a class of global fractional order projective dynamical systems involving set valued perturbations this paper studies a class of global fractional order projective dynamical systems involving set valued perturbations in real separable hilbert spaces we prove that the set of solutions for this type of systems is nonempty and closed under some suitable conditions furthermore we show that the set of solutions is continuous with respect to initial value in the sense of hausdorff metric finally an interesting numerical example is given to illustrate the validity of the main theorem presented in this paper

brief note on the variation of constants formula for fuzzy differential equations this note gives a theory of state transition matrices for linear systems of fuzzy differential equations this is used to give a fuzzy version of the classical variation of constants formula a simple example of a time independent control system is used to illustrate the methods while similar problems to the crisp case arise for time dependent systems in time independent cases the calculations are elementary solutions of eigenvalue eigenvector problems in particular for nonnegative or nonpositive matrices the problems at each level set can easily be solved in matlab to give the level sets of the fuzzy solution

q epsilon tensor for quantum and braided spaces the machinery of braided geometry introduced previously is used now to construct the e totally antisymmetric tensor on a general braided vector space determined by r matrices this includes natural q euclidean and q minkowski spaces the formalism is completely covariant under the corresponding quantum group such as soq 4 or soq 1 3 the hodge operator and differentials are also constructed in this approach

evolution equations and levy processes on quantum groups in this chapter we consider stochastic processes on quantum groups that are related to evolution equations of the form partial t u lu with some difference differential operator l for the equations considered in section 7 1 u is an element of a quantum or braided group a we recall that solutions of these equations can be given as appell systems or shifted moments of the associated process and show how these can be calculated explicitly on the q affine group and on a braided analogue of the heisenberg weyl group

rogue waves rational solutions the patterns of their zeros and integral relations the focusing nonlinear schrodinger equation which describes generic nonlinear phenomena including waves in the deep ocean and light pulses in optical fibres supports a whole hierarchy of recently discovered rational solutions we present recurrence relations for the hierarchy the pattern of zeros for each solution and a set of integral relations which characterizes them

two point boundary value problems of undamped uncertain dynamical systems in this paper we study two point boundary value problems of a class of uncertain dynamical systems i e undamped uncertain dynamical systems we introduce the concept of large solutions and small solutions for two point boundary value problems of undamped uncertain dynamical systems and under some conditions establish the existence and uniqueness of large solutions and small solutions and settle the relationship between large solutions and small solutions

other 2n 2 parameters solutions of the nls equation and 2n 1 highest amplitude of the modulus of the n th order ap breather in this paper we construct new deformations of the akhmediev peregrine ap breather of order n or apn breather with real parameters other families of quasirational solutions of the nonlinear schrodinger nls equation are obtained we evaluate the highest amplitude of the modulus of the ap breather of order n we give the proof that the highest amplitude of the apn breather is equal to we get new formulas for the solutions of the nls equation which are different from these already given in previous works new solutions for the order 8 and their deformations according to the parameters are explicitly given we simultaneously get triangular configurations and isolated rings moreover the appearance for certain values of the parameters and of new configurations of concentric rings are underscored

tau method for the numerical solution of a fuzzy fractional kinetic model and its application to the oil palm frond as a promising source of xylose the oil palm frond a lignocellulosic material is a high yielding energy crop that can be utilized as a promising source of xylose it holds the potential as a feedstock for bioethanol production due to being free and inexpensive in terms of collection storage and cropping practices the aim of the paper is to calculate the concentration and yield of xylose from the acid hydrolysis of the oil palm frond through a fuzzy fractional kinetic model the approximate solution of the derived fuzzy fractional model is achieved by using a tau method based on the fuzzy operational matrix of the generalized laguerre polynomials the results validate the effectiveness and applicability of the proposed solution method for solving this type of fuzzy kinetic model a new fractional kinetic equation under uncertainty was addressed to depict the chemical reaction arising in palm oil frond an efficient numerical simulation based on a tau method was derived to solve the proposed kinetic equation different cases were solved to demonstrate the validity and efficiency of the proposed technique

fuzzy fractional functional integral and differential equations this paper is devoted to considering the existence and uniqueness results for fuzzy fractional functional integral equations employing the contraction principle moreover the approach is followed to prove the existence and uniqueness of solutions for the fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives finally the solution to fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives by a modified fractional euler method mfem is presented the method is illustrated by solving some examples

