The Euler--Poincaré equations, firstly introduced by Henri Poincaré in 1901, arise from the application of Lagrangian mechanics to systems on Lie groups that exhibit symmetries, particularly in the contexts of classical mechanics and fluid dynamics. These equations have been extended to various settings, such as semidirect products, advected parameters, and field theory, and have been widely applied to mechanics and physics. In this paper, we introduce the discrete Euler--Poincaré reduction for discrete Lagrangian systems on Lie groups with advected parameters and additional dynamics, utilizing the group difference map technique. Specifically, the group difference map is defined using either the Cayley transform or the matrix exponential. The continuous and discrete Kelvin--Noether theorems are extended accordingly, that account for Kelvin--Noether quantities of the corresponding continuous and discrete Euler--Poincaré equations. As an application, we show both continuous and discrete Euler--Poincaré formulations about the dynamics of underwater vehicles, followed by numerical simulations. Numerical results illustrate the scheme's ability to preserve geometric properties over extended time intervals, highlighting its potential for practical applications in the control and navigation of underwater vehicles, as well as in other domains.

Fault tolerance in quantum protocols requires contributions from error-correcting codes and their suitable decoders. Quantum Low-Density Parity Check (QLDPC) codes are one of the most explored quantum codes that have good coding rate and efficient decoders. Iterative message passing-based decoders, although fast, fail to produce suitable success rates due to the colossal degeneracy and short cycles intrinsic to these codes. In this work we present a strategy to improve the performance of the Belief Propagation (BP) decoding, specifically the min-sum algorithm. We propose a collaborative decoding framework that integrates message passing with stabilizer check node removals. We further introduce the concept of ``qubit separation" and show that the improved decoding performance is directly related to the generation of highly separated trapped data qubits. To guide a more selective removal of check nodes that constrain the separation of the trapped data qubits, we introduce information measurements (IMs) for the data qubits and their adjacent stabilizer checks. We evaluate the performance of the proposed collaborative decoder on Generalized Hypergraph Product (GHP) codes and demonstrate that appropriate decoder configurations mitigate trapping sets in min-sum decoding without significant overhead.

We propose a test of the conditional independence of random variables $X$ and~$Y$ given~$Z$ under the additional assumption that $X$ is stochastically nondecreasing in~$Z$. The well-documented hardness of testing conditional independence means that some further restriction on the null hypothesis parameter space is required. In contrast to existing approaches based on parametric models, smoothness assumptions, or approximations to the conditional distribution of $X$ given $Z$ and/or $Y$ given $Z$, our test requires only the stochastic monotonicity assumption. Our procedure, called \textnormal{\texttt{PairSwap-ICI}}, determines the significance of a statistic by randomly swapping the $X$ values within ordered pairs of~$Z$ values. The matched pairs and the test statistic may depend on both $Y$ and $Z$, providing the analyst with significant flexibility in constructing a powerful test. Our test offers finite-sample Type~I error control, and provably achieves high power against a large class of alternatives. We validate our theoretical findings through a series of simulations and real data experiments.

Today, there is no clear legal test for regulating the use of variables that proxy for race and other protected classes and classifications. This Article develops such a test. Decision tools that use proxies are narrowly tailored when they exhibit the weakest total proxy power. The test is necessarily comparative. Thus, if two algorithms predict loan repayment or university academic performance with identical accuracy rates, but one uses zip code and the other does not, then the second algorithm can be said to have deployed a more equitable means for achieving the same result as the first algorithm. Scenarios in which two algorithms produce comparable and non-identical results present a greater challenge. This Article suggests that lawmakers can develop caps to permissible proxy power over time, as courts and algorithm builders learn more about the power of variables. Finally, the Article considers who should bear the burden of producing less discriminatory alternatives and suggests plaintiffs remain in the best position to keep defendants honest - so long as testing data is made available.

Avalanche-like plastic bursts in crystalline materials follow power law statistics, but the scaling exponents and cutoff parameters vary widely in the literature ($α$ ranging from 1 to 2.2), hindering predictive modeling. Since distributions do not follow Gaussian behavior, the average of plastic kinetics is not correctly defined. Larger-scale models that rely on average behavior are therefore fundamentally flawed. {We performed extensive three-dimensional Dislocation Dynamics simulations} of FCC Cu deformation across three orders of magnitude in dislocation density ($ρ= 5 \times 10^{10} \ \text{to} \ 2 \times 10^{12} \ \text{m}^{-2}$) under constant strain rates. Our results demonstrate that the power law exponent ($α\approx 1.6 \pm 0.1$ ) is invariant to both dislocation density and loading direction, resolving previous inconsistencies. However, dislocation density strongly controls the power law truncation scaling ($Δγ_{max} \propto \ b/\sqrtρ$) and the distribution of avalanche triggering stresses. We quantify correlations between slip system activities and show how individual system contributions evolve with avalanche size. These findings reconcile experimental scatter in avalanche statistics and provide quantitative scaling laws for mesoscale-to-continuum plasticity models.

With the introduction of data protection regulations, the need for innovative privacy-preserving approaches to process and analyse sensitive data has become apparent. One approach is the Personal Health Train (PHT) that brings analysis code to the data and conducts the data processing at the data premises. However, despite its demonstrated success in various studies, the execution of external code in sensitive environments, such as hospitals, introduces new research challenges because the interactions of the code with sensitive data are often incomprehensible and lack transparency. These interactions raise concerns about potential effects on the data and increases the risk of data breaches. To address this issue, this work discusses a PHT-aligned security and audit pipeline inspired by DevSecOps principles. The automated pipeline incorporates multiple phases that detect vulnerabilities. To thoroughly study its versatility, we evaluate this pipeline in two ways. First, we deliberately introduce vulnerabilities into a PHT. Second, we apply our pipeline to five real-world PHTs, which have been utilised in real-world studies, to audit them for potential vulnerabilities. Our evaluation demonstrates that our designed pipeline successfully identifies potential vulnerabilities and can be applied to real-world studies. In compliance with the requirements of the GDPR for data management, documentation, and protection, our automated approach supports researchers using in their data-intensive work and reduces manual overhead. It can be used as a decision-making tool to assess and document potential vulnerabilities in code for data processing. Ultimately, our work contributes to an increased security and overall transparency of data processing activities within the PHT framework.

Axions and axion-like particles can be probed through gravitational waves indirectly, often referred to as "audible axions". The usual concept of audible axion relies on the coupling between the axions and the gauge fields. Here we consider an axion-like mechanism with coupling to the Nieh-Yan term. This interaction leads to the direct and efficient production of gravitational waves during the radiation-dominated era, originating from the tachyonic instability of the gravitational perturbations with the Nieh-Yan term. We calculate the energy spectral density of the chiral gravitational wave background and the comoving energy density of axion-like fields. Based on the numerical results, we explore the parameter space of axion masses and decay constants for detectable gravitational wave signals, either in pulsar timing arrays or space-based gravitational wave detections.

In this paper, we investigate three schemes for implementing Controlled-Z (CZ) gates between individual ytterbium (Yb) rare-earth ions doped into yttrium orthovanadate (YVO$_4$ or YVO). Specifically, we investigate the CZ gates based on magnetic dipolar interactions between Yb ions, photon scattering off a cavity, and a photon interference-based protocol, with and without an optical cavity. We introduce a theoretical framework for precise computations of state and gate infidelities, accounting for noise effects. We then compute the state fidelity for each scheme to evaluate the feasibility of their experimental implementation. Based on these results, we compare the performance of the two-qubit gate schemes and discuss their respective advantages and disadvantages. We conclude that the probabilistic photon interference-based scheme offers the best fidelity scaling with cooperativity and is superior with the current technology of Yb values, while photon scattering is nearly deterministic but slower with less favourable fidelity scaling as a function of cooperativity. The cavityless magnetic dipolar scheme provides a fast, deterministic gate with decent fidelities if close ion localization can be realized. While focusing on $^{171}$Yb$^{3+}$:YVO system as a case study, the theoretical tools and approaches developed in this work are broadly applicable to other spin qubit systems.

For the N-body problem we prove that any two hyperbolic rays having the same limit shape define the same Busemann function. We localize a region of differentiability for these functions, of which we know that they are viscosity solutions of the stationary Hamilton-Jacobi equation. As a first corollary, we deduce that every hyperbolic motion of the $N$-body problem must become, after some time, a calibrating curve for the Busemann function associated to its limit shape. This implies that every hyperbolic motionof the $N$-body problem is eventually a minimizer, that is, it must contain a geodesic ray of the Jacobi-Maupertuis metric. Since the viscosity solutions of the Hamilton-Jacobi equation are almost everywhere differentiable, we also deduce the generic uniqueness of geodesic rays with a given limit shape without collisions. That is to say, if the limit shape is given, then for almost every initial configuration the geodesic ray is unique.

