on fuzzy linear representations of finite groups abstract a linear representation of a finite group g in a finite dimensional vector space v induces through zadeh s extension principle a function 9 from igto into igl v where gl v is the group of all linear automorphisms of v if w is a fuzzy subspace of v the group of all fuzzy linear automorphisms of w gl w is a subgroup of gl v w is said to be stable under the action of a fuzzy subgroup a of g if 9 a is a subset of gl w i e 9 a is zero at every f inside gl v and outside gl q if w is stable under the action of a then its support subspace is stable under the action of the support subgroup of a in the crisp sense finally we show that if there are two stable fuzzy subspaces one of them is contained in the other then the smaller one has a fuzzy direct summand in the bigger which is also a stable fuzzy subspace

a new class of solvable dynamical systems a new class of dynamical systems are presented together with their solutions some of these models are isochronous namely their generic solutions are all completely periodic with the same period others are characterized by friction all solutions vanishing in the remote future and still others are asymptotically isochronous approaching an isochronous behavior in the remote future

k fuzzy ideals in semirings abstract we introduce the k fuzzy ideal of a semiring also we construct an extension of a fuzzy ideal in a k semiring and study the quotient structure of a k semiring by a fuzzy ideal

high temperature expansion of the emptiness formation probability for the isotropic heisenberg chain recently gohmann klumper and seel have derived novel integral formulae for the correlation functions of the spin 1 2 heisenberg chain at finite temperature we have found that the high temperature expansion hte technique can be effectively applied to evaluate these integral formulae actually as for the emptiness formation probability p n of the isotropic heisenberg chain we have found a general formula of the hte for p n with arbitrary up to o j t 4 if we fix a magnetic field to a certain value we can calculate the hte to much higher order for example the order up to o j t 42 has been achieved in the case of p 3 when h 0 we have compared these hte results with the data by quantum monte carlo simulations they exhibit excellent agreement

high resolution quantization theory and the vector quantizer advantage the authors consider how much performance advantage a fixed dimensional vector quantizer can gain over a scalar quantizer they collect several results from high resolution or asymptotic in rate quantization theory and use them to identify source and system characteristics that contribute to the vector quantizer advantage one well known advantage is due to improvement in the space filling properties of polytopes as the dimension increases others depend on the source s memory and marginal density shape the advantages are used to gain insight into product transform lattice predictive pyramid and universal quantizers although numerical prediction consistently overestimated gains in low rate 1 bit sample experiments the theoretical insights may be useful even at these rates

fuzzy relations and fuzzy groups abstract given a group s we consider fuzzy relations on s that is maps from s s into 0 1 of particular interest is to investigate conditions under which the fuzzy relation becomes a fuzzy subgroup on s s we prove that if is a fuzzy subset of s and is the strongest fuzzy relation on s that is a fuzzy relation on then is a fuzzy subgroup if and only if is a fuzzy subgroup a number of other results are obtained about the interrelationships between fuzzy relations on s including the weakest fuzzy relation and fuzzy subgroups on s s

asymptotically optimal model estimation for quantization using high rate theory approximations we introduce flexible practical quantizers based on possibly non gaussian models in both the constrained resolution cr and the constrained entropy cases we derive model estimation criteria optimizing asymptotic with increasing rate quantizer performance we show that in the cr case the optimal criterion is different from the maximum likelihood criterion commonly used for that purpose and introduce a new criterion that we call constrained resolution minimum description length cr mdl we apply these principles to the generalized gaussian scaled mixture model which is accurate for many real world signals we provide an explanation of the reason why the cr mdl improves quantization performance in the cr case and show that cr mdl can compensate for a possible mismatch between model and data distribution thus this criterion is of a great interest for practical applications our experiments apply the new quantization method to controllable artificial data and to the commonly used modulated lapped transform representation of audio signals we show that both the cr mdl criterion and a non gaussian modeling have significant advantages

fuzzy direct sums of fuzzy rings abstract in this paper we introduce the concepts of the fuzzy weak direct sum and the fuzzy complete direct sum of fuzzy subrings of commutative rings we prove some basic properties analogous to those of ordinary rings

lagrangian vector quantization with combined entropy and codebook size constraints in this paper the lagrangian formulation of variable rate vector quantization is extended to quantization with simultaneous constraints on entropy and codebook size including variable and fixed rate quantization as special cases the formulation leads to a lloyd quantizer design algorithm and generalizations of gersho s approximations characterizing optimal performance for asymptotically large rate a variation of gersho s approach is shown to yield rigorous results partially characterizing the asymptotically optimal performance

self replicating pulses and sierpinski gaskets in excitable media in our previous papers we have shown by computer simulations that a sierpinski gasket pattern appears in a bonhoeffer van der pol type reaction diffusion system in this paper we show another class of regular self similar structure which is found in four different excitable reaction diffusion systems this result strongly implies that the existence of the self similar spatiotemporal evolution is universal in excitable reaction diffusion media

length functions and fuzzy groups fuzzy sets were introduced by zadeh 7 in 1965 and fuzzy groups by rosenfeld 4 in 1971 fuzzy groups were also considered by anthony and sherwood i in 1979 and in 1981 the theory was developed by das 2 as in the paper of das 2 we also show how a fuzzy subgroup of a group g is determined by a chain of subgroups of g however we wish to point out that fuzzy subgroups are constructed in a way which is analogous to that of constructing certain length functions of g wilkens s 61 thus in section 3 we consider the theory of length functions and we show the connection between certain length functions and fuzzy subgroups of a group in section 4 we apply the structural results of section 3 on finite solvable groups and we gain a generalization of a theorem of das 2 our notation is standard however we use the symbol e to denote the identity element of a group g and i denotes a real number we start with a preliminary section where we briefly analyse the definition of a fuzzy subgroup given by rosenfeld 4 naturally we use this definition of a fuzzy subgroup throughout the paper

extended supersymmetry for the bound states of the generalized hulthen potential hierarchy using the associated hypergeometric differential equation we analytically solve the bound states corresponding to a hierarchy of the radial potential v0 e r 1 e r c e r 1 e r 2 as a generalization of the hulth n potential then an analytic solution corresponding to a special case for which the parameter c is expected to be in terms of l l 1 is also derived meanwhile without introducing a superpotential and in the framework of supersymmetric quantum mechanics it is shown that these bound states can be calculated by two different algebraic methods based on these two approaches it is noted that the bound states realize an extended supersymmetry structure

spontaneous symmetry breaking in the non linear schrodinger hierarchy with defect we introduce and solve the one dimensional quantum non linear schrodinger nls equation for an n component field defined on the real line with a defect sitting at the origin the quantum solution is constructed using the quantum inverse scattering method based on the concept of reflection transmission rt algebras recently introduced the symmetry of the model is generated by the reflection and transmission defect generators defining a defect subalgebra we classify all the corresponding reflection and transmission matrices this provides the possible boundary conditions obeyed by the canonical field and we compute these boundary conditions explicitly finally we exhibit a phenomenon of spontaneous symmetry breaking induced by the defect and identify the unbroken generators as well as the exact remaining symmetry

the transition from regular to irregular motions explained as travel on riemann surfaces we introduce and discuss a simple hamiltonian dynamical system interpretable as a three body problem in the complex plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on riemann surfaces the interest of this phenomenology illustrating the onset in a deterministic context of irregular motions is underlined by its generality suggesting its eventual relevance to understand natural phenomena and experimental investigations here only some of our main findings are reported without detailing their proofs a more complete presentation will be published elsewhere

two results on the asymptotic performance of quantizers a necessary and sufficient condition is presented for the normalized asymptotic r th power distortion of a mismatched multidimensional quantizer to converge to a certain optimum constant known as bennett s integral the dimensionality of our results is allowed to approach infinity in order to make some universal source coding comparisons to quantization theory

high resolution source coding for non difference distortion measures multidimensional companding entropy coded vector quantization is studied using high resolution multidimensional companding over a class of nondifference distortion measures for distortion measures which are locally quadratic a rigorous derivation of the asymptotic distortion and entropy coded rate of multidimensional companders is given along with conditions for the optimal choice of the compressor function this optimum compressor when it exists depends on the distortion measure but not on the source distribution the rate distortion performance of the companding scheme is studied using an asymptotic expression for the rate distortion function which parallels the shannon lower bound for difference distortion measures it is proved that the high resolution performance of the scheme is arbitrarily close to the rate distortion limit for large quantizer dimensions if the compressor function and the lattice quantizer used in the companding scheme are optimal extending an analogous statement for entropy coded lattice quantization and mse distortion the companding approach is applied to obtain a high resolution quantizing scheme for noisy sources

the painleve property for quasihomogenous systems and a many body problem in the plane we investigate a many body problem in the plane introduced by calogero and intensively studied by calogero francoise and sommacal an ad hoc complexification transforms the many body problem to a system of second order autonomous complex equations depending on some complex constants that describe the two body interactions we investigate the sets of two body interaction constants that make the complexified equation have the painleve property this is its solutions are given by single valued meromorphic functions in this case the original system has only periodic isochronous solutions we exhibit a family of settings where the system displays this property and show that it is not present in the three and four body problems that do not fall within our class for this we introduce a necessary condition for the presence of the painleve property in some quasihomogenous systems

from cellular automaton to difference equation a general transformation method which preserves time evolution patterns we propose a general method to construct a partial difference equation which preserves any time evolution patterns of a cellular automaton the method is based on inverse ultradiscretization with filter functions

intertwining relations for the matrix calogero like models supersymmetry and shape invariance we investigate the intertwining relations for n particle calogero like models with internal degrees of freedom starting from the well known dunkl polychronakos operators we construct a new type of local without exchange operation differential operator these operators intertwine the matrix hamiltonians corresponding to irreducible representations of the permutation group sn in particular cases this method allows us to construct a new class of exactly solvable dirac like equations and a new class of matrix models with shape invariance we establish a connection with the approach of multi dimensional supersymmetric quantum mechanics

