scholarly journals Wrapping the cube and other polyhedra

2012 ◽  
Vol 468 (2145) ◽  
pp. 2652-2666 ◽  
Author(s):  
T. Tarnai ◽  
F. Kovács ◽  
P. W. Fowler ◽  
S. D. Guest
Keyword(s):  
Regular Graph ◽  
Infinite Series ◽  
Complex Geometry ◽  
Convex Polyhedra ◽  
Octahedral Symmetry ◽  
Symmetry Constraint ◽  
Advanced Composite ◽  
Double Covers ◽  
Two Parameter

An infinite series of twofold, two-way weavings of the cube, corresponding to ‘wrappings’, or double covers of the cube, is described with the aid of the two-parameter Goldberg–Coxeter construction. The strands of all such wrappings correspond to the central circuits (CCs) of octahedrites (four-regular polyhedral graphs with square and triangular faces), which for the cube necessarily have octahedral symmetry. Removing the symmetry constraint leads to wrappings of other eight-vertex convex polyhedra. Moreover, wrappings of convex polyhedra with fewer vertices can be generated by generalizing from octahedrites to i -hedrites, which additionally include digonal faces. When the strands of a wrapping correspond to the CCs of a four-regular graph that includes faces of size greater than 4, non-convex ‘crinkled’ wrappings are generated. The various generalizations have implications for activities as diverse as the construction of woven-closed baskets and the manufacture of advanced composite components of complex geometry.

scholarly journals A Two–Parameter Model of Snow–Avalanche Motion

1980 ◽  
Vol 26 (94) ◽  
pp. 197-207 ◽  
Author(s):  
R. Perla ◽  
T. T. Cheng ◽  
D.M. McClung
Keyword(s):  
Complex Geometry ◽  
A Priori ◽  
Snow Avalanche ◽  
Objective Criterion ◽  
Starting Position ◽  
Mathematical Properties ◽  
Distance Data ◽  
Two Parameters ◽  
Two Parameter ◽  
Selection Of

AbstractVoellmy’s (1955) method for computing the run-out distance of a snow avalanche includes an unsatisfactory feature: the a priori selection of a midslope reference where the avalanche is assumed to begin decelerating from a computed steady velocity. There is no objective criterion for selecting this reference, and yet the choice critically determines the computed stopping position of the avalanche. As an alternative, a differential equation is derived in this paper on the premise that the only logical reference is the starting position of the avalanche. The equation is solved numerically for paths of complex geometry. Solutions are based on two parameters: a coefficient of friction μ; and a ratio of avalanche mass–to–drag, M⁄D. These are analogous to the two parameters in Voellmy’s model, μ and ξH. Velocity and run-out distance data are needed to estimate μ and M⁄D to useful precision. The mathematical properties of two–parameter models are explored, and it is shown that some difficulties arise since similar results are predicted by dissimilar pairs of μ and M⁄D.

Added Moment of Inertia of Rolling Ship Sections

1982 ◽  
Vol 26 (02) ◽  
pp. 89-93
Author(s):  
P. P. Hsu ◽  
L. Landweber
Keyword(s):  
Free Surface ◽  
Infinite Series ◽  
Laurent Series ◽  
Low Frequency ◽  
Moment Of Inertia ◽  
Section Area ◽  
Family Of Curves ◽  
Section Rolling ◽  
The Given ◽  
Two Parameter

An expression for the added moment of inertia of a ship section rolling at low frequency at a free surface, in terms of the coefficients of the Laurent series of the function which maps the given section into a circle, has been derived. The method is applied to the two-parameter family of Lewis forms and the results are presented as a family of curves which gives the coefficient of the added moment of inertia as a function of the thickness ratio and the section-area coefficient of a form. A second application is to a square section, for which the Laurent expansion is an infinite series.

scholarly journals A Two–Parameter Model of Snow–Avalanche Motion

1980 ◽  
Vol 26 (94) ◽  
pp. 197-207 ◽  
Author(s):  
R. Perla ◽  
T. T. Cheng ◽  
D.M. McClung
Keyword(s):  
Complex Geometry ◽  
A Priori ◽  
Snow Avalanche ◽  
Objective Criterion ◽  
Starting Position ◽  
Mathematical Properties ◽  
Distance Data ◽  
Two Parameters ◽  
Two Parameter ◽  
Selection Of

Abstract Voellmy’s (1955) method for computing the run-out distance of a snow avalanche includes an unsatisfactory feature: the a priori selection of a midslope reference where the avalanche is assumed to begin decelerating from a computed steady velocity. There is no objective criterion for selecting this reference, and yet the choice critically determines the computed stopping position of the avalanche. As an alternative, a differential equation is derived in this paper on the premise that the only logical reference is the starting position of the avalanche. The equation is solved numerically for paths of complex geometry. Solutions are based on two parameters: a coefficient of friction μ; and a ratio of avalanche mass–to–drag, M⁄D. These are analogous to the two parameters in Voellmy’s model, μ and ξH. Velocity and run-out distance data are needed to estimate μ and M⁄D to useful precision. The mathematical properties of two–parameter models are explored, and it is shown that some difficulties arise since similar results are predicted by dissimilar pairs of μ and M⁄D.

