branching index sum rules for simple lie algebras let l and l0 be a simple lie algebra and its sub lie algebra respectively then a given irreducible representation of l decomposes into a direct sum of irreducible components of l0 which is called the branching rule the general dynkin indices introduced earlier satisfy many sum rules for the branching rule these are found to be strong enough to uniquely determine the branching rule for many cases we have studied the sum rules are especially useful for cases of exceptional lie algebras

discrete series representations for the orbit spaces arising from two involutions of real reductive lie groups leth gbe real reductive lie groups a discrete series representation for a homogeneous spaceg his an irreducible representation ofgrealized as a closedg invariant subspace ofl2 g h the condition for the existence of discrete series representations forg hwas not known in general except for reductive symmetric spaces this paper offers a sufficient condition for the existence of discrete series representations forg hin the setting thatg his a homogeneous submanifold of a symmetric space g hwhereg g h we prove that discrete series representations are non empty for a number of non symmetric homogeneous spaces such assp 2n r sp n0 c gl n1 c gl nk c nj n ando 4m n u 2m j 0 2j n

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

structure constants for lie algebras an orthogonal basis in root space related to the weights of the smallest representation is used to provide a list of the algebraic conditions which the structure constants n must satisfy for all simple lie algebras a particular explicit set of solutions for all the n is given

localization and standard modules for real semisimple lie groups i the duality theorem in this paper we relate two constructions of representations of semisimple lie groups constructions that appear quite different at first glance homogeneous vector bundles are one source of representations if a real semisimple lie group go acts on a vector bundle e m over a quotient space m go ho then go acts also on the space of sections c m e and on any subspace vcc m e defined by a go invariant system of differential equations ordinary induction so called cohomological induction and the construction of representations by quantizat ion all fit into the framework of homogeneous vector bundles for any complex semisimple lie algebra g there is an equivalence of categories due to beilinson bernstein 1 between modules on the one hand and sheaves of modules over the flag variety x of g on the other in the context of real semisimple lie groups this equivalence of categories associates infinitesimal representations to orbits in the flag variety of the complexified lie algebra orbits not of the group go itself but of the complexification of maximal compact subgroup ko c go we shall show that these beilinson bernstein modules are naturally dual to modules attached to certain homogeneous vector bundles in the special case of a compact group both the beilinson bernstein construction and the construction via homogeneous vector bundles reduce to the borel weil bott theorem our duality theorem is then a particular instance of serre duality

the semisimple subalgebras of lie algebras techniques for obtaining generators of nonregular maximal subalgebras of lie algebras alternative to those of dynkin are developed they exploit the use of orthogonal bases in weight space which are related to quark weights the projection from an algebra g to its nonregular subalgebras g is related to an orthogonal matrix the roots of g which project onto roots of g can be simply specified in the orthogonal bases the phases of the expansion coefficients of generators of g in terms of generators of g are specified in such a manner that for a given g the entire set g of maximal subalgebras have consistent phases the condition e e is satisfied for all generators of g the generators of all maximal nonregular subalgebras of all exceptional algebras are exhibited

poincare series for non riemannian locally symmetric spaces abstract we initiate the spectral analysis of pseudo riemannian locally symmetric spaces g h beyond the classical cases where h is compact automorphic forms or is trivial analysis on symmetric spaces for any non riemannian reductive symmetric space x g h on which the discrete spectrum of the laplacian is nonempty and for any discrete group of isometries whose action on x is sufficiently proper we construct l 2 eigenfunctions of the laplacian on x x for an infinite set of eigenvalues these eigenfunctions are obtained as generalized poincare series i e as projections to x of sums over the orbits of eigenfunctions of the laplacian on x we prove that the poincare series we construct still converge and define nonzero l 2 functions after any small deformation of inside g for a large class of groups thus the infinite set of eigenvalues we construct is stable under small deformations this contrasts with the classical setting where the nonzero discrete spectrum varies on the teichmuller space of a compact riemann surface we actually construct joint l 2 eigenfunctions for the whole commutative algebra of invariant differential operators on x

single boson realizations of so 3 and so 2 1 realizations of the lie algebras so 3 and so 2 1 are developed which are functions of integer powers of creation and destruction operators for a single boson i e adagger m and a m as well as a function of the associated single boson number operator n identical to adagger a three types of realizations of the lie algebra elements together with the corresponding casimir operators are developed which depend on arbitrary m periodic functions of n a generic form is achieved in terms of devised m quanta operators the boson representation space is infinite dimensional but may contain as many as two imbedded invariant subspaces a distinction is made between irreducible representations of a lie algebra which lead to global group representations and those which lead only to local non integrable group representations it is shown that the two non unitary types of representations developed are equivalent to the unitary one for m 1 specific choices for the m periodic functions lead to realizations known in the literature

restrictions of quaternionic representations in a paper by gross and wallach 1996 j reine angew math 481 73 123 the k types of the continuations of the quaternionic discrete series of a quaternionic lie group g are associated with projective orbits o of certain subgroups in g c in this paper we will show that the restrictions of the representations to quaternionic subgroups are closely related with the intersection of the zariski closure of o with hyperplanes we apply this to the minimal representations of the exceptional groups of real rank 4 and investigate the correspondences of certain compact dual pairs

discrete decomposability of the restriction of a_q λ with respect to reductive subgroups ii micro local analysis and asymptotic k support let g c g be real reductive lie groups this paper offers a criterion on the triplet g g ir that the irreducible unitary representation ir of g splits into a discrete sum of irreducible unitary representations of a subgroup g when restricted to g each of finite multiplicity furthermore we shall give an upper estimate of the multiplicity of an irreducible unitary representation of g occurring in wrig

differential operators on homogeneous spaces iii characteristic varieties of harish chandra modules and of primitive ideals letg be a semi simple complex algebraic group with lie algebra and flag varietyx g b for each primitive idealj with trivial central character in the enveloping algebrau we define a characteristic variety in the cotangent bundle ofx which projects under the springer resolution mapt x onto the closure of a nilpotent orbit we prove that this characteristic variety is theg saturation of the characteristic variety of a highest weight module with annihilatorj we conjecture that it is irreducible forg sl n our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of weyl group representations as achieved by a joseph the presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space and to study more generally characteristic varieties of harish chandra modules

a weakly nonlinear energy stable scheme for the strongly anisotropic cahn hilliard equation and its convergence analysis abstract in this paper we propose and analyze a weakly nonlinear energy stable numerical scheme for the strongly anisotropic cahn hilliard model in particular a highly nonlinear and singular anisotropic surface energy makes the pde system very challenging at both the analytical and numerical levels to overcome this well known difficulty we perform a convexity analysis on the anisotropic interfacial energy and a careful estimate reveals that all its second order functional derivatives stay uniformly bounded by a global constant this subtle fact enables one to derive an energy stable numerical scheme moreover a linear approximation becomes available for the surface energy part and a detailed estimate demonstrates the corresponding energy stability its combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear energy stable scheme for the whole system in particular such an energy stability is in terms of the interfacial energy with respect to the original phase variable and no auxiliary variable needs to be introduced this has important implications for example in the case that the method needs to satisfy a maximum principle more importantly with a careful application of the global bound for the second order functional derivatives an optimal rate convergence analysis becomes available for the proposed numerical scheme which is the first such result in this area meanwhile for a cahn hilliard system with a sufficiently large degree of anisotropy a willmore or biharmonic regularization has to be introduced to make the equation well posed for such a physical model all the presented analyses are still available the unique solvability energy stability and convergence estimate can be derived in an appropriate manner in addition the fourier pseudo spectral spatial approximation is applied and all the theoretical results could be extended for the fully discrete scheme finally a few numerical results are presented which confirm the robustness and accuracy of the proposed scheme

existence and uniqueness of global weak solutions to a cahn hilliard stokes darcy system for two phase incompressible flows in karstic geometry abstract we study the well posedness of a coupled cahn hilliard stokes darcy system which is a diffuse interface model for essentially immiscible two phase incompressible flows with matched density in a karstic geometry existence of finite energy weak solution that is global in time is established in both 2d and 3d weak strong uniqueness property of the weak solutions is provided as well

uniquely solvable and energy stable decoupled numerical schemes for the cahn hilliard stokes darcy system for two phase flows in karstic geometry we propose and analyze two novel decoupled numerical schemes for solving the cahn hilliard stokes darcy chsd model for two phase flows in karstic geometry in the first numerical scheme we explore a fractional step method operator splitting to decouple the phase field cahn hilliard equation from the velocity field stokes darcy fluid equations to further decouple the stokes darcy system we introduce a first order pressure stabilization term in the darcy solver in the second numerical scheme so that the stokes system is decoupled from the darcy system and hence the chsd system can be solved in a fully decoupled manner we show that both decoupled numerical schemes are uniquely solvable energy stable and mass conservative ample numerical results are presented to demonstrate the accuracy and efficiency of our schemes

kirillov s orbit method the case of discrete series representations let be a discrete series representation of a real reductive lie group g and let g be a reductive subgroup of g in this paper we give a geometric expression of the g multiplicities in g when the representation is g admissible

a robust solver for a second order mixed finite element method for the cahn hilliard equation abstract we develop a robust solver for a second order mixed finite element splitting scheme for the cahn hilliard equation this work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the cahn hilliard equation the key ingredient of the solver is a preconditioned minimal residual algorithm with a multigrid preconditioner whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter the dependence on the interfacial width parameter is also mild

branching laws for verma modules and applications in parabolic geometry i abstract we initiate a new study of differential operators with symmetries and combine this with the study of branching laws for verma modules of reductive lie algebras by the criterion for discretely decomposable and multiplicity free restrictions of generalized verma modules t kobayashi 2012 22 we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds then we apply a new method f method relying on the fourier transform to find singular vectors in generalized verma modules which significantly simplifies and generalizes many preceding works in certain cases it also determines the jordan holder series of the restriction for singular parameters the f method yields an explicit formula of such unique operators for example giving an intrinsic and new proof of juhl s conformally invariant differential operators juhl 2009 16 and its generalizations to spinor bundles this article is the first in the series and the next ones include their extension to curved cases together with more applications of the f method to various settings in parabolic geometries

finite dimensional global attractor of the cahn hilliard brinkman system abstract our aim in this paper is to study the asymptotic behavior of the cahn hilliard brinkman system we obtain the existence of a finite fractal dimensional global attractor a in h 4 v i by asymptotical a priori estimates for the cahn hilliard brinkman system

finite multiplicity theorems for induction and restriction abstract we find upper and lower bounds of the multiplicities of irreducible admissible representations of a semisimple lie group g occurring in the induced representations ind h g from irreducible representations of a closed subgroup h as corollaries we establish geometric criteria for finiteness of the dimension of hom g ind h g induction and of hom h h restriction by means of the real flag variety g p and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety

optimal distributed control of the cahn hilliard brinkman system with regular potential abstract the main objective of this paper is to study the optimal distributed control of the three dimensional cahn hilliard brinkman system with regular potential we apply recently proved well posedness and regularity results in order to establish the existence of optimal controls as well as the first order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow

symmetrised powers of rotation group representations the results given permit the unambiguous evaluation of all possible kronecker products of the irreducible representations tensor and spinor of on and son for n 2 nu and n 2 nu 1 a complete resolution of the second and third powers of the basic spinor representations of so2 nu and so2 nu 1 is given together with a prescription for analysing the fourth power of these representations detailed application is made to the enumeration of properties of so10 relevant to grand unified theories and sufficient information given to resolve the fourth power of any representation of so10

a novel second order linear scheme for the cahn hilliard navier stokes equations abstract in this paper we consider the cahn hilliard equation coupled with the incompressible navier stokes equation usually known as the cahn hilliard navier stokes chns system the chns system has been widely embraced to investigate the dynamics of a binary fluid mixture by utilizing the modified leap frog time marching method we propose a novel numerical algorithm for solving the chns system in an efficient and accurate manner this newly proposed scheme has several advantages first of all the proposed scheme is linear in time and space such that only a linear algebraic system needs to be solved at each time marching step making it extremely efficient also the existence and uniqueness of numerical solutions are guaranteed for any time step size in addition the scheme is unconditionally energy stable with second order accuracy in time and spectral accuracy in space such that relatively large temporal and spatial mesh sizes can be used to obtain reliable numerical solutions the rigorous proofs for the unconditional energy stable property and solution existence and uniqueness are given furthermore we present several numerical examples to test the proposed numerical algorithm and illustrate its accuracy and efficiency the differences of coarsening dynamics between the cahn hilliard equation and the cahn hilliard navier stokes equations have been investigated as well

branching rules and even dimensional rotation groups so2k unambiguous methods are developed for calculating branching rules for the classical subgroups of the even dimensional rotation group so2k complete results are given for the subgroups suk u1 so2k 2 u1 so2p so2q and so2p 1 so2q 1 a number of examples relevant to problems in supergravity and unification theories are given a complete resolution of the antisymmetric powers of the basic spinor irrep of so10 is given and the results extended to so11

realisation of lie algebras and of representations of lie groups in terms of harmonic oscillators the algebras and irreducible representations irreps of compact lie groups including the exceptional groups are realised in terms of sets of bose oscillator sho creation and annihilation operators in particular not only the tensor irreps but also the spinor irreps of orthogonal groups of all ranks can be constructed using bose rather than the more customary fermi oscillators

degenerate whittaker functionals for real reductive groups in this paper we establish a connection between the associated variety of a representation and the existence of certain it degenerate whittaker functionals for both smooth and k finite vectors for all quasi split real reductive groups thereby generalizing results of kostant matumoto and others

preconditioned steepest descent methods for some nonlinear elliptic equations involving p laplacian terms abstract we describe and analyze preconditioned steepest descent psd solvers for fourth and sixth order nonlinear elliptic equations that include p laplacian terms on periodic domains in 2 and 3 dimensions the highest and lowest order terms of the equations are constant coefficient positive linear operators which suggests a natural preconditioning strategy such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena in particular liquid crystals thin film epitaxial growth and phase transformations the analyses of the schemes involve the characterization of the strictly convex energies associated with the equations we first give a general framework for psd in hilbert spaces based on certain reasonable assumptions of the linear pre conditioner a geometric convergence rate is shown for the nonlinear psd iteration we then apply the general theory to the fourth and sixth order problems of interest making use of sobolev embedding and regularity results to confirm the appropriateness of our pre conditioners for the regularized p lapacian problems our results include a sharper theoretical convergence result for p laplacian systems compared to what may be found in existing works we demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods numerical simulations for some important physical application problems including thin film epitaxy with slope selection and the square phase field crystal model are carried out to verify the efficiency of the scheme

modified fourth order casimir invariants and indices for simple lie algebras the fourth order indices for lie algebras have been defined and studied by patera sharp and winternitz we show that it may be more convenient to modify the original definition and that the modified fourth order indices are intimately related to eigenvalues of symmetrized fourth order casimir invariants explicit expressions for these quantities are given and we also find a quartic trace identity involving the generic element of these lie algebras we discuss the triality principle for the lie algebra d4 in connection with identical vanishing of the modified fourth order index for this algebra

convergence analysis and error estimates for a second order accurate finite element method for the cahn hilliard navier stokes system in this paper we present a novel second order in time mixed finite element scheme for the cahn hilliard navier stokes equations with matched densities the scheme combines a standard second order crank nicolson method for the navier stokes equations and a modification to the crank nicolson method for the cahn hilliard equation in particular a second order adams bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the cahn hilliard equation we show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the pde system specifically the discrete phase variable is shown to be bounded in ell infty left 0 t l infty right and the discrete chemical potential bounded in ell infty left 0 t l 2 right for any time and space step sizes in two and three dimensions and for any finite final time t we subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions

picard lefschetz theory and characters of a semisimple lie group the paper applies picard lefschetz theory to the distribution characters of infinite dimensional representations of semisimple lie groups and analyzes their asymptotic behaviour at the identity

indices of representations of simple superalgebras indices and anomaly numbers for representations of basic classical lie superalgebras are defined and their explicit expressions are derived in terms of kac dynkin labels useful properties of indices and anomalies are determined and several examples are given a similar analysis is made for superindices and superanomalies and it is demonstrated how all these objects form a helpful tool in decomposing tensor products or in constructing branching rules for representations

error estimates for time discretizations of cahn hilliard and allen cahn phase field models for two phase incompressible flows we carry out rigorous error analysis for some energy stable time discretization schemes developed in shen and yang siam j sci comput 32 3 1159 1179 2010 for a cahn hilliard phase field model and in shen and yang chin ann math ser b 31 5 743 758 2010 for an allen cahn phase field model

geometric realization of discrete series for semisimple symmetric spaces let g be a connected semisimple lie group a an involution of g and h the connected component of the fixed point set of a containing the identity then g h is a semisimple symmetric space we assume g is a real form of a complex lie group and rank g h rank k k c h where k is a a stable maximal compact subgroup of g flensted jensen using his duality principle and poisson transform constructed countably many discrete series in u g h f j i then using deep results of hyperfunction theory and microlocal analysis oshima and matsuki gave a detailed characterization of these discrete series including a proof of flensted jensen s conjecture c 0 om based on these results we give the discrete series a geometric interpretation in locally symmetric spaces f c where g k and f cg is a cocompact discrete group such that f c h c h is cocompact our geometric interpretation is the following first of all to a flensted jensen function which is the generating function of a discrete series v we associate a harmonic form cb z on g with values in a finite dimensional g module v right g translates of coefficients of 5z realizes the contragredient t of r and tba is left invariant under h and under right k action it transforms according to the minimal k type with highest weight of 3 is the pull back of a v valued from coz on and v determines a locally constant bundle e on f so that if convergence holds

general indices of simple lie algebras and symmetrized product representations in many branches of physics it is important to know the decomposition of a product representation n times of identical representations of a simple lie algebra into irreducible components with a given young tableau symmetry we show that the notion of representation indices introduced elsewhere is a very useful tool in dealing with this problem we calculate explicit formula for general pth order indices d p for all classical simple lie algebras sixth order indices for exceptional lie algebras are also discussed

branching rules for e8 so16 branching rules for e8 down arrow so16 are derived from those of e8 down arrow su2 e7 down arrow su8 and so16 down arrow u1 su8 by noting that both group chains share a common u1 su8 subgroup additional branching rules are deduced from kronecker products of e8 and so16 irreps in each case the rules distinguish unambiguously between conjugate irreps of so16 an alternative labelling scheme for the irreps of so2k based on its maximal u1 suk subgroup is outlined

wave front sets of reductive lie group representations iii abstract let g be a real reductive algebraic group and let x be a homogeneous space for g with a non zero invariant density we give an explicit description of a zariski open dense subset of the asymptotics of the tempered support of l 2 x under additional hypotheses this result remains true for vector bundle valued harmonic analysis on x these results follow from an upper bound on the wave front set of an induced lie group representation under a uniformity condition

an efficient unconditionally energy stable local discontinuous galerkin scheme for the cahn hilliard brinkman system in this paper we present an efficient and unconditionally energy stable fully discrete local discontinuous galerkin ldg method for approximating the cahn hilliard brinkman chb system which is comprised of a cahn hilliard type equation and a generalized brinkman equation modeling fluid flow the semi discrete energy stability of the ldg method is proved firstly due to the strict time step restriction t o x 4 of explicit time discretization methods for stability we introduce a semi implicit scheme which consists of the implicit euler method combined with a convex splitting of the discrete cahn hilliard energy strategy for the temporal discretization the unconditional energy stability of this fully discrete convex splitting scheme is also proved obviously the fully discrete equations at the implicit time level are nonlinear and to enhance the efficiency of the proposed approach the nonlinear full approximation scheme fas multigrid method has been employed to solve this system of algebraic equations we also show the nearly optimal complexity numerically numerical experiments based on the overall solution method of combining the proposed ldg method convex splitting scheme and the nonlinear multigrid solver are given to validate the theoretical results and to show the effectiveness of the proposed approach for the chb system

error analysis of a mixed finite element method for a cahn hilliard hele shaw system we present and analyze a mixed finite element numerical scheme for the cahn hilliard hele shaw equation a modified cahn hilliard equation coupled with the darcy flow law this numerical scheme was first reported in feng and wise siam j numer anal 50 1320 1343 2012 with the weak convergence to a weak solution proven in this article we provide an optimal rate error analysis a convex splitting approach is taken in the temporal discretization which in turn leads to the unique solvability and unconditional energy stability instead of the more standard ell infty 0 t l 2 cap ell 2 0 t h 2 l 0 t l2 l2 0 t h2 error estimate we perform a discrete ell infty 0 t h 1 cap ell 2 0 t h 3 l 0 t h1 l2 0 t h3 error estimate for the phase variable through an l 2 l2 inner product with the numerical error function associated with the chemical potential as a result an unconditional convergence for the time step tau in terms of the spatial resolution h is derived the nonlinear analysis is accomplished with the help of a discrete gagliardo nirenberg type inequality in the finite element space gotten by introducing a discrete laplacian delta h h of the numerical solution such that delta h phi in s h h sh for every phi in s h sh where s h sh is the finite element space

numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach summary we present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model which is derived from the variation of a free energy functional consisting of a temperature dependent bulk potential and a conformational entropy with a gradient dependent anisotropic coefficient we introduce a novel invariant energy quadratization approach to transform the free energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations based on the reformulated equivalent governing system we develop a first and a second order semi discretized scheme in time for the system in which all nonlinear terms are treated semi explicitly the resulting semi discretized equations consist of a linear elliptic equation system at each time step where the coefficient matrix operator is positive definite and thus the semi discrete system can be solved efficiently we further prove that the proposed schemes are unconditionally energy stable convergence test together with 2d and 3d numerical simulations for dendritic crystal growth are presented after the semi discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes copyright 2016 john wiley sons ltd

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

branching rules for irreducible representations of e8 into d8 the branching rules for irreducible representations of e8 into d8 are calculated using the knowledge of the kronecker products for those two algebras tables of kronecker products for both e8 and d8 algebras are also included

analysis of a mixed finite element method for a cahn hilliard darcy stokes system in this paper we devise and analyze a mixed finite element method for a modified cahn hilliard equation coupled with a nonsteady darcy stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts the time discretization is based on a convex splitting of the energy of the equation we prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable we prove that the discrete phase variable is bounded in l infty left 0 t l infty right and the discrete chemical potential is bounded in l infty left 0 t l 2 right for any time and space step sizes in two and three dimensions and for any finite final time t we subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions

on the triangle anomaly number of su n representations a simple geometrical interpretation of the triangle anomaly number of su n representations is demonstrated the number is equal to the sum of cubes of projections of weights on the u 1 direction u 1 is the group in the reduction su n u 1 su n 1 properties of the anomaly number are then consequences of properties of weight systems explicit formulas are derived for the anomaly numbers for reducible representations corresponding to general plethysms based on an arbitrary representation of su n generalizations to anomaly numbers corresponding to degrees other than three are discussed

anomaly coefficients their calculation and congruences a new method for the calculation of anomaly coefficients is presented for su n some explicit and general expressions are given for these in particular certain congruences are discovered and investigated among the leading anomaly coefficients as an application of these congruences the absence of global six dimensional gauge anomalies is shown

character calculations for semisimple lie algebras a recursive method for calculation of characters of semisimple lie algebras is outlined by slight modifications algorithms are obtained for the computation of weight multiplicities kronecker products and branching rules as well as symmetrized kronecker powers

a diffuse interface model and semi implicit energy stable finite element method for two phase magnetohydrodynamic flows abstract in this paper we propose a diffuse interface model and finite element approximation for two phase magnetohydrodynamic mhd flows with different viscosities and electric conductivities an energy stable scheme which is based on the finite element method for the spatial discretization and first order semi implicit scheme combined with convex splitting method for the temporal discretization is proposed to solve this new model the numerical scheme is proved to be mass conservative and energy law preserving by leray schauder fixed point theorem the existence of solutions to the numerical scheme is shown the uniqueness of the numerical solutions is obtained utilizing the stability of the numerical scheme and the compactness method the existence of the weak solutions to the two phase mhd model is established as well furthermore given more regularity on the weak solution the convergence of the numerical scheme is derived finally numerical experiments are provided to verify the theoretical results and validate the proposed model

analysis of a novel finite element method for a modified cahn hilliard hele shaw system abstract in this paper a novel finite element method for solving a modified cahn hilliard hele shaw system is proposed the time discretization is based on the convex splitting of the energy functional in the modified cahn hilliard equation i e the high order nonlinear term and the linear term in the chemical potential are treated explicitly and implicitly respectively designing in this way leads to solving a linear system at each time step which is much efficient compared to solving a nonlinear system resulting from most existing schemes the proposed scheme is proved to be unconditionally energy stable and optimally convergent for the phase variable numerical results are presented to support our theoretical analysis

unitary highest weight modules the unitary highest weight modules of a semisimple lie group g have been classified s 93 unfortunately this classification does not fit well with a more general picture of the unitary dual of g we give such a classification for g a classical group using derived functors and the philosophy of unipotent representations in particular these representations are associated to coadjoint orbits of g let 8 lie g 0 a cartan involution of 6 8 c a derived functor module is defined by the data q i u a o stable parabolic subalgebra of 0 and 17 an irreducible unitary representation of l the stabilizer of l in g considering z7 as a representation of i write z7 a rc where a e c c center of i denotes dual and rt z7l r we define a rc i rc cf definition 6 1 now fix n and vary 1 under certain restrictions on i which amount to a positivity condition on the corresponding bundle over g l a i rr is known to be irreducible and unitary as ii varies within this range the langlands parameters of a a rr vary smoothly however one cannot obtain all unitary highest weight modules under these restrictions the orbit method suggests how to obtain the remaining unitary highest weight modules the modules a rr are associated to coadjoint orbits as follows if rc is trivial we say a rr is associated to the elliptic orbit 0 g ii not all unitary highest weight modules may be obtained with rc trivial an example of which is the oscillator representation of 2n r the metapletic group we define 2n irreducible unitary highest weight modules rc 1 2 n k r is 113 oool 8708 87 7 50

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

analysis on the minimal representation of o p q i realization via conformal geometry abstract this is the first in a series of papers devoted to an analogue of the metaplectic representation namely the minimal unitary representation of an indefinite orthogonal group this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of kirillov kostant we begin by applying methods from conformal geometry of pseudo riemannian manifolds to a general construction of an infinite dimensional representation of the conformal group on the solution space of the yamabe equation by functoriality of the constructions we obtain different models of the unitary representation as well as giving new proofs of unitarity and irreducibility the results in this paper play a basic role in the subsequent papers where we give explicit branching formulae and prove unitarization in the various models

discrete decomposability of the restriction of aq λ with respect to reductive subgroupsiii restriction of harish chandra modules and associated varieties let h g be real reductive lie groups and an irreducible unitary representation of g we introduce an algebraic formulation discretely decomposable restriction to single out the nice class of the branching problem breaking symmetry in physics in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction h this paper offers basic algebraic properties of discretely decomposable restrictions especially for a reductive symmetric pair g h and for the zuckerman vogan derived functor module and proves that the sufficient condition invent math 94 is in fact necessary a finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum an application to the restriction h of discrete series for a symmetric space g h is also given

characters dimensions and branching rules for covariant irreps of u n m the theory of symmetric functions is used to obtain simple expressions for the characters and dimensions of the covariant irreps of the supergroup u n m schur function expressions are given for a number of important branching rules

classification of discretely decomposable aq λ with respect to reductive symmetric pairs abstract we give a classification of the triples g g q such that zuckerman s derived functor g k module a q for a stable parabolic subalgebra q is discretely decomposable with respect to a reductive symmetric pair g g the proof is based on the criterion for discretely decomposable restrictions by the first author and on berger s classification of reductive symmetric pairs

the weyl orbits of g2 f4 e6 and e7 simple methods are described for constructing the weyl orbits of exceptional lie algebras and for identifying the orbit of an arbitrary weight the methods make use of convenient bases in weight space they are applied to g2 f4 e6 and e7 a complete table of all weights of all orbits of f4 is given the weights of the 37 orbits of smallest weight length of e6 and e7 are given in tabular form the depth structures of the orbits are discussed

discrete decomposability of the restriction of a q λ with respect to reductive subgroups and its applications letg g be real reductive lie groups and q a stable parabolic subalgebra of lie g this paper offers a sufficient condition on g g q that the irreducible unitary representation mathop a q limits ofg with non zero continuous cohomology splits into a discrete sum of irreducible unitary representations of a subgroupg each of finite multiplicity as an application to purely analytic problems new results on discrete series are also obtained for some pseudo riemannian non symmetric spherical homogeneous spaces which fit nicely into this framework some explicit examples of a decomposition formula are also found in the cases wherea q is not necessarily a highest weight module

general indices of representations and casimir invariants a modified definition of the index of degree p of a finite dimensional representation of a simple lie algebra is given the definition applies equally to even and odd p the correspondence between the earlier definitions of indices even p and anomaly numbers odd p is pointed out as well as the relation to casimir invariants of the algebra a closed formula for the fifth order index is derived

extension and replacement bases for semisimple lie algebras two simple prescriptions are given for obtaining sets of orthogonal bases for semisimple lie algebras the first method allows one to obtain the irreducible representations of all the simple lie algebras starting from su 2 and dynkin s method for constructing representations from simple roots and highest weights the second method relates algebras of the same rank several examples are discussed a method is given for listing the bases that may be obtained from the prescriptions

duflo s conjecture for the branching problem to the iwasawa an subgroup of a semisimple lie group the purpose of this paper is to prove duflo s conjecture for g pi an where g is a real connected simple lie group of hermitian type and pi is a discrete series of g and an is the maximal exponential solvable subgroup for an iwasawa decomposition g kan this is essentially deduced from the following general theorem which we prove in this paper let g be a real connected semisimple lie group with lie algebra mathfrak g then a strongly elliptic g coadjoint orbit mathcal o is holomorphic if and only if text p mathcal o is an open an coadjoint orbit here text p mathfrak g longrightarrow mathfrak a oplus mathfrak n is the natural projection

distinguished nilpotent orbits kostant pairs and normalizers of lie algebras abstract a pair of lie algebras g g 1 will be called a kostant pair if g is semisimple g 1 is reductive in g and the restriction of the killing form b g to g 1 is nondegenerate we study the class of such nonsymmetric pairs and obtain some useful and new structural results we study the structure of the normalizers n g g 1 and as a consequence we obtain some corresponding worthy results about algebraic groups in particular we consider an interesting case when g 1 is a distinguished sl 2 subalgebra of g combined with the research due to v l popov we observe that the notions of self normalizing reductive subalgebras of a semisimple lie algebra and projective self dual algebraic subvarieties of the usual nilpotent cones are closely related

fock model and segal bargmann transform for minimal representations of hermitian lie groups abstract for any hermitian lie group g of tube type we construct a fock model of its minimal representation the fock space is defined on the minimal nilpotent k c orbit x in p c and the l 2 inner product involves a k bessel function as density here k g is a maximal compact subgroup and g c k c p c is a complexified cartan decomposition in this realization the space of k finite vectors consists of holomorphic polynomials on x the reproducing kernel of the fock space is calculated explicitly in terms of an i bessel function we further find an explicit formula of a generalized segal bargmann transform which intertwines the schrodinger and fock model its kernel involves the same i bessel function using the segal bargmann transform we also determine the integral kernel of the unitary inversion operator in the schrodinger model which is given by a j bessel function

a deposition model coupling stokes and darcy s equations with nonlinear deposition abstract in this work we investigate a filtration process whereby particulate is deposited in the flow domain causing the porosity of the region to decrease the fluid flow is modeled as a coupled stokes darcy flow problem and the deposition in the darcy domain is modeled using a nonlinear equation for the porosity existence and uniqueness of a solution to the governing equations is established additionally the nonnegativity and boundedness of the porosity is shown a finite element approximation scheme that preserves the nonnegativity and boundedness of the porosity is investigated accompanying numerical experiments support the analytical findings

standard young tableaux and character generators of classical lie groups character generators are derived for each of the classical groups su k so 2k 1 sp 2k and so 2k in order to do this use is made of various generalisations of standard young tableaux one set of these is in one to one correspondence with the contributions to the characters of each irreducible representation and the other involving shifted young diagrams provides a means of enumerating those maximal chains in the fundamental posets of each group which serve to generate via their descent subchains all possible multichains the character generators are then written down in terms of a suitable labelling of the poset elements illustrative examples are given

a multilevel decoupling method for the navier stokes darcy model abstract this paper considers a multilevel decoupling method for the coupled navier stokes darcy model describing a free flowing fluid over a porous medium the method utilizes a sequence of meshes on which a low dimensional fully coupled nonlinear problem is solved only on a very coarse initial mesh on subsequent finer meshes the approximate solution in each flow region is obtained by solving a linear decoupled problem and performing a correction step the correction step in each domain is achieved by solving a linear system that differs from the original decoupled system only in the right hand side we prove optimal error estimates and demonstrate that for a sequence of meshes with spacing h j h j 1 2 the decoupling method is computationally efficient and achieves the same order of approximation as the fully coupled method