On matrix analogs of Fermat’s little theorem

The theorem proved in this paper gives a congruence for the traces of powers of an algebraic integer for the case in which the exponent of the power is a prime power. The theorem implies a congruence in Gauss’ form for the traces of the sums of powers of algebraic integers, generalizing many familiar versions of Fermat’s little theorem. Applied to the traces of integer matrices, this gives a proof of Arnold’s conjecture about the congruence of the traces of powers of such matrices for the case in which the exponent of the power is a prime power.     

Floer–Novikov cohomology and the flux conjecture

We study Floer–Novikov cohomology with local coefficients and prove the flux conjecture for general closed symplectic manifolds. Received: February 2005, Revised: May 2006, Accepted: May 2006 Partially supported by the Grant-in-Aid for Scientific Research No. 14003419, Japan Society for the Promotion of Sciences.     

A descent principle for the Dirac-dual-Dirac method

Let G be a torsion-free discrete group with a finite-dimensional classifying space BG. We show that G has a dual-Dirac morphism if and only if a certain coarse (co-)assembly map is an isomorphism. Hence the existence of a dual-Dirac morphism for such groups is a metric, that is, coarse, invariant. We get results for groups with torsion as well.     

Proper actions on cohomology manifolds

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most , then the group is effectively a Lie group.

     

Correspondences Between 2-Brauer Characters of Solvable Groups

Let G be a finite solvable group and let p be a prime. Let P ∈ Syl _p (G) and N = N _G (P). We prove that there exists a natural bijection between the 2-Brauer irreducible characters of p′-degree of G and those of N _G (P).     

Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256 (9) (2009) 2967-3034]

In lines 8-11 of Lu (2009) [18, p. 2977] we wrote: “For integer m?3, if M is Cm-smooth and Cm−1-smooth L:R×TM→R satisfies the assumptions (L1)-(L3), then the functional Lτ is C²-smooth, bounded below, satisfies the Palais-Smale condition, and all critical points of it have finite Morse indexes and nullities (see [1, Prop. 4.1, 4.2] and [4])”. However, as proved in Abbondandolo and Schwarz (2009) [2] the claim that Lτ is C²-smooth is true if and only if for every (t,q) the function v?L(t,q,v) is a polynomial of degree at most 2. So the arguments in Lu (2009) [18] are only valid for the physical Hamiltonian in (1.2) and corresponding Lagrangian therein. In this note we shall correct our arguments in Lu (2009) [18] with a new splitting lemma obtained in Lu (2011) [20].     

Set Systems with L‐Intersections and k‐Wise L‐Intersecting Families

In this paper, by employing linear algebra methods we obtain the following main results:

• (i) Let and be two disjoint subsets of such that Suppose that is a family of subsets of such that for every pair and for every i. Then Furthermore, we extend this theorem to k‐wise L‐intersecting and obtain the corresponding result on two cross L‐intersecting families. These results show that Snevily's conjectures proposed by Snevily (2003) are true under some restricted conditions. This result also gets an improvement of a theorem of Liu and Hwang (2013).
• (ii) Let p be a prime and let and be two subsets of such that or and Suppose that is a family of subsets of [n] such that (1) for every pair (2) for every i. Then This result improves the existing upper bound substantially.

     

Nakai's conjecture for varieties smoothed by normalization

The notion of D-simplicity is used to give a short proof that varieties whose normalization is smooth satisfy Ishibashi's extension of Nakai's conjecture to arbitrary characteristic. This gives a new proof of Nakai's conjecture for curves and Stanley-Reisner rings.