Higher order differential lift equations over manifolds

LetM be a differentiable manifold modeled on a Banach space overK=R or C. LetT ^k(M) be thekth iterated tangential extension ofM, and letk ^M be thekth Bowman (=restricted tangential) extension ofM. It is shown that there is an embedding ϕ_k:k ^{→T k}(M), and that such embeddings constitute a natural transformation of functors. LetQ be a subset/submanifold inT ^k(M), and letV:Q→T(Q) be a differentiable vector field. CallV k-suitable if everyK-curveg inQ satisfyingg′=V° g has the formg=f ^[k], wheref ^[k] denotes thekth iterated differential lift of aK-curvef inM. It is shown thatV isk-suitable if and only if: (a) , where is a subset/submanifold ink ^M, and (b) , where isk-suitable relative to restricted tangentialK-curve liftsf ^(k). Interpretive consequences for motion problems are discussed.     

An approximate expression for the galvanomagnetic tensor

Using the iterative solution to the Boltzmann equation for electrons in d.c. electric and magnetic fields, an expression for the resistivity tensor can be obtained in the form of an infinite series. This series can be approximated by retaining only the first two terms. In the cases where relaxation times exist — in the sense that the collision term in the Boltzmann equation can be written asg(k)/τ(k), whereτ(k) is the relaxation time, andf (k) = f _E(ɛ k) + [∂f _E(εk)/∂ε]·g(k) the distribution function for electrons with wavevectork — this approximation is exact. For polyvalent metals in the one-OPW approximation, the complete galvanomagnetic tensor can be obtained using this approximation and the result differs from that obtained by using a time of relaxation given by an expression suggested byZiman. A calculation for a simple model Fermi surface, with screened Coulomb scattering, is carried out and the results compared with those of the relaxation time approximation.     

Distribution function for large velocities of a two-dimensional gas under shear flow

The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degreek⩾4 diverge for shear rates larger than a critical valuea _c ^(k) , which behaves for largek asa _c ^(k) ∼k ⁻¹. This divergence is consistent with an algebraic tail of the formf(V) ∼V ^−4-σ(a), where σ is a decreasing function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.     

Study of analogue states in52Cr through proton capture by51V

γ-Ray yield function has been studied for the proton capture by vanadium in the proton energy range 720–1300 keV. Isobaric analogues of low lying states in⁵²V have been identified. At two resonances theγ-decay andγ-ray angular distributions have been obtained and the branching ratios and the multipole mixing ratios have been deduced. The analogue-antianalogue M1 transition in⁵²Cr is found to be strongly hindered as in other f_7/2 nuclei. TheQ-value obtained for this reaction is (10500±2.8) keV and the Coulomb displacement energy is (8.06±0.01) MeV. An upper limit of 2 meV has been obtained for theα- decay strength of the 11.395 MeV state in⁵²Cr.     

Structural phase transitions in Pd0.8 Si0.2 and Pd0.75 Si0.20 Ag0.05 alloys observed by pac

The phase transition in the alloys Pd_0.8 Si_0.2 and Pd_0.75 Si_0.20 Ag_0.05 have been investigated through the quadrupole interaction of¹¹¹Cd impurities. The quadrupole interactions were measured by means of the TDPAC technique from room temperature up to about 870 K. The variation of the quadrupole interaction with temperature in the alloy PdSiAg shows aT ^3/2 dependence below and above 629 K, with coefficientsB=5.43(25)·10⁻⁵ K^−3/2 andB=3.70(15)·10⁻⁵ K^−3/2, respectively. This demonstrates that the alloy undergoes a phase transition around 629 K. The existence of two electric field gradients observed in the alloy PdSi,V _zz (1)=3.47(54)·10¹⁷ V/cm² andV _zz (2)=2.29(36)·10¹⁷ V/cm², indicates that there are two different¹¹¹Cd sites. The corresponding fractionsf ₁ andf ₂ strongly depend on temperature. Below 520 K, most¹¹¹Cd nuclei are subject to the higher EFGV _zz (1) (f ₁≈70%), whereas above 520 Kf ₁ falls rapidly to zero andV _zz (2) becomes dominant. The temperature dependences of thef ₁ andf ₂ reveal a picture of the phase transition between the two crystal structures.     

Studies of parity and time reversal symmetries in neutron scattering from165Ho

We describe searches for parity and time reversal violations in the scattering of polarized neutrons from polarized and aligned¹⁶⁵Ho targets. We have completed a search with 7.1 and 11.0 MeV neutrons for P_oddT_odd terms in the elastic scattering forward amplitude of the form s. (I×K), wheres is the neutron spin,I is the target spin andk is the neutron momentum vector. The target was a single crystal of holmium, polarized horizontally along itsb axis by a 1 Tesla magnetic field. The neutrons were polarized vertically. Differences in the neutron transmission were measured for neutrons with spins parallel (antiparallel) toI×k. The P,T violating analyzing powers were found to be consistent with zero at the few 10⁻³ level: ρ_P,T(7.1 MeV)=−0.88 (±2.02) x 10⁻³, ρ_P,T(11.0 MeV)=−0.4 (±2.88) x 10⁻³. We have also attempted to find enhancements with MeV neutrons in P-violation due to the term s\k. We are preparing an aligned target cryostat for investigations of P_evenT_odd terms {bd(I\k)(I×k)\s} in neutron scattering. The target will be a single crystal cylinder of¹⁶⁵Ho cooled to 100 mK in a bath of liquid helium and rotated by a shaft from a room temperature stepping motor. The cylinder will be oriented vertically and the alignment (c) axis oriented horizontally. Warming or rotation of the sample allows one to separate effects that mimic the sought-after time reversal violating term.     

Motion in the complex plane of the poles of the quantum-mechanical partial amplitude for a bond

Details are given of the analytic features of this amplitudef _g(l, k) in the g plane for a very large class of potentials that satisfy conditions (1). It is shown that there is a region (l, k²) around the point g = 0 that is free from singularities inf _g(l, k), so the Mittag-Leffler method can be applied to findf _g(l, k) and hence also the total amplitude T_g(k, t) for any g to any specified degree of accuracy with reference to the information contained in the coefficients of a finite number of terms of the series given by perturbation methods forf _g(l, k).This paper was read in substance on 8th April 1963 at Uzhgood University [1] and on 29 June 1963 at the All-Union Symposium on the Theory of Elementary Particles at Novosibirsk.     

Tests of the infinite mass effective theory and properties of the charming and beautiful mesons

Using an operator product expansion (OPE) in the inverse of the heavy quark massM _Q and a dispersive approach, we rederive theB andB ^* couplings and the Isgur-Wise function predicted by the infinite mass effective theory (IMET). We find that, at the subtraction pointM _Q and using VDM for the spectral function, these observables are governed by the universal light quark condensate. We also find that the corrections induced by finite values ofM _c andM _b are large and are due to the splitting _b between the ground state and its radial excitation. We review the determinations of the different decay constants, couplings and masses from QCD spectral sum rules (QSSR). Forf _B andf _D , we conclude the optimum average:f _B =(1.59±0.09±0.27)f andf _D =(1.31±0.12)f , where the main error is due to the quark mass, while the first one inf _B is due to the non-perturbative terms and to the choice of the continuum thresholdt _c from the onset of sum rule variable stability until thet _c -stability regions. We also find that the vector coupling _V satisfies approximately the IMETM _Q ^3/2 mass dependence scaling law, whilef _P obeys in units of GeV:f P/f(4.6±0.4)M _Q ^–1/2 (1–1.56/M _Q +0.88/M _Q ² ), forM _Q M _c . We also predict the flavour independence:M _P -M _Q 0.6-0.7 GeV andM _V -M _Q 0.7-0.8 GeV of the mass differences between the mesons and the corresponding quark. Finally we find that theBDev _e form factorf ₊(0)(0.55±0.10) deviates from the naive expectation (f ₊(0)=1).     

