Papers

We prove Steinness over $\breve{\mathbf{Q}}_p$ for the basic local Shimura tower attached to $G=\mathrm{GL}_n$, $\mu=(1,1,0^{n-2})$, with basic Newton slope $2/n$, for every odd $n\ge3$, at every finite level and for every prime $p$. The proof proceeds by constructing global analytic functions from the crystalline period map, proving compactness of their sublevel sets via a perfectoid normalization of Tate lattices, and showing that the resulting function map to affine space is finite. The key geometric inputs are the Fargues–Fontaine realization of the universal cover, a normalized determinant on the rank-two locus, and an integral PEL realization over $W(\overline{\mathbf{F}}_p)$ used to produce bounded global Hodge generators. The integral constructions include the prime $p=2$.

We resolve an existing question concerning dynamical and structural stability of the E {\times} B staircases in magnetically confined fusion plasma. The problem has stirred considerable interest in the literature due to a unique combination of its scientific and practical aspects. Here, we report a new approach to the problem by combining the paradigmatic Hasegawa-Wakatani model of electrostatic drift-wave turbulence with an idea of interaction renormalization, imported from quantum field theory. It is shown based on the Schrödinger-Newton equation with renormalized interaction constant that the plasma staircase is robust and stable as soon as the nonadiabaticity parameter, {\delta}, goes above a certain critical value, and is unstable otherwise. This critical value is an exact result of the interaction model and is found to be {\delta}c = {\pi}. For {\delta} < {\delta}c , the unstable system expands in the radial direction, resulting in a radial second moment that grows with time t as t^(2/3). The results obtained bear important implications with regard to the implementation of arrays of semipermeable transport barriers in tokamaks, suggesting a threshold condition for their generation.

We study finite-time singularity formation for a nonlocal Boussinesq system arising in cold-plasma dynamics. We show that smooth, symmetric initial data with strictly positive total density and a sufficiently negative velocity slope at the origin lead to finite-time blow-up. More precisely, we give an explicit upper bound for the lifespan and show that the velocity gradient becomes unbounded.

For an odd prime $p$, let $\Gamma(R)$ denote the zero-divisor graph associated with the finite local ring $R= \mathbb{F}_p[x]/\langle x^4 \rangle$, having $p^3 - 1$ vertices and size $|E(\Gamma(R))| = \frac{1}{2}(3p^4 - 4p^3 - p^2 + 2)$. Using an equitable 4-cell partition induced by the ideal filtration $\langle u^3 \rangle \subset \langle u^2 \rangle \subset \langle u \rangle$, we prove that $\operatorname{Spec}_A(\Gamma(R))$ decomposes into discrete levels $\lambda \in \{0, -1\}$ with multiplicities $p^3 - p^2 - 1$ and $p^2 - 3$, respectively, alongside three irrational roots of an irreducible cubic polynomial $P_3(\lambda) = 0$, yielding adjacency energy $\epsilon(\Gamma(R)) = 4p^2 + \mathcal{O}(p)$. In contrast, the Laplacian spectrum is entirely integral: $\operatorname{Spec}_L(\Gamma(R)) = \{0^1, (p - 1)^{p^3 - p^2}, (p^2 - 1)^{p^2 - p - 1}, (p^3 - 1)^{p - 1}\}$. Consequently, the tree entropy satisfies $z(\Gamma(R)) = \ln p - \frac{1}{p} + \frac{\ln p}{p} + \mathcal{O}(\frac{\ln p}{p^2})$, with small-failure unreliability bounded by $p^2(p - 1)(1 - q)^{p-1}$. In linear consensus $\dot{x} = -Lx$, error trajectories decouple into three modal timescales $\{ (p^3 - 1)^{-1}, (p^2 - 1)^{-1}, (p - 1)^{-1} \}$ with algebraic connectivity $\alpha(\Gamma(R)) = p - 1$ and steady-state $H_2$ error variance $H_2^2 = \frac{Kf(\Gamma(R))}{2n^2} = \frac{1}{2p} + \mathcal{O}(p^{-3})$. Furthermore, the Kirchhoff index yields a vanishing average resistance distance $\overline{r}(\Gamma(R)) = \frac{2}{p} + \mathcal{O}(p^{-3}) \to 0$, while unbiased Markovian random walks induce an asymptotic traffic concentration vector $\Pi = (\frac{1}{3}, 0, \frac{1}{3}, \frac{1}{3})$ with quadratic per-node core routing stress $\pi_{\mathrm{core}}/\pi_{\mathrm{periph}} \sim p^2 + p + 1$.

In this paper, we study the characterizations of Schatten \(p\)-class Toeplitz operators on smoothly bounded convex domains of finite type in \(\C^n\). On one hand, by using discrete Kobayashi lattice, we characterize the Schatten \(p\)-class Toeplitz operators by employing a probabilistic selection method for \(0<p<1\). At the same time, we show that the characterizations obtained by Kobayashi lattice are equivalent to the Berezin transform for \(p > \frac{n}{n+1}\). On the other hand, as an application, the criterion gives a geometric characterization for Schatten \(p\)-class weighted composition operators for \(0<p<2\) and also gives an analytic characterization for \(p>\frac{2n}{n+1}\). Our main results give an answer to the question that find a geometric or analytic characterization of the Schatten \(p\)-class weighted composition operators whenever \(0<p<2\) raised by Xiao-Yang-Yuan \cite{Xiao2026}.

We study a natural class of normal functions on the moduli space of pointed curves and determine their rank at every point. We prove that the rank is maximal outside the corresponding stratum of Abelian differentials, while along this stratum it drops by exactly one. As a consequence, we describe the local geometry of the associated double ramification locus.

In this paper, we continue to investigate Campanato spaces via semigroups on von Neumann algebras. One of the main results is their coincidence with Lipschitz spaces for all regularity indices $\alpha > 0$ and for all analytic semigroups on von Neumann algebras not necessarily being finite; the desired self-improving property of Campanato spaces has also been verified. As a consequence, we demonstrate that the column Campanato spaces are isomorphic to the row ones for all $\alpha > 0$. This not only removes several restrictions in the previous work \cite{HJ24} by the first two authors, but also extends the previous results to any analytic semigroup, and thus resolves several problems left open in \cite{HJ24}. Both the results and the proof are new even in the commutative setting.

Artificial intelligence can increasingly produce plausible, sophisticated mathematical material faster than a developing graduate student can understand or verify it. A sophisticated result or paper draft therefore becomes weaker evidence of the student's own mathematical development. This creates a formation gap between output and personal capacity, and a trust gap between a convincing argument and warranted acceptance. The formation gap can persist even when the student understands the output: understanding a supplied argument does not by itself establish the capacity to initiate and direct inquiry. These gaps are not the whole story. AI can also help students explore examples, compare approaches, enter unfamiliar areas, and undertake ambitious research. The task is to design an apprenticeship that realizes these possibilities while developing substantive mathematical command. The central purpose of a mathematics PhD is to form mathematicians capable of original, independent, and responsible inquiry, including inquiry conducted with AI. This document develops that objective through four connected capacities: competence, judgment, independence, and responsibility. It distinguishes a work's contribution to mathematics from the evidence it provides of a student's formation; explains how a known answer can initiate rather than end creative inquiry; and proposes changes in learning activities, assessment, doctoral originality, advising, and institutional support. Purposeful independent work and ambitious AI-assisted research are complementary parts of the model. Its recommendations include proportionate contribution statements, recognition of advising costs, and staged pilots evaluating both mathematical ability and effective human–AI collaboration. The aim is not to preserve an inherited sequence of training, but to improve mathematical formation as mathematical practice changes.

Among lattices normalized to have packing radius one, the known shallowest depths of the deep holes in $7$-, $8$-, $9$- and $10$-dimensional lattices are $\sqrt{7/3}$, $\sqrt2$, $\sqrt{5/2}$ and $\sqrt{8/3}$, given by $E_7^*$, $E_8$, the laminated lattices $\Lambda_9$ and $\Lambda_{10}$, respectively. This paper presents new lattices in these dimensions with shallower deep holes. It also answers two problems of J. Martinet about short representatives of the classes of $\Lambda/2\Lambda$ in $7,8,9,10$ and $12$ dimensions.

