Papers

Nonlinear Dynamic Inversion (NDI) has become a standard methodology in flight control law design. It offers an intuitive approach to decouple commanded variable responses, handle system nonlinearities, and adapt to operating conditions. NDI also comes with a well-structured architecture that reduces design effort by addressing various functional aspects in separate components, and that allows straightforward integration of extended capabilities, such as envelope protection, control saturation handling, and compensating for faults or damage. A development that has resulted in considerable renewed attention is the use of (angular) acceleration sensors to partially replace inverse model equations. Known as incremental NDI, or INDI, this development offers reduced sensitivity to modeling errors and lower control law complexity. Incremental NDI, however, lacks useful design degrees of freedom in the previously present inverse model equations and underlying feedback signal synthesis, and comes with pitfalls in design aspects like control allocation, disturbance rejection, and inter-disciplinary couplings. This has given rise to recently developed hybrid and mathematically restructured approaches. The aim of this article is to give an up-to-date, structured overview of the various evolved forms of NDI from conceptual, historical, architectural, and mathematical perspectives. It hereby intends to provide useful considerations for future flight control law developments by reviewing its various forms and potentials from methodological, design, and application points of view.

Let $\beta$ be a coset incidence system and fix a type $j$. We use the orbits of the group of $\beta$ on pairs of distinct $j$-elements to define a Coxeter graph. The action on the $j$-elements induces an action on the corresponding Coxeter group, so the twisting construction for coset incidence systems can be applied. The resulting coset geometry, called the $j$-diagonals twisting, is always a regular hypertope if $\beta$ is a regular hypertope. Using this, we show that finite regular hypertope whose diagram is a tree with all but one of the labels equal to four always exist. We also define an extension operation that combines this Coxeter graph with another given Coxeter graph. For regular polytopes, these constructions recover the twisting extensions of McMullen and Schulte.

Advances in neural theorem provers have been impressive, but the successes obscure a broader vision of what AI can do for mathematics and how mathematicians can engage with AI. This essay advances a more expansive and optimistic point of view.

This is the text of my diploma thesis on Euclidean Rings from 1989, along with an English translation. Its main result is the generalization of Lenstra's idead of exceptional sequences to k-stage Euclidean rings.

In this work, we study the oracle complexity of finding an $\epsilon$-stationary point for nonconvex-strongly-convex (NC-SC) bilevel optimization using only first-order oracles. Existing methods achieving the best-known complexity guarantees typically rely on double-loop, penalty-based procedures. We propose a novel single-loop algorithm based on a constrained reformulation in which lower-level stationarity is imposed as a constraint. Specifically, we construct a regularized Lagrangian by introducing a quadratic regularizer and restricting the dual variable to a bounded domain, and then apply Smoothed Gradient Descent Ascent [Zhang et al., 2020], with Hessian-vector products approximated via finite differences of gradients. We refer to the resulting deterministic and stochastic algorithms as SGHA and Stoc-SGHA, respectively. In the deterministic setting, SGHA achieves an oracle complexity of $O(\bar{\kappa}_y^{5}\epsilon^{-2})$, where $\bar{\kappa}_y$ denotes the relevant condition number. In the stochastic setting, Stoc-SGHA achieves an oracle complexity of $O\left(\bar{\kappa}_y^{17}\epsilon^{-6}\rho^{-3}\right)$ with probability at least $1-\rho$ for any $\rho\in(0,1)$, and an oracle complexity of $O\left(\bar{\kappa}_y^{17}\epsilon^{-6}\right)$ in expectation under an additional bounded-iterate assumption. Moreover, under an additional stochastic smoothness assumption imposed only on the lower-level objective, the stochastic oracle complexity of Stoc-SGHA improves to $O\left(\bar{\kappa}_y^{11}\epsilon^{-4}\rho^{-2}\right)$ with high probability and $O\left(\bar{\kappa}_y^{11}\epsilon^{-4}\right)$ in expectation, matching the $\epsilon$-dependence of the lower bounds.

