Open Subset Research Articles

A Unique Continuation Result for a 2D System of Nonlinear Equations for Surface Waves

In this paper, we establish a result of unique continuation for a special two-dimensional nonlinear system that models the evolution of long water waves with small amplitude in the presence of surface tension. More precisely, we will show that if (eta ,Phi ) = (eta (x,y, t),Phi (x,y, t)) is a solution of the nonlinear system, in a suitable function space, and (eta ,Phi ) vanishes on an open subset Omega of mathbb {R}^2 times [-T,T], then (eta ,Phi )equiv 0 in the horizontal component of Omega . To state such property, we use a Carleman-type estimate for a differential operator mathcal {L} related to the system. We prove the Carleman estimate using a particular version of the well known Treves’ inequality.

PFH spectral invariants on the two-sphere and the large scale geometry of Hofer’s metric

We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer’s metric. Namely: (1) we resolve the Kapovich–Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. We also find, as a corollary, that the group of area-preserving diffeomorphisms of the open disc, equipped with an area form of finite area, is not perfect. Central to all of our proofs are new sequences of spectral invariants over the two-sphere, defined via periodic Floer homology.

AN EXISTENCE AND UNIQUENESS OF THE SOLUTION OF SEMILINEAR MONOTONE ELLIPTIC EQUATION WITH THE DATA IN STUMMEL CLASSES

Let be a bounded open subset of , , be a function in Stummel classes , where , and
 be a semilinear monotone elliptic equation, where is symmetric matrix, elliptic, bounded, and is non decreasing and Lipschitz. By proving a weighted estimation for a function in Stummel class with its weight in , which allows us to use Stampacchia’s lemma, we obtained the existence and uniqueness of the solution of this equation.

Geodesics in the space of relatively Kähler metrics

AbstractWe derive the geodesic equation for relatively Kähler metrics on fibrations and prove that any two such metrics with fibrewise constant scalar curvature are joined by a unique smooth geodesic. We then show convexity of the log‐norm functional for this setting along geodesics, which yields simple proofs of Dervan and Sektnan's uniqueness result for optimal symplectic connections and a boundedness result for the log‐norm functional. Next, we associate to a fibration degeneration a unique geodesic ray defined on a dense open subset. Calculating the limiting slope of the log‐norm functional along a globally defined smooth geodesic ray, we prove that fibrations admitting optimal symplectic connections are polystable with respect to a large class of fibration degenerations that are smooth over the base. We give examples of such degenerations in the case of projectivised vector bundles and isotrivial fibrations.

Recursive Polish spaces

This paper is concerned with the proper way to effectivize the notion of a Polish space. A theorem is proved that shows the recursive Polish space structure is not found in the effectively open subsets of a space {mathcal {X}}, and we explore strong evidence that the effective structure is instead captured by the effectively open subsets of the product space mathbb {N}times {mathcal {X}}.

A remark on quantitative unique continuation from subsets of the boundary of positive measure

The question of uniqueness of harmonic functions in a domain Ω ⊂ R d \Omega \subset \mathbb {R}^d with boundary ∂ Ω \partial \Omega , satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset of the boundary ω \omega of positive d − 1 d-1 dimensional measure has attracted a lot of attention (see e.g. V. Adolfsson and L. Escauriaza [Comm. Pure Appl. Math. 50 (1997), pp. 935–969]; F.-H. Lin [Comm. Pure Appl. Math. 44 (1991), pp. 287–308]; X. Tolsa [Comm. Pure Appl. Math. 76 (2023), pp. 305–336]). It is essentially a consequence of Carleman estimates when ω \omega contains an open subset of the boundary (uniqueness for second order elliptic operators across an hypersurface). The main open questions (about uniqueness) concern now Lipschitz domains and variable coefficients. Here, using results by Logunov and Malinnikova [Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimensions two and three, Birkhäuser/Springer, Cham, 2018, pp. 333-344; Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018, World Scientific Publishing, Hackensack, NJ, 2018, pp. 2391–2411], we consider the simpler case of W 2 , ∞ W^{2, \infty } domains but prove quantitative uniqueness both for Dirichlet and Neumann boundary conditions. As an application, we deduce quantitative estimates for the Dirichlet and Neumann Laplace eigenfunctions on a W 2 , ∞ W^{2, \infty } domain with boundary.

Finite-Time Stabilization and Impulse Control of Heat Equation with Dynamic Boundary Conditions

In this paper, we study the impulse controllability of a multi-dimensional heat equation with dynamic boundary conditions in a bounded smooth domain. Using a recent approach based on finite-time stabilization, we show that the system is impulse null controllable at any positive time via impulse controls supported in a nonempty open subset of the physical domain. Furthermore, we infer an explicit estimate for the exponential decay of the solution. The proof of the main result combines a logarithmic convexity estimate and some spectral properties associated to dynamic boundary conditions. In our setting, the nature of the equations, which couple intern-boundary phenomena, makes it necessary to go into quite sophisticated estimates incorporating several boundary terms.

