text:
\section{Motivation and Problem Definition}
Traffic networks -- consisting of highways, streets, and other kinds of roadways -- provide convenient and economical conveyance of passengers and goods. The basic activity in transportation is a trip, defined by its origin/destination, departure time/arrival time and travel route. A myriad of trips interact on the network to produce an intricate pattern of traffic flows. Since traffic conditions in many major metropolitan areas are becoming increasingly congested, affecting the operational efficiency of whole networks as well as the travel cost of each trip, traffic flow models are becoming more important in traffic engineering and the transportation policy making process. For example, well-developed traffic models are used in developing advanced ramp metering methods as well as in determining dynamic traffic assignment (JWL \& Zhang, 2000a\nocite{JWL2000a}).

There have been two approaches in mathematical modeling of traffic flow. One approach, from a microscopic view, studies individual movements of vehicles and interactions between vehicle pairs. This approach considers driving behavior and vehicle pair dynamics. But the size of the problem in a microscopic model becomes mathematically intractable when a considerable volume of traffic flow is considered. One example of the microscopic approach is the GM family of car-following models developed in the 1960's (e.g., Gazis et al., 1961\nocite{gazis1961}). The other approach studies the macroscopic features of traffic flows such as flow rate $q$, traffic density $\rho$ and travel speed $v$. The basic relationship between the three variables is: $q=\rho v$. Macroscopic models are more suitable for modeling traffic flow in complex networks since less supporting data and computation are needed.

In this thesis macroscopic traffic flow models are studied both theoretically and numerically.
Traffic flows are classified according to traffic conditions, roadway conditions and traffic network structure. Traffic flows are in equilibrium when the travel speed of these flows is uniquely determined as a function of traffic density, otherwise they are in non-equilibrium. Traffic flows are considered inhomogeneous when the roadway has different parameters at different locations. Link flows are flows on road links, and network flows are traffic flows on networks of roadways. Network flows differ from link flows in that vehicles in the former have different characteristics which affect traffic dynamics, such as the origins or destinations.

Different types of traffic flow are described by different models.
For equilibrium link flow, the celebrated LWR model was developed by Lighthill and Whitham (1955) and Richards (1956). The LWR model has been solved for the homogeneous roadway rigorously. There have been empirical solutions to the inhomogeneous  LWR model. In this work a rigorous procedure to solve the inhomogeneous LWR model is developed. (JWL \& Zhang, 2000b\nocite{JWL2000b}). The LWR model is a first-order model in the sense of PDE system order. In this thesis we also discuss the PW model (Payne, 1971\nocite{payne}; Whitham, 1974\nocite{Whitham}) and Zhang's model (1998\nocite{Zhang1998}, 1999a\nocite{Zhang1999a}) for non-equilibrium link flow. Finally we introduce a multi-commodity model when traffic flow is disaggregated by origins, destinations or departure times.

All the models we consider are based on conservation of traffic flow, i.e., the increment of vehicles in a section is equal to the difference between upstream in-flux and downstream out-flux in unit time. Mathematically, every model except the multi-commodity model, which is a discrete model, can be written as a continuous system of hyperbolic conservation laws. Solutions to the Riemann problem for all the continuous models are studied analytically, and all the models including the multi-commodity model are solved numerically with Godunov type of methods.

\section{Background and Research Overview}
\subsection{Traffic Flow Models}
In what follows the term ``traffic flow model" means a macroscopic traffic flow model. Many continuum traffic flow models can be described by a system of hyperbolic PDEs. The first of these models was the LWR model. This model relies on the assumption that there exists an equilibrium speed-density relationship $v=\vs (\r)$. Like other dynamic continuum flow models the LWR model is based on the mass conservation, i.e., traffic conservation, and is described by a first-order, nonlinear PDE:
\bqn
\r_t+f(\r)_x&=&0, \label{org:lwr}
\eqn
in which $f(\r)=\r \vs(\r)$ is the traffic flow rate. Equation \refe{org:lwr} is in conservation form. It is also called a kinematic wave model since it shows wave properties analogous to those of gases. There are many numerical methods to solve the LWR model. One approach is to solve the Riemann problem and apply a Godunov method for this model. Both solutions to the Riemann problem and the Godunov method are well-developed for hyperbolic conservation laws (Smoller, 1983\nocite{smoller1983}). Another approach is to use the demand and supply functions (Lebacque, 1996\nocite{lebacque1996}; Daganzo, 1995\nocite{Daganzo95a}), which turns out to be variants of Godunov's method.

The PW model, derived based on microscopic car-following models, discards the equilibrium assumption. It is a second-order system of hyperbolic conservation laws with a source term:
\bqn
\r_t+(\r v)_x&=&0,\label{org:pw1}\\
v_t+vv_x+\frac{c_0^2}\r \r_x &=&\frac {\vs (\r)-v}{\tau},\label{org:pw2}
\eqn
in which the constant $c_0$ is the traffic sound speed and $\tau$ is the relaxation time.  Equation \refe{org:pw1} is the continuity equation, and \refe{org:pw2} is the momentum equation. The PW model relates to driver behavior models better than the LWR model because it accounts for drivers' anticipation and inertia. However it's been shown that the LWR model is an asymptotic approximation of the PW model (Schochet, 1988\nocite{schochet1988}). The PW model better captures non-equilibrium wave phenomena in traffic flow. Another property of the PW model is that it is unstable under certain situations. Since there are no known analytical solutions to the PW model,  we use numerical methods to solve it in this research. All of these methods are developed from Godunov's method.

Also based on microscopic models, Zhang (1998\nocite{Zhang1998}) developed a new non-equilibrium traffic flow theory :
\bqn
\r_t +(\r v)_x&=&0, \label{org:zhang1}\\
v_t+vv_x+\frac{(\r \vs'(\r))^2}{\r} \r_x &=& \frac {\vs(\r)-v}{\tau}. \label {org:zhang2}
\eqn
In the momentum equation \refe{org:zhang2}, a varying sound speed $c=\r \vs'(\r)$ has been introduced. It has been shown that this new model avoids ``wrong-way travel" which is exhibited in the PW model (Zhang, 1998\nocite{Zhang1998}),  and the model is always stable. Wave solutions to this model were discussed in (Zhang, 1999a\nocite{Zhang1999a}), and finite difference approximations were studied in (Zhang, 2000a\nocite{Zhang2000a}). In this research we perform the numerical simulations of this model.

For a roadway with inhomogeneities such as a change in number of lanes, curvature and slopes, the LWR model can still be used, but the equilibrium speed-density relationship varies with location. By introducing an inhomogeneity function $a(x)$ which is a profile of the roadway at the location $x$, we can write the inhomogeneous LWR model as
\bqn
\r_t+f(a,\r)_x&=&0.
\eqn
Here the traffic flow rate $f(a,\r)$ is a function of the inhomogeneity $a(x)$. By writing
\bqn
a_t&=&0,
\eqn
the inhomogeneous LWR model is a non-strictly hyperbolic system. There have been empirical methods to solve the inhomogeneous LWR model (Lebacque, 1995\nocite{lebacque1995}; Daganzo, 1995a\nocite{Daganzo95a}). In this research, we develop a rigorous procedure to solve the LWR model based on the work by Isaacson et al. (1992\nocite{isaacson_t1992}) and Lin et al. (1995\nocite{lin_t_w1995}). We find the solutions that are consistent with those given by Lebacque. However,  our method can be extended to solve higher-order inhomogeneous models while those of Lebacque and Daganzo cannot.

Multi-commodity models are discussed in (Daganzo, 1994\nocite{Daganzo94} and 1995\nocite{Daganzo95}; Jayakrishnan, 1991\nocite{jayakrishnan91}; Vaughan et al., 1984\nocite{vaughan84}). In these models traffic flow is disaggregated by origins, destinations or departure times. All of these models are based on traffic conservation.  A First-In-First-Out (FIFO) discipline is assumed in all of these multi-commodity models. The model studied by Vaughan et al. is a continuous model. The models studied by Jayakrishnan and Daganzo are discrete models. In these two discrete models vehicles that are close to each other (in the sense of location or time) and have the same origin or destination or some other common characteristics are combined as a macroparticle. The macroparticles in a zone are ordered by time or location. Macroparticles are moved according to traffic conditions, which are solved with a link flow model. In the thesis, we introduce a new multi-commodity model that is more efficient in moving macroparticles.

\subsection{Hyperbolic Systems of Conservation Laws and Godunov Methods}
A system of hyperbolic conservation laws (Smoller, 1983\nocite{smoller1983}) takes the following form:
\bqn
u_t+f(u)_x&=&0,\label{conse}
\eqn
where $u=(u_1,\cdots, u_n)\in {\mathbb R}^n, n\geq 1$, and $(x,t)\in {\mathbb R}\times {\mathbb R}_+$. The $n$ eigenvalues of the differential of $f(u)$, $D f(u)$, are denoted as $\lambda_1,\cdots,\lambda_n$. The solutions related to $i$-th eigenvalue are called $i$-family wave solutions. If the eigenvalues are distinct, the system \refe{conse} is a strictly hyperbolic system. To solve \refe{conse}, initial and boundary conditions are needed. The Riemann problem is to solve \refe{conse} with the following jump initial condition:
\bqn
u(x,t=0)&=&\cas{{ll}u_l, & x<0\\u_r, &x> 0},
\eqn
where $u_l$ and $u-r$ are constants.

It is well-known that the weak solutions to the Riemann problem exist and are unique for the system \refe{conse} under the so-called ``Lax's entropy condition" (Lax, 1972\nocite{lax1972}). The system admits discontinuous solutions, i.e., shock waves, and the wave speed $s$ is determined by Rankine-Hugoniot condition:
\bqn
s[u]&=[f(u)],
\eqn
where $[u]=u_l-u_r$, and similarly, $[f(u)]=f(u_l)-f(u_r)$. For valid $i$-family shock wave solutions, the entropy inequalities hold, i.e., $\lambda_i(u_l)>s>\lambda_i(u_r)$. There are continuous solutions $u=u(\xi)$, $\xi=x/t$, which satisfy the ordinary differential equations
\bqn
-\xi u_{\xi}+f(u)_{\xi}&=&0.
\eqn

For most general initial and boundary conditions, there are no analytical solutions to \refe{conse}, hence, one must use numerical methods to solve it. The Godunov method is one of the most efficient numerical methods, which combines solutions to the Riemann problem and conservation laws. In a Godunov method, the space region $[a,b]$ is divided into $N$ grids $x_0=a, x_1,\cdots, x_{N-1}, x_N=b$; the time scale $[t_0, t_1]$ are partitioned into $M$ time steps $t^0=t_0, t^1, \cdots, t^{M-1}, t^M=t_1$. The hyperbolic system of conservation laws \refe{conse} can be approximated by the finite difference equations:
\bqn
\frac{ U_i^{j+1}-U_i^j}{\dt}+\frac {f(U^{\ast}_{i-1/2})-f(U^{\ast}_{i+1/2})}{\dx}&=&0, \label{diff}
\eqn
where $U_i^j$ is the average of $u(x,t)$ in grid $[x_{i-1/2},x_{i+1/2}]$ at time step $t_j$, similarly $U_i^{j+1}$ is the average at time step $t_{j+1}$; $U^{\ast}_{i-1/2}$ is the average of $u(x,t)$ during time interval $[t_j, t_{j+1}]$ at grid boundary $x_{i-1/2}$, similarly $U^{\ast}_{i+1/2}$ is the average at boundary $x_{i+1/2}$. The boundary flux $f(U^{\ast}_{i-1/2})$ is calculated by solving a Riemann problem at each cell edge with the following initial conditions:
\bqs
u(x,t^j)&=&\cas{{ll}U^j_{i-1}, &x<x_{i-1/2}\\U^j_i, &x> x_{i+1/2}}.
\eqs
Equations \refe{diff} says that the increment of $u$ is equal to the difference between both boundary fluxes, which is the general idea of conservation.

In this thesis our second-order models are not exact conservation laws since they have a source term, therefore our methods  have been extended to address the effect of source terms.

\section{Layout of the Thesis}
In chapter 2, we study the homogeneous LWR model through theoretical discussions and numerical simulations. This is the first step for us to understand traffic flow models and associated numerical methods. In chapter 3, Godunov-type methods are developed for Zhang's model. The Riemann problem is solved in detail, and computational results are discussed. In chapter 4, the PW model is studied with several different Godunov-type finite difference methods, and the stability of the model is tested. In chapter 5, the Riemann problem for the inhomogeneous LWR model is solved rigorously. In chapter 6, a multi-commodity model based on the LWR model is studied. In chapter 7, possible extensions of this research are discussed.