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

degenerate determinant representation of solutions of the nonlinear schrodinger equation higher order peregrine breathers and multi rogue waves we present a new representation of solutions of the focusing nls equation as a quotient of two determinants this work is based on a recent paper in which we have constructed a multi parametric family of solutions of this equation in terms of wronskians this formulation was written in terms of a limit involving a parameter here we give a very compact formulation without presence of a limit this is a completely new result which gives a very efficient procedure to construct families of quasi rational solutions of the nls equation with this method we construct peregrine breathers of orders n 4 6 and multi rogue waves associated by deformation of parameters

quantum lorentz and braided poincare groups quantum lorentz groups h admitting quantum minkowski space v are selected the natural structure of a quantum space is introduced defining a quantum group structure on g only for triangular h q 1 we show that it defines a braided quantum group structure on g for q 1

a system of fractional order interval projection neural networks in this paper a system of fractional order interval projection neural networks is introduced and investigated under some suitable assumptions the existence and uniqueness of the equilibrium point of this type of interval projection neural networks is proved moreover exponential stability of this type of neural networks is obtained also in the last section we give two numerical examples to demonstrate the validity of our results

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

the gamma poincare quantum group from quantum group contraction we propose a contraction of the de sitter quantum group leading to a poincare quantum group in any dimensions the method relies on the coaction of the de sitter quantum group on a non commutative space and the deformation parameter q is sent to one the bicrossproduct structure of this gamma poincare quantum group is exhibited and shown to be dual to the one of the kappa poincare hopf algebra at least in two dimensions

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

covariant realization of quantum spaces as star products by drinfeld twists covariance of a quantum space with respect to a quantum enveloping algebra ties the deformation of the multiplication of the space algebra to the deformation of the coproduct of the enveloping algebra since the deformation of the coproduct is governed by a drinfeld twist the same twist naturally defines a covariant star product on the commutative space however this product is in general not associative and does not yield the quantum space it is shown that there are certain drinfeld twists which realize the associative product of the quantum plane quantum euclidean four space and quantum minkowski space these twists are unique up to a central two coboundary the appropriate formal deformation of real structures of the quantum spaces is also expressed by these twists

general high order rogue waves and their dynamics in the nonlinear schrodinger equation general high order rogue waves in the nonlinear schrodinger equation are derived by the bilinear method these rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions it is shown that the general n th order rogue waves contain n 1 free irreducible complex parameters in addition the specific rogue waves obtained by akhmediev et al akhmediev et al 2009 phys rev e 80 026601 doi 10 1103 physreve 80 026601 correspond to special choices of these free parameters and they have the highest peak amplitudes among all rogue waves of the same order if other values of these free parameters are taken however these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns

on a system of fuzzy fractional differential inclusions with projection operators abstract the aim of this paper is to introduce and study a class of fuzzy fractional differential inclusions with projection operators which captures the desired features of both fuzzy differential inclusions and fractional order projective dynamical systems within the same framework the existence of solutions for the open situation is proved by using continuous selection theorem moreover the existence of solutions for the closed situation is showed by employing lipschitzean selection theorem and fixed point theorem respectively furthermore an application and a numerical example are also given

metrics on the real quantum plane using the frame formalism we determine some possible metrics and metric compatible connections on the noncommutative differential geometry of the real quantum plane by definition a metric maps the tensor product of two 1 forms into a function on the quantum plane it is symmetric in a modified sense namely in the definition of symmetry one has to replace the permutator map with a deformed map fulfilling some suitable conditions correspondingly also the definition of the hermitian conjugate of the tensor product of two 1 forms is modified but reduces to the standard one if coincides with the permutator the metric is real with respect to such modified structure

stability and periodicity in fuzzy differential equations formulations of fuzzy differential equations des in terms of the hukuhara derivative do not reproduce the rich and varied behavior of crisp des another interpretation in terms of differential inclusions expressed level setwise overcomes much of this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies studies lyapunov stability of fuzzy des and the periodicity of the fuzzy solution set for both the time dependent and autonomous cases

two parameters wronskian representation of solutions of nonlinear schrodinger equation eighth peregrine breather and multi rogue waves in this paper we present a representation of solutions of the one dimensional focusing nonlinear schrodinger equation as a quotient of two wronskians depending on two parameters here we give the complete proof of this representation we have already constructed peregrine breathers and their two parameter deformations until order 7 with this method the construction of the explicit analytical expressions of peregrine breather of order 8 was made for the first time when parameters a or b are equal to 0 we recover the peregrine breather of order 8 we obtain multi rogue waves by deformation of parameters a and b these expressions enable us to understand the evolution of the solutions in the case of order 8 it is shown for high values of parameters a or b the appearance of peregrine breather of order 6

six parameters deformations of fourth order peregrine breather solutions of the nonlinear schrodinger equation we construct solutions of the focusing nonlinear schrodinger nls equation as a quotient of two determinants this formulation gives in the case of the order 4 new deformations of the peregrine breather with 6 real parameters we construct families of quasi rational solutions of the nls equation and describe the apparition of multi rogue waves with this method we construct the analytical expressions of deformations of the peregrine breather of order 4 with 6 real parameters and plot different types of rogue waves

on the classification of quantum poincare groups using the general theory of 10 quantum poincare groups without dilatations are described and investigated the description contains a set of numerical parameters which satisfy certain polynomial equations for most cases we solve them and give the classification of quantum poincare groups each of them corresponds to exactly one quantum minkowski space the poincare series of these objects are the same as in the classical case we also classify possibler matrices for the fundamental representation of the group

differential hopf algebra structure of the quantum standard complex we are investigating the quantum standard complex k q g d of the quantum enveloping algebra uq g for lie algebras g associated with the root systems an bn cn and dn it is a quantum version of the standard koszul complex associated to a lie algebra as applied for instance in the brs quantization procedure in connection with spin representations using techniques from the theory of braided monoidal categories we obtain a differential hopf algebra structure on the complex k q g d

braided geometry of the conformal algebra we show that the action of the special conformal transformations of the usual undeformed conformal group is the q 1 scaling limit of the braided adjoint action or r commutator of q minkowski space on itself we also describe the q deformed conformal algebra in r matrix form and its quasi structure

families of quasi rational solutions of the nls equation and multi rogue waves we construct a multi parametric family of the solutions of the focusing nonlinear schr dinger equation nls from the known results describing the multi phase almost periodic elementary solutions given in terms of riemann theta functions we give a new representation of their solutions in terms of wronskians determinants of order 2n composed of elementary trigonometric functions when we perform a special passage to the limit when all the periods tend to infinity we obtain a family of quasi rational solutions this leads to efficient representations for the peregrine breathers of orders n 1 2 3 first constructed by akhmediev and his co workers and also allows us to obtain a simpler derivation of the generic formulas corresponding the three or six rogue wave formation in the frame of the nls model first explained by v b matveev in 2010 our formulation allows us to isolate easily the second or third order peregrine breathers from generic solutions and also to compute the peregrine breathers of orders 2 and 3 easily with respect to other approaches in the cases n 2 3 we obtain the comfortable formulas to study the deformation of a higher peregrine breather of order 2 to the three rogue wave or order 3 to the six rogue wave solutions via the variation of the free parameters of our construction

braided momentum in the q poincare group the q poincare group of m schlieker et al z phys c 53 79 1992 is shown to have the structure of a semidirect product and coproduct b soq 1 3 where b is a braided quantum group structure on the q minkowski space of four momentum with braided coproduct p p 1 1 p here the necessary b is not a usual kind of quantum group but one with braid statistics similar braided vectors and covectors v r v r exist for a general r matrix the abstract structure of the q lorentz group is also studied

rational solitons of wave resonant interaction models integrable models of resonant interaction of two or more waves in 1 1 dimensions are known to be of applicative interest in several areas here we consider a system of three coupled wave equations which includes as special cases the vector nonlinear schrodinger equations and the equations describing the resonant interaction of three waves the darboux dressing construction of soliton solutions is applied under the condition that the solutions have rational or mixed rational exponential dependence on coordinates our algebraic construction relies on the use of nilpotent matrices and their jordan form we systematically search for all bounded rational mixed rational exponential solutions and find a broad family of such solutions of the three wave resonant interaction equations

cross products by braided groups and bosonization let h be a braided cocommutative hopf algebra in a braided monoidal category c and b a hopf algebra in c on which h acts we construct a cross product hopf algebra b formula h in c as an application we show that every b in a certain class can be converted to an equivalent ordinary hopf algebra by a process of bosonization the class includes for example all super hopf algebras the constructions respect any quasitriangular structures on b as a corollary we show that if h r is an ordinary quasitriangular hopf algebra then the smash product by the adjoint action of h on itself had formula h can be given the structure of a hopf algebra we prove a similar result for any hopf algebra to which a quasitriangular hopf algebra maps

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

on the addition of quantum matrices an addition law is introduced for the usual quantum matrices a r by means of a coaddition t t 1 1 t it supplements the usual comultiplication t t t and together they obey a codistributivity condition the coaddition does not form a usual hopf algebra but a braided one the same remarks apply for rectangular m n quantum matrices as an application left invariant vector fields are constructed on a r and other quantum spaces they close in the form of a braided lie algebra as another application the wave functions in the lattice approximation of kac moody algebras and other lattice fields can be added and functionally differentiated

a noncommutative hopf structure on c sl 2 c as a quantum lorentz group we introduce a classical frechet hopf algebra a containing the quantum deformation of the enveloping lorentz algebra of ogievestky and wess et al as a dense hopf subalgebra we also show that a is isomorphic to a quasitriangular twisted topological tensor product hopf algebra by duality we recover on the topological dual space of a the non commutative algebra structure introduced by podles and woronowicz finally this non commutative product can be extended to c sl 2 c which then becomes a topological hopf algebra for its natural frechet topology

