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Randall R. Holmes - One of the best experts on this subject based on the ideXlab platform.
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Dimensions of the simple restricted modules for the restricted contact Lie algebra
1994Co-Authors: Randall R. HolmesAbstract:It is shown that, with a few exceptions, the simple restricted modules for a restricted contact Lie algebra are induced from those for the homogeneous Component of degree zero. In [3], Shen constructed the simple restricted modules for the restricted Witt, special and hamiltonian Lie algebras which comprise three of the four classes of restricted Lie algebras of Cartan type. His methods, however, do not apply to the algebras in the fourth class, namely, the contact algebras. Here, this remaining case is considered and it is shown that, with a few exceptions, the simple restricted modules are induced from simple restricted modules (extended trivially to positive Components) for the homogeneous Component of degree zero in the usual grading of the algebra. As this Component is isomorphic to the direct sum of a symplectic algebra and the trivial algebra, the problem of determining, say, the dimensions of the simple restricted modules is then reduced to the classical situation for which Lusztig has a conjecture (see [3], p. 294). The author would like to thank Dan Nakano for introducing him to Lie algebras of Cartan type and for the infectious enthusiasm with which he discusses their properties. 1 Notation and Statement of Main Theorem The notation will be, for the most part, the same as that in [4] and this reference can also be consulted for the precise definition and fundamental properties of the contact Lie algebras. Let F be an algebraically closed field of characteristic p> 2 and let n = 2r + 1 with r ∈ N. For 1 ≤ k ≤ n let εk be the n-tuple with Jth Component δjk (Kronecker delta). Set A = {a = ∑ k akεk | 0 ≤ ak < p} ⊂ Z n. 1 The underlying vector space of the restricted contact Lie algebra K (denoted K(2r + 1, 1) in [4]) has as basis {x(a) | a ∈ A} if n + 3 ≡ 0 (mod p) and {x(a) | a ∈ A, a = ∑ (p − 1)εk} otherwise. (The x (a) are standard basis vectors for a divided power algebra.
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Cartan invariants for the restricted toral rank two contact Lie algebra
1994Co-Authors: Randall R. HolmesAbstract:Abstract. Restricted modules for the restricted toral rank two contact Lie algebra are considered. Contragredients of the simple modules, Cartan invariants, and dimensions of the simple modules and their projective covers are determined. Let L be a finite dimensional restricted Lie algebra. All L-modules in this paper are assumed to be left, restricted and finite dimensional over the defining field. Each simple L-module has a projective cover; the multiplicities of the composition factors of the various projective covers are called Cartan invariants. Here, we use the method of [3] to compute the Cartan invariants for the restricted toral rank two contact Lie algebra K(3, 1). To carry out the computation, one needs to know the simple modules and their multiplicities as composition factors of certain induced modules. In [4]–which considered restricted contact Lie algebras of arbitrary toral rank–it was shown that these multiplicities are generically one, that is, the induced modules are, with a few exceptions, simple (see 1.1 below). Although it is not known at the time of this writing, it is expected that (for arbitrary toral rank) the few exceptional induced modules will not be simple. At least this is the case for the algebra K(3, 1) as will be shown in this paper (see 6.1). In addition to the Cartan invariants for K(3, 1) we will compute the dimensions of the simple modules which will give in turn the dimensions of their projective covers. Also, we determine the contragredient of each simple module. I thank the referee for the improvement in 2.3(2) and its proof as well as for other useful comments. 1. Statement of Main Results Let F be an algebraically closed field of characteristic p> 2 and let n = 2r + 1 with r ∈ N. For 1 ≤ k ≤ n let εk be the n-tuple with Jth Component δjk (Kronecker delta). Set A = {a
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Simple restricted modules for the restricted contact Lie algebras
Proceedings of the American Mathematical Society, 1992Co-Authors: Randall R. HolmesAbstract:It is shown that, with a few exceptions, the simple restricted modules for a restricted contact Lie algebra are induced from those for the homogeneous Component of degree zero. In [3], Shen constructed the simple restricted modules for the restricted Witt, special and hamiltonian Lie algebras which comprise three of the four classes of restricted Lie algebras of Cartan type. His methods, however, do not apply to the algebras in the fourth class, namely, the contact algebras. Here, this remaining case is considered and it is shown that, with a few exceptions, the simple restricted modules are induced from simple restricted modules (extended trivially to positive Components) for the homogeneous Component of degree zero in the usual grading of the algebra. As this Component is isomorphic to the direct sum of a symplectic algebra and the trivial algebra, the problem of determining, say, the dimensions of the simple restricted modules is then reduced to the classical situation for which Lusztig has a conjecture (see [3], p. 294). The author would like to thank Dan Nakano for introducing him to Lie algebras of Cartan type and for the infectious enthusiasm with which he discusses their properties. 1 Notation and Statement of Main Theorem The notation will be, for the most part, the same as that in [4] and this reference can also be consulted for the precise definition and fundamental properties of the contact Lie algebras. Let F be an algebraically closed field of characteristic p > 2 and let n = 2r+ 1 with r ∈ N. For 1 ≤ k ≤ n let ek be the n-tuple with Jth Component δjk (Kronecker delta). Set A = {a = ∑ k akek | 0 ≤ ak < p} ⊂ Zn.
