Papers

The vulnerability and unavailability of global navigation satellite system (GNSS) signals in dense urban environments have motivated the use of low earth orbit (LEO) signals of opportunity (SoOP) for positioning, navigation, and timing (PNT). However, multipath and non-ideal noise can significantly degrade delay–Doppler estimation and, consequently, navigation accuracy. This paper proposes a hybrid generalized likelihood ratio test (GLRT) and off-grid sparse Bayesian learning (SBL) framework for LEO-SoOP navigation under multipath and non-Gaussian, temporally correlated noise. First, a GLRT-based framework is developed, demonstrating reliable detection under ideal Gaussian noise but degraded estimation under heavy-tailed or colored noise and unresolved multipath. Next, an off-grid SBL framework is introduced for high-resolution separation and estimation of line-of-sight (LOS) delay–Doppler parameters. The estimated LOS parameters are subsequently tracked and incorporated into an extended Kalman filter (EKF) to obtain the navigation solution. Simulation results demonstrate that, in urban environments, the proposed method reduces delay RMSE by up to 77% and position RMSE by 88.5% compared with GLRT-based estimation. These results demonstrate the potential of the proposed framework for robust and accurate LEO-SoOP navigation.

In this paper, we investigate the differentiability of Korevaar–Schoen $p$-energy forms and the associated $p$-energy measures. We obtain higher-order derivatives by virtue of explicit realizations of Korevaar–Schoen $p$-energy forms and the associated $p$-energy measures as subsequential pointwise limits of certain double integrals.

Explicit R-matrices of quantum groups are routinely transcribed from the literature, yet whether they satisfy an identity depends on basis order, tensor placement, the coproduct and, for superalgebras, the Koszul sign rule. quantum-group is a Python package that builds such matrices exactly with SymPy and returns every identity as an exact residual matrix, whose nonzero entries locate the convention that disagrees. It covers U_q(sl_2) modules, tensor products and the fundamental R-matrix, and the graded Yang–Baxter equation for GL_q(2|1). Applied to the thesis it was written for, it exposed an R-matrix that satisfies the Yang–Baxter equation but intertwines the opposite coproduct.

Phylogenetic networks generalize phylogenetic trees as models of evolutionary history, allowing lineages to merge as well as to diverge. For many types of genetic data the root position of such a network cannot be recovered, so that only a semi-directed network can be inferred: a mixed graph in which only the edges entering a reticulation vertex are directed. A common strategy for inferring such a network is to first infer the subnetwork it induces on each set of $k\geq 3$ of its leaves, called a $k$-net, and then to assemble these pieces. This can only succeed if the $k$-nets determine the network, in which case that network is said to be encoded by its $k$-nets. Semi-directed networks of level-1 and 2, those whose biconnected components contain at most one, respectively two, reticulations, are known to be encoded by their $4$-nets, or quarnets, whereas level-3 networks are not. Even so, in this paper we show that level-3 semi-directed networks are encoded by their $5$-nets, or quinnets, and we characterize the limitation of quarnets exactly: we show that a single previously reported counterexample captures the only obstruction, every other level-3 network being encoded by its quarnets. Our proofs rest on a collection of encoding results for individual structural features of a network, which we establish for networks of arbitrary level and which are of independent interest.

In current work, we delve into the study of complex Hessian equations on compact Kähler manifolds. By imposing a constraint on both the bisectional curvature and the oscillation of the solution, we demonstrate that the Laplacian estimate can be linearly controlled by its gradient estimate. Consequently, we are able to derive gradient estimate for the solution using a standard interpolation argument, without the need to employ the blow-up technique. As an application, we demonstrate that when the right-hand side of the complex Hessian equations approximate 1 in the $L^1$ norm, the desired gradient estimate is obtained.

The scenario approach is a well-established statistical framework for data-driven decision-making. In particular, in data-driven optimization, the scenario approach unveils how the problem structure governs out-of-sample generalization, and offers a principled basis for assessing and certifying the reliability of the optimal solution as per constraint satisfaction. Despite its strong theoretical development and wide applicability, no software toolbox has been available to date that enables user-friendly, data-driven convex optimization within the scenario-approach framework. In this paper, we introduce Scen-Opt, an open-source software tool that integrates convex programming with data samples while providing statistical guarantees grounded in scenario theory. Scen-Opt is implemented in Python, supporting data-driven linear, quadratic, and semidefinite programming, and offers a Python-based web application with an intuitive and reactive graphical user interface (GUI) built using modern web technologies. Scen-Opt can be used directly through its online interface or installed locally, accommodating both manual input and data-file uploads (CSV, JSON, TXT, TSV, MAT, Excel, NPY, NPZ, Parquet). Built on a Python backend with a modern JavaScript frontend, Scen-Opt offers a highly user-friendly experience and efficient usability across desktops, laptops, tablets, and mobile devices. In this paper, Scen-Opt is applied to a set of representative benchmarks, demonstrating its practical effectiveness for data-driven convex optimization with guaranteed performance.

