Papers

Vector-valued multiscale representations are essential when signals or fields take values in $\mathbb{R}^m$ and component interactions carry meaningful information. Most multiwavelet and super-wavelet constructions are formulated in scalar Hilbert-space settings and typically produce channelwise scalar coefficients followed by recombination. We develop an intrinsic framework for vector-valued wavelets on $L^2(\mathbb{R}^d,\mathbb{R}^m)$ by endowing this space with a natural $\mathbb{M}_m(\mathbb{R})$-valued inner product, thereby turning it into a Hilbert $\mathbb{M}_m(\mathbb{R})$-module. This module viewpoint yields matrix-valued coefficients that encode cross-component interactions and provides canonical reconstruction through a Parseval-type identity. Within this setting, we introduce a constructive lifting procedure that builds separable multivariate vector-valued wavelet bases in $L^2(\mathbb{R}^d,\mathbb{R}^m)$ from scalar wavelet bases while preserving compact support, vanishing moments, and regularity.

We develop a purely algebraic framework for index-type theorems based on the Rees construction for filtered differential graded algebras (FDGAs). Alongside the classical Rees module we introduce a smooth variant $C^\omega_{\mathcal{R}}A$, adapted to analytic arguments, and we study traces and their pointwise and coefficient-wise extensions to Rees algebras. The main result (Corollary 3.11) is an abstract index theorem: given a compatible datum of filtered differential graded associative algebras with traces $A$, $B$, $G$ with a morphism $\phi\colon A\to B$, an action $\rho\colon G\otimes A\to A$ and a morphism $i\colon G\to B$, together with a graded-central element, one has $\mathrm{tr}_B(e^{f_0+f_1}) = \mathrm{tr}_{\widehat{\mathrm{gr}}A}(e^{\gamma}h)$, where the left-hand side is the trace in $B$ and the right-hand side the trace in the Laurent series associated graded $\widehat{\mathrm{gr}}A$ of $A$. Here $f_0+f_1$ is a curvature-type element, i.e., an element of the form $d_B\beta+\beta^2$ for some odd-degree element $\beta$, while $\gamma$ and $h$ are certain elements in $\widehat{\mathrm{gr}}A$. The formalism is modelled on the Getzler rescaling technique and on the derivation of the localization formula for the loop space Chern character by Ludewig and Yi.

Terrestrial base stations (BSs) are typically configured with fixed downtilt to serve ground users, resulting in weak illumination of low-altitude airspace even under line-of-sight (LoS) propagation. In this paper, we establish a channel model that incorporates BS and intelligent reflecting surface (IRS) radiation patterns for three-dimensional (3D) low-altitude coverage while preserving the existing BS configuration. We formulate a budget-constrained IRS deployment problem that jointly determines candidate-site selection, IRS orientations, and phase shifts to maximize the worst-case signal-to-noise ratio (SNR) over the 3D low-altitude airspace. The selected sites and optimized IRS parameters remain fixed after deployment, yielding a quasi-static IRS configuration. We characterize the illumination geometry between the fixed-downtilt BS and rooftop candidates by deriving the nonnegative installation-height range satisfying the BS main-lobe condition. The separation between the mapped main-lobe height boundaries grows linearly with horizontal BS-to-site distance and decreases inversely with the number of BS antennas. We further derive an analytical lower bound on the regional worst-case normalized array gain achievable through IRS phase design over served directions with different direction spans. The resulting sufficient direction span decreases inversely with the square root of the number of IRS elements when the same worst-case normalized gain guarantee is maintained. We develop a mixed-integer alternating optimization (AO) algorithm to solve the resulting problem. Simulation results validate the analytical characterizations and show that the proposed scheme achieves higher worst-case SNR than benchmarks across different deployment budgets.

For any $\gamma \in \left[ {0,\frac{1}{3}} \right)$, we construct $\gamma$-Hölder-continuous martingale solutions to the two-dimensional stochastic incompressible Euler equations. These martingale solutions exhibit dissipative behavior and intermittency, thereby establishing the flexible part of the intermittent Onsager theorem. The proof rests on a stochastic and intermittent variant of the Newton–Nash iteration combined with the Littlewood–Paley decomposition scheme. This composite iteration scheme incorporates new stochastic pressure, Reynolds stress and intermittent perturbations to formalize the stochastic and intermittent fluctuations.

