Maps from M2({\mathbb{F}}) to M3({\mathbb{F}}) that are multiplicative with respect to the Jordan triple product

We characterize the Jordan triple product preserving maps M₂( ${\mathbb{F}}$ ) to M₃( ${\mathbb{F}}$ ) for algebraically closed fields ${\mathbb{F}}$ , char ${\mathbb{F} \neq 2}$ .     

Permanent Versus Determinant over a Finite Field

Let $ \mathbb{F} $ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and for the whole matrix space M _n ( $ \mathbb{F} $ ). It is known that for n = 2, there are bijective linear maps Φ on $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, ?), where Φ is an arbitrary bijective map on matrices and ? : $ \mathbb{F} $ → $ \mathbb{F} $ is an arbitrary map such that per A = ?(det Φ(A)) for all matrices A from the spaces $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ), respectively. Moreover, for the space M _n ( $ \mathbb{F} $ ), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $ \mathbb{F} $ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.     

About the stability of the tangent bundle of {\mathbb{P}^n} restricted to a surface

Let X be a smooth projective surface over ${\mathbb{C}}$ and let L be a line bundle on X generated by its global sections. Let ${\phi _L:X\longrightarrow{{\mathbb{P}}^r}}$ be the morphism associated to L; we investigate the μ?stability of ${\phi _L^*T_{{\mathbb{P}}^r}}$ with respect to L when X is either a regular surface with p _g  = 0, a K3 surface or an abelian surface. In particular, we show that ${\phi_L^*T_{{\mathbb{P}}^r}}$ is μ? stable when X is K3 and L is ample and when X is abelian and L ² ≥ 14.     

Representation of Nelson algebras by rough sets determined by quasiorders

In this paper, we show that every quasiorder R induces a Nelson algebra ${{\mathbb R}{\mathbb S}}$ such that the underlying rough set lattice RS is algebraic. We note that ${{\mathbb R}{\mathbb S}}$ is a three-valued ?ukasiewicz algebra if and only if R is an equivalence. Our main result says that if ${{\mathbb A}}$ is a Nelson algebra defined on an algebraic lattice, then there exists a set U and a quasiorder R on U such that ${{\mathbb A} \cong {\mathbb R}{\mathbb S}}$ .     

On a problem concerning the Banach algebra generated by the maps n\mapsto\lambda^{\binom{n}{k}}

In Jabbari and Namioka (Milan J. Math. 78:503?C522, 2010), the authors characterized the spectrum M(W) of the Weyl algebra W, i.e. the norm closure of the algebra generated by the family of functions $\{n\mapsto x^{n^{k}}; x\in\mathbb{T}, k\in\mathbb{N}\}$ , ( $\mathbb{T}$ the unit circle), with a closed subgroup of $E(\mathbb{T})^{\mathbb{N}}$ where $E(\mathbb{T})$ denotes the family of the endomorphisms of the multiplicative group $\mathbb{T}$ . But the size of M(W) in $E(\mathbb{T})^{\mathbb{N}}$ as well as the induced group operation were left as a problem. In this paper, we will give a solution to this problem.     

Central limit theorems for moving average processes*

Let $ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $ be a stationary sequence of real random variables with E ξ ₀ = 0 and infinite variance. Furthermore, assume that $ {{\left( {{c_n}} \right)}_{{n\in \mathbb{Z}}}} $ is a sequence of real numbers and $ {X_n}=\sum {_{{j\in \mathbb{Z}}}{c_j}{\xi_{n-j }}} $ is a moving average processes driven by $ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $ . By using a decomposition of the moving average processes, a central limit theorem for the partial sums $ \sum\nolimits_{k=1}^n {{X_k}} $ is established. As applications, we obtain some central limit theorems for stationary dependent sequences $ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $ , such as associated sequence, martingale difference, and so on.     

The subvariety of commutative residuated lattices represented by twist-products

Given an integral commutative residuated lattice L, the product L × L can be endowed with the structure of a commutative residuated lattice with involution that we call a twist-product. In the present paper, we study the subvariety ${\mathbb{K}}$ of commutative residuated lattices that can be represented by twist-products. We give an equational characterization of ${\mathbb{K}}$ , a categorical interpretation of the relation among the algebraic categories of commutative integral residuated lattices and the elements in ${\mathbb{K}}$ , and we analyze the subvariety of representable algebras in ${\mathbb{K}}$ . Finally, we consider some specific class of bounded integral commutative residuated lattices ${\mathbb{G}}$ , and for each fixed element ${{\bf L} \in \mathbb{G}}$ , we characterize the subalgebras of the twist-product whose negative cone is L in terms of some lattice filters of L, generalizing a result by Odintsov for generalized Heyting algebras.     

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 ${\mathbb{C}}$ .     

Constant Angle Property and Canonical Principal Directions for Surfaces in {\mathbb{M}^{2}(c) \times {\mathbb{R}_{1}}}

In this paper, we study surfaces in Lorentzian product spaces ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ . We classify constant angle spacelike and timelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ . Moreover, complete classifications of spacelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ and timelike surfaces in ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ with a canonical principal direction are obtained. Finally, a new characterization of the catenoid of the 3rd kind is established, as the only minimal timelike surface with a canonical principal direction in Minkowski 3–space.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

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 ${\lambda_1,{\ldots},\lambda_{N^n}}$ .     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Sets Computing the Symmetric Tensor Rank

Let ${\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}$ denote the degree d Veronese embedding of ${\mathbb{P}^{r}}$ . For any ${P\, \in \, \mathbb{P}^{N}}$ , the symmetric tensor rank sr(P) is the minimal cardinality of a set ${\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}$ spanning P. Let ${\mathcal{S}(P)}$ be the set of all ${A \subset \mathbb{P}^{r}}$ such that ${\nu_{d}(A)}$ computes sr(P). Here we classify all ${P \,\in\, \mathbb{P}^{n}}$ such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of ${\nu_{d}(\mathbb{P}^{r})}$ . For such tensors ${P\, \in\, \mathbb{P}^{N}}$ , we prove that ${\mathcal{S}(P)}$ has no isolated points.     

