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1.
Wu Shengjian 《数学学报(英文版)》1994,10(2):168-178
Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then
- λ is finite;
- for every arbitrary numberk 1>1, there existsk 2>1 such thatT(k 1 r,f)≤k 2 T(r,f) for allr≥r 0. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then
- every deficient values off(z) is also its asymptotic value;
- every asymptotic value off(z) is also its deficient value;
- λ=μ;
- $\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} $
2.
Jean-Philippe Labrousse 《Rendiconti del Circolo Matematico di Palermo》1980,29(2):161-258
The present paper is concerned with the study of a new class of linear operators on a Hilbert space: the class of quasi-Fredholm operators, which contains many operators already studied in the litterature (in particular semi-Fredholm operators). An operatorA is said to be quasi-Fredholm of degreed, if the following conditions are satisfied:
- For alln greater thand, R(A n )∩N(A)=R(A d )∩N(A);
- N(A)∩R(A d ) is closed inH;
- R(A)+N(A d ) is closed inH.
- A is quasi-Fredholm iff there exists a direct decomposition ofH into the sum of two subspacesH 1 andH 2 which are invariant underA and such that the restriction ofA toH 1 is quasi-Fredholm of degree 0 and the restriction ofA toH 2 is nilpotent (Kato decomposition).
- A is quasi-Fredholm iff there exists a neighborhoodD of 0 in C such that for all λ≠0 in that neighborhoodA?λI has a generalized inverse which is meromorphic inD?{0} (The generalized inverse is holomorphic inD iffA is of degree 0).
3.
Liu Yanpei 《数学学报(英文版)》1989,5(1):64-79
The purpose of this paper which is a sequel of “ Boolean planarity characterization of graphs ” [9] is to show the following results.
- Both of the problems of testing the planarity of graphs and embedding a planar graph into the plane are equivalent to finding a spanning tree in another graph whose order and size are bounded by a linear function of the order and the size of the original graph, respectively.
- The number of topologically non-equivalent planar embeddings of a Hamiltonian planar graphG is τ(G)=2 c(H)?1, wherec (H) is the number of the components of the graphH which is related toG.
4.
M. Pellegrini 《Discrete and Computational Geometry》1994,12(1):203-221
We show some combinatorial and algorithmic results concerning finite sets of lines and terrains in 3-space. Our main results include:
- An $O(n^3 2^{c\sqrt {\log n} } )$ upper bound on the worst-case complexity of the set of lines that can be translated to infinity without intersecting a given finite set ofn lines, wherec is a suitable constant. This bound is almost tight.
- AnO(n 1.5+ε) randomized expected time algorithm that tests whether a directionv exists along which a set ofn red lines can be translated away from a set ofn blue lines without collisions. ε>0 is an arbitrary small but fixed constant.
- An $O(n^3 2^{c\sqrt {\log n} } )$ upper bound on the worst-case complexity of theenvelope of lines above a terrain withn edges, wherec is a suitable constant.
- An algorithm for computing the intersection of two polyhedral terrains in 3-space withn total edges in timeO(n 4/3+ε+k 1/3 n 1+ε+klog2 n), wherek is the size of the output, and ε>0 is an arbitrary small but fixed constant. This algorithm improves on the best previous result of Chazelleet al. [5].
5.
Svetoslav Ivanov Nenov 《Annali dell'Universita di Ferrara》1996,42(1):121-125
The existence and the uniqueness (with respect to a filtration-equivalence) of a vector flowX on ? n ,n≥3, such that:
- X has not any stationary points on ? n ;
- all orbits ofX are bounded;
- there exists a filtration forX are proved in the present note.
6.
Michiro Kondo 《Mathematica Slovaca》2014,64(5):1093-1104
We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X,
- If s is a state, then X/ker(s) is an MV-algebra.
- If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.
- s is a state-morphism on X.
- ker(s) is a maximal filter of X.
- s is extremal on X.
7.
