mathcal{B}^k(\epsilon)} |\phi_{\hbar}^{(u)}(x)|^2 \, {\rm d}u \leq C_{2}$$ , uniformly for ${x \in M}$ and ${\hbar \in (0,h_0]}$ . We also give an application to eigenfunction restriction bounds in Theorem 1.3.     

Characterization of (generalized) Lie derivations on {\mathcal{J}}-subspace lattice algebras by local action

Let ${\mathcal{L}}$ be a ${\mathcal{J}}$ -subspace lattice on a Banach space X over the real or complex field ${\mathbb{F}}$ with dim X ≥ 2 and Alg ${\mathcal{L}}$ be the associated ${\mathcal{J}}$ -subspace lattice algebra. For any scalar ${\xi \in \mathbb{F}}$ , there is a characterization of any linear map L : Alg ${\mathcal{L} \rightarrow {\rm Alg} {\mathcal{L}}}$ satisfying ${L([A,B]_\xi) = [L(A),B]_\xi + [A,L(B)]_\xi}$ for any ${A, B \in{\rm Alg} {\mathcal{L}}}$ with AB = 0 (rep. ${[A,B]_ \xi = AB - \xi BA = 0}$ ) given. Based on these results, a complete characterization of (generalized) ξ-Lie derivations for all possible ξ on Alg ${\mathcal{L}}$ is obtained.