Kuramoto’s original model describes the dynamics and synchronization behavior of a set of interacting oscillators represented by their phases. The system can also be pictured as a set of particles moving on a circle in two dimensions, which allows a direct generalization to particles moving on the surface of higher dimensional spheres. One of the key features of the 2D system is the presence of a continuous phase transition to synchronization as the coupling intensity increases. Ott and Antonsen proposed an ansatz for the distribution of oscillators that allowed them to describe the dynamics of the order parameter with a single differential equation. A similar ansatz was later proposed for the D-dimensional model by using the same functional form of the 2D ansatz and adjusting its parameters. In this article, we develop a constructive method to find the ansatz, similarly to the procedure used in 2D. The method is based on our previous work for the 3D Kuramoto model where the ansatz was constructed using the spherical harmonics decomposition of the distribution function. In the case of motion in a D-dimensional sphere, the ansatz is based on the hyperspherical harmonics decomposition. Our result differs from the previously proposed ansatz and provides a simpler and more direct connection between the order parameter and the ansatz.
The Kuramoto model describes the synchronization of coupled oscillators that have different natural frequencies. Among the many generalizations of the original model, Kuramoto and Sakaguchi (KS) proposed a frustrated version that resulted in dynamic behavior of the order parameter, even when the average natural frequency of the oscillators is zero. Here, we consider a generalization of the frustrated KS model that exhibits new transitions to synchronization. The model is identical in form to the original Kuramoto model but written in terms of unit vectors and with the coupling constant replaced by a coupling matrix. The matrix breaks the rotational symmetry and forces the order parameter to point in the direction of the eigenvector with the highest eigenvalue, when the eigenvalues are real. For complex eigenvalues, the module of order parameter oscillates while it rotates around the unit circle, creating active states. We derive the complete phase diagram for the Lorentzian distribution of frequencies using the Ott–Antonsen ansatz. We also show that changing the average value of the natural frequencies leads to further phase transitions where the module of the order parameter goes from oscillatory to static.
<p>Most models of soil C dynamics can be expressed by the differential vector equation:</p> <p><strong>dC(t)/dt = f(t).K.C(t) + b(t)</strong></p> <p>where each element of the vector C(t) represents a carbon compartment with intrinsic decomposition rate (usually fast, slow and passive); K is the transition matrix between the compartments (decomposition rates and decomposition partitioning); the scalar function f(t) is a forcing function of the decomposition rates modifiers (e.g. soil moisture and temperature); and b(t) is the vector with rates of external C inputs for each compartment. Considering the case where only total soil carbon is measured, only the sum of C in all compartments can be used for model evaluation, calibration and data assimilation. Also, in most compartmental models there are too many parameters to be adjusted, leading to identifiability problems. Although some parameters can be constrained according to the model&#8217;s assumptions, identifiability is still problematic except for the simplest compartmental models. By working on the differential equation, it is possible to deduce an explicit representation of the total carbon trajectory, in a way that the number of necessary empirical parameters is reduced, without loss of generality or need of further assumptions. In this work we propose such a representation for the total carbon trajectory whose generality embraces implicitly the mechanism of models as Century, RothC and CQESTR. The solution requires less parameters than the original models do but still allows mapping the original model parameters and decomposition modifiers functions onto the solution. Additionally, we show how the main processes of decomposition of soil organic matter can be represented by the terms of the solution found. Finally, we present the solution behavior under extreme conditions of temperature, humidity and initial stocks. We expect our general framework to help improving model&#8217;s calibration and data assimilation procedures.</p>
How come such a successful theory like Quantum Mechanics has so many mysteries? The history of this theory is replete with dubious interpretations and controversies. The knowledge of its predictions, however, caused the amazing technological revolution of the last hundred years. In its very beginning Einstein pointed out that there was something missing due to contradictions with the relativity theory. So, even though Quantum Mechanics explains all the physical phenomena, due to its mysteries, there were many attempts to find a way to "complete" it, e.g. hidden-variable theories. In this paper, we discuss some of these mysteries, with special attention to the concepts of physical reality imposed by quantum mechanics, the role of the observer, prediction limits, definition of collapse, and how to deal with correlated states (the basic strategy for quantum computers and quantum teleportation). The discussion is carried out by accepting that there is nothing important missing. We are just restricted by the limitations imposed by quantum mechanics. The mysteries are cleared out by a proper interpretation of these limitations. This is done by introducing two interpretation rules within the Copenhagen interpretation.
Why does such a successful theory like Quantum Mechanics have so many mysteries? The history of this theory is replete with dubious interpretations and controversies, and yet a knowledge of its predictions, however, contributed to the amazing technological revolution of the last hundred years. In its very beginning Einstein pointed out that there was something missing, due to contradictions with the relativity theory. So, even though Quantum Mechanics explains all the nanoscale physical phenomena, there were many attempts to find a way to "complete" it, e.g. hidden-variable theories. In this paper, we discuss some of those enigmas, with special attention to the concepts of physical reality imposed by quantum mechanics, the role of the observer, prediction limits, a definition of collapse, and how to deal with correlated states (the basic strategy for quantum computers and quantum teleportation). That discussion is carried out within the framework of accepting that there is in fact nothing important missing, rather we are just restricted by the limitations imposed by quantum mechanics. The mysteries are thus explained by a proper interpretation of those limitations, which is achieved by introducing two interpretation rules within the Copenhagen paradigm.
Este projeto de iniciação científica tem por objetivo estudar a Relatividade Especial através de um caminho mais voltado à matemática, de uma maneira elegante e rigorosa.A formulação da Relatividade Especial em termos geométricos torna o estudo de tal teoria especialmente atraente a estudantes de física matemática. De fato, um sólido curso de álgebra linear já é suficiente para dominar os conceitos geométricos do espaço de Minkowski e, a partir daí, entender suas implicações físicas no contexto da teoria da relatividade propriamente dita. Os fenômenos de dilatação temporal, contração de Lorentz e (falsos) paradoxos de simultaneidade seguem de maneira direta e elegante do estudo do grupo de Lorentz e suas simetrias. Além disso, a teoria da Relatividade Especial possui uma formulação matemática elegante e relativamente simples. A referência principal a ser usada aqui será o livro The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity de G. L. Naber, que tem exatamente a proposta discutida acima. Nas palavras do próprio autor: “It is the intention of this monograph to provide an introduction to the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics.”
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