Greatest Common Divisor Research Articles

Parallel complexity for nilpotent groups

Recently, Macdonald et al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in [Formula: see text]. Here, we follow their approach and show that all these problems are complete for the uniform circuit class [Formula: see text] — even if an [Formula: see text]-generated nilpotent group of class at most [Formula: see text] is part of the input but [Formula: see text] and [Formula: see text] are fixed constants. In particular, unary encoded systems of a bounded number of linear equations over the integers can be solved in [Formula: see text]. In order to solve these problems in [Formula: see text], we show that the unary version of the extended gcd problem (compute greatest common divisors and express them as linear combinations) is in [Formula: see text]. Moreover, if we allow a certain binary representation of the inputs, then the word problem and computation of normal forms is still in uniform [Formula: see text], while all the other problems we examine are shown to be [Formula: see text]-Turing-reducible to the binary extended gcd problem.

Influence of Context on Greatest Common Divisor Problem Solving: A Qualitative Study

This paper presents results from a study about problem solving related to the concept of the greatest common divisor with secondary school students. The perspective of the analysis is the meaningful learning of the constructivist theory. The main objectives are to assess the students’ competence in the resolution of such problems and analyze if the difficulties in the acquisition of this competence are influenced by the kinds of magnitudes or the context of the problem. The results suggest that some contexts generate more difficulties to perform the use of the greatest common divisor. Moreover, some erroneous patterns have been detected. On one hand, students tend to relate and confuse the concepts of greatest common divisor and lowest common multiple. On the other hand, they have a predisposition to simplify problems, performing only the operation to obtain the greatest common divisor, and without thinking that additional arithmetic operations can be performed.

Prime Cordial Labeling of Generalized Petersen Graph under Some Graph Operations

A graph is a connection of objects. These objects are often known as vertices or nodes and the connection or relation in these nodes are called arcs or edges. There are certain rules to allocate values to these vertices and edges. This allocation of values to vertices or edges is called graph labeling. Labeling is prime cordial if vertices have allocated values from 1 to the order of graph and edges have allocated values 0 or 1 on a certain pattern. That is, an edge has an allocated value of 0 if the incident vertices have a greatest common divisor (gcd) greater than 1. An edge has an allocated value of 1 if the incident vertices have a greatest common divisor equal to 1. The number of edges labeled with 0 or 1 are equal in numbers or, at most, have a difference of 1. In this paper, our aim is to investigate the prime cordial labeling of rotationally symmetric graphs obtained from a generalized Petersen graph P(n,k) under duplication operation, and we have proved that the resulting symmetric graphs are prime cordial. Moreover, we have also proved that when we glow a Petersen graph with some path graphs, then again, the resulting graph is a prime cordial graph.

Orbital angular momentum of superpositions of optical vortices after passing through a sector diaphragm

In optical communications, it is desirable to know some quantities describing a light field, that are conserved on propagation or resistant to some distortions. Typically, optical vortex beams are characterized by their orbital angular momentum (OAM) and/or topological charge (TC). Here, we study what happens with the OAM of a superposition of two or several optical vortices (with different TCs) when it is distorted by a hard-edge sector aperture. We discover several cases when such perturbation does not violate the OAM of the whole superposition. The first case is when the incident beam consists of two vortices of the same power. The second case is when the aperture half-angle equals an integer number of π divided by the difference between the topological charges. For more than two incident beams, this angle equals an integer number of π divided by the greatest common divisor of all possible differences between the topological charges. For two incident vortex beams with real-valued radial envelopes of the complex amplitudes, the OAM is also conserved when there is a ±(pi)/2 phase delay between the beams. When two beams with the same power pass through a binary radial grating, their total OAM is also conserved.

On the number of epi-, mono- and homomorphisms of groups

Abstract It is well known that the number of homomorphisms from a group to a group is divisible by the greatest common divisor of the order of and the exponent of . We study the question of what can be said about the number of homomorphisms satisfying certain natural conditions like injectivity or surjectivity. A simple non-trivial consequence of our results is the fact that in any finite group the number of generating pairs such that is divisible by the greatest common divisor of fifteen and the order of the group .

