Papers

In this work we introduce a numerical framework for the simulation of biological capsules based on meshes that adapt iteration-wise to the shape of the membrane. The discretization leverages a Hybrid High Order method (HHO) to solve a Stokes problem with a source term distributed on the interface between the internal and the external phases. The HHO discretization supports meshes composed of general polygons, permitting to use fitted meshes that can be easily generated by cutting the elements intersected by the interface. Moreover, polygonal meshes allow arbitrarily accurate interface discretizations without introducing bulk degrees of freedom. The presentation of the simulation algorithm is completed with a proof-of-concept numerical test on a well-established benchmark: the deformation of an elastic capsule immersed in a shear flow.

We study automorphisms of Hausel's medium algebras by identifying them with equivariant cohomology algebras of affine Schubert varieties of a complex reductive group $G$. The automorphisms in question are then given by the action of either complex automorphisms or real structures of $G$ on affine Schubert varieties. We focus on describing the fixed points of those automorphisms on the spectra of medium algebras, and prove that the torsion-free quotients of the corresponding coinvariant algebras are isomorphic to equivariant cohomology rings of fixed loci in affine Schubert varieties. In the case of real structures, this naturally leads to and involves the study of affine Grassmannians for quasi-anisotropic (also called quasi-compact) real reductive groups, closely related to recent work of O'Brien. We also discuss relations to automorphisms of Higgs bundle moduli and to symmetric pair variants of medium algebras.

Retuning feedback controllers in critical processes is challenging when experiments are costly, open-loop excitation is infeasible, and candidate controllers must be screened before deployment. The existing Adaptive Sampling for Counterfactual Explanations (ACE) method provides a sample-efficient approach for finding minimal controller-parameter changes that recover prescribed closed-loop performance, but it does not explicitly account for stability of candidate controllers proposed during the search, which may be destabilizing under plant uncertainty. This paper introduces UQ-ACE, a stability-aware retuning method based on Bayesian SINDy. In UQ-ACE, closed-loop trajectories are used to infer a posterior distribution over local dynamical models, from which candidate controllers are screened for stability before implementation. Extensive simulations show that UQ-ACE achieves valid retunings with few closed-loop experiments while preventing deployment of unstable candidates.

We study learning an unknown element $g$ of a group acting with finite orbits on a countable set, from examples $(x,gx)$. After a sample on $A$, the consistent hypotheses are the restrictions of a coset of the pointwise stabilizer of $A$, and the sample forces exactly the labels of points this stabilizer fixes; for a Galois group, they form the field generated by the data. Through this correspondence, multiclass dimensions become group invariants: the DS and graph dimensions equal the height, the Littlestone dimension the longest irredundant sequence length, and the teaching dimension the minimal base size. Among deterministic learners, symmetry makes empirical risk minimization optimal: on nondegenerate classes every consistent learner, whatever its tie-breaking, attains the optimal realizable sample complexity up to constant factors, and every consistent online learner the optimal mistake bound. Agnostically, every ERM's sample complexity exceeds the known lower bound by at most the DS-to-Natarajan ratio, up to constants. For finite Galois extensions of countable fields, the Natarajan dimension equals a group invariant, the coset-cube number, which is at least the number of complemented factors of any chief series of any subgroup. Hence the dimensions coincide for every solvable Galois group, while for the groups $A_5$ and $S_5$ of the quintic they are $2<3$ and $3<4$, and $A_5$ is the smallest group with a gap. For the Galois group $S_n$ of the general equation of degree $n$, they coincide exactly when $n\le 4$, i.e., in characteristic zero, exactly when the equation is solvable by radicals; the implication is one-way, as the simple group $\mathrm{PSL}(2,8)$ has no gap. For linear, orthogonal and unitary maps of $\mathbb R^d$ or $\mathbb C^d$ both dimensions equal $d$, and a two-fibered pseudo-cube is a disjoint union of Natarajan cubes exactly when its monodromy group is abelian.

We study alternating Hamilton paths and cycles in the comparability graphs of finite posets. We prove that if a finite poset $P$ with at least two elements admits an alternating Hamilton path, then $P\times B_r$ admits an alternating Hamilton cycle for every $r\ge2$, where $B_r$ is the Boolean lattice. As a corollary, we settle a conjecture of Katona [Discrete Appl. Math., 2013]. Furthermore, we characterize the divisor lattices admitting alternating Hamilton paths and cycles.

In this paper, reverse translation and convolution estimates in the frame of $\mu^{th}$ order Mehler-Fock transform are obtained. Continuity properties of these operators on the test function space $\mathfrak{H}_{l,r,n}$ have been discussed. Moreover, Sobolev space $S_{r,p}^{N,\alpha}$ has been introduced and again continuity properties of these operators in this Sobolev space have been discussed.

Let $W_n$ be the Lie algebra of polynomial vector fields in $n$ variables over a field of characteristic zero, and let $s_k$ denote the alternating sum of compositions of $k$ vector fields. The identity $s_{n^2+2n-1}=0$ has been asserted for arbitrary $n$. We give an explicit counterexample in dimension five, verified by an exact integer computation of a $34$-commutator. Together with the known vanishing of $s_{35}$, this shows that $35$ is the least degree of a standard identity on $W_5$. We then prove that $s_{n^2+2n-1}=0$ for every even $n\geq2$. The proof uses Lie algebra homology with coefficients in the polynomial algebra modulo constants and a lemma on invariant tensors. We describe the computation and provide the source code and verification results.

We study the topological entropy of saturated sets for continuous flows, with emphasis on singular hyperbolic dynamics. For a $C^r$-generic ($r \geq 2$) vector field $X$ carrying a geometric Lorenz attractor $\Lambda$, we prove that for every nonempty compact connected set $\mathcal{K} \subseteq \mathcal{M}(\Phi,\Lambda)$ and every open set $U \subseteq M$ with $U \cap \Lambda \neq \emptyset$, $h_{\mathrm{top}}(\Phi,G_{\mathcal{K}}(\Phi) \cap \operatorname{Trans}(\Phi,\Lambda) \cap U)=\inf_{\mu \in \mathcal{K}} h_\mu(\phi_1)$, where $\Phi=(\phi_t)_{t \in \mathbb{R}}$ is the flow induced by $X$ and $\operatorname{Trans}(\Phi,\Lambda)$ denotes the set of points whose positive orbits are dense in $\Lambda$. The same formula holds $C^1$-generically for singular hyperbolic attractors. In contrast with the uniformly hyperbolic case, we also prove that, for vector fields in a dense subset of $\mathcal{X}^r(M^3)$, geometric Lorenz attractors admit a positive-entropy invariant measure with no generic points. An analogous conclusion holds for vector fields in a dense subset of $\mathcal{X}^1(M)$ whose non-trivial singular hyperbolic attractors contain singularities and satisfy $\dim E^{cu}=2$. These examples show that the discrete-time saturated entropy theory does not extend directly to singular flows.

This paper classifies the possible inertias that are realized by an excess matrix of a finite metric space with at most nine points. For the classification, we make use of relationships between finite metric spaces and non-obtuse and acute sets in classical geometry. As an application, we prove that the small-scale limit of the magnitude function of any six-point metric space is at least one whenever this limit exists.

We introduce a class of extensions of unital operator systems satisfying a certain Toeplitz type property. We show that these extensions induce corresponding extensions of the $C^*$-algebras they generate. We prove that hyperrigidity is preserved under these extensions, yielding new examples through iteration. We introduce the quantum double suspension of operator systems as a special class of these extensions and show that its iterates yield the Hong-Szymański quantum double suspensions at the $C^*$-algebra level, encompassing, in particular, the Vaksman-Soibelman quantum spheres. We construct hyperrigid operator systems underlying these quantum spheres. As a further application, we show that every counterexample to Arveson's hyperrigidity conjecture gives rise to a sequence of counterexamples. We identify the $C^*$-envelopes of these extensions and use this identification to obtain a $K$-theoretic condition ensuring that each such sequence contains infinitely many pairwise non-isomorphic counterexamples. We also observe that the validity of the conjecture in the commutative setting is linked to its validity for an infinite family of noncommutative operator systems obtained through successive extensions. Finally, we compute the propagation number, an invariant of operator systems introduced by Connes and van Suijlekom, for these extensions.

