Papers

Let $q$ be an odd prime power, and let $A_n=(a_{ij})\in\mathbb F_q^{n\times n}$ be a random matrix whose entries are independent and uniformly distributed on $\mathbb F_q$. The permanent of $A_n$ is defined by $\operatorname{per}(A_n)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}$, where $S_n$ denotes the symmetric group on $[n]$. Ghasemi, Gross, and Kopparty conjectured the zero-mass asymptotic $\Pr[\operatorname{per}(A_n)=0]=1/q+o(1)$ for every fixed odd prime power $q$, and Hunter, Kwan, and Sauermann subsequently stated its equivalent full-distribution formulation: for every fixed $q$ and every $x\in\mathbb F_q$, \[ \lim_{n\to\infty}\Pr[\operatorname{per}(A_n)=x]=\frac1q. \] In this paper, we prove this conjecture. More precisely, we prove that there is an absolute constant $C>0$ such that \[\frac12\sum_{x\in\mathbb F_q}\left|\Pr[\operatorname{per}(A_n)=x]-\frac1q\right|\le C\frac{\log n}{n}\] for every odd prime power $q$ and every $n\ge 7$. The estimate is uniform in $q$, so the conclusion remains valid for every sequence $q=q(n)$ of odd prime powers.

Chemotactic motion is described by run-and-tumble kinetic models at microscopic scales and by Keller–Segel equations at macroscopic scales. We develop variational loss functionals for estimating the two components $T_0(x)$ and $T_1(x)$ of a turning kernel $T_\epsilon=T_0+\epsilon T_1$, where $T_0$ determines the leading-order turning rate and diffusion, while $T_1$ governs the macroscopic chemotactic drift. Under suitable regularity and data-informativeness assumptions, we establish conditional scale-uniform forward-stability estimates showing that a small loss leads to a small discrepancy between the forward solutions generated by the true and estimated kernels across the kinetic and diffusive regimes. Combined with sparse inversion, the method accurately recovers smooth, nonsmooth, and strongly heterogeneous kernels and remains robust under measurement noise.

The rapid growth of the low-altitude economy calls for integrated sensing and communication (ISAC) networks capable of robust flight monitoring. This article advocates wireless imaging as a unifying sensing paradigm that enables ISAC networks to function as comprehensive low-altitude guardians. We present a hierarchical imaging framework that enhances sensing capability from wide-area snapshot imaging to dynamic trajectory-aware imaging and target-centric fine-grained characterization. At the wide-area level, low-altitude sensing is reformulated as a spatial imaging problem, where distributed base stations and communication users collaboratively construct a holistic view of the aerial space. Building on this foundation, multi-frame dynamic imaging exploits temporal correlations to support robust trajectory tracking and prediction under mobility induced challenges such as occlusions. For security-critical scenarios, the framework further enables fine-grained imaging of flying and hovering uncrewed aerial vehicles, providing detailed characterization beyond conventional point-target abstractions. Additionally, we propose a novel evaluation metric named imaging coverage to examine the sensing fidelity of the proposed framework. Illustrative case studies demonstrate the potential of imaging in ISAC networks to support wide-area monitoring, motion-aware tracking, and fine-grained target analysis.

We prove the claim in the title and complete Parker and Rowley's classification (J. Algebra 2001) of which sporadic simple groups are completions of the Goldschmidt G3-amalgam.

Recent work in bilateral logic has concerned itself with whether and to what extent strong negation, which "toggles" between assertion and denial, is definable in bilateral systems that lack it. This paper presents a number of results about the definability of strong negation in bilateral systems. First, I show that a constructive Nelson-style Sheffer stroke defines the strong negation of A as (A|(A|A))|A. The same stroke defines the constructive Nelson implication, thus providing a single-connective basis for the strong-negation-and-implication fragment of N4. Second, I provide an exhaustive characterization of the eight combinations of a single "aggregative" connective and a single Nelson-style connective that suffice to jointly define strong negation, with neither defining it individually. One such combination is the material conditional together with constructive co-implication. Third, I show that a constructive and connexive Wansing-style Sheffer stroke (recently referred to as "connexive exclusion") likewise defines the strong negation of A as (A|(A| A))|A and also defines connexive implication. Fourth, I show that, unlike with the Nelson-style connectives, there are no combinations of Wansing-style and aggregative connectives that jointly define strong negation if neither connective defines it individually.