We prove a finite-entropy criterion for the entropic conditional central limit theorem. Let $(\xi_i,\eta_i)_{i\geq 1}$ be independent copies of a pair $(\xi,\eta)$, and set $W_n=n^{-1/2}\sum_{i=1}^n \xi_i$ and $\boldsymbol{\eta}_n=(\eta_1,\ldots,\eta_n)$. Under the assumptions that $\mathbb{E}\operatorname{Var}(\xi\mid\eta)<\infty$ and that the conditional law of $\xi$ given $\eta$ is absolutely continuous almost surely, we show that $\mathbb{E}h(W_n\mid\boldsymbol{\eta}_n)$ converges to the Gaussian entropy $\frac12\log(2\pi e\sigma^2)$, where $\sigma^2=\mathbb{E}\operatorname{Var}(\xi\mid\eta)$, if and only if $\mathbb{E}h(W_{n_0}\mid\boldsymbol{\eta}_{n_0})>-\infty$ for some $n_0$. The main technical ingredient is a continuity theorem for Fisher information under Gaussian smoothing, which allows us to replace the finite expected conditional Fisher-information assumption by a necessary and sufficient finite-entropy condition.

We study non-negative solutions of the indefinite sublinear Dirichlet problem $$ \left \{\ \begin{array}{ll} -\Delta u=\lambda u+m(x)|u|^{\alpha -1}u&\quad\hbox{ in $\Omega$},\\ u=0&\quad \hbox{ on $\partial \Omega$}, \end{array} \right . $$ where $\Omega \subset \mathbb{R}^{N}$ is a smooth bounded domain, $0<\alpha <1$, $m\in L^{\infty }(\Omega )$ is a sign-changing or non-positive weight, and $\lambda \in \mathbb{R}$.

We study constructive Kempe factorizations for rational curves on $\operatorname{SO}_4(\mathbb{R})$. Motivated by motion-polynomial factorization and rational matrix curves on real classical groups, we prove that every rational curve of degree $2d$ with $d\ge1$ first factors into $d$ quadratic rational curves and then into a product of at most $2d$ planar rotation curves, where each planar rotation curve fixes a two-dimensional plane pointwise. The construction proceeds by extracting left and right isoclinic polynomial parts via Cayley's factorization, factoring the corresponding quaternion polynomials, and pairing linear factors with equal norm polynomials. The resulting algorithms are explicit and are illustrated by examples.

We provide a new instance of singular relative Langlands duality, underlying a Rankin-Selberg integral on $\mathrm{GSO}_6$ due to Bump-Friedberg-Ginzburg. We conclude that this integral represents an essentially self-dual object in the relative Langlands program, and we demonstrate that the Langlands dual automorphic integral computes a finite sum of $L$-functions, reflecting stacky structure on the spectral side.

In 1997, Belinsky conjectured that, for convex subsequences, the logarithmic growth condition of Carleson, Trigub, and Zagorodniĭ is necessary and sufficient for the arithmetic means of subsequential Fourier partial sums to converge at every Lebesgue point of every integrable function. We disprove the sufficiency part of this conjecture. More precisely, we construct a strictly convex increasing sequence $(a_m)$ satisfying $a_m\leq 7m^8$ and a function $f\in L^1(\mathbb T)$ for which $0$ is a Lebesgue point, $f(0)=0$, and the means $m^{-1}\sum_{k=1}^m S_{a_k}f(0)$ are unbounded.

Given two quadratic functions \( f(x) = x^T Ax + 2a^T x + a_0 \) and \( g(x) = x^T Bx + 2b^T x + b_0 ,\) each associated with either the strict inequality ($<0$); non-strict inequality ($\leq 0$); or the equality ($=0$), it is a fundamental question to ask whether or not the joint system has a solution. For homogeneous quadratic systems ($a=b=0,~a_0=b_0=0$), starting from Finsler's lemma in 1936 until Yuan's alternative lemma in 1990, all combinations of the unsolvability for $\{x\in \mathbb{R} ^n\mid x^T Ax \mathbin{\star } 0\}\cap \{x\in \mathbb{R} ^n\mid x^T Bx \mathbin{\#} 0\}\subset \{0\}$, where $\star $ and $\#$ can be any of $\{<,\leq ,=\}$, have been shown to possess either a positive definite or a positive semi-definite matrix pencil of $A$ and $B.$ Extensions to nonhomogeneous quadratic systems $\{x\in \mathbb{R} ^n\mid f(x) \mathbin{\star } 0\}\cap \{x\in \mathbb{R} ^n\mid g(x) \mathbin{\#} 0\}=\emptyset $ have been done for several cases already. Two challenging cases remain open: the nonhomogeneous Calabi Theorem which determines when $\{f(x)=0\}\cap \{g(x)=0\}=\emptyset $; and the nonhomogeneous (strict) Finsler lemma to determine whether $\{f(x)\leq 0\}\cap \{g(x)=0\}=\emptyset .$ The paper provides the answers to both, in theorems and algorithms.

