free products of compact quantum groups we construct and study compact quantum groups from free products ofc algebras in this connection we discover two mysterious classes of natural compact quantum groups a u m anda o m thea u m s respectivelya o m s are non isomorphic to each other for differentm s and are not obtainable by the ordinary quantization method we also clarify some basic concepts in the theory of compact quantum groups

division theorem combined with the riccati equation for solving some nonlinear schrodinger like equations abstract in this paper division theorem is employed for obtaining the first integral with polynomial coefficients of two dimensional autonomous planar system which can construct the periodic and solitary wave solutions of nonlinear partial differential equations by using the solutions of the riccati equation the power of this manageable method is presented by applying it to some nonlinear schrodinger like equations this approach can also be applied to other nonlinear differential equations

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

a new general sufficient condition for almost sure stability of jump linear systems in this paper we study the almost sure stability of discrete time jump linear systems with a finite state markov form process a general condition for almost sure stability which is a necessary and sufficient condition for scalar one dimensional systems is derived many simpler testable sufficient conditions for almost sure stability are derived from this sufficient condition

ccap for universal discrete quantum groups we show that the discrete duals of the free orthogonal quantum groups have the haagerup property and the completely contractive approximation property analogous results hold for the free unitary quantum groups and the quantum automorphism groups of finite dimensional c algebras the proof relies on the monoidal equivalence between free orthogonal quantum groups and su q 2 quantum groups on the construction of a sufficient supply of bounded central functionals for su q 2 quantum groups and on the free product techniques of ricard and xu our results generalize previous work in the kac setting due to brannan on the haagerup property and due to the second author on the ccap

the extended tanh method for abundant solitary wave solutions of nonlinear wave equations the extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations the obtained solutions include solitons kinks and plane periodic solutions the study is an extension to the remarkable development by malfliet 1 the extended tanh method presents a wider applicability for handling nonlinear wave equations

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 boundary of universal discrete quantum groups exactness and factoriality we study the c algebras and von neumann algebras associated with the universal discrete quantum groups they give rise to full prime factors and simple exact c algebras the main tool in our work is the study of an amenable boundary action yielding the akemann ostrand property finally this boundary can be identified with the martin or the poisson boundary of a quantum random walk

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

symmetries of levy processes on compact quantum groups their markov semigroups and potential theory quantum markov semigroups qms i e strongly continuous semigroups of unital completely positive maps on compact quantum groups are studied we show that translation invariant qmss on the universal or reduced c algebra of a compact quantum group are in one to one correspondence with levy processes on its hopf algebra we use the theory of levy processes on involutive bialgebras to characterize symmetry properties of the associated qms it turns out that the qms is self adjoint resp kms symmetric if and only if the generating functional of the levy process is invariant under the antipode resp the unitary antipode furthermore we study levy processes whose marginal states are invariant under the adjoint action finally some related aspects of the potential theory as dirichlet form derivation and spectral triple are investigated we discuss how the above results apply to compact groups group c algebras of discrete groups free orthogonal quantum groups on and twisted suq 2 quantum group

the tanh coth method for solitons and kink solutions for nonlinear parabolic equations abstract the tanh coth method is used to derive solitons and kink solutions for some of the well known nonlinear parabolic partial differential equations the equations include the fisher equation newell whithead equation allen cahn equation fitzhugh nagumo equation fisher s equation and the burgers fisher equation the new tanh coth approach provides abundant solitons and kink solutions in addition to the existing ones the power of this manageable method is confirmed

the effect of perturbed advection on a class of solutions of a non linear reaction diffusion equation in this work the traveling wave solutions of a one dimensional reaction diffusion equation with advection are studied the traveling wave solutions are obtained using the g g expansion method the shock thickness and spectral stability have been discussed for the obtained solution in the parameter interval the essential spectra of the perturbed and linearized differential operator about the traveling antikink and kink solutions at the equilibrium states are obtained the point spectrum is calculated using evans function with lie midpoint method and magnus method it is shown that for a symmetric potential well the traveling kink and antikink solutions which connect the stable equilibrium states of the system are stable it is observed that the perturbation on the advection exhibits contrasting effect on the solution properties shock thickness and the eigenvalue of kink and antikink solutions variation of the reaction coefficient leads to instability of the solutions unlike the diffusion coefficient which enhances the stability on the other hand the variation of reaction and diffusion coefficients show the monotonic effect on the shock thickness of the traveling kink and antikink solutions this study is expected to be useful in analyzing the slow or fast invasion and stability of the population movement in different steady states

travelling wave solutions of the cahn allen equation by using first integral method abstract in this paper we established travelling wave solutions of the nonlinear equation the first integral method was used to construct travelling wave solutions of the cahn allen equation the obtained results include periodic and solitary wave solutions the power of this manageable method is confirmed

new approach for exact solutions of time fractional cahn allen equation and time fractional phi 4 equation abstract in this article new exact analytical solutions of some nonlinear evolution equations nlees arising in science engineering and mathematical physics namely time fractional cahn allen equation and time fractional phi 4 equation are developed using tanh method by means of fractional complex transform the obtained results are demonstrated by graphs for the new solutions

time fractional cahn allen and time fractional klein gordon equations lie symmetry analysis explicit solutions and convergence analysis abstract this research analyzes the symmetry analysis explicit solutions and convergence analysis to the time fractional cahn allen ca and time fractional klein gordon kg equations with riemann liouville rl derivative the time fractional ca and time fractional kg are reduced to respective nonlinear ordinary differential equation of fractional order we solve the reduced fractional odes using an explicit power series method the convergence analysis for the obtained explicit solutions are investigated some figures for the obtained explicit solutions are also presented

unique cartan decomposition for ii1 factors arising from arbitrary actions of free groups we prove that for any free ergodic probability measure preserving action mathbb f n curvearrowright x mu of a free group on n generators mathbb f n 2 leq n leq infty the associated group measure space ii1 factor l infty x rtimes mathbb f n has l x as its unique cartan subalgebra up to unitary conjugacy we deduce that group measure space ii1 factors arising from actions of free groups with different number of generators are never isomorphic we actually prove unique cartan decomposition results for ii1 factors arising from arbitrary actions of a much larger family of groups including all free products of amenable groups and their direct products

the hyperbolic allen cahn equation exact solutions using the first integral method a general set of analytical solutions is obtained for the hyperbolic allen cahn equation the solutions are presented by i the class of continual solutions described by profiles for traveling waves of the order parameter and ii the class of singular solutions which exhibit unbounded discontinuity in the profile of the order parameter at the origin of the coordinate system it is shown that the solutions include the previous analytical results for the parabolic allen cahn equation as a limited class of functions in which the inertial effects are omitted

amenability and co amenability of algebraic quantum groups ii abstract we continue our study of the concepts of amenability and co amenability for algebraic quantum groups in the sense of a van daele and our investigation of their relationship with nuclearity and injectivity one major tool for our analysis is that every non degenerate representation of the universal c algebra associated to an algebraic quantum group has a unitary generator which may be described in a concrete way

an analytic family of uniformly bounded representations of free groups harmonic analysis on a free group f has attracted considerable attention in the last ten years or so there seem to be two reason for that one is the discovery of deep analogy of certain aspects of harmonic analysis on a free group and harmonic analysis on sl 2 r cf e g fundamental works of p cartier 4 a figa talamanca and m a picardello 8 the other being the interest in the c algebra generated by the regular representation of f cf a connes 5 j cuntz 6 u haagerup 10 m pimsner and d voiculescu 14

the baum connes conjecture for free orthogonal quantum groups we prove an analogue of the baum connes conjecture for free orthogonal quantum groups more precisely we show that these quantum groups have a element and that 1 it follows that free orthogonal quantum groups are k amenable we compute explicitly their k theory and deduce in the unimodular case that the corresponding reduced c algebras do not contain nontrivial idempotents our approach is based on the reformulation of the baum connes conjecture by meyer and nest using the language of triangulated categories an important ingredient is the theory of monoidal equivalence of compact quantum groups developed by bichon de rijdt and vaes this allows us to study the problem in terms of the quantum group suq 2 the crucial part of the argument is a detailed analysis of the equivariant kasparov theory of the standard podle

the first integral method for the time fractional kaup boussinesq system with time dependent coefficient the first integral method fim is applied to reach the different type solutions of the time fractional kaup boussinesq fkb system with time dependent coefficient it is obtained the envelope bell shaped triangular and kink soliton solutions of this equation the method is an effective one to reach the different types of solutions of fractional nonlinear partial differential equations fnpde

the first integral method to study the burgers korteweg de vries equation in this paper applying the theory of commutative algebra we propose a new approach which we currently call the first integral method to study the burgers korteweg de vries equation

exact solutions of some nonlinear systems of partial differential equations by using the first integral method abstract in recent years many approaches have been utilized for finding the exact solutions of nonlinear systems of partial differential equations in this paper the first integral method introduced by feng is adopted for solving some important nonlinear systems of partial differential equations including kdv kaup boussinesq and wu zhang systems analytically by means of this method some exact solutions for these systems of equations are formally obtained the results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations

ergodic coactions with large multiplicity and monoidal equivalence of quantum groups we construct new examples of ergodic coactions of compact quantum groups in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter these examples are obtained using a bijective correspondence between certain ergodic coactions on c algebras and unitary fiber functors on the representation category of a compact quantum group we classify these unitary fiber functors on the universal orthogonal and unitary quantum groups the associated c algebras and von neumann algebras can be defined by generators and relations but are not yet well understood

irreducible unitary representations of quantum lorentz group a complete classification of irreducible unitary representations of a one parameter deformations q l 2 c 0 q 1 ofsl 2 c is given it shows that in spite of a popular belief the representation theory fors q l 2 c is not a smooth deformation of the one forsl 2 c

amenability of hopf von neumann algebras and kac algebras abstract let m be a hopf von neumann algebra the operator predualm ofmis a completely contractive banach algebra with multiplicationm m m m we call m operator amenableif the completely contractive banach algebram isoperator amenable i e for every operatorm bimodulev every completely bounded derivation fromm into the dualm bimodulev is inner there is a weaker notion of amenability introduced by d voiculescu we say that a hopf von neumann algebra m isvoiculescu amenableif there exists a left invariant mean onm we show that if a hopf von neumann algebra m is operator amenable then it is voiculescu amenable for kac algebras there is astrong voiculescu amenability we show that for discrete kac algebras these amenabilities are all equivalent in fact if we letk m be a discrete kac algebra and letk m be its compact dual kac algebra then the following are equivalent 1 kis operator amenable 2 kis voiculescu amenable 3 the von neumann algebramis hyperfinite 4 kis strong voiculescu amenable 5 kis operator amenable 6 m has a bounded approximate identity

an algebraic framework for group duality abstract a hopf algebra is a pair a whereais an associative algebra with identity and a homomorphism formatoa asatisfying certain conditions if we drop the assumption thatahas an identity and if we allow to have values in the so called multiplier algebram a a we get a natural extension of the notion of a hopf algebra we call this a multiplier hopf algebra the motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group also for these multiplier hopf algebras there is a natural notion of left and right invariance for linear functionals called integrals in hopf algebra theory we show that if such invariant functionals exist they are unique up to a scalar and faithful for a regular multiplier hopf algebra a i e with invertible antipode with invariant functionals we construct in a canonical way the dual it is again a regular multiplier hopf algebra with invariant functionals it is also shown that the dual of is canonically isomorphic with the original multiplier hopf algebra a it is possible to generalize many aspects of abstract harmonic analysis here one can define the fourier transform one can prove plancherel s formula because any finite dimensional hopf algebra is a regular multiplier hopf algebra and has invariant functionals our duality theorem applies to all finite dimensional hopf algebras then it coincides with the usual duality for such hopf algebras but our category of multiplier hopf algebras also includes in a certain way the discrete quantum groups and the compact quantum groups our duality includes the duality between discrete quantum groups and compact quantum groups in particular it includes the duality between compact abelian groups and discrete abelian groups one of the nice features of our theory is that we have an extension of this duality to the non abelian case but within one category this is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type we prove that also these are dual to each other we treat an example that is sufficiently general to illustrate most of the different features of our theory it is also possible to construct the quantum double of drinfel d within this category this provides a still wider class of examples so we obtain many more than just the compact and discrete quantum within this setting

completely bounded multipliers of the fourier algebra of a simple lie group of real rank one the fourier algebra a g of a locally compact group g is the space of matrix coefficients of the regular representation and is the predual of the yon neumann algebra vn g generated by the regular representation of g on l 2 g a multiplier m of a g is a bounded operator on a g given by pointwise multiplication by a function on g also denoted m we say m is a completely bounded multiplier ofa g if the transposed operator on vn g is completely bounded definition below it may be possible to find a net ofa g functions m i ie i say such that mi tends to

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations