Papers

We introduce a version of the Monge–Ampère equation on finite graphs, motivated by nonlinear graph-based interpolation and semi-supervised learning. The operator is defined as the product of discrete analogs of the Hessian eigenvalues, obtained via local order statistics of function values at neighboring vertices. We derive an equivalent Bellman-type formulation of the inhomogeneous Dirichlet problem, establish a comparison principle and uniqueness in the strictly graph-convex class, and investigate existence via Perron's method, identifying certain graph-theoretic obstructions. We also study the homogeneous equation, for which the problem reduces to a nonlinear interpolation rule involving the smallest discrete eigenvalue. Finally, we propose numerical schemes for both the homogeneous and inhomogeneous problems.

We study the spatially homogeneous Boltzmann equation without angular cutoff for very soft potentials satisfying the inverse power law relation $\gamma+4s=1$. Our main results establish the existence, uniqueness, stability and regularization estimates of solutions in the critical Sobolev space $ H^{-(\gamma + 2s + \frac{3}{2})} $ with a logarithmic correction. Combined with the recently established monotonicity of the Fisher information, the solutions extend globally in time. Our primary tools are energy estimates based on a simultaneous dyadic localization in the phase and frequency variables, together with sharp commutator estimates between the collision operator and the localization operators.

We study the radial extremal function $\varphi(|{\,\cdot\,}|)$ arising in the problem of finding the sharp Nikolskii constant $\mathcal C_d$ in $\mathit{PW}_1^1(\mathbb R^d)$ for arbitrary dimension $d\ge1$. We prove a factorization $\varphi=\Phi_1\Phi_2$, where $\Phi_1$ and $\Phi_2$ are entire functions of exponential type $1/2$ satisfying a functional equation and second-order differential equations with polynomial coefficients. As a result, the original extremal problem is reduced to a one-dimensional spectral problem depending on at most $d+1$ parameters. We also obtain a zeta interpretation of the coefficients of the polynomial appearing in the functional equation and a multiplicative equilibrium condition for the zeros of the extremal function. These results can be used to construct several algorithms for computing $\mathcal C_d$.

Graph neural networks (GNNs) have emerged as a powerful framework for learning from graph-structured data. However, their efficient training remains challenging, particularly in distributed computing environments. This challenge arises from the use of message passing, which couples all graph nodes, leading to expensive optimization steps, high memory requirements, and substantial communication overhead. To alleviate these limitations, we propose a novel domain-decomposition (DD) variant of AG2m, an AdaGrad method enhanced with second-order curvature information and momentum, denoted by DD-AG2m. The proposed DD-AG2m alternates between AG2m optimization on the original (global) graph and AG2m optimization on the partitioned graphs. To incorporate global information at reduced cost, we further introduce a two-level variant (2DD-AG2m) that performs global optimization steps on a coarse graph obtained by randomly subsampling nodes within each subdomain. Numerical experiments spanning graph classification, node-level regression, and spatiotemporal forecasting tasks demonstrate that the proposed DD methods reduce the computational cost required to achieve the same predictive performance by a factor of 4-8. Moreover, for the fixed computational cost, they improve the predictive performance of GNNs by up to 22% compared with the baseline AG2m.

We study singularity formation for classical solutions of the parabolic Bernoulli free boundary problem and the associated self-similar profiles. Inspired by Ricci and mean curvature flows, we introduce a Type I assumption on the blow-up rate at a singular free boundary point, under which the parabolic rescalings are compact and every tangent flow is a non-trivial self-shrinking solution of the elliptic profile equation. The proof establishes convergence of the positivity sets and recovers the Bernoulli condition in the limit, despite the absence of a minimizing structure. In the subcritical regime, every tangent flow is a double plane. We next classify all radial self-shrinkers. Besides the known ball and exterior profiles, there is a unique annular profile. We obtain dimension-uniform bounds and sharp asymptotics for its inner and outer radii, including a quantitative outward bias of its midpoint. These delicate estimates yield a complete classification of the signs of the linear spectrum for the annular solution in every dimension. As a consequence, the ball is dynamically stable modulo ambient symmetries, whereas the annulus is dynamically unstable, with genuine unstable modes of angular degrees $0,1,2,3$. We finally prove, from the spectral nondegeneracy of the compact radial profiles, that whenever one tangent flow is a ball or an annular solution, the tangent flow is unique. One borderline spectral sign in dimensions $2\leq n \leq 4000$ is verified by a rigorous computer-assisted argument using interval arithmetic.

