Papers

Let $G=(V,E)$ be a possibly infinite, locally finite graph with non-negative Ollivier–Ricci curvature and degrees bounded by $d<\infty$. We prove that there exists a constant $C_d$ such that the continuous-time random walk displacement and log-volume growth satisfy \[ \mathbb{E}_x \mathrm{dist}(x,X_t)^2 \le t \exp\left[C_d \sqrt{\log t \log\log t}\right], \] \[ \log \mathrm{Vol}(B(x,r)) \le \exp\left[C_d \sqrt{\log r \log\log r}\right], \] for every $x\in V$ and all $r,t \ge e^e$.

These notes provide a self-consistent summary of the stochastic approach to anomalous diffusion based on Continuous Time Random Walks (CTRWs) and their scaling limits. An introduction to CTRWs and their relationship to the theory of semi-Markov processes is provided. A general technique to study scaling limits of CTRWs is then described, and the semi-Markov property of the limiting processes is discussed. With this at hand, the connection of limit processes with non-local (fractional-type) equations is introduced, and the most recent (and general) contributions, going far beyond fractional equations, are also described. Indeed, the theory presented here includes very general non-local evolution equations as abstract Cauchy problems as well as pointwise non-local fractional diffusion equations in bounded domains.

In this paper, we establish asymptotic formulas for the first and second twisted moments of $r$-th order Hecke $L$-functions over global fields that contain the $2r$-th roots of unity, for $r\ge 3$. We focus primarily on algebraic number fields. As a consequence, we establish a positive proportion of non-vanishing central values for these $L$-functions, specifically for families of both square-free and $r$-th power-free ideals. Our approach is based on the machinery of multiple Dirichlet series.

We present stellarator optimization algorithms for designing the edge magnetic structure in vacuum fields, together with the vacuum vessel. First, we introduce a numerical method that robustly computes periodic-orbit fixed points of any type (elliptic, hyperbolic, or parabolic), which could form the basis of a divertor. To couple divertor and vacuum vessel design, we introduce parametric families of vacuum vessels for which point-to-vessel distances, and their derivatives, can be computed efficiently. The resulting algorithms use signed distance functions to enforce coil-vessel clearance while allowing coils to be placed on or off the vessel. Using these methods, we jointly optimize modular coils and the vacuum vessel to realize a wide range of magnetic topologies for diverting exhaust, including standard X-point divertors, and single- and double-null configurations. For the first time, we show that precise snowflake divertors can be achieved in stellarators. Using this framework, we generate a number of quasi-axisymmetric stellarator designs with compatible vacuum vessels and diverse divertor architectures, which we consider to be candidates for a next-generation STAR Lite prototype.

In the theory of spectral algebraic geometry, few objects receive as much study as the spectrum of topological modular forms. In this paper, we compute the strict units of topological modular forms, defined as the connective cover of the mapping spectrum from $\mathbb{Z}$ to the units spectrum.

We develop a global differential calculus on the $L^2$-Wasserstein space over a closed Riemannian manifold, based on derivations of cylinder functions rather than on the standard pointwise approach. Within this framework we define some fundamental geometric tools. In particular, we show that the Levi-Civita connection on the base manifold lifts to the unique torsion-free connection compatible with the 'extended' Otto metric. The corresponding Riemann tensor is exactly the lift of the base Riemann tensor, which shows that - in this framework - the correction terms of the classical gradient formalism are not intrinsic curvature terms, but arise from the projection onto the measure-dependent gradient distribution. This allows us to revisit with a global and purely differential approach the smooth computations by J. Lott, Comm. Math. Phys., 277(2):423-437, 2007, reaching partially different conclusions.

