The (2, 2, 0) group inverse problem

We characterize the existence of the group inverse of a two by two matrix with zero (2, 2) entry, over a ring by means of the existence of the inverse of a suitable function of the other three entries. Some special cases are derived.     

Представление функций классовL p [0, 1], 0<p<1, ортогональными рядами

u . uuu u     

Recursive algorithm for the reliability of a connected-(1, 2)-or-(2, 1)-out-of-(m, n):F lattice system

The connected-(1, 2)-or-(2, 1)-out-of-(m, n):F lattice system is included by the connected-X-out-of-(m, n):F lattice system defined by Boehme et al. [Boehme, T.K., Kossow, A., Preuss, W., 1992. A generalization of consecutive-k-out-of-n:F system. IEEE Transactions on Reliability 41, 451–457]. This system fails if and only if at least one subset of connected failed components occurs which includes at least a (1, 2)-matrix (that is, a row and two columns) or a (2, 1)-matrix(that is, two rows and a column) of failed components. This system is applied to two-dimensional network problems with adjacent constraints, and various systems, for example, a supervision system, etc.     

Classification of small (0, 1) matrices

Denote by A_n the set of square (0, 1) matrices of order n. The set A_n, n ? 8, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of regular (0, 1) matrices of order 8 is 10160459763342013440. Let D_n, S_n denote the set of absolute determinant values and Smith normal forms of matrices from A_n. Denote by a_n the smallest integer not in D_n. The sets D₉ and S₉ are obtained; especially, a₉ = 103. The lower bounds for a_n, 10 ? n ? 19 (exceeding the known lower bound a_n ? 2f_n − 1, where f_n is nth Fibonacci number) are obtained. Row/permutation equivalence classes of A_n correspond to bipartite graphs with n black and n white vertices, and so the other applications of the classification are possible.     

B0,I0,B1,I1,C0-半群

利用半群的理想和双理想呈现的包含关系定义了新的半群类B0,I0,B1,I1,C0-半群,讨论其性质,证明了正则半群S是C0-半群的充要条件是S是矩形带;B0,I0,B1,I1,C0-半群各自的直积半群和关于同余的商半群保持B0,I0,B1,I1,C0性.     

N-soliton solutions for shallow water waves equations in (1 + 1) and (2 + 1) dimensions

A variety of shallow water waves equations in (1 + 1) and (2 + 1) dimensions are investigated. We first show that these models are completely integrable. We next determine multiple-soliton solutions for each equation. The simplified Hirota’s bilinear method developed by Hereman will be employed to achieve this goal. A comparison between dispersion relations and the phase shifts will be conducted. (But possess the same coefficients for the polynomials of exponentials.)     

Soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m, n) equations

We consider the nonlinear dispersive K(m,n) equation with the generalized evolution term and derive analytical expressions for some conserved quantities. By using a solitary wave ansatz in the form of sech^p function, we obtain exact bright soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m,n) equations with the generalized evolution terms. The results are then generalized to multi-dimensional K(m,n) equations in the presence of the generalized evolution term. An extended form of the K(m,n) equation with perturbation term is investigated. Exact bright soliton solution for the proposed K(m,n) equation having higher-order nonlinear term is determined. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients.     

Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter

In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation σ field and the tail σ field of the signal, in the stationary filtering problem. This property can be viewed as the permissibility of the interchange of the order of the operations of maximum and countable intersection for certain σ-fields. Under suitable conditions, it is shown that the above property is equivalent to various desirable properties of the filter such as

(a) uniqueness of invariant measure for the signal,

(b) uniqueness of invariant measure for the pair (signal, filter),

(c) a finite memory property of the filter,

(d) a property of finite time dependence between the signal and observation σ fields and

(e) asymptotic stability of the filter.

Previous works on the asymptotic stability of the filter for a variety of filtering models then identify a rich class of filtering problems for which the above equivalent properties hold.     

Russian text ignored Lp, 0 < p < 1

Certain weak forms of normality are discussed and are shown to coincide with normality in an appropriate setting by devising a strong form of semilocal connectedness. It is shown that every compact Hausdorff locally connected space is strongly semilocally connected. This improves a result of Mrówka and Pervin. Properties of s-continuous mappings are studied and in the process improvements of certain results of Pervin and Levine, Friedler, and Kohli are obtained.     

Maximal three-independent subsets of {0, 1, 2} n

We consider a variant of the classical problem of finding the size of the largest cap in ther-dimensional projective geometry PG(r, 3) over the field IF₃ with 3 elements. We study the maximum sizef(n) of a subsetS of IF ₃ ⁿ with the property that the only solution to the equationx ₁+x₂+x₃=0 isx ₁=x₂=x₃. Letc _n=f(n)^1/n andc=sup{c ₁, c₂, ...}. We prove thatc>2.21, improving the previous lower bound of 2.1955 ...     

New explicit solutions for (2 + 1)-dimensional soliton equation

In this Letter, we study (2 + 1)-dimensional soliton equation by using the bifurcation theory of planar dynamical systems. Following a dynamical system approach, in different parameter regions, we depict phase portraits of a travelling wave system. Bell profile solitary wave solutions, kink profile solitary wave solutions and periodic travelling wave solutions are given. Further, we present the relations between the bounded travelling wave solutions and the energy level h. Through discussing the energy level h, we obtain all explicit formulas of solitary wave solutions and periodic wave solutions.     

Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inL _p[−π,π], algebraic polynomial approximation inL _p[−1,1], algebraic polynomial approximation inL _p(S), and entire function of exponential type approximation inL _p(R), and characterizeK-functionals for certain pairs of function spaces including (L _p[−π,π],B _s ^a(L _p[−π,π])), (L _p(R),_s ^a(L_p(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.     

Moduli of smoothness andK-functionals inL p, 0<p<1

It is shown that forL _p, 0p<1, the=">K-functional betweenL _p andW _p ^r is identically zero. Useful measures that are equivalent to the moduli of smoothness are found. The equivalence results that are given are valid for 0p.Communicated by Vilmos Totik.