共查询到20条相似文献,搜索用时 31 毫秒
1.
We consider two mass points of masses m
1=m
2= moving under Newton's law of gravitational attraction in a collision elliptic orbit while their centre of mass is at rest.
A third mass point of mass m
3≈ 0, moves on the straight line L, perpendicular to the line of motion of the first two mass points and passing through their centre of mass. Since m
3≈ 0, the motion of the masses m
1 and m
2 is not affected by the third mass, and from the symmetry of the motion it is clear that m
3 will remain on the line L. So the three masses form an isosceles triangle whose size changes with the time. The elliptic collision restricted isosceles
three-body problem consists in describing the motion of m
3.
In this paper we show the existence of a Bernoulli shift as a subsystem of the Poincaré map defined near a loop formed by
two heteroclinic solutions associated with two periodic orbits at infinity. Symbolic dynamics techniques are used to show
the existence of a large class of different motions for the infinitesimal body.
Accepted July 6, 2000?Published online February 14, 2001 相似文献
2.
We consider the motion of an infinitesimal particle under the gravitational field of (n+1) bodies in ring configuration, that consist of n primaries of equal mass m placed at the vertices of a regular polygon, plus another primary of mass m0=βm located at the geometric center of the polygon. We analyze the phase flow, determine the equilibria of the system, their linear stability and the bifurcations depending on the mass of the central primary (parameter β).This study is extended to the case when the central body is an ellipsoid or a radiation source. In this case, the topology of the problem is modified. 相似文献
3.
For a given m=(m1,...,mn)(R+)n, let p and q(R3)n be two central configurations for m. Then we call p and q equivalent and write pq if they differ by an SO(3) rotation followed by a scalar multiplication as well as by a permutation of bodies. Denote by L(n,m) the set of equivalent classes of n-body collinear central configurations in R3 for any given mass vector m=(m1,...,mn)(R+)n. The main discovery in this paper is the existence of a union H3 of three non-empty algebraic surfaces in the mass half space (m1,m2–m1,m3–m2)R+×R2 besides the planes generated by equal masses, which decreases the number of collinear central configurations. The union H3 in R+×R 2 is explicitly constructed by three 6-degree homogeneous polynomials in three variables such that, for any mass vector m=(m1,m2,m3)(R+)3, # L(3,m)=3, if m1, m2, and m3 are mutually distinct and (m1,m2–m1,m3–m2)H3, # L(3,m)=2, if m1, m2, and m3 are mutually distinct and (m1,m2–m1,m3–m2)H3, # L(3,m)=2, if two of m1, m2, and m3 are equal but not the third, # L(3,m)=1, if m1=m2=m3. We give also a sharp upper bound on #L(n,m) for any positive mass vector m(R+)n. 相似文献
4.
Victor A. Galaktionov Sergei I. Shmarev Juan L. Vazquez 《Archive for Rational Mechanics and Analysis》1999,149(3):183-212
We present a method of analysis which allows us to establish the interface equation and to prove Lipschitz continuity of interfaces and solutions which appear in a large class of nonlinear parabolic equations and conservation laws posed in one space dimension. Its main feature is intersection comparison with travelling waves. The method is explained on the following study case: We consider the Cauchy problem for the diffusion-absorption model: ut = ( um )xx - up, u 3 0, u_t = \left( u^m \right)_{xx} - u^p, \quad u\ge 0, in the range of parameters m > 1, 0 < p < 1, m+p 3 2m>1,\ 0p 3 1p\ge 1, or the purely diffusive equation ut=(um)xx, m > 1u_t=(u^m)_{xx},\ m>1, where the support of the solution expands with time and the motion is governed by Darcy's law, in the strong absorption range there might appear shrinking interfaces and the interface evolution obeys a different mechanism. Previous methods have failed to provide an adequate analysis of the interface motion and regularity in such a situation. 相似文献
5.
Alessio Figalli 《Archive for Rational Mechanics and Analysis》2010,195(2):533-560
Given two densities f and g, we consider the problem of transporting a fraction ${m \in [0,\min\{\|f\|_{L^1},\|g\|_{L^1}\}]}Given two densities f and g, we consider the problem of transporting a fraction m ? [0,min{||f||L1,||g||L1}]{m \in [0,\min\{\|f\|_{L^1},\|g\|_{L^1}\}]} of the mass of f onto g minimizing a transportation cost. If the cost per unit of mass is given by |x − y|2, we will see that uniqueness of solutions holds for m ? [||fùg||L1,min{||f||L1,||g||L1}]{m \in [\|f\wedge g\|_{L^1},\min\{\|f\|_{L^1},\|g\|_{L^1}\}]} . This extends the result of Caffarelli and McCann in Ann Math (in print), where the authors consider two densities with disjoint supports. The free boundaries of the active
regions are shown to be (n − 1)-rectifiable (provided the supports of f and g have Lipschitz boundaries), and under some weak regularity assumptions on the geometry of the supports they are also locally
semiconvex. Moreover, assuming f and g supported on two bounded strictly convex sets
W,L ì \mathbb Rn{{\Omega,\Lambda \subset \mathbb {R}^n}} , and bounded away from zero and infinity on their respective supports, C0,aloc{C^{0,\alpha}_{\rm loc}} regularity of the optimal transport map and local C
1 regularity of the free boundaries away from W?L{{\Omega\cap \Lambda}} are shown. Finally, the optimal transport map extends to a global homeomorphism between the active regions. 相似文献
6.
In this article, we present an improved third-order finite difference weighted essentially nonoscillatory (WENO) scheme to promote the order of convergence at critical points for the hyperbolic conservation laws. The improved WENO scheme is an extension of WENO-ZQ scheme. However, the global smoothness indicator has a little different from WENO-ZQ scheme. In this follow-up article, a convex combination of a second-degree polynomial with two linear polynomials in a traditional WENO fashion is used to compute the numerical flux at cell boundary. Although the same three-point information is adopted by the improved third-order WENO scheme, the truncation errors are smaller than some other third-order WENO schemes in L∞ and L2 norms. Especially, the convergence order is not declined at critical points, where the first and second derivatives vanish but not the third derivative. At last, the behavior of improved scheme is proved on a variety of one- and two-dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with other third-order WENO schemes. 相似文献
7.
The stability of the laminar helical flow of pseudoplastic liquids has been investigated with an indirect method consisting in the measurement of the rate of mass transfer at the surface of the inner rotating cylinder. The experiments have been carried out for different values of the geometric parameter = R
1/R
2 (the radius ratio) in the range of small values of the Reynolds number,Re < 200. Water solutions of CMC and MC have been used as pseudoplastic liquids obeying the power law model. The results have been correlated with the Taylor and Reynolds numbers defined with the aid of the mean viscosity value. The stability limit of the Couette flow is described by a functional dependence of the modified critical Taylor number (including geometric factor) on the flow indexn. This dependence, general for pseudoplastic liquids obeying the power law model, is close to the previous theoretical predictions and displays destabilizing influence of pseudoplasticity on the rotational motion. Beyond the initial range of the Reynolds numbers values (Re>20), the stability of the helical flow is not affected considerably by the pseudoplastic properties of liquids. In the range of the monotonic stabilization of the helical flow the stability limit is described by a general dependence of the modified Taylor number on the Reynolds number. The dependence is general for pseudoplastic as well as Newtonian liquids.Nomenclature
C
i
concentration of reaction ions, kmol/m3
-
d = R
2 –R
1
gap width, m
-
F
M
()
Meksyn's geometric factor (Eq. (1))
-
F
0
Faraday constant, C/kmol
-
i
l
density of limit current, A/m3
-
k
c
mass transfer coefficient, m/s
-
n
flow index
-
R
1,R
2
inner, outer radius of the gap, m
-
Re = V
m
·2d·/µ
m
Reynolds number
-
Ta
c
=
c
·d3/2·R
1
1/2
·/µ
m
Taylor number
-
Z
i
number of electrons involved in electrochemical reaction
-
= R
1/R
2
radius ratio
-
µ
apparent viscosity (local), Ns/m2
-
µ
m
mean apparent viscosity value (Eq. (3)), Ns/m2
-
µ
i
apparent viscosity value at a surface of the inner cylinder, Ns/m2
-
density, kg/m3
-
c
angular velocity of the inner cylinder (critical value), 1/s 相似文献
8.
9.
José A. Carrillo Marco Di Francesco Giuseppe Toscani 《Archive for Rational Mechanics and Analysis》2006,180(1):127-149
We investigate the long time asymptotics in L1+(R) for solutions of general nonlinear diffusion equations ut = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities
ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second
moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points
of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d2. In the particular case of ϕ(u) ~ um at first order when u ~ 0, we also obtain an optimal rate of convergence in L1 towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent
m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved
by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation
of the porous medium equation. 相似文献
10.
Alexander Mielke 《Journal of Dynamics and Differential Equations》1992,4(3):419-443
We consider the equation a(y)uxx+divy(b(y)yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L2() of the operatorAu= (1/a) divy(byu)+(c/a) in the case ofb changing sign. This leads to the abstract problem uxx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, ux) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x)
Ce
-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear. 相似文献
11.
Daniel M. Tartakovsky 《Transport in Porous Media》1994,17(3):257-269
Non-linear diffusion and velocity-dependent dispersion problems are under consideration. The necessary and sufficient conditions allowing the comparison of solutions to the two dimensional convection-dispersion equations with different coefficients are obtained. These conditions provide a framework within which solutions to the complex non-linear problems mentioned above can be estimated by solutions to the problems possessing analytical solvability.Nomenclature
c(x, y, t)
concentration of solute in solution,ML
–3
-
C(h)=d/dh
moisture capacity function
-
D,D
ij
hydrodynamic dispersion coefficient, a second order tensor,L
2
T
–1
-
D
L
longitudinal hydrodynamic dispersion coefficient,L
2
T
–1
-
D
m
molecular diffusion coefficient,L
2
T
–1
-
D
T
transverse hydrodynamic coefficient,L
2
T
–1
-
G
flow domain for the unsaturated flow problem
-
G
z
, G
w
flow domain and complex potential domain, respectively, for the hydrodynamic dispersion problem
-
h
piezometric head,L
-
I
n
given mass flux normal to the boundary,MLT
–1
-
k
hydraulic conductivity,LT
–1
-
K(h)
unsaturated hydraulic conductivity,LT
–1
-
L
continuously differentiable function with respect to all arguments
-
m
porosity
-
n(x,t)
outer normal vector to the boundary
-
t
time,T
-
V(x, y, t)
seepage velocity vector withV=V,LT
–1
-
x
Cartesian coordinate system
-
x
horizontal coordinate,L
-
y
vertical coordinate (elevation),L
-
(x),(x,t)
given functions in initial and boundary conditions (3), (4)
-
1(,)
angle between vectors 1c andV
-
boundary of the flow domain
-
L
,
T
longitudinal and transverse dispersivities, respectively,L
-
water mass density,ML
–3
-
v
i
components of a unit vector in the direction of the outward normal to the boundary
-
=–kh
velocity potential
- =/m
-
stream function defined such thatw=+i is the complex potential
-
=/m 相似文献
12.
Chaotic motions of a rigid rotor in short journal bearings 总被引:65,自引:0,他引:65
In the present paper the conditions that give rise to chaotic motions in a rigid rotor on short journal bearings are investigated and determined. A suitable symmetry was given to the rotor, to the supporting system, to the acting system of forces and to the system of initial conditions, in order to restrict the motions of the rotor to translatory whirl. For an assigned distance between the supports, the ratio between the transverse and the polar mass moments of the rotor was selected conveniently small, with the aim of avoiding conical instability. Since the theoretical analysis of a system's chaotic motions can only be carried out by means of numerical investigation, the procedure here adopted by the authors consists of numerical integration of the rotor's equations of motion, with trial and error regarding the three parameters that characterise the theoretical model of the system: m, the half non-dimensional mass of the rotor, , the modified Sommerfeld number relating to the lubricated bearings, and , the dimensionless value of rotor unbalance. In the rotor's equations of motion, the forces due to the lubricating film are written under the assumption of isothermal and laminar flow in short bearings. The number of numerical trials needed to find the system's chaotic responses has been greatly reduced by recognition of the fact that chaotic motions become possible when the value of the dimensionless static eccentricity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabew7aLnaaBaaaleaacaWGZbaabeaaaaa!4046!\[\varepsilon _s \] is greater than 0.4. In these conditions, non-periodic motions can be obtained even when rotor unbalance values are not particularly high (=0.05), whereas higher values (>0.4) make the rotor motion periodic and synchronous with the driving rotation. The present investigation has also identified the route that leads an assigned rotor to chaos when its angular speed is varied with prefixed values of the dimensionless unbalance . The theoretical results obtained have then been compared with experimental data. Both the theoretical and the experimental data have pointed out that in the circumstances investigated chaotic motions deserve more attention, from a technical point of view, than is normally ascribed to behaviours of this sort. This is mainly because such behaviours are usually considered of scarce practical significance owing to the typically bounded nature of chaotic evolution. The present analysis has shown that when the rotor exhibits chaotic motions, the centres of the journals describe orbits that alternate between small and large in an unpredictable and disordered manner. In these conditions the thickness of the lubricating film can assume values that are extremely low and such as to compromise the efficiency of the bearings, whereas the rotor is affected by inertia forces that are so high as to determine severe vibrations of the supports.Nomenclature
C
radial clearance of bearing (m)
-
D
diameter of bearing (m)
-
e
dimensional eccentricity of journal (m)
-
e
s
value of e corresponding to the static position of the journal
-
E
dimensional static unbalance of rotor (m)
-
f
x, f
y =F
x/(P), F
y/(P):
non-dimensional components of fluid film force
-
F
x,
F
y dimensional components of fluid film force (N)
-
g
acceleration of gravity (m/s2)
-
L
axial length of bearing (m)
-
m
% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabg2da9maalaaabaGaeqyYdC3aaWbaaSqa% beaacaaIYaaaaaGcbaGaeqyYdC3aa0baaSqaaGabciaa-bdaaeaaca% WFYaaaaaaakiabg2da9maalaaabaGaeqyYdC3aaWbaaSqabeaacaaI% YaaaaOGaam4qaaqaaiabeo8aZjaadEgaaaaaaa!4C14!\[ = \frac{{\omega ^2 }}{{\omega _0^2 }} = \frac{{\omega ^2 C}}{{\sigma g}}\]: half non-dimensional mass of rotor
-
M
half mass of rotor (kg)
-
n
angular speed of rotor (in r.p.m.=60/2)
-
t
time 相似文献
13.
We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case.
The entropy solutions we address assume their initial data only in the sense of weak-star in L
∞ as t→0+ and satisfy the entropy inequality in the sense of distributions for t>0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and
the initial layers do not appear. We also discuss applications to the zero relaxation limit for hyperbolic systems of conservation
laws with relaxation.
Accepted: October 26, 1999 相似文献
14.
Summary The development of a compressible boundary layer over a wedge impulsively set into motion is studied in this paper. The initial motion is independent of the leading edge effect and the solutions are those of a Rayleigh-type problem. The motion tends to an ultimate steady state of Falkner-Skan type. The equations governing the transient boundary layer from the initial steady state to the terminal steady-state change their character after certain time due to the leading edge effect and thereafter solution depends on both the end conditions. Numerical solutions are obtained through the second-order accuracy upwind scheme. The effects of the Falkner-Skan parameter and the surface temperature on the transient flow and heat transfer are also studied. It has been found that the flow separation does not occur form–0.0707 when
w
= 1.5 (hot wall), andm–0.118 when
0.5
(cold wall). 相似文献
15.
The effects of span position of winglet vortex generator on local heat/mass transfer over a three-row flat tube bank fin 总被引:3,自引:0,他引:3
The naphthalene sublimation method was used to study the effects of span position of vortex generators (VGs) on local heat transfer on three-row flat tube bank fin. A dimensionless factor of the larger the better characteristics, JF, is used to screen the optimum span position of VGs. In order to get JF, the local heat transfer coefficient obtained in experiments and numerical method are used to obtain the heat transferred from the fin. A new parameter, named as staggered ratio, is introduced to consider the interactions of vortices generated by partial or full periodically staggered arrangement of VGs. The present results reveal that: VGs should be mounted as near as possible to the tube wall; the vortices generated by the upstream VGs converge at wake region of flat tube; the interactions of vortices with counter rotating direction do not effect Nusselt number (Nu) greatly on fin surface mounted with VGs, but reduce Nu greatly on the other fin surface; the real staggered ratio should include the effect of flow convergence; with increasing real staggered ratio, these interactions are intensified, and heat transfer performance decreases; for average Nu and friction factor (f), the effects of interactions of vortices are not significant, f has slightly smaller value when real staggered ratio is about 0.6 than that when VGs are in no staggered arrangement.
A
cross section area of flow passage [m2]
-
A
mim
minimum cross section area of flow passage [m2]
-
a
width of flat tube [m]
-
b
length of flat tube [m]
-
B
pT
lateral pitch of flat tube: B
pT = S
1/T
p
-
d
h
hydraulic diameter of flow channel [m]
-
D
naph
diffusion of naphthalene [m2/s]
-
f
friction factor: f = pd
h/(Lu
2
max/2)
-
h
mass transfer coefficient [m/s]
-
H
height of winglet type vortex generators [m]
-
j
Colburn factor [–]
-
JF
a dimensionless ratio, defined in Eq. (23) [–]
-
L
streamwise length of fin [m]
-
L
PVG
longitudinal pitch of vortex generators divided by fin spacing: L
pVG = l
VG/T
p
-
l
VG
pitch of in-line vortex generators [m]
-
m
mass [kg]
-
m
mass sublimation rate of naphthalene [kg/m2·s]
-
Nu
Nusselt number: Nu = d
h/
-
P
pressure of naphthalene vapor [Pa]
-
p
non-dimensional pitch of in-line vortex generators: p = l
VG/S
2
- Pr
Prandtl number [–]
-
Q
heat transfer rate [W]
- R
universal gas constant [m2/s2·K]
- Re
Reynolds number: Re = ·u
max·d
h/
-
S
1
transversal pitch between flat tubes [m]
-
S
2
longitudinal pitch between flat tubes [m]
-
Sc
Schmidt number [–]
-
Sh
Sherwood number [–]: Sh = hd
h/D
naph
-
Sr
staggered ratio [–]: Sr = (2Hsin – C)/(2Hsin)
-
T
p
fin spacing [m]
-
T
temperature [K]
-
u
max
maximum velocity [m/s]
-
u
average velocity of air [m/s]
-
V
volume flow rate of air [m3/s]
-
x,y,z
coordinates [m]
-
z
sublimation depth[m]
-
heat transfer coefficient [W/m2·K]
-
heat conductivity [W/m·K]
-
viscosity [kg/m2·s]
-
density [kg/m3]
-
attack angle of vortex generator [°]
-
time interval for naphthalene sublimation [s]
-
fin thickness, distance between two VGs around the tube [m]
-
small interval
- C
distance between the stream direction centerlines of VGs
- p
pressure drop [Pa]
- 0
without VG enhancement
- 1, 2, I, II
fin surface I, fin surface II, respectively
- atm
atmosphere
- f
fluid
- fin
fin
- local
local value
- m
average
- naph
naphthalene
- n,b
naphthalene at bulk flow
- n,w
naphthalene at wall
- VG
with VG enhancement
- w
wall or fin surface 相似文献
16.
In this article, we present two improved third‐order weighted essentially nonoscillatory (WENO) schemes for recovering their design‐order near first‐order critical points. The schemes are constructed in the framework of third‐order WENO‐Z scheme. Two new global smoothness indicators, τL3 and τL4, are devised by a nonlinear combination of local smoothness indicators (ISk) and reference values (ISG) based on Lagrangian interpolation polynomial. The performances of the proposed schemes are evaluated on several numerical tests governed by one‐dimensional linear advection equation or one‐ and two‐dimensional Euler equations. Numerical results indicate that the presented schemes provide less dissipation and higher resolution than the original WENO3‐JS and subsequent WENO3‐N scheme. 相似文献
17.
The Newtonian circular restricted four-body problem is considered. We obtain nonlinear algebraic equations determining equilibrium
solutions in the rotating frame and find six possible equilibrium configurations of the system. Studying the stability of
equilibrium solutions, we prove that the radial equilibrium solutions are unstable, while the bisector equilibrium solutions
are stable in Lyapunov’s sense if the mass parameter satisfies the conditions μ ∈ (0, μ0, where μ0 is a sufficiently small number, and μ ≠ μj, j = 1, 2, 3. We also prove that, for μ = μ1 and μ = μ3, the resonance conditions of the third order and the fourth order, respectively, are satisfied and, for these values of μ,
the bisector equilibrium solutions are unstable and stable in Lyapunov’s sense, respectively. All symbolic and numerical calculations
are done with the Mathematica computer algebra system.
Published in Neliniini Kolyvannya, Vol. 10, No. 1, pp. 66–82, January–March, 2007. 相似文献
18.
In high shear rate capillary rheometry the combined effect of pressure dependent viscosity and dissipative heating becomes significant. Analytical expressions are derived to treat curved Bagley plots and throttle experiments. End effects are taken into account by using an effective length over radius ratio. The non-adiabatic case is described using a lump heat transfer coefficient ? following Hay et al. (1999). The latter enters into the dissipative heating coefficient % MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfKttLearuavTnhis1MBaeXatLxBI9gBam % XvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2DaeHbuLwB % Lnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFf % euY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9 % q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqaba % WaaqaafaaakeaacqaH1oqzdaWgaaWcbaGaemiCaahabeaakiabg2da % 9iabeg8aYnaaCaaaleqabaGaeyOeI0IaeGymaedaaOWaaeWaaeaacq % WGJbWydaWgaaWcbaGaemiCaahabeaakiabgUcaRmaalyaabaGaeu4M % dWeabaGafmyBa0MbaiaaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacq % GHsislcqaIXaqmaaaaaa!4D6C! ep = r - 1 ( cp + L \mathord