Rotary transformers are commonly used in wound rotor resolvers to transfer excitation signals to the rotating winding without mechanical contact. In many analyses, the rotary transformer is modeled as an ideal transformer, where the voltage transfer ratio is assumed to be equal to the turns ratio. However, in miniature rotary transformers used in compact resolver systems, leakage inductance can become comparable to the magnetizing inductance due to reduced core dimensions and unavoidable air gaps, leading to deviations from the ideal voltage transfer behavior. This paper presents an accurate equivalent circuit model for miniature rotary transformers employed in wound rotor resolvers. The proposed model analytically derives the magnetizing and leakage inductances using a magnetic equivalent circuit that accounts for flux fringing and air gap effects. The model is validated through three dimensional finite element analysis and experimental measurements on a fabricated prototype under both no load and resolver excitation conditions. The results demonstrate improved prediction accuracy of the secondary voltage compared with conventional models, enabling more reliable characterization of excitation transfer in compact resolver systems.

We give upper bounds for the Poincaré and Logarithmic Sobolev constants for doubly weighted Brownian motion on manifolds with sticky reflecting boundary diffusion under curvature assumptions on the manifold and its boundary. We therefor use an interpolation approach based on energy interactions between the boundary and the interior of the manifold and the weighted Reilly formula. Along the way we also obtain a lower bound on the first nontrivial doubly weighted Steklov eigenvalue and an upper bound on the norm of the doubly weighted boundary trace operator on Sobolev functions. We also consider the case of weighted Brownian motion with pure sticky reflection.

We investigate two unconditionally energy stable invariant energy quadratization (IEQ) finite element methods (FEMs) [Chen et al. Numerical Algorithms, DOI: 10.1007/s11075-024-01910-z, 2024] for solving the Cahn-Hilliard-Navier-Stokes (CHNS) equations. The time discretization of these IEQ-FEMs is based on the first- and second-order backward differentiation methods. \textcolor{black}{The auxiliary energy function introduced by the IEQ approach, modeling the square root of the nonlinear part of the energy, does not belong to the finite element space used for the spatial discretization.} These methods offer distinct advantages. Consequently, we propose a new hybrid IEQ-FEM that combines the strengths of both schemes, offering computational efficiency and unconditional energy stability in the finite element space. We provide rigorous proofs of mass conservation and energy dissipation for the proposed IEQ-FEMs. Several numerical experiments are presented to validate the accuracy, efficiency, and solution properties of the proposed method.

Dissipative tunneling remains a cornerstone effect in quantum mechanics. In chemistry, it plays a crucial role in governing the rates of chemical reactions, often modeled as the motion along the reaction coordinate from one potential well to another. The relative positions of energy levels in these wells strongly influence the reaction dynamics. Chemical research will benefit from a fully adjustable, asymmetric double-well equipped with precise measurement capabilities of the tunneling rates. In this paper, we show a quantum simulator system that consists of a continuously driven Kerr parametric oscillator with a third order non-linearity that can be operated in the quantum regime to create a fully tunable asymmetric double-well. Our experiment leverages a low-noise, all-microwave control system with a high-efficiency readout, based on a tunnel Josephson junction circuit, of the which-well information. We explore the reaction rates across the landscape of tunneling resonances in parameter space. We uncover two new and counter-intuitive effects: (i) a weak asymmetry can significantly decrease the activation rates, even though the well in which the system is initialized is made shallower, and (ii) the width of the tunneling resonances alternates between narrow and broad lines as a function of the well depth and asymmetry. We predict by numerical simulations that both effects will also manifest themselves in ordinary chemical double-well systems in the quantum regime. Our work is a first step for the development of analog molecule simulators of proton transfer reactions based on quantum parametric processes.

In recent years, GPUs have become the preferred accelerators for HPC and ML applications due to their parallelism and fast memory bandwidth. While GPUs boost computation, inter-GPU communication can create scalability bottlenecks, especially as the number of GPUs per node and cluster grows. Traditionally, the CPU managed multi-GPU communication, but advancements in GPU-centric communication now challenge this CPU dominance by reducing its involvement, granting GPUs more autonomy in communication tasks, and addressing mismatches in multi-GPU communication and computation. This paper provides a landscape of GPU-centric communication, focusing on vendor mechanisms and user-level library supports. It aims to clarify the complexities and diverse options in this field, define the terminology, and categorize existing approaches within and across nodes. The paper discusses vendor-provided mechanisms for communication and memory management in multi-GPU execution and reviews major communication libraries, their benefits, challenges, and performance insights. Then, it explores key research paradigms, future outlooks, and open research questions. By extensively describing GPU-centric communication techniques across the software and hardware stacks, we provide researchers, programmers, engineers, and library designers insights on how to exploit multi-GPU systems at their best.

In this paper, we propose a circuit design for implementing quantum walks on complex networks. Quantum walks are powerful tools for various graph-based applications such as spatial search, community detection, and node classification. Although many quantum-walk-based graph algorithms have been extensively studied, specific quantum circuits for implementing these algorithms have not yet been provided. To address this issue, we present a circuit design for implementing the discrete-time quantum walk on complex networks. We investigate the functionality of our circuit using the small-sized Watts-and-Strogatz model as the complex network model, comparing it with theoretical calculations. This work offers a new approach to constructing quantum circuits for implementing quantum walks on arbitrary complex networks.

We prove that if the Carathéodory metric on a strictly pseudoconvex domain with a smooth boundary is locally Kähler near the boundary, then the domain is biholomorphic to a ball. We also establish a local rigidity theorem for domains with Bergman metrics of constant holomorphic sectional curvature, and highlight this relationship with the Lu constant.

This paper studies mean-field control with joint law dependence under dynamic expectation constraints and/or dynamic state-control-law constraints. We pioneer the establishment of the stochastic maximum principle (SMP) and the derivation of the backward SDE (BSDE) from the perspective of constrained optimization using the method of Lagrangian multipliers. We first propose to embed the constrained mean-field control (C-MFC) with joint-law dependence into some abstract optimization problems with constraints on Banach spaces, for which we develop the generalized Fritz-John (FJ) optimality conditions. We then prove the stochastic maximum principle (SMP) for C-MFC by transforming the FJ conditions into an equivalent stochastic first-order condition associated with a general type of constrained forward-backward SDEs (FBSDEs). Contrary to the existing literature, we treat the McKean-Vlasov SDE as an infinite-dimensional equality constraint such that the BSDE induced by the FJ first-order optimality conditions can be interpreted as the generalized Lagrange multiplier. We also employ the methodology to stochastic control and mean field game problems under dynamic constraints.

We study the effect of gravitational clustering at small scales on larger scales by studying mode coupling between virialised halos. We build on the calculation by Peebles (1974) where it was shown that a virialised halo does not contribute any mode coupling terms at small wave numbers $k$. Using a perturbative expansion in wave number, we show that this effect is small and arises from the deviation of halo shapes from spherical and also on tidal interactions between halos. We connect this with the impact of finite mass resolution of cosmological N-Body simulations on the evolution of perturbations at early times. This difference between the expected evolution and the evolution obtained in cosmological N-Body simulations can be quantified using such an estimate. We also explore the impact of a finite shortest scale up to which the desired power spectrum is realised in simulations. Several simulation studies have shown that this effect is small in comparison with the effect of perturbations at large scales on smaller scales. It is nevertheless important to study these effects and develop a general approach for estimating their magnitude. This is especially relevant in the present era of precision cosmology. We provide basic estimates of the magnitude of these effects and their power spectrum dependence. We find that the impact of small scale cutoff in the initial power spectrum and discreteness increases with $(n+3)$, with $n$ being the index of the power spectrum. In general, we recommend that cosmological simulation data should be used only if the scale of non-linearity, defined as the scale where the linearly extrapolated {\it rms} amplitude of fluctuations is unity, is larger than the average inter-particle separation.

