the binary weight distribution of the extended 2m 2m 4 code of the reed solomon code over gf 2m with generator polynomial x α x α2 x α3 abstract consider an n k linear code with symbols from gf 2 m if each code symbol is represented by a binary m tuple using a certain basis for gf 2 m we obtain a binary nm km linear code called a binary image of the original code in this paper we present a lower bound on the minimum weight of a binary image of a cyclic code over gf 2 m and the weight enumerator for a binary image of the extended 2 m 2 m 4 code of the reed solomon code over gf 2 m with generator polynomial x x 2 x 3 and its dual code where is a primitive element in gf 2 m

positive definite functions on coxeter groups with applications to operator spaces and noncommutative probability a new class of positive definite functions related to colour length function on arbitrary coxeter group is introduced extensions of positive definite functions called the riesz coxeter product from the riesz product on the rademacher abelian coxeter group to arbitrary coxeter group is obtained applications to harmonic analysis operator spaces and noncommutative probability are presented characterization of radial and colour radial functions on dihedral groups and infinite permutation group are shown

testing by verifying walsh coefficients testing of logic networks by verifying the walsh coefricients of the outputs is explored measurement of one of these can detect arbitrarily many input leads stuck and just two measurements requiring little hardware can detect any single stuck at fault in appropriately designed networks

the ito clifford integral abstract we present a theory of non commutative stochastic integration analogous to the ito theory it is shown that wick products of fermi fields define martingales and that stochastic integrals with respect to these are defined for adapted operatorvalued square integrable integrands for square integrable martingales associated with an arbitrary probability gage space a stochastic integral is defined and a doob meyer decomposition for supermartingales obtained

group theoretic signature analysis a new class of data compression techniques called group theoretic signature analysis gtsa for testing a logic network is proposed and the fault coverage of the method is analyzed this method is a generalization of accumulator compression testing built in self test for processor environments is feasible with gtsa

counter based compaction delay and stuck open faults in this paper we study the properties of all major counter based compaction schemes when the circuit under test is affected by delay or stuck open faults we present an error model that accurately describes the behavior of such circuits the error model inherits from the asymmetric error model using this model we compute exact aliasing probability for any test session length we also determine the asymptotic aliasing probability examples that compare aliasing in counter based compaction with aliasing in lfsrs indicate that the latter is more predictable

accumulator compression testing a new test data reduction technique called accumulator compression testng act is proposed act is an extension of syndrome testing it is shown that the enumeration of errors missed by act for a unit under test is equivalent to the number of restricted partitions of a number asymptotic results are obtained for independent and dependent error modes comparison is made between signature analysis sa and act theoretical results indicate that with act a better control over fault coverage can be obtained than with sa experimental results are supportive of this indication built in self test for processor environments may be feasible with act however for general vlsi circuits the complexity of act may be a problem as an adder is necessary

test pattern generation based on reed muller coefficients reed muller coefficients are used to generate a test pattern selection procedure for detecting single stuck at faults this procedure is based on the heuristics deduced from the way in which the spectral coefficients are affected by such faults the changes that the spectral coefficients undergo are also compared with the fault models used most frequently to model defects in combinational circuits the proposed pattern generating procedure does not need a simulation of the circuit for each of the possible stuck at faults and its complexity is proportional to the number of gates in the circuit the proportionality constant increases exponentially as the number of inputs in the circuit increases to evaluate the performance of the proposed method its application to some benchmark circuits including the alu 74181 is presented

non commutative probability and non commutative processes beyond the heisenberg algebra a probability space is a pair a where a is an algebra and is a state on the algebra in classical probability a is the algebra of linear combinations of indicator functions on the sample space and in quantum probability a is the heisenberg or clifford algebra however other algebras are of interest in noncommutative probability after a short review of the framework of classical and quantum probability other noncommutative probability spaces are discussed in particular those associated with noncommutative space time a probability space is a pair a where a is an algebra and is a state on the algebra in classical probability a is the algebra of linear combinations of indicator functions on the sample space and in quantum probability a is the heisenberg or clifford algebra however other algebras are of interest in noncommutative probability after a short review of the framework of classical and quantum probability other noncommutative probability spaces are discussed in particular those associated with noncommutative space time

