scholarly journals A Type of Alternant

1953 ◽  
Vol 9 (1) ◽  
pp. 20-27
Author(s):  
F. W. Ponting
Keyword(s):  
Image Position ◽  
Usual Notation

We definewhere ap ≠ aq when p ≠ q. If N = Σλi, then the partition (λ1, λ2, …, λn) of N with λ1 ≥ λ2 ≥ … ≥ λn is denoted by (λ) and we setAll partitions will be in descending order and the usual notation for repeated parts will be used.

Double Meijer transformations of certain hypergeometric functions

1968 ◽  
Vol 64 (2) ◽  
pp. 425-430 ◽  
Author(s):  
H. M. Srivastava ◽  
J. P. Singhal
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Image Position ◽  
Usual Notation

Following the usual notation for generalized hypergeometric functions we let(a) denotes the sequence of A parametersthat is, there are A of the a parameters and B of the b parameters. Thus ((a))m has the interpretationwith a similar interpretation for ((b))m; Δ(k; α) stands for the set of k parametersand for the sake of brevity, the pair of parameters like α + β, α − β will be written as α ± β, the gamma product Γ(α + β) Γ(α − β) as Γ(α ± β), and so on.

On a group of 1440 birational transformations of four variables that arises in considering the projective equivalence of double sixes

1926 ◽  
Vol 23 (2) ◽  
pp. 103-108
Author(s):  
W. Burnside
Keyword(s):  
The Other ◽  
Birational Transformations ◽  
Actual Position ◽  
Projective Equivalence ◽  
Image Position ◽  
Usual Notation

The lines of a double-six will here be represented by the usual notationwhere two lines whose symbols are in the same line or same column of this scheme are non-intersectors and all other pairs of lines intersect. Any six of the lines, no two of whose symbols are in the same column, and just three are in the same row, are generators of a quadric, and the actual position in space of each of the other six is determined by the two points in which it intersects this quadric.

Some expansions of generalized Whittaker functions

1965 ◽  
Vol 61 (4) ◽  
pp. 895-896 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Whittaker Functions ◽  
Image Position ◽  
Usual Notation

In the usual notation letwhere (a)n = a(a + 1)(a + 2)…(a + n − 1), (a)0 = 1.Dr L. J. Slater ((3), page 628) gave an expansion of the formfor all values of n and t, real or complex.

Partitions of products

1993 ◽  
Vol 58 (3) ◽  
pp. 860-871 ◽  
Author(s):  
Carlos A. Di Prisco ◽  
James M. Henle
Keyword(s):  
Finite Number ◽  
Axiom Of Choice ◽  
Inaccessible Cardinal ◽  
Partition Relation ◽  
Partition Relations ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Usual Notation

We will consider some partition properties of the following type: given a function F: ωω →2, is there a sequence H0, H1, … of subsets of ω such that F is constant on ΠiεωHi? The answer is obviously positive if we allow all the Hi's to have exactly one element, but the problem is nontrivial if we require the Hi's to have at least two elements. The axiom of choice contradicts the statement “for all F: ωω→ 2 there is a sequence H0, H1, H2,… of subsets of ω such that {i|(Hi) ≥ 2} is infinite and F is constant on ΠHi”, but the infinite exponent partition relation ω(ω)ω implies it; so, this statement is relatively consistent with an inaccessible cardinal. (See [1] where these partition properties were considered.)We will also consider partitions into any finite number of pieces, and we will prove some facts about partitions into ω-many pieces.Given a partition F: ωω → k, we say that H0, H1…, a sequence of subsets of ω, is homogeneous for F if F is constant on ΠHi. We say the sequence H0, H1,… is nonoverlapping if, for all i ∈ ω, ∪Hi > ∩Hi+1.The sequence 〈Hi: i ∈ ω〉 is of type 〈α0, α1,…〉 if, for every i ∈ ω, ∣Hi∣ = αi.We will adopt the usual notation for polarized partition relations due to Erdös, Hajnal, and Rado.means that for every partition F: κ1 × κ2 × … × κn→δ there is a sequence H0, H1,…, Hn such that Hi ⊂ κi and ∣Hi∣ = αi for every i, 1 ≤ i ≤ n, and F is constant on H1 × H2 × … × Hn.

Some expansions in products of hypergeometric functions

1966 ◽  
Vol 62 (2) ◽  
pp. 245-247 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Image Position ◽  
Usual Notation

In the usual notation letwhere .Also, for the generalized hypergeometric function pFq(x) let us employ a contracted notation and writeThroughout the present paper i will run from 1 to p, I from 1 to P, and so on. Thus ((a) )m is to be interpreted as,and similar interpretations hold for ((A))m, etc.

An integral of hypergeometric type

1952 ◽  
Vol 48 (4) ◽  
pp. 578-582 ◽  
Author(s):  
L. J. Slater
Keyword(s):  
Hypergeometric Series ◽  
Basic Series ◽  
Hypergeometric Type ◽  
Image Position ◽  
Usual Notation

In a recent paper (2) I gave some integrals which represented general transformations of hypergeometric series of both ordinary and basic types. In this paper I give the most general form of this kind of integral, from which all the previously known general basic transformations can be deduced.In the usual notation for basic series,where x may be a function of n, for example, and Π is written for ‘Idem (a; b)' means that the expression immediately preceding is to be repeated with b written in place of a and a written in place of b.

The integration of generalized hypergeometric functions

1966 ◽  
Vol 62 (4) ◽  
pp. 761-764 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Image Position ◽  
Usual Notation

In the usual notation for generalized hypergeometric functions we letwhereand (a) denotes the sequence of parametersThroughout the present paper we shall suppose that there are A of the a parameters, Bof the b parameters, and so on. Thus ((a))m is to be interpreted asand similar interpretations hold for ((b))m, etc.

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.

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