Infinite order differential operators in spaces of entire functions

Differential operators ?(Δ_θ,ω), where ? is an exponential type entire function of a single complex variable and Δ_θ,ω=(θ+ωz)D+zD², D=∂/∂z, , θ?0, , acting in the spaces of exponential type entire function are studied. It is shown that, for ω?0, such operators preserve the set of Laguerre entire functions provided the function ? also belongs to this set. The latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane . The operator exp(aΔ_θ,ω), a?0 is studied in more details. In particular, it is shown that it preserves the set of Laguerre entire functions for all . An integral representation of exp(aΔ_θ,ω), a>0 is obtained. These results are used to obtain the solutions to certain Cauchy problems employing Δ_θ,ω.     

Whitney's extension problem for multivariate -functions

We prove that the trace of the space to an arbitrary closed subset is characterized by the following ``finiteness' property. A function belongs to the trace space if and only if the restriction to an arbitrary subset consisting of at most can be extended to a function such that

The constant is sharp.

The proof is based on a Lipschitz selection result which is interesting in its own right.

     

Scalar Domination and Normal Fluctuations in N-Vector Quantum Anharmonic Crystals

A lattice model of N-dimensional quantum anharmonic oscillators with a polynomial anharmonicity and a ferroelectric pair interaction is considered. For arbitrary N , correlation inequalities, showing that the temperature Green functions of this model are dominated by the corresponding Green functions of the scalar (N=1) model, are proven. These inequalities are then used to prove that the fluctuations of displacements of particles remain normal at all temperatures provided the model parameters obey a certain condition. In particular this means that the smallest distance between the energy levels of the corresponding one-dimensional isolated oscillator should be large enough or its mass should be small enough.     

Power losses in the international linear collider 20 mrad extraction line at 1 TeV

We have performed a detailed study of the power losses in the post-collision extraction line of a TeV e ⁺ e ⁻ collider with a crossing angle of 20 mrad at the interaction point. Five cases were considered: four luminosity configurations for ILC and one for CLIC. For all of them, the strong beam-beam effects at the interaction point lead to an emittance growth for the outgoing beam, as well as to the production of beamstrahlung photons and e ⁺ e ⁻ coherent pairs. The power losses along the extraction line, which are due to energy deposition by a fraction of the disrupted beam, of the beamstrahlung photons and of the coherent pairs, were estimated in the case of ideal collisions, as well as with a vertical position or angular offset at the interaction point.      

Self-collimation and beam splitting in low-index photonic crystals

We study self-collimation and beam splitting in low-refractive-index photonic crystals created within chalcogenide glass. We propose a beam splitter structure that allows direct experimental verification of photonic-crystal effects at optical wavelengths in a straightforward and definitive manner. The beam splitter provides angular separation of 90° using a highly compact spatial footprint, thus delivering direct application in highly integrated photonic devices.     

Effectivity of Brauer–Manin obstructions on surfaces

We study Brauer–Manin obstructions to the Hasse principle and to weak approximation on algebraic surfaces over number fields. A technique for constructing Azumaya algebra representatives of Brauer group elements is given, and this is applied to the computation of obstructions.     

Choiceless polynomial time

Turing machines define polynomial time (PTime) on strings but cannot deal with structures like graphs directly, and there is no known, easily computable string encoding of isomorphism classes of structures. Is there a computation model whose machines do not distinguish between isomorphic structures and compute exactly PTime properties? This question can be recast as follows: Does there exist a logic that captures polynomial time (without presuming the presence of a linear order)? Earlier, one of us conjectured a negative answer. The problem motivated a quest for stronger and stronger PTime logics. All these logics avoid arbitrary choice. Here we attempt to capture the choiceless fragment of PTime. Our computation model is a version of abstract state machines (formerly called evolving algebras). The idea is to replace arbitrary choice with parallel execution. The resulting logic expresses all properties expressible in any other PTime logic in the literature. A more difficult theorem shows that the logic does not capture all of PTime.     

The Effect of Blood Contained in the Samples on the Metabolomic Profile of Mouse Brain Tissue: A Study by NMR Spectroscopy

(1) Recently, metabolic profiling of the tissue in the native state or extracts of its metabolites has become increasingly important in the field of metabolomics. An important factor, in this case, is the presence of blood in a tissue sample, which can potentially lead to a change in the concentration of tissue metabolites and, as a result, distortion of experimental data and their interpretation. (2) In this paper, the metabolomic profiling based on NMR spectroscopy was performed to determine the effect of blood contained in the studied samples of brain tissue on their metabolomic profile. We used 13 male laboratory CD-1^® IGS mice for this study. The animals were divided into two groups. The first group of animals (n = 7) was subjected to the perfusion procedure, and the second group of animals (n = 6) was not perfused. The brain tissues of the animals were homogenized, and the metabolite fraction was extracted with a water/methanol/chloroform solution. Samples were studied by high-frequency ¹H-NMR spectroscopy with subsequent statistical data analysis. The group comparison was performed with the use of the Student’s test. We identified 36 metabolites in the brain tissue with the use of NMR spectroscopy. (3) For the major set of studied metabolites, no significant differences were found in the brain tissue metabolite concentrations in the native state and after the blood removal procedure. (4) Thus, it was shown that the presence of blood does not have a significant effect on the metabolomic profile of the brain in animals without pathologies.     

Einstein solvmanifolds with a simple Einstein derivation

The structure of a solvable Lie group admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).