analytical properties of ultradiscrete burgers equation and rule 184 cellular automaton in this paper we propose an ultradiscrete burgers equation of which all the variables are discrete the equation is derived from a discrete burgers equation under an ultradiscrete limit and reduces to an ultradiscrete diffusion equation through the cole hopf transformation moreover it becomes a cellular automaton ca under appropriate conditions and is identical to rule 184 ca in a specific case we show shock wave solutions and asymptotic behaviours of the ca exactly via the diffusion equation finally we propose a particle model expressed by the ca and discuss a mean flux of particles

general solution of a three body problem in the plane we provide and discuss the general solution of a hamiltonian three body problem in the plane characterized by newtonian equations of motion with rotation and translation invariant velocity dependent one body and two body forces the model features a nonnegative real parameter when it does not vanish all solutions are completely periodic with period t 2 when it vanishes both unbounded and confined motions are possible with a rather rich phenomenology of possible behaviour in the latter case including completely periodic motions and limit cycles

on prime and primary fuzzy ideals and their radicals abstract the merits and demerits of various definitions of prime and primary fuzzy ideals are discussed and new definitions are suggested it is shown that the new definitions are more satisfactory the nil radical is defined as against the prime radical of a fuzzy ideal some theorems regarding the nil radical are proved it is shown that the two radicals coincide when the grade membership lattice is totally ordered

a technique to identify solvable dynamical systems and another solvable extension of the goldfish many body problem we take advantage of the simple approach recently discussed which associates to solvable matrix equations solvable dynamical systems interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles starting from a solvable matrix evolution equation we obtain the corresponding many body model and note that in one case the auxiliary variables can be altogether eliminated obtaining thereby an also hamiltonian extension of the goldfish model the solvability of this novel model and of its isochronous variant is exhibited a related as well solvable model is also introduced as well as its isochronous variant finally the small oscillations of the isochronous models around their equilibrium configurations are investigated and from their isochronicity certain diophantine relations are evinced

novel solvable variants of the goldfish many body model a recent technique to identify solvable many body problems in two dimensional space yields via a new twist new many body problems of goldfish type some of these models are isochronous namely their generic solutions are completely periodic with a fixed period independent of the initial data the investigation of the behavior of some of these isochronous systems in the vicinity of their equilibrium configurations yields some amusing diophantine relations

mismatch in high rate entropy constrained vector quantization bucklew s 1984 high rate vector quantizer mismatch result is extended from fixed rate coding to variable rate coding using a lagrangian formulation it is shown that if an asymptotically high rate optimal sequence of variable rate codes is designed for a k dimensional probability density function pdf g and then applied to another pdf f for which f g is bounded then the resulting mismatch or loss of performance from the optimal possible is given by the relative entropy or kullback leibler 1968 divergence i f spl par g it is also shown that under the same assumptions an asymptotically optimal code sequence for g can be converted to an asymptotically optimal code sequence for a mismatched source f by modifying only the lossless component of the code applications to quantizer design using uniform and gaussian densities are described including a high rate analog to the shannon rate distortion result of sakrison 1975 and lapidoth 1997 showing that the gaussian is the worst case for lossy compression of a source with known covariance by coupling the mismatch result with composite quantizers the worst case properties of uniform and gaussian densities are extended to conditionally uniform and gaussian densities which provides a lloyd clustering algorithm for fitting mixtures to general densities

the replica mean field theory for the glass matrix model we study the dynamical phase transition of the glass matrix model by using the replica method for deterministic models in the glass matrix model n dynamical variables are defined by p component vectors which make states of a particle each component of a vector is assumed to take 1 to perform replica calculation auxiliary variables are introduced to control the order of the sum of replicated partition function these variables work like quenched random variables of the usual replica method using the approximation for small p n and assuming the homogeneity of the component space we find that the replica theory is similar to that of the previously studied glassy spin models we study the dynamical solution which is defined by imposing the marginally stable condition for a one step replica symmetry breaking ansatz to find the marginally stable condition we study the fluctuation modes in replica spaces and component space the results of simulated annealing is presented to compare with the analytic result the transition temperatures suggested by the two approaches are consistent however the agreement is modest in the sense that simulated annealing shows richer behavior than the replica results mainly due to the lower energy states which appear by the slow annealing schedule

a nonlinear schrodinger equation with two symmetric point interactions in one dimension we consider a time dependent one dimensional nonlinear schrodinger equation with a symmetric double well potential represented by two dirac s among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the hamiltonian then we establish the corresponding strichartz type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

the neatest many body problem amenable to exact treatments a goldfish various formulations findings and conjectures are reviewed which relate to a many body problem that is arguably the neatest nontrivial such model amenable to exact treatments

a further solvable three body problem in the plane the solution is provided of a three body problem in the plane which is the third of a trio recently identified as likely to display a particularly simple time evolution hence to be amenable to exact treatment this conjecture already validated by providing the solution of the first two of these three models is now completely proven by exhibiting the solution of the third this finding also demonstrates the conjectured super painleve character of certain nonlinear ordinary differential equations namely the fact that their general solution is an entire function of the independent variable

fuzzy partitions and fuzzy quotient rings abstract the cosets of a fuzzy ideal m in a ring a form another ring a m called the fuzzy quotient ring of a and m we show that a m is always a fuzzy partition of a and a fuzzy partition of a will be a fuzzy quotient ring with respect to some fuzzy ideal if and only if its associated fuzzy similarity relation is compatible with the ring operations we also discuss relationships between the two operations of the fuzzy quotient ring a m and the extensions over ia of the two operations of a by zadeh s extension principle we study relationships between the level ring structures of a fuzzy quotient ring and the level relations of its associated fuzzy similarity relations we introduce and study a new notion of the factorization of a ring homomorphism through fuzzy quotients finally a relationship between fuzzy prime ideals and fuzzy similarity relation has been established

susyqm and other symmetries in quantum mechanics the relation of supersymmetric quantum mechanics susyqm is discussed with two other symmetry based approaches symmetry and a differential realization of the su 1 1 and su 2 algebra it is demonstrated that symmetry imposes conditions on the even and odd parts of the real and imaginary components of the superpotential w x and these are expressed in terms of a system of first order linear differential equations which is homogeneous when the factorization energy is real and inhomogeneous when it is complex the formal solution of this system is presented for various special cases as well as for the general case it is shown that a trivial solution of this system corresponds to the unbroken symmetry for the scarf ii potential the formalism of susyqm is also linked with that of the potential algebra approach and it is demonstrated that the j and j ladder operators of some su 1 1 or su 2 algebras act on series of degenerate levels of different potentials essentially in the same way as the a and a shift operators of susyqm examples are presented for su 1 1 and su 2 potential algebras as well as for spectrum generating algebras of the same type possible generalizations of this construction are also pointed out

spectrum of prime fuzzy ideals abstract a topological space called a spectrum of prime fuzzy ideals and denoted by fspec r is obtained by introducing a topology on the set of prime fuzzy ideals of a ring r the prime fuzzy ideals used here introduced by the author elsewhere are not necessarily two valued the topology introduced is compact and the usual nonfuzzy prime spectrum spec r is a subspace of fspec r and dense in it situations under which fuzzy spectrums of different rings are homeomorphic are discussed in particular fspec r is proved to be homeomorphic to fspec r frad r where frad r is the fuzzy prime radical of r the correspondence associating a ring r to the topological space fspec r is shown to define a contravarient functor from the category of rings with unity into the category of compact topological spaces

a characterization of fuzzy subgroups abstract an ordinary subgroup of a group g is 1 a subset of g 2 closed under the group operation in a fuzzy subgroup it is precisely these two notions that lose their deterministic character a fuzzy subgroup of a group g associates with each group element a number the larger the number the more certainly that element belongs to the fuzzy subgroup the closure property is captured by the inequality x y t x y in a rosenfeld s original definition t was the function minimum however any t norm t provides a meaningful generalization of the closure property two classes of fuzzy subgroups are investigated the fuzzy subgroups in one class are subgroup generated those in the other are function generated each fuzzy subgroup in these classes satisfies the above inequality with t given by t a b max a b 1 0 while the two classes look different each fuzzy subgroup in either is isomorphic to one in the other it is shown that a fuzzy subgroup satisfies the above inequality with t minimum if and only if it is subgroup generated of a very special type finally these notions are applied to some abstract pattern recognition problems

bennett s integral for vector quantizers this paper extends bennett s 1948 integral from scalar to vector quantizers giving a simple formula that expresses the rth power distortion of a many point vector quantizer in terms of the number of points point density function inertial profile and the distribution of the source the inertial profile specifies the normalized moment of inertia of quantization cells as a function of location the extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increase precise conditions are given for the convergence of distortion suitably normalized to bennett s integral previous extensions did not include the inertial profile and consequently provided only bounds or applied only to quantizers with congruent cells such as lattice and optimal quantizers the new version of bennett s integral provides a framework for the analysis of suboptimal structured vector quantizers it is shown how the loss in performance of such quantizers relative to optimal unstructured ones can be decomposed into point density and cell shape losses as examples these losses are computed for product quantizers and used to gain further understanding of the performance of scalar quantizers applied to stationary memoryless sources and of transform codes applied to gaussian sources with memory it is shown that the short coming of such quantizers is that they must compromise between point density and cell shapes

integrable and solvable many body problems in the plane via complexification a simple prescription allows us to transform by appropriate complexification any one dimensional many body problem describing motions on a line with newtonian acceleration equals forces equations of motion featuring forces which depend analytically on the positions and possibly on the velocities of the particles and which are moreover scaling or translation invariant into a two dimensional many body problem describing motions in the plane with rotation invariant equations of motion if the original one dimensional model is hamiltonian the two dimensional model is also hamiltonian in fact bi hamiltonian in this manner starting from known integrable or solvable one dimensional many body problems integrable or solvable many body problems in the plane generally featuring much richer dynamics are obtained several examples are exhibited finally another more direct prescription is outlined to transform by complexification almost any one dimensional many body problem into a rota

