Papers

Low-rank tensor methods are an important tool in the numerical treatment of equations with a high-dimensional state space. Nearest neighbor interaction systems like the Ising model or more general Markov jump processes, as well as 1D finite-state quantum systems are examples of such problems. While low-rank tensor train/matrix product state models have been shown to be highly efficient for the simulation of such systems, providing theoretical justification for this remains a challenging task. One approach for obtaining estimates on required ranks for certain accuracies is to investigate the rank increase in Krylov subspace methods for solving the problem at hand. In the context of area laws for ground states of 1D spin systems, nontrivial results on rank-increasing properties of nearest neighbor operator polynomials have been obtained in work of Arad et al. [arXiv:1301.1162] by studying the partial commutativity of local operators. In the present work, this technique is applied to polynomial methods for definite linear equations and dissipative linear ODEs with nearest neighbor structure. This allows to derive corresponding low-rank approximability statements for solutions of such problems which are independent of the system size. Numerical simulations of high-dimensional nearest neighbor systems illustrate the theoretical findings.

Global space-time spectral methods give spectral accuracy in time but typically require the whole space-time history to be resolved and stored on a single tensor-product domain $T \times \Omega$. We record that in an endpoint-benign Legendre or Chebyshev-$T$ time basis, whose polynomials all equal one at the right endpoint, the final time slice of a space-time block is recovered exactly by summing the stored coefficients along the time index. This peel-and-pass step is a special case of a Jacobi endpoint identity, which also gives derivative formulae for higher-order equations. Writing such higher-order equations as first-order systems preserves the benign value-passing structure. The result is a sparse space-time spectral element method that advances block by block, stores only one block, and needs far fewer time coefficients per solve for long-time problems. We prove the identities, give resident-memory, solve-cost and error-propagation models, and demonstrate the method on $(1{+}1)$D heat, wave and Klein–Gordon equations, and on $(2{+}1)$D fractional heat on the disk with weighted Zernike polynomials in space.

We study solution discovery in the token-sliding model from a logical and cost-value optimization perspective. In solution discovery, we are given a graph, an initial placement of $k$ tokens, and a movement budget $b$. The task is to find a reachable target configuration satisfying a prescribed condition. Our results are inspired by Partial Vertex Cover Discovery, where the condition is that the~$k$ tokens cover at least $t$ edges of the input graph. This objective is not merely a sum of independent occupied vertex contributions: each selected vertex contributes its degree, but edges with both endpoints selected have to be subtracted once. To capture this phenomenon, we introduce FO Value Discovery, an optimization problem in which the value of a selected tuple is given by unary vertex weights together with first-order definable correction terms. We further generalize the setting to FO Cost-Value Decision, where vertices carry both costs and values, and the task is to decide whether there is a tuple whose first-order value expression reaches a prescribed value threshold while respecting a cost bound. Finally, we study the parameterized complexity of Partial Vertex Cover Discovery and Vertex Cover Discovery. As a consequence of the logical meta-theorems, we obtain fixed-parameter tractability of Partial Vertex Cover Discovery on several graph classes, including classes of locally bounded cliquewidth. We also show that Partial Vertex Cover Discovery is W[1]-hard parameterized by $k+b$ and fixed-parameter tractable on $d$-degenerate graphs parameterized by $k+d$. For Vertex Cover Discovery, we prove NP-hardness on planar graphs, W[1]-hardness parameterized by the clique cover number, even when a clique cover is supplied with the input, and W[1]-hardness with respect to parameter cutwidth.

Subdivisions of complete graphs play a central role in combinatorics, having deep connections to structural, extremal, and topological aspects of graph theory. A celebrated conjecture of Mader, proved independently by Bollobás and Thomason and by Komlós and Szemerédi, states that every graph of average degree of order $h^2$ contains a subdivision of $K_h$. In this paper, we consider the induced variant of this problem. A theorem of Kühn and Osthus implies that, for every fixed graph $H$ and every $s\ge 1$, graphs of sufficiently large average degree contain either a copy of $K_{s,s}$ or an induced subdivision of $H$. However, even for $H=K_h$, the best previous quantitative bounds were far from optimal. We prove nearly tight bounds for forcing induced subdivisions of $K_h$. We show that every $K_{s,t}$-free graph of average degree $\Omega_{s,t}(h^{2(s-1)}\log^{7(s-1)} h)$ contains an induced subdivision of $K_h$, and that every $C_{2k}$-free graph with $k \geq 3$ and average degree $\Omega_k(h\log^5 h)$ contains an induced subdivision of $K_h$. These bounds substantially improve the previously known results and are nearly optimal in both settings. They also hold if $K_h$ is replaced by any other graph on $h$ vertices.

Recently, Frankl and Wang considered a product version of the classical Hilton-Milner theorem. They conjectured that, if $\mathcal{F} \subset \binom{[n]}{k}$ and $\mathcal{G} \subset \binom{[n]}{\ell}$ are non-trivial cross-intersecting families with $n \geq 2k > 2\ell \geq 4$, the maximum of $|\mathcal{F}||\mathcal{G}|$ is attained by the natural Hilton-Milner-type configurations. In this paper, we present two main results concerning this conjecture. Firstly, we show that the conjecture does not hold in general. By introducing a two-center construction, we prove that for every fixed integer $\ell \geq 3$ and all sufficiently large $k$, the conjecture is false in a linear range $2k+1 \leq n \leq (c_\ell - \epsilon)k$ for any $0 < \epsilon < c_\ell - 2$, where $c_\ell > 2$ is an explicit constant. Secondly, we prove that the conjecture holds when $n > 100\ell k^2$ and $3 \leq \ell < k$, and we completely characterize the extremal families. Our proofs rely on the size of minimal covers and analyzing the structural properties of $2$-cover graphs.

On its absolute-integrability domain, the positive $\beta$-integral is integration with respect to a finite positive purely atomic measure. After normalisation, its Chebyshev functional is a covariance, and its $L^p$-spaces are canonically isometric to direct sums of weighted sequence spaces. On the natural product- and square-integrability domains, the previously formulated $\beta$-Grüss inequalities reduce to Korkine's identity, Hölder's inequality, Cauchy-Schwarz, and elementary variance bounds. On the induced countably atomic probability space, the optimal fixed-grid coefficient is $\kappa_\beta=\sup_A P_\beta(A)(1-P_\beta(A))\leq 1/4$, the countably atomic counterpart of the classical finite weighted coefficient; it may be strictly smaller than $1/4$. The same reduction corrects a coefficient previously claimed to be best possible and identifies a missing sign hypothesis in a related convexity estimate. For the Riemann–Stieltjes $\beta$-integral, within the class of finite induced signed measures, the $\beta$-Lipschitz condition is equivalent to $|\nu_u|\leq L\mu_\beta$. This reduces the principal centred signed estimate to total variation and yields its exact fixed-grid coefficient $2\kappa_\beta$. Finally, truncation of the two atomic orbits gives positive quadrature rules with explicit tail masses. A fixed-point correction yields computable Hölder error bounds, while the uncorrected geometrically graded rule accommodates integrable singularities at the fixed point.

Roaming reactions are organized not by potential-energy saddles but by transition states that are unstable invariant objects in phase space, periodic orbits in the two degrees of freedom studied here. To ask what controls roaming, we modify the Chesnavich model of a barrierless ion–molecule dissociation: its orientation-dependent angular hindrance is replaced by a transverse-stiffness ridge whose angular frequency peaks at an interior radius, and the classical dynamics are studied at a fixed energy just above the dissociation threshold. Comparing two ensembles that differ only in this angular interaction (same radial channel, energy, and inward initial conditions) isolates its effect. The ridge gates entry into the inner well, cutting inner capture from $57\%$ to $15\%$ and returning most of the incoming flux directly to reactants; it does not eliminate roaming but relocates it outward, suppressing it inside the ridge and switching it on farther out. The model retains analogues of the original model's three transition states (tight, free-rotor, and outer orbiting orbits), which we locate as unstable periodic orbits. The tight orbit spans a dividing surface that coincides, within numerical accuracy, with the variational minimum-flux surface, and it carries no barrier along the reaction coordinate: a deep entropic bottleneck placed at an interior radius by the stiffness maximum. Its entropic character is shared with the original model. Strength-matched monotone controls show that the gating tracks the hindrance strength at the bottleneck radius; what the interior maximum supplies is placement, concentrating that strength where it gates most effectively. The trajectories it admits roam nonstatistically, with nonexponential gap-time distributions: the entropic bottleneck governs how much is captured, not the dynamics that follow.

Let $p$ be a prime number. When $p$ is odd, we study finite groups in which every maximal subgroup is either non-abelian simple or $p$-nilpotent, as well as those in which every maximal subgroup is either non-abelian simple or $p$-decomposable. We prove that every non-simple, non-solvable group satisfying the first condition is $p$-nilpotent, and every non-simple, non-solvable group satisfying the second condition is $p$-decomposable. In addition, we determine the possible occurrence of non-abelian simple maximal subgroups in these cases. These results provide a substantial partial answer to two questions posed by V.S. Monakhov and I.N. Tyutyanov in the Kourovka Notebook concerning the non-abelian composition factors of such groups. For non-abelian simple groups, we determine those satisfying the corresponding conditions within the alternating and sporadic families. The case of simple groups of Lie type is left open. Finally, for $p=2$, we obtain a complete classification of the non-solvable finite groups whose maximal subgroups are either non-abelian simple or $2$-nilpotent.

Landau–Siegel zeros are hypothetical zeros of Dirichlet $L$-functions that are close to the point $s=1$. A classic theorem of Page shows at most one such zero can exist among all Dirichlet $L$-functions with conductor $\leq Q$. We show that one can significantly refine Page's theorem under the assumption that all non-real zeros of Dirichlet $L$-functions lie outside a shrinking neighborhood of $s=1$.

The simulation of high-frequency acoustic wave propagation in unbounded domains with local heterogeneous materials and high-contrast interfaces poses significant challenges to numerical methods. The volume-surface integral equation (VSIE) method is an attractive approach as it automatically satisfies the radiation condition at infinity via Green's functions, handles heterogeneous materials via Newton potentials, and models scattering at high-contrast interfaces via surface integral operators. However, its effectiveness in practical simulations has been limited by high computational costs, sensitivity to sharp interfaces, and insufficient computational verification. This study extends the applicability of VSIE by deriving integral formulations for nested heterogeneous materials with parameter jumps at interfaces. We also develop extensive benchmarks against coupled finite-element and boundary-element methods to verify the VSIE's accuracy and mesh convergence. The various benchmarks using open-source software demonstrate the effectiveness of VSIE for large-scale acoustic simulations.

We construct early-warning signals for impending critical transitions in non-Markovian systems. We analyze stochastic forcings such as fractional Brownian motion, fractional Ornstein-Uhlenbeck processes and red noise in fast-slow systems exhibiting such transitions. We show that the effectiveness of indicators such as autocovariance, autocorrelation, and spectral density depends on several properties of the underlying system. In particular, we compare the influence of the Hurst index and the bifurcation type. We prove that the rotatory dynamics associated with a Hopf bifurcation substantially alters the scaling laws of these observables. Finally, we provide practical guidelines for implementing these signals and validate them on both theoretical and applied models.

We show that the genuinely enumerative count of algebraic elliptic curves in any toric variety agrees with the count of the corresponding well-spaced tropical curves, weighted by explicit combinatorial multiplicities. This provides a complete genus-$1$ generalization of the celebrated Nishinou–Siebert correspondence theorem in genus $0$. The proof is algebro-geometric and relies on logarithmic deformation theory together with an explicit enumeration of logarithmic maps with fixed tropicalization.

Adaptive control learns the plant online; neural-operator control learns the control gains offline. We bring the two together for a class of nonlinear hyperbolic PDEs whose dynamics are governed by an unknown Volterra series of arbitrarily many kernels. An observer-based passive identifier learns a truncation of this series online. The infinite-dimensional map that synthesizes the backstepping kernels from the parameter estimates – a cascade of PDEs on simplex domains of increasing dimension, prohibitive to solve in real time – is approximated once, offline, by a neural operator. The closed loop then carries two learning processes in series: online learning of the plant feeds an offline-learned PDE solver, whose output is the online control gains. We prove closed-loop stability and asymptotic regulation of the plant state, observer state, and input, on a basin that recovers the exact-kernel basin as the neural-operator accuracy improves. With a single Lyapunov function we absorb at once the perturbations – all vanishing – of truncating an infinite Volterra series, of identifying the plant online, and of approximating the gains.

Let $G$ be a group, let $A$ be a finite alphabet, and let $\phi: G \to G$ be an endomorphism. We study the topological dynamics of the pullback map $\phi^* : A^G \to A^G$, given by $\phi^*(x)=x\circ\phi$, a canonical example of a generalized cellular automaton. In the one-dimensional case, where $G=\mathbb Z$ and $\phi_k(n)=kn$, we prove a sharp dichotomy: $\phi_k^*$ is equicontinuous precisely for $k\in\{-1,0,1\}$, and cofinitely sensitive otherwise. Although the fixed identity coordinate prevents transitivity on the full shift, the restriction to the natural invariant components is topologically mixing exactly when $k\notin\{-1,0,1\}$. We then extend the analysis to countable groups, showing that $\phi^*$ is equicontinuous if and only if every element of $G$ is eventually periodic under $\phi$, while the existence of a non-eventually-periodic element is equivalent to cofinite sensitivity and to the absence of equicontinuous points. Finally, we characterize Bernoulli measure preservation and strong mixing on the punctured configuration space in terms of injectivity and eventual periodicity.

We describe Lie and Jordan isomorphisms of algebras of triangular matrices over associative rings.

We prove stability for the affine Sobolev inequality for exponents $p\geq 2$ with best possible norm and best possible stability exponent. We also show a corresponding result for critical points of the functional in the absence of bubbling. An important ingredient in our proof is the classification of positive energy solutions to the critical affine $p$-Laplace equation.

This paper investigates the N-elliptic localized solutions of the Foka-Lenells equation. Based on the corresponding Lax pair, the Weierstrass elliptic functions are adopted to construct the elliptic function solutions and the fundamental solution matrix of the equation. The N-elliptic localized solutions are further derived via the N-fold Darboux-Backlund transformation. By virtue of the Cauchy determinant expressed with sigma functions, the asymptotic behaviors of the obtained solutions are systematically analyzed along and between their propagation directions, and the symmetry properties of these solutions are established.

Periodic CMV matrices are unitary matrices that can be specified by $O(n)$ data. Their eigenvalues can be computed by standard methods, storing them as conventional matrices (using $O(n^{2})$ data) in $O(n^{3})$ time. Here a fast method that computes the eigenvalues in $O(n^{2})$ time (using $O(n)$ data) is presented.

For a large family of polynomials $P(X)\in \mathbb{Z}[X]$, we prove central limit theorems for $\sum_{n\le N} f(P(n))$ for both Rademacher and extended Rademacher multiplicative functions $f$. To achieve this, we establish a paucity phenomenon in counting solutions to \[P(n_1)P(n_2)P(n_3)P(n_4) = \square, \quad 1\le n_1, n_2, n_3, n_4 \le N.\] Results of Hooley, Evertse–Silverman, and Reuss play an important role in the proof. Our estimates are sharpest for $\deg P = 2$, thanks to the rich theory of Pell–Fermat equations.

In this note, we present RationalUnivariateRepresentation$.$jl (https://newrur.gitlabpages.inria.fr/RationalUnivariateRepresentation.jl/), a Julia package for computing rational univariate representations of zero-dimensional polynomial systems. The package uses dense linear algebra and Gaussian elimination for the FGLM-like stage. The purpose of this contribution is to advocate for this choice and explain the implementation details that turn the algorithm into practical software. In particular, we show that our implementation can compute guaranteedly correct parametrizations of ideals with thousands of solutions within seconds.

If $G$ and $H$ are two cubic multi-graphs, then an $H$-coloring of $G$ is a mapping $f: E(G)\rightarrow E(H)$, such that for every $v\in V(G)$ there is a vertex $x\in V(H)$, such that $f(\partial_G(v))=\partial_H(x)$. If $G$ admits an $H$-coloring then it is common to write $H\prec G$. The Petersen coloring conjecture predicts that for any bridgeless cubic graph $G$ one has $P_{10}\prec G$. Here $P_{10}$ is the Petersen graph. Let $f: E(G)\rightarrow E(H)$ be any mapping. Define: $V(f)=\{v\in V(G):\exists x\in V(H), f(\partial_G(v))=\partial_H(x)\}$. Let $S_{10}$ be the smallest cubic multi-graph that has no perfect matching. It has ten vertices. Define $S_{12}$ as the cubic graph that is obtained from $S_{10}$, by replacing its unique vertex $z$ adjacent to three bridges with a triangle. In this paper we show that (1) for every cubic multi-graph $G$ with a perfect matching, there is a mapping $f:E(G)\rightarrow E(S_{12})$, such that $|V(f)|\geq \frac{4}{5}\cdot |V(G)|$, and (2) for every cubic multi-graph $G$, there is a mapping $f:E(G)\rightarrow E(S_{10})$, such that $|V(f)|\geq \frac{5}{6}\cdot |V(G)|$. Our second result improves the $\frac{4}{5}$-bound by Hakobyan and the second author from 2018.

Let $R$ be a standard graded algebra over an infinite field $\mathsf k$, and let $\mathbb{B}_{\mathbb{Q}}(R)$ and $\mathbb{B}_{\mathbb{Q}}^{\mathrm{pure}}(R)$ denote the rational cones spanned by the Betti tables of all finitely generated $R$-modules and of those with pure resolutions, respectively. We establish several necessary conditions for the equality $\mathbb{B}_{\mathbb{Q}}(R) = \mathbb{B}_{\mathbb{Q}}^{\mathrm{pure}}(R)$. When $\operatorname{edim}(R)\ge 2$, we prove that $\mathsf k$ has a pure resolution if and only if it has a linear resolution, and consequently, if the extremal rays of $\mathbb{B}_{\mathbb{Q}}(R)$ are pure, then $R$ is Koszul and good (in the sense of Roos). We show that if $R$ has depth zero, it must be Artinian for the equality of the two cones to hold. For rings with linear pairs of exact zerodivisors, we show that the equality of the cones implies that the $h$-polynomial has degree at most $2$, and use it to characterize generic Gorenstein Artin algebras satisfying $\mathbb{B}_{\mathbb{Q}}(R) = \mathbb{B}_{\mathbb{Q}}^{\mathrm{pure}}(R)$. We also characterize algebras whose extremal rays are exactly the Betti tables of shifts of $R/\mathfrak m^j$ and of pure modules $M$ with $\operatorname{codim}(M)=\operatorname{pdim}(M)$: apart from polynomial rings, these are precisely Cohen–Macaulay algebras of dimension at most one with minimal multiplicity. In addition, we obtain a characterization of Cohen–Macaulay algebras of minimal multiplicity in terms of the extremal rays of the Betti cone of maximal Cohen–Macaulay modules.

We study the solvability of the divergence equation $$ \operatorname{div} \u = f $$ in bounded $C^2$ domains under homogeneous Dirichlet boundary conditions for data $f\in C^{0,\alpha}(\Omega)$ satisfying the compatibility condition $ \int_\Omega f =0. $ We construct a solution $\u$ such that for every $0<\beta<\alpha$ $$ \u\in C^{1,\beta}(\Omega)^n $$ satisfies $$ \|\u\|_{C^{1,\beta}(\Omega)} \le C\|f\|_{C^{0,\alpha}(\Omega)}. $$ The proof combines localization techniques with a boundary flattening procedure reducing the problem to a model half-cube.

Boolean hyperdoctrines provide an algebraic semantics for classical first-order logic with equality. In the definition of a Boolean hyperdoctrine, the Beck–Chevalley condition captures the commutativity of substitutions with quantifiers and with equality. Often, a generalization of these conditions is considered, which requires the commutativity of an appropriate square for every pullback square in the base category. A Boolean hyperdoctrine satisfying this condition is called full. Our contribution is twofold. On the negative side, we exhibit a non-full Boolean hyperdoctrine. On the positive side, we show that every Boolean hyperdoctrine $\mathsf{FinSet} \to \mathsf{BA}$ over $\mathsf{FinSet}^{\mathrm{op}}$ is full.

Two developments are unfolding at once: artificial intelligence systems have begun to produce genuine research-level mathematics, and the United States is weakening the pipeline that produces humans capable of understanding what such systems are doing. This essay argues that, taken together, these developments amount to a strategic error. Mathematical capacity, which is the trained ability to verify, interpret, and challenge mathematical reasoning, is not a byproduct of theorem production but a form of infrastructure, built over generations by institutions that cannot be reconstituted on demand. Drawing on the May 2026 AI disproof of a longstanding Erdős conjecture on the planar unit distance problem and on recent disruptions to federal support for the mathematical sciences, the essay makes the case for treating mathematical capacity as a strategic asset on a par with semiconductor capability. It further proposes, among other measures, that AI systems performing consequential reasoning be required to expose their decision-critical claims in formal, machine-checkable form, converting part of AI reasoning from opaque persuasion into auditable structure.

We introduce the microlocal Bernstein–Sato polynomial of a function on a possibly singular ambient variety, extending the theory of Saito. We show that, contrary to the smooth ambient setting, these polynomials are not generally equal to the reduced $b$-functions obtained by removing the trivial root. We define the minimal exponent and use it to study the singularities of the divisor and the Hodge filtration on local cohomology. Our main results include a generalization of Saito's theorem relating the minimal exponent to rational singularities, a characterization of purity of local cohomology, a Thom–Sebastiani formula for the minimal exponent, and a linear combination formula for Bernstein–Sato polynomials of ideals. When the ambient variety is a complete intersection with rational singularities, we provide effective algorithms for these Bernstein–Sato polynomials and implement them in Macaulay2.

Rolling covariance estimates feed two objects that are routinely treated as market structure. The first is the dominant eigenspace, monitored through the projector movement $\widehat D_{K,t}=\|\widehat P_{K,t}-\widehat P_{K,t-1}\|_F$; the second comprises scalar spectral functionals such as the absorption ratio and the leading-eigenvalue share. Both fluctuate under estimation noise, and shrinkage changes the law of that noise, so reading their movements as structural change requires calibration. For the eigenspace, we derive a first-order null law for $\widehat D_{K,t}$ between overlapping windows that share most of their data and show that it transfers without change to rotation-equivariant shrinkage estimators. A distribution-free Davis-Kahan band gauges whether the eigenspace is identified, an estimator-aware bootstrap provides the calibrated test, and a companion power analysis gives an approximate design rule for the smallest detectable rotation. For the scalar functionals, we show that first-order immunity to elliptical kurtosis holds for scale-invariant functionals and only for them, so that one estimated scalar calibrates the projector null and the absorption-ratio and leading-share intervals across the elliptical family. In high dimensions, where shrinkage cleaning biases the absorption ratio, we give a trace-preserving spike-debiased estimator that removes the bias. The results are verified by simulation under a known population covariance; an equity-panel appendix shows the procedures as diagnostics when the population is unknown.

A standard way to control expression swell in computer algebra is to use multi-modular or evaluation-interpolation methods. In computations involving Gröbner bases, these techniques typically require repeatedly computing Gröbner bases of specializations of the same ideal. These repeated computations can be accelerated through precomputation, notably using Traverso's tracing. We present Groebner$.$jl (https://github.com/sumiya11/Groebner.jl), a Julia implementation of the F4 algorithm that exposes Traverso's tracing through a reusable public interface. The implementation supports SIMD-friendly coefficient types, such as tuples of machine integers, which Julia compiles to efficient code with little manual intervention. This lets other Julia software leverage tracing to obtain speedups in applications such as structural identifiability of ordinary differential equation models and polynomial system solving.

This paper investigates feature learning within the framework of the deep Ritz method for solving the stationary Schrödinger equation with Neumann boundary conditions. We first analyze the convergence of Riemannian gradient descent in an agnostic setting, where the hypothesis function is restricted to a single-index model while the PDE solution is arbitrary. We prove that gradient descent reaches an approximate global minimum: after T = O(log(1/\epsilon)) iterations, the loss is within \epsilon of a constant multiple of the optimal loss. We then examine the loss landscape when the source term of the PDE itself follows a single-index model, considering hypothesis functions given by either a single-index model or a two-neuron multi-index model. In the single-index case, we show that the minimum Ritz energy is attained at the feature vector aligned with that of the source term. In the two-neuron case, we study the landscape of regularized Ritz losses and characterize how a second feature emerges, given that the first feature is aligned with the source, as the regularization parameter varies. Finally, numerical experiments are presented to validate the feature emergence theory in the two-neuron setting.

This paper develops the asymptotic theory for high-dimensional panel data regressions in settings with cross-sectionally dependent errors driven by common shocks. We consider a factor-augmented sparse-group LASSO estimator that combines MIDAS aggregation with latent factors. The estimator can take advantage of the mixed-frequency group structure in the time-series dimension. Theory shows that it can outperform the standard LASSO estimator both for prediction and estimation while allowing for cross-sectional dependence.