Family Of Invariants Research Articles

The finite generation ideal for Daigle and Freudenburg’s counterexample to Hilbert’s fourteenth problem

We compute the finite generation ideal for Daigle and Freudenburg’s counterexample to Hilbert’s fourteenth problem. This ideal helps to understand how far the ring of invariants is from being finitely generated. Our calculations show that the finite generation ideal is the radical of an ideal generated by three infinite families of invariants. We show that these three families together with an additional invariant form a SAGBI-basis of the ring of invariants. We use the properties of our SAGBI-basis in our computation of the finite generation ideal.

Knot concordance invariants from Seiberg–Witten theory and slice genus bounds in 4-manifolds

We construct a new family of knot concordance invariants [Formula: see text], where [Formula: see text] is a prime number. Our invariants are obtained from the equivariant Seiberg–Witten–Floer cohomology, constructed by the author and Hekmati, applied to the degree [Formula: see text] cyclic cover of [Formula: see text] branched over [Formula: see text]. In the case [Formula: see text], our invariant [Formula: see text] shares many similarities with the knot Floer homology invariant [Formula: see text] defined by Hom and Wu. Our invariants [Formula: see text] give lower bounds on the genus of any smooth, properly embedded, homologically trivial surface bounding [Formula: see text] in a definite [Formula: see text]-manifold with boundary [Formula: see text].

Curvature Sets Over Persistence Diagrams

We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers k≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 0$$\\end{document} and n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 1$$\\end{document} we consider the dimension k Vietoris–Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and denote them as Dn,kVR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{D}}_{n,k}^{\ extrm{VR}}$$\\end{document}. We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parameters n and k, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which D4,1VR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{D}}_{4,1}^{\ extrm{VR}}$$\\end{document} fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a space X with cardinality 2k+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2k+2$$\\end{document} with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction.

The Bohr radius and its modifications for linearly invariant families of analytic functions

In 1914, Harald Bohr published an article in which he considered the class B of all analytic functions in the unit disk |z|<1, bounded in absolute value by 1. The article contains proofs that for any function f(z)=∑n=0∞anzn from B, the inequality ∑n=0∞|anzn|≤1 holds in the disk of radius 1/3 centered at the origin, and the value 1/3 is optimal. Also, Bohr himself in this article proved the corresponding result in a circle of radius 1/6 and adds, following the request of Wiener, Wiener's later proof for the disk of radius 1/3. Since then, the constant 1/3 in this problem has been called the Bohr radius. This was followed by a series of papers dealing with analogues of the Bohr radius or its estimates in other classes of functions. In this article, we give estimates for the Bohr radius in some classes of analytic functions in D, associated with linearly invariant families of finite order.

Strong computable type

A compact set has computable type if any homeomorphic copy of the set which is semicomputable is actually computable. Miller proved that finite-dimensional spheres have computable type, Iljazović and other authors established the property for many other sets, such as manifolds. In this article we propose a theoretical study of the notion of computable type, in order to improve our general understanding of this notion and to provide tools to prove or disprove this property. We first show that the definitions of computable type that were distinguished in the literature, involving metric spaces and Hausdorff spaces respectively, are actually equivalent. We argue that the stronger, relativized version of computable type, is better behaved and prone to topological analysis. We obtain characterizations of strong computable type, related to the descriptive complexity of topological invariants, as well as purely topological criteria. We study two families of topological invariants of low descriptive complexity, expressing the extensibility and the null-homotopy of continuous functions. We apply the theory to revisit previous results and obtain new ones.

A diffeomorphism invariant family of metric-affine actions for loop cosmologies

In loop quantum cosmology (LQC) the big bang singularity is generically resolved by a big bounce. This feature holds even when modified quantization prescriptions of the Hamiltonian constraint are used such as in mLQC-I and mLQC-II. While the later describes an effective description qualitatively similar to that of standard LQC, the former describes an asymmetric evolution with an emergent Planckian de-Sitter pre-bounce phase even in the absence of a potential. We consider the potential relation of these canonically quantized non-singular models with effective actions based on a geometric description. We find a 3-parameter family of metric-affine f(ℛ) theories which accurately approximate the effective dynamics of LQC and mLQC-II in all regimes and mLQC-I in the post-bounce phase. Two of the parameters are fixed by enforcing equivalence at the bounce, and the background evolution of the relevant observables can be fitted with only one free parameter. It is seen that the non-perturbative effects of these loop cosmologies are universally encoded by a logarithmic correction that only depends on the bounce curvature of the model. In addition, we find that the best fit value of the free parameter can be very approximately written in terms of fundamental parameters of the underlying quantum description for the three models. The values of the best fits can be written in terms of the bounce density in a simple manner, and the values for each model are related to one another by a proportionality relation involving only the Barbero-Immirzi parameter.

