Papers

We adapt the theory of Ruelle-Pollicott resonances to annealed random dynamical systems generated by independent and identically distributed families of maps. Introducing annealed transfer and Koopman operators, we define resonances as elements of the point spectrum of the associated operators and establish a decorrelation formula relating these resonances to the asymptotic decay of annealed correlations. We then study several classes of systems for which the theory can be made explicit. First, we consider a family of piecewise expanding Markov maps of the interval, we construct Banach spaces adapted to the dynamics and obtain a complete description of the annealed resonance spectrum. Then we investigate an inverse spectral problem, proving realisability results for prescribed collections of complex numbers as resonances of suitably constructed annealed dynamical systems.

In this paper, we study the time-periodic problem for the Vlasov-Poisson-Boltzmann (VPB) system with a given time-periodic external force in the whole space $\mathbb{R}^3$. The force is allowed to be non-potential. Around the global Maxwellian, we prove the global existence of small solutions in a hybrid function space that combines the low-frequency Besov framework for the forced Boltzmann equation with a corresponding control of the self-consistent electric field. The main novelty lies in the treatment of the nonlinear Vlasov force $-\nabla_x\phi \cdot \nabla_vf + \frac{1}{2}(v \cdot \nabla_x\phi)f$ at low frequencies. Rather than treating it as a generic source term, we exploit the Poisson equation and macroscopic balance laws to recover the structural cancellation required for the VPB semi-group estimates, which combined with high-frequency energy estimates and weighted microscopic propagation, yields a closed global well-posedness theory. We further prove the asymptotic stability of small solutions driven by the same force. When the external force is time-periodic, Serrin's method yields a unique time-periodic solution with the same period, together with its stability. As a direct consequence, our result also gives the existence and stability of stationary solutions when the external force is time-independent.

The higher-order Craig-Bampton (HCB) method improves reduced-model accuracy through successive residual-mode enrichments. This study proposes an a priori estimator for the eigenvalue error remaining in the first-order HCB model (HCB-1) without using the full-order eigensolution. The estimator is defined as the relative difference between the HCB-1 and HCB-2 eigenvalues. Under symmetric Ritz approximation assumptions, ordered positive eigenvalues, nested augmented spaces, and preservation of retained eigenvalues by the System Equivalent Reduction Expansion Process (SEREP), the estimator is shown to be non-negative and to provide a lower bound on the relative HCB-1 eigenvalue error. A projection-based expression and an exact error identity are also derived, clarifying the conditions under which the estimator is accurate and showing that the lower-bound property alone does not guarantee a prescribed tolerance. Numerical results for a plate, an elbow pipe, and a reactor pressure vessel show estimation-to-error ratios of about 0.85-0.99, while a cost assessment reports the additional expense associated with the HCB-2 reference solution.

The generalized Busemann–Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ having larger volume of all $m$-dimensional sections necessarily have larger volume. When $m\geq 4$, this is known to be false, but the cases $m=2, 3$ for $n\geq 5$ have remained open since the 1990s. In this paper, we resolve these cases. Together with the known results, the generalized Busemann–Petty problem is completely solved: the answer is affirmative for $m=1, 2, 3$, and negative for $m\geq 4$.

