Papers

In this paper, which originates from Chapter 4 of the author's PhD thesis [1, Chap. 4], we investigate the global well-posedness of a class of partially diffusive hyperbolic systems in critical homogeneous Besov spaces. Our main objective is to extend the analysis developed in our previous work [2] to a functional framework in which the high-frequency com- ponents of the solution are controlled in $L^p$-based spaces with $p > 2$. This broader framework allows us to establish global existence under weaker smallness assumptions on the initial data and provides more precise information on the qualitative behavior of the resulting solutions. In particular, our analysis yields refined regularity. As an application, our global existence theorem applies to the multidimensional compressible Fourier-Navier-Stokes system

We construct hypocoercive Kolmogorov-type operators on the Euclidean motion group $\mathrm{SE}(3)=\mathbb{R}^3\rtimes\mathrm{SO}(3)$ using ZXZ Euler angles and study their long-time behaviour using the abstract hypocoercivity method. The operator $L=S-A$ decomposes into a symmetric part $S$, corresponding to a degenerate diffusion on the rotation component $\mathrm{SO}(3)$, and an antisymmetric part $A$, which captures transport and torque effects induced by the Lie group structure. The associated invariant probability measure is $\mu_\Phi=\gamma_\Phi\otimes\nu_3$, where $\gamma_\Phi$ is a probability measure on $\mathbb{R}^3$ with density $Z(\Phi)^{-1}e^{-\Phi}$ with respect to the Lebesgue measure, and $\nu_3$ is the normalized Haar measure on $\mathrm{SO}(3)$. As a consequence, if $\Phi\in C^\infty(\mathbb{R}^3)$ is bounded from below, $\gamma_\Phi$ satisfies a Poincaré inequality with constant $\Lambda>0$, and there exists $c<\infty$ such that \begin{equation*} |\nabla_\xi^2\Phi(\xi)| \leq c\bigl(1+|\nabla_\xi\Phi(\xi)|\bigr), \quad \xi\in\mathbb{R}^3, \end{equation*} then the semigroup generated by the closure of $L$ converges exponentially fast to equilibrium in $L^2(\mathrm{SE}(3),\mu_\Phi)$.

We study open billiard flows in $\mathbb{R}^3$ generated by reflections in the exterior of at least three ellipsoids with $C^3$ boundaries. Each obstacle is congruent to $$\tilde{K} = \{(x,y,z)\in \mathbb{R}^3 : x^2+y^2+(\beta z)^2 \leq r^2\},$$ with parameter $\beta>1$, small radii, and centers $O_i$ lying in the plane $\Gamma=\{z=0\}$. The configuration satisfies the no-eclipse condition (H). For any Gibbs measure on the non-wandering set of the billiard ball map, we prove that if $\beta>1/c_1^2$, for some constant $c_1>0$ determined by the geometry of the configuration, the two positive Lyapunov exponents are distinct and positive: $\lambda_1(\beta)>\lambda_2(\beta)>0$. Moreover, $\lambda_2$ is independent of $\beta$ and corresponds to the restriction of the billiard flow to the plane $\Gamma$.

Branson's $Q$-curvature is a natural fourth-order counterpart of scalar curvature in conformal geometry. In analogy with the classical recovery of the Ricci tensor from the linearized scalar curvature, Lin and Yuan introduced a symmetric $2$-tensor $J_g$ through the formal adjoint of the linearized $Q$-curvature and established its explicit decomposition in terms of the Bach tensor and auxiliary curvature terms. In this paper, we study the associated fourth-order geometric flow $\partial_t g = -2J_g$ on closed Riemannian manifolds of dimension $n \ge 4$. We establish short-time existence and uniqueness using a biharmonic DeTurck gauge and derive global and local integral smoothing estimates under curvature bounds. In dimension four, we prove long-time existence assuming a uniform Sobolev inequality and sufficiently small total $L^2$ curvature energy throughout the maximal interval of existence. Under uniform curvature and Sobolev bounds, time integrability of the $L^2$-norm of $J$ implies smooth convergence in fixed coordinates to a $J$-flat metric. Explicit Einstein and product solutions illustrate the evolution and show that small curvature energy alone does not prevent degeneration. Positive initial Yamabe constant and positive total $Q$-curvature give a uniform positive Yamabe lower bound throughout the smooth four-dimensional flow, whereas in higher dimensions total $Q$-curvature can decrease even near flat metrics. Finally, we prove that every closed four-dimensional $J$-soliton is trivial, its metric is Bach-flat with constant $Q$-curvature, and its soliton vector field is Killing.

