Convexity Estimates Research Articles

Superharmonic functions related to convexity estimates for a class of semilinear elliptic problems

In this paper, we consider homogeneous Dirichlet problems for a class of semilinear elliptic equations involving the Laplacian with power-type nonlinearities in bounded convex domains of Rn, and find some superharmonic functions which can be used to give convexity estimates for the solutions directly.

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.

Impulse controllability for the heat equation with inverse square potential and dynamic boundary conditions

Abstract In this paper, we investigate the null approximate impulse controllability of the heat equation with an inverse square potential subject to dynamic boundary conditions in the ball $B(0, R_{0})$ of radius $R_{0}=\left (\frac{4}{3}\right )^{\frac{3}{2}}$. To that purpose, we use the Carleman commutator approach to show a logarithmic convexity estimate traducing an observability inequality at one instant of time.

Local convexity estimates for mean curvature flow

Abstract We develop a local version of Huisken–Stampacchia iteration, using it to obtain local versions of the sharp asymptotic curvature pinching estimates for mean curvature flow. The local estimates we obtain do not depend on the quality of noncollapsing of the solution and the method adopted applies in a host of other settings.

Convexity Estimates for the Solution of the Elliptical Partial Differential Equation det D 2 u = eu

The Monge-Ampère equation det D 2 u = eu is completely nonlinear and elliptic, the convexity estimates for the solution of the elliptical partial differential equation is very important. We establish a differential inequality by constructing an auxiliary function and give two differential estimates for the solution of the equation det D 2 u = eu with 0 boundary value condition.

Estimates for L-functions in the Critical Strip Under GRH with Effective Applications

Assuming the generalized Riemann hypothesis, we provide explicit upper bounds for moduli of log {mathcal {L}(s)} and mathcal {L}'(s)/mathcal {L}(s) in the neighbourhood of the 1-line when mathcal {L}(s) are the Riemann, Dirichlet and Dedekind zeta-functions. To do this, we generalize Littlewood’s well-known conditional result to functions in the Selberg class with a polynomial Euler product, for which we also establish a suitable convexity estimate. As an application, we provide conditional and effective estimates for the Mertens function.

Sketching Low-Rank Matrices With a Shared Column Space by Convex Programming

In many practical applications including remote sensing, multi-task learning, and multi-spectrum imaging, data are described as a set of matrices sharing a common column space. We consider the joint estimation of such matrices from their noisy linear measurements. We study a convex estimator regularized by a pair of matrix norms. The measurement model corresponds to block-wise sensing and the reconstruction is possible only when the total energy is well distributed over blocks. The first norm, which is the maximum-block-Frobenius norm, favors such a solution. This condition is analogous to the notion of low-spikiness in matrix completion or column-wise sensing. The second norm, which is a tensor norm on a pair of suitable Banach spaces, induces low-rankness in the solution together with the first norm. We demonstrate that the joint estimation provides a significant gain over the individual recovery of each matrix when the number of matrices sharing a column space and the ambient dimension of the shared column space are large relative to the number of columns in each matrix. The convex estimator is cast as a semidefinite program and an efficient ADMM algorithm is derived. The empirical behavior of the convex estimator is illustrated using Monte Carlo simulations and recovery performance is compared to existing methods in the literature.

Convex Estimation of Sparse-Smooth Power Spectral Densities From Mixtures of Realizations With Application to Weather Radar

Convex Estimation of Sparse-Smooth Power Spectral Densities From Mixtures of Realizations With Application to Weather Radar

Impulse null approximate controllability for heat equation with dynamic boundary conditions

<p style='text-indent:20px;'>The main purpose of this article is to prove a logarithmic convexity estimate for the solution of a linear heat equation subject to dynamic boundary conditions in a bounded convex domain. As an application, we prove the impulsive null approximate controllability for an impulsive heat equation with dynamic boundary conditions.</p>

Convexity estimates for high codimension mean curvature flow

We consider the flow by mean curvature of smooth n-dimensional submanifolds of Rn+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{n+k}$$\\end{document}, k≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\ge 2$$\\end{document}, which are compact and quadratically pinched. We establish that such flows are asymptotically convex, that is, the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. This generalises the convexity estimate of Huisken–Sinestrari to higher codimensions. By combining our estimate with work of Naff, we conclude that singularity models for the flow are convex ancient solutions of codimension one.

Convexity estimates for hypersurfaces moving by concave curvature functions

We study fully nonlinear geometric flows that deform strictly k-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the k-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve k-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.

Remarks on convexity estimates for solutions of the torsion problem

Remarks on convexity estimates for solutions of the torsion problem

Convexity estimate for translating solitons of concave fully nonlinear extrinsic geometric flows in [formula omitted

The main result of this paper is a convexity estimate for translating solitons of extrinsic geometric flows which evolve under a 1-homogeneous concave function in the principal curvatures. In addition, we show examples of these hypersurfaces in Rn+1 for particular functions.

