Finite Set Of Integers Research Articles

Equidistribution of numerical semigroup gaps modulo m

For a positive integer m, a finite set of integers is said to be equidistributed modulo m if the set contains an equal number of elements in each congruence class modulo m. In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup S is equidistributed modulo m. Of particular interest is the case when the nonzero elements of an Apéry set of S form an arithmetic sequence. We explicitly describe such numerical semigroups S and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo m.

Generalizing the distribution of missing sums in sumsets

Given a finite set of integers A, its sumset is A+A:={ai+aj|ai,aj∈A}. We examine |A+A| as a random variable, where A⊂In=[0,n−1], the set of integers from 0 to n−1, so that each element of In is in A with a fixed probability p∈(0,1). Recently, Martin and O'Bryant studied the case in which p=1/2 and found a closed form for E[|A+A|]. Lazarev, Miller, and O'Bryant extended the result to find a numerical estimate for Var(|A+A|) and bounds on the number of missing sums in A+A, mn;p(k):=P(2n−1−|A+A|=k). Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for E[|A+A|] and Var(|A+A|) for all p∈(0,1) and establish good bounds for E[|A+A|] and mn;p(k). We continue to investigate mn;p(k) by studying mp(k)=limn→∞⁡mn;p(k), proven to exist by Zhao. Lazarev, Miller, and O'Bryant proved that, for p=1/2, m1/2(6)>m1/2(7)<m1/2(8). This distribution is not unimodal, and is said to have a “divot” at 7. We report results investigating this divot as p varies, and through both theoretical and numerical analysis, prove that for p≥0.68 there is a divot at 1; that is, mp(0)>mp(1)<mp(2). Finally, we extend the graph-theoretic framework originally introduced by Lazarev, Miller, and O'Bryant to correlated sumsets A+B where B is correlated to A by the probabilities P(i∈B|i∈A)=p1 and P(i∈B|i∉A)=p2. We provide some preliminary results using the extension of this framework.

When Does F(α d ) = F(α)?

Suppose that α is algebraic over a field . A standard exercise in a first course in field theory is to show that if the degree of the minimal polynomial of α over is odd, then . In this note, we generalize the sufficient conditions on the minimal polynomial for α so that for any particular integer . Then, given any finite set of integers , this generalization allows us to construct irreducible polynomials f, with , such that for all .

Determinants of Normalized Bohemian Upper Hessenberg Matrices

A matrix is Bohemian if its elements are taken from a finite set of integers. An upper Hessenberg matrix is normalized if all its subdiagonal elements are ones, and hollow if it has only zeros along the main diagonal. All possible determinants of families of normalized and hollow normalized Bohemian upper Hessenberg matrices are enumerated. It is shown that in the case of hollow matrices the maximal determinants are related to a generalization of Fibonacci numbers. Several conjectures recently stated by Corless and Thornton follow from these results.

The Erdős–Moser Sum-free Set Problem

AbstractWe show that there is an absolute $c&gt;0$ such that if $A$ is a finite set of integers, then there is a set $S\subset A$ of size at least $\log ^{1+c}|A|$ such that the restricted sumset $\{s+s^{\prime }:s,s^{\prime }\in S\text{ and }s\neq s^{\prime }\}$ is disjoint from $A$. (The logarithm here is to base $3$.)

Elliptic key cryptography with Beta Gamma functions for secure routing in wireless sensor networks

The security of data communicated through wireless networks is a challenging issue due to the presence of malicious and unauthenticated users whose intention is either to disrupt the communication or to know context of the data communicated to perform data theft and prevent the communication made by genuine users. To address this problem, a new secure routing algorithm with a novel encryption scheme is proposed in this paper using the cyclic group based Elliptic curve cryptography in which an encryption key is formed from the points taken from the Beta and Gamma functions in the elliptic key cryptography. For this purpose, the elliptic key is encrypted using the proposed encryption and decryption schemes. Further, it is proved that the proposed model decreases the computational complexity and increases the security, when it is compared to the other signature schemes. This scheme has been tested both analytically and using computer implementation of the proposed secure routing algorithm for wireless sensor networks. It is proved that this proposed scheme is more reliable in providing security than the existing algorithms. The cyclic group used in this work makes the number of elements to be from a finite set of integers and hence the computational complexity is reduced and finite. The experiments have been carried out using Python coding and by transmitting different number of words through the wireless network. It is found that overall complexity for encryption and communication is less in this proposed algorithm than the existing Elliptic Curve Cryptography algorithm.

On Freiman's Theorems concerning the sum of two finite sets of integers

On Freiman's Theorems concerning the sum of two finite sets of integers

Direct and inverse theorems on signed sumsets of integers

Let G be an additive abelian group and h be a positive integer. For a nonempty finite subset A={a0,a1,…,ak−1} of G, we leth+_A:={Σi=0k−1λiai:(λ0,…,λk−1)∈Zk,Σi=0k−1|λi|=h}, be the h-fold signed sumset of A.The direct problem for the signed sumset h+_A is to find a nontrivial lower bound for |h+_A| in terms of |A|. The inverse problem for h+_A is to determine the structure of the finite set A for which |h+_A| is minimal. In this article, we solve both the direct and inverse problems for |h+_A|, when A is a finite set of integers.

