About the maximal rank of 3-tensors over the real and the complex number field

Tensor data are becoming important recently in various application fields. In this paper, we consider the maximal rank problem of 3-tensors and extend Atkinson and Stephens’ and Atkinson and Lloyd’s results over the real number field. We also prove the assertion of Atkinson and Stephens: ${{\rm max.rank}_{\mathbb{R}}(m,n,p) \leq m+\lfloor p/2\rfloor n}$ , ${{\rm max.rank}_{\mathbb{R}}(n,n,p) \leq (p+1)n/2}$ if p is even, ${{\rm max.rank}_{\mathbb{F}}(n,n,3)\leq 2n-1}$ if ${\mathbb{F}=\mathbb{C}}$ or n is odd, and ${{\rm max.rank}_{\mathbb{F}}(m,n,3)\leq m+n-1}$ if m < n where ${\mathbb{F}}$ stands for ${\mathbb{R}}$ or ${\mathbb{C}}$ .     

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 ${\hat{F}_{u, N}.}$      

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 ${\delta}$ .     

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 ${\mathbb{P\ast P}}$ .     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

Quasilinear elliptic equations in \mathbb{R }^{N} via variational methods and Orlicz-Sobolev embeddings

In this paper we prove the existence of a nontrivial non-negative radial solution for the quasilinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\nabla \cdot \left[\phi ^{\prime }(|\nabla u|^2)\nabla u \right] +|u|^{\alpha -2}u =|u|^{s-2} u,&x\in \mathbb{R }^{N},\\ u(x) \rightarrow 0, \quad \text{ as} |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where $N\ge 2, \phi (t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t, 1< p<q<N, 1<\alpha \le p^* q^{\prime }/p^{\prime }$ and $\max \{q,\alpha \}< s<p^*,$ being $p^*=\frac{pN}{N-p}$ and $p^{\prime }$ and $q^{\prime }$ the conjugate exponents, respectively, of $p$ and $q$ . Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.     

A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.     

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.     

The Arboreal Jump Number of an Order

Let ${\mathcal P}$ be a partial order and ${\mathcal A}$ an arboreal extension of it (i.e. the Hasse diagram of ${\mathcal A}$ is a rooted tree with a unique minimal element). A jump of ${\mathcal A}$ is a relation contained in the Hasse diagram of ${\mathcal A}$ , but not in the order ${\mathcal P}$ . The arboreal jump number of ${\mathcal A}$ is the number of jumps contained in it. We study the problem of finding the arboreal extension of ${\mathcal P}$ having minimum arboreal jump number—a problem related to the well-known (linear) jump number problem. We describe several results for this problem, including NP-completeness, polynomial time solvable cases and bounds. We also discuss the concept of a minimal arboreal extension, namely an arboreal extension whose removal of one jump makes it no longer arboreal.     

Prescribed diagonal Schouten tensor in locally conformally flat manifolds

We consider the pseudo-euclidean space ${(\mathbb{R}^n, g)}$ , with n ≥  3 and ${g_{ij} = \delta_{ij} \varepsilon_i, \varepsilon_i = \pm 1}$ and tensors of the form ${T = \sum \nolimits_i \varepsilon_i f_i (x) dx_i^2}$ . In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric ${\bar{g}}$ , conformal to g, so that ${A_{\bar g}=T}$ , where ${A_{\bar g}}$ is the Schouten Tensor of the metric ${\bar g}$ . The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric ${\bar g}$ is complete on ${\mathbb{R}^n}$ . Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics ${\bar g}$ , conformal to g, such that ${\sigma_2 ({\bar g })}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })}}$ is equal to a given function. We prove that for some functions, f ₁ and f ₂, there exist complete metrics ${\bar{g} = g/{\varphi^2}}$ , such that ${\sigma_2 ({\bar g }) = f_1}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })} = f_2}$ .     

Toric surfaces,K-stability and Calabi flow

Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$ . Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $ \int _{\partial P} u d \sigma < C_2, $ then there exists a constant $C_3$ depending only on $C_1, C_2$ and $P$ such that the diameter of $X$ is bounded by $C_3$ . Moreoever, we can show that there is a constant $M > 0$ depending only on $C_1, C_2$ and $P$ such that Donaldson’s $M$ -condition holds for $u$ . As an application, we show that if $(X,P)$ is (analytic) relative $K$ -stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.     

