Classic Example Research Articles

Higher-order interactions lead to ‘reluctant’ synchrony breaking

To model dynamical systems on networks with higher-order (non-pairwise) interactions, we recently introduced a new class of ordinary differential equations (ODEs) on hypernetworks. Here, we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking steady-state branch ‘reluctant’ if it is tangent to a synchrony space, but does not lie inside it. We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady-state branch.

Quantization in fibering polarizations, Mabuchi rays and geometric Peter–Weyl theorem

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter–Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact connected Lie groups K. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of K×K-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action.Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for K with the Borel–Weil description of irreducible representations of K.

Discontinuities Cause Essential Spectrum on Surfaces

AbstractTwo-dimensional maps with discontinuities are considered. It is shown that, in the presence of discontinuities, the essential spectrum of the transfer operator is large whenever it acts on a Banach space with norm that is stronger than $$L^\infty $$ L ∞ or $$BV$$ BV . Three classes of examples are introduced and studied, both expanding and partially expanding. In two dimensions, there is complication due to the geometry of the discontinuities, an issue not present in the one-dimensional case and which is explored in this work.

On the classification of low-degree ovoids of Q+(5,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q^+(5,q)$$\\end{document}

Ovoids of the Klein quadric Q+(5,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q^+(5,q)$$\\end{document} of PG(5,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PG}(5,q)$$\\end{document} have been studied in the last 40 years, also because of their connection with spreads of PG(3,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PG}(3,q)$$\\end{document} and hence translation planes. Beside the classical example given by a three-dimensional elliptic quadric (corresponding to the regular spread of PG(3,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PG}(3,q)$$\\end{document}) many other classes of examples are known. First of all the other examples (beside the elliptic quadric) of ovoids of Q(4, q) give also examples of ovoids of Q+(5,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q^+(5,q)$$\\end{document}. To every ovoid of Q+(5,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q^+(5,q)$$\\end{document} two bivariate polynomials f1(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_1(x,y)$$\\end{document} and f2(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_2(x,y)$$\\end{document} can be associated. Another important class of ovoids of Q+(5,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q^+(5,q)$$\\end{document} is given by the ones associated to a flock of a three-dimensional quadratic cone and in this case f1(x,y)=y+g(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_1(x,y)=y+g(x)$$\\end{document}. In this paper, we classify such ovoids of Q+(5,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q^+(5,q)$$\\end{document} with the additional properties that max{deg(f1),deg(f2)}<(16.31q)313-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\max \\{\\deg (f_1),\\deg (f_2)\\}<(\\frac{1}{6.31}q)^{\\frac{3}{13}}-1$$\\end{document}, that is f1(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_1(x,y)$$\\end{document} and f2(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_2(x,y)$$\\end{document} have “low degree" compared with q.

Some Aspects of Differential Topology of Subcartesian Spaces

In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

A survey and exposition of sub-Hausdorff separation axioms

This paper surveys and develops sub-Hausdorff axioms. It augments known relationships involving T 0 (Kolmogorov), S 0 (quasi-sobriety), S 1 (sobriety), T 1 (Frchet), and T 2 (Hausdorff ) by examining a large suite of sub-Hausdorff axioms. Emphasis is given to hereditary S 0, hereditary S 1, deleted S 0, deleted S 1, (locally) strong S 0, (locally) strong S 1, Tsc , and pre-T 2. Properties of the Kolmogorov functor are applied to preservation, reflection, and characterization of certain sub-Hausdorff axioms. Classes of examples are constructed to illustrate main ideas, show non-reversibility of key implications, and correct published errors. Many theorems are given both point-set based and spectrum based proofs for additional insights. Links to domain theory and the topology of digital displays are noted. Key relationships are summarized in two tables.

On finitely generated Engel branch groups

AbstractWe construct finitely generated Engel branch groups, answering a question of Fernández‐Alcober, Noce and Tracey on the existence of such objects. In particular, the groups constructed are not nilpotent, yielding the second known class of examples of finitely generated non‐nilpotent Engel groups following a construction by Golod from 1969. To do so, we exhibit groups acting on rooted trees with growing valency on which word lengths of elements are contracting very quickly under section maps. Our methods apply in principle to a wider class of iterated identities, of which the Engel words are a special case.

Mourre theory and spectral analysis of energy-momentum operators in relativistic quantum field theory

A central task of theoretical physics is to analyse spectral properties of quantum mechanical observables. In this endeavour, Mourre’s conjugate operator method emerged as an effective tool in the spectral theory of Schrödinger operators. This paper introduces a novel class of examples from relativistic quantum field theory that are amenable to Mourre’s method. By assuming Lorentz covariance and the spectrum condition, we derive a limiting absorption principle for the energy-momentum operators and provide new proofs of the absolute continuity of the energy-momentum spectra. Moreover, under the assumption of dilation covariance, we show that the spectrum of the relativistic mass operator is purely absolutely continuous in (0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(0,\\infty )$$\\end{document}.

Achiral words

A word w in a free group is achiral if for every group G, G w = G w − 1 , where Gw is the image of the word map w on G. We give few classes of examples of achiral words. We prove that all words in F 2 of length at most 7 are achiral. We provide a criterion to decide whether a word in F 2 is chiral or achiral. As a result, we give shortest chiral word in F 2 which is of length 8 and we answer Cocke and Ho’s query regarding whether Engel words are achiral.

Optimal transport for types and convex analysis for definable predicates in tracial W⁎-algebras

We investigate the connections between continuous model theory, free probability, and optimal transport/convex analysis in the context of tracial von Neumann algebras. In particular, we give an analog of Monge-Kantorovich duality for optimal couplings where the role of probability distributions on Cn is played by model-theoretic types, the role of real-valued continuous functions is played by definable predicates, and the role of continuous function Cn→Cn is played by definable functions. In the process, we also advance the understanding of definable predicates and definable functions by showing that all definable predicates can be approximated by “C1 definable predicates” whose gradients are definable functions. As a consequence, we show that every element in the definable closure of W⁎(x1,…,xn) can be expressed as a definable function of (x1,…,xn). We give several classes of examples showing that the definable closure can be much larger than W⁎(x1,…,xn) in general.

Applications of higher‐dimensional Heegaard Floer homology to contact topology

AbstractThe goal of this paper is to set up the general framework of higher‐dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient condition for the Weinstein conjecture to hold. We discuss several classes of examples including those coming from analyzing a close cousin of symplectic Khovanov homology and the analog of the Plamenevskaya invariant of transverse links.

One-dimensional continuous-time quantum Markov chains: qubit probabilities and measures

Quantum Markov chains (QMCs) are positive maps on a trace-class space describing open quantum dynamics on graphs. Such objects have a statistical resemblance with classical random walks, while at the same time they allow for internal (quantum) degrees of freedom. In this work we study continuous-time QMCs on the integer line, half-line and finite segments, so that we are able to obtain exact probability calculations in terms of the associated matrix-valued orthogonal polynomials and measures. The methods employed here are applicable to a wide range of settings, but we will restrict ourselves to classes of examples for which the Lindblad generators are induced by a single positive map, and such that the Stieltjes transforms of the measures and their inverses can be calculated explicitly.

Enhancing the description of multi-time-scale geophysical phenomena: Incorporating finite time Scale separation and feedback, illustrated with the case of a 1D variable of interest.

Stochastic approaches play a vital role in weather, climate, and, more in general, geophysics systems, addressing processes and scales beyond the resolution of deterministic models. Similar to equilibrium/non-equilibrium thermodynamics, intricate fast and local dynamics may not always be the primary focus. Practical applications often prioritize observables capturing phenomena at dominant temporal and spatial scales. Developing models for these "large-scale" observables, resulting from averaging fast and local contributions, can be simplified into Low Order Models (LOMs) with reduced degrees of freedom described by ordinary differential equations. Unresolved degrees of freedom are introduced as stochastic components, exhibiting either Markovian or non-Markovian characteristics. The challenge lies in deriving dependable stochastic differential equations representing the statistics of real large-scale, slow features in the climate/ocean system. While paralleling material physics, it is crucial to recognize that direct transfer of tools and outcomes is hindered by the non-Hamiltonian nature of climate/geophysical LOMs and the impracticality of a Markovian treatment of noise due to wide-ranging time scales. A critical examination of the conventional statistical mechanics approach, customized for such LOMs, becomes essential. To this end, we propose utilizing an approach based on the operator cumulant method, which has been recently revisited and generalized, along with the linear response method in a non-Hamiltonian setting. Formal results are then derived, and applications to some typical classes of examples are presented to clarify this approach.

Radio relics in massive galaxy cluster mergers in the TNG-Cluster simulation

Radio relics are diffuse synchrotron sources in the outskirts of merging galaxy clusters energized by the merger shocks. In this paper, we present an overview of the radio relics in massive cluster mergers identified in the new TNG-Cluster simulation. This is a suite of magnetohydrodynamical cosmological zoom-in simulations of 352 massive galaxy clusters with M500c = 1014.0 − 15.3 M⊙ sampled from a 1 Gpc-sized cosmological box. The simulations were performed using the moving-mesh code AREPO with the galaxy formation model and high numerical resolution consistent with the TNG300 run of the IllustrisTNG series. We post-processed the shock properties obtained from the on-the-fly shock finder to estimate the diffuse radio emission generated by cosmological shockwaves for a total of ∼300 radio relics at redshift z = 0 − 1. TNG-Cluster returned a variety of radio relics with diverse morphologies, encompassing classical examples of double radio relics, single relics, and “inverted” radio relics that are convex to the cluster center. Moreover, the simulated radio relics reproduced both the abundance and statistical relations of observed relics. We find that extremely large radio relics (&gt; 2 Mpc) are predominantly produced in massive cluster mergers with M500c ≳ 8 × 1014 M⊙. This underscores the significance of simulating massive mergers to study giant radio relics similar to those found in observations. We released a library of radio relics from the TNG-Cluster simulation, which will serve as a crucial reference for upcoming next-generation surveys.

Differential Calculi on Quantum Principal Bundles Over Projective Bases

We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of covariant bimodules together with a morphism of sheaves -the differential- such that the Leibniz rule and surjectivity hold locally. The main class of examples is given by covariant calculi over quantum flag manifolds, which we provide via an explicit Ore extension construction. In a second step we introduce principal covariant calculi by requiring a local compatibility of the calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms of exact sequences. In this case Hopf–Galois extensions of algebras lift to Hopf–Galois extensions of exterior algebras with compatible differentials. In particular, the examples of principal (covariant) calculi on the quantum principal bundles Oq(SL2(C))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_q(\ extrm{SL}_2(\\mathbb {C}))$$\\end{document} and Oq(GL2(C))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_q(\ extrm{GL}_2(\\mathbb {C}))$$\\end{document} over the projective space P1(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{P}^1(\\mathbb {C})$$\\end{document} are discussed in detail.

Newton–Okounkov bodies and minimal models for cluster varieties

Let Y be a (partial) minimal model of a scheme V with a cluster structure (of type A, X or of a quotient of A or a fibre of X). Under natural assumptions, for every choice of seed we associate a Newton–Okounkov body to every divisor on Y supported on Y∖V and show that these Newton–Okounkov bodies are positive sets in the sense of Gross, Hacking, Keel and Kontsevich [31]. This construction essentially reverses the procedure in [31] that generalizes the polytope construction of a toric variety to the framework of cluster varieties.In a closely related setting, we consider cases where Y is a projective variety whose universal torsor UTY is a partial minimal model of a scheme with a cluster structure of type A. If the theta functions parametrized by the integral points of the associated superpotential cone form a basis of the ring of algebraic functions on UTY and the action of the torus TPic(Y)⁎ on UTY is compatible with the cluster structure, then for every choice of seed we associate a Newton–Okounkov body to every line bundle on Y. We prove that any such Newton–Okounkov body is a positive set and that Y is a minimal model of a quotient of a cluster A-variety by the action of a torus.Our constructions lead to the notion of the intrinsic Newton–Okounkov body associated to a boundary divisor in a partial minimal model of a scheme with a cluster structure. This notion is intrinsic as it relies only on the geometric input, making no reference to the auxiliary data of a valuation or a choice of seed. The intrinsic Newton–Okounkov body lives in a real tropical space rather than a real vector space. A choice of seed gives an identification of this tropical space with a vector space, and in turn of the intrinsic Newton–Okounkov body with a usual Newton–Okounkov body associated to the choice of seed. In particular, the Newton–Okounkov bodies associated to seeds are related to each other by tropicalized cluster transformations providing a wide class of examples of Newton-Okounkov bodies exhibiting a wall-crossing phenomenon in the sense of Escobar–Harada [18].This approach includes the partial flag varieties that arise as minimal models of cluster varieties (for example full flag varieties and Grassmannians). For the case of Grassmannians, our approach recovers, up to interesting unimodular equivalences, the Newton–Okounkov bodies constructed by Rietsch–Williams in [56].

Dynamical phase transitions in graph cellular automata.

Discrete dynamical systems can exhibit complex behavior from the iterative application of straightforward local rules. A famous class of examples comes from cellular automata whose global dynamics are notoriously challenging to analyze. To address this, we relax the regular connectivity grid of cellular automata to a random graph, which gives the class of graph cellular automata. Using the dynamical cavity method and its backtracking version, we show that this relaxation allows us to derive asymptotically exact analytical results on the global dynamics of these systems on sparse random graphs. Concretely, we showcase the results on a specific subclass of graph cellular automata with "conforming nonconformist" update rules, which exhibit dynamics akin to opinion formation. Such rules update a node's state according to the majority within their own neighborhood. In cases where the majority leads only by a small margin over the minority, nodes may exhibit nonconformist behavior. Instead of following the majority, they either maintain their own state, switch it, or follow the minority. For configurations with different initial biases towards one state we identify sharp dynamical phase transitions in terms of the convergence speed and attractor types. From the perspective of opinion dynamics this answers when consensus will emerge and when two opinions coexist almost indefinitely.

Higher Airy Structures, 𝒲 Algebras and Topological Recursion

We define higher quantum Airy structures as generalizations of the Kontsevich–Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of higher quantum Airy structures as modules of W ( g ) \mathcal {W}(\mathfrak {g}) algebras at self-dual level, with g = g l N + 1 \mathfrak {g}= \mathfrak {gl}_{N+1} , s o 2 N \mathfrak {so}_{2 N } or e N \mathfrak {e}_N . We discuss their enumerative geometric meaning in the context of (open and closed) intersection theory of the moduli space of curves and its variants. Some of these W \mathcal {W} constraints have already appeared in the literature, but we find many new ones. For g l N + 1 \mathfrak {gl}_{N+1} our result hinges on the description of previously unnoticed Lie subalgebras of the algebra of modes. As a consequence, we obtain a simple characterization of the spectral curves (with arbitrary ramification) for which the Bouchard–Eynard topological recursion gives symmetric ω g , n \omega _{g,n} s and is thus well defined. For all such cases, we show that the topological recursion is equivalent to W ( g l ) \mathcal {W}(\mathfrak {gl}) constraints realized as higher quantum Airy structures, and obtain a Givental-like decomposition for the corresponding partition functions.

Locally conformally product structures

A locally conformally product (LCP) structure on compact manifold [Formula: see text] is a conformal structure [Formula: see text] together with a closed, non-exact and non-flat Weyl connection [Formula: see text] with reducible holonomy. Equivalently, an LCP structure on [Formula: see text] is defined by a reducible, non-flat, incomplete Riemannian metric [Formula: see text] on the universal cover [Formula: see text] of [Formula: see text], with respect to which the fundamental group [Formula: see text] acts by similarities. It was recently proved by Kourganoff that in this case [Formula: see text] is isometric to the Riemannian product of the flat space [Formula: see text] and an incomplete irreducible Riemannian manifold [Formula: see text]. In this paper, we show that for every LCP manifold [Formula: see text], there exists a metric [Formula: see text] such that the Lee form of [Formula: see text] with respect to [Formula: see text] vanishes on vectors tangent to the distribution on [Formula: see text] defined by the flat factor [Formula: see text], and use this fact in order to construct new LCP structures from a given one by taking products. We also establish links between LCP manifolds and number field theory, and use them in order to construct large classes of examples, containing all previously known examples of LCP manifolds constructed by Matveev–Nikolayevsky, Kourganoff and Oeljeklaus–Toma (OT-manifolds).

Robust Kalman filters under epistemic uncertainty for non‐Gaussian systems with multiplicative noise

AbstractThis article proposes two robust Kalman filters to solve the issue of inaccurate modeling in multiplicative noise systems due to epistemic limitations. First, we construct all conceivable state/measurement transition probability densities as an ambiguity set. This ambiguity set chooses the Wasserstein distance or the moment‐based metric as the distance metric. Besides, this set is an inequality set with a chosen tolerance, which can be seen as a non‐negative radius ball. Then, by combining the robust solution of the least favorable model in that ball with the alternating direction method of multipliers or an efficient direct solution method, we propose two robust Kalman filters based on the minimum mean square error criterion. A classical example is provided to verify the effectiveness of the proposed robust filters in comparison to existing state‐of‐the‐art filters.