共查询到20条相似文献，搜索用时 15 毫秒
1.
Christian Wagner. 《Mathematics of Computation》1996,65(214):785-800
We outline the determination of all imaginary quadratic fields with class number 5, 6 or 7.
2.
We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x
2+Ny
2 for a squarefree integer N.
相似文献
3.
Anitha Srinivasan. 《Mathematics of Computation》1998,67(223):1285-1308
In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field is presented, which computes the class number in expected time . The algorithm is a random version of Shanks' algorithm. One of the main steps in algorithms to compute the class number is the approximation of . Previous algorithms with the above running time , obtain an approximation for by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for without assuming the Riemann Hypothesis, by using a new technique that we call the `Random Summation Technique'. As a result, we are able to compute the regulator deterministically in expected time . However, our estimate of on the running time of our algorithm to compute the class number is not effective.
4.
5.
Let d≡ 5 mod 8 be a positive square-free integer and let h(d) be the class number of the real quadratic field ℚ(√d). Let p be a divisor of d = pq and let
. Assume that
is prime or equal to 1 for all integers x with 0≤x<W. Under the assumption that the Riemann hypothesis is true, we prove that if
, then h(d) < 2. Furthermore we show that h(d)< 2 implies d < 4245. In the case when there exists at least one split prime less than W, we prove the following results without any assumptions on the Riemann hypothesis. If
then h< 2 or h = 4. If
, then h≤ 2, h = 4 or h = 2t−2, where t is the number of primes dividing d. In the case when h = 2t−2 we have
, where φ = 2 or 4.
2000 Mathematics Subject Classification: Primary–11R29 相似文献
6.
本文用 Ｓｉｅｇｅｌ－Ｔａｔｕｚａｗａ定理证明了：当ｎ＞１．２×１０~１１时,至多有两个正 整数ｎ。使方程ｘｕ＋ｙｚ＋ｚｘ＝ｎ无适合（ｘ,ｙ,ｚ）＝１且０＜ｘ＜ｙ＜ｚ的解（ｘ,ｙ,ｚ）, 并给出类数为２的二次域与多项式表素数的一个结果． 相似文献
7.
We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimization problems. Finally, we also discuss the existence of a natural combinatorial structure on the set of similarity classes of well-rounded sublattices of the hexagonal lattice, induced by the action of a certain matrix monoid. 相似文献
8.
Detlev W. Hoffmann 《Mathematische Nachrichten》1998,190(1):149-168
Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices iW(φk) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN , we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ?(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ? ? φ ∈ In F and give some bounds on m depending on n which assure that they have the same splitting pattern. 相似文献
9.
设K是一个包含21/2的四次数域.设O?K是一个满足21/2∈O的序模.作者证明O的毕达哥拉斯数至多是5.这证实了Krásensky,Ra?ka和Sgallová的一个猜想.本文的证明用到了Beli关于二进局部域上二次格的范生成元基理论. 相似文献
10.
H. Väliaho 《Journal of Optimization Theory and Applications》1982,38(1):143-145
A short proof is given of the necessary and sufficient conditions for the positivity and nonnegativity of a quadratic form subject to linear constraints. 相似文献
11.
Mélanie Raczek 《代数通讯》2013,41(1):342-360
For a field F of characteristic different from 2, containing a square root of -1, endowed with an F×2-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant, and Hasse–Witt invariant can be written in the Witt ring as the sum of at most (d2)/8 3-fold Pfister forms. 相似文献
12.
Alfred J. van der Poorten. 《Mathematics of Computation》2003,72(244):1935-1946
This note is a detailed explanation of Shanks-Atkin NUCOMP--composition and reduction carried out ``simultaneously'--for all quadratic fields, that is, including real quadratic fields. That explanation incidentally deals with various ``exercises' left for confirmation by the reader in standard texts. Extensive testing in both the numerical and function field cases by Michael J Jacobson, Jr, reported elsewhere, confirms that NUCOMP as here described is in fact efficient for composition both of indefinite and of definite forms once the parameters are large enough to compensate for NUCOMP's extra overhead. In the numerical indefinite case that efficiency is a near doubling in speed already exhibited for discriminants as small as .
13.
Wai Kiu Chan Byeong-Kweon Oh 《Transactions of the American Mathematical Society》2003,355(6):2385-2396
An integral quadratic form of variables is said to be -regular if globally represents all quadratic forms of variables that are represented by the genus of . For any , it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of variables that are -regular. We also investigate similar finiteness results for almost -regular and spinor -regular quadratic forms. It is shown that for any , there are only finitely many equivalence classes of primitive positive definite spinor or almost -regular quadratic forms of variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).
14.
关于方程xy+yz+zx=n的正整数解 总被引：1，自引：1，他引：0
本文在广义Ｒｉｅｍａｎｎ猜想成立的条件下证明了：当且仅当正整数ｎ＝１,２,４,６,１０,１８,２２,３０,４２,５８,７０,７８,１０２,１３０,１９０,２１０,３３０,４６２时,方程ｘｙ＋ｙｚ＋ｚｘ＝ｎ无正整数解（ｘ,ｙ,ｚ）． 相似文献
15.
This paper deals with two different asymptotically fast algorithms for the computation of ideal sums in quadratic orders. If the class number of the quadratic number field is equal to 1, these algorithms can be used to calculate the GCD in the quadratic order. We show that the calculation of an ideal sum in a fixed quadratic order can be done as fast as in up to a constant factor, i.e., in where bounds the size of the operands and denotes an upper bound for the multiplication time of -bit integers. Using Schönhage-Strassen's asymptotically fast multiplication for -bit integers, we achieve
16.
LetA
1 andA
2 be two symmetric matrices of ordern×n. According to Yuan, there exists a convex combination of these matrices which is positive semidefinite, if and only if the functionxR
n
max {x
T
A
1
x,x
T
A
2
x} is nonnegative. We study the case in which more than two matrices are involved. We study also a related question concerning the maximization of the minimum eigenvalue of a convex combination of symmetric matrices.This research was partially supported by Dirección General de Investigación Científica y Técnica (DGICYT) under Project PB92-0615. 相似文献
17.
Hendrik Vogt 《Mathematische Nachrichten》2002,237(1):125-128
When defining an elliptic operator –∇ · (a∇) via the form method, one normally imposes pointwise conditions on the matrix valued function a in order to get positivity, ellipticity and sectoriality of the form. In this note we show that the pointwise conditions on a are equivalent to the corresponding global ones on the form. 相似文献
18.
B. M. Kim 《Commentarii Mathematici Helvetici》2000,75(3):410-414
In this paper, we will prove there are infinitely many integers n such that n
2— 1 is square-free and admits universal octonary diagonal quadratic forms.
Received: November 2, 1998. 相似文献
19.
For two given ternary quadratic formsf(x, y, z) andg( x, y, z), letr(f, n) andr(g, n) be the numbers of representations of n represented byf( x, y, z) and g( x, y, z) respectively. In this paper we study the following problem: when will we haver(f, n) =r(g, n) orr( f, n)≠r(g, n). Our method is to use elliptic curves and the corresponding new forms. 相似文献
20.