Counterexample to a problem of Geoghegan-West

Let be a Banach space and its general linear group. Let denote the operator norm and ``' the pointwise convergence topology on . Is the identity map a homotopy equivalence? The answer is negative. One of the possible counterexamples is a well-known James space ---the ``space of counterexamples in Banach spaces theory'.

     

Simultaneous Pell Equations

Let and be positive integers with . We shall call the simultaneous Diophantine equations

simultaneous Pell equations in and . Each such pair has the trivial solution but some pairs have nontrivial solutions too. For example, if and , then is a solution. Using theorems due to Baker, Davenport, and Waldschmidt, it is possible to show that the number of solutions is always finite, and it is possible to give a complete list of them. In this paper we report on the solutions when .

     

On the Finite Screw Cylindroid Represented as a 2-system of Screws

The problem of displacing a line with a definite point on it from one spatial position to another is studied by utilizing the concept of screw matrix. It is known that all the available finite twists (screws) associated with this displacement form a ruled surface, the so-called finite screw cylindroid. If the definition of the pitch given by Parkin is used, then the finite screw cylindroid can be regarded as a 2-system of screws. This brings to one's mind the question as to whether there exist different appropriate measures for pitch other than Parkin's under which all the available finite twists form a 2-system. This question is answered in this paper. By deriving a general expression of the pitch for these available finite twists under the said condition, it is shown that Parkin's pitch plus an arbitrary constant is the only possible measure of pitch under which the finite screw cylindroid represents a 2-system of screws. However, since adding a constant to the pitches of all screws of any 2-system still gives a 2-system, constant term may be omitted. It is also shown that the determined 2-system of screws can be described as a linear combination of two special basis screws which are called in this paper the = 0 and the = screws.     

Tiling Proofs of Recent Sum Identities Involving Pell Numbers

In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of a number of the identities. Received July 20, 2006     

Eddy diffusivity in convective hydromagnetic systems

An eigenvalue equation, for linear instability modes involving large scales in a convective hydromagnetic system, is derived in the framework of multiscale analysis. We consider a horizontal layer with electrically conducting boundaries, kept at fixed temperatures and with free surface boundary conditions for the velocity field; periodicity in horizontal directions is assumed. The steady states must be stable to short (fast) scale perturbations and possess symmetry about the vertical axis, allowing instabilities involving large (slow) scales to develop. We expand the modes and their growth rates in power series in the scale separation parameter and obtain a hierarchy of equations, which are solved numerically. Second order solvability condition yields a closed equation for the leading terms of the asymptotic expansions and respective growth rate, whose origin is in the (combined) eddy diffusivity phenomenon. For about 10% of randomly generated steady convective hydromagnetic regimes, negative eddy diffusivity is found.     

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

Rayleigh functions σl(ν) are defined as series in inverse powers of the Bessel function zeros λν,n≠0,

     

Riordan 矩阵的$(m,r,s)$-Halves

Riordan矩阵的垂直一半和水平一半已经被许多学者分别研究过.本文给出了Riordan矩阵的$(m,r,s)$-halves的定义.利用此定义能够统一的讨论Riordan矩阵的垂直一半和水平一半.作为应用,通过对Pascal和Delannoy矩阵的$(m,r,s)$-halves的研究,可以得到了一些与Fibonacci, Pell和Jacobsthal序列相关的等式.     

Exact solutions for the massless plane symmetric scalar field in general relativity,with cosmological constant

The Klein–Gordon equations are solved for the case of a plane-symmetric static massless scalar field in general relativity with cosmological constant, generalizing the solutions found by Taub, Novotny and Horsky, and Singh. A separate class of solutions is obtained in which the metrics reduce to flat space in the limit that .The static solutions can be used to generate time-dependent cosmological solutions, one of which exhibits rapid inflation followed by continued exponential expansion at all later times.