Tootfinder

Triangle-free subsets of the Hypercube
Padmini Mukkamala
https://arxiv.org/abs/2506.18782 https://arxiv.org/pdf/2506.18782

Triangle-free subsets of the Hypercube
We study the problem of determining the largest subset of vertices of the $n$ dimensional hypercube without three vertices at a Hamming distance of exactly $r$ from each other. In particular, we provide a lower bound of $\frac{c2^n}{e^r2^{\frac r2} (\frac nr)^{\frac{3r}{4}}}$ and an upper bound of $O(\frac{r2^n}{n+1})$ when $2r \le n$. In particular, when $r$ is a constant, the lower bound is $\frac{c2^n}{n^{3r/4}}$, while the upper bound is $\frac{c'2^n}{n}$.

EHCube4P: Learning Epistatic Patterns Through Hypercube Graph Convolution Neural Network for Protein Fitness Function Estimation
Muhammad Daud, Philippe Charton, Cedric Damour, Jingbo Wang, Frederic Cadet
https://arxiv.org/abs/2506.16921

EHCube4P: Learning Epistatic Patterns Through Hypercube Graph Convolution Neural Network for Protein Fitness Function Estimation
Understanding the relationship between protein sequences and their functions is fundamental to protein engineering, but this task is hindered by the combinatorially vast sequence space and the experimental noise inherent in fitness measurements. In this study, we present a novel framework that models the sequence landscape as a hypercube $H(k,2)$ and integrates wavelet-based signal denoising with a graph convolutional neural network (GCN) to predict protein fitness across rugged fitness landsca…

Cycle lengths in the percolated hypercube
Michael Anastos, Sahar Diskin, Joshua Erde, Mihyun Kang, Michael Krivelevich, Lyuben Lichev
https://arxiv.org/abs/2506.16858

Cycle lengths in the percolated hypercube
Let $Q^d_p$ be the random subgraph of the $d$-dimensional binary hypercube obtained after edge-percolation with probability $p$. It was shown recently by the authors that, for every $\varepsilon > 0$, there is some $c = c(\varepsilon)>0$ such that, if $pd\ge c$, then typically $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. We strengthen this result to show that, under the same assumptions, typically $Q^d_p$ contains cycles of all even lengths between $4$ and $(1-\varepsilon)2…

Hamiltonian paths extending a set of matchings in hypercubes
Abid Ali, Lina Ba, Weihua Yang
https://arxiv.org/abs/2506.21432 https://…

Hamiltonian paths extending a set of matchings in hypercubes
The hypercube \( Q_n \) contains a Hamiltonian path joining \( x \) and \( y \) (where $x$ and $y$ from the opposite partite set) containing \( P \) if and only if the induced subgraph of \( P \) is a linear forest, where none of these paths have \( x \) or \( y \) as internal vertices nor both as endpoints. Dvořák and Gregor answered a problem posed by Caha and Koubek and proved that for every \( n \geq 5 \), there exist vertices \( x \) and \( y \) with a set of \( 2n - 4 \) edges in \( Q_n…

Dynamical Phase Diagram of the REM under independent spin-flips
Chokri Manai, Simone Warzel
https://arxiv.org/abs/2506.15353 https://…

Dynamical Phase Diagram of the REM under independent spin-flips
We study the energy landscape of the Random Energy model (REM) integrated along trajectories of the simple random walk on the hypercube. We show that the quenched cumulant generating function of the time integral of the REM energy undergoes phase transitions in the large $N$ limit for trajectories of any time extent, and identify phases distinguished by the activity and value of the time integral. This is achieved by relating the dynamical behavior to the spectral properties of Hamiltonians ass…

Hamiltonian laceability with a set of faulty edges in hypercubes
Abid Ali, Weihua Yang
https://arxiv.org/abs/2506.21391 https://arxiv…

Hamiltonian laceability with a set of faulty edges in hypercubes
Faulty networks are useful because link or node faults can occur in a network. This paper examines the Hamiltonian properties of hypercubes under certain conditional faulty edges. Let consider the hypercube \( Q_n \), for \( n \geq 5 \) and set of faulty edges \( F \) such that \( |F| \leq 4n - 17 \). We prove that a Hamiltonian path exists connecting any two vertices in \( Q_n - F \) from distinct partite sets if they verify the next two conditions: (i) in $Q_n - F$ any vertex has a …

Optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiments
Xu He, Junpeng Gong, Zhaohui Li
https://arxiv.org/abs/2506.04582

Optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiments
Computer simulations serve as powerful tools for scientists and engineers to gain insights into complex systems. Less costly than physical experiments, computer experiments sometimes involve large number of trials. Conventional design optimization and model fitting methods for computer experiments are inefficient for large-scale problems. In this paper, we propose new methods to optimize good lattice point sets, using less computation to construct designs with enhanced space-filling properties …

Khovanov-Rozansky cycle calculus for bipartite links
A. Anokhina, E. Lanina, A. Morozov
https://arxiv.org/abs/2506.08721 https://arxi…

Khovanov-Rozansky cycle calculus for bipartite links
Bipartite calculus is a direct generalization of Kauffman planar expansion from $N=2$ to arbitrary $N$, applicable to the restricted class of knots which are entirely made of antiparallel lock tangles. Whenever applicable, it allows a straightforward generalization of the Khovanov calculus without a need of the technically complicated matrix factorization used for arbitrary $N$ in the Khovanov-Rozansky (KR) approach. The main object here is the $3^{n}$-dimensional hypercube with $n$ being the n…

This https://arxiv.org/abs/2506.02120 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csNE_…

Random-key genetic algorithms
A random-key genetic algorithm is an evolutionary metaheuristic for discrete and global optimization. Each solution is encoded as a vector of N random keys, where a random key is a real number randomly generated in the continuous interval [0, 1). A decoder maps each vector of random keys to a solution of the optimization problem being solved and computes its cost. The benefit of this approach is that all genetic operators and transformations can be maintained within the unitary hypercube, regar…

This https://arxiv.org/abs/2401.12526 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Refined generalization analysis of the Deep Ritz Method and Physics-Informed Neural Networks
In this paper, we derive refined generalization bounds for the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). For the DRM, we focus on two prototype elliptic partial differential equations (PDEs): Poisson equation and static Schrödinger equation on the $d$-dimensional unit hypercube with the Neumann boundary condition. Furthermore, sharper generalization bounds are derived based on the localization techniques under the assumptions that the exact solutions of the PDEs lie …

Hamiltonian connectivity of some base-cobase graphs
Leonardo Mart\'inez-Sandoval, Kolja Knauer
https://arxiv.org/abs/2506.15049 https://

Hamiltonian connectivity of some base-cobase graphs
There has been wide interest in understanding which properties of base graphs of matroids extend to base-cobase graphs of matroids. A significant result of Naddef and Pulleyblank (1984) shows that the $1$-skeleton of any $(0,1)$-polytope is either a hypercube, or Hamiltonian-connected, i.e. there is a Hamiltonian path connecting any two vertices. In particular, this is true for base graphs of matroids. A natural question raised by Farber, Richter, and Shank (1985) is whether this extends to bas…

Random-key genetic algorithms
Mariana A. Londe, Luciana S. Pessoa, Carlos E. Andrade, Jos\'e F. Gon\c{c}alves, Mauricio G. C. Resende
https://arxiv.org/abs/2506.02120

Random-key genetic algorithms
A random-key genetic algorithm is an evolutionary metaheuristic for discrete and global optimization. Each solution is encoded as a vector of N random keys, where a random key is a real number randomly generated in the continuous interval [0, 1). A decoder maps each vector of random keys to a solution of the optimization problem being solved and computes its cost. The benefit of this approach is that all genetic operators and transformations can be maintained within the unitary hypercube, regar…

2-Homogeneous bipartite distance-regular graphs and the quantum group $U^\prime_q(\mathfrak{so}_6)$
Paul Terwilliger
https://arxiv.org/abs/2506.02190 https…

2-Homogeneous bipartite distance-regular graphs and the quantum group $U^\prime_q(\mathfrak{so}_6)$
We consider a 2-homogeneous bipartite distance-regular graph $Γ$ with diameter $D \geq 3$. We assume that $Γ$ is not a hypercube nor a cycle. We fix a $Q$-polynomial ordering of the primitive idempotents of $Γ$. This $Q$-polynomial ordering is described using a nonzero parameter $q \in \mathbb C$ that is not a root of unity. We investigate $Γ$ using an $S_3$-symmetric approach. In this approach one considers $V^{\otimes 3} = V \otimes V \otimes V$ where $V$ is the standard module of $Γ$. W…