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1.
It is well known that the number of isolated singular points of a hypersurface of degree d in ℂPm does not exceed the Arnol’d number Am(d), which is defined in combinatorial terms. In the paper it is proved that if b
m−1
±
(d) are the inertia indices of the intersection form of a nonsingular hypersurface of degree d in ℂPm, then the inequality Am(d)<min{b
m−1
+
(d), b
m−1
−
(d)} holds if and only if (m−5)(d−2)≥18 and (m,d)≠(7,12). The table of the Arnol’d numbers for 3≤m≤14, 3≤d≤17 and for 3≤m≤14,
d=18, 19 is given. Bibliography: 6 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 180–190.
Translated by O. A. Ivanov and N. Yu. Netsvetev. 相似文献
2.
Effective algebraic degeneracy 总被引:1,自引:0,他引:1
We show that for every smooth projective hypersurface X⊂ℙ
n+1 of degree d and of arbitrary dimension n
≥2, if X is generic, then there exists a proper algebraic subvariety Y
⊊
X such that every nonconstant entire holomorphic curve f
:ℂ→X has image f(ℂ) which lies in Y, as soon as its degree satisfies the effective lower bound
d\geqslant 2n5d\geqslant 2^{n^{5}}
. 相似文献
3.
Let Gn,k denote the oriented grassmann manifold of orientedk-planes in ℝn. It is shown that for any continuous mapf: Gn,k → Gn,k, dim Gn,k = dim Gm,l = l(m −l), the Brouwer’s degree is zero, providedl > 1,n ≠ m. Similar results for continuous mapsg: ℂGm,l → ℂGn,k,h: ℍGm,l → ℍGn,k, 1 ≤ l < k ≤ n/2, k(n — k) = l(m — l) are also obtained. 相似文献
4.
We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree ≤n and denominator degree ≤m
n
for meromorphic functions f on a compact set E of ℂ where m
n
=o(n/log n) as n→∞. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of
the maximal Green domain E
ρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary of E
ρ(f). The paper extends results for polynomial approximation and rational approximation with fixed degree of the denominator.
As applications, Padé approximation and real rational best approximants are considered. 相似文献
5.
Zbigniew Jelonek 《manuscripta mathematica》2003,110(2):145-157
Let be a polynomial dominant mapping and let deg f
i
≤d. We prove that the set K(f) of generalized critical values of f is contained in the algebraic hypersurface of degree at most D=(d+s(m−1)(d−1))
n
, where . This implies in particular that the set B(f) of bifurcations points of f is contained in the hypersurface of degree at most D=(d+s(m−1)(d−1))
n
. We give also an algorithm to compute the set K(f) effectively.
Received: 11 June 2001 / Revised version: 1 July 2002 Published online: 24 January 2003
The author is partially supported by the KBN grant 2 PO3A 017 22.
Mathematics Subject Classification (2000): 14D06, 14Q20, 51N10, 51N20, 15A04 相似文献
6.
In this paper we consider special elements of the Fock space #x2131;
n
. That is the space of entire functionsf:ℂ:
n
→ℂ, such that the followingL
2- condition is satisfied:
. Here we show that there exists an entire functiong:ℂ
n
→ℂ such that for every one-dimensional subspace Π⊂ℂ
n
and for all 0<∈<2 we have
, but in the limit case ∈=0 we have
. This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore
the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely
for spaces of entire functions. 相似文献
7.
Zhou Songping 《分析论及其应用》1989,5(1):11-14
In 1980, M. Hasson raised a conjecture as follows: Let N≥1, then there exists a function f0(x)∈C
[−1,1]
2N
, for N+1≤k≤2N, such that p
n
(k)
(f0,1)→f
0
(k)
(1), n→∞, where pn(f,x) is the algebraic polynomial of best approximation of degree ≤n to f(x). In this paper, a, positive answer to this conjecture
is given. 相似文献
8.
Dejan Kolarič 《Journal d'Analyse Mathématique》2009,107(1):239-250
We investigate the notion of CR transversality of a generic holomorphic map f: ℂ
n
→ ℂ
m
to a smooth CR submanifold M of ℂ
m
. We construct a stratification of the set of non-CR transversal points in the preimage M′ = f
−1 (M) by smooth submanifolds, consisting of points where the CR dimension of M′ is constant. We show the existence of a Whitney stratification for sets which are locally diffeomorphic to the product of
an open set and an analytic set.
Work on this paper was supported by ARRS, Republic of Slovenia. 相似文献
9.
Let f ∈ L
w
1
[−1, 1], let r
n,m(f) be the best rational L
w
1
-approximation for f with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let m = m(n), and let lim
n → ∞ (n-m(n)) = ∞. In this case, we show that the counting measures of certain subsets of sign changes of f-r
n,m
(f) converge weakly to the equilibrium measure on [−1, 1] as n → ∞. Moreover, we prove estimates for discrepancy between these counting measures and the equilibrium measure.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 283–287, February, 2006. 相似文献
10.