Yang Yang - One of the best experts on this subject based on the ideXlab platform.
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High-order bound-preserving finite difference methods for miscible displacements in porous media
'Elsevier BV', 2020Co-Authors: Guo Hui, Liu Xinyuan, Yang YangAbstract:In this paper, we develop high-order bound-preserving (BP) finite difference (FD) methods for the coupled system of compressible miscible displacements. We consider the problem with multi-Component fluid mixture and the (volumetric) concentration of the Jth Component, cj, should be between 0 and 1. It is well known that cj does not satisfy a maximum-principle. Hence most of the existing BP techniques cannot be applied directly. The main idea in this paper is to construct the positivity-preserving techniques to all c\u27js and enforce ∑jc j = 1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure dp/dt as a source in the concentration equation and choose suitable “consistent” numerical fluxes in the pressure and concentration equations. Recently, the high-order BP discontinuous Galerkin (DG) methods for miscible displacements were introduced in [4]. However, the BP technique for DG methods is not straightforward extendable to high-order FD schemes. There are two main difficulties. Firstly, it is not easy to determine the time step size in the BP technique. In finite difference schemes, we need to choose suitable time step size first and then apply the flux limiter to the numerical fluxes. Subsequently, we can compute the source term in the concentration equation, leading to a new time step constraint that may not be satisfied by the time step size applied in the flux limiter. Therefore, it would be very difficult to determine how large the time step is. Secondly, the general treatment for the diffusion term, e.g. centered difference, in miscible displacements may require a stencil whose size is larger than that for the convection term. It would be better to construct a new spatial discretization for the diffusion term such that a smaller stencil can be used. In this paper, we will solve both problems. We first construct a special discretization of the convection term, which yields the desired approximations of the source. Then we can find out the time step size that suitable for the BP technique and apply the flux limiters. Moreover, we will also construct a special algorithm for the diffusion term whose stencil is the same as that used for the convection term. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique
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High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes
'Elsevier BV', 2019Co-Authors: Chuenjarern Nattaporn, Xu Ziyao, Yang YangAbstract:© 2018 Elsevier Inc. In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-Component fluid mixture and the (volumetric) concentration of the Jth Component, cj, should be between 0 and 1. There are three main difficulties. Firstly, cj does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in Zhang and Shu (2010) [44] cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all cj′s and enforce ∑jcj=1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure dp/dt as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to cj. To construct the BP technique, we will not approximate cj directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique
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High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes
Journal of Computational Physics, 2019Co-Authors: Nattaporn Chuenjarern, Yang YangAbstract:Abstract In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-Component fluid mixture and the (volumetric) concentration of the Jth Component, c j , should be between 0 and 1. There are three main difficulties. Firstly, c j does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in Zhang and Shu (2010) [44] cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all c j ′ s and enforce ∑ j c j = 1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure d p / d t as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to c j . To construct the BP technique, we will not approximate c j directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.
Keith M. Kendrick - One of the best experts on this subject based on the ideXlab platform.
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p values of Student’s t test of the entropy ratios of the instantaneous amplitude of ith and Jth Component of IMFs of the uterine EMG signals.
2015Co-Authors: Peng Ren, Shuxia Yao, Pedro A. Valdes-sosa, Keith M. KendrickAbstract:p values of Student’s t test of the entropy ratios of the instantaneous amplitude of ith and Jth Component of IMFs of the uterine EMG signals.
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p values of Student’s t test of the entropy ratios of the instantaneous frequency of ith and Jth Component of IMFs of the uterine EMG signals.
2015Co-Authors: Peng Ren, Shuxia Yao, Pedro A. Valdes-sosa, Keith M. KendrickAbstract:p values of Student’s t test of the entropy ratios of the instantaneous frequency of ith and Jth Component of IMFs of the uterine EMG signals.
F Thomas Bruss - One of the best experts on this subject based on the ideXlab platform.
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Multiple buying or selling with vector offers
Journal of Applied Probability, 1997Co-Authors: Thomas S. Ferguson, F Thomas BrussAbstract:We consider a generalization of the house-selling problem to selling k houses. Let the offers, X1, X2, ⋯, be independent, identically distributed k-dimensional random vectors having a known distribution with finite second moments. The decision maker is to choose simultaneously k stopping rules, N1, ⋯, Nk, one for each Component. The payoff is the sum over j of the Jth Component of XN, minus a constant cost per observation until all stopping rules have stopped. Simple descriptions of the optimal rules are found. Extension is made to problems with recall of past offers and to problems with a discount.
Nattaporn Chuenjarern - One of the best experts on this subject based on the ideXlab platform.
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High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes
Journal of Computational Physics, 2019Co-Authors: Nattaporn Chuenjarern, Yang YangAbstract:Abstract In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-Component fluid mixture and the (volumetric) concentration of the Jth Component, c j , should be between 0 and 1. There are three main difficulties. Firstly, c j does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in Zhang and Shu (2010) [44] cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all c j ′ s and enforce ∑ j c j = 1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure d p / d t as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to c j . To construct the BP technique, we will not approximate c j directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.
