New infrared cut-off for the holographic scalar fields models of dark energy

Introducing a new infrared cut-off for the holographic dark-energy, we study the correspondence between the quintessence, tachyon, K-essence and dilaton energy density with this holographic dark energy density in the flat FRW universe. This correspondence allows to reconstruct the potentials and the dynamics for the scalar fields models, which describe accelerated expansion.     

Global Regularity for the Navier-Stokes-Maxwell System with Fractional Diffusion

In this paper, we study the global regularity for the Navier-Stokes-Maxwell system with fractional diffusion. Existence and uniqueness of global strong solution are proved for \(\alpha \geqslant \frac {3}{2}\). When 0 < α < 1, global existence is obtained provided that the initial data \(\|u_{0}\|_{H^{\frac {5}{2}-2\alpha }}+\|E_{0}\|_{H^{\frac {5}{2}-2\alpha }}+\|B_{0}\|_{H^{\frac {5}{2}-2\alpha }}\) is sufficiently small. Moreover, when \(1<\alpha <\frac {3}{2}\), global existence is obtained if for any ε >?0, the initial data \(\|u_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}+\|E_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}+\|B_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}\) is small enough.     

Quintessence Reissner Nordström Anti de Sitter Black Holes and Joule Thomson Effect

In this work we investigate corrections of the quintessence regime of the dark energy on the Joule-Thomson (JT) effect of the Reissner Nordström anti de Sitter (RNAdS) black hole. The quintessence dark energy has equation of state as p _q = ωρ _q in which \(-1<\omega <-\frac {1}{3}\). Our calculations are restricted to ansatz: ω = ??1 (the cosmological constant regime) and \(\omega =-\frac {2}{3}\) (quintessence dark energy). To study the JT expansion of the AdS gas under the constant black hole mass, we calculate inversion temperature T _i of the quintessence RNAdS black hole where its cooling phase is changed to heating phase at a particular (inverse) pressure P _i . Position of the inverse point {T _i , P _i } is determined by crossing the inverse curves with the corresponding Gibbons-Hawking temperature on the T-P plan. We determine position of the inverse point versus different numerical values of the mass M and the charge Q of the quintessence AdS RN black hole. The cooling-heating phase transition (JT effect) is happened for M > Q in which the causal singularity is still covered by the horizon. Our calculations show sensitivity of the inverse point {T _i , P _i } position on the T-P plan to existence of the quintessence dark energy just for large numerical values of the AdS RN black holes charge Q. In other words the quintessence dark energy dose not affect on position of the inverse point when the AdS RN black hole takes on small charges.     

Heuristic Relative Entropy Principles with Complex Measures: Large-Degree Asymptotics of a Family of Multi-variate Normal Random Polynomials

Let \(z\in \mathbb {C}\), let \(\sigma ^2>0\) be a variance, and for \(N\in \mathbb {N}\) define the integrals

$$\begin{aligned} E_N^{}(z;\sigma ) := \left\{ \begin{array}{ll} {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}}\! (x^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x^2}}{\sqrt{2\pi }}dx&{}\quad \text{ if }\, N=1,\\ {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}^N}\! \prod \prod \limits _{1\le k<l\le N}\!\! e^{-\frac{1}{2N}(1-\sigma ^{-2}) (x_k-x_l)^2} \prod _{1\le n\le N}\!\!\!\!(x_n^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x_n^2}}{\sqrt{2\pi }}dx_n &{}\quad \text{ if }\, N>1. \end{array}\right. \!\!\! \end{aligned}$$

