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1.
Lucian Badescu 《Proceedings of the American Mathematical Society》2008,136(5):1505-1513
Let be a submanifold of dimension of the complex projective space . We prove results of the following type.i) If is irregular and , then the normal bundle is indecomposable. ii) If is irregular, and , then is not the direct sum of two vector bundles of rank . iii) If , and is decomposable, then the natural restriction map is an isomorphism (and, in particular, if is embedded Segre in , then is indecomposable). iv) Let and , and assume that is a direct sum of line bundles; if assume furthermore that is simply connected and is not divisible in . Then is a complete intersection. These results follow from Theorem 2.1 below together with Le Potier's vanishing theorem. The last statement also uses a criterion of Faltings for complete intersection. In the case when this fact was proved by M. Schneider in 1990 in a completely different way.
2.
Let denote the polynomial ring in variables over a field with each . Let be a homogeneous ideal of with and the Hilbert function of the quotient algebra . Given a numerical function satisfying for some homogeneous ideal of , we write for the set of those integers such that there exists a homogeneous ideal of with and with . It will be proved that one has either for some or .
3.
Stefano Meda Peter Sjö gren Maria Vallarino 《Proceedings of the American Mathematical Society》2008,136(8):2921-2931
We prove that if is in , is a Banach space, and is a linear operator defined on the space of finite linear combinations of -atoms in with the property that then admits a (unique) continuous extension to a bounded linear operator from to . We show that the same is true if we replace -atoms by continuous -atoms. This is known to be false for -atoms.
4.
It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .
5.
Clinton P. Curry John C. Mayer E. D. Tymchatyn 《Proceedings of the American Mathematical Society》2008,136(11):4045-4055
We show that a plane continuum is indecomposable iff has a sequence of not necessarily distinct complementary domains satisfying the double-pass condition: for any sequence of open arcs, with and , there is a sequence of shadows , where each is a shadow of , such that . Such an open arc divides into disjoint subdomains and , and a shadow (of ) is one of the sets .
6.
Natasha Dobrinen 《Proceedings of the American Mathematical Society》2008,136(5):1815-1821
Suppose are models of ZFC with the same ordinals, and that for all regular cardinals in , satisfies . If contains a sequence for some ordinal , then for all cardinals in with regular in and , is stationary in . That is, a new -sequence achieves global co-stationarity of the ground model.
7.
Let be the set of all positive integers , where are primes and possibly two, but not all three of them are equal. For any , define a function by where is the largest prime factor of . It is clear that if , then . For any , define , for . An element is semi-periodic if there exists a nonnegative integer and a positive integer such that . We use ind to denote the least such nonnegative integer . Wushi Goldring [Dynamics of the function and primes, J. Number Theory 119(2006), 86-98] proved that any element is semi-periodic. He showed that there exists such that , ind, and conjectured that ind can be arbitrarily large.
In this paper, it is proved that for any we have ind , and the Green-Tao Theorem on arithmetic progressions in the primes is employed to confirm Goldring's above conjecture.
8.
Wendy Lowen 《Proceedings of the American Mathematical Society》2008,136(9):3045-3050
For a scheme , we construct a sheaf of complexes on such that for every quasi-compact open , is quasi-isomorphic to the Hochschild complex of (Lowen and Van den Bergh, 2005). Since is moreover acyclic for taking sections on quasi-compact opens, we obtain a local to global spectral sequence for Hochschild cohomology if is quasi-compact.
9.
Mark Elin Marina Levenshtein Simeon Reich David Shoikhet 《Proceedings of the American Mathematical Society》2008,136(12):4313-4320
We present a rigidity property of holomorphic generators on the open unit ball of a Hilbert space . Namely, if is the generator of a one-parameter continuous semigroup on such that for some boundary point , the admissible limit - , then vanishes identically on .
10.
Xian-Jin Li 《Proceedings of the American Mathematical Society》2008,136(6):1945-1953
An explicit Dirichlet series is obtained, which represents an analytic function of in the half-plane except for having simple poles at points that correspond to exceptional eigenvalues of the non-Euclidean Laplacian for Hecke congruence subgroups by the relation for . Coefficients of the Dirichlet series involve all class numbers of real quadratic number fields. But, only the terms with for sufficiently large discriminants contribute to the residues of the Dirichlet series at the poles , where is the multiplicity of the eigenvalue for . This may indicate (I'm not able to prove yet) that the multiplicity of exceptional eigenvalues can be arbitrarily large. On the other hand, by density theorem the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on .
11.
