A remark on Donovan’s conjecture

It is shown that the difference between Donovans conjecture and the weaker conjecture bounding Cartan numbers of blocks of finite groups by the defect of the blocks can be expressed in terms of the relationship between pairs of Galois conjugate blocks. A consequence is that for principal blocks the two conjectures are equivalent.Received: 11 August 2003     

A remark on Barth’s connectivity theorem

It has been remarked by Hartshorne, that Barth’s theorem for a smooth projective X follows from the strong Lefschetz theorem for the cohomology of X. Using the strong Lefschetz theorem for intersection cohomology, we give an extension of Barth’s theorem to singular X. This naturally raises several questions concerning possible Barth theorems on the level of intersection cohomology.     

A remark on Nesterenko’s theorem for Ramanujan functions

We study the algebraic independence of values of the Ramanujan q-series $A_{2j+1}(q)=\sum_{n=1}^{\infty}n^{2j+1}q^{2n}/(1-q^{2n})$ or S _2j+1(q) (j≥0). It is proved that, for any distinct positive integers i, j satisfying $(i,j)\not=(1,3)$ and for any $q\in \overline{ \mathbb{Q}}$ with 0<|q|<1, the numbers A ₁(q), A _2i+1(q), A _2j+1(q) are algebraically independent over $\overline{ \mathbb{Q}}$ . Furthermore, the q-series A _2i+1(q) and A _2j+1(q) are algebraically dependent over $\overline{ \mathbb{Q}}(q)$ if and only if (i,j)=(1,3).     

A remark on Gagola’s Theorem

In this paper, we generalize Gagola’s Theorem [1]. Firstly we obtain several new identities. With the help of these identities, we prove a conclusion similar with Gagola’s under some more general conditions. Finally, we get a result regarding the control of p-transfer by Tate’s Theorem.     

A remark on the conjecture of Erdös, Faber and Lovász

The celebrated Erd?s, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν. We show that the conjecture is equivalent to the following assumption: For any graph , where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G. As we will see for any graph G = (V, E), where denotes the complement of G. Hence, at least G or fulfills the conjecture.      

A note on Hardy’s theorem and rotations

Semigroup Forum - We give a very short proof, using the Hermite semigroup, to a generalized version of Hardy’s theorem due to Hogan and Lakey. We characterize $$fin L^2({mathbb {R}}^n)$$...     

A note on Barnette’s conjecture

A well-known conjecture of Barnette states that every 3-connected cubic bipartite planar graph has a Hamiltonian cycle, which is equivalent to the statement that every 3-connected even plane triangulation admits a 2-tree coloring, meaning that the vertices of the graph have a 2-coloring such that each color class induces a tree. In this paper we present a new approach to Barnette’s conjecture by using 2-tree coloring.A Barnette triangulation is a 3-connected even plane triangulation, and a B-graph is a smallest Barnette triangulation without a 2-tree coloring. A configuration is reducible if it cannot be a configuration of a B-graph. We prove that certain configurations are reducible. We also define extendable, non-extendable and compatible graphs; and discuss their connection with Barnette’s conjecture.     

A remark on Jeśmanowicz’ conjecture for the non-coprimality case

Let a, b, c be relatively prime positive integers such that a² + b² = c². Je?manowicz' conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation (aN)^x + (bN)^y = (cN)^z has no positive solution (x, y, z) other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.     

A remark on Kac’s scattering length formula

The scattering length formula was formulated and proved in special cases by Kac in 1974 and 1975.It was discussed by a series of authors,including Taylor 1976,Tamura 1992 and Takahashi 1990.The formula was proved by Takeda 2010 in symmetric case and by He 2011 assuming weak duality.In this article,we shall use the powerful tool of Kutznetsov measures to prove this formula in the general framework of right Markov processes without further assumptions.     

A remark on the Abel’s and Dirichlet’s criterions concerning generalizations to monotonicity

The Abel’s and Dirichlet’s criterions for convergence of series in analysis are very basic classical results and both require the monotonicity condition. In this note we show that the monotonicity condition in these criterions can be generalized to RBV condition, while cannot be generalized to quasimonotonicity.     

A note on Stone’s,Baire’s,Ky Fan’s and Dugundji’s theorem in tvs-cone metric spaces

Recently, Ayse Sonmez [A. Sonmez, On paracompactness in cone metric spaces, Appl. Math. Lett. 23 (2010) 494–497] proved that a cone metric space is paracompact when the underlying cone is normal. Also, very recently, Kieu Phuong Chi and Tran Van An [K.P. Chi, T. Van An, Dugundji’s theorem for cone metric spaces, Appl. Math. Lett. (2010) doi:10.1016/j.aml.2010.10.034] proved Dugundji’s extension theorem for the normal cone metric space. The aim of this paper is to prove this in the frame of the tvs-cone spaces in which the cone does not need to be normal. Examples are given to illustrate the results.     

Atiyah’s L 2-index theorem,Kan-Thurston construction,and weak Bass conjecture

We conjecture that for a group G of type FP, the L ²-Euler characteristic of a group G is the same as the ordinary Euler characteristic of G, and show that this conjecture is closely related with the weak Bass conjecture. We also present a class of groups satisfying this conjecture. Ourmethod combines the Kan-Thurston construction, Atiyah’s L ²-index theorem, and a result of Berrick, Chatterji, and Mislin.     

A remark on the discriminant of Hill’s equation and Herglotz functions

We establish a link between the basic properties of the discriminant of periodic second-order differential equations and an elementary analysis of Herglotz functions. Some generalizations are presented using the language of self-adjoint extensions.     

A note on Linnik’s theorem on quadratic non-residues

Archiv der Mathematik - We present a short and purely combinatorial proof of Linnik’s theorem: for any $$varepsilon &gt;0$$ there exists a constant $$C_varepsilon $$ such that for any...     

Remarks on Harada’s conjecture

An open conjecture by Harada from 1981 gives an easy characterization of the p-blocks of a finite group in terms of the ordinary character table. Kiyota and Okuyama have shown that the conjecture holds for p-solvable groups. In the present work we extend this result using a criterion on the decomposition matrix. In this way we prove Harada’s Conjecture for several new families of defect groups and for all blocks of sporadic simple groups. In the second part of the paper we present a dual approach to Harada’s Conjecture.