Spherical Symmetric Solution in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) Model Around Charged Black Hole

A static, asymptotically flat, spherically symmetric solutions is investigated in f(R) theories of gravity for a charged black hole. We have studied the weak field limit of f(R) gravity for the some f(R) model such as f(R)=R+ε h(R). In particular, we consider the case lim  _R→0 h(R)/h′(R)→0 and find the space time metric for f(R)=R+[(m⁴)/(R)]f(R)=R+{\mu^{4}\over R} and f(R)=R ^1+ε theories of gravity far away a charged mass point.     

Unifying dark energy and dark matter with the modified Ricci model

In this paper, two modified Ricci models are considered as the candidates of unified dark matter–dark energy. In model one, the energy density is given by r_MR=3M_pl(aH²+b[(H)\dot])\rho_{\mathrm{MR}}=3M_{\mathrm{pl}}(\alpha H^{2}+\beta\dot{H}), whereas, in model two, by r_MR=3M_pl(\fraca6 R+g[(H)\ddot]H^-1)\rho_{\mathrm{MR}}=3M_{\mathrm{pl}}(\frac{\alpha}{6} R+\gamma\ddot{H}H^{-1}). We find that they can explain both dark matter and dark energy successfully. A constant equation of state of dark energy is obtained in model one, which means that it gives the same background evolution as the wCDM model, while model two can give an evolutionary equation of state of dark energy with the phantom divide line crossing in the near past.     

Minimum Polynomial for Single-Mode Parabose Parasupercharge and Hamiltonian

We calculate the minimum polynomial φ(x,y) of parasupercharge Q and Hamiltonian H for single-mode parabose parasupersymmetry (P-PSUSY). Suppose that φ(x,y) satisfies the homogeneity ∀ λ∈ℝ,φ(λ x,λ ² y)=λ ⁿ φ(x,y), then the parafermionic order p _f is restricted to either 1, 2, or 4. Under the P-PSUSY, the homogeneity of the φ(x,y) is equivalent to the parasuperconformality of Q and H. The physical meaning of the parasuperconformality is discussed, in connection with the spin of the elementary particle.     

Reconstruction of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">T</Emphasis>) gravity according to holographic dark energy

We develop the reconstruction of the f(T) gravity model according to the holographic dark energy. T is the torsion scalar and its initial value from the teleparallel gravity is imposed for fitting the initial value of the function f(T). The evolutionary nature of the holographic dark energy is essentially based on two important parameters, Ω _V  and ω _V , respectively, the dimensionless dark energy and the parameter of the equation of state, related to the holographic dark energy. The result shows a polynomial function for f(T), and we also observe that, when Ω _V →1 at the future time, ω _V may cross −1 for some values of the input parameter b. Another interesting aspect of the obtained model is that it provides a unification scenario of dark matter with dark energy.     

Bulk Viscous Bianchi Type I Barotropic Fluid Cosmological Model with Varying <Emphasis Type="Bold">Λ</Emphasis> and Functional Relation on Hubble Parameter

Bianchi Type I bulk viscous barotropic fluid cosmological with varying Λ is investigated. We have also assumed a functional relation on Hubble parameter as H(R)=a(R ⁻ⁿ +1), n>1, a>0, H the Hubble constant, R being scale factor and H = [(R)\dot]/RH = \dot{R}/R. The physical and geometrical aspects of the model related with astronomical observations are discussed.     

Synthesis and characterization of ZnxHg1?xSeyS1?y quantum dots

This article describes the synthesis of highly water-soluble Zn _x Hg_1−x Se _y S_1−y quantum dots (QDs) in aqueous solution through a simple photo-assisted reaction between ZnSe QDs and mercury(I) nitrate dihydrate [Hg₂(NO₃)₂·2H₂O]. In order to deduce the optimal synthesis conditions, we varied several parameters, including the concentrations of mercaptosuccinic acid (MSA) and Hg₂(NO₃)₂·2H₂O, the illumination time, and the reaction temperature. When irradiated at temperatures below 80 °C, the ZnSe QDs reacted with the S²⁻ ions formed rapidly from MSA and the Hg²⁺ ions formed from Hg₂ ²⁺ ions to form Zn _x Hg_1−x Se _y S_1−y QDs through a process of photo-etching and surface combination. Under different conditions, we prepared a series of Zn _x Hg_1−x Se _y S_1−y QDs that emit fluorescence at the maximum wavelengths ranging from 405 to 760 nm. Inductively coupled plasma-mass spectrometry and transmission electron microscopy/energy dispersive spectrometry revealed that the content of Hg in the Zn _x Hg_1−x Se _y S_1−y QDs was greater when the synthesis was conducted at higher temperature. The Zn_0.88Hg_0.12Se_0.44S_0.56 QDs exhibit improved photostability than crude ZnSe QDs and possess long lifetimes (τ₁ ~ 38 ns and τ₂ ~ 158 ns).     

