Papers

This paper develops a general framework for the well-posedness, regularity, and strong approximation of the stochastic reaction–diffusion equation (SRDE) with superlinear drift and diffusion coefficients. We first extend the well-posedness results in W. Liu and M. Röckner, J. Funct. Anal., 2902–2922, 2010 and W. Liu, J. Differential Equations, 572–592, 2013 to the case of superlinear diffusion in the Gelfand triple \(V \hookrightarrow H \hookrightarrow V^*\), with \(V\) equipped with the norm \(\|\cdot\|_V\), and derive a moment estimate by establishing a new Itô formula for \(\|X\|_V^p\) with general \(p \ge 2\). We then apply this abstract result to the SRDE, establish higher spatial regularity \(\dot H^{1+\gamma}\) for any \(\gamma \in [0,1]\) whenever the initial datum lies in the same Sobolev space, and obtain temporal Hölder regularity. Finally, we construct a family of tamed finite element methods (tamed-FEMs) for the SRDE under general assumptions on the tamed functions, derive their long-time unconditional stability, and establish optimal strong convergence rates. To our knowledge, this is the first strong approximation result for SPDEs with superlinear diffusion coefficients.

We study the equilibrium structure and long-time behavior of a nonlinear Fokker–Planck equation with diffusion and nonlocal interactions generated by squared simplex volumes. For $1\leq n<d$, the stationary states are isotropic Gaussian measures, and every solution converges to the equilibrium determined by its conserved mean. At $n=d$, the free energy fixes only the covariance determinant, allowing a continuum of anisotropic Gaussian equilibria when $d\geq2$. We establish global well-posedness for arbitrary initial probability measures with finite second moment, identify stationary states with the global minimizers of the free energy, and prove convergence in quadratic Wasserstein distance with explicit exponential rates.

In this paper we study a linear drift perturbed by a superposition of $m$ independent fractional Brownian motions with known Hurst parameters and a common scale, observed at $N$ equidistant times. Inference for such models is usually asymptotic; we show that here it is exact. We derive the maximum likelihood estimators of the drift $\theta$ and of the scale $\alpha^{2}$ in closed form and obtain their exact finite-sample joint law: $\widehat\theta$ is Gaussian, $N\widehat\alpha^{\,2}/\alpha^{2}$ is chi-square with $N-1$ degrees of freedom, and the two are independent. As this law is free of every model parameter, we deduce Student and chi-square confidence intervals and tests of exact level for every $N\ge2$, whatever the Hurst vector. We also prove that the estimators are uniformly minimum variance unbiased with $\widehat\theta$ attaining the Cramér–Rao bound at every $N$, that both are strongly consistent and asymptotically normal, and that the drift estimators form, in law, a Brownian motion run along their own variance scale. A sharp non-asymptotic bound shows that the accuracy of the drift is governed by the length of the observation window and not by the mesh, and a Monte Carlo study confirms exact coverage, even at small sample sizes, and quantifies what is lost when the Hurst vector is misspecified.

Let $\Gamma$ be a countable discrete $C^*$-simple group. Let $\mu$ be a $\Gamma$-invariant Borel probability measure on $\text{Sub}(\Gamma)$, i.e., an IRS. We show that $\mu$-almost every subgroup $H$ is $C^*$-simple.

We study complex-symbol Toeplitz operators on weighted Fock spaces. The weight has a uniformly positive and bounded real Hessian. Local analytic distance is unchanged when an entire function is added to the symbol. We combine this distance with a complex ball average or the Berezin transform to control the local size of the symbol. Under a matching Lorentz condition on the distance, we characterize membership in every Schatten–Lorentz class. The local symbol exponent can be any $1\le q<\infty$ and is independent of the two Schatten–Lorentz indices. The proof uses positive Toeplitz estimates, Hankel estimates, and a local decomposition that preserves the initial domain. Two examples show why the local exponent and the secondary Lorentz index must be kept. We also obtain mixed mapping criteria and regular singular-value decay estimates. An error operator with a lower-order counting function does not change the leading coefficient. This gives explicit Weyl constants for quadratic weights. A radial example shows that uniform Hessian bounds alone do not imply such a constant.

