Papers

We characterize bivariate measures of concordance of degree at most two in the sense that the measures evaluated at copula-mixture segments are polynomials of degree at most two. Our main device is the canonical polarization, which converts quadraticity of a functional into separate affinity, thereby allowing an integral representation for affine functionals to be applied sectionwise. Using this technique, we prove that measures of degree at most two are in bijection with copula-indexed measure fields satisfying several properties including square-group equivariance. The field at independence determines the affine linearization, while its displacement determines the purely quadratic part. Moreover, exact degree two is equivalent to field non-constancy. Leveraging our characterization result, we provide a construction of degree-two measures of concordance based on affine copula operators that are square-group equivariant. As a more concrete example, we derive a necessary and sufficient condition for a weighted averaging operator of square-group transformed copulas to generate a measure of concordance of degree two.

Supply chains are interconnected networks of processes and operations producing and delivering high-value products. These chains are increasingly subjected to structural changes from the energy transition and other external factors. To address this, this work develops mixed-integer programming formulations to identify optimal reconfigurations that preserve industrial operations and profitability. We propose products and spatial neighborhoods to restrict the feasible search space and enable fast heuristic solutions. Furthermore, this restriction combines structural and product-based information, thus allowing to explore and define multiple reconfiguration scenarios. We demonstrate the approach using an agricultural waste case study, showing its ability to quickly produce good quality solutions.

In this paper, we investigate the convergence of an implicit defect-correction solver for a viscous discretization based on the alpha-damping scheme for unstructured grids. We show that significantly more robust iterative convergence is achieved by evaluating the damping term at the midpoint between two adjacent cell centers (edge midpoint) rather than at the face centroid. A one-dimensional Fourier analysis reveals that the implicit solver tends to be stable when the damping term in the residual is smaller than that used to construct the Jacobian. This observation suggests that the solver can be stabilized by effectively reducing the magnitude of the damping term in the residual - an effect achieved by the edge-midpoint evaluation. Robust convergence is demonstrated numerically for two-dimensional viscous-flow problems on highly irregular mixed-element and triangular grids.

For a bounded operator \(T\) between Banach spaces, we introduce the fixed-copy exponent sets \[ \begin{aligned} \operatorname{Fix}_{\ell}(T) &:= \{r\ge1:T\text{ fixes a copy of }\ell^r\},\\ \operatorname{Fix}_{L}(T) &:= \{r\ge1:T\text{ fixes a copy of }L^r(0,1)\}. \end{aligned} \] For \(1\le p,q<\infty\), we completely determine both sets for every bounded composition operator \(C_\varphi:H^p\to H^q\). In particular, \(C_\varphi\) is strictly singular if and only if \(\operatorname{Fix}_{\ell}(C_\varphi)=\varnothing\). The classification also gives complete characterizations of the \(\ell^r\)-singular and \(L^r(0,1)\)-singular subclasses for every \(r\ge1\). As a further consequence, it completely resolves Problems~4.3(1) and~4.3(2) posed by Laitila, Nieminen, Saksman, and Tylli. The proofs introduce a new localization method for producing fixed copies from boundary lower estimates. Its main ingredient is a localization theorem independent of composition operators: for every measurable \(E\subset\mathbb T\) with \(m(E)>0\) and \(1\le p<r\le2\), it constructs a single copy of \(L^r(0,1)\) in \(H^p\) on which lower \(L^s(E)\) estimates hold simultaneously for all \(1\le s\le p\), with constants depending on \(E\) only through \(m(E)\). The classification combines this method with pullback measure criteria, known fixed-copy results, and classical subspace restrictions. For \(r<2\), the localization theorem is proved using stable integrals and analytic lifting; the endpoint \(r=2\) is handled by an \(E\)-adapted lacunary construction.

In this paper, we study the Cauchy problem for the compressible Navier-Stokes system in $\mathbb{R}^3$. Suppose that the viscosity coefficients satisfy $0<\max\{\mu, \nu=\lambda+2\mu\}<1$, and set $\varepsilon=\min\{\mu, \nu=\lambda+2\mu\}$. We establish the global existence of classical solutions when the initial perturbations of the density and the curl-free part of the velocity are smaller than $\varepsilon^{\frac12+}$ (up to a logarithmic loss), while the divergence part of the initial velocity is smaller than $\varepsilon$. This improves the classical global existence result of Matsumura-Nishida \cite{MaN80}, which requires all the initial data to be smaller than $\varepsilon (<1)$. We expect that this result is representative of general Shizuta-Kawashima systems arising in physical applications. The improvement of the index from $1$ to $\frac12+$ relies on exploiting the hidden Kawashima-type dissipation for the density and controlling the spacetime trace norm of the solution at the scale $\sqrt{\varepsilon}$. These two ingredients are obtained through a weighted trace inequality and a Morawetz-type inequality for the perturbed sound speed and the divergence of the velocity.

Let $B$ denote Bloch's constant. We prove \[ B \ge \frac{\sqrt{3}}{4}+0.0153 = 0.448312701\ldots, \] improving the lower bounds by Chen–Gauthier ($\sqrt{3}/4+2\cdot10^{-4}$) and Xiong ($\sqrt{3}/4+3\cdot10^{-4}$). The proof refines Bonk's method through computer-assisted estimates rigorously verified using interval arithmetic.

We find some significant special cases of subtraction games with infinite subtraction sets, whose outcome sequences are periodic, but whose SG value sequences are bounded and aperiodic.

We develop a quantum algorithm for slow analytic Hamiltonians $\widetilde H(t)=H(t/T)$ with $\|H(s)\|\leq\alpha$ that achieves nearly additive query complexity and low gate overhead. Our main technical contribution is a periodic Gevrey extension of $H(s)$, together with Fourier component decay and truncation bounds that enable an efficient finite-dimensional simulation. Combined with Floquet embedding and optimal time-independent Hamiltonian simulation technique, this gives query complexity $\widetilde{\mathcal O}\!\left(\alpha T+\log(1/\varepsilon)\right)$ and additional gate complexity $\widetilde{\mathcal O}\!\left((\alpha T+\log(1/\varepsilon))^2\log(1/\varepsilon)\right)$, assuming coherent access to $H'(s)$ and endpoint derivatives. For slow analytic control Hamiltonians, only block encodings of the time-independent control operators are required, with the same query complexity and lower gate overhead. Our method also extends to Gevrey Hamiltonians and improves the precision dependence for simulating slow analytic semi-dissipative linear differential equations.

It is well known in random matrix literature that the limiting spectral distribution of a Wigner matrix is the semi circular law while the limiting spectral distributions of other patterned matrices like Toeplitz, Hankel, symmetric circulant and reverse circulant matrices have unbounded supports. One fundamental difference between the Wigner matrices and the other matrices mentioned above is that the Wigner matrices have $O(n^2)$ independent random variables while the others have $O(n)$ independent random variables. In this paper, we show that this is not true in general. In particular, we form matrices with $O(n)$ independent random variables whose empirical spectral distributions are arbitrary close to the semi circular law.

We study an inverse problem for the heat equation in a medium that generates internal heat at a rate proportional to the heat density. The goal consists of determining the heat-generation coefficient $V$ from the knowledge of the initial-to-final map, which assigns to every initial heat density its final heat density. We state and prove a uniqueness result for the heat operator in a region modelled by $\mathbb{R}^n$ with $n \geq 2$. We assume that $V$ is bounded and decays super-exponentially. Our approach relies on constructing exponentially-growing solutions for the corresponding heat operator. A key contribution of our approach is to provide a weighted $L^2$-estimate, which follows from the moment generating function of a Gaussian random variable. This extends the initial-to-final-state inverse problem, previously studied for the Schrödinger equation, to the parabolic setting.

This work presents a classification of the maximal totally geodesic submanifolds of the complex manifold M = SU(n+2)/S(U(1) x U(1) x U(n)) \cong F_{1,2}(\mathbb{C}^{n+2}), n\geq 2, equipped with its standard homogeneous Riemannian metric. Using the natural fibration S^2 \to M \to G_2(\mathbb{C}^{n+2}) and curvature computations derived from O'Neill's formulas, we identify curvature-invariant subspaces of the tangent representation. As a consequence, we obtain a complete description of the maximal totally geodesic submanifolds of M. These include twistor spaces of lower-dimensional complex Grassmannians, real flag manifolds, products of complex projective spaces, and twistor spaces of quaternionic projective spaces.

In 1987, Elkies proved the striking result that every elliptic curve over $\mathbb{Q}$ has infinitely many supersingular primes. Motivated by this theorem and its connection with the Lang-Trotter conjecture, we study the analogous problem for Frobenius fields of elliptic curves. For certain families of non-CM elliptic curves $E/\mathbb{Q}$ and imaginary quadratic fields $K$, we prove that there exist infinitely many primes $p$ for which the Frobenius field of $E$ at $p$ equals $K$. More precisely, letting $\pi_E(x,K)$ denote the number of such primes with $p\leq x$, we establish the unconditional bound $\pi_E(x, K)\gg_{E, K, \epsilon} (\log\log x)^{1-\epsilon}$ for every $\epsilon>0$. We also prove unconditional power-saving upper bounds for a restricted counting function associated with $\pi_E(x,K)$. The approach combines Deuring's theory of complex multiplication, properties of singular moduli, and arithmetic intersection theory on modular curves.

For each $d\in\N_{\geq 2}$ we construct a Delone set $Y$ in $\R^{d}$ for which every Lipschitz bijection from $Y$ to $\Z^{d}$ has a very irregular inverse. For example, the inverse fails to be Lipschitz, even at just a single point. Further, we clarify the relationship between several notions of regularity for bijections between Delone sets, studied in the literature.

This paper considers European multi-asset option pricing under Lévy and affine characteristic-function models. The main obstruction is the curse of dimensionality: direct multidimensional COS pricing forms tensor-product coefficient arrays whose size grows exponentially with the number of assets. We study and extend COS-TT-CHF, a low-rank construction that uses TT-cross to compress sampled characteristic-function tensors into tensor-train COS coefficients for arithmetic basket and min/max option pricing. Once built, the compressed representation gives fast post-setup strike-grid and selected component Delta/Vega calculations. The numerical study compares with adaptive-quadrature Fourier benchmarks, direct COS, a tensor-Fourier min-option benchmark, and quasi-Monte Carlo (QMC) references based on randomized Sobol points. The reported timings show a low-dimensional crossover against direct COS as the benchmark moves from $d=2$ to $d=4$, favorable timings against the tensor-Fourier min-option benchmark from $d=3$ onward, and favorable timings against the QMC common-Heston reference already at $d=2$. The reported tests reach $d=30$ for GBM and $d=20$ for VG, NIG, and common-Heston benchmark families, with accuracy, rank, runtime, control-sensitivity, and component Delta/Vega diagnostics reported throughout.

Let G(p, .) be the normalized minimal positive Green function on a complete nonparabolic Riemannian manifold with nonnegative Ricci curvature, and set b = G^(1/(2-n)). Colding established the following rigidity result in dimension three: if |nabla b|^2 admits a C^2 regularity across the pole, then the manifold must be isometric to Euclidean space. We show that rigidity conclusions corresponding to the pole regularity condition alone do not hold in higher dimensions: for any n >= 4, we can construct explicit complete nonflat rotationally symmetric manifolds with nonnegative sectional curvature and Euclidean volume growth, such that |nabla b|^2 extends smoothly across the pole. For any odd integer n >= 3, we impose finite-order pointwise curvature conditions at the pole (where this condition is vacuous when n = 3), so that the Green function satisfies the asymptotic expansion G(p,x) = r^(2-n) + H_p + O_1(r). Under this curvature assumption, if |nabla b|^2 extends to a C^(n-2) function near the pole, then the manifold must be isometric to R^n, and this regularity threshold is also sharp. As special cases: in dimension three, if |nabla b|^2 is in C^1, then the manifold must be isometric to Euclidean space; in dimension five, it suffices to assume that the scalar curvature at the pole satisfies Scal(p) = 0 and |nabla b|^2 is in C^3. In contrast, for any even dimension n >= 4, there exist complete nonflat examples that are Euclidean in a neighborhood of the pole and admit a smooth extension of |nabla b|^2 across the pole.

In this paper, we consider the connectivity of a random network of N mobile devices in three dimensions (3D), where the location of each device or node has a Gaussian distribution in each dimension. For each pair of nodes, the probability of connectivity is related to the nodes' separation by a Gaussian connectivity function. The fundamental analytical tool for studying such systems is the probability of a given network state, derived and expressed in terms of its graph Laplacian. Leveraging this result, we obtain results for the connectivity of small networks, the probability of a complete network (where all nodes are connected to all other nodes), and the probability of an isolated node, which gives an approximation to the connectivity probability of larger networks. The general results are then simplified for special cases and limiting scenarios.

Near-field holotomography combines coherent diffraction imaging with tomographic acquisition to recover the three-dimensional complex refractive index of a specimen. Since the measured diffraction intensities are generated by a nonlinear object transmission and wave propagation process, the resulting inverse problem is intrinsically nonlinear and ill-posed. We study a direct variational reconstruction framework based on a fully nonlinear wave propagation model that avoids both intermediate phase retrieval and linearization under the weak-object approximation. We analyze the forward operator in appropriate Banach spaces, establish its Fréchet differentiability, and derive explicit gradient expressions for variational reconstruction. To improve quantitative reconstruction of weak absorption features, we further develop a phase-guided Bregman TV regularization framework that exploits structural correlations between phase and absorption components. This enables multi-material reconstruction without imposing a globally fixed ratio between the two components. We perform numerical studies on synthetic phantoms and experimental data. The results demonstrate stable three-dimensional reconstructions and improved recovery of the absorption contrast compared to state-of-the-art methods.

We study the asymptotic behavior of the initial-boundary value problem for the radially symmetric Navier–Stokes–Korteweg (NSK) equations defined on the exterior domain $\Omega = \{x\in\R^n~|~|x|> 1\}$. In particular, we consider the impermeable wall problem, where the velocity at the boundary $\{x\in\R^n~|~|x|=1\}$ is set to be zero. We show that, if the initial data is a small perturbation of the stationary solution, and the boundary data are sufficiently small, then there exists a global-in-time strong solution to the radially symmetric NSK equations, and it converges to the stationary solution time-asymptotically. Our method is based on elementary energy estimates with a combination of carefully designed energy functionals.

This article presents a fast, high-order Nyström solver for the two-dimensional Lippmann–Schwinger equation arising from time-harmonic scattering by penetrable, piecewise-smooth heterogeneous media. Relying on high-order evaluation of the Newtonian potential on unstructured grids adapted to interfaces of discontinuity, the methodology achieves high-order accuracy using existing fast algorithms such as the fast multipole method. As an iterative method the solver exhibits rapid convergence when coupled to a preconditioning strategy that exploits a class of structured-grid solvers—fast solution methods offering quasi-linear time and memory complexity and nearly-constant iteration counts, but long limited in accuracy. The preconditioning strategy couples the Nyström discretization—given by high-order quadrature nodes over a (curved) unstructured mesh conforming to the support of the spatially varying contrast—to a uniform Cartesian grid underlying the fast preconditioner via a pair of transfer operators. The resulting preconditioner inherits the frequency-robust behavior of its Cartesian counterpart without sacrificing the geometric flexibility and high-order accuracy of the unstructured discretization. We prove that invertibility of the proposed preconditioner holds under explicit conditions on the mesh sizes and on the Cartesian preconditioner. Numerical experiments demonstrate that the preconditioned system requires significantly fewer GMRES iterations than its unpreconditioned counterpart, with iteration counts almost independent of mesh size and wavenumber, and illustrate the method's robustness for inhomogeneities with piecewise-smooth refractive indices and jump discontinuities across interfaces.

A Delannoy–Steinhaus triangle is obtained from a finite sequence by a recurrence governed by the Delannoy numbers. We introduce this construction over $\mathbb{Z}/2\mathbb{Z}$ and study its weight distribution. The relevant Delannoy coefficients are all odd, which reduces every entry to the parity of a consecutive interval of the generating sequence. Encoding these interval parities by prefix parities yields a weight formula depending only on the numbers of zeros and ones in the prefix-parity sequence. We use this formula to determine the complete weight spectrum and the exact multiplicity of each weight. As a consequence, we characterize and enumerate the balanced triangles: a balanced triangle generated by a binary sequence of length $n$ exists if and only if $n+1$ is a perfect square. We also determine the canonical-vector weights, the minimum nonzero weight, the {second-smallest nonzero weight}, the maximum weight, and the average weight.

For a semi-arithmetic Fuchsian group $\Gamma$ of arithmetic dimension $r$ with a generalized modular embedding, EGMM holds for $r\leq2$, and for $r\geq3$ under an explicit strong contraction condition, via a multi-dimensional Schwarz-Pick bound and a geometry-of-numbers estimate. A covering construction turns every known example into an infinite explicit family with the same exponent via principal congruence towers. We add new $r\leq2$ examples (genus-$3$/$4$ Prym eigenform loci, a different commensurability class from the known triangle groups and McMullen genus-two family) and attempt a direct flat-geometric bound on the contraction constant $\delta$ for the latter via its Lyapunov exponents ($\delta=2/3$ on ergodic average, by Bainbridge exact computation), explaining why an average bound does not yet certify the pointwise one \eqref{eqstrong-contraction} needs.

In this paper, we investigate Chocolate Games. In Chocolate Games, each player cuts a chocolate bar into two chocolate bars and eats one of them so that they do not eat an indicated bitter block. The player who eats the bitter part loses the game. This game can be considered as a generalization of Nim, and previous studies consider the condition that for what kind of the shape of chocolate bar, the Sprague–Grundy value of the position can be calculated by Nim–sum (XOR) of the width and height of the chocolate bar. Higher dimensional cases were also studied. In this paper, we show that the “Tasty condition,” which was introduced in previous work for increasing staircase chocolate bars, can also be used for various other shapes of chocolate bars as a sufficient condition, or in some cases a necessary and sufficient condition, for the Sprague–Grundy value to be given by the Nim-sum of the sizes of the chocolate bar in each dimension.

We address the inverse problem of reactive transport in heterogeneous porous media, where unknown model parameters must be inferred from sparse experimental observations. The problem is complicated by strong nonlinearities, spatial heterogeneity, and limited data availability. We propose a Conditioned-on-Design Bayesian Physics-Informed Neural Network (CoDe-BPINN), which combines a domain-decomposed PINN solver with a Bayesian inference network that learns the conditional distribution of model parameters given experimental design variables. The framework is trained by maximizing a physics-informed Evidence Lower Bound (ELBO), enabling simultaneous reconstruction of spatiotemporal concentration fields, probabilistic parameter estimation, and uncertainty quantification. We demonstrate the approach using laboratory experiments on contaminant transport through a multilayer porous column with an iodinated contrast medium. The model accurately reproduces breakthrough dynamics while revealing systematic parameter dependence on flow conditions. In particular, it identifies a nonlinear decrease in effective sorption capacity with increasing flow rate and porosity, consistent with kinetic limitations and reduced adsorbent mass. The Bayesian formulation also uncovers a strong negative correlation between sorption affinity and sorption capacity, quantifying the intrinsic non-identifiability of the inverse problem. CoDe-BPINN provides a robust framework for parameter inference and uncertainty quantification in data-scarce reactive transport problems.

The maximum size of a weak $P$-free family $\mathcal{F}\subseteq 2^{[n]}$ is denoted by $La(n,P)$. Let $e(P)$ denote the maximum integer $k$ such that the union of any $k$ consecutive layers of $2^{[n]}$ is weak $P$-free. In recent years, multiple examples of posets with $e(P)<\pi^-(P):=\liminf_{n\to\infty} \frac{La(n,P)}{\binom{n}{\lfloor \frac{n}{2}\rfloor}}$ have been found. We add several further posets with $e(P)=1$ to this list. We define a family $\mathcal{F}\subseteq 2^{[n]}$ of size at least $(1.22+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ that is weak $O_6$-free, where $O_6$ is the six-element crown poset. We also show an infinite set of posets $P$ with $1=e(P)<\pi^-(P)$ that are minimal with respect to this property. Finally, we consider how far apart $e(P)$ and $\pi^-(P)$ can be. We prove that for every fixed finite poset $P$ with $e(P)=1$, there is a constant $\delta_P>0$ such that $La(n,P)\le(2-\delta_P+o(1))\binom{n}{\lfloor n/2\rfloor}$. The value 2 is optimal: explicit vertex-edge incidence posets with $e(P)=1$ have $\pi^-(P)$ values tending to $2$. In contrast, for every $K>0$ we construct a finite poset $P$ with $e(P)=2$ and lower density greater than $K$.

We consider the perturbation of a Zoll contact form on some prescribed domain of the underlying manifold by multiplying it with a positive function which is constant of value $1$ outside the domain. For every sufficiently $C^0$-small such perturbation we find a periodic Reeb orbit of the perturbed contact form intersecting the domain. As an application, starting from a Zoll Riemannian manifold, we demonstrate that for every exact magnetic field, with $C^0$-small magnetic potential vanishing outside some given domain, there exists a periodic magnetic geodesic intersecting this domain. The theorem is a rather direct consequence of a result which is of independent interest: For Hamiltonians sufficiently $C^0$-close to a defining Hamiltonian, we show existence of a gradient flow line of the corresponding Rabinowitz action functional with a constraint on the position of the cylinder component at $(0,0) \in \mathbb{R} \times \mathbb{S}^1$. This relies on a homotopy stretching argument for the Rabinowitz action functional, in the course of which we have to derive some delicate estimates to ensure compactness of the appearing moduli spaces.

For a mixed finite element method, a Dirac source is first a failure of duality, not of regularity: the conservation equation is tested against a Lebesgue space, and a Dirac measure lies in the dual of none. We therefore remove the measure from the conservation law by a divergence-form splitting. An explicit field whose divergence is the Dirac measure is subtracted from the physical flux, and the modified flux is taken as the mixed unknown, so that only the regular part of the load remains in the conservation equation. Equivalently, and independently of any discretization, the Dirac problem is rewritten as an elliptic equation whose data are in divergence form, generated by a field of L^p. The subtracted field depends on the location of the source alone and not on the coefficient. No coefficient-dependent singular solution and no discrete delta is needed, only the load vector of the RT_0-P_0 system changes, and the source may sit anywhere relative to the mesh: at a vertex, on a coefficient interface, or inside an element. Unless the splitting is matched to the operator at the source, the modified flux lies in L^p for every p<2 but not in L^2, so the flux error analysis has to leave the Hilbert scale. We prove a quasi-best approximation bound for the flux, and with it that on a quasi-uniform family the flux error is exactly of order h^(2/p-1): the matching lower bound comes already from the single element carrying the source. Grading the mesh there restores first-order complexity, N^(-1/2) in the number of elements, and the adaptive computations attain it. The scalar variable is limited only by piecewise constant approximation of the solution, which it attains. We also prove a residual norm equivalence in the Lebesgue scale, yielding a computable L^p estimator, reliable and locally efficient for the mixed flux together with a recovered potential.

In this note, we give a complete characterization of square matrices of size $n$ over a commutative field which can be expressed as a product of a nilpotent matrix and an involutive matrix and matrices which can be expressed as a product of a nilpotent matrix and an idempotent matrix, in terms of invariant factors.

Let $\|\cdot\|_p$ denote the Schatten $p$-norm and let $|A|=(A^*A)^{1/2}$. For $2\leq p<\infty$, let $x_{p,m}>1$ be the unique solution of $x_{p,m}^p=2x_{p,m}+m-1$, and set \[ C_{p,m}=\frac{\sqrt{x_{p,m}(x_{p,m}+m-1)}}{(x_{p,m}^p+m-1)^{1/p}}. \] We prove the sharp inequality \[ \|A_1+\cdots+A_m\|_p\leq C_{p,m}\bigl\||A_1|+\cdots+|A_m|\bigr\|_p \] for arbitrary complex matrices of arbitrary size. Equivalently, if $q=p/(p-1)$ and $R,X_1,\cdots,+X_m$ are positive semidefinite, then \[ \|RX_1\|_1+\cdots\|RX_m\|_1 \leq C_{p,m}\|R\|_q\|X_1+\cdots X_m\|_p. \] For $1<p<2$, we also show that the formula proposed for the optimal constant fails. We give both a numerical counterexample and a systematic analytic construction.oposed for the optimal constant fails. We give both a numerical counterexample and a systematic analytic construction.

We prove that, under certain homological conditions, the attaching map of the top cell of a Poincaré duality complex is inert when localised away from a finite set of primes. This improves on a result of Félix and Tanré in these cases. As an additional application of the methods, we give a loop space decomposition of simply-connected $6$-dimensional Poincaré duality complexes satisfying certain hypotheses. We also show that, under the hypotheses of the inertness theorem, the $(n-1)$-skeleton of an $n$-dimensional Poincaré duality complex satisfies the hyperbolic form of Moore's Conjecture after localising away from an explicit finite set of primes, and use this to obtain new examples of \(p\)-local maps between spheres that are not inert.

We derive convergence rates for the approximation of the Fréchet distribution $\mathcal{F}(\alpha)$ with parameter $\alpha > 0$ by sequences of renormalized maxima in the extreme value theorem. Our proofs rely on the application of the infinitesimal generator approach to Stein's method to max-stable distributions, using the family of Markov semi-groups recently introduced in \cite{CostacequePhD, Costaceque24}. We develop two different approaches to compute rates of convergence; the first one relies on the second-order regular variation assumption, while the second one requires the existence of a density function for the base distribution. In particular, with the first approach, our bounds are expressed using the Kolmogorov distance, and the Wasserstein distance when $\alpha > 1$. The second approach allows also rates for a smooth Hölder distance when $\alpha \in (0,1)$. In both cases, we also obtain convergence rates for moments when they exist.