Papers

This paper is concerned with the long-time dynamics of a coupled nonlinear suspension bridge system with past history, in which the deck and the main cable interact through the nonlinear restoring force \(k[u-v]^{+}\). Building on the well-posedness and the solution semigroup established in \cite{Mukiawa2025}, we first show the exponential stability of the associated linear semigroup by means of the Gearhart–Prüss theorem together with a detailed resolvent estimate. We then combine this exponential stability with the local Lipschitz property of the nonlinearity, the fact that the nonlinearity acts only on the elastic component, the smoothing of the elastic component, and the exponential decay of the memory kernels to conclude that the solution semigroup is quasi-stable in the sense of Chueshov–Lasiecka, and therefore asymptotically smooth. Together with the dissipativity recalled from \cite{Mukiawa2025}, this leads to a compact global attractor in the natural energy space. In particular, our approach may be viewed as closing a gap in the proof of asymptotic smoothness in \cite{Mukiawa2025}, where a non-decaying initial-difference term was absorbed into an arbitrarily small constant without justification.

The real-axis jump of the prime-zeta function gives an integral form of Gram's series for Riemann's $R$-function. Applying the same operation to the classical almost-prime generating functions defines a smooth family $R_k$, with $R_1=R$. We give the construction and a convergence argument, and explain why its expansion at every fixed logarithmic order is the Selberg–Delange expansion. The distinction lies in the smaller real-axis contributions retained by the definition, rather than in new asymptotic coefficients. A finite Perron decomposition also identifies the additional contour terms required for exact recovery of the count.

Let $(\mathcal N,\nu)$ be a finite tracial von Neumann algebra and set $\mathcal{K}_r(P,Q)=\nu((P^{1/2}QP^{1/2})^r)$. We prove that $$ [\mathcal{K}_r(P_i,P_j)]_{i,j=1}^3\ge0, \qquad 1/2\le r\le1. $$ The range $1/2<r<1$ is new even for matrices, and order three is sharp for $1/2\le r<1$. This single mechanism resolves two problems. First, if $(\mathcal R,\omega)$ is finite tracial and $H=[H_{ij}]_{i,j=1}^3\in M_3(\mathcal R)_+$, then $$ [\omega(|H_{ij}|^p)]_{i,j=1}^3\ge0, \qquad 1\le p\le2, $$ settling the Lin–van den Driessche conjecture on Schatten norm compression. Second, for a semifinite tracial von Neumann algebra $(\mathcal M,\tau)$, define $$ d_t(A,B)^2=\frac{\tau(A)+\tau(B)}2- \|B^{t/4}A^{t/4}\|_{2/t}^{2/t}. $$ We prove that $d_t$ is a complete metric on $L^1(\mathcal M,\tau)_+$ for $1\le t\le2$, including the previously open range $1<t<2$ between the Hellinger and Bures endpoints, and that $$ d_t\text{ is a metric on }L^1(\mathcal M,\tau)_+ \quad\Longleftrightarrow\quad t\in[1,2]\ \text{or}\ \mathcal M\text{ is abelian}. $$ Its topology and Cauchy sequences agree with those of $L^1$. The same classification holds for unital tracial $C^*$-algebras and, with the known $t=0$ obstruction, resolves Problem~2 of Komálovics and Molnár.

We develop a theory of ending currents for properly discontinuous isometric actions of hyperbolic groups on Gromov hyperbolic spaces, generalizing Thurston's ending laminations. Several natural generalizations are considered, which agree in the 3-dimensional hyperbolic setting but are otherwise distinct. We illustrate these invariants and their differences by several examples, including closed surface group actions on hyperbolic group extensions arising from Teichmüller theory and a geometrically finite action of a closed surface group on $\mathbb{H}^4$ with a single cusp that corresponds to a non-simple curve on the surface. We also discuss the conditions under which ending currents characterize geometrically finite actions, obtaining several equivalent conditions and some surprises, illustrated by an example of a geometrically infinite action whose ending currents are all supported on a single orbit of peripheral subgroups.

Let $(M^n,g)$, $n\ge2$, be a compact connected Riemannian manifold with smooth nonempty boundary, $\operatorname{Ric}_M\ge(n-1)Kg$ for $K>0$, and nonnegative outward boundary mean curvature. We derive an explicit algebraic lower bound $B_\eta$ for the first Dirichlet eigenvalue from Kasue's mixed inradius model. With $R$ the inradius and $s=4KR^2/\pi^2$, the bound involves one square root and approximates the model eigenvalue with relative error $O_n(s^4)$ as $s\downarrow0$ for fixed $n$. The construction uses the first two moments of a positive representation of the averaged model potential and a sharp pointwise corrector estimate. For the chosen spectral transfer, the first two moments determine an optimal bound. We also show that the model eigenvalue is the geometric infimum at fixed $(n,K,R)$. For $0<s<1$, it is approached by smooth collapsing balls with sectional curvature at least $K$ and totally geodesic boundary. At the maximal inradius, the model value is attained by the round hemisphere. The bound strictly improves the variance refinement throughout $0<s\le1$. We also give a direct algebraic comparison with an optimized power-test benchmark.

We prove that the three-element flat semiring $S_7$ is strongly nonfinitely based within the class of additively idempotent semirings. More generally, let $\mathcal N$ be the subvariety of $\Var(S_7)$ defined by $x^2y\approx x^2$. For every finite ai-semiring $A$, each variety $\mathcal W$ satisfying $\mathcal N\subseteq\mathcal W\subseteq\Var(A)$ has infinite axiomatic rank.

We prove that the geodesic flow of a closed real-analytic Riemannian manifold has zero topological entropy if it admits globally defined, real-analytic Liouville integrals. This proves a longstanding conjecture proposed by Bolsinov and Taimanov.

We establish tight bounds for the width and maximal order type of the h-preorder on Q-labeled trees in terms of the corresponding invariants of well-quasi-order Q.

A challenge in nonlinear filtering is to incorporate nonlinear effects into approximations of the conditional mean while retaining a finite-dimensional recursive algorithm. We derive a second-order asymptotic expansion of the filtering mean with respect to the system-noise intensity. We apply this expansion to the Lorenz–63 model, a prototypical three-dimensional chaotic dynamical system. The first-order approximation is computed using a Kalman–Bucy filter for the system linearized around the deterministic reference trajectory. The second-order correction incorporates the Hessian of the Lorenz–63 drift and is computed through a closed recursive system with 3+9+18=30 independent scalar components. The full system, including the reference trajectory and the first-order mean and covariance, has 42 independent scalar components. To support stable tracking of chaotic trajectories over long time intervals, we introduce recentering of the expansion and redistribution of Gaussian components between expansion orders. In the numerical experiments, we compare the proposed algorithm with an ensemble transform Kalman filter (ETKF) as a practical benchmark and a particle filter (PF) as a numerical reference. Under the full-observation settings, the proposed algorithm achieves a lower root-mean-square error relative to the truth and a smaller distance to the 10,000-particle PF reference than the 20-member ETKF. These results suggest that the proposed second-order extension of Kalman filtering has the potential to balance filtering accuracy and computational cost.

The Ungar game, introduced by Defant et al., is a combinatorial game played on a finite lattice, which originates from a geometric transformation introduced by Ungar to resolve Scott's slope problem. When played on a distributive lattice $L$, the Ungar game reduces to repeatedly choosing and removing a non-empty set of maximal elements from a finite poset $P$, where $L$ is represented as the lattice $J(P)$ of the order ideals of $P$ by Birkhoff's representation theorem. Defant et al. raised the open question of whether determining the outcome of the Ungar game on $J(P)$ is PSPACE-complete with respect to $|P|$. In this paper, we settle this open question in the affirmative. More precisely, we prove that the problem is PSPACE-complete even when restricted to posets $P$ with maximum chain length $l(P) \le 3$ such that the underlying graph of the Hasse diagram of $P$ is bipartite, where the length of a chain is defined as the number of elements in the chain minus one. Furthermore, we show that the problem is NP-complete for $l(P) \le 2$, whereas the case $l(P) \le 1$ is known to belong to LOGSPACE.

We determine the minimum volume product of centrally symmetric polytopes obtained by adding facet apices to a cube while fixing their total outward height. Heights and tangential coordinates vary subject to a product condition on the polar-cap denominators. In every dimension at least two, the minimum has an explicit formula, attained by concentrating the height at one opposite pair of corner apices. For positive height in dimensions at least three, every minimizing array has this structure. The product condition strictly extends coordinatewise quadratic constraints. We also obtain two bounds without that condition: a four-dimensional Mahler bound for a specified height range, and a correction measuring volume generated jointly by two apices. The latter certifies a continuous family outside the preceding sufficient conditions. The proof combines known cap volumes with a sharp aggregate height estimate and a double-pyramid construction. Lean~4 verifies the actual-volume minimum, the necessary equality structure and both extensions.

Let $\delta=xz-yt$. We give an explicit tame factorization of the Anick automorphism $(x+\delta t,y+\delta z,z,t)$ of $F[x,y,z,t]$. After a linear change of coordinates, the factorization consists of sixteen elementary automorphisms, all fixing $z$. The proof uses a commutator construction with variables already present in the algebra. A complete list of the elementary factors is included in Section 5. The explicit polynomial factorization identities are defined over $\ZZ[1/2]$; no conclusion in characteristic $2$ is asserted.

At the start of every session, LLM agents load a fixed context file, such as $\texttt{AGENTS.md}$. Each loaded token in the file is charged again in every later round of the session, and these files can degrade performance as they grow in size. However, in practice, human or automated curators usually grow these files by appending. We formulate context curation as a capacitated assortment problem. Instructions consume tokens under a finite attention capacity; adding an instruction never raises the compliance of the others, while retained instructions incur a per-session setup cost. We prove an upper bound on the optimal file size, regardless of the number of available candidate instructions, and that appending every instruction with positive standalone value can be arbitrarily worse in net value than selecting an optimal subset. A token budget also limits the loss when the token price is underestimated. We then examine what can be learned from past sessions and how this information can guide decisions to add or remove instructions. Feedback is inherently censored: the benefits and harms of loaded instructions are observable, whereas missing instructions generate feedback only when their absence causes harm. In this setting, we show that deleting instructions ignored by agents can inevitably remove helpful ones. We characterize how much evidence should be collected before adding an instruction. Besides, we bound regret when human reviewers can inspect only a limited number of edits per period. Empirical experiments further show that irrelevant rules drawn from real context files reduce language-model compliance.

When two quantum systems are prepared independently and each is driven to an idealized limit, the expectation of a bipartite observable may depend on which system is idealized first. We call an observable stable when it does not. We show that stability is equivalent to two operational properties. First, every idealized preparation of one system can be simulated to arbitrary accuracy, uniformly over all probes, by finite mixtures of actual preparations, provided that the probing system is idealized first. Second, any two idealized preparations have a unique, symmetric independent product. Under a uniform form of stability, we prove, the number of preparations that $n$ adaptive probes can tell apart at a fixed margin is bounded by a polynomial in $n$, and responses can be learned with a bounded number of mistakes; otherwise, at some margin, protocols with $n$ probes tell apart $2^n$ preparations. The swap test, Hong–Ou–Mandel coincidences and discrete equality tests turn out to be uniformly stable, with bounds independent of the dimension, while order comparisons of discrete spectra unbounded above are unstable. The proofs rest on the classical fact that a single double limit condition appears, under different names, in topology (Grothendieck), functional analysis (Krivine–Maurey), algebra (Arens), probability (Simons) and combinatorics (Pták), and in model theory as Shelah's stability, where the independent product corresponds to Harrington's lemma. Model theory thus serves as a Rosetta stone, pairing each physical question with its mathematical formulation.

We study discrete networks of $N+1$ identical phase oscillators with positive reciprocal weights and smooth odd periodic interactions, motivated by discrete models of Huygens synchronisation. The interaction laws define an intrinsic potential, and removing the common phase gives a gradient map on the $N$-torus. A weighted graph Laplacian yields explicit step sizes for strict Lyapunov decrease and global invertibility. Our main result is a Morse–Smale criterion for maps $z\mapsto z+hK\sin z$: if $K$ is symmetric, nonsingular and weakly diagonally dominant, the map is Morse–Smale for $0<h<1/\lambda_{\max}(K)$, and this interval is optimal. The proof combines invariant coordinate slabs with a maximum principle for the adjoint variational equation. The criterion applies to unit-weight paths of any length and to the three-leaf star, and a weighted version covers paths with discretely concave weights. By structural stability, it extends to every graph containing a spanning path with sufficiently weak additional links. Conversely, we construct infinitely many trees with Morse potentials and an almost-global sink, whose maps nevertheless have non-transverse heteroclinic intersections.

Underactuated systems with nontrivial geometry are abundant in practical robotic problems. However, little is known about constructive safety-critical control design in the presence of underactuation. In this work, we provide a constructive framework for synthesizing safe control architectures and control barrier functions for a class of quadrotor-like underactuated systems. By studying the geometry of their underactuation, we characterize when these systems have natural layered architectures useful for safety-critical control design. Using these architectures, we devise a constructive, hierarchical CBF synthesis procedure and provide conditions under which it succeeds.

Hybrid automatic repeat request with Chase combining (HARQ-CC) improves the reliability of slow fluid antenna multiple access (sFAMA), but its transmission limit must be jointly optimized with user concurrency. This paper proposes a quality-of-service (QoS)-aware joint design of concurrency level U and HARQ limit C for downlink HARQ-CC-aided sFAMA. Based on the selected-port signal-to-interference ratio (SIR) distribution, we derive multi-round decoding failure probabilities and develop a two-stage optimization that maximizes feasible concurrency and then selects the minimum C satisfying outage and mean-service constraints. We prove that the first feasible C minimizes service duration for a given U, and that feasible U values form an initial segment, enabling optimal bisection search. Numerical results agree with Monte Carlo simulations under full Jakes spatial covariance. The proposed adaptive FAS design supports six concurrent users, compared with three to five for fixed-HARQ FAS and two for HARQ-adaptive fixed-position-antenna (FPA), demonstrating that joint concurrency and HARQ adaptation expands the QoS-feasible region of FAS.

For the $H^{2|2}$ model on $\mathbb Z^2$, we prove that, at energy zero, fractional moment of Green functions, $\mathbb E[ G(x,y)^s]$ decays exponentially in the distance between $x$ and $y$, for every constant edge weight $W>0$ and every $0<s<1/2$. Here $G$ is the infinite-volume Green function at energy zero of the associated random Schrödinger operator. The main ingredient is a quantitative lower bound on the derivative of $\mathbb E[e^{u_x/2}]$ with respect to the edge weights, where $u$ is the pinned effective field.

We show that González Merino and Schymura's Conjecture 4.8, a sufficient condition for Makai's reverse isodiametric conjecture, fails in all sufficiently large dimensions. Its three-dimensional case follows from Aliev's proof. Pełczyński and Szarek's construction for enclosing parallelepipeds implies this failure. Using random frames, we also determine the sharp exponential determinant guarantee for every fixed ratio greater than one between the number of vectors and the dimension.

We prove that the $Q$th symmetric product of an absolute $\lambda$-Lipschitz retract, equipped with the bottleneck matching metric, is an absolute $(3Q-2)\lambda$-Lipschitz retract. This answers a question of De Pauw about the target spaces of Almgren's $Q$-valued maps. We also prove that the space of nonempty subsets with at most $n$ points, equipped with the Hausdorff metric, is an absolute $(6n-5)\lambda$-Lipschitz retract. The main tool is a reconstruction theorem for bounded products of metric stars with finitely many rays, modulo a finite permutation group: these quotients are absolute $3$-Lipschitz retracts, independently of the group and the number of coordinates. A variant of this theorem, in which unordered sets of sign vectors take the place of the group action, handles finite subset spaces.

Off-policy evaluation (OPE) estimates the value of a target policy from logged data, but limited behavior-policy coverage can force high-variance reweighting or reward-model extrapolation. Counterfactual annotations can add evidence about unobserved actions, yet practical sources, including domain experts and large language models (LLMs), may be costly, biased, or noisy. We study budgeted acquisition of such annotations for contextual-bandit OPE. Given source-specific costs and error profiles, we formulate an integer allocation problem over context-action pairs and annotation sources to minimize the component of estimator variance that depends on the annotation plan. We characterize when annotations are valuable through a first-annotation threshold and local annotation-value regimes. For the coupled multi-source problem, we develop a majorization-minimization algorithm with dynamic-programming subroutines that monotonically improves the objective. Experiments in synthetic clinical and LLM-annotated education bandits show that our allocation method reduces fixed-profile mean squared error (MSE) by 20.58% and 10.77%, respectively, relative to no annotation.

Let $k>t\ge 2$ be fixed integers, and let $G$ be a permutation group on a set of $v$ points. A $G$-invariant packing is free if every block has $|G|$ distinct images under $G$. We give two conditions on $G$ under which, as $v\to\infty$, there is a free $G$-invariant $t$-$(v,k,1)$ packing with $(1-o(1))\binom{v}{t}/\binom{k}{t}$ blocks. This is a $G$-invariant form of the Erdős–Hanani theorem on asymptotically optimal packings. For semiregular groups $G$, packings with these properties exist if and only if all but $o(v^t)$ of the $t$-subsets of points have trivial setwise stabiliser. This property is satisfied by every semiregular group for $t\ge 3$, and by every cyclic semiregular group for $t=2$. As a consequence, the Johnson bound is asymptotically attained at all lengths by optical orthogonal codes of constant weight $w$ and constant correlation $\lambda$ with $w\ge\lambda+2$, by their multidimensional and signature pattern versions, and by cyclic and quasi-cyclic constant-weight codes. The same method gives a form of the theorem for group divisible packings that are invariant under a cyclic group. This determines the asymptotic maximum size of two-dimensional optical orthogonal codes with at most one pulse per wavelength.

This work is devoted to the study of blow-up phenomena for solutions of \[-\Delta u=\lambda V(x)|x|^{2N}e^v,\quad \mbox{in}\quad B_1,\quad v=0\quad \mbox{on}\quad \partial B_1. \] where $B_1$ denotes the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth function, $N\in \mathbb N$ and $\lambda>0$ is a small parameter. The main goal is to determine conditions on the potential $V$ guaranteeing the existence of families of solutions that blow up at the origin as $\lambda\to 0^+$. Moreover, we establish higher order criteria that distinguish between simple and non-simple blow-up behavior at $x=0$. In particular, for $N-1\in 2^{\mathbb N}$, we construct a non-simple blow-up sequence $v_k$, corresponding to coefficient function $V_k$ whose derivatives vanish up to the maximal order allowed by the upper bound of Wei-Zhang, thereby proving that this bound is sharp.

In this paper we study Koopman operators associated with finite periodic realizations of elementary cellular automata. The spectra of these operators encompass various properties of the underlying dynamical systems, independently of any initial condition. A characterization of these spectra is proposed, leading to a spectral classification of all elementary cellular automata. This work is thought as a preliminary step for studying the spectra of Koopman operators for infinite realizations of elementary cellular automata.

In this paper, we prove a conditional result that yields an asymptotic formula with an explicit main term and a power-saving error term for the generalized shifted ternary additive divisor sum $D_{\mathcal{I},\mathcal{J}}(X,1)=\sum_{n \leq X} \tau_{\mathcal{I}}(n) \tau_{\mathcal{J}}(n + 1)$, where $\mathcal{I}=\{\alpha_1,\alpha_2,\alpha_3\}$ and $\mathcal{J} = \{\beta_1, \beta_2, \beta_3\}$, with $|\alpha_i|, |\beta_i|\leq \delta$ for $i=1,2,3$, and $0<\delta<1/44$. The proof of our main theorem relies on assuming an averaged level of distribution for the generalized ternary additive divisor function $\tau_{\mathcal{J}}$ in arithmetic progressions up to the level $2/3$.

Massive discrete Dirac-type operators on periodic planar graphs are investigated. Using Floquet–Bloch decomposition, we first show that neglecting the faces leads to spurious flat bands that are unrelated to the graph's geometry. Then, for the correct Dirac–Hodge operator $D_m$, we establish a spectral reduction expressing $\sigma(D_m)$ through the graph Laplacian on vertices $-\Delta_V$ and the Laplacian on faces $-\Delta_F$. As a consequence, $D_m$ carries a flat band precisely when one of these scalar Laplacians does. We finally apply this reduction to the study of flat bands of the operator $D_m$ on Archimedean lattices and contrast the outcome with that on the Lieb lattice.

Natural gas networks must accommodate variable demand and supply while maintaining pressure across pipelines whose stored gas couples operating decisions through time. Probabilistic surrogates can simplify compressor scheduling, but their predictive uncertainty must remain reliable where optimization uses it. We combine direct learning of the minimum pressure margin with assessment at optimizer-selected schedules. Using a warped Gaussian process, we express the posterior pressure requirement as one inequality in its predictive mean and standard deviation. We then parameterize block and Fourier schedules so scalar predictions and analytic derivatives replace trajectory states and time integration in whole-day optimization. In steady state, a surrogate that passes held-out accuracy and coverage checks nevertheless falls below the minimum pressure on 71% of its own dispatches at a nominal 1% tolerance, as the optimizer is drawn to schedules where the margin is overestimated. This diagnostic motivates an enrichment loop that audits selected dispatches against the physics and updates the surrogate by exact conditioning, with fitted quantities held fixed. The loop restores steady-state minimum-pressure risk control at every studied tolerance, with conditioning about four orders of magnitude faster than refitting. In transient operation, we complement this enrichment with empirical threshold calibration to address residual hourly optimism, bringing hourly violation rates closer to their declared tolerances with little compressor work overhead.

We prove that a smooth Dunkl harmonic function in the upper half-space whose Dunkl gradient satisfies a Carleson measure condition is the Poisson integral of a function in $\mathrm{BMO}(dw)$, defined using Euclidean balls. The result holds for arbitrary finite root systems and nonnegative multiplicities. The proof uses an estimate for increments from the origin, obtained from a finite-dimensional system indexed by the reflection group, together with the atomic characterization of the Dunkl Hardy space.

String attractors provide a compact way of representing the complete factor structure of a word: a set of positions is an attractor if every distinct factor has at least one occurrence crossing one of the selected positions. Although the minimum attractor size is known for several classical families of words, describing all optimal attractors is typically much more difficult, since it requires understanding the geometry of all factor occurrences rather than constructing a single optimal solution. We give a complete description for the finite Thue–Morse words. Earlier work proved that four positions are necessary and sufficient for every order $n\geq 4$, but the collection of all smallest attractors remained unknown. For every $n\geq 6$, writing $h=2^{n-3}$, we prove that the smallest attractors are exactly the two reflected families where the four offsets are chosen independently from $\{0,1\}$. Hence there are exactly $32$ smallest attractors for every $n\geq6$. The initial cases are genuinely exceptional: $t_5$ has $40$ smallest attractors and $t_4$ has $87$. We also study the attractor condition independently of optimality. For every $n\geq5$, we characterize the complete antichain of inclusion-minimal factor coverages of $t_n$. It consists precisely of the coverages of $aa$, $bb$, and the eight minimal unique substrings of every generation $t_m$, $4\leq m\leq n$. Thus there are exactly $8n-22$ canonical constraints, forming an irredundant exact certificate for attractors of arbitrary cardinality. When attention is restricted to four-position sets, this linear-size system collapses to a constant one: it is enough to test the $24$ minimal unique substrings coming from three consecutive generations, and sixteen of these already force the two optimal families. We also show that three generations are necessary within this natural consecutive-generation hierarchy.

In this note, we study the problem of designing a feedback law that globally steers a system to a prescribed target configuration. Even if the system is fully actuated, topological obstructions generally prevent the existence of globally asymptotically stabilizing continuous feedback laws. We revisit this problem in a stochastic setting by allowing noise to enter through the control channels. Using a criterion for asymptotic stability in the large that combines local Lyapunov stability with positive recurrence, we constructively show that one can construct elementary feedback laws that achieve global asymptotic stability in the large, of the target equilibrium point in connected Euclidean domains with obstacles and manifolds without boundary. For Euclidean domains with obstacles, we also show that the method extends naturally to the problem of finding the minimizer of a strongly convex function with non-convex constraints. Numerical experiments illustrate the effectiveness of the approach for Euclidean domains with circular obstacles and the two dimensional sphere. Additionally, we study the role noise strength when there is a non-convex obstacle, in which case the system might show metastability.