scholarly journals On the centre of spherical curvature of a curve

1937 ◽  
Vol 30 ◽  
pp. xi-xii
Author(s):  
C. E. Weatherburn
Keyword(s):  
Position Vector ◽  
Perpendicular Bisector ◽  
Arc Length ◽  
Normal Plane ◽  
Adjacent Point ◽  
Current Point ◽  
Image Position ◽  
Spherical Curvature ◽  
Unit Vectors ◽  
Polar Line

The position of the centre S of spherical curvature at a point P of a given curve C may be found in the following manner, regarding S as the limiting position of the centre of a sphere through four adjacent points P, P1, P2, P3 on the curve, as these points tend to coincidence at P. The centre of a sphere through P and P1 lies on the plane which is the perpendicular bisector of the chord PP1 and so on. Thus the centre of spherical curvature is the limiting position of the intersection of three normal planes at adjacent points. Let s be the arc-length of the curve C, r the position vector of the point P, and t, n, b unit vectors in the directions of the tangent, principal normal and binormal at P. Then if s is the position vector of the current point on the normal plane at P, the equation of this plane isSince r and t are functions of s, the limiting position of the line of intersection of the normal planes at P and an adjacent point (i.e. the polar line) is determined by (1) and the equation obtained by differentiating this with respect to s, viz.

The form of the tangent-developable at points of zero torsion on space curves

1980 ◽  
Vol 88 (3) ◽  
pp. 403-407 ◽  
Author(s):  
J. P. Cleave
Keyword(s):  
Power Series ◽  
Space Curve ◽  
Arc Length ◽  
Space Curves ◽  
Normal Plane ◽  
Tangent Developable ◽  
Cuspidal Edge ◽  
Curvature And Torsion ◽  
Image Position ◽  
Classical Derivation

A tangent-developable is a surface generated by the tangent lines of a space curve. The intersection of a tangent-developable with the normal plane at a point P of the curve generally has a cusp at that point. Thus the tangent-developable of a space curve has a cuspidal edge along the curve. The classical derivation of this phenomenon takes the trihedron (t, n, b) at P as coordinate axes to which the curve is referred. Then the intersection of the part of the tangent-developable generated by tangent lines at points close to P with the normal plane at P (i.e. the plane through P containing n and b) is given parametrically by power serieswhere K, T are the curvature and torsion, respectively, of the curve at P and s is arc-length measured from P ((2) p. 68). It is tacitly understood in this analysis that curvature and torsion are both defined and non-zero.

The role of differential phase contrast in electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 176-177
Author(s):  
E.M. Waddell ◽  
J.N. Chapman ◽  
R.P. Ferrier
Keyword(s):  
Phase Contrast ◽  
Practical Method ◽  
Position Vector ◽  
Differential Phase ◽  
Pure Phase ◽  
Differential Phase Contrast ◽  
The Difference ◽  
Image Position ◽  
First Moment

Dekkers and de Lang (1977) have discussed a practical method of realising differential phase contrast in a STEM. The method involves taking the difference signal from two semi-circular detectors placed symmetrically about the optic axis and subtending the same angle (2α) at the specimen as that of the cone of illumination. Such a system, or an obvious generalisation of it, namely a quadrant detector, has the characteristic of responding to the gradient of the phase of the specimen transmittance. In this paper we shall compare the performance of this type of system with that of a first moment detector (Waddell et al.1977).For a first moment detector the response function R(k) is of the form R(k) = ck where c is a constant, k is a position vector in the detector plane and the vector nature of R(k)indicates that two signals are produced. This type of system would produce an image signal given bywhere the specimen transmittance is given by a (r) exp (iϕ (r), r is a position vector in object space, ro the position of the probe, ⊛ represents a convolution integral and it has been assumed that we have a coherent probe, with a complex disturbance of the form b(r-ro) exp (iζ (r-ro)). Thus the image signal for a pure phase object imaged in a STEM using a first moment detector is b2 ⊛ ▽ø. Note that this puts no restrictions on the magnitude of the variation of the phase function, but does assume an infinite detector.

Crystallochemical classifications of phyllosilicates based on the unified system of projection of chemical composition: I. The mica group

Clay Minerals ◽  
1990 ◽  
Vol 25 (1) ◽  
pp. 73-81 ◽  
Author(s):  
A. Wiewióra
Keyword(s):  
Chemical Composition ◽  
Divalent Cations ◽  
Layer Charge ◽  
Vector Representation ◽  
Similar Formula ◽  
Crystallochemical Formula ◽  
Trivalent Cations ◽  
Image Position ◽  
Composition I ◽  
Unit Vectors

AbstractA unified system of vector representation of chemical composition is proposed for the phyllosilicates based on projection of the composition, as given by crystallochemical formula, onto a field with orthogonal axes chosen for octahedral divalent cations, R2+, and Si (X, Y, respectively), and oblique axes for octahedral trivalent cations, R3+, and vacancies, □, (V, Z, respectively). Point coordinates for each set of axes were used to define the direction and length of the unit vectors for phyllosilicates belonging to different groups. Parallel to these fundamental directions the composition isolines were drawn in the projection fields. Applied to micas, this system enables control of the chemical composition by the general crystallochemical formula covering all varieties of Li-free dioctahedral and trioctahedral micas:where z (number of vacancies) = (y-x+ m)/2; m (layer charge) =1; u+y+z = 3. There is a similar formula for vacancy-free lithian micas:where w = m — x+y;m=1; u+y+w = 3, and for Li-free brittle micas:where z = (y — x+m)/2; m = 2; u+y+z = 3. Projection fields were used to classify micas.

Arcs with increasing chords

1972 ◽  
Vol 72 (2) ◽  
pp. 205-207 ◽  
Author(s):  
D. G. Larman ◽  
P. McMullen
Keyword(s):  
Absolute Constant ◽  
Complex Analysis ◽  
Private Communication ◽  
Arc Length ◽  
Euclidean Length ◽  
Jordan Arc ◽  
Image Position

Let f:[0, 1]→R2 be a Jordan arc, and for t, u ∈ [0, 1] let d(t, u) = d(f(t), f(u)) denote the Euclidean length of the chord between f(t) and f(u), and l(t, u) = l(f(t), f(u)) the corresponding arc-length, when this is defined. We say that f has the increasing chord property if d(t2, t3) ≤ d(t1, t4) whenever 0 ≤ t1 ≤ t2 ≤ t3 ≤ t4 ≤ 1. In connexion with a problem in complex analysis, K. Binmore has asked (private communication, see (1)) whether there exists an absolute constant K such that.

scholarly journals An elementary renewal theorem for random compact convex sets

1995 ◽  
Vol 27 (4) ◽  
pp. 931-942 ◽  
Author(s):  
Ilya S. Molchanov ◽  
Edward Omey ◽  
Eugene Kozarovitzky
Keyword(s):  
Support Function ◽  
Convex Sets ◽  
Compact Convex ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Minkowski Sums ◽  
Image Position ◽  
Unit Vectors

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

scholarly journals A special simplex

1966 ◽  
Vol 6 (1) ◽  
pp. 18-24 ◽  
Author(s):  
Sahib Ram Mandan
Keyword(s):  
Position Vector ◽  
Image Position

Let S = A0 hellip An be an n-simplex and Aih the foot of its altitude from its vertex Ai to its opposite prime face Si; O, G the circumcentre and centroid of S and Oi, Gi of Si. Representing the position vector of a point P, referred to O, by p, Coxeter [2] defines the Monge pointM of S Collinear with O and G by the relation so that the Monge point Mi of Si is given by If the n+1 vectors a are related by oi be given by Aih is given by Since Aih lies in Si, If Ti be a point on MiAih such that i.e. That is, MTi is parallel to ooi or normal to Si at Ti:. Or, the normals to the prime faces Si of S at their points Ti concur at M. In fact, this property of M has been used to prove by induction [3] that an S-point S of S lies at M. Thus M = 5, M = S or .

On the Linear and Vector Function

1897 ◽  
Vol 21 ◽  
pp. 310-312
Author(s):  
Tait
Keyword(s):  
Vector Function ◽  
Early Stage ◽  
Present Communication ◽  
Real Roots ◽  
Image Position ◽  
The Right ◽  
Homogeneous Strain ◽  
Unit Vectors

In a paper read to the Society in May last, I treated specially the case in which the Hamiltonian cubic has all its roots real. In that paper I employed little beyond the well-known methods of Hamilton, but some of the results obtained seemed to indicate a novel and useful classification of the various forms of the Linear and Vector Function. This is the main object of the present communication.It is known that we may always writeand that three terms of the sum on the right are sufficient, and in general more than is required, to express any linear and vector function. In fact, all necessary generality is secured by fixing, once for all, the values of α, β, γ, or of α1, β1, γ1, leaving the others arbitrary:—subject only to the condition that neither set is coplanar. Thus as a particular case we may write eitherIn either case we secure the nine independent scalar coefficients which are required for the expression of the most general homogeneous strain. But forms like these are relics of the early stage of quaternion development, and (as Hamilton expressly urged) they ought to be dispensed with as soon as possible.2. A linear and vector function is completely determined if we know its effects on each of any system of three non-coplanar unit-vectors, say α, β, γ. If its cubic have three real roots, these vectors may, if we choose be taken as the directions which it leaves unaltered; if but one, we may take a corresponding system in the formα, βcosa ± ιγsinα,where ι is But it is preferable to keep the simpler form α, β, γ, with the understanding that β and γ may be bi-vectors, of the form just written.

On the Radius of Curvature for Convex Analytic Functions

1970 ◽  
Vol 22 (3) ◽  
pp. 486-491 ◽  
Author(s):  
Paul Eenigenburg
Keyword(s):  
Real Axis ◽  
Jordan Curve ◽  
Convex Functions ◽  
Analytic Functions ◽  
Tangent Line ◽  
Positive Real Axis ◽  
Radius Of Curvature ◽  
Arc Length ◽  
Positive Real ◽  
Image Position

Definition 1.1. Let be analytic for |z| < 1. If ƒ is univalent, we say that ƒ belongs to the class S.Definition 1.2. Let ƒ ∈ S, 0 ≦ α < 1. Then ƒ belongs to the class of convex functions of order α, denoted by Kα, provided(1)and if > 0 is given, there exists Z0, |Z0| < 1, such thatLet ƒ ∈ Kα and consider the Jordan curve ϒτ = ƒ(|z| = r), 0 < r < 1. Let s(r, θ) measure the arc length along ϒτ; and let ϕ(r, θ) measure the angle (in the anti-clockwise sense) that the tangent line to ϒτ at ƒ(reiθ) makes with the positive real axis.

scholarly journals DIRECTIONAL MAXIMAL OPERATORS AND RADIAL WEIGHTS ON THE PLANE

2013 ◽  
Vol 89 (3) ◽  
pp. 397-414
Author(s):  
HIROKI SAITO ◽  
HITOSHI TANAKA
Keyword(s):  
Maximal Operator ◽  
Unit Vector ◽  
Norm Inequality ◽  
Maximal Operators ◽  
Weighted Norm Inequality ◽  
Weighted Norm ◽  
Radial Weight ◽  
Orthogonality Principle ◽  
Image Position ◽  
Unit Vectors

AbstractLet $\Omega $ be the set of unit vectors and $w$ be a radial weight on the plane. We consider the weighted directional maximal operator defined by $$\begin{eqnarray*}{M}_{\Omega , w} f(x): = \sup _{x\in R\in \mathcal{B} _{\Omega }}\frac{1}{w(R)} \int \nolimits \nolimits_{R} \vert f(y)\vert w(y)\hspace{0.167em} dy,\end{eqnarray*}$$ where ${ \mathcal{B} }_{\Omega } $ denotes the set of all rectangles on the plane whose longest side is parallel to some unit vector in $\Omega $ and $w(R)$ denotes $\int \nolimits \nolimits_{R} w$. In this paper we prove an almost-orthogonality principle for this maximal operator under certain conditions on the weight. The condition allows us to get the weighted norm inequality $$\begin{eqnarray*}\Vert {M}_{\Omega , w} f\mathop{\Vert }\nolimits_{{L}^{2} (w)} \leq C\log N\Vert f\mathop{\Vert }\nolimits_{{L}^{2} (w)} ,\end{eqnarray*}$$ when $w(x)= \vert x\hspace{-1.2pt}\mathop{\vert }\nolimits ^{a} $, $a\gt 0$, and when $\Omega $ is the set of unit vectors on the plane with cardinality $N$ sufficiently large.

scholarly journals Null 2-type Chen surfaces

1995 ◽  
Vol 37 (2) ◽  
pp. 233-242 ◽  
Author(s):  
Shi-Jie Li
Keyword(s):  
Mean Curvature ◽  
Spectral Decomposition ◽  
Harmonic Functions ◽  
Curvature Vector ◽  
Minimal Submanifolds ◽  
Dimensional Euclidean Space ◽  
Position Vector ◽  
Mean Curvature Vector ◽  
The Mean ◽  
Image Position

Let M be an n-dimensional connected submanifold in an mdimensional Euclidean space Em. Denote by δ the Laplacian of M associated with the induced metric. Then the position vector x and the mean curvature vector H of Min Em satisfyThis yields the following fact: a submanifold M in Em is minimal if and only if all coordinate functions of Em, restricted to M, are harmonic functions. In other words, minimal submanifolds in Emare constructed from eigenfunctions of δ with one eigenvalue 0. By using (1. 1), T. Takahashi proved that minimal submanifolds of a hypersphere of Em are constructed from eigenfunctions of δ with one eigenvalue δ (≠0). In [3, 4], Chen initiated the study of submanifolds in Em which are constructed from harmonic functions and eigenfunctions of δ with a nonzero eigenvalue. The position vector x of such a submanifold admits the following simple spectral decomposition:for some non-constant maps x0and xq, where A is a nonzero constant. He simply calls such a submanifold a submanifold of null 2-type.

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