Coefficient Of Ln Research Articles

On the fixed points of the map x↦xx modulo a prime, II

We study number theoretic properties of the map x↦xx(modp), where x∈{1,2,…,p−1}, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes p<N for which the map only has the trivial fixed point x=1. A key technical result, possibly of independent interest, is the existence of subsets Nq⊂{2,3,…,q−1} such that almost all k-tuples of distinct integers n1,n2,…,nk∈Nq are multiplicatively independent (if k is not too large), and |Nq|=q⋅(1+o(1)) as q→∞. For q a large prime, this is used to show that the number of solutions to a certain large and sparse system of Fq-linear forms {Ln}n=2q−1 “behaves randomly” in the sense that |{v∈Fqd:Ln(v)=1,n=2,3,…,q−1}|∼qd(1−1/q)q∼qd/e. (Here d=π(q−1) and the coefficients of Ln are given by the exponents in the prime power factorisation of n.)

Integral estimates for derivatives of univalent functions

Brennan’s conjecture for a conformal mapping f of the unit disk is proved under a certain condition on the Taylor coefficients of ln f′. Brennan’s conjecture is also proved in the case where 1/f′ expands into a series of simple fractions absolutely converging to zero. An estimate for the approximation of 1/f′ by simple fractions is obtained.

Implementation of Radiation, Ablation, and Free Energy Minimization in Hypersonic Simulations

Aj–Ej = curve fit constants for Kj T to define equilibrium relation for nonbase species j ~ ci;abl = elemental mass fraction of element i in ablation products Dj = effective diffusion coefficient for species j, LrefV1 F = transformation matrix from species continuity to elemental continuity equations Fi;j = element of F defined in Eq. (1) f = species flux vector frad = fraction of divergence of radiative flux introduced into flow solver, Eq. (19) Gj;k = coefficient of ln k in partial equilibrium relation for nonbase species j [Eq. (10)] Hn = enthalpy of species n, J=kmole Kj T = equilibrium constant for reaction j, Eq. (11) Lref = reference length used to nondimensionalize distance, m ~ Mi = molecular weight of element i, kg=kmole Mj = molecular weight of species j, kg=kmole _ m = blowing rate of ablation products per unit area, 1V1 Ne = number of elements Nexchange = frequency for updating ghost cells Nrad = frequency for updating divergence of radiative flux in interior Ns = number of species pn = partial pressure of species n, atm p = pressure, 1V 2 1 q = vector of dependent variables, species densities, 1 q = heating rate,W=m R = universal gas constant, 8314.3, J=K kmole R = modified gas constant Sn = entropy of species n, J=K kmole T = temperature, K t = time, Lref=V1 _ w = vector of chemical source terms, 1V1=Lref x, y = distance in Cartesian coordinates, Lref u, v = velocities in x and y directions, respectively, V1 ~ Vi = diffusion velocity of element i, V1 Vj = diffusion velocity of species j, V1 V1 = reference velocity in the freestream, m=s Z = 10; 000=T used in Eq. (11) i;j = number of atoms of element i in species j m;j = stoichiometric coefficient of reactant m in nonbase species relation j n;j = stoichiometric coefficient of product n in nonbase species relation j F j = change in free energy in Eqs. (7) and (8), J=kmole = surface emissivity = damping coefficient on surface temperature updates ~ i = density of element i, 1 j = density of species j, 1 1 = reference density in the freestream, kg=m 3 = Stefan-Boltzmann constant, 5:66961 10 8 W= m K j = mole fraction of species j

Strong-coupling expansion of cusp anomaly from quantum superstring

We consider the world surface in AdS5 that ends on two intersecting null lines at the boundary. The corresponding superstring partition function describes the expectation value of the Wilson line with a null cusp in dual large N maximally supersymmetric gauge theory and thus determines the cusp anomaly function f(l) of the gauge coupling l or the string tension √λ/2π. The first two coefficients in its strong-coupling or string inverse tension expansion were determined in hep-th/0210115 (a0 = 1) and in arXiv:0707.4254 (a1 = − 3 ln 2). Here we find that the 2-loop coefficient is a2 = −K where K is the Catalan's constant. This is in agreement (expected on the general grounds) with the previous results for f(l) as the coefficient of ln S term in the energy of the closed spinning string in AdS5. The string theory value for a2 is in agreement with the numerical result in hep-th/0611135 and the recent analytic result in arXiv:0708.3933 for the coefficients in strong-coupling Ssolution of the BES equation. We explicitly verify the cancellation of all 2-loop logarithmic divergences thus demonstrating the quantum consistency of the AdS5×S5 superstring action at this order. We also discuss the structure of the three and higher string loop corrections to the cusp anomaly function giving a 2d QFT diagrammatic interpretation to the result of arXiv:0708.3933 for the solution of the BES equation following from the Bethe ansatz prescription for the spectrum of the theory.

Electron weak localization in disordered films

The logarithmic temperature dependence of resistivity, commonly observed in disordered films, has generally been interpreted as evidence for electron weak localization, with its slope indicative of the inelastic scattering mechanism. In this work, we show that the 2D quantum percolation (QP) model, pertaining to disordered metallic films, predicts a sample-size dependent ln L conductance correction that is three times larger than that for the Anderson model. Moreover, when the film has a finite thickness, the coefficient of ln L decreases to about 2 3 of its 2D value for both the QP and the Anderson models. Since for disordered metallic films the QP model is more realistic than the Anderson model, which pertains to doped metallic films, it follows that many prior experimental results on metallic films have to be re-interpreted in regard to their implications about the inelastic scattering mechanism(s). These results have direct implications for the interpretation of experimental data.

