on adjunctions between lim sl top and sl lim abstract consider l 1 be a commutative strictly two sided quantale with the underlying lattice l being meet continuous two adjunctions one is between limit spaces and stratified l limit spaces and the other is between stratified l limit spaces and stratified l topological spaces are established the first adjunction can be viewed as an extension of lowen s adjunction between the category of topological spaces and stratified 0 1 topological spaces the second is an extension of an adjunction between limit spaces and stratified l topological spaces established in u hohle and t kubiak hohle kubiak semigroup forum 2007

several types of enriched l m fuzzy convergence spaces abstract in this paper several types of enriched l m fuzzy convergence spaces are introduced including enriched l m fuzzy kent convergence spaces enriched l m fuzzy limit spaces pretopological enriched l m fuzzy convergence spaces topological enriched l m fuzzy convergence spaces and enriched l m fuzzy choquet convergence spaces these concepts generalize the concepts of kent convergence spaces of limit spaces of pretopological convergence spaces of topological convergence spaces and of choquet convergence spaces in general topology to the setting of l m fuzzy topology also their categorical properties and their mutual categorical relations are investigated

generalized hermite polynomials associated with functions of parabolic cylinder associated hermite polynomials he n n z which generalize the usual scaled hermite polynomials he n z corresponding to n 0 are introduced for the purpose to represent the raising and lowering of the indices of functions of the parabolic cylinder d n z in finite integer steps properties of these polynomials such as recursion relations explicit representations and the differential equation are derived the generation of the associated hermite polynomials from the usual hermite polynomials by differential operators representable by means of the confluent hypergeometric function is given an application for the explicit calculation of the functions of the parabolic cylinder for negative integer indices is discussed other applications are visible for the investigation of the zeros of the functions of the parabolic cylinder

stratified hutton uniform spaces in this paper some categorical properties of hutton uniform spaces are investigated at first the initial and final structures for hutton uniform spaces are discussed and it is proved that the canonical functor from the construct of hutton uniform spaces to that of l topological spaces preserves initial sources secondly the notion of stratified hutton uniform spaces is introduced and it is demonstrated that every stratified hutton uniformity induces a stratified l topology and that the relationship between stratified hutton uniform spaces and hutton uniform spaces is like that between stratified l topological spaces and l topological spaces finally it is showed that the construct of uniform spaces can be embedded in the construct of stratified hutton uniform spaces as a concretely both reflective and co reflective full sub construct

limit structures and many valued topologies abstract this paper presents a canonical construction of limit spaces from many valued topological spaces in particular there exists a limit structure on l0 0 1 b which characterizes pointwise almost everywhere convergence

monoidal closedness of the category of stratified l semiuniform convergence spaces abstract for an integral commutative unital quantale l we show that there exists a tensor product between stratified l semiuniform convergence spaces and with respect to this tensor product that the category of stratified l semiuniform convergence spaces and uniformly continuous maps is a symmetric monoidal category as a result we obtain the monoidal closedness of the category of stratified l semiuniform convergence spaces and uniformly continuous maps

the ceo problem multiterminal source coding we consider a new problem in multiterminal source coding motivated by the following decentralized communication estimation task a firm s chief executive officer ceo is interested in the data sequence x t sub t 1 sup spl infin which cannot be observed directly perhaps because it represents tactical decisions by a competing firm the ceo deploys a team of l agents who observe independently corrupted versions of x t sub t 1 sup spl infin because x t is only one among many pressing matters to which the ceo must attend the combined data rate at which the agents may communicate information about their observations to the ceo is limited to say r bits per second if the agents were permitted to confer and pool their data then in the limit as l spl rarr spl infin they usually would be able to smooth out their independent observation noises entirely then they could use their r bits per second to provide the ceo with a representation of x t with fidelity d r where d spl middot is the distortion rate function of x t in particular with such data pooling d can be made arbitrarily small if r exceeds the entropy rate h of x t suppose however that the agents are not permitted to convene agent i having to send data based solely on his own noisy observations y sub i t we show that then there does not exist a finite value of r for which even infinitely many agents can make d arbitrarily small furthermore in this isolated agents case we determine the asymptotic behavior of the minimal error frequency in the limit as l and then r tend to infinity

on l m fuzzy convergence spaces this paper presents a definition of l m fuzzy convergence spaces it is shown that the category l m fc of l m fuzzy convergence spaces which embeds the category l m ftop of l m fuzzy topological spaces as a reflective subcategory is a cartesian closed topological category further it is proved that the category of topological pretopological l m fuzzy convergence spaces is isomorphic to the category of topological l m fuzzy quasi coincident neighborhood spaces moreover the relations among l m fuzzy convergence spaces pretopological l m fuzzy convergence spaces and topological l m fuzzy convergence spaces are investigated in the categorical sense

on stratified l convergence spaces in the classical theory of convergence spaces fischer s diagonal axiom ensures that a generalized convergence space is topological in this note we present a lattice valued fischer diagonal axiom and show that this axiom leads to the conclusion that a stratified l convergence space is l topological

pretopological and topological lattice valued convergence spaces we show that the classical axiom which characterizes pretopological convergence spaces splits into two axioms in the general heyting algebra valued case furthermore we present a generalization of kowalski s diagonal condition to the lattice valued case

the role of jacobi polynomials in the theory of hermite and laguerre 2d polynomials using an alternative definition of usual hermite polynomials two problems in the theory of general hermite and laguerre 2d polynomials can be separated with advantage for the further treatment the introduction of a general 2d matrix in the linear transformation of powers of the components of a 2d vector and the generation of hermite or laguerre polynomials by applying an integral operator to these powers the jacobi polynomials appear in the finite dimensional irreducible representations of the two dimensional general linear group gl 2 c

on the relationship between limit spaces many valued topological spaces and many valued preorders let l 1 be a residuated complete lattice with the underlying lattice l a meet continuous lattice this paper presents a systematic investigation of the interrelationship between the categories of limit spaces l topological spaces and l preorders the results exhibit a close connection between these different mathematical structures

separability dynamics of two mode gaussian states in parametric conversion and amplification we give a simplified form of simon s separability criterion for two mode gaussian states showing that for systems whose unitary evolution is governed by arbitrary time dependent quadratic hamiltonians the separability dynamics is completely described in terms of the determinant of the cross covariance matrix as concrete examples we consider the evolution of the inverse negativity coefficient which gives a quantitative estimation of the degree of entanglement for two initially uncoupled modes each being in a squeezed thermal state in the cases of parametric converter parametric amplifier and for a cavity whose boundary oscillates in resonance with two field modes

fuzzy uniform structures and continuous t norms we introduce and discuss a notion of fuzzy uniform structure that provides a direct link with the classical theory of uniform spaces more exactly for each continuous t norm we prove that the category of all fuzzy uniform spaces in our sense and fuzzy uniformly continuous mappings is isomorphic to the category of uniform spaces and uniformly continuous mappings by means of a covariant functor moreover we also describe the inverse functor and then we discuss completeness and completion of these fuzzy uniform structures it follows from our results that each fuzzy uniform structure in our sense induces a hutton 0 1 uniformity and conversely each hutton 0 1 uniformity induces a fuzzy uniform structure in our sense

multi dimensional hermite polynomials in quantum optics we study a class of optical circuits with vacuum input states consisting of gaussian sources without coherent displacements such as down converters and squeezers together with photo detectors and passive interferometry beamsplitters polarization rotations phase shifters etc we show that the outgoing state leaving the optical circuit can be expressed in terms of socalled multi dimensional hermite polynomials and give their recursion and orthogonality relations we show how quantum teleportation of single photon polarization states can be modelled using this description

pointwise uniformities in fuzzy set theory a theory of pointwise quasi uniformities is based on fuzzy sets many theorems corresponding to the usual theorems in general topology are generalized on fuzzy sets in particular each fuzzy topological space is pointwise quasi uniformizable a pointwise uniformity is characterized in terms of a kind of complete regularity a uniformity on the fuzzy unit interval is constructed

random sampling of deterministic signals statistical analysis of fourier transform estimates this paper considers the estimation of the fourier transform of continuous time deterministic signals from a finite number n of discrete time nonuniform observations we first extend the recent results mean and variance of tarczynski and allay by providing both distributional properties as well as rates of almost sure convergence for the simple random sampling scheme of tarczynski and allay the primary focus is to consider and investigate the properties of a class of stratified random sampling estimates we establish the statistical properties of the estimates including precise expressions and rate of convergence of the mean square errors we also establish the asymptotic normality of the estimates useful for confidence interval calculations further we optimize over the class of the estimates in order to obtain the best performance in particular we show that for functions with a first order continuous derivative the mean square estimation error decays precisely at the fast rate of 1 n sup 3 this rate is significantly higher than the rate of 1 n for simple random sampling estimates the analytical results are illustrated by numerical examples the effect of observation errors on the performance is also investigated

the lower and upper p topological p regular modifications for lattice valued convergence spaces in this paper the notions of lower and upper topological regular modifications with respect to a stratified l generalized convergence structure and those with respect to a stratified l strong generalized convergence structure are defined and studied the basic properties and characterizations are presented in particular it is proved that the lower modifications behave reasonably well relative to final structures whereas the upper modifications exhibit comparable behavior relative to initial structures

connectedness and local connectedness for lattice valued convergence spaces we generalize preuss e connectedness to lattice valued convergence spaces and prove the basic theory for connected sets including the product theorem we further give a suitable definition of local e connectedness and study its properties

laguerre 2d functions and their application in quantum optics a set of orthonormalized and complete functions of two real variables in complex representation involving laguerre polynomials as a substantial part is introduced and is referred to as the set of laguerre two dimensional 2d functions the properties of the set of laguerre 2d functions are discussed and the fourier and radon transforms of these functions are calculated the laguerre 2d functions form a basis for a realization of the five dimensional lie algebra to the heisenberg weyl group for a two mode system real representation of this set of functions by a sum over products of hermite functions involving jacobi polynomials at the zero argument as coefficients is derived and it leads to new connections between laguerre and hermite polynomials in both directions the set of laguerre 2d functions is the most appropriate set of functions for the fock state representation of quasi probabilities in quantum optics the wigner quasi probability in fock state representation is up to a factor and an argument scaling directly given for each matrix element by a corresponding laguerre 2d function the properties of orthonormality and completeness of the laguerre 2d functions provide the fock state matrix elements of the density operator directly from the quasi probabilities the perina mista representation of the glauber sudarshan quasi probability can be represented with advantage by the laguerre 2d functions the fock state matrix elements of the displacement operator and the scalar product of displaced fock states are closely related to laguerre 2d functions

diagonal conditions and continuous extension theorem abstract in this paper we obtain a generalization of kowalsky diagonal condition and that of fischer diagonal condition respectively namely kowalsky diagonal condition and fischer diagonal condition we show that our fischer diagonal condition assures a complete mv algebra valued convergence space proposed in this paper is strong l topological and kowalsky diagonal condition assures a principle or pretopological complete mv algebra valued convergence space is strong l topological also as applications we give a dual form of our fischer diagonal condition and obtain a concept of regular convergence space in addition we present an extension theorem for continuous maps from a dense subspace to a regular convergence space to show that our diagonal conditions works indeed

uncoded transmission is exactly optimal for a simple gaussian sensor network a single memoryless gaussian source is observed by many terminals subject to independent gaussian observation noises the terminals are linked to a fusion center via a standard gaussian multiple access channel the fusion center needs to recover the underlying gaussian source with respect to mean squared error in this correspondence a theorem of witsenhausen is shown to imply that an optimal communication strategy is uncoded transmission i e each terminal s channel input is merely a scaled version of its noisy observation

quantitative domains via fuzzy sets part i continuity of fuzzy directed complete posets this paper deals with quantitative domain theory via fuzzy sets it examines the continuity of fuzzy directed complete posets dcpos for short based on complete residuated lattices first we show that a fuzzy partial order in the sense of fan and zhang and an l order in the sense of belohlavek are equivalent to each other then we redefine the concepts of fuzzy directed subsets and continuous fuzzy dcpos we also define and study fuzzy galois connections on fuzzy posets we investigate some properties of continuous fuzzy dcpos we show that a fuzzy dcpo is continuous if and only if the fuzzy double downward arrow operator has a right adjoint we define fuzzy auxiliary relations on fuzzy posets and approximating fuzzy auxiliary relations on fuzzy dcpos we show that a fuzzy dcpo is continuous if and only if the fuzzy way below relation is the smallest approximating fuzzy auxiliary relation

linear fuzzy neighborhood spaces some of the properties of linear fuzzy neighborhood spaces are studied a special class of these spaces is the family of all fuzzy linear spaces whose topology is given by probabilistic seminorms as in the classical case it is shown that a linear fuzzy neighborhood space is probabilistic normable iff it is hausdorff and it has an n bounded convex fuzzy neighborhood of zero

field estimation from randomly located binary noisy sensors the estimation of bounded multivariate fields from 1 bit quantized and dithered noisy observations is considered we consider two models for random sensor deployment based on regular monte carlo simple random sampling and stratified sampling we propose linear estimators and for both sensors deployment methods we establish exact expressions for the bias and variance of the estimates including integrated mean square errors we show in particular that estimates of the field on the basis of stratified sensor locations always outperform estimates based on regular monte carlo sensor locations for both estimation schemes we also establish central limit theorems which can be used to compute the probability of events involving the estimates including confidence intervals

asymptotic formulae for two variable hermite polynomials new asymptotic formulae for the two dimensional hermite polynomials with large values of indices are found the applications to the photon distribution functions in squeezed one mode mixed quantum states are considered

note on stratified l ordered convergence structures in the paper we point out that the proof of theorem 5 5 in fang 2010 1 which says the category of stratified l ordered convergence spaces is cartesian closed is not correct by an alternative method the cartesian closedness of the category of stratified l ordered convergence spaces is confirmed

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

incomplete 2d hermite polynomials properties and applications incomplete forms of two variable two index hermite polynomials are introduced their link with laguerre polynomials is discussed and it is shown that they are a useful tool to study quantum mechanical harmonic oscillator entangled states the possibility of developing the theory of complete 2d hermite polynomials from the point of view of the incomplete forms is analyzed too the orthogonality properties of the associated harmonic oscillator functions are finally discussed

hutton 0 1 quasi uniformities induced by fuzzy quasi metric spaces it is well known that given a probabilistic metric space menger space with continuous t norm t there is a hausdorff topology associated this association factorizes through strong uniformities or e l uniformities similarly any fuzzy metric space x m can be endowed with a hausdorff topology t m in the case of fuzzy quasi metric spaces a t 1 topology and again this association factorizes through quasi uniform spaces in this paper we associate to each fuzzy quasi metric space a hutton 0 1 quasi uniformity u m this allows us to give a factorization of the previous association via hutton 0 1 quasi uniformities it is also proved that the topology t m is exactly the image under lowen s functor i of the 0 1 topology induced by u m as a consequence we get a class of hutton 0 1 quasi uniformities which are probabilistic metrizable

