非线性扩散方程和Thompson方程的无穷维Lie代数(英文)

本文研究非线性扩散方程(1.1)和Thompson方程(2.1)的对称结构,提出了一种产生无穷维Lie代数的简单途径:仅从递推算子的基本关系式出发,无须进行Frechet导数的冗长计算就可得到某种“抽象”的仿射Lie代数。通常经由Frechet导数产生的Lie代数只是上述的仿射Lie代数的具体实现。     

一类Hamilton算子,遗传对称及可积系

本文构造了一类矩阵微分Hamilton算子并且生成了新的遗传对称和相应的可积系,进一步提出了一个新可积族的双Hamilton结构和公共遗传强对称算子。     

On the Frame–Stewart algorithm for the multi-peg Tower of Hanoi problem

It is proved that seven different approaches to the multi-peg Tower of Hanoi problem are all equivalent. Among them the classical approaches of Stewart and Frame from 1941 can be found.     

Poloidal–toroidal decomposition in a finite cylinder: II. Discretization, regularization and validation

The Navier–Stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility. Regularity of the solutions is ensured in several ways: First, the potentials are represented using a spectral basis which is analytic at the cylindrical axis. Second, the non-physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile. Third, the nonlinear term is evaluated in such a way as to eliminate singularities. The resulting pseudo-spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated. Our solutions are shown to agree with exact polynomial solutions and with previous calculations of axisymmetric vortex breakdown and of onset of non-axisymmetric helical spirals. Parallelization by azimuthal wavenumber is shown to be highly effective.     

An explicit solution to Post's Problem over the reals

In the BSS model of real number computations we prove a concrete and explicit semi-decidable language to be undecidable yet not reducible from (and thus strictly easier than) the real Halting Language. This solution to Post's Problem over the reals significantly differs from its classical, discrete variant where advanced diagonalization techniques are only known to yield the existence   of such intermediate Turing degrees. Then we strengthen the above result and show as well the existence of an uncountable number of incomparable semi-decidable Turing degrees below the real Halting Problem in the BSS model. Again, our proof will give concrete such problems representing these different degrees. Finally we show the corresponding result for the linear BSS model, that is over (R,+,-,<)$(\mathbb{R},+,-,<)$ rather than (R,+,-,×,÷,<)$(\mathbb{R},+,-,\times ,÷,<)$.     

Factorisation of recursion operators of some Lagrangian systems

We observe that recursion operator of an S-integrable hyperbolic equation that degenerates into a Liouvile-type equation admits a particular factorisation. This observation simplifies the construction of such operators. We use it to find a new quasi-local recursion operator for a triplet of scalar fields. The method is also illustrated with examples of the sinh-Gordon, the Tzitzeica and the Lund-Regge equations.     

Influence of Glycine Adsorption on Segregation of Ni-Cu (110) Surface

An atomic group model of the disordered binary alloy NixCu1-x (x=0.4) was constructed to investigate surface segregation. According to the model, the electronic structure of the NixCu1-x alloy surface was calculated by the Recursion method when glycine atoms are adsorbed on the NixCu1-x (110) surface under the condition of 0.33 coverage. The calculation results indicate that Cu is segregated on the surface of the NixCu1-x alloy, and the chemisorption of glycine restrains the segregation. In addition, the chemical adsorption of glycine greatly changes the density of states of the alloy surface near the Fermi level, and there is electric charge transfer between the alloy surface and the glycine.     

Corrigendum to “The four-dimensional Martínez Alonso–Shabat equation: Differential coverings and recursion operators” [J. Geom. Phys. 85 (2014) 75–80]

We correct an error in Theorem 1 from our article stated in the title. The ensuing results are added as well.     

Tight frame expansions of multiscale reproducing kernels in Sobolev spaces

Multiscale kernels are a new type of positive definite reproducing kernels in Hilbert spaces. They are constructed by a superposition of shifts and scales of a single refinable function and were introduced in the paper of R. Opfer [Multiscale kernels, Adv. Comput. Math. (2004), in press]. By applying standard reconstruction techniques occurring in radial basis function- or machine learning theory, multiscale kernels can be used to reconstruct multivariate functions from scattered data. The multiscale structure of the kernel allows to represent the approximant on several levels of detail or accuracy. In this paper we prove that multiscale kernels are often reproducing kernels in Sobolev spaces. We use this fact to derive error bounds. The set of functions used for the construction of the multiscale kernel will turn out to be a frame in a Sobolev space of certain smoothness. We will establish that the frame coefficients of approximants can be computed explicitly. In our case there is neither a need to compute the inverse of the frame operator nor is there a need to compute inner products in the Sobolev space. Moreover we will prove that a recursion formula between the frame coefficients of different levels holds. We present a bivariate numerical example illustrating the mutiresolution and data compression effect.     

There is no safe pairing function over an arbitrary structure

In [1] the class of safe recursive functions over an arbitrary structure is defined. We prove that in this class, one cannot define a total pairing function independently of the structure. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)