We prove existence, uniqueness, and stability for fully coupled McKean-Vlasov forward-backward SDEs with jumps whose drift, diffusion, jump, and driver coefficients may depend Lipschitz-continuously, in quadratic Wasserstein distance, on the joint law of the full solution tuple $\Theta=(X,Y,Z,U)$: forward state, backward variable, Brownian integrand, and $L^2(\nu)$-valued jump integrand. The terminal function may depend Lipschitz-continuously on $X_T$ and its law. The system is driven by a Brownian motion and an independent compensated Poisson random measure with arbitrary $\sigma$-finite intensity, so infinite jump activity is admitted. Both the Lipschitz and monotonicity hypotheses are imposed only along diagonal tuple-law pairs $(\Theta,\mathrm{Law}(\Theta))$; we show that expected diagonal monotonicity is strictly weaker than pointwise monotonicity. Under a jump-extended $G$-monotonicity condition we establish an a priori continuous-dependence estimate, uniqueness, and existence on every prescribed finite horizon, by monotone continuation in the coupling strength from a small-coupling base case. A mean-field dealer-market example realises the $U$-law dependence non-perturbatively: its law interaction is monotone at every interaction strength, and its mark measure has infinite activity.

We give a full characterization of the fixed points of Branching Brownian motion with critical and supercritical drifts under no additional assumptions besides it being locally finite almost surely. In particular, we do not assume finite intensity (Kabluchko, 2012) or the finite top particle (Chen-Garban-Shekhar, 2023) conditions. We also give a full characterization of the domain of attraction of the fixed points of BBM.

The sewing operation is the fundamental algebraic structure of the (infinite-dimensional) Segal moduli spaces as the conformal cobordism category. We prove a universal property of local $\mathbb{R}_+$-extensions of the sewing operation, characterizing them by one real parameter: their central charge. Our result is an analogue of “Mumford's theorem,” that appeared in Segal's monograph [Seg88, Seg04], classifying complex extensions, also known as one-dimensional modular functors. Algebraically, the real extensions are characterized by real-valued cocycles on pairs of surfaces under sewing. We identify the disk-disk cocycle as the universal Liouville action, also known as loop Loewner energy. The real determinant line bundle encodes the trace anomaly of euclidean two-dimensional conformal field theories (CFT) and the restriction function of Schramm-Loewner evolution (SLE) loop measures. It serves as the guiding example in our setup and proof strategy, providing the universal $\mathbb{R}_+$-extension. Our results thus give a mathematical explanation for the appearance of the central charge in CFT and SLE, and prove the conjectured uniqueness of Brownian loop measures as the canonical restriction function for SLE.

We introduce a logarithmic velocity alignment model for collective motion on Riemannian manifolds. The interaction law is defined by the covariant time derivative of logarithmic displacement vectors and therefore provides an intrinsic geometric analogue of pairwise velocity-difference coupling. Under two-sided nonpositive sectional-curvature bounds and a compatibility condition between the interaction kernel and the curvature-induced growth of logarithmic terms, we derive a kinetic-energy dissipation estimate and prove interaction-weighted asymptotic velocity alignment, assuming that the logarithmic interactions remain globally well-defined and that transported velocity discrepancies have uniformly controlled time variation. In hyperbolic space, both assumptions are verified directly from the geometry and the energy estimates, yielding an unconditional interaction-weighted alignment result within the stated class of global solutions.

We study representations of integers using $k$-generalized Fibonacci numbers. For $k\geq 2$, we first consider signed representations of zero with coefficients in $\{-1,0,1\}$ and give a recursive description of their number. The resulting counting sequences satisfy linear recurrences whose characteristic polynomials are determined explicitly. In the Fibonacci and Tribonacci cases, these recurrences reveal unexpected connections between the corresponding representation counts and Tribonacci and Fibonacci sequences, respectively. We then consider representations with coefficients in $\{0,1\}$ in the Tribonacci case. Using a random inhomogeneous Tribonacci recurrence, we construct a binary-tree model in which representation multiplicities are encoded by a family of polynomials satisfying the product formula $Q_n(x)=\prod_{k=2}^{n-1}(1+x^{T_k})$. This product admits a probabilistic interpretation in terms of weighted Bernoulli sums. After normalization by $T_n$, these sums converge in distribution to the Bernoulli convolution $\sum_{j=1}^{\infty}\varepsilon_j\rho^{-j}$, where $\rho$ is the Tribonacci constant and the $\varepsilon_j$ are independent Bernoulli random variables. The limiting distribution satisfies a natural self-similarity relation, linking the representation problem to self-similar measures associated with Tribonacci scaling.