T. S. H. Driessen 《Mathematical Methods of Operations Research》1986,30(1):A49-A64
For any natural numbersk andn, the subclass ofk-convexn-person games is introduced. In casek=n, the subclass consists of the convexn-person games. Ak-convexn-person game is characterized in several ways in terms of the core and certain marginal worth vectors. The marginal worth vectors of a game are described in terms of an upper bound for the core and the corresponding gap function. It is shown that thek-convexity of ann-person gamev is equivalent to
- all marginal worth vectors ofv belong to the core ofv; or
- the core ofv is the convex hull of the set consisting of all marginal worth vectors ofv; or
- the extreme points of the core ofv are exactly the marginal worth vectors ofv.
8.
Haruko Okamura 《Graphs and Combinatorics》1990,6(2):179-185
We prove:
- Fork ≥ 2 andα = 0, 1, every (4k + 2α)-edge-connected graph is weakly (3k + 2α)-linked.
- IfG is ak-edge-connected graph (k ≥ 2),s, t are vertices andf is an edge, then there exists a pathP betweens andt such thatf ? E(P) andG ? E(P) ? f is (k ? 2)-edge-connected, whereE(P) denotes the edge set ofP.
9.
Ramanujan graphs 总被引:2,自引:0,他引:2
A large family of explicitk-regular Cayley graphsX is presented. These graphs satisfy a number of extremal combinatorial properties.
- For eigenvaluesλ ofX eitherλ=±k or ¦λ¦≦2 √k?1. This property is optimal and leads to the best known explicit expander graphs.
- The girth ofX is asymptotically ≧4/3 log k?1 ¦X¦ which gives larger girth than was previously known by explicit or non-explicit constructions.
10.
We show that the problem of constructing a perfect matching in a graph is in the complexity class Random NC; i.e., the problem is solvable in polylog time by a randomized parallel algorithm using a polynomial-bounded number of processors. We also show that several related problems lie in Random NC. These include:
- Constructing a perfect matching of maximum weight in a graph whose edge weights are given in unary notation;
- Constructing a maximum-cardinality matching;
- Constructing a matching covering a set of vertices of maximum weight in a graph whose vertex weights are given in binary;
- Constructing a maximums-t flow in a directed graph whose edge weights are given in unary.
11.
H. D. Macpherson 《Periodica Mathematica Hungarica》1986,17(3):211-233
It is shown that ifG is a permutation group on an infinite setX, andG is (k?1)-transitive but notk-transitive (wherek ≥ 5), then the following hold:
- G is not (k + 3)-homogeneous.
- IfG is (k + 2)-homogeneous, then the group induced byG on ak-subset ofX is the alternating groupA k .
12.
13.
Sylvie Guerre 《Israel Journal of Mathematics》1986,56(3):361-380
Let 1≦q<p<2. We construct a bounded sequence (X n ) n∈N inL q which defines a typeσ onL q , such that:
- (X n ) n∈N is equivalent to the unit vector basis ofl p .
- The l-conic classK 1(σ) generated byσ is not relatively compact for the topology of uniform convergence on bounded sets ofL q .
- (X n ) n∈N has no almost exchangeable subsequence after any change of density.
14.
Horst Herrlich 《Applied Categorical Structures》1996,4(1):1-14
In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:
- C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
- Equivalent are:
- the axiom of choice,
- A-compactness = D-compactness,
- B-compactness = D-compactness,
- C-compactness = D-compactness and complete regularity,
- products of spaces with finite topologies are A-compact,
- products of A-compact spaces are A-compact,
- products of D-compact spaces are D-compact,
- powers X k of 2-point discrete spaces are D-compact,
- finite products of D-compact spaces are D-compact,
- finite coproducts of D-compact spaces are D-compact,
- D-compact Hausdorff spaces form an epireflective subcategory of Haus,
- spaces with finite topologies are D-compact.
- Equivalent are:
- the Boolean prime ideal theorem,
- A-compactness = B-compactness,
- A-compactness and complete regularity = C-compactness,
- products of spaces with finite underlying sets are A-compact,
- products of A-compact Hausdorff spaces are A-compact,
- powers X k of 2-point discrete spaces are A-compact,
- A-compact Hausdorff spaces form an epireflective subcategory of Haus.
- Equivalent are:
- either the axiom of choice holds or every ultrafilter is fixed,
- products of B-compact spaces are B-compact.
- Equivalent are:
- Dedekind-finite sets are finite,
- every set carries some D-compact Hausdorff topology,
- every T 1-space has a T 1-D-compactification,
- Alexandroff-compactifications of discrete spaces and D-compact.
