On spacelike hypersurfaces of constant sectional curvature lorentz manifolds

Let $x:M^n\to {\overline{M}}^{n+1}$ be an n-dimensional spacelike hypersurface of a constant sectional curvature Lorentz manifold $\overline{M}$. Based on previous work of S. Montiel, L. Alías, A. Brasil and G. Colares studied what can be said about the geometry of M when $\overline{M}$ is a conformally stationary spacetime, with timelike conformal vector field K. For example, if $M^n$ has constant higher order mean curvatures $H_r$ and $H_{r+1}$, they concluded that $M^n$ is totally umbilical, provided $H_{r+1}\ne 0$ on it. If div($K$) does not vanish on $M^n$ they also proved that $M^n$ is totally umbilical, provided it has, a priori, just one constant higher order mean curvature.In this paper, we compute $L_r(S_r)$ for such an immersion, and use the resulting formula to study both r-maximal spacelike hypersurfaces of $\overline{M}$, as well as, in the presence of a constant higher order mean curvature, constraints on the sectional curvature of M that also suffice to guarantee the umbilicity of M. Here, by $L_r$ we mean the linearization of the second order differential operator associated to the r-th elementary symmetric function $S_r$ on the eigenvalues of the second fundamental form of x.     

Regularities of many-body systems interacting by a two-body random ensemble

The ground states of all even–even nuclei have angular momentum, I, equal to zero, $I=0$, and positive parity, $\pi =+$. This feature was believed to be a consequence of the attractive short-range interaction between nucleons. However, in the presence of two-body random interactions, the predominance of $I^{\pi }=0^{+}$ ground states (0 g.s.) was found to be robust both for bosons and for an even number of fermions. For simple systems, such as d bosons, sp bosons, sd bosons, and a few fermions in single-$j$ shells for small $j$, there are a few approaches to predict and/or explain spin I ground state (I g.s.) probabilities. An empirical approach to predict I g.s. probabilities is available for general cases, such as fermions in a single-$j$ ($j>\frac{7}{2}$) or many-$j$ shells and various boson systems, but a more fundamental understanding of the robustness of 0 g.s. dominance is still out of reach. Further interesting results are also reviewed concerning other robust phenomena of many-body systems in the presence of random two-body interactions, such as the odd–even staggering of binding energies, generic collectivity, the behavior of average energies, correlations, and regularities of many-body systems interacting by a displaced two-body random ensemble.     

Triangle for the entropic index q of non-extensive statistical mechanics observed by Voyager 1 in the distant heliosphere

Tsallis [Physica A 340 (2004) 1) identified a set of numbers, the “q-triplet” ≡ {q_stat, q_sen, q_rel}, for a system described by non-extensive statistical mechanics. The deviation of the q's from unity is a measure of the departure from thermodynamic equilibrium. We present observations of the q-triplets derived from two sets of daily averages of the magnetic field strength B observed by Voyager 1 in the solar wind near 40 A.U. during 1989 and near 85 A.U. during 2002, respectively. The results for 1989 do not differ significantly from those for 2002. We find $q_{\mathrm{stat}}=1.75\pm 0.06$, $q_{\mathrm{sen}}=-0.6\pm 0.2$, and $q_{\mathrm{rel}}=3.8\pm 0.3$.     

Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems

This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space $Q$ makes use of Dirac structures on $T^{1}Q$ that are induced from a constraint distribution on $Q$ as well as natural symplectomorphisms between the spaces $T^{1}TQ$, $T{T}^{1}Q$, and $T^1{T}^{1}Q$. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L–C circuit, a degenerate Lagrangian system with holonomic constraints.     

First-principles study on the phase transitions,crystal stabilities and thermodynamic properties of TiN under high pressure

The structural and thermodynamic properties of titanium nitride (TiN) have been investigated by merging first-principles calculations and particle-swarm algorithm. The three phases are identified for TiN, including the B1, the $P{6}_3/mmc$, and the B2 phases. A new phase of anti-TiP structure with the space group $P{6}_3/mmc$ has been predicted. The calculated phase transition from the B1 to the $P{6}_3/mmc$ occurs at 270 GPa. The vibrational, elastic, and thermodynamic properties for the three phases have been calculated and discussed.     

Exact PsTd invariant and PsTd symmetric breaking solutions,symmetry reductions and Bäcklund transformations for an AB–KdV system

In natural and social science, many events happened at different space–times may be closely correlated. Two events, A (Alice) and B (Bob) are defined as correlated if one event is determined by another, say, $B=\stackrel{?}{f}A$ for suitable $\stackrel{?}{f}$ operators. A nonlocal AB–KdV system with shifted-parity ($P_s$, parity with a shift), delayed time reversal ($T_d$, time reversal with a delay) symmetry where $B=\stackrel{?}{P_s}\stackrel{?}{T_d}A$ is constructed directly from the normal KdV equation to describe two-area physical event. The exact solutions of the AB–KdV system, including $P_s{T}_d$ invariant and $P_s{T}_d$ symmetric breaking solutions are shown by different methods. The $P_s{T}_d$ invariant solution show that the event happened at A will happen also at B. These solutions, such as single soliton solutions, infinitely many singular soliton solutions, soliton–cnoidal wave interaction solutions, and symmetry reduction solutions etc., show the AB–KdV system possesses rich structures. Also, a special Bäcklund transformation related to residual symmetry is presented via the localization of the residual symmetry to find interaction solutions between the solitons and other types of the AB–KdV system.     

Possible competing order-induced Fermi arcs in cuprate superconductors

We investigate the scenario of competing order (CO) induced Fermi arcs and pseudogap in cuprate superconductors. For hole-type cuprates, both phenomena as functions of temperature and doping level can be accounted for if the CO vanishes at $T^1$ above the superconducting transition $T_c$ and the CO wave-vector Q is parallel to the antinodal direction. In contrast, the absence of these phenomena and the non-monotonic $d$-wave gap in electron-type cuprates may be attributed to $T^1<T_c$ and a CO wave-vector Q parallel to the nodal direction.     

Shannon and Fisher entropies for a hydrogen atom under soft spherical confinement

We calculate Shannon and Fisher entropies in the position and momentum space, and some complexity measures for a variationally described hydrogen atom confined in soft and hard spherical boxes of varying dimension $r_c$ and selected values of strength $U_0$. We include calculations for a free particle trapped in impenetrable boxes. It is found that the Shannon entropy $S_r$ becomes negative for small cavity radii and large values of $U_0$, due to the highly localized nature of the particle. For soft confinement and small cavity dimensions, the entropies change very rapidly over short radial intervals.     

Brouillage thermique d'un gaz cohérent de fermions

It is generally assumed that a condensate of paired fermions at equilibrium is characterized by a macroscopic wavefunction with a well-defined, immutable phase. In reality, all systems have a finite size and are prepared at non-zero temperature; the condensate has then a finite coherence time, even when the system is isolated in its evolution and the particle number N is fixed. The loss of phase memory is due to interactions of the condensate with the excited modes that constitute a dephasing environment. This fundamental effect, crucial for applications using the condensate of pairs' macroscopic coherence, was scarcely studied. We link the coherence time to the condensate phase dynamics, and we show with a microscopic theory that the time derivative of the condensate phase operator ${\hat{\theta }}_0$ is proportional to a chemical potential operator that we construct including both the pair-breaking and pair-motion excitation branches. In a single realization of energy E, ${\hat{\theta }}_0$ evolves at long times as $-2{\mu }_{\mathrm{mc}}(E)t/ħ$, where ${\mu }_{\mathrm{mc}}(E)$ is the microcanonical chemical potential; energy fluctuations from one realization to the other then lead to a ballistic spreading of the phase and to a Gaussian decay of the temporal coherence function with a characteristic time $\propto N^{1/2}$. In the absence of energy fluctuations, the coherence time scales as N due to the diffusive motion of ${\hat{\theta }}_0$. We propose a method to measure the coherence time with ultracold atoms, which we predict to be tens of milliseconds for the canonical ensemble unitary Fermi gas.