Papers

Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$ for every $x\ge1$. If $|\alpha-a/q|\le q^{-2}$, $(a,q)=1$, and $3\le R\le q\le N/R$, we prove \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha) \ll_A \frac{N}{\log N} +\frac{N}{\sqrt R}\sqrt{\log\log(3R)} \] with effective implied constants. Montgomery and Vaughan proved this with second term $NR^{-1/2}(\log R)^{3/2}$, and, for $1$-bounded functions, Bachman replaced it by $NR^{-1/2}\sqrt{\log R\log\log R}$. We remove the factor $\sqrt{\log R}$ from Bachman's second term while retaining the original coefficient hypotheses of Montgomery and Vaughan. A more precise estimate records the distance from a rational number. The proof combines the Brun-Titchmarsh inequality on short intervals with maximal Fourier estimates derived from the Carleson-Hunt theorem; the local bounds permit arbitrary prime-dependent prefixes. We also prove sharpness of the square-root displacement dependence.

Diagonally implicit Runge-Kutta (DIRK) methods are a prominent class of numerical methods for solving stiff systems of ordinary differential equations (ODEs). Stiffness does not only impose stability challenges on Runge-Kutta methods; it can also degrade the order of convergence. This so-called order reduction phenomenon occurs when assumptions used for classical convergence analysis, e.g., an asymptotically small step size, fail to hold. In a prior paper by the authors, sharp order conditions and global error bounds for Runge-Kutta methods were developed, which hold uniformly with respect to stiffness when applied to a wide class of semilinear ODEs. In this work, those conditions are leveraged to construct the first DIRK methods of order four and five which satisfy these conditions and thus do not exhibit order reduction. Numerical results demonstrate that for a broad class of relevant nonlinear test problems, these new methods successfully mitigate order reduction, accurately estimate local error via an embedding for adaptive step size control, and can outperform classical DIRK methods.

This paper considers the lowest-order first type Nédélec edge element method (EEM) on Cartesian grids for the three-dimensional time-harmonic Maxwell equations with a large wave number. New polynomial preserving recovery (PPR) operators are proposed for the curl of the edge element solution and for the solution itself, respectively. Under the condition that $\kappa^3 h^2 C_{\mathrm{sol}}$ is sufficiently small, second-order superconvergence estimates are proved for both the recovered curl and the recovered solution, where $\kappa$ is the wave number, $h$ is the mesh size, and $C_{\mathrm{sol}}$ is a stability constant associated with the Maxwell solution operator. In particular, the analysis shows that the proposed PPR procedures cannot mitigate the well-known pollution effect inherent to the EEM. To reduce the pollution error, we further propose a new continuous interior penalty edge element method (CIP-EEM) that incorporates an additional normal-jump penalty term. It is shown that by appropriately choosing the penalty parameters, the new CIP-EEM can improve the phase error by two orders in $\kappa h$. Numerical experiments are presented to confirm the theoretical superconvergence results and to demonstrate that the CIP-EEM can effectively reduce the pollution error in the high-frequency regime.

The Kolmogorov complexity of a string characterize how complex it is to describe the string. If every prefix of a real $x$ is more complex to describe than every prefix (of the same length) of real $y$, then it is seen as $x$ is more complex to describe than $y$. It is wondered if there is a real $x$ so that no other reals are strictly more complex (to describe) than $x$. The behavior of Kolmogorov complexity functions generated by reals (namely $n\mapsto$ the minimal description length of the real) is quite chaos. Therefore, it is widely believed that there are many reals that are maximally complex to describe. For instance, it is conjectured that all random enough reals have maximal $K$-degree. In this paper, it is shown that there is no real with maximal $K$-degree. Actually, for almost all real $x$, we can uniformly computably find another real whose $K$-degree is strictly above $x$. The same construction also derives the following: (a) There is no maximal $C $-degree. (b) Every $n$-random real $x$ admits an $n$-random real $y$ such that $y>_Kx$ and $y>_C x$; moreover, the difference of $K$-complexity and $C$-complexity of $y,x$ (namely $K(y\upharpoonright n)-K(x\upharpoonright n)$ and $C(y\upharpoonright n)-C(x\upharpoonright n)$) do not tend to infinity.

