Greatest Common Divisor Research Articles

Anisotropic higher rank $\mathbb{Z}_N$ topological phases on graphs

We study unusual gapped topological phases where they admit \mathbb{Z}_NℤN fractional excitations in the same manner as topologically ordered phases, yet their ground state degeneracy depends on the local geometry of the system. Placing such phases on 2D lattice, composed of an arbitrary connected graph and 1D line, we find that the fusion rules of quasiparticle excitations are described by the Laplacian of the graph and that the number of superselection sectors is related to the kernel of the Laplacian. Based on this analysis, we further show that the ground state degeneracy is given by \bigl[N\times \prod_{i}\text{gcd}(N, p_i)\bigr]^2[N×∏igcd(N,pi)]2, where p_ipi’s are invariant factors of the Laplacian that are greater than one and gcd stands for the greatest common divisor. We also discuss braiding statistics between quasiparticle excitations.

A spectral force representation and its physical implication for vortex shedding past a stationary or an oscillating circular cylinder at low Reynolds number

Vortex shedding is an ubiquitous phenomenon behind a bluff body (such as circular cylinder) and becomes more complicated when the body is also in oscillation. It is apparent that periodic behavior must be accompanied by the time-varying force, such as lift and drag (coefficients) with known distinguished cases (say, at Re=200) of low-frequency modulation (LFM), sub-harmonic synchronization (SHS), and normal harmonic synchronization (NHS). In a classical analysis, the force spectrum is often analyzed by the Fourier transform or some more recent methods, and typically, a quite complex frequency spectrum is obtained owing to the inherent nonlinearity in the flow system. In the present study, we extend the principal frequency analysis [Lu et al., “An EMD-based principal frequency analysis with applications to nonlinear mechanics,” Mech. Syst. Signal Process. 150, 107300 (2021)] to the principal spectrum analysis (PSA) with both its amplitude and phase in a composite functional form and provide a spectral representation (SR) of the force coefficients only in terms of the characteristic frequencies. In particular, we consider the unsteady laminar flow past a stationary circular cylinder or an oscillating circular cylinder (with frequency f0), while the resulting vortex shedding frequency is denoted by fVS. The spectral representation via the proposed PSA can reveal nonlinear interactions of the two characteristic frequencies (f0 and fVS) in influencing the force coefficients and distinguish direct and interactive modes in which f0 and fVS interact with each other. As a matter of fact, the successively shed vortices are not identical in the strength (amplitude) nor in the phase function. The spectral representation further enables us to identify complicated vorticity activity near around the bluff body: the periodicity of the strength of the shed vortices and the phase shift in the successive vortex shedding—all at the integer multiples of the greatest common-divisor (gcd) of the (two) characteristic frequencies. The gcd frequency of ⟨f0, fVS⟩ is identified as the genuine (slow, long-term) frequency of the entire vortex shedding process in contrast to the (fast, short-term) vortex shedding frequency. It turns out in this scheme of classification by the PSA-SR that all the distinguished types of the above-mentioned LFM, SHS, and NHS can be considered to be gcd-frequency synchronization.

Improving of learning outcomes in greatest common divisor and lowest common multiple mater by developed safari numbered model

The purpose of this study was to determine the improvement in learning outcomes of the Greatest Common Divisor (GCD) and Lowest Common Multiple (LCM) by developing the Safari Numbered Model (SNM). Further, the activeness of students was also measured. The method used in this study was a quasi-experimental. Data collection was carried out using a test instrument to obtain students' mathematics learning outcomes and a questionnaire to measure the activeness of the students. Data analysis is carried out using the Kolmogorov-Smirnov (KS) test and a two-way ANOVA statistical test. Based on the results of the research, it can be concluded that the implementation of SNM can improve the learning outcomes in GCD and LCM. But the student’s activeness does not affect learning outcomes because the student has been active since the beginning of the lesson.

