Basis Of Vector Space Research Articles

Very nilpotent basis and n-tuples in Borel subalgebras

A (vector space) basis B of a Lie algebra is said to be very nilpotent if all the iterated brackets of elements of B are nilpotent. In this Note, we prove a refinement of Engel's Theorem. We show that a Lie algebra has a very nilpotent basis if and only if it is a nilpotent Lie algebra. When g is a semisimple Lie algebra, this allows us to define an ideal of S ( ( g n ) ⁎ ) G whose associated algebraic set in g n is the set of n-tuples lying in a same Borel subalgebra.

Taylor Formula on step two Carnot Groups

In the setting of step two Carnot groups we give an explicit representation of Taylor polynomial in terms of a suitable basis of the real vector space of left invariant differential operators, acting pointwisely on monomials like the ordinary Euclidean iterated derivations.

An invariant for matrices and sets of points in prime characteristic

There is polynomial function X q in the entries of an m × m(q ? 1) matrix over a field of prime characteristic p, where q = p h is a power of p, that has very similar properties to the determinant of a square matrix. It is invariant under multiplication on the left by a non-singular matrix, and under permutations of the columns. This gives a way to extend the invariant theory of sets of points in projective spaces of prime characteristic, to make visible hidden structure. There are connections with coding theory, permanents, and additive bases of vector spaces.

On the equivalence between real mutually unbiased bases and a certain class of association schemes

Mutually unbiased bases (MUBs) in complex vector spaces play several important roles in quantum information theory. At present, even the most elementary questions concerning the maximum number of such bases in a given dimension and their construction remain open. In an attempt to understand the complex case better, some authors have also considered real MUBs, mutually unbiased bases in real vector spaces. The main results of this paper establish an equivalence between sets of real mutually unbiased bases and 4-class cometric association schemes which are both Q -bipartite and Q -antipodal. We then explore the consequences of this equivalence, constructing new cometric association schemes and describing a potential method for the construction of sets of real MUBs.

Vector space bases associated to vanishing ideals of points

In this paper we discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give new complexity bounds. As an application, we drastically improve the computational algebra approach to the reverse engineering of gene regulatory networks.

The unitary-group formulation of coupled-cluster many-electron theory

Coupled-cluster many-electron theory (CCMET) has been developed in the second-quantized formulation with and without spin projection. In this article CCMET is developed in the unitary-group formulation where the group is U(p) and p is the number of orbitals in the basis set. The zero-order ground state is the highest-weight state of an irreducibly invariant subspace of U(p) and the excitation operators are infinitesimal generators. These irreducible spaces are uniquely labeled by Young diagrams which the Pauli principle limits to no more than one column for fermion orbitals and no more than two columns for freeon (spin-free) orbitals. In the latter case the spin is one-half the difference in the lengths of the two columns. We employ both the Gel'fand and the generator bases for the irreducible invariant vector spaces. Matrix elements for the former are evaluated by techniques due to Gel'fand, Biedenharn, Louck, Paldus, and Shavitt and for the latter directly by Lie algebraic and diagrammatic techniques as a simple function of the weight components of the highest-weight state. For freeon or fermion orbitals the results are equivalent to those obtained by the second-quantized formulation with or without spin projection.

Function bases for topological vector spaces

Our main interest in this work is to characterize certain operator spaces acting on some important vector-valued function spaces such as $(V_{a}) _{c_{0}}^{a\in{\mathbb A}}$, by introducing a new kind basis notion for general Topological vector spaces. Where ${\mathbb A}$ is an infinite set, each $V_{a}$ is a Banach space and $( V_{a}) _{c_{0}}^{a\in{\mathbb A}}$ is the linear space of all functions $x\colon{\mathbb A} \rightarrow\bigcup V_{a}$ such that, for each $\varepsilon> 0$, the set $\{ a\in{\mathbb A}:\Vert x_{a}\Vert > \varepsilon\} $ is finite or empty. This is especially important for the vector-valued sequence spaces $( V_{i}) _{c_{0}}^{i\in{\mathbb N}}$ because of its fundamental place in the theory of the operator spaces (see, for example,[H. P. Rosenthal, {\it The complete separable extension property}, J. Oper. Theory, 43, No. 2, (2000), 329-374]).

Similarity of signal processing effect between Hankel matrix-based SVD and wavelet transform and its mechanism analysis

It is pointed out that signal processing effect of singular value decomposition (SVD) is very similar to that of wavelet transform when Hankel matrix is used. It is proved that a signal can be decomposed into the linear sum of a series of component signals by Hankel matrix-based SVD, and essentially what the component signals reflect are projections of original signal on the orthonormal bases of m-dimensional and n-dimensional vector spaces. The similarity mechanism of signal processing between SVD and wavelet transform is analyzed from the angle of basis of vector space and characteristic of Hankel matrix. The orthogonality of the component signals got by SVD and wavelet transform is also studied. It is discovered that singularity of signal can also be detected by Hankel matrix-based SVD, and compared with wavelet transform, there are two characteristics in SVD for singularity detection, one is that the order of vanishing moment of SVD component signals is increased progressively and the one of the nth SVD component signal is ‘ n−1’, so singular points with different Lip index can all be detected, the other is that the width of impulse indicating the position of singularity will always keep the same throughout all SVD components and this width is determined by the column number of Hankel matrix.

Quadratic and cubic invariants of unipotent affine automorphisms

Let K be an arbitrary field of characteristic zero, P n : = K [ x 1 , … , x n ] be a polynomial algebra, and P n , x 1 : = K [ x 1 −1 , x 1 , … , x n ] , for n ⩾ 2 . Let σ ′ ∈ Aut K ( P n ) be given by x 1 ↦ x 1 − 1 , x 2 ↦ x 2 + x 1 , … , x n ↦ x n + x n − 1 . It is proved that the algebra of invariants, F n ′ : = P n σ ′ , is a polynomial algebra in n − 1 variables which is generated by [ n 2 ] quadratic and [ n − 1 2 ] cubic (free) generators that are given explicitly. Let σ ∈ Aut K ( P n ) be given by x 1 ↦ x 1 , x 2 ↦ x 2 + x 1 , … , x n ↦ x n + x n − 1 . It is well known that the algebra of invariants, F n : = P n σ , is finitely generated (theorem of Weitzenböck [R. Weitzenböck, Über die invarianten Gruppen, Acta Math. 58 (1932) 453–494]), has transcendence degree n − 1 , and that one can give an explicit transcendence basis in which the elements have degrees 1 , 2 , 3 , … , n − 1 . However, it is an old open problem to find explicit generators for F n . We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants P n , x 1 σ is a polynomial algebra over K [ x 1 , x 1 −1 ] in n − 2 variables which is generated by [ n − 1 2 ] quadratic and [ n − 2 2 ] cubic (free) generators that are given explicitly. The coefficients of these quadratic and cubic invariants throw light on the ‘unpredictable combinatorics’ of invariants of affine automorphisms and of SL 2 -invariants.

Lattice-ordered Fields Determined by d-elements

Most results on the structure of lattice-ordered fields require that the field have a positive multiplicative identity. We construct a functor from the category of lattice-ordered fields with a vector space basis of d-elements to the full subcategory of such fields with positive multiplicative identities. This functor is a left adjoint to the forgetful functor and, in many cases, allows us to write all compatible lattice orders in terms of orders with positive multiplicative identities. We also use these results to characterize algebraically those extensions of totally ordered fields that have vl-bases of d-elements.

A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra

Given an n-dimensional Lie algebra g over a field k ⊃ Q , together with its vector space basis X 1 0 , … , X n 0 , we give a formula, depending only on the structure constants, representing the infinitesimal generators, X i = X i 0 t in g ⊗ k k [ [ t ] ] , where t is a formal variable, as a formal power series in t with coefficients in the Weyl algebra A n . Actually, the theorem is proved for Lie algebras over arbitrary rings k ⊃ Q . We provide three different proofs, each of which is expected to be useful for generalizations. The first proof is obtained by direct calculations with tensors. This involves a number of interesting combinatorial formulas in structure constants. The final step in calculation is a new formula involving Bernoulli numbers and arbitrary derivatives of coth ( x / 2 ) . The dimensions of certain spaces of tensors are also calculated. The second method of proof is geometric and reduces to a calculation of formal right-invariant vector fields in specific coordinates, in a (new) variant of formal group scheme theory. The third proof uses coderivations and Hopf algebras.

An algebraic relation between consimilarity and similarity of complex matrices and its applications

An antilinear operator in complex vector spaces is an important operator in the study of modern quantum theory, quantum and semiclassical optics, quantum electronics and quantum chemistry. Consimilarity of complex matrices arises as a result of studying an antilinear operator referred to different bases in complex vector spaces, and the theory of consimilarity of complex matrices plays an important role in the study of quantum theory. This paper, by means of a real representation of a complex matrix, studies the relation between consimilarity and similarity of complex matrices, sets up an algebraic bridge between consimilarity and similarity and turns the theory of consimilarity into that of ordinary similarity. This paper also gives some applications of consimilarity of complex matrices.

