Empty Boundary Research Articles

Sensitivity Matrix Update Method for Electrical Resistance Tomography Based on Error-Constrained Cross Fusion Residual Attention Network

Abstract Electrical resistance tomography (ERT) is an imaging technique of conductivity distribution. The image reconstruction algorithm based on the empty field sensitivity matrix is widely used because it provides an approximate solution to the complex ERT inverse problems. However, due to the soft field effect, the sensitivity matrix changes with the media distribution, leading to significant errors in image reconstruction. To address this issue, an error-constrained deep learning scheme for bubble flow, namely cross fusion residual attention network (CFRA-Net) is designed to update the sensitivity matrix during gas-liquid-solid fluidized bed operation. CFRA-Net employs a multi-head self-attention mechanism to enhance bubble features in boundary measurements, a multilevel cross fusion module to fuse empty field strength and boundary measurement information, and a novel serial-channel residual spatial attention block to boost the feature extraction capability of the model. A weighted loss function is designed to give more weight to the gas phase distribution region. Additionally, this paper establishes a dataset for gas-liquid-solid three-phase flow, considering background field conductivity variations. Validation and reliability of the proposed method are assessed through ablation, comparison, and noise experiments. The experimental results demonstrate that CFRA-Net achieves rapid updates of the sensitivity matrix, high imaging accuracy, and robust noise immunity.

A central limit theorem for a classical gas

For a class of translation-invariant pair potentials ϕ in (mathbb{R}^{d},zlambda ) satisfying a stability and regularity condition, we choose z so small that the associated collection mathcal{ G}(phi,zlambda ) of Gibbs processes contains at least the stationary process G, which is a Gibbs process in the sense of DLR and is given by the limiting Gibbs process with empty boundary conditions. Using an abstract version of the method of cluster expansions and Dobrushin’s approach to the central limit theorem, we present a central limit theorem for the particle numbers of G.

Sutured manifolds and ℓ2-Betti numbers

Abstract Using the virtual fibering theorem of Agol, we show that a sutured 3-manifold $(M, R_{+},R_{-},\gamma)$ is taut if and only if the $\ell^{2}$-Betti numbers of the pair $(M,R_{-})$ are zero. As an application, we can characterize Thurston norm minimizing surfaces in a 3-manifold N with empty or toroidal boundary by the vanishing of certain $\ell^{2}$-Betti numbers.

Relative Reshetikhin–Turaev Invariants, Hyperbolic Cone Metrics and Discrete Fourier Transforms I

We propose the Volume Conjecture for the relative Reshetikhin–Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern–Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient 3-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston’s result that all compact oriented 3-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen–Yang’s Volume Conjecture for the Reshetikhin–Turaev invariants of closed oriented hyperbolic 3-manifolds. We also introduce a family of topological operations (the change-of-pair operations) that connect all pairs of a closed oriented 3-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin–Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.

Hilbert complexes with mixed boundary conditions—Part 2: Elasticity complex

We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are also proved. This paper extends recent results on the de Rham Hilbert complex with mixed boundary conditions from Pauly and Schomburg (2021, 2022) and recent results on the elasticity Hilbert complex with empty or full boundary conditions from Pauly and Zulehner (2020, 2022).

PRESCRIBED VIRTUAL HOMOLOGICAL TORSION OF 3-MANIFOLDS

AbstractLet M be an irreducible $3$ -manifold M with empty or toroidal boundary which has at least one hyperbolic piece in its geometric decomposition, and let A be a finite abelian group. Generalizing work of Sun [20] and of Friedl–Herrmann [7], we prove that there exists a finite cover $M' \to M$ so that A is a direct factor in $H_1(M',{\mathbb Z})$ .

Growth of quantum $6j$-symbols and applications to the volume conjecture

We prove the Turaev–Viro invariants volume conjecture for a “universal” class of cusped hyperbolic $3$-manifolds that produces all $3$-manifolds with empty or toroidal boundary by Dehn filling. This leads to two-sided bounds on the volume of any hyperbolic $3$-manifold with empty or toroidal boundary in terms of the growth rate of the Turaev–Viro invariants of the complement of an appropriate link contained in the manifold. We also provide evidence for a conjecture of Andersen, Masbaum and Ueno (AMU conjecture) about certain quantum representations of surface mapping class groups. A key step in our proofs is finding a sharp upper bound on the growth rate of the quantum $6j$-symbol evaluated at $q=e\frac{2\pi i}{r}$.

