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1.
Let and be relatively prime monic irreducible polynomials in (). In this paper, we give an elementary proof for the following law of quadratic reciprocity in : where is the Legendre symbol.
2.
Jiecheng Chen Dashan Fan Meng Wang Xiangrong Zhu 《Proceedings of the American Mathematical Society》2008,136(9):3145-3153
We study the oscillatory hyper-Hilbert transform
along the curve , where are some real positive numbers. We prove that if , then is bounded on whenever . Furthermore, we also prove that is bounded on when . Our work improves and extends some known results by Chandarana in 1996 and in a preprint. As an application, we obtain an boundedness result for some strongly parabolic singular integrals with rough kernels.
along the curve , where are some real positive numbers. We prove that if , then is bounded on whenever . Furthermore, we also prove that is bounded on when . Our work improves and extends some known results by Chandarana in 1996 and in a preprint. As an application, we obtain an boundedness result for some strongly parabolic singular integrals with rough kernels.
3.
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.
4.
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.
5.
David Alonso-Gutié rrez 《Proceedings of the American Mathematical Society》2008,136(9):3293-3300
Let be the symmetric convex hull of independent random vectors uniformly distributed on the unit sphere of . We prove that, for every , the isotropy constant of is bounded by a constant with high probability, provided that .
6.
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.
7.
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 .
8.
Tomoaki Ono 《Proceedings of the American Mathematical Society》2008,136(9):3079-3087
Let be a tower of commutative rings where is a regular affine domain over an algebraically closed field of prime characteristic and is a regular domain. Suppose has a -basis over and . For a subset of whose elements satisfy a certain condition on linear independence, let be a set of maximal ideals of such that is a -basis of over . We shall characterize this set in a geometrical aspect.
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.
M. Z. Garaev 《Proceedings of the American Mathematical Society》2008,136(8):2735-2739
Let be the field of prime order It is known that for any integer one can construct a subset with such that One of the results of the present paper implies that if with then
11.
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.
12.
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.
13.
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 .
14.
A. Abdollahi 《Proceedings of the American Mathematical Society》2008,136(9):3185-3193
Let be a conformal automorphism on the unit disk and be the composition operator on the Dirichlet space induced by . In this article we completely determine the point spectrum, spectrum, essential spectrum and essential norm of the operators and self-commutators of , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.
15.
Bruce Ebanks 《Proceedings of the American Mathematical Society》2008,136(11):3911-3919
The main result is an improvement of previous results on the equation for a given function . We find its general solution assuming only continuous differentiability and local nonlinearity of . We also provide new results about the more general equation for a given function . Previous uniqueness results required strong regularity assumptions on a particular solution . Here we weaken the assumptions on considerably and find all solutions under slightly stronger regularity assumptions on .
16.
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.
17.
Natan Kruglyak Eric Setterqvist 《Proceedings of the American Mathematical Society》2008,136(7):2505-2513
It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions in , then we have the sharp estimate for In other words, for each and each integer . for all .
It is also shown, via a connection between the operator and Laguerre functions, that
18.
David Chillag 《Proceedings of the American Mathematical Society》2008,136(6):1961-1966
We show that if is a finite group, a conjugacy class of and , are the distinct elements in the multiset (here is the value of on any element of ), then This is a dual to a generalization of a theorem of Blichfeldt stating that if is a finite group, a generalized character and are the distinct values of , then
We also observe that in Blichfeldt's congruence can be replaced, with a minor adjustment, by any rational value of . A similar change can be done to the first congruence above.
19.
Dorin Bucur Alessandro Giacomini Paola Trebeschi 《Proceedings of the American Mathematical Society》2008,136(7):2535-2545
For , we prove that all the functions of satisfy the Whitney property; i.e., if is such that (in the sense of capacity) on a connected set , then is constant on .
20.
M. Drissi M. El Hodaibi E. H. Zerouali 《Proceedings of the American Mathematical Society》2008,136(5):1609-1617
Let be a Banach space and let be the class that consists of all operators such that for every , the range of has a finite-codimension when it is closed. For an integer , we define the class as an extension of . We then study spectral properties of such operators, and we extend some known results of multi-cyclic operators with .