Dimer Coverings on the Sierpinski Gasket

We present the number of dimer coverings N _d (n) on the Sierpinski gasket SG _d (n) at stage n with dimension d equal to two, three, four or five. When the number of vertices, denoted as v(n), of the Sierpinski gasket is an even number, N _d (n) is the number of close-packed dimers. When the number of vertices is an odd number, no close-packed configurations are possible and we allow one of the outmost vertices uncovered. The entropy of absorption of diatomic molecules per site, defined as , is calculated to be ln (2)/3 exactly for SG ₂. The numbers of dimers on the generalized Sierpinski gasket SG _d,b (n) with d=2 and b=3,4,5 are also obtained exactly with entropies equal to ln (6)/7, ln (28)/12, ln (200)/18, respectively. The number of dimer coverings for SG ₃ is given by an exact product expression, such that its entropy is given by an exact summation expression. The upper and lower bounds for the entropy are derived in terms of the results at a certain stage for SG _d (n) with d=3,4,5. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of with d=3,4,5 can be evaluated with more than a hundred significant figures accurate. This paper is written during the Lung-Chi Chen visit to PIMS, University of British Columbia. The author thanks the institute for the hospitality.     

Spanning Trees on the Sierpinski Gasket

We present the numbers of spanning trees on the Sierpinski gasket SG _d (n) at stage n with dimension d equal to two, three and four. The general expression for the number of spanning trees on SG _d (n) with arbitrary d is conjectured. The numbers of spanning trees on the generalized Sierpinski gasket SG _d,b (n) with d = 2 and b = 3,4 are also obtained.     

Effects of the velocity dependence of the collision frequency on the Dicke line narrowing

The effect of the velocity (v) dependence of the transport collision frequency ν_trv on the Dicke line narrowing is analyzed in terms of the strong-collision model generalized to velocity-dependent collision frequencies (the so-called kangaroo model). This effect has been found to depend on the mass ratio of the resonance (M) and buffer (M _b) particles, β = M _b/M: it is at a minimum for β ? 1 and reaches a maximum for β ? 3. A power-law particle interaction potential, U(r) ∝ r ^?n , is used as an example to show that, compared to ν_trv (v) = const (n = 4), the line narrows if ν_trv (v) decreases with increasing v (n < 4) and broadens if ν _trv (v) increases with v(n > 4). At β ? 3, the line width can increase [compared to ν_trv (v) = const] by 5 and 12% for the potentials with n = 6 and n ? 10, respectively; for the potentials with n = 1 (Coulomb potential) and n = 3, it can decrease by more than half and 6%, respectively. The line profile I(Ω) has been found to be weakly sensitive to ν_trv (v) at some detuning Ω_c of the radiation frequency Ω. Dicke line narrowing is used as an example to analyze the collisional transport of nonequilibrium in the resonance-particle velocity distribution in a laser field. The transport effect is numerically shown to be weak. This allows simpler approximate one-dimensional quantum kinetic equations to be used instead of the three-dimensional ones to solve spectroscopic problems in which it is important to take into account the velocity dependence of the collision frequency when the phase memory is preserved during collisions.     

The higgs boson mass from precision electroweak data

We present a new global fit to precision electroweak data, including new low- and high-energy data and analyzing the radiative corrections arising from the minimal symmetry breaking sectors of the Standard Model (SM) and its supersymmetric extension (MSSM). It is shown that present data favor a Higgs mass ofO(M _z):M _H=76 _?50 ⁺¹⁵² GeV. We confront our analysis with (meta) stability and perturbative bounds on the SM Higgs mass, and the theoretical upper bound on the MSSM Higgs mass. Present data do not discriminate significantly between the SM and MSSM Higgs mass ranges. We comment in passing on the sensitivity of the Higgs mass determination to the values ofα(M _z) andα _s(M _z).     

