We shall investigate and arrive at a certain functional property of the double series \[ \sum\limits_{n,r\geq 1}\frac{1}{\sqrt{x^2n^2+r^2+w^2}\left( e^{2 πy\sqrt{x^2n^2+r^2+w^2}}-1\right)}. \]
We complete the classification of B-facets of a 4-dimensional Newton polyhedron, filling a gap in the classification of arXiv:1309.0630, found by the authors of arXiv:2209.03553.
AbstractWe show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the lattice isomorphism problem, which is known to be in the complexity class SZK. We achieve this by deploying a result on characteristic vectors by Elkies that gained attention in the context of 4-manifolds and Seiberg-Witten equations, but seems rather unnoticed in the algorithmic lattice community. On the way, we also show that for a given Gram matrix $$G \in \mathbb {Q}^{n \times n}$$ G ∈ Q n × n , we can efficiently find a rational lattice that is embedded in at most four times the initial dimension n, i.e. a rational matrix $$B \in \mathbb {Q}^{4n \times n}$$ B ∈ Q 4 n × n such that $$B^\intercal B = G$$ B ⊺ B = G .
Question when rectangle can be tiled with similar copies of rectangles witch quetient of sides quadratic irrationalities. New proof of one part F. Sharov's theorem. Other close result.
We give an algebraic proof of the determinant formulas for factorial Grothendieck polynomials obtained by Hudson--Ikeda--Matsumura--Naruse and by Hudson--Matsumura.
An arc in $\Z^2_n$ is defined to be a set of points no three of which are collinear. We describe some properties of arcs and determine the maximum size of arcs for some small $n$.
We offer streamlined proofs of fundamental theorems regarding the index theory for partial self-maps of an infinite set that are bijective between cofinite subsets.
The reciprocal Pascal matrix has entries $\binom{i+j}{j}^{-1}$. Explicit formullae for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, $q$-analogues are also presented.
The classification of Grassmannian cluster algebras resembles that of regular polygonal tilings. We conjecture that this resemblance may indicate a deeper connection between these seemingly unrelated structures.
We prove that a random labeled (unlabeled) tree is balanced. We also prove that random labeled and unlabeled trees are strongly $k$-balanced for any $k\geq 3$.
This article is about discrete periodicities and their combinatorial structure. It describes the unique structure caused by the alteration of a pattern in a repetition. That alteration of a pattern could be "heard" as the disturbance that one can hear when a record is scratched and jumps.
We give a simple proof of the "tree-width duality theorem" of Seymour and Thomas that the tree-width of a finite graph is exactly one less than the largest order of its brambles.
Given a k k –scheme X X that admits a tilting object T T , we prove that the Hochschild (co-)homology of X X is isomorphic to that of A = End X ( T ) A=\operatorname {End}_{X}(T) . We treat more generally the relative case when X X is flat over an affine scheme Y = Spec R Y=\operatorname {Spec} R , and the tilting object satisfies an appropriate Tor-independence condition over R R . Among applications, Hochschild homology of X X over Y Y is seen to vanish in negative degrees, smoothness of X X over Y Y is shown to be equivalent to that of A A over R R , and for X X a smooth projective scheme we obtain that Hochschild homology is concentrated in degree zero. Using the Hodge decomposition of Hochschild homology in characteristic zero, for X X smooth over Y Y the Hodge groups H q ( X , Ω X / Y p ) H^{q}(X,\Omega _{X/Y}^{p}) vanish for p > q p > q , while in the absolute case they even vanish for p ≠ q p\neq q . We illustrate the results for crepant resolutions of quotient singularities, in particular for the total space of the canonical bundle on projective space.