In this paper, we prove that any two dimensional complete translating soliton in ${\mathbb R}^4$ with finite total curvature is a plane. It is in fact a consequence of our main result.

We consider a discrete-time financial market model where, in addition to finitely many dynamically traded assets, there are also (possibly infinitely many) static options to choose from. We introduce the concept of small cones of random variables and present a sufficient condition for the attainable positions in the market to be closed in probability. A fundamental theorem of asset pricing is shown in the present context. Utility maximization will also be considered. We will provide economically meaningful examples of infinite dimensional small cones to demonstrate the pertinence of our approach.

We introduce the second order Zarankiewicz number $z_2(m,n)$ for irreducible doubly simple biquadratic forms with $|E_1|=z(m,n)$, together with the intermediate recursive-line and signed parameters $z_{RL}(m,n)$ and $z_{SL}(m,n)$. They satisfy the unconditional hierarchy \[ \operatorname{BSR}(m,n) \ge z_2(m,n) \ge z_{SL}(m,n) \ge z_{RL}(m,n) \ge z_{wL}(m,n) \ge z(m,n), \] where $z_{RL}$ is defined by the strengthened recursive rectangle criterion $(RW3^+)$ together with the conditions $(S)$ and $C_4$-freeness of $G_1$, and $z_{SL}$ is defined by the signed criterion $(RW3^\pm)$. We show that $(RW3^+)$ is sound and strictly weaker than the literal weak cross-cell test $(W3)$ on the weak-admissible class, and that $(RW3^\pm)$ is likewise sound. At the smallest four-column cases we obtain \[ z_2(5,4)=z_{SL}(5,4)=z_{RL}(5,4)=13>12=z_{wL}(5,4), \] and \[ z_2(6,4)=z_{SL}(6,4)=z_{RL}(6,4)=16>14=z_{wL}(6,4). \] In three columns this yields \[ z_2(m,3)=z_{RL}(m,3)=2m \qquad \text{for all } m\ge 3, \] with strict separation from $z_{wL}(m,3)$ for every $m\ge10$, and exact gap $\lfloor(m-3)/3\rfloor$ for $m\ge16$. Further finite computations give $z_{RL}(5,5)=17$, $z_{RL}(7,4)=19$, and $z_{RL}(7,7)\ge32>28=z_{wL}(7,7)$. Along $N=2p$ with $p$ an odd prime, we obtain the cubic asymptotic separation \[ z_2\!\left(\binom{N}{2},N\right)-z_{wL}\!\left(\binom{N}{2},N\right)\ge \left(\frac{1}{16}-o(1)\right)N^3. \] For the exceptional incidence case $p=3$ we prove $z_{SL}(15,6)=z_2(15,6)=60$. Concurrent work of Chen and Chen, using the recursive-line definition introduced here, incorporates the finite recursive-line values of this manuscript and establishes further exact four-column values together with an eventual formula for all $m\ge15$. The conjectural equality $z_2=z_{RL}$ is supported by the exact two-column, three-column, four-column, and odd-prime incidence families.

In this paper we continue the investigation of generalizations of the strong law of large numbers to operator means which has been initiated in the earlier paper "Strong law of large numbers for generalized operator means", Adv. Math. 457 (2024), 109933 of the authors. In particular, we clarify an inaccuracy in the statement and proof of the strong law for means associated with uniformly exponentially contracting flows in Thompson metric spaces. We then significantly improve the variant of this strong law given in the mentioned paper for operator means by relaxing the restrictive required moment condition to $L\log^+(L)$ by proving almost sure convergence of implicit stochastic approximations for monotone operators under this condition.

We establish ground-state representations and critical Hardy inequalities in the discrete $L^p$ setting. In particular, we construct critical Hardy weights for the discrete Dirichlet Laplacian on the half-line and the discrete fractional Laplacian on the integers for all $p\in(1,\infty)$. Our approach uses Sobolev–Bregman forms, for which the ground-state representations are exact identities and the corresponding Hardy weights are expressed through linear operators acting on powers of positive superharmonic functions.

$\newcommand{\X}{\mathbb{X}}\newcommand{\SC}{\mathcal{C}}$ We establish the existence of a "maximally spread out" Borel probability measure on a totally bounded subset $\SC$ of a (quasi)-Banach space $\X$ under two mild conditions: (i) a growth condition on the covering numbers $N(\SC, \epsilon)$ of $\SC$, and (ii) a technical topological condition that is in particular satisfied whenever $\SC\subset\X$ is closed, bounded, and convex. More formally, condition (i) requires that the so-called lower power-exponential Minkowski dimension of $\SC$, i.e., \[s_\ast:=\liminf_{\epsilon\downarrow 0}\frac{\log\log N(\SC,\epsilon)}{\log(1/\epsilon)}\] satisfies $s_\ast>0$. Under these conditions, we construct a Borel probability measure $\mu$ on $\X$ that is critical for $\SC$, or maximally spread out, meaning that the associated outer measure $\mu^\ast$ satisfies $\mu^\ast(\X\setminus\SC)= 0$ and furthermore satisfies for every $0<s<s_\ast$ the small-ball condition \[\mu^\ast(B(x,r))\le\exp\bigl(-c(s)\cdot(1/r)^s\bigr)\quad\text{ for all }x\in\X\text{ and }0<r<r_0(s).\] The existence of such a critical measure in particular implies that the so-called power-exponential Hausdorff dimension of $\SC$ introduced in [J.~Topol.~Anal.~4(2):203–235, 2012] coincides with the lower power-exponential Minkowski dimension. Previous work [Found.~Comput.~Math.~23(1):329–392, 2023] shows that such a critical measure gives rise to a phase transition regarding lossy compression and approximation by quantized neural networks of elements of $\SC$, provided that the $\liminf$ in the definition of $s_\ast$ exists as an actual limit. There, critical measures were constructed for unit balls of certain Besov and Sobolev spaces considered as subsets of $L^2$. In contrast, our construction is completely general. In particular, our results apply to function spaces of dominating mixed smoothness.

Starting from localized energy estimates, we prove finite speed of propagation for kinetic solutions to stochastic porous media equations with nonlinear conservative noise, the existence and uniqueness of which has recently been established. In particular, we propose a novel iteration technique which allows us to obtain a Stampacchia-type inequality involving one single integral quantity, despite the possibly different scaling behaviors of the porous media and the noise term. This allows us to apply stochastic filtering arguments developed for the case of linear source-type noise. Using related ideas, we identify flatness conditions on initial data which guarantee locally the occurrence of a waiting time phenomenon, i.e., the onset of forward propagation of the solution's support is locally delayed. The condition for the latter matches the one for the deterministic porous media equation up to a logarithmic correction in the case of critical nonlinearity in the noise, but it requires more and more flatness of the initial data as the nonlinearity tends towards linear conservative noise. This is in line with the expected behavior: In the case of linear conservative noise, instantaneous forward motion is possible due to the effects of stochastic transport, no matter how flat the initial profile is.

For a graph $H$, let $\operatorname{ex}(Q_n, H)$ be the largest number of edges in a subgraph of the hypercube $Q_n$ of dimension $n$ that contains no subgraph isomorphic to $H$. We define $\pi_\square(H)=\lim_{n\to\infty}\operatorname{ex}(Q_n,H)/|E(Q_n)|$ as the Turán density of $H$ in the hypercube. Although determining $\pi_\square(H)$ for general $H$ remains widely open, Conlon found a large class of graphs with zero Turán density in a hypercube. In this note, we address the case when $\pi_{\square} (H)>0$. If a graph $H$ cannot be embedded in an edge layer of a hypercube, then $\pi_{\square} (H)\geq 1/2$, as can be seen by taking every other edge layer of $Q_n$. Previously known examples of layered graphs of positive Turán density in a hypercube include graphs containing cycles of length $6$ or $10$, as well as examples of girth $8$. Also, to the best of our knowledge, all previously known examples of graphs with Turán densities arbitrarily close to one are not layered. We show that for every $g \geq 3$, there are layered graphs of girth greater than $g$ whose Turán densities in a hypercube are arbitrarily close to one. Moreover, these graphs have an induced embedding into an edge layer of a hypercube.

