On Hua-Tuan’s conjecture

Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.     

On De Jong’s conjecture

Let X be a smooth projective curve over a finite field F _q . Let ρ be a continuous representation π(X) → GL _n (F), where F = F _l ((t)) with F _l being another finite field of order prime to q. Assume that is irreducible. De Jong’s conjecture says that in this case is finite. As was shown in the original paper of de Jong, this conjecture follows from an existence of an F-valued automorphic form corresponding to ρ is the sense of Langlands. The latter follows, in turn, from a version of the Geometric Langlands conjecture. In this paper we sketch a proof of the required version of the geometric conjecture, assuming that char(F) ≠ 2, thereby proving de Jong’s conjecture in this case.     

On Hua-Tuan’s conjecture Ⅱ

Let G be a group of order pn, p a prime. For 0 m n, sm(G) denotes the number of subgroups of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan had ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1+p, 1+p+p2 or 1+p+2p2(mod p3). The conjecture has a negative answer. In this paper, we further investigate the conjecture and propose two new conjectures.     

On Ueno’s conjecture K

We show that if X is a smooth complex projective variety with Kodaira dimension 0 then the Kodaira dimension of a general fiber of its Albanese map is at most . J. A. Chen was partially supported by NCTS, TIMS, and NSC of Taiwan. C. D. Hacon was partially supported by NSF research grant no: 0456363 and an AMS Centennial Scholarship. We would like to thank J. Kollár, R. Lazarsfeld, C.-H. Liu, M. Popa, P. Roberts, and A. Singh for many useful comments on the contents of this paper.     

On Urysohn’s ℝ-Tree

In the short note of 1927, Urysohn constructed the metric space R that is nowhere locally separable. There is no publication with indications that R is a (noncomplete) ?-tree that has valency c at each point. The author in 1989, as well as Polterovich and Shnirelman in 1997, constructed ?-trees isometric to R unaware of the paper by Urysohn. In this paper the author considers various constructions of the ?-tree R and of the minimal complete ?-tree of valency c including R, as well as the characterizations of ?-trees, their properties, and connections with ultrametric spaces.

     

On mockenhoupt’s conjecture in the Hardy-Littlewood majorant problem

The Hardy-Littlewood majorant problem has a positive answer only for even integer exponents p, while there are counterexamples for all p {ie91-1} 2?. Montgomery conjectured that even among the idempotent polynomials there must exist counterexamples, i.e. there exist a finite set of characters and some ± signs with which the signed character sum has larger p ^th norm than the idempotent obtained with all the signs chosen + in the character sum. That conjecture was proved recently by Mockenhaupt and Schlag. However, Mockenhaupt conjectured that even the classical {ie91-2} three-term character sums, used for p = 3 and k = 1 already by Hardy and Littlewood, should work in this respect. That remained unproved, as the construction of Mockenhaupt and Schlag works with four-term idempotents. In our previous work we proved this conjecture for k = 0, 1, 2, i.e. in the range 0 < p < 6, p {ie91-3} 2?. Continuing this work here we demonstrate that even k = 3, 4 cases hold true. Several refinement in the technical features of our approach include improved fourth order quadrature formulae, finite estimation of G′²/G (with G being the absolute value square function of an idempotent), valid even at a zero of G, and detailed error estimates of approximations of various derivatives in subintervals, chosen to have accelerated convergence due to smaller radius of the Taylor approximation     

On Oliver’s p-group conjecture: II

Let p be an odd prime and S a finite p-group. B. Oliver’s conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup \mathfrakX(S){\mathfrak{X}(S)} of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver’s conjecture as a statement about the representation theory of the factor group S/\mathfrakX(S){S/\mathfrak{X}(S)}. We now verify the conjecture for a wide variety of groups S/\mathfrakX(S){S/\mathfrak{X}(S)}.     

On simplices in diameter graphs in ℝ4

Agraph G is a diameter graph in ?^d if its vertex set is a finite subset in ?^d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ?⁴ contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ?⁴, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

     

Goldbach’s conjecture in max-algebra

The Goldbach conjecture is one of the best known open problems in number theory. It claims that every even integer greater than 2 can be written as the sum of two primes. The present paper formulates a max-algebraic claim that is equivalent to Goldbach’s conjecture. The max-algebraic analogue allows examination of the conjecture by the methods of max-algebra. A max-algebra is an algebraic structure in which classical addition \(+\) and multiplication \(\times \) are replaced by the operations maximum \(\oplus \) and addition \(\otimes \), in other words \(a\oplus b=\max \{a,b\}\) and \(a\otimes b=a+b\).     

On Sarnak’s conjecture and Veech’s question for interval exchanges

Using a criterion due to Bourgain [10] and the generalization of the self-dual induction defined in [19], for each primitive permutation we build a large family of k-interval exchanges satisfying Sarnak’s conjecture, and, for at least one permutation in each Rauzy class, smaller families for which we have weak mixing, which implies a prime number theorem, and simplicity in the sense of Veech.     

On Sullivan��s conjecture on cycles in 4-free and 5-free digraphs

For a simple digraph G, let β(G) be the size of the smallest subset X■E(G) such that G-X has no directed cycles, and let γ(G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This paper proves that β(G) ≤ 0.3819γ(G) if G is a 4-free digraph, and β(G) ≤ 0.2679γ(G) if G is a 5-free digraph. These improve the results of Sullivan in 2008.     

On Schur’s theorem and its converses for n-Lie algebras

An n-Lie algebra analogue of Schur’s theorem and its converse as well as a Lie algebra analogue of Baer’s theorem and its converse are presented. Also, it is shown that, an n-Lie algebra with finite dimensional derived subalgebra and finitely generated central factor is isoclinic to some finite dimensional n-Lie algebra.     

On Borsuk's Problem in ℝ3

Mathematical Notes -