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Let F be a field of characteristic 2. The aim of this paper is to give a complete proof of the norm theorem for singular F-quadratic forms which are not totally singular, i.e., we give necessary and sufficient conditions for which a normed irreducible polynomial of becomes a norm of such a quadratic form over the rational function field . This completes partial results proved on this question in [8]. Combining the present work with the papers [1] and [7], we obtain the norm theorem for any type of quadratic forms in characteristic 2. 相似文献
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Let q be a perfect power of a prime number p and be an elliptic curve over given by the equation . For a positive integer n we denote by the number of rational points on E (including infinity) over the extension . Under a mild technical condition, we show that the sequence contains at most 10200 perfect squares. If the mild condition is not satisfied, then is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range and . 相似文献
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Jacques Darné 《Journal of Pure and Applied Algebra》2019,223(12):5484-5525
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Motivated by Ramsey-type questions, we consider edge-colorings of complete graphs and complete bipartite graphs without rainbow path. Given two graphs and , the -colored Gallai–Ramsey number is defined to be the minimum integer such that and for every , every rainbow -free coloring (using all colors) of the complete graph contains a monochromatic copy of . In this paper, we first provide some exact values and bounds of . Moreover, we define the -colored bipartite Gallai–Ramsey number as the minimum integer such that and for every , every rainbow -free coloring (using all colors) of the complete bipartite graph contains a monochromatic copy of . Furthermore, we describe the structures of complete bipartite graph with no rainbow and , respectively. Finally, we find the exact values of (), (where is a subgraph of ), and by using the structural results. 相似文献
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We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form , where is a finite-dimensional Lie algebra over a field of characteristic zero, and I is a finite-codimensional ideal of generated by monomials. In particular, when is semisimple and is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in . As an application of our main result, we show that the center of the universal enveloping algebra of acts trivially on all irreducible finite-dimensional representations provided I has codimension at least two. 相似文献
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Guillaume Lecué Matthieu Lerasle 《Stochastic Processes and their Applications》2019,129(11):4385-4410
New robust estimators are introduced, derived from median-of-means principle and Le Cam’s aggregation of tests. Minimax sparse rates of convergence are obtained with exponential probability, under weak moment’s assumptions and possible contamination of the dataset. These derive from general risk bounds of the following informal structure In this result, the number of outliers may be as large as (number of data)(minimax rate) without affecting the rates. As an example, minimax rates of recovery of -sparse vectors in holding with exponentially large probability, are deduced for median-of-means versions of the LASSO when the noise has moments for some , the entries of the design matrix have moments and the dataset is corrupted by up to outliers. 相似文献
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Dapeng Zhan 《Stochastic Processes and their Applications》2019,129(1):129-152
We show that, for , the integral of the laws of two-sided radial SLE curves through different interior points against a measure with SLE Green’s function density is the law of a chordal SLE curve, biased by the path’s natural length. We also show that, for , the integral of the laws of extended SLE curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLE curve, biased by the path’s capacity length restricted in that set. Another result is that, for , if one integrates the laws of two-sided chordal SLE curves through different force points on against a measure with density on , then one also gets a law that is absolutely continuous w.r.t. that of a chordal SLE curve. To obtain these results, we develop a framework to study stochastic processes with random lifetime, and improve the traditional Girsanov’s Theorem. 相似文献
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Let be the finite field of order q. Let G be one of the three groups , or and let W be the standard n-dimensional representation of G. For non-negative integers m and d we let denote the representation of G given by the direct sum of m vectors and d covectors. We exhibit a minimal set of homogeneous invariant polynomials such that for all cases except when and or . 相似文献