Book VII of Euclid's Elements represents a remarkable mathematical achievement, marking the birth of number theory. Although it falls short of explicitly stating the Fundamental Theorem of Arithmetic (that every natural number is uniquely a product of primes), it contains all the tools necessary for its proof: the Principle of the Least (equivalent to Mathematical Induction) and arithmetical anthyphairesis for finding the greatest common divisor. Our work presents novel arguments that Book VII evolved directly from early Pythagorean arithmetized music. The accounts attributing this arithmetization to Pythagoras' acoustical experiments were shown to be fictitious by Vincenzo Galilei. In contrast, the alternative experiments and Hippasus' 4-chord (bronze cylinders of heights 6, 8, 9, 12) are validated as physically correct by Euler's Law for pipes. This experimental foundation led to the arithmetization of musical intervals and to the discovery of musical (multiplicative) anthyphairesis. Applying Aristotle's Topics 158b24-29 Principle, we reconstruct how musical anthyphairesis was transferred - through an inductive ladder of ratios starting with multiple and epimoric ratios recounted by Theon of Smyrna - to its arithmetical (additive) counterpart. Furthermore, we show that the mathematical peculiarities in the definitions and proofs of Book VII find a convincing explanation only through their musical origin. Ultimately, Hippasus emerges as the pivotal figure who discovered musical anthyphairesis (which evolved into Philolaus' Fragment 6) and transferred arithmetical anthyphairesis to geometry, crucial for Pythagorean incommensurability and the principles of the Infinite and Finite.

We obtain a complete description of the class of entrywise preservers of Lorentz-Gram matrices. This resolves, for the case of constant negative curvature, the classification of entrywise preservers obtained by Schoenberg in the zero-curvature (Euclidean) and constant-positive-curvature (spherical) settings. These preservers admit a Lévy–Khintchine-type representation and their asymptotic characteristics are related to Krein's classification of screw lines in Lobachevsky space. Connections with complete Nevanlinna–Pick kernels and Bochner subordination are also obtained.

We show that every increasing union of Borel graphs of pointwise volume growth at most $\exp(O(r^\gamma))$ is hyperfinite, where $\gamma \approx 0.1523$ is the unique real root of $(1 - \gamma)^3 - 4\gamma = 0$. This implies a positive answer to the special case of Weiss's question on the hyperfiniteness of Borel actions of countable amenable groups, for countable amenable groups locally of volume growth $\exp(O(r^\gamma))$ for $\gamma$ as above. We also show that every bounded degree Borel graph of subexponential volume growth has Borel Følner tilings, improving a result of Downarowicz and Zhang for graphs generated by free Borel actions of groups of subexponential volume growth. Our main tools are the development of a Borel analogue of the two-parameter dimension growth function of a metric space as introduced by Dranishnikov and Sapir, and ball carving algorithms from theoretical computer science. We end with a pair of conjectures relating Borel amenability, Borel subexponential dimension growth, and hyperfiniteness, which would imply a positive answer to Weiss's question.

A curious property of integers, first observed by D.~R. Kaprekar in 1949, is as follows: write the digits of a 4-digit number in both descending and ascending order, and then find the positive difference between these integers. Iterating this process on any 4-digit number - except for multiples of 1111 - eventually produces the number 6174. The process ends here, since 6174 is a fixed point of the procedure. In general, if we start with any integer and iterate this process, it either ends at a fixed point (as with 4 digits) or enters a cycle of numbers, called a Kaprekar cycle. In 2011, Dolan classified all fixed points of this process. We study Kaprekar cycles of length 2. We find general properties of numbers in a Kaprekar 2-cycle, and we classify all 2-cycles such that the two integers have the same smallest digit.

We investigate the behavior of the singular spectrum of self-adjoint operators under families of Hermitian perturbations, even non-compact ones. Under mild assumptions we have the “shift” of the singular spectrum for almost all values of the parameter. Moreover, if we consider trace class perturbations, we observe the “shift” outside a countable set of the values of the parameter. While similar results were known for finite rank perturbations, the extensions to trace class perturbations are far from easy, and the proofs require significant new ideas. In addition, some of our results hold (surprisingly enough!) even for all positive and bounded perturbations. This is a consequence of our generalized Aleksandrov disintegration theorem established in this paper. We use some advanced techniques, involving Sz.-Nagy–Foia\c s theory and the operator ${\bf A}_2$ condition.

