The evolution of the cluster size distribution in a coagulation system

The coagulation equation with kernelK _ij =A+B(i+j)+C _ij and arbitrary initial conditions is studied analytically and a simple expression for the solution is found. For monodisperse initial conditions, we recover the known size distribution expressed in terms of a degeneracy factorN _k, which is determined by a recursion relation. For polydisperse initial conditions, a similar solution form is found, which includes a degeneracy factorN _kl, also determined by a recursion relation. The physical meaning ofN _kl and the recursion relation is given. A method to get explicit expressions forN _k andN _kl is illustrated. Finally, the pre-gel solution is given explicitly and a general method to get the post-gel solution is proposed.     

Ansätze for self-dual Yang-Mills fields

A sequenceA ₁,A ₂, ... of ansätze for generating self-dual solutions of the Yang-Mills equations is presented. For eachn,A _n produces a solution depending on two arbitrary functions of three variables. As an application, we see thatA ₂ generates a static Yang-Mills-Higgs 2-monopole solution.     

On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity

The Bardeen-Cooper-Schrieffer integral equation with a positive kernel is studied in full generality. It is shown that, there exists a unique finite transition temperature, T _c so that, if T _c ,the equation possesses a positive solution, representing the onset of the superconducting phase, while if T>T _c ,the only solution of the equation is the trivial one, indicating the occurrence of the normal phase. Moreover, it is demonstrated that such a positive solution may be approximated by a sequence of solutions of the equation restricted on bounded domains. This latter result provides a useful computational scheme for the problem.     

AG 2 II algebraically general twistfree vacuum spacetime

Several authors, e.g., Kerr and Debney (1970), Lun (1978), have obtained severalG ₂ II algebraically special vacuum solutions. NoG ₂ II algebraically general vacuum solutions in explicit form have been found before. In this paper, we start from a system of first order partial differential equations, obtained by using a triad formalism, which determines twistfree vacuum metrics with a spacelike Killing vector. The method of group-invariant solutions is then used and aG ₂ II algebraically general twistfree vacuum solution is obtained. The solution also admits a homothetic Killing vector and is non-geodesic. It is believed to be new. The following explicit solutions are also obtained: (1) A Petrov type II with aG ₁-group of motions solution which belongs to Kundt's class. (2) A Petrov type III,G ₃ Robinson-Trautman solution. All these solutions are known.     

Atom in an elliptically polarized resonance field: Exact stationary solution

An analytical method is developed for finding the exact stationary solution for the density matrix of atoms in a monochromatic resonance field of arbitrary ellipticity and intensity. A solution for the transitions F _g=F → F _e=F+1 (F _g and F _e are the angular momenta of the ground and excited states) is obtained in an analytical invariant form. The properties of this solution are investigated. Pis'ma Zh. éksp. Teor. Fiz. 64, No. 1, 8–12 (10 July 1996)     

Transition from pulled to pushed premixed turbulent flames due to countergradient transport

The influence of countergradient transport on the speed of a statistically stationary, planar, 1D premixed flame that propagates in frozen turbulence is studied theoretically and numerically by considering the normalised magnitude N_B of the countergradient flux to be an input parameter. Spectra of admissible flame speeds are analytically determined and explicit travelling wave solutions are found for two algebraic relations widely used to close the mean rate of product creation. A problem of selecting the physically relevant solution that is approached for sufficiently steep initial conditions is addressed. It is argued that, if N_B is larger than an analytically determined critical number N^cr_B, then the type of the physically relevant solution is drastically changed. If N_B < N^cr_B, the physically relevant solution is of pulled wave type, i.e. its speed is controlled by processes localised to the leading edge of the flame brush and can be determined within the framework of a linear analysis at the leading edge. If N_B > N^cr_B, the physically relevant solution is of pushed wave type, i.e. its speed is controlled by processes in the entire flame brush. Analytical expressions for the speed of the physically relevant solution as a function of N_B and the density ratio are obtained. For N_B > N^cr_B, the mean flame brush thickness and the spatial profile of the Favre-averaged combustion progress variable are also determined analytically. These results are validated by numerical simulations. Both analytical expressions and numerical data indicate that (i) both turbulent flame speed and thickness are decreased when N_B is increased and (ii) the direction of total scalar flux (i.e. the sum of countergradient and gradient contributions) is strongly affected not only by N_B, but also by the shape of the dependence of the mean rate of product creation on the mean combustion progress variable.     

Mechanism of NdSr2 ? xCaxMn2O7 (0 ≤ x ≤ 2.0) solid solution formation

The mechanisms of formation of NdSr_{2− x} Ca _x Mn₂O₇ (0 ≤ x ≤ 2.0) solid solution have been studied. The dependence of its volume magnetic susceptibility on the calcium concentration is obtained. The stability of this solid solution is compared with that of similar solid solutions in lanthanum-containing systems.     

SO q (4) quantum mechanics

In this paper we extend our previously discovered exact solution for an SU(2) Yang-Mills-Higgs theory, to the general group SU(N+1). Using the first-order formalism of Bogomolny, an exact, spherically symmetric solution for the gauge and scalar fields is found. This solution is similar to the Schwarzschild solution of general relativity, in that the gauge and scalar fields become infinite on a spherical shell of radiusr ₀=K. However in the Schwarzschild case the singularity at the event horizon is a coordinate singularity while for the present solution the singularity is a true singularity. It is speculated that this solution may give a confinement mechanism for non-Abelian gauge theories, since any particle which carries the SU(N+1) charge would become permanently trapped inside the regionr<r ₀.     