These are expected values of the polynomials \(P_N^{}(z)=\prod _{1\le n\le N}(X_n^2+z^2)\) whose 2N zeros \(\{\pm i X_k\}^{}_{k=1,\ldots ,N}\) are generated by N identically distributed multi-variate mean-zero normal random variables \(\{X_k\}^{N}_{k=1}\) with co-variance \(\mathrm{{Cov}}_N^{}(X_k,X_l)=(1+\frac{\sigma ^2-1}{N})\delta _{k,l}+\frac{\sigma ^2-1}{N}(1-\delta _{k,l})\). The \(E_N^{}(z;\sigma )\) are polynomials in \(z^2\), explicitly computable for arbitrary N, yet a list of the first three \(E_N^{}(z;\sigma )\) shows that the expressions become unwieldy already for moderate N—unless \(\sigma = 1\), in which case \(E_N^{}(z;1) = (1+z^2)^N\) for all \(z\in \mathbb {C}\) and \(N\in \mathbb {N}\). (Incidentally, commonly available computer algebra evaluates the integrals \(E_N^{}(z;\sigma )\) only for N up to a dozen, due to memory constraints). Asymptotic evaluations are needed for the large-N regime. For general complex z these have traditionally been limited to analytic expansion techniques; several rigorous results are proved for complex z near 0. Yet if \(z\in \mathbb {R}\) one can also compute this “infinite-degree” limit with the help of the familiar relative entropy principle for probability measures; a rigorous proof of this fact is supplied. Computer algebra-generated evidence is presented in support of a conjecture that a generalization of the relative entropy principle to signed or complex measures governs the \(N\rightarrow \infty \) asymptotics of the regime \(iz\in \mathbb {R}\). Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.

     

Cosmological analysis of scalar field models in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) gravity

This paper determines the existence of Noether symmetry in non-minimally coupled f(R, T) gravity admitting minimal coupling with scalar field models. We consider a generalized spacetime which corresponds to different anisotropic and homogeneous universe models. We formulate symmetry generators along with conserved quantities through Noether symmetry technique for direct and indirect curvature–matter coupling. For dust and perfect fluids, we evaluate exact solutions and construct their cosmological analysis through some cosmological parameters. We conclude that decelerated expansion is obtained for the quintessence model with a dust distribution, while a perfect fluid with dominating potential energy over kinetic energy leads to the current cosmic expansion for both phantom as well as quintessence models.     

Dark Energy Models in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) Theory with Variable Deceleration Parameter

In this communication we have investigated Bianchi type-II dark energy (DE) cosmological models with and without presence of magnetic field in modified f(R, T) gravity theory as proposed by Harko et al. (Phys. Rev. D, 84, 024020, 2011). The exact solution of the field equations is obtained by setting the deceleration parameter q as a time function along with suitable assumption the scale factor \(a(t)= [sinh(\alpha t)]^{\frac {1}{n}}\), α and n are positive constant. We have obtained a class of accelerating and decelerating DE cosmological models for different values of n and α. The present study believes that the mysterious dark energy is the main responsible force for accelerating expansion of the universe. For our constructed models the DE candidates cosmological constant (Λ) and the EoS parameter (ω) both are found to be time varying quantities. The cosmological constant Λ is very large at early time and approaches to a small positive value at late time whereas the EoS parameters is found small negative at present time. Physical and kinematical properties of the models are discussed with the help of pictorial representations of the parameters. We have observed that our constructed models are compatible with recent cosmological observations.     

Large-Scale Corrections to the CMB Anisotropy from Asymptotic de Sitter Mode

In this study, large-scale effects from asymptotic de Sitter mode on the CMB anisotropy are investigated. Besides the slow variation of the Hubble parameter onset of the last stage of inflation, the recent observational constraints from Planck and WMAP on spectral index confirm that the geometry of the universe can not be pure de Sitter in this era. Motivated by these evidences, we use this mode to calculate the power spectrum of the CMB anisotropy on the large scale. It is found that the CMB spectrum is dependent on the index of Hankel function ν which in the de Sitter limit \(\nu \rightarrow \frac {3}{2}\), the power spectrum reduces to the scale invariant result. Also, the result shows that the spectrum of anisotropy is dependent on angular scale and slow-roll parameter and these additional corrections are swept away by a cutoff scale parameter H ? M_? < M _P .     

Quantifying Contextuality of Empirical Models in Terms of Trace-Distance

In order to quantify contextuality of empirical models, the quantity of contextuality (QoC) of empirical models is introduced in terms of the trace-distance. Let Q C(e) denote the QoC of an empirical model e. The following conclusions are proved. (i) An empirical model e is non-contextual if and only if Q C(e)=0, and then it is contextual if and only if Q C(e)>0; (ii) the QoC function QC is convex, contractive and continuous. Finally, the QoC of some famous models is computed, including PM-isotropic boxes P M ^α , M-isotropic boxes M ^α , C H _n -isotropic boxes \(CH_{n}^{\alpha }\) as well as K box, where α∈[0,1]. Moreover, P M ^α is non-contextual if and only if \(\alpha \in [\frac {1}{6},\frac {5}{6}]\); M ^α is non-contextual if and only if \(\alpha \in [0,\frac {4}{5}]\); when n is even, \(CH_{n}^{\alpha }\) is non-contextual if and only if \(\alpha \in [\frac {1}{n},\frac {n-1}{n}]\), and when n is odd, \(CH_{n}^{\alpha }\) is non-contextual if and only if \(\alpha \in [0,\frac {n-1}{n}]\). The most important thing is that it is very easy to compare the QoC of any two isotropic boxes discussed in the above.     

Universal Components of Random Nodal Sets

We give, as L grows to infinity, an explicit lower bound of order \({L^{\frac{n}{m}}}\) for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of P with eigenvalues below L. Here, P denotes an elliptic self-adjoint pseudo-differential operator of order \({m > 0}\), bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed n-dimensional manifold M equipped with some Lebesgue measure. In fact, for every closed hypersurface \({\Sigma}\) of \({\mathbb{R}^n}\), we prove that there exists a positive constant \({p_\Sigma}\) depending only on \({\Sigma}\), such that for every large enough L and every \({x \in M}\), a component diffeomorphic to \({\Sigma}\) appears with probability at least \({p_\Sigma}\) in the vanishing locus of a random section and in the ball of radius \({L^{-\frac{1}{m}}}\) centered at x. These results apply in particular to Laplace–Beltrami and Dirichlet-to-Neumann operators.     

Inflation Driven by q-de Sitter

We propose a generalised de Sitter scale factor for the cosmology of early and late time universe, including single scalar field is called as inflaton. This form of scale factor has a free parameter q is called as nonextensivity parameter. When q = 1, the scale factor is de Sitter. This scale factor is an intermediate form between power-law and de Sitter. We study cosmology of such families. We show that both kinds of dark components, dark energy and dark matter simultaneously are described by this family of solutions. As a motivated idea, we investigate inflation in the framework of q-de Sitter. We consider three types of scenarios for inflation. In a single inflation scenario, we observe that, inflation ended without any specific ending inflation ?_end, the spectral index and the associated running of the spectral index are n_s ? 1 ～ ?2??, α_s ≡ 0. To end the inflation: we should have \(q=\frac {3}{4}\). We deduce that the inflation ends when the evolution of the scale factor is a(t) = e_3/4(t). With this scale factor there is no need to specify ?_end. As an alternative to have inflation with ending point, We will study q-inflation model in the context of warm inflation. We propose two forms of damping term Γ. In the first case when Γ = Γ₀, we show the scale invariant spectrum, (Harrison-Zeldovich spectrum, i.e. n_s = 1) may be approximately presented by (\(q=\frac {9}{10},~~N=70\)). Also there is a range of values of R and n_s which is compatible with the BICEP2 data where \(q=\frac {9}{10}\). In case Γ = Γ₁V(?), it is observed that small values of a number of e-folds are assured for small values of q parameter. Also in this case, the scale-invariant spectrum may be represented by \((q,N) = (\frac {9}{10},70)\). For \(q=\frac {9}{10}\) a range of values of R and n_s is compatible with the BICEP2 data. Consequently, the proposal of q-de Sitter is consistent with observational data. We observe that the non-extensivity parameter q plays a significant role in inflationary scenario.     

