In this paper, by using the arithmetic properties of character sums over finite fields, we investigate some cyclotomic matrices involving Kloosterman sums over finite fields. For example, let $$K_q(u)=\sum_{x\in\mathbb{F}_q\setminus\{0\}}e^{\frac{2πi}{p}{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}\left(x+\frac{u}{x}\right)}$$ be the Kloosterman sum over $\mathbb{F}_q$, where $q=p^f$ is an odd prime power. We prove that matrix $[K_q(s_i+s_j)]_{1\le i,j\le (q-1)/2}$ is singular whenever $q\ge 11$, where $s_1,s_2,\cdots,s_{(q-1)/2}$ are exactly all non-zero squares over $\mathbb{F}_q$.

We prove that the optimal pebbling number of the $n$-dimensional hypercube is \[ o(Q_n)\,=\,Θ\!\left(\left(\frac43\right)^n\right). \]

Under coprimality assumptions on certain submatrices, Rosenbrock's Theorem relates the Smith form of a matrix $P$ over an elementary divisor domain $\mathcal{R}$ to the Smith-McMillan form of a matrix $G$ over the field of fractions of $\mathcal{R}$, where $G$ is a Schur complement in $P$. If $\FR$ is not an elementary divisor domain, Rosenbrock's Theorem is not directly applicable in its original form, because not every matrix is unimodularly equivalent to a matrix in Smith form. In this paper, we state an ideal-theoretic version of Rosenbrock's Theorem that is meaningful over any integral domain, and we show that it is equivalent to the classic formulation over an elementary divisor domain. Moreover, we give a characterization of Prüfer domains as those integral domains over which the ideal-theoretic version of Rosenbrock's Theorem holds for every matrix satisfying the assumptions. In particular, the theorem does not hold for every admissible matrix over $\mathbb{C}[x_1,\dots,x_d]$ when $d \geq 2$, but we show that it holds Zariski-generically when $d \leq 3$. Finally, we prove that, if $\mathcal{R}$ is an integral domain such that every right invertible matrix can be completed to a unimodular matrix, then every matrix $P$ that satisfies the assumptions of the ideal-theoretic Rosenbrock's Theorem and realizes the same Schur complement $G$ shares the same ideal-theoretic generalization of the Smith form.

We investigate the ill-posedness mechanism of the hard-sphere Boltzmann equation in $H_{x}^{s}$ Sobolev space. Via a direct construction, we prove a strong-weak type ill-posedness result in the low-regularity regime $s<1$, establishing a sharp threshold in connection to the local $s>1$ well-posedness result [11]. Instead of originating from the large-velocity growth of the collision kernel, this illposedness is generated by the loss term and dispersive effects. Consequently, we prove a dispersion-driven nonlinear instability mechanism for the hard-sphere Boltzmann equation, and provide a capstone of the ill-posedness series [18,20].

We develop the Boltzmann-Wasserstein (BW) distance, a temperature-dependent metric on the space of quantum theories, defined as the optimal $W_2$ distance between Boltzmann-weighted energy spectra. Computing it is an optimisation over wormholes: each unitary identification of the two energy bases defines a coupling of the two boundaries in the doubled Hilbert space, and the optimum - the comonotone partition function $C_{\max}$, which pairs states by rank - is the dominant wormhole connecting the two theories. For semiclassical theories differing by a small entropy shift, the normalised BW distance collapses to a squared horizon-area comparator, $\tilde{\mathcal{W}}^2 \approx (δA/4G)^2/8$, with the two areas evaluated at equal energy. When the Hamiltonians differ by an operator $V$, the BW distance equals a long-time average of the real-time thermal two-point function of $V$; when the thermal one-point function of $V$ vanishes - for instance for $V$ odd under an unbroken discrete global symmetry - a four-point representation appears at the next order. On the gravity side we construct the classical saddle that computes $C_{\max}$: a Schwinger-Keldysh wormhole built from two Euclidean caps sharing a single horizon, joined by Lorentzian segments that adiabatically interpolate between the two theories. Its on-shell action reproduces the spectral saddle of $C_{\max}$ - both the saddle-point conditions and the on-shell value - and the Lorentzian segments are essential: a purely Euclidean interpolation is exponentially suppressed. The saddle captures only the rearrangement of the spectrum; the perturbative representations retain in addition the variance of the matrix elements of $V$, invisible to the classical geometry. We work out two examples - two BTZ black holes with different cosmological constants and a $T\bar{T}$ deformation of BTZ.

We present an image-based workflow for representative elementary volume (REV) sizing in chemically mapped granular microstructures, applied to Arabian dune-sand samples characterized by mineralogical and textural heterogeneity. The REV is treated as a finite-window convergence scale within approximately stationary material domains, rather than as a global length assigned to a non-stationary image. Full-resolution backscattered-electron (BSE) gray-level maps are screened by local mean and standard-deviation compatibility to identify stationary domains. Candidate windows are sampled only inside these domains, and the representative support is selected using a persistent mean--spectral criterion requiring both the apparent-mean residual and the low-wavenumber covariance-spectrum residual to remain within tolerance over the non-reference tail. Ensemble reproducibility is used as an auxiliary check. Applied to seven full-resolution BSE images of dune-sand microstructures, the strict stationary-domain criterion gives $(L_{\rm REV}=1536~\mathrm{pixels})$, corresponding to $(\ell_{\rm REV}\approx2.01~\mathrm{mm})$ for a BSE pixel size of $(1.31~μ\mathrm{m})$. Property-level homogenization on QEMSCAN-derived numerical maps independently supports this millimetre-scale estimate: the converted support is $(L_{\rm REV}^{\rm prop}=201.2)$ pixels and is snapped to the nearest tested size, $(L_{\rm REV}^{\rm prop}=204)$ pixels $(\ell_{\rm REV}^{\rm prop}=2.04~\mathrm{mm})$. This length lies in the large-window regime of the apparent conductivity, stiffness, and directional Young-modulus curves. The workflow provides a reproducible route for REV sizing while making explicit its dependence on stationarity, image field, window sequence, and target observable.

In this paper we study the Stokes phenomenon of the quantum confluent hypergeometric supersystem, certain meromorphic linear system of ordinary differential equation with a second order pole, associated to the Lie superalgebra $\mathfrak{gl}_{m|n}$. We prove that its Stokes supermatrices satisfy the Yang-Baxter equation, and thus give rise to the quantum supergroup $U_q(\mathfrak{gl}(m|n))$.

The square-and-multiply algorithm, also known as binary exponentiation or repeated squaring, is a technique for fast exponentiation commonly used in modern cryptography and computational number theory. Despite its prominence, the historical origins of the algorithm are not known with certainty. This paper critically examines the origins and formalization of the algorithm through primary source analysis. We focus on Jamshid al-Kashi's fifteenth-century Miftah al-Hisab where the algorithm is articulated explicitly as a general method and claimed by al-Kashi as his own innovation. To contextualize this, we trace earlier instances of successive squaring in the works of al-Uqlidisi and al-Biruni, who applied these techniques for specific calculations, but did not formalize them into a general procedure. The earliest known work on this method of computation is found in Pingala's prosodic studies in ancient India (c. 200 BCE). Even though it was not fully developed as a general technique, Pingala's work seems to contain the conceptual foundation of the algorithm which is to employ the binary representation of a positive integer. By mapping this intellectual progression, the paper illustrates the historical background of an algorithm that is prominent in modern computation.

We study the distribution of indecomposable elements in Euclidean lattices. A positive element in a lattice is called indecomposable if it cannot be represented as a sum of two other positive nonzero elements. The set of all indecomposables in a lattice forms the Hilbert basis for the positive lattice semigroup. We classify lattices that contain only finitely many indecomposables versus those that contain infinitely many. In the two-dimensional case, we prove that every positive element in a lattice can be represented as a positive integer linear combination of at most two indecomposables, which is a certain variation of the discrete Carathéodory's property. In the case of lattices coming from fractional ideals in real quadratic number fields, we obtain an explicit counting estimate for the number of indecomposables with bounded norm, showing logarithmic growth.

Pseudo-angular momentum (PAM) underlies optical selection rules for chiral phonons, but whether it also constrains thermally populated finite-wave-number phonon-phonon scattering has remained unresolved. We show that rotational or screw eigenphase conservation imposes a PAM residue rule on cubic anharmonic vertices, revealing a hidden selection rule for heat transport. In screw-symmetric helical Te, an exact platform, implementing this rule as a projector in first-principles Boltzmann transport leaves spectra and force constants unchanged but removes roughly two thirds of kinematically allowed triplets, suppresses resistive Umklapp relaxation, and enhances lattice thermal conductivity by a factor of 5.30 at 300 K, remaining above fivefold up to 400 K. A bulk chiral-crystal benchmark further shows that explicit eigenphase organization can increase the calculated lattice thermal conductivity by about 24%, comparable to the reported first-principles underestimation of experiment. These results establish PAM conservation as an anharmonic selection principle for chiral-phonon heat transport and as a fundamental principle to guide the prediction and control of thermal conductivity in chiral crystals and nanoscale phononics.