Gaussian curvature estimates for the convex level sets of minimal graphs revisited

The main aim of the present paper is to give a simplified and easy-to-follow proof of the convexity estimates of level sets of minimal graphs. This proof greatly simplifies the original one in Wang and Zhang (2012) [36]. The innovation lies in that we find new curvature functions which are exactly supersolutions of the minimal surface equations.

Target Localization in Wireless Sensor Networks Based on Received Signal Strength and Convex Relaxation.

A new positioning algorithm based on RSS measurement is proposed. The algorithm adopts maximum likelihood estimation and semi-definite programming. The received signal strength model is transformed to a non-convex estimator for the positioning of the target using the maximum likelihood estimation. The non-convex estimator is then transformed into a convex estimator by semi-definite programming, and the global minimum of the target location estimation is obtained. This algorithm aims at the known problem and then extends its application to the case of unknown. The simulations and experimental results show that the proposed algorithm has better accuracy than the existing positioning algorithms.

Exploiting perturbed and coalescent anchor node geometry with semidefinite relaxation for sensor network localization

Localization in wireless sensor networks (WSNs) suffer from performance issues whenever the anchor nodes (which are aware of their location) are subjected to motion from their usual position. Moreover, accurate localization demands more anchor nodes which is a scarce resource and needs to be used judiciously. In the current work, we propose a novel framework that addresses these two prime concerns by harnessing the inter relationship of anchor node geometry. For an unknown source node surrounded by anchor nodes, the anchors lying on the inner boundary of the deployment geometry may be carrying closely related information about source node, leading to redundancy and inefficient utilization. By anticipating the level of correlation between these anchors, localization can be made more frugal. Rigorous mathematical analysis is carried out to derive lower bounds on estimated locations. Based on fisher information from two proposed models, a convex estimation objective function is formulated using semidefinite programming (SDP) approach to validate the theoretical proceedings. Based on the findings, the proposed method is able to successfully extract useful information about the unknown source node location with limited number of anchor nodes, hence achieving superior localization.

Hypercontractivity on the unit circle for ultraspherical measures: linear case

In this paper we extend complex uniform convexity estimates for \mathbb{C} to \mathbb{R}^n and determine best constants. Furthermore, we provide the link to log-Sobolev inequalities and hypercontractivity estimates for ultraspherical measures.

Discrete Carleman estimates and three balls inequalities

We prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schrödinger operators. These quantitatively connect the discrete setting in which the unique continuation property fails and the continuum setting in which the unique continuation property is known to hold under suitable regularity assumptions. As a key auxiliary result which might be of independent interest we present a Carleman estimate for these discrete operators.

A unified approach for optimal dose delivery and trajectory optimization for the treatment of prostate cancer

For a Fluence map optimization problem, we present a unique iterative reweighting algorithm-based method to achieve optimality. The purpose of planning is to determine a dose distribution in such a manner that it should cover the maximum part of the target without affecting the functionality of organ at risk. We suggest a unique methodology to solve dose-volume bounds while maintaining their non-convexity, as compared to earlier approaches that dedicated to convex estimation. The suggested approach is cooperative to competent procedures centered on partial minimization and certainly adjusts to tackle maximum-dose bounds. Fluence mapping is used for the inverse planning which determines the intensity profile of each beam. For the analysis, 09 equally separated beams with dose-volume limits on the organ at risk have used to treat malignancies in the prostate. Cumulative dose-volume histogram is used for the treatment plans quality measurement. We considered 4-different samples from CORT dataset named prostate, Liver, TG119 phantom, and Head & Neck to make a judgment about the generalizability of the performance of the proposed algorithm, and after getting this we finally compared our result with other method using prostate sample to validate our approach. After comparing with other methods centered on dose volume limits we found that in our approach maximum dose can deliver within minimum time across the planned target. For the optimization of trajectory we kept, the radiation delivery rate fixed to its extreme level and employed the independence property of each single leaf pair equally. By making this provision for all total T delivery times, trade-off between distribution time and fluency mapping quality is produced.

Some Geometric Properties of a Hyperbolic Univalent Function

In this paper, we analyze several aspects of a hyperbolic univalent function related to convexity properties, by assuming to be the univalent holomorphic function maps of the unit disk onto the hyperbolic convex region ( is an open connected subset of). This assumption leads to the coverage of some of the findings that are started by seeking a convex univalent function distortion property to provide an approximation of the inequality and confirm the form of the lower bound for . A further result was reached by combining the distortion and growth properties for increasing inequality . From the last result, we wanted to demonstrate the effect of the unit disk image on the condition of convexity estimation by proving the two inequalities of
 , and .