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

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

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

Response of a particle in a one-dimensional lattice to an applied force: Dynamics of the effective mass

We study the behaviour of the expectation value of the acceleration of a particle in a one-dimensional periodic potential when an external homogeneous force is suddenly applied. The theory is formulated in terms of modified Bloch states that include the interband mixing induced by the force. This approach allows us to understand the behaviour of the wavepacket, which responds with a mass that is initially the bare mass, and subsequently oscillates around the value predicted by the effective mass. If Zener tunneling can be neglected, the expression obtained for the acceleration of the particle is valid over timescales of the order of a Bloch oscillation, which are of interest for experiments with cold atoms in optical lattices. We discuss how these oscillations can be tuned in an optical lattice for experimental detection.Comment: 15 pages, 12 figure

Lax-Phillips scattering theory for PT-symmetric \rho-perturbed operators

The S-matrices corresponding to PT-symmetric \rho-perturbed operators are defined and calculated by means of an approach based on an operator-theoretical interpretation of the Lax-Phillips scattering theory

Magnetization pinning in conducting films demonstrated using broadband ferromagnetic resonance

The broadband microstrip ferromagnetic resonance technique has been applied for detection and characterization of a magnetic inhomogeneity in a film sample. In the case of a 100nm thick Permalloy film an additional magnetically depleted top sub-layer, practically unidentifiable by the conventional ferromagnetic resonance setup, has been detected and characterized. These results have been confirmed by Brillouin light scattering spectroscopy revealing the fact that the optical properties of the additional sub-layer do not differ much from those of the bulk of the film. Subsequent characterization of a large number of other presumably single-layer films with thicknesses in the range 30-100nm using the same ferromagnetic resonance technique also revealed the same effect

On asymptotic stability of the Skyrmion

We study the asymptotic behavior of spherically symmetric solutions in the Skyrme model. We show that the relaxation to the degree-one soliton (called the Skyrmion) has a universal form of a superposition of two effects: exponentially damped oscillations (the quasinormal ringing) and a power law decay (the tail). The quasinormal ringing, which dominates the dynamics for intermediate times, is a linear resonance effect. In contrast, the polynomial tail, which becomes uncovered at late times, is shown to be a \emph{nonlinear} phenomenon.Comment: 4 pages, 4 figures, minor changes to match the PRD versio

Astrocytes and Müller Cell Alterations During Retinal Degeneration in a Transgenic Rat Model of Retinitis Pigmentosa

Purpose: Retinitis pigmentosa includes a group of progressive retinal degenerative diseases that affect the structure and function of photoreceptors. Secondarily to the loss of photoreceptors, there is a reduction in retinal vascularization, which seems to influence the cellular degenerative process. Retinal macroglial cells, astrocytes, and Müller cells provide support for retinal neurons and are fundamental for maintaining normal retinal function. The aim of this study was to investigate the evolution of macroglial changes during retinal degeneration in P23H rats. Methods: Homozygous P23H line-3 rats aged from P18 to 18 months were used to study the evolution of the disease, and SD rats were used as controls. Immunolabeling with antibodies against GFAP, vimentin, and transducin were used to visualize macroglial cells and cone photoreceptors. Results: In P23H rats, increased GFAP labeling in Müller cells was observed as an early indicator of retinal gliosis. At 4 and 12 months of age, the apical processes of Müller cells in P23H rats clustered in firework-like structures, which were associated with ring-like shaped areas of cone degeneration in the outer nuclear layer. These structures were not observed at 16 months of age. The number of astrocytes was higher in P23H rats than in the SD matched controls at 4 and 12 months of age, supporting the idea of astrocyte proliferation. As the disease progressed, astrocytes exhibited a deteriorated morphology and marked hypertrophy. The increase in the complexity of the astrocytic processes correlated with greater connexin 43 expression and higher density of connexin 43 immunoreactive puncta within the ganglion cell layer (GCL) of P23H vs. SD rat retinas. Conclusions: In the P23H rat model of retinitis pigmentosa, the loss of photoreceptors triggers major changes in the number and morphology of glial cells affecting the inner retina

Factorization of Seiberg-Witten Curves and Compactification to Three Dimensions

We continue our study of nonperturbative superpotentials of four-dimensional N=2 supersymmetric gauge theories with gauge group U(N) on R^3 x S^1, broken to N=1 due to a classical superpotential. In a previous paper, hep-th/0304061, we discussed how the low-energy quantum superpotential can be obtained by substituting the Lax matrix of the underlying integrable system directly into the classical superpotential. In this paper we prove algebraically that this recipe yields the correct factorization of the Seiberg-Witten curves, which is an important check of the conjecture. We will also give an independent proof using the algebraic-geometrical interpretation of the underlying integrable system.Comment: laTeX, 14 pages, uses AMSmat