In this paper, we include some new results for the Lucas calculus. A Lucas-Pantograph type exponential function is introduced. Additionally, we define Lucas-Pantograph type trigonometric functions, and some of their most notable identities are given: parity, sum and difference formulas, Pythagorean identities, double-angle identities, and some special values. Lucas-Pantograph type hyperbolic functions are also introduced.

We compute the stable Faltings height of the hyperelliptic curve $X_n\colon y^2=x^{n}-1$ for every odd integer $n\ge 3$ in terms of special values of Euler's gamma function. In particular, we prove the bounds $$-0.975n< h_{\mathrm{Fal}}(X_n)-\tfrac{n}{8}\log n<\tfrac{9}{64}n\log\log n-0.263n.$$ As an application, we bound the Faltings height of any abelian variety with complex multiplication by the canonical CM-type of the $n$-th cyclotomic field by $\frac{n}{8}\log n+\frac{9}{64}n\log\log n-0.136n$.

Let $R = W(k)$ be the ring of Witt vectors over an algebraically closed field $k$ of characteristic $p > 2$. Let $M$ be a three-dimensional regular integral flat projective $R$-scheme such that $H^0(M,\mathcal{O}_M) = R$ and the anticanonical sheaf $ω_M^{-1}$ is ample. We show that $M$ is globally $+$-regular if the closed fiber $M_k$ is reduced.

We study a series of dynamical concepts for self-maps in the primal topology induced by them. Among the concepts studied are non-wandering points, limit points, recurrent points, minimal sets, transitive points and self-maps, topologically ergodic self-maps, weakly mixing self-maps, strongly mixing self-maps, Lyapunov stable self-maps, chaotic self-maps in the sense of Auslander-Yorke, chaotic self-maps in the sense of Devaney, asymptotic pairs, proximal pairs, and syndetically proximal pairs. Some results are given in the more general context of continuous self-maps in an Alexandroff topological space. We prove that a continuous self-map of an Alexandroff space is always Lyapunov stable.

We prove the existence of non-hydrodynamic solutions to the linear density-dependent BGK equation in $d$ dimensions. Specifically, we show the existence of an initial condition for any Knudsen number $τ$ for which the dissipation rate of the macroscopic mass density diverges $\sim 1/τ$. Our results rely on a detailed spectral analysis of the linear BGK operator, an explicit solution formula for the time-dependent problem using a combination of Fourier series with the Laplace transform and subsequent contour integration arguments from complex analysis.

This paper investigates the derived and spectral analogs of logarithmic geometry. We develop the deformation theory for animated log rings and $\mathbb{E}_\infty$-log rings and examine the corresponding theories of derived and spectral log stacks. Furthermore, we define moduli stacks for derived and spectral log structures and establish their representability. As an application, we will construct $\infty$-root stacks in the derived and spectral settings and study the associated geometric properties.

An integer program is called ideal if its continuous relaxation coincides with its convex hull allowing the problem to be solved as a continuous program and offering substantial computational advantages. Proving idealness analytically can be extraordinarily tedious -- even for small formulations -- such proofs often span many pages of intricate case analysis which motivates the development of automated verification methods. We develop a general-purpose framework for certifying idealness in Mixed Binary Linear Programs (MBLPs), formulating the verification problem as a linear program when the data is fixed and as a nonconvex quadratic program when the data is parametric. We apply this framework to study several formulations of the rectangle packing problem that are conjectured to be pairwise-ideal, obtaining computational proofs where analytic proofs were previously unknown or impractical. As our second contribution, we introduce and model a novel generalization of the rectangle packing problem that enforces edge clearances between selected rectangles. We present both existing and novel MBLP formulations which arise from different encodings of the underlying disjunctive constraints. We perform some computational experiments on these formulations under a strip-packing objective to determine the importance of pairwise-idealness in practice.

