Inverse dispersion method for calculation of complex photonic band diagram and PT\cal{PT}-symmetry

We suggest an inverse dispersion method for calculating photonic band diagram for materials with arbitrary frequency-dependent dielectric functions. The method is able to calculate the complex wave vector for a given frequency by solving the eigenvalue problem with a non-Hermitian operator. The analogy with $\cal{PT}$-symmetric Hamiltonians reveals that the operator corresponds to the momentum as a physical quantity and the singularities at the band edges are related to the branch points and responses for the features on the band edges. The method is realized using plane wave expansion technique for two-dimensional periodical structure in the case of TE- and TM-polarization. We illustrate the applicability of the method by calculation of the photonic band diagrams of an infinite two-dimension square lattice composed of dielectric cylinders using the measured frequency dependent dielectric functions of different materials (amorphous hydrogenated carbon, silicon, and chalcogenide glass). We show that the method allows to distinguish unambiguously between Bragg and Mie gaps in the spectra.Comment: 8 pages, 5 figure

On Chow weight structures for cdhcdh-motives with integral coefficients

The main goal of this paper is to define a certain Chow weight structure $w_{Chow}$ on the category $DM_c(S)$ of (constructible) $cdh$-motives over an equicharacteristic scheme $S$. In contrast to the previous papers of D. H\'ebert and the first author on weights for relative motives (with rational coefficients), we can achieve our goal for motives with integral coefficients (if $\operatorname{char}S=0$; if $\operatorname{char}S=p>0$ then we consider motives with $\mathbb{Z}[\frac{1}{p}]$-coefficients). We prove that the properties of the Chow weight structures that were previously established for $\mathbb{Q}$-linear motives can be carried over to this "integral" context (and we generalize some of them using certain new methods). In this paper we mostly study the version of $w_{Chow}$ defined via "gluing from strata"; this enables us to define Chow weight structures for a wide class of base schemes. As a consequence, we certainly obtain certain (Chow)-weight spectral sequences and filtrations for any (co)homology of motives.Comment: To appear in Algebra i Analiz (St. Petersburg Math Journal). arXiv admin note: substantial text overlap with arXiv:1007.454

Surface tension of small bubbles and droplets and the cavitation threshold

In this paper, using an unified approach, estimates are given of the magnitude of the surface tension of water for planar and curved interfaces in the pairwase interaction approximation based on the Lennard-Jones potential. It is shown that the surface tensions of a bubble and droplet have qualitatively different dependences on the curvature of the surface: for the bubble, as the radius of the surface's curvature decreases, the surface tension decreases, whereas it increases on the droplet. The corresponding values of the Tolman corrections are also determined. In addition, it is shown that the dependence of the surface tension on the surface's curvature is important for evaluating the critical negative pressure for the onset of cavitation

Selected problems

This is a renovated list of open problems, to appear in: "Affine Algebraic Geometry" conference Proceedings volume in Contemporary Mathematics series of the Amer. Math. Soc. Ed. by Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu

The fractal theory of the Saturn Ring

The true reason for partition of the Saturn ring as well as rings of other planets into great many of sub-rings is found. This reason is the theorem of Zelikin-Lokutsievskiy-Hildebrand about fractal structure of solutions to generic piece-wise smooth Hamiltonian systems. The instability of two-dimensional model of rings with continues surface density of particles distribution is proved both for Newtonian and for Boltzmann equations. We do not claim that we have solved the problem of stability of Saturn ring. We rather put questions and suggest some ideas and means for researches.Comment: 19 pages, 1 figur

A bijective proof of Loehr-Warrington's formulas for the statistics \mbox{ctot}_{\frac{q}{p}} and \mbox{midd}_{\frac{q}{p}}

Loehr and Warrington introduced partitional statistics \mbox{ctot}_{\frac{q}{p}}(D) and \mbox{midd}_{\frac{q}{p}}(D) and provided formulas for these statistics in terms of the boundary graph of the Young diagram $D$. In this paper we give a bijective proof of Loehr-Warrington's formulas using the following simple combinatorial observation: given a Young diagram $D$ and two numbers $a$ and $l,$ the number of boxes in $D$ with the arm length $a$ and the leg length $l$ is one less than the number of boxes with the same properties in the complement to $D.$ Here the complement is taken inside the positive quadrant or, equivalently, a very large rectangle.Comment: Title, abstract, and introduction changed. Last section (Hilbert schemes) expanded. Multiple minor correction. 13 pages, 5 figure

Semi-derived and derived Hall algebras for stable categories

Given a Frobenius category $\mathcal{F}$ satisfying certain finiteness conditions, we consider the localization of its Hall algebra $\mathcal{H(F)}$ at the classes of all projective-injective objects. We call it the {\it "semi-derived Hall algebra"} $\mathcal{SDH(F, P(F))}.$ We discuss its functoriality properties and show that it is a free module over a twisted group algebra of the Grothendieck group $K_0(\mathcal{P(F)})$ of the full subcategory of projective-injective objects, with a basis parametrized by the isomorphism classes of objects in the stable category $\underline{\mathcal{F}}$. We prove that it is isomorphic to an appropriately twisted tensor product of $\mathbb{Q}K_0(\mathcal{P(F)})$ with the derived Hall algebra (in the sense of To\"{e}n and Xiao-Xu) of $\underline{\mathcal{F}},$ when both of them are well-defined. We discuss some situations where the semi-derived Hall algebra is defined while the derived Hall algebra is not. The main example is the case of $2-$periodic derived category of an abelian category with enough projectives, where the semi-derived Hall algebra was first considered by Bridgeland who used it to categorify quantum groups.Comment: 13 page

On fields of definition of arithmetic Kleinian reflection groups

We show that degrees of the real fields of definition of arithmetic Kleinian reflection groups are bounded by 35.Comment: 6 pages, to appear in Proc. Amer. Math. So

Martingale-Coboundary Representation for a Class of Random Fields

A stationary random sequence admits under some assumptions a representation as the sum of two others: one of them is a martingale difference sequence, and another is a so-called coboundary. Such a representation can be used for proving some limit theorems by means of the martingale approximation. A multivariate version of such a decomposition is presented in the paper for a class of random fields generated by several commuting non-invertible probability preserving transformations. In this representation summands of mixed type appear which behave with respect to some groupof directions of the parameter space as reversed multiparameter martingale differences (in the sense of one of several known definitions) while they look as coboundaries relative to the other directions. Applications to limit theorems will be published elsewhere.Comment: 20 pages; http://www.esi.ac.at/Preprint-shadows/esi2069/htm

Analytical Framework for Credit Portfolios

Analytical, free of time consuming Monte Carlo simulations, framework for credit portfolio systematic risk metrics calculations is presented. Techniques are described that allow calculation of portfolio-level systematic risk measures (standard deviation, VaR and Expected Shortfall) as well as allocation of risk down to individual transactions. The underlying model is the industry standard multi-factor Merton-type model with arbitrary valuation function at horizon (in contrast to the simplistic default-only case). High accuracy of the proposed analytical technique is demonstrated by benchmarking against Monte Carlo simulations.Comment: 16 pages, 2 figure