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.     

Entropy function approach to charged BTZ black hole

We find solution to the metric function f(r) = 0 of charged BTZ black hole making use of the Lambert function. The condition of extremal charged BTZ black hole is determined by a non-linear relation of M _e (Q) = Q ²(1 − ln Q ²). Then, we study the entropy of extremal charged BTZ black hole using the entropy function approach. It is shown that this formalism works with a proper normalization of charge Q for charged BTZ black hole because AdS₂ × S¹ represents near-horizon geometry of the extremal charged BTZ black hole. Finally, we introduce the Wald’s Noether formalism to reproduce the entropy of the extremal charged BTZ black hole without normalization when using the dilaton gravity approach.     

Free Energies of Dilute Bose Gases: Upper Bound

We derive an upper bound on the free energy of a Bose gas at density ϱ and temperature T. In combination with the lower bound derived previously by Seiringer (Commun. Math. Phys. 279(3): 595–636, 2008), our result proves that in the low density limit, i.e., when a ³ ϱ≪1, where a denotes the scattering length of the pair-interaction potential, the leading term of Δf, the free energy difference per volume between interacting and ideal Bose gases, is equal to 4pa(2r²-[r-r_c]²₊)4\pi a(2\varrho^{2}-[\varrho-\varrho_{c}]^{2}_{+}). Here, ϱ _c (T) denotes the critical density for Bose–Einstein condensation (for the ideal Bose gas), and [⋅]₊=max {⋅,0} denotes the positive part.     

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.     

Ionization of Coulomb Systems in {\mathbb{R}^3} by Time Periodic Forcings of Arbitrary Size

We analyze the long time behavior of solutions of the Schrödinger equation ${i\psi_t=(-\Delta-b/r+V(t,x))\psi}We analyze the long time behavior of solutions of the Schr?dinger equation iy_t=(-D-b/r+V(t,x))y{i\psi_t=(-\Delta-b/r+V(t,x))\psi}, x ? \mathbbR³{x\in\mathbb{R}^3}, r =  |x|, describing a Coulomb system subjected to a spatially compactly supported time periodic potential V(t, x) =  V(t +  2π/ω, x) with zero time average.     

Potentials of pointlike particles in gauge theory with a dilaton

I examine the potential of a pointlike particle carrying SU (N _c) charge in a gauge theory with a dilaton. The potential depends on boundary conditions imposed on the dilaton: For a dilaton that vanishes at infinity the resulting potential is a regulatized Coulomb potential of the form (r+r _ϕ)⁻¹, withr _ϕ, inversely proportional to the decay constant of the dilaton. Another natural constraint on the dialaton ϕ is independence of (1/g ²) exp(ϕ/f_ϕ) from the gauge couplingg. This requirement yields a confining potential proportional tor.     

General Dynamical Equations for Dingle’s Space-Times Filled with a Charged Non-perfect Fluid

In this paper we have assumed charged non-perfect fluid as the material content of the space-time. The expression for the “mass function-M(r,y,z,t)” is obtained for the general situation and the contributions from the Ricci tensor in the form of material energy density ρ, pressure anisotropy [\fracp₂+p₃2-p₁][\frac{p_{2}+p_{3}}{2}-p_{1}] , electromagnetic field energy ℰ and the conformal Weyl tensor, viz. energy density of the free gravitational field ε (=\frac-3Y₂4p)(=\frac{-3\Psi_{2}}{4\pi}) are made explicit. This work is an extension of the work obtained earlier by Rao and Hasmani (Math. Today XIIA:71, 1993; New Directions in Relativity and Cosmology, Hadronic Press, Nonantum, 1997) for deriving general dynamical equations for Dingle’s space-times described by this most general orthogonal metric,

Static charged spheres with anisotropic pressure in general relativity

We report a generalization of our earlier formalism [Pramana, 54, 663 (1998)] to obtain exact solutions of Einstein-Maxwell’s equations for static spheres filled with a charged fluid having anisotropic pressure and of null conductivity. Defining new variables: w=(4π/3)(ρ+ε)r ², u=4πξr ², v _r=4πp _r r ², v _⊥=4πp _⊥ r ²[ρ, ξ(=−(1/2)F ₁₄ F ¹⁴), p _r, p _⊥ being respectively the energy densities of matter and electrostatic fields, radial and transverse fluid pressures whereas ε denotes the eigenvalue of the conformal Weyl tensor and interpreted as the energy density of the free gravitational field], we have recast Einstein’s field equations into a form easy to integrate. Since the system is underdetermined we make the following assumptions to solve the field equations (i) u=v _r=(a ²/2κ)r ⁿ⁺², v _⊥=k ₁ v _r, w=k ₂ v _r; a ², n(>0), k ₁, k ₂ being constants with κ=((k ₁+2)/3+k ₂) and (ii) w+u=(b ²/2)r ⁿ⁺², u=v _r, v _⊥−v _r=k, with b and k as constants. In both cases the field equations are integrated completely. The first solution is regular in the metric as well as physical variables for all values of n>0. Even though the second solution contains terms like k/r ² since Q(0)=0 it is argued that the pressure anisotropy, caused by the electric flux near the centre, can be made to vanish reducing it to the generalized Cooperstock-de la Cruz solution given in [14]. The interior solutions are shown to match with the exterior Reissner-Nordstrom solution over a fixed boundary. Dedicated to Prof. F A E Pirani.     

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.     

Dielectric relaxation and ionic conductivity studies of Ag<Subscript>2</Subscript>ZnP<Subscript>2</Subscript>O<Subscript>7</Subscript>

The complex impedance of the Ag₂ZnP₂O₇ compound has been investigated in the temperature range 419–557 K and in the frequency range 200 Hz–5 MHz. The Z′ and Z′ versus frequency plots are well fitted to an equivalent circuit model. Dielectric data were analyzed using complex electrical modulus M* for the sample at various temperatures. The modulus plot can be characterized by full width at half-height or in terms of a non-exponential decay function f( \textt ) = exp( - \textt/t )^b \phi \left( {\text{t}} \right) = \exp {\left( { - {\text{t}}/\tau } \right)^\beta } . The frequency dependence of the conductivity is interpreted in terms of Jonscher’s law: s( w) = s_\textdc + \textAwⁿ \sigma \left( \omega \right) = {\sigma_{\text{dc}}} + {\text{A}}{\omega^n} . The conductivity σ _dc follows the Arrhenius relation. The near value of activation energies obtained from the analysis of M″, conductivity data, and equivalent circuit confirms that the transport is through ion hopping mechanism dominated by the motion of the Ag⁺ ions in the structure of the investigated material.     

The 2009 world average of <Emphasis Type="Italic">α</Emphasis><Subscript><Emphasis Type="Bold">s</Emphasis></Subscript>

Measurements of α _s, the coupling strength of the Strong Interaction between quarks and gluons, are summarised and an updated value of the world average of a_s(M_Z⁰)\alpha_{\mathrm{s}}(M_{\mathrm{Z}^{0}}) is derived. Special emphasis is laid on the most recent determinations of α _s. These are obtained from τ-decays, from global fits of electroweak precision data and from measurements of the proton structure function F₂, which are based on perturbative QCD calculations up to O(a_s⁴)\mathcal{O}(\alpha_{\mathrm{s}}^{4}); from hadronic event shapes and jet production in e⁺e⁻ annihilation, based on O(a_s³)\mathcal{O}(\alpha_{\mathrm{s}}^{3}) QCD; from jet production in deep inelastic scattering and from ϒ decays, based on O(a_s²)\mathcal{O}(\alpha_{\mathrm{s}}^{2}) QCD; and from heavy quarkonia based on unquenched QCD lattice calculations. A pragmatic method is chosen to obtain the world average and an estimate of its overall uncertainty, resulting in

On the Massive Wave Equation on Slowly Rotating Kerr-AdS Spacetimes

The massive wave equation ${\square_{g}\psi - \alpha \frac{\Lambda}{3}\psi = 0}The massive wave equation \square_gy- a\fracL3y = 0{\square_{g}\psi - \alpha \frac{\Lambda}{3}\psi = 0} is studied on a fixed Kerr-anti de Sitter background (M,g_M,a,L){\left(\mathcal{M},g_{M,a,\Lambda}\right)}. We first prove that in the Schwarzschild case (a = 0), ψ remains uniformly bounded on the black hole exterior provided that a < \frac94{\alpha < \frac{9}{4}}, i.e. the Breitenlohner-Freedman bound holds. Our proof is based on vectorfield multipliers and commutators: The total flux of the usual energy current arising from the timelike Killing vector field T (which fails to be non-negative pointwise) is shown to be non-negative with the help of a Hardy inequality after integration over a spacelike slice. In addition to T, we construct a vectorfield whose energy identity captures the redshift producing good estimates close to the horizon. The argument is finally generalized to slowly rotating Kerr-AdS backgrounds. This is achieved by replacing the Killing vectorfield T = ∂ _t with K=?_t + l?_f{K=\partial_t + \lambda \partial_\phi} for an appropriate λ ~ a, which is also Killing and–in contrast to the asymptotically flat case–everywhere causal on the black hole exterior. The separability properties of the wave equation on Kerr-AdS are not used. As a consequence, the theorem also applies to spacetimes sufficiently close to the Kerr-AdS spacetime, as long as they admit a causal Killing field K which is null on the horizon.     

Evaluation of Effective Resistances in Pseudo-Distance-Regular Resistor Networks

The effective resistance or two-point resistance between two nodes of a resistor network is the potential difference that appears across them when a unit current source is applied between the nodes as terminals. This concept arises in problems which deal with graphs as electrical networks including random walks, distributed detection and estimation, sensor networks, distributed clock synchronization, collaborative filtering, clustering algorithms and etc. In the previous paper (Jafarizadeh et al. in J. Math. Phys. 50:023302, 2009) a recursive formula for evaluation of effective resistances on the so-called distance-regular networks was given based on the Christoffel-Darboux identity. In this paper, we consider more general networks called pseudo-distance-regular networks or QD type networks, where we use the stratification of these networks and show that the effective resistances between a given node, say α, and all of the nodes β belonging to the same stratum with respect to α, are the same. Then, based on the spectral techniques, for those α,β’s which satisfy L^-1_aa=L^-1_bbL^{-1}_{\alpha\alpha}=L^{-1}_{\beta\beta} (L ⁻¹ is the pseudo-inverse of the Laplacian of the network), an analytical formula for effective resistances R_ab^(m)R_{\alpha\beta^{(m)}} (the equivalent resistance between terminals α and β, so that β belongs to the m-th stratum with respect to α) is given in terms of the first and second orthogonal polynomials associated with the network. From the fact that in distance-regular networks, L^-1_aa=L^-1_bbL^{-1}_{\alpha\alpha}=L^{-1}_{\beta\beta} is satisfied for all nodes α,β of the network, the effective resistances R_ab^(m)R_{\alpha\beta^{(m)}} for m=1,2,…,d (d is diameter of the network which is the same as the number of strata) are calculated directly, by using the given formula.     

Convergence Radii for Eigenvalues of Tri-Diagonal Matrices

Consider a family of infinite tri-diagonal matrices of the form L + zB, where the matrix L is diagonal with entries L _kk  = k ², and the matrix B is off-diagonal, with nonzero entries B _k,k+1 = B _k+1,k  = k ^α , 0 ≤ α < 2. The spectrum of L + zB is discrete. For small |z| the nth eigenvalue E _n (z), E _n (0) = n ², is a well-defined analytic function. Let R _n be the convergence radius of its Taylor’s series about z = 0. It is proved that

N-dimensional non-abelian dilatonic,stable black holes and their Born–Infeld extension

We find large classes of non-asymptotically flat Einstein–Yang–Mills–Dilaton and Einstein–Yang–Mills–Born–Infeld–Dilaton black holes in N-dimensional spherically symmetric spacetime expressed in terms of the quasilocal mass. Extension of the dilatonic YM solution to N-dimensions has been possible by employing the generalized Wu-Yang ansatz. Another metric ansatz, which aided in finding exact solutions is the functional dependence of the radius function on the dilaton field. These classes of black holes are stable against linear radial perturbations. In the limit of vanishing dilaton we obtain Bertotti–Robinson type metrics with the topology of AdS ₂×S ^N–2. Since connection can be established between dilaton and a scalar field of Brans–Dicke type we obtain black hole solutions also in the Brans–Dicke–Yang–Mills theory as well.     

Analysis of shape isomer yields of <Superscript><Emphasis Type="Bold">237</Emphasis></Superscript>Pu in the framework of dynamical–statistical model

Data on shape isomer yield for α + ²³⁵U reaction at E_a ^lab =   E_\alpha ^{\rm lab} =\,\,20–29 MeV are analysed in the framework of a combined dynamical–statistical model. From this analysis, information on the double humped fission barrier parameters for some Pu isotopes has been obtained and it is shown that the depth of the second potential well should be less than the results of statistical model calculations.     

A Class of Well Behaved Charged Analogues of Schwarzchild’s Interior Solution

A class of well behaved charged analogues of Schwarzchild’s interior solution has been obtained using a particular electric intensity. The solutions of this class are utilized to depict a superdense star model with surface density 2×10¹⁴ g cm⁻³. The solution obtained is new and the pressure (p), density (c ² ρ), velocity of sound and (p/(c ² ρ)) are monotonically decreasing towards the pressure free interface. Moreover the adiabatic constant is found to be more than (4/3) which is necessary for stability under radial perturbation. Also the electric intensity increases monotonically towards the surface. The well behaved model has the maximum mass M=1.740793M _Θ , Radius 12.130308 km. The redshift at the center and on the surface is given by z ₀=0.384261 and z _a =0.292489. Out of the models of superdense star obtained couple of models represent Vela Pulsar for (i) α ²=1.03, b=0.33, , Radius=10.8566 km, M=1.18331M _Θ , I=0.642601×10⁴⁵, (ii) α ²=1.1, b=0.3, , Radius=11.197533 km, M=1.311438M _Θ , I=0.774508×10⁴⁵. All the solutions mentioned above are reducible to Schwarzchild interior solution in the absence of charge.     

Synchrotron and annihilation channels for axion production in an external electromagnetic field

We consider axion formation processes in the synchrotron (e ⁻→e ⁻ a) and annihilation (e ⁻ e ⁺→a) channels in a constant crossed field F _μν F^μν=Fμν*F ^μν =0, which approximates constant fields of other configurations in the ultrarelativistic asymptotic limit. The probability and intensity of axion emission are obtained, and we analyze the energy and field asymptotics. A comparison with the characteristic neutrino channel yields the constraints on the axion mass and the energy scale for Peccei-Quinn symmetry breaking. Possible astrophysical applications are discussed. Zh. éksp. Teor. Fiz. 112, 25–31 (July 1997)     

Mass operator of an axion in a crossed field

The dominant one-loop electron contribution to the mass operator of an axion in a crossed field in the asymptotic limits of the parameters q ²/m _e ² and is calculated. The corresponding electromagnetic mass of the axion is compared with the quantum-chromodynamic mass due to mixing with π. Expressions are derived for the probability of pair creation a→e ⁺ e ⁻, and the fundamental conclusion is reached that refractive effects are present in the propagation of an axion in an external electromagnetic field. Zh. éksp. Teor. Fiz. 113, 1558–1565 (May 1998)