generalized chapman kolmogorov equation a unifying approach to the description of anomalous transport in external fields the generalized chapman kolmogorov equation v m kenkre e w montroll and m f shlesinger j stat phys 9 45 1973 is discussed it is demonstrated that this equation unifies recently proposed kinetic equations of fractional order that describe anomalous transport in external fields as well as continuous time random walks the conditions under which the individual models can be established are discussed

representation of multivalued functions using the direct cover method an efficient method for representing multivalued functions is described the method employs an algorithm which generates an efficient cover for a given function directly i e without resorting to the intermediate step of creating a table of prime implicants data are presented to show that the covers generated are as efficient in terms of cover size as prime implicant based methods more importantly however the direct cover method is shown to require much less computation time than prime implicant based methods thus making it practical for functions with a large number of input variables and or as the radix of implementation increases the algorithm is introduced by applying it to functional representations employing the traditional max and min operation next a modified form of the algorithm is presented for use with the sum and product operators more appropriate to i2l and other current summation technologies

asymptotic solution of the wang uhlenbeck recurrence time problem a langevin particle is initiated at the origin with positive velocity its trajectory is terminated when it returns to the origin in 1945 wang and uhlenbeck posed the problem of finding the joint probability density function pdf of the recurrence time and velocity naming it the recurrence time problem we show that the short time asymptotics of the recurrence pdf is similar to that of the integrated brownian motion solved in 1963 by mckean we recover the long time t ensuremath 3 2 decay of the first arrival pdf of diffusion by solving asymptotically an appropriate variant of mckean s integral equation

bounds on the average number of products in the minimum sum of products expressions for multiple value input two valued output functions the authors derive an upper and a lower bound on the average number of products in the minimum sum of products expressions sopes the upper bound is obtained by using the minimization results of functions with fewer variables the lower bound is based on an assumption so it is incorrect until this assumption is proven a lower bound l sub p n u and an upper bound u sub p n u on s sub p n u are derived where s sub p n u is the average number of products in minimum sope for p valued input two valued output functions n is the number of the inputs and u is the number of minterms the values of s sub p n u are obtained by minimizing randomly generated functions and they are compared to the calculated values of u sub p n u and l sub p n u these bounds are useful for estimating the size of programming logic arrays

heuristic minimization of mvl functions a direct cover approach a heuristic method to obtain near minimal covers of p valued switching functions is introduced first we describe transform tools useful in the processing of mvl functions they are p adic shifting weighting and implicant detecting transformations based on these tools a direct cover algorithm is presented that uses local information for heuristic decision making the heuristics are taken from weight coefficients calculated for canonical terms and implicants the method allows to assign cost factors to implicants further the algorithms can be modified easily so as to correspond to various connectives e g max plus

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

exact solution of smoluchowski s equation for reorientational motion in maier saupe potential abstract the analytic treatment of the non inertial rotational diffusion equation i e of smoluchowski s one se in a symmetric genuinely double well maier saupe uniaxial potential of mean torque is considered such potential may find applications to reorientations of the fragments of structure in polymers and proteins we obtain the exact solution of se via the confluent heun s function the solution is uniformly valid for any barrier height we apply the obtained solution to the calculation of the mean first passage time and the longitudinal correlation time and obtain their precise dependence on the barrier height in the intermediate to high barrier low temperature region the results of our approach are in full agreement with those of the approach developed by coffey kalmykov dejardin and their coauthors in the low barrier high temperature region our results noticeably distinguish from the predictions of the literature formula and give appreciably greater values for the transition rates from the potential well the reason is that the above mentioned formula is obtained in the stationary limit we conclude that for very small barrier heights the transient dynamics plays a crucial role and has to be taken into account explicitly when this requirement is satisfied as e g at the calculation of the longitudinal correlation time we obtain absolute identity of our results with the literature formula in the whole range of barrier heights the drawbacks of our approach are its applicability only to the symmetric potential and its inability to yield an analytical expression for the smallest non vanishing eigenvalue

synthesis of multivalued multithreshold functions for ccd implementation basic multivalued building blocks constructed using ccd charge coupled devices technology are presented they are used to realize simple threshold functions existing techniques for decomposition of multivalued multithreshold functions into simpler subfunctions are reviewed usage of the ccd technology in implementation of these subfunctions is discussed two new decomposition techniques are proposed their aim is to obtain realizations that are better suited to ccd technology in terms of the maximum number of thresholds per element and the generation of negative weights synthesis techniques based on the proposed decompositions are presented and compared although 4 valued logic functions are stressed the techniques are in principle applicable to functions of any radix

results for a turbulent system with unbounded viscosities weak formulations existence of solutions boundedness and smoothness we consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities various notions of weak solution are considered and compared we establish existence and regularity results in particular we study the boundedness of weak solutions we also establish an existence result for a classical solution

relaxation dynamics of a particle in the presence of an external potential exact solution in terms of matrix continued fractions exact expressions for the laplace transform of the after effect function arising from the solution of the langevin or underlying fokker planck equation for brownian motion in an external potential are obtained as a sum of products of infinite matrix continued fractions this is accomplished by reducing the scalar multiterm recurrence relations associated with the fokker planck equation to matrix three term recurrence relations the solution is illustrated by considering the problem of dielectric relaxation of a single axis rotator subjected to both constant and crystalline anisotropy fields

escape by diffusion from a double well potential across a barrier abstract escape by diffusion from a double well potential across a barrier is studied on the basis of the smoluchowski equation in one dimension by comparison with exact results for a piecewise parabolic potential a reduced description is constructed in terms of a set of rate equations for the occupation probabilities of the two wells the rate equations contain memory terms and a source term for the rate of return from the outer space the reduced description yields a quite accurate approximation for the two example potentials studied

quantization and roundoff noises in fixed point fir digital filters the errors in a fixed point finite impulse response fir filter due to quantization analog to digital conversion and roundoff are considered expressions for the exact moments of the filter output noise are derived it is well known that when the input signal satisfies certain conditions the popular additive white noise model can be valid in describing the quantization noise the characteristics of multiplicative roundoff noises however differ from what this model predicts even under conditions where the roundoff noises are white hence the additive white noise model does not provide accurate results on the characteristics of the output error in an fir filter using the exact formulas for the moments the author computes the exact power spectrum of the filter output error these results agree well with those obtained from simulation

automated synthesis of combinational logic using theorem proving techniques recent findings indicate that logic design still constitutes 50 percent of the total design effort and yet few automated tools are being used in industry the use of an automated theorem proving system in the logic design process presents an intriguing addition to traditional design automation tools this paper deals with the automated synthesis of combinational logic the method consists of a theorem proving implementation of a systematic uniform procedure for the synthesis of an arbitrary switching function any multiple valued including binary function can be synthesized in a top down fashion using the functional blocks of the designer s choice the method is general enough to allow for the choice of an arbitrary logic system and radix additional constraints of modularity technology dependence fault tolerance and others may be imposed upon the design it may also be possible to accommodate into this approach formal design verification design for testability functional level modeling and formal analysis of race and hazard conditions

multiple valued logic its status and its future multiple valued logic in which the number of discrete logic levels is not confined to two has been the subject of much research over many years the practical objective of this work has been to increase the information content of the digital signals in a system to a higher value than that provided by binary operation in this tutorial survey paper we will review the historical developments in this field both in circuit realizations and in methods of handling multiple valued design data and consider the present state of the art and future expectations

diffusion in a bistable potential abstract the problem of diffusion of a particle in a bistable potential is studied on the basis of the one dimensional smoluchowski equation for the space and time dependent probability distribution the potential is modeled as two parabolic wells separated by a parabolic barrier for the model potential the smoluchowski equation is solved exactly by a laplace transform with respect to time for the initial condition that at time zero the probability distribution is given by a thermal equilibrium distribution in one of the wells in the limit of a high barrier the rate of transition to the other well is given by an asymptotic result due to kramers for a potential barrier of moderate height there are significant corrections to the asymptotic result

modular decomposition of combinatorial multiple values circuits a decomposition approach to the modular design of multiple valued logic functions is discussed systematic procedures to utilize a fixed set of building blocks from which an arbitrary function can be designed are illustrated the building blocks are composed of t gates multiplexers the first step is the partitioning of all logic functions into classes representative building blocks for each class are then developed finally optimization techniques are described that reduce the number of building blocks required in the design this approach is in principle applicable to functions in any radix and will always yield a design for the target function examples are presented to illustrate the approach for ternary functions

on the design of 4 valued digital systems the availability of 4 valued i2l circuits presents a new perspective for digital systems designers this correspondence explores several aspects of digital system design using 4 valued logic elements an approach to designing the ith stage of an alu employing these elements is presented the realization of rudimentary alu operations is shown using the t gate as a universal building block system design based on both post and boolean 4 valued logics is discussed

an algorithm to derive the complement of a binary function with multiple valued inputs a recursive algorithm to obtain a complement of a sum of products expression for a binary function with p valued inputs is presented it produces at most pn 2 products for n variable functions whereas a conventional elementary algorithm produces o tn n 1 t 2 products where t 2p 1 it is 10 20 times faster than the elementary one when p 2 and n 8 for large practical problems it produces many fewer products than the disjoint sharp algorithm used by mini appplications of the algorithm to pla minimization are also presented

processors for generalized stack filters new processor structures for generalized stack filters are proposed they can be implemented using different numbers of boolean function units the class of pipeline parallel structures for generalized stack filters is simple and modular in structure and suitable for vlsi implementation coder and decoder networks are developed for the mutual transform of binary weighted and unary weighted codes as the thresholding and addition units of the processors the area time complexity a spl middot t of the proposed filters is o k sup 3 due to the feedback instead of o 2 sup k for the threshold decomposition structure where k is the number of bits for sample representation

vector continued fractions using a generalized inverse a real vector space combined with an inverse involution for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale the choice of sign for different components of the vector inverse permits construction of vector analogues of the jacobi continued fraction these vector jacobi fractions are related to vector and scalar valued polynomial functions of the vectors which satisfy recurrence relations similar to those of orthogonal polynomials the vector jacobi fraction has strong convergence properties which are demonstrated analytically and illustrated numerically

first passage times for the uhlenbeck ornstein process a numerical solution is obtained for the laplace transformed backward kramers equation from which the mean first passage time may be obtained the main difficulties are associated with a the parabolic nature of the time development operator and b the existence of a double structure in the solution near the absorbing barrier both of these difficulties are resolved by computational methods derived from boundary layer theory the reliability of the method is assessed by comparing its results with an earlier analytic solution for the case of a uniform force field the authors also present the results for a harmonic force field for which no analytic solution is yet known

stochastic resonance in a trapping overdamped monostable system the response of a trapping overdamped monostable system to a harmonic perturbation is analyzed in the context of stochastic resonance phenomenon we consider the dynamics of a brownian particle moving in a piecewise linear potential with a white gaussian noise source based on linear response theory and laplace transform technique analytical expressions of signal to noise ratio snr and signal power amplification spa are obtained we find that the snr is a nonmonotonic function of the noise intensity while the spa is monotonic theoretical results are compared with numerical simulations

discrete interval truth values logic and its application in this paper we focus on functions defined on a special subset of the power set of 0 1 r 1 the elements in the subset will be called discrete interval truth values and operations on the truth values the operations discussed in this paper will be called regular because they are one of the extensions of the regularity which was first introduced by kleene in his ternary logic mukaidono investigated some properties of ternary functions which can be represented by the regular operations he called such ternary functions regular ternary logic functions regular ternary logic functions are useful for representing and analyzing ambiguities such as transient states and or initial states in binary logic circuits that boolean functions cannot cope with furthermore they are also applied to studies of fail safe systems for binary logic circuits in this paper we will discuss an extension of regular ternary logic functions to functions on the discrete interval truth values first we will suggest an extension of the regularity in the sense of kleene into operations on the discrete interval truth values we will then present some mathematical properties of functions on the discrete interval truth values consisting of regular operations and one application of these functions

the kinetic boundary layer for the fokker planck equation with absorbing boundary the fokker planck equation for the distribution of position and velocity of a brownian particle is a particularly simple linear transport equation its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly by a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall and that approach a linearly increasing position space density far from the wall various aspects of these approximate solutions are discussed in particular we find that the extrapolated asymptotic density reaches zero at a distance xm beyond the wall we find xm 1 46 in units of the velocity persistence length of the brownian particle this study was motivated by certain problems in the theory of diffusion controlled reactions and the results might be used to test approximate theories employed in that field

bi decomposition of multi valued logical functions and its applications abstract the bi decomposition of multi valued logical mvl functions including disjoint and non disjoint cases is considered using a semi tensor product an mvl function can be expressed in its algebraic form based on this form straightforward verifiable necessary and sufficient conditions are provided for each case respectively the constructive proofs also lead to constructing corresponding decompositions using these results the implicit function theorem ift of k valued functions as a special bi decomposition is obtained finally as an application the normalization of dynamic algebraic d a boolean networks is investigated using ift of k valued functions

a learning multiple valued logic network algebra algorithm and applications we propose a multiple valued logic mvl network with functional completeness and develop its learning capability the mvl network consists of layered arithmetic piecewise linear processors since the arithmetic operations of the network are basically a wired sum and a piecewise linear operation their implementations should be rather simple and straightforward furthermore the mvl network can be trained by the traditional backpropagation algorithm directly the algorithm trains the networks using examples and appears to be available for most mvl problems of interest

exact solution for the extended debye theory of dielectric relaxation of nematic liquid crystals abstract the exact solution for the transverse i e in the direction perpendicular to the director axis component of a nematic liquid crystal and the corresponding correlation time t is presented for the uniaxial potential of martin et al symp faraday soc 5 1971 119 the corresponding longitudinal i e parallel to the director axis quantities t may be determined by simply replacing magnetic quantities by the corresponding electric ones in our previous study of the magnetic relaxation of single domain ferromagnetic particles coffey et al phys rev e 49 1994 1869 the calculation of is accomplished by expanding the spatial part of the distribution function of permanent dipole moment orientations on the unit sphere in the fokker planck equation in normalised spherical harmonics this leads to a three term recurrence relation for the laplace transform of the transverse decay functions the recurrence relation is solved exactly in terms of continued fractions the zero frequency limit of the solution yields an analytic formula for the transverse correlation time t which is easily tabulated for all nematic potential barrier heights a simple analytic expression for t which consists of the well known meier saupe formula mol cryst 1 1966 515 with a substantial correction term which yields a close approximation to the exact solution for all and the correct asymptotic behaviour is also given the effective eigenvalue method is shown to yield a simple formula for t which is valid for all it appears that the low frequency relaxation process for both orientations of the applied field is accurately described in each case by a single debye type mechanism with corresponding relaxation times t t

multiple valued logic charge coupled devices a new method to implement multiple valued logic in large scale integrated circuits is introduced the data are represented by discrete amounts of charge in a charge coupled device in this paper the design principles and realizations in four valued logic of a minimum and maximum circuit a complement circuit a literal a successor and an adder are presented also the designs of a binary to quaternary converter and vice versa are discussed charge coupled devices offer low power consumption high packing density and the possibility to perform most ic compatible multiple valued logic in any arbitrary radix

exact solution of kramers problem for piecewise parabolic potential profiles abstract this paper presents a general procedure for deducing exact time characteristics of brownian diffusion in the high damping limit in piecewise parabolic pontential profiles our approach to solve this problem is based on the laplace transform method for obtaining the desired time characteristics for some concrete examples of piecewise parabolic profiles exact values of metastable state decay times and bistable system relaxation times are found these results are valid for any height of the potential barrier and coincide with kramer s results for a high potential barrier conditions for the validity of kramer s formula are also derived

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

on the size of plas required to realize binary and multiple valued functions upper and lower bounds are shown for the average number of product terms required in the minimal realization as a function of the number of nonzero output values the variance in addition to the bounds allows conclusions to be drawn about how pla size determines the set of realizable functions although the bounds are most accurate when there are few nonzero values they are adequate for analyzing commercially available plas the analysis shows that for all but one commercially available pla the number of nonzero values is a statistically meaningful criterion for determining whether or not a given function is likely to be realized

novel relations and new properties of confluent heun s functions and their derivatives of arbitrary order the present paper reveals important properties of the confluent heun s functions we derive a set of novel relations for confluent heun s functions and their derivatives of arbitrary order specific new subclasses of confluent heun s functions are introduced and studied a new alternative derivation of confluent heun s polynomials is presented

analytic solution for a boundary value problem in the theory of brownian motion a closed form formula for the solution of the first boundary value problem for the kolmogorov equation is found by constructing explicitly its green function

numerical solution of two dimensional fokker planck equations we present a robust finite difference scheme for the integration of the fokker planck fp equation with two variables plus time the scheme is checked with problems where analytical solutions exist and it is compared with finite element codes as well as with simulations of the associated stochastic differential equation sde the precision and stability of the scheme are verified

on a turbulent system with unbounded eddy viscosities in this system the functions and a are real valued functions of k which represent eddy viscosities this system is a mathematical subproduct of the large scale one degree closure reynolds system used by engineers oceanographs meteorologists and others for simulating turbulent flows the reader can find details concerning modelization in 10 13 and 3 the variable k is the turbulent kinetic energy and the variable u an idealization of the velocity of the flow the interest in studying the mathematical system s lies in a better understanding of the interaction between an eddy diffusion

escape times for rigid brownian rotators in a bistable potential from the time evolution of the green function and the characteristic time of the probability evolution abstract the greatest relaxation time for an assembly of three dimensional rigid rotators in an axially symmetric bistable potential is obtained exactly in terms of continued fractions as a sum of the zero frequency decay functions averages of the legendre polynomials of the system this is accomplished by studying the entire time evolution of the green function transition probability by expanding the time dependent distribution as a fourier series and proceeding to the zero frequency limit of the laplace transform of that distribution the procedure is entirely analogous to the calculation of the characteristic time of the probability evolution the integral of the configuration space probability density function with respect to the position co ordinate for a particle undergoing translational diffusion in a potential a concept originally used by malakhov and pankratov physica a 229 1996 109 this procedure allowed them to obtain exact solutions of the kramers one dimensional translational escape rate problem for piecewise parabolic potentials the solution was accomplished by posing the problem in terms of the appropriate sturm liouville equation which could be solved in terms of the parabolic cylinder functions the method as applied to rotational problems and posed in terms of recurrence relations for the decay functions i e the brinkman approach c f blomberg physica a 86 1977 49 as opposed to the sturm liouville one demonstrates clearly that the greatest relaxation time unlike the integral relaxation time which is governed by a single decay function albeit coupled to all the others in non linear fashion via the underlying recurrence relation is governed by a sum of decay functions the method is easily generalized to multidimensional state spaces by matrix continued fraction methods allowing one to treat non axially symmetric potentials where the distribution function is governed by two state variables

stable and efficient spectral methods in unbounded domains using laguerre functions stable and efficient spectral methods using laguerre functions are proposed and analyzed for model elliptic equations on regular unbounded domains it is shown that spectral galerkin approximations based on laguerre functions are stable and convergent with spectral accuracy in the usual not weighted sobolev spaces efficient accurate and well conditioned algorithms using laguerre functions are developed and implemented numerical results indicating the spectral convergence rate and effectiveness of these algorithms are presented

the prospects for multivalued logic a technology and applications view advances in multiple valued logic mvl have been inspired in large part by advances in integrated circuit technology multiple valued logic has matured to the point where four valued logic is now part of commercially available vlsi ic s besides reduction in chip area mvl offers other benefits such as the potential for circuit test this paper describes the historical and technical background of mvl and areas of present and future application it is intended as well to serve as a tutorial for the nonspecialist

worst and best irredundant sum of products expressions in an irredundant sum of products expression isop each product is a prime implicant pl and no product can be deleted without changing the function among the isops for some function f a worst isop wsop is an isop with the largest number of pls and a minimum isop msop is one with the smallest number we show a class of functions for which the minato morreale isop algorithm produces wsops since the ratio of the size of the wsop to the size of the msop is arbitrarily large when it the number of variables is unbounded the minato morreale algorithm can produce results that are very far from minimum we present a class of multiple output functions whose wsop size is also much larger than its msop size for a set of benchmark functions we show the distribution of isops to the number of pls among this set are functions where the msops have almost as many pls as do the wsops these functions are known to be easy to minimize also there are benchmark functions where the fraction of isops that are msops is small and msops have many fewer pls than the wsops such functions are known to be hard to minimize for one class of functions we show that the fraction of isops that are msops approaches 0 as n approaches infinity suggesting that such functions are hard to minimize

meaningful special classes of ternary logic functions regular ternary logic functions and ternary majority functions special ternary logic functions namely regular functions suitable for treating ambiguity and majority functions based on an extended majority principle are considered the one to one correspondence between n ary monotone regular ternary logic functions and n 1 ary monotone boolean functions is investigated as well as the one to one correspondence between n ary monotone ternary majority functions and n 1 ary monotone binary threshold functions the ternary majority functions are enumerated

multiple valued decomposition of generalized boolean functions and the complexity of programmable logic arrays generalized boolean functions are shown to be useful for the design of programmable logic arrays pla s and the complexity of three types of pla s is obtained by the theory of multiple valued decomposition a two level pla consists of an and array and an or array and they are cascaded to perform a two level and or circuit a pla with decoders consists of decoders an and array and an or array a three level pla consists of a d array an and array and an or array and they are cascaded to perform a three level or and or circuit it is shown that a generalized boolean function f x1 x2 xr x bni b where b 0 1 is represented by a generalized boolean expression of 2ni valued variables x i and f can be directly realized by a pla with decoders or a three level pla to realize a function of n variables n 2r the following sizes are shown to be sufficient for a two level pla n 2n for a pla with two bit decoders 4 n 4 2n for a three level pla 2n 3n l 2n 2n2especially in the case of pla with two bit decoders the following sizes are shown to be necessary and sufficient for an arbitrary symmetric function 3 2 n 3n and for a parity function n 2n

generating the group of reversible logic gates reversible logic plays a fundamental role both in ultra low power electronics and in quantum computing it is therefore important to have an insight into the structure of the group formed by the reversible logic gates and their cascading into reversible circuits such insight is gained from constructing chains of maximal subgroups the subgroup of control gates plays a prominent role as it is a sylow 2 subgroup

a model for two coupled turbulent fluids part iii numerical approximation by finite elements this paper introduces a scheme for the numerical solution of a model for two turbulent flows with coupling at an interface we consider a variational formulation of the coupled model where the turbulent kinetic energy equation is formulated by transposition we prove the convergence of the approximation to this formulation for 2d flows by piecewise affine triangular elements our main contribution is to prove that the standard galerkin finite element approximation of the laplace equation approximates in l2 norm its solution by transposition for data with low smoothness we include some numerical tests for simple geometries that exhibit the behaviour predicted by our analysis

synthesis of discrete functions using i 2 l technology algebraic methods for designing multiple valued i2l circuits are presented a new operation namely the truncated difference is introduced and proved to be an efficient tool for both analyzing and synthesizing circuits

design of a multiple valued systolic system for the computation of the chrestenson spectrum this correspondence deals with the computation of the chrestenson spectrum of an n ary n place p valued function by means of a systolic system the design of a systolic system in a multiple valued environment is discussed in details and some aspects are compared to binary realizations

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

an optimization technique for the design of multiple valued pla s an optimization technique for the design of two types of multiple valued plas is described in a type i pla the multiple valued function is realized directly whereas in a type ii pla output encoding is used to encode the binary output of the pla in both types multiple function literal circuits are used for the purpose of minimization it is shown that the proposed technique leads to a considerably reduced size of pla when compared to the earlier techniques

relationship between p valued majority functions and p valued threshold functions in a previous paper trans iece japan vol j63 d p 493 500 1980 vol j64 d p 172 3 1981 and vol e 67 p 47 8 1984 the authors defined a new class of multiple valued logic functions called multiple valued majority functions the authors clarifies the distinction of multiple valued majority functions from multiple valued threshold functions through the difference between a number function and an inner product of an input vector and a weight vector

a drop of ink falls from my pen it comes to earth i know not when the authors obtain for a brownian particle in a uniform force field the mean and asymptotic first passage times as functions of the particle s initial position and velocity with the recurrence times given as a special case they discuss the region of phase space for which the diffusion model of brownian motion provides an adequate approximation and conclude that there is no possibility of obtaining the recurrence times within that model they find that the nature of the boundary value problem is profoundly altered when the motion is treated as a process in phase rather than configuration space because the time development operator is then parabolic rather than elliptic they argue that such a change in the treatment of brownian motion places it within the sphere of transport theory rather than diffusion theory and that consequently results such as ours have relevance to the study of phenomena such as radiative transfer and neutron transport

the hermite spectral method for gaussian type functions although hermite functions were widely used for many practical problems numerical experiments with the standard normalized hermite functions psi n v worked poorly in the sense that too many hermite functions are required to solve differential equations in order to obtain accurate numerical solutions it is necessary to choose a scaling factor alpha and use psi n alpha v as the basis functions in this paper the scaling factors are given for functions that are of gaussian type which have finite supports m m the scaling factor used is max 0 leqq j leqq n gamma j m where gamma j j 0 n are the roots of psi n 1 v and n 1 is the number of the truncated terms used the numerical results show that after using this scaling factor only reasonable numbers of the hermite functions are required to solve differential equations

generating essential primes for a boolean function with multiple valued inputs detecting essential primes is important in multiple valued logic minimization in this correspondence we present a fast algorithm that can generate all essential primes without generating a prime cover of the boolean function a new consensus operation called asymmetric consensus acons is defined in terms of acons we prove a necessary and sufficient condition for detecting essential primes for a boolean function with multiple valued inputs the detection of essential primes can be performed by using a tautology checking algorithm we exploit the unateness of a boolean function to speed up tautology checking the notion of unateness considered is more general than that has appeared in the literature

combined hermite spectral finite difference method for the fokker planck equation the convergence of a class of combined spectral finite difference methods using hermite basis applied to the fokker planck equation is studied it is shown that the hermite based spectral methods are convergent with spectral accuracy in weighted sobolev space numerical results indicating the spectral convergence rate are presented a velocity scaling factor is used in the hermite basis and is shown to improve the accuracy and effectiveness of the hermite spectral approximation with no increase in workload some basic analysis for the selection of the scaling factors is also presented

probability distribution function for reorientations in maier saupe potential abstract exact analytic solution for the probability distribution function of the non inertial rotational diffusion equation i e of the smoluchowski one in a symmetric maier saupe uniaxial potential of mean torque is obtained via the confluent heun s function both the ordinary maier saupe potential and the double well one with variable barrier width are considered thus the present article substantially extends the scope of the potentials amenable to the treatment by reducing smoluchowski equation to the confluent heun s one the solution is uniformly valid for any barrier height we use it for the calculation of the mean first passage time also the higher eigenvalues for the relaxation decay modes in the case of ordinary maier saupe potential are calculated the results obtained are in full agreement with those of the approach developed by coffey kalmykov dejardin and their coauthors in the whole range of barrier heights

comparison of methods for the calculation of superparamagnetic relaxation times a general expression for the correlation time of the decay of the magnetization of an assembly of singledomain noninteracting ferromagnetic particles is given in terms of the inverse of the fokker planck operator the results of moro and nordio g moro and p l nordio mol phys 56 255 1985 given in the context of dielectric relaxation are recovered when the fokker planck operator is axially symmetric their result is a particular example of szabo s calculation of the correlation times of the autocorrelation functions of the legendre polynomials by means of a generalization of the theory of first passage times a szabo j chem phys 72 4620 1980 likewise the results of garanin ischenko and panina nd a garanin v v ischenko and l v panina teor mat fiz 82 242 1990 theor math phys 82 169 1990 o for the integral relaxation time i e the area under the curve of the normalized decay of the magnetization are regained in the axially symmetric case where it is possible to integrate the fokker planck equation directly it is shown by manipulation of kummer s functions that the exact integral expression for the correlation time for simple uniaxial anisotropy derived by coffey et al w t coffey d s f crothers yu p kalmykov e s massawe and j t waldron phys rev e 49 1869 1994 by representing the fokker planck equation as a differential recurrence relation is identical to the integral relaxation time originally derived by garanin et al by direct integration of the fokker planck equation s1063 651x 96 08811 3

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

on the optimal design of multiple valued plas a description is given of the design and analysis of three types of multivalued plas programmable logic arrays type 1 plas realize functions directly in the form of the max of min of literal functions and constants in type 2 plas the body of the pla is binary and the output is encoded as a multiple valued logic value type 3 plas are the same as type 2 plas except for the use of 2 bit decoders and a permutation network on the input using the number of columns required to realize a given function as a measure to compare plas it is shown that type 3 plas are superior to type 2 which in turn are superior to type 1

a hardware friendly arithmetic method and efficient implementations for designing digital fuzzy adders a new hardware friendly mathematical method for realizing low complexity universal adder cells as well as its efficient hardware implementations is proposed in this paper this method can be used in binary logic multiple valued logic mvl and specifically digital fuzzy systems the proposed mathematical method can be implemented in both voltage and current modes the voltage mode hardware implementation is very simple and is based on input capacitors and mvl or analog inverters and buffers in addition the current mode hardware implementation leads to simple and efficient structures for digital fuzzy systems simulations are carried out for ternary logic as well as for digital fuzzy logic with high precision by using 180nm standard cmos technology and at 1 8v supply voltage simulation results demonstrate that the proposed designs have excellent functionality and are very suitable for implementing mvl and fuzzy arithmetic circuits

input variable assignment and output phase optimization of pla s a pla minimization system having the following features is presented 1 minimization of both two level pla s and pla s with two bit decoders 2 optimal input variable assignment to the decoders 3 optimal output phase assignment and 4 essential prime implicants detection without generating all the prime implicants

a numerical approach to study the kramers equation for finite geometries boundary conditions and potential fields the kramers equation for the phase space function which models the dynamics of an underdamped brownian particle is the subject of our study numerical solutions of this equation for natural boundaries unconfined geometries have been well reported in the literature but not much has been done on the kramers equation for finite confining geometries which require a set of additional constraints imposed on the phase space function at physical boundaries in this paper we present numerical solutions for the kramers equation with a variety of potential fields namely constant linear harmonic and periodic in the presence of fully absorbing and fully reflecting boundary conditions bcs the choice of the numerical method and its implementation take into consideration the type of bcs in order to avoid the use of ghost points or artificial conditions we study and assess the conditions under which the numerical method converges various aspects of the solutions for the phase space function are presented with figures and discussed in detail

substitution systems and nonextensive statistics abstract substitution systems evolve in time by generating sequences of symbols from a finite alphabet at a certain iteration step the existing symbols are systematically replaced by blocks of n k symbols also within the alphabet with n k a natural number being the length of the k th block of the substitution the dynamics of these systems leads naturally to fractals and self similarity by using b calculus garcia morales 2012 universal maps for deterministic substitution systems both of constant and non constant length are formulated in 1d it is then shown how these systems can be put in direct correspondence with tsallis entropy a second law of thermodynamics is also proved for these systems in the asymptotic limit of large words