Alex Grossmann

PAINLESS NONORTHOGONAL EXPANSIONS

In a Hilbert space H, discrete families of vectors {hj} with the property that f=∑j〈hj‖ f〉hj for every f in H are considered. This expansion formula is obviously true if the family is an orthonormal basis of H, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non‐Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.

Transforms associated to square integrable group representations. I. General results

Let G be a locally compact group, which need not be unimodular. Let x→U(x) (x∈G) be an irreducible unitary representation of G in a Hilbert space H(U). Assume that U is square integrable, i.e., that there exists in H(U) at least one nonzero vector g such that ∫‖(U(x)g,g)‖2 dx<∞. We give here a reasonably self‐contained analysis of the correspondence associating to every vector f∈H(U) the function (U(x)g,f) on G, discussing its isometry, characterization of the range, inversion, and simplest interpolation properties. This correspondence underlies many properties of generalized coherent states.

A class of explicitly soluble, local, many‐center Hamiltonians for one‐particle quantum mechanics in two and three dimensions. I

We derive an explicit formula for the resolvent of a class of one‐particle, many‐center, local Hamiltonians. This formula gives, in particular, a full description of a model molecule given by point interactions at n arbitrarily placed fixed centers in three dimensions. It also gives a three−dimensional analog of the Kronig–Penney model.

Partial inner product spaces. I. General properties

Vector spaces V with a natural “partial inner product” are very common in analysis and in mathematical physics. In this series of papers, we study them on their own terms; the only input is the domain of the partial inner product and the partial inner product itself. Our main assumption is that the linear forms associated to “infinitely good elements” of V, separate points in V. We show in this paper that V carries then a well-defined “self-dual” structure. Examples are discussed.

An integral transform related to quantization

We study in some detail the correspondence between a function f on phase space and the matrix elements. (Qf)(a,b) of its quantized Qf between the coherent states ‖ a〉 and ‖ b〉. It is an integral transform: Qf(a,b) = F{a,b ‖ v}f(v) dv, which resembles in many ways the integral transform of Bargmann. We obtain the matrix elements of Qf between harmonic oscillator states as the Fourier coefficients of f with respect to an explicit orthonormal system.

Decoding the nucleoid organisation of Bacillus subtilis and Escherichia coli through gene expression data

BackgroundAlthough the organisation of the bacterial chromosome is an area of active research, little is known yet on that subject. The difficulty lies in the fact that the system is dynamic and difficult to observe directly. The advent of massive hybridisation techniques opens the way to further studies of the chromosomal structure because the genes that are co-expressed, as identified by microarray experiments, probably share some spatial relationship. The use of several independent sets of gene expression data should make it possible to obtain an exhaustive view of the genes co-expression and thus a more accurate image of the structure of the chromosome.ResultsFor both Bacillus subtilis and Escherichia coli the co-expression of genes varies as a function of the distance between the genes along the chromosome. The long-range correlations are surprising: the changes in the level of expression of any gene are correlated (positively or negatively) to the changes in the expression level of other genes located at well-defined long-range distances. This property is true for all the genes, regardless of their localisation on the chromosome.We also found short-range correlations, which suggest that the location of these co-expressed genes corresponds to DNA turns on the nucleoid surface (14–16 genes).ConclusionThe long-range correlations do not correspond to the domains so far identified in the nucleoid. We explain our results by a model of the nucleoid solenoid structure based on two types of spirals (short and long). The long spirals are uncoiled expressed DNA while the short ones correspond to coiled unexpressed DNA.

Schrödinger Scattering Amplitude. I

The Schrodinger scattering amplitude for a fixed potential is studied as a function of the three components of the initial momentum, the three components of the final momentum, and the square root of the energy.

Rate matrices for analyzing large families of protein sequences.

We propose and study a new approach for the analysis of families of protein sequences. This method is related to the LogDet distances used in phylogenetic reconstructions; it can be viewed as an attempt to embed these distances into a multidimensional framework. The proposed method starts by associating a Markov matrix to each pairwise alignment deduced from a given multiple alignment. The central objects under consideration here are matrix-valued logarithms L of these Markov matrices, which exist under conditions that are compatible with fairly large divergence between the sequences. These logarithms allow us to compare data from a family of aligned proteins with simple models (in particular, continuous reversible Markov models) and to test the adequacy of such models. If one neglects fluctuations arising from the finite length of sequences, any continuous reversible Markov model with a single rate matrix Q over an arbitrary tree predicts that all the observed matrices L are multiples of Q. Our method exploits this fact, without relying on any tree estimation. We test this prediction on a family of proteins encoded by the mitochondrial genome of 26 multicellular animals, which include vertebrates, arthropods, echinoderms, molluscs, and nematodes. A principal component analysis of the observed matrices L shows that a single rate model can be used as a rough approximation to the data, but that systematic deviations from any such model are unmistakable and related to the evolutionary history of the species under consideration.

Partial inner product spaces. II. Operators

We study linear operators between nondegenerate partial inner product spaces and their relationships to selfadjoint operators in a “middle” Hilbert space.

An integral transform related to quantization. II. Some mathematical properties

We study in more detail the mathematical properties of the integral transform relating the matrix elements between coherent states of a quantum operator to the corresponding classical function. Explicit families of Hilbert spaces are constructed between which the integral transform is an isomorphism.