Elliptic scaling functions as compactly supported multivariate analogs of the B-splines

In the paper, we present a family of multivariate compactly supported scaling functions, which we call as elliptic scaling functions. The elliptic scaling functions are the convolution of elliptic splines, which correspond to homogeneous elliptic differential operators, with distributions. The elliptic scaling functions satisfy refinement relations with real isotropic dilation matrices. The elliptic scaling functions satisfy most of the properties of the univariate cardinal B-splines: compact support, refinement relation, partition of unity, total positivity, order of approximation, convolution relation, Riesz basis formation (under a restriction on the mask), etc. The algebraic polynomials contained in the span of integer shifts of any elliptic scaling function belong to the null-space of a homogeneous elliptic differential operator. Similarly to the properties of the B-splines under differentiation, it is possible to define elliptic (not necessarily differential) operators such that the elliptic scaling functions satisfy relations with these operators. In particular, the elliptic scaling functions can be considered as a composition of segments, where the function inside a segment, like a polynomial in the case of the B-splines, vanishes under the action of the introduced operator.Comment: To appear in IJWMI

INSURANCE AS AN EFFECTIVE MECHANISM TO MINIMIZE RISKS AT THE ENTERPRISE

Purpose: The article analyzes the state of domestic insurance in agricultural enterprises, which is a complex type of property insurance, subspecies of which are insurance of crops, animals, commodity aquaculture, real estate and income of agricultural producers. Methodology: Generally accepted methods and techniques of economic research were used in the study process: monographic (in the process of studying risk management theoretical foundations), statistical and economic (when studying trends of AIC enterprise development and functioning), design-constructive (when justifying and calculating indicators of enterprise functioning), abstract and logical (when generalizing conceptual and methodological approaches in identifying, analyzing and assessing risks), comparative analysis (synthesis of native and foreign risk management experience), various risk assessment methodologies. Result: The economic risk passport is understood as a set of information about the risk area, risk criteria, as well as for instructions on the application of the necessary methods to manage or minimize the risk. The article presented a liquidity loss risk passport with one of the measures to minimize it - self-insurance. Applications: This research can be used for universities, teachers, and students. Novelty/Originality: In this research, the model of Insurance as an Effective Mechanism to Minimize Risks at the Enterprise is presented in a comprehensive and complete manner

Quasi-Gaussian Statistics of Hydrodynamic Turbulence in 3/4+\epsilon dimensions

The statistics of 2-dimensional turbulence exhibit a riddle: the scaling exponents in the regime of inverse energy cascade agree with the K41 theory of turbulence far from equilibrium, but the probability distribution functions are close to Gaussian like in equilibrium. The skewness \C S \equiv S_3(R)/S^{3/2}_2(R) was measured as \C S_{\text{exp}}\approx 0.03. This contradiction is lifted by understanding that 2-dimensional turbulence is not far from a situation with equi-partition of enstrophy, which exist as true thermodynamic equilibrium with K41 exponents in space dimension of $d=4/3$. We evaluate theoretically the skewness \C S(d) in dimensions ${4/3}\le d\le 2$, show that \C S(d)=0 at $d=4/3$, and that it remains as small as \C S_{\text{exp}} in 2-dimensions.Comment: PRL, submitted, REVTeX 4, 4 page

Parametric generation of second sound in superfluid helium: linear stability and nonlinear dynamics

We report the experimental studies of a parametric excitation of a second sound (SS) by a first sound (FS) in a superfluid helium in a resonance cavity. The results on several topics in this system are presented: (i) The linear properties of the instability, namely, the threshold, its temperature and geometrical dependencies, and the spectra of SS just above the onset were measured. They were found to be in a good quantitative agreement with the theory. (ii) It was shown that the mechanism of SS amplitude saturation is due to the nonlinear attenuation of SS via three wave interactions between the SS waves. Strong low frequency amplitude fluctuations of SS above the threshold were observed. The spectra of these fluctuations had a universal shape with exponentially decaying tails. Furthermore, the spectral width grew continuously with the FS amplitude. The role of three and four wave interactions are discussed with respect to the nonlinear SS behavior. The first evidence of Gaussian statistics of the wave amplitudes for the parametrically generated wave ensemble was obtained. (iii) The experiments on simultaneous pumping of the FS and independent SS waves revealed new effects. Below the instability threshold, the SS phase conjugation as a result of three-wave interactions between the FS and SS waves was observed. Above the threshold two new effects were found: a giant amplification of the SS wave intensity and strong resonance oscillations of the SS wave amplitude as a function of the FS amplitude. Qualitative explanations of these effects are suggested.Comment: 73 pages, 23 figures. to appear in Phys. Rev. B, July 1 st (2001

The variational Poisson cohomology

It is well known that the validity of the so called Lenard-Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.Comment: 130 pages, revised version with minor changes following the referee's suggestion

Conformal and Affine Hamiltonian Dynamics of General Relativity

The Hamiltonian approach to the General Relativity is formulated as a joint nonlinear realization of conformal and affine symmetries by means of the Dirac scalar dilaton and the Maurer-Cartan forms. The dominance of the Casimir vacuum energy of physical fields provides a good description of the type Ia supernova luminosity distance--redshift relation. Introducing the uncertainty principle at the Planck's epoch within our model, we obtain the hierarchy of the Universe energy scales, which is supported by the observational data. We found that the invariance of the Maurer-Cartan forms with respect to the general coordinate transformation yields a single-component strong gravitational waves. The Hamiltonian dynamics of the model describes the effect of an intensive vacuum creation of gravitons and the minimal coupling scalar (Higgs) bosons in the Early Universe.Comment: 37 pages, version submitted to Gen. Rel. Gra