Piecewise-linear maps with heterogeneous chaos

Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos), While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two more explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.Comment: 16 pages, 9 figure

Hausdorff dimension of heteroclinic intersections for some partially hyperbolic sets

We introduce a $C^2$-open set of diffeomorphisms of $\mathbb R^3$ which have two transitive hyperbolic sets, one is of index 1 (the dimension of the unstable subbundle) and the other is of index 2. We prove that: the unstable set of the first hyperbolic set and the stable set of the second are of Hausdorff dimension nearly 2; the intersection of these unstable and stable sets contains a set of Hausdorff dimension nearly 1.Comment: 28 pages, 8 figure

The dynamics of the heterochaos baker maps

The heterochaos baker maps are piecewise affine maps of the unit square or cube introduced by Saiki et al. (2018), to provide a hands-on, elementary understanding of complicated phenomena in systems of large degrees of freedom. We review recent progress on a dynamical systems theory of the heterochaos baker maps, and present new results on properties of measures of maximal entropy and the underlying Lebesgue measure. We address several conjectures and questions that may illuminate new aspects of heterochaos and inspire future research.Comment: 37 pages, 10 figure

Quantum noise in ideal operational amplifiers

We consider a model of quantum measurement built on an ideal operational amplifier operating in the limit of infinite gain, infinite input impedance and null output impedance and with a feddback loop. We evaluate the intensity and voltage noises which have to be added to the classical amplification equations in order to fulfill the requirements of quantum mechanics. We give a description of this measurement device as a quantum network scattering quantum fluctuations from input to output ports.Comment: 4 pages, 2 figures, RevTe

Fluctuation-dissipation theorem and quantum tunneling with dissipation at finite temperature

A reformulation of the fluctuation-dissipation theorem of Callen and Welton is presented in such a manner that the basic idea of Feynman-Vernon and Caldeira -Leggett of using an infinite number of oscillators to simulate the dissipative medium is realized manifestly without actually introducing oscillators. If one assumes the existence of a well defined dissipative coefficient $R(\omega)$ which little depends on the temperature in the energy region we are interested in, the spontanous and induced emissions as well as induced absorption of these effective oscillators with correct Bose distribution automatically appears. Combined with a dispersion relation, we reproduce the tunneling formula in the presence of dissipation at finite temperature without referring to an explicit model Lagrangian. The fluctuation-dissipation theorem of Callen-Welton is also generalized to the fermionic dissipation (or fluctuation) which allows a transparent physical interpretation in terms of second quantized fermionic oscillators. This fermionic version of fluctuation-dissipation theorem may become relevant in the analyses of, for example, fermion radiation from a black hole and also supersymmetry at the early universe.Comment: 19 pages. Phys. Rev. E (in press

Ulam type stability problems for alternative homomorphisms

We introduce an alternative homomorphism with respect to binary operations and investigate the Ulam type stability problem for such a mapping. The obtained results apply to Ulam type stability problems for several important functional equations.ArticleJOURNAL OF INEQUALITIES AND APPLICATIONS. 2014:228 (2014)journal articl

A note on alpha-vacua and interacting field theory in de Sitter space

We set up a consistent renormalizable perturbation theory of a scalar field in a nontrivial alpha vacuum in de Sitter space. Although one representation of the effective action involves non-local interactions between anti-podal points, we show the theory leads to causal physics, and we prove a spectral theorem for the interacting two-point function. We construct the renormalized stress energy tensor and show this develops no imaginary part at leading order in the interactions, consistent with stability.Comment: 22 pages, 2 figures, latex. v4 some clarifications, some typos fixe

Black Hole Thermodynamics in Horava Lifshitz Gravity and the Related Geometry

Recently, Ho$\breve{r}$ava proposed a non-relativistic renormalizable theory of gravity which is essentially a field theoretic model for a UV complete theory of gravity and reduces to Einstein gravity with a non-vanishing cosmological constant in IR. Also the theory admits a Lifshitz scale-invariance in time and space with broken Lorentz symmetry at short scale. On the other hand, at large distances higher derivative terms do not contribute and the theory coincides with general relativity. Subsequently, Cai and his collaborators and then Catiuo et al have obtained black hole solutions in this gravity theory and studied the thermodynamic properties of the black hole solution. In the present paper, we have investigated the black hole thermodynamic for two choices of the entropy function - a classical and a topological in nature. Finally, it is examined whether a phase transition is possible or not.Comment: 8 figure

Long-range attraction between particles in dusty plasma and partial surface tension of dusty phase boundary

Effective potential of a charged dusty particle moving in homogeneous plasma has a negative part that provides attraction between similarly charged dusty particles. A depth of this potential well is great enough to ensure both stability of crystal structure of dusty plasma and sizable value of surface tension of a boundary surface of dusty region. The latter depends on the orientation of the surface relative to the counter-ion flow, namely, it is maximal and positive for the surface normal to the flow and minimal and negative for the surface along the flow. For the most cases of dusty plasma in a gas discharge, a value of the first of them is more than sufficient to ensure stability of lenticular dusty phase void oriented across the counter-ion flow.Comment: LATEX, REVTEX4, 7 pages, 6 figure

VERTICAL AND HORIZONTAL FORCES DURING CUTIING IN BASKETBALL UNDER DIFFERENT CONDITIONS

The purpose of this study is to evaluate ground reaction force responses in professional basketball athletes while executing this sport's typical cutting maneuver with and without ankle bracing: taping, aircast-type orthosis and basketball shoes. Eight athletes were dynamically analyzed during a basketball cutting maneuver with a force platform. We collected vertical and medial-lateral forces under these three conditions and analyzed force peaks of foot contact with the ground and propulsion and growth gradient for these forces. Results show that bracing did not significantly change Fymax1 and GCFymax1; significantly reduced Fymax2 and GG Fymax2. With respect to the medial-lateral component, there were no significant differences in relation to force magnitudes between the three study conditions. However, GG Fzmax1 was significantly greater for the sport shoe condition than for the taping condition. Bracing decreased ground reaction force at some instances, but increased in others