Noether symmetry of <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">T</Emphasis>) cosmology with quintessence and phantom scalar fields

In this paper, we investigate the Noether symmetries of F(T) cosmology involving matter and dark energy. In this model, the dark energy is represented by a canonical scalar field with a potential. Two special cases for dark energy are considered, including phantom energy and quintessence. We obtain F(T)～T ^3/4, and the scalar potential V(?)～? ² for both models of dark energy and discuss quantum picture of this model. Some astrophysical implications are also discussed.     

Weyl’s Type Estimates on the Eigenvalues of Critical Schrödinger Operators

We consider on a bounded domain \(\Omega \subset {\mathbb{R}}^N\) , the Schrödinger operator ? Δ ? V supplemented with Dirichlet boundary solutions. The potential V is either the critical inverse square potential V(x) = (N ? 2)²/4|x|^?2 or the critical borderline potential V(x) =  (1/4)dist(x, ?Ω)^?2. We present explicit asymptotic estimates on the eigenvalues of the critical Schrödinger operator in each case, based on recent results on improved Hardy–Sobolev type inequalities.     

On a Semiclassical Formula for Non-Diagonal Matrix Elements

Let H(?)=?? ²d²/dx ²+V(x) be a Schrödinger operator on the real line, W(x) be a bounded observable depending only on the coordinate and k be a fixed integer. Suppose that an energy level E intersects the potential V(x) in exactly two turning points and lies below V _∞=lim?inf?_|x|→∞ V(x). We consider the semiclassical limit n→∞, ?=? _n →0 and E _n =E where E _n is the nth eigenenergy of H(?). An asymptotic formula for 〈n|W(x)|n+k〉, the non-diagonal matrix elements of W(x) in the eigenbasis of H(?), has been known in the theoretical physics for a long time. Here it is proved in a mathematically rigorous manner.     

Resolvent Expansion and Time Decay of the Wave Functions for Two-Dimensional Magnetic Schrödinger Operators

We consider two-dimensional Schrödinger operators H(B, V) given by Eq. (1.1) below. We prove that, under certain regularity and decay assumptions on B and V, the character of the expansion for the resolvent (H(B, V) ? λ)^?1 as λ → 0 is determined by the flux of the magnetic field B through \({\mathbb{R}^2}\) . Subsequently, we derive the leading term of the asymptotic expansion of the unitary group e ^{?i t H(B, V)} as t → ∞ and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e ^{?i t H(0, V)}.     

Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains

On a smooth bounded domain \(\Omega \subset {\bf {\rm R}}^N\) we consider the Schrödinger operators ? Δ ? V, with V being either the critical borderline potential V(x) =  (N ? 2)²/4 |x|^?2 or V(x) =  (1/4) dist(x, ?Ω)^?2, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Schrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a series of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincaré. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies.     

Inflation of the early cold Universe filled with a nonlinear scalar field and a nonideal relativistic Fermi gas

We consider a possible scenario for the evolution of the early cold Universe born from a fairly large quantum fluctuation in a vacuum with a size a ₀ ? l _P (where l _P is the Planck length) and filled with both a nonlinear scalar field φ, whose potential energy density U(φ) determines the vacuum energy density λ, and a nonideal Fermi gas with short-range repulsion between particles, whose equation of state is characterized by the ratio of pressure P(n _F ) to energy density ε(n _F ) dependent on the number density of fermions n _F . As the early Universe expands, the dimensionless quantity ν(n _F ) = P(n _F )/ε(n _F ) decreases with decreasing n _F from its maximum value ν_max = 1 for n _F → ∞ to zero for n _F → 0. The interaction of the scalar and gravitational fields, which is characterized by a dimensionless constant ξ, is proportional to the scalar curvature of four-dimensional space R = κ[3P(n _F )–ε(n _F )–4λ] (where κ is Einstein’s gravitational constant), and contains terms both quadratic and linear in φ. As a result, the expanding early Universe reaches the point of first-order phase transition in a finite time interval at critical values of the scalar curvature R = R _c =–μ²/ξ and radius a _c ? a ₀. Thereafter, the early closed Universe “rolls down” from the flat inflection point of the potential U(φ) to the zero potential minimum in a finite time. The release of the total potential energy of the scalar field in the entire volume of the expanding Universe as it “rolls down” must be accompanied by the production of a large number of massive particles and antiparticles of various kinds, whose annihilation plays the role of the Big Bang. We also discuss the fundamental nature of Newton’ gravitational constant G _N .     

On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

For a two-dimensional Schrödinger operator H _{α V}  = ?Δ ?αV with the radial potential V(x) = F(|x|), F(r) ≥ 0, we study the behavior of the number N _?(H _{α V} ) of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N _?(H _{α V} ) = O(α) and for the validity of the Weyl asymptotic law.     

Zur Chronologie des alten Ägypten

The velocityv of the propagation of discharge along the anode of a self-quenchingG—M-counter is a function of total pressureP, pressure of the quenching gasP _D, radius of the cathoder _a and of the anoder _i andV _ü the difference between working- and starting-potential. For the mixtures argon-methylal, argon-alcohol and helium-alcohol isv=v ₀·exp[k·(V _ü/V _e)^1/2] withv ₀ the velocity at the starting potentialV _e v ₀=(a+b·P _D/P)·V _n ^1/2 ·exp [(c?d·P_D/P·V _n ^?1/2 ] andV _n=V _e·(lnr _a/r _i)^?1.k, a, b, c andd are characteristical constants of the filling gas.     

Spectra and mean energies of prompt neutrons from <Superscript>238</Superscript>U fission induced by primary neutrons of energy in the region <Emphasis Type="Italic">E</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><20 MeV

The time-of-flight technique is used to measure the ratios R(E, E _n )=N(E, E _n )/N_Cf(E) of the normalized (to unity) spectra N(E, E _n ) of neutrons accompanying the neutron-induced fission of ²³⁸U at primary-neutron energies of E _n =6.0 and 7.0 MeV to the spectrum N_Cf(E) neutrons from the spontaneous fission of ²⁵²Cf. These experimental data and the results of their analysis are discussed together with data that were previously obtained for the neutron-induced fission of ²³⁸U at the primary energies of E _n =2.9, 5.0, 13.2, 14.7, 16.0, and 17.7 MeV.     

Untersuchungen über die Energieabhängigkeit von Kernreaktionen mit Neutronen im Energiebereich zwischen 12 und 19 MeV

Continuing earlier investigations we studied the energy dependence of the cross sections of the following nuclear reactions produced by neutrons in the energy range from 12 to 19 MeV: P³¹(n, 2n)P³², P³¹(n,α)Al²⁸, Cu⁶⁵(n, 2n)Cu⁶⁴, Cu⁶⁵(n, p)Ni⁶⁵, Zn⁶⁴(n, 2n)Zn⁶³, Zn⁶⁴(n, p)Cu⁶⁴, V⁵¹(n, p)Ti⁵¹, Br⁷⁹(n, α)As⁷⁶, O¹⁶(n, α)C¹³. The results were compared with cross sections calculated according to the statistical theory, considering the competition ofγ-ray emission and particle (predominantly neutron) emission from the excited residual nucleus.     

Percolation and the electron–electron interaction in an array of antidots

A square lattice of microcontacts with a period of 1 μm in a dense low-mobility two-dimensional electron gas is studied experimentally and numerically. At the variation of the gate voltage V _g , the conductivity of the array varies by five orders of magnitude in the temperature range T from 1.4 to 77 K in good agreement with the formula σ(V _g ) = (V _g ?V _g ^* (T))^β with β = 4. The saturation of σ(T) at low temperatures is absent because of the electron–electron interaction. A random-lattice model with a phenomenological potential in microcontacts reproduces the dependence σ(T, V _g ) and makes it possible to determine the fraction of microcontacts x(V _g , T) with conductances higher than σ. It is found that the dependence x(V _g ) is nonlinear and the critical exponent in the formula σ ∝ ? (x - 1/2) ^t in the range 1.3 < t(T, V _g ) < β.     

Exchange and correlation in finite-temperature TDDFT

We discuss the finite-temperature generalization of time-dependent density functional theory (TDDFT). The theory is directly analogous to that at temperature T = 0. For example, the finite-T TDDFT exchange-correlation kernel f_xc(T, n) in the local density approximation can again be expressed as a density derivative of the exchange correlation potential f_xc(T, n) = [?v_xc(T, n)∕?n]δ(r ? r′), where n = N∕V is the electron number density. An approximation for the kernel f_xc(T, n) is obtained from the finite-T generalization of the retarded cumulant expansion applied to the homogeneous electron gas. Results for f_xc and the loss function are presented for a wide range of temperatures and densities including the warm dense matter regime, where T ≈ T_F, the electron degeneracy temperature. The theory also permits a physical interpretation of the exchange and correlation contributions to the theory.     

