Square Matrix Of Size Research Articles

KOMPUTASI ENKRIPSI DAN DEKRIPSI MENGGUNAKAN ALGORITMA HILL CIPHER

Hill Cipher is a symmetric cryptographic algorithm that can be used to encrypt and decrypt. Hill Cipher has several advantages including the key used to encrypt and decrypt is a square matrix of size n x n which has a multiplicative inverse because the inverse is the key used to decrypt. This study discusses the design of encryption and decryption program using with Java. The resulting program is capable of encrypting and decrypting with key 2 x 2, 3 x 3 , and 4 x 4.

FPGA, GPU, and CPU implementations of Jacobi algorithm for eigenanalysis

Parallel implementations of Jacobi algorithm for eigenanalysis of a matrix on most commonly used high performance computing (HPC) devices such as central processing unit (CPU), graphics processing unit (GPU), and field-programmable gate array (FPGA) are discussed in this paper. Their performances are investigated and compared. It is shown that CPU, even with multi-threaded implementation, is not a feasible option for large dense matrices. For the GPU implementation, performance impact of the global memory access patterns on the GPU board and the memory coalescing are emphasized. Three memory access methods are proposed. It is shown that one of them achieves 81.6% computational performance improvement over the traditional GPU methods, and it runs 68.5 times faster than a single-threaded CPU for a dense symmetric square matrix of size 1,024. Furthermore, FPGA implementation is presented and its performance running on chips from two major manufacturers are reported. A comparison of GPU and FPGA implementations is quantified and ranked. It is reported that FPGA design delivers the best performance for such a task while GPU is a strong competitor requiring less development effort with superior scalability. We predict that emerging big data applications will benefit from real-time and high performance computing implementations of eigenanalysis for information inference and signal analytics in the future.

Some applications of the Kronecker product in Hubbard representation

The properties of the Kronecker product are revisited in terms of Hubbard operators. The simplest representation of a Hubbard operator Xi,jn is a square matrix of size n with an entry equal to 1 and zero elsewhere. This framework simplifies the calculation of the Kronecker product of arbitrary matrices no matter the size or the number of the involved factors. Some applications are presented, these include the algebra of permutation matrices, the Hadamard matrix, the XXX Heisenberg model and the interaction of an atom with radiation fields.

Regular Steinhaus graphs of odd degree

A Steinhaus matrix is a binary square matrix of size n which is symmetric, with a diagonal of zeros, and whose upper-triangular coefficients satisfy a i , j = a i − 1 , j − 1 + a i − 1 , j for all 2 ⩽ i < j ⩽ n . Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states that even Steinhaus graphs, i.e. those with all vertex degrees even, have doubly-symmetric Steinhaus matrices. In 1979 Dymacek conjectured that the complete graph on two vertices K 2 is the only regular Steinhaus graph of odd degree. Using Dymacek’s theorem, we prove that if ( a i , j ) 1 ⩽ i , j ⩽ n is a Steinhaus matrix associated with a regular Steinhaus graph of odd degree then its sub-matrix ( a i , j ) 2 ⩽ i , j ⩽ n − 1 is a multi-symmetric matrix, that is a doubly-symmetric matrix where each row of its upper-triangular part is a symmetric sequence. We prove that the multi-symmetric Steinhaus matrices of size n whose Steinhaus graphs are regular modulo 4, i.e. where all vertex degrees are equal modulo 4, only depend on ⌈ n 24 ⌉ parameters for all even numbers n , and on ⌈ n 30 ⌉ parameters in the odd case. This result permits us to verify Dymacek’s conjecture up to 1500 vertices in the odd case.

Multi-Fourier spectra descriptor and augmentation with spectral clustering for 3D shape retrieval

We propose a new method of similarity search for 3D shape models, given an arbitrary 3D shape as a query. The method features the high search performance enabled in part by our unique feature vector called Multi-Fourier Spectra Descriptor (MFSD), and in part by augmenting the feature vector with spectral clustering. The MFSD is composed of four independent Fourier spectra with periphery enhancement. It allows us to faithfully capture the inherent characteristics of an arbitrary 3D shape object regardless of the dimension, orientation, and original location of the object when it is first defined. Given a 3D shape database, the augmentation with spectral clustering is done first by computing the p-minimum spanning tree of the whole data set, where p is a number usually much less than m, the size of the whole 3D shape data set. We then define the affinity matrix, which is a square matrix of size m by m, where each element of the matrix denotes the distance between two shape objects. The distance is computed in advance by traversing the p-minimum spanning tree. The eigenvalue decomposition is then applied to the affinity matrix to reduce dimensionality of the matrix, followed by grouping into k clusters. The cluster information is kept for augmenting the search performance when a query is given. With a series of benchmark data sets, we will demonstrate that our approach outperforms previously known methods for 3D shape retrieval.

Optimal algorithms for parallel Givens factorization on a coarse-grained PRAM

We study the complexity of the parallel Givens factorization of a square matrix of size n on a shared memory architecture composed with p identical processors (coarse grained EREW PRAM). We show how to construct an asymptotically optimal algorithm. We deduce that tje time complexity is equal to: ... (formule)... and that the minimum number of processors in order to compute the Givens factorization in asymptotically optimal time (2n + o(n)) is equal to p opt = n/(2 + √2) + o(n). These results complete previous analysis presented in the case where the number of processors is unlimited

A Bauer-Fike theorem for the joint spectrum of commuting matrices

Let A=( A 1,…, A m ) and B=( B 1,…, B m ) be m-tuples commuting n by n self-adjoint matrices. We obtain a number ε=‖ Cliff( A− B)‖ such that within a distance ε of each joint eigenvalue of A there is a joint eigenvalue of B. The Clifford operator Cliff( A− B) of A− B can be represented by a square matrix of size 2 m n and is defined using Clifford algebras. When m = 1, ‖ Cliff( A− B)‖ = ‖ A− B‖, the operator bound norm of A - B. Similar results are obtained for arbitrary commuting matrices A j and simultaneously diagonalizable matrices B j .

The van der waerden conjecture: two proofs in one year

One of the famous open problems in combinatorial theory was the van der Waerden conjecture on permanents of doubly stochastic matrices. After more than fifty years in which it managed to resist attacks it has finally been proved. As so often happens two mathematicians gave independent proofs nearly at the same time. At the end of 1980 the news that G. P. Egoritsjev (F. H. EropbiyeB) [2] had found a proof spread quickly and within a few months translations and expositions of the proof were circulating (cf. D. E. Knuth [5], J. H. van Lint [6]). It came as quite a shock when Matemati~eski Zametki 29 No. 6 appeared a few months ago with a paper by D. I. Falikman ()I. I/I. ~a~lnI<Man) [3], submitted 14.5. 1979 (!), with a completely different proof of the conjecture. Perhaps even more surprising is the fact that the two proofs have as a common feature that they use an elegant inequality, due to A. D. Alexandroff and W. Fenchel, which until quite recently seemed to be unknown to combinatorialists. From the literature it is obvious that geometers certainly know about the inequality. In this note we shall describe and compare the two new proofs and the ideas that led to their discovery. Let A be a square matrix of size n with entries a 6 (1 <~ i, j ~< n). We define the permanent of A (notation: perA) by

On Downton's carrier-borne epidemic

SUMMARY The stochastic version of Downton's (1968) carrier-borne epidemic is considered. From the time-dependent solution, obtained by Gani's (1967) technique, the distribution of the total size of the continuous time epidemic is found, in a form suitable for computer calcu- lations. Two additional results obtained seem useful and practical. An (n + 1) x (n + 1) matrix with first column the distribution of the size of an epidemic initiated by n susceptibles and a carriers is shown to give the distributions of epidemics initiated by 0, ..., n - 1 susceptibles and a carriers; this matrix is also shown to be the a-fold product of a similar matrix for epidemics initiated by just olne carrier. The first of the additional results has an analogue for the time-dependent solution. The purpose of this note is to consider the stochastic version of Downton's (1968) carrier- borne epidemic. The time-dependent distribution of the size of the continuous time epidemic is obtained. The method used is similar to that of Gani (1967) and uses Laplace transform methods. From the time-dependent distribution, the distribution of the total size of the epidemic is derived. Both these results are given in matrix form. An advantage of this, which has not been sufficiently stressed, is that the distribution of the total size of the epidemic may be easily evaluated by matrix operations suitable for computer calculations. We also find two results which seem to make this method for obtaining the distribution of the total size of the epidemic more useful than others. When the last stage of the matrix operations is omitted, we obtain a square matrix of size n + 1. This matrix has for its ith column (i = 1, ..., n + 1) the distribution of the total size of a carrier-borne epidemic initiated by n + 1 - i susceptibles and a carriers. We show that this matrix is the ath power of a similar matrix for epidemics initiated by just one carrier. Downton's carrier-borne epidemic model postulates a closed population of size n +a. Initially there are n susceptibles and a carriers mixing freely, the carriers infecting the susceptibles. A proportion of the infected individuals are recognized as infectives and they are removed from circulation immediately. The remainder of the infected individuals are not recognized as infectives and they become carriers until, like the initial carriers, they are detected and removed from circulation. Since no new carriers are introduced into the population, the epidemic ceases as soon as the number of carriers or susceptibles reaches zero. Downton discusses both the deterministic and stochastic models; for the stochastic model he obtains an exact expression for the probability distribution of the number of survivors. The method used by Downton is a generalization of that of Whittle (1955), which involves establishing recurrence relationships for the required probabilities.