Relaxation of the greatest common divisor condition for the controllability of bilinear systems

When a bilinear system has an input matrix that has rank of one, bilinear controllability can be expressed in terms of the controllability and observability of an associated linear system, defined by the rank-one decomposition. Bilinear controllability also requires a greatest common divisor (GCD) condition to hold. We show in the absence of the GCD condition that the system is nearly controllable, i.e., controllable except for a set of Lebesgue measure zero. The presence of sparsity in the system dynamics structure then motivates a new definition of bilinear controllability in which initial and final conditions must satisfy certain combinations of non-zero state values.

NEW ARITHMETIC FUNCTION RELATED TO THE LEAST COMMON MULTIPLE

The author’s last two papers concerned the study of arithmetic functions related to the greatest common divisor. In the present paper, similarly to the two previous papers we will define new multiplicative arithmetic functionsrelated to the least common multiple and we will study several interesting properties of them. Also, we will discuss the values of two special functionsof them at perfect numbers.

Centralizers of differential operators of rank h

In the paper “Burchnall-Chaundy bundles” ((1998) [12]) the second author conjectured that the centralizer of a differential operator L=x−nδ(δ−m)(δ−2m)...(δ−m(n−1)) where δ=xddx is generated by operators L and B=x−mδ(δ−n)(δ−2n)...(δ−n(m−1)) and therefore has rank equal to the greatest common divisor h of m and n. In this note we will show that this is indeed the case if the ground field K has characteristic zero. Here we restrict ourselves to purely algebraic considerations; the reader interested in geometric aspects is advised to see the paper mentioned above.

Realization of zero-divisor graphs of finite commutative rings as threshold graphs

Let R be a finite commutative ring with unity, and let $$G = (V, E)$$ be a simple graph. The zero-divisor graph, denoted by $$\Gamma (R)$$ is a simple graph with vertex set as R, and two vertices $$x, y \in R$$ are adjacent in $$\Gamma (R)$$ if and only if $$xy=0$$ . In [5], the authors have studied the Laplacian eigenvalues of the graph $$\Gamma (\mathbb {Z}_{n})$$ and for distinct proper divisors $$d_1, d_2, \cdots , d_k$$ of n, they defined the sets as, $$\mathcal {A}_{d_i} = \{x \in \mathbb {Z}_{n} : (x, n) = d_i\}$$ , where (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets $$\mathcal {A}_{d_i}$$ , $$1 \le i \le k$$ are actually orbits of the group action: $$Aut(\Gamma (R)) \times R \longrightarrow R$$ , where $$Aut(\Gamma (R))$$ denotes the automorphism group of $$\Gamma (R)$$ . Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that $$\Gamma (R)$$ is a connected threshold graph if and only if $$R\cong {F}_{q}$$ or $$R\cong {F}_2 \times {F}_{q}$$ . We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold.

Privacy preserving via multi-key homomorphic encryption in cloud computing

As the world grapples with the COVID-19 and its variants, multi-user collaboration by means of cloud computing is ubiquitous. How to make better use of cloud resources while preventing user privacy leakage has become particularly important. Multi-key homomorphic encryption(MKHE) can effectively deal with the privacy disclosure issue during the multi-user collaboration in the cloud computing setting. Firstly, we improve the DGHV homomorphic scheme by modifying the selection of key and the coefficients in encryption, so as to eliminate the restriction on the parity of the ciphertext modulus in the public key. On this basis, we further propose a DGHV-type MKHE scheme based on the number theory. In our scheme, an extended key is introduced for ciphertext extension, and we prove that it is efficient in performance analysis. The semantic security of our schemes is proved under the assumption of error-free approximate greatest common divisor and the difficulty of large integer factorization. Furthermore, the simulation experiments show the availability and computational efficiency of our MKHE scheme. Therefore, our scheme is suitable for the multi-user scenario in cloud environment.

Periodic measures and Wasserstein distance for analysing periodicity of time series datasets

In this article, we establish the probability foundation of the periodic measure approach in analysing periodicity of a dataset. It is based on recent work of random periodic processes. While random periodic paths provide a pathwise model for time series datasets with a periodic pattern, their law is a periodic measure and gives a statistical description and the ergodic theory offers a scope of statistical analysis. The connection of a sample path and the periodic measure is revealed in the law of large numbers (LLN). We prove first the period is actually a deterministic number and then for discrete processes, Bézout’s identity comes in naturally in the LLN along an arithmetic sequence of an arbitrary increment. The limit is a periodic measure whose period is equal to the greatest common divisor between the test period and the true period of the random periodic process. This leads to a new scheme of detecting random periodicity of a dataset and finding its period, as an alternative to the Discrete Fourier Transformation (DFT) and periodogram approach. We find that in some situations, the classical method does not work robustly, but the new one can work efficiently. We prove that the periodicity is quantified by the Wasserstein distance, in which the convergence of empirical distributions is established.

