TITLE: Witten Index and letter partition function
QUESTION [7 upvotes]: I haven't seen any reference which explains these things and I am not sure of all the steps of the argument or the equations. I am trying to reproduce here a sequence of arguments that I have mostly picked up from discussions and I would like to know of references for the background details and explanations.

It seems that in general the Witten Index (${\cal I}$) can be written as $ Tr (-1)^F \prod_i x_{i}^{C_i}$ where $x_is$ are like the fugacities of the conserved quantity $C_is$ and $C_is$ form a complete set of simultaneously measurable operators.
I would like to know the motivation for the above and especially about calling the above an ``index" even if it depends on the fugacities which doesn't seem to be specified by the Lagrangian of the theory or by the underlying manifold? Even with this dependence does it reproduce some intrinsic property of either the underlying manifold or the theory?

Say $\phi$ and $\psi$ are the bosonic and fermionic component fields of a superfield and ${\cal D}$ be the superderivative. Now when the above tracing is done over operators of the kind $Tr({\cal D}^n\phi)$ and $Tr({\cal D}^n\psi)$ it is probably called a "single trace letter partition function (STLP)" and when it also includes things like products of the above kind of stuff it is called a ``multi-trace letter partition function (MTLP)"

I would like to know about the exact/general definitions/references for the above terminologies and  the motivations behind them.

Probably for adjoint fields, if $f(x)$ is a STLP then apparently in some limit ("large N"?) one has, MTLP = $\prod _{n=1} ^{\infty} \frac{1}{1-f(x^n)}$

One also defines something called the ``full STLP (FSTLP)" where one probably includes terms also of the kind $Tr({\cal D}^n \phi ^m {\cal D}^p \psi ^q)$ and then the MTLP can be gotten from that as MTLP = $exp [ \sum _{m=1} ^ {\infty} \frac{1}{m} FSTLP]$

The upshot all this is probably to show that MTLP is the same as Witten Index (in some limit?).

I don't understand most of the above argument and I would be happy to know of explanations and expository references which explain the above concepts (hopefully beginner friendly!).
(I see these most often come up in the context of superconformal theories and hence references along that might be helpful especially about the representations of the superconformal group.)

REPLY [3 votes]: This appears to be related to twisted or broken supersymmetry.  For unbroken supersymmetry the total Hamiltonian ${\cal H}~=~N{\cal H}_b~+~N{\cal H}_f$ for $N$ strings with the bosonic operators $a,~a^\dagger$ and the fermionic operators $b,~b^\dagger$ for N strings is:
$$
{\cal H}~=~{N\over 2}\sum_k (\omega(k)a_k^\dagger a_k~+~{N\over 2}\sum_k \omega(k)b_k^\dagger b_k.
$$
There exists an operator $\lambda$ so that $\omega(k)/2~=~\omega log(\lambda_k)$.  Thus by supersymmetry we may write the Hamiltonian as
$$
{\cal H}~=~N\sum_k (log(\lambda_k)a_k^\dagger a_k~+~N\sum_k log(\lambda_k)b_k^\dagger b_k.
$$
Now Wick rotate the integrand in the path integral $e^{-it H_b}$ under the substitution $it~\rightarrow~\beta$ and find that
$$
Tre^{-NH_b\beta}~=~\sum_{i=1}^\infty\lambda_i^{-N\omega\beta}~=~\zeta(N\omega\beta).
$$
This is a generalized zeta function, and for both bosons and fermions,
$$
Tr\big((-1)^{N_b}e^{-N{\cal H}_b\beta}e^{-N{\cal H}_f\beta}\big)~=~e^{-N({N}_b~-~{N}_f)\omega\beta}~=~1
$$
since ${N}_b~=~{N}_f$ in unbroken supersymmetry.  As such we then have that
$$
Tr\big((-1)^{N_F}e^{-N{\cal H}_f\beta}\big)~=~{1\over{\zeta(N\omega\beta)}}.
$$
and the fermion partition function may be written with the Mobius function $\mu$ as
$$
Tr\big((-1)^{N_f}e^{-NH_f\beta}\big)~=~\sum_i{{\mu(\lambda_i)}\over{\lambda_i}^{N\omega\beta}}.
$$
The supersymmetry is partially broken, only half the stringy bosons have superpairs then the bosonic Hamiltonian takes the form
$$
{\cal H}_B~=~2N\sum_klog(\lambda_k)c^\dagger_k c_k~+~N\sum_k log(\lambda_k)f_k^\dagger f_k.
$$
where $c_k~=~a_ka_k$ and $c_k^\dagger~=~a_k^\dagger a_k^\dagger$.  The operators $f_k$ and $f^\dagger_k$ are fermionic operators \lq\lq recovered\rq\rq$~$from the bosonic Hamiltonian.  Now if we compute $Z_F(\beta)$ we find that
$$
Z_F(\beta)~=~ Tr\big((-1)^{N_F}e^{-N\sum_k log(\lambda_k)f^\dagger_k f_k\beta} \big)\times 1
$$
$$
~=~ Tr\big((-1)^{N_F}e^{-N\sum_k log(\lambda_k)f^\dagger_k f_k\beta} e^{-2N\sum_k log(\lambda_k)c^\dagger_k c_k\beta}e^{-2N\sum_k log(\lambda_k)b^\dagger_k b_k\beta}\big)
$$
$$
~=~Tr\big((-1)^{N_f}e^{-\big(2N{\cal H}_F~+~N{\cal H}_B\big)\beta}\big).
$$
where the unity involves the old fermion and boson operators for unbroken supersymmetry.  Now $Z_F(\beta)$ is
$$
Z_F(\beta)~=~ \sum_{n=1}^\infty{{\mu(n)}\over{n^{2N\omega\beta}}}~=~ {{\zeta(N\omega\beta)}\over{\zeta(2N\omega\beta)}}.
$$
Here the $c,~c^\dagger$ boson fields have superpartners and the $f,~f^\dagger$ fields do not.


