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Torsion classes of finite type and spectra
Get PDFGiven a commutative ring (respectively a positively graded
commutative ring A=\ps_{j\geq 0}A_j which is finitely generated as
an -algebra), a bijection between the torsion classes of finite
type in \Rfp (respectively tensor torsion classes of finite type
in \QGr A) and the set of all subsets Y\subseteq\spec R
(respectively Y\subseteq\Proj A) of the form
, with \spec R\setminus Y_i
(respectively \Proj A\setminus Y_i) quasi-compact and open for all
, is established. Using these bijections, there are
constructed isomorphisms of ringed spaces
(\spec R,\cc O_{R})\lra{\sim}(\spec(\Rfp),\cc O_{\Rfp})
and
(\Proj A,\cc
O_{\Proj A})\lra{\sim}(\spec(\QGr A),\cc O_{\QGr A}),
where (\spec(\Rfp),\cc O_{\Rfp}) and (\spec(\QGr A),\cc O_{\QGr
A}) are ringed spaces associated to the lattices L_{\serre}(\Rfp)
and L_{\serre}(\QGr A) of torsion classes of finite type. Also, a
bijective correspondence between the thick subcategories of perfect
complexes \perf(R) and the torsion classes of finite type in
\Rfp is established
Vector Spaces of Linearizations for Matrix Polynomials
Get PDFThe classical approach to investigating polynomial eigenvalue problems is linearization,
where the polynomial is converted into a larger matrix pencil with the same eigenvalues. For
any polynomial there are innitely many linearizations with widely varying properties, but in practice
the companion forms are typically used. However, these companion forms are not always entirely
satisfactory, and linearizations with special properties may sometimes be required.
Given a matrix polynomial P, we develop a systematic approach to generating large classes of
linearizations for P. We show how to simply construct two vector spaces of pencils that generalize the
companion forms of P, and prove that almost all of these pencils are linearizations for P. Eigenvectors
of these pencils are shown to be closely related to those of P. A distinguished subspace is then
isolated, and the special properties of these pencils are investigated. These spaces of pencils provide
a convenient arena in which to look for structured linearizations of structured polynomials, as well
as to try to optimize the conditioning of linearizations [7], [8], [12]
Integrable matrix equations related to pairs of compatible associative algebras
No full textWe study associative multiplications in semi-simple associative algebras over {\bb C} compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of \skew5\tilde{A}_{2 k-1}, \tilde{D}_{k}, \tilde{E}_{6}, \tilde{E}_{7} , and \tilde{E}_{8} -type. In this paper we investigate in detail the multiplications of the \skew5\tilde{A}_{2 k-1} -type and integrable matrix ODEs and PDEs generated by them
The cohomology of certain 2-local finite groups
No full textThere exist spaces BSol(q) which are the classifying spaces of a family of 2-local finite groups based on certain fusion system over the Sylow 2-subgroups of Spin7(q). In this paper we calculate the cohomology of BSol(q) as an algebra over the Steenrod algebra . We also provide the calculation of the cohomology algebra over of the finite group of Lie type G2(q)
The Conjugacy Problem in Amalgamated Products I: Regular Elements and Black Holes
No full textWe discuss the time complexity of the word and conjugacy problems for free products of two groups with
amalgamation over a subgroup. We stratify the set of elements of the fre product with respect to the complexity of the word and conjugacy problems and show that for the generic stratum the conjugacy search problem is decidable under some reasonable assumptions about the groups groups involved. Moreover, the decision algorithm is fast on
the generic stratum
Conjugate connections and differential equations on infinite dimensional manifolds
Get PDFOn a smooth manifold M, the vector bundle structures of the
second order tangent bundle, T^2M, bijectively correspond to
linear connections. In this paper we classify such structures for
those Frechet manifolds which can be considered as projective
limits of Banach manifolds. We investigate also the relation
between ordinary differential equations on Frechet spaces and
the linear connections on their trivial bundle. Such equations
arise in theoretical physics
The Weak Euler Scheme for Stochastic Differential Delay Equations
Get PDFWe develop a weak numerical Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The weak Euler scheme has
order of convergence 1, as in the case of stochastic ordinary
differential equations (SODEs) (i.e., without delay).The result
holds for SDDEs with multiple finite fixed delays in the drift and
diffusion terms. Although the set-up is non-anticipating,
our approach uses the Malliavin calculus and the
anticipating stochastic analysis techniques of Nualart
and Pardoux
A boundary point lemma for Black-Scholes type equations
No full textWe prove a sharp version of the Hopf boundary point lemma for Black-Scholes type equations. We also investigate the existence and the regularity of the spatial derivative of the solutions at the spatial boundary
Structured Linearizations for Matrix Polynomials
Get PDFThe classical approach to investigating polynomial eigenvalue problems is linearization, where the underlying matrix polynomial is converted into a larger matrix pencil
with the same eigenvalues. For any polynomial there are infinitely many linearizations with widely varying properties, but in practice the companion forms are typically used. However, these companion forms are not always entirely satisfactory, and
linearizations with special properties may sometimes be required.
Given a matrix polynomial P, we develop a systematic approach to generating
large classes of linearizations for P. We show how to simply construct two vector
spaces of pencils that generalize the companion forms of P, and prove that almost all
of these pencils are linearizations for P. Eigenvectors of these pencils are shown to
be closely related to those of P. A distinguished subspace, denoted DL(P), is then
isolated, and the special properties of these pencils are investigated. These spaces of
pencils provide a convenient arena in which to look for structured linearizations of
structured polynomials, as well as to try to optimize the conditioning of linearizations.
Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial; perhaps the most well-known are symmetric and
Hermitian polynomials. In this thesis we also identify several less well-known types
of structured polynomial (e.g., palindromic, even, odd), explore the relationships
between them, and illustrate their appearance in a variety of applications. Special
classes of linearizations that respect the structure of these polynomials, and therefore
preserve symmetries in their spectra, are introduced and investigated. We analyze
the existence and uniqueness of such linearizations, and show how they may be systematically constructed.
The infinitely many linearizations of any given polynomial P can have widely
varying eigenvalue condition numbers. We investigate the conditioning of linearizations from DL(P), looking for the best conditioned linearization in that space and
comparing its conditioning with that of the original polynomial. We also analyze the
eigenvalue conditioning of the widely used first and second companion linearizations,
and find that they can potentially be much more ill conditioned than P. Our results
are phrased in terms of both the standard relative condition number and the condition number of Dedieu and Tisseur for the problem in homogeneous form, this latter
condition number having the advantage of applying to zero and infinite eigenvalues
Tomographic reconstruction of stress from photoelastic measurements using elastic regularization.
Get PDFIn this paper we consider the problem of recovering the stress tensor field of a three dimensional object from
measurements of the polarization state of transmitted light. In contrast to the ray transform approach suggested
by Sharafutdinov, which uses the inversion of planar Radon transforms to recover a single component of the
deviatoric stress normal to a plane, we study the simultaneous reconstruction of all components of the deviatoric
stress in each voxel using a matrix approximation to the truncated transverse ray transform. This approach allows us to employ partial differential operators related to linear elasticity in a regularizing penalty term resulting in a well posed problem. We note that the hydrostatic stress is determined by the deviatoric stress (up to
an additive constant) from the equilibrium equation, and that our numerical results confirm that the full stress tensor can be recovered using elastic regularization