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Given a collection of historical sports rankings, can one tell which player is the greatest of all time (i.e., the GOAT)? In this work, we design a data-driven random walk on the symmetric group to obtain a stationary distribution over player rankings, spanning across different time periods in sports history. We combine this distribution with a notion of stochastic dominance to obtain a partial order over the players. Compared to existing methods, our approach is distinct in that i) using historical rankings captures the evolution of value systems and facilitates player comparisons when head-to-head data is unavailable, and i) aggregating into a partial order formally comes to terms with the possibility that, while some player comparisons can be established conclusively, others are "too close to call." We implement our methods using publicly available data from the Association of Tennis Professionals (ATP) and the Women's Tennis Association (WTA). Our main findings indicate that Steffi Graf and Serena Williams are the ones that come ahead as the GOATs of the WTA. Likewise, the "Big Three," that is Novak Djokovic, Roger Federer, and Rafael Nadal, are the ones that come ahead as the GOATs of the ATP. As a secondary finding, we note major differences in terms of career and dominance longevity for top players across the associations. While initially motivated by this application in sports analytics, our methods can also be applied to other practical domains where deriving rankings from historical data can inform operational decisions, such as route planning logistics.

In this expository article I describe classical results in the combinatorics of parking functions. Its English-Spanish translation is included. -- -- En este art\'iculo de difusi\'on matem\'atica describo resultados cl\'asicos en la combinatoria de funciones de parqueo. Su traducci\'on espa\~nol-ingl\'es est\'a incluida., Comment: To appear in publication

Given a Coxeter group $W$ with Coxeter system $(W,S)$, where $S$ is finite. We provide a complete characterization of Boolean intervals in the weak order of $W$ uniformly for all Coxeter groups in terms of independent sets of the Coxeter graph. Moreover, we establish that the number of Boolean intervals of rank $k$ in the weak order of $W$ is ${i_k(\Gamma_W)\cdot|W|}\,/\,2^{k}$, where $\Gamma_W$ is the Coxeter graph of $W$ and $i_k(\Gamma_W)$ is the number of independent sets of size $k$ of $\Gamma_W$ when $W$ is finite. Specializing to $A_n$, we recover the characterizations and enumerations of Boolean intervals in the weak order of $A_n$ given in arXiv:2306.14734. We provide the analogous results for types $C_n$ and $D_n$, including the related generating functions and additional connections to well-known integer sequences.

The influence of the interlayer cations, sonication and grinding on surface area of the vermiculite from Santa Olalla has been studied. For untreated samples the surface area increases with the size of the exchanged cation and only the samples with the largest cations (NH+ and K+) show some microporosity. Sonication produces an increase of the surface area but the amount of micropores remains almost unchanged. Grinding leads to a significant increase of the surface area and of the microporosity. In the ground samples the percentage of micropores contributing to the surface area is influenced by the interlayer cation and the extent of N2 penetration by the water kept in the sample at a given degassing temperature. From transmission electron microscopy, thermogravimetry and XRD (d00l diffraction peaks) it can be concluded that grinding produces imperfections in crystal ordering, some alteration in the crystal edges, and a significant release of loosely bound water at temperatures up to 450 8C. These effects are responsible for the high surface area and microporosity measured by the N2 adsorption.

Cement hydration products were studied as influenced by the hydration conditions (hydration time in liquid phase; relative humidity, RH, in gaseous phase). The formation of calcium hydroxide (portlandite, P) and its transformation to calcium carbonates is mainly discussed here.

An urban planner might design the spatial layout of transportation amenities so as to improve accessibility for underserved communities -- a fairness objective. However, implementing such a design might trigger processes of neighborhood change that change who benefits from these amenities in the long term. If so, has the planner really achieved their fairness objective? Can algorithmic decision-making anticipate second order effects? In this paper, we take a step in this direction by formulating processes of neighborhood change as instances of no-regret dynamics; a collective learning process in which a set of strategic agents rapidly reach a state of approximate equilibrium. We mathematize concepts of neighborhood change to model the incentive structures impacting individual dwelling-site decision-making. Our model accounts for affordability, access to relevant transit amenities, community ties, and site upkeep. We showcase our model with computational experiments that provide semi-quantitative insights on the spatial economics of neighborhood change, particularly on the influence of residential zoning policy and the placement of transit amenities.

Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter $\ell\geq 0$, we then consider the subset of interval rational parking functions in which each car parks at most $\ell$ spots away from their initial preference. We call these $\ell$-interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers $m\geq n$ and $\ell$. We also establish formulas for the number of nondecreasing $\ell$-interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between $\ell$-interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing $\ell=1$, and establish that the set of $1$-interval rational parking functions with $n$ cars and $m$ spots are in bijection with the set of barred preferential arrangements of $[n]$ with $m-n$ bars. This readily implies enumerative formulas. Further, in the case where $\ell=1$, we recover the results of Hadaway and Harris that unit interval parking functions are in bijection with the set of Fubini rankings, which are enumerated by the Fubini numbers.

Autonomous flight of flapping-wing robots is a major challenge for robot perception. Most of the previous sense-and-avoid works have studied the problem of obstacle avoidance for flapping-wing robots considering only static obstacles. This paper presents a fully onboard dynamic sense-and-avoid scheme for large-scale ornithopters using event cameras. These sensors trigger pixel information due to changes of illumination in the scene such as those produced by dynamic objects. The method performs event-by-event processing in low-cost hardware such as those onboard small aerial vehicles. The proposed scheme detects obstacles and evaluates possible collisions with the robot body. The onboard controller actuates over the horizontal and vertical tail deflections to execute the avoidance maneuver. The scheme is validated in both indoor and outdoor scenarios using obstacles of different shapes and sizes. To the best of the authors' knowledge, this is the first event-based method for dynamic obstacle avoidance in a flapping-wing robot., Comment: 8 pages, 10 figures, Journal paper. For associated video, see "this http URL" https://www.youtube.com/watch?v=cBMcw5jRnfU&list=PL-Kzs2T7Hx3K-IDKsgUwPUnzHmk8Pcsek

In this paper, we study the total displacement statistic of parking functions from the perspective of cooperative game theory. We introduce parking games, which are coalitional cost-sharing games in characteristic function form derived from the total displacement statistic. We show that parking games are supermodular cost-sharing games, indicating that cooperation is difficult (i.e., their core is empty). Next, we study their Shapley value, which formalizes a notion of "fair" cost-sharing and amounts to charging each car for its expected marginal displacement under a random arrival order. Our main contribution is a polynomial-time algorithm to compute the Shapley value of parking games, in contrast with known hardness results on computing the Shapley value of arbitrary games. The algorithm leverages the permutation-invariance of total displacement, combinatorial enumeration, and dynamic programming. We conclude with open questions around an alternative solution concept for supermodular cost-sharing games and connections to other areas in combinatorics., Comment: 16 pages

Graphical designs are a framework for sampling and numerical integration of functions on graphs. In this note, we introduce a method to address the trade-off between graphical design sparsity and accuracy. We show how to obtain sparse graphical designs via linear programming and design objective functions that aim to maximize their accuracy. We showcase our approach using yellow taxicab data from New York City.

Let $\mathfrak{S}_n$ denote the symmetric group and let $W(\mathfrak{S}_n)$ denote the weak order of $\mathfrak{S}_n$. Through a surprising connection to a subset of parking functions, which we call unit Fubini rankings, we provide a complete characterization and enumeration for the total number of Boolean intervals in $W(\mathfrak{S}_n)$ and the total number of Boolean intervals of rank $k$ in $W(\mathfrak{S}_n)$. Furthermore, for any $\pi\in\mathfrak{S}_n$, we establish that the number of Boolean intervals in $W(\mathfrak{S}_n)$ with minimal element $\pi$ is a product of Fibonacci numbers. We conclude with some directions for further study., Comment: 17 pages, 4 figures. To appear in Journal of Combinatorics

