Papers

In this paper, we obtain two complete lattices that are sober with respect to the Scott topology, whereas the Scott space of their product is not a sober space. Thus, we answer a question posed by Miao, Xi, Jia, Li and Zhao in [8].

In this paper, we study nonlinear Schrödinger equations arising in the study of Markovian open quantum systems. These equations are derived from the normalization of linear stochastic Schrödinger equations. First, we prove the global well-posedness of the linear equation in the energy space, which is less regular than the usual space for this equation. Then, we show the indistinguishability of solutions to the nonlinear SPDE under a relaxation of the usual monotonicity assumption. Finally, by proving the pathwise uniqueness and normalizing the solutions to the linear equation, we are able to prove the well-posedness of solutions to the required class of nonlinear stochastic equations.

Persistent homology can be differentiated and incorporated into learning pipelines, but no analogous framework exists for zigzag persistence, which is needed when the underlying topological structure evolves non-monotonically over time. We develop such a framework for sequences of simplicial complexes obtained by thresholding time-dependent filtering values on a fixed complex. By assigning persistence diagram endpoints the real-valued times at which linearly interpolated filtering values cross the threshold, we transfer the continuity of the filtering values to the diagram points. This yields smooth local lifts of the resulting persistence-diagram-valued map, from which we derive differentials almost everywhere under mild regularity conditions on the parametrization of the filtering values. We prove local Lipschitz continuity outside an explicit measure-zero exclusion set; standard stochastic subgradient convergence guarantees therefore do not apply directly. We argue that, even without such guarantees, this exclusion set is small enough in practice to allow effective optimization. We test this empirically in two experiments: sensor network coverage optimization and dynamic graph classification.

Let $A$ and $B$ be strongly Morita equivalent $C^*$-algebras, implemented by a nonzero imprimitivity bimodule $X={}_AX_B$ with its canonical operator-space structure. Denote by $Z_X$ their common multiplier centre, canonically identified through $X$. The coefficient action gives a complete contraction $\Theta_X:A\otimes_{Z_X,h}B\to\operatorname{CB}(X)$, where $A\otimes_{Z_X,h}B$ is the central Haagerup tensor product obtained by balancing over $Z_X$ and $\operatorname{CB}(X)$ is the space of completely bounded maps on $X$. We show that $\Theta_X$ is injective exactly when every Glimm ideal of $A$ is $2$-primal, and isometric, equivalently completely isometric, exactly when every such ideal is primal. For each positive integer $\ell$, norm preservation for sums of at most $\ell$ elementary tensors is equivalent to $(\ell^2+1)$-primality of every Glimm ideal. These conditions may equivalently be imposed on $B$. This extends the corresponding theorems of Somerset and Archbold–Somerset–Timoney to imprimitivity bimodules. We also show that, after matrix stabilization, the optimal constants bounding the central Haagerup norm by the completely bounded norm coincide for $X$, $A$ and $B$, at every finite tensor length and on the completed products, and agree with the corresponding constants after compact stabilization.

Cyclic coordinate descent is widely used to train support vector machines, but global linear convergence alone gives limited guidance about the rate on a particular instance. We study exact cyclic coordinate minimization for box-constrained convex quadratics and develop global contraction bounds that capture both coordinate order and boundary geometry. Our convergence certificate combines two ingredients: interactions between successive coordinate updates, encoded by the triangular part of the normalized Hessian, and objective growth, captured by a matrix lower bound on the optimality gap. It remains valid for cases with singular Hessians, nonunique minimizers, and updates clipped at the boundary. An error-bound specialization of our analysis strictly sharpens the Wang-Lin factor when both use the same error-bound constant. In the positive-definite case, the analysis shows that the worst-case contraction of one Gauss-Seidel sweep remains a global upper bound under box constraints, with equality for an interior optimizer. We also demonstrate the limitations of Hessian-only bounds by constructing a two-coordinate family with a fixed singular normalized Hessian, whose worst-case contraction approaches one as its boundary margin vanishes. The theory is complemented by practical procedures for bounding the worst-case contraction on individual instances and numerical evaluation on several classical SVM datasets.

We present multitable and its Rust reference implementation: a stable hash table both materially faster at equal physical memory and more flexible than the SwissTable in its Rust's hashbrown implementation. As an arithmetic mean over 84 configurations it delivers $\mathbf{2.1\times}$ hashbrown's throughput when both hash the same raw bytes and $\mathbf{1.9\times}$ when hashbrown is keyed on native integers, its best case; on negative lookups alone, $3.2\times$ and $2.9\times$. Multitable reaches any physical load factor up to and including one ($0.9999$ demonstrated), exactly for the requested capacity, compared to hashbrown which doubles at $0.777$ for 4-byte keys and values. At $75\%$ saturation of hashbrown (assumed average case of its rigid ladder) and multitable sized to $0.97$ physical load factor, hashbrown takes $66\%$ more space. The lookup probe count has no cliff as the load factor approaches one. Bucket size, physical load factor, and failure budget are parameters, and the multitable can be grown without rehashing. We implement two variants of multitable: plain and filtered. At equal physical memory on an Apple M2 Pro the filtered multitable leads hashbrown in all $84$ insert, hit, and miss configurations. Multitable is more memory-efficient, at equal mixed-lookup throughput on the map of $4$-byte keys and values the filtered multitable needs up to $12\%$ fewer bytes than hashbrown, and the plain multitable is $18\%$ smaller, holding $\mathbf{22\%}$ more keys in the same memory.

We develop a multi-view linear-programming framework for locally recoverable codes with arbitrary fixed availability $a$. For any retained order $1 \le s \le a$, the selected helper sets, the recovered coordinate, and their complement form an $(s+2)$-part partition. Recording the Hamming distance on all blocks preserves both compatibility among the selected repair alternatives and their coupling with the remaining coordinates. The resulting joint distribution satisfies centered counting identities, product-Krawtchouk positivity, and collision inequalities from the local-distance condition; for fixed $s$, these constraints give a polynomial-size relaxation for arbitrary, possibly nonlinear, codes over any finite field. Retaining one view recovers the three-block model of our companion paper. We develop the first genuinely multi-view case, $s=2$, in detail and specialize the exact computations to availability $a=2$. The resulting four-block LP projects to both the one-view three-block model and a globally conditioned two-view relaxation, making explicit the information lost by each coarsening. Exact rational primal-dual certificates together with checked constructions prove $M_{\max}(2,8,4,2,2,2)=8$, $M_{\max}(3,7,3,2,2,2)=27$, and $M_{\max}(4,7,3,2,2,2)=64$; in all three cases the four-block bound is strictly stronger than both coarsenings.

We prove an annealed central limit theorem for the weight of the maximum weight matching on uniformly random simple graphs with prescribed, uniformly bounded degrees and i.i.d. exponential edge weights. In particular, the result applies to random $d$-regular graphs for every fixed $d \ge 2$. The proof separates the fluctuations arising from the edge weights from those arising from the graph. The correlation decay estimate of Lam and Sen (arXiv:2511.18861) yields Gaussian fluctuations for the former. The main difficulty is to analyze the fluctuations of the conditional mean of the optimal weight given the graph. To address this, we prove a stronger perturbative correlation decay estimate that, together with a variance bound, reduces the problem to the central limit theorem of Barbour and Röllin [Ann. Appl. Probab. 29(2) (2019)] for local statistics of the configuration model.

Let $H=\langle a_1,\ldots,a_n\rangle$ be a numerical semigroup. A conjecture on numerical semigroup rings predicts that the defining ideal $I_H$ has a determinantal presentation if and only if the set of pseudo-Frobenius numbers of $H$ has form $\{h+\alpha,h+2\alpha,\ldots,h+(n-1)\alpha\}$ for $h\ge 0, \alpha>0$. We prove this conjecture under the condition that $h$ has a unique factorization in $H$. We also give characterization this unique-factorization condition. As an application, we consider the family of affine-orbit numerical semigroups determined by $m=cA_{n-1}+1$, $1\le c\le a$. We prove that the relevant element $h$ has a unique factorization and the defining ideal is generated by the 2-minors of a $2\times n$-matrix. In particular, the Eagon–Northcott complex of the matrix mentioned above gives the graded minimal free resolution of $k[H]$.

Nikodym constructed a measure-zero Borel set in the plane that contains a punctured line segment passing through each point in the unit square. Fenton and Falconer asked whether a similar construction is possible when “lines through a point” are replaced by “circles or circular arcs with a prescribed center.” This was answered in the negative by Bourgain and Marstrand. We study generalizations of this question, in which “lines through a point” or “circles with prescribed center” are replaced by more general analytic families of plane curves. As a key ingredient, we prove new maximal function and local smoothing estimates for families of curves satisfying a multi-parameter analogue of Sogge's cinematic curvature condition; this allows us to establish the non-existence of Nikodym-type sets in certain specific settings. The result also resolves a local smoothing conjecture by the sixth author. In the other direction, we use a recent nonlinear generalization of Falconer's prescribed projection theorem to construct Nikodym-type sets in other specific settings. Combining these ingredients with tools from tame geometry, we precisely characterize when Nikodym-type sets exist for analytic families of plane curves.

This survey presents heat kernel regularization as a method for quantitative stability of Kantorovich potentials for the quadratic transport cost. We give a complete new heat kernel proof on Heisenberg groups equipped with the Carnot–Carathéodory distance. For source densities bounded above and away from zero on bounded John domains, we obtain $L^2$ stability with rate $W_1^{1/2}$ for arbitrary targets in a fixed compact set, including atomic measures. We also establish a new $L^2$ stability estimate on finite-dimensional RCD spaces, with the optimal dimension-independent exponent $1/2$.

Despite the availability of excellent vaccines and antiviral medicines, hepatitis B virus (HBV) remains a major public health problem. In this work, we develop a mathematical vaccination-treatment model for HBV transmission to investigate threshold conditions for illness persistence as well as the level of intervention. The analysis gives the characterisation of both disease-free and endemic equilibria and determines the basic reproduction number, $\mathcal{R}_0$, by the next-generation matrix approach. The model indicates that $\mathcal{R}_0=1$ is a critical threshold dividing disease-free from endemic states and characterises a forward transcritical bifurcation regarding the transmission parameter. Numerical equilibrium continuation and stability assessments confirm the analytical results. A two-parameter study of transmission and vaccination rates identifies threshold geometries and assesses impacts of important parameters using sensitivity analysis. The model is calibrated with India-specific data, such as HBsAg prevalence, HBV incidence and mortality rates and fits epidemiological targets closely through limited nonlinear calibration. Additionally, multi-start optimisation and practical identifiability profiling are used to investigate parameter uncertainty. Robustness analyses reveal that the calibrated $\mathcal{R}_0^*=1.1744$ is associated with a super-threshold regime; however, parameter flexibility permits both sub-threshold and super-threshold regimes. This combined analytical and data-driven framework connects epidemic thresholds, bifurcation structures, intervention efficacy, and parameter uncertainty. It provides a quantitative framework for analysing HBV persistence and evaluating control options, especially in the absence of epidemiological evidence.

We consider harmonic approximation on the torus $\mathbb{T}^d$, the sphere $\mathcal{S}^d$, and the rotation group $\mathrm{SO}(3)$. While several global approaches are available for computing a harmonic expansion from scattered data, their stability and, especially, their runtime depend strongly on the geometry of the nodes, in particular on a sufficiently small fill distance. Even a single large hole in the data may cause instability and long runtimes. We propose harmonic approximation via moving least squares (HAMLS), a global harmonic approximation scheme that avoids the ill-conditioned global solve and works well in both the overdetermined and the underdetermined settings. The main idea is an intermediate transition from the scattered data to a quadrature grid, which is realized via moving least squares (MLS). This replaces one large global problem by many small local ones that can be regularized individually. The global harmonic approximation is then obtained via quadrature using a fast Fourier transform. This makes the global stage fast, stable and non-iterative. For sufficiently smooth functions sampled at well-distributed nodes with small fill distance $h$, we bound the resulting $\mathrm{L}^2$-error by the error of the harmonic approximation obtained from exact values on the quadrature grid, plus a term of order $h^{K+1}$, where $K$ is the polynomial degree employed in the MLS step. Numerical experiments on $\mathcal{S}^2$ with ill-distributed nodes show that HAMLS achieves errors comparable to those of global least-squares approximation, while being significantly faster, especially for larger bandwidths.

Radchenko and Wheeler (RW) recently proved a finite pentagon relation for real quadratic special values of the modular quantum dilogarithm and used it to establish the rank-$1$ twisted convolution identity conjectured by the current authors. RW gave an explicit argument in the principal case and remarked that their proof holds for all rank-$1$ admissible tuples. We extend their proof to all rank-$r$ admissible tuples and provide an explicit dictionary between the modular quantum dilogarithm and the Shintani-Faddeev modular cocycle conventions in the respective papers. Thus, we establish that, if $d,r$ are positive integers such that $r<\frac{d-1}{2}$ and $\frac{d^2-1}{r(d-r)} \in \mathbb{Z}$, then there exist ghost $r$-SICs: i.e., configurations of $d^2$ rank-$r$ subspaces in $\mathbb{C}^d$ that satisfy a non-Hermitian equichordal condition. Under the Stark conjecture, these configurations are Galois conjugate to Hermitian equichordal configurations called $r$-SICs (or rank-$r$ SIC-POVMs).

The extreme therapeutic resistance of malignant gliomas motivates the development of multimodal treatment strategies. We present a mechanistic mathematical framework to investigate the coupled dynamics of radiotherapy, temozolomide, and chimeric antigen receptor (CAR) T-cell therapy. The model is formulated as a nine-dimensional impulsive dynamical system that captures proliferative-quiescent tumor states, treatment-specific resistance, tissue damage, TMZ pharmacokinetics, and the lymphotoxic interferences exerted by conventional regimens on engineered T cells. Mathematical analysis establishes the existence of biologically meaningful solutions, maps invariant structures, and identifies threshold conditions under which sustained therapy locally stabilizes the tumor-free equilibrium. After study-matched benchmarking against clinical trials, including the standard Stupp protocol, we deploy heterogeneous virtual patient cohorts to evaluate five combined CAR-T–Stupp sequences. At the population level, the immunotherapeutic adjunct yields a consistent but modest survival extension, increasing median overall survival by $0.8$–$0.9$ months with minimal sensitivity to temporal ordering or scheduling perturbations. Crucially, these population-level medians mask profound individual heterogeneity: under combined protocols, $19\%$–$22\%$ of virtual patients extend survival by more than $60$ days, and $\sim 3\%$ gain over a year. A low intrinsic tumor proliferation rate $r_1$ emerges as the primary determinant of response, bolstered by robust T-cell expansion and attenuated microenvironmental suppression. These findings suggest that the clinical promise of multimodal integration depends on identifying responsive tumor phenotypes for personalized treatments rather than searching for a single universally optimal timeline.

A deep classifier is defined not only by the decision it produces, but also by the sequence of transformations through which that decision is formed. Treating this evolution as a dynamical system across layers provides a natural framework for asking how decision geometry emerges through depth and how far back we can trace a boundary's dynamical signature. We model a feed-forward classifier as a finite, nonautonomous discrete dynamical system, with layers playing the role of discrete time steps. We study the Finite-Time Maximum Lyapunov Exponent (FTMLE) of the data samples' dynamical trajectory through depths of the classifier. The FTMLE measures the rate of convergence/divergence of nearby trajectories. We move the observation endpoint backward from probabilities to logits and then to hidden representations. For Gaussian classes, we prove that probability-level FTMLE carries a clear geometric signature of the decision boundary, with its dominant direction aligned with the boundary normal. Moving one step backward to the logits, we prove this relationship is no longer universal but depends critically on how the classifier is trained, particularly on the choice of loss function. Moving further backward to the hidden representation, the connection becomes more conditional: boundary-related FTMLE can persist, but only under identifiable structural conditions. We propose geometry-aware fine-tuning for restructuring the classifier's hidden FTMLE, and propose conditions for guaranteed concentration of high hidden FTMLE near the decision boundary. Through our numerical results, we show the generality and validity of our theoretical results. Understanding the evolution of data samples as traveling through the layers of classifier provides a principled foundation for identifying where boundary-relevant sensitivity emerges and for developing layer-aware regularization strategies.

In 1943, R. Salem asked whether the Fourier-Stieltjes transform of the Minkowski question-mark function vanishes at infinity. This problem was answered affirmatively by Jordan and Sahlsten in 2016 as a consequence of their general results on Gibbs measures for the Gauss map. In this note we give a self-contained proof for ?(x). Several aspects of our argument are substantially different, making a very short proof possible in the ?(x) case. Moreover, we obtain an explicit polynomial decay estimate, with exponent at least 0.0472. We then prove the existence of the correlation dimension of the Minkowski measure, thereby improving the previous upper bound for the optimal Fourier decay exponent from 0.4373 to 0.4223.

In this paper, we introduce a new set of $\mathbb{L}_2$-orthogonal polynomials on the hemisphere $\mathbb{S}^d_+:=\{\mathbf{x}\in\mathbb{R}^{d+1}:\|\mathbf{x}\|=1, \mathbf{x}\cdot\mathbf{e}_{d+1}\geq0\}$ and establish $\mathbb{L}_1$ Marcinkiewicz-Zygmund inequalities on the hemisphere for even and odd spherical polynomials. Hemispherical $t$-designs provide equal weight cubature rules on the hemisphere which are exact for polynomials up to degree $t$. We propose variational characterizations of hemispherical $t$-designs by the $\mathbb{L}_2$-orthogonal polynomials. Moreover, based on the Marcinkiewicz-Zygmund inequalities and the $\mathbb{L}_2$-orthogonal polynomials, we prove that for each $n\geq C_d t^d$, there exists a hemispherical $t$-design on the hemisphere with $n$ points, where $C_d$ is a constant depending only on $d$.

Nonlinear partial differential equations (PDEs) underpin computational fluid dynamics, yet resolving nonlinear transport on fine grids remains computationally demanding. Quantum linear-evolution algorithms offer a possible route to large-scale simulation but cannot directly propagate nonlinear coupling. Here we develop a Frozen–Oseen quantum homotopy method (FOQHAM) framework that links transport-aware auxiliary-operator selection to the size of the resulting linear representation. We freeze the full Fréchet derivative at a prescribed flow profile, retaining transport and profile-gradient coupling and close each prescribed finite-order homotopy hierarchy as a fixed affine linear system by product lifting. We formulate its propagation through Linear Combinations of Hamiltonian Simulations (LCHS), without outer homotopy iterations or profile updates. For spatially semidiscrete equations, we establish local approximation bounds and exact closure of the finite-order hierarchy. Classical tests on Burgers, Korteweg–de Vries, Zakharov–Kuznetsov, and one- and two-dimensional isothermal compressible Navier–Stokes equations provide numerical evidence of accurate non-iterative approximations in the tested regimes. The framework thus connects nonlinear PDE approximation to quantum linear evolution and offers a conditional path toward quantum acceleration.

In this paper, we establish Edgeworth expansions of extreme eigenvalue distributions for two types random unitary ensembles at the spectral edges and reveal certain universal structures for the correction terms. More precisely, we show the correlation kernels for the Gaussian-type unitary ensembles and Laguerre-type unitary ensembles admit full expansions at the soft edge and the hard edge, respectively. The coefficients of the correction terms are given by finite sums of Airy function or the Bessel functions of the first kind and their derivatives with polynomial coefficients. By lifting the kernel expansion to the associated Fredholm determinant, we further obtain full expansions for the largest eigenvalue distribu- tion of the Gaussian-type unitary ensembles and the smallest eigenvalue distribution of the Laguerre-type unitary ensembles. In addition, the first correction term therein involves the derivatives of the leading term.

Electromagnetic (EM)-reconfigurable antennas provide multiple candidate radiation patterns per element, thereby introducing an additional EM-domain degree of freedom. Integrating radiation-pattern reconfigurability, realized as EM-domain precoding, with conventional hybrid analog-digital precoding yields tri-hybrid multiple-input multiple-output (MIMO) precoding, which can substantially improve the spectral efficiency of wideband multi-user MIMO orthogonal frequency-division multiplexing (OFDM) systems. However, the joint design of EM, analog, and digital precoding remains challenging. To address this challenge, we propose a tri-hybrid precoding network (Tri-PNet) based on Conformer, an emerging neural architecture that combines the local modeling strength of convolutional neural networks with the global dependency modeling of Transformers. Furthermore, two representative radiation-pattern modes, i.e., the non-regular mode and the 3rd Generation Partnership Project (3GPP) Technical Report (TR) 38.901 mode, are investigated. Tri-PNet is trained in an unsupervised manner to jointly learn EM, analog, and digital precoding by maximizing the average sum spectral efficiency. Its radiation-pattern selection network (RPSNet) employs a Conformer encoder to capture both local and global frequency-domain correlations, whereas its hybrid analog-digital precoding network (HPNet) combines cross-attention and dual-path processing with singular-value-decomposition (SVD) and zero-forcing (ZF) priors. Simulation results under both radiation-pattern modes demonstrate that Tri-PNet outperforms random EM precoding and conventional hybrid MIMO without EM precoding, approaches the greedy EM precoding search scheme with substantially lower online complexity, and remains robust to imperfect channel state information (CSI).

In this paper, we investigate the componentwise linear property of support-two monomial ideals. Our first main result shows that if $I$ is a support-two monomial ideal, then its underlying simple graph $G_I$ is co-chordal; equivalently, by Fröberg's theorem, $\sqrt{I}$ admits a linear resolution. This phenomenon is quite rare for general monomial ideals. In fact, there exist monomial ideals with linear resolutions whose radicals fail to have linear resolutions, even when the radical is the edge ideal of a graph. For any monomial ideal $I$, one always has $\mu(I)\geq \mu(\sqrt{I})$. In the literature, support-two monomial ideals satisfying $\mu(I)=\mu(\sqrt{I})$ are of special interest, as they include edge ideals of simple graphs, weighted oriented graphs, edge-weighted graphs, and vertex-weighted graphs. We refer to such ideals as minimal support-two monomial ideals. We explicitly characterize all minimal support-two monomial ideals, as well as their powers, that admit linear resolutions. Next, we classify the linearity of non-minimal ones. Consequently, we obtain a complete classification of linear support-two monomial ideals, which shows that the property of being linear does not depend on the characteristics of the base field for support-two monomial ideals.

We study equivalences between quotient categories of module categories associated with Gabriel topologies. We establish necessary and sufficient conditions for a Morita context to induce an equivalence between quotient categories and characterize such equivalences in terms of Morita contexts after passing to rings of quotients. Motivated by Artin–Zhang's noncommutative Serre theorem, we introduce a notion of ampleness relative to a Gabriel topology and use it to characterize quotient categories of finitely generated modules under suitable noetherian hypotheses. As an application, we establish a sufficient condition under which the noncommutative Auslander theorem holds for actions of finite-dimensional Hopf algebras on AS-regular algebras without assuming that the Hopf algebras are semisimple.

For a map $F:\Omega\subset\mathbb{R}^n\to\mathbb{R}^m$, let $\Theta(F)$ be the $n$-form on $\Omega\times\mathbb{R}^m$ introduced by Tsai and Wang [arXiv:2604.04336]. If $\lambda_1,\ldots,\lambda_{r}$ are the nonzero singular values of $d F$ at any fixed point, we prove that $$ \Theta(F) \text{ has comass one if and only if }~ \mathcal{S}(\lambda):=\sum_{i=1}^{r}\frac{\lambda_i^2}{1+\lambda_i^2}\leq1. $$ Note that if $\text{rank} d F\le1$, $\mathcal{S}$ is always less than $1$. We also prove a dichotomy that if $\mathcal{S}\leq 1$, then either $\mathcal{S} < 1$ or $\mathcal{S}\equiv1$. When $\mathcal{S}<1$, the graph tangent plane is the only calibrated plane, generalizing the corresponding result for hypersurfaces. When $\mathcal{S}\equiv1$, we prove that $F$ is affine or of rank $2$.

We prove a sharp Gaussian approximation for the invariant law of constant-stepsize SGD with bounded additive noise generated by an exogenous uniformly ergodic Markov chain. For a smooth, strongly convex objective with a Lipschitz Hessian and nondegenerate long-run noise covariance, the centered iterate normalized by the square root of the stepsize is $O(\sqrt{\alpha})$-close in 1-Wasserstein distance to its limiting Gaussian. The proof combines blockwise Gaussian comparison with long-run contraction. A four-state example gives a matching lower bound although the one-time noise marginal is symmetric and every nonzero-lag autocovariance vanishes. In this example, an adjacent third-order mixed moment produces the leading correction.

Let $f$ be a nonzero holomorphic germ at $0 \in \mathbb C^n$ with $f(0)=0$, and let $\chi$ be a $C^2$ non-decreasing convex function on the left half-line. We establish sharp necessary and sufficient conditions for the local Sobolev regularity of the plurisubharmonic function $v=\chi(\log|f|).$ The criteria for the $L^p$-integrability of the classical Laplacian and for $W^{1,p}$-regularity are given by a weighted integral involving $\chi''$ and $\chi'$, respectively, and depend on $f$ only through the smallest multiplicity of $\operatorname{Div}(f)$. We also obtain the $W^{2,p}_{\mathrm{loc}}$ criterion for $1<p<\infty$. At the endpoint $p=1$, we prove that \[ v\in W^{2,1}_{\mathrm{loc}} \quad\Longleftrightarrow\quad \chi'\in L^1((-\infty,A)), \] equivalently, $v$ is locally bounded. As applications, we obtain counterexamples to Calderón-Zygmund theory and characterize a class of functions in the local Monge–Ampère domain. % and exhibit a natural family in $W^{2,1}_{\mathrm{loc}}$ whose limit fails to belong to $W^{2,1}_{\mathrm{loc}}$.

In this article, we compute the limit Hilbert–Kunz multiplicities and limit F-signatures of all the simple singularities and the Hilbert–Kunz multiplicities of the Fermat hypersurfaces, generalizing the well-known sec z + tan z formula by Gessel and Monsky for the A1 singularities, using the theory of h-functions. We also obtain a functional equation relating the phi-function of a hypersurface to its reflection. Using limit F-invariants, we generalize both a characterization of regular local rings through F-invariants and the known cases of the Watanabe–Yoshida conjecture to characteristic zero.

To a quiver of finite type one can associate two abelian categories: the category of finite dimensional modules over the corresponding affine quiver Hecke algebra and the Coulomb category of Koszul-perverse coherent sheaves. In this paper we define an exact functor from the former to the latter. This functor sends a simple to a simple or zero and is (almost) essentially surjective on simples. This implies that simple Koszul-perverse sheaves categorify the dual canonical basis.

Faulty syndrome measurements can corrupt an otherwise correct quantum-error correction step. Bivariate bicycle (BB) codes contain dependent stabilizer checks, so every valid syndrome obeys additional parity constraints, or metachecks. We study how far this built-in redundancy can identify measurement faults and when the remaining ambiguity is unavoidable, while separately checking the code's logical structure. A logical decomposition is used as a preliminary safety check: it identifies a $k/2$-dimensional annihilator subspace and a $k/2$-dimensional colon quotient, and the minimum-weight logical need not be visible from the annihilator side alone. On the measurement side, translation symmetry partitions syndrome locations into classes that carry identical metacheck information. This gives an exact characterization of the leading single-fault ambiguity, a bound on how many fault locations can be distinguished, and a family-level repair limit when the encoded dimension stays bounded while the block length grows. Under a static-data assumption, the same calculation gives the minimum number of checks that must be remeasured to remove every single-fault ambiguity. Exact finite-code calculations illustrate both regimes: all single measurement faults are distinguishable in a 72-qubit BB code, whereas the 144-qubit Gross code merges its 72 syndrome locations into 36 indistinguishable pairs. In a 108-qubit example, one logical component first appears at weight 12 while the other contains a weight-10 logical. Sustained phenomenological experiments show that joint data–measurement decoding is more robust than a separated repair stage on the more ambiguous codes. The resulting tests apply to general two-block BB codes, including non-coprime periods and repeated-root cases.

This paper examines how row-wise renormalization affects Muon, focusing on the gap between NorMuon's worst-case guarantees and its practical performance (Li et al.). Despite its growing adoption and promising performance in large language model (LLM) pretraining, NorMuon's worst-case guarantees remain poorly understood. One fundamental question is: Does row normalization yield provable convergence gains, potentially through its interaction with approximate polar computation and exponential moving-average momentum? Our results show that row normalization introduces a dimension-dependent factor in the worst-case iteration complexity under the operator-norm geometry, which persists even with exact polar computation and any fixed momentum parameters. Indeed, we establish an algorithm-dependent lower bound and a matching upper bound in deterministic settings, and extend our upper bound analysis to stochastic settings. Both upper-bound analyses allow approximate polar computation. Experiments show that NorMuon is slower than Muon on synthetic problems inspired by our worst-case construction, yet outperforms Muon in LLM pretraining. These findings sharpen the puzzle of why row normalization helps in practice and complement the recent findings of Dewulf et al.