Little Fermat Theorem Applying For Problems about Division

This article provides solutions to some divisibility problems using Fermat's little theorem. To have beautiful solutions for each of those problems, mathematicians have combined knowledge of: Theory of divisibility and division with remainder, greatest common divisor, least common multiple, prime numbers, congruences, exponentiation, etc. This helps students think positively and flexibly about their existing knowledge and skills, and present concise and creative solutions.

An Alternative Approach to Find the Greatest Common Divisor in Polynomials

An Alternative Approach to Find the Greatest Common Divisor in Polynomials

Greatest-common-divisor dependency of juggling sequence rotation efficient performance

AbstractIn previous experimental study with three-way-reversal and juggling sequence rotation algorithms, using 20,000,000 elements for type LONG in Java, the average execution times have been shown to be 49.66761ms and 246.4394ms, respectively. These results have revealed appreciable low performance in the juggling algorithm despite its proven optimality. However, the juggling algorithm has also exhibited efficiency with some offset ranges. Due to this pattern of the juggling algorithm, the current study is focused on investigating source of the inefficiency on the average performance. Samples were extracted from the previous experimental data, presented differently and analyzed both graphically and in tabular form. Greatest common divisor values from the data that equal offsets were used. As emanating from the previous study, the Java language used for the rotation was to simulate ordering of tasks for safety and efficiency in the context of real-time task scheduling. Outcome of the investigation shows that juggling rotation performance competes favorably with three-way-reversal rotation (and even better in few cases) for certain offsets, but poorly with the rests. This study identifies the poorest performances around offsets in the neighborhood ofsquare rootof the sequence size. From the outcome, the study therefore strongly advises application developers (especially for real-time systems) to be mindful ofwhereandhowto in using juggling rotation.

Computability, Notation, and de re Knowledge of Numbers

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects.

Handheld laser scanning microscope catheter for real-time and in vivo confocal microscopy using a high definition high frame rate Lissajous MEMS mirror.

A handheld confocal microscope using a rapid MEMS scanning mirror facilitates real-time optical biopsy for simple cancer diagnosis. Here we report a handheld confocal microscope catheter using high definition and high frame rate (HDHF) Lissajous scanning MEMS mirror. The broad resonant frequency region of the fast axis on the MEMS mirror with a low Q-factor facilitates the flexible selection of scanning frequencies. HDHF Lissajous scanning was achieved by selecting the scanning frequencies with high greatest common divisor (GCD) and high total lobe number. The MEMS mirror was fully packaged into a handheld configuration, which was coupled to a home-built confocal imaging system. The confocal microscope catheter allows fluorescence imaging of in vivo and ex vivo mouse tissues with 30 Hz frame rate and 95.4% fill factor at 256 × 256 pixels image, where the lateral resolution is 4.35 μm and the field-of-view (FOV) is 330 μm × 330 μm. This compact confocal microscope can provide diverse handheld microscopic applications for real-time, on-demand, and in vivo optical biopsy.

ANALISIS KESALAHAN SISWA KELAS IV SD MENYELESAIKAN SOAL KPK DAN FPB BERDASARKAN PROSEDUR NEWMAN

AbstractThis study aims to analyze the types of errors and the factors that cause students to make mistakes in solving the LCM (Least Common Multiple) and GCD (Greatest Common Divisor) questions based on Newman's procedures in grade IV State Elementary School 01 Balai Karangan. The research method used in this research is descriptive qualitative. The research subjects in this study were the fourth grade students of State Elementary School 01 Balai Karangan, totaling 28 students. Data collection techniques in this study were tests and interviews. Based on the results of the study, the errors made by students were (1) misunderstood the question as much 55,36%, (2) transformation error as much 58,93%, (3) process skill error as much 83,04%, and (4) errors in writing the final answer as much 88,39%. While the factors that cause students to make mistakes are internal factors, namely from the lack of ability of the students themselves and external factors, namely from lack of attention, guidance, and motivation from teachers and families. Keyword: Error Analysis, Prosedur Newman, Error Type, Causative Factor, LCM and GCD.

EXTENDED EUCLIDEAN ALGORITHM OF AUNU BINARY POLYNOMIALS OF CARDINALITY ELEVEN

Binary polynomials representation of Aunu permutation patterns has been used to perform arithmetic operations on words and sub-words of the polynomials using addition, multiplication, and division modulo two. The polynomials were also found to form some mathematical structures such as group, ring, and field. This paper presents the extension of our earlier work as it reports the Aunu binary polynomials of cardinality eleven and how to find their greatest common divisor (gcd) using the extended Euclidean algorithm. The polynomials are pairly permuted and the results found showed that one polynomial is a factor of the other polynomial or one polynomial is relatively prime to the other and some gave different results. This important feature is of combinatorial significance and can be investigated further to formulate some theoretic axioms for this class of Aunu permutation pattern. Binary polynomials have important applications in coding theory, circuit design, and the construction of cryptographic primitives.

Anti-aliasing phase reconstruction via a non-uniform phase-shifting technique.

The conventional phase-shifting techniques commonly suffer from frequency aliasing because of their number of phase shifts below the critical sampling rate. As a result, fringe harmonics induce ripple-like artifacts in their reconstructed phase maps. For solving this issue, this paper presents an anti-aliasing phase-measuring technique. Theoretical analysis shows that, with phase-shifting, the harmonics aliased with the fundamental frequency component of a fringe signal depend on the greatest common divisor (GCD) of the used phase shifts. This fact implies a possibility of removing such aliasing effects by selecting non-uniform phase shifts that together with 2π have no common divisors. However, even if we do so, it remains challenging to separate harmonics from the fundamental fringe signals, because the systems of equations available from the captured fringe patterns are generally under-determined, especially when the number of phase shifts is very few. To overcome this difficulty, we practically presume that all the points over the fringe patterns have an identical characteristic of harmonics. Under this constraint, using an alternate iterative least-squares fitting procedure allows us to estimate the fringe phases and the harmonic coefficients accurately. Simulation and experimental results demonstrate that this proposed method enables separating high order harmonics from as few as 4 fringe patterns having non-uniform phase shifts, thus significantly suppressing the ripple-like phase errors caused by the frequency aliasing.

The p-Frobenius and p-Sylvester numbers for Fibonacci and Lucas triplets.

In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $ a_1, a_2, \dots, a_l $ be positive integers such that their greatest common divisor is one. For a nonnegative integer $ p $, denote the $ p $-Frobenius number by $ g_p (a_1, a_2, \dots, a_l) $, which is the largest integer that can be represented at most $ p $ ways by a linear combination with nonnegative integer coefficients of $ a_1, a_2, \dots, a_l $. When $ p = 0 $, the $ 0 $-Frobenius number is the classical Frobenius number. When $ l = 2 $, the $ p $-Frobenius number is explicitly given. However, when $ l = 3 $ and even larger, even in special cases, it is not easy to give the Frobenius number explicitly. It is even more difficult when $ p > 0 $, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers [1] or of repunits [2] for the case where $ l = 3 $. In this paper, we show the explicit formula for the Fibonacci triple when $ p > 0 $. In addition, we give an explicit formula for the $ p $-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most $ p $ ways. Furthermore, explicit formulas are shown concerning the Lucas triple.

On coprime percolation, the visibility graphon, and the local limit of the GCD profile

Colour an element of Zd white if its coordinates are coprime and black otherwise. What does this colouring look like when seen from a “uniformly chosen” random point of Zd? More generally, label every element of Zd by its greatest common divisor: what do the labels look like around a “uniform” random point of Zd? We answer these questions and generalisations of them, which includes a result a “local/graphon” convergence. We also investigate the percolative properties of the colouring under study.

