Hermitian unitary matrices with modular permutation symmetry

We study Hermitian unitary matrices $\mathcal{S}\in\mathbb{C}^{n,n}$ with the following property: There exist $r\geq0$ and $t>0$ such that the entries of $\mathcal{S}$ satisfy $|\mathcal{S}_{jj}|=r$ and $|\mathcal{S}_{jk}|=t$ for all $j,k=1,\ldots,n$, $j\neq k$. We derive necessary conditions on the ratio $d:=r/t$ and show that these conditions are very restrictive except for the case when $n$ is even and the sum of the diagonal elements of $\S$ is zero. Examples of families of matrices $\mathcal{S}$ are constructed for $d$ belonging to certain intervals. The case of real matrices $\mathcal{S}$ is examined in more detail. It is demonstrated that a real $\mathcal{S}$ can exist only for $d=\frac{n}{2}-1$, or for $n$ even and $\frac{n}{2}+d\equiv1\pmod 2$. We provide a detailed description of the structure of real $\mathcal{S}$ with $d\geq\frac{n}{4}-\frac{3}{2}$, and derive a sufficient and necessary condition of their existence in terms of the existence of certain symmetric $(v,k,\lambda)$-designs. We prove that there exist no real $\mathcal{S}$ with $d\in\left(\frac{n}{6}-1,\frac{n}{4}-\frac{3}{2}\right)$. A parametrization of Hermitian unitary matrices is also proposed, and its generalization to general unitary matrices is given. At the end of the paper, the role of the studied matrices in quantum mechanics on graphs is briefly explained.Comment: revised version, 21 page

Minimal symmetric Darlington synthesis

We consider the symmetric Darlington synthesis of a p x p rational symmetric Schur function S with the constraint that the extension is of size 2p x 2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multiplicity of I-SS*. A constructive characterization of all such extensions is provided in terms of a symmetric realization of S and of the outer spectral factor of I-SS*. The authors's motivation for the problem stems from Surface Acoustic Wave filters where physical constraints on the electro-acoustic scattering matrix naturally raise this mathematical issue

On Stieltjes integral transforms involving Γ\Gamma-functions

After some methodological remarks on the theory of Stieltjes transforms, a systematic classification of transforms involving Gamma-functions is presented. As a consequence, many new transforms are established and much simpler proofs for a few known transforms are obtained

On Hilberg's Law and Its Links with Guiraud's Law

Hilberg (1990) supposed that finite-order excess entropy of a random human text is proportional to the square root of the text length. Assuming that Hilberg's hypothesis is true, we derive Guiraud's law, which states that the number of word types in a text is greater than proportional to the square root of the text length. Our derivation is based on some mathematical conjecture in coding theory and on several experiments suggesting that words can be defined approximately as the nonterminals of the shortest context-free grammar for the text. Such operational definition of words can be applied even to texts deprived of spaces, which do not allow for Mandelbrot's ``intermittent silence'' explanation of Zipf's and Guiraud's laws. In contrast to Mandelbrot's, our model assumes some probabilistic long-memory effects in human narration and might be capable of explaining Menzerath's law.Comment: To appear in Journal of Quantitative Linguistic