scholarly journals On a New Formula for Fibonacci’s Family m-step Numbers and Some Applications

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 805 ◽  
Author(s):  
Monther Rashed Alfuraidan ◽  
Ibrahim Nabeel Joudah
Keyword(s):  
Number Theory ◽  
Time Complexity ◽  
Matrix Multiplication ◽  
Square Roots ◽  
Computational Number Theory ◽  
The Matrix ◽  
Theory Application ◽  
Constant Coefficients ◽  
New Formula

In this work, we obtain a new formula for Fibonacci’s family m-step sequences. We use our formula to find the nth term with less time complexity than the matrix multiplication method. Then, we extend our results for all linear homogeneous recurrence m-step relations with constant coefficients by using the last few terms of its corresponding Fibonacci’s family m-step sequence. As a computational number theory application, we develop a method to estimate the square roots.

scholarly journals Dreibens Modulo 7: A New Formula for Primality Testing

Elements ◽  
2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Arthur Diep-Nguyen
Keyword(s):  
Number Theory ◽  
Linear Algebra ◽  
Abstract Algebra ◽  
Additional Insight ◽  
Primality Testing ◽  
New Formula ◽  
Insight Into ◽  
Rule Out

In this paper, we discuss strings of 3’s and 7’s, hereby dubbed “dreibens.” As a first step towards determining whether the set of prime dreibens is infinite, we examine the properties of dreibens when divided by 7. by determining the divisibility of a dreiben by 7, we can rule out some composite dreibens in the search for prime dreibens. We are concerned with the number of dreibens of length n that leave a remainder i when divided by 7. By using number theory, linear algebra, and abstract algebra, we arrive at a formula that tells us how many dreibens of length n are divisible by 7. We also find a way to determine the number of dreibens of length n that leave a remainder i when divided by 7. Further investigation from a combinatorial perspective provides additional insight into the properties of dreibens when divided by 7. Overall, this paper helps characterize dreibens, opens up more paths of inquiry into the nature of dreibens, and rules out some composite dreibens from a prime dreiben search.

scholarly journals The Explicit Laplace Transform for the Wishart Process

2014 ◽  
Vol 51 (3) ◽  
pp. 640-656 ◽  
Author(s):  
Alessandro Gnoatto ◽  
Martino Grasselli
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Laplace Transform ◽  
Explicit Formula ◽  
Time Integral ◽  
Variation Of Constants ◽  
The Matrix ◽  
Wishart Process ◽  
New Formula ◽  
Original Approach

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation-of-constants method, the linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.

scholarly journals PENERAPAN METODE MULTI ATTRIBUTE UTILITY THEORY (MAUT) DALAM PEMETAAN TINGKAT DAMPAK BENCANA BANJIR DI KABUPATEN BANTUL

Telematika ◽  
2020 ◽  
Vol 17 (1) ◽  
pp. 26
Author(s):  
Afif Irfan Abdurrahman ◽  
Bambang Yuwono ◽  
Yuli Fauziah
Keyword(s):  
Utility Theory ◽  
Matrix Multiplication ◽  
Flood Disaster ◽  
Accuracy Rate ◽  
Affected Area ◽  
A Value ◽  
Multi Attribute Utility Theory ◽  
The Matrix ◽  
The Difference ◽  
The Impact

Flood disaster is a dangerous disaster, an event that occurs due to overflow of water resulting in submerged land is called a flood disaster. Almost every year Bantul Regency is affected by floods due to high rainfall. The flood disaster that struck in Bantul Regency made the Bantul District Disaster Management Agency (BPBD) difficult to handle so that it needed a mapping of the level of the impact of the flood disaster to minimize the occurrence of floods and provide information to the public.This study will create a system to map the level of impact of floods in Bantul Regency with a decision support method namely Multi Attribute Utility Theory (MAUT). The MAUT method stage in determining the level of impact of flood disasters through the process of normalization and matrix multiplication. The method helps in determining the areas affected by floods, by managing the Indonesian Disaster Information Data (DIBI). The data managed is data on criteria for the death toll, lost victims, damage to houses, damage to public facilities, and damage to roads. Each criteria data has a value that can be used to determine the level of impact of a flood disaster. The stages for determining the level of impact of a disaster require a weighting calculation process. The results of the weighting process display the scoring value which has a value of 1 = low, 2 = moderate, 3 = high. To assist in determining the affected areas using the matrix normalization and multiplication process the process is the application of the Multi Attribute Utility Theory (MAUT) method.This study resulted in a mapping of the level of impact displayed on google maps. The map view shows the affected area points and the level of impact of the flood disaster in Bantul Regency. The mapping produced from the DIBI data in 2017 produced the highest affected area in the Imogiri sub-district. The results of testing the data can be concluded that the results of this study have an accuracy rate of 95% when compared with the results of the mapping previously carried out by BPBD Bantul Regency. The difference in the level of accuracy is because the criteria data used are not the same as the criteria data used by BPBD in Bantul Regency so that the accuracy rate is 95%.

scholarly journals Computing the Atom Graph of a Graph and the Union Join Graph of a Hypergraph

