We study properties of eigenvalues of a matrix associated with a randomly chosen partial automorphism of a regular rooted tree. We show that asymptotically, as the numbers of levels goes to infinity, the fraction of non-zero eigenvalues converges to zero in probability.

Eugenia Kochubinska

We study some combinatorial properties of partial wreath $k$-th power of the semigroup $\mathcal{IS}_d$. In particular, we calculate its order, the number of idempotents and the number of D-classes.

Eugenia Kochubinska

A simple physical mechanism of stimulated light scattering on nanoscale objects in water suspension similar to Langmuir waves mechanism in plasma is proposed. The proposed mechanism is based on a dipole interaction between the light wave and the non-compensated electrical charge that inevitably exists on a nanoscale object (a virus or a nanoparticle) in water environment. The experimental data for tobacco mosaic virus and polystyrene nanospheres are presented to support the suggested physical mechanism. It has been demonstrated that stimulated amplification spectral line frequencies observed experimentally are well explained by the suggested mechanism. In particular, the absence of lower frequency lines and the generation lines shift when changing the pH are due to ion friction appearing in the ionic solution environment. The selection rules observed experimentally also confirm the dipole interaction type. It has been shown that microwave radiation on nanoscale object acoustic vibrations frequency should appear under such scattering conditions. We demonstrate that such conditions also allow for local selective heating of nanoscale objects by dozens to hundreds degrees K. This effect is controlled by the optical irradiation parameters and can be used for affecting selectively certain types of viruses.

V. B. Oshurko O. V. Karpova A. N. Fedorov M. A. Davydov A. F. Bunkin S. M. Pershin M. Ya. Grishin

We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (E_Q,T_Q) and extend this assignation to a functor. We exhibit a relationship between the slice construction on symmetric multicategories, and the `free operad' monad construction on suitable monads. We use this to give an explicit description of the relationship between Baez-Dolan and Leinster opetopes.

Eugenia Cheng

We continue our previous modifications of the Baez-Dolan theory of opetopes to modify the Baez-Dolan definition of universality, and thereby the category of opetopic n-categories and lax functors. For the case n=2 we exhibit an equivalence between this category and the category of bicategories and lax functors. We examine notions of strictness in the opetopic theory.

Eugenia Cheng

We address the fact that composition in an opetopic weak n-category is in general not unique and hence is not a well-defined operation. We define composition with a given k-cell in an n-category by a span of (n-k)-categories. We characterise such a cell as universal if its composition span gives an equivalence of (n-k)-categories.

Eugenia Cheng

We give a precise description of combed trees in terms of Kelly-Mac Lane graphs. We show that any combed tree is uniquely expressed as an allowable Kelly-Mac Lane graph of a certain shape. Conversely, we show that any such Kelly-Mac Lane graph uniquely defines a combed tree.

Eugenia Cheng

This paper follows from two earlier works. In the first we gave an explicit construction of opetopes, the underlying cell shapes in the theory of opetopic n-categories; at the heart of this construction is the use of certain trees. In the second we gave a description of trees using Kelly-Mac Lane graphs. In the present paper we apply the latter to the former, to give a construction of opetopes using Kelly-Mac Lane graphs.

Eugenia Cheng

We study harmonic functions which admit a certain majorant in the unit ball in $\R^m $. We prove that when the majorant fulfills a doubling condition, the extremal growth or decay may occur only along small sets of radii, and we give precise estimates of these exceptional sets.

Kjersti Solberg Eikrem Eugenia Malinnikova

We present the status of recent results from the BaBar and Belle experiments on the measurement of the angle gamma from the Dalitz plot analyses of B0->Kspi+pi- and B0->K+pi-pi0.

Eugenia Maria Teresa Irene Puccio