The shape of $$ \mathbb {Z}/\ell \mathbb {Z}$$-number fields

Let \(\ell \) be a prime and let \(L/ \mathbb {Q}\) be a Galois number field with Galois group isomorphic to \( \mathbb {Z}/\ell \mathbb {Z}\). We show that the shape of L, see Definition 1.2, is either \(\frac{1}{2}\mathbb {A}_{\ell -1}\) or a fixed sub-lattice depending only on \(\ell \); such a dichotomy in the value of the shape only depends on the type of ramification of L. This work is motivated by a result of Bhargava and Shnidman, and a previous work of the first named author, on the shape of \( \mathbb {Z}/3 \mathbb {Z}\) number fields.     

Some results on linear codes over the ring $$\mathbb {Z}_4+u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4$$

In this paper, we mainly study the theory of linear codes over the ring \(R =\mathbb {Z}_4+u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4\). By using the Chinese Remainder Theorem, we prove that R is isomorphic to a direct sum of four rings. We define a Gray map \(\Phi \) from \(R^{n}\) to \(\mathbb {Z}_4^{4n}\), which is a distance preserving map. The Gray image of a cyclic code over R is a linear code over \(\mathbb {Z}_4\). We also discuss some properties of MDS codes over R. Furthermore, we study the MacWilliams identities of linear codes over R and give the generator polynomials of cyclic codes over R.     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).     

There is exactly one $${\mathbb {Z}}_2{\mathbb {Z}}_4$$-cyclic 1-perfect code

Let \(\mathcal{C}\) be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive code of length \(n > 3\). We prove that if the binary Gray image of \(\mathcal{C}\) is a 1-perfect nonlinear code, then \(\mathcal{C}\) cannot be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic code except for one case of length \(n=15\). Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive 1-perfect code gives a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive extended 1-perfect code. We also prove that such a code cannot be \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic.     

Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

In this article, we consider the following fractional Hamiltonian systems:

$$\begin{aligned} {_{t}}D_{\infty }^{\alpha }({_{-\infty }}D_{t}^{\alpha }u) + \lambda L(t)u = \nabla W(t, u), \;\;t\in \mathbb {R}, \end{aligned}$$

where \(\alpha \in (1/2, 1)\), \(\lambda >0\) is a parameter, \(L\in C(\mathbb {R}, \mathbb {R}^{n\times n})\) and \(W \in C^{1}(\mathbb {R} \times \mathbb {R}^n, \mathbb {R})\). Unlike most other papers on this problem, we require that L(t) is a positive semi-definite symmetric matrix for all \(t\in \mathbb {R}\), that is, \(L(t) \equiv 0\) is allowed to occur in some finite interval \(\mathbb {I}\) of \(\mathbb {R}\). Under some mild assumptions on W, we establish the existence of nontrivial weak solution, which vanish on \(\mathbb {R} \setminus \mathbb {I}\) as \(\lambda \rightarrow \infty ,\) and converge to \(\tilde{u}\) in \(H^{\alpha }(\mathbb {R})\); here \(\tilde{u} \in E_{0}^{\alpha }\) is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval \(\mathbb {I}\). Furthermore, we give the multiplicity results for the above fractional Hamiltonian systems.

     

Estimates for coefficients of certain <Emphasis Type="Italic">L</Emphasis>-functions

Let \(\pi _{\varphi }\) (or \(\pi _{\psi }\)) be an automorphic cuspidal representation of \(\text {GL}_{2} (\mathbb {A}_{\mathbb {Q}})\) associated to a primitive Maass cusp form \(\varphi \) (or \(\psi \)), and \(\mathrm{sym}^j \pi _{\varphi }\) be the jth symmetric power lift of \(\pi _{\varphi }\). Let \(a_{\mathrm{sym}^j \pi _{\varphi }}(n)\) denote the nth Dirichlet series coefficient of the principal L-function associated to \(\mathrm{sym}^j \pi _{\varphi }\). In this paper, we study first moments of Dirichlet series coefficients of automorphic representations \(\mathrm{sym}^3 \pi _{\varphi }\) of \(\text {GL}_{4}(\mathbb {A}_{\mathbb {Q}})\), and \(\pi _{\psi }\otimes \mathrm{sym}^2 \pi _{\varphi }\) of \(\text {GL}_{6}(\mathbb {A}_{\mathbb {Q}})\). For \(3 \le j \le 8\), estimates for \(|a_{\mathrm{sym}^j \pi _{\varphi }}(n)|\) on average over a short interval have also been established.     