Let G be a nonempty finite simple graph of order n, and let m(G) be the upper median of its degree sequence. We prove that the 2-domination number satisfies gamma_2(G) <= n - m(G) + 1. This proves Graffiti.pc Conjecture 387. In fact, the argument establishes the inequality for every nonempty finite simple graph, so the connectedness hypothesis in the original formulation is unnecessary. The proof uses the complement graph and a minimally linearly dependent family of polynomials encoding selected nonneighborhoods.

Parastatistics is an exotic type of exchange statistics beyond fermions and bosons. Paraparticles transform in higher dimensional representations of the exchange symmetry group, analogous to non-Abelian anyons, yet consistently defined in any dimension. Although paraparticles have long been proposed, they were widely believed to be physically equivalent to fermions or bosons. Nevertheless, a recent paper proposed a different theory, called $R$-parastatistics, and demonstrated that nontrivial $R$-paraparticles can emerge as quasiparticles in condensed matter systems, and are observably distinct from both fermions and bosons. This paper develops the theoretical foundation and several extensions of $R$-parastatistics, with particular emphasis on its observable consequences. Central to this paper is a general theory of local observables extending the basic family introduced before. First, we define local observables that distinguish particle types. Second, we formulate local observables at special point defects that probe the internal indices of $R$-paraparticles, crucial for observing $R$-parastatistics and for the proposed applications in quantum information. Third, we introduce local observables that create or annihilate particle-antiparticle pairs, important for building a relativistic quantum field theory for $R$-paraparticles. We further introduce generalized hidden symmetries that act on internal indices of $R$-paraparticles while preserving the local observable algebra, providing a basis for proving local indistinguishability and for connecting to a categorical description of $R$-paraparticles. This work sets a solid theoretical foundation for understanding the fundamental physical properties of $R$-paraparticles and pave the way for finding them in nature.

We present a novel slack-free, penalty-based framework for reformulating constrained binary optimization as Quadratic Unconstrained Binary Optimization (QUBO) on near-term quantum annealing hardware. Given a user-chosen penalty function that most naturally captures a constraint—typically non-quadratic, such as a Heaviside-function surrogate—and a target probability measure over the Boolean hypercube, our method returns the weighted least-squares projection of the chosen penalty function onto the subspace spanned by linear and quadratic Walsh–Fourier characters that correspond to physically realizable couplings on the target hardware graph. Within this restricted family, the resulting quadratic surrogate is uniquely and optimally determined by the normal equations: unlike state-of-the-art approaches, it introduces no per-constraint penalty coefficients to tune and avoids dense all-pairs couplings by construction. Two practical consequences follow. First, the projected penalty respects device connectivity, reducing chain lengths and physical-qubit overhead after minor embedding. Second, we show empirically that this hardware-native surrogate can outperform denser full-pairwise projections, despite being drawn from a strictly smaller approximation space. This advantage widens once the QUBO is embedded and sampled on quantum annealers, yielding samples with the lowest worst-case and mean objective gaps compared to unbalanced penalization and a hardware-blind projection onto all quadratic terms.

Knowledge graph learning offers a powerful framework for representing and inferring structured knowledge, with broad practical applications. Two obstacles limit current methods: labeled triples are scarce for each entity and relation, which makes expressive models hard to train, and scoring functions are designed ad hoc, which limits their generalizability and leaves them without theoretical grounding. We address both obstacles with a theoretically grounded framework, with end-to-end guarantees, that extends and subsumes existing methods. The framework proceeds in two stages: multi-view unsupervised pretraining that integrates heterogeneous sources of entity side information into a shared representation, followed by supervised learning across multiple relation types. We establish a unified nonasymptotic risk bound that combines the subspace recovery error of pretraining and the sample complexity of the labeled data in a single guarantee, quantifying formally how large-scale unlabeled data from multiple views benefits downstream knowledge prediction. Synthetic experiments validate each component of the theory, and experiments on large-scale real-world knowledge graph benchmarks confirm the effectiveness of our approach.

