From the neutrino to the edge of the universe

Two recent findings necessitate a closer look at the existing standard models of particle physics and cosmology. These are the discovery of neutrino oscillation, and hence a non-zero mass on the one hand and, on the other, observations of distant supernovae which indicate that contrary to popular belief, the universe would continue to expand for ever, possibly accelerating in the process. In this paper it is pointed out that relatively recent studies which indicate a stochastic, quantum vacuum underpinning and a fractal structure for space time, reconcile both of the recent observations, harmoniously.     

On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix

In this paper, the authors cosider the derivation of the exact distributions of the ratios of the extreme roots to the trace of the Wishart matrix. Also, exact percentage points of these distributions are given and their applications are discussed.     

On the comparison between the APV and the NPV computed via the WACC

The valuation of levered investment in the practice is made with the WACC approach, even if the superior technique of the APV is available. The paper shows that the APV can be interpreted as the arbitrage free value of the portfolio made by an investment and a supporting loan. Therefore the WACC evaluation, generally different from the APV, allows for arbitrage. We provide various conditions which completely characterize the sign of the error as a functions of all the variables entering the model.     

About the distance between the distributions of the sums of the random polynomial forms

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania Translated from Lietuvos Matematikos Rinkinys, Vol. 32, No. 4, pp. 443–453, October–December, 1992.     

用常微分方程模型分析预防和隔离措施对SARS发病率的影响

通过建立常微分方程模型 ,分析了预防和隔离措施对 SARS发病率的影响 ,并把计算结果与实际统计数据进行了比较 ,结果表明 ,及时高效的预防和隔离措施能够有效地控制 SARS的传播 .     

On the dependence of the growth rate on the length of the defining relator

Let { } be a sequence of finitely presented groups with generating setA={a1, …, am}, and letRk be the symmetrized set of words over the alphabetA∪A−1 obtained from the defining words and their inverses by all cyclic shifts. We shall assume that the words inRk are cyclically irreducible, and their lengths tend to ∞ ask increases. In the paper, it is proved that ifRk satisfies the small cancellation conditionC'(1/6) and the number of relators increases not very rapidly with increasingk, then the growth rate ψ(Gk) tends to 2m−1 ask→∞. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 611–617, April, 1999.     

On the relation between the A-polynomial and the Jones polynomial

This paper shows that the noncommutative generalization of the A-polynomial of a knot, defined using Kauffman bracket skein modules, together with finitely many colored Jones polynomials, determines the remaining colored Jones polynomials of the knot. It also shows that under certain conditions, satisfied for example by the unknot and the trefoil knot, the noncommutative generalization of the A-polynomial determines all colored Jones polynomials of the knot.

     

On choosing the location of the sources in the MFS

The satisfactory location for the sources outside the closure of the domain of the problem under consideration remains one of the major issues in the application of the method of fundamental solutions (MFS). In this work we investigate this issue by means of two algorithms, one based on the satisfaction of the boundary conditions and one based on the leave-one-out cross validation algorithm. By applying these algorithms to several numerical examples for the Laplace and biharmonic equations in a variety of geometries in two and three dimensions, we obtain locations of the sources which lead to highly accurate results, at a relatively low cost.     

On the summation of the Laguerre series by the Euler-Knopp method in the problem of inverting the Laplace transform

A method for inverting the Laplace transform based on expanding the original in Laguerre polynomials as $ f(t) = \sum\limits_{k = 0}^\infty {a_k L_k (bt)} $ is suggested. The representation of the Laguerre series by a linear-fractional mapping is reduced to a power series of the form $ \sum\nolimits_{k = 0}^\infty {a_k z^k } A method for inverting the Laplace transform based on expanding the original in Laguerre polynomials as is suggested. The representation of the Laguerre series by a linear-fractional mapping is reduced to a power series of the form , which is summed by the well-known Euler-Knopp method. The summation parameter is chosen in the complex plane so that the new expansion of the original corresponding to the Euler-Knopp transformation converge at a maximum rate. On the basis of geometric representations, the influence of the requirement that the Euler-Knopp transformation must be regular on the choice of the summation parameter is discussed. Numerical experiments are performed, which demonstrate the high efficiency of the method of choosing a complex parameter suggested in this paper. Original Russian Text ? M.M. Kabardov, 2008, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2008, No. 4, pp. 84–89.     

宏观因素影响下的系统中元件重要性研究

为研究复杂系统在工作环境中其组成元件对系统安全运行的重要性,将汪培庄先生的因素空间理论与笔者提出的空间事故树理论相结合,构造了一套元件重要性研究方法.构建系统T={U,C,D},将元件作为研究对象集合U,系统工作的宏观环境作为因素集C,元件重要性排序集作为D.对宏观环境中的工作时间a1和温度a_2进行划分形成不同的状态区域S_q,计算在S_q中元件xj的失效权重γ(AS_q(x_j))和在S_q中系统T的失效权重δ(AS_q(T))),从而得到x_j在S状态下的等效失效权重Z(AS_q(x_j)),研究状态S_q下的原件重要性排序D_η,及元件x_j失效性对a_1及a_2的敏感性.使用一个实际的电气系统维修情况统计资料,使用上述方法进行了研究,结果表明:不同工作环境下元件对系统的重要程度是不同的.元件对温度和使用时间是敏感的,并得到了在10³0°且5030°且50⁷5d环境下工作系统可靠性是最高的结论.在给定工作环境下,重要性大的元件多储备,重要性小的元件少储备,以满足系统维修需要,并指导实际工程.     