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).

Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

We consider the Klein-Gordon operator on an n-dimensional asymptotically anti-de Sitter spacetime (M,g) together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on ∂M of order up to 2. Using techniques from b-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.

Existence and structure of symmetric Beltrami flows on compact 3-manifolds

We show that for almost every given symmetry transformation of a Riemannian manifold there exists an eigenvector field of the curl operator, corresponding to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a smooth, compact, oriented Riemannian 3-manifold (M¯,g) with (possibly empty) boundary and a smooth flow of isometries ϕt:M¯→M¯ we show that, if M¯ has non-empty boundary or if the infinitesimal generator is not purely harmonic, there is a smooth vector field X, tangent to the boundary, which is an eigenfield of curl and satisfies (ϕt)⁎X=X, i.e. is invariant under the pushforward of the symmetry transformation. We then proceed to show that if the quantities involved are real analytic and (M¯,g) has non-empty boundary, then Arnold's structure theorem applies to all eigenfields of curl, which obey a symmetry and appropriate boundary conditions. More generally we show that the structure theorem applies to all real analytic vector fields of non-vanishing helicity which obey some nontrivial symmetry. A byproduct of our proof is a characterisation of the flows of real analytic Killing fields on compact, connected, orientable 3-manifolds with and without boundary.

Zero Set Structure of Real Analytic Beltrami Fields

In this paper, we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected 3-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected 1-dimensional submanifolds with (possibly empty) boundary and tame knots. Further, we consider the question of how complicated these tame knots can possibly be. To this end, we prove that on the standard (open) solid toroidal annulus in {mathbb {R}}^3, there exist for any pair (p, q) of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric, there exists a real analytic Beltrami field, corresponding to the eigenvalue +1 of the curl operator, whose zero set is precisely given by a standard (p, q)-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.

On distinct finite covers of 3-manifolds

Every closed orientable surface S has the following property: any two connected covers of S of the same degree are homeomorphic (as spaces). In this, paper we give a complete classification of compact 3-manifolds with empty or toroidal boundary which have the above property. We also discuss related group-theoretic questions.

On Maxwell’s Equations on Globally Hyperbolic Spacetimes with Timelike Boundary

We study Maxwell's equation as a theory for smooth $k$-forms on globally hyperbolic spacetimes with timelike boundary as defined by Ak\'e, Flores and Sanchez. In particular we start by investigating on these backgrounds the D'Alembert - de Rham wave operator $\Box_k$ and we highlight the boundary conditions which yield a Green's formula for $\Box_k$. Subsequently, we characterize the space of solutions of the associated initial and boundary value problem under the assumption that advanced and retarded Green operators do exist. This hypothesis is proven to be verified by a large class of boundary conditions using the method of boundary triples and under the additional assumption that the underlying spacetime is ultrastatic. Subsequently we focus on the Maxwell operator. First we construct the boundary conditions which entail a Green's formula for such operator and then we highlight two distinguished cases, dubbed $\delta\mathrm{d}$-tangential and $\delta\mathrm{d}$-normal boundary conditions. Associated to these we introduce two different notions of gauge equivalence and we prove that in both cases, every equivalence class admits a representative abiding to the Lorenz gauge. We use this property and the analysis of the operator $\Box_k$ to construct and to classify the space of gauge equivalence classes of solutions of the Maxwell's equations with the prescribed boundary conditions. As a last step and in the spirit of future applications in the framework of algebraic quantum field theory, we construct the associated unital $*$-algebras of observables proving in particular that, as in the case of the Maxwell operator on globally hyperbolic spacetimes with empty boundary, they possess a non-trivial center.

Compactn-Manifolds via (n+ 1)-Colored Graphs: a New Approach

We introduce a representation via (n+1)-colored graphs of compact n-manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is a generalization to arbitrary dimension of the one recently given by Cristofori and Mulazzani in dimension three, and it is dual to the one given by Pezzana in the 1970s. In this context we establish some results concerning the topology of the represented manifolds: suspensions, fundamental groups, connected sums and moves between graphs representing the same manifold. Classification results of compact orientable 4-manifolds representable by graphs up to six vertices are obtained, together with some properties of the G-degree of 5-colored graphs relating this approach to tensor models theory.

Liquid Drops, Soap Bubbles and Surfaces with Constant Mean Curvature

This is a review of the theory of compact surfaces with constant mean curvature with special attention on surfaces with non empty boundary. We analyze the relation between the geometry of the boundary curve with the one of the surface that spans. This will allow to give a mathematical support to describe the geometry of liquid drops and soap bubbles.

Reciprocity as an Ever-Present Dual Property of Everything

Reciprocity may be understood as relation of action and reaction in the sense of Hegel’s philosophical definition. Quoting Kant, freedom and ethical necessities are reciprocally limited. In this contribution, a more mathematical than philosophical reflection about reciprocity as an ever-present dual property of everything was given. As a crystallographer, the author is familiar with the action of Fourier transforms and the relation between a crystal lattice and its reciprocal lattice, already pointing to the duality between particles and waves. A generalization of the reciprocity term was stimulated by results of the famous Information Relativity (IR) theory of Suleiman with its proven physical manifestation of matter-wave duality, compared to the set-theoretical E-Infinity theory developed by El Naschie, where the zero set represents the pre-quantum particle, and the pre-quantum wave is assigned to the empty set boundary surrounding the pre-particle. Expectedly, the most irrational number of the golden mean is involved in these thoughts, because this number is intimately connected with its inverse. An important role plays further Hardy’s maximum quantum entanglement probability as the fifth power of φ and its connection to the dark matter. Remembering, the eleven dimensions in Witten’s M-theory may be decomposed into the Lucas number L5 = 11 = φ−5 – φ5. Reciprocity is indeed omnipresent in our world as piloting waves that accompany all observable earthen and cosmic matter. As a side effect of the IR theory some fundamental constants such as the gyromagnetic factor of the electron, Sommerfeld’s fine-structure constant as well as the charge of the electron must be marginally changed caused by altered relativistic corrections. Consequences also arise for our vision about the evolution of life and consciousness.

Primary singularities of vector fields on surfaces

Unless another thing is stated one works in the $$C^\infty $$ category and manifolds have empty boundary. Let X and Y be vector fields on a manifold M. We say that Y tracks X if $$[Y,X]=fX$$ for some continuous function $$f:M\rightarrow \mathbb {R}$$ . A subset K of the zero set $${\mathsf {Z}} (X)$$ is an essential block for X if it is non-empty, compact, open in $${\mathsf {Z}}(X)$$ and its Poincare-Hopf index does not vanishes. One says that X is non-flat at p if its $$\infty $$ -jet at p is non-trivial. A point p of $${\mathsf {Z}}(X)$$ is called a primary singularity of X if any vector field defined about p and tracking X vanishes at p. This is our main result: consider an essential block K of a vector field X defined on a surface M. Assume that X is non-flat at every point of K. Then K contains a primary singularity of X. As a consequence, if M is a compact surface with non-zero characteristic and X is nowhere flat, then there exists a primary singularity of X.

Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary

Abstract We set out to obtain estimates of the Laplacian Spectrum of Riemannian manifolds with non-empty boundary. This was achieved using standard doubled manifold techniques. In simple terms, we pasted two copies of the same manifold along their common boundary thereby obtaining a Riemannian manifold with empty boundary and with a C 0 −metric. This made it possible to adapt some estimates of the spectrum dependent on the volume or genus of the manifold as calculated in recent years by several authors. In order to extend further estimates that depend on the curvature, it is necessary to regularize the metric of the doubled manifold so that the new metric is isometric to that of each copy and such that the curvature has a finite lower bound. Controlling the curvature in this way also makes estimates of topological invariants available.

Non-LERFness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups

We will show that for any noncompact arithmetic hyperbolic m-manifold with m>3, and any compact arithmetic hyperbolic m-manifold with m>4 that is not a 7-dimensional one defined by octonions, its fundamental group is not locally extended residually finite (LERF). The main ingredient in the proof is a study on abelian amalgamations of hyperbolic 3-manifold groups. We will also show that a compact orientable irreducible 3-manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is LERF.

Asymptotic dimension and the disk graph I

For an aspherical oriented 3-manifold M and a subsurface X of the boundary of M with empty or incompressible boundary, we use surgery to identify a graph whose vertices are disks with boundary in X and which is quasi-isometrically embedded in the curve graph of X.