Adams and Siegel (1982) proposed an extra stage cube interconnection network that tolerates one switch failure with one extra stage. We extend their results and discover a class of extra stage interconnection networks that tolerate multiple switch failures with a minimal number of extra stages. Adopting the same fault model as Adams and Siegel, the faulty switches can be bypassed by a pair of demultiplexer/multiplexer combinations. It is easy to show that, to maintain point to point and broadcast connectivities, there must be at least S extra stages to tolerate I switch failures. We present the first known construction of an extra stage interconnection network that meets this lower-bound. This 12-dimensional multistage interconnection network has n+f stages and tolerates I switch failures. An n-bit label called mask is used for each stage that indicates the bit differences between the two inputs coming into a common switch. We designed the fault-tolerant construction such that it repeatedly uses the singleton basis of the n-dimensional vector space as the stage mask vectors. This construction is further generalized and we prove that an n-dimensional multistage interconnection network is optimally fault-tolerant if and only if the mask vectors of every n consecutive stages span the n-dimensional vector space.

Certain deterministic nonlinear systems may show chaotic behavior. We consider the motion of qualitative information and the practicalities of extracting a part from chaotic experimental data. Our approach based on a theorem of Takens draws on the ideas from the generalized theory of information known as singular system analysis. We illustrate this technique by numerical data from the chaotic region of the chaotic experimental data. The method of the singular-value decomposition is used to calculate the eigenvalues of embedding space matrix. The corresponding concrete algorithm to calculate eigenvectors and to obtain the basis of embedding vector space is proposed in this paper. The projection on the orthogonal basis generated by eigenvectors of timeseries data and concrete paradigm are also provided here. Meanwhile the state space reconstruction technology of different kinds of chaotic data obtained from dynamical system has also been discussed in detail.

Mutual dimensional properties of basis-equivalent (b-equivalent) and weakly l-equivalent topological spaces are studied. It is shown that the b-equivalence does not preserve bicompactness; in particular, b-equivalent topological spaces can be non-l-equivalent. Bibliography: 18 titles.

On local bases of fuzzy topological vector spaces

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), we shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [8].

Geometric properties, invariants, and the Toeplitz structure of minimal bases of rational vector spaces

It is shown that the axiom of choice follows in a weaker form than the Zermelo - Fraenkel set theory from the assertion: every set of generators G of a vector space V over the two element field includes a basis L for V. It is also shown that: for every family A = { A i : i ∈ k } \mathcal {A}=\{A_i:i\in k\} of non empty sets there exists a family F = { F i : i ∈ k } \mathcal {F=}\{F_i:i\in k\} of odd sized sets such that, for every i ∈ k i\in k , F i ⊆ A F_i\subseteq A iff in every vector space B B over the two-element field every subspace V ⊆ B V\subseteq B has a complementary subspace S S iff every quotient group of an abelian group each of whose elements has order 2 has a set of representatives.

On fuzzy bases of vector spaces

Bases of fuzzy vector spaces

Young tableaux and linear independence of standard monomials in multiminors of a multimatrix

Generalized rodeletive correspondence between multitableaux and multimonomials

Additive bases of vector spaces over prime fields

Generalized roinsertive correspondence between multitableaux and multimonomials

A set of N rectangular blocks of unit density form the basis of a linear vector space. This space will be a Hilbert space with reproducing kernel, when equipped with the L2 norm (and inner product). The kernel, K(P, Q), will be the product sum of the (maximally N) orthonormalized base functions evaluated at two points P, Q, in space. Multiplied by an appropriate scaling constant, this function will be a covariance function for a random density (anomaly) function. Using covariance propagation, auto- and cross-covariance functions for and with quantities related to the (anomalous) gravity field created by the random density function may be computed. When blocks of varying volume are used, the density variance will vary within the structure described by the blocks. For disjoint blocks, the variance is inversely proportional to the volume. The use of overlapping blocks generally creates correlations between density values in different blocks, a possibility which does not exist when using disjoint blocks. Computational examples are presented which show that it is possible to create realistic covariance models for the gravity anomaly, and that the cross-correlation between surface gravity and surface density may obtain values in the interval from 0.4 to 0.7. These values are more realistic than values obtained in earlier studies using quasi-harmonic density models.

Module and vector space bases for spline spaces

Fuzzy vector spaces

Biorthonormal systems, partial fractions, and Hermite interpolation

On weak and Schauder ?-bases in topological vector spaces

This paper presents a new approach to disturbance decoupling problems by measurement feedback (DDPM). The novelty in this approach is that it yields a characterization of the sets of all solutions to DDPM and DDPM with internal stability and that the set of fixed modes of DDPM with respect to dynamic output feedback is easily identified. The central object used in deriving these results is a minimal polynomial basis of a rational vector space which conveniently represents the problem data.

Discrete-time Adaptive Control for Periodically Time-varying Systems