Quintic function

About: Quintic function is a research topic. Over the lifetime, 1677 publications have been published within this topic receiving 26780 citations.

Almost sure local well-posedness for the supercritical quintic NLS

Abstract: This paper studies the quintic nonlinear Schrodinger equation on ℝ d with randomized initial data below the critical regularity H ( d − 1 ) ∕ 2 for d ≥ 3 . The main result is a proof of almost sure local well-posedness given a Wiener randomization of the data in H s for s ∈ ( 1 2 ( d − 2 ) , 1 2 ( d − 1 ) ) . The argument further develops the techniques introduced in the work of A. Benyi, T. Oh and O. Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global well-posedness.

On the field intersection problem of generic polynomials: a survey

Abstract: Let k be a eld of characteristic 2. We survey a general method of the eld intersection problem of generic polynomials via formal Tschirnhausen transformation. We announce some of our recent results of cubic, quartic and quintic cases the details of which are to appear elsewhere. In this note, we give an explicit answer to the problem in the cases of cubic and dihedral quintic by using multi-resolvent polynomials.

Analytic Continuation of Hypergeometric Functions in the Resonant Case

Abstract: We perform the analytic continuation of solutions to the hypergeometric differential equation of order $n$ to the third regular singularity, usually denoted $z=1$, with the help of recurrences of their Mellin--Barnes integral representations. In the resonant case, there are necessarily logarithmic solutions. We apply the result to Picard-Fuchs equations of certain one--parameter families of Calabi--Yau manifolds, known as the mirror quartic and the mirror quintic.

Global classification of a class of cubic vector fields whose canonical regions are period annuli

Abstract: We study cubic vector fields with inverse radial symmetry, i.e. of the form ẋ = δx - y + ax2 + bxy + cy2 + σ(dx - y)(x2 + y2), ẏ = x + δy + ex2 + fxy + gy2 + σ(x + dy) (x2 + y2), having a center at the origin and at infinity; we shortly call them cubic irs-systems. These systems are known to be Hamiltonian or reversible. Here we provide an improvement of the algorithm that characterizes these systems and we give a new normal form. Our main result is the systematic classification of the global phase portraits of the cubic Hamiltonian irs-systems respecting time (i.e. σ = 1) up to topological and diffeomorphic equivalence. In particular, there are 22 (resp. 14) topologically different global phase portraits for the Hamiltonian (resp. reversible Hamiltonian) irs-systems on the Poincare disc. Finally we illustrate how to generalize our results to polynomial irs-systems of arbitrary degree. In particular, we study the bifurcation diagram of a 1-parameter subfamily of quintic Hamiltonian irs-systems. Moreover, we indicate how to construct a concrete reversible irs-system with a given configuration of singularities respecting their topological type and separatrix connections.

Efficient computation of root numbers and class numbers of parametrized families of real abelian number fields

Abstract: Let {K m } be a parametrized family of simplest real cyclic cubic, quartic, quintic or sextic number fields of known regulators, e.g., the so-called simplest cubic and quartic fields associated with the polynomials P m (x) = x3 - mx2 - (m + 3)x + 1 and P m (x) = x 4 - mx 3 - 6x 2 + mx + 1. We give explicit formulas for powers of the Gaussian sums attached to the characters associated with these simplest number fields. We deduce a method for computing the exact values of these Gaussian sums. These values are then used to efficiently compute class numbers of simplest fields. Finally, such class number computations yield many examples of real cyclotomic fields Q(ζ p )+ of prime conductors p > 3 and class numbers h + p greater than or equal to p. However, in accordance with Vandiver's conjecture, we found no example of p for which p divides h + p .