15.
THINNINGOFPOINTPROCESSES,REVISITEDHESHENGWU(何声武)(DepartmentofMathematicalStatistics,EastChinaNormalUniversityShanghai200062,C... 相似文献
16.
V. É. Geit 《Mathematical Notes》1971,10(5):768-776
We prove the following: for every sequence {Fv}, Fv ? 0, Fv > 0 there exists a functionf such that
- En(f)?Fn (n=0, 1, 2, ...) and
- Akn?k? v=1 n vk?1 Fv?1?Ωk (f, n?1) (n=1, 2, ...).
17.
Jon Lee 《Mathematical Programming》1990,47(1-3):441-459
Thespectrum spec( ) of a convex polytope is defined as the ordered (non-increasing) list of squared singular values of [A|1], where the rows ofA are the extreme points of . The number of non-zeros in spec( ) exceeds the dimension of by one. Hence, the dimension of a polytope can be established by determining its spectrum. Indeed, this provides a new method for establishing the dimension of a polytope, as the spectrum of a polytope can be established without appealing to a direct proof of its dimension. The spectrum is determined for the four families of polytopes defined as the convex hulls of:
- The edge-incidence vectors of cutsets induced by balanced bipartitions of the vertices in the complete undirected graph on 2q vertices (see Section 6).
- The edge-incidence vectors of Hamiltonian tours in the complete undirected graph onn vertices (see Section 6).
- The arc-incidence vectors of directed Hamiltonian tours in the complete directed graph ofn nodes (see Section 7).
- The edge-incidence vectors of perfect matchings in the complete 3-uniform hypergraph on 3q vertices (see Section 8).
18.
Marcel Erné 《Algebra Universalis》1981,13(1):1-23
This paper deals with the question under which circumstances filter-theoretical order convergence in a product of posets may be computed componentwise, and the same problem is treated for convergence in the order topology (which may differ from order convergence). The main results are:
- Order convergence in a product of posets is obtained componentwise if and only if the number of non-bounded posets occurring in this product is finite (1.5).
- For any product of posets, the projections are open and continuous with respect to the order topologies (2.1).
- A productL of chainsL i has topological order convergence iff all but a finite number of the chains are bounded. In this case, the order topology onL agrees with the product topology (2.7).
- If (L i :j ∈J) is a countable family of lattices with topological order convergence and first countable order topologies then order topology of the product lattice and product topology coincide (2.8).
- LetP 1 be a poset with topological order convergence and locally compact order topology. Then for any posetP 2, the order topology ofP 1?P 2 coincides with the product topology (2.10).
- A latticeL which is a topological lattice in its order topology is join- and meet-continuous. The converse holds whenever the order topology ofL?L is the product topology (2.15).
19.
A seagull in a graph is an induced three-vertex path. When does a graph G have k pairwise vertex-disjoint seagulls? This is NP-complete in general, but for graphs with no stable set of size three we give a complete solution. This case is of special interest because of a connection with Hadwiger’s conjecture which was the motivation for this research; and we deduce a unification and strengthening of two theorems of Blasiak [2] concerned with Hadwiger’s conjecture. Our main result is that a graph G (different from the five-wheel) with no three-vertex stable set contains k disjoint seagulls if and only if
- |V (G)|≥3k
- G is k-connected
- for every clique C of G, if D denotes the set of vertices in V (G)\C that have both a neighbour and a non-neighbour in C then |D|+|V (G)\C|≥2k, and
- the complement graph of G has a matching with k edges.
20.
Mendel David 《Israel Journal of Mathematics》1971,9(1):34-42
LetH be a separable infinite-dimensional Hilbert space and letC be a normal operator andG a compact operator onH. It is proved that the following four conditions are equivalent.
- C +G is a commutatorAB-BA with self-adjointA.
- There exists an infinite orthonormal sequencee j inH such that |Σ j n =1 (Ce j, ej)| is bounded.
- C is not of the formC 1 ⊕C 2 whereC 1 has finite dimensional domain andC 2 satisfies inf {|(C 2 x, x)|: ‖x‖=1}>0.
- 0 is in the convex hull of the set of limit points of spC.