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1.
To any integral algebra with valuation an abelian group is associated, which measures how much the uniqueness of the division with remainder is violated. The analogy with the divisor class group is discussed. Examples of such groups are computed in cases of formal local rings of some cusps on an algebraic curve. 相似文献
2.
An example is given of division algebras D
1 and D
2 of odd prime degree p over a field K such that D
1 and D
2 have no common subfield properly containing F, but D
1
⊗i
⊗
K
D
2 is not a division algebra for 1≤ i≤ p−1.
Supported in part by the NSF. 相似文献
3.
Let k be any field and G a finite group. Given a cohomology class α∈ H
2( G,k
*), where G acts trivially on k
*, one constructs the twisted group algebra k
αG. Unlike the group algebra kG, the twisted group algebra may be a division algebra (e.g. symbol algebras, where G⋞ Z
n×Z n). This paper has two main results: First we prove that if D= k
α
G is a division algebra central over k (equivalenty D has a projective k-basis) then G is nilpotent and G’ the commutator subgroup of G, is cyclic. Next we show that unless char( k)=0 and
, the division algebra D= k
α
G is a product of cyclic algebras. Furthermore, if D
p is a p-primary factor of D, then D
p is a product of cyclic algebras where all but possibly one are symbol algebras. If char( k)=0 and
, the same result holds for D
p, p odd. If p=2 we show that D
2 is a product of quaternion algebras with (possibly) a crossed product algebra ( L/k,β), Gal( L/k)⋞ Z
2×Z 2n. 相似文献
4.
The genus gen( D) of a finite-dimensional central division algebra D over a field F is defined as the collection of classes [ D′] ∈ Br( F), where D′ is a central division F-algebra having the same maximal subfields as D. For any prime p, we construct a division algebra of degree p with infinite genus. Moreover, we show that there exists a field K such that there are infinitely many nonisomorphic central division K-algebras of degree p and any two such algebras have the same genus. 相似文献
5.
Using elementary methods we prove a theorem of Rost, Serre, and Tignol that any division algebra of degree 4 over a -field containing is cyclic. Our methods also show any division algebra of degree 8 over a -field containing is cyclic. 相似文献
9.
We develop necessary and sufficient conditions for central simple algebras to have involutions of the first kind, and to be
tensor products of quaternion subalgebras. The theory is then applied to give an example of a division algebra of degree 8
with involution (of the first kind), without quaternion subalgebras, answering an old question of Albert; another example
is a division algebra of degree 4 with involution (*) has no (*)-invariant quaternion subalgebras.
The research of the second author is supported by the Anshel Pfeffer Chair.
The third author would like to express his gratitude to Professor J. Tits for many stimulating conversations. 相似文献
14.
Let F be the class of all 1-periodic real functions with absolutely convergent Fourier series expansion and let ( xn) n ≧ 0 be the van der Corput sequence. In this paper results on the boundedness of
are given. We give a criterion on the convergence rate of the Fourier coefficients of f such that the above sum is bounded independently of N. Further we show that our result is also best possible.
The first author is supported by the Australian Research Council under its Center of Excellence Program. The second author
is supported by the Austrian Research Foundation (FWF), Project S9609 that is part of the Austrian National Research Network
“Analytic Combinatorics and Probabilistic Number Theory”.
Received: 14 December 2005 相似文献
15.
The usual Sobolev inequality in n, n ? 3, asserts that , with Sn being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ? n. Two kinds of inequalities are established: (i) If ? = 0 on ?Ω, then with and with . (ii) If ? ≠ 0 on ?Ω, then with . Some further results and open problems in this area are also presented. 相似文献
16.
Let be an algebraically closed field of characteristic 0. We prove that any division algebra over whose ramification locus lies on a quartic curve is cyclic. 相似文献
18.
Improved Hardy inequalities, involving remainder terms, are established both in the classical and in the limiting case. The relevant remainders depend on a suitable distance from the families of the “virtual” extremals. 相似文献
19.
Understanding the solution of a problem may require the reader to have background knowledge on the subject. For instance, finding an integer which, when divided by a nonzero integer leaves a remainder; but when divided by another nonzero integer may leave a different remainder. To find a smallest positive integer or a set of integers following the given conditions, one may need to understand the concept of modulo arithmetic in number theory. The Chinese Remainder Theorem is a known method to solve these types of problems using modulo arithmetic. In this paper, an efficient remainder rule has been proposed based on basic mathematical concepts. These core concepts are as follows: basic remainder rules of divisions, linear equation in slope intercept form, arithmetic progression and the use of a graphing calculator. These are easily understood by students who have taken prealgebra or intermediate algebra. 相似文献
20.
In this note we consider the Sobolev inequality where is the best Sobolev constant and is the space obtained by taking the completion of with the norm . We prove here a refined version of this inequality, where is a positive constant, the distance is taken in the Sobolev space , and is the set of solutions which attain the Sobolev equality. This generalizes a result of Bianchi and Egnell (A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18-24), which was posed by Brezis and Lieb (Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86). regarding the classical Sobolev inequality A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation where and is the unique radial function in with . We will show that the eigenvalues of the above equation are discrete: and the corresponding eigenfunction spaces are 相似文献
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