Let $\mathcal{U} \Subset \mathbb{C}$ be fixed, $\Phi(s)=e^s-s-1$, $F(\nu)=\frac{\nu}{2+|\nu|}$, and let $f^{F(\nu)}$ denote the principal solution of the corresponding Beltrami equation. The identity $K_{F(\nu)}=1+|\nu|$ identifies compactly supported David coefficients with exponential-Orlicz parameters. We prove that, on the open subset of $L^\Phi_{\mathcal{U}}(\mathbb{C})$ where a sufficiently high finite exponential moment is available, the principal solution map is locally real $C^{1,1}$ with values in $W^{1,2}_{\mathrm{loc}}(\mathbb{C})$. The derivative in a direction $\eta \in L^\Phi_{\mathcal{U}}(\mathbb{C})$ is the principally normalized solution of $\bar{\partial} V - F(\nu) \partial_z V = DF_\nu(\eta) \partial_z f^{F(\nu)}$. The proof uses a pullback by the base principal solution. The key estimate is the pointwise cancellation $\frac{\|DF_\nu\|_{\mathrm{op}}}{1-|F(\nu)|^2} \le \frac{1}{2}$, which converts the linearized equation into a $\bar{\partial}$-equation whose source is controlled directly by the $L^\Phi$-norm of the direction. Combined with the principal degenerate $L^2$-resolvent and the optimal Jacobian regularity for exponentially integrable distortion, this yields a uniform quadratic remainder estimate. At the origin one obtains $D \mathrm{Sol}_0[\eta]=(1/2)\mathcal{C}\eta$ in $W^{1,2}_{\mathrm{loc}}$.

We study a coupled magneto-quasistatic model for an eddy current brake. The model leads to a nonlinear infinite-dimensional differential-algebraic system in which the nonlinearity is structural rather than caused by nonlinear material laws. It is generated by the closed electromechanical feedback loop: the angular momentum determines the velocity of the conducting disk, the velocity and the magnetic field generate motional eddy currents, and these currents induce a Lorentz torque acting back on the rotor. We introduce a finite-dissipation solution concept and prove global existence and uniqueness of weak solutions for initial data given by the magnetic field and the angular momentum, and for inputs given by the supplied coil current and the externally applied torque.

A long-standing conjecture in Hermitian geometry says that a compact Hermitian manifold with constant Chern holomorphic sectional curvature $c$ is Kähler for $c\neq 0$ and Chern flat for $c=0$. Although the conjecture has been established in complex dimension two, it remains open in general in higher dimensions. We verify the conjecture for compact balanced threefolds when $c\leq 0$. For compact locally conformally Kähler manifolds, Chen, Chen, and Nie established the case $c\leq 0$, while Huang and Wan recently settled the remaining case. Inspired by the approach of Huang and Wan, we investigate a generalization of the conjecture for canonical metric connections and establish it for connected compact locally conformally Kähler manifolds.

In this paper, we propose a low-complexity ordered-reliability-bits Chase (ORB-Chase) decoding algorithm for BCH codes. The proposed algorithm differs from the traditional Chase algorithm in two key aspects. First, it employs the logical weight as a metric to generate test error patterns (TEPs). Second, it introduces an integer-based early termination criterion that ensures computation can stop at the earliest possible stage if the maximum-likelihood codeword is identified, thereby minimizing unnecessary computational effort. Simulation results for (127, 113, 5) BCH codes and (256, 239, 6) eBCH codes demonstrate that the ORB-Chase algorithm achieves near-ML performance with significantly fewer test patterns compared to the Chase algorithm. Moreover, the average number of Berlekamp-Massey (BM) decoding calls decreases rapidly as $E_b / N_0$ increases, achieving a reduction of up to 98.1% compared to the Chase algorithm at the same BLER performance.

Understanding how experts make decisions and being able to transfer that knowledge is important, especially in complex engineering applications. It is highly valuable for training novices, improving the performance of human-machine systems, and potentially enabling fully autonomous systems that perform as well as human experts. However, an expert's decision-making strategy, developed through years of experience, is often not directly accessible, since the implicit preferences and decision rules involved can be difficult to specify explicitly. This has motivated the use of observed decisions made by the expert to learn an interpretable model that captures the expert's decision-making process. In this work, we develop an inverse optimization approach to jointly learn the decision-maker's preferences (or perceived costs) and the decision rules governing their choices. We demonstrate the general applicability of our approach using three case studies that consider a shift assignment problem, a production planning problem, and a real-world routing problem, respectively. Across these case studies, modeling both perceived costs and decision rules leads to better predictions, highlighting the value of the proposed framework and its greater flexibility in capturing and replicating expert decision making.

The genus polynomial of a graph is the generating polynomial for the number of nonequivalent embeddings of the graph on each orientable surface. In this paper, we address three questions on genus polynomials for wheel graphs: the computation of genus polynomials, the unimodality and the asymptotic normality of their coefficients. We derive an explicit formula for the genus polynomial of wheel graphs by combining methods of the joint tree model and characters theory, and then prove its real-rootedness. This stronger result implies the log-concavity, unimodality, and asymptotic normality of its coefficients. Thus, we confirm the unimodality conjecture for the genus distribution of wheel graphs and provide a positive answer to the asymptotic normality question posed by Zhang, Peng, and Chen (Adv. in Appl. Math. 127 (2021), 102175).

This paper establishes essentially optimal asymptotic bounds for Kolmogorov widths under holomorphic mappings between complex Banach spaces. Given a compact parameter set whose Kolmogorov widths decay algebraically with rate s, we prove that the widths of its image under a holomorphic mapping decay algebraically with every rate t<s, thereby answering an open question raised by Cohen and DeVore. As an application, we obtain a sharp characterization of the approximability of solution manifolds associated with inf-sup stable parametrized PDEs. We also construct an explicit example showing that the arbitrarily small loss in the algebraic decay exponent is unavoidable. Finally, we provide similar characterization of asymptotic bounds for Kolmogorov widths in the exponentially decaying regime. Our analysis involves multilinear Taylor expansion in Banach spaces and a novel block dyadic expansion-truncation technique.

This paper presents a novel higher-order multi-scale (HOMS) computational framework for efficient, high-accuracy, and low-cost simulation of nonlinear hygro-thermo-mechanical (H-T-M) coupling problems in heterogeneous structures. The inherent nonlinearity in the investigated model stems primarily from temperature- or moisture-dependent material properties, and this model also accounts for temperature-dependent internal heat source and moisture sink terms induced by exothermic, moisture-consuming chemical reactions (e.g., hydration). The main contributions of this work are as follows. First, a high-accuracy multi-scale asymptotic model incorporating higher-order correction terms is constructed for nonlinear H-T-M coupling problems in heterogeneous structures with highly spatial inhomogeneity, using the multi-scale asymptotic approach together with Taylor series expansions. Second, rigorous error estimates in both point-wise and integral senses are derived for the multi-scale asymptotic solutions, which theoretically demonstrate the necessity and superiority of the proposed HOMS method. Third, an efficient two-stage numerical algorithm with off-line and on-line stages is developed, based on finite difference and finite element methods, and its convergence is also proved rigorously. Finally, two- and three-dimensional numerical experiments are performed to assess the computational performance of the proposed HOMS approach, showing excellent numerical accuracy and robustness with low computational overhead.

The analytic properties of the Dirichlet-to-Neumann operator reflect the geometry of the underlying manifold. As a guiding example of this relationship, consider the Euclidean unit ball. There, the Dirichlet-to-Neumann operator has an explicitly computable spectrum, consisting of the nonnegative integers with multiplicity, and commutes with the boundary Laplacian. In this paper, we study the extent to which these properties distinguish the ball among various classes of manifolds. We show that, in every dimension, Euclidean balls are uniquely determined by their Steklov spectrum among bounded Euclidean domains with smooth and possibly disconnected boundary, answering a well-known open question. We then classify all smooth compact connected oriented surfaces with nonempty boundary whose Dirichlet-to-Neumann operator commutes with the boundary Laplacian. Finally, in dimensions three and above, we prove that, for smooth metrics conformal to the Euclidean ball, the Dirichlet-to-Neumann operator commutes with the Laplacian on the round sphere if and only if the conformal factor is radial.

