On generalized perfect difference sets constructed from Sidon sets

We examine the ideals of nowhere dense sets in three topologies on the set of positive integers, namely Furstenberg’s, Rizza’s and the common division topology. We mainly concentrate on inclusions between these ideals, we present a diagram showing these and we explore all possible inclusions between them. We present a formula for the closure of a set in the common division topology. We answer a question posed by Kwela and Nowik (Topol Appl. 248:149–163, 2018) by constructing a set in {{mathcal {I}}}_G {setminus } ({{mathcal {I}}}_K cup {{mathcal {I}}}_F). Therefore, the main diagram of comparison between the ideals of nowhere dense sets in various topologies from the article by M. Kwela and A. Nowik is completed.

The graphs \(\mathcal{G}_F=\{(x,F(x)); x\in \mathbb {F}_2^n\}\) of those (n, n)-functions \(F:\mathbb {F}_2^n\mapsto \mathbb {F}_2^n\) that are almost perfect nonlinear (in brief, APN; an important notion in symmetric cryptography) are, equivalently to their original definition by K. Nyberg, those Sidon sets (an important notion in combinatorics) S in \(({\mathbb {F}}_2^n\times {\mathbb {F}}_2^n,+)\) such that, for every \(x\in {\mathbb {F}}_2^n\), there exists a unique \(y\in {\mathbb {F}}_2^n\) such that \((x,y)\in S\). Any subset of a Sidon set being a Sidon set, an important question is to determine which Sidon sets are maximal relatively to the order of inclusion. In this paper, we study whether the graphs of APN functions are maximal (that is, optimal) Sidon sets. We show that this question is related to the problem of the existence/non-existence of pairs of APN functions lying at distance 1 from each others, and to the related problem of the existence of APN functions of algebraic degree n. We revisit the conjectures that have been made on these latter problems.KeywordsAlmost perfect nonlinear functionSidon set in an Abelian groupSymmetric cryptography

For h ge 2 and an infinite set of positive integers A, let R_{A,h}(n) denote the number of representations of the positive integer n as the sum of h distinct terms from A. A set of positive integers A is called a B_h[g] set if every positive integer can be written as the sum of h not necessarily distinct terms from A at most g different ways. We say a set A is a basis of order h if every positive integer can be represented as the sum of h terms from A. Recently, Vu [17] proved the existence of a thin basis of order h formed by perfect powers. In this paper, we study weak B_{h}[g] sets formed by perfect powers. In particular, we prove the existence of a set A formed by perfect powers with almost possible maximal density such that R_{A,h}(n) is bounded by using probabilistic methods.

This theorem proves the equivalence of two conjectures on perfect Singer difference sets, one of which is concerned with the actual computation of the set, and the other being of interest in the theory of multipliers and fixed sets.

In view of the problems that the encoding complexity of quasi-cyclic low-density parity-check (QC-LDPC) codes is high and the minimum distance is not large enough which leads to the degradation of the error-correction performance, the new irregular type-II QC-LDPC codes based on perfect cyclic difference sets (CDSs) are constructed. The parity check matrices of these type-II QC-LDPC codes consist of the zero matrices with weight of 0, the circulant permutation matrices (CPMs) with weight of 1 and the circulant matrices with weight of 2 (W2CMs). The introduction of W2CMs in parity check matrices makes it possible to achieve the larger minimum distance which can improve the error- correction performance of the codes. The Tanner graphs of these codes have no girth-4, thus they have the excellent decoding convergence characteristics. In addition, because the parity check matrices have the quasi-dual diagonal structure, the fast encoding algorithm can reduce the encoding complexity effectively. Simulation results show that the new type-II QC-LDPC codes can achieve a more excellent error-correction performance and have no error floor phenomenon over the additive white Gaussian noise (AWGN) channel with sum-product algorithm (SPA) iterative decoding.

In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

In this paper we study the semigroup $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also we prove that every Baire topology $\tau$ on $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N}),\tau)$ is a semitopological semigroup is discrete, describe the closure of $(\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N}),\tau)$ in a topological semigroup and construct non-discrete Hausdorff semigroup topologies on $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$.

Network-on-chip architecture is a new paradigm shift for designing IP core based system on chip and also referred as network based communication subsystem which is recently looked as an innovative approach to provide a highly scalable, high computational and communication performance. Energy consumption of network based communication subsystems is becoming the valuable parameter in the design of system which further needs to be optimized. In the recent development of IP core architecture, it is necessary to propose new approach for design methodologies to minimize the communication energy for network based communication subsystems. We have addressed the Rectangular Perfect Difference Network topology for network based communication subsystems for providing optimum bandwidth utilization with lesser number of routing hops and at the most two hops in the communication to achieve the best energy performance. In this paper, we propose Rectangular PDN topology for network based communication subsystems for minimization of communication energy using the mathematical representation of Perfect Difference Set (PDS). We have proposed the analytical model with lower energy consumption for chordal Ring Perfect Difference Network Topology and Rectangular Perfect Difference Network Topology. The proposed analytical model for network based communication subsystems using Perfect Difference Network topology results is simulated and validated for different Network topology having order of n=7. The link energy model and router energy model are validated against simulation results for Rectangular PDN topology of network based communication subsystems. The overall average energy consumption for transfer of data through router from one IP to another IP for Rectangular PDN Topology for network n=7 for perfect difference set of {0, 1, 3} having order δ=2; is compared with overall average energy consumption for 2X2 CLICHÉ architecture

Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are infinitely many elements x in X with d_A(e,x)=h. It is proved that if P is a nonempty finite set of prime numbers and A is the set of positive integers whose prime factors all belong to P, then the diameter of the metric space (\Z,d_A) is infinite. Let \lambda_A(h) denote the smallest positive integer x with d_A(e,x)=h. It is an open problem to compute \lambda_A(h) and estimate its growth rate.

