Papers

We construct a symmetric conference matrix of order $86$, the smallest order for which existence was open. Equivalently, there exist a conference graph with parameters $(85,42,20,21)$, a regular two-graph on $86$ points, and a real equiangular tight frame of $86$ vectors in $\mathbb{R}^{43}$. The matrix is a $12\times12$ array of $7\times7$ circulants with two border rows, and it admits a group of automorphisms of order $21$. Its $86$ descendants give exactly ten nonisomorphic strongly regular graphs.

We prove that every finitely generated group with rapid decay and trivial amenable radical is C*-simple.

Consider placing the numbers $1, \ldots, n$ consecutively around an $n$-cycle. We ask for a given $n$, which numbers $k$ can be written as a sum of consecutive numbers on the $n$-cycle. If all numbers between $1$ and $T_n$, the $n$th triangular number, can be written as a sum this way we say the $n$-cycle is complete. Our main result shows that for $n\geq 8$, $n$-cycles are incomplete.

Let $\mathcal P_1$ denote the class of 1-planar graphs and let $tC_5$ be the disjoint union of $t$ copies of $C_5$. For every fixed $t\ge3$ and all sufficiently large $n$, we determine the unique $n$-vertex $tC_5$-free graph in $\mathcal P_1$ with maximum adjacency spectral radius, answering Problem 1 of Li, Wang and Zhao. The proof first gives a structural description of every extremizer. After two dominating vertices are removed, the remainder consists of copies of the seven-vertex graph $B=K_1\vee2K_3$ together with at most one bounded connected core. This follows from a two-family covering theorem for 1-planar $K_2$-joins and the sharp packing–defect inequality \[ 12v(Q)-7e(Q)\ge1-10\nu_5(Q). \] The same inequality yields an exact edge-extremal result for remainders with bounded pentagon packing. A normalized resolvent then cancels the repeated $B$-components, and finite moment comparisons force all packing and all nonzero defect into one core and identify that core uniquely in each residue class modulo $7$. The case $t=3$ is the pentagon-free boundary case and is completed by one exact finite component lemma.

Let $n\ge2$ be a fixed integer, and let $K,x,y$ be positive integers satisfying $2\le K\le y<x$. Consider a polynomial \[ f(u)=\alpha u^n+\alpha_{n-1}u^{n-1}+\cdots+\alpha_1u+\alpha_0 \] with real coefficients. We study the quantity \[ V_n(K,x,y) = \sum_{k=1}^{K} \left| \sum_{x-y<p\le x}e(kf(p)) \right|, \qquad e(t)=e^{2\pi it}, \] where the inner sum is over primes. For $\alpha=\frac{a}{q}+\frac{\theta}{q^2}$, $a\in\mathbb Z$, $q\in\mathbb N$, $(a,q)=1$, $|\theta|\le1$, we establish the bound \[ \begin{aligned} V_n(K,x,y) &\ll Ky\Biggl[ \frac{1}{\sqrt{K\log(2y)}}+ \min\left\{ \Delta^{\frac{1}{2^n}} \bigl(\log(2y)\bigr)^{\frac{n^2-1}{2^n}}, \Delta^{\frac{1}{3\cdot 2^{n-2}}} \bigl(\log(2y)\bigr)^{\frac{n^3-1}{3\cdot 2^{n-1}}} \right\} \Biggr], \end{aligned} \] where \[ \Delta= \frac{1}{q}+\frac{1}{y}+\frac{q}{Ky^n}. \] The bound is uniform in the position of the interval, and the logarithmic factors depend on its length. The proof uses the nonnegative Fejér kernel, successive differencing, and the second and third moments of the generalized divisor function. The Brun-Titchmarsh inequality accounts for the number of primes in the interval and provides an additional logarithmic saving in the term arising from averaging. We obtain sufficient conditions for a saving of any fixed power of the logarithm of the interval length, as well as an analogous mean estimate for sums over all integers.

We show that Hitchin-Wolf's harmonic map parametrization of Teichmüller space, the Schwarzian holonomy map, and Uhlenbeck's almost-Fuchsian holonomy map all arise as restrictions of a single locally biholomorphic map into the $\mathrm{PSL}(2,\mathbb{C})$ character variety. The map admits a right inverse defined on an open subset of quasi-Fuchsian space, encoding harmonic maps together with the Bers embedding. The construction goes through complex harmonic maps to the space of oriented geodesics of $\mathbb{H}^3$. Along the way, we further develop the theory of complex harmonic maps, with a focus on the natural $\mathbb{C}^*$-action and the relation to complex projective structures. This leads to an independent application: we obtain a new geometric interpretation of certain real twistor lines in the rank-two Deligne-Hitchin moduli space and we recover a theorem of Gothen-Silva.

A declared task can sometimes certify a reduced source before a downstream encoder, codebook, rate, distortion target, or optimizer is chosen. This paper studies when one such reduction preserves the complete downstream problem family, a property termed Task-Sufficient Contraction. The reduced source is fixed by the task before the later operating point is selected. An exact contraction allows the later problem to be solved on that source with the same result as if the full source had been retained. For a machine with a fixed set of possible actions and a fixed loss, the paper identifies a consumer-specific source by merging states only when every available action has the same regret in both. For finite action sets, replacing the richer source by this reduced source preserves the complete one-step rate-regret curve, even though the reduction is fixed before the distortion target is chosen. A second result gives an exact characterization for quadratic loss on affine feasible-action sets: the canonical reduced source is the projection onto the directions in which feasible actions can differ. Under a fixed energy budget, this becomes centered load, while retaining only the optimal water-filled action is too coarse. Earlier Information Bottleneck, semantic rate-distortion, and goal-oriented quantization results are then used to distinguish exact, architecture-conditioned, approximate, failed, and corrected contractions. The framework suggests a way for heterogeneous machines to exchange what a receiving task needs without first aligning their full internal representations.

Finding low-energy configurations of interacting particles and approximating probability distributions lead to the minimization of potential-interaction energies in Wasserstein space. These energies can be nonconvex, making it important to exploit second-order information while controlling the reliability of local approximations. We study trust-region optimization of smooth potential-interaction energies on the Wasserstein space of probability measures with finite second moment. The method uses a quadratic model along pushforward curves, an $L^2(\rho)$ step radius, and a Steihaug-Toint subsolver with an explicit self-adjoint second-variation operator. A ratio test determines acceptance and guides the radius update. Under a lower energy bound and globally bounded Hessians of the potential and interaction kernel, we prove that the objective is nonincreasing, the Wasserstein-gradient norms converge to zero, and an $\varepsilon$-stationary iterate is reached within $O(\varepsilon^{-2})$ total outer trials, including rejected trials. If the potential is quadratically coercive, every weak accumulation point is stationary. The analysis applies to arbitrary initial measures with finite second moment. For empirical measures, the iteration is a finite-dimensional trust-region method in the $L^2(\rho_N)$ inner product, with complexity constants independent of particle number and dimension when the initial objective gaps are uniformly bounded. Numerical experiments include a smooth soft-particle energy, maximum-mean-discrepancy minimization for non-Gaussian targets, component ablations, and scaling studies in particle number and dimension.

A classical phenomenon distinguishing infinite-dimensional holomorphy from its finite-dimensional counterpart is that monomial expansions of holomorphic functions need not converge absolutely. In this paper, we systematically investigate monomial convergence on complex Banach sequence spaces. We establish general structural results and give a negative answer to Lempert's problem. We then determine the domains of absolute monomial convergence for holomorphic functions on the classical Banach sequence spaces and obtain a complete classification for continuous homogeneous polynomials. As a further application, we establish a sharp comparison between two notions of convergence radius for homogeneous Taylor expansions on $\ell^1$. The methods developed here are of independent interest and may be applied in other places.

We study community recovery in the sparse symmetric stochastic block model with $q$ communities, average degree $d$ and signal strength $\lambda$ through statistical decision theory and Fisher information, and obtain three characterizations of the Kesten-Stigum threshold $d\lambda^2=1$ and of the information-computation gap below it. First, on each community-size profile the minimax risk of any class of rules closed under averaging and vertex relabeling equals its Bayes risk under the uniform prior; the posterior mean is the unique Bayes rule and is admissible, and the Bayes risk of degree-$D$ polynomial rules is the trivial risk times $1-\mathrm{Corr}_D^2$. Combined with known low-degree and information-theoretic results, this gives the gap as a worst-case statement: for $q\ge 5$ there is a window below the threshold in which no low-degree rule beats the trivial risk asymptotically, while an exponential-time rule does on a set of labelings of probability $1-o(1)$. Second, the Fisher information about $\lambda$ carried by cycle counts is a series with terms of order $k(d\lambda^2)^k$, convergent exactly when $d\lambda^2<1$; below the threshold the relative error of every unbiased cycle-based estimator of $\lambda^k$ stays above an explicit constant, and every cycle-count test has success probability bounded below one. Third, the derivative of belief propagation at its uninformative fixed point multiplies a random perturbation by $|\lambda|\sqrt{d}$ per iteration, and one EM step taken there leaves $\lambda$ unchanged. A signal-to-noise computation recovers the condition $d\lambda^{1/\chi}>1$ of Chin et al. for $q=n^\chi$ communities and identifies personalized PageRank as a walk count with suboptimal weights. Experiments on networks with up to $3\times 10^5$ vertices confirm the threshold for $q=2$, the hard window for $q=5$, and the many-community scaling.

Emergent generalized symmetries and the holographic principle play very important roles in understanding quantum phase transitions. In recent years, tensor-network renormalization with generalized symmetry and the corresponding fixed-point tensor formulation have been proposed as a discrete spacetime framework for reformulating conformal field theory. Nevertheless, a complete holographic formulation within this framework is still lacking, and the current formulation does not directly accommodate sector partition functions that are modular covariant in general. In this work, we formulate $\mathcal{R}$-TNR, which builds the categorical constraints of a fusion category $\mathcal{R}$ into local tensor spaces and preserves these constraints under Loop-TNR. For suitable choices of $\mathcal{R}$, the initial tensors constructed here flow under $\mathcal{R}$-Loop-TNR to stable gapless fixed points. In the Ising, tricritical Ising, and three-state Potts conformal field theory examples, a broad range of initial parameters leads to the same gapless fixed point in each example. The method resolves conformal towers by anyon sector and yields scaling dimensions, conformal spins, and magnitudes of sector-resolved structure constants with extremely high accuracy compared to previous methods. These results realize a topological bootstrap based on stable symmetry-topological-order in which finite categorical and geometric input, together with an initial tensor that need not be fine-tuned, suffices to recover the infrared conformal field theory and its sector-resolved conformal data.

A gap between achievable rates and converse bounds persists for energy-constrained bosonic communication in the presence of excess noise. We derive a unified adaptive weak-converse bound for thermal attenuation, noisy amplification, and additive Gaussian noise under an unconditional mean transmitted-photon-number constraint. The bound applies to arbitrary adaptive protocols with quantum memories and constrains the two-way quantum, entanglement-distribution, private, and secret-key capacities. It vanishes throughout the entanglement-breaking region and approaches the corresponding PLOB bound at infinite energy. At finite energy, it improves the evaluated Gaussian squashed-entanglement and PLOB bounds in relevant parameter regimes, while comparison with hashing rates quantifies the remaining gap to achievability. The result provides a common energy-dependent benchmark for noisy bosonic Gaussian channels and extends finite-energy converse methods beyond the quantum-limited setting.

In this paper, we study joint deformations of pairs $(X,E)$, where $X$ is a compact complex manifold and $E\rightarrow X$ is a holomorphic vector bundle. We prove that $(X,E)$ has unobstructed deformations if $X$ is a Fujiki manifold with torsion canonical bundle and $H^2(X,\textrm{End}^0E)=0$, where $\textrm{End}^0 E$ denotes the trace-free endomorphism bundle. When the canonical bundle is trivial, the Fujiki assumption can be replaced by three weak $\partial\bar\partial$-conditions. We construct examples satisfying these conditions whose Frölicher spectral sequences do not degenerate at $E_1$. These results provide non-Kähler extensions of the theorems of Li–Pan and Iacono–Manetti. Without this vanishing assumption, we obtain both unobstructed and obstructed pairs. In Thomas's example, we prove unobstructedness of the pair, although the bundle has obstructed deformations with the manifold fixed. On the other hand, we prove that every strict projective Calabi–Yau manifold of dimension at least three admits a simple bundle with obstructed joint deformations. We also construct obstructed pairs with Hermitian flat bundles on complex tori. These results answer two questions raised by Felten.

We study Hamiltonian elliptic systems with prescribed bilinear mass $\int_{\mathbb{R}^N}uv=a>0$. We establish a Gagliardo–Nirenberg inequality in the crossed gradient pairing and bilinear mass, on scaling-invariant component-bounded classes. It identifies the critical curve $1/p+1/q=N/(N+2)$ and the restricted energy trichotomy. For power nonlinearities, we establish uniqueness up to common translations and nondegeneracy of positive profiles throughout the Sobolev-subcritical hyperbola. Exact scaling then classifies positive normalised solutions and identifies the unique critical mass. In dimension two, we develop a bilinear exponential Gagliardo–Nirenberg estimate with a sharp gradient threshold and a truncated scalar refinement with the optimal quartic leading coefficient. Adapting variational ideas from [Cassani-Tarsi, Calc.Var.PDE (2015)] we combine a mass-normalising quotient, reduction over the full negative fibres and compactness below the concentration threshold to obtain least-energy positive radial solutions for pure exponential nonlinearities at every prescribed mass below the cubic limiting mass. Local perturbative branches and a small-frequency exponential branch yield explicit mass-response formulas.

We study the inference of a planted dense component in a sparse random graph from a single spreading process. The graph has an Erdős-Rényi background with edge probability $p_n$ and contains a planted dense component of size $n^{\alpha}$, with $\alpha > 1/2$, whose internal edge density $\xi>0$ is constant. The edge set is unobserved; the data only consist of the successive infection times from a single realization of a continuous-time SI process with independent and exponentially distributed transmission times. We show that the planted dense component leaves a detectable signature in the spreading process: after the exploration enters the dense component, it undergoes a short phase of accelerated growth. By analyzing this phase, we localize its onset and endpoint. These localization results yield consistent estimators of the component-size exponent $\alpha$, the background edge density $p_n$, and the internal edge density $\xi$ from the infection times alone. When the identities of the infected vertices are also observed, we further establish consistent recovery of the planted dense component.

We develop the first feedback design for rapid stabilization of the Kuramoto–Sivashinsky equation with a spatially varying anti-diffusion coefficient. For constant coefficients, the single-input Fredholm design of Coron and Lü (2015) excludes a discrete set of values at which repeated unstable eigenvalues cause a loss of controllability. We overcome this obstruction by introducing a second boundary input and assigning the two inputs distinct roles. The key idea, inspired by Heymann's Lemma, is to use the boundary value $u(0,t)$ entirely for a pre-feedback that renders the modified plant controllable through the curvature input $u_{xx}(0,t)$. The latter input then stabilizes the plant through a Fredholm backstepping transformation. We show that two inputs suffice for controllability and are necessary when the plant has an unstable double eigenvalue. However, the Fredholm kernel still must be approximated for implementation. Hence, to enable kernel and gain approximation, we prove continuity of the coefficient-to-gain design map on compact admissible design classes. Unlike Volterra-based continuity proofs using successive approximations, our proof uses the modal representation to control the spectral data, the inverse coefficient system, and the tails of the kernel and gain series. This yields a single neural operator approximation of the gain to any prescribed $L^2$ accuracy across the class. Finally, we establish rapid local stabilization of the nonlinear closed-loop system under both the exact gains and sufficiently accurate approximations. We conclude with numerical results that illustrate prescribed decay rates and the computational cost of the approximations. In particular, we train a Fourier neural operator that achieves typical relative gain errors of approximately $0.1\%$ and stabilizes all held-out cases tested, including a plant with an unstable double eigenvalue.

We study the Yamabe–Escobar equation prescribing positive constant scalar curvature and minimal boundary. The compactness conjecture asks whether being conformally equivalent to the round hemisphere is the only obstruction for its set of positive solutions to be compact. In this paper we show that this is true for dimensions $3\leq n\leq 14$. We show that this dimensional threshold $n_*=14$ is sharp by constructing in $n\geq 15$ nonumbilic and non-locally-conformally-flat examples of metrics in the hemisphere for which a noncompact sequence of solutions exists. We do the same for the special classes of locally conformally flat metrics, $n_*^{LCF}=18$, and the metrics with umbilic boundary, $n_*^{umb} =20$. In all three settings we also prove that the Weyl–umbilicity vanishing conjecture holds for $4\leq n\leq n_*$ and construct counterexamples in $n>n_*$ by perturbing the noncompactness examples. To the best of the author's knowledge, these are the first locally conformally flat noncompactness constructions for the positive-scalar-curvature, minimal-boundary case of the Yamabe–Escobar equation, outside the conformal class of the round hemisphere.

We study metastability and long-time propagation-of-chaos for mean-field interacting diffusions on the circle with multiple local free-energy minima. Working modulo rotations, we identify sharp exponential time scales for metastability and the validity and breakdown of the McKean-Vlasov approximation. Assuming finitely many critical rotation orbits and distinct local-minimum energies, we show that, for initial data in the attraction basin of a nonglobal minimum, metastability and propagation-of-chaos hold uniformly over deterministic times up to $e^{N(D-\varepsilon)}$ and fail at every deterministic sequence growing faster than $e^{N(D+\varepsilon)}$. Here $N$ is the particle number and $D$ the depth of the maximal Freidlin-Wentzell energy cycle having that minimum as its bottom. The same threshold governs running averages of the squared approximation error in a stronger pathwise sense, with corresponding metastability and propagation-of-chaos bounds for suitably aligned time-averaged empirical measures. When the global-minimum orbit is unique, we establish uniform-in-time propagation-of-chaos in its attraction basin without assuming finitely many critical orbits. The proofs develop the Freidlin-Wentzell cycle analysis for the empirical measure process and exploit Dawson and Gärtner's fixed-horizon large deviation estimates to analyze long excursions between metastable wells. We give a checkable Fourier criterion for finiteness of the critical orbits. We also investigate connections with phase transitions in the free-energy landscape, and discuss applications to the noisy Kuramoto and noisy transformer models. We provide, to our knowledge, the first sharp exponential-time analysis of the Dawson-Gärtner `tunnelling phenomenon' for general smooth interactions, with two-sided probability bounds and matching propagation-of-chaos thresholds.

For a smooth projective curve $C/\mathbb{Q}$ of genus $\geq 2$ and $L/\mathbb{Q}$ an extension, we write $C(L)_{\text{new}}=\{P\in C(L):\mathbb{Q}(P)=L\}$. Recent work of Khawaja and Siksek conjectures that this set is empty for $100\%$ of degree $n$ number fields $L$, when ordered by absolute discriminant. Moreover, they bring evidence towards this conjecture when $C$ is a degree $n$ cover of $\mathbb{P}^1$. We complement their work by proving analogous results for degree $n$ covers $\psi:C\to E$ of elliptic curves $E$. Our main result shows that, under suitable hypotheses, the number of distinct absolute discriminants at most $X$ of primitive degree $n$ fields $L$ with $C(L)_{\text{new}}\neq\varnothing$ is $O(X^{1/2})$ or $O(X/(\log X)^{\alpha})$, for some $\alpha>0$. In degrees $2,3,4$ and $5$ we show that these fields have density $0$ among all fields of the same degree (in degree $4$, also among the primitive ones). The novelty is for degrees $4$ and $5$, where we use work of Bhargava–Shankar–Wang and McGown–Thorne–Tucker to count fields with specified local constraints. Moreover, we give concrete examples of $8$ bielliptic modular curves $X_0(N)$, for which $X_0(N)(L)_{\text{new}}=\emptyset$ for $100 \%$ of quadratic fields $L$. Lastly, we point out modular covers of degrees $3$ and $5$ in the LMFDB for which similar conclusions hold.

We establish an Alexandrov–Fenchel type inequality in which one of the reference bodies is replaced by a function whose Hessian satisfies a balancing condition on its eigenvalues, without being required to be positive semidefinite. The proof hinges on a new extension of Alexandrov's inequality for mixed discriminants. As applications, we derive a log-concavity principle for linear functionals of area measures, answering a question of Colesanti, Hug, and Saorín-Gómez, as well as Brunn–Minkowski type inequalities for intrinsic volumes of mean section bodies, answering a question of Schuster.

