Linear recurring sequences and subfield subcodes of cyclic codes

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .     

A unified approach for minimizing composite norms

We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem $$\begin{aligned} \begin{array}{ll} \min \limits _{X\in \mathbb{R }^{m\times n}}&\mu _1\Vert \sigma (\mathcal{F }(X)-G)\Vert _\alpha +\mu _2\Vert \mathcal{C }(X)-d\Vert _\beta ,\\ \text{ subject} \text{ to}&\mathcal{A }(X)-b\in \mathcal{Q }, \end{array} \end{aligned}$$ where $\sigma (X)$ denotes the vector of singular values of $X \in \mathbb{R }^{m\times n}$ , the matrix norm $\Vert \sigma (X)\Vert _{\alpha }$ denotes either the Frobenius, the nuclear, or the $\ell _2$ -operator norm of $X$ , the vector norm $\Vert .\Vert _{\beta }$ denotes either the $\ell _1$ -norm, $\ell _2$ -norm or the $\ell _{\infty }$ -norm; $\mathcal{Q }$ is a closed convex set and $\mathcal{A }(.)$ , $\mathcal{C }(.)$ , $\mathcal{F }(.)$ are linear operators from $\mathbb{R }^{m\times n}$ to vector spaces of appropriate dimensions. Basis pursuit, matrix completion, robust principal component pursuit (PCP), and stable PCP problems are all special cases of the composite norm minimization problem. Thus, FALC is able to solve all these problems in a unified manner. We show that any limit point of FALC iterate sequence is an optimal solution of the composite norm minimization problem. We also show that for all $\epsilon >0$ , the FALC iterates are $\epsilon $ -feasible and $\epsilon $ -optimal after $\mathcal{O }(\log (\epsilon ^{-1}))$ iterations, which require $\mathcal{O }(\epsilon ^{-1})$ constrained shrinkage operations and Euclidean projection onto the set $\mathcal{Q }$ . Surprisingly, on the problem sets we tested, FALC required only $\mathcal{O }(\log (\epsilon ^{-1}))$ constrained shrinkage, instead of the $\mathcal{O }(\epsilon ^{-1})$ worst case bound, to compute an $\epsilon $ -feasible and $\epsilon $ -optimal solution. To best of our knowledge, FALC is the first algorithm with a known complexity bound that solves the stable PCP problem.     

Planar Beurling Transform and Bergman Type Spaces

In this paper we describe the actions of the operator $S_\mathbb{D }$ or its adjoint $S_\mathbb{D }^*$ on the poly-Bergman spaces of the unit disk $\mathbb{D }.$ Let $k$ and $j$ be positive integers. We prove that $(S_\mathbb{D })^{j}$ is an isometric isomorphism between the true poly-Bergman subspace $\mathcal{A }_{(k)}^2(\mathbb{D })\ominus N_{(k),j}$ onto the true poly-Bergman space $\mathcal{A }_{(j+k)}^2(\mathbb{D }),$ where the linear space $N_{(k),j}$ have finite dimension $j.$ The action of $(S_\mathbb{D })^{j-1}$ on the canonical Hilbert base for the Bergman subspace $\mathcal{A }^2(\mathbb{D })\ominus \mathcal{P }_{j-1},$ gives a Hilbert base $\{ \phi _{ j , k } \}_{ k }$ for $\mathcal{A }_{(j)}^2(\mathbb{D }).$ It is shown that $\{ \phi _{ j , k } \}_{ j, k }$ is a Hilbert base for $L^2(\mathbb{D },d A)$ such that whenever $j$ and $k$ remain constant we obtain a Hilbert base for the true poly-Bergman space $\mathcal{A }_{(j)}^2(\mathbb{D })$ and $\mathcal{A }_{(-k)}^2(\mathbb{D }),$ respectively. The functions $\phi _{ j , k }$ are polynomials in $z$ and $\overline{z}$ and are explicitly given in terms of the $(2,1)$ -hypergeometric polynomials. We prove explicit representations for the true poly-Bergman kernels and the Koshelev representation for the poly-Bergman kernels of $\mathbb{D }.$ The action of $S_\Pi $ on the true poly-Bergman spaces of the upper half-plane $\Pi $ allows one to introduce Hilbert bases for the true poly-Bergman spaces, and to give explicit representations of the true poly-Bergman and poly-Bergman kernels.     

