K2 and K3 of the circle

Let k be $\text{Z}\text{[}\text{1}\text{2}\text{]}$, $\text{Q}$ or $\text{R}$, and set $\text{A =}\text{k[x,y]}\text{(x}{}^2\text{+ y}{}^2\text{? 1)}$. We compute K₂(A) and K₃(A). Our method is to construct a map $\text{? : K}{}_1\text{(k[i])→K}{}_{\mathrm{1\; +\; 1}}\text{(A)}$ and compare this to a localization sequence.We give three applications. We show that ? accounts for the primitive elements in K₂(A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec(A). Third, we can iterate the construction to compute the K-theory of the torus ring A ?_kA.     

Stability for Quadratic K1

The general quadratic group GQ _2n and its elementary subgroup EQ _2n are analogs in the theory of quadratic forms of the general linear group GL _n and its elementary subgroup E _n . This article proves that the stabilization map GQ _2n /EQ _2n GQ _2(n+1) /EQ _2(n+1) is an isomorphism whenever n +1 and S denotes the -stable rank of rings with anti-involution. As a corollary, a result is obtained which has been anticipated since the late 1960s: over rings of finite Bass–Serre dimension d, the stabilization map is an isomorphism whenever n d + 2.     

Triangular imbedding of Kn − K6

For n = 12s + 9 (s ≥ 4) we imbed K_n ? K₆ in an orientable surface of genus 12s² + 11s.     

When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

We discuss subsetsS of ℝⁿ such that every real valued functionf onS is of the formf(x₁, x₂, ..., x_n) =u ₁(x₁) +u ₂(x₂) +...+u _n(x_n), and the related concepts and situations in analysis.     

On the average depth of a prefix of the Dycklanguage D1

If D₁ is the Dycklanguage with one type of bracket then the level of a bracket in a word w ∈ D₁ is defined as the number of preceding opening brackets minus the number of preceding closing brackets. The depth of a Dyckword w is the maximum level of a bracket appearing in w. In this paper we derive an explicit expression for the average depth of a prefix of length n of the Dycklanguage D₁ and show that the average depth is given for all ε>0 by

$\text{(ln2)}\text{2πn}\text{?}\text{3}\text{2}\text{+}\text{1}\text{8}\text{2π}\text{n}\text{[4ln2?(?1)}{}^{\mathrm{n}}\text{(2ln2?1)]+O(}\text{ln n}\text{n}{}^{\mathrm{1.5?\; ?}}$

. The variance is asymptotically for all ?>0

$\text{(}\text{π}\text{3}\text{?2 ln}{}^2\text{2)πn+}\text{5π}{}^2\text{18}\text{?2πln}{}^2\text{2+}\text{1}\text{12}\text{?π(?1)}{}^{\mathrm{n}}\text{(}\text{π}\text{24}\text{? ln}{}^2\text{2+}\text{1}\text{2}\text{ln2)+O(}\text{ln n}\text{n}{}^{\mathrm{1??}}$

. Furthermore, we derive several enumeration results describing the distribution of the number of certain prefixes of length n.     

Dealing with the multiplicity of solutions of the ℓ1 and ℓ∞ regression models

It is well known that the least absolute value (?_∞) and the least sum of absolute deviations (?₁) algorithms produce estimators that are not necessarily unique. In this paper it is shown how the set of all solutions of the ?₁ and ?_∞ regression problems for moderately large sample sizes can be obtained. In addition, if the multiplicity of solutions wants to be avoided, two new methods giving the same optimal ?₁ and ?_∞ values, but supplying unique solutions, are proposed. The idea consists of using two steps: in the first step the optimal values of the ?₁ and ?_∞ errors are calculated, and in the second step, in case of non-uniqueness of solutions, one of the multiple solutions is selected according to a different criterion. For the ?_∞ the procedure is used sequentially but removing, in each iteration, the data points with maximum absolute residual and adding the corresponding constraints for keeping these residuals, and this process is repeated until no change in the solution is obtained. In this way not only the maximum absolute residual values are minimized in the modified method, but also the maximum absolute residual values of the remaining points sequentially, until no further improvement is possible. In the ?₁ case a least squares criterion is used but restricted to the ?₁ residual condition. Thus, in the modified ?₁ method not only the ?₁ residual is minimized, but also the sum of squared residuals subject to the ?₁ residual. The methods are illustrated by their application to some well known examples and their performances are tested by some simulations, which show that the lack of uniqueness problem cannot be corrected for some experimental designs by increasing the sample size.     