TITLE: Is a conditional probability measure dominated by its unconditional counterpart?
QUESTION [1 upvotes]: A dilettante here trying to better understand the basic definition/properties of conditional probabilities.
In my setting I have a probability triple $(S, \Sigma, P)$ and a random vector $(X,Y)$ living on that space and mapping to a measurable space with a product space structure $(\textbf{X $\times$ Y}, \mathcal{X} \otimes \mathcal{Y})$.
Let $\mu$ be the "law"/"push-forward measure" associated with $(X,Y)$ and let $\mu_X$, $\mu_Y$ be the (marginal) laws of $X$ and $Y$ defined through "projection", e.g. $\mu_X(B) = \mu(B\times \textbf{Y})$ for $B \in \mathcal{X}$. Suppose conditional probability measures denoted by $\mu_{X | Y=y}$ are well-defined (regular) for $y \in \textbf{Y}$.
My question: Is it true that $\mu_{X | Y=y}$ is absolutely continuous w.r.t. $\mu_X$ for almost all $y$? Intuitively, it would make sense to me that if X being in some range is "impossible", getting more information through $Y$ cannot suddenly make this "possible".
I have been attempting a contradiction argument and assume there is a set $A \in \mathcal{Y}$ with $\mu_Y(A) >0$ s.t. for all $y \in A$ there is a $B_y$ with $\mu_X(B_y) = 0$ but $\mu_{X|Y=y}(B_y) >0$. I have tried to get a contradiction by using the definition of conditional probabilities, in particular that for $C \in \mathcal{X}$ we have $\int_A \mu_{X | Y=y}(C)dy = \mu(C \times A)$, but this has not been fruitful for me so far.

REPLY [2 votes]: This is not true. Consider $(X, Y) = (Z,Z) \in \mathbb{R}^2$ where $Z$ is a $\mathcal{N}(0,1)$ random variable. Then $\mu_X = \mathcal{N}(0,1)$ but $\mu_{X \mid Y=y} = \delta_{y}$, the Dirac measure at point $y$.
The problem is that absolute continuity is not always about having the same range, but rather about having a density.


TITLE: How to find the measure of an angle on the interior of an isosceles triangle?
QUESTION [1 upvotes]: The problem is as follows:

On the interior of an isosceles triangle $\triangle{ABC}$ where $\angle{B}=110^{\circ}$ it is situated a point $M$ such as $AB=MC$ and $\angle{BAM}=5^{\circ}$. Using this information find $\angle{MCA}$.

The choices in my book are as follows:
$\begin{array}{cc}
1.&10^{\circ}\\
2.&15^{\circ}\\
3.&20^{\circ}\\
4.&25^{\circ}\\
5.&30^{\circ}\\
\end{array}$
According to the official answers sheet the answer is choice 4. But how would you get there?.
I've been looking at this figure and I am out of ideas. Can someone help me with a sketch for this problem and how to solve it?.
This problem should be solved relying only in euclidean geometry constructions, is there a way to do that to solve this?. Since I am not good with that I will really appreciate someone could help me here.

REPLY [0 votes]: Here is a further solution based on the symmetries of a regular polygon. In this case, a regular $36$-gon $0123\dots$, needed to cover the $5^\circ$ angle from the problem. The prime-notation means reflection w.r.t. the symmetry line $0\ 18$. We start with the following picture, which realizes the given $\Delta ABC$ as $\Delta 707'$ with vertices among those of a $36$-gon:

Claim: The chords $1'7$, $0\;12$, $1\; 14'$, $27'$ are concurrent in a point $M$.
Proof: The angle bisectors in $\Delta 0\, 2\, 14'$ are concurrent, these are the last three chords in the list. Let us denote by $M$ this incenter.
It is also the orthocenter of $\Delta 1\, 7'\,12$, built from the other three vertices involved in the chords.

(Why does $1'7$ go through this point, too?! The simplest solution i found also constructs the point from the other solutions, but in a different setting with a different argument.)
Angles between chords in the picture are easily computed,
for instance the angle
$\widehat{0M7'}$
between $27'$ and $0\;12$ is the mean of the arcs measured they delimit, $\frac 12((0'-7')+(12-2))\cdot10^\circ=85^\circ$.
This is the same measure as for $\widehat{M07'}$, so $\Delta 7'0M$ isosceles.
Let now $M^*$ be the reflection of $M$ w.r.t. $77'$. Since $M\in 7'2$, the reflection $M^*$ is on the reflected line $7'12$. We have $7'M^*=7'M=7'0$, and the angle $\widehat{07'M^*}=\widehat{07'12}=\frac 12\cdot (12-0)\cdot 10^\circ=60^\circ$. So $\Delta 07'M^*$ is equilateral:
$$
\tag{$\dagger$}
0M^*=7'M^*=7'M=7'0=70\ ,
$$
and $M^*$ is also on $0\; 17$.
We compare now the triangles
$$
\begin{aligned}
&\Delta 0MM^* &&\text{ and}\\
&\Delta 0M7 \ .\\
\end{aligned}
$$
$07$ is a common side, $0M=0\;12$ is the angle bisector of
$\widehat{70M^*}=\widehat{7\;0\;17}$, given two congruent angles in $0$, and $(\dagger)$ gives a third side, $0M^*=07$. We compute now
$$
\widehat{M70} =
\widehat{MM^*0} =
\widehat{M^*\;0\; 18} =
\widehat{17\;0\; 18} =
5^\circ\ ,
$$
so $M$ is also on the chord $71'$, the one building a $5^\circ$ degree angle with $70$ inside $\Delta 707'=\Delta ABC$.
This finishes the proof.
$\square$

Note: From $\Delta 0MM^* \cong\Delta 0M7$ we obtain $M7=MM*$, so $\Delta 7MM^*$ is isosceles in $M$, which together with the knowledge of
$\widehat{7MM^*}=2\cdot\widehat{7\;M\;12}=2\cdot 30^\circ=60^\circ$ makes it equilateral. We get $7,M^*,13'$ colinear. As already observed, $M^*\in 7'12$, and it seems that $M^*$ is also on a fourth chord, $14'6$.


TITLE: harmonic function with Laplace equation
QUESTION [1 upvotes]: Let be A an open and bounded set in $\mathbb{R}^{n}$ with border $\partial A$.  A function $f:A\to \mathbb{R} $ is harmonic in $A$ if $f$ is continuos in $\bar{A}$ and satisfies Laplace ecuation $\displaystyle\sum_{k=1}^{n}{\frac{\partial^2 f}{\partial x_k^2}{(x)}}=0$ for all $x \in A$.
Prove that if $f,g$ are harmonic functions  such that $f(x)=g(x)$ for all $x \in \partial A $ then $f(x)=g(x)$ for all $x \in \bar{A}$. Prove that if $f,g$ are harmonic and $f(x)= \psi(x)$ and $g(x)=\phi(x)$ for all $x\in \partial A$, then
$sup_{x\in A}{|{f(x)}-{g(x)}|}$ $=sup_{y\in \partial A}{|{\psi(y)}-\phi(y)|}$
I tried to show that the maximum and the minimum of $f$ and $g$ were equals but i couldn't use that in the proof.
I need your help with some alternative to prove this.

REPLY [1 votes]: You are in the right track. It suffices to show that if $\triangle f=0$ and $f|_{\partial A}=0$, then $f=0$. The maximum principle says that $$\sup_A f =\sup_{\partial A} f =0,$$so $f\leq 0$ on $\overline{A}$. Since $-f$ is also harmonic and vanishes at $\partial A$, using that $\sup(-f)=-\inf f$ we also conclude that $-f\leq 0$, which is to say $f\geq 0$. So $f=0$. This takes care of the first part.
The second part is also immediate from the maximum principle: the difference $f-g$ is harmonic and equals $\phi-\psi$ on the boundary, so $$\sup_A (f-g) =\sup_{\partial A}(\phi-\psi).$$
Reverse the roles of $f$ and $g$ to get the conclusion with the absolute values.


text:
\begin{document}

\title{Grounded Hyperspheres as Squashed Wormholes}
\author{H Alshal$^{1,2}$\thanks{{\small alshal@cu.edu.eg}} \ and T Curtright$^{2}
$\thanks{{\small curtright@miami.edu}}\\$^{1}$Department of Physics, Cairo University, Giza, 12613, Egypt\\$^{2}$Department of Physics, University of Miami, Coral Gables, FL 33124-8046, USA}
\date{5 December 2018}
\maketitle

\begin{abstract}
We compute exterior Green functions for equipotential, grounded\ hyperspheres
in $N$-dimensional electrostatics by squashing Riemannian wormholes,\ where an
image charge is placed in the branch of the wormhole opposite the branch
containing the source charge, thereby providing a vivid geometrical approach
to a method first suggested in 1897 by Sommerfeld. \ We compare and contrast
the strength and location of the image charge in the wormhole approach with
that of the conventional Euclidean solution where an image charge of reduced
magnitude is located inside the hypersphere. \ While the two approaches give
mathematically equivalent Green functions, we believe they provide strikingly
different physics perspectives.

\end{abstract}

\begin{center}
\bigskip

In tribute to Richard Feynman (1918-1988) and Arnold Sommerfeld (1868-1951)
\end{center}

\section{Introduction}

\href{https://en.wikipedia.org/wiki/Richard_Feynman}{Feynman} constantly
emphasized that insight could be gained by approaching a problem from a
different point of view \cite{Mehra}. \ We follow that philosophy here to
construct a Green function, $G_{o}$,\ for the $N$-dimensional electrostatics
of an equipotential, grounded hypersphere --- i.e. a \textquotedblleft
conducting\textquotedblright\ hypersphere. \ We stress the geometrical aspects
of a method first employed in the late 19th century by
\href{https://en.wikipedia.org/wiki/Arnold_Sommerfeld}{Sommerfeld}
\cite{Eckert}, albeit not for this specific problem. \ Although the method was
introduced some 120 years ago \cite{Sommerfeld}, we believe it suggests
insights that are not widely appreciated. \ While Sommerfeld's method has been
employed during the intervening century to solve a handful of otherwise
difficult electrostatic and heat conduction problems
\cite{Hobson,Waldmann,DavisReitz,DavisReitzAgain,Duffy}, we believe that a
stronger emphasis on its geometrical aspects is worthwhile and justifies
applying the method to a broader class of problems, even those which are not
difficult to solve by other means. \ To that end we reconsider grounded
hyperspheres in $N$ \emph{spatial} dimensions.

Although the Green function with homogeneous Dirichlet boundary conditions is
known for the grounded hypersphere and can be obtained using well-known
techniques, here we compute $G_{o}$ by a novel method that uses
\textquotedblleft wormholes\textquotedblright\ \cite{MorrisThorne,JamesEtAl}
\ in an $N$-dimensional Riemannian space. \ We build appropriate Green
functions for the invariant Laplacian, $\nabla^{2}$, acting on various
wormhole geometries, initially by imposing boundary conditions only
asymptotically, to obtain $G$, and finally by requiring that the Green
function also vanish at the narrowest part of the wormhole's \textquotedblleft
throat\textquotedblright\ to obtain $G_{o}$. \

To be more specific, we consider $N$-dimensional versions of the isotropic
Ellis wormhole \cite{Ellis}, whose equatorial slices have radii given by
$r\left(  w\right)  =\sqrt{R^{2}+w^{2}}$ where $R$ is a constant and
$-\infty\leq w\leq+\infty$. \ We construct these manifolds so that the region
near $w=0$ is a curved bridge\ \cite{EinsteinRosen}\ that connects two
distinct branches of the manifold, with those branches approaching two
separate copies of $N$-dimensional Euclidean space, $\mathbb{E}_{N}$,
asymptotically as $w\rightarrow\pm\infty$ (please see the Figures). \ We build
appropriate Green functions for $\nabla^{2}$ acting on these specific
geometries, once again by first imposing boundary conditions only as
$w\rightarrow\pm\infty$, to obtain $G$, and then by requiring that the Green
function also vanish at $w=0$, i.e. at radius $R$, to obtain $G_{o}$.

The \textquotedblleft grounded\textquotedblright\ Green function $G_{o}$ is
constructed by placing a source and its negative image at exactly the
\emph{same radius} and in precisely the \emph{same direction} on the manifold,
but on \emph{opposite} branches --- somewhat striking but nevertheless very
natural positions in this context. \

Next we introduce deformations of the Ellis wormhole whose radii are given by
the $p$-norm $r\left(  w\right)  =\left(  R^{p}+\left\vert w\right\vert
^{p}\right)  ^{1/p}$. \ We compute Green functions $G$ and $G_{o}$ for this
family of manifolds as well. \ Finally, we consider the $p\rightarrow1$ limit
of this family of manifolds and Green functions. \

In the $p\rightarrow1$ limit, where $r\left(  w\right)  =R+\left\vert
w\right\vert $ is the so-called Manhattan\ norm, both branches of the manifold
are completely \textquotedblleft squashed flat\textquotedblright. \ That is to
say, the manifold degenerates into two distinct Euclidean spaces joined
together along a hyperspherical \textquotedblleft doorway\textquotedblright
\ of radius $R$. (For equatorial slices of the manifolds as $p$ approaches
$1$, please see the Figures.) \ However, the \emph{interior} of the
hypersphere is \emph{excluded} from either branch. \ Moreover, in this limit
it is clear that by restricting $G_{o}$ to have source and field points on
just one of the distinct branches, a Green function for the grounded
equipotential hypersphere is obtained. \ Minor rearrangements of the terms
shows that $G_{o}$ is in exact agreement with the usual symmetrized Green
function for the grounded hypersphere, as obtained by considering a single
copy of $\mathbb{E}_{N}$ with an image charge placed inside the hypersphere.

\section{Electrostatics in $N$ Euclidean dimensions}

The point-particle electric potential in an $N$-dimensional Euclidean space,
$\mathbb{E}_{N}$, is well-known to vary inversely with distance as the
$\left(  N-2\right)  $-th power. \ For a unit point charge located at the
origin,
\begin{equation}
\Phi_{\mathbb{E}_{N}}\left(  \overrightarrow{r}\right)  =\frac{k_{N}}{r^{N-2}
}\ ,\ \ \ \ \ \nabla^{2}\Phi_{\mathbb{E}_{N}}\left(  \overrightarrow{r}
\right)  =-~\delta^{N}\left(  \overrightarrow{r}\right)  \ ,\ \ \ \ k_{N}
\equiv\frac{1}{\left(  N-2\right)  \Omega_{N}}\label{PointCharge}
\end{equation}
The total hyper-angle $\Omega_{N}$ (i.e. the area of the \emph{unit radius}
sphere, $S_{N-1}$, embedded in $N$ dimensions) is given by
\begin{equation}
\Omega_{N}=\int_{S_{N-1}}d\Omega=\frac{2\pi^{N/2}}{\Gamma\left(  N/2\right)
}\ .\label{HyperAngle}
\end{equation}
where $d\Omega$ is the standard measure on $S_{N-1}$. \ For example,
$\Omega_{1}=2$, $\Omega_{2}=2\pi$, $\Omega_{3}=4\pi$, $\Omega_{4}=2\pi^{2}$,
etc. \ The case $N=2$ is handled as a limit, to obtain
\begin{equation}
\Phi_{\mathbb{E}_{2}}\left(  \overrightarrow{r}\right)  =-\frac{1}{2\pi}
\ln\left(  r/R\right)  \ ,\text{\ \ \ \ \ }\nabla^{2}\Phi_{\mathbb{E}_{2}
}\left(  \overrightarrow{r}\right)  =-~\delta^{2}\left(  \overrightarrow{r}
\right)  \ ,\label{PointOnSphere}
\end{equation}
up to a constant $R$ that sets the distance scale. \ For more details in this
particular case, see \cite{CurtrightEtAl}.

Consequently, a Green function for $\mathbb{E}_{N}$ is
\begin{equation}
G_{\mathbb{E}_{N}}\left(  \overrightarrow{r_{1}};\overrightarrow{r_{2}
}\right)  =\frac{k_{N}}{\left\vert \overrightarrow{r_{1}}
-\overrightarrow{r_{2}}\right\vert ^{N-2}}\ ,\ \ \ \ \ \nabla^{2}
G_{\mathbb{E}_{N}}\left(  \overrightarrow{r_{1}};\overrightarrow{r_{2}
}\right)  =-~\delta^{N}\left(  \overrightarrow{r_{1}}-\overrightarrow{r_{2}
}\right)  \label{EuclideanGreen}
\end{equation}
This result is translationally invariant, that is to say, $G_{\mathbb{E}_{2}
}\left(  \overrightarrow{r_{1}};\overrightarrow{r_{2}}\right)  $ depends only
on the difference $\overrightarrow{r_{1}}-\overrightarrow{r_{2}}$. \ Moreover,
this choice for the Green function incorporates boundary conditions at spatial
infinity that mimic the behavior in (\ref{PointCharge}). \ So any sufficiently
localized charge distribution $\rho\left(  \overrightarrow{r}\right)  $ gives
rise to the usual linear superposition,
\begin{equation}
\Phi\left(  \overrightarrow{r_{1}}\right)  =\int G_{\mathbb{E}_{N}}\left(
\overrightarrow{r_{1}};\overrightarrow{r_{2}}\right)  ~\rho\left(
\overrightarrow{r_{2}}\right)  ~d^{N}r_{2}\label{LinearSuperposition}
\end{equation}
where we assume the integral is well-defined and finite. \ For a localized
charge distribution, $\Phi\left(  \overrightarrow{r_{1}}\right)
\underset{r_{1}\rightarrow\infty}{\sim}\frac{k_{N}Q}{r_{1}^{N-2}}+O\left(
\frac{1}{r_{1}^{N-1}}\right)  $ where $Q=\int\rho\left(  \overrightarrow{r_{2}
}\right)  ~d^{2}r_{2}$ is the total charge. \ Finally, we note that the Green
function (\ref{EuclideanGreen}) in $N$ dimensions can be expanded in terms of
\href{https://en.wikipedia.org/wiki/Gegenbauer_polynomials}{Gegenbauer
polynomials}, $C_{l}^{\left(  \alpha\right)  }\left(  \cos\theta\right)  $,
namely,
\begin{equation}
\frac{1}{\left\vert \overrightarrow{r_{1}}-\overrightarrow{r_{2}}\right\vert
^{N-2}}=\sum_{l=0}^{\infty}\frac{\left(  r_{<}\right)  ^{l}}{\left(
r_{>}\right)  ^{l+N-2}}~C_{l}^{\left(  \frac{N-2}{2}\right)  }\left(
\widehat{r_{1}}\cdot\widehat{r_{2}}\right)  \label{GegenbauerExpansion}
\end{equation}
where $\widehat{r_{1}}\cdot\widehat{r_{2}}\equiv\cos\theta$ and $r_{>}$ or
$r_{<}$ is the $\max$ or $\min$ of $r_{1}$ and $r_{2}$, respectively. \ This
is a straightforward generalization of the well-known $N=3$ case that involves
the \href{https://en.wikipedia.org/wiki/Legendre_polynomials}{Legendre
polynomials} $P_{l}\left(  \cos\theta\right)  $.

\section{Electrostatics in $N$ curved dimensions}

On a Riemannian manifold, described by a metric $g_{\mu\nu}$, distance
increments are given by $ds$ where
\begin{equation}
\left(  ds\right)  ^{2}=g_{\mu\nu}~dx^{\mu}dx^{\nu}
\end{equation}
Summation over integer $\mu$ and $\nu$ is implicitly understood, with
$1\leq\mu,\nu\leq N$ for an $N$-dimensional manifold. \ The invariant
Laplacian on such a manifold is given by
\begin{equation}
\nabla^{2}=\frac{1}{\sqrt{g}}~\partial_{\mu}\left(  \sqrt{g}~g^{\mu\nu
}\partial_{\nu}\right)
\end{equation}
where $g\equiv\det g_{\mu\nu}$ and $g^{\mu\nu}$ is the matrix inverse of
$g_{\mu\nu}$. \

Consider now an infinite isotropic manifold with
\begin{equation}
\left(  ds\right)  ^{2}=\left(  dw\right)  ^{2}+r^{2}\left(  w\right)  \left(
d\widehat{r}\right)  ^{2} \label{GeneralIsotropicWormhole}
\end{equation}
The variable $w$ takes on values $-\infty\leq w\leq+\infty$, the unit vectors
$\widehat{r}$ represent the points on $S_{N-1}$ in terms of the standard
angular parameterization, and $r\left(  w\right)  $ is assumed to be a
positive, non-vanishing \textquotedblleft radius\textquotedblright\ function
that has a minimum at $w=0$ and becomes infinite as $w\rightarrow\pm\infty$.
\ For fixed angles, radial displacements on the manifold are determined just
by $ds=\pm dw$. \

We refer to $w>0$ and $w<0$ as the \textquotedblleft upper\textquotedblright
\ and \textquotedblleft lower\textquotedblright\ branches of the manifold,
respectively, and following Einstein and Rosen \cite{EinsteinRosen}, we call
the region near $w=0$ the \textquotedblleft bridge\textquotedblright\ between
the two branches. \ For visualization purposes, please see the examples shown
in the Figures.

The metric, its inverse, and the determinant $g$ have well-known, standard
expressions for this manifold. \ In any case, the form of the metric leads to
the invariant Laplacian
\begin{equation}
\nabla^{2}=\frac{1}{r\left(  w\right)  ^{N-1}}~\partial_{w}\left(  r\left(
w\right)  ^{N-1}\partial_{w}\right)  -\frac{1}{r\left(  w\right)  ^{2}}
~L^{2}\ ,\ \ \
\end{equation}
where all the $N-1$ angular derivatives are contained in $L_{jk}
\equiv-i\left(  x_{j}\partial_{k}-x_{k}\partial_{j}\right)  $ with
$L^{2}\equiv\sum_{1\leq j<k\leq N}L_{jk}L_{jk}$. \

The eigenfunctions of $L^{2}$ form a complete set on $S_{N-1}$, the
\href{https://en.wikipedia.org/wiki/Spherical_harmonics#Higher_dimensions}{hyperspherical
harmonics} $Y_{lm_{1}m_{2}\cdots m_{N-2}}$, analogous to the familiar
\href{https://en.wikipedia.org/wiki/Spherical_harmonics}{spherical harmonics}
$Y_{lm}$ on $S_{2}$. \ The $Y_{lm_{1}m_{2}\cdots m_{N-2}}$ depend on the $N-1$
angles parameterizing all points on the hypersphere, but do not depend on $w$.
\ Thus, with unit vectors $\widehat{r_{1}}$ and $\widehat{r_{2}}$ representing
two points on $S_{N-1}$,
\begin{equation}
\sum_{l,m_{1},m_{2},\cdots,m_{N-2}}Y_{lm_{1}m_{2}\cdots m_{N-2}}\left(
\widehat{r_{1}}\right)  Y_{lm_{1}m_{2}\cdots m_{N-2}}^{\ast}\left(
\widehat{r_{2}}\right)  =\delta^{N-1}\left(  \widehat{r_{1}}-\widehat{r_{2}
}\right)  \ ,\ \ \ \int\delta^{N-1}\left(  \widehat{r_{1}}-\widehat{r_{2}
}\right)  d\Omega\left(  \widehat{r_{1}}\right)  =1
\end{equation}
More details about hyperspherical harmonics may be found in
\href{https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias
books}, although notation and conventions may differ from those used here.
\ Acting on $Y_{lm_{1}m_{2}\cdots m_{N-2}}$ the $L^{2}\ $eigenvalues are given
by
\begin{equation}
L^{2}Y_{lm_{1}m_{2}\cdots m_{N-2}}=l\left(  l+N-2\right)  Y_{lm_{1}m_{2}\cdots
m_{N-2}}
\end{equation}
for $l=0,1,2,\cdots$, generalizing the well-known $N=3$ case. \ As a further
generalization of $N=3$ results, there is an addition formula for
hyperspherical harmonics resulting in a Gegenbauer polynomial. \ For fixed $l$
in $N$ dimensions,
\begin{equation}
\frac{2l+N-2}{N-2}~C_{l}^{\left(  \frac{N-2}{2}\right)  }\left(
\widehat{r_{1}}\cdot\widehat{r_{2}}\right)  =\Omega_{N}\sum_{m_{1}
,m_{2},\cdots,m_{N-2}}Y_{lm_{1}m_{2}\cdots m_{N-2}}\left(  \widehat{r_{1}
}\right)  Y_{lm_{1}m_{2}\cdots m_{N-2}}^{\ast}\left(  \widehat{r_{2}}\right)
\end{equation}
where our orthonormalization convention is
\begin{equation}
\int Y_{lm_{1}\cdots m_{N-2}}\left(  \widehat{r}\right)  Y_{l^{\prime}
m_{1}^{\prime}\cdots m_{N-2}^{\prime}}^{\ast}\left(  \widehat{r}\right)
~d\Omega=\delta_{ll^{\prime}}\delta_{m_{1}m_{1}^{\prime}}~\cdots
~\delta_{m_{N-2}m_{N-2}^{\prime}}
\end{equation}
\newpage

Elementary harmonic functions on the manifold have the form
\begin{equation}
h_{lm_{1}m_{2}\cdots m_{N-2}}=h_{l}\left(  w\right)  Y_{lm_{1}m_{2}\cdots
m_{N-2}}\left(  \widehat{r}\right)
\end{equation}
where the radial functions satisfy the equation
\begin{align}
\frac{l\left(  l+N-2\right)  }{r\left(  w\right)  ^{2}}~h_{l}\left(  w\right)
&  =\frac{1}{r\left(  w\right)  ^{N-1}}\frac{d}{dw}\left(  r\left(  w\right)
^{N-1}\frac{d}{dw}h_{l}\left(  w\right)  \right)  \label{RadialHarmonicEqn}\\
&  =\frac{1}{r\left(  w\right)  }\frac{d}{dw}\left(  r\left(  w\right)
\frac{d}{dw}h_{l}\left(  w\right)  \right)  +\left(  N-2\right)
\frac{r^{\prime}\left(  w\right)  }{r\left(  w\right)  }\frac{d}{dw}
h_{l}\left(  w\right)
\end{align}
Suppose $h_{l}^{\left(  1\right)  }$ and $h_{l}^{\left(  2\right)  }$ are two
solutions of this second-order differential equation. \ Then by
\href{https://en.wikipedia.org/wiki/Abel's_identity}{Abel's identity} their
\href{https://en.wikipedia.org/wiki/Wronskian}{Wronskian} is
\begin{equation}
W\left[  h_{l}^{\left(  1\right)  }\left(  w\right)  ,h_{l}^{\left(  2\right)
}\left(  w\right)  \right]  \equiv h_{l}^{\left(  1\right)  }\left(  w\right)
~\overleftrightarrow{\frac{d}{dw}}~h_{l}^{\left(  2\right)  }\left(  w\right)
=\frac{c_{l}}{r^{N-1}\left(  w\right)  }\label{GeneralWronskian}
\end{equation}
where $c_{l}=r^{N-1}\left(  0\right)  W\left[  h_{l}^{\left(  1\right)
}\left(  0\right)  ,h_{l}^{\left(  2\right)  }\left(  0\right)  \right]  $ is
a constant. \ For the cases of interest, $r\left(  w\right)  $ is monotonic in
$\left\vert w\right\vert $ as well as symmetric under $w\rightarrow-w$ , and
therefore if $h_{l}\left(  w\right)  $ is a solution to
(\ref{RadialHarmonicEqn}), so is $h_{l}\left(  -w\right)  $. \

For the rest of the discussion, we assume both $r\left(  w\right)  =r\left(
-w\right)  $ and $r\left(  w\right)  \underset{w\rightarrow\pm\infty}{\sim
}\left\vert w\right\vert $, in which case the asymptotic behavior of the two
radial functions is familiar\footnote{Compare (\ref{AsymptoticHarmonics}) to
the usual radial equation and its solutions on $\mathbb{E}_{N}$,
namely,\ $\frac{1}{r^{N-1}}\frac{d}{dr}\left(  r^{N-1}\frac{d}{dr}h_{l}\left(
r\right)  \right)  =\frac{l\left(  l+N-2\right)  }{r^{2}}h_{l}\left(
r\right)  $, which is solved by $h_{l}\left(  r\right)  =C_{1}r^{l}
+C_{2}r^{2-N-l}$.} since to leading order (\ref{RadialHarmonicEqn}) reduces to
\begin{equation}
\frac{1}{\left\vert w\right\vert ^{N-1}}\frac{d}{dw}\left(  \left\vert
w\right\vert ^{N-1}\frac{d}{dw}h_{l}\left(  w\right)  \right)
\underset{r\rightarrow\pm\infty}{\sim}\frac{l\left(  l+N-2\right)
}{\left\vert w\right\vert ^{2}}~h_{l}\left(  w\right)
\label{AsymptoticHarmonics}
\end{equation}
with simple power law solutions
\begin{equation}
h_{l}\left(  w\right)  \underset{w\rightarrow\pm\infty}{\sim}\left\vert
w\right\vert ^{l}\text{ \ \ or \ \ \ \ }\frac{1}{\left\vert w\right\vert
^{l+N-2}}
\end{equation}
For $N>2$ we identify the exact solution that falls off when $w\rightarrow
+\infty$ but does not fall off when $w\rightarrow-\infty$ as $h_{l}^{\left(
+\right)  }$. \ Thus, with a normalization chosen for convenience,
\begin{equation}
h_{l}^{\left(  +\right)  }\left(  w\right)  \underset{w\rightarrow
+\infty}{\sim}\frac{1}{\left\vert w\right\vert ^{l+N-2}}\ ,\ \ \ h_{l}
^{\left(  +\right)  }\left(  w\right)  \underset{w\rightarrow-\infty}{\sim
}b_{l}~\left\vert w\right\vert ^{l}
\end{equation}
where $b_{l}$ is another constant. \ For the cases of interest, for which
$r\left(  w\right)  =r\left(  -w\right)  $, it follows that $h_{l}^{\left(
-\right)  }\left(  w\right)  \equiv h_{l}^{\left(  +\right)  }\left(
-w\right)  $ is an independent exact solution that has the asymptotic behavior
\begin{equation}
h_{l}^{\left(  -\right)  }\left(  w\right)  \underset{w\rightarrow
-\infty}{\sim}\frac{1}{\left\vert w\right\vert ^{l+N-2}}\ ,\ \ \ h_{l}
^{\left(  -\right)  }\left(  w\right)  \underset{w\rightarrow+\infty}{\sim
}b_{l}~\left\vert w\right\vert ^{l}
\end{equation}
Independence follows from the Wronskian
\begin{align}
W\left[  h_{l}^{\left(  -\right)  }\left(  w\right)  ,h_{l}^{\left(  +\right)
}\left(  w\right)  \right]   &  \equiv h_{l}^{\left(  -\right)  }\left(
w\right)  ~\overleftrightarrow{\frac{d}{dw}}~h_{l}^{\left(  +\right)  }\left(
w\right)  \underset{w\rightarrow+\infty}{\sim}b_{l}\left(  w^{l}~\frac{\left(
-l-N+2\right)  }{w^{l+N-1}}-\frac{1}{w^{l+N-2}}~w^{l-1}l\right) \nonumber\\
&  \underset{w\rightarrow+\infty}{\sim}b_{l}~\frac{\left(  2-N-2l\right)
}{w^{N-1}}
\end{align}
as expected from the asymptotic form of (\ref{GeneralWronskian}) with
$c_{l}=\left(  2-N-2l\right)  b_{l}$.

For $N>2$ then, a Green function that vanishes as $\left\vert w\right\vert
\rightarrow\infty$ has the form
\begin{align}
G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)   &
=\sum_{l,m_{1},m_{2},\cdots,m_{N-2}}\frac{1}{c_{l}}~h_{l}^{\left(  +\right)
}\left(  w_{>}\right)  h_{l}^{\left(  -\right)  }\left(  w_{<}\right)
Y_{lm_{1}m_{2}\cdots m_{N-2}}\left(  \widehat{r_{1}}\right)  Y_{lm_{1}
m_{2}\cdots m_{N-2}}^{\ast}\left(  \widehat{r_{2}}\right) \nonumber\\
&  =\frac{1}{\left(  2-N\right)  \Omega_{N}}\sum_{l=0}^{\infty}\frac{1}{b_{l}
}~h_{l}^{\left(  +\right)  }\left(  w_{>}\right)  h_{l}^{\left(  -\right)
}\left(  w_{<}\right)  ~C_{l}^{\left(  \frac{N-2}{2}\right)  }\left(
\cos\theta\right)  \label{GGreenFcn}
\end{align}
where $w_{>}$ or $w_{<}$ is the $\max$ or $\min$ of $w_{1}$ and $w_{2}$,
respectively. \ When $r\left(  w\right)  =r\left(  -w\right)  $ it follows
that
\begin{equation}
G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)  =G\left(
w_{2},\widehat{r}_{2};w_{1},\widehat{r}_{1}\right)  \label{Symmetry}
\end{equation}
and, due to the isotropy of the manifold, the sum over the various $m_{k}$
labeling the hyperspherical harmonics always reduces to a function of just a
single angle $\theta$, namely, $C_{l}^{\left(  \frac{N-2}{2}\right)  }\left(
\cos\theta\right)  $ with $\cos\theta\equiv\widehat{r_{1}}\cdot\widehat{r_{2}
}$. \

Completeness of the $Y_{lm_{1}m_{2}\cdots m_{N-2}}$ on the hypersphere
$S_{N-1}$ and the radial discontinuity given by
\begin{equation}
\lim_{\varepsilon\rightarrow0}\left.  \frac{d}{dw_{1}}\left(  r\left(
w_{1}\right)  ^{N-1}h_{l}^{\left(  +\right)  }\left(  w_{1}\right)  \right)
h_{l}^{\left(  -\right)  }\left(  w_{2}\right)  \right\vert _{w_{1}
=w_{2}+\varepsilon}-\lim_{\varepsilon\rightarrow0}\left.  h_{l}^{\left(
+\right)  }\left(  w_{2}\right)  \frac{d}{dw_{1}}\left(  r\left(
w_{1}\right)  ^{N-1}h_{l}^{\left(  -\right)  }\left(  w_{1}\right)  \right)
\right\vert _{w_{1}=w_{2}-\varepsilon}=-c_{l}~
\end{equation}
produces the expected invariant Dirac delta in the equation obeyed by $G$,
namely,
\begin{equation}
\nabla^{2}G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)
=-\frac{1}{r\left(  w\right)  ^{N-1}}~\delta\left(  w_{1}-w_{2}\right)
~\delta^{N-1}\left(  \widehat{r_{1}}-\widehat{r_{2}}\right)
\label{GreenFunctionEquation}
\end{equation}

If either $w_{>}\rightarrow+\infty$ or $w_{<}\rightarrow-\infty$, the
individual terms vanish in the sum (\ref{GGreenFcn}), so that the Green
function satisfies the homogeneous boundary condition $G=0$, assuming
convergence of the sum. \ However, when $w_{2}=0$ the value of $G$ is not so
obvious. \ Nevertheless, a simple linear combination can be selected to
construct a Green function that vanishes at $w_{2}=0$, namely,
\begin{equation}
G_{o}\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)  =G\left(
w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)  -G\left(  w_{1}
,\widehat{r}_{1};-w_{2},\widehat{r}_{2}\right)  \label{GoGreenFcn}
\end{equation}
Note that $G_{o}$ is manifestly an odd function of $w_{2}$ and is also an odd
function of $w_{1}$ as a consequence of (\ref{Symmetry}).

This $G_{o}$ has a simple interpretation as the potential at $\left(
w_{1},\widehat{r}_{1}\right)  $ due to a point charge source at $\left(
w_{2},\widehat{r}_{2}\right)  $ and a negative image charge of that point
source at $\left(  -w_{2},\widehat{r}_{2}\right)  $. \ The source and image
charges are therefore of \emph{equal} magnitude but opposite sign, and are
positioned symmetrically but on \emph{opposite} branches of the manifold.
\ So, if both $w_{1}$ and $w_{2}$ are restricted to one branch of the
manifold, (\ref{GreenFunctionEquation}) will still hold on that branch, since
the image will produce an additional Dirac delta only on the other branch.
\ Therefore $G_{o}$ is an appropriate Green function to solve $\nabla^{2}
\Phi=-\rho$ on the upper branch of the manifold with the condition $\Phi=0$ on
the inner boundary of that branch, i.e. on the hypersphere at $w=0$.

The radial equation (\ref{RadialHarmonicEqn}) is perhaps more transparent
after changing variable to
\begin{equation}
u=\int_{0}^{w}\frac{d\varpi}{r\left(  \varpi\right)  } \label{uVariable}
\end{equation}
with implicit inverse $w\left(  u\right)  $. \ Note that $\operatorname*{sgn}
u=\operatorname*{sgn}w$, and if $r\left(  w\right)  \underset{w\rightarrow
\pm\infty}{\sim}\left\vert w\right\vert $ then $u\underset{w\rightarrow
\pm\infty}{\rightarrow}\pm\infty$. \ With this variable change, the radial
parts of harmonic functions satisfy the equation
\begin{equation}
\frac{d^{2}}{du^{2}}h_{l}+\left(  N-2\right)  r^{\prime}\left(  w\left(
u\right)  \right)  \frac{d}{du}h_{l}=l\left(  l+N-2\right)  h_{l}
\end{equation}
This may be cast into
\href{https://en.wikipedia.org/wiki/Sturm-Liouville_theory}{Sturm-Liouville}
form on the interval $-1\leq t\leq+1$\ upon changing variables to $t=\tanh u$,
and multiplying by the integrating factor \
\begin{equation}
\mu\left(  t\right)  =\exp\left(  \int_{0}^{t}\frac{\left(  N-2\right)
r^{\prime}\left(  w\left(  \operatorname{arctanh}\tau\right)  \right)
}{\left(  1-\tau^{2}\right)  }d\tau\right)
\end{equation}
The result is
\begin{equation}
\frac{d}{dt}\left(  \mu\left(  t\right)  \left(  1-t^{2}\right)  \frac{d}
{dt}h_{l}\right)  =\frac{l\left(  l+N-2\right)  }{1-t^{2}}~\mu\left(
t\right)  h_{l}
\end{equation}
So written, the reader can easily
\href{https://search.yahoo.com/search;_ylt=AwrE19d45RNbb1AAJ9lXNyoA;_ylc=X1MDMjc2NjY3OQRfcgMyBGZyA3lmcC10BGdwcmlkA2htcEFhNXN6VC5xbmY3NGNHWlNlS0EEbl9yc2x0AzAEbl9zdWdnAzIEb3JpZ2luA3NlYXJjaC55YWhvby5jb20EcG9zAzAEcHFzdHIDBHBxc3RybAMwBHFzdHJsAzM2BHF1ZXJ5A3N0dXJtJTIwbGlvdXZpbGxlJTIwZ3JlZW4lMjBmdW5jdGlvbgR0X3N0bXADMTUyODAzMDU5Ng--?p=sturm+liouville+green+function&fr2=sb-top&fr=yfp-t&fp=1}{find
in the mathematical literature} several detailed discussions of Green
functions for this equation.

\section{The Ellis wormhole in $N$ dimensions}

The so-called Ellis wormhole \cite{Ellis} is defined by
(\ref{GeneralIsotropicWormhole}) with
\begin{equation}
r\left(  w\right)  =\sqrt{R^{2}+w^{2}}
\end{equation}
or $w=\pm\sqrt{r^{2}-R^{2}}$ on the upper\ and lower\ branches of the
wormhole, respectively. \ In this case
\begin{align}
u\left(  w\right)   &  =\int_{0}^{w}\frac{dv}{\sqrt{R^{2}+v^{2}}
}=\operatorname{arcsinh}\left(  \frac{w}{R}\right)  =\ln\left(  \frac{w}
{R}+\sqrt{1+\frac{w^{2}}{R^{2}}}\right) \\
w\left(  u\right)   &  =R\sinh u\ ,\ \ \ r\left(  w\left(  u\right)  \right)
=R\cosh u\ ,\ \ \ r^{\prime}\left(  w\left(  u\right)  \right)  =\frac
{w\left(  u\right)  }{\sqrt{R^{2}+w^{2}\left(  u\right)  }}=\tanh u
\end{align}
and therefore
\begin{equation}
\frac{d^{2}}{du^{2}}h_{l}+\left(  N-2\right)  \left(  \tanh u\right)  \frac
{d}{du}h_{l}=l\left(  l+N-2\right)  h_{l} \label{EllisDiffEqnND}
\end{equation}
This has exact solutions
\begin{equation}
h_{l}^{\left(  1\right)  }\left(  u\right)  =\frac{\left(  1+\tanh u\right)
^{\frac{1}{2}\left(  N+l-2\right)  }}{\left(  1-\tanh u\right)  ^{\frac{1}
{2}l}}\left.  _{2}F_{1}\right.  \left(  2-\frac{1}{2}N,\frac{1}{2}
N-1;2-\frac{1}{2}N-l;\frac{1-\tanh u}{2}\right)  \label{1stEllisRadial}
\end{equation}
\begin{equation}
h_{l}^{\left(  2\right)  }\left(  u\right)  =\frac{\left(  1-\tanh u\right)
^{\frac{1}{2}\left(  N+l-2\right)  }}{\left(  1+\tanh u\right)  ^{\frac{1}
{2}l}}\left.  _{2}F_{1}\right.  \left(  2-\frac{1}{2}N,\frac{1}{2}N-1;\frac
{1}{2}N+l;\frac{1-\tanh u}{2}\right)  \label{2ndEllisRadial}
\end{equation}
when written in terms of Gauss hypergeometric functions,
\begin{equation}
\left.  _{2}F_{1}\right.  \left(  a,b;c;z\right)  =\sum_{k=0}^{\infty}
\frac{\left(  a\right)  _{k}\left(  b\right)  _{k}}{\left(  c\right)  _{k}
}\frac{z^{k}}{k!}
\end{equation}
where $\left(  a\right)  _{k}=\Gamma\left(  a+k\right)  /\Gamma\left(
a\right)  $, etc. \ It follows that $h_{l}^{\left(  1\right)  }$ is
well-behaved as $u\rightarrow-\infty$ for all even $N>2$, but \emph{not} for
odd $N$, where it is necessary to take a linear combination of $h_{l}^{\left(
1\right)  }$ and $h_{l}^{\left(  2\right)  }$ to find good behavior. \ On the
other hand, $h_{l}^{\left(  2\right)  }$ is well-behaved as $u\rightarrow
+\infty$ for all $N>2$. \ (For $N=2$, see \cite{CurtrightEtAl}.)

Evidently the simplest case beyond two dimensions is $N=4$, for which the
general solution of (\ref{EllisDiffEqnND}) is given by
\begin{equation}
h_{l}\left(  u\right)  =\frac{1}{\cosh u}~\left(  c_{1}\exp\left[  \left(
l+1\right)  u\right]  +c_{2}\exp\left[  -\left(  l+1\right)  u\right]
\right)
\end{equation}
for any constants $c_{1}$ and $c_{2}$. \ That is to say,
\begin{gather}
h_{l}^{\left(  1,2\right)  }\left(  u\right)  =\frac{1}{\cosh u}~\exp\left[
\pm\left(  l+1\right)  u\right]  \ ,\ \ \ h_{l}^{\left(  1\right)  }\left(
-\infty\right)  =0\ ,\ \ \ h_{l}^{\left(  2\right)  }\left(  +\infty\right)
=0\\
W\left[  h_{l}^{\left(  1\right)  }\left(  u\right)  ,h_{l}^{\left(  2\right)
}\left(  u\right)  \right]  \equiv h_{l}^{\left(  1\right)  }\left(  u\right)
~\overleftrightarrow{\frac{d}{du}}~h_{l}^{\left(  2\right)  }\left(  u\right)
=-\frac{2\left(  l+1\right)  }{\cosh^{2}u}
\end{gather}
Or, in terms of the variable $t=\tanh u=\frac{w}{\sqrt{R^{2}+w^{2}}}$, for
$N=4$\ the two independent solutions (\ref{1stEllisRadial}) and
(\ref{2ndEllisRadial}) become
\begin{gather}
h_{l}^{\left(  1,2\right)  }\left(  t\right)  =\frac{\left(  1\pm t\right)
^{l+1}}{\left(  1-t^{2}\right)  ^{l/2}}\ ,\ \ \ h_{l}^{\left(  1\right)
}\left(  -1\right)  =0\ ,\ \ \ h_{l}^{\left(  2\right)  }\left(  +1\right)
=0\label{N=4EllisHarmonics}\\
W\left[  h_{l}^{\left(  1\right)  }\left(  t\right)  ,h_{l}^{\left(  2\right)
}\left(  t\right)  \right]  \equiv h_{l}^{\left(  1\right)  }\left(  t\right)
~\overleftrightarrow{\frac{d}{dt}}~h_{l}^{\left(  2\right)  }\left(  t\right)
=-2\left(  l+1\right)
\end{gather}

In terms of these harmonic functions for $N=4$, the Green function
(\ref{GGreenFcn}) is
\begin{align}
&  G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)
\label{N=4EllisGreenSeriesSummed}\\
&  =\frac{1}{4\pi^{2}\sqrt{\left(  R^{2}+w_{>}^{2}\right)  \left(  R^{2}
+w_{<}^{2}\right)  }}\sum_{l=0}^{\infty}~\left(  \sqrt{\frac{\left(
w_{<}+\sqrt{R^{2}+w_{<}^{2}}\right)  \left(  w_{>}-\sqrt{R^{2}+w_{>}^{2}
}\right)  }{\left(  w_{>}+\sqrt{R^{2}+w_{>}^{2}}\right)  \left(  w_{<}
-\sqrt{R^{2}+w_{<}^{2}}\right)  }}\right)  ^{l+1}~C_{l}^{\left(  1\right)
}\left(  \widehat{r_{1}}\cdot\widehat{r_{2}}\right) \nonumber\\
&  =\frac{1}{4\pi^{2}\sqrt{\left(  R^{2}+w_{>}^{2}\right)  \left(  R^{2}
+w_{<}^{2}\right)  }}\frac{1}{\sqrt{\frac{\left(  w_{>}+\sqrt{R^{2}+w_{>}^{2}
}\right)  \left(  w_{<}-\sqrt{R^{2}+w_{<}^{2}}\right)  }{\left(  w_{<}
+\sqrt{R^{2}+w_{<}^{2}}\right)  \left(  w_{>}-\sqrt{R^{2}+w_{>}^{2}}\right)
}}+\sqrt{\frac{\left(  w_{<}+\sqrt{R^{2}+w_{<}^{2}}\right)  \left(
w_{>}-\sqrt{R^{2}+w_{>}^{2}}\right)  }{\left(  w_{>}+\sqrt{R^{2}+w_{>}^{2}
}\right)  \left(  w_{<}-\sqrt{R^{2}+w_{<}^{2}}\right)  }}-2\widehat{r_{1}
}\cdot\widehat{r_{2}}}\nonumber
\end{align}
In the last line we have used the sum
\begin{equation}
\sum_{l=0}^{\infty}x^{l}~C_{l}^{\left(  1\right)  }\left(  \cos\theta\right)
=\frac{1}{1+x^{2}-2x\cos\theta}
\end{equation}
This result for the $N=4$\ Green function is more compactly written as
\begin{equation}
G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)  =\frac{1}
{4\pi^{2}r_{1}r_{2}\left(  \sqrt{\frac{\left(  r_{1}+w_{1}\right)  \left(
r_{2}-w_{2}\right)  }{\left(  r_{2}+w_{2}\right)  \left(  r_{1}-w_{1}\right)
}}+\sqrt{\frac{\left(  r_{2}+w_{2}\right)  \left(  r_{1}-w_{1}\right)
}{\left(  r_{1}+w_{1}\right)  \left(  r_{2}-w_{2}\right)  }}-2\widehat{r_{1}
}\cdot\widehat{r_{2}}\right)  } \label{N=4EllisGreen}
\end{equation}
where $r_{1}\equiv\sqrt{R^{2}+w_{1}^{2}}$ and $r_{2}\equiv\sqrt{R^{2}
+w_{2}^{2}}$. \ Note that $G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}
_{2}\right)  =G\left(  w_{2},\widehat{r}_{2};w_{1},\widehat{r}_{1}\right)  $.
\ The $N=4$ Green function for the grounded wormhole is then defined as in
(\ref{GoGreenFcn}), so that $G_{o}\left(  w_{1},\widehat{r}_{1};w_{2}
,\widehat{r}_{2}\right)  =-G_{o}\left(  -w_{1},\widehat{r}_{1};w_{2}
,\widehat{r}_{2}\right)  $. \

Asymptotically, with both points on the upper branch of the manifold,
\begin{equation}
G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)  \underset{w_{1}
,w_{2}\gg R}{\sim}\frac{1}{4\pi^{2}\left\vert \overrightarrow{r_{1}
}-\overrightarrow{r_{2}}\right\vert ^{2}}+O\left(  \frac{R}{r_{1,2}^{3}
}\right)  \label{N=4Asymptotics}
\end{equation}
As should be expected, the leading term here is in agreement with
(\ref{EuclideanGreen}) for $N=4$. \

A contour plot of $G$ versus $w_{1}$ and $\theta\equiv\arccos\left(
\widehat{r_{1}}\cdot\widehat{r_{2}}\right)  $ is shown for the $N=4$ Ellis
wormhole in Figure 10, with $R=1$ and unit source at $\left(  w_{2}
,\widehat{r_{2}}\right)  =\left(  1,\widehat{r_{2}}\right)  $. \ A similar
plot of $G_{o}$ versus $w_{1}$ and $\theta$ is shown in Figure 11. \ In
addition to the unit source at $\left(  w_{2},\widehat{r_{2}}\right)  =\left(
1,\widehat{r_{2}}\right)  $, $G_{o}$\ incorporates a negative image of that
source at $\left(  w_{2},\widehat{r_{2}}\right)  =\left(  -1,\widehat{r_{2}
}\right)  $.

\section{The $p$-norm wormholes}

Consider next a continuous deformation of the Ellis wormhole. \ Define a class
of \textquotedblleft$p$-norm radial functions\textquotedblright\ with $R$ a
constant radius, $p\geq1$ a real number, and
\begin{equation}
r\left(  w\right)  =\left(  R^{p}+\left\vert w\right\vert ^{p}\right)  ^{1/p}
\label{pNormRadius}
\end{equation}
Note that indeed $r\left(  w\right)  \underset{\left\vert w\right\vert \gg
R}{\sim}\left\vert w\right\vert $, so the asymptotic behavior of harmonic
functions on these manifolds falls within the scope of the discussion
following (\ref{AsymptoticHarmonics}). \ Also note the case $p=2$ is the Ellis
wormhole. \ Equatorial slices of the manifolds defined by (\ref{pNormRadius})
and (\ref{GeneralIsotropicWormhole}) for various values of $p$ are shown in
the Figures\ as 3D embeddings of 2D surfaces, to obtain curved surfaces of
revolution about the $z$-axis. \

As $p\rightarrow1$, or else as $p\rightarrow\infty$, this class of manifolds
continuously interpolates between the Ellis wormhole, with its smoothly curved
bridge, and two flattened copies of $\mathbb{E}_{N}$ each of which is missing
an $N$-ball $\mathbb{B}_{N}$ of radius $R$. \ That is to say, varying $p$ away
from $p=2$ interpolates between the Ellis wormhole and a pair of
$\mathbb{E}_{N}-\mathbb{B}_{N}\left(  R\right)  $ manifolds, either as
$p\rightarrow1$ or else as $p\rightarrow\infty$. \ Nonetheless, the two copies
of $\mathbb{E}_{N}-\mathbb{B}_{N}\left(  R\right)  $ are joined together,
either by a single $S_{N-1}$ of radius $R$, as $p\rightarrow1$, or by a tube
composed of such $S_{N-1}$s, as $p\rightarrow\infty$. \ For the first of these
limits, we say the two copies of $\mathbb{E}_{N}-\mathbb{B}_{N}\left(
R\right)  $ are \textquotedblleft creased\textquotedblright\ together on a
single hypersphere of radius $R$, that hypersphere providing an open
\textquotedblleft doorway\textquotedblright\ to go from one copy of
$\mathbb{E}_{N}-$ $\mathbb{B}_{N}\left(  R\right)  $ to the other.

For $G$ and $G_{o}$, in this short section on generic $p$-norm manifolds we
are content to refer to the previous general discussion of the Green
functions. \ To supplement that discussion, we only point out that the
variable $u$ defined in (\ref{uVariable}) is explicitly given for the $p$-norm
radial functions by Gauss hypergeometric functions, namely,
\begin{equation}
u\left(  w\right)  =\int_{0}^{w}\frac{1}{\left(  R^{p}+\left(  \varpi
^{2}\right)  ^{p/2}\right)  ^{1/p}}~d\varpi=\frac{w}{R}\left.  _{2}
F_{1}\right.  \left(  \frac{1}{p},\frac{1}{p};1+\frac{1}{p};-\left(
\frac{w^{2}}{R^{2}}\right)  ^{\frac{1}{2}p}\right)
\end{equation}

\section{The squashed wormhole and its Green functions}

Flattening the two branches of the wormhole, to obtain two independent copies
of $\mathbb{E}_{N}-\mathbb{B}_{N}\left(  R\right)  $ joined together on a
common hypersphere, is achieved by taking the $p=1$ Manhattan
norm.\footnote{In the opposite extreme, when $p\rightarrow\infty$ an
equatorial slice of the $p$-norm wormhole becomes the right-circular
cylindrical tube of \cite{JamesEtAl}, p 488, Eqn (3).}\
\begin{equation}
r\left(  w\right)  =R+\left\vert w\right\vert
\end{equation}
Then so long as $w\neq0$ (\ref{RadialHarmonicEqn}) is solved by familiar
functions,
\begin{equation}
h_{l}\left(  w\right)  =r\left(  w\right)  ^{l}\text{ \ \ or \ \ \ \ }r\left(
w\right)  ^{2-N-l}
\end{equation}
It suffices to consider two situations for the source and field point
locations, either with $w_{1}$ and $w_{2}$ on the same branch of the manifold,
or with $w_{1}$ and $w_{2}$ on opposite branches. \

Suppose $w_{>}$ is always on the upper branch, say. \ Then (\ref{GGreenFcn})
will have different forms for the two possible source and field point
locations.
\begin{align}
G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)   &  =k_{N}
\sum_{l=0}^{\infty}\frac{\left(  r\left(  w_{<}\right)  \right)  ^{l}}{\left(
r\left(  w_{>}\right)  \right)  ^{l+N-2}}~C_{l}^{\left(  \frac{N-2}{2}\right)
}\left(  \widehat{r_{1}}\cdot\widehat{r_{2}}\right)  \text{ \ \ \ \ if both
}w_{>}>0\text{ \& }w_{<}>0\label{GUpUp}\\
&  =k_{N}\sum_{l=0}^{\infty}\frac{\left(  R\right)  ^{2l+N-2}}{\left(
r\left(  w_{>}\right)  r\left(  w_{<}\right)  \right)  ^{l+N-2}}
~C_{l}^{\left(  \frac{N-2}{2}\right)  }\left(  \widehat{r_{1}}\cdot
\widehat{r_{2}}\right)  \text{ \ \ \ \ if }w_{>}>0\text{ but}\ w_{<}<0
\label{GUpLow}
\end{align}
The two forms are chosen so that $G\rightarrow0$ as $w_{>}\rightarrow+\infty$
or as $w_{<}\rightarrow-\infty$, and so that $G$ is continuous as
$w_{<}\rightarrow0$, i.e. at $r\left(  w_{<}\right)  =R$. \

For the first situation with\ $w_{1}>0$ and $w_{2}>0$, in light of
(\ref{GegenbauerExpansion}) the sum in (\ref{GUpUp}) gives
\begin{align}
\left.  G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)
\right\vert _{\substack{w_{1}>0\\w_{2}>0}}  &  =\frac{k_{N}}{\left(
r^{2}\left(  w_{1}\right)  +r^{2}\left(  w_{2}\right)  -2r\left(
w_{1}\right)  r\left(  w_{2}\right)  \widehat{r_{1}}\cdot\widehat{r_{2}
}\right)  ^{\frac{N-2}{2}}}\label{Gflat++}\\
&  =\frac{k_{N}}{\left\vert \overrightarrow{r_{1}}-\overrightarrow{r_{2}
}\right\vert ^{N-2}}
\end{align}
where in the last expression we have identified $\overrightarrow{r_{1}
}=\left(  R+\left\vert w_{1}\right\vert \right)  \widehat{r}_{1}$ and
$\overrightarrow{r_{2}}=\left(  R+\left\vert w_{2}\right\vert \right)
\widehat{r}_{2}$ to obtain a symmetrical form that can also be used if both
$w_{1}<0$ and $w_{2}<0$. \

For the second situation with $w_{1}>0$ and $w_{2}<0$, the sum in
(\ref{GUpLow}) gives
\begin{align}
\left.  G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)
\right\vert _{\substack{w_{1}>0\\w_{2}<0}}  &  =\frac{R^{N-2}k_{N}}{\left(
r^{2}\left(  w_{1}\right)  r^{2}\left(  w_{2}\right)  +R^{4}-2R^{2}r\left(
w_{1}\right)  r\left(  w_{2}\right)  \widehat{r_{1}}\cdot\widehat{r_{2}
}\right)  ^{\frac{N-2}{2}}}\label{Gflat+-}\\
&  =\frac{R^{N-2}k_{N}}{\left(  r_{1}^{2}r_{2}^{2}+R^{4}-2R^{2}
\overrightarrow{r_{1}}\cdot\overrightarrow{r_{2}}\right)  ^{\frac{N-2}{2}}}
\end{align}
where in the last expression we have again identified $\overrightarrow{r_{1}
}=\left(  R+\left\vert w_{1}\right\vert \right)  \widehat{r}_{1}$ and
$\overrightarrow{r_{2}}=\left(  R+\left\vert w_{2}\right\vert \right)
\widehat{r}_{2}$ to obtain a symmetrical form that can also be used if
$w_{1}<0$ and $w_{2}>0$. \

So, as might have been anticipated, $G\left(  w_{1},\widehat{r}_{1}
;w_{2},\widehat{r}_{2}\right)  =G\left(  w_{2},\widehat{r}_{2};w_{1}
,\widehat{r}_{1}\right)  $ in all situations. \ Taken together, (\ref{Gflat++}
) and (\ref{Gflat+-}) give\ $G$ as a solution to (\ref{GreenFunctionEquation})
for any field and source point locations on the squashed manifold. \ A contour
plot of $G$ versus $w_{1}$ and $\theta\equiv\arccos\left(  \widehat{r_{1}
}\cdot\widehat{r_{2}}\right)  $ is shown for the squashed wormhole in Figure
12, with $N=4$, $R=1$, and unit source at $\left(  w_{2},\widehat{r_{2}
}\right)  =\left(  1,\widehat{r_{2}}\right)  $. \

Once again a simple linear combination can be taken to construct a Green
function that vanishes at $w_{2}=0$, as in (\ref{GoGreenFcn}). \ Given that
$r\left(  w\right)  =r\left(  -w\right)  $, when both $w_{1}>0$ and $w_{2}>0$
this grounded Green function for the squashed wormhole is explicitly
\begin{equation}
\left.  G_{o}\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)
\right\vert _{\substack{w_{1}>0\\w_{2}>0}}=\frac{k_{N}}{\left\vert
\overrightarrow{r_{1}}-\overrightarrow{r_{2}}\right\vert ^{N-2}}-\frac
{R^{N-2}k_{N}}{\left(  r_{1}^{2}r_{2}^{2}+R^{4}-2R^{2}\overrightarrow{r_{1}
}\cdot\overrightarrow{r_{2}}\right)  ^{\frac{N-2}{2}}}
\label{GreenGroundedSquashedUpper}
\end{equation}
Recall that $G_{o}$ is not only manifestly an odd function of $w_{2}$ but is
also an odd function of $w_{1}$ as a consequence of (\ref{GoGreenFcn}) and the
symmetry $G\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)
=G\left(  w_{2},\widehat{r}_{2};w_{1},\widehat{r}_{1}\right)  $. \ Therefore,
when the field point is on the lower branch of the squashed wormhole with the
source on the upper branch,
\begin{equation}
\left.  G_{o}\left(  w_{1},\widehat{r}_{1};w_{2},\widehat{r}_{2}\right)
\right\vert _{\substack{w_{1}<0\\w_{2}>0}}=-\frac{k_{N}}{\left\vert
\overrightarrow{r_{1}}-\overrightarrow{r_{2}}\right\vert ^{N-2}}+\frac
{R^{N-2}k_{N}}{\left(  r_{1}^{2}r_{2}^{2}+R^{4}-2R^{2}\overrightarrow{r_{1}
}\cdot\overrightarrow{r_{2}}\right)  ^{\frac{N-2}{2}}}
\label{GreenGroundedSquashedLower}
\end{equation}
Again, for emphasis, this $G_{o}$ has the interpretation as the potential at
$\left(  w_{1},\widehat{r}_{1}\right)  $ due to a point charge source at
$\left(  w_{2},\widehat{r}_{2}\right)  $ and a negative image charge of that
point source at $\left(  -w_{2},\widehat{r}_{2}\right)  $. \ The source and
image charges are therefore of \emph{equal} magnitude but opposite sign, and
are positioned in an \emph{natural} way on \emph{opposite} branches of the
manifold. \

At the risk of being repetitive, if both $w_{1}$ and $w_{2}$ are restricted to
one branch of the manifold, (\ref{GreenFunctionEquation}) will still hold on
that branch, since the image will produce an additional Dirac delta only on
the other branch. \ Therefore $G_{o}$ is an appropriate Green function to
solve $\nabla^{2}\Phi=-\rho$ on the upper branch of the manifold with the
condition $\Phi=0$ on the inner boundary of that branch, i.e. on the
hypersphere at $w=0$.

When both field and source points are on the upper branch of the wormhole,
this result for $G_{o}$ is exactly the usual result for the Green function in
the region exterior to a grounded hypersphere, as obtained by considering only
\emph{one copy} of $\mathbb{E}_{N}$ and putting a unit source at
$\overrightarrow{r_{2}}$ outside the hypersphere along with a negative image
source of \emph{reduced strength} $-\left(  R/r_{2}\right)  ^{N-2}$ at an
\textquotedblleft inversion point\textquotedblright\ $\overrightarrow{r}
_{\text{image}}=\frac{R^{2}}{r_{2}^{2}}~\overrightarrow{r_{2}}$ that
lies\ \emph{inside} the hypersphere, namely,
\begin{equation}
G_{0}\left(  \overrightarrow{r_{1}};\overrightarrow{r_{2}}\right)
=\frac{k_{N}}{\left\vert \overrightarrow{r_{1}}-\overrightarrow{r_{2}
}\right\vert ^{N-2}}-\frac{R^{N-2}}{r_{2}^{N-2}}\frac{k_{N}}{\left\vert
\overrightarrow{r_{1}}-\frac{R^{2}}{r_{2}^{2}}~\overrightarrow{r_{2}
}\right\vert ^{N-2}}=G_{0}\left(  \overrightarrow{r_{2}};\overrightarrow{r_{1}
}\right)  \label{StandardGo}
\end{equation}
Thus, either image procedure gives the same $G_{o}$ Green function when
restricted to this single copy of $\mathbb{E}_{N}-\mathbb{B}_{N}\left(
R\right)  $. In particular, for $N=4$, on the upper branch of the wormhole,
\begin{equation}
G_{0}\left(  \overrightarrow{r_{1}},\overrightarrow{r_{2}}\right)  =\frac
{1}{4\pi^{2}\left\vert \overrightarrow{r_{1}}-\overrightarrow{r_{2}
}\right\vert ^{2}}-\frac{R^{2}}{r_{2}^{2}}\frac{1}{4\pi^{2}\left\vert
\overrightarrow{r_{1}}-\frac{R^{2}}{r_{2}^{2}}~\overrightarrow{r_{2}
}\right\vert ^{2}}=\frac{1}{4\pi^{2}\left(  r_{1}^{2}+r_{2}^{2}
-2\overrightarrow{r_{1}}\cdot\overrightarrow{r_{2}}\right)  }-\frac{R^{2}
}{4\pi^{2}\left(  r_{1}^{2}r_{2}^{2}+R^{4}-2R^{2}\overrightarrow{r_{1}}
\cdot\overrightarrow{r_{2}}\right)  }
\end{equation}
A contour plot of $G_{o}$ for the $N=4$ squashed wormhole is shown in Figure
13, for $\overrightarrow{r_{1}}$ on both upper and lower branches, with a unit
source\ on the upper branch at $\left(  w_{2},\theta_{2}\right)  =\left(
1,0\right)  $ and its negative image on the lower branch at $\left(
-w_{2},\theta_{2}\right)  =\left(  -1,0\right)  $.

\section{Relating image charge distributions by inversion}

Coordinate inversion for a single copy of $\mathbb{E}_{N}$\ maps the interior
of the hypersphere to the exterior, and vice versa, and therefore inversion
should be expected to relate the standard image method, where the so-called
\href{https://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin}{Kelvin}
image is placed inside the hypersphere, to the Sommerfeld method for the
squashed wormhole. \ Indeed, inversion of the source position is the technique
that is normally invoked to locate Kelvin images for grounded hypersphere
Green functions on $\mathbb{E}_{N}$. \

The inversion mapping is defined by
\begin{equation}
\overrightarrow{\mathfrak{r}}=\frac{R^{2}}{r^{2}}~\overrightarrow{r}
\end{equation}
Radial distances change under the inversion, $\mathfrak{r}=R^{2}/r$, but
angles do not, $\widehat{\mathfrak{r}}=\widehat{r}$. \ Under an inversion the
Laplacian does \emph{not} transform into a geometric factor multiplying just
the Laplacian, \emph{except when }$N=2$. \ In other dimensions, the Laplacian
mixes with the \emph{scale operator} under an inversion, as follows.
\begin{equation}
\nabla_{r}^{2}=\left(  \frac{\mathfrak{r}^{2}}{R^{2}}\right)  ^{2}\left(
\nabla_{\mathfrak{r}}^{2}+\frac{2\left(  2-N\right)  }{\mathfrak{r}^{2}
}~D_{\mathfrak{r}}\right)  \ ,\ \ \ \ \ D_{\mathfrak{r}}
=\overrightarrow{\mathfrak{r}}\cdot\overrightarrow{\nabla}_{\mathfrak{r}}
\label{InversionOnLaplacian}
\end{equation}
This statement may be understood by considering harmonic functions, upon
noting that\ under inversions $r$ effectively becomes $1/r$, and only when
$N=2$ do both $r^{l}$ and $r^{-l}$ appear as factors in harmonic functions.
\ For other $N$ the factors are $r^{l}$ and $r^{2-N-l}$.

Moreover, $G$ itself is \emph{not} invariant under the inversion, \emph{except
when }$N=2$. \ This is obvious on dimensional grounds, since $G\left(
\overrightarrow{r},0\right)  \propto1/r^{N-2}\underset{\text{inversion}
}{\longrightarrow}\mathfrak{r}^{N-2}/R^{2N-4}\propto\left(  \mathfrak{r}
^{2N-4}/R^{2N-4}\right)  G\left(  \overrightarrow{\mathfrak{r}},0\right)  $.
\ More precisely, in $N$ spatial dimensions,
\begin{equation}
G\left(  \overrightarrow{r_{1}};\overrightarrow{r_{2}}\right)
\underset{\text{inversion}}{\longrightarrow}\frac{\mathfrak{r}_{1}
^{N-2}~\mathfrak{r}_{2}^{N-2}}{R^{2N-4}}~G\left(  \overrightarrow{\mathfrak{r}
_{1}};\overrightarrow{\mathfrak{r}_{2}}\right)
\end{equation}
Note the symmetry under $\overrightarrow{r_{1}}\leftrightarrow
\overrightarrow{r_{2}}$\ is maintained under $\overrightarrow{\mathfrak{r}
_{1}}\leftrightarrow\overrightarrow{\mathfrak{r}_{2}}$. \ The complete
transformation of the differential equation for the Green function in $N$
dimensions is
\begin{align}
\nabla_{r_{1}}^{2}G\left(  \overrightarrow{r_{1}};\overrightarrow{r_{2}
}\right)   &  =-\frac{1}{\sqrt{g}}~\delta^{N}\left(  \overrightarrow{r_{1}
}-\overrightarrow{r_{2}}\right)  \underset{\text{inversion}}{\longrightarrow
}\\
\left(  \frac{\mathfrak{r}_{1}^{2}}{R^{2}}\right)  ^{2}\left(  \nabla
_{\mathfrak{r}_{1}}^{2}+\frac{2\left(  2-N\right)  }{\mathfrak{r}_{1}^{2}
}~D_{\mathfrak{r}_{1}}\right)  \left(  \frac{\mathfrak{r}_{1}^{N-2}
~\mathfrak{r}_{2}^{N-2}}{R^{2N-4}}~G\left(  \overrightarrow{\mathfrak{r}_{1}
};\overrightarrow{\mathfrak{r}_{2}}\right)  \right)   &  =-\frac
{\mathfrak{r}_{1}^{N+1}}{R^{2N}}~\delta\left(  \mathfrak{r}_{1}-\mathfrak{r}
_{2}\right)  ~\delta^{N-1}\left(  \widehat{\mathfrak{r}}_{1}
-\widehat{\mathfrak{r}}_{2}\right)
\end{align}
Again the $N=2$ case is especially simple. \ For $N=2$, up to a common factor,
the equation is unchanged in form by the inversion.

In view of these results, it is not difficult to map only the lower branch of
the squashed wormhole into the interior of the hypersphere while leaving the
upper branch unchanged, thereby obtaining a single copy of $\mathbb{E}_{N}$
that includes both the exterior and the interior of the hypersphere. \ In the
course of this inversion, the image charge is moved to its more conventional
position within the hypersphere. \ We leave the details as an exercise for the reader.

\section{On the grounded conducting disk in 3D}

\href{https://en.wikipedia.org/wiki/E._W._Hobson}{Hobson} used Sommerfeld's
method and a clever coordinate choice to find the Green function for an
equipotential circular disk in three Euclidean dimensions \cite{Hobson}. \ In
this approach, the disk serves as a doorway between two copies of
$\mathbb{E}_{3}$, with the unit source and field point located in one copy of
$\mathbb{E}_{3}$,\ and an equal strength, negative image of the source
obviously placed in the same position as the source except in the second copy
of $\mathbb{E}_{3}$. \ Nearly forty years later, Waldmann (a student of
Sommerfeld) offered another solution to this problem \cite{Waldmann} by
mapping the half-plane to a finite radius disk and then transforming
Sommerfeld's 1897 result for the grounded half-plane Green function. \ Another
thirty-four years after that, Davis and Reitz independently solved the same
problem, again using Sommerfeld's method but with an emphasis on the use of
complex analysis to construct directly the Green function on the two copies of
$\mathbb{E}_{3}$ \cite{DavisReitz}. \ In this regard, their approach is more
in line with Sommerfeld's original analysis, wherein complex variables also
play a central role.

Neither of these treatments invoke Riemannian geometry as we have done here
for hyperspheres. \ However it is possible in principle to consider the
conducting disk in $\mathbb{E}_{3}$ as a squashed oblate spheroid, and thereby
obtain the Green function for the disk by taking a limit of Green functions on
branched manifolds connected by spheroidal generalizations of the Ellis
wormhole, analogous to the $p$-norm wormholes used above. \ This more
geometrical treatment will be discussed elsewhere \cite{AlshalCurtright}.

\section{Conclusions}

We have obtained the Green function for grounded hyperspheres in $N$ spatial
dimensions by first constructing Green functions on Riemannian manifolds
(rather unfortunately, in our opinion, but commonly known as \textquotedblleft
wormholes\textquotedblright) and then by squashing these manifolds to produce
two copies of flat Euclidean space creased together along a hypersphere of
radius $R$. \ The distribution of source and image charges on the final
squashed manifold illustrates Sommerfeld's generalization of
\href{https://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin}{Thomson}
's method.

Sommerfeld knew that his generalized method could be used to solve a large
variety of problems \cite{Sommerfeld}, writing to
\href{https://en.wikipedia.org/wiki/Felix_Klein}{Klein} in the spring of 1897
(see \cite{Eckert} page 80):

\begin{quote}
\textquotedblleft The number of boundary value problems solvable by means of
my elaborated Thomson's method of images is very great.\textquotedblright
\end{quote}

\noindent But he does not seem to have pursued this during the next
half-century, perhaps because more interesting mathematics and physics
questions captured his attention.

In our opinion, the most prescient aspect of Sommerfeld's nineteenth century
work lies in its suggestion that physical problems in electromagnetic theory
may be simplified and perhaps more easily understood through the study of
Riemannian geometries, a view that developed much later in general relativity.
\ However, like \href{https://en.wikipedia.org/wiki/Bernhard_Riemann}{Riemann}
before him, in 1897 Sommerfeld had no reason to include time along with the
spatial dimensions of his envisioned manifolds, thus making his work
premature. \

Nevertheless, considering its application of Riemann's ideas from geometry and
complex analysis to higher dimensional branched manifolds, we believe
Sommerfeld's work should be recognized as a legitimate precursor to the
wormhole studies that appeared a few decades later
\cite{Flamm,EinsteinRosen,Ellis} and continue to the present day
\cite{MorrisThorne,JamesEtAl,Lobo}. \ We hope our paper encourages readers to
share this opinion.\bigskip

\noindent\textbf{Acknowledgements} \ It has been our pleasure to reconsider
this elementary subject during the year of the Feynman Centennial and the
Sommerfeld Sesquicentennial. \ This work was supported in part by a University
of Miami Cooper Fellowship, and by a Clark Way Harrison Visiting Professorship
at Washington University in Saint Louis.

\bigskip

config: arxiv


TITLE: Confirming an $LDL^t$ factorization
QUESTION [0 upvotes]: Hoping someone can confirm my work here. I'm trying to find the $LDL^t$ factorization (http://www.mathworks.com/help/dsp/ref/ldlfactorization.html) of the matrix
$$
A = \left(\begin{array}{ccc} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{array}\right).
$$
I'm omitting my work because it's really long, but I got
$$
A = \overbrace{\left(\begin{array}{ccc} 1 & 0 & 0 \\ -1/2 & 1 & 0 \\ 0 & -2/3 & 1 \end{array}\right)}^{L} \overbrace{\left(\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 3/2 & 0 \\ 0 & 0 & 4/3 \end{array}\right)}^{D} \overbrace{\left(\begin{array}{ccc} 1 & -1/2 & 0 \\ 0 & 1 & -2/3 \\ 0 & 0 & 1 \end{array}\right)}^{L^t}.
$$
Any chance could confirm my work? Thanks!

REPLY [1 votes]: The goal is to factorize:
$$A = LDL^T$$
We want:
$$\begin{bmatrix}2 & -1 & 0\\-1 & 2 & -1 \\ 0 & -1 & 2\end{bmatrix} = \begin{bmatrix}1 & 0 & 0\\l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1\end{bmatrix} \begin{bmatrix}d_{11} & 0 & 0\\ 0 & d_{22} & 0 \\ 0 & 0 & d_{33}\end{bmatrix} \begin{bmatrix}1 & l_{21} & l_{31}\\0 & 1 & l_{32} \\ 0 & 0 & 1\end{bmatrix}$$
The basic iteration (you can find in numerical analysis texts or "Matrix Computations" by Golub and Van Loan) we want for an $LDL^T$ factorization is given by:
$$\displaystyle d_{jj} = a_{jj} - \sum_{k=1}^{j-1} l^2_{jk}d_{kk} \\ \displaystyle l_{ij} = \left(a_{ij} - \sum_{k=1}^{j-1} l_{ik}d_{kk}l_{jk}\right) \times \dfrac{1}{d_{jj}}$$
We have:

$d_{11} = a_{11}  = 2$
$l_{11} = 1$
$l_{21} = \dfrac{a_{21}}{d_{11}} = -\dfrac{1}{2}$
$l_{31} = \dfrac{a_{31}}{d_{11}} = \dfrac{0}{2} = 0$
$\displaystyle d_{22} = a_{22} - \sum_{k=1}^{1} l_{jk}^2d_{kk} = 2 - l^2_{21}d_{11} = 2 - \left(-\dfrac{1}{2}\right)^2(2) = \dfrac{3}{2}$
$l_{22} = 1$
$\displaystyle l_{32} = \left(a_{32} - \sum_{k=1}^{1}l_{3k}^2d_{kk}l_{2k}  \right ) \times \dfrac{1}{d_{22}} = (-1 - 0) \times \dfrac{2}{3} = -\dfrac{2}{3}$
$\displaystyle d_{33} = a_{33} - \sum_{k=1}^{2} l_{jk}^2d_{kk} = a_{33} - (l_{31}^2d_{11} + l_{32}^2d_{22}) = 2-\left(0 + \left(-\dfrac{2}{3}\right)^2\left(\dfrac{3}{2}\right)\right) = \dfrac{4}{3}$

This gives us:
$$\begin{bmatrix}2 & -1 & 0\\-1 & 2 & -1 \\ 0 & -1 & 2\end{bmatrix} = \begin{bmatrix}1 & 0 & 0\\-\dfrac{1}{2} & 1 & 0 \\ 0 & -\dfrac{2}{3} & 1\end{bmatrix} \begin{bmatrix} 2 & 0 & 0\\ 0 & \dfrac{3}{2} & 0 \\ 0 & 0 & \dfrac{4}{3}\end{bmatrix} \begin{bmatrix}1 & -\dfrac{1}{2} & 0\\0 & 1 & -\dfrac{2}{3} \\ 0 & 0 & 1\end{bmatrix}$$
This agrees with your result and also was verified to produce your original matrix when multiplied out.


TITLE: High dimensional generalized Poincare hypothesis without the h-cobordism theorem?
QUESTION [7 upvotes]: The generalized PL Poincare hypothesis states that in dimension $n$ there is a unique PL manifold that has the homotopy-type of $S^n$.   It's known to be true in all dimensions except perhaps $n=4$.
For $n \geq 5$ the standard argument uses rather explicit handle manipulations to come up with the proof (proving the h-cobordism theorem in the process), which is somewhat spiritually-similar to (although more complicated than) the proofs in dimension $n=2$.
The proof in dimension $n=3$ is very different than the above two cases.
But I wonder, perhaps there is a proof that avoids the h-cobordism theorem, perhaps there is a more direct proof?   Has there been much discussion of this in the literature?
One thought would be to find an appropriately simplified proof of the Farrell fibering theorem (when a manifold fibers over $S^1$), one that perhaps allows you to reduce $S^n$ recognition into a homotopy-unknot recognition problem.
I imagine back in the 60's and 70's there was some discussion of these topics but I wouldn't know where to look.

REPLY [2 votes]: The Poincare conjecture predates the h-cobordism theorem, and the original proof of it in Smale's paper does not prove the h-cobordism theorem (though the handle manipulations in it are what inspired the h-cobordism theorem, so this is maybe not a good answer).
A very different way of proving the Poincare conjecture is Stallings's proof using "engulfing".  A nice modern source for this is Chapter 9 of Ferry's notes here.


TITLE: Find a rank 1 matrix that better approximates a given matrix
QUESTION [0 upvotes]: I have been working on a problem that asks the following: given a matrix
$$\begin{pmatrix}
0 & 1 & 3\\
3 & 1 & 0
\end{pmatrix},$$
I need to find a matrix $M$ such that $rank(M) = 1$ that better approximates A. That is, I have to find a matrix of type
$$M = \begin{pmatrix}
a & b & c\\
ra & rb & rc
\end{pmatrix},$$
such that the (real) values $a, b, c, r$ are such that it minimizes the following expression:
$(a^2) + (1-b)^2 + (3-c)^2 + (3-ra)^2 + (1-rb)^2 + (rc)^2$ (that is, the Frobenius norm of the difference $A-M$).
I know that I can do it by finding the minimmum of a function on 4 variables. But I was trying to find a better and clever way to do this, but I wasn't able to do so.
I would appreciate if you could come up with any ideas that could make this problem more easy to solve.
(Edit 1) Maybe an idea could arise out of a decomposition of this matrix into a product (for instance, LU decomposition or something like this).

REPLY [2 votes]: The EYM theorem states that this approximation may be obtained using the singular value decomposition. In particular, we proceed as follows.
Begin by computing the singular value decomposition $A = U \Sigma V^T$, where
$$
A = \pmatrix{0 & 1 & 3\\
3 & 1 & 0}, \quad U = \frac 1{\sqrt{2}}\pmatrix{1&-1\\1&1}, \quad \Sigma = \pmatrix{\sqrt{11}\\ & 3}, \quad
\\ V = \pmatrix{
3/\sqrt{22} & 1/\sqrt{2} & 1/\sqrt{11}\\
\sqrt{2/11} & 0 & -3/\sqrt{11}\\
3/\sqrt{22} & -1/\sqrt{2} & 1/\sqrt{11}}.
$$
Then, compute the rank-1 "truncated SVD" $M = U \hat \Sigma V$, where
$$
\hat \Sigma = \pmatrix{\sqrt{11}&0\\0&0}.
$$
Equivalently, we find
$$
M = \sqrt{11} \cdot \frac 1{\sqrt{2}}\pmatrix{1\\1}\pmatrix{3/\sqrt{22} & \sqrt{2/11} & 3/\sqrt{22}}
\\ = \left(\begin{matrix}1.5 & 1.0 & 1.5\\1.5 & 1.0 & 1.5\end{matrix}\right).
$$


TITLE: Condition for the unseparability of Banach Spaces
QUESTION [0 upvotes]: A basic question but I can't quite resolve it:
Why is the following equivalent to unseparability of a Banach space X:
For some uncountable set S $\subseteq$ X, there exists $\delta$ > 0 such that for any $x_1$, $x_2$ $\in$ S, norm of $x_1$ - $x_2$ $\ge$ $\delta$
I realise this means there any no cauchy sequences in S, hence no convergent sequences other than those which are eventually just constant.

REPLY [0 votes]: Hint: Consider family of open balls $$\left\{\mathrm{Ball}\left(x,\frac{\delta}{2}\right):x\in S\right\}$$


TITLE: Identifying error in proof that $\mathbb{Q}_{\mathbb{Z}}[x]$ is a PID but not a UFD
QUESTION [2 upvotes]: In the book Rings, Fields and Groups by R B J T Allenby I stumbled upon a "proof" that a counterexample exists to a previously proven theorem in the following exercise:
Let $\mathbb{Q}_{\mathbb{Z}}[x]=\{f:f\in \mathbb{Q}[x]\ \text{and f has integer constant term}\}$ . Given $f,g\in \mathbb{Q}_{\mathbb{Z}}[x]$ let $d$ be a common divisor of maximum possible degree in $\mathbb{Q}_{\mathbb{Z}}[x]$. Show that $f_0=f/d$ and $g_0=g/d$ (both in $\mathbb{Q}_{\mathbb{Z}}[x]$) have gcd 1 in $\mathbb{Q}[x]$. Deduce that $z=mf_0+ng_0$ for suitable $z\in \mathbb{Z}$ and $m$ and $n$ in $\mathbb{Q}_{\mathbb{Z}}[x]$. Write $f_0=f_1+\alpha$, $g_0=g_1+\beta$ where $\alpha , \beta$ are the constant terms in $f̣_0$ and $g_0$. Show that $\alpha=f_0 - (\frac{f̣_1}{z})z$ lies in the ideal $[f_0,g_0]$ in $\mathbb{Q}_{\mathbb{Z}}[x]$ and similarly for $\beta$. Set $c=(\alpha,\beta)$ in $\mathbb{Z}$. Deduce $[f_0,g_0]=[c]$ and hence that $[f,g]=[cd]$, a principal ideal in $\mathbb{Q}_{\mathbb{Z}}[x]$. Deduce from $x=\frac{1}{2}x\cdot 2=\frac{1}{4}x\cdot 2^2=...$ that $x$ cannot be expressed as a product of irreducibles in $\frac{1}{2}x*2$ so that $\frac{1}{2}x*2$ is not a UFD. This surely contradicts 3.7.14 (every PID is a UFD)? [This example is due to P M Cohn]
I'm assuming that the error lies in the part $x=\frac{1}{2}x\cdot 2=\frac{1}{4}x\cdot 2^2=...$ But I'm having a hard time actually figuring out what it is, since to me neither $2^n$ nor $\frac{1}{2^n}x$ is a unit in $\mathbb{Q}_{\mathbb{Z}}[x]$. The book did not give a solution to this excercise and instead referred to the book Algebra 1:st Edition Vol 1, 1974 by P M Cohn, a book which I do not own.
Any help on what might be the error would be very useful.
(less important)
I also never managed to prove that $f_0,g_0$ have gcd 1 in $\mathbb{Q}[x]$, only that its degree is 0. So any help with that would also be great. Thanks

REPLY [1 votes]: $\DeclareMathOperator{\QZ}{\mathbb{Q}_\mathbb{Z}}
\DeclareMathOperator{\Q}{\mathbb{Q}}
\DeclareMathOperator{\Z}{\mathbb{Z}}$
First, I'll show that if $d$ is a common divisor of greatest degree for $f, g\in \QZ[x]$, then $d$ is a greatest common divisor of $f$ and $g$ in $\Q[x]$. Such an element $d$ divides $f$ and $g$ in $\Q[x]$, so it divides their GCD. Further, if $d'$ is a greatest common divisor of $f$ and $g$ in $\Q[x]$, then $nd' \in \QZ[x]$ for some $n \in \Z$, so the degree of $d$ is at least the degree of $d'$.

The given proof shows by induction that any finitely generated ideal of $\QZ[x]$ is principal. However, the ring $\QZ[x]$ is not Noetherian. Consider the ascending chain of ideals
$$ (x) \subseteq \left(\frac{1}{2}x\right) \subseteq \left( \frac{1}{2^2}x\right) \subseteq \cdots$$
Each inclusion is proper, as can be seen by taking the degree 1 part of each ideal. Hence, the union of these ideals is an ideal of $\QZ[x]$ which is not principal.


text:
\section{Discrete Space satisfies all Separation Properties}
Tags: Discrete Topology, Separation Axioms

\begin{theorem}
Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ fulfils all [[Definition:Separation Axioms|separation axioms]]:
:$T$ is a [[Definition:Kolmogorov Space|$T_0$ (Kolmogorov) space]]
:$T$ is a [[Definition:Fréchet Space (Topology)|$T_1$ (Fréchet) space]]
:$T$ is a [[Definition:Hausdorff Space|$T_2$ (Hausdorff) space]]
:$T$ is a [[Definition:Semiregular Space|semiregular space]]
:$T$ is a [[Definition:Completely Hausdorff Space|$T_{2 \frac 1 2}$ (completely Hausdorff) space]]
:$T$ is a [[Definition:T3 Space|$T_3$ space]]
:$T$ is a [[Definition:Regular Space|regular space]]
:$T$ is an [[Definition:Urysohn Space|Urysohn space]]
:$T$ is a [[Definition:T3 1/2 Space|$T_{3 \frac 1 2}$ space]]
:$T$ is a [[Definition:Tychonoff Space|Tychonoff (completely regular) space]]
:$T$ is a [[Definition:T4 Space|$T_4$ space]]
:$T$ is a [[Definition:Normal Space|normal space]]
:$T$ is a [[Definition:T5 Space|$T_5$ space]]
:$T$ is a [[Definition:Completely Normal Space|completely normal space]]
:$T$ is a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]]
:$T$ is a [[Definition:Perfectly Normal Space|perfectly normal space]]
\end{theorem}

\begin{proof}
We have that a [[Standard Discrete Metric induces Discrete Topology]].
Then we use [[Metric Space fulfils all Separation Axioms]].
{{qed}}
\end{proof}

config: wiki


TITLE: Calculate the area inside the loop
QUESTION [1 upvotes]: The question is - how to calculate the area inside the loop $(x+y)^3=axy$, using double integral?
In this case parameter $a$ is positive. It's easy to imagine how it looks like, depending on $a$ value. For example $a=1$:

or $a=100$

No idea how to do that.

REPLY [2 votes]: HINT:
In polar coordinates, the loop is described by
$$r(\theta)=\frac{a\sin(\theta)\cos(\theta)}{(\cos(\theta)+\sin(\theta))^3}$$
$\theta \in [0,\pi/2]$.  The area, $A$, is then given by
$$A=\int_0^{\pi/2}\int_0^{r(\theta)}r\,dr\,d\theta \tag 1$$

SPOILER ALERT:  Scroll over the highlighted area to reveal the solution

From $(1)$, we have $$\begin{align}A&=\frac{a^2}2\int_0^{\pi/2} \frac{\sin^2(\theta)\cos^2(\theta)}{(\cos(\theta)+\sin(\theta))^6}\,d\theta\\\\&=\frac{a^2}2 \int_0^{\pi/2}\frac{\tan^2(\theta)\sec^2(\theta)}{(1+\tan(\theta))^6}\,d\theta\\\\&=\frac{a^2}2\int_0^\infty \frac{x^2}{(1+x)^6}\,dx\\\\&=\frac{a^2}{60}\end{align}$$