Hypersurfaces in nearly Kaehler manifold $$\pmb {\mathbb {S}}^{\varvec{3}}\varvec{\times } \pmb {\mathbb {S}}^{\varvec{3}}$$

In this paper, we initiate the study of contact and minimal hypersurfaces in nearly Kaehler manifold \({\mathbb {S}}^3\times {\mathbb {S}}^3\) with a conformal vector field. There are three almost contact metric structures on a hypersurface of \({\mathbb {S}}^3\times {\mathbb {S}}^3\), and we will give some important properties of them. Besides, we study the influence of the conformal vector field on the almost contact metric structures and use it to characterize the hypersurfaces in \({\mathbb {S}}^3\times {\mathbb {S}}^3\).     

Some aspects of shift-like automorphisms of $$\pmb {\mathbb {C}}^{\varvec{k}}$$

The goal of this article is two fold. First, using transcendental shift-like automorphisms of \(\mathbb C^k, ~k \ge 3\) we construct two examples of non-degenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of \(\mathbb C^k, ~k \ge 3\) whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of Dixon–Esterle in \(\mathbb C^2.\) The second example shows the existence of a Fatou–Bieberbach domain in \(\mathbb C^k,~k \ge 3\) that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay–Rudin. In the second part we compute the order and type of entire mappings that parametrize one dimensional unstable manifolds for shift-like polynomial automorphisms and show how they can be used to prove a Yoccoz type inequality for this class of automorphisms.     

Decay rates for the magneto-micropolar system in $$\varvec{L^2(}\pmb {\mathbb {R}}^{\varvec{n)}}$$

In this paper, the large time decay of the magneto-micropolar fluid equations on \(\mathbb {R}^n\) (\( n=2,3\)) is studied. We show, for Leray global solutions, that \( \Vert ({\varvec{u}},{\varvec{w}},{\varvec{b}})(\cdot ,t) \Vert _{{L^2(\mathbb {R}^n)}} \rightarrow 0 \) as \(t \rightarrow \infty \) with arbitrary initial data in \( L^2(\mathbb {R}^n)\). When the vortex viscosity is present, we obtain a (faster) decay for the micro-rotational field: \( \Vert {\varvec{w}}(\cdot ,t) \Vert _{{L^2(\mathbb {R}^n)}} = o(t^{-1/2})\). Some related results are also included.     

Some infinite families of Ramsey ($${\varvec{P}}_\mathbf{3},{\varvec{P}}_{{\varvec{n}}}$$)-minimal trees

For any given two graphs G and H, the notation \(F\rightarrow \) (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if \(F\rightarrow \) (G, H) but \(F-e\nrightarrow (G,H)\), for every \(e \in E(F)\). The class of all Ramsey (G, H)-minimal graphs is denoted by \(\mathcal {R}(G,H)\). In this paper, we construct some infinite families of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n=8\) and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n\ge 10\).     

On sums of polynomial-type exceptional units in $$\varvec{\mathbb {Z}}/\varvec{n\mathbb {Z}}$$Z/nZ

A unit u in a commutative ring with unity R is called exceptional if $$u-1$$ is also a unit. We introduce the notion of a polynomial version of this (abbreviated as $$f\hbox {-exunits}$$) for any $$f(X) \in \mathbb {Z}[X]$$. We find the number of representations of a non-zero element of $$\mathbb {Z}/n\mathbb {Z}$$ as a sum of two f-exunits for an infinite family of polynomials f of each degree $$\ge 1$$. We also derive the exact formulae for certain infinite families of linear and quadratic polynomials. This generalizes a result proved by Sander (J Number Theory 159:1–6, 2016).     

On unit power integral bases of $$\mathbb{Z}\left[ {\sqrt[4]{m}} \right]$$

Let m ≠ 0 be an integer which is not a perfect square and consider number fields of the form \(\mathbb{Q}\left[ {\sqrt[4]{m}} \right]\). We characterize all orders of the form \(\mathbb{Z}\left[ {\sqrt[4]{m}} \right]\) which admit a unit power integral basis, i.e., there exists a unit ε such that 1, ε, ε ² and ε ³ is an integral basis of \(\mathbb{Z}\left[ {\sqrt[4]{m}} \right]\).     

Hyperelliptic parametrizations of $$\pmb {\mathbb {Q}}$$ Q -curves

The Ramanujan Journal - In this paper, our aim is to find the radii of starlikeness and convexity of the Ramanujan type function for three different kinds of normalization by using their...     

The sum of digits of primes in $${\mathbb{Z}}$$[i]

We study the distribution of the complex sum-of-digits function s _q with basis q = –a±i, \({a \in \mathbb{Z}^+}\) for Gaussian primes p. Inspired by a recent result of Mauduit and Rivat (http://iml.univ-mrs.fr/~rivat/publications.html) for the real sum-of-digits function, we here get uniform distribution modulo 1 of the sequence (αs _q (p)) provided \({\alpha \in \mathbb{R} \setminus \mathbb{Q}}\) and q is prime with a ≥ 28. We also determine the order of magnitude of the number of Gaussian primes whose sum-of-digits evaluation lies in some fixed residue class mod m.     

A Difference Set in \left( {\mathbb{Z}/4\mathbb{Z}} \right)^3 \times \mathbb{Z}/5\mathbb{Z}

We present a (320, 88, 24)-difference set in , the existence of which was previously open. This new difference set improves a theorem of Davis-Jedwab with the removal of the exceptional case. It also enables us to state a theorem of Schmidt on Davis-Jedwab difference sets more neatly.     

Diophantine quadruples in $$\mathbb {Z}[i][X]$$ Z [ i ] [ X ]

Periodica Mathematica Hungarica - In this paper, we prove that every Diophantine quadruple in $$\mathbb {Z}[i][X]$$ is regular. More precisely, we prove that if $$\{a, b, c, d\}$$ is a set of four...     

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

In this work, we focus on cyclic codes over the ring \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (\mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂)/(xⁿ-1){({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} in general.     

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.     

Recognition of PSL($$\mathbf {2, 2}^{{\varvec{a}}}$$) by the orders of vanishing elements

Here, we show that the simple groups PSL\((2, 2^a)\), \(a\ge 2\), are characterized by the orders of vanishing elements.     

Convergence of polyharmonic splines on semi-regular grids \mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}} for {\boldsymbol{a}\rightarrow\mathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ?, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.     

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     

Invariant surfaces with constant mean curvature in $${\mathbb{H}}^2 \times {\mathbb{R}}$$

We classify the profile curves of all surfaces with constant mean curvature in the product space , which are invariant under the action of a 1-parameter subgroup of isometries. The author was supported by INdAM (Italy) and Fapesp (Brazil).     

A Construction of Partial Difference Sets in {\mathbb{Z}}_{p^{2 \times } } {\mathbb{Z}}_{p^{2 \times \cdot \cdot \cdot \times } } {\mathbb{Z}}_{p^2 }

In this paper, we give a construction of partial difference sets in _p ² x _p ² x ... x _p ²using some finite local rings.Dedicated to Hanfried Lenz on the occasion of his 80th birthdayThe work of this paper was done when the authors visited the University of Hong Kong.     

Some permutations over $$\pmb {\mathbb {F}}_p$$ F p concerning primitive roots

The Ramanujan Journal - Let p be an odd prime and let $${\mathbb {F}}_p$$ denote the finite field with p elements. Suppose that g is a primitive root of $${\mathbb {F}}_p$$ . Define the permutation...