We study the stationary spectrum of a coherently driven single-mode Kerr cavity in Bargmann–Fock space. For positive Kerr coupling the Hamiltonian is self-adjoint, bounded below, and has compact resolvent. For nonzero drive, a minimal-solution continued fraction yields an exact scalar spectral condition, with all eigenvalues simple. A Liouville transformation identifies the eigenvalue equation with a degenerate double-confluent Heun problem and classifies its singularities. An Olver-type Volterra construction gives normalized sectorial solutions at infinity with explicit error bounds. In the strong-drive regime, the leading turning-point energy coefficient is proved, whereas the next coefficient remains conditional on a stated global-connection conjecture; independent Bogoliubov and complex-WKB calculations provide consistency checks. A formal weak-drive expansion is verified from the Bargmann recurrence, and its upper-edge tridiagonal series is summed in closed form. Exploratory WKB iteration portraits are included only as visual aids and do not establish the conjectural global connection.

Let $T(x)=\prod_{j\ge 0}(1-x^{2^j})=\sum_{n\ge 0}t(n)x^n$ be the Thue–Morse generating function, and write $T(x)^m=\sum_{n\ge 0}t_m(n)x^n$ for a positive integer $m$. For a nonzero integer $a$, let $\nu_2(a)$ be the exponent of $2$ in $a$ and put $\operatorname{odd}(a)=a/2^{\nu_2(a)}$. For every $s\ge 1$, we prove that the sequence $(\operatorname{odd}(t_m(n))\bmod 2^s)_{n\ge 0}$ is $2$-automatic when $m=2^r$, $r\ge 1$, or $m=3\cdot 2^r$, $r\ge 2$. In both families, $\nu_2(t_m(n))=\nu_2\binom{n+m-1}{m-1}$. The valuation identity for $m=2^r$ is known; our proof recovers it and also establishes the automaticity assertion. For $m=6$ we prove $\nu_2(t_6(n))=\nu_2\binom{n+5}{5}+\mathbf{1}_{n\equiv 3\pmod 4}$ and show that its odd parts modulo every $2^s$ are $2$-automatic.

Our aim is to find all the prime numbers $p$ such that $p-x^2$ has at most two different prime factors, for all the odd integers $x$ such that $x^2<p$. We solve entirely the cases $p\equiv 1,5,7 \pmod 8$. The case $p\equiv 3 \pmod 8$ is solved with one possible exception.

In this paper we prove some one-dimensional symmetry results for stable solutions to quasilinear anisotropic elliptic equations in $\mathbb R^2$ and $\mathbb R^3$. In this regard, we give an extension of a formula obtained in \cite{SZ1,SZ2, FSV}. The techniques used are inspired by the works of \cite{Fhdr,FSV}.

The derived delooping level is a recently introduced homological invariant that provides an upper bound for the finitistic dimension of the opposite algebra. In this paper, we employ two reduction techniques-cleft extensions and recollements-to study the finiteness of the derived delooping level of finite-dimensional algebras over a field. By applying the theory of cleft extensions to bound quiver algebras, we establish arrow-removal operations that preserve the finiteness of the derived delooping level. In parallel, using recollement techniques, we develop vertex-removal operations with the same finiteness-preserving property. We conclude with several examples illustrating the applicability and effectiveness of these reduction methods.

In this paper, we introduce an amnesic elephant random walk with polynomially decaying step sizes. The walk retains the loss of memory effect of amnesic ERW, however its step sizes decay polynomially over time. We study the effect of step-size exponent and memory parameter on the long-time behaviour of the walk. Two critical thresholds are identified that determine its phase diagram. We obtain almost sure convergence results, law of iterated logarithm, asymptotic normality and mean square displacement rate of the walk across different parameter regimes. The polynomial decay of step sizes gives rise to a subdiffusive regime while classical diffusion is recovered in the absence of decay. Also, we identify localization regimes in which the walk converges almost surely to a finite random variable.

Cladograms of size $n$ are unrooted binary trees whose leaves are labelled from 1 to $n$. Aldous introduced in 2000 a Markov chain on cladograms, a step of which consists of removing a leaf uniformly at random and reinserting it on a uniformly chosen edge. We introduce a coupling for this walk, which follows multiple colored subtrees in parallel. We go around the lack of independence of the colored components by finding a relevant statistic, namely the sum of the squares of the sizes of all colored components, which we prove has a drift. We deduce that the mixing time of the walk is of order $n^2$, solving a conjecture of Aldous.

