Topic
Quintic function
About: Quintic function is a research topic. Over the lifetime, 1677 publications have been published within this topic receiving 26780 citations.
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TL;DR: In this article, it was shown that the number of quintic fields having bounded discriminant at most X is a constant times X. In contrast with the quartic case, the authors of this paper show that a density of 100% of all quintic rings and fields, when ordered by absolute discriminant, has Galois closure with full Galois group $S_5.
Abstract: We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an interpretation of the various constants appearing in these theorems in terms of local masses of quintic rings and fields.
243 citations
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01 Jan 1995TL;DR: A homogeneous polynomial equation in five variables determines a quintic 3-fp;d in ℂP4.
Abstract: A homogeneous polynomial equation in five variables determines a quintic 3-fp;d in ℂP4. Hodge numbers of a nonsingular quintic are know to be: h p, p = 1, p = 0, 1, 2, 3 (Kahler form and its powers), h3, 0 = h0,3 = 1 (a quintic happens to bear a holomorphic volume form), h2,1 = h1, 2 = 101 = 126 - 25 (it is the dimension of the space of all quintics modulo projective transformations, and h2,1 is responsible here for infinitesimal variations of the complex structure) and all the other h p,q = 0.
224 citations
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214 citations
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TL;DR: In this paper, the authors give a mathematical account of a recent string theory calcula- tion which predicts the number of rational curves on the generic quintic three-fold, using the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new q-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the mirror symmetry.
Abstract: We give a mathematical account of a recent string theory calcula- tion which predicts the number of rational curves on the generic quintic three- fold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new q-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the "mirror symmetry" phe- nomenon recently observed by string theorists. DEPARTMENT OF MATHEMATICS, DUKE UNIVERSITY, DURHAM, NORTH CAROLINA 27706 E-mail address: drm@math.duke.edu This content downloaded from 157.55.39.224 on Wed, 14 Dec 2016 04:59:36 UTC All use subject to http://about.jstor.org/terms
203 citations
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TL;DR: The work of N.A. and V.V. is supported by the Australian Photonics Cooperative Research Centre (APCRC) and the authors are grateful to Dr Adrian Ankiewicz for the critical reading of this manuscript.
Abstract: Soliton solutions of the one-dimensional (1D) complex Ginzburg-Landau equations (CGLE) are analyzed. We have developed a simple approach that applies equally to both the cubic and the quintic CGLE. This approach allows us to find an extensive list of soliton solutions of the CGLE, and to express all these solutions explicitly. In this way, we were able to classify them clearly. We have found and analyzed the class of solutions with fixed amplitude, revealed its singularities, and obtained a class of solitons with arbitrary amplitude, as well as some other special solutions. The stability of the solutions obtained is investigated numerically.
189 citations