Open Dense Set Research Articles

Density of the boundary regular set of 2d area minimizing currents with arbitrary codimension and multiplicity

In the present work, we consider area minimizing currents in the general setting of arbitrary codimension and arbitrary boundary multiplicity. We study the boundary regularity of 2d area minimizing currents, beyond that, several results are stated in the more general context of (C0,α0,r0)-almost area minimizing currents of arbitrary dimension m and arbitrary codimension taking the boundary with arbitrary multiplicity. Furthermore, we do not consider any type of convex barrier assumption on the boundary, in our main regularity result which states that the regular set, which includes one-sided and two-sided points, of any 2d area minimizing current T is an open dense set in the boundary.

Staircase patterns in Hirzebruch surfaces

The ellipsoidal capacity function of a symplectic four manifold X measures how much the form on X must be dilated in order for it to admit an embedded ellipsoid of eccentricity z . In most cases there are just finitely many obstructions to such an embedding besides the volume. If there are infinitely many obstructions, X is said to have a staircase. This paper gives an almost complete description of the staircases in the ellipsoidal capacity functions of the family of symplectic Hirzebruch surfaces H_{b} formed by blowing up the projective plane with weight b . We describe an interweaving, recursively defined, family of obstructions to symplectic embeddings of ellipsoids that show there is an open dense set of shape parameters b that are blocked, i.e. have no staircase, and an uncountable number of other values of b that do admit staircases. The remaining b -values form a countable sequence of special rational numbers that are closely related to the symmetries discussed in Magill–McDuff (arXiv:2106.09143). We show that none of them admit ascending staircases. Conjecturally, none admit descending staircases. Finally, we show that, as long as b is not one of these special rational values, any staircase in H_{b} has irrational accumulation point. A crucial ingredient of our proofs is the new, more indirect approach to using almost toric fibrations in the analysis of staircases by Magill (arXiv:2204.12460). In particular, the structure of the relevant mutations of the set of almost toric fibrations on H_{b} is echoed in the structure of the set of blocked b -intervals.

Congruence Representations via Soft Ideals in Soft Topological Spaces

This article starts with a study of the congruence of soft sets modulo soft ideals. Different types of soft ideals in soft topological spaces are used to introduce new weak classes of soft open sets. Namely, soft open sets modulo soft nowhere dense sets and soft open sets modulo soft sets of the first category. The basic properties and representations of these classes are established. The class of soft open sets modulo the soft nowhere dense sets forms a soft algebra. Elements in this soft algebra are primarily the soft sets whose soft boundaries are soft nowhere dense sets. The class of soft open sets modulo soft sets of the first category, known as soft sets of the Baire property, is a soft σ-algebra. In this work, we mainly focus on the soft σ-algebra of soft sets with the Baire property. We show that soft sets with the Baire property can be represented in terms of various natural classes of soft sets in soft topological spaces. In addition, we see that the soft σ-algebra of soft sets with the Baire property includes the soft Borel σ-algebra. We further show that soft sets with the Baire property in a certain soft topology are equal to soft Borel sets in the cluster soft topology formed by the original one.

Open loci results for commutative DG-rings

Given a commutative noetherian non-positive DG-ring with bounded cohomology which has a dualizing DG-module, we study its regular, Gorenstein and Cohen-Macaulay loci. We give a sufficient condition for the regular locus to be open, and show that the Gorenstein locus is always open. However, both of these loci are often empty: we show that no matter how nice $\mathrm{H}^0(A)$ is, there are examples where the Gorenstein locus of $A$ is empty. We then show that the Cohen-Macaulay locus of a commutative noetherian DG-ring with bounded cohomology which has a dualizing DG-module always contains a dense open set. Our results imply that under mild hypothesis, eventually coconnective locally noetherian derived schemes are generically Cohen-Macaulay, but that even in very nice cases, they need not be generically Gorenstein.

High degeneracy of effective Hamiltonian in two dimensions

Consider the effective Hamiltonian \overline{H}(p) associated with the mechanical Hamiltonian H(p,x)=\frac12|p|^2+V(x) . We prove that for generic V , \overline{H} is piecewise one-dimensional in a dense open set in two dimensions using Aubry–Mather theory.

On the Equations Defining Some Hilbert Schemes

We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme \(\text {Hilb}^{2} \mathbb {P}^{2}\) of two points on \(\mathbb {P}^{2}\) and the dense open set parametrizing non-planar clusters in the punctual Hilbert scheme \(\text {Hilb}^{4}_{0}(\mathbb {A}^{3})\) of clusters of length four on \(\mathbb {A}^{3}\) with support at the origin. We find explicit equations in projective, respectively affine, embeddings for these spaces. In particular, we answer a question of Bernd Sturmfels who asked for a description of the latter space that is amenable to further computations. While the explicit equations we find are controlled in a precise way by the representation theory of SL3, our arguments also rely on computer algebra.

