A polynomial analogue of the twin prime conjecture

We consider the problem of counting the number of (not necessarily monic) `twin prime pairs' of degree , where is a polynomial of degree . We formulate an asymptotic prediction for the number of such pairs as and then prove an explicit estimate confirming the conjecture in those cases where is large compared with . When has degree , our theorem implies the validity of a result conditionally proved by Hayes in 1963. When has degree zero, our theorem refines a result of Effinger, Hicks and Mullen.

     

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 Hanna Neumann conjecture using Dicks' method

The Hanna Neumann conjecture states that the intersection of two nontrivial subgroups of rankk+1 andl+1 of a free group has rank at mostkl+1. In a recent paper [3] W. Dicks proved that a strengthened form of this conjecture is equivalent to his amalgamated graph conjecture. He used this equivalence to reprove all known upper bounds on the rank of the intersection. We use his method to improve these bounds. In particular we prove an upper bound of 2kl–k–l+1 for the rank of the intersection above (k,l2) improving the earlier 2kl-min(k, l) bound of [1].We prove a special case of the amalgamated graph conjecture in the hope that it may lead to a proof of the general case and thus of the strengthened Hanna Neumann conjecture.Oblatum 6-II-1995 & 19-VI-1995Supported by the NSF grants No. CCR-92-00788 CCR-95-03254 and the (Hungarian) National Scientific Research Fund (OTKA) grant No. F014919. The author was visiting the Computation and Automation Institute of the Hungarian Academy of Sciences while part of this research was done.     

An analogue of nakai's conjecture

Let R be an associative ring with 1 and let T be a hereditary torsion theory in the category of left R-modules. In defining the localizatio n of R respect to x, the concept of T-injective module arises (see [5] , [11]) . We can consider the family E _T of all short exact sequences of left R-modules respect to which every T-injective left R-module is injective . E _T proper class in the sense defined in [ 9] . In this paper we characterize proper classe s which are of the form E _T for some hereditary torsion theory x. On the other hand, we give some conditions on x, which imply that E _T has enough projectives , and we show an example where E _T does not have enough projectives.     

Locating the first nodal linein the Neumann problem

The location of the nodal line of the first nonconstant Neumann eigenfunction of a convex planar domain is specified to within a distance comparable to the inradius. This is used to prove that the eigenvalue of the partial differential equation is approximated well by the eigenvalue of an ordinary differential equation whose coefficients are expressed solely in terms of the width of the domain. A variant of these estimates is given for domains that are thin strips and satisfy a Lipschitz condition.

     

On an Archimedean analogue of Tate's conjecture

We consider an Archimedean analogue of Tate's conjecture, and verify the conjecture in the examples of isospectral Riemann surfaces constructed by Vignéras and Sunada. We prove a simple lemma in group theory which lies at the heart of T. Sunada's theorem about isospectral manifolds.     

The iterated transfer analogue of the new doomsday conjecture

A strong general restriction is given on the stable Hurewicz image of the classifying spaces of elementary abelian -groups. In particular, this implies the iterated transfer analogue of the new doomsday conjecture.

     

On an analogue of a conjecture of Gross

The paper is concerned with a problem in the theory of congruence function fields which is analogous to a conjecture of Gross in Iwasawa Theory. Z_p-extensions K_∞/K₀ of congruence function fields K₀ of characteristic p≠2 involving no new constants are considered such that the set S of ramified primes is finite and these primes are fully ramified. Is the set of S-classes invariant under Gal(K_∞/K₀) finite ? Gross' conjecture asserts that a similar question has an affirmative answer for the class of cyclotomic Z_p- extensions of CM-type if S is the set of p-primes and the classes considered are minus S-classes. Using a formula of Witt for the norm residue symbol in cyclic p-extensions of local fields of characteristic p, a necessary and sufficient condition for the validity of the analogue of Gross' conjecture is given for a class of extensions K_∞/K₀. It is shown by examples that the analogue of Gross' conjecture is not always true.     

On the number of nodal solutions to a singularly perturbed Neumann problem

We show that for ε small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ² and f grows superlinearly. (A typical f(u) is f(u)= a₁ u₊^p – a₁ u_-^p, a₁, a₂ >0, p, q>1.) More precisely, for any positive integer K, there exists ε_K>0 such that for 0<ε<ε_K, the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed.     

Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture

The hypersurfaces of degree d in the projective space Pn correspond to points of PN, where . Now assume d=2e is even, and let X(n,d)⊆PN denote the subvariety of two e-fold hyperplanes. We exhibit an upper bound on the Castelnuovo regularity of the ideal of X(n,d), and show that this variety is r-normal for r?2. The latter result is representation-theoretic, and says that a certain GLn+1-equivariant morphism

Sr(S2e(Cn+1))→S₂(Sre(Cn+1))     

An analogue of the Descartes-Euler formula for infinite graphs and Higuchi's conjecture

Let be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let be the set of vertices, and for every , let denote the (Gaussian) curvature of : minus the sum of incident polygon angles. Descartes showed that whenever may be realized as the surface of a convex polytope in . More generally, if is made of finitely many polygons, Euler's formula is equivalent to the equation where is the Euler characteristic of . Our main theorem shows that whenever converges and there is a positive lower bound on the distance between any pair of vertices in , there exists a compact closed 2-manifold and an integer so that is homeomorphic to minus points, and further .

In the special case when every polygon is regular of side length one and for every vertex , we apply our main theorem to deduce that is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless is a prism, antiprism, or the projective planar analogue of one of these that . This resolves a recent conjecture of Higuchi.

     

An analogue of Berry's conjecture for the phase in fractal obstacle scattering

In this paper, the authors consider the high-frequency asymptoticsof the phase s() of acoustic waves scattered by an obstacleRⁿ with fractal boundary. Under certain conditions, it is provedthat if is –Minkowski measurable with –Minkowskimeasure µ then there exists a positive constant C_n, dependingonlyon n and such that where     

Fibred coarse embeddings,a-T-menability and the coarse analogue of the Novikov conjecture

The connection between the coarse geometry of metric spaces and analytic properties of topological groupoids is well known. One of the main results of Skandalis, Tu and Yu is that a space admits a coarse embedding into Hilbert space if and only if a certain associated topological groupoid is a-T-menable. This groupoid characterisation then reduces the proof that the coarse Baum–Connes conjecture holds for a coarsely embeddable space to known results for a-T-menable groupoids. The property of admitting a fibred coarse embedding into Hilbert space was introduced by Chen, Wang and Yu to provide a property that is sufficient for the maximal analogue to the coarse Baum–Connes conjecture and in this paper we connect this property to the traditional coarse Baum–Connes conjecture using a restriction of the coarse groupoid and homological algebra. Additionally we use this results to give a characterisation of the a-T-menability for residually finite discrete groups.     

Zhelobenko invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra and $ \mathfrak{h} $ a Cartan subalgebra. The Clifford algebra C( $ \mathfrak{g} $ ) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2?m?+?1 is just the zero weight vector of the simple (2?m?+?1)-dimensional module of the principal s-triple obtained from the Langlands dual $ {\mathfrak{g}^\vee } $ . Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford algebra conjecture is a question concerning the Harish-Chandra map for the enveloping algebra U( $ \mathfrak{g} $ ) composed with evaluation at the half sum ?? of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.     

A note on Greenberg's conjecture and the abc conjecture

For any totally real number field and any prime number , Greenberg's conjecture for asserts that the Iwasawa invariants and are both zero. For a fixed real abelian field , we prove that the conjecture is ``affirmative' for infinitely many (which split in if we assume the abc conjecture for .