The classic upper bound on the chromatic number of $d$-degenerate graphs is $d+1$, shown to be tight by complete graphs. A natural question is whether this bound remains tight if one forbids large cliques. Classic constructions of Tutte and Zykov from the early 50s show that there exist $d$-degenerate $(d+1)$-chromatic graphs that are triangle-free, however these constructions grow rapidly with $d$. Motivated by this and addressing a problem posed by the second author at the Oberwolfach Graph Theory workshop, we prove that the minimum order $f(d)$ of a $d$-degenerate triangle-free graph of chromatic number $d+1$ satisfies $e^{Ω(d)}\le f(d)\le e^{O(d^2\log d)}.$ The lower bound follows from a novel upper bound on the chromatic number of triangle-free graphs: Every triangle-free $d$-degenerate graph $G$ on $n \le e^{O(d)}$ vertices satisfies $$χ(G)\le O\left(\frac{d}{\log\left(d/\log n\right)}\right).$$ We extend this to a more general result about degenerate graphs with sparse neighborhoods, which has applications to many graph coloring problems: For example, we prove that every counterexample to Hadwiger's conjecture with parameter $t$ must have a complete bipartite subgraph with one exponentially large side ($K_{a,b}$ where $a=(\log t)^{1/2-o(1)}$ and $b=e^{t^{1-o(1)}}$) or a small and very dense subgraph (of order $\le t$ with $t^{2-o(1)}$ edges) in some neighborhood. For the upper bound on $f(d)$ we establish a surprising connection between $f(d)$ and the on-line-chromatic number $g(n)$ of $n$-vertex triangle-free graphs. We also give an asymptotic improvement of the previous best upper bound for $g(n)$ due to Lovász, Saks and Trotter from 1989. Along the way we disprove a generalization of Harris' fractional coloring conjecture to graphs of bounded clique number and raise numerous problems which open up interesting directions to explore for future research.

In this paper we study the Wahl map for the normalization of a $δ$-nodal curve $C$ on a Hirzebruch surface $\mathbb{F}_{n}$ for $n\geq 0$. Let $σ:X\rightarrow \mathbb{F}_{n}$ be the blow up of $\mathbb{F}_{n}$ along the $δ$ nodes of $C$ and let $\widetilde{C}$ be the normalization of $C$ under $σ$. Let $K_{X}$ be the canonical bundle of $X$ and let $Ω^{1}_{X}$ be the sheaf of $1$-holomorphic forms on $X$. We give conditions for the surjectivity of the map $Φ_{X,\mathcal{O}_{X}(K_{X}+\widetilde{C})}: \bigwedge^{2}H^{0}(X,\mathcal{O}_{X}(K_{X}+\widetilde{C}))\rightarrow H^{0}(X,Ω^{1}_{X}(2K_{X}+2\widetilde{C}))$. Using this surjectivity, we analyze the Wahl map $Φ_{\widetilde{C}}:\bigwedge^{2}H^{0}(\widetilde{C},Ω^{1}_{\widetilde{C}})\rightarrow H^{0}(\widetilde{C},(Ω^{1}_{\widetilde{C}})^{\otimes 3})$ and compute the corank of $Φ_{\widetilde{C}}$ in various cases. We prove that the corank of the Wahl map for the normalization of a $δ$-nodal curve on $\mathbb{F}_{n}$ is $h^{0}(\mathbb{F}_{n},\mathcal{O}_{\mathbb{F}_{n}}(-K_{\mathbb{F}_{n}}))$, that verifies a conjecture by Wahl. Furthermore, as an application of our results, we demonstrate that, under certain conditions, a $δ$-nodal curve on a Hirzebruch surface $\mathbb{F}_{n}$ cannot be embedded as $δ-$nodal curve on a different Hirzebruch surface $\mathbb{F}_{m}$, for $n\neq m$.

As the binomial edge ideal of a graph is always generated by homogeneous quadratic polynomials corresponding to the edges of the graph, the question of when a binomial edge ideal defines a Koszul algebra has been studied by many authors ever since the class of ideals was first defined. Several partial results are known, including a characterization of those binomial edge ideals that possess a quadratic Gröbner basis. However, a complete characterization of the graphs determining Koszul binomial edge ideals has remained elusive. Inspired by our recent work characterizing when the graded Möbius algebras of graphic matroids are Koszul, we answer the question once and for all by proving that a graph defines a Koszul binomial edge ideal if and only if it is strongly chordal and claw-free.