Feature-based multi-product pricing uses customer characteristics to identify demand heterogeneity and tailor prices across products. Choice model trees segment customers through interpretable feature rules and fit a demand model within each leaf. Existing methods typically construct these trees greedily, selecting one myopic split at a time. We develop optimal choice model trees with multinomial logit leaves (OCMT-MNL), jointly optimizing the tree and leaf models within a prescribed depth. Our exact dynamic program derives closed-form Fenchel lower bounds during constrained Newton iterations and propagates them across nested and disjoint customer subsets, avoiding new fits and resuming unfinished fits without repeating completed work. In synthetic experiments, it reduces exact leaf fits by 99.98% and leaf evaluations by 86.13%, achieving up to 7.15-fold speedups over unpruned dynamic programming. One-dimensional lookup tables translate offline estimation into real-time pricing, with a revenue-loss bound quadratic in grid spacing under the fitted model. Compared with greedy trees, OCMT-MNL achieves lower revenue loss with fewer leaves on synthetic data and better predictive fit on real data. In a 23-week randomized experiment on ancillary seat pricing across 48 airline markets and 190,220 passengers, OCMT-MNL increases seat revenue per passenger by a statistically significant 11.3% over static pricing.

This paper is the first in a two-part series, seeking to i) extend the theory of mass erasure developed by Duquesne and Winkel to a setting with infinite measures, and ii) apply this theory to obtain invariance principles for Galton–Watson and Lévy forests in the most general setting where the trees may be supercritical and not satisfy Grey's condition. We consider $\mathbb{R}$-trees equipped with a suitable subclass of boundedly finite measures, which we call discretely infinite measures. Let $(T, d, \rho)$ be a complete and separable $\mathbb{R}$-tree, let $\mu$ be a discretely infinite measure on $T$, and let $h > 0$. The $h$-mass-erased subtree is obtained by removing all fringe subtrees with $\mu$-mass strictly less than $h$, and we prove that all $h$-mass-erased subtrees have a discrete branching structure. The mass-erased subtrees may be equipped with associated measures, which gives rise to a family of operators $(\mathscr{E}_h)_{h \geq 0}$ which forms a semigroup on the space of discretely infinite measures. The operators may be lifted to an appropriate space of Gromov-vague isometry classes. Given a sequence $\boldsymbol{\mu}_n = [T_n, d_n, \rho_n, \mu_n]$, $n \in \mathbb{N}$, of suitable isometry classes, we say that they converge vaguely in the sense of mass erasure if their mass erasures $(\mathscr{E}_h \boldsymbol{\mu}_n)_{n \in \mathbb{N}}$ converge in the Gromov-vague topology for all $h > 0$. We prove that by imposing suitable tightness conditions on $(\boldsymbol{\mu}_n)_{n \in \mathbb{N}}$, Gromov-vague convergence implies vague convergence in the sense of mass erasure. Under appropriate tightness, we identify conditions which relate the two notions of convergence, and prove that either mode of convergence implies local Gromov–Hausdorff convergence of the mass-erased subtrees. The applications to Galton–Watson and Lévy forests are considered in part two.

We study an effective model of topological insulators joined along a bent interface. The Hamiltonian is a Dirac operator $\mathcal{D} = D_1\sigma_1 + D_2\sigma_2 + \kappa(x) \sigma_3$ on $L^2(\mathbb{R}^2,\mathbb{C}^2)$, where $\kappa(x)$ represents a domain wall with a corner: while initially straight, it undergoes a bend when passing from one side of $\mathbb{R}^2$ to the other. For most energies in a spectral window centered at $0$, we construct a reflectionless edge state of $\mathcal{D}$: a distorted plane wave with transmission coefficient of modulus $1$. This implies that $\mathcal{D}$ admits modes confined near $\kappa^{-1}(\{0\})$, that travel through the bend with no loss neither to the bulk nor to reflection. Our result explains experimental observations from the physics literature, as well as our own numerical simulations.

