$${{\mathbb {Z}}}_2$$-double cyclic codes

A binary linear code C is a \({\mathbb {Z}}_2\)-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the \({\mathbb {Z}}_2[x]\)-module \({\mathbb {Z}}_2[x]/(x^r-1)\times {\mathbb {Z}}_2[x]/(x^s-1).\) We determine the structure of \({\mathbb {Z}}_2\)-double cyclic codes giving the generator polynomials of these codes. We give the polynomial representation of \({\mathbb {Z}}_2\)-double cyclic codes and its duals, and the relations between the generator polynomials of these codes. Finally, we study the relations between \({{\mathbb {Z}}}_2\)-double cyclic and other families of cyclic codes, and show some examples of distance optimal \({\mathbb {Z}}_2\)-double cyclic codes.     

Inverse Spectral Problem for Analytic {{(\mathbb{Z}/2 \mathbb{Z})^{n}}} -Symmetric Domains in {{\mathbb{R}^{n}}}

We prove that bounded real analytic domains in ${\mathbb{R}^{n}}$ , with the symmetries of an ellipsoid and with one axis length fixed, are determined by their Dirichlet or Neumann eigenvalues among other bounded real analytic domains with the same symmetries and axis length. Some non-degeneracy conditions are also imposed on the class of domains. It follows that bounded, convex analytic domains are determined by their spectra among other such domains. This seems to be the first positive result for the well-known Kac problem, “Can one hear the shape of a drum?”, in higher dimensions.     

New complete embedded minimal surfaces in $${{\mathbb {H} ^2\times \mathbb {R}}}$$

We construct three kinds of complete embedded minimal surfaces in \({\mathbb {H}^2\times \mathbb {R}}\) . The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These two are conjugate surfaces just as the helicoid and the catenoid are in \({\mathbb {R}^3}\) . The third one is a finite total curvature surface which is conformal to \({\mathbb {S}^2\setminus\{p_1,\ldots,p_k\}, k\geq3.}\)     

Explicit div-curl inequalities in bounded and unbounded domains of $${{\mathbb {R}}}^3$$

We give a constructive proof of some functional inequalities related to the div and curl operators in bounded and unbounded domains of \({{\mathbb {R}}}^3\). Our new innovation consists in giving explicit constants in several geometric configurations. These inequalities are of a first use in solving div-curl systems and vector potential problems arising in physics.     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

Trajectory of an Observer Tracking Object Motion around Convex Obstacles in $${{\mathbb{R}}^{2}}$$ and $${{\mathbb{R}}^{3}}$$

Doklady Mathematics - An autonomous object moves at a constant speed along the shortest path, while bypassing an ordered collection of pairwise disjoint convex sets. The object is tracked by an...     

Density of translates in weighted $${{\varvec{L}}}^{{\varvec{p}}}$$ spaces on locally compact groups

Let G be a locally compact group, and let \(1\leqslant p < \infty \). Consider the weighted \(L^p\)-space \(L^p(G,\omega )=\{f:\int |f\omega |^p<\infty \}\), where \(\omega :G\rightarrow \mathbb {R}\) is a positive measurable function. Under appropriate conditions on \(\omega \), G acts on \(L^p(G,\omega )\) by translations. When is this action hypercyclic, that is, there is a function in this space such that the set of all its translations is dense in \(L^p(G,\omega )\)? Salas (Trans Am Math Soc 347:993–1004, 1995) gave a criterion of hypercyclicity in the case \(G=\mathbb {Z}\). Under mild assumptions, we present a corresponding characterization for a general locally compact group G. Our results are obtained in a more general setting when the translations only by a subset \(S\subset G\) are considered.     

The finite Littlewood problem in $${{\mathbb {F}}}_p$$

Let p be a large prime number and f(x) be an integer-valued function defined in \({\mathbb F}_p\). The Littlewood problem in \({{\mathbb {F}}}_p\) is to establish non-trivial lower bounds for the \(\ell _1\) norm of exponential sums involving f(x). In the present paper, we establish new lower bounds for exponential sums including polynomials, powers of any primitive root and subgroups of \(\mathbb {F}_p^*.\)     

Type $${{\varvec{D}}}_{{\varvec{n}}}^\mathbf{(1)}$$ rigged configuration bijection

We establish a bijection between the set of rigged configurations and the set of tensor products of Kirillov–Reshetikhin crystals of type \(D^{(1)}_n\) in full generality. We prove the invariance of rigged configurations under the action of the combinatorial R-matrix on tensor products and show that the bijection preserves certain statistics (cocharge and energy). As a result, we establish the fermionic formula for type \(D_n^{(1)}\). In addition, we establish that the bijection is a classical crystal isomorphism.     