Achieving Transfer from Mathematics Learning

The question of transfer is a special challenge in mathematics teaching because the wide range and fragmentation of the curricula have in many cases fostered an instrumental understanding, which makes transfer difficult for the students. Although promoting a relational learning has been a huge step forward in achieving transfer, understanding usually remains at the technical level of learning. Fostering critical thinking and metacognition raises learning to the psychological level, as students are encouraged to analyse their own thinking. Despite this, our hypothesis is that transfer will only be achieved when students are helped to reach a personal dimension, being encouraged to discover their own way of approaching the global reality of their lives beyond the subject. Learning, for instance, the greatest common divisor should be an opportunity to discover that, as numbers can be presented by their prime factors, people can be recognised by their features and interests. As such, looking for the greatest common divisor should not differ from discovering common interests with friends. Integrating specific and general learning will make transfer no longer unattainable. Personalising learning means discovering how one specific learning impacts on the personal way of understanding reality (oneself, others, and the world), thus making transfer natural.

On The Foundations of Fuzzy Number Theory and Fuzzy Diophantine Equations

Despite the great and rapid progress in the study of Fuzzy Logic and its applications in various scientific fields, it has not yet been used to build a consistent number theory like classical number theory. This research provides for the first time a conception of the concepts of number theory based on fuzzy logic and fuzzy membership functions, where it defines the division process, the fuzzy congruence, the greatest common divisor between integers with a fuzzy membership function. On the other hand, it presents many famous Diophantine equations formulated using fuzzy sets, in addition to many properties of fuzzy number theoretical systems, through many related theorems and accompanying illustrative examples. Also, in this research, we are raising many open research questions related to fuzzy number theory, which we believe will represent the future of progress in the study of this new mathematical branch.

The GCD for finite geometric series

The Greatest Common Divisor is given for a pair of finite geometric eries.

Full-Length Row-Multiplier QC-LDPC Codes With Girth Eight and Short Circulant Sizes

As a special case of a quasi-cyclic (QC) low-density parity-check (LDPC) code, a full-length row-multiplier (FLRM) QC-LDPC code is described by a compact exponent matrix based on two sequences of integers. The codes designed by a framework known as greatest-common-divisor (GCD) method, belong to a salient class of FLRM QC-LDPC codes, which can eliminate cycles of length up to six by carefully selecting a special sequence subject to a set of simple inequalities. However, the GCD method ensures the absence of these cycles only if circulant sizes are larger than a certain threshold. By combining the existing GCD method, novel sequences and a new analysis method (based on new lemmas of circulants and integers) for modulo equations, a group of novel FLRM QC-LDPC codes free of 4-cycles and 6-cycles are explicitly proposed for column weights from three to five in this paper, which possess circulant sizes much smaller than the forgoing threshold. Simulations show that the new FLRM QC-LDPC codes with shorter lengths perform almost the same as the existing FLRM QC-LDPC codes with longer lengths, and that the novel FLRM QC-LDPC codes noticeably outperform their counterparts with (nearly) identical lengths.

Square Difference Prime Labeling for Duplication of Graphs

Let G(V, E) be a graph with pvertices and qedges. Let f∶ V (G) → {0,1,2,…, p-1} be a bijection such that the induced function f*: E(G) → N defined by f_sqdp* (uv)=|[f(u) ]2-[f(v) ]2 | for everyuv∈E(G).If f_sqdp* is injective, then f_sqdp*is calledsquare difference labeling of G.A graph Gwhich admits square difference labeling is called square difference graph. The greatest common incidence number (gcin) of a vertex v of degree v > 1 is defined as the greatest common divisor (g.c.d) of the labels of the incident edges on v. A square difference labeling fis said to be a square difference prime labeling if for each vertex v of degree >1 then gcin(v) = 1. In this paper we investigate the square difference prime labelling of Petal graph and duplication of petal graph

The structure of the Smith normal form of the greatest common divisor and the least common multiple of the third order matrices over Bezout domains of stable range 1.5

The structure of the Smith normal form of the greatest common divisor and the least common multiple of the third order matrices over Bezout domains of stable range 1.5

Odd Prime Labeling of Graphs Related to Circular Ladder

In a graph \(G\) with point set \(V\) a mapping $f$ is said to be an odd prime labeling if \(f\) is a one-to-one function from point set \(V\) to \({\{1,3,5,2|V|-1}\}\) satisfying the condition that for each line \(uv\) in \(G\) the greatest common divisor of the labels of the end points \(f(u),f(v)\) is one. Investigated in this paper the odd prime labeling of circular ladder related graphs and we prove that the graphs such as \(CL(n)\), \(SCL(n)\), \(CL(n)\bigodot K_{1}\), \(CL(n)\bigodot \bar{K}_{2}\), \(CL(n) \bigodot\bar{K}_{3}\) are all odd prime graphs.

Analysis of Students' Errors in Solving the Least Common Multiple and Greatest Common Divisor Word Problems

Students' ability to solve word problems has an essential role in learning mathematics. However, some junior high school students still have difficulty in solving word problems. The difficulties experienced by students cause errors when solving mathematical word problems. This qualitative study aims to analyze students' errors in solving the Least Common Multiple and Greatest Common Divisor word problems. Fifty-nine students of Integrated Islamic Junior High School Al-Fahmi in Palu were involved in this study. Two students were selected to participate in the interview process. The interview was conducted to obtain in-depth information about how students solve word problems and the types of errors experienced by students. Students' errors in solving word problems were analyzed based on Newman's theory. The errors experienced by students are comprehension errors, transformation errors, process skills errors, and encoding errors

A formula on Stirling numbers of the second kind and its application to the unstable K-theory of stunted complex projective spaces

A formula on Stirling numbers of the second kind S(n,k) is proved. As a corollary, for odd n and even k, it is shown that k!S(n,k) is a positive multiple of the greatest common divisor of j!S(n,j) for k+1≤j≤n. Also, as an application to algebraic topology, some isomorphisms of unstable K1-groups of stunted complex projective spaces are deduced.

Low-Latency Design and Implementation of the Squaring in Class Groups for Verifiable Delay Function Using Redundant Representation

A verifiable delay function (VDF) is a function whose evaluation requires running a prescribed number of sequential steps over a group while the result can be efficiently verified. As a kind of cryptographic primitives, VDFs have been adopted in rapidly growing applications for decentralized systems. For the security of VDFs in practical applications, it is widely agreed that the fastest implementation for the VDF evaluation, sequential squarings in a group of unknown order, should be publicly provided. To this end, we propose a possible minimum latency hardware implementation for the squaring in class groups by algorithmic and architectural level co-optimization. Firstly, low-latency architectures for large-number division, multiplication, and addition are devised using redundant representation, respectively. Secondly, we present two hardware-friendly algorithms which avoid time-consuming divisions involved in calculations related to the extended greatest common divisor (XGCD) and design the corresponding low-latency architectures. Besides, we schedule and reuse these computation modules to achieve good resource utilization by using compact instruction control. Finally, we code and synthesize the proposed design under the TSMC 28nm CMOS technology. The experimental results show that our design can achieve a speedup of 3.6x compared to the state-of-the-art implementation of the squaring in the class group. Moreover, compared to the optimal C++ implementation over an advanced CPU, our implementation is 9.1x faster.

The sandpile group of a thick cycle graph

The majority of graphs whose sandpile groups are known are either regular or simple. We give an explicit formula for a family of non-regular multi-graphs called thick cycles . A thick cycle graph is a cycle where multi-edges are permitted. Its sandpile group is the direct sum of cyclic groups of orders given by quotients of greatest common divisors of minors of its Laplacian matrix. We show these greatest common divisors can be expressed in terms of monomials in the graph’s edge multiplicities.