On the distribution of $ k $-full lattice points in $ \mathbb{Z}^2 $

<abstract><p>Let $ \mathbb{Z}^2 $ be the two-dimensional integer lattice. For an integer $ k\geq 2 $, we say a non-zero lattice point in $ \mathbb{Z}^2 $ is $ k $-full if the greatest common divisor of its coordinates is a $ k $-full number. In this paper, we first prove that the density of $ k $-full lattice points in $ \mathbb{Z}^2 $ is $ c_k = \prod_{p}(1-p^{-2}+p^{-2k}) $, where the product runs over all primes. Then we show that the density of $ k $-full lattice points on a path of an $ \alpha $-random walk in $ \mathbb{Z}^2 $ is almost surely $ c_k $, which is independent on $ \alpha $.</p></abstract>

New properties of an arithmetic function

Recently the author and Derbal introduced and studied some elementary properties of arithmetic functions related the greatest common divisor. New properties of them are given in this paper.

Notes on Hong's conjecture on nonsingularity of power LCM matrices

<abstract><p>Let $ a, n $ be positive integers and $ S = \{x_1, ..., x_n\} $ be a set of $ n $ distinct positive integers. The set $ S $ is said to be gcd (resp. lcm) closed if $ \gcd(x_i, x_j)\in S $ (resp. $ [x_i, x_j]\in S $) for all integers $ i, j $ with $ 1\le i, j\le n $. We denote by $ (S^a) $ (resp. $ [S^a] $) the $ n\times n $ matrix having the $ a $th power of the greatest common divisor (resp. the least common multiple) of $ x_i $ and $ x_j $ as its $ (i, j) $-entry. In this paper, we mainly show that for any positive integer $ a $ with $ a\ge 2 $, the power LCM matrix $ [S^a] $ defined on a certain class of gcd-closed (resp. lcm-closed) sets $ S $ is nonsingular. This provides evidences to a conjecture raised by Shaofang Hong in 2002.</p></abstract>

Submergence depth modeling of oil well reservoirs and applications

In oilfield production, it is important to be able to acquire the reservoir status of oil wells in real time, especially for low or ultra-low permeability oil wells. In this paper, based on the reservoir percolation characteristics of the oil layer and combined with the oil well permeability, liquid pressure and oil well pressure formulas, the non-homogeneous linear differential equation for oil well pressure is derived, based on which a submergence depth model of oil well reservoirs is established. The parameter merging is applied to reduce the parameter dimension of the submergence depth model; this approach substantially reduces the parameter dimension and application complexity and effectively improves the model application range. The model not only conforms to the reservoir percolation law of the oil layer but also applies to the balance laws of other substances with potential energy balancing ability. The application methods studied (namely, the extraction of the greatest common divisor, the weighted dichotomy and least-squares curve fitting) successively reduce the calculation error and expand the application scope. The least-squares curve fitting method causes the submergence depth error to reach the minimum of the 2-norm, and the number of iterations can be accurately assessed by the convergence speed of halving the numerical calculation error in every iteration. This method is suitable not only for the zero submergence depth case in the initial state but also for non-zero submergence depth cases. The experimental results show that the proposed submergence depth model for oil well reservoirs accurately reflects the change laws of submergence depth and time. Combined with the proposed application methods, the corresponding submergence depth can be obtained at any time point. The model and application methods provide strong support for formulating scientific oil production plans and implementing reasonable production methods for oil wells.

Blind Estimation of Self-Synchronous Scrambler Using Orthogonal Complement Space in DSSS Systems

In a non-cooperative context, a receiver has to estimate the communication parameters without any prior knowledge of the transmitter, which is highly demanding. Estimating a self-synchronous scrambler is even more challenging because the scrambling sequence of the self-synchronous scrambler is affected by the input sequence. This paper proposes an improved algorithm for blind estimation of a self-synchronous scrambler using the orthogonal property without any bias condition of the received signals in direct sequence spread spectrum (DSSS) systems. We first examine the linear relation of a scrambling sequence using the repetitive property of the spreading code used in a DSSS system. Using the obtained linear relation and the basis of the orthogonal complement space, we then acquire the feedback polynomial candidates of the scrambler. Finally, by calculating the greatest common divisor polynomial of the feedback polynomial candidates, we estimate the correct feedback polynomial. Through computer simulations, we verify that the proposed method achieves superior estimation performance compared to the existing method. Furthermore, we show that the proposed method has practically acceptable computational complexity. For these reasons, it is expected that the proposed method can be applied to blind estimation of a self-synchronous scrambler in a practical non-cooperative system.