Some non- linear problems in theoretical physics

An immense variety of problems in theoretical physics are of the non-linear type. Non~linear partial differential equations (NPDE) have almost become the rule rather than an exception in diverse branches of physics such as fluid mechanics, field theory, particle physics, statistical physics and optics, and the construction of exact solutions of these equations constitutes one of the most vigorous activities in theoretical physics today. The thesis entitled ‘Some Non-linear Problems in Theoretical Physics’ addresses various aspects of this problem at the classical level. For obtaining exact solutions we have used mathematical tools like the bilinear operator method, base equation technique and similarity method with emphasis on its group theoretical aspects. The thesis deals with certain methods of finding exact solutions of a number of non-linear partial differential equations of importance to theoretical physics. Some of these new solutions are of relevance from the applications point of view in diverse branches such as elementary particle physics, field theory, solid state physics and non-linear optics and give some insight into the stable or unstable behavior of dynamical Systems The thesis consists of six chapters.

First-principles electronic theory of non-collinear magnetic order in transition-metal nanowires

The structural, electronic and magnetic properties of one-dimensional 3d transition-metal (TM) monoatomic chains having linear, zigzag and ladder geometries are investigated in the frame-work of first-principles density-functional theory. The stability of long-range magnetic order along the nanowires is determined by computing the corresponding frozen-magnon dispersion relations as a function of the 'spin-wave' vector q. First, we show that the ground-state magnetic orders of V, Mn and Fe linear chains at the equilibrium interatomic distances are non-collinear (NC) spin-density waves (SDWs) with characteristic equilibrium wave vectors q that depend on the composition and interatomic distance. The electronic and magnetic properties of these novel spin-spiral structures are discussed from a local perspective by analyzing the spin-polarized electronic densities of states, the local magnetic moments and the spin-density distributions for representative values q. Second, we investigate the stability of NC spin arrangements in Fe zigzag chains and ladders. We find that the non-collinear SDWs are remarkably stable in the biatomic chains (square ladder), whereas ferromagnetic order (q =0) dominates in zigzag chains (triangular ladders). The different magnetic structures are interpreted in terms of the corresponding effective exchange interactions J(ij) between the local magnetic moments μ(i) and μ(j) at atoms i and j. The effective couplings are derived by fitting a classical Heisenberg model to the ab initio magnon dispersion relations. In addition they are analyzed in the framework of general magnetic phase diagrams having arbitrary first, second, and third nearest-neighbor (NN) interactions J(ij). The effect of external electric fields (EFs) on the stability of NC magnetic order has been quantified for representative monoatomic free-standing and deposited chains. We find that an external EF, which is applied perpendicular to the chains, favors non-collinear order in V chains, whereas it stabilizes the ferromagnetic (FM) order in Fe chains. Moreover, our calculations reveal a change in the magnetic order of V chains deposited on the Cu(110) surface in the presence of external EFs. In this case the NC spiral order, which was unstable in the absence of EF, becomes the most favorable one when perpendicular fields of the order of 0.1 V/Å are applied. As a final application of the theory we study the magnetic interactions within monoatomic TM chains deposited on graphene sheets. One observes that even weak chain substrate hybridizations can modify the magnetic order. Mn and Fe chains show incommensurable NC spin configurations. Remarkably, V chains show a transition from a spiral magnetic order in the freestanding geometry to FM order when they are deposited on a graphene sheet. Some TM-terminated zigzag graphene-nanoribbons, for example V and Fe terminated nanoribbons, also show NC spin configurations. Finally, the magnetic anisotropy energies (MAEs) of TM chains on graphene are investigated. It is shown that Co and Fe chains exhibit significant MAEs and orbital magnetic moments with in-plane easy magnetization axis. The remarkable changes in the magnetic properties of chains on graphene are correlated to charge transfers from the TMs to NN carbon atoms. Goals and limitations of this study and the resulting perspectives of future investigations are discussed.

Non-commutative symbolic coding

We give a non-commutative generalization of classical symbolic coding in the presence of a synchronizing word. This is done by a scattering theoretical approach. Classically, the existence of a synchronizing word turns out to be equivalent to asymptotic completeness of the corresponding Markov process. A criterion for asymptotic completeness in general is provided by the regularity of an associated extended transition operator. Commutative and non-commutative examples are analysed.

Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers-Kronig relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

Non-genomic actions of estrogens and their interaction with genomic actions in the brain

Ligands for the nuclear receptor superfamily have at least two mechanisms of action: (a) classical transcriptional regulation of target genes (genomic mechanisms); and (b) non-genomic actions, which are initiated at the cell membrane, which could also impact transcription. Though transcriptional mechanisms are increasingly well understood, membrane-initiated actions of these ligands are incompletely understood. This has led to considerable debate over the physiological relevance of membrane-initiated actions of hormones versus genomic actions of hormones, with genomic actions predominating in the endocrine field. There is good evidence that the membrane-limited actions of hormones, particularly estrogens, involve the rapid activation of kinases and the release of calcium and that these are linked to physiologically relevant scenarios in the brain. We show evidence in this review, that membrane actions of estrogens, which activate these rapid signaling cascades, can also potentiate nuclear transcription in both the central nervous system and in non-neuronal cell lines. We present a theoretical scenario which can be used to understand this phenomenon. These signaling cascades may occur in parallel or in series but subsequently, converge at the modification of transcriptionally relevant molecules such as nuclear receptors and/or coactivators. In addition, other non-cognate hormones or neurotransmitters may also activate cascades to crosstalk with estrogen receptor-mediated transcription, though the relevance of this is less clear. The idea that coupling between membrane-initiated and genomic actions of hormones is a novel idea in neuroendocrinology and provides us with a unified view of hormone action in the central nervous system.

Membrane-initiated non-genomic signaling by estrogens in the hypothalamus: cross-talk with glucocorticoids with implications for behavior

The estrogen receptor and glucocorticoid receptor are members of the nuclear receptor superfamily that can signal using both non-genomic and genomic transcriptional modes. Though genomic modes of signaling have been well characterized and several behaviors attributed to this signaling mechanism, the physiological significance of non-genomic modes of signaling has not been well understood. This has partly been due to the controversy regarding the identity of the membrane ER (mER) or membrane GR (mGR) that may mediate rapid, non-genomic signaling and the downstream signaling cascades that may result as a consequence of steroid ligands binding the mER or the mGR. Both estrogens and glucocorticoids exert a number of actions on the hypothalamus, including feedback. This review focuses on the various candidates for the mER or mGR in the hypothalamus and the contribution of non-genomic signaling to classical hypothalamically driven behaviors and changes in neuronal morphology. It also attempts to categorize some of the possible functions of non-genomic signaling at both the cellular level and at the organismal level that are relevant for behavior, including some behaviors that are regulated by both estrogens and glucocorticoids in a potentially synergistic manner. Lastly, it attempts to show that steroid signaling via non-genomic modes may provide the organism with rapid behavioral responses to stimuli.

Aging and stationary properties of non-equilibrium symmetrical three-state models

We consider a non-equilibrium three-state model whose dynamics is Markovian and displays the same symmetry as the three-state Potts model, i.e. the transition rates are invariant under the cyclic permutation of the states. Unlike the Potts model, detailed balance is, in general, not satisfied. The aging and the stationary properties of the model defined on a square lattice are obtained by means of large-scale Monte Carlo simulations. We show that the phase diagram presents a critical line, belonging to the three-state Potts universality class, that ends at a point whose universality class is that of the Voter model. Aging is considered on the critical line, at the Voter point and in the ferromagnetic phase.

The Stern-Gerlach Phenomenon According to Classical Electrodynamics

We present a description of the Stem-Gerlach type experiments using only the concepts of classical electrodynamics and the Newton`s equations of motion. The quantization of the projections of the spin (or the projections of the magnetic dipole) is not introduced in our calculations. The main characteristic of our approach is a quantitative analysis of the motion of the magnetic atoms at the entrance of the magnetic field region. This study reveals a mechanism which modifies continuously the orientation of the magnetic dipole of the atom in a very short time interval, at the entrance of the magnetic field region. The mechanism is based on the conservation of the total energy associated with a magnetic dipole which moves in a non uniform magnetic field generated by an electromagnet. A detailed quantitative comparison with the (1922) Stem-Gerlach experiment and the didactical (1967) experiment by J.R. Zacharias is presented. We conclude, contrary to the original Stern-Gerlach statement, that the classical explanations are not ruled out by the experimental data.

Coherent states of non-relativistic electron in the magnetic-solenoid field

In the present work we construct coherent states in the magnetic-solenoid field, which is a superposition of the Aharonov-Bohm field and a collinear uniform magnetic field. In the problem under consideration there are two kinds of coherent states, those which correspond to classical trajectories which embrace the solenoid and those which do not. The constructed coherent states reproduce exactly classical trajectories, maintain their form under the time evolution and form a complete set of functions, which can be useful in semiclassical calculations. In the absence of the solenoid field these states are reduced to the well known in the case of uniform magnetic field Malkin-Man`ko coherent states.

