parallel high radix nonrestoring division an algorithm for high radix nonrestoring division is proposed which combines a cost efficient quotient estimation technique with collapsing of the division into one operation each iteration the quotient estimation technique is a direct combinatorial algorithm which allows a more generalized partial remainder than similar studies in addition the successor partial remainder is calculated directly from the previous partial remainder the intermediate steps of calculating the next quotient estimate multiplying it by the divisor and then subtracting the result from the previous remainder are telescoped into one step this is possible due to the use of simple direct combinatorial equations for the quotient estimate thus one operation is used to generate the next remainder from the previous remainder this operation is similar to a multiplication with a reduced carry propagate adder this algorithm reduces the latency of each iteration to less than the delay of one multiplication thus high radix nonrestoring division is possible with a low latency and no need for an off chip rom to hold a quotient estimate

invariant grids for reaction kinetics abstract in this paper we construct low dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold mim mim is based on a formulation of the condition of invariance as an equation and its solution by newton iterations a grid based version of mim is developed the method of invariant grids we describe the newton method and the relaxation method for the invariant grids construction the problem of the grid correction is fully decomposed into the problems of the grid s nodes correction the edges between the nodes appear only in the calculation of the tangent spaces this fact determines high computational efficiency of the method of invariant grids the method is illustrated by two examples the simplest catalytic reaction michaelis menten mechanism and the hydrogen oxidation the algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed generalizations to open systems are suggested the set of methods covered makes it possible to effectively reduce description in chemical kinetics

signed digit addition and related operations with threshold logic assuming signed digit number representations we investigate the implementation of some addition related operations assuming linear threshold networks we measure the depth and size of the networks in terms of linear threshold gates we show first that a depth 2 network with o n size weight and fan in complexities can perform signed digit symmetric functions consequently assuming radix 2 signed digit representation we show that the two operand addition can be performed by a threshold network of depth 2 having o n size complexity and o 1 weight and fan in complexities furthermore we show that assuming radix 2n 1 signed digit representations the multioperand addition can be computed by a depth 2 network with o n sup 3 size with the weight and fan in complexities being polynomially bounded finally we show that multiplication can be performed by a linear threshold network of depth 3 with the size of o n sup 3 requiring o n sup 3 weights and o n sup 2 log n fan in

division using a logarithmic exponential transform to form a short reciprocal two trees are used sequentially to calculate an approximation to 1 a where 1 a 2 these trees calculate the logarithm and exponential and the division reciprocation process can be described by 1 a e lna for bit skip accuracy of six to 10 this logarithmic exponential method uses significantly less hardware with respect to the scheme in 3 and the delays may be greater or less than those of 3 depending on the method used and the minimum bit skip

radix 2 division with over redundant quotient selection in this paper we present a new radix 2 division algorithm that uses a recurrence employing simple 3 to 2 digit carry free adders to perform carry free addition subtraction for computing the partial remainders in radix 2 signed digit form the quotient digit during any iteration of the division recursion is generated from the two most significant radix 2 digits of the partial remainder and independent of the divisor in over redundant radix 2 digit form i e with digits which belong to the digit set 2 1 0 1 2 the over redundant quotient digits are then converted to the conventional radix 2 digits belonging to the set 1 0 1 by using a reduction technique this division algorithm is well suited for ieee 754 standard operands belonging to the range 1 2 and is slightly faster than previously proposed radix 2 designs such as the radix 2 srt which do not employ input scaling since the quotient selection for such algorithms is a function of more than two most significant radix 2 digits of the partial remainder in comparison with the designs that employ input scaling the proposed design although slightly slower saves hardware required for scaling purposes

a generalization of circulant hadamard and conference matrices abstract we study the existence and construction of circulant matrices c of order n 2 with diagonal entries d 0 off diagonal entries 1 and mutually orthogonal rows these matrices generalize circulant conference d 0 and circulant hadamard d 1 matrices we demonstrate that matrices c exist for every order n and for d chosen such that n 2 d 2 and we find all solutions c with this property furthermore we prove that if c is symmetric or n 1 is prime or d is not an odd integer then necessarily n 2 d 2 finally we conjecture that the relation n 2 d 2 holds for every matrix c which generalizes the circulant hadamard conjecture we support the proposed conjecture by computing all the existing solutions up to n 50

binary and quadriphase sequences with optimal autocorrelation properties a survey time discrete signals with good autocorrelation properties are used in various applications in communications engineering from an implementation point of view usually sequences with a small phase alphabet and a maximal energy efficiency i e a uniform envelope are most favorable for this reason known methods as well as several new construction techniques of binary and quadriphase sequences with optimal or best known autocorrelation properties are discussed in this correspondence in many cases the achievable correlation properties can be improved significantly if a single zero element per sequence is accepted these almost binary and almost quadriphase sequences are considered as well the optimality criteria used include the maximum absolute sidelobe and the merit factor of the periodic and the odd periodic autocorrelation function respectively for sequence lengths of up to 44 in the binary case and up to 32 in the quadriphase case the best known parameters obtained by computer search are compared with the constructed results

over redundant digit sets and the design of digit by digit division units over redundant digit sets are defined as those ranging from spl minus s to s with s spl ges b b being the radix this paper presents new techniques for the direct computation of division that use an over redundant digit set for representing the quotient instead of simply redundant ones used previously in particular general criteria for synthesizing the digit selection rules and remainder updating are given for any radix and index of redundancy a methodology combining the use of over redundant digit sets with the prescaling of the divisor is also studied in order to achieve radix b division units with trivial digit selection functions it is also shown for the specific case of radix 4 that using a prescaling slightly wider than in a radix 4 unit by m d ercegovac and t lang 1990 possible to avoid the digit selection table the paper also presents a modified algorithm for on the fly conversion of the result into the irredundant form the proposed methodology can be considered as an alternative to existing division techniques

prescribed absolute values character sums and spectrum integrality abstract let g be a group and let reg be the complex left regular representation of g we consider the following problem for which inverse closed subsets s g the spectrum of the matrix g s reg g is integral that is all of the eigenvalues of the matrix are integers for abelian g a complete characterization due to bridges and mena is known here we are interested in the case where g is the dihedral group d 2 n which turns out to be a far reaching generalization of the abelian case we obtain a complete characterization when n is an odd prime and present partial results when n is a prime power

unified mixed radix 2 4 redundant cordic processor we present a unified mixed radix cordic algorithm with carry save arithmetic with a constant scale factor the pipelined architecture of the processor is determined by a unique sequence of microrotations for the two modes of operation rotation and vectoring in circular and hyperbolic coordinates the combination of radix 2 and radix 4 microrotations allows us to reduce the latency and size of the pipeline significantly the unified algorithm is based on the correcting microrotation method which we have extended to the vectoring mode in hyperbolic coordinates we have also generalized the use of radix 4 microrotations to the two operation modes and coordinate systems

nonexistence of abelian difference sets lander s conjecture for prime power orders in 1963 ryser conjectured that there are no circulant hadamard matrices of order 4 and no cyclic difference sets whose order is not coprime to the group order these conjectures are special cases of lander s conjecture which asserts that there is no abelian group with a cyclic sylow p subgroup containing a difference set of order divisible by p we verify lander s conjecture for all difference sets whose order is a power of a prime greater than 3

planar functions relative difference sets and character theory planar functions from2nto3nare studied in this paper by investigating the character values of the corresponding relative difference sets we obtain some nonexistence results of planar functions in particular we show that there is no planar functions fromzpqto4pq wherepandqare any primes we also show that except for four undecided cases there is no planar function from5nto6nifnis not a prime andn 50 000

new bounds and optimal binary signature sets part ii aperiodic total squared correlation we derive new bounds on the aperiodic total squared correlation atsc of binary antipodal signature sets for any number of signatures k and any signature length l we then present optimal designs that achieve the new bounds for several k l cases as interesting arguably side results we show that individual maximal merit factor sequences for example barker sequences are single user atsc optimal while neither the familiar gold nor the kasami set designs are atsc optimal in general the atsc optimal signature set designs provided in this work are in this sense better suited for asynchronous and or multipath code division multiplexing applications

cyclotomic integers and finite geometry the most powerful method for the study of finite geometries with regular or quasiregular automorphism groups g is to translate their definition into an equation over the integral group ring 2 g and to investigate this equation by applying complex representations of g for the definitions and the basic facts see section 2 if g is abelian this approach boils down to proving or disproving the existence of elements of 2 g with 0 1 coefficients whose character values are cyclotomic integers of certain prescribed absolute values up to now there have been two general methods to tackle this problem namely hall s multiplier concept and turyn s selfconjugacy approach however both methods need severe technical assumptions and thus are not applicable to many classes of problems despite many efforts over a period of more than 30 years no general method has been found to overcome these difficulties in this paper we present a new approach to the study of combinatorial structures via group ring equations which works without any restrictive assumptions in order to understand the method of the present paper it will be instructive to briefly discuss the self conjugacy concept first turyn 51 demonstrated that the character method for the study of group ring equations works very nicely under the so called self conjugacy condition an integer n is called self conjugate modulo m if all prime ideals above n in the mth cyclotomic field q m are invariant under complex conjugation under this condition it is possible to find all cyclotomic integers in q m of absolute value nt 2 for any positive integer t it is the complete knowledge of the cyclotomic integers of prescribed absolute value which makes the character method work so well under the self conjugacy condition since turyn s fundamental work 51 there have been dozens of papers extending and refining his approach however all these results are restricted to the case of self conjugacy and that is a very severe restriction indeed namely the probability that n is selfconjugate modulo m decreases exponentially fast in the number of distinct prime divisors of n and m see remark 2 2 this means that the self conjugacy method fails in almost all cases one may ask if it is possible to extend turyn s method in order to get rid of the self conjugacy assumption it turns out that in general this is impossible at least with present day methods the required complete knowledge of the cyclotomic integers of prescribed absolute value would yield an

