We give short elementary expositions of combinatorial proofs of some variants of Euler's partition identity that were first addressed analytically by George Andrews, and later combinatorially by ...others. The method using certain matrices to concisely explain these bijections, based on ideas first used in a previous manuscript by the author, enables us to also give new generalizations of two of these results.
An Even Number of Common Neighbors Silwal, Sandeep; Pathak, Aritro
The American mathematical monthly,
01/2020, Letnik:
127, Številka:
1
Journal Article
The Quadratic Density Hales Jewett conjecture with \(2\) letters states that for large enough \(n\), every dense subset of \(\{0,1\}^{n^{2}}\) contains a combinatorial line where the wildcard set is ...of the form \(\gamma \times \gamma\) where \(\gamma \subset \{1,2,\dots n\}\). We show in an elementary quantitative way that every dense subset of \(\{0,1\}^{n^{2}}\), for sufficiently large \(n\), contains two elements such that the set of coordinate points where they differ, which we term the difference set of these two elements, is of the form \(\gamma_{1}\times \gamma_{2}\) where \(\gamma_1, \gamma_2\) are both nonempty subsets of \(\{1,2,\dots n\}\). Further we give several non-trivial examples of dense vector subspaces of \(\{0,1\}^{n^{2}}\), where in each case the wildcard set of the combinatorial line that can be obtained has restrictions on its size and shape.
We extend known methods to establish upper bounds on the least character non-residues contingent on different zero-free regions within the critical strip, in particular on bounded rectangles within ...the critical strip along the \(\sigma=1\) line at arbitrary heights. This relates to earlier conditional results on least character non-residues, and recent results of Granville and Soundararajan on character sums.
We show optimal upper and lower pointwise estimates for the Green's function for elliptic operators that are the sum of a Laplacian term and a singular drift term that behaves like the inverse ...distance to the boundary, in not necessarily bounded chord-arc domains in \(\R^{n}\) with \(n\geq 3\). In particular, such a term does not belong to the Lorentz space or the Kato space considered in other recent work of Sakellaris and Mourgoglou. We also do not make any assumptions on the divergence of the drift term. Adopting standards arguments, while assuming the Bourgain estimate on the elliptic measure, this also shows doubling of the corresponding elliptic measure, in the chord-arc domains.
The questions of the measure and finding open intervals in certain sets of sums and products of elements of the middle third Cantor set (or a variant of it), have generated considerable interest ...recently. A broad general framework that makes it possible to deal with these questions was outlined by Astels. The question of finding the measure of the product of the middle third Cantor set was considered recently in an article by Athreya, Reznick and Tyson. Astels' methods apply to product sets upon considering a generalized logarithmic Cantor set, while Athreya, Reznick and Tyson's methods become difficult in dealing with sums or products of \(m\)'th powers as \(m\) becomes large. With a new elementary dynamical technique, we can deal with both the questions of sums and products in a satisfactory way, and the proofs are of the same level of complexity as that of an elementary proof of the fact that the sum of two copies of the middle third Cantor set is \(0,2\). Further, some observations are made on the question of intersections of one central Cantor set with an affine image of another central Cantor set both with arbitrary parameters.
Starting with a trivial periodic flow on \(\mathbb{S}M\), the unit tangent bundle of a genus two surface, we perform a Dehn-type surgery on the manifold around a tubular neighborhood of a curve on ...\(\mathbb{S}M\) that projects to a self-intersecting closed geodesic on \(M\), to get a surgered flow which restricted to the surgery region is ergodic with respect to the volume measure. The surgered flow projects to a map on the surgery track that can be taken to be a linked twist map with oppositely oriented shears which generates the ergodic behavior for sufficiently strong shears in the surgery.
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