Analysis of the vector form factors f<Superscript>+</Superscript><Subscript>Kπ</Subscript>(Q<Superscript>2</Superscript>) and f<Superscript>-</Superscript><Subscript>Kπ</Subscript>(Q<Superscript>2</Superscript>) with light-cone QCD sum rules

In this article, we calculate the vector form factors f⁺ _Kπ(Q²) and f^- _Kπ(Q²) within the framework of the light-cone QCD sum rule approach. The numerical values of f⁺ _Kπ(Q²) are compatible with existing theoretical calculations, and the central value of f⁺ _Kπ(0) (f⁺ _Kπ(0)=0.97) is in excellent agreement with the values from chiral perturbation theory and lattice QCD. The values of |f^- _Kπ(0)| are very large compared to the theoretical calculations and experimental data, and they cannot give any reliable prediction. At large momentum transfer with Q² > 5 GeV², the form factors f⁺ _Kπ(Q²) and |f^- _Kπ(Q²)| can either show the asymptotic behavior of or decrease more quickly than ; more experimental data are needed to select the ideal sum rules. PACS 12.38.Lg; 12.38.Bx; 12.15.Hh     

Magnetic and electric hyperfine interactions in the system EuSm

Magnetic and electric hyperfine interactions in the system EuSm were investigated with the isomeric 11/2^– state in¹⁴⁵Eu by applying the time-differential perturbed angular distribution (TDPAD) method. The temperature dependence of paramagnetism was studied between 90 K and 1000 K by measuring the magnetic hyperfine interaction frequency _L=g_NN¹ (T)B_ext. The paramagnetic correction factor strictly follows the Curie-Weiss relation =1+C/(T-), withC=–50(2) K and =–29(5) K. This is compatible with a hyperfine field ofB _int(0)=–25(1) T, a valence of two for Eu in Sm, and antiferromagnetic order at low temperatures.The temperature dependence of the electric quandrupole coupling constant v_Q, investigated between 100 K and 400 K, can be reproduced by a linear temperature variation v_Q(T)=v_Q(0) (1-AT), with v_Q(0)=16.2(4) MHz andA=7.2(8)·10^–4 K^–1.The paramagnetic relaxation time _rel of the nuclear alignment is proportional to the temperature of the sample, with _rel T^–1=3.7(2) ·10^–9s K^–1.This leads to the Korringa relation _J T=const=5.1(5)·10^–11s K for the relaxation time of the 4f electronic spinJ. Assuming that the relaxation ofJ is mainly caused by exchange interaction between conduction electrons and localized 4f electrons at the Eu site, an exchange integral of |J _eff|=0.10(2) eV can be deduced.     

Application of the quantum mechanical hypervirial theorems to even-power series potentials

The class of the even-power series potentials,V(r)=-D+∑ _k-0 ^∞ V_kλ^kr^2k+2,V ₀=ω²>0is studied with the aim of obtaining approximate analytic expressions for the nonrelativistic energy eigenvalues, the expectation values for the potential and kinetic energy operators, and the mean square radii of the orbits of a particle in its ground and excited states. We use the hypervirial theorems (HVT) in conjunction with the Hellmann-Feynman theorem (HFT), which provide a very powerful scheme for the treatment of the above and other types of potentials, as previous studies have shown. The formalism is reviewed and the expressions of the above-mentioned quantities are subsequently given in a convenient way in terms of the potential parameters, the mass of the particle, and the corresponding quantum numbers, and are then applied to the case of the Gaussian potential and to the potentialV(r)=−D/cosh²(r/R). These expressions are given in the form of series expansions, the first terms of which yield, in quite a number of cases, values of very satisfactory accuracy.     

η(1475) and <Emphasis Type="Italic">f</Emphasis><Subscript>1</Subscript>(1420) resonances in the decay processes <Emphasis Type="Italic">J</Emphasis>/ψ → γ(ρρ, γρ<Superscript>0</Superscript>, γϕ) and in γγ* collisions

A scenario that removes the contradiction between the suppression of the η(1475) → γγ decay width and the strong coupling of η(1475) to the ρρ, ωω, and γρ⁰ channels and which leads to a nontrivial prediction for the manifestation of η(1475) in γγ*(Q ²) collisions is considered. Data on the dependence of the cross section for the reaction γγ*(Q ²) → K[`(K)]pK\bar K\pi on the photon virtuality in the energy range 1.35–1.55 GeV are explained here by the production of an η(1475) resonance in contrast to their standard interpretation in terms of the f <sub>1</sub>(1420) resonance. Experimental verification of the present explanation requires determining the spin-parity of resonance contributions, R, in the reactions γγ*(Q <sup>2</sup>) R → R → K[`(K)]pK\bar K\pi and J/ψ → γR → γ(γρ⁰, γϕ).     

Magnetic relaxation in a three-dimensional ferromagnet with weak quenched random-exchange disorder

Isothermal remanent magnetization decay,M _r(t), and ‘in-field’ growth of zero-field-cooled magnetization,M _ZFC(t), with time have been measured over four decades in time at temperatures ranging from 0.25T _c to 1.25T _c (whereT _c is the Curie temperature, determined previously for the same sample from static critical phenomena measurements) for a nearly ordered intermetallic compound Ni₃Al, which is an experimental realization of a three-dimensional (d = 3) ferromagnet with weak quenched random-exchange disorder. None of the functional forms ofM _r(t) predicted by the existing phenomenological models of relaxation dynamics in spin systems with quenched randomness, but only the expressions and closely reproduce such data in the present case. The most striking features of magnetic relaxation in the system in question are as follows: Aging effects are absent in bothM _r t andM _ZFC(t) at all temperatures in the temperature range covered in the present experiments. A cross-over in equilibrium dynamics from the one, characteristic of a pured = 3 ferromagnet with complete atomic ordering and prevalent at temperatures away from T_c, to that, typical of ad = 3 random-exchange ferromagnet, occurs asT → T_c. The relaxation times τ₁(T)(τ₁ ^′(T)) and τ₂(T)(τ₂ ^′(T)) exhibit logarithmic divergence at critical temperatures and ; and both increase with the external magnetic field strength,H, such that at any given field value, . The exponent characterizing the logarithmic divergence in τ ₁ ^′ (T) and τ ₂ ^′ T possesses a field-independent value of ≃16 for both relaxation times. Of all the available theoretical models, the droplet fluctuation model alone provides a qualitative explanation for some aspects of the present magnetic relaxation data     

A note on traversal time for tunneling

A simple approach to study the traversal time for tunneling is given. By using the WKB wave function to evaluate the velocity field of particles in the barrier region, an expression for the traversal time τ=εdx[m/2(V(x)-E)]^1/2 is obtained in conformity with the recent results.     

Elastic shape memory effect in In-Pb alloys in the temperature range 0.48–180 K

Pseudoelasticity caused by pseudotwinning in short-range ordered In-Pb alloys (6, 8 and 11.6 at. % Pb) is studied in the temperature range 0.48–180 K. The mechanical hysteresis parameters, namely, the thermodynamic stress τ _T which provides the reversibility of plastic deformation and the frictional stress τ _f which characterizes the resistance offered by crystal lattice and its defects to twin boundaries motion are estimated. It is found that athermal processes determine the reversible deformation: the mechanical parameters τ _T and τ _f do not depend on temperature and strain rate. The stress τ _T increases and the stress τ _f decreases with increasing Pb content. One of the main conditions of the exhibition of superelasticity is the fulfillment of the inequality τ _T >τ _f .     

A new perturbative approach to the classical anharmonic oscillator

The periodic motion of the classical anharmonic oscillator characterized by the potentialV(x)=1/2x ²+λ/2k x ^2k is considered. The period is first determined to all orders inλ in a perturbative series. Making use of this, the solution of the nonlinear equation of motion is then expressed in the form of a Fourier series. The Fourier coefficients are obtained by solving simple algebraic relations. Secular terms are inherently absent in this perturbative scheme. Explicit solution is presented for generalk up to the second order, from which the Duffing and the sextic oscillator results follow as special cases.     

Signature of chaos in power spectrum

We investigate the nature of the numerically computed power spectral densityP(f, N, τ) of a discrete (sampling time interval,τ) and finite (length,N) scalar time series extracted from a continuous time chaotic dynamical system. We highlight howP(f, N, τ) differs from the true power spectrum and from the power spectrum of a general stochastic process. Non-zeroτ leads to aliasing;P(f, N, τ) decays at high frequencies as [πτ/sinπτf]², which is an aliased form of the 1/f ² decay. This power law tail seems to be a characteristic feature of all continuous time dynamical systems, chaotic or otherwise. Also the tail vanishes in the limit ofN → ∞, implying that the true power spectral density must be band width limited. In striking contrast the power spectrum of a stochastic process is dominated by a term independent of the length of the time series at all frequencies.     

