Papers

Ferguson's 1973 introduction of the Dirichlet process marked a breakthrough in Bayesian nonparametric statistics. For the first time, a prior on the space of probability measures fulfilled two key desiderata: large support and analytical tractability. In this paper, we review three complementary constructions of the Dirichlet process, whose roots can be traced back to Ferguson: through finite-dimensional distributions, via normalization of a gamma process, and through predictive distributions. Each perspective not only deepens the understanding of the Dirichlet process but also provides a template for generalizations, from normalized random measures with independent increments to Gibbs–type priors and beyond. Over the past fifty years, the Dirichlet process has become the cornerstone of Bayesian nonparametric methodology and applications, while simultaneously inspiring the expansion of the landscape of nonparametric priors. Since de Finetti laid out the Bayesian nonparametric framework in the 1930s, the key obstacle had been the absence of a tractable nonparametric prior. Ferguson's contribution overcame this challenge, providing a solution to a decades-long open problem. In recognition of this decisive advance, it seems appropriate to refer to the Dirichlet process as the Ferguson–Dirichlet process.

In this paper, we analyse the existence of solutions and finite element approximation of a steady-state two-dimensional induction heating problem. One of the main difficulties of the problem is its right-hand side which, at a first sight, is only integrable. Using a priori regularity results for the PDEs involved it is shown that the natural weak formulation of the problem can be justified. Then, we study the finite element approximation and prove that the standard Galerkin FEM converges in convex domains and under suitable conditions on the mesh. We improve on this result by applying the recently-proposed bound-preserving method (BPM) to the heat equation, and show that this method converges to a solution of the problem under less stringent conditions on the domain and the mesh. As these analyses are carried out without any assumption on regularity of the solutions, then the convergence of the finite element method also proves existence of solutions. Several numerical experiments confirm the theoretical results, and showcase the improvement provided by the use of the bound-preserving method over the standard finite element method.

We present a method for computing the exact output distribution of small programs with random inputs. Specifically, we are interested in inline programs manipulating sensor data such as \eg GPS or inertial measurement sensors whose inputs have a known or well-modelled distribution. These programs typically only include relatively few variables, arithmetic operations, square roots and if-else statements. This small syntax allows us to recast the problem of computing the exact output distribution as a cylindrical algebraic decomposition problem followed by symbolic and/or numerical integration. We present this method in detail and show with two prototypes that it can successfully be applied to benchmarks from the literature on floating-point arithmetic and small programs from open-source sensor libraries.

The CR Paneitz operator is closely related to several important problems in CR geometry. In this paper, we study the CR Paneitz operator on non-embeddable three-dimensional tori. Under mild assumptions, we show that it possesses infinitely many negative eigenvalues. We also provide concrete examples satisfying the assumptions.

We prove an off-diagonal expansion of the kernel of the Toeplitz operator whose symbol is the indicator function of a compact domain with smooth boundary in a complete symplectic manifold of bounded geometry. Using our approach, we extend two results to the non-compact setting: the first concerns the asymptotics of the trace of polynomials in this operator, and the second establishes a Weyl law for this Toeplitz operator.

We construct an equivariant E-theory and a Baum-Connes assembly map at the level of Fell bundles of inverse semigroups over separable C*-algebras. This generalizes previous work of several authors, and allows to discuss E-theoretic matters in the context of Cartan pairs; maximal and essential C*-algebras of non-Hausdorff groupoids; and Fell bundles over discrete groups and étale groupoids, among others. In order to do this we establish several functoriality properties for maximal, reduced and essential cross-sectional C*-algebras associated with a (saturated) Fell bundle of an inverse semigroup. This allows to discuss when these algebras give rise to short exact sequences, generalizing the classical case of discrete groups. We also introduce the adequate notion of “proper” Fell bundle, or “proper” action of an inverse semigroup, and prove a weak containment property for these. Using these functoriality properties and these proper actions we then introduce (maximal, reduced and/or essential) equivariant E-theory by means of adequately equivariant asymptotic morphisms, and construct a Baum-Connes assembly map that is both natural and reasonably well-behaved.

