Narrow Results

In this paper, a new model has been developed to reflect the similarity in the formulation between two-fluid plasma theory and the Maxwell equations of electromagnetism. For this aim, a mathematical tool based on complexified Cayley–Dickson octonions has been proposed to unify the behaviors of both the electromagnetic and fluidic character of plasma. Moreover, derived equations have been associated with their corresponding $8\times 8$ matrices simply and consistently.

In this paper, we develop four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a shared-memory multiprocessor architecture. We particularly consider the SVD of unstructured sparse matrices in which the number of rows may be substantially larger or smaller than the number of columns. We emphasize Lanczos, block-Lanczos, subspace iteration and the trace minimization methods for determining a select number of smallest singular triplets (singular values and corresponding left- and right-singular vectors) for sparse matrices. The target architectures for implementations of such methods include the Alliant FX/80 and the Cray-2S/4–128. This algorithmic research is particularly motivated by recent information-retrieval techniques in which approximations to large sparse term-document matrices are needed, and by nonlinear inverse problems arising from seismic reflection tomography applications.

In this paper, we study the optimal or best approximation of any linear operator by low rank linear operators, especially, any linear operator on the ${\ell }_p$-space, $p\in [1,\infty )$, under ${\ell }_p$ norm, or in Minkowski distance. Considering generalized singular values and using techniques from differential geometry, we extend the classical Schmidt–Mirsky theorem in the direction of the ${\ell }_p$-norm of linear operators for some $p$ values. Also, we develop and provide algorithms for finding the solution to the low rank approximation problems in some nontrivial scenarios. The results can be applied to, in particular, matrix completion and sparse matrix recovery.

In this paper a novel method for circuit analysis is proposed. It is based on using symbolic analysis in matrix form, which is especially appropriate for repetitive similar calculations of the same circuit. This method, which applies hyper-complex numbers (hypernions), was first developed by the authors for analyzing the non-sinusoidal operation of electrical circuits. Now, the method has been extended to the analysis of electronic/switching circuits in which the sources and/or the parameters are step-wisely changed as a result of switching. Such circuits are common in different kinds of DC-DC converters (such as Buck, Boost, Cuk, etc.). The proposed method gives a new approach to the analysis of the above circuits by opening the possibility of treating them in a general-analytical form, just like in regular electrical circuits having a constant configuration and constant parameters. The computation of such kinds of circuits by using the proposed method becomes very simple, since the circuit does not have to be analyzed many times, each time for a different configuration, but all at once by performing a parallel computation. The theoretical presentation is accompanied by numerical examples.

The faithfulness of the orthogonal group case of Brauer's representation of the Brauer centralizer algebras restricted to their Temperley–Lieb subalgebras, which was established by Vaughan Jones, is here proved in a new, elementary and self-contained, manner.

Contrary to the commutative case, the set of linear recurring sequences with values in a module over a noncommutative ring is no more a module for the usual operations. We show the stability of these operations when the ring is a matrix ring or a division ring. In the case of a finite dimensional division ring over its center, we give an algorithm for the determination of a recurrence relation for the sum of two linear recurring sequences.

An $R$-module $V$ over a semiring $R$ lacks zero sums (LZS) if $x+y=0$ implies $x=y=0$. More generally, a submodule $W$ of $V$ is “summand absorbing” (SA), if, for all $x,y\in V$, $x+y\in W\Rightarrow x\in W,y\in W.$ These relate to tropical algebra and modules over (additively) idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of SA submodules of a given LZS module, especially, those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this paper, we consider which submodules are SA and describe their explicit generation.

We deduce continuity and Schatten–von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in $(0,\infty ]$. We use these results to deduce continuity and Schatten–von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces, or in appropriate Hörmander classes.

Refinement equations of the type play an exceptional role in the theory of wavelets, subdivision algorithms and computer design. It is known that the regularity of their compactly supported solutions (refinable functions) depends on the spectral properties of special N-dimensional linear operators T₀, T₁ constructed by the coefficients of the equation. In particular, the structure of kernels and of common invariant subspaces of these operators have been intensively studied in the literature. In this paper, we give a complete classification of the kernels and of all the root subspaces of T₀ and T₁, as well as of their common invariant subspaces. This result answers several open questions stated in the literature and clarifies the structure of the space spanned by the integer translates of refinable functions. This also leads to some results on the moduli of continuity of refinable functions and wavelets in various functional spaces. In particular, it is proved that the Hölder exponent of those functions is sharp whenever it is not an integer.

Using the attractive properties of octonion algebra, an alternative formulation has been proposed for the Maxwell-type equations of compressible fluids. Although the origins of electromagnetic theory and fluid mechanics are completely different, a series of suitable and elegant 8-dimensional equations have been derived in a form similar to electromagnetic, gravitational counterparts previously given in relevant literature. Moreover, the corresponding matrix representations of derived expressions have also been presented after describing the connections between the algebraic properties of Cayley–Dickson octonions and some pseudo-real matrices.

We identify a solvable dynamical system — interpretable to some extent as a many-body problem — and point out that — for an appropriate assignment of its parameters — it is entirely isochronous, namely all its nonsingular solutions are completely periodic (i.e., periodic in all degrees of freedom) with the same fixed period (independent of the initial data). We then identify its equilibrium configurations and investigate its behavior in their neighborhood. We thereby identify certain matrices — of arbitrary order — whose eigenvalues are all rational numbers: a Diophantine finding.

In this chapter, we first discuss the basic concepts of linear algebra and linear combination and its distribution. Then we discuss the concepts of vectors, matrices, and their operations. Linear-equation system and its solution are also explored in detail. Based upon this information, we discuss discriminant analysis, factor analysis, and principal component analysis. Some applications of these three analyses are also demonstrated.

This paper describes a topological structure of a certain group of iterations of functions. This problem arose in the transformation theory of differential equations. By contrast to analytic methods having mostly been used in this area, an algebraic approach is dominated here.