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Let f be an analytic function in a convex domain D?C. A well-known theorem of Ozaki states that if f is analytic in D, and is given by f(z)=zp+n=p+1anzn for zD, and
Re{eiαf(p)(z)}>0,(zD),
for some real α, then f is at most p-valent in D. Ozaki's condition is a generalization of the well-known Noshiro–Warschawski univalence condition. The purpose of this paper is to provide some related sufficient conditions for functions analytic in the unit disk D={zC:|z|<1} to be p-valent in D, and to give an improvement to Ozaki's sufficient condition for p-valence when zD.  相似文献   

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We define a family KV(g,n+1) of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with n+1 boundary components. The problem KV(0,3) is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to KV(g,n+1) for arbitrary g and n. The key point is the solution to KV(1,1) based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman–Turaev Lie bialgebra g(g,n+1). In more detail, we show that every solution to KV(g,n+1) induces a Lie bialgebra isomorphism between g(g,n+1) and its associated graded grg(g,n+1). For g=0, a similar result was obtained by G. Massuyeau using the Kontsevich integral. For g1, n=0, our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.  相似文献   

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Let K be the algebraic closure of a finite field Fq of odd characteristic p. For a positive integer m prime to p, let F=K(x,y) be the transcendence degree 1 function field defined by yq+y=xm+x?m. Let t=xm(q?1) and H=K(t). The extension F|H is a non-Galois extension. Let K be the Galois closure of F with respect to H. By Stichtenoth [20], K has genus g(K)=(qm?1)(q?1), p-rank (Hasse–Witt invariant) γ(K)=(q?1)2 and a K-automorphism group of order at least 2q2m(q?1). In this paper we prove that this subgroup is the full K-automorphism group of K; more precisely AutK(K)=Δ?D where Δ is an elementary abelian p-group of order q2 and D has an index 2 cyclic subgroup of order m(q?1). In particular, m|AutK(K)|>g(K)3/2, and if K is ordinary (i.e. g(K)=γ(K)) then |AutK(K)|>g3/2. On the other hand, if G is a solvable subgroup of the K-automorphism group of an ordinary, transcendence degree 1 function field L of genus g(L)2 defined over K, then |AutK(K)|34(g(L)+1)3/2<682g(L)3/2; see [15]. This shows that K hits this bound up to the constant 682.Since AutK(K) has several subgroups, the fixed subfield FN of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in AutK(K) is large enough. This possibility is worked out for subgroups of Δ.  相似文献   

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Let R be an associative ring with unit and denote by K(R-Proj) the homotopy category of complexes of projective left R-modules. Neeman proved the theorem that K(R-Proj) is ?1-compactly generated, with the category K+(R-proj) of left bounded complexes of finitely generated projective R-modules providing an essentially small class of such generators. Another proof of Neeman's theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in K(R-Proj) vanishes in the Bousfield localization K(R-Flat)/K+(R-proj).  相似文献   

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We propose an algorithm for determining the primality of numbers M=Apn+wn where wnp11(modpn) and A<pn and give example when p=7. The pth reciprocity law is involved. If p is viewed as a constant, the algorithm runs in O((logM)2(loglogM)3) bit operations. Our results and those of P. Berrizbeitia et al. [7], [6] resolve an open problem proposed by F. Lemmermeyer in 2000 [15].  相似文献   

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Let Fq be a field of q elements, where q is a power of an odd prime p. The polynomial f(y)Fq[y] defined byf(y):=(1+y)(q+1)/2+(1y)(q+1)/2 has the property thatf(1y)=ρ(2)f(y), where ρ is the quadratic character on Fq. This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity.  相似文献   

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