Money and Money Substitutes: Reply

The central question of our earlier paper [2] is whether there are or are not significant differences in portfolio substitutability between money and each of several interest-bearing assets. Our tentative conclusion is that there are not. Professor Brox [1] notes that the wealth maximand we proposed is inconsistent with the sign of the interest-rate variable in our regression equations. Specifically, he shows that the sign of the coefficient of ln [1/(1 + ri(t))] in our equation (6) is positive rather than negative, when (6) is properly derived using our equations (1) and (3). In order to deduce our original equation (6) with a negative sign for ln [1/(1 + ri(t))], the wealth maximand must be written as equation (1) in Brox's comment, rather than in the way we originally proposed (see Brox's equation (2)). Brox has correctly pinpointed an algebraic slip; but neither the analysis nor our original conclusions should thereby be affected. For as we emphasized in the earlier analysis [2, p. 187], the coefficient of ln [1/(1 + rit))] must be negative to yield negatively sloped asset demand curves. It bears emphasizing that the duality principle stated in our paper remains applicable, with a minor empirical modification. Whereas Chetty assumed utility maximization subject to a budget constraint [given by Brox's equation (1)], we adopted the hypothesis of wealth maximization subject to a transactions technology constraint. With the wealth maximand suitably defined, these alternative hypotheses yield equivalent derived demand equations, except that Xi(t) in our model should correctly be dated (t + 1). This change should make practically no difference in the estimates obtained when the model is applied to quarterly data. It seems that a more substantive issue is the structure of the derived asset demand equations. Wealth-maximizing behavior requires that the term ln [1/(1 + ri(t))] bear a negative sign in equation (6). This term is inversely related to the opportunity cost of holding money. Clearly, an increase in the opportunity cost of holding current assets as money, given by an increase in ri(t) must induce a portfolio substitution of Xi for money. The original interpretation of our statistical findings is grounded in this reasoning, and it is not a terec 3y Brox's comment.

Universality of the rate of growth of all hadron-nucleon total cross sections in the range 10 to 1500 GeV/c

The energy dependence of total cross section of hadron-nucleon collisions has been studied in terms of two components: the first component decreases with increasing energy of the incident hadron whereas the second component increases like ln s. The coefficient of ln s is found to be the same within 5% for p ±p, π ±p and K ±p collisions. The rate of growth of the rising component i.e., d σ /d(ln, s), of inelastic cross sections, deduced from σ tot and σ el, of pp and πp are found to be the same within 1.5 standard deviations; an attempt has been made to understand this near equality from factorization of the pomeron.

Kondo‐Type Resistivity Anomaly in Noble Metal Alloys with 3d Metals

AbstractThe electrical resistivities of the f.c.c. Au‐V, Au‐Mn, Au‐Fe, and Au‐Ni alloys with high concentration of 3d‐element impurities have been measured from 4 to 1000 K. The results were analyzed together with other data for Au‐Cr, Au‐Co, Ag‐Mn, Cu‐Mn, and Cu‐Ni alloys. At high temperatures, the decrease of magnetic resistivity with increasing temperature is observed for Au‐V, Au‐Cr, Au‐Mn, Au‐Fe, Au‐Co, Ag‐Mn, and Cu‐Mn alloys and is expressed empirically by the equation Rmag(T) = ‐ c (Y + Z c) ln (T2 + + X2 c2)1/2 + c (R1 + R2 c), except for Au‐Co alloys, where c is the concentration of 3d element, T the temperature, and X, Y, Z, R1, and R2 are constants. Yc coincides with the coefficient of ln T in each dilute alloy. R1, c coincides with the temperature independent term in each dilute alloy. Xc is nearly of the same order as the RKKY interaction strength. Au‐Ni and Cu‐Ni alloys do not show such a decrease. The maximum of magnetic resistivity observed for Au‐Cr, Au‐Mn, Au‐Fe, Ag‐Mn, and Cu‐Mn alloys is explained by the destruction of very short range ordering, such as pairs and triplets, remaining far above the Néel temperature.

Multiplicities in lepton-hadron processes

The average multiplicity in deep inelastic electro- and neutrinoproduction at large ω( ω∼ s/ Q 2 + 1) is related in Feynman's version of the parton model to the average multiplicities in high-energy electron-positron annihilation and hadron-hadron scattering. The relation is: 〈n(s, Q 2)〉 eP,ν P ∼ C e + e − ln( Q 2 M 1⊥ 2 ) + C h ln(ω − 1) , where C e + e − and C h are, respectively, the coefficients of ln( s/ M 1⊥ 2) in the multiplicities from e +-e − and P-P in to hadrons, and M 1⊥ is an average transverse mass.

The bjorken-johnson-low limit in perturbation theory

The Bjorken-Johnson-Low limit is analysed in perturbation theory with currents constructed from fermion fields with a neutral gluon interaction. Diagrams with many loops can be treated. For those which are not divergent it is shown that the limit is violated in well-defined circumstances by terms involving powers of logarithms. The maximum power of ln (−q02), and its numerical coefficient, are directly related to the degree of divergence, and its coefficient, of an associated vertex part. The terms of orderq0−1 are shown to correspond to renormalized vertex parts with the vertex given by the canonical commutator together with a series of dynamical corrections to it. Some simple diagrams requiring renormalization are also analysed. Coefficients of ln (−q02) can again be related to those of divergencies in self-energy and vertex parts except that there is now a minus sign multiplying them.