Results 21 to 30 of about 45 (44)
Realizability of tropical pluri‐canonical divisors
Journal of the London Mathematical Society, Volume 110, Issue 6, December 2024.Abstract Consider a pair consisting of an abstract tropical curve and an effective divisor from the linear system associated to k$k$ times the canonical divisor for k∈Z⩾1$k \in \mathbb {Z}_{\geqslant 1}$. In this article, we give a purely combinatorial criterion to determine if such a pair arises as the tropicalization of a pair consisting of a smooth ...
Felix Röhrle, Johannes Schwab
wiley +1 more source
Colouring versus density in integers and Hales–Jewett cubes
Journal of the London Mathematical Society, Volume 110, Issue 5, November 2024.Abstract We construct for every integer k⩾3$k\geqslant 3$ and every real μ∈(0,k−1k)$\mu \in (0, \frac{k-1}{k})$ a set of integers X=X(k,μ)$X=X(k, \mu)$ which, when coloured with finitely many colours, contains a monochromatic k$k$‐term arithmetic progression, whilst every finite Y⊆X$Y\subseteq X$ has a subset Z⊆Y$Z\subseteq Y$ of size |Z|⩾μ|Y|$|Z ...
Christian Reiher +2 more
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A discrete mean value of the Riemann zeta function
Mathematika, Volume 70, Issue 4, October 2024.Abstract In this work, we estimate the sum ∑0<ℑ(ρ)⩽Tζ(ρ+α)X(ρ)Y(1−ρ)$$\begin{align*} \sum _{0 < \Im (\rho) \leqslant T} \zeta (\rho +\alpha)X(\rho) Y(1\!-\! \rho) \end{align*}$$over the nontrivial zeros ρ$\rho$ of the Riemann zeta function where α$\alpha$ is a complex number with α≪1/logT$\alpha \ll 1/\log T$ and X(·)$X(\cdot)$ and Y(·)$Y(\cdot)$ are ...
Kübra Benli, Ertan Elma, Nathan Ng
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Special cubic zeros and the dual variety
Journal of the London Mathematical Society, Volume 110, Issue 3, September 2024.Abstract Let F$F$ be a diagonal cubic form over Z$\mathbb {Z}$ in six variables. From the dual variety in the delta method of Duke–Friedlander–Iwaniec and Heath‐Brown, we unconditionally extract a weighted count of certain special integral zeros of F$F$ in regions of diameter X→∞$X \rightarrow \infty$.
Victor Y. Wang
wiley +1 more source
On a conjecture of Graham on the p$p$‐divisibility of central binomial coefficients
Mathematika, Volume 70, Issue 3, July 2024.Abstract Let p1,…,pr$p_1,\ldots,p_r$ be pairwise distinct primes. From a theorem of Kummer, each prime pj$p_j$ can divide 2nn${2n \atopwithdelims ()n}$ at most 1+logn/logpj$1 + \log n / \log p_j$ times. We show that, for all ε>0$\varepsilon > 0$, if p1,…,pr$p_1,\ldots,p_r$ are sufficiently large in terms of r$r$ and ε$\varepsilon$, then there exist ...
Ernie Croot +2 more
wiley +1 more source
Badly approximable grids and k$k$‐divergent lattices
Mathematika, Volume 70, Issue 3, July 2024.Abstract Let A∈Matm×n(R)$A\in \operatorname{Mat}_{m\times n}(\mathbb {R})$ be a matrix. In this paper, we investigate the set BadA⊂Tm$\operatorname{Bad}_A\subset \mathbb {T}^m$ of badly approximable targets for A$A$, where Tm$\mathbb {T}^m$ is the m$m$‐torus. It is well known that BadA$\operatorname{Bad}_A$ is a winning set for Schmidt's game and hence
Nikolay Moshchevitin +2 more
wiley +1 more source
Mathematical Methods in the Applied Sciences, Volume 47, Issue 3, Page 1218-1235, February 2024.If an initial frame of vectors {ei}$$ \left\{{e}_i\right\} $$ is related to a final frame of vectors {fi}$$ \left\{{f}_i\right\} $$ by, in geometric algebra (GA) terms, a rotor, or in linear algebra terms, an orthogonal transformation, we often want to find this rotor given the initial and final sets of vectors.
Anthony Lasenby +2 more
wiley +1 more source
Buildings, valuated matroids, and tropical linear spaces
Journal of the London Mathematical Society, Volume 109, Issue 1, January 2024.Abstract Affine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL$\mathrm{PGL}$ parameterizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite‐dimensional vector space, up to homothety.
Luca Battistella +4 more
wiley +1 more source
Correlation of multiplicative functions over Fq[x]$\mathbb {F}_q[x]$: A pretentious approach
Mathematika, Volume 70, Issue 1, January 2024.Abstract Let Mn$\mathcal {M}_n$ denote the set of monic polynomials of degree n over a finite field Fq$\mathbb {F}_q$ of q elements. For multiplicative functions ψ1,ψ2$\psi _1,\psi _2$, using the recently developed “pretentious method,” we establish a “local‐global” principle for correlation functions of the form ∑f∈Mnψ1(f+h1)ψ2(f+h2)asn→∞(qremains ...
Pranendu Darbar, Anirban Mukhopadhyay
wiley +1 more source
Some of the next articles are maybe not open access.
Looking into Pascal's Triangle: Combinatorics, Arithmetic, and Geometry
Mathematics Magazine, 1987Peter Hilton, Jean Pedersen
exaly +2 more sources