Two Phases of the Non-Commutative Quantum Mechanics with the Generalized Uncertainty Relations

We consider the quantum mechanics on the noncommutative plane with the generalized uncertainty relations \({\Delta } x_{1} {\Delta } x_{2} \ge \frac {\theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac {\bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar }{2}, {\Delta } x_{1} {\Delta } p_{2} \ge \frac {\eta }{2}\). We show that the model has two essentially different phases which is determined by \(\kappa = 1 + \frac {1}{\hbar ^{2} } (\eta ^{2} - \theta \bar {\theta })\). We construct a operator \(\hat {\pi }_{i}\) commuting with \(\hat {x}_{j} \) and discuss the harmonic oscillator model in two dimensional non-commutative space for three case κ > 0, κ = 0, κ < 0. Finally, we discuss the thermodynamics of a particle whose hamiltonian is related to the harmonic oscillator model in two dimensional non-commutative space.     

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

The higher spin Dirac operator \(\mathcal{Q}_{k,l}\) acting on functions taking values in an irreducible representation space for \(\mathfrak{so}(m)\) with highest weight \((k+\frac{1}{2},l+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2})\), with k, l?∈?\(\mathbb{N}\) and \(k\geqslant l\), is constructed. The structure of the kernel space containing homogeneous polynomial solutions is then also studied.     

Emergence and expansion of cosmic space as due to <Emphasis Type="Italic">M</Emphasis>0-branes

Recently, Padmanabhan (arXiv:1206.4916 [hep-th]) discussed that the difference between the number of degrees of freedom on the boundary surface and the number of degrees of freedom in a bulk region causes the accelerated expansion of the universe. The main question arising is: what is the origin of this inequality between the surface degrees of freedom and the bulk degrees of freedom? We answer this question in M-theory. In our model, first M0-branes are compactified on one circle and N D0-branes are created. Then N D0-branes join each other, grow, and form one D5-branes. Next, the D5-brane is compactified on two circles and our universe’s D3-brane, two D1-branes and some extra energies are produced. After that, one of the D1-branes, which is closer to the universe’s brane, gives its energy into it, and this leads to an increase in the difference between the numbers of degrees of freedom and the occurring inflation era. With the disappearance of this D1-brane, the number of degrees of freedom of boundary surface and bulk region become equal and inflation ends. At this stage, extra energies that are produced due to the compactification cause an expansion of the universe and deceleration epoch. Finally, another D1-brane dissolves in our universe’s brane, leads to an inequality between degrees of freedom, and there occurs a new phase of acceleration.     

A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model

We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite-range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime \({\beta < \beta_c}\), and the mean-field lower bound \({\mathbb{P}_\beta[0\longleftrightarrow \infty ]\ge (\beta-\beta_c)/\beta}\) for \({\beta > \beta_c}\). For finite-range models, we also prove that for any \({\beta < \beta_c}\), the probability of an open path from the origin to distance n decays exponentially fast in n. For the Ising model, we prove finiteness of the susceptibility for \({\beta < \beta_c}\), and the mean-field lower bound \({\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}}\) for \({\beta > \beta_c}\). For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for \({\beta < \beta_c}\).     

Correspondence of F-Essence with Holographic and New Agegraphic Dark Energy Models

In this work, we consider a non-flat universe filled with Fermionic field. First, we have considered the holographic dark energy and new agegraphic dark energy in the framework of F-essence cosmology and investigated the consequences for their co-existence. The correspondence of F-essence with the above types of dark energy models have been investigated. The natures of K and Y for these correspondence of F-essence with the above dark energies have been analyzed.     

Instability of Interacting Ghost Dark Energy Model in an Anisotropic Universe

A new dark energy model called “ghost dark energy” was recently suggested to explain the observed accelerating expansion of the universe. This model originates from the Veneziano ghost of QCD. The dark energy density is proportional to Hubble parameter, ρ _Λ = α H, where α is a constant of order \({\Lambda }^{3}_{QCD}\) and Λ _{Q C D} ～ 100M e V is QCD mass scale. In this paper, we investigate about the stability of generalized QCD ghost dark energy model against perturbations in the anisotropic background. At first, the ghost dark energy model of the universe with spatial BI model with/without the interaction between dark matter and dark energy is discussed. In particular, the equation of state and the deceleration parameters and a differential equation governing the evolution of this dark energy model are obtained. Then, we use the squared sound speed \({v_{s}^{2}}\) the sign of which determines the stability of the model. We explore the stability of this model in the presence/absence of interaction between dark energy and dark matter in both flat and non-isotropic geometry. In conclusion, we find evidence that the ghost dark energy might can not lead to a stable universe favored by observations at the present time in BI universe.     

Bound States for a Klein–Gordon Particle in Vector Plus Scalar Generalized Hulthén Potentials in D Dimensions

The Green’s function associated with a Klein–Gordon particle moving in a D-dimensional space under the action of vector plus scalar q-deformed Hulthén potentials is constructed by path integration for \({q \geq 1}\) and \({\frac{1}{\alpha} \ln q < r < \infty}\). An appropriate approximation of the centrifugal potential term and the technique of space-time transformation are used to reduce the path integral for the generalized Hulthén potentials into a path integral for q-deformed Rosen–Morse potential. Explicit path integration leads to the radial Green’s function for any l state in closed form. The energy spectrum and the correctly normalized wave functions, for a state of orbital quantum number \({l \geq 0}\), are obtained. Eventually, the vector q-deformed Hulthén potential and the Coulomb potentials in D dimensions are considered as special cases.     

Viability of variable generalised Chaplygin gas: a thermodynamical approach

The viability of the variable generalised Chaplygin gas (VGCG) model is analysed from the standpoint of its thermodynamical stability criteria with the help of an equation of state, \(P = - \frac{B}{\rho ^{\alpha } }\), where \(B = B_{0}V^{-\frac{n}{3}}\). Here \(B_{0}\) is assumed to be a positive universal constant, n is a constant parameter and V is the volume of the cosmic fluid. We get the interesting result that if the well-known stability conditions of a fluid is adhered to, the values of n are constrained to be negative definite to make \( \left( \frac{\partial P}{\partial V}\right) _{S} <0\) & \( \left( \frac{\partial P}{\partial V}\right) _{T} <0\) throughout the evolution. Moreover the positivity of thermal capacity at constant volume \(c_{V}\) as also the validity of the third law of thermodynamics are ensured in this case. For the particular case \(n = 0\) the effective equation of state reduces to \(\Lambda \)CDM model in the late stage of the universe while for \(n <0\) it mimics a phantom-like cosmology which is in broad agreement with the present SNe Ia constraints like VGCG model. The thermal equation of state is discussed and the EoS parameter is found to be an explicit function of temperature only. Further for large volume the thermal equation of state parameter is identical with the caloric equation of state parameter when \( T \rightarrow 0\). It may also be mentioned that like Santos et al. our model does not admit of any critical points. We also observe that although the earlier model of Lu explains many of the current observational findings of different probes it fails to explain the crucial tests of thermodynamical stability.     

Low-regularity Schrödinger maps,II: global well-posedness in dimensions <Emphasis Type="Italic">d</Emphasis> ≥ 3

In dimensions d ≥ 3, we prove that the Schrödinger map initial-value problem

$ \left\{ \begin{array}{l} \partial_ts=s\times\Delta_x s\hbox{ on }\mathbb{R}^d\times\mathbb{R};\\ s(0)=s_0 \end{array} \right. $

is globally well-posed for small data s ₀ in the critical Besov spaces \({\dot{B}_Q^{d/2}(\mathbb{R}^d;\mathbb{S}^2)}\), \({Q\in\mathbb{S}^2}\).