A study of operands in terms of maximal generalized orbits

An operand X of a monoid S is called saturated if every generalized orbit in X is contained in a union of others. Every operand has a natural decomposition as a union of an operand admitting an irredundant cover by maximal generalized orbits and of a saturated operand. There is a descending chain of suboperands of an operand which leads to the definition of the saturation length of an operand. S has no saturated operands if and only if S satisfies the ascending chain condition on orbits.     

Discrete fourier transform computation using prime Ramanujan numbers

Ramanujan numbers were introduced in [2] to implement discrete fourier transform (DFT) without using any multiplication operation. Ramanujan numbers are related to π and integers which are powers of 2. If the transform sizeN, is a Ramanujan number, then the computational complexity of the algorithms used for computing isO(N ²) addition and shift operations, and no multiplications. In these algorithms, the transform can be computed sequentially with a single adder inO(N ²) addition times. Parallel implementation of the algorithm can be executed inO(N) addition times, withO(N) number of adders. Some of these Ramanujan numbers of order-2 are related to the Biblical and Babylonian values of π [1]. In this paper, we analytically obtain upper bounds on the degree of approximation in the computation of DFT if JV is a prime Ramanujan number.     

Prospective Elementary Teachers’ Understanding of Order of Operations

This study investigates how well 381 prospective elementary, early childhood, and special education majors solved four arithmetic problems that required using the order of operations. Self‐reported data show these students to be relatively able mathematically and confident in their ability, with no substantial dislike of mathematics. The percentage of answers that were incorrect that is attributable to order of operations ranged from 21.7% to 78.5%. Overall, fewer than half the subjects answered more than two questions correctly. Of those subjects who performed multiplication before addition, which indicates some knowledge of order of operations, 30.9% performed addition before subtraction and 38.0% performed multiplication before division rather than from left to right, which suggests that instead of using the correct order of operations, these students used the common mnemonic PEMDAS or “Please excuse my dear Aunt Sally “literally, performing multiplication before division and performing addition before subtraction, rather than from left‐to‐right. Furthermore, 78.5% of subjects used the incorrect order of operations to compute ?3².     

Fuzzy Operands of Semigroups

We first obtain results of fuzzy operands of semigroups. Then some nice interplays between operands $M$ of a semigroup and its fuzzy suboperands are obtained in terms of decomposition. These are then used to revisit one important result of semigroup operands viz. “there exists a bijection between the set of all decompositions of a right operand $M$ over a semigroup $S$ and the set of a special type of operand equivalences on $M$”.     

Unitäre Transformationen großer Matrizen

Summary Strassen [2] has described a method for the multiplication of (N, N)-matrices which needs O (N ^2.8...) basic operations. Here algorithms are given forQR-decomposition and unitary transformation of arbitrary complex matrices to upper Hessenberg form and for unitary triangularization of hermitean matrices, which by use of a fast matrix multiplication with time bound O (N ) have nearly the same speed.     

Towards Minimal Addition Chains Using Ant Colony Optimisation

Modular exponentiation is one of the most important operations in almost all modern cryptosystems. It is performed using a series of modular multiplications. This operation is time consuming for large operands as is always the case in cryptography. Hence fast public-key cryptography software or hardware requires optimisation of the time consumed by a single modular multiplication and/or the reduction of the total number of modular multiplications required. This paper introduces a novel idea based on the principles of ant colony optimisation for finding a minimal addition chain that allows one to reduce the number of modular multiplications so that modular exponentiation can be implemented efficiently. The best addition chain reached by the ant system is compared to the one used in the m-ary and sliding window methods as well as with the best addition chain evolved by genetic algorithms. We demonstrate that the ant system significantly outperforms all these methods for any exponent size. ^★★ Research supported by FAPERJ () and CNPq ().     