On Weighted Average Interpolation with Cardinal Splines

Given a sequence of data \(\{ y_{n} \} _{n \in \mathbb{Z}}\) with polynomial growth and an odd number \(d\), Schoenberg proved that there exists a unique cardinal spline \(f\) of degree \(d\) with polynomial growth such that \(f ( n ) =y_{n}\) for all \(n\in \mathbb{Z}\). In this work, we show that this result also holds if we consider weighted average data \(f\ast h ( n ) =y_{n}\), whenever the average function \(h\) satisfies some light conditions. In particular, the interpolation result is valid if we consider cell-average data \(\int_{n-a}^{n+a}f ( x ) dx=y_{n}\) with \(0< a\leq 1/2\). The case of even degree \(d\) is also studied.     

Algebras of Convolution-Type Operators with Piecewise Slowly Oscillating Data on Weighted Lebesgue Spaces

Let \({\mathcal B}_{p,w}\) be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space \(L^p(\mathbb {R},w)\), where \(p\in (1,\infty )\) and w is a Muckenhoupt weight. We study the Banach subalgebra \(\mathfrak {A}_{p,w}\) of \({\mathcal B}_{p,w}\) generated by all multiplication operators aI (\(a\in \mathrm{PSO}^\diamond \)) and all convolution operators \(W^0(b)\) (\(b\in \mathrm{PSO}_{p,w}^\diamond \)), where \(\mathrm{PSO}^\diamond \subset L^\infty (\mathbb {R})\) and \(\mathrm{PSO}_{p,w}^\diamond \subset M_{p,w}\) are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of \(\mathbb {R}\cup \{\infty \}\), and \(M_{p,w}\) is the Banach algebra of Fourier multipliers on \(L^p(\mathbb {R},w)\). For any Muckenhoupt weight w, we study the Fredholmness in the Banach algebra \({\mathcal Z}_{p,w}\subset \mathfrak {A}_{p,w}\) generated by the operators \(aW^0(b)\) with slowly oscillating data \(a\in \mathrm{SO}^\diamond \) and \(b\in \mathrm{SO}^\diamond _{p,w}\). Then, under some condition on the weight w, we complete constructing a Fredholm symbol calculus for the Banach algebra \(\mathfrak {A}_{p,w}\) in comparison with Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 74:377–415, 2012) and Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 75:49–86, 2013) and establish a Fredholm criterion for the operators \(A\in \mathfrak {A}_{p,w}\) in terms of their symbols. A new approach to determine local spectra is found.     

Lattices from codes over $$\mathbb {Z}_q$$: generalization of constructions D, $$ D'$$ D′ and $$\overline{ D}$$ D¯

In this paper, we extend the lattice Constructions D, \(D'\) and \(\overline{D}\) (this latter is also known as Forney’s code formula) from codes over \(\mathbb {F}_p\) to linear codes over \(\mathbb {Z}_q\), where \(q \in \mathbb {N}\). We define an operation in \(\mathbb {Z}_q^n\) called zero-one addition, which coincides with the Schur product when restricted to \(\mathbb {Z}_2^n\) and show that the extended Construction \(\overline{D}\) produces a lattice if and only if the nested codes are closed under this addition. A generalization to the real case of the recently developed Construction \(A'\) is also derived and we show that this construction produces a lattice if and only if the corresponding code over \(\mathbb {Z}_q[X]/X^a\) is closed under a shifted zero-one addition. One of the motivations for this work is the recent use of q-ary lattices in cryptography.     

