Papers

The Connes-van Suijlekom and Connes-Consani-Moscovici truncations of the Weil quadratic form, at a prime cutoff c>1 and frequency band N, produce finite Galerkin matrices whose spectra are the finite-rank window on Weil positivity. We prove two exact finite theorems about this truncation. First, every real even Galerkin coefficient vector v determines, in closed form, a band-limited Guinand-Weil test function g_v whose zero sum over the nontrivial zeros of zeta equals the quadratic value <v, Q v> exactly: every value of the truncated form is an exact sum over the zeros. The construction factors through an exact source quotient of dimension 2N+1 and admits a non-collapsing pole-neutral subfamily. Second, beyond the Galerkin band the omitted archimedean tail is a totally positive Cauchy-Stieltjes increment. This yields a two-sided certification rule with an explicit budget B_T ~ (2N+1) rho log(T) / (pi^2 T), where T is the archimedean cutoff and rho = 2 pi / log c: finite-cutoff positivity certifies cutoff-free positivity, a finite-cutoff eigenvalue below -B_T certifies a cutoff-free negative, and a negative eigenvalue in the band [-B_T, 0) certifies nothing. Resolving a spectral scale of 10^-59 at c=100 by brute cutoff would require T of order 10^63; a cutoff-free interval LDL^T factorization resolves it directly. The dictionary is verified over the first 512 zeros of zeta and by three independent computational routes; all scripts and artifacts ship with the paper. The paper makes no Riemann Hypothesis, prime-counting, next-prime, or factoring claim.

We propose a kinetic model of opinion dynamics under selective media influence on both graphon-based and real-world networks. The media action, inspired by Hallin's theory of spheres, is incorporated through a model predictive control strategy designed to steer agents' opinions toward a desired target opinion. For the resulting Boltzmann-type description, we analyse the evolution of the moments and, in the quasi-invariant interaction limit, derive a Fokker–Planck-type equation together with a characterisation of its stationary states. We also prove exponential convergence to equilibrium in the Fourier metric. Numerical experiments are performed on networks generated by a Gaussian graphon and on real-world, single-issue Twitter networks, allowing us to investigate the role of control and interaction parameters, as well as the impact of the subset of agents subject to media influence. Using real-world social networks data to initialise opinions and infer the interaction structure, we then compare the dynamics obtained on the original networks with those produced by Gaussian graphons fitted to their adjacency matrices, thereby assessing the descriptive power of the graphon approach for real-world opinion dynamics.

This paper establishes the $p$-adic rigidity of certain suitably refined noncuspidal automorphic Saito$\unicode{x2013}$Kurokawa representations of $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$, in the sense that they cannot be interpolated in a nontrivial positive dimensional $p$-adic family. The results provide a $\mathrm{GSp}_4$ analogue of Bellaïche's rigidity theorems for $\mathrm{U}(2,1)$ and identify an obstruction to $p$-adic variation on the $\mathrm{GSp}_4$ eigenvariety.

Continuation methods help trace Pareto sets in multi-objective optimization but are inherently local: a single run traces a single connected branch, requiring multiple restarts to recover disconnected components of Pareto fronts. We show that, for unconstrained bi-objective problems under weighted-sum scalarization, these disconnects can be artifacts of singularities in the scalarization parameter, where the weight $\lambda$ diverges as the objective gradients become collinear. Recasting Pareto optimality as a nonlinear system, we apply pseudo-arclength continuation to follow the Pareto-critical set, and show that suitable singular reparameterizations allow crossing these singularities in systematically, recovering disconnected branches in a single run. A coordinate-wise projective compactification further provides a unified framework for parameter and decision-space variables. We demonstrate the approach on the ZDT3 benchmark and modifications.

In this paper we will investigate Pettis integrability of $\mathcal{C}^{\circ}_{\Phi}(\mathcal{H})$-valued functions. We will study weakly$^*$ integrable $\mathcal{B}(\mathcal{H})$-valued functions and establish sufficient conditions for such functions to be Pettis integrable as $\mathcal{C}^{\circ}_{\Phi}(\mathcal{H})$-valued functions. In addition, we prove the inequality $$\left\|\int_E\mathscr{A}^*\mathscr{B}d\mu \right\|_{\Phi^{(p)}} \leqslant \|\mathscr{A}\|_{L^q_s}\cdot\left\|\sqrt[p]{\int_E|\mathscr{B}|^pd\mu}\right\|_{\Phi^{(p)}},$$ where $\Phi^{(p)}$ is $p$-modification of the function $\Phi$ and the functions $\mathscr{A}$ and $\mathscr{B}$ belong to the suitable spaces of operator-valued functions. Finally, under some additional integrability assumptions on $\mathscr{B}$ we provide similar estimates of the Pettis norm.

We describe and analyze new methods for approximating manifold-to-manifold maps using only scattered data information. To this end, we first study kernel-based approximation methods for scalar-valued functions defined on a manifold and derive a new sampling inequality and error estimates for functions from Sobolev spaces. After that, these methods are combined with a closest point projection to reconstruct manifold-to-manifold maps. The new methods are analyzed and error estimates are derived. Finally, numerical examples are given to verify the theoretical findings.

This paper presents a development of a nonlinear extension of the Parareal-Incomplete Optimized Schwarz Waveform Relaxation (OSWR) method, aimed at efficiently simulating coupled reaction-diffusion systems, with particular attention to those encountered in cardiac electrophysiology. The strategy leverages the synergy of spatial and temporal decomposition to tackle the computational challenges of large-scale, nonlinear simulations. The proposed approach is accompanied by a comprehensive convergence analysis. Extensive numerical experiments confirm the convergence of the nonlinear Parareal-Incomplete OSWR method. The results showcase its robust convergence behavior across various space-time subdomains, underscoring the method's reliability and effectiveness, especially in complex simulations involving multiple subdomains.

The Fisher-KPP equation is the canonical reaction-diffusion equation used in the study of invasive phenomena in mathematical biology. While moving boundaries have already been included to introduce biologically realistic sharp interfaces, these flux-based conditions assume loss of mass as a byproduct of boundary movement. We consider a mass-conserving boundary condition at a free boundary of the Fisher-KPP equation to allow interface movement without population consumption. We use a combination of phase plane analysis, perturbation analysis, and numerical simulation to show that the model supports both invading and receding travelling wave solutions with distinct wave speeds and front densities. Prescribing the free boundary velocity as a linear function of the front density, we show the existence of multiple stable and unstable travelling wave solutions, as well as receding population blow-up. Our results indicate that in regions where multiple steady states exist, those with larger wave speeds are stable. This is corroborated by considering the parameter regime associated with the uniform steady state solution and performing linear stability analysis. The framework we develop is readily extendable to more general boundary velocity functions, offering a means of studying increasingly complex and biologically relevant boundary dynamics.

We prove two strict total-positivity results by isolating two strictification mechanisms. The first is a spectral Darboux mechanism: an induction converts positivity and ordered endpoint asymptotics for a one-dimensional spectral family into positive Wronskians and hence into strict total positivity. As an application, the modified-Bessel kernel $K(x,s)=I_s(x)$, $x>0$, $s\ge 0$, is strictly totally positive of infinite order. This proves the real-order determinant positivity asked for by Buchstaber and Glutsyuk after their nonnegative-integer-order theorem. The second mechanism is discrete Toeplitz smoothing: every two-sided Polya-frequency sequence is a pointwise limit of totally positive Polya-frequency sequences. This gives a product-topology answer to Question 12.2 of Belton, Guillot, Khare, and Putinar. The density statement is in the product topology on $\mathbb{R}^{\mathbb{Z}}$; no uniform, weighted, or norm-density assertion is made.

We establish $L^2$-exponential strong ergodicity (strong mixing) with an explicit rate of convergence for a class of degenerate diffusions with multiplicative noise and with singular drift in both the noisy and noise-free components. This class includes diffusions with an additional inert drift given by the gradient of a singular potential, as well as singular generalized stochastic Hamiltonian systems. Cases in which the diffusion is confined to a proper, bounded or unbounded subset of $\mathbb{R}^{d_1+d_2}$ are included. Concrete examples of admissible potentials are provided. To obtain these results, we use an analytical approach and study the long-time behavior of the strongly continuous contraction semigroup generated by the formal Kolmogorov backward operator. Using the theory of generalized Dirichlet forms, these objects are then identified with the transition semigroup and generator of the unique weak solution to the original stochastic differential equation. The existence and uniqueness of this solution are established under near-minimal conditions.

Given a fiber bundle $Y\rightarrow M\xrightarrow{p}B$ and a fiber map $f: M\rightarrow M$ over $B$, we introduce the notion of fiberwise homotopy minimal periods, denoted by $H_BPer(f).$ This invariant records the periods that occur among representatives of the fiberwise homotopy class of $f.$ We investigate the case in which both the base and the fiber are circles. Up to isomorphism, the corresponding total spaces are the torus and the Klein bottle. Using Nielsen theory for fiber maps and Nielsen-type periodic numbers, we obtain a complete classification of $H_{S^1}Per(f)$ for fiber maps of the torus and the Klein bottle over $S^1$. In particular, we identify the exceptional cases in which some periods can be removed by fiberwise homotopy, including the case $H_{S^1}Per(f)=\mathbb{N}\setminus{2}.$

Interlacing of the real roots of a weighted matching polynomial for a graph $G$ and that of a vertex-deleted subgraph is classical and well-known. In the context of strict interlacing of distinct roots, a demonstrated graph construction gives rise to a new classification of graphs, called ${ \rm SRSI}$ graphs, which include graphs that contain a Hamilton path. Graphs with a perfect (or nearly perfect) matching are shown to exhibit the SRSI$_w(v)$ property with respect to a particular weighting and for specific vertices, and all such graphs are characterized via this graph construction. As a consequence, we also characterize the trees that possess the SRSI property for all edge weightings and all vertices.

Consistent families of locally defined probability distributions that do not admit a joint global distribution are known as contextual, with primary examples arising in quantum theory. In this paper, we study such families of distributions using the theory of simplicial distributions, and further develop the theory for possibilistic distributions defined over the Boolean semiring. We characterize possibilistic collapses of simplicial distributions geometrically using bundle scenarios. Using this characterization together with a new connectivity condition on the total space of a bundle scenario, we provide a criterion for detecting extremal simplicial distributions. In parallel, we develop an analogous theory for presheaves on simplicial complexes, describe possibilistic collapses of empirical models on them using event scenarios together with a categorical extremality condition, and relate the two frameworks via a comparison result. We provide examples of contextual simplicial distributions that arise from our criteria on scenarios of interest in quantum foundations, such as Bell scenarios and boundaries of standard simplices, the latter connecting to Vorob'ev's classical theorem on acyclic complexes.

This paper presents a guidance framework for lunar powered descent and landing that combines sequential convex programming (SCP) with real-time online model identification. A nonconvex energy-optimal landing problem is developed and then reformulated into a sequence of convex second-order cone programs (SOCPs) through a change of variables, successive linearization, and a lossless second-order cone relaxation of the thrust direction constraint. An online identification layer, built from a recursive least squares (RLS) filter with exponential forgetting and an exponential moving average (EMA) smoother, estimates unknown gravitational, thrust-scale, and mass-gauging perturbations from noisy navigation measurements and injects a corrected bias term into the dynamics constraint of each convex subproblem at every guidance cycle. Building on this architecture and my prior work in this area, a baseline SCP algorithm and a receding-horizon online SCP algorithm with model identification are developed. Also, I try to explore some theoretical foundations, establishing the losslessness of the convex relaxation, the mean-square stability and convergence of the identification filters, the guaranteed convergence of the SCP iteration, and explicit convergence radius and convergence rate results. Numerical simulations across four perturbation scenarios of increasing complexity are implemented in MATLAB using YALMIP and the ECOS solver. The results show that the proposed online algorithm consistently reduces landing position and velocity error and better tracks the true propellant consumption relative to an uncorrected nominal trajectory, while retaining the predictable convergence and real-time computational properties of convex optimization.

In this article, we consider the parabolic Anderson model driven by a Lévy white noise with finite variance in dimension 1, and we study the asymptotic behaviour of the spatial average of the solution. The main result shows that, with appropriate normalization and centering, the spatial integral converges in distribution to the standard normal distribution, and gives an estimate for the rate of this convergence in the Wasserstein distance, the Kolmogorov distance, and the Fortet-Mourier distance. We also prove the functional limit theorem corresponding to this result.

This paper develops an Optimality-Informed Neural Network (OINN) approach for the energy-optimal, free-final-time powered descent of a lunar lander from any initial position, velocity, and mass within a bounded operating envelope to a fixed landing site with zero terminal velocity. Building on a recent framework that jointly embeds Pontryagin's minimum principle and the Hamilton-Jacobi-Bellman equation for general nonlinear optimal control, the proposed OINN approach specializes that idea to a lunar landing problem with free time of flight and fixed terminal state. Every boundary and transversality condition is hard-encoded into the network architecture by construction, the closed-form Pontryagin-optimal thrust magnitude and direction law is substituted directly rather than learned, and the remaining state, costate, and an auxiliary value-function output are trained against a physics-residual loss formed entirely from the necessary conditions of optimality, with no precomputed optimal trajectories required. A preliminary theoretical analysis is explored, establishing a stochastic-optimization stationarity guarantee for the offline training procedure, an explicit bound translating the achieved training residual into bounds on touchdown position, touchdown velocity, and flight-time error, and a fixed, input-independent onboard computational and memory cost suitable for real-time deployment. Numerical simulations evaluate the trained policy, with no retraining, against an independently solved indirect-method boundary-value problem at six representative initial states spanning the operating envelope and against eighty additional Monte Carlo simulation runs, demonstrating close agreement with the indirect-method solution and consistently small dynamics and transversality residuals throughout the envelope.

For a delta-matroid, the maximum twist width theorem states that the maximum width over all twists can be reached along a non-decreasing sequence of intermediate twist widths. In this paper we study analogous monotone maximum width sequences for partial twualities generated by twist and loop complementation. We prove that, for each non-twist partial-twuality operation on a vf-safe delta-matroid, there exists a subset attaining the corresponding maximum partial-twuality width whose elements can be ordered so that the successive intermediate widths are non-decreasing. Together with the known twist case, this gives a monotone maximum width theorem for all five nontrivial partial-twuality operations on vf-safe delta-matroids. We also prove feasible-set attainment results for the operations $\ast\times$ and $\ast\times\ast$. Finally, we translate these results to ribbon graphs, obtaining monotone sequences for maximum partial-twuality Euler genera and spanning quasi-tree attainment results for the corresponding ribbon graph operations.

In this paper we consider solutions $\boldsymbol{u}$ of the three-dimensional Navier-Stokes system and investigate sharpness of the a priori bound \begin{align*} \frac{d}{dt}\|\boldsymbol{u}\|_q^q \leq C\|\boldsymbol{u}\|_q^{q\frac{q-1}{q-3}}, \qquad q > 3. \end{align*} This bound is closely related to the Ladyzhenskaya-Prodi-Serrin conditions characterizing classical solutions of the Navier-Stokes system. Velocity fields maximizing the rate of growth $(d/dt)\|\boldsymbol{u}\|_q^q$ under certain constraints are found as solutions of a suitable optimization problem which is solved numerically using a Riemannian conjugate gradient approach. The results obtained for different $q$ and increasing values of $\|\boldsymbol{u}\|_q$ indicate that the bound is indeed sharp, up to a numerical prefactor, and therefore cannot be fundamentally improved. Additionally, the results also suggest that the rate of growth $(d/dt)\|\boldsymbol{u}\|_q^q$ diverges as $q\to 3$.

Define an $(a,a)$-Carmichael number to be a squarefree natural number $n$ such that $p\mid n$ implies $p-a\mid n-a$. For such a number $n$ with prime factors $p_1,\cdots,p_m$, define $$K=GCD[p_1-a,\cdots,p_m-a],$$ and let $C_\nu(X,a)$ denote the number of $(a,a)$-Carmichael numbers up to $X$ such that $K=\nu$. Assuming a strong conjecture on the first prime in an arithmetic progression, we prove that for any integer $a$ and for any natural number $\nu$ with $(\nu,a)=1$ and $a$ and $\nu$ having opposite parity, $$C_\nu(X,a)\geq X^{1-(2+o(1))\frac{\log\log\log \log X}{\log\log\log X}}.$$ This is a departure from many traditional constructions of Carmichael numbers, which generally require $K$ to grow along with $n$.

This paper introduces a transition-set morphometry for quantifying inter-scale contacts in digital sandstones. Pore volume, a medial pore-throat subsystem, and fracture porosity are treated as distinct geometric subsystems. Their finite-neighborhood contacts are represented by transition sets such as $T_{VL}$, $T_{VF}$, and, when a residual skeleton is resolved, $T_{LF}$. The method measures the density, finite-scale dimension, and weighted contact measures of these sets. A matrix-mode test was performed on 21 digital sandstone samples from the Imperial College micro-CT collection and Digital Porous Media Portal dataset DRP-317. Permeability, expressed as $\log_{10}K$, served as an external response for assessing how much structural information the descriptors carry. The transition dimension $d_{VL}$ alone produced little independent predictive gain, whereas contact measures combining transition-set density with local pore radius were more informative. The adaptive measure $\widehat C_{VL}^{(d+1)}=\mu_{VL}(\overline D_{VL}/D_*)^{d_{VL}+1}$ increased leave-one-out $R^2$ from 0.752 to 0.881 in the combined sample and from 0.287 to 0.807 within DRP-317. Fractured datasets DRP-5, DRP-31, and DRP-285 were then used to test the transfer of the same transition-set construction to fracture-related contacts. The results support transition sets as reproducible morphometric descriptors of inter-scale contact.

Efficient representations of multipartite quantum states play a fundamental role in quantum information theory, providing both conceptual insight and practical tools for characterizing entanglement. Motivated by the axiomatic framework for graph states [Phys. Rev. A 85, 062313 (2012)] and its subsequent extension to hypergraph states [Phys. Rev. A 87, 022311 (2013)], we introduce an axiomatic construction of matroid states, a new family of multipartite quantum states associated with matroids. Our constructions are based on a set of axioms analogous to those that define graph and hypergraph states, yielding a consistent quantum representation of arbitrary matroids. Two ways of constructing matroid states are proposed: the first is defined in terms of circuits, and the second in terms of independent sets. In both approaches, we establish the existence of universal global operators that satisfy desirable properties such as locality, symmetry, commutativity, and are associated with the combinatorial structure of matroids. Furthermore, we establish a hierarchy connecting graph, matroid, and hypergraph states within a unified framework. Additionally, we show how to obtain an arbitrary graph state by applying suitable families of matroid states, whose corresponding operators are the generators of the stabilizer subgroup of the graph state. These results identify matroid theory as a natural combinatorial language for the efficient description of multipartite quantum states and open new perspectives for the investigation of quantum entanglement and related combinatorial structures.

We prove that, for every fixed integer $g\geq 1$, the largest cardinality of a $B_2[g]$ subset of the first $n$ squares is at least a positive constant, depending only on $g$, times $$ n^{\frac{2g}{2g+1}}(\log n)^{\frac{2-2^g}{2g+1}}, $$ for all sufficiently large $n$. For $g=1$, this recovers the theorem of Lefmann and Thiele on Sidon subsets of the first squares. The proof follows their hypergraph method, but replaces the $4$-uniform hypergraph encoding two representations as a sum of two squares by a $2(g+1)$-uniform hypergraph encoding $g+1$ such representations. The main point is to verify that this higher-uniformity hypergraph has few edges and few $2$-cycles; the lower bound then follows from an independence theorem for uncrowded hypergraphs due to Duke, Lefmann and Rödl.

We present theoretical insights into $H^{-1}$ residual loss formulations of physics-informed neural networks (PINNs) for learning solutions of partial differential equations (PDEs). Standard PINN formulations use a multi-term loss functional consisting of interior and boundary loss terms that are based on $L^2$-residuals and discretized as mean square errors (MSE). Imbalanced magnitudes of these terms cause numerical stiffness phenomena, resulting in ill-conditioning and slow convergence. In this work, we analyze discretizations of the $H^{-1}$-norm that are used in the context of elliptic PDEs with arbitrary, nonzero Dirichlet boundary conditions. We prove that these $H^{-1}$ discretizations rebalance the PDE loss, improve conditioning, and mitigate stiffness effects compared with the standard MSE discretization. We validate our theoretic results through operator-level experiments with randomly sampled residuals and PINN experiments for the Poisson and stationary incompressible Navier-Stokes equations. These experiments confirm the numerical effectiveness of the proposed rebalancing for elliptic PDEs and, more broadly, for problems with elliptic behavior.

In a recent paper \cite{q}, Quinn-Gregson fully classified ultrahomogeneous mono-unary algebras of arbitrary cardinalities. In particular, he proved that every 1-ultrahomogeneous mono-unary algebra is ultrahomogeneous, and every locally finite mono-unary algebra with finitely many 1-orbits is $\omega$-categorical. It is also known that every 1-homomorphism homogeneous mono-unary algebra is homomorphism-homogeneous \cite{ed}. Very naturally, Quinn-Gregson asked if these three results can be generalized to unary algebras. This short note answers the first and the third question negatively by providing two simple counterexamples. We also show that the second question has a positive answer in “tree-like" cases which covers his result to mono-unary algebras and has a small combinatorial implication.

In this paper, we provide a complete isomorphism classification of the spaces $C_p(K)$ of real-valued continuous functions endowed with the topology of pointwise convergence for separable compact lines $K$ of weight $\omega_1$, under the assumption of Baumgartner's Axiom $\mathsf{BA}$. More specifically, we show that, up to linear homeomorphism, there are exactly two function spaces $C_p(K)$ for such $K$. We also construct an example of a separable compact line $K$ of weight $2^{\omega}$ neither of whose spaces of continuous functions, $C_p(K)$ and $C_w(K)$, is homeomorphic to its square.

We establish the boundedness of exponential Schrödinger-Calderón-Zygmund operators on weighted $BMO_\rho^\alpha(w)$ spaces via a $T1$ criterion, where the weights belong to classes that capture the exponential decay of the operators, and $\rho$ is a critical radius function. Specifically, we prove that the boundedness of such an operator $T$ on $BMO_\rho^\alpha(w)$ is equivalent to a natural oscillation condition on $T1$ over sub-critical balls. The weight classes considered, introduced in connection with $\rho$, include and extend the classical $A_p^\rho$ weights, and are well-adapted to the exponential decay of the kernels. As applications, we derive weighted endpoint estimates for several operators associated to the generalized Schrödinger operator $\mathcal{L}_\mu=-\Delta+\mu$, including Riesz transforms, Laplace transform-type multipliers, maximal operators for the heat and Poisson semigroups, Littlewood-Paley functions and fractional integral operators. When $d\mu(x)=V(x)dx$, the results above extend the known endpoint estimates to larger classes of weights.

We present an embedded (in $G/B$) degeneration of Lusztig varieties (which generalize type $A$ Hessenberg varieties) to certain unions of Richardson varieties, giving a simultaneous reproof (and extension) of results of Anderson–Tymoczko, Harada–Horiguchi–Masuda–Park, and Kim. Although torus-equivariant, the degeneration is not Gröbner. In the case that the Lusztig variety is the permutahedral toric variety, this degeneration provides a subdivision of the permutahedron into Bruhat interval polytopes, and we prove a more general result showing equivariant degenerations of projective toric varieties produce subdivisions of the moment polytope (as was shown in the Gröbner case by Sturmfels). A Gröbner degeneration would result in a regular subdivision, and despite our degeneration not being Gröbner we show in types $A,B,C$ that our subdivisions of the permutahedron are indeed regular.