A Uniqueness Theorem for Bounded Analytic Functions on the Polydisc

Given n, N ≥ 1 we construct a set of points ${\lambda_1,{\ldots},\lambda_{N^n}\in{\mathbb D}^n}$ such that for each rational inner function f on ${{\mathbb D}^n}$ of degree less than N the Pick problem on ${{\mathbb D}^n}$ with data ${\lambda_1,{\ldots},\lambda_{N^n}}$ and ${f(\lambda_1),{\ldots},f(\lambda_{N^n})}$ has a unique solution. In particular, we construct a 1-dimensional inner variety V and show that the points ${\lambda_1,{\ldots},\lambda_{N^n}}$ may be chosen almost arbitrarily in ${V\cap{\mathbb D}^n}$ . Our results state that f is uniquely determined in the Schur class of ${{\mathbb D}^n}$ by its values on ${\lambda_1,{\ldots},\lambda_{N^n}}$ .     

Variance asymptotics for random polytopes in smooth convex bodies

Let $K \subset \mathbb R ^d$ be a smooth convex set and let $\mathcal{P }_{\lambda }$ be a Poisson point process on $\mathbb R ^d$ of intensity ${\lambda }$ . The convex hull of $\mathcal{P }_{\lambda }\cap K$ is a random convex polytope $K_{\lambda }$ . As ${\lambda }\rightarrow \infty $ , we show that the variance of the number of $k$ -dimensional faces of $K_{\lambda }$ , when properly scaled, converges to a scalar multiple of the affine surface area of $K$ . Similar asymptotics hold for the variance of the number of $k$ -dimensional faces for the convex hull of a binomial process in $K$ .     

A covering lemma for {K(\mathbb {R})}

The Dodd–Jensen Covering Lemma states that “if there is no inner model with a measurable cardinal, then for any uncountable set of ordinals X, there is a ${Y\in K}$ such that ${X\subseteq Y}$ and |X| = |Y|”. Assuming ZF+AD alone, we establish the following analog: If there is no inner model with an ${\mathbb {R}}$ –complete measurable cardinal, then the real core model ${K(\mathbb {R})}$ is a “very good approximation” to the universe of sets V; that is, ${K(\mathbb {R})}$ and V have exactly the same sets of reals and for any set of ordinals X with ${|{X}|\ge\Theta}$ , there is a ${Y\in K(\mathbb {R})}$ such that ${X\subseteq Y}$ and |X| = |Y|. Here ${\mathbb {R}}$ is the set of reals and ${\Theta}$ is the supremum of the ordinals which are the surjective image of ${\mathbb {R}}$ .     

Fixed point homomorphisms for parameterized maps

Let X be an ANR (absolute neighborhood retract), ${\Lambda}$ a k-dimensional topological manifold with topological orientation ${\eta}$ , and ${f : D \rightarrow X}$ a locally compact map, where D is an open subset of ${X \times \Lambda}$ . We define Fix(f) as the set of points ${{(x, \lambda) \in D}}$ such that ${x = f(x, \lambda)}$ . For an open pair (U, V) in ${X \times \Lambda}$ such that ${{\rm Fix}(f) \cap U \backslash V}$ is compact we construct a homomorphism ${\Sigma_{(f,U,V)} : H^{k}(U, V ) \rightarrow R}$ in the singular cohomologies H* over a ring-with-unit R, in such a way that the properties of Solvability, Excision and Naturality, Homotopy Invariance, Additivity, Multiplicativity, Normalization, Orientation Invariance, Commutativity, Contraction, Topological Invariance, and Ring Naturality hold. In the case of a ${C^{\infty}}$ -manifold ${\Lambda}$ , these properties uniquely determine ${\Sigma}$ . By passing to the direct limit of ${\Sigma_{(f,U,V)}}$ with respect to the pairs (U, V) such that ${K = {\rm Fix}(f) \cap U \backslash V}$ , we define a homomorphism ${\sigma_{(f,K)} : {H}_{k}({\rm Fix}(f), Fix(f) \backslash K) \rightarrow R}$ in the ?ech cohomologies. Properties of ${\Sigma}$ and ${\sigma}$ are equivalent each to the other. We indicate how the homomorphisms generalize the fixed point index.     

Optimality conditions for weak {\psi} -sharp minima in vector optimization problems

In this paper, we develop the characterization of weak ${\psi}$ -sharp minimizer by means of an oriented distance function and investigate the weak ${\psi}$ -sharp minimizer of the composition of two functions. Moreover, we establish sufficient conditions of the weak ${\psi_1}$ -sharp and ${\psi_2}$ -sharp Pareto minimality for vector optimization problem with strictly differentiable and twice strictly differentiable objective function, respectively. Our results extend the corresponding ones in the literature.     

Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.