1. IntroductionLet f: Re -- R be a differelltiable fUnction. f reaChes its extremes on the setJ = {x E R"lfx(x) = 0}, (1.1)where,x(X) = (V,..., V)". (1.2)If jx can be observed exactly at any x e R", then there are various numerical methods toconstruct {xh}, xk E Re such that the distance d(xk, J) between uk and J tends to zero ask -- co. However, in many application problems jx can only be observed with noise, i.e.,the observation at time k 1 isYk 1 = fi(~k) (k 1, (1'3)where xk is … 相似文献
11.
E. A. Rovba 《Mathematical Notes》1997,61(2):221-226
Summation rational positive operatorsD
4n−n(x; f) of the Jackson type are constructed on the real axis. The corresponding approximations of continuous functionsf onℝwith coinciding finite limits limx→−∞
f(x) and limx→+∞
f(x) are estimated.
Translated fromMatematischeskie Zametki, Vol. 61, No. 2, pp. 270–277, February, 1997.
Translated by N. K. Kulman 相似文献
12.
Fang Liping 《数学学报(英文版)》1998,14(1):139-144
Letf be a holomorphic self-map of the punctured plane ℂ*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI
0(f)={z ∈ ℂ*:f
n
(z) → 0,n → ∞} andI
∞(f)={z ∈ ℂ*:f
n
(z) → 0,n → ∞}. We try to find the relation betweenI
0(f),I
∞(t) andJ(f). It is proved that both the boundary ofI
0(f) and the boundary ofI
∞)f) equal toJ(f),I
0(f) ∩J(f) ≠ θ andI
∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI
0(f) andI
∞(f) are not doubly-bounded.
Supported by the National Natural Science Foundation of China 相似文献
13.
Jie-hua MAI~ Tai-xiang SUN~ 《中国科学A辑(英文版)》2007,50(12):1818-1824
Let G be a graph and f:G→G be continuous.Denote by R(f) andΩ(f) the set of recurrent points and the set of non-wandering points of f respectively.LetΩ_0(f) = G andΩ_n(f)=Ω(f|_(Ω_(n-1)(f))) for all n∈N.The minimal m∈NU {∞} such thatΩ_m(f)=Ω_(m 1)(f) is called the depth of f.In this paper,we show thatΩ_2 (f)=(?) and the depth of f is at most 2.Furthermore,we obtain some properties of non-wandering points of f. 相似文献
14.
V. D. Zalizko 《Ukrainian Mathematical Journal》2007,59(1):28-44
The Jackson inequality
relates the value of the best uniform approximation E
n
(f) of a continuous 2π-periodic function f: ℝ → ℝ by trigonometric polynomials of degree ≤ n − 1 to its third modulus of continuity ω
3(f, t). In the present paper, we show that this inequality is true if continuous 2π-periodic functions that change their convexity
on [−π, π) only at every point of a fixed finite set consisting of an even number of points are approximated by polynomials
coconvex to them.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 29–43, January, 2007. 相似文献
15.
Soon Mo JUNG 《数学学报(英文版)》2006,22(2):583-586
The stability problems of the exponential (functional) equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved: Let X be a real (or complex) normed space. A mapping f : X → C is exponential if and only if f(x + y) - f(x)f(y) → 0 as ||x|| + ||y|| → ∞ under some suitable conditions. 相似文献
16.
We describe sequences of zeros of functionsf≢0 that are analytic in the half-plane ℂ+={z:Rez> and satisfy the condition
where 0≤σ<+∞ and η is a positive function continuously differentiable on [0; +∞) and such thatxη′(x)/η(x)→0 asx→+∞.
Drohobych Pedagogic University, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 7, pp. 904–909,
July, 1999. 相似文献
17.
W.M. Priestly 《Semigroup Forum》1998,56(3):301-322
t , for t ≥ 0, be a strongly continuous Markovian semigroup acting on C(X), where X is a compact Hausdorf space, and let D
denote the domain of its infinitesimal generator Z. Suppose D contains a (perhaps finite) family of functions f separating
the points of X and satisfying Zf2 = 2fZf. If either
(1) there exists δ > 0 such that (Tt f)2∈ D if 0 ≤ t ≤δ for each f in this family; or
(1′) for some core D′ of Z, g ∈ D′ implies g2∈ D, then the underlying Markoff process on X is deterministic. That is, there exists a semiflow — a semigroup (under composition)
of continuous functions φt from X into X — such that Ttf(x) = f(φt (x)). If the domain D should be an algebra then conditions (1) and (1′) hold trivially. Conversely, if we have a separating
family satisfying Zf2 = 2fZf then each of these conditions implies that D is an algebra. It is an open question as to whether these conditions
are redundant. If the functions φt are homeomorphisms from X onto X, then of course we have a Markovian group induced by a flow. This result is obtained by
first providing general results about the null-space N of the (function-valued) positive semidefinite quadratic form defined
by < f, g > = Z(fg) - fZg - gZf. The set N can be defined for any generator Z of a strongly continuous Markovian semigroup
and is equivalently given by
N = {f ∈ D| f2∈ D and Zf2 = 2fZf}
= {f ∈ D| Tt(f2)-(Ttf)2 is o(t2) in C(X)}.
