Papers

Neural PDE training yields a finite checkpoint archive, yet its logged energy errors are inaccessible without the exact solution, while loss-based selection does not necessarily recover the logged energy oracle. For admissible neural approximations of symmetric coercive variational problems, we introduce a reference-free selection rule based on minimizing a computable conforming Riesz monitor. The exact residual-energy identity and conforming projection make the monitor an unconditional lower bound converging monotonically to each logged energy error under nested conforming refinement; under saturation, hierarchical enrichment yields a computable upper estimate and hence a lower-upper bracket. A key finding is that archive selection is order-sensitive: unresolved checkpoint-dependent components can reverse the oracle-non-oracle ranking at finite resolution, so checkpointwise recovery alone is insufficient. For finite archives, we prove uniform recovery, yielding convergence to the logged-oracle error and, without saturation, logged-oracle selection at sufficiently fine auxiliary resolution. Under saturation, the bracket gives a computable near-oracle bound and certifies unique logged-oracle selection upon interval separation. We also bound logging-resolution loss and certify oracle inclusion over prescribed comparison trajectories. The resulting criterion replaces inaccessible exact-error minimization by computable, training-independent post-training selection on the intrinsic energy-error scale, requiring only the computed candidates and the variational problem. Experiments on diffusion and elasticity, including a non-manufactured perforated plate, demonstrate energy-scale calibration, oracle-level selection, and modest post-processing cost.

We give a characterization of the existence of Lipschitz surjections between dust-like self-similar sets of the same dimension under the assumption that the similarity ratios are commensurable. Our results apply more generally to graph-directed fractals.

We develop a measure-theoretic analogue of matroid theory on standard atomless measure spaces. Motivated by the quotient-convergence framework for submodular set functions, our aim is to identify suitable measurable objects on the limit side of finite matroid theory and to develop their basic structural and optimization theory. We prove that measurable matroids admit equivalent descriptions by independent sets, bases, rank functions, and closure operators. The class includes normalized finite matroids, cycle matroids of graphings, and measurable analogues of partition, nested, lattice path, transversal, and matching matroids. We establish measurable analogues of truncation, elongation, direct sum, duality, and minors. We prove measurable versions of Edmonds' matroid intersection theorem and the Edmonds–Fulkerson matroid union theorem, together with an attainment theorem for common bases. The intersection theorem retains the classical min–max form, with the maximum replaced in general by a supremum. Applied to partition matroids, it gives an exact Hall-deficiency formula for measurable matchings in bipartite graphings and a min–max theorem for continuous bipartite $b$-matchings, with applications to capacity-constrained transport and prescribed cross-sections. We give a rank-expansion criterion under which the supremum is attained. For bipartite graphings, this criterion specializes to the measurable perfect matching theorem of Lyons and Nazarov. For measurable union, the Edmonds–Fulkerson rank formula remains valid. As a main application of measurable union and its attainment theorem, we prove measurable Nash-Williams–Tutte theorems for graphings, characterizing approximate coverings and packings by hyperfinite essential spanning forests and obtaining exact decompositions under strengthened rank inequalities.

Elastic wave propagation in fractured media is relevant to applications such as analysis of seismic waves and non-destructive characterization of materials. Understanding attenuation and scattering behavior arising from wave-fracture interaction is important for interpreting observations at both field and laboratory scales. This work presents a computational method for elastic wave propagation in fractured media based on a mixed-dimensional discrete fracture-matrix representation. Fracture deformation is governed by four models of increasing complexity, ranging from widely used spring-based formulations to frictional fracture contact models. Many previous studies are restricted to simplified wave fields, single fractures or subsets of the relevant fracture deformation mechanisms. In contrast, the proposed methodology enables fully coupled simulation of elastic wave propagation with fracture deformation models that account for elastic normal deformation, frictional contact and fracture opening and closure. The elastic wave equation is discretized in space using the cell-centered finite volume method Multi-Point Stress Approximation with weak symmetry and in time using the Newmark method. The spatial discretization is locally conservative and applicable to general polyhedral grids, making it well suited for media containing fractures, material heterogeneities and anisotropy. The proposed methodology is verified through numerical convergence analyses and is subsequently applied to wave propagation and fracture deformation in two- and three-dimensional media containing multiple intersecting fractures, demonstrating the influence of fracture constitutive behavior on wave transmission, scattering and fracture deformation.

Torus computed tomography (Torus CT) reconstructs the Fourier coefficients of an image from X-ray data along closed geodesics on the flat torus. We extend Torus CT to experimental X-ray tomographic data by converting the data to a parallel-beam geometry and mapping the measured projection angles to the geodesic directions on the torus. In addition, we introduce two extensions of the Torus CT framework: the Star TCT method, which recovers additional Fourier coefficients beyond the standard frequency box using the same data, and torus backprojection (TBP), which sums the data over the measured directions. We prove that filtered TBP converges to the Tikhonov-regularized solution for general data up to an explicit error term that vanishes for data which is invariant along fixed geodesics. We also show that the direction-wise averaging of the zero mode is an unbiased estimator and its mean squared error decays like $N^{-2}$. We evaluate the methods on simulated phantoms and experimental X-ray data of a walnut using an updated implementation. The results show that torus-based reconstruction methods are applicable to experimental data, avoid the streak artifacts of filtered backprojection (FBP), and yield competitive reconstruction errors on both simulated and experimental data.

We determine the exact strong converse exponent of quantum soft covering under the sandwiched Rényi divergence for all orders $\alpha\in[\frac{1}{2},\infty)$. For $\alpha\in[\frac{1}{2},1)$, the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for $\alpha\in[1,\infty)$, it is characterized by the order-$\alpha$ sandwiched Rényi mutual information. We also determine the exact strong converse exponent of privacy amplification against quantum side information under the sandwiched Rényi divergence for $\alpha\in(2,\infty)$, expressed in terms of the corresponding order-$\alpha$ sandwiched Rényi conditional entropy. To the best of our knowledge, these results provide the first exact characterization of the strong converse exponent of quantum soft covering and the first precise operational interpretation of the two-parameter club-sandwiched mutual information in the quantum setting. The key ingredient is that we establish the exponential rate of the $K$-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for $\alpha\in[\frac{1}{2},1)$.

We study the dynamics of the exponential map on the complex plane. The set $\Lambda_{\mathbf{c}}$ of all points sharing a given itinerary $\mathbf{c}$ is non-empty if and only if $\mathbf{c}$ is an exponentially bounded itinerary. For such itineraries, $\Lambda_{\mathbf{c}}$ also contains a curve of escaping points, and hence its Hausdorff dimension is at least~$1$. We prove that for every exponentially bounded itinerary this dimension is in fact equal to~$1$. In comparison, for certain itineraries, the set $\Lambda_{\mathbf{c}}$ exhibits highly complicated topological structures, such as indecomposable continua.

For a finite set $A$ in an abelian group, let \[ E(A)=\#\{(a_1,a_2,a_3,a_4)\in A^4:a_1+a_2=a_3+a_4\}. \] We obtain an estimate uniform in $d$ that compares the normalized additive energy of $\mathbb{Z}^d \cap B_d(R)$ with the continuous energy of $B_d(R)$ . If $R_d/\sqrt d\to\infty$, then \[ \lim_{d\to\infty} \left( \frac{E\bigl(\mathbb{Z}^d\cap B_d(R_d)\bigr)} {\lvert \mathbb{Z}^d\cap B_d(R_d)\rvert^3} \right)^{1/d} =\frac{4\sqrt{3}}{9}. \] As an application, consider \[ A_n = R_n\mathbf{1}_d + \bigl(\mathbb{Z}^d\cap B_d(R_n)\bigr), \] where $d=d(n)\to\infty$ satisfy $\log d=o(\log n)$, and $R_n=\lfloor(n-1)/2\rfloor$. Then $A_n\subset\{0,1,\ldots,n-1\}^d$ and \[ \log E(A_n) =3\log|A_n|-d\log\frac{3\sqrt3}{4}+o(d). \] In particular, taking $d=\lfloor(\log n)^{1/2}\rfloor$ gives an explicit construction answering a question of Shao \cite{Shao2026}. We also prove that in Gram-matrix coordinates, the exponential rate of the continuous ball energy is determined by a fixed dimensional determinant maximization whose extremizer is the Gram matrix of a regular tetrahedron.