Tonia Ricciardi 《Proceedings of the American Mathematical Society》2008,136(8):2771-2783
We obtain an estimate for the Hölder continuity exponent for weak solutions to the following elliptic equation in divergence form: where is a bounded open subset of and, for every , is a symmetric matrix with bounded measurable coefficients. Such an estimate ``interpolates' between the well-known estimate of Piccinini and Spagnolo in the isotropic case , where is a bounded measurable function, and our previous result in the unit determinant case . Furthermore, we show that our estimate is sharp. Indeed, for every we construct coefficient matrices such that is isotropic and has unit determinant, and such that our estimate for reduces to an equality, for every .
12.
Manuel Gonzá lez Mostafa Mbekhta Mourad Oudghiri 《Proceedings of the American Mathematical Society》2008,136(10):3521-3528
For a bounded operator acting on a complex Banach space, we show that if is not surjective, then is an isolated point of the surjective spectrum of if and only if , where is the quasinilpotent part of and is the analytic core for . Moreover, we study the operators for which . We show that for each of these operators , there exists a finite set consisting of Riesz points for such that and is connected, and derive some consequences.
13.
Sam Lichtenstein 《Proceedings of the American Mathematical Society》2008,136(10):3419-3428
Suppose that (resp. ) is a modular form of integral (resp. half-integral) weight with coefficients in the ring of integers of a number field . For any ideal , we bound the first prime for which (resp. ) is zero ( ). Applications include the solution to a question of Ono (2001) concerning partitions.
14.
Qing-Ming Cheng 《Proceedings of the American Mathematical Society》2008,136(9):3309-3318
Let be an -dimensional compact hypersurface with constant scalar curvature , , in a unit sphere . We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral of the mean curvature . In this paper, we first study the eigenvalue of the Jacobi operator of . We derive an optimal upper bound for the first eigenvalue of , and this bound is attained if and only if is a totally umbilical and non-totally geodesic hypersurface or is a Riemannian product , .
15.
Phil Hanlon 《Proceedings of the American Mathematical Society》2008,136(11):3741-3749
Let be a simple graph with nodes. The coloring complex of , as defined by Steingrimsson, has -faces consisting of all ordered set partitions, in which at least one contains an edge of . Jonsson proved that the homology of the coloring complex is concentrated in the top degree. In addition, Jonsson showed that the dimension of the top homology is one less than the number of acyclic orientations of .
In this paper, we show that the Eulerian idempotents give a decomposition of the top homology of into components . We go on to prove that the dimensions of the Hodge pieces of the homology are equal to the absolute values of the coefficients of the chromatic polynomial of . Specifically, if we write , then .
16.
Istvá n Juhá sz Zoltá n Szentmikló ssy 《Proceedings of the American Mathematical Society》2008,136(8):2979-2984
All spaces below are Tychonov. We define the projective - character of a space as the supremum of the values where ranges over all (Tychonov) continuous images of . Our main result says that every space has a -base whose order is ; that is, every point in is contained in at most -many members of the -base. Since for compact , this is a significant generalization of a celebrated result of Shapirovskii.
17.
Adam Osekowski 《Proceedings of the American Mathematical Society》2008,136(8):2951-2958
Let be a nonnegative supermartingale and be a predictable process with values in . Let denote the stochastic integral of with respect to . The paper contains the proof of the sharp inequality where . A discrete-time version of this inequality is also established.
18.
Yan-Ni Chen Hong-Ke Du Hai-Yan Zhang 《Proceedings of the American Mathematical Society》2008,136(10):3483-3492
Let and be two idempotents on a Hilbert space. In 2005, J. Giol in [Segments of bounded linear idempotents on a Hilbert space, J. Funct. Anal. 229(2005) 405-423] had established that, if is invertible, then and are homotopic with In this paper, we have given a necessary and sufficient condition that where denotes the minimal number of segments required to connect not only from to , but also from to in the set of idempotents.
19.
Dmitri Akhiezer Annett Pü ttmann 《Proceedings of the American Mathematical Society》2008,136(5):1649-1657
Let be a holomorphically separable irreducible reduced complex space, a connected compact Lie group acting on by holomorphic transformations, a Weyl involution, and an antiholomorphic map satisfying and for . We show that if is a multiplicity free -module, then maps every -orbit onto itself. For a spherical affine homogeneous space of the reductive group we construct an antiholomorphic map with these properties.
20.
Rui Miguel Saramago 《Proceedings of the American Mathematical Society》2008,136(8):2699-2709
We use Dieudonné theory for periodically graded Hopf rings to determine the Dieudonné ring structure of the -graded Morava -theory , with an odd prime, when applied to the -spectrum (and to ). We also expand these results in order to accomodate the case of the full Morava -theory .