We develop a Cartesian grid method for advection–diffusion equations with Robin boundary conditions on moving domains. The moving-domain problem is reformulated as an interface problem on a box, with an unknown density introduced on the moving interface to enforce the Robin condition. The bulk equation is discretized by a cell-centered finite-difference scheme on the Cartesian grid, while interface corrections are obtained from local problems in a narrow band around the interface. The resulting method requires only modest computational geometry, avoids remeshing and cut cells, and is compatible with geometric multigrid and matrix-free GMRES. The GMRES iteration count is essentially independent of the mesh size, and the computational cost scales linearly with the number of bulk degrees of freedom. For the one-dimensional scheme, first-order convergence in time and second-order convergence in space are proved. Numerical examples in one and two dimensions, including manufactured solutions and an active transport problem without an exact solution, demonstrate the accuracy and efficiency of the method.

We extend the nice treatment of V. Bardenova, E. Insko, K. Johnson, and S. Sullivan of the combinatorics of two-lane mergings to the multi-lane case.

We construct quantum Turing patterns in Lindblad lattice dynamics and establish a rigorous theory of their nonlinear order and quantum fluctuations. For an explicit family of completely positive lattice generators with finite-range couplings, the first-moment equations undergo a supercritical instability at a nonzero wave number and admit analytic site- and bond-centered commensurate stripe branches. These branches are locally asymptotically stable in their reflection-fixed period-cell spaces, and projected coherent states exhibit extensive Bragg order on every bounded time interval in the semiclassical limit. We prove $O(N^{-1/2})$ convergence of microscopic covariances to a nonautonomous Gaussian Lyapunov flow, transferring strict partial-transpose uncertainty violations to sufficiently large $N$. In the homogeneous Gaussian sector, a single dimensionless ratio controls both the Turing stability determinant and the logarithmic negativity of opposite momenta, relating wavelength selection directly to quantum entanglement. Differential transport shifts the strongest opposite-momentum correlations from the infrared to the selected Turing scale. Numerical continuation and two-dimensional simulations display stripe, spot, and labyrinth morphologies whose Fourier modes and fluctuation spectra concentrate at the same selected wave numbers.

The de Rham-Witt cohomology of a smooth proper variety in characteristic $p$ contains a canonical piece called the domino, which is not finitely generated over the Witt vectors and carries the nonzero differentials of the slope spectral sequence. Beyond dimension one, dominoes were unclassified. We classify the two-dimensional ones and put a domino of any dimension into a normal form. To each domino we attach two unipotent groups, one formal and one perfect, and prove that their isogeny partitions agree. In degree two we recover the domino of a Mazur-Ogus variety from its crystalline cohomology, compute it for two families of supersingular abelian varieties, and bound the exponent of the $p$-primary Brauer group in terms of the $a$-number, for every prime $p$. This answers a question of Grammatica-Skorobogatov-Yang.

Bhargava's factorial construction attaches to a subset $S$ of a Dedekind domain a sequence of factorial ideals determined by local $p$-orderings. We introduce determinant-admissibility, a framework for factorial calculi whose local ordering data admit canonical carry, orbit, or renewal models together with compatible relative determinants. For such sets, the natural analogue of the Gamma function is generally not a scalar-valued function but a completed determinant line. After clearing rational spectral multiplicities, its integer fibers recover a finite tensor power of the Bhargava factorial ideals. Over number fields this line carries Archimedean metrics, while over global function fields it is completed at the distinguished places at infinity. In this formulation, Stirling-type formulas arise from large-parameter asymptotics of metrized determinants, and reflection formulas arise from dualities of the corresponding determinant complexes. We develop the theory through classical factorials, quadratic polynomial images, geometric and $q$-factorials, Polya–Ostrowski factorial ideals, Carlitz–Goss factorials, and a Bhargava–Barnes lift whose second-level determinant recovers the optimal Vandermonde energy of $n$-optimal sets. The set of cubes provides the first genuinely renewal-theoretic example. For primes $p \equiv 1 \pmod{3}$, we construct an exact three-state carry determinant, while projection through the quadratic character $\chi_{-3}$ yields the Archimedean half-determinant \[ \Gamma_{C,\infty}(z) = \sqrt{\Gamma(z)\Gamma(3z-2)} \] and a natural square-root determinant line. The remaining finite-place contribution is governed by nonlinear renewal systems at primes $p \equiv 2 \pmod{3}$ and by a ramified system at $p=3$; constructing their canonical global relative determinant is the principal open analytic problem.

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $\mathbb{V}_0=\mathbb{F}_q^n$ be the $n$-dimensional vector space over $\mathbb{F}_q$, for an integer $n\geq2$. We introduce the spectral incidence graph of $\mathbb{V}_0$, denoted by $\mathbf{SIG}(\mathbb{V}_0)$, a bipartite graph whose two vertex classes consist of the one-dimensional subspaces of $M_n(\mathbb{F}_q)$ generated by matrices having at least one eigenvector in $\mathbb{V}_0$, and the one-dimensional subspaces of $\mathbb{V}_0$ (i.e. the projective space $\mathbb{P}^{n-1} (\mathbb{F}_q)$), respectively. A matrix vertex $\langle M\rangle$ and a point vertex $\langle v\rangle$ are adjacent if and only if $v$ is an eigenvector of $M$. We determine several structural parameters of $\mathbf{SIG}(\mathbb{V}_0)$, including its twin classes, connectivity, domination number, diameter, vertex degrees, and number of edges. We show that every automorphism of $\mathbf{SIG}(\mathbb{V}_0)$ preserves the two parts of the bipartition. We prove that the kernel of the induced action of $Aut(\mathbf{SIG}(\mathbb V_0))$ on the projective part $\mathbb{P}^{n-1} (\mathbb{F}_q)$ is precisely $\prod_{\mathcal C\in\mathcal T} S_{\mathcal C}$, where $\mathcal T$ denotes the set of twin classes of matrix vertices and $S_{\mathcal{C}}$ is the symmetric group on $\mathcal{C}$. We then determine the full automorphism group of $\mathbf{SIG}(\mathbb{V}_0)$. For $n\geq3$, we prove that $Aut(\mathbf{SIG}(\mathbb V_0)) \cong \left(\prod_{\mathcal C\in\mathcal T} S_{\mathcal C}\right) \rtimes P\Gamma L_n(q)$. For $n=2$, we obtain $Aut(\mathbf{SIG}(\mathbb V_0)) \cong \left( \prod_{i=1}^{q+1}S_q\times \prod_{i=1}^{\frac{q(q+1)}{2}}S_q \right) \rtimes S_{q+1}.$

Let p be an odd prime and K/Qp a finite unramified extension of degree f > 1. Let Z(r) be the reduced special fiber of the Emerton-Gee stack of two-dimensional crystalline representations of Hodge type r of the absolute Galois group of K. We study the collection of stacks Z(r) as r varies over p-bounded Hodge types, as a set partially ordered under inclusion. We prove that aside from two degenerate cases, simple inclusions can be classified in terms of three operations on Hodge types, two of which have standard automorphic interpretations. We also prove, with one exception, that inclusions can be detected at the level of an inclusion of closed points (equivalently, semisimple mod p Galois representations). GPT-5.5 Pro was used extensively in the course of this work.

We extend the Caffarelli-Hardt-Simon perturbation argument for truncated regular minimal cones to the special Lagrangian setting and prove a bridge principle for regular special Lagrangian cones in the spirit of Nathan Smale. Our bridge principle yields a general existence theorem for conically singular special Lagrangian submanifolds with prescribed regular tangent cones: for any finite list of such cones in $\mathbb C^m$ having the same Lagrangian angle and suitably arranged, there exists a connected special Lagrangian submanifold with boundary and isolated conical singularities whose tangent cones at its singularities are precisely the prescribed cones. When applied to the graphical special Lagrangian cone in $\mathbb{C}^5$ recently discovered by Bhattacharya, Orriols, and Skorobogatova, we obtain infinite families of $C^{1,1}$ viscosity solutions to the special Lagrangian equation having any finite number of isolated singularities.

Alon and Füredi proved that at least $n$ affine hyperplanes are required to cover $\{0,1\}^n\setminus\{\textbf{0}\}$ while avoiding the origin, and that this bound is sharp. We study how small the largest Boolean intersection among the hyperplanes can be in a cover attaining this minimum. Let $F(n)$ denote the minimum possible value of $\max_{H\in\mathcal H} |H\cap\{0,1\}^n|$ over all families $\mathcal{H}$ of $n$ affine hyperplanes covering $\{0,1\}^n\setminus{\mathbf{0}}$ and avoiding the origin. We give an explicit construction, proving that $F(n)=(1+o(1))\frac{2^n}{n},$ and hence asymptotically attain the averaging lower bound.

Our main results are an upper bound on the asymptotic resurgence of the facet ideal of a graphic matroid in terms of the number of vertices and a lower bound in terms of the circumference. These bounds coincide for Hamiltonian graphs, which form the majority of graphs on $n$ vertices as $n$ tends to infinity. For simple $2$-connected graphs on up to nine vertices, we compute in Sage that the asymptotic resurgence of the facet ideal of a non-Hamiltonian graphic matroid is given either by our upper or lower bound.

We propose a stabilized version of the fully discrete Dziuk's method for the curve-shortening flow of a closed planar curve with piecewise linear parametric finite elements. With a carefully designed stabilization term, we are able to show a surprising discrete tangential stability of the Barrett–Garcke–Nürnberg (BGN) type under the parabolic scaling $\tau\simeq h^2$—a feature hidden at the continuous level. Together with a new super-approximation result for the reversely averaged normal vector of linear elements, this Dziuk-type discrete tangential stability yields optimal $L^2$ convergence.

Two central challenges in diffusion-based sampling are the theoretical one of understanding their remarkable effectiveness even in high-dimensional settings, and the practical one of designing algorithms with certified performance guarantees. We show that these questions are intimately connected via the denoising growth complexity ($\mathsf{DGC}$). It is a geometric measure defined by a log-time weighted integral of the derivative of the denoising mean-squared error along the Gaussian heat flow. We show how the $\mathsf{DGC}$ increments lead to a simple and explicit bound on the KL error of an Euler scheme applied to the stochastic innovations representation. The bound is local along the path: each step is controlled by the corresponding $\mathsf{DGC}$ increment and its relative stepsize. This structure allows us to derive KL sampling guarantees for optimized stepsize schedules, both in a simpler single-block setting and in a more refined $K$-block setting. The $\mathsf{DGC}$ function has a natural martingale structure, which we exploit to develop fully data-certified versions of these algorithms. It also admits information-theoretic upper bounds in terms of covariance, rate distortion, metric entropy, and the Poincar'e constant, thereby recovering and sharpening a range of existing diffusion-sampling guarantees, as well as giving new results. In log heat-time, the fine partition limit is governed by an integral involving the square root of the $\mathsf{DGC}$ density, whereas a single-block schedule depends on its ordinary integral. This comparison precisely characterizes when adaptation to data geometry yields substantial computational gains, including logarithmic-to-constant separations for simple Gaussian mixture models.

Let $G$ be an automatic group with associated regular language $L$. We describe a procedure for constructing an automaton which recognises elements of a given submonoid or rational subset $K$ of $G$. This builds on work of Kharlampovich, Miasnikov and Weil, on the case where $K$ is a subgroup of $G$. Our construction succeeds, after sufficiently many iterations, whenever $K$ satisfies a certain convexity property, which we call $L$-proximity. We show how to test whether the construction is complete in the case that $K$ is a submonoid; we have no such test for the general case of a rational subset $K$. We focus particularly on the case of a surface group $G$ of genus $g>1$, where $L$ is the language of geodesic words in the standard generators. We use small cancellation theory to obtain a method for constructing $L$-recognisable submonoids of $G$.

Paths $X \colon [0,T] \to \mathbb R^d$ are traditionally stored in finite memory as time series. Recent research has underscored the benefits of instead representing them as collections of iterated integrals $\{\int_{0 < u_1 < \ldots < u_n < T} \mathrm{d} X_{u_1} \otimes \cdots \otimes \mathrm{d} X_{u_n}\}_{n = 0}^N$. These two encodings can be viewed as the extrema on a two-parameter spectrum of representations of the path as degree-$N$ signatures on $m$ intervals in a partition of $[0,T]$. We ask the question of which such representation takes up the least amount of memory, measured as number of real values needed to store the truncated log-signature, subject to the constraint of it being able to approximate solutions to linear controlled differential equations (CDEs) $\mathrm{d} Y = AY \mathrm{d} X$ with $|A| \leq r$ at accuracy at least $\varepsilon$. Estimating the error in terms of the length of $X$, we find that the optimal representation generally lies strictly in between the two naive choices $N = 1$ or $m = 1$, and derive its asymptotics as $r \to \infty$ and $\varepsilon \to 0^+$. Similar considerations can be made when estimating the error in terms of the $p$-variation norm of $X$: in this regime we prove an error bound of the degree-$N$ Euler scheme for linear CDEs with decay in both $m$ and (factorially) in $N$ with the other arbitrarily fixed. We conclude by setting up the analogous problem for SDEs, with the error measured in $L^2$, and derive a similar $L^2$-Euler error estimate for Itô SDEs with drift. We include an empirical study of the optimisation problem, which we demonstrate for toy examples of $p$-rough paths and for fractional Brownian motion.

This paper develops data-driven conditions for certifying internal stability and induced-$\ell_2$ performance of discrete-time Lurye systems. The Lurye system is an interconnection of a nominal LTI system in feedback with a static, memoryless nonlinearity. The nonlinearity satisfies a known set of quadratic constraints on the inputs and outputs. Existing conditions for Lurye systems require a state-space realization of the nominal LTI dynamics. Our first data-driven result instead formulates the stability and performance conditions using finite input, output, and state trajectories of the nominal LTI block. Our second condition removes the need for measured state trajectories by reconstructing the state sequence, up to a similarity transformation, from input/output data. This state reconstruction is performed using deterministic subspace-identification techniques. Both data-driven conditions are expressed as convex semidefinite programs. These conditions, given sufficiently exciting inputs, recover the corresponding model-based Lurye condition in the noiseless setting. The proposed methods are illustrated via a simple example with a sector-bounded nonlinearity. Both proposed methods obtain the same induced-$\ell_2$ gain bound as the model-based approach while using only trajectory data from the nominal system.

Gill (arXiv:2402.13376) introduced the probabilistic automatic complexity $A_P(w)$ of a finite string $w$: the least number of states of a probabilistic finite automaton (PFA) for which $w$ is the unique most probably accepted string of its length. He asked whether $A_P$ is unbounded, noting that no string with $A_P > 3$ was known (Question 4.14 of that paper). We answer the question by proving that $A_P(w)\le 3$ for every string $w$ over every finite alphabet. The witnessing three-state automaton is explicit: its reduced dynamics tracks the pair $(u,u^2)$, where $u$ is the reversed base-$b$ value of the input, and its acceptance functional is a downward parabola peaked at the value of the target string. Combined with Gill's classification of the binary strings with $A_P=2$, this completely determines $A_P$ on binary strings.

In low-rank numerical methods for Vlasov dynamics, the SVD-type truncation procedure may introduce negative entries into the numerical solution. Such negative values are unphysical because the solution is a probability distribution function. We design optimization-based post-processing algorithms to recover nonnegativity while preserving the macroscopic quantities (density, momentum, and energy) pointwise. The preservation of the macroscopic quantities is written as an orthogonality constraint on the correction term. For a convex formulation based on squared nuclear norm minimization, we show that the proximal operator with the orthogonality constraint is characterized by an implicit singular value thresholding equation, and the threshold can be computed efficiently by bisection. Based on this result, we develop five algorithms for the convex formulation: Douglas–Rachford splitting, restarted dual FISTA, restarted dual accelerated gradient descent, dual PR+ conjugate gradient, and dual L-BFGS. We also consider a non-convex formulation with an explicit rank constraint and develop a tangent-space accelerated alternating projection algorithm that only requires a \(2r \times 2r\) SVD per iteration. Numerical results for a Landau damping test case show that the proposed algorithms give comparable correction quality. Among them, the tangent-space accelerated alternating projection is the most cost-efficient, increasingly so as the problem size grows. We further demonstrate the correction as a positivity limiter inside a time-dependent conservative low-rank Vlasov solver, where it removes the negativity introduced by the SVD-type truncation while preserving the conserved mass, momentum, and energy.

Martinsson and Steiner recently proved that the fractional chromatic number of any $d$-degenerate triangle-free graph $G$ satisfies $\chi_f(G) = O\left(\frac{d}{\log d}\right)$. They further conjectured a sharp leading constant $1 + o(1)$. In this paper, we confirm their upper bound conjecture for graphs having girth at least $5$. Our proof is constructive: it gives an efficient randomized algorithm that, with high probability, computes a fractional coloring of weight at most $(1 + o(1))\frac{d}{\log d}$ in such graphs. Furthermore, we establish their conjectured lower bound in a stronger form: for any constant $g \ge 4$, there exist $d$-degenerate graphs having girth at least $g$ with $\chi_f(G) \ge (1 - o(1))\frac{d}{\log d}$. This lower bound is achieved by analyzing a random graph based on the uniform attachment model. Notably, our results reveal that this model lacks the typical computational complexity barriers found in Erdős-Rényi graphs, where there is a conjectured factor-$2$ algorithmic gap for this problem.

We find sharp upper bounds for the initial higher-order Schippers-Schwarzian derivatives $\sigma_3(f)(0)$ and $\sigma_4(f)(0)$ for several subclasses of univalent and starlike functions in the unit disk. The functions in these classes are defined by subordination to domains bounded by an exponential-sine curve, a cardioid, or a petal-shaped curve. We determine the sharp bounds and find the corresponding extremal functions for each invariant in these subclasses. As an application, we also obtain sharp bounds for the initial Grunsky coefficients $\omega_{1,1}$ and $\omega_{1,2}$.

It was asked in [ACTT23] if every faithful, discrete, ergodic pmp action of a mixed identity-free (MIF) group is essentially free. In this short note we give some positive evidence by showing that for a MIF group, any faithful generalized Bernoulli action and any affine action on a compact abelian group is essentially free. This gives some classes of discrete ergodic actions with this property beyond the totally ergodic or compact case.

The problems of characterizing the graphs $G$ which are generically rigid in ${\mathbb R}^d$, or more generally, determining the rank function of the $d$-dimensional rigidity matroid ${\cal R}_d(G)$ of an arbitrary graph $G$, have been solved when $d\leq 2$ but are major open problems in discrete geometry when $d\geq 3$. In this paper we shall concentrate on the case when $d=3$. We first revisit a conjecture of Dress from 1987 that the rank of the ${\cal R}_3$-closure of a graph $G$ is determined by its maximal complete subgraphs of size at least five. We show that his conjectured value for the rank of the closure gives an upper bound on the actual value. We also deduce that the truth of this conjecture would imply a good characterization of the rank of ${\cal R}_3(G)$ for all graphs $G$. The rank formula in Dress's conjecture leads us to consider the family of $K_t$-covered graphs, i.e., graphs in which every edge belongs to a complete subgraph $K_t$, for some $t\geq 3$. This family contains several well-studied graph classes such as body-pin graphs, combinatorial zeolites, and molecular graphs. We introduce a new notion of rank contributions of vertices in an arbitrary matroid on the edge set of a graph $G$, and use it to obtain lower bounds on the rank contributions of vertices in ${\cal R}_3(G)$ and ${\cal C}^1_2(G)$ when $G$ is $K_t$-covered. We use these bounds to show that a conjectured min-max formula for the rank of body-pin graphs in ${\cal R}_3$ holds for the $C_2^1$-cofactor matroid (which is conjectured by Whiteley to be equal to ${\cal R}_3$), and to obtain new sufficient connectivity conditions for the (global) rigidity of $K_4$- and $K_5$-covered graphs in ${\mathbb R}^3$.