We present a variational–adjoint interpretation of quantum mechanics in which the interaction between forward and adjoint wavefunctions defines a general sensitivity kernel of the system. This Fréchet interaction density quantifies how perturbations in the wavefunction (or system parameters) influence a chosen observable. The familiar Born probability density appears as a special case when the adjoint wavefunction is chosen as the complex conjugate of the forward wavefunction, for which the interaction density becomes real and non-negative. Within this framework, probability is a particular positive-definite form of sensitivity described by the Fréchet interaction density. In general, the variational–adjoint interpretation also produces Fréchet sensitivity kernels associated with other quantum observables, including momentum, energy, and spin. This suggests that the Born probability density belongs to a broader class of Fréchet sensitivity kernels associated with quantum observables. The proposed interpretation also provides a connection with time-symmetric interpretations of quantum mechanics and possible future applications in quantum control and quantum metrology.

Let $G$ be a finite simple graph with at least one edge. We prove the sharp vanishing theorem \[ \kappa_{\min}^{\mathrm{LLY}}(G)>\frac12 \quad\Longrightarrow\quad \PathH_1(G;\R)=0. \] Equivalently, nonzero first GLMY path homology forces an edge of Lin–Lu–Yau curvature at most $1/2$. The threshold $1/2$ is sharp and is attained by $C_5$. The proof combines the cycle-space description of first GLMY path homology with the limit-free Laplacian characterization of Lin–Lu–Yau curvature. As a secondary consequence of the curvature-preserving universal-cover method, we prove that if $G$ is connected and $\kappa_{\min}^{\mathrm{LLY}}(G)>0$, then $\pi_1(\Xshort{5}(G),o)$ is finite, where $\Xshort{5}(G)$ is obtained by filling every simple cycle of length at most five. Equivalently, the normal subgroup generated by based simple $5$-cycle loops has finite index in $\pi_1^{\mathrm{GLMY}}(G,o)$. In higher degrees the situation is different: for each integer $r\geq1$, the Cartesian product $T_r=C_5^{\square r}$ has curvature $1/(2r)$ on every edge and, for every field $\F$, \[ \PathH_p(T_r;\F)\cong\F^{\binom rp}\qquad(0\leq p\leq r), \] so strict positivity of Lin–Lu–Yau curvature does not force higher-dimensional GLMY path homology to vanish.

We study the time-asymptotic stability of solutions to the inflow problem for the one-dimensional Navier–Stokes–Fourier system on the half-line. We consider the most generic wave pattern: the superposition of a degenerate boundary layer, a rarefaction, a viscous contact wave, and a viscous shock. More precisely, if the boundary data belongs to the subsonic region, and the initial perturbation and strengths of the boundary layer, viscous contact wave, and viscous shock are sufficiently small, then the solution to the inflow problem converges to the corresponding superposition, up to a time-dependent shift for a shock. The rarefaction wave, however, is allowed to have arbitrarily large strength. To control the viscous shock, we employ the method of $a$-contraction with shifts. A notable feature of our analysis is that this method can be applied even when the rarefaction wave has large amplitude. In particular, this resolves, in a generic setting, the open problem of the stability of inflow wave patterns containing a viscous shock for Navier–Stokes–Fourier system.

