a boundary value problem for second order fuzzy differential equations abstract in this paper we interpret a two point boundary value problem for a second order fuzzy differential equation by using a generalized differentiability concept we present a new concept of solutions and utilizing the generalized differentiability we investigate the problem of finding new solutions some examples are provided for which the new solutions are found

first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

periodic boundary value problems for first order linear differential equations with uncertainty under generalized differentiability we study the existence of solutions to a class of first order linear fuzzy differential equations subject to periodic boundary conditions from the point of view of generalized differentiability the objective of this paper is to show that fuzzy differential equations under generalized differentiability can be used in the study of periodic phenomena by considering a combination of two types of derivatives with a switching point we provide sufficient conditions which guarantee that the piecewise defined solutions match adequately and illustrate through some examples the process of construction and calculation of solutions

quantum fluctuation relations for the lindblad master equation an open quantum system interacting with its environment can be modeled under suitable assumptions as a markov process described by a lindblad master equation in this work we derive a general set of fluctuation relations for systems governed by a lindblad equation these identities provide quantum versions of jarzynski hatano sasa and crooks relations in the linear response regime these fluctuation relations yield a fluctuation dissipation theorem fdt valid for a stationary state arbitrarily far from equilibrium for a closed system this fdt reduces to the celebrated callen welton kubo formula

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

quantum entropy production as a measure of irreversibility we consider conservative quantum evolutions possibly interrupted by macroscopic measurements when started in a nonequilibrium state the resulting path space measure is not time reversal invariant and the weight of time reversal breaking equals the exponential of the entropy production the mean entropy production can then be expressed via a relative entropy on the level of histories this gives a partial extension of the result for classical systems that the entropy production is given by the source term of time reversal breaking in the path space measure

fluctuation theorems for quantum master equations a quantum fluctuation theorem for a driven quantum subsystem interacting with its environment is derived based solely on the assumption that its reduced density matrix obeys a closed evolution equation i e a quantum master equation qme quantum trajectories and their associated entropy heat and work appear naturally by transforming the qme to a time dependent liouville space basis that diagonalizes the instantaneous reduced density matrix of the subsystem a quantum integral fluctuation theorem a steady state fluctuation theorem and the jarzynski relation are derived in a similar way as for classical stochastic dynamics

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives in this paper a class of uncertain linear dynamical systems called fuzzy fractional linear dynamical systems is investigated the aim is to find control inputs to keep the states of the fuzzy fractional dynamical systems near the zero in an optimal manner the optimality criterion is in a form of a granular fuzzy integral whose integrand is a quadratic function of the state variables and control inputs the fuzzy fractional dynamical system is described using fuzzy fractional differential equations ffdes in order to achieve the aim an effective approach for solving ffdes should be at disposal due to some restrictions imposed by the previous approaches dealing with ffdes a new approach is proposed the proposed approach is based on the granular derivative and the so called relative distance measure fuzzy interval arithmetic new definitions of fuzzy fractional derivatives and integral called left and right granular riemann liouville fuzzy fractional derivatives left and right granular caputo fuzzy fractional derivatives and the left and right granular fuzzy fractional integral are also presented in addition the concepts of granular fuzzy partial derivative and granular fuzzy chain rule are introduced by the approximations of the granular fuzzy fractional integral and the granular caputo fuzzy fractional derivative the approximation solution to the ffdes is obtained consequently based on the new concepts and theorems the solution to the fuzzy fractional quadratic regulator problem is given by a theorem this paper closes with an example of regulating the motion of boeing 747 in longitudinal direction with the presence of uncertainty in the initial conditions and the coefficients

generalized boltzmann equation for lattice gas automata in this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi detailed balance starting from a coupled bbgky hierarchy for then particle distribution functions cluster expansion techniques are used to derive approximate kinetic equations in zeroth approximation the standard nonlnear boltzmann equation is obtained the next approximation yields the ring kinetic equation similar to that for hard sphere systems describing the time evolution of pair correlations the ring equation is solved to determine the nonvanishing pair correlation functions in equilibrium for two models that violate semidetailed balance one is a model of interacting random walkers on a line the other one is a two dimensional fluid type model on a triangular lattice the numerical predictions agree very well with computer simulations

quantum work fluctuation theorem exact case study for a perturbed oscillator we analytically explore the fluctuation of the work induced entropy production of an externally perturbed quantum harmonic oscillator interacting with several reservoirs the quantum fluctuation of the work amounts to a non gaussian fluctuation of the entropy production which interpolates the gaussian and the poissonian distributions at high and low temperature regimes also it is shown that the corresponding fluctuation theorem symmetry is rigorously satisfied

a random laplace transform method for solving random mixed parabolic differential problems this paper deals with the explicit solution of random mixed parabolic equations in unbounded domains by using the random laplace transform to second order stochastic processes the mean square random laplace operational calculus is stated and its application to the random parabolic equation together with previous results of the underlying random ordinary differential equations allow us to obtain an explicit solution of the problem a numerical example which includes simulations illustrates the developed method

