topological representation for fuzzy topological spaces abstract let l be a completely distributive complete lattice with universal bounds 0 and 1 such that 0 1 let us denote by x l the of set all prime proper filters of l to every l fuzzy topological space x one can associate a topological x l t one obtains a covariant functor from the category cg l into the category of topological spaces we discuss further how far one can go in this direction following ideas in 2

when is a quantity additive and when is it extensive abstract the difference between the terms additivity and extensivity as well as their respective negations is critically analyzed and illustrated with a few examples the concepts of subadditivity pseudo additivity and pseudo extensivity are also defined

thermodynamic versus statistical nonequivalence of ensembles for the mean field blume emery griffiths model abstract we illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean field blume emery griffiths beg model as a test model the novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states for this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy by computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the beg model we show that for values of the mean energy where the microcanonical entropy is nonconcave the microcanonical distributions of states are nowhere realized in the canonical ensemble moreover we show that for values of the mean energy where the microcanonical entropy is strictly concave the equilibrium microcanonical distributions of states can be put in one to one correspondence with equivalent canonical equilibrium distributions of states our numerical computations illustrate general results relating thermodynamic and statistical equivalence and nonequivalence of ensembles proved by ellis et al j stat phys 101 2000 999

a note on the hypergraph functor the purpose of this note is to show that in the case of spatial locales l the hypergraph functor has a right adjoint functor

statistical mechanics of classical systems with distinguishable particles the properties of classical models of distinguishable particles are shown to be identical to those of a corresponding system of indistinguishable particles without the need for ad hoc corrections an alternative to the usual definition of the entropy is proposed the new definition in terms of the logarithm of the probability distribution of the thermodynamic variables is shown to be consistent with all desired properties of the entropy and the physical properties of thermodynamic systems the factor of 1 n in the entropy connected with gibbs paradox is shown to arise naturally for both distinguishable and indistinguishable particles these results have direct application to computer simulations of classical systems which always use distinguishable particles such simulations should be compared directly to experiment in the classical regime without correcting them to account for indistinguishability

validity of the third law of thermodynamics for the tsallis entropy bento et al phys rev e 91 022105 2015 recently stated that the tsallis entropy violates the third law of thermodynamics for 0 1 is already inherent in the fundamental incomplete structure of the deformed logarithms and exponentials underlying the tsallis entropy then we provide the complete deformed functions and show that the tsallis entropy conforms to the third law of thermodynamics for both superadditive q 1 regimes finally we remark that the tsallis entropy does not require the use of an escort averaging scheme once it is expressed in terms of the complete deformed functions

point set lattice theoretic topology this essay attempts to survey in a coherent way certain aspects of point set lattice theoretic or poslat topology by which we mean fuzzy topology grounded in notions of sets functions powersets and powerset operators the latter two being lattice theoretic in nature and examined from a lattice theoretic point of view using methods of category theory connections with other related issues and developments are frequently given

construction of microcanonical entropy on thermodynamic pillars a question that is currently highly debated is whether the microcanonical entropy should be expressed as the logarithm of the phase volume volume entropy also known as the gibbs entropy or as the logarithm of the density of states surface entropy also known as the boltzmann entropy rather than postulating them and investigating the consequence of each definition as is customary here we adopt a bottom up approach and construct the entropy expression within the microcanonical formalism upon two fundamental thermodynamic pillars i the second law of thermodynamics as formulated for quasistatic processes q t is an exact differential and ii the law of ideal gases pv k b nt the first pillar implies that entropy must be some function of the phase volume the second pillar singles out the logarithmic function among all possible functions hence the construction leads uniquely to the expression s k b ln that is the volume entropy as a consequence any entropy expression other than that of gibbs e g the boltzmann entropy can lead to inconsistencies with the two thermodynamic pillars we illustrate this with the prototypical example of a macroscopic collection of noninteracting spins in a magnetic field and show that the boltzmann entropy severely fails to predict the magnetization even in the thermodynamic limit the uniqueness of the gibbs entropy as well as the demonstrated potential harm of the boltzmann entropy provide compelling reasons for discarding the latter at once

