Papers

In this work, we develop a novel hybrid Schwarz method, termed as edge multiscale space based hybrid Schwarz (EMs-HS), for solving the Helmholtz problem with large wavenumbers. The problem is discretized using $H^1$-conforming nodal finite element methods on meshes of size $h$ decreasing faster than $k^{-3/2}$ such that the discretization error can be bounded by the data uniformly with respect to $k$. EMs-HS consists of a one-level Schwarz preconditioner (RAS-imp) and a coarse solver in a multiplicative way. The RAS-imp preconditioner solves local problems on overlapping subdomains with impedance boundary conditions in parallel, and combines the local solutions using partition of unity. On the PDE level, this coarse space corresponds to an edge multiscale space proposed in {[S. Fu, G. Li, R. Craster, and S. Guenneau, Multiscale Model. Simul., 19(4):1684-1709, 2021]}. The approximation properties of this coarse space to the discrete solutions from the primal and adjoint problems are derived by a novel conforming Galerkin finite element method based upon a nonstandard variational formulation. This work represents the first theoretical analysis of an edge multiscale space at the discrete level. We prove that the EMs-HS preconditioner leads to a convergent fixed-point iteration uniformly for large wavenumbers, by rigorously analyzing the approximation properties of the coarse space to the global Helmholtz harmonic extension part and to the solution of the adjoint problem. We provide extensive numerical results on the sharpness of the theoretical findings and also demonstrate the method on challenging heterogeneous models.

We study inference for continuous linear functionals of solutions to ill-posed inverse problems, including Euclidean parameters arising in nonparametric instrumental variables, proximal causal inference, and shadow-variable models. Although pointwise root-$n$ inference is possible under standard conditions, we show that this need not translate into locally reliable inference. Under a model imposing only minimal regularity conditions, the target is Hellinger discontinuous when both inverse problems are ill-posed: arbitrarily small changes in the data-generating law can produce arbitrarily large changes in the target, ruling out informative locally uniform inference. Imposing a uniform bound on a minimum-norm solution restores Hellinger continuity under regularity conditions, but does not by itself ensure stable root-n inference. If, in addition, both inverse problems are ill-posed and the solution whose norm is uniformly bounded is nonunique at a law $P_0$, then under regularity conditions the target fails to be pathwise differentiable; consequently, asymptotically linear estimators exhibit nonvanishing root-$n$ bias under some local alternatives, causing Wald confidence intervals to undercover asymptotically. Under regularity conditions, pathwise differentiability is restored if the uniformly bounded solution is unique at $P_0$. It is also restored, regardless of uniqueness, under the stronger requirement that the corresponding solution satisfy a uniformly bounded $\beta$-source condition for some $\beta$>0. We relate these regimes to very weak, weak, and strong identification in instrumental variables models.

We study positive limit cycles of the Abel equation $$\dot{x}=A(t)x^{3}+B(t)x^{2},$$ where $A$ and $B$ are $2\pi$-periodic functions. By developing a half-period convexity method, we establish a new criterion ensuring that the equation has at most one positive limit cycle. Moreover, if such a limit cycle exists, it is hyperbolic. As applications, we consider two classes of planar polynomial systems that can be reduced to Abel equations. For a subfamily of cubic rigid systems, we prove that there is at most one limit cycle, thereby giving a partial answer to the first problem in Gasull's list of 33 open problems. For quadratic systems, we derive a new uniqueness criterion for limit cycles surrounding the origin and provide explicit parameter regions under which the criterion is satisfied.

We prove that contact $(+1)$-surgery along a Legendrian $n$-sphere $\Lambda\subset(\mathbb{S}^{2n+1},\xi_{\mathrm{std}})$ with \[ \mathrm{tb}(\Lambda)\ne (-1)^{\frac{n(n-1)}{2}+1} \] has vanishing contact homology. In dimension three, this produces new examples of tight algebraically overtwisted contact manifolds, vastly extending the authors' previous results: If a Legendrian knot in $(\mathbb{S}^3,\xi_{\mathrm{std}})$ admits an exact, oriented Lagrangian filling that is not a disk, then contact $(+1/k)$-surgery on it is algebraically overtwisted and tight for every positive integer $k$. Our proofs use versions of symplectic cohomology and wrapped Floer cohomology defined by augmentations of a chain-level contact homology algebra.

We study the local asymptotic behavior of nonnegative type I blow-up solutions to semilinear heat equations with nonlinearities that need not be scale invariant. Using a quasi-similarity transformation adapted to the associated ordinary differential equation, we establish locally uniform convergence to a constant profile on the parabolic spatial scale near every interior blow-up point. Under a quantitative assumption on the nonlinearity at infinity, our results cover regularly varying nonlinearities in the Sobolev subcritical and critical ranges and, in one and two space dimensions, rapidly growing nonlinearities including double exponentials. A key ingredient in this analysis is a positive lower bound for the normalized solution at interior blow-up points. In fact, under weaker assumptions on the nonlinearity, we prove this bound in arbitrary space dimension without assuming radial symmetry.

A sequence of eigenvalue problems requires solving $A_i\vec{x}_i= \lambda_i\vec{x}_i$ where $i \in \{1, \dots, N\}$, $N \gg 1$ and (usually) $\lambda_i$ is the largest or smallest eigenvalue of $A_i$. Often, the matrices $A_i$ come from a discretization of a smooth map depending on a parameter $p \rightarrow A(p)$, that is $A_i = A(p_i)$, so the matrices $A_i$ change gradually along the sequence. Under this latter assumption, we introduce an efficient algorithm for solving sequences of eigenvalue problems. Our algorithm combines the ideas of reduced order model and row subsampling, as done in [E. Jones and Y. Nakatsukasa, arXiv, 2025] for linear systems. In addition, in our approach, the same set of rows is guaranteed to work well independently of the value of the parameter $p$ or the matrix $A(p)$ as long as the snapshot matrix is unchanged and given that the row subset was chosen well at the beginning. Furthermore, we implement an update strategy of the snapshot matrix. Along these results, we present theoretical analysis of the snapshot matrix for eigenvalue problems, showing an exponential decay of the singular values, and we give bounds on the error on eigenvectors for unseen value of the parameter. This analysis generalizes that in [M. Guido, D. Kressner, and P. Ricci, J. Sci. Comput., 2024] and shows that, regardless of the distribution of the snapshot points, the convergence is always (asymptotically) exponential in the number of snapshot points if the function $A(p)$ admits an analytical extension. We further generalize our algorithm for computing largest or smallest singular values and relative singular vectors and to generalized eigenvalue problems. We conclude with numerical results showing the efficiency of our approach.

The shape of a number field $K$ of degree $n\geq2$ is the similarity class of the lattice obtained by projecting $\mathscr{O}_K$ orthogonally away from the rational line in Minkowski space. It defines a point in the space of shapes of rank-$(n-1)$ lattices, $\mathscr{S}_{n-1}:=\mathrm{GL}_{n-1}(\mathbb{Z})\backslash \mathrm{GL}_{n-1}(\mathbb{R})/\mathrm{GO}_{n-1}(\mathbb{R})$, where $\mathrm{GO}_{n-1}(\mathbb{R})$ is the group of orthogonal similitudes. Let $G$ be a fixed nontrivial finite abelian group, and let $K/\mathbb{Q}$ vary over tamely ramified Galois extensions with Galois group isomorphic to $G$. We study the variation of the shapes of such number fields in $\mathscr S_{n-1}$, where $n=|G|$.