Near-field to far-field transformation is central to large-aperture antenna testing, yet two coupled challenges remain: costly phase acquisition at millimeter-wave bands and violations of the centering assumption under offset mounting. Existing methods address these issues separately, requiring either dense full-field data or offset vectors. We tackle both jointly by exploiting a key observation: amplitude fields under different offsets are coordinate-transformed views of the same near field. The challenge is to recover the center-aligned field from offset amplitudes without a phase or offset vector. We propose AntennaFlow, a three-stage framework: a contrastively learned encoder that maps offset views to an offset-invariant embedding, a deterministic flow-matching transport that maps offset amplitudes to center-aligned ones, and the Simplified Extrapolation Technique, whose Green-function Taylor expansion is valid only for centered fields. Experiments show that AntennaFlow enables fast, phaseless, offset-vector-free NF–FF reconstruction from sparse amplitude-only measurements, consistently outperforming existing baselines while preserving physical consistency.

Let $\mathcal S(T)$ be the adjacency spectral center of a tree $T$, and let $\mathcal C(T)$ be its characteristic set. We determine the largest possible separation $d(T) := \operatorname{dist}_T(\mathcal{S}(T), \mathcal{C}(T))$ among trees of every order $n\ge3$. Writing $\Delta_n:=\max_{|V(T)|=n}d(T)$, we prove $\Delta_n=0\quad(3\le n\le11),\quad \Delta_{12}=1$, and $\Delta_n=\left\lfloor\frac{n-11}{2}\right\rfloor \quad(n\ge13)$. The argument rests on a simple opposition between two rooted-tree weights. An endpoint-rooted path minimizes adjacency spectral radius, but maximizes bottleneck Perron value. A one-sided replacement by a path therefore cannot decrease the distance between the two centers. Quantitatively, this gives the sharp estimate $|V(T)|\ge 2d(T)+11 \quad(d(T)\ge2)$. A preliminary six-vertex barrier shows that disjoint center sets require at least twelve vertices, and the four-leaf broom is extremal for every $n\ge12$.

We survey recent developments on the null energy condition (NEC) from the perspective of optimal transport, with emphasis on non-smooth Lorentzian geometry. After recalling the role of classical energy conditions in general relativity, we discuss the relation between Ricci curvature and entropy convexity in optimal transport. The main focus is the authors' optimal-transport characterization of the NEC. In the smooth setting, the NEC is equivalent to displacement concavity of the Shannon entropy power along suitable null-geodesic transport plans supported on null hypersurfaces and measured with respect to rigged measures. We review the geometric ingredients of this characterization and applications, including a weighted Hawking area theorem. We then turn to the synthetic theory introduced in a second paper of the authors. Synthetic null hypersurfaces are defined using achronal boundaries, gauge functions, and reference measures in topological causal spaces. In this framework, the displacement-concavity characterization becomes a synthetic curvature condition, $\mathsf{NC}^{e}(N)$, meaningful without differentiable structure. We discuss compatibility with the smooth theory, invariance under natural changes of gauge and reference measure, and stability under convergence. Finally, we present a synthetic version of Penrose's singularity theorem for continuous Lorentzian metrics. These results indicate that geometric and causal consequences of the NEC persist beyond the smooth setting and highlight optimal transport as a framework connecting curvature, causality, and singularity formation in Lorentzian geometry.

Langevin-based Markov chain Monte Carlo (MCMC) algorithms use gradient information to improve sampling, particularly in high dimensions. Classical optimal scaling theory for these algorithms has largely focused on the Metropolis-Hastings (MH) acceptance rule. However, there has been a recent surge in acceptance rules beyond MH for applications spanning differential privacy, stochastic MCMC, diffusion models, and molecular dynamics. We develop optimal scaling results for Langevin proposals employed with generalized acceptance rules belonging to a suitable class. For high-dimensional targets, we recover the usual $O(d^{-1/3})$ scaling, while different acceptance rules lead to different optimal acceptance probabilities.

Let $\mathcal{M}$ be a finite von Neumann algebra with normalized center-valued trace $T_{\mathcal{M}}$. We prove that every $A\in\mathcal{M}$ with $T_{\mathcal{M}}(A)=0$ admits a representation $A=BC-CB$, where $B,C\in\mathcal{M}$ satisfy $\|B\|\|C\|\le K\|A\|$ for an absolute constant $K$ independent of $\mathcal{M}$.

