Geometric Hypotheses Research Articles

Computing eigenvalues of the Laplacian on rough domains

We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.

Spectral inequality for Dirac right triangles

We consider a Dirac operator on right triangles, subject to infinite-mass boundary conditions. We conjecture that the lowest positive eigenvalue is minimized by the isosceles right triangle under the area or perimeter constraints. We prove this conjecture under extra geometric hypotheses relying on a recent approach of Briet and Krejčiřík [J. Math. Phys. 63, 013502 (2022)].

Existence of solutions to a class of quasilinear elliptic equations

This paper is concerned with the existence and nonexistence of nontrivial solutions for a class of quasilinear elliptic equations in RN involving the p-Laplacian. By utilizing a change of variables, the quasilinear equations are reduced to semilinear equations, whose associated functionals satisfy the geometric hypotheses of the mountain pass theorem. We establish the existence of nontrivial solutions via Mountain-Pass theorem. Furthermore, by using a Pohozaev type identity we prove the nonexistence of nontrivial solution under certain conditions.

A geometric criterion for equation [formula omitted] having at most m isolated periodic solutions

This paper is devoted to the investigation of generalized Abel equation x˙=S(x,t)=∑i=0mai(t)xi, where ai∈C∞([0,1]). A solution x(t) is called a periodic solution if x(0)=x(1). In order to estimate the number of isolated periodic solutions of the equation, we propose a hypothesis (H) which is only concerned with S(x,t) on m straight lines: There exist m real numbers λ1<⋯<λm such that either (−1)i⋅S(λi,t)≥0 for i=1,⋯,m, or (−1)i⋅S(λi,t)≤0 for i=1,⋯,m. By means of Lagrange interpolation formula, we prove that the equation has at most m isolated periodic solutions (counted with multiplicities) if hypothesis (H) holds, and the upper bound is sharp. Furthermore, this conclusion is also valid under some weaker geometric hypotheses. Applying our main result for the trigonometrical generalized Abel equation with coefficients of degree one, we give a criterion to obtain the upper bound for the number of isolated periodic solutions. This criterion is “almost equivalent” to hypothesis (H) and can be much more effectively checked.

A geometric characterization of the symmetrized bidisc

The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\} \] has interesting geometric properties. While it has a plentiful supply of complex geodesics and of automorphisms, there is nevertheless a unique complex geodesic $\mathcal{R}$ in $G$ that is invariant under all automorphisms of $G$. Moreover, $G$ is foliated by those complex geodesics that meet $\mathcal{R}$ in one point and have nontrivial stabilizer. We prove that these properties, together with two further geometric hypotheses on the action of the automorphism group of $G$, characterize the symmetrized bidisc in the class of complex manifolds.

Laue’s theorem revisited: Energy–momentum tensors, symmetries, and the habitat of globally conserved quantities

The energy–momentum tensor for a particular matter component summarises its local energy–momentum distribution in terms of densities and current densities. We re-investigate under what conditions these local distributions can be integrated to meaningful global quantities. This leads us directly to a classic theorem by Max von Laue concerning integrals of components of the energy–momentum tensor, whose statement and proof we recall. In the first half of this paper, we do this within the realm of Special Relativity (SR) and in the traditional mathematical language using components with respect to affine charts, thereby focusing on the intended physical content and interpretation. In the second half, we show how to do all this in a proper differential-geometric fashion and on arbitrary spacetime manifolds, this time focusing on the group-theoretic and geometric hypotheses underlying these results. Based on this we give a proper geometric statement and proof of Laue’s theorem, which is shown to generalise from Minkowski space (which has the maximal number of isometries) to spacetimes with significantly less symmetries. This result, which seems to be new, not only generalizes but also clarifies the geometric content and hypotheses of Laue’s theorem. A series of three appendices lists our conventions and notation and summarises some of the conceptual and mathematical background needed in the main text.

Agustín de Betancourt’s Double-Acting Steam Engine: Geometric Modeling and Virtual Reconstruction

In this paper, the geometric modeling and virtual reconstruction of the double-acting steam engine designed by Agustín de Betancourt in 1789 are shown. For this, the software Autodesk Inventor Professional is used, which has allowed us to obtain its geometric documentation. The material for the research is available on the website of the Betancourt Project of the Canary Orotava Foundation for the History of Science. Almost all parts of the steam engine are drawn on the sheets, but due to the absence of scale and space, it is insufficient to obtain an accurate and reliable 3D CAD (Computer-Aided Design) model. For this reason a graphic scale has been adopted so that the dimensions of the elements are coherent. Also, it has been necessary to make some dimensional and geometric hypotheses, as well as restrictions of movement (degrees of freedom). Geometric modeling has made it possible to know that the system is balanced with the geometric center of the rocker arm shaft, and presents an energetic symmetry whose axis is the support of the parallelogram where the shaft rests: calorific energy to the left and mechanical energy to the right, with the rocker arm acting as a transforming element from one to the other.

The Crepant Transformation Conjecture for toric complete intersections

Let X and Y be K-equivalent toric Deligne–Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y, with a Fourier–Mukai transformation between the K-groups of X and Y, via an equivariant version of the Gamma-integral structure on quantum cohomology. We prove similar results for toric complete intersections. We impose only very weak geometric hypotheses on X and Y: they can be non-compact, for example, and need not be weak Fano or have Gorenstein coarse moduli space. Our main tools are the Mirror Theorems for toric Deligne–Mumford stacks and toric complete intersections, and the Mellin–Barnes method for analytic continuation of hypergeometric functions.

An Exact Redatuming Procedure for the Inverse Boundary Value Problem for the Wave Equation

Redatuming is a data processing technique to transform measurements recorded in one acquisition geometry to an analogous data set corresponding to another acquisition geometry, for which there are no recorded measurements. We consider a redatuming problem for a wave equation on a bounded domain, or on a manifold with boundary, and model data acquisition by a restriction of the associated Neumann-to-Dirichlet map. This map models measurements with sources and receivers on an open subset $\Gamma$ contained in the boundary of the manifold. We model the wavespeed by a Riemannian metric and suppose that the metric is known in some coordinates in a neighborhood of $\Gamma$. Our goal is to move sources and receivers into this known near boundary region. We formulate redatuming as a collection of unique continuation problems and provide a two-step procedure to solve the redatuming problem. We investigate the stability of the first step in this procedure, showing that it enjoys conditional Hölder stability under suitable geometric hypotheses. In addition, we provide computational experiments that demonstrate our redatuming procedure.

On the proof of the thin sandwich conjecture in arbitrary dimensions

In this paper we show the validity, under certain geometric conditions, of Wheeler’s thin sandwich conjecture for higher dimensional theories of gravity. We extend the results for the 3-dimensional case in Phys. Rev. D 48, 3596–3599 (1993) in two ways. On the one hand, we show that the results presented in Phys. Rev. D 48, 3596–3599 (1993) are valid in arbitrary dimensions, and on the other hand, we show that the geometric hypotheses needed for the proofs can always be satisfied, which constitutes in itself a new result for the 3-dimensional case. In this way, we show that on any compact n-dimensional manifold, n≥3, there is an open set in the space of all possible initial data where the thin sandwich problem is well-posed.

Agustin de Betancourt’s Wind Machine for Draining Marshy Ground: Approach to Its Geometric Modeling with Autodesk Inventor Professional

The present study shows the process followed in making the three-dimensional model and geometric documentation of a historical invention of the renowned Spanish engineer Agustin de Betancourt y Molina, which forms part of his rich legacy. Specifically, this was a wind machine for draining marshy ground, designed in 1789. The present research relies on the computer-aided design (CAD) techniques using Autodesk Inventor Professional software, based on the scant information provided by the only two drawings of the machine, making it necessary to propose a number of dimensional and geometric hypotheses as well as a series of movement restrictions (degrees of freedom), to arrive at a consistent design. The results offer a functional design for this historic invention.

A General, Synthetic Model for Predicting Biodiversity Gradients from Environmental Geometry.

Latitudinal and elevational biodiversity gradients fascinate ecologists, and have inspired dozens of explanations. The geometry of the abiotic environment is sometimes thought to contribute to these gradients, yet evaluations of geometric explanations are limited by a fragmented understanding of the diversity patterns they predict. This article presents a mathematical model that synthesizes multiple pathways by which environmental geometry can drive diversity gradients. The model characterizes species ranges by their environmental niches and limits on range sizes and places those ranges onto the simplified geometries of a sphere or cone. The model predicts nuanced and realistic species-richness gradients, including latitudinal diversity gradients with tropical plateaus and mid-latitude inflection points and elevational diversity gradients with low-elevation diversity maxima. The model also illustrates the importance of a mid-environment effect that augments species richness at locations with intermediate environments. Model predictions match multiple empirical biodiversity gradients, depend on ecological traits in a testable fashion, and formally synthesize elements of several geometric models. Together, these results suggest that previous assessments of geometric hypotheses should be reconsidered and that environmental geometry may play a deeper role in driving biodiversity gradients than is currently appreciated.

Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

A new asymptotic approximation of the dynamic equations in the two-dimensional classical theory of thin-elastic shells is established for a circular cylindrical shell. It governs long wave vibrations in the vicinity of the lowest cut-off frequency. At a fixed circumferential wavenumber, the latter corresponds to the eigenfrequency of in-plane vibrations of a thin almost inextensible ring. It is stressed that the well-known semi-membrane theory of cylindrical shells is not suitable for tackling a near-cut-off behaviour. The dispersion relation within the framework of the developed formulation coincides with the asymptotic expansion of the dispersion relation originating from full two-dimensional shell equations. Asymptotic analysis also enables refining the geometric hypotheses underlying various ad hoc set-ups, including the assumption on vanishing of shear and circumferential mid-surface deformations used in the semi-membrane theory. The obtained results may be of interest for dynamic modelling of elongated cylindrical thin-walled structures, such as carbon nanotubes.

Estimation of effective usable and burial volumes of artificial reefs and the prediction of cost-effective management

In this study, effective usable and burial volumes were used to estimate the current state of nine cube-type artificial reefs (AR) constructed between 1987 and 2002. Three volume coefficients (k1, k2, and k3) were proposed for estimation of the total loss, effective usable volume, and burial volume of ARs, based on the recorded facility standard volume and seabed condition at the time of construction. Five parameters were redefined to clarify their use; two measures and three statistically obtainable coefficients were proposed from geometric hypotheses. Field data obtained from coastal waters near Busan in South Korea were used to demonstrate the estimation and prediction of these parameters. We determined that the burial volume of an AR can be predicted with less than 20% error using the mean values of k1 = 1.33, k2 = 1.36, and k3 = 1.37 for a rigid seabed and k1 = 1.32, k2 = 1.72, and k3 = 1.83 for a soft seabed. The effective usable volume can be predicted with less than 11% error using the mean values, seabed condition, and facility standard volume determined at the design stage. We found that it is unnecessary to use multi-beam echo, side scan sonar, and divers' investigation for prediction; hence, a cost-effective method was developed for the future management of ARs.

Soliton solutions to a class of relativistic nonlinear Schrödinger equations

By using a change of variables, we get new equations, whose respective associated functionals are well defined in H1(RN) and satisfy the geometric hypotheses of the mountain pass theorem. Using this fact, we obtain a nontrivial solution.

Positive solution for quasilinear Schrödinger equations with a parameter

In this paper, we study the following quasilinear Schrödinger equations of the form \begin{eqnarray} -\Delta u+V(x)u-[\Delta(1+u^2)^{\alpha/2}]\frac{\alpha u}{2(1+u^2)^{(2-\alpha)/2}}=\mathrm{g}(x,u)， \end{eqnarray} where $1 \le \alpha \le 2$, $N \ge 3$, $V\in C(R^N, R)$ and $\mathrm{g}\in C(R^N\times R, R)$. By using a change of variables, we get new equations, whose respective associated functionals are well defined in $H^1(R^N)$ and satisfy the geometric hypotheses of the mountain pass theorem. Using the special techniques, the existence of positive solutions is studied.

Soliton Solutions for Quasilinear Schrödinger Equations

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Quasilinear asymptotically periodic Schrödinger equations with critical growth

It is established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in \({\mathbb{R}^N}\) with critical growth. Applying a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in \({H^1(\mathbb{R}^N)}\) and satisfy the geometric hypotheses of the Mountain Pass Theorem. The Concentration–Compactness Principle and a comparison argument allow to verify that the problem possesses a nontrivial solution.

Quasilinear asymptotically periodic Schrödinger equations with subcritical growth

In this article the existence of one solution for a class of asymptotically periodic equations in the euclidean space is established. The basic tools employed here are the Mountain Pass Theorem and the Concentration–Compactness Principle. By using a change of variable, the quasilinear equation is reduced to a semilinear equation, whose respective associated functional is well defined in H 1 ( R N ) and satisfies the geometric hypotheses of the Mountain Pass Theorem.

Experimental validation of FastSLAM algorithm integrated with a linear features based map

In this paper the Simultaneous Localization And Mapping (SLAM) problem in unknown indoor environments is addressed. A probabilistic approach integrating FastSLAM algorithm and a line feature map is developed and validated. Experimental validation is performed by a smart wheelchair equipped with proprioceptive and exteroceptive sensors in an office like environment where loop closing is achieved without any dedicated algorithm. Geometric hypotheses of orthogonal line features are considered to enhance the performance of the algorithm in the considered environment. The proposed approach results in a computationally efficient solution to the SLAM problem and the high quality sensor measurements allow to maintain a good localization of the mobile base and a compact representation of the environment.