TITLE: Find the coefficient of $x^{21}$ in $(1+x^{1} + x^{2} + x^{3} + x^{4} + x^{5} + x^{6})^6$
QUESTION [0 upvotes]: Find the coefficient of $x^{21}$ in  $(1+x^{1} + x^{2} + x^{3} + x^{4} + x^{5} + x^{6})^6$
Any direction or hint would be appreciated

REPLY [4 votes]: Notice that it is
$$\left(\frac{x^7-1}{x-1}\right)^6$$
Now, using binomial theorem:
$$(x+y)^{n}=\sum_{k=0}^n\binom{n}{k}x^{n-k}y^k$$
you get two polynomials, divide them until you reach $x^{21}$, you'll find that the solution is $7872$.


TITLE: Suppose H is a subring of R such that for every $r∈H$ we have $r^2=e$. Prove that $H$ is commutative.
QUESTION [2 upvotes]: So I know that since H is a subring of $\mathbb{R}$ that H is then by definition:
1) closed under addition
2) closed under multiplication
3) have the same multiplicative identity as $\mathbb{R}$
Am I missing anything? I think I need to only reference fact 2, which states that H is closed under multiplication for this proof.
I tried looking up on the internet how to show that a subring is commutative and did not find anything. Is the answer as simple as saying that since $\mathbb{R}$ is commutative then the subring H, is itself by definition commutative?
Or do I go about it by showing that lets say for example:
Let $r_1, r_2 \in H$ then $r_1 \times r_2 = e = r_1 \times r_2$ which is then commutative?

REPLY [2 votes]: I assume you are using $e$ to denote the multiplicative identity of the ring $R$. I will follow custom and write $1$ rather than $e$. If $H$ is a subring of a ring $R$, then the additive identity $0$ is a member of $H$. As $0^2 = 0$, if $0^2 = 1$ we have $0 = 1$ which implies $x = 1x = 0x = 0$ for every $x \in R$. Thus your hypothesis implies that $R$ (and $H$) is the trivial ring with just one element $0 = 1$. The trivial ring is commutative.


TITLE: comparing two concave functions with different inequalities
QUESTION [0 upvotes]: I am deriving some bounds involving two functions, which are $f(x)=\log(x)$ and $f(x)=-x\times\log(x)$. I found in several books that these two functions are concave. I have draw them down using python to quickly check. For that reason, by applying Jensen's inequality the next inequality should hold:
$f(x)+f(y)\leq f(x+y)$
which holds in case of $f(x)=\log(x)$ but not for $f(x)=-x\times\log(x)$. What am I missing?
Thank you.

REPLY [0 votes]: Here is a possible way I think, comment if anything wrong.
Note that, for any $x>1$ we have $(x+y)^{x+y}=(x+y)^x(x+y)^y>x^xy^y$
So, $$(x+y)^{-(x+y)}<x^{-x}y^{-y}$$ $$\implies -(x+y)\log(x+y)<-x\log x-y\log y$$
$$\implies f(x+y)<f(x)+f(y)$$


TITLE: Divergent Series
QUESTION [9 upvotes]: Thinking about divergent series and ways of "summing" them, they seem to fall into two categories (roughly):

Series like $\sum_{k=1}^\infty \frac{1}{k}$, which defy all kinds of regularization or summing methods.
Series which can be summed, in one way or another, like $\sum_{k=1}^\infty k^s$, which for at least $s \geq 1$ can be seen as $\zeta(-s)$.

My question is:

Can the groundfields $\mathbb{R}$ or $\mathbb{C}$ be extented in some ways (like the Hyperreals $\mathbb{R}^*$, or the surreals $\mathbb{S}_\mathbb{R}$ or the surcomplex numbers $\mathbb{S}_\mathbb{C}$), such that you can sum all the divergent series (at least the second type from above) in a more rigorous fashion than the bag of tricks that are usually used to explain these results.
This sounds like a "sort of" completion of the reals with all the summable but divergent series in it. I would like to know if such a beast exist, what its properties are and if the $\sum_{k=1}^\infty k^s=\zeta(-s)$ results can be explained from this.

REPLY [2 votes]: The answer to your question depends on what properties you want these sums to have, and what properties you want the extended field to have.
If you want to extend the reals in such a way that they satisfy the elementary axioms of algebra (commutativity of addition, etc.), but you want the extended system to include infinite quantities, then you are going to come out with exactly the hyperreal number system that you've already mentioned. (See [*] below for a more formal statement of this.)
But the hyperreals are not necessarily going to have the other properties that you might like. In particular, we would like to be able to say whether or not two divergent series have the same sum. But the standard constructive approach to the hyperreals involves taking equivalence classes of sequences, with the equivalence relation defined by an ultrafilter. Since ultrafilters cannot be explicitly constructed, you can't say in general whether divergent sequences are "approaching" the same hyperreal number. What you can say is whether two convergent sequences are approaching the same real number.
This is a lot of technical stuff about the hyperreals, but it arises from more general considerations about how field axioms work. Basically the thing that stops us from doing what you'd like to do in a completely satisfactory way is that the field axioms only deal with finitely many operations. For example, if associativity of addition applied to infinite sums, then we could rewrite the divergent sum $1-1+1-1+\ldots$ as $1+(-1+1)+(-1+1)+\ldots=1$. Because the field axioms lack this kind of power over infinite processes, you can't really make infinite sums behave like finite ones. The ambiguity in the associativity of the sum $1-1+1+\ldots$ is very much like the ambiguity involved in specifying an ultrafilter.
If you use a system like the surreals, then you have to give up the ability to talk about transcendental functions and do analysis.
[*] The hyperreals are the extension of the reals that you get when you demand the transfer principle for all statements in first-order logic but take the converse of the Archimedean property as an axiom. You also get stuff like transcendental functions, which it's not immediately obvious that you get for free from the transfer principle. The hyperreals are not unique in ZFC, but the different models don't differ in ways that make a difference in this discussion. The hyperreals lack the completeness property of the reals: http://en.wikipedia.org/wiki/Completeness_of_the_real_numbers


TITLE: Probability of at least one 2-hop neighbor
QUESTION [2 upvotes]: Consider a bipartition $\{A_1, A_2, ..., A_n\}$ and $\{B_1, B_2, ..., B_m\}$. No edges exists between $(A_i, A_j)$ or $(B_i, B_j)$, for any $i$ and $j$.
The probability that the edge $(A_i, B_j)$ exists is $p$. Node $A_i$ is said to be a 2-hop neighbor of $A_1$ if there exists a connected path $A_1 \rightarrow B_j \rightarrow A_i$. (Also, we don't consider $A_1$ as a 2-hop neighbor of itself.)
What is the probability that there exists at least one 2-hop neighbor of $A_1$, and that at the same time, for every 2-hop neighbor of $A_1$, the 2-hop path is unique?

REPLY [0 votes]: Let $X_1,X_2,\dots X_n$ be the sets of  neighbours of $A_1,A_2,\dots,A_n$.
We need to calculate the probability each of these events happens: $|X_1\cap X_j|=1$.
If we condition this to the size of $X_1$ being equal to $k$ then the events are independent.
The probability that $|X_1\cap X_2|=1$ is clearly $kp(1-p)^{k-1}$.
The probability that $X_1$ has size $k$ is $\binom{m}{k}p^k(1-p)^{m-k}$
So the answer is $p\sum\limits_{k=0}^m k\binom{m}{k}p^k(1-p)^{m-k}(1-p)^{k-1}=p(1-p)^{m-1}\sum\limits_{k=0}^mk\binom{m}{k}p^k$.
Notice that $f(x)=(x+1)^m=\sum\limits_{k=0}^m\binom{m}{k}x^k$ so $pf'(p)=\sum\limits_{k=0}^m k\binom{m}{k}p^k$.
Hence what we want is $p^2(1-p)^{m-1}f'(p)$


text:
\section{Analysis -- Ideal Case}
\label{sec:analysis}
In this section, we follow \cite{schiessl2018misoglobecom} and outline the analytical approach to determine $\pvk(w)$. Specifically, in Sec.~\ref{ssec:queueing}, we present a summary of the delay analysis through stochastic network calculus in a transform domain \cite{alzubaidy2016ton}.
In Sec.~\ref{ssec:queueing_rr} we show how these results can be used when the users are only scheduled once per superframe. In Sec.~\ref{ssec:delay_analysis}, we analytically obtain the stochastic network calculus bounds for the ideal case.
We assume that all users are subject to the same channel characteristics and delay requirements, and we drop the subscript $k$ to shorten the notation.

\subsection{Stochastic Network Calculus (SNC)}
\label{ssec:queueing}
This section closely follows our previous works \cite{schiessl2016imperfectcsi,schiessl2018misoglobecom} and provides a summary of stochastic network calculus \cite{fidler2006network,alzubaidy2016ton}.

The delay $\Delay(t)$ in (\ref{eq:def_delay}) is defined in terms of the arrival and departure processes.
However, the distribution of the delay can be found directly from the statistics of the arrival and service processes.
We follow \cite{alzubaidy2016ton} and describe these processes in the exponential domain, also referred to as \emph{SNR domain}. The arrival and service processes in the bit domain, $\Abit(\ts)$ and $\Sbit(\ts)$, are converted to the SNR domain (denoted by calligraphic letters) as
\begin{equation}
\Asnr(\ts) \defined e^{\Abit(\ts)}\,,\quad\Ssnr(\ts) \defined e^{\Sbit(\ts)}
\,.
\end{equation}
We assume constant arrivals with $\Abit(\ts)=\alpha$.
Consider for now a service process $\Sbit(\ts)$ that is independent and identically distributed (i.i.d.) between time slots.
Then, an upper bound on the delay violation probability $\pv(w)$ can be obtained in terms of the Mellin transforms of $\Asnr$ and $\Ssnr$.
The Mellin transform $\Mellin_\Xsnr(\s)$ of a nonnegative random variable $\Xsnr$ is defined as \cite{alzubaidy2016ton}
\begin{equation}
\Mellin_\Xsnr(\s)\defined\mathbb{E}\left[\Xsnr^{\s-1}\right]
\end{equation}
for a parameter $\s \in \mathbb{R}$.
For the analysis, we choose $\s>0$ and define the kernel \cite{alzubaidy2016ton,schiessl2015delay}
\begin{align}
\Mfun{\s,\wnormal} &\defined \lim\limits_{t\to\infty} \sum_{u=0}^{t} \Mellin_{\Asnr}(1+\s)^{t-u} \cdot \Mellin_{\Ssnr}(1-\s)^{t+\wnormal-u}
\,.
\end{align}
Under the condition $\Mellin_{\Asnr}(1+\s)\Mellin_{\Ssnr}(1-\s) < 1$, the kernel converges:
\begin{align}
\Mfun{\s,\wnormal}&= \frac{\Mellin_{\Ssnr}(1-\s)^{\wnormal}}{1-\Mellin_{\Asnr}(1+\s)\Mellin_{\Ssnr}(1-\s)}
\;.
\label{eq:snc_kernel}
\end{align}
For any parameter $\s>0$, the kernel $\Mfun{\s,\wnormal}$ provides an upper bound on the delay violation probability $\pv(w)$ \cite{alzubaidy2016ton, schiessl2015delay}. This holds also in steady-state, i.e., in the limit $t\to\infty$.
The tightest upper bound can be found by iterating over the parameter $\s>0$:
\begin{equation}
\pv(\wnormal) \leq \inf_{\s>0}\left\{ \Mfun{\s,\wnormal} \right\}
\;.
\label{eq:pdelay_bound}
\end{equation}

\subsection{SNC and Scheduling}
\label{ssec:queueing_rr}
The delay analysis through stochastic network calculus as shown in Sec.~\ref{ssec:queueing} cannot be applied directly because $\Sbit(\ts)$ is zero in the time slots where the user is not scheduled, i.e., $\Sbit(\ts)$ is not i.i.d. between time slots. However, stochastic network calculus can be applied on the superframe level. The service that a user receives in superframe $i$ is denoted as $\Sbit^{(\suf)}(i)$, and is i.i.d. between superframes, because each user is scheduled exactly once per superframe of length $\suf$. The arrival process on the superframe level is given as $\Abit^{(\suf)}(i)= \alpha T$ bits, and the Mellin transform of the process $\Asnr$ in the SNR domain is $\Mellin_{\Asnr^{(T)}}(\s) = e^{\alpha T(\s-1)}$.

In case $\wtotal/\suf$ is an integer, it follows directly from \eqref{eq:pdelay_bound} that:
\begin{align}
\pv(\wtotal) & \le \inf_{\s>0}\left\{  \Mfunsuper{\s,\frac{\wtotal}{\suf}} \right\}
\label{eq:snc_pv_bound_singlegroup}
\,.
\end{align}
When the condition $\Mellin_{\Asnr^{(\suf)}}(1+\s)\Mellin_{\Ssnr^{(\suf)}}(1-\s) < 1$ holds, the kernel $\Mfunsuper{\cdot}$ converges to
\begin{align}
\Mfunsuper{\s,\frac{\wtotal}{\suf}}&= \frac{\Mellin_{\Ssnr^{(\suf)}}(1-\s)^{\frac{\wtotal}{\suf}}}{1-\Mellin_{\Asnr^{(\suf)}}(1+\s)\Mellin_{\Ssnr^{(\suf)}}(1-\s)}
\,.
\label{eq:snc_kernel2}
\end{align}
The bound depends on the Mellin transform of the service $\Ssnr^{(\suf)}$ per superframe in the SNR-domain, which is connected to the bit-domain service process as $\Ssnr^{(\suf)}=e^{\Sbit^{(\suf)}}$.
In the bit-domain, each user experiences a service of $\Sbit^{(\suf)}=\nd R Z$ bits per superframe, where $Z=0$ when a transmission error occurs and $Z=1$ otherwise. Therefore:
\begin{align}
\Mellin_{\Ssnr^{(\suf)}}(\s)=\expected{\left(e^{\nd \rate Z}\right)^{\s-1}}
\,.
\end{align}

In case $\wtotal/\suf$ is not an integer, some users (denoted as group 1) will be served $\lceil \wtotal/\suf\rceil$ times before the deadline, while others (group 2) will only be served $\lfloor\wtotal/\suf\rfloor$ times. For the sake of fairness, we assume that the users are assigned randomly to the slots. Then, the probability of being in the second group is $p_2=\frac{\mod(\wtotal,\suf)}{\suf}$, and $p_1 = 1-p_2$. Thus, the overall bound on the delay violation probability is given by \cite{schiessl2018misoglobecom}
\begin{align}
\pv(\wtotal) \le p_1 \Mfunsuper{\s,\left\lceil\frac{\wtotal}{\suf}\right\rceil} + p_2 \Mfunsuper{\s,\left\lfloor \frac{\wtotal}{\suf}\right\rfloor}
\;.
\label{eq:snc_pv_bound_multigroups}
\end{align}

