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1.
The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f
(k)(z)| < |h(z)|; (b) f
(k)(z) ≠ h(z). Then F is normal on D. 相似文献
2.
The h-super connectivity κh and the h-super edge-connectivity λh are more refined network reliability indices than the conneetivity and the edge-connectivity. This paper shows that for a connected balanced digraph D and its line digraph L, if D is optimally super edge-connected, then κ1(L) = 2λ1 (D), and that for a connected graph G and its line graph L, if one of κ1 (L) and λ(G) exists, then κ1(L) = λ2(G). This paper determines that κ1(B(d, n) is equal to 4d- 8 for n = 2 and d ≥ 4, and to 4d-4 for n ≥ 3 and d ≥ 3, and that κ1(K(d, n)) is equal to 4d- 4 for d 〉 2 and n ≥ 2 except K(2, 2). It then follows that B(d,n) and K(d, n) are both super connected for any d ≥ 2 and n ≥ 1. 相似文献
3.
Let F(z) = Re(P(z)) + h.o.t be such that M = (F = 0) defines a germ of real analytic Levi-flat at 0 ∈ ℂ
n
, n ≥ 2, where P (z) is a homogeneous polynomial of degree k with an isolated singularity at 0 ∈ ℂ
n
and Milnor number μ. We prove that there exists a holomorphic change of coordinate ϕ such that ϕ(M) = (Re(h) = 0), where h(z) is a polynomial of degree μ + 1 and j
0
k
(h) = P. 相似文献
4.
Sergiy Maksymenko 《Annals of Global Analysis and Geometry》2006,29(3):241-285
Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ
1 or the circle S
1, and D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h
−1 for h ∊ D(M) and f ∊ C^∞ (M,P). Let f: M → P be an arbitrary Morse mapping, Σ
f
the set of critical points of f, D(M,Σ
f
) the subgroup of D(M) preserving Σ
f
, and S(f), S (f,Σ
f
), O(f), and O(f,Σ
f
) the stabilizers and the orbits of f with respect to D(M) and D(M,Σ
f
). In fact S(f) = S(f,Σ
f
).In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σ
f
). It is proved that except for few cases the connected components of S(f) and O(f,Σ
f
) are contractible, π
k
O(f) = π
k
M for k ≥ 3, π2 O(f) = 0, and π1 O(f) is an extension of π1 D(M) ⊕ Z
k
(for some k ≥ 0) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f.We also generalize the methods of F. Sergeraert to give conditions for a finite codimension orbit of a tame smooth action of a tame Lie group on a tame Fréchet manifold to be a tame Fréchet manifold itself. In particular, we obtain that O(f) and O(f, Σ
f
) are tame Fréchet manifolds.
Communicated by Peter Michor Vienna
Mathematics Subject Classifications (2000): 37C05, 57S05, 57R45. 相似文献
5.
Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f
(k) share 0, and |f(z)| ≥ M whenever f
(k)(z) = h(z), then F is normal in D. The condition that f and f
(k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f
(k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f
(k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc. 相似文献
6.
Yair Glasner 《Transformation Groups》2009,14(4):787-800
Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F k ) on the set of marked, k-generated, dense subgroups $ {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\} Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F
k
) on the set of marked, k-generated, dense subgroups
Dk,A: = { h ? \textHom( Fk,A )| [`(
á f( Fk )
ñ )] = A } {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\}
. We prove the ergodicity of this action for the following two families of simple, totally disconnected, locally compact groups:
In contrast, a recent result of Minsky’s shows that the same action fails to be ergodic for A = PSL2(C) and, when k is even, also for A = PSL2(R). Therefore, if
k \geqslant 4 k \geqslant 4 is even and K is a local field (with char(K) ≠ 2), the action of Aut(F
k
) on
Dk,\textPS\textL2(K) {D_{k,{\text{PS}}{{\text{L}}_2}(K)}} is ergodic if and only if K is non-Archimedean.
Ergodicity implies that every “measurable property” either holds or fails to hold for almost every k-generated dense subgroup of A. 相似文献
• | A = PSL2(K) where K is a non-Archimedean local field (of characteristic ≠ 2); |
• | A = Aut0(T q+1)—the group of orientation-preserving automorphisms of a q + 1 regular tree, for q \geqslant 2.q \geqslant 2. |
7.
A criterion of normality based on a single holomorphic function 总被引:1,自引:0,他引:1
Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f
(k)(z)| ≤ c, where c is a constant. Then F is normal on D. 相似文献
8.
D. R. Heath-Brown 《Proceedings Mathematical Sciences》1994,104(1):13-29
LetF(x) =F[x1,…,xn]∈ℤ[x1,…,xn] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ
n
;F(x)=0, |x|⩽X}, where
. It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X
n−2+2/n
for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields.
However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process.
Dedicated to the memory of Professor K G Ramanathan 相似文献
9.