In this article, we are concerned about the velocity tracking optimal control problem for 3D critical convective Brinkman-Forchheimer equations defined on a simply connected bounded domain $\mathbb{D}\subset\mathbb{R}^3$ with $\mathrm{C}^2$-boundary $\partial\mathbb{D}$. The control is introduced through an external force. The objective is to optimally minimize a velocity tracking cost functional, for which the velocity vector field is oriented towards a target velocity. Most importantly, we are concerned about the first-order necessary optimality conditions for above-mentioned optimal control problem which is the main challenging task of this article. To overcome the difficulties related to the differentiability of the control-to-state mapping, consequence of the lack of regularity of the state variable on bounded domains, we first establish some intermediate optimality conditions and then pass to the limit.

Principal component analysis (PCA) is a tool to capture factors that explain variation in data. Across domains, data are now collected across multiple contexts (for example, individuals with different diseases, cells of different types, or words across texts). While the factors explaining variation in data are undoubtedly shared across subsets of contexts, no tools currently exist to systematically recover such factors. We develop multi-context principal component analysis (MCPCA), a theoretical and algorithmic framework that decomposes data into factors shared across subsets of contexts. Applied to gene expression, MCPCA reveals axes of variation shared across subsets of cancer types and an axis whose variability in tumor cells, but not mean, is associated with lung cancer progression. Applied to contextualized word embeddings from language models, MCPCA maps stages of a debate on human nature, revealing a discussion between science and fiction over decades. These axes are not found by combining data across contexts or by restricting to individual contexts. MCPCA is a principled generalization of PCA to address the challenge of understanding factors underlying data across contexts.

This book presents a comprehensive regularity theory for solutions of elliptic, parabolic, and kinetic equations. The foundation of this theory was laid by E. De Giorgi's groundbreaking resolution of Hilbert's nineteenth problem in 1956. The innovative tools he developed to tackle this problem proved to be remarkably versatile. In 1957, just one year later, J. Nash independently developed analogous techniques for parabolic equations, concurrently with De Giorgi's research. By the year 2000, these techniques had been extended to address elliptic and parabolic equations featuring integral diffusion, such as the fractional Laplacian. More recently, the theory has evolved to encompass kinetic equations, accommodating both local and integral diffusions. This book aims to present these results in a unified and coherent manner, beginning with the classical elliptic framework and progressing through to the most recent advancements in kinetic equations.

In this article we explain the connection between the famous problemfrom the IMO 1988 and elements of small norms in quadratic number fields with parametrized units.

This study investigates a stochastic version of a class of non-Newtonian fluids governed by third-grade fluid equations, which exhibit complex and highly nonlinear dynamics. In particular, we address the random dynamics and asymptotic behavior of stochastic third-grade fluid equations (STGFEs) driven by a \emph{linear multiplicative Itô-type white noise} on general domains $\mathbb{Q}\subseteq\mathbb{R}^d$, $d\in\{2,3\}$. We first prove that the non-autonomous STGFEs generate a continuous non-autonomous random dynamical system $Φ$, and we establish the existence of a pullback absorbing set. Using compact Sobolev embeddings on bounded domains and uniform tail estimates on unbounded domains, we show the pullback asymptotic compactness of $Φ$, which leads to the existence of pullback random attractors that are compact and attracting in $\mathbb{L}^2(\mathbb{Q})$. As a consequence, we demonstrate the existence of an invariant measure associated with the STGFEs and, exploiting the linear multiplicative structure of the noise along with the exponential stability of solutions, we prove uniqueness of the invariant measure in the case of zero external forcing. These results are entirely new for STGFEs on general domains, and, in particular, the existence of pullback random attractors with linear multiplicative noise is obtained here for the first time. We further note that, unlike Stratonovich noise, which is widely used in the literature to study random attractors, Itô noise is more appropriate for domains that do not satisfy the Poincaré inequality. Overall, this work resolves several open problems regarding random attractors, invariant measures, and ergodicity for stochastic third-grade fluids on general unbounded domains $\mathbb{Q}\subseteq\mathbb{R}^d$, $d\in\{2,3\}$.

