Constructing Pandiagonal Snowflake Magic Squares of Odd Order

The constructional methods of pandiagonal snowflake magic squares of orders 4m are established in paper [3]. In this paper, the constructional methods of pandiagonal snowflake magic squares of odd orders n are established with n = 6m+l, 6m+5 and 6m+3, m is an odd positive integer and m is an even positive integer 9|6m + 3. It is seen that the number sets for constructing pandiagonal snowflake magic squares can be extended to the matrices with symmetric partial difference in each direction for orders 6m + 1 , 6m + 5; to the trisection matrices with symmetric partial difference in each direction for order 6m + 3.

Nonequidistant two-dimensional antenna arrays based on magic squares

New methods of synthetizing nonequidistant sparse antenna arrays based on the properties of magic squares are studied.The methods of construction and algorithms of synthesis of two-dimensional antennas based on them providing a high degreeof dilution and sufficiently small side radiation are proposed.The methods for construction of such antennas and their maincharacteristics are considered.

PANSYSTEMS METHODOLOGY AND CONSTRUCTION OF MAGIC SQUARES

Some patterns of refined epitomes of pansystems methodology were revealed roles and the related of them in problem-solving, modeling, algorithm-generating and theory-constructing were introduced. An important application of pansystems methodology is to give some methods of constructing the typical pansymmetric-magic squares: 1. a method of recursively constructing magic squares of order n ( n greater than or equal to 5) ; 2. when magic squares of order m( m greater than or equal to 3) and magic squares of order n ( n greater than or equal to 3) are given of formula of obtaining magic squares of order nm; 3. when magic squares of order m ( m greater than or equal to 3) are given, a method of obtaining magic squares of order 2m.

A Constructive Method of Pandiagonal Magic Squares

By means of this approach, a constructive method of pandiagonal magic squares is proposed. Pandiagonalmagic squares of order mn can be generated via two ones which are orders m and n, respectively.

Construction of Normal Bimagic Squares of Order 2u

An n×n matrix A consisting of nonnegative integers is a general magic square of order n if thesum of elements in each row,column,and main diagonal is the same.A general magic square A of order n iscalled a magic square,denoted by MS(n),if the entries of A are distinct.A magic square A of order n is normalif the entries of A are n^2 consecutive integers.Let A^*d denote the matrix obtained by raising each elementof A to the d-th power.The matrix A is a d-multimagic square,dcnoted by MS(n,d),if A^*e is an MS(n)for 1≤e≤d.In this paper we investigate the existence of normal bimagic squares of order 2u and prove that thereexists a normal bimagic square of order 2u,where u and 6 are coprime and u≥5.

A Block Cipher Algorithm Based on Magic Square for Secure E-bank Systems

Nowadays the E-bank systems witnessed huge growth due to the huge developments in the internet and technologies.The transmitted information represents crucial information that is exposed to various kinds of attacks.This paper presents a new block cipher technique to provide security to the transmitted information between the customers and the ebank systems.The proposed algorithm consists of 10 rounds,each round involves 5 operations.The operations involve Add round key,Sub bytes,Zigzag method,convert to vector,and Magic Square of order 11.The purpose of this algorithm is to make use of the complexity of the Magic Square algorithm,the speed of addition operation,the confusion provided by the zigzag,using these operations with Galois field 28 GF(28),and repeating these operations for several rounds to build fast high secure encryption algorithm.This algorithm is designed to provide fast with high complexity and security which is suitable to encrypt the data that is transmitted over the internet.Speed,complexity,and The National Institute of Standards and Technology Framework NIST suite tests were done.The complexity of the proposed algorithm is=((256)32)r+1∗((256)89)r+1+(256)121.The proposed technique gives higher speed and security in the encryption and decryption phases,according to the results of the experiments.The degree of randomness has grown by 31.8 percent.Due to a decrease in the time of encrypting and decrypting,as well as the usage of the central processing unit(CPU),efficiency is improved.The encryption process throughput is enhanced by 13%,while the decryption process throughput is increased by 11.6 percent with the recommended approach.

Ways of constructing optimal magic cube of order n when (n,2·3·5·7)=1

An optimal magic cube of order n is a magic cube whose row sums, column sums and oblique sums of 9 n layers are n(n 3+1)/2. The author proved that optimal magic cubes of order n may be constructed as long as n and 2, 3, 5, 7 are relatively prime, and a formula for making optimal magic cubes by using optimal Latin squares and optimal magic squares was given.

The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays

Let N = {0, 1, ···, n-1}. A strongly idempotent self-orthogonal row Latin magic array of order n（SISORLMA（n） for short） based on N is an n × n array M satisfying the following properties：（1） each row of M is a permutation of N, and at least one column is not a permutation of N;（2） the sums of the n numbers in every row and every column are the same;（3） M is orthogonal to its transpose;（4） the main diagonal and the back diagonal of M are 0, 1, ···, n-1 from left to right. In this paper, it is proved that an SISORLMA（n）exists if and only if n ？ {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ？ {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠2.

Performance enhancement of partially shaded solar PV array using novel shade dispersion technique

Solar photo voltaic array （SPVA） generates a smaller amount of power than the standard rating of the panel due to the partial shading effect. Since the modules of the arrays receive different solar irradiations, the P-V characteristics ofphotovoltaic （PV） arrays contain multiple peaks or local peaks. This paper presents an innovative method （magic square） in order to increase the generated power by configuring the modules of a shaded photo- voltaic array. In this approach, the physical location of the modules in the total cross tied （TCT） connected in the solar PV array is rearranged based on the magic square arrangement pattern. This connection is done without altering any electrical configurations of the modules in the PV array. This method can distribute the shading effect over the entire PV array, without concentrating on any row of modules and can achieve global peaks. For different types of shading patterns, the output power of the solar PV array with the proposed magic square configuration is compared with the traditional configurations and the performance is calculated. This paper presents a new reconfiguration technique for solar PV arrays, which increases the PV power under different shading conditions. The proposed technique facilitates the distribution of the effect of shading over the entire array, thereby, reducing the mismatch losses caused by partial shading. The theoretical calculations are tested through simulations in Matlab/ Simulink to validate the results. A comparison of power loss for different types of topologies under different types of shading patterns for a 4 × 4 array is also explained.

Representation theory of Dynkin quivers. Three contributions

The representations of the Dynkin quivers and the corresponding Euclidean quivers are treated in many books. These notes provide three building blocks for dealing with representations of Dynkin （and Euclidean） quivers. They should be helpful as part of a direct approach to study representations of quivers, and they shed some new light on properties of Dynkin and Euclidean quivers.