In a world where valuable artifacts are increasingly created, completed, or processed by LLMs, the central economic question is not only what the LLM can produce, but what value remains in the inputs (i.e., the prompts) we provide to it. Given a prompt, hint, critique, problem statement, or partial solution that helps an LLM produce an artifact $z$—a proof, program, design, or scientific hypothesis—how should we measure the value of that input? Intuitively, an input is valuable when it makes the target artifact easier for the model to generate: either by increasing its sampling probability, or by reducing the thinking time needed to find it. We propose a computational Levin–Kolmogorov complexity approach to this problem, by appropriately replacing the universal Turing machine in the classical definitions by the LLM itself. Concretely, we introduce an LLM-relative notion of probabilistic Levin–Kolmogorov complexity $pKt$—treating the model's thinking as the random tape of the program, and charging logarithmically for it in Levin's manner—and define prompt value as algorithmic mutual information with respect to $pKt$. This captures the intuition above: a prompt having $b$ bits of value for an artifact $z$ makes $z$ $2^b$ times “easier to obtain”, by multiplying the success probability by $2^b$, by dividing the required computation by $2^b$, or by any corresponding tradeoff between probability and computation. In contrast to the classical notion of algorithmic mutual information, ours is efficiently estimable. We additionally show that, under a natural reproduction experiment, a prompt value of \(b\) bits means that reproducing \(z\) without the prompt has median token cost \(2^b\) times that of reproducing it with the prompt.

We study the two-dimensional stationary incompressible Navier–Stokes equations on $\mathbb R^2$ with fractional dissipation $(-\Delta)^s$. In the full range $s \in (0,1)$, we prove that every smooth solution satisfying the natural energy condition $u\in\dot{\mathrm H}^s(\mathbb R^2;\mathbb R^2)$ has $u\equiv0$ and constant pressure. This is a fractional counterpart of the planar finite-Dirichlet theorem of Gilbarg and Weinberger at $s=1$. The proof uses different arguments in three ranges. For $0<s<\frac13$, we combine an $\mathrm L^2$-estimate derived from the equation with a stream-function truncation argument. For $\frac13\leq s\leq\frac23$, we use a localized energy estimate whose boundary terms are supported on expanding annuli. For $\frac23<s<1$, we establish regularity and decay via a Lorentz-space bootstrap and then apply the maximum principle to the vorticity. We also treat the stationary damped Euler system at $s=0$ by combining the Bernoulli identity with a cut-off argument under an annular growth condition that includes $u\in \mathrm L^r(\mathbb R^2)$ for every $1\le r\le2$.

In collection operations, accumulating payload progressively slows the vehicle, imposing a cumulative penalty on routing efficiency. An onboard drone can offset this penalty by retrieving outlying items, thereby shortening the makespan and increasing operational profit. However, travel time remains load-dependent, and each item collected by the ground vehicle shifts the arrival times that govern the drone's launch and rendezvous points. This paper introduces the Travelling Thief Problem with Drone (TTP-D), which maximises the collected profit, net of a time-based rental cost, by jointly optimising item selection, vehicle routing, and flight synchronisation. We formulate a mixed-integer linear program that solves small instances to optimality, and develop both metaheuristics and an attention-based Deep Reinforcement Learning (DRL) policy for larger instances. We further propose a learner-initialised hybrid solver, in which the DRL policy constructs an initial solution that a short annealing run subsequently refines. On two benchmark sets, this hybrid recovers most of the metaheuristic baseline's quality at a fraction of its computational budget, although the largest instances still require the baseline at its full budget. Finally, a sensitivity analysis reveals that the rental ratio is the primary driver of profitability, whereas the fleet parameters affect profit only at the margin.

Let $G$ be a finite group. The enhanced power graph of $G$, denoted by $\mathcal{E}(G)$, is the graph with vertex set $G$ in which two vertices $u$ and $v$ are adjacent if and only if there exists an element $w \in G$ such that both $u$ and $v$ belong to $\langle w \rangle$. The proper enhanced power graph of $G$, denoted by $\mathcal{E}^{**}(G)$, is the subgraph of $\mathcal{E}(G)$ induced by the non-dominating vertices. The main objective of this paper is to investigate finite groups whose proper enhanced power graph is claw-free, that is, contains no induced subgraph isomorphic to the complete bipartite graph $K_{1,3}$. We first prove that $\mathcal{E}(G)$ is claw-free if and only if $G$ is cyclic. The set of dominating vertices of $\mathcal{E}(G)$ forms a cyclic subgroup of the center of $G$, namely the cyclicizer $\cyc(G)$ of $G$. This allows us to give a precise description of the structure of $G/\cyc(G)$ when $\mathcal{E}^{**}(G)$ is claw-free. If $G$ is solvable but not nilpotent, then $G$ is metacyclic, or $G/\cyc(G)$ is either a Frobenius group or a $2$-Frobenius group. If $G$ is non-solvable, then $G/\cyc(G)$ is isomorphic to $\PSL(2,q)$ or $\PGL(2,q),$ and this allows us to give a complete classification of the non-solvable groups whose proper enhanced power graph is claw-free.

Let $\mathcal{L}_n \subset \mathcal{M}_2$ be the locus of genus-two curves admitting a maximal degree-$n$ elliptic subcover, cut out in $\mathbb{P}(2,4,6,10)$ by an irreducible weighted-homogeneous polynomial $F_n \in \mathbb{Z}[J_2,J_4,J_6,J_{10}]$. Let $\nu(n)$ be the degree of $X_1(n) \to X(1)$, let $G_{n^2}$ be the Siegel modular form of level one with divisor the Humbert surface $H_{n^2}$, and let $k(H_{n^2})$ be its weight. We prove that the meromorphic Siegel modular form $F_n(\tau)$ obtained from $F_n$ has a pole of order exactly $\nu(n)$ along the product locus, that $\chi_{10}^{\nu(n)} F_n(\tau)$ is a constant multiple of $G_{n^2}$, and that $\deg_w F_n = k(H_{n^2}) - 10\nu(n)$ for every $n \geq 2$, even or odd. We determine the restriction of $G_{n^2}$ to the product locus as an explicit product of modular polynomials and, for $n \geq 3$, its leading Fourier-Jacobi coefficient as a product of theta functions over the torsion points of exact order $n$, and we characterize $G_{n^2}$, up to scalar, as the unique form of its weight vanishing on a single torsion divisor. These data convert the computation of $F_n$ from elimination into a linear problem of the size the formula prescribes, which we carry out for $n=5$.

We construct a new flag-transitive $2$-$(496,4,1)$ design with automorphism group $\mathrm{P\Gamma L}_2(32)$. This corrects an omission in the classification of the finite flag-transitive linear spaces.

A graph is 2-planar if it can be drawn in the plane such that each edge is crossed by at most two other edges. It is known that for a 2-planar graph $G$, $|E(G)| \le 5|V(G)| - 10$. When the equality holds, we call $G$ an optimal 2-planar graph. This paper investigates the matching extendability of optimal 2-planar graphs. By local optimality, we prove that every 4-connected optimal 2-planar graph $G$ of even order is 1-extendable, and give a criterion for $G$ to be 2-extendable. We also prove that no optimal 2-planar graph is 5-extendable and construct a 4-extendable optimal 2-planar graph based on the dodecahedron. Finally, we show that every 6-connected optimal 2-planar graph of even order with at least $2m+2$ vertices is distance 3 $m$-extendable for any $m \ge 0$.