Development of metal borohydrides for hydrogen storage

A metal borohydride M(BH₄)_n is a potential candidate for hydrogen storage materials because of its high gravimetric hydrogen density. The important research issues for M(BH₄)_n are to control the thermodynamic stability and to achieve the faster reaction kinetics. To clarify the thermodynamic stability, M(BH₄)_n (M=Mg, Ca∼Mn, Zn, Al, Y, Zr and Hf; n=2-4) were synthesized by mechanical milling and its thermal desorption properties were investigated. The hydrogen desorption temperature T_d of M(BH₄)_n decreases with increasing Pauling's electronegativities χ_P of M. Because Mn, Zn, and Al borohydrides (χ_P?1.5) desorb borane, they are too unstable for hydrogen storage applications. The enthalpy changes of desorption reaction ΔH_des can be estimated by using our predicted heat of formation of M(BH₄)_n ΔH_boro and reported data for decomposed products ΔH_prod, which are useful indicators for searching M(BH₄)_n with appropriate stability for hydrogen storage material. In the latter case, microwave processing was adopted for achieving fast reaction kinetics. Among metal borohydrides, LiBH₄ was rapidly heated above 380 K by microwave irradiation, 13.7 mass% of hydrogen was desorbed by microwave irradiation. The composites of LiBH₄ with B or C desorbed hydrogen within 3 min. Microwave heating aids in realizing faster kinetics of the hydrogen desorption reaction.     

Stabilizer of a function in the gage group

It is proved that, for the dimension d of the stabilizer of an analytic function z(x, y) in the gage pseudogroup G = {z(x, y) → c(z(a(x), b(y))}, there are precisely four possibilities: (1) d = ∞ and the complexity of z is zero, (2) d = 3 and the complexity of z is equal to one, (3) d = 1 and z is equivalent the function r(x + y) ? x of complexity two, (4) d = 0 in all remaining cases.     

Classical stability of direct products of spheres in gravitational systems

Classical stability of Einstein spaces S^d₁ ×?×S^d_n(d_j ? 2) against all fluctuations is investigated in euclidean gravity with a cosmological constant. It is shown that S^d is classically stable, while S^d₁ ×?× S^d_n(n ? 2) is classically unstable. As a generalization of this analysis it is proved that a compact Einstein space B₁ ×?× B_n(n ? 2) which is a direct product of each Einstein space is classically unstable. Non-Einstein spaces M² × S⁴ (M² × S² × S²) are also considered in six- dimensional Einstein-Maxwell theory and are shown to be classically stable (unstable).     

Spectral Estimates for Periodic Jacobi Matrices

 We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ²(ℤ) of the form (Hψ) _n =a _n−1 ψ _n−1 +b _nψ _n +a _nψ _n+1 , where a _n=a _n+q and b _n=b _n+q are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding to H. We consider k(z) as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent over the gaps with the normalised traces of powers of H. Received: 17 April 2002 / Accepted: 1 October 2002 Published online: 13 January 2003 Communicated by B. Simon     

Nonperturbative contribution to the fermion propagator and dynamical electroweak symmetry breaking

We examine the field-theoretical contribution of fermion-antifermion condensates arising from a weak-SU(2) doublet of condensing fermions to electroweak vacuum polarization functions. For the custodial-SU(2) case of equal condensates and masses, we find that the condensate contributions to vacuum polarization functions uphold the electroweak signature relationm _w=m _zcosθ_w, and that these contributions are decoupled entirely from oblique radiative corrections. If only the upper member of the doublet forms a fermion-antifermion condensate, the relationm _w=m _zcosθ_w is again upheld in the limit that the mass of the lower member of the doublet is small compared to that of the upper member. For this case, the upper-member's fermion-antifermion condensate is shown to enter oblique radiative corrections. In the absense of an explicit Higgs mechanism, identification of this doublet with (t, b) is shown to be excluded by present empirical bounds onS, T, andU parameters. Further phenomenological consequences of fermion-antifermion condensate contributions to theW-Z mass matrix are discussed, both in the absense and in the presence of an explicit Higgs mechanism.     

First passage time of multiple Brownian particles on networks with applications

In this article, we study some theoretical and technological problems with relation to multiple Brownian particles on networks. We are especially interested in the behavior of the first arriving Brownian particle when all the Brownian particles start out from the source s simultaneously and head to the destination h randomly. We analyze the first passage time (FPT) Y_sh(z) and the mean first passage time (MFPT) 〈Y_sh(z)〉 of multiple Brownian particles on complex networks. Equations of Y_sh(z) and 〈Y_sh(z)〉 are obtained. On a variety of commonly encountered networks, we observe first passage properties of multiple Brownian particles from different aspects. We find that 〈Y_sh(z)〉 drops substantially when particle number z increases at the first stage, and converges to d_sh, the distance between the source and the destination when z→∞. The distribution of FPT Prob{Y_sh(z)=t},t=0,1,2,… is also analyzed in these networks. The distribution curve peaks up towards t=d_sh when z increases. Consequently, if particle number z is set appropriately large, the first arriving Brownian particle will go along the shortest or near shortest paths between the source and the destination with high probability. Simulations confirm our analysis. Based on theoretical studies, we also investigate some practical problems using multiple Brownian particles, such as communication on P2P networks, optimal routing in small world networks, phenomenon of asymmetry in scale-free networks, information spreading in social networks, pervasion of viruses on the Internet, and so on. Our analytic and experimental results on multiple Brownian particles provide useful evidence for further understanding and properly tackling these problems.     

