Papers

A simple complex of groups is a functor from a poset to the category of groups satisfying certain properties. They were introduced by Bridson and Haefliger as an abstract way to encode isotropy subgroups of actions on posets. A simplex complex of groups which embeds in a group $K$ has an associated $K$-poset. We will explain how to obtain information about the classifying space of $K$ through the $K$-equivariant homotopy type of the construction of Bridson-Haefliger, and we will also obtain finiteness properties of cohomology.

Recent work has constructed higher-dimensional analogs of non-invertible symmetries similar to 1+1d Kramers-Wannier duality. Although their continuum descriptions often treat purely gravitational topological terms as inessential counterterms, these terms can have an essential lattice manifestation: they distinguish states prepared by finite-depth quantum circuits (FDQCs) from those entangled by nontrivial quantum cellular automata (QCAs). Motivated by this mismatch, we show that QCAs associated with gravitational topological responses arise in several related settings: (1) lattice realizations of projective $\mathrm{SL}(2,\mathbb{Z}_N)$ relations generated by topological operations on symmetries; (2) squares of dualities that generalize the relation between fermionization and Kramers-Wannier duality; (3) lattice implementations of QCAs through higher-form gauging; and (4) invertible phases protected by generalized time-reversal symmetries. We derive new projective $\mathrm{SL}(2,\mathbb{Z}_N)$ relations whose projective phases are gravitational topological responses constructed from Stiefel-Whitney classes. We furthermore give a general protocol for preparing the associated QCA-entangled states using finite-depth unitary circuits, measurements, and error correction. These results unify the study of gravitational topological responses in field theories, higher dimensional dualities, and quantum cellular automata.

In 1961 Higman proved that every finitely generated recursively presented group embeds into a finitely presented group. In 2018 Leary proved that every finitely generated group embeds into a group of type $FP_2$. One naturally wonders whether analogous results hold for the higher finiteness properties $F_n$ and $FP_n$ ($3 \leq n \leq \infty$), i.e., whether every finitely generated recursively presented group embeds into a group of type $F_n$ and whether every finitely generated group embeds into a group of type $FP_n$. We prove that a positive answer to the $FP_n$ question that moreover preserves recursive presentability would imply a positive answer to the $F_n$ question. We also investigate the output groups from Higman's and Leary's proofs, which in both cases arise from the so-called “Higman rope trick”, and find that they are never of type $FP_3(\mathbb{Q})$ (hence also never $FP_3$ nor $F_3$). Thus, any approach to the questions of higher finiteness properties must go via a different route than the Higman rope trick.

Due to the complexity of neural network loss landscapes, optimization theory is forced to rely on idealized models, and there is generally a tradeoff between how theoretically tractable the model is, and how accurately it describes the true optimization dynamics. In this work, we stress test the simplest possible model of optimization – the quadratic model – and show that it can be surprisingly predictive in an LLM setting with 150M parameters and 3B training tokens. Specifically, we show that Taylor expanding the model and the loss function at intermediate checkpoints through training can accurately predict the optimization dynamics over windows that can last up to 10% of training. Having established this agreement, we then turn to analyzing the structure of these local quadratic optimization problems through two lenses: the Hessian spectrum and local stability. Using Lanczos quadrature with extremely deep probes, we are able to estimate the Hessian spectrum deep into the tail, and we find a surprising amount of structure in both the eigenvalues and eigenvectors, which depends on the batch size, preconditioner, and training time. We also empirically test local linear stability at intermediate checkpoints and compare it to theoretical predictions to demonstrate that optimization in LLMs typically occurs at a stochastic edge of stability, whose nature is also determined by batch size. Our results indicate the quadratic model may be a theoretically tractable proxy for pretraining optimization dynamics.

