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1.
E. A. Kudryavtseva 《Moscow University Mathematics Bulletin》2012,67(1):1-10
Let M be a smooth closed orientable surface and F = F
p,q,r
be the space of Morse functions on M having exactly p points of local minimum, q ≥ 1 saddle critical points, and r points of local maximum, moreover, all the points are fixed. Let F
f
be a connected component of a function f ∈ F in F.We construct a surjection π
0(F) → ℤ
p+r−1 by means of the winding number introduced by Reinhart (1960). In particular, |π0(F)| = ∞, and the component F
f
is not preserved under the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one
of which is a saddle point. Let D be the group of orientation preserving diffeomorphisms of M leaving fixed the critical points, D
0 be the connected component of id
M
in D, and D
f
⊂ D be the set of diffeomorphisms preserving F
f
. Let H
f
be the subgroup of D
f
generated by D
0 and all diffeomorphisms h ∈ D preserving some function f
1 ∈ F
f
, and let H
f
abs be its subgroup generated by D
0 and the Dehn twists about the components of level curves of the functions f
1 ∈ F
f
. We prove thatH
f
abs ⊊ D
f
for q ≥ 2 and construct an epimorphism D
f
/H
f
abs → ℤ2
q−1 by means of the winding number. A finite polyhedral complex K = K
p,q,r
associated with the space F is defined. An epimorphism μ: π
1(K) → D
f
/H
f
and finite generating sets for the groups D
f
/D
0 and D
f
/H
f
in terms of the 2-skeleton of the complex K are constructed. 相似文献
2.
Jing YANG Shi Xin LUO Ke Qin FENG 《数学学报(英文版)》2006,22(3):833-844
Assume that m ≥ 2, p is a prime number, (m,p(p - 1)) = 1,-1 not belong to 〈p〉 belong to (Z/mZ)^* and [(Z/mZ)^*:〈p〉]=4.In this paper, we calculate the value of Gauss sum G(X)=∑x∈F^*x(x)ζp^T(x) over Fq,where q=p^f,f=φ(m)/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G(x) belongs to the decomposition field K of p in Q(ζm) and K is an imaginary quartic abelian unmber field.When the Galois group Gal(K/Q) is cyclic,we have studied this cyclic case in anotyer paper:"Gauss sums of index four:(1)cyclic case"(accepted by Acta Mathematica Sinica,2003).In this paper we deal with the non-cyclic case. 相似文献
3.
Christiane Kraus 《Constructive Approximation》2011,33(2):191-217
The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions
in the complex plane to real analytic functions in ℝ
N
. In particular, we investigate the polynomial approximation behavior for functions F:L→ℂ, L={(Re z,Im z):z∈K}, of the structure F=g[`(h)]F=g\overline{h}, where g and h are holomorphic in a neighborhood of a compact set K⊂ℂ
N
. To this end the maximal convergence number ρ(S
c
,f) for continuous functions f defined on a compact set S
c
⊂ℂ
N
is connected to a maximal convergence number ρ(S
r
,F) for continuous functions F defined on a compact set S
r
⊂ℝ
N
. We prove that ρ(L,F)=min {ρ(K,h)),ρ(K,g)} for functions F=g[`(h)]F=g\overline{h} if K is either a closed Euclidean ball or a closed polydisc. Furthermore, we show that min {ρ(K,h)),ρ(K,g)}≤ρ(L,F) if K is regular in the sense of pluripotential theory and equality does not hold in general. Our results are based on the theory
of the pluricomplex Green’s function with pole at infinity and Lundin’s formula for Siciak’s extremal function Φ. A properly chosen transformation of Joukowski type plays an important role. 相似文献
4.
We study some properties of sets of differences of dense sets in ℤ2 and ℤ3 and their interplay with Bohr neighbourhoods in ℤ. We obtain, inter alia, the following results.
(i) | If E ⊂ ℤ2, $
\bar d
$
\bar d
(E) > 0 and p
i
, q
i
∈ ℤ[x], i = 1, ..., m satisfy p
i
(0) = q
i
(0) = 0, then there exists B ⊂ ℤ such that $
\bar d
$
\bar d
(B) > 0 and
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