A Local Cauchy Integral Formula for Slice-Regular Functions

AbstractWe prove a local Cauchy-type integral formula for slice-regular functions. The formula is obtained as a corollary of a general integral representation formula where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two axially monogenic functions.

Stabilization and observer design for the wave equation

In this paper, we present some results on the observer’s design for the wave equation in a bounded, regular domain of Rn. We study the multidimensional case with measurements of state and velocity in an open subset. We obtain an exponential convergence of the error and we calculate the rate of decay, that is we stabilize the equation with a feedback law that acts in the interior of the domain.

On the dimension drop conjecture for diagonal flows on the space of lattices

Let X=G/Γ, where G is a Lie group and Γ is a lattice in G, let U be an open subset of X, and let {gt} be a one-parameter subgroup of G. Consider the set of points in X whose gt-orbit misses U; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when G is a simple Lie group of real rank 1. In this paper we prove this conjecture for the case G=SLm+n(R), Γ=SLm+n(Z) and gt=diag(ent,…,ent,e−mt,…,e−mt), in fact providing an effective estimate for the codimension. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on SLm+n(R)/SLm+n(Z). We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation.

Floer theory of disjointly supported Hamiltonians on symplectically aspherical manifolds

We study the Floer-theoretic interaction between disjointly supported Hamiltonians by comparing Floer-theoretic invariants of these Hamiltonians with the ones of their sum. These invariants include spectral invariants, boundary depth and Abbondandolo-Haug-Schlenk's action selector. Additionally, our method shows that in certain situations the spectral invariants of a Hamiltonian supported in an open subset of a symplectic manifold are independent of the ambient manifold.

Barr-exact categories and soft sheaf representations

It has long been known that a key ingredient for a sheaf representation of a universal algebra A consists in a distributive lattice of commuting congruences on A. The sheaf representations of universal algebras (over stably compact spaces) that arise in this manner have been recently characterised by Gehrke and van Gool (J. Pure Appl. Algebra, 2018), who identified the central role of the notion of softness.In this paper, we extend the scope of this theory by replacing varieties of algebras with Barr-exact categories, thus encompassing a number of “non-algebraic” examples. Our approach is based on the notion of K-sheaf: intuitively, whereas sheaves are defined on open subsets, K-sheaves are defined on compact ones. Throughout, we consider sheaves on complete lattices rather than spaces; this allows us to obtain point-free versions of sheaf representations whereby spaces are replaced with frames.These results are used to construct sheaf representations for the dual of the category of compact ordered spaces, and to recover Banaschewski and Vermeulen's point-free sheaf representation of commutative Gelfand rings (Quaest. Math., 2011).

Cohen strongly p-summing holomorphic mappings on Banach spaces

Let E and F be complex Banach spaces, U be an open subset of E and 1le ple infty . We introduce and study the notion of a Cohen strongly p-summing holomorphic mapping from U to F, a holomorphic version of a strongly p-summing linear operator. For such mappings, we establish both Pietsch Domination/Factorization Theorems and analyse their linearizations from (the canonical predual of ) and their transpositions on Concerning the space formed by such mappings and endowed with a natural norm we show that it is a regular Banach ideal of bounded holomorphic mappings generated by composition with the ideal of strongly p-summing linear operators. Moreover, we identify the space with the dual of the completion of tensor product space endowed with the Chevet–Saphar norm g_p.

Determining the Coefficients of the Thermoelastic System from Boundary Information

Given a compact Riemannian manifold (M,g) with smooth boundary ∂M, we give an explicit expression for the full symbol of the thermoelastic Dirichlet-to-Neumann map Λg with variable coefficients λ,μ,α,β∈C∞(M¯). We prove that Λg uniquely determines partial derivatives of all orders of these coefficients on the boundary ∂M. Moreover, for a nonempty smooth open subset Γ⊂∂M, suppose that the manifold and these coefficients are real analytic up to Γ. We show that Λg uniquely determines these coefficients on the whole manifold M¯.

Expected local topology of random complex submanifolds

Let n ≥ 2 n\geq 2 and r ∈ { 1 , ⋯ , n − 1 } r\in \{1, \cdots , n-1\} be integers, M M be a compact smooth Kähler manifold of complex dimension n n , E E be a holomorphic vector bundle with complex rank r r and equipped with a Hermitian metric h E h_E , and L L be an ample holomorphic line bundle over M M equipped with a metric h h with positive curvature form. For any d ∈ N d\in \mathbb N large enough, we equip the space of holomorphic sections H 0 ( M , E ⊗ L d ) H^0(M,E\otimes L^d) with the natural Gaussian measure associated to h E h_E , h h and its curvature form. Let U ⊂ M U\subset M be an open subset with smooth boundary. We prove that the average of the ( n − r ) (n-r) -th Betti number of the vanishing locus in U U of a random section s s of H 0 ( M , E ⊗ L d ) H^0(M,E\otimes L^d) is asymptotic to ( n − 1 r − 1 ) d n ∫ U c 1 ( L ) n {n-1 \choose r-1} d^n\int _U c_1(L)^n for large d d . On the other hand, the average of the other Betti numbers is o ( d n ) o(d^n) . The first asymptotic recovers the classical deterministic global algebraic computation. Moreover, such a discrepancy in the order of growth of these averages is new and contrasts with all known other smooth Gaussian models, in particular the real algebraic one. We prove a similar result for the affine complex Bargmann-Fock model.

Existence of solutions for a nonlocal elliptic system with critical and subcritical exponential growth

In this paper, the existence of nontrivial solutions to the following nonlocal elliptic system: is studied by variational method, where is a bounded open subset of with smooth boundary , and the nonlinearity and have critical or subcritical exponential growth in the sense of Trudinger–Moser inequality.

The enumerative geometry of cubic hypersurfaces: point and line conditions

The set of smooth cubic hypersurfaces in Pn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {P}}}^n$$\\end{document} is an open subset of a projective space. A compactification of the latter which allows to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points is termed a 1–complete variety of cubic hypersurfaces, in analogy with the space of complete quadrics. Imitating the work of Aluffi for plane cubic curves, we construct such a space in arbitrary dimensions by a sequence of five blow-ups. The counting problem is then reduced to the computation of five total Chern classes. In the end, we derive the desired numbers in the case of cubic surfaces.

Pseudo-Riemann’s quartics in Finsler’s geometry—two-dimensional case

Finsler’s geometry usually describes an extension of Riemmann’s geometry into a direction-dependent geometric structure. Historically, the well-known Riemann’s quartic length element example served as the inspiration for this construction. Surprisingly, the covariant Fresnel equation—a fundamental dispersion relation in solid-state electrodynamics—emerges as the exact same quartic expression. As a result, Riemann’s quartic length expression can be regarded of as a mathematical representation of a well-known physical phenomenon. In this study, we offer numerous Riemann’s quartic examples that show Finsler’s geometry, even in the situation of a positive definite Euclidean signature space, is too restrictive for many applications. The strong axioms of Finsler’s geometry are violated in a substantially greater number of distinctive subsets for the spaces having an indefinite (Minkowski) signature. We suggest a weaker characterization of Finsler’s structure based on explicitly calculated two-dimensional examples. In tangential vector space, this concept permits singular subsets. Only open subsets of a manifold’s tangent bundle are required to satisfy the strong axioms of Finsler’s geometry. We demonstrate the distinctive unique subsets of the Riemann’s quartic in two dimensions and briefly discuss their possible physical origins.

A non-Archimedean analogue of Campana's notion of specialness

Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let $\mathscr{X}$ be a $K$-analytic space (in the sense of Huber). In this work, we pursue a non-Archimedean characterization of Campana's notion of specialness. We say $\mathscr{X}$ is $K$-analytically special if there exists a connected, finite type algebraic group $G/K$, a dense open subset $\mathscr{U}\subset G^{\text{an}}$ with $\text{codim}(G^{\text{an}}\setminus \mathscr{U}) \geq 2$, and an analytic morphism $\mathscr{U} \to \mathscr{X}$ which is Zariski dense. With this definition, we prove several results which illustrate that this definition correctly captures Campana's notion of specialness in the non-Archimedean setting. These results inspire us to make non-Archimedean counterparts to conjectures of Campana. As preparation for our proofs, we prove auxiliary results concerning the indeterminacy locus of a meromorphic mapping between $K$-analytic spaces, the notion of pseudo-$K$-analytically Brody hyperbolic, and extensions of meromorphic maps from smooth, irreducible $K$-analytic spaces to the analytification of a semi-abelian variety.

F−open and F−closed sets in Topological Spaces

An open (resp., closed) subset A of a topological space (X, T ) is called F-open (resp., F-closed) set if cl(A)\A (resp., A\int(A)) is finite set. In this work, we study the main properties of these definitions and examine the relationships between F-open and F-closed sets with other kinds such as regularly open, regularly closed, closed, and open sets. Then, we establish some operators such as F-interior, F-closure, and F-derived...etc., using F-open and F-closed sets. At the end of this work, we introduce definitions of F-continuous function, F-compact space, and other related properties.