Generalizations of some conjectures of Sun on the relations between N(a, b, c, d; n) and T(a, b, c, d; n)

The Ramanujan Journal - We derive some new finite sums involving the sequence $$s_{2}\left( n\right) ,$$ the sum-of-digits of the expansion of n in base 2. These functions allow us to generalize...     

Generalizations of theorems of J. Aczél and R. Ger and T. Kochanek

Periodica Mathematica Hungarica - We present a result which generalizes a theorem proved by Aczél (Bull Am Math Soc 54:392–400, 1948). Hence we obtain generalizations of a theorem proved...     

Molecular dynamics of thermal vibration effects: Ar + Ni(1 0 0) collision system

In this work constant energy molecular dynamics simulations are achieved for investigating sputtering process of Ar + Ni(1 0 0) collision system. The Ni crystal is imitated by an embedded-atom potential, whereas the interaction between the projectile and the surface is modeled by re-parameterized ZBL potential. Seven hundred eighty-four Ar atoms carrying 20–50 eV are sent towards a Ni(1 0 0) surface at normal incidence. Each projectile meets with different initial coordinates (phase) of the nickel atoms because of thermal vibrations in the slab. Effects of the different initial phases of the surface are compared with the available theoretical and experimental data.     

On generating (2 + 1)-dimensional hierarchies of evolution equations

Two isospectral problems are constructed with the help of a 6-dimensional Lie algebra. By using the Tu scheme, a (1 + 1)-dimensional expanding integrable couplings of the KdV hierarchy is obtained and the corresponding Hamiltonian structure is established. In addition, the 2-order matrix operators proposed by Tuguizhang are extended to the case where some 4-order matrices are given. Based on the extension, a new hierarchy of 2 + 1 dimensions is obtained by the Hamiltonian operator of the above (1 + 1)-dimensional case and the TAH scheme. The new hierarchy of 2 + 1 dimensions can be reduced to a coupled (2 + 1)-dimensional nonlinear equation and furthermore it can be reduced to the (2 + 1)-dimensional KdV equation which has important physics applications. The Hamiltonian structure for the (2 + 1)-dimensional hierarchy is derived with the aid of an extended trace identity. To the best of our knowledge, generating the (2 + 1)-dimensional equation hierarchies by virtue of the TAH scheme has not been studied in detail except to previous little work by Tu et al.     

Optimal solutions of a (3 + 1)-dimensional B-Kadomtsev-Petviashvii equation

We explored and specialized new Lie infinitesimals for the (3 + 1)-dimensional B-Kadomtsev-Petviashvii (BKP) using the commutation product, which results a system of nonlinear ODEs manually solved. Through two stages of Lie symmetry reduction, (3 + 1)-dimensional BKP equation is reduced to nonsolvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the integration and Riccati equation methods, we investigate new analytical solutions for these ODEs. Back substituting to the original variables generates new solutions for BKP. Some selected solutions illustrated through three-dimensional plots.     

Tridiagonal pairs of shape (1, 2, 1)

Let denote a field and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A^*:V→V that satisfies the following conditions: (i) each of A,A^* is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^*V_iV_i-1+V_i+V_i+1 for 0id, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^* such that for 0iδ, where and ; (iv) there is no subspace W of V such that AWW, A^*WW, W≠0,W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and that for 0id the dimensions of coincide; we denote this common value by ρ_i. The sequence is called the shape of the pair. In this paper we assume the shape is (1,2,1) and obtain the following results. We describe six bases for V; one diagonalizes A, another diagonalizes A^*, and the other four underlie the split decompositions for A,A^*. We give the action of A and A^* on each basis. For each ordered pair of bases among the six, we give the transition matrix. At the end we classify the tridiagonal pairs of shape (1,2,1) in terms of a sequence of scalars called the parameter array.     