Oort's conjecture asserts that for any $g \geq 2$ and any prime $p$, every geometric generic member in the supersingular locus $\mathcal{S}_g$ has automorphism group $\{ \pm 1 \}$. This has been proved very recently by Viehmann in full generality, with previously known counterexamples for $p=2$ and $g=2,3$. We construct, for any $g \geq 3$, a closed subvariety of dimension $g-1$ which contains an open dense subset $\mathcal{U}$ of $a$-invariant $g-2$ such that $\mathcal{U}$ meets every irreducible component of $\mathcal{S}_g$ and every geometric point in $\mathcal{U}$ has automorphism group $\{ \pm 1 \}$ if $p >2$. This gives an independent proof of Oort's conjecture for $p>2$. When $g>3$, so $a = g-2 > 1$, this provides complementary information to the $a=1$ locus investigated in Viehmann's work on how the automorphism groups interact with the geometry.

Statistical inverse problems have garnered significant attention in recent years due to the growing importance of statistical learning theory and functional analytic approaches in the fields of machine learning and artificial intelligence. In this paper, we investigate the stable approximation of the element $u^{\dagger}$ that satisfies the equation $Au = g$, where $A$ is a linear operator that maps a Banach space into an appropriate function space. The function $g$ is observed only through independently and identically distributed data points that are corrupted by noise and assumed to follow an unknown distribution $\rho$. We employ the Tikhonov regularization scheme, leveraging statistical learning techniques and the framework of reproducing kernel Banach spaces to estimate the solution. We establish convergence and derive the convergence rate of the estimated solution with respect to the true solution as the number of data points increases, with the rate expressed in probabilistic terms. The theoretical findings are further supported by numerical experiments that demonstrate the effectiveness of the proposed approach.

We generalize the Stone space of ultrafilters on Boolean algebras and prove a generalization of Stone duality which is applicable to locally compact Polish spaces. Using this, we obtain complete invariants for simplicial complexes up to PL-homeomorphism and for non-compact $2$- and $3$-manifolds up to homeomorphism. We prove that the homeomorphism relation on non-compact $2$-manifolds without boundary, the homeomorphism relation on non-compact $3$-manifolds with or without boundary, the homeomorphism relation on open subsets of $\mathbb{R}^2$ and $\mathbb{R}^3$, and conjugacy of Cantor sets in $\mathbb{R}^3$ are classifiable by countable structures. Together with known lower bounds, this implies that these relations are Borel bireducible with isomorphism of countable graphs. We also show that PL-homeomorphism of Heine-Borel simplicial complexes and PL-homeomorphism of PL $n$-manifolds, for every $n$, are classifiable by countable structures.

Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $\chi \in {\rm Irr}(G)$ in the principal $p$-block of $G$ such that $1<\chi(1)_p\le m(P)$, giving one inequality of the Eaton-Moretó conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moretó conjecture for principal blocks.

In this paper, we define and analyze the quantized maximal correlation, an extension of the notion of maximal correlation restricted to functions taking values in sets of bounded cardinality. We derive an upper bound on the quantized maximal correlation by showing that the correlation between any quantized functions of $X$ and $Y$ is related to the MMSE distortion in quantization of a particular linear combination of random variables. Following this, we leverage rate-distortion techniques and anti-concentration inequalities to further bound this MMSE, which results in explicit bounds on the quantized maximal correlation. Unlike the quantized maximal correlation itself, which does not generally tensorize, our bounds on the mean squared error do tensorize, resulting in a dimension-free upper bound on the quantized maximal correlation for product distributions. Our results also lead to improved bounds on the isoperimetric constants of reversible Markov chains and product chains, strengthening classical results such as those by Alon and Milman.

We present a Runge-Kutta method of weak order 2 for the numerical time integration of stochastic delay differential equations. This scheme extends the class of second order Runge-Kutta methods introduced by A. Rößler in [SIAM J. Numer. Anal., 47(3):1713-1738, 2009] for stochastic ordinary differential equations. The proposed integrator is applicable to equations with discrete commensurable delays and is particularly efficient for problems involving multiple noise terms. Experimental confirmation of the weak order 2 is provided and MATLAB codes are freely available.

The usual dual of an infinite-dimensional order-unit space need not carry an order unit, and this is the main reason why duality for operator systems is notoriously hard. Rather than trying to compute a canonical dual object, we introduce a relative notion of duality for operator systems: Two systems are in duality when a pairing of their underlying vector spaces induces conic pairings at every matrix level. We develop basic properties of this notion, showing that beyond finite-dimensional operator systems, also separable $B(H)$ and $C^*$-algebras with a faithful trace admit duals, while some other operator systems do not. We prove that subsystems, quotients and suitable tensor products of systems with duals again admit duals. As consequences, we show that finite-level maximality is dual to finite-level minimality, and we establish weakly continuous and weak$^*$ closed realizations into $\ell^\infty$-products of matrix algebras.

We prove a concavity inequality for the Hessian quotient operator $\sigma_k/\sigma_{k-1}$ using a change of basis for symmetric polynomials. This removes an additional structural concavity assumption previously imposed in the interior $C^2$ estimate for convex solutions of the $\sigma_k/\sigma_{k-1}$ equation. The method also yields several useful inequalities for elementary symmetric polynomials.

Drawing on Walter Benjamin's account of technological reproducibility and aura in relation to art, together with insights from Yehuda Rav and Ludwig Wittgenstein, this speculative essay reflects on the impact of AI systems on the practice of pure mathematics.

Let Phi be an irreducible reduced crystallographic root system of rank r at least 2, let N = |Phi^+| be the number of positive roots, and let h be its Coxeter number. For the normalized Witten zeta function xi_Phi, we determine the first distinct pole below the leading pole 2/h. It is located at q_2(Phi) = (r-1)/(N-1) = 2(r-1)/(rh-2), is simple, and receives contributions precisely from the codimension-one faces of the dominant chamber. Its residue is zeta_R(q_2)/(N-1) times the sum of the corresponding wall periods, where zeta_R denotes the Riemann zeta function. These periods are finite and positive, so the residue is strictly negative. We also prove a Stokes relation for projective hyperplane-arrangement periods. Let A be an essential central real arrangement in R^n with weights lambda_H strictly between 0 and 1. Suppose that the sum of lambda_H over all H in A equals n, and that for every nonzero proper intersection flat X the sum of lambda_H over those H containing X is strictly less than the codimension of X. Then the vector of positive chamber periods lies in the kernel of the Varchenko matrix with weights exp(pi i lambda_H). Applying this relation, we evaluate the relevant wall periods in types F4 and D5. A two-orbit reduction and Dixon's 3F2(1) summation give a gamma-product evaluation in type F4, while a four-orbit reduction and Selberg's integral give a gamma-product evaluation of the complete wall sum in type D5. Consequently, we obtain exact formulas for the normalized and ordinary Witten zeta residues at 3/23 and 4/19.

We consider splitting methods for partial differential equations involving unbounded operators. For non-time-reversible dynamics, such as dissipative systems, negative splitting coefficients are generally not admissible because they require stepping backward in time, leading to an order barrier when all coefficients are required to be positive. We introduce generalized modified splitting methods to overcome this barrier. To analyze their convergence, we develop the corresponding multi-index series formalism, which provides a systematic framework for deriving order conditions. Using this formalism, we derive the order conditions and construct a generalized modified splitting method of order $6$. We also provide Python scripts that automate the generation and verification of order conditions, as well as the construction of new generalized modified splitting methods. Finally, we establish connections between the introduced multi-index series and related series formalisms from the literature, in particular, word series, Lie-Butcher series, and multi-index Butcher series.

We develop a unified analysis of inexact stochastic Riemannian proximal optimization for finite-sum nonsmooth composite problems over compact embedded submanifolds. The framework accommodates variance-reduced gradient estimators, projected momentum, and inexact tangent-space proximal solves under a single conditional error-dissipation condition, verified for projection-based SVRG, SARAH/SPIDER, SAGA, and SAG. A computable Fenchel-dual residual criterion, with tolerance prescribed before sampling and inner iterations, enables explicit control of the inner work. We establish conditional expected descent, subsequential stationarity, and an \(O(\epsilon^{-2})\) outer complexity. With SARAH/SPIDER and accumulative regularization, iRPMVR attains \(O(n+\sqrt n\,\epsilon^{-2})\) component-gradient and \(O(\epsilon^{-3})\) proximal-operator complexities. We further develop an abstract KL principle for conditional expected descent with memory and summable tails using only the ordinary pointwise KL property. A counterexample shows that a power-type expected-KL implication used in earlier stochastic analyses can fail. The principle yields almost-sure finite length, whole-sequence convergence, and deterministic KL rates.

In this article, we study an inhomogeneous critical nonlinear equation involving the sub-Laplacian on the Heisenberg group $\mathbb H^n$. We prove the multiplicity of positive solutions for the critical problem \begin{align*} \mathcal{L}_{\mathbb H^n} u=|u|^{2^\star-2}u+f(\xi) \quad \text{in } \mathbb H^n, \qquad u>0,\quad u\in S^{1,2}(\mathbb H^n), \end{align*} where $ \mathcal{L}_{\mathbb H^n} $ is the sub-Laplacian on $\mathbb H^n$, $2^\star=\frac{2Q}{Q-2}$, $Q=2n+2$, $n\geq 1$, $S^{1,2}(\mathbb H^n)$ is the homogeneous Sobolev space on $\mathbb H^n$, and $f$ is a nontrivial nonnegative functional in the dual space $(S^{1,2}(\mathbb H^n))'$ satisfying a suitable smallness condition. The above mentioned equation appeared as a perturbation of the CR Yamabe equation on the Heisenberg group. A major difficulty comes from the lack of compactness of the critical Folland-Stein embedding into critical Lebesgue space. To overcome this, we establish a Palais-Smale profile decomposition for the associated energy functional. The obtained Palais-Smale profile decomposition identifies the precise energy levels at which lack of compactness may occur via energy quantization, and shows that every noncompact Palais-Smale sequence decomposes into a finite superposition of weakly interacting bubbles. As a key analytic ingredient, we establish an improved Folland-Stein-Sobolev inequality involving the Morrey norm, which serves as a fundamental interpolation inequality and plays a crucial role in detecting the concentration of noncompact Palais-Smale sequences.

Allocation schemes that greedily maximize a readiness metric over the actuator fiber bundle of an overactuated multirotor produce commands that jump between disconnected optimal strata, demanding actuator rates no motor can deliver; effort-minimizing schemes are continuous but cannot guarantee that wrench-rate authority stays above any certified level. We reconcile the two by treating authority as a forward-invariant quantity: a control barrier function on the log-determinant of the drag-aware actuator-authority co-metric, enforced at torque level by a quadratic program in the allocation null space. A single design inequality renders the certified set compact and strictly interior to the actuator box, with the readiness cost of any rotor deactivation given in closed form as $\ln(n/(n{-}m))$ for symmetric designs. An explicit alignment ratio states when tracking can stay exact, otherwise the wrench error is bounded by $\mathcal{O}(\rho^{-1/2})$, and a robust variant handles motor-parameter uncertainty with a closed-form floor shift independent of the airframe matrix. On a hexarotor and a fully-actuated octorotor the closed-form gap matches simulation to machine precision. Over $300$ randomized missions, greedy maximization violates the certified floor in up to $51\%$ of authority-scarce missions, while the proposed filter never leaves the certified set and, in the hardest moment reversal, commits a $79\times$ smaller wrench error. The robust variant holds the floor on all $150$ perturbed plants. A high-fidelity simulation with measured propeller and motor data shows that the floor holds under unmodeled aerodynamics, electrical limits, and delay, and where it stops applying.

We study skew products of Hénon maps fibered over suitable metric spaces and investigate the analytic structure of their escaping sets. Our study extends the description of escaping sets for Hénon maps developed by Hubbard and Oberste-Vorth to skew products and provides a framework for studying their rigidity. For a compact base space, we construct an intermediate covering space of each fiberwise escaping set. When the base is the closed unit disk and the family depends holomorphically on the parameter, we obtain an analogous description of the global escaping set. We further use this covering space construction to study the relationship between biholomorphic equivalences of their global escaping sets and the underlying dynamics. Finally, for skew products over the non-compact base $\mathbb{C}$, we consider the corresponding escaping region and investigate its analytic structure.