Two dimensional dynamical systems which admit Lie and Noether symmetries

We prove two theorems which relate the Lie point symmetries and the Noether symmetries of a dynamical system moving in a Riemannian space with the special projective group and the homothetic group of the space respectively. The theorems are applied to classify the two dimensional Newtonian dynamical systems, which admit a Lie point/Noether symmetry. Two cases are considered, the non-conservative and the conservative forces. The use of the results is demonstrated for the Kepler - Ermakov system, which in general is non-conservative and for potentials similar to the H\`enon Heiles potential. Finally it is shown that in a FRW background with no matter present, the only scalar cosmological model which is integrable is the one for which 3-space is flat and the potential function of the scalar field is exponential. It is important to note that in all applications the generators of the symmetry vectors are found by reading the appropriate entry in the relevant tables.Comment: 25 pages, 17 table

Quantum chemical investigation on the reaction mechanism of tertiary phosphines with unsaturated carboxylic acids: An insight into kinetic data

The structures of intermediates and transition states in the reaction of tertiary phosphines with unsaturated carboxylic acids have been calculated at the B3LYP level of theory using the 6-31+G(d,p) basis set. Analysis of the results shows that [1,3]-intramolecular migration of carboxylic proton to carbanionic center of generated zwitterionic intermediate is strongly kinetically unfavorable, and external proton-donor source is essential to complete quaternization. A molecular cluster of the intermediate with one molecule of water has been modeled for intermolecular reaction pathway, but even in this case, the proton transfer remains to be the rate-determining step that is in a good agreement with previous kinetic investigations on this reaction. The data obtained for this reaction have much in common with recent studies on the mechanisms of the Morita-Baylis-Hillman reaction and phosphine-catalyzed [3+2] cycloaddition, which revealed paramount importance of proton-transfer steps. Copyright © 2012 Wiley Periodicals, Inc

Quantum-chemical study of the effects of noncovalent interactions on the nuclear magnetic screening constants of pyrimidine base associates

The effects of weak noncovalent interactions on the nuclear magnetic screening (NMS) constants (σ 1H), (σ 13C) and charge distribution (q t ) on atoms in van der Waals model associates of unsubstituted and substituted pyrimidines and substituted uracil are considered. The NMS constants were calculated by the UB3LYP/6-31G(d,p) with GIAO functions. The correlation dependences of the 1H and 13C σ constants on the charge q on atoms whre constructed. It were shown that they can be represented as polynomials that include the terms that are linear and quadratic relative to the charge. The relations obtained in this way are similar in form and close in magnitude to the coefficients of the known Buckingham and Augspurger functions that describe the electric field effects on the nuclear magnetic screening constants. It was found that the coefficients in these polynomials have a definite physical sense in that they characterize nuclear magnetic screening and the "screening polarizability" tensor in the unperturbed molecule and associate, respectively. The NMS constants and charge distribution in pyrimidine base associates and accordingly the coefficients that reflect their values in polynomials depend on the form, size, and composition of the associate and can vary significantly depending on the position of the pyrimidine base in the associate. © 2007 Springer Science+Business Media, Inc

Using the Noether symmetry approach to probe the nature of dark energy

We propose to use a model-independent criterion based on first integrals of motion, due to Noether symmetries of the equations of motion, in order to classify the dark energy models in the context of scalar field (quintessence or phantom) FLRW cosmologies. In general, the Noether symmetries play an important role in physics because they can be used to simplify a given system of differential equations as well as to determine the integrability of the system. The Noether symmetries are computed for nine distinct accelerating cosmological scenarios that contain a homogeneous scalar field associated with different types of potentials. We verify that all the scalar field potentials, presented here, admit the trivial first integral namely energy conservation, as they should. We also find that the exponential potential inspired from scalar field cosmology, as well as some types of hyperbolic potentials, include extra Noether symmetries. This feature suggests that these potentials should be preferred along the hierarchy of scalar field potentials. Finally, using the latter potentials, in the framework of either quintessence or phantom scalar field cosmologies that contain also a non-relativistic matter(dark matter) component, we find that the main cosmological functions, such as the scale factor of the universe, the scalar field, the Hubble expansion rate and the metric of the FRLW space-time, are computed analytically. Interestingly, under specific circumstances the predictions of the exponential and hyperbolic scalar field models are equivalent to those of the $\Lambda$CDM model, as far as the global dynamics and the evolution of the scalar field are concerned. The present analysis suggests that our technique appears to be very competitive to other independent tests used to probe the functional form of a given potential and thus the associated nature of dark energy.Comment: Accepted for publication in Physical Review D (13 pages

