A pseudohedron

2004 ◽  
Vol 88 (512) ◽  
pp. 226-229
Author(s):  
H. Martyn Cundy
Keyword(s):  
The Other ◽  
Regular Polygons ◽  
Spherical Trigonometry ◽  
Equilateral Triangles ◽  
Infinite Set

The other day I received a long tube from a Canadian stranger containing a large poster featuring over a hundred polyhedra, including ‘all 92 Johnson polyhedra’. This term, though probably unfamiliar this side of the pond, was not completely unknown to me; it means convex polyhedra, excluding the regular and Archimedean ones, all of whose faces are regular polygons. Of course, as usual, we have to exclude those naughty polyhedra whose faces go around in pairs collecting squares (prisms) or equilateral triangles (antiprisms) and don’t know when to stop. The word convex is vital, otherwise there would be another infinite set. A lot of them are rather trivial, like sticking pyramids on the faces of a dodecahedron, but they include the deltahedra and many other interesting members. But they have at least one imitator who didn't quite make the grade. Trying to discover why, and how to coach him so that he would, I found that my spherical trigonometry was getting rather rusty so I set out to make one and see what was happening. I thought perhaps other readers would like to share this piece of antiresearch.

scholarly journals Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles - II

10.37236/815 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
A. M. d'Azevedo Breda ◽  
Patrícia S. Ribeiro ◽  
Altino F. Santos
Keyword(s):  
Equilateral Triangle ◽  
Isosceles Triangle ◽  
Subsequent Paper ◽  
Euclidean Sphere ◽  
Regular Polygons ◽  
Combinatorial Description ◽  
Equilateral Triangles

The study of dihedral f-tilings of the Euclidean sphere $S^2$ by triangles and $r$-sided regular polygons was initiated in 2004 where the case $r=4$ was considered [5]. In a subsequent paper [1], the study of all spherical f-tilings by triangles and $r$-sided regular polygons, for any $r\ge 5$, was described. Later on, in [3], the classification of all f-tilings of $S^2$ whose prototiles are an equilateral triangle and an isosceles triangle is obtained. The algebraic and combinatorial description of spherical f-tilings by equilateral triangles and scalene triangles of angles $\beta$, $\gamma$ and $\delta$ $(\beta>\gamma>\delta)$ whose edge adjacency is performed by the side opposite to $\beta$ was done in [4]. In this paper we extend these results considering the edge adjacency performed by the side opposite to $\delta$.

Stranger things: multiverse, string cosmology, physical eschatology

2019 ◽  
pp. 464-496
Author(s):  
Milan M. Ćirković
Keyword(s):  
Big Bang ◽  
String Cosmology ◽  
The Other ◽  
Research Activity ◽  
Eternal Inflation ◽  
The Future ◽  
Anthropic Reasoning ◽  
M Theory ◽  
Infinite Set

The period (roughly) 1990-today is characterized by a big watershed and branching of cosmology into multiple and hitherto unexpected directions. On one side, the generic chaotic/eternal inflation has provided physical grounds for rather wild speculative ideas about the multiverse: the possibly infinite set of cosmological domains (‘universes’). In order to determine how observed features of our universe are (im)probable in the multiverse context requires application of anthropic reasoning which is still controversial in many circles. On the other side, we encounter applications of other speculative physical theories, like the string/M-theory to cosmology, resulting in unusual hypotheses like those of the pre-Big Bang cosmologies. In this period we have also witnessed the birth of physical eschatology as the true ‘cosmology of the future’. This chapter will attempt a survey of these and related developments, with necessary qualifications which accompany any ongoing, evolving research activity.

LXXIV. Spherical trigonometry reduced to plane

1752 ◽  
Vol 47 ◽  
pp. 441-444 ◽  
Keyword(s):  
The Other ◽  
Spherical Trigonometry ◽  
Single Word

It is observable, that the analogies of spherical trigonometry, exclusive of the terms co-fine and co-tangent, are applicable to plane, by only changing the expression, sine or tangent of side, into the single word, side: so that the business of plane trigonometry, like a corollary to the other, is thence to be inferr’d.

Potentiel d'erreur et seconde quantification: application à la corrélation

1985 ◽  
Vol 63 (7) ◽  
pp. 1788-1796 ◽  
Author(s):  
Alexandre Laforgue
Keyword(s):  
Potential Method ◽  
The Other ◽  
Second Quantization ◽  
Correlation Operator ◽  
Correlation Problem ◽  
Quantization Formalism ◽  
Electronic Configurations ◽  
Recurrent Process ◽  
Error Potential ◽  
Infinite Set

We reexamine an error potential method. The so-called "perturbative error correction" is analysed on the pattern of the correlation problem. In the whole C.I. space the idea of a perturbation correction of each w.f., cannot lead to a linear operator. But a restriction of this process to the uncorrelated q-configurations only permits proposing a q-operator which is of linear form in the whole C.I. space. Summing over q we obtain a Cler operator adapted to an r-repetitive method which performs the whole C.I. without diagonalization. On the other hand Cler is determined by a rapidly convergent, recurrent process. Its r-times product has as a limit a correlation operator adapted to all the electronic configurations. Now, the second quantization formalism expands under a polynomial form both the cler and the correlation operator. Both these operators can be expanded in clusters. Then the excitation matrix appears to be a sum of an infinite set of r-matrices which can be determined by an ordinary perturbation.

Theories incomparable with respect to relative interpretability

1962 ◽  
Vol 27 (2) ◽  
pp. 195-211 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Natural Numbers ◽  
Infinite Set

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

scholarly journals MHV amplitudes and BCFW recursion for Yang-Mills theory in the de Sitter static patch

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Emil Albrychiewicz ◽  
Yasha Neiman ◽  
Mirian Tsulaia
Keyword(s):  
Scattering Problem ◽  
The Other ◽  
Tree Level ◽  
De Sitter ◽  
Recursion Relations ◽  
Yang Mills ◽  
The Past ◽  
Infinite Set ◽  
Mills Field ◽  
Helicity Structure

Abstract We study the scattering problem in the static patch of de Sitter space, i.e. the problem of field evolution between the past and future horizons of a de Sitter observer. We formulate the problem in terms of off-shell fields in Poincare coordinates. This is especially convenient for conformal theories, where the static patch can be viewed as a flat causal diamond, with one tip at the origin and the other at timelike infinity. As an important example, we consider Yang-Mills theory at tree level. We find that static-patch scattering for Yang-Mills is subject to BCFW-like recursion relations. These can reduce any static-patch amplitude to one with N−1MHV helicity structure, dressed by ordinary Minkowski amplitudes. We derive all the N−1MHV static-patch amplitudes from self-dual Yang-Mills field solutions. Using the recursion relations, we then derive from these an infinite set of MHV amplitudes, with arbitrary number of external legs.

Bending of Clamped Plates

1947 ◽  
Vol 14 (1) ◽  
pp. A55-A62
Author(s):  
W. B. Stiles
Keyword(s):  
Exact Solution ◽  
Normal Modes ◽  
Point Load ◽  
Simultaneous Equations ◽  
Central Point ◽  
The Other ◽  
Rectangular Plates ◽  
Vibrating Membrane ◽  
Square Plate ◽  
Infinite Set

Abstract The exact solution of thin rectangular plates clamped on all or part of the boundary requires the solution of two infinite sets of simultaneous equations in two sets of unknowns. A method of obtaining an approximate solution based upon minimization of energy and requiring the solution of the first i equations of a single infinite set of simultaneous equations is described and illustrated in this paper. The approximation functions are derived from functions representing the normal modes of a freely vibrating membrane for the same region. Solutions are obtained for a rectangular clamped plate supporting a uniform or a central point load and for a square plate clamped on two adjacent edges and pinned on the other two edges with either a uniform or a central point load. Analytical results are compared with experimentally determined deflections and stresses.

Concerning Cross Track Distance at Mid-Longitude (3)

2005 ◽  
Vol 58 (1) ◽  
pp. 152-153
Author(s):  
Paul Hickley
Keyword(s):  
The Other ◽  
Spherical Trigonometry ◽  
Map Projection ◽  
Solid Geometry ◽  
Cross Track

I am grateful to both Dr Ponsonby and Sqn Ldr Hoare for their responses to my original article and would like to thank them for replying. It is interesting that they have come up with such different approaches, one based on solid geometry (but not spherical trigonometry) and the other based on a map projection, which both give exact or near-exact answers.

Cell Growth Problems

1967 ◽  
Vol 19 ◽  
pp. 851-863 ◽  
Author(s):  
David A. Klarner
Keyword(s):  
Cell Growth ◽  
Equivalence Class ◽  
Equivalence Classes ◽  
Lattice Points ◽  
The Other ◽  
Square Lattice ◽  
Cut Set ◽  
Point Set ◽  
Infinite Set ◽  
Interior Points

The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. An n-omino is a union of n cells which is connected and has no finite cut set.The set of all n-ominoes, Rn is an infinite set for each n; however, we are interested in the elements of two finite sets of equivalence classes, Sn and Tn, which are defined on the elements of Rn as follows: Two elements of Rn belong to the same equivalence class (i) in Sn, or (ii) in Tn, if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.

Wavefront Analysis in the Nonseparable Elastodynamic Quarter-Plane Problems, I—Part 1: The General Method

1982 ◽  
Vol 49 (4) ◽  
pp. 797-807 ◽  
Author(s):  
J. Miklowitz
Keyword(s):  
Integral Equations ◽  
Formal Solution ◽  
Classical Case ◽  
The Other ◽  
New Method ◽  
Infinite Layer ◽  
Quarter Plane ◽  
Edge Conditions ◽  
Long Time ◽  
Infinite Set

This work concerns the transient response in elastodynamic quarter-plane problems. In particular, the work focuses on the nonseparable problems, developing and using a new method to solve basic quarter-plane cases involving nonmixed edge conditions. Of initial interest is the classical case in which a uniform step pressure is applied to one edge, along with zero shear stress, while the other edge is traction free. The new method of solution is related to earlier work on nonseparable wavequide (semi-infinite layer) problems in which long time, low-frequency response was the objective. The basic idea in the earlier work was the exploitation of a solution boundedness condition in the form of an infinite set of integral equations. These equations were generated by eliminating the residues at an infinite set of poles stemming from the zeros of the Rayleigh-Lamb frequency equation. These poles were associated with exponentially unbounded contributions. The solution of the integral equations determined the unknown edge conditions in the problem, hence giving the long time solution. In the new method for the quarter-plane problems the analog of the Rayleigh-Lamb frequency equations is the Rayleigh function that yields just three zeros, i.e., say the squares of cR1, cR2, and cR3 speeds for the Rayleigh surface waves. Of these six roots (and corresponding poles) only one is physical, i.e., cR1. The other five are eliminated since they correspond to: (1) exponentially unbounded waves, or (2) waves with speeds cR2 > cd, cR3 > cd, all inadmissible. They analogously (to the layer case) give four integral equations for the edge unknowns, hence the formal solution for the problems. To date the method has been successful in obtaining all the wavefront expansions for the uniform step pressure case mentioned earlier. The dilatational and equivoluminal wavefronts were obtained for the interior and both edges and the Rayleigh wavefronts for both edges. This asymptotic solution has been verified and hence establishes the credibility of the new method.

Sign in / Sign up

Export Citation Format

Share Document