Boundary Hyperplane Research Articles

Singular extension of critical Sobolev mappings under an exponential weak-type estimate

Given m∈N∖{0} and a compact Riemannian manifold N, we construct for every map u in the critical Sobolev space Wm/(m+1),m+1(Sm,N), a map U:B1m+1→N whose trace is u and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity.

Available wrench set robustness under hybrid joint-space control to uncertain wrench for a cable-driven parallel robot

For a overconstrained cable-driven parallel robot (CDPR), the number of cables is more than the number of the moving platform degrees of freedom. Therefore, a hybrid joint-space control method is used, where as many cables as the moving platform degrees of freedom are kinematically controlled, and the redundant ones are tension controlled. However, the selection of which cables to tension control is not straightforward. This work aims at evaluating the maximum value of the resistance to the wrench deviation, while tension-controlling a chosen cable set, by computing a performance index called available wrench set robustness (AWSR). In this case, the wrench deviation refers to the difference between the actual wrench and the nominal model wrench, which is related to the wrench uncertainties influenced by elements such as varying mass, shifted centre of mass and environmental disturbances. Firstly, owing to the inhomogeneous wrench deviation, a normalization method was proposed. Based on this, geometrically, the AWSR is defined as the minimum distance from the nominal wrench to the hyperplane boundary of available wrench set. As an application example of a planar CDPR with four cables and a spatial CDPR with eight cables, a comparison with previous methods illustrated that the proposed method significantly increased the range of wrench deviation, and this avoids situations where the configuration is not feasible using the traditional method. Therefore, the proposed method is more suitable for uncertain wrenches.

On the hemispherical transform in the half-space and related Radon transforms

We consider an integral transform which maps functions on the Euclidean half-space to integrals of these functions over hemispheres centered on the boundary hyperplane. The main results include sharp [Formula: see text]-[Formula: see text] estimates for this transform and new explicit inversion formulas under minimal assumptions for functions. The main idea is an intriguing connection between the hemispherical transform, the Radon transform over paraboloids, and the transversal Radon transform over hyperplanes meeting the last coordinate axis.

Differential-Difference Elliptic Equations with Nonlocal Potentials in Half-Spaces

We investigate the half-space Dirichlet problem with summable boundary-value functions for an elliptic equation with an arbitrary amount of potentials undergoing translations in arbitrary directions. In the classical case of partial differential equations, the half-space Dirichlet problem for elliptic equations attracts great interest from researchers due to the following phenomenon: the solutions acquire qualitative properties specific for nonstationary (more exactly, parabolic) equations. In this paper, such a phenomenon is studied for nonlocal generalizations of elliptic differential equations, more exactly, for elliptic differential-difference equations with nonlocal potentials arising in various applications not covered by the classical theory. We find a Poisson-like kernel such that its convolution with the boundary-value function satisfies the investigated problem, prove that the constructed solution is infinitely smooth outside the boundary hyperplane, and prove its uniform power-like decay as the timelike independent variable tends to infinity.

Эллиптические уравнения со сдвигами произвольных направлений в полупространстве

In this paper, we investigate the half-space Dirichlet problem for elliptic differential-difference equations with superpositions of differential operators and translation operators acting in arbitrary directions parallel to the boundary hyperplane. The summability assumption is imposed on the boundary-value function of the problem. The specified equations, substantially generalizing classical elliptic partial differential equations, arise in various models of mathematical physics with nonlocal and (or) heterogeneous properties or the process or medium: multi-layer plates and envelopes theory, theory of diffusion processes, biomathematical applications, models of nonlinear optics, etc. The theoretical interest to such equations is caused by their nonlocal nature: they connect values of the desired function (and its derivatives) at different points (instead of the same one), which makes many classical methods unapplicable. For the considered problem, we establish the solvability in the sense of generalized functions, construct Poisson-like integral representations of solutions, and prove the infinite smoothness of the solution outside the boundary hyperplane and its uniform convergence to zero (together with all its derivatives) as the timelike variable tends to infinity. We find a power estimate of the velocity of the specified extinction of the solution and each its derivative.

Physics-informed neural networks method in high-dimensional integrable systems

In this paper, the physics-informed neural networks (PINNs) are applied to high-dimensional system to solve the [Formula: see text]-dimensional initial-boundary value problem with [Formula: see text] hyperplane boundaries. This method is used to solve the most classic (2+1)-dimensional integrable Kadomtsev–Petviashvili (KP) equation and (3+1)-dimensional reduced KP equation. The dynamics of (2+1)-dimensional local waves such as solitons, breathers, lump and resonance rogue are reproduced. Numerical results display that the magnitude of the error is much smaller than the wave height itself, so it is considered that the classical solutions in these integrable systems are well obtained based on the data-driven mechanism.

A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the non local Gaussian perimeter taken into consideration.

Discrimination of phthalate species using a simple phage-based colorimetric sensor in conjunction with hierarchical support vector machine

Here, the analytical potential of an M13 bacteriophage-based color sensor for discrimination of 4 phthalates with similar molecular structures (bis-(2-ethylhexyl)-phthalate (BEHP), dibutyl phthalate (DBP), diethyl phthalate (DEP), and benzyl-butyl-phthalate (BBP)) was investigated. The pattern and magnitude of the RGB color changes were different depending on the functional groups present in the phthalate structures. For example, BEHP possessing a long alkyl chain resulted in a minute color change, while the variation of color was substantially large when BBP containing an additional benzene ring was measured. Since a tryptophan-histidine-tryptophan residue possessing indole and imidazole was present on the self-assembled phages, the π-π interaction of benzene in BBP with the sensor surface produced a considerably greater color change. To evaluate the multi-modally varying color signals due to diverse interactions of the phthalates with the sensor and to discriminate them, support vector machine (SVM), which can construct a boundary hyperplane among complexly scattered sample groups, was used. In addition, hierarchical SVM (H-SVM) was adopted to deal with multi-class discrimination. The use of H-SVM improved the discrimination accuracy up to 90.1%, compared to 87.1% using SVM. The demonstrated color sensor is versatile and can be potentially adopted as an on-site screening tool. Strategies to improve the accuracy further for real applications are also discussed.

On the Dirichlet Problem for Differential-Difference Elliptic Equations in a Half-Plane

The Dirichlet problem is considered in a half-plane (with continuous and bounded boundaryvalue function) for the model elliptic differential-difference equation $$ {u}_{xx}+a{u}_{xx}\left(x+h,y\right)+{u}_{yy}=0,\mid a\mid <1. $$ Its solvability is proved in the sense of generalized functions, the integral representation of the solution is constructed, and it is proved that everywhere but the boundary hyperplane this solution satisfies the equation in the classic sense as well.

A generalized normal flow method with online step controls for pushover analysis of nonlinear softening structures

This paper introduces a robust path-following method for pushover analysis of softening structures due to geometric and/or material nonlinearities. The proposed method is a generalized form of the conventional normal flow method (NFM) that provides dimensional consistency in the correction iterations and improves the convergence properties of the algorithm. The generalized NFM is further enhanced by two online step control processes, namely, the Convex Step-size Control (CSC) process and the Hyper-Plane Step Control (HPSC) process. The CSC process introduces a set of inequality constraints on the step size to form a convex solution space and (i) enhance the convergence properties of the algorithm by preventing overshooting in highly nonlinear areas, and (ii) improve robustness of the path-tracing process by controlling the resolution of the traced equilibrium path (or spacing/distance between successive equilibrium points) making it unlikely to miss (highly nonlinear) sections of the path. The HPSC process introduces a hyper-plane boundary in the solution space to prevent backward path-tracing. The performance of the proposed generalized NFM with the CSC and HPSC processes is validated through examples with structures exhibiting snap-back and snap-through response due to geometric and material nonlinearities. The proposed method is also compared with existing path-following methods typically used for the pushover analysis of nonlinear structures.

Stability of inverse problems in an infinite slab with partial data

ABSTRACTIn this paper, we study the stability of two inverse boundary value problems in an infinite slab with partial data. These problems have been studied by Li and Uhlmann for the case of the Schrödinger equation and by Krupchyk, Lassas, and Uhlmann for the case of the magnetic Schrödinger equation. Here, we quantify the method of uniqueness proposed by Li and Uhlmann and prove a log–log stability estimate for the inverse problems associated to the Schrödinger equation. The boundary measurements considered in these problems are modeled by partial knowledge of the Dirichlet-to-Neumann map: in the first inverse problem, the corresponding Dirichlet and Neumann data are known on different boundary hyperplanes of the slab; in the second inverse problem, they are known on the same boundary hyperplane of the slab.

Integral representations of a class of harmonic functions in the half space

In this article, motivated by the classic Hadamard factorization theorem about an entire function of finite order in the complex plane, we firstly prove that a harmonic function whose positive part satisfies some growth conditions, can be represented by its integral on the boundary of the half space. By using Nevanlinna's representation of harmonic functions and the modified Poisson kernel of the half space, we further prove a representation formula through integration against a certain measure on the boundary hyperplane for harmonic functions not necessarily continuous on the boundary hyperplane whose positive parts satisfy weaker growing conditions than the first question. The result is further generalized by involving a parameter m dealing with the singularity at the infinity.

Backstepping observers for linear PDEs on higher-dimensional spatial domains

The extension of the backstepping-based state observer design is considered for systems governed by parabolic PDEs defined on a higher-dimensional cuboidal spatial domain with the measured output being restricted to a single boundary hyperplane. Thereby, a suitable coordinate and state transformation is employed, which allows for the application of the backstepping method for the determination of a Luenberger-type state observer with observer corrections entering the PDE and the boundary condition. This ensures that the observer error dynamics follows the behavior of a predefined exponentially stable target system. In view of a practical realization of the proposed observer approach, the idealized system output, previously assumed as measurable on the entire boundary hyperplane, is reconstructed from a limited set of optimally located measurements by means of a least squares approximation. The observer error convergence and the applicability of the proposed approach are confirmed analytically and are evaluated in numerical simulations.

Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data

In this paper we study inverse boundary value problems with partial data for the bi-harmonic operator with first order perturbation. We consider two types of subsets of Rn(n≥3), one is an infinite slab, the other is a bounded domain. In the case of a slab, we show that, from Dirichlet and Neumann data given either on the different boundary hyperplanes of the slab or on the same boundary hyperplane, one can uniquely determine the magnetic potential and the electric potential.In the case of a bounded domain, we show the unique determination of the magnetic potential and the electric potential from partial Dirichlet and Neumann data under two different assumptions. The first assumption is that the magnetic and electric potentials are known in a neighborhood of the boundary, in this situation we obtain the uniqueness result when the Dirichlet and Neumann data are only given on two arbitrary open subsets of the boundary. The second assumption is that the Dirichlet and Neumann data are known on the same part of the boundary whose complement is a part of a hyperplane, we also establish the unique determination result in this local data case.

Robust uniform persistence in discrete and continuous nonautonomous systems

This is an extension of the work in Salceanu (2011) [14] to nonautonomous systems of difference and differential equations on the positive cone of Rm that exhibit a positively invariant boundary hyperplane X. It is shown that when a compact subset of X, which attracts all orbits in X, is a robust uniform weak repeller, robust uniform persistence for the complementary dynamics (i.e., the dynamics in Rm∖X) is obtained. Additional assumptions are made, to deal with the nonautonomous nature of the systems. Some particular cases that often occur in applications are discussed and then sufficient conditions for the robust uniform persistence of the disease in two epidemic models from the literature are given.

Symmetry results for decay solutions of semilinear elliptic systems on half spaces

In this paper, we prove that the decaying positive solutions of semi-linear elliptic systems on the half space with Neumann type condition is symmetric with respect to a line orthogonal to the boundary hyperplane. The monotonicity result is also obtained. Our methods employ the Alexandrov–Serrin method of moving planes combined with Hopf’s lemma at a corner. Our result can be applied to the Lane–Emden system and the stationary Schrödinger system.

Adaptive Cluster Distance Bounding for High-Dimensional Indexing

We consider approaches for similarity search in correlated, high-dimensional data sets, which are derived within a clustering framework. We note that indexing by “vector approximation” (VA-File), which was proposed as a technique to combat the “Curse of Dimensionality,” employs scalar quantization, and hence necessarily ignores dependencies across dimensions, which represents a source of suboptimality. Clustering, on the other hand, exploits interdimensional correlations and is thus a more compact representation of the data set. However, existing methods to prune irrelevant clusters are based on bounding hyperspheres and/or bounding rectangles, whose lack of tightness compromises their efficiency in exact nearest neighbor search. We propose a new cluster-adaptive distance bound based on separating hyperplane boundaries of Voronoi clusters to complement our cluster based index. This bound enables efficient spatial filtering, with a relatively small preprocessing storage overhead and is applicable to euclidean and Mahalanobis similarity measures. Experiments in exact nearest-neighbor set retrieval, conducted on real data sets, show that our indexing method is scalable with data set size and data dimensionality and outperforms several recently proposed indexes. Relative to the VA-File, over a wide range of quantization resolutions, it is able to reduce random IO accesses, given (roughly) the same amount of sequential IO operations, by factors reaching 100X and more.

A Fast Nonparametric Noncausal MRF-Based Texture Synthesis Scheme Using a Novel FKDE Algorithm

In this paper, a new algorithm is proposed for fast kernel density estimation (FKDE), based on principal direction divisive partitioning (PDDP) of the data space. A new framework is also developed to apply FKDE algorithms (both proposed and existing), within nonparametric noncausal Markov random field (NNMRF) based texture synthesis algorithm. The goal of the proposed FKDE algorithm is to use the finite support property of kernels for fast estimation of density. It has been shown that hyperplane boundaries for partitioning the data space and principal component vectors of the data space are two requirements for efficient FKDE. The proposed algorithm is compared with the earlier algorithms, with a number of high-dimensional data sets. The error and time complexity analysis, proves the efficiency of the proposed FKDE algorithm compared to the earlier algorithms. Due to the local simulated annealing, direct incorporation of the FKDE algorithms within the NNMRF-based texture synthesis algorithm, is not possible. This work proposes a new methodology to incorporate the effect of local simulated annealing within the FKDE framework. Afterward, the developed texture synthesis algorithms have been tested with a number of different natural textures, taken from a standard database. The comparison in terms of visual similarity and time complexity, between the proposed FKDE based texture synthesis algorithm with the earlier algorithms, show the efficiency.

On stabilization of solutions of singular elliptic equations

We study linear and quasi-linear elliptic equations containing the Bessel operator with respect to a selected variable (so-called special variable). The well-posedness of the nonclassical Dirichlet problem (with the additional condition of evenness with respect to the special variable) in the half-space is proved, an integral representation of the solution is constructed, and a necessary and sufficient condition of stabilization is established. The stabilization is understood as follows: the solution has a finite limit as the independent variable tends to infinity along the direction orthogonal to the boundary hyperplane.

A Presentation for the Chow Ring of

We give a presentation for the Chow ring of the moduli space of degree 2 stable maps from 2-pointed rational curves to the projective line. Also, integrals of all degree 4 monomials in the hyperplane pullbacks and boundary divisors of this ring are computed using equivariant intersection theory. Finally, the presentation is used to give a new computation of the (previously known) values of the genus zero, degree 2, 2-pointed gravitational correlators of the projective line.