Change of bases in the Steenrod algebra modp
We study the triangular property for changes of additive bases in the Steenrod algebra. Bibliography: 9 titles. (paper)
97 MATHEMATICS AND COMPUTING↗
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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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Accelerating Scientific Computing in the Post-Moore’s Era
Novel uses of graphical processing units for accelerated computation revolutionized the field of high-performance scientific computing by providing specialized workflows tailored to algorithmic requirements. As the era of Moore’s law draws to a close, many new non–von Neumann processors are emerging as potential computational accelerators, including those based on the principles of neuromorphic computing, tensor algebra, and quantum information. While development of these new processors is continuing to mature, the potential impact on accelerated computing is anticipated to be profound. We discuss how different processing models can advance computing in key scientific paradigms: machine learning and constraint satisfaction. Significantly, each of these new processor types utilizes a fundamentally different model of computation, and this raises questions about how to best use such processors in the design and implementation of applications. While many processors are being developed with a specific domain target, the ubiquity of spin-glass models and neural networks provides an avenue for multi-functional applications. Furthermore, this also hints at the infrastructure needed to integrate next-generation processing units into future high-performance computing systems.
97 MATHEMATICS AND COMPUTING↗
Simple 14-Dimensional Lie Algebras in Characteristic Two
Using theory of deformations of Lie algebra of type G{sub 2}, isomorphisms between the known simple 14-dimensional Lie algebras over a field of even characteristic and Lie algebras of Cartan type of S or H are constructed.
97 MATHEMATICS AND COMPUTING↗
The Existence of Root Subgroup Translated by a Given Element into its Opposite
Let Φ be a simply laced root system, K an algebraically closed field, and G = G{sub ad}(Φ,K) the adjoint group of type Φ over K. Then for every nontrivial element g ∈ G there exists a root element x of the Lie algebra of G such that x and gx are opposite.
97 MATHEMATICS AND COMPUTING↗
Rings whose p-ranks do not exceed 1
We consider associative torsion-free rings of finite rank whose p-ranks do not exceed 1. For these rings, certain analogues of Wedderburn's theorem on finite-dimensional algebras are found. Bibliography: 11 titles. (paper)
97 MATHEMATICS AND COMPUTING↗
Spherical Parameter Estimation and MPDV Applications
A new algebraic least square approach is presented for estimating effective zero center for verification of MPDV(Multichannel Photon Doppler Velocimetry) probe designs. Unlike previous algebraic least square methods for spherical parameter estimation, the new method can be shown to be concise in mathematic formulations, easy of computations, invariant under rigid -body transformations, no need for numerical approximations, and precise in dealing with precise data.
97 MATHEMATICS AND COMPUTING↗
Supercharacters of Unipotent and Solvable Groups
The notion of the supercharacter theory was introduced by P. Diaconis and I. M. Isaaks in 2008. In this paper, we present a review of the main notions and facts of the general theory and discuss the construction of the supercharacter theory for algebra groups and the theory of basic characters for unitriangular groups over a finite field. Based on his earlier papers, the author constructs the supercharacter theory for finite groups of triangular type. The structure of the Hopf algebra of supercharacters for triangular groups over finite fields is also characterized.
97 MATHEMATICS AND COMPUTING↗
On the third cohomology of algebraic groups of rank two in positive characteristic
We evaluate the third cohomology of simple simply connected algebraic groups of rank 2 over an algebraically closed field of positive characteristic with coefficients in simple modules. It is assumed that the characteristic p of the field is greater than 3 for SL{sub 3}, greater than 5 for Sp{sub 4}, and greater than 11 for G{sub 2}. It follows from the main result that the dimensions of the cohomology spaces do not exceed the rank of the algebraic group in question. To prove the main results we study the properties of the first-quadrant Lyndon-Hochschild-Serre spectral sequence with respect to an infinitesimal subgroup, namely, the Frobenius kernel of the given algebraic group. Bibliography: 49 titles.
