Papers

In higher category theory, lax colimits are often understood to be a more useful and powerful generalization of usual (homotopy) colimits, which can be recovered from the lax colimit by a suitable localization. However, lax colimits do not provide the correct notion of gluing from the geometric perspective. Indeed, they are incompatible with the notion of categorical dimension, the Gray tensor product, and other basic geometric operations. In this paper, we develop the theory of oriented colimits, which correct the defects of lax colimits, and agree with lax colimits in dimension less than or equal to one. In order to study oriented colimits, we introduce a version of the Grothendieck construction which is compatible with enrichment in the Gray tensor product. We prove that the Grothendieck construction induces an equivalence between cartesian fibrations and presheaves of $(\infty,\infty)$-categories, which is enriched in the Gray tensor product of $(\infty,\infty)$-categories. Oriented colimits simultaneously generalize the concept of lax colimits and the Gray tensor product, and differ from lax colimits in much the same way in which the Gray tensor product differs from the cartesian product. We demonstrate the necessity of oriented colimits by showing that various fundamental constructions in higher category theory fail to be lax colimits but are instances of oriented colimits. As applications, we classify higher-categorical principal bundles, represent higher dimensional adjunctions by bicartesian fibrations of $(\infty,\infty)$-categories, and obtain higher categorical versions of Quillen's Theorems A and B, which admit very natural formulations in our framework.

We construct an autonomous Lipschitz fast dynamo on the flat three-torus whose particle flow has zero topological entropy and whose ideal induction equation has no exponential growth. For every sufficiently small positive magnetic diffusivity, the induction operator has an eigenvalue with real part bounded below by a positive constant independent of the diffusivity. A corresponding non-zero real-valued magnetic field satisfies an exact exponential growth law in $\mathrm{L}^2$. For the same velocity field, every ideal solution grows at most linearly in $\mathrm{L}^2$. Moreover, the particle flow and its inverse have Lipschitz constants growing at most linearly in time. The velocity field is real-valued, divergence-free, differentiable everywhere and smooth away from one circle, but is not $\mathrm{C}^1$.

We show that the category of twisted/untwisted representations of the Heisenberg conformal net $\text{Heis}$ is a continuous Tambara-Yamagami category for the group $\mathbb{R}$. We compute the associators and the $\mathbb{Z}/2$-crossed braiding. By taking a $\mathbb{Z}/2$-equivariantization, we obtain an explicit computation of the braided continuous tensor category of representations of the fixed-points conformal net $\text{Heis}^{\mathbb{Z}/2}$. This provides the first explicit computation of a category of representations of a conformal net containing irreducible representations whose tensor product is a direct integral of irreducible representations.

We develop a local-to-global differential framework for additive decomposition problems involving multiplicative subgroups of prime fields. Starting from Hanson–Petridis-type auxiliary polynomials, we use degree bounds, in the spirit of Stepanov's method, to lift local coefficient relations at their roots to global differential identities. This yields a unified treatment of \[ H=A+B,\qquad H\cup\{0\}=A-A,\qquad H=A\mathbin{\widehat{+}} A,\qquad H\cup\{0\}=A\mathbin{\widehat{+}} A, \] where $H$ is a proper multiplicative subgroup. This circle of problems is motivated by Sárközy's conjecture on the additive irreducibility of nonzero quadratic residues and its generalizations to multiplicative subgroups. Rudnev and Tyrrell recently classified all decompositions $H=A+B$, building on the approach introduced by Hanson–Petridis and further developed by Kalmynin. Our framework gives a new polynomial proof of the Rudnev–Tyrrell classification and substantially streamlines the existing proofs: it gives an independent proof of Kalmynin's equal-size theorem and reduces the classification to direct coefficient comparisons, avoiding the residue-theoretic input and more elaborate arithmetic analysis of earlier proofs. It also yields a streamlined proof of Kalmynin's resolution of a conjecture of Lev and Sonn on $H\cup\{0\}=A-A$. For the two restricted-sumset problems, we obtain complete classifications, substantially improving earlier results of Shkredov and Yip. We also establish some stability refinements.

We study the dependence on time of the spatial regularity of solutions to the periodic Benjamin–Ono equation with real, mean-zero initial data of bounded variation. For every datum with a nonzero jump, we prove that continuity and essential boundedness at an irrational time $t=2\pi\alpha$ are equivalent to the arithmetic condition $\sum_j\log(q_{j+1})/\sqrt{q_j}<\infty$, where $q_j$ are the continued-fraction denominators of $\alpha$. We also determine the exact Hölder regularity and the Besov scale for spatial exponents $2<p\leq\infty$. We derive a high-frequency energy formula and prove the same sharp local criteria on every spatial interval at every irrational time for an open dense set of BV data. At rational times, piecewise smooth data give finite sums of logarithmic and jump kernels modulo a continuous remainder. For arbitrary BV data with a jump, the nonlinear and linear solutions have different logarithmic Poisson coefficients at every sufficiently large reduced denominator, uniformly in the numerator. The proofs combine the smoothing of Tao's gauge transform with estimates for quadratic Fourier series. A finite averaging argument isolates an atom of the initial gauge measure while retaining the real nonlinear reconstruction.

For a non-decreasing sequence $S=(a_1,a_2,\ldots,a_r)$ of positive integers, an $S$-packing coloring of a graph $G$ is a partition of $V(G)$ into sets $A_1,\ldots,A_r$ such that any two distinct vertices in $A_i$ are at distance greater than $a_i$, for every $i\in\{1,\ldots,r\}$. Gastineau and Togni [Discrete Math. 339 (2016), 2461–2470] asked whether every subcubic graph, except the Petersen graph, is $(1,1,2,3)$-packing colorable. In this paper, we prove that every claw-free subcubic graph is $(1,1,2,3)$-packing colorable. Moreover, we show that every connected claw-free subcubic graph, except a single graph $\mathcal{H}$, is $(1,1,3,3,3)$-packing colorable, thereby confirming a conjecture of the first two authors. Both results are best possible. Our proofs rely on a structural framework based on the skeleton and core graphs of a claw-free subcubic graph, together with a Hall-type matching argument that reduces the construction of suitable $3$-packings to a matching problem in an auxiliary bipartite graph.

Multi-level apportionment allocates integer seats through a hierarchy of groups. Schmidt-Kraepelin, Suksompong, and Wijaya proved that, at every fixed house size, lower and upper quota can be satisfied simultaneously; they also constructed house-monotone rules satisfying either quota separately. They left open whether one rule can satisfy lower quota, upper quota, and house monotonicity together, even when quota is required only relative to the root. We give a negative answer. For a full binary comb with $D$ equally entitled leaves, every house-monotone allocation sequence induces a sequence of seat recipients. Quota for the nested comb groups would force every grid-aligned prefix discrepancy to be below one. A midpoint embedding then bounds the full interval discrepancy by this quantity plus $1/2$, contradicting Schmidt's logarithmic lower bound. Conversely, a binary van der Corput seat schedule has comb-prefix error at most $\log D/(3\log 2)+1$. Thus the optimal worst-case error on the comb is $\Theta(\log D)$, and for sufficiently large finite $D$ no house-monotone quota rule exists. The proof isolates a static–dynamic gap: each house size admits a quota-feasible allocation, but the feasible allocations cannot be embedded into one monotone path. In quantization language, the result characterizes the order of the embedded-quantization penalty for progressive one-hot rounding on the comb.

We investigate a broad class of mixed local and nonlocal degenerate $p$-Laplace equations with general right-hand sides. The degeneracy is governed by Muckenhoupt $A_p$-weights, yielding a highly nonuniform elliptic framework in which both the local and nonlocal operators may degenerate simultaneously. We establish a comprehensive local regularity theory, including local boundedness and lower semicontinuity of weak subsolutions, weak Harnack inequalities for weak supersolutions, Harnack inequalities, and local Hölder continuity of weak solutions. Our approach combines weighted analytic techniques with the De Giorgi–Nash–Moser iteration method, adapted to the mixed local–nonlocal setting. To the best of our knowledge, this is the first systematic regularity theory for mixed local and nonlocal equations with Muckenhoupt weights. In particular, our results are new even for homogeneous linear equations ($p=2$) under the natural assumption $w\in A_2$, and therefore substantially extend the existing regularity theory for mixed local–nonlocal equations to a degenerate weighted framework with general right-hand sides.

We study dual volume approximation of the Euclidean ball by polytopes with a prescribed number of $k$-dimensional faces. This continues the authors' previous work on intrinsic volume approximation of the ball by polytopes with a fixed number of $k$-faces, and develops the corresponding dual radial theory. Our first main result gives a nonasymptotic lower bound for the volume deficit of an inscribed polytope $P_M\subset B_d$ with at most $M$ $k$-faces, for $0\leq k\leq \lfloor d/2\rfloor$. The estimate has the form \[\operatorname{vol}_d(B_d\setminus P_M) \geq \frac{1-e^{-1}}{2}\kappa_d\min\left\{ 1,\frac{d}{2}\left(\frac{\omega_d}{4\kappa_{d-1}}\right)^{\frac{2}{d-1}} M^{-\frac{2}{d-1}}\right\},\] and, in the vertices case $k=0$, in the large-$M$ regime it recovers the order of the lower bound of Gordon, Reisner and Schütt (J. Approx. Theory, 1997). We also prove the polar counterpart for the mean width excess of circumscribed polytopes with at most $M$ $k$-faces, for $\lceil d/2\rceil-1\leq k\leq d-1$. More generally, using the analytic extension of the dual volume deviations introduced by Besau, Hoehner and Kur (Int. Math. Res. Not., 2021), we obtain nonasymptotic lower bounds for all $q\in\mathbb R$, including $q=0$, in both the inscribed and circumscribed models.

For $0<\alpha\leq1$, define $(\mathcal T_\alpha f)(z) :=\int_{B(z,\alpha(1-|z|))}f(\zeta)\,dA(\zeta)$ for $z\in\mathbb D$, where $dA$ is planar Lebesgue measure. We prove that $\mathcal T_\alpha$ is injective on $C(\mathbb D)\cap L^\infty(\mathbb D)$ for $0<\alpha<1$, and that $\mathcal T_1$ is injective on $L^1(\mathbb D)$. In contrast, for each $0<\alpha<1$ there is an injective linear map from $C_c^\infty((0,\alpha))$ into the kernel of $\mathcal T_\alpha$ on $C^\infty(\mathbb D)$; every nonzero function in its image is necessarily unbounded near $\partial\mathbb D$. Under the area-measure interpretation, these results give a complete answer to Hayman–Lingham Problem~7.29, attributed there to L.~Zalcman. The proof combines generalized Abel equations, an Euler–Poisson–Darboux energy argument, and Volterra continuation.

Imaginary Liouville theory was recently proposed as a path integral construction of a non-rational conformal field theory (CFT) with central charge less than one (Usciati et al. 2026). In the present work, we establish Ward identities and Belavin-Polyakov-Zamolodchikov differential equations in this framework, assuming a precise conjecture on the decay of the Laplace transform of Imaginary Gaussian multiplicative chaos. These results provide a first step towards a rigorous implementation of the conformal bootstrap program for conformal field theories with central charge lower than one. This work is the first to investigate the intrinsic properties of Imaginary Liouville theory beyond exact computations.

We present a simple Gaussian approximation to the finite-sample distribution of the classical ridge regression estimator. Our approximation captures the fact that, in finite samples, the ridge regression estimator trades off bias and variance to reduce estimation and prediction error. Our approximation is based on nonstandard asymptotics where $i)$ we let the estimator's regularization parameter grow proportionally to the sample size; and $ii)$ we treat the population regression coefficients as local to the reference vector that defines the estimator's direction of shrinkage. In contrast to other asymptotic approximations in the literature, we allow for general forms of heteroskedasticity and autocorrelation in the data generating process (at the cost of considering a low-dimensional model where the number of covariates is not allowed to grow with the sample size). We use our simple Gaussian approximation to propose two new strategies to select the regularization parameter for the ridge regression estimator. The suggested strategies select the regularization parameter to minimize either average or worst-case excess prediction risk, where risk is computed using our suggested Gaussian approximation.

This paper is concerned with one-bit mean estimation, where each independent sample is represented by a single binary message. We consider distributions on $\mathbb{R}$ with mean in $[-\lambda,\lambda]$ and absolute $k$-th central moment at most $\sigma^k$, where $k>1$ is fixed. For this class, previous work attained the optimal sample complexity for general queries using a two-stage protocol. The first stage localizes the mean. The second-stage queries are chosen after localization and refine the estimate around the decoded center. We show that this interaction can be avoided by constructing a randomized fully non-adaptive protocol that fixes all queries before observing the data and matches the optimal adaptive sample complexity. For target accuracy $\epsilon$ and confidence $1-\delta$, its sample complexity scales as \[ \log\frac{\lambda}{\sigma} + \begin{cases} (\sigma/\epsilon)^2\log(1/\delta), & k>2,\\ (\sigma/\epsilon)^2\log(\sigma/\epsilon)\log(1/\delta), & k=2,\\ (\sigma/\epsilon)^{k/(k-1)}\log(1/\delta), & 1<k<2, \end{cases} \] up to constants depending only on $k$. In the range covered by the known lower bound, this rate is minimax optimal even among fully adaptive protocols. This gives a negative answer to the COLT 2026 open problem asking whether interaction is necessary for order-optimal one-bit mean estimation with general queries \citep[Open Problem~1]{lau2026open}.

In a recent breakthrough, OpenAI proved that the $k$-color Ramsey number of the triangle $C_3$ grows super-exponentially, more precisely, they proved that $R_k(C_3)\ge k^{k/3-o(k)}$. In this short note, we present a modification of their recursive construction that works for multicolor Ramsey numbers of fixed odd cycles. More precisely, for $p\ge 1$, let $\mathcal{O}_p=\{C_3,C_5,\ldots,C_{2p+1}\}$. We show that \[ R_k(\mathcal{O}_p)\ge (\log^{(p-1)}k)^{k/3-o(k)} \] for every fixed $p$, where $\log^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This immediately implies that for every fixed odd cycle, the multicolor Ramsey number is superexponential in the number of colors. The presented proof was found autonomously by ChatGPT 5.6 Pro/Sol.

We study the long-time behavior of the one-dimensional defocusing cubic nonlinear Schrödinger equation with initial data in $L^2(\mathbb R)$. We first prove that the standard fixed-profile modified-scattering ansatz fails at the unweighted $L^2$ endpoint for the one-dimensional defocusing cubic nonlinear Schrödinger equation. More precisely, there exists a real-valued datum $$u_*\in L^1(\mathbb R)\cap\bigcap_{m\in\mathbb N}H^m(\mathbb R), \qquad xu_*\notin L^2(\mathbb R).,$$ for which the corrected Fourier profile has no strong $L^2$ limit. Secondly, for every $L^2$ datum, we prove generalized modified scattering: in the physical ray coordinates $x=2tz$, the rescaled solution admits a strong $L^2$ asymptotic representation whose amplitude is the square root of the conserved transmission density and whose unit complex phase is determined by the initial datum.

Countable subsets of solutions of higher-order nonlinear ODEs and elliptic PDEs with indefinite non-coercive operators from the reaction-diffusion, thin film and dynamical system (DS) theories are obtained via a gluing/matching argument. In particular we study some classic and quasilinear degenerate ODEs in $\mathbb{R}$, with boundary conditions at infinity $F(\infty)=0$, with non-odd nonlinearities such as $$ \begin{matrix} F^{(4)} =-F+F^2,\, \,\, F^{(4)}=-F+F^2{\rm e}^{F-1}, \, (|F''|F'')''=-F + F^2, \\ F^{(4)} =-|F|F+F^2, \,\,\,F^{(4)} =-F^3+F^4, \,\,\, F^{(6)}=F-F^2, \end{matrix} $$ etc, as well as equations with odd and non-smooth nonlinearities like $$F^{(4)}=-F+F^3, \,F^{(4)}=-F-(|F|F-F)'', \,\, F^{(4)}=- \frac F{\sqrt{|F|}}-(F^3-F)'',\,\mbox{etc.}$$ Some of these ODEs are Hamiltonian and were studied in detail in the DS theory. On the basis of nonlinear operators, elliptic PDEs and variational theory, related polyharmonic elliptic equations in $\mathbb{R}^N$, $F(\infty)=0$, such as $$\Delta^2 F=-F+F^2, \quad \Delta^2 F=-F -\Delta (F^2-F), \quad \Delta^3 F=F-F^2, \quad \mbox{etc.};$$ are also shown to admit countable families of solutions. For such equations with non-odd functionals associated Lusternik–Schnirel'man (L–S) genus/category variational theory guaranteeing existence of a sequence of critical points does not apply. These ODEs and elliptic PDEs (e.g., in the radial setting) are shown to admit at least two basic countable families ${\mathcal F}_{1,2}$ of positively dominant solutions connected with two periodic orbits $\Gamma_{\rm max/min}$. Patterns obtained by gluing together via exponentially decaying tails of arbitrary finite samples from $\Gamma$'s form a countable subset of homoclinics in $\mathbb{R}^4$ of an arbitrary complexity.

We show that the methods of rewriting theory to establish coherence theorems can applied at the level of quasicategories. More precisely, for any category C which admits a presentation by a convergent rewrite system, we show that the corresponding weak (infinity,1)-category admits an Amick-Groves-Squier style presentation whose generators correspond to (some of) the critical branchings of the rewrite system. This follows entirely from reinterpreting K. Brown's simplicial proof of the usual homological Anick-Groves-Squier presentation in terms of the Joyal model structure instead of the Kan-Quillen model structure. We give several applications of this to the theory of quasicategories, including a relatively general coherence theorem for loop-free planar category theoretic diagrams and a new proof that pushout of Dwyer maps are homotopy pushouts.

Conlon and Lee asked for strongly dominating graphs beyond norming graphs and even paths. We construct a two-parameter family of pairwise non-isomorphic $2$-connected strongly dominating graphs that are not seminorming, and hence lie outside the two classes of examples previously identified for signed strong domination. The construction uses cyclic amalgamation of two-rooted blocks. For root-reversible blocks, we characterize the generation of all even cyclic amalgams by local even-Schatten inequalities for transfer operators. We determine this criterion for $K_{2,m}$, with the roots in the part of size $m$: it holds exactly when $m$ is even. We also classify the connected outerplanar strongly dominating graphs and the connected root-reversible outerplanar blocks satisfying the universal cyclic criterion.

The Poisson-Boltzmann (PB) theory is a cornerstone of implicit solvent models for electrostatic analysis, and has found a great success in various biomolecular applications. However, in calculating polar solvation energy, one should consider that the structure of the protein changes upon transition from vacuum to water phases. To address this, here we report for the first time a generalized PB framework capable of accommodating nonrigid conformational changes without suffering from self-energy artifacts. For regularized PB models, in which the charge singularities are captured by the Green's functions, self-energies in the water and vacuum states will be analytically canceled. For non-regularized PB solvers, such as APBS and DelPhi, a simple thermodynamic cycle is proposed for nonrigid proteins by adding a Coulombic correction in vacuum. The generalized PB theory is validated using a perturbed two-atom system and a diverse set of proteins with different structures in vacuum and water, demonstrating its accuracy and robustness, regardless of the choice of sharp-interface and diffuse-interface PB models and different numerical solvers.

A famous conjecture of Erdős and Hajnal (1969) states that for every integer $g\ge 4$ there is a smallest function $f_g:\mathbb{N}\to\mathbb{N}$ such that every graph of chromatic number at least $f_g(k)$ contains a subgraph of chromatic number $k$ and girth at least $g$. So far, this has only been proved for $g=4$ by Rödl (1977), with $f_4(k)$ bounded by a tower of height $\Theta(k^2\log k)$. We exhibit a surprising connection between finding high-chromatic subgraphs of large odd-girth (avoiding short odd cycles) and lower-bounding multicolor Ramsey numbers of odd cycles. Using this connection, we prove that for every odd $g\ge 5$ there is a function $h_g:\mathbb{N}\to\mathbb{N}$ growing as a power tower of height $\frac{g-3}{2}$ such that every graph of chromatic number at least $h_g(k)$ contains a subgraph of chromatic number at least $k$ and odd-girth at least $g$. This proves a conjecture of Mohar and Wu (2018), addresses a question of Erdős and Hajnal (1975), and for $g=5$ improves Rödl's bound on $f_4(k)$ to a single-exponential. We extend this to a much more general meta-theorem which applies to many graph parameters: if $f$ is the fractional chromatic number, the Hall ratio, or the strict vector chromatic number (Lovász-Theta-function of the complement), then for every $k,g\in\mathbb{N}$, every graph $G$ with sufficiently large $f(G)$ contains a subgraph $G'$ of odd-girth at least $g$ with $f(G')\ge k$. The key Ramsey-theoretic ingredient is a new lower bound on Ramsey numbers of odd cycles. For $p\ge 1$, let $\mathcal{O}_p=\{C_3,C_5,\ldots,C_{2p+1}\}$. We show that $R_k(\mathcal{O}_p)\ge(\log^{(p-1)}k)^{k/3-o(k)}$ for every fixed $p$, where $\log^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This yields the first superexponential lower bound on multicolor Ramsey numbers of fixed odd cycles, and extends the recent breakthrough by OpenAI for triangles.

We study automorphic functions for square-zero extensions $C$ of curves $\overline{C}$ over finite fields, a study initiated by Braverman-Kazhdan-Polishchuk in [BKP23]. More precisely, we study the cuspidality and Hecke-finiteness of the functions in the orbit decomposition introduced in loc. cit. for split connected reductive groups $G$, generalizing some of the results for $G=\mathrm{PGL}_2$. As a result, for $G=\mathrm{PGL}_3$, we prove a new case of a conjecture in [BK23] concerning the finite-dimensionality of the space of unramified Hecke-finite functions. We also introduce a formulation of support bounds for spherical cuspidal and Hecke-finite functions in terms of the Harder-Narasimhan stratification of $G$-bundles on the reduced curve. Using representation-theoretic constructions together with their geometric interpretations in terms of $G$-bundles on $C$ and certain twisted $G$-Higgs bundles on $\overline{C}$, we compute the optimal bounds in several cases and, in particular, determine the optimal bound for $G=\mathrm{PGL}_3$.

We determine the two cross-stage morphisms arising from adjacent Stiefel fiber sequences in the special linear tower. Over characteristic-zero fields, the first is the Wood morphism at even stages and vanishes at odd stages; the second is induced stably by the motivic Hopf element at even stages and is zero at odd stages. Complex realization consequently classifies oriented rank-$(n-2)$ vector bundles on the split quadric $Q_{2n-1}$ for $n\geq5$. We then apply these calculations to give exact corank-two splitting and efficient generation criteria for projective modules.

We characterize the finite sample behavior of the log-likelihood ratio statistic in binary logistic regression, uniformly over both the design and the target parameter. For $n\geq d\geq 3$, we determine, up to universal constants, its worst case $(1-\delta)$ quantile over all fixed collections of design vectors and all target parameters: \[ d\log\left(\frac{e n}{d}\right)+\log\left(\frac{1}{\delta}\right). \] This is a nonasymptotic analogue of the Wilks $\chi^2_d$ phenomenon and requires no regularity assumptions on the design. The low dimensional cases exhibit unusual behavior. The worst case quantile in dimension $d=2$ is sharply of order \[ \log\log\log n+\log\left(\frac{1}{\delta}\right). \] The worst case quantile in dimension $d=1$ is of order $\log(1/\delta)$, with no dependence on $n$. Finally, i.i.d. Gaussian design vectors recover the classical Wilks scale. In the regime $n\gtrsim d+\log(1/\delta)$, we prove the sharp bound \[ d+\log\left(\frac{1}{\delta}\right). \] Unlike existing asymptotic results, our bounds are uniform over the target parameter, which may depend on $n$, $d$, and $\delta$.

A simple min-max theorem is formulated and proved for the smallest modification (measured in $l_1$-norm) of an input cost function $w_0$ that makes a target arborescence $F_0$ of a digraph a cheapest arborescence. The constructive proof gives rise to a polynomial time algorithm for computing both a minimizer cost function on the primal side and a maximizer dual object in the min-max formula.

For an elliptic curve $E$ defined over a field $K$ of characteristic $0$ with $\operatorname{End}_K \! E \cong \mathbb{Z}$, we classify which isogeny graphs $\mathcal{G}(E/K)$ can occur. We first show that $\mathcal{G}(E/K)$ decomposes as a weak Cartesian product of its $p$-primary isogeny graphs, one for each prime $p$, thereby reducing the problem to classifying $p$-primary isogeny graphs. We then show that each such graph is isomorphic, as an edge-weighted graph, to a member of an explicit family of edge-weighted graphs $\mathcal{H}_{p^k}^r$ and $\mathcal{H}_{p^{\infty,+}}^r$, every member of which occurs as a $p$-primary isogeny graph except for $\mathcal{H}_{2^k}^0$ for $k\ge 2$. The proof relies on a detailed study of the $p$-adic Galois representation attached to $E$, through which we identify each graph with a subgroup of $\operatorname*{GL}\nolimits_{2}(\mathbb{Z}_{p})$. More generally, we identify subgroups of $\operatorname*{GL}\nolimits_{2}(\widehat{\mathbb{Z}})$ for each possible isogeny graph and describe their corresponding modular curves, completing, in the genus $0$ case, the explicit parameterization of isogeny graphs via parameterized isogenous families of elliptic curves. We also introduce the $p$-blooming invariant $\mathfrak{I}_p(E/K)$, an isogeny class invariant determining the value of $r$ in the $p$-primary isogeny graph, and show that elliptic curves over fields with a real embedding attain the smallest possible value. As applications, we characterize the isogeny graphs of elliptic curves with potential complex multiplication; give an algorithm for determining the isogeny graph from the adelic Galois representation; recover the classification of rational isogeny graphs; and, under GRH, classify the isogeny graphs occurring over certain number fields.

We study policy optimization for gain-scheduled linear quadratic regulation, where one schedule of gains, interpolated through fixed weighting functions, is optimized against a family of plants. The resulting cost can develop spurious local minima, and existing convergence certificates are either local or severely conservative. We establish an exact identity: when the gradient of the cost is evaluated with the minimizer's closed-loop covariances, the scheduled cost is star-convex about the minimizer. The identity holds on the entire feasible set, for any parametrization of the schedule. Convergence is governed by a single dimensionless ratio. Wherever the ratio satisfies a threshold condition, gradient descent converges linearly to the optimum on entire sublevel regions at an explicit rate; at every spurious stationary point the condition necessarily fails. Experiments that maximize the ratio directly show the threshold to be an active boundary of the landscape. This extended version contains the complete proofs and additional numerical studies omitted from the letter for space.

In this paper, we establish quantitative Denjoy–Koksma type estimates for higher-dimensional quasi-periodic torus rotations. For Diophantine frequency vectors, we establish quantitative estimates on the discrepancy between Birkhoff averages and spatial averages for observables with various Besov-type regularities. The Besov framework is naturally adapted to the small-divisor structure of the problem, allowing a sharp analysis of resonant frequencies and yielding optimal rates throughout the entire parameter range. These estimates yield, to the best of our knowledge, the sharpest currently available convergence rates for Hölder continuous observables. As applications, we obtain substantially improved quantitative homogenization results for Hamilton–Jacobi equations in spatially quasi-periodic settings, as well as nearly optimal statistical regularity estimates for invariant measures under perturbations.

Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying c-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. More generally, for a Gaussian to log-concave coupling, the ordinary rectified flow can reach a fixed point after one step from independent initialization. In contrast, under suitable regularity and uniform-integrability assumptions, iterative c-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a Hölder ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative c-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension \(d \ge 3\) and a nearly parametric rate for \(d=1,2\).

For piecewise constant fan subsolutions to the isentropic Euler equations, the entropy and action rates of any associated convex integration solution, relative to a common reference solution, depend only on the fan and are computed explicitly. Moreover, these explicit expressions may be decomposed into a kinetic energy mismatch and an internal energy mismatch, from which we characterize agreement of Dafermos' entropy rate criterion with action rate criteria. The known disagreement for the Chiodaroli–Kreml two-shock data is recovered. For the planar contact reference, we prove that both criteria strictly prefer every convex integration solution associated with a strictly dissipative admissible fan to the classical contact discontinuity. Certified computations locate the Krupa–Székelyhidi solutions in this region, where the criteria agree, while Horimoto's analytical construction provides such fans for every strictly increasing pressure law.

Very often, accuracy of analysis and forecasting (multiple coefficient of regression and residual means) obtained for a sample used to formulate a regression model is not equal to the accuracy achieved for another homogeneous sample. Indeed, accuracy of analysis and forecasting based on another sample is much worse. This is explained by the discrepancy between a postulated and a real model. This discrepancy is caused by the redundancy of the number of terms of an approximating series. It describes a phenomenon under study with the sample noise. To filter this noise it is necessary to correctly choose the terms of an approximating series and to determine their number, which in turn depends on sample size. It is possible to include in this approximating series not only independent variables but also various functions of the value of the independent variables (e.g., squares, cubes, algorithms, etc.). This paper gives the procedure for selection of an approximating series.