Precise Quadrupole Moment of the Doubly Closed Shell Plus One Proton Nucleus 41Sc

In order to precisely determine the |eqQ(⁴¹Sc)/h| of ⁴¹Sc(I ^π=7/2⁻, T _1/2=0.596 s) implanted in a TiO₂ single crystal the electric field gradient tensor and anisotropic chemical shift were determined by detecting the Fourier Transformed Pulse-NMR of ⁴⁵Sc(I ^π=7/2⁻, stable) in the crystal at a high field of 7.0 T and 9.4 T. Combining the new |eqQ(⁴¹Sc)/h| value with the Q(⁴⁵Sc)=−(23.6±0.2) fm², also renewed by using the known atomic hyperfine interaction constants and Sternheimer polarization effect, the |Q(⁴¹Sc;I ^π=7/2⁻,T _1/2=0.596 s)|=(15.6±0.3) fm² has been precisely determined. The value is compared with the theoretical Q(⁴¹Sc) given by a shell model code. This revised version was published online in September 2006 with corrections to the Cover Date.     

Maximum Solutions of Normalized Ricci Flow on 4-Manifolds

We consider the maximum solution g(t), t ∈ [0,  + ∞), to the normalized Ricci flow. Among other things, we prove that, if (M, ω) is a smooth compact symplectic 4-manifold such that and let g(t), t ∈ [0, ∞), be a solution to (1.3) on M whose Ricci curvature satisfies that |Ric(g(t))| ≤ 3 and additionally χ(M) = 3τ (M) > 0, then there exists an , and a sequence of points {x _j,k ∈ M}, j = 1, . . . , m, satisfying that, by passing to a subsequence,

Black holes with metric and fields of the same form

We present two rotating black hole solutions with axion ξ, dilaton f{\phi} and two U(1) vector fields. Starting from a non-rotating metric with three arbitrary parameters, which we have found previously, and applying the “Newman–Janis complex coordinate trick” we get a rotating metric g _μν with four arbitrary parameters namely the mass M, the rotation parameter a and the charges electric Q _E and magnetic Q _M . Then we find a solution of the equations of motion having this g _μν as metric. Our solution is asymptotically flat and has angular momentum J = M a, gyromagnetic ratio g = 2, two horizons, the singularities of the solution of Kerr, axion and dilaton singular only when r = a cos θ = 0 etc. By applying to our solution the S-duality transformation we get a new solution, whose axion, dilaton and vector fields have one more parameter. The metrics, the vector fields and the quantity l = x+ie^-2f{\lambda=\xi+ie^{-2\phi}} of our solutions and the solution of: Sen for Q _E , Sen for Q _E and Q _M , Kerr–Newman for Q _E and Q _M , Kerr, Reference Kyriakopoulos [Class. Quantum Grav. 23:7591, 2006, Eqs. (54–57)], Shapere, Trivedi and Wilczek, Gibbons–Maeda–Garfinkle–Horowitz–Strominger, Reissner–Nordstr?m, Schwarzschild are the same function of a, and two functions ρ ² = r(r + b) + a ² cos² θ and Δ = r(r + b) − 2Mr + a ² + c, of a, b and two functions for each vector field, and of a, b and d respectively, where a, b, c and d are constants. From our solutions several known solutions can be obtained for certain values of their parameters. It is shown that our two solutions satisfy the weak the dominant and the strong energy conditions outside and on the outer horizon and that all solutions with a metric of our form, whose parameters satisfy some relations satisfy also these energy conditions outside and on the outer horizon. This happens to all solutions given in the “Appendix”. Mass formulae for our solutions and for all solutions which are mentioned in the paper are given. One mass formula for each solution is of Smarr’s type and another a differential mass formula. Many solutions with metric, vector fields and λ of the same functional form, which include most physically interesting and well known solutions, are listed in an “Appendix”.