关于m-HPK(n1,n2,n3,n4)-残差图

本文研究了3-维超平面完备残差图以及m 重3-维超平面完备残差图. 利用容斥原理以及集合的运算性质等方法, 分别获得了3-维超平面完备残差图和m 重3-维超平面完备残差图的最下阶以及二者的唯一极图, 将文献[1] 中定义的残差图从平面推广到超平面上.     

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 Word Length of Commutators in GLn (R)*

Van der Kallen proved that the elementary group E_n(C[x]) does not have bounded word length with respect to the set of all elementary transvections. Later, Dennis and Vaserstein showed that the same is true, even with respect to the set of all commutators. The natural question is: Does the set of all commutators in E_n(R) have bounded word length with all elementary transvections as generators? The article provides a positive answer over a finite-dimensional commutative ring R.     

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.     

蕴含三类导出子图的可图序列

对非负整数序列π=（d1,d2……,dn),0≤di≤n-1,本分别给出了它蕴含导出子图为几乎处处完全图,完全图去掉一个Hamilton圈的边,完全k-部图可图（即蕴含aw^1,Aw^2和Ar,r2…,rk-可图）的判别准则。     

Automorphisms of Chevalley groups of types Al, Dl, El over local rings without 1/2

In this paper, we prove that every automorphism of a Chevalley group of type B _l , l ≥ 2, over a commutative local ring with 1/2 is standard, i.e., it is a composition of ring, inner, and central automorphisms.     

K2, K3 and nilpotent ideals

In this note is given an explicit computation method to decide about the minimality of a word in the free group of rank two. A cohomological use of this method is to yield words defining one relator duality groups.     

The Ramsey numbers r(Pm, Kn)

This note evaluates the Ramsey numbers r(P_m,K_n), and discusses developments in 0 generalized Ramsey theory for graphs.     

Automorphisms of elementary adjoint Chevalley groups of types Al, Dl, and El over local rings with 1/2

It is proved that every automorphism of an elementary adjoint Chevalley group of type A _l , D _l , or E _l over a local commutative ring with 1/2 is a composition of a ring automorphism and conjugation by some matrix from the normalizer of that Chevalley group in GL(V) (V is an adjoint representation space).     

Invariants of the adjoint action in the nilradical of a parabolic subalgebra of types Bn, Cn, and Dn

We study invariants of the adjoint action of the unipotent group in the nilradical of a parabolic subalgebra of types B _n , C _n , D _n . We introduce the notion of expanded base in the set of positive roots and construct an invariant for every root of the expanded base. We prove that these invariants are algebraically independent. We also give an estimate of the transcendence degree of the invariant field. Bibliography: 6 titles.     

Wr,p(R)-splines

In [3] Golomb describes, for 1 < p < ∞, the H^r,p(R)-extremal extension $\text{F}{}_1$ of a function $\text{?:E → R}$ (i.e., the H^r,p-spline with knots in E) and studies the cone $\text{H}{}_{1E}{}^{\mathrm{r,p}}$ of all such splines. We study the problem of determining when $\text{F}{}_1$ is in W^r,p ≡ H^r,p ∩ L^p. If $\text{F}{}_1\text{? W}{}^{\mathrm{r,p}}$, then $\text{F}{}_1$ is called a W^r,p-spline, and we denote by $\text{W}{}_{1E}{}^{\mathrm{r,p}}$ the cone of all such splines. If E is quasiuniform, then $\text{F}{}_1\text{? W}{}^{\mathrm{r,p}}$ if and only if . The cone $\text{W}{}_{1E}{}^{\mathrm{r,p}}$ with E quasiuniform is shown to be homeomorphic to l^p. Similarly, $\text{H}{}_{1E}{}^{\mathrm{r,p}}$ is homeomorphic to h^r,p. Approximation properties of the W^r,p-splines are studied and error bounds in terms of the mesh size $\text{¦ E ¦}$ are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of $\text{¦ E ¦}$ are derived.     