We study continuous-time projection neural networks for variational inequalities on closed convex sets. A positive definite matrix that depends on the state preconditions the operator, and its inverse defines the projection metric. Existing convergence analyses of this flow cover Hessian-generated inverse metrics and state-dependent scalar metrics. In the first case, a Bregman distance eliminates metric-derivative terms. We treat a matrix metric whose inverse is of neither kind. We prove joint regularity of the projection in its argument and metric. Under common spectral bounds, the Euclidean Lipschitz estimate improves from the squared bound to the bound itself. For Lipschitz strongly monotone operators and Lipschitz metrics, we prove local exponential convergence with explicit rate and radius. On compact feasible sets, an explicit metric-variation bound yields global exponential convergence. A twice continuously differentiable, uniformly positive definite metric and a strongly monotone linear operator produce an annulus of periodic orbits. The inverse metric violates Hessian integrability throughout the annulus. Deterministic computations confirm the analytic formulas and quantify slack in both convergence certificates.

We study the maximum size of an independent set in the graph of a simplicial sphere. Let $\beta(d,n)$ denote this maximum over all simplicial $(d-1)$-spheres on $n$ vertices, and let $\alpha(d,n)$ denote the maximum restricted to flag $(d-1)$-spheres. For every fixed $d\geq4$, we prove $\beta(d,n)=n-\Theta(n^{1/\lfloor d/2\rfloor})$. For flag spheres, we show $\alpha(d,n)\geq n-4\sqrt n+O(1)$ for all $d\geq4$ and determine the correct asymptotic order $\alpha(d,n)=n-\Theta(\sqrt n)$ for dimensions $d=4,5$. We also investigate the independence sets of Bier spheres and show that, in contrast to our other results, for this very large family of spheres, the independence number cannot be larger than $\left\lfloor\frac{n}{2}\right\rfloor.$

We present an open-source software package that implements a provably convergent Benders-type decomposition algorithm for multistage stochastic integer programs. In addition to standard cut families, such as Benders, strengthened Benders, and Lagrangian cuts, the algorithm incorporates rectified linear unit (ReLU) cuts, which provide convergence guarantees for general mixed-integer state variables. However, the dual problems used to generate these cuts often admit multiple optimal solutions. Although each solution yields a valid cut that separates the incumbent, the resulting cuts can differ in how well they approximate the subproblem cost. To strengthen these cuts, our package implements and evaluates two cut-selection strategies based on normalization and regularization of the dual problem. We also incorporate an alternating-cut criterion that uses cheaper Benders cuts when they are effective and invokes more expensive tight cuts only when necessary. Computational experiments on four classes of multistage stochastic integer programs benchmark these methods and provide insights on how problem structure affects their practical performance.

Quasi-static electroporoelasticity couples Maxwell's equations with Biot's poroelasticity through electrokinetic coupling between the electric field and the pressure gradient. By introducing the total pressure, the electroporoelasticity equations are reformulated as a five-field system to address poroelastic locking in the nearly incompressible regime. For the resulting five-field system, a monolithic weak formulation is derived, together with a continuous stability estimate. A second-order backward differentiation formula (BDF2) time discretization and a mixed finite- element spatial discretization are then introduced, yieling to a fully discrete monolithic scheme. Building on this scheme, we develop an iterative decoupling method that alternates between an electromagnetic subproblem and a poroelastic subproblem, and prove its geometric convergence to the monolithic solution with an explicit mesh-independent contraction factor under the physical coupling condition. An algebraically equivalent reduced form is also presented, in which the electromagnetic block is solved only once per time step, while the poroelastic block is solved iteratively with electric-field correction updates obtained from the pressure-gradient feedback. Numerical experiments verify the theoretical predictions and demonstrate the locking-free performance.

We compare the periodic orbit types of the Lorenz, horseshoe, and Rössler systems through their associated templates. Lorenz links are carried by the Lorenz template, while horseshoe links are carried by the horseshoe template. For the standard template model considered here, the Rössler template is identified, up to inversion symmetry, with the template of the horseshoe mechanism. Thus, in this paper, Rössler links and horseshoe links are treated as belonging to the same template class. We show that the relationship between Lorenz links and horseshoe/Rössler links has two complementary sides. First, we prove that the overlap between the two families is nontrivial by constructing infinite families of links which can be embedded in both templates. This verifies, for these families, a conjecture stated by Kofman that horseshoe links should also be Lorenz links. On the other hand, we prove that the two families are far from being the same. Holmes and Williams showed that many Lorenz torus knots cannot be embedded in the horseshoe template: if the torus knot \(T(p,q)\), with \(p<q\), is a horseshoe knot, then \(3p\leq 2q\). We show that this phenomenon is much broader. We extend the Holmes–Williams obstruction from torus knots to torus links, and we construct infinitely many hyperbolic Lorenz knots and links, as well as infinitely many satellite Lorenz knots and links, which cannot be embedded in the horseshoe template. Consequently, these examples are Lorenz links which are not horseshoe links and hence not Rössler links.

The Yau-Yau filter reformulates nonlinear state estimation as probability-density propagation governed by the Forward Kolmogorov equation (FKE). Applying it to target tracking, however, requires efficient FKE approximation on a finite computational domain. This paper proposes a Recentered-Domain Yau-Yau Filter (RD-YYF), which solves the FKE within a fixed-size local window centered at the latest state estimate. This design concentrates numerical resolution near the dominant posterior density. Offline, physics-informed neural networks (PINNs) generate FKE solution snapshots, while principal component analysis constructs a low-dimensional representation of density evolution. A lightweight residual surrogate maps the initial-condition coefficients and domain center to the terminal-solution coefficients. Online, the pretrained surrogate predicts density evolution within the recentered window, followed by observation update and state estimation. Experiments on two geometrically constrained target-tracking examples show that RD-YYF achieves lower tracking errors than the extended Kalman filter (EKF) and particle filter (PF), while retaining efficient per-timestep inference. Ablation results indicate that domain recentering improves density approximation in high-probability regions and accelerates offline PINN convergence. These results demonstrate the potential of RD-YYF for efficient nonlinear target tracking

In this paper, we prove the existence of Lyapunov exponents for the geodesic flow on complete Riemannian manifolds without conjugate points whose sectional curvature is bounded below. Under suitable additional curvature assumptions, we establish Ruelle's inequality. Furthermore, assuming the same geometric conditions, we obtain the Pesin entropy formula for $C^1$-Hölder geodesic flows on finite-volume manifolds without conjugate points.

We develop and analyze a family of divergence-free kernel interpolation methods for tangential vector fields on the unit sphere. Starting from scalar radial kernels in Euclidean space, we construct tangent-valued, surface divergence-free matrix kernels without repeatedly applying surface differential operators. The construction separates the geometric enforcement of the divergence-free constraint from the choice of scalar generator and admits lower-order variants with reduced regularity requirements compared with classical potential-based methods. Using vector spherical harmonics, we derive explicit kernel representations and characterize their Fourier multipliers. We also introduce an inverse Laplace-Beltrami construction that preserves the multipliers of the underlying scalar zonal kernel. For interpolation at scattered nodes, we establish a lower bound for the smallest eigenvalue of the interpolation matrix and derive pointwise and Sobolev error estimates, including superconvergence for targets smoother than the native space. Numerical experiments corroborate the theoretical convergence and stability results, while additional examples on nonspherical surfaces illustrate the applicability of the kernel formula beyond the sphere.

Every closed orientable 3-manifold admits both a Heegaard splitting and a coorientable codimension-one foliation. We address the foliation realization problem: constructing a foliation directly from a Heegaard diagram of arbitrary genus. Using Gabai's sutured manifold theory, we introduce the notion of a meridional sutured handlebody and prove that every such handlebody decomposes, via basic sutured decomposition operations, into a Reeb component together with product disks. We then present a three-step construction: (i) building two meridional sutured handlebodies from a given Heegaard diagram, (ii) gluing them along compatible disk regions by reversing disk decompositions, and (iii) filling the remaining cavities with a meridional sutured handlebody and some Reeb components. The construction applies to Heegaard diagrams of any genus.

Recently, novel integrability techniques, and most notably Birkhoff coordinates followed by an explicit formula, have been introduced to solve the Benjamin–Ono (BO) equation, leading to major theoretical and computational advances. We propose to perturb this explicit BO formula via a novel splitting method for numerically solving a class of PDEs, which includes the Intermediate Long Wave (ILW) equation, as well as other quasilinear equations that are not necessarily integrable. Using the Birkhoff coordinates of BO we establish first-order convergence in the $H^s$-norm ($s \ge 0$) while requiring only one additional derivative on the initial data $u_0 \in H^{s+1}_{0}$, significantly improving upon the regularity requirements of typical splitting methods for nonlinear dispersive equations. Furthermore, we prove that in the deep-water limit, the scheme for the ILW equation converges to the BO solution. Computational advantages of these schemes are shown in simulations: unlike classical splitting methods, our approach does not require a restrictive quadratic time step condition to nearly preserve energy over long time scales, thus rendering efficient and accurate long-time simulations feasible. As an application, we numerically explore the soliton resolution conjecture for both the ILW and a KdV–BO equation.

Iterate convergence of Bregman proximal methods (BPMs) has long remained open, especially for nonconvex objectives. Recently, \citet{chen2026skl} made progress by establishing iterate convergence for a BPM via the so-called scaled Kurdyka-Łojasiewicz (SKŁ) property, but only for the Shannon entropy kernel and linearly constrained problems. In this paper, we develop a unified iterate convergence framework that applies to a broad group of kernels and composite objective functions. Our approach extends the analytical tools in \cite{chen2026skl}, in particular the SKŁ property, which plays a central role in ensuring convergence of the generated sequences. By introducing kernel-dependent parameterization functions, we show that the extended SKŁ property holds for all continuous subanalytic functions, particularly when the kernel has a closed domain. We then verify that the assumptions of the framework are satisfied by standard BPMs under mild regularity conditions, thereby establishing their iterate convergence for a wide range of objective functions. Furthermore, based on the parameterization functions, we show that the continuous-time BPM (mirror flow) converges to a stationary point for o-minimal definable objective functions, yielding the first trajectory convergence result for mirror flow without imposing convexity assumptions on the objective function or isolation assumptions on stationary points. Taken together, these discrete- and continuous-time convergence results provide a unified trajectory convergence theory for BPMs.

Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$, where $0 \in \text{int} G$ and $G \subset \text{int} K$. Given $\alpha > 0$, the Hilbert packing number $M_H(G, K; \alpha)$ is the maximum cardinality of a set of points in $G$, each pair of which is separated by a distance of at least $\alpha$ in the Hilbert geometry defined by $K$. The Hilbert convexified packing number $\widehat{M}_H(G, K; \alpha)$ is the maximum length of a sequence of points in $G$, such that each point is separated by distance at least $\alpha$ from the convex hull of its predecessors. We prove a dimension-free primal-polar bound for convexified packing in Hilbert geometry. Letting $G^\circ$ and $K^\circ$ denote the polar bodies, we show that there exist absolute constants $C, c > 0$ such that, for every $\alpha > 0$, \[ \widehat{M}_H(G, K; \alpha) ~ \leq ~ C \cdot \widehat{M}_H(K^\circ, G^\circ; c\alpha)^2 M_H(K^\circ, G^\circ; c\alpha). \] As a direct corollary, we have \[ \widehat{M}_H(G, K; \alpha) ~ \leq ~ C \cdot M_H(K^\circ, G^\circ; c\alpha)^3. \] Thus, the primal convexified packing number is bounded by a fixed polynomial in the ordinary packing number for the reversed polar bodies, with absolute constants independent of the dimension. This is motivated by the duality conjecture for packing and covering numbers, which relates covering $G$ by $K$ to covering $K^\circ$ by $G^\circ$. Our results represent a key step in extending the work of Artstein, Milman, Szarek, and Tomczak-Jaegermann from normed spaces to Hilbert geometries.

Quantum state tomography (QST) has attracted considerable attention due to its fundamental role in quantum information processing. In this paper, we develop a unified theoretical framework for analyzing the sample complexity of structured QST under noisy observations arising from state preparation noise, measurement noise, and finite-shot statistical noise. The proposed framework applies to a broad family of structured quantum-state classes, including general mixed states, sparse states, low-rank states, matrix product states (MPSs), matrix product operators (MPOs), projected entangled-pair states (PEPSs), and projected entangled-pair operators (PEPOs), while further introducing physically consistent structured models—including low-rank and sparse states, low-rank MPOs (LR-MPOs), and low-rank PEPOs (LR-PEPOs)—that simultaneously exploit low-dimensional structures and preserve the physical constraint. Within this framework, we derive unified non-asymptotic sample complexity guarantees for two constrained least-squares estimators under noisy observations: a noise-aware estimator that incorporates the calibrated noise model and a noise-unaware estimator based on the ideal Born measurement model. For the noise-aware estimator, we derive unified trace-norm recovery guarantees that explicitly characterize the dependence of the sample complexity on three fundamental quantities: the complexity of the underlying structured state class, the complexity of the measurement ensemble, and the state preparation and measurement noise levels. For the noise-unaware estimator, we establish a unified non-asymptotic recovery guarantee consisting of a statistical error term and an additional deterministic bias term arising from the mismatch between the assumed reconstruction model and the noisy observation process.

Nguyen~\cite{Nguyen2025} recently proved that every $\{P_5,C_5\}$-free graph $G$ satisfies $\chi(G)\leq \omega(G)^{40}$. Building on his framework, we introduce two refinements, namely a sharper cutset decomposition using the $C_5$-free condition and an improved density-increment argument. These yield a polynomial $\chi$-binding function with exponent $24$, improving the previous bound of $40$.

We introduce a notion of completeness for two-dimensional second-gradient elastic continua and propose a microstructural route toward its synthesis. A continuum is said to be complete if the Hessian of the stored energy with respect to the second-gradient variable is locally positive definite, so that every nonzero admissible increment of the placement second gradient is quadratically controlled in the highest-order part of the energy about each configuration. Starting from the Principle of Virtual Work, we derive the constitutive relations, equilibrium equations, and admissible boundary interactions for a broad class of fibrous second-gradient continua whose stored energies depend on fiber stretch, stretch gradients, and curvature. The theory is then applied to several continua motivated by pantographic microstructures. Classical pantographic sheets are shown to be incomplete, while bi-pantographic fabrics enlarge the class of components of the second gradient detected by the energy but remain incomplete. Finally, we formulate a tri-pantographic continuum associated with a proposed three-family architecture and prove that it is complete. The examples illustrate how microstructural architecture influences both the completeness properties of an effective continuum and the pointwise form of its higher-order boundary interactions.

Let $p$ be an odd prime number and $\zeta_p$ a primitive $p$-th root of unity. By computing special arithmetic elements modulo prime ideals, we have investigated the Iwasawa invariants and exceptional pairs $(p,\chi \omega_p^k)$ of $\mathbf{Q}(\sqrt{d},\zeta_p)$. Here $\chi$ denotes the Dirichlet character associated to $\mathbf{Q}(\sqrt{d})$, and $\omega_p$ the Teichmüller character at $p$. First, we determine $\varLambda$-isomorphism classes of Iwasawa modules, and describe structural differences between exceptional and non-exceptional pairs in terms of unramified extensions outside $p$. Next, we report 15 new exceptional pairs in the range $|d|<200$ (resp.~$|d|<10$) and $p <2,\!000,\!000$ (resp.~$p <30,\!000,\!000$), including $(p,k,d)=(28,\!679,\!999,\;2,\!284,\!521,-8)$ for which the $\chi \omega_p^{1-k}$-part of the Iwasawa $\lambda$-invariant is equal to two.

Using the theory of stochastic integration in duals of nuclear spaces with respect to cylindrical martingale-valued measures, a vector-valued Itô formula is proved for generalized Itô processes defined with respect to these integrals. The abstract result is then applied to prove an Itô formula for Lévy-Itô processes defined with respect to Lévy processes taking values in the dual of a reflexive nuclear space.