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Flanders� theorem for many matrices under commutativity assumptions
Get PDFWe analyze the relationship between the Jordan canonical form of products, in different orders, of square matrices . Our results extend some classical results by H. Flanders. For the eigenvalue zero, we obtain an upper bound on the difference between the sizes of Jordan blocks for any two permuted products, and we show that this bound is attainable. For we show that, moreover, the bound is exhaustive
The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential
Get PDFA new matrix function corresponding to the scalar unwinding number of
Corless, Hare, and Jeffrey is introduced.
This matrix unwinding function, , is shown to
be a valuable tool for deriving
identities involving the matrix logarithm and
fractional matrix powers,
revealing, for example, the
precise relation between and .
The unwinding function is also shown to be closely connected with the matrix
sign function.
An algorithm for computing the unwinding function based on the
Schur--Parlett method with a special reordering is proposed.
It is shown that matrix argument reduction using the function
,
which has eigenvalues with imaginary parts in the interval
and for which \e^A = \e^{\mathrm{mod}(A)},
can give significant computational savings in the evaluation
of the exponential by scaling and squaring algorithms
On the local Langlands correspondence for non-tempered representations
No full textLet be a reductive -adic group. We study how a local Langlands correspondence for
irreducible tempered -representations can be extended to a local Langlands correspondence
for all irreducible smooth representations of . We prove that, under a natural condition
involving compatibility with unramified twists, this is possible in a canonical way.
To this end we introduce analytic R-groups associated to non-tempered essentially
square-integrable representations of Levi subgroups of . We establish the basic properties
of these new R-groups, which generalize Knapp-Stein R-groups
The Irrelevant Information Principle for Collective Probabilistic Reasoning
No full textWithin the framework of discrete probabilistic uncertain reasoning a large literature exists justifying the maximum entropy inference process, ME, as being optimal in the context of a single agent whose subjective probabilistic knowledge base is consistent. In particular Paris and Vencovska completely characterised the ME inference process by means of an attractive set of axioms which an inference process should satisfy.
More recently the second author extended the Paris-Vencovska axiomatic approach to inference processes in the context of several agents whose subjective probabilistic knowledge bases, while individually consistent, may be collectively inconsistent. In particular he defined a natural multi--agent extension of the inference process ME called the social entropy process, SEP. However, while SEP has been shown to possess many attractive properties, those which are known are almost certainly insufficient to uniquely characterise it. It is therefore of particular interest to study those Paris-Vencovska principles valid for ME whose immediate generalisations to the multi-agent case are not satisfied by SEP. One of these principles is the Irrelevant Information Principle, a powerful and appealing principle which very few inference processes satisfy even in the single agent context. In this paper we will investigate whether SEP can satisfy an interesting modified generalisation of this principle
Depth and the local Langlands correspondence
No full textLet G be an inner form of a general linear group or a special linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths
Higher Order Frechet Derivatives of Matrix Functions and the Level-2 Condition Number
No full textThe Fr\'echet derivative of a matrix function
controls the sensitivity of the function to small perturbations
in the matrix.
While much is known about the properties of
and how to compute it,
little attention has been given to higher order Fr\'echet derivatives.
We derive sufficient conditions for the
th Fr\'echet derivative to exist and be continuous
in its arguments
and we develop algorithms for computing the th derivative and its
Kronecker form.
We analyze the level-2 absolute condition number of a
matrix function (``the condition number of the condition number'')
and bound it in terms of the second Fr\'echet derivative.
For normal matrices and the exponential we show that in the 2-norm
the level-1 and level-2 absolute condition numbers are equal
and that the relative condition numbers
are within a small constant factor of each other.
We also obtain an exact relationship between the level-1 and level-2
absolute condition numbers for
the matrix inverse and arbitrary nonsingular matrices,
as well as a weaker connection for Hermitian matrices
for a class of functions that includes the logarithm and square root.
Finally, the relation between the level-1 and level-2 condition numbers
is investigated more generally through numerical experiments
FREE CENTRE-BY-(ABELIAN-BY-EXPONENT 2) GROUPS
No full textWe study free centre-by-(abelian-by-exponent 2) groups. Our
main result is a complete description of the centre. It is isomorphic to a direct sum of a free abelian group and a torsion subgroup. The latter is a direct sum of cyclic groups of order two and cyclic groups of order four. We
exhibit a generating set consisting of elements of innite order, order 2, and order 4, such that the centre is the direct sum of cyclic subgroups generated by those generators. Our approach makes essential use of homological methods
On Local Fusion Graphs of Finite Coxeter Groups
No full textGiven a finite group G and G-conjugacy class of involutions X, the local fusion graph F(G,X) has X as its vertex set, with x,y in X joined by an edge if, and only if, x is not equal to y and the product xy has odd order. In this note we investigate such graphs when G is a finite Coxeter group, addressing questions of connectedness and diameter. In particular, our results show that local fusion graphs may have an arbitrary number of connected components, each with arbitrarily large diameter
Large-Scale Metabolic Models: From Reconstruction to Differential Equations
Get PDFGenome-scale kinetic models of metabolism are important for rational design of the metabolic engineering required for industrial biotechnology applications. They allow one to predict the alterations needed to optimize the flux or yield of the compounds of interest, while keeping the other functions of the host organism to a minimal, but essential, level. We define a pipeline for the generation of genome-scale kinetic models from reconstruction data. To build such a model, inputs of all concentrations, fluxes, rate laws, and kinetic parameters are required. However, we propose typical estimates for these numbers when experimental data are not available. While little data are required to produce the model, the pipeline ensures consistency with any known flux or concentration data, or any kinetic constants. We apply the method to create genome-scale models of Escherichia coli and Saccharomyces cerevisiae. We go on to show how these may be used to expand a detailed model of yeast glycolysis to the genome level
An assessment of sustainable housing affordability using a multiple criteria decision making method
Get PDFHousing affordability is a complex issue that must not only be assessed in terms economic viability. In order to increase quality of life and community sustainability the environmental and social sustainability of housing must also be taken into consideration.
The paper considers the application of a methodology that can be applied to assess the affordability of different housing locations in a sustainable manner, taking into account a range of economic, environmental and social criteria. The COPRAS method of multi-criteria decision making (MCDM) is selected and applied to three residential areas as an example of how sustainable housing affordability can be assessed using a MCDM method. The outcome of the study reveals that considering a range of social and environmental criteria can greatly affect the calculation of an areas affordability, in comparison to focusing solely on financial attributes. COPRAS was found to be an effective method for the assessment and could be applied in other regions or internationally