The well-known inequality of Fiedler ensures that the spectral gap $r-\lambda_2(G)$ of a connected non-complete $r$-regular graph $G$ is bounded from above by the vertex connectivity $\kappa(G)$ of $G$. We prove that, for integers $t\geq2$ and $r>2t^2$, every connected $r$-regular graph with vertex connectivity at most $2t$ has spectral gap at most $\frac{1}{2}(r+t+2-\sqrt{(r-t+2)^2-4t(t-1)})$. We show that this bound is nearly optimal for each fixed $t \ge 2$ and sufficiently large $r > 2t^2$ such that $r+1$ is divisible by $t$. We further prove that if $r>(2t-1)(2t-2)$, then every connected $r$-regular graph whose vertex connectivity is odd and no more than $2t-1$ has spectral gap at most $r-\min\bigl\{\xi(r,t),\nu(r,t)\bigr\}$ for some explicitly given functions $\xi(r,t)$ and $\nu(r,t)$. If $r\geq6t^2$, then this minimum equals $\xi(r,t)=\frac{r(2r-4t+5)}{2(r-t+2)}$. In particular, when $\kappa(G)=2t-1$, this gives $r-\lambda_2(G)<t=(\kappa(G)+1)/2$, which improves Fiedler's bound by nearly a factor of two. Our upper bounds in their parameter ranges improve two known bounds on the spectral gap of regular graphs.

A nonfatal propulsion fault can invalidate the nominal ascent trajectory while leaving sufficient vehicle capability to reach the original orbit or a degraded target. Deterministic replanning can restore nominal feasibility, but residual uncertainty in the identified propulsion capability may still lead to path-constraint violations and terminal injection dispersion. This paper develops a fast chance-constrained stochastic ascent trajectory replanner that explicitly accounts for persistent propulsion uncertainty. The uncertainty model combines persistent random uncertainties shared across the prediction horizon with interval-wise independent disturbances. For a fixed nominal trajectory, system level synthesis is used to parameterize the closed-loop responses. A Cholesky-based transformation of the system responses converts the resulting cross-time correlated feedback design problem into column-wise affine recursions that share a common Riccati recursion. A task-oriented terminal weight is further derived from the minimum-energy post-injection orbital correction in phase-free modified equinoctial elements, thereby relating terminal covariance shaping directly to correction demand. The stochastic replanning problem is solved by alternating deterministic trajectory replanning and feedback controller design, with covariance-based chance-constraint backoffs coupling the two updates. Nonlinear Monte Carlo simulations demonstrate reduced equivalent correction demand, effective chance-constraint satisfaction, and fast online replanning over a range of uncertainty levels and fault conditions.

We develop two methods in quantitative homological algebra: A declarative approach to integral cohomological filling problems and a quantitatively controlled replacement of normed chain complexes by their homology complex. Both methods are connected through so-called chain extensions. As applications, we obtain transparent proofs for uniform integral simplicial cohomological fillings for subdivisions of simplicial complexes, for uniform integral cohomological fillings of certain finite coverings, and for lower bounds of the $\ell^1$-semi-norm on product spaces.

Direct approaches to discrete parabolic Carleman estimates often rely on weighted identities tailored to the discrete operator. We prove an abstract transfer theorem that derives such estimates for fully discrete backward Euler schemes with conforming spatial reconstructions from a continuous Carleman inequality together with suitable weighted approximation and compatibility properties. For a class of time-independent positive self-adjoint operators, the key construction assigns to each discrete state a shifted continuous auxiliary problem whose discrete solution is exactly the prescribed state. Weighted space-time error estimates, uniform over the full discrete load space, then transfer the inequality while preserving the residual and observation terms of the original scheme. This separates the PDE-specific Carleman analysis from the numerical verification required for each discretization. We verify the hypotheses for the heat equation discretized by consistent-mass $P_1$ finite elements on locally graded meshes in two and three dimensions and by standard Cartesian finite differences in any fixed dimension. For the heat equation, the transferred estimate is valid for Carleman parameters up to the scale $\min\{h^{-4/5},(\delta t)^{-2/5}\}$. For bounded real-valued space-time potentials, the resulting estimates yield relaxed observability and approximate null controls with uniformly bounded cost and exponentially small terminal errors under independent spatial and temporal refinement; weak accumulation points of the reconstructed controls are continuous null controls.

We construct smooth time-periodic Hamiltonians on the monotone manifolds \(S^2\times S^2\) and \(\CP^2\) for which the Clifford torus meets its Hamiltonian image in exactly two isolated degenerate points.

The proximal point algorithm (PPA) stands as a fundamental approach for solving monotone inclusion problems. Notably, several key convex optimization algorithms have been proven to be specific instances of PPA. Given the importance of the PPA, there has been growing interest in developing its accelerated variants. However, for some specific cases, the PPA converges faster than the accelerated PPAs. In this paper, we mainly study an adaptive version of symplectic accelerated PPA, called adaptive symplectic proximal point algorithm (ASPPA). We first prove that the convergence rate of ASPPA with respect to the square norm term is $O(1/k^2)$, which is the same as the convergence rate of some accelerated PPAs. Also, we show that ASPPA exists exponential convergence rate when solving strongly monotone inclusion problem. Our numerical experiments indicate that the performance of ASPPA is very good.

We prove that any counter-example to the two dimensional Jacobian Conjecture with integral coefficients and determinant of its Jacobian matrix equals one, has only paths of asymptotic values that can contain only finitely many points of the planar integral lattice. The second part of our paper gives some nontrivial applications of the theory of certain Theta functions to the two dimensional Jacobian Conjecture over the real field, and over the complex field.

Finite-state interval Markov-chain (IMC) abstractions provide sound verification and performance bounds for continuous-state stochastic systems by enclosing cell-to-cell transition probabilities and costs in intervals. Finite-state Markovian abstractions are generally lossy, as state aggregation often destroys the Markov property. Recently, memory-dependent Markov-chain (MC) abstractions of stochastic systems have been developed to mitigate this by recording recently visited cells. In this paper, we develop the memory-dependent extension of IMC abstractions. Building on the insight that memory narrows the family of state distributions that could occur inside the current cell, we prove that memory tightens the local intervals, at the price of a larger abstraction. We also formulate no-memory IMC abstractions, tightening classic IMC abstraction intervals without enlarging the state space. To characterize spatial refinement relative to memory, we derive two cost-guarantee tightness upper bounds. The bounds depend, respectively, on cell size, and on the worst-case remaining distributional ambiguity after filtering over the remembered past. We then specialize our result to linear-Gaussian systems, for which we derive a closed-form upper bound on this ambiguity. Numerical examples show that the memory-dependent and no-memory constructions can produce tighter expected-cost intervals than comparable partition refined models.

We study the effect of deleting a horizontal divisor from an arithmetic scheme on its étale fundamental group. Assume that the scheme is regular along the divisor, that each prime component contains a normal crossings point of residue characteristic $2$, and that the generic fibers of the boundary components remain geometrically integral after finite étale base change. We prove that the kernel of the induced surjection on fundamental groups is topologically perfect. In particular, the inclusion induces an isomorphism on maximal prosolvable quotients. We apply this result to projective hyperplane complements and to the extension of representations over complete local coefficient rings.

