Difficult dialogs about degenerate cases: A proof script study

2018 ◽  
Vol 52 ◽  
pp. 61-76 ◽  
Author(s):  
Stacy Brown
Keyword(s):  
Proof Script ◽  
Degenerate Cases

scholarly journals Some Remarks on the Regular Splitting of Quasi-Polynomials with Two Delays. Characterization of Double Roots in Degenerate Cases

2020 ◽  
Vol 53 (2) ◽  
pp. 4386-4391
Author(s):  
Alejandro Martínez-González ◽  
César-Fernando Méndez-Barrios ◽  
Silviu-Iulian Niculescu
Keyword(s):  
Regular Splitting ◽  
Two Delays ◽  
Degenerate Cases

scholarly journals On Desargues Theorem

1944 ◽  
Vol 34 ◽  
pp. 17-19
Author(s):  
J. H. M. Wedderburn
Keyword(s):  
Analytical Methods ◽  
Geometrical Proof ◽  
Analytical Proof ◽  
Desargues Theorem ◽  
Degenerate Cases

The usual proofs of Desargues Theorem employ either metrical or analytical methods of projection from a point outside the plane; and if it is attempted to translate the analytical proof by the von Stuadt-Reye methods, the result is very long and there is trouble with coincidences. It is the object of this note to give a short geometrical proof which, in addition to the usual axioms of incidence and extension, uses only the assumption that a projectivity which leaves three points on a line unchanged also leaves all points on it unchanged. Degenerate cases are excluded as having no interest.

Chiral spiral cyclic twins

2022 ◽  
Vol 78 (1) ◽  
Author(s):  
Wolfgang Hornfeck
Keyword(s):  
Long Range ◽  
Orientational Order ◽  
Two Dimensional ◽  
Point Sets ◽  
Degenerate Cases

A formula is presented for the generation of chiral m-fold multiply twinned two-dimensional point sets of even twin modulus m > 6 from an integer inclination sequence; in particular, it is discussed for the first three non-degenerate cases m = 8, 10, 12, which share a connection to the aperiodic crystallography of axial quasicrystals exhibiting octagonal, decagonal and dodecagonal long-range orientational order and symmetry.

On The Factorization of Partial Differential Equations

1987 ◽  
Vol 39 (4) ◽  
pp. 825-834 ◽  
Author(s):  
W. Dale Brownawell
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nevanlinna Theory ◽  
Algebraic Differential Equation ◽  
List Type ◽  
Variable Result ◽  
Several Variables ◽  
Function Of Several Variables ◽  
Degenerate Cases

In [4] N. Steinmetz used Nevanlinna theory to establish remarkably versatile theorems on the factorization of ordinary differential equations which implied numerous previous results of various authors. (Here factorization is taken in the sense of function composition as introduced by F. Gross in [2].) The thrust of Steinmetz’ central results on factorization is that if g(z) is entire and f(z) is meromorphic in C such that the composite fog satisfies an algebraic differential equation, then so do f(z) and, degenerate cases aside, g(z). In addition, the more one knows about the equation for fog (e.g. degree, weight, autonomy), the more one can conclude about the equations for f and g.In this note we generalize Steinmetz’ work to show the following:a) Steinmetz’ two basic results, Satz 1 and Korollar 1 of [4] can be seen as one-variable specializations of a single two variable result, andb) the function g(z) can itself be allowed to be a function of several variables.

scholarly journals Balanced Polymorphisms With Unlinked Loci

1963 ◽  
Vol 16 (1) ◽  
pp. 1 ◽  
Author(s):  
PAP Moran
Keyword(s):  
Stationary States ◽  
Genetic Population ◽  
Degenerate Cases

The stationary states of a genetic population whose fitness is controlled by two independent loci, with two alleles each, are considered. It is shown that apart from degenerate cases there are at most five such states. It is also shown that there are at most three stationary states which are stable. Examples are given where these bounds are attained.

Weak‐contrast reflection–transmission coefficients in a generally anisotropic background

Geophysics ◽  
2003 ◽  
Vol 68 (6) ◽  
pp. 2063-2072 ◽  
Author(s):  
Luděk Klimeš
Keyword(s):  
Simple Form ◽  
Seismic Data ◽  
Elastic Moduli ◽  
Anisotropic Media ◽  
Transmitted Wave ◽  
Arbitrary Orientation ◽  
Cartesian Coordinates ◽  
Transmission Coefficients ◽  
Degenerate Cases

Explicit equations for approximate linearized reflection–transmission coefficients at a generally oriented weak‐contrast interface separating two generally and independently anisotropic media are presented. The equations are derived also for all singular directions and are thus valid in degenerate cases (e.g., in an isotropic background). The equations are expressed in general Cartesian coordinates, with arbitrary orientation of the interface. The explicit equations for linearized reflection–transmission coefficients have a very simple form—much simpler than the equations published previously. The equations for all reflection–transmission coefficients, with the exception of the unconverted transmitted wave, have a common form. The form of the equations is very suitable for inversion and for analyzing the sensitivity of seismic data to discontinuities in individual elastic moduli. The factors of proportionality of the contrasts of elastic moduli and density are expressed in terms of the slowness and polarization vectors of the corresponding generated wave and incidentwave.

Bifurcations near a multiple eigenvalue of the rectangular plate problem

1986 ◽  
Vol 103 (1-2) ◽  
pp. 35-62
Author(s):  
Zoltán Sadovský
Keyword(s):  
Aspect Ratio ◽  
Rectangular Plate ◽  
Small Perturbation ◽  
Perturbation Parameter ◽  
Multiple Eigenvalue ◽  
Bifurcation Problem ◽  
Operator Equations ◽  
Double Eigenvalue ◽  
Elastic Rectangular Plate ◽  
Degenerate Cases

SynopsisWe consider the bifurcation problem of the Föppl–Kármán equations for a thin elastic rectangular plate near a multiple eigenvalue allowing for a small perturbation parameter related to the aspect ratio of the plate. The first step in the study is to introduce equivalent operator equations in the energy spaces of the problem which explicitly contain the perturbation parameter. By dealing partially with a general formulation, we obtain the main results for the double eigenvalue and Z2 ⊓ Z2 symmetry of bifurcation equations. We are chiefly interested in the degenerate cases of bifurcation equations.

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