Papers

Consider a sequence of metrics $g_i$ on the torus whose members have uniform lower bounds on their first stable systoles and Ricci curvatures, and have a uniform upper bound on their diameters. If the $L^1$ norm of the negative part of the scalar curvatures vanishes along this sequence, then we show there is a subsequence of the metrics which converges in the measured-Gromov-Hausdorff topology to a flat metric on the torus. Something analogous holds for sequences of asymptotically flat spin Riemannian manifolds with a uniform lower bound on Ricci curvature and nonnegative scalar curvature: if the ADM masses of the distinguished ends tend to zero along the sequence, then the manifolds converge to Euclidean space in the pointed measured Gromov-Hausdorff sense.

Let $Q = ax^2+bxy+cy^2$ be a primitive positive definite integral binary quadratic form with discriminant $\Delta = b^2-4ac$. It is known that $Q$ admits a local obstruction; that is, there exist $k,l \in \mathbb{Z}$ such that $Q \not\equiv l \pmod k$. We study the minimal obstruction modulus $\kappa_Q := \min \{k \in \mathbb{Z}_{\geq 1} \mid \text{there exists } l \text{ such that } Q \not\equiv l \pmod k \}$, and we determine $\kappa_Q$ completely, treating the cases $\Delta \equiv 0 \pmod 4$ and $\Delta \equiv 1 \pmod 4$ separately. We also determine the analogous invariants for primitive ternary diagonal forms.

In 1991 Rickard asked whether every derived equivalence of algebras over a common field is standard. We construct a series of examples of non-standard derived equivalences, and thus answer the question negatively. Moreover, we give a necessary and sufficient condition for the question to be true.

In this paper, we investigate the following critical Trudinger–Moser inequality on $\mathbb R^2$ under sharp $L^p$-perturbations: $$ S(\lambda,p) := \sup_{\substack{u\in H^{1}(\mathbb R^{2})\\ \int_{\mathbb R^2}(|\nabla u|^2+|u|^2)\,dx\le 1}} \int_{\mathbb R^2} \left(e^{4\pi u^2}-1-\lambda |u|^p\right)\,dx . $$ For $2<p\le 4$, we prove the existence of a critical value $\lambda^{\ast}\in(0,+\infty)$ such that $S(\lambda,p)$ is attained when $\lambda<\lambda^{\ast}$ and is not attained when $\lambda>\lambda^{\ast}$. Moreover, we show that the nonattainment in this range is caused by a vanishing phenomenon. For $p=2$, combining our analysis with the nonexistence results for $L^2$-perturbed Trudinger–Moser inequalities obtained in \cite{Chenluzhu}, we establish the existence of two finite thresholds $\lambda_{\ast}>-\infty$ and $\lambda^{\ast}<+\infty$ such that $S(\lambda,2)$ is attained when $\lambda_{\ast}<\lambda<\lambda^{\ast}$, and is not attained when $\lambda<\lambda_{\ast}$ or $\lambda>\lambda^{\ast}$. In contrast, for $p>4$, we prove that $S(\lambda,p)$ is attained for all admissible values of $\lambda$. Our results indicate that, in the whole-space setting, the $L^p$-perturbation term affects the existence and nonexistence of extremals through either concentration or vanishing phenomena, which is fundamentally different from the bounded-domain case, where existence or nonexistence is governed solely by concentration phenomena. These results provide a complete characterization of how sharp $L^p$ perturbations determine the existence and nonexistence of extremals for critical Trudinger–Moser inequalities on the entire $\mathbb R^2$. The resulting existence and nonexistence theory exhibits a threshold structure with respect to the $L^p$ pertubation reminiscent of the classical Brezis–Nirenberg phenomenon in the whole space $\mathbb R^2$.

We establish a Schauder estimate for a nonlocal Cauchy problem with drift. The leading operator is the generator of a non-degenerate $\alpha$-stable process with $\alpha\in(0,1)$, and the drift is $\beta$-Hölder continuous with $\beta \in (1-\alpha,1)$. The proof relies on a refined conic Littlewood–Paley decomposition and a one-dimensional one-sided dyadic maximum principle.

Universally optimal (UO) codes were introduced by H. Cohn and A. Kumar in 2007 and later extended to the discrete setting by Cohn and Y. Zhao. They minimize the “energy” among all codes of the same size for a certain class of potential functions. So far only a small number of specific functionals have been linked to information theory problems. We add one more, showing that UO codes optimize $\alpha$-mutual information for $\alpha=2$ in the context of wiretap channels with a noiseless main channel. This implies that UO codes are optimal for transmission over this type of wiretap channels.

We study decision-focused learning (DFL) in shortest-path network interdiction (SPNI) games, a Stackelberg game where an interdictor (leader) strengthens the networks' arcs against attacks, while an evader (follower) who is uncertain about costs of attacking network arcs relies on a machine-learned predictor to identify the shortest path. While DFL is highly effective as an end-to-end optimization framework, we show that it faces a fundamental structural failure when employed in this game setting: its training objective admits a broad decision-equivalence class of cost estimators that achieve zero nominal loss yet fail under interdiction, reversing DFL's usual advantage over a naive prediction-focused learning (PFL) approach. To address this, we propose Adversarial DFL (A-DFL), which replaces nominal training samples with interdicted scenarios to collapse the harmful equivalence class. Experiments on synthetic and real-world networks confirm that A-DFL restores DFL's advantage in this game setting, enabling effective end-to-end optimization.

We investigate the optimal Bianchi-Egnell-type quantitative stability constant for the critical nonlocal Sobolev inequality on the Heisenberg group $\mathbb{H}^{n}$, \begin{equation}\label{nS} S_{HL}(Q,\mu)\left(\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}} \frac{|u(\xi)|^{Q^{\ast}_{\mu}}|u(\eta)|^{Q^{\ast}_{\mu}}} {|\eta^{-1}\xi|^{\mu}}\,d\xi d\eta\right)^{\frac{1}{Q^{\ast}_{\mu}}} \leq \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}}u|^{2}d\xi, \qquad u\in S^{1,2}(\mathbb{H}^{n}), \end{equation} where $Q=2n+2$, $n\geq1$, $0<\mu<Q$, and $Q^{\ast}_{\mu}=(2Q-\mu)/(Q-2)$. Let $\mathfrak{M}$ denote the manifold of Jerison-Lee bubbles, and let $H_{NS}$ be the infimum of the quotient between the deficit in \eqref{nS} and $\mathrm{dist}(u,\mathfrak{M})^{2}$. The central issue is that the Euclidean nonlocal problem and the local Folland-Stein-Sobolev problem each possess their own spectral and compactness structures, whereas the present problem couples the HLS interaction with the noncommutative conformal geometry of $\mathbb{H}^{n}$. Under the stated conditions on $(n,\mu)$, these strict thresholds, together with a Heisenberg-group profile decomposition and a nonlocal Br{e}zis-Lieb splitting, imply that $H_{NS}$ is attained. The minimizer then yields the strict comparison $H_{NS}>H_{BE}$ with the optimal stability constant for the local Folland-Stein-Sobolev inequality. We further prove that the sharp universal upper constant for the deficit-to-distance comparison is $1$ and characterize equality. Finally, for the associated Euler-Lagrange equation, we formulate the corresponding residual quotient and derive a strict single-bubble upper bound; this critical-point statement requires a separate expansion and does not follow from attainment of $H_{NS}$.

What should a machine learning model learn when data is missing during training? We look at the learning process from a dynamical systems perspective, cast data missingness as a structured loss of actuation that limits controllability of the parameter error dynamics, and ultimately derive adaptation mechanisms with Lyapunov stability characteristics that throttle model updates in ways that preserve learning coherence under partial, intermittent observability. Under recurrent excitation, our analysis provides ISS-type residual-to-state bounds with respect to a bounded closed-loop mismatch between the loss residual and the preconditioned update geometry. We evaluate the efficacy of our directional observability-aware adaptive learning approach on multimodal contexts, reinforcing its premise in promoting learning coherence and stability even in pathologically sparse domains and problems.

In this paper, we develop provable parameter-free fixed-point algorithms for contractive mappings, with an emphasis on automatically exploiting hidden contractivity without requiring prior knowledge of the contraction factor. Our first method is a parameter-free variant of the Halpern fixed-point iteration with no line-search, bisection, or prior estimate of the contraction factor, while retaining essentially the same per-iteration computational cost as classical fixed-point schemes. We establish explicit linear convergence rates for both the fixed-point residual and the distance to the unique fixed point. The second algorithm is also a parameter-free Halpern method, but it requires a line-search procedure to find a local estimate for the contraction factor, which may need a few additional operator evaluations. This method also enjoys explicit linear convergence guarantees. We further extend these ideas in two directions. First, using the relation between Halpern iterations and Nesterov-type accelerated fixed-point schemes, we derive parameter-free Nesterov-type (three-term) variants that inherit linear convergence in the contractive setting. Second, by combining the proposed fixed-point schemes with Tikhonov regularization, we obtain new methods for solving cocoercive equations and establish an iteration complexity of $\mathcal{O}({\epsilon^{-1}\log(\epsilon^{-1})})$ for computing an $\epsilon$-solution, provided that a valid lower bound on the cocoercivity constant is available. Numerical experiments on several examples demonstrate that the proposed algorithms are competitive with, and often outperform, existing adaptive fixed-point methods. In particular, the methods successfully exploit contractive behavior when it is present on the selected nonexpansive test problems.

In this paper, we develop a systematic framework for deriving explicit error bounds for semidefinite feasibility problems without assuming strict feasibility (Slater's condition), a setting in which existing results are limited. Our main technical contribution is the introduction of radial-type Hölder error bounds, where the error bound constant depends explicitly on the norm of the reference matrix through radial modulus functions. By combining facial reduction with recently developed facial residual functions, we obtain explicit descriptions of these modulus functions, yielding qualitative radial-type Hölder error bounds without imposing any constraint qualifications. Our results complement the classical work of Sturm by providing explicit estimates for the constants involved in Sturm's local Hölder error bounds over bounded sets with a given size. We further analyze the asymptotic behavior of these bounds as the dimension of the underlying matrix space grows, identifying regimes in which they can be asymptotically tight up to a dimension-free constant. As an application, we establish explicit error bounds for the optimality system of semidefinite programs by reformulating them as feasibility problems, a setting where Slater's condition typically fails. Under the generically satisfied strict complementarity condition, we derive radial-type error bounds without assuming the usual solution uniqueness requirement, and demonstrate their asymptotic tightness through an explicit example.

For any ideal $I$ (whose projective dimension need not be finite) in a local Noetherian ring $R$, we show that, if the conormal module $I/I^2$ has a free summand given by $F$, the natural map from $\bigwedge F$ to $\text{Tor}_{*}^{R}(R/I,R/I)$ splits as algebras. To achieve this, we use dg algebra techniques and remodel results of André and Iyengar, proving them in setting of semi-free extensions, replacing the need for semi-free $\Gamma$-extensions. In this new setting, we show that free summands of the conormal module correspond to central elements of the relative homotopy Lie algebra $\pi^*(\varphi)$ and, lastly, we provide an explicit splitting morphism in lower degrees.

Let $Q\geq 1$ be large, and $\delta \in(0,1)$ be small. Denote by $\mathcal C \subset \mathbb R^3$ a sufficiently smooth curve with non-vanishing curvature and torsion. How many rational points $\mathbf{a}/q$ of height $q\in[1, Q]$ are $\delta/q$-near $\mathcal C$? This manuscript provides an essentially optimal answer, thereby addressing a problem stated by Beresnevich and Kleinbock, for space curves. We show that the folklore conjectures are incorrect for certain manifolds with codimension $\ge 2$, including the moment curve $(t,t^2,t^3)$. The reason is a {hitherto} hidden `major arc' type obstruction. We also establish matching upper bounds, up to endpoints. Our argument combines purely Fourier analytical techniques with the planar counting results by Vaughan and Velani.

Closed-form waveforms of the modified Emden oscillators $\ddot x+(q+2)kx^q\dot x+k^2x^{2q+1}+\omega^2x=0$ of arbitrary natural power $q$, and of their damped $c$-deformed versions, are obtained through a generalized commutative factorization that reduces them to Bernoulli equations. We show that commutative factorizations of Liénard equations are gauge-conjugated forms of the nonlocal linearizing transformation of Chandrasekar, Senthilvelan, Kundu and Lakshmanan, the factorization constant being half the difference of the roots of the associated linear oscillator. These oscillators display an odd–even dichotomy: for odd $q$ they are isochronous, with period $2\pi/\omega$ independent of the amplitude, within an explicitly bounded range of the integration constant, while for even $q$ the oscillations decay algebraically, as $t^{-1/q}$. The dichotomy is confirmed by an elliptic polar representation, in which the phase rotates uniformly for every $q$, and, for even $q$, by the Bendixson–Dulac criterion applied to the divergence of the vector field. We present a dissipative Lagrangian of Lurie–Rayleigh type and, for $q=1$, an autonomous Sundman linearization that yields a rational first integral, a Jacobi last multiplier and a non-dissipative Lagrangian; for every $q$, $\omega^2x^2+(\dot x+kx^{q+1})^2$ is a Darboux polynomial providing a global Jacobi last multiplier. Explicit waveforms and phase portraits are given for $q=1,\dots,4$, with and without damping, and a recent application to magnon dynamics in the magnetic semiconductor CrSBr is discussed.

Programmable microcavities support grand-canonical photon gases. We show that bosonic state counting creates an entropic staircase of most-probable total occupations. For detuning density continuous and nonzero at the chemical-potential threshold, the active-cell fraction is asymptotically linear at low temperature and the conditional law has a universal limit; for uniform disorder the law is exact over a finite temperature interval. For two modes it is the Gauss–Kuzmin distribution, linking photonic thermodynamics and metric number theory. We outline finite-array and dye-microcavity tests.

We prove a sharp lower bound for smooth nonconvex stochastic optimization with uniformly bounded gradient noise. In the \(K=1\) fresh-sample model, every randomized adaptive algorithm requires $$\Omega\left( \frac{\Delta L}{\epsilon^2} + \frac{\Delta L\sigma^2}{\epsilon^4} \right)$$ queries to find a point with expected gradient norm at most \(\epsilon\). This matches the standard upper bound and, to the best of our knowledge, resolves the question raised by [Arjevani et al. 2023] of whether almost-surely bounded oracle error permits a better rate than bounded variance. The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only forchecking the proof and revising and polishing the manuscript.

Let $\pi\colon X\to\mathbb P^1$ be a Jacobian elliptic surface over $\mathbb C$, and set $\chi=\chi(\mathcal O_X)\ge3$, so that $\kappa(X)=1$. Assume that the Mordell–Weil group of $\pi$ is finite and that $\pi$ has at least one reducible fiber, the reducible fibers being of types $I_{n_1},\ldots,I_{n_s}$. We prove that the zero section and the components of the reducible fibers generate the closed Mori cone if and only if \[ \sum_{i=1}^s\frac{\lfloor n_i^2/4\rfloor}{n_i}\le\chi. \] If $\sum_i n_i\le2\chi+3$, then $X$ is a Mori dream surface. The proof combines an explicit description of the facets of the cone generated by the curves visible in the fibration with Artin's criterion applied to the null loci of the dual nef rays. We also show that, in Kodaira dimension one, finiteness of both the Mordell–Weil group and the automorphism group does not mply polyhedrality of the Mori cone. In the polyhedral range, we construct a Jacobian elliptic surface with $(\chi,n)=(3,11)$, Picard number $12$, and a big and nef divisor which is not semiample; in particular, this surface is not a Mori dream surface. Finally, for every integer $\rho\ge2$, we construct a Jacobian elliptic surface of Kodaira dimension one and Picard number $\rho$ that is a Mori dream surface.

For Gamow's liquid drop model, we improve the nonexistence bound of Frank-Killip-Nam from 8 to 7.5. The key additional input is a geometric perimeter inequality arising from the capillarity problem. This yields a quantitative gain in the averaged slicing argument and shows that the variational problem admits no minimizer for $A\geq 7.5$.

We consider the probability that a discrete random matrix $M_n(\xi)$ is strongly non-singular, meaning all its leading principal submatrices are non-singular. This property is equivalent to the existence of an LU factorization. We show that for any discrete random variable $\xi$ with finite support and $|\xi|_\infty < 1$, there is a constant probability that $M_n(\xi)$ is strongly non-singular with a growth factor bounded by $n^{5/2+\delta}$. Furthermore, we provide a tight asymptotic lower bound for this probability as $|\xi|_\infty \to 0$. Finally, we provide exact counts for strongly non-singular binary matrices up to $n=9$ and use these to derive improved upper bounds for the Bernoulli case.

For $n>2$ and $k>1$, define the polynomial \[F_{n,k}(x) = \Phi_n(1 + x + \cdots + x^k),\] where $\Phi_n$ denotes the $n$-th cyclotomic polynomial. The cyclotomic conjecture proposed by Gimbert (1999) exactly describes the irreducibility of $F_{n,k}(x)$ over $\mathbb{Q}$ in terms of $n$ and $k$. Conde, Gimbert, González, Miller and Miret (2014) established that the cyclotomic conjecture, if true, would imply the non-existence of almost Moore digraphs - a well-known open question concerning the directed degree-diameter problem. In this article, we prove the cyclotomic conjecture and, as a consequence, show that there are no almost Moore digraphs with maximum out-degree $d$ and diameter $k$ for any $d>1$ and $k>2$.

An explicit construction of a free nearring as well as the free product (= coproduct) of two nearrings are given in the variety of all, not necessarily zero-symmetric, nearrings.

We develop a variational calculus for curves on Fréchet manifolds and apply it to the Lagrangian formulation of the Einstein evolution equations on the manifold $\Mm$ of Riemannian metrics of a compact manifold. For mechanical Lagrangians associated with weak pseudo-Riemannian metrics, we establish the Euler–Lagrange equations, energy conservation, and an infinitesimal Noether theorem that does not require the symmetry vector field to generate a flow. Applied to the DeWitt Lagrangian, this framework identifies the momentum constraint with the vanishing of a Noether momentum map and gives a direct proof of its propagation. We also derive the pointwise transport identity \[ \partial_t\bigl(\Ham_\lambda\,\dv_g\bigr) = 2\alpha\,\delta_g(\delta_g^-k)\,\dv_g\,, \] which implies propagation of the Hamiltonian constraint. Finally, we prove a Maupertuis– Jacobi theorem for weak pseudo-Riemannian metrics on Fréchet manifolds and show that, away from the zero set of the total scalar-curvature potential, constrained Einstein evolutions are reparametrized unit-speed geodesics of the conformal DeWitt metric $-4\alpha S_\lambda G^-$.

Optimization under zeroth-order (i.e., bandit) feedback is central to many engineering problems where the analytic forms of objectives and/or constraints are unavailable. In modern applications, such as online control and online learning, optimization problems often evolve with time, requiring adaptive optimization methodologies. Yet, existing methods in this seting are largely confined to adaptations of methodologies developed for time-invariant or first-order optimization, and thus often rely on gradient surrogates that fail to fully exploit the zeroth-order structure of the available information. In this paper, we propose a randomized two-point direct-search algorithm for nonconvex time-varying optimization and derive iteration-complexity bounds under both constant and diminishing probing ratios. The resulting analysis yields explicit stationarity bounds in terms of the temporal variability of the problem and possible oracle errors. Our complexity bounds recover the complexity of existing zeroth-order methods in the time-invariant setting, while extending direct- search methods beyond static settings. As an illustrative application, we show that the methodology is naturally suited to solve optimal (equilibrium-selection) control problems for dynamical systems. In this setting, the analysis yields explicit stationarity bounds in terms of the temporal variability of the problem, measured through the effects of plant dynamics and exogenous disturbance variations.

Let $F$ be a non-Archimedean local field of characteristic zero and residue characteristic different from $2$, and let $(G,H)$ be an equal-rank split symmetric pair with $G$ split semisimple and adjoint and with an $F$-split maximal $\theta$-split torus. Motivated by the singular contribution near the nilpotent fiber on the $\theta$-fixed side of the infinitesimal local relative trace formula, we construct finite descent charts for regular families in $\mathfrak h$ approaching that fiber. On each chart, a Cartan lift defined over $F$ conjugates the sparse Jacobi family \[ Q(q)=\sum_{\alpha\in\Delta}(\gamma_\alpha n_{-\alpha}+q_\alpha n_\alpha) \] into $\mathfrak h$. The family $Q(q)$ is regular for every $q$ and takes the principal nilpotent value $Q(0)$. The restriction of the adjoint quotient to this family has generic rank equal to the number of odd exponents of $G$. For split symmetric pairs this rank equals $\operatorname{rank}H$; hence in equal rank the map is generically finite étale. Kummer theory classifies the twisted square-root covers on which the Cartan lifts are defined, and these covers exhaust the rational branches. On every branch, with $q_\alpha=z_\alpha^2$, the quotient Jacobian cancels the relative Weyl discriminant exactly: \[ \frac{ds}{|D_H^G|^{1/2}}=C\prod_{\alpha\in\Delta}d^\times z_\alpha. \] Thus the normalized $H^\circ$-quotient density is a multiplicative Haar measure on the parameter torus, uniformly across all rational twists. After a finite clopen refinement, the associated Iwasawa heights are piecewise affine in the valuations $\operatorname{val}(z_\alpha)$. For $(\operatorname{Sp}_{2n},\operatorname{GL}_n)$, the construction is explicit in Hurwitz coordinates, and the Cartan lift factors through commuting long-root $\operatorname{SL}_2$-subgroups.

In this paper, we study the equation which we call Manton's exotic vortex equation. The equation arose in Manton's study of a deformation of the classical vortex equation [Ma]. We first investigate the existence problem for Manton's exotic vortex equation and establish existence results. We then construct the moduli space of solutions. Finally, we establish a dimensional reduction for Manton's exotic vortex equation. More precisely, we establish a correspondence between solutions of Manton's exotic vortex equation and pseudo-Hermitian-Yang–Mills connections with pseudo-Hermitian metrics of signature (1,1).

On finite fields of characteristic $p$, PN (perfect nonlinear) functions for odd $p$ and APN (almost perfect nonlinear) functions for even $p$ are well-known classes of highly nonlinear functions. GAPN (generalized almost perfect nonlinear) functions were introduced as a generalization of APN functions for even $p$ to all $p$. One of the main targets of studies on such highly nonlinear functions is their classification. While $p$-exceptional monomial PN and $2$-exceptional monomial APN functions have been classified, the corresponding problem for GAPN functions remains open. Here, a polynomial over $\mathbb{F}_p$ is called a $p$-exceptional PN (resp. APN, resp. GAPN) function if it is a PN (resp. APN, resp. GAPN) function on $\mathbb{F}_{p^n}$ for infinitely many positive integers $n$. In this paper, we give a geometric characterization of $p$-exceptional monomial GAPN functions. To this end, for a prime power $q$, we establish a geometric characterization of non-zero polynomials in $\mathbb{F}_{q}[x,y]$ having no absolutely irreducible factor defined over $\mathbb{F}_{q}$.

