Tootfinder

Basis partitions and their signature
Krishnaswami Alladi
https://arxiv.org/abs/2507.14734 https://arxiv.org/pdf/2507.14734

Basis partitions and their signature
Basis partitions are minimal partitions corresponding to successive rank vectors. We show combinatorially how basis partitions can be generated from primary partitions which are equivalent to the Rogers-Ramanujan partitions. This leads to the definition of a signature of a basis partition that we use to explain certain parity results. We then study a special class of basis partitions which we term as complete. Finally we discuss basis partitions and minimal basis partitions among partitions wit…

Metas neue VR-Prototypen loten die Grenzen der Technik aus
Meta wird im August voraussichtlich drei neue Headset-Prototypen vorstellen, die bei Sichtfeld und Pixeldichte mit dem menschlichen Auge konkurrieren können.

Initiating and Replicating the Observations of Interactional Properties by User Studies Optimizing Applicative Prototypes
Guillaume Rivi\`ere
https://arxiv.org/abs/2507.13923

Initiating and Replicating the Observations of Interactional Properties by User Studies Optimizing Applicative Prototypes
The science of Human-Computer Interaction (HCI) is populated by isolated empirical findings, often tied to specific technologies, designs, and tasks. This paper proposes a formalization of user interaction observations (instead of user interfaces) and an associated revealing method (interaction loop diffraction). The resulting interactional properties that are studied in a calibrated manner, are well suited to replication across various conditions (prototypes, technologies, tasks, and user prof…

"Paradoxer" Akku-Prototyp:
Beim Entladen lädt er sich selbst wieder auf
Eine Batterie, die sich beim Entladen teilweise selbst wieder auflädt - Mehr....
https://winfuture.de/news,152233.html

"Paradoxer" Akku-Prototyp: Beim Entladen lädt er sich selbst wieder auf
Eine Batterie, die sich beim Entladen teilweise selbst wieder auflädt - ohne Ladegerät, ohne Kabel. Was wie ein Paradoxon klingt, wurde nun im Labor experimentell nachgewiesen. Möglich wird das durch einen physikalischen Effekt: bipolare Elektrochemie.

Asymptotics for moments of the minimal partition excludant in congruence classes
Shane Chern, Ernest X. W. Xia
https://arxiv.org/abs/2507.14431 https://

Asymptotics for moments of the minimal partition excludant in congruence classes
The minimal excludant statistic, which denotes the smallest positive integer that is not a part of an integer partition, has received great interest in recent years. In this paper, we move on to the smallest positive integer whose frequency is less than a given number. We establish an asymptotic formula for the moments of such generalized minimal excludants that fall in a specific congruence class. In particular, our estimation reveals that the moments associated with a fixed modulus are asympt…

It does have use cases (we use it for prototyping spatial experiences at work) but for mainstream use that is spot on. The tech doesn't work good enough to combat all the negative aspects that come with its usage.
https://mastodon.social/@anon_opin/114890146242778709

Got some batteries, and now my prototype solar node is ready for being tested in the field.
#meshtastic

karate: Zachary Karate Club
Network of friendships among members of a university karate club. Includes metadata for faction membership after a social partition. Note: there are two versions of this network, one with 77 edges and one with 78, due to an ambiguous typo in the original study. (The most commonly used is the one with 78 edges.).
This network has 34 nodes and 77 edges.
Tags: Social, Offline, Unweighted

On Ward Numbers and Increasing Schr\"oder Trees
Elena L. Wang, Guoce Xin
https://arxiv.org/abs/2507.15654 https://arxiv.org/pdf/…

On Ward Numbers and Increasing Schröder Trees
The Ward numbers $W(n,k)$ combinatorially enumerate set partitions with block sizes $\geq 2$ and phylogenetic trees (total partition trees). We prove that $W(n,k)$ also counts \emph{increasing Schröder trees} by verifying they satisfy Ward's recurrence. We construct a direct type-preserving bijection between total partition trees and increasing Schröder trees, complementing known type-preserving bijections to set partitions (including Chen's decomposition for increasing Schröder trees). Weig…

Combinatorial proof of a congruence for partitions into two sizes of part
Eli R. DeWitt, William J. Keith
https://arxiv.org/abs/2507.13566 https://

Combinatorial proof of a congruence for partitions into two sizes of part
Previous work showed that, for $ν_2(n)$ the number of partitions of $n$ into exactly two part sizes, one has $ν_2(16n + 14) \equiv 0 \pmod{4}$. The earlier proof required the technology of modular forms, and a combinatorial proof was desired. This article provides the requested proof, in the process refining divisibility to finer subclasses. Some of these subclasses have counts closely related to the divisor function $d(16n + 14)$, and we offer a conjecture on a potential rank statistic.