Differential-geometric Newton method for the best rank-(R1, R2, R3) approximation of tensors

An increasing number of applications are based on the manipulation of higher-order tensors. In this paper, we derive a differential-geometric Newton method for computing the best rank-(R ₁, R ₂, R ₃) approximation of a third-order tensor. The generalization to tensors of order higher than three is straightforward. We illustrate the fast quadratic convergence of the algorithm in a neighborhood of the solution and compare it with the known higher-order orthogonal iteration (De Lathauwer et al., SIAM J Matrix Anal Appl 21(4):1324–1342, 2000). This kind of algorithms are useful for many problems. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. Research supported by: (1) Research Council K.U.Leuven: GOA-Ambiorics, CoE EF/05/006 Optimization in Engineering (OPTEC), (2) F.W.O.: (a) project G.0321.06, (b) Research Communities ICCoS, ANMMM and MLDM, (3) the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, “Dynamical systems, control and optimization”, 2007–2011), (4) EU: ERNSI. M. Ishteva is supported by a K.U.Leuven doctoral scholarship (OE/06/25, OE/07/17, OE/08/007), L. De Lathauwer is supported by “Impulsfinanciering Campus Kortrijk (2007–2012)(CIF1)” and STRT1/08/023.     

Almansi-Type Decomposition Theorem for Bi-k-regular Functions in the Clifford Algebra Cl2n+2,0(R)*

Almansi-type decomposition theorem for bi-k-regular functions defined in a star-like domain Ω Rⁿ⁺¹ × Rⁿ⁺¹ centered at the origin with values in the Clifford algebra Cl_2n+_2,0(R) is proved. As a corollary, Almansi-type decomposition theorem for biharmonic functions of degree k is given.     

ON DEGENERATE WEAKLY （K1, K2）-QUASIREGULAR MAPPINGS

The authors first give the definition of degenerate weakly （K1,K2）-quasiregular mappings using the technique of exterior power and exterior differential forms, and then, using the method of McShane extension, a useful inequality is obtained, which can be used to derive the self-improving regularity.     

Very tight embeddings of subspaces ofLp, 1 =p< 2, intolp n

We prove that for 1 p < r < 2, every n-dimensional subspace E of L _r, in particular l _r ⁿ , well-embeds into l _p ^m for some m (1 + $$\epsilon$$)n, where well depends on p, r, and the arbitrary $$\epsilon$$ > 0, but not on n.     

Toeplitz algebras, subnormal tuples and rigidity on reproducing C[z1,…,zd]-modules

This paper mainly considers Toeplitz algebras, subnormal tuples and rigidity concerning reproducing C[z₁,…,z_d]-modules. By making use of Arveson's boundary representation theory, we find there is more rigidity in several variables than there is in single variable. We specialize our attention to reproducing C[z₁,…,z_d]-modules with -invariant kernels by examining the spectrum and the essential spectrum of the d-tuple {M_z1,…,M_zd}, and deducing an exact sequence of C∗-algebras associated with Toeplitz algebra. Finally, we deal with Toeplitz algebras defined on Arveson submodules and rigidity of Arveson submodules.     

Some New Results Related to Compact Matrix Operators in the Class((l_p)T,l_∞)

The main goal of this paper is to establish necessary and sufficient conditions for a matrix A ∈((lp)T,l∞), where T is an arbitrary triangle, 1 ≤ p ≤∞, to be a compact operator. In the past,only sufficient conditions were established in almost all of those cases, by using the Hausdorff measure of noncompactness. We improve those results by applying another method for the characterizations of compact linear operators between BK spaces.     

Planar kernel and grundy with d≤3, dout≤2, din≤2 are NP-complete

It is proved that the questions whether a finite diagraph G has a kernel K or a Sprague—Grundy function g are NP-complete even if G is a cyclic planar digraph with degree constraints d_out(u)≤2, d_in(u)≤2 and d(u)≤3. These results are best possible (if P ≠ NP) in the sense that if any of the constraints is tightened, there are polynomial algorithms which either compute K and g or show that they do not exist. The proof uses a single reduction from planar 3-satisfiability for both problems.