An algebra with binary operations $+,\cdot,\uparrow$ and constant $1$ is called an HSI algebra if it satisfies the basic commutative semiring laws for $+,\cdot,1$ on~$\mathbb{N}$ as well as the familiar index laws for exponentiation $\uparrow$. These basic axioms, known as the “High School Identities” are known to be incomplete, and an algebra satisfying $\HSI$ but failing an equation valid on ${\bf N}:=\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$ is called a Gurevič algebra. It is currently unknown if there is an algorithm to recognise finite Gurevič algebras, and the best current result is that there exists a 12-element Gurevič algebra, and that none of the five 2-element HSI algebras are Gurevič algebras. We explain how the work of Alex Wilkie can be used to provide an algorithm for deciding validity of the equational laws of $\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$, and use this to show that multiplicatively-idempotent HSI algebras satisfy all valid laws of $\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$. As consequences of this result, we show that all 44 HSI algebras on 3 elements lie in the variety of $\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$ (that is, are not Gurevič algebras), as well as 597 of the 657 models on 4 elements and 11158 of the 13577 models on 5 elements. A further consequence is that every Brouwerian lattice lies in the variety of $\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$ (up to a simple term equivalence), showing that the variety of $\langle \mathbb{N};+,\cdot,\uparrow,1\rangle$ has continuum many subvarieties. The constant-free signature is also explored, and it is shown that in this case all $2$-element models of the constant-free High School Laws satisfy all valid constant-free laws in $\langle \mathbb{N};+,\cdot,\uparrow\rangle$.

Linear complementary pairs (LCPs) and linear complementary dual (LCD) codes have important applications in orthogonal direct-sum masking (ODSM), which provides effective countermeasures against side-channel attacks and fault-injection attacks. While LCD codes have been extensively investigated, comparatively fewer results are available for general LCPs. In this paper, we further investigate LCPs of twisted Reed–Solomon (TRS) codes. We derive necessary conditions for two TRS codes to form an LCP and establish several sufficient conditions and explicit constructions. We also study LCD codes constructed from TRS codes and investigate the security parameters of the resulting LCPs. Furthermore, under suitable conditions, we obtain MDS LCPs of TRS codes.

A complete characterization of the stability-preserving exponent set for fractional Hadamard powers of a monic Hurwitz-stable polynomial f of degree five is presented. The stability problem is reduced to the analysis of a single scalar function depending only on three parameters formed from the coefficients of the polynomial f. This reduction leads to a unique stability threshold $p_*(f) \in (0,1)$ such that the $p$-th Hadamard power of $f$ is Hurwitz stable if and only if $p > p_*(f)$. Consequently, the stability-preserving exponent set is precisely $(p_*(f),\infty)$. This threshold depends smoothly on the parameters and provides a global coordinate on the admissible parameter region. Finally, smooth dependence is illustrated by a one-parameter family of Hurwitz-stable polynomials whose complex-conjugate zeros approach the imaginary axis.

We investigate the regularity of viscosity solutions to a class of nonlocal Hamilton-Jacobi equations driven by x-dependent integro-differential operators and coercive superlinear Hamiltonians. We first establish Hölder regularity for bounded viscosity solutions under general structural and continuity assumptions on the underlying Lévy measures, without imposing any ellipticity condition on the nonlocal operator. The Hölder exponent is given explicitly in terms of the order $\sigma$ $\in$ (0, 2) of the operator and the growth exponent m > 1 of the Hamiltonian. In particular, our approach applies to arbitrary superlinear Hamiltonians, including the delicate regime 1 < m < $\sigma$ < 2, and yields an improved regularity exponent when $\sigma$ $\in$ (1, 2). Assuming in addition a weak ellipticity condition on the nonlocal operator, we prove that viscosity solutions are globally Lipschitz continuous. The proof combines the Hölder regularity supplied by the coercive Hamiltonian with the regularizing effect of the nonlocal diffusion through an Ishii-Lions argument, allowing us to treat Hamiltonians with arbitrary superlinear growth. Finally, we provide a counterexample showing that, in the absence of ellipticity, Lipschitz regularity may fail if the spatial dependence of the Lévy measures is merely Hölder continuous, thereby illustrating the sharpness of our continuity assumptions.