Composed products and factors of cyclotomic polynomials over finite fields

Let q = p ^s be a power of a prime number p and let ${\mathbb {F}_{\rm q}}$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over ${\mathbb {F}_{\rm q}}$ . In particular we obtain the explicit factorization of the cyclotomic polynomial ${\Phi_{2^nr}}$ over ${\mathbb {F}_{\rm q}}$ where both r ≥ 3 and q are odd, gcd(q, r) = 1, and ${n\in \mathbb{N}}$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of ${\Phi_r}$ over ${\mathbb {F}_{\rm q}}$ is given to us as a known. Let ${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$ be the factorization of ${n \in \mathbb{N}}$ into powers of distinct primes p _i , 1 ≤ i ≤ s. In the case that the multiplicative orders of q modulo all these prime powers ${p_i^{e_i}}$ are pairwise coprime, we show how to obtain the explicit factors of ${\Phi_{n}}$ from the factors of each ${\Phi_{p_i^{e_i}}}$ . We also demonstrate how to obtain the factorization of ${\Phi_{mn}}$ from the factorization of ${\Phi_n}$ when q is a primitive root modulo m and ${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$ Here ${\phi}$ is the Euler’s totient function, and ord _n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over ${\mathbb {F}_{\rm q}}$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334–336, 1984).     

Line transversals to blown up closed balls

The number ?? _d (k) is defined as the minimum ???>?0 such that the following holds: For any finite family ${\mathcal {F}=\{B_1,B_2, \ldots , B_n\}}$ of closed balls in ${{\mathbb{R}}^d}$ such that every k elements of ${\mathcal {F}}$ have a common line transversal, the elements of the blown up family ${\lambda\mathcal {F}=\{\lambda B_1,\lambda B_2, \ldots , \lambda B_n\}}$ have a common line transversal. In this paper we show that ${\lambda_d(d+1)\leq4, \lambda_2(4)\leq 2\sqrt 2}$ and ??₂(3)?<?2.88.     

Mass Transport and Uniform Rectifiability

In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance W ₂ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance W ₂ which asserts that if??? and ?? are probability measures in ${{\mathbb{R}^n}}$ , ${{\varphi}}$ is a radial bump function smooth enough so that ${{\int \varphi d \mu \gtrsim 1}}$ , and??? has a density bounded from above and from below on supp( ${{\varphi}}$ ), then ${{W_2(\varphi \mu, a\varphi \nu) \leq cW_2(\mu, \nu)}}$ , where ${{a = \int \varphi d\mu/ \int \varphi d\nu}}$ .     

Actions infinitésimales dans la correspondance de Langlands locale p-adique

Let V be a two-dimensional absolutely irreducible ${\overline{\mathbb Qp}}$ -representation of ${{\rm Gal}(\overline{\mathbb Qp}/\mathbb Qp)}$ and let ${\prod(V)}$ be the ${{\rm GL}_2(\mathbb Qp)}$ Banach representation associated by Colmez??s p-adic Langlands correspondence. We establish a link between the action of the Lie algebra of ${{\rm GL}_2(\mathbb Qp)}$ on the locally analytic vectors ${\prod(V)^{\rm an}}$ of ${\prod(V)}$ , the connection ${\nabla}$ on the ${(\varphi, \Gamma)}$ -module associated to V and the Sen polynomial of V. This answers a question of Harris, concerning the infinitesimal character of ${\prod(V)^{\rm an}}$ . Using this result, we give a new proof of a theorem of Colmez, stating that ${\prod(V)}$ has nonzero locally algebraic vectors if and only if V is potentially semi-stable with distinct Hodge?CTate weights.     

{SL(2, \mathbb{R})} -invariant probability measures on the moduli spaces of translation surfaces are regular

In the moduli space ${{\mathcal {H}}_g}$ of normalized translation surfaces of genus g, consider, for a small parameter ρ > 0, those translation surfaces which have two non-parallel saddle-connections of length ? ρ. We prove that this subset of ${{\mathcal {H}}_g}$ has measure o(ρ ²) w.r.t. any probability measure on ${{\mathcal {H}}_g}$ which is invariant under the natural action of ${SL(2,\mathbb{R})}$ . This implies that any such probability measure is regular, a property which is important in relation with the recent fundamental work of Eskin–Kontsevich–Zorich on the Lyapunov exponents of the KZ-cocycle.     

The Algebra of {{\mathbf{SL}}_3}\left( \mathbb{C} \right) Conformal Blocks

We construct and study a family of toric degenerations of the Cox ring of the moduli of quasi-parabolic principal SL₃( $ \mathbb{C} $ ) bundles on a smooth, marked curve (C, $ \vec{p} $ ): Elements of this algebra have a well known interpretation as conformal blocks, from the Wess-Zumino-Witten model of conformal field theory. For the genus 0; 1 cases we find the level of conformal blocks necessary to generate the algebra. In the genus 0 case we also find bounds on the degrees of relations required to present the algebra. As a consequence we obtain a toric degeneration for the projective coordinate ring of an effective divisor on the moduli $ {{\mathcal{M}}_{{C,\vec{p}}}}\left( {\mathrm{S}{{\mathrm{L}}_3}\left( \mathbb{C} \right)} \right) $ of quasi-parabolic principal SL₃( $ \mathbb{C} $ ) bundles on (C, $ \vec{p} $ ). Along the way we recover positive polyhedral rules for counting conformal blocks.