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The Hurst effect plays an important role in many areas such as physics, climate and finance. It describes the anomalous growth of range and constrains the behavior and predictability of these systems. The Hurst effect is frequently taken to be synonymous with Long-Range Dependence (LRD) and is typically assumed to be produced by a stationary stochastic process which has infinite memory. However, infinite memory appears to be at odds with the Markovian nature of most physical laws while the stationary assumption lacks robustness. Here we use Lorenz's paradigmatic chaotic model to show that regime behavior can also cause the Hurst effect. By giving an alternative, parsimonious, explanation using non stationary Markovian dynamics, our results question the common belief that the Hurst effect necessarily implies a stationary infinite memory process. We also demonstrate that our results can explain atmospheric variability without the infinite memory previously thought necessary and are consistent with climate model simulations.
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion[note 1] with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean space-time.In such quantum theories, solutions to the equations of motion may be thought of as critical points of the action. The critical points of the action may be local maxima of the action, local minima, or saddle points. For example, the classical path (or classical equation of motion) is the path that minimizes the action and is therefore a global minimum. Instantons are important in quantum field theory because:they appear in the path integral as the leading quantum corrections to the classical behavior of a system, and they can be used to study the tunneling behavior in various systems such as a Yang–Mills theory.
in Lorentzian spacetime and thus not relevant to density perturbations. But their properties are important in establishing the meaning of the Euclidean path integral. If negative modes are present, the Euclidean path integral is not well defined, but may nevertheless be useful in an approximate description of the decay of an unstable state. When gravitational dynamics is included, counting negative modes requires a careful treatment of the conformal factor problem. We demonstrate that for an appropriate choice of coordinate on phase space, the second order Euclidean action is bounded below for normalized perturbations and has a finite number of negative modes. We prove that there is a negative mode for many gravitational Instantons of the Hawking-Moss or Coleman-De Luccia type, and discuss the associated spectral flow. We also investigate Hawking-Turok constrained Instantons, which occur in a generic inflationary model. Implementing the regularization and constraint proposed by Kirklin, Turok and Wiseman, we find that those Instantons leading to substantial inflation do not possess negative modes. Using an alternate regularization and constraint motivated by reduction from five dimensions, we find a negative mode is present. These investigations shed new light on the suitability of Euclidean quantum gravity as a potential description of our universe. Comment: 16 pages, compressed and RevTex file, including one postscript figure file
Abstract: The Kasner solutions of the Einstein equations within the General Relativity, i.e. (0, 0, 1) and (2/3, 2/3, -1/3), concern the gravitationally massless superluminal entanglement of the Einstein-spacetime components. For entanglement are responsible the superluminal zero-helicity vector particles/entanglons. The new interpretation of the Einstein formula E = mcc leads to conclusion that Nature using the matter, i.e. the Einstein-spacetime components, “copies” the non-gravitational objects in bigger scales so there appear the cores of baryons, electrons and new cosmology. In such phase transitions appear the spinors applied in the Quantum Physics. The entanglons appeared during the inflation in the first phase transition of the modified Higgs field and both fields, i.e. the modified Higgs field and the field composed of the exchanged entanglons in some structure satisfy all initial conditions for the Kasner solutions. The Kasner solutions define many properties of the spinors. The Kasner solution (0, 0, 1) describes spinning loop composed of the exchanged entanglons that entangle the Einstein-spacetime components whereas the generalized Kasner solution (2/3, 2/3, -1/3) describes binary system of tori composed of the exchanged entanglons. The two tori have parallel spins and the directions of spin overlap but their internal helicities are opposite. We can partially unify the gravity and quantum physics via the Kasner solutions. The basic mathematical method applied in Quantum Physics, i.e. the action of some orthogonal groups on column vector that leads to the spin representations, is some generalization of action of matrix of rotation of a circle around the z-axis on circle on xy plane defined parametrically and written as a column – such action leads to the parametric equations for torus. The Kasner solutions as well lead to the holography. The symmetrical decays of multi-loops lead to the Theory of Chaos.