QCD asymptotics and kinematic thresholds in deep inelastic scattering

The Nachtmann moments of deep inelastic scattering structure functions are required by kinematics to contain a so far neglected threshold factor which is dependent on both n and q². Its presence significantly affects the “moment analysis” in the usual QCD phenomenology and it resolves the difficulties connected with improper threshold behavior of the “ξ-scaling” analysis of structure functions.     

Random grafting statistics in graft distribution analysis

Graft distribution functions have been derived from random grafting statistics. Among the functions, the weight fraction of ungrafted backbone chains, the molecular weight distribution of the ungrafted backbone chains and the GPC apparent molecular weight distribution of the graft copolymer have been found to agree with experimental values determined for a graft copolymer system in which grafting was expected to be random. The other functions, which are not directly measurable, are therefore probably also correct. In analytical work the entire set of graft distribution functions may be computed for a graft copolymer system from the following experimental data: (1) molecular weight distribution of the starting backbone chains; (2) the chemical composition of the mixture of the graft copolymer and ungrafted backbone; (3) the graft side-chain molecular weight distribution, which may be assumed to be identical to that of the ungrafted homopolymer separable from the reaction mixture.     

Determination of long-chain branching in model polystyrene samples by the GPC-sedimentation method

The GPC–sedimentation velocity method proposed earlier for the determination of long-chain branches was examined quantitatively by using model star-shaped and comb-shaped polystyrene samples of known degree of branching. The results showed that separation by sedimentation velocity was more sensitive to branching variations than separation by GPC. The results showed also that at low and moderate levels of branching the GPC–sedimentation velocity method reflected accurately the amounts of branching in the model samples. At high levels of branching, the method underestimated the amount of branching. The discrepancy, however, appears to have been caused by the inadequacy of the theories used in the interpretation of the raw data and not by any inherent problem in the method itself. The relative sensitivities of the GPC-sedimentation, GPC-viscosity, and GPC–light scattering methods are discussed.     

The observation of abnormal signals in laser flash photolysis: a type of probable synchronized nuclear spin signals

In laser flash photolysis (LFP) work, noise-like signals were observed together with transient absorption decay spectra in time domain. Analyzed results show that it is a valuable resonance spectrum of excited state molecules, in which four molecular cases are given here. We proposed that this kind signal might originate from nuclear or nuclear and electron spin resonance features in the excited molecules based on radio wave frequency spectrum levels and the significant interaction with static magnetic field.     

Simulation of harmonic oscillators on the lattice

This work deals with the simulation of a two-dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second-order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions. For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state-of-the-art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination. For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two-dimensional lattice and check its energy conservation.     

Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

Using a geometric flow, we study the following prescribed scalar curvature plus mean curvature problem: Let \((M,g_0)\) be a smooth compact manifold of dimension \(n\ge 3\) with boundary. Given any smooth functions f in M and h on \(\partial M\), does there exist a conformal metric of \(g_0\) such that its scalar curvature equals f and boundary mean curvature equals h? Assume that f and h are negative and the conformal invariant \(Q(M,\partial M)\) is a negative real number, we prove the global existence and convergence of the so-called prescribed scalar curvature plus mean curvature flows. Via a family of such flows together with some additional variational arguments, we prove the existence and uniqueness of positive minimizers of the associated energy functional and give a confirmative answer to the above problem. The same result also can be obtained by sub–super-solution method and subcritical approximations.     

First eigenvalues of geometric operators under the Yamabe flow

Suppose \((M,g_0)\) is a compact Riemannian manifold without boundary of dimension \(n\ge 3\). Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of \(g_0\) with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.     

Max-min of polynomials and exponential diophantine equations

In this paper we study the maximum-minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|?B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.