the solvability of fuzzy fractional partial differential equations under caputo gh differentiability the foundation of the concepts of fuzzy fractional integral and caputo gh partial for fuzzy valued multivariable functions is defined as a result fuzzy fractional partial differential equations are considered and the appropriateness of local boundary value problems for hyperbolic equations is proved we present two new results on the existence of two kinds of gh weak solutions of these problems the first result is based on the banach fixed point theorem with the lipschitz condition of functions on the right hand side of equations the second result is based on the nonlinear alternative of the schauder type for fuzzy valued continuous functions without the lipschitz condition moreover we indicate the boundedness and continuous dependence of solutions on the initial data of the problems our models are embedded in the sense of caputo gh differentiability some examples are presented to illustrate the results

examples of braided groups and braided matrices matrix braided groups are developed as an analog of the coordinate functions on a group or supergroup the 1 in the super case is replaced by braid statistics there are braided group analogs of all the classical simple lie groups as well as braided matrix groups and braided matrices b r for every regular solution r of the quantum yang baxter equations a direct verification of b r is provided and some of the simplest examples are computed in detail

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

q deformed poincare algebra theq differential calculus for theq minkowski space is developed the algebra of theq derivatives with theq lorentz generators is found giving theq deformation of the poincare algebra the reality structure of theq poincare algebra is given the reality structure of theq differentials is also found the real laplacian is constructed finally the comultiplication counit and antipode for theq poincare algebra are obtained making it a hopf algebra

day to day dynamic network disequilibria and idealized traveler information systems in this paper we present tatonnement models for calculating static wardropian user equilibria on congested networks with fully general demand and cost structures we present both a qualitative analysis of stability and numerical studies which show that such an approach provides a reliable means for determining static user equilibria we also describe circumstances for which these models depict day to day adjustments from one realizable disequilibrium state to another and how these adjustment processes differ depending on the quality of the information being provided by abstract traveler information systems specifically we demonstrate that such dynamic adjustment processes settle down to equilibria both when information is complete and when it is incomplete

random fuzzy fractional integral equations theoretical foundations abstract this paper presents mathematical foundations for studies of random fuzzy fractional integral equations which involve a fuzzy integral of fractional order we consider two different kinds of such equations their solutions have different geometrical properties the equations of the first kind possess solutions with trajectories of nondecreasing diameter of their consecutive values on the other hand the solutions to equations of the second kind have trajectories with nonincreasing diameter of their consecutive values firstly the existence and uniqueness of solutions is investigated this is showed by using a method of successive approximations an estimation of error of n th approximation is given also a boundedness of the solution is indicated to show well posedness of the considered theory we prove that solutions depend continuously on the data of the equations some concrete examples of random fuzzy fractional integral equations are solved explicitly

kappa deformation of poincare superalgebra with classical lorentz subalgebra and its graded bicrossproduct structure the kappa deformed d 4 poincare superalgebra written in hopf superalgebra form is transformed to the basis with classical lorentz subalgebra generators we show that in such a basis the kappa deformed d 4 poincare superalgebra can be written as a graded bicrossproduct we show that the kappa deformed d 4 superalgebra acts covariantly on a kappa deformed chiral superspace

c statistical quantum groups and weyl algebras c statistical quantum groups and algebras are obtained by a process of transmutation they are like super quantum groups and algebras but with statistics given now by a complex number among the examples it is shown that the weyl algebra of canonical quantization may be viewed equivalently as a c statistical plane related examples are the noncommutative torus and a double loop variable quantization of photons these results lead toward a reformulation of ordinary quantum mechanics as a classical theory with c statistics in this reformulation the role of the 1 factors of super statistics is played by the free particle time evolution

quantum and braided linear algebra quantum matrices a r are known for every r matrix obeying the quantum yang baxter equations it is also known that these act on vectors given by the corresponding zamalodchikov algebra this interpretation is developed in detail distinguishing between two forms of this algebra v r vectors and v r covectors a r v r21 v r is an algebra homomorphism i e quantum matrices are realized by the tensor product of a quantum vector with a quantum covector while the inner product of a quantum covector with a quantum vector transforms as a scaler it is shown that if v r and v r are endowed with the necessary braid statistics then their braided tensor product v r v r is a realization of the braided matrices b r introduced previously while their inner product leads to an invariant quantum trace introducing braid statistics in this way leads to a fully covariant quantum braided linear algebra the braided groups obtained from b r act on themselves by conjugation in a way impossible

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h

differential inclusions for fuzzy maps we introduce the problems of differential inclusions for fuzzy maps and study the existence of solutions to these problems by the continuous selection theorem and fixed point theorems respectively