We establish a polynomial analogue of Lindenstrauss' theorem in full generality. More precisely, we prove that, for every real or complex Banach space X and every positive integer n, the n-homogeneous scalar-valued polynomials whose Aron-Berner extensions to the bidual X** attain their norms form a dense subset of the space of continuous n-homogeneous polynomials on X. This resolves a problem posed by Aron, Garcia and Maestre in 2003, extending their result for quadratic polynomials to arbitrary degrees without imposing any structural assumptions on the underlying Banach space. We further establish the corresponding theorem for vector-valued polynomials with arbitrary Banach space codomains. Our approach relies on a new perturbation scheme based on averaged linear contractions, which provides simultaneous control of the multilinear variables and yields norm attainment through iterated limits in the bidual.

We prove that for every $\varepsilon >0$, there is $\delta >0$ such that for every probability measure $\mu$ on a free group $\mathbb{F}$, if the operator norm of the action of $\mu$ on $L^2\mathbb{F}$ by convolution exceeds $\varepsilon$, then there is a coset of an abelian subgroup of $\mathbb{F}$ with $\mu$-mass at least $\delta$.

This paper examines the stability analysis of two-dimensional incompressible fluids in $L^2_{uloc}$ spaces. Extending the classical stability framework for Euler equations in Yudovich-type settings, we establish the convergence of Navier-Stokes solutions toward Euler solutions as viscosity $\nu \to 0$, under the weakened assumptions $u^\nu_0 \in L^\infty$ and $\omega^\nu_0 \in L^\infty$ without imposing decay conditions at infinity. We first derive a priori estimates for the Navier-Stokes equations in $L^\infty$ spaces. Then, using harmonic analysis tools including Littlewood-Paley decomposition and Bony's paraproduct calculus, we prove the $L^\infty$-convergence of the velocity field and $L^2_{uloc}$-convergence of the vorticity as $\nu \to 0$.

We introduce a strict monoidal groupoid associated with a given commutative ring $K$. It acts on the Clifford extension systems of $K$-central algebras which are graded by a group $G$, as well as on the generalized collective characters, both of type $\phi \in$ Hom$(G,$ Aut$(K))$. These actions partition the two sets above with respect to graded Morita equivalence. By that we establish a seven term exact sequence of type $\phi$, which boils down to the celebrated Chase-Harrison-Rosenberg sequence when $\phi$ is Galois.

We give a proof of the convexity of the upper bild of a quaternionic matrix, a theorem of So and Thompson. After reduction to $2\times2$ matrices, we introduce real coordinates on a sphere and obtain two representations of the upper bild over the closed unit ball, involving a convex function and a concave function. Minimizing the first and maximizing the second over parameters with a fixed real part yield a convex lower boundary function and a concave upper boundary function, from which convexity follows.

Let $f$ be a dominant rational self-map of a normal projective variety over an algebraically closed field of arbitrary characteristic, or a dominant meromorphic self-map of a compact Kähler manifold. We prove that in every codimension, the degree sequence of $f$ differs from its dynamical-degree growth by at most a polynomial factor, with an explicit universal bound on the polynomial exponent. In particular, subexponential but superpolynomial degree growth cannot occur. At a strict dynamical-degree break, we prove the sharper asymptotic \[ \mathrm{deg}_{i,\omega}(f^n)=C_{f,\omega,i}\,\lambda_i(f)^n+O(\eta^n), \] for every $\sqrt{\lambda_{i-1}(f)\lambda_{i+1}(f)}<\eta<\lambda_i(f)$. This extends the degree-asymptotic theorem of Dang–Favre from codimension one in characteristic zero to arbitrary codimension and arbitrary characteristic, and recovers the surface theorem of Boucksom–Favre–Jonsson. Both results rely on a uniform estimate for the temporal discrete logarithmic Laplacian of the degree sequences, obtained by applying the Ptolemy inequality to ordered Hodge–Lorentz packages on varying finite-time orbit spaces.

Dynamics of biochemical systems are dictated by the stoichiometry and kinetics of reaction networks. These systems may admit positive steady states that could undergo qualitative change in response to varying rate parameters. These changes are called steady-state bifurcations, and they correspond to biologically meaningful processes. Classical bifurcation analysis relies on exact kinetics and rate parameters, which are often unknown or difficult to measure for complex systems. To address this, structural bifurcation analysis (SBA) was introduced to predict bifurcations using only the system's associated reaction network. In this work, the computational package NetBiF (Network Bifurcation Finder) is introduced to implement SBA. NetBiF is applied across diverse biochemical systems, including fundamental metabolic pathways, signaling and gene regulatory networks, and a large-scale carbon cycle model, to predict steady-state bifurcations. Crucially, for biochemical systems governed by power law kinetics, the implementation of NetBiF also classifies the steady-state bifurcations as saddle-node, transcritical, or pitchfork via a structural analog of Sotomayor's theorem. To this end, explicit formulations for the bifurcation criteria were derived for these biochemical systems. Determining and classifying bifurcation behaviors provide insight into biochemical systems' abilities to adapt to changing and fluctuating conditions. Collectively, the study presents NetBiF as a scalable, structure-based, and consistent computational tool to implement SBA. Our findings enable researchers to systematically study dynamical behaviors of biochemical systems through their reaction network structures without requiring explicit kinetic information.

We study the contributions of half-BPS surface defects to supersymmetric Casimir energy and the universal logarithmic term of supersymmetric Rényi entropy in four- and six-dimensional SCFTs. Using equivariant integration of defect anomaly polynomials and its extension to supersymmetric Rényi backgrounds, we obtain closed expressions for these observables in terms of defect 't Hooft anomalies. For Gukov–Witten defects preserving transverse rotations in four-dimensional $\mathcal{N} = 4$ super Yang–Mills theories, both observables depend on a single R-symmetry anomaly. For $\mathcal{N} = (0, 4)$ defects in six-dimensional $\mathcal{N} = (1, 0)$ theories, additional terms arise from the gravitational anomaly. We provide complementary checks using chiral algebras, five-dimensional localization, and holographic probe branes at large $N$, while localized chiral fermions reproduce the gravitational-anomaly contributions. The framework also recovers Wilson-surface results in six-dimensional $\mathcal{N} = (2, 0)$ theories.

It is observed that the dimension of the space of invariant spinors on the manifold $\mathrm{SU}(n)/T^{n-1}$ of full flags in $\mathbb{C}^n$ equals the number $\mathrm{RT}(n)$ of labelled regular tournaments of order $n$, for $n$ odd. This space is decomposed under a double cover of the Weyl group, and realized as a pair of eigenspaces, first for the Dirac operator, then for a twisted version of it which acts equivariantly; this is used to show that multiplicities of irreducible representations are even for $n\equiv 1\pmod 4$.

We study the optimization inverse spectral problem (OISP) for the first $m$ eigenvalues of the Schrödinger operator on a compact metric graph. We prove the existence of a solution for every strictly increasing finite set of prescribed eigenvalues. On a graph without cycles, we obtain a representation of the optimal correction as a linear combination of squared eigenfunctions, provided that the selected eigenvalues are simple and separated from the next eigenvalue. On a graph with cycles, under the same simplicity and separation assumptions, we obtain a strong limiting representation of the optimal potential in terms of eigenfunctions of approximating operators. For a graph with a cycle, we give a counterexample with simple eigenvalues and linearly dependent squared eigenfunctions. We also prove that linear dependence of the squares of nonzero restrictions to an edge of eigenfunctions corresponding to distinct eigenvalues forces the potential on that edge to have an analytic representative

Let $M$ be a finite-dimensional complete Riemannian manifold of bounded geometry. We consider a time-dependent vector field $X : \mathbb{R} \times M \to TM$ with $X_t:=X(t,\cdot) \in \mathfrak{X}^r(M)$ for all $t \in \mathbb{R}$ that satisfies some further uniform continuity assumptions. We suppose that $X$ admits a globally defined integral curve $\gamma : \mathbb{R} \to M$. To any isolated subset $\Omega$ of the Sacker-Sell spectrum of $t \mapsto \nabla X_t(\gamma(t))$, Sacker-Sell theory associates a subbundle $\mathscr{N}^\Omega$ of $T_\gamma M$ invariant under $\nabla X_t$. Under additional nonresonance conditions on $\Omega$, we prove, by an application of the parameterization method (Cabré et al., 2003), the existence of an invariant topological submanifold $\mathscr{W}^\Omega \subset \mathbb{R} \times M$ with $C^r$ fibers $W_t^\Omega:= \mathscr{W}^\Omega \cap \{t\} \times M$ that is unique in a sufficiently smooth subclass of invariant manifolds uniformly tangent to $\mathscr{N}^\Omega$ (made precise later). We moreover show regularity of $\mathscr{W}^\Omega$ with respect to parameters, and characterize the attraction towards $\mathscr{W}^\Omega$ via a foliation in the case $\Omega$ is the slowest part of the spectrum.

We propose the study of the class of groups that fix a point for every continuous action on any convex compact space. Oliver completed the corresponding program of classifying the finite groups that have this fixed-point property in finite dimensions. Turning to infinite groups, the Schauder–Tychonoff theorem establishes that the integers have the fixed-point property, but this has not been shown for any other infinite, finitely generated group. We establish a local transfer principle and use it to show, as a proof of concept, that $\mathbf{Z} \times (\mathbf{Z}/p\mathbf{Z})$ belongs to this class of groups.

We settle two conjectures of Berndt and Straub asserting that all nonzero zeros of the generalized Ramanujan polynomials $R_k(x;\chi,\psi)$ and $S_{2n}(x;\chi,\chi)$ lie on the unit circle for real nonprincipal Dirichlet characters. We prove their conjecture for $R_k$. For $S_{2n}$, we prove that at most four nonzero zeros lie off the unit circle, all purely imaginary. If $\chi_{163}=\left(\frac{\cdot}{163}\right)$ is the primitive quadratic character modulo $163$, then $S_6(x;\chi_{163},\chi_{163})$ has four exceptional zeros, showing that this bound is sharp. This disproves the second conjecture and establishes that the `Riemann hypothesis for period polynomials' can fail for Eisenstein newforms. Further, we introduce a transcendental invariant whose sign determines whether exceptional zeros of $S_{2n}(x;\chi,\chi)$ occur in all sufficiently large degrees, and show that these zeros converge to transcendental limits off the unit circle. For primitive odd characters, we relate this invariant and these limits to Fekete polynomials.

We study non-additive rank-metric codes contained in the space of symmetric matrices over a finite field. Our main focus is on symmetric $2$-codes. For odd $q$, we introduce a quotient construction for $\mathrm{Sym}_3(q)$ in which rank-one differences are described by isotropic vectors of a nondegenerate quadratic form on $\mathbb F_q^4$. This leads to a correspondence with partial ovoids of the elliptic quadric $Q^-(5,q)$. Using a construction of Cossidente and Pavese, we obtain a non-additive symmetric $2$-code in $\mathrm{Sym}_3(q)$ of cardinality $q^4+2q^3$, improving the previously known general lower bound. We also give an explicit matrix description of this family. Lastly, we present a block-based construction for symmetric $2$-codes that extends the three-dimensional improvement to all odd dimensions. In particular, for every odd $n\geq3$ and every prime power $q>2$, we obtain \begin{equation*} A_q^sym(n,2) \ge q^{(n^2-1)/2}+q^{(n^2-3)/2}+q^{(n^2-9)/2}, \end{equation*} which is strictly larger than the maximum cardinality of an additive symmetric $2$-code with the same parameters. This yields the first infinite family (as the matrix dimension $n$ varies) of non-additive symmetric $2$-codes whose size is larger than the largest achievable size of an additive code with the same parameters; before this, such a gap had only been established for $n=3$.

In this paper, we study the estimation of the signal subspace dimension $k$ in a $p$-variate signal-plus-noise complex Gaussian model, when the noise covariance matrix is assumed to be diagonal or Toeplitz. We consider the high-dimensional asymptotic regime in which both the data dimension $p$ and the available sample size $n$ diverge to infinity at the same rate, while we also allow $k$ to diverge at a slower speed. In this context, the classical estimates of the literature, consistent in the high-dimensional regime, often rely on the a priori knowledge of an upper bound $k_*$ of $k$, which can be misspecified in practice. Our proposed approach relies on building estimates of the tracial moments of the noise covariance matrix using the $p-k_*$ smallest eigenvalues of the sample covariance matrix (SCM), which are in turn used to build a threshold above which the $k$ largest eigenvalues are isolated. The resulting estimate of $k$ is proved to be consistent in the high-dimensional regime, even for the misspecified situations where $k_* < k$. Numerical experiments in different challenging scenarios illustrate the improvement of the proposed approach over the aforementioned methods.

We propose a smoothed augmented Lagrangian method (ALM) for solving mathematical programs with vertical complementarity constraints (VCCs). The augmented Lagrangian function involving the VCCs is smoothed by the Lasry–Lions double envelope, which is an extension of the Moreau envelope that enjoys enhanced smoothness properties. This construction yields a smooth unconstrained subproblem within the ALM framework that can be efficiently solved by many fast optimization algorithms. In the limiting setting, we establish global convergence to Mordukhovich- and Clarke-stationary points. For the convergence rate, we show that, under appropriate assumptions, the smoothed ALM achieves Q-linear convergence. We further provide a worst-case iteration complexity analysis for computing an approximate stationary point of the original problem or of an associated feasibility problem. Preliminary numerical experiments demonstrate the effectiveness of the proposed method.

Coordinate descent solvers for regularized generalized linear models (GLMs) differ in how they update the intercept. This seemingly routine choice can substantially affect convergence when the response is imbalanced. For direct coordinate descent on the original GLM loss, we compare a conservative step based on worst-case curvature, a safeguarded Newton step based on current curvature, and repeated Newton steps that solve the intercept subproblem. The conservative update can leave much of the interaction between the intercept and the coefficients unresolved, producing a slowdown when that interaction controls convergence. A single Newton step avoids this attenuation and, in our experiments, follows nearly the same outer trajectory as exact minimization while requiring fewer inner steps. Simulated and real-data experiments, including comparisons across six production solvers and a controlled intervention within skglm, support this explanation. We also show how the same mechanism appears in quadratic-approximation methods and extends to multinomial models with rare classes. We recommend a safeguarded Newton update for direct coordinate descent and full intercept optimization within frozen local quadratic surrogates.

We study Strichartz estimates for orthonormal families of initial data when time is integrated against an $\alpha$-dimensional Frostman measure $\mu$, $0<\alpha\le1$, with coefficients measured in $\ell^\beta$. For the fractional Schrödinger, wave and Klein–Gordon equations, we obtain orthogonality gains in the admissible region and necessary conditions on $\beta$. For the fractional Schrödinger equation the exponents are sharp up to an endpoint; the optimality of the threshold $\beta=q/2$ requires $\mu$ to be Ahlfors regular. For the Schrödinger equation in dimensions $d\ge2$, we prove a Lorentz endpoint estimate and, if $d>2\alpha$, the critical exponent $\beta=q/2$ at every pair $2\le q,r<\infty$ with \[\frac d2(\frac12-\frac1r)\ge\frac\alpha q, \quad r>\frac{dq}{d-\alpha};\] on $r=\infty$ we obtain a restricted weak-type estimate. Since the constants depend on $\mu$ only through its Frostman constant, Lebesgue measure yields global-in-time critical estimates in this range when $d\ge3$. The proofs combine Frostman fractional integration, a spectral near–far argument at the endpoint, and asymmetric Hilbert–Schmidt bounds with orthogonal frequency-block summation. We also obtain further estimates in the low-dimensional cases $d=1$ (fractional Schrödinger) and $d=2$ (wave and Klein–Gordon) with $\alpha>1/2$, and formulate the consequences for densities of freely evolving states.

Let $2<p<\infty$. We prove that every infinite-dimensional closed subspace $F$ of a noncommutative $L_p$-space contains a subspace completely isomorphic to $\ell_p$, $C_p$, $R_p$, or $C_p\cap R_p$ and completely complemented in $F$. This answers the question of Junge and Xu on the existence of completely symmetric basic sequences.

We introduce Contractor Filters (CFs), a set-membership architecture for nonlinear continuous-time state estimation with bounded uncertainties and sporadic outliers. CFs represent feasible states symbolically by compositions of contractors rather than rematerializing a geometric enclosure after each update. This provides an abstraction layer between the set-theoretic filtering equations and their numerical realization, so prediction, correction, ambiguity, and outlier handling can be expressed within one algebra without prescribing a state geometry. We derive an exact nominal recursion and a guaranteed robust enclosure, with validated flow inclusions preserving soundness. Optional sub-paving compression then provides finite outer representations. CFs are evaluated against a robust particle filter on range-only and terrain-based localization under outliers, probabilistic model mismatch, association ambiguity, and multimodality. CFs retain the simulated truth in all tested configurations, while particle-filter reliability degrades under persistent bias and spatial aliasing. Contractor strength and compression budget control the trade-off between paving tightness and computational cost, while preserving the enclosure guarantee.