Abstract. Starting from formal deformation quantization we use an explicit formula for a star product on the Poincaré disk 𝔻 n $\mathbb {D}_n$ to introduce a Fréchet topology making the star product continuous. To this end a general construction of locally convex topologies on algebras with countable vector space basis is introduced and applied. Several examples of independent interest are investigated as, e.g., group algebras over finitely generated groups and infinite matrices. In the case of the star product on 𝔻 n $\mathbb {D}_n$ the resulting Fréchet algebra is shown to have many nice features: it is a strongly nuclear Köthe space, the symmetry group SU ( 1 , n ) $\mathrm {SU}(1, n)$ acts smoothly by continuous automorphisms with an inner infinitesimal action, and evaluation functionals at all points of 𝔻 n $\mathbb {D}_n$ are continuous positive functionals.

We present a new method to explicitly define Abelian functions associated with algebraic curves, for the purpose of finding bases for the relevant vector spaces of such functions. We demonstrate the procedure with the functions associated with a trigonal curve of genus four. The main motivation for the construction of such bases is that it allows systematic methods for the derivation of the addition formulae and differential equations satisfied by the functions. We present a new 3-term 2-variable addition formulae and a complete set of differential equations to generalise the classic Weierstrass identities for the case of the trigonal curve of genus four.

Cayley graph expanders and groups of finite width

Very nilpotent basis and n-tuples in Borel subalgebras

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.

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

Vector space bases associated to vanishing ideals of points

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.

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

Quadratic and cubic invariants of unipotent affine automorphisms

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

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.

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.

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.

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.

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.

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.