可合对称张量相等的条件

本文讨论了由一般对称化算子诱导的两非零可合对称张量相等的必要充分条件与可合对称张量为零的必要充分条件。     

Construction of orthonormal bases in higher symmetry classes of tensors

We present a method for constructing an orthonormal basis for a symmetry class of tensors from an orthonormal basis of the underlying vector space. The basis so obtained is not composed of decomposable symmetrized tensors. Indeed, we show that, for symmetry classes of tensors whose associated character has degree higher than one, it is impossible to construct an orthogonal basis of decomposable symmetrized tensors from any basis of the underlying vector space. We end with an open problem on the possibility of a symmetry class having an orthonormal basis of decomposable symmetrized tensors.     

Orthogonal bases of Brauer symmetry classes of tensors for the dihedral group

In this article necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the dihedral group.     

Orthogonality of cosets relative to irreducible characters of finite groups

Studied is an assumption on a group that ensures that no matter how the group is embedded in a symmetric group, the corresponding symmetrized tensor space has an orthogonal basis of standard (decomposable) symmetrized tensors.     

A high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors

We present a high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors, and this is a counter-example of the claim presented in [1].     

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

Upper and lower bounds for recurrent and recursively decomposable parallel processor-networks

A “recursively decomposable network” can be partitioned into a finite number of subnetworks that are smaller “versions of itself,” where the subnetworks are themselves recursively decomposable. Several interesting notions of such networks emerge depending on the collections of parameters chosen to model the relative behavioral characteristics that make a subnetwork a version of another. Examples of such parameters are permutation time, bandwidth, latency, topology, wires, degree, and size. This paper introduces and studies the class of networks that are recursively decomposable relative to bandwidth limitations, and the subclass of “recurrent networks” that are recursively decomposable relative to topology limitations. We show that an N-node recursively decomposable into halves network with bandwidth-inefficiency function β(x) must be at least $\frac{N}{2}\sum\nolimits_{i = l}^{\lg N}{\frac{1}{{\beta(2^i)}}}$ wires. This implies that the linear array, hypercube, and completely connected networks are all exactly optimal. The above lower bound is generalized to networks that are recursively decomposable, but not necessarily into halves. We show that the bound is tight up to a constant factor by exhibiting recurrent networks matching the above lower bounds. Our lower bound results both tighten and generalize a result of Meertens: no recurrent, fixed degree network can permute in O(log N) time. © 1996 John Wiley & Sons, Inc. ©1996 John Wiley & Sons, Inc.     

On mean values of multiplicative functions on the symmetric group

Mean values of nonnegative multiplicative functions defined on the symmetric group are explored in the paper. The result gives a sharp quantitative upper bound for their Cesàro mean. An approach that originated in number theory is adopted. It can be further applied for mappings defined on general decomposable structures, in particular, for estimating mean values with respect to multiplicative measures defined on additive partitions of a natural number.     

A Lagrangean based branch-and-cut algorithm for global optimization of nonconvex mixed-integer nonlinear programs with decomposable structures

In this work we present a global optimization algorithm for solving a class of large-scale nonconvex optimization models that have a decomposable structure. Such models, which are very expensive to solve to global optimality, are frequently encountered in two-stage stochastic programming problems, engineering design, and also in planning and scheduling. A generic formulation and reformulation of the decomposable models is given. We propose a specialized deterministic branch-and-cut algorithm to solve these models to global optimality, wherein bounds on the global optimum are obtained by solving convex relaxations of these models with certain cuts added to them in order to tighten the relaxations. These cuts are based on the solutions of the sub-problems obtained by applying Lagrangean decomposition to the original nonconvex model. Numerical examples are presented to illustrate the effectiveness of the proposed method compared to available commercial global optimization solvers that are based on branch and bound methods.     

Generalized Wave Operators in Von Neumann Algebras

Let M?B(H)be a countable decomposable properly infinite von Neumann algebra with a faithful normal semifinite tracial weight r where B(H)is the set of all bound...     

