towards a matrix theory for unbounded operator matrices sur l espace produit r l 2 r on considere l operateur 7b a a 0 ou d d dx est defini sur d a w 1 r 0 est la mesure de dirac en 0 c l 2 r est un operateur de r dans l 2 r et 7b a a un domaine d 7b a r w 1 r on introduit une fonction caracteristique non commutative donnant une facon efficace pour calculer le spectre de 7b a on obtient des resultats sur la generation de semi groupes par des matrices operateur

an introduction to mathematical models of coagulation fragmentation processes a discrete deterministic mean field approach abstract we summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity an integer multiple of some basic unit size in particular we discuss smoluchowski s equation for aggregation the becker doring model of simultaneous aggregation and fragmentation and more general models involving coagulation and fragmentation

fuzzy s irresolute mappings in this paper the concepts of fuzzy s irresolute semi s irresolute and s precontinuous mappings in fuzzy topological spaces in sostak sense are introduced and studied some of their characteristic properties are considered also fuzzy s connectedness is introduced and studied with the help of r fuzzy strongly semi open sets

semi groupes sous markoviens engendres par des operateurs matriciels we characterize generators of sub markovian semigroups onlp by a version of kato s inequality this will be used to show under precise assumptions that the semigroup generated by a matrix operatora aij 1 i j n on lp n is sub markovian if and only if the semigroup generated by the sum of each rowai1 ain 1 i n is sub markovian the corresponding result on c0 x n characterizes dissipative operator matrices

on improved bounds on the decoding error probability of block codes over interleaved fading channels with applications to turbo like codes we derive here improved upper bounds on the decoding error probability of block codes which are transmitted over fully interleaved rician fading channels coherently detected and maximum likelihood ml decoded we assume that the fading coefficients during each symbol are statistically independent due to a perfect channel interleaver and that perfect estimates of these fading coefficients are provided to the receiver the improved upper bounds on the block and bit error probabilities are derived for fully interleaved fading channels with various orders of space diversity and are found by generalizing some previously introduced upper bounds for the binary input additive white gaussian nose awgn channel the advantage of these bounds over the ubiquitous union bound is demonstrated for some ensembles of turbo codes and low density parity check ldpc codes and it is especially pronounced in a portion of the rate region exceeding the cutoff rate our generalization of the duman and salehi bound duman and salehi 1998 duman 1998 which is based on certain variations of gallager s 1965 bounding technique is demonstrated to be the tightest reported upper bound we therefore apply it to calculate numerically upper bounds on the thresholds of some ensembles of turbo like codes referring to the optimal ml decoding for certain ensembles of uniformly interleaved turbo codes the upper bounds derived here also indicate good match with computer simulation results of efficient iterative decoding algorithms

on the spectrum of evolution operators generated by hyperbolic systems abstract we prove that the asymptotic behavior of semigroups generated by mixed initial value problems for hyperbolic systems in one space variable is determined by the zeroes of the characteristic equation the proof is based on the following reduction principle we construct a reduced system which is much simpler than the given one such that the difference of the semigroups is a compact operator the construction of the reduced system is presented in the non autonomous case because it has implifications in the study of systems which are time periodic

on characterizations of some covering properties in l fuzzy topological spaces in sostak s sense in this work various characterizations of some weaker and stronger forms of l fuzzy compactness are obtained in l fuzzy topological spaces in sostak s sense where l is a fuzzy lattice

on the application of substochastic semigroup theory to fragmentation models with mass loss a linear integro differential equation modelling multiple fragmentation with inherent mass loss is investigated by means of substochastic semigroup theory the existence of a semigroup is established and under natural conditions on certain coefficients the generator of this semigroup is identified this yields in particular a validation of the formal mass loss rate equation for the model

multivariable extremum seeking for pde dynamic systems in this article we address the design and analysis of multivariable newton based extremum seeking es for static locally quadratic maps subject to actuation dynamics governed by diffusion partial differential equations pdes multi input systems with distinct diffusion coefficients in each individual input channel are dealt with the phase compensation of the dither signals is handled as a trajectory generation problem and the inclusion of a multivariable diffusion feedback controller with a perturbation based averaging based estimate of the hessian s inverse allow to obtain local exponential convergence results to a small neighborhood of the optimal point the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the diffusion pdes in addition the generalization of the results for different classes of parabolic reaction advection diffusion pdes wave equations and or first order hyperbolic transport dominated pdes is also discussed as for ordinary differential equations case the proposed newton approach removes the dependence of the algorithm s convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable unlike the gradient algorithm a numerical example illustrates the performance of our multivariable es for compensating pde based systems

k fan s theorem in fuzzifying topology in this paper we offer an application of ukasiewicz logic in topology by ukasiewicz logic we introduce a new definition of connectivity in fuzzifying topology and prove k fan s theorem of connectivity in fuzzifying topology

strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss abstract in the paper we shall present a survey of recent results on substochastic semigroups and provide new methods for determining their honesty these methods are applied to the fragmentation equation with mass loss yielding sufficient conditions for the existence of conservative and shattering solutions our results provide a mathematical framework that clarifies the discussion of phys rev a 43 1991 656 phys rev a 41 1990 5755 j phys a 24 1991 3967 on shattering fragmentation in such models showing among others that there occurs an unexpected mass loss associated with shattering which is not accounted for by the discrete and continuous mass loss contrary to the conjecture of phys rev a 43 1991 656 phys rev a 41 1990 5755

new performance bounds for turbo codes we derive a new upper bound on the word and bit error probabilities of turbo codes with maximum likelihood decoding by using the gallager bound since the derivation of the bound for a given interleaver is intractable we assume uniform interleaving as in the derivation of the standard union bound for turbo codes the result is a generalization of the transfer function bound and remains useful for a wider range of signal to noise ratios particularly for some range below the channel cutoff rate the new bound is also applicable to other linear codes

on i fuzzy topologies derived from a decreasing family of i topologies the aim of this paper is to study i fuzzy topologies fuzzy topologies in sostak s sense derived from decreasing families of i topologies fuzzy topologies in chang s sense defined on a set x

new solutions to the fragmentation equation a new general class of exact explicit scaling solutions to the fragmentation equation is given this class is described by a breakage rate a x xlambda and the daughter distribution function yb x mod y a gamma xy gamma 2 1 a delta x y delta 2 and includes as special cases all previously known scaling solutions to the fragmentation equation for a subset of this class with gamma lambda and a delta delta lambda the complete time dependent solution for a monodisperse initial condition is also given

three topological structures of smooth topological spaces abstract various concepts related to smooth topological spaces have been introduced and relations among them studied by several authors 2 6 in this paper we introduce the notions of smooth quasi smooth and weak smooth structure showing that various properties of a smooth topological space can be expressed in terms of these structures in particular the definitions and the results of 2 6 may be expressed in terms of the smooth and quasi smooth structures then we will set out the fundamental concepts of these three structures and the relations between them

strongly differentiable solutions of the discrete coagulation fragmentation equation we examine an infinite system of ordinary differential equations that models the binary coagulation and multiple fragmentation of clusters in contrast to previous investigations our analysis does not involve finite dimensional truncations of the system instead we treat the problem as an infinite dimensional differential equation posed in an appropriate banach space and apply perturbation results from the theory of strongly continuous semigroups of operators the existence and uniqueness of physically meaningful solutions are established for uniformly bounded coagulation rates but with no growth restrictions imposed on the fragmentation rates

on a class of continuous coagulation fragmentation equations abstract a model for the dynamics of a system of particles undergoing simultaneously coalescence and breakup is considered each particle being assumed to be fully identified by its size existence of solutions to the corresponding evolution integral partial differential equation is shown for product type coagulation kernels with a weak fragmentation the failure of density conservation or gelation is also investigated in some particular cases

kinetics of fragmentation a general discussion of the kinetics of continuous irreversible fragmentation processes is presented for a linear process where particle breakup is driven by an external force the authors develop a scaling theory to describe the evolution of the cluster size distribution they treat the general case where the breakup rate of a cluster of mass x varies as xlambda when lambda 0 corresponding to larger clusters more likely to break up the scaled cluster size distribution phi x decays with the scaled mass x as x 2 exp axlambda as x for small mass phi x has the log normal form exp a ln2 x if the breakup kernel has a small size cutoff while phi x has a power law tail in the absence of a cutoff they also show that a conventional scaling picture applies only for the case lambda 0 for lambda 0 they develop an alternative formulation for the cluster size distribution in which the typical mass scale is determined by the initial condition in this regime they also investigate the nature of a shattering transition where mass is lost to a dust phase of zero mass particles they also study the kinetics of a nonlinear collision induced fragmentation process they analyse the asymptotic behaviour of a simple minded class of models in which a two particle collision results in either 1 both particles splitting into two equal pieces 2 only the larger particle splitting in two or 3 only the smaller particle splitting they map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations scaling is found to hold for different ranges of homogeneity index for the three models

pseudospectral laguerre approximation of transport fragmentation equations we develop a laguerre type pseudo spectral scheme for solving transport fragmentation equations the method converges rapidly for certain type of fragmentation problems and works under mild restrictions on the growth rate of the coefficients of the equation rigorous stability and convergence analyses are provided numerical simulations illustrate the performance of the scheme

tighter bounds on the error probability of fixed convolutional codes we introduce new upper bounds for the bit and burst error probabilities when a convolutional code is used for transmission over a binary input additive white gaussian noise awgn channel and the decoder performs maximum likelihood decoding the bounds are tighter than existing bounds and nontrivial also for signal to noise ratios below the computational cutoff rate

bounds on the decoding error probability of binary linear codes via their spectra bounds on the error probability of maximum likelihood decoding of a binary linear code are considered the bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel the bounds derived are applied to a code with an average spectrum and the result is compared to the random coding exponent the author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound for the case of awgn channel the author shows that berlekamp s 1980 tangential bound can be improved but even this improved bound does not coincide with the random coding bound although it can be very close to it

fuzzy proximities and fuzzy uniformities abstract a new definition of fuzzy proximity is given and some of its properties are studied we also give a definition of fuzzy uniformity whose underlying fuzzy proximity is a fuzzy proximity in our sense

shattering and non uniqueness in fragmentation models an analytic approach abstract fragmentation equations have features which are rather unexpected in such a relatively simple linear framework these are shattering attributed to a phase transition creating a dust of zero size particles and the existence of multiple solutions the existence of these phenomena indicates that the fragmentation equation alone does not offer a full description of the dynamics of the problem in this paper we present a survey of recent results explaining both features from a unified dynamical systems point of view we also use the developed theory to discuss a number of conjectures and observations made in the literature of the subject

matrix methods for wave equations in analogy to a characterisation of operator matrices generating c0 semigroups due to r nagel 13 we give conditions on its entries in order that a 2 2 operator matrix generates a cosine operator function we apply this to systems of wave equations to second order initial boundary value problems and to overdamped wave equations

abstract time dependent transport equations abstract we consider the abstract time dependent linear transport equation as an initial boundary value evolution problem in the banach spaces lp 1 p

fuzzy proximities structures and fuzzy grills abstract in this paper definitions of basic proximities of fuzzy sets and basic fuzzy proximities are given and a classification scheme for proximities of fuzzy sets as well as fuzzy proximities are introduced

convergence in i fuzzy topological spaces the purpose of this work is to combine the fuzzy q neighborhood system of fuzzy points introduced by sostak with the convergence structures of lee et al using the l filter approach of gahler and eklund

substochastic semigroups and densities of piecewise deterministic markov processes abstract necessary and sufficient conditions are given for a substochastic semigroup on l 1 obtained through the kato voigt perturbation theorem to be either stochastic or strongly stable we show how such semigroups are related to piecewise deterministic markov process provide a probabilistic interpretation of our results and apply them to fragmentation equations

complete second order linear differential operator equations in hilbert space and applications in hydrodynamics we study the cauchy problem for a complete second order linear differential operator equation in a hilbert space h of the form d 2 u dt 2 f ik du dt bu f u 0 u 0 u 1 0 u 1 problems of this kind arise e g in hydrodynamics where the coefficients f k and b are unbounded selfadjoint operators it is assumed that f is the dominating operator in the cauchy problem above i e d f d b d f d k we also suppose that f and b are bounded from below but the operator coefficients are not assumed to commute the main results concern the existence of strong solutions to the stated cauchy problem and applications of these results to the cauchy problem associated with small motions of some hydrodynamical systems

