Inconsistency of inaccessibility

The work presents the brief exposition of the proof (in ZF) of inaccessible cardinals nonexistence. To this end in view there is used the apparatus of subinaccessible cardinals and its basic tools -- reduced formula spectra and matrices and matrix functions and others. Much attention is devoted to the explicit and substantial development and cultivation of basic ideas, serving as grounds for all main constructions and reasoningsComment: 8 pages, e-mail [email protected]. arXiv admin note: substantial text overlap with arXiv:1010.195

Imbedded Singular Continuous Spectrum for Schr\"odinger Operators

We construct examples of potentials $V(x)$ satisfying $|V(x)| \leq \frac{h(x)}{1+x},$ where the function $h(x)$ is growing arbitrarily slowly, such that the corresponding Schr\"odinger operator has imbedded singular continuous spectrum. This solves one of the fifteen "twenty-first century" problems for Schr\"odinger operators posed by Barry Simon. The construction also provides the first example of a Schr\"odinger operator for which M\"oller wave operators exist but are not asymptotically complete due to the presence of singular continuous spectrum.Comment: 30 pages, 2 figure

Twist of cholesteric liquid crystal cell with substrates of different anchoring strengths

We consider director configurations of cholesteric liquid crystal (CLC) cells with two plane confining substrates. Exact solutions of the Euler-Lagrange equations for out-of-plane orientations of the easy axes that correspond to inhomogeneous conical structures of CLC director are derived. We study dependence of the CLC twist wavenumber on the free twisting number assuming that anchoring energies at the substrates are either equal or different. In both cases this dependence is found to be generally discontinuous with hysteresis loops and bistability effects involved. For CLC cells with identical substrates and planar anchoring conditions the jump-like behaviour only disappears in the weak anchoring limit. Contrastingly, when the anchoring strengths are different, there is the finite value of anchoring below which the dependence becomes continuous. Another effect is the appearance of the gap between the adjacent twist wavenumber intervals representing locally stable director configurations. We calculate the critical value of anchoring asymmetry and present the results of numerical calculations.Comment: 15 pages, 8 figure

Saddle-splay term induced orientational instability in nematic liquid crystal cells and director fluctuations at substrates

We analyze stability of the planar orientational structure in a nematic liquid crystal (NLC) cell with planar anchoring conditions at both substrates. Specifically, we study the instabilities of the ground state caused by surface elasticity at large saddle-splay elastic constant, $K_{24}$. For relatively small $K_{24}$ violating the Ericksen inequalities the theory predicts that the critical fluctuation mode of the wavelength, $\lambda_c$, will render the structure unstable when the thickness of the cell is below its critical value, $d_c$. The parity of the critical mode changes as the twist-splay ratio $K_2/K_1$ is passing through the unity. Further increase of $K_{24}$ beyond the second threshold value, $4K_1K_2/(K_1+K_2)$, leads to the instability with respect to the short wavelength fluctuations regardless of the cell thickness. We compute the critical thickness and the critical wavelength as functions of $K_{24}$, the twist-splay ratio and the azimuthal anchoring strength.Comment: extended version, uses revtex4, 29 pages, 13 figure

On the associative homotopy Lie algebras and the Wronskians

Representations of the Schlessinger-Stasheff's associative homotopy Lie algebras in the spaces of higher-order differential operators are analyzed; in particular, a remarkable identity for the Wronskian determinants is obtained. The W-transformations of chiral embeddings, related with the Toda equations, of complex curves into the Kaehler manifolds are shown to be endowed with the homotopy Lie algebra structures. Extensions of the Wronskian determinants that preserve the properties of the Schlessinger-Stasheff's algebras are constructed for the case of $n\geq1$ independent variables.Comment: 18 pages, no figures. To appear in: Fundamental'naya i Prikladnaya Matematika (English transl.: Journal of Mathematical Sciences

Kinetics of photoinduced anisotropy in azopolymers: models and mechanisms

We consider the effect of photoinduced optical anisotropy (POA) in azopolymers. Using a unified approach to the kinetics of photo-reorientation we discuss the assumptions underlying the known theoretical models of POA and formulate a tractable phenomenological model in terms of angular redistribution probabilities and order parameter correlation functions. The model takes into account biaxiality effects and long term stability of POA in azopolymers. It predicts that under certain conditions two different mechanisms, photoorientation and photoselection, will dominate POA depending on the wavelength of pumping light. By using available experimental data, we employ the model to compute dependencies of principal absorption coefficients on the illumination time. Our calculations clearly indicate the different regimes of POA and the numerical results are found to be in good agreement with the experimental data.Comment: 10 pages, 2 figures, uses revtex

Absolutely continuous spectrum of Stark operators

We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is $|q(x)| \leq C(1+|x|)^{-{1/4}-\epsilon};$ in the smoothness direction, a sufficient condition in H\"older classes is $q \in C^{{1/2}+\epsilon}(\reals)$. On the other hand, we show that there exist potentials which both satisfy $|q(x)| \leq C(1+|x|)^{-1/4}$ and belong to $C^{1/2}(\reals)$ for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.Comment: 29 page

Quenching of Reaction by Cellular Flows

We consider a reaction-diffusion equation in a cellular flow. We prove that in the strong flow regime there are two possible scenario for the initial data that is compactly supported and the size of the support is large enough. If the flow cells are large compared to the reaction length scale, propagating fronts will always form. For the small cell size, any finitely supported initial data will be quenched by a sufficiently strong flow. We estimate that the flow amplitude required to quench the initial data of support $L_0$ is $A>CL_0^4\ln(L_0)$. The essence of the problem is the question about the decay of the $L^\infty$ norm of a solution to the advection-diffusion equation, and the relation between this rate of decay and the properties of the Hamiltonian system generated by the two-dimensional incompressible fluid flow

On weakly non-local, nilpotent, and super-recursion operators for N=1 super-equations

We consider nonlinear, scaling-invariant N=1 boson + fermion supersymmetric systems whose right-hand sides are homogeneous differential polynomials and satisfy some natural assumptions. We select the super-systems that admit infinitely many higher symmetries generated by recursion operators; we further restrict ourselves to the case when the dilaton dimensions of the bosonic and fermionic super-fields coincide and the weight of the time is half the weight of the spatial variable. We discover five systems that satisfy these assumptions; one system is transformed to the purely bosonic Burgers equation. We construct local, nilpotent, triangular, weakly non-local, and super-recursion operators for their symmetry algebras.Comment: 6 pages, no figures. Proc. Int. Workshop `Supersymmetries and Quantum Symmetries-05,' JINR, Dubna, 27-31 July 2005. MSC 2000: 35Q53, 37K05, 37K10, 81T4

Spectral Properties of Schr\"odinger Operators with Decaying Potentials

We review recent advances in the spectral theory of Schr\"odinger operators with decaying potentials. The area has seen spectacular progress in the past few years, stimulated by several conjectures stated by Barry Simon starting at the 1994 International Congress on Mathematical Physics in Paris. The one-dimensional picture is now fairly complete, and provides many striking spectral examples. The multidimensional picture is still far from clear and may require deep original ideas for further progress. It might hold the keys for better understanding of a wide range of spectral and dynamical phenomena for Schr\"odinger operators in higher dimensions.Comment: 25 page