132 Results Found

Secret key exchange is a necessary function for modern cryptography. The integer partition function is a mathematical function that arises from number theory. New methods for computing the integer partition function were developed and evaluated in th...

After conducting deep research on all existing reversible data hiding (RDH) methods based on Alattar’s integer transform, we discover that the frequently-used method in obtaining the difference value list of an image block may lead to high embe...

In this work we present strategies for parallelization of algorithms for representing integers as a sum of positive integers using OpenMP (Open Multi-Processing). We consider different types of algorithms—a non-recursive algorithm, a recursive...

We study the set of integer partitions as a probability space that generates distributions and, in the asymptotic limit, obeys thermodynamics. We view ordered integer partition as a configuration of cluster masses and associate them with the distribu...

Let $V$ be a finite set of positive integers with sum equal to a multiple of the integer $b$ . When does $V$ have a partition into $b$ parts so that all parts have equal sums? We develop algorithmic constructions which yield positive,...

Brauer configuration algebras are path algebras induced by appropriated multiset systems. Since their structures underlie combinatorial data, the general description of some of their algebraic invariants (e.g., their dimensions or the dimensions of t...

An even Eisenstein integer is a multiple of an Eisenstein prime of the least norm. Otherwise, an Eisenstein integer is called odd. An Eisenstein integer that is not an integer multiple of another one is said to be primitive. Such integers can be used...

Seaweeds or seaweed Lie algebras are subalgebras of the full-matrix algebra $\mathrm{Mat}\left(n\right)$ introduced by Dergachev and Kirillov to give an example of algebras for which it is possible to compute the Dixmier index via combinatorial methods. It is worth noting...

A diffusion-taking value in probability-measures on a graph with vertex set V, ${\sum }_{i\in V}{x}_i{\delta }_i$ is studied. The masses on each vertex satisfy the stochastic differential equation of the form $d{x}_i={\sum }_{j\in N\left(i\right)}\sqrt{x_i{x}_j}d{B}_{ij}$ on the simplex, where $\{$...

A heterogeneous system-on-chip (SoC) integrates multiple types of processors on the same chip. It has great advantages in many aspects, such as processing capacity, size, weight, cost, power, and energy consumption, which result in it being widely ad...

Three optimization problems based on node-colored undirected graphs are the subject of the present study. These problems model real-world applications in several domains, such as cybersecurity, bioinformatics, and social networks, although they have...

In this work, alternative product formulations are represented as a bipartite graph, and mixed-integer programming is used to find a graph partitioning that identifies the most dissimilar formulations within a potentially large design space. This inv...

We propose a linear time algorithm, called G2DLP, for generating 2D lattice L(n₁, n₂) paths, equivalent to two-item multiset permutations, with a given number of turns. The usage of turn has three me...

Confidentiality and efficient bandwidth utilization require a combination of compression and encryption of digital images. In this paper, a new method for joint image compression and encryption based on set partitioning in hierarchical trees (SPIHT)...

A novel approach for a robust computation of positional relations of two-dimensional geometric features is presented which guarantees reliable results, provided that the initial data is valid. The method is based on the use of integer coordinates and...

The Dedekind $\eta$-function plays an important role in number theory, particularly in the study of modular forms, q-series, and partition identities. In this paper, we investigate several level-12 $\eta$-function identities and examine their combinato...

Theta function identities play an important role in the theory of modular forms and partitions, with Ramanujan’s q-series providing a fundamental analytical framework. This work derives several new identities of theta functions at level 16 by e...

The paper is a reflection on the author’s work with students enrolled in undergraduate and graduate elementary mathematics content and methods courses. Two specific pedagogical issues make up the focus of the paper. The first issue deals with demonst...

The domatic partition problem consists of partitioning a given graph into a maximum number of disjoint dominating sets. This problem is related with the domatic number problem, which consists of quantifying this maximum number of disjoint dominating...

In this paper, we present a fast and effective method for Bidimensional Empirical Mode Decomposition (BEMD) of large images. The proposed approach is based on the integration and adaptation of the partition of unity (PU) algorithm with compactly supp...

The Drop when seen (DWS) technique, an online network coding strategy is capable of making a broadcast transmission over erasure channels more robust. This throughput optimal strategy reduces the expected sender queue length. One major issue with the...

Motivated by studies of data retrieval in polymer-based storage systems, we consider the problem of reconstructing a multiset of binary strings that have the same length and the same weight from the compositions of their prefixes and suffixes of ever...

Boltzmann macrostates, which are in 1:1 correspondence with the partitions of integers, are investigated. Integer partitions, unlike entropy, uniquely characterize Boltzmann states, but their use has been limited. Integer partitions are well known to...

Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence ${\left\{N_S(m,n)\right\}}_m$ is unimodal, where $N_S(m,n)$ is the number of S-partitions of size n with crank m weight by the spt...

Sylvester’s theorem states that every number can be decomposed into a sum of consecutive positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number as a sum of consecutive integers. The first generaliz...

We consider field theory formulation for directed polymers and interfaces in the presence of quenched disorder. We write a series representation for the averaged free energy, where all the integer moments of the partition function of the model contri...