On Submanifolds with 2-Type Pseudo-Hyperbolic Gauss Map in Pseudo-Hyperbolic Space

In this paper, we study pseudo-Riemannian submanifolds of a pseudo-hyperbolic space \(\mathbb H^{m-1}_s (-1) \subset \mathbb E^m_{s+1}\) with 2-type pseudo-hyperbolic Gauss map. We give a characterization of proper pseudo-Riemannian hypersurfaces in \(\mathbb H^{n+1}_s (-1) \subset \mathbb E^{n+2}_{s+1}\) with non-zero constant mean curvature and 2-type pseudo-hyperbolic Gauss map. For \(n=2\), we prove classification theorems. In addition, we show that the hyperbolic Veronese surface is the only maximal surface fully lying in \(\mathbb H^4_2 (-1) \subset \mathbb H^{m-1}_2 (-1)\) with 2-type pseudo-hyperbolic Gauss map. Moreover, we prove that a flat totally umbilical pseudo-Riemannian hypersurface \(M^n_t\) of the pseudo-hyperbolic space \(\mathbb {H}^{n+1}_t(-1) \subset \mathbb E^{n+2}_{t+1}\) has biharmonic pseudo-hyperbolic Gauss map.     

Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation

We study the one-dimensional periodic derivative nonlinear Schrödinger equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion \({\textstyle \int }h_k\), \(k\in {\mathbb {Z}}_{+}\). In each \({\textstyle \int }h_{2k}\) the term with the highest regularity involves the Sobolev norm \(\dot{H}^{k}({\mathbb {T}})\) of the solution of the DNLS equation. We show that a functional measure on \(L^2({\mathbb {T}})\), absolutely continuous w.r.t. the Gaussian measure with covariance \(({\mathbb {I}}+(-\varDelta )^{k})^{-1}\), is associated to each integral of motion \({\textstyle \int }h_{2k}\), \(k\ge 1\).     

On limit theorems for Banach-space-valued linear processes

Let \( {\left( {{\epsilon_i}} \right)_{i \in \mathbb{Z}}} \) be i.i.d. random elements in a separable Banach space \( \mathbb{E} \), and let \( \mathop {\left( {{a_i}} \right)}\nolimits_{i \in \mathbb{Z}} \) be continuous linear operators from \( \mathbb{E} \) to a Banach space \( \mathbb{F} \) such that \( \sum\nolimits_{i \in \mathbb{Z}} {\left\| {{a_i}} \right\|} \) is finite. We prove that the linear process \( \mathop {\left( {{X_n}} \right)}\nolimits_{n \in \mathbb{Z}} \) defined by \( {X_n}: = \sum\nolimits_{i \in \mathbb{Z}} {{a_i}} \left( {{\epsilon_{n - i}}} \right) \) inherits from \( \mathop {\left( {{\epsilon_i}} \right)}\nolimits_{i \in \mathbb{Z}} \) the central limit theorem and functional central limit theorems in various Banach spaces of \( \mathbb{F} \)-valued functions, including Hölder spaces.     

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and

$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$

is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.

     

Map determined by rank-$$\varvec{}$$ matrice for relatively mall $$\varvec{}$$

Let n and s be integers such that \(1\le s<\frac{n}{2}\), and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\). Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\). In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\), then \(g(x)=f(x)+g(0)\), \(x\in M_n(\mathbb {K})\), for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\). Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \).     

One-sided invertibility of discrete operators and their applications

For \(p\in [1,\infty ]\), we establish criteria for the one-sided invertibility of binomial discrete difference operators \({{\mathcal {A}}}=aI-bV\) on the space \(l^p=l^p(\mathbb {Z})\), where \(a,b\in l^\infty \), I is the identity operator and the isometric shift operator V is given on functions \(f\in l^p\) by \((Vf)(n)=f(n+1)\) for all \(n\in \mathbb {Z}\). Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators \(A=aI-bU_\alpha \) on the Lebesgue space \(L^p(\mathbb {R}_+)\) for every \(p\in [1,\infty ]\), where \(a,b\in L^\infty (\mathbb {R}_+)\), \(\alpha \) is an orientation-preserving bi-Lipschitz homeomorphism of \([0,+\infty ]\) onto itself with only two fixed points 0 and \(\infty \), and \(U_\alpha \) is the isometric weighted shift operator on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f= (\alpha ^\prime )^{1/p}(f\circ \alpha )\). Applications of binomial discrete operators to interpolation theory are given.     