Cubic m-polar fuzzy topology with multi-criteria group decision-making

<abstract><p>The concept of cubic m-polar fuzzy set (CmPFS) is a new approach to fuzzy modeling with multiple membership grades in terms of fuzzy intervals as well as multiple fuzzy numbers. We define some fundamental properties and operations of CmPFSs. We define the topological structure of CmPFSs and the idea of cubic m-polar fuzzy topology (CmPF topology) with P-order (R-order). We extend several concepts of crisp topology to CmPF topology, such as open sets, closed sets, subspaces and dense sets, as well as the interior, exterior, frontier, neighborhood, and basis of CmPF topology with P-order (R-order). A CmPF topology is a robust approach for modeling big data, data analysis, diagnosis, etc. An extension of the VIKOR method for multi-criteria group decision making with CmPF topology is designed. An application of the proposed method is presented for chronic kidney disease diagnosis and a comparative analysis of the proposed approach and existing approaches is also given.</p></abstract>

Stability and collapse of the Lyapunov spectrum for Perron–Frobenius operator cocycles

In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron–Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron–Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That is, under perturbations, the Lyapunov exponents become 0 with multiplicity 1, and $-\\infty$ with infinite multiplicity. This is superficially similar to the finite-dimensional phenomenon, discovered by Bochi \[4], that away from the uniformly hyperbolic setting, small perturbations can lead to a collapse of the Lyapunov spectrum to zero. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different. We study stability of the Perron–Frobenius cocycles arising from general random Blaschke products. We give a necessary and sufficient criterion for stability of the Lyapunov spectrum in terms of the derivative of the random Blaschke product at its random fixed point, and use this to show that an open dense set of Blaschke product cocycles have hyperbolic Perron–Frobenius cocycles. In the final part, we prove a relationship between the Lyapunov spectrum of a single cocycle acting on two different Banach spaces, allowing us to draw conclusions for the same cocycles acting on $C^r$ function spaces.

Global Łojasiewicz inequalities on comparing the rate of growth of polynomial functions

We present a global version of the Łojasiewicz inequality on comparing the rate of growth of two polynomial functions in the case the mapping defined by these functions is (Newton) non-degenerate at infinity. In addition, we show that the condition of non-degeneracy at infinity is generic in the sense that it holds in an open dense semi-algebraic set of the entire space of input data.

On the central geometry of nonnoetherian dimer algebras

Let Z be the center of a nonnoetherian dimer algebra on a torus. Although Z itself is also nonnoetherian, we show that it has Krull dimension 3, and is locally noetherian on an open dense set of Max Z . Furthermore, we show that the reduced center Z / nil Z is depicted by a Gorenstein singularity, and contains precisely one closed point of positive geometric dimension.

On the density of \u201cwild\u201d initial data for the compressible Euler system

We consider a class of “wild” initial data to the compressible Euler system that give rise to infinitely many admissible weak solutions via the method of convex integration. We identify the closure of this class in the natural L^1-topology and show that its complement is rather large, specifically it is an open dense set.

Toric geometry of G2–manifolds

We consider $G_2$-manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of $T^3$-actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three- and four-form, we derive a Gibbons-Hawking type ansatz giving the geometry on an open dense set in terms a symmetric $3\times 3$-matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to $G_2$. We prove that the multi-moment maps exhibit the full orbit space topologically as a smooth four-manifold containing a trivalent graph as the image of the set of special orbits and describe these graphs in some complete examples.

Toric Geometry of Spin(7)-Manifolds

Abstract We study $ \operatorname{Spin}(7) $-manifolds with an effective multi-Hamiltonian action of a four-torus. On an open dense set, we provide a Gibbons–Hawking type ansatz that describes such geometries in terms of a symmetric $ 4\times 4 $-matrix of functions. This description leads to the 1st known $ \operatorname{Spin}(7) $-manifolds with a rank $ 4 $ symmetry group and full holonomy. We also show that the multi-moment map exhibits the full orbit space topologically as a smooth four-manifold, containing a trivalent graph in $ \mathbb{R}^4 $ as the image of the set of the special orbits.

The persistence of impulse controllability

This paper shows that within the space of all LTI systems, equipped with the Zariski topology, the set of impulse controllable systems contains an open dense set of systems; in other words, impulse controllable systems are generic. This genericity persists for many closed subsets of LTI systems of interest, such as the class of singular descriptor systems.

On the Generic Structure and Stability of Stackelberg Equilibria

We consider a noncooperative Stackelberg game, where the two players choose their strategies within domains $$X\subseteq {{\mathbb {R}}}^m$$ and $$Y\subseteq {{\mathbb {R}}}^n$$ . Assuming that the cost functions F, G for the two players are sufficiently smooth, we study the structure of the best reply map for the follower and the optimal strategy for the leader. Two main cases are considered: either $$X=Y=[0,1]$$ , or $$X={{\mathbb {R}}}, Y={{\mathbb {R}}}^n$$ with $$n\ge 1$$ . Using techniques from differential geometry, including a multi-jet version of Thom’s transversality theorem, we prove that, for an open dense set of cost functions $$F\in {{\mathcal {C}}}^2$$ and $$G\in {{\mathcal {C}}}^3$$ , the Stackelberg equilibrium is unique and is stable w.r.t. small perturbations of the two cost functions.

C-projective compactification; (quasi-)Kähler metrics and CR boundaries

For complete complex connections on almost complex manifolds we introduce a natural definition of compactification. This is based on almost c-projective geometry, which is the almost complex analogue of projective differential geometry. The boundary at infinity is a (possibly non-integrable) CR structure. The theory applies to almost Hermitian manifolds which admit a complex metric connection of minimal torsion, which means that they are quasi-K\"ahler in the sense of Gray-Hervella; in particular it applies to K\"ahler and nearly K\"ahler manifolds. Via this canonical connection, we obtain a notion of c-projective compactification for quasi-K\"ahler metrics of any signature. We describe an asymptotic form for metrics that is necessary and sufficient for c-projective compactness. This metric form provides local examples and, in particular, shows that the usual complete K\"ahler metrics associated to smoothly bounded, strictly pseudoconvex domains in ${\Bbb C}^n$ are c-projectively compact. For a smooth manifold with boundary and a complete quasi-K\"ahler metric $g$ on the interior, we show that if its almost c-projective structure extends smoothly to the boundary then so does its scalar curvature. We prove that in this case (and under some natural assumptions on the extension of $J$ to the boundary) $g$ is almost c-projectively compact if and only if this scalar curvature is non-zero on an open dense set of the boundary. In that case it is, along the boundary, locally constant and hence nowhere zero there. Finally we describe the asymptotics of the curvature, showing, in particular, that the canonical connection satisfies an asymptotic Einstein condition. Key to much of the development is a certain real tractor calculus for almost c-projective geometry, and this is developed in the article.

Existence and genericity of finite topological generating sets for homeomorphism groups

We show that the topological groups $Diff_{+}^{1}(I)$ and $Diff_{+}^{1}(\mathbb{S}^1)$ of orientation-preserving $C^1$-diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite topological generating sets in related groups. We show that the generic pair of elements in the homeomorphism group $Homeo_+(I)$ generate a dense subgroup of $Homeo_+(I)$. By contrast, if $M$ is any compact connected manifold with boundary other than the interval, we observe that an open dense set of pairs from the associated boundary-fixing homeomorphism group $Homeo(M,\partial M)$ will generate a discrete subgroup. We make similar observations for homeomorphism groups of manifolds without boundary including $\mathbb{S}^1$.

Weierstrass weight of the hyperosculating points of generalized Fermat curves

Let (S,H) be a generalized Fermat pair of the type (k,n). If F⊂S is the set of fixed points of the non-trivial elements of the group H, then F is exactly the set of hyperosculating points of the standard embedding S↪Pn. We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.

Matching for generalised [formula omitted]-transformations

We investigate matching for the family Tα(x)=βx+α(mod1), α∈[0,1], for fixed β>1. Matching refers to the property that there is an n∈N such that Tαn(0)=Tαn(1). We show that for various Pisot numbers β, matching occurs on an open dense set of α∈[0,1] and we compute the Hausdorff dimension of its complement. Numerical evidence shows more cases where matching is prevalent.

Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps

The notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say about the Hausdorff and packing dimension of the fibers of prevalent continuous maps?Let K be an uncountable compact metric space. We prove that a prevalent f∈C(K,Rd) has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that a prevalent f∈C(K,Rd) has graph of maximal Hausdorff dimension, generalizing a result of Bayart and Heurteaux. We obtain similar results for the packing dimension.We show that for a prevalent f∈C([0,1]m,Rd) the set of y∈f([0,1]m) for which dimH⁡f−1(y)=m contains a dense open set having full measure with respect to the occupation measure λm∘f−1, where dimH and λm denote the Hausdorff dimension and the m-dimensional Lebesgue measure, respectively. We also prove an analogous result when [0,1]m is replaced by any self-similar set satisfying the open set condition.We cannot replace the occupation measure with Lebesgue measure in the above statement: We show that the functions f∈C[0,1] for which positively many level sets are singletons form a non-shy set in C[0,1]. In order to do so, we generalize a theorem of Antunović, Burdzy, Peres and Ruscher. As a complementary result we prove that the functions f∈C[0,1] for which dimH⁡f−1(y)=1 for all y∈(min⁡f,max⁡f) form a non-shy set in C[0,1].We also prove sharper results in which large Hausdorff dimension is replaced by positive measure with respect to generalized Hausdorff measures, which answers a problem of Fraser and Hyde.