We develop a method that uses Picard iterations to accelerate Newton continuation for the semi-implicit finite-difference discretization of forward-backward partial differential equation (PDE) systems arising in mean field games (MFGs). We first investigate the computational properties of the Picard and Newton methods, which are widely used separately in the MFG literature but whose comparative cost and robustness across different regimes remain insufficiently explored and documented. The Picard method uses an outer fixed-point iteration that alternates a forward Fokker-Planck solve and a backward Hamilton-Jacobi-Bellman solve. The Newton method instead applies Newton's method directly to the coupled nonlinear space-time system. Across one- and two-dimensional MFG benchmarks, Picard offers substantial computational savings in favorable regimes, but may fail at sufficiently low viscosity or require strong damping under temporal shocks, making Newton continuation preferable. With parameter continuation in the viscosity parameter, the Newton method is more robust in these regimes, at the cost of larger coupled linear systems. We relate these trade-offs to the residuals, Jacobian blocks, and sparsity structures produced by separable, local nonseparable, and nonlocal Hamiltonians. We then demonstrate how to combine inexpensive Picard iterations with Newton continuation in a hybrid method to reduce the total computational cost for a two- dimensional double-well problem.

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

Menshikov and Volkov [Electron. J. Probab. 13 (2008)] studied recurrence and transience of a class of Markovian random walks on $\mathbb R_+$ whose conditional drift depends on both time and position and is of order $\rho x^\alpha t^{-\beta}$ with $\rho>0$. The case on the critical line $2\beta-\alpha=1$, with $\alpha\in(-1,1)\setminus\{0\}$, remained open. We prove recurrence in this remaining case. Furthermore, we establish recurrence and transience criteria that complete the classification for $-1<\alpha<1$ and $\beta\ge 0$, without assuming the Markov property and under weaker assumptions on the increments than those imposed by Menshikov and Volkov.

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin–Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni–Zhang's conjecture on the nonvanishing of local relative characters.

We study a continuous-time opinion model in which agents interact in pairs and each agent has a fixed anchor. At each interaction, both opinions are updated according to a possibly nonlinear rule with random inputs. Under suitable stability and moment assumptions, we establish uniform-in-time propagation of chaos: as the population grows, any fixed number of agents become asymptotically independent, with a common law solving a nonlinear Boltzmann equation. The proof combines a graphical approximation on finite time intervals with exponential convergence of the finite system and its nonlinear limit to their respective stationary laws. We examine two applications. For an anchored variant of the Friedkin-Johnsen model with random coefficients, we give conditions for sub-Gaussian or power-law tails of stationary deviations from the anchors. Occasional overreaction can produce power-law tails with Gaussian anchors and noise, even as the nonlinear law converges exponentially fast to stationarity. We also study an anchored model of biased assimilation, determining when a nearby opinion moves toward or away from neutrality with the anchor and incoming evidence held at neutral values. Simulations illustrate the tail predictions and show how a single peak in the smoothed empirical opinion profile can split into two.

Exploring high dimensional energy landscape is a challenging problem in many applications where both ground states and transition states offer important information about the underlying physical systems. To facilitate the computational exploration, we propose a cubic-regularized second-order method for locating critical points of prescribed index, including both energy minima (with an index $0$) and transition states (saddle points with positive index). At each step of the iteration, the proposed method involves several ingredients, such as updating the eigen-directions of the Hessian according to the prescribed index, reflecting the gradient and the Hessian along selected unstable eigen-directions, if any, and constructing the next step by solving a cubic-regularized subproblem. Under suitable assumptions, we show that, when the eigenspace and the cubic subproblem are solved exactly, the iteration converges locally and quadratically to a critical point of the prescribed index. We further establish convergence results in the inexact setting, where both the eigenspace computation and the solution of the cubic subproblem are performed approximately. We show that the resulting iteration retains local linear convergence, with the convergence rate depending on the accuracy of the inexact computations. Moreover, we prove that the fixed points of the algorithm are precisely critical points of the prescribed index. In addition, we introduce an adaptive strategy for updating the cubic regularization parameter to improve robustness and computational efficiency of the proposed algorithm. Numerical experiments confirm the predicted convergence behavior, demonstrate the effectiveness of the adaptive strategy, and illustrate the capability of the proposed method for locating critical points and constructing solution landscapes.