In this paper, we establish the existence and uniqueness of solutions for mean-field stochastic differential equations (MF-SDEs in short) and backward stochastic differential equations (MF-BSDE) driven by anomalous sub-diffusions $\{B_{L_t}; t\geq 0\}$ with random coefficients, respectively. Here $B$ is a Brownian motion on ${\mathbb R}^d$ and $L$ is the inverse of a subordinator $S$ with drift $\kappa >0$ that is independent of $B$. We further study the stochastic maximum principles (SMPs) for control problems of the stochastic systems modelled by the MF-SDEs using a convex variational method. A linear quadratic control example is given in the last section of this paper, for which both the SMP and the sufficient SMP established in this paper are utilized to show explicitly that it admits a unique stochastic optimal control.

We extend the theory of $s$-energy to $\mathbb{Q}_p^n$. We give an exact representation for the $s$-energy in terms of the Fourier coefficients of a finite Borel measure with support in $\mathbb{Z}_p^n$, which is stronger than the approximate representation in $\mathbb{R}^n$. We also compute the Fourier dimension of the $p$-adic Riesz product measure, even in the case $p=2$, which has never been studied before.

We study a fractional Brezis–Nirenberg problem on a bounded domain with finitely many Hardy potentials centered at distinct points $a_1,\dots,a_k$, with masses $\mu_1,\dots,\mu_k\geq 0$. We assume $\sum_{i=1}^k\mu_i<\Lambda_{N,s}$ and consider $N>4s$ and $0<\mu_{\max}<\overline{\mu}_{N,s}$. The interaction between the poles allows existence for $\lambda=0$ and for some negative values of $\lambda$. At a pole $a_j$, this interaction is described by $B_j(\lambda,\mathbf a,\boldsymbol\mu)=\lambda+\sum_{i\neq j}\frac{\mu_i}{|a_i-a_j|^{2s}}$. This leads to a threshold $\lambda_{\mathrm{geo}}\leq 0$. We prove that the critical Rayleigh quotient is attained for $\lambda_{\mathrm{geo}}<\lambda<\lambda_1(\boldsymbol\mu,\mathbf a)$, and hence the problem admits a positive least-energy solution. If at least two masses are positive, then $\lambda_{\mathrm{geo}}<0$, so the result includes $\lambda=0$ and a nonempty interval of negative values of $\lambda$. We also prove a direct compactness estimate for weakly vanishing sequences by localizing the Gagliardo seminorm. The argument avoids both profile decomposition and the Caffarelli-Silvestre extension.

Given a polynomial \(g\) of degree \(\delta>0\) over a finite field, we study monic polynomials \(f\) of degree \(n\) that can be written as \(f=h+g^k\), where \(h\) is a monic irreducible polynomial of degree \(n\) and \(k\in\mathbb{N}\) with \(\delta k<n\). Following Erdős, we show that some \(f\) admit at least \(c\log n\) such representations. If \(\delta\rho_g<1\), where \(\rho_g=\varphi_q(g)/|g|\), then a positive proportion of degree-\(n\) polynomials admits at least two such representations. More generally, for sums of powers \(g^{\lfloor k_i^{r_i}\rfloor}\), we prove that a positive proportion of polynomials of degree \(n\) is representable whenever \(\sum_{i=1}^{t}r_i^{-1}\ge 1\), while certain residue classes modulo some irreducible polynomial contain no representable polynomial.

