conjugacy classes in finite permutation groups via homomorphic images the lifting of results from factor groups to the full group is a standard technique for solvable groups this paper shows how to utilize this approach in the case of non solvable normal subgroups to compute the conjugacy classes of a finite group

a comment on generalized reed muller forms as a tool to detect symmetries this comment relates to two published articles by tsai et al ibid vol 45 1 1996 and vol 46 2 1997 that used four types of two variable boolean symmetries and explained their transitivity conditions and relations with reed muller transform we show that some research in this area had been done earlier about which the authors are unaware

cone kernel representation versus instantaneous power spectrum the cone kernel representation ckr and the instantaneous power spectrum ifs are two time frequency representations tfrs where the cross terms are localized in the region of auto terms exploring their relationship for discrete sequences we show that the ifs kernel is only two extreme terms of the ckr kernel further comparing other properties of these two tfrs viz invertibility aliasing and noise robustness we show that there is aliasing in some transform domains for signals sampled at nyquist rate and that the ckr has better noise robustness than the ifs

a comment on boolean functions classification via fixed polarity reed muller form a correction to the classification of the space of n variable boolean functions proposed in the paper cited in c c tsai et al 1997 is reported here and commented upon

on the convergence of parallel asynchronous block iterative computations abstract this paper considers the convergence problem of parallel asynchronous block iterative computation schemes a new mathematical state space model for a class of nonlinear time varying parallel iterative schemes is proposed using this model which generalizes several models of the chazan miranker type together with large scale systems and liapunov techniques it is shown that the well known quasidominance condition on a certain aggregated matrix guarantees exponential convergence of this class of methods

a minimal universal test set for self test of exor sum of products circuits a testable exor sum of products esop circuit realization and a simple universal test set which detects all single stuck at faults in the internal lines and the primary inputs outputs of the realization are given since esop is the most general form of and exor representations our realization and test set are more versatile than those described by other researchers for the restricted grm fprm and pprm forms of and exor circuits our circuit realization requires only two extra inputs for controllability and one extra output for observability the cardinality of our test set for an n input circuit is n 6 for built in self test bist applications we show that our test set can be generated internally as easily as a pseudorandom pattern and that it provides 100 percent single stuck at fault coverage in addition our test set requires a much shorter test cycle than a comparable pseudoexhaustive or pseudorandom test set

minimization of and exor expressions using rewrite rules conditions for generating optimal two level and exor representations using rewrite rules are considered four results are presented first it is shown that a necessary condition for obtaining minimality is a temporary increase in the size of expressions during minimization second a sufficient condition for obtaining minimality that consists of adding certain two rules to rule sets proposed in the literature is given third transformations that allow the minimization of an expression to proceed by minimizing a transformed expression instead are defined fourth it is determined experimentally that the above three theoretical results lead to better benchmarks results as well

derivation of reduced test vectors for bit parallel multipliers over gf 2 m this paper presents an algebraic testing method for detecting stuck at faults in the polynomial basis pb bit parallel bp multiplier circuits over gf 2m the proposed technique derives the test vectors from the expressions of the inner product ip variables without any requirement of the atpg tool this low complexity testing method requires 2m 1 test vectors for detecting single stuck at faults in the and part and multiple stuck at faults in the exor part of the multiplier circuits the test vectors are independent of the multiplier s structure as proposed in t a gulliver et al 1991 but are dependent on m for the multiplier circuits the test set is found to be smaller in size than the atpg generated test set the test set provides 100 percent single stuck at fault coverage

constructing differentially 4 uniform permutations over bbf _ 2 2k via the switching method many block ciphers use permutations defined on f 22k with low differential uniformity high nonlinearity and high algebraic degree as their s boxes to provide confusion it is well known that for a function on f 2n the lowest differential uniformity is 2 and the functions achieving this lower bound are called almost perfect nonlinear apn functions however due to the lack of knowledge on apn permutations on f 22k differentially 4 uniform permutations are usually chosen as s boxes for example the currently endorsed advanced encryption standard chooses one such function the multiplicative inverse function as its s box by a recent survey on differentially 4 uniform permutations over f 22k there are only five known infinite families of such functions and most of them have small algebraic degrees in this paper we apply the powerful switching method to discover many ccz inequivalent infinite families of such functions on f 22k with optimal algebraic degree where k is an arbitrary positive integer this greatly expands the list of differentially 4 uniform permutations and hence provide more choices for the s boxes furthermore lower bounds for the nonlinearity of the functions obtained in this paper are presented and they imply that some infinite families have high nonlinearity