We introduce a new Euclidean quantum model with zero-spin scaled scalar fields on a spacetime hypercubic lattice, which we refer to as the hybrid model. The model coupling parameters are the hopping parameter $\kappa^2>0$ and the field interaction coupling $\alpha>0$. Using the technique of lattice scaled fields, with a noncanonical scaling that corresponds to a partial renormalization (wave function renormalization), we show that the partition function of this model satisfies both thermodynamic and ultraviolet stability bounds, with constants which are uniform in the lattice volume $\Lambda_s$ and the lattice spacing $a\in(0,1]$. As a consequence, the model has a finite free energy per degree of freedom in the thermodynamic limit $\Lambda_s\nearrow\infty$, at least in the subsequential sense. Subsequently, we can also take the ultraviolet or continuum limit $a\searrow 0$. Besides the finiteness property of the free energy, for sufficiently small values of the model coupling parameters $\kappa^2$ and $\alpha$, we show the model satisfies Osterwalder-Schrader positivity and construct the underlying physical quantum mechanical Hilbert space, and, using a polymer expansion, we show correlations also remain bounded in the thermodynamic limit. Finally, we perturbatively compute an infinite volume truncated four-point correlation showing it is nonzero. Noting that, for fixed $a$, perturbation series in $\kappa^2$ and $\alpha$ converges, these results are sufficient to show that our hybrid model makes sense as a lattice quantum field theory in spacetime dimension $d=3,4$, with possible extensions to other higher values of $d$.

Bachmann and van Ittersum conjectured that every polynomial $f\in\mathbb{Q}[x,y,z]$ satisfying $$ f(x,y,z)=f(y,z,x),\qquad f(x,y,z)+f(-y,x-y,z)+f(y-x,-x,z)=0 $$ is a sum of polynomials in two variables. We prove this conjecture over every field of characteristic zero and determine all homogeneous solutions. Moreover, restriction to $z=0$ identifies the degree $n$ solution space with the space of binary period polynomials of degree $n$ normalized to vanish on both coordinate axes. The solution corresponding to such a polynomial $q$ is $$ f(x,y,z)=q(x,y)+q(y,z)+q(z,x). $$ There are no nonzero homogeneous solutions of odd degree. For even degrees $n$, the symmetric and alternating solution spaces are each isomorphic over $\mathbb{C}$ to the space of weight $n+2$ cusp forms on $SL_2(\mathbb{Z})$. The proof connects the polynomial relations with the cohomology of arithmetic groups through the rational Tits building for $GL_3$. Cusp forms enter both the description of the solutions and the proof that there are no others. The main theorems and the essential argument are formalized unconditionally in Lean~4 by AxiomProver.

In this paper, we study the Chow group of zero-cycles $\text{CH}_0(S)$ for certain Enriques quotients $S$ of the Kummer surface associated to the product of two geometrically non-isogenous elliptic curves $E_1$ and $E_2$. We show that over fields of characteristic different from $2$ or $3$, the degree-zero subgroup $A_0(S) \subset \text{CH}_0(S)$ is 16-torsion. Over p-adic fields, we compute the transcendental Brauer group $\text{Br}(S)/\text{Br}_1(S)$. Using the Brauer-Manin pairing, we construct examples of Enriques surfaces for which $A_0(S)$ attains nontrivial 2-torsion elements arising from push-forward from the abelian surface $E_1\times E_2$.

In recent work (arXiv:2610.10341), the authors introduced a framework for studying the vanishing viscosity limit for sequences of weak solutions of the multi-dimensional compressible Navier-Stokes equations to planar shocks of the compressible Euler equations. In this paper, we introduce these methods and apply them to the multi-dimensional viscous Burgers' equation. We show that arbitrarily large viscous shocks enjoy $L^2$-stability among perturbations whose gradients in the directions transverse to the shock are controlled in an appropriate manner. As an application, we study the rate of convergence to the inviscid shock in the vanishing viscosity limit.

The torn paper channel breaks a codeword into fragments and returns all fragments without their original order. The noisy torn paper channel additionally introduces bit substitution errors to each fragment. Recovering the message therefore requires accounting for both bit errors and uncertainty about the fragment order. We establish a capacity upper bound for the noisy torn paper channel that matches the achievable rate of Ravi, Vahid, and Shomorony and resolves their conjecture. Our results also determine the capacity of the noisy shuffling channel of Shomorony and Heckel, in which all fragments have the same length, and resolve their open question.

We construct computable a posteriori error estimators for parametric families of nonlinear elliptic problems by measuring the residual of a variational formulation in the dual norm of a closed subspace of the test space. A single abstract theorem shows that the resulting estimator is bounded above and below by the error in the trial norm, i.e., it is a variationally correct loss function. Two instances are covered. The first considers an $L_2$-like test space, yielding a nonlinear first-order system least-squares loss function that certifies arbitrary, possibly meshless, approximations. The second breaks the test space, yielding a nonlinear discontinuous Petrov-Galerkin loss function computed by element-local problems. All assumptions are verified for quasilinear diffusion, including monotone fluxes and semilinear problems, and a nonlocal Kirchhoff model. Numerical studies train neural networks approximating the parameter-to-solution maps with the latter loss and measure the ratio of loss to squared error.

Let $p$ be a prime $\ge 5$. We determine the semisimplifications of the mod $p$ reductions of irreducible two-dimensional crystalline representations of the Galois group of ${\mathbb Q}_p$ of slope $2$ and weights $k\ge3p+4$ with $k \not \equiv 6 \bmod (p-1)$. The proof relies on the compatibility between the $p$-adic and the mod $p$ Local Langlands Correspondences for $\mathrm{GL}_2(\mathbb{Q}_p)$.

We study the linearization, with respect to zeroth-order perturbations, of the local Dirichlet-to-Neumann map of a formally self-adjoint elliptic operator $P=-\Delta_g+Q$ on a compact Riemannian manifold $(M,g)$ of dimension $n\geq 2$ with boundary, where $Q$ is of order at most one. Near a boundary point where the boundary, metric, and $P$ are analytic, we show that the linearized map is a classical analytic pseudodifferential operator of order $-1$ if and only if the zeroth-order perturbation $q$ extends analytically across the boundary. We also prove that local boundary data in an arbitrary open analytic boundary patch determines $q$ locally. In particular, if the measurement set meets every connected component of $M$, the local linearized DN map is injective on $L^\infty$ perturbations. Finally, we obtain a global completeness result for products of pairs of solutions whose boundary traces are supported in the measurement set. The global recovery proceeds by local steps using Runge approximation.

We give matching two-sided estimates for heat kernels obtained from Schrödinger operators $\Delta+W$ on weighted graphs. The graphs we consider are transient and admit two-sided sub-Gaussian heat kernel estimates for the Laplacian $\Delta$, and the potentials $W$ decay to zero at infinity. We offer two proofs for one of our main results: one using the Feynman-Kac formula, and one without using it. Our results apply to fractal-type graphs such as graphical Sierpinski carpets, among other examples.

We examine the metamathematical significance of the Levy scheme $\Lambda$, which asserts, for each natural number $n$, the existence of an $n$-Mahlo cardinal $\kappa$ such that $V_\kappa\prec_{\Sigma_n}V$. Several initially unrelated constructions single out $\mathsf{ZFC}+\Lambda$: comparisons with the model theory of arithmetic and finite set theory; class theories in which the class of ordinals has the tree property; set theories with a proper class of indiscernibles; and interpretations of ordinary set theory in Cantorian parts of models of the extension $\mathsf{NFUA}$ of Quine–Jensen set theory $\mathsf{NFU}$. Taken together, these results provide a precise sense in which $\mathsf{ZFC}+\Lambda$ is an inevitable extension of $\mathsf{ZFC}$ if one attempts to generalize certain metamathematical results about the universe of all finite sets to the entire universe of sets.