Ramanujan series for Epstein zeta functions

In the spirit of Ramanujan, we derive exponentially fast convergent series for Epstein zeta functions \(E^{\varGamma _0(N)}(z,s)\) on the Hecke congruence groups \( \varGamma _0(N),N\in \mathbb {Z}_{>0}\), where z is an arbitrary point in the upper half-plane \( \mathfrak {H}\) and \(s\in \mathbb {Z}_{>1}\). These Ramanujan series can be reformulated as integrations of modular forms, in the framework of Eichler integrals. Particular cases of these Eichler integrals recover part of the recent results reported by Wan and Zucker (arXiv:1410.7081v1, 2014).     

Symbol <Emphasis Type="Italic">p</Emphasis>-algebras of prime degree and their <Emphasis Type="Italic">p</Emphasis>-central subspaces

We prove that the maximal dimension of a p-central subspace of the generic symbol p-algebra of prime degree p is \({p+1}\). We do it by proving the following number theoretic fact: let \({\{s_1,\dots,s_{p+1}\}}\) be \({p+1}\) distinct nonzero elements in the additive group \({G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})}\), then every nonzero element \({g \in G}\) can be expressed as \({d_1 s_1+\dots+d_{p+1} s_{p+1}}\) for some non-negative integers \({d_1,\dots,d_{p+1}}\) with \({d_1+\dots+d_{p+1}\leq p-1}\).     

On kernels and nuclei of rank metric codes

For each rank metric code \(\mathcal {C}\subseteq \mathbb {K}^{m\times n}\), we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When \(\mathcal {C}\) is \(\mathbb {K}\)-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When \(\mathbb {K}\) is a finite field \(\mathbb {F}_q\) and \(\mathcal {C}\) is a maximum rank distance code with minimum distance \(d<\min \{m,n\}\) or \(\gcd (m,n)=1\), the kernel of the associated translation structure is proved to be \(\mathbb {F}_q\). Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over \(\mathbb {F}_q\) must be a finite field; its right nucleus also has to be a finite field under the condition \(\max \{d,m-d+2\} \geqslant \left\lfloor \frac{n}{2} \right\rfloor +1\). Let \(\mathcal {D}\) be the DHO-set associated with a bilinear dimensional dual hyperoval over \(\mathbb {F}_2\). The set \(\mathcal {D}\) gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to \(\mathbb {F}_2\). Also, its middle nucleus must be a finite field containing \(\mathbb {F}_q\). Moreover, we also consider the kernel and the nuclei of \(\mathcal {D}^k\) where k is a Knuth operation.     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).     

A Spanning Set and Potential Basis of the Mixed Hecke Algebra on Two Fixed Strands

The mixed braid groups \(B_{2,n}, \ n \in \mathbb {N}\), with two fixed strands and n moving ones, are known to be related to the knot theory of certain families of 3-manifolds. In this paper, we define the mixed Hecke algebra \(\mathrm {H}_{2,n}(q)\) as the quotient of the group algebra \({\mathbb Z}\, [q^{\pm 1}] \, B_{2,n}\) over the quadratic relations of the classical Iwahori–Hecke algebra for the braiding generators. We further provide a potential basis \(\Lambda _n\) for \(\mathrm {H}_{2,n}(q)\), which we prove is a spanning set for the \(\mathbb {Z}[q^{\pm 1}]\)-additive structure of this algebra. The sets \(\Lambda _n,\ n \in \mathbb {Z}\) appear to be good candidates for an inductive basis suitable for the construction of Homflypt-type invariants for knots and links in the above 3-manifolds.