analytical solutions for a nonlinear diffusion equation with convection and reaction abstract nonlinear diffusion equations with the convection and reaction terms are solved by using a power law ansatz this kind of equations typically appears in nonlinear problems of heat and mass transfer and flows in porous media the solutions that we introduce in this work are analytical at least in the convection case the result recovers its linear form as a special limit in the reaction case we define a class of nonlinearity to discuss the evolution of general solutions we also add the verhulst like dynamics and global regulation we think this method based on this kind of ansatz can also be applied to solve other nonlinear partial differential equations

basic hopf algebras and quantum groups this paper investigates the structure of basic finite dimensional hopf algebras h over an algebraically closed field k the algebra h is basic provided h modulo its jacobson radical is a product of the field k in this case h is isomorphic to a path algebra given by a finite quiver with relations necessary conditions on the quiver and on the coalgebra structure are found in particular it is shown that only the quivers gamma g w given in terms of a finite group g and sequence w w 1 w 2 ldots w r of elements of g in the following way can occur the quiver gamma g w has vertices v g g in g and arrows a i g colon v g 1 to v w ig 1 mid g in g w i in w where the set w 1 w 2 ldots w r is closed under conjugation with elements in g and for each g in g the sequences w and gw 1g 1 gw 2g 1 ldots gw rg 1 are the same up to a permutation we show how k gamma g w is a kg bimodule and study properties of the left and right actions of g on the path algebra furthermore it is shown that the conditions we find can be used to give the path algebras k gamma g w themselves a hopf algebra structure for an arbitrary field k the results are also translated into the language of coverings finally a new class of finite dimensional basic hopf algebras are constructed over a not necessarily algebraically closed field most of which are quantum groups the construction is not characteristic free all the quivers gamma g w where the elements of w generates an abelian subgroup of g are shown to occur for finite dimensional hopf algebras the existence of such algebras is shown by explicit construction for closely related results of cibils and rosso see ci r

newton s laws of motion in the form of a riccati equation we discuss two applications of a riccati equation to newton s laws of motion the first one is the motion of a particle under the influence of a power law central potential v r kr ensuremath epsilon for zero total energy we show that the equation of motion can be cast in the riccati form we briefly show here an analogy to barotropic friedmann robertson lemaitre cosmology where the expansion of the universe can be also shown to obey a riccati equation a second application in classical mechanics where again the riccati equation appears naturally are problems involving quadratic friction we use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems

monotone and convex restrictions of continuous functions abstract suppose that f belongs to a suitably defined complete metric space c of holder functions defined on 0 1 we are interested in whether one can find large in the sense of hausdorff or lower upper minkowski dimension sets a 0 1 such that f a is monotone or convex concave some of our results are about generic functions in c like the following one we prove that for a generic f c 1 0 1 0 2 for any a 0 1 such that f a is convex or concave we have dim h a dim m a max 0 1 on the other hand we also have some results about all functions belonging to a certain space for example the previous result is complemented by the following one for 1 2 for any f c 0 1 there is always a set a 0 1 such that dim h a 1 and f a is convex or concave on a

two characterizations of finite quasi hopf algebras abstract let h be a finite dimensional quasibialgebra we show that h is a quasi hopf algebra if and only if the monoidal category of its finite dimensional left modules is rigid if and only if a structure theorem for hopf modules over h holds

finiteness conditions co frobenius hopf algebras and quantum groups abstract some finiteness conditions for infinite dimensional coalgebras particularly right or left semiperfect coalgebras or co frobenius hopf algebras are studied as well examples of co frobenius hopf algebras are constructed via a hopf algebra structure on an ore extension of a group algebra and it is shown that bicrossproducts of co frobenius hopf algebras are themselves co frobenius

population dynamics with global regulation the conserved fisher equation we introduce and study a conserved version of the fisher equation within a population biology context this model describes spatially extended populations in which the total number of individuals is fixed due to either biotic or environmental factors we find a rich spectrum of dynamical phases including a pseudotraveling wave and in the presence of the allee effect a phase transition from a locally constrained high density state to a low density fragmented state

a wave function for stock market returns the instantaneous return on the financial times stock exchange ftse all share index is viewed as a frictionless particle moving in a one dimensional square well but where there is a non trivial probability of the particle tunneling into the well s retaining walls our analysis demonstrates how the complementarity principle from quantum mechanics applies to stock market prices and of how the wave function presented by it leads to a probability density which exhibits strong compatibility with returns earned on the ftse all share index in particular our analysis shows that the probability density for stock market returns is highly leptokurtic with slight though not significant negative skewness moreover the moments of the probability density determined under the complementarity principle employed here are all convergent in contrast to many of the probability density functions on which the received theory of finance is based

an approach to quasi hopf algebras via frobenius coordinates abstract we study quasi hopf algebras and their subobjects over certain commutative rings from the point of view of frobenius algebras we introduce a type of radford formula involving an anti automorphism and the nakayama automorphism of a frobenius algebra then view several results in quantum algebras from this vantage point in addition separability and strong separability of quasi hopf algebras are studied as frobenius algebras

galois extensions for co frobenius hopf algebras abstract forhan infinite dimensional co frobenius hopf algebra over a fieldk andaanh comodule algebra the smash producta h ratis linked to the ring of coinvariantsacohby a morita context we use the morita setting to show that for co frobeniush equivalent conditions fora acohto be galois parallel the case ofhfinite dimensional

a family of analytical solutions of a nonlinear diffusion convection equation abstract despite its popularity in many engineering fields the nonlinear diffusion convection equation has no general analytical solutions this work presents a family of closed form analytical traveling wave solutions for the nonlinear diffusion convection equation with power law nonlinearities this kind of equations typically appears in nonlinear problems of flow and transport in porous media the solutions that are addressed are simple and fully analytical three classes of analytical solutions are presented depending on the type of the nonlinear diffusion coefficient increasing decreasing or constant it has shown that the structure of the traveling wave solution is strongly related to the diffusion term the main advantage of the proposed solutions is that they are presented in a unified form contrary to existing solutions in the literature where the derivation of each solution depends on the specific values of the diffusion and convection parameters the proposed closed form solutions are simple to use do not require any numerical implementation and may be implemented in a simple spreadsheet the analytical expressions are also useful to mathematically analyze the structure and properties of the solutions

convexity conditions and intersections with smooth functions a continuous function that agrees with each member of a family sof smooth functions in a small set must itself possess certain desirable properties we study situations that arise when sconsists of the family of polynomials of degree at most n as well as certain larger families and when the small sets of agreement are finite the conclusions of our theorems involve convexity conditions for example if a continuous function f agrees with each polynomial of degree at most n in only a finite set then f is n 1 convex or n 1 concave on some interval we consider also certain variants of this theorem provide examples to show that certain improvements are not possible and present some applications of our results

a quiver quantum group we construct quantum groups at a root of unity and we describe their monoidal module category using techniques from the representation theory of finite dimensional associative algebras

deriving general solutions of the heat transfer and the fokker planck boundary value problems through modified separation methods abstract we present modified separation methods to solve the heat equation which provide the general solutions in terms of a non separable basis this basis not only covers the set of the hermite polynomials but also a more general set of confluent hypergeometric functions which are known from the similarity solution of the heat equation modifying the closed form solutions of the heat equation with an exponential weight we derive corresponding partial differential equations which include simple cases of the fokker planck equation their closed form solutions are thus provided

constructing quantum groups and hopf algebras from coverings abstract given a group graded free associative algebra we show that in many cases the path algebra associated to the covering coming from the grading has a hopf algebra structure our structure on the path algebra is that of a quantum group far most of these constructions adding more restrictions we create some finite dimensional quotients which inherit the structure of a hoof algebra usually a quantum group from the path algebra

on the central charge of a factorizable hopf algebra abstract for a semisimple factorizable hopf algebra over a field of characteristic zero we show that the value that an integral takes on the inverse drinfel d element differs from the value that it takes on the drinfel d element itself by at most a fourth root of unity this can be reformulated by saying that the central charge of the hopf algebra is an integer if the dimension of the hopf algebra is odd we show that these two values differ by at most a sign which can be reformulated by saying that the central charge is even we give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs to formulate our results we use the language of modular data

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

quiver approaches to quasi hopf algebras we provide a quiver setting for quasi hopf algebras generalizing the hopf quiver theory as applications we obtain some general structure theorems in particular the quasi hopf analog of the cartier theorem and the cartier gabriel decomposition theorem

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

co frobenius hopf algebras integrals doi koppinen modules and injective objects abstract we investigate hopf algebras with non zero integral from a coalgebraic point of view categories of doi koppinen modules are studied in the special case where the defining coalgebra is left and right semiperfect and several pairs of adjoint functors are constructed as applications we give a very short proof for the uniqueness of the integrals and provide information about injective objects in the category of doi koppinen modules

frobenius and separable functors for the category of entwined modules over cowreaths ii applications abstract let h be a quasi hopf algebra we apply results obtained in 8 to give necessary and sufficient conditions for the forgetful functor from doi hopf modules two sided hopf modules or yetter drinfeld modules over h to representations of the underlying algebra to be frobenius resp separable we show that in some situations these conditions reduce to the unimodularity and or co semisimplicity of the quasi hopf algebra h

bosonization for dual quasi bialgebras and preantipode abstract to every dual quasi bialgebra h and every bialgebra r in the category of yetter drinfeld modules over h one can associate a dual quasi bialgebra called bosonization in this paper using the fundamental theorem we characterize as bosonizations the dual quasi bialgebras with a projection onto a dual quasi bialgebra with a preantipode as an application we investigate the structure of the graded coalgebra gr a associated to a dual quasi bialgebra a with the dual chevalley property e g a is pointed

bialgebra cohomology deformations and quantum groups abstract we introduce cohomology and deformation theories for a bialgebra a over a commutative unital ring k such that the second cohomology group is the space of infinitesimal deformations our theory gives a natural identification between the underlying k modules of the original and the deformed bialgebra certain explicit deformation formulas are given for the construction of quantum groups i e hopf algebras that are neither commutative nor cocommutative whether or not they arise from quantum yang baxter operators these formulas yield in particular all glq n and slq n as deformations of gl n and sl n using a hodge decomposition of the underlying cochain complex we compute our cohomology for gl n with this we show that every deformation of gl n is equivalent to one in which the comultiplication is unchanged not merely on elements of degree one but on all elements settling in the strongest way a decade old conjecture and in which the quantum determinant as an element of the underlying k module is identical with the usual one

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

general method to solve the heat equation abstract general solutions of the heat equation are presented in terms of the koopman darmois family of exponential functions which include both the separable solution and the fundamental solution in particular we derive a new closed form solution which may not be obtained via the separation of variables or via an integral transform it is demonstrated that the new solution describes the time evolution of the distribution of random walkers under an absorbing boundary

relative hopf modules for dual quasi hopf algebras univ bucharest fac math ro 70109 bucharest 1 romania limburgs univ ctr dept math b 3590 diepenbeek belgium bulacu d univ bucharest fac math str acad 14 ro 70109 bucharest 1 romania

general solutions of the heat equation abstract we derive a general solution of the heat equation through the use of the similarity reduction method the obtained solution is expressed as linearly combined kernel solutions in terms of the hermite polynomials which appears to provide an explanation of non gaussian behavior observed in various cases as examples we consider a few typical boundary conditions and construct corresponding solutions demonstrating the versatile applicability of our scheme it is thus revealed that the heat equation carries many solutions under given boundary conditions the entropy borne by a non gaussian solution is also computed and shown to approach in the long time limit the maximum one corresponding to the fundamental gaussian solution

monomial hopf algebras abstract let k be a field of characteristic 0 containing all roots of unity we classified all the hopf structures on monomial k coalgebras or in dual version on monomial k algebras

emergence of heavy tailed skew distributions from the heat equation abstract it is well known that the symmetric gaussian function called the fundamental solution serves as the green s function of the heat equation in reality on the other hand distribution functions obtained empirically often differ from the gaussian function this study presents a new solution of the heat equation satisfying localized initial conditions like the gaussian fundamental solution the new solution corresponds to a hetero mixture distribution which generalizes the gaussian distribution function to a skewed and heavy tailed distribution and thus provides a candidate for the empirical distribution functions

a family of evolution equations with nonlinear diffusion verhulst growth and global regulation exact time dependent solutions abstract a family of evolution equations describing a power law nonlinear diffusion process coupled with a local verhulst like growth dynamics and incorporating a global regulation mechanism is considered these equations admit an interpretation in terms of population dynamics and are related to the so called conserved fisher equation exact time dependent solutions exhibiting a maximum nonextensive q entropy shape are obtained

quivers quasi quantum groups and finite tensor categories we study finite quasi quantum groups in their quiver setting developed recently by the first author we obtain a classification of finite dimensional pointed majid algebras of finite corepresentation type or equivalently a classification of elementary quasi hopf algebras of finite representation type over the field of complex numbers by the tannaka krein duality principle this provides a classification of the finite tensor categories in which every simple object has frobenius perron dimension 1 and there are finitely many indecomposable objects up to isomorphism some interesting information of these finite tensor categories is given by making use of the quiver representation theory

a quantum model for the stock market abstract beginning with several basic hypotheses of quantum mechanics we give a new quantum model in econophysics in this model we define wave functions and operators of the stock market to establish the schrodinger equation for stock price based on this theoretical framework an example of a driven infinite quantum well is considered in which we use a cosine distribution to simulate the state of stock price in equilibrium after adding an external field into the hamiltonian to analytically calculate the wave function the distribution and the average value of the rate of return are shown

ulam zahorski problem on free interpolation by smooth functions let f be a function belonging to cn o 1 is it possible to find a smoother function g e cn 1 or at least cn 8 which has infinitely many points of contact of maximal order n with f or at least arbitrarily many such points with fixed norm iigiicn e it turns out that for n 0 and 1 the answer is positive but if n 2 it is negative this gives a complete solution to the ulam zahorski question on free interpolation on perfect sets

doubles of quasi quantum groups in dr1 drinfeld showed that any finite dimensional hopf algebra extends to a quasitriangular hopf algebra the quantum double of based on the construction of a so called diagonal crossed product developed by the authors in hn we generalize this result to the case of quasi hopf algebras as for ordinary hopf algebras as a vector space the quasi quantum double is isomorphic to where denotes the dual of we give explicit formulas for the product the coproduct the r matrix and the antipode on and prove that they fulfill drinfeld s axioms of a quasitriangular quasi hopf algebra in particular becomes an associative algebra containing as a quasi hopf subalgebra on the other hand is not a subalgebra of unless the coproduct on is strictly coassociative it is shown that the category of finite dimensional representations of coincides with what has been called the double category of modules by s majid m2 thus our construction gives a concrete realization of majid s abstract definition of quasi quantum doubles in terms of a tannaka krein like reconstruction procedure the whole construction is shown to generalize to weak quasi hopf algebras with now being linearly isomorphic to a subspace of

quantum brownian motion model for the stock market abstract it is believed by the majority today that the efficient market hypothesis is imperfect because of market irrationality using the physical concepts and mathematical structures of quantum mechanics we construct an econophysical framework for the stock market based on which we analogously map massive numbers of single stocks into a reservoir consisting of many quantum harmonic oscillators and their stock index into a typical quantum open system a quantum brownian particle in particular the irrationality of stock transactions is quantitatively considered as the planck constant within heisenberg s uncertainty relationship of quantum mechanics in an analogous manner we analyze real stock data of shanghai stock exchange of china and investigate fat tail phenomena and non markovian behaviors of the stock index with the assistance of the quantum brownian motion model thereby interpreting and studying the limitations of the classical brownian motion model for the efficient market hypothesis from a new perspective of quantum open system dynamics

integrals for dual quasi hopf algebras applications abstract a classical result in the theory of hopf algebras concerns the uniqueness and existence of integrals for an arbitrary hopf algebra the integral space has dimension 1 and for a finite dimensional hopf algebra this dimension is exactly one we generalize these results to quasi hopf algebras and dual quasi hopf algebras in particular it will follow that the bijectivity of the antipode follows from the other axioms of a finite dimensional quasi hopf algebra we give a new version of the fundamental theorem for quasi hopf algebras we show that a dual quasi hopf algebra is co frobenius if and only if it has a non zero integral in this case the space of left or right integrals has dimension one

quiver hopf algebras abstract in this paper we study subhopfalgebras of graded hopf algebra structures on a path coalgebra k q c we prove that a hopf structure on some subhopfquivers can be lifted to a hopf structure on the whole hopf quiver if q is a schurian hopf quiver then we classified all simple pointed subhopfalgebras of a graded hopf structure on k q c we also prove a dual gabriel theorem for pointed hopf algebras

general solution of the black scholes boundary value problem abstract the black scholes formula for a european option price which resulted in the 1997 nobel prize in economic sciences is known to be the unique solution of the boundary value problem consisting of the black scholes partial differential equation and the terminal condition defined by the european call option this has been one of the most popular tools of finance in theory as well as in practice here we present infinitely many solutions of the boundary value problem involving hermite polynomials this indicates that the black scholes boundary value problem violates the law of one price which is one of the fundamental concepts in economics