Poisson vertex algebras in the theory of Hamiltonian equations

We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms \omega_j, j in Z_+, of the variational complex \Omega. If these forms are exact, i.e. \omega_j are variational derivatives of some local functionals \int h_j, then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex \Omega is exact, provided that the algebra of functions V is "normal"; in particular, for arbitrary V, any closed form in \Omega becomes exact if we add to V a finite number of antiderivatives. We demonstrate on the examples of KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.Comment: 95 page

Exact solutions for vibrational levels of the Morse potential via the asymptotic iteration method

Exact solutions for vibrational levels of diatomic molecules via the Morse potential are obtained by means of the asymptotic iteration method. It is shown that, the numerical results for the energy eigenvalues of $^{7}Li_{2}$ are all in excellent agreement with the ones obtained before. Without any loss of generality, other states and molecules could be treated in a similar way

Rational matrix pseudodifferential operators

The skewfield K(d) of rational pseudodifferential operators over a differential field K is the skewfield of fractions of the algebra of differential operators K[d]. In our previous paper we showed that any H from K(d) has a minimal fractional decomposition H=AB^(-1), where A,B are elements of K[d], B is non-zero, and any common right divisor of A and B is a non-zero element of K. Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-zero element of K[d]. In the present paper we study the ring M_n(K(d)) of nxn matrices over the skewfield K(d). We show that similarly, any H from M_n(K(d)) has a minimal fractional decomposition H=AB^(-1), where A,B are elements of M_n(K[d]), B is non-degenerate, and any common right divisor of A and B is an invertible element of the ring M_n(K[d]). Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-degenerate element of M_n(K [d]). We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures.Comment: 20 page

Meeting the Sustainable Development Goals leads to lower world population growth

Here we show the extent to which the expected world population growth could be lowered by successfully implementing the recently agreed-upon Sustainable Development Goals (SDGs). The SDGs include specific quantitative targets on mortality, reproductive health, and education for all girls by 2030, measures that will directly and indirectly affect future demographic trends. Based on a multidimensional model of population dynamics that stratifies national populations by age, sex, and level of education with educational fertility and mortality differentials, we translate these goals into SDG population scenarios, resulting in population sizes between 8.2 and 8.7 billion in 2100. Because these results lie outside the 95% prediction range given by the 2015 United Nations probabilistic population projections, we complement the study with sensitivity analyses of these projections that suggest that those prediction intervals are too narrow because of uncertainty in baseline data, conservative assumptions on correlations, and the possibility of new policies influencing these trends. Although the analysis presented here rests on several assumptions about the implementation of the SDGs and the persistence of educational, fertility, and mortality differentials, it quantitatively illustrates the view that demography is not destiny and that policies can make a decisive difference. In particular, advances in female education and reproductive health can contribute greatly to reducing world population growth

Palladate precatalysts for the formation of C-N and C-C bonds

A series of imidazolium-based palladate precatalysts has been synthesized and the catalytic activity of these air- and moisture-stable complexes evaluated as a function of the nature of the imidazolium counterion. These precatalysts can be converted under catalytic conditions to Pd-NHC species capable of enabling the Buchwald-Hartwig aryl amination and the alpha-arylation of ketones. Both reactions can be carried out efficiently under very mild operating conditions. The effectiveness of the protocol was tested on functionality-laden substrates

Radio Astronomical Polarimetry and the Lorentz Group

In radio astronomy the polarimetric properties of radiation are often modified during propagation and reception. Effects such as Faraday rotation, receiver cross-talk, and differential amplification act to change the state of polarized radiation. A general description of such transformations is useful for the investigation of these effects and for the interpretation and calibration of polarimetric observations. Such a description is provided by the Lorentz group, which is intimately related to the transformation properties of polarized radiation. In this paper the transformations that commonly arise in radio astronomy are analyzed in the context of this group. This analysis is then used to construct a model for the propagation and reception of radio waves. The implications of this model for radio astronomical polarimetry are discussed.Comment: 10 pages, accepted for publication in Astrophysical Journa