On the Relation between the Scalar Moment Problem and the Matrix Moment Problem on *-Semigroups

There is a countable cancellative commutative *-semigroup S withzero (in fact, a *-subsemigroup of G × N₀ for some abelian group G carrying the inverse involution) such that the answer to the question “if f is a function on S , with values in M_d(C) (the square matrices of order d) and such that $\sum^{n}_{j,k=1} \lbrak f(s^*_k s_j)\xi_j, \xi_k \rbrak \ge 0$ for all n in N, s₁, . . . , s_n in S , and $\xi_1$, . . . , $\xi_n$ in C^d, does it follow that $f(s) = \int_{S^*}\sigma (s) d\mu(\sigma) (s \memb S)$ for some measure $\mu$ (with values in M_d(C)₊ , the positive semidenite matrices) on the space S of hermitian multiplicative functions on S?” is “yes” if d = 1 but “no” if d = 2 (hence also for d > 2).     

Normal Cayley graphs of finite groups

LetG be a finite group and let S be a nonempty subset of G not containing the identity element 1. The Cayley (di) graph X = Cay(G, S) of G with respect to S is defined byV (X)=G, E (X)={(g,sg)|g∈G, s∈S} A Cayley (di) graph X = Cay (G,S) is said to be normal ifR(G) ◃A = Aut (X). A group G is said to have a normal Cayley (di) graph if G has a subset S such that the Cayley (di) graph X = Cay (G, S) is normal. It is proved that every finite group G has a normal Cayley graph unlessG≅ℤ₄×ℤ₂ orG≅Q ₈×ℤ ₂ ^r (r⩾0) and that every finite group has a normal Cayley digraph, where Z_m is the cyclic group of orderm and Q₈ is the quaternion group of order 8. Project supported by the National Natural Science Foundation of China (Grant No. 10231060) and the Doctorial Program Foundation of Institutions of Higher Education of China.     

几乎优越扩张与同调维数

设环S是环R的几乎优越扩张．本文证明了R和S具有相同的f．f．P．维数以及finitistic维数．若M_S是右S－模,则FP－id（M_S）＝FP－id（M_R）．若G是有限群,R是G分次环且|G|^－1∈R,则Smash积R＃G^*和R具有相同的f．f．P．维数,finitistic维数,以及FP－整体维数．     

关于有限Abelian群的生成子集的基数

假若G =Z_m1 Z_m2… Z_mr 为 (m₁, m₂,…, m_r)型Abelian群, 其中Z_mi 为 m_i 阶的循环群且1≤i≤ r, m₁ |m₂|…| m_r, S 为G 的满足0∈ S=-S 的生成子集. 如果 |S|>|G|/ρ, 其中ρ≥l m_r /2l且m_r=e(G) 为群 G 的所有元素的阶的最小公倍数, 则ρS=G. 更进一步作者推广了Klopsch与lev ^[1]的一个结论,有:若 G=Z₂Z_m 为 (2, m) 型 Abelian 群(m ≥8), 则 t_m/2(G)=0.     

Local generalized empirical estimation of regression

Let f(x) be the density of a design variable X and m(x) = E[Y\X = x] the regression function. Then m(x) - G(x)/f(x), where G(x) = m(x)f(x). The Dirac δ-function is used to define a generalized empirical function Gn (x) for G(x) whose expectation equals G(x). This generalized empirical function exists only in the space of Schwartz distributions, so we introduce a local polynomial of order p approximation to Gn(.) which provides estimators of the function G(x) and its derivatives. The density f(x) can be estimated in a similar manner. The resulting local generalized empirical estimator (LGE) of m(x) is exactly the Nadaraya-Watson estimator at interior points when p = 1, but on the boundary the estimator automatically corrects the boundary effect. Asymptotic normality of the estimator is established. Asymptotic expressions for the mean squared errors are obtained and used in bandwidth selection. Boundary behavior of the estimators is investigated in details. We use Monte Carlo simulations to show that the     