Let X be a Banach space, A : D(A) X → X the generator of a compact C0- semigroup S(t) : X → X, t ≥ 0, D a locally closed subset in X, and f : (a, b) × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u'(t) = Au(t) + f(t, u(t - q)), t ∈ [to, to + T], with initial condition uto = φ ∈C([-q, 0]; X), is the tangency condition lim infh10 h^-1d(S(h)v(O)+hf(t, v(-q)); D) = 0 for almost every t ∈ (a, b) and every v ∈ C([-q, 0]; X) with v(0), v(-q)∈ D. 相似文献
10.
On Group Chromatic Number of Graphs 总被引:2,自引:0,他引:2
Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For f ∈ F(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G)↦A such that for every directed edge uv from u to v, c(u)−c(v) ≠ f(uv). G is A-colorable under the orientation D if for any function f ∈ F(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ≥m, and is denoted by χg(G). In this note we will prove the following results. (1) Let H1 and H2 be two subgraphs of G such that V(H1)∩V(H2)=∅ and V(H1)∪V(H2)=V(G). Then χg(G)≤min{max{χg(H1), maxv∈V(H2)deg(v,G)+1},max{χg(H2), maxu∈V(H1) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K3,3-minor, then χg(G)≤5. 相似文献
11.
Zbigniew Lonc 《Graphs and Combinatorics》1992,8(4):333-341
Anh-uniform hypergraph generated by a set of edges {E
1,...,E
c} is said to be a delta-system Δ(p,h,c) if there is ap-element setF such that ∇F|=p andE
i⌢E
j=F,∀i≠j.
The main result of this paper says that givenp, h andc, there isn
0 such that forn≥n
0 the set of edges of a completeh-uniform hypergraphK
n
h can be partitioned into subsets generating isomorphic delta-systems Δ(p, h, c) if and only if
. This result is derived from a more general theorem in which the maximum number of delta-systems Δ(p, h, c) that can be packed intoK
n
h and the minimum number of delta-systems Δ(p, h, c) that can cover the edges ofK
n
h are determined for largen. Moreover, we prove a theorem on partitioning of the edge set ofK
n
h into subsets generating small but not necessarily isomorphic delta-systems. 相似文献
12.
Summary For P∈ F2[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σn≧0 p(A,n)zn ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ d|n, d∈A d, namely, the periodicity of the sequences (σ(A,2kn) mod 2k+1)n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula
to σ(A,2kn) mod 2k+1 will be established, from which it will be shown that the weight σ(A1,2kzi) mod 2k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$
is moved on some other orbit zj when A1 is replaced by A2 with A1= A(P1) and A2= A(P2) P1 and P2 being irreducible in F2[z] of the same odd order. 相似文献
13.
We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(Kr,r,n) such that every n-term graphic sequence π = (d1,d2,...,dn) with term sum σ(π) = d1 + d2 + ... + dn ≥ σ(Kr,r,n) is potentially Kr,r-graphic, where Kr,r is an r × r complete bipartite graph, i.e. π has a realization G containing Kr,r as its subgraph. In this paper, the values σ(Kr,r,n) for even r and n ≥ 4r2 - r - 6 and for odd r and n ≥ 4r2 + 3r - 8 are determined. 相似文献
14.
Here we prove the following result.
Theorem 1.1.Let X be an integral projective curve of arithmetic genus g and k≧ ≧4 an integer. Assume the existence of L ∈ Pick
(X) with h
0
(X, L)=2 and L spanned. Fix a rank 1 torsion free sheaf M on X with h
0(X,M)=r+1≧2, h1
(X, M)≧2 and M spanned by its global sections. Set d≔deg(M) and s≔max {n≧0:h
0 (X, M ⊗(L*)⊗n)>0}. Then one of the following cases occur:
We find also other upper bounds onh
0 (X, F).
(a) | M≊L ⊗r; |
(b) | M is the subsheaf of ω X⊗(L*)⊗t, t:=g−d+r−1, spanned by H0(X, ωX⊗(L*)⊗t); |
(c) | there is a rank 1 torsion free sheaf F on X with 1≦h 0(X, F)≦k−2 such that M≊L⊗s⊗F. Moreover, if we fix an integer m with 2≦m≦k−2 and assume r#(s+1)k−(ns+n+1) per every 2≦n≦m, we have h0 (X, F)≦k−m−1. |
Sunto In questo lavoro si dimostra il seguente teorema. Teorem 1.1.Sia X una curva proiettiva ridotta e irriducibile di genere aritmetico g e k≥4 un intero. Si supponga l'esistenza di L ε Pick (X) con h 0 (X, L)=2 e L generato. Si fissi un fascio senza torsione di rango uno M su X con h0 (X, M)=r++1≥2, h1 (X, M) ≧2 e M generato dalle sue sezioni globali. Si ponga d≔deg(M) e s≔max{n≧0:h 0(X, M ⊗(L*)⊗n)>0}. Allora si verifica uno dei casi seguenti:相似文献Si ricavano anche altre maggiorazioni suh 0,(X, F).
(a) M≊L ⊗r; (b) M è il sottofascio di ω X⊗(L*)⊗t, t:=g−d+r−1 generato da H0 (X, ωX⊗(L*)⊗t); (c) esiste un fascio senza torsione di rango un F su X con 1≦h 0 (X, F) <=k−2 tale che M ≊L ⊗8 ⊗ F. Inoltre, se si fissa un intero m con 2≦m≦k−2 e si suppone r#(s+1) k−(ns+n+1) per ogni 2≦n≦m, si ottiene h 0 (X, F)≦k−m−1.
15.
Yosef Stein 《Israel Journal of Mathematics》1989,68(1):109-122
LetK be an algebraically closed field of characteristic zero. ForA ∈K[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏
i=1
n(λ)
A
iλ
k
μ whereA
iλ are distinct primes. Let ϱλ(A) =n(λ) − 1 and let ρ(A) = Σλɛσ(A)ϱλ(A). The main result is the following:
Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA. 相似文献
16.
Fernando Giménez 《Israel Journal of Mathematics》1990,71(2):239-255
LetM be a Kaehler manifold of real dimension 2n with holomorphic sectional curvatureK
H≥4λ and antiholomorphic Ricci curvatureρ
A≥(2n−2)λ, andP is a complex hypersurface. We give a bound for the quotient (volume ofP)/(volume ofM) and prove that this bound is attained if and only ifP=C
P
n−1(λ) andM=C
P
n(λ). Moreover, we give some results on the volume of of tubes aboutP inM.
Work partially supported by a DGICYT Grant No. PS87-0115-CO3-01. 相似文献
17.
A. N. Dyogtev 《Algebra and Logic》1996,35(2):80-85
Let A⊆N={0,1,2,...} and β be an n-ary Boolean function. We call A a β-implicatively selector (β-IS) set if there exists an
n-ary selector general recursive function f such that (∀x1,...,xn)(β(χ(x1),...,χ(xn))=1⟹f(x1,...,xn)∈A), where χ is the characteristic function of A. Let F(m), m≥1, be the family of all d
m+1
*
-IS sets, where
, F(0)=N, and F(∞) is the class of all subsets in N. The basic result of the article says that the family of all β-IS sets coincides with one
of F(m), m≥0, or F(∞), and, moreover, the inclusions F(0)⊂F(1)⊂...⊂F(∞) hold.
Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 145–153, March–April, 1996. 相似文献
18.
An optimal bound on the tail distribution of the number of recurrences of an event in product spaces
Let X
1
,X
2
,... be independent random variables and a a positive real number. For the sake of illustration, suppose A is the event that |X
i+1
+...+X
j
|≥a for some integers 0≤i<j<∞. For each k≥2 we upper-bound the probability that A occurs k or more times, i.e. that A occurs on k or more disjoint intervals, in terms of P(A), the probability that A occurs at least once.
More generally, let X=(X
1
,X
2
,...)Ω=Π
j
≥1Ω
j
be a random element in a product probability space (Ω,ℬ,P=⊗
j
≥1
P
j
). We are interested in events AB that are (at most contable) unions of finite-dimensional cylinders. We term such sets sequentially searchable. Let L(A) denote the (random) number of disjoint intervals (i,j] such that the value of X
(i,j]
=(X
i+1
,...,X
j
) ensures that XA. By definition, for sequentially searchable A, P(A)≡P(L(A)≥1)=P(𝒩−ln
(P(Ac))
≥1), where 𝒩γ denotes a Poisson random variable with some parameter γ>0. Without further assumptions we prove that, if 0<P(A)<1, then P(L(A)≥k)<P(𝒩−ln
(P(Ac))
≥k) for all integers k≥2.
An application to sums of independent Banach space random elements in l
∞
is given showing how to extend our theorem to situations having dependent components.
Received: 8 June 2001 / Revised version: 30 October 2002 Published online: 15 April 2003
RID="*"
ID="*" Supported by NSF Grant DMS-99-72417.
RID="†"
ID="†" Supported by the Swedish Research Council.
Mathematics Subject Classification (2000): Primary 60E15, 60G50
Key words or phrases: Tail probability inequalities – Hoffmann-Jo rgensen inequality – Poisson bounds – Number of event recurrences – Number of
entrance times – Product spaces 相似文献
19.
Let P(G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete
tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng [Lau, G. C., Peng, Y. H.: Chromatic uniqueness of certain complete t-partite graphs. Ars Comb., 92, 353–376 (2009)] show that K(p − k, p − i, p) for i = 0, 1 are chromatically unique if p ≥ k + 2 ≥ 4. In this paper, we show that if 2 ≤ i ≤ 4, the complete tripartite graph K(p − k, p − i, p) is chromatically unique for integers k ≥ i and p ≥ k
2/4 + i + 1. 相似文献
20.
Tatsuo Nishitani 《Annali dell'Universita di Ferrara》2006,52(2):395-430
Abstract The well posedness of the Cauchy problem for the operator P=Dt2–Dxa(t,x)nDx,
with data on t=0 is studied assuming a ∈ CN(
(R)), s0>1 and sufficiently close to 1, a(t,x)≥ 0. Well posedness is proved in Gevrey classes γ(s), for
, n≥ n0.
Keywords: Partial differential equations, Cauchy problem, Well posedness 相似文献