Laplace Sequences of Surfaces in Projective Space and Two-Dimensional Toda Equations

We find that the Laplace sequences of surfaces of period n in projective space P _n–1 have two types, while type II occurs only for even n. The integrability condition of the fundamental equations of these two types have the same form

From the Becker–Döring to the Lifshitz–Slyozov–Wagner Equations

Connections between two classical models of phase transitions, the Becker–Döring (BD) equations and the Lifshitz–Slyozov–Wagner (LSW) equations, are investigated. Homogeneous coefficients are considered and a scaling of the BD equations is introduced in the spirit of the previous works by Penrose and Collet, Goudon, Poupaud and Vasseur. Convergence of the solutions to these rescaled BD equations towards a solution to the LSW equations is shown. For general coefficients an approach in the spirit of numerical analysis allows to approximate the LSW equations by a sequence of BD equations. A new uniqueness result for the BD equations is also provided.     

Letter: A Solution of the Weyl–Lanczos Equations for the Schwarzschild Space-Time

The spin coefficient form of the Weyl–Lanczos equations is analysed for the Schwarzschild space-time. The solution obtained yields an alternative form of Lanczos coefficients to the one currently known for this particular metric.     

Multiplication for solutions of the equation

Linear first-order systems of partial differential equations (PDEs) of the form f=Mg, where M is a constant matrix, are studied on vector spaces over the fields of real and complex numbers. The Cauchy–Riemann equations belong to this class. We introduce on the solution space a bilinear *-multiplication, playing the role of a nonlinear superposition principle, that allows for algebraic construction of new solutions from known solutions. The gradient equation f=Mg is a simple special case of a large class of systems of PDEs, admitting a *-multiplication of solutions. We prove that any gradient equation has the exceptional property that the general analytic solution can be expressed as *-power series of certain simple solutions.     

Trigonometric Solutions of the WDVV Equations from Root Systems

By introduction of an additional variable and addition of a Weyl invariant correction term to the perturbative prepotential in five-dimensional Seiberg-Witten theory we construct solutions of the Witten–Dijkgraaf–Verlinde–Verlinde equations of trigonometric type for all crystallographic root systems.     

Two Hierarchies of New Differential-Difference Equations Related to the Darboux Transformations of the Kaup–Newell Hierarchy

Two hierarchies of new nonlinear differential-difference equations with one continuous variable and one discrete variable are constructed from the Darboux transformations of the Kaup–Newell hierarchy of equations. Their integrable properties such as recursion operator, zero-curvature representations, and bi-Hamiltonian structures are studied. In addition, the hierarchy of equations obtained by Wu and Geng is identified with the hierarchy of two-component modified Volterra lattice equations.     

Generalization of the Knizhnik–Zamolodchikov Equations

In this Letter, we introduce a generalization of the Knizhnik–Zamolodchikov equations from affine Lie algebras to a wide class of conformal field theories (not necessarily rational). The new equations describe correlations functions of primary fields and of a finite number of their descendents. Our proposal is based on Nahm's concept of small spaces which provide adequate substitutes for the lowest energy subspaces in modules of affine Lie algebras. We explain how to construct the first order differential equations and investigate properties of the associated connections, thereby preparing the grounds for an analysis of quantum symmetries. The general considerations are illustrated in examples of Virasoro minimal models.     

q-Deformation of the Krichever–Novikov Algebra

Using q-operator product expansions between U(1) current fields and the corresponding energy-momentum tensors, we furnish the q-analogues of the generalized Heisenberg algebra and the Krichever–Novikov algebra.     

An Asymptotic Expansion for Bloch Functions on Riemann Surfaces of Infinite Genus and Almost Periodicity of the Kadomcev–Petviashvilli Flow

This article describes the solution of the Kadomcev–Petviashvilli equation with C¹⁰ real periodic initial data in terms of an asymptotic expansion of Bloch functions. The Bloch functions are parametrized by the spectral variety of a heat equation (heat curves) with an external potential. The mentioned spectral variety is a Riemann surface of in general infinite genus; the Kadomcev–Petviashvilli flow is represented by a one-parameter-subgroup in the real part of the Jacobi variety of this Riemann surface. It is shown that the KP-I flow with these initial data propagates almost periodically.     

The Becker–Döring Equations and the Lifshitz–Slyozov Theory of Coarsening

In this paper the relation between the kinetic set of Becker–Döring (BD) equations and the classical Lifshitz–Slyozov (LS) theory of coarsening is studied. A model that resembles the LS theory but keeps some of the nucleation effects is derived. For this model a solution is described that shows how the kinetic effects explain the particular solution selected in the LS theory. By means of a renormalization procedure, a discrete group of transformations is shown to play an important role in describing the structure of the solution near the critical size of the LS theory.     

带加速因子的线性方程组通用性迭代解法程序实现

本文用C语言实现了带加速因子的线性方程组通用性迭代解法，并通过若干测试用例对该算法进行了验证。测试结果表明：该算法对任意相容线性代数方程组均收敛，且收敛速度较快。     

Some Comments on a Recently Derived Approximated Solution of the Einstein Equations for a Spinning Body with Negligible Mass

Recently, an approximated solution of the Einstein equations for a rotating body whose mass effects are negligible with respect to the rotational ones has been derived by Tartaglia. At first sight, it seems to be interesting because both external and internal metric tensors have been consistently found, together an appropriate source tensor; moreover, it may suggest possible experimental checks since the conditions of validity of the considered metric are well satisfied at Earth laboratory scales. However, it should be pointed out that reasonable doubts exist if it is physically meaningful because it is not clear if the source tensor related to the adopted metric can be realized by any real extended body. Here we derive the geodesic equations of the metric and analyze the allowed motions in order to disclose possible unphysical features which may help in shedding further light on the real nature of such approximated solution of the Einstein equations.     

Letter: Generation of Solutions of the Einstein Equations by Means of the Kaluza–Klein Formulation

It is shown that starting from a solution of the Einstein–Maxwell equations coupled to a scalar field given by the Kaluza–Klein theory, invariant under a one-parameter group, one can obtain a one-parameter family of solutions of the same equations.     

联立方程组新解法测定小儿退烧片的含量

本文采用联立方程组新解法,不经提取分离,直接测定小儿退烧片中阿斯匹林和扑热息痛的含量。阿斯匹林和扑热息痛的平均回收率和变异系数分别为９９．９５％,１．７４％和１００．１％,０．９３％（ｎ＝１０）,方法简便、快速、准确。     

Existence and Nonlinear Stability of Stationary States of the Schrödinger–Poisson System

We consider the Schrödinger–Poisson system in the repulsive (plasma physics) Coulomb case. Given a stationary state from a certain class we prove its nonlinear stability, using an appropriately defined energy-Casimir functional as Lyapunov function. To obtain such states we start with a given Casimir functional and construct a new functional which is in some sense dual to the corresponding energy-Casimir functional. This dual functional has a unique maximizer which is a stationary state of the Schrödinger–Poisson system and lies in the stability class. The stationary states are parameterized by the equation of state, giving the occupation probabilities of the quantum states as a strictly decreasing function of their energy levels.     

On the infrared behavior of Landau gauge Yang–Mills theory

We discuss the properties of ghost and gluon propagators in the deep infrared momentum region of Landau gauge Yang–Mills theory. Within the framework of Dyson–Schwinger equations and the functional renormalization group we demonstrate that it is only a matter of infrared boundary conditions whether infrared scaling or decoupling occurs. We argue that the second possibility is at odds with global BRST symmetry in the confining phase. For this purpose we improve upon existing truncation schemes in particular with respect to transversality and renormalization.     

Solution calorimetry of organic nonelectrolytes as a tool for investigation of intermolecular interactions

The paper is mainly the review and generalization of the previous publications of the authors. It demonstrates that solution calorimetry method gives the opportunities of more detailed understanding of various aspects of intermolecular interactions in solution. We are assured that prerequisite to such an understanding is the successive analysis of various solute–solvent systems from the simplest ones which include alkanes as a solute or as a solvent to the most complex systems with solvent self‐association via hydrogen bonding. Particular findings discussed in this paper are (i) an inconspicuous contribution of electrostatic solute–solvent interaction to the solvation enthalpy and, accordingly, the dominating contribution of dispersion interactions for nonspecifically solvated solutes; (ii) new, very general method for the extraction of specific interaction enthalpy from the enthalpy of solvation; (iii) new method of determination of self‐association enthalpies for the solvents associated via hydrogen bonding; (iv) new method for determination of cooperative hydrogen bonding enthalpies of proton acceptors with associated species of alcohols; (v) the unique method of experimental determination of the hydrophobic effect enthalpy. Copyright © 2007 John Wiley & Sons, Ltd.     

Single Peak Soliton and Periodic Cusp Wave of the Generalized Schrödinger-Boussinesq Equations

In this paper, we study peakon, cuspon, smooth soliton and periodic cusp wave of the generalized Schrödinger-Boussinesq equations. Based on the method of dynamical systems, the generalized Schrödinger-Boussinesq equations are shown to have new the parametric representations of peakon, cuspon, smooth soliton and periodic cusp wave solutions. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.     

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged, leads to a linear system of equations for the interface configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin’s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system’s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix–vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve.