Papers

In this paper, we consider the weighted Hardy–Littlewood operator $\mathcal{H}_{\varphi}$ and its commutators in the Dunkl setting. We introduce mixed-norm radial-angular Dunkl central Morrey spaces, together with their $\lambda$-central counterparts. For the operator $\mathcal{H}_{\varphi}$, we derive the necessary and sufficient conditions on the non-negative weight $\varphi$ that ensure its boundedness on these spaces, and explicitly determine the exact operator norms. For the commutator $\mathcal{H}_{\varphi,b}$, we establish the necessary and sufficient conditions on the weight $\varphi$ such that the commutator is bounded for all symbols $b$ in the mixed-norm radial-angular Dunkl central bounded mean oscillation space $\mathrm{CMO}_{\vec p, \vec q, k}(\mathbb{R}^n)$. Furthermore, for all symbols $b$ in the mixed-norm radial-angular Dunkl $\lambda$-central bounded mean oscillation space $\mathrm{CMO}_{\vec p, \vec q,\lambda, k}(\mathbb{R}^n)$ with positive index $\lambda>0$, we also obtain a valid sufficient condition for boundedness.

In this paper, we prove a uniqueness result for the partial data Calderón problem with quasilinear conductivity in two dimensions. The proof is based on higher-order linearization and the use of CGO solutions in dimension two that vanish on part of the boundary. Since derivatives of the solutions appear in the integral identity, we need improved remainder estimates, for which we introduce a modification in the choice of the phase, analogous to limiting Carleman weights. We also analyze how combinations of phases produce specific patterns in the products of solutions, which allows us to apply both stationary and nonstationary phase arguments to recover the conductivity.

We study simplicity and splitting phenomena in finite skew braces under cyclicity assumptions on Sylow subgroups. We first classify finite simple skew braces whose multiplicative group is a \(Z\)-group. If the additive group is soluble, then the skew brace is either trivial of prime order or isomorphic to one of the two simple skew braces of order \(12\) with additive group \(A_4\) and multiplicative group \(C_3\rtimes C_4\). If the additive group is insoluble, then it is necessarily isomorphic to \(\operatorname{PSL}_2(p)\) for some prime \(p\geq5\). This conclusion is sharp, since such examples exist for every prime \(p\geq5\). We then consider the more general situation in which only a Sylow subgroup corresponding to the smallest prime divisor \(p\) of the order is assumed to be cyclic. Under suitable hypotheses on the additive or multiplicative Sylow \(p\)-subgroup, we prove that the skew brace contains a Hall \(p'\)-ideal and splits as a semidirect product of this ideal with a Sylow \(p\)-subbrace. As a consequence, every finite simple skew brace satisfying one of these hypotheses is trivial of prime order. Moreover, as a consequence of our splitting theorem, we verify Byott's solvability conjecture for finite skew braces whose additive group has a cyclic Sylow $2$-subgroup.

We show that scalar discrete-series unitary irreducible representations (UIRs) $\Pi_{p,0}$ ($p=1,2,\cdots$) of the de Sitter (dS) group $\mathrm{SO}_0(1,4)$ admit a dS-covariant Krein realization on the dS hyperboloid, endowed with a dS-invariant non-degenerate Klein-Gordon (KG) sesquilinear form, in which the group action is indecomposable and organizes naturally into a Gupta-Bleuler triplet. The positive- and negative-norm sectors are already present in the underlying Krein space, whereas a null sector emerges only at an intermediate stage, where the induced KG form becomes degenerate and its radical leads canonically to the physical quotient carrying the UIR $\Pi_{p,0}$. We further show that suitable limits of the bulk theory at the “future” and “past” conformal boundaries ${\mathcal{I}}^\pm$ give rise to dS-invariant boundary realizations endowed with induced kernel inner products. While the bulk negative-norm sector admits no independent boundary counterpart, the boundary realization retains the physical and gauge structures inherited from the bulk. The resulting boundary module nevertheless remains indecomposable, with its physical quotient carrying the discrete-series representation $\Pi_{p,0}$. The antipodal symmetry provides a natural relation between the realizations on ${\mathcal{I}}^+$ and ${\mathcal{I}}^-$, ensuring the consistency of the boundary construction and its geometric interpretation. At the heart of the analysis lies a Fourier-type bulk-boundary transform that provides a dS-covariant identification of the bulk and boundary physical sectors, establishing a one-to-one intertwining correspondence between the bulk and boundary realizations of $\Pi_{p,0}$ while preserving reflection positivity.