zero aliasing for modeled faults when using built in self test bist for testing vlsi circuits the circuit response to an input test sequence which may consist of thousands to millions of bits is compacted into a signature which consists of only tens of bits usually a linear feedback shift register lfsr is used for response compaction via polynomial division the compacting function is a many to one function and as a result some erroneous responses may be mapped to the same signature as the good response this is known as aliasing in this paper we deal with the selection of a feedback polynomial for the compacting lfsr such that an erroneous response resulting from any modeled fault is mapped to a signature that is different from that for the good response such lfsrs are called zero aliasing lfsrs only zero aliasing lfsrs with primitive or irreducible feedback polynomials are considered due to their suitability for bist test pattern generation

the q meixner distributions associated with a q deformed symmetric fock space in this paper we consider the self adjoint operator of the linear sum of the double q creation the double q annihilation the q number and the scalar operators on a q deformed symmetric fock space and investigate its distribution by the self adjointness of the operator the theory of orthogonal polynomials can help us to determine the distributions well it will be found that the distributions are in the types of the q meixner family

comparative analysis of different implementations of multiple input signature analyzers signature analysis is an accepted method of obtaining data compression for built in testing applications the author deals with a unified approach to the analysis of multiple input signature analyzers by considering them as finite state switching circuits this approach is used to investigate and compare different implementations and it is shown that there is a large range of alternatives to achieve a given characteristic polynomial particular emphasis is placed on the two most common implementations it is shown that both perform polynomial division with each input of one circuit being equivalent to a combination of inputs of the other the approach also gives a method of expressing aliasing patterns for both implementations which leads to a study of the differences with regard to certain error detection capabilities

the derivative of the maximal function in this note we show that the local hardy littlewood maximal operator is bounded in the sobolev space thus the maximal function often has partial derivatives we also show that the maximal operator preserves the zero boundary values in sobolev s sense

on random pattern test length the testing of large logic networks with random patterns is examined work by previous workers for single faults is extended to a class of multiple fault situations not only is the problem of fault detection in the presence of nonmasking multiple faults treated but the question of distinguishing between them is also examined it is shown that a test that merely exposes each fault has a high probability of distinguishing between the faults the relationships between quality diagnostic resolution and random pattern test length are developed the results have application to self test schemes that use random patterns as stimuli

analysis and design of linear finite state machines for signature analysis testing the authors present a theoretical investigation of the aliasing error probability aep in signature analysis testing by means of linear finite state machines lfsms the equations of the resulting markov chain model of the lfsm are solved to determine an exact expression of the aep as a function of the main lfsm features and of the relevant parameters of the testing environment this expression is used to prove criteria for the synthesis of lfsms with minimum asymptotic and transient aep a fundamental lower bound on the aep is presented which represents the performance limit of any lfsm with respect to aliasing minimization it is shown that the aep in machines realizing counters mod 2 sup k 1 is the closest to such a bound in particular periodically reaching it

fock representation of the renormalized higher powers of white noise and the centreless virasoro or witt zamolodchikov w lie algebra the identification of the lie algebra of the renormalized higher powers of white noise rhpwn and the analytic continuation of the second quantized centreless virasoro or witt zamolodchikov w lie algebra of conformal field theory and high energy physics was recently established in 5 based on results obtained in 3 and 4 in the present paper we show how the rhpwn fock kernels must be truncated in order to be positive semi definite and we obtain a fock representation of the two algebras we show that the truncated renormalized higher powers of white noise trhpwn fock spaces of order 2 host the continuous binomial and beta processes

regularity of the centered fractional maximal function on radial functions abstract we study the regularity properties of the centered fractional maximal function m more precisely we prove that the map f m f is bounded and continuous from w 1 1 r d to l q r d in the endpoint case q d d if f is a radial function for d 1 the radiality assumption can be removed this corresponds to the counterparts of known results for the non centered fractional maximal function the main new idea consists in relating the centered and non centered fractional maximal function at the derivative level

a combinatorial approach to x tolerant compaction circuits test response compaction for integrated circuits ics with scan based design for testability dft support in the presence of unknown logic values xs is investigated from a combinatorial viewpoint the theoretical foundations of x codes employed in an x tolerant compaction technique called x compact are examined through the formulation of a combinatorial model of x compact novel design techniques are developed for x codes to detect a specified maximum number of errors in the presence of a specified maximum number of unknown logic values while requiring only small fan out the special class of x codes that results leads to an avoidance problem for configurations in combinatorial designs general design methods and nonconstructive existence theorems to estimate the compaction ratio of an optimal x compactor are also derived

markov processes and generalized schrodinger equations starting from the forward and backward infinitesimal generators of bilateral time homogeneous markov processes the self adjoint hamiltonians of the generalized schrodinger equations are first introduced by means of suitable doob transformations then by broadening with the aid of the dirichlet forms the results of the nelson stochastic mechanics we prove that it is possible to associate bilateral and time homogeneous markov processes to the wave functions stationary solutions of our generalized schrodinger equations particular attention is then paid to the special case of the levy schrodinger ls equations and to their associated levy type markov processes and to a few examples of cauchy background noise

an example of a generalized brownian motion we present an example of a generalized brownian motion it is given by creation and annihilation operators on a twisted fock space ofl2 these operators fulfill for a fixed 1 1 the relationsc f c g c g c f f g 1 f g l2 we show that the distribution of these operators with respect to the vacuum expectation is a generalized gaussian distribution in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula we also indicate that our brownian motion is one component of ann dimensional brownian motion which is invariant under the quantum groups u n of woronowicz with v2

quadratic exponential vectors we give a necessary and sufficient condition for the existence of a quadratic exponential vector with test function in l2 rd l rd we prove the linear independence and totality in the quadratic fock space of these vectors using a technique different from the one used by accardi et al quantum probability and infinite dimensional analysis vol 25 p 262 2009 we also extend to a more general class of test functions the explicit form of the scalar product between two such vectors

notes on multiple input signature analysis many results regarding the probability of aliasing for multiple input compactors have been derived under error assumptions that are not very realistic for vlsi circuits recently the value of aliasing probability has been proven to tend to 2 sup k where k is the number of binary memory elements of the linear compactor this result is based on the assumption that the compactor is characterized by an irreducible polynomial and that the no error vector has a probability different from zero in these notes the above result is generalized more specifically it is proved that it is valid if any two error vectors neither of which needs to be the no error vector have probabilities of occurrence different from zero to make the error model complete the situation in which exactly one error vector has a probability different from zero is also considered for the latter type of error distributions the test lengths at which aliasing occurs are determined simple proofs for the results are provided they are based on standard linear algebra notions and well known theorems

a note on aliasing probability for multiple input signature analyzer recently d k pradhan s k gupta and m g karpovsky 1990 affirmed that the codes generated by x alpha x beta were not reed solomon where alpha and beta were roots of the distinct primitive polynomials of degree m it was assumed that beta alpha sup 8 where s 1 and 2 sup m 1 were mutually prime the authors show that under the above assumption the respective codes are reed solomon codes

the quadratic fock functor we construct the quadratic analog of the boson fock functor while in the first order linear case all contractions on the 1 particle space can be second quantized the semigroup of contractions that admit a quadratic second quantization is much smaller due to the nonlinearity the encouraging fact is that it contains as proper subgroups i e the contractions all the gauge transformations of second kind and all the a e invertible maps of rd into itself leaving the lebesgue measure quasi invariant in particular all diffeomorphism of rd this allows quadratic two dimensional quantization of gauge theories of representations of the witt group in fact it continuous analog of the zamolodchikov hierarchy and much more within this semigroup we characterize the unitary and the isometric elements and we single out a class of natural contractions