For multivariate scale and location–scale models with independent components, we extend the univariate results of Zhou and Nayak (2012) and derive optimum equivariant estimators under the generalized Pitman closeness criterion. We first show, by a counterexample, that in the multivariate case the Pitman closeness comparison within the class of equivariant estimators is not transitive, so that a Pitman closest equivariant estimator does not exist in general. We then enlarge the transformation group by the coordinate permutations—equivalently, impose the formal equivariance principle of Berger (1985) across isomorphic component problems, in the spirit of the separable rules of Robbins' (1951) compound decision theory—and show that within the resulting restricted class an optimum is restored. A multivariate median lemma based on strictly convex losses then yields explicit Pitman closest equivariant estimators of the scale parameters, powers of the scale parameters, and the location parameters, given by median-adjusted versions of any given equivariant estimator. Applications to the multivariate uniform and multivariate normal distributions are worked out in detail. Monte Carlo experiments for the Rayleigh distribution and for a competing risks model with Rayleigh component lifetimes confirm that the proposed estimators dominate the maximum likelihood and Bayes estimators under the Pitman closeness criterion, and a real industrial data set on ball bearing failure times illustrates the feasibility of the method in practice.

Burson and Eichhorn proposed infinite and finite coefficientwise non-negativity conjectures for products arising from copartitions. We obtain several further results. First, both products admit decompositions into elementary local blocks, and the finite conjecture reduces exactly to its diagonal specialization. The finite conjecture is proved when the first truncation parameter is one or when the two residue parameters are equal. We also prove that every finite product has non-negative coefficients below the third shifted signed exponent. For fixed finite parameters, the coefficient sequence is shown to be eventually quasipolynomial; an explicit period bound, an explicit onset for the quasipolynomial formula, and the leading term on every residue class are obtained. This yields eventual strict positivity, apart from forced odd-degree zeros in one solved diagonal case. For the infinite conjecture, we prove non-negativity under the broader condition $a\equiv b\pmod m$, which includes the known case $a=b$. Finally, for arbitrary $b\mid a$, we improve the uniform initial non-negative range from degrees below $a+m$ to degrees below $a+2m$.

We devise a polynomial-time $3.159$-approximation algorithm for the metric unsplittable Capacitated Vehicle Routing Problem. We build on the Relative Greedy Algorithm suggested by Traub (2025), which can be considered as a variant of the LP rounding algorithm of Friggstad, Mousavi, Rahgoshay, and Salavatipour (2025). Our main ingredient is the Average Greedy Algorithm, a new algorithm that controls both tour costs and the coverage of clients with high demand. This additional control enables a sharper averaging argument for the cost of subsequent greedy choices. Similarly to Zhao and Xiao (2026), combining the Average Greedy with variants of tour partitioning and a matching algorithm yields the final approximation guarantee.

Lyapunov constants are a crucial tool for analyzing bifurcation behavior in switching systems. This paper proposes an efficient algebraic symbolic algorithm for computing Lyapunov constants in planar switching systems. By mapping the vector fields into the complex domain, we construct a normal form algorithm based on Laurent poly?nomials. The classical Poincaré map method requires repeated symbolic integration, which may lead to substantial computational difficulties at high orders. Our ap?proach improves computational efficiency by replacing continuous integration with direct algebraic operations. We apply this approach to investigate two switching models. For a generalized switching quartic Liénard system, the algebraic method extracts the center conditions and proves the existence of 10 small-amplitude limit cycles, establishing a new lower bound for its cyclicity. Furthermore, a physically motivated circuit-level modification of the Alpazur oscillator is investigated. We show that at most one small-amplitude limit cycle can bifurcate from the center and that this upper bound is attainable, illustrating the applicability of the proposed algorithm to a practical switching circuit. This application shows that our method provides a systematic framework for analyzing and interpreting complex dynamical phenomena arising in practical piecewise smooth systems.

We continue to develop a theory of (Besov and) Triebel–Lizorkin spaces associated with weighted subcoercive operators on a real connected Lie group, focusing on the Triebel–Lizorkin space $F^{\infty,q}_\alpha(\beta)$, which requires a more technical treatment.

We continue to develop a theory of Besov and Triebel-Lizorkin spaces associated with weighted subcoercive operators on a real connected Lie group, focusing on the algebra properties, pointwise multipliers, localized norms, and characterization by differences.