We present an expository proof of the irrationality of $\zeta\left(3\right)$ using modular forms of level 6. By constructing a suitable Eichler integral, we obtain a power series with controlled denominators and sufficiently large radius of convergence. Beukers' irrationality criterion then implies that $\zeta\left(3\right)$ is irrational.

We study a continuous-time dynamic pricing problem for a data platform that purchases raw data from privacy-sensitive providers and sells data products to consumers. The platform controls both the acquisition price offered to providers and the selling price charged to consumers. Provider and consumer arrivals are modeled by point processes whose intensities depend on the platform's current data stock, capturing feedback between data accumulation and market participation. We formulate the platform's problem as an infinite-horizon stochastic control problem with a jump-diffusion state process and derive the associated nonlinear integro-differential HJB equation. We prove that the value function is the unique viscosity solution, establish classical regularity under suitable conditions, and verify the optimal feedback pricing policy. Finally, we conduct numerical analyses to examine the influences of model parameters on the optimal pricing policies.

The planted subgraph detection problem asks whether a random graph contains a hidden structured subgraph. In the classical formulation, the entire adjacency matrix is observed and one distinguishes between an Erdős–Rényi random graph and one obtained by planting a copy of a prescribed graph inside an Erdős–Rényi random graph. The statistical and computational limits of this problem under full observation are now well understood, even for arbitrary planted subgraphs. In this paper, we investigate an information-limited version of the problem in which the planted structure is an arbitrary sequence of graphs $\Gamma=(\Gamma_n)_{n\geq1}$, where $\Gamma_n$ is embedded in an ambient graph on $n$ vertices, but the observer does not have access to the full adjacency matrix. Instead, information is acquired through a limited number of non-adaptive edge queries. We study the minimum query complexity required for reliable detection. We derive general information-theoretic lower bounds and complementary algorithmic upper bounds on the query complexity as functions of the query budget and structural properties of the planted graph. The proposed algorithms exploit three distinct structural mechanisms: dense local motifs, high-degree vertices, and global edge density. We establish matching bounds, up to polylogarithmic factors, for several broad families of planted graphs, including clique-like, bounded-cover, and hub-dominated graph classes. Our framework substantially generalizes existing query-complexity results for planted clique and planted dense subgraph models, providing a unified treatment of arbitrary planted subgraphs under restricted graph access.

Results by Jackson and Frank imply that every 2k-arc-strong digraph D contains a spanning k-arc-strong oriented subdigraph. This is best possible, even for very dense digraphs. A digraph is semicomplete if at least one of the arcs xy,yx is present for every pair of distinct vertices x,y. A tournament has exactly one of xy,yx for every such pair. Clearly every semicomplete digraph D contains a spanning tournament T which is obtained by deleting one arc from every 2-cycle of D. We prove that every (2k-1)-arc-strong semicomplete digraph on at least 2k+1 vertices contains a spanning k-arc-strong tournament. Both bounds 2k-1 and 2k+1 are best possible. The proof uses Frank's general orientation theorem for graphs as well as counting arguments based on the semicomplete structure.

In Einstein's seminal work [Ann. Physik, 17 (1905), 549-560], he pointed out that the temperature of a fluid influences the motion of suspended particles dramatically. To describe the effect of the temperature in this physical process more precisely, Boudin et al. [ESAIM Proc., 28 (2009), 195-210] introduced a new fluid-particle interaction model containing of the non-isentropic compressible Euler equations for the fluid and a nonlinear Vlasov-Fokker-Planck type equation for the particles. By adding some viscous and heat conductive terms to the fluid part of this model, Mu and Wang [Calc. Var. Partial Differential Equations, 59 (2020), Paper no. 110] established the global existence of classical solutions near an equilibrium state. In this paper, through establishing the uniform a priori estimates with respect to the viscosity and heat conductivity coefficients and taking the combined zero viscosity and heat conductivity limits, we show that the model introduced by Boudin et al. still admits a global classical solution and enjoys optimal decay rates thereby improving Mu and Wang's results and confirming Einstein's predications. Our work indicates that the presence of particles indeed emanates new dissipation effects on the non-isentropic compressible fluid-particle model via the differences between the macroscopic velocity of the particles and the fluid velocity, and the macroscopic temperature of the particles and the fluid temperature, which is significantly different from the case of pure non-isentropic compressible Euler equations. To achieve these goals, we have developed new ideas and techniques to surmount substantial obstacles caused by the absence of viscosity and heat conductivity, and the nonlinear interactions between the fluid and particles.

