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We characterize mixed-level orthogonal arrays it terms of algebraic designs in a special multigraph. We prove a mixed-level analog of the Bierbrauer--Friedman (BF) bound for pure-level orthogonal arrays and show that arrays attaining it are radius-$1$ completely regular codes (equivalently, intriguing sets, equitable $2$-partitions, perfect $2$-colorings) in the corresponding multigraph. For the case when the numbers of levels are powers of the same prime number, we characterize, in terms of multispreads, additive mixed-level orthogonal arrays attaining the BF bound. For pure-level orthogonal arrays, we consider versions of the BF bound obtained by replacing the Hamming graph by its polynomial generalization and show that in some cases this gives a new bound. Keywords: orthogonal array, algebraic $t$-design, mixed orthogonal array, completely-regular code, equitable partition, intriguing set, Hamming graph, Bierbrauer-Friedman bound, additive code.

We consider various definitions of degrees of discrete functions and establish relations between the number of relevant (essential) variables and degrees of two- and three-valued functions. Keywords: relevant variable, sensitivity, degree of Boolean function.

We describe the Ruijsenaars' action-angle duality in classical many-body integrable systems through the spectral duality transformation relating the classical spin chains and Gaudin models. For this purpose, the Lax matrices of many-body systems are represented in the multi-pole (Gaudin-like) form by introducing a fictitious spectral parameter. This form of Lax matrices is also interpreted as classical-classical version of quantum-classical duality., Comment: 24 pages, reference added

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to $1$. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if $d$ is even, then the permanent of a $d$-dimensional polystochastic matrix of order $4$ is positive, and for odd $d$, we give a complete characterization of $d$-dimensional polystochastic matrices with zero permanent., Comment: 17 pages

A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each line is equal to $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $\Omega_n^d$. In the present paper, we compare known bounds on the number of vertices of the polytope $\Omega_n^d$ and prove that the number of vertices of $\Omega_3^d$ is doubly exponential on $d$., Comment: 8 pages, Section 2 is transferred from arXiv:2311.06905

Real-world recommender systems often need to balance multiple objectives when deciding which recommendations to present to users. These include behavioural signals (e.g. clicks, shares, dwell time), as well as broader objectives (e.g. diversity, fairness). Scalarisation methods are commonly used to handle this balancing task, where a weighted average of per-objective reward signals determines the final score used for ranking. Naturally, how these weights are computed exactly, is key to success for any online platform. We frame this as a decision-making task, where the scalarisation weights are actions taken to maximise an overall North Star reward (e.g. long-term user retention or growth). We extend existing policy learning methods to the continuous multivariate action domain, proposing to maximise a pessimistic lower bound on the North Star reward that the learnt policy will yield. Typical lower bounds based on normal approximations suffer from insufficient coverage, and we propose an efficient and effective policy-dependent correction for this. We provide guidance to design stochastic data collection policies, as well as highly sensitive reward signals. Empirical observations from simulations, offline and online experiments highlight the efficacy of our deployed approach., Comment: Accepted as a full paper in the 2024 ACM Conference on Recommender Systems (RecSys '24)

It is proved that 1) the indicator function of some onefold or multifold independent set in a regular graph is a perfect coloring if and only if the set attain the Delsarte--Hoffman bound; 2) each transversal in a uniform regular hypergraph is an independent set attaining the Delsarte--Hoffman bound in the vertex adjacency multigraph of this hypergraph; 3) combinatorial designs with parameters $t$-$(v,k,\lambda)$ and similar $q$-designs, difference sets, Hadamard matrices, and bent functions are equivalent to perfect colorings of special graphs and multigraphs, in particular, it is true in the cases of the Johnson graphs $J(n,k)$ for $(k-1)$-$(v,k,\lambda)$ designs and the Grassmann graphs $J_2(n,2)$ for bent functions. Keywords: perfect coloring, equitable partition, transversal of hypergraph, combinatorial design, $q$-design, difference set, bent function, Johnson graph, Grassmann graph, Delsarte--Hoffman bound, Comment: This is improved version of the paper published in Siberian Mathematical Journal. We fix some misprints and a gap in the proof of Theorem 2

Existence of strongly bound water molecules on silicate surfaces, above the desorption temperature of water ice, has been first predicted by computational studies and recently demonstrated by laboratory experiments. Such trapped water may be present in various astrophysical environments and there is now evidence for its presence in the diffuse interstellar medium and in extraterrestrial particles. We present here new results of a laboratory study of the phenomenon of trapping (strong bonding) of water molecules by silicates. We show that the efficiency of trapping is strongly dependent on the properties and composition of the surface. Our results point out that the presence of trapped water should be due to the hydrophilic properties of the silicate surface and that the nature of trapping is physical (physisorption rather than chemisorption). We demonstrate that water can be trapped on silicates up to the temperatures of about 470 K, which speaks for the presence of wet silicate grains in the terrestrial planet formation zone in planet-forming disks. Studying the thermal and UV stability of trapped water, we conclude that the detection of trapped water in the diffuse ISM speaks for its efficient continuous formation. We discuss our results as relevant to fundamental scientific questions, such as the oxygen depletion problem, the origin of water on Earth, and the formation of rocky planets., Comment: Accepted for publication in ApJ