Quicksort is a classical divide-and-conquer sorting algorithm. It is a comparison sort that makes an average of $2(n+1)H_n - 4n$ comparisons on an array of size $n$ ordered uniformly at random, where $H_n = \sum_{i=1}^n\frac{1}{i}$ is the $n$th harmonic number. Therefore, it makes $n!\left[2(n+1)H_n - 4n\right]$ comparisons to sort all possible orderings of the array. In this article, we prove that this count also enumerates the parking preference lists of $n$ cars parking on a one-way street with $n$ parking spots resulting in exactly $n-1$ lucky cars (i.e., cars that park in their preferred spot). For $n\geq 2$, both counts satisfy the second order recurrence relation $ f_n=2nf_{n-1}-n(n-1)f_{n-2}+2(n-1)! $ with $f_0=f_1=0$., Comment: 8 pages, and 2 figures, to appear in The American Mathematical Monthly

The Compact Muon Solenoid (CMS) experiment prepares its Phase-2 upgrade for the high-luminosity era of the LHC operation (HL-LHC). Due to the increase of occupancy, trigger latency and rates, the full electronics of the CMS Drift Tube (DT) chambers will need to be replaced. In the new design, the time bin for the digitisation of the chamber signals will be of around 1~ns, and the totality of the signals will be forwarded asynchronously to the service cavern at full resolution. The new backend system will be in charge of building the trigger primitives of each chamber. These trigger primitives contain the information at chamber level about the muon candidates position, direction, and collision time, and are used as input in the L1 CMS trigger. The added functionalities will improve the robustness of the system against ageing. An algorithm based on analytical solutions for reconstructing the DT trigger primitives, called Analytical Method, has been implemented both as a software C++ emulator and in firmware. Its performance has been estimated using the software emulator with simulated and real data samples, and through hardware implementation tests. Measured efficiencies are 96 to 98\% for all qualities and time and spatial resolutions are close to the ultimate performance of the DT chambers. A prototype chain of the HL-LHC electronics using the Analytical Method for trigger primitive generation has been installed during Long Shutdown 2 of the LHC and operated in CMS cosmic data taking campaigns in 2020 and 2021. Results from this validation step, the so-called Slice Test, are presented., Comment: 16 pages, 11 figures

Public transit is an essential infrastructure enabling access to employment, healthcare, education, and recreational facilities. While accessibility to transit is important in general, some sections of the population depend critically on transit. However, existing public transit is often not designed equitably, and often, equity is only considered as an additional objective post hoc, which hampers systemic changes. We present a formulation for transit network design that considers different notions of equity and welfare explicitly. We study the interaction between network design and various concepts of equity and present trade-offs and results based on real-world data from a large metropolitan area in the United States of America., Comment: Accepted in the non-archival track at the ACM Conference on Equity and Access in Algorithms, Mechanisms, and Optimization (EAAMO), 2023

We introduce parking assortments, a generalization of parking functions with cars of assorted lengths. In this setting, there are $n\in\mathbb{N}$ cars of lengths $\mathbf{y}=(y_1,y_2,\ldots,y_n)\in\mathbb{N}^n$ entering a one-way street with $m=\sum_{i=1}^ny_i$ parking spots. The cars have parking preferences $\mathbf{x}=(x_1,x_2,\ldots,x_n)\in[m]^n$, where $[m]:=\{1,2,\ldots,m\}$, and enter the street in order. Each car $i \in [n]$, with length $y_i$ and preference $x_i$, follows a natural extension of the classical parking rule: it begins looking for parking at its preferred spot $x_i$ and parks in the first $y_i$ contiguously available spots thereafter, if there are any. If all cars are able to park under the preference list $\mathbf{x}$, we say $\mathbf{x}$ is a parking assortment for $\mathbf{y}$. Parking assortments also generalize parking sequences, introduced by Ehrenborg and Happ, since each car seeks for the first contiguously available spots it fits in past its preference. Given a parking assortment $\mathbf{x}$ for $\mathbf{y}$, we say it is permutation invariant if all rearrangements of $\mathbf{x}$ are also parking assortments for $\mathbf{y}$. While all parking functions are permutation invariant, this is not the case for parking assortments in general, motivating the need for characterization of this property. Although obtaining a full characterization for arbitrary $n\in\mathbb{N}$ and $\mathbf{y}\in\mathbb{N}^n$ remains elusive, we do so for $n=2,3$. Given the technicality of these results, we introduce the notion of minimally invariant car lengths, for which the only invariant parking assortment is the all ones preference list. We provide a concise, oracle-based characterization of minimally invariant car lengths for any $n\in\mathbb{N}$. Our results around minimally invariant car lengths also hold for parking sequences.

We introduce a new parking procedure called MVP parking in which $n$ cars sequentially enter a one-way street with a preferred parking spot from the $n$ parking spots on the street. If their preferred spot is empty, they park there. Otherwise, they park there and the car parked in that spot is bumped to the next unoccupied spot on the street. If all cars can park under this parking procedure, we say the list of preferences of the $n$ cars is an MVP parking function of length $n$. We show that the set of (classical) parking functions is exactly the set of MVP parking functions although the parking outcome (order in which the cars park) is different under each parking process. Motivating the question: Given a permutation describing the outcome of the MPV parking process, what is the number of MVP parking functions resulting in that given outcome? Our main result establishes a bound for this count which is tight precisely when the permutation describing the parking outcome avoids the patterns 321 and 3412. We then consider special cases of permutations and give closed formulas for the number of MVP parking functions with those outcomes. In particular, we show that the number of MVP parking functions which park in reverse order (that is the permutation describing the outcome is the longest word in $\mathfrak{S}_n$, which does not avoid the pattern 321) is given by the $n$th Motzkin number. We also give families of permutations describing the parking outcome for which the cardinality of the set of cars parking in that order is exponential and others in which it is linear., Comment: 20 pages, 1 table, 7 figures

The displacement of a parking function measures the total difference between where cars want to park and where they ultimately park. In this article, we prove that the set of parking functions of length $n$ with displacement one is in bijection with the set of ideal states in the famous Tower of Hanoi game with $n+1$ disks and $n+1$ pegs, both sets being enumerated by the Lah numbers., Comment: 7 pages; 4 figures (5 image files); Final version to appear in The American Mathematical Monthly

A water company decides to expand its network with a set of water lines, but it cannot build them all at once. However, it starts reaping benefits from a partial expansion. In what order should the company build the lines? We formalize a class of permutatorial problems with combinatorial/continuous subproblems capturing applications of incremental deployment. We show that, for additive/linear objective functions, efficient polyhedral methods for the subproblems extend to the permutatorial problem. Our main technical ingredient is the permutahedron.

The development of automatic perception systems and techniques for bio-inspired flapping-wing robots is severely hampered by the high technical complexity of these platforms and the installation of onboard sensors and electronics. Besides, flapping-wing robot perception suffers from high vibration levels and abrupt movements during flight, which cause motion blur and strong changes in lighting conditions. This paper presents a perception dataset for bird-scale flapping-wing robots as a tool to help alleviate the aforementioned problems. The presented data include measurements from onboard sensors widely used in aerial robotics and suitable to deal with the perception challenges of flapping-wing robots, such as an event camera, a conventional camera, and two Inertial Measurement Units (IMUs), as well as ground truth measurements from a laser tracker or a motion capture system. A total of 21 datasets of different types of flights were collected in three different scenarios (one indoor and two outdoor). To the best of the authors' knowledge this is the first dataset for flapping-wing robot perception., Comment: 8 pages, 22 figures, Video: "this https URL https://www.youtube.com/watch?v=ymCRnlWxX24&t=35s"