We study supervised regression with neural ODEs (NODEs) from a control-theoretic perspective to derive explicit population-risk bounds. We focus on a widely used class of non-autonomous models with constant parameters and explicit time dependence, which we call semi-autonomous NODEs (SA-NODEs). We constructively prove that SA-NODEs are capable of \emph{exact} interpolation of admissible finite datasets, and even satisfy a stronger property that we call \emph{simultaneous cell controllability} (SCC): their flows can map prescribed disjoint cells into arbitrarily small target balls. This property is the mechanism that upgrades interpolation into quantitative generalization, by allowing SA-NODEs to emulate piecewise-constant nonparametric estimators. Consequently, our risk bounds recover the rates of histogram and nearest-neighbor estimators, provided the network width satisfies a conservative scaling with the sample size. Numerical experiments show that trained SA-NODEs achieve competitive -- often lower -- test errors than these baselines. Finally, we show that the explicit time dependence is essential. Although two-layer autonomous NODEs can interpolate geometrically nondegenerate datasets, structural obstructions prevent them from achieving SCC. These limitations, further confirmed numerically, support the view that SA-NODEs provide a minimal effective architecture for learning.

Stochastic inflation is a powerful technique for calculating the probability distribution function (PDF) of large inflationary perturbations, which may collapse to form Primordial Black Holes. The PDF, $P({\cal N})$, of the stochastic number of e-folds, ${\cal N}$, satisfies an adjoint Fokker-Planck Equation. We develop a new self-contained eigenvalue technique which can be used to determine $P({\cal N})$. First we apply this method to the simple case of quantum diffusion along a flat potential without any classical drift. We recover the expression for the PDF that has previously been found using characteristic functions, with an exponential tail, and a power-law behaviour, $P({\cal N}) \propto {\cal N}^{-3/2}$, in the intermediate regime between the peak and the tail of the PDF. Finally we apply the method to constant drift inflation, in the narrow- and broad-well limits. In the narrow-well limit, there is an analytic solution and the PDF is similar to the drift-free case, with a mildly suppressed tail. In the broad-well limit, determining the full set of eigenvalues and eigenfunctions requires a piecewise construction of the spectrum, and the broad-well PDF is qualitatively different, with an enhanced peak and a strongly suppressed tail.

Transport measurements that probe electrical resistivity of a material under varying external stimuli, such as temperature, magnetic field, optical illumination, and gate voltage, are among the most important experimental techniques in condensed matter physics. These measurements provide critical insights into the fundamental electronic properties of materials. In recent years, they have facilitated the discovery and exploration of intriguing physical phenomena (e.g., superconductivity and quantum oscillations) and unique device functionalities (e.g., photoresponse and electrostatic gating effects) in various emerging materials, particularly in the form of single crystals. However, unlike large-scale wafers or thin films, newly synthesized single crystals often pose substantial challenges in establishing reliable electrical contacts due to their irregular geometries, limited dimensions, inherent structural characteristics, and potential susceptibility to degradation. In this review, we highlight recent technological advancements in the fabrication of high-quality, lithographically defined multi-terminal electrodes on both exfoliable and non-exfoliable single crystals for transport measurements. Our work provides a practical guide for researchers seeking to select appropriate contact-fabrication strategies tailored to unique characteristics of emerging crystals.

Two-state stochastic models, where motion alternates between distinct dynamical modes, are widely observed in complex systems. Here we study the Two-State Random Walk (TSRW), which switches between a continuous-time random walk (CTRW) rest state and a standard L'evy walk (LW) motion state, each with power-law distributed sojourn times. Using anomalous diffusion decomposition, we show that TSRWs exhibit a generic coexistence of Joseph (correlation), Noah (heavy-tailed increments), and Moses (aging) effects. Strikingly, although classical L'evy walks alone possess only the Joseph effect, both Noah and Moses effects emerge in TSRWs solely due to stochastic switching with the CTRW phase. Our results demonstrate that coupling between dynamical states can fundamentally reshape the mechanisms driving anomalous diffusion, offering a minimal yet powerful framework for transport in heterogeneous and intermittently switching environments.

Distributed applications increasingly support local-first collaboration over shared data, allowing multiple users to perform updates concurrently without global coordination. Such collaboration requires careful design to capture the intended semantics of the concurrent interactions. We introduce a declarative framework for specifying and reasoning about the semantics of conflict-free replicated data types (CRDTs) and CRDT-based applications in Datalog. The framework models CRDT semantics as executable logic programs over operation contexts, making concurrency explicit and compositional, and thus amenable to automated analysis. As one application, we use property-based testing to compare implementations. To the best of our knowledge, this is the first work to systematically use Datalog as a foundation for prototyping and analyzing complex CRDTs and their compositions. We evaluate our methodology using a collaborative graph data editing case study and report experimentation results assessing correctness validation and scalability with an increasing number of operations and replicas.

We present a new automated procedure for selecting fitting methodologies in the determination of parton distribution functions (PDFs), based on a hyperoptimisation algorithm using ensemble regression. Building on the k-folding approach previously employed by the NNPDF collaboration, we introduce a systematic strategy to generate ensembles of hyperparameter configurations and define a new k-folding metric that consistently incorporates full PDF uncertainties. The algorithm outputs an ensemble of statistically equivalent fitting methodologies, which are combined to produce a single PDF set whose uncertainties consistently account for hyperparameter variation. We assess the impact of this approach by comparing results obtained with identical data and theoretical inputs but different fitting methodologies, determined using the previous and the new hyperoptimisation procedures. The new method broadly confirms earlier NNPDF results, while yielding moderately larger uncertainties in regions where data provide limited constraints.

Charged impurities in quantum fluids have unveiled new classes of strongly correlated many-body states across condensed matter, ultracold gases, and hybrid atom-ion platforms. While previous studies have primarily focused on their ground-state and static properties, much less is known about their finite-momentum behavior, which governs transport, dissipation, and quasiparticle stability. Here, we investigate the momentum-dependent properties of a charged Bose polaron using a diagrammatic approach within second-order perturbation theory, explicitly accounting for the finite-range nature of the ion-atom interaction. We show that the interaction range introduces a characteristic momentum scale at which many-body dressing and dissipation are maximized, leading to a non-monotonic behavior of the damping rate and quasiparticle energy. In the high-momentum regime, we uncover a scaling law $Γ_p \sim 1/p$, signaling the suppression of many-body dressing and the recovery of quasi-free impurity dynamics, in stark contrast to the divergent behavior predicted by contact-interaction perturbative treatments.

Landscape dynamics are key drivers of the movement and distribution of sylvatic hematophagous disease vectors and their (wild) animal hosts. Their habitats are undergoing increasing change, particularly fragmentation, through anthropogenic activity. In this article, we present and analyse a novel mathematical model that explicitly combines anthropization-induced landscape dynamics with the population dynamics of hematophagous vectors and (wild) animals dynamics. We develop a phenomenological and analytically tractable two-patch model in which the migration terms between the patches nonlinearly depend on the anthropization level of the patches. Our model analysis comprising analytical stability analysis and numerical bifurcation analysis provides information on how changes in model parameters, especially anthropization levels, shape the long-term dynamics in the model. Precisely, we find that low anthropogenic activity allows for a vector-animal coexistence state, while high anthropization leads to a vector extinction state. However, we establish that for intermediate anthropization levels, the transition between the two states is not necessarily monotonic, but may instead occur via a sequence of concurrent bifurcations along the anthropization axis.

We present a few ideas on the Riemann Hypothesis based on properties of analytic functions in the complex plane. In particular, we focus on the real and imaginary parts of the Riemann xi ($ξ$) function whose zeros coincide with those of the zeta ($ζ$) function within the critical strip. We discuss the forms of the zero contour lines of the two conjugate harmonic functions (the real and imaginary parts of xi) and consider where their intersections could conceivably occur. Those intersections would be the roots of both $ξ$ and $ζ$ functions. The question of whether a zero could occur away from the critical line becomes equivalent to whether the solutions of a pair of Laplace's equations with well-defined boundary conditions in some semi-infinite strip can possess zero contour lines that intersect within that strip.