Magnetic composites used for the conversion of electrical energy often incorporate ferromagnetic inclusions insulated from each other to mitigate eddy current losses. Numerical models for these composites must be robust enough to address potential convergence issues arising from the presence of nonlinear magnetic inclusions and presence of significant confined eddy currents in the cell. This paper introduces an $\mathbf{h}$-conforming multiscale formulation for magnetic composites in a periodic framework. The proposed method uses the Heterogeneous Multiscale Method (HMM) with two mesoscale problems: a magnetoquasistatic problem for upscaling the homogenized magnetic flux density $\mathbf{B}_M$ and a magnetostatic problem for upscaling the macroscale incremental reluctivity $(\partial \mathbf{B}_M/\partial \mathbf{H}_M)$. Additionally, the method uses relaxed Newton–Raphson schemes at both macro and mesoscale levels to mitigate the well-known NR convergence issue linked to nonlinear BH constitutive laws in $\mathbf{h}$-conforming formulations. The accuracy and performance of the formulation are evaluated using 2D and 3D idealized periodic soft magnetic composites with linear and nonlinear BH curves. Furthermore, the paper demonstrates that the magnetoquasistatic mesoscale problem can be replaced by a magnet a

We extend the functional determinant approach (FDA), previously restricted to static impurities, to the case of finite-mass impurities by using the mass-gap description of Fermi polarons [Phys. Rev. Lett. 135, 193401 (2025)]. The quadratic structure of the mass-gap model enables the exact evaluation of many-body spectra and Ramsey dynamics for mobile impurities. We show that this mass-gap FDA smoothly interpolates between the Fermi-edge singularity for infinitely heavy impurities and the emergence of quasiparticle weight for finite impurity mass. Our results demonstrate that the mass-gap FDA provides a transparent and computationally efficient framework to describe the quantum dynamics of mobile impurities.

A universal cover is a convex set in the plane that contains a congruent copy of every planar set of diameter one. Lebesgue asked in 1914 for one of least area, and the value is not known. We prove that every convex universal cover has area at least 0.8344, improving on 0.832, published in 2005, and 0.833, in a 2026 preprint, both of which come from a disc together with an equilateral triangle and a regular pentagon. Our test sets are instead curves of constant width: the disc, the Reuleaux triangle and the Reuleaux pentagon. Each contains the regular polygon it is built on, so the family is strictly larger at the same number of bodies and the same number of placement parameters, and we show that the classical configuration admits an arrangement whose hull has area below 0.8336, so no bound drawn from those three sets by this argument reaches ours. Curves of constant width were proposed for this role, and explored numerically, by Gibbs in 2014; what is added here is a proof. It consists of an analytic reduction followed by one finite computation. The reduction bounds the hull area from below over an entire box of placements at once, by eroding each Reuleaux polygon to a fixed set contained in every placement that box allows. The computation is an exhaustive subdivision of the resulting five-dimensional space, recorded as a certificate of 486,799,600 nodes and checked by a verifier independent of the search, with a rigorous bound on its floating point error some thousands of times smaller than the margin the verification attains.

Automated material handling systems in semiconductor fabs are operated by a material control system (MCS) that must schedule a relay route for every transport command online, before execution. This is a data-driven scheduling problem in which route cost is dominated in the upper tail by queueing at heterogeneous, partially observable relay equipment, so route selection requires estimating both delivery time and congestion risk at the decision moment. This paper proposes a transport-network-aware dynamic congestion representation (TN-DCR). Built on a static directed transport graph induced by historically observed relay segments, TN-DCR combines structural route priors, multi-window network-wide congestion context, route-level bottleneck exposure, and an inductive graph-aware route embedding, all constructed under a prediction-time-safety invariant that admits only information observed strictly before the prediction moment. The representation feeds separate queue- and transfer-time regressors and an ordinal multi-label classifier producing calibrated multi-threshold exceedance scores, with an empirical-Bayes stock-key residual correction reducing systematic queue-time underprediction. The predictions serve as costs in a risk-constrained route-scheduling rule that minimizes predicted delivery time subject to a bound on extreme-congestion probability, embedding the learned predictors within a lightweight operations-research decision model. In a controlled closed-loop evaluation, mean delivery time falls by 16.4% and internal resource waiting time by 22.6% while throughput remains essentially unchanged.

This paper concerns the grow-up rate of solutions to a semilinear heat equation in the unit ball with the exponential nonlinearity and an inhomogeneous term $f$. When $f=0$, it is known that the large-time behavior of a solution changes at the critical dimension $N=10$, and the grow-up phenomenon occurs for the case $N\ge 10$. For the inhomogeneous case with $N\ge 10$, the present author and a coauthor showed in [12] that the grow-up phenomenon disappears once $f$ exceeds a threshold. In this paper, we provide a quantitative characterization of the disappearance of the grow-up phenomenon by obtaining the grow-up rate in the critical dimension. Our result shows that a qualitatively different type of grow-up behavior occurs in the critical dimension compared with the case $N\ge 11$ studied in [12]. The difference is caused by a change in the outer behavior of the solution. Even in the case $f=0$, our result is new in that it provides a rigorous justification of the formal computation by Galaktionov and King [10].

We construct a smooth, equivariantly formal, simply connected torus $8$-manifold over a contractible $4$-manifold with the boundary a Brieskorn homology $3$-sphere having a nontrivial fundamental group. We prove extension to a torus graph for GKM graphs of GKM$_{4}$ manifolds in complexity $2$, and $3$ under a finite fundamental group assumption (that is not satisfied, in general).

Large-scale industrial robot fleets share constrained physical infrastructure, making vehicle travel times dependent on safety separation, intersection access, downstream blocking, and station contention. We study this problem in overhead hoist transport (OHT) systems, a representative ceiling-mounted material-handling system used in semiconductor fabs. Static shortest-path routing cannot account for these time-varying traffic costs, whereas tabular Q-routing adapts online but learns each destination–node–action value independently, limiting information sharing across sparsely visited routing contexts and making startup behavior sensitive to inaccurate value estimates. We propose Neural Double Q-routing, which replaces destination-indexed tables with a shared state–action value network. The network is warm-started through return-to-go regression on mixed simulator-generated routing trajectories and then refined online using Double-Q updates, local congestion correction, and event-stratified structured replay. Across nine matched fleet-size–arrival-rate settings with 100, 150, and 200 OHTs, the proposed framework reduces mean completion time relative to tabular Double Q-routing by $0.8\%$–$8.8\%$. It achieves the lowest mean completion time among all compared methods in the six 150- and 200-OHT settings, whereas Dijkstra remains best in the three 100-OHT settings. Completed-task counts remain within $1\%$ of tabular Double Q-routing in eight of nine settings, and 95th-percentile completion time decreases in eight settings. In two matched startup scenarios, offline initialization increases the number of completed tasks by up to $23\%$ and reduces tail completion time by up to $15\%$.

Quantum non-local games with quantum inputs or outputs have been formulated in several different languages, including rank-one and quantum XOR games, support maps between projection lattices, probabilistic quantum hypergraphs, and Frobenius-algebraic rule operators on finite quantum sets. We give a unified operator-algebraic comparison of these models by assigning to each rule its winning transformation space: the operator space of transformations accepted with certainty. We introduce \(\mathcal R\)-projection-test quantum games with finite-dimensional input and output von Neumann algebras and an arbitrary, possibly infinite-dimensional, referee von Neumann algebra \(\mathcal R\). Their winning transformation spaces are precisely operator spaces with a natural bimodule structure, equivalently rectangular quantum relations. We compare this formalism with projection-lattice games, hypergraph quantum games, and the graphical rule-operator definition. Projection-lattice games capture exactly the reflexive winning bimodules. Hypergraph quantum games admit value-preserving projection-test realisations, and, after passing to perfect transformations, describe the same reflexive part as projection-lattice games. Using Daws' technique, we also translate rule operators to projections in tensor products of finite-dimensional von Neumann algebras. This identification places graphical rules in the same operator-bimodule framework and preserves both values and perfectness. Finally, we compare concurrency with the synchronicity conditions of Goldberg and of Bochniak–Kasprzak–Sołtan. The framework is illustrated by classical, rank-one, quantum XOR, colouring, and quantum graph homomorphism and isomorphism games.

These notes were prepared for a series of lectures delivered at the 43rd Finnish Summer School on Probability and Statistics, held in Lammi, Finland, from May 26 to 30, 2025. They provide an introduction to McKean-Vlasov stochastic differential equations and their connections with non-linear Fokker-Planck equations and mean-field interacting particle systems. Particular attention is devoted to density-dependent equations and to kinetic-type models with degenerate noise. For the latter, we introduce the underlying non-Euclidean geometry and the associated anisotropic and intrinsic Hölder spaces, and present well-posedness and regularity results based on Schauder estimates and superposition principles.

Under mild assumptions, the main results of this paper characterize a (real or complex) symmetric operator space affiliated with a semi-finite atomless von Neumann algebra $(\mathcal M, \tau)$ possessing the Daugavet property as $L_1(\mathcal M,\tau)$ or $\mathcal M$ (up to equivalent norms, or even proportional norms). Our results are new even for complex symmetric function spaces, which extend results of \cite{AKM12, KMMW13, AKM15} concerning the Daugavet property for real symmetric function spaces.

We prove the refined Koornwinder–Kostenko–Teschl conjecture. For the normalized weighted Jacobi function and all $n\in\mathbb N\_0$, $\alpha,\beta\ge0$, and $-1\le x\le1$, $$ |g\_n^{(\alpha,\beta)}(x)| \le \left[ \frac{(n+1)(n+\alpha+\beta+1)} {(n+\alpha+1)(n+\beta+1)} \right]^{1/4}\le1. $$ The proof combines a central contour estimate with Sturm–Sonin localization and an exact inverse moment. It also yields an $n$-uniform extreme-lobe theorem and a sharp canonical-product first-lobe principle. Applied to the discrete Laguerre evolution, the estimate gives the optimal positive-parameter decay. For $-1<\alpha\le0$, a complementary one-sided Jacobi inequality gives the exact norm $\|\mathrm{e}^{-\mathrm{i} tH_\alpha}\|_{\ell^1\to\ell^\infty} =(1+t^2)^{-(1+\alpha)/2}$. Thus the large-time decay exponent is $\min\{1,1+\alpha\}$ for every $\alpha>-1$. Bessel, Laguerre, and Darboux scaling limits show that the temporal and fixed-diagonal exponents are optimal.

Let $V$ be a DVR with quotient field $K$ and perfect residue field, $v$ be the valuation on $K$ associated with $V$, $\widehat K$ be the completion of $K$, and $\mathbb{K}$ be the completion of an algebraic closure $\overline{\widehat{K}}$ of $\widehat K$. We show that a DVR of the rational function field $K(X)$ which is a residually algebraic extension of $V$ is necessarily of the form $V_{\alpha}=\{\phi\in K(X)\mid v(\phi(\alpha))\geq0\}$, for an element $\alpha$ of $\mathbb{K}$ transcendental over $K$, and that $\alpha$ is algebraic over $\widehat K$ if and only if the residue field extension is finite. Not every such $V_{\alpha}$ is a DVR, however, and we characterize the $\alpha \in\mathbb{K}$ for which $V_{\alpha}$ is a DVR: they are the elements which can be approximated by algebraic elements in $\overline{\widehat{K}}$ with bounded ramification indexes. Combining the two results, we obtain a complete description of the extensions of $V$ to $K(X)$ which are DVRs and residually algebraic over $V$, together with a criterion for each of the two cases to occur. The proofs rest on a bound for the ramification index in a compositum, valid with no tameness assumption and under a separability hypothesis on one residue field extension only; we show that the inequality cannot be improved to a divisibility and that this hypothesis cannot be dropped. We also show that the hypothesis of discreteness cannot be omitted. Furthermore, we show that the set of $\alpha\in\mathbb K$ for which $V_{\alpha}$ is a DVR is a subfield of $\mathbb K$, which sits properly between $\overline{\widehat{K}}$ and $\mathbb K$, and corresponds to those elements $\alpha$ for which the value group of $\widehat K(\alpha)$ is discrete.