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 .

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.

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.

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.

Continuous data dependence for the equations of classical elastodynamics

1969 ◽  
Vol 66 (2) ◽  
pp. 481-491 ◽  
Author(s):  
R. J. Knops ◽  
L. E. Payne
Keyword(s):  
Smooth Surface ◽  
Body Force ◽  
Three Dimensional ◽  
Elastic Solid ◽  
Time Variable ◽  
Position Vector ◽  
Governing Equations ◽  
Linear Elastic ◽  
Explicit Mention ◽  
Image Position

Consider a linear elastic solid occupying a bounded regular three-dimensional region B with smooth surface ∂B. The components of displacement ui referred to cartesian axes xi are then well known to satisfy the system of governing equationsin which t denotes the time variable, x = (x1, x2, x3) denotes the position vector, ρ(x) is the non-homogeneous density, assumed positive,. i(x, t) are the Cartesian components of body force per unit mass, and cijkl(x) are the non-homogeneous elasticities, which apart from certain smoothness conditions stated later, are assumed to possess the symmetryThroughout this paper, all suffixes range over the values 1, 2, 3 and the usual converition of summing over repeated indices is adopted. Except where it is in the interest of clarity we avoid explicit mention of the dependence of functions on their arguments.

scholarly journals An integral formula for hypersurfaces in space forms

1995 ◽  
Vol 37 (3) ◽  
pp. 337-341 ◽  
Author(s):  
Theodoros Vlachos
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Distance Function ◽  
Integral Formula ◽  
Constant Sectional Curvature ◽  
Position Vector ◽  
Space Forms ◽  
Image Position ◽  
Simply Connected

Let be an n+ 1-dimensional, complete simply connected Riemannian manifold of constant sectional curvature c and We consider the function r(·) = d(·, P0) where d stands for the distance function in and we denote by grad r the gradient of The position vector (see [1]) with origin P0 is defined as where ϕ(r)equalsr, if c = 0, c< 0 or c <0 respectively.

scholarly journals Forced motion of a charged particle in a magnetic field

1966 ◽  
Vol 6 (4) ◽  
pp. 385-398
Author(s):  
L. J. Gleeson
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Charged Particle ◽  
Constant Magnetic Field ◽  
Unit Vector ◽  
Electrical Charge ◽  
Position Vector ◽  
Forced Motion ◽  
Image Position

We consider the motion of a particle of mass m and electrical charge e, moving in a constant magnetic field Bk, where k is a unit vector, and acted upon by a force mf(t). The position vector r(t) of this particle is governed by the differential equation where .

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

Optimizing conditions for x-ray microchemical analysis in analytical electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 548-549 ◽  
Author(s):  
N. J. Zaluzec
Keyword(s):  
Solid Angle ◽  
Analytical Electron Microscopy ◽  
Incident Beam ◽  
Thin Foil ◽  
Experimental Conditions ◽  
X Ray ◽  
Microchemical Analysis ◽  
Analytical Electron ◽  
Image Position ◽  
Incident Beam Energy

The ultimate sensitivity of microchemical analysis using x-ray emission rests in selecting those experimental conditions which will maximize the measured peak-to-background (P/B) ratio. This paper presents the results of calculations aimed at determining the influence of incident beam energy, detector/specimen geometry and specimen composition on the P/B ratio for ideally thin samples (i.e., the effects of scattering and absorption are considered negligible). As such it is assumed that the complications resulting from system peaks, bremsstrahlung fluorescence, electron tails and specimen contamination have been eliminated and that one needs only to consider the physics of the generation/emission process.The number of characteristic x-ray photons (Ip) emitted from a thin foil of thickness dt into the solid angle dΩ is given by the well-known equation

Sign in / Sign up

Export Citation Format

Share Document