Decomposition rank and \mathcal{Z} -stability

We show that separable, simple, nonelementary, unital C^*-algebras with finite decomposition rank absorb the Jiang–Su algebra Z\mathcal{Z} tensorially. This has a number of consequences for Elliott’s program to classify nuclear C^*-algebras by their K-theory data. In particular, it completes the classification of C^*-algebras associated to uniquely ergodic, smooth, minimal dynamical systems by their ordered K-groups.     

On the rational recursive sequencex_{n + 1} = \frac{{\alpha x_n + \beta x_{n - 1} + \gamma x_{n - 2} + \delta x_{n - 3} }}{{Ax_n + Bx_{n - 1} + Cx_{n - 2} + Dx_{n - 3} }}

The main objective of this paper is to study the boundedness character, the periodic character and the global stability of the positive solutions of the following difference equation $x_{n + 1} = \frac{{\alpha x_n + \beta x_{n - 1} + \gamma x_{n - 2} + \delta x_{n - 3} }}{{Ax_n + Bx_{n - 1} + Cx_{n - 2} + Dx_{n - 3} }},n = 0,1,2.....$ where the coefficientsA, B, C, D, α, β, γ, δ, and the initial conditionsx _-3,x _-2,x _-1,x ₀ are arbitrary positive real numbers.     

r模同态的{1}-逆与{2}-逆

本文探讨模同态广义逆在环模理论中的应用．利用模同态的{1}-逆与{2}-逆，分别给出了一类环及一类重要模的特征刻画．     

On the difference equationx_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}}

In this paper we consider the difference equation $$x_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}},$$ wherea, b, c are nonegative real numbers,k, l, m are nonnegative integers andg(x) is a nonegative real function. The oscillatory and periodic character, the boundedness and the stability of positive solutions of the equation is investigated. The existence and nonexistence of two-period positive solutions are investigated in details. In the last section of the paper we consider a generalization of the equation.     

On the recursive sequencex_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation $$x_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}, n = 0, 1,...$$ wherem, k ∈ N, 2k > 2m?1,a, b, c are nonnegative real numbers andp < 1, are investigated.     

涉及无限族非扩张映象$\{T_n\}_{n=1}^\infty$的迭代算法$x_{n 1}=\alpha_{n 1}f(x_n) (1-\alpha_{n 1})T_{n 1}x_n$

Under the framework of uniformly smooth Banach spaces, Chang proved in 2006 that the sequence {xn} generated by the iteration xn＋1 =αn＋1f（xn） ＋ （1 - αn＋1）Tn＋1xn converges strongly to a common fixed point of a finite family of nonexpansive maps {Tn}, where f ： C → C is a contraction. However, in this paper, the author considers the iteration in more general case that {Tn} is an infinite family of nonexpansive maps, and proves that Chang＇s result holds still in the setting of reflexive Banach spaces with the weakly sequentially continuous duality mapping.     

Conservative fragments of {{S}^{1}_{2}} and {{R}^{1}_{2}}

Conservative subtheories of ${{R}^{1}_{2}}$ and ${{S}^{1}_{2}}$ are presented. For ${{S}^{1}_{2}}$ , a slight tightening of Je?ábek??s result (Math Logic Q 52(6):613?C624, 2006) that ${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$ is presented: It is shown that ${T^{0}_{2}}$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this ${\forall\Sigma^{b}_{1}}$ -theory, we define a ${\forall\Sigma^{b}_{0}}$ -theory, ${T^{-1}_{2}}$ , for the ${\forall\Sigma^{b}_{0}}$ -consequences of ${S^{1}_{2}}$ . We show ${T^{-1}_{2}}$ is weak by showing it cannot ${\Sigma^{b}_{0}}$ -define division by 3. We then consider what would be the analogous ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ based on Pollett (Ann Pure Appl Logic 100:189?C245, 1999. It is shown that this theory, ${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$ , also cannot ${\Sigma^{b}_{0}}$ -define division by 3. On the other hand, we show that ${{S}^{0}_{2}+open_{\{||id||\}}}$ -COMP is a ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ . Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium?? 98, 262?C279, 2000) and show that ${\hat{C}^{0}_{2}}$ is ${\forall\hat\Sigma^{b}_{1}}$ -conservative over a theory based on open _cl-comprehension.     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

$$ H^{p}_{w} - L^{p}_{w} $$ Boundedness of Marcinkiewicz Integral

The Marcinkiewicz integral is essentially a Littlewood-Paley g-function, which plays a very important role in harmonic analysis. In this paper we give weaker smoothness conditions assumed on Ω to imply the boundedness of the Marcinkiewicz integral operator μ_Ω, where w belongs to the Muckenhoupt weight class.     

On the functional equation $${\varvec{f}}({\varvec{x}})+{\varvec{f}}({\varvec{y}})=\mathbf{max} \{{\varvec{f}}({\varvec{xy}}),{\varvec{f}}({\varvec{xy}}^{-{\varvec{1}}})\}$$ on groups

We analyse the functional equation

$$\begin{aligned} f(x)+f(y)=\max \{f(xy),f(xy^{-1})\} \end{aligned}$$

for a function \(f:G\rightarrow \mathbb R\) where G is a group. Without further assumption it characterises the absolute value of additive functions. In addition \(\{z\in G\mid f(z)=0\}\) is a normal subgroup of G with abelian factor group.

     

On the Number of Solutions of the Equation a_{1}x_{1}^{m_{1}}+\cdots +a_{n}x_{n}^{m_{n}}=bx_{1}\cdots x_{n} in a Finite Field

We obtain an explicit formula for the number of solutions of a special equation in a finite field under a certain restriction on the exponents. Mathematics Subject Classifications (2000) primary: 11G25, secondary: 11T24.     

An Exotic {T_{1}\mathbb{S}^4} with Positive Curvature

We construct a metric with positive sectional curvature on a 7-manifold which supports an isometry group with orbits of codimension 1. It is a connection metric on the total space of an orbifold 3-sphere bundle over an orbifold 4-sphere. By a result of S. Goette, the manifold is homeomorphic but not diffeomorphic to the unit tangent bundle of the 4-sphere.     

Iterative algorithm for solving a class of general Sylvester-conjugate matrix equation \sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C

This paper is concerned with iterative solution to general Sylvester-conjugate matrix equation of the form $\sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C$ . An iterative algorithm is established to solve this matrix equation. When this matrix equation is consistent, for any initial matrices, the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.     

On solutions of the simultaneous Pell equations $$ x^{2}-\left( a^{2}-1\right) y^{2}=1$$ and $$y^{2}-pz^{2}=1$$y2-pz2=1

Let \(a\ge 2\) be an integer and p prime number. It is well-known that the solutions of the Pell equation have recurrence relations. For the simultaneous Pell equations

$$\begin{aligned}&x^{2}-\left( a^{2}-1\right) y^{2} =1 \\&y^{2}-pz^{2} =1 \end{aligned}$$

assume that \(x=x_{m}\) and \(y=y_{m}\). In this paper, we show that if \(m\ge 3\) is an odd integer, then there is no positive solution to the system. Moreover, we find the solutions completely for \(5\le a\le 14\) in the cases when \(m\ge 2\) is even integer and \(m=1\).