This paper introduces and investigates a new Lagrange interpolation process based on the zeros of anti-Gauss Jacobi polynomials. Fundamental properties of anti-Gauss nodes, including their asymptotic distribution, are established, together with estimates for the associated polynomials and their derivatives. These results provide the basis for the construction of an interpolation process whose weighted Lebesgue constants exhibit logarithmic growth, ensuring optimal approximation properties. Compared with previously known interpolation schemes based on Jacobi nodes, the proposed process achieves optimal Lebesgue constants for a shifted range of endpoint weight parameters, allowing the use of smaller endpoint weight exponents. Convergence estimates are established for functions in suitable weighted Sobolev spaces, and numerical experiments support the theoretical findings.

We prove that the wavefunction of a crystal at a high-symmetry momentum, $\Psi_{\boldsymbol{k}_*}(\boldsymbol{r})$, has symmetry-enforced zeros at certain positions in the unit cell, using a new invariant fixed uniquely by the symmorphic symmetry representation of the wavefunction. This allows one to infer the topology of an electronic band by probing zeros of the charge density, and in turn to connect scanning tunnelling microscopy to the group representation theory of bandstructure. We apply the theorem to 1H transition metal dichalcogenides, where it detects the obstructed atomic limit of WSe$_2$, the Haldane model, where it detects the Chern number modulo three, and the Bernevig-Hughes-Zhang model, where it detects the $\mathbb{Z}_2$ index. In addition, the zeros have important consequences for interaction effects: in kagome metals, they fix the sublattice structure of Van Hove wavefunctions, and in twisted bilayer graphene, they explain the qualitative interaction-induced reshaping of the flat bands.

Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a $\mathbb{Z}_p$ one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality $S$, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT $T$. In the continuum, they form a central extension of $PSL(2,\mathbb{Z}_4)$ for $p=2$, and of $SL(2,\mathbb{Z}_p)$ for odd primes $p$, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For $p=2$ the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a $\mathbb{Z}_p$ 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For $p=2$ they are generated by the non-trivial semion QCA, and for odd prime $p$ they are generated by the non-trivial $\mathbb{Z}_p$ Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.

Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and link quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, remains challenging: although a topological action yields a Hamiltonian with commuting terms that realizes the desired ground state, it does not by itself specify an automorphism of the full operator algebra. We develop a unified algebraic construction that extends such Hamiltonians to complete separator-flipper algebras on the full tensor-product Hilbert space, thereby defining the corresponding QCAs microscopically. In 3 spatial dimensions, our framework unifies all previously known QCA constructions associated with the $\mathbb{Z}_8\times\mathbb{Z}_2$ subgroup of the Witt group, including the $U(1)_2$ and $U(1)_4$ QCAs. The same algebraic structure yields new infinite families of generalized $U(1)_2$ and $U(1)_4$ non-Clifford QCAs in spatial dimensions $d=4k-1$. We also reformulate the 4-dimensional $w_2w_3$ QCA and use it to construct QCAs from TQFTs associated with arbitrary products of Stiefel–Whitney classes. For example, we explicitly construct two infinite families: $w_2^nw_3^m$ QCAs in dimension $d=2n+3m-1$ and $w_2w_{4k-1}$ QCAs in dimension $d=4k$. For the 5-dimensional $w_3^2$ and $w_2^3$ QCAs, our construction gives finite-depth quantum circuits, consistent with the cobordism classification. More generally, we show that our construction yields a finite-depth quantum circuit whenever the corresponding Stiefel-Whitney number vanishes on every closed oriented manifold. Overall, these results convert invertible TQFTs into microscopic QCAs, provide higher-dimensional constructions beyond the Clifford setting, and open a systematic route to characterizing their stable structures and boundary anomalies.

I argue that dimensional analysis can appropriately be thought of as a gauge theory. This picture naturally leads to the usual quantity calculus through inherent properties of Lie groups, Lie algebras, and representation theory. The gauge theory interpretation is perhaps strongest for non-constant units. I explain how Stevens's classification of scales of measurement can be understood by choices of different gauge groups. Finally, I reinterpret and rephrase the Buckingham-{\Pi} Theorem in a way that also applies to typical gauge theories. Counting the number of ”{\Pi}”s becomes a well-known problem of invariant theory.

This paper investigates the optimal surplus management problem of an insurance company operating in a financial market with stochastic interest rates and jump-driven liabilities. The insurer dynamically allocates its surplus between a risky stock and a risk-free zero-coupon bond while facing insurance claims modeled by a compound Poisson process with exponentially distributed claim sizes. The short term interest rate follows a Cox-Ingersoll-Ross (CIR) process, which captures mean-reverting dynamics commonly observed in term structure models. The insurer maximizes the expected exponential utility of terminal surplus. Using stochastic control techniques, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation. Although the exponential utility structure suggests an exponential affine representation, the interaction between the interest rate hedge and the surplus state generates quadratic surplus terms in the HJB equation. To obtain a tractable formulation, we adopt a normalized surplus projection method, which provides an approximate reduction of the full three-dimensional problem to a nonlinear system of partial differential equations (which is subsequently numerically validated). The optimal investment policy admits an economically meaningful decomposition consisting of a myopic demand component and an interest rate hedging component. Numerical experiments illustrate how the optimal strategy and the surplus distribution depend on interest rate volatility, claim intensity, and risk aversion. The results highlight the importance of jointly modeling stochastic interest rates and insurance liability risk when designing optimal investment policies for insurance companies.

Digital twins are often used to predict how a system would respond to an intervention. In systems with feedback, a twin must reproduce an equilibrium counterfactual, and a twin developed in one domain may fail after mechanisms change. We study when these predictions can be validated and transported. For equilibrium causal games, we give conditions on the mechanisms, equilibrium selection, and intervention design under which agreement with experimental distributions identifies the counterfactual of interest. We show why agreement of means and covariances is insufficient for distributional queries. We then introduce cyclic selection diagrams and derive criteria for direct reuse and for hybrid models that combine invariant source mechanisms with target information. An impossibility result constructs systems that agree under every experiment in a finite design but disagree on the target counterfactual, showing that validation requires structural assumptions. For linear models, we derive intervention requirements that depend on the mechanisms that changed, the observation model, and graph support. When point identification fails, we characterize the remaining range of query values. We also provide statistical tests for reconstructed means and covariances and illustrate the theory in synthetic feedback systems.

We study the problem of reconstructing the probability measure of a multi-group version of the Curie-Weiss or mean-field model of ferromagnetism from a sample of the voting behaviour the population. While originally used to study phase transitions in statistical mechanics, the Curie-Weiss or mean-field model has been applied to study phenomena where many agents interact with each other, in particular in case of a heterogeneous population with identifiable subpopulations. The degree of social cohesion within social groups manifests in the way the members of the group influence each others' decisions as well as how they behave under outside influence from voters belonging to another group. Contrary to single-group Curie-Weiss models, here we have a larger number of coupling parameters which have to be estimated. While the maximum likelihood estimator of the coupling parameters has desirable statistical properties in theory, computational challenges make applications to larger populations impractical. Therefore, we analyse an estimator based on asymptotic approximations to the behaviour of the Curie-Weiss model valid for large populations. Due to the wide applicability of models such as Curie-Weiss, the estimator is potentially useful in disciplines such as political science, sociology, automated voting, and preference aggregation.

We develop an inverse theory for time–frequency localization operators, formulated as a free-boundary problem in which the localization domain is unknown and its boundary is recovered from prescribed spectral data. The guiding principle is that an eigenfunction can itself be regarded as geometric data determining a localization domain, with strong consequences for the associated variational problem. We obtain five main results. First, for every polynomial $f_0$ sufficiently close to the Gaussian and every $\lambda\in(0,1)$, we construct a real-analytic domain $U_\lambda$ such that $f_0$ is an eigenfunction of the corresponding localization operator with eigenvalue $\lambda$, providing a general inverse construction of localization domains. Second, we recover the null-set invariant Abreu–D"orfler characterization of disks as the only simply connected localization domains having a Hermite polynomial as eigenfunction. Third, we show that simple connectivity is essential: a weighted Hele–Shaw flow yields an infinite-dimensional family of non-radial, doubly connected analytic domains for which the Gaussian is an eigenfunction. Fourth, we prove the optimality of the exponent $1/2$ in the quantitative stability inequality of G'omez–Guerra–Ramos–Tilli. Finally, we prove that every local maximizer of the Gaussian Faber–Krahn problem is a disk and, using a Fock-space concentration–compactness profile decomposition, obtain a new proof of the Nicola–Tilli concentration theorem.

Let $\mathcal{F}\subseteq\binom{[n]}k$, and let $\partial_{k\to\ell}\mathcal{F}$ be the family of all $\ell$-sets containing a member of $\mathcal{F}$. Writing $\mu_k(\mathcal{F})=|\mathcal{F}|/\binom nk$, $p=k/n$, and $q=\ell/n$, we prove $$ \mu_\ell(\partial_{k\to\ell}\mathcal{F})\geq \begin{cases}\mu_k(\mathcal{F})^{\frac{\log q}{\log p}}, &0\leq \mu_k(\mathcal{F})\leq p^2,\\ q\left[1-\left(1-\frac{\mu_k(\mathcal{F})}{p}\right)^{ \frac{\log(1-q)}{\log(1-p)}} \ \ \right], &p^2\leq \mu_k(\mathcal{F})\leq p,\\ 1-(1-\mu_k(\mathcal{F}))^{\frac{\log(1-q)}{\log(1-p)}}, &p\leq \mu_k(\mathcal{F})\leq1. \end{cases} $$ This yields a dimension-free closed-form lower bound for the finite Kruskal–Katona profile and each branch is asymptotically sharp. As the main application, we settle both the uniform and biased product conjectures of Frankl and Tokushige. If $r\geq2$, $0\leq k_i\leq(r-1)n/r$, and $\mathcal{F}_i\subseteq\binom{[n]}{k_i}$ are $r$-cross-intersecting, then $$ \prod_{i=1}^r\mu_{k_i}(\mathcal{F}_i)\leq\prod_{i=1}^r\frac{k_i}{n}. $$ In biased product setting, we prove that for $r$-cross-intersecting families $\mathcal{A}_i\subseteq 2^{[n]}$ and $0\leq p_i\leq(r-1)/r$, $\prod_{i=1}^r\mu_{p_i}(\mathcal{A}_i)\leq\prod_{i=1}^rp_i.$ We further determine all equality cases in both settings.

Barzilai–Borwein (BB) method has shown strong practical performance in continuous optimization, yet its convergence dynamics remains poorly understood. In particular, a central unresolved question is whether BB converges superlinearly for almost every strictly convex quadratic problem and initialization. We provide a negative answer to this question. Specifically, for every finite dimension $n\geq4$, we construct a nonempty open, hence positive-Lebesgue-measure, family of strictly convex quadratic problems and initial points for which the long Barzilai–Borwein method (BB1) converges but cannot converge root-superlinearly. More precisely, with the explicit constants $\rho_{\min}=10^{-6},\rho_{\max}=0.61$, every spectral component of the gradient is bounded above and below by the corresponding geometric sequence. Consequently, the gradient norm and the energy norm of the error satisfy two-sided geometric estimates with the same rates, while the objective gap satisfies the corresponding estimates with squared rates. In particular, all three quantities are bounded below by geometric sequences, ruling out superlinear convergence. The construction is highly nontrivial, based on a computer-assisted proof of a nonresonant, attracting seven-cycle of the projectivized BB dynamics in dimension four.

A central problem in harmonic analysis is to understand how frequency resolution constrains stability under nonlinear changes of coordinates. Motivated by Mallat's wavelet scattering theory, we establish a sharp resolution–robustness principle for nonlinear representations built from two-dimensional wave-packet systems, showing that the transverse geometry of the frequency channels determines the critical Sobolev regularity. Whenever the channels are thinner than in the wavelet regime, arbitrarily small smooth deformations can move high-frequency mass between adjacent channels, producing instability already at the level of the linear filter bank and the first scattering layer. We match this obstruction with scale-sharp commutator estimates propagated through the full scattering cascade, obtaining Lipschitz stability at the critical regularity threshold. Together, the results provide the first intrinsic deformation-stability theory for non-wavelet Euclidean scattering transforms and identify a microlocal transport mechanism governing robustness across nonlinear multiscale representations.

We determine the possible generic degrees of real polynomial Keller maps with non-dense image, where density is understood in the Euclidean topology. For every dimension $n\geq3$, these degrees are exactly the even integers $d\geq4$. They are realized in dimension three by an explicit family $G_d\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$, and hence in every higher dimension by stabilization. In dimension two, any such degree is an even integer at least six, and no such maps exist if the planar Jacobian conjecture holds; in dimension one, none exist. For the family $G_d$, we describe the image exactly and show that no $G_d$ omits a half-space. We also show that the maximal cardinality of a real fibre is not determined by the generic degree and is not uniformly bounded in this class.

We study tight Hamilton cycles in linearly quasirandom $3$-graphs. An $n$-vertex $3$-graph $H$ is $(p,\mu)$-dense if $e_H(X,Y,Z)\ge p|X||Y||Z|-\mu n^3$ for all $X,Y,Z\subseteq V(H)$. Araújo, Piga and Schacht asked whether the conditions $p,\alpha>1/4$ and $\delta_2(H)\ge\alpha n$ force a tight Hamilton cycle. We give a negative answer to this question. More generally, we determine the asymptotically sharp minimum-codegree threshold for the existence of a tight Hamilton cycle for every density $p\in(0,1)$. The resulting threshold is a discontinuous piecewise-defined function with four distinct regimes, and matching constructions show that every piece is best possible. The proof combines the absorption method and a fixed-length connecting lemma above density $1/3$ with a canonical-component Hamilton framework at and below density $1/3$.

We compute the cardinalities of the integral algebraic K-theory groups of connective Morava K-theory $k(n)$ in all degrees that are not congruent to $0$ or $1$ modulo $2p-2$. After base-changing the coefficients of $k(n)$ to the algebraic closure $\overline{\mathbb{F}}_p$, we determine the corresponding cardinalities in all degrees and show that the groups vanish in even degrees. Our approach uses what we call the orbit filtration on topological cyclic homology arising from the May filtration on topological Hochschild homology. Combining this with an analysis of the topological cyclic homology of formal DGAs of the form $\mathbb{F}[x_{2m}]$, where $\mathbb{F}$ is a perfect field of characteristic $p$, we prove strong cardinality results for the topological cyclic homology of $\mathbb{E}_1$-rings with homotopy $\mathbb{F}[x_{2m}]$. For Morava K-theories over finite fields, we further use the motivic filtration of Hahn–Raksit–Wilson. As an application of our methods, we compute the cardinalities of the algebraic K-theory groups of the truncation $W(\overline{\mathbb{F}}_p)/p^n$ of the $p$-typical Witt vectors of $\overline{\mathbb{F}}_p$; in particular, they vanish in even degrees.

A restricted-budget version of the random graph process, introduced by Frieze, Krivelevich, and Michaeli in 2025, studies the construction of structures by an online player who can purchase only a limited number of random edges. In this paper, we transfer this framework from random graphs to random subsets of integers, focusing on the construction of $k$-term arithmetic progressions. A player, Builder, is presented with a sequence of $t$ integers drawn uniformly at random from $[n]$. As the elements are revealed one by one, Builder must immediately and irrevocably decide whether to select the current integer, subject to a maximum budget of $b$ selected elements in total. We establish the optimal thresholds for this process, proving that for $t = \omega(n^{1-2/k})$, a budget of $b = \Theta((n/t)^{\frac{k-2}{2}})$ is both necessary and sufficient for Builder to successfully construct a $k$-term arithmetic progression with high probability.

Consider the partial linear model $Y = \mu_0(X) + \beta_0 \cdot T + \varepsilon$ and $T = \pi_0(X) + u$ in the structure-agnostic setting, where we are blind to the structure $\mu_0$ and $\pi_0$ and estimate the nuisances by a black-box hypothesis class. The learnability of the class is characterized by the estimation error $\delta_s$ in the absence of model misspecification and its $L_2$ mis-specification error $\delta_{a, \mu}$ and $\delta_{a, \pi}$ for $\mu_0$ and $\pi_0$, respectively. We propose a novel estimator of the target linear coefficient $\theta_0 = \beta_0$ with error rate \[ \frac{1}{\sqrt{n}} + \delta_{a, \mu} \cdot \delta_{a, \pi} + [\delta_s]^2. \] A matching lower bound is also established, implying that this rate is unimprovable. Compared with the product rate yielded by double machine learning (DML), our estimator removes the suboptimal term $\max(\delta_{a, \mu}, \delta_{a, \pi})\cdot \delta_s$ at no extra cost or assumption. Building on the underlying insights, which are neither tailored to the one-learner setting nor the partial linear model, we propose Transductive Adversarial Moment-calibrated Editing (TAME), which locally edits debiasing weights induced by black-box regression estimates on the inference sample through adversarial conditional moment calibration. TAME can be combined with any initial black-box estimates and can strictly improve on DML guarantees when the nuisance difficulties are imbalanced. We discuss how to fully exploit the advantages introduced by TAME, including the gains from using two learners, the resulting under-smoothing principle for model selection, and extensions to other linear functional estimation problems.

We say that a convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. We prove that this position is unique up to orthogonal transformations, answering a question of Schmuckenschlaeger from 2011. This is a corollary of a new log-convexity property of the first eigenvalue under positive definite linear deformations. While the centrally symmetric case follows from the Gaussian B-theorem, the extension to arbitrary convex bodies requires a quantitative analysis of conditioned Brownian motion. As consequences, we obtain a new proof of the Polya-Szego theorem for triangles and its analogue for simplices, and show that regular polygons minimize the first eigenvalue within their linear orbits of fixed volume. We also prove related convexity results for the first eigenvalue of the Ornstein-Uhlenbeck operator, the inverse inradius and the planar Cheeger constant. In a different direction, we show using similar ideas that the Gaussian conjugate Rogers-Shephard inequality due to Milman-Nakamura-Tsuji yields improved Schmuckenschlaeger-type bounds for intersections and Minkowski sums of centrally symmetric convex bodies.

For a compact, connected, oriented Einstein four-manifold, we prove that, if the largest eigenvalue of the self-dual Weyl curvature $W^+$ is everywhere simple, then, after at worst passing to a double cover, the metric is conformally Kähler with positive scalar curvature; more generally, this result holds for metrics with harmonic self-dual Weyl curvature. For Einstein metrics satisfying a uniform simplicity hypothesis on the largest eigenvalue, we further prove that either $W^+\equiv 0$, or $W^+$ nowhere vanishes and the previous conclusion holds. We also obtain extensions to complete Ricci-flat four-manifolds and an optimal pinching theorem for the holomorphic sectional curvature of compact Kähler–Einstein surfaces. The proof combines LeBrun's conformal normalization with the resulting weighted divergence equation and new first-order identities. A zero-capacity argument allows this method to be used across the zero set of $W^+$ and at infinity in the noncompact case. Finally, K3 surfaces and multicentered Gibbons–Hawking gravitational instantons show that our assumptions are sharp.

Under the Generalized Riemann Hypothesis, we prove a weighted central limit theorem for the joint distribution of four Dirichlet $L$–functions at the central point, twisted by the family of primitive characters to a large prime modulus. As an application, we show that a positive proportion of the characters in the family yield four central values that are simultaneously large, and a positive proportion yield values that are simultaneously nonzero and small.

