Compressibility of two-dimensional pores having n -fold axes of symmetry

2006 ◽  
Vol 462 (2071) ◽  
pp. 1933-1947 ◽  
Author(s):  
Thushan C Ekneligoda ◽  
Robert W Zimmerman
Keyword(s):  
Scaling Law ◽  
Mapping Function ◽  
Closed Form Expression ◽  
Fold Axis ◽  
Two Dimensional ◽  
Form Expression ◽  
Pore Compressibility ◽  
Special Cases ◽  
Axis Of Symmetry ◽  
Axes Of Symmetry

The complex variable method and conformal mapping are used to derive a closed-form expression for the compressibility of an isolated pore in an infinite two-dimensional, isotropic elastic body. The pore is assumed to have an n -fold axis of symmetry, and be represented by at most four terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. The results are validated against some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from three and four terms of the Schwarz–Christoffel mapping function for regular polygons, the compressibilities of a triangle, square, pentagon and hexagon are found (to at least three digits). Specific results for some other pore shapes, more general than the quasi-polygons obtained from the Schwarz–Christoffel mapping, are also presented. An approximate scaling law for the compressibility, in terms of the ratio of perimeter-squared to area, is also tested. This expression gives a reasonable approximation to the pore compressibility, but may overestimate it by as much as 20%.

Shear compliance of two-dimensional pores possessing N -fold axis of rotational symmetry

2008 ◽  
Vol 464 (2091) ◽  
pp. 759-775 ◽  
Author(s):  
Thushan C Ekneligoda ◽  
Robert W Zimmerman
Keyword(s):  
Rotational Symmetry ◽  
Scaling Law ◽  
Mapping Function ◽  
Closed Form Expression ◽  
Fold Axis ◽  
Two Dimensional ◽  
Form Expression ◽  
Special Cases ◽  
Fourfold Symmetry ◽  
Shear Compliance

We use the complex variable method and conformal mapping to derive a closed-form expression for the shear compliance parameters of some two-dimensional pores in an elastic material. The pores have an N -fold axis of rotational symmetry and can be represented by at most three terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. We validate our results against the solutions of some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from two and three terms of the Schwarz–Christoffel mapping function for regular polygons, we find the shear compliance of a triangle, square, pentagon and hexagon. We explicitly verify the fact that the shear compliance of a symmetric pore is independent of the orientation of the pore relative to the applied shear, for all cases except pores of fourfold symmetry. We also show that pores having fourfold symmetry, or no symmetry, will have shear compliances that vary with cos 4 θ . An approximate scaling law for the shear compliance parameter, in terms of the ratio of perimeter squared to area, is proposed and tested.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

scholarly journals A Note on a Triple Integral

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2056
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Polynomial Functions ◽  
Fundamental Constants ◽  
Form Expression ◽  
Zero Distribution ◽  
Special Cases ◽  
Almost All

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

scholarly journals Fundamental problems for infinite plate with a curvilinear hole having finite poles

2001 ◽  
Vol 7 (6) ◽  
pp. 485-501 ◽  
Author(s):  
M. A. Abdou ◽  
A. A. El-Bary
Keyword(s):  
Complex Variable ◽  
Arbitrary Shape ◽  
Infinite Plate ◽  
Mapping Function ◽  
Curvilinear Hole ◽  
Two Dimensional ◽  
Rational Mapping ◽  
Special Cases ◽  
Elasticity Problems ◽  
Different Shapes

In the present paper Muskhelishvili's complex variable method of solving two-dimensional elasticity problems has been applied to derive exact expressions for Gaursat's functions for the first and second fundamental problems of the infinite plate weakened by a hole having many poles and arbitrary shape which is conformally mapped on the domain outside a unit circle by means of general rational mapping function. Some applications are investigated. The interesting cases when the shape of the hole takes different shapes are included as special cases.

The Gerber-Shiu Discounted Penalty Function for Risk Process with Double Markovian Environment

2010 ◽  
Vol 108-111 ◽  
pp. 1103-1108
Author(s):  
Wen Guang Yu
Keyword(s):  
Penalty Function ◽  
Risk Process ◽  
Closed Form Expression ◽  
Standard Wiener Process ◽  
Form Expression ◽  
Expected Discounted Penalty Function ◽  
Premium Rate ◽  
Special Cases ◽  
Discounted Penalty Function ◽  
Expected Discounted Penalty

In this paper, we study the Gerber-Shiu discounted penalty function. We shall consider the case where the discount interest process and the occurrence of the claims are driven by two distinguished Markov process, respectively. Moreover, in this model we also consider the influence of a premium rate which varies with the level of free reserves. Using backward differential argument, we derive the integral equation satisfied by the expected discounted penalty function via differential argument when interest process in every state is perturbed by standard Wiener process and Poisson process. In some special cases, closed form expression for these quantities are obtained.

scholarly journals Exploring Novel Surface Representations via an Experimental Ray-Tracer in CGA

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Hugo Hadfield ◽  
Sushant Achawal ◽  
Joan Lasenby ◽  
Anthony Lasenby ◽  
Benjamin Young
Keyword(s):  
Closed Form Expression ◽  
Conformal Transformations ◽  
Square Root ◽  
Form Expression ◽  
Point Pair ◽  
Special Cases ◽  
Pure Grade ◽  
Surface Representations ◽  
Geometric Primitives ◽  
Vector Element

AbstractConformal Geometric Algebra (CGA) provides a unified representation of both geometric primitives and conformal transformations, and as such holds significant promise in the field of computer graphics. In this paper we implement a simple ray tracer in CGA with a Blinn–Phong lighting model, before putting it to use to examine ray intersections with surfaces generated from the direct interpolation of geometric primitives. General surfaces formed from these interpolations are rendered using analytic normals. In addition, special cases of point-pair interpolation, which might find use in graphics applications, are described and rendered. A closed form expression is found for the derivative of the square root of a scalar plus 4-vector element with respect to a scalar parameter. This square root derivative is used to construct an expression for the derivative of a pure-grade multivector projected to the blade manifold. The blade manifold projection provides an analytical method for finding the normal line to the interpolated surfaces and its use is shown in lighting calculations for the ray tracer and in generating vertex normals for exporting the evolved surfaces as polygonal meshes.

Gust Loading on a Thin Aerofoil

1971 ◽  
Vol 22 (3) ◽  
pp. 301-310 ◽  
Author(s):  
B. D. Mugridge
Keyword(s):  
Approximate Solution ◽  
Closed Form ◽  
Incompressible Flow ◽  
Velocity Component ◽  
Closed Form Expression ◽  
Sound Power ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Form Expression ◽  
Final Expression

SummaryA closed-form expression is derived which gives an approximate solution to the lift generated on a two-dimensional thin aerofoil in incompressible flow with a normal velocity component of the form exp [i(ωt–xx+yy)]. The inaccuracy of the solution when compared with other published work is compensated by the simplicity of the final expression, particularly if the result is required for the calculation of the sound power radiated by an aerofoil in a turbulent flow.

On the approximation of finite loop sources by two‐dimensional line sources

Geophysics ◽  
1984 ◽  
Vol 49 (7) ◽  
pp. 1027-1029 ◽  
Author(s):  
M. N. Nabighian ◽  
M. L. Oristaglio
Keyword(s):  
Half Space ◽  
Line Source ◽  
Opposite Polarity ◽  
Closed Form Expression ◽  
Late Time ◽  
Two Dimensional ◽  
Form Expression ◽  
Time Domain Electromagnetics ◽  
Receiving Coil ◽  
Finite Loop

An appealing feature of time‐domain electromagnetics is that the transient response simplifies considerably at late time, usually tending to a power‐law or exponential decay. In this note, we point out an interesting discrepancy between the late‐time asymptotics of a finite loop source over a half‐space and its natural two‐dimensional (2-D) approximation, which is two line sources of opposite polarity lying on a half‐space. Expressions for the transient responses of both loop (Wait and Ott, 1972) and line sources (Oristaglio, 1982) have been derived before; they show that at late times the voltage induced in a horizontal receiving coil decays as [Formula: see text] for a loop source and [Formula: see text] for a line source. Here we show that the slower decay for the line source is inherently a 2-D effect. To do this, we derive a closed‐form expression for the transient voltage induced by a finite wire of length 2L on a half‐space—a new result, for which we can separately examine the limits [Formula: see text] and [Formula: see text] Surprisingly, these limits are not interchangeable. First taking L to be infinite and then doing the late‐time asymptotic expansion yields the [Formula: see text] decay of a line source; in contrast, first doing the late‐time expansion gives a decay of [Formula: see text] for the finite wire, which is formally unchanged as the length goes to infinity.

scholarly journals A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Aili Zhang ◽  
Zhang Liu
Keyword(s):  
Risk Model ◽  
Poisson Model ◽  
Risk Process ◽  
Closed Form Expression ◽  
Form Expression ◽  
Ruin Time ◽  
Scale Functions ◽  
Surplus Process ◽  
Dividend Payments ◽  
Special Cases

This paper focuses on the De Finetti’s dividend problem for the spectrally negative Lévy risk process, where the dividend is deducted from the surplus process according to the racheting dividend strategy which was firstly introduced in Albrecher et al. (2018). A major feature of the racheting strategy lies in which the dividend rate never decreases. Unlike the conventional studies, the closed form expression for the expected, accumulated, and discounted dividend payments until the draw-down time (rather than the ruin time) is obtained in terms of the scale functions corresponding to the underlying Lévy process. The optimal barrier for the ratcheting strategy is also studied, where the dividend rate can be increased. Finally, two special cases, where the scale functions are explicitly known, i.e., the Brownian motion with drift and the compound Poisson model, are considered to illustrate the main result.

Sign in / Sign up

Export Citation Format

Share Document