Nonexistence results for classes of 3 × 3 elliptic systems

We consider the system

     

Classification of qt=P(x, t, q, qx, qxx)

The classification of q_t = P ( x ,  t ,  q ,  q_x ,  q_xx ) is given by using linearization. Two types of integrable equations which are generalizations of the integrable equations of Fokas and Svinolupov are given. Results are compared with Fokas's symmetry and the Mikhailov–Shabat–Sokolov formal symmetry approaches.     

Exhaustive determination of (1023, 511, 255)-cyclic difference sets

An exhaustive search for (1023, 511, 255)-cyclic difference sets has been conducted. A total of 10 non-equivalent (1023, 511, 255)-cyclic difference sets have been found, all of which are members of previously known or conjectured infinite families. A fast and effective autocorrelation test method was utilized that can also facilitate the testing of longer sequences.

     

d- Dimensional Dual Hyperovals in PG(d + n, 2) for d + 1 ≤ n ≤ 3d − 7

Assume that d ≥  4. Then there exists a d -dimensional dual hyperoval in PG(d +  n, 2) for d +  1  ≤  n ≤  3 d −  7.     

关于极大强单调算子的不精确邻近点算法的收敛性分析

该文研究集值映象方程0∈T(z)的解的迭代逼近，其中T是极大强单调算子．设{x^k}与{e^k}是由不精确邻近点算法x^{k+1}+c_kT(x^{k+1})> x^k+e^{k+1}生成的序列，满足‖e^{k+1}‖≤η_k‖x^{k+1}_x^k‖, ∑^∞_{k=0}(η_k-1)<+∞且inf_(k≥0) η_k=μ≥1.在适当的限制下证明了，{x^k}收敛到T的一个根当且仅当 lim inf_{k→+∞} d(x^k,Z)=0，其中Z是方程0∈T(z)的解集     

Chaotic behaviors in the (2 + 1)-dimensional breaking soliton system

Starting from the extended tanh-function method based on mapping method, the variable separation solutions of the (2 + 1)-dimensional breaking soliton system are derived. By further studying, we find that these variable separation solutions, which seem independent, actually depend on each other. Based on the derived variable separation solution, chaotic behaviors, i.e. periodic solution with chaotic behavior and chaotic peaked and compact line solitons, are investigated.     

Two-dimensional symmetry reduction of (2 + 1)-dimensional nonlinear Klein–Gordon equation

In this paper, one-dimensional optimal system of group invariant solutions of (2 + 1)-dimensional Klein–Gordon system is constructed. Then the classification of group invariant solutions is given out and the corresponding two-dimensional symmetry reduced equations are obtained. At last some symmetry transformations are gained in detail. Especially, we obtain the most general solution from a given solution by use of six variable one-parameter subgroups transformations.     

Soliton-like and periodic form solutions to (2 + 1)-dimensional Toda equation

In this paper, we present a further extended tanh method for constructing exact solutions to nonlinear difference-differential equation(s) (NDDEs) and Lattice equations. By using this method via symbolic computation system MAPLE, we obtain abundant soliton-like and period-form solutions to the (2 + 1)-dimensional Toda equation. Solitary wave solutions are merely a special case in one family. This method can also be used to other nonlinear difference differential equations.     

A (2 + 1)-dimensional integrable hierarchy and its extending integrable model

A loop algebra is constructed, whose subalgebra is first used to present a Lax pair. By making use of the Tu scheme by Tu Guizhang, a generalized (2 + 1)-dimensional KN hierarchy is worked out. Further, based on the associated relations between the subalgebras in the above loop algebra, an extending integrable model of the generalized (2 + 1)-dimensional KN hierarchy as above is produced.     

Symbolic computation of exact solutions for the (2 + 1)-dimensional generalized stochastic KP equation

In this paper, the F-expansion method is extended and applied to construct the exact solutions of the (2 + 1)-dimensional generalized Wick-type stochastic Kadomtsev–Petviashvili equation by the aid of the symbolic computation system Maple. Some new stochastic exact solutions which include kink-shaped soliton solution, singular soliton solution and triangular periodic solutions are obtained via this method and Hermite transformation.