adaptive finite element methods for parabolic problems vi analytic semigroups we continue our work on adaptive finite element methods with a study of time discretization of analytic semigroups we prove optimal a priori and a posteriori error estimates for the discontinuous galerkin method showing in particular that analytic semigroups allow long time integration without error accumulation

quantum spinors and spin groups from quantum clifford algebras a general construction of multiparametric quantum spinors and corresponding quantum groups associated to pseudo euclidean spaces is presented and their homomorphism to the respective groups is discussed this construction is based on a quantum clifford algebra and is described in detail for involutive pure twists intertwining braids for general braid operators that admit abstract volume elements a procedure is also given for deriving quantum analogues of these groups

rieffel deformation via crossed products we start from rieffel data a where a is a c algebra is an action of an abelian group on a and is a 2 cocycle on the dual group using landstad theory of crossed product we get a deformed c algebra a in the case of rn we obtain a very simple proof of invariance of k groups under the deformation in the general case we also get a very simple proof that nuclearity is preserved under the deformation we show how our approach leads to quantum groups and investigate their duality the general theory is illustrated by an example of the deformation of sl 2 c a description of it in terms of noncommutative coordinates is given

kawada ito kelley theorem for quantum semigroups abstract idempotent states on locally compact quantum semigroups with weak cancellation properties are shown to be haar states on a certain sub object described by an operator system with comultiplication we also give a characterization of the situation when this sub object is actually a compact quantum subgroup in particular we reproduce classical results on idempotent probability measures on locally compact semigroups with cancellation

wkb equivalent potentials for q deformed harmonic and anharmonic oscillators wkb equivalent potentials wkb ep s giving the same wkb spectrum as the q deformed harmonic oscillator are determined in analytic form for q being complex the wkb ep resembles the poschl teller morse or woods saxon potentials widely used in nuclear hypernuclear and molecular physics for q real the wkb ep still goes to infinity at large x while at small x it resembles a sextic oscillator the wkb ep s of the anharmonic oscillators with uq 2 or suq 1 1 symmetries used for the description of vibrational molecular spectra are also calculated they correspond to q deformed versions of the modified poschl teller potential

quantum planes and quantum cylinders from poisson homogeneous spaces quantum planes and a new quantum cylinder are obtained as quantization of poisson homogeneous spaces of two different poisson structures on classical euclidean group e 2

c algebraic partial compact quantum groups abstract in this paper we introduce c algebraic partial compact quantum groups which are quantizations of topological groupoids with discrete object set and compact morphism spaces these c algebraic partial compact quantum groups are generalizations of hayashi s compact face algebras to the case where the object set can be infinite they form the c algebraic counterpart of an algebraic theory of partial compact quantum groups developed in an earlier paper by the author and t timmermann the correspondence between which will be dealt with in a separate paper as an interesting example to illustrate the theory we show how the dynamical quantum su 2 group as studied by etingof varchenko and koelink rosengren fits into this framework

on the stability of the l 2 projection in h 1 ω we prove the stability in h1 of the l2 projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions we give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions in particular stability of the l2 projection in h1 holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically

a noncommutative de finetti theorem for boolean independence abstract we introduce a family of quantum semigroups and its natural coactions on noncommutative polynomials we define three invariance conditions for the joint distribution of sequences of selfadjoint noncommutative random variables associated with these coactions for one of the invariance conditions we show that the joint distribution of an infinite sequence of noncommutative random variables satisfy it is equivalent to the fact that the sequence of the random variables is identically distributed and boolean independent with respect to the conditional expectation onto its tail algebra this is a boolean analogue of de finetti theorem on exchangeable sequences in the end of the paper we also discuss the other two invariance conditions which lead to some trivial results

isoq 3 and isoq 2 1 we prove the embedding of isoq 3 isu qex 2 and isoq 2 1 islqex 2 r as algebras and give a hilbert space representation of isu qex 2

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 local global principle for regular operators in hilbert c modules abstract hilbert c modules are the analogues of hilbert spaces where a c algebra plays the role of the scalar field with the advent of kasparov s celebrated kk theory they became a standard tool in the theory of operator algebras while the elementary properties of hilbert c modules can be derived basically in parallel to hilbert space theory the lack of an analogue of the projection theorem soon leads to serious obstructions and difficulties in particular the theory of unbounded operators is notoriously more complicated due to the additional axiom of regularity which is not easy to check in this paper we present a new criterion for regularity in terms of the hilbert space localizations of an unbounded operator we discuss several examples which show that the criterion can easily be checked and that it leads to nontrivial regularity results

exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in banach spaces new exponentially convergent algorithms for the operator exponential generated by a strongly positive operator a in a banach space x are proposed these algorithms are based on representations by a dunford cauchy integral along paths enveloping the spectrum of a combined with a proper quadrature involving a short sum of resolvents where the choice of the integration path dramatically affects desired features of the algorithms a parabola and a hyperbola are analyzed as the integration paths and scales of estimates of dependence on the smoothness of initial data i e of the initial vector and of the inhomogeneous right hand side are obtained one of the algorithms possesses an exponential convergence rate for the operator exponential e at for all t ge 0 including the initial point this allows one to construct an exponentially convergent algorithm for inhomogeneous initial value problems the algorithm is parallelizable it turns out that the resolvent must be modified in order to get numerically stable algorithms near the initial point the efficiency of the proposed method is demonstrated by numerical examples

basis of the identities of the matrix algebra of order two over a field of characteristic p 2 abstract in this paper we prove that the polynomial identities of the matrix algebra of order 2 over an infinite field of characteristic p 2 admit a finite basis we exhibit a finite basis consisting of four identities and in almost all cases for p we describe a minimal basis consisting of two identities the only possibilities for p where we do not exhibit minimal bases of these identities are p 3 and p 5 we show that when p 3 one needs at least three identities and we conjecture a minimal basis in this case in the course of the proof we construct an explicit basis of the vector space of the central commutator polynomials modulo the ideal of the identities of the matrix algebra of order two

analysis of fully discrete fem for miscible displacement in porous media with bear scheidegger diffusion tensor fully discrete galerkin finite element methods are studied for the equations of miscible displacement in porous media with the commonly used bear scheidegger diffusion dispersion tensor begin aligned d mathbf u gamma d m i mathbf u bigg alpha t i alpha l alpha t frac mathbf u otimes mathbf u mathbf u 2 bigg end aligned previous works on optimal order l infty 0 t l 2 norm error estimate required the regularity assumption nabla x partial td mathbf u x t in l infty 0 t l infty omega while the bear scheidegger diffusion dispersion tensor is only lipschitz continuous even for a smooth velocity field mathbf u in terms of the maximal l p regularity of fully discrete finite element solutions of parabolic equations optimal error estimate in l p 0 t l q norm and almost optimal error estimate in l infty 0 t l q norm are established under the assumption of d mathbf u being lipschitz continuous with respect to mathbf u

green function on the quantum plane a green function which can be called q analogous of the hankel function on the quantum plane eq2 eq 2 u 1 is constructed

discrete maximal regularity and the finite element method for parabolic equations maximal regularity is a fundamental concept in the theory of partial differential equations in this paper we establish a fully discrete version of maximal regularity for parabolic equations on a polygonal or polyhedral domain varomega we derive various stability results in the discrete l p 0 t l q varomega norms for the finite element approximation with the mass lumping to the linear heat equation our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of the result of dore and venni as an application optimal order error estimates in those norms are proved furthermore we study the finite element approximation for semilinear heat equations with locally lipschitz continuous nonlinear terms and offer a new method for deriving optimal order error estimates some interesting auxiliary results including discrete gagliardo nirenberg and sobolev inequalities are also presented

constructing the polynomlal identities and central identities of degree 9 of 3 3 matrices abstract we present an algorithm for computing an independent generating set for the multilinear identities and the multilinear central identities of the m m matrices over a field o of characteristic zero or a large enough prime then we use it to construct all the multilinear identities and all the multilinear central identities of degree m 3 o

on idempotent states on quantum groups idempotent states on a compact quantum group are shown to yield group like projections in the multiplier algebra of the dual discrete quantum group this allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the haar state on a finite quantum hypergroup a natural order structure on the set of idempotent states is also studied and some examples discussed

from multiplicative unitaries to quantum groups ii abstract it is shown that all important features of a c algebraic quantum group a defined by a modular multiplicative w depend only on the pair a rather than the multiplicative unitary operator w the proof is based on thorough study of representations of quantum groups as an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary without assuming existence of haar weights

discrete maximal parabolic regularity for galerkin finite element methods the main goal of the paper is to establish time semidiscrete and space time fully discrete maximal parabolic regularity for the time discontinuous galerkin solution of linear parabolic equations such estimates have many applications they are essential for example in establishing optimal a priori error estimates in non hilbertian norms without unnatural coupling of spatial mesh sizes with time steps

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

maximal regularity of fully discrete finite element solutions of parabolic equations we establish the maximal ell p regularity for fully discrete finite element solutions of parabolic equations with time dependent lipschitz continuous coefficients the analysis is based on a dis

stability analyticity and almost best approximation in maximum norm for parabolic finite element equations we consider semidiscrete solutions in quasi uniform finite element spaces of order o hr of the initial boundary value problem with neumann boundary conditions for a second order parabolic differential equation with time independent coefficients in a bounded domain in r n we show that the semigroup on l defined by the semidiscrete solution of the homogeneous equation is bounded and analytic uniformly in h we also show that the semidiscrete solution of the inhomogeneous equation is bounded in the space time l norm modulo a logarithmic factor for r 2 and we give a corresponding almost best approximation property

groetsch s representation of moore penrose inverses and ill posed problems in hilbert c modules abstract the explicit representation lim 0 1 t t 1 t for the moore penrose inverse of an operator t between hilbert spaces has been given by c w groetsch 1975 6 we obtain his formula for the moore penrose inverse of an unbounded operator between hilbert c modules ill posed problems with unbounded operator between hilbert c modules are also discussed

finite volume methods for a keller segel system discrete energy error estimates and numerical blow up analysis we consider the finite volume approximation for a non linear parabolic elliptic system which describes the aggregation of slime molds resulting from their chemotactic features called a simplified keller segel system first we present a linear finite volume scheme that satisfies both positivity and mass conservations which are important features of the original system we derive some inequalities on the discrete free energy then under some assumptions on the regularity of solution admissible mesh and a priori estimates of the discrete solution we establish error estimates in l p lp norm with a suitable p 2 p 2 for the two dimensional case in the last part of this paper we restrict our attention to the radially symmetric solution of chemotaxis system and we derive some inequalities concerned with the blow up phenomenon of numerical solution several numerical experiments are presented to verify the theoretical results

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

sums of regular self adjoint operators in hilbert c modules abstract we introduce a notion of weak anticommutativity for a pair s t of self adjoint regular operators in a hilbert c module e we prove that the sum s t of such pairs is self adjoint and regular on the intersection of their domains a similar result then holds for the sum s 2 t 2 of the squares we show that our definition is closely related to the connes skandalis positivity criterion in kk theory as such we weaken a sufficient condition of kucerovsky for representing the kasparov product our proofs indicate that our conditions are close to optimal

on c algebra of a semigroup of partial isometries abstract a c algebra associated to strongly continuous one parameter semigroups of partial isometries is introduced as a groupoid c algebra c g a one to one correspondence between non degenerate representations of c g and the semigroups is established it is proved also that c g contains an ideal j isomorphic to c 0 0 k and c g j c 0 r where k is the algebra of compact operators the c algebra c g is shown to be generated by one unbounded affiliated element whose image under each non degenerate representation is the infinitesimal generator of the corresponding semigroup

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

rieffel deformation of group coactions let g be a locally compact group gamma subset g an abelian subgroup and let be a continuous 2 cocycle on the dual group hat gamma let b be a c algebra and delta b in rm mor b b otimes rm c 0 g a continuous right coaction using rieffel deformation we can construct a quantum group rm c 0 g tilde psi otimes psi delta psi and the deformed c algebra b the aim of this paper is to present a construction of the continuous coaction delta b psi of the quantum group rm c 0 g tilde psi otimes psi delta psi on b the transition from the coaction b to its deformed counterpart delta b psi is nontrivial in the sense that delta b psi contains complete information about b in order to illustrate our construction we apply it to the action of the lorentz group on the minkowski space obtaining a c algebraic quantum minkowski space

operator representations of the real twisted canonical commutation relations the covariant differential calculus on the quantum vector space cnq leads to q deformations of the canonical commutation relations and of the weyl algebra if the deformation parameter q is of modulus one integrable well behaved representations of this twisted canonical commutation relations by self adjoint operators on a hilbert space are defined and studied all faithful irreducible integrable representations of the real q weyl algebra are classified up to unitary equivalence

quantumsu 2 ande 2 groups contraction procedure in 3 it was shown in the framework of deformed enveloping algebras that quantumsu 2 ande 2 groups are related by the contraction procedure we consider the same problem on thec level as a result we find a number of formulae coupling the comultiplications in quantumsu 2 ande 2 in particular we show that the comultiplications in both groups are implemented by partial isometries an unexpected feature of quantume 2 is discovered and the corresponding strange behavior of quantumsu 2 is described

q integrals on the quantum complex plane q integration is defined for the complex quantum plane the existence of classical and grassmanian limits of q integration is proved quantum versions of cauchy s and stokes theorems are formulated

crossed products of c algebras and spectral analysis of quantum hamiltonians we study spectral properties of a hamiltonian by analyzing the structure of certain c algebras to which it is affiliated the main tool we use for the construction of these algebras is the crossed product of abelian c algebras generated by the classical potentials by actions of groups we show how to compute the quotient of such a crossed product with respect to the ideal of compact operators and how to use the resulting information in order to get spectral properties of the hamiltonians this scheme provides a unified approach to the study of hamiltonians of anisotropic and many body systems including quantum fields

harmonic analysis on the quantum lorentz group using a continuation of 6j symbols of su q 2 with complex spins we give a new description of the unitary representations of sl q 2 and find explicit expressions for the characters of unitary representations of sl q 2 we prove a plancherel theorem for the quantum lorentz group

a locally compact quantum group analogue of the normalizer of su 1 1 in sl 2 c s l woronowicz proved in 1991 that quantum su 1 1 does not exist as a locally compact quantum group results by l i korogodsky in 1994 and more recently by woronowicz gave strong indications that the normalizer of su 1 1 in sl 2 is a much better quantization candidate than su 1 1 itself in this paper we show that this is indeed the case by constructing a new example of a unimodular locally compact quantum group depending on a parameter 0 q 1 that is a deformation of after defining the underlying von neumann algebra of we use a certain class of q hypergeometric functions and their orthogonality relations to construct the comultiplication the coassociativity of this comultiplication is the hardest result to establish we define the haar weight and obtain simple formulas for the antipode and its polar decomposition as a final result we produce the underlying c algebra of the proofs of all these results depend on various properties of q hypergeometric 1 1 functions

embeddable quantum homogeneous spaces abstract we discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups on the von neumann algebra level we recall an interesting duality for such objects studied earlier by m izumi r longo s popa for compact kac algebras and by m enock in the general case of locally compact quantum groups a definition of a quantum homogeneous space is proposed along the lines of the pioneering work of vaes on induction and imprimitivity for locally compact quantum groups the concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces among various examples we single out the quantum analog of the quotient of the cartesian product of a quantum group with itself by the diagonal subgroup analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles the former turns out to be an interesting application of the duality mentioned above

discrete maximal l p regularity for finite element operators let a h h 0 be a family of elliptic finite element operators let i 0 t and consider the problem u h t a hu h t f h t t in i u h 0 0 in this paper we show that for 1 0 is said to have discrete maximal l p regularity

stability and analyticity in maximum norm for simplicial lagrange finite element semidiscretizations of parabolic equations with dirichlet boundary conditions stability and analyticity estimates in maximum norm are shown for spatially discrete finite element approximations based on simplicial lagrange elements for the model heat equation with dirichlet boundary conditions the bounds are logarithm free and valid in arbitrary dimension and for arbitrary polynomial degree the work continues an earlier study by schatz et al 5 in which neumann boundary conditions were considered

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

jordan trialgebras and post jordan algebras abstract we compute minimal sets of generators for the s n modules n 4 of multilinear polynomial identities of arity n satisfied by the jordan product and the jordan diproduct resp pre jordan product in every triassociative resp tridendriform algebra these identities define jordan trialgebras and post jordan algebras jordan analogues of the lie trialgebras and post lie algebras introduced by dotsenko et al pei et al vallette loday we include an extensive review of analogous structures existing in the literature and their interrelations in order to identify the gaps filled by our two new varieties of algebras we use computer algebra linear algebra over finite fields representation theory of symmetric groups to verify in both cases that every polynomial identity of arity 6 is a consequence of those of arity 4 we conjecture that in both cases the next independent identities have arity 8 imitating the glennie identities for jordan algebras we formulate our results as a commutative square of operad morphisms which leads to the conjecture that the squares in a much more general class are also commutative

duality and conditional expectations in the nakajima mori zwanzig formulation we develop a new operator algebraic formulation of the nakajima mori zwanzig nmz method of projections the new theory is built upon rigorous mathematical foundations and it can be applied to both classical and quantum systems we show that a duality principle between the nmz formulation in the space of observables and in the state space can be established analogous to the heisenberg and schrodinger pictures in quantum mechanics based on this duality we prove that under natural assumptions the projection operators appearing in the nmz equation must be conditional expectations the proposed formulation is illustrated in various examples

generic 2 2 matrices in positive characteristic let r m k be the k algebra generated by the generic 2 2 matrices y 1 y m over a unitary commutative associative ring k our main result is that with a small class of exceptions for m a positive integer and p a prime the kernel of the natural homomorphism r m r m p coincides with pr m the only exceptions are for m 5 and p 2 when we give an explicit multilinear polynomial identity of degree 5 for the matrix algebra m 2 2 which does not follow from the spolynomial identities of m 2 2 this improves on schelter s construction of a non multilinear identity of this sort of degree 6 and drensky and tsiganchev s existence result for a multilinear identity such as we have found

a stable time discretizations preserve maximal parabolic regularity it is shown that for a parabolic problem with maximal l p regularity for 1 p infty the time discretization by a linear multistep method or runge kutta method has maximal ell p regularity uniformly in the stepsize if the method is a stable and satisfies minor additional conditions in particular the implicit euler method the crank nicolson method the second order backward difference formula bdf and the radau iia and gauss runge kutta methods of all orders preserve maximal regularity the proof uses weis characterization of maximal l p regularity in terms of r boundedness of the resolvent a discrete operator valued fourier multiplier theorem by blunck and generating function techniques that have been familiar in the stability analysis of time discretization methods since the work of dahlquist the a alpha stable higher order bdf methods have maximal ell p regularity under an r boundedness condition in a larger sector as an illustration of the use of maximal regularity

applications of discrete maximal l p regularity for finite element operators in this paper we present applications of discrete maximal lp regularity for finite element operators more precisely we show error estimates of order h2 for linear and certain semilinear problems in various lp norms discrete maximal regularity allows us to prove error estimates in a very easy and efficient way moreover we also develop interpolation theory for fractional powers of finite element operators and extend the results on discrete maximal lp regularity formerly proved by the author

non compact quantum groups associated with abelian subgroups letg be a lie group for any abelian subalgebra mathfrak h of the lie algebra g ofg and any r in mathfrak h lambda mathfrak h the difference of the left and right translates ofr gives a compatible poisson bracket ong we show how to construct the corresponding quantum group in thec algebra setting the main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onc algebras

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

closed quantum subgroups of locally compact quantum groups abstract we investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group two definitions one due to s vaes and one due to s l woronowicz are analyzed and relations between them discussed among many reformulations we prove that the former definition can be phrased in terms of quasi equivalence of representations of quantum groups while the latter can be related to an old definition of podle from the theory of compact quantum groups the cases of classical groups duals of classical groups compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context a deep relationship with the quantum group generalization of the herz restriction theorem from classical harmonic analysis is also established in particular in the course of our analysis we give a new proof of the herz restriction theorem

maximum norm stability and maximal l p lp regularity of fems for parabolic equations with lipschitz continuous coefficients in this paper we study the semi discrete galerkin finite element method for parabolic equations with lipschitz continuous coefficients we prove the maximum norm stability of the semigroup generated by the corresponding elliptic finite element operator and prove the space time stability of the parabolic projection onto the finite element space in l infty q t l qt and l p 0 t l q omega lp 0 t lq 1

quantum semigroups from synchronous games we show that the c algebras associated with synchronous games give rise to certain quantum families of maps between the input and output sets of the game in particular situations e g for graph endomorphism games these quantum families have a natural quantum semigroup structure and if the condition of preservation of a natural state is added they are in fact compact quantum groups we show that the c algebras associated with synchronous games give rise to certain quantum families of maps between the input and output sets of the game in particular situations e g for graph endomorphism games these quantum families have a natural quantum semigroup structure and if the condition of preservation of a natural state is added they are in fact compact quantum groups

on the existence of maximum principles in parabolic finite element equations in 1973 h fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space in particular he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute and this is known to generalize in two space dimensions to triangulations of delauney type in this note we consider more general parabolic equations and first show that a maximum principle cannot hold for the standard spatially semidiscrete problem we then show that for the lumped mass method the above conditions on the triangulation are essentially sharp this is in contrast to the elliptic case in which the requirements are weaker we also study conditions for the solution operator acting on the discrete initial data with homogeneous lateral boundary conditions to be a contraction or a positive operator

differential calculus onisoq n quantum poincare algebra and q gravity we present a general method to deform the inhomogeneous algebras of thebn cn dn type and find the corresponding bicovariant differential calculus the method is based on a projection frombn 1 cn 1 dn 1 for example we obtain the bicovariant inhomogeneousq algebraisoq n as a consistent projection of the bicovariant q algebrasoq n 2 this projection works for particular multiparametric deformations ofso n 2 the so called minimal deformations the case ofisoq 4 is studied in detail a real form corresponding to a lorentz signature exists only for one of the minimal deformations depending on one parameterq the quantum poincare lie algebra is given explicitly it has 10 generators no dilatations and contains theclassical lorentz algebra only the commutation relations involving the momenta depend onq finally we discuss aq deformation of gravity based on the gauging of thisq poincare algebra the lagrangian generalizes the usual einstein cartan lagrangian

quantum groups with projection on von neumann algebra level abstract we introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study their properties our approach is slightly different from the one introduced in the thesis of s roy 16 and unlike the investigations of roy we work within the von neumann algebraic picture this allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups in particular we show existence of a braided comultiplication on the algebra spanned by slices of braided multiplicative unitary

quantum algebras and q special functions related to coherent states maps of the disc the quantum algebras generated by the coherent states maps of the disc are investigated it is shown that the analytic realization of these algebras leads to a generalized analysis which includes standard analysis as well as q analysis the applications of the analysis to star product quantizations and q special functions theory are given among others the meromorphic continuation of the generalized basic hypergeometric series is found and a reproducing measure is constructed when the series is treated as a reproducing kernel

a central polynomial of low degree for 4 4 matrices abstract we have found a central polynomial of degree 13 for the 4 4 matrix algebra over a field of characteristic 0 this result agrees with the conjecture that the minimal degree of such polynomials for n n matrices is n 2 3 n 2 2 the polynomial has been obtained by explicitly exhibiting an essentially weak polynomial identity of degree 9 for 4 4 matrices

examples of non compact quantum group actions abstract we present two examples of actions of non regular locally compact quantum groups on their homogeneous spaces the homogeneous spaces are defined in a way specific to these examples but the definitions we use have the advantage of being expressed in purely c algebraic language we also discuss continuity of the obtained actions finally we describe in detail a general construction of quantum homogeneous spaces obtained as quotients by compact quantum subgroups

on the h 1 h1 stability of the l_2 l2 projection onto finite element spaces we study the stability in the h 1 h 1 seminorm of the l 2 l 2 projection onto finite element spaces in the case of nonuniform but shape regular meshes in two and three dimensions and prove in particular stability for conforming triangular elements up to order twelve and conforming tetrahedral elements up to order seven

general de finetti type theorems in noncommutative probability we prove general de finetti type theorems for classical and free independence the de finetti type theorems work for all non easy quantum groups which generalize a recent work of banica curran and speicher we determine maximal distributional symmetries which means the corresponding de finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one in addition we define boolean quantum semigroups in analogous to the easy quantum groups by universal conditions on matrix coordinate generators and an orthogonal projection then we show a general de finetti type theorem for boolean independence