efficient multiple range random walk algorithm to calculate the density of states we present a new monte carlo algorithm that produces results of high accuracy with reduced simulational effort independent random walks are performed concurrently or serially in different restricted ranges of energy and the resultant density of states is modified continuously to produce locally flat histograms this method permits us to directly access the free energy and entropy is independent of temperature and is efficient for the study of both 1st order and 2nd order phase transitions it should also be useful for the study of complex systems with a rough energy landscape

on the low frequency asymptotics in electromagnetic theory on considere les equations de maxwell harmoniques par rapport au temps avec des conditions aux limites on etudie le comportement asymptotique a basse frequence

towards the theory of m approximate systems fundamentals and examples the concept of an m approximate system is introduced basic properties of the category of m approximate systems and in a natural way defined morphisms between them are studied it is shown that categories related to fuzzy topology as well as categories related to rough sets can be described as special subcategories of the category of m approximate systems

global bethe lattice consideration of the spin 1 ising model the spin 1 ising model with bilinear and biquadratic exchange interactions and single ion crystal field is solved on the bethe lattice using exact recursion equations the general procedure of investigation of critical properties is discussed and a full set of phase diagrams are constructed for both positive and negative biquadratic couplings a comparison is made with the results of other approximation schemes

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

categories isomorphic to the kubiak sostak extension of tml abstract in this paper we study wang guo jun s topological molecular lattices wang s tml in a kubiak sostak sense and construct a category the kubiak sostak extension of tml denoted by ftml then we prove that ftml is topological over cd op and tml is bireflective full subcategory of ftml finally we introduce several important categories to characterize ftml

statistical entropy of open quantum systems dissipative quantum systems are frequently described within the framework of the so called system plus reservoir approach in this work we assign their description to the maximum entropy formalism and compare the resulting thermodynamic properties with those of the well established approaches due to the non negligible coupling to the heat reservoir these systems are nonextensive by nature and the former task may require the use of nonextensive parameter dependent informational entropies in doing so we address the problem of choosing appropriate forms of those entropies in order to describe a consistent thermodynamics for dissipative quantum systems nevertheless even having chosen the most successful and popular forms of those entropies we have proven our model to be a counterexample where this sort of approach leads us to wrong results

on fuzzy convergence abstract we propose and examine two concepts of convergence for sequences of elements of a set x with respect to a fuzzy topology on x such concepts are obtained by interpreting in a multivalued logic two different first order formulas expressing the classical convergence

fuzzy points and local properties of fuzzy topological spaces abstract we prove that three theorems of c k wong on local properties of fuzzy topology are wrong thereby we discuss his notions of fuzzy point c 1 separability and local compactness

a categorical accommodation of various notions of fuzzy topology different definitions of fuzzy topology have been stated and developed in the literature but have not been satisfactorily related to each other and to ordinary topology to remedy this a new fuzzy topological category fuzz is defined it is a significant generalization of previous definitions it is the simplest setting yet constructed into which ordinary topology and these previous definitions may be placed by identifying each with a different subcategory of fuzz it therefore allows us to conveniently analyze the relative merits of previous definitions give coherence to known results and indicate appropriate directions for future development and it generates and may be viewed as both including and included in categories of fuzzy topological spaces which exhibit higher order fuzziness

finite bath fluctuation theorem we demonstrate that a finite bath fluctuation theorem of the crooks type holds for systems that have been thermalized via weakly coupling them to a bath with energy independent finite specific heat we show that this theorem reduces to the known canonical and microcanonical fluctuation theorems in the two respective limiting cases of infinite and vanishing specific heat of the bath the result is elucidated by applying it to a two dimensional hard disk colliding elastically with few other hard disks in a rectangular box with perfectly reflecting walls

suitability in fuzzy topological spaces abstract an l fuzzy topological space is said to be suitable if it possesses a nontrivial crisp closed subset basic properties of and sufficient conditions for suitable spaces are derived characterizations of the suitable subspaces of the fuzzy unit interval the fuzzy open unit interval and the fuzzy real line are obtained suitability is l fuzzy productive nondegenerate 1 hausdorff spaces are suitable the fuzzy unit interval the fuzzy open unit interval and the fuzzy real line are not suitable and no suitable subspace of the fuzzy unit interval the fuzzy open unit interval or the fuzzy real line is a fuzzy retract of the fuzzy unit interval the fuzzy open unit interval or the fuzzy real line respectively without restrictions there cannot be a fuzzy extension theorem

thermodynamics with generalized ensembles the class of dual orthodes abstract we address the problem of the foundation of generalized ensembles in statistical physics the approach is based on boltzmann s concept of orthodes these are the statistical ensembles that satisfy the heat theorem according to which the heat exchanged divided by the temperature is an exact differential this approach can be seen as a mechanical approach alternative to the well established information theoretic one based on the maximization of generalized information entropy our starting point are the tsallis escort ensembles which have been previously proved to be orthodes and have been proved to interpolate between canonical and microcanonical ensembles here we shall see that the tsallis escort ensembles belong to a wider class of orthodes that include the most diverse types of ensembles all such ensembles admit both a microcanonical like parametrization via the energy and a canonical like one via the parameter for this reason we name them dual one central result used to build the theory is a generalized equipartition theorem the theory is illustrated with a few examples and the equivalence of all the dual orthodes is discussed

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

the definition of the thermodynamic entropy in statistical mechanics abstract a definition of the thermodynamic entropy based on the time dependent probability distribution of the macroscopic variables is developed when a constraint in a composite system is released the probability distribution for the new equilibrium values goes to a narrow peak defining the entropy by the logarithm of the probability distribution automatically makes it a maximum at the equilibrium values so it satisfies the second law it also satisfies the postulates of thermodynamics objections to this definition by dieks and peters are discussed and resolved

on the concept of fuzzy point abstract we examine in a general setting the question of a suitable definition of the point concept in the space of the l subsets of a given set namely an axiomatic treatment of this question is proposed we find necessary and sufficient conditions for the existence of a good definition of point in particular we prove that if l 0 1 a definition of point with a good behaviour with respect to infinite unions and intersections is impossible

generating fuzzy topologies with semi closure operators let x be an l fuzzy topological space for a e l 1 and a x x ec a a if and only if g fuzzy open and g x a imply there is a e a with g a 0 under certain restrictions on a which are always satisfied if l is linearly ordered c a is a semi closure operator this paper contains necessary and sufficient conditions under which a collection of semi closure operators generates a fuzzy topology in general such a collection does not generate a unique topology the class of topologies generated is shown to be closed under suprema and its largest member is characterized the topology for the fuzzy unit interval is characterized among those generated by its a closure operators

negative temperatures and the definition of entropy the concept of negative temperature has recently received renewed interest in the context of debates about the correct definition of the thermodynamic entropy in statistical mechanics several researchers have identified the thermodynamic entropy exclusively with the volume entropy suggested by gibbs and have further concluded that by this definition negative temperatures violate the principles of thermodynamics we disagree with these conclusions we demonstrate that volume entropy is inconsistent with the postulates of thermodynamics for systems with non monotonic energy densities while a definition of entropy based on the probability distributions of macroscopic variables does satisfy the postulates of thermodynamics our results confirm that negative temperature is a valid extension of thermodynamics

on a structural observation in generalized electromagnetic theory are satisfied equations 0 2 describe the electromagnetic properties of the medium where 0 1 is considered if this medium is confined to a certain region g of r3 then the electromagnetic properties of its complent cc are usually represented via boundary conditions on dg we shall deal here with the boundary condition appearing if cc is assumed to be a perfect conductor i e nxe t o on c7g 13 0 0 3

ensemble inequivalence in random graphs we present a complete analytical solution of a system of potts spins on a random k regular graph in both the canonical and microcanonical ensembles using the large deviation cavity method ldcm the solution is shown to be composed of three different branches resulting in a non concave entropy function the analytical solution is confirmed with numerical metropolis and creutz simulations and our results clearly demonstrate the presence of a region with negative specific heat and consequently ensemble inequivalence between the canonical and microcanonical ensembles

a lattice of continuities for fuzzy topological spaces abstract several continuities are defined for fuzzy topological spaces and characterizations of these continuities are obtained it is shown that these continuities form a lattice ordered by implication whose supremum is fuzzy continuity and whose infimum is the continuity of general topology

point spaces for fuzzy topological spaces abstract we define a category f of fuzzy topological spaces whose objects are the pairs l where l is any distributive lattice with universal bounds 0 and 1 0 1 and is a fuzzy topology on l and whose morphisms u l 1 l 2 are relations such that u 1 l 2 l 1 is a lattice homomorphism we define a covariant functor f t where t is the category of fuzzy topological spaces the points of the topological spaces l are called fuzzy points of l and allow us to define local topological concepts which depend on a base for a topology we have natural functors from the hutton category h from the chang goguen category c g l from the lowen category l and from the rodabaugh category fuzz into f

on fuzzy topological spaces abstract in this paper a few separation properties and some aspects of subspace fuzzy topology have been studied where both the crisp and the fuzzy elements have been taken into consideration since the conventional definition of compactness is not quite meaningful in hausdorff fuzzy spaces as introduced by us a new more natural definition of proper compactness is given and a few properties resulting from this are established

a large deviation perspective on ratio observables in reset processes robustness of rate functions we study large deviations of a ratio observable in discrete time reset processes the ratio takes the form of a current divided by the number of reset steps and as such it is not extensive in time a large deviation rate function can be derived for this observable via contraction from the joint probability density function of current and number of reset steps the ratio rate function is differentiable and we argue that its qualitative shape is robust i e it is generic for reset processes regardless of whether they have short or long range correlations we discuss similarities and differences with the rate function of the efficiency in stochastic thermodynamics

comparison of canonical and microcanonical definitions of entropy abstract for more than 100 years one of the central concepts in statistical mechanics has been the microcanonical ensemble which provides a way of calculating the thermodynamic entropy for a specified energy a controversy has recently emerged between two distinct definitions of the entropy based on the microcanonical ensemble 1 the boltzmann entropy defined by the density of states at a specified energy and 2 the gibbs entropy defined by the sum or integral of the density of states below a specified energy a critical difference between the consequences of these definitions pertains to the concept of negative temperatures which by the gibbs definition cannot exist in this paper we call into question the fundamental assumption that the microcanonical ensemble should be used to define the entropy we base our analysis on a recently proposed canonical definition of the entropy as a function of energy we investigate the predictions of the boltzmann gibbs and canonical definitions for a variety of classical and quantum models our results support the validity of the concept of negative temperature but not for all models with a decreasing density of states we find that only the canonical entropy consistently predicts the correct thermodynamic properties while microcanonical definitions of entropy including those of boltzmann and gibbs are correct only for a limited set of models for models which exhibit a first order phase transition we show that the use of the thermodynamic limit as usually interpreted can conceal the essential physics

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

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

thermodynamic laws in isolated systems the recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful analysis of the conceptual foundations underlying microcanonical thermostatistics here we provide a detailed comparison of the most commonly considered microcanonical entropy definitions focusing specifically on whether they satisfy or violate the zeroth first and second laws of thermodynamics our analysis shows that for a broad class of systems that includes all standard classical hamiltonian systems only the gibbs volume entropy fulfills all three laws simultaneously to avoid ambiguities the discussion is restricted to exact results and analytically tractable examples

hypergraph functor and attachment using an arbitrary variety of algebras the paper introduces a fuzzified version of the notion of attachment in a complete lattice of guido to provide a common framework for the concept of hypergraph functor considered by different authors in the literature the new notion also gives rise to a category of variable basis topological spaces which is a proper supercategory of the respective category of rodabaugh

fuzzy grills and a characterization of fuzzy proximity in his classical paper lo zadeh introduced the fundamental concept of fuzzy sets this idea was then applied to various other fields of mathematics including topology resulting in the study of fuzzy topology by chang i fuzzy uniformity by lowen 7 and fuzzy proximity by katsaras 5 thron is presented a new approach to poximity structures using the theory of grills an important concept introduced by choquet 3 the importance of grills in the proximity theory is noticed sufficiently in various other papers e g cf 2 4 also recently katsaras 6 introduced and studied filters ultrafilters and clusters in the new setup of fuzzy proximity on a set in the present paper we introduce fuzzy stacks and fuzzy grills and obtain a characterization of fuzzy proximity using fuzzy grills in section 2 fuzzy stacks and fuzzy grills and in section 3 on the lines of basic proximity or c proximity as described in 8 9 the concept of fuzzy basic proximity on a set are introduced also the fuzzy basic proximal neighborhood of a fuzzy set with respect to a fuzzy basic proximity is defined and some of the properties of sets of all fuzzy basic proximal neighborhoods of fuzzy sets are obtained finally in section 4 a necessary and sufficient condition for a fuzzy basic proximity to be a fuzzy proximity is obtained

route from discreteness to the continuum for the tsallis q entropy the existence and exact form of the continuum expression of the discrete nonlogarithmic q entropy is an important open problem in generalized thermostatistics since its possible lack implies that nonlogarithmic q entropy is irrelevant for the continuous classical systems in this work we show how the discrete nonlogarithmic q entropy in fact converges in the continuous limit and the negative of the q entropy with continuous variables is demonstrated to lead to the csiszar type q relative entropy just as the relation between the continuous boltzmann gibbs expression and the kullback leibler relative entropy as a result we conclude that there is no obstacle for the applicability of the q entropy to the continuous classical physical systems

complete analysis of ensemble inequivalence in the blume emery griffiths model we study inequivalence of canonical and microcanonical ensembles in the mean field blume emery griffiths model this generalizes previous results obtained for the blume capel model the phase diagram strongly depends on the value of the biquadratic exchange interaction k the additional feature present in the blume emery griffiths model at small values of k as for the blume capel model lines of first and second order phase transitions between a ferromagnetic and a paramagnetic phase are present separated by a tricritical point whose location is different in the two ensembles at higher values of k the phase diagram changes substantially with the appearance of a triple point in the canonical ensemble which does not find any correspondence in the microcanonical ensemble moreover one of the first order lines that starts from the triple point ends in a critical point whose position in the phase diagram is different in the two ensembles this line separates two paramagnetic phases characterized by a different value of the quadrupole moment these features were not previously studied for other models and substantially enrich the landscape of ensemble inequivalence identifying new aspects that had been discussed in a classification of phase transitions based on singularity theory finally we discuss ergodicity breaking which is highlighted by the presence of gaps in the accessible values of magnetization at low energies it also displays new interesting patterns that are not present in the blume capel model

tsallis power laws and finite baths with negative heat capacity it is often stated that heat baths with finite degrees of freedom i e finite baths are sources of tsallis distributions for classical hamiltonian systems by using well known fundamental statistical mechanics expressions we rigorously show that tsallis distributions with fat tails are possible only for finite baths with constant negative heat capacity while constant positive heat capacity finite baths yield decays with sharp cutoff with no fat tails however the correspondence between tsallis distributions and finite baths holds at the expense of violating the equipartition theorem for finite classical systems at equilibrium we comment on the implications of the finite bath for the recent attempts towards a q generalized central limit theorem

domain dependent dirac s delta and derivatives with an application to electromagnetic boundary integral representations the domain dependent versions of derivatives and dirac s delta are defined in distributional sense these operations enable to obtain domain dependent fundamental solutions and global boundary integral representation formulae a global representation formula is defined everywhere also on the boundary and includes the jump relations of the boundary the use of the domain dependent objects can be interpreted as taking the boundary limit in prior to integrating by parts when deriving the familiar boundary integral equations as an application the representation formulae are obtained for the solutions of the helmholtz equation and the maxwell equations

higher separation axioms in l topologically generated i l topological spaces the main result of this paper is a theorem about inserting a pair of semicontinuous l real valued functions which extends the insertion theorem of kubiak comment math univ carolinae 34 1993 357 362 from l 0 1 to an arbitrary meet continuous lattice l endowed with an order reversing involution with this result it is shown that the normality type separation axioms in top l are preserved by the functor which takes an l topological space x to the i l topological space l x obtained by providing the set x with the i l topology consisting of all lower semicontinuous functions from x to i l the same is proved for the case of the regularity axiom

gibbs volume entropy is incorrect we show that the expression for the equilibrium thermodynamic entropy contains an integral over a surface in phase space and in so doing we confirm that negative temperature is a valid thermodynamic concept this demonstration disproves the claims of several recent papers that the gibbs entropy which contains an integral over a volume in phase space is the correct definition and that thermodynamics cannot be extended to include negative temperatures we further show that the gibbs entropy fails to satisfy the postulates of thermodynamics and that its predictions for systems with nonmonotonic energy densities of states are incorrect

generalized fuzzy points let x be a set and f x the lattice offuzzy subsets of x 6 in s in order to give a local approach to fuzzy topologies wong defines the fuzzy point concept he calls fuzzy point every fuzzy set f with a e x and c1 1 defined puttingf x 0 if x a andf x a if x a moreover a befonging relation between a fuzzy point f and a fuzzy set f is defined by setting f ef if and only if f a a if we set x f a e x 0 a 1 then for every x y e x and a g e f x

representations of fuzzy topologies we make use of the point space concept 2 to give some representations of fuzzy topologies by classical topologies we show that fuzzy convergence and fuzzy continuity with respect given fuzzy topologies are reducible to the convergence and the continuity with respect to suitable associated topologies

phase diagrams and tricritical effects in the beg model abstract the critical properties of the spin 1 ising model on the bethe lattice with dipolar and quadrupolar exchange interactions are studied exactly this method is based on recursion relations which are exact in the infinite dimensional case and as a result the phase diagrams are constructed the line ending in the tricritical point is defined exactly it is shown that when the quadrupolar exchange interaction is large enough the tricritical point disappears the tricritical exponents are calculated exactly

fuzzy points and attachment in this paper a binary relation between l sets called attachment is defined by means of a suitable family of completely coprime filters in l examples are given using several kinds of lattice ordered algebras and a class of attachments is determined whose elements generalize pu liu s quasi coincidence relation mapping any l set on x to the set of the attached l points one gets a frame map from the l powerset l x to the powerset p s x of the set s x of l points on x this allows one to define functors from the category l top of l topological spaces either to the category top of topological spaces or to the category topsys of topological systems a comparison with the hypergraph functor is also included

in defense of negative temperature this pedagogical comment highlights three misconceptions concerning the usefulness of the concept of negative temperature being derived from the usual often termed boltzmann definition of entropy first both the boltzmann and gibbs entropies must obey the same thermodynamic consistency relation second the boltzmann entropy does obey the second law of thermodynamics third there exists an integrating factor of the heat differential with both definitions of entropy

on a topological model for the category of fuzzy topological spaces i we study global properties of fuzzy topological spaces which depend only on open sets fuzzy continuous functions and the relations we obtain many known theorems as purely topological theorems

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

fuzzy addition in the l fuzzy real line abstract under the hypothesis l is a chain we construct a binary operation on the l fuzzy real line r l which reduces to the usual addition on r if is restricted to the embedded image of r in r l which yields a partially ordered abelian cancellation semigroup with identity and which is jointly fuzzy continuous on r l we show is unique i e it is the only extension of addition to r l which is consistent we study the relationship between and other fuzzy continuous extensions of the usual addition we also show that fuzzy translation is a weak fuzzy homeomorphism and under certain conditions a fuzzy homeomorphism

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators