Papers

This paper is devoted to the study of Bowen's dimensional entropy on subsets for actions of amenable groups. We prove three main results. (1) First, topological conditional entropy is characterized by the dimensional entropy of stable sets (Theorem 1.1), answering a question of Dou, Wang and the second author of the present paper raised in [Fund. Math., 2025]. We remark that our Theorem 1.1 is the first characterization of topological conditional entropy via Bowen's dimensional entropy of stable sets even for $\mathbb{Z}$-actions. (2) Second, we establish a dimensional entropy inequality for factor maps (Theorem 1.2). It relates dimensional entropy of a set to that of its image and topological entropy of fibres, and may be viewed as the dimensional-entropy counterpart of the factor-map inequality for packing topological entropy due to Dou, Zheng, and Zhou proved as Theorem 1.4 in [Ergodic Theory Dynam. Systems, 2023]. (3) Third, the relative topological entropy of a factor map is determined by the dimensional entropy of the fibres (Theorem 1.3). Notably, our proof of this formula (Theorem 1.3) is purely topological, in contrast to the recent measure-theoretic approach of Dou, Wang and Zhou based on relative Shannon–McMillan–Breiman theorems. These results (Theorem 1.2 and 1.3) not only generalize the work of Oprocha and the second author of the present paper [Nonlinearity, 2011] from single transformations to amenable group actions, but also provide a purely topological and self-contained proof of a fibre entropy characterization recently obtained through measure-theoretic arguments.

We investigate the realizations of Coxeter permutahedra which are also Coxeter matroid polytopes; these are polytopes of the form $\mathrm{conv}(W \cdot \mathbf{a})$ where $W$ is a finite Coxeter group acting on $\mathbb{R}^n$ and $\mathbf{a}$ is generic. Our main focus is how the geometric properties of $\mathrm{conv}(W \cdot \mathbf{a})$ change as $\mathbf{a}$ changes, with particular attention to persistent simplices, triangulations, and subdivisions.

In this paper, we extend Stein's method to the symmetric matrix normal distribution. In particular, we obtain a Stein characterization of the symmetric matrix normal distribution involving the extended generator of the symmetric matrix Ornstein-Uhlenbeck process, present a semigroup representation of the solution of the corresponding Stein equation, and establish regularity estimates for the solution. This framework of Stein's method for symmetric matrix normal approximation complements the recent theory of Stein's method for matrix normal approximation, and we make an explicit connection between these frameworks. We apply this theory to derive a Wasserstein distance bound for the symmetric matrix normal approximation of the Wishart distribution.

Finite fields are important algebraic structures that have a wide range of applications in fields such as coding theory and cryptography. But the standard construction of finite field extensions through polynomial quotients is computationally opaque, especially when we want to identify a degree-$2$ extension of $F_8$ and a degree-$3$ extension of $F_4$. In this short note, we present a coherent family of representations by matrices $\rho_q^n\colon F_{q^n} \to F_q^{n\times n}$ for all prime powers $q$ and all degrees $n \ge 1$. These maps are chosen so that concatenating $\rho_{q^n}^m$ and $\rho_q^n$ recovers $\rho_q^{nm}$ up to row and column permutations. As a consequence, the images of $\rho_2^6$ can be partitioned into four $3 \times 3$ blocks or nine $2 \times 2$ blocks to visualize the subfield chains $F_{64} / F_8 / F_2$ and $F_{64} / F_4 / F_2$ at the same time. A variant $\varrho$ is also discussed, wherein the Frobenius automorphism is represented by a cyclic shift of rows and columns. From an educational point of view, these rhos give explicit and self-contained mental models of finite fields; subfields, trace, norm, minimal polynomial, and Frobenius all become visible through matrix algebra accessible to most students. From a theoretical point of view, the construction exhibits structural implications of Conway polynomials and the normal basis theorem.

In a previous paper, we introduced the notion of hyper swap structures, a novel class of hyperalgebras that naturally generalizes swap structures semantics. In this paper we introduce the concept of hyper Boolean algebras based on Morgado hyperlattices, proving some basic properties. From this, we show that several paraconsistent logics in the hierarchy of Logics of Formal Inconsistency (LFIs) can be naturally characterized in terms of hyper swap structures semantics generated by hyper Boolean algebras. Finally, for each of these LFIs we obtain a Kalman-style functor which establishes an equivalence between the category of hyper Boolean algebras and a category of hyper algebras for the corresponding LFI having the hyper swap structures as representative objects.