Robots are becoming an increasingly common part of scientific work within laboratory environments. In this paper, we investigate the problem of designing \emph{schedules} for completing a set of tasks at fixed locations with multiple robots in a laboratory. We represent the laboratory as a graph with tasks placed on fixed vertices and robots represented as agents, with the constraint that no two robots may occupy the same vertex at any given timestep. Each schedule is partitioned into a set of timesteps, corresponding to a walk through the graph (allowing for a robot to wait at a vertex to complete a task), with each timestep taking time equal to the time for a robot to move from one vertex to another and each task taking some given number of timesteps during the completion of which a robot must stay at the vertex containing the task. The goal is to determine a set of schedules, with one schedule for each robot, minimising the number of timesteps taken by the schedule taking the greatest number of timesteps within the set of schedules. We show that this problem is NP-complete for many simple classes of graphs, the problem of determining the fastest schedule, defined by the number of time steps required for a robot to visit every vertex in the schedule and complete every task assigned in its assigned schedule. Explicitly, we provide this result for complete graphs, bipartite graphs, star graphs, and planar graphs. Finally, we provide positive results for line graphs, showing that we can find an optimal set of schedules for $k$ robots completing $m$ tasks of equal length of a path of length $n$ in $O(kmn)$ time, and a $k$-approximation when the length of the tasks is unbounded.

The problem of ponderomotive separation of ions with different charge-to-mass ratios under the influence of Alfven waves, which permanently exist in the magnetosphere in the form of geomagnetic pulsations, is posed. Formulas are derived for partial ponderomotive forces acting on light and heavy (metal) ions. In the quasi-hydrodynamic approximation, it was found that the Clarke number, which characterizes the metallicity of the plasma, is maximum at the minimum of the magnetic field on the field line along which the Alfv\'en wave propagates. Key words: ponderomotive forces, Alfv\'en waves, heavy ions, Clarke number., Comment: 10 pages

Denote by Q_d the d-dimensional hypercube. Addressing a recent question we estimate the number of ways the vertex set of Q_d can be partitioned into vertex disjoint smaller cubes. Among other results, we prove that the asymptotic order of this function is not much larger than the number of perfect matchings of Q_d. We also describe several new (and old) questions., Comment: Proofs slightly shortened and referee comments addressed

Shock breakout emission is light that arises when a shockwave, generated by core-collapse explosion of a massive star, passes through its outer envelope. Hitherto, the earliest detection of such a signal was at several hours after the explosion, though a few others had been reported. The temporal evolution of early light curves should reveal insights into the shock propagation, including explosion asymmetry and environment in the vicinity, but this has been hampered by the lack of multiwavelength observations. Here we report the instant multiband observations of a type II supernova (SN 2023ixf) in the galaxy M101 (at a distance of 6.85+/-0.15 Mpc), beginning at about 1.4 hours after the explosion. The exploding star was a red supergiant with a radius of about 440 solar radii. The light curves evolved rapidly, on timescales of 1-2 hours, and appeared unusually fainter and redder than predicted by models within the first few hours, which we attribute to an optically thick dust shell before it was disrupted by the shockwave. We infer that the breakout and perhaps the distribution of the surrounding dust were not spherically symmetric., Comment: 32 pages, 9 figures, and 3 tables. Online publication in Nature (Dec. 13th 2023)

Visual localization is a critical task in mobile robotics, and researchers are continuously developing new approaches to enhance its efficiency. In this article, we propose a novel approach to improve the accuracy of visual localization using Structure from Motion (SfM) techniques. We highlight the limitations of global SfM, which suffers from high latency, and the challenges of local SfM, which requires large image databases for accurate reconstruction. To address these issues, we propose utilizing Neural Radiance Fields (NeRF), as opposed to image databases, to cut down on the space required for storage. We suggest that sampling reference images around the prior query position can lead to further improvements. We evaluate the accuracy of our proposed method against ground truth obtained using LIDAR and Advanced Lidar Odometry and Mapping in Real-time (A-LOAM), and compare its storage usage against local SfM with COLMAP in the conducted experiments. Our proposed method achieves an accuracy of 0.068 meters compared to the ground truth, which is slightly lower than the most advanced method COLMAP, which has an accuracy of 0.022 meters. However, the size of the database required for COLMAP is 400 megabytes, whereas the size of our NeRF model is only 160 megabytes. Finally, we perform an ablation study to assess the impact of using reference images from the NeRF reconstruction.

Imagine a polygon-shaped platform $P$ and only one static spotlight outside $P$; which direction should the spotlight face to light most of $P$? This problem occurs in maximising the visibility, as well as in limiting the uncertainty in localisation problems. More formally, we define the following maximum cover problem: "Given a convex polygon $P$ and a Field Of View (FOV) with a given centre and inner angle $\phi$; find the direction (an angle of rotation $\theta$) of the FOV such that the intersection between the FOV and $P$ has the maximum area". In this paper, we provide the theoretical foundation for the analysis of the maximum cover with a rotating field of view. The main challenge is that the function of the area $A_{\phi}(\theta)$, with the angle of rotation $\theta$ and the fixed inner angle $\phi$, cannot be approximated directly. We found an alternative way to express it by various compositions of a function $A_{\theta}(\phi)$ (with a restricted inner angle $\phi$ and a fixed direction $\theta$). We show that $A_{\theta}(\phi)$ has an analytical solution in the special case of a two-sector intersection and later provides a constrictive solution for the original problem. Since the optimal solution is a real number, we develop an algorithm that approximates the direction of the field of view, with precision $\varepsilon$, and complexity $\mathcal{O}(n(\log{n}+(\log{\varepsilon})/\phi))$.