In this paper, we study two notions of rigidity, one of conformal submersions and the other of quasi Einstein manifolds, with an attempt to relate the two notions. Note that a smooth submersion between Riemannian manifolds is called conformal if it restricts to a conformal isometry on the horizontal distribution. A conformal submersion is said to be rigid if it reduces to a Riemannian submersion up to homothety. On the other hand, quasiEinstein manifolds are generalizations of Einstein manifolds that are of interest both in Riemannian geometry and theoretical physics. A Riemannian manifold $(M, g)$ is called quasi-Einstein if its Ricci tensor satisfies the identity: $R i c_g+ H e s s(f)-\frac{1}{m} d f \otimes d f=λg$ for some $f \in C^{\infty}(M)$ and constants $λ\in \mathbb{R}$ and $0<m \leq \infty$. A quasi-Einstein manifold is said to be rigid if it reduces to an Einstein manifold. In this paper, we employ certain techniques involving conformal submersions to establish rigidity results for a class of closed quasiEinstein manifolds with $λ>0$. In particular, we study curvature conditions that force conformal submersions to be rigid, also leading to the rigidity of a related class of quasi-Einstein manifolds.

We study asymptotic frequentist coverage and approximately Gaussian properties of Bayes posterior credible sets in nonlinear inverse problems when a Gaussian prior is placed on the parameter of the PDE. The aim is to ensure valid frequentist coverage of Bayes credible intervals when estimating continuous linear functionals of the parameter. Our results show that Bayes credible intervals have conservative coverage under certain smoothness assumptions on the parameter and a compatibility condition between the likelihood and the prior, regardless of whether an efficient limit exists or Bernstein von-Mises (BvM) theorem holds. In the latter case, our results yield a corollary with more relaxed sufficient conditions than previous works. The theory is illustrated with a PDE that arises in predicting the transport of radioactive waste from underground repositories and optimizing oil recovery from subsurface fields: an elliptic inverse problem for Darcy flow. In this case, a near-$1/\sqrt{N}$ contraction rate and conservative coverage results are obtained for linear functionals that were shown not to be estimable efficiently.

Measurements of the Hubble-Lemaître constant ($H_0$) require us to estimate the distance and recession velocity of galaxies independently. Gravitational clustering that leads to the formation of galaxies and the large scale structure leaves its imprints in the form of peculiar velocities of galaxies. In general, it is not possible to disentangle the peculiar velocity component from the recession velocities of galaxies, and this introduces an uncertainty in the determination of $H_0$. Using N-body simulations, we quantify the impact of peculiar velocities on the $H_0$ estimation. We consider observers to be located in dark matter halos and compute the distribution of the estimated value of $H_0$ across all such observers. We find that the dispersion of this distribution is large at small scales, and it diminishes as we go to large separations, reaching the level of the quoted statistical error in Planck and SH0ES measurements well beyond $\sim$135 Mpc/h and $\sim$220 Mpc/h, respectively. Measurements at smaller scales are susceptible to errors arising from peculiar motions, and this error can propagate to measurements at larger scales in the distance ladder. Notably, we observe a negative correlation between the local over-density around an observer and the deviation of the local and the global value of $H_0$. We show that deviations more significant than 5% of the global values can be encountered frequently at scales of up to 40 Mpc/h, and this is considerably larger than the statistical errors on local estimates. We also analyse the cumulative effect of such errors on mock measurements of $H_0$ as measured from Milky Way-sized halos. We find that this error is sensitive to the lowest distance at which we use measurements. The distribution of $H_0$ in mock measurements has a large tail, and deviations of a few percent from the global value cannot be ruled out.

We prove Brezis--Browder type results for fractional Sobolev spaces and quantitative type estimates for $s$-harmonic functions. Furthermore, we give sufficient conditions for distributional solutions to the fractional Poisson's equation $(-Δ)^su=T$ on $\mathbb{R}^d$ to be of the form $$u(x)=\int_{\mathbb{R}^d}\frac{T(y)}{|x-y|^{d-2s}}dy+l,\quad l\in \mathbb{R}.$$

We study how Rankin-Selberg periods and distinction problems interact with integral structures in spherical Whittaker type representations. Using this representation-theoretic framework, we settle a conjecture of Loeffler by showing that the local Euler factors appearing in the construction of the motivic Rankin-Selberg Euler system for a product of modular forms are integrally optimal; i.e. any construction of this type with any choice of integral input data in the recipe of Loeffler-Skinner-Zerbes, would give local factors appearing in tame norm relations at $p$, which are integrally divisible by the Euler factor $\mathcal{P}_p^{'}(\mathrm{Frob}_p^{-1})$ modulo $p-1$. We also interpret this as an integrality result on the unramified part of the period associated to the Rankin-Selberg convolution of two modular forms.

Building upon factor decomposition to overcome the curse of dimensionality inherent in multivariate volatility processes, we develop a factor model-based multivariate stochastic volatility (fMSV) framework. We propose a two-stage estimation procedure for the fMSV model: in the first stage, estimators of the factor model are obtained, and in the second stage, the MSV component is estimated using the estimated common factor variables. We derive the asymptotic properties of the estimators, taking into account the estimation of the factor variables. The prediction performances are illustrated by finite-sample simulation experiments and applications to portfolio allocation.

In Beyond5G and 6G networks, a common theme is that sensing will play a more significant role than ever before. Over this trend, Integrated Sensing and Communications (ISAC) is focused on unifying the sensing functionalities and the communications ones and to pursue direct tradeoffs between them as well as mutual performance gains. We frame the resource tradeoff between the SAC functionalities within an economic setting. We model a service provision by one operator to the users, the utility of which is derived from both SAC functionalities. The tradeoff between the resources that the operator assigns to the SAC functionalities is analyzed from the point of view of the service prices, quantities and profits. We demonstrate that equilibrium quantities and prices exist. And we provide relevant recommendations for enforcing regulatory limits of both power and bandwidth.

Starting from the Ginzburg--Landau model in a planar simply connected domain, with a local compactly supported applied magnetic field, we derive an effective model in the strong field limit, defined on a non-simply connected domain. The effective model features oscillations in the Little-Parks and Aharonov--Bohm spirit. We discuss also a similar question for the lowest eigenvalue of the magnetic Laplacian.

We establish an explicit correspondence between the Drinfeld current algebra presentation for the two-parameter quantum affine algebra $U_{r, s}(\mathrm{C}_n^{(1)})$ and the $R$-matrix realization á la Faddeev, Reshetikhin and Takhtajan.

Let $f$ and $g$ be spectrally normalized holomorphic newforms of even weight $k \geq2$ on $Γ_0(q)$. If $f\neq g$, then assume that $q$ is squarefree. For a nice test function $ψ$ supported on $Γ_0(1)\backslash\mathbb{H}$, we establish the best known bounds (uniform in $k$, $q$, and $ψ$) for \[ \int_{Γ_0(q)\backslash\mathbb{H}}ψ(z)f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^2}-\mathbf{1}_{f = g}\frac{3}π\int_{Γ_0(1)\backslash\mathbb{H}}ψ(z)\frac{dx dy}{y^2}.\] When $f=g$, our results yield an effective holomorphic variant of quantum unique ergodicity, refining work of Holowinsky-Soundararajan and Nelson-Pitale-Saha. When $f \neq g$, our results extend and improve the effective decorrelation result of Huang for $q=1$. To prove our results, we refine Soundararajan's weak subconvexity bound for Rankin-Selberg $L$-functions.

We provide examples of groups with transcendental spectral radius: We first construct finitely presented examples, using links between decidability of the Word Problem and semi-computability of the spectral radius. This argument extends to the exponential growth rate and the asymptotic entropy. We also construct a finitely generated example with decidable Word Problem, using classical small-cancellation theory. Along the way, we prove that $C'(1/6)$ groups satisfy the Rapid Decay property, and deduce some properties on their spectral radii of independent interest.

We introduce the game Slow $A$-Nim which generalizes a number of recently studied games. Slow $A$-Nim is played on $n$ stacks of tokens, and the set $A$ indicates the number of stacks a player can play on. Once a player has decided on the number $a$ of stacks, s/he will select any $a$ stacks and then remove one token from each stack. The last player to move wins. We give results on the $\mathcal{P}$-positions of Slow $A$-Nim for several infinite families. The results for $A = \{n-1\}$, which is the game Slow Exact $k$-Nim for $k=n-1$ extend recent results for small values of $n$. The other two families, $A=\{n-1,n\}$ and $A=\{1,n\}$ have not been previously studied. The $\mathcal{P}$-positions for $A = \{n-1\}$ and $A = \{n-1,n\}$ are closely related and have a very elegant description in terms of reduced positions, that is, positions for which unplayable tokens are disregarded. We also provide some general results that will be useful in the study of other sets $A$.