We study the Fischer-Muszély functional equation for the positive semidefinite and the positive definite cones of unital $C^*$-algebras. We show that any bijection between the positive semidefinite cones satisfying the Fischer-Muszély equality extends to a Jordan $*$-isomorphism followed by multiplication on both sides by a positive element. As a corollary, we obtain a similar result for the positive definite cones of unital $C^*$-algebras.

In this partially expository paper, we present a novel construction of differentially closed fields of characteristic $0$: Let $\mathcal{K}_{\mathrm{dense}}$ be the differential ring of all meromorphic functions whose domain is a (not necessarily connected) dense open subset of $\mathbb{C}$ modulo agreement on dense open sets (i.e., $f$ and $g$ are considered equal if there is a dense open $U \subseteq \mathbb{C}$ such that $f|_U = g|_U$). We show that every ring ideal of $\mathcal{K}_{\mathrm{dense}}$ is a differential ideal and that for every maximal ideal $\mathfrak{m}$, the quotient $\mathcal{K}_{\mathrm{dense}}/\mathfrak{m}$ is a differentially closed field. We also show that $\mathcal{K}_{\mathrm{dense}}/\mathfrak{m}$ is saturated and has cardinality of the continuum, implying that any two such quotients are isomorphic as differential fields. We then discuss how to motivate this construction in terms of set-theoretic forcing, Boolean-valued models, and $\neg\neg$-sheaves on $\mathbb{C}$, taking the opportunity to present an impressionistic expository account of these ideas. Finally, we discuss some immediate generalizations of this construction involving the real and $p$-adic numbers and ask some questions about them.

We prove an asymptotic formula for a weighted Riesz mean of Hurwitz class numbers and real quadratic class numbers. To do this, we introduce L-functions for weight $\frac {1}{2} $ sesquiharmonic Maass forms of moderate growth and prove a formula for the Riesz means of the corresponding generalized mock modular forms, generalizing a recent result of the first author with Diamantis, Gupta, Rolen, and Thalagoda for mock modular forms. We then apply this formula to a sesquiharmonic Maass form that was first introduced by Duke, Imamoḡlu, and Tóth.

Thermal simulation is becoming increasingly critical in modern IC design and manufacturing. Feynman–Kac random walk methods can estimate localized temperatures without solving the global field, but in practical settings without Dirichlet boundaries they require very long paths to control the truncation residual. This paper presents FasterRW, a continuation-value framework for accelerating Feynman–Kac thermal random walks. We prove a stopping-time residual identity showing that the expected tail after truncation equals the current attenuation weight multiplied by the temperature at the stopping location. This identity enables a prior-corrected single-point estimator: a cheap, noisy prior replaces the unknown continuation value, making the truncation bias scale with prior error rather than absolute temperature and yielding an error-controlled threshold-selection rule. For multi-point queries, FasterRW further converts cross-query pass-through tails into pseudo-samples and refines the resulting observations in prior-error coordinates through kriging. Compared with prior state-of-the-art methods, FasterRW achieves up to $28\times$ acceleration while maintaining accuracy. Our code is open-sourced at https://github.com/ShiningSord/FastRW.

Conically averaged mappings, a generalization of averaged mappings, are important in a wide range of Optimization Algorithms. In this paper, we propose the modulus of conical averagedness to classify conical averaged mappings. Introducing the monotone and comonotone values of generalized monotone mappings, we investigate their connections to the modulus of conical averagedness. In the linear setting, we completely characterize conically averaged matrices, and derive explicit and pleasing formulae for computing their modulus of averagedness. As applications, we compute the Dixmier and Friedrichs angles between two subspaces. Nonlinear results are established as extensions of the linear case. Conical averagedness of proximal and reflection mappings of hypoconvex functions are also studied.

Bayesian and multiplicative-weights updates reweight experts, models, or actions from sequential feedback. We show that the regret of any such update obeys an exact information-accounting identity. On each round, the learner's excess loss to any chosen comparator is the sum of an immediate cost for the uncertainty exposed by the round and a reduction in the information distance from the learner's current weights to the comparator. The cumulative cost defines a pathwise uncertainty clock, the intrinsic time of the realized sequence. Summing one-step balances yields two exact adaptive decompositions of cumulative regret, one for each natural way of composing the update across rounds. Because the decompositions are exact, favorable stochastic or low-noise regimes appear as self-bounding properties of the realized intrinsic time. The accounting also fixes a learning rate, inverse in the square root of intrinsic time. That schedule is competitive with adaptive baselines in selected online-learning settings. The same calculus covers Hedge, optimistic and side-information variants, continuous priors, boosting, online convex optimization, contextual bandits, and repeated games: the pathwise account is the same in every case.

The Karhunen–Loève transform (KLT) diagonalizes the covariance of a second-order process and is optimal for mean-square truncation. Which classical transform it reduces to is governed by the symmetry commutant of the covariance: when the kernel commutes with a group action, the KLT eigenfunctions are the irreducible representation functions of that group, recovering the Fourier, cosine, Mellin, and spherical-harmonic systems. We study the inverse question. Given a covariance $R$ and a finite-dimensional space of candidate generators, the generator nearest to commuting with $R$, the minimizer of $\delta(A,R)=\|[R,A]\|_F/(\|R\|_F\|A\|_F)$, is the smallest-eigenvalue solution of a double-commutator eigenvalue problem $\mathrm{ad}_R^2(A^\ast)=\lambda A^\ast$, a Hermitian generalized eigenvalue problem of size the number of generators, independent of dimension. The framework recovers hidden transforms as well as classical ones: a variational characterization turns the existence of a commuting generator into a spectral condition, and a tridiagonal commutant-uniqueness result yields the prolate spheroidal, cosine, and discrete orthogonal-polynomial bases as exact recoveries, with matrix-valued extensions, and produces a continuum of transforms interpolating between and beyond the classical families. When symmetry is approximate, the coding penalty of the symmetry-adapted blockwise transform equals the multi-information among the sectors, an exact threshold between the fixed and data-driven transforms. We further give a graph-automorphism characterization of permutation structure, a sequential deflation for non-Abelian symmetry, and stability bounds under estimation error. As an application, the KLT of a two-paradigm covariance is synthesized from its two known generators, without forming the mixed covariance, reaching the full-data transform's compaction from few observations.

Value-of-information (VOI) analysis is usually conducted under a single probability measure. However, in practice, the available evidence often pins the measure down only to a set. Consequently, under a set of probability measures, VOI requires different formulations. First, we explicate a rule-specific VOI that fixes a decision rule for acting under imprecision (such as Gamma-maximin) and measures what the information is worth to a decision maker who uses that rule. Second, we derive a fixed-measure envelope that evaluates the classical VOI functional over all admissible precise measures. We formalize this distinction and explicate its consequences for the expected perfect, partial, and sample information. The expected value of perfect information is concave over the credal set. Hence, when the set is generated by finitely many measures, its lower envelope endpoint is obtained exactly from the generators, while its upper endpoint may be interior and is computed by a finite linear program. The Gamma-maximin value, in contrast, can exceed the entire envelope, so a rule-specific value is not recovered from the envelope's endpoints. A continuity bound limits how much the VOI can change as the measure varies, and we identify when the partial- and sample-information endpoints can still be obtained from the generators. Because the single-measure VOI must itself be estimated, the procedure we give combines standard estimators for it with a search over the credal set. By using a worked decision problem, we show how the two quantities separate conclusions that hold across every admissible measure from conclusions that depend on one unidentified choice of measure.

Accurate and timely channel state information (CSI) is essential for next-generation wireless systems, yet existing works treat CSI compression and CSI prediction as separate problems, both in academia and in current 3GPP studies. Consequently, channel aging remains insufficiently addressed within standardized CSI feedback pipelines. In this article, we propose a unified compression-prediction framework that integrates Contrastive Predictive Coding (CPC) directly into the 3GPP-compliant CSI compression architecture. Instead of predicting high-dimensional CSI matrices, our approach forecasts future latent representations and jointly optimizes reconstruction fidelity and temporal predictive coherence via a combined 1-SGCS and InfoNCE objective. This design enables temporal representation learning without increasing feedback overhead. We present two variants: CPC-before-Compression, which performs autoregressive modeling on encoded features prior to quantization, and CPC-after-Compression, which shifts temporal modeling to the base-station to reduce the complexity of users' devices. Evaluations on 3GPP-compliant datasets from Nokia, Oppo, and CATT show that CPC-before-Compression achieves over 90% reconstruction accuracy with 32x lower decoder GFLOPs than the 3GPP baseline, while CPC-after-Compression preserves an identical encoder footprint and the same 64-bit feedback overhead. By unifying compression and prediction within a standardized pipeline, the proposed framework provides an age-aware, computationally efficient CSI feedback solution. The source code is publicly available at: https://github.com/AhmedRadwan02/cpc-3gpp

In this paper, we study the group randomness of 0-1 real sequences derived from a binary linear code by investigating the spectral behaviour of a suitable normalization of the Gram matrix of a $p \times n$ random matrix whose rows are uniformly drawn from those 0-1 real sequences, where $y=p/n \in (0,1)$ is fixed. We show that as $n \to \infty$, its empirical spectral distribution converges to the Marchenko-Pastur law at a rate at least of the order $n^{-1/4}$ with high probability, and the fluctuation of its largest eigenvalue is asymptotically Gaussian with mean $p+1+y$ and variance $4y$, provided that the dual distance of the code is at least 5.

Rate-distortion (RD) theory and its related formulations play a central role in understanding efficient information representation, but computing these quantities remains challenging in high-dimensional settings. Classical iterative methods such as the Blahut-Arimoto algorithm become impractical in high-dimensional domains due to the curse of dimensionality and the intractability of mutual-information terms. Recent advances in neural modeling and differentiable optimization offer a promising alternative through a learn-to-compute paradigm, in which probability distributions and objective functionals are represented by flexible neural parameterizations. This survey presents an overview of neural approaches for evaluating the RD-type objectives. We present three representative families of methods: variational inference, neural mutual-information estimation, and dual-form optimization. By reviewing their theoretical principles, algorithmic techniques, and consistency properties, we elucidate how these methods collectively transform classical RD-type problems into scalable differentiable objectives suitable for deep learning, though challenges remain in large-scale applications. Together, these perspectives offer promising avenues for scaling information-theoretic computation to complex, high-dimensional machine learning systems.

Electroelastic shells are widely used in soft actuators, sensors, and energy harvesters owing to their large electrically induced deformations. However, the accurate simulation of their complex nonlinear multiphysics coupling, including bifurcation and post-buckling responses, remains challenging. This work presents an isogeometric Kirchhoff-Love shell formulation for the nonlinear analysis of electroelastic thin structures undergoing finite deformations. The formulation incorporates geometrically nonlinear kinematics, Maxwell-stress-induced electromechanical coupling, material incompressibility, and initial prestretch. Catmull–Clark subdivision surfaces are employed to ensure the C1 continuity required by Kirchhoff–Love shell theory. Consistent tangent operators are derived analytically, and a static condensation procedure is introduced to satisfy the plane-stress constraint. To trace bifurcation and post-buckling equilibrium paths, a staged Newton–Raphson algorithm with arc-length continuation and eigenmode perturbation is adopted. Numerical examples involving spherical membranes, prestretched circular plates, and toroidal membranes demonstrate the capability of the proposed framework to accurately capture large deformations, symmetry-breaking instabilities, and post-buckling responses under coupled electromechanical loading.

Let $g \ge1$. A set $\mathcal{A}$ of nonnegative integers is a Sidon set if for each $d>0$ there is at most one pair $(a,b) \in \mathcal{A} \times \mathcal{A}$ with $d=a-b$. If there are at most $g$ pairs, then $\mathcal{A}$ is a $g$-Golomb ruler. We prove that if $\mathcal{A}$ is a $g$-Golomb ruler, then \[\liminf_{n\to\infty} \frac{ | \mathcal{A}\cap[0,n) | }{\sqrt{n/\log n}} \le \frac{2\sqrt g }{\sqrt{\log 2}},\] generalizing and sharpening results of Erdős and Cilleruelo. There is a $g$-Golomb ruler $\mathcal{G}$ with \[\frac{\sqrt g }{\sqrt2} \le \limsup_{n\to\infty} \frac{ | \mathcal{G}\cap[0,n) | }{\sqrt n} \le \sqrt{g } ,\] generalizing a result of Krückeberg.

We study the exponential rate $r(\alpha,\lambda)$ of the energy $\mathcal{E}_N$ needed to steer a far site, at distance $N$, of an Aubry–André chain $H_\lambda$ via one boundary actuator with closed-loop margin $\alpha$. An exact eigenbasis reduction writes $\mathcal{E}_N$ as a Cauchy quadratic form $\tilde b^\top C^{-1}\tilde b$ in the boundary-amplitude ratios, whose rate is the off-spectral Lyapunov exponent of the transfer cocycle at the reflected band edge $z^\star=2E_{\min}-2\alpha-E_{\max}$, giving $r(\alpha,\lambda)=\gamma_\lambda(z^\star)$. The rate lies in a bracket of width $\log_+(\lambda/2)$ whose ends coincide for $\lambda\le2$, the spectrum having logarithmic capacity $\max(1,\lambda/2)$. We prove the identity unconditionally, for every phase, on the whole metallic–critical range $0<\lambda\le2$: for $\lambda<2$ through subcritical almost reducibility as the sole external input, and at $\lambda=2$ because the Green's function there equals the Lyapunov exponent. For $\lambda>2$ the upper bound is unconditional, and the lower bound takes localization as its only external input: an inverse-free cocycle form makes $\mathcal{E}_N$ a cancellation-free positive sum, and a Christoffel–Darboux identity collapses its coefficients to $|c_k|=Q(\delta_k)(\hat\psi^{(k)}_N)^2$, where band-edge near-degeneracies cancel. With a three-distance lemma this yields $r=\gamma_\lambda(z^\star)$ at every Diophantine frequency and almost every phase, with gap $O(N^{-2/(2+\tau)})$ for type $\tau$ ($N^{-2/3}$ at bounded type), unconditionally for $\lambda\ge\lambda_1$ and under a polynomial-prefactor localization hypothesis for $2<\lambda<\lambda_1$. The relative gap $1-r/\gamma_\lambda(z^\star)$ vanishes at both ends of the localized phase, with $g_{\mathbb{C}\setminus\Sigma_\lambda}(z^\star)\to\operatorname{arccosh}3$ as $\lambda\to\infty$.

Let $Q^{(k)}_N$ be an $N\times N$ matrices with entries satisfying CAR, normalized to have variance $1/\sqrt{N}$ with respect to the trace of the CAR algebra. We show that, although the operator norm of the real part of an individual matrix $Q^{(k)}_N$ converges as $N\to\infty$ to the semicircular limit, the family of matrices does not converge to the free probability limit strongly. In fact, even the operator space structure of the linear spans of the real and imaginary parts of $Q^{(k)}_N$'s, $k=1,\dots,M$, does not converge to the semicircular limit.

We study $[\phi_t , X]$, the maximal space of strong continuity for a semigroup of composition operators induced by a semigroup $\{\phi_t\}_{t\ge0}$ of analytic self-maps of the unit disk, when $X$ is BMOA, $H^\infty$ or the disk algebra. In particular, we show that $[\phi_t,\text{BMOA}] \neq \text{BMOA}$ for all nontrivial semigroups. We also prove, for every semigroup $\{\phi_t\}_{t\ge0}$, that $\lim_{t \to 0^+} \phi_t(z) = z$ not just pointwise, but in $H^{\infty}$ norm. This provides a unified proof of known results about $[\phi_t , X]$ when $X \in \{H^p, A^p, \mathcal B_0, \text{VMOA}\}$.

Considering pointwise and sup-norm estimation, we analyze the non-asymptotic behavior of local averaging estimators for Lipschitz regression functions. Building on a general deviation bound for estimators based on a VC family of localizing sets, we introduce the notion of shape-regular local maps, where averaging is performed over sets with an almost isotropic geometry. Our main message is a characterization: shape regularity is both necessary and sufficient to attain optimal rates, up to logarithmic factors. Necessity is established non-asymptotically through an explicit anisotropic example, sharpening a phenomenon previously understood only heuristically in asymptotic theory. We then draw two consequences. First, the simple $k$-nearest neighbor rule is shape-regular by construction and attains the optimal rate, even on unbounded supports. Second, and perhaps surprisingly, the popular random-split condition for trees – known to ensure consistency and vanishing cell diameters – does not guarantee optimal rates: for blind tree constructions, the cell aspect ratio diverges exponentially with depth, so that shape regularity fails with positive probability. This identifies the absence of a geometric correction mechanism, rather than a slowly shrinking diameter, as the obstruction to optimality. Motivated by this gap, we propose a tree construction that enforces shape regularity through a simple constraint on admissible splits, and prove a uniform deviation inequality showing that it restores the optimal rate for Lipschitz functions.

We formulate a continuity equation for the Schrödinger equation in the complex space. We define a complex momentum by normalizing the complex current by the particle density. This momentum is a quantum analog of the classical, kinematic momentum analytically continued into the complex plane. The kinematic momentum and the gradient of the wavefunction's phase each represent a fluid-like flow in the complex plane; the phase-gradient flow is incompressible. The zeros of the wavefunction give rise to simple poles in the momentum. The poles manifest as irrotational vortexes in the phase-gradient flow, while critical points of the wavefunction present as rigid body-like rotational flows of the kinematic momentum. A discrete nature of elementary excitations comes about inherently because the quantity of the poles is automatically integer. An exact quantization condition is subsequently formulated, which reduces to the Bohr-Sommerfeld condition in the semiclassical limit. We establish a priori that the Bohr-Sommerfeld condition must be exact for the Harmonic Oscillator. We show that the kinetic energy is a sum of contributions of the average value and fluctuations, respectively, of the kinematic momentum. The zero-point vibrations within bound states are solely due to the fluctuations of the momentum and manifest as rigid-body flows at infinity. The momentum poles – and hence the wavefunction's zeros – can be viewed as emergent, consistent with the remarkable property of quantum entanglement exhibited by standing wave solutions of the Schrödinger equation.

Our main result is a characterization of $g$ for which the operator $S_g(f)(z) = \int_0^z f'(w)g(w)\, dw$ is bounded below on the Bloch space. We point out analogous results for the Hardy space $H^2$ and the Bergman spaces $A^p$ for $1 \leq p < \infty$. We also show the companion operator $T_g(f)(z) = \int_0^z f(w)g'(w) \, dw$ is never bounded below on $H^2$, Bloch, nor BMOA, but may be bounded below on $A^p$.

More than forty years ago, Andersen and Hoffman independently proved that every symmetric Latin rectangle can be extended to a symmetric Latin square with prescribed diagonal entries. We generalize this theorem as follows. Let $k\leq n^2$, and let $M$ be an $n\times n$ array whose top-left $r\times r$ subarray is filled with symbols from $\{1,2,\ldots,k\}$. Suppose that, for each $i\in\{1,\ldots,r\}$ and each symbol, the number of occurrences of that symbol in row $i$ equals its number of occurrences in column $i$, and that each remaining diagonal entry is either empty or already contains a symbol from $\{1,\ldots,k\}$. We establish necessary and sufficient conditions for completing $M$ so that the resulting array is symmetric off the prescribed $r\times r$ subarray, each symbol occurs a specified total number of times in $M$, and, for every symbol, its numbers of occurrences in any two rows (respectively, columns) differ by at most one. Restricted to symmetric arrays, our theorem generalizes results of Cruse (1974), Goldwasser et al. (2015), and Bahmanian and Hilton (2025). It also extends Baranyai's theorem for complete graphs (1973) by characterizing when a partial coloring of $K_r$ with a loop on every vertex can be extended to an almost regular coloring of $K_n$ with a loop on every vertex, where $n\geq r$.

Robust Markov decision processes provide a principled framework for protecting sequential decision-making against transition-law misspecification and have attracted substantial recent research interest. As in non-robust stochastic optimal control, an important question is whether the robust value function is sufficiently regular for approximation and learning. This paper studies Lipschitz regularity of optimal value functions for Wasserstein robust stochastic optimal control on possibly unbounded Polish state spaces under an infinite-horizon discounted reward criterion. We consider two robustness formulations: a kernel-robust model, in which the adversary perturbs the next-state distribution within a Wasserstein ball around a nominal transition kernel, and a noise-robust model, in which the adversary perturbs the driving noise of the state transition dynamics. In the non-robust setting, Lipschitz rewards and Lipschitz transition dynamics do not, in general, imply a Lipschitz value function. In contrast, we show that in these Wasserstein robust formulations, Lipschitz assumptions on the model primitives yield Lipschitz robust value functions. Thus, Wasserstein robustness not only protects against misspecification but also regularizes the Bellman fixed point, providing stability relevant to discretization, value-function approximation, estimation, and learning.

We study the maximal displacement of a one-dimensional subcritical branching random walk with offspring distribution $\{p_k\}$ and step size $X$ such that $m := \sum_{k=1}^\infty k p_k \in (0,1)$. Let $M_n$ denote the maximal position of all particles alive at time $n$ and let $M := \sup_{n \in \mathbb{N}} M_n$. First, we show that \[ \lim_{x \to +\infty} \frac{e^{\lambda x^b}}{\ell(x) x^a } \, \mathbb{P}(M > x) = \frac{1 - p_0}{1 - m} \] whenever $\mathbb{P}(X > x) = \ell(x) x^a e^{-\lambda x^b}$ for some slowly varying function $\ell$, $b \in [0,1)$, and under further assumptions on $a$. Next, we prove that \[ \lim_{x \to +\infty} \frac{e^{\lambda x^b+\gamma x}}{\ell(x) x^a } \, \mathbb{P}(M > x) \quad \text{exists and belongs to } (0, \infty) \] provided that $\sum_{k=1}^\infty k (\log k) p_k < \infty$ and for some $x_*>0$, $\mathbb{P}(X > x) = \int_x^\infty \ell(y) y^a e^{-\lambda y^b - \gamma y} \, \mathrm{d}y$ for all $x > x_*$. Here, $\ell$ is a slowly varying function, $m \mathbb{E}(e^{\gamma X}) < 1$, $b \in [0,1)$, and $a$ satisfies certain conditions.

We study the family of complex rational functions known as Generalized McMullen maps, F(z) = z^n + a/z^n+b, for integer n at least 3 fixed, and complex parameters a, b with a nonzero. In prior work by the same authors, we provided a combinatorial model for a large class of maps whose Julia sets contain both infinitely many homeomorphic copies of quadratic Julia sets conjugate to the “basilica”, and infinitely many subsets homeomorphic to a set which is obtained by starting with the basilica, then changing a finite number of pairs of external ray landing point identifications, following an algorithm we described. In this article, we generalize beyond the basilica, and provide a catalog of additional types of hyperbolic Julia sets of Generalized McMullen maps, where the “baby” Julia set can be any rabbit, aeroplane, or Kokopelli quadratic Julia set; that is, where the c-value can be taken from any bulb attached to the main cardioid of the Mandelbrot set, or from the main cardioid of any principal baby Mandelbrot set (no renormalizations).

The cart-pole swing-up is a canonical benchmark for nonlinear control of underactuated systems, yet an end-to-end guarantee linking the global swing-up maneuver to the local stabilizer is seldom formalized. We present a reachability analysis of a switched energy-based/LQR controller that certifies convergence to the upright equilibrium from a compact set of initial conditions. The swing-up design exploits the phase-space geometry of the conservative pendulum: the upright equilibrium lies on the homoclinic orbit, and an energy-shaping law drives the energy error to zero, steering the pendulum onto this orbit; convergence follows from LaSalle's invariance principle. An augmented Lyapunov function additionally regulates the steady-state cart velocity to zero, and we prove almost-global convergence of the resulting closed-loop system. A local LQR with a certified ellipsoidal region of attraction stabilizes the upright equilibrium, and we verify numerically that the swing-up phase delivers the state into this region, formalizing the handoff. Numerical simulations corroborate the theoretical analysis.

Prediction-powered inference (PPI) refers to a two-level situation where the statistician observes a set of $(x,y)$ pairs and another set of $x$s with the responses $y$ missing. Also available is some independent background data from which a prediction rule $f(x)$ has been produced, perhaps by a machine learning algorithm; $f(x)$ approximates $E\{y\mid x\}$ but there is no guarantee of its accuracy for the situation at hand. Angelopoulos et al. (2023a) developed an algorithm that makes use of all the data, including the unlabeled $x$s, for the estimation of a parameter of interest. A different algorithm is proposed here, using the bootstrap to avoid asymptotics, that is shown to have advantages of efficiency and generality. It is similar in spirit to the original PPI paper by Wang, McCormick and Leek (2020). Prediction-powered inference raises questions about the information available in unlabeled data, with some surprises here, particularly concerning the estimation of the expected value of $y$.