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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

Discretization-Accuracy Convergence for Full Algebraic Multigrid

Full multigrid (FMG) is well known for converging to the level of discretization accuracy in a single cycle for a wide class of partial differential equations when the multigrid hierarchy is derived from problem geometry. When applying an FMG cycle to a hierarchy generated by algebraic multigrid (AMG), however, this scalable convergence to discretization accuracy is usually lost. Furthermore, this paper examines the cause of this loss and explores some improvements to standard AMG interpolation that can restore single-cycle convergence to discretization accuracy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Topos models for physics and topos theory

What is the role of topos theory in the topos models for quantum theory as used by Isham, Butterfield, Döring, Heunen, Landsman, Spitters, and others? In other words, what is the interplay between physical motivation for the models and the mathematical framework used in these models? Concretely, we show that the presheaf topos model of Butterfield, Isham, and Döring resembles classical physics when viewed from the internal language of the presheaf topos, similar to the copresheaf topos model of Heunen, Landsman, and Spitters. Both the presheaf and copresheaf models provide a “quantum logic” in the form of a complete Heyting algebra. Although these algebras are natural from a topos theoretic stance, we seek a physical interpretation for the logical operations. Finally, we investigate dynamics. In particular, we describe how an automorphism on the operator algebra induces a homeomorphism (or isomorphism of locales) on the associated state spaces of the topos models, and how elementary propositions and truth values transform under the action of this homeomorphism. Also with dynamics the focus is on the internal perspective of the topos.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2013

A diagrammatic approach to the categorical coherent state

In this paper, we study the categorification of the coherent states, which is equivalent to the categorification of corresponding displacement operators. Based on the categorification of Heisenberg algebras, we construct some complexes in a homotopy category which can be considered as the categorical analogues of the displacement operators. Using the diagrammatic calculus, we find that the properties of the categorical displacement operators coincide with those in normal quantum mechanics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2013

The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl{sub -1}(2)

The algebra H of the dual -1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl{sub -1}(2). The dual -1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q{yields}-1 limit of the dual q-Hahn polynomials. The Hopf algebra sl{sub -1}(2) has four generators including an involution, it is also a q{yields}-1 limit of the quantum algebra sl{sub q}(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of u(2) with an involution as additional generator, is first derived from the recurrence relation of the -1 Hahn polynomials. It is then shown that H can be realized in terms of the generators of two added sl{sub -1}(2) algebras, so that the Clebsch-Gordan coefficients of sl{sub -1}(2) are dual -1 Hahn polynomials. An irreducible representation of H involving five-diagonal matrices and connected to the difference equation of the dual -1 Hahn polynomials is constructed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Some applications of mathematics in theoretical physics - A review

Mathematics is a very beautiful subject−very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like−differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical tools are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Multidimensional smooth loops with universal elasticity

Let E-tilde be a universal (isotopically invariant) identity that is derived from the elasticity identity E:(xy)x=x(yx). One of the authors has previously shown that a) each local loop of dimension r with identity E-tilde (briefly, a loop E-tilde ) is a smooth middle Bol loop of dimension r; b) smooth two-dimensional loops E-tilde are Lie groups; c) up to isotopy, there exist only two three-dimensional loops E-tilde : the loops E{sub 1} and E{sub 2}. In this paper, the loops E{sub 1} and E{sub 2} are extended to the multidimensional case. The fact that each smooth loop E-tilde of dimension r corresponds to a unique multidimensional three-web on a manifold of dimension 2r is key to our work. In addition, the class of loops under investigation is characterized by the fact that the torsion tensor of the corresponding web has rank 1 (that is, the algebra generated by this tensor has a one-dimensional derived algebra). This enables us to express the differential equations of the problem in an invariant form. The system of equations thus obtained was found to be amenable to integration in the most general case, and the equations of the required loops have been obtained in local coordinates. Bibliography: 17 titles.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method

Here, we propose two multilevel spectral techniques for constructing coarse discretization spaces for saddle-point problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar self-adjoint second order elliptic equations on general unstructured grids. We use element agglomeration algebraic multigrid (AMGe), which employs coarse elements that can have nonstandard shape since they are agglomerates of fine-grid elements. The coarse basis associated with each agglomerated coarse element is constructed by solving local eigenvalue problems and local mixed finite element problems. This construction leads to stable upscaled coarse spaces and guarantees the inf-sup compatibility of the upscaled discretization. Also, the approximation properties of these upscaled spaces improve by adding more local eigenfunctions to the coarse spaces. The higher accuracy comes at the cost of additional computational effort, as the sparsity of the resulting upscaled coarse discretization (referred to as operator complexity) deteriorates when we introduce additional functions in the coarse space. We also provide an efficient solver for the coarse (upscaled) saddle-point system by employing hybridization, which leads to a symmetric positive definite (s.p.d.) reduced system for the Lagrange multipliers, and to solve the latter s.p.d. system, we use our previously developed spectral AMGe solver. Numerical experiments, in both two and three dimensions, are provided to illustrate the efficiency of the proposed upscaling technique.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Derivatives at the boundary for analytic Lipschitz functions

We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent α, with 0<α<1, in the vicinity of an exceptional boundary point where all such functions exhibit some kind of smoothness. Specifically, we consider the relation between the abstract idea of a bounded point derivation on the algebra of such functions and the classical complex derivative evaluated as a limit of difference quotients. We show that whenever such a bounded point derivation exists at a boundary point b, it may be evaluated by taking a limit of classical difference quotients, approaching from a set having full area density at b. Bibliography: 13 titles. (paper)

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Medical diagnoses using three-way fuzzy concept lattice and their Euclidean distance

Searching the closest symptoms of a particular disease in the given patient is a major concern for medical diagnoses expert. In most cases, this problem arises due to large number of incomplete, uncertain, or inconsistent information generated in medical diagnoses data set. Recently, some of the researchers tried to pay attention towards precise measurement of uncertainty and vagueness in medical data set using the properties of applied abstract algebra. In this process, researchers noticed that calculus of three-way decision space is more useful to characterize the uncertainty and vagueness in medical data set based on truth, indeterminacy, and falsity membership-values. However, searching some relevant query based on user requirement to diagnoses the disease is a major issue in three-way decision space. To encounter this problem, current paper tried to analyze the medical data set using the properties of single-valued neutrosophic graph-based concept lattice. In addition, another method is proposed to select some of the interesting three-way fuzzy concepts at user-defined granulation for their computed Euclidean distance with an illustrative example.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Regularization and index reduction for linear differential–algebraic systems

In this paper, a necessary and sufficient condition for the existence of a proportional semistate feedback is established such that the closed loop system is regular and of index at most two. The condition is characterized by a rank condition that involves only the original system coefficient matrices. Also, a new rank condition ensuring that a given linear time-invariant descriptor system is regular and of index at most some specific value is also derived. The developed theory is illustrated through physical and numerical examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

On the Additive Structure and Asymptotics of Codimensions c{sub n} in the Algebra F{sup (5)}

In this paper, we investigate the additive structure of the algebra F{sup (5)}, i.e., a relatively free, associative, countably-generated algebra with the identity [x{sub 1}, . . . , x{sub 5}] = 0 over an infinite field of characteristic ≠2, 3. We study the space of proper multilinear polynomials in this algebra and means of basis construction in one of its basic subspaces. As an additional result, we obtain estimations of codimensions c{sub n} = dimP{sub n}/P{sub n}∩ T{sup (5)}, where P{sub n} is the space of multilinear polynomials of degree n in F{sup (5)} and T{sup (5)} is the T-ideal generated by the long commutator [x{sub 1}, . . . , x{sub 5}].

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

The Limit Shape of a Probability Measure on a Tensor Product of Modules of the B{sub n} Algebra

We study a probability measure on the integral dominant weights in the decomposition of the Nth tensor power of the spinor representation of the Lie algebra so(2n + 1). The probability of a dominant weight λ is defined as the dimension of the irreducible component of λ divided by the total dimension 2{sup nN} of the tensor power. We prove that as N →∞, the measure weakly converges to the radial part of the SO(2n+1)-invariant measure on so(2n+1) induced by the Killing form. Thus, we generalize Kerov’s theorem for su(n) to so(2n + 1).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Geometric properties of commutative subalgebras of partial differential operators

We investigate further algebro-geometric properties of commutative rings of partial differential operators, continuing our research started in previous articles. In particular, we start to explore the simplest and also certain known examples of quantum algebraically completely integrable systems from the point of view of a recent generalization of Sato's theory, developed by the first author. We give a complete characterization of the spectral data for a class of 'trivial' commutative algebras and strengthen geometric properties known earlier for a class of known examples. We also define a kind of restriction map from the moduli space of coherent sheaves with fixed Hilbert polynomial on a surface to an analogous moduli space on a divisor (both the surface and the divisor are part of the spectral data). We give several explicit examples of spectral data and corresponding algebras of commuting (completed) operators, producing as a by-product interesting examples of surfaces that are not isomorphic to spectral surfaces of any (maximal) commutative ring of partial differential operators of rank one. Finally, we prove that any commutative ring of partial differential operators whose normalization is isomorphic to the ring of polynomials k[u,t] is a Darboux transformation of a ring of operators with constant coefficients. Bibliography: 39 titles.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Continuous and Smooth Envelopes of Topological Algebras. Part 1

Since the first optical instruments were invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way of formalizing this idea in mathematics is the construction that assigns to an arbitrary object A in a category K its envelope Env{sub Φ}{sup Ω}A in a given class of morphisms (a class of representations) Ω with respect to a given class of morphisms (a class of observation tools) Φ. It turns out that if we take a sufficiently wide category of topological algebras as K, then each choice of the classes Ω and Φ defines a “projection of functional analysis into geometry,” and the standard “geometric disciplines,” like complex geometry, differential geometry, and topology, become special cases of this construction. This gives a formal scheme of “categorical construction of geometries” with many interesting applications, in particular, “geometric generalizations of the Pontryagin duality” (to the classes of noncommutative groups). In this paper we describe this scheme in topology and in differential geometry.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Continuous and Smooth Envelopes of Topological Algebras. Part 2

Since the first optical instruments were invented, the idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way of formalizing this idea in mathematics is the construction that assigns to an arbitrary object A in a category K its envelope Env{sub Φ}{sup Ω} A in a given class of morphisms (a class of representations) Ω with respect to a given class of morphisms (a class of observation tools) Φ. It turns out that if we take a sufficiently wide category of topological algebras as K, then each choice of the classes Ω and Φ defines a “projection of functional analysis into geometry,” and the standard “geometric disciplines,” like complex geometry, differential geometry, and topology, become special cases of this construction. This gives a formal scheme of “categorical construction of geometries” with many interesting applications, in particular, “geometric generalizations of the Pontryagin duality” (to the classes of noncommutative groups). In this paper we describe this scheme in topology and in differential geometry.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Noncommutative reciprocity laws on algebraic surfaces: the case of tame ramification

We prove noncommutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws establish that some central extensions of globally constructed groups split over certain subgroups constructed by points or projective curves on a surface. For a two-dimensional local field with a last finite residue field, the local central extension which is constructed is isomorphic to the central extension which comes from the case of tame ramification of the Abelian two-dimensional local Langlands correspondence suggested by Kapranov. Bibliography: 9 titles.

97 MATHEMATICS AND COMPUTING↗