We performed a time-resolved spectroscopic study of the VUV/UV argon scintillation as a function of pressure and electric field, by means of a wavelength sensitive detector operated with different radioactive sources. Our work conveys new evidence of distinctive features of the argon light which are in contrast with the general assumption that, for particle detection purposes, the scintillation can be considered to be largely monochromatic at 128 nm (second continuum). The wavelength and the time-resolved analysis of the photon emission reveal that the dominant component of the argon scintillation during first tens of ns is in the range [160, 325] nm. This light is consistent with the third continuum emission from highly charged argon ions/molecules. This component of the scintillation is field-independent up to 25 V/cm/bar and shows a very mild dependence with pressure in the range [1,16] bar. The dynamics of the second continuum emission is dominated by the excimer formation time, whose variation as a function of the pressure has been measured. Additionally, the time and pressure-dependent features of electron-ion recombination, in the second continuum band, have been measured. This study opens new paths toward a novel particle identification technique based on the spectral information of the noble-elements scintillation light., Comment: 20 pages, 15 figures

The request-trip-vehicle assignment problem is at the heart of a popular decomposition strategy for online vehicle routing. In this framework, assignments are done in batches in order to exploit any shareability among vehicles and incoming travel requests. We study a natural ILP formulation and its LP relaxation. Our main result is an LP-based randomized rounding algorithm that, whenever the instance is feasible, leverages mild assumptions to return an assignment whose: i) expected cost is at most that of an optimal solution, and ii) expected fraction of unassigned requests is at most $1/e$. If trip-vehicle assignment costs are $\alpha$-approximate, we pay an additional factor of $\alpha$ in the expected cost. We can relax the feasibility requirement by considering the penalty version of the problem, in which a penalty is paid for each unassigned request. We find that, whenever a request is repeatedly unassigned after a number of rounds, with high probability it is so in accordance with the sequence of LP solutions and not because of a rounding error. We additionally introduce a deterministic rounding heuristic inspired by our randomized technique. Our computational experiments show that our rounding algorithms achieve a performance similar to that of the ILP at a reduced computation time, far improving on our theoretical guarantee. The reason for this is that, although the assignment problem is hard in theory, the natural LP relaxation tends to be very tight in practice., Comment: SIAM Conference on Applied and Computational Discrete Algorithms (ACDA21)

Large liquid argon detectors offer one of the best avenues for the detection of galactic weakly interacting massive particles (WIMPs) via their scattering on atomic nuclei. The liquid argon target allows exquisite discrimination between nuclear and electron recoil signals via pulse-shape discrimination of the scintillation signals. Atmospheric argon (AAr), however, has a naturally occurring radioactive isotope, $^{39}$Ar, a $\beta$ emitter of cosmogenic origin. For large detectors, the atmospheric $^{39}$Ar activity poses pile-up concerns. The use of argon extracted from underground wells, deprived of $^{39}$Ar, is key to the physics potential of these experiments. The DarkSide-20k dark matter search experiment will operate a dual-phase time projection chamber with 50 tonnes of radio-pure underground argon (UAr), that was shown to be depleted of $^{39}$Ar with respect to AAr by a factor larger than 1400. Assessing the $^{39}$Ar content of the UAr during extraction is crucial for the success of DarkSide-20k, as well as for future experiments of the Global Argon Dark Matter Collaboration (GADMC). This will be carried out by the DArT in ArDM experiment, a small chamber made with extremely radio-pure materials that will be placed at the centre of the ArDM detector, in the Canfranc Underground Laboratory (LSC) in Spain. The ArDM LAr volume acts as an active veto for background radioactivity, mostly $\gamma$-rays from the ArDM detector materials and the surrounding rock. This article describes the DArT in ArDM project, including the chamber design and construction, and reviews the background required to achieve the expected performance of the detector., Comment: 13 pages, 8 figures. Corresponding author: E. S\'anchez Garc\'ia

