Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture

Summary We show that Walter Neumann's strengthened form of Hanna Neumann's conjecture on the best possible upper bound for the rank of the intersection of two subgroups of a free group is equivalent to a conjecture on the best possible upper bound for the number of edges in a bipartite graph with a certain weak symmetry condition. We illustrate the usefulness of this equivalence by deriving relatively easily certain previously known results.Oblatum 30-VIII-1993     

Towards the Willmore conjecture

We develop a variety of approaches, mainly using integral geometry, to proving that the integral of the square of the mean curvature of a torus immersed in must always take a value no less than . Our partial results, phrased mainly within the -formulation of the problem, are typically strongest when the Gauss curvature can be controlled in terms of extrinsic curvatures or when the torus enjoys further properties related to its distribution within the ambient space (see Sect. 3). Corollaries include a recent result of Ros [20] confirming the Willmore conjecture for surfaces invariant under the antipodal map, and a strengthening of the expected results for flat tori. The value arises in this work in a number of different ways – as the volume (or renormalised volume) of or , and in terms of the length of shortest nontrivial loops in subgroups of SO(4). Received April 26, 1999 / Accepted January 14, 2000 / Published online June 28, 2000     

The Coherence of One-Relator Groups with Torsion and the Hanna Neumann Conjecture

A conjecture is proposed, bounding the number of cycles withlabel Wⁿ in a labeled directed graph. Some partial results towardsthis conjecture are established. As a consequence, it is provedthat a₁, a₂,...|Wⁿ is coherent for n 4. Furthermore, it iscoherent for n 2, provided that the strengthened Hanna Neumannconjecture holds. 2000 Mathematics Subject Classification 20F06,05C38.     

Subgroups of Free Groups: a Contribution to the Hanna Neumann Conjecture

We prove that the strengthened Hanna Neumann conjecture, on the rank of the inter-section of finitely generated subgroups of a free group, holds for a large class of groups characterized by geometric properties. One particular case of our result implies that the conjecture holds for all positively finitely generated subgroups of the free group F(A) (over the basis A), that is, for subgroups which admit a finite set of generators taken in the free monoid over A.     

A toy Neumann analogue of the nodal line conjecture

We introduce an analogue of Payne’s nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of the domain. The assertion here is that any eigenfunction associated with the first nontrivial eigenvalue of the Neumann Laplacian on a domain \(\Omega \) with rotational symmetry of order two (i.e. \(x\in \Omega \) iff \(-x\in \Omega \)) “should normally” be rotationally antisymmetric. We give both positive and negative results which highlight the heuristic similarity of this assertion to the nodal line conjecture, while demonstrating that the extra structure of the problem makes it easier to obtain stronger statements: it is true for all simply connected planar domains, while there is a counterexample domain homeomorphic to a disk with two holes.     

A sufficient condition for the Hanna Neumann property of submonoids of a free monoid

Using an automata-theoretic approach, Giambruno and Restivo have investigated the intersection of two finitely generated submonoids of the free monoid over a finite alphabet. In particular, they have obtained the Hanna Neumann property for a special class of submonoids generated by finite prefix sets. This work continues their work and provides a sufficient condition for the Hanna Neumann property for the entire class of submonoids generated by finite prefix sets. In this connection, a general rank formula for the submonoids which are accepted by semi-flower automata is also obtained.     

Towards a proof of the 24-cell conjecture

This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in 4-dimensional Euclidean space. We consider two long-standing open problems: the uniqueness of maximum kissing arrangements in 4 dimensions and the 24-cell conjecture. Note that a proof of the 24-cell conjecture also proves that the lattice packing D₄ is the densest sphere packing in 4 dimensions.     

Towards an effective version of Mazur conjecture

In this Note we present an axiomatic exposition of Cornut–Vatsal's main ingredient in their proof of Mazur conjecture. Our aim is to extract clean ergodic statements and discuss a possible strategy towards effectivity.     

Towards generalizing MacDougall’s conjecture on vertex-magic total labelings

MacDougall’s conjecture states that every regular graph of degree at least 2 has a vertex-magic total labeling (VMTL) with the lone exception of $2K_3$. Since there is enormous empirical evidence supporting this conjecture, it is reasonable to seek generalizations. Thus we ask the more general question: to what extent does the degree sequence of a graph determine the existence or nonexistence of a VMTL? We provide beginning steps towards answering this question, and related questions, by providing infinite families of degree sequences, and for each sequence, a graph with a VMTL and another graph without a VMTL.     

Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

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The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s,E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre (1986) [Mes] showed the first zero occurs by , where NE is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller . We significantly improve on Mestre's result by averaging over a one-parameter family of elliptic curves E over Q(T). We assume GRH, Tate's conjecture if E is not a rational surface, and either the ABC or the Square-Free Sieve Conjecture if the discriminant has an irreducible polynomial factor of degree at least 4. We find non-trivial upper and lower bounds for the average number of normalized zeros in intervals on the order of (which is the expected scale). Our results may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyer conjecture, as well as the Katz-Sarnak density conjecture from random matrix theory (as the number of zeros predicted by random matrix theory lies between our upper and lower bounds). These methods may be applied to additional families of L-functions.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3EVYPNi_LG0.     

Towards the generalized purely wild inertia conjecture for product of alternating and symmetric groups

We obtain new evidence for the Purely Wild Inertia Conjecture posed by Abhyankar and for its generalization. We show that this generalized conjecture is true for any product of simple Alternating groups in odd characteristics, and for any product of certain Symmetric or Alternating groups in characteristic two. We also obtain important results towards the realization of the inertia groups which can be applied to more general set up. We further show that the Purely Wild Inertia Conjecture is true for any product of perfect quasi p-groups (groups generated by their Sylow p-subgroups) if the conjecture is established for individual groups.     

Modified Rayleigh conjecture method and its applications

The Rayleigh conjecture about convergence up to the boundary of the series representing the scattered field in the exterior of an obstacle D$D$ is widely used by engineers in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), which is an exact mathematical result. In this paper we present the theoretical basis for the MRC method for 2D and 3D obstacle scattering problems, for static problems, and for scattering by periodic structures. We also present successful numerical algorithms based on the MRC for various scattering problems. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods. Various direct and inverse scattering problems require finding global minima of functions of several variables. The Stability Index Method (SIM) combines stochastic and deterministic method to accomplish such a minimization.     

A spectral method for elliptic equations: the Neumann problem

Let Ω be an open, simply connected, and bounded region in ℝ ^d , d ≥ 2, and assume its boundary ∂Ω is smooth. Consider solving the elliptic partial differential equation − Δu + γu = f over Ω with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball B, and then a spectral method is given that uses a special polynomial basis. In the case the Neumann problem is uniquely solvable, and with sufficiently smooth problem parameters, the method is shown to have very rapid convergence. Numerical examples illustrate exponential convergence.     

Jacobi spectral Galerkin method for elliptic Neumann problems

This paper is concerned with fast spectral-Galerkin Jacobi algorithms for solving one- and two-dimensional elliptic equations with homogeneous and nonhomogeneous Neumann boundary conditions. The paper extends the algorithms proposed by Shen (SIAM J Sci Comput 15:1489–1505, 1994) and Auteri et al. (J Comput Phys 185:427–444, 2003), based on Legendre polynomials, to Jacobi polynomials with arbitrary α and β. The key to the efficiency of our algorithms is to construct appropriate basis functions with zero slope at the endpoints, which lead to systems with sparse matrices for the discrete variational formulations. The direct solution algorithm developed for the homogeneous Neumann problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Neumann data are accounted for by means of a lifting. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.     

Towards a better understanding of wall-driven square cavity flows using the lattice Boltzmann method

Wall-driven flow in square cavities has been studied extensively, yet it appears some main flow characteristics have not been fully investigated. Previous research on the classic lid-driven cavity (S1) flow has produced the critical Reynolds numbers separating the laminar steady and unsteady flows. Wall-driven cavities with two opposing walls moving at the same speed and the same (S2p) or opposite (S2a) directions have seldom been studied in the literature and no critical Reynolds numbers characterizing transitional flows have ever been investigated. After validating the LBM code for the three configurations studied, extensive numerical simulations have been undertaken to provide approximate ranges for the critical Hopf and Neimark-Sacker bifurcations for the classic and two two-sided cavity configurations. The threshold for transition to chaotic motion is also reported. The symmetries of the solutions are monitored across the various bifurcations for the two-sided wall driven cavities. The mirror-symmetry of the base solution for case S2p is lost at the Hopf bifurcation. The exact same scenario occurs with the pi-rotational symmetry of the base state for case S2a.     

Generalizations of the image conjecture and the Mathieu conjecture

We first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of Rⁿ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion.     

Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.