Discrete Orbits,Recurrence and Solvable Subgroups of $${{\mathrm{Diff}}\, ({{\mathbb {C}}}^2, 0)}$$

We discuss the local dynamics of a subgroup of \({{\mathrm{Diff}}\, ({{\mathbb {C}}}^2, 0)}\) possessing locally discrete orbits as well as the structure of the recurrent set for more general groups. It is proved, in particular, that a subgroup of \({{\mathrm{Diff}}\, ({{\mathbb {C}}}^2, 0)}\) possessing locally discrete orbits must be virtually solvable. These results are of considerable interest in problems concerning integrable systems.     

Growth in solvable subgroups of {{\mathrm{GL}}}_r({\mathbb {Z}}/p{\mathbb {Z}})

Let \(K={\mathbb {Z}}/p{\mathbb {Z}}\) and let \(A\) be a subset of \({{\mathrm{GL}}}_r(K)\) such that \(\langle A \rangle \) is solvable. We reduce the study of the growth of \(A\) under the group operation to the nilpotent setting. Fix a positive number \(C\ge 1\) ; we prove that either \(A\) grows (meaning \(|A_3|\ge C|A|\) ), or else there are groups \(U_R\) and \(S\) , with \(U_R\unlhd S \unlhd \langle A\rangle \) , such that \(S/U_R\) is nilpotent, \(A_k\cap S\) is large and \(U_R\subseteq A_k\) , where \(k\) depends only on the rank \(r\) of \({{\mathrm{GL}}}_r(K)\) . Here \(A_k = \{x_1 x_2 \cdots x_k : x_i \in A \cup A^{-1} \cup \{1\}\}\) . When combined with recent work by Pyber and Szabó, the main result of this paper implies that it is possible to draw the same conclusions without supposing that \(\langle A \rangle \) is solvable.     

On Linear Codes over $${\mathbb{Z}}_{2^{s}}$$

In an earlier paper the authors studied simplex codes of type α and β over and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over The above codes are also shown to satisfy the chain condition.A part of this paper is contained in his Ph.D. Thesis from IIT Kanpur, India     

Markov chains on $${{\mathbb {Z}}^+}$$ Z + : analysis of stationary measure via harmonic functions approach

Queueing Systems - We suggest a method for constructing a positive harmonic function for a wide class of transition kernels on $${{\mathbb {Z}}^+}$$ . We also find natural conditions under which...     

Problem of Safely Tracking an Object Avoiding Observation in $${{\mathbb{R}}^{2}}$$

Doklady Mathematics - For the problem of an autonomous object moving under hostile observation, the observer’s positions are characterized in which the object following any route can choose a...     

Scattering of wave maps from {{\mathbb {R}^{2+1}}} to general targets

We show that smooth, radially symmetric wave maps U from ${\mathbb {R}^{2+1}}$ to a compact target manifold (N, where ? _r U and ? _t U have compact support for any fixed time, scatter. The result will follow from the work of Christodoulou and Tahvildar-Zadeh, and Struwe, upon proving that for ${(\lambda^{\prime} \in (0,1),}$ energy does not concentrate in the set $$K_{\frac{5}{8}T,\frac{7}{8}T}^{\lambda^{\prime}} = \{(x,t) \in \mathbb{R}^{2+1} \vert \quad|x| \leq \lambda^{\prime} t, t \in [(5/8)T,(7/8)T] \}.$$      

Boundedness of Singular Integrals on Multiparameter Weighted Hardy Spaces \text{{\textit{H}}}^\text{{\textit{p}}}_{\text{{\textit{w}}}}\ (\mathbb{R}^{\text{{\textit{n}}}}\times \mathbb{R}^{\text{{\textit{m}}}})

We apply the discrete version of Calderón??s identity and Littlewood?CPaley?CStein theory with weights to derive the $(H^p_w, H^p_w)$ and $(H^p_w, L^p_w) (0<p\le 1)$ boundedness for multiparameter singular integral operators. It turns out that even in the one-parameter case, our results substantially improve the known ones in the literature where w????A ₁ was needed. Our results in the multiparameter setting can be regarded as a natural extension of $L^p_w$ boundedness for p?>?1 for w????A _p to the case of weighted Hardy spaces $H^p_w$ for p????1, but under a weaker assumption that w belongs to the class of product A _???? weights with respect to rectangles in product spaces.