Cross-fitting is routine in much of applied research. While conventional confidence intervals that ignore cross-fold dependence are asymptotically valid in several settings, they undercover in many applications that share a common form of nonregularity: from the classic cross-validation problem of testing whether a fitted model outperforms another, to testing for heterogeneous treatment effects with machine learning, to estimating the value of a potentially non-unique optimal treatment regime. Exploiting a new locality condition, I show that a large class of cross-fitting estimators still satisfies a central limit theorem despite the nonregularity, but with an asymptotic variance that must be adjusted for the cross-fold correlation. Then, I propose a method for estimating this correlation and construct new confidence intervals that attain asymptotically nominal coverage. Finally, I show that the proposed confidence intervals attain approximately nominal coverage in a simulation study with random forests and neural networks.

We characterize lower curvature bounds on metric surfaces by the Bishop–Gromov inequality for metric balls.

We study weighted bilinear Hardy-Steklov operators on the half-line for $1<p_1,p_2<\infty$ and $0<q<\infty$. The operator-theoretic core is an exact output-localisation principle. For a bounded bilinear map into $L^q$, compact middle truncations identify compactness with vanishing of a directed output-tail invariant and, with constant one, identify that invariant with the essential distance and a radius-based Hausdorff measure of noncompactness. Under the additional hypothesis of local finite-rank approximability, the same invariant also equals the finite-rank distance and governs the rank-approximation conclusions. We verify these hypotheses for the Hardy-Steklov map and then translate the tail invariant through the five weighted boundedness regions. In Regions $I,\mathrm{II}_1,\mathrm{II}_2,\mathrm{III}_1$, boundedness must be supplemented by lower- and upper-endpoint vanishing, and the essential distance is comparable to the larger endpoint defect. In the strict subcritical Region $\mathrm{III}_2$, boundedness alone implies compactness. This yields a sharp structural transition between the equality boundary $P=Q$ and the summable regime $P>Q$. The quasi-Banach range $0<q<1$ is treated throughout with the powered metric $d_q(f,g)=\|f-g\|_q^q$.

In this paper, we investigate refined Bohr-type inequalities involving Schwarz mappings for holomorphic mappings defined on the unit ball \( B_X \) of a complex Banach space. We first establish refined Bohr inequalities for scalar-valued holomorphic mappings \( F : B_X \to \mathbb{U} \). We then extend some results to vector-valued holomorphic mappings involving Schwarz mappings of the form \( F : B_X \to \overline{\mathbb{U}}^{\,n} \), \( n \ge 2 \), where improved Bohr radii are obtained under suitable norm conditions. Furthermore, we study the Bohr radius involving Schwarz mappings for holomorphic mappings whose ranges are contained in the closure of the unit ball of a JB$^*$-triple. Our results generalize the results of Jia et al. \cite{JLP}, Liu et al. \cite{LLP2021} and Liu et al. \cite{LSX2018}.

We present a root isolation algorithm for an analytic function. The method combines interval subdivision with a heuristic test based on local cubic Hermite interpolation. Whenever the function is sufficiently close to its cubic Hermite interpolant at a few sample points, we assume that the critical points of the cubic interpolant give a valid root isolation for the original function. Such an algorithm cannot, in general, provide a rigorous certification of the computed intervals. Nevertheless, extensive numerical experiments show that it is highly efficient and almost always produces the correct results. We hope that a probabilistic or exact algorithm can be derived from this work.

We study the focusing inhomogeneous nonlinear Schrödinger equation with an inverse-square potential $$ i\partial_t u+\Delta u-a|x|^{-2}u+|x|^{-b}|u|^{\alpha}u=0, \qquad (t,x)\in\mathbb{R}\times\mathbb{R}^{N}, $$ in the mass-supercritical and energy-subcritical regime $$ \frac{4-2b}{N}<\alpha<\frac{4-2b}{N-2}, $$ where $N\ge3$ and $0<b<2$. Two main lines of research have developed largely independently for this model: scattering theory below the ground-state threshold and the variational analysis of solutions with prescribed mass. In this work, we establish a connection between these two approaches.

Random sampling in DNA sequencing produces repeated reads, increasing retrieval latency and sequencing cost. We study the coverage-depth problem for full-message recovery in distributed DNA storage under noiseless uniform sampling, where strands are partitioned among $M$ containers and one strand is independently sampled with replacement from each container per round. For arbitrary linear codes and ordered partitions, we derive exact formulas for the recovery-time distribution and expectation. We prove that MDS codes, whenever they exist, are optimal for every fixed partition, and establish a universal lower bound on the expected total read cost together with its equality conditions. For MDS codes, we identify container-size regimes that yield genuine savings in total reads and regimes that provide only parallelism without changing the asymptotic sequencing cost. For simplex codes, we prove that the $q$-ary simplex code is, up to isomorphism, the unique single-container minimizer among codes with the same parameters, resolving a recent conjecture by Bertuzzo, Ravagnani, and Yaakobi. We further construct a partition attaining the minimum total read cost and derive bounds for intermediate and balanced partitions. These results clarify when distributed sampling reduces latency alone and when it also reduces sequencing cost.

We show that $\mathbb{Z}_2$ is a quantum normal subgroup of $SU_q(2)\times SU_q(2)$ and that the quotient group is isomorphic to $SO_q(4)$, thereby establishing that $SU_q(2)\times SU_q(2)$ is a double cover of $SO_q(4)$.

An extension with temporal modulation is presented of the canonical 2 + 1-dimensional solitonic Dym equation as originally introduced by Konopelchenko and Dubrovsky. Application is made of admitted Painleve' II symmetry reduction to derive exact solution to a class of associated 2 + 1-dimensional nonlinear moving boundary problems of generalized Stefan-type.

We establish local and global spectral bounds for Hermitian $A_\alpha$-matrices of mixed graphs. Using the first three spectral moments, we obtain an upper bound for the largest eigenvalue as the largest real zero of an explicit cubic polynomial. Vertexwise estimates for the extreme eigenvalues yield new lower bounds for the spectral spread, including a bound that strictly improves an existing degree-based estimate for ordinary graphs. A local two-dimensional compression produces a spread bound involving neighbour-degree data and gain-weighted triangles. This bound is exact for every mixed orientation of a star and is independent of a known Zagreb-index bound. We also derive two complementary upper bounds for sums of the smallest eigenvalues. As further consequences, the spectral estimates provide a computable convergence guarantee for Richardson graph filtering and a stability certificate for residual graph-neural-network layers on directed networks. Numerical examples illustrate the sharpness and mutual incomparability of the proposed bounds.

This paper investigates non-parametric regression estimation when the explanatory variable takes values in a separable Hilbert space and observations are sampled over an increasing regular spatial lattice. Under a rigorous field-to-field independence setup between covariates and errors, we explore the structural and asymptotic concentration properties of the functional $k$-nearest-neighbour ($k$-NN) estimator. By establishing a sharp pathwise deterministic sandwiching framework for the data-driven random bandwidth, we successfully decouple the local infinite-dimensional small-ball profile from the polynomial covariance decay of the neighborhood indicators. Pointwise convergence rates are derived across short-range, critical, and long-range spatial regimes, revealing a combined penalty term that reflects both covariate spatial interaction and response error memory. Furthermore, uniform consistency over compact subsets is established for unbounded sub-Gaussian error processes through metric entropy and indicator boundary shell chaining. Structured polynomial simulations confirm our theoretical rates and exemplify the precise mechanics of the spatial long-range bottleneck.

This paper provides a complete characterization of the spectrum, point spectrum, continuous spectrum, and residual spectrum of bounded convolution operators acting on the non-separable space $\sigma_\infty$, the dual of the Hahn sequence space. Because standard duality and separability-based techniques fail in this setting, a linear functional approach is employed with the fine spectral decomposition of the unilateral shift operator. Furthermore, as a major application of this methodology, we establish a complete description of the fine spectrum for operator polynomials acting on general sequence spaces without requiring separability. This result provides a unified framework that subsumes classical spectral determinations for general banded triangular matrices across various sequence spaces and enables the explicit computation of fine spectrum on non-separable spaces.

We consider an abstract wave-type evolution equation with a propagation speed depending only on time. We assume that this propagation speed is the product of a monotone, but possibly degenerate, shape function and an oscillating factor bounded from above and from below by positive constants. We investigate how degeneracy and oscillations interact in determining the derivative loss of solutions. After reducing the problem to a family of ordinary differential equations through spectral analysis, we derive frequency-dependent energy estimates. In the purely degenerate case, monotonicity of the shape function prevents excessive loss, and solutions lose at most one derivative. In the presence of oscillations, the resulting derivative loss is described by a minimization problem in which the degenerate and oscillatory effects genuinely interact. The proof is based on a decomposition of the time interval into frequency-dependent regions, where different adapted energies are used.

For a tree $T$, let $g(T)=\lambda_1(T)-\lambda_2(T)$ be the difference between its two largest adjacency eigenvalues. A balanced double comet is obtained by attaching equally many leaves to the two endpoints of a path. Jovović, Koledin and Stanić conjectured that such a tree attains the minimum adjacency spectral gap among trees of any fixed order. We prove that every minimizing tree of order $n\ge200$ is a balanced double comet. We also show that, for any finite irreducible reversible continuous-time Markov chain, the inverse spectral gap differs from the effective resistance between two states times the stationary variance of their hitting probability by at most the inverse Dirichlet gap for killing at those states. For Perron chains, we give an exact two-vertex Schur-complement formula for this resistance–variance quantity.

We prove that the irrationality exponent of $\zeta(2)=\pi^2/6$ satisfies $\mu(\zeta(2))<5.0193784$. The best published bound is 5.095412 (Zudilin, 2014); a bound 5.0495243 with a machine-checked proof was made public in September 2026 by J. Kleid. We use Zudilin's two hypergeometric constructions at a new choice of parameters. The two families of linear forms in 1 and $\zeta(2)$ so obtained coincide for every sufficiently large index: this coincidence, an instance of an identity conjectured by Zudilin, is established by comparing an arithmetic bound for the denominators of the difference with an analytic bound for its size, without creative telescoping. All numerical constants are certified in ball arithmetic.

Consensus-based optimization (CBO) is a robust and flexible zero-order, multi-particle class of methods devised to provide provable solutions to high-dimensional global optimization problems, including truly nonconvex and/or nonsmooth cases. The algorithmic framework is grounded in a principled trade-off between stochastic exploration of the search space and deterministic contraction toward a consensus point. This thesis is concerned with three recently established variants of CBO. The $\delta$-CBO method is similar to the classic CBO method, with the difference that the diffusion function is constant. The consensus freezing (CF) scheme can be understood as a piecewise CBO method. The overall time horizon is divided into intervals in which the consensus point is fixed. The consensus hopping (CH) scheme can be understood as a deterministic iterative scheme that jumps at each iteration to a Gaussian distribution centered at the consensus point of the previous distribution. The central contribution of this thesis is the rigorous derivation of propagation of chaos results for all three schemes. For each variant, we establish that the empirical measure of the interacting particle system converges, as the number of particles $N$ approaches infinity, to the solution of the corresponding mean-field equation. Two regimes are treated: first, on a high-probability set on which the empirical measures are uniformly bounded; second, in the unrestricted setting, where moment bounds and a careful treatment of the inter-particle dependence yield convergence without imposing conditioning on favorable events. The propagation of chaos estimates are derived by the coupling method. Together, these results provide quantitative propagation of chaos estimates for all three CBO variants and establish that the particle systems faithfully approximate the idealized mean-field dynamics in the large-particle limit.

In their seminal work, Baum-Douglas-Taylor showed that under a certain compactness assumption, local boundary conditions for first-order elliptic operators over compact manifolds with boundary define cycles for relative K-homology. Using a systematic approach to regular boundary conditions, we generalize their result to elliptic operators of arbitrary order and to non-compact manifolds with potentially non-compact boundary. In addition, we show that under stronger regularity conditions, local boundary conditions even define cycles for absolute K-homology. In contrast to the relative classes, the resulting absolute K-homology classes are in general not independent of the choice of boundary condition.