On the Non‐Hierarchical Preconditioner Constructions

In this paper a new cyclic matrix representation of the H^–1/2 norm is presented. Its application as Schur complement preconditioning matrix requires only matrix‐vector multiplication. The computational cost of this matrix‐vector multiplication is O (N · log(N)) arithmetic operations, where N is the number of unknowns. The efficiency of the construction to elliptic problems has been verified by numerical tests. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fast algorithm for numerical solutions to Fortet's equation

A fast algorithm for computation of default times of multiple firms in a structural model is presented. The algorithm uses a multivariate extension of the Fortet's equation and the structure of Toeplitz matrices to significantly improve the computation time. In a financial market consisting of M1 firms and N discretization points in every dimension the algorithm uses O(nlogn·M·M!·N^M(M-1)/2) operations, where n is the number of discretization points in the time domain. The algorithm is applied to firm survival probability computation and zero coupon bond pricing.     

Fixed-Size Quadruples for a New, Hardware-Oriented Representation of the 4D Clifford Algebra

Clifford algebra (geometric algebra) offers a natural and intuitive way to model geometry in fields as robotics, machine vision and computer graphics. This paper proposes a new representation based on fixed-size elements (quadruples) of 4D Clifford algebra and demonstrates that this choice leads to an algorithmic simplification which in turn leads to a simpler and more compact hardware implementation of the algebraic operations. In order to prove the advantages of the new, quadruple-based representation over the classical representation based on homogeneous elements, a coprocessing core supporting the new fixed-size Clifford operands, namely Quad-CliffoSor (Quadruple-based Clifford coprocesSor) was designed and prototyped on an FPGA board. Test results show the potential to achieve a 23× speedup for Clifford products and a 33× speedup for Clifford sums and differences compared to the same operations executed by a software library running on a general-purpose processor.     

Products of operators of multidimensional structured model of systems

The author has previously defined the concept of a general system in terms of operators and operands. An operand is a mapping defined on a subset of an m-fold Cartesian product instead of the usual set and collection of k-ary relations on it. An operator is a kind of mapping between two collections of operands. Here subsystems, extensions, and the notion of P-semiexactness is studied. In particular we derive conditions such that P-semiexactness of a composition of operators, and of one factor, implies P-semiexactness of the other factor.     

A study of operands in terms of maximal generalized orbits

A left operand X of a monoid S is called saturated if every generalized orbit (g.o.) in X is contained in a union of others. Every operand has a natural decomposition as a union of an operand admitting an irredundant cover by maximal g.o. ′s and of a saturated operand. There is a descending chain of suboperands of an operand, defined in terms of maximal g.o. ′s, which leads to the definition of the saturation length of an operand. S has no saturated operands if and only if S satisfies the a.c.c. on orbits. Full proofs will be given in [2]. Communicated by A. H. Clifford     

Some remarks on multiplication ideals,ii

Let R bea commutative ring with identity. An R-module (ideal of R) A is called a multiplication module (ideal) if for each submodule N of A there exists an ideal I of R with N = I A. We give several characterizations of multiplication modules. Using the method of idealization we show how to reduce questions concerning multiplication modules to multiplication ideals. For example, we show that if S is a commutative R-algebra and ψ: M→an R-module homomorphism where M is a multiplication R-module and N is an S-module, then Sψ(M) is a multiplication S-module.     

A method of estimating computational complexity based on input conditions for N-vehicle problem

This paper proposes a method of estimating computational complexity of problem through analyzing its input condition for N-vehicle exploration problem. The N-vehicle problem is firstly formulated to determine the optimal replacement in the set of permutations of 1 to N. The complexity of the problem is factorial of N （input scale of problem）. To balance accuracy and efficiency of general algorithms, this paper mentions a new systematic algorithm design and discusses correspondence between complexity of problem and its input condition, other than just putting forward a uniform approximation algorithm as usual. This is a new technique for analyzing computation of NP problems. The method of corresponding is then presented. We finally carry out a simulation to verify the advantages of the method： 1） to decrease computation in enumeration; 2） to efficiently obtain computational complexity for any N-vehicle case; 3） to guide an algorithm design for any N-vehicle case according to its complexity estimated by the method.     

Speeding up Huff form of elliptic curves

This paper presents faster inversion-free point addition formulas for the curve \(y (1+ax^2) = cx (1+dy^2)\). The proposed formulas improve the point doubling operation count record (I, M, S, D, a are arithmetic operations over a field. I: inversion, M: multiplication, S: squaring, D: multiplication by a curve constant, a: addition/subtraction) from \(6\mathbf{{M}}+ 5\mathbf{{S}}\) to \(8\mathbf{{M}}\) and mixed addition operation count record from \(10\mathbf{{M}}\) to \(8\mathbf{{M}}\). Both sets of formulas are shown to be 4-way parallel, leading to an effective cost of \(2\mathbf{{M}}\) per either of the group operations.     

Fuzzy real algebra: Some results

In this paper, the possibility to perform easily most of the extended n-ary operations on fuzzy subsets of the real line is shown. A general algorithm is given. These results are particularized for usual operations such as addition, subtraction, multiplication, division, ‘max’ and ‘min’ operations for normalized convex fuzzy subsets of the real line, i.e. fuzzy numbers. A three parameters representation for fuzzy numbers is shown to be very convenient to perform usual operations. Lastly, interpretative comments about fuzzy real algebra are given and possible applications pointed out.     

An economical backup strategy for floppy disks

A variety of data files are preserved on floppy disks. Document files created using a word processor as well as a personal computer are, for instance, stored on floppy disks. Nevertheless, such flies are occasionally lost due to human errors, the life of the floppy disk and a failure of hardware devices, which are comprised in a word processor or a personal computer. This is called a floppy disk failure. One of the simplest methods for protecting us from such serious losses is to backup files on another floppy disk periodically. Frequent backup operations would spend considerable cost in operations themselves although they could reduce the loss at a floppy disk failure. On the contrary, rare backup operations would make the loss at a floppy disk failure very large although they could considerably save cost in backup operations. These observations reveal the significance of determining an adequate backup timing of files.The present study considers an economical backup strategy for floppy disks, which suggests backing up files when each 1/N of the total memory of a floppy disk is consumed. The long-run average cost of the proposed backup strategy is formulated as an objective function. It is shown that there always exists an economical integer N* that minimizes the long-run average cost. Numerical examples are also presented.     

On computing bounds for the least singular value of a triangular matrix

An algorithm for rapid computation of a lower bound for the least singular value of a triangular matrix is presented. It requiresO(N) operations whereN is the order of the matrix, and is based on the Perron-Frobenius theory of non-negative matrices. The input data are the diagonal elements and the off-diagonal elements of maximum modulus in each row.     

On N-Summations, I

This paper continues recent work on N-summations (see [8, 9, 12]). More specifically, it addresses the issue of existence of N-summations both for cone semirings and for prenormed semitopological semimodules. In the case of a cone semiring C we assume N-order completeness plus compatibility of N-joins with addition and multiplication to make the class of summarily bounded elements of C ^N into an N-summation for C. In the case of a prenormed semitopological semimodule M we use certain completeness properties of semitopologies on M to make the class of Cauchy elements of M ^N into an N-summation for M. Results on semitopologies and their connection with closure operators are contained in the Appendix.     

模糊一次方程求解的实际研究

传统的加法,减法,乘法和除法运算都属于自然运算,因此不能用于区间数（interval）的分析中,故需要进一步研究区间数的特殊的运算规律。本文介绍模糊集合理论中的区间数进行加、减、乘、除等特殊的算术运算规律为基础,主要介绍并提出模糊一次方程bx＋a=c的求解过程及方法,并给出其模糊方程的解。     

Log-det approximation based on uniformly distributed seeds and its application to Gaussian process regression

Maximum likelihood estimation (MLE) of hyperparameters in Gaussian process regression as well as other computational models usually and frequently requires the evaluation of the logarithm of the determinant of a positive-definite matrix (denoted by C hereafter). In general, the exact computation of is of O(N³) operations where N is the matrix dimension. The approximation of could be developed with O(N²) operations based on power-series expansion and randomized trace estimator. In this paper, the accuracy and effectiveness of using uniformly distributed seeds for approximation are investigated. The research shows that uniform-seed based approximation is an equally good alternative to Gaussian-seed based approximation, having slightly better approximation accuracy and smaller variance. Gaussian process regression examples also substantiate the effectiveness of such a uniform-seed based log-det approximation scheme.