Considering an $n$-dimensional compact Riemannian manifold with a boundary that satisfies a Serrin-type problem, we prove sharp upper and lower bounds for the area of such a boundary. Then, we present the main result of this paper: a divergence formula for a special vector field on a given Riemannian manifold. This divergence formula is closely related to sub-static manifolds and related metrics, e.g., $V$-static, static, and electrostatic manifolds. We show Minkowski-type inequalities for $V$-sub-static manifolds under different Neumann boundary conditions. Moreover, we prove an area-charge inequality for compact (and noncompact) electrostatic manifolds. The main application of the divergence formula presented in this work generalizes the classical result of Boucher–Gibbons–Horowitz: we prove that an asymptotically hyperbolic space of a static spacetime satisfying the null convergence condition must be the hyperbolic space.

Direct policy optimization is widely used in reinforcement learning and control, but generally leads to nonconvex optimization problems. For state-feedback $H_\infty$ control, the policy objective is also nonsmooth, despite possessing a benign landscape whose hidden convexity can be revealed by the recently developed extended convex lifting framework. Motivated by recent advances in hidden convex optimization, we study zeroth-order optimization of nonsmooth, nonconvex problems admitting a convex lifting. We propose a zeroth-order proximal point algorithm: An inexact proximal-point outer loop constructs strongly convex subproblems, while an inner loop approximately solves each subproblem using only function evaluations. With probability at least $1-\delta$, our proposed algorithm returns an $\epsilon$-optimal solution using $\widetilde{O}\left(d\epsilon^{-3}\right)$ function evaluations, while all iterates remain feasible without explicit projection. Finally, we verify that the assumptions underlying our analysis hold for discrete-time state-feedback $H_\infty$ policy optimization, yielding an oracle complexity of $\widetilde{O}\left(n_u n_x\epsilon^{-3}\right)$ for attaining a prescribed objective value gap, where $n_u\times n_x$ is the dimension of the feedback gain to be optimized over.

We study the class of transversal field models with pair interactions, including the quantum Curie-Weiss and the transversal field Ising model, as well as quantum spin glasses based on the Sherrington-Kirkpatrick (SK) and Edwards-Anderson models. Our main result is a general correlation inequality based on a Gaussian convolution estimate, which is derived from an extension of the Ding-Song-Sun inequality and an application of Brascamp-Lieb convexity techniques. The results apply for sufficiently strong transversal fields and, for quantum spin glasses, yield a high-field regime without replica order. Combined with the previously established low-field replica-symmetry-breaking regime, this gives distinct low- and high-field overlap phases. As a corollary of the logarithmic Sobolev inequalities for the underlying functional integrals derived here, we show the concentration of the self- and replica-overlap, and determine the high-field asymptotics of the pressure in the quantum SK model.

We give a necessary and sufficient triangular condition characterizing the toroidal pseudodifferential operators on $\mathbb T^d$ that preserve the positive-frequency cone $\mathbb N_0^d$ and hence act on holomorphic boundary values on the polydisk. For symbols of type $(1,0)$, we prove boundedness on holomorphic Besov and Triebel–Lizorkin spaces throughout the quasi-Banach range. For $S^m_{\rho,\delta}$, $0\leq\delta<\rho\leq1$, we obtain the critical loss $d(1-\rho)|1/p-1/2|$, together with endpoint estimates. A positive-cone oscillatory multiplier proves sharpness of the loss and necessity of the Besov endpoint condition $q\leq t$. As an application, we prove well-posedness for a first-order holomorphic differential operator on these scales.

We study the disc counting statistics of the real Ginibre ensemble, whose spectrum consists of real and non-real eigenvalues, the latter occurring in complex-conjugate pairs. As the matrix dimension increases, we derive the asymptotic behaviour of the joint cumulants of the numbers of real and non-real eigenvalues contained in a centred disc in the bulk, edge, and exterior regimes of the circular law. As consequences, we establish a joint central limit theorem and obtain the conjectured asymptotics for the number variance. Furthermore, our results reveal a universality phenomenon: although the fluctuations of the real and non-real eigenvalues separately retain dependence on the symmetry class, their combined fluctuations exhibit the same limiting behaviour as in the previously established complex and symplectic Ginibre ensembles.