We formulate binary fragmentation as a discrete stochastic process in which an integer mass k splits into two integer fragments j, $k-j$, with rate proportional to the fragmentation kernel $F_{j,k-j}$. We construct the ensemble of all distributi...

Techniques and methods of linear optimization underwent a significant improvement in the 20th century which led to the development of reliable mixed integer linear programming (MILP) solvers. It would be useful if these solvers could handle mixed int...

Current work provides an algebraic and geometric technique for building topological quantum codes. From the lattice partition derived of quotient lattices ${\Lambda }^{\prime }/\Lambda$ of index m combined with geometric technique of the projections of vec...

An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by $v{a}_k^{\equiv }\left(G\right)$, is the smallest integer t such that G can be...

If the vertex set of a graph G can be partitioned into k subsets $V_1,V_2,\dots ,V_k$, and the induced subgraph on each subset $V_i$ is a forest whose maximum degree is at most $d_i$ ($i=1,\dots ,k$), then this partition is called an $({\mathcal{F}}_{d_1},\dots ,{\mathcal{F}}_{d_k})$-partit...

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to...

Refinery planning optimization is a challenging problem as regards handling the nonconvex bilinearity, mainly due to pooling operations in processes such as crude oil distillation and product blending. This work investigated the performance of severa...

The spin-one Ising model on the honeycomb lattice has never been solved exactly in spite of its simplicity. Even its exact critical temperature is not known. The exact integer values for the density of states of the spin-one Ising model on the $L\&time$...

Collatz conjecture is also known as 3X + 1 conjecture. For verifying the conjecture, we designed an algorithm that can output reduced dynamics (occurred 3 × x + 1 or x/2 computations from a starting integer to the first integer smaller than the...

The differential box-counting (DBC) method is useful for determining the fractal dimension of grayscale images. It is simple to learn and implement and has been extensively utilized. However, this approach has several problems, such as over- or under...

In this article, some properties of the zero-divisor graph $\Gamma \left(Z_n\right)$ are investigated when n is a square-free positive integer. It is shown that the zero-divisor graph $\Gamma \left(Z_n\right)$ of ring $Z_n$ is a $(2^k-2)$-partite graph when the prime decomposit...

We study two types of probability measures on the set of integer partitions of n with at most m parts. The first one chooses the partition with a chance related to its largest part only. We obtain the limiting distributions of all of the parts togeth...

For each n-dimensional real Banach space X, each positive integer m, and each bounded set $A\subseteq X$ with diameter greater than 0, let ${\beta }_X(A,m)$ be the infimum of $\delta \in (0,1]$ such that $A\subseteq X$ can be represented as the union of m subsets of...

In order to ensure the security of power grids and control the level of short-circuit currents, a multi-objective optimization method for power grid partitioning is proposed. This method takes into consideration both short-circuit currents and multi-...

Let X be a partitioned matrix and let B its equitable quotient matrix. Consider a simple, undirected, connected graph G of order n. In this paper, we employ a technique based on quotient matrices derived from block-partitioned structures to establish...

From basic principles, we review some fundamentals of entropy calculations, some of which are implicit in the literature. We mainly deal with microcanonical ensembles to effectively compare the counting of states in continuous and discrete settings....

Dong, Goldschmidt and Martin (2006) (DGM) showed that, for $0<\alpha <1,$ and $\theta >-\alpha$, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson&ndash...

Let k denote an integer greater than 2, let G denote a k-partite graph, and let S denote the set of all maximal k-partite cliques in G. Several open questions concerning the computation of S are resolved. A straightforward and highly-scalable modific...

To address the problem of regional coverage path planning for unmanned aerial vehicle swarms (UAVs), this study proposes an algorithm based on objective programming partitions (OPP) and rule-based spanning tree coverage (RSTC). Aiming at the shortcom...

Fully homomorphic encryption (FHE) offers a promising solution for privacy-preserving machine learning by enabling arbitrary computations on encrypted data. However, the efficient evaluation of non-linear functions—such as the ReLU activation f...

In this paper, the authors derive some believed-to-be new recursion and explicit formulas for the generalized Ramanujan numbers ${\tau }_s\left(n\right)$$(s\in \mathbb{N}\cup \{-1\left\}\right)$, where, as usual, $\mathbb{N}$ is the set of positive integers. The authors consider the associate...

In this paper, the implementation of the Benders decomposition method to solve the Adjustable Robust Counterpart for Internet Shopping Online Problem (ARC-ISOP) is discussed. Since the ARC-ISOP is a mixed-integer linear programming (MILP) model, the...

Let $\mathcal{R}$ be a commutative ring with nonzero identity and $k\ge 2$ be a fixed integer. The k-zero-divisor hypergraph ${\mathcal{H}}_k\left(\mathcal{R}\right)$ of $\mathcal{R}$ consists of the vertex set $Z(\mathcal{R},k)$, the set of all k-zero-divisors of $\mathcal{R}$, and the hyperedges of the form $\{a_1,a_2,a_3,\dots ,a_k\}$, where $a_1,a_2$...

Dynamically partially reconfigurable (DPR) technology based on FPGA is applied extensively in the field of high-performance computing (HPC) because of its advantages in processing efficiency and power consumption. To make full use of the advantages o...