Algorithms ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 347
Author(s):  
Anne Berry ◽  
Geneviève Simonet
Keyword(s):  
Real Number ◽  
Time Complexity ◽  
Matrix Multiplication ◽  
Efficient Algorithms ◽  
Join Graph ◽  
Current Value ◽  
Minimal Separator

The atom graph of a graph is a graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all possible atom trees of this graph. We provide two efficient algorithms for computing this atom graph, with a complexity in O(min(nωlogn,nm,n(n+m¯)) time, where n is the number of vertices of G, m is the number of its edges, m¯ is the number of edges of the complement of G, and ω, also denoted by α in the literature, is a real number, such that O(nω) is the best known time complexity for matrix multiplication, whose current value is 2,3728596. This time complexity is no more than the time complexity of computing the atoms in the general case. We extend our results to α-acyclic hypergraphs, which are hypergraphs having at least one join tree, a join tree of an hypergraph being defined by its hyperedges in the same way as an atom tree of a graph is defined by its atoms. We introduce the notion of union join graph, which is the union of all possible join trees; we apply our algorithms for atom graphs to efficiently compute union join graphs.

scholarly journals On Solutions of the Matrix Equation A ∘ l X  = B with respect to M M -2 Semitensor Product

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Jin Wang
Keyword(s):  
Matrix Equation ◽  
Matrix Multiplication ◽  
Sufficient Condition ◽  
Mathematical Tool ◽  
Necessary And Sufficient Condition ◽  
The Matrix ◽  
Necessary And Sufficient

M M -2 semitensor product is a new and very useful mathematical tool, which breaks the limitation of traditional matrix multiplication on the dimension of matrices and has a wide application prospect. This article aims to investigate the solutions of the matrix equation A ° l X = B with respect to M M -2 semitensor product. The case where the solutions of the equation are vectors is discussed first. Compatible conditions of matrices and the necessary and sufficient condition for the solvability is studied successively. Furthermore, concrete methods of solving the equation are provided. Then, the case where the solutions of the equation are matrices is studied in a similar way. Finally, several examples are given to illustrate the efficiency of the results.

The algebra of differential forms on a full matric bialgebra

1993 ◽  
Vol 114 (1) ◽  
pp. 111-130 ◽  
Author(s):  
A. Sudbery
Keyword(s):  
Vector Space ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Differential Forms ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Algebra Of Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Necessary And Sufficient

AbstractWe construct a non-commutative analogue of the algebra of differential forms on the space of endomorphisms of a vector space, given a non-commutative algebra of functions and differential forms on the vector space. The construction yields a differential bialgebra which is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. We give necessary and sufficient conditions for such an algebra to exist, show that it is uniquely determined by the differential algebra on the vector space, and show that it is a non-commutative superpolynomial algebra in the matrix elements and their differentials (i.e. that it has the same dimensions of homogeneous components as in the classical case).

Fast 3D Block Parallelisation for the Matrix Multiplication Prefix Problem

2010 ◽  
pp. 39-50 ◽  
Author(s):  
K. Waldherr ◽  
T. Huckle ◽  
T. Auckenthaler ◽  
U. Sander ◽  
T. Schulte-Herbrüggen
Keyword(s):  
Matrix Multiplication ◽  
The Matrix

scholarly journals An Accelerator Architecture of Changeable-Dimension Matrix Computing Method for SVM

Electronics ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 143 ◽  
Author(s):  
Ruidong Wu ◽  
Bing Liu ◽  
Ping Fu ◽  
Junbao Li ◽  
Shou Feng
Keyword(s):  
Machine Learning ◽  
Learning Algorithms ◽  
Matrix Multiplication ◽  
Critical Time ◽  
Machine Learning Algorithms ◽  
Support Vector ◽  
Practical Applications ◽  
The Matrix ◽  
Field Programmable ◽  
Computing Method

Matrix multiplication is a critical time-consuming processing step in many machine learning applications. Due to the diversity of practical applications, the matrix dimensions are generally not fixed. However, most matrix calculation methods, based on field programmable gate array (FPGA) currently use fixed matrix dimensions, which limit the flexibility of machine learning algorithms in a FPGA. The bottleneck lies in the limited FPGA resources. Therefore, this paper proposes an accelerator architecture for matrix computing method with changeable dimensions. Multi-matrix synchronous calculation concept allows matrix data to be processed continuously, which improves the parallel computing characteristics of FPGA and optimizes the computational efficiency. This paper tests matrix multiplication using support vector machine (SVM) algorithm to verify the performance of proposed architecture on the ZYNQ platform. The experimental results show that, compared to the software processing method, the proposed architecture increases the performance by 21.18 times with 9947 dimensions. The dimension is changeable with a maximum value of 2,097,151, without changing hardware design. This method is also applicable to matrix multiplication processing with other machine learning algorithms.

Stationary waiting-time distributions in the GI/PH/1 queue

1981 ◽  
Vol 18 (4) ◽  
pp. 901-912 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Waiting Time ◽  
Embedded Markov Chain ◽  
Asymptotic Results ◽  
Waiting Time Distributions ◽  
Structured Systems ◽  
Queue Discipline ◽  
The Matrix ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Stationary Waiting Time

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.

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