Explicit Heegner points: Kolyvagin's conjecture and non-trivial elements in the Shafarevich-Tate group

Kolyvagin used Heegner points to associate a system of cohomology classes to an elliptic curve over Q and conjectured that the system contains a non-trivial class. His conjecture has profound implications on the structure of Selmer groups. We provide new computational and theoretical evidence for Kolyvagin's conjecture. More precisely, we explicitly approximate Heegner points over ring class fields and use these points to give evidence for the conjecture for specific elliptic curves of rank two. We explain how Kolyvagin's conjecture implies that if the analytic rank of an elliptic curve is at least two then the Zp-corank of the corresponding Selmer group is at least two as well. We also use explicitly computed Heegner points to produce non-trivial classes in the Shafarevich-Tate group.     

The Laplacian energy of random graphs

Gutman et al. introduced the concepts of energy E(G) and Laplacian energy EL(G) for a simple graph G, and furthermore, they proposed a conjecture that for every graph G, E(G) is not more than EL(G). Unfortunately, the conjecture turns out to be incorrect since Liu et al. and Stevanovi? et al. constructed counterexamples. However, So et al. verified the conjecture for bipartite graphs. In the present paper, we obtain, for a random graph, the lower and upper bounds of the Laplacian energy, and show that the conjecture is true for almost all graphs.     

Global estimates and energy identities for elliptic systems with antisymmetric potentials

We derive global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions. Moreover, we obtain new energy identities in such norms for sequences of solutions of these systems. The results apply to harmonic maps into general target manifolds and surfaces with prescribed mean curvature. In particular, the results confirm a conjecture of Rivière in the two-dimensional setting.     

Estimation of the time a perturbed Lagrangian system stays in an assigned region

The problem of estimating the mean time a weakly perturbed dynamical system stays in a fixed region of the phase variables is investigated. The motion is described by Lagrange's equations with an attractive force potential and in the presence of additive dissipative forces. The corresponding Cauchy problem is obtained in Hamiltonian variables for a non-linear first-order partial differential equation. Its classical positive solution specifies the action functional and the estimate sought for the time interval. The structure of the equations that allows of an explicit solution in terms of expressions for the kinetic and potential energy, as well as dissipative and dispersion matrices for a random Wiener-type perturbation, is established. The phenomenon of the escape of a phase point from different parts of the boundary of the region is investigated. Interesting problems of estimating the time for the inversion of the inner gimbal of a gyroscope, the time taken to reach an assigned level or a potential barrier of a multidimensional oscillatory system that has central symmetry, and the time a non-linear system with two degrees of freedom takes to escape over a potential barrier for a Henon–Heiles potential are investigated as examples.     

Counterexamples to the Connectivity Conjecture of the Mixed Cells

In [4] a conjecture concerning the connectivity of mixed cells of subdivisions for Minkowski sums of polytopes was formulated. This conjecture was, in fact, originally proposed by Pedersen [3]. It turns out that a positive confirmation of this conjecture can substantially speed up the algorithm for the ``dynamical lifting' developed in [4]. In the mean time, when the polyhedral method is used for solving polynomial systems by homotopy continuation methods [2], the positiveness of this conjecture also plays an important role in the efficiency of the algorithm. Very unfortunately, we found that this conjecture is inaccurate in general. In Section 1 a counterexample is presented for a general subdivision. In Section 2 another counterexample shows that even restricted to ``regular' subdivisions induced by liftings, this conjecture still fails to be true. Received September 18, 1996, and in revised form February 17, 1997.     

An Ansatz for the asymptotics of hypergeometric multisums

Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the asymptotic expansion of such sequences is to find a recurrence satisfied by them, convert it into a differential equation satisfied by their generating series, and analyze the singularities in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set, for which we conjecture (and in some cases prove) that it contains all the singularities of the generating series. Our construction of this finite set is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. The finite set can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive K-theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples.     

Comparing quantum and classical approaches in statistical physics

We discuss hypotheses pertaining to the behavior of the mean energy of an equilibrium system in the quantum and classical theories. We justify the hypotheses for the cases of the harmonic oscillator and the potential well. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 516–520, June, 2000.     

Asymptotic behaviour and bifurcation in competitive Lotka-Volterra Systems

A conjecture about global attraction in autonomous competitive Lotka-Volterra systems is clarified by investigating a special system with a circular matrix. Under suitable assumptions, this system meets the condition of the conjecture but Hopf bifurcation occurs in a particular instance. This shows that the conjecture is not true in general and the condition of the conjecture is too weak to guarantee global attraction of an equilibrium. Sufficient conditions for global attraction are also obtained for this system.     

Modeling Nondegenerate Bifurcations of Closures of Solutions for Integrable Systems with Two Degrees of Freedom by Integrable Topological Billiards

It is well known that surgeries of closures of solutions for integrable nondegenerate Hamiltonian systems with two degrees of freedom at a level of constant energy are classified by the so-called 3-atoms. These surgeries correspond to singular leaves of the Liouville foliation of three-dimensional isoenergetic surfaces. In this paper we prove the Fomenko conjecture that all such surgeries are modeled by integrable topological two-dimensional billiards (billiard books).     

渐近欧氏流形上带有阻尼和位势项的波动方程的生命跨度估*

本文研究了渐近欧氏流形上带有阻尼和位势的半线性波动方程的有限时间破裂以及解的生命跨度上界估计,其半线性项是形如c₁ |u_t|^p + c₂ |u|^p的混合项. 该问题与Strauss猜测和Glassey猜测紧密相关.     

Conservation laws for conformally invariant variational problems

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) in divergence form. These divergence-free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE (see [Hel]). It enables us also to establish new results. In particular we solve a conjecture by E. Heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by S. Hildebrandt [Hil1] claiming that critical points of continuously differentiable elliptic conformally invariant Lagrangian in two dimensions are continuous.     

Primes between consecutive squares

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. The aim of this paper is to provide a bound for the exceptional set for this conjecture, unconditionally and under the assumption of some classical hypothesis. We also provide a conditional proof of the conjecture assuming an hypothesis about the behavior of Selberg's integral in short intervals.     

Poincare‘—Cartan积分不变量的推广和Dirac猜想

该文将Poincare－Cartan积分不变量推广到显含时间的高阶微商奇异拉氏量系统，研究了该不变量与正则方程、正则变换之间的联系，讨论了广义Poincare－Cartan积分不变量与Dirac猜想的关系，以一个例子说明，对高阶微商奇异拉氏量系统，Dirac猜想是无效的．     

The Alperin and dade conjectures for the simple Conway’s third group

This paper is part of a program to study Alperin’s weight conjecture and Dade’s conjecture on counting ordinary characters in blocks for several finite groups. The classifications of radical subgroups and certain radical chains and their local structures of the simple Conway’s third group have been obtained by using the computer algebra system CAYLEY. The Alperin weight conjecture and the Dade final conjecture have been confirmed for the group.     

Interaction of Moving Localized Oscillations with a Local Inhomogeneity in Nonlinear Hamiltonian Klein–Gordon Lattices

We study the interaction of moving localized oscillations with a local inhomogeneity in a discrete nonlinear Hamiltonian system. We conjecture that resonance with a static nonlinear localized oscillation centered at the local inhomogeneity is a necessary condition for observing the trapping phenomenon. Analytic calculations and numerical simulations agree well with our hypothesis.     

势阱深度对双稳态电磁发电系统发电性能的影响研究

电磁式振动能量捕获技术从单稳态系统发展到多稳态系统,拓宽了响应频带,增大了输出电压,能够获得较好的发电性能.以附加线性振子的双稳态电磁式振动能量捕获器为研究对象,主要研究了势阱深度对双稳态系统发电性能的影响,并基于最优发电性能下的势阱深度,研究了双稳态系统结构参数中质量比与调频比对系统发电性能的影响.通过数值仿真结果说明,在外部激励频率为低频时:势阱深度较大时,双稳态系统的振子只能在一个阱内发生小幅振动运动;当势阱深度小到一定程度时,双稳态系统的振子跨过势垒在两个阱间内发生大幅混沌运动或周期运动,其优于小幅振动运动时的平均输出功率.通过数值模拟,得到双稳态系统具有较高的发电性能下的最优质量比、调频比以及阻尼比参数.     

A conjecture on small embeddings of partial Steiner triple systems

A well‐known, and unresolved, conjecture states that every partial Steiner triple system of order u can be embedded in a Steiner triple system of order υ for all υ ≡ 1 or 3, (mod 6), υ ≥ 2u + 1. However, some partial Steiner triple systems of order u can be embedded in Steiner triple systems of order υ <2u + 1. A more general conjecture that considers these small embeddings is presented and verified for some cases. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 313–321, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10017     

A note on the Gautschi-type method for oscillatory second-order differential equations

The Gautschi-type method has been proposed by Hochbruck and Lubich for oscillatory second-order differential equations. They conjecture that this method allows for a uniform error bound independent of the size of the system. The conjecture is proved in this note.     

A positive radial product formula for the Dunkl kernel

It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial here means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.

     

The exceptional set for the distribution of primes between consecutive powers

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. This paper is concerned with the distribution of prime numbers between two consecutive powers of integers, as a natural generalization of the afore-mentioned conjecture.