Quintic function

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

Stable Soliton Propagation in a System with Spectral Filtering and Nonlinear Gain

Abstract: Stable soliton propagation in a system with linear and nonlinear gain and spectral filtering is investigated. Different types of exact analytical solutions of the cubic and the quintic complex Ginzburg-Landau equation (CGLE) are reviewed. The conditions to achieve stable soliton propagation are analyzed within the domain of validity of soliton perturbation theory. We derive an analytical expression defining the region in the parameter space where stable pulselike solutions exist, which agrees with the numerical results obtained by other authors. An analytical expression for the soliton amplitude corresponding to the quintic CGLE is also obtained. We show that the minimum value of this amplitude depends only on the ratio between the linear gain and the quintic gain saturating term.

The Equation that Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry (APS Colloquium, 2006)

Abstract: For thousands of years, mathematicians solved progressively more difficult algebraic equations, from the simple quadractic to the more complex quartic equation, yielding important insights along the way. Then they were stumped by the quintic equation, which resisted solutions for three centuries, until two great prodigies independently proved that quaintic equations cannot be solved by simple formula. These geniuses, a young Norwegian named Niels Henrik Abel and an even younger Frenchman named Evariste Galois, both died tragically. Galois' work gave rise to group theory, the "language" that defines symmetry. Group theory explains much about the aesthetics of our world, from the choosing of mates to Rubik's cube, Bach's musical compositions, the physics of subatomic particles, and the popularity of Anna Kournikova. Some of the mysteries surrounding Galois' death, which have lingered for more than 170 years, are finally resolved in The Equation that Couldn't Be Solved. Livio will discuss this first popular-level book to explore group theory, not through abstract formulas but in a dramatic account of the lives and work of some of the greatest mathematicians in history.

An Effective Theory of GW and FJRW Invariants of Quintics Calabi-Yau Manifolds

Abstract: This is the second part of the project toward an effective algorithm to evaluate all genus Gromov-Witten invariants of quintic Calabi-Yau threefolds. In this paper, the localization formula is derived, and algorithms toward evaluating these Gromov-Witten invariants are derived.

Stability criterion for bright solitary waves of the perturbed cubic-quintic Schroedinger equation

Abstract: The stability of the bright solitary wave solution to the perturbed cubic-quintic Schroedinger equation is considered. It is shown that in a certain region of parameter space these solutions are unstable, with the instability being manifested as a small positive eigenvalue. Furthermore, it is shown that in the complimentary region of parameter space there are no small unstable eigenvalues. The proof involves a novel calculation of the Evans function, which is of interest in its own right. As a consequence of the eigenvalue calculation, it is additionally shown that N-bump bright solitary waves bifurcate from the primary wave.

An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type

Abstract: Due to the work of Arthur and Clozel [AC], the automorphic induction for cyclic Galois extensions of prime degree is well understood in great generality. It is not the case for non-normal extensions, even for monomial representations, i.e., the ones induced from grossencharacters. The only examples we have at the moment are: first, non-normal cubic automorphic induction due to Jacquet, Piatetski-Shapiro and Shalika [JPSS]. They obtained the automorphic induction as a consequence of the converse theorem on GL3. The second example is that of Harris [H]. He constructed automorphic induction for special class of algebraic Hecke characters of (suitable) non-normal extensions with solvable Galois closure. In this note, we give an example of automorphic induction for nonnormal quintic extension whose Galois closure is not solvable (Theorem 6.3). In fact, the Galois group is A5, the alternating group on 5 letters. The key observation due to Ramakrishnan is that symmetric fourth of the two dimensional icosahedral representation is equivalent to a suitable twist (by a character) of the five dimensional monomial representation of A5 . Our result follows immediately by combining results of K. Buzzard, M. Dickinson, N. Shepherd-Barron and R. Taylor [BDST] or R. Taylor’s result [Ta2] on modularity of certain icosahedral representations and our result on the functoriality of symmetric fourth [Ki]. We also prove the modularity of all symmetric powers of cuspidal representations of icosahedral type (Theorem 6.4).