A normalized analytic function $f(z)=z+\sum_{k=n+1}^{\infty}a_kz^k$ defined on the unit disk $\mathbb{D}$ is called Ma–Minda starlike or Ma–Minda convex if $zf'(z)/f(z)$ or $1+zf''(z)/f'(z)$, respectively, is subordinate to a Ma–Minda function $\varphi$. In this paper, we investigate the radii of Ma–Minda starlikeness and convexity for functions satisfying $|f''(z)|<M$ in $\mathbb{D}$. We first develop a unified approach for determining these radii and then apply it to obtain explicit radius estimates for several important choices of the Ma–Minda function~$\varphi$.

Wind farm layout optimization decides where to place turbines to maximize annual energy production, and wake effects decide how much of that energy is produced. Accurate wake models are too slow for a search. We build on EndoWake, an endogenous wake model in which the wind speed at every grid cell is a variable linked to its upwind neighbors by linear constraints, so that wake field, siting decisions and project constraints share one mixed-integer model. Its four parameters are calibrated once against the single-wake field of a Reynolds-averaged Navier-Stokes (RANS) simulation, and a GPU scores over a million layouts per second per wind direction. Comparing EndoWake-guided searches with a PyWake-based pipeline at Lillgrund, under a RANS judge that no search calls, revealed that the fidelity of a wake model is not the quality of the layouts optimized under it. A search is drawn to the gaps between sampled wind directions, the sirens; a diagnostic of the field finds a second artifact, an optimistic band off each wake edge, the mermaid cells, not reported before to our knowledge. Judged at random directions, the two pipelines are statistically indistinguishable. Our main result puts this speed to use: EndoWake screens the candidate moves of a local search on the GPU, and PyWake confirms every accepted move. On twenty unseen starts, with one GPU and thirty CPU processes, this screen-and-confirm search matches the PyWake value of a search on PyWake alone 7.9 times faster, and 14.9 times faster on the annual energy.

In the 1960s, Erdős and Gallai conjectured that the edges of every $n$-vertex graph can be decomposed into $O(n)$ cycles and edges. We prove this conjecture. Equivalently, every Eulerian graph on $n$ vertices can be decomposed into $O(n)$ cycles, which confirms Hajós' conjecture up to a constant factor.

For a proper cocompact action of a finitely generated discrete group $\Gamma$ on a spin manifold $M$, the higher localised $\widehat{A}$-genera $\widehat{A}_g(M,\omega)$ introduced earlier by the author and Mathai were shown to be an obstruction to $\Gamma$-invariant metrics of positive scalar curvature, for $\omega$ arising from the subring of $H^*(\underline{B}\Gamma,\mathbb{R})$ generated by elements of degree at most $2$, under the assumptions that the fixed-point set $M^g$ is connected and that $g$ is regular with respect to a certain real group $2$-cocycle. We show that both assumptions can be dropped.

Let $E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ be a flat torus and $G(z)=G(z;\tau)$ be the Green function on $E_{\tau}$ with the singularity at $0$. Consider the multiple Green function $G_{n}$ on $(E_{\tau})^{n}$: \[ G_{n}(z_{1},\cdots,z_{n}):=\sum_{i<j}G(z_{i}-z_{j})-n\sum_{i=1}^{n}G(z_{i}). \] A critical point is called trivial if $\{z_{1},\cdots,z_{n}\}=\{-z_{1},\cdots,-z_{n}\}$ in $E_{\tau}$. Lin and Wang (Ann. Math. 2010) proved that $G_1=-G$ has exactly three trivial critical points and at most one pair of nontrivial critical points (depends on the choice of $\tau$). For general $n\geq 2$, Chai, Lin ang Wang (Camb. J. Math. 2015) proved that $G_n$ has at most $2n+1$ trivial critical points, but how many nontrivial critical points might exist remained completely open there. In this paper, we prove that $G_n$ has at most $n$ pairs of nontrivial critical points, and this upper bound is optimal at least for $n=2,3$. The key idea is to show that nontrivial critical points (if exist) are always non-degenerate and contribute the same degree $(-1)^{n+1}$. Applications to the curvature equation $\Delta u+e^{u}=8\pi n \delta_{0}$ and the integral Lamé equation $y''=[n(n+1)\wp(z)+B]y$ will be given. In particular, we show that the kernel of the linearized operator is one-dimensional for any solution of the curvature equation.

Experiments on binary run-length differentiability lead to sharp computer-assisted criteria for shuffle squares. Every $C^3$-word of length greater than $16$ is a shuffle square exactly when both letter multiplicities are even. All $34$ nonempty even-Parikh exceptions are smooth and persist in every higher differentiability class. For $C^2$ the sharp threshold is $48$, with $212$ nonempty exceptions. At level $C^1$ no global parity threshold exists, but every even-Parikh non-shuffle-square of length at least $36$ has proper nonempty shuffle-square prefixes and suffixes. Exactly $230$ nonempty even-Parikh $C^1$-words have no nonempty shuffle-square prefix. The full tree avoiding such prefixes eventually consists of $422$ periodic rays. Consequently, a nonempty Kolakoski prefix is a shuffle square exactly when both multiplicities are even, apart from lengths $4$ and $8$. We also characterize classes of morphisms reflecting shuffle squares. For doubly binary words, we determine the exact deletion distance and largest twins, and prove sharp bounds for single local repairs. Exact recurrences, residual-state checks, and separate Python programs make the finite computations reproducible.

Let $p:X\rightarrow\Delta$ be a proper holomorphic submersion whose total space $X$ is a Kähler manifold. We prove that $\dim_{\mathbb C}H^0(X_t,mK_{X_t})$ is independent of $t$ for every integer $m\geq1$. This was conjectured by Y.-T. Siu in 2002.

In this article, we study the global well-posedness for the quadratic Schrödinger system on $\mathbb{T}^6$. To achieve this, we utilize the concentration-compactness/rigitdity argument initiated by Ionescu and Pausader \cite{IPT3}. Our result is also regarded as a periodic analogue of Gao-Meng-Xu-Zheng \cite{GMXZ}.

We study the Jacobi spectrum and low-index rigidity of complex Hopf fibrations. We first give a representation-theoretic description of the Jacobi operator of a homogeneous Riemannian fibration and apply it to \[ \pi:(\mathbb{S}^{2n+1},g_\epsilon)\longrightarrow(\mathbb{C}P^n,h), \] where $g_\epsilon$ is a Berger metric and $h$ is the Fubini–Study metric. For $n\geq3$, we determine all negative and zero eigenvalues and obtain explicit formulas for the Morse index and nullity for every $\epsilon>0$. In particular, for the unit round metric, $\mathrm{Ind}(\pi)=2n+2$ and $\mathrm{Null}(\pi)=n(3n+5)$. We then extend Rivière's rigidity argument from $\mathbb{S}^3\to\mathbb{S}^2$ to higher dimensions. If $\phi:\mathbb{S}^{2n+1}\to\mathbb{C}P^n$ is harmonic and $\mathrm{Ind}(\phi)=2n+2$, then $d\phi_x\circ d\phi_x^\ast$ commutes with the complex structure of $\mathbb{C}P^n$ at every point. Consequently, up to an isometry of the domain, $\phi$ factors through the Hopf fibration as $\phi=P\circ\pi\circ f$, where $P:\mathbb{C}P^n\to\mathbb{C}P^n$ is holomorphic.

We introduce complementary determinantal edge ideals. For a graph $G$ on $n$ vertices, the complementary determinantal edge ideal $J_{c}(G)$ is generated by the maximal minors of a generic $(n-2)\times n$ matrix obtained by deleting the two columns indexed by each edge of $G$. We completely characterize the graphs for which $J_{c}(G)$ is Cartwright–Sturmfels with respect to the grading by columns. We prove that this property is equivalent to the existence of a multilinear universal Gröbner basis and characterize the graphs satisfying these equivalent conditions. In particular, the ideals satisfying these equivalent conditions are radical.

Let $\textit{spex}_{\mathcal{P}_1}(n,F)$ be the maximum spectral radius among all $n$-vertex $F$-free $1$-planar graphs. Define $F_t$ as the friendship graph formed by $t$ triangles sharing exactly one common vertex. Tait and Tobin (2017)~\cite{Tait2017} used the fundamental structure of spectral extremal graphs to determine the unique planar graph with maximum spectral radius for sufficiently large order. Subsequently, Zhang, Wang and Wang (2024)~\cite{Zhang2024} characterized the corresponding extremal graph in the class of $1$-planar graphs. In this paper, we focus on $F_t$-free $1$-planar graphs and establish a structural theorem for their spectral extremal graphs for all $t\geq1$ and sufficiently large $n$. More precisely, every extremal graph is connected and contains a copy of $K_{2,n-2}$, and for $t\geq2$ the two distinguished vertices are adjacent and the subgraph induced by the remaining vertices is a bipartite graph. Based on this structure result together with the drawing properties of $K_{3,6}$, we determine $\textit{spex}_{\mathcal{P}_1}(n,F_t)$ and characterize its unique extremal graph.

Over a field of characteristic zero, we characterize the Hilbert functions that determine the degree reverse lexicographic generic initial ideal among Artinian quotients in four variables with the $2$-strong Lefschetz property. Our criterion is a finite numerical test that strictly extends the quasi-symmetry condition of Harima and Wachi. The proof parametrizes the possible generic initial ideals by flags of lower sets in the Borel orders on their common three-variable section. We also give nonrecursive formulas for the standard monomials and minimal generators of the almost reverse lexicographic ideal, and specialize them to generic complete intersections in at most four variables without restrictions on the degrees.

We consider the emergence of rapidly propagating, exponentially small-amplitude oscillations in the impulsively started fifth-order Korteweg–de Vries equation (KdV5), and demonstrate that these waves originate due to Stokes' phenomenon in a small-time boundary layer. We rescale the problem in time and construct an auxiliary linear PDE (lKdV5) that reproduces the relevant exponential asymptotic structure, and proceed by two main methods. Firstly, the Fourier transform solution of the auxiliary PDE is analysed via the method of steepest descents to extract the relevant asymptotic contributions and the locations of the Stokes curves. Secondly, we reproduce these results using a direct factorial-over-power analysis. In this analysis, we make use of the initial and far-field conditions to avoid performing the local asymptotic matching that is typically required by exponential asymptotic methods and is frequently impossible for nonlinear PDEs. A similar factorial-over-power analysis is then applied to the full nonlinear PDE, which cannot be studied using steepest descent methods. The results agree well with numerical simulations, demonstrating both the rapid propagation of non-decaying oscillations and transient ripples whose amplitudes decay in the far field, and revealing the complex Stokes structure (including both normal and higher-order Stokes curves) that governs their emergence. Such a complex Stokes structure in a temporal boundary layer is a novel phenomenon.

We study parabolic stochastic partial differential equations on metric measure spaces $(\mathbb{X}, d,m)$ of the form $$ \partial_t u(t,x) = \mathcal{L}^* u(t,x) + b(t,x,u(t,x)) + \sigma(t,x,u(t,x)) \dot{W}(t,x),\quad t>0,\, x \in \mathbb X, $$ where $\mathcal{L}$ is the generator of a Markov process which possesses transition densities, and $\dot{W}$ is a Gaussian noise that is white in time and possibly with spatial correlation. We assume the coefficients $b$ and $\sigma$ are Lipschitz and satisfy the linear growth condition. We formulate general and checkable assumptions on $(\mathbb{X},\mathcal{L},\dot{W})$ that ensure existence and uniqueness of probabilistically strong, continuous, tempered mild solutions. We then prove comparison principles (including a strong comparison principle) and strict positivity, relative to initial conditions. Our framework allows non-symmetric heat kernels and includes diffusion-type and stable-type examples, such as metric graphs and fractal spaces with sub-Gaussian heat kernel estimates.

Let $x:M^n\to\mathbb{S}^{n+q}$ be a minimal immersion of a complete connected manifold with flat normal bundle, where $n\ge3$ and $q\ge2$. We prove that if the squared norm of the second fundamental form $S$ satisfies $S<2n$, then $M^n$ lies in $\mathbb{S}^{n+1}$. The standard minimal products of three spheres show that the upper bound $2n$ is optimal. We also prove a codimension reduction theorem under the condition $S+\lambda_2\le 2n+\frac{2n}{3n-2}$, where $\lambda_2$ is the second largest eigenvalue of the Gram matrix of the shape operators, implying that $M^n$ lies in $\mathbb{S}^{n+1}$.

Motivated by the open question of Bianchi, Hachem, and Schechtman (Bianchi et al., 2024, Remark 4), we show that in the non-weakly convex definable setting, omnidirectional noise with conditional fourth-moment control does not suffice for universal almost-sure avoidance of sharply repulsive critical points. We demonstrate this failure by constructing a globally Lipschitz, coercive, definable objective that is not weakly convex. For this objective and its sufficiently small linear perturbations, stochastic subgradient descent with independent noise uniformly distributed on a ball in $\mathbb{R}^2$ converges with probability $1$ to a nonminimal sharply repulsive critical point from every initial point in a specified ball. A two-cycle argument establishes uniform bounds on the rescaled iterates, yielding convergence to the sharply repulsive critical point. In contrast, for locally Lipschitz definable objectives, Gaussian noise ensures simultaneous almost-sure avoidance of any finite collection of sharply repulsive points. Through a construction of difference quotient functions, we show that convergence to any such point with positive probability would produce a stationary distribution with expected descent $0$, whereas a global positive lower bound on the Fréchet subgradient norms of the limiting functions forces strictly positive expected descent under the same distribution. This contradiction proves avoidance, with the everywhere positive Gaussian density playing a key role.

The eigenstate quantum-mechanical bootstrap can converge to the physical moment set while remaining nonexact at every finite order. We prove this for a one-dimensional quasi-exactly solvable sextic oscillator at explicitly known eigenvalues, including excited levels. At every finite order, the lower and upper bounds on $u^2$ remain strictly below and above its physical value, even after adjoining arbitrary finite polynomial Gram tests and all eigenstate-annihilator relations visible in the retained moment space. Hence neither side of the centered observable admits a finite sum-of-squares certificate modulo the eigenstate annihilator. The gap persists under any finite set of additional localizing constraints whose weights are strictly positive on polynomial vectors. We then prove convergence for polynomial Schrödinger operators satisfying explicit sum-of-squares confinement and derivative bounds. The eigenstate relations give moment estimates uniform in the truncation order; these yield a regular Schrödinger representation supported on the prescribed eigenspace. Finally, a four-mode extension combines this fixed-observable obstruction with feasible position moments having no positive representing measure at every finite order. At level three, this construction yields nonphysical feasible points with strictly positive definite moment matrices.

We disprove the Clunie–Sheil-Small conjecture $\bigl||a_n|-|b_n|\bigr|\le n$ for normalized univalent harmonic functions $f=h+\overline g$ in the unit disk $\mathbb{D}$, where $a_n$ and $b_n$ are the Taylor coefficients of $h$ and $g$, respectively. For every $K>1$, we construct a harmonic $K$-quasiconformal map which has $|a_n|-|b_n|\asymp n^{1+\varepsilon_K}$ as $n\to\infty$, with $\varepsilon_K>0$. Nevertheless, this map satisfies both individual coefficient bounds in the conjecture. Next, we show that the same map belongs to the harmonic Hardy space $h^p$ if and only if $0<p<1/(2+\varepsilon_K)$. Thus, harmonic quasiconformal maps do not retain the full conformal Hardy range $p<1/2$. Interestingly, the latter range is recovered when the oscillation of the analytic dilatation on disks of fixed hyperbolic radius tends to zero near the boundary. This holds, in particular, when the dilatation has finite Dirichlet energy. For general $K$-quasiconformal maps, we impose the additional assumption that their Beltrami coefficients belong to the Sobolev space $W^{1,s}(\mathbb{D})$ for some $s\ge1$. Every such map has bounded $p$-integral means for $0<p<1/(2K)$ when $1\le s<2$, and for $0<p<1/2$ when $s\ge2$. Both these ranges are sharp. For $s\ge2$, each map also has the same critical Hardy exponent as a conformal map onto the same image.

We construct cochain models for Serre fibrations over path-connected bases with coefficients in an arbitrary field. Our construction refines Brown's twisted tensor product to produce a differential graded module model over the singular cochain algebra of the base. If the homology of the fibre is finite dimensional in each degree and on which the action of the fundamental group of the base is degreewise nilpotent, then this model is a semifree resolution generated by the cohomology of the fibre. This extends the classical construction for simply connected bases. As applications, we compare the resulting model with arbitrary semifree resolutions, showing that every such resolution recovers the cohomology of the fibre after base change to the basepoint, and obtain an explicit bound on the page at which an associated spectral sequence collapses.

We study discrete-time quantum walks on weighted graphs, where every edge in an edge cut set is assigned a small weight $\epsilon>0$. The parameter $\epsilon$ represents the strength of the connections through the cut edges: as $\epsilon \to 0$, these connections vanish, and the graph decomposes into the connected components obtained by removing the cut edges. We refer to these weighted cut edges as weak edges. We show that, for sufficiently small $\epsilon$ and on the time scale $t=\Theta(\epsilon^{-1/2})$, the finding probabilities between the connected components are asymptotically described by a continuous-time wave equation on a reduced graph, whose vertices represent the connected components and whose edges represent the weak edges connecting them. The wave equation is governed by a symmetric weighted Laplacian determined by the reduced graph and the number of arcs contained in the components. Consequently, finding probabilities of leading-term are independent of the detailed internal structures of the components. We further show that this wave equation has the same form as Newton's equation of motion for a classical spring-mass system on the reduced graph.

We prove a sharp quantitative form of the van der Corput property for squares. For all sufficiently large $N$, we construct a real trigonometric polynomial $P(\theta)=a_0+\sum_{1\le n\le\sqrt N}a_n\cos(2\pi n^2\theta)$ satisfying \begin{equation}\tag{1}\label{eq:abstract-vdc} P(\theta)\ge0\quad(\theta\in\mathbb{R}/\mathbb{Z}),\qquad P(0)=1, \end{equation} with \[0\le a_0\le\exp(-c\sqrt{\log N\log\log N}),\] where $c>0$ is an absolute constant. As a consequence, if $A\subseteq\mathbb{N}\cap[1,N]$ has no nonzero square difference, then \[|A|\le N\exp(-c\sqrt{\log N\log\log N}),\] improving upon a recent result of Green and Sawhney. Our construction builds on previous work of Slijepčević, Slijepčević–Ninčević, and Fan–Lott, and the proof of the upper bound is elementary and self-contained. We also show that every real trigonometric polynomial of the above form satisfying \eqref{eq:abstract-vdc} has \[a_0\ge\exp(-C\sqrt{\log N\log\log N})\] for an absolute constant $C>0$. Thus the van der Corput property cannot be used to obtain a power-saving bound for the Furstenberg–Sárközy theorem.

Decentralized optimization on the Stiefel manifold has broad applications in machine learning and signal processing. We propose PLADOS, a parameter-free landing algorithm for decentralized optimization on the Stiefel manifold. Each iteration combines local stochastic gradient updates with a single retraction-free communication step over the communication graph. A key convex-like property of the intersection of the nonconvex Stiefel manifold and consensus constraints is characterized through a restricted secant inequality for the penalty-only scheme, ensuring local contraction of the manifold consensus error without introducing additional parameters that require tuning. For $n$ nodes using independent local batches of size $b$, we prove that PLADOS achieves linear speedup with respect to the number of nodes with an asymptotic convergence rate of $O(\sigma/\sqrt{nbK}+(n\xi^2)^{1/3}/K^{2/3})$ after $K$ iterations, where $\sigma^2>0$ is the sampling variance and $\xi^2$ quantifies gradient heterogeneity. Experiments demonstrate stability across tested stepsizes and networks, time efficiency, and linear speedup with respect to the number of nodes.

For even $n$, let $P_n$ be the probability that independent uniform strict preference lists on $n$ participants admit a stable perfect matching. We prove \[ P_n\sim\frac{e\,2^{1/4}\Gamma(3/4)}{\sqrt\pi}\,n^{-1/4}. \] This establishes Mertens's conjectured exponent of decay, with a leading constant different from his original numerical prediction. The proof starts from Mertens's exact alternating sum over stable permutations. To preserve its cancellation, we construct a common approximation for all cycle structures with the same number of vertices in cycles longer than two. By symmetry, the integrated first-order correction is the same for every such cycle structure, and the remaining errors can be summed in absolute value. The enumeration then reduces the probability to a one-dimensional sum with positive terms.

This paper proves local propagation of momentum moments of order $k\ge5/2$ for the three-dimensional relativistic Vlasov–Poisson system. In the Coulomb case, this improves the previous result which requires integer moments of order $k\ge3$. The proof adapts the critical density method developed in our recent work on the classical Vlasov–Poisson equation. We introduce a weighted kinetic tensor adapted to the relativistic velocity and obtain the critical density estimate. This estimate then allows us to close the moment argument by a short-time bootstrap.