On linear functional equations modulo $${\mathbb {Z}}$$ Z

Aequationes mathematicae - In this paper, we consider the condition $$\sum _{i=0}^{n+1}\varphi _i(r_ix+q_iy)\in {\mathbb {Z}}$$ for real valued functions defined on a linear space V. We derive...     

Modular independence and generator matrices for codes over $${\mathbb {Z}_m}$$

We introduce a new notion of modular independence to define bases and the generator matrices for the codes over the ring of integers of general modulus m. We define standard forms for such generator matrices, and discuss how to find such forms and the parity check matrices.      

On the maximal unramified pro-2-extension of certain cyclotomic $$\mathbb {Z}_2$$ Z 2 -extensions

Periodica Mathematica Hungarica - In this paper, we establish a necessary and sufficient criterion for a finite metabelian 2-group G whose abelianized $$G^{ab}$$ is of type $$(2, 2^m)$$ , with...     

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...     

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).     

Type II Codes over $$\mathbb{Z}/2k\mathbb{Z}$$, Invariant Rings and Theta Series

We consider type II codes over finite rings . It is well-known that their gth complete weight enumerator polynomials are invariant under the action of a certain finite subgroup of , which we denote H_k,g. We show that the invariant ring with respect to H_k,g is generated by such polynomials. This is carried out by using some closely related results concerning theta series and Siegel modular forms with respect to .     

On the linearity and classification of $${\mathbb {Z}}_{p^s}$$ Z p s -linear generalized hadamard codes

Designs, Codes and Cryptography - $${\mathbb {Z}}_{p^s}$$ -additive codes of length n are subgroups of $${\mathbb {Z}}_{p^s}^n$$ , and can be seen as a generalization of linear codes over...     

On a problem of partitions of $$\mathbb {Z}_{m}$$ Z m with the same representation functions

Periodica Mathematica Hungarica - For any positive integer m, let $$\mathbb {Z}_{m}$$ be the set of residue classes modulo m. For $$A\subseteq \mathbb {Z}_{m}$$ and $$\overline{n}\in \mathbb...     

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     

On the Enumeration of Labeled Series-Parallel $$k $$ -Cyclic $$2 $$ -Connected Graphs

Journal of Applied and Industrial Mathematics - We deduce an explicit formula for the number of labeled series-parallel $$k $$ -cyclic $$n $$ -vertex $$2 $$ -connected graphs and find the...     

Actions of SL $$(n,\mathbb{Z})$$ on Homology Spheres

Any continuous action of SL , where n > 2, on a r-dimensional mod 2 homology sphere factors through a finite group action if r < n −1. In particular, any continuous action of SL on the n-dimensional sphere factors through a finite group action.Mathematics Subject Classiffications (2000). Primary 57S25; Secondary 37C85, 57S17     

Pro- $$\mathcal {C}$$ C congruence properties for groups of rooted tree automorphisms

We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro- $$\mathcal {C}$$ completions of the group, where $$\mathcal {C}$$ is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the $$\mathcal {C}$$ -congruence subgroup property ( $$\mathcal {C}$$ -CSP) if its pro- $$\mathcal {C}$$ completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the $$\mathcal {C}$$ -CSP. In the case where $$\mathcal {C}$$ is also closed under extensions (for instance the class of all finite p-groups for some prime p), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the p-CSP.     

Ramanujan-type $$1/\pi $$ -series from bimodular forms

The Ramanujan Journal - We develop an approach to establish $$1/\pi $$ -series from bimodular forms. Utilizing this approach, we obtain new families of 2-variable $$1/\pi $$ -series associated to...     

$${\mathbb{Z}}_2^k$$ -Manifolds are isospectral on forms

We obtain a simple formula for the multiplicity of eigenvalues of the Hodge-Laplace operator, Δ _f , acting on sections of the full exterior bundle over an arbitrary compact flat Riemannian n-manifold M with holonomy group , 1 ≤ k ≤ n − 1. This formula implies that any two such manifolds having isospectral lattices of translations are isospectral with respect to Δ _f . As a consequence, we construct a large family of pairwise Δ _f -isospectral and nonhomeomorphic n-manifolds of cardinality greater than . Supported by Conicet, Secyt-UNC.     

$${\cal K}$$-Convexity1 in $$\Re^{n}$$

We generalize the concept of K-convexity to an n-dimensional Euclidean space. The resulting concept of -convexity is useful in addressing production and inventory problems where there are individual product setup costs and/or joint setup costs. We derive some basic properties of -convex functions. We conclude the paper with some suggestions for future research. Support from Columbia University and University of Texas at Dallas is gratefully acknowledged. Helpful comments from Qi Feng are appreciated.     

Actions of the groups $${\mathbb C}$$ and $${\mathbb C^*}$$ on Stein varieties

In this paper we present recent results concerning global aspects of and -actions on Stein surfaces. Our approach is based on a byproduct of techniques from Geometric Theory of Foliations (holonomy, stability), Potential theory (parabolic Riemann surfaces, Riemann-Koebe Uniformization theorem) and Several Complex Variables (Hartogs’ extension theorems, Theory of Stein spaces). Our main motivation comes from the original works of M. Suzuki and Orlik-Wagreich. Some of their results are extended to a more general framework. In particular, we prove some linearization theorems for holomorphic actions of and on normal Stein analytic spaces of dimension two. We also add a list of questions and open problems in the subject. The underlying idea is to present the state of the art of this research field.      

Klingen $${\mathfrak {p}}^2$$ p 2 vectors for $$\mathrm{GSp}(4)$$ GSp ( 4 )

The Ramanujan Journal - We calculate the dimensions of the spaces of invariant vectors under the Klingen congruence subgroup of level $${\mathfrak {p}}^2$$ for all irreducible, admissible...     

CR-Admissible $$\mathbb{{Z}}_{2}$$-gradations and CR-symmetries

Complementing the results of (Lotta and Nacinovich, Adv. Math. 191(1): 114–146, 2005), we show that the minimal orbit M of a real form G of a semisimple complex Lie group in a flag manifold is CR-symmetric (see (Kaup and Zaitsev Adv. Math. 149(2):145–181, 2000)) if and only if the corresponding CR algebra admits a gradation compatible with the CR structure.      

Infinite number of eigenvalues of $$2{\times}2$$ operator matrices: Asymptotic discrete spectrum

Theoretical and Mathematical Physics - We study an unbounded $$2{\times}2$$ operator matrix $$ \mathcal{A} $$ in the direct product of two Hilbert spaces. We obtain asymptotic formulas for the...     

Multiplicative updates for convolutional NMF under $$\beta $$ β -divergence

Optimization Letters - We generalize the convolutional NMF by taking the $$\beta $$ -divergence as the contrast function and present the exact multiplicative updates for its factors in closed form....