a short proof of a matrix decomposition with applications abstract we give a very short proof of the main result of j beni tez a new decomposition for square matrices electron j linear algebra 20 2010 207 225 also we present some consequences of this result

first and second order operator splitting methods for the phase field crystal equation in this paper we present operator splitting methods for solving the phase field crystal equation which is a model for the microstructural evolution of two phase systems on atomic length and diffusive time scales a core idea of the methods is to decompose the original equation into linear and nonlinear subequations in which the linear subequation has a closed form solution in the fourier space we apply a nonlinear newton type iterative method to solve the nonlinear subequation at the implicit time level and thus a considerably large time step can be used by combining these subequations we achieve the first and second order accuracy in time we present numerical experiments to show the accuracy and efficiency of the proposed methods

characterization of discontinuities in high dimensional stochastic problems on adaptive sparse grids in this paper we present a set of efficient algorithms for detection and identification of discontinuities in high dimensional space the method is based on extension of polynomial annihilation for discontinuity detection in low dimensions compared to the earlier work the present method poses significant improvements for high dimensional problems the core of the algorithms relies on adaptive refinement of sparse grids it is demonstrated that in the commonly encountered cases where a discontinuity resides on a small subset of the dimensions the present method becomes optimal in the sense that the total number of points required for function evaluations depends linearly on the dimensionality of the space the details of the algorithms will be presented and various numerical examples are utilized to demonstrate the efficacy of the method

matrices a such that aa a a are nonsingular abstract in this paper we study the class of square matrices a such that aa a a is nonsingular where a stands for the moore penrose inverse of a among several characterizations we prove that for a matrix a of order n the difference aa a a is nonsingular if and only if r a r a c n 1 where r denotes the range space also we study matrices a such that r a r a

groupoid equivariant prequantization in their 2005 paper c laurent gengoux and p xu define prequantization for pre hamiltonian actions of quasi presymplectic lie groupoids in terms of s1 central extensions of lie groupoids the definition requires that the quasi presymplectic structure be exact i e the closed 3 form on the unit space of the lie groupoid must be exact in the present paper we define prequantization for pre hamiltonian actions of not necessarily exact quasi presymplectic lie groupoids in terms of dixmier douady bundles the definition is a natural adaptation of e meinrenken s treatment of prequantization for quasi hamiltonian lie group actions with group valued moment map the definition given in this paper is shown to be compatible with the definition of laurent gengoux and xu when the underlying quasi preysmplectic structure is exact properties related to morita invariance and symplectic reduction are established

classical solutions of the quantum yang baxter equation the classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a poisson lie group we work directly with a poisson lie groupg and its associated symplectic groupoid the classical analog of the quantumr matrix is a lagrangian submanifold open image in new window in the cartesian square of the symplectic groupoid for any symplectic leafs ing open image in new window induces a symplectic automorphism ofs s which satisfies the set theoretic yang baxter equation when combined with the flip map exchanging components and suitably implanted in each cartesian powers n generates a symplectic action of the braid groupb n ons n application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants but work on this last step is still in progress

unified products for leibniz algebras applications abstract let g be a leibniz algebra and e a vector space containing g as a subspace all leibniz algebra structures on e containing g as a subalgebra are explicitly described and classified by two non abelian cohomological type objects hl g 2 v g provides the classification up to an isomorphism that stabilizes g and hl 2 v g will classify all such structures from the view point of the extension problem here v is a complement of g in e a general product called the unified product is introduced as a tool for our approach the crossed resp bicrossed products between two leibniz algebras are introduced as special cases of the unified product the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem the description and the classification of all complements of a given extension g e of leibniz algebras are given as a converse of the factorization problem they are classified by another cohomological object denoted by ha 2 h g where is the canonical matched pair associated to a given complement h several examples are worked out in details

