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1.
Jack M. Shapiro 《代数通讯》2013,41(6):1879-1898
For a field F with trivial involution we have the Karoubi L-groups ±1Ln(F). For 0≤n≤2 these groups are intimately related to subgroups of the classical Witt ring of quadratic forms. -1L 2(F) also has a presentation by symbols due to Matsumoto. In terms of this data we make explicit calculations for two cup product maps that appear in the L-theory of fields.  相似文献   

2.
Suppose that in a domain R(, B) of variables (r, ): (0 r , 1 +B(r–r 0 ) 2–B(r–r0), where > 0, B > 0, 1 < 0 < 2 are numbers) a metric ds2 = dr2 +G(r, )d 2 and a function k(r, ) are given. The problem of isometrically immersing ds2 in E 4 with prescribed Gaussian torsion is considered. The following is proved: The class C 5 metric ds 2 is locally realized in the form of a class C 3 surface F 2 whose Gaussian torsion is the prescribed class C 3 function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.  相似文献   

3.
In [Michailov I.M., On Galois cohomology and realizability of 2-groups as Galois groups, Cent. Eur. J. Math., 2011, 9(2), 403–419] we calculated the obstructions to the realizability as Galois groups of 14 non-abelian groups of order 2 n , n ≥ 4, having a cyclic subgroup of order 2 n−2, over fields containing a primitive 2 n−3th root of unity. In the present paper we obtain necessary and sufficient conditions for the realizability of the remaining 8 groups that are not direct products of smaller groups.  相似文献   

4.
It is proved that an integrable functionf can be approximated by the Kantorovich type modification of the Szász—Mirakjan and Baskakov operators inL 1 metric in the optimal order {n –1} if and only if 2 f is of bounded variation where and , respectively.  相似文献   

5.
This paper establishes the estimates of L 3/2 norm of the vector fields in a bounded domain with vanishing tangential component on the boundary, in terms of the L 1 norm of the curl, the negative exponent Sobolev norm of the divergence, and on some quantities depending on the topology of the domain. As the similar proof we also obtain the estimates of L p norm of the vector fields in terms of the negative exponent Sobolev norms of the curl and divergence.  相似文献   

6.
In this paper we study the regularity of flow maps of H 3/2-vector fields on the circle in terms of fractional Sobolev spaces. This problem is motivated by the understanding of the geometry of Bers’s universal Teichmüller space.  相似文献   

7.
We study the manifold of complex Bloch-Floquet eigenfunctions for the zero level of a two-dimensional nonrelativistic Pauli operator describing the propagation of a charged particle in a periodic magnetic field with zero flux through the elementary cell and a zero electric field. We study this manifold in full detail for a wide class of algebraic-geometric operators. In the nonzero flux case, the Pauli operator ground state was found by Aharonov and Casher for fields rapidly decreasing at infinity and by Dubrovin and Novikov for periodic fields. Algebraic-geometric operators were not previously known for fields with nonzero flux because the complex continuation of “magnetic” Bloch-Floquet eigenfunctions behaves wildly at infinity. We construct several nonsingular algebraic-geometric periodic fields (with zero flux through the elementary cell) corresponding to complex Riemann surfaces of genus zero. For higher genera, we construct periodic operators with interesting magnetic fields and with the Aharonov-Bohm phenomenon. Algebraic-geometric solutions of genus zero also generate soliton-like nonsingular magnetic fields whose flux through a disc of radius R is proportional to R (and diverges slowly as R → ∞). In this case, we find the most interesting ground states in the Hilbert space L 2 (ℝ 2 ).  相似文献   

8.
9.
Lately, I. Miyada proved that there are only finitely many imaginary abelian number fields with Galois groups of exponents ≤2 with one class in each genus. He also proved that under the assumption of the Riemann hypothesis there are exactly 301 such number fields. Here, we prove the following finiteness theorem: there are only finitely many imaginary abelian number fields with one class in each genus. We note that our proof would make it possible to find an explict upper bound on the discriminants of these number fields which are neither quadratic nor biquadratic bicyclic. However, we do not go into any explicit determination.  相似文献   

10.
Let be an imaginary biquadratic number field with Clk,2, the 2-class group of k, isomorphic to Z/2Z × Z/2mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1. 2000 Mathematics Subject Classification Primary—11R29  相似文献   

11.
A Locally Correctable Code (LCC) is an error correcting code that has a probabilistic self-correcting algorithm that, with high probability, can correct any coordinate of the codeword by looking at only a few other coordinates, even if a δ fraction of the coordinates is corrupted. LCCs are a stronger form of LDCs (Locally Decodable Codes) which have received a lot of attention recently due to their many applications and surprising constructions.In this work, we show a separation between linear 2-query LDCs and LCCs over finite fields of prime order. Specifically, we prove a lower bound of the form p Ω(δd) on the length of linear 2-query LCCs over F p , that encode messages of length d. Our bound improves over the known bound of 2 Ω(δd) [8,10,6] which is tight for LDCs. Our proof makes use of tools from additive combinatorics which have played an important role in several recent results in theoretical computer science.We also obtain, as corollaries of our main theorem, new results in incidence geometry over finite fields. The first is an improvement to the Sylvester-Gallai theorem over finite fields [14] and the second is a new analog of Beck's theorem over finite fields.The paper also contains an appendix, written by Sergey Yekhanin, showing that there do exist nonlinear LCCs of size 2 O(d) over F p , thus highlighting the importance of the linearity assumption for our result.  相似文献   

12.
Suppose that G is an infinite group generated by obliquereflections with respect to hyperplanes in the real spaceE m and that theµ j-planes IIµj = IIdj IIj (j = G(u) -orbits of directions of symmetryu (hyperplanes of symmetry conjugate to vectors IIdj pass through j-planesII j). A complete solution of the problem of the mutual position of three IIj is given. Systems of generatrices of the rings of invariants of a series of groups G are found.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 42–51, 1991.  相似文献   

13.
14.
Using Becker's results we obtain here a simple first order axiomatization, looking like those by Artin-Schreier and also written in the language of fields, for the theory of Rolle fields (i.e. fields with the Rolle's property for every order). In fields having a finite number of orders, we characterize Rolle fields as those which are pythagorean at level 4 and do not admit any algebraic extension of odd degree. Then we give an axiomatization for Rolle fields having exactly 2n orders (n≥0); in fact, for n=0 we recover an axiomatization of the theory of real-closed fields and for n=1 we get exactly an axiomatization given for the theory of chain-closed fields by the author in [G1]. Finally we prove that a Rolle field with exactly 2n orders is the intersection of n+1 real closures of the field.   相似文献   

15.
We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ2≃ℂ. We describe all representations of \mathfrakg\mathfrak{g} via vector fields in J 02=ℝ3(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants.  相似文献   

16.
The aim of this paper is to classify (lócally) all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classification of all non-equivalent transitive Lie algebras of vector fields in ℝ2 according to P.J. Olver [7].  相似文献   

17.
In this paper we will apply Biró's method in [A. Biró, Yokoi's conjecture, Acta Arith. 106 (2003) 85-104; A. Biró, Chowla's conjecture, Acta Arith. 107 (2003) 179-194] to class number 2 problem of real quadratic fields of Richaud-Degert type and will show that there are exactly 4 real quadratic fields of the form with class number 2, where n2+1 is a even square free integer.  相似文献   

18.
19.
We classify the foliations associated to Hamiltonian vector fields on C2, with an isolated singularity, admitting a semi-complete representative. In particular we also classify semi-complete foliations associated to the differential equation .  相似文献   

20.
Let X be a homogeneous polynomial vector field of degree 2 on $ \mathbb{S}^2 $ \mathbb{S}^2 . We show that if X has at least a non-hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on $ \mathbb{S}^2 $ \mathbb{S}^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 th Hilbert’s problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on $ \mathbb{S}^2 $ \mathbb{S}^2 of degree n.  相似文献   

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