In classical and quantum statistics, high-dimensional unknown parameters are abundant and it is often prudent to make minimal assumptions about them using so-called semiparametric models. To attack a wide range of semiparametric problems in one broad stroke, we present a unified treatment of statistical efficiency for classical and quantum semiparametric models, generalizing the Cramér-Rao and Helstrom bounds beyond finite-dimensional parameters. We introduce the fundamental concepts in abstract and geometric terms before applying them to many examples, covering general classical and quantum models as well as the paradigmatic special cases of Gaussian and Poisson fields. We give an in-depth treatment of channels in the semiparametric efficiency theory and advocate the use of the singular value decomposition to elucidate the statistical effects of channels. To demonstrate the utility of the formalism, we apply it to coherent and incoherent optical imaging problems, assuming an arbitrary field or intensity on the object plane without parametric assumptions. Our formalism enables us to compute classical and quantum limits to coherent and incoherent imaging resolution in statistical terms. For subdiffraction incoherent imaging, we demonstrate that spatial-mode demultiplexing can be far superior to direct imaging in estimating generalized Fourier coefficients and come closer to the quantum limits. We envision our theory becoming an essential tool for both classical and quantum statistics with useful applications to sensing and imaging, whenever minimal assumptions about a high-dimensional parameter should be made.

We give a recursive construction of an LKTS(7q + 2) from an LKTS(q + 2) and an auxiliary permutation structure on an abelian group of order q, which we call a cubic orthomorphism. We prove that cubic orthomorphisms exist whenever q is a product of prime powers congruent to 1 modulo 6. The construction yields, in particular, LKTSs of orders 93, 177, 219, 261, and 303.

We derive statistical limit theorems for sample-based approximations of infinite-horizon discounted stochastic optimal control problems in discrete time. Our first result is a functional central limit theorem for the sample-based value function under a uniqueness-type condition on population optimal policies. The limiting law is a mean-zero Gaussian process characterized by a linear fixed point equation that resembles a dynamic programming principle. We compare these asymptotics with those obtained from sample-based policy optimization and illustrate that their limiting variances can be different. We also derive a limit theorem for models with nonunique optimal policies, where the limiting law may be non-Gaussian. Applications to inventory control and renewable harvesting illustrate the theory.

The Shannon capacity $\Theta(G)$ of a graph $G$ quantifies the maximum rate at which information can be transmitted with zero error over a noisy channel. It is lower bounded by $\alpha(G^d)^{1/d}$ for any $d$, where $\alpha(G^d)$ is the independence number of the $d$-th strong product of $G$. We construct independent sets of size $134753$ in $C_7^{10}$, $21909$ in $C_{11}^{6}$, $62530$ in $C_{13}^{6}$, and $8076974$ in $C_{15}^{8}$, improving the best known lower bounds for the Shannon capacity of these graphs to $\Theta(C_7)\geq 134753^{1/10}>3.258020$, $\Theta(C_{11})\geq 21909^{1/6}>5.289773$, $\Theta(C_{13})\geq 62530^{1/6}>6.300109$, and $\Theta(C_{15})\geq 8076974^{1/8}>7.301399$. We also improve the best known lower bounds on the independence numbers of several individual strong products of odd cycles that do not improve the Shannon capacity lower bound. The constructions were discovered through iterative interactions with a Large Language Model (LLM), illustrating the potential of LLMs for finding explicit combinatorial constructions.

We develop a technique based on sectional transfer operators and the regularity of their fixed points to study rates of correlation decay for partially hyperbolic skew-products. To illustrate the range of applicability of this technique, we apply it to a family of Viana like maps with non-uniformly contracting fibres, a family of skew-product maps with non-uniformly expanding base dynamics and to a generalised family of heterochaos baker maps. The latter partially hyperbolic systems have recently been an active topic of research due to their connection with the Dyck shift.

We identify and study the interval Garside groups arising from the restriction of the absolute order on a Coxeter group to a lattice $[1,w]_T$, where $w$ is an involution. Those involutions $w$ for which $[1,w]_T$ is a lattice were previously classified by the second author; every such involution lies in the center of the parabolic subgroup generated by $[1,w]_T$. Except in type $B_n$, the obtained groups are isomorphic to (decomposable) right-angled Artin groups. We also investigate the situation for some finite complex reflection groups, mostly in rank two, taking for $w$ a (not necessarily involutive) central element.