Supersymmetric AdS_4 vacua in N=3 gauged supergravity

We study the maximally supersymmetric AdS backgrounds of matter-coupled N=3 gauged supergravity in four dimensions. We find that to admit supersymmetric AdS vacua, the gauge group can only be of the form G_0×H?SO(3,n) with G_0 =SO(3),SO(3,1) or SL(3,R) and H a compact group of dimension n+3-dim(G_0). We also show that these AdS vacua have no moduli, namely they correspond to critical points in field space.     

The role of the neutron electric form factor ind(e,e′ N)N including polarization observables

The influence of the neutron electric form factor on various observables in two-body break-up of deuterons by electrons such as differential cross section, beam, target and beam-target asymmetries and outgoing nucleon polarization as well is investigated for different kinematic regions. The electron-deuteron vector asymmetryA _ed ^V and the outgoing nucleon polarization component P′_x(n) are the most promising observables in and off the quasi-free region for a determination ofG _En. Also the single polarization observablesA _d ^T and P_y ⁰(p) and the double polarization observable P′_z(n) show significant influences fromG _En.     

Estimates of critical lengths and critical temperatures for classical and quantum lattice systems

Local Ward identities are derived which lead to the mean-field upper bound for the critical temperature for certain multicomponent classical lattice systems (improving by a factor of two an estimate of Brascamp-Lieb). We develop a method for accurately estimating lattice Green's functionsI _d yielding 0.3069<I ₄<0.3111 and the global bounds (d?1/2)^?<I _d <(d?1)^? for alld?4. The estimate forI _d implies the existence of a critical length for classical lattice systems with fixed length spins. Forv-component spins with fixed lengthb on the lattice ? ^d ,v=1, 2, 3, 4, the critical temperature for spontaneous magnetization satisfies $$\frac{{2Jb^2 }}{k}\frac{{d - 1}}{v}< T{}_c \leqslant \frac{{2Jb^2 }}{k}\frac{d}{v} for d \geqslant 4$$ ford4 Using GHS or generalized Griffiths' inequalities, we find that the upper bounds on the critical temperature extend to certain classical and quantum systems with unbounded spins. Absence of symmetry breakdown at high temperature for quantum lattice fields follows from bounding the energy density by a multiple ofkT. Path space techniques for finite degrees of freedom show that the high-temperature limit is classical.     

Large Time Behavior of¶Schrödinger Heat Kernels and Applications

We obtain global in time and qualitatively sharp bounds for the heat kernel G of the Schrödinger operator mj +V. The potential V satisfies V(x)~- C/d(x)^b near infinity with b ] (0, 2). When V S 0 our result can be described as follows: G is bounded from above and below by the multiples of standard Gaussians with a weight function. If b>2 then the weight is bounded between two positive {\it constants}; if b=2, the weight is bounded between two positive functions of t, d(x) and d(y), which have polynomial decay; if b<2, the weight is bounded between two positive functions of t, d(x) and d(y), which have exponential decay. Up to now satisfactory bounds for heat kernels can only be found when b>2 or b<0. An application to a semilinear elliptic problem is also given.     

Critical dynamics of the sine-Gordon model ind=2−ε dimensions

The renormalization scheme of Amit, Goldschmidt and Grinstein is extended tod=2?ε dimensions. The exponent ν of the correlation lengthv ^?1=2ε+O(ε²) is in agreement with the result of Kosterlitz for the Coulomb gas. The exponent η of the correlation function of the sine-Gordon field is η=ε+O(ε²). The scaling form of the dynamical structure factorS(q,ω) of the dynamic sine-Gordon ModelA is studied ind=2?ε dimensions. The dynamic exponentz is found to bez=2+(b?1)ε+O(ε²) for ε≧0. The constantb is given by the integral $$b = \int\limits_0^\infty {dss^{ - 2} \exp } \left( { - 2\int\limits_s^\infty {dxx^{ - 1} e^{ - x} } } \right) = 2,371544...$$      

