Differential equations on variable time scales

We introduce a class of differential equations on variable   time scales with a transition condition between two consecutive parts of the scale. Conditions for existence and uniqueness of solutions are obtained. Periodicity, boundedness and stability of solutions are considered. The method of investigation is by means of two successive reductions: B$B$-equivalence of the system [E. Akalín, M.U. Akhmet, The principles of B-smooth discontinuous flows, Computers and Mathematics with Applications 49 (2005) 981–995; M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163–178; M.U. Akhmet, N.A. Perestyuk, The comparison method for differential equations with impulse action, Differential Equations 26 (9) (1990) 1079–1086] on a variable time scale to a system on a time scale, a reduction to an impulsive differential equation [M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163–178; M.U. Akhmet, M. Turan, The differential equations on time scales through impulsive differential equations, Nonlinear Analysis 65 (2006) 2043–2060]. Appropriate examples are constructed to illustrate the theory.     

Geometric approach to stable homotopy groups of spheres. The Adams–Hopf invariants

In this paper, a geometric approach to stable homotopy groups of spheres based on the Pontryagin–Thom construction is proposed. From this approach, a new proof of the Hopf-invariant-one theorem of J. F. Adams for all dimensions except 15, 31, 63, and 127 is obtained. It is proved that for n > 127, in the stable homotopy group of spheres Π _n , there is no element with Hopf invariant one. The new proof is based on geometric topology methods. The Pontryagin–Thom theorem (in the form proposed by R. Wells) about the representation of stable homotopy groups of the real, projective, infinite-dimensional space (these groups are mapped onto 2-components of stable homotopy groups of spheres by the Kahn–Priddy theorem) by cobordism classes of immersions of codimension 1 of closed manifolds (generally speaking, nonoriented) is considered. The Hopf invariant is expressed as a characteristic class of the dihedral group for the self-intersection manifold of an immersed codimension-1 manifold that represents the given element in the stable homotopy group. In the new proof, the geometric control principle (by M. Gromov) for immersions in the given regular homotopy classes based on the Smale–Hirsch immersion theorem is required. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 3–15, 2007.     

On two definitions of a narrow operator on Köthe–Bochner spaces

It is already known that the Cesàro matrices of orders one and two are coposinormal operators on \(\ell ^2\). Here it is shown that the Cesàro matrices of all orders are coposinormal; the proof employs posinormality, achieved by means of a diagonal interrupter, and makes use of the Zeilberger’s algorithm and computational assistance by Maple?.     

Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons

Paraxial beams whose transverse structure rotates upon free propagation (spiral beams) can be treated as analogs of azimuthons recently found in nonlinear media [Phys. Rev. Lett.95, 203904 (2005)]. These linear azimuthons have essentially a nonlocalized character and can possess an almost arbitrary rotation rate independent of the angular momentum of the beam. Such beams can be assimilated into fluent mechanical bodies with intrinsic mass flows determined by transverse energy redistribution over the beam cross section.     

Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields

There exists a substantial body of theory that predicts mutual screening of signed topological singularities (topological charges) in random optical fields (speckle patterns). Such screening appears to be rather mysterious because there are neither energetic nor entropic reasons for its existence. We present the first experimental confirmation of mutual screening by the stationary points of the intensity, the canonical optical scalar field, and of mutual screening by C points in elliptically polarized light, the generic optical vector field. We also elucidate specific aspects of the geometry and topology of these fields that we argue give rise to screening.     

Mapping phases of singular scalar light fields

We implement experimentally a simple method for accurate measurements of phase distributions of scalar light fields. The method is based on the polarimetric technique for recording the polarization maps of vector fields, where coaxial superposition of orthogonally polarized reference and signal beams allows the signal phase to be reconstructed from the polarization map of the total field. We demonstrate this method by resolving topologically neutral pairs of closely positioned vortices in a speckle field and recovering the positions of vortices within a Laguerre-Gaussian beam with the topological charge three.     

Optical polarization singularities and elliptic stationary points

Polarization singularities and elliptic stationary points (collectively, elliptic critical points) were measured experimentally via the complex Stokes field S1 + iS2, where S1 and S2 are Stokes parameters. This new, easily implemented method yielded detailed, high-resolution experimental data for all elliptic critical points. These data confirm with high precision the elliptic-field topological sign rule, loop rules, and Stokes singularity relations introduced recently.     

Domination problem for narrow orthogonally additive operators

The “Up-and-down” theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result in operator theory. We prove an analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result is used to prove a theorem of domination for order narrow positive abstract Uryson operators from a vector lattice E to a Banach lattice F with an order continuous norm.     

Reversible Photoswitching of a Spin‐Crossover Molecular Complex in the Solid State at Room Temperature

Spin‐crossover metal complexes are highly promising magnetic molecular switches for prospective molecule‐based devices. The spin‐crossover molecular photoswitches developed so far operate either at very low temperatures or in the liquid phase, which hinders practical applications. Herein, we present a molecular spin‐crossover iron(II) complex that can be switched between paramagnetic high‐spin and diamagnetic low‐spin states with light at room temperature in the solid state. The reversible photoswitching is induced by alternating irradiation with ultraviolet and visible light and proceeds at the molecular level.     

There is No Low Maximal D.C.E. Degree

We show that for any computably enumerable (c.e.) set A and any set L, if L is low and , then there is a c.e. splitting such that . In Particular, if L is low and n‐c.e., then is n‐c.e. and hence there is no low maximal n‐c.e. degree.