We study high-order analogues of the classical Heun operator of Fuchs index one, \[ \dq=\sum_{i=1}^k Q_i(z)\frac{d^i}{dz^i}, \qquad \deg Q_i\le i+1, \qquad \deg Q_k=k+1. \] For a fixed degree $n$ we consider the linear Van Vleck polynomials $V$ for which $\dq+V$ has a polynomial solution of degree $n$, and we form the spectral polynomial $Sp_n$ whose zeros are the zeros of these Van Vleck polynomials. The main result is a finite-band determinant representation and the resulting universality theorem: after normalization, all fixed power sums of the zeros of $Sp_n$ have limits given by explicit constant-term formulae depending only on the leading coefficient $Q_k$. The lower coefficients of $\dq$ enter only lower order correction terms. Combining this with the localization theorem for Van Vleck roots, we strengthen the usual germ-at-infinity conclusion to locally uniform convergence of the normalized Cauchy transforms and logarithmic potentials on the whole exterior of the convex hull of the zeros of $Q_k$. We also prove a determinacy criterion: if the spectral roots are asymptotically confined to a compact set with empty interior and connected complement, then the finite-band moments determine the actual weak limit. In particular, when the zeros of $Q_k$ are collinear the root-counting measures of $Sp_n$ converge weakly to a probability measure supported on the corresponding segment; this limit is independent of all lower coefficients of $\dq$. Finally, we prove holonomicity of the exterior Cauchy transform and derive Picard–Fuchs equations for the WKB periods, with an explicit third-order equation in the first non-classical case $k=3$. The paper ends with a precise mother-body conjecture for the genuinely complex case, clearly separated from the unconditional results.

For a graph $G$ with line graph $L(G)$, $\chi(L(G)^2)$ and $\omega(L(G)^2)$ are called the strong chromatic index and strong clique index of $G$, respectively. A well-known conjecture of Erdős and Nešetřil (1985) posits that $\chi(L(G)^2)\le \frac{5}{4}\Delta(G)^2$. Related to that, Faudree, Gyárfás, Schelp and Tuza (1990) conjectured that $\omega(L(G)^2) \le \frac{5}{4}\Delta(G)^2$. We show that $\omega(L(G)^2) \le \frac{2607}{1987}\Delta(G)^2 < \frac{21}{16}\Delta(G)^2$ improving the upper bound $\frac{4}{3}\Delta(G)^2$ of Faron and Postle. Indeed, we make progress towards a stronger conjecture of Faron and Postle in terms of Ore-degree. For positive integers $\Delta$ and $t$, let $h_t(\Delta)$ denote the smallest integer such that any graph $G$ with size at least $h_t(\Delta)$ and maximum degree $\Delta(G)\le \Delta$, contains two edges with distance at least $t$. An old problem of Erdős and Nešetřil (1986) concerns estimating the quantity $h_t(\Delta)$ and can be thought of as the edge-version of the degree-diameter problem. Chung, Gyárfás, Tuza and Trotter established the sharp inequality $h_2(\Delta)\le \frac{5}{4}\Delta^2+1$. We disprove two conjectures of Cambie, Cames van Batenburg, Joannis de Verclos and Kang concerning the next open case $h_3(\Delta)$.

We investigate, theoretically and numerically, a class of elastically safeguarded augmented Lagrangian methods for nonlinear optimization problems with inequality and equality constraints. Safeguarded augmented Lagrangian methods are known to exhibit substantially stronger global convergence guarantees than their non-safeguarded counterparts, making them attractive in practice. A persistent limitation, however, is that existing methods rely on a fixed safeguard, whose a priori selection can be difficult and inherently limits adaptivity, e.g., with respect to problem scaling. We propose an adaptive safeguarding mechanism that allows the safeguard to grow dynamically, overcoming these drawbacks while preserving the desirable global convergence properties of variants with a fixed safeguard. We further establish complexity bounds in terms of worst-case iteration counts. Numerical experiments comparing ALGENCAN, a state-of-the-art solver with rigid safeguard, against an elastically safeguarded variant thereof confirm that elastic safeguarding yields consistent practical benefits.