On the boundary of the attainable set of the dirichlet spectrum

Denoting by ${\varepsilon\subseteq\mathbb{R}^2}$ the set of the pairs ${(\lambda_1(\Omega),\,\lambda_2(\Omega))}$ for all the open sets ${\Omega\subseteq\mathbb{R}^N}$ with unit measure, and by ${\Theta\subseteq\mathbb{R}^N}$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that ${\partial\varepsilon}$ has horizontal tangent at its lowest point ${(\lambda_1(\Theta),\,\lambda_2(\Theta))}$ .     

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 ${A \in \mathfrak{A}_{p,w}}$ .     

Postulation of Disjoint Unions of Lines and Double Lines in {\mathbb{P}^3}

A double line ${C \subset \mathbb{P}^3}$ is a connected divisor of type (2, 0) on a smooth quadric surface. Fix ${(a, c) \in \mathbb{N}^2\ \backslash\ \{(0, 0)\}}$ . Let ${X \subset \mathbb{P}^3}$ be a general disjoint union of a lines and c double lines. Then X has maximal rank, i.e. for each ${t \in \mathbb{Z}}$ either ${h^1(\mathcal{I}_X(t)) = 0}$ or ${h^0(\mathcal{I}_X(t)) = 0}$ .     

Ranks of backward shift invariant subspaces of the Hardy space over the bidisk

Let $\{\varphi _n(z)\}_{n\ge 0}$ be a sequence of inner functions satisfying that $\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)$ for every $n\ge 0$ and $\{\varphi _n(z)\}_{n\ge 0}$ has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace $\mathcal{M }$ of $H^2(\mathbb{D }^2)$ . The ranks of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }_z$ and $\mathcal{F }^*_z$ respectively are determined, where $\mathcal{F }_z$ is the fringe operator on $\mathcal{M }\ominus w\mathcal{M }$ . Let $\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }$ . It is also proved that the rank of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }^*_z$ equals to the rank of $\mathcal{N }$ for $T^*_z$ and $T^*_w$ .     

Extending structures for Lie algebras

Let $\mathfrak{g }$ be a Lie algebra, $E$ a vector space containing $\mathfrak{g }$ as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on $E$ such that $\mathfrak{g }$ is a Lie subalgebra of $E$ . A general product, called the unified product, is introduced as a tool for our approach. Let $V$ be a complement of $\mathfrak{g }$ in $E$ : the unified product $\mathfrak{g } \,\natural \, V$ is associated to a system $(\triangleleft , \, \triangleright , \, f, \{-, \, -\})$ consisting of two actions $\triangleleft $ and $\triangleright $ , a generalized cocycle $f$ and a twisted Jacobi bracket $\{-, \, -\}$ on $V$ . There exists a Lie algebra structure $[-,-]$ on $E$ containing $\mathfrak{g }$ as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras $(E, [-,-]) \cong \mathfrak{g } \,\natural \, V$ . All such Lie algebra structures on $E$ are classified by two cohomological type objects which are explicitly constructed. The first one $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ will classify all Lie algebra structures on $E$ up to an isomorphism that stabilizes $\mathfrak{g }$ while the second object $\mathcal{H }^{2} (V, \mathfrak{g })$ provides the classification from the view point of the extension problem. Several examples that compute both classifying objects $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ and $\mathcal{H }^{2} (V, \mathfrak{g })$ are worked out in detail in the case of flag extending structures.     