GENERALIZED LAMBERT SERIES, RAABE’S COSINE TRANSFORM AND A GENERALIZATION OF RAMANUJAN’S FORMULA FOR

2018 ◽  
Vol 239 ◽  
pp. 232-293 ◽  
Author(s):  
ATUL DIXIT ◽  
RAJAT GUPTA ◽  
RAHUL KUMAR ◽  
BIBEKANANDA MAJI
Keyword(s):  
Eisenstein Series ◽  
Infinite Series ◽  
Hyperbolic Functions ◽  
Lambert Series ◽  
Cosine Transform ◽  
Special Cases ◽  
Ramanujan’S Formula ◽  
Modular Relation ◽  
Two Parameter ◽  
Comprehensive Study

A comprehensive study of the generalized Lambert series $\sum _{n=1}^{\infty }\frac{n^{N-2h}\text{exp}(-an^{N}x)}{1-\text{exp}(-n^{N}x)},0<a\leqslant 1,~x>0$, $N\in \mathbb{N}$ and $h\in \mathbb{Z}$, is undertaken. Several new transformations of this series are derived using a deep result on Raabe’s cosine transform that we obtain here. Three of these transformations lead to two-parameter generalizations of Ramanujan’s famous formula for $\unicode[STIX]{x1D701}(2m+1)$ for $m>0$, the transformation formula for the logarithm of the Dedekind eta function and Wigert’s formula for $\unicode[STIX]{x1D701}(1/N),N$ even. Numerous important special cases of our transformations are derived, for example, a result generalizing the modular relation between the Eisenstein series $E_{2}(z)$ and $E_{2}(-1/z)$. An identity relating $\unicode[STIX]{x1D701}(2N+1),\unicode[STIX]{x1D701}(4N+1),\ldots ,\unicode[STIX]{x1D701}(2Nm+1)$ is obtained for $N$ odd and $m\in \mathbb{N}$. In particular, this gives a beautiful relation between $\unicode[STIX]{x1D701}(3),\unicode[STIX]{x1D701}(5),\unicode[STIX]{x1D701}(7),\unicode[STIX]{x1D701}(9)$ and $\unicode[STIX]{x1D701}(11)$. New results involving infinite series of hyperbolic functions with $n^{2}$ in their arguments, which are analogous to those of Ramanujan and Klusch, are obtained.

Some results of the Barbier-Chalonge-Divan system of stellar classification

1966 ◽  
Vol 24 ◽  
pp. 77-90 ◽  
Author(s):  
D. Chalonge
Keyword(s):  
Early Type ◽  
Parameter System ◽  
Early Type Stars ◽  
Two Parameter

Several years ago a three-parameter system of stellar classification has been proposed (1, 2), for the early-type stars (O-G): it was an improvement on the two-parameter system described by Barbier and Chalonge (3).

Subjective Time Preference and Willingness to Pay for an Energy-Saving Durable Good

2001 ◽  
Vol 32 (3) ◽  
pp. 133-141 ◽  
Author(s):  
Gerrit Antonides ◽  
Sophia R. Wunderink
Keyword(s):  
Willingness To Pay ◽  
Energy Saving ◽  
Subjective Time ◽  
Durable Good ◽  
Discount Function ◽  
Hyperbolic Discount ◽  
The One ◽  
Two Parameter ◽  
Difference Effect ◽  
Better Than

Summary: Different shapes of individual subjective discount functions were compared using real measures of willingness to accept future monetary outcomes in an experiment. The two-parameter hyperbolic discount function described the data better than three alternative one-parameter discount functions. However, the hyperbolic discount functions did not explain the common difference effect better than the classical discount function. Discount functions were also estimated from survey data of Dutch households who reported their willingness to postpone positive and negative amounts. Future positive amounts were discounted more than future negative amounts and smaller amounts were discounted more than larger amounts. Furthermore, younger people discounted more than older people. Finally, discount functions were used in explaining consumers' willingness to pay for an energy-saving durable good. In this case, the two-parameter discount model could not be estimated and the one-parameter models did not differ significantly in explaining the data.

Hydrodynamics of polymer solutions via two-parameter scaling

1994 ◽  
Vol 4 (8) ◽  
pp. 1299-1310 ◽  
Author(s):  
Ralph H. Colby ◽  
Michael Rubinstein ◽  
Mohamed Daoud
Keyword(s):  
Polymer Solutions ◽  
Two Parameter
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