global existence for semilinear parabolic systems on etudie des problemes aux valeurs limites et initiales paraboliques semilineaires de la forme u t a t u f t x u du d 2m 1 u dans t 0 b t u 0 sur t 0 u t 0 u 0 sur ou est un domaine de r n

on several types of degree of fuzzy compactness abstract sostak 1985 defined a fuzzy topology on a set x as a fuzzy subset of the family i x of fuzzy subsets of x satisfying some axioms the concept of the degree of compactness of a fuzzy set was first introduced by sostak 1985 in this paper we discuss the degrees of almost compactness near compactness countable compactness light compactness and strong compactness of a fuzzy set also we investigate the behavior of the degrees of compactness under several types of fuzzy continuous functions

eventual norm continuity for neutral semigroups on banach spaces in this paper we prove under suitable hypotheses eventually norm continuity and compactness of the solution semigroups of certain neutral differential equations in banach spaces our approach is based on a general perturbation theorem obtained from closed loop systems of infinite dimensional control systems with unbounded control and observation operators in fact the solution semigroup can be viewed as the semigroup of an appropriate closed loop system in product state spaces

fuzzy topology redefined abstract in this paper we give a new definition of fuzzy topology by introducing a concept of gradation of openness of fuzzy subsets and then study fuzzy continuity

on several types of compactness in smooth topological spaces abstract the smooth closure and smooth interior of a fuzzy set w r t a smooth topology were defined by gayyar et al 1994 and some relations between a few types of compactness were established in the presence of strong restrictions in this paper by constructing new definitions of smooth closure and smooth interior which have more desirable properties than those of gayyar et al 1994 we prove that several hypothesis in the results of gayyar et al 1994 can be weakened and show that the relations which hold between various types of compactness in fuzzy topological spaces in chang s sense cfts for short di concilio and gerla 1984 haydar es 1987 can be extended to smooth topological spaces

variations on the gallager bounds with some applications by generalizing the framework of the second version of the duman salehi bound encompassing also deterministic and random codes we demonstrate its rather broad features and show that this variation provides the natural bridge between the 1961 and 1965 gallager bounds this approach entails a natural geometric interpretation encompassing also a large class of e0cient recent bounds or their cherno2 versions which are demonstrated to be special cases of the generalized second version of the duman salehi bound implications and applications of these observations are pointed out referring to known bounds as well as a novel extended version of the shulman feder bound c 2001 elsevier science b v all rights reserved

wellposedness and wave propagation for a class of integrodifferential equations in banach space in recent years there has been an increasing interest in the mathematical treatment of problems which originate in the theory of viscoelastic materials in such problems the constitutive relations for these materials incorporate the history of the strain or the strain rate and in the linear theory lead to linear volterra integrodifferential equations with infinite memory in a banach space x the purpose of this paper is to examine the problems of wellposedness and wave propagation for one such equation in particular we are concerned with equations of the form x t 4 x r j f t s x s ds al

asymptotic stability of the solution of the m mb 1 queueing model by using positive c 0 semigroup theory we study the asymptotic stability of the solution of a bulk queueing model m m b 1

a variational principle for block operator matrices and its application to the angular part of the dirac operator in curved spacetime abstract the operator associated to the angular part of the dirac equation in the kerr newman background metric is a block operator matrix with bounded diagonal and unbounded off diagonal entries the aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above in the last section these analytic bounds are compared with numerical values from the literature

a mixed discrete continuous fragmentation model abstract motivated by the occurrence of shattering mass loss observed in purely continuous fragmentation models this work concerns the development and the mathematical analysis of a new class of hybrid discrete continuous fragmentation models once established the model which takes the form of an integro differential equation coupled with a system of ordinary differential equations is subjected to a rigorous mathematical analysis using the theory and methods of operator semigroups and their generators most notably by applying the theory relating to the kato voigt perturbation theorem honest substochastic semigroups and operator matrices the existence of a unique differentiable solution to the model is established this solution is also shown to preserve nonnegativity and conserve mass

on nonautonomous functional differential equations abstract in this paper we prove a theorem on boundary perturbation of nonautonomous cauchy problems and then apply this result to show the existence and uniqueness of classical solutions of the nonautonomous banach space valued functional differential equation formula

the discrete coagulation fragmentation equations existence uniqueness and density conservation the discrete coagulation fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones these models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys our results relate to existence uniqueness density conservation and continuous dependence and they generalise the corresponding results in ref 2 for the becker doring equations for which the processes are restricted to clusters gaining or shedding one particle examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation fragmentation equations

fuzzy topology fuzzy closure operator fuzzy compactness and fuzzy connectedness abstract in this paper definitions of fuzzy closure operator fuzzy compactness and fuzzy connectedness are given in a redefined fuzzy topological space and then some characteristic properties of fuzzy closure operator product theorems of fuzzy compactness invariance of fuzzy compactness and fuzzy connectedness under gradation preserving maps and invariance of fuzzy connectedness under the fuzzy closure operator are studied

gradation of openness fuzzy topology abstract in an earlier paper we have given a new definition of fuzzy topology by introducing a concept of gradation of openness of fuzzy subsets in order to make the concept more appropriate in this paper we modify the definition of gradation function and then study subspace of fuzzy topological spaces gradation preserving maps and category of fuzzy topological spaces and gradation preserving maps

general solutions and scaling violation for fragmentation with mass loss explicit solutions of a linear rate equation lead to improved understanding of fragmenlatian with discrete and continuous mass loss discrete mass loss provides a general expression for the overall mass 10 rate which accounts for mass loss in the shattering regime where runaway fragmentation for small panicles producer a phase of zero mass panicles an explicit solution far the recession regime where small panicles lose large fractionsoftheir mass to surface recession shows that the total number ofpaniclesincreases with time until the typical panicles become small enough to lose all of their mass and disappear rather than to break a general series solution is presented for fragmentation with continuous and discrete mass loss a proof that continuous mass loss precludes dynamic scaling is presented

spectral enclosures for a class of block operator matrices abstract we prove new spectral enclosures for the non real spectrum of a class of 2 2 block operator matrices with self adjoint operators a and d on the diagonal and operators b and b as off diagonal entries one of our main results resembles gershgorin s circle theorem the enclosures are applied to j frame operators

matrices of operators and regularized semigroups where p is a matr ix of polynomials in n variables and ia1 ia is a family of c o m m u t i n g generators of bounded linear strongly cont inuous groups of operators intuitively one wants to take the exponent ia l of the matr ix of opera to r s p a since the spec t rum of ak is real for all k this means we want the complex matr ix pi j x to be contained in a left half plane in some sense for all x in r the strongest sense is to have the numerical range of the complex matr ix conta ined in a half plane r e z 0 xer a sense that is easier to calculate or at least estimate is to have the spectrum of p d x conta ined in a left half plane for all x r the system 1 1 is then said to be petrowsky correct see 9 the intuit ion we just described is not quite true in general wi thout some

an explicit solution to a discrete fragmentation model the discrete binary fragmentation equation is solved explicitly far a model where tnenetrateoibreakupoiaparlicie oishek o equals k lj k l andthedaughter ri e distribution b lx equals 2 k 1 this system is closely related to a model of polymer degradation considered by simha in which ax i and b is as above in the continuum limit both of these models go over to a continuous fragmentation model in which all particles break with an equal rate a x i and the daughter size distribution is uniform b yl x z x which is at the borderline of the shattering transition

mathematical analysis of finite volume preserving scheme for nonlinear smoluchowski equation abstract this study presents the convergence analysis of the recently developed finite volume preserving scheme forestier coste and mancini 2012 for approximating a coalescence or smoluchowski equation the idea of the finite volume scheme is to preserve the total volume in the system by modifying the coalescence kernel using the notion of overlapping bins cells the consistency of the finite volume scheme is examined thoroughly in order to prove second order convergence on uniform non uniform smooth and locally uniform grids independently of the aggregation kernel the theoretical observations of order of convergence is verified using the experimental order of convergence for analytically tractable kernels

controlling number of particles in fragmentation equations abstract one of the most interesting features of fragmentation models is the possibility of breaching the mass conservation principle through shattering that is the formation of a dust of zero size particles a similar phenomenon may occur in the evolution of the number of particles in the system which to some extent is intertwined with the evolution of its total mass to investigate these phenomena we consider the fragmentation equation in the space of densities yielding both a finite number of particles and a finite mass of the ensemble and show in particular that in a non shattering fragmentation one can typically control the total number of particles in the system on the other hand both mass and particles in the shattering fragmentation can disappear from the system it is conjectured that in such a case the fragmentation equation alone does not offer the full description of the dynamics of the problem

on fuzzy bitopological spaces abstract in this paper we have used the fuzzy topologies 1 and 2 to generate a family s which is a supra fuzzy topology on x using the family s we introduce and study the concepts of separation axioms continuity resp openness closedness of a mapping and compactness for a fuzzy bitopological space x 1 2 our definitions preserve much of the correspondence between concepts of fuzzy bitopological spaces and the associated fuzzy topological spaces we then investigate the relationship between these concepts and their correspondence with the fuzzy bitopological spaces kandil and el shafee 1991

random coding techniques for nonrandom codes this work provides techniques to apply the channel coding theorem and the resulting error exponent which was originally derived for totally random block code ensembles to ensembles of codes with less restrictive randomness demands as an example the random coding technique can even be applied for an ensemble that contains a single code for a specific linear code we get an upper bound for the error probability which equals gallager s 1968 random coding bound up to a factor determined by the maximum ratio between the weight distribution of the code and the expected random weight distribution

the spectrum of unbounded operator matrices with non diagonal domain abstract systems of linear evolution equations involving non diagonal boundary conditions yield operator matrices having non diagonal domains in products of banach spaces we show how to compute the spectrum of these matrices by using virtual matrix elements and characteristic operator functions

polynomial operator matrices as semigroup generators the 2 2 case on considere des systemes d equations d evolution non lineaires ecrits sous la forme u t au t ou u est une fonction a valeurs dans un espace produit e n et a a ij n m est une matrice dont les a ij sont des operateurs lineaires sur e on etudie le cas ou les a ij sont des polynomes p ij a par rapport a un seul operateur a sur e et on se restreint au cas des matrices 2 2

on a coagulation and fragmentation equation with mass loss a nonlinear integro differential equation that models a coagulation and multiple fragmentation process in which continuous and discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators under the assumptions that the coagulation kernel is constant the fragmentation rate function is linearly bounded and the continuous mass loss rate function is locally lipschitz global existence and uniqueness of solutions that lose mass in accordance with the model are established in the case when no coagulation is present and the fragmentation process is binary with constant fragmentation kernel and constant continuous mass loss an explicit formula is given for the associated substochastic semigroup

neighborhood structures of smooth topological spaces abstract smooth topological spaces and their fundamental concepts have been discussed in the literature 2 4 7 in this paper we present the notions of base and two sorts of neighborhood structure of smooth topological spaces and give some of their properties which are results in 3 extended to smooth topological spaces

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions

coagulation and fragmentation with discrete mass loss a nonlinear integro differential equation that models a coagulation and multiple fragmentation process in which discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators under the assumptions that the coagulation kernel click to view the mathml source is bounded and the fragmentation rate function a satisfies a linear growth condition global existence and uniqueness of solutions that lose mass in accordance with the model are established in the case when no coagulation is present and the fragmentation process is governed by power law kernels an explicit formula is given for the substochastic semigroup associated with the resulting mass loss fragmentation equation

gradations of uniformity and gradations of proximity abstract in 1992 a new definition of a fuzzy topological space was given by introducing the concept of a gradation of openness in this paper a new definition of a fuzzy uniform resp proximity space is given by introducing the concept of a gradation of uniformity resp proximity on a non empty set x how to construct a gradation of openness induced by a gradation of uniformity resp proximity is explained the connexions between gradations of proximity and gradations of uniformity are investigated

on l fuzzy topological spaces in this paper we study l fuzzy topological spaces where l represents a completely distributive lattice we shall investigate the level decomposition of l fuzzy topologies on x and the corresponding l fuzzy continuous maps in addition we shall establish the representation theorems of l fuzzy topologies on x