Papers

Let K be an algebraically closed field of characteristic zero. Many of the geometric properties of the plane curves in K^2 with one place at infinity can be obtained from the arithmetic of numerical semigroups associated with these curves. In this survey, we recall the main properties of these semigroups as well as the geometric results obtained from these properties. We also address some problems.

In this article, we investigate spectral properties of compact $V$-static manifolds, namely, compact manifolds with boundary whose metrics are critical points of the volume functional under a scalar curvature constraint. We derive sharp estimates for the first Steklov eigenvalue and the entire fourth-order Steklov spectrum. We further obtain a Lichnerowicz-type lower bound for the first eigenvalue of the drifted Laplacian naturally associated with the $V$-static potential. In the corresponding equality cases, we obtain rigidity results characterizing the Euclidean ball and the hemisphere.

We give an approximately uniform sampler for satisfying assignments of constraint satisfaction problems that satisfy $4\mathrm e p(\Delta+1)^2\le1$, where $p$ is the largest constraint-violation probability under the uniform product distribution, and $\Delta$ is the maximum degree of the dependency graph. The algorithm invokes the recent efficient approximate counting algorithm of Liu, Wang, Yin, Zhang, and Zhou~\cite{CountingLLL} as a subroutine and returns a satisfying assignment sampled within total-variation distance $\varepsilon$ of the uniform distribution in $(n+m/\varepsilon)^{O(k\Delta\log D)}$ time, where $n$ and $m$ are the numbers of variables and constraints, $D$ is the common domain size, and $k$ bounds the constraint arity. We also give an asymmetric product-form condition under which both counting and sampling are efficient. To supply the counts needed in the additional regime, we extend the marginal-expansion proof of Liu et al.

We prove that the Dirichlet-to-Neumann map uniquely determines a smooth real Schrödinger potential on any prescribed compact connected smooth Riemannian manifold of dimension at least three with nonempty smooth boundary. The proof combines local continuation of Green kernels with uniform estimates for the associated integral representations on shrinking spheres. These estimates yield local recovery of the potential, and a continuation argument extends this recovery throughout the manifold.

Computing the total variation (TV) distance between succinctly represented high-dimensional distributions is generally intractable. We give an FPRAS for TV distance between mixtures of product distributions and, more generally, for a natural class of structured probabilistic circuits. Our main technique is a novel application of domain reduction: Given a family of feature vectors indexed by assignments, we use Lewis-weight sampling to replace the assignment domain by a polynomial-size weighted subset that simultaneously approximates the sum of absolute values of every linear projection. For mixtures of product distributions, we construct such reduced domains incrementally over the coordinates, obtaining the first FPRAS with running time polynomial in both the dimension and the number of mixture components. We then extend the approach to smooth, structured-decomposable probabilistic circuits with a common structured architecture.

We consider the integrable dynamics of a planar Kepler billiard in the plane bounded by a branch of a hyperbola focused at the Kepler center. As a sequel to our previous work \cite{JZ2}, we use the existence of the foci-caustic circle to identify an elliptic curve on which the dynamics is linearized and we analyze Cayley's condition on $n$-periodic orbits in this setting. By taking various limits also we discuss the dynamics of planar Kepler billiards with boundary being a focused parabolic arc, as well as a straight line. In this last case, our approach offers a different way to deduce the elliptic curve as in \cite{Felder}.

We study the stationary focusing nonlinear Schrödinger equation on the product $\mathbb{R}^N \times \mathcal{G}$ of the Euclidean space with a compact metric graph, in the mass-constrained variational setting. Such a product is a hybrid structure of a new type: all its faces are $(N+1)$-dimensional and are glued along interfaces of codimension one, so that the energy space is a genuine Sobolev space, while both the metric and the topology of the graph enter the variational problem. We first develop the functional framework, giving two equivalent descriptions of $H^1(\mathbb{R}^N \times \mathcal{G})$, introducing partial rearrangements in each of the two variables together with the corresponding Pólya–Szegő inequalities, and proving Gagliardo–Nirenberg inequalities in a localized form, with a comparison of the optimal constants with those of $\mathbb{R}^{N+1}$ and of the half-space. We then study the mass-constrained problem. Ground states exist for every mass when $2<p<2_*:=2+4/(N+1)$. At the critical exponent $p=2_*$, they exist below a graph-dependent threshold lying between one half of the Euclidean critical mass and the full Euclidean critical mass. The latter value is attained when the graph admits a cycle covering, whereas the threshold is exactly halved in the presence of a terminal edge. In both cases, the threshold is sharp. For $2_*<p<2+4/N$ global minimizers do not exist, but we prove the existence of local minimizers below a further mass threshold, for which we give an explicit lower bound. Finally, we describe the dimensional crossover: below a critical mass the minimizers do not depend on the graph variable, and we characterize the threshold below which the semi-trivial solution is a local minimizer in terms of the first nonzero eigenvalue of the Kirchhoff Laplacian on $\mathcal{G}$.

We study the expected number of zeros of the Short-Time Fourier Transform (STFT) with a Gaussian window for signals degraded by complex Gaussian colored noise. We provide an exact formula for this quantity and investigate its asymptotic behavior. From a computational perspective, we propose an algorithmic method to recover a smoothed power spectral density (PSD) directly from the spatial distribution of the transform's zeros. Our results formalize the commonly-held heuristic that detection algorithms based on spectrogram zeros, though designed for white noise, also perform adequately under moderately colored noise.

We show that a compact mean curvature flow whose singularities have multiplicity-one cylindrical tangent flows admits a smooth Morse resolution. The resolution agrees with the spacetime track outside any prescribed neighborhood of its singular set and all its critical points have the expected index. No nondegeneracy or isolatedness assumption is required. In the mean-convex case the result follows by smoothing the global arrival function.

We assemble a finite-dimensional \(\mathcal{VU}\)-calculus for composite nonsmooth functions whose outer function is convex: the chain rule, separable and convex sums, a strictly differentiable perturbation of a lower semicontinuous (lsc) term, the model \(\delta_X+f_0+\theta\circ F\), and a finite maximum of \(C^1\) functions. The same algebra yields an \(\varepsilon\)-\(\mathcal{VU}\) chain rule for a proper outer approximation of the subdifferential. On the set \(\mathcal{R}_{h,F}\) of points at which the convex chain rule holds, the subspaces \(\mathcal{V}f\) and \(\mathcal{U}f\), an orthonormal frame of \(\mathcal{U}f\), and the \(\mathcal{U}\)-gradient \(\bar g_u\) are written in terms of the factors. If \(f\) is \(C^1\)-partly smooth at \(\bar{x}\), that gradient is \(\bar g_u=U_f^\top\nabla_{\mathcal{M}}f(\bar{x})\). If \(f\) is \(C^2\)-partly smooth and \(0\in\ri\partial f(\bar{x})\), the convex \(\mathcal{U}\)-Hessian \(H_U\) (when \(f\) is convex) or the local matrix \(H_\varepsilon\) (when \(f\) is prox-regular at \(\bar{x}\) for \(0\)) equals the Gram matrix \(U_f^\top\nabla^2_{\mathcal{M}}f(\bar{x})\,U_f\); the same matrix, written along the active manifold in a continuous frame, depends continuously on the base point. If in addition \(f\) is \(C^2\)-partly smooth at \(\bar{x}\), then under prox-regularity and subdifferential continuity at \(\bar{x}\) for \(0\), \(H_\varepsilon\succ 0\) is equivalent to tilt stability of \(\bar{x}\) and to strong metric regularity of \(\partial f\) at \((\bar{x},0)\), with \(\lip\bigl((\partial f)^{-1}\bigr)(0\mid\bar{x})=\|H_\varepsilon^{-1}\|\). The calculus and the test are illustrated on a hinge composite and two elementary tests of \(H_U\succ 0\).

Fu, Guil Asensio, Herzog and Torrecillas asked whether a complete ideal cotorsion pair of object ideals in an exact category always induces a complete cotorsion pair of objects. We show that it need not, even in a Hom-finite, weakly idempotent complete Frobenius exact category. Our example consists of bounded complexes of finite-dimensional vector spaces with even total cohomology dimension. We characterize special object approximations by parity and identify the obstruction as a non-split idempotent in the stable category. For cohereditary pairs, a criterion of Saorín and Šťovíček gives an affirmative answer when the stable category is idempotent complete.

We consider the problem of noisy gradient-free minimization of the k-th order partial derivative of a $\beta$-Hölder function supported on a d-dimensional cube. We show that T ^{($\beta$+d+k)/(2$\beta$+d)} log(T )^{(\beta-k)/(2\beta+d)} is a non-asymptotic minimax rate of the T step cumulative regret for all $\beta$ \ge 0. In the special case k = 0, our results cover the problem of noisy gradient-free minimization of $\beta$-Hölder functions, closing the existing gap between the known upper and lower bounds. We show that a minimizer of a suitably chosen local polynomial estimator is rate-optimal. The minimax optimal upper bound is achieved under the passive design, that is, when the query points are i.i.d. Thus, there is no advantage in considering sequential designs when it is only known that f is a $\beta$-Hölder function with no additional property. We propose an algorithm feasible in polynomial time that constructs a proxy of the minimizer of the local polynomial estimator. The procedure requires computing the estimator on auxiliary random points. The resulting polynomial time algorithm matches the lower bound.

In this article, we prove the sharp Strichartz estimates for the Schrödinger equation on the one-dimensional torus. More precisely, via obtaining a sharp weak $L^{6}$ estimate with no derivative loss, we obtain sharp $L^{6}$ Strichartz estimate, and sharp no loss $L^{p}$ Strichartz estimate for any $2<p<6$. The latter answers a question by Bourgain \cite{Bourgain93}.

Let $(G,X)$ be a Shimura datum of abelian type satisfying the hypotheses of the main theorem, and suppose that the associated Shimura varieties have hyperspecial good reduction at $p$. Their special fibres are stratified by the $\sigma$-conjugacy classes $b\in B(G_{\mathbb{Q}_p},\mu_h^{-1})$. For an individual Newton stratum, this paper constructs a stabilized formula for the alternating traces of Frobenius–Hecke correspondences on its compactly supported $\ell$-adic cohomology, with coefficients in an $\ell$-adic local system. The formula, obtained by modifying the Langlands–Kottwitz method, expresses these Lefschetz numbers as elliptic stable geometric distributions on endoscopic groups of $G$, placing the cohomological traces in a form suitable for comparison with automorphic spectral data. We also give a group-theoretic criterion determining which elliptic endoscopic groups can contribute to a given Newton stratum. For certain unitary Shimura varieties, we exhibit an intermediate Newton stratum with no non-trivial endoscopic contribution, although there are non-trivial endoscopic contributions in the cohomology of the ambient Shimura variety.

We introduce a codimension one construction of coarse Ricci curvature in Lorentzian geometry. Using the $1$-Lorentz-Wasserstein distance $\ell_1$, we compare probability measures supported on small spacelike hypersurfaces through nearby events and recover, in a precise asymptotic regime, the ambient Ricci curvature in future-directed unit timelike directions, with a universal dimensional prefactor reflecting this codimension one construction. The construction echoes the Raychaudhuri equation, which relates the evolution of spatial volume expansion along timelike geodesics to Ricci curvature. Our quantitative estimates rely on transport maps with controlled displacement, constructed through a Moser-type flow. At leading order, the slice construction is insensitive to smooth weights. By smearing the spacelike slices into thin timelike tubes and calibrating the temporal and spatial scales, we recover the timelike Bakry-Émery tensor $\mathrm{Ric}+\mathrm{Hess}\,V$ associated with the weighted reference measure $\smash{\mathfrak{m}=\mathrm{e}^{-V}\,\mathrm{vol}_g}$.

We study conserved and dissipated quantities associated with contact canonoid transformations and their extension to possibly singular smooth maps. The defining condition is imposed on the pulled-back contact one-form, without requiring it to remain contact. A polynomial construction produces first integrals from one-forms having the same dissipation rate as the original contact form. Two normalizations extend this construction to maps whose associated Hamiltonians have different dissipation rates. On the set where the associated Hamiltonian is nonzero, we construct a smooth invariant endomorphism whose traces are conserved. This construction includes zeros of the original Hamiltonian, where the endomorphism vanishes. We relate the tensor traces to the coefficients of the Hamiltonian-normalized polynomial and obtain an upper bound of n functionally independent trace invariants in dimension 2n + 1. A five-dimensional singular map attains this bound. Examples include singular maps for the damped free particle, a transformation with unequal dissipation rates, the damped harmonic oscillator, and motion under gravity with linear friction.

{Let $\lambda_i = \nu_i(F)$ denote the three singular values of the deformation gradient $F \in {\rm GL}^+(3)$.} We consider the family of incompressible isotropic energies $ W_\psi(F)=\sum_i \psi\left(|\!\log\lambda_i|\right)$ with $\psi:[0,\infty)\mapsto\mathbb{R}$. Set $g(s)=\psi\left({\rm arcosh}\frac{s}{2}\right)$ for all $s\geq2$. If $g$ has a convex and non-decreasing extension $\bar g$ to $[0,\infty)$, then $W_\psi$ is the restriction to ${\rm SL}(3)$ of the explicit polyconvex function \ {$ F\mapsto\sum_i \bar g\left(\nu_i(F+{\rm Cof} F)\right). $} The proof uses the identity $\nu_i(F+{\rm Cof} F)=\lambda_i+\lambda_i^{-1}$, up to permutation, on ${\rm SL}(3)$ and Ball's convexity theorem for functions of the singular values. We also give a direct proof along rank-one lines contained in ${\rm SL}(3)$ and derive a convenient one-dimensional differential sufficient condition. In particular, $ F\mapsto\sum_i e^{\log^2\!\lambda_i} $ is rank-one convex on ${\rm SL}(3)$ and possesses the stated polyconvex extension. A simple-shear computation shows that scalar convexity in $\log \lambda_i$ alone is insufficient; the quadratic Hencky energy $\sum_i \log^2\lambda_i$ is not rank-one convex on ${\rm SL}(3)$.

Common finite element discretizations for the incompressible Stokes equations with a continuous pressure approximation – the MINI element, the Taylor–Hood element, or stabilized equal-order elements – are not pressure-robust: the velocity error is polluted by the pressure best-approximation error, multiplied by the inverse viscosity. The established remedy replaces the test function in the momentum equation by a divergence-preserving reconstruction operator; for continuous pressure approximations this construction is cumbersome, requiring the solution of local problems (Lederer, Linke, Merdon & Schöberl, SIAM J. Numer. Anal., 2017). We propose a simpler alternative: interpolating the force by a Nédélec interpolation operator of suitable order instead of reconstructing the test function. Exploiting the commuting diagram property linking Nédélec and Lagrange interpolation operators, we show that the resulting method is pressure-robust for any $H^1$-conforming, continuous-pressure discretization. We give the complete analysis for the MINI element, and then for stabilized equal-order $P^kP^k$ elements of arbitrary degree $k$, proving that the interpolation-induced consistency error is pressure-robust and converges of optimal order. We validate the theory by several numerical examples.

Sign-based methods reduce communication costs in distributed environments, but aggregating local signs can introduce bias when data are heterogeneous. As a result, existing sign-based variance reduction methods fail to obtain the optimal convergence rates. In this paper, we solve this problem and obtain optimal rates for both nonconvex stochastic and finite-sum optimization. We first give a counterexample showing that majority voting can fail to approach stationary points even with exact local gradients. Motivated by this limitation, we propose tracking the global gradient at the server through unbiased compression of recursive gradient increments. As a result, we can obtain the convergence rates of $O(\sqrt{d/K}+\sqrt d (a/(nK))^{1/3})$ for the $\ell_1$-norm and $O(\sqrt{a/K}+\sqrt a/(nK)^{1/3})$ for the $\ell_2$-norm. Here, $K$ is the iteration number, $n$ is the number of workers, $d$ is the dimension, and $a=1+\omega$, with $\omega$ denoting the compressor's relative variance. For finite-sum problems with $M$ components, we combine periodic exact gradient refreshes with compressed component-gradient differences. The resulting total sample complexities are $O(M+d\sqrt{aM}\epsilon^{-2})$ and $O(M+a\sqrt M\ epsilon^{-2})$ for $\ell_1$ and $\ell_2$ gradient norms at most $\epsilon$, matching the corresponding bounds in centralized settings.

We develop and analyze a fully discrete, globally divergence-free hybridizable discontinuous Galerkin (HDG) method for a gradient-based Smagorinsky model. The method combines backward Euler time stepping, interior-penalty discretizations of molecular and nonlinear eddy diffusion, and an upwind convective flux. The discrete velocity is $H(\operatorname{div})$-conforming and pointwise divergence-free, which yields pressure robustness. For sufficiently large penalty parameters, we prove unconditional energy stability and existence of a discrete solution, and establish uniqueness under additional smallness conditions. A velocity error estimate is derived without explicit inverse powers of the molecular viscosity. The nonlinear facet residuals are controlled using local trace-approximation estimates and a viscosity-independent facet penalty. We retain the dependence of the discrete Gronwall factor on the filter scale and the mesh size; a mesh-uniform bound follows under suitable solution regularity, a fixed time-step margin, and the scaling $\delta=O(h)$ on quasi-uniform meshes. The reported manufactured-solution results are consistent with the resulting pre-asymptotic error bounds. Further flow examples illustrate the dissipative behavior of the method and are distinguished from the boundary conditions and parameter range covered by the analysis.

The tree-independence number $tree\text{-}\alpha(G)$, the induced matching treewidth $tree\text{-}\mu(G)$, and the sim-width $simw(G)$ are graph parameters defined in terms of tree or branch decompositions. We establish two polynomial bounds for the tree-independence number of induced $K_{t,t}$-free graphs, one in terms of sim-width and the other in terms of induced matching treewidth. Abrishami et al. (SIDMA, 2025) and Brettell et al. (EJC, 2025) asked whether bounded sim-width, together with the exclusion of an induced $K_{t,t}$, implies bounded tree-independence number. We answer this question by proving that, for integers $t\geq 2$ and $s\geq 1$, every induced $K_{t,t}$-free graph $G$ with $simw(G)\leq s$ satisfies $tree\text{-}\alpha(G)=O_t\left((s+1)^{2t^2-2t}\right)$. This also proves a polynomial strengthening of a conjecture of Bešter Štorgel et al. (arXiv, 2026) concerning induced $K_{1,t}$-free graphs and improves a theorem of Alon et al. (arXiv, 2025) by reducing the exponent from $3t^2+1$ to $2t^2-2t$. Alon et al. (arXiv, 2025) asked whether, for fixed induced matching treewidth, the tree-independence number is polynomially bounded in $t$. Using a VC-dimension argument, we answer this question affirmatively by showing that, for integers $\mu\geq 1$ and $t\geq 2$, every induced $K_{t,t}$-free graph $G$ with $tree\text{-}\mu(G)\leq\mu$ satisfies $tree\text{-}\alpha(G)=t^{O_\mu(1)}$.

In this paper, we derive a new lower bound for the Hausdorff dimension of 3D Brownian cut points by proving an explicit upper bound ($<0.9999$) for $\xi_3(1,1)$, the intersection exponent for two independent Brownian motions in 3D. Consequently, the growth exponent of 3D loop-erased random walk is at least $1.0001$.

Let $G=D_{\langle2\rangle}([0,1])$ be equipped with either of its two Dlab orders. There exists an order automorphism $\alpha$ of $G$ such that, for every rank-one subgroup $H\leq\mathbb R_{>0}^{\times}$, every one of the six corresponding Dlab groups $A$ listed below, equipped with any of its linear orders, every order-preserving embedding $e:G\hookrightarrow A$, and every $u\in A$, there exists $f\in G$ such that $e(\alpha(f))\neq u^{-1}e(f)u$.

The Hajnal-Szemerédi theorem guarantees an equitable $(r + 1)$-coloring of every graph of maximum degree at most $r$. We prove a version of this result for more general prescribed color class sizes, subject only to a natural upper bound.

Let $(A,\mathfrak{m}), (B,\mathfrak{n}) $ be Gorenstein local rings and let $\underline{CM}(A)$ be its stable category of finitely generated maximal Cohen-Macaulay $A$-modules. Suppose we have an equivalence $\Phi \colon\underline{CM}(A) \rightarrow \underline{CM}(A)$ as triangulated categories. We show (1) If $A, B$ are not hypersurfaces then $dim A = dim B$. (2) If $M$ is a maximal \CM \ $A$-module then $$\text{curv}_A(M) = \text{curv}_B (\Phi(M)),$$ here $\text{curv}_A(M) = \limsup_n \sqrt[n]{\ell(\text{Tor}^A_n(M,k))}$. (3) $A$ satisfies Serre's condition $R_i$ if and only if $B$ satisfies $R_i$. (4) $A$ is a complete intersection on the punctured spectrum of $A$ if and only if $B$ is a complete intersection on the punctured spectrum of $B$. We also show that $\Phi$ imposes constraints of residue field of $B$ in terms of residue field of $A$ and vice-versa. Finally if $I$ is an ideal in $A$ such that the extended Rees algebra $\mathcal{R}(I) = A[It, t^{-1}]$ is Gorenstein then we construct a triangulated functor $\Psi \colon \underline{CM}^\mathbb{Z}(\mathcal{R}(I)) \rightarrow \underline{CM}(A)$ where $\underline{CM}^\mathbb{Z}(\mathcal{R}(I))$ is the stable category of all graded maximal Cohen-Macaulay $\mathcal{R}(I)$-modules. We show that $\Psi$ induces an equivalence $\underline{CM}^\mathbb{Z}(\mathcal{R}(I))/\ker \Psi \rightarrow \underline{CM}(A)$. We give some applications of this map.

In this paper, we study the nonlinear Schrödinger equation with a nonlinearity combining a focusing logarithmic term and a smooth $L^2$-subcritical function. Our main result is the construction of multisoliton solutions for this equation. These special configurations behave asymptotically, for large times, as the superposition of several solitons moving with distinct velocities. The proof relies on an iterative construction method combined with localized energy estimates adapted to the mixed structure of the nonlinearity.

Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a Hörmander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, defined by conical Littlewood–Paley square functions. Let $W_2^s$ denote the classical Sobolev space in $\mathbb{R}^N$. More precisely, if a multiplier $m$ satisfies \[ \sup_{t>0} \|\psi(\cdot)m(t\cdot)\|_{W_2^s}<\infty \] for a nonzero radial cutoff $\psi\in C_c^\infty(\mathbb R^N\setminus\{0\})$ and \[ s>\mathbf N\left(\frac1p-\frac12\right), \] where $\mathbf N$ is the homogeneous dimension, then the Dunkl multiplier \[ \mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff) \] extends to a bounded operator on $H^p_{\mathrm{Dunkl}}$.

Let $E$ be a Hilbert C*-module and $\mathcal{L}(E)$ the C*-algebra of all bounded adjointable operators on $E$. An operator $T \in \mathcal{L}(E)$ is a Kasparov cycle if $T^*T - 1$ and $TT^*-1$ are compact operators on $E$. If $\mathcal{L}(E)\rightarrow \mathcal{L}(E)$ is a prime C*-algebra, and $\varphi:\mathcal{L}(E)\rightarrow \mathcal{L}(E)$ is a linear map which is unital and surjective up to compact operators, and preserves Kasparov cycles in both directions, then the induced map $\psi:\mathcal{L}(E)/\mathcal{K}(E) \rightarrow \mathcal{L}(E)/\mathcal{K}(E)$ is either a $*$-automorphism or a $*$-anti-automorphism. Our work extends the main result of [J. Math. Anal. Appl. 354 (2009), 625-629] to the set of Kasparov cycles, showing that the assumption “$\mathcal{L}(E)/\mathcal{K}(E)$ has real rank zero” is redundant in our results.

The present article investigates the well–posedness of a discrete coagulation–annihilation model originally introduced by E. Ben-Naim and P. Krapivsky in Physical Review E (1995). We first establish the existence of solutions for a broad class of coagulation and annihilation rates, including rates that may be unbounded. These results are obtained by combining uniform estimates with Helly's selection theorem and the Rellich–Kondrachov compact embedding theorem. Under appropriate structural assumptions on these rates, we further prove uniqueness and continuous dependence of solutions on the initial data, thereby establishing well–posedness of the model. Finally, for a specific class of coagulation and annihilation rates, we additionally study the differentiability of solutions and characterize their long–time asymptotic behaviour. These results provide a rigorous mathematical framework for the analysis of the model and contribute to a deeper understanding of its well–posedness, regularity, and long-time dynamics.

This paper studies (co)products and additive structures under Gabriel–Zisman localization $Q:\mathcal A\longrightarrow\mathcal A[S^{-1}]$. Examples show that localization may destroy existing (co)products or fail to preserve (co)products that remain, even when $Q$ is additive. For a set-indexed family $(\mathcal A_i,S_i)_{i\in I}$, necessary and sufficient conditions are given for the canonical functor $\big(\prod_{i\in I}\mathcal A_i\big)\big[\big(\prod_{i\in I}S_i\big)^{-1}\big]\longrightarrow\prod_{i\in I}\mathcal A_i[S_i^{-1}]$ to be full or faithful. These criteria yield compatibility with $I$-indexed (co)products for localizations admitting a calculus of fractions or arising from model categories, provided the relevant (co)products exist and the designated classes of morphisms are closed under them.