Hamilton算子和R3中相似运动

四元数是一个可除环·　它可表达为R3中通过原点的平面中的四元数乘积的域·　利用Hamilton算子,定义了在该平面上的相似运动,并讨论了这些运动的新特性·     

The Generalized Holditch Theorem for the Homothetic Motions on the Planar Kinematics

W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let E/E be a 1-parameter closed planar Euclidean motion with the rotation number and the period T. Under the motion E/E, let two points A = (0, 0), B = (a + b, 0) E trace the curves k _A, k _B E and let F _A, F _B be their orbit areas, respectively. If F _X is the orbit area of the orbit curve k of the point X = (a, 0) which is collinear with points A and B then

平面运动学位似运动Steiner公式的推广

对单参数闭平面复意义位似运动，给出了Muller提出的Steiner公式和混合面公式的表达式。利用推广的Steiner公式得到了位似运动的Holditch定理．作为特例，给出了Muller位似比h≡1时的结果。     

Convex Bodies with Homothetic Sections

We prove that if K is a convex body in Eⁿ⁺¹, n2, and p₀ is apoint of K with the property that all n-sections of K throughp₀ are homothetic, then K is a Euclidean ball.     

A New Expression for Higher Order Accelerations and Poles Under the One Parameter Planar Hyperbolic Homothetic Motions

The one parameter planar hyperbolic homothetic motion was introduced in Ersoy and Akyigit (Adv Appl Clifford Algebras 21:297–313, 2011). We give a formula for higher order accelerations and poles under this motion. In the case of the homothetic rate \({h\equiv 1}\) we obtain the higher order accelerations and poles under one parameter planar hyperbolic motion which was given by Sahin and Yüce (Math Probl Eng 2014, 2014). Also, the higher order velocities and accelerations are analyzed by taking the angle of the rotation instead of the parameter of the motion.     

Covering a Triangle with Positive and Negative Homothetic Copies

Let be a triangle and let be a set of homothetic copies of . We prove that implies that there are positive and negative signs and there exist translates of that cover .     

Covering a Triangle with Sequences of its Homothetic Copies

Every triangle with unit area can be translatively covered with any sequence of its homothetic copies with total area greater than or equal to 4.     

One-Parameter Homothetic Motions and Euler-Savary Formula in Generalized Complex Number Plane $${\mathbb{C}_{J}}$$

In this paper, we introduce one-parameter homothetic motions in the generalized complex number plane (\({\mathfrak{p}}\)-complex plane)

$$\mathbb{C}_{J}=\left\{x+Jy:\,\,\, x,y \in \mathbb{R},\quad J^2=\mathfrak{p},\quad \mathfrak{p} \in \{-1,0,1\} \right\} \subset \mathbb{C}_\mathfrak{p}$$

where

$$\mathbb{C}_\mathfrak{p}=\{x+Jy:\,\,\, x,y \in \mathbb{R}, \quad J^2=\mathfrak{p}\}$$

such that \({-\infty < \mathfrak{p} < \infty}\). The velocities, accelerations and pole points of the motion are analysed. Moreover, three generalized complex number planes, of which two are moving and the other one is fixed, are considered and a canonical relative system for one-parameter planar homothetic motion in \({\mathbb{C}_{J}}\) is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, during the one-parameter homothetic motions, is obtained with the aim of this canonical relative system.

     

Non-Jumping Numbers for 4-Uniform Hypergraphs

Let r≥2 be an integer. A real number α ∈ [0,1) is a jump for r if for any Open image in new window >0 and any integer m, m≥r, any r-uniform graph with n>n₀( Open image in new window ,m) vertices and at least Open image in new window edges contains a subgraph with m vertices and at least Open image in new window edges, where c=c(α) does not depend on Open image in new window and m. It follows from a theorem of Erd?s, Stone and Simonovits that every α ∈ [0,1) is a jump for r=2. Erd?s asked whether the same is true for r≥3. Frankl and Rödl gave a negative answer by showing that Open image in new window is not a jump for r if r≥3 and l>2r. Following a similar approach, we give several sequences of non-jumping numbers generalizing the above result for r=4.     

The Value of Bernoulli Polynomials at Rational Numbers

We give a short elementary proof of a result of Almkvist andMeurman [1] on an integrality property of the values taken bythe Bernoulli polynomials at a rational number. We use a lemmaon the divisibility properties of certain binomial coefficientswhich seems to be of independent interest.     

Boundary Harnack Principle and Martin Boundary at Infinity for Subordinate Brownian Motions

In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of κ-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.     

Barotropic Instability of the Bickley Jet at High Reynolds Numbers

We investigate the linear stability of the Bickley jet in the framework of the beta-plane approximation. Because singular inviscid neutral modes exist in the retrograde case     , it is necessary to add viscosity to interpret them. One of these modes was found in closed form by Howard and Drazin [1] . However, its critical point is at the center of the jet and it was therefore not possible for these authors to ascertain the relationship of this mode to the stability problem or to discuss how to continue the eigenfunction across the singularity.
The viscous critical layer problem associated with this singularity is considerably more difficult than the usual one (which leads to integrals of the Airy function) because     and, consequently, a second-order turning point is involved. Our analysis shows that the Howard–Drazin mode is degenerate in the domain where it is valid as a limit of the viscous problem (wavenumber  α²≤ 9/2  ), that is, it corresponds to both an odd and an even mode. This conclusion is confirmed by direct numerical solution of the Orr–Sommerfeld equation which shows, in addition, that viscosity is destabilizing along portions of the stability boundary. For a retrograde jet, instability is found to occur beyond the inviscid critical value of β, that is, in the region where the flow would be stable according to the Rayleigh–Kuo condition.     

Stability of Preconditioned Finite Volume Schemes at Low Mach Numbers

A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments that the preconditioned scheme combined with an explicit time integrator is unstable if the time step Δt does not satisfy the requirement to be as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to , M → 0, though producing unphysical results. A comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis is presented, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M^–2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. The theoretical results are afterwards confirmed by numerical experiments. AMS subject classification (2000) 35L65, 35C20, 76G25