Abstract. We show that the ordinary derivative of a real analytic function of one variable can be realized as a Grassmann-Berezin-type integration over the Zeon algebra, the Z-integral. 1. Introduction. The intersection of methods coming from distinct disciplines, such as combinatorics, physics, and graph theory, has been a subject of considerable interest. Clearly, once a common language is established among distinct disciplines, one can gain insight into the structure of each one, first enhancing our understanding through basic manipulations and then, ultimately, pointing toward further constructions. In this manner, some remarkable results have been achieved, including operator methods in graph enumeration problems [1,2,3,4], partitions and quantum field theory [5,6], normal ordering of operators [7,8,9,10], coherent states and combinatorics [11,12], supersymmetry and combinatorics [13], spinorial formulation of graphs [14,15], the Lagrange-Good formula via methods of quantum field theory [16], Heisenberg algebras and Fibonacci numbers [17], and generalizations of the Kirchhoff matrix-tree theorem via Grassmann algebras (see [18] and references therein). Of special relevance is the Zeon algebra [19,20,21,22], comprising important results in the realm of enumerative combinatorics, such as counting problems in lattice partitions [19,20] and derangements [22].In this work, seeking such cross-fertilization of ideas, we revisit a realization of ordinary differentiation of a real analytic function of one variable as a GrassmannBerezin (G-B) integral [23,24,25,26,27,28] over the Zeon algebra introduced in [29]
We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d-dimensional lattice ferromagnetic classical N -component vector spin systems at high temperature (0 < β 1). Each system is characterized by a single site "a priori" spin probability distribution. The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe-Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe-Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.
Neste trabalho revisamos alguns dos sistemas não-lineares mais paradigmáticos e desvendamos algumas das suas surpreendentes interligações. Os problemas de interesse, descrito matematicamente pelas equações de sine-Gordon, Toda e KdV, generalizam modelos físicos conhecidos, como o pêndulo simples, o sistema massa-mola e as ondas lineares emágua, respectivamente. Depois de discutirmos as peculiaridades decorrentes da presença de não-linearidades nos modelos, esclarecemos como os sistemas apresentados são relacionados uns aos outros, indicando a existência de uma família de equações que compartilham propriedades como a integrabilidade. Nós mostramos como a equação de KdV pode ser convenientemente discretizada a fim de preservar tais características importantes. Além de apresentarmos as ligações estreitas que esta equação tem com a cadeia de Toda e com a equação sine-Gordon, nós também investigamos outros procedimentos capazes de gerar um sistema integrável discreto a partir do modelo KdV com a discretização de Hirota. Palavras-chave: KdV, Sine-Gordon, Cadeia de Toda.In this report we review some of the most paradigmatic nonlinear systems and unveil some of their suprising interconnections. The problems of interest, described mathematically by the equations of sine-Gordon, Toda and KdV, generalize well known physical models, the simple pendulum, the mass on a spring and the linear waves, respectively. After discussing the differences arising from the presence of nonlinearities in the models, we clarify how the systems presented are related to each other, indicating existence of a family of equations sharing integrability properties. We show how the KdV equation can be conveniently discretized in order to preserve important properties. Besides presenting the close connections this equation has with respect to the Toda lattice and the sine-Gordon equation, we also investigate other procedures capable of generating a discrete integrable system from the KdV model, as the Hirota discretization. Keywords: KdV, Sine-Gordon, Toda Lattice. IntroduçãoA maior parte dos fenômenos que ocorrem na natureza envolvem efeitos não-lineares. No entanto, os cursos de graduação tendem a focar nos aspectos lineares destes fenômenos. Dessa forma, em nossa formação, somos levados a pensar que os sistemas não-lineares são tão complicados que nunca podem ser tratados de forma analítica. Assim, pode parecer que o profissional daárea deve resignar-se, apenas, a simulações numéricas, como ocorre com a previsão do tempo, por exemplo. * Endereço de correspondência: paulo.assis@ufg.br.Diante de tal cenário, neste trabalho trazemos uma introdução a alguns problemas não-lineares que podem ser tratados exatamente, com obtenção de soluções analíticas, e que, embora possam parecerà primeira vista completamente independentes, estão relacionados entre si de uma maneira profunda por meio de simetrias. Sabe-se que a presença de simetrias num problema está associadaà existência de leis de conservação [1,2], que podem permitir ou facilitar a solução do proble...
Resumo Reconhecendo a carência de pesquisas nacionais acerca da estratégia de pedir ajuda, o presente estudo objetivou conhecer o uso e a concepção de pedir ajuda em 60 estudantes de ambos os sexos de 2ª a 4ª série do ensino fundamental. Os instrumentos de coleta de dados foram uma prancha contendo uma situação problema hipotética e uma entrevista estruturada desenvolvidos para o presente estudo. Os dados foram examinados qualitativamente por meio da análise de conteúdo. Resultados indicam que a estratégia de pedir ajuda é bastante relatada pelos participantes para a realização de atividades acadêmicas ou cotidianas. O pedido de ajuda é expresso verbalmente. Os familiares são as principais pessoas a quem os estudantes recorrem no momento de dificuldade. Os dados são discutidos em termos de reconhecer a importância da estratégia de pedir ajuda e da necessidade de novos estudos que aprofundem o conhecimento acerca dessa estratégia. Palavras-chave: psicologia cognitiva, estratégia de pedir ajuda, estratégia de aprendizagem. Abstract The national research about help seeking strategy is slim. In this sense, this study investigated the conceptions about help seeking of 60 students of both sexes from 2 nd to 4 th grade education. Data collection instruments were both a problem-solving hypothetical situation related to help seeking and a structured interview specially developed for this study. The data were examined qualitatively by means of content analysis. Results indicated that help seeking strategy is frequently reported by students to do their academic activities or daily. The request for assistance is expressed verbally. Family members ate the key persons to whom the students seek at the time of difficulty. Data are discussed not only in terms of the need of acknowledging the importance of help seeking strategy, but also in terms of the need to deepen the knowledge about this strategy.
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