In this paper, we study the ${\rm v}$-number of the symbolic-ordinary discrepancy modules $M_t(I) = I^{(t)} / I^t$ over standard graded Noetherian rings. We show that this coincides with the ${\rm v}$-number of $I^t$ at embedded primes. Moreover, under mild assumptions on $I$, we show that the ${\rm v}$-number of $N_t(I) = I^{(t)} / I^{(t+1)}$ is the same as the ${\rm v}$-number of $I^{(t+1)}$. Furthermore, we provide combinatorial formulas for these ${\rm v}$-numbers when $t = 2, 3$ and $I$ is the edge ideal of a graph. Consequently, we classify all graphs for which ${\rm v}(I^2) = {\rm v}(I^{(2)})$.

We study the application of exponential time integration methods, coupled with Fourier pseudospectral space discretization, to the hyperbolic Korteweg-de Vries (KdVH) system. We investigate the asymptotic preserving (AP) properties of such discretizations, showing that, in general, Lawson methods are not asymptotic preserving, because the auxiliary (derivative-approximating) variables do not satisfy the limit equilibrium manifold, whereas exponential time differencing (ETD) methods are AP for all components, including both the solution variable and the auxiliary variables. We also present an efficient numerical implementation based on an exact formula for the matrix exponential of this system, and compare it with previously-proposed ImEx Runge-Kutta time integration, showing that exponential methods can be competitive in this context.

Graph neural networks are widely used for drug–target affinity (DTA) prediction, and discrete Ricci curvature has recently been used to characterize molecular graph geometry. Existing curvature-aware DTA approaches mainly use static curvature on the drug graph while representing proteins primarily with sequence-derived features. This leaves pair-adaptive use of graph geometry underexplored, which may limit adaptation to unseen entities in cold-start settings relevant to practical screening. We present CurvFlow-DTA, which replaces a single static curvature representation with weighted Forman curvature flow on both molecular and protein residue–residue contact graphs. A label-independent flow trajectory is precomputed for each entity, and a pair-conditioned selector determines the horizons read by a dual-branch Flow-GINE. A frozen ESM-2 supplies residue-level representations and contact scores used to construct the protein graph. Inference requires only SMILES strings and protein sequences, without a bound complex structure. On Davis and KIBA, CurvFlow-DTA improves on the protocol-matched Ricci-GraphDTA baseline in every warm and cold-start setting. Warm-split mean squared error (MSE) decreases by $19.9\%$ on Davis and $18.9\%$ on KIBA. Across the six cold-start comparisons, MSE decreases by $14.3$–$27.4\%$, with higher concordance index (CI) throughout. Within our compiled set of literature baselines, CurvFlow-DTA achieves the lowest MSE on both warm benchmarks and across four out of six cold-start evaluation settings.

Four-dimensional conformal gravity, twistor theory, and ambitwistor theory are reviewed within the unified framework of holomorphic conformal Cartan geometry. Starting from the spinorial representation of the geometry, the normal $\mathfrak{sl}(4,\mathbb{C})$-valued Cartan connection, its curvature $2$-form, and the associated tractor bundles are systematically formulated. Evaluated on the real Lorentzian slice, the quadratic curvature action is shown to reproduce four-dimensional Weyl–Bach conformal gravity upon fixing a Weyl gauge. Local twistor and dual twistor bundles are constructed as associated spin tractor bundles, illustrating how parallel transport yields the classical incidence relations in flat space and chiral integrability constraints in curved backgrounds: anti-self-duality for $\alpha$-surfaces and self-duality for $\beta$-surfaces. Furthermore, by pairing local twistors and dual twistors via an invariant canonical pairing, ambitwistor space is shown to parameterize complex null geodesics and symmetrically accommodate both left- and right-handed Weyl curvatures. This framework establishes a direct gauge-theoretic bridge linking conformal Cartan geometry to twistor and ambitwistor theories.

In this paper we extend the results from Yang and Li(2017), in a non-renewal driven bivariate risk model. Concretely, we are interested in the asymptotic behavior of the joint tail of the discounted aggregate claims over finite and infinite time horizon in a bivariate risk model with two arbitrarily dependent counting processes. We additionally suppose that the sequences of claim-sizes of the two lines of business are independent, but each of them contains weakly dependent terms. In our main results, on finite and on infinite time horizon, we assume two (different) general conditions for the counting processes, that are satisfied by a wide spectrum of processes beyond the renewal ones. In the case of finite horizon we suppose that the claim distributions from the two lines, belong to the subexponential distribution class, while in the case of infinite horizon we restrict ourselves to the consistently varying and positively decreasing distribution class. More explicit asymptotic expressions are derived in the case we are restricted on the regularly varying class for the claim distributions. We note that our results indicate the presence of multivariate non-linear single big jump principle for the discounted aggregate claims.

Let $\pi:S\twoheadrightarrow R$ be a split ring epimorphism with nilpotent kernel $J$, and let $R\hookrightarrow S$ be a fixed ring-theoretic section. We prove a change-of-rings formula which is considerably stronger than an equality of global invariants: if $X$ is a left $S$-module satisfying $\Tor_i^S(R,X)=0$ for every $i>0$, then \[ \pd_S X=\pd_R(R\otimes_S X) \] in $\mathbb N\cup\{0,\infty\}$. In particular, for every flat left $S$-module $F$, \[ \pd_SF=\pd_R(F/JF), \] and $F/JF$ is flat over $R$. Reduction and induction of scalars therefore show that the same numerical set of projective dimensions is attained by flat modules over $S$ and over $R$. Consequently, left $n$-perfectness is invariant under every split-by-nilpotent extension. The right-handed analogue holds as well. For a commutative ring $R$ and an arbitrary $R$-module $M$, this applies to the idealization $S=R\ltimes M$ without any projectivity, flatness, finite generation, or finite flat-dimension assumption on $M$. We obtain a list of equivalent characterizations of the condition that $R\ltimes M$ is $n$-perfect, together with \[ \cotD(R\ltimes M)=\cotD(R). \] We further derive exact preservation of the flat-projective-dimension spectrum, strict $n$-perfectness, pointwise localization, arbitrary iterated idealizations, $n$-trivial extensions, truncated polynomial rings, and upper triangular matrix rings.

Motivated by the result that a complex $n\times n$ matrix $A$ being unitarily equivalent to a real matrix, we extend the conclusion to the quaternion skew field in this paper, we present a necessary and sufficient condition for that a quaternion $n\times n$ matrix $A$ is unitarily equivalent to a complex matrix. To state the truth more clearly, we put forward the concept which we call $\boldsymbol{i}$-conjugate. Furthermore, we study the concepts related to $\boldsymbol{i}$-conjugate and their properties, such as unitary $\boldsymbol{i}$-congruence, $\boldsymbol{i}$-conjugate normality and $\boldsymbol{i}$-Hermicity in $M_{n}(\mathbb{H})$ as generalizations of the conventional unitary congruence, conjugate normality and Hermicity of matrices in $M_{n}(\mathbb{C})$. Finally, we present a new type of polar decomoposition of quaternion matrices.

We give a unified arithmetic and geometric treatment of the Hall-Janko simple group and the icosian Leech lattice. Let $\mathcal{I}$ be the standard icosian maximal order over $K=\mathbb{Q}(\sqrt5)$. Following the mass-formula and Venkov framework for the Niemeier and Leech lattices, we classify positive definite unimodular Hermitian $\mathcal{I}$-lattices of quaternionic rank $3$. Hashimoto's mass formula, the Siegel-Weil average, and Gundlach's structure theorem determine the scalar theta series of every nonsplit class. A two-place harmonic separation of the Leech minimal shell proves that its three Hermitian components are quaternionic projective $5$-designs. For the norm-$2$ shell, integrality and projective moments recover the complete angle scheme and its $525$ orthogonal frames. The finite code of one frame then saturates the residual mass, giving exactly the split and Tits classes and an independent computation of the order of $\operatorname{Aut}_{\mathcal{I}}(L_{\mathrm T})\cong2.J_2$. Finally, a quaternionic line system with the prescribed five projective inner products has at most $315$ points, with equality precisely for the Hall-Janko configuration up to $P\operatorname{Sp}(3)$. Equality in Hoggar's bound supplies the design property; Cohen-Tits uniqueness, golden-angle fission, and the centered cubic moment give geometric rigidity.

We classify compact Kähler–Einstein surfaces whose metric is induced by a holomorphic isometric immersion into a finite-dimensional complex projective space. No symmetry assumption and no bound on the codimension are imposed. We prove that the only such surfaces are \[ (\PP^2,m g_{\FS}) \quad\text{and}\quad \bigl(\PP^1\times\PP^1,m(g_{\FS}\oplus g_{\FS})\bigr), \qquad m\in\mathbb Z_{>0}, \] realized respectively by the Veronese and Segre–Veronese embeddings. The main new ingredient is a codimension-independent exclusion of the entire Fano-index-one branch, combining a common anticanonical root construction with Gram–Gauss rank estimates and, in degree five, an equivariant curvature argument. Consequently, every connected compact Kähler–Einstein surface whose metric is induced by a holomorphic isometric immersion into a finite-dimensional complex projective space is homogeneous.

We study data-driven policy construction for dynamical systems using trajectories generated by surrogate finite-horizon Model Predictive Control (MPC), with the goal of approximating an infinite-horizon optimal control policy. We interpret these trajectories as non-expert demonstrations and propose a memory-based, nonparametric policy that constructs a Lipschitz-regularized upper envelope of the finite- horizon surrogate value function, which serves as an explicit upper bound on the value of the resulting policy, thereby providing a computable performance certificate while bypassing explicit optimization at runtime. We establish relative optimality guarantees with respect to the infinite-horizon optimal value function and derive sufficient MPC horizon conditions, together with sample complexity bounds, for achieving a prescribed relative error. Numerical experiments on a rocket landing task validate the theoretical predictions and demonstrate near-optimal performance.

Let $\phi$ be a quaternion of modulus $1$. In this article, we study some topics related to $\phi$-conjugation for quaternionic matrices, including $\phi$-Hermitian matrices, $\phi$-conjugate normal matrices, unitary $\phi$-congruence and $\phi$-HSH decomposition (decomposition of a $\phi$-Hermitian matrix and a skew $\phi$-Hermitian matrix). In particular, we generalize the Autonne-Takagi factorization of quaternion $\phi$-Hermitian matrices for all unit quaternion $\phi$. This gives an affirmative answer to a problem proposed by R. Horn and F. Zhang in the paper “A generalization of the complex Autonne-Takagi factorization to quaternion matrices, Linear Multilinear A. 60: 1239–1244, 2012”.

Let \[ A_\omega=-\partial_x^2+\rho\dot B_x(\omega), \qquad \rho\ne0, \] be the Dirichlet Schrödinger operator on $(0,\pi)$ with a spatial Gaussian white-noise potential, realized pathwise via quasi-derivatives. Here $\dot B_x$ denotes the distributional derivative of Brownian motion with respect to the spatial variable $x$. For $\kappa\ne0$, we consider the cubic nonlinear Schrödinger equation \[ i u_t=A_\omega u+\kappa |u|^2u. \] We first prove a zero-set theorem for locally real-analytic functions on classical Wiener space. As a consequence, on a single event of probability one, no nontrivial finitely supported integer combination of the random eigenvalues vanishes, and the quartic twist matrix is nonsingular for every finite tangential set. After fixing a path in this event, we combine these qualitative nondegeneracy properties with a partial quartic Birkhoff normal form adapted to the KAM scheme. For every finite nonempty tangential set \(J\) of cardinality \(b\), we obtain a Cantor family of linearly stable, real-analytic, small-amplitude invariant \(b\)-tori. The family is parametrized by a Cantor subset of \([\nu,2\nu]^b\) whose relative measure tends to one as \(\nu\to0\).

