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Minimization of Haar wavelet series and Haar spectral decision diagrams for discrete functions

COMPUTERS & ELECTRICAL ENGINEERING 2005年第3期31卷 203-216页

作者： Stankovic, RS Jankovic, D Falkowski, BJ Nanyang Technol Univ Sch Elect & Elect Engn Nanyang Ave 639798 Singapore Fac Elect Dept Comp Sci Nish Serbia Monteneg

In this paper, a minimization of Haar wavelet series for simplification of circuits and Haar based decision diagrams representing discrete multiple-valued functions is proposed. The minimization is performed by permutation of indices of generalized Haar functions. Experimental results show that this method provides reasonable reduction in the number of non-zero coefficients. The Haar series reduced this way can be useful in the circuit synthesis for realization of multiple-valued functions. The same algorithm can be also used to reduce the number of paths in decision diagrams related to the Haar wavelet transforms. In many cases, this reduction provides smaller size of such decision diagrams. (c) 2005 Elsevier Ltd. All rights reserved.

关键词： switching theory discrete transforms discrete functions wavelet transforms Haar transform decision diagrams applications of design theory

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Non-Abelian groups in optimization of decision diagrams representations of discrete functions

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FORMAL METHODS IN SYSTEM DESIGN 2001年第3期18卷 209-231页

作者： Stankovic, RS Niš Yugoslavia

This paper is devoted to the reduction of decision diagram (DD) representations of discrete functions by using the non-Abelian groups and Fourier DDs on these groups. The number of levels in a DD can be reduced through decomposition of the domain group of the represented function into large subgroups. That approach however increases the number of nodes per levels which may decrease the global reduction possibilities in the DD. At the same time, the nodes with a considerable number of outgoing edges an required. In applications, the complexity, thus the prise, of a node is proportional to the number of outgoing edges. From an inspection of optimization methods fur DDs representations, it follows that a solution of this problem can hardly be found with DDs on Abelian groups. Therefore. we are proposing the use of Fourier DDs on finite non-Abelian groups. In that way, the matrix-valued decision diagrams are introduced, since in Fourier DDs on non-Abelian groups some of constant nodes are matrices. Three DDs permit two-steps optimization. First we determine the optimal structure of the corresponding decision tree by the reduction of which a DD is derived. The optimization is done with respect to the combination of the number of nodes and levels we may require depending on the application intended. Then, we do the optimization of the representations of matrix-valued constant nodes by ordinary DDs of smaller size. In that way, the complex-valued and integer-valued Fourier DDs are derived depending on the chosen group representations for these applications where the matrix-valued Fourier DDs may not be allowed. With thus derived Fourier DDs, the reduction of both number of levels and non-terminal nodes map be achieved. Efficiency of representations with the Fourier DDs on non-Abelian groups is illustrated by the example of DDs representations of n-bit multipliers.

关键词： discrete functions decision diagrams non-Abelian groups Fourier transform

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GENERAL INTERPOLATION FORMULAS FOR SPACES OF discrete functions WITH NONUNIFORM MESHES

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Journal of Computational Mathematics 1995年第1期13卷 70-93页

作者： Zhou Yu-lin(Institute of Applied Physics and Computational Mathematics, Beijing, China ) 北京应用物理和计算数学所北京

The unequal meshsteps are unavoidable in general for scientific and engineering computations especially in large Scale computations. The analysis of difference schemes with nonuniform meshes is very rare even by use of fully heuristic methods. For the purpose of the systematic and theoretical study of the finite difference method with nonuniform meshes for the problems of partial differential equations, the general interpolation formulas for the spaces of discrete functions of one index with unequal meshsteps are established in the present work. These formulas give the connected relationships among the norms of various types, such as' the sum of powers of discrete values, the discrete maximum modulo, the discrete Holder and Lipschitz coefficients.

关键词： discrete functions Interpolation Mathematical constants Partial differential equations Difference quotients Real lines Finite difference methods Mathematical theorems Ratios Engineering

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On additivity and affinity coefficients of discrete functions

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discrete Mathematics and Applications 2012年第1期22卷 1-22页

作者： Glukhov, M.M. Zakrevskii, P.N.

We introduce and investigate the coefficients of additivity and affinity of functions on finite groups. Particular attention is devoted to Boolean functions and bent-functions. © de Gruyter 2012.

关键词： coefficients functions additivity ADDITIVITY Boolean functions discrete functions Coefficients

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The Numerical Integration of discrete functions on a Triangular Element

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Journal of Modern Transportation 2001年第1期18卷 50-42,51-58页

作者：陆宏轮仇文革关宝树 Department of Applied Mechanics and Engineering Southwest Jiaotong University School of Civil Engineering Southwest Jiaotong University

With the application of Hammer integral formulas of a continuous function on a triangular element, the numerical integral formulas of some discrete functions on the element are derived by means of decomposition and re... 详细信息

关键词： numerical integration discrete functions finite element method base function triangular element

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LOGIC PROPERTIES OF UNATE discrete AND SWITCHING functions

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IEEE TRANSACTIONS ON COMPUTERS 1977年第12期26卷 1202-1212页

作者： THAYSE, A DESCHAMPS, JP MBLE Research Laboratory

The total and local unateness of discrete and of switching functions are studied from a theoretical point of view. One shows that the local unateness leads to the concept of hazard-free transition for a discrete function. Unate covers for discrete functions are defined: they are either the smallest unate functions larger than a discrete function, or the largest unate functions smaller than a discrete function. These concepts play a key role in hazard-free design of multiple-valued networks. Three-level types of multiple-valued networks using MIN and MAX gates are presented. These networks improve, from a hazard point of view the well known two-level networks presented by Eichel-berger in the frame of switching theory.

关键词： discrete functions hazards logic design multiple-valued logic switching function unateness.

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DIRECT IMPLEMENTATION OF discrete AND RESIDUE-BASED functions VIA OPTIMAL ENCODING - A PROGRAMMABLE ARRAY LOGIC APPROACH

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IEEE TRANSACTIONS ON COMPUTERS 1983年第10期32卷 961-968页

作者： PAPACHRISTOU, CA Department of Electrical and Computer Engineering University of Cincinnati Abstract Authors References Cited By Keywords Metrics Similar Download Citation Email Print Request Permissions

This paper presents a technique for direct truth table implementation of residue-based functions by an encoding scheme that employs programmable array logic (PAL) technology. The scheme models the basic associative memory operation, i.e., the detection of matchings between input patterns and prestored information in the PAL"s. The complexity of this model is related to the amount of stored logic, i.e., the P-terms in the logic arrays. A linear programming approach is proposed for the encoding of the residue set with the objective of minimizing the complexity of addition and multiplication, modulo M, simultaneously. It is shown that the addition is more complex than the multiplication modulo M, with both (two-operand) operations being upper bounded by O(M2). Results produced using the optimal encoding compare favorably to corresponding results regarding the usual binary representation of residues. Practical constraints are also considered such as limitations on the number of pins, the number of P-terms, and the chip area, with the latter shown to be more efficiently utilized in the PAL scheme than in a ROM-or PLA-based implementation. The encoding technique is also applicable to the functions of discrete logic, in general.

关键词： Addition and multiplication mod M PLA"s ROM"s VLSI associative memories discrete functions optimal residue encoding programmable array logic (PAL) residue-based functions

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ON discrete QUASICONVEXITY CONCEPTS FOR SINGLE VARIABLE SCALAR functions

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TAIWANESE JOURNAL OF MATHEMATICS 2009年第2B期13卷 837-853页

作者： Cambini, Riccardo Riccardi, Rossana Univ Pisa Dept Stat & Appl Math Fac Econ I-56124 Pisa Italy

The aim of this paper is to propose quasiconvexity concepts for discrete single variable functions and state some related optimality conditions. Four classes of discrete quasiconvex single variable functions are intro... 详细信息

关键词： Integer programming Quasiconvexity discrete functions

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Representation of Multiple-Valued functions with Flat Vilenkin-Chrestenson Spectra by Decision Diagrams

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JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING 2014年第5-6期23卷 485-501页

作者： Stankovic, Stanislav Stankovic, Milena Astola, Jaakko Tanzpere Univ Technol Dept Signal Proc FI-33101 Tampere Finland Univ Nis Fac Elect Engn Nish 18000 Serbia

In this paper we examine the relationship between multiple-valued bent functions and Vilenkin-Chrestenson spectral invariant operations and Vilenkin-Chrestenson decision diagrams. In binary domain bent functions are a class of discrete functions with a highest degree of nonlinearity and form an essential part of cryptographic systems. Multiple-valued bent functions are an extension of bent functions to higher order finite fields. These functions are defined in terms of properties of their Vilenkin-Chrestenson spectra. We demonstrate that the application of spectral invariant operations to a given multiple-valued bent function does not alter the structure of the corresponding Vilenkin-Chrestenson decision diagrams. We exploit this property to efficiently represent whole sets of multiple-valued bent functions using a single Vilenkin-Chrestenson decision diagram. Furthermore, we present a decision diagram based method of construction of multiple-valued bent functions of arbitrary size.

关键词： Agrostis representations discrete functions Function decision diagrams multi-value Construction methods

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A Proposal of Equivalence Classes in Maximally Asymmetric functions and Their Application to Benchmark Generation 54

A Proposal of Equivalence Classes in Maximally Asymmetric Fu...

引用

54th IEEE International Symposium on Multiple-Valued Logic (ISMVL)

作者： Kometani, Rie Nagayama, Shinobu Lukac, Martin Inagi, Masato Wakabayashi, Shin'ichi Hiroshima City Univ Dept Comp & Network Engn Hiroshima Japan

ISBN: (纸本)9798350343090;9798350343083

This paper proposes some operations on maximally asymmetric functions. The proposed operations are unary, and produce another function from a maximally asymmetric function. Since the proposed operations can keep the asymmetry of functions, the produced function is maximally asymmetric as well if the original function is maximally asymmetric. Based on the operations, we can divide the set of maximally asymmetric functions into equivalence classes that have functions able to be produced from the same original function (a representative) by those operations. By using the operations and some representatives of the equivalence classes, we can efficiently generate many benchmarks of maximally asymmetric functions without generating functions from scratch. This paper also shows that the number of the equivalence classes for some examples is very small. This means that most of the benchmarks can be generated by applying only the operations.

关键词： Maximally asymmetric functions discrete functions characteristic analyses benchmark generation equivalence classes

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