Generalizations of the Q-prime curvature via renormalized characteristic forms

The Q-prime curvature is a local pseudo-Einstein invariant on CR manifolds defined by Case and Yang, and Hirachi. Its integral, the total Q-prime curvature, gives a non-trivial global CR invariant. On the other hand, Marugame has constructed a family of global CR invariants via renormalized characteristic forms, which contains the total Q-prime curvature. In this paper, we introduce a generalization of the Q-prime curvature for each renormalized characteristic form, and show that its integral coincides with Marugame's CR invariant. We also study generalizations of the critical CR GJMS operator and the P-prime operator, which are related to the transformation laws of our new curvatures under conformal change.

The Best Interests of The Child Self-Report (BIC-S): Psychometric Properties of the Adapted Version of the BIC-S used as a Monitoring Instrument to Measure the Quality of The Children’s Rearing Environment From a Children’s Rights Perspective

BackgroundIn line with the legal duty to monitor the compliance of policy and practice with the Convention on the Rights of the Child, the Netherlands’ Ombudsman for Children collects data concerning children’s views about their rearing environment and well-being. This Children’s Rights Monitor uses the Best Interests of the Child Self-Report (BIC-S). The psychometric properties of the BIC-S need to be further investigated.MethodFor the 2018 Children’s Rights Monitor, 1639 children (age: M = 12.05 SD = 2.70) completed the BIC-S (quality of rearing environment) and value their life on a scale of 1 to 10 (well-being). Mokken Scale Analysis was applied to determine the construct validity, and a Pearson correlation coefficient between well-being and the quality of rearing environment was used to determine the convergent validity of the BIC-S.ResultsThe results of the Mokken Scale Analysis reveal an invariant, strong, and reliable family scale (H = 0.60; Rho = 0.88) and an invariant, moderate, and reliable society scale (H = 0.45; Rho = 0.81). Two conditions (safe wider physical environment and adequate examples in society) should be viewed as separate items. Strong and significant correlations are observed between well-being, on the one hand, and the family and society scales on the other (respectively, r = 0.54 and 0.63).ImplicationsResults of this study point to a reliable and valid BIC-S for measuring the quality of the rearing environment. This instrument can be used to bring policy, practice, and decision-making in line with Children’s Rights.

On the Hamilton–Poisson structure and solitons for the Maxwell–Lorentz equations with spinning particle

We construct the Hamilton–Poisson structure for the Maxwell–Lorentz equations with spinning extended charged particle. We calculate the integral kernel of the corresponding Poisson bracket. The calculation relies on the Hamilton least action principle which is proved using the Lie–Poincaré calculus. The Hamilton–Poisson structure turns out to be degenerate and admits a functional family of Casimir invariants. The structure is applied to the calculation of the solitons corresponding to the uniform motion and rotation of the particle. We calculate effective masses and momenta of inertia of the soltions.As an example, the Poisson structure is applied to the construction of the momentum map corresponding to the symmetry group of translations. All calculations are rigorously justified by the proof of the well-posedness and smooth approximations of solutions. The investigation is inspired by the problems of stability of solitons.

Two Novel Classes of Arbitrary High-Order Structure-Preserving Algorithms for Canonical Hamiltonian Systems

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.

Non-stationary dynamical systems; Shadowing theorem and some applications

In the present paper, we mean a sequence of maps along a sequence of spaces by a non-stationary dynamical system. We use an Anosov family as a generalization of an Anosov map, which is a sequence of diffeomorphisms along a sequence of compact Riemannian manifolds, so that the tangent bundles split into expanding and contracting subspaces, with uniform bounds for the contraction and the expansion. Also, we introduce the shadowing property on non-stationary dynamical systems. Then, we prepare the necessary conditions for the existence of the shadowing property to prove the shadowing theorem in nonstationary dynamical systems. The shadowing theorem is a known result in dynamical systems, which states that any dynamical system with a hyperbolic structure has the shadowing property. Here, we prove that the shadowing theorem is established on any invariant Anosov family in a non-stationary dynamical system. Then, as in some applications of the shadowing theorem, we check the stability of Anosov families, and also we peruse the stability of isolated invariant Anosov families in non-stationary dynamical systems.

On minimal asymptotically nonexpansive mappings

&lt;abstract&gt;&lt;p&gt;In this paper we present the following two results: 1.- A characterization of the renorming invariant family of asymptotically nonexpansive mappings defined on a convex, closed and bounded set of a Banach space; 2.- A comparison of the renorming invariant family of asymptotically nonexpansive mappings with the renorming invariant family of nonexpansive mappings. Additionally, a series of examples are shown for general and particular cases.&lt;/p&gt;&lt;/abstract&gt;

Invariant family of leaf measures and the Ledrappier–Young property for hyperbolic equilibrium states

AbstractThe manifold M is a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$ , and f is a $C^{1+\beta }$ ( $\kern0.3pt\beta&gt;0$ ) diffeomorphism of M. $\varphi $ is a Hölder continuous potential on M. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by $\varphi $ ), which sit on local unstable leaves. We present two main applications. First, given an ergodic homoclinic class $H_\chi (p)$ , we prove that $\varphi $ admits a local equilibrium state on $H_\chi (p)$ if and only if $\varphi $ is ‘recurrent on $H_\chi (p)$ ’ (a condition tested by counting periodic points), and one of the leaf measures gives a positive measure to a set of positively recurrent hyperbolic points; and if an equilibrium measure exists, the said invariant and absolutely continuous family of measures constitute as its conditional measures. An immediate corollary is the local product structure of hyperbolic equilibrium states. Second, we prove a Ledrappier–Young property for hyperbolic equilibrium states: if $\varphi $ admits a conformal family of leaf measures and a hyperbolic local equilibrium state, then the leaf measures of the invariant family (respective to $\varphi $ ) are equivalent to the conformal measures (on a full measure set). This extends the celebrated result by Ledrappier and Young for hyperbolic Sinai–Ruelle–Bowen measures, which states that a hyperbolic equilibrium state of the geometric potential (with pressure zero) has conditional measures on local unstable leaves which are absolutely continuous with respect to the Riemannian volume of these leaves.

Annular Link Invariants from the Sarkar–Seed–Szabó Spectral Sequence

For a link in a thickened annulus A×I, we define a Z⊕Z⊕Z filtration on Sarkar–Seed–Szabó’s perturbation of the geometric spectral sequence. The filtered chain homotopy type is an invariant of the isotopy class of the annular link. From this, we define a two-dimensional family of annular link invariants and study their behavior under cobordisms. In the case of annular links obtained from braid closures, we obtain a necessary condition for braid quasi-positivity and a sufficient condition for right-veeringness, as well as Bennequin-type inequalities.

Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond

Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of a crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a $2\pi$-quantized change in the Berry-Zak phase between a pair of rotation-invariant lines in the bulk, three-dimensional Brillouin zone; because this change is reversed on the complementary section of the Brillouin zone, we refer to this new invariant as a returning Thouless pump (RTP). We find that the RTP is associated to anomalous values for the angular momentum of surface states, which guarantees metallic in-gap states for open boundary condition with sharply terminated hoppings; more generally for arbitrarily terminated hoppings, surface states are characterized by Berry-Zak phases that are quantized to a rational multiple of $2\pi$. The RTP adds to the family of topological invariants (the Hopf and Chern numbers) that are known to classify two-band Hamiltonians in Wigner-Dyson symmetry class A. Of these, the RTP and Hopf invariants are delicate, meaning that they can be trivialized by adding a particular trivial band to either the valence or the conduction subspace. Not all trivial band additions will nullify the RTP invariant, which allows its generalization beyond two-band Hamiltonians to arbitrarily many bands; such generalization is a hallmark of symmetry-protected delicate topology.

Persistence Steenrod modules

It has long been envisioned that the strength of the barcode invariant of filtered cellular complexes could be increased using cohomology operations. Leveraging recent advances in the computation of Steenrod squares, we introduce a new family of computable invariants on mod 2 persistent cohomology termed Sq^k-barcodes. We present a complete algorithmic pipeline for their computation and illustrate their real-world applicability using the space of conformations of the cyclo-octane molecule.

Canonical Forms for Reachable Systems over Von Neumann Regular Rings

If (A,B) is a reachable linear system over a commutative von Neumann regular ring R, a finite collection of idempotent elements is defined, constituting a complete set of invariants for the feedback equivalence. This collection allows us to construct explicitly a canonical form. Relations are given among this set of idempotents and various other families of feedback invariants. For systems of fixed sizes, the set of feedback equivalent classes of reachable systems is put into 1-1 correspondence with an appropriate partition of Spec(R) into open and closed sets. Furthermore, it is proved that a commutative ring R is von Neumann regular if and only if every reachable system over R is a finite direct sum of Brunovsky systems, for a suitable decomposition of R.

Inner geometry of complex surfaces: a valuative approach

Given a complex analytic germ $(X, 0)$ in $(\mathbb C^n, 0)$, the standard Hermitian metric of $\mathbb C^n$ induces a natural arc-length metric on $(X, 0)$, called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity $(X,0)$ by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of $(X,0)$. We deduce in particular that the global data consisting of the topology of $(X,0)$, together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of $(X,0)$, completely determine all the inner rates on $(X,0)$, and hence the local metric structure of the germ. Several other applications of our formula are discussed in the paper.

Non-semisimple 3-manifold invariants derived from the Kauffman bracket

We recover the family of non-semisimple quantum invariants of closed oriented 3-manifolds associated with the small quantum group of \mathfrak{sl_2} using purely combinatorial methods based on Temperley–Lieb algebras and Kauffman bracket polynomials. These invariants can be understood as a first-order extension of Witten–Reshetikhin–Turaev invariants, which can be reformulated following our approach in the case of rational homology spheres.

Quantitative Heegaard Floer cohomology and the Calabi invariant

Abstract We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of hameomorphisms constructed by Oh and Müller, and we construct an infinite-dimensional family of quasi-morphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on ‘classical’ Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms).