a multiple valued reed muller transform for incompletely specified functions this paper considers the reed muller transform for incompletely specified multiple valued logic functions which is obtained as the finite field polynomial representation a new algorithm for dealing with single variable functions is presented it is applicable to finite fields of small sizes two to four which is of interest because only these fields are readily implementable with today s mvl technology it is shown how such or any similar single variable algorithm can be used to obtain a fast n variable reed muller transform based on this transform a heuristic scheme is derived for dealing with incompletely specified functions it has better computational properties than other methods and achieves the best results when applied to functions with a large number of unspecified points

output of a neuronal population code in the population coding framework we consider how the response distributions affect output distribution a general theory for the output of neuronal population code is presented when the spike train is a renewal process under a given condition on the response distribution the most probable value of the output distribution is the center of input preferred values whereas in the other cases the most improbable value of the output distribution is the center of input preferred values or there are no most probable states depending on the exact form of the response distributions the variance of the output distributions can either enlarge or reduce the tuning width of the tuning curves

fisher information and temporal correlations for spiking neurons with stochastic dynamics population coding accuracy can be studied using fisher information here the fisher information and correlation functions are determined analytically for a network of coupled spiking neurons with a more general than poisson stochastic dynamics it is shown that stimulus driven temporal correlations between neurons always increase the fisher information whereas stimulus independent correlations need not do so additionally we find that for subthreshold stimuli there is some nonzero level of noise for which network coding is optimal we also find that the fisher information is larger for purely excitatory than for purely inhibitory networks but only in a limited range of values of synaptic coupling strengths in most cases the dependence of the fisher information on time is linear except for excitatory networks with strong synaptic couplings and for strong stimuli in the latter case this dependence shows two distinct regimes fast and slow for excitatory networks short term synaptic depression can improve the coding accuracy significantly whereas short term facilitation can lower the coding accuracy for inhibitory networks coding accuracy is insensitive to short term synaptic dynamics

a method for modulo 2 minimization a minimization approach is presented for a modulo 2 expansion of a switching function this method uses mixed polarity representation for the variables a local optimization procedure is presented which requires modest computation and is easily expanded the procedure applied to the 32 des select functions of six variables is reported

minimization algorithms for multiple valued programmable logic arrays the performance of various heuristic algorithms for minimizing realizations of multiple valued functions by the charge coupled device ccd and cmos programmable logic arrays plas of h g kerkhoff and j t butler 1986 and j g samson 1988 respectively is analyzed the functions realized by the plas are in sum of products form where the sum is ordinary addition truncated to the highest logic value and the product represents the min operation of functions of the input variables that are the interval literal operations three heuristics proposed by g pomper and j a armstrong 1981 p w besslich 1986 and g w dueck and d m miller 1987 are compared over sets of random and random symmetric functions an exact minimization method that is a tree search using backtracking is described a reduction in the search space is achieved by considering constrained implicant sets and by eliminating some implicants altogether even with this improvement the time required for exact minimization is extremely high when compared to all three heuristics the case involving only prime implicants is considered and it is shown that such implicants have marginal value compared to constrained implicant sets the basis of comparison is the average number of product terms

super critical free surface flow over a trapezoidal obstacle a numerical method based on series truncation is presented to solve the problem of an irrotational inviscid incompressible and steady flow over a two dimensional trapezoidal obstacle with obstacle height w lying on the bottom of the running stream the suggested numerical method for the solution of the fully nonlinear problem is presented for which the flow is super critical both upstream and downstream it is found that solutions exist for symmetric trapezoidal for arbitrary size the results are plotted for different trapezoidal shapes and different values of froude number f 1 the effect of the froude number the bottom height and the shape of the bottom are discussed

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

approximate determination of the transmission and reflection coefficients for water wave flow over a topography the objective of this paper is to present a semi analytical method for the calculation of the transmitted and the reflected parts of an incident harmonic progressive wave in a homogeneous fluid layer over a general topography the domain of definition of the problem is properly divided into suitable subdomains elementary solutions of the problem satisfying the field equation and a subset of the boundary conditions as well are obtained in each subdomain by the combined use of the techniques of separation of variables and finite integral transforms these solutions are given in the form of infinite series involving an infinite number of unknown coefficients the series are truncated and only a finite number of the coefficients is determined so as to satisfy approximately the remaining boundary conditions numerical illustrations indicate that the method yields highly accurate results and that the accuracy of the procedure is improved by increasing the number of the unknown coefficients taken to account with the presence of certain optimum degrees of freedom

logic networks with a minimum number of nor nand gates for parity functions of n variables design of logic networks in single rail input logic with a minimum number of nor gates for parity functions of an arbitrary number of variables is described this is partly based on minimum networks for parity functions of a small number of variables which are designed by the integer programming logic design method although it is generally difficult to design minimum networks for functions of an arbitrarily large number of variables we have previously designed minimum networks for adders of an arbitrary number of variables the minimum networks for parity functions of an arbitrary number of variables discussed in this paper is another case many unique properties of minimum nor networks for parity functions are shown minimum networks with nand gates for parity functions can be easily obtained from those with nor gates because of duality relationship between nand and nor

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

on the boundary integral formulation of the plane theory of elasticity computational aspects a numerical scheme is proposed for the solution of the system of field equations and boundary conditions of the static linear plane strain problem of the theory of elasticity in stresses the work is based on a previous analytical approach by the authors within the boundary integral method for a homogeneous and isotropic elastic material occupying a simply connected region the differential and integral operators appearing in the resulting system of integro differential equations for the determination of the boundary values of four unknown harmonic functions are discretized and the problem at this stage is reduced to finding the solution of a linear system of algebraic equations formulae are provided for the determination of the displacements and stresses at the boundary as an illustration the proposed numerical scheme is applied to find the solution of the following problems i a nearly circular boundary subject to a uniform pressure or to a uniform normal displacement ii a square subject to a uniform pressure or to different prescribed normal displacements this choice of the problems has allowed to assess the efficiency of the method through comparison with known analytical solutions and to test the accuracy of the results under different smoothness properties of the boundary and boundary conditions whenever new the obtained results were thoroughly discussed future work in this field will be devoted to the extension of the method to include multiply connected regions and mixed boundary conditions and to investigate the prediction and isolation of the singularities in the solution caused by boundary irregularities

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

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

generalized partially mixed polarity reed muller expansion and its fast computation generalized partially mixed polarity reed muller gpmprm expansion a canonical subfamily of exclusive sum of products esop is presented an efficient algorithm in two dimensional data flow is proposed for computation of the gpmprm forms mcnc benchmark experimental results show that the minimal gpmprm forms of these functions on the average have similar number of terms to their sum of products sop counterparts while there are many functions for which the gpmprm circuits are much smaller

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

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

strip a strip based neural network growth algorithm for learning multiple valued functions we consider the problem of synthesizing multiple valued logic functions by neural networks a genetic algorithm ga which finds the longest strip in v spl sube k sup n is described a strip contains points located between two parallel hyperplanes repeated application of ga partitions the space v into certain number of strips each of them corresponding to a hidden unit we construct two neural networks based on these hidden units and show that they correctly compute the given but arbitrary multiple valued function preliminary experimental results are presented and discussed

majority based reversible logic gates reversible logic plays an important role in the synthesis of circuits for quantum computing in this paper we introduce families of reversible gates based on the majority boolean function mbf and we prove their properties in reversible circuit synthesis these gates can be used to synthesize reversible circuits of minimum scratchpad register width for arbitrary reversible functions we show that given a mbf f with 2k 1 inputs f can be implemented by a reversible logic gate with 2k 1 inputs and 2k 1 outputs i e without any constant inputs we also demonstrate new gates from this family with very efficient quantum realizations for majority based applications they can be used to synthesize any reversible function of the same width in conjunction with inverters and feynman 2 qubit controlled not gates the gate universality problem is formulated in terms of elementary group theory and solved using the algebraic software gap

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

solving boolean equations using rosop forms boolean equations are important tools in digital logic previous algorithms for solving boolean equations are based on the boolean algebra of disjoint sop forms in this paper we develop a new boolean algebra with more efficient boolean operation algorithms called the reduced ordered sop rosop forms which are canonical representations rosops are closely related to the well known obdd data structure the results here also show the algebraic structure of obdds

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

multidimensional encoding strategy of spiking neurons neural responses in sensory systems are typically triggered by a multitude of stimulus features using information theory we study the encoding accuracy of a population of stochastically spiking neurons characterized by different tuning widths for the different features the optimal encoding strategy for representing one feature most accurately consists of narrow tuning in the dimension to be encoded to increase the single neuron fisher information and broad tuning in all other dimensions to increase the number of active neurons extremely narrow tuning without sufficient receptive field overlap will severely worsen the coding this implies the existence of an optimal tuning width for the feature to be encoded empirically only a subset of all stimulus features will normally be accessible in this case relative encoding errors can be calculated that yield a criterion for the function of a neural population based on the measured tuning curves

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 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

trefftz method fitting boundary conditions the paper presents various ways of fitting the boundary conditions in the t complete functions method the authors point out the distinct advantages of the orthogonal collocation in comparison to the equidistant collocation and the integral fit the convergence of the collatz error measures and the conditioning of the solution matrices are investigated in detail

nonlinear transient gravity waves due to an initial free surface elevation over a topography the behavior of nonlinear two dimensional transient gravity waves inside an incompressible nonviscous and homogeneous fluid is predicted these waves are the response of an initial free surface elevation in an infinite channel with a general topography a perturbation technique is used for the investigation in contrast to the limitations made on the horizontal extent of the topography by the shallow water theory the present approach necessitates small vertical extent of the topography relative to its horizontal extent and to the mean water s depth as well the same limitation is assumed satisfied by the initial free surface elevation solutions up to the second order are obtained discussed and illustrated the asymptotic behavior of the solutions is obtained and discussed the expressions for the right going and the left going waves are separated the case of a very shallow fluid is deduced and analyzed for such a model which is free from the limitations assumed by the shallow water theory the validity of the obtained solution is tested by comparing the lowest streamline with the bottom topography the difference between the linear and the nonlinear theories and the effect of the bottom topography on the resulting flow are illustrated

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

derivation of human chromatic discrimination ability from an information theoretical notion of distance in color space the accuracy with which humans detect chromatic differences varies throughout color space for example we are far more precise when discriminating two similar orange stimuli than two similar green stimuli in order for two colors to be perceived as different the neurons representing chromatic information must respond differently and the difference must be larger than the trial to trial variability of the response to each separate color photoreceptors constitute the first stage in the processing of color information many more stages are required before humans can consciously report whether two stimuli are perceived as chromatically distinguishable therefore although photoreceptor absorption curves are expected to influence the accuracy of conscious discriminability there is no reason to believe that they should suffice to explain it here we develop information theoretical tools based on the fisher metric that demonstrate that photoreceptor absorption properties explain about 87 of the variance of human color discrimination ability as tested by previous behavioral experiments in the context of this theory the bottleneck in chromatic information processing is determined by photoreceptor absorption characteristics subsequent encoding stages modify only marginally the chromatic discriminability at the photoreceptor level

towards a theory of early visual processing we propose a theory of the early processing in the mammalian visual pathway the theory is formulated in the language of information theory and hypothesizes that the goal of this processing is to recode in order to reduce a generalized redundancy subject to a constraint that specifies the amount of average information preserved in the limit of no noise this theory becomes equivalent to barlow s redundancy reduction hypothesis but it leads to very different computational strategies when noise is present a tractable approach for finding the optimal encoding is to solve the problem in successive stages where at each stage the optimization is performed within a restricted class of transfer functions we explicitly find the solution for the class of encodings to which the parvocellular retinal processing belongs namely linear and nondivergent transformations the solution shows agreement with the experimentally observed transfer functions at all levels of signal to noise

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

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

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

using decision trees to derive the complement of a binary function with multiple valued inputs an algorithm to minimize decision trees of boolean functions with multiple valued inputs is presented the recursive algorithm is used to obtain a complement of a sum of products expression for a binary function with multiple valued inputs in the case where each input is p valued the algorithm produces at most pn l products for n variable functions whereas sasao s algorithm produces pn 2 products this upper bound on the number of products is the best possible

negation trees a unified approach to boolean function complementation boolean function complementation is a basic operation of boolean algebra for functions given in sop form the complementation methods include the demorgan s law sharp disjoint sharp unate complementation recursive and sasao s method this paper accomplishes three purposes 1 it exposes an underlying unification of the existing complementation algorithms it is proven that a unate complementation and sharp are the same as the demorgan law algorithm b sasao s algorithm is the same as disjoint sharp c disjoint sharp is a special case of the recursive method 2 it proposes negation trees for the representation of functions and their complements 3 it gives faster algorithms for finding complements of functions in sop form based on negation trees

parameter extraction from population codes a critical assessment in perceptual systems a stimulus parameter can be extracted by determining the center of gravity of the response profile of a population of neural sensors likewise at the motor end of a neural system center of gravity decoding also known as vector decoding generates a movement direction from the neural activation profile we evaluate these schemes from a statistical perspective by comparing their statistical variance with the minimum variance possible for an unbiased parameter extraction from the noisy neuronal ensemble activation profile center of gravity decoding can be statistically optimal this is the case for regular arrays of sensors with gaussian tuning profiles that have an output described by poisson statistics and for arrays of sensors with a sinusoidal tuning profile for the angular parameter estimated however there are also many cases in which center of gravity decoding is highly inefficient this includes the important case where sensor positions are very irregular finally we study the robustness of center of gravity decoding against response nonlinearities at different stages of an information processing hierarchy we conclude that in neural systems instead of representing a parameter explicitly it is safer to leave the parameter coded implicitly in a neuronal ensemble activation profile

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

a computer algorithm for minimizing reed muller canonical forms a computer algorithm is presented that uses geometrical operations to minimize multioutput reed muller expansions of up to ten variables

a canonical neural circuit for cortical nonlinear operations a few distinct cortical operations have been postulated over the past few years suggested by experimental data on nonlinear neural response across different areas in the cortex among these the energy model proposes the summation of quadrature pairs following a squaring nonlinearity in order to explain phase invariance of complex v1 cells the divisive normalization model assumes a gain controlling divisive inhibition to explain sigmoid like response profiles within a pool of neurons a gaussian like operation hypothesizes a bell shaped response tuned to a specific optimal pattern of activation of the presynaptic inputs a max like operation assumes the selection and transmission of the most active response among a set of neural inputs we propose that these distinct neural operations can be computed by the same canonical circuitry involving divisive normalization and polynomial nonlinearities for different parameter values within the circuit hence this canonical circuit may provide a unifying framework for several circuit models such as the divisive normalization and the energy models as a case in point we consider a feedforward hierarchical model of the ventral pathway of the primate visual cortex which is built on a combination of the gaussian like and max like operations we show that when the two operations are approximated by the circuit proposed here the model is capable of generating selective and invariant neural responses and performing object recognition in good agreement with neurophysiological data

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

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

neuronal tuning to sharpen or broaden sensory and motor variables are typically represented by a population of broadly tuned neurons a coarser representation with broader tuning can often improve coding accuracy but sometimes the accuracy may also improve with sharper tuning the theoretical analysis here shows that the relationship between tuning width and accuracy depends crucially on the dimension of the encoded variable a general rule is derived for how the fisher information scales with the tuning width regardless of the exact shape of the tuning function the probability distribution of spikes and allowing some correlated noise between neurons these results demonstrate a universal dimensionality effect in neural population coding

maximally informative stimuli and tuning curves for sigmoidal rate coding neurons and populations a general method for deriving maximally informative sigmoidal tuning curves for neural systems with small normalized variability is presented the optimal tuning curve is a nonlinear function of the cumulative distribution function of the stimulus and depends on the mean variance relationship of the neural system the derivation is based on a known relationship between shannon s mutual information and fisher information and the optimality of jeffrey s prior it relies on the existence of closed form solutions to the converse problem of optimizing the stimulus distribution for a given tuning curve it is shown that maximum mutual information corresponds to constant fisher information only if the stimulus is uniformly distributed as an example the case of sub poisson binomial firing statistics is analyzed in detail

reduction for the nonlinear problem of fluid waves to a system of integro differential equations with an oceangraphical application a numerical procedure for the solution of the nonlinear problem of irrotational wave propagation inside a finite or an infinite homogeneous fluid mass is proposed this procedure is applied to calculate the fluid gravity waves resulting from certain prescribed varying pressure applied to the free surface of an infinite fluid mass with finite or infinite depth these waves also are calculated analytically within the frame of the linear theory of motion the variation of the fluid s constant depth for this application is found to have no influence on the resulting flow a slight agreement between the numerical solution of the nonlinear problem and the analytical solution of the corresponding linearized problem is noticed in a narrow interval of time following the start of the motion in the course of time a significant divergence between the two theories is found and the nonlinear theory is therefore indispensable for the theoretical prediction of this phenomenon the proposed procedure can be applied to problems with more complicated geometry

implementation of trefftz method for the solution of some elliptic boundary value problems the purpose of this paper is to implement the well known trefftz method for the approximate solution of some elliptic boundary value problems for the two dimensional laplace s and the biharmonic operators through the use of polar harmonics as trial functions four applications treating four different situations are presented the dirichlet problem for the steady state temperature distribution in an infinite elliptic cylinder giving rise to poisson s or laplace s equations with discontinuous boundary function for the complete and the truncated ellipses cases 1 2 and the first and the second fundamental problems of the plane strain theory of elastostatics for an infinite cylinder with rectangular normal cross section cases 3 4 for two of the examples cases 1 3 the results could be checked against known exact solutions to show in particular that the method can sometimes give the exact solution while for the other two cases 2 4 the method has provided solutions for new problems it is shown that the use of polar harmonics possibly together with other harmonics belonging to different coordinate systems can handle many situations in an efficient way i e by using few trial functions

optimal short term population coding when fisher information fails efficient coding has been proposed as a first principle explaining neuronal response properties in the central nervous system the shape of optimal codes however strongly depends on the natural limitations of the particular physical system here we investigate how optimal neuronal encoding strategies are influenced by the finite number of neurons n place constraint the limited decoding time window length t time constraint the maximum neuronal firing rate fmax power constraint and the maximal average rate f max energy constraint while fisher information provides a general lower bound for the mean squared error of unbiased signal reconstruction its use to characterize the coding precision is limited analyzing simple examples we illustrate some typical pitfalls and thereby show that fisher information provides a valid measure for the precision of a code only if the dynamic range fmint fmaxt is sufficiently large in particular we demonstrate that the optimal width of gaussian tuning curves depends on the available decoding time t within the broader class of unimodal tuning functions it turns out that the shape of a fisher optimal coding scheme is not unique we solve this ambiguity by taking the minimum mean square error into account which leads to fiat tuning curves the tuning width however remains to be determined by energy constraints rather than by the principle of efficient coding

boolean matrix transforms for the minimization of modulo 2 canonical expansions fast transforms for computing modulo 2 ring sum canonical expansions of a boolean function are described using kronecker products of elementary boolean matrices these transforms unify and generalize existing ones in the literature previous algorithms which employ such transforms for finding a minimal fixed polarity expansion fpe are reviewed analyzed and extended to a minimal fixed basis expansion fbe

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

efficient sensory encoding and bayesian inference with heterogeneous neural populations the efficient coding hypothesis posits that sensory systems maximize information transmitted to the brain about the environment we develop a precise and testable form of this hypothesis in the context of encoding a sensory variable with a population of noisy neurons each characterized by a tuning curve we parameterize the population with two continuous functions that control the density and amplitude of the tuning curves assuming that the tuning widths vary inversely with the cell density this parameterization allows us to solve in closed form for the information maximizing allocation of tuning curves as a function of the prior probability distribution of sensory variables for the optimal population the cell density is proportional to the prior such that more cells with narrower tuning are allocated to encode higher probability stimuli and that each cell transmits an equal portion of the stimulus probability mass we also compute the stimulus discrimination capabilities of a perceptual system that relies on this neural representation and find that the best achievable discrimination thresholds are inversely proportional to the sensory prior we examine how the prior information that is implicitly encoded in the tuning curves of the optimal population may be used for perceptual inference and derive a novel decoder the bayesian population vector that closely approximates a bayesian least squares estimator that has explicit access to the prior finally we generalize these results to sigmoidal tuning curves correlated neural variability and a broader class of objective functions these results provide a principled embedding of sensory prior information in neural populations and yield predictions that are readily testable with environmental physiological and perceptual data

approximate solution of a flow over a ramp for large froude number abstract an approximate method is presented to solve the problem of steady free surface flow of an ideal fluid over a semi infinite ramp in the bottom schwartz christoffel transformation is used to map the region of flow in the complex potential plane onto the upper half plane the hilbert transformation as well as the perturbation technique are used as a basis for the approximate solution of the problem for large froude number and small inclination angle of the ramp general equations in integral form for any order of approximation are obtained solution up to first order approximation is discussed and illustrated elevation of the free surface for different ramp heights different inclination angles of the ramp and different froude numbers are plotted an approximate formula of maximum elevation of the free surface in terms of the ramp heights and its inclination angle is found