On a Class of Enumeration Problems in Additive Arithmetics

Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form [Formula: see text] Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M = 32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these could be proved using the theory of modular functions. Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form [Formula: see text] In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

(1) Bd(E) and B(E) are isometrically isomorphic for finite E ⊂ Γ. Bd(E) = `∞(E) characterizes I0 sets E and B(E) = `∞(E) characterizes Sidon sets E. [In general, Sidon sets are distinct from I0 sets. Within the group of integers Z, the set {2}n ⋃ {2+n}n is helsonian (hence Sidon) but not I0.] (2) Both are Fσ in 2 (as is also the class of finite unions of I0 sets). (3) There is an analogue for I0 sets of the sup-norm partition construction used with Sidon sets. (4) A set E is Sidon if and only if, there is some r ∈ R and positive integer N such that, for all finite F ⊂ E, there is some H ⊂ F with |H| ≥ r|F | and H is an I0 set of degree at most N . [Here |S| denotes the cardinality of S; two different but comparable definitions of degree for I0 sets are made below.] (5) IF all Sidon subsets of Z are finite unions of I0 sets, the number of I0 sets required is bounded by some function of the Sidon constant. This is also true in the category of all discrete abelian groups.

Let IN be the set of positive integers, $${\cal B}$$ = {b1 < ⋅s < bh}⊂IN, N∊IN and N≥ bh. $${\cal A}$$ = $${\cal A}$$ 0( $${\cal B}$$ ,N) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sarkozy) such that $${\cal A}$$ {1,..., N} = $${\cal B}$$ and p( $${\cal A}$$ ,n)≡ 0 (mod 2) for n∊IN and n > N, where p( $${\cal A}$$ ,n) denotes the number of partitions of n with parts in $${\cal A}$$ . Let us denote by σ ( $${\cal A}$$ ,n) the sum of the divisors of n belonging to $${\cal A}$$ . In a paper jointly written with J.-L. Nicolas, we have recently proved that, for all k≥ 0, the sequence (σ( $${\cal A}$$ ,2k n))n≥ 1mod 2k+1 is periodic with an odd period qk. In this paper, we will characterize for any fixed odd positive integer q, the sets $${\cal B}$$ and the integers N such that q0 = q, and those for which qk = q for all k≥ 0. Moreover, a set $${\cal A}$$ = $${\cal A}$$ 0( $${\cal B}$$ ,N) is constructed with the property that its period, i.e. the period of (σ( $${\cal A}$$ ,n))n≥ 1mod 2, is 217, and for which the counting function is asymptotically equal to that of $${\cal A}$$ 0({1,2,3,4,5},5) which is a set of period 31.

Let $(u_n)_{n \geq 0}$ be a nondegenerate linear recurrence of integers, and let $\mathcal{A}$ be the set of positive integers $n$ such that $u_n$ and $n$ are relatively prime. We prove that $\mathcal{A}$ has an asymptotic density, and that this density is positive unless $(u_n / n)_{n \geq 1}$ is a linear recurrence.

The principal result of this paper is the introduction of a readily verifiable sufficient condition for the existence of an invariant measure on transient Markov chains. Specifically, subject to mild additional regularity conditions, it is enough to check that for every compact set $A$ in the state space $X$, there exists a compact $C \supset A$ such that \begin{equation*}\tag{0.1} \lim_{y\in F, y\rightarrow\infty} \frac{P(y, A)}{P(y, C)} = 0\quad\text{if} \bar{F} \text{is not compact and} F = \{y: P(y, A) > 0\},\end{equation*} $P(y, A)$ denote the 1-step transition probabilities of the Markov chain. (0.1) is readily implied by simple conditions on the generating functional of the offspring distribution (Theorems 2.1 and 2.2) for discrete time multi-type Markov processes. These conditions together with some additional regularity-hypotheses are verified for (i) A 1-dimensional neutron branching model of Harris, (ii) Discrete-time age dependent branching processes, (iii) Galton-Watson processes with immigration. A generalization (to discrete-time temporally homogeneous Markov processes with $\sigma$-compact metric state space) of a condition of T. Harris (1957) for existence of invariant measures on transient Markov chains is also given. This condition is, unfortunately, difficult to check in specific examples. The sufficient condition (0.1) involves only 1-step transition probabilities, as opposed to the $n$-step transitions incorporated in the Harris condition, which is also necessary. Throughout this paper, all subsets of a topological space considered are assumed Borel measurable. The closure, interior, boundary and complement of a set $A$ are denoted by $\bar{A}, A^0, \partial A$ and $A^c$ respectively, and $I_A$ denotes the indicator function of $A$. The sets of positive integers, nonnegative integers, are denoted by $I$ and $I_0$ respectively. For any $\sigma$-compact metric space $X$, the following notations are used: $M(X) =$ set of regular measures (Borel measures finite on compact sets) on $X$. $B(X) =$ set of bounded measurable functions on $X$ with sup norm. $B'(X) = \{s \in B(X): \|s\| < 1\}$. $C(X) =$ set of bounded continuous functions on $X$. $C_{00}(X) = \{s\in C(X): s$ has compact support$\}$. $C_0(X) = \{s \in C(X):\lim_{x\rightarrow\infty}s(x) = 0\}.$ $C'(X) = C(X) \cap B'(X)$, similarly for $C_0'(X)$ and $C'_{00}(X)$. For any $\mathscr{L} \subset B(X), \mathscr{L}^+ = \{s \in \mathscr{L}: s \geqq 0\}$. If $\mu \in M(X)$ and $\in B(X)$, then $\mu f = \int_xf(x)\mu(dx)$ when defined.