This work disproves the 1-leaky positive semidefinite edge-deletion conjecture and replaces it with a sharp theorem. For every leak level $\ell$ and edge $e$, one has $|Z^+_{(\ell)}(G)-Z^+_{(\ell)}(G-e)|\le 2$. More generally, if two graphs differ only on edges with both endpoints in $S$, their parameters differ by at most $|S|$. An endpoint-sensitive refinement recovers an increase of at most one whenever some minimum set for $G-e$ contains an endpoint of $e$. Both signs are sharp for every positive leak level. Joining two copies of $K_{\ell+1}$ by a bridge gives $Z^+_{(\ell)}(G)=2\ell$ and $Z^+_{(\ell)}(G-e)=2\ell+2$. For every $\ell\ge 2$, a connected clique-leaf pair of order $2\ell+3$ gives the opposite difference. The remaining positive-difference one-leak case is attained by connected graphs on nine vertices with $Z^+_{(1)}(H)=4$ and $Z^+_{(1)}(G)=6$. Explicit forcing sequences, fort certificates, and an exact verifier check the finite extremal example and stress-test the general results.

We consider minimum-time control for the linear system $$ \dot{z}(t)=Az(t)+Bu(t),\qquad \|u\|_{L^\infty(0,\infty;\mathbb{R}^m)}\leq 1, $$ with $A\in\mathbb R^{n\times n}$ and $B\in\mathbb R^{n\times m}$. While the individual point-target and ball-target problems are classical, we study a different question: when are their optimal controls exactly compatible, in the sense that, for every nonzero initial state $x$ and all sufficiently small $\varepsilon>0$, the optimal control for the tolerance ball $\overline{B}_\varepsilon(0)$ is precisely the restriction of the point-target optimal control? We prove that this {\it{exact truncation}} property is equivalent to the rigidity conditions $$ B B^\top=\beta I_n,\qquad A+A^\top=2aI_n, \qquad \beta>0,\; a\leq 0, $$ and also to Euclidean radiality of the point-target minimum-time function. Thus exact truncation holds precisely when the sublevel sets of the point-target minimum-time function are Euclidean balls centered at the origin, matching the geometry of the tolerance targets. Moreover, the local property automatically extends to every $0<\varepsilon<|x|$, and the resulting optimal times and point-target optimal feedback are both explicit.

The hypersensitivity and turnpike phenomena in the optimal control of an inverted pendulum are investigated from a dynamical-systems perspective. We show that, for a fixed terminal time and a fixed terminal state optimal control problem, (1) the hypersensitivity originates from the fractal structure of the set of initial adjoint variables in the associated Hamiltonian dynamics, (2) the turnpike arises from slow dynamics in the vicinity of a degenerate center manifold, and (3) the escape channels are formed by normally hyperbolic invariant manifolds (NHIMs). As a consequence, small perturbations in the initial adjoint variables lead to qualitatively distinct extremal trajectories, resulting in severe numerical instability in trajectory optimization. Both the fractal structure and the invariant sets are characterized numerically and analytically.

Equilibria of the Hamiltonian system associated with Pontryagin's minimum principle exhibit an exact doubling of the natural spectrum and, under a simple pairing condition, a Jordan block at every simple purely imaginary eigenvalue. Consequently, the classical Lyapunov Center Theorem does not apply to the augmented system, and no periodic orbit with nonzero optimal control bifurcates at linear order. We establish this mechanism in general and show that an optimal-control-induced periodic family emerges at cubic order in a Lindstedt–Poincaré expansion. The mechanism is illustrated in closed form for the pendulum and evaluated numerically for the planar $L_2$ equilibrium of Hill's restricted three-body problem, where the third-order approximation is validated against an independently computed family of periodic orbits.

Gödelian incompleteness yields arithmetical sentences \(A\) such that \(\mathsf{PA}+A\) and \(\mathsf{PA}+\neg A\) are both consistent. Are such extensions equally legitimate? I propose graded epistemic arithmetical non-pluralism: justification for choosing between them varies with the set-theoretic strength of \(A\). I defend \(\mathsf{PA}+\mathsf{Con}(\mathsf{PA})\) and show that Friedman's concrete incompleteness results pose a serious challenge to non-pluralism about first-order arithmetic. Resolving the selection problem for such sentences turns on justifying large cardinals. Defending graded non-pluralism thus engages Gödel's programme and the justification of very large cardinals.