Papers

Let $G$ be a connected semisimple real Lie group, $\Gamma$ an irreducible lattice in $G$ and $X = G/\Gamma$. Let $F = \{g_t: t\ge 0\}$ be a non-quasiunipotent one-parameter subsemigroup of $G$. Then it is known that the set of points in $X$ with bounded $F$-trajectories has full Hausdorff dimension. In addition, if $U$ is the expanding horospherical subgroup relative to $g_1$, then for any $x \in X$ the set of points $u \in U$ such that the $F$-trajectory of $ux$ is bounded has full Hausdorff dimension. In this paper we take $U$ to be a horospherical subgroup of $G$ and apply Shi's equidistribution theorem for elements of the expanding cone with respect to $U$ to describe a class of subsets $F$ in $G$, not presupposing the group structure, for which the above full Hausdorff dimension statements also hold. As an application, we prove that the set of badly approximable matrices in the set-up of Diophantine approximations with quasimultiplicative weight functions has full Hausdorff dimension.

We study distributionally robust linear chance-constrained problems in which uncertainty is modeled by a Gaussian mixture model (GMM). Finite-support distributionally robust (FDR) formulations, widely used in data-driven robust optimization, robustify over empirical mixture support points and therefore primarily stress-test the fitted nominal mixture. This can be insufficient when service reliability depends on structural misspecification of the nominal mixture-support parameters. To address this limitation, we describe the ambiguity set of distributions by developing a novel formulation of a Wasserstein-2 metric that uses the Bures-Wasserstein (BW) metric over probability measures with finite second moments. Unlike FDR, which generally sets finitely many empirical support points a priori, the proposed ambiguity set allows the worst-case distribution to endogenously determine both how many mixture components receive mass and where their means and covariances lie within a continuous support. For the resulting ambiguity set, under mild regularity conditions, we prove strong duality for the inner worst-case chance-constraint problem and derive its semi-infinite reformulation. We then develop an adaptive cutting-surface algorithm, which endogenously determines the locations of mixture components receiving mass, and the mean and covariances of the Gaussian distributions at these locations. The algorithm attains any prescribed optimality gap in finitely many iterations, while a block-alternating local search identifies new components. A case study using the electric-vehicle charging-station energy-allocation problem demonstrates the framework's practical value in achieving any reliability targets. CDR also induces structural changes in energy allocations, unlike FDR, whose allocations remain close to the nominal solution.

In this paper, we study the local polynomial convexity of certain smooth real surfaces in \(\mathbb{C}^2\) with isolated CR singularity at the origin with higher-order of degeneracy. Under the assumption that the surface can be pulled back to a union of finitely many pairwise transverse totally real surfaces by a proper holomorphic map from $\mathbb{C}^2$ to $\mathbb{C}^2$, we obtain a normal form for such surfaces near the origin as $\{(z,w)\in\mathbb{C}^2: w= \overline{z}^k+o(|z|^{k})\}$ or $M_t := \left\{ (z,w)\in\mathbb{C}^2 : w=(z+t\overline{z})^k+o(|z|^k) \right\}$, for some \(t>0\), where the parameter $t$ is a local biholomorphic invariant. We focus on the surfaces with order of degeneracy $k\geq 3$. We prove that $M_t$ is locally polynomially convex at the origin if $t>cosec\left(\frac{\pi}{k}\right)$. On the other hand, for $0<t<\frac{1}{k-2}$, we will also show that $M_t$ fails to be locally polynomially convex at the origin; and furthermore, a $(2k-3)$-parameter family of analytic discs attached to $M_t$ for $0<t<\min\left\{\sin\left(\frac{\pi}{k}\right),\frac{1}{k-2}\right\}$.

This paper further studies the model companion of an endomorphism acting on a vector space, possibly with extra structure. Let $T$ be a model-complete theory that $\varnothing$-defines an infinite $K$-vector space $\mathbb{V}$. In previous work, we introduced a family $\{T^C_\theta : C \in \mathcal{C}\}$ of extensions of the theory $T_\theta := T \cup \{\text{``$\theta$ is an endomorphism of $\mathbb{V}$''}\}$ that parameterizes all consistent extensions of the form $$ T_\theta \cup \left\{\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(\rho_{j, k, l}[\theta]) = \sum\nolimits_{k}\bigcap\nolimits_{l} \operatorname{Ker}(\eta_{j, k, l}[\theta]) : j \in \mathcal{J}\right\}, $$ where all sums and intersections are finite, all the $\rho[\theta]$'s and $\eta[\theta]$'s are polynomials over $K$ with $\theta$ plugged in, and $\mathcal{J}$ is some possibly infinite index set. We also presented a sufficient condition that implies that every $T^C_\theta$ has a model companion $T\theta^C$. We simplify our axiomatization of $T\theta^C$ and the criterion for its existence for theories “close to the theory of $K$-vector spaces”. We apply this to the explicit case where $T$ is the pure theory of $K$-vector spaces and characterize all $\varnothing$-definable endomorphisms of $\mathbb{V}$ in this case. Given an existentially closed model $(\mathcal{M}, \theta) \models T^C_\theta$ and a polynomial $\rho\in K[X]$, we show that $(\mathcal{M},\operatorname{Ker}(\rho[\theta]))$ is, unless $\operatorname{Ker}(\rho[\theta]) = \{0\}$ or $\operatorname{Ker}(\rho[\theta]) = \mathbb{V}$, an existentially closed model of $T_V := T \cup \{\text{``$V$ is a vector subspace of $\mathbb{V}$''}\}$. In the same vein, we present a criterion for when $(\mathcal{M}, \rho[\theta])$ is again an existentially closed model of $T^{C'}_\theta$ for some $C' \in \mathcal{C}$.

In this article we propose a theory of Chow-Witt groups, in the paradigm of Quillen's arrow based approach to $K$-theory. We refer to this style of arguments as arrowtic paradigm. Other than establishing the machinery needed to work in this paradigm, we establish the homotopy invariance property of the Arrowtic Chow-Witt groups. We also compute the arrowtic Chow-Witt groups of the projective spaces, over fields.

We study viscosity solutions for a class of degenerate one-phase free boundary problems of the form $w\Delta w = h(\nabla w)$. We assume the existence of a star-shaped domain $D$ such that $h < 0$ in $D$, $h = 0$ on $\partial D$, and $h > 0$ in $\bar{D}^{c}$. This class of degenerate one-phase free boundary problems arises when a canonical transformation is performed to a semilinear equation $\Delta u = f(u)$, and $f$ behaves like $-\gamma u^{-(\gamma + 1)}$ for some $\gamma \in (0,2)$. In this case, known as the Alt-Phillips equation for negative power potentials, $h(\rho) = c(|\rho|^2 - 1)$. We show existence of a viscosity solution, Lipschitz regularity, and regularity of the free boundary at flat points. Additionally, we show that as $\gamma$ converges to $2$, the free boundary converges to a minimal surface.

We present a trajectory planning method that balances information collection and control effort to improve cislunar spacecraft-to-spacecraft absolute tracking. Expanding use of cislunar space requires alternative navigation and tracking procedures that minimize reliance on ground-based support. Among efforts to address this need, spacecraft-to-spacecraft tracking exploits nonlinear dynamical tracers encoded in a series of relative measurements to infer absolute states of both an observer and a target. The geometry between spacecraft operating under this mode can significantly influence tracking performance. This work considers this geometrical impact by designing observer trajectories that jointly balance control effort and an information-theoretic quantification of the expected spacecraft-to-spacecraft tracking performance. Our methods are designed for multiple low-thrust observation platforms of various sensing modalities and can incorporate multiple space object targets in planning. By leveraging information gain in the optimal control problem, we report almost an order of magnitude improvement in the expected navigation and tracking errors for an optical observer operating in a Distant Retrograde Orbit (DRO). This work demonstrates the feasibility and value of information-optimal low-thrust spacecraft trajectory design for current and upcoming cislunar missions.

We study Brenier's relaxed least-action problem in the unit disc $D:=\{x\in\mathbb{R}^2:\ |x|<1\}$ at the critical time $T=\pi$, with the endpoint pair $i_D$ (the identity on $D$) and $-i_D$. We prove that the critical energy shell \[ S^3=\{(x,v)\in\mathbb{R}^2\times\mathbb{R}^2: |x|^2+|v|^2=1\} \] supports stationary action-minimising generalised incompressible flows that are not invariant under physical rotations. This answers the question posed by Bernot, Figalli, and Santambrogio. Our construction relies on the normalised surface measure $\sigma$ on $S^3$ and the Hopf quotient \[ \Pi=(N,M,L):S^3\to\mathbb S^2_{1/2}, \qquad \mathbb S^2_{1/2}:=\{(n,m,\ell)\in\mathbb{R}^3: n^2+m^2+\ell^2=1/4\}. \] Here, \[ N=\frac12(x_1^2+v_1^2-x_2^2-v_2^2),\quad M=x_1x_2+v_1v_2,\quad L=x_1v_2-x_2v_1 . \] This quotient is a first integral of the harmonic-oscillator flow. Let $\tau:=\Pi_\#\sigma$, let $g$ be a bounded Borel function on $\mathbb S^2_{1/2}$ whose $L^\infty(\tau)$-class is odd under $(n,m,\ell)\mapsto(n,m,-\ell)$, and let $\delta\in\mathbb{R}$. If $1+\delta g\circ\Pi\ge0$ $\sigma$-a.e., then \[ d\mu_{\delta,g}=\pi(1+\delta g\circ\Pi)\,d\sigma \] has Lebesgue spatial marginal and is stationary. The induced path measure is then minimising. We also determine exactly which members of this tilted family are rotationally invariant. If $\delta\ne0$, this is equivalent to axisymmetry of the $L^\infty(\tau)$-class of $g$, modulo $\tau$-null sets. In particular, taking $g(n,m,\ell)=\ell m$ with $0<|\delta|<8$ gives strictly positive non-rotationally invariant minimisers.

Let $x_{0}$ be a positive integer, let $x_{n}=\lceil 3x_{n-1}/2\rceil$, and let $w_{n}=2x_{n+1}-3x_{n}\in\{0,1\}$ be the associated word studied by Dubickas; for $x_{0}=1$ the orbit is A061419 and $w$ is the minimal word $g_{3/2}$ of the rational base number system of Akiyama, Frougny and Sakarovitch. The orbit encodes a real constant $K=\lim_{n}x_{n}(2/3)^{n}$, equal for $x_{0}=1$ to $\omega_{3/2}=K(3)=1.6222705028\ldots$, whose irrationality has been open since 1977. We prove that if $w$ is automatic then $K$ is transcendental; equivalently, an algebraic $K$ forces $w$ to be non-automatic. Further we prove that either the complexity of $w$ exceeds every linear bound or $K$ is irrational. All results are formally verified in the Lean~4 proof assistant, on three cited axioms.

For the nearest-neighbour Edwards–Anderson Ising model on the discrete torus, we prove existence of the quenched thermodynamic limit and almost-sure self-averaging of the free-energy density. The only moment assumption in the main argument is $\E|J|<\infty$, and no symmetry or centering of the coupling law is needed. The proof avoids periodic subadditivity: the wrap-around bonds form a surface-order perturbation, while a tiling argument and the strong law of large numbers give the free-boundary limit. We also obtain the quantitative bounds $(O(L^{-1})$ for the disorder-averaged finite-volume correction. A precise common probability space is specified, since an almost-sure statement across volumes is otherwise not well defined.

We study a low-rank solver for the stochastic unsteady incompressible Navier–Stokes equations with uncertain viscosity. The problem is discretized by a stochastic Galerkin method and written in an all-at-once (monolithic) form where the time steps are informed by a sequential solve of the mean problem. The solution vector is represented in Tensor Train (TT) format, while the system matrices are represented in CANDECOMP/PARAFAC (CP) format as needed. We propose preconditioners based on the low-rank CP approximations that retain additional stochastic and nonlinear information compared to mean-based preconditioners. Numerical experiments for a benchmark channel-flow problem are presented to show the effectiveness of the low-rank approximation, the relative tolerance strategy, and the proposed preconditioners.

An Euler magic matrix is an integer matrix $M$ with $MM^{t}=\gamma I$ whose squared entries sum to $\gamma$ along both main diagonals; it is proper if its squared entries are pairwise distinct. Euler constructed an order-$4$ proper example, and Müller settled orders $3$ (none exist) and $8$, leaving order $5$ as the smallest open case. We construct such a matrix, by rotating one of Müller's "near-misses" under a mirror-symmetric coordinate pair so that the two diagonal conditions collapse to a single rational equation; the same invariant suggests a uniform approach to the odd orders.

Recently, stochastic pantograph models have gained an intensive attention and have been used in different fields such as finance, biology, control and stochastic neural networks. It is also more preferable to incorporate jumps during the study of stochastic differential equations. In this paper, stochastic pantograph model with Poisson random measure is studied. The compensated split-step theta technique is applied to the considered model. The numerical scheme exhibits a non divergent attitude and converges to the solution of our model under assumptions addressed later on. Furthermore, the almost sure exponential stability of the numerical scheme is investigated via utilizing the discrete semi-martingale convergence theorem. Finally, theoretical findings are manifested via some numerical examples.

The boundary correction to the free energy of a disordered Ising model depends on the boundary condition, the normalization, and the finite-volume sequence. We give a one-dimensional i.i.d. counterexample showing that the surface free-energy density need not be independent of the van Hove sequence and that the corresponding random correction need not converge in probability. For bounded couplings in the Dobrushin uniqueness regime on cubic boxes in dimensions $d\geq2$, we prove convergence of the normalized free-to-fixed boundary free-energy difference in expectation, almost surely, and in every $L^p$, $1\leq p<\infty$. For Gaussian boundary couplings, we derive an exact finite-volume interpolation identity and explain why it does not by itself imply a low-temperature surface limit. For a seam-flip free-energy difference $D_L$ across a set $S_L$ of independent symmetric bonds of variance $v$, we prove $\Var(D_L)\leq4v\abs{S_L}$; one-dimensional examples show that symmetry and finite moments alone do not determine a stiffness exponent, and the low-temperature Gaussian Edwards–Anderson problem remains open.

Let $k=k_0(t_1,t_2,\ldots)$ be a flexible field of characteristic different from $2$, let $\mathbb X$ be the mod-$2$ isotropic sphere, and set $E=\mathbb X \wedge MBP$. We prove $$Pic\bigl(SH(k/k)^c_{cell}\bigr)\cong\mathbb Z^2; $$ thus every tensor-invertible finite cellular isotropic spectrum is a unique bigraded suspension of $\mathbb X$. More generally, $E_{**}$ is conservative on finite cellular objects, and concentration on one diagonal forces a finite direct sum of suspended isotropic spheres. A bounded diagonal weight structure recovers the exact weights and minimal-complex terms from $E_{**}$. For every nonzero finite cellular $M$, the kernel of $$End(M)\longrightarrow End(E\wedge_{\mathbb X}M)$$ is a composition-nilpotent ideal, with exponent at most $d(M)(2L(M)-1)$, where $L(M)$ is the diagonal width and $d(M)$ the maximal number of distinct Tate degrees on one diagonal. This yields detection of composition nilpotence and canonical Fitting decompositions.

This paper presents a safety-critical distributed nonlinear model predictive control (DNMPC) framework for cooperative payload transportation by teams of quadrupedal robots. The proposed approach models the robotic team and the shared payload as a dynamically coupled networked system with rigid holonomic coupling constraints arising from cooperative transportation. To enable distributed real-time optimization, the centralized finite-horizon optimal control problem is decomposed into parallel local NMPC subproblems coordinated through the alternating direction method of multipliers (ADMM). The resulting distributed framework enforces consensus over both payload-state and interaction-wrench trajectories while explicitly incorporating acceleration-level holonomic coupling constraints within the distributed predictive control formulation. Safety-critical obstacle avoidance constraints for both the robotic agents and payload are enforced using higher-order control barrier functions (HOCBFs). The framework is validated through numerical simulations with teams of two, three, and four quadrupedal robots transporting shared payloads in cluttered environments. Real-time experiments on two- and three-robot teams demonstrate safe and robust transportation under payload uncertainty and external disturbances. Compared with centralized NMPC, the proposed framework achieves up to 23% reduction in average NLP solve time while maintaining comparable closed-loop performance. Ablation studies further demonstrate robustness to communication delays and show that explicit payload-state consensus and holonomic constraints substantially improve payload tracking and distributed coordination over existing wrench-only consensus formulations.

Master–slave synchronization reconstructs the evolving state of a reference (or master) system by continuously correcting a second system, the slave, using observations of the master. In applications, the observations are often contaminated by noise, so the corrections injected into the slave are themselves uncertain. In this paper we aim at quantifying the effect of noisy observations on master–slave synchronization for a family of evolution equations that combine a quadratic transport term with a polynomial linear differential operator. The family includes the Burgers, Kuramoto–Sivashinsky, Kawahara, Benney–Lin and Nikolaevskiy equations, among others. Under periodic boundary conditions, a Fourier–Galerkin approximation of order $K$ yields a finite-dimensional master–slave system in which white observation noise of intensity $\sigma$ is modeled through an Itô stochastic differential equation. In this framework, we first establish local exponential stability of noiseless synchronization for sufficiently strong scalar coupling. Then we compare the stochastic synchronization error with its noiseless counterpart resulting from the same initial slave state, isolating in this way the effect of noise. Assuming bounded reference paths, the mean-square deviation between the noiseless and noisy synchronization errors is $\mathcal O(\sigma^2)$ on finite time intervals. Under additional regularity assumptions, we also bound the error between the finite-resolution stochastic slave model and the infinite-dimensional deterministic master field, explicitly separating spectral approximation error, deterministic synchronization error and observation noise. Numerical experiments for the Kawahara and Nikolaevskiy equations confirm the predicted noise scaling.

We study entropy density for full shifts over amenable residually finite groups. Entropy density means that every invariant measure can be approximated in the weak$^*$ topology, together with its entropy, by measures from a distinguished family. In symbolic dynamics it is known, by a result of Weiss, that uniquely ergodic measures are entropy dense among ergodic measures for the shift action of $\mathbb Z$. We extend this result to full shifts over amenable residually finite groups: invariant measures supported on uniquely ergodic subsystems are entropy dense in the collection of ergodic invariant measures. This applies in particular to full shifts over finitely generated abelian groups. The proof uses Cortez–Petite Følner tilings and a block-replacement construction adapted to finite-index subgroups.

We consider incompressible FENE dumbbells coupled to a reversible Markov operator on a compact continuum of internal states. At zero centre-of-mass diffusion we prove that state averaging is an exact factor map and that a Lagrangian state lift is its unique inverse on the natural energy-solution classes. Thus existence, multiplicity, and uniqueness of the state-resolved system are precisely those of its scalar FENE projection; the internal-state dynamics creates no additional large-data nonuniqueness. The lift is driven by a trace-free matrix fibre evolution and permits nonlinear local activities, including rates with linear dependence on the singular Kramers stress.We also prove a sequential form of this structure. For state-resolved regularizations with relative-entropy-prepared initial fibres and no viscous dissipation defect, the complete densities converge strongly without reconstructing them after the scalar limit. Consequently the full drag,state-dependent activity, singular stress, and non-atomic Jeffreys production all pass to the limit. The argument combines stability of regular Lagrangian flows with a relative-entropy estimate for simultaneously varying matrix drifts and jump rates. A stationary oscillation shows that the preparation cannot follow from the natural entropy bounds alone. With positive centre-of-mass diffusion we independently construct global large-data weak solutions and identify the same nonlinear terms. An explicit infinite-rank kernel proves that these results are not finite-species reductions.

Over-the-air (OTA) computation has recently gained significant attentions as an effective approach to enhance the communication efficiency of wireless federated learning (FL). By enabling simultaneous transmission and aggregation of local model updates, OTA-FL can substantially reduce both latency and bandwidth consumption. However, a key challenge lies in the imperfect aggregation of global models caused by channel state information (CSI) uncertainty, which introduces distortion to the final learning performance. To address this issue, we study the long-term mean squared error (MSE) minimization problem for OTA-FL under imperfect CSI conditions. Through convergence analysis, we establish an upper bound for the time-averaged MSE, thereby revealing the effect of aggregation errors accumulated throughout multiple communication rounds on the overall training performances. Based on this analysis, an optimization framework is developed to minimize the long-term MSE via the joint design of (i) transmit coefficients at the local devices and (ii) receive combining vectors at the parameter server (PS). Since this alternating optimization approach requires non-causal CSI, a Lyapunov-based optimization method is further introduced to handle causal CSI scenarios. By incorporating virtual queues to characterize long-term energy consumption, the proposed method effectively decouples temporal dependencies and allows transmit coefficients to be optimized based on the causal CSI of each aggregation round. Comprehensive evaluations on Fashion-MNIST, CIFAR-10 and CIFAR-100 datasets have demonstrated that the proposed algorithms can significantly reduce the degradation of test accuracy caused by imperfect CSI. Comparisons with other benchmark schemes further verify the superiority of our proposed algorithms.

This paper addresses the multi-period log-truck routing and scheduling problem ($\mathcal{LTRSP}$), a key operational activity in the forestry industry, where transportation accounts for more than one-third of total operational costs. The Canadian forestry sector faces significant logistical difficulties driven by vast geographic distances, seasonal variability, volatile markets, and environmental considerations. Our research tackles a long-standing open question in the forestry operations literature, namely the absence of an exact and scalable formulation of the forestry vehicle routing problem integrating the full range of operational constraints \cite{ronnqvist2015operations}: How can we model and solve an exact formulation of the forestry VRP problem? In response, we analyze business rules specific to the forestry sector to construct a routing network reflecting real operational practices, and then propose a comprehensive improved mixed-integer linear programming (MILP) formulation incorporating all known forestry operational constraints, with a detailed justification of key modeling choices such as time discretization, along with a decomposition-based solution methodology tailored for large-scale, multi-period industrial instances. To address the inherent combinatorial complexity, we combine state-of-the-art solvers with a metaheuristic decomposition strategy based on Relax&Fix and Fix&Optimize. Computational experiments on historical data from a Canadian forest company demonstrate near-optimal results within practical computation times, with significant financial gains and reduced greenhouse gas emissions.

We consider defining risk probability in stochastic control problems under distribution ambiguity. Current approaches for chance-constrained control typically assume that the true state distribution is known and Gaussian distributed. These assumptions are not amenable to many real-world engineering applications where system dynamics are nonlinear and only approximately modeled. In this work, we define a distribution ambiguity set and, with a variational expression for exponential integrals, bound the expected risk value under an unknown distribution that resides within a relative entropy distance of a nominal Gaussian reference distribution. Our bound recovers the reference risk value in the zero-divergence limit. A method is presented to determine the relative entropy distance defining the ambiguity set that is a function of the reference covariance evolution and second-order dynamical truncation errors. The resulting contributions provide a framework for handling distributional ambiguity in nonlinear covariance steering problems. A stochastic spacecraft guidance example is presented to demonstrate our contributions.

A vertex-reinforced random walk steps to a neighbour with probability proportional to $1+\beta n^{a}$, where $n$ counts previous visits to that neighbour and $a\in(0,1)$ sets the memory strength. On the rooted $b$-ary tree the exponential growth of the vertex set drives the walk outward while the reinforcement pulls it back. We report a sharp condensation transition of the occupation measure at a finite $\beta_c(a,b)$: below it the occupation spreads and the range grows linearly; above it a single vertex holds an $O(1)$ fraction of the time, stable in the observation time, while the range keeps growing very slowly, at a rate better described by $\log t$ than by any power. We do not find the range to be bounded, and keep this condensation distinct from finite-range localization. Four estimators locate the same threshold, which shows no systematic drift out to $t=3\times10^{7}$. In a frozen environment the walk is reversible, with edge conductances $c_{uv}=w_{u}w_{v}$, $w_{v}=1+\beta n_{v}^{a}$, and measure $\mu_{v}\propto w_{v}\sum_{u\sim v}w_{u}$ describing the condensed core, whose neighbour coupling we test directly. Reversibility places the escape at the frontier within the branching-number criterion for biased walks on trees, predicting $\beta_c\propto b-1$; the measured lines for $b=2,3,4$ collapse under division by $b-1$ to a few percent (bootstrap). The value $a=1/2$ that governs the walk on $\mathbb{Z}$ enters only as the marginal exponent of the condensed profile. Near $\beta_c$ the occupancy is non-self-averaging and bimodal, a coexistence-type phenomenology.

Estimating entropy from samples is fundamental in information theory and property testing. Shannon entropy measures average uncertainty and can be estimated to constant additive accuracy over a $k$-symbol alphabet using $\Theta(k/\log k)$ samples. Min-entropy depends only on the most likely symbol. Both are special cases of order-$\alpha$ R'{e}nyi entropy, $H_\alpha$. We characterize the sample complexity of estimating min-entropy and R'{e}nyi entropy for $k$ and integer $\alpha>1$; our lower bounds also hold for noninteger $\alpha\ge1.001$. We prove that min-entropy estimation to constant additive accuracy has sample complexity $\Theta(k\log k)$. The upper bound uses the largest empirical frequency and concentration via dyadic grouping. The matching lower bound hides a slightly heavier symbol at a uniformly random location. Thus, min-entropy requires $\Theta(\log^2 k)$ more samples than Shannon entropy and corrects a previously stated $\Theta(k/\log k)$ characterization. For every integer $2\le\alpha\le c_0\log k$, we prove the matching fixed-accuracy bound $\Theta_{c_0}(\alpha k^{1-1/\alpha})$. Previous results gave $\Omega_\alpha(k^{1-1/\alpha})$ for fixed integer $\alpha>1$ and $O_{c_0}(\alpha^2k^{1-1/\alpha})$ for all integer $\alpha>1$. Our upper bound analyzes an unbiased falling-factorial estimator based on $\alpha$-way collisions, while a hidden-heavy-coordinate construction gives the matching lower bound and shows that the factor $\alpha$ is unavoidable. For every real $1.001\le\alpha\le c_0\log k$, we prove the uniform lower bound $\Omega_{c_0}(\alpha k^{1-1/\alpha})$. Finally, since $0\le H_\alpha(p)-H_\infty(p)\le\log k/(\alpha-1)$, min-entropy uniformly approximates $H_\alpha$ when $\alpha$ is a sufficiently large multiple of $\log k$. Combining this reduction with our min-entropy bounds gives $\Theta_\varepsilon(k\log k)$ sample complexity in the high-order regime.

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. A Hessenberg pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a Hessenberg fashion. A Hessenberg system $\Phi$ an `oriented' version of a Hessenberg pair. It is known that $\Phi$ is determined up to isomorphism by its parameter array; this consists of the eigenvalue sequence of $\Phi$, the dual eigenvalue sequence of $\Phi$, and a sequence of nonzero scalars $\{\phi_i\}_{i=1}^d$ called the split sequence of $\Phi$. We are interested in some types of Hessenberg matrices, said to be circular, quasi-circular, tridiagonal, and irreducible tridiagonal. We are interested in the families of Hessenberg systems for which the associated Hessenberg matrices have one of the above types. A Hessenberg system of irreducible tridiagonal type is often called a Leonard system. In this case the associated Hessenberg pair satisfies two relations, called the tridiagonal relations. We are interested in the family of Hessenberg systems for which the associated Hessenberg pair satisfies the tridiagonal relations. We are also interested in the family of Hessenberg systems for which the eigenvalue sequence and dual eigenvalue sequence satisfy a linear three-term recurrence. In the present paper we have two main goals. First, we show how the above families of Hessenberg systems are related to each other. Second, we describe each family in terms of the parameter array.

The tuned mass damper inerter (TMDI) is a passive vibration absorber that suppresses structural vibrations by leveraging the inertance property of inerter devices. Previous studies have demonstrated TMDI effectiveness in wind turbines, but relied on simplified structural models. To date, no study has implemented multi-directional TMDIs within aero-hydro-servo-elastic tools such as OpenFAST to enable independent vibration control across multiple turbine components, including the tower, blades, and substructure. This paper addresses this gap by implementing the TMDI in OpenFAST. The governing equations of motion are derived and implemented within the Structural Control (StC) Module, supporting arbitrary TMDI connectivity and independent multi-directional configurations across turbine components. Verification against independent numerical models confirms the implementation's accuracy across all supported configurations. An application study examines the IEA 15MW reference wind turbine equipped with a tower-top TMDI under combined wind-wave loading. Optimal tuning is performed using a simplified two-degree-of-freedom model, with modal properties extracted from an OpenSeesPy finite element model of the IEA 15MW. Results show that TMDIs with secondary mass up to 100 times smaller than a conventional TMD achieve matching or superior vibration suppression in the fore-aft and side-side directions, in terms of peak and standard deviation of tower-top displacement and acceleration. Notably, TMDI stroke is markedly reduced compared to the TMD, better suiting the spatially constrained wind turbine environment. These results support the practical merit of lightweight TMDIs for wind turbine vibration control, with the OpenFAST implementation enabling performance-driven design under realistic loading conditions.

The Lee-Yang theory is based on the theorem concerning a property of the ferromagnetic Ising model partition function proved by T. D. Lee and C. N. Yang in 1952. It provides powerful tools for understanding the very nature of phase transitions and critical phenomena in various spin and similar models. Sometimes, this theory is applied to models for which the theorem's validity has not been proved. By the main result of E. H. Lieb and A. D. Sokal, Commun. Math. Phys. 80, 153 (1981), for a given ferromagnetic model, this validity is guaranteed by the same property of the single-spin partition function. Due to the anisotropy term $\sum_i \Delta S_i^2$, the single-spin partition function of the Blume-Capel model fails to have the Lee-Yang property for $\beta \Delta> \ln 2$. In this article, we show that the ferromagnetic interaction in such a model can induce the Lee-Yang property, even for these values of $\beta \Delta$. To the best of our knowledge, it is the first result of this kind.

Persistence diagrams are fundamental descriptors in topological data analysis. Many statistical and machine learning methods require persistence diagrams to be mapped into vector spaces or compared through kernels. Although many persistence diagram vectorizations and persistence kernels have been proposed, there is no general framework for determining which persistence diagram vectorizations are synthesizable. Spectral synthesis asks whether the exponential monomials contained in the translation-invariant variety generated by a function span a dense subspace of that variety. We introduce uniformly discrete spectral synthesis for persistence diagram vectorizations using Fourier analysis on groups of virtual persistence diagrams. When the space of persistence intervals is separable, we prove sufficient conditions for uniformly discrete spectral synthesis and derive approximation results for virtual persistence diagrams with bounded total multiplicity. We then classify many persistence diagram vectorizations and persistence kernels from the literature by uniformly discrete spectral synthesis.

Recent works of Galligo, Najnudel, and Vu (2025) and Najnudel and Vu (2026) study repeated differentiation for polynomials of the form $P(z)=p(z^m)$, where $p$ is a deterministic polynomial of degree $n$ with real, non-negative roots, in the regime where $m$ and $n$ are large. If $m\gg \log(n)$ and the root distribution of $P$ converges to a compactly supported, radial probability measure $\mu_0$, these works show that for $0\le t<1$, the root distribution of the $\lfloor nmt\rfloor$-th derivative of $P$ converges to a compactly supported probability measure $\mu_t$ given by an explicit formula for its radial quantile function. We give a substantially simplified proof of this result and also extend the result from repeated differentiation to repeated applications of the differential operator $z^a(d/dz)^b$. We also compute the limiting root distribution in the case when $m$ is fixed and $n$ tends to infinity.

We prove a conjecture of Lurie characterizing $(n+1)$-dualizability in higher Morita categories of $\mathbb{E}_n$-algebras in terms of dualizability over certain factorization homologies. A key ingredient is a higher Morita category based on the recently developed framework of pointless factorization algebras of Karlsson and the first author. We also verify an invertibility conjecture of Brochier–Jordan–Safranov–Snyder as an immediate corollary of our main result. Moreover, we prove a relative version of the dualizability conjecture, yielding a new criterion for relative/twisted field theories. We give some examples, including Dirichlet and Neumann relative theories.