Crystal structure and ferromagnetism of (n-C3H7)4N[CoFe(dto)3] (dto=C2O2S2)

Recently, we have discovered a new type of first order phase transition around 120 K for (n-C₃H₇)₄N[Fe^IIFe^III(dto)₃] (dto=C₂O₂S₂), where the charge transfer transition between Fe^II and Fe^III occurs reversibly. In order to elucidate the origin of this peculiar first order phase transition. Detailed information about the crystal structure is indispensable. We have synthesized the single crystal of (n-C₃H₇)₄N[Co^IIFe^III(dto)₃] whose crystal structure is isomorphous to that of (n-C₃H₇)₄N[Fe^IIFe^III(dto)₃], and determined its detailed crystal structure. Crystal data: space group P6₃, a=b=10.044(2) Å, c=15.960(6) Å, α=β=90°, γ=120°, Z=2 (C₁₈H₂₈NS₆O₆FeCo). In this complex, we found a ferromagnetic transition at T_c=3.5 K. Moreover, on the basis of the crystal data of (n-C₃H₇)₄N[Co^IIFe^III(dto)₃], we determined the crystal structure of (n-C₃H₇)₄N[Fe^IIFe^III(dto)₃] by simulation of powder X-ray diffraction results.     

Modes of natural vibration for magnetic domains

We analyze wall-vibration modes for the case of plane parallel stripe domains in a uniaxial film whose easy axis is normal to the film plane, using Landau-Lifshitz equations carried to the limit of vanishing wall thickness. We take into account long-range dipole interactions and wall-moment twist due to stray fields from magnetic charges on the film surfaces. The small-amplitude wall displacement q(k, z) depends on the position coordinate z normal to the film plane, and on a two dimensional wave vector k parallel to the film plane. Numerically computed natural frequencies v_n(k) depend on the number of nodes n(=0, 1, 2 …) in the dependence of q on z. Surface and bulk modes are distinguished by the z-dependence of computed eigenmodes q_n(k, z). The spectrum of computed natural frequencies compares favorably with available experimental data.     

The hyperfine splittings in heavy-light mesons and quarkonia

Hyperfine splittings (HFS) are calculated within the Field Correlator Method, taking into account relativistic corrections. The HFS in bottomonium and the B _q (q = n, s) mesons are shown to be in full agreement with experiment if a universal coupling α _HF = 0.310 is taken in perturbative spinspin potential. It gives M(B*) −M(B) = 45.7(3) MeV, M(B _s ^* ) − M(B _s ) = 46.7(3) MeV (n _f = 4), while in bottomonium Δ_HF(b $ \bar b $ \bar b ) = M(Υ(9460)) − M(η _b (1S)) = 63.4 MeV for n _f = 4 and 71.1 MeV for n _f = 5 are obtained; just the latter agrees with recent BaBar data. For unobserved excited states we predict M(Υ(2S))−M(η _b (2S)) = 36(2)MeV,M(Υ(3S))−M(η _b (3S)) = 28(2)MeV, and also M(B _c ^*) = 6334(4) MeV, M(B _c (2S)) = 6868(4) MeV, M(B _c ^* (2S)) = 6905(4) MeV. The mass splittings between D(2³ S ₁) − D(2¹ S ₀), D _s (2³ S ₁) − D _s (2¹ S ₀) are predicted to be ∼75 MeV, which are significantly smaller than in several other studies but agree with the mass splitting between recently observed D(2533) and D*(2610).     

Wave scattering by many small particles embedded in a medium

Theory of wave scattering by many small bodies is developed under various assumptions concerning the ratio , where a is the characteristic dimension of a small body and d is the distance between neighboring bodies d=O(aκ₁), 0<κ₁<1. On the boundary Sm of every small body an impedance-type condition is assumed on Sm, 1?m?M, ζm=hma−κ, 0<κ, hm are constants independent of a. The behavior of the field in the region in which M=M(a)?1 small particles are embedded is studied as a→0 and M(a)→∞. Formulas for the refraction coefficient of the limiting medium are derived under the assumptions: (a) κ₁=(2−κ)/3, 0<κ?1, and (b) κ₁=1/3, κ>1. A method for creating materials with a desired refraction coefficient is proposed and justified theoretically on the basis of the above results.