In the general case N is an algebra closed under composition with any C1-function φ from the reals to the reals, and Z(φ[f]) = (Zf)φ′[f] if f ∈ N. This "chain rule" on N (on which Z must act as
a derivation) is a special case of a theorem for C2-functions φ which holds more generally for all f in d, viz.,
Z(φ[f] = (Zf) φ′[f] + ? <f, f> φ″[f],
Provided Z is a local operator and D is an algebra. In this case the form < f, g > itself enjoys the relation
< φ[f], ψ[g] > = φ′ [f] ψ′[g] < f, g >,
for C2functions φ and ψ. Some of the results and their proofs continue to hold when the setting is switched from the commutative
C*-algebra C(X) to a general (noncommutative) C*-algebra A. In the norm continuous case we obtain a sharp characterization
of Markovian semigroups that are groups: Let Tt = etz , defined for t ≥ 0, be a Markovian semigroup acting on a C*-algebra A that is norm continuous, i.e., ||Tt - I|| ⇒ 0 as t ⇒ 0 +. Assume Z(a2) = a(Za) + (Za) a for some (perhaps finite) set of self-adjoint elements a that generate a Jordan algebra dense among the
self-adjoint elements of A. The etz , -∞ < t < ∞, is a group of Markovian operators. 相似文献
18.
Semeon Bogatyi Daciberg L. Gonçalves Elena Kudryavtseva Heiner Zieschang 《Central European Journal of Mathematics》2003,1(2):184-197
Let V be a closed surface, H⊑π1(V) a subgroup of finite index l and D=[A
1,...,A
m
] a collection of partitions of a given number d≥2 with positive defect v(D). When does there exist a connected branched covering f:W→V of order d with branch data D and
f∶W→V
It has been shown by geometric arguments [4] that, for l=1 and a surface V different from the sphere and the projective plane, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D)э0 mod 2. In the case l>1, the corresponding branched covering exists if and only if v(D)э0 mod 2, the number d/l is an integer, and each partition A
i
∈D splits into the union of l partitions of the number d/l. Here we give a purely algebraic proof of this result following the approach of Hurwitz [11].
The realization problem for the projective plane and l=1 has been solved in [7,8]. The case of the sphere is treated in [1, 2, 12, 7]. 相似文献
19.
Kâzim Ilhan Ikeda 《Proceedings Mathematical Sciences》2003,113(2):99-137
This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK
k. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren
G is the break in the upper ramification filtration ofG = Gal(E/K) defined by
. Next, we study the basic properties of the idealf(E/K) inO
k in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]).
After reviewing the Artin charactera
G : G → ℂ ofG := Gal(E/K) and Artin representationsA
g G → G →GL(V) corresponding toa
G : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5)
where Χgr
: G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then
where Gal(Eker(ρ)/Eker(ρ)•) is any maximal abelian normal subgroup of Gal(Eker(ρ)/K) containing Gal(Eker(ρ)
/K)′, and the break nG/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The
proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf.
[1]) and on metabelian local class field theory (cf. [8]).
We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A
G ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then
Kazim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V
n denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations
ω : (G/N)• → ℂΧ satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂδ} = ω =(G/N) and
where δ runs over R((G/N)•/(G/N)), a fixed given complete system of representatives of (G/N)•/(G/N), by declaring that ω1 ∼ ω2 if and only if ω1
= ω
2,δ for some δ ∈ R((G/N)•/(G/N)).
Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3. 相似文献
20.
Philippe Bonnet 《Transformation Groups》2007,12(4):619-630
Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism
Φ, we denote by k(X)Φ its field of invariants, i.e., the set of rational functions f on X such that f o Φ = f. Let n(Φ) be the transcendence degree
of k(X)Φ over k. In this paper we study the class of automorphisms Φ of X for which n(Φ) = dim X - 1. More precisely, we show that
under some conditions on X, every such automorphism is of the form Φ = ϕg, where ϕ is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application,
we determine the conjugacy classes of automorphisms of the plane for which n(Φ) = 1. 相似文献