For any primes $p \neq \ell$, $\ell$-isogeny graphs of ordinary elliptic curves defined over $\mathbb{F}_{p^2}$ have a typical structure called $\ell$-volcanoes, and the structure is the core of Sutherland's supersingularity testing algorithm for elliptic curves. In this paper, by exploiting the properties of $3$-isogenies between twisted Hessian curves, we show that when $p \equiv 2 \pmod{3}$ and $\ell = 3$, every ordinary twisted Hessian curve defined over $\mathbb{F}_p$ lies on the surface of the $3$-volcano. As an application, we give an improved version of Sutherland's supersingularity testing algorithm specialized to twisted Hessian curves defined over $\mathbb{F}_p$ with $p \equiv 2 \pmod{3}$. We also give a generalization of the known fact that any supersingular $j$-invariant is a cube in $\mathbb{F}_{p^2}$; we show that for any twisted Hessian curve $H(a,d)$ defined over $\mathbb{F}_{p^2}$, its $j$-invariant is not a cube in $\mathbb{F}_{p^2}$ if and only if $H(a,d)$ is ordinary and lies on the floor of a $3$-volcano.

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

This is the second part of a two-part work concerning boundary layer solutions to the coupled Chemotaxis-Navier-Stokes system in the two-dimensional half-space. In the present work, we address the convergence of boundary layer solutions to singular chemotaxis-fluid equations under slip boundary conditions with respect to the chemical diffusion-viscosity parameter $\varepsilon$ in the two-dimensional half-plane. More precisely, we show that the boundary layer for $\varepsilon>0$ (viscous convection coefficient) converges to the superposition of the outer layer (solution with $\varepsilon=0$) and the inner layer as $\varepsilon\rightarrow0$. The outer and inner profiles are explicitly derived as in the first part\cite{WWZ}. Furthermore, the well-posedness results of the coupled Chemotaxis-Navier-Stokes system in conormal Sobolev spaces will be presented in Appendix. They answer the question mentioned in the first part of the two-part work. This study could help the understanding of the chemotactic movement of aerobic bacteria to the water-air surface observed experimentally in fluids, and enrich the theoretical results of boundary layer in chemotactic fluid models.

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive Hölder class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-Hölder continuity of the integrated density of states for the associated quasiperiodic Schrödinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

The PCA algorithm is sensitive to changes in measurement scale. Measuring one variable of a system in inches rather than centimeters, say, alters both its principal axes and principal eigenvalues. Although this scale dependence is generally complicated, we show here that it nevertheless obeys a strict invariance property: under a continuous scale adjustment, the initial state's $k$-th largest principal component (ordered by eigenvalue) continuously evolves into the final state's $k$-th largest principal component, for each $k$. In this sense, we can say that the modes of PCA are "order-stable" with respect to changes in measurement scale. A special case occurs when scaling along directions that are orthogonal to some modes. Here, apparent eigenvalue crossings can occur. However, we show that we can interpret these apparent crossings as cases where the modes instantaneously swap their orientation, in this way maintaining the required order stability.

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

Montgomery recently proved that for sufficiently large $n$, every Latin square of order $n$ has a partial transversal with $n-1$ cells, and every Steiner triple system of order $n$ has a matching with $\lfloor n/3\rfloor-1$ edges, thus confirming the Ryser–Brualdi–Stein conjecture for even $n$ and the conjecture of Brouwer. We prove sharp enumerative refinements of these results: there is an absolute constant $c>0$ such that, for sufficiently large $n$, 1) every Latin square of order $n$ has $ \left((1\pm n^{-c})\frac{n}{\mathrm {e}^2}\right)^n$ partial transversals with $n-1$ cells; 2) every Steiner triple system of order $n$ has $ \left((1\pm n^{-c})\frac{n}{2\mathrm {e}^2}\right)^{\lfloor n/3\rfloor}$ matchings with $\lfloor n/3\rfloor-1$ edges. The first estimate confirms predictions of Montgomery and Kelly.

We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, in the spirit of Banchoff-Pohl, together with an estimate for Jacobi fields along geodesic chords. The inequality persists for boundaries of isoperimetric regions in geodesic balls, whose mean curvature is constant only on the free part. An isoperimetric-profile argument, after Kleiner, completes the proof. Our method also gives a new proof in dimension $3$.

This paper deals with certain classes of modules under module-finite extensions. Let $\varphi: R\hookrightarrow S$ be a module-finite extension between commutative Noetherian local rings. We investigate the transfer of Artinian module structures and attached primes between $R$ and $S$. We clarify the behavior of local cohomology modules as well as certain structures of finitely generated $S$-modules under the restriction of scalars to $R$ via $\varphi$. We show that $R$ is a quotient of a Cohen-Macaulay local ring if and only if so is $S$. As an application, we characterize the structure of Nagata's idealization. Using Macaulayfication of algebraic varieties and idealization, we give an example to illustrate the results.

In this paper, we study averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection. First, we derive a general averaging principle applicable to such equations under minimal assumptions. Subsequently, since the coefficients of the obtained averaged equation still depend on the small scaling parameter $\e$, we impose either periodic or asymptotic conditions on the coefficients, thereby obtain two distinct averaged equations whose coefficients are independent of $\e$ and establish two averaging principles. Stopping times and Khasminskii's time discretization schemes play an important role. Finally, a concrete example is provided to illustrate the applicability and validity of the theoretical results.

We introduce and study a scalar $q$-deformation of the cross-ratio on $\mathbb P^1(\mathbb Q)$. Our construction is based on the notion of $q$-deformed rational numbers due to Morier-Genoud and the author. The $q$-cross-ratio is invariant under $\mathrm{PSL}(2,\mathbb{Z})$, while elements of determinant $-1$ of $\mathrm{PGL}(2,\mathbb{Z})$ act by $q\mapsto q^{-1}$. A principal result is its relation to $q$-deformed Coxeter friezes associated with rational polygons. The expansion at $q=e^h$ yields an algebraically independent sequence of modular invariants and relative invariants, although this sequence does not separate modular orbits. We compute the first two nonconstant coefficients of this expansion explicitly.

Dynamic allocation decisions couple present resource use to evolving internal conditions, delayed returns, and future costs. We represent this interaction by four probability localizations linked through regime-indexed, graph-constrained column-stochastic operators. Pre-action state or context selects a locally affine model, while action-dependent changes update subsequent regimes, yielding a causal switched representation of nonlinear evolution. We characterize operator identifiability relative to the graph, the stochastic constraints, and the sampled embedding, separating coefficient recovery from predictive equivalence on the decision domain. Decision making is then formulated through implementable return–cost acceptability regions. Finite-horizon error propagation supplies conservative classification margins, and simultaneous intervals distinguish model-relative near-optimality from certified $\epsilon$-optimality over a declared finite policy class. Regime-indexed stochastic feedback is admitted when it satisfies the same certification test. Reproducible synthetic laboratories for personal preparation, supplier participation, and customer retention illustrate exact, operator-supplied, and noisy feedback cases. Multinomial experiments show improving recovery of the feedback function and fewer unresolved decisions with increasing sample size, while unrestricted off-policy recovery remains limited. The contribution is a structure-preserving representation–identification–decision workflow, not a domain-specific physiological or commercial calibration.

In several storage systems, including DNA storage and flash memory, errors affect neighbouring symbols jointly, and the Hamming metric does not adequately capture such error patterns. The symbol-pair read channel, introduced by Cassuto and Blaum~\cite{cassuto2011codes}, addresses this by reading consecutive pairs of symbols rather than individual symbols. Motivated by this, we introduce function-correcting symbol-pair codes with data protection (FCSPC-DP), which guarantee reliable recovery of a desired function of the message while simultaneously protecting the message itself against symbol-pair errors. We derive bounds on the optimal redundancy of such codes and establish a relationship with joint-pair distance matrices. We also give explicit constructions of FCSPC-DP for locally pair-bounded functions and symbol-pair weight functions. We introduce the pair-separation constant of a function, the minimum symbol-pair distance between messages sharing a function value, and show that when it is sufficiently large, data protection requires no additional redundancy: the optimal redundancy coincides with that of the corresponding code without data protection. Considering the symbol-pair analogue of the $\alpha$-distance graph, we introduce two code invariants, the generation profile and the disconnection threshold, and use them to characterise a code's protection properties. Relating the two metrics through these invariants yields upper and lower bounds on the symbol-pair threshold in terms of its Hamming counterpart, both of which are attained. We further extend the classical Plotkin and sphere-packing bounds to this setting.

It is shown that every Eulerian digraph on $n$ vertices with more than $(t-1)n$ arcs contains every oriented tree with $t$ edges. The digraphs have no loops or repeated arcs, but opposite arcs are permitted. The bound is sharp for each fixed oriented tree, as witnessed by disjoint unions of complete bidirected graphs. Previously, such tight bounds were not known, even just for directed paths. This can be considered as a directed analog of the recently proved Erdős-Sós conjecture. The result was proved by GPT-6 Astra.

We propose a mathematical theory of pragmatic information that connects communication, control, and decision-making. Its central notion is the isoteleia mapping, which formalizes equifinality: distinct semantic paths that lead to the same optimal action are treated as pragmatically equivalent. This mapping yields a three-tier hierarchy of syntactic, semantic, and pragmatic information, in which each successive abstraction removes distinctions that are irrelevant to the task. We then define pragmatic entropy, up/down pragmatic mutual information, channel capacity, and rate-distortion, and prove lossless source coding, channel coding, and rate-distortion theorems that extend Shannon's results. These measures quantify decision uncertainty, reliable transmission, and task-oriented compression at the level of terminal actions. We further introduce pragmatic value of information (VoI) and pragmatic cost of information (CoI) as decision-theoretic duals to rate-distortion and capacity, and develop a Lagrangian dual framework for cross-layer optimization. The resulting pragmatic efficiency bound $\mathcal{E}_p(\lambda)=\sup_R[\Phi_p(R)-\lambda\mathrm{CoI}_p(R)]$ characterizes the maximum net utility attainable by a resource-constrained intelligent system under a given resource price, yielding a behavioral capacity that extends Shannon's symbol-level capacity to goal-directed action. Extensions to continuous messages provide closed-form expressions for Gaussian channels and sources, while dynamic settings are addressed through a Bellman equation for sequential decision-making. The framework supports task-oriented communication, networked control, autonomous systems, and embodied AI by shifting emphasis from symbol fidelity to the effectiveness of information in guiding actions. In this way, it offers a common language for systems that extract value from information under resource constraints.

Stress localization in heterogeneous solids is governed by the bimaterial interface, where the displacement field remains $C^0$-continuous, while in-plane stresses jump due to the stiffness mismatch. Coordinate-based physics-informed neural networks (PINNs) represent this jump via a prescribed regularization width or a weighted interface penalty, making their accuracy sensitive to how phase-contrast changes are handled. This work presents a variational, label-free physics-informed graph neural network (PI-GNN) in which the heterogeneity is carried by the discretization rather than by the trial field. The solver operates on a conforming adaptive mesh graph, assigns constitutive behavior per element, and minimizes the discrete total potential energy as a single unweighted objective in which only first derivatives appear. The discrete energy on piecewise-linear elements coincides with the finite element (FE) Ritz functional. Dirichlet conditions are enforced by construction, with no penalty term, no interface weight, and no prescribed transition width. Using one fixed architecture, optimizer, and loss across small-strain elasticity and finite-strain Neo-Hookean hyperelasticity in two and three dimensions, the von Mises error remains below $3.58\%$ across a stiffness-contrast sweep spanning $(E_{\mathrm{inc}}/E_{\mathrm{mat}}\in[10^{-2},10^{2}])$, where a strong-form PINN degrades to $5.58\%$, and its displacement error reaches $7.66\%$ against $0.49\%$ for the PI-GNN. A trained network halves the ($\sigma_{xx}$) error of an energy-based PINN ($5.01\%$ versus $10.94\%$). Training cost exceeds a single FE solve by more than an order of magnitude, so the construction is a variationally consistent, penalty-free interface representation for parametric surrogates and inverse identification rather than a replacement for a one-off FE analysis.

EGGROLL (Sarkar et al., 2026) makes evolution strategies (ES) practical for LLMs by replacing dense Gaussian weight perturbations with low-rank Gaussian products, often of rank one. This choice is computationally attractive but geometrically severe: Each rank-one perturbation lies in a zero-volume subset of the ambient matrix space, despite having identity covariance. We characterize the EGGROLL update mean field at finite rank and nonzero perturbation radius as a resolvent applied to the gradient of the perturbation-smoothed objective. This transformation can make the mean field nonconservative and reverse the local stability of an optimum. EGGROLL nevertheless recovers the gradient exactly on quadratic objectives at every rank and radius. In finite populations, the additional sampling variance of rank-one perturbations relative to dense Gaussian ES decays inversely with matrix width under a local affine model, and is only $0.098\%$ at width $4096$. Finally, we introduce LOO-ROLL, a leave-one-out estimator that replaces EGGROLL's two antithetic evaluations per direction by one. At equal evaluation cost, LOO-ROLL halves estimator MSE in transformer blocks. Across fourteen post-training settings up to 14B parameters, matched-time comparisons with EGGROLL yield eleven improvements in individual paired tests and no significant loss. At 1.7B, 8B, and 14B parameters, matched-time gains are $2.9$, $14.1$, and $7.9$ percentage points on GSM8K and $12.2$, $8.4$, and $8.1$ points on MATH-500.

Bernshteyn, Kostochka, and Zhu (2020) introduced the notion of fractional DP-coloring, which generalizes both fractional coloring and fractional list coloring. Among several foundational results, they proved that every $d$-degenerate bipartite graph $G$ satisfies $\chi_f^{\mathrm{DP}} \le (1 + o(1))\frac{d}{\log d}$, and that this bound is optimal—a stark contrast to ordinary fractional coloring. In this paper, we extend this upper bound to all $d$-degenerate triangle-free graphs, proving that $\chi_f^{\mathrm{DP}} \le (4 + o(1))\frac{d}{\log d}$. This generalizes a recent result of Martinsson and Steiner (2025) for ordinary fractional coloring. We derive this result as a corollary of a more general upper bound concerning locally sparse graph orderings. Specifically, a $d$-degenerate graph $G$ is left $k$-locally-sparse if it admits a degeneracy ordering in which, for every vertex $v$, the subgraph induced by its back-neighbors contains at most $k$ edges. We show that if a $d$-degenerate graph $G$ is left $\frac{d^2}{f}$-locally-sparse, then \[ \chi_f^{\mathrm{DP}}(G) \le (8 + o(1))\frac{d}{\log f}. \] This immediately yields an identical upper bound on the ordinary fractional chromatic number $\chi_f(G)$, improving upon the leading constants of previously known bounds. Additionally, we establish the asymptotic sharpness of this result up to the leading constant. For any $1 \ll f \le d^2$, we construct $d$-degenerate graphs that are left $\frac{d^2}{f}$-locally-sparse and satisfy $\chi_f(G) \ge (1 - o(1))\frac{d}{\log f}$. Finally, as applications of our main theorem, we obtain improved upper bounds on the fractional DP-chromatic number of $d$-degenerate $K_{1,t,t}$-free graphs, as well as $K_{t,t,t}$-free graphs with maximum degree $\Delta$. Notably, these bounds improve upon existing results even in the setting of ordinary fractional coloring.

We study Turán problems for balanced even cycles in simple signed graphs, where signed subgraphs are considered up to switching. For every balanced bipartite signed graph, the signed and ordinary Turán numbers differ by at most a factor of two. Our main structural results concern the underlying graphs that admit a signing in which every $2k$-cycle is unbalanced. We characterize these graphs by the absence of an odd dependence among their $2k$-cycle incidence vectors, give a cohomological formulation, and construct subgraph-minimal obstructions of arbitrarily large order. In particular, there is no finite forbidden-subgraph characterization. We also give an exact closed-walk criterion for cycles in double covers and derive a direct signed breadth-first-search upper bound. As applications, we prove \[ \hex(n,C_{+4})=\left(\frac{\sqrt2}{2}+o(1)\right)n^{3/2} \] and study the signed hexagon number $R_6(n)=\hex(n,\{C_{-3},C_{+6}\})$. We characterize the underlying graphs counted by $R_6$ and express it as an extremal problem for ordinary $C_6$-free graphs with a prescribed involution. For every sufficiently large $n$, we construct examples with $\Omega(n^{4/3})$ edges, and we give an equivariant construction attaining the coefficient obtained from the Füredi–Naor–Verstraëte lower bound by double-cover transfer. Finally, we give $n$-vertex $C_{+10}$-free signed graphs with $\Omega(n^{6/5})$ edges and use octagon examples to illustrate the limitations of theta-freeness as a signing criterion.