Given a BV theory on a space of fields split into two subspaces ("infrared" and "ultraviolet"), one has the BV pushforward map $P_*$, sending observables to observables of the effective theory on the infrared space. This note proves that $P_*$ is a quasi-isomorphism of BV complexes, by realizing it as a part of a strong deformation retraction constructed using the homological perturbation lemma. Two proofs are given: (i) comparing Feynman diagrams for $P_*$ with "cable diagrams" arising from homological perturbation theory and (ii) using topological quantum mechanics. This construction gives a formula for the quasi-inverse $i_\mathrm{int}$ of $P_*$ - the map lifting observables of the effective theory to the full theory. The topological quantum mechanics perspective - and its realization as an AKSZ theory - allows one to write $i_\mathrm{int}$ as a path integral (realizing cable diagrams for $i_\mathrm{int}$ as Feynman diagrams) and to study its classical limit.

The bootstrap method for quantum mechanics is a powerful tool for computing the energy eigenvalues of a Hamiltonian. However, we point out that this method suffers from an ambiguity problem: it fails to yield the correct spectrum when the potential contains different types of functions, such as polynomial and exponential terms. Similarly, the bootstrap method may break down when evaluating the expectation values of operators of different types. This issue can arise in a wide range of systems, including statistical models and matrix models. We propose three possible resolutions to this problem.

We present a set of metrics and methods for testing and comparing a range of modern gradient-free Markov Chain Monte Carlo (MCMC) samplers against the commonly used Metropolis-Hastings (MH) algorithm. The goal is to quantify key performance metrics, including sampler ergodicity, robustness and overall likelihood performance. To provide a controlled and interpretable testbed, we use the Rosenbrock function and Neal's funnel as representative unimodal cases, while Gaussian random likelihood landscapes in three, five, and eight dimensions serve as multimodal test scenarios. The samplers considered include affine-invariant moves from the literature, such as the stretch and walk moves, the differential evolution move, and the snooker move. We additionally introduce two novel variations: a modified stretch move that incorporates a Principal Component Analysis (PCA) transformation, and a hybrid blend move that combines features of both differential evolution and stretch dynamics. Beyond sampler evaluation, we demonstrate reconstructing likelihood landscapes from sampled points using a quadtree algorithm. Additionally, we explore the use of optimisation algorithms to refine the best parameter set in terms of its likelihood, and find consistent improvements in log-likelihood values, with the post-sampling gain becoming more significant in higher-dimensional problems. Our comparative results of sampler testing show that the differential evolution algorithm, when tuned to a target acceptance fraction of 25%, consistently outperforms all other samplers in terms of ergodicity, robustness, and likelihood performance.

A set of quantum data can look classical in every local test and still fail to admit a single classical explanation of the whole composite system. We formulate this failure as global contextuality. Here global means global in the physical sense of the whole multipartite system, not the local/global terminology of sheaf theory. Each party's local statistics are noncontextual and each measured multipartite context admits a generalized local hidden-variable description, but the GLHV block descriptions cannot be promoted to a single noncontextual hidden-variable model for the whole system. Three bipartite constructions exhibit this separation. A polarization-path construction gives a direct global obstruction. A qubit-qutrit KCBS construction gives an algebraic scenario-level example, with explicit formulas for the unconditional KCBS operator, the correlation-polytope constraints, and the postselected violation. A flagged qutrit Werner-local state gives a state-level example: the state is entangled and local for all projective measurements, its local qutrit marginals do not violate KCBS, yet postselection rules out a single GNCHV model. We also spell out the classical composition lemma: classical conditional hidden variables can be absorbed into a larger hidden variable, whereas quantum contextual data need not allow such a factorization. Within the general Bell-type framework considered here, with arbitrary parties and arbitrary local compatible contexts, but no cross-party joint measurements, the absence of local contextuality and GLHV-type generalized Bell nonlocality does not imply the existence of a global noncontextual hidden-variable model. Global contextuality is thus a compositional obstruction to classical explanation.

The material point method (MPM) is a hybrid particle-grid method widely used for simulating large deformation with history-dependent behavior. Standard MPM often relies on a dense background grid, which can be highly inefficient when material occupies a small fraction of the computational domain. Such sparsity is common in many large-scale problems, from geophysical mass flows over large terrain domains to visual-computing applications. Here, we introduce a unified sparse background-grid framework for large-scale MPM simulation. The framework treats sparse grid construction as a general active-node indexing problem. We develop two architecture-specific implementations to realize the same sparse framework: a scan-based strategy for CPUs and a hash-based strategy for GPUs. Through benchmark problems and a large-scale landslide simulation, we show that the framework provides the same results as standard dense MPM while reducing computational time and memory usage by one to two orders of magnitude in strongly sparse cases.

