Discriminants, symmetrized graph monomials, and sums of squares

Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. In the 20-th century this construction was studied by several authors. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the derivative of a univariate polynomial, and an interesting example of P. and A. Lax of a graph with 4 edges whose symmetrized graph monomial is non-negative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations

Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems

We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on $\mathbb{R}^+$ by an $\omega$-parametric analytic family better than order of $(\omega\ln\omega)^{-(m+1)}$. Next, we prove an estimation of the eigenvalues and characteristic values of a Sturm-Liouville operator and some properties of the solution of a certain integral equation. This allows us to deduce from [Henkin-Novikova] some positive results about the best reconstruction formula by giving an almost optimal formula of order of $\omega^{-m}$.Comment: 40 page

Laser-like Instabilities in Quantum Nano-electromechanical Systems

We discuss negative damping regimes in quantum nano-electromechanical systems formed by coupling a mechanical oscillator to a single-electron transistor (normal or superconducting). Using an analogy to a laser with a tunable atom-field coupling, we demonstrate how these effects scale with system parameters. We also discuss the fluctuation physics of both the oscillator and the single-electron transistor in this regime, and the degree to which the oscillator motion is coherent.Comment: 4+ pages, 1 figure; reference to the work of Dykman and Krivoglaz adde

Microbial Similarity between Students in a Common Dormitory Environment Reveals the Forensic Potential of Individual Microbial Signatures.

The microbiota of the built environment is an amalgamation of both human and environmental sources. While human sources have been examined within single-family households or in public environments, it is unclear what effect a large number of cohabitating people have on the microbial communities of their shared environment. We sampled the public and private spaces of a college dormitory, disentangling individual microbial signatures and their impact on the microbiota of common spaces. We compared multiple methods for marker gene sequence clustering and found that minimum entropy decomposition (MED) was best able to distinguish between the microbial signatures of different individuals and was able to uncover more discriminative taxa across all taxonomic groups. Further, weighted UniFrac- and random forest-based graph analyses uncovered two distinct spheres of hand- or shoe-associated samples. Using graph-based clustering, we identified spheres of interaction and found that connection between these clusters was enriched for hands, implicating them as a primary means of transmission. In contrast, shoe-associated samples were found to be freely interacting, with individual shoes more connected to each other than to the floors they interact with. Individual interactions were highly dynamic, with groups of samples originating from individuals clustering freely with samples from other individuals, while all floor and shoe samples consistently clustered together.IMPORTANCE Humans leave behind a microbial trail, regardless of intention. This may allow for the identification of individuals based on the "microbial signatures" they shed in built environments. In a shared living environment, these trails intersect, and through interaction with common surfaces may become homogenized, potentially confounding our ability to link individuals to their associated microbiota. We sought to understand the factors that influence the mixing of individual signatures and how best to process sequencing data to best tease apart these signatures

The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation

We establish the existence of a real solution y(x,T) with no poles on the real line of the following fourth order analogue of the Painleve I equation, x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert problem, we obtain the asymptotics for y(x,T) as x\to\pm\infty.Comment: 27 pages, 5 figure

First-order symmetric-hyperbolic Einstein equations with arbitrary fixed gauge

We find a one-parameter family of variables which recast the 3+1 Einstein equations into first-order symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an arbitrary factor times a power of the determinant of the 3-metric; under certain assumptions, the exponent can be chosen arbitrarily, but positive, with no implication of gauge-fixing.Comment: 5 pages; Latex with Revtex v3.0 macro package and style; to appear in Phys. Rev. Let

Dissipationless Spin Current in Anisotropic p-Doped Semiconductors

Recently, dissipationless spin current has been predicted for the p-doped semiconductors with spin-orbit coupling. Here we investigate the effect of spherical symmetry breaking on the dissipationless spin current, and obtain values of the intrinsic spin Hall conductivity for realistic semiconductor band structures with cubic symmetry

``Good Propagation'' Constraints on Dual Invariant Actions in Electrodynamics and on Massless Fields

We present some consequences of non-anomalous propagation requirements on various massless fields. Among the models of nonlinear electrodynamics we show that only Maxwell and Born-Infeld also obey duality invariance. Separately we show that, for actions depending only on the F_\mn^2 invariant, the permitted models have $L \sim \sqrt{1 + F^2}$. We also characterize acceptable vector-scalar systems. Finally we find that wide classes of gravity models share with Einstein the null nature of their characteristic surfaces.Comment: 11 pages, LaTeX, no figure

A hadron model with breaking of spatial homogeneity of vacuum

A possible breaking of spatial homogeneity of vacuum due to the interaction between quark and Bose-field is analyzed. It is shown that in this case quark can be in a localized state (like wave packet). Energetic conditions for such a spontaneous symmetry breaking are found in suggested model. Possible consequences of such symmetry breaking, in particular, the origin of deep inelastic processes and quark confinement phenomenon are discussed.Comment: 4 page