An equivalent formulation of the Riemann hypothesis recently led to a partition expansion naturally indexed by diagonals of the Young lattice. Segovia isolated the hook families $(r,1^m)$ on these diagonals and computed their limiting contributions $\rho_r$, while observing that non-hook families provide a missing contribution. We introduce a bivariate finite generating function that packages all Young shapes on every fixed-excess diagonal at once. For each fixed $r\geq1$, we obtain a diagonal generating polynomial $D_r(n;z)$ and prove \[ A_r(n)\sim C_r\,n\log\log n, \] where $C_r$ is the $(r-1)$st coefficient of an explicit convergent infinite product. Moreover, \[ C_r=\sum_{\nu\vdash r-1} C_\nu , \] giving a canonical decomposition over the partitions of the excess $r-1$. The one-part contribution is Segovia's hook constant $\rho_r$, while the remaining terms give all non-hook corrections simultaneously. We then refine these constants by introducing coefficients $C_{r,d}$ that record simultaneously the Young-lattice excess $r-1$ and the number $d$ of non-unit rows. Row sums recover the fixed-excess constants $C_r$, while column sums recover the depth decomposition in an Abel-regularized multiple-zeta expansion of $e^\gamma$. More precisely, each partition $\nu\vdash r-1$ is identified with an Abel-regularized multiple-zeta block of depth $\ell(\nu)$. Thus the same array $(C_{r,d})$ organizes the decomposition simultaneously by Young-lattice excess and multiple-zeta depth. Our results concern the combinatorial and asymptotic structure of this decomposition, rather than the Riemann hypothesis itself.

We consider the compression of kernel matrices on the unit interval $[0,1]$ by interpolets that have sufficiently many vanishing moments. We define a compression rule which discards most matrix coefficients without compromising the accuracy offered by the underlying discretization. Since interpolets can be scaled such that the compressed kernel matrices are well conditioned, we derive a fully discrete scheme that solves a kernel interpolation problem under consideration in linear overall complexity. We finally generalize this approach to the unit $n$-cube $[0,1]^n$ by means of the sparse grid combination technique. Numerical experiments are carried out to validate the theoretical findings.

We study the existence and classification of local $(-1)$-homogeneous axisymmetric solutions $(u, p)$ of the three-dimensional incompressible stationary Navier-Stokes equations near a singular ray. We focus on such solutions with Type II behavior, which satisfy $0<\limsup_{x\in\bS^2, x\to P}|u|/|\ln \text{dist}(x, P)|<\infty$, where $P$ is the south or north pole. Apart from the Landau solutions, these are the least singular nontrivial $(-1)$-homogeneous axisymmetric solutions. We construct a four-parameter family of local $(-1)$-homogeneous axisymmetric solutions with at most logarithmic growth near the singular ray and represent them as convergent series whose coefficients are determined recursively. Conversely, we prove that every local $(-1)$-homogeneous axisymmetric solution satisfying $|u|=o(\text{dist} (x, P)^{-1})$ as $x\to P$ on $\bS^2$ belongs to this family. In particular, this family contains Type II solutions with nonzero swirl, in contrast to the global $(-1)$-homogeneous axisymmetric Type II solutions in $\R^3\setminus\{x'=0\}$, which have no swirl. We also establish derivative estimates for the constructed solutions and identify the singular force generated by these solutions across the singular ray in the sense of distributions.

Model specification tests play a crucial role in evaluating the appropriateness of probability models for estimation and inference. Existing methods for the detection of model misspecification such as the chi-square goodness-of-fit (GOF) tests and more recently the M2 statistic \autocite{MaydeuOlivares2005,MaydeuJoe2014} tend to result in test statistics with excessive degrees of freedom for models with larger numbers of parameters. An alternative approach is based upon the Information Matrix (IM) equality. The IM equality asserts that if a probability model is correctly specified, the asymptotic covariance matrix of the maximum likelihood estimators can be asymptotically estimated using a methodology based upon either the first or second derivatives of the log-likelihood function. Using a contrapositive argument, White (1982) \nocite{Wh82} proposed a misspecification test methodology based upon comparing these two alternative covariance matrix estimators. Extending this work, Presnell and Boos (2004) \nocite{Presnell2004} showed how to develop a misspecification test which only requires one degree of freedom regardless of the complexity of the model or data. In this paper, we extend prior work and additionally apply methods of Golden et al. (2013, 2016) \nocite{golden2013}\nocite{Golden2016} to derive and evaluate misspecification tests for Cognitive Diagnostic Models (CDMs) which only require 1 or 2 degrees of freedom regardless of model or data complexity. Analytic formulas for the tests are derived so they can be applied without requiring computationally intensive bootstrap simulation methods. Our simulation studies show the asymptotic statistical tests have good level (type 1 error) and power performance for CDM models and data which might be encountered in practice.