The Euler characteristic transform (ECT) is a computable topological signature for shapes embedded in Euclidean space. Existing ECT stability results control changes in the transform through deterministic geometric perturbation bounds, but many data-generating mechanisms produce perturbations that are random and unbounded in the worst case. In such settings a uniform deterministic bound may be unavailable or too pessimistic. This paper formulates a probabilistic stability-transfer principle for one-dimensional ECT stability: a deterministic stability estimate is combined with a tail bound for the induced geometric error to obtain a high-probability ECT error bound. We instantiate this principle using the stability theorem of Marsh and Beers for embeddings of finite one-dimensional CW complexes. The resulting consequences cover ambient displacement fields conditionally on admissibility and supremum-tail estimates, and finite landmark perturbations with reconstruction. For landmark perturbations, bounded, sub-Gaussian, sub-exponential, and polynomial-tail noise assumptions give explicit high-probability ECT stability bounds. The framework isolates a geometric stochastic component that arises in shape analysis pipelines across biological morphology, medical imaging, molecular structure analysis, and remote sensing.

Inspired by graph theory, Neumann Boundary Value Problems on $p$-adic analytic manifold are formulated. Existence and uniqueness of weak solutions is proven in the ultracontractive case. For this task, a $p$-adic Gauss-Green identity and an extended eigenvalue formula for wavelets with small supports on the manifold are proven. As an application, the closeness to a zero of an analytic function on the manifold is shown to be detectable via a Neumann Boundary Value Problem.

In this paper, we consider a Bernoulli random graph~\(G\) on~\(n\) vertices with non-uniform edge probabilities, where each vertex has an independent mark and each edge is equipped with an independent positive weight. The cost of an edge depends on the weight as well the marks of the endvertices and we estimate the growth of the maximum and minimum cost of a spanning tree containing all the vertices. For edge weights with heavy tails and vertex marks distributed uniformly in the unit square, we obtain a phase transition in terms of the tail decay exponent~\(s:\) If~\(s\) is large, then the maximum cost is essentially determined by the vertex locations and if~\(s\) is small, then the edge weights crucially influence the maximum cost. We derive a similar result for minimum cost spanning trees and use martingale difference based methods to establish the~\(L^2-\)convergence of the extremal cost, appropriately scaled and centred.

For every fixed $k\ge 3$, every finite nonempty clutter $\mathcal{H}$ can be realized exactly as the family of inclusion-minimal terminal traces of the $k$-bad odd cycles, namely the odd cycles $C$ satisfying $\chi(G[V(C)])\ge k+1$. The host graph $G$ may be chosen $2$-connected, with $T=V(\mathcal{H})$ independent, and with $\chi(G)=k+1$, $\omega(G)=k$, and $|E(G)|\le k|V(G)|$. Given any integer $B\ge 1$, one realization simultaneously preserves the transversal number, the fractional packing number, and every integer $c$-packing number for $1\le c\le B$. Thus chromatic richness on odd-cycle spans has the full packing-covering complexity of arbitrary finite set systems, even in sparse graphs whose chromatic number exceeds their clique number by one. As a consequence, for every $b,N\ge 1$ and every $\varepsilon>0$, there is such a graph with $\tau_k(G)=N$, $\nu_k^{1/c}(G)=c$ for every $1\le c\le b$, and $\nu_k^*(G)<1+\varepsilon$. Hence every finite-congestion and fractional Erdős–Pósa property fails for $k\ge 3$, with an asymptotically optimal fractional obstruction. This contrasts sharply with $k=2$, where the relevant cycles are the ordinary odd cycles: the integral property fails, while Reed's theorem yields every congestion level at least two and the fractional property. We also characterize the shortest $k$-bad odd cycles by a single fixed witness graph.