higher order matrix susy transformations in two dimensional quantum mechanics the iteration procedure of supersymmetric transformations for the two dimensional schrodinger operator is implemented by means of the matrix form of factorization in terms of matrix 2 2 supercharges two different types of iterations are investigated in detail the particular case of diagonal initial hamiltonian is considered and the existence of solutions is demonstrated explicit examples illustrate the construction

finite temperature properties of the dirac operator under local boundary conditions we study the finite temperature free energy and fermion number for dirac fields in a one dimensional spatial segment under two different members of the family of local boundary conditions defining a self adjoint euclidean dirac operator in two dimensions for one of such boundary conditions compatible with the presence of a spectral asymmetry we discuss in detail the contribution of this part of the spectrum to the zeta regularized determinant of the dirac operator and thus to the finite temperature properties of the theory

not so fuzzy fuzzy ideals abstract a characterization of rings which are both artinian and noetherian is obtained they are proved to be the rings whose fuzzy ideals assume only finitely many distinct values prime fuzzy ideals have been defined and conditions have been obtained under which they have only two distinct values

integrable systems of quartic oscillators in ordinary three dimensional space a class of completely integrable and indeed solvable hamiltonian many body problems are exhibited characterized by rotation invariant newtonian equations of motion acceleration equals force with linear and cubic forces in ordinary three dimensional space the corresponding hamiltonians are of normal type with the kinetic energy quadratic in the canonical momenta and the potential energy quadratic and quartic in the canonical coordinates

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation

variance mismatched fixed rate scalar quantization of laplacian sources the paper investigates the mean squared error mse distortion of a variance mismatched fixed rate scalar quantizer that is optimal or asymptotically optimal in the minimum mse sense for a laplacian density with standard deviation q but is applied to another with standard deviation p a nitadori like formula is discovered for an optimal quantizer when p q 1 2 also it is shown that the distortion expressions derived rigorously for asymptotically optimal quantile quantizers essentially confirm the heuristically obtained previous result that the distortion decreases as 1 n3 for the heavy mismatch of 3 2 as lnn n2 for the critical mismatch of 3 2 and as 1 n2 for the mild mismatch of 1 and the resulting distortion decreases as in n n2 1 p

high rate vector quantization for detection we investigate high rate quantization for various detection and reconstruction loss criteria a new distortion measure is introduced which accounts for global loss in best attainable binary hypothesis testing performance the distortion criterion is related to the area under the receiver operating characteristic roc curve specifically motivated by sanov s theorem we define a performance curve as the trajectory of the pair of optimal asymptotic type i and type ii error rates of the most powerful neyman pearson test of the hypotheses the distortion measure is then defined as the difference between the area under the curve auc of the optimal pre encoded hypothesis test and the auc of the optimal post encoded hypothesis test as compared to many previously introduced distortion measures for decision making this distortion measure has the advantage of being independent of any detection thresholds or priors on the hypotheses which are generally difficult to specify in the code design process a high resolution zador gersho type of analysis is applied to characterize the point density and the inertial profile associated with the optimal high rate vector quantizer the analysis applies to a restricted class of high rate quantizers that have bounded cells with vanishing volumes the optimal point density is used to specify a lloyd type algorithm which allocates its finest resolution to regions where the gradient of the pre encoded likelihood ratio has greatest magnitude

fuzzy homomorphisms of rings abstract in this paper we introduce the concept of fuzzy homomorphisms of rings we prove for fuzzy homomorphisms analogues of the homomorphism theorems of ordinary ring homomorphisms

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

a complex deformation of the classical gravitational many body problem that features many completely periodic motions a complex deformation of the newtonian equations of motion of the classical gravitational many body problem is introduced namely a many body problem that features a parameter and that reduces when this parameter vanishes to the standard equations of motion of newtonian gravitation for an arbitrary number of pointlike bodies with arbitrary masses and it is shown that when this parameter is instead positive 0 there is an open set of complex initial data such that all the complex motions originating from it are completely periodic with period t 2 and that the infinite measure of this set is a finite nonvanishing fraction of the measure of the entire set of initial data

tl subrings and tl ideals part 1 basic concepts abstract in 25 27 we put forward a rudimentary frame of the theory of tl subalgebras in 26 we established a theory of tl subgroups and tl subgroups in this paper we will continue our work in concern with basic ideas provided in 25 27 introduce the notions of tl subrings and tl ideals of a ring and study some of their properties where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l

a class of integrable hamiltonian systems whose solutions are perhaps all completely periodic we show that the dynamical system characterized by the complex equations of motion qj i qj k 1 k jnqjqkf qj qk j 1 n with f x x x is hamiltonian and integrable and we conjecture that all its solutions qj t j 1 n are completely periodic with a period that is a finite integral multiple of t 2 here n is an arbitrary positive integer is an arbitrary nonvanishing real constant y y is the weierstrass function with arbitrary semiperiods and are two arbitrary constants special cases are f x 2 coth x 1 r2 sinh2 x f x 2 coth x f x 2 sinh x f x 2 x 1 2x2 and of course f x 2 x these findings as well as the conjecture which is shown to be true in some of these special cases are based on the possibility to recast these equations of motion in the modified lax form l i l l m with l and m appropriate n n matrix functions of the n dynamical variables qj and of their time derivatives qj

