377 results on '"Mathematics"'
Search Results
2. The vanishing rate of Weil–Petersson sectional curvatures
- Author
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Scott A. Wolpert
- Subjects
Geodesic ,Hyperbolic geometry ,Riemann surface ,Mathematical analysis ,Curvature ,Moduli space ,symbols.namesake ,Differential geometry ,Metric (mathematics) ,symbols ,Mathematics::Differential Geometry ,Geometry and Topology ,Sectional curvature ,Mathematics - Abstract
The Weil–Petersson metric for the moduli space of Riemann surfaces has negative sectional curvature. Surfaces represented in the complement of a compact set in the moduli space have short geodesics. At such surfaces the Weil–Petersson metric is approximately a product metric. An almost product metric has sections with almost vanishing curvature. We bound the sectional curvature away from zero in terms of the product of lengths of short geodesics on Riemann surfaces. We give examples and an expectation for the actual vanishing rate.
- Published
- 2021
3. Periodic magnetic geodesics on Heisenberg manifolds
- Author
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Jonathan Epstein, Ruth Gornet, and Maura B. Mast
- Subjects
Mathematics - Differential Geometry ,Geodesic ,Mathematical analysis ,Dynamical Systems (math.DS) ,Riemannian geometry ,Magnetic field ,symbols.namesake ,Differential Geometry (math.DG) ,Differential geometry ,Flow (mathematics) ,Metric (mathematics) ,FOS: Mathematics ,Heisenberg group ,symbols ,Mathematics::Differential Geometry ,Geometry and Topology ,Mathematics - Dynamical Systems ,Invariant (mathematics) ,Analysis ,Mathematics - Abstract
We study the dynamics of magnetic flows on Heisenberg groups. Let $H$ denote the three-dimensional simply connected Heisenberg Lie group endowed with a left-invariant Riemannian metric and an exact, left-invariant magnetic field. Let $\Gamma$ be a lattice subgroup of $H,$ so that $\Gamma\backslash H$ is a closed nilmanifold. We first find an explicit description of magnetic geodesics on $H$, then determine all closed magnetic geodesics and their lengths for $\Gamma \backslash H$. We then consider two applications of these results: the density of periodic magnetic geodesics and marked magnetic length spectrum rigidity., Comment: 32 pages
- Published
- 2021
4. Symmetric solutions of the singular minimal surface equation
- Author
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Nico Groh and Ulrich Dierkes
- Subjects
021103 operations research ,Minimal surface ,010102 general mathematics ,Mathematical analysis ,0211 other engineering and technologies ,02 engineering and technology ,01 natural sciences ,Stability (probability) ,Alpha (programming language) ,Differential geometry ,Mathematik ,Geometry and Topology ,0101 mathematics ,Analysis ,Mathematics - Abstract
We classify all rotational symmetric solutions of the singular minimal surface equation in both cases $$\alpha α < 0 and $$\alpha >0$$ α > 0 . In addition, we discuss further geometric and analytic properties of the solutions, in particular stability, minimizing properties and Bernstein-type results.
- Published
- 2021
5. On the fluid ball conjecture
- Author
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Benedito Leandro, Hiuri Fellipe dos Santos Reis, and Fernando Coutinho
- Subjects
Mathematics - Differential Geometry ,Conjecture ,Equation of state (cosmology) ,010102 general mathematics ,Mathematical analysis ,FOS: Physical sciences ,Perfect fluid ,Rigidity (psychology) ,Mathematical Physics (math-ph) ,01 natural sciences ,Physics::Fluid Dynamics ,Differential Geometry (math.DG) ,Differential geometry ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,Ball (mathematics) ,Boundary value problem ,0101 mathematics ,Divergence (statistics) ,Mathematical Physics ,Analysis ,Mathematics - Abstract
The fluid ball conjecture states that a static perfect fluid space-time is spherically symmetric. In this paper we construct a Robinson’s divergence formula for the static perfect fluid space-time. Inspired by this conjecture, a rigidity result for the spatial factor of a static perfect fluid space-time satisfying some boundary conditions is proved, provided that an equation of state holds.
- Published
- 2021
6. Spectral Determinant on Euclidean Isosceles Triangle Envelopes
- Author
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Victor Kalvin
- Subjects
Differential geometry ,Mathematical analysis ,Isosceles triangle ,Euclidean geometry ,Geometry and Topology ,Function (mathematics) ,Computer Science::Computational Geometry ,Equilateral triangle ,Laplace operator ,Critical point (mathematics) ,Mathematics ,Envelope (waves) - Abstract
We study extremal properties of the determinant of Friedrichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. We deduce an explicit closed formula for the determinant. Small-angle asymptotics show that the determinant grows without any bound as an angle of a triangle envelope goes to zero. We prove that the equilateral triangle envelope (the most symmetrical geometry) always gives rise to a critical point of the determinant and finds the critical value. When the area is below a certain value (approximately 1.92), the equilateral envelope minimizes the determinant. When the area exceeds this value, the equilateral envelope is a local maximum of the determinant.
- Published
- 2021
7. Non-Parametric Shape Design of Free-Form Shells Using Fairness Measures and Discrete Differential Geometry
- Author
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Kentaro Hayakawa and Makoto Ohsaki
- Subjects
GAUSS MAP ,Shape design ,Mechanical Engineering ,SHELL ,Mathematical analysis ,Nonparametric statistics ,FAIRNESS MEASURE ,Building and Construction ,DIFFERENTIAL GEOMETRY ,SHAPE OPTIMIZATION ,FORM-FINDING ,Arts and Humanities (miscellaneous) ,Fairness measure ,Free form ,Discrete differential geometry ,Civil and Structural Engineering ,Mathematics - Abstract
A non-parametric approach is proposed for shape design of free-form shells discretized into triangular mesh. The discretized forms of curvatures are used for computing the fairness measures of the surface. The measures are defined as the area of the offset surface and the generalized form of the Gauss map. Gaussian curvature and mean curvature are computed using the angle defect and the cotangent formula, respectively, defined in the field of discrete differential geometry. Optimization problems are formulated for minimizing various fairness measures for shells with specified boundary conditions. A piecewise developable surface can be obtained without a priori assignment of the internal boundary. Effectiveness of the proposed method for generating various surface shapes is demonstrated in the numerical examples.
- Published
- 2021
8. Low-Dimensional Pinned Distance Sets Via Spherical Averages
- Author
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Terence L. J. Harris
- Subjects
Mathematical analysis ,Mathematics::Classical Analysis and ODEs ,Distance measures ,symbols.namesake ,42B10, 28A78 ,Differential geometry ,Dimension (vector space) ,Mathematics - Classical Analysis and ODEs ,Fourier analysis ,Condensed Matter::Superconductivity ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Geometry and Topology ,GEOM ,Mathematics - Abstract
An inequality is derived for the average $t$-energy of pinned distance measures, where $0 < t < 1$. This refines Mattila's theorem on distance sets to pinned distance sets, and gives an analogue of Liu's theorem for pinned distance sets of dimension smaller than 1., 5 pages. Accepted version
- Published
- 2021
9. Stable approximations for axisymmetric Willmore flow for closed and open surfaces
- Author
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Harald Garcke, John W. Barrett, and Robert Nürnberg
- Subjects
Numerical Analysis ,Mean curvature ,65M60, 65M12, 35K55, 53C44 ,Willmore flow / Helfrich flow / axisymmetry / parametric finite elements / stability / tangential movement / spontaneous curvature / ADE model / clamped boundary conditions / Navier boundary conditions / Gaussian curvature energy / line energy ,Applied Mathematics ,Mathematical analysis ,Numerical Analysis (math.NA) ,Curvature ,Computational Mathematics ,Willmore energy ,symbols.namesake ,Hypersurface ,Flow (mathematics) ,Differential geometry ,Modeling and Simulation ,FOS: Mathematics ,Gaussian curvature ,symbols ,Mathematics - Numerical Analysis ,Mathematics::Differential Geometry ,Boundary value problem ,Analysis ,Mathematics - Abstract
For a hypersurface in ${\mathbb R}^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods., 52 pages, 19 figures
- Published
- 2021
10. Angular Extents and Trajectory Slopes in the Theory of Holomorphic Semigroups in the Unit Disk
- Author
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Pavel Gumenyuk, Manuel D. Contreras, Santiago Díaz-Madrigal, Universidad de Sevilla. Departamento de Matemática Aplicada II, and Universidad de Sevilla. FQM133: Grupo de Investigación en Análisis Funcional
- Subjects
Planar domain ,Koenigs function ,Holomorphic function ,Slope problem ,Primary: 30D05, Secondary: 30C35, 30C45, 34M15 ,Dynamical Systems (math.DS) ,30C45 ,01 natural sciences ,Domain (mathematical analysis) ,Image (mathematics) ,34M15 ,Corollary ,0103 physical sciences ,FOS: Mathematics ,Semigroups of holomorphic functions ,Mathematics - Dynamical Systems ,Complex Variables (math.CV) ,0101 mathematics ,Mathematics ,Mathematics - Complex Variables ,Mathematics::Complex Variables ,Semigroup ,010102 general mathematics ,Mathematical analysis ,Function (mathematics) ,30D05 [Primary] ,Unit disk ,Differential geometry ,010307 mathematical physics ,Geometry and Topology ,30C35 [Secondary] - Abstract
We study relationships between the asymptotic behaviour of a non-elliptic semigroup of holomorphic self-maps of the unit disk and the geometry of its planar domain (the image of the Koenigs function). We establish a sufficient condition for the trajectories of the semigroup to converge to its Denjoy–Wolff point with a definite slope. We obtain as a corollary two previously known sufficient conditions. Ministerio de Economía y Competitividad (Spain) / European Union (FEDER) PGC2018-094215-B-100 Junta de Andalucía (Spain) FQM-133
- Published
- 2021
11. Sharp upper diameter bounds for compact shrinking Ricci solitons
- Author
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Jia-Yong Wu
- Subjects
Mathematics - Differential Geometry ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Ricci soliton ,Differential Geometry (math.DG) ,Differential geometry ,0103 physical sciences ,FOS: Mathematics ,SPHERES ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Entropy (arrow of time) ,Analysis ,Scalar curvature ,Mathematics ,Logarithmic sobolev inequality - Abstract
We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman's entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument., 13 pages, coefficient of Theorem 1.1 improved, accepted by AGAG
- Published
- 2021
12. On Finite Energy Solutions of 4-harmonic and ES-4-harmonic Maps
- Author
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Branding, Volker
- Subjects
Mathematics - Differential Geometry ,ES-4-harmonic maps ,Harmonic (mathematics) ,53C43 ,01 natural sciences ,Article ,4-harmonic maps ,symbols.namesake ,Mathematics - Analysis of PDEs ,Nonexistence result ,0103 physical sciences ,FOS: Mathematics ,Order (group theory) ,0101 mathematics ,Mathematics ,58E20 ,Euclidean space ,010102 general mathematics ,Mathematical analysis ,Harmonic maps ,Harmonic map ,Nonlinear system ,Differential Geometry (math.DG) ,Elliptic partial differential equation ,Differential geometry ,Fourier analysis ,symbols ,010307 mathematical physics ,Geometry and Topology ,Analysis of PDEs (math.AP) - Abstract
4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied harmonic map equation which are both given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. In this article we prove that finite energy solutions of both 4-harmonic and ES-4-harmonic maps from Euclidean space must be trivial. However, the energy that we require to be finite is different for 4-harmonic and ES-4-harmonic maps pointing out a first difference between these two variational problems.
- Published
- 2021
13. Travel Time Tomography in Stationary Spacetimes
- Author
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Gunther Uhlmann, Yang Yang, and Hanming Zhou
- Subjects
010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Rigidity (psychology) ,Function (mathematics) ,Stationary spacetime ,01 natural sciences ,Domain (mathematical analysis) ,Differential geometry ,0103 physical sciences ,Metric (mathematics) ,010307 mathematical physics ,Geometry and Topology ,Diffeomorphism ,0101 mathematics ,Mathematics - Abstract
In this paper, we consider the boundary rigidity problem on a cylindrical domain in $${\mathbb {R}}^{1+n}$$ , $$n\ge 2$$ , equipped with a stationary (time-invariant) Lorentzian metric. We show that the time separation function between pairs of points on the boundary of the cylindrical domain determines the stationary spacetime, up to some time-invariant diffeomorphism, assuming that the metric satisfies some a-priori conditions.
- Published
- 2021
14. A shape optimization problem for the first mixed Steklov–Dirichlet eigenvalue
- Author
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Dong-Hwi Seo
- Subjects
010102 general mathematics ,Mathematical analysis ,Mathematics::Spectral Theory ,01 natural sciences ,Geometric proof ,Dirichlet eigenvalue ,Differential geometry ,Bounded function ,0103 physical sciences ,Homogeneous space ,010307 mathematical physics ,Geometry and Topology ,Shape optimization problem ,0101 mathematics ,Analysis ,Shell theorem ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We consider a shape optimization problem for the first mixed Steklov–Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton’s shell theorem.
- Published
- 2021
15. Compact traveling waves for anisotropic curvature flows with driving force
- Author
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H. Monobe and Hirokazu Ninomiya
- Subjects
Differential geometry ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Traveling wave ,Algebraic geometry ,Curvature ,Anisotropy ,Mathematics - Published
- 2021
16. On the boundary injectivity radius of Buser–Colbois–Dodziuk-Margulis tubes
- Author
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Di Cerbo and F Luca
- Subjects
010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Radius ,Curvature ,01 natural sciences ,Upper and lower bounds ,Differential geometry ,Dimension (vector space) ,0103 physical sciences ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Analysis ,Mathematics - Abstract
We give a lower bound on the boundary injectivity radius of the Margulis tubes with smooth boundary constructed by Buser, Colbois, and Dodziuk. This estimate depends on the dimension and a curvature bound only.
- Published
- 2021
17. Rigidity of Einstein Metrics as Critical Points of Some Quadratic Curvature Functionals on Complete Manifolds
- Author
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Guangyue Huang, Yu Chen, and Xingxiao Li
- Subjects
Mathematics - Differential Geometry ,010102 general mathematics ,Mathematical analysis ,Curvature ,01 natural sciences ,Sobolev space ,symbols.namesake ,Rigidity (electromagnetism) ,Quadratic equation ,Differential Geometry (math.DG) ,Differential geometry ,0103 physical sciences ,FOS: Mathematics ,symbols ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Einstein ,Constant (mathematics) ,Ricci curvature ,Mathematics - Abstract
In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the trace-less Ricci curvature and the Sobolev constant, accordingly., Comment: All comments are welcome
- Published
- 2021
18. ON THE SOLVABILITY OF THE MONGE – AMPERE EQUATION ON , RELATED TO THE PROBLEM OF DIFFERENTIAL GEOMETRY
- Author
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Tatyana Aleksandrovna Yuryeva
- Subjects
Differential geometry ,Mathematics::Complex Variables ,Mathematical analysis ,Monge–Ampère equation ,General Medicine ,Mathematics - Abstract
The paper provides a proof of the closed nature of the solution of a family of Monge –Ampere differential one-parameter equations. The obtained result is used in the study of the unique solvability of the differential equation under study.
- Published
- 2021
19. Reflection principle for lightlike line segments on maximal surfaces
- Author
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Hiroki Fujino and Shintaro Akamine
- Subjects
Mathematics - Differential Geometry ,Minimal surface ,Maximal surface ,010102 general mathematics ,Mathematical analysis ,53A10, 53B30, 31A05, 31A20 ,01 natural sciences ,General Relativity and Quantum Cosmology ,Line segment ,Differential Geometry (math.DG) ,Differential geometry ,0103 physical sciences ,Euclidean geometry ,FOS: Mathematics ,Gravitational singularity ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Invariant (mathematics) ,Analysis ,Mathematics - Abstract
As in the case of minimal surfaces in the Euclidean 3-space, the reflection principle for maximal surfaces in the Lorentz-Minkowski 3-space asserts that if a maximal surface has a spacelike line segment $L$, the surface is invariant under the $180^\circ$-rotation with respect to $L$. However, such a reflection property does not hold for lightlike line segments on the boundaries of maximal surfaces in general. In this paper, we show some kind of reflection principle for lightlike line segments on the boundaries of maximal surfaces when lightlike line segments are connecting shrinking singularities. As an application, we construct various examples of periodic maximal surfaces with lightlike lines from tessellations of $\mathbb{R}^2$., 16 pages, 15 figures
- Published
- 2020
20. Estimates for the constant in two nonlinear Korn inequalities
- Author
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Maria Malin and Cristinel Mardare
- Subjects
Inequality ,Euclidean space ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematical analysis ,Shell theory ,01 natural sciences ,Tensor field ,Nonlinear system ,Differential geometry ,Mechanics of Materials ,0103 physical sciences ,General Materials Science ,010307 mathematical physics ,0101 mathematics ,Constant (mathematics) ,Nonlinear elasticity ,Mathematics ,media_common - Abstract
A nonlinear Korn inequality estimates the distance between two immersions from an open subset of [Formula: see text] into the Euclidean space [Formula: see text], [Formula: see text], in terms of the distance between specific tensor fields that determine the two immersions up to a rigid motion in [Formula: see text]. We establish new inequalities of this type in two cases: when k = n, in which case the tensor fields are the square roots of the metric tensor fields induced by the two immersions, and when k = 3 and n = 2, in which case the tensor fields are defined in terms of the fundamental forms induced by the immersions. These inequalities have the property that their constants depend only on the open subset over which the immersions are defined and on three scalar parameters defining the regularity of the immersions, instead of constants depending on one of the immersions, considered as fixed, as up to now.
- Published
- 2020
21. Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces
- Author
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Ao Sun
- Subjects
Mathematics - Differential Geometry ,Mean curvature flow ,Closed manifold ,010102 general mathematics ,Mathematical analysis ,Riemannian manifold ,Submanifold ,01 natural sciences ,symbols.namesake ,Monotone polygon ,Differential Geometry (math.DG) ,Differential geometry ,Fourier analysis ,0103 physical sciences ,FOS: Mathematics ,symbols ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,53C44, 35K08 ,0101 mathematics ,Mathematics::Symplectic Geometry ,Ricci curvature ,Mathematics - Abstract
Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature., Comment: 16 pages, comments are welcomed! The paper is revised according to the comments of the journal referees, and some references are added. Accepted by The Journal of Geometric Analysis
- Published
- 2020
22. Asymptotically Hyperbolic Manifolds with Boundary Conjugate Points but no Interior Conjugate Points
- Author
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C. Robin Graham and Nikolas Eptaminitakis
- Subjects
Condensed Matter::Quantum Gases ,010102 general mathematics ,Mathematical analysis ,Conjugate points ,Boundary (topology) ,Mathematics::Geometric Topology ,01 natural sciences ,symbols.namesake ,Differential geometry ,Fourier analysis ,0103 physical sciences ,symbols ,Physics::Atomic Physics ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points
- Published
- 2020
23. Antithesis of the Stokes Paradox on the Hyperbolic Plane
- Author
-
Chi Hin Chan and Magdalena Czubak
- Subjects
media_common.quotation_subject ,Hyperbolic geometry ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Stokes flow ,Infinity ,01 natural sciences ,Domain (mathematical analysis) ,Physics::Fluid Dynamics ,Mathematics - Analysis of PDEs ,Differential geometry ,0103 physical sciences ,Euclidean geometry ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,Boundary value problem ,0101 mathematics ,Stokes' paradox ,Analysis of PDEs (math.AP) ,media_common ,Mathematics - Abstract
We show there exists a nontrivial $H^1_0$ solution to the steady Stokes equation on the 2D exterior domain in the hyperbolic plane. Hence we show there is no Stokes paradox in the hyperbolic setting. We also show the existence of a nontrivial solution to the steady Navier-Stokes equation in the same setting, whereas the analogous problem is open in the Euclidean case., Comment: 35 pages
- Published
- 2020
24. On Ecker’s local integral quantity at infinity for ancient mean curvature flows
- Author
-
Keita Kunikawa
- Subjects
Mean curvature flow ,Mean curvature ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Differential geometry ,0103 physical sciences ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Entropy (arrow of time) ,Analysis ,Mathematics - Abstract
We point out that Ecker’s local integral quantity agrees with Huisken’s global integral quantity at infinity for ancient mean curvature flows if Huisken’s one is finite on each time-slice. In particular, this means that the finiteness of Ecker’s integral quantity at infinity implies the finiteness of the entropy at infinity.
- Published
- 2020
25. A Note on the Evolution of the Whitney Sphere Along Mean Curvature Flow
- Author
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Celso Viana
- Subjects
Computer Science::Machine Learning ,Mean curvature flow ,010102 general mathematics ,Mathematical analysis ,Tangent ,Multiplicity (mathematics) ,Computer Science::Digital Libraries ,01 natural sciences ,Statistics::Machine Learning ,symbols.namesake ,Differential geometry ,Fourier analysis ,0103 physical sciences ,Computer Science::Mathematical Software ,symbols ,Equivariant map ,SPHERES ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Lagrangian ,Mathematics - Abstract
We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in $${{\mathbb {C}}^n}$$ C n satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.
- Published
- 2020
26. Well-Posedness of Weinberger’s Center of Mass by Euclidean Energy Minimization
- Author
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Richard S. Laugesen
- Subjects
010102 general mathematics ,Mathematical analysis ,Centroid ,Moment of inertia ,Energy minimization ,01 natural sciences ,Measure (mathematics) ,symbols.namesake ,Differential geometry ,Fourier analysis ,0103 physical sciences ,symbols ,Shape optimization ,010307 mathematical physics ,Geometry and Topology ,Center of mass ,0101 mathematics ,Mathematics - Abstract
The center of mass of a finite measure with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure.
- Published
- 2020
27. Regularity of Solutions to the Quaternionic Monge–Ampère Equation
- Author
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Marcin Sroka and Sławomir Kołodziej
- Subjects
Dirichlet problem ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Hölder condition ,Monge–Ampère equation ,pluripotential theory ,01 natural sciences ,Stability (probability) ,Omega ,Boundary values ,Monge-Ampere equation ,symbols.namesake ,Differential geometry ,Fourier analysis ,0103 physical sciences ,symbols ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,subharmonic functions ,Mathematics - Abstract
The regularity of solutions to the Dirichlet problem for the quaternionic Monge–Ampère equation is discussed. We prove that the solution to the Dirichlet problem is Hölder continuous under some conditions on the boundary values and the quaternionic Monge–Ampère density from $$L^p(\Omega )$$Lp(Ω) for $$p>2$$p>2. As a step towards the proof, we provide a refined version of stability for the weak solutions to this equation.
- Published
- 2020
28. Boundary Hölder Gradient Estimates for the Monge–Ampère Equation
- Author
-
Qian Zhang and Ovidiu Savin
- Subjects
010102 general mathematics ,Mathematical analysis ,Regular polygon ,Boundary (topology) ,Monge–Ampère equation ,01 natural sciences ,Omega ,Domain (mathematical analysis) ,symbols.namesake ,Differential geometry ,Fourier analysis ,Bounded function ,0103 physical sciences ,symbols ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics - Abstract
We investigate global Holder gradient estimates for solutions to the Monge–Ampere equation $$\begin{aligned} {\mathrm {det}}\;D^2 u=f\quad {\mathrm {in}}\;\Omega , \end{aligned}$$where the right-hand side f is bounded away from 0 and $$\infty $$. We consider two main situations when (a) the domain $$\Omega $$ is uniformly convex and (b) $$\Omega $$ is flat.
- Published
- 2020
29. Spectral estimates and discreteness of spectra under Riemannian submersions
- Author
-
Panagiotis Polymerakis
- Subjects
Mathematics - Differential Geometry ,Mean curvature ,Spectral theory ,Base space ,010102 general mathematics ,Spectrum (functional analysis) ,Mathematical analysis ,Space (mathematics) ,01 natural sciences ,Spectral line ,Mathematics - Spectral Theory ,010101 applied mathematics ,Differential Geometry (math.DG) ,Differential geometry ,58J50, 35P15, 53C99 ,Bounded function ,FOS: Mathematics ,Mathematics::Differential Geometry ,Geometry and Topology ,0101 mathematics ,Spectral Theory (math.SP) ,Analysis ,Mathematics - Abstract
For Riemannian submersions, we establish some estimates for the spectrum of the total space in terms of the spectrum of the base space and the geometry of the fibers. In particular, for Riemannian submersions of complete manifolds with closed fibers of bounded mean curvature, we show that the spectrum of the base space is discrete if and only if the spectrum of the total space is discrete., Comment: 18 pages
- Published
- 2020
30. Radius Estimates for Alexandrov Space with Boundary
- Author
-
Jian Ge and Ronggang Li
- Subjects
Mathematics - Differential Geometry ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Rigidity (psychology) ,Radius ,Space (mathematics) ,01 natural sciences ,Convexity ,symbols.namesake ,Differential Geometry (math.DG) ,Differential geometry ,Fourier analysis ,0103 physical sciences ,FOS: Mathematics ,symbols ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Convex function ,Mathematics - Abstract
In this note, we study the radius of positively curved or non-negatively curved Alexandrov space with strictly convex boundary, with convexity measured by the Base-Angle defined by Alexander and Bishop. We also estimate the volume of the boundary of non-negatively curved spaces as well as the rigidity case, which can be thought as a non-negatively curved version of a recent result of Grove-Petersen.
- Published
- 2019
31. Computing the Riemannian curvature of image patch and single-cell RNA sequencing data manifolds using extrinsic differential geometry
- Author
-
Sahand Hormoz, Shu Wang, and Duluxan Sritharan
- Subjects
0301 basic medicine ,Laplace-Beltrami ,Cells ,Space (mathematics) ,Curvature ,03 medical and health sciences ,0302 clinical medicine ,data manifold ,Humans ,differential geometry ,Klein bottle ,Mathematics ,Multidisciplinary ,Sequence Analysis, RNA ,Second fundamental form ,Applied Mathematics ,Mathematical analysis ,Biological Sciences ,Manifold ,Ambient space ,Biomechanical Phenomena ,Biophysics and Computational Biology ,030104 developmental biology ,Differential geometry ,Physical Sciences ,Embedding ,Mathematics::Differential Geometry ,Single-Cell Analysis ,Transcriptome ,Riemannian curvature ,single-cell transcriptomics ,030217 neurology & neurosurgery - Abstract
Significance High-dimensional datasets are becoming increasingly prevalent in many scientific fields. A universal theme connecting these high-dimensional datasets is the ansatz that data points are constrained to lie on nonlinear low-dimensional manifolds, whose structure is dictated by the natural laws governing the data. While tools have been developed for estimating global properties of these data manifolds, estimating the Riemannian curvature, a local property, has not been considered. Computing curvature of data manifolds offers both detailed criteria with which to evaluate models of these complex data (e.g., a Klein bottle model of image patches) and a way to explore detailed geometric features that cannot simply be visualized by the naked eye (e.g., in single-cell RNA-sequencing data)., Most high-dimensional datasets are thought to be inherently low-dimensional—that is, data points are constrained to lie on a low-dimensional manifold embedded in a high-dimensional ambient space. Here, we study the viability of two approaches from differential geometry to estimate the Riemannian curvature of these low-dimensional manifolds. The intrinsic approach relates curvature to the Laplace–Beltrami operator using the heat-trace expansion and is agnostic to how a manifold is embedded in a high-dimensional space. The extrinsic approach relates the ambient coordinates of a manifold’s embedding to its curvature using the Second Fundamental Form and the Gauss–Codazzi equation. We found that the intrinsic approach fails to accurately estimate the curvature of even a two-dimensional constant-curvature manifold, whereas the extrinsic approach was able to handle more complex toy models, even when confounded by practical constraints like small sample sizes and measurement noise. To test the applicability of the extrinsic approach to real-world data, we computed the curvature of a well-studied manifold of image patches and recapitulated its topological classification as a Klein bottle. Lastly, we applied the extrinsic approach to study single-cell transcriptomic sequencing (scRNAseq) datasets of blood, gastrulation, and brain cells to quantify the Riemannian curvature of scRNAseq manifolds.
- Published
- 2021
32. Efficient Conformal Parameterization of Multiply-Connected Surfaces Using Quasi-Conformal Theory
- Author
-
Gary P. T. Choi
- Subjects
Computational Geometry (cs.CG) ,FOS: Computer and information sciences ,Mathematics - Differential Geometry ,Surface (mathematics) ,Work (thermodynamics) ,Conformal map ,01 natural sciences ,Theoretical Computer Science ,Computer Science - Graphics ,FOS: Mathematics ,Complex Variables (math.CV) ,0101 mathematics ,Mathematics ,Numerical Analysis ,Mathematics - Complex Variables ,Plane (geometry) ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Annulus (mathematics) ,Unit disk ,Graphics (cs.GR) ,010101 applied mathematics ,Computational Mathematics ,Differential Geometry (math.DG) ,Computational Theory and Mathematics ,Differential geometry ,Computer Science - Computational Geometry ,Focus (optics) ,Software - Abstract
Conformal mapping, a classical topic in complex analysis and differential geometry, has become a subject of great interest in the area of surface parameterization in recent decades with various applications in science and engineering. However, most of the existing conformal parameterization algorithms only focus on simply-connected surfaces and cannot be directly applied to surfaces with holes. In this work, we propose two novel algorithms for computing the conformal parameterization of multiply-connected surfaces. We first develop an efficient method for conformally parameterizing an open surface with one hole to an annulus on the plane. Based on this method, we then develop an efficient method for conformally parameterizing an open surface with $k$ holes onto a unit disk with $k$ circular holes. The conformality and bijectivity of the mappings are ensured by quasi-conformal theory. Numerical experiments and applications are presented to demonstrate the effectiveness of the proposed methods.
- Published
- 2021
33. Convergence Weibull Distribution to Normal Distribution by using Differential Geometry
- Author
-
Wasan Abbas Jasim
- Subjects
Normal distribution ,Differential geometry ,Mathematical analysis ,Convergence (routing) ,General Engineering ,Weibull distribution ,Mathematics - Published
- 2020
34. Inverse Spectral Theory for Perturbed Torus
- Author
-
Hiroshi Isozaki and Evgeny Korotyaev
- Subjects
Spectral theory ,010102 general mathematics ,Mathematical analysis ,Torus ,Inverse problem ,Space (mathematics) ,01 natural sciences ,Manifold ,Differential geometry ,0103 physical sciences ,010307 mathematical physics ,Geometry and Topology ,Isomorphism ,0101 mathematics ,Mathematics::Symplectic Geometry ,Rotation (mathematics) ,Mathematics - Abstract
We consider an inverse problem for Laplacians on rotationally symmetric manifolds, which are finite for the transversal direction and periodic with respect to the axis of the manifold, i.e., Laplacians on tori. We construct an infinite dimensional analytic isomorphism between the space of profiles (the radius of the rotation) of the torus and the spectral data as well as the stability estimates: those for the spectral data in terms of the profile and conversely, for the profile in term of the spectral data.
- Published
- 2019
35. Simultaneous determination of two coefficients in the Riemannian hyperbolic equation from boundary measurements
- Author
-
Mourad Bellassoued and Zouhour Rezig
- Subjects
Mathematics::Operator Algebras ,Mathematical analysis ,Boundary (topology) ,Mathematics::Spectral Theory ,Riemannian manifold ,Inverse problem ,Wave equation ,Dirichlet distribution ,symbols.namesake ,Mathematics - Analysis of PDEs ,Differential geometry ,FOS: Mathematics ,symbols ,Geometry and Topology ,Electric potential ,Hyperbolic partial differential equation ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we consider the inverse problem of determining on a compact Riemannian manifold the electric potential and the absorption coefficient in the wave equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n \geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uni, arXiv admin note: text overlap with arXiv:1510.04247
- Published
- 2019
36. Second-Order Regularity for Parabolic p-Laplace Problems
- Author
-
Andrea Cianchi and Vladimir Maz'ya
- Subjects
Laplace transform ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,35K20 ,Boundary (topology) ,Space (mathematics) ,01 natural sciences ,Domain (mathematical analysis) ,Sobolev space ,Mathematics - Analysis of PDEs ,Differential geometry ,Bounded function ,0103 physical sciences ,FOS: Mathematics ,p-Laplacian ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of $p$-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.
- Published
- 2019
37. Isoparametric functions and nodal solutions of the Yamabe equation
- Author
-
Guillermo Henry
- Subjects
Mathematics - Differential Geometry ,010102 general mathematics ,Mathematical analysis ,Function (mathematics) ,01 natural sciences ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Differential geometry ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,NODAL ,Constant (mathematics) ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We prove existence results for nodal solutions of the Yamabe equation that are constant along the level sets of an isoparametric function., Comment: 23 pages, minor changes. To appear in Annals of Global Analysis and Geometry
- Published
- 2019
38. A Weak Reverse Hölder Inequality for Caloric Measure
- Author
-
Steve Hofmann and Alyssa Genschaw
- Subjects
Dirichlet problem ,010102 general mathematics ,Mathematical analysis ,Open set ,Boundary (topology) ,Harmonic measure ,01 natural sciences ,Measure (mathematics) ,Omega ,symbols.namesake ,Mathematics - Analysis of PDEs ,Differential geometry ,Fourier analysis ,0103 physical sciences ,FOS: Mathematics ,symbols ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Following a result of Bennewitz–Lewis for non-doubling harmonic measure, we prove a criterion for non-doubling caloric measure to satisfy a weak reverse Holder inequality on an open set $$\Omega \subset \mathbb {R}^{n+1}$$, assuming as a background hypothesis only that the essential boundary of $$\Omega $$ satisfies an appropriate parabolic version of Ahlfors–David regularity (which entails some backwards in time thickness). We also show that the weak reverse Holder estimate is equivalent to solvability of the initial Dirichlet problem with “lateral” data in $$L^p$$, for some $$p
- Published
- 2019
39. The Gauss–Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications
- Author
-
Wojciech Domitrz and M. Zwierzyński
- Subjects
Mathematics - Differential Geometry ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Tangent ,Curvature ,01 natural sciences ,symbols.namesake ,Differential Geometry (math.DG) ,Differential geometry ,Fourier analysis ,Gauss–Bonnet theorem ,0103 physical sciences ,FOS: Mathematics ,symbols ,Primary 57R45, Secondary 53A05 ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Link (knot theory) ,Geodesic curvature ,Mathematics - Abstract
In [31,32,33] the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary) were proved. We establish the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with boundary. We apply this theorem to investigate global properties of maps between surfaces with boundary. As a corollary of our results we obtain Fukuda-Ishikawa's theorem. We also study geometry of the affine extended wave fronts for planar closed non singular hedgehogs (rosettes). In particular, we find a link between the total geodesic curvature on the boundary and the total singular curvature of the affine extended wave front, which leads to a relation of integrals of functions of the width of a resette., 28 pages, 8 figures
- Published
- 2019
40. Local representation and construction of Beltrami fields
- Author
-
Naoki Sato and Michio Yamada
- Subjects
Curl (mathematics) ,Representation theorem ,Solenoidal vector field ,Eikonal equation ,Mathematical analysis ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Mathematics::Spectral Theory ,Condensed Matter Physics ,01 natural sciences ,Physics - Plasma Physics ,010305 fluids & plasmas ,Euler equations ,Plasma Physics (physics.plasm-ph) ,symbols.namesake ,Computer Science::Graphics ,Orthogonal coordinates ,Differential geometry ,0103 physical sciences ,symbols ,010306 general physics ,Mathematical Physics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
A Beltrami field is an eigenvector of the curl operator. Beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence. By application of the Lie-Darboux theorem of differential geoemtry, we prove a local representation theorem for Beltrami fields. We find that, locally, a Beltrami field has a standard form amenable to an Arnold-Beltrami-Childress flow with two of the parameters set to zero. Furthermore, a Beltrami flow admits two local invariants, a coordinate representing the physical plane of the flow, and an angular momentum-like quantity in the direction across the plane. As a consequence of the theorem, we derive a method to construct Beltrami fields with given proportionality factor. This method, based on the solution of the eikonal equation, guarantees the existence of Beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal. We construct several solenoidal and non-solenoidal Beltrami fields with both homogeneous and inhomogeneous proportionality factors., 5 figures
- Published
- 2019
41. Sharp Li–Yau-Type Gradient Estimates on Hyperbolic Spaces
- Author
-
Chengjie Yu and Feifei Zhao
- Subjects
Hyperbolic space ,010102 general mathematics ,Mathematical analysis ,Riemannian manifold ,01 natural sciences ,Upper and lower bounds ,symbols.namesake ,Differential geometry ,Fourier analysis ,0103 physical sciences ,symbols ,Heat equation ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics::Symplectic Geometry ,Ricci curvature ,Heat kernel ,Mathematics - Abstract
In this paper, motivated by the works of Bakry et al. in finding sharp Li–Yau-type gradient estimates for positive solutions of the heat equation on complete Riemannian manifolds with nonzero Ricci curvature lower bound, we first introduce a general form of Li–Yau-type gradient estimate and show that the validity of such an estimate for any positive solutions of the heat equation reduces to the validity of the estimate for the heat kernel of the Riemannian manifold. Then, a sharp Li–Yau-type gradient estimate on the three-dimensional hyperbolic space is obtained by using the explicit expression of the heat kernel, and some optimal Li–Yau-type gradient estimates on general hyperbolic spaces are obtained.
- Published
- 2019
42. Fixed Point of Surface Transformation and a Fixed Point Theorem for Triangular Surface Mapping
- Author
-
Şükrü Ilgün
- Subjects
Surface mapping ,Functional analysis ,Differential geometry ,Surface transformation ,Mathematical analysis ,Fixed-point theorem ,Fixed point ,Surface (topology) ,Space mapping ,Mathematics - Abstract
In this study, it was shown that the existence of fixed points of some surface transformations was defined as an example according to the theorems in functional analysis in differential geometry. It was shown that fi real-valued coordinate functions of F triangular space mapping defined from En to Em has a single fixed point, if
- Published
- 2018
43. Inverse Problem of Travel Time Difference Functions on a Compact Riemannian Manifold with Boundary
- Author
-
Teemu Saksala and Maarten V. de Hoop
- Subjects
Atlas (topology) ,010102 general mathematics ,Mathematical analysis ,Inverse ,Inverse problem ,Riemannian manifold ,01 natural sciences ,Travel time ,symbols.namesake ,Differential geometry ,Fourier analysis ,0103 physical sciences ,symbols ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics - Abstract
We show that the travel time difference functions, between common interior points and pairs of points on the boundary, determine a compact Riemannian manifold with smooth boundary up to Riemannian isometry if the boundary satisfies a certain visibility condition. This corresponds with the inverse microseismicity problem. In the proof of this result, we also construct an explicit smooth atlas from the travel time difference functions.
- Published
- 2018
44. Uniformly Compressing Mean Curvature Flow
- Author
-
Wenhui Shi and Dmitry Vorotnikov
- Subjects
Mean curvature flow ,Geodesic ,010102 general mathematics ,Mathematical analysis ,Geometric flow ,Submanifold ,Space (mathematics) ,01 natural sciences ,Mathematics - Analysis of PDEs ,Flow (mathematics) ,Differential geometry ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Balanced flow ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints on the evolving surfaces. We introduce a related geometric flow which is free of these drawbacks. Our flow can be viewed as a formal gradient flow on a certain submanifold of the Wasserstein space of probability measures endowed with Otto’s Riemannian structure. We obtain a number of analytic results concerning well-posedness and long-time stability which are, however, restricted to the 1D case of evolution of loops.
- Published
- 2018
45. On the nonlinear Hadamard-type integro-differential equation
- Author
-
Li, Chenkuan
- Subjects
Mathematics::Functional Analysis ,Research ,010102 general mathematics ,Mathematical analysis ,Banach space ,34A08 ,Absolute continuity ,34A12 ,01 natural sciences ,010101 applied mathematics ,Banach’s contraction principle ,Nonlinear system ,Differential geometry ,Integro-differential equation ,Hadamard transform ,Babenko’s approach ,Uniqueness ,0101 mathematics ,Contraction principle ,Multivariate Mittag-Leffler function ,Hadamard-type fractional integral ,Hadamard-type fractional derivative ,Mathematics - Abstract
This paper studies uniqueness of solutions for a nonlinear Hadamard-type integro-differential equation in the Banach space of absolutely continuous functions based on Babenko’s approach and Banach’s contraction principle. We also include two illustrative examples to demonstrate the use of main theorems.
- Published
- 2021
46. Curve Flows with a Global Forcing Term
- Author
-
Friederike Dittberner
- Subjects
Mathematics - Differential Geometry ,Forcing (recursion theory) ,Curve-shortening flow ,Plane (geometry) ,Mathematical analysis ,Computer Science::Digital Libraries ,Convexity ,Constrained curve flow, Area preserving curve shortening flow, Length preserving curve flow, Curve flow, Forcing term, Geometric flow ,Differential Geometry (math.DG) ,Differential geometry ,Bounded function ,FOS: Mathematics ,Computer Science::Mathematical Software ,Total curvature ,Gravitational singularity ,Geometry and Topology ,ddc:510 ,53-XX, 58J35 ,Mathematics - Abstract
We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. Firstly, we prove an analogue to Huisken's distance comparison principle for curve shortening flow for initial curves whose local total curvature does not lie below $-\pi$, and show that this condition is sharp. Secondly, for bounded forcing terms, we exclude singularities in finite time. Thirdly, for immortal flows of closed curves whose forcing terms provide non-vanishing enclosed area and bounded length, we show convexity in finite time and smooth and exponential convergence to a circle., Comment: 2 figures
- Published
- 2021
47. Constant Mean Curvature Surfaces for the Bessel Equation
- Author
-
Eduardo Mota
- Subjects
symbols.namesake ,Mean curvature ,Differential geometry ,Mathematical analysis ,symbols ,Riemann sphere ,Mathematics::Differential Geometry ,Constant (mathematics) ,Bessel function ,Mathematics - Abstract
In this note we construct a family of immersions with constant mean curvature of the twice-punctured Riemann sphere into \(\mathbb {R}^3\) from the Bessel equation.
- Published
- 2021
48. Multipoint formulas for phase recovering from phaseless scattering data
- Author
-
Roman Novikov, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), and École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Helmholtz equation ,Scattering ,010102 general mathematics ,Mathematical analysis ,Phase (waves) ,Monochromatic scattering data ,Schrödinger equation ,01 natural sciences ,symbols.namesake ,Phase recovering ,Differential geometry ,Dimension (vector space) ,Phaseless inverse scattering ,Fourier analysis ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,0103 physical sciences ,symbols ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,010307 mathematical physics ,Geometry and Topology ,Monochromatic color ,0101 mathematics ,Mathematics - Abstract
We give formulas for phase recovering from appropriate monochromatic phaseless scattering data at 2n points in dimension $$d=3$$ and in dimension $$d=2$$ . These formulas are recurrent and explicit and their precision is proportional to n. By this result we continue studies of Novikov (Bulletin des Sciences Mathematiques 139(8):923–936, 2015), where formulas of such a type were given for $$n=1$$ , $$d\ge 2$$ .
- Published
- 2021
49. Variance of Lattice Point Counting in Thin Annuli
- Author
-
Leonardo Colzani, Giacomo Gigante, Bianca Gariboldi, Colzani, L, Gariboldi, B, and Gigante, G
- Subjects
Mathematics::Dynamical Systems ,60D05, 42B05, 11P21 ,fourier series ,Poisson random variable ,lattice point counting ,thin annuli ,01 natural sciences ,Physics::Fluid Dynamics ,symbols.namesake ,Dimension (vector space) ,Settore MAT/05 - Analisi Matematica ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,MAT/05 - ANALISI MATEMATICA ,Mathematics ,Fourier serie ,Mathematics::Complex Variables ,010102 general mathematics ,Mathematical analysis ,Variance (accounting) ,Mathematics::Geometric Topology ,Condensed Matter::Soft Condensed Matter ,Differential geometry ,Fourier analysis ,Mathematics - Classical Analysis and ODEs ,Thin annuli ,symbols ,010307 mathematical physics ,Geometry and Topology ,Lattice point counting ,Integer (computer science) - Abstract
We give asymptotic estimates of the variance of the number of integer points in translated thin annuli in any dimension.
- Published
- 2021
50. Geodesic random walks, diffusion processes and Brownian motion on Finsler manifolds
- Author
-
Ilya Pavlyukevich, Vladimir S. Matveev, and Tianyu Ma
- Subjects
Mathematics - Differential Geometry ,Geodesic ,01 natural sciences ,Mathematics - Analysis of PDEs ,Mathematics::Probability ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Metric Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Brownian motion ,Mathematics ,010308 nuclear & particles physics ,010102 general mathematics ,Mathematical analysis ,Probability (math.PR) ,Random walk ,Differential geometry ,Diffusion process ,Differential Geometry (math.DG) ,Bounded function ,Metric (mathematics) ,Geometry and Topology ,Finsler manifold ,Mathematics::Differential Geometry ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric., Comment: 32 pages, 3 figures. Comments from reads are welcome
- Published
- 2021
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