Semidefinite representations for finite varieties

We consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities. When the polynomial equations have a finite set of complex solutions, we can reformulate this problem as a semidefinite programming problem. Our semidefinite representation involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space ℝ[x 1, . . . ,x n ]/I, where I is the ideal generated by the polynomial equations in the problem. Moreover, we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by Lasserre. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem to optimality.

Intensity-invariant nonlinear filtering for detection in camouflage

We introduce a method based on an orthonormal vector space basis representation to detect camouflaged targets in natural environments. The method is intensity invariant so that camouflaged targets are detected independently of the illumination conditions. The detection technique does not require one to know the exact camouflage pattern, but only the class of patterns (e.g., foliage, netting, woods). We use nonlinear filtering and the calculation of several correlations. The nonlinearity of the filtering process also allows high discrimination against false targets. Several experiments confirm the target detectability where strong camouflage might delude even human viewers.

Semantic graphs and associative memories

Graphs have been increasingly utilized in the characterization of complex networks from diverse origins, including different kinds of semantic networks. Human memories are associative and are known to support complex semantic nets; these nets are represented by graphs. However, it is not known how the brain can sustain these semantic graphs. The vision of cognitive brain activities, shown by modern functional imaging techniques, assigns renewed value to classical distributed associative memory models. Here we show that these neural network models, also known as correlation matrix memories, naturally support a graph representation of the stored semantic structure. We demonstrate that the adjacency matrix of this graph of associations is just the memory coded with the standard basis of the concept vector space, and that the spectrum of the graph is a code invariant of the memory. As long as the assumptions of the model remain valid this result provides a practical method to predict and modify the evolution of the cognitive dynamics. Also, it could provide us with a way to comprehend how individual brains that map the external reality, almost surely with different particular vector representations, are nevertheless able to communicate and share a common knowledge of the world. We finish presenting adaptive association graphs, an extension of the model that makes use of the tensor product, which provides a solution to the known problem of branching in semantic nets.

Intensity-carrying modes important for vibrational polarizabilities and hyperpolarizabilities of molecules: derivation from the algebraic properties of formulas and applications.

The intensity-carrying mode (ICM) theory is developed for analyzing the vibrational motions that mainly contribute to vibrational polarizabilities and hyperpolarizabilities, which are important for describing intermolecular electrostatic interactions and nonlinear optical properties of molecules. The ICMs are derived from dipole derivatives, polarizability derivatives, and first hyperpolarizability derivatives by using algebraic properties of intensity formulas. The way to obtain explicit forms of ICMs, including the optimization method of the basis of the ICM vector space, is discussed in detail. One- and two-dimensional models are constructed on the basis of the ICMs. The theory is applied to three molecules (a push-pull type polyene, a streptocyanine dye cation, and a symmetric neutral polyene) taken as typical examples. It is shown that the ICM theory provides a reasonable picture on the vibrational polarization properties of these molecules. On the basis of this result, the validity of the valence-bond charge transfer (VB-CT) model, which is a one-dimensional model and is widely used to describe the electronic and vibrational properties of dye molecules, is also discussed.

The Theorem of the k -1 Happy Divorces

Given a binary relation R between the elements of two sets X and Y and a natural number k, it is shown that there exist k injective maps f1, f2,...,fk: \( X \hookrightarrow Y \) with \( \# \{f_1(x), f_2(x),...,f_k(x)\}=k \quad{\rm and}\quad (x,f_1(x)), (x, f_2(x)),...,(x, f_k(x)) \in R \) for all \( x \in X \) if and only if the inequality \( k \cdot \# A \leq \sum_{y \in Y} min(k, \#\{a \in A \mid (a,y) \in R\}) \) holds for every finite subset A of X, provided \( \{y \in Y \mid (x,y) \in R\} \) is finite for all \( x \in X \).¶Clearly, as suggested by this paper's title, this implies that, in the context of the celebrated Marriage Theorem, the elements x in X can (simultaneously) marry, get divorced, and remarry again a partner from their favourite list as recorded by R, for altogether k times whenever (a) the list of favoured partners is finite for every \( x \in X \)and (b) the above inequalities all hold.¶In the course of the argument, a straightforward common generalization of Bernstein's Theorem and the Marriage Theorem will also be presented while applications regarding (i) bases in infinite dimensional vector spaces and (ii) incidence relations in finite geometry (inspired by Conway's double sum proof of the de Bruijn-Erdos Theorem) will conclude the paper.

Tolerating multiple faults in multistage interconnection networks with minimal extra stages

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.

The state space reconstruction technology of different kinds of chaotic data obtained from dynamical system

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.

Topological characteristics of free bases of topological groups and topological vector spaces

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.