Fourier Transforms of Bounded Bilinear Forms on C＊（S1）×C＊（S2） of Foundation ＊-semigroups S1 and S2

The author gives a characterization of the Fourier transforms of bounded bilinear forms on C＊（S1）×C＊（S2） of two foundation semigroups S1 and S2 in terms of Jordan ＊-representations, hemimogeneous random fields, and as well as weakly harmonizable random fields of S1 and S2 into Hilbert spaces.     

Cyclic codes over the rings Z 2?+ uZ 2 and Z 2?+ uZ 2?+ u 2 Z 2

In this paper, we study cyclic codes over the rings Z ₂ + uZ ₂ and Z ₂ + uZ ₂ + u ² Z ₂ . We find a set of generators for these codes. The rank, the dual, and the Hamming distance of these codes are studied as well. Examples of cyclic codes of various lengths are also studied.      

Regularizing effects for ut + Aϑ(u) = 0 in L1

Various initial-boundary value problems and Cauchy problems can be written in the form $\text{du}\text{dt}\text{+ A?(u) = 0}$, where ?:R→$\text{R}$ is nondecreasing and A is the linear generator of strongly continuous nonexpansive semigroup e^?tA in an L¹ space. For example, if A = ?Δ (subject, perhaps, to suitable boundary conditions) we obtain equations arising in flow in a porous medium or plasma physics (depending on the choice of ?) while if $\text{A =}\text{?}\text{?x}$ acting in L¹($\text{R}$) we have a scalar conservation law. In this paper we show that if M, m > 0 and m?′² ? ν??′' ? M?′², where ν ? {1,?1}, then (roughly speaking), the norm of $\text{t}\text{du}\text{dt}$ may be estimated in terms of the initial data u₀ in L¹. Such estimates give information about the regularity of solutions, asymptotic behaviour, etc., in applications. Side issues, such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator A?, are also dealt with. These considerations are of independent interest.     

关于L2(En+1+,(dxdy)/(yn+1)中的柱面函数空间的正交直和分解

本文利用小波变换给出了L²(E_n+1⁺,(dxdy)/(yⁿ⁺¹)中的柱面函数空间的一种正交直和分解.在这种分解下定义了Toeplitz-Hankel型算子,得到了类似的Schatten-Von Neumann性质.     

SMALL INTO ISOMORPHISM FROM （C0） INTO C（Ω） TYPE

If T is an isomorphism of （c0） into C（Ω） （where Ω is a sequentially compact and paracompact space, or a compact metric space in particular）, which satisfies the condition ｜｜T｜｜·｜｜T^-1｜｜ ≤ 1 ＋ε for some ε ∈ （0,1/5）, then T/｜｜T｜｜ is close to an isometry with an error less than 9ε. The proof of this article is simple without using the dual space or adjoint operator.     

环F3+vF3上的循环码与常循环码

本文研究了环F₃+vF₃上的循环码与常循环码.通过环F₃+vF₃与域F₃上的循环码之间关系,证明了环F₃+vF₃上循环码是由一个多项式生成的.最后,用类似的方法,得到了环F₃+vF₃上v-常循环码也是由一个多项式生成的.     

W0^1p（x） Versus C^1 Local Minimizers for a Functional with Critical Growth

Let Ω IR^N, （N ≥ 2） be a bounded smooth domain, p is Holder continuous on Ω^-,
1 〈 p^- ：= inf pΩ（x） ≤ p＋ = supp（x） Ω〈∞,
and f：Ω^-× IR be a C^1 function with f（x,s） ≥ 0, V （x,s） ∈Ω × R^＋ and sup ∈Ωf（x,s） ≤ C（1＋s）^q（x）, Vs∈IR^＋,Vx∈Ω for some 0〈q（x） ∈C（Ω^-） satisfying 1 〈p（x） 〈q（x） ≤p^＊ （x） -1, Vx ∈Ω ^- and 1 〈 p^- ≤ p^＋ ≤ q- ≤ q＋. As usual, p＊ （x） = Np（x）/N-p（x） if p（x） 〈 N and p^＊ （x） = ∞- if p（x） if p（x） 〉 N. Consider the functional I： W0^1,p（x） （Ω） →IR defined as
I（u） def= ∫Ω1/p（x）｜△｜^p（x）dx-∫ΩF（x,u^＋）dx,Vu∈W0^1,p（x）（Ω）,
where F （x, u） = ∫0^s f （x,s） ds. Theorem 1.1 proves that if u0 ∈ C^1 （Ω^-） is a local minimum of I in the C1 （Ω^-） ∩C0 （Ω^-）） topology, then it is also a local minimum in W0^1,p（x） （Ω）） topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation （P） defined below.     

基于全牛顿求解P*(κ)阵水平线性互补问题的内点算法

提出一种求解P_*(k)阵水平线性互补问题的全牛顿内点算法,全牛顿算法的优势在于每次迭代中不需要线性搜寻.当给定适当的中心路径邻域的阈值和更新势垒参数,证明算法中心邻域的全牛顿是局部二次收敛的,最后给出算法迭代复杂性O(√n)log(n+1+k)/εμ₀.     

A combinatorial interpretation of the recurrence fn+1 = 6fn − fn−1

Bonin et al. (1993) recalled an open problem related to the recurrence relation verified by NSW numbers. The recurrence relation is the following: f_n+1 = 6f_n − f_n−1, with f₁ = 1 and f₂ = 7, and no combinatorial interpretation seems to be known. In this note, we define a regular language L whose number of words having length n is equal to f_n+1. Then, by using L we give a direct combinatorial proof of the recurrence.     

Some group actions on K(x 1 , x 2 , x 3 )

LetK be any field which may not be algebraically closed,K(x ₁ ,x ₂ ,x ₃ ) be the rational function field of three variables overK, and σ:K(x ₁ ,x ₂ ,x ₃ ) → K(x ₁ ,x ₂ ,x ₃ ) be aK-automorphism defined by wherea _i ,b _i ,c _i ,d _i ∈K anda _i d _i −b _i c _i ≠0. Let ,f _i (T)=T ² −(a _i +d _i )T+(a _i d _i −b _i c _i )∈K[T] be the “characteristic polynomial” of σ _i . Theorem:Assume that charK≠2.Then the fixed field K(x ₁ ,x ₂ ,x ₃ ) ^<σ> is not rational (=purely transcendental) over K if and only if (i) for each 1≤i≤3, f _i (T) is irreducible; (ii) the Galois group of f ₁ (T)f ₂ (T)f ₃ (T) over K is of order 8; and (iii) for each 1≤i≤3,ord (σ _[itn] )is an even integer.     

有一个实现包含Kr +1的可图序列

设K _{r +1}是一个r +1个顶点的完全图. 一个可图序列π =(d₁, d₂,…, d_n)称为是蕴含K _r+1 -可图的, 如果π有一个实现包含 K _{r +1}作为子图. 该文进一步研究了蕴含K _{r+1 -}可图序列的一些新的条件, 证明了这些条件包含文献[14,10,11]中的一些主要结果和当n≥5r/2 +1时,σ(K _r+1, n)之值(此值在文献[2]中被猜测, 在文献[6,7,8,3]中被证实). 此外, 确定了所有满足n≥5, d₅≥4 且不蕴含K₅ -可图序列π=(d₁, d₂,…, d_n)的集合.     

关于p叶近于凸性的充分条件

Let Ap(n)(p, n∈N={1,2,...}) denote the class of functions of the form f(z)=zp+ap+nzp+n+... which are analytic in the unit disc E={z:|z|＜1}. By using the method of differential subordinations we give some sufficient conditions for a function f(z)∈Ap(n) to be a certain subclass Rp(n,k) of p-valently close-to-convexity functions.     

基于混合l_2/l_1范数极小化方法的块稀疏信号重构条件

研究压缩感知中的块稀疏信号重构问题,主要对混合l_2/l_1极小化方法建立了一类改进的可重构条件.具体地说,本文证明若测量矩阵满足条件δ_k+θ_(k,k)1,则混合l_2/l_1极小化方法可精确重构(无噪声情形)或鲁棒重构(有噪声情形)原始块k-稀疏信号.进而表明本文给出的新条件弱于现有文献所给出的条件.