aliasing probability for multiple input signature analyzer single and multiple multiple input signature register misr aliasing probability expressions are presented for arbitrary test lengths a framework based on algebraic codes is developed for the analysis and synthesis of misr based test response compressors for bist this framework is used to develop closed form expressions for the aliasing probability of misr for arbitrary test length an error model based on q ary symmetric channel is proposed using more realistic assumptions results are presented that provide the weight distributions for q ary codes q 2 sup m where the circuit under test has m outputs these results are used to compute the aliasing probability for the misr compression technique for arbitrary test lengths this result is extended to compression using two different misrs it is shown that significant improvements can be obtained by using two signature analyzers instead of one the weight distribution of a class of codes of arbitrary length is also given

on the probability of undetected error for linear block codes the problem of computing the probability of undetected error is considered for linear block codes used for error detection the recent literature is first reviewed and several results are extended it is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual using the dual code formulation the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes several bounds on the undetected error probability are then presented we conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained an efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results

on martingale inequalities in non commutative stochastic analysis we develop a non commutativelpstochastic calculus for the clifford stochastic integral anl2theory of which has been developed by barnett streater and wilde the main results are certain non commutativelpinequalities relating clifford integrals and their integrands these results are applied to extend the domain of the clifford integral froml2tol1integrands and we give applications to optional stopping of clifford martingales proving an analog of a theorem of burkholder the stopped clifford processfthas zero expectation providedet in proving these results we establish a number of results relating the clifford integral to the differential calculus in the clifford algebra in particular we show that the clifford integral is given by the divergence operator and we prove an explicit martingale representation theorem both of these results correspond closely to basic results for stochastic analysis on wiener space thus furthering the analogy between the clifford process and brownian motion

endpoint sobolev and bv continuity for maximal operators abstract in this paper we investigate some questions related to the continuity of maximal operators in w 1 1 and bv spaces complementing some well known boundedness results letting m be the one dimensional uncentered hardy littlewood maximal operator we prove that the map f m f is continuous from w 1 1 r to l 1 r in the discrete setting we prove that m b v z b v z is also continuous for the one dimensional fractional hardy littlewood maximal operator we prove by means of counterexamples that the corresponding continuity statements do not hold both in the continuous and discrete settings and for the centered and uncentered versions

on the regularity of maximal operators supported by submanifolds abstract we prove that the maximal operators supported by submanifolds are bounded and continuous on the triebel lizorkin spaces f s p q r d and besov spaces b s p q r d for 0 s 1 and 1 p q as a corollary we also show that these operators are bounded and continuous on the fractional sobolev spaces w s p r d for 0 s 1 and 1 p our main results represent significant improvements as well as natural extensions of what was known previously

on the one mode quadratic weyl operators abstract it has been known in 7 that one mode quadratic weyl operators are well defined as unitary operators acting on the quadratic fock space these operators were defined by their action on the finite particles space however their action on the domain of the quadratic exponential vectors is still unknown in this paper we provide a new reformulation w r t 1 of the quadratic exponential vectors and we compute the action of the one mode quadratic weyl operators on the set of these exponential vectors then we prove the independence of the one mode quadratic weyl operators parameterized by the principal domain associated with the quadratic heisenberg group obtained in 7 this significant contribution to the program of developing the quadratic white noise calculus constitutes a step toward the c representation of the renormalized square of white noise algebra

signature analysis for multiple output circuits a scheme of a good signature analysis by a linear feedback shift register lfsr is presented it works for k output circuits even if k is greater than the register length it is built according to rules which are presented in the correspondence taking into account error models which are introduced the rules are derived from a property which is formally shown for one kind of lfsr however some of them apply to other lfsr schemes too

a unified view of test compression methods a unified treatment of the various techniques to reduce the output data from a unit under test is given the characteristics of time compression schemes with respect to errors detected are developed the use of two or more of these methods together is considered methods to design efficient test compression structures for built in tests are proposed the feasibility of the proposed approach is demonstrated by simulation results

classical and quantum probability we follow the development of probability theory from the beginning of the last century emphasizing that quantum theory is really a generalization of this theory the great achievements of probability theory such as the theory of processes generalized random fields estimation theory and information geometry are reviewed their quantum versions are then described

pointwise behaviour of m1 1 sobolev functions our main objective is to study hajlasz type sobolev functions with the exponent one on metric measure spaces equipped with a doubling measure we show that a discrete maximal function is bounded in the hajlasz space with the exponent one this implies that every such function has lebesgue points outside a set of capacity zero we also show that every hajlasz function coincides with a holder continuous hajlasz function outside a set of small hausdorff content our proofs are based on sobolev space estimates for maximal functions

can nonlinear compactors be better than linear ones this paper presents a new analytical method for estimating compaction quality based on the entropy maximization of the entropy of the signature results in an increase of compaction quality the paper studies the influence of the architecture of a compactor on the entropy growth this approach is suitable for nonlinear compactors widely known linear feedback shift registers lfsrs and the counting compactors as well useful theorems to determine entropy growth during the test have been given and proven furthermore the class of nonlinear compactors based on shift registers named cyclic feedback shift registers cfsrs which guarantee the maximal entropy growth and the aliasing probability of 2 sup n is found it is shown that they are considerably better than the simplest linear compactor feedback shift register and in many cases are quite as good as linear compactors lfsrs having primitive polynomials on the opposite side compactors based on counters are far worse than cfsrs the family of cfsrs is much greater than the already explored family of lfsrs and some cfsrs require less overhead

c quadratic quantization in the first part of the paper we introduce a new parametrization for the manifold underlying quadratic analogue of the usual heisenberg group introduced in accardi et al infin dimens anal quantum probab relat top 13 551 587 2010 which makes the composition law much more transparent in the second part of the paper the new coordinates are used to construct an inductive system of algebras each of which is isomorphic to a finite tensor product of copies of the one mode quadratic weyl algebra we prove that the inductive limit algebra is factorizable and has a natural localization given by a family of sub algebras each of which is localized on a bounded borel subset of mathbb r moreover we prove that the family of quadratic analogues of the fock states defined on the inductive family of algebras is projective hence it defines a unique state on the limit algebra finally we complete this algebra under the minimal regular c norm thus obtaining a c algebra

accumulator based compaction of test responses an accumulator based compaction abc scheme for parallel compaction of test responses is introduced the asymptotic and transient coverage drop introduced by accumulators with binary and 1 s complement adders is studied using markov chain models it is proven that the asymptotic coverage drop in abc with binary adders is 2 sup k where k is the number of bits in the adder that the fault can reach in abc with 1 s complement adders the asymptotic coverage drop for a fairly general class of faults is 2n 1 sup 1 where n is the total number of bits the analysis of transient behavior relates the coverage drop with the probability of fault injection the size of the accumulator and the length of the test experiment the process is characterized by damping factors derived for various values of these parameters

geometry stochastic calculus and quantum fields in a noncommutative space time the algebras of nonrelativistic and of classical mechanics are unstable algebraic structures their deformation towards stable structures leads respectively to relativity and to quantum mechanics likewise the combined relativistic quantum mechanics algebra is also unstable its stabilization requires the noncommutativity of the space time coordinates and the existence of a fundamental length constant the new relativistic quantum mechanics algebra has important consequences on the geometry of space time on quantum stochastic calculus and on the construction of quantum fields some of these effects are studied in this paper

on the variation of maximal operators of convolution type abstract in this paper we study the regularity properties of two maximal operators of convolution type the heat flow maximal operator associated to the gauss kernel and the poisson maximal operator associated to the poisson kernel in dimension d 1 we prove that these maximal operators do not increase the l p variation of a function for any p 1 while in dimensions d 1 we obtain the corresponding results for the l 2 variation similar results are proved for the discrete versions of these operators

weak differentiability for fractional maximal functions of general lp functions on domains abstract let r n be bounded a domain we prove under certain structural assumptions that the fractional maximal operator relative to maps l p w 1 p for all p 1 when the smoothness index 1 in particular the results are valid in the range p 1 n n 1 that was previously unknown as an application we prove an endpoint regularity result in the domain setting