The CMS drift tubes (DT) muon detector, built for withstanding the LHC expected integrated and instantaneous luminosities, will be used also in the High Luminosity LHC (HL-LHC) at a 5 times larger instantaneous luminosity and, consequently, much higher levels of radiation, reaching about 10 times the LHC integrated luminosity. Initial irradiation tests of a spare DT chamber at the CERN gamma irradiation facility (GIF++), at large ($\sim$O(100)) acceleration factor, showed ageing effects resulting in a degradation of the DT cell performance. However, full CMS simulations have shown almost no impact in the muon reconstruction efficiency over the full barrel acceptance and for the full integrated luminosity. A second spare DT chamber was moved inside the GIF++ bunker in October 2017. The chamber was being irradiated at lower acceleration factors, and only 2 out of the 12 layers of the chamber were switched at working voltage when the radioactive source was active, being the other layers in standby. In this way the other non-aged layers are used as reference and as a precise and unbiased telescope of muon tracks for the efficiency computation of the aged layers of the chamber, when set at working voltage for measurements. An integrated dose equivalent to two times the expected integrated luminosity of the HL-LHC run has been absorbed by this second spare DT chamber and the final impact on the muon reconstruction efficiency is under study. Direct inspection of some extracted aged anode wires presented a melted resistive deposition of materials. Investigation on the outgassing of cell materials and of the gas components used at the GIF++ are underway. Strategies to mitigate the ageing effects are also being developed. From the long irradiation measurements of the second spare DT chamber, the effects of radiation in the performance of the DTs expected during the HL-LHC run will be presented., Comment: 10 pages, 7 figures, to be published in JINST, editor I. Gonz\'alez Caballero

We show how two electrically coupled semiconductor lasers having optical feedback can present simultaneous anti-phase correlated fast power fluctuations, and strong in-phase synchronized spikes of chaotic power drops. This quite counter intuitive phenomenon is demonstrated experimentally and confirmed by numerical solutions of a deterministic dynamical system of rate equations. The occurrence of negative and positive cross correlation between parts of a complex system according to time scales, as proved in our simple arrangement, is relevant for the understanding and characterization of collective properties in complex networks., Comment: 11 pages, 19 figures

This paper presents a general method for obtaining radius of the corresponding circumference to a cyclical polygon $n$ sides given the lengths of said sides, using the notion of complex number. As of radius $r$, obtained, can then be calculated polygon area in question applying known expressions which requires, after $n=4$, a powerful calculation tool. Are also given like elements regarding non convex polygons cyclic, although in this respect it deepens less and is not considered, course, the calculation of areas.

Several mechanisms have been proposed to explain the transient seis- mic emission, i.e., sunquakes, from some solar flares. Some theories associate high-energy electrons and/or white-light emission with sunquakes. High-energy charged particles and their subsequent heating of the photosphere and/or chro- mosphere could induce acoustic waves in the solar interior. We carried out a correlative study of solar flares with emission in hard-X rays (HXRs), enhanced continuum emission at 6173{\AA}, and transient seismic emission. We selected those flares observed by RHESSI (Reuven Ramaty High Energy Solar Spectroscopic Imager) with a considerable flux above 50 keV between January 1, 2010 and June 26, 2014. We then used data from the Helioseismic and Magnetic Imager onboard the Solar Dynamic Observatory (SDO/HMI) to search for excess visible continuum emission and new sunquakes not previously reported. We found a total of 18 sunquakes out of 75 investigated. All of the sunquakes were associated with a enhancement of the visible continuum during the flare time. Finally, we calculated a coefficient of correlation for a set of dichotomic variables related to these observations. We found a strong correlation between two of the standard helioseismic detection techniques, and between sunquakes and visible continuum enhancements. We discuss the phenomenological connectivity between these physical quantities and the observational difficulties of detecting seismic signals and excess continuum radiation., Comment: Submitted to Solar Physics