a fast radix 4 division algorithm and its architecture in this paper we present a fast radix 4 division algorithm for floating point numbers this method is based on svoboda s division algorithm and the radix 4 redundant number system the algorithm involves a simple recurrence with carry free addition and employs prescaling of the operands in the proposed divider implementation each radix 4 digit belonging to set 3 3 of the quotient and partial remainder is encoded using two radix 2 digits belonging to the set 1 0 1 and this leads to hardware simplicity the quotient digits are determined by observing three most significant radix 2 digits of the partial remainder and independent of the divisor the architecture presented for the proposed algorithm is faster than previously proposed radix 4 dividers which require at least four digits of the partial remainder to be observed to determine quotient digits

low latency floating point division and square root unit digit recurrence algorithms are widely used in actual microprocessors to compute floating point division and square root these iterative algorithms present a good trade off in terms of performance area and power we present a floating point division and square root unit which implements a radix 64 floating point division and a radix 16 floating point square root to have an affordable implementation each radix 64 division iteration and radix 16 square root iteration are made of simpler radix 4 iterations 3 radix 4 iterations in division and 2 in square root speculation is used between consecutive radix 4 iterations to get a reduced timing there are three different parts in digit recurrence implementations initialization digit iterations and rounding the digit iteration is the iterative part and it uses the same logic for several cycles division and square root share partially the initialization and rounding stages whereas each one has different logic for the digit iterations the result is a low latency floating point divider and square root requiring 11 6 and 4 cycles for double single and half precision division with normalized operands and result and 15 8 and 5 cycles for square root one or two additional cycles are needed in case of subnormal operand s or result

a triangular discontinuous galerkin method for non newtonian implicit constitutive models of rarefied and microscale gases the discontinuous galerkin dg method has been popular as a numerical technique for solving the conservation laws of gas dynamics in the present study we develop an explicit modal dg scheme for multi dimensional conservation laws on unstructured triangular meshes in conjunction with non newtonian implicit nonlinear coupled constitutive relations nccr special attention is given to how to treat the complex non newtonian type constitutive relations arising from the high degree of thermal nonequilibrium in multi dimensional gas flows within the galerkin framework the langmuir velocity slip and temperature jump conditions are also implemented into the two dimensional dg scheme for high knudsen number flows as a canonical scalar case newtonian and non newtonian convection diffusion burgers equations are studied to develop the basic building blocks for the scheme in order to verify and validate the scheme we applied the scheme to a stiff problem of the shock wave structure for all mach numbers and to the two dimensional hypersonic rarefied and low speed microscale gas flows past a circular cylinder the computational results show that the nccr model yields the solutions in better agreement with the direct simulation monte carlo dsmc data than the newtonian linear navier stokes fourier nsf results in all cases of the problem studied

a remark on reducing iteration time when result digit is zero for radix 2 srt division and square root with redundant remainders in a previous paper by p montuschi and l ciminiera ibid vol 42 no 2 p239 246 feb 1993 an architecture for shared radix 2 division and square root has been presented whose main characteristic is the ability to avoid any addition subtraction when the digit 0 has been selected here we emphasize the characteristics of the digit selection mechanism used by montuschi and ciminiera by presenting a small modification of the digit selection hardware which has the benefit to further reduce the computation delay with respect to the time estimated in that work

exact summation of the chapman enskog expansion from moment equations the nonlinear recurrence chapman enskog solution for the linearized grad ten moment equations is resummed exactly using this solution the stability of higher order hydrodynamics in various approximations is discussed

higher radix square root with prescaling a scheme for performing higher radix square root based on prescaling of the radicand is presented to reduce the complexity of the result digit selection the scheme requires several steps namely multiplication for prescaling the radicand square root multiplication for prescaling for the division and division online algorithms are used to reduce the overall time and pipelining to reuse the different modules an estimate of the execution time for a radix 256 unit for double precision square root and a comparison with other implementations indicate that the proposed approach is an alternative to consider when designing a square root unit

new binary sequences with optimal autocorrelation magnitude new binary sequences of period n 4 2m 1 for even m ges 4 are found where the sequences are described by a 4 x 2m 1 array structure the new sequences are almost balanced and have four valued autocorrelation i e n 0 plusmn4 which is optimal with respect to autocorrelation magnitude the complete autocorrelation distribution and the exact linear complexity of the sequences are mathematically derived finally it is shown that the sequences are implemented by a combination of linear feedback shift registers and a simple logic

on the fly conversion of redundant into conventional representations an algorithm to convert redundant number representations into conventional representations is presented the algorithm is performed concurrently with the digit by digit generation of redundant forms by schemes such as srt division it has a step delay roughly equivalent to the delay of a carry save adder and simple implementation the conversion scheme is applicable in arithmetic algorithms such as nonrestoring division square root and on line operations in which redundantly represented results are generated in a digit by digit manner from most significant to least significant

choices of operand truncation in the srt division algorithm the paper presents an analysis of the number of partial remainder digits and divisor bits that must be examined in the srt division algorithm the number of examined digits is found to be the same for both signed digit and 2s complement partial remainder representations and appears to increase as 3log sub 2 r approximately where r is the radix of the divider in some cases it proves advantageous to examine a fractional number of remainder digits by inspecting more positive than negative bits

simple radix 4 division with operands scaling a radix 4 division algorithm with operands scaling is proposed the algorithm uses a recurrence with redundant addition carry save or signed digit and combines simple scaling with a quotient selection function that depends only on the estimate of the partial remainder and is independent of the divisor the scheme results in a significant speedup with respect to both the radix 2 and radix 4 without scaling

very high radix square root with prescaling and rounding and a combined division square root unit an algorithm for square root with prescaling and selection by rounding is developed and combined with a similar scheme for division since division is usually more frequent than square root the main concern of the combined implementation is to maintain the low execution time of division while accepting a somewhat larger execution time for square root the algorithm is presented in detail including the mathematical development of bounds for the first square root digit and for the scaling factor the proposed implementation is described evaluated and compared with other combined div sqrt units the comparisons show that the proposed scheme potentially produces a significant speed up for division whereas for square root the speed up is small

on aperiodic and periodic complementary binary sequences we give direct and recursive constructions for aperiodic and periodic complementary sequences using these constructions many missing entries in the table of bomer and antweiler 1990 can be filled

reduced description in the reaction kinetics models of complex reactions in thermodynamically isolated systems often demonstrate evolution towards low dimensional manifolds in the phase space for this class of models we suggest a direct method to construct such manifolds and thereby to reduce the eective dimension of the problem the approach realizes the invariance principle of the reduced description it is based on iterations rather than on a small parameter expansion it leads to tractable linear problems and is consistent with thermodynamic requirements the approach is tested with a model of catalytic reaction c 2000 elsevier science b v all rights reserved

hilbert s 6th problem exact and approximate hydrodynamic manifolds for kinetic equations the problem of the derivation of hydrodynamics from the boltz mann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions we review a few in stances where such hydrodynamic manifolds were found analytically both as the result of summation of the chapman enskog asymptotic expansion and by the direct solution of the invariance equation these model cases comprising grad s moment systems both linear and nonlinear are studied in depth in order to gain understanding of what can be expected for the boltzmann equa tion particularly the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short wave limit and the viscosity modifica tion at high divergence of the flow velocity are indicated as severe obstacles to the resolution of hilbert s 6th problem furthermore we review the derivation of the approximate hydrodynamic manifold for the boltzmann equation using newton s iteration and avoiding smallness parameters and compare this to the exact solutions additionally we discuss the problem of projection of the boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts finally a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the boltzmann equation in the future

binary sequences with merit factor greater than 6 34 the maximum known asymptotic merit factor for binary sequences has been stuck at a value of 6 since the 1980s several authors have suggested that this value cannot be improved in this paper we construct an infinite family of binary sequences whose asymptotic merit factor we conjecture to be greater than 6 34 we present what we believe to be compelling evidence in support of this conjecture the numerical experimentation that led to this construction is a significant part of the story

birational mappings and matrix subalgebra from the chiral potts model we study birational transformations of the projective space originating from lattice statistical mechanics specifically from various chiral potts models associating these models with stable patterns and signed patterns we give general results which allow us to find all chiral q state spin edge potts models when the number of states q is a prime or the square of a prime as well as several q dependent family of models we also prove the absence of monocolor stable signed pattern with more than four states this demonstrates a conjecture about cyclic hadamard matrices in a particular case the birational transformations associated with these lattice spin edge models show complexity reduction in particular we recover a one parameter family of integrable transformations for which we give a matrix representation when the parameter has a suitable value

sufficient conditions for a conjecture of ryser about hadamard circulant matrices let h be a hadamard circulant matrix of order n 4h2 where h 1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial n t are bounded by a constant independent of n then for all the n n th primitive roots w of 1 p w n is not an algebraic integer in the cyclotomic field k q w where p t is the representer polynomial of h and is the euler function this implies that p w is not a real number