2,666 results on '"Regular polygon"'
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2. On Monge-Ampère equations with nonlinear gradient terms – Infinite boundary value problems
- Author
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Ahmed Mohammed and Giovanni Porru
- Subjects
Nonlinear system ,Mathematics::Complex Variables ,Applied Mathematics ,Bounded function ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Regular polygon ,Boundary (topology) ,Uniqueness ,Boundary value problem ,Ampere ,Analysis ,Mathematics - Abstract
The paper investigates the existence, asymptotic boundary behavior and uniqueness of convex solutions to the Monge-Ampere equation that take infinite values on the boundary of smooth and bounded open convex sets in R n .
- Published
- 2021
3. Rectifiable Curves in Proximally Smooth Sets
- Author
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Grigory Ivanov and Mariana S. Lopushanski
- Subjects
Statistics and Probability ,Numerical Analysis ,Smoothness ,Geodesic ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,Hilbert space ,Banach space ,Type (model theory) ,Convexity ,Moduli ,symbols.namesake ,symbols ,Geometry and Topology ,Analysis ,Mathematics - Abstract
In this article we study some geometric properties of proximally smooth sets. First, we introduce a modification of the metric projection and prove its existence. Then we provide an algorithm for constructing a rectifiable curve between two sufficiently close points of a proximally smooth set in a uniformly convex and uniformly smooth Banach space, with the moduli of smoothness and convexity of power type. Our algorithm returns a reasonably short curve between two sufficiently close points of a proximally smooth set, is iterative and uses our modification of the metric projection. We estimate the length of the constructed curve and its deviation from the segment with the same endpoints. These estimates coincide up to a constant factor with those for the geodesics in a proximally smooth set in a Hilbert space.
- Published
- 2021
4. Volume ratios for Cartesian products of convex bodies
- Author
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A. Khrabrov
- Subjects
symbols.namesake ,Algebra and Number Theory ,Surface-area-to-volume ratio ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,symbols ,Cartesian product ,Analysis ,Volume (compression) ,Mathematics - Abstract
The paper is devoted to the behavior of volume ratios, the modified Banach–Mazur distance, and the vertex index for sums of convex bodies. It is shown that sup d ( A ⊕ K , B ⊕ L ) ≥ sup ∂ ( A ⊕ K , B ⊕ L ) ≥ c ⋅ n 1 − k + k ′ 2 n , \begin{equation*} \sup d (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq \sup \partial (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq c \cdot n^{1-\frac {k+k’}{2n}}, \end{equation*} if K ⊂ R n \mathrm {K}\subset \mathbb {R}^n and L ⊂ R k \mathrm {L}\subset \mathbb {R}^k are convex and symmetric (the supremum is taken over all symmetric convex bodies A ⊂ R n − k \mathrm {A}\subset \mathbb {R}^{n-k} and B ⊂ R n − k ′ ) \mathrm {B}\subset \mathbb {R}^{n-k’}) . Furthermore, some examples are discussed that show that the available extimates of the vertex index in terms of the volume ratio are not sharp.
- Published
- 2021
5. Boundary layer of the Boltzmann equation in 2-dimensional convex domains
- Author
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Lei Wu
- Subjects
Numerical Analysis ,Boundary layer ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,Boltzmann equation ,Analysis ,Mathematics - Published
- 2021
6. Optimal gradient estimates for the perfect conductivity problem with C1,α inclusions
- Author
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Longjuan Xu, Yu Chen, and Haigang Li
- Subjects
Pointwise ,Smoothness (probability theory) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,01 natural sciences ,Upper and lower bounds ,Domain (mathematical analysis) ,Convexity ,010101 applied mathematics ,Line (geometry) ,0101 mathematics ,Convex function ,Mathematical Physics ,Analysis ,Mathematics - Abstract
In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly conducting inclusions are located very close to each other. To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two folds: First, we weaken the smoothness of the inclusions from C 2 , α to C 1 , α . To obtain a pointwise upper bound of the gradient, we follow an iteration technique which is first used to deal with elliptic systems in a narrow domain by Li-Li-Bao-Yin (QAM 2014). However, when the inclusions are of C 1 , α , we can not use W 2 , p estimates for elliptic equations any more. In order to overcome this new difficulty, we take advantage of De Giorgi-Nash estimates and Campanato's approach to apply an adapted version of the iteration technique with respect to the energy. A lower bound in the shortest line between two inclusions is also obtained to show the optimality of the blow-up rate. Second, when two inclusions are only convex but not strictly convex, we prove that blow-up does not occur any more. The establishment of the relationship between the blow-up rate of the gradient and the order of the convexity of the inclusions reveals the mechanism of such concentration phenomenon.
- Published
- 2021
7. Finite difference modified WENO schemes for hyperbolic conservation laws with non‐convex flux
- Author
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Asha K. Dond and Rakesh Kumar
- Subjects
Conservation law ,Finite volume method ,Mechanics of Materials ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Computational Mechanics ,Regular polygon ,Finite difference ,Flux ,Computer Science Applications ,Mathematics - Published
- 2021
8. On a Class of Generalized Curve Flows for Planar Convex Curves
- Author
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Li Ma and Huaqiao Liu
- Subjects
Class (set theory) ,Plane (geometry) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Zero (complex analysis) ,01 natural sciences ,010104 statistics & probability ,Planar ,Flow (mathematics) ,Convergence (routing) ,Point (geometry) ,0101 mathematics ,Mathematics - Abstract
In this paper, the authors consider a class of generalized curve flow for convex curves in the plane. They show that either the maximal existence time of the flow is finite and the evolving curve collapses to a round point with the enclosed area of the evolving curve tending to zero, i.e., $$\mathop {\lim}\limits_{t \to T} A(t) = 0$$ , or the maximal time is infinite, that is, the flow is a global one. In the case that the maximal existence time of the flow is finite, they also obtain a convergence theorem for rescaled curves at the maximal time.
- Published
- 2021
9. Multiplicity of positive solutions to a critical fractional equation with Hardy potential and concave–convex nonlinearities
- Author
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Yansheng Shen
- Subjects
Computational Mathematics ,Numerical Analysis ,Applied Mathematics ,Fractional equations ,Mathematical analysis ,Regular polygon ,Multiplicity (mathematics) ,Fractional Laplacian ,Analysis ,Mathematics - Abstract
The aim of this paper is to study the following fractional critical problem with Hardy potential and concave–convex nonlinearities (−Δ)su−μu|x|2s=λuq+u2s∗−1,u>0 in Ω,u=0in RN∖Ω, where (−Δ)s is the ...
- Published
- 2021
10. Singular matrices arising in the MFS from certain boundary and pseudo-boundary symmetries
- Author
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Andreas Karageorghis
- Subjects
Collocation ,Discretization ,Applied Mathematics ,Mathematical analysis ,Astrophysics::Instrumentation and Methods for Astrophysics ,General Engineering ,Regular polygon ,Boundary (topology) ,02 engineering and technology ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Computational Mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Polygon ,Method of fundamental solutions ,Boundary value problem ,0101 mathematics ,Astrophysics::Galaxy Astrophysics ,Analysis ,Mathematics - Abstract
We consider the application of the method of fundamental solutions (MFS) to certain two–dimensional second order boundary value problems (BVPs). We assume that the domain of the problem under consideration is a regular polygon and that the sources in the MFS are placed uniformly on a circle concentric to and surrounding the polygon. For certain uniform distributions of the collocation points we prove that, for general classes of regular polygons, the MFS discretization leads to singular coefficient matrices.
- Published
- 2021
11. Error estimates in weak Galerkin finite element methods for parabolic equations under low regularity assumptions
- Author
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Naresh Kumar and Bhupen Deka
- Subjects
Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,010103 numerical & computational mathematics ,01 natural sciences ,Parabolic partial differential equation ,Finite element method ,010101 applied mathematics ,Euler method ,Computational Mathematics ,symbols.namesake ,Discrete time and continuous time ,Norm (mathematics) ,symbols ,Piecewise ,0101 mathematics ,Galerkin method ,Mathematics - Abstract
In this paper, we consider the weak Galerkin finite element approximations of second order linear parabolic problems in two dimensional convex polygonal domains under the low regularities of the solutions. Optimal order error estimates in L 2 ( L 2 ) and L 2 ( H 1 ) norms are shown to hold for both the spatially discrete continuous time and the discrete time weak Galerkin finite element schemes, which allow using the discontinuous piecewise polynomials on finite element partitions with the arbitrary shape of polygons with certain shape regularity. The fully discrete scheme is based on first order in time Euler method. We have derived O ( h r + 1 ) in L 2 ( L 2 ) norm and O ( h r ) in L 2 ( H 1 ) norm when the exact solution u ∈ L 2 ( 0 , T ; H r + 1 ( Ω ) ) ∩ H 1 ( 0 , T ; H r − 1 ( Ω ) ) , for some r ≥ 1 . Numerical experiments are reported for several test cases to justify our theoretical convergence results.
- Published
- 2021
12. Mean value theorems for polynomial solutions of linear elliptic equations with constant coefficients in the complex plane
- Author
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Olga D. Trofymenko
- Subjects
Statistics and Probability ,Polynomial ,Constant coefficients ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mean value ,Regular polygon ,Computer Science::Computational Geometry ,01 natural sciences ,010305 fluids & plasmas ,Elliptic curve ,0103 physical sciences ,0101 mathematics ,Complex plane ,Mathematics - Abstract
We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon.
- Published
- 2021
13. Positive solutions for fractional Laplacian system involving concave-convex nonlinearities and sign-changing weight functions
- Author
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Maoding Zhen
- Subjects
Computational Mathematics ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,Fractional Laplacian ,Sign changing ,Analysis ,Mathematics - Published
- 2021
14. Approximate expression of the Prandtl membrane analogy in linear elastic pure torsion of open thin-walled cross sections and regular polygons
- Author
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Enrique Hernández-Montes, Antonio Palomares, and Luisa María Gil-Martín
- Subjects
Applied Mathematics ,Mechanical Engineering ,Prandtl number ,Mathematical analysis ,Linear elasticity ,Regular polygon ,Torsion (mechanics) ,02 engineering and technology ,021001 nanoscience & nanotechnology ,Condensed Matter Physics ,Membrane analogy ,law.invention ,symbols.namesake ,020303 mechanical engineering & transports ,Quadratic equation ,0203 mechanical engineering ,Mechanics of Materials ,law ,Modeling and Simulation ,symbols ,Piecewise ,General Materials Science ,Cartesian coordinate system ,0210 nano-technology ,Mathematics - Abstract
This paper presents a new general methodology to obtain an approximate analytical expression of the Saint-Venant's torsion. The shear stress in each of the principal Cartesian directions is obtained by the derivation of the stress function, whose analytical expression is obtained from the Prandtl analogy. The proposed methodology uses two variables quadratic piecewise functions to define the Prandtl membrane. This document shows that the approximate procedure is especially suitable for steel shapes, giving values very close to those obtained with more precise methods. The main advantage of the presented methodology is its simplicity, which makes it useful for both pedagogic and practical applications. Several examples are developed.
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- 2021
15. Averaging of Hamilton-Jacobi equations along divergence-free vector fields
- Author
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Taiga Kumagai and Hitoshi Ishii
- Subjects
Dirichlet problem ,Applied Mathematics ,Mathematical analysis ,Degenerate energy levels ,Regular polygon ,Hamilton–Jacobi equation ,Nonlinear system ,symbols.namesake ,Dirichlet boundary condition ,symbols ,Discrete Mathematics and Combinatorics ,Vector field ,Hamiltonian (quantum mechanics) ,Analysis ,Mathematics - Abstract
We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by divergence-free vector fields. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians ( \begin{document}$ G $\end{document} below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian \begin{document}$ G $\end{document} .
- Published
- 2021
16. Linear instability for periodic orbits of non-autonomous Lagrangian systems
- Author
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Ran Yang, Li Wu, and Alessandro Portaluri
- Subjects
Path (topology) ,Geodesic ,FOS: Physical sciences ,General Physics and Astronomy ,Dynamical Systems (math.DS) ,Linear instability ,Instability ,Periodic orbits, Non-autonomous Lagrangian functions, Linear instability, Maslov index, Spectral flow ,symbols.namesake ,FOS: Mathematics ,Periodic orbits ,Mathematics - Dynamical Systems ,Mathematics::Symplectic Geometry ,Legendre polynomials ,Mathematical Physics ,Mathematics ,58E10, 53C22, 53D12, 58J30 ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Riemannian manifold ,Non-autonomous Lagrangian functions ,Orientation (vector space) ,Maslov index ,Poincaré conjecture ,Spectral flow ,symbols - Abstract
Inspired by the classical Poincar\'e criterion about the instability of orientation preserving minimizing closed geodesics on surfaces, we investigate the relation intertwining the instability and the variational properties of periodic solutions of a non-autonomous Lagrangian on a finite dimensional Riemannian manifold. We establish a general criterion for a priori detecting the linear instability of a periodic orbit on a Riemannian manifold for a (maybe not Legendre convex) non-autonomous Lagrangian simply by looking at the parity of the spectral index, which is the right substitute of the Morse index in the framework of strongly indefinite variational problems and defined in terms of the spectral flow of a path of Fredholm quadratic forms on a Hilbert bundle., Comment: 32 pages, no figures
- Published
- 2021
17. Minimality of polytopes in a nonlocal anisotropic isoperimetric problem
- Author
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Marco Bonacini, Ihsan Topaloglu, and Riccardo Cristoferi
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Polytope ,Computer Science::Computational Geometry ,01 natural sciences ,010101 applied mathematics ,Perimeter ,Rigidity (electromagnetism) ,Mathematics - Analysis of PDEs ,49Q10, 49Q20, 49J10, 49K21 ,FOS: Mathematics ,Minification ,0101 mathematics ,Isoperimetric inequality ,Anisotropy ,Analysis ,Mathematics ,Analysis of PDEs (math.AP) ,Energy functional - Abstract
We consider the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. We show that, in the small mass regime, if the Wulff shape of the anisotropic perimeter has certain symmetry properties, then it is the unique global minimizer of the total energy. In dimension two this applies to convex polygons which are reflection symmetric with respect to the bisectors of the angles. We further prove a rigidity result for the structure of (local) minimizers in two dimensions.
- Published
- 2021
18. Nonunique Weak Solutions in Leray--Hopf Class for the Three-Dimensional Hall-MHD System
- Author
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Mimi Dai
- Subjects
Computational Mathematics ,Class (set theory) ,Pure mathematics ,Hall effect ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,Magnetohydrodynamics ,Analysis ,Mathematics - Abstract
Nonunique weak solutions in Leray--Hopf class are constructed for the three-dimensional magneto-hydrodynamics with Hall effect. We adapt the widely appreciated convex integration framework develope...
- Published
- 2021
19. Recovering the Initial Data of the Wave Equation from Neumann Traces
- Author
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Florian Dreier and Markus Haltmeier
- Subjects
Trace (linear algebra) ,medicine.diagnostic_test ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,Boundary (topology) ,Computed tomography ,Iterative reconstruction ,Directional derivative ,Wave equation ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,medicine ,0101 mathematics ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We study the problem of recovering the initial data (f, 0) of the standard wave equation from the Neumann trace (the normal derivative) of the solution on the boundary of convex domains in arbitrary spatial dimension. Among others, this problem is relevant for tomographic image reconstruction including photoacoustic tomography. We establish explicit inversion formulas of the back-projection type that recover the initial data up to an additive term defined by a smoothing integral operator. In the case that the boundary of the domain is an ellipsoid, the integral operator vanishes, and hence we obtain an analytic formula for recovering the initial data from Neumann traces of the wave equation on ellipsoids.
- Published
- 2021
20. Exact distribution for the generalized F tests
- Author
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Roman Zmy, Monte da Caparica, Miguel Fonseca, and Ao Tiago Mexia
- Subjects
Distribution (mathematics) ,Mathematical analysis ,Degrees of freedom (statistics) ,Regular polygon ,Applied mathematics ,Exact distribution ,Quotient ,Statistical hypothesis testing ,Mathematics - Abstract
Generalized F statistics are the quotients of convex combinations of central chi-squares divided by their degrees of freedom. Exact expressions are obtained for the distribution of these statistics when the degrees of freedom either in the numerator or in the denominator are even. An example is given to show how these expressions may be used to check the accuracy of Monte-Carlo methods in tabling these distributions. Moreover, when carrying out adaptative tests, these expressions enable us to estimate the p-values whenever they are available.
- Published
- 2023
21. A note on the selfsimilarity of limit flows
- Author
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Beomjun Choi, Robert Haslhofer, and Or Hershkovits
- Subjects
Mathematics - Differential Geometry ,Sequence ,Mean curvature flow ,Partial differential equation ,010308 nuclear & particles physics ,Applied Mathematics ,General Mathematics ,Open problem ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,01 natural sciences ,Physics::Fluid Dynamics ,Mathematics - Analysis of PDEs ,Singularity ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Gravitational singularity ,Limit (mathematics) ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if and only if there are only finitely many spherical singularities. More generally, using the solution of the mean convex neighborhood conjecture for neck singularities, we establish a local version of this equivalence for neck singularities in arbitrary dimension. In particular, we see that the ancient ovals occur as limit flows if and only if there is a sequence of spherical singularities converging to a neck singularity., Comment: 5 pages
- Published
- 2020
22. Travelling wave solutions in a negative nonlinear diffusion–reaction model
- Author
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Yifei Li, Peter van Heijster, Robert Marangell, and Matthew J. Simpson
- Subjects
Computer Science::Machine Learning ,Travelling wave solutions ,92D25 ,35B35 ,Thermal diffusivity ,Models, Biological ,Wiskundige en Statistische Methoden - Biometris ,Computer Science::Digital Libraries ,01 natural sciences ,Article ,Domain (mathematical analysis) ,010305 fluids & plasmas ,Diffusion ,Statistics::Machine Learning ,0103 physical sciences ,Traveling wave ,0101 mathematics ,Mathematical and Statistical Methods - Biometris ,Spectral stability ,Phase plane analysis ,Physics ,Nonlinear diffusion ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Geometric methods ,Function (mathematics) ,PE&RC ,Agricultural and Biological Sciences (miscellaneous) ,92C17 ,Linear map ,Nonlinear system ,Nonlinear Dynamics ,35K57 ,Modeling and Simulation ,Computer Science::Mathematical Software ,Sign (mathematics) - Abstract
We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion–reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, $$c^*$$ c ∗ , and investigate its relation to the spectral stability of a desingularised linear operator associated with the travelling wave solutions.
- Published
- 2020
23. On Particle-Size Distribution of Convex Similar Bodies in $${\mathbb {R}}^3$$
- Author
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Gabriela Baluchová and Jozef Kiselak
- Subjects
Statistics and Probability ,Distribution (number theory) ,Applied Mathematics ,Probability (math.PR) ,Mathematical analysis ,Regular polygon ,02 engineering and technology ,Condensed Matter Physics ,Integral equation ,45E99, 45D05, 60D05 ,Modeling and Simulation ,Particle-size distribution ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Geometry and Topology ,Computer Vision and Pattern Recognition ,Uniqueness ,Mathematics - Probability ,Mathematics - Abstract
We have solved an old problem posed by Santalo of determining the size distribution of particles derived from the size distribution of their sections. We give an explicit form of particle-size distributions of convex similar bodies for random planes and random lines, which naturally generalize famous Wicksell’s corpuscle problem. The results are achieved by applying the method of model solutions for solving well-known Santalo’s integral equations. We give a partial result related to the question of the existence and uniqueness of these solutions. We also emphasize that the original form of solution of Wicksell’s problem is insufficient. We finally illustrate our approach in several examples.
- Published
- 2020
24. Reflected BSDEs with jumps in time-dependent convex càdlàg domains
- Author
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Youssef Ouknine, M’hamed Eddahbi, and Imade Fakhouri
- Subjects
Statistics and Probability ,Class (set theory) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Type (model theory) ,01 natural sciences ,010104 statistics & probability ,Stochastic differential equation ,Reflection (mathematics) ,Mathematics::Probability ,Modeling and Simulation ,Uniqueness ,0101 mathematics ,Normal ,Mathematics - Abstract
In the first part of the paper, we study the unique solvability of multidimensional reflected backward stochastic differential equations (RBSDEs) of Wiener–Poisson type with reflection in the inward spatial normal direction of a time-dependent adapted cadlag convex set D = { D t , t ∈ [ 0 , T ] } . The existence result is obtained by approximating the solutions of this class of RBSDEs by solutions of BSDEs with reflection in discretizations of D , while the uniqueness is established by using Ito’s formula. In the second part of the paper, we show that the solutions of our RBSDEs can be approximated via a non-standard penalization method.
- Published
- 2020
25. TWO DIMENSIONAL ZONOIDS AND CHEBYSHEV MEASURES
- Author
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Stefano Bianchini, Raphaël Cerf, Carlo Mariconda, and Cerf, Raphael
- Subjects
Pure mathematics ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,Boundary (topology) ,52A10 ,Chebyshev filter ,Measure (mathematics) ,[MATH.MATH-CA] Mathematics [math]/Classical Analysis and ODEs [math.CA] ,Convexity ,Range (mathematics) ,Vector measure ,Settore MAT/05 - Analisi Matematica ,41A50 ,1991 Mathematics Subject Classification. 46G10 ,26A51 ,Convex function ,Analysis ,Mathematics - Abstract
We give an alternative proof to the well known fact that each convex compact centrally symmetric subset of R 2containing the origin is a zonoid, i.e., the range of a two dimensional vector measure, and we prove that a two dimensional zonoid whose boundary contains the origin is strictly convex if and only if it is the range of a Chebyshev measure. We give a condition under which a two dimensional vector measure admits a decomposition as the difference of two Chebyshev measures, a necessary condition on the density function for the strict convexity of the range of a measure and a characterization of two dimensional Chebyshev measures.
- Published
- 2022
26. K-mean convex and K-outward minimizing sets
- Author
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Annalisa Cesaroni and Matteo Novaga
- Subjects
Mean curvature ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,k-means clustering ,nonlocal minimal surfaces ,minimizing movements ,mean convexity ,Nonlocal curvature flows ,level set flow ,Convexity ,Level set ,Flow (mathematics) ,Convergence (routing) ,Limit (mathematics) ,Mathematics - Abstract
We consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level set formulation and the minimizing movement scheme for the nonlocal flow. When the initial set is outward minimizing, we also show the convergence of the (time integrated) nonlocal perimeters of the discrete evolutions to the nonlocal perimeter of the limit flow.
- Published
- 2022
27. On the Spherical Quasi-convexity of Quadratic Functions on Spherically Subdual Convex Sets
- Author
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Lianghai Xiao, Orizon P. Ferreira, and Sandor Nemeth
- Subjects
021103 operations research ,Control and Optimization ,Applied Mathematics ,Lorentz transformation ,Mathematical analysis ,0211 other engineering and technologies ,Regular polygon ,010103 numerical & computational mathematics ,02 engineering and technology ,Quadratic function ,Management Science and Operations Research ,Characterization (mathematics) ,01 natural sciences ,Convexity ,symbols.namesake ,Theory of computation ,symbols ,0101 mathematics ,Mathematics - Abstract
In this paper, the spherical quasi-convexity of quadratic functions on spherically subdual convex sets is studied. Sufficient conditions for spherical quasi-convexity on spherically subdual convex sets are presented. A partial characterization of spherical quasi-convexity on spherical Lorentz sets is given, and some examples are provided.
- Published
- 2020
28. On a length-preserving inverse curvature flow of convex closed plane curves
- Author
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Dong-Ho Tsai, Laiyuan Gao, and Shengliang Pan
- Subjects
Plane curve ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Inverse ,Curvature ,01 natural sciences ,Convexity ,Physics::Fluid Dynamics ,010101 applied mathematics ,Flow (mathematics) ,Global flow ,0101 mathematics ,Analysis ,Mathematics - Abstract
This paper deals with a 1 / κ α -type length-preserving nonlocal flow of convex closed plane curves for all α > 0 . Under this flow, the convexity of the evolving curve is preserved. For a global flow, it is shown that the evolving curve converges smoothly to a circle as t → ∞ . Some numerical blow-up examples and a sufficient condition leading to the global existence of the flow are also constructed.
- Published
- 2020
29. Parallel coordinates in three dimensions and sharp spectral isoperimetric inequalities
- Author
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Anastasia V. Vikulova
- Subjects
Conjecture ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,01 natural sciences ,010305 fluids & plasmas ,Mathematics - Spectral Theory ,0103 physical sciences ,FOS: Mathematics ,Ball (mathematics) ,Diffeomorphism ,0101 mathematics ,Isoperimetric inequality ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Parallel coordinates ,Mathematics - Abstract
In this paper we show how the method of parallel coordinates can be extended to three dimensions. As an application, we prove the conjecture of Antunes, Freitas and Krej\v{c}i\v{r}\'ik \cite{AFK} that "the ball maximises the first Robin eigenvalue with negative boundary parameter among all convex domains of equal surface area" under the weaker restriction that the boundary of the domain is diffeomorphic to the sphere and convex or axiconvex. We also provide partial results in arbitrary dimensions.
- Published
- 2020
30. Non-uniqueness of weak solutions to 2D hypoviscous Navier-Stokes equations
- Author
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Peng Qu and Tianwen Luo
- Subjects
Applied Mathematics ,Weak solution ,010102 general mathematics ,Non uniqueness ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Regular polygon ,Type (model theory) ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Scheme (mathematics) ,FOS: Mathematics ,Compressibility ,0101 mathematics ,Navier–Stokes equations ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Through an adaption of the convex integration scheme in the two dimensional case, a result of the h-principle type is presented for the two-dimensional hypoviscous incompressible Navier-Stokes equations. It is shown that the C t 0 L x 2 weak solutions can possess compact temporal supports and thus are not unique in general.
- Published
- 2020
31. Constant diameter and constant width of spherical convex bodies
- Author
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Denghui Wu and Huhe Han
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Metric Geometry (math.MG) ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics - Metric Geometry ,FOS: Mathematics ,Pi ,Discrete Mathematics and Combinatorics ,Convex body ,Mathematics::Differential Geometry ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
In this paper we show that a spherical convex body $C$ is of constant diameter $\tau$ if and only if $C$ is of constant width $\tau$, for $0, Comment: 6 pages
- Published
- 2020
32. Nodal solution for Kirchhoff-type problems with concave-convex nonlinearities
- Author
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Zeng-Qi Ou and Bin Chen
- Subjects
Numerical Analysis ,Kirchhoff type ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,0101 mathematics ,Nehari manifold ,NODAL ,Analysis ,Mathematics - Abstract
In this paper, we study the existence of nodal solutions for the Kirchhoff-type problem with concave-convex nonlinearities − a + b ∫ Ω | ∇ u | 2 d x Δ u = λ | u | q − 1 u + | u | p − 1 u i n Ω , ...
- Published
- 2020
33. Hyperelastic deformations and total combined energy of mappings between annuli
- Author
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David Kalaj
- Subjects
Mathematics - Complex Variables ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Annulus (mathematics) ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Distortion (mathematics) ,Operator (computer programming) ,Hyperelastic material ,Euclidean geometry ,FOS: Mathematics ,Complex Variables (math.CV) ,0101 mathematics ,Linear combination ,Analysis ,Mathematics - Abstract
We consider the so called combined energy of a deformation between two concentric annuli and minimize it, provided that it keep order of the boundaries. It is an extension of the corresponding result of Euclidean energy. It is intrigue that, the minimizers are certain radial mappings and they exists if and only if the annulus on the image domain is not too thin, provided that the original annulus is fixed. This in turn implies a Nitsche type phenomenon. Next we consider the combined distortion and obtain certain related results which are dual to the results for combined energy, which also involve some Nitche type phenomenon. {The main part of the paper is concerned with the total combined energy, a certain integral operator, defined as a convex linear combination of the combined energy and combined distortion, of diffeomorphisms between two concentric annuli $\A(1,r)$ and $\B(1,R)$. First we construct radial minimizers of total combined energy, then we prove that those radial minimizers are absolute minimizers on the class of all mappings between the annuli under certain constraint. This extends the main result obtained by Iwaniec and Onninen in \cite{arma}.}, Comment: 30 pages, 6 figures
- Published
- 2020
34. A Quasiconcavity Property for the Heat Equation in a Convex Ring
- Author
-
Jingjing Suo
- Subjects
Statistics and Probability ,Computational Mathematics ,Ring (mathematics) ,Quasiconvex function ,Property (philosophy) ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,Boundary (topology) ,Heat equation ,Probability density function ,Brownian motion ,Mathematics - Abstract
We give an exposition of a result of Borell (Commun Math Phys 86:143–147, 1982) that the probability function that Brownian motion hits the inner boundary before time t and before hitting the outer boundary is a space-time quasiconcave function.
- Published
- 2020
35. An overdetermined problem of anisotropic equations in convex cones
- Author
-
Liangjun Weng
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Regular polygon ,01 natural sciences ,Domain (mathematical analysis) ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Overdetermined system ,Maximum principle ,Bounded function ,Convex cone ,Boundary value problem ,0101 mathematics ,Anisotropy ,Analysis ,Mathematics - Abstract
In this paper, we study some overdetermined boundary value problems for anisotropic elliptic PDEs in a bounded domain Ω in a convex cone of R n . By using some integral identities and maximum principle, we prove the corresponding Wulff shape characterizations, which includes the classical Serrin's overdetermined boundary value problem.
- Published
- 2020
36. Harmonic mappings with analytic part convex in one direction
- Author
-
J. K. Prajapat, Sudhananda Maharana, and M. Manivannan
- Subjects
symbols.namesake ,Algebra and Number Theory ,Mathematics::Complex Variables ,Fourier analysis ,Special functions ,Applied Mathematics ,Norm (mathematics) ,Mathematical analysis ,Regular polygon ,symbols ,Geometry and Topology ,Analysis ,Mathematics - Abstract
In this paper, we study a family of sense-preserving harmonic mappings whose analytic part is convex in one direction. We first establish the bounds on the pre-Schwarzian norm. Next, we obtain radius of fully starlike and radius of fully convex for this family of harmonic mappings.
- Published
- 2020
37. Existence of multiple positive solutions for a truncated Kirchhoff-type system involving weight functions and concave–convex nonlinearities
- Author
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Yupeng Qin and Qingjun Lou
- Subjects
geography ,Algebra and Number Theory ,geography.geographical_feature_category ,Partial differential equation ,Nehari manifold ,Kirchhoff type ,Applied Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,lcsh:QA1-939 ,01 natural sciences ,Multiple positive solutions ,010101 applied mathematics ,Kirchhoff system ,Bounded function ,Ordinary differential equation ,Mountain pass theorem ,Mountain pass ,0101 mathematics ,Analysis ,Mathematics - Abstract
We consider the combined effect of concave–convex nonlinearities on the number of solutions for an indefinite truncated Kirchhoff-type system involving the weight functions. When $\alpha+ \betaα+β<4, since the concave-convex nonlinearities do not satisfy the mountain pass geometry, it is difficult to obtain a bounded Palais–Smale sequence by the usual mountain pass theorem. To overcome the problem, we properly introduce a method of Nehari manifold and then establish the existence of multiple positive solutions when the pair of the parameters is under a certain range.
- Published
- 2020
38. Anisotropic flow of convex hypersurfaces by the square root of the scalar curvature
- Author
-
Ki-Ahm Lee, Lami Kim, and Hyunsuk Kang
- Subjects
Euclidean space ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,01 natural sciences ,Manifold ,010101 applied mathematics ,Hypersurface ,Square root ,Flow (mathematics) ,Mathematics::Differential Geometry ,0101 mathematics ,Convex function ,Analysis ,Mathematics ,Scalar curvature - Abstract
We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor ψ when the strictly convex initial hypersurface in Euclidean space is suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere. For a general case in dimension two, it is shown that, with a volume preserving rescaling, the limit profile satisfies a soliton equation.
- Published
- 2020
39. Entropy and drift for Gibbs measures on geometrically finite manifolds
- Author
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Giulio Tiozzo and Ilya Gekhtman
- Subjects
Entropy (statistical thermodynamics) ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Geometric Topology (math.GT) ,Dynamical Systems (math.DS) ,Random walk ,01 natural sciences ,Mathematics - Geometric Topology ,Metric space ,Corollary ,FOS: Mathematics ,Mathematics - Dynamical Systems ,60G50, 37D35, 53D25, 60J50 ,0101 mathematics ,Critical exponent ,Mathematics - Probability ,Quotient ,Mathematics - Abstract
We prove a generalization of the fundamental inequality of Guivarc'h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of CAT(-1) metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is equivalent to the hitting measure. As a corollary, if the action is not convex cocompact, any hitting measure is singular to any Gibbs density., Comment: 29 pages
- Published
- 2020
40. Global and Local Pointwise Error Estimates for Finite Element Approximations to the Stokes Problem on Convex Polyhedra
- Author
-
Dmitriy Leykekhman, Boris Vexler, and Niklas Behringer
- Subjects
Pointwise ,Numerical Analysis ,65N30, 65N15 ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,Finite element approximations ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Finite element solution ,01 natural sciences ,Stability (probability) ,Finite element method ,Computational Mathematics ,Polyhedron ,FOS: Mathematics ,Stokes problem ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Abstract
The main goal of the paper is to show new stability and localization results for the finite element solution of the Stokes system in $W^{1,\infty}$ and $L^{\infty}$ norms under standard assumptions on the finite element spaces on quasi-uniform meshes in two and three dimensions. Although interior error estimates are well-developed for the elliptic problem, they appear to be new for the Stokes system on unstructured meshes. To obtain these results we extend previously known stability estimates for the Stokes system using regularized Green's functions.
- Published
- 2020
41. Boundary Blow-Up Analysis of Gradient Estimates for Lamé Systems in the Presence of $m$-Convex Hard Inclusions
- Author
-
Haigang Li and Zhiwen Zhao
- Subjects
Applied Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Regular polygon ,Boundary (topology) ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Mathematics - Analysis of PDEs ,Point (geometry) ,0101 mathematics ,Analysis ,Mathematics ,Stress concentration - Abstract
In high-contrast elastic composites, it is vitally important to investigate the stress concentration from an engineering point of view. The purpose of this paper is to show that the blowup rate of the stress depends not only on the shape of the inclusions, but also on the given boundary data, when hard inclusions are close to matrix boundary. First, when the boundary of inclusion is partially relatively parallel to that of matrix, we establish the gradient estimates for Lam\'{e} systems with partially infinite coefficients and find that they are bounded for some boundary data $\varphi$ while some $\varphi$ will increase the blow-up rate. In order to identify such novel blowup phenomenon, we further consider the general $m$-convex inclusion cases and uncover the dependence of blow-up rate on the inclusion's convexity $m$ and the boundary data's order of growth $k$ in all dimensions. In particular, the sharpness of these blow-up rates is also presented for some prescribed boundary data.
- Published
- 2020
42. Traveling front of polyhedral shape for a nonlocal delayed diffusion equation
- Author
-
Jia Liu
- Subjects
polyhedral shape ,Diffusion equation ,Applied Mathematics ,Mathematical analysis ,Front (oceanography) ,Regular polygon ,traveling front ,Stability result ,Stability (probability) ,Exponential stability ,QA1-939 ,reaction-diffusion equation ,nonlocal delayed ,Diffusion (business) ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematics - Abstract
This paper is concerned with the existence and stability of traveling fronts with convex polyhedral shape for nonlocal delay diffusion equations. By using the existence and stability results of V-form fronts and pyramidal traveling fronts, we first show that there exists a traveling front V(x, y, z) with polyhedral shape of nonlocal delay diffusion equation associated with z = h(x, y). Moreover, the asymptotic stability and other qualitative properties of such traveling front V(x, y, z) are also established.
- Published
- 2020
43. Differential inclusions with mean derivatives having extreme right-hand sides and optimal control
- Author
-
Olga O. Zheltikova and Yuri E. Gliklikh
- Subjects
Differential inclusion ,Applied Mathematics ,Mathematical analysis ,Regular polygon ,Extreme right ,Extreme point ,Optimal control ,Analysis ,Mathematics - Abstract
First we prove the existence of solutions of some special stochastic differential inclusion with mean derivatives having lower semi-continuous right-hand sides that may not be convex. Then we show ...
- Published
- 2019
44. Superharmonicity of curvature function for the convex level sets of harmonic functions
- Author
-
Wei Zhang and Xi-Nan Ma
- Subjects
symbols.namesake ,Subharmonic function ,Harmonic function ,Applied Mathematics ,Mathematical analysis ,Curvature function ,Mathematics::Analysis of PDEs ,Gaussian curvature ,symbols ,Regular polygon ,Analysis ,Mathematics - Abstract
We prove that the combination of the norm of gradient and the Gaussian curvature for the convex level sets of harmonic function is superharmonic.
- Published
- 2021
45. Minimal brake orbits of first-order convex Hamiltonian systems with anisotropic growth
- Author
-
Chungen Liu and Xiaofei Zhang
- Subjects
Physics ,Numerical Analysis ,Partial differential equation ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Regular polygon ,Anisotropic growth ,Hamiltonian system ,Dual (category theory) ,Computational Mathematics ,Variational principle ,Brake ,Analysis - Abstract
Using dual variational principle, we concern the problem of existence of minimal periodic brake orbits of some first-order convex Hamiltonian systems with anisotropic growth.
- Published
- 2021
46. Extremal Problems for Convex Curves with a Given Self Chebyshev Radius
- Author
-
Yulia Nikonorova, Vitor Balestro, Horst Martini, and Yurii Nikonorov
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Boundary (topology) ,Radius ,Computer Science::Computational Geometry ,01 natural sciences ,Chebyshev filter ,010101 applied mathematics ,Perimeter ,Mathematics (miscellaneous) ,Euclidean geometry ,Mathematics::Metric Geometry ,0101 mathematics ,Mathematics - Abstract
The paper is devoted to some extremal problems for convex curves and polygons in the Euclidean plane referring to the self Chebyshev radius. In particular, we determine the self Chebyshev radius for the boundary of an arbitrary triangle. Moreover, we derive the maximal possible perimeter for convex curves and boundaries of convex n-gons with a given self Chebyshev radius.
- Published
- 2021
47. Error estimates for two-scale composite finite element approximations of parabolic equations with measure data in time for convex and nonconvex polygonal domains
- Author
-
Rajen Kumar Sinha and Tamal Pramanick
- Subjects
Numerical Analysis ,Approximations of π ,Applied Mathematics ,Composite number ,Mathematical analysis ,Regular polygon ,Finite element approximations ,010103 numerical & computational mathematics ,01 natural sciences ,Parabolic partial differential equation ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,Norm (mathematics) ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In this exposition we study two-scale composite finite element approximations of parabolic problems with measure data in time for both convex and nonconvex polygonal domains. This research is motivated by the work of Hackbusch and Sauter [Numer. Math., 75 (1997) 447–472] on the composite finite element approximations of elliptic boundary value problems. The main features of the composite finite element method is that, it not only uses minimal dimension of the approximation space but also handle the domain boundary in a flexible and systematic manner, which is very advantageous for domains with complicated geometry. Both spatially semidiscrete and fully discrete approximations of the proposed method are analyzed. In the case of convex domains, we derive error estimate of order O ( H Log ˜ 1 / 2 ( H / h ) + k 1 / 2 ) in the L 2 ( 0 , T ; L 2 ( Ω ) ) -norm, where H and h denote the coarse-scale and fine-scale mesh size, respectively, and k is the time step. Further, an error estimate of order O ( H s Log ˜ s / 2 ( H / h ) + k 1 / 2 ) , 1 / 2 ≤ s ≤ 1 is shown to hold in the L 2 ( 0 , T ; L 2 ( Ω ) ) -norm for nonconvex domains. Numerical experiment confirms the theoretical findings and reveals the potential of the composite finite element method.
- Published
- 2019
48. A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results
- Author
-
Rafael de la Llave, Tere M-Seara, Marian Gidea, Universitat Politècnica de Catalunya. Departament de Matemàtiques, and Universitat Politècnica de Catalunya. SD - Sistemes Dinàmics de la UPC
- Subjects
Integrable system ,General Mathematics ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,01 natural sciences ,Hamiltonian system ,010104 statistics & probability ,symbols.namesake ,FOS: Mathematics ,Sistemes hamiltonians ,Normally hyperbolic invariant manifold ,Mathematics - Dynamical Systems ,Hamiltonian systems ,0101 mathematics ,Mathematical Physics ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Matemàtiques i estadística [Àrees temàtiques de la UPC] ,Torus ,Mathematical Physics (math-ph) ,Invariant (physics) ,Nonlinear Sciences - Chaotic Dynamics ,37J40, 37C50, 37C29, 37B30 ,symbols ,A priori and a posteriori ,Chaotic Dynamics (nlin.CD) ,Hamiltonian (quantum mechanics) - Abstract
We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the `outer dynamics' along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined map, referred to as the `scattering map'. We find pseudo-orbits of the scattering map that keep advancing in some privileged direction. Then we use the recurrence property of the `inner dynamics', restricted to the normally hyperbolic invariant manifold, to return to those pseudo-orbits. Finally, we apply topological methods to show the existence of true orbits that follow the successive applications of the two dynamics. This method differs, in several crucial aspects, from earlier works. Unlike the well known `two-dynamics' approach, the method we present relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, as its invariant objects (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets) are not used at all. The method applies to unperturbed Hamiltonians of arbitrary degrees of freedom that are not necessarily convex. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, as well as to establish diffusion in generic systems. We include several applications, such as bridging large gaps in a priori unstable models in any dimension, and establishing diffusion in cases when the inner dynamics is a non-twist map.
- Published
- 2019
49. A Monge–Ampère problem with non-quadratic cost function to compute freeform lens surfaces
- Author
-
J.H.M. ten Thije Boonkkamp, Wilbert L. IJzerman, N.K. Yadav, and Center for Analysis, Scientific Computing & Appl.
- Subjects
Least-squares method ,Differential equation ,Monge–Ampère equation ,01 natural sciences ,Theoretical Computer Science ,law.invention ,law ,Freeform lens surfaces ,Transport boundary conditions ,Boundary value problem ,0101 mathematics ,Mathematics ,Numerical Analysis ,Non-quadratic cost function ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Regular polygon ,Optical design ,Function (mathematics) ,Inverse problem ,Ray ,010101 applied mathematics ,Lens (optics) ,Computational Mathematics ,Computational Theory and Mathematics ,Software - Abstract
In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost function. The lens can transfer a given emittance of the source into a desired illuminance at the target. The freeform lens design problem can be formulated as a Monge–Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction. Our least-squares algorithm is capable to handle a non-quadratic cost function, and provides two solutions corresponding to either convex or concave lens surfaces.
- Published
- 2019
50. Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type
- Author
-
Brian White and Or Hershkovits
- Subjects
Mean curvature flow ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Regular polygon ,Gravitational singularity ,Type (model theory) ,Mathematics - Published
- 2019
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