fuzzy ideals and fuzzy subalgebras of lie algebras abstract a fuzzy idea l of a lie algebra l factors it into a lie algebra l l called a fuzzy quotient lie algebra this fuzzy algebra lie algebra is always a fuzzy partition of l and a fuzzy partition of l will be a fuzzy quotient lie algebra if and only if its associated fuzzy similarity relation is compatible with the lie algebra operations relationship between the operations of the lie algebras l i t and zadeh s extension principle are studied we also study homomorphisms of lie algebras and the relationship between them and fuzzy ideals and fuzzy subalgebras of the domains and the co domains of these homomorphisms finally solvability of fuzzy ideals and studied in defining these solvable fuzzy ideals we use zadeh s extension principle

entropy density and mismatch in high rate scalar quantization with renyi entropy constraint properties of scalar quantization with th power distortion and constrained renyi entropy of order are investigated for an asymptotically high rate optimal sequence of quantizers the contribution to the renyi entropy due to source values in a fixed interval is identified in terms of the entropy density of the quantizer sequence this extends results related to the well known point density concept in optimal fixed rate quantization a dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion the distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined this extends bucklew s fixed rate and gray et al s variable rate mismatch results to general values of the entropy order parameter

the action principle for a system of differential equations we consider the problem of constructing an action functional for physical systems whose classical equations of motion cannot be directly identified with euler lagrange equations for an action principle two ways of constructing the action principle are presented from simple consideration we derive the necessary and sufficient conditions for the existence of a multiplier matrix which can endow a prescribed set of second order differential equations with the structure of the euler lagrange equations an explicit form of the action is constructed if such a multiplier exists if a given set of differential equations cannot be derived from an action principle one can reformulate such a set in an equivalent first order form which can always be treated as the euler lagrange equations of a certain action we construct such an action explicitly there exists an ambiguity not reduced to a total time derivative in associating a lagrange function with a given set of equations we present a complete description of this ambiguity the general procedure is illustrated by several examples

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

correspondence theorem for fuzzy ideals abstract the ring r j of cosets of a fuzzy ideal j of a ring r is defined and shown to be isomorphic to a factor ring of r in a natural way if f r r is an epimorphism of rings a one to one order preserving correspondence is established between the fuzzy ideals of r and those of r which are constant on the kernel of f this generalizes the correspondence theorem of non fuzzy ideals

new two dimensional integrable quantum models from susy intertwining supersymmetrical intertwining relations of second order in the derivatives are investigated for the case of supercharges with deformed hyperbolic metric gik diag 1 a2 several classes of particular solutions of these relations are found the corresponding hamiltonians do not allow the conventional separation of variables but they commute with symmetry operators of fourth order in momenta for some of these models the specific susy procedure of separation of variables is applied

general technique to produce isochronous hamiltonians we introduce a new technique characterized by an arbitrary positive constant with which we associate the period t 2 to modify a hamiltonian so that the new hamiltonian thereby obtained is entirely isochronous namely it yields motions all of which except possibly for a lower dimensional set of singular motions are periodic with the same fixed period t in all their degrees of freedom this technique transforms real autonomous hamiltonians into modified hamiltonians which are also real and autonomous and it is widely applicable for instance to the most general many body problem characterized by newtonian equations of motion acceleration equal force provided it is translation invariant the modified hamiltonians are of course not translation invariant but for 0 they reduce up to marginal changes to the unmodified hamiltonians they were obtained from hence when this technique is applied to translation invariant hamiltonians yielding in their center of mass systems chaotic motions with a natural time scale much smaller than t the corresponding modified hamiltonians shall display a chaotic behavior for quite some time before the isochronous character of the motions takes over we moreover show that the quantized versions of these modified hamiltonians feature equispaced spectra

path integral for quantum operations in this paper we consider a phase space path integral for general time dependent quantum operations not necessarily unitary we obtain the path integral for a completely positive quantum operation satisfied lindblad equation quantum markovian master equation we consider the path integral for quantum operation with a simple infinitesimal generator

isomorphism theorems and fuzzy k ideals of k semirings we define and study the fuzzy maximal k ideal of a k semiring also we characterize the quotient k semiring r a of a k semiring r by a fuzzy maximal k ideal a and we have some isomorphism theorems in commutative k semirings and further we have some properties of the quotient k semiring r a of all cosets of a fuzzy semiprimary semiprime primary prime k ideal a in a commutative k semiring r

goldfishing by gauge theory a new solvable many body problem of goldfish type is identified and used to revisit the connection between two different approaches to solvable dynamical systems an isochronous variant of this model is identified and investigated alternative versions of these models are presented the behavior of the alternative isochronous model near its equilibrium configurations is investigated and a remarkable diophantine result as well as related diophantine conjectures are thereby obtained

connection between the spectrum and the moments of the ground state density in n dimensional space we show that a series of recurrent inequalities derived in n 3 have the same formal expressions in any dimension n 2 they are derived from the multipole sum rules and provide us with upper bounds for the moments of the ground state density depending only on the lowest multipole excitation energy these bounds are transformed into approximate recurrent relations by means of an empirical correction factor the 1 r potential and the harmonic oscillator play a key role in establishing this factor which is exact for these two potentials by construction for a large class of potentials we show that this factor tends to 1 as n in such cases at the large n limit the lowest state for each multipole excitation exhausts the sum rule it thus acquires the characteristics of the one phonon excitation typical of the harmonic oscillator

controllability of open quantum systems with kraus map dynamics this paper presents a constructive proof of complete kinematic state controllability of finite dimensional open quantum systems whose dynamics are represented by kraus maps for any pair of states pure or mixed on the hilbert space of the system we explicitly show how to construct a kraus map that transforms one state into another moreover we prove by construction the existence of a kraus map that transforms all initial states into a predefined target state such a process may be used for example in quantum information dilution thus in sharp contrast to unitary control kraus map dynamics allows for the design of controls which are robust to variations in the initial state of the system the capabilities of non unitary control for population transfer between pure states illustrated for an example of a two level system by constructing a family of non unitary kraus maps to transform one pure state into another the problem of dynamic state controllability of open quantum systems i e controllability of state to state transformations given a set of available dynamical resources such as coherent controls incoherent interactions with the environment and measurements is also discussed

strong ellipticity and spectral properties of chiral bag boundary conditions we prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds from a knowledge of the heat kernel in an infinite cylinder some basic properties of the zeta function are analysed on cylindrical product manifolds of arbitrary even dimension

goldfishing a new solvable many body problem a recent technique allows one to identify and investigate solvable dynamical systems naturally interpretable as classical many body problems being characterized by equations of motion of newtonian type generally in two dimensional space in this paper we tersely review results previously obtained in this manner and present novel findings of this kind mainly solvable variants of the goldfish many body model including models that feature isochronous classes of completely periodic solutions different formulations of these models are presented the behavior of one of these isochronous dynamical systems in the neighborhood of its equilibrium configuration is investigated and in this manner some remarkable diophantine findings are obtained

fuzzy subgroup some characterizations ii abstract we consider an analysis of fuzzy subgroups in terms of the corresponding family of level subgroups given a finite chain of subgroups of a group g we prove that there exists a fuzzy subgroup of g whose level subgroups are exactly the subgroups of this chain as a corollary we obtain an interpretation of the number of chains of subgroups of a group g in which a subgroup h is a member when the group g is a supersolvable group some further interpretation of this number is obtained

large gaps in point coupled periodic systems of manifolds we study a free quantum motion on periodically structured manifolds composed of elementary two dimensional cells c onnected either by linear segments or through points where the two cells touch the general theory is illustrated with numerous examples in which the elementary components are spherical surfaces arranged into chains in a straight or zigzag way or two dimensional square lattice carpets we show that the spectra of such systems have an infinite number of gaps and that the latter dominate the spectrum at high energies

on modified versions of some solvable ordinary differential equations due to chazy we introduce modified versions featuring a deformation parameter of four explicitly solvable nonlinear autonomous ordinary differential equations odes introduced almost a century ago by chazy chazy j 1909 c r acad sci paris 148 157 9 when the deformation parameter vanishes 0 the modified odes reduce to those introduced by chazy when the deformation parameter does not vanish and is real say positive 0 then all the nonsingular solutions of these modified odes considered as functions of the real independent variable t say time are periodic with period t 2 for two of these modified odes there is also a small subset of these solutions that are also periodic with a smaller period

quantization on the grassmann manifold the problem of quantization in an euclidean space with unitary constraints can be formulated as an unconstrained problem on a grassmann manifold such constraints arise in areas such as wireless communication with multiple antennas at the transmitter and at the receiver due to the constraints the distortion rate analysis developed for euclidean spaces cannot be applied directly this paper extends gersho s asymptotic large rate small distortion distortion bounds to the case when the source is distributed on the complex grassmann manifold the special structure of the grassmann manifold and the distortion measures defined on it differentiate this problem from the traditional vector quantization in euclidean spaces

on a modified version of a solvable ode due to painleve we introduce a modified version of an ode originally introduced by painleve and we show that it possesses a lot of periodic solutions we also use a simplified version of it to verify the efficacy of an iterative approach used elsewhere in more general contexts to evince analogous properties of periodicity of systems of nonlinear odes

fuzzy normal subgroups and fuzzy cosets abstract we introduce here the notion of 1 i a fuzzy normal subgroup and 2 ii a fuzzy coset we prove that for a group g a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of g further we show that the level subgroups of a fuzzy normal subgroup are all normal we obtain several results which are analogs of some basic theorems of group theory in particular a fuzzy version of lagrange s theorem is established