Semiclassical and quantum motions on the non-commutative plane

We study the canonical and the coherent state quantizations of a particle moving in a magnetic field on the non-commutative plane. Using a theta-modified action, we perform the canonical quantization and analyze the gauge dependence of the theory. We compare coherent states quantizations obtained through Malkin-Man`ko states and circular squeezed states. The relation between these states and the ""classical"" trajectories is investigated, and we present numerical explorations of some semiclassical quantities. (C) 2009 Elsevier B.V. All rights reserved.

Approximations of modal logics: K and beyond

Inspired by the recent work on approximations of classical logic, we present a method that approximates several modal logics in a modular way. Our starting point is the limitation of the n-degree of introspection that is allowed, thus generating modal n-logics. The semantics for n-logics is presented, in which formulas are evaluated with respect to paths, and not possible worlds. A tableau-based proof system is presented, n-SST, and soundness and completeness is shown for the approximation of modal logics K, T, D, S4 and S5. (c) 2008 Published by Elsevier B.V.

Non-destructive determination of photosynthetic pigments in the leaves of castor oil plants

Objetivou-se com o presente trabalho, estabelecer a relação entre os pigmentos fotossintéticos extraídos em DMSO e as leituras obtidas no clorofilômetro portátil ClorofiLOG® 1030, gerando modelos matemáticos capazes de predizer os teores de clorofila e de carotenóides em folhas de mamoneira. O trabalho foi conduzido na Empresa Brasileira de Pesquisa Agropecuária (EMBRAPA) Algodão, situada em Campina Grande, Estado da Paraíba, em outubro de 2010. Para a análise indireta, foi utilizado um equipamento portátil, sendo realizada a leitura em discos foliares com diferentes tonalidades de verde, sendo feita, nesses mesmos discos, a determinação da clorofila pelo método clássico. Para a extração da clorofila, utilizaram-se 5 mL de dimetilsulfóxido (DMSO), a qual foi mantida em banho-maria a 70ºC, por 30 minutos, e retirou-se 3 mL da alíquota para leitura em espectrofotômetro nos comprimentos de onda de 470, 646 e 663 nm. Os dados foram submetidos à análise da variância e regressão polinomial. A leitura obtida no clorofilômetro portátil foi a variável dependente, e os pigmentos fotossintéticos determinados pelo método clássico foi a variável independente. Os resultados indicaram que o clorofilômetro portátil ClorofiLOG® 1030, associado a modelos matemáticos, permitiu estimar a concentração dos pigmentos fotossintéticos, exceto a clorofila b, com alta precisão, com economia de tempo e com reagentes normalmente utilizados nos procedimentos convencionais.

A Model for Bundled Conductors considering a Non-Homogeneous Distribution of the Current through Subconductors

This paper describes an alternative procedure to obtain an equivalent conductor from a bundled conductor, taking into account the distribution of the current in subcondutors of the bundle. Firstly, it is introduced a brief background about the concept of Geometric Mean Radius (GMR) and how this methodology is applied to define an equivalent conductor and its electric parameters. Emphasizing that the classical procedure, using GMR, is limited to premise which the current is equally distributed through subconductors. Afterwards, it is described the development of proposed method and applications for an equivalent conductor obtained from a conventional transmission line bundled conductor and from an equivalent conductor based on a bundle with compressed SF(6) insulation system, where the current is unequally distributed through subconductors.

SU(2,R)(q) symmetries of Non-Abelian Toda theories

The classical and quantum algebras of a class of conformal NA-Toda models are studied. It is shown that the SL(2,R)(q) Poisson brackets algebra generated by certain chiral and antichiral charges of the nonlocal currents and the global U(1) charge appears as an algebra of the symmetries of these models. (C) 1998 Elsevier B.V. B.V. All rights reserved.

Non-relativistic spacetimes with cosmological constant

Recent data on supernovae favour high values of the cosmological constant. Spacetimes with a cosmological constant have non-relativistic kinematics quite different from Galilean kinematics. de Sitter spacetimes, vacuum solutions of Einstein's equations with a cosmological constant, reduce in the non-relativistic limit to Newton-Hooke spacetimes, which are non-metric homogeneous spacetimes with non-vanishing curvature. The whole non-relativistic kinematics would then be modified, with possible consequences to cosmology, and in particular to the missing-mass problem.