The variance of nonparametric estimators is typically insensitive to the regularity of the object being estimated. We establish such a property for the spectra of graph Laplacian matrices at a fixed bandwidth $h>0$. Specifically, given $n$ i.i.d. samples from a probability measure $\mu$ on a Polish metric space, we compare the eigenvalues of the empirical weighted Laplacian operator $\Delta_{\mu_n}^h$ to those of the population counterpart $\Delta_\mu^h$ under a spectral gap condition, bounding the relative error by $1/\sqrt{nv_\mu(h)}$ for eigenvalues of order smaller than $h^{-2}$, where $v_\mu(h)$ is the smallest mass of a ball of radius $h$. This bound requires very weak regularity conditions on $\mu$: it is satisfied if $\mu$ belongs to the class of coarse PI measures that we introduce. This class contains measures on metric graphs, spaces with sufficiently regular boundaries, corners, or branch points, together with discretizations or thickenings of these at scale $O(h)$. Even for measures having densities of regularity $s>2$ on manifolds (the only known case so far), our bound improves on the state-of-the-art by shaving off logarithmic factors.

An explicit sequence of tetrahedra generated by the longest-edge bisection algorithm is shown to degenerate. The example violates shape regularity and both the minimum- and maximum-angle conditions, demonstrating that arbitrary tie-breaking among longest edges does not guarantee nondegeneration.

In this paper, we prove that the fluctuations of the graph distance and the effective resistance on the trace of a random walk in four and five dimensions converge in distribution to a stable law. In previous work, the first and second authors proved that the corresponding fluctuations converge to a Gaussian distribution in dimensions six and higher. Taken together, these results reveal a phase transition between dimensions five and six. Our proof develops a novel coupling with long range percolation, and we expect this technique to find applications in a broad class of related models.

We derive energy scaling laws for thin elastic sheets with topological defects — disclinations and dislocations — for a fully nonlinear 3D model. For disclinations, the scaling laws are tight in the thickness parameter, and improve upon previous results by applying simultaneously to positive and negative disclinations (e-cones) and by giving an explicit dependence on the defect parameter; this latter dependence is, however, still not tight. For thin bodies with dislocations, these are, to the best of our knowledge, the first rigorous bounds for models of finite thickness, are tight when the Burgers vector is not large with respect to the thickness, and relate to a well-known conjecture from the physics literature about the scaling. A main tool is modeling these bodies in the framework of non-Euclidean elasticity, as bodies with a curl-free pre-strain; the curl-freeness allows us to obtain geometric rigidity estimates for the lower bounds.

Let $A$ be a finite-dimensional representation-finite algebra over an algebraically closed field $k$, and let $M$ be a finite-dimensional $A$-module such that $\operatorname{End}_A(M)$ has finite global dimension. We prove that $\operatorname{End}_A(M)$ has Cartan determinant one if $A$ has finite global dimension or $\operatorname{char}k\neq2$; in particular, the Cartan determinant conjecture holds for representation-finite algebras over an algebraically closed field.

We construct two examples clarifying the position of enriched $P$-contractions among related classes of mappings on normed spaces. Our first example is a $P$-contraction operator on a finite subset of a normed linear space that is not an enriched contraction. In the second example we have a continuous operator on the whole of a normed linear space that is a $P$-contraction but not an enriched contraction. These examples answer in negative a question raised in \cite{AltunHancerAtes2024} asking whether every enriched $P$-contraction is an enriched contraction.

Let \(\DCPO\) be the category of dcpos and Scott-continuous maps, and let \(\Rtwo\) be the reflective hull of the two-element chain \(2\). We prove that \(\Rtwo\) is the least non-discrete proper reflective full subcategory of \(\DCPO\). It is closed under limits and Skula-closed sub-dcpos and contains every sober dcpo. We show that \(\Rtwo\) is the maximal \(\Gamma\)-faithful full subcategory of \(\DCPO\) that strictly contains weakly dominated dcpos. Finally, we give the concrete construction of \(2\)-reflection internally by a transfinite iteration and a criterion for reflective full subcategories of \(\DCPO\), analogous to the Keimel–Lawson conditions for \(T_0\) topological spaces.

