Some applications of Gabaiʼs internal hierarchy

Any Haken 3-manifold (possibly with boundary consisting of tori) can be transformed into a surface×S¹$\text{surface}\times S^1$ by a series of splitting and regluing along incompressible surfaces. This fact was proved by Gabai as an application of his sutured manifold theory. The first half of this paper provides a few technical details in the proof. In the second half of this paper, some applications of Gabai?s theorem to Heegaard Floer homology are given. We refine the known results about the Thurson norm and fibrations. We also give some classification results for Floer simple knots in manifolds with positive _b1$b_1$.     

Some finite generalizations of Euler’s pentagonal number theorem

Euler’s pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler’s pentagonal number theorem.     

A tiling proof of Euler’s Pentagonal Number Theorem and generalizations

The Ramanujan Journal - In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider...     

Some Elementary Consequences of Perelman’s Canonical Neighborhood Theorem

In this purely expository note, we deduce a few known direct consequences of Perelman’s canonical neighborhood theorem for 3-dimensional Ricci flow and compactness theorem for 3-dimensional κ-solutions. These corollaries regard elementary properties of 3-dimensional singularity models and κ-solutions.     

Ramanujanʼs Master Theorem for Riemannian symmetric spaces

Ramanujan?s Master Theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of compact and noncompact reductive Riemannian symmetric spaces inside a common complexification, we prove an analogue of Ramanujan?s Master Theorem for the spherical Fourier transform of a spherical Fourier series. This extends the results proven by Bertram for Riemannian symmetric spaces of rank-one.     

Some generalizations of Lindelöf and paracompact spaces

In this paper notions ofm-Lindelöf, meta-m-Lindelöf, para-m-Lindelöf andm-closure preserving property are defined, wherem is any infinite cardinal. The main results are the following:

1. A topological space ism-Lindelöf if and only if it is meta-m-Lindelöf and it ism-Lindelöf in the sense of complete accumulation point.
2. A regular topological space is paracompact if and only if it is para-m-Lindelöf and it hasm-closure preserving property for somem.

     

Several generalizations of Tverberg’s theorem

In a generalization of Radon’s theorem, Tverberg showed that each setS of at least (d+1) (r ? 1)+1 points inR ^d has anr-partition into (pair wise disjoint) subsetsS =S ₁ ∪ … ∪S _r so that \(\bigcap\nolimits_i^r {\underline{\underline {}} } _1 \) convS _i # Ø. This note considers the following more general problems: (1) How large mustS σR ^d be to assure thatS has anr-partitionS=S ₁∪ … ∪S _r so that eachn members of the family {convS _i ～ _i-1 ^r have non-empty intersection, where 1<=n<=r. (2) How large mustS ∪R ^d be to assure thatS has anr-partition for which \(\bigcap\nolimits_i^r {\underline{\underline {}} } _1 \) convS _r is at least 1-dimensional.     

Nakajimaʼs remark on Hennʼs proof

We fill up a gap in Hennʼs proof concerning large automorphism groups of function fields of degree 1 over an algebraically closed field of positive characteristic.     

A generalization of Menonʼs identity

In this note we give a generalization of the well-known Menon?s identity. This is based on applying the Burnside?s lemma to a certain group action.     

A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \(\mathbf {RCA}_0\) to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements.     

On a generalization of Deuringʼs results

Using the Dieudonné theory we will study a reduction of an abelian variety with complex multiplication at a prime. Our results may be regarded as generalization of the classical theorem due to Deuring for CM-elliptic curves. We will also discuss a sufficient condition for a prime at which the reduction of a CM-curve is maximal.