We classify all open substacks admitting a proper good moduli space inside the stack $\mathcal{G}_{1,n}$ of log-canonically polarized Gorenstein curves of genus one with $n$ distinct marked points: they are exactly the stacks $\overline{\mathcal{M}}_{1,n}(c)$ defined by Bozlee–Kuo–Neff. Restricting to the case where $c$ is pure of level $m$, we then give a numerical criterion for the projectivity of the good moduli spaces: these turn out to be asymptotically almost never projective as $n$ tends to infinity.

In this paper, we study analytic multiplier ideal sheaves of toric plurisubharmonic functions on normal affine toric complex analytic spaces equipped with torus-invariant rational divisors, assuming that a positive integer multiple of the log canonical divisor is Cartier. We describe these sheaves in terms of Newton convex bodies and show that they are globally generated by finitely many meromorphic monomials. Our results generalize the description of An and the formulas of Howald and Blickle for monomial ideals, and recover the descriptions of Rashkovskii and Guenancia on affine space. We also extend this description to weights obtained by subtracting positive rational multiples of monomial ideal weights.

We study sliced Wasserstein barycenters from a variational perspective in Wasserstein space, with emphasis on the analysis problem in the barycentric coding model: given a query measure and a finite dictionary of probability measures, recover simplex-constrained barycentric coordinates. We derive an explicit first-variation formula for the sliced Wasserstein barycenter functional with respect to the classical 2-Wasserstein geometry. The resulting gradient is expressed as an average of 1D monotone transport displacements over the sphere. The stationarity equation yields a Gram-matrix criterion for estimating barycentric coordinates through a quadratic program, while its fixed-point form leads to a synthesis iteration to new barycentric measures. For Gaussian templates, we show that every global sliced Wasserstein barycenter is Gaussian; however, the functional may admit non-Gaussian critical points. We also provide a certificate for global optimality built from the 1D transport potentials. Numerical experiments demonstrate accurate coordinate recovery on synthesized queries and illustrate the use of these coordinates for data representation and stationarity residuals for reliability assessment.

This work investigates the number of limit cycles bifurcating from the periodic orbits of a linear center when it is perturbed by piecewise holomorphic systems with three zones separated by two external circles. We analyze how different geometric configurations of these zones influence the number of limit cycles. Moreover, we show the existence of at least sixteen limit cycles in fourth-degree piecewise polynomial holomorphic systems under certain configurations.

This study investigates the asymptotic behavior of two dependence measures defined on networks, Moran's $I$ statistic and Newman's assortativity, under the null hypothesis that a Gaussian node attribute $Y$ is independent of the network structure. We demonstrate that the structure of the network directly affects the convergence rate to normality of these measures as the size of the network increases. We further establish that, in some instances, the mean values of these dependence measures under the null hypothesis remain non-negligible asymptotically and must therefore be explicitly accounted for when calculating the test statistics. Applications to a variety of simulated and real networks also reveal that the normal approximation performs well only when the network is not strongly heterogeneous. Network topology determines both the convergence rate to normality and whether the limiting distribution is Gaussian. In dense networks whose degree heterogeneity does not vanish, we further show that assortativity can fail to be a valid test statistic even though Moran's $I$ remains well behaved, whereas a dominating node invalidates both.

We present an exposition of an argument, discovered by GPT-6 Pro, that gives an optimal bound for the polynomial Littlewood-Offord problem. Namely, let $F$ be a degree-$d$ multilinear polynomial that contains $r$ degree-$d$ monomials involving disjoint sets of variables. Then, for i.i.d. Rademacher random variables $\xi_1, \ldots, \xi_n$, we have $\mathbb{P}[F(\xi_1, \ldots, \xi_n) = 0] = O_d(r^{-1/2})$. This improves upon the previous bound of $(\log r)^{O_d(1)} r^{-1/2}$ due to Meka, O. Nguyen, and Vu, and resolves a conjecture attributed to H. Nguyen and Vu. The key part of the proof is an estimate for the total influence of bounded-degree rational functions, which resolves a recent conjecture of Kothari, Kovacs-Deak, Wang, and Yang.

Given any fixed integer $k \ge 2$ and sufficiently large $d$, we show that the largest possible fractional chromatic number of a $k$-uniform $d$-degenerate uncrowded hypergraph $H$ (i.e., with girth at least $5$) satisfies \[ \chi_f(H) = (1 + o_d(1)) \left((k-1)\,\frac{d}{\log d}\right)^{\frac{1}{k-1}}. \] In fact, we prove that this holds for $k$-uniform $d$-degenerate hypergraphs of girth at least $g$, for any given $g \ge 5$. As a corollary, we obtain improved bounds on the fractional chromatic number of $d$-degenerate linear hypergraphs. This work builds upon a recent result by Allen, Dhawan, and Noel, extending it from graphs to hypergraphs. In addition to overcoming the new difficulties that arise in the hypergraph setting, our approach yields a simpler proof even in the original graph case. Our proof of the upper bound uses a simpler iterative procedure for sampling independent sets. We also establish bounds for fractional colorings with local demands, a framework introduced by Kelly and Postle, verifying a recent conjecture of Yu and Zhang. As a consequence, we obtain a degree-sequence bound on the independence number of uncrowded hypergraphs with a leading constant matching the shattering threshold. For the matching lower bound, we use a hypergraph variant of the uniform attachment model and harmonic vertex weights to bound the fractional chromatic number via linear programming duality, then remove all short cycles by deleting vertices of negligible total weight. We also replace the Catalan-number argument used in the graph case with a matrix-norm estimate, simplifying the analysis.

Motivated by recent results on the order of connected graphs with positive Lin–Lu–Yau Ricci curvature under forbidden minor or forbidden subgraph conditions and minimum degree assumptions, we prove that, for every integer $t\ge 5$, every connected graph $G$ with no $K_t$ minor, minimum degree at least $t-1$ and positive Lin–Lu–Yau Ricci curvature on every edge satisfies \[ \Delta(G)=O(t^5\log^{3/2}t) \quad\text{and}\quad |V(G)|<2t\Delta(G)^6=O(t^{31}\log^9 t). \] The minimum degree condition $t-1$ is best possible. Moreover, the bound on the maximum degree $\Delta(G)$ extends to locally finite graphs and, consequently, every connected locally finite graph satisfying these conditions is finite.

We study a discrete Lagrangian model for a one-dimensional chain of rigid rods coupled by elastic springs. We characterize the geometry of its zero-energy constraint and distinguish the spinner and flipper regimes. Under graph-type hypotheses, selected zero-energy chains converge, after rescaling, to trajectories of an autonomous ordinary differential equation. For the physical linkage, we identify a simple alternating selection criterion under which chains form a unique monotone heteroclinic family in the spinner regime. In the flipper regime, we determine the topology of the constraint set and the number of successors, but leave global branch selection open. We also derive one- and two-component continuum residual expansions, distinguishing consistency from convergence of time-dependent solutions. Reproducible numerical experiments test the selected spinner recurrence and its asymptotic multiplier; for a simplified one-component model with smooth periodic data, they recover first- and second-order discrepancies for the leading and corrected continuum equations, respectively.

Battery energy storage systems must respect strict state-of-charge (SOC) limits to prevent overcharge and deep discharge, a task complicated by measurement noise and costly-to-model nonlinear dynamics. Data-enabled predictive control (DeePC) resolves the costly modeling issue by predicting future behavior directly from data. However, its regularization robustifies only the prediction and not the constraints. Scenario-based DeePC (Scenario-DeePC) extends DeePC with the scenario approach, building constraint robustness fully data-driven from observed prediction errors rather than an assumed disturbance distribution. This paper presents the first real-world deployment of Scenario-DeePC, on a grid-connected battery system at the NEST research facility, whose SOC estimate exhibits abrupt, irregular recalibration jumps in addition to ordinary noise. Compared to standard DeePC, Scenario-DeePC achieves comparable tracking performance with substantially fewer constraint violations. Its adaptive scenario buffer further tightens constraint handling automatically, improving robustness to unpredictable recalibration events.