Energy levels of the Schrödinger equation for some rational potentials in two-dimensional space using the inner product technique

The energy levels of a two-dimensional system are calculated for the rational potential,V(x, y; λ, g)=x ²+y ²+λ[x ²/(1+gx ²)+y ²/(1+gy ²)+a _xxx⁴+a _xyx² y ²+a _yyy⁴] using the inner product technique over a wide range of values of the perturbation parameters (g, λ) and for various eigenstates.     

Solutions of Doubly and Higher Order Iterated Equations

Michael Fisher once studied the solution of the equation f(f(x))=x ²–2. We offer solutions to the general equation f(f(x))=h(x) in the form f(x)=g(ag ^–1(x)) where a is in general a complex number. This leads to solving duplication formulas for g(x). For the case h(x)=x ²–2, the solution is readily found, while the h(x)=x ²+2 case is challenging. The solution to these types of equations can be related to differential equations.     

On the Energy Growth of Some Periodically Driven Quantum Systems with Shrinking Gaps in the Spectrum

We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E _n ∼n ^α , with 0<α<1. In particular, the gaps between successive eigenvalues decay as n ^α−1. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate ‖V(t) _m,n ‖≤ε|m−n|^−p max {m,n}^−2γ for m≠n, where ε>0, p≥1 and γ=(1−α)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and ε is small enough. More precisely, for any initial condition Ψ∈Dom(H ^1/2), the diffusion of energy is bounded from above as 〈H〉 _Ψ (t)=O(t ^σ ), where . As an application we consider the Hamiltonian H(t)=|p| ^α +ε v(θ,t) on L ²(S ¹,dθ) which was discussed earlier in the literature by Howland.     

A very strong difference property for semisimple compact connected lie groups

Let G be a topological group. For a function f: G → ℝ and h ∈ G, the difference function Δ _h f is defined by the rule Δ _h f(x) = f(xh) − f(x) (x ∈ G). A function H: G → ℝ is said to be additive if it satisfies the Cauchy functional equation H(x + y) = H(x) + H(y) for every x, y ∈ G. A class F of real-valued functions defined on G is said to have the difference property if, for every function f: G → ℝ satisfying Δ _h f ∈ F for each h ∈ G, there is an additive function H such that f − H ∈ F. Erdős’ conjecture claiming that the class of continuous functions on ℝ has the difference property was proved by N. G. de Bruijn; later on, F. W. Carroll and F. S. Koehl obtained a similar result for compact Abelian groups and, under the additional assumption that the other one-sided difference function ∇ _h f defined by ∇ _h f(x) = f(xh) − f(x) (x ∈ G, h ∈ G) is measurable for any h ∈ G, also for noncommutative compact metric groups. In the present paper, we consider a narrower class of groups, namely, the family of semisimple compact connected Lie groups. It turns out that these groups admit a significantly stronger difference property. Namely, if a function f: G → ℝ on a semisimple compact connected Lie group has continuous difference functions Δ _h f for any h ∈ G (without the additional assumption concerning the measurability of the functions of the form ∇ _h f), then f is automatically continuous, and no nontrivial additive function of the form H is needed. Some applications are indicated, including difference theorems for homogeneous spaces of compact connected Lie groups.     

Super-Diffusivity in a Shear Flow Model¶from Perpetual Homogenization

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dy _t =dω _t −∇Γ(y _t ) dt, y ₀=0 and d=2. Γ is a 2 &\times; 2 skew-symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales Γ₁₂=−Γ₂₁=h(x ₁), with h(x ₁)=∑ _{n =0} ^∞γ _n h ⁿ (x ₁/R _n ), where h ⁿ are smooth functions of period 1, h ⁿ (0)=0, γ _n and R _n grow exponentially fast with n. We can show that y ^t has an anomalous fast behavior (?[|y ^t |²]∼t ^1+ν with ν > 0) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization. Received: 1 June 2001 / Accepted: 11 January 2002     

Temperature dependences of the high-field magnetization of dilute frustrated ferrimagnetic spinels

Results are presented of an investigation of the magnetic properties of dilute frustrated ferrimagnetic spinels Li_0.5Fe_2.5−x Ga_xO₄ (x=0.8–1.2), which characterize the main parameters of the ferrimagnetic state and provide evidence of local violation of collinear spin ordering and frustrations. In particular, measurements were made of the concentration dependences of the magnetic moment n ₀(x) and the Curie point T _c (x), the magnetization isotherms σ _T (H) at T=4.2 K and H⩽10 kOe, and also the low-and high-field magnetization polytherms σ _H (T). It was established that for x⩾0.8 in fields exceeding the technical saturation field H _s ∼2 kOe, the temperature dependences of the high-field magnetization σ _H (T) between 4.2 and 230 K cannot be described by the Bloch T ^3/2 law whereas this law is satisfied for undiluted Li spinel (x=0). Over the entire temperature range (4.2–230 K) the experimental curves σ _H (T) may be approximated by σ _H (T)=σ ₀(1−AT ^3/2 − BT ^5/2) for x=0.8–1.0 and σ _H (T)=σ ₀[1−CT ^3/2exp(μ(H−H ₀)/k _B T)] for x=1.1, 1.2, where μH ₀∼15 K is the internal field produced by competition between exchange interactions and frustrations. Fiz. Tverd. Tela (St. Petersburg) 40, 1075–1079 (June 1998)     

Spectral coincidences between CO-Laser and nitric-oxide. A reinvestigation

The distribution of the localized levelsg(ε) in the forbidden gap of amorphous silicon is calculated from the dependence of the Fermi energy on the doping concentration of phosphorus donors and boron acceptors. The minimum ofg(ε) is verified by this method to be in the order of 10¹⁷ to 10¹⁸ cm⁻³eV⁻¹, whereas the maxima ofg(ε), which have been reported in the literature, are not confirmed.     

A Note on Classical Ground State Energies

The pair-specific ground state energy ε _g (N):=ℰ _g (N)/(N(N−1)) of Newtonian N body systems grows monotonically in N. This furnishes a whole family of simple new tests for minimality of putative ground state energies ℰ _g ^x (N) obtained through computer experiments. Inspection of several publicly available lists of such computer-experimentally obtained putative ground state energies ℰ _g ^x (N) has yielded several dozen instances of ℰ _g ^x (N) which failed one of these tests; i.e., for those N one concludes that ℰ _g ^x (N)>ℰ _g (N) strictly. Although the correct ℰ _g (N) is not revealed by this method, it does yield a better upper bound on ℰ _g (N) than ℰ _g ^x (N) whenever ℰ _g ^x (N) fails a monotonicity test. The surveyed N-body systems include in particular N point charges with 2- or 3-dimensional Coulomb pair interactions, placed either on the unit 2-sphere or on a 2-torus (a.k.a. Thomson, Fekete, or Riesz problems).     

Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods

We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ² subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H ₀+εP(ωt) for ε small. Here H ₀ is the one-dimensional Schr?dinger operator p ²+V, V(x)∼|x|^α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|^β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients. Received: 3 October 2000 / Accepted: 20 December 2000     

A Construction of Blow up Solutions for Co-rotational Wave Maps

The existence of co-rotational finite time blow up solutions to the wave map problem from ${\mathbb{R}^{2+1} \to N}The existence of co-rotational finite time blow up solutions to the wave map problem from \mathbbR²⁺¹ ? N{\mathbb{R}^{2+1} \to N} , where N is a surface of revolution with metric d ρ ² + g(ρ)² dθ², g an entire function, is proven. These are of the form u(t,r)=Q(l(t)t)+R(t,r){u(t,r)=Q(\lambda(t)t)+\mathcal{R}(t,r)} , where Q is a time independent solution of the co-rotational wave map equation −u _tt  + u _rr  + r ⁻¹ u _r  = r ⁻² g(u)g′(u), λ(t) = t ^−1-ν, ν > 1/2 is arbitrary, and R{\mathcal{R}} is a term whose local energy goes to zero as t → 0.     

Calculation of the electronic structure and spectra for bacteriochlorophyll analogs

Geometric structures and excited electronic states for free bases of bacteriochlorin (H₂BC) and tetraazabacteriochlorin (H₂TABC) as well as for their magnesium complexes (MgBC and MgTABC), analogs of bacteriopheophytin a (H₂BPhea) and bacteriochlorophyll a (MgBPhea), have been calculated by a DFT method and by an INDO/Sm method (the INDO/S method with parameterization modified by the authors), respectively. The factors responsible for the observed bathochromic shift of the long-wavelength Q _x (0–0) band of MgBPhea relative to H₂BPhea, \updelta E_Qx @ - 300  \textc\textm ^{- 1} {{\updelta }}{E_{{Q_x}}} \cong - 300\;{\text{c}}{{\text{m}}^{ - 1}} , have been clarified. Contributions of one- and two-electron interactions to the resulting shift of the Q _x (0–0) band have been analyzed in detail for the H₂BC/MgBC, H₂TABC/MgTABC, and porphine (H₂P)/Mg porphine (MgP) pairs. It is shown that the bathochromic shift under consideration for the tetrahydro derivatives is caused by a decrease of the orbital energy gap ε₁–ε₋₁ between the lowest unoccupied and highest occupied molecular orbitals. The variation of δ(ε₁–ε₋₁) is large and amounts to –1660 and –920 cm^–1 for the H₂BC/MgBC and H₂TABC/MgTABC pairs, respectively. The two-electron contributions, both into the energy of electronic configurations and due to the superposition of the configurations, produce a compensating hypsochromic effect such that the shifts \updelta E_Qx {{\updelta }}{E_{{Q_x}}} are –260 and –150 cm^–1 for the H₂BC/MgBC and H₂TABC/MgTABC pairs, respectively. It is also shown that the calculated electronic spectra for the considered molecules agree quantitatively with the experimental absorption spectra.     

NMR-ON measurements of187WFe,182,183,186ReNi,186ReFe and203PbFe

The magnetic hyperfine splitting frequencies of¹⁸⁷WFe,¹⁸²Re(j ^π=2⁺)Ni,¹⁸³ReNi,¹⁸⁶ReNi,¹⁸⁶ReFe and²⁰³PbFe in a zero external magnetic field have been determined by the NMR-ON method at about 7 mK as 225.56(6), 130.9(1), 98.17(4), 136.6(4), 1007.0(3) and 58.43(3) MHz, respectively. With the knowng-factors ofg(¹⁸⁶Re, 1⁻)=1.739(3) andg(²⁰³Pb, 5/2⁻)=0.27456(20), the following hyperfine fields were deduced:B _HF(¹⁸⁶ReNi)=−103.05(35) kG;B _HF(¹⁸⁶ReFe)=−759.7(13) kG;B _HF(²⁰³PbFe)=+279.18(25) kG. Taking hyperfine anomalies into account, theg-factor of¹⁸³Re was deduced as |g(¹⁸³Re, 5/2⁺)|=1.267(6). With the assumption of Knight shift factorK=0, theg-factors of¹⁸²Re and¹⁸⁷W and the hyperfine field of¹⁸⁷WFe were determined as |g(¹⁸²Re, 2⁺)|=1.63(5), |g(¹⁸⁷W, 3/2⁻)|=0.414(10) andB _HF(¹⁸⁷WFe) =−714(18) kG. The large hyperfine anomaly was deduced to be¹⁸³W Δ¹⁸⁷W =−0.124(22).     

Constraints on accelerating universe using ESSENCE and Gold supernovae data combined with other cosmological probes

We use recent data: the 192 ESSENCE type Ia supernovae (SNe Ia), the 182 Gold SNe Ia, the three-year WMAP, the SDSS baryon acoustic peak, the X-ray gas mass fraction in clusters and the observational H(z) data, to constrain models of the accelerating universe. Combining the 192 ESSENCE data with the observational H(z) data to constrain the parameterized deceleration parameter, we obtain the best-fit values of the transition redshift and current deceleration parameter z _T=0.632_−0.127^+0.256 and q ₀=−0.788_−0.182^+0.182. Furthermore, using the ΛCDM model and two model-independent equations of state of the dark energy, we find that the combined constraint from the 192 ESSENCE data and four other cosmological observations gives smaller values for Ω _0m and q ₀, but a larger value for z _T than the combined constraint from the 182 Gold data with four other observations. Finally, according to the Akaike information criterion it is shown that the recently observed data equally support three dark energy models: ΛCDM, w _de(z)=w ₀ and w _de(z)=w ₀+w ₁ln (1+z).     

Higher-order theories of gravity matched with large-scale structure and cosmological observations

The so-called f(R)-gravity could, in principle, explain the accelerated expansion of the Universe without adding unknown forms of dark energy/dark matter, but more simply extending the general relativity by generic functions of the Ricci scalar. However, as a part of several phenomenological models, there is no final f(R)-theory capable of fitting all the observations and addressing all the issues related to the presence of dark energy and dark matter. Astrophysical observations are pointing out huge amounts of “dark matter” and “dark energy” needed to explain the observed large-scale structures and cosmic accelerating expansion. Up to now, no experimental evidence has been found, at a fundamental level, to explain such mysterious components. The problem could be completely reversed considering dark matter and dark energy as “shortcomings” of general relativity.