For an odd prime $p$, we give an explicit formula for the wreath transfer $\mathrm{tr} \colon H_*(\Sigma_{p^2};\mathbb{F}_p)\longrightarrow H_*(\Sigma_p\wr\Sigma_p;\mathbb{F}_p)$ associated with the standard inclusion $\Sigma_p\wr\Sigma_p\le\Sigma_{p^2}$. The proof is a direct algebraic computation, using Cartan's classical double-coset formula together with a generating-function argument. As an immediate consequence, we obtain an elementary proof of the validity of the mixed Adem relations in Goodwillie calculus.

We establish quantitative two-time spatial decorrelation for the Kardar–Parisi–Zhang fixed point with flat initial data. For every $s,t>0$,there exist constants $C,c>0$ such that \[ \big|{\rm Cov}(\mathfrak{h}(t,x),\mathfrak{h}(s,0))\big| \le C\exp\{-c|x|^3\},\qquad |x|\ge1. \] Unlike the fixed-time covariance, which is governed directly by the Airy$_1$ process, the two-time covariance involves the nonlinear variational evolution of the entire earlier height profile. Our proof combines cubic-exponential mixing of the Airy$_1$ process with a uniform localization estimate for intermediate optimizers in the directed landscape. As a consequence, the centered spatial averages, normalized by $N^{1/2}$, converge in finite-dimensional distributions to a centered Gaussian process whose covariance is the space-integrated two-time correlation of the flat KPZ fixed point.

We consider the initial value problem for the vorticity equation in the endpoint critical Sobolev space $W^{d,1}(\mathbb{R}^{d})$ for $d = 2, 3$. In two dimensions, we prove global propagation of the $W^{2,1}(\mathbb{R}^{2})$ regularity of the vorticity. In three dimensions, for axisymmetric flows without swirl, we propagate $W^{3,1}(\mathbb{R}^{3})$ regularity of the vorticity for all times. These are in stark contrast to existing strong ill-posedness results in critical Sobolev spaces $W^{d/p,p}(\mathbb{R}^{d})$ for all $1 < p < \infty$, which were based on axisymmetric flows without swirl when $d = 3$.

A counting problem arising naturally from game theory leads to the problem of estimating the coefficients $t_{ij}$ of a particular bivariate rational function. Restricting to the diagonal sequences $t_{j+k,k}$ (with $k$ fixed) allows us to study these sequences using elementary one-variable methods.

We study the expressivity of shallow polynomial neural networks (PNNs) with monomial activation functions over finite fields. For a given architecture, we define a neuromanifold as the image of the map from all possible network weights into the product of polynomial rings. We quantify the expressivity by the cardinality of the neuromanifold, and derive a natural lower and upper bound. This leads to counting rational points over finite fields, a problem closely linked to the Weil conjectures. Finally, we present an architecture that exhibits a striking difference in the neuromanifolds when considered over a characteristic zero versus a finite-characteristic field, illustrating the critical role of field characteristic in the notion of expressivity.

The exponential growth of AI is accelerating the deployment of data centers (DCs), placing unprecedented strain on power infrastructures. In response, major IT corporations are increasingly adopting on-site solar generation to reduce grid dependence and meet sustainability targets. However, the true impacts of this strategy remain ambiguous. While DCs are flexible assets capable of temporal load-shifting, anchoring them to self-generated power may inadvertently constrain their grid responsiveness. To evaluate these trade-offs, we propose a receding-horizon optimization (RHO) framework coordinating job scheduling, grid interactions, and on-site solar generation for a stand-alone DC. Our findings reveal a critical paradox: although solar integration increases energy self-sufficiency and reduces overall DC emissions, it inherently limits the facility's capacity to absorb low-cost, low-carbon electricity from the grid. This implies a fundamental tension between individual corporate sustainability goals and system-wide grid decarbonization.

In this paper, we establish transportation-cost inequalities for invariant measures of stochastic reaction-diffusion equations driven by multiplicative space-time white noise. In particular, the transportation-cost inequalities are established with respect to the $L^1$ metric, the $L^2$ metric and the uniform metric under different dissipativity strengths of the drift. We first obtain a global-in-time and spatially uniform moment estimate for stochastic convolutions driven by space-time white noise, which is of independent interest. We then establish Lipschitz continuity of the solutions to the associated stochastic controlled equations with respect to the controls with a time-independent Lipschitz constant. Applying this Lipschitz continuity estimate, we finally prove transportation-cost inequalities for the invariant measures of stochastic reaction-diffusion equations.

Many multiple-testing procedures control the false discovery rate (FDR) by comparing the two tails of a null distribution. At a fixed cutoff, marginal symmetry makes this natural. Mirror and knockoff+ thresholds select the cutoff from the same data, so the standard finite-sample guarantee uses a stronger property: conditional on magnitudes and nonnull scores, null signs are independent fair coins. Failure can be severe without this property. Models satisfying positive regression dependence on a subset (PRDS) can have exactly uniform null $p$-values and large FDR. Under every fixed positive Gaussian equicorrelation, the all-null FDR converges to one half. Opposing loadings in Gaussian factor models can make FDR and power arbitrarily close to one; near-total failure also occurs for exchangeable, pairwise-uncorrelated scores. At a nominal input level $q<1/2$, no deterministic rule based only on the two tail counts can both reject and control FDR uniformly over our class if more discoveries or fewer controls cannot make rejection harder. We give finite-sample repairs based on joint sign information. Conditional sign odds may be bounded outside an exceptional event or averaged over negative controls; neither route uniformly dominates, and the integrated bounds are sharp. Independent calibration data or a specified Gaussian joint model yield valid adjusted levels. Simulations show that integration retains more power under diffuse Gaussian dependence, whereas exceptional-event calibration is more powerful when very large odds occur only for rare aligned signs; unadjusted FDR exceeds the target in both settings. Covariance alone is insufficient outside a specified joint model. Thus the relevant boundary is not marginal symmetry but joint information that remains valid after adaptive cutoff selection.

Imagine that you are in front of a cockpit with several on-off buttons. If you were to thoroughly test it, you would need to try a prohibitive number of configurations. But since most bugs in practice can be isolated to interactions among few components, having tests that cover every possible pairwise configuration is a good start. However, this is a problem that goes from easy to NP-hard as soon as some pairwise configurations are forbidden. In this paper, we revisit unconstrained combinatorial testing with pairwise coverage on binary parameters and contrast it with the constrained case, showing and conjecturing properties that either are upheld or change from one to the other. In particular, we discuss the extent to which it remains a good idea – and sometimes indeed optimal – to have every button almost as many times on as off to minimize testing. We propose the first exact algorithm based on integer programming and a faster heuristic that often produces optimal solutions, both outperforming or competitive with their baselines.

Base-2 digital nets are practical high-dimensional integration rules: the sample budget $N=2^m$ can be chosen independently of the ambient dimension, and the generating matrices provide algebraic control of projections and Walsh-dual weights. They are therefore well suited to problems whose error is governed by weighted or low-dimensional projection structure. However, when one restricts attention to a smooth low-dimensional projected component, a low-dimensional cubature rule with a comparable number of nodes can be substantially more accurate than the projected digital-net points. This raises the question of whether low-dimensional cubature accuracy can be inserted into a high-dimensional digital-net rule without forming the full tensor product. We answer this question by a simple coordinate embedding: read the leading $p$ binary digits of each coordinate as an index into $2^p$ equal-weight cubature nodes, and replace the coordinate by the indexed node. When a projection forms the full $p$-bit grid, the transformed rule coincides on that projection with the corresponding product cubature rule; small projected $t$-values provide sufficient conditions for such full-grid recovery. For general integrands, the error separates into the corresponding product cubature error and a residual digital-net term. Experiments with scrambled Sobol' nets in dimension $50$ illustrate this mechanism and show finite-budget improvements for the smooth low-order and coordinate-decaying test functions considered here.

We prove that every infinite subgroup of a higher rank irreducible lattice in a semi-simple Lie group has finite index in the lattice, proving the Margulis-Zimmer conjecture for these lattices. The method of proof combines the method used in a work of Shalom and Willis and also some work of Raghunathan and Venkataramana.

This paper studies transformations that increase efficiency in solving equilibrium systems without information loss. Our approach exploits order-theoretic structure commonly found in economic problems to obtain conditions under which high-dimensional systems can be transformed into low-dimensional systems while preserving exact relationships between their solutions. The transformations can also be used for purposes other than dimensionality reduction, such as simplifying analysis and facilitating stochastic approximation routines. The theoretical ideas are illustrated using applications from economics and finance. In a real option problem, we demonstrate speed gains of up to 70,000 times.

We consider \[ \ddot x+f(x,\omega t)\dot x+g(x,\omega t)=0, \] where \[ f(x,\theta)=\sum_{j=0}^{m}a_j(\theta)x^{2j+1},\qquad g(x,\theta)=x^{2n+1}+\sum_{j=0}^{n-1}b_j(\theta)x^{2j+1}. \] The coefficient functions are real analytic and even on the torus and the frequency vector $\omega$ is Diophantine. If $n\geq2(m+1)$, we construct codimension-one reversible KAM tori accumulating at infinity and prove that all solutions are bounded. The main point is a finite normal-form procedure. After the reversible polynomial reduction, a logarithmic Fourier cut-off is introduced. At the $v$-th step a truncated homological equation is solved on a non-resonant action interval and the new error satisfies an explicit finite-step recurrence. Thus an arbitrarily small negative power of the large action is reached after finitely many steps. Finally, the Largrangian stability and the existence of quasi-periodic solutions are proved by the reversible KAM theorem.

We propose a reduced order modeling (ROM) framework for 1D conservative PDEs based on the cumulative distribution transform (CDT). The CDT maps nonnegative, equal-mass states into a Hilbert space in which 1D Wasserstein distances become weighted $L^2$ distances and translations become affine shifts. This makes the transform especially suited for transport-dominated dynamics, where Eulerian linear-subspace ROMs often suffer from slow decay of Kolmogorov widths. We study this phenomenon for scalar conservative dynamics by analyzing the solution manifold in CDT coordinates. For linear transport, the transformed solution manifold is contained in the 2-dimensional space spanned by the transformed initial datum and the constant function, and has zero Kolmogorov $2$-width. For nonlinear hyperbolic conservation laws, we prove two complementary types of estimates: robust $O(n^{-1})$ bounds that rely only on the conservative transport structure and remain meaningful after shock formation, and sharper $O(n^{-2})$ bounds in smooth pre-shock regimes. For conservative advection-diffusion, we show that the CDT trajectory remains within distance $O(\sqrt{DT})$ of the pure-transport plane, and we also obtain sharper $O(D^2T^2)$ estimates under additional regularity or away from initial layers. In both cases, the zero 2-width behavior of linear transport is recovered as the diffusion coefficient tends to zero. Motivated by these estimates, we develop a CDT-POD numerical scheme: snapshots are mapped to CDT space, Proper Orthogonal Decomposition (POD) is performed in transformed coordinates, and the inverse CDT is used to reconstruct physical states. Numerical experiments for several transport-dominated dynamics show that CDT-POD can capture solution manifolds with substantially fewer modes than Eulerian POD.

The convergence of a finite element discretization for the two-dimensional Ricci flow is proved. In this method, the Ricci flow on a two-dimensional surface is formulated into solution-driven metric evolution, with the metric evolution driven by the Gauss curvature. The Gauss curvature satisfies a parabolic equation that in turn depends on the metric, thereby enhancing the parabolic structure of the problem. The solution-driven metric evolution formulation is discretized by the finite element method, and the convergence of finite element approximations is proved by adapting the matrix-vector formulation developed in the literature initially for studying solution-driven surface evolution in extrinsic curvature flow. In addition to its convergence, the proposed method also preserves important geometric structures of the Ricci flow at the discrete level, such as area conservation and the Gauss-Bonnet theorem. Extensive numerical experiments are presented to demonstrate the convergence of the proposed method as well as the simulation of Ricci flow.

Let $X_1, X_2, \ldots, X_n$ be independent random vectors. For a directed graph $G=(V,E)$ with vertex set $V=\{1,2,\ldots,n\}$ and a collection of bivariate kernels $\{h_e:e\in E\}$, we consider \[ U=\sum_{e=(i,j)\in E} h_e(X_i,X_j). \] This framework generalizes incomplete U-statistics by allowing the random vectors to be non-identically distributed, the kernels to be asymmetric and edge-dependent, and the sampling structure to be specified by an arbitrary graph. We derive several concentration inequalities for $U-\mathbb{E}U$. The main proof strategy exploits edge-coloring results from graph theory and relates the tail behavior of $U$ to the chromatic index of $G$. This approach is elementary, transparent, and readily adaptable to broader settings, including U-statistics of order $m>2$ and statistics involving doubly indexed random vectors.

In this article, we establish Hölder regularity for viscosity solutions to a class of degenerate fully nonlinear elliptic equations of the form \[ F(D^2u,Du)=f(x)~~\text{in}~~B_1, \] where the operator is elliptic only in regions where the Hessian is sufficiently large. Such equations arise naturally in free boundary problems and models with partial ellipticity. The proof combines a modified cusp function with a decomposition of the contact set in a point-to-measure argument. As a consequence, interior Hölder continuity follows under natural structural assumptions.

We propose a linear variable-step exponential time-differencing method for the incompressible Navier–Stokes equations in vorticity–streamfunction formulation on a two-dimensional periodic box. The method consists of a second-order scheme and an embedded first-order variant, yielding a natural mechanism for adaptive time stepping and a posteriori error control. Each time step requires only uniquely solvable linear problems: two heat equation solves, efficiently handled by Fourier methods in the periodic setting, and one linear scalar auxiliary-variable equation, evaluated via Laplace transform and Talbot's numerical inverse transform. The construction combines the ETD framework, a mean-reverting scalar auxiliary variable (mr-SAV), and second-order extrapolation of the nonlinear term. The mean-reverting correction enables long-time stability while preserving full linearity, distinguishing the method from related mr-SAV schemes that require nonlinear algebraic solves. We prove unconditional long-time stability: for uniformly bounded $L^2$ forcing, the discrete vorticity remains bounded in $L^\infty(0,\infty;L^2)$ for all Reynolds numbers and time-step sizes. Numerical experiments

We study stationary measures of the log-gamma polymer on a finite diagonal strip and in a half-space. We establish the phase diagram for the stationary measure on the strip. To this end we develop a representation of {the Laplace transform of} this stationary measure by independent $\mathrm{Beta}_{II}$ random variables. We also present an analytic approach that extends Barraquand's contour integral representation of the Laplace transform. We prove that, as the strip width tends to infinity, the stationary measure of the log-gamma polymer on the strip converges to the stationary measure of the half-space log-gamma polymer. Finally, we derive a contour integral formula for the Laplace transform of the stationary measure of the half-space log-gamma polymer.

Let $A:\mathbb{F}_2^n\to\mathbb{F}_2^m$ be a binary linear map with fixed coordinate bases, let $C_A=\ker A$, and let $\lambda_A(y)$ be the minimum Hamming weight of a preimage of the syndrome $y$. We define $\operatorname{Shat}_{q,s}(A)$ as the least common check support of a $q$-dimensional syndrome subspace whose every nonzero element has coset-leader weight at least $s$. It therefore distinguishes release of $q$ independent syndromes from release of a subspace with no easy linear combination. Deleting check coordinates $F$ releases $\ker A_{\bar{F}}/\ker A$, canonically isomorphic to $(\operatorname{im} A)[F]$. Finiteness implies $R_q(C_A)\ge \mathsf{N}_2(q,s)$, where $\mathsf{N}_2(q,s)$ is the shortest length of a binary code of dimension $q$ and distance at least $s$; profile-Griesmer bounds independently control common check support. The hierarchy is coordinate-relabeling invariant but can change under a change of check basis. For the pair-repetition code $C_n=\{(x,x):x\in\mathbb{F}_2^n\}$, the standard realization $H_0=[I_n\ I_n]$ has $\operatorname{Shat}_{q,s}(H_0)=\mathsf{N}_2(q,s)$ whenever feasible. For every $q\ge 1$ and $s\ge 2$, with $n=\mathsf{N}_2(q,s)$, a row-equivalent realization of the same code has value $q$. For a simplicial coboundary map $A=\delta_k$, check erasure is top-face erasure and the released quotient is emergent cohomology. At $s=1$ the hierarchy reduces to generalized Hamming weights and is Tutte-determined; for $s\ge 2$, even identical labeled cut codes can have different values.

For a constructible sheaf on a smooth scheme over a field of positive characteristic, the singular support and the characteristic cycle are defined on the cotangent bundle. We study if a construction using the microlocalization similar to that in a transcendental setting due to Kashiwara–Schapira works in the algebraic setting and prove partial positive results.