Proton-proton interaction with high multiplicity at energy of 70 GeV (proposal)

The goal of the proposed experiment is to investigate the collective behavior of particles in the process of multiple hadron production in pp interaction pp → n _π π + 2N at the beam energy E_lab = 70 GeV. The domain of high multiplicity n _π = 30–40, or z = n/\(\bar n\) = 4–6, will be studied. Near the threshold of reaction n _π → 69, z → z_th = 8.2, all particles acquire small relative momentum Δq < 1/R, where R is the dimension of the particle production region. As a consequence of multiboson interference, a number of collective effects may show up: (a) a drastic increase in the partial cross section σ(n) of production of n identical particles is expected, compared with commonly accepted extrapolation; (b) the formation of jets consisting of identical particles may occur as a result of the multiboson Bose-Einstein correlation (BEC) effect; (c) a large fluctuation of charged n(π⁺,π^?) and neutral n(π⁰) components and onset of centauros or chiral condensate effects are anticipated; (d) an increase in the rate of direct γ as a result of the bremsstrahlung in the partonic cascade and annihilation of π⁺π^? → nγ in dense and cold pionic gas or condensate is expected. In the domain of high multiplicity z ≥ 5, a major part of the c.m. energy \(\sqrt s = 11.6\) GeV is materialized, leading to a high-density thermalized hadronic system. Under this condition, a phase transition to cold quark-gluon plasma (QGP) may occur. The search for QGP signatures like large intermittency in the phase-space particle distribution and an enhanced rate of direct photons will be performed. The experimental setup is designed for detection of rare high-multiplicity events. The experiment is to be carried out at the extracted proton beam of the IHEP U-70 accelerator. The required beam intensity is ～10⁷ s^?1. Under the assumption that the partial cross section σ(n _π = 35) = 10-1 nb, the anticipated counting rate is 10-1 events/h. The multiboson BEC enhancement may drastically increase the counting rate.     

The trinification model <Emphasis Type="Italic">SU</Emphasis>(3)<Superscript>3</Superscript> from orbifolds for fuzzy spheres

In this review, we consider an N = 4 supersymmetric SU(3N) gauge theory defined on the Minkowski spacetime. Then we apply an orbifold projection leading to an N = 1 supersymmetric SU(N)³ model, with a truncated particle spectrum. Then, we present the dynamical generation of (twisted) fuzzy spheres as vacuum solutions of the projected field theory, breaking the SU(N)³ spontaneously to a chiral effective theory with unbroken gauge group the trinification group, SU(3)³.     

The reactions N14(n, α)B11 and N14(n,t)C12 observed with a gridded ionization chamber

The cross sections for the reactions N¹⁴(n, α)B¹¹ and N¹⁴(n, t)C¹² have been measured in the neutron energy range 4.0 to 6.4 MeV and at 2.5 MeV. Mono-energetic neutrons were produced in the D(d, n) He³ reaction using a gas target. The (n, α) and (n, t) disintegrations were detected in a gridded ionization chamber filled with an argonnitrogen mixture. The response of the chamber under different operation conditions is described. The excitation functions, measured with a neutron energy resolution of 40 to 50 keV, are given for theα ₀ group from the N¹⁴(n,α)B¹¹ reaction over the entire neutron energy range and for theα ₁ group and the t₀ group from N¹⁴(n, t) C¹² for neutron energies above 4.3 and 5.6 MeV, respectively.     

Non-extensivity of the chemical potential of polymer melts

Following Flory’s ideality hypothesis, the chemical potential of a test chain of length n immersed into a dense solution of chemically identical polymers of length distribution P(N) is extensive in n . We argue that an additional contribution \( \delta\) \( \mu_{{{\rm c}}}^{}\)(n) ～ +1/\( \rho\) \( \sqrt{{n}}\) arises (\( \rho\) being the monomer density) for all P(N) if n ? 〈N〉 which can be traced back to the overall incompressibility of the solution leading to a long-range repulsion between monomers. Focusing on Flory-distributed melts, we obtain \( \delta\) \( \mu_{{{\rm c}}}^{}\)(n) \( \approx\) (1 - 2n/〈N〉)/\( \rho\) \( \sqrt{{n}}\) for n ? 〈N〉² , hence, \( \delta\) \( \mu_{{{\rm c}}}^{}\)(n) \( \approx\) -1/\( \rho\) \( \sqrt{{n}}\) if n is similar to the typical length of the bath 〈N〉 . Similar results are obtained for monodisperse solutions. Our perturbation calculations are checked numerically by analyzing the annealed length distribution P(N) of linear equilibrium polymers generated by Monte Carlo simulation of the bond fluctuation model. As predicted we find, e.g., the non-exponentiality parameter K _p \( \equiv\) 1 - 〈N ^p〉/p!〈N〉^p to decay as K _p \( \approx\) 1/\( \sqrt{{\langle N \rangle }}\) for all moments p of the distribution.     

Censored Glauber Dynamics for the Mean Field Ising Model

We study Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss Model. It is well known that at high temperature (β<1) the mixing time is Θ(nlog?n), whereas at low temperature (β>1) it is exp?(Θ(n)). Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed β>1, the mixing-time of this model is Θ(nlog?n), analogous to the high-temperature regime of the original dynamics. Furthermore, they showed cutoff for the original dynamics for fixed β<1. The question whether the censored dynamics also exhibits cutoff remained unsettled.In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Curie-Weiss model. Namely, we found a scaling window of order \(1/\sqrt{n}\) around the critical temperature β _c =1, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging.In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if β=1+δ for some δ>0 with δ ² n→∞, then the mixing-time has order (n/δ)log?(δ ² n). The cutoff constant is (1/2+[2(ζ² β/δ?1)]^?1), where ζ is the unique positive root of g(x)=tanh?(β x)?x, and the cutoff window has order n/δ.     

Internal friction in alloys due to plasticity of the diffusion phase transformation

An analytic expression is obtained for the time dependence Q ^?1(t) of internal friction associated with plasticity of a phase transformation. Time dependences Q ^?1(t) of internal friction of the Pb-62Sn and Pb-1.9Sn alloys (wt.%) alloys were studied in the regime of continuous excitation of resonant flexural vibrations. The measurements of the Q ^?1(t) dependences for 1 h at room temperature and a fixed strain amplitude ε₀ ≈ 7 and 19 min) for the Pb-62Sn alloy. For the Pb-1.9Sn alloy under the same conditions, an exponential decrease followed by an internal friction peak (at t _m ≈ 7 min) is observed. It is shown numerically that the above singularities of internal friction are formed by processes of intermittent phase decomposition of Pb-Sn alloys in the cyclic stress field produced by an external load. Experimental data on Q ^?1(t) are used for reconstructing the kinetic curves describing the decomposition (conversion) ratio as a function of time and for calculating the corresponding values of parameters K and n of the Avrami kinetic equation for the Pb-62Sn alloy.     

Charge transfer over localized states in a TlS single crystal

It is revealed that TlS single crystals exhibit a variable range hopping conduction along a normal to their natural layers at temperatures T ≤ 230 K in a dc electric field and a nonactivated hopping conduction at low temperatures in strong electric fields. Estimates are made for the density of states near the Fermi level (N _F = 2.8 × 10²⁰ eV^?1 cm^?3 and their energy spread (ΔW = 0.02 eV), the localization radius (a = 33 Å), the average jump distance in the region of activated (R _av(T) = 40 Å) and nonactivated (R _av(F) = 78 Å) hopping conduction, and also the drop in the charge carrier potential energy along the jump distance in an electric field F: eFR = 0.006 and 0.009 eV at F = 7.50 × 10³ and 1.25 × 10⁴ V/cm, respectively.     

Über eine Näherung für Gitterelektronen in äußeren Feldern

The Hamilton operator of an electron in a periodic lattice potential under influence of external electric and magnetic fields with potentialsV(r) andA(r) resp. is often replaced by an approximate operatorW ⁰ (?i?+A(r))+V(r) for one single energy bandW ⁰(k) which means a renormalization of the kinetic energy by the lattice. The validity of this replacement is examined and the magnitude of its error is roughly estimated. Neglecting other bands one obtains an error term proportional to the derivative of the electric field strengthF, if one takes a suitable position of the “raster” of the replacement operator, and to the square of the magnetic field strengthB resp. The decoupling from the other energy bands leads to error terms proportionalF ² andB ² resp. which however in the general case increase rapidly in the vicinity of overlapping energy bands.