Given partial information about a set, we are interested in fully recovering the original set from what is given. If a set encodes itself robustly, any partial information about the set suffices to fully recover the information about the original set. Jockusch defined a set $A$ to be introenumerable if each infinite subset of $A$ can enumerate $A$, and this is an example of a set which encodes itself. There are several other notions capturing self-encoding differently. We say $A$ is uniformly introenumerable if each infinite subset of $A$ can uniformly enumerate $A$, whereas $A$ is introreducible if each infinite subset of $A$ can compute $A$. We investigate properties of various notions of self-encoding and prove new results on their interactions. Greenberg, Harrison-Trainor, Patey, and Turetsky showed that we can always find some uniformity from an introenumerable set. We show that this can be reversed: we can construct an introenumerable set by patching up uniformity. This gives a rise to a new method of constructing a nontrivial introenumerable or introreducible set.

In this paper we describe a general framework for constructing examples of locally linear semistrict monoidal 2-categories covering many examples appearing in link homology theory. The main input datum is a closed foam evaluation formula. As examples, we rigorously construct semistrict monoidal 2-categories based on gl(N)-foams, which underlie the general linear link homology theories, and further examples based on Bar-Natan's decorated cobordisms, related to Khovanov homology. These monoidal 2-categories are typically non-semisimple, have duals for all objects, adjoints for all 1-morphisms, and carry a canonical spatial duality structure expressing oriented 3-dimensional pivotality and sphericality.

Accurate localization of maritime targets by unmanned aerial vehicles (UAVs) remains challenging in GPS-denied environments. UAVs equipped with gimballed electro-optical sensors are typically used to localize targets, however, reliance on these sensors increases mechanical complexity, cost, and susceptibility to single-point failures, limiting scalability and robustness in multi-UAV operations. This work presents a new trajectory optimization framework that enables cooperative target localization using UAVs with fixed, non-gimballed cameras operating in coordination with a surface vessel. This estimation-aware optimization generates dynamically feasible trajectories that explicitly account for mission constraints, platform dynamics, and out-of-frame events. Estimation-aware trajectories outperform heuristic paths by reducing localization error by more than a factor of two, motivating their use in cooperative operations. Results further demonstrate that coordinated UAVs with fixed, non-gimballed cameras achieve localization accuracy that meets or exceeds that of single gimballed systems, while substantially lowering system complexity and cost, enabling scalability, and enhancing mission resilience.

We consider an electrostatic system whose spatial factor is conformal to an $n$-dimensional Euclidean space. We provide a complete characterization of the most general ansatz, thereby reducing the associated electrostatic system of partial differential equations to an ordinary differential equation system. We prove that there are only two possibilities: either the cosmological constant is nonzero, in which case the solutions are necessarily invariant under rotations or translations, or the cosmological constant vanishes, and the solutions belong to the Majumdar-Papapetrou class with a degree of freedom associated with an invariant $(n-1)$-dimensional subgroup. As a result, we introduce a new solution to the electrovacuum system in the Majumdar-Papapetrou class that is invariant under an $(n-1)$-dimensional group of dilations.

Kruse and Wu [Math. Comp. 88 (2019) 2793--2825] proposed a fully discrete randomized Galerkin finite element method for semilinear stochastic evolution equations (SEEs) driven by additive noise and showed that this method attains a temporal strong convergence rate exceeding order $\frac{1}{2}$ without imposing any differentiability assumptions on the drift nonlinearity. They further discussed a potential extension of the randomized method to SEEs with multiplicative noise and introduced the so-called drift-randomized Milstein-Galerkin finite element fully discrete scheme, but without providing a corresponding strong convergence analysis. This paper aims to fill this gap by rigorously analyzing the strong convergence behavior of the drift-randomized Milstein-Galerkin finite element scheme. By avoiding the use of differentiability assumptions on the nonlinear drift term, we establish strong convergence rates in both space and time for the proposed method. The obtained temporal convergence rate is $O(Δt^{1-\varepsilon_0})$, where $Δt$ denotes the time step size and $\varepsilon_0$ is an arbitrarily small positive number. Numerical experiments are reported to validate the theoretical findings.

Reduced-rank linear discriminant analysis (RRLDA) is a foundational method of dimension reduction for classification that has been useful in a wide range of applications. The goal is to identify an optimal subspace to project the observations onto that simultaneously maximizes between-group variation while minimizing within-group differences. The solution is straight forward when the number of observations is greater than the number of features but computational difficulties arise in both the high-dimensional setting, where there are more features than there are observations, and when the data are very large. Many works have proposed solutions for the high-dimensional setting and frequently involve additional assumptions or tuning parameters. We propose a fast and simple iterative algorithm for both classical and high-dimensional RRLDA on large data that is free from these additional requirements and that comes with guarantees. We also explain how RRLDA-RK provides implicit regularization towards the least norm solution without explicitly incorporating penalties. We demonstrate our algorithm on real data and highlight some results.

Enormous fluid antenna systems (E-FAS), the system concept that utilizes position reconfigurability in the large scale, have emerged as a new architectural paradigm where intelligent surfaces are repurposed from passive smart reflectors into multi-functional electromagnetic (EM) interfaces that can route guided surface waves over walls, ceilings, and building facades, as well as emit space waves to target receivers. This expanded functionality introduces a new mode of signal propagation, enabling new forms of wireless communication. In this paper, we provide an analytical performance characterization of an E-FAS-enabled wireless link. We first develop a physics-consistent end-to-end channel model that couples a surface-impedance wave formulation with small-scale fading on both the base station (BS)-surface and launcher-user segments. We illustrate that the resulting effective BS-user channel remains circularly symmetric complex Gaussian, with an enhanced average power that explicitly captures surface-wave attenuation and junction losses. For single-user cases with linear precoding, we derive the outage probability and ergodic capacity in closed forms, together with high signal-to-noise ratio (SNR) asymptotics that quantify the gain of E-FAS over purely space-wave propagation. For the multiuser case with zero-forcing (ZF) precoding, we derive the distribution of the signal-to-interference-plus-noise ratio (SINR) and obtain tractable approximations for the ergodic sum-rate, explicitly revealing how the E-FAS macro-gain interacts with the BS spatial degrees of freedom (DoF). In summary, our analysis shows that E-FAS preserves the diversity order dictated by small-scale fading while improving the coding gain enabled by cylindrical surface-wave propagation.

We study first-order phase transitions in continuum Gibbs point processes with saturated interactions. These interactions form a broad class of Hamiltonians in which the local energy in regions of high particle density depends only on the number of points. Building on ideas of Pirogov-Sinai-Zahradnik theory and its adaptations to the continuum, we develop a general method for establishing the existence of two distinct infinite-volume Gibbs measures with different intensities in this setting, demonstrating a first-order phase transition. Our approach extends previous results obtained for the Quermass model and applies in particular to a new class of diluted pairwise interactions introduced in this work.

We construct a statistic-swapping involution on the Cartesian product of the generalized symmetric group $S(k,n)$ with the symmetric group $S_{kn}$, which swaps the number of fixed points in the generalized symmetric group element with the number of $k$-cycles in the symmetric group element. This gives a combinatorial proof for a probabilistic observation: the distribution of fixed points on $S(k,n)$ matches the distribution of $k$-cycles on $S_{kn}$.

Suppose a Bayesian agent seeks to traverse a graph. Each time she crosses an edge, she pays a price. The first time she reaches a node, there is a payoff. She has an opponent who can reduce the payoffs. This paper uses adversarial risk analysis to find a solution to her route selection problem. It shows how the traveler is advantaged by having an accurate subjective distribution over the costs/payoffs and by having a Bayesian prior for her opponent's strategic choices. The results are relevant to military convoy routing, corporate competition, and certain games.

We establish a weighted simultaneous Khintchine-type theorem, both convergence and divergence, for all nondegenerate manifolds, which answers a problem posed in [Math. Ann., 337(4):769-796, 2007]. This extends the main results of [Acta Math., 231:1-30, 2023] and [Ann. of Math. (2), 175(1):187-235, 2012] in the weighted set-up. As a by-product of our method, we also obtain a multiplicative Khintchine-type convergence theorem for all nondegenerate manifolds, which is a simultaneous analogue of the celebrated result of Bernik, Kleinbock, and Margulis for dual approximation.

We investigate the complexity of the Hamiltonian in the pure $p$-spin spin glass model accompanied with a polynomial-type potential on $\mathbb{R}^N$. In this Hamiltonian, the Gaussian field is anisotropic, and the potential lacks rotational invariance. Our main result derives the logarithmic limit for the expected number of critical points in terms of a variational formula. As a consequence, by identifying the critical location of the phase transition from our representation, we provide an upper bound for the ground state energy of the model.

A minimal equicontinuous action of a group $G$ on a Stone space $X$ is called a subodometer. If such a subodometer arises from a group rotation, we refer to it as an odometer. For subodometers $(X,G)$ we show that the hyperspace $\mathcal{H}(X)$ - given by all closed subsets of $X$ and the Vietoris topology - decomposes into subodometers. We show that an infinite subodometer is an odometer if and only if $\mathcal{H}(X)$ decomposes into factors of $(X,G)$. Similarly, we consider $\mathcal{M}(X)$, the space of regular Borel probability measures equipped with the weak-* topology. We show that for a subodometer $(X,G)$ also the connected space $\mathcal{M}(X)$ decomposes into subodometers. We prove that an infinite subodometer $(X,G)$ is an odometer if and only if $\mathcal{M}(X)$ decomposes into factors of $(X,G)$. For this, we study different notions of regular recurrence. Furthermore, we study the disjointness of minimal actions to subodometers and show that this disjointness can be detected from the pairwise disjointness of finite factors. Using this we prove that a minimal action is disjoint from all subodometers if and only if it has a connected maximal equicontinuous factor.

This paper investigates the asymptotic behavior of suitably time-modulated Hawkes processes with heavy-tailed kernels in a nearly unstable regime. We show that, under appropriate scaling, both the intensity processes and the rescaled Hawkes processes converge to a mean-reverting, time-inhomogeneous rough fractional square-root process and its integrated counterpart, respectively. In particular, when the original Hawkes process has a stationary first moment (constant marginal mean), the limiting process takes the form of a time-inhomogeneous rough fractional Cox-Ingersoll-Ross (CIR) equation with a constant mean-reversion parameter and a time-dependent diffusion coefficient. This class of equations is particularly appealing from a practical perspective, especially for the so-called $\textit{fake stationary rough Heston}$ model. We further investigate the properties of such limiting scaled time-inhomogeneous Volterra equations, including moment bounds, path regularity and maximal inequality in the $L^p$ setting for every $p>0$.

In this article, we solve the twisted conjugacy problem with respect to endomorphisms for solvable Baumslag--Solitar groups $BS(1,n)$, i.e., we propose an algorithm which, given two elements $u,v \in BS(1,n)$ and an endomorphism $ψ\in End(BS(1,n))$, decides whether $v=(xψ)^{-1} u x$ for some $x\in BS(1,n)$. Also, we connect the outer fixed points of a given endomorphism $ψ$ with $φ$-twisted conjugacy problem for two words $u, v \in BS(1,n)$, where $φ\in End(BS(1,n))$ and $u, v$ depend on $ψ$. Furthermore, we define the weakly (outer) fixed points and discuss its interplay with the endo-twisted conjugacy problem in $BS(1, n)$.

Biological systems, particularly the human brain, achieve remarkable energy efficiency by abstracting information across multiple hierarchical levels. In contrast, modern artificial intelligence and communication systems often consume significant energy overheads in transmitting low-level data, with limited emphasis on abstraction. Despite its implicit importance, a formal and computational theory of information abstraction remains absent. In this work, we introduce the Degree of Information Abstraction (DIA), a general metric that quantifies how well a representation compresses input data while preserving task-relevant semantics. We derive a tractable information-theoretic formulation of DIA and propose a DIA-based information abstraction framework. As a case study, we apply DIA to a large language model (LLM)-guided video transmission task, where abstraction-aware encoding significantly reduces transmission volume by $99.75\%$, while maintaining semantic fidelity. Our results suggest that DIA offers a principled tool for rebalancing energy and information in intelligent systems and opens new directions in neural network design, neuromorphic computing, semantic communication, and joint sensing-communication architectures.

We establish the convergence theory of multiplicative Diophantine approximation for all non-degenerate, smooth manifolds. We also settle said convergence theory for all affine subspaces satisfying a highly generic and essentially optimal Diophantine condition. This answers a question of Beresnevich and Velani from 2005, while simultaneously sharpening results of Kleinbock and Margulis on the strong extremality of non-degenerate manifolds, and of Kleinbock on the strong extremality of affine subspaces.

We compute the dual Steenrod algebra for Bredon homology with constant coefficients $\underline{\mathbb Z}$ and $\underline{\mathbb Z}/2$ in the category of modules over $MU^{((G))}$, the norm to $G=C_{2^n}$ of $MU_{\mathbb R}$. Using this and an equivariant version of the Greenlees--Serre spectral sequence, we give a spectral sequence computing the $RO$-graded homotopy of the Eilenberg--Mac Lane spectrum $H\underline{\mathbb F}_2\otimes H\underline{\mathbb Z}$.

We investigate instability phenomena for linear evolution equations within the framework of $C_0$--semigroups on infinite--dimensional spaces. We show that Devaney chaos, being formulated in purely topological terms, may depend on the choice of topology and therefore fail to capture intrinsic dynamical behavior. To address this issue, we introduce a trajectory--based notion of relative chaos, defined with respect to a reference solution and measured in a fixed, physically meaningful norm. This criterion is independent of topological refinements and is shown to be strictly weaker than classical Devaney chaos. Its relevance is illustrated on boundary--driven reaction--diffusion--transport semigroups.

Multiscale simulations utilizing high-fidelity, microscopic Monte Carlo models to provide the nonlinear response for continuum models can easily become computationally intractable. Surrogate models for the high-fidelity Monte Carlo models can overcome this but come with some challenges. One such challenges arise by the sampling noise in the underlying Monte Carlo data, which leads to uncontrolled errors possibly corrupting the surrogate even though it would be highly accurate in the case of noise-free data. Another challenge arises by the 'curse of dimensionality' when the response depends on many macro-variables. These points are addressed by a novel noise-balanced sparse grids interpolation approach which, in a quasi-optimal fashion, controls the amount of Monte Carlo sampling for each data point. The approach is complemented by a multilevel on-the-fly construction during the multiscale simulation. Besides its efficiency, a particularly appealing feature is the ease of use of the approach with only a single hyperparameter controlling the whole surrogate construction - from the surrogate's accuracy with guaranteed convergence to which data needs to be created with which accuracy. The approach is demonstrated on challenging examples from heterogeneous catalysis, coupling microscopic kinetic Monte Carlo models into macroscopic reactor simulations.

Let $(M,h)$ be a compact Kähler manifold. Under a suitable spectral positivity assumption we prove that $M$ is simply connected, projective, uniruled and $h^{p,0}(M)=\{0\}$ for each $p>0$. Then, in the second part of this paper, we focus on Riemannian manifolds and we provide an appropriate spectral positivity assumption which guarantees that a compact and oriented even dimensional Riemannian manifold $(M,g)$ is a simply connected rational homology sphere.

For $p>1$, we study subordination phenomena for local and non-local regular $p$-energies on metric measure spaces. Under suitable geometric assumptions, we show that if a local regular $p$-energy satisfies a Poincaré inequality together with a cutoff Sobolev inequality with scaling function $Ψ$, then all associated stable-like non-local $p$-energies with scaling functions strictly below $Ψ$ are regular and satisfy the corresponding non-local cutoff Sobolev inequalities. Moreover, if a stable-like non-local regular $p$-energy with scaling function $Ψ$ satisfies the corresponding non-local cutoff Sobolev inequality, then the same conclusion holds for all associated stable-like non-local $p$-energies with scaling functions below $Ψ$. These results provide a non-linear extension of the classical subordination principle beyond the Dirichlet form framework.

In this paper, we study deformation quantization of symplectic vector fields à la Fedosov. We show that each symplectic vector field can be quantized to a derivation of the deformed star algebra. Moreover, we show that this quantization yields a non-abelian $2$-cocycle on the Lie algebra of symplectic vector fields with values in the deformed star algebra. Therefore, we can quantize any Lie algebra action by symplectic vector fields.

Integral discriminants provide a simple and fundamental model for non-Gaussian integrals, associated with homogeneous polynomials of degree r in n variables. We argue that, in this context, the study of correlators is equally if not more important. In this paper, we study a natural class of correlators in this model -- the invariant correlators. We suggest a general method to compute invariant correlators using differential operators that act on the partition function. This method allows to compute general invariant correlators in terms of the fundamental invariants. Moreover, in some cases the correlators appear to be simply polynomials in the invariants. This could be an interesting manifestation of superintegrability phenomenon in the theory of integral discriminants.

Near-field multi-antenna wireless communication has attracted growing research interest in recent years. Despite this development, most of the current literature on antennas and reflecting structures relies on simplified models, whose validity for real systems remains unclear. In this paper, we introduce a physically consistent near-field model, which we use to evaluate commonly used models. Our results indicate that common models are sufficient for basic beamfocusing, but fail to accurately predict the sidelobes and frequency dependence of reflecting structures.

We study the star hull of planar Brownian motion and bridge. Roughly speaking, this is the smallest starshaped set (with respect to the origin) that contains the trace of the path. In particular, we prove that the expected areas of the star hulls are $\frac{3π}{8}$ and $\fracπ{4}$ for planar Brownian motion and bridge, respectively. Our proofs rely on a detailed analysis of the first hitting time and place of a horizontal ray $\mathcal{R}_ρ: = [ρ,\infty)\times\{0\}$ by planar Brownian motion starting at the origin. After deriving a remarkably simple Laplace transform of this joint law, we uncover via a probabilistic argument a surprising conditional structure: conditionally on the first hitting place being the point $(x,0)\in \mathcal{R}_ρ$, the hitting time is distributed as the first passage time to the level $x$ of one-dimensional Brownian motion starting at $0$.

We study the graph-theoretic properties of the trace of random walks on pseudorandom graphs. We show that for any $\varepsilon>0$, there exists a constant $C$ such that the cover time of an $(n,d,λ)$-graph $G$ with $d/λ\ge C$ is at most $(1+\varepsilon)n\log n$, meaning the expected number of steps needed to reach all vertices at least once is at most $(1+\varepsilon)n\log n$ regardless of the starting vertex. Furthermore, we prove that with high probability, the trace of a random walk of length $(1+\varepsilon)n\log n$ on $G$ is Hamiltonian, regardless of the starting vertex. These results also hold for random $d$-regular graphs with sufficiently large $d$. These findings answer two questions proposed by Frieze, Krivelevich, Michaeli, and Peled [PLMS, 2018]. Notably, our results imply a bound on a stronger version of the cover time: with high probability, all vertices are covered after $(1+\varepsilon)n\log n$ steps, regardless of the starting vertex. Our proofs rely on the spectral properties of the adjacency matrix and the graph expansion. All results are asymptotically optimal.

In this paper, we derive the continuous space-time equations of motion of a three-dimensional geometrically exact rod, or the Cosserat rod, incorporating planar cross-sectional deformation. We then adopt the Lie group variational integrator technique to obtain a discrete model of the rod incorporating both rotational motion and cross-sectional deformation as well. The resulting discrete model possesses several desirable features: it ensures volume conservation of the discrete elements by considering cross-sectional deformation through a local dilatation factor, it demonstrates the beneficial properties associated with the variational integrator technique, such as the preservation of the rotational configuration, and energy conservation with a bounded error. An exhaustive set of numerical results under various initial conditions of the rod demonstrates the efficacy of the model in replicating the physics of the system.

We define trajectory predictive control (TPC) as a family of output-feedback indirect data-driven predictive control (DDPC) methods that represent the output trajectory of a discrete-time system as a linear function of the recent input/output history and the planned input trajectory. This paper shows that for different choices of the trajectory predictor, TPC encompasses a wide variety of DDPC methods, including subspace predictive control (SPC), closed-loop SPC, $γ$-DDPC, causal-$γ$-DDPC, transient predictive control, and others. This paper introduces a trajectory predictor that corresponds to a linear state-space model with the recent input/output history as the state. With this state-space predictor, TPC is a special case of linear model predictive control and therefore inherits its mature theory. In numerical experiments, TPC performance approaches the limit of oracle $H_2$-optimal control with perfect knowledge of the underlying system model. For TPC with small training datasets, the state-space predictor outperforms other predictors because it has fewer parameters.

We prove a Riemannian positive mass theorem for asymptotically flat spin manifolds with hypersurface singularities. Unlike earlier results, some components of the singular set may be mean-concave, provided that other components of the singular set are sufficiently mean-convex. Our proof uses initial data sets where a suitably chosen second fundamental form transfers convexity defects between different singularity components.