on a new boolean function with applications consider a hypercube of 2 sup n points described by n boolean variables and a subcube of 2 sup m points m spl les n as is well known the boolean function with value 1 in the points of the subcube can be expressed as the product and of n m variables the standard synthesis of arbitrary functions exploits this property we extend the concept of subcube to the more powerful pseudocube the basic set is still composed of 2 sup m points but has a more general form the function with value 1 in a pseudocube called pseudoproduct is expressed as the and of n m exor factors each containing at most m 1 variables subcubes are special cases of pseudocubes and their corresponding pseudoproducts reduce to standard products an arbitrary boolean function can be expressed as a sum of pseudoproducts spp this expression is in general much shorter than the standard sum of products as demonstrated on some known benchmarks the logical network of an n bit adder is designed in spp as a relevant example of application of this new technique a class of symmetric functions is also defined particularly suitable for spp representation

reduction of average path length in binary decision diagrams by spectral methods this paper deals with analytic methods for the calculation and reduction of the average path lengths apls in binary decision diagrams bdds usually information theoretic measures and information theoretic techniques are used to construct bdds of minimal apls specifically the mutual information between a boolean function and its variables and the shannon fano prefix coding are utilized this paper deals with the problem of the apl reduction by using spectral techniques particularly methods based on the properties of the walsh spectrum of a boolean function and its autocorrelation function are discussed it is shown that the apl is a linear function of the autocorrelation values that is the apl depends on the boolean function s properties in the time domain autocorrelation rather on the boolean function s properties in the frequency domain walsh spectrum in addition it is shown that information theoretic criteria like the mutual information or the conditional entropy are equivalent to frequency domain criteria consequently existing information theoretic approaches for apl reduction that are based on mutual information criterion or the walsh transform coefficients may derive a suboptimal apl the representation of the apl as a function of the autocorrelation values opens a way to determine the optimal ordering of the input variables analytically two procedures for apl reduction by ordering and by using linear combinations of the input variables are presented 1 minimization by using the autocorrelation values and 2 minimization by using the mutual information between the boolean function and a linear function of the input variables the time domain approach may derive a linearized bdd of a lower apl whereas the information theoretic approach has a lower computational complexity with comparable performance experimental results show the efficiency of the suggested techniques

symmetric and threshold boolean functions are exhaustive the worst case number of variable evaluations testing cost of boolean functions is examined following up on a result by rivest and vuillemin we show that all symmetric as well as all linearly separable boolean functions are exhaustive that is have a pessimal worst case testing cost

boolean functions classification via fixed polarity reed muller forms in this paper we present a new method to characterize completely specified boolean functions the central theme of the classification is the functional equivalence a k a boolean matching two boolean functions are equivalent if there exists input permutation input negation or output negation that can transform one function to the other we have derived a method that can efficiently identify equivalence classes of boolean functions the well known canonical fixed polarity reed muller fprm forms are used as a powerful analysis tool the necessary transformations to derive one function from the other are inherent in the fprm representations to identify uniquely each equivalence class a set of well known characteristics of boolean functions and their variables including linearity symmetry total symmetry self complement and self duality are employed it is shown that all the equivalence classes of four variable functions are uniquely identified where majority of the classes have a single fprm form as their representative the boolean matching has applications in technology mapping and in design of standard cell libraries

