Papers

We discuss the algebraic structure of observables in Schrödinger CFTs. These operators have zero mass (or particle number) and generically transform in staggered ”pyramid representations” built from ”alien operators,” as we explain with the doubled state-operator correspondence. We comment on implications for the space of non-relativistic CFTs, thermal physics, and generalize the exceptional symmetry conservation laws of Bekaert, Meunier, and Moroz, and Golkar and Son. We show that alien operators are analogous to double-twist operators in Lorentzian CFT, with systematic cross-channel corrections from massless particles when they exist.

We study the $p$-colored top-$m$-to-random shuffle on $C_p\wr S_n$ when the block size $m=m_n$ grows with $n$. Let $E_{k_n}^{(m_n)}$ be the number of labels never touched after $k_n$ independent uniform $m_n$-subset draws, and set $b_n=n-m_n$, $q_n=b_n/n$, and $\lambda_n=nq_n^{k_n}$. We prove that if $\lambda_n\to\lambda\in(0,\infty)$ and $b_n\to\infty$, then $E_{k_n}^{(m_n)}\Rightarrow\mathrm{Poisson}(\lambda)$. Combining this with the exact nested-set reduction for colored top-$m$-to-random shuffles, we obtain growing-block total variation, separation, and integrated likelihood-ratio profiles. In particular, if $Q_{n,p}^{(m_n)}$ is the one-step law and $U_{n,p}$ is uniform on $C_p\wr S_n$, then the separation distance from $(Q_{n,p}^{(m_n)})^{*k_n}$ to $U_{n,p}$ tends to $1-e^{-\lambda}(1+\lambda)$ for $p=1$ and to $1-e^{-\lambda}$ for $p\ge2$. The criterion applies to small blocks, proportional blocks, and near-full blocks.

In this paper, we consider operators that are compositions of an isometric isomorphism and a left multiplier on a Banach algebra, and we provide necessary and sufficient conditions for these operators to have a dense set of periodic elements. As an application of this result, we characterize generalized weighted shifts with a dense set of periodic elements on the standard Hilbert module over C*-algebra of compact operators on a separable Hilbert space. As another application, we characterize generalized weighted shifts with a dense set of periodic elements on the standard Hilbert module over commutative non-unital C*-algebra. Finally, in the last section of the paper, we provide extensions of these results from the case of a sequence generated by powers of a single operator to families of operators indexed over a countable set.

In this paper, we introduce the class of Stable Positive Integral Deferred Correction (SPIDeC) methods for the numerical integration of positive dynamical systems. The proposed framework embeds a deferred correction mechanism within an exponential-type Volterra reformulation of the underlying differential problem. The resulting multiplicative structure guarantees the unconditional preservation of both positivity and equilibria, independently of the integration stepsize. Arbitrarily high-order accuracy is systematically achieved through successive explicit-in-sweep corrections applied to a low-order base approximation. From a stability viewpoint, the SPIDeC integrators are L-stable and exactly reproduce the continuous semigroup generated by diagonal linear operators. Furthermore, when Gauss–Radau quadrature nodes are employed, the associated discrete flow asymptotically approaches a logarithmically contractive map as the number of sweeps increases, ensuring stability. Numerical experiments are provided to validate the theoretical analysis and illustrate the practical performance of the proposed methods.

In this paper, we investigate the multiplicity of normalized solutions to a nonlinear Dirac equation with localized nonlinearities on noncompact metric graphs under the \(L^2\)-constraint, as well as the asymptotic behavior of these solutions in the nonrelativistic limit. First, we establish the existence of multiple normalized bound states. Moreover, we explore the nonrelativistic limit and show that, as the speed of light tends to infinity, the solutions converge to those of a nonlinear Schrödinger equation. Our results including the mass-subcritical, mass-critical and, in particular, mass-supercritical regimes.

Theorems on uniqueness and continuous dependence on the initial condition of solutions to stochastic Navier-Stokes equations driven by Wiener processes and Poisson martingale measures are presented. These theorems generalise some results from \cite{GK2026}.

For a fixed integer $k \ge 3$, we study the multiplicative functions $f\colon\mathbb{N}\to\mathbb{C}$ satisfying \[ f\Bigl(\sum_{i=1}^{2k} x_i^2\Bigr) = \sum_{j=1}^{k} f\bigl(x_{2j-1}^2 + x_{2j}^2\bigr) \] for all positive integers $x_1,\dots,x_{2k}$. This extends a theorem of Park on sums of two nonzero squares, which established the $k=2$ case. For $k=3$ and $k=4$, we prove that every such $f$ with $f(2)\neq 0$ is the identity function on $\mathbb{N}$. For $k \ge 5$, we show that such a function $f$ must be either the identity function on $\mathbb{N}$, or $f(n) = 0$ for all $n > 2k + 21$.

Under appropiate hypotheses on the external force acting on an incompressible flow diffusing through a porous medium, we show that there is a unique stationary solution to the diffusive porous media equation. Moreover, we show that this solution is asymptotically stable and estimate the decay rate of any perturbation towards this steady state.

Rado's Conjecture (RC) is a compactness principle for a certain class of partial orders, namely trees $T$ of height $\omega_1$ without cofinal branches, postulating that a partial order $P$ from this class can be decomposed into at most countably many antichains if and only if all its suborders of size $\omega_1$ can be decomposed into at most countably many antichains. Rado's Conjecture is thus an uncountable version of Mirsky's theorem asserting that for every natural number $n$, every infinite partial order $P$ can be decomposed into at most $n$ many antichains if and only if all its finite suborders can be decomposed into at most $n$ many antichains. Todorcevic showed that RC is consistent modulo a strongly compact cardinal. RC implies $2^\omega \le \omega_2$, and has powerful consequences such as the Singular Cardinal Hypothesis, the failure of $\square(\kappa)$ for every regular $\kappa \ge \omega_2$ (and hence in particular the Projective Determinacy), and the Strong Chang Conjecture. It is also known that it is incompatible with Martin Axiom. We show that RC is consistent with $2^\omega = \omega_2$ and the cardinal invariants in Cichon diagram corresponding to forcing with the random algebra, i.e., $\mathfrak{d} = \omega_1$, $\mathrm{cov}(\mathcal{N}) = \omega_2$, $\mathrm{non}(\mathcal{N}) = \omega_1$. This provides a new pattern of cardinal invariants known to be consistent with RC. To prove the theorem, we first observe that random algebras do not specialize non-special trees of height $\omega_1$ without cofinal branches. Then we use the random algebra $\mathcal{B}_\kappa$ for a strongly compact $\kappa$ to define a new version of Mitchell forcing which yields the required result.

This work is concerned with the nonlinear information in the linear sampling method for the inverse medium scattering problem. In addition to the well-known capability in shape characterization, we demonstrate that the imaging indicator represents nonlinear information about the unknown contrast. We further demonstrate additional representations for this nonlinear information using the generalized linear sampling method. We shed light on this new result using the Born model and analytical examples.

We construct a form of the $D_4^-$-singularity of fronts in $\R^3$ which uses coordinate transformation on the source and isometry on the target. As an application, we compute differential geometric invariants near the $D_4^-$-singularity, and give a Gauss-Bonnet type theorem for one-parameter generic fronts.

This paper characterizes the stabilized second James-Hopf invariant by means of three axioms. Specifically, we show that it is the unique natural transformation satisfying the Cartan formula, vanishing on suspensions. The proof combines the natural stable splitting of the James construction with Goodwillie calculus.

Let $G/\mathbb{Q}_p$ be a connected, split, reductive group over $\mathbb{Q}_p$. In this paper I show that if $K$ and $L$ are anabelomorphic $p$-adic fields i.e. $K$ and $L$ have topologically isomorphic absolute Galois groups, then the stacks of Langlands parameters (for the fields $K$ and $L$) considered in [Fargues and Scholze, 2024], are also isomorphic (Theorem 2.2.1). This leads to Conjecture 3.3.1 which provides a precise relationship between the main conjecture of [Fargues and Scholze, 2024] and anabelomorphy of $p$-adic fields considered in [Joshi, 2020a]. I establish my conjecture for a split torus in Theorem 4.1.

