Open Subset Research Articles

Linearly S-closed spaces

We introduce the class of linearly S-closed spaces as a proper subclass of linearly H-closed spaces. This property lies between S-closedness and countable S-closedness. A space is called linearly S-closed if and only if any semi-open chain cover posses a member dense in the space. It is shown that in the class of extremally disconnected spaces the class of linearly H-closed spaces and linearly S-closed spaces coincide. We gave characterizations of these spaces in terms of s-accumulation points of chain filter bases and complete s-accumulation points of families of open subsets. While regular S-closed spaces are compact there is a non compact, regular, linearly S-closed space. It is shown that a Hausdorff, first countable, linearly S-closed space is extremally disconnected. Moreover, in the class of first countable, regular, compact spaces the notions of S-closedness, linearly S-closedness and extremally disconnectedness are equivalent. Some cardinality bounds for this class of spaces are obtained. Several examples are provided to illustrate our results.

Zariski Topologies on Graded Ideals

Abstract In this paper, we show there are strong relations between the algebraic properties of a graded commutative ring R and topological properties of open subsets of Zariski topology on the graded prime spectrum of R. We examine some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense, and irreducible. We also present a characterization for the radical of a graded ideal in R by using topological properties.

Study of some properties of complement of open subset inclusion graph of a topological space

In the recent paper, authors introduced a graph topological structure, called as open subset inclusion graph 𝚥(𝜏) of a topological space (𝑋, 𝜏) on a finite set 𝑋 and discussed some important properties of this graph. In this paper, we discuss some properties of the graph 𝚥(𝜏) 𝑐 . It is shown that, if 𝜏 is a discrete topology defined on nonempty set 𝑋 with |𝑋| ≤ 3, then the graph 𝚥(𝜏) 𝑐 is bipartite, and if |𝑋| = 2, then the graph 𝚥(𝜏) 𝑐 is regular & complete bipartite. Moreover, if 𝜏 is a discrete topology defined on nonempty set 𝑋 with |𝑋| = 2 or |𝑋| = 3 then it is shown that the graph 𝚥(𝜏) 𝑐 is Hamiltonian, vertex-transitive, edge-transitive and has a perfect matching. We also provide exact value of the independence number, vertex connectivity and edge connectivity of the graph 𝚥(𝜏) 𝑐 of a discrete topology defined on nonempty set𝑋 with |𝑋| = 2 or |𝑋| = 3. Main finding of this work is that, if (𝑋, 𝜏) is a discrete topological space with |𝑋| = 2 or |𝑋| = 3 then it is shown that 𝚥(𝜏) 𝑐 is distance-transitive graph and distance regular graph.

Existence of minimal hypersurfaces with non-empty free Boundary for generic metrics

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a compact manifold with boundary $(M^{n+1},\partial M)$, $3\leq (n + 1)\leq 7$, we prove that, for any open subset $V$ of $\partial M$, there exists a compact, properly embedded free boundary minimal hypersurface intersecting $V$.

Robust existence of nonhyperbolic ergodic measures with positive entropy and full support

We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic measures which are ergodic, fully supported and have positive entropy. To do so, we formulate abstract conditions sufficient for the construction of an ergodic, fully supported measure $\mu$ which has positive entropy and is such that for a continuous function $\phi\colon X\to\mathbb{R}$ the integral $\int\phi\,d\mu$ vanishes. The criterion is an extended version of the control at any scale with a long and sparse tail technique coming from the previous works.

Degenerate elliptic problem with a singular nonlinearity

ABSTRACT In this paper, we prove existence and regularity results for solutions of some nonlinear Dirichlet problems for an elliptic equation defined by a degenerate coercive operator and a singular right-hand side (1) where Ω is bounded open subset of and f is a non-negative function that belongs to some Lebesgue space.

Partial Regularity for Local Minimizers of Variational Integrals With Lower-Order Terms

Abstract We consider functionals of the form $$\begin{equation*} \mathcal{F}(u):=\int_\Omega\!F(x,u,\nabla u)\,\mathrm{d} x, \end{equation*}$$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded. The integrand $F\colon\Omega\times\mathbb{R}^N\times\mathbb{R}^{N\times n}\to\mathbb{R}$ is assumed to satisfy the classical assumptions of a power p-growth and the corresponding strong quasiconvexity. In addition, F is Hölder continuous with exponent $2\beta\in(0,1)$ in its first two variables uniformly with respect to the third variable and bounded below by a quasiconvex function depending only on $z\in\mathbb{R}^{N\times n}$. We establish that strong local minimizers of $\mathcal{F}$ are of class $\operatorname{C}^{1,\beta}$ in an open subset $\Omega_0\subseteq\Omega$ with $\mathcal{L}^n(\Omega\setminus\Omega_0)=0$. This partial regularity also holds for a certain class of weak local minimizers at which the second variation is strongly positive and satisfying a bounded mean oscillation (BMO) smallness condition. This extends the partial regularity result for local minimizers by Kristensen and Taheri (2003) to the case where the integrand depends also on u. Furthermore, we provide a direct strategy for this result, in contrast to the blow-up argument used for the case of homogeneous integrands.