Decomposable subspaces,linear sections of Grassmann varieties,and higher weights of Grassmann codes

We consider the question of determining the maximum number of points on sections of Grassmannians over finite fields by linear subvarieties of the Plücker projective space of a fixed codimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties. We recover most of the known results as well as prove some new results. A basic tool used is a characterization of decomposable subspaces of exterior powers, that is, subspaces in which every nonzero element is decomposable. Also, we use a generalization of the Griesmer–Wei bound that is proved here for arbitrary linear codes.     

Planar graphs decomposable into a forest and a matching

He, Hou, Lih, Shao, Wang, and Zhu showed that a planar graph of girth 11 can be decomposed into a forest and a matching. Borodin, Kostochka, Sheikh, and Yu improved the bound on girth to 9. We give sufficient conditions for a planar graph with 3-cycles to be decomposable into a forest and a matching.     

Orthogonal splitting and class numbers of quadratic forms

Orthogonal splitting for lattices on quadratic spaces over algebraic number fields is studied. It is seen that if the rank of a lattice is sufficiently large, then its spinor genus must contain a decomposable lattice. Also, splitting theory is used to obtain a lower bound for the class number of a lattice (in the definite case) in terms of its rank, via the partition function.     

Choice of strategies for extrapolation with symmetrization in the constant stepsize setting

Symmetrization has been shown to be efficient in solving stiff problems. In the constant stepsize setting, we study four ways of applying extrapolation with symmetrization. We observe that for stiff linear problems the symmetrized Gauss methods are more efficient than the symmetrized Lobatto IIIA methods of the same order. However, for two-dimensional nonlinear problems, the symmetrized 4-stage Lobatto IIIA method is more efficient. In all cases, we observe numerically that passive symmetrization with passive extrapolation is more efficient than active symmetrization with active extrapolation.     

Estimates of the Carathéodory metric on the symmetrized polydisc

Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.     

Dominions in decomposable varieties

Dominions, in the sense of Isbell, are investigated in the context of decomposable varieties of groups. An upper and lower bound for dominions in such a variety is given in terms of the two varietal factors, and the internal structure of the group being analyzed. Finally, the following result is established: If a variety has instances of nontrivial dominions, then for any proper subvariety of , also has instances of nontrivial dominions. Received September 2, 1998; accepted in final form October 24, 1999.     

Decomposition of partial orders

A decomposition theory for partial orders which arises from the split decomposition of submodular functions is introduced. As a consequence of this theory, any partial order has a unique decomposition consisting of indecomposable partial orders and certain highly decomposable partial orders. The highly decomposable partial orders are completely characterized. As a special case of partial orders, we consider lattices and distributive lattices. It occurs, that the highly decomposable distributive lattices are precisely the Boolean lattices.     

Robust decomposable Markov decision processes motivated by allocating school budgets

Motivated by an application to school funding, we introduce the notion of a robust decomposable Markov decision process (MDP). A robust decomposable MDP model applies to situations where several MDPs, with the transition probabilities in each only known through an uncertainty set, are coupled together by joint resource constraints. Robust decomposable MDPs are different than both decomposable MDPs, and robust MDPs and cannot be solved by a direct application of the solution methods from either of those areas. In fact, to the best of our knowledge, there is no known method to tractably compute optimal policies in robust, decomposable MDPs. We show how to tractably compute good policies for this model, and apply the derived method to a stylized school funding example.     

The duals of almost completely decomposable groups

In this paper, we classify the direct products of one-dimensional compact connected abelian groups by cardinal invariants dualizing Baer’s classification theorem of completely decomposable groups. Almost completely decomposable groups are finite rank torsion-free abelian groups which contain a completely decomposable group of finite index. An isomorphism theorem for their Pontrjagin dual groups is given by using the dual concept of a regulating subgroup.