97 MATHEMATICS AND COMPUTING↗
Design Considerations for GPU-based Mixed Integer Programming on Parallel Computing Platforms
Mixed Integer Programming (MIP) is a powerful abstraction in combinatorial optimization that finds real-life application across many significant sectors. The recent proliferation of graphical processing unit (GPU)-based accelerated computing architectures in large-scale parallel computing or supercomputing presents new opportunities as well as challenges in the advancement of MIP solver technology to effectively use the new accelerated computing platforms and scale to large parallel systems. Here, we recount the conventional processor-based strategies and focus on configurations where the most promising intersection lies between parallel MIP solver approaches and the specific strengths of accelerated parallel platforms. We note that the best potential lies in solving problems whose individual matrix sizes (of the linear program relaxation) fit entirely within one accelerator's memory and whose branch-and-bound (or branch-and-cut) trees cannot be fully contained within a small number of computational nodes. Additionally, we identify ideal features of computational linear algebra support on GPU accelerators that would help advance this direction of scalable parallel solution of MIP problems on GPU-based accelerated computing architectures.
Perumalla, Kalyan↗
The topology of integrable systems with incomplete fields
Liouville's theorem holds for Hamiltonian systems with complete Hamiltonian fields which possess a complete involutive system of first integrals; such systems are called Liouville-integrable. In this paper integrable systems with incomplete Hamiltonian fields are investigated. It is shown that Liouville's theorem remains valid in the case of a single incomplete field, while if the number of incomplete fields is greater, a certain analogue of the theorem holds. An integrable system on the algebra sl(3) is taken as an example. Bibliography: 11 titles.
97 MATHEMATICS AND COMPUTING↗
Quadratic pseudospectrum for identifying localized states
Here we examine the utility of the quadratic pseudospectrum for understanding and detecting states that are somewhat localized in position and energy, in particular, in the context of condensed matter physics. Specifically, the quadratic pseudospectrum represents a method for approaching systems with incompatible observables {A j |1 ≤ j ≤ d} as it minimizes collectively the errors $\parallel$A j v - λ j v$\parallel$ while defining a joint approximate spectrum of incompatible observables. Moreover, we derive an important estimate relating the Clifford and quadratic pseudospectra. Finally, we prove that the quadratic pseudospectrum is local and derive the bounds on the errors that are incurred by truncating the system in the vicinity of where the pseudospectrum is being calculated.
97 MATHEMATICS AND COMPUTING↗
Integral boundary conditions in phase field models
Modeling the chemical, electric and thermal transport as well as phase transitions and the accompanying mesoscale microstructure evolution within a material in an electronic device setting involves the solution of partial differential equations often with integral boundary conditions. Employing the familiar Poisson equation describing the electric potential evolution in a material exhibiting insulator to metal transitions, we exploit a special property of such an integral boundary condition, and we properly formulate the variational problem and establish its well-posedness. Next, we compare our method with the commonly-used Lagrange multiplier method that can also handle such boundary conditions. Numerical experiments demonstrate that our new method achieves optimal convergence rate in contrast to the conventional Lagrange multiplier method. Furthermore, the linear system derived from our method is symmetric positive definite, and can be efficiently solved by Conjugate Gradient method with algebraic multigrid preconditioning.
97 MATHEMATICS AND COMPUTING↗
Fast Transient Simulations for Multi-Segment Transmission Lines with a Graphical Model
This paper studies a computationally efficient algebraic graph theory engine for simulating time-domain one-dimensional waves in a multi-segment transmission line, such as for reflectometry applications. Efficient simulation of time-domain signals in multi-segment transmission lines is challenging because the number of propagation paths (and therefore the number of operations) increases exponentially with each new interface. We address this challenge through the use of a frequency-domain, algebraic graphical model of wave propagation, which is then converted to the time domain via the Fourier transform. We use this model to achieve an exact, stable, and computationally efficient (O(NQ), where N is the number of segments and Q is the bandwidth) approach for studying one-dimensional wave propagation. Our approach requires the reflection and transmission coefficients for each interface and each segment's complex propagation constant. Furthermore, we compare our simulation results with known analytical solutions.
42 ENGINEERING↗
Reducing Parallel Communication in Algebraic Multigrid through Sparsification
Algebraic multigrid (AMG) is an $\mathcal{O}(n)$ solution process for many large sparse linear systems. A hierarchy of progressively coarser grids which utilize complementary relaxation and interpolation operators is constructed. High-energy error is reduced by relaxation, while low-energy error is mapped to coarse-grid matrices and reduced there. However, large parallel communication costs often limit parallel scalability. Furthermore, as the multigrid hierarchy is formed, each coarse matrix is formed through a triple matrix product. The resulting coarse grids often have significantly more nonzeros per row than the original fine-grid operator, thereby generating high parallel communication costs associated with sparse matrix-vector multiplication (SpMV) on coarse levels. In this paper, we introduce a method that systematically removes entries in coarse-grid matrices after the hierarchy is formed, leading to improved communication costs. We sparsify by removing weakly connected or unimportant entries in the matrix, leading to improved solve time. The main trade-off is that if the heuristic identifying unimportant entries is used too aggressively, then AMG convergence can suffer. To counteract this, the original hierarchy is retained, allowing entries to be reintroduced into the solver hierarchy if convergence is too slow. This enables a balance between communication cost and convergence, as necessary. In this paper we present new algorithms for reducing communication and present a number of computational experiments in support.
97 MATHEMATICS AND COMPUTING↗
On the Varieties of Commutative Metabelian Algebras
The paper presents new results on varieties of commutative metabelian algebras over a field of zero characteristic. We study the structure of the multilinear part of the variety of all commutative metabelian algebras as a module of the symmetric group. Two almost nilpotent varieties are introduced and studied in this class of algebras. We prove the nonexistence of other almost nilpotent commutative metabelian varieties of subexponential growth.
97 MATHEMATICS AND COMPUTING↗
On the τ-Compactness of Products of τ -Measurable Operators Adjoint to Semi-Finite Von Neumann Algebras
Let M be the von Neumann algebra of operators in a Hilbert space H and τ be an exact normal semi-finite trace on M. We obtain inequalities for permutations of products of τ-measurable operators. We apply these inequalities to obtain new submajorizations (in the sense of Hardy, Littlewood, and Pólya) of products of τ -measurable operators and a sufficient condition of orthogonality of certain nonnegative τ-measurable operators. We state sufficient conditions of the τ –compactness of products of self-adjoint τ -measurable operators and obtain a criterion of the τ -compactness of the product of a nonnegative τ-measurable operator and an arbitrary τ -measurable operator. We present an example that shows that the nonnegativity of one of the factors is substantial. We also state a criterion of the elementary nature of the product of nonnegative operators from M . All results are new for the *-algebra B(H) of all bounded linear operators in H endowed with the canonical trace τ = tr.
97 MATHEMATICS AND COMPUTING↗
Exponential growth of codimensions of identities of algebras with unity
The asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly 1 when an outer unity is adjoined to the original algebra. It is shown that the PI-exponents of unital algebras can take any value greater than 2, and the exponents of finite-dimensional unital algebras form a dense subset in the domain [2,∞). Bibliography: 34 titles.
97 MATHEMATICS AND COMPUTING↗
Non-Galerkin Coarse Grids for Algebraic Multigrid
Algebraic multigrid (AMG) is a popular and effective solver for systems of linear equations that arise from discretized partial differential equations. And while AMG has been effectively implemented on large scale parallel machines, challenges remain, especially when moving to exascale. Particularly, stencil sizes (the number of nonzeros in a row) tend to increase further down in the coarse grid hierarchy, and this growth leads to more communication. Therefore, as problem size increases and the number of levels in the hierarchy grows, the overall efficiency of the parallel AMG method decreases, sometimes dramatically. This growth in stencil size is due to the standard Galerkin coarse grid operator, $P^T A P$, where $P$ is the prolongation (i.e., interpolation) operator. For example, the coarse grid stencil size for a simple three-dimensional (3D) seven-point finite differencing approximation to diffusion can increase into the thousands on present day machines, causing an associated increase in communication costs. We therefore consider algebraically truncating coarse grid stencils to obtain a non-Galerkin coarse grid. First, the sparsity pattern of the non-Galerkin coarse grid is determined by employing a heuristic minimal “safe” pattern together with strength-of-connection ideas. Second, the nonzero entries are determined by collapsing the stencils in the Galerkin operator using traditional AMG techniques. The result is a reduction in coarse grid stencil size, overall operator complexity, and parallel AMG solve phase times.
97 MATHEMATICS AND COMPUTING↗