Mössbauer effect study of Cs2Fe(57Co)O4

We have found that both Fe^VI and Fe^IV states form as a result of the electron capture decay of⁵⁷Co in solid cesium oxoferrate(VI). The daughter Fe^VI state is stable only in a perfect tetrahedral oxygen environment in a solid phase, and does not exist in alkaline media. The most probable mechanism for its formation is chemical substitution through electronically excited isotopic exchange.     

Multinomial coefficients modulo a prime

We say that the multinomial coefficient (m.c.) has order and power . Let be the number of m.c. that are not divisible by and have order with powers which are not larger than . If and

then for any integer

     

Euler solutions of pseudodifferential equations

We consider a homogeneous pseudodifferential equation on a cylinderC=×X over a smooth compact closed manifoldX whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators overX. When assuming the symbol to be independent on the variablet , we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity point of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler's theorem on the exponential solutions. Our setting is model for the analysis on manifolds with conical points sinceC can be thought of as a stretched manifold with conical points att=– andt=. When compared with the general theory, our approach is constructive while highlighting all the features of this latter.     

On the conjectures of J. Thompson and O. Ore

If is a finite simple group of Lie type over a field containing more than elements (for twisted groups we require , except for , , and , where we assume ), then is the square of some conjugacy class and consequently every element in is a commutator.

     

Global regular solutions for the 3D Kawahara equation posed on unbounded domains

An initial boundary value problem for the 3D Kawahara equation posed on a channel-type domain was considered. The existence and uniqueness results for global regular solutions as well as exponential decay of small solutions in the H²-norm were established.     

On the Number of Irreducible Points in Polyhedra

An integer point in a polyhedron is called irreducible iff it is not the midpoint of two other integer points in the polyhedron. We prove that the number of irreducible integer points in n-dimensional polytope P is at most \(O(m^{\lfloor \frac{n}{2}\rfloor }\log ^{n-1}\gamma )\), where n is fixed and P is given by a system of m linear inequalities with integer coefficients not exceeding (by absolute value) \(\gamma \). This bound is tight. Using this result we prove the conjecture asserting that the teaching dimension in the class of threshold functions of k-valued logic in n variables is \(\varTheta (\log ^{n-2} k)\) for any fixed \(n\ge 2\).     

Multi-particle dynamical systems and polynomials

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived.     

Some aspects of AFM nanomechanical probing of surface polymer films

We analyzed how the approach developed for the microindentation of non-uniform elastic solids can be adapted to analyze the atomic force microscopy (AFM) probing of ultrathin (1-100 nm thick) polymer films on a solid substrate, as well as polymer films with a multilayered structure. We suggested that recent Johnson's modification of the contact mechanics model that included a viscoelastic contribution could also be utilized to analyze rate-dependent loading data for polymer surfaces. The graded model proposed for microindentation experiments was modified allowing to account not only for variable elastic moduli within different layers but also for the gradient of properties between layers within a transition zone. Two examples of a recent application of this model for molecularly thick hyperbranched polymer monolayers (<3 nm thick) and tri-layered polymer films (20-40 nm thick) tethered to a solid substrate were presented and discussed. In both cases, complex shapes of both loading curves and elastic modulus depth profiles obtained from experimental AFM data were successfully fitted by the graded model with realistic structural parameters.