On fixed-point-free permutation properties in groups and semigroups

We say that a semigroup S is (fixed-point-free, for short f.p.f.) permutable, if, for some integer n and for every x₁,..., x_n in S, there exists a non-trivial (fixed-point-free) permutation on {1,..., n}, such that:

Nonsmooth analysis of vector-valued mappings with contributions to nondifferentiable programming

We define order Lipschitz mappings from a Banach space to an order complete vector lattice and present a nonsmooth analysis for such functions. In particular, we establish properties of a generalized directional derivative and gradient and derive results concerning a calculus of generalized gradients (i.e., calculation of the generalized gradient of f when f = f₁ + f₂, f = f · ₂, etc.). We show the relevance of the above analysis to nondifferentiaile programming by deriving optimality conditions for problems of the form min f(x) subject to x [euro] S. For S arbitrary we state the results in terms of cones of displacement of the feasible region at the optimal point; when S ={x ? A|g(x) ? B}, we obtain Kuhn-Tucker type results.     

GENERALISED MAXIMAL INDEPENDENCE AND CLIQUE NUMBERS OF GRAPHS

Abstract

A set B of vertices of a graph G = (V,E) is a k-maximal independent set (kMIS) if B is independent but for all ?-subsets X of B, where ? ? k—1, and all (? + 1)-subsets Y of V—B, the set (B—X) u Y is dependent. A set S of vertices of C is a k-maximal clique (kMc) of G iff S is a kMIS of [Gbar]. Let β_k, (G) (w_k(G) respectively) denote the smallest cardinality of a kMIS (kMC) of G—obviously β_k(G) = w_k([Gbar]). For the sequence m₁ ? m₂ ?…? m_n = r of positive integers, necessary and sufficient conditions are found for a graph G to exist such that w_k(G) = m_k for k = 1,2,…,n and w(G) = r (equivalently, β_k(G) = m_k for k = 1,2,…,n and β(G) = r). Define s_k(?,m) to be the largest integer such that for every graph G with at most s_k(?,m) vertices, β_k(G) ? ? or w_k(G) ? m. Exact values for s_k(?,m) if k ≥ 2 and upper and lower bounds for s₁(?,m) are de termined.     

On Jensen's functional equation

Summary Letf be a map from a groupG into an abelian groupH satisfyingf(xy) + f(xy ^–1) = 2f(x), f(e) = 0, wherex, y G ande is the identity inG. A set of necessary and sufficient conditions forS(G, H) = Hom(G, H) is given whenG is abelian, whereS(G, H) denotes all the solutions of the functional equation. The case whenG is non-abelian is also discussed.     

GEODETIC GAMES FOR GRAPHS

ABSTRACT

Let S be a subset of the vertex set V(G) of a nontrivial connected graph G. The geodetic closure (S) of S is the set of all vertices on geodesics between two vertices in S. The first player A chooses a vertex v₁ of G. The second player B then picks v₂ ≠ v₁ and forms the geodetic closure (S₂) = ({v₁, v₂}). Now A selects v₃ ε V—S₂ and forms (S₃) = ({v₁, v₂, v₃}), etc. The player who first selects a vertex v_n such that (S_n) = V wins the achievement game, but loses the avoidance game. These geodetic achievement and avoidance games are solved for several families of graphs by determining which player is the winner.     

Estimates of Conformal Radius and Distortion Theorems for Univalent Functions

A simple proof of the recent result by E. G. Emel'yanov concerning the maximum of the conformal radius r(D,1) for a family of simply connected domains with a fixed value r(D,0) is given. A similar problem is solved for a family of convex domains. Exact estimates for functionals of the form are obtained for families of functions inverse to elements of the classes S and S_m, where S={f:f is regular and univalent in the disk {z:|z| < 1} and f(0)=f'(0)-1=0} and S_M= for . Bibliography: 7 titles.     