Orlicz affine isoperimetric inequalities for star bodies

The original goal of this paper is to extend the affine isoperimetric inequality and Steiner type inequality of Orlicz projection bodies (which originated to Lutwak, Yang, and Zhang [32]), from convex bodies to Lipschitz star bodies (whose radial functions are locally Lipschitz).In order to achieve it, we investigate the graph functions of the given Lipschitz star body K: Along almost all directions u, we can define the graph functions on an open dense subset of the orthogonal projection of K onto u⊥.

A non-local inverse problem with boundary response

The problem of interest in this article is to study the (non-local) inverse problem of recovering a potential based on the boundary measurement associated with the fractional Schrödinger equation. Let 0<a<1 , and let u solve \begin{cases} ((-\Delta)^a + q )u = 0 &\operatorname{in} \Omega,\\ \operatorname{supp}\, u\subseteq \overline{\Omega}\cup \overline{W} ,\\ \overline{W} \cap \overline{\Omega}=\emptyset. \end{cases} We show that, by making an exterior to boundary measurement as \big(u|_{W}, \frac{u(x)}{d(x)^a}\big|_{\Sigma}\big) , it is possible to determine q uniquely in \Omega , where \Sigma\subseteq\partial\Omega is a non-empty open subset and d(x)=d(x,\partial\Omega) denotes the boundary distance function. We also discuss a local characterization of large a -harmonic functions in a ball or in the half space and its applications, which include boundary unique continuation and a local density result.

Delay Differential Equations with Differentiable Solution Operators on Open Domains in C((-∞, 0], Rn) and Processes for Volterra Integro-Differential Equations

For autonomous delay differential equations x'(t)=f(xt){x'(t)=f(x_t)} we construct a continuous semiflow of continuously differentiable solution operators x0xt{x_0 \to x_t}, t0{t \le 0}, on open subsets of the Frechet space C((-,0],Rn){C((-\infty, 0], R^n)}. For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application, we obtain processes which incorporate all solutions of Volterra integro-differential equations x'(t)=∫0tk(t,s)h(x(s))ds{x'(t)={\int_0}^t k(t,s) h(x(s)) ds}.

The Minimum Principle for Convex Subequations

A subequation, in the sense of Harvey–Lawson, on an open subset \(X\subset \mathbb R^n\) is a subset F of the space of 2-jets on X with certain properties. A smooth function is said to be F-subharmonic if all of its 2-jets lie in F, and using the viscosity technique one can extend the notion of F-subharmonicity to any upper-semicontinuous function. Let \(\mathcal P\) denote the subequation consisting of those 2-jets whose Hessian part is semipositive. We introduce a notion of product subequation \(F\#\mathcal P\) on \(X\times \mathbb R^{m}\) and prove, under suitable hypotheses, that if F is convex and f(x, y) is \(F\#\mathcal P\)-subharmonic then the marginal function $$\begin{aligned} g(x):= \inf _y f(x,y) \end{aligned}$$is F-subharmonic. This generalises the classical statement that the marginal function of a convex function is again convex. We also prove a complex version of this result that generalises the Kiselman minimum principle for the marginal function of a plurisubharmonic function.

Carleman estimates and unique continuation property for N-dimensional Benjamin–Bona–Mahony equations

We study the unique continuation property for the N-dimensional BBM equations using Carleman estimates. We prove that if the solution of this equation vanishes in an open subset, then this solution is identically equal to zero in the horizontal component of the open subset.

Maximally singular weak solutions of Laplace equations

It is known that there exists an explicit function F in L2(Ω), where Ω is a given bounded open subset of ℝN, such that the corresponding weak solution of the Poisson boundary value problem −Δu=F(x), u∈H01(Ω), is maximally singular; that is, the singular set of u has Hausdorff dimension equal to (N−4)+. This constant is optimal, i.e., the largest possible. Here, we show that much more is true: when N≥5, there exists F∈L2(Ω) such that the corresponding weak solution has the pointwise concentration of singular set of u, in the sense of the Hausdorff dimension, equal to N−4 at all points of Ω. We also consider the problem of generating weak solutions with the property of contrast; that is, we construct solutions u that are regular (more specifically, of class Cloc2,α for arbitrary α∈(0,1)) in any prescribed open subset Ωr of Ω, while they are maximally singular in its complement Ω∖Ωr. We indicate several open problems.

Approximation and Gâteaux differentiability of convex function in Banach spaces

AbstractThis paper shows that if f is a convex continuous function on a Gâteaux differentiability space X, then for any separable closed subspace E of X, there exists a sequence of continuous convex functions such that (1) on X; (2) is Gâteaux differentiable at all points of a dense open subset of X; (3) on E. Moreover, if X is separable, then there exists a continuous convex function sequence such that (1) and (2) are true and on X.

Orbit spaces for torus actions on Hessenberg varieties

In this paper we study effective actions of the compact torus on smooth compact manifolds of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to , the complement to the union of disjoint open subsets of the -sphere. The results obtained are applied to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices of step type. Bibliography: 23 titles.

Feynman–Kac Formulas for Dirichlet–Pauli–Fierz Operators with Singular Coefficients

We derive Feynman-Kac formulas for Dirichlet realizations of Pauli-Fierz operators generating the dynamics of nonrelativistic quantum mechanical matter particles, which are minimally coupled to both classical and quantized radiation fields and confined to an arbitrary open subset of the Euclidean space. Thanks to a suitable interpretation of the involved Stratonovich integrals, we are able to retain familiar formulas for the Feynman-Kac integrands merely assuming local square-integrability of the classical vector potential and the coupling function in the quantized vector potential. Allowing for fairly general coupling functions becomes relevant when the matter-radiation system is confined to cavities with inward pointing boundary singularities.

Open discrete shrinkings in function spaces

Given a sequence U={Un:n∈ω} of non-empty open subsets of a space X, a family {Vn:n∈ω} is a shrinking of U if Vn is a non-empty open set and Vn⊂Un for every n∈ω. A space X has discrete shrinking property if every sequence of non-empty open subsets of X has a discrete shrinking. We show that if L is a non-metrizable locally convex topological vector space and Lw is the set L with the weak topology of the space L, then Lw has the discrete shrinking property. In particular, a space X is uncountable if and only if Cp(X) has the discrete shrinking property. The same statement is not true for Cp(X,[0,1]) so we study when the discrete shrinking property holds in Cp(X,[0,1]). We prove, among other things, that Cp(X,[0,1]) features discrete shrinking property if X is an essentially uncountable space.

Regulous Functions over Real Closed Fields

Abstract Let $X$ be a quasi-projective algebraic variety over a real closed field $R$, and let $f \colon U \to R$ be a function defined on an open subset $U$ of the set $X(R)$ of $R$-rational points of $X$. Assume that either the function $f$ is locally semialgebraic or the field $R$ is uncountable. If for every irreducible algebraic curve $C \subset X$ the restriction $f|_{U \cap C}$ is continuous and admits a rational representation, then $f$ is continuous and admits a rational representation. There are also suitable versions of this theorem with algebraic curves replaced by algebraic arcs. Heretofore, results of such a type have been known only for $R={\mathbb{R}}$. The transition from ${\mathbb{R}}$ to $R$ is not automatic at all and requires new methods.

On the uniqueness of inverse problems for the reduced wave equation with unknown embedded obstacles

In this paper, we consider inverse problems associated with the reduced wave equation on a bounded domain for the case where unknown obstacles are embedded in the domain Ω. We show that, if both the leading and 0-order coefficients in the equation are a priori known to be piecewise constant functions, then both the coefficients and embedded obstacles can be simultaneously recovered in terms of the local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the boundary ∂Ω. The method depends on a well-defined coupled PDE-system constructed for the reduced wave equations in a sufficiently small domain and the singularity analysis of solutions near the interface for the model.

Open Subsets in a Stein Space with Singularities

Serre proved that a domain $Y$ in $\mathbb{C}^n$ is Stein if and only if $H^i(Y,\mathcal{O}_Y) = 0$ for all $i \gt 0$. Laufer showed that if $Y$ is an open subset of a Stein manifold of dimension $n$ and $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$, then $Y$ is Stein. Vâjâitu generalized these theorems to singular Stein space of dimension $n$. In this paper, we consider singular Stein spaces $X$ with arbitrary dimension and give necessary and sufficient conditions for an open subset $Y$ in $X$ to be Stein. We show that if $Y$ is an open subset of a reduced Stein space $X$ with arbitrary dimension and singularities, then $Y$ is Stein if and only if $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$. Without cohomology condition, if $X-Y$ is a closed subspace of $X$, then we show that the geometric condition of the boundary $X-Y$ determines the Steinness of $Y$. More precisely, we show that if $X$ is normal and the boundary $X-Y$ is the support of an effective $\mathbb{Q}$-Cartier divisor, or $X-Y$ is of pure codimension $1$ and does not contain any singular points of $X$, then $Y$ is Stein.