analysis of signal reconstruction with jittered sampling we consider the linear estimation of square integrable signals from jittered noisy samples we establish an expression for the integrated mean square error for correlated jitter and noise processes we also find the optimal weights of the linear filter for the case of independent identically distributed i i d noise and jitter processes and a suboptimal filter we derive the asymptotic distribution of the estimate of the signal numerical examples are presented

p topologicalness and p regularity for lattice valued convergence spaces in this paper p topologicalness and p regularity for jager s stratified l generalized convergence spaces and those for boustique et al s stratified l convergence spaces are studied through two different methods one method generalizes the well known fischer s dual diagonal condition the other method uses the neighborhood closure of stratified l filter in addition the p topologicalness and p regularity w r t initial constructions and final constructions are also discussed

consistent estimation of continuous time signals from nonlinear transformations of noisy samples a signal cannot in general be reconstructed from its sign i e from its hard limited version however by the deliberate addition of noise to samples of the signal prior to hard limiting it is shown that the signal can be estimated consistently from its hard limited noisy samples as the sampling rate tends to infinity in fact such estimates are shown to converge with probability one to the signal and to be asymptotically normal although the estimates are in general nonlinear they can be made linear by a proper choice of the noise distribution these rather unexpected results hold for all bounded and uniformly continuous signals in addition to the hard limiter such results are also established for certain monotonic and nonmonotonic nonlinearities

on limit relations between some families of bivariate hypergeometric orthogonal polynomials in this paper we deal with limit relations between bivariate hypergeometric polynomials we analyze the limit relation from trinomial distribution to bivariate gaussian distribution obtaining the limit transition from the second order partial difference equation satisfied by bivariate hypergeometric kravchuk polynomials to the second order partial differential equation verified by bivariate hypergeometric hermite polynomials as a consequence the limit relation between both families of orthogonal polynomials is established a similar analysis between bivariate hahn and bivariate appell orthogonal polynomials is also presented

on fuzzy uniform spaces on donne des resultats sur les espaces flous uniformes on montre que pour toute uniformite u sur un ensemble x correspond une uniformite flou u

fuzzy counterparts of hull operators and interval operators in the framework of l convex spaces abstract for fuzzy counterparts of hull operators and interval operators two types of fuzzy hull operators and one type of fuzzy interval operators are proposed firstly the concept of l hull operators is introduced and the resulting category is shown to be isomorphic to that of l convex spaces secondly considering the graded inclusions of l subsets the notion of l ordered hull operators is presented which is shown to be categorically isomorphic to strong l convex structures finally the concept of l interval operators is introduced and it is shown that there is a galois correspondence between the category of l interval spaces and that of l convex spaces in particular the category of arity 2 l convex spaces can be reflectively embedded into that of l interval spaces

l approximation spaces and l fuzzy quasi uniform spaces we introduce the notion of generalized l upper and generalized l lower approximation spaces under a family of information systems we define l upper and l lower quasi approximation operators in fuzzy rough sets and l fuzzy quasi uniform spaces as a view point of the entourage approach in a strictly two sided commutative bi quantale we investigate the relations between l quasi approximation spaces and l fuzzy quasi uniform spaces

lattice valued convergence associated with cns spaces abstract in this paper in the context of meet continuous quantale we present a notion of conical topological lattice valued convergence space which provides the theory of lattice valued convergence associated with such conical neighborhood spaces

on the category of fixed basis frame valued topological groups this article gives results on fixed basis frame valued neighborhood topological groups and stratified neighborhood topological groups includes some characterization theorems and presents the uniformizability of stratified neighborhood topological groups within the unified approach of gutierrez garcia deprada vicente and sostak it is shown that for a frame l the category l ngrp is topological over grp with respect to the forgetful functor

on many valued stratified l fuzzy convergence spaces based on complete residuated lattices the relations between stratified l fuzzy convergence spaces and enriched l fuzzy topological spaces are studied it is shown that as a reflective subcategory the category of enriched l fuzzy topological spaces can embedded in the category of stratified l fuzzy convergence spaces also the specialization l preorder of stratified l fuzzy convergence spaces is investigated

fock expansion of multimode pure gaussian states the fock expansion of multimode pure gaussian states is derived starting from their representation as displaced and squeezed multimode vacuum states the approach is new and appears to be simpler and more general than previous ones starting from the phase space representation given by the characteristic or wigner function fock expansion is performed in terms of easily evaluable two variable hermite kampe de feriet polynomials a relatively simple and compact expression for the joint statistical distribution of the photon numbers in the different modes is obtained in particular this result enables one to give a simple characterization of separable and entangled states as shown for two mode and three mode gaussian states

an enriched category approach to many valued topology in this paper we discuss a basic question in fuzzy set theory what is a many valued topology on a set x an answer is presented by interpreting the axioms two logical formulas for classical topological spaces in the fuzzy setting making use of the intrinsic fuzzy partial order on the fuzzy power set of x as preparation a systematic investigation of fuzzy partial orders is presented from the viewpoint of enriched category theory

on the reflective and coreflective subcategory of stratified l cech closure spaces abstract in this paper it is shown that the category of cech closure spaces can be embedded in the category of stratified l cech closure spaces as a coreflective full subcategory furthermore if the underlying lattice of the complete residuated lattice l is completely distributive the embedding is also reflective finally some related commutative diagrams are obtained

stratified l ordered quasiuniform limit spaces abstract by making use of the intrinsic lattice valued inclusion orders of l subsets and that of stratified l filters a concept of stratified l ordered quasiuniform limit structure is proposed which is asymmetric and available to establish a framework of asymmetric structures then the category of stratified l ordered quasiuniform limit spaces is introduced and its categorical properties are shown the next objective of the paper is to propose a concept of stratified l ordered principle quasiuniform limit space and we prove that the category of stratified l quasiuniform spaces can be embedded into the category of stratified l quasiuniform limit spaces as a bireflective subcategory moreover two ways of inducing stratified l ordered limit structures from stratified l ordered quasiuniform structures are presented and the method of obtaining stratified 0 1 ordered quasiuniform structures from 0 1 ordered probabilistic quasiuniform limit structures and fuzzy quasi metrics is introduced in the paper

fuzzy topological vector spaces ii abstract some of the properties of fuzzy topological vector spaces are investigated also there are given necessary and sufficient conditions for a family of fuzzy sets in a vector space e to be the family of all neighborhoods of zero for a fuzzy linear topology

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

stratified l ordered convergence structures in this paper a new kind of lattice valued convergence structures on a universal set called stratified l ordered convergence structures are presented by modifying the axiom for stratified l generalized convergence structures in the fuzzy setting so as to make use of the intrinsic fuzzy inclusion order on the fuzzy power set the category of stratified l ordered convergence spaces described here is shown to be a reflective full subcategory in the category of stratified l generalized convergence spaces and hence it is topological and cartesian closed as preparation a further investigation of stratified l filters is presented from the viewpoint that latticed valued filters should be compatible with the intrinsic fuzzy inclusion order on the fuzzy power set

universal decentralized estimation in a bandwidth constrained sensor network consider a situation where a set of distributed sensors and a fusion center wish to cooperate to estimate an unknown parameter over a bounded interval u u each sensor collects one noise corrupted sample performs a local estimation and transmits a message to the fusion center while the latter combines the received messages to produce a final estimate this correspondence investigates optimal local estimation and final fusion schemes under the constraint that the communication from each sensor to the fusion center must be a one bit message such a binary message constraint is well motivated by the bandwidth limitation of the communication links fusion center and by the limited power budget of local sensors in the absence of bandwidth constraint and assuming the noises are bounded to the interval u u additive independent but otherwise unknown the classical estimation theory suggests that a total of o u2 epsi2 sensors are necessary and sufficient in order for the sensors and the fusion center to jointly estimate the unknown parameter within epsi root mean squared error mse it is shown in this correspondence that the same remains true even with the binary message constraint furthermore the optimal decentralized estimation scheme suggests allocating 1 2 of the sensors to estimate the first bit of the unknown parameter 1 4 of the sensors to estimate the second bit and so on

relationships between l ordered convergence structures and strong l topologies by making use of the intrinsic fuzzy inclusion orders on the fuzzy power set the relationships between l ordered convergence spaces and strong l topological spaces are researched where l is a commutative unital quantale it is shown that the category of strong l topological spaces can be embedded in the category of l ordered convergence spaces as a reflective subcategory as a result we observe that there is a galois correspondence between the category of l ordered convergence spaces and the category of strong l topological spaces further it is proved that the class of all strong l topological l ordered convergence spaces precisely is the class of all strong l topological spaces and the class of spaces with non idempotent l ordered interior operators is characterized as a subclass of the class of l ordered convergence spaces

lattice valued spaces completions abstract the concept of a convergence space has recently been introduced and studied these spaces are related to the top level space of a lattice valued convergence space in order to consider completions cauchy spaces are defined and investigated in the present work each cauchy space possesses a completion and moreover each topological cauchy space has a topological completion completions of uniform limit spaces are also introduced

totally bounded fuzzy syntopogenous structures abstract some of the properties of the totally bounded fuzzy syntopogenous structures and the totally bounded fuzzy quasi uniformities are studied it is also proved that the restriction to a subset y of the fuzzy real line of the usual fuzzy uniformity is totally bounded iff y is bounded

lattice valued convergence spaces extending the lattice context we define a category of stratified l generalized convergence spaces for the case where the lattice is an enriched cl premonoid we then investigate some of its categorical properties and those of its subcategories in particular the stratified l principal convergence spaces and the stratified l topological convergence spaces for some results we need to introduce a new condition on the lattice which is always true in the case where the lattice is a frame but not always true in the more general case as examples where we may apply the more general lattice context we examine the stratified l topological spaces and probabilistic limit spaces we show that the category of stratified l topological spaces is a reflective subcategory of our category and that the category of probabilistic limit spaces under a t norm is both a reflective and a coreflective subcategory of our category if we choose the lattice context appropriately

general hermite and laguerre two dimensional polynomials general hermite and laguerre two dimensional 2d polynomials which form a complex three parameter unification of the special hermite and laguerre 2d polynomials are defined and investigated the general hermite 2d polynomials are related to the two variable hermite polynomials but are not the same the advantage of the newly introduced hermite and laguerre 2d polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the two variable hermite polynomials two different sets of such polynomials are introduced which are biorthonormal to each other the matrix which plays a role in the new definition of hermite and laguerre 2d polynomials is in a considered sense the square root of the matrix which plays a role in the definition of two variable hermite polynomials two essentially different explicit representations of the hermite and laguerre 2d polynomials are derived where the first involves jacobi polynomials as coefficients in superpositions of special hermite or laguerre 2d polynomials and the second is a superposition of products of two hermite polynomials with decreasing indices and with coefficients related to the special laguerre 2d polynomials generating functions are derived for the hermite and laguerre 2d polynomials

an invariant class of wave packets for the wigner transform abstract generalised hagedorn wave packets appear as exact solutions of schrodinger equations with quadratic possibly complex potential and are given by a polynomial times a gaussian we show that the wigner transform of generalised hagedorn wave packets is a wave packet of the same type in phase space the proofs build on a parametrisation via lagrangian frames and a detailed analysis of the polynomial prefactors including a novel laguerre connection our findings directly imply the recently found tensor product structure of the wigner transform of hagedorn wave packets

implicit summation formulae for hermite and related polynomials in this article we derive some implicit summation formulae for hermite and related polynomials by using different analytical means on their respective generating functions

the reconstruction of analog signals from the sign of their noisy samples the reconstruction of a continuous time signal s t from the sign of its deliberately contaminated samples is considered sequential generally nonlinear estimates of s t are established and their performance is studied error bounds and convergence rates are derived the signal s t need not be band limited the convergence rates obtained here are faster than those obtained previously for nonsequential estimates the degradation in the reconstruction of the signal due to transmission over a noisy channel is also investigated and bounds on the additional error are obtained

a comparison of various uniformities in fuzzy topology it is known that the construct fts of stratified fuzzy topological spaces contains non trivial concretely both reflective and coreflective full subconstructs since each of these subconstructs is both initially and finally closed in fts it gives rise to a perfectly viable and autonomous theory of fuzzy topology in this paper this idea is applied to analyze the interrelations between the various notions of uniformity in fuzzy set theory the results show that these notions can be put into a coherent and clear picture if we interpret them as notions of uniformity for certain subconstructs of fts that are concretely both reflective and coreflective or some superconstructs of fuzzy topological spaces

information rates of densely sampled data distributed vector quantization and scalar quantization with transforms for gaussian sources this paper establishes rates attainable by several lossy schemes for coding a continuous parameter source to a specified mean squared error distortion based on sampling at asymptotically large rates first a densely sampled spatiotemporal stationary gaussian source is distributively encoded the berger tung bound to the distributed rate distortion function and three convergence theorems are used to obtain an upper bound expressed in terms of the source spectral density to the smallest attainable rate at asymptotically large sampling rates the bound is tighter than that recently obtained by kashyap both indicate that with ideal distributed lossy coding dense sensor networks can efficiently sense and convey a field in contrast to the negative result obtained by marco for encoders based on scalar quantization and slepian wolf distributed lossless coding the second scheme is transform coding with scalar coefficient quantization a new generalized transform coding analysis as well as the aforementioned convergence theorems is used to find the smallest attainable rate at asymptotically large sampling rates in terms of the source spectral density and the operational rate distortion function of the family of quantizers which in contrast to previous analyses need not be convex the result shows that when a transform is used scalar quantization need not cause the poor performance found by marco as a corollary the final result pursues an approach originally proposed by berger to show that the inverse water pouring formula for the rate distortion function can be attained at high sampling rates by transform coding with ideal vector quantization to encode the coefficients also established in the paper are relations between operational rate distortion and distortion rate functions for a continuous parameter source and those for the discrete parameter source that results from sampling

on stratified l convergence spaces pretopological axioms and diagonal axioms this paper focuses on pretopological and diagonal axioms for stratified l convergence spaces recently defined by the authors in 2011 this notion is slightly stronger than jager s stratified l generalized convergence spaces in particular one pretopological axiom is stated without resorting to the notion of neighborhood l filters some generalizations of kowalsky s and fischer s diagonal conditions are presented the relationship between these axioms are investigated

meet continuous lattices limit spaces and l topological spaces in this paper a systematic investigation of the relationship between meet continuous lattices limit spaces and l topological spaces is given it is a continuation of the investigation on this topic by hohle 2000 2001 the relationship between the lowen functors and the functors introduced by hohle 2000 2001 is made clear

transformations of laguerre 2d polynomials with applications to quasiprobabilities laguerre 2d polynomials are defined and their properties are investigated the laguerre 2d functions introduced in 1 2 are related to the laguerre 2d polynomials in such a way that they also include the weight function for the orthonormalization of the laguerre 2d polynomials a one parameter group of transformations applicable to certain classes of polynomials and discrete sets of functions is investigated and applied in particular to hermite polynomials and to laguerre 2d polynomials these transformations allow us to represent the polynomials of the corresponding classes by superpositions of the same kind of polynomials with stretched arguments they contain limiting cases with delta functions and their derivatives and lead to regularized representations of the delta functions and their derivatives as demonstrated for hermite and laguerre 2d polynomials applications of the laguerre 2d polynomials and 2d functions and their transformations to problems of quantum optics as the representation of quasiprobabilities in the fock state basis and by normally and otherwise ordered moments are considered the inversion of these representations is obtained in all cases a restricted design of quasiprobabilities should become possible

a common framework for lattice valued uniform spaces and probabilistic uniform limit spaces we study a category of lattice valued uniform convergence spaces where the lattice is enriched by two algebraic operations this general setting allows us to view the category of lattice valued uniform spaces as a reflective subcategory of our category and the category of probabilistic uniform limit spaces as a coreflective subcategory

subcategories of lattice valued convergence spaces we study several subcategories of heyting algebra valued convergence spaces these categories generalize the well known categories of kent convergence spaces of limit spaces of pretopological spaces of pseudo topological spaces and of topological limit spaces to the many valued setting moreover a functorial mechanism is described for changing the basis lattice

gahler s neighborhood condition for lattice valued convergence spaces we generalize a neighborhood condition from the category of convergence spaces to the category of lattice valued convergence spaces for a space in the category of convergence spaces this condition is equivalent to being a topological space it turns out that there are two meaningful generalizations of this condition to the category of lattice valued convergence spaces the first one guarantees that a space in a certain subcategory is a lattice valued topological space the other one is a levelwise condition which is equivalent to a generalization of fischer s diagonal condition the latter generalization is preserved under the embedding of the categories of convergence approach spaces and of probabilistic limit spaces into the category of lattice valued convergence spaces

resilience properties of redundant expansions under additive noise and quantization representing signals using coarsely quantized coefficients of redundant expansions is an interesting source coding paradigm the most important practical case of which is oversampled analog to digital a d conversion signal reconstruction from quantized redundant expansions and the accuracy of such representations are problems which are not well understood and we study them in this paper for uniform scalar quantization in finite dimensional spaces to give a more global perspective we first present an analysis of the resilience of redundant expansions to degradation by additive noise in general and then focus on the effects of uniform scalar quantization the accuracy of signal representations obtained by applying uniform scalar quantization to coefficients of redundant expansions measured as the mean squared euclidean norm of the reconstruction error has been previously shown to be lower bounded by an 1 r sup 2 expression we establish some general conditions under which the 1 r sup 2 accuracy can actually be attained and under those conditions prove a 1 r sup 2 upper error bound for a particular kind of structured expansions which includes many popular frame classes we propose reconstruction algorithms which attain the 1 r sup 2 accuracy at low numerical complexity these structured expansions moreover facilitate efficient encoding of quantized coefficients in a manner which requires only a logarithmic bit rate increase in redundancy resulting in an exponential error decay in the bit rate results presented in this paper are immediately applicable to oversampled a d conversion of periodic bandlimited signals

modes in n dimensional first order systems properties of the extended hermite polynomials are studied it is proved apparently for the first time that these functions times gaussian functions are the modes of an n dimensional first order system thus generalizing an earlier result due to hardy sheng and siegman for one dimension misalignment nonlosslessness and the exact normalization constant are explicitedly included in the treatment phase conjugators are also analyzed

lattice valued preuniform convergence spaces using the intrinsic lattice valued inclusion orders of l subsets and of stratified l filters the concept of a stratified l preuniform convergence structure is proposed this convergence structure is asymmetric and can be used to establish a framework of asymmetric lattice valued space structures the category of stratified l preuniform convergence spaces is introduced and some categorical properties are presented a reflective subcategory of stratified l preuniform convergence spaces is found that is categorically isomorphic to strong stratified l convergence spaces originally called stratified l ordered convergence spaces several subcategories of stratified l preuniform convergence spaces are established and relations between the different subcategories are presented we conclude that stratified l preuniform convergence structures introduced here could play a role in the framework of lattice valued asymmetric space structures in lattice valued topology

generalized zernike or disc polynomials we investigate generalized zernike or disc polynomials p m n z z which are orthogonal 2d polynomials in the unit disc 0 zz 1 is a free parameter these polynomials can be expressed by jacobi polynomials of transformed arguments in connection with a simple angle dependence a limiting procedure leads to laguerre 2d polynomials lm n z z furthermore we introduce the corresponding orthonormalized disc functions the disc polynomials and disc functions obey two differential equations a first order and a second order one with a certain degree of freedom and the operators of lowering and raising of the indices are found these operators can be closed to a lie algebra su 1 1 su 1 1 new generating functions are derived from an operational representation which is alternative to the rodrigues type representation the one dimensional analogue of the disc polynomials which are orthogonal polynomials in the interval 0 r 1 with weight factors 1 r2 x are ultraspherical or gegenbauer polynomials in a new standardization the lowering and raising operators to the corresponding orthonormalized functions form a simple su 1 1 lie algebra this is given in the appendix in sketched form

o n symmetries sum rules for generalized hermite polynomials and squeezed states quantum optics has been dealing with coherent states squeezed states and many other non classical states the associated mathematical framework makes use of special functions as hermite polynomials laguerre polynomials and others in this connection we here present some formal results that follow directly from the group o n of complex transformations motivated by the squeezed states structure we introduce the generalized hermite polynomials which include as particular cases the hermite polynomials as well as the heat polynomials using generalized raising operators we derive new sum rules for the which are covariant under o n transformations the and the associated sum rules become useful for evaluating wigner functions in a straightforward manner as a byproduct we use one of these sum rules on the operator level to obtain raising and lowering operators for the laguerre polynomials and show that they generate an sl 2 r su 1 1 algebra

new relations for two dimensional hermite polynomials the effective formulas reducing the two dimensional hermite polynomials to the classical one dimensional orthogonal polynomials by jacobi gegenbauer legendre laguerre and hermite are given new one parameter generating functions for the diagonal multidimensional hermite polynomials are derived the factorial moments and cumulants of the distribution functions related to the hermite polynomials of two variables with equal indices are expressed in terms of the legendre and chebyshev polynomials asymptotical formulas for the two dimensional polynomials with large values of indices and zero arguments are found the applications to the squeezed one mode states and to the time dependent quantum harmonic oscillator are considered

fuzzy uniform spaces in this paper we introduce the notion of fuzzy uniform spaces in section 2 we give some basic results and show how a fuzzy topology is derived from a fuzzy uniformity in sections 3 and 4 we prove that the notions which are introduced are good extensions and we prove that the category of uniform spaces is nicely injected in the category of fuzzy uniform spaces finally in section 5 we prove some basic results

lattice valued semiuniform convergence spaces in this paper two kinds of lattice valued semiuniform convergence spaces are proposed namely stratified l semiuniform convergence spaces and stratified l ordered semiuniform convergence spaces respectively it is shown that i the category of stratified l semiuniform convergence spaces is topological ii the category of stratified l ordered semiuniform convergence spaces is a bireflective full subcategory of the category of stratified l semiuniform convergence spaces and hence it is topological iii both the category of stratified l semiuniform convergence spaces and that of stratified l ordered semiuniform convergence spaces are cartesian closed iv the category of stratified l semiuniform convergence spaces is extensional v both the category of stratified l semiuniform convergence spaces and that of stratified l ordered semiuniform convergence spaces are closed under the formation of products of quotient mappings in case that l is the two point chain both coincide with the category of semiuniform convergence spaces in the classical case

on l m fuzzy quasi uniform spaces an l m fuzzy topology is a graded extension of topological spaces handling m valued families of l fuzzy subsets of a referential where l and m are completely distributive lattices when m reduces to the set 2 0 1 a 2 m fuzzy topology is called a fuzzifying topology after ying sostak introduced the notion l m fuzzy uniform spaces the aim of this paper is to study the relationship between 2 m fuzzy quasi uniform spaces and l m fuzzy quasi uniform spaces as well as the relationship between 2 m fuzzy quasi uniform spaces and pointwise l m fuzzy quasi uniform spaces the extension of shi s l quasi uniform space in a kubiak sostak sense it is shown that the category of 2 m fuzzy quasi uniform spaces can be embedded in the category of stratified l m fuzzy quasi uniform spaces as a both reflective and coreflective full subcategory and the former category can also be embedded in the category of pointwise l m fuzzy quasi uniform spaces

bivariate raising and lowering differential operators for eigenfunctions of a 2d fourier transform we define a two dimensional 2d fourier transform that self reproduces a one parameter family of bivariate hermite functions these are eigenfunctions of a hamiltonian differential operator of second order whose exponential is that transform we find explicit forms of the bivariate raising and lowering partial differential operators of first degree for the eigenfunctions of this 2d fourier transform

hermite and laguerre 2d polynomials abstract we define hermite 2d polynomials hm n u x y and laguerre 2d polynomials l m n u z z as functions of two variables with an arbitrary 2d matrix u as parameter and discuss their properties and their explicit representation recursion relations and generating functions for these polynomials are derived the advantage of the introduced hermite and laguerre 2d polynomials in comparison to the related usual two variable hermite polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the last one has to introduce two different kinds of such polynomials which are biorthogonal to each other