We study the higher-order flows of the modified Burgers' equation generated by its standard recursion operator. In particular, we show that the second-order positive flow takes the form of a non-local integro-differential equation. After reducing the equation to a traveling-wave form, we investigate its asymptotic energy behavior. The resulting analysis shows that the non-local energy flux prevents the formation of both topological kink and bell-shaped solitary waves. We show that the non-local dissipation permanently breaks the asymptotic equilibrium between nonlinear convection and linear dispersion, forcing an irreversible energy imbalance. Our analysis shows that the solitary waves are unstable. The non-local term causes continuous energy loss through radiation. As a result, the traveling wave cannot keep a stable and symmetric shape.

This paper explores the algebraic invariants and ergodic behavior of the finite-index inclusions forming commuting squares of $\text{II}_1$ factors. We begin by establishing how regularity, intermediate-subfactor lattices, and Weyl groups transfer across the commuting square under relative commutant and irreducibility conditions. Furthermore, we provide a complete characterization, via a novel factorization theorem, of the unitary normalizers of the upper inclusion when the lower inclusion is regular. Applying these findings to crossed-product inclusions induced by discrete groups, we introduce a relative eigenbasis property that preserves regular inclusions. Finally, we contrast this by proving that a relative weak mixing condition on the group action forces the crossed-product inclusion to be singular.

Bond-based peridynamics provides a non-local framework for modelling fracture without requiring spatial derivatives of the displacement field. However, when spatially varying horizons are used together with non-uniform discretisations, the classical single-horizon bond-based peridynamics formulation leads to asymmetric interactions between material points. These asymmetric interactions violate balance laws and can introduce non-physical artefacts such as ghost forces and spurious wave reflections. In this work, we develop a variational formulation for bond-based peridynamics with spatially varying horizons. Starting from the Lagrange-d'Alembert principle, we derive the governing equations of motion and show that the dual-horizon peridynamics formulation emerges naturally from the variation of the internal energy. Building on this variational structure, we construct asynchronous variational integrators that allow different time step sizes in different regions of the domain. This is particularly useful for dynamic fracture simulations with local refinement, where small time steps are required only near regions of high resolution or expected crack growth. Numerical examples involving wave propagation, a pre-cracked plate under tension, and the Kalthoff-Winkler impact experiment demonstrate that the proposed framework removes spurious reflections caused by non-uniform horizons, preserves physically consistent fracture patterns, and achieves results comparable to uniformly refined simulations. At the same time, the asynchronous variational integrator reduces the number of internal force evaluations compared to the standard velocity-Verlet method. The proposed approach therefore provides a consistent variational foundation and an efficient time-integration strategy for bond-based peridynamic simulations with spatially varying horizons.

We completely classify conformal metrics on the unit ball $(\mathbb{B}^{n+1},|\mathrm{d} x|^2)$, $n\geq4$, with positive constant $Q$-curvature, positive constant $T$-curvature, and minimal boundary. After normalizing the $Q$-curvature, there is a unique conformal metric for each $T$-curvature value in $[0,+\infty)$, up to conformal diffeomorphism. For positive $T$-curvature, these metrics are not Einstein and yield a new family of bubble profiles, distinct from the Aubin–Talenti bubble family except when $T=0$. This new phenomenon has no analogue in either the second-order boundary Yamabe problem or the constant $Q$-curvature problem on closed manifolds. To our knowledge, this is the first classification result for a fourth-order boundary value problem with nonlinear terms both in the interior and on the boundary.

For every $n\geq 3$ we construct a bounded discontinuous map $u\in W^{1,n}(\mathbb{B}^n,\mathbb{R}^{n+2})$ solving an $n$-Laplace system with an antisymmetric potential $\Omega\in L^n$. This shows that a recent regularity result on $n$-Laplace systems with antisymmetric potentials in Lorentz spaces by the authors is sharp in the sense that we cannot move from Lorentz spaces to classical Lebesgue spaces. This gives in particular a negative answer to a question by Rivière.