unified treatment of the quantum fluctuation theorem and the jarzynski equality in terms of microscopic reversibility there are two related theorems which hold even in far from equilibrium namely fluctuation theorem and jarzynski equality fluctuation theorem states the existence of symmetry of fluctuation of entropy production while the jarzynski equality enables us to estimate the free energy change between two states by using irreversible processes on the other hand the relationship between these theorems was investigated by crooks phys rev e 60 2721 1999 for the classical stochastic systems in this paper we derive quantum analogues of fluctuation theorem and jarzynski equality in terms of microscopic reversibility in other words the quantum analog of the work by crooks is presented also for the quasiclassical langevin system microscopically reversible condition is confirmed

arithmetic operations on independent random variables a numerical approach dealing with imprecise quantities is an important problem in scientific computation model parameters are known only approximately typically in the form of probability density functions unfortunately there are currently no methods of taking uncertain parameters into account which would at the same time be easy to apply and highly accurate an important special case is operations on independent random variables which occur frequently in obtaining confidence intervals for physical measurements and statistical estimators in this paper we investigate the possibility of implementing arithmetic operations on independent random variables numerically equivalently the problem can be viewed as propagating approximated probability density functions through arithmetic operations we introduce a very broad family of distributions which is closed under the four arithmetic operations and taking powers furthermore we show that the densities of the distributions in the family can be effectively approximated using ch

initial value problems for higher order fuzzy differential equations we consider nth order fuzzy differential equations with initial value conditions we prove the existence and uniqueness of solution for nonlinearities satisfying a lipschitz condition we apply the obtained results to the particular case of linear fuzzy problems

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

on possibilistic representations of fuzzy intervals it is acknowledged that a fuzzy interval has two equivalent representations given in terms of the so called left and right sides of the membership function lr representation or in terms of the lower and upper branches defining the endpoints of the cuts lu representation in this paper we suggest an additional representation of fuzzy intervals called acf representation using an average cumulative function instead of the membership function based on possibility theory we illustrate how to build the new representation and we state its basic properties the main result is that the average cumulative ac function can be uniquely defined for any fuzzy interval and it is possible to move from one representation to the others through appropriate transformations an interesting link can be established between acf representation and quantile functions with a possible statistical interpretation useful in real application we also recommend a parametric form of the ac function

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

entropy production for quantum markov semigroups an invariant state of a quantum markov semigroup is an equilibrium state if it satisfies a quantum detailed balance condition in this paper we introduce a notion of entropy production for faithful normal invariant states of a quantum markov semigroup on mathcal b mathsf h as a numerical index measuring how far they are from equilibrium the entropy production rate is defined as the derivative of the relative entropy of the one step forward and backward evolution in analogy with the classical probabilistic concept we prove an explicit trace formula expressing the entropy production rate in terms of the completely positive part of the generator of a norm continuous quantum markov semigroup showing that it turns out to be zero if and only if a standard quantum detailed balance condition holds

neutral fuzzy fractional functional differential equations abstract in this paper a class of fuzzy fractional functional differential equations of neutral type is investigated the existence uniqueness and iterative formula in the case of linear equations are obtained meanwhile the continuous dependence of solutions on initial data parameters and functions involved in right hand side of equation is also discussed several examples are given to illustrate the applications of our results

hydrodynamics and time correlation functions for cellular automata hydrodynamic excitations in lattice gas cellular automata are described in terms of equilibrium time correlation functions for the local conserved variables for large space and time scales the linearized hydrodynamic equations are obtained to navier stokes order exact expressions for the associated susceptibilities and transport coefficients are identified in terms of correlation functions the general form of the time correlation functions for conserved densities in the hydrodynamic limit is given and illustrated by some examples suitable for comparison with computer simulation the transport coefficients are related to time correlation functions for the conserved fluxes in a way analogous to the green kubo expressions for continuous fluids the general results are applied for a one component fluid and several types of binary diffusion also discussed are the effects of unphysical slow modes such as staggered particle or momentum densities

a class of linear differential dynamical systems with fuzzy matrices abstract this paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices we use a complex number representation of the level sets of the fuzzy system and obtain the solution by employing such representation it is applicable to practical computations and has also some implications for the theory of fuzzy differential equations we then present some properties of the 2 dimensional dynamical systems and their phase portraits some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear we finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems

existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions abstract in this paper the existence and uniqueness of a fuzzy solution for the semilinear fuzzy integrodifferential equation is established via the banach fixed point analysis approach and using the fuzzy number whose values are normal convex upper semicontinuous and compactly supported interval in e n

time reversal and entropy there is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of time reversal symmetry and the entropy production in such processes we explain on an elementary mathematical level the relations between entropy production phase space contraction and time reversal starting from a deterministic dynamics both closed and open systems in the transient and in the steady regime are considered the main result identifies under general conditions the statistical mechanical entropy production as the source term of time reversal breaking in the path space measure for the evolution of reduced variables this provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful in equalities we also discuss the markov approximation important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior

collective diffusion in a lattice gas automaton we report on non mean field and ring kinetic theory calculations of both the momentum autocorrelation function and the collective diffusion coefficient in a diffusive lattice gas automaton for both quantities the ring approximation is calculated exactly a saddle point method yields a leadingt 2 and a subleadingt 5 2 long time tail in the momentum autocorrelation function the ring kinetic corrections to the mean field value of the diffusion coefficient are in good agreement with computer simulations

variation of constant formula for first order fuzzy differential equations in this paper we study first order linear fuzzy differential equations by using the generalized differentiability concept and we present the general form of their solutions we also correct and complete some previous results finally some examples are given to illustrate our results

z differential equations this paper is devoted to make a framework for studying a class of uncertain differential equations called z differential equations in order to achieve the purpose we first introduce four basic operations on z numbers based on semigranular function then the limit and continuity concepts of a z number valued function are given under a definition of a metric on the space of z numbers moreover the concepts of z differentiability z integral and z laplace transform of a z number valued function are introduced in addition by giving some theories proved in this paper a basis for calculus z calculus is established we further give theories based on which existence and uniqueness of z differential equations are investigated a conceptual unity between z differential equations and z numbers is also shown the conceptual unity demonstrates that a z differential equation may be expressed as a bimodal differential equation combining a fuzzy differential equation fde and a random differential equation moreover the concept of a bimodal cut called s mu cut is introduced and its relation to other new concepts such as acceptable time and acceptable information area is explained using an example the application of z differential equations in medicine is clarified it is demonstrated that z differential equations outperform fdes in making a decision under uncertainty

an algorithm for the convolution of legendre series an mathcal o n 2 algorithm for the convolution of compactly supported legendre series is described the algorithm is derived from the convolution theorem for legendre polynomials and the recurrence relation satisfied by spherical bessel functions combining with previous work yields an mathcal o n 2 algorithm for the convolution of chebyshev series numerical results are presented to demonstrate the improved efficiency over the existing algorithm

granular differentiability of fuzzy number valued functions in this paper using the concept of horizontal membership functions a new definition of fuzzy derivative called granular derivative is proposed based on granular difference moreover a new definition of fuzzy integral called granular integral is defined and its relation with the granular derivative is given a new definition of a metric granular metric on the space of type 1 fuzzy numbers and a concept of continuous fuzzy functions are also presented restrictions associated to previous approaches hukuhara differentiability strongly generalized hukuhara differentiability generalized hukuhara differentiability generalized differentiability zadeh s extension principle and fuzzy differential inclusions dealing with fuzzy differential equations fdes are expressed it is shown that the proposed approach does not have the drawbacks of the previous approaches it is also demonstrated how this approach enables researchers to solve fdes more conveniently than ever before moreover we showed that this approach does not necessitate that the diameter of the fuzzy function be monotonic it is also proved that the result of each of the four basic operations on fuzzy numbers introduced based on the proposed approach leads to a fuzzy number moreover the condition for the existence of the granular derivative of a fuzzy function is provided by a theorem additionally by two examples it is shown that the existence of the granular derivative of a fuzzy function does not imply the existence of the generalized hukuhara differentiability of the fuzzy function and vice versa the terms doubling property and unnatural behavior in modeling phenomenon are also introduced furthermore using some examples the paper proceeds to elaborate on the efficiency and effectiveness of the proposed approach moreover as an application of the proposed approach the response of boeing 747 to impulsive elevator input is obtained in the presence of uncertain initial conditions and parameters

a new approach for the optimal fuzzy linear time invariant controlled system with fuzzy coefficients in this article a new technique for solving the linear time invariant dynamical system with fuzzy parameters is proposed this approach is used to solve a fuzzy controlled system by using a cuts and presentation of numbers in a more compact form by moving to the field of complex numbers by considering a new feature of the hamiltonian function and using the pontryagin maximum principle the solution of the fuzzy optimal controlled system is considered

microscopic origins of irreversible macroscopic behavior time asymmetric behavior as embodied in the second law of thermodynamics is observed in individual macroscopic systems it can be understood as arising naturally from time symmetric microscopic laws in accord with the ideas of thompson maxwell and boltzmann alternative explanations based on equating irreversible macroscopic behavior with the mixing type of behavior already present in the time evolution of ensembles probability distributions of certain systems having only a few degrees of freedom are in my opinion unnecessary misguided and misleading

fractional differential systems a fuzzy solution based on operational matrix of shifted chebyshev polynomials and its applications in this paper a new formula of fuzzy caputo fractional order derivatives 0 v leq 1 in terms of shifted chebyshev polynomials is derived the proposed approach introduces a shifted chebyshev operational matrix in combination with a shifted chebyshev tau technique for the numerical solution of linear fuzzy fractional order differential equations the main advantage of the proposed approach is that it simplifies the problem alike in solving a system of fuzzy algebraic linear equations an approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors is derived furthermore we also discuss the convergence of the proposed method from the fuzzy perspective experimentally we show the strength of the proposed method in solving a variety of fractional differential equation models under uncertainty encountered in engineering and physical phenomena i e viscoelasticity oscillations and resistor capacitor rc circuits comparisons are also made with solutions obtained by the laguerre polynomials and the fractional euler method

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned