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The eigenvalue problem for the Laplacian on bounded, planar, convex domains with mixed boundary conditions is considered, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its complement. Given two different such choices of boundary conditions for the same domain, we prove inequalities between their lowest eigenvalues. As a special case, we prove parts of a conjecture on the order of mixed eigenvalues of triangles.

P. Turán was the first to derive lower estimations on the uniform norm of the derivatives of polynomials p of uniform norm 1 on the disk D:={z∈C:|z|≤1} and the interval I:=[−1,1], under the normalization condition that the zeros of the polynomial p in question all lie in D or I, resp. Namely, in 1939 he proved that with n:=degp tending to infinity, the precise growth order of the minimal possible derivative norm is n for D and n for I. Already the same year J. Erőd considered the problem on other domains. In his most general formulation, he extended Turán's order n result on D to a certain general class of piecewise smooth convex domains. Finally, a decade ago the growth order of the minimal possible norm of the derivative was proved to be n for all compact convex domains. Turán himself gave comments about the above oscillation question in Lq norm on D. Nevertheless, till recently results were known only for D, I and so-called R-circular domains. Continuing our recent work, also here we investigate the Turán-Erőd problem on general classes of domains. © 2017 Elsevier Inc.

Lam, Hiu Fung., Thesis (M.Phil.)--Chinese University of Hong Kong, 2013., Includes bibliographical references (leaves 58-60)., Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web., s also in Chinese., http://library.cuhk.edu.hk/record=b5884340, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Lam, Hiu Fung., Thesis (M.Phil.)--Chinese University of Hong Kong, 2013., Includes bibliographical references (leaves 58-60)., Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web., s also in Chinese., http://library.cuhk.edu.hk/record=b5884340, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

This thesis examines the history and some major results of the Gauss Circle Problem. The goal of the Gauss Circle Problem is to determine the best bound for the error between the number of lattice points inside a disk and that disks area, otherwise known as the lattice point discrepancy. First we state some of the required definitions and properties from Fourier analysis that will be used throughout. In particular, we establish asymptotic results for oscillatory integrals and more specifically for Bessel functions. After examining the geometrical method for precisely counting the number of lattice points inside a disk of radius R, we use the Poisson Summation Formula and the Bessel function results to prove initial bounds on the lattice point discrepancy. We present two such results, employing a similar technique for both, and then apply oscillatory integral asymptotics to extend this method and establish a lattice point discrepancy result for strongly convex domains.

This thesis examines the history and some major results of the Gauss Circle Problem. The goal of the Gauss Circle Problem is to determine the best bound for the error between the number of lattice points inside a disk and that disks area, otherwise known as the lattice point discrepancy. First we state some of the required definitions and properties from Fourier analysis that will be used throughout. In particular, we establish asymptotic results for oscillatory integrals and more specifically for Bessel functions. After examining the geometrical method for precisely counting the number of lattice points inside a disk of radius R, we use the Poisson Summation Formula and the Bessel function results to prove initial bounds on the lattice point discrepancy. We present two such results, employing a similar technique for both, and then apply oscillatory integral asymptotics to extend this method and establish a lattice point discrepancy result for strongly convex domains.

This paper proposes a system that relates objects in an image using occlusion cues and arranges them according to depth. The system does not rely on a priori knowledge of the scene structure and focuses on detecting special points, such as T-junctions and highly convex contours, to infer the depth relationships between objects in the scene. The system makes extensive use of the binary partition tree as hierarchical region-based image representation jointly with a new approach for candidate T-junction estimation. Since some regions may not involve T-junctions, occlusion is also detected by examining convex shapes on region boundaries. Combining T-junctions and convexity leads to a system which only relies on low level depth cues and does not rely on semantic information. However, it shows a similar or better performance with the state-of-the-art while not assuming any type of scene. As an extension of the automatic depth ordering system, a semi-automatic approach is also proposed. If the user provides the depth order for a subset of regions in the image, the system is able to easily integrate this user information to the final depth order for the complete image. For some applications, user interaction can naturally be integrated, improving the quality of the automatically generated depth map., Peer Reviewed, Postprint (published version)

We introduce a notion of k-convexity and explore some properties of polygons that have this property. In particular, 2-convex polygons can be recognized in O(n log n) time, and k-convex polygons can be triangulated in O(kn) time., Postprint (published version)

We consider the Dirichlet problem for strongly elliptic systems of order 2m in convex domains. Under a positivity assumption on the Poisson kernel it is proved that the weak solution has bounded derivatives up to order m provided the outward unit normal has no big jumps on the boundary. In the case of second order symmetric systems in plane convex domains the boundedness of the first derivatives is proved without the assumption on the normal.

Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics., MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE., Bibliography: leaves 270-272., by Stephen Kinyon Young., Ph.D.

Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics., MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE., Bibliography: leaves 270-272., by Stephen Kinyon Young., Ph.D.

© Nasibullin R.G. 2017.We prove exact Hardy type inequalities with the weights depending on a Bessel function. We obtain one-dimensional Lp-inequalities and provide an example of extending these inequalities for the case of convex domains with a finite inner radius. The proved statements are generalization for the case of arbitrary p ≥ 2 of the corresponding inequality proved by F.G. Avkhadiev and K.-J. Wirths for p = 2.

© 2016, Pleiades Publishing, Ltd.We obtain a new sharp Hardy type inequality with an additional term. Using the Bessel functions we prove one dimensional inequality and their multidimensional analogs in domains with a finite inradius. The weight functions depend on the Bessel functions and Lamb’ s constants.

© Nasibullin R.G. 2017. We prove exact Hardy type inequalities with the weights depending on a Bessel function. We obtain one-dimensional L p -inequalities and provide an example of extending these inequalities for the case of convex domains with a finite inner radius. The proved statements are generalization for the case of arbitrary p ≥ 2 of the corresponding inequality proved by F.G. Avkhadiev and K.-J. Wirths for p = 2.

© Nasibullin R.G. 2017.We prove exact Hardy type inequalities with the weights depending on a Bessel function. We obtain one-dimensional Lp-inequalities and provide an example of extending these inequalities for the case of convex domains with a finite inner radius. The proved statements are generalization for the case of arbitrary p ≥ 2 of the corresponding inequality proved by F.G. Avkhadiev and K.-J. Wirths for p = 2.

© 2016, Pleiades Publishing, Ltd.We obtain a new sharp Hardy type inequality with an additional term. Using the Bessel functions we prove one dimensional inequality and their multidimensional analogs in domains with a finite inradius. The weight functions depend on the Bessel functions and Lamb’ s constants.