Papers

This article introduces crystallographic wraiths, new combinatorial invariants attached to Euclidean lattices and linked to the geometry of Voronoï cells. They are based on geometric properties of shortest linear dependencies among relevant Voronoï vectors. Moreover, it explores the new theory of crystallographic wraiths, proves that only finitely many occur in each dimension, and extends the construction to a larger class of combinatorial objects. As an application, it provides a new combinatorial proof of Voronoï's finiteness theorem for perfect lattices.

We compute parabolic $\mathrm{SL}_2(\mathbb{C})$-character schemes of knots using the parabolic quandle. To this end, we introduce sign-refined arc-colorings and show that their sign data encode the obstruction classes of the induced parabolic representations. We also establish a correspondence between the schemes defined by sign-refined arc-colorings and the parabolic character scheme. This yields a practical diagrammatic method for computing complete lists of parabolic characters, together with their multiplicities and obstruction classes. Using this method, we verify a conjecture of Bénard and Detcherry for all small knots with at most $12$ crossings.

We consider the defocusing nonlinear Schrödinger equation for step-like data connecting two, in general different, genus-one periodic states. Starting from the Jost solutions associated with the left and right backgrounds, we determine the corresponding scattering data and reorganize them through a scalar conjugation and a suitable symmetrization. The resulting formulation identifies the inverse problem with a full dark-soliton gas Riemann–Hilbert problem supplemented by a radiative jump on the real axis. In this way, the effects of the two periodic backgrounds are separated, and a precise link is established between step-like periodic scattering and the full-gas construction. Assuming the absence of discrete eigenvalues and appropriate regularity of the scattering data, we establish the existence and uniqueness of the associated Riemann–Hilbert problem.

We introduce the $h$- and $v$-topologies in the context of logarithmic geometry and discuss their applications to log étale cohomology, log differential forms, and log motives.

Estimating the parameters of an achievement scalarising function is a challenging task in multiobjective optimisation. Achievement scalarising functions are widely used in several contexts, including preference-based optimisation methods and algorithmic procedures for generating or selecting efficient solutions. However, their effectiveness strongly depends on the appropriate specification of their parameters, such as weights and reference points. In this paper, we propose an ordinal-regression-based methodology to estimate the parameters of an achievement scalarising function from preference information provided by the DM. In particular, the proposed approach allows us to infer both the weights associated with the objectives and the reference points characterising the scalarising function. The methodology is illustrated through a didactic example based on a multiobjective knapsack problem, which shows step by step how the preference information is translated into model parameters. The example illustrates that ordinal regression provides a flexible and interpretable framework for calibrating achievement scalarising functions and supporting the elicitation of decision-maker preferences in multiobjective optimisation.

We propose an integral framework for the anticyclotomic Iwasawa theory of CM elliptic curves $E$ at primes $p$ ramified in the CM field. The $\varepsilon$-constants of the geometric specialisations of the associated $p$-adic conjugate symplectic self-dual deformation equidistribute between $\pm 1$ within every layer of the anticyclotomic tower, and none of the geometric specialisations are trianguline at $p$. We define signed Selmer groups via the Lagrangian local conditions arising from the local sign decomposition established in the prequel \cite{BKNO}, and our central result is the formulation and proof of an integral Iwasawa main conjecture relating one of them to the $p$-adic $L$-function $\mathscr{L}_p(E)$ constructed there. We further show that $\mathscr{L}_p(E)$ interpolates the central Hecke $L$-values of the twists with $\varepsilon$-constant $+1$, including twists of arbitrary infinity type, and relate its values at twists with $\varepsilon$-constant $-1$ to the $p$-adic logarithm of certain Selmer elements. This provides the first Iwasawa main conjecture in terms of a $p$-adic $L$-function and Selmer groups for a $p$-adic deformation admitting no trianguline geometric specialisation. The proofs rest on our resolution of a Rubin-type conjecture for the underlying local deformation, together with a theory of plus/minus local points along the anticyclotomic tower, based on the Gaussian plus/minus cyclotomic polynomials rooted in Gauss' Disquisitiones Arithmeticae.

We establish a spectral representation theorem for compact operators that are self-adjoint with respect to the duality mapping on smooth Banach spaces. The result is a natural analogue of the classical spectral theorem for compact self-adjoint operators on Hilbert spaces.

Condorcet's paradox is a foundational result in social choice theory, showing that no matter which candidate wins an election, a majority of voters may prefer some losing candidate. Worse still, even if the election can choose a committee of $k$ winners, some loser may beat every winner in a majority vote. Recent work showed that this obstruction can be sidestepped by relaxing the majority threshold. For all $\varepsilon > 0$, any election can select a committee of $O(1/\varepsilon^2)$ winners such that no loser is preferred to every winner by $\frac12 + \varepsilon$ fraction of voters. We present a simple construction, found by GPT-5.6 Sol Ultra, which proves that this result is tight up to a constant factor.

For a closed, connected 2-manifold $M$, and for a triangulation $T$ of $M$, we write $V(T)$ in place of the vertex set associated with $T$. A cyclic coloration of a triangulation $T$ of $M$ refers to a face coloring of $T$ such that: For each $v \in V(T)$, the faces incident to $v$ have distinct colors. Chen and Lawrencenko [Yokohama Math. J., 1999] conjectured that there exists a constant $C(M)$ (depending only on $M$) such that $|V(T)| + C(M)$ colors suffice for there to exist a cyclic coloration of a triangulation $T$ of $M$. We prove this conjecture, using a greedy algorithm related to the Euler-Poincaré formula for surface triangulations. Chen and Lawrencenko also conjectured that: If $M$ is not the projective plane and $T$ is a triangulation of $M$ that is minimal with respect to the number of vertices, then $\xi(T) = |V(T)| = V_{\min}(M)$, where $V_{\min}(M)$ denotes the minimum possible number of vertices among all triangulations of the 2-manifold $M$, and where $\xi(T)$ denotes the minimal value $k$ such that $T$ admits a cyclic coloration with $k$ colors. We disprove this latter conjecture via an explicit counterexample, using an 8-vertex triangulation of the Klein bottle with 16 faces. It appears that both of the Chen-Lawrencenko conjectures have remained open, prior to our work.

Motivated by developments in metric regularity theory for set-valued mappings, this paper introduces several directional regularity notions that are weaker than the corresponding classical regularity properties. We investigate the relationships among these concepts and establish a classification framework based on directional minimal time functions. Particular attention is devoted to directional orbital regularity and directional orbital Aubin continuity, which are shown to provide a natural setting for the analysis of fixed point phenomena. Using these notions, we derive approximate and exact fixed point theorems for set-valued mappings under directional assumptions. The obtained results are expressed in terms of minimal time functions and orbital approximation procedures, allowing for anisotropic and asymmetric behaviors that are not captured by the classical nondirectional framework. We further apply the developed theory to directional coincidence point results, coupled fixed point theorems, and Milyutin-type perturbation stability properties. The proposed approach provides a unified directional perspective on fixed point theory and variational analysis. In particular, it recovers several known results from the literature as special cases and yields new extensions under substantially weaker assumptions than classical contraction or global regularity conditions.

Let the symmetric group $S_n$ act on the space of $n \times r$ real matrices by permuting rows, so orbits are multisets of $n$ points in $\mathbb{R}^r$. The basic $r$-symmetric tropical polynomials form a family of $\binom{n+r}{r}-1$ nonconstant invariants of degree at most $n$ that separates orbits and embeds the orbit space bi-Lipschitzly. We prove that this family generates the semifield of all $r$-symmetric tropical rational functions, answering a question raised in [J. Pure Appl. Algebra 223 (2019) 72-85]. Derksen showed that the invariant semifield of any permutation group $G \le S_N$ is generated in degree at most $N p_1 \cdots p_{|G|}$ ($p_i$ the $i$th prime), which for the row action is $nr p_1 \cdots p_{n!}$; the present result replaces this by generators of degree at most $n$. The generating expression is a finite minimum over the ways of re-assembling a multiset from its sorted columns, with penalties from the basic values that, via the bi-Lipschitz inequality, dominate a wrong re-assembly. The same penalties describe the image of the basic coordinate map as the zero set of a single tropical rational function and yield an expression algorithm. Subfamilies of the basic family containing the single-column values generate if and only if they separate. For any permutation group $G \le S_N$ the same mechanism generates the invariant semifield in degree at most $\max\{N, \binom{N}{2}\}$, a quadratic bound independent of the group order; combined with a genericity theorem of Cahill, Iverson, Mixon, and Packer, it yields $2N+1$ invariant tropical polynomials that separate orbits and $3N$ that generate, with at least $N$ necessary for each task. The quadratic bound is optimal: every $A_N$-invariant tropical polynomial of degree less than $\binom{N}{2}$ is $S_N$-invariant, so every separating family for the alternating group $A_N$ contains a member of degree at least $\binom{N}{2}$.

In this largely expository paper we provide a deep and explicit exploration and exposition of the tangent structure on the category of schemes $\mathbf{Sch}_{/S}$ whose tangent functor $T(X) = T_{X/S}$ is the relative tangent scheme of Grothendieck described in \emphÉléments de Géométrie Algébrique} 4. In particular we provide explicit descriptions of the ways that the bifibration of quasicoherent sheaves and bifbration of quesicoherent sheaves of algebras over schemes may be built from the ways in which the bifibrations of modules and commutative algebras over commutative rings interact. We also show the ways in which these interactions give rise to an explicit description of the standard tangent structure on the category of schemes in terms of sheaves of Kähler differentials, properties of the relative spectrum functor, and more. Finally, we show that quasi-coherent sheaves can be reconstructed from their category of differential bundles by showing that for quasi-separated schemes $X$ and $Y$, there is an isomorphism $X \cong Y$ if and only if there is an equivalence of categories $\mathbf{DBun}(X) \simeq \mathbf{DBun}(Y)$.

This paper studies entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built via a CSS-like stabilizer construction from pairs of classical locally recoverable codes (cLRCs), without requiring dual-containment. We define such codes through local recovery channels, give a sufficient stabilizer criterion for the construction, and derive Singleton-, Griesmer-, Plotkin-, and sphere-packing-like converse bounds on the parameters of the resulting pure CSS-like EA-qLRCs, along with a Cadambe–Mazumdar-like bound that, as in the classical case, lacks a closed form, plus a comparison of their relative tightness across finite-length and asymptotic regimes. We give necessary and sufficient conditions for a pure CSS-like EA-qLRC to attain the Singleton-like bound with equality; for the single-code case $\mathcal{C}_1=\mathcal{C}_2=\mathcal{C}$, this reduces to a simple condition on the hull dimension $s=\dim(\mathcal{C}\cap\mathcal{C}^\perp)$, which also fixes the entanglement count via $c=n-k-s$. We present CSS-like EA-qLRC constructions from classical LRC families—Tamo–Barg and cyclic codes—and characterize when these attain the Singleton-like bound, showing the cyclic families yield optimal codes while the Tamo–Barg construction, though valid, attains the bound only in the degenerate regime $k \le r$, where locality is vacuous. We complement these constructions with two Gilbert–Varshamov-like achievability bounds, via a classical parity-check augmentation and a sharper concatenated-code construction, and show both hold unconditionally for field size $q>3$ via a monomial-equivalence argument. Finally, we unify all bounds—converse and achievability alike—under a common maximally entangled regime, giving a single comparison of the achievable and forbidden rate–distance–locality region for CSS-like EA-qLRCs.

For Anosov diffeomorphisms the geometric approach for constructing Sinai-Ruelle-Bowen (SRB) measures works by pushing forward under the dynamics the leaf volume on a local unstable manifold or any measure that is equivalent to the leaf volume. Such a measure is called a reference measure. In applications one often needs to use a reference measure that is absolutely continuous with respect to leaf volume. In this paper we show that a reference measure can be chosen from a substantially more general class of measures which are the restriction of the leaf volume to any measurable subset of a local unstable manifold of positive leaf volume. Moreover, we do this in a more general settings of diffeomorphisms which are effectively hyperbolic on a set of positive volume. Such diffeomorphisms were introduced by Climenhaga, Dolgopyat and Pesin in \cite{CDP} where they used the geometric approach to construct SRB measures. Our results applies also to diffeomorphisms that are uniformly partially hyperbolic or more generally possess a dominated splitting.

Eilers showed that for AD algebras of real rank zero with bounded torsion in $\mathrm{K}_1$, the coefficient transformations $\kappa$ are redundant in the classification by ordered scaled total $K$-theory. In this paper we treat the unbounded torsion case and prove that, in contrast, $\kappa$ becomes necessary. Specifically, we construct two non-isomorphic unital AD algebras of real rank zero, $E_0$ and $E_1$, such that their ordered scaled total $K$-theory invariants agree when the $\kappa$-maps are forgotten, i.e., \[ \bigl( \underline{\mathrm{K}}(E_0), \underline{\mathrm{K}}(E_0)_+, [1_{E_0}] \bigr)_{\underline{\mathrm{K}}_{\langle\kappa\rangle}} \cong \bigl( \underline{\mathrm{K}}(E_1), \underline{\mathrm{K}}(E_1)_+, [1_{E_1}] \bigr)_{\underline{\mathrm{K}}_{\langle\kappa\rangle}} \] but are not isomorphic under the full $\Lambda$-module structure: \[ \bigl( \underline{\mathrm{K}}(E_0), \underline{\mathrm{K}}(E_0)_+, [1_{E_0}] \bigr)_{\Lambda} \not\cong \bigl( \underline{\mathrm{K}}(E_1), \underline{\mathrm{K}}(E_1)_+, [1_{E_1}] \bigr)_{\Lambda} .\] This completes the picture for the necessity of all three operations $\rho$, $\beta$, and $\kappa$ in this context.

Classical Barron spaces are function spaces specifically designed for shallow neural networks mostly with ReLU, $\mathrm{ReLU}^k$ (RePU) or Lipschitz continuous activation functions. In the present work, we introduce a generalized Barron space \( B_{\sigma}^{\varphi} \) for shallow neural networks with a generic activation function possessing certain smoothness properties. The subscript \( \sigma \) denotes the activation function, while the superscript \( \varphi \) controls the smoothness of the generalized Barron spaces, defined via a \( \varphi \)-weighted integral norm imposed on neural network parameter measures. Under certain assumptions on \( \varphi \) and \( \sigma \), we show that \( B_{\sigma}^{\varphi} \) can be continuously embedded into Sobolev spaces. We also explore the relationships among various function spaces for shallow neural networks, demonstrating that our definition encompasses most conventional ones. As applications of the proposed generalized Barron spaces, we derive approximation rates within these spaces and establish error bounds for numerical differentiation with regularization penalized by the newly introduced generalized Barron norm. Numerical examples confirm that the proposed spaces allow the construction of neural networks with varying degrees of smoothness while using the same activation function.

In this paper, we extend a weak adversary neural network pushforward method for solving time dependent Boltzmann equation and a weak formulation of the collision operator is proposed where an invertible neural pushforward mapping is used to generating samples given by the distribution governed by the Boltzmann equation. The training of the pushforward mapping is learnt by enforcing the weak form of the Boltzmann equation. Numerical results have demonstrated the effectiveness of the proposed method.

We exhibit nineteen explicit vectors in $\mathbb{C}^6$, with Gaussian-integer entries, that do phase retrieval: the intensities $|\langle a_j,x\rangle|^2$, $j=1,\dots,19$, determine every $x\in\mathbb{C}^6$ up to a unimodular scalar. This gives $m{\mathbb{C}}(6)\le 19=4d-5$, one below the generic count $4d-4$; combined with the lower bound $m{\mathbb{C}}(6)\ge 18$ of Wang and Xu, the minimal measurement number in dimension six is 18 or 19. The claim is verified by an exact rational computation, archived and rerunnable: after an elementary reduction, phase retrieval is equivalent to the emptiness of the real zero set of a system of five quadratic equations in five unknowns, and the archived eliminant of that system is a squarefree integer polynomial of degree fourteen with no real roots.

Two-stage distributionally robust optimization chooses a here-and-now decision and a wait-and-see decision policy that adapts to uncertainty realizations. Decision-rule methods specify the form of this adaptive policy in advance, whereas decomposition-generation methods construct second-stage value information iteratively. We connect these approaches through extreme points of the second-stage dual problem. Each extreme point defines an affine value piece, and a compatible primal basis can define an affine policy piece. Solving the first-stage problem may require only a subset of these pieces. We specialize an extreme-point generation method to solve the first-stage problem. A separate linear-programming procedure adds pieces until it recovers the recourse value over the uncertainty set, and the pieces can also be used to recover the optimal recourse policy. Building on the algorithm output, we develop a posteriori exactness test for conventional decision rules. We give extensions for degeneracy, rank-deficient recourse, and structured random recourse. The proposed algorithm is computationally efficient on the reported instances, and the results show that completing the second-stage value across the uncertainty set requires more extreme points than solving the first-stage problem alone.

For a finite hypergraph H, the complements of the models of its associated definite Horn CNF are exactly the stopping sets of H; hence the complements of its coatoms are the inclusion-minimal nonempty stopping sets. We study their output-sensitive enumeration from the hypergraph incidence representation. Our main result is a representation of arbitrary Horn model posets whose incidence size is linear in the incidence length of the normalized Horn input. Given a Horn CNF $\Gamma$, we construct a hypergraph $C(\Gamma)$ of rank at most three whose proper-model poset is inclusion-order isomorphic to the model poset of $\Gamma$; equivalently, each source model has a unique extension to a proper target model. Thus maximal models of $\Gamma$ correspond bijectively to target coatoms. Combining this representation with the maximal-model construction of Kavvadias, Sideri, and Stavropoulos shows that coatom enumeration is not in OutputP unless P=NP, even when all hyperedges have size two or three. Incidence splitting reduces maximum element frequency to three while preserving the stopping-set poset, and a local replacement of two-element hyperedges yields the same lower bound for three-uniform hypergraphs of maximum element frequency at most three. These thresholds are conditionally sharp for arbitrary-order enumeration: rank at most two and maximum element frequency at most two both admit output-linear total-time generation; in the frequency-two case, a polynomial-delay, polynomial-space algorithm is also available. In contrast, coatom extension is NP-complete already for three-uniform hypergraphs of exact element frequency two.

For a graph $G$, let avm($G$) denote the average size of its maximal matchings. Engbers and Erey initiated the extremal study of this parameter and asked for extensions from trees and unicyclic graphs to $k$-cyclic graphs. In this paper, we determine the maximum value of avm($G$) over all connected bicyclic graphs with $n$ vertices and $n+1$ edges. If $n\ge 5$ is odd, then \[ \text{avm}(G)\le \frac{n-1}{2}, \] and we characterize all graphs attaining equality. For $n=6$, the maximum value is 13/5, attained uniquely by $\Theta(1,3,3)$. If $n\ge 8$ is even, then \[ \text{avm}(G)\le \frac{n}{2}-1+\frac{2}{n-4}. \] Equality holds precisely for the graph obtained from two copies of $C_4$ joined by an edge by attaching $(n-8)/2$ pendant 2-paths to one endpoint of the joining edge, and, when $n\ge 10$, for the graph obtained from two copies of $C_4$ joined by a path of length 2 by attaching one leaf and $(n-10)/2$ pendant 2-paths to the internal vertex of the joining path. The proofs combine structural characterizations of odd-order extremal graphs with counting and switching arguments based on perfect matchings.

Let $G$ be a finite connected multigraph whose edges receive independent weights from one atomless law, and let $\operatorname{MST}(G)$ be the resulting random minimum spanning tree. Its law is not pairwise negatively correlated: Lyons, Peres and Schramm exhibited two positively correlated edges, and we give such an example on a simple graph. We prove that positive correlation is nevertheless uniformly controlled: $\mathbf{P}(e,f\in T)\leq 8\mathbf{P}(e\in T)\mathbf{P}(f\in T)$, answering a question of R. Lyons recorded by Tang and Zhang. After conditioning on all other weights, Harris's inequality gives conditional negative correlation; two bottleneck distances and a sharp second-moment estimate control the remaining environmental covariance. For $K_n$ we prove pairwise negative correlation for every $n\geq 3$. The key finite identity is $\mathbf{E}[\mathrm{deg}(x)^2]=10(n-1)/n-4\mathbf{E}[L_n]$, where $L_n$ is the total weight of the minimum spanning tree under rate-one exponential weights. Known expansions for $\mathbf{E}[L_n]$ then give the rate of convergence to $10-4\zeta(3)$ and the limits of both pair-correlation ratios. Finally, an explicit $K_4$ family shows that no universal constant survives when the independent edge laws need not be identical.

In this paper we apply the method of Kosygina, Rezakhanlou and Varadhan (CPAM 2006) to establish stochastic homogenization of Hamilton-Jacobi-Bellman (HJB) equation with a vanishing non-local integro-differential operator and a convex super-linearly growing Hamiltonian in stationary ergodic random medium. Their method, first designed for homogenization of HJB equations with vanishing Laplacian operator, relies on stochastic optimal control representation of the solution, a key construction of approximate super-correctors and the technique of linking diffusion process to abstract diffusion in random media. We show how the procedures can be carried out for HJB equations with jump-diffusion. In particular, for the construction of approximate super-correctors, we represent the non-local integro-differential operator as the divergence of a regular integral operator acting on the gradient.

We establish two pointwise estimates for fractional difference operators, tracking explicitly the dependence of the constants on the smoothness index $s\in(0,1)$. Using these, within the framework of ball Banach function spaces we obtain two fractional Gagliardo–Nirenberg inequalities, including the BMO endpoint case. Furthermore, we establish endpoint asymptotic results as $s\to0^+$ and $s\to1^-$, proving that the asymptotic factors appearing in these inequalities have optimal order. Under the additional assumption that the underlying function space is rearrangement invariant, we show that the optimal Gagliardo–Nirenberg target spaces are precisely those given by the Calderón–Lozanovskiĭ space. This completely characterizes the rearrangement invariant target spaces for which the corresponding Gagliardo–Nirenberg inequalities hold, thereby answering an open question posed by K. Leśnik, T. Roskovec, and F. Soudský. These results can be applied to various function spaces; in particular, they are completely new in the off-diagonal and BMO cases.

Keevash, Lenz, and Mubayi proved a spectral Erdős–Ko–Rado theorem, showing that, for sufficiently large $n$, the complete $t$-star uniquely maximizes the adjacency-tensor spectral radius among all $t$-intersecting $k$-uniform families. In this paper, we establish a spectral Hilton–Milner–Frankl theorem for nontrivial $t$-intersecting families in the explicit range $1\le t\le k-2$ and $n\ge 100\cdot 2^k k^7$. More precisely, we prove that, for every nontrivial $t$-intersecting $k$-uniform family $\mathcal F$, the spectral radius satisfies \[ \rho(\mathcal F)\le \max\{\rho(\mathcal H_{n,k,t}),\rho(\mathcal A_{n,k,t})\}, \] where $\mathcal H_{n,k,t}$ and $\mathcal A_{n,k,t}$ are the two extremal families appearing in the classical Hilton–Milner–Frankl theorem. Moreover, equality holds only for the extremal candidates attaining the maximum, up to isomorphism. We further compare the two candidates asymptotically. For each fixed $t$, the unique real solution $x=x_t$ of \[ (t+2)^{x-t-1}(t+1)^{t+1}=(x-t+1)^{x-1} \] determines, as $k$ varies, which of $\mathcal H_{n,k,t}$ and $\mathcal A_{n,k,t}$ has the larger asymptotic spectral radius.

We study Multi-Level Aggregation with Maximum Delay (MLA-MD) on finite weighted rooted trees. Each service pays for a rooted subtree and for the maximum waiting time among the requests it clears. For the offline problem, we give an exact polynomial-time algorithm that computes both the optimum and an optimal service schedule. For the online problem, we give a deterministic $2$-competitive algorithm and, against an oblivious adversary, a randomized $e/(e-1)$-competitive algorithm. Both competitive ratios are optimal, even on rooted paths. We also extend the offline result and both online guarantees to service systems with submodular costs.

By working in a setting of azimuthal and $z$-directional invariance, we reduce the existence of electrically neutral, magnetically driven, equilibria of the two-species Vlasov-Maxwell (VM) equation to a second-order ODEs in the radial variable. In contrast to previous works, we consider specific situations that correspond physically to $z$-pinch, $\theta$-pinch, and screw-pinch equilibria in plasma physics. Semi-explicit solutions in these situations are provided.

Covariate adjustment improves the efficiency of treatment-effect analyses in randomized clinical trials, provided the adjustment targets the correct quantity. For time-to-event endpoints, two marginal targets are of primary interest: the log-rank test for the presence of a treatment effect and the marginal hazard ratio for its magnitude. Existing covariate adjustment approaches reach these targets by different ways. Augmentation adjusts the log-rank score by regressing derived outcomes on the baseline covariates within each arm. Weighting instead reweights the two arms to balance the covariates before the survival comparison is formed: inverse probability weighting does so through a fitted propensity model, while calibration weighting solves directly for weights that match covariate means. In this manuscript, we first develop balancing weighting for time-to-event endpoints, covering both calibration weights (stable balancing weights and entropy balancing) and propensity score weights, and prove that any balancing-regular weighting is first-order equivalent to the augmented log-rank score and to the root of the marginal Cox score. All three routes therefore deliver the same estimator to first order, and calibration reaches it without fitting any model. The weighted procedures thereby inherit the validity and guaranteed efficiency gain of the augmentation approach. In addition, we show that the efficiency gain grows with the prognostic strength of the adjustment covariates, while the practical caveat lies in variance estimation, for which we give recommendations to guard against finite-sample Type I error inflation. We further confirm our results through simulation studies and an analysis of the REWIND cardiovascular trial.

A graphic arrangement $\A_G$ associated with a simple graph $G$ is a classical and well-studied object in the theory of hyperplane arrangements. In this note, we show that, for a connected graph $G$, a slight modification of the logarithmic vector field $D(\A_G)$ of $\A_G$ is isomorphic to the face ring of a certain simplicial poset. This allows us to give formulas for several algebraic invariants of $D(\A_G)$, such as its Hilbert series, local cohomology, projective dimension, and Castelnuovo–Mumford regularity, in terms of combinatorial and topological information about the corresponding simplicial poset. As a by-product, we also give an explicit vector space basis of $D(\A_G)$.

A $q$-deformed real number, or “$q$-real”, was defined by Morier-Genoud and the second author. When $x\in\mathbb{R}$ such that $x\geq0$, the $q$-analogue $[x]_q$ is a power series with integer coefficients in one formal variable~$q$. In general a $q$-real is a formal Laurent series. The main goal of this paper is to study the coefficients of $q$-reals as functions on~$\mathbb{R}$ and give a combinatorial interpretation of these coefficients. This allows us to prove a conjecture studied by several authors stating that the $q$-deformed golden ratio has the smallest radius of convergence among the radii of the $q$-reals associated with positive real numbers. This is a $q$-analogue of the classical Hurwitz theorem. Our approach is combinatorial. We prove that for every real number $x$ in the interval $(1,2)$ the absolute value of each coefficient of the power series representing the $q$-real $[x]_q$ is dominated by the absolute value of the corresponding coefficient of the $q$-deformed golden ratio. The main notion is a certain collection of ordered rooted trees associated with a $q$-real. We prove that the golden ratio corresponds to a universal class of trees.