Let $\Omega\subset\mathbb H^2$ be a bounded smooth horoconvex domain and let $\psi_1>0$ be its first Dirichlet eigenfunction. We prove that \[ \operatorname{Hess}_{\mathbb H^2}(-\log\psi_1)>0 \] throughout $\Omega$, with no restriction on the diameter or the first eigenvalue. The proof is by contradiction. A degenerate Hessian would yield a shifted translation Killing derivative with a singular interior zero. Then the boundary-zero theorem of Grossi and Provenzano shows that the shifted Killing derivative has exactly two zeros on the boundary. A nodal-domain argument on the surface rules this out. As an application we prove that every superlevel set of $\psi_1$ is horoconvex: every level curve has geodesic curvature at least $1$. The Hessian bound makes the shifted construction available for Killing fields with nonvanishing rotation part, and yields the pointwise inequality $|(\operatorname{Hess} u)^{-1}J\nabla u|\le1$ for $u=-\log\psi_1$, where $J$ is rotation by $\pi/2$; a boundary-zero count for translation fields with arbitrary axis completes the argument.

We obtain novel integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous initial-boundary-value as well as interface problems for the Barenblatt-Zheltov-Kochina pseudo-parabolic equation of Sobolev-Galpern type formulated on the real line, half-line and finite interval. This fundamental partial differential equation (PDE) of mathematical physics emerges in a wide variety of natural phenomena and applied sciences including continuum mechanics, thermodynamics, chemical engineering, solid-state electronics, semi-conductor devices, battery research, and nanotechnology. A suitable implementation of a modern methodology, known as Unified Transform Method, is in force throughout this study, with particular challenges arising due to the higher-order mixed-derivative term of the PDE and the generality of the problems under consideration altogether. The boundary and interface conditions appear to be non-standard but are naturally dictated by the structure of the PDE itself. Our explicit analytical formulae directly lend themselves to future explorations of the solutions' qualitative properties such as asymptotic behavior, spatio-temporal dynamics, regularity and well-posedness. This work is expected to be of utility also in the investigation of nonlinear counterparts as well as towards the study of phase-transition phenomena and free-boundary problems, where the interface evolves dynamically according to energy balance laws.

In this paper, we show the $L^2$ contraction property of the planar oscillatory or monotone dispersive shock profiles of the dissipative Kadomtsev-Petviashvili (KP) equation modelling water waves and the multi-dimensional Korteweg-de Vries (KdV) Burgers equation under arbitrarily large perturbations in two space dimensions, up to Lipschitz time-dependent shifts. This stability result extends the results of two recent papers by Chen, Eun, Kang, and Shen on $L^2$ contraction for the KdV-Burgers equation.

We provide a short proof of a Ky Fan-type majorization relation for the singular values of a sum of binary tensor products of matrices. This generalizes Alhejji's result (arXiv:2410.18254) from two summands to arbitrary sums and from positive matrices to arbitrary matrices. As an application, we show a majorization relation between the singular values of a completely positive map and those of its Kraus operators.

We consider the $k$-generalized Korteweg-de Vries equation \begin{equation*} \partial_{t}v+\partial_{x}^{3}v +\partial_{x}(v^{k+1})=0, \qquad (t,x)\in\mathbb R\times\mathbb R, \qquad k\in\mathbb Z_{+}, \end{equation*} emphasizing the modified case $k=2$ and the cubic case $k=3$. We prove that solutions from low-regularity, possibly singular, data $u_{0}$ become real analytic in $(t,x)$ for all $t\neq0$, whenever $u_0$ satisfies a Nelson-type condition \begin{equation*} \sum_{k=0}^{\infty}\frac{\alpha^{k}}{k!}\,\big\|(x\partial_x)^{k}u_0\big\|_{X}<\infty, \end{equation*} for some $\alpha>0$. For the cubic equation, this includes data such as $u_0=x_{+}^{\lambda}$, singular at the origin; for mKdV even discontinuous data yield analytic solutions e.g $u_{0}(x)=\sgn(x)e^{-x^{2}}$.For mKdV we work in the sharp well-posedness space $X=\widehat H^{r}_{s}(\mathbb R)$, $r\in(1,2]$, $s\geq\frac12-\frac1{2r}$, with $r=2$ recovering analyticity on $H^s(\mathbb R)$, $s\geq\frac14$, the best mKdV space in the sense of Kato; for the cubic equation we work in $X=H^{s}(\mathbb R)$, $s>-\frac16$, approaching the critical exponent $s=-\frac16$ from above. Analyticity thus holds on the largest known data class for which mKdV is well-posed, and on data approaching the corresponding threshold for the cubic equation, extending a known smoothing effect for KdV ($k=1$) to the modified and cubic nonlinearities and to a broader class of singular profiles, avoiding pseudo-differential calculus via Lorentz-space refinements replacing Bourgain-space localization. The mechanism is dispersive: analyticity is generated by the flow, symmetrically in time, and singular profiles become instantaneously analytic for $t\neq0$.

