417 research outputs found
Lagrangian submanifolds in affine symplectic geometry
We uncover the lowest order differential invariants of Lagrangian
submanifolds under affine symplectic maps, and find out what happens when they
are constant.Comment: 23 pages, no figure
A Nonconforming Finite Element Approximation for the von Karman Equations
In this paper, a nonconforming finite element method has been proposed and
analyzed for the von Karman equations that describe bending of thin elastic
plates. Optimal order error estimates in broken energy and norms are
derived under minimal regularity assumptions. Numerical results that justify
the theoretical results are presented.Comment: The paper is submitted to an international journa
Generalized Newton's Method based on Graphical Derivatives
This paper concerns developing a numerical method of the Newton type to solve
systems of nonlinear equations described by nonsmooth continuous functions. We
propose and justify a new generalized Newton algorithm based on graphical
derivatives, which have never been used to derive a Newton-type method for
solving nonsmooth equations. Based on advanced techniques of variational
analysis and generalized differentiation, we establish the well-posedness of
the algorithm, its local superlinear convergence, and its global convergence of
the Kantorovich type. Our convergence results hold with no semismoothness
assumption, which is illustrated by examples. The algorithm and main results
obtained in the paper are compared with well-recognized semismooth and
-differentiable versions of Newton's method for nonsmooth Lipschitzian
equations
Discrete phase-space structure of -qubit mutually unbiased bases
We work out the phase-space structure for a system of qubits. We replace
the field of real numbers that label the axes of the continuous phase space by
the finite field \Gal{2^n} and investigate the geometrical structures
compatible with the notion of unbiasedness. These consist of bundles of
discrete curves intersecting only at the origin and satisfying certain
additional properties. We provide a simple classification of such curves and
study in detail the four- and eight-dimensional cases, analyzing also the
effect of local transformations. In this way, we provide a comprehensive
phase-space approach to the construction of mutually unbiased bases for
qubits.Comment: Title changed. Improved version. Accepted for publication in Annals
of Physic
Aspects of higher degree forms with symmetries
Bibliography: pages 113-119.In Chapter One we develop a basis for studying higher degree alternating forms. The concepts and results we present are mostly obvious analogues of Harrison's treatment of higher degree symmetric forms. We explain antisymmetrization; discuss the derivative of an alternating form and its corresponding anticommutative polynomial; define alternating spaces and their direct sum; establish decomposition and cancellation results for alternating spaces; and construct a Witt-Grothendieck group of alternating spaces. In Chapter Two we discuss hyperbolic alternating space. We compute the centre, algebraic isometry group and its corresponding Lie algebra, and prove a descent result. There are important parallels with Keet's results for hyperbolic symmetric spaces, as well as significant differences, especially in the methods we employ. In Chapter Three we develop a framework for the study of two aspects of forms of general Young symmetry type: their hyperbolics, and a generalization of the Weil-Siegel duality between symmetric and alternating bilinear forms. We introduce notions like nondegeneracy, derivative of a form, and derivative and integral symmetry types, and are then able to construct a hyperbolic space which is cofinal for spaces equipped with a form of the same symmetry type, and show that symmetry types are Siegel duals in our generalized sense if they have the same derivative symmetry type. In Chapter Four we present a few results and observations concerning nondegeneracytype conditions on symmetric forms. These include: an extension of Harrison's proof that nonsingularity implies nonzero Hessian to forms of arbitrary degree; a discussion of s-nondegeneracy and s-regularity; and a relation between a strong nondegeneracy condition on forms of even degree and the catalecticant, a classical invariant
Real, complex and quaternionic toric spaces
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1993.Includes bibliographical references (p. 61-62).by Richard A. Scott.Ph.D
The moduli space of stable quotients
A moduli space of stable quotients of the rank n trivial sheaf on stable
curves is introduced. Over nonsingular curves, the moduli space is
Grothendieck's Quot scheme. Over nodal curves, a relative construction is made
to keep the torsion of the quotient away from the singularities. New
compactifications of classical spaces arise naturally: a nonsingular and
irreducible compactification of the moduli of maps from genus 1 curves to
projective space is obtained. Localization on the moduli of stable quotients
leads to new relations in the tautological ring generalizing Brill-Noether
constructions.
The moduli space of stable quotients is proven to carry a canonical 2-term
obstruction theory and thus a virtual class. The resulting system of descendent
invariants is proven to equal the Gromov-Witten theory of the Grassmannian in
all genera. Stable quotients can also be used to study Calabi-Yau geometries.
The conifold is calculated to agree with stable maps. Several questions about
the behavior of stable quotients for arbitrary targets are raised.Comment: 50 page
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