aliasing probability calculations for arbitrary compaction under independently selected random test vectors this paper discusses a systematic methodology for calculating the exact aliasing probability associated with schemes that use an arbitrary finite state machine to compact the response of a combinational circuit to a sequence of independently selected random test input vectors the proposed approach identifies the strong influence of fault activation probabilities on the probability of aliasing and uses an asymmetric error model to simultaneously track the states of two fictitious compactors one driven by the response of the fault free combinational circuit and one driven by the response of the faulty combinational circuit by deriving the overall markov chain that describes the combined behavior of these two compactors we are able to calculate the exact aliasing probability for any test sequence length in particular for long enough sequences the probability of aliasing is shown to only depend on the stationary distribution of the markov chain the insights provided by our analysis are used to evaluate the testing performance of simple examples of nonlinear compactors and to demonstrate regimes where they exhibit lower aliasing probability than linear compactors with the same number of states finally by establishing connections with previous work that evaluated aliasing probability in linear compactors our analysis clarifies the role played by the entropy of the stationary distribution of the compactor states

on powers of gaussian white noise classical gaussian white noise in communications and signal processing is viewed as the limit of zero mean second order gaussian processes with a compactly supported flat spectral density as the support goes to infinity the difficulty of developing a theory to deal with nonlinear transformations of white noise has been to interpret the corresponding limits in this paper we show that a renormalization and centering of powers of band limited gaussian processes is gaussian white noise and as a consequence homogeneous polynomials under suitable renormalization remain white noises

self adjointness and boundedness in quadratic quantization we construct a counter example showing for the quadratic quantization the identity t t is not necessarily true we characterize all operators on the one particle algebra whose quadratic quantization are self adjoint operators on the quadratic fock space finally we discuss the boundedness of the quadratic quantization

fock space associated to coxeter groups of type b abstract in this article we construct a generalized gaussian process coming from coxeter groups of type b it is given by creation and annihilation operators on an q fock space which satisfy the commutation relation b q x b q y q b q y b q x x y i x y q 2 n where x y are elements of a complex hilbert space with a self adjoint involution x x and n is the number operator with respect to the grading on the q fock space we give an estimate of the norms of creation operators we show that the distribution of the operators b q x b q x with respect to the vacuum expectation becomes a generalized gaussian distribution in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions our generalized gaussian distribution is associated to the orthogonal polynomials called the q meixner pollaczek polynomials yielding the q hermite polynomials when 0 and free meixner polynomials when q 0

boundedness and unboundedness results for some maximal operators on functions of bounded variation abstract we characterize the space bv i of functions of bounded variation on an arbitrary interval i r in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator m r from bv i into the sobolev space w 1 1 i by restriction the corresponding characterization holds for w 1 1 i we also show that if u is open in r d d 1 then boundedness from bv u into w 1 1 u fails for the local directional maximal operator m t v the local strong maximal operator m t s and the iterated local directional maximal operator m t d m t 1 nevertheless if u satisfies a cone condition then m t s bv u l 1 u boundedly and the same happens with m t v m t d m t 1 and m r

strong solutions for differential equations in abstract spaces abstract let e f be a locally convex space we denote the bounded elements of e by e b x e x f sup f x in this paper we prove that if b e b is relatively compact with respect to the f topology and f i e b e b is a measurable family of f continuous maps then for each x 0 e b there exists a norm differentiable i e differentiable with respect to the f norm local solution to the initial valued problem u t t f t u t u t 0 x 0 all of this machinery is developed to study the lipschitz stability of a nonlinear differential equation involving the hardy littlewood maximal operator

sharp sobolev inequalities in lorentz spaces for a mean oscillation abstract we exhibit the optimal constant for sobolev inequalities in lorentz spaces for a mean oscillation and its relation with a boundedness of the hardy littlewood maximal operator in sobolev spaces some applications to a scale invariant form of hardy s inequality in a limiting case are also considered

spectral fault signatures for single stuck at faults in combinational networks a method is described for the derivation of fault signatures for the detection of stuck at faults in single output combinational networks these signatures consist of a set of values derived from the network any single stuck at fault causes at least one value to change the fault signatures developed are a generalization of syndrome testing the technique is developed in the rademacher walsh spectral domain but is easily implemented using conters and basic gates

a new framework for designing and analyzing bist techniques and zero aliasing compression a general framework for shift register based signature analysis is presented and a mathematical model for this framework based on coding theory is developed there are two key features of this formulation first it allows for uniform treatment of lfsr misr and multiple misr based signature analyzer in addition using this formulation a new compression scheme for multiple output cut is proposed this scheme referred to as multiinput lfsr has the potential to achieve better aliasing than other schemes such as the multiple misr scheme of comparable hardware complexity several results on aliasing are presented and certain known results are shown to be direct consequences of the formulation also developed are error models that take into account the circuit topology and the effect of faults at the outputs using these models exact closed form expressions for aliasing probability are developed a closed form aliasing expression for misr under an independent error model is provided

sobolev regularity of polar fractional maximal functions abstract we study the sobolev regularity on the sphere s d of the uncentered fractional hardy littlewood maximal operator m at the endpoint p 1 when acting on polar data we first prove that if q d d 0 d and f is a polar w 1 1 s d function we have m f q d f 1 we then prove that the map f m f is continuous from w 1 1 s d to l q s d when restricted to polar data our methods allow us to give a new proof of the continuity of the map f m f from w rad 1 1 r d to l q r d moreover we prove that a conjectural local boundedness for the centered fractional hardy littlewood maximal operator m implies the continuity of the map f m f from w 1 1 to l q in the context of polar functions on s d and radial functions on r d

spectral fault signatures for internally unate combinational networks a method is described for the derivation of fault signatures for certain classes or irredundant combinational networks these signatures consist of a set of values derived from the network any stuck at fault causes at least one of the values to change the signatures provide complete fault detection for all single stuck at faults they are usually short and never contain more than n 1 values for an n input network

functions of bounded variation the derivative of the one dimensional maximal function and applications to inequalities we prove that if f i r r is of bounded variation then the uncentered maximal function mf is absolutely continuous and its derivative satisfies the sharp inequality dmf l 1 i df i this allows us to obtain under less regularity versions of classical inequalities involving derivatives

optimal zero aliasing space compaction of test responses many built in self testing bist schemes compress the test responses from a k output circuit to q signature streams where q spl lt k a process termed space compaction the effectiveness of such a compaction method can be measured by its compaction ratio c k q a high compaction ratio can introduce aliasing which occurs when a faulty test response maps to the fault free signature we investigate the problem of designing zero aliasing space compaction circuits with maximum compaction ratio c sub max we introduce a graph representation of test responses to study the space compaction process and relate space compactor design to a graph coloring problem given a circuit under test a fault model and a test set we determine q sub min which yields c sub max k q sub min this provides a fundamental bound on the cost of signature based bist we show that q sub min spl les 2 for all the iscas 85 benchmark circuits we develop a systematic design procedure for the synthesis of space compaction circuits and apply it to a number of iscas 85 circuits finally we describe multistep compaction which allows zero aliasing to be achieved with any q even when q sub min 1

parallel signature analysis design with bounds on aliasing this paper presents parallel signature design techniques that guarantee the aliasing probability to be less than 2 l where l is the test length using y signature samples a parallel signature analysis design is proposed that guarantees the aliasing probability to be less than y l sup y 2 inaccuracies and incompleteness in previously published bounds on the aliasing probability are discussed simple bounds on the aliasing probability are derived for parallel signature designs using primitive polynomials

syndrome and transition count are uncorrelated logic circuit testing in the testing of logic circuits two proposed data compression methods use the number of ones syndrome and the number of sequence changes transition count an enumeration n m k t of the number of length m binary sequences having syndrome value k and transition count t is developed examination of this result reveals that the parallel compression of these two methods has small overlap in error masking an asymptotic expression for n m k t is developed

reformulating the schrodinger equation as a shabat zakharov system we reformulate the second order schrodinger equation as a set of two coupled first order differential equations a so called shabat zakharov system sometimes called a zakharov shabat system there is considerable flexibility in this approach and we emphasize the utility of introducing an auxiliary condition or gauge condition that is used to cut down the degrees of freedom using this formalism we derive the explicit but formal general solution to the schrodinger equation the general solution depends on three arbitrarily chosen functions and a path ordered exponential matrix if one considers path ordering to be an elementary process then this represents complete quadrature albeit formal of the second order linear ordinary differential equation

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

quantum independent increment processes on superalgebras on introduit la notion de processus quantique independant a increment stationnaire sur des superalgebres et on demontre un theoreme de reconstruction qui etablit une correspondance un a un entre ces processus et leurs generateurs infinitesimaux

on induced scalar products and unitarization infinitesimal unitarity of representations of the simple real lie algebras su 2 su 1 1 and so 4 is discussed with respect to scalar products induced by sesquilinear forms on their universal enveloping algebras sesquilinear forms are explicitly calculated the verma modules their irreducible quotients their irreducible submodules and their infinitesimal unitarizability are discussed

simple bounds on serial signature analysis aliasing for random testing it is shown that the aliasing probability is bounded above by 1 epsilon l approximately 1 l epsilon small for large l for test lengths l less than the period l sub c of the signature polynomial for test lengths l that are multiples of l sub c the aliasing probability is bounded above by 1 for test lengths l greater than l sub c and not a multiple of l sub c the aliasing probability is bounded above by 2 l sub c 1 these simple bounds avoid any exponential complexity associated with the exact computation of the aliasing probability simple bounds also apply to signature analysis based on any linear finite state machine including linear cellular automaton from these simple bounds it follows that the aliasing probability in a signature analysis design using beta intermediate signatures is bounded by 1 epsilon sup beta beta sup beta l sup beta for beta

quadratic bosonic and free white noises we discuss the meaning of renormalization used for deriving quadratic bosonic commutation relations introduced by accardi alv and find a representation of these relations on an interacting fock space also we investigate classical stochastic processes which can be constructed from noncommutative quadratic white noise we postulate quadratic free white noise commutation relations and find their representation on an interacting fock space

non gaussian malliavin calculus on real lie algebras abstract the non commutative malliavin calculus on the heisenberg weyl algebra is extended to the affine algebra a differential calculus and a non commutative integration by parts are established as an application we obtain sufficient conditions for the smoothness of wigner type laws of non commutative random variables with gamma or continuous binomial marginals

double phase image restoration abstract in this paper we explore the potential of the double phase functional in an image processing context to this end we study minimizers of the double phase energy for functions with bounded variation and show that this energy can be obtained by convergence or relaxation of regularized functionals a central tool is a capped fractional maximal function of the derivative of bv functions

renormalized squares of white noise and other non gaussian noises as levy processes on real lie algebras it is shown how the relations of the renormalized squared white noise defined by accardi lu and volovich alv99 can be realized as factorizable current representations or levy processes on the real lie algebra 2 this allows to obtain its ito table which turns out to be infinite dimensional the linear white noise without or with number operator is shown to be a levy process on the heisenberg weyl lie algebra or the oscillator lie algebra furthermore a joint realization of the linear and quadratic white noise relations is constructed but it is proved that no such realizations exist with a vacuum that is an eigenvector of the central element and the annihilator classical levy processes are shown to arise as components of levy processes on real lie algebras and their distributions are characterized in particular the square of white noise analogue of the quantum poisson process is shown to have a 2 probability density and the analogue of the field operators to have a density proportional to where is the usual function and m 0 a real parameter

quasi free quantum stochastic integrals for the car and ccr abstract a theory of quantum martingales and quantum stochastic integrals in quasi free representations of the car and ccr is presented for the car the results generalize some of those developed in barnett streater and wilde j funct anal 48 1982 172 212 j london math soc 27 1983 373 384 and for the ccr the results contain the standard ito theory of stochastic integration with respect to brownian motion as a special case