Conditions for the imbedding of the function class W f 1 (G,A) in the space C(G)

The function class W _f ¹ (G, A) is defined. A general problem concerning necessary and sufficient conditions under which this class can be imbedded in the space C(G) of functions continuous on G is posed, and the special case of this problem in which the function f(x₁, x₂,..., x_n) involved in the definition of w _f ¹ (G, A) on is solved.Translated from Matematicheskie Zametki, Vol. 9, No. 6, pp. 639–650, June, 1971.In conclusion, the author wishes to thank V. M. Tikhomirov for posing the problem and for his criticism and advice. Thanks are also due to S. B. Stechkin for his interest in the work.     

Gelfer函数与Yamashita问题

Ｇｅｌｆｅｒ函数与Ｙａｍａｓｈｉｔａ问题邹中柱（湖南怀化师范专科学校，湖南怀化４１８００８）１９９１年７月１９日收到，１９９２年１１月２４日收到修改稿．若函数在单位圆盘中解析，且对于一切都有则称函数［２］，记其全体为Ｇ，并用Ｇ表示其单叶子类．显然，当...     

Smoothness of weak solutions of elliptic systems in the sense of Douglis-Nirenberg

Regularity properties of functions from the domain of definition of the operator L which is the closure in (L₂(G))^N of the map $$u = lu (u \in C ^\infty (bv) = \{ v \in (C^\infty (G))^N : bv = 0\} ),$$ where ?u = f (in G), bu=? (on ?G) is an elliptic boundary problem for a system which is elliptic in the sense of Douglis-Nirenberg.     

General problem of the polynomial approximation in jordan domains

Let G and Ω. be two Radon domains in ?, let ?:?G-?Ω be a conformai mapping, and let L_τ, τ>O be the level curves of the Green function of the domain ? ¯Ω. We completely describe the class of functions f, analytic in G, which can be approximated by polynomials of degree n{P_n} ₁ ^∞ such that $$\left| {f(Z) + P_n (Z)} \right| \leqslant const \cdot \rho [\varphi ({\rm Z}),L_{\begin{array}{*{20}c} 1 \\ n \\ \end{array} } ]^s ,Z \in \partial G$$ . It is shown that this class coincides with the “relative Holder class of order S,”, generated by Ω. For Ω=G and Ω={Z:|Z|<1} one obtains V. K. Dzyadyk's approximation and the uniform approximation, respectively.     

Neighbor distinguishing total choice number of sparse graphs via the Combinatorial Nullstellensatz

Acta Mathematicae Applicatae Sinica, English Series - Let G = (V,E) be a graph and ϕ: V ∪E → {1, 2, · · ·, k} be a total-k-coloring of G. Let f(v)(S(v)) denote the...     

Steiner Numbers in Graphs

Abstract

The Steiner distance d(S) of a set S of vertices in a connected graph G is the minimum size of a connected subgraph of G that contains S. The Steiner number s(G) of a connected graph G of order p is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = p—1. A smallest set S of vertices of a connected graph G of order p for which d(S) = p—1 is called a Steiner spanning set of G. It is shown that every connected graph has a unique Steiner spanning set. If G is a connected graph of order p and k is an integer with 0 ≤ k ≤ p—1, then the kth Steiner number sk(G) of G is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = k. The sequence s_o(G),s₁ (G),…,8_p-1(G) is called the Steiner sequence of G. Steiner sequences for trees are characterized.     

Almansi decomposition for Dunkl operators

Let Ω be a G-invariant convex domain in ℝ^N including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (Δh)ⁿf = 0 for some integer n. Here_333-01is the Dunkl Laplacian, and D_j is the Dunkl operator attached to the Coxeter group G,

$$\mathcal{D}_j f(x) = \frac{\partial }{{\partial x_j }}f(x) + \sum\limits_{v \in R_ + } {\kappa _v \frac{{f(x) - f(\sigma _v x)}}{{\left\langle {x,v} \right\rangle }}} v_j ,$$

where K_v is a multiplicity function on R and σ_v is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form

$$f(x) = f_0 (x) + \left| x \right|^2 f_1 (x) + \cdots + \left| x \right|^{2(n - 1)} f_{n - 1} (x), \forall x \in \Omega ,$$

where f_j are Dunkl harmonic functions, i.e. Δ_hf_j = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.