density matrix elements and entanglement entropy for the spin 1 2 xxz chain at δ 1 2 we have analytically obtained all the density matrix elements up to six lattice sites for the spin 1 2 heisenberg xxz chain at 1 2 we use the multiple integral formula of the correlation function for the massless xxz chain derived by jimbo and miwa as for the spin spin correlation functions we have newly obtained the fourth and fifth neighbour transverse correlation functions we have calculated all the eigenvalues of the density matrix and analyse the eigenvalue distribution using these results the exact values of the entanglement entropy for the reduced density matrix up to six lattice sites have been obtained we observe that our exact results agree quite well with the asymptotic formula predicted by the conformal field theory

fuzzy ideals and fuzzy bi ideals in fuzzy semigroups abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra such as fuzzy left right ideal and fuzzy bi ideal in fuzzy semigroup through the new approach of fuzzy space and fuzzy group introduced by dib 1994 our notion of fuzzy ideal and fuzzy bi ideal includes the classical concepts of fuzzy ideal and fuzzy bi ideal of ordinary semigroup many counterexamples are also given

on isochronous bruschi ragnisco ruijsenaars toda lattices equilibrium configurations behaviour in their neighbourhood diophantine relations and conjectures isochronous versions of the bruschi ragnisco ruijsenaars toda lattice and of some of its also integrable variants are introduced their equilibrium configurations are found when they exist and by investigating the motions of these systems near equilibrium some diophantine relations are obtained as well as some insight into the solution of those of these integrable models whose solutions are not yet known

lattice and trellis quantization with lattice and trellis bounded codebooks high rate theory for memoryless sources high rate lattice and trellis quantizers for nonuniform sources are introduced and analyzed the performance of these quantizers is determined by two separable quantities the granular gain and the boundary gain which are determined by the shapes of the granular cells and of the support region respectively the granular gain and boundary gain are the duals of shaping and coding gain in data transmission applications using this duality it is shown for gaussian sources that the ultimate achievable boundary gain with high rate lattice bounded lattice codebooks is the same as the ultimate gain that can be obtained from variable rate entropy coding it is observed that if lattice codebooks can achieve the ultimate granular gain of 0 255 b per dimension then lattice bounded lattice codebooks can approach the rate distortion limit the performance of lattice quantizers is compared to that of optimum vector quantizers

some results on l fuzzy modules abstract an l fuzzy quotient module of a module is redefined and a modular complete pseudo complemented lattice structure is given to the set of all l fuzzy modules of a given module some isomorphisms between the lattices or posets of l fuzzy modules are obtained finally a galois connection is established between two lattices of l fuzzy modules of homomorphic modules

asymptotic distribution of the errors in scalar and vector quantizers high rate or asymptotic quantization theory has found formulas for the average squared length more generally the qth moment of the length of the error produced by various scalar and vector quantizers with many quantization points in contrast this paper finds an asymptotic formula for the probability density of the length of the error and in certain special cases for the probability density of the multidimensional error vector itself the latter can be used to analyze the distortion of two stage vector quantization the former permits one to learn about the point density and cell shapes of a quantizer from a histogram of quantization error lengths histograms of the error lengths in simulations agree well with the derived formulas also presented are a number of properties of the error density including the relationship between the error density the point density and the cell shapes the fact that its qth moment equals bennett s integral a formula for the average distortion of a scalar or vector quantizer and the fact that for stationary sources the marginals of the multidimensional error density of an optimal vector quantizer with large dimension are approximately i i d gaussian

third neighbour and other four point correlation functions of spin 1 2 xxz chain the correlation functions of the spin 1 2 xxz chain in the ground state were expressed in the form of multiple integrals for 1 1 and 1 in particular adjacent four point correlation functions were given as certain four dimensional integrals we show that these integrals can be reduced to polynomials with respect to specific one dimensional integrals the results give the polynomial representation of the third neighbour correlation functions

a characterization of l fuzzy prime ideals abstract in this paper the concepts of a l fuzzy prime and a l fuzzy completely prime ideal are given and some fundamental lemmas are proved in particular bijections among some of the l fuzzy prime ideals of two homomorphic rings are constructed also a characterization of a l fuzzy prime ideal of a ring is given

nonlinear harmonic oscillators the existence is noted of assemblies of an arbitrary number of complex oscillators or equivalently of an arbitrary even number of real oscillators characterized by newtonian equations of motion acceleration equal force with one body velocity dependent linear forces and many body velocity independent cubic forces all the nonsingular solutions of which are isochronous completely periodic with the same period as for the singular solutions as usual they emerge in the context of the initial value problem from a closed domain in phase space having lower dimensionality

asymptotic formulas for variance mismatched fixed rate scalar quantization of a gaussian source asymptotic formulas are derived for the mean squared error mse distortion and the snr of a variance mismatched fixed rate scalar quantizer designed mse optimally for a gaussian source it is discovered that in the case that the standard deviation ratio of the applied to density to the designed for density is greater than 3 2 the snr in db increases at large rate r as 9 03r 2 15 1 1 2 log10r plus a constant that depends only on