We resolve a conjecture of Guy David and Svitlana Mayboroda. We show that if $\Omega \subset \mathbb{R}^n$ is a uniform domain with $(n-1)$-dimensional Ahlfors David regular boundary $E = \partial \Omega$, and if $E$ satisfies the Weak Geometric Lemma, then $E$ satisfies the Bilateral Weak Geometric Lemma. In particular, $E$ is uniformly rectifiable.

Most existing seizure detection algorithms require extensive pre-processing of the data and rely on heuristic or currently unexplainable machine learning approaches. These approaches often struggle with balancing detection sensitivity and specificity in the presence of variable seizure morphologies, interictal epileptiform discharges, and artefacts. Here, we consider an alternative approach: our seizure detection algorithm, which is based on the concept of critical transitions and overcomes the aforementioned limitations. Specifically, we perform a receiver-operating-characteristic analysis to quantify the performance of our algorithm in terms of its agreement with expert annotations of seizure onset and offset times in the voltage recordings of seizure activity in epileptic rodents with different seizure morphologies. We demonstrate how performance depends on algorithm parameters and varies across different rodent recording sessions. We determine the optimal set of algorithm parameters for each recording session, with near expert-level performance achieved in most cases. Finally, we derive a single general set of algorithm parameters applicable across all recording sessions. The algorithm maintains its high performance in this general setting, demonstrating its versatility, robustness across varying seizure morphologies, and potential to complement machine learning algorithms.

In this work, we propose a general discretization strategy for solving the London equation for type-II superconductors in the whole space $\mathbb{R}^3$. To compute the magnetic field $H_0$, we reformulate the problem for the magnetic potential as a transmission problem and discretize it through a nonstandard FEM-BEM coupling. This formulation accounts for both the bounded interior domain and the unbounded exterior domain without introducing an artificial truncation. We then compute the vector field $B_0$, which arises from the Helmholtz-Hodge decomposition of the magnetic potential in the superconducting sample. This field enters the isoflux problem, which identifies the curves along which vortex nucleation first becomes energetically favorable in the Ginzburg–Landau model of superconductivity. We recast the equations for $B_0$ using the mixed formulation of Kikuchi, in which the divergence-free constraint is imposed weakly, and discretize the resulting problem using a classical $H(\operatorname{curl})$-conforming finite element discretization. We validate our discretization strategy through convergence tests and conclude with an application to the isoflux problem. For a ball under a constant applied magnetic field, the unique maximizer is the diameter aligned with the field. For ellipsoids under a constant applied magnetic field aligned with their major axis, our computations provide numerical evidence of a different behavior in sufficiently elongated, cigar-shaped geometries: off-axis competitors reminiscent of U-shaped vortex configurations attain a larger isoflux ratio than the major axis. Since the major axis is therefore not a maximizer, any off-axis maximizer generates, by rotational symmetry, a continuous family of equivalent configurations, implying non-uniqueness and the presence of a degenerate rotational direction in the isoflux problem.

This paper investigates the relationship between coding theory and extremal combinatorics by representing codes in general metric spaces as independent sets in proximity graphs. We provide a generalized framework for the Gilbert-Varshamov (GV) bound applicable to codes over any finite metric space and explore the conditions under which global combinatorial parameters can force the existence of codes exceeding this bound. Central to our analysis is the introduction of Ramsey-Sidorenko and independence-forcing graphs. We establish density thresholds for various graph families and utilize the Karush–Kuhn–Tucker conditions to analyze entropy optimization in the Hamming case. Furthermore, we derive upper bounds on code sizes using fractional packings in vertex-transitive and nonedge-transitive graphs. Our findings demonstrate that local subgraph statistics alone are insufficient to surpass the GV bound in the Hamming case, suggesting that improvements must stem from large-scale structural properties of the space.

For a cone $K\subseteq \mathbb{R}^n$, a real symmetric matrix $A$ is called $K$-copositive if $x^\top A x\geq 0$ for every $x\in K.$ This class of matrices plays a central role in copositive optimization and linear complementarity problems. However, a complete characterization of linear maps that preserve the $K$-copositive cone is unknown, even for $K:=\mathbb{R}^n_+$. In this paper, we develop a new approach to copositivity preservers that uses only order-theoretic arguments. We consider a smaller class of copositivity preservers, called Riesz* homomorphisms, and develop a general technique to deduce the structure of these preservers directly from a representation theorem of Riesz* homomorphisms on $S_n$. Following are the main outcomes of this paper: 1) We obtain a representation theorem for Riesz* homomorphisms on the partially ordered vector space of all real symmetric matrices endowed with the cone of all $K$-copositive matrices. 2) As a corollary of our representation theorem, we recover the main results of [Shitov, Proc. Amer. Math. Soc., 2021] and [Gowda et al., Linear Alg. Appl., 2013], providing a unified framework for studying cone automorphisms. 3) We introduce the notion of a $(K_1,K_2)$-unisigned matrix $P\in M_{m\times n}$, defined by the algebraic condition $P[K_1]\subseteq K_2\cup (-K_2)$, for cones $K_1\subseteq \mathbb{R}^n$ and $K_2\subseteq \mathbb{R}^m$. We also provide a characterization of such matrices. 4) We prove that a linear map of the standard form ($A\mapsto P^\top AP$; for $P\in M_{m\times n}$) preserves copositivity if and only if $P$ is $(K_2,K_1)$-unisigned, correcting a recent characterization of such maps preserving the $\mathbb{R}^n_+$-copositivity.

Quantum locally recoverable codes (qLRCs) allow a single qudit erasure to be corrected by accessing only a small number of other qudits. Standard CSS and Hermitian constructions, however, impose dual-containing or self-orthogonal constraints on the underlying classical codes, thereby restricting the well-structured classical LRCs (cLRCs) that can be used to construct qLRCs. To relax these constraints, we introduce entanglement-assisted quantum locally recoverable codes (EAQLRCs) by assuming that halves of the pre-shared maximally entangled pairs are noiseless. We characterize sufficient support conditions on extended stabilizers under which entanglement-assisted stabilizer codes have locality $r$ and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to $\ell$-intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.

We study the skew product systems $T: \mathbb{S}^1\times\mathbb{R}^d \to \mathbb{S}^1 \times\mathbb{R}^d$, \begin{equation*} T(x,y)=(\ell x, A y+\phi(x)), \end{equation*} where $\ell\geq 2,$ $A \in GL_d(\mathbb{R})$ with $\rho(A)<1$, and $\phi \in C^r(\mathbb{S}^1, \mathbb{R}^d)$, with $r>(d+3)/2. We allow the fibers to have any dimension and $A$ to be non-conformal. We show that: when $|\det(A)|\ell<1$, for almost every $\phi$, the Hausdorff dimension of the solenoid attractor and the SRB measure equal 1+ the affinity dimension; when $|\det(A)|\ell>1$, for almost every $\phi$, the SRB measure is absolutely continuous. In the latter case we investigate the Sobolev regularity of the density of the SRB measure. We introduce a derivative dispersion condition, replacing the usual transversality condition.

We study an interval-activation frog model on \(\mathbb Z\) with i.i.d.\ initial numbers of frogs \((\eta_x)_{x\in\mathbb Z}\), satisfying \(0<\mathbb{E}[\eta_0]<\infty\). Frogs at the origin are initially active and all others are sleeping. Each frog performs a symmetric integer-valued random walk and has a random lifetime \(L\) determined by an i.i.d.\ survival parameter \(\pi\in(0,1)\), with \(\mathbb{P}(L\ge k\mid \pi=p)=p^k\). Every jump activates all sleeping frogs at the integer sites between its endpoints. Let \(D^\to\) denote the maximal rightward displacement of a single frog before death. We derive survival and extinction criteria from the tail behavior of \(D^\to\). If \(\mathbb{P}(|\xi_1|\ge n)\sim n^{-\alpha}L_\xi(n)\), with \(L_\xi\) slowly varying, then survival holds with positive probability for \(0<\alpha<1\), while for \(\alpha=1\) both survival and almost sure extinction may occur. For \(1<\alpha<2\), assume \(\mathbb{P}(|\xi_1|>n)\sim c_\xi n^{-\alpha}\); in the finite-variance case assume \(\mathbb{E}[\xi_1]=0\) and \(\operatorname{Var}(\xi_1)=\sigma^2\in(0,\infty)\). Setting \(r=\alpha\) in the stable case and \(r=2\) in the finite-variance case, if the law of \(\pi\) has density \(f_\pi(u)\sim(1-u)^{\beta-1}\ell((1-u)^{-1})\) as \(u\uparrow1\), then, for \(0<\beta<1\), \(n\mathbb{P}(D^\to\ge n)\sim C_\beta n^{1-r\beta}\ell(n^r)\), with explicit \(C_\beta\). Hence the sharp off-critical threshold is \(\beta_c=1/r\): survival holds for \(\beta<1/r\), extinction holds almost surely for \(\beta>1/r\), and explicit sufficient conditions on the critical line leave a factor-four gap.

In this paper, we formulate three communication tasks for empirical optimal transport: distributed coupling sampling, cost-evaluable coupling output, and scalar value-certified sampling. Our main result is a field-code compiler: any communicated transport field approximating an optimal empirical Monge map to error $\eta$ can be completed by sparse target-cell residuals into an exact-marginal value-certified sampler with scalar certificate $W_1(\mu,\nu)\leq U\leq W_1(\mu,\nu)+2\Delta$, where $\Delta$ is the public target-partition diameter. The certificate accuracy is controlled by $\Delta$ alone. The field error $\eta$ controls residual communication under a cell-margin condition; without a margin, $\eta$ alone does not bound residuals. We instantiate the compiler via adaptive local-affine and tensor-product spline codes with $d(m+1)^db$ field bits in the spline case, plus residual lists charged separately. For lower bounds, exact Gap-Hamming embeddings prove certified output is hard, including a smooth cell-packing diffeomorphism family requiring $\Omega(\varepsilon^{-2d/(d+4)})$ communication for any cost-evaluable, cost-certified, or value-certified protocol. The same gadgets admit zero-communication samplers, formally separating the sampler and certificate-bearing output models. These results identify the transport field as the right communicated object whenever a field code is available, primarily as a residual-sparsity tool.

We study online convex optimization with one unbiased stochastic subgradient per round and noise having a finite $p$-th central moment, where $p\in(1,2]$ is unknown. For a bounded convex domain of diameter $D$, subgradients bounded by $G$, noise scale $\sigma$, and comparator path length $P_T$, let $\Lambda_T=1+P_T/D$. A single algorithm, using none of $G,\sigma,p,P_T$, attains expected dynamic regret $O_p\left(\min\{GD\sqrt{T\Lambda_T}+\sigma DT^{1/p}\Lambda_T^{(p-1)/p},\,GDT\}\right)$ against every fixed comparator sequence. Restarted AdaGrad experts produce the noise-path exponent $(p-1)/p$, and a prior favoring longer restart intervals removes horizon-dependent logarithmic overhead. We give an explicit bound uniform in $p$; its logarithm-free form has noise coefficient $O(1+\log(p/(p-1)))$, while the static-regret constant is universal. The analysis requires only marginal noise moments and permits dependent errors. Complete pathwise proofs retain both the expert-loss range and the gradient energies preceding comparator movement. Matching lower bounds hold on every bounded convex domain of positive diameter, under the same gradient-only information model. Together with a path-budget-tuned upper bound, they characterize the minimax rate with universal constants, including its linear-regret saturation.