Theoretical guarantees for kernel regression are typically formulated in terms of smoothness, but practical accuracy depends critically on how the design resolution compares with the target and kernel lengthscales. We develop a finite-sample theory for Matérn regression on periodic domains with quasi-uniform designs, covering noiseless interpolation and noisy kernel ridge regression. Our minimax result shows that accurate recovery requires a design dense enough to resolve the target lengthscale and sufficient information at that scale to overcome noise. We prove that Matérn interpolation additionally requires the design to resolve the kernel lengthscale: if the kernel is too short relative to the point spacing, an additional error remains even when the target is well resolved. For noisy Matérn regression, we derive a three-term fixed-ridge risk characterization consisting of target-scale bias, kernel-scale bias, and variance, and show that optimizing over the ridge parameter yields four distinct contributions. When the target is no more than twice as smooth as the kernel, choosing a kernel lengthscale longer than the target does not worsen the oracle risk, whereas choosing one too short can. Thus these resolution conditions determine when accurate recovery becomes possible, while smoothness determines how rapidly the error decreases thereafter.

We consider families of lines that cover every point of the integer lattice $\mathbf{Z}^2$ in the plane, subject to the constraint that no two lines of different direction in the family meet at a lattice point. Restricting to lattice lines (lines containing at least two, hence infinitely many, lattice points, equivalently of rational direction), we show that the set of directions occurring in such a covering can be made dense in the space of line directions. The construction is a recursive splitting of $\mathbf{Z}^2$ into nested rank-2 sublattice cosets, each handed off to a freshly chosen direction; the key technical point is a steering lemma showing that at every stage of the recursion a new direction arbitrarily close to any prescribed target can still be realized, via an elementary sieve bound.

The observable of the Talbot effect is the intensity behind the grating: the density of the evolving field, not the field itself. Its fractality was previously known only for step data with rational jumps. We prove the general case. For arbitrary real data of bounded variation with at least one jump, the density of the free Schrödinger evolution on the torus is fractal at almost every time, with graph of upper box dimension exactly three halves. Its critical Sobolev mass diverges logarithmically, at a rate given in closed form by Wiener's jump statistic of the datum and independent of the positions and phases of the jumps. The same rate is halved along time traces and doubled for the Airy flow, so the critical mass probes the dispersion relation. Transported by the rank calculus of derangetropy operators, the theory equips every probability law with a quantum carpet: Gauss-sum revivals on quantile cells at rational times, a universal fractal at almost every other. The spectral growth rate of a single measured intensity profile returns the jump content of the grating, a prediction open to test in optical and matter-wave interferometry.

We introduce a natural class of hyperbolic surfaces called flippered surfaces that generalize crowned hyperbolic surfaces (i.e.: worldsheets for open strings). We develop their Teichmüller and moduli-space theory, construct generalized Weil-Petersson volume forms and Chekhov's action, and prove that the resulting generalized Mirzakhani volumes are finite. We establish three geometric recursion formulae-neck chopping, disk excision, and crown extraction-which express these volumes in terms of those of topologically simpler surfaces. For the fundamental polygonal and annular cases, we derive integral representations involving conical Legendre functions, as well as explicit formulae in terms of elliptic integrals and polylogarithms. We further describe the arithmetic structure of the Taylor coefficients of these volumes, show that suitable specializations are Kontsevich-Zagier periods, and prove identities at the imaginary boundary length $2\pi\sqrt{-1}$ that generalize the Do-Norbury paraphrasing of string and dilaton-type equations.

We consider the $2W\times 2W$ transfer matrices associated to the block Anderson model with GOE potential blocks. We make the classical Lyapunov exponent theory quantitative in two ways. First, we prove a quantitative limit theorem for the top Lyapunov exponent. Second, we prove every gap between Lyapunov exponents is at least $c/W$. As a corollary, this implies the localization length of this $1$d block Anderson model is at most $CW^2$. The proof uses Furstenberg type formulas for the Lyapunov exponents, and Malliavin calculus style arguments in the symplectic group to show the product of sufficiently many transfer matrices has a sufficiently smooth density. The main technical input for the latter is a least singular value estimate for a structured random matrix.

Several important fans studied at the interface of combinatorics and algebraic geometry arise from barycentric subdivisions of other fans; the central example is the braid arrangement. The collection of faces of the standard simplex forms a basis for the deformation cone of the braid arrangement, i.e. the cone of generalized permutahedra. We reinterpret this simplicial basis as the collection of deep truncations of the standard simplex, and systematically abstract this perspective to produce polytopal bases for the deformation cones of barycentric subdivisions of simplicial projective fans. We further demonstrate that these bases restrict to bases for deformation cones of fans obtained by a sequence of stellar subdivisions induced by a building set. When the starting polytope is smooth with all edge lengths equal to one, we show that we can upgrade our basis to a collection of flat truncations. We then investigate two extensions of the braid arrangement where we are able to further upgrade our flat truncation basis to an indecomposable polytopal basis: 1. the barycentric subdivision of the normal fan of a product of standard simplices and 2. the barycentric subdivision of the braid arrangement. We discuss connections to, and implications for, Archimedean solids and regular polytopes, permutahedral plates, root polytopes, permutoassociahedra, simple permutonestohedra, cosmohedra, omnitruncations of Coxeter permutahedra, bipermutahedra, $\pi$-colored fans, and tropical $\alpha$ and $\beta$ classes.

Let $X$ be a noetherian scheme with the resolution property, and let $p:Y\to X$ be a projective morphism. Suppose that $R^{i}p_{*}=0$ for $i>2$ and that $ \mathcal {O}_{X}\longrightarrow Rp_{*}\mathcal {O}_{Y} $ is an isomorphism. We show that derived pushforward induces an equivalence \[ D^{b}(Y)/\operatorname{Ker}(Rp_{*})\simeq D^{b}(X). \] As an application, we prove a characteristic-free form of the Bondal–Orlov localization conjecture for quasi-projective threefolds.

Let $\mathcal{B}$ be the class consisting of the six-vertex bipartite graphs that possess a perfect matching and their complements. It is proved that every $\mathcal{B}$-free graph $G$ satisfies $\alpha(G)+\omega(G)\ge |V(G)|-1$. This establishes Conjecture 3.1 of Litjens, Polak and Sivaraman (B-Free Graphs Conjecture). For a smallest counterexample, Hall-type exchange arguments show that two maximum stable sets, and likewise two maximum cliques, differ in at most two vertices. A core-corona matching lemma then forces $|\alpha(G)-\omega(G)|\le 2$. Double counting between suitably dense and sparse vertices reduces the problem to twenty-one binary feasibility systems on at most fourteen vertices. Their infeasibility is verified by two independent exact encodings, with a separate exhaustive validation of the forbidden-family constraints.

Suppose $X,Y$ are independent random variables with values in a compact abelian group $(G,+)$. We examine the following two entropy power-type inequalities: $h(X+Y)\geq \frac{1}{2}h(X)+\frac{1}{2}h(Y)$ and $h(X+Y)\geq \max\{h(X),h(Y)\}$, where the entropy $h(Z)$ of a $G$-valued random variable $Z$ is defined in terms of its density with respect to Haar measure on $G$. For groups that are either connected or finite with no nontrivial subgroups, we precisely characterize the cases of equality and establish explicit, quantitative stability estimates in terms of relative entropy for these two inequalities. The main tools are a generalization of an entropic inequality obtained by Green, Manners and Tao (2023) for discrete entropy, and a harmonic-analytic estimate for the chi-squared contraction coefficient in connected compact groups. As an application, we derive exponential convergence rates to the uniform distribution in relative entropy for random walks on connected compact abelian groups.

This article establishes the quasi-optimality of adaptive nonconforming finite element methods for a class of optimal control problems involving Dirac measures governed by the biharmonic equation. The nonconforming Morley finite elements are employed for discretising both the state and adjoint variables. A modified right-hand side through a companion operator that maps Morley finite elements toaconformingspacehelpstoovercomethechallengeinhandlingpointsourcesontheright-hand side. A priori and a posteriori error estimates for the optimal control problems are derived. Further,optimal convergence rates for adaptive finite element methods are established using an axiomatic framework: by proving key properties such as stability, reduction, discrete reliability, and quasi-orthogonality. Numerical experiments for three types of optimal control problems are discussed extensively and they validate the theoretical results.

Let $n\ge2$ and let $\Omega\subset\mathbb R^n$ be an arbitrary bounded open convex set. The author prove that, for $p>n$, the Dirichlet problem \[ \det D^2u=(-u)^p\quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial\Omega, \qquad u>0\quad\text{in }\Omega \] has at most one convex Alexandrov solution. The proof is based on the affine behavior of the Monge–Ampère energy and on a power-concavity property of the $L^{p+1}$ mass along the Legendre path connecting two solutions. At the homogeneous exponent $p=n$, the same argument shows that any two nonzero solutions with the same coefficient are positive multiples of one another.

The Hadamard maximal determinant problem seeks an $n\times n$ matrix $X$ with entries in $\{\pm 1\}$, which maximizes the determinant $\vert \det X \vert$ for a given order $n \in \mathbb{N}$. We report matrices attaining new record determinants for orders $n\equiv 3\pmod{4}$ between $103$ and $119$, as well as $n=51$. We provide a proof of optimality within a specific family of circulant–block constructions for $X$. For these orders, we recast the Hadamard problem as a sequence reconstruction problem from a pair of periodic autocorrelation functions, subject to certain arithmetic conditions. We describe several learning–based strategies for reconstruction and provide a comparison with classical simulated annealing–style approaches as a benchmark.

We investigate diffusion approximations for a class of multivariate inhomogeneous finite-velocity random motions motivated by models of active particles. A particle alternates cyclically among prescribed velocity fields $V_1,\dots,V_p$ with random run times and evolves according to either a flight dynamics, consisting of piecewise-linear motion, or a gliding dynamics, in which the particle follows the corresponding velocity flow. Under a Kac-type scaling that couples vanishing run times with diverging particle speed, we prove weak convergence of both processes to multidimensional diffusions. It turns out that the order of successive cyclic motions affects the diffusion limit. In addition to the second-order term generated by fluctuations of the random run times, the limiting drift contains directional derivatives $DV_i[V_j]$ of the underlying vector fields. In Stratonovich form, part of this drift is expressed through their Lie brackets $[V_i,V_j]$. Thus, the generic non-commutativity of the microscopic motions survives the diffusive scaling and generates a macroscopic drift which depends on the cyclic order. The limiting drift also distinguishes the flight and gliding dynamics, despite their being driven by the same vector fields and run times. Several examples, including run-and-reverse motion and cyclic dynamics generated by multiple vector fields, illustrate how the cyclic switching protocol and the geometry of the underlying velocity fields shape the limiting diffusions.

For a minimal nilrotation on a compact connected nilmanifold, we prove that the polynomial diagonal orbit closure associated with any finite family of polynomials with integer coefficients vanishing at the origin is connected. This resolves a conjecture of Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson. Combined with their equivalence theorem, our result yields polynomial multiple recurrence in every prescribed residue class in topological dynamics, provided that the corresponding power of the transformation is minimal. Furthermore, we independently establish the measure-theoretic counterpart of this recurrence phenomenon. Finally, we construct a totally minimal nilsystem for which the lower central series identity proposed by Leibman fails.

We develop an extension-by-approximation principle for holomorphic Runge embeddings of increasing union of Stein manifolds into Stein manifolds with density property. The basic hypothesis is the existence, on each stage of the exhaustion, of a Runge isotopy which compresses the stage and whose terminal map extends holomorphically to the next stage. The resulting global embedding of the union may be chosen with Runge image, and every Runge embedding of a fixed stage can be approximated uniformly on compact subsets by the Runge embeddings of the union. We apply this principle to domains that are invariant under positive time part of holomorphic $(R,+)$-actions, to Stein manifolds carrying a semicomplete holomorphic vector field with globally attracting fixed point. It also gives a Runge embedding of $(\mathbb{C}^n\setminus \{z\in\mathbb{C}^n: f(z)=0\})\times \mathbb{C}$ in $\mathbb{C}^{n+1}$, which generalizes previous result of Runge embedding of $(\mathbb{C}^*)^n\times\mathbb{C}$ into $\mathbb{C}^{n+1}$. We also construct Stein globalization of an injective holomorphic semigroup action to holomorphic $(R,+)$-action. Finally, the abstract Loewner range of a Herglotz vector field is shown to admit a same-dimensional Runge embedding whenever the initial domain admits a Runge embedding into a Stein domain with density property; this yields a corresponding solution of the Loewner PDE with values in $\mathbb{C}^n$. We also give an example of non-Runge complete hyperbolic domain which admits $\mathbb{C}^n$-valued solution of the Loewner PDE.

We construct a theory of Fourier transform for rational étale motives on derived vector bundles over derived Artin stacks over a field of positive characteristic, defined as the integral transform with kernel the Artin-Schreier motive. The basic properties are established: involutivity, base change, functoriality, duality, monodromicity, constructibility, and compatibilities with various natural transformations.