Tootfinder

Constructions of $t$-designs from weighing matrices and walk-regular graphs
Gary Greaves, Sho Suda
https://arxiv.org/abs/2402.17528 https://

Constructions of $t$-designs from weighing matrices and walk-regular graphs
We provide a method to construct $t$-designs from weighing matrices and walk-regular graphs. One instance of our method can produce a $3$-design from any (symmetric or skew-symmetric) conference matrix, thereby providing a partial answer to a question of Gunderson and Semeraro JCTB 2017. We explore variations of our method on some matrices that satisfy certain combinatorial restrictions. In particular, we show that there exist various infinite families of partially balanced incomplete blo…

On the algebra generated by three commuting matrices: combinatorial cases
Ron Cherny, Matthew Satriano, Yohan Song
https://arxiv.org/abs/2402.16334 https:/…

On the algebra generated by three commuting matrices: combinatorial cases
Gerstenhaber proved in 1961 that the unital algebra generated by a pair of commuting $d\times d$ matrices over a field has dimension at most $d$. It is an open problem whether the analogous statement is true for triples of matrices which pairwise commute. We answer this question for special classes of triples of matrices arising from combinatorial data.

The Fubini--Study metric on an `odd' Grassmannian is rigid
Stuart James Hall
https://arxiv.org/abs/2403.18757 https://arxiv.org/p…

The Fubini--Study metric on an `odd' Grassmannian is rigid
Following the ideas of Gasqui and Goldschmidt, we give an explicit description of the infinitesimal Einstein deformations admitted by the Fubini--Study metric on complex Grassmannians $G_{m}(\mathbb{C}^{n+m})$ with $m,n\geq 2$. The deformations were first shown to exist by Koiso in the 1980s but it has remained an open question as to whether they can be integrated to give genuine deformations of the Fubini--Study metric. We show that when $n+m$ is odd, the answer is no.

Tight Lower Bounds for Block-Structured Integer Programs
Christoph Hunkenschr\"oder, Kim-Manuel Klein, Martin Kouteck\'y, Alexandra Lassota, Asaf Levin
https://arxiv.org/abs/2402.17290

Tight Lower Bounds for Block-Structured Integer Programs
We study fundamental block-structured integer programs called tree-fold and multi-stage IPs. Tree-fold IPs admit a constraint matrix with independent blocks linked together by few constraints in a recursive pattern; and transposing their constraint matrix yields multi-stage IPs. The state-of-the-art algorithms to solve these IPs have an exponential gap in their running times, making it natural to ask whether this gap is inherent. We answer this question affirmative. Assuming the Exponential Tim…

Constructions of $t$-designs from weighing matrices and walk-regular graphs
Gary Greaves, Sho Suda
https://arxiv.org/abs/2402.17528 https://

Constructions of $t$-designs from weighing matrices and walk-regular graphs
We provide a method to construct $t$-designs from weighing matrices and walk-regular graphs. One instance of our method can produce a $3$-design from any (symmetric or skew-symmetric) conference matrix, thereby providing a partial answer to a question of Gunderson and Semeraro JCTB 2017. We explore variations of our method on some matrices that satisfy certain combinatorial restrictions. In particular, we show that there exist various infinite families of partially balanced incomplete blo…

Undecidability of the stabilizer and zero-in-the-corner problems for matrix groups
Emmanuel Breuillard, Georgi Kocharyan
https://arxiv.org/abs/2403.14875 h…

Undecidability of the stabilizer and zero-in-the-corner problems for matrix groups
The show that the upper-left-corner problem and upper-right-corner problem for matrix groups with rational entries are undecidable. To reach this aim, we answer a question of Dixon from 1985 by proving the undecidability of the stabilizer problem for matrix groups.

Russland: Hafturteil in Abwesenheit gegen Meta-Sprecher
Ein russisches Militärgericht hat den Sprecher des US-Internetriesen Meta, Andy Stone, in Abwesenheit wegen "Terrorismus-Verherrlichung" zu sechs Jahren Straflager verurteilt. Wie die russische Nachrichtenagentur Interfax berichtete, stufte der Richter den Facebook-Mutterkonzern in seinem Urteil als "extremistisch" ein. Stones Anwältin Walentina Filippenkowa sagte Interf…
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Ukraine-Liveblog: ++ Fernsehturm in Charkiw nach Angriff eingestürzt ++
Nach einem Angriff auf die ukrainische Stadt Charkiw ist dort ein Fernsehturm teilweise eingestürzt. Russland hat eigenen Angaben zufolge ein Dorf in der Ostukraine eingenommen. Die Entwicklungen vom Montag zum Nachlesen.

Rethinking LLM Memorization through the Lens of Adversarial Compression
Avi Schwarzschild, Zhili Feng, Pratyush Maini, Zachary C. Lipton, J. Zico Kolter
https://arxiv.org/abs/2404.15146

Rethinking LLM Memorization through the Lens of Adversarial Compression
Large language models (LLMs) trained on web-scale datasets raise substantial concerns regarding permissible data usage. One major question is whether these models "memorize" all their training data or they integrate many data sources in some way more akin to how a human would learn and synthesize information. The answer hinges, to a large degree, on $\textit{how we define memorization}$. In this work, we propose the Adversarial Compression Ratio (ACR) as a metric for assessing memorization in L…

Context Quality Matters in Training Fusion-in-Decoder for Extractive Open-Domain Question Answering
Kosuke Akimoto, Kunihiro Takeoka, Masafumi Oyamada
https://arxiv.org/abs/2403.14197

Context Quality Matters in Training Fusion-in-Decoder for Extractive Open-Domain Question Answering
Retrieval-augmented generation models augment knowledge encoded in a language model by providing additional relevant external knowledge (context) during generation. Although it has been shown that the quantity and quality of context impact the performance of retrieval-augmented generation models during inference, limited research explores how these characteristics affect model training. This paper explores how context quantity and quality during model training affect the performance of Fusion-i…

Undecidability of the stabilizer and zero-in-the-corner problems for matrix groups
Emmanuel Breuillard, Georgi Kocharyan
https://arxiv.org/abs/2403.14875 h…

Undecidability of the stabilizer and zero-in-the-corner problems for matrix groups
The show that the upper-left-corner problem and upper-right-corner problem for matrix groups with rational entries are undecidable. To reach this aim, we answer a question of Dixon from 1985 by proving the undecidability of the stabilizer problem for matrix groups.