A quantum chemical study of an interaction between collagen fragments and calcium ions using calculations of model complexes

© 2015 Springer Science+Business Media New York. Quantum chemical calculations revealed that the complexation between proline-containing collagen fragments and calcium ions involves a preferential interaction between Ca2+ ions and the oxygen and nitrogen atoms of proline in the glycine - proline - alanine fragment of the amino acid sequence of collagen

Deformation quantization of linear dissipative systems

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product $*_t$ and an ``extended'' operator of time derivative $D_t=\partial_t+...$, differentiating the $\ast_t$-product. As in the usual case, the $\ast_t$-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional $\mathrm{Tr}_t$. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.Comment: 14 pages, typos correcte

Production method of nanostructured wood-polymer composition with microwave application

ABSTRACT: Introduction. Currently, wood-polymer compositions (WPC) are widely used in the national economy and construction. The composition of WPC varies widely depending on the further purpose. Improving the binding quality in the wood-polymer system is one of the promising areas for enhancing operational characteristics. Organic and inorganic substrates nanostructured with individual substances, including metal particles, are used as binding components. In the petrochemical industry, most high-capacity productions use catalysts based on active carriers like heavy metals when developing targeted products for various purposes. After several stages of regeneration, recovering these heavy metals becomes impossible. Consequently, spent catalysts accumulate in sedimentation tanks and sludge collectors, lacking an efficient method for disposal and secondary use. One of the components included in the composition of spent catalysts is chromium (+6), which belongs to carcinogenic metals. Numerous disposal methods are currently inadequate for neutralizing this metal on an industrial scale, which is of interest for research. Methods and materials. The study is aimed at converting carcinogenic chromium (+6) into non-carcinogenic chromium (+3) by ultrahigh frequency exposure (microwave), which will open up opportunities for its use as a chromium-containing nanocomplex binding a tree-polymer. Results and discussions. The ultrahigh-frequency effect on the mixture of wood-polymer composition and spent chromium (+6) causes an increase in the penetration depth of high–frequency waves, characterized by a uniform distribution of energy over the entire area of the composite, which is explained by the reduction of chromium (VI) oxide into chromium (III) oxide, and there is also a change in the color of the nanostructured wood-polymer composition (WP – compositions) from yellow to malachite. Conclusion. This study, which consists in the application of microwave exposure to the wood-nanoparticle-polymer system, confirms the receipt of a durable construction product and its use in the construction of roofs, facade boards, sidewalks, piers, port facilities, et

Conditional linearizability criteria for a system of third-order ordinary differential equations

We provide linearizability criteria for a class of systems of third-order ordinary differential equations (ODEs) that is cubically semi-linear in the first derivative, by differentiating a system of second-order quadratically semi-linear ODEs and using the original system to replace the second derivative. The procedure developed splits into two cases, those where the coefficients are constant and those where they are variables. Both cases are discussed and examples given

Constraints and analytical solutions of f(R)f(R) theories of gravity using Noether symmetries

We perform a detailed study of the modified gravity $f(R)$ models in the light of the basic geometrical symmetries, namely Lie and Noether point symmetries, which serve to illustrate the phenomenological viability of the modified gravity paradigm as a serious alternative to the traditional scalar field approaches. In particular, we utilize a model-independent selection rule based on first integrals, due to Noether symmetries of the equations of motion, in order to identify the viability of $f(R)$ models in the context of flat FLRW cosmologies. The Lie/Noether point symmetries are computed for six modified gravity models that include also a cold dark matter component. As it is expected, we confirm that all the proposed modified gravity models admit the trivial first integral namely energy conservation. We find that only the $f(R)=(R^{b}-2\Lambda)^{c}$ model, which generalizes the concordance $\Lambda$ cosmology, accommodates extra Lie/Noether point symmetries. For this $f(R)$ model the existence of non-trivial Noether (first) integrals can be used to determine the integrability of the model. Indeed within this context we solve the problem analytically and thus we provide for the first time the evolution of the main cosmological functions such as the scale factor of the universe and the Hubble expansion rate.Comment: Accepted for publication in Physical Review D (10 pages), typos corrected and some new references include