\subsection{Delay Analysis -- Ideal Case}
\label{ssec:delay_analysis}

In the ideal case with perfect CSI and long codewords, the rate is given as $R=\log_2(1+\power\xi)$, and no errors occur ($Z=1$ with prob. $1$). In this case, we can obtain the Mellin transform of $\Ssnr^{(\suf)}$ in closed form.
The variable $\xi$ is a central $\chi^2$ variable with $2m= 2(\Nt-K+1)$ degrees of freedom as outlined in Sec.~\ref{ssec:system_data_phy}.
Thus, we obtain the following result:
\begin{result}
Given the transmit power $\fixpower$ and the number of scheduled users $K$, the Mellin transform of the service process is given as:
\begin{align}
\Mellin_{\Ssnr\sufss}(1-\s) &=\sum\limits_{\LL=0}^{m-1}\frac{\binom{m-1}{\LL} (-1)^{\LL}}{\Gamma(m)\fixpower^{\LL+\salt}}  e^{\frac{1}{\fixpower}}
\Gamma\left(m-\mu-\salt,\frac{1}{\fixpower}\right)
\label{eq:mellin_service_constpower_series}
\,,
\end{align}
where $m=\Nt-K+1$, $\salt\defined\frac{\s \nd}{\ln 2}$, and $\Gamma(s,x)$ is the upper incomplete Gamma function
\begin{align}
\Gamma(s,x) = \int_{x}^{\infty}t^{s-1}e^{-t}dt
\,.
\label{eq:incomplete_gamma}
\end{align}
\end{result}
\begin{proof}
Outline: Start with
\begin{align}
\Mellin_{\Ssnr\sufss}(1-\s) &=\expected{\left(e^{\nd R}\right)^{-\s}} = \int\limits_0^\infty (1+\fixpower\xi)^{-\salt} f_m(\xi)d\xi
\,,
\end{align}
where $f_m(\xi)$ in \eqref{eq:pdf_central_chisquare_scaled} contains the term $\xi^{m-1}$, which is expanded using the binomial theorem:
\begin{align}
\xi^{m-1} &=
\frac{1}{\fixpower^{m-1}}\sum\limits_{\LL=0}^{m-1}\binom{m-1}{\LL}\left(1+\fixpower\xi\right)^{m-1-\LL}(-1)^{\LL}
\,.
\end{align}
The result is obtained from further algebraic derivations. The details can be found in \cite{schiessl2018misoglobecom}.
\end{proof}
Thus, given the arrival rate $\alpha$ in bits per time slot and a specific choice of superframe length $\suf$, the upper bound \eqref{eq:snc_pv_bound_multigroups} on $\pv(w)$ can be obtained analytically through \eqref{eq:mellin_service_constpower_series}.

config: arxiv


TITLE: Show that if a vector subspace of a Hilbert space is closed, then it is a Hilbert subspace.
QUESTION [1 upvotes]: Show that if a vector subspace of a Hilbert space is closed, then it is a Hilbert subspace.
Here is what I have so far. Any comments or hints are greatly appreciated.
Let $H$ be a Hilbert space and $S$ be the vector subspace of $H$. Since $H$ is closed that means $H$ contains all of its limit points. Let $(u_n)$ be a Cauchy sequence in $S$. Then $(u_n)\to U\in H$ (since $S$ is closed) where $U$ is a limit point of $S$. Thus $S$ is a Hilbert subspace.

REPLY [1 votes]: Remember, just because a sequence is Cauchy in a metric space, doesn't necessarily mean that it converges, so I don't think you want to rely on the fact that S contains it's limit points for the proof, rather you probably want to use the fact that H is a Hilbert space and thus complete, meaning all cauchy sequences in H end up converging