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

Bochner–Riesz Means with Respect to a Generalized Cylinder

We study the L ^p boundedness of the generalized Bochner–Riesz means S ^λ which are defined as $$S^{\lambda}f(x) = \mathcal{F}^{-1} \left[\left(1 - \rho \right)_{+}^{\lambda} \widehat{f} \right](x)$$ where ${\rho(\xi) = {\rm max}\{|\xi_{1}|, \ldots, |\xi_{\ell}|\}}$ for ${\xi = (\xi_{1},\ldots, \xi_{\ell}) \in \mathbb{R}^{{d}_{1}} \times \cdots \times \mathbb{R}^{{d}_{\ell}}}$ and ${\mathcal{F}^{-1}}$ is the inverse Fourier transform.     

New open-book decompositions in singularity theory

In this article, we study the topology of real analytic germs ${F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)}$ given by ${F(x,y,z)=\overline{xy}(x^p+y^q)+z^r}$ with ${p,q,r \in \mathbb{N}, p,q,r \geq 2}$ and (p, q)?=?1. Such a germ gives rise to a Milnor fibration ${\frac{F}{\mid F \mid}\colon \mathbb{S}^5\setminus L_F \to \mathbb{S}^1}$ . We describe the link L _F as a Seifert manifold and we show that in many cases the open-book decomposition of ${\mathbb{S}^5}$ given by the Milnor fibration of F cannot come from the Milnor fibration of a complex singularity in ${\mathbb{C}^3}$ .     

2-Nilpotent real section conjecture

We show a $2$ -nilpotent section conjecture over $\mathbb{R }$ : for a geometrically connected curve $X$ over $\mathbb{R }$ such that each irreducible component of its normalization has $\mathbb{R }$ -points, $\pi _0(X(\mathbb{R }))$ is determined by the maximal $2$ -nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that for $X$ smooth and proper, $X(\mathbb{R })^{\pm }$ is determined by the maximal $2$ -nilpotent quotient of $\mathrm{Gal }(\mathbb{C }(X))$ with its $\mathrm{Gal }(\mathbb{R })$ action, where $X(\mathbb{R })^{\pm }$ denotes the set of real points equipped with a real tangent direction, showing a $2$ -nilpotent birational real section conjecture.     

The modal logic of {\beta(\mathbb{N})}

Let ${\beta(\mathbb{N})}$ denote the Stone–?ech compactification of the set ${\mathbb{N}}$ of natural numbers (with the discrete topology), and let ${\mathbb{N}^\ast}$ denote the remainder ${\beta(\mathbb{N})-\mathbb{N}}$ . We show that, interpreting modal diamond as the closure in a topological space, the modal logic of ${\mathbb{N}^\ast}$ is S4 and that the modal logic of ${\beta(\mathbb{N})}$ is S4.1.2.     

On p-tuples of the Grassmann manifolds

We provide a matrix invariant for isometry classes of p-tuples of points in the Grassmann manifold ${G_{n}\left(\mathbb{K}^{d}\right) }$ ( ${\mathbb{K=\mathbb{R}}}$ or ${\mathbb{C}}$ ). This invariant fully characterizes the p-tuple. We use it to classify the regular p-tuples of ${G_{2}\left(\mathbb{R}^{d}\right) }$ , ${G_{3}\left( \mathbb{R}^{d}\right) }$ and ${G_{2}\left( \mathbb{C}^{d}\right) }$ .     

Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H _∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ .     

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 ${\mathbb {R}}$ .     

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$ with a general function ${\varphi(x, r)}$ defining the Morrey-type norm. Here ${\Pi \subseteq \Omega}$ is an arbitrary subset in Ω including the extremal cases ${\Pi = \{x_0\}, x_0 \in \Omega}$ and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$ -theorem for the potential operator I ^α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${\varphi(x, r)}$ . No monotonicity type condition is imposed on ${\varphi(x, r)}$ . In case ${\varphi}$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ${\varphi}$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces