scholarly journals Local homeo- and diffeomorphisms: invertibility and convex image

1994 ◽  
Vol 49 (3) ◽  
pp. 377-398 ◽  
Author(s):  
Gaetano Zampieri ◽  
Gianluca Gorni
Keyword(s):  
Euclidean Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Connected Subset ◽  
Local Homeomorphism ◽  
Local Diffeomorphisms ◽  
Open Connected Subset ◽  
One To One ◽  
Necessary And Sufficient

We prove a necessary and sufficient condition for a local homeomorphism defined on an open, connected subset of a Euclidean space to be globally one-to-one and, at the same time, for the image to be convex. Among the applications we give a practical sufficiency test for invertibility for twice differentiable local diffeomorphisms defined on a ball.

THEORY OF NON-DESTRUCTIVE INSPECTION OF CERAMICS BASED ON FLAW-SIZE DISTRIBUTION

2008 ◽  
Vol 22 (09n11) ◽  
pp. 1489-1495
Author(s):  
YOHTARO MATSUO
Keyword(s):  
Stress State ◽  
Size Distribution ◽  
Flaw Size ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Surface Strengthening ◽  
One To One ◽  
Necessary And Sufficient ◽  
Non Destructive ◽  
Machining Damage

Two kinds of theory for non-destructive inspection of ceramics were reviewed, which were derived by the authors using flaw-size distribution. In the sufficient condition theory (Realistic NDI), we had discussed about the screening flaw-size and a screening region. In the necessary and sufficient condition theory (Ideal NDI), we found that the screening flaw-size should be changed depending on stress state and specimen configuration. It was proved that there is one-to-one correspondence between Ideal NDI and a proof test. Applications to machining damage problems and surface strengthening of ceramics were also presented.

A continuous generalization of the transversal property

1984 ◽  
Vol 95 (1) ◽  
pp. 21-23
Author(s):  
P. Komjáth
Keyword(s):  
Finite Subset ◽  
Measure Space ◽  
Choice Function ◽  
Finite Measure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Sets ◽  
Real Numbers ◽  
One To One ◽  
Necessary And Sufficient

A transversal for a set-system is a one-to-one choice function. A necessary and sufficient condition for the existence of a transversal in the case of finite sets was given by P. Hall (see [4, 3]). The corresponding condition for the case when countably many countable sets are given was conjectured by Nash-Williams and later proved by Damerell and Milner [2]. B. Bollobás and N. Varopoulos stated and proved the following measure theoretic counterpart of Hall's theorem: if (X, μ) is an atomless measure space, ℋ = {Hi: i∈I} is a family of measurable sets with finite measure, λi (i∈I) are non-negative real numbers, then we can choose a subset Ti ⊆ Hi with μ(Ti) = λi and μ(Ti ∩ Ti′) = 0 (i ≠ i′) if and only if μ({U Hi: iεJ}) ≥ Σ{λi: iεJ}: for every finite subset J of I. In this note we generalize this result giving a necessary and sufficient condition for the case when I is countable and X is the union of countably many sets of finite measure.

scholarly journals On the Differential Equation Governing Torqued Vector Fields on a Riemannian Manifold

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1941
Author(s):  
Sharief Deshmukh ◽  
Nasser Bin Turki ◽  
Haila Alodan
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Rigidity Results ◽  
Necessary And Sufficient

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere Sn(c). A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space En whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space En, this type of torqued vector fields could not be extended globally to En. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.

scholarly journals Canal surfaces in 4-dimensional Euclidean space

2016 ◽  
Vol 7 (1) ◽  
pp. 83-89 ◽  
Author(s):  
Betül Bulca ◽  
Kadri Arslan ◽  
Bengü Bayram ◽  
Günay Öztürk
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Four Dimensions ◽  
Canal Surfaces ◽  
Curvature Ellipse ◽  
Normal Vectors ◽  
Necessary And Sufficient

In the present study, we consider canal surfaces imbedded in an Euclidean space of four dimensions. The curvature properties of these surface are investigated with respect to the variation of the normal vectors and curvature ellipse. We also give some special examples of canal surfaces in E^4. Further, we give necessary and sufficient condition for canal surfaces in E^4 to become superconformal. Finally, the visualization of the projections of canal surfaces in E^3 are presented.

scholarly journals Framed slant helices in Euclidean 3-space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Osman Zeki Okuyucu ◽  
Mevlüt Canbirdi
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Slant Helix ◽  
Spherical Images ◽  
Necessary And Sufficient

AbstractIn this paper, we define framed slant helices and give a necessary and sufficient condition for them in three-dimensional Euclidean space. Then, we introduce the spherical images of a framed curve. Also, we examine the relations between a framed slant helix and its spherical images. Moreover, we give an example of a framed slant helix and its spherical images with figures.

Multidimensional Iterative Interpolation

1991 ◽  
Vol 43 (2) ◽  
pp. 297-312 ◽  
Author(s):  
Gilles Deslauriers ◽  
Jacques Dubois ◽  
Serge Dubuc
Keyword(s):  
Euclidean Space ◽  
Discrete Subgroup ◽  
Interpolation Process ◽  
Sufficient Condition ◽  
Interpolation Function ◽  
Necessary And Sufficient Condition ◽  
Schwartz Distributions ◽  
Algebraic Properties ◽  
Iterative Interpolation ◽  
Necessary And Sufficient

AbstractWe define an iterative interpolation process for data spread over a closed discrete subgroup of the Euclidean space. We describe the main algebraic properties of this process. This interpolation process, under very weak assumptions, is always convergent in the sense of Schwartz distributions. We find also a convenient necessary and sufficient condition for continuity of each interpolation function of a given iterative interpolation process.

scholarly journals On generating points of a lattice in the region

1964 ◽  
Vol 6 (3) ◽  
pp. 141-155 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Image Position ◽  
Necessary And Sufficient

A lattice An in n-dimensional Euclidean space En consists of the aggregate of all points with coordinates (xx,…, xn), wherefor some real ars (r, s = 1,…, n), subject to the condition ∥ αrs ∥nn ╪ 0. The determinant Δn of Λn, is denned by the relation , the sign being chosen to ensure that Δn > 0.If A1…, An are the n points of Λn having coordinates (a11, a21…, anl),…, (a1n, a2n,…, ann), respectively, then every point of Λn may be expressed in the formand Ai,…, An, together with the origin O, are said to generate Λn. This particular set of generating points is not unique; it may be proved that a necessary and sufficient condition that n points of Λn should generate the lattice is that the n × n determinant formed by their x coordinates should be ±Δn, or, equivalently, that the n×n determinant formed by their corresponding u-coordinates should be ±1.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

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