We study singularity formation for the heat flow of harmonic maps from $\R^d$ into $\mathbb{S}^d$ in supercritical dimensions $d \in \{3,4,5,6\}$. It is well known that in each of these dimensions there exist infinitely many self-similar solutions that provide examples of loss of regularity in finite time. In this paper, we extend the results of \cite{BieDon18}, \cite{BieDonSch17} for $d=3$ to higher space dimensions $d \in \{4,5,6\}$ and prove the existence of a monotonically increasing self-similar profile $f_0$, which is asymptotically stable under small corotational perturbations. To construct the solution and resolve the spectral problem, we use rigorous computer assistance. As a byproduct of our stability analysis, we also obtain finite-codimension stability of arbitrary self-similar profiles within the corotational class.

We study the asymptotic expansion of p-Green functions and their derivatives near the pole. In the Euclidean setting, we strengthen a theorem from [arXiv:2203.01206] by establishing improved integrability properties and estimates for the first and second derivatives. In the more general Riemannian setting, we derive a refined asymptotic expansion of the p-Green function near its pole, thereby improving the corresponding result proved in [arXiv:2512.14591].

In 1967 Efron showed that the Kaplan-Meier estimator can be defined as a fixed point of a certain mapping, calling this property “self-consistency”. He also proposed an iterative algorithm for approximating this fixed point and suggested its convergence. The purpose of the present note is to prove this fact.

We develop a structure-preserving neural variational framework for random Young-diagram ensembles, with representations adapted to the structure and scaling of each measure. The method is validated on the Plancherel, uniform, minimal-difference, and fixed-\(q\) \(q\)-Plancherel ensembles, using known asymptotic profiles only for post-training comparison. We then study a quartically deformed hook-length ensemble without assuming an analytical saddle shape. Large-\(n\) neural profiles are compared with finite-size MAP profiles obtained from exact-action searches and with mean profiles obtained from corner-transfer Metropolis–Hastings sampling. Increasing the deformation suppresses the leading rows and broadens the support, while the neural, discrete, and sampled mean profiles agree at the percent level. These results provide numerical evidence for a deformation-dependent macroscopic saddle family.

Let $C$ and $W$ be two integer sets. If $C+W=\mathbb{Z}$, then we say that $C$ is an additive complement to $W$. If no proper subset of $C$ is an additive complement to $W$, then we say that $C$ is a minimal additive complement to $W$. In this paper, we give an affirmative answer to one problem of Ma and Chen [On a problem of minimal additive complements of integers, J. Number Theory 284(2026), 178-187.]

Given two $m$-ary predicates $P,Q$, an $n$-ary polymorphism is a tuple $(f_1,\dots,f_m)$ of functions such that $x^{(1)},\dots,x^{(n)} \in P$ implies $(f_1(y_1),\dots,f_m(y_m)) \in Q$, where $y_i = (x^{(1)}_i,\dots,x^{(m)}_i)$. This generalizes the usual definition in universal algebra, in which $P = Q$ and $f_1 = \cdots = f_m$. In earlier work, we studied when all polymorphisms of a single predicate are "trivial": either all depend on a single coordinate (common to all of them), or they constitute a "certificate" for the predicate. We showed that it suffices to check this condition for $2$-ary polymorphisms, and even for $1$-ary polymorphisms, modulo an explicit list of obstructions. In this paper we generalize the first result to the $P,Q$ setting, for a relaxed notion of certificate. We also generalize the second result in the promise setting, in which $P,Q$ range over the same alphabets and $P \subseteq Q$.

Our previous work established the multi-set allocation occupancy (MAO) distribution theory, including the norms, inequalities, and limit theory, under constraints on set sizes alone. By further conditioning on all pairwise intersections while leaving triple and higher-order membership patterns random, we develop the second-order extension. This extension naturally identifies the earlier marginal-size-only framework as the first-order MAO theory. Let $N$ be the population size, $T$ the number of labelled sets, and $M$ a feasible symmetric matrix whose diagonal and off-diagonal entries specify marginal sizes and pairwise intersections. Write $a=(a_B)_{B\subseteq[T]}$ for the atomic membership counts, where $a_B$ is the number of elements belonging to exactly the sets indexed by $B$, and let $A(N,M)$ be the resulting feasible atomic region. With labelled multiplicity $w(a)=N!/\left(\prod_{B\subseteq[T]}a_B!\right)$, $Z_{N,M}=\sum_{a\in A(N,M)}w(a)$, we define the second-order exact-$t$ and at-least-$t$ MAO norms by the direct atom formulas $\lVert t^r\rVert_T=\left(\sum_{a\in A(N,M)}\left(\sum_{|B|=t}a_B\right)_r w(a)\right)/\left((N)_r Z_{N,M}\right)$, $\lVert [t,T]^r\rVert_T=\left(\sum_{a\in A(N,M)}\left(\sum_{|B|\geq t}a_B\right)_r w(a)\right)/\left((N)_r Z_{N,M}\right)$. The moments can be calculated exactly based on the norms: If $X_{=t}$ and $X_{\geq t}$ denote the numbers of elements with membership degree exactly $t$ and at least $t$, respectively, then for every $\nu\geq1$, $\mathbb{E}_{N,M}[X_{=t}^{\nu}]=\sum_{i=1}^{\nu}S(\nu,i)\lVert t^i\rVert_T$, $\mathbb{E}_{N,M}[X_{\geq t}^{\nu}]=\sum_{i=1}^{\nu}S(\nu,i)\lVert [t,T]^i\rVert_T$. We further developed Poisson and normal limiting theory, verified by numerical approximation. The model provides a systematic route to higher-order MAO theories by prescribing intersections up to any chosen order.

Although entropy-based summation-by-parts (SBP) discretizations of hyperbolic conservation laws are widely used for their robustness and stability properties, there are very few results on their convergence. We extend a recent convergence analysis of Worku, Del Rey Fernández, and Zingg (2026, DOI: 10.48550/arXiv.2603.18369) in two ways. First, instead of allowing only hyperbolic conservation laws whose fluxes are homogeneous and have globally bounded second derivatives (a restriction essentially to linear or quadratic fluxes), we consider general hyperbolic systems with strictly convex entropy and source terms depending on time and space. Second, instead of requiring a special class of SBP operators, we consider a general framework of diagonal-norm SBP operators on curved meshes, including finite differences, continuous and discontinuous Galerkin methods. Since the error analysis is based on a discrete relative entropy, it is restricted to smooth solutions. To enable a unified treatment of conservation laws, we restrict the analysis to periodic boundary conditions. Numerical results demonstrate that the predicted convergence rates are sharp in general, but can be improved for special cases such as discontinuous Galerkin methods with even polynomial degree and multi-block finite difference methods. An optimal analysis is expected to require more sophisticated arguments specialized to the class of methods instead of the general framework of SBP operators used in this work.

In this work, we settle the boundedness of a vast class of elliptic Calabi-Yau 4-folds. In particular, we show that any elliptic Calabi-Yau 4-fold that is not crepant to the quotient of a product $Y \times E$, where $E$ is an elliptic curve and $Y$ a Calabi-Yau 3-fold, belongs to finitely many algebraic families. In particular, the statement applies whenever the elliptic fibration of the Calabi-Yau 4-fold is not isotrivial. We also provide partial evidence for the boundedness of the middle Betti number of Calabi-Yau 3-folds and study the index of fibered $K$-trivial 4-folds.