Gröbner bases can be used for computing the Hilbert basis of a numerical submonoid. By using these techniques, we provide an algorithm that calculates a basis of a subspace of a finite-dimensional vector space over a finite prime field given as a matrix kernel. AMS Subject Classification: 13P10, 94B05

Every frame in Hilbert space contains a subsequence equivalent to an orthogonal basis. If a frame is n-dimensional then this sub-sequence has length (1 − ε)n. On the other hand, there is a frame which does not contain bases with brackets.

We establish the first previously unknown case of the EisenbudHarris conjecture in Castelnuovo theory concerning algebraic curves of high genus in Pn. The problem is reduced to a question about zero-dimensional schemes Γ ⊂ P in symmetric position with certain constrains on the Hilbert function. The method of Gröbner bases is then applied to study the homogeneous ideal of Γ.

The concepts of basis and frame are studied in the classical literature of functional analysis, Fourier analysis, and wavelet theory in a wide range. In this paper, we consider an operator-theoretic approach to discrete frame theory on a separable Hilbert space. For this purpose, we define a special type of frames and bases, called wavelet-type frames and wavelet-type bases, obtained by acting ...

A classical theorem of Robbins states that the edges of a graph may be oriented, in such a way that an oriented path exists between any source and destination, if and only if the graph is both connected and twoconnected (it cannot be disconnected by the removal of an edge). In this paper, an algebraic version of Robbins' result becomes a lemma on Hilbert bases for free abelian groups, which is ...

We study sets of pure states in a Hilbert space of dimension d which are mutually unbiased (MU), that is, the moduli of their scalar products are equal to zero, one, or 1/ √ d. These sets will be called a MU constellation, and if four MU bases were to exist for d = 6, they would give rise to 35 different MU constellations. Using a numerical minimisation procedure, we are able to identify only 1...

A configuration of lattice vectors is supernormal if it contains a Hilbert basis for every cone spanned by a subset. We study such configurations from various perspectives, including triangulations, integer programming and Gröbner bases. Our main result is a bijection between virtual chambers of the configuration and virtual initial ideals of the associated binomial ideal.

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and analysed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl p...

We prove that the set of non-identity generalized Pauli operators on the Hilbert space of N d-level quantum systems, d an odd prime, naturally forms a symplectic polar space W2N−1(d) of rank N and order d. This generalizes the solution (by the author) of a recent conjecture posed by Saniga-Planat (which covers the case d= 2). As an application, we give a new short proof for the existence of max...

The primary goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present linear reconstruction algorithms for tight frames associated with projective 2-designs in fini...