Controlling the $\mathcal W_\infty$ Wasserstein distance by the $\mathcal W_p$ Wasserstein distance is interesting both for theorical and numerical applications. A first paper on this problem was written several years ago [3]. Some year later [14] framed it in the same inequality for more general costs which increase with the distance. In this paper, we prove this type of inequality for optimal transport problems with pointwise cost which is a decreasing function of the distance. We show, in particular, that there is a general framework that encompasses all the cases above.

In this article, we investigate the long-term dynamics of a class of two- and three-dimensional non-Newtonian fluids of differential type, known as third-grade fluids. We first show that when the external forcing is sufficiently small, the global attractor of the underlying system (which characterizes its asymptotic behavior) reduces to a single point. We then consider the system under stochastic perturbations, specifically infinite-dimensional additive white noise. In this random setting, we do not find conclusive evidence that the corresponding random attractor remains a single point, as in the deterministic case. However, we are able to estimate the rate at which the random attractor approaches the deterministic singleton attractor as the intensity of the stochastic noise tends to zero.

We introduce a notion of non-commutative joint independence for multiple algebras in a non-commutative probability space. The pairwise relationships between these algebras are encoded by a graph with two edge sets -- a combinatorial structure we call a bigraph -- and naturally encompass the five fundamental types of independence: tensor, free, (anti)monotone, and Boolean. It subsumes the BMT independence of Arizmendi--Mendoza--Vazquez-Becerra (when all pairwise relationships are Boolean, (anti)monotone, or tensor) and the $ε$ or $Λ$-independence of Mlotkowski (when the pairwise relationships are tensor and free). We present explicit combinatorial moment formulas, a Hilbert space construction, and natural associativity relations within this setting. Furthermore, we demonstrate that bigraph independence emerges in the asymptotic behavior of tensor product random matrix models with respect to a vector state, encompassing the Charlesworth--Collins model for $\varepsilon$-independence as a special case and offering a random matrix perspective on BMT independence.

We propose a penalty-based smoothing framework for convex nonsmooth functions with a supremum structure. The regularization yields a differentiable surrogate with controlled approximation error, a single-valued dual maximizer, and explicit gradient formulas. We then study an accelerated inertial dynamic with vanishing damping driven by a time-dependent regularized function whose parameter decreases to zero. Under mild integrability and boundedness conditions on the regularization schedule, we establish an accelerated $\mathcal{O}(t^{-2})$ decay estimate for the regularized residual and, in the regime $α>3$, a sharper $o(t^{-2})$ decay together with weak convergence of trajectories to a minimizer of the original nonsmooth problem via an Opial-type argument. Applications to multiobjective optimization (through Chebyshev/max scalarization) and to distributionally robust optimization (via entropic regularization over ambiguity sets) illustrate the scope of the framework.

We formulate the flow of thick fluids as evolution variational and quasi-variational inequalities, with a variable threshold on the absolute value of the deformation rate tensor. In the variational case, we show the existence and uniqueness of strong and weak solutions in the viscous case and also the existence of strong and weak solutions in the inviscid case. These problems correspond to solve, respectively, the Navier-Stokes and the Euler equations with an additional generalised Lagrange multiplier associated with the threshold on the deformation rate tensor. Applying the continuous dependence of strong and weak solutions to the variational inequalities for the Navier-Stokes with constraints on the derivatives, and on their respective generalised Lagrange multipliers, we can solve the case of the variable threshold depending on the solution itself that correspond to quasi-variational problems. \vspace{2mm} $$ \text{Dedicated to Vsevolod Alekseevich Solonnikov, {\em in memoriam}}$$

In this paper, we present a method capable of ensuring practical prescribed-time control with guaranteed performance for a class of nonlinear systems in the presence of time-varying parametric and dynamic uncertainties, and uncertain control coefficients. Our design consists of two key steps. First, we construct a performance-rate function that freezes at and after a user-specified time T, playing a crucial role in achieving desired precision within prescribed time T and dealing with unmodeled dynamics. Next, based on this function and a sigma-modification strategy in which the leakage term starts to vanish at t > T, we develop an adaptive dynamic surface control framework to reduce control complexity, deal with uncertainties, ensure prescribed performance, practical prescribed-time convergence to a specific region, and ultimately achieve asymptotic convergence. The effectiveness of the proposed control method is validated through numerical simulations.

Cluster-randomized trials (CRTs) are widely used to evaluate group-level interventions and increasingly collect multiple outcomes capturing complementary dimensions of benefit and risk. Investigators often seek a single global summary of treatment effect, yet existing methods largely focus on single-outcome estimands or rely on model-based procedures with unclear causal interpretation or limited robustness. We develop a unified potential outcomes framework for generalized treatment effects with multiple outcomes in CRTs, accommodating both non-prioritized and prioritized outcome settings. The proposed cluster-pair and individual-pair causal estimands are defined through flexible pairwise contrast functions and explicitly account for potentially informative cluster sizes. We establish nonparametric estimation via weighted clustered U-statistics and derive efficient influence functions to construct covariate-adjusted estimators that integrate debiased machine learning with U-statistics. The resulting estimators are consistent and asymptotically normal, attain the semiparametric efficiency bounds under mild regularity conditions, and have analytically tractable variance estimators that are proven to be consistent under cross-fitting. Simulations and an application to a CRT for chronic pain management illustrate the practical utility of the proposed methods.

In this note, we promote an infinite Kadison transitivity theorem on massive $C^*$-algebras, including the Calkin algebra. This transitivity stems from the analog of countable degree-1 saturation on pure states which is inherited from these algebras via excision. We show this saturation to be equivalent to several order-theoretic properties on the quantum filter associated to the state, in particular the property of being a quantum P-point. While we show their existence is independent from ZFC, under basic set theoretic assumptions, we produce a plethora of these states. Finally, we find an irreducible representation of the Calkin algebra which fails infinite transitivity.

We introduce a new dependence order, termed the conditional convex order, whose minimal and maximal elements characterize independence and perfect dependence. Moreover, it characterizes conditional independence, satisfies information monotonicity, and exhibits several invariance properties. Consequently, it is an ordering for the strength of functional dependence of a random variable Y on a random vector X. As we show, various recently studied dependence measures -- including Chatterjee's rank correlation, Wasserstein correlations, and rearranged dependence measures -- are increasing in this order and inherit their fundamental properties from it. We characterize the conditional convex order by the Schur order and by the concordance order, and we verify it in settings such as additive error models, the multivariate normal distribution, and various copula-based models. Our results offer a unified perspective on the behavior of dependence measures across statistical models.

Risk measures such as Conditional Value-at-Risk (CVaR) focus on extreme losses, where scarce tail data makes model error unavoidable. To hedge misspecification, one evaluates worst-case tail risk over an ambiguity set. Using Extreme Value Theory (EVT), we derive first-order asymptotics for worst-case tail risk for a broad class of tail-risk measures under standard ambiguity sets, including Wasserstein balls and $ϕ$-divergence neighborhoods. We show that robustification can alter the nominal tail asymptotic scaling as the tail level $β\to0$, leading to excess risk inflation. Motivated by this diagnostic, we propose a tail-calibrated ambiguity design that preserves the nominal tail asymptotic scaling while still guarding against misspecification. Under standard domain of attraction assumptions, we prove that the resulting worst-case risk preserves the baseline first-order scaling as $β\to0$, uniformly over key tuning parameters, and that a plug-in implementation based on consistent tail-index estimation inherits these guarantees. Synthetic and real-data experiments show that the proposed design avoids the severe inflation often induced by standard ambiguity sets.