The Ablowitz–Ladik system on a finite set of integers

We show how to solve initial-boundary value problems for integrable nonlinear differential–difference equations on a finite set of integers. The method we employ is the discrete analogue of the unified transform (Fokas method). The implementation of this method to the Ablowitz–Ladik system yields the solution in terms of the unique solution of a matrix Riemann–Hilbert problem, which has a jump matrix with explicit -dependence involving certain functions referred to as spectral functions. Some of these functions are defined in terms of the initial value, while the remaining spectral functions are defined in terms of two sets of boundary values. These spectral functions are not independent but satisfy an algebraic relation called global relation. We analyze the global relation to characterize the unknown boundary values in terms of the given initial and boundary values. We also discuss the linearizable boundary conditions.

On the complexity of finding and counting solution-free sets of integers

Given a linear equation L, a set A of integers is L-free if A does not contain any ‘non-trivial’ solutions to L. This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving L-free sets of integers. The main questions we consider involve deciding whether a finite set of integers A has an L-free subset of a given size, and counting all such L-free subsets. We also raise a number of open problems.

Finding a Low-dimensional Piece of a Set of Integers

We show that a finite set of integers $A \subseteq \mathbb{Z}$ with $|A+A| \le K |A|$ contains a large piece $X \subseteq A$ with Freĭman dimension $O(\log K)$, where large means $|A|/|X| \ll \exp(O(\log^2 K))$. This can be thought of as a major quantitative improvement on Freĭman's dimension lemma, or as a weak Freĭman--Ruzsa theorem with almost polynomial bounds. The methods used, centered around an additive energy increment strategy, differ from the usual tools in this area and may have further potential. Most of our argument takes place over $\mathbb{F}_2^n$, which is itself curious. There is a possibility that the above bounds could be improved, assuming sufficiently strong results in the spirit of the Polynomial Freĭman--Ruzsa Conjecture over finite fields.

NXOR- or XOR-based robust template decomposition for cellular neural networks implementing an arbitrary Boolean function via support vector classifiers

Robust template design for cellular neural networks (CNNs) implementing an arbitrary Boolean function is currently an active research area. If the given Boolean function is linearly separable, a single robust uncoupled CNN can be designed preferably as a maximal margin classifier to implement the Boolean function. On the other hand, if the linearly separable Boolean function has a small geometric margin or the Boolean function is not linearly separable, a popular approach is to find a sequence of robust uncoupled CNNs implementing the given Boolean function. In the past research works using this approach, the control template parameters and thresholds are usually restricted to assume only a given finite set of integers. In this study, we try to remove this unnecessary restriction. NXOR- or XOR-based decomposition algorithm utilizing the soft margin and maximal margin support vector classifiers is proposed to design a sequence of robust templates implementing an arbitrary Boolean function. Several illustrative examples are simulated to demonstrate the efficiency of the proposed method by comparing our results with those produced by other decomposition methods with restricted weights.

I-Mark: A new subtraction division game

Given two finite sets of integers S⊆N∖{0} and D⊆N∖{0,1}, the impartial combinatorial game i-Mark(S,D) is played on a heap of tokens. From a heap of n tokens, each player can move either to a heap of n−s tokens for some s∈S, s≤n, or to a heap of n/d tokens for some d∈D if d divides n. Such games can be considered as an integral variant of Mark-type games, introduced by Elwyn Berlekamp and Joe Buhler and studied by Aviezri Fraenkel and Alan Guo, for which it is allowed to move from a heap of n tokens to a heap of ⌊n/d⌋ tokens for any d∈D.Under normal convention, it is observed that the Sprague–Grundy sequence of the game i-Mark(S,D) is aperiodic for any sets S and D. However, we prove that in many cases this sequence is almost periodic and that the sequence of winning positions is periodic. Moreover, in all these cases, the Sprague–Grundy value of a heap of n tokens can be computed in time O(log⁡n).We also prove that under misère convention the outcome sequence of these games is purely periodic.

Strongly restricted permutations and tiling with fences

We identify bijections between strongly restricted permutations of {1,2,…,n} of the form π(i)−i∈W, where W is any finite set of integers which is independent of i and n, and tilings of an n-board (a linear array of n square cells of unit width) using square tiles and (12,g)-fence tiles where g∈Z+. A (12,g)-fence is composed of two pieces of width 12 separated by a gap of width g. The tiling approach allows us to obtain the recurrence relation for the number of permutations when W={−1,d1,…,dr} where dr>0 and the remaining dl are non-negative integers which are independent of i and n. This is a generalization of a previous result. Terms in this recurrence relation, along with terms in other recurrences we obtain for more complicated cases, can be identified with certain groupings of interlocking tiles. The ease of counting tilings gives rise to a straightforward way of obtaining identities concerning the number of occurrences of patterns such as fixed points or excedances in restricted permutations. We also use the tilings to obtain the possible permutation cycles.

EZADS inputs which produce half-factorial block monoids

We study the factorization properties of block monoids $$\fancyscript{B}(\mathbb {Z}/q\mathbb {Z},S)$$ resulting from the EZADS construction given in Chapman and Smith (Period Math Hungar 64(2): 227–246, 2012). This construction, which is inspired by a paper of Erdos and Zaks (J Number Theory 36 (1): 89–94, 1990), takes a finite set of integers (called an EZADS input) and produces a weakly half-factorial set called an EZADS. We are primarily interested in determining which EZADS inputs will produce a half-factorial EZADS. In Sect. 3 we show that if an EZADS input produces a half-factorial EZADS, then so will any of its subsets. In Sect. 4 we derive a bound which significantly simplifies the problem of determining whether an EZADS is half-factorial. In Sect. 5, we show how this bound may be used to reformulate the problem in terms of continuous quantities, then in Sect. 6 we apply these ideas to study four-element EZADSs. We describe a finite algorithm which, for fixed $$m$$ , can be used to classify all half-factorial EZADSs in the form $$\{\overline{1},\overline{ab},\overline{mb},\overline{ma}\}\subseteq \mathbb {Z}/mab\mathbb {Z}$$ . Finally in Sect. 7 we show that a special sequence of EZADSs studied in Chapman and Smith (Period Math Hungar 64(2): 227–246, 2012) are always half-factorial. We close by describing possible extensions of our work and related conjectures.

On a Sumset Problem for Integers

Let $A$ be a finite set of integers. We show that if $k$ is a prime power or a product of two distinct primes then \[ |A+k\cdot A|\geq(k+1)|A|-\lceil k(k+2)/4\rceil \] provided $|A|\geq (k-1)^{2}k!$, where $A+k\cdot A=\{a+kb:\ a,b\in A\}$. We also establish the inequality $|A+4\cdot A|\geq5|A|-6 $ for $|A|\geq5$.

Density of primes in l-th power residues

Given a prime number l, a finite set of integers S = {a 1, ...,a m } and m many l-th roots of unity $\zeta_l^{r_i}, i=1, \ldots ,m$ we study the distribution of primes p in ℚ(ζ l ) such that the l-th residue symbol of a i with respect to p is $\zeta_l^{r_i}, \mbox{ for all } i$ . We find out that this is related to the degree of the extension $$\mathbb{Q}(a_1^{\frac{1}{l}}, \ldots ,a_m^{\frac{1}{l}})/\mathbb{Q}$$ . We give an algorithm to compute this degree. Also we relate this degree to rank of a matrix obtained from S = {a 1, ...,a m }. This latter argument enables one to describe the degree $$\mathbb{Q}(a_1^{\frac{1}{l}}, \ldots ,a_m^{\frac{1}{l}})/\mathbb{Q}$$ in much simpler terms.

The L1-norm of exponential sums in d

AbstractLet A be a finite set of integers and FA(x) = ∑a∈A exp(2πiax) be its exponential sum. McGehee, Pigno and Smith and Konyagin have independently proved that ∥FA∥1 ≥ c log|A| for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper we present lower bounds on the L1-norm of exponential sums of sets in the d-dimensional grid d. We show that ∥FA∥1 is considerably larger than log|A| when A ⊂ d has multidimensional structure. We furthermore prove similar lower bounds for sets in , which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno and Smith and Konyagin.

Robust Template Decomposition without Weight Restriction for Cellular Neural Networks Implementing Arbitrary Boolean Functions Using Support Vector Classifiers

If the given Boolean function is linearly separable, a robust uncoupled cellular neural network can be designed as a maximal margin classifier. On the other hand, if the given Boolean function is linearly separable but has a small geometric margin or it is not linearly separable, a popular approach is to find a sequence of robust uncoupled cellular neural networks implementing the given Boolean function. In the past research works using this approach, the control template parameters and thresholds are restricted to assume only a given finite set of integers, and this is certainly unnecessary for the template design. In this study, we try to remove this restriction. Minterm- and maxterm-based decomposition algorithms utilizing the soft margin and maximal margin support vector classifiers are proposed to design a sequence of robust templates implementing an arbitrary Boolean function. Several illustrative examples are simulated to demonstrate the efficiency of the proposed method by comparing our results with those produced by other decomposition methods with restricted weights.

Fraction Sets for Basic Digit Sets

A finite set of integers $D$ with $0\in D$ is basic for the base $b\in \mathbb Z$ if every $z\in\mathbb Z$ can be written uniquely as an integer in the base $b$ with digits from $D$. Since such a base representation is unsigned, basic sets must have a negative base $b$ or some negative integers in $D$. The fraction set for $(b,D)$ is the set of all representable numbers with integer part zero. We show that if $(b,D)$ is basic and $D$ consists of consecutive digits, then the fraction set is an interval of length one whose endpoints have redundant representations. Furthermore, we show that if $D$ does not consist of consecutive integers, then $F$ is disconnected.