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α?p/q|<q ^?μ has infinitely many solutions (p,q)∈?×?. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n?1. We then use this result to prove that for b?2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.     

On the Aleksandrov-Fenchel inequality and the stability of the sphere

LetF be a closed convex hypersurface in Euclideand-space with almost constantq-th mean curvatureH _q (q=1, ...,d–1). The deviation ofF from a suitable sphere is estimated explicitely in terms of geometric quantities ofF. The proof depends on a new stability result on the Aleksandrov-Fenchel inequality, which improves a theorem of Schneider.     

一类树并的补图的色唯一性

彻底解决了一类不可约树并的补图是色唯一的 ,并得到了一些图的伴随多项式的最小根的重要规律 .     

Hydroelastic phenomena in the neighborhood of the bifurcation of the aorta

The mechanical behavior of the vessels and the blood near a bifurcation is analyzed. A single-layer homogeneous shell is taken as a model of the aorta on the assumption that the intima is much less stiff than the other layers. In analyzing the blood flow in large vessels, the blood is treated as a viscous Newtonian liquid, whose motion is described by the Navier-Stokes equations and the continuity equation.Zhukovskii Air Force Engineering Academy, Moscow; Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 164–166, January–February, 1971.     

HIV-1的表型间变异与免疫因子相互作用的动力学模型

该文建立了一个描述两种不同的 HIV-1 表型与细胞因子相互作用的动力学模型. 作者用K_m单调系统理论研究了 HIV-1 中两种不同表型:噬巨嗜细胞型 (NSI) 和嗜淋巴细胞型 (SI) 与两种在HIV感染过程中的重要指标性细胞因子:IL-2 和 CAF的发展趋势. 在 HIV-1 感染过程中,两种细胞表型与两种细胞因子之间形成了一种负反馈环. 用 Hill 函数表达这种负反馈作用. 结果表明模型的平衡态的数量为奇数个,它们之间满足一种K_m 偏序,并且第奇数个平衡态是渐近稳定的,而第偶数平衡态是不稳定的. 此外还得到了各个平衡态的吸引域. 其生物学的意义为:即使系统存在较低水平的平衡态, 感染后的病毒载量仍会趋向于一个较高的水平. 这个结论和临床研究的发现是一致的.     

On the Decomposition of the Riesz Operator and the Expansion of the Riesz Semigroup

For a Riesz operator T on a reflexive Banach space X with nonzero eigenvalues denote by E(λ_i; T) the eigen-projection corresponding to an eigenvalue λ_i. In this paper we will show that if the operator sequence is uniformly bounded, then the Riesz operator T can be decomposed into the sum of two operators T_p and T_r: T = T_p + T_r, where T_p is the weak limit of T_n and T_r is quasi-nilpotent. The result is used to obtain an expansion of a Riesz semigroup T(t) for t ≥ τ. As an application, we consider the solution of transport equation on a bounded convex body.     

On the Ramsey number of the triangle and the cube

The Ramsey number r(K ₃,Q _n ) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K _N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erd?s conjectured that r(K ₃,Q _n )=2 ⁿ⁺¹?1 for every n∈?, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K ₃,Q _n )?7000·2 ⁿ . Here we show that r(K ₃,Q _n )=(1+o(1))2 ⁿ⁺¹ as n→∞.     

On the algorithm of the solution of the signorini problem

An algorithm is proposed for solving the Signorini problem /1/ in the formulation of a unilateral variational problem for the boundary functional in the zone of possible contact /2/. The algorithm is based on a dual formulation of Lagrange maximin problems for whose solution a decomposition approach is used in the following sense: a Ritz process in the basis functions that satisfy the linear constraint of the problem, the differential equation in the domain, is used in solving the minimum problem (with fixed Lagrange multipliers); the maximum problem is solved by the method of descent (a generalization of the Frank-Wolf method) under convexity constraints on the Lagrange multipliers. The algorithm constructed can be conisidered as a modification of the well-known algorithm to find the Udzawa-Arrow-Hurwitz saddle points /3, 4/. The convergence of the algorithm is investigated. A numerical analysis of the algorithm is performed in the example of a classical contact problem about the insertion of a stamp in an elastic half-plane under approximation of the contact boundary by isoparametric boundary elements. The comparative efficiency of the algorithm is associated with the reduction in the dimensionality of the boundary value problem being solved and the possibility of utilizing the calculation apparatus of the method of boundary elements to realize the solution.     

Interpreting the Action of the Endomorphism Monoid of the Rationals

Algebra and Logic - In a first order language we interpret the action of the monoid M of embeddings of (?,&lt;) on the set ? inside (M,°). A similar result is proved for the...     

Hume, the Occult, and the Substance of the School

I have not been able to locate any critique of Hume on substance by a Schoolman, at least in English, dating from Hume's period or shortly thereafter. I have, therefore, constructed my own critique as an exercise in ??post facto history??. This is what a late eighteenth-century/early nineteenth-century Scholastic could, would, and should have said in response to Hume's attack on substance should they have been minded to do so. That no one did is somewhat mysterious. My critique is precisely in the language of the period, using solely the conceptual resources available to a Schoolman at that time. The arguments, however, are as sound now as they were then, and in this sense the paper performs a dual role??contributing to the defence of substance contra Hume, and filling, albeit two hundred?years or so too late, a gap in the historical record.