We study the $\alpha$-Ricci flow $(g(t),\alpha(t))$, which is a Ricci flow coupled with a harmonic 1-form heat flow, on the 2-torus $M$. We prove global existence for this flow as well as a Type III curvature estimate. The long-time behavior of the $\alpha$-Ricci flow depends on the cohomology class $[\alpha] \in H^1(M,\mathbb{R})$, which is invariant under the flow. If $[\alpha]$ is rational, then as $t\to\infty$, $(M,t^{-1}g(t))$ Gromov–Hausdorff converges to $S^1$ with length $\ell([\alpha])/\sqrt 2$ where $\ell([\alpha])$ is the primitive length of $[\alpha]$. On the universal cover, the rescaled flow $(s^{-1}\tilde g(st),\tilde\alpha(st))$ smoothly converges to an expander $(g^*(t),\alpha^*(t))$ on $\mathbb{R}^2$ as $s\to \infty$ where $\alpha^*(t)$ is nonzero and constant in spacetime and $g^*(t)$ is flat and expanding in the $\alpha^*$ direction. If $[\alpha]$ is irrational, then there is a sequence $s_i\to\infty$ such that $(M,s_i^{-1}g(s_it))$ Gromov–Hausdorff converges to a point. Moreover, on the universal cover, $(s_i^{-1}\tilde g(s_it),\tilde\alpha(s_it))$ smoothly converges to $(g^*(t),\alpha^*(t))$.

We give an infinite family of non-diffeomorphic exotic $\mathbb{R}^4$'s, each of which is the union of two copies of a standard open tubular neighborhood of the circle, which intersect in a standard open tubular neighborhood of the wedge of two circles. This gives the simplest known decomposition of an exotic $\mathbb{R}^4$ into two open sets, making progress on Question 4.43 (A) of Kirby's 1995 problem list. The decompositions are obtained by studying fibered slice disk complements in certain Casson-Gordon homotopy 4-balls, for which we provide similar decompositions. We distinguish the $\mathbb{R}^4$'s using end Floer homology.

Generators of continuous-time quantum Markovian evolution, known as Lindblad operators, play a key role in the theory of open quantum systems. The question of how to define and characterize random Lindblad operators has attracted considerable attention in recent years. Purely dissipative random Lindbladians have been extensively studied using toolboxes of free probability and non-Hermitian random matrix theories, which helped, among other results, to uncover a characteristic lemon-like spectral support. In this work, we focus on random Lindbladians with a Hamiltonian component. We analyze three ensembles of increasing complexity, designed to illustrate the joint effects of dissipative and Hamiltonian dynamics. While the first two ensembles allow for an exact analytic treatment with standard probabilistic and free-probability methods, the generic case of independently random Hamiltonian and dissipative components requires a novel framework that combines operator-valued free probability with sequential noise averaging. For this ensemble, our approach preserves the underlying tensor-product structure of the Lindblad operator and derives a general equation for the spectral boundary. We characterize the effect of Hamiltonian-induced spectral squeezing and explain how the spectral support changes with the relative Hamiltonian strength and the rank of the dissipative component.

We study the Ahlfors-regular conformal dimension of spaces with locally expanding self-covering maps and, more generally, limit spaces of contracting self-similar groups. We show that the conformal dimension is characterized by combinatorial moduli associated with the Schreier graphs of these groups. We compare these moduli with contraction coefficients of self-similar group actions and with algebras of functions on the limit spaces satisfying asymptotic $p$-energy conditions. We also give conditions on the group when the conformal dimension is equal to the infimum of visual metrics on the self-similarity complex.

The aim of this note is to introduce natural geometry, in which the rectilinear plane is a primitive. This plane is admitted together with its rectitude, and its rectitude states that a Lambert quadrilateral has its fourth angle right. The fourth angle of a Lambert quadrilateral may be right, acute or obtuse, and this trichotomy distinguishes three planes. Starting from the rectitude, we prove the uniqueness of the parallel by a chain of neutral steps: the rectangle, the triangle whose angles sum to $\pi$, and Legendre's theorem \cite{legendre}, all borrowed from the classical literature. That uniqueness is equivalent to the fifth postulate, which we thus reach in Euclid's own wording: the equivalence is read in the natural direction, and it is not a derivation of the fifth postulate from the other four, since the independence of that postulate is not in question. In the rectilinear plane the parallels are characterised as exactly the images of a line under a point-reflection, and equivalently as the pairs of equidistant lines. The other two planes are then placed beside the rectilinear one, and the same definitions serve for all three.

Let $G$ be a finite group, $M$ a $G$-module, and $F$ a field. A continuous group homomorphism $i : \Gamma_L=\text{Gal}(L_{\text{sep}}/L) \rightarrow G$ yields a homomorphism $i^* : H^d(G,M) \rightarrow H^d(L,M)$ for every $d\in \mathbb{N}$. An element $u \in H^d(G,M)$ is negligible over $F$ if $u\in\ker(i^*)$ for every $i$ and every field extension $L/F$. If $M=\mathbb{F}_p$ for a prime $p$ with a trivial $G$-action, an element $u$ is eventually negligible over $F$ if $u^k$ is negligible over $F$ for some $k\geq1$. The (eventually) negligible elements over $F$ form an ideal of the cohomology ring $H^*(G,\mathbb{F}_p)$, called the (eventually) negligible ideal. We prove an analog of Quillen-Venkov for eventually negligible elements. An element $u\in H^*(G,\mathbb{F}_p)$ is eventually negligible over $F$ if and only if the restriction of $u$ to the cohomology of each elementary abelian $p$-subgroup of $G$ is eventually negligible over $F$. Paired with applications of a Gysin-type long exact sequence, we identify generators of the negligible ideal over $\mathbb{Q}$ of the mod-2 cohomology of dihedral groups, cyclic $2$-groups, and generalized quaternion groups. We prove that the negligible ideal over $\mathbb{Q}$ of the mod-2 cohomology of every finite irreducible Coxeter group is radical. The result was previously known for Weyl groups; we prove it for the remaining non-Weyl types $I_2(m)$, $H_3$, and $H_4$.

Reduced-order models (ROMs) make model predictive control (MPC) of high-dimensional nonlinear systems tractable, but the resulting modeling error can lead to constraint violations on the real system. We introduce CoRE-MPC, a robust MPC formulation that plans with a nonlinear ROM and guarantees recursive feasibility and constraint satisfaction for the full-order system. It can be applied with a broad class of nonlinear model reduction techniques and the resulting MPC does not require evaluating the full-order model online, thus ensuring computational efficiency. The key insight is in how the ROM affects the dynamics of the prediction error. We account for this with two error radii that are propagated alongside the ROM and provide a guaranteed bound on the prediction error. The approach is validated in simulation on an 80-state oscillator chain, a Burgers equation, and a 9,768-state finite-element soft robot, where a two-state ROM with data-driven bounds keeps all 1,628 nodes clear of a wall at a median solve time below 1 ms for a 20 ms control period. Across these examples, CoRE-MPC is 13 to 61 times faster than full-order robust MPC at only 27 to 35% higher closed-loop cost, whereas existing certified schemes are either infeasible or at least twice as conservative. Code: [released upon publication].

We investigate Conjecture 7.4 of Guo and Krattenthaler \cite{GK}, which predicts that for every positive integer $n$, the polynomial \[ P_n(q) = \frac{(1-q)^2}{(1-q^{10n-1})(1-q^{15n-1})}\qbin{30n}{5n} \] has non-negative coefficients, except for $q^1$ and $q^{125n^2-25n+3}$, which equal $-1$. We study this problem through three main perspectives: First, by linking the boundary coefficients to the second differences of the partition function, we settle the first and last $25n+1$ coefficients via a novel partition injection. Second, by introducing a “defect” generalization of Andrews' classical lemma, we reduce the conjecture to an almost-unimodality property and employ a uniform saddle-point analysis to prove it analytically for all large $n \ge 2\cdot 10^{29}$. Finally, we connect these quotients to graph binomial coefficients, providing a natural orbit-counting interpretation via Burnside's lemma on directed cycles and paths.

The Briot–Bouquet theorem describes the meromorphic solutions of autonomous first-order algebraic differential equations. We prove Eremenko's conjecture extending it to all orders: if $f$ is meromorphic in $\mathbb{C}$ and $P(f,f^{(n)})=0$ for a nonzero polynomial $P$, then $f$ is rational, elliptic, a rational function of $e^{cz}$ with $c\neq 0$, or satisfies $f^{(n)}=af+b$. In particular, every meromorphic solution with $P$ irreducible of degree at least two belongs to the Weierstrass class. Solutions with poles were classified by Eremenko, Liao and Ng. We settle the remaining entire case with a rigidity theorem for polynomial compositions: if $h$ is nonconstant entire and $D^nR(h)=S(h)$, where $D=d/dz$ and $\deg R=\deg S\ge 2$, then $h(z)=Ae^{bz}+Be^{-bz}+C$. No growth assumption is needed. The proof turns the cancellation of leading terms into a contraction of indicator diagrams and obtains incompatible zero counts near an exposed edge. As an arithmetic application, an algebraic Laurent expansion at one algebraic point and an algebraic value at another force a solution to lie in $\overline{\mathbb{Q}}(z)$. Hence any two distinct derivatives of a function in Mahler's class (transcendental meromorphic functions with algebraic Laurent coefficients at every algebraic point) are algebraically independent over $\mathbb{C}$.

Given a complete and non-compact Riemannian manifold of dimension $3\le n\le 7$, we find conditions for the existence of compact exhaustions $\{U_i\}_{i=1}^\infty$ such that all boundaries $\partial U_i$ have the same sign of mean curvature. In particular, we show that in these dimensions, a manifold with one end satisfying a no tentacle condition always has a weakly mean convex, weakly mean concave, or minimal exhaustion, according to the behaviour of the area profile of separating hypersurfaces. This is a version of a result sketched by Gromov, under a condition which is weaker than his thickness at infinity. In more general situations, we study the existence of exhaustions which are weakly mean concave or weakly mean convex apart from an arbitrarily small $L^1$ error. That is, we find sufficient conditions for the existence of exhaustions $\{U_i\}_{i=1}^\infty$ which satisfy $\int_{\partial U_i}H^+\to0$, $\int_{\partial U_i}H^-\to0$, or $\int_{\partial U_i}|H|\to0$, where $H^-$, $H^+$, and $|H|$ are the negative part, positive part, and the absolute value of the mean curvature respectively. As an application of our methods, we study the existence of almost $c$-convex exhaustions, that is, exhaustions whose boundaries satisfy $\int_{\partial U_i}(H-c)^-\to0$ for some $c>0$.

In this note we explore the following simple question: How does the classic gambler's ruin phenomenon change if we allow the gambler financial leverage? We show that, with a moderate amount of leverage, it is possible for an (idealized) gambler to sustain seemingly improbable profits over a remarkably long time horizon without ever going bankrupt, even if the gambler makes only fair bets. With a leverage ratio of $L = 10$ (less than what Lehman Brothers Holdings had from 1995–2008), the probability of multiplying one's money by $1000\times$ without going bankrupt is roughly $50\%$. More generally, given $L$-fold leverage, and given the goal of multiplying one's money by a factor of $r > 1$, the optimal gambling strategy has probability roughly $r^{-1/L}$ of success. The results in this note are designed as part of the course development for the CMU randomized algorithms course.

As its 40th anniversary nears, we present a history of the International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC), through interviews with figures from its past, present, and future.

We construct finite Gamma convolutions approximating the first hitting time of $1$ by a Bessel process of index $\nu>-1$ started at the origin. The scale measure in the classical exponential representation of this law is obtained by squaring the positive half of a dilated modified Lommel polynomial orthogonality measure. Its Jacobi recurrence yields an $N$-factor Gamma convolution that uniquely matches the first $2N$ cumulants, the maximal number attainable in this class. From an exact Gaussian quadrature remainder we derive supergeometric convergence of the Laplace transforms on compact subsets of a common slit plane and geometric convergence of the distribution functions uniformly on the real line. The approximants have total Gamma shape $N(N+\nu+1)$, and their Laplace transforms and negative moments decrease to those of the exit law.