scholarly journals Deforming convex bodies in Minkowski geometry

2021 ◽  
Author(s):  
V. Rovenski ◽  
P. Walczak
Keyword(s):  
Recent Work ◽  
Finsler Geometry ◽  
Convex Bodies ◽  
Positive Function ◽  
Euclidean Norm ◽  
Minkowski Geometry ◽  
Linearly Independent ◽  
Minkowski Norms ◽  
Smooth Positive Function ◽  
Axis Passing

We introduce and study certain deformation of Minkowski norms in [Formula: see text] determined by a set of [Formula: see text] linearly independent 1-forms and a smooth positive function of [Formula: see text] variables. In particular, the deformation of a Euclidean norm [Formula: see text] produces a Minkowski norm defined in our recent work; its indicatrix is a rotation hypersurface with a [Formula: see text]-dimensional axis passing through the origin. For [Formula: see text], our deformation generalizes the construction of [Formula: see text]-norms which form a rich class of “computable” Minkowski norms and play an important role in Finsler geometry. We characterize such pairs of a Minkowski norm and its image that Cartan torsions of the two norms either coincide or differ by a [Formula: see text]-reducible term. We conjecture that for [Formula: see text] any Minkowski norm can be approximated by images of a Euclidean norm.

scholarly journals Non-degeneracy of least-energy solutions for an elliptic problem with nearly critical nonlinearity

2010 ◽  
Vol 140 (1) ◽  
pp. 203-222
Author(s):  
Futoshi Takahashi
Keyword(s):  
Recent Result ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Positive Function ◽  
Coefficient Function ◽  
Critical Nonlinearity ◽  
Smooth Bounded Domain ◽  
Least Energy Solutions ◽  
Least Energy ◽  
Smooth Positive Function

We consider the problem −Δu = c0K(x)upε, u > 0 in Ω, u = 0 on δΩ, where Ω is a smooth, bounded domain in ℝN, N ≥ 3, c0 = N(N − 2), pε = (N + 2)/(N − 2) − ε and K is a smooth, positive function on . We prove that least-energy solutions of the above problem are non-degenerate for small ε > 0 under some assumptions on the coefficient function K. This is a generalization of the recent result by Grossi for K ≡ 1, and needs precise estimates and a new argument.

scholarly journals A Pinching Theorem for Compact Minimal Submanifolds in Warped Products I×fSm(c)

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1445
Author(s):  
Xin Zhan ◽  
Zhonghua Hou
Keyword(s):  
Minimal Submanifolds ◽  
Positive Function ◽  
Open Interval ◽  
Type I ◽  
Warped Products ◽  
Euclidean Sphere ◽  
Smooth Positive Function

Let Sm(c) be a Euclidean sphere of curvature c>0 and R be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type I×fSm(c), where f:I→R+ is a smooth positive function on an open interval I of R. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products Sm(c)×R to Riemannian warped products I×fSm(c).

scholarly journals On the Irreducibility of Convex Bodies

1959 ◽  
Vol 11 ◽  
pp. 256-261 ◽  
Author(s):  
A. C. Woods
Keyword(s):  
Euclidean Space ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Star Body ◽  
Closed Set ◽  
Absolute Value ◽  
Linearly Independent ◽  
The Absolute ◽  
Set Of Points ◽  
Rational Integral

We select a Cartesian co-ordinate system in ndimensional Euclidean space Rn with origin 0 and employ the usual pointvector notation.By a lattice Λ in Rn we mean the set of all rational integral combinations of n linearly independent points X1, X2, … , Xn of Rn. The points X1 X2, … , Xn are said to form a basis of Λ. Let {X1, X2, … , Xn) denote the determinant formed when the co-ordinates of Xi are taken in order as the ith row of the determinant for i = 1,2, … , n. The absolute value of this determinant is called the determinant d(Λ) of Λ. It is well known that d(Λ) is independent of the particular basis one takes for Λ.A star body in Rn is a closed set of points K such that if X ∈ K then every point of the form tX where — 1 < t < 1 is an inner point of K.

XVII.—Invariant Matrices and the Geometry of Numbers

1957 ◽  
Vol 64 (3) ◽  
pp. 223-238
Author(s):  
K. Mahler
Keyword(s):  
Recent Work ◽  
Linear Group ◽  
Affine Space ◽  
Matrix Representation ◽  
Convex Bodies ◽  
Linear Mapping ◽  
Geometry Of Numbers ◽  
Arithmetical Property ◽  
The Real ◽  
Invariant Matrices

SynopsisWith every matrix representation of the (real) full linear group can be associated a multi-linear mapping of one affine space, Rn, into another, RN. This mapping is studied from the viewpoint of the geometry of numbers of convex bodies, and a general arithmetical property of such mappings is proved. The result generalizes my recent work on compound convex bodies.

scholarly journals The determination of caloric morphisms on Euclidean domains

2000 ◽  
Vol 158 ◽  
pp. 133-166 ◽  
Author(s):  
Katsunori Shimomura
Keyword(s):  
Heat Equation ◽  
Positive Function ◽  
Smooth Mapping ◽  
Direct Sum ◽  
Smooth Positive Function ◽  
Euclidean Domains

AbstractLetDbe a domain in ℝm+1andEbe a domain in ℝn+1. A pair of a smooth mappingf:D → Eand a smooth positive function ϕ onDis called a caloric morphism if ϕ˙uofis a solution of the heat equation inDwheneveruis a solution of the heat equation inE. We give the characterization of caloric morphisms, and then give the determination of caloric morphisms. In the case ofm < n, there are no caloric morphisms. In the case ofm = n, caloric morphisms are generated by the dilation, the rotation, the translation and the Appell transformation. In the case ofm > n, under some assumption onf, every caloric morphism is obtained by composing a projection with a direct sum of caloric morphisms of ℝn+1.

scholarly journals On the principal Ricci curvatures of a Riemannian 3-manifold

2019 ◽  
Vol 19 (2) ◽  
pp. 251-262
Author(s):  
Amir Babak Aazami ◽  
Charles M. Melby-Thompson
Keyword(s):  
Vector Field ◽  
Positive Constant ◽  
Ricci Tensor ◽  
Unit Length ◽  
Positive Function ◽  
Universal Cover ◽  
Riemannian Metric ◽  
Ricci Solitons ◽  
Topological Obstruction ◽  
Smooth Positive Function

Abstract We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form (−μ, f, f), where μ is a positive constant and f is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form (0, λ, λ), where λ is a positive constant, then its universal cover must split isometrically. If the manifold is closed, scalar-flat, and its zero eigenspace contains a unit length vector field that is geodesic and divergence-free, then the manifold must be flat. Our techniques also apply to the study of Ricci solitons in dimension three.

SPECTRAL STUDY OF A COUPLED COMPACT-NONCOMPACT PROBLEM IN A NON-SELF-ADJOINT CASE

1994 ◽  
Vol 04 (05) ◽  
pp. 607-624
Author(s):  
A. BENKADDOUR ◽  
J. SANCHEZ-HUBERT ◽  
A. RIDHA
Keyword(s):  
Porous Medium ◽  
Essential Spectrum ◽  
Neumann Boundary ◽  
Positive Function ◽  
Acoustic Vibration ◽  
Resolvent Set ◽  
First Case ◽  
Free Air ◽  
Impedance Condition ◽  
Smooth Positive Function

We consider a coupled problem of acoustic vibration of air in a porous medium Ω p that is in contact, by a plane interface Γ (x1=0), with free air in some region Ω f . The porous medium is made of infinitely narrow thin channels parallel to the x1-axis. In Refs. 1–3 and 5, problems of this type were considered with homogeneous Neumann boundary conditions on the exterior boundary of the whole domain Ω, Ω= Ω f ∪Γ∪Ω p . In this paper, we study the problem in the mentioned configuration but with the impedance condition ∂u/∂n=u/Z (where Z is a given complex number) on a part of the boundary of the porous medium Ω p defined by x1=–ℓ(x2, x3), where ℓ is a smooth positive function. Let us denote by A the operator associated with the coupled eigenvalue problem –Au=ω2u, and by Ap(x2, x3) the corresponding problem in a channel x2= const , x3= const . As in Ref. 1, two cases appear according to the values of ω2. In the first case ω2 is not an eigenvalue of Ap(x2, x3), and then we show that ω2 is either a point of the resolvent set or an isolated eigenvalue with finite multiplicity of A. In the second case ω2 is an eigenvalue of Ap(a2, a3) for some value (a2, a3) of (x2, x3), and then ω2 is a point of the essential spectrum of A. The novelty of this paper is the study of the spectrum of the operator A which is non self-adjoint and has a noncompact resolvent. We find a complex essential spectrum of A and we study its topological structure.

What sentimentalists should say about emotion

2019 ◽  
Vol 42 ◽  
Author(s):  
Charlie Kurth
Keyword(s):  
Moral Judgment ◽  
Recent Work ◽  
Relevant Information ◽  
Moral Cognition ◽  
Multilevel Structure

Abstract Recent work by emotion researchers indicates that emotions have a multilevel structure. Sophisticated sentimentalists should take note of this work – for it better enables them to defend a substantive role for emotion in moral cognition. Contra May's rationalist criticisms, emotions are not only able to carry morally relevant information, but can also substantially influence moral judgment and reasoning.

The Sun as a Magnetic Star

1976 ◽  
Vol 32 ◽  
pp. 457-463
Author(s):  
John M. Wilcox ◽  
Leif Svalgaard
Keyword(s):  
Recent Work ◽  
Rotation Axis ◽  
Polar Field ◽  
Magnetic Star ◽  
Rotational Axis ◽  
Field Variation ◽  
The Sun ◽  
Arbitrary Angle ◽  
Solar Magnetism ◽  
Oblique Rotator

SummaryThe sun as a magnetic star is described on the basis of recent work on solar magnetism. Observations at an arbitrary angle to the rotation axis would show a 22-year polar field variation and a 25-day equatorial sector variation. The sector variation would be similar to an oblique rotator with an angle of 90° between the magnetic and rotational axis.

Microstructural comparison of synthetic bioceramics with natural bioceramics

1991 ◽  
Vol 49 ◽  
pp. 38-39
Author(s):  
Shulin Wen ◽  
Jingwei Feng ◽  
A. Krajewski ◽  
A. Ravaglioli
Keyword(s):  
Recent Work ◽  
Human Body ◽  
Human Tissues ◽  
Main Constituent ◽  
Laboratory Furnace ◽  
Chinese Adult ◽  
Human Enamel ◽  
Biological Compatibility ◽  
Synthetic Hydroxyapatite ◽  
Synthetic Work

Hydroxyapatite bioceramics has attracted many material scientists as it is the main constituent of the bone and the teeth in human body. The synthesis of the bioceramics has been performed for years. Nowadays, the synthetic work is not only focused on the hydroapatite but also on the fluorapatite and chlorapatite bioceramics since later materials have also biological compatibility with human tissues; and they may also be very promising for clinic purpose. However, in comparison of the synthetic bioceramics with natural one on microstructure, a great differences were observed according to our previous results. We have investigated these differences further in this work since they are very important to appraise the synthetic bioceramics for their clinic application.The synthetic hydroxyapatite and chlorapatite were prepared according to A. Krajewski and A. Ravaglioli and their recent work. The briquettes from different hydroxyapatite or chlorapatite powders were fired in a laboratory furnace at the temperature of 900-1300°C. The samples of human enamel selected for the comparison with synthetic bioceramics were from Chinese adult teeth.

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