13,416 research outputs found
A Spectral Theory for Tensors
In this paper we propose a general spectral theory for tensors. Our proposed
factorization decomposes a tensor into a product of orthogonal and scaling
tensors. At the same time, our factorization yields an expansion of a tensor as
a summation of outer products of lower order tensors . Our proposed
factorization shows the relationship between the eigen-objects and the
generalised characteristic polynomials. Our framework is based on a consistent
multilinear algebra which explains how to generalise the notion of matrix
hermicity, matrix transpose, and most importantly the notion of orthogonality.
Our proposed factorization for a tensor in terms of lower order tensors can be
recursively applied so as to naturally induces a spectral hierarchy for
tensors.Comment: The paper is an updated version of an earlier versio
Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
We introduce the framework of path-complete graph Lyapunov functions for
approximation of the joint spectral radius. The approach is based on the
analysis of the underlying switched system via inequalities imposed among
multiple Lyapunov functions associated to a labeled directed graph. Inspired by
concepts in automata theory and symbolic dynamics, we define a class of graphs
called path-complete graphs, and show that any such graph gives rise to a
method for proving stability of the switched system. This enables us to derive
several asymptotically tight hierarchies of semidefinite programming
relaxations that unify and generalize many existing techniques such as common
quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov
functions. We compare the quality of approximation obtained by certain classes
of path-complete graphs including a family of dual graphs and all path-complete
graphs with two nodes on an alphabet of two matrices. We provide approximation
guarantees for several families of path-complete graphs, such as the De Bruijn
graphs, establishing as a byproduct a constructive converse Lyapunov theorem
for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has
gone through two major rounds of revision. In particular, a section on the
performance of our algorithm on application-motivated problems has been added
and a more comprehensive literature review is presente
Ranges of bimodule projections and reflexivity
We develop a general framework for reflexivity in dual Banach spaces,
motivated by the question of when the weak* closed linear span of two reflexive
masa-bimodules is automatically reflexive. We establish an affirmative answer
to this question in a number of cases by examining two new classes of
masa-bimodules, defined in terms of ranges of masa-bimodule projections. We
give a number of corollaries of our results concerning operator and spectral
synthesis, and show that the classes of masa-bimodules we study are operator
synthetic if and only if they are strong operator Ditkin
Native defects in the CoTi ( Si, Ge, Sn) full Heusler alloys: formation and influence on the thermoelectric properties
We have performed first-principles investigations on the native defects in
the full Heusler alloys CoTi ( one of the group IV elements Si, Ge,
Sn), determining their formation energies and how they influence the transport
properties. We find that Co vacancies (Vc) in all compounds and the
Ti anti-site exhibit negative formation energies. The smallest
positive values occur for Co in excess on anti-sites (Co or Co)
and for Ti. The most abundant native defects were modeled as dilute alloys,
treated with the coherent potential approximation in combination with the
multiple-scattering theory Green function approach. The self-consistent
potentials determined this way were used to calculate the residual resistivity
via the Kubo-Greenwood formula and, based on its energy dependence, the Seebeck
coefficient of the systems. The latter is shown to depend significantly on the
type of defect, leading to variations that are related to subtle, spin-orbit
coupling induced, changes in the electronic structure above the half-metallic
gap. Two of the systems, Vc and Co, are found to exhibit a
negative Seebeck coefficient. This observation, together with their low
formation energy, offers an explanation for the experimentally observed
negative Seebeck coefficient of the CoTi compounds as being due to
unintentionally created native defects
Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis
A method for synthesizing quantum gates is presented based on interpolation
methods applied to operators in Hilbert space. Starting from the diagonal forms
of specific generating seed operators with non-degenerate eigenvalue spectrum
one obtains for arity-one a complete family of logical operators corresponding
to all the one-argument logical connectives. Scaling-up to n-arity gates is
obtained by using the Kronecker product and unitary transformations. The
quantum version of the Fourier transform of Boolean functions is presented and
a Reed-Muller decomposition for quantum logical gates is derived. The common
control gates can be easily obtained by considering the logical correspondence
between the control logic operator and the binary propositional logic operator.
A new polynomial and exponential formulation of the Toffoli gate is presented.
The method has parallels to quantum gate-T optimization methods using powers of
multilinear operator polynomials. The method is then applied naturally to
alphabets greater than two for multi-valued logical gates used for quantum
Fourier transform, min-max decision circuits and multivalued adders
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