This paper studies the core problems in the $L_p$ dual Brunn-Minkowski theory, encompassing the $L_p$ Minkowski problem and $L_p$ Brunn-Minkowski inequality for dual quermassintegrals. For the case $0<p<q\leq n$, we establish $C^0$ estimates for the $L_p$ dual Minkowski problem without symmetric assumptions, thereby resolving a related problem proposed by Böröczky-Chen-Liu-Saroglou in the smooth sense. We further prove the uniqueness of smooth solutions under appropriate conditions, provided the density function is sufficiently close to a constant in the Hölder norm. Finally, exploiting the fact that the uniqueness of the Minkowski type problem is equivalent to the validity of the Brunn-Minkowski inequality in a certain sense, we study the $L_p$ Brunn-Minkowski inequality for dual quermassintegrals for origin-symmetric convex bodies with $p<q$.

In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new class of problems for these and related additive bases. The problems are designed, in part, to be susceptible to solution by AI.

In this work, we formulate a two-species Lotka--Volterra competition model on time scales. To derive the global dynamics of solutions of this planar model, we introduce a dynamic augmented phase plane analysis that extends the traditional phase plane analysis known to be a powerful tool in the analysis of planar differential equations. In the case of a constant graininess, the method is consistent with the augmented phase portrait introduced for the discrete space. However, for non-constant graininess, this augmented phase plane is time-dependent and therefore dynamic. Despite the dynamic character, we are able to identify time-independent information that allows the conclusion of the global dynamics of the introduced Lotka--Volterra competition model on time scales. The dynamic phase plane method, albeit only introduced in the context of this particular two-species competition model, it can be extended to other planar systems on time scales, providing a novel technique to study the global dynamics of planar systems on time scales.

We optimize the trade-off between privacy and utility in the high-privacy regime. We adopt local differential privacy (LDP) and its quantum extension, quantum local differential privacy (QLDP), for privacy protection, and investigate utility functions including the Holevo information (which reduces to the mutual information in the classical case) and the error exponents in symmetric and asymmetric hypothesis testing. These utility functions have classical and quantum optimal values, which are denoted by $C$ and $Q$, respectively, in this abstract for simplicity. In this paper, we provide optimal LDP and QLDP mechanisms achieving the classical and quantum optimal values in the high-privacy regime, and prove that the asymptotic ratio $Q/C$ in this regime takes the same value regardless of the utility function. Our results reveal quantum advantages (more precisely, $Q/C\ge3/2$) for the above utility functions when the protected private data are $n$-ary with $n\ge3$.

We consider the problem of determining feasible systems from a finite set of simulated alternatives with respect to probability constraints, where the observations from stochastic simulations are Bernoulli distributed. Most statistically valid procedures for feasibility determination focus on constraints on the means of normally distributed observations. Although these procedures can be adapted to Bernoulli-distributed data by treating batch means as basic observations, achieving approximate normality often requires a large batch size, potentially leading to the unnecessary waste of observations in reaching a decision. This paper proposes a procedure that utilizes the Bernoulli-distributed observations directly to determine feasibility. In addition, we incorporate subjective constraints, allowing for multiple thresholds for each constraint. We demonstrate that our proposed procedure is statistically valid and that it outperforms an existing feasibility determination procedure for subjective constraints originally developed for normally distributed observations. Furthermore, we propose two heuristic feasibility check approaches for thresholds that are sequentially added by decision makers, allowing thresholds to be tightened when many systems are feasible or relaxed when no feasible system exists. We show by experiments that the proposed procedures can efficiently provide feasibility decisions to systems with respect to all thresholds considered.

We show that if $T$ is a Dini-continuous Calderón--Zygmund operator satisfying $T(1)=0$, then the usual sparse domination for $T$ can be sharpened by replacing local averages with local mean oscillations. This extends a result of Benea and Bernicot for smoother kernels to the more general Dini-continuous setting. As an application, we characterize the Calderón--Zygmund operators for which a pointwise Sobolev-type inequality holds: this is the case if and only if $T(1)\in L^\infty$. This answers a recent question of Hoang, Moen and Pérez.