Study of fission track registration efficiency of γ-irradiated polyester track detector in solution medium

The fission track registration efficiency (K _wet) of the Garware polyester track (GPF) detector in solution has been determined with respect to the Lexan detector whose K _wet is very well known in the literature. The K _wet is found to be (1.2 ± 0.1) 10⁻³ cm. The track registration efficiency of the GPF detector in solution is better than the efficiencies of other nuclear track detectors reported in the literature. This implies that the threshold energies for track registration are different for different detectors and that GPF detector has a lower threshold-energy value. The effects of gamma irradiation in the dose range of 5–51 Mrad on the fission track registration efficiency (K _wet) of this track detector from the solution medium are also investigated. The results show that the K _wet values for the gamma-irradiated detectors in the above dose range in 2M HNO₃ solution medium decreases by ∼7–15%. The changes induced in the gamma-irradiated detectors as a function of gamma dose have also been studied by bulk-etch rate measurements.     

q-Painlevé VI Equation Arising from q-UC Hierarchy

We study the q-difference analogue of the sixth Painlevé equation (q-P_VI) by means of tau functions associated with the affine Weyl group of type D₅. We prove that a solution of q-P_VI coincides with a self-similar solution of the q-UC hierarchy. As a consequence, we obtain in particular algebraic solutions of q-P_VI in terms of the universal character which is a generalization of the Schur polynomial attached to a pair of partitions.     

Generating conjecture and some einstein-maxwell fields of high symmetry

For stationary cylindrically symmetric solutions of the Einstein-Maxwell equation we have shown that the “charged” solutions of McCrea, Chitre et al. (CGN), Van den Bergh and Wils (VW) can be obtained from the seed metrics using generating conjecture. The McCrea “charged” solution has as a seed vacuum metric the Van Stockum solution with a Killing vector (0, 0, 1, 0). The CGN “charged” solution and the VW “charged” solution have the static seed metrics connected by the complex substitutiont → iz, z → it and the Killing vector which is a simple linear combination of∂ _ϕ and∂ _t Killing vectors (VW), respectively∂ _ϕ and∂ _z Killing vectors (CGN).     

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,⁽¹⁸⁾ showed a way to relate the measure solution {P _t } _t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P _t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.     

On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour

Hamiltonian perturbations of the simplest hyperbolic equation u _t + a(u) u _x = 0 are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE.     

Algebraic and Geometric Properties of Matrix Solutions of Nonlinear Wave Equations

We improve the construction of exact matrix solutions for nonlinear wave equations by using unitary anti-Hermitian and anticommuting matrices. We prove the theorem that constructs the matrix functions u _n satisfying the nonlinear wave equation for a set of special potentials. In this case, the graph of complex solution u ₁ has a soliton-like form with a finite number of coils. Exponential representation of matrix solutions u _n is associated with continuous rotations that can be used for describing intrinsic rotations and state changes of elementary particles. We also prove the theorem on the decomposition of continuous rotation (described by solution u ₂) onto three simultaneous rotations about coordinate vectors. Each of the three constructed matrix solutions u ₃ is also decomposed into the triplet of elementary matrix solutions.     

3ω法加热/测温膜中温度波解析及其在微/纳米薄膜导热系数测量中的应用

给出了3ω法测试系统中描述薄膜表面加热/测温膜中温度波动的级数形式解，并将复数温度波动的实部和虚部分开表示.利用该解分析了交流加热频率、加热膜宽度和材料热物性的组合参数对加热膜温度波动幅度的影响.并根据此解对测量原理的数学模型进行了修正，建立了相应的3ω测试系统，首先测定了厚度为500 nm SiO₂薄膜的导热系数，验证了实验系统的合理性.加大了测试频率，利用级数模型在高频段直接得到SiO₂薄膜的导热系数，结合低频段的数据同时确定了Si基体的导热系数.利用级数解分析测试了激光晶体Nd:YAG〈111〉面上多层ZrO₂/SiO₂增透膜的导热系数，测试的ZrO₂薄膜的导热系数比体材料小.进行了不确定度分析.结果表明，提出的分析方法可以有效研究微器件表面薄膜结构的导热性能. 关键词： ω法')" href="#">3ω法 微/纳米薄膜 导热系数 微尺度加热膜     

On frequency and temperature dependence of limiting efficiency in direct utilization of quasi-monochromatic radiation

The solution to the problem on limiting (thermodynamically permissible) efficiency η_m of direct conversion of energy of quasi-monochromatic radiation into work known from the literature leads to physically unacceptable results (η_m<0) in the range of small frequencies ν and temperatures T _ν of converted radiation. It is shown that the noted feature is a consequence of the approximate character of the mentioned solution obtained without taking into account background heat radiation: the exact solution obtained for η_m is valid for all ν and T _ν ≥ 0 and turns into the solution known earlier for $\overset{\lower0.5em\hbox{$\overset{\lower0.5em\hbox{ Fiziki, Vol. 70, No. 3, 2000, pp. 34–36.     

A Meinardus Theorem with Multiple Singularities

We solve the quantum version of the A ₁ T-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A ₁ Q-system and generalize it to the fully non-commutative case. We give the relation between the quantum T-system and the quantum lattice Liouville equation, which is the quantized Y-system.     

Heat Kernel Expansions on the Integers

In the case of the heat equation u _t =u _xx +Vu on the real line, there are some remarkable potentials V for which the asymptotic expansion of the fundamental solution becomes a finite sum and gives an exact formula.We show that a similar phenomenon holds when one replaces the real line by the integers. In this case the second derivative is replaced by the second difference operator L ₀. We show if L denotes the result of applying a finite number of Darboux transformations to L ₀ then the fundamental solution of u _t =Lu is given by a finite sum of terms involving the Bessel function I of imaginary argument.