11,307 research outputs found
From discretization to regularization of composite discontinuous functions
Discontinuities between distinct regions, described by different equation sets, cause difficulties for PDE/ODE solvers. We present a new algorithm that eliminates integrator discontinuities through regularizing discontinuities. First, the algorithm determines the optimum switch point between two functions spanning adjacent or overlapping domains. The optimum switch point is determined by searching for a ājump pointā that minimizes a discontinuity between adjacent/overlapping functions. Then, discontinuity is resolved using an interpolating polynomial that joins the two discontinuous functions.
This approach eliminates the need for conventional integrators to either discretize and then link discontinuities through generating interpolating polynomials based on state variables or to reinitialize state variables when discontinuities are detected in an ODE/DAE system. In contrast to conventional approaches that handle discontinuities at the state variable level only, the new approach tackles discontinuity at both state variable and the constitutive equations level. Thus, this approach eliminates errors associated with interpolating polynomials generated at a state variable level for discontinuities occurring in the constitutive equations.
Computer memory space requirements for this approach exponentially increase with the dimension of the discontinuous function hence there will be limitations for functions with relatively high dimensions. Memory availability continues to increase with price decreasing so this is not expected to be a major limitation
The Representation of Natural Numbers in Quantum Mechanics
This paper represents one approach to making explicit some of the assumptions
and conditions implied in the widespread representation of numbers by composite
quantum systems. Any nonempty set and associated operations is a set of natural
numbers or a model of arithmetic if the set and operations satisfy the axioms
of number theory or arithmetic. This work is limited to k-ary representations
of length L and to the axioms for arithmetic modulo k^{L}. A model of the
axioms is described based on states in and operators on an abstract L fold
tensor product Hilbert space H^{arith}. Unitary maps of this space onto a
physical parameter based product space H^{phy} are then described. Each of
these maps makes states in H^{phy}, and the induced operators, a model of the
axioms. Consequences of the existence of many of these maps are discussed along
with the dependence of Grover's and Shor's Algorithms on these maps. The
importance of the main physical requirement, that the basic arithmetic
operations are efficiently implementable, is discussed. This conditions states
that there exist physically realizable Hamiltonians that can implement the
basic arithmetic operations and that the space-time and thermodynamic resources
required are polynomial in L.Comment: Much rewrite, including response to comments. To Appear in Phys. Rev.
On the Chow rings of classifying spaces for classical groups
We show how the stratification method, introduced by Vezzosi in his study of
PGL_3, provides a unified approach to the known computations of the Chow rings
of the classifying spaces of GL_n, SL_n, Sp_n, O_n and SO_n.Comment: 24 page
Double Schubert polynomials and degeneracy loci for the classical groups
We propose a theory of double Schubert polynomials P_w(X,Y) for the Lie types
B, C, D which naturally extends the family of Lascoux of Schutzenberger in type
A. These polynomials satisfy positivity, orthogonality, and stability
properties, and represent the classes of Schubert varieties and degeneracy loci
of vector bundles. When w is a maximal Grassmannian element of the Weyl group,
P_w(X,Y) can be expressed in terms of Schur-type determinants and Pfaffians, in
analogy with the type A formula of Kempf and Laksov. An example, motivated by
quantum cohomology, shows that there are no Chern class formulas for degeneracy
loci of ``isotropic morphisms'' of bundles.Comment: 34 pages, LaTeX; final versio
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