Reachability analysis for dynamical systems seeks to compute a set containing all reachable states at a given time. Compared to ordinary differential equations (ODEs), the analysis of nonlinear reaction–diffusion PDEs with parametric uncertainties remains largely underexplored, due to the infinite-dimensional state space and the variety of solutions under different parameters. We address this through a three-step procedure: 1) Finite Element Methods (FEM)s to discretise the space and generate a finite-dimensional FEM-based model, 2) Proper Orthogonal Decomposition (POD) to build a Reduced-Order Model (ROM), and 3) set-based reachability-analysis methods applied to the ROM. We propose a framework that enables us to derive explicit upper bounds on the approximation errors introduced at each stage of the pipeline. In particular, we quantify the discrepancy between trajectories of the original PDE and those of the FEM-based discretization, as well as the error between the FEM-based model and the reduced-order model. Importantly, these bounds are shown to hold uniformly over the considered set of parameters. By combining these error estimates, we obtain an over-approximation of the reachable set of the original PDE. The approach is illustrated on the Allen–Cahn equation and a logistic growth PDE.

"Colored" knot polynomials satisfy difference equation w.r.t. the highest weights of the underlying representation – which in the case of symmetrically colored Jones are named "quantum $A$-polynomials". In the double scaling quasiclassical (Kashaev) limit, when representation size $r\sim \hbar^{-1}$, there are different phases – in one of them the classical action vanishes and in another one it is a deformation of hyperbolic volume (of a knot complement in $S^3$). This corresponds to a splitting of the non-homogeneous version of the quantum $A$-polynomial into two pieces, which we illustrate by more examples than just a figure-eight knot $4_1$ in the original paper. From the point of view of quasiclassics, hyperbolic volume is just an integration constant, which is not fully determined by the $A$-polynomial equation – and actually remains ambiguous in this formalism. As a byproduct, we expect that classical $A$-polynomial at $L=1$ becomes proportional to Alexander: $A^{\cal K}(1,M)\sim \Delta^{\cal K}(M)$ – this seems true, but $A$ should be consistent with the polynomiality of non-homogeneous quantum ${\cal A}$-polynomial, what sometime implies that it is not minimal.

We consider weighted-sum scalarizations for an abstract multi-objective minimization problem defined by the vector-valued map $U\ni u\mapsto \big( f_1(u),\ldots, f_N(u)\big)$, where $U$ is an arbitrary nonempty set and no topology, convexity, compactness, or lower semicontinuity assumption is imposed. Using the open simplex as parameter space for positive weights, we show that the Trade-off Invariance Principle for scalarizations yields a generic uniqueness property in the objective space. Namely, for almost every weight vector, all minimizers of the corresponding weighted-sum scalarization have the same objective vector, although the minimizers themselves need not be unique. Moreover, excluding again a null-measure subset, all minimizing sequences determine the same limiting objective vector, independently of the chosen sequence. We also give a geometric interpretation of these results in the attainable objective set: for almost every positive weight vector, the scalarization exposes at most one nondominated point. Furthermore, minimizing sequences determine at most one asymptotically exposed objective vector in the closure of the attainable set.

Although various boundary integral formulations are available for the Helmholtz transmission problem, the weakly singular Burton–Miller equation is promising because it possesses favorable properties for numerical computation. This paper rigorously shows that the weakly singular Burton–Miller equation is well-posed. We establish bounded invertibility on $C(\Gamma)\times C(\Gamma)$ for $C^2$ boundaries by proving injectivity and applying the Fredholm alternative. For the prescribed analytic incident field, the integral equation further yields $u\in C^{1,\beta}(\Gamma)$ and $q\in C^{0,\beta}(\Gamma)$, for $0<\beta<1$. We also provide a proof that its fictitious eigenvalues are associated with an exterior Dirichlet problem and an interior impedance problem.

In earlier work \cite{sharma2025} we developed deterministic analytical bounds on the T-eigenvalues of symmetric positive definite (SPD) third-order tensors under the Kilmer–Martin T-product: the trace–determinant (TDet) bounds via the AM–GM inequality, and the trace-dependent (TDep) bounds generalizing Samuelson's inequality. While these bounds are cheap and guaranteed-valid, their relative gap grows as $\sqrt{d-1}$ in the tensor dimension $d = np$, limiting their usefulness for large tensors. This paper develops randomized estimators for the extreme T-eigenvalues of T-SPD tensors that complement the deterministic bounds. We adapt the Halko–Martinsson–Tropp framework \cite{halko2011} to the T-product setting and introduce four methods: (i) a randomized power method that produces a lower bound on $\lambda_1$ with exponential convergence; (ii) a randomized subspace iteration with a tensor-analogue HMT error bound; (iii) a two-sided rigorous bracket combining the randomized lower bound with the deterministic TDep upper bound; and (iv) a Hutchinson-based fully randomized TDep bound for matvec-only settings.

We consider a one-parameter family of isotropic planar Gaussian processes \[ X_\sigma(t) =B_t+\sigma t Z,\qquad 0\le t\le 1,\quad 0\le \sigma\le 1, \] where $B$ is a standard ($0$-to-$0$) planar Brownian bridge on $[0,1]$, and $Z\sim \mathrm N(0,I)$ is a standard Gaussian random vector independent of $B$. The family interpolates between standard planar Brownian bridge ($\sigma=0$) and standard planar Brownian motion ($\sigma=1$). As the main result of the paper we compute the expected perimeter and area of the convex hull of the random set $\left\{X_\sigma(t) \colon 0\le t\le 1\right\}$ as closed formulas in terms of $\sigma$, and recover the classical Brownian bridge and Brownian motion values at $\sigma=0$ and $\sigma=1$. We also consider the convex hull spanned by multiple independent processes of this type and the possibilities for closed formulas in special cases. The key observation in our argument is that the isotropy property reduces the expected perimeter and area to one-dimensional quantities through the support function and Cauchy's formulas.

We study the homotopy coherent analogue of the Nielsen realization problem for smooth 4-manifolds. Using family Seiberg-Witten theory, we prove that for K3-type 4-manifolds and their blow-ups, certain order two subgroups cannot be realized even homotopy coherently. This in particular gives an alternative proof that the classical Nielsen realization fails for these manifolds. In contrast, we also construct non-kinetic homotopy coherent actions that are not induced by genuine group actions.

We study the asymptotic behavior of Kähler-Einstein potentials on stable varieties near the singularities. Using iterated logarithmic functions associated with a defining function of the non-klt locus, we obtain refined lower bounds for the Kähler-Einstein potential, improving previous estimates of Di Nezza-Guedj-Guenancia and Datar-Fu-Song. Under additional assumptions on the log resolution, we also establish upper bounds. The proofs are based on the construction of explicit subsolutions and supersolutions for degenerate complex Monge-Ampère equations together with refined integrability estimates in pluripotential theory.

We prove new error bounds for finite Carleman truncations of polynomial ordinary differential equations. The analysis works directly in the original monomial basis and for selected observables, such as state coordinates. Using a Dyson–Duhamel expansion, we separate the degree-preserving linear part from the degree-raising nonlinear part and track how truncation errors can propagate back to the observable. The resulting bounds are degree-aware and retain logarithmic-norm information from the original linear dynamics. We obtain explicit finite-degree estimates and geometric convergence over certified time horizons. Comparisons with existing bounds, in particular those of Forets–Pouly, are given on the Stuart–Landau and Van der Pol systems.

We show that random linear codes (RLCs) possess nearly optimal discrepancy-type properties in a broad range of settings. Our main results are two general discrepancy theorems: one controls all translates of a fixed test, and the other controls large families of Fourier-pseudorandom tests. Two motivating examples follow: First, RLCs behave essentially like unstructured random codes for list-decoding from errors above capacity. More precisely, an RLC $C\subseteq \mathbb{F}_q^n$ of rate $1 - \frac{1}{n}\log_q|B_\rho| + \varepsilon$, where $|B_\rho|$ is the volume of a radius-$\rho$ Hamming ball in $\mathbb{F}_q^n$, satisfies $|C \cap B| = (1\pm o(1)) \frac{|C|\cdot |B|}{q^n}$ simultaneously for all radius-$\rho$ Hamming balls $B$ with high probability. This significantly strengthens the best prior results, which follow from work by Blinovsky (1987, 1990) and Levitin (1988) on the covering properties of RLCs. Second, over prime fields, RLCs behave essentially like unstructured random codes for zero-error list-recovery, and list-recovery from erasures, above capacity. More precisely, for a prime $q>2$ and input list size $2\leq \ell\leq q-1$, an RLC $C\subseteq \mathbb{F}_q^n$ of rate $1-\log_q \ell+\varepsilon$ will satisfy $|C \cap S| = (1\pm o(1)) \frac{|C|\cdot \ell^n}{q^n}$ simultaneously for all combinatorial rectangles $S=S_1\times S_2\times\cdots\times S_n$, where $|S_i|=\ell$ for all $i$, with high probability. An analogous result also holds when we can bound $|S_i|$ only for some of the $i$'s. We use this to show the abundance of locally leakage-resilient $n$-party linear ramp secret sharing schemes with any linear reconstruction threshold and sublinear threshold gap $O(n/\log n)$ over fields of polynomial size $q=\Theta(n^\gamma)$ for a constant $\gamma\in(0,1/5)$. Prior work was stuck at reconstruction thresholds above $n/2$ for both threshold and ramp schemes.

We study the problem of estimating the arrival times of vertices in a uniform random recursive tree from its unlabeled structure. We adopt a pointwise perspective and analyze the distribution of the relative estimation error, and derive tail bounds that are uniform in both the vertex and the tree size. For the ranking induced by Jordan centrality, the probability that the estimate exceeds the true arrival time by a factor $S$ decays on the order of $1/S$, while the probability of underestimating the arrival time by a factor $1/S$ decays exponentially in $S$. We introduce a refined centrality measure whose overestimation tail decays on the order of $(\log S)/S^{2}$, at the cost of a heavier lower tail of order $1/S^{2}$. These results reveal a tradeoff between upper- and lower-tail performance in arrival-time estimation that is invisible to the previously studied risk functional. Nevertheless, the refined centrality measure attains the optimal order of the risk for all its parameter values.

We study the usual square function estimate associated with the Cauchy single-layer kernel in the plane, without assuming Ahlfors-David regularity. We prove that a finite Radon measure with positive and finite upper density is rectifiable if it satisfies the usual square function estimate and a weak flatness condition. We also prove that, under the same finiteness and density hypotheses, the weak flatness condition follows when the support is contained in a locally two-sided NTA curve. As a corollary, rectifiability follows when the support is contained in a quasicircle.

We study the Fefferman–Szegő metric on egg domains \[ \mathcal D_{2m}=\{(z,w)\in\mathbb C^2: |z|^2+|w|^{2m}<1\},\qquad\qquad\qquad m\in\mathbb Z^+. \] Our first main result establishes the existence of the Fefferman–Szegő kernel on $\mathcal{D}_{2m}$ by verifying that the Fefferman weight lies in the Muckenhoupt class $A_2(\partial\mathcal{D}_{2m})$. We then derive an explicit closed-form expression for this kernel, demonstrate that its blowup occurs precisely on the boundary diagonal, and determine its boundary asymptotic behaviour. Using this kernel, we compute the associated Fefferman–Szegő metric and its Ricci curvature. As applications, we prove several rigidity results: the metric is Kähler–Einstein if and only if $m=1$; proportionality to the Bergman metric or to some complete Kähler metric $g_m^{\mathcal D_{2m}}$ is also equivalent to $m=1$. Finally, we establish the vanishing of the $L^2$-cohomology outside the middle dimension for the Fefferman–Szegő metric.

This paper investigates perfect state transfer (PST) in discrete-time quantum walks constructed via the shunt decomposition method. The walks are defined on a graph $G$ and its associated quotient graph $G/\pi$, induced by an equitable partition $\pi$. Through the shunt decomposition of $G$, we derive an explicit relation between the shift operator of the parent graph $G$ and that of its quotient graph $G/\pi$. We construct a reflection operator based on the characteristic matrix, which establishes a connection between the transition operator of the parent graph and that of its lower-dimensional quotient graph. We then prove that PST occurs on $G$ if and only if it occurs on $G/\pi$. Furthermore, we express the unitary evolution operator of the quotient graph in terms of Chebyshev polynomials of the first kind, from which we derive explicit criteria for PST. As an application, we establish PST on the cycle graph $C_{n}$ at time $k = n/2$, and lift the result to the parent graph $C_{2n}$ via the equitable partition $\pi$. We further show that if an equitable partition $\pi$ of $G$ induces a quotient isomorphic to $K_n^{\circlearrowleft}$, the complete digraph on $n$ vertices with a loop at every vertex, then PST occurs at step $k = n$, and the walk is periodic at $k = 2n$. This framework is applied to two families of graphs, which are the complete bipartite digraph $K_{n,n}^{\rightleftharpoons}$ and the circulant graph $\operatorname{Circ}(2n, S)$, where $S$ consists of all odd residues modulo $2n$ and $n = 2^s$ for some $s \geq 1$, establishing PST in their respective line digraphs. Collectively, these results also answer the question posed by Godsil and Zhan concerning which shunt decompositions or embeddings of a graph admit PST.

A soliton gas theory for the derivative nonlinear Schrödinger equation is developed, focusing on the Gerdjikov-Ivanov equation and using its gauge equivalence with the Kaup-Newell and Chen-Lee-Liu equations. Starting from the reflectionless inverse scattering problem, we formulate pure N-soliton solutions through a meromorphic Riemann-Hilbert problem and pass to the continuum limit as the discrete spectrum condenses on the planar spectral domains. Under a suitable scaling of the norming constants, the limit yields a compactly supported dbar-problem. For domains admitting a Schwarz function, in particular elliptic domains, this problem reduces to a Riemann-Hilbert problem on the associated mother body. For the elliptic soliton gas, the large-x and long-time asymptotic behaviors are established: the solution decays as x->+infty, approaches an elliptic finite-gap background as x->-infty, and exhibits stratified long-time sectors described by one-, two-, and three-phase Riemann theta functions. A distinctive feature of derivative nonlinear Schrodinger equation is the z->-z symmetry, which induces a quotient reduction of the effective Abelian geometry via lambda=z^2. We further derive a kinetic equation for the effective velocity of the elliptic soliton gas and an Its-Izergin-Korepin-Slavnov type Fredholm determinant representation of the continuum tau-function.

Let $p$ be an odd prime and let $S\subseteq \Z_p$ be symmetric with $0\notin S$. Let $\Cay(\Z_p,S)$ be the undirected Cayley graph on $\Z_p$ in which $x$ and $y$ are adjacent if and only if $x-y\in S$. For $1\le \ell\le (p-1)/2$, define \[ \ex_{\Cay}(C_{2\ell+1},\Z_p)=\max\{|S|: S=-S,\ 0\notin S,\ \Cay(\Z_p,S)\text{ contains no }C_{2\ell+1}\}. \] Confirming a conjecture of Cashman and Kelley, we prove that if $p=2\ell+1$, then $\ex_{\Cay}(C_{2\ell+1},\Z_p)=0$, while if $p>2\ell+1$, then \[ \ex_{\Cay}(C_{2\ell+1},\Z_p)=2\floor{\frac{p+2\ell+1}{2(2\ell+1)}}. \] The proof combines a sharp additive zero-sum odd-girth argument with weak odd pancyclicity to transfer the result from odd-girth exclusion to fixed odd-cycle exclusion. We also give a canonical extremal family, an exact extremality criterion in terms of odd zero-sum avoidance, and an example showing that extremizers need not be dilates of the canonical construction.

We prove the Chern version of the constant holomorphic sectional curvature conjecture for compact locally conformal Kähler manifolds. More precisely, let $(M^n,h)$, $n\geq2$, be a compact locally conformal Kähler manifold whose Chern holomorphic sectional curvature is a constant $c$. We show that $h$ is necessarily Kähler and therefore is a complex space form metric of holomorphic sectional curvature $c$. In particular, when $c=0$, the metric is Kähler flat. This removes the nonpositivity assumption from a theorem of Chen, Chen, and Nie. The proof derives a curvature identity on the universal Kähler cover and shows that the covering metric is Bochner–Kähler. The globally conformally Kähler case is then treated by compact Bochner–Kähler rigidity, while the strict LCK case is excluded by Kamishima's uniformization theorem and the automorphy of the conformal factor.

We develop a sparse spectral method for solving partial differential equations on a class of two-dimensional geometries bounded by algebraic curves. The numerical method uses generalised bivariate Koornwinder polynomials which form a complete orthogonal basis, but one which is not graded in terms of polynomial degree. The polynomials are built from new families of univariate semiclassical orthogonal polynomials whose associated operator matrices (Jacobi matrices, raising matrices and differentiation matrices) are computed with optimal linear complexity in the number of basis functions. When used to discretise partial differential equations the resulting matrices are sparse enabling efficient numerical solution. Moreover, we develop fast transforms from values on a grid to expansion coefficients. The efficiency and accuracy of the resulting spectral method are illustrated through a series of numerical experiments on geometries whose boundaries are smooth and piecewise smooth including non-convex geometries. We observe algebraic convergence for geometries with corners, which accelerates to exponentially fast (spectral) convergence when the boundary is smooth.

We review here two different viewpoints on the Berry-Keating operator $H_{BK}$, whose connection to the Riemann hypothesis remains an intriguing and not yet fully understood question, despite considerable attention in the recent literature. In particular, we propose two somehow complementary views to $H_{BK}$: the first is based on a purely Hilbertian point of view, on dilation operators and on the Mellin transform. The second is a distributional approach, with a specific view to ladder operators, generalized eigenstates of $H_{BK}$, and generalized coherent states.

We revisit the bound states of the Morse potential in the momentum representation. After the ground-state factor is extracted, the remaining factors are finite rational functions of the momentum variable. These functions are eigenfunctions of a second-order difference operator and are identified with the symmetric specialization of a finite family of biorthogonal rational functions introduced by Koepf and Masjed-Jamei. They also satisfy a generalized eigenvalue problem in the degree variable, thereby placing the Morse momentum wavefunctions within the framework of rational bispectrality and $R_{II}$-type systems. Finally, after extraction of their poles, the same wavefunctions are expressed in terms of Meixner–Pollaczek polynomials with degree-dependent parameters. This gives a simple description of their zeros. The Morse potential thus provides a concrete quantum-mechanical realization of finite biorthogonal rational functions.

The $n$-queens completion threshold $qc(n)$ is the largest integer $k < n$ such that any placement of $k$ mutually non-attacking queens on an $n \times n$ chessboard can be completed to an $n$-queens configuration by adding $n - k$ queens. For all sufficiently large $n$, we improve the previously best-known upper bound on $qc(n)$ from $qc(n) \leq 0.241n$ to $qc(n) \leq 0.216n$, by constructing a non-completable partial configuration of fewer than $0.216n$ queens.

We study the distance function from a point target in the complement of a compact obstacle endowed with a smooth Riemannian metric. We prove that the obstacle necessarily generates singularities of the distance function: every sufficiently high level set contains a singular point. We also show that every singular point outside the obstacle belongs to a nontrivial Lipschitz arc of singularities, thereby extending to the constrained setting classical propagation results for Hamilton–Jacobi equations. Finally, we provide examples showing that these results are essentially sharp, including a nonconvex obstacle for which the distance function is differentiable at every boundary point.

Line planning, the problem of deciding which lines to operate and at what frequency, is a fundamental step in public transport planning. To accurately model passenger routing, the problem is traditionally defined on a change-and-go network (CGN), which captures transfers between lines exactly. However, this network grows large quickly and is hard to solve at scale. We compare the CGN against three more compact models, differing with respect to how transfers are approximated, and characterize how they relate in terms of solution quality and modeling accuracy. We develop state-of-the-art solution methods tailored to each model, and evaluate all four across 972 instances based on the Dutch and Swiss railway networks. Contrary to the CGN's canonical status, we find that it is competitive only on small or easy instances and often fails to find any feasible solution on large networks. Instead, a compact direct connection model performs best overall, finding the best solution on over 83% of instances. Our results indicate that carefully designed approximations, rather than exact transfer modeling, are the more promising foundation for large-scale line planning.

We study a stochastic mean-field interacting particle system whose state space is $\Y = \Tt^d \times \cP(U)$, where the first component represents a spatial variable and the second one is a probability measure over a compact metric space $U$. The dynamics are driven by locally Lipschitz drift operators: the spatial component evolves according to a Brownian diffusion, while the measure-valued component is perturbed by a projected cylindrical noise acting in the Arens–Eells space. We first establish existence and uniqueness of strong solutions for both the $N$-particle system and the associated nonlinear McKean–Vlasov equation under locally Lipschitz and linear growth assumptions on the drift coefficients. We then prove propagation of chaos: as $N\to\infty$, the empirical measure converges in expectation in Wasserstein–1 distance towards the unique McKean–Vlasov solution. Further, we investigate exponential convergence of the nonlinear McKean–Vlasov dynamics towards a unique invariant measure.