Choreography solutions of the $n$-body problem on $S^2$
We try to prove the existence of choreography solutions for the $n-$body problem on $S^2$. For the three-body problem, we show the existence of the 8-shape orbit on $S^2$.
math.DS↗
arXiv subjects
Shiqing Zhang
Publications and source records attributed to Shiqing Zhang.
Merging and Evolution: Improving Convolutional Neural Networks for Mobile Applications
Compact neural networks are inclined to exploit "sparsely-connected" convolutions such as depthwise convolution and group convolution for employment in mobile applications. Compared with standard "fully-connected" convolutions, these convolutions are more computationally economical. However, "sparsely-connected" convolutions block the inter-group information exchange, which induces severe performance degradation. To address this issue, we present two novel operations named merging and evolution to leverage the inter-group information. Our key idea is encoding the inter-group information with a narrow feature map, then combining the generated features with the original network for better representation. Taking advantage of the proposed operations, we then introduce the Merging-and-Evolution (ME) module, an architectural unit specifically designed for compact networks. Finally, we propose a family of compact neural networks called MENet based on ME modules. Extensive experiments on ILSVRC 2012 dataset and PASCAL VOC 2007 dataset demonstrate that MENet consistently outperforms other state-of-the-art compact networks under different computational budgets. For instance, under the computational budget of 140 MFLOPs, MENet surpasses ShuffleNet by 1% and MobileNet by 1.95% on ILSVRC 2012 top-1 accuracy, while by 2.3% and 4.1% on PASCAL VOC 2007 mAP, respectively.
cs.CV↗
On mountain pass theorem and its application to periodic solutions of some nonlinear discrete systems
We obtain a new quantitative deformation lemma, and then gain a new mountain pass theorem. More precisely, the new mountain pass theorem is independent of the functional value on the boundary of the mountain, which improves the well known results (\cite{AR,PS1,PS2,Qi,Wil}). Moreover, by our new mountain pass theorem, new existence of nontrivial periodic solutions for some nonlinear second-order discrete systems is obtained, which greatly improves the result in \cite{Z04}.
math.DS↗
On the symmetric central configurations for the planar 1+4-body problem
We study the planar symmetric central configurations of the $1+4$-body problem where the symmetry axis does not contain any infinitesimal masses. Under certain assumptions we find analytically some central configurations, and also get some numerical results of symmetric central configurations where infinitesimal masses are not necessarily equal.
math-ph↗
Generalized Mountain Pass Lemma Related with a Closed Subset and Locally Lipschitz Functionals
The classical Mountain Pass Lemma of Ambrosetti-Rabinowitz has been studied, extended and modified in several directions, notable examples would certainly include the generalization to locally Lipschitz functionals in K.C. Chang, analysis of the structure of the critical set in the mountain pass theorem by Hofer and Pucci-Serrin and Tian, the extension by Ghoussoub-Preiss to closed subsets in a Banach space, and variations found in the recent Peral . In this paper, we utilize the generalized gradient of Clarke and Ekeland's variational principle to generalize the Ghoussoub-Preiss's Theorem in the setting of locally Lipschitz functionals.
math.FA↗
Four-body Central Configurations with Adjacent Equal Masses
For any convex non-collinear central configuration of the planar Newtonian 4-body problem with adjacent equal masses $m_1=m_2\neq m_3=m_4$, with equal lengths for the two diagonals, we prove it must possess a symmetry and must be an isosceles trapezoid; furthermore, which is also an isosceles trapezoid when the length between $m_1$ and $ m_4$ equals the length between $m_2$ and $ m_3$.
math.DS↗
New Periodic Solutions for Newtonian $n$-Body Problems with Dihedral Group Symmetry and Topological Constraints
In this paper, we prove the existence of a family of new non-collision periodic solutions for the classical Newtonian $n$-body problems. In our assumption, the $n=2l\geq4$ particles are invariant under the dihedral rotation group $D_l$ in $\mathbb{R}^3$ such that, at each instant, the $n$ particles form two twisted $l$-regular polygons. Our approach is variational minimizing method and we show that the minimizers are collision-free by level estimates and local deformations.
math-ph↗
Action Minimizing Solutions of The One-Dimensional $N$-Body Problem With Equal Masses
When we use variational methods to study the Newtonian $N$-body problem, the main problem is how to avoid collisions. C.Marchal got a remarkable result, that is, a path minimizing the Lagrangian action functional between two given configurations is always a true (collision-free) solution, so long as the dimension $d$ of physical space $\mathbb{R}^d$ satisfies $d\geq2$. But Marchal's idea can't apply to the case of the one-dimensional physical space. In this paper, we will study the fixed-ends problem for the one-dimensional Newtonian $N$-body problem with equal masses to supplement Marchal's result. More precisely, we first get the isolated property of collision moments for a path minimizing the action functional between two given configurations, then, if the particles at two endpoints have the same order, the path minimizing the action functional is always a true (collision-free) solution; otherwise, although there must be collisions for any path, we can prove that there are at most $N! - 1$ collisions for any action minimizing path.
math-ph↗
A Note on Homoclinic Orbits for Second Order Hamiltonian Systems
In this paper, we study the existence for the homoclinic orbits for the second order Hamiltonian systems. Under suitable conditions on the potential $V$, we apply the direct method of variations and the Fourier analysis to prove the existence of homoclinc orbits.
math.DS↗
New Periodic Solutions of Singular Hamiltonian Systems with Fixed Energies
By using the variational minimizing method with a special constraint and the direct variational minimizing method without constraint, we study second order Hamiltonian systems with a singular potential $V\in C^2(R^n\backslash O,R)$ and $V\in C^1(R^2\backslash O,R)$ which may have an unbounded potential well, and prove the existence of non-trivial periodic solutions with a prescribed energy. Our results can be regarded as some complements of the well-known Theorems of Benci-Gluck-Ziller-Hayashi and Ambrosetti-Coti Zelati and so on.
math-ph↗
Periodic Solutions of Non-Autonomous Second Order Hamiltonian Systems
We try to generalize a result of M. Willem on forced periodic oscillations which required the assumption that the forced potential is periodic on spatial variables. In this paper, we only assume its integral on the time variable is periodic, and so we extend the result to cover the forced pendulum equation. We apply the direct variational minimizing method and Rabinowtz's saddle point theorem to study the periodic solution when the integral of the potential on the time variable is periodic.
math.CA↗
New Quantitative Deformation Lemma and New Mountain Pass Theorem
In this paper, we obtain a new quantitative deformation Lemma so that we can obtain more critical points, especially for supinf critical value $c_1$, $x=φ^{-1}(c_1)$ is a new critical point. For $infmax$ critical value $c_2$, we can obtain two new critical points $x = 0$ (valley point) and $x = e$(peak point) ,comparing with Willem's variant of the mountain pass theorem of Ambrosetti-Rabinowitz,in which $φ(e)\leqφ(0)<c_2$, but in our new mountain pass theorem, $ φ(e)=c_2$.
math.FA↗
Periodic Solutions for $N$-Body-Type Problems
We consider non-autonomous $N$-body-type problems with strong force type potentials at the origin and sub-quadratic growth at infinity, and using Ljusternik-Schnirelmann theory, we prove the existence of unbounded sequences of critical values for the Lagrangian action corresponding to non-collision periodic solutions.
math-ph↗
Central Configurations Formed By Two Twisted Regular Polygons
In this paper, we study the necessary conditions and sufficient conditions for the central configurations formed by two twisted regular polygons (one N-regular polygon and one L-regular polygon). We wish to extend the results of the symmetrical central configurations formed by two twisted N-regular polygons, however, it will be proved that there are not more central configurations in a more general setting than the central configurations considered for some more particular situations before.
math-ph↗
Some Applications of Generalized Mountain Pass Lemma
The Ghoussoub-Preiss's generalized Mountain Pass Lemma with Cerami-Palais-Smale type condition is a generalization of classical MPL of Ambrosetti-Rabinowitz, we apply it to study the existence of the periodic solutions with a given energy for some second order Hamiltonian systems with symmetrical and non-symmetrical potentials.
math.FA↗
Saari's Conjecture for Elliptical Type $N$-Body Problem and An Application
By using an arithmetic fact, we will firstly prove Saari's conjecture in a particular case, which is called the Elliptical Type N-Body Problem, and then we apply it to prove that the variational minimal solution of the planar Newtonian N-body problem is precisely a relative equilibrium solution whose configuration minimizes the function $IU^2$, it's worth noticing that we don't need the hypothesis of Finiteness of Central Configurations. In the Planetary Restricted Problem (which ignore all the mutual gravitational interactions between the planets), the corresponding Saari's conjecture is stated and proved.
math-ph↗
New Periodic Solutions for Some Planar $N+3$-Body Problems with Newtonian Potentials
For some planar Newtonian $N+3$-body problems, we use variational minimization methods to prove the existence of new periodic solutions satisfying that $N$ bodies chase each other on a curve, and the other 3 bodies chase each other on another curve. From the definition of the group action in equations $(3.1)-(3.3)$, we can find that they are new solutions which are also different from all the examples of Ferrario and Terracini (2004)$[22]$.
math-ph↗
Notes on a Theorem of Benci-Gluck-Ziller-Hayashi
We use constrained variational minimizing methods to study the existence of periodic solutions with a prescribed energy for a class of second order Hamiltonian systems with a $C^2$ potential function which may have an unbounded potential well. Our result can be regarded as complementary to the well-known theorem of Benci-Gluck-Ziller and Hayashi.
math.CA↗