models of parallel chaotic iteration methods abstract we consider two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations ax b in the first model each processor can carry out an arbitrary number of local iterations before the next global approximation to the solution is formed in the second model any processor can update the global approximation which resides in the central processor at any time this model is a generalization of a sequential iterative scheme due to ostrowski called the free steering group jacobi iterative scheme and a chaotic relaxation point iterative scheme due to chazan and miranker we show that when a is a monotone matrix and all the splittings are weak regular both models lead to convergent schemes

generalized multisplitting asynchronous iteration abstract we give a generalized asynchronous iteration based on the multisplitting of linear or nonlinear operators to solve systems of equations the condition for convergence can be imposed only on the components which actually enter the computation the results contain synchronous multisplitting iterations as a special case

easily testable realizations for generalized reed muller expressions this paper presents a design method of easily testable and exor networks it is an improvement of reddy 1972 and saluja reddy s 1975 methods and has the following features the network uses generalized reed muller expressions grms instead of positive polarity reed muller expressions pprms the average number of products for grms is less than half of that for pprms and is less than that of sum of products expressions sops the network consists of a literal part an and part an exor part and a check part the exor part can be a tree instead of a cascade thus the network is faster the test detects multiple stuck at faults under the assumption that the faults occur at most one part either the literal part the and part the exor part or the check part

partially asynchronous parallel algorithms for network flow and other problems the problem of computing a fixed point of a nonexpansive function f is considered sufficient conditions are provided under which a parallel partially asynchronous implementation of the iteration x f x converges these results are then applied to i quadratic programming subject to box constraints ii strictly convex cost network flow optimization iii an agreement and a markov chain problem iv neural network optimization and v finding the least element of a polyhedral set determined by a weakly diagonally dominant leontief system finally simulation results illustrating the attainable speedup and the effects of asynchronism are presented

generalized reed muller forms as a tool to detect symmetries in this paper we present a new method for detecting groups of symmetric variables of completely specified boolean functions the canonical generalized reed muller grm forms are used as a powerful analysis tool to reduce the search space we have developed a set of signatures that allow us to identify quickly sets of potentially symmetric variables our approach allows for detecting symmetries of any number of inputs simultaneously totally symmetric functions can be detected very quickly the traditional definitions of symmetry have also been extended to include more types this extension has the advantage of grouping input variables into more classes experiments have been performed on mcnc benchmark cases and the results verify the efficiency of our method

on closedness and test complexity of logic circuits the concept of closedness of a set of logic functions under stuck type faults is introduced all sets of logic functions closed under stuck type faults are classified for the sets of logic functions closed under stuck type faults the test complexity and the universal test sets are considered it is shown that for each class of linear functions or functions and and functions both the minimum numbers of multiple fault detection tests and multiple fault location tests are exactly n 1 where n is the number of inputs of the circuits and that there exists universal test sets with n 1 tests to detect and locate all multiple faults in such circuits

convergence of sequential and asynchronous nonlinear paracontractions we establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces applications to the solution of linear systems of equations with convex constraints are outlined a first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented

some properties of the generalized time frequency representation with cone shaped kernel the cone shaped kernel generalized time frequency representation gtfr of zhao atlas and marks zam has been shown empirically to generate quite good time frequency representation in comparison to other approaches the authors analyze some specific properties of this gtfr and compare them to other tfrs asymptotically the gtfr is shown to produce results identical to that of the spectrogram for stationary signals interference terms normally present in many gtfrs are shown to be attenuated drastically by the use of the zam gtfr the ability of the zam gtfr to track frequency hopping is shown to be close to that of the wigner distribution when a signal is subjected to white noise the zam gtfr produces an unbiased estimate of the zam gtfr of the signal without noise in many other gtfrs the power spectral density of the noise is superimposed on the gtfr of the signal it is also shown that in discrete form the zam gtfr is generally invertible

minimum variance time frequency distribution kernels for signals in additive noise we derive the minimum variance time frequency distribution kernel for signals in additive circular complex gaussian white noise processes it is shown that the kernel that minimizes the average variance and simultaneously satisfies the time frequency constraints for the noise only ease remains optimum in the presence of frequency modulated fm signals

easily testable multiple valued logic circuits derived from reed muller circuits s m reddy 1972 showed that the binary circuits realizing reed muller canonical form are easily testable in this paper we extend reddy s result to multiple valued logic circuits employing more than two discrete levels of signal the electronic fabrication of such circuits became feasible due to the recent advances in integrated circuit technology we show that in the multiple valued case several new phenomena occur which allow us to asymptotically reduce the upper bound on the number of tests required for fault detection but make the generation of tests harder

parallel nonlinear multisplitting methods linear multisplitting methods are known as parallel iterative methods for solving a linear systemax b we extend the idea of multisplittings to the problem of solving a nonlinear system of equationsf x 0 our nonlinear multisplittings are based on several nonlinear splittings of the functionf in a parallel computing environment each processor would have to calculate the exact solution of an individual nonlinear system belonging to his nonlinear multisplitting and these solutions are combined to yield the next iterate although the individual systems are usually much less involved than the original system the exact solutions will in general not be available therefore we consider important variants where the exact solutions of the individual systems are approximated by some standard method such as newton s method several methods proposed in literature may be regarded as special nonlinear multisplitting methods as an application of our systematic approach we present a local convergence analysis of the nonlinear multisplitting methods and their variants one result is that the local convergence of these methods is determined by an induced linear multisplitting of the jacobian off

asynchronous two stage iterative methods parallel block two stage iterative methods for the solution of linear systems of algebraic equations are studied convergence is shown for monotone matrices and for h matrices two different asynchronous versions of these methods are considered and their convergence investigated

time varying spectral estimation using the instantaneous power spectrum ips the spectral content of time varying signals is examined using a modified version of the instantaneous power spectrum ips the short duration periodogram and the pseudo wigner ville distribution pwd have been used extensively to display the time varying spectral information a disadvantage of the periodogram is its inherently poor resolution due to the requirement of matching the resolution width to the spectral dynamics of the signal potential problems associated with pwd are spectral cross terms and an inability to resolve spectral components at the endpoints of the analysis segment a modified version of ips is shown to be less sensitive to these types of problems in addition ips can work with real valued signals sampled at the traditional nyquist rate

fault detection in combinational networks by reed muller transforms a new approach for fault detection in combinational networks based on reed muller rm transforms is presented an upper bound on the number of rm spectral coefficients required to be verified for detection of multiple stuck at faults and single bridging faults at the input lines of an n input network is shown to be n the time complexity time required to test a network for detection of multiple terminal faults and the storage required for storing the test are determined an upper bound is found for the minimum number of test patterns required to detect a fault the authors present standard tests based on this result with a simple test generation procedure and upper bounds on minimal numbers of test patterns

some convergence results for asynchronous algorithms nous presentons dans cet article des resultats de convergence des algorithmes asynchrones bases essentiellement sur la notion classique de contraction nous generalisons en particulier tous les resultats de convergence de ces algorithmes qui font l hypothese de contraction en norme vectorielle qui recemment a ete tres souvant utilisee par ailleurs l hypothese de contraction en norme vectorielle peut se trouver difficile voire impossible a verifier pour certains problemes qui peuvent etre cependant abordes dans le cadre de la contraction classique que nous adoptons in this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction we generalize in particular all convergence results for those algorithms which are based on the vectorial norm hypothesis in wide spread use recently certain problems for which the vectorial norm hypothesis can be difficult or even impossible to verify can nontheless be tackled within the scope of the classical contraction that we adopte

a general asynchronous block iterative model with related convergence conditions abstract this paper formulates a general time varying asynchronous block iterative model a convergence condition for asynchronous block iterations based on this model is given compared to existing conditions for similar and shown to be strictly weaker

on the complexity of mod 2l sum pla s consideration is given to the realization of logic functions by using plas with an exclusive or exor array where a function is represented by mod 2 exor sum of products esops and both true and complemented variables are used the authors propose a new pla structure using an exor array they derive upper bounds on the number of products of this type of pla that are useful for estimating the size of a pla as well as for assessing the minimality of the solutions obtained by heuristic esop minimization algorithms computer simulation using randomly generated functions shows that plas with the exor array require on the average fewer products than conventional plas for symmetric functions the authors conjecture that the plas with an exor array require at most as many products as the conventional plas the proposed plas can be made easily testable by adding a small amount of hardware

optimization of polynomial expressions by using the extended dual polarity reed muller expressions and their various extensions and generalizations for binary and multiple valued logic functions are an important class of discrete function representations that are often used in practical applications these expressions can be uniformly viewed as discrete polynomial expressions over finite fields gf 2 and gf q or the field of rational numbers in the case of expressions with integer valued coefficients the optimization of them in the number of product terms count is performed by selecting either positive or negative literals polarities for variables in the functions to be represented since there are no ways to select in advance the polarity for variables that will result in most compact expression for a given function all possible expressions have to be generated and the simplest of them selected this is a task computationally very demanding the complexity of which is o q n times c where c is the time to calculate a particular polarity since the reduction of the first factor may lead to missing the most compact expression the reduction of c is the single option to speed up the procedure in this paper we propose an approach to the solution of this problem by exploiting the notion of extended dual polarity which provides a simple way of ordering polarities to obtain an effective way of finding the optimal one by reducing the time to move between them the method still implies exhaustive search but it is an optimized search which may be expressed in very simple rules resulting in efficient implementation experimental results illustrate the effectiveness of the proposed method

computing affine equivalence classes of boolean functions by group isomorphism affine equivalence classification of boolean functions has significant applications in logic synthesis and cryptography previous studies for classification have been limited by the large set of boolean functions and the complex operations on the affine group although there are many research on affine equivalence classification for parts of boolean functions in recent years there are very few results for the entire set of boolean functions the best existing result has been achieved by harrison with 15768919 affine equivalence classes for 6 variable boolean functions this paper presents a concise formula for affine equivalence classification of the entire set of boolean functions as well as a formula for affine classification of boolean functions with distinct on set size respectively the method outlined in this paper greatly simplifies the affine group s action by constructing an isomorphism mapping from the affine group to a permutation group by this method we can compute the affine equivalence classes for up to 10 variables experiment results indicate that our scheme for calculating the affine equivalence classes for more than 6 variables is a significant advancement over previous published methods

advances in parallel algorithms supercomputers pipelining and farming parallel computing paradigms neural network paradigm parallel task driven versus process driven approaches parallel branch and bound method parallel divide and conquer approach parallel tree and graph searching parallel algebraic transformation approach parallel approximate and probabilistic methods parallel dynamic programming method parallel exploration of maze parallel graph reduction algorithms parallel algorithms for matrix operations parallel pattern recognition methods parallel stable marriage problem parallel travelling salesman problem neural network approach computational complexity of parallel algorithms performance evaluation of parallel algorithms

constructions of quadratic and cubic rotation symmetric bent functions in this paper we consider constructions of rotation symmetric bent functions which are of the forms fc x i 1m 1 ci j 0n 1 xjxi j cm j 0m 1 xjxm j and ft x i 0n 1 xixt ixm i xixt i i 0m 1 xixm i where n 2m ci 0 1 the subscript u of xu in the previous expressions is taken as u modulo n for each case a necessary and sufficient condition is obtained to the best of our knowledge this class of cubic rotation symmetric bent functions is the first example of an infinite class of nonquadratic rotation symmetric bent functions

asynchronous block iterative methods for almost linear equations abstract this paper gives convergence conditions for asynchronous block iterative methods for the solution of the almost linear equation ax f x where a is a linear operator f a block diagonal lipschitz continuous operator and x a vector in terms of a splitting of a and the lipschitz constant of f the methods used are a combination of the contraction mapping approach using a vectorial norm and a large scale systems approach using vector difference inequalities the load flow equations for a power system are almost linear in the above sense and considerable speedup can be obtained on a four transputer machine

affine equivalence of quartic monomial rotation symmetric boolean functions in prime power dimension in this paper we analyze and exactly compute the number of affine equivalence classes under permutations for quartic monomial rotation symmetric functions in prime and prime power dimensions