Tootfinder

State Transfer on Unitary Cayley Graphs and Quadratic Unitary Cayley Graphs
Akash Kalita, Bikash Bhattacharjya
https://arxiv.org/abs/2508.18068 https://arx…

State Transfer on Unitary Cayley Graphs and Quadratic Unitary Cayley Graphs
The unitary Cayley graph, denoted $X_n$, is the graph with vertex set ${\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u$ for some $u$ with $1 \leq u \leq n-1$ and $\gcd(u,n) = 1$. The quadratic unitary Cayley graph, denoted $G_n$, is the graph with vertex set ${\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u^2$ or $a-b=-u^2$ for some $u$ with $1 \leq u \leq n-1$ and $\gcd(u,n) = 1$. In this paper, we classify all $X_n$ admittin…

Mindone: Quadratisches KI-Smartphone mit Gratis-Internet für KI-Tools
Ikko bringt mit dem Mindone ein fast quadratisches Smartphone mit Fokus auf KI-Funktionen und kostenlosem Internet für KI-Anwendungen auf den Markt.

ILRe: Intermediate Layer Retrieval for Context Compression in Causal Language Models
Manlai Liang, Mandi Liu, Jiangzhou Ji, Huaijun Li, Haobo Yang, Yaohan He, Jinlong Li
https://arxiv.org/abs/2508.17892

ILRe: Intermediate Layer Retrieval for Context Compression in Causal Language Models
Large Language Models (LLMs) have demonstrated success across many benchmarks. However, they still exhibit limitations in long-context scenarios, primarily due to their short effective context length, quadratic computational complexity, and high memory overhead when processing lengthy inputs. To mitigate these issues, we introduce a novel context compression pipeline, called Intermediate Layer Retrieval (ILRe), which determines one intermediate decoder layer offline, encodes context by streamin…

#FensterFreitag #LocHal

Quadratic points on modular curves $X_0(N)$ for $N\leq 100$
Filip Najman, Ivan Novak
https://arxiv.org/abs/2508.14480 https://arxiv.org/pdf/2508.14480

Quadratic points on modular curves $X_0(N)$ for $N\leq 100$
We determine the quadratic points on the modular curves $X_0(N)$ for $N\leq 100$ for which this has not been previously done, namely the cases $$N\in\{66,70,78,82,84,86,87,88,90,96,99\}.$$ We accomplish this by improving on the ``going down method," which uses the fact that we have a moduli description of all the (infinitely many) quadratic points on $X_0(n)$ for some divisor $n$ of $N$.

Acyclic and complete coloring of digraphs with the minimum and maximum possible numbers of colors
Mika Olsen, Christian Rubio-Montiel, Alejandra Silva Ramirez
https://arxiv.org/abs/2508.16043

Acyclic and complete coloring of digraphs with the minimum and maximum possible numbers of colors
The dichromatic and diachromatic numbers of a digraph are the minimum and maximum numbers of colors, respectively, in acyclic and complete colorings of the digraph. In this paper, we construct, for all $r \leq t$, non-symmetric digraphs with dichromatic number $r$ and diachromatic number $t$. Moreover, we discuss the existence of asymmetric digraphs with dichromatic number $ r $ and diachromatic number $ t \geq r $, establishing a quadratic upper bound $ b(r) \leq t $.

Quadratic Embedding Constants of Cartesian Products and Joins of Graphs
Projesh Nath Choudhury, Raju Nandi
https://arxiv.org/abs/2508.10834 https://arxiv.o…

Quadratic Embedding Constants of Cartesian Products and Joins of Graphs
The quadratic embedding constant (QEC) of a finite, simple, connected graph originated from the classical work of Schoenberg [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938] on Euclidean distance geometry. In this article, we study the QEC of graphs in terms of two graph operations: the Cartesian product and the join of graphs. We derive a general formula for the QEC of the join of an arbitrary graph with a regular graph and with a complete multipartite graph. We then provide quadratic…

On the quadratic complexity of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC_{2}}$-dimension
C. Terry, J. Wolf
https://arxiv.org/abs/2510.12767 https://

On the quadratic complexity of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC_{2}}$-dimension
In prior work, we showed that subsets of $\mathbb{F}_{p}^{n}$ of $\mathrm{VC_{2}}$-dimension at most $k$ are well approximated by a union of atoms of a quadratic factor of complexity $(\ell,q)$, where the complexity $\ell$ of the linear part and the complexity $q$ of the quadratic part are both bounded in terms of $k$, $p$, and the desired level of approximation $μ$. A key tool in the proof of this result was an arithmetic regularity lemma for the Gowers $U^3$-norm by Green and Tao, which resu…

On the Walsh spectra of quadratic APN functions
Sophie Hannah B\'en\'eteau, Nicolas Goluboff, Lukas K\"olsch, Divyesh Vaghasiya
https://arxiv.org/abs/2510.12008 htt…

On the Walsh spectra of quadratic APN functions
APN functions play a central role as building blocks in the design of many block ciphers, serving as optimal functions to resist differential attacks. One of the most important properties of APN functions is their linearity, which is directly related to the Walsh spectrum of the function. In this paper, we establish two novel connections that allow us to derive strong conditions on the Walsh spectra of quadratic APN functions. We prove that the Walsh transform of a quadratic APN function $F$ op…

Necklaces, permutations, and periodic critical orbits for quadratic polynomials
Matthew Baker, Andrea Chen, Sophie Li, Matthew Qian
https://arxiv.org/abs/2508.12924 https://

Necklaces, permutations, and periodic critical orbits for quadratic polynomials
Let $G_n$ denote the $n^{\rm th}$ Gleason polynomial, whose roots correspond to parameters $c$ such that the critical point $0$ is periodic of exact period $n$ under iteration of $z^2 + c$, and let $\bar{G}_n$ denote the reduction of $G_n$ modulo $2$. Buff, Floyd, Koch, and Parry made the surprising observation that the number of real roots of $G_n$ is equal to the number of irreducible factors of $\bar{G}_n$ for all $n$. We provide a bijective proof for this result by first providing explicit …