quantization of poisson manifolds from the integrability of the modular function we discuss a framework for quantizing a poisson manifold via the quantization of its symplectic groupoid combining the tools of geometric quantization with the results of renault s theory of groupoid c algebras this setting allows very singular polarizations in particular we consider the case when the modular function is multiplicatively integrable i e when the space of leaves of the polarization inherits a groupoid structure if suitable regularity conditions are satisfied then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the bohr sommerfeld conditions

poisson lie group of pseudodifferential symbols we introduce a lie bialgebra structure on the central extension of the lie algebra of differential operators with scalar or matrix coefficients on the line and on the circle this defines a poisson lie structure on the dual group of pseudodifferential symbols of an arbitrary real or complex order we show that the usual second benney gl n kdv or gl n adler gelfand dickey and kp poisson structures are naturally realized as restrictions of this poisson structure to submanifolds of this universal poisson lie group moreover the reduced sl n versions of these manifolds orw n algebras in physical terminology can be viewed as certain subspaces of the quotient of this poisson lie group by the dressing action of the group of functions on the circle or as a result of a poisson reduction finally we define an infinite set of commuting functions on the poisson lie group that give the standard families of hamiltonians when restricted to the submanifolds mentioned above the poisson structure and hamiltonians on the whole group interpolate between the poisson structures and hamiltonians of benney kp and kdv flows we also discuss the geometrical meaning ofw as a limit of poisson algebraswe as e 0

applications of cs decomposition in linear combinations of two orthogonal projectors abstract we study the spectrum and the rank of a linear combination of two orthogonal projectors we characterize when this linear combination is ep diagonalizable idempotent tripotent involutive nilpotent generalized projector and hypergeneralized projector also we derive the moore penrose inverse of a linear combination of two orthogonal projectors in a particular case the main tool used here is the cs decomposition

algebraic structure of multiparameter quantum groups let g be a connected semi simple complex lie group we define and study the multi parameter quantum group cq p g in the case where q is a complex parameter that is not a root of unity using a method of twisting bigraded hopf algebras by a cocycle 2 we develop a unified approach to the construction of cq p g and of the multi parameter drinfeld double dq p using a general method of deforming bigraded pairs of hopf algebras we construct a hopf pairing between these algebras from which we deduce a peterweyl type theorem for cq p g we then describe the prime and primitive spectra of cq p g generalizing a result of joseph in the one parameter case this description was conjectured and established in the sl n case by the first and second authors in 15 16 it was proved in the general case by joseph in 18 19 in particular the orbits in prim cq p g under the natural action of the maximal torus h are indexed as in the one parameter case by the elements of the double weyl group w w unlike the one parameter case there is not in general a bijection between symp g and prim cq p g however in the case when

convergence of a fast explicit operator splitting method for the epitaxial growth model with slope selection a fast explicit operator splitting method for the epitaxial growth model with slope selection has been presented in cheng et al j comput phys 303 2015 pp 45 65 the original problem is split into linear and nonlinear subproblems for the linear part the pseudospectral method is adopted for the nonlinear part a 33 point difference scheme is constructed here we give a compact center difference scheme involving fewer points for the nonlinear subproblem in addition we analyze the convergence rate of the algorithm the global error order mathcal o tau 2 h 4 in discrete l 2 norm is proved theoretically and verified numerically some numerical experiments show the robustness of the algorithm for small coefficients of the fourth order term for the one dimensional case in addition coarsening dynamics are simulated in large domains and the 1 3 power laws are observed for the two dimensional case

multilevel sparse kernel based interpolation a multilevel kernel based interpolation method suitable for moderately high dimensional function interpolation problems is proposed the method termed multilevel sparse kernel based interpolation mlski for short uses both levelwise and directionwise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel based interpolants with different scaling in each direction the multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level on each level anisotropic radial basis functions are used for solving a number of small interpolation problems which are subsequently linearly combined to produce the interpolant mlski can be viewed as an extension of d boolean interpolation which is closely related to ideas in sparse grid and hyperbolic crosses literature to kernel ba

the rate of convergence for the method of alternating projections ii abstract the purpose of the paper is threefold 1 to develop a useful error bound for the method of alternating projections which is relatively easy to compute and remember 2 to exhibit a counterexample to a conjecture of kayalar and weinert 3 to show that in the case of at least three subspaces any error bound which only depends on the angles between the various subspaces involved can never be sharp

idempotency of linear combinations of three idempotent matrices two of which are commuting abstract the considerations of the present paper were inspired by baksalary o m baksalary idempotency of linear combinations of three idempotent matrices two of which are disjoint linear algebra appl 388 2004 67 78 who characterized all situations in which a linear combination p c 1 p 1 c 2 p 2 c 3 p 3 with c i i 1 2 3 being nonzero complex scalars and p i i 1 2 3 being nonzero complex idempotent matrices such that two of them p 1 and p 2 say are disjoint i e satisfy condition p 1 p 2 0 p 2 p 1 is an idempotent matrix in the present paper by utilizing different formalism than the one used by baksalary the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of p 1 and p 2 to the commutativity condition p 1 p 2 p 2 p 1

prime spectra of quantum semisimple groups we study the prime ideal spaces of the quantized function algebras rq g for g a semisimple lie group and q an indeterminate our method is to examine the structure of algebras satisfying a set of seven hypotheses and then to demonstrate using work of joseph hodges and levasseur that the algebras rq g satisfy this list of assumptions rings satisfying the assumptions are shown to satisfy normal separation and therefore jategaonkar s strong second layer condition for such rings much representation theoretic information is carried by the graph of links of the prime spectrum and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions homogeneity is a key feature it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms bounds on the ranks of these groups are obtained in the case of rq g in the final section the results are specialized to the case g sln c where detailed calculations can be used to illustrate the general results as a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine n space satisfy our axiom scheme when the group generated by the parameters is torsionfree 0 introduction and background 0 1 let p and q be prime ideals of a noetherian ring r we say that p is linked to q and write p q if there is an ideal a of r with pq c a c p n q such that p n q a is a nonzero torsionfree left r p module and a torsionfree right r q module this condition can be simplified in case r satisfies the second layer condition see below in that case p q if and only if p n q pq is faithful as a left r p module and as a right r q module 39 10 1 4 the graph of links of r is the directed graph whose vertices are the points of spec r the set of prime ideals of r with a directed edge from p to q if and only if p q the primes in the connected component of the graph of links containing p constitute the clique of p denoted clique p there is a close connection between the graph of links of r and the representation theory of r cf 20 4 this connection is particularly strong when r satisfies the strong second layer condition which requires that whenever u is a cyclic uniform r module with the annihilator received by the editors november 4 1994 and in revised form september 5 1995 1991 mathematics subject classification primary 16d30 16d60 16p40 17b37 the research of the second author was partially supported by a grant from the national science foundation usa part of the work was carried out while he visited the university of glasgow mathematics department during october 1993 supported by the edinburgh and london mathematical societies work on a revised version of the paper was completed in summer 1995 during a visit by both authors to the department of mathematics of the university of washington whom both thank for its hospitality the travel costs of the first author were in part covered by a grant from the carnegie trust for the universities of scotland 1996 american mathematical society

a class of quadratic matrix algebras arising from the quantized enveloping algebra uq a2n 1 a natural family of quantized matrix algebras is introduced it includes the two best studied such located inside uq a2n 1 it consists of quadratic algebras with the same hilbert series as polynomials in n2 variables we discuss their general properties and investigate some members of the family in great detail with respect to associated varieties degrees centers and symplectic leaves finally the space of rank r matrices becomes a poisson submanifold and there is an associated tensor category of rank r matrices

on poisson actions of compact lie groups on symplectic manifolds let g p be a compact simple poisson lie group equipped with a poisson structure p and m omega be a symplectic manifold assume that m carries a poisson action of g p and there is an equivariant moment map in the sense of lu and weinstein which maps

quantum lobachevsky planes we classify all sl 2 r covariant poisson structures on the lobachevsky plane with respect to all multiplicative poisson structures on sl 2 r and describe quantizations for all these poisson structures

leaf preserving quantizations of poisson su 2 are not coalgebra homomorphisms although it has been found that some deformation quantizations of the poissonsu 2 preserve symplectic leaves and some preserve the group i e coalgebra operation this paper shows that a quantization ofsu 2 cannot be both leaf preserving and group preserving

free motion on the poisson su n group is the phase space of the poisson su n we explicitly calculate the symplectic structure of define an analogue of the hamiltonian of the free motion on su n and solve the corresponding equations of motion velocity is related to the momentum by a nonlinear legendre transformation

quantization of the poisson su 2 and its poisson homogeneous space the 2 sphere we show that deformation quantizations of the poisson structures on the poisson lie groupsu 2 and its homogeneous space the 2 sphere are compatible with woronowicz s deformation quantization ofsu 2 s group structure and podles deformation quantization of 2 sphere s homogeneous structure respectively so in a certain sense the multiplicativity of the lie poisson structure onsu 2 at the classical level is preserved under quantization

generalized quantization scheme for central extensions of lie algebras we present the method for finding nonlinear poisson lie group structures on the vector spaces and for their quantization explicit quantization formulas are proposed for arbitrary central extension of a lie algebra

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

a poisson structure on compact symmetric spaces we present some basic results on a natural poisson structure on any compact symmetric space the symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold

the poisson geometry of su 1 1 we study the natural poisson structure on the group su 1 1 and related questions in particular we give an explicit description of the ginzburg weinstein isomorphism for the sets of admissible elements we also establish an analog of thompson s conjecture for this group

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

quantum line bundles on s2 and method of orbits for suq 2 the quantum sphere is covered by two quantum complex planes quantum line bundle on s2 is introduced by specifying a transition function the structure of the quantum complex planes enables one to construct representations of suq 2 acting in spaces of holomorphic sections in the quantum case dressing transformation replaces coadjoint action

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

upper bound on the coarsening rate for an epitaxial growth model we study a specific example of energy driven coarsening in two space dimensions the energy is u 2 1 u 2 2 the evolution is the fourth order pde representing steepest descent this equation has been proposed as a model of epitaxial growth for systems with slope selection numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t1 3 and the energy per unit area decays like t 1 3 we prove a weak one sided version of the latter statement the time averaged energy per unit area decays no faster than t 1 3 our argument follows a strategy introduced by kohn and otto in the context of phase separation combining i a dissipation relation ii an isoperimetric inequality and iii an ode lemma the interpolation inequality is new and rather subtle our proof is by contradiction relying on recent compactness results for the aviles giga energy 2003 wiley periodicals inc

the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

a multigrid method for adaptive sparse grids sparse grids have become an important tool to reduce the number of degrees of freedom of discretizations of moderately high dimensional partial differential equations however the reduction in degrees of freedom comes at the cost of an almost dense and unconventionally structured system of linear equations to guarantee overall efficiency of the sparse grid approach special linear solvers are required we present a multigrid method that exploits the sparse grid structure to achieve an optimal runtime that scales linearly with the number of sparse grid points our approach is based on a novel decomposition of the right hand sides of the coarse grid equations that leads to a reformulation in so called auxiliary coefficients with these auxiliary coefficients the right hand sides can be represented in a nodal point basis on low dimensional full grids our proposed multigrid method directly operates in this auxiliary coefficient representation circumventing most of the computationally cumbersome sparse gr

symplectic structures associated to lie poisson groups the lie poisson analogues of the cotangent bundle and coadjoint orbits of a lie group are considered for the natural poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described thus the construction of the kirillov symplectic form is generalized for lie poisson groups

error analysis of a mixed finite element method for the molecular beam epitaxy model this paper investigates the error analysis of a mixed finite element method with crank nicolson time stepping for simulating the molecular beam epitaxy mbe model the fourth order differential equation of the mbe model is replaced by a system of equations consisting of one nonlinear parabolic equation and an elliptic equation then a mixed finite element method requiring only continuous elements is proposed to approximate the resulting system it is proved that the semidiscrete and fully discrete versions of the numerical schemes satisfy the nonlinearity energy stability property which is important in the numerical implementation moreover detailed analysis is provided to obtain the convergence rate numerical experiments are carried out to validate the theoretical results

optimal rate convergence analysis of a second order scheme for a thin film model with slope selection abstract in this paper an energy stable second order mixed finite element scheme is proposed and analyzed for the thin film epitaxial growth model with slope selection we employ second order backward differentiation formula bdf scheme with a second order stabilized term which guarantees the long time energy stability to approximate the continuous model in terms of the convergence analysis the key difficulty to derive an optimal rate spatial estimate is associated with the appearance of the gradient operator in the nonlinear terms which may lead to a loss of optimal accuracy order to overcome this well known difficulty we make use of some auxiliary techniques over triangular elements and obtain an optimal convergence rate o h q 1 t 2 in comparison with o h q t 2 rate from a standard projection estimate furthermore we use an efficient preconditioned steepest descent psd solver for the numerical implementation a few numerical examples are presented to validate the stability and convergence

a finite element method for density estimation with gaussian process priors a variational problem characterizing the density estimator defined by the maximum a posteriori method with gaussian process priors is derived it is shown that this problem is well posed and can be solved with newton s method numerically the solution is approximated by a galerkin finite element method with piecewise multilinear functions on uniform grids error bounds for this method are given and numerical experiments are performed for one two and three dimensional examples

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

postprocessed two scale finite element discretizations part i this paper studies a simple postprocessing of two scale finite element discretizations of elliptic partial differential operators where boundary value problems and eigenvalue problems on tensor product domains omega subset mathbb r d are both examined and conforming piecewise d linear elements are used for d geq2 suppose that the two scale method is applied using a coarse mesh and some univariate fine meshes satisfying h mathcal o h 2 3 where h and h are the fine and coarse mesh widths it is shown for both classes of problem that the postprocessed two scale finite element solution achieves the same order of accuracy in the h 1 omega norm as the postprocessed standard finite element solution but the former solution uses only mathcal o h 2d 1 3 degrees of freedom compared with the mathcal o h d degrees of freedom required by the latter

lie poisson structure on some poisson lie groups poisson lie groups appeared in the work of drinfel d see e g drl dr2 as classical objects corresponding to quantum groups going in the other direction we may say that a poisson lie group is a group of symmetries of a phase space that are allowed to twist in a certain sense the symplectic or poisson structure the poisson structure on the group controls this twisting in a precise way quantizing both the phase space and the symmetries one may obtain a quantum group acting on a quantum phase space in recent work lu and ratiu lr used so called standard poisson structures on a compact semisimple lie group k and on its poisson dual k in order to give a new proof of the nonlinear convexity theorem of kostant ko their method is analogous to the famous symplectic proof of the linear convexity theorem given by atiyah a and guillemin and stemnberg gs the nonlinear convexity theorem like the linear one follows from a very general result on convexity of the image of the momentum map a gs however in lr it is applied not to a coadjoint orbit in t but to a symplectic leaf in k the main result of this paper is that the standard poisson structure on the poisson dual k to a compact semisimple poisson lie group k is actually isomorphic to the linear one on t this theorem seems to be related to some facts in the theory of quantum groups namely for generic q the universal enveloping algebra u t and its quantum deformation uq t are isomorphic as algebras though not as coalgebras of course since the quantum version is not cocommutative in particular there is a bijective correspondence between their representations a direct connection between our work and its quantum analogues though is still to be found the present work supplies a positive answer to question 5 1 in lr and strongly depends on that paper to simplify reading we keep the notations of lr wherever possible on one hand but give all necessary definitions on the other our work is also related to du in which the nonlinear convexity theorem is reduced to the linear one by a deformation argument not unlike the one that we use in 5 the paper is organized as follows in 2 we define poisson lie groups discuss

quantized algebras of functions on homogeneous spaces with poisson stabilizers let g be a simply connected semisimple compact lie group with standard poisson structure k a closed poisson lie subgroup 0 q 1 we study a quantization c gq kq of the algebra of continuous functions on g k using results of soibelman and dijkhuizen stokman we classify the irreducible representations of c gq kq and obtain a composition series for c gq kq we describe closures of the symplectic leaves of g k refining the well known description in the case of flag manifolds in terms of the bruhat order we then show that the same rules describe the topology on the spectrum of c gq kq next we show that the family of c algebras c gq kq 0 q 1 has a canonical structure of a continuous field of c algebras and provides a strict deformation quantization of the poisson algebra mathbb c g k finally extending a result of nagy we show that c gq kq is canonically kk equivalent to c g k

poisson structures on affine spaces and flag varieties ii the standard poisson structures on the flag varieties g p of a complex reductive algebraic group g are investigated it is shown that the orbits of symplectic leaves in g p under a fixed maximal torus of g are smooth irreducible locally closed subvarieties of g p isomorphic to intersections of dual schubert cells in the full flag variety g b of g and their zariski closures are explicitly computed two different proofs of the former result are presented the first is in the framework of poisson homogeneous spaces and the second one uses an idea of weak splittings of surjective poisson submersions based on the notion of poisson dirac submanifolds for a parabolic subgroup p with abelian unipotent radical in which case g p is a hermitian symmetric space of compact type it is shown that all orbits of the standard levi factor l of p on g p are complete poisson subvarieties which are quotients of l equipped with the standard poisson structure moreover it is proved that the poisson structure on g p vanishes at all special base points for the l orbits on g p constructed by richardson rohrle and steinberg

quantum flag manifolds quantum symmetric spaces and their associated universal k matrices abstract let u be a connected simply connected compact lie group with complexification g let u and g be the associated lie algebras let be the dynkin diagram of g with underlying set i and let u q u be the associated quantized universal enveloping algebra of u for some 0 q distinct from 1 let o q u be the coquasitriangular quantized function hopf algebra of u whose drinfeld double o q g r we view as the quantized function algebra of g considered as a real algebraic group we show how the datum of an involution of and a invariant function i r can be used to deform o q g r into a algebra o q id g r by a modification of the drinfeld double construction we then show how by a generalized theory of universal k matrices a specific subalgebra of o q id g r admits homomorphisms into both u q u and o q u the images being coideal subalgebras of respectively u q u and o q u we illustrate the theory by showing that two main classes of examples arise by such coideals namely quantum flag manifolds and quantum symmetric spaces except possibly for certain exceptional cases in the former case this connects to work of the first author and neshveyev while for the latter case we heavily rely on recent results of balagovic and kolb

multi index stochastic collocation for random pdes in this work we introduce the multi index stochastic collocation method misc for computing statistics of the solution of a pde with random data misc is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data we propose an optimization procedure to select the most effective mixed differences to include in the misc estimator such optimization is a crucial step and allows us to build a method that provided with sufficient solution regularity is potentially more effective than other multi level collocation methods already available in literature we then provide a complexity analysis that assumes decay rates of product type for such mixed differences showing that in the optimal case the convergence rate of misc is only dictated by the convergence of the deterministic solver applied to a one dimensional problem we show the effectiveness of misc with some computational tests comparing it with other related methods available in the literature such as the multi index and multilevel monte carlo multilevel stochastic collocation quasi optimal stochastic collocation and sparse composite collocation methods

idempotency of linear combinations of two idempotent matrices abstract a complete solution is established to the problem of characterizing all situations where a linear combination of two different idempotent matrices p 1 and p 2 is also an idempotent matrix including naturally three such situations known in the literature viz if the combination is either the sum p 1 p 2 or one of the differences p 1 p 2 p 2 p 1 and appropriate additional conditions are fulfilled the solution asserts that in the particular case where p 1 and p 2 are complex matrices such that p 1 p 2 is hermitian these three situations exhaust the list of all possibilities and that this list extends when the above assumption on p 1 and p 2 is violated a statistical interpretation of the idempotency problem considered in this note is also pointed out

multi index monte carlo when sparsity meets sampling we propose and analyze a novel multi index monte carlo mimc method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients the mimc method is both a stochastic version of the combination technique introduced by zenger griebel and collaborators and an extension of the multilevel monte carlo mlmc method first described by heinrich and giles inspired by giles s seminal work we use in mimc high order mixed differences instead of using first order differences as in mlmc to reduce the variance of the hierarchical differences dramatically this in turn yields new and improved complexity results which are natural generalizations of giles s mlmc analysis and which increase the domain of the problem parameters for which we achieve the optimal convergence mathcal o mathrm tol 2 o tol 2 moreover in mimc the rate of increase of required memory with respect to mathrm tol tol is independent of the number of directions up to a logarithmic term which allows far more accurate solutions to be calculated for higher dimensions than what is possible when using mlmc we motivate the setting of mimc by first focusing on a simple full tensor index set we then propose a systematic construction of optimal sets of indices for mimc based on properly defined profits that in turn depend on the average cost per sample and the corresponding weak error and variance under standard assumptions on the convergence rates of the weak error variance and work per sample the optimal index set turns out to be the total degree type in some cases using optimal index sets mimc achieves a better rate for the computational complexity than the corresponding rate when using full tensor index sets we also show the asymptotic normality of the statistical error in the resulting mimc estimator and justify in this way our error estimate which allows both the required accuracy and the confidence level in our computational results to be prescribed finally we include numerical experiments involving a partial differential equation posed in three spatial dimensions and with random coefficients to substantiate the analysis and illustrate the corresponding computational savings of mimc

the combination technique and some generalisations the combination technique has repeatedly been shown to be an effective tool for the approximation with sparse grid spaces little is known about the reasons of this effectiveness and in some cases the combination technique can even break down it is known however that the combination technique produces an exact result in the case of a projection into a sparse grid space if the involved partial projections commute the performance of the combination technique is analysed using a projection framework and the c s decomposition error bounds are given in terms of angles between the spanning subspaces or the projections onto these subspaces based on this analysis modified combination coefficients are derived which are optimal in a certain sense and which can substantially extend the applicability and performance of the combination technique

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

ginzburg weinstein via gelfand zeitlin let u n be the unitary group and u n the dual of its lie al gebra equipped with the kirillov poisson structure in their 1983 paper guillemin sternberg introduced a densely defined hamil tonian action of a torus of dimension n 1 n 2 on u n with mo ment map given by the gelfand zeitlin coordinates a few years later flaschka ratiu described a similar multiplicative gelfand zeitlin system for the poisson lie group u n by the ginzburg weinstein theorem u n is isomorphic to u n as a poisson manifold flaschka ratiu conjectured that one can choose the ginzburg weinstein diffeomorphism in such a way that it intertwines the linear and nonlinear gelfand zeitlin systems our main result gives a proof of this conjecture and produces a canonical ginzburg weinstein diffeomorphism

a family of poisson structures on hermitian symmetric spaces we investigate the compatibility of symplectic kirillov kostant souriau structure and poisson lie structure on coadjoint orbits of semisimple lie group we prove that they are compatible for an orbit compact lie group iff the orbit is hermitian symmetric space we prove also the compatibility statement for non compact hermitian symmetric space as an example we describe a structure of symplectic leaves oncp n for this family these leaves may be considered as a perturbation of schubert cells possible applications to infinite dimensional examples are discussed