The Polyak stepsize has been proven to be a fundamental stepsize in convex optimization, giving near optimal gradient descent rates across a wide range of assumptions. The universality of the Polyak stepsize has also inspired many stochastic variants, with theoretical guarantees and strong empirical performance. Despite the many theoretical results, our understanding of the convergence properties and shortcomings of the Polyak stepsize or its variants is both incomplete and fractured across different analyses. We propose a new, unified, and simple perspective for the Polyak stepsize and its variants as gradient descent on a surrogate loss. We show that each variant is equivalent to minimize a surrogate function with stepsizes that adapt to a guaranteed local curvature. Our general surrogate loss perspective is then used to provide a unified analysis of existing variants across different assumptions. Moreover, we show a number of negative results proving that the non-convergence results in some of the upper bounds is indeed real.

We study nonlinear Dirac equations (NLDE) on periodic quantum graphs endowed with Kirchhoff-type vertex conditions. Our main goal is to establish existence and multiplicity of bound states, which arise as critical points of the associated NLDE action functional. The underlying Dirac operator has a spectral gap around the origin, so the corresponding functional is strongly indefinite, and in addition the Palais--Smale condition fails due to the noncompactness and the periodic structure of the graph. To overcome these difficulties, we combine the spectral properties of the periodic Dirac operator with critical point theorems for strongly indefinite functionals and a concentration--compactness analysis adapted to periodic quantum graphs, and derive existence and multiplicity results for bound states with frequencies lying in the spectral gap.

A continuous-aperture array (CAPA)-based integrated sensing and communications (ISAC) framework is proposed for both downlink and uplink scenarios. Within this framework, continuous operator-based signal models are employed to describe the sensing and communication processes. The performance of communication and sensing is analyzed using two information-theoretic metrics: the communication rate (CR) and the sensing rate (SR). 1) For downlink ISAC, three continuous beamforming designs are proposed: i) the communications-centric (C-C) design that maximizes the CR, ii) the sensing-centric (S-C) design that maximizes the SR, and iii) the Pareto-optimal design that characterizes the Pareto boundary of the CR-SR region. A low-complexity signal subspace-based approach is proposed to derive the closed-form optimal beamformers for the considered designs. On this basis, closed-form expressions are derived for the achievable CRs and SRs, and the downlink rate region achieved by CAPAs is characterized. 2) For uplink ISAC, the C-C and S-C successive interference cancellation-based methods are proposed to manage inter-functionality interference. Using the subspace approach closed-form expressions for the optimal detectors as well as the achievable CRs and SRs are derived. The uplink SR-CR region is characterized based on the time-sharing technique. Numerical results demonstrate that, for both downlink and uplink, CAPA-based ISAC achieves higher CRs and SRs as well as larger CR-SR regions compared to conventional spatially discrete array-based ISAC.

Dirichlet's Lemma states that every primitive quadratic Dirichlet character $χ$ can be written in the form $χ(n) = (\fracΔn)$ for a suitable quadratic discriminant $Δ$. In this article we define a group, the separant class group, that measures the extent to which Dirichlet's Lemma fails in general number fields $F$. As an application we will show that over fields with trivial separant class groups, genus theory of quadratic extensions can be made as explicit as over the rationals.

The paper is devoted to provide Michael-Simon-type $L^p$-logarithmic-Sobolev inequalities on complete, not necessarily compact $n$-dimensional submanifolds $Σ$ of the Euclidean space $\mathbb R^{n+m}$. Our first result, stated for $p=2$, is sharp, it is valid on general submanifolds, and it involves the mean curvature of $Σ$. It implies in particular the main result of S. Brendle [Comm. Pure Appl. Math.}, 2022]. In addition, it turns out that equality can only occur if and only if $Σ$ is isometric to the Euclidean space $\mathbb R^{n}$ and the extremizer is a Gaussian. The second result is a general $L^p$-logarithmic-Sobolev inequality for $p\geq 2$ on Euclidean submanifolds with constants that are codimension-free in case of minimal submanifolds. In order to prove the above results - especially, to deal with the equality cases - we elaborate the theory of optimal mass transport on submanifolds between measures that are not necessarily compactly supported. Applications are provided to sharp hypercontractivity estimates of Hopf-Lax semigroups on submanifolds. The first hypercontractivity estimate is for general submanifolds with bounded mean curvature vector, the second one is for self-similar shrinkers endowed with the natural Gaussian measure. The equality cases are characterized here as well.