The specification number $\sigma_n(f)$ of a Boolean threshold function $f$ on $n$ variables is the least number of points whose $f$-values determine $f$ uniquely among all threshold functions. Its essential points form the unique minimum such set. We develop Zuev's geometric interpretation: the threshold functions are the chambers of a central hyperplane arrangement in the $(n+1)$-dimensional space of weights and thresholds, and the essential points of a function correspond exactly to the facets of its chamber, so the specification number is the chamber's facet number. The lower bound $\sigma_n(f)\ge n+1$ becomes the fact that a pointed full-dimensional cone has at least $n+1$ facets, with equality for simplicial chambers. The average specification number $\overline\sigma_n$ becomes an average facet count. We evaluate this average exactly via the resonance arrangement and bound it through a theorem of Fukuda, Tamura, and Tokuyama, obtaining $\overline\sigma_n\le 2n$; hence $\overline\sigma_n=\Theta(n)$. This settles a question of Gutekunst, Mészáros, and Petersen. The method also extends to polynomial threshold functions. The same geometry links threshold functions with a threshold zonotope, whose vertices are modified Chow vectors. Its one-skeleton is the one-inclusion graph, and a vertex's degree is the specification number of that function. Finally, we treat the operations of Lozin et al. on functions of minimum specification number. Adding a variable and extending on a variable both take the product of a chamber closure with a half-line, preserving simpliciality. For the symmetric-variables extension we give an exact thresholdness criterion and show that minimum specification number is preserved whenever the extension is a threshold function. We also resolve a question they pose concerning a fourth operation.

This paper develops an a posteriori error analysis framework for decoupled neural approximations of fully coupled forward–backward stochastic differential equations (FBSDEs). It provides an a posteriori error-analysis for the idealized discrete adapted trajectory. The main feature of the proposed formulation is the use of an auxiliary control process in the forward coefficients, which may differ from the backward component approximated by the neural network. This decoupling is useful in practical deep learning implementations, but it creates a control mismatch that must be included in the error analysis. We first establish a continuous-time stability estimate for fully coupled FBSDEs under perturbations of the drift, diffusion, generator, terminal condition, and auxiliary control input. We then transfer this estimate to the discrete-time setting and derive computable a posteriori error bounds depending only on the terminal defect, the pathwise residual, and the control mismatch. When the auxiliary control is identified with the backward approximation, the mismatch term vanishes and the bound reduces to the standard two-term form. Numerical experiments on a linear–quadratic FBSDE with an explicit reference solution and a multidimensional Burgers-type FBSDE without a reference solution illustrate the diagnostic role of the proposed indicators and the contribution of the mismatch penalty to the consistency and reproducibility of the numerical approximations.

Strictly proper scoring rules identify the true conditional class distribution at population level, but their curvature can alter optimization and finite-sample behavior. We study three multiclass objectives: a class-aware quadratic Bregman score (CAPM), a strongly convex generator with constrained log-cosh ridges (HPG), and an HPG objective with an annealed probability-margin penalty (APMS). CAPM is treated as a structured instance of established quadratic scoring-rule theory. We derive conditional-regret, curvature, range, and logit-gradient bounds for CAPM and HPG, and prove exact penalty-range and conditional-target displacement bounds for APMS. Controlled five-seed experiments use Digits, Wisconsin breast cancer, and synthetic confusion and long-tail problems under clean labels, symmetric and pair-flip corruption, class imbalance, calibration evaluation, input corruption, and first-order adversarial perturbations. The candidates are close to cross-entropy on clean data and show descriptive gains in some noisy-label cells, but the five-seed comparisons are interpreted descriptively rather than as significance evidence. The selected noisy-label baselines perform better on Digits with 40% symmetric label noise, and explicit prior-adjustment methods perform better in the 30:1 synthetic long-tail experiment. Ablations do not show a consistent benefit from the candidate-specific graph, ridge, or margin components. The mathematical analysis establishes the stated properties, and the experiments delimit the empirical evidence; together they do not support a claim of general superiority.

In the present note, certain scenarios of potential Type II blowups of solutions to the Navier-Stokes equations are considered on the local level. They generalise particular scenarios described in the previous papers of the author. The main features of the approach, adopted in the note, are a zoom based on the Euler scaling and Liouville type theorems for the Euler equations in classes motived by a particular scenario of the Type II blowup.

We study weak convergence rates of numerical approximations for stochastic Volterra integral equations (SVIEs), a class of non-Markovian models that arises naturally in stochastic volatility modeling and other fields. The intrinsic non-Markovian nature prevents the direct application of classical weak error techniques developed for finite-dimensional Markov processes. To overcome this difficulty, we combine a Markovian lifting technique with a domino argument, Taylor expansions, and Fréchet differential calculus for path-dependent functionals, and establish a fundamental weak convergence theorem for nonsingular SVIEs, providing a unified approach to the weak error analysis for a broad class of numerical approximations. As applications, we derive the first-order weak convergence rate for the stochastic theta method and the Wong–Zakai approximation. Our results relax existing assumptions for Euler-type schemes by removing the boundedness requirement on the diffusion coefficient. Furthermore, to the best of our knowledge, this work provides the first weak convergence result for Wong–Zakai approximations of SVIEs. Numerical experiments for a stochastic volatility model corroborate the theoretical convergence rate.

This paper investigates a novel quasi-singularity formation phenomenon in the isentropic compressible Euler equations in three dimensions. For any prescribed finite set of points $\{z_j\}_{j=1}^N \subset \mathbb{R}^3$ and an arbitrarily large parameter $M \gg 1$, we explicitly construct real-analytic initial data generated by Herglotz-type wave fields. We prove that the corresponding classical solution $$ (p,v)\in C^1([0,T]\times\mathbb{R}^3;\mathbb{R}\times\mathbb{R}^3), $$ with $$ T = C_0 \, M^{-\frac{3\gamma+1}{\gamma-1}}, $$ where $\gamma > 1$ is the adiabatic index and $C_0 > 0$ is a constant independent of $M$. Throughout the time interval $[0, T]$, the velocity field, its spatial gradient, and the pressure gradient exhibit simultaneous localized amplification at each prescribed point $z_j$. More precisely, each component of the velocity field is bounded below by $M$, each component of the pressure gradient is of size at least $M^{\frac{2\gamma}{\gamma-1}}$, and the velocity gradient exhibits a prescribed anisotropic sign pattern with entries of size at least $M$ at every $z_j$. All implicit constants in these estimates are independent of $M$ and depend solely on the fixed physical parameters and the locations $\{z_j\}_{j=1}^N$. This mechanism generates highly localized, shock-like gradient concentrations and implosion-like spatial profiles within the smooth regime. The underlying mechanism relies on the construction of the Herglotz kernel via Helmholtz far-field patterns, together with delicate quantitative energy estimates for the quasilinear hyperbolic system.

We establish handle attachment formulas for the Khovanov skein lasagna module with 1-dimensional inputs over $\mathbb{Q}$, defined recently by Ren, Wedrich, Willis, Zhang, and the second author. For a $4$-manifold built out of $1$- and $2$-handles, the invariant can be computed in terms of a cabled colimit of Rozansky-Willis homologies, modulo a new relation which we call the lasso relation. We then present some explicit calculations for disk bundles over $S^{2}$, as well as a partial vanishing result for $4$-manifolds of the form $\Sigma_{g}\times D^{2}$, $g\geq 1$.

The KKL Theorem, a seminal result in boolean function analysis, characterizes the structure of low-influence (non-expanding) functions on the hypercube. While recent years have seen breakthrough results across a variety of areas relying on analogs of the KKL Theorem beyond the cube (e.g., on product spaces, Grassmann graphs), further progress has been inhibited by our poor understanding of the phenomenon across more general domains. Motivated in this context, Bafna, Hopkins, Kaufman, and Lovett (STOC 2022) and Gur, Lifshitz, and Liu (STOC 2022) proved a generalized KKL-type Theorem for spectral high dimensional expanders (HDX). Their results, however, remain highly restricted due to strong quantitative expansion requirements on the underlying complex. In this work, we introduce a simple local-to-global method for analyzing low influence functions on simplicial complexes. Using this method we prove a local-to-global KKL-type Theorem: any simplicial complex whose links satisfy a KKL-Theorem also satisfies such a result globally. Building on Gotlib and Kaufman (RANDOM 2023), we also prove a weaker dimension-dependent KKL-type Theorem for simplicial complexes with any non-trivial (two-sided) expansion. As concrete applications of our framework, we give the first characterization of non-expanding functions on `combinatorial' HDX such as dense clique complexes and a corresponding Kruskal-Katona Theorem, as well as a small-set expansion theorem for the Ramanujan Complexes of Lubotzky, Samuels, and Vishne (EJC '05).

Let $\mathcal{P}$ be a set of $n$ points in $\mathbb{R}^2$, with a convex hull of size $O(n/\log n)$. We prove that $\Omega(12.24^n)$ plane graphs can be drawn on $\mathcal{P}$, the first non-trivial bound for this problem. We also show that a random plane graph, uniformly chosen from the set of all plane graphs of $\mathcal{P}$, has at most $n/12.24$ isolated vertices. This improves upon a previous bound of $n/10.18$. Our analysis is based on studying the expected vertex potentials in a random plane graph. The potential of a vertex is its degree plus the number of vertices visible from it. We show that this quantity can be used to study numbers of plane graphs.

Repeatedly solving parametric PDEs is essential for uncertainty quantification, design optimization and inverse problems, but conventional neural operators require expensive non-convex training. We introduce PCA–RaNN, a randomized latent neural operator that combines PCA-based dimensionality reduction with fixed random features and a closed-form least-squares readout. It recasts latent operator learning as fixed-feature linear regression, reducing training time by one to three orders of magnitude across benchmarks while maintaining competitive accuracy. We introduce an energy-matched scaling rule and a lightweight two-parameter BFGS refinement to correct suboptimal feature scales. Ensemble averaging reduces predictive variance. On Burgers, Darcy, Navier–Stokes and backward heat equation benchmarks, PCA–RaNN provides a favorable speed–accuracy trade-off against operator-learning baselines. The ensemble supports split-conformal prediction intervals, and the linear readout enables rapid online adaptation via recursive least squares without retraining hidden features. This provides an efficient, uncertainty-aware surrogate for many-query scientific workflows.

In this paper, we develop a fractional stochastic neural network with residual dynamics driven by fractional Brownian motion. By introducing a discrete stochastic maximum principle for the network, we construct the corresponding adjoint recursion. For deterministic network parameters, we prove mean square convergence of projected samplewise stochastic gradient descent. Numerical experiments include a closed form convergence test, noisy regression with uncertainty quantification, long memory time series generation and image classification under structured perturbations. The results identify settings in which fractional drivers improve long memory recovery or robustness relative to Brownian and deterministic baselines.

We study operator learning for random obstacle-to-solution maps arising from elliptic variational inequalities with finite-band self-affine random obstacle fields. Instead of introducing an explicit truncated stochastic parametrization of the random input, we learn the map directly from sampled obstacle realizations on a fixed grid. This problem is challenging because the solution is governed not only by the obstacle field itself, but also by the induced contact set and free-boundary geometry. We introduce a post-training least-squares readout refit for the Fourier neural operator (FNO). After the FNO is trained end to end, its nonlinear backbone is frozen and the final affine readout is recomputed by solving the induced linear least-squares problem over all training samples and grid points. The refit yields the empirical squared-error optimal readout for the learned frozen features while leaving the nonlinear representation unchanged. We compare vanilla DeepONet, POD-DeepONet, a two-stage DeepONet baseline, FNO, and FNO with least-squares readout refit (FNO-LS) on two obstacle ensembles with different amplitude levels. Numerical results show that FNO-LS achieves the strongest overall performance among the tested models, particularly for higher-amplitude obstacles with more complex contact geometry. The method improves average field accuracy, contact-set recovery, and obstacle-violation metrics at low additional cost, especially when the FNO backbone is informative but not fully converged. These results suggest that least-squares readout refit is a simple and effective post-training enhancement for learning random obstacle-to-solution maps.

Heterogeneous and in particular multi-altitude low Earth orbit (LEO) satellite constellations exhibit complex spatial and temporal structures, which require new modeling tools for their performance analysis. In this paper, we develop an orbit-aware stochastic geometry framework modeling today's LEO satellites on various orbits and various altitudes. In particular, we characterize such a system as the superposition of multiple Walker point processes and formulate it as a dynamical system determined by an initial condition and the rotation speeds of satellites and Earth. We show that when the speeds are rationally commensurable, the proposed satellite system is periodic. Then, we show that the system is ergodic when the speeds are rationally independent, establishing a theoretical link between time averages of the system and the expectation of it under the invariant measure. We derive the nearest-satellite distance distribution of a typical receiver at a given latitude and analyze the signal to interference-plus-noise ratio (SINR) coverage probability of the typical receiver. We then derive the ergodic throughput of the downlink communication to the typical receiver. Overall, the proposed framework offers a rigorous and tractable tool for analyzing downlink performance in Walker-type heterogeneous LEO satellite networks.

A graph homomorphism is an integer-valued function on the vertex set of a graph that assigns values differing by exactly one to adjacent vertices. We consider uniformly random homomorphisms on general finite trees, conditioned to take the value zero at all leaves, and study the distribution of the value at the root. Our main result is a stochastic comparison, both from above and below, between the absolute values of the homomorphism value at the root and certain discrete Gaussian-like random variables. In particular, we obtain a subgaussian tail bound valid for all deviations, a matching lower bound that holds up to a certain threshold, and upper and lower variance bounds that differ by a constant factor. These bounds depend solely on the effective resistance between the root and the leaves in the associated electrical network. As a consequence, in the setting of infinite locally finite trees, we obtain that the homomorphism model is localized on transient trees and delocalized on recurrent trees. Analogous results are obtained for random integer-valued Lipschitz functions. Our results extend previous results of Benjamini–Häggström–Mossel on homomorphisms on regular trees, of Peled–Samotij–Yehudayoff on Lipschitz functions on regular trees, and of Lammers–Toninelli on homomorphisms on trees of minimum degree at least three.

The main objective of this paper is to give a positive answer to the natural question proposed by Ashot Minasyan: Is the fundamental group of finite graph of conjugacy separable groups with finite edge groups conjugacy separable?

This paper presents reproducible experimental evidence on unit-distance graph density. Our approach is based on a novel algorithmic exploration of the rational plane for the generation of unit-distance graphs. An efficient algorithm for this utility must perform a local-breadth search on a bounded and finite set of elements and generate a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation. To this end, we show that our approach accomplishes this purpose by overcoming the limitations of grid-based structures, used in the literature for generating unit-distance graphs. When applied to large unit-distance graphs, our method generates large graphs with a scaling exponent of 1.17, that surpasses recent theoretical predictions on graph density growth.

This paper addresses the problem of achieving both coarse and precise privacy in state estimation. Coarse privacy forces the eavesdropper's total mean-square error (MSE) to infinity, but errors along certain confidential directions may remain bounded. This motivates precise privacy, which additionally drives the MSE along prescribed directions to infinity. For coarse privacy, an analytical transformation is established, preserving the user's optimality and driving the eavesdropper's total MSE to infinity at a polynomial-exponential rate. A stochastic intermittent encryption scheme is further developed, and an explicit lower bound on the encryption probability is derived to guarantee divergence. For precise privacy, by analyzing the behavior of the Riccati equation on the unobservable subspace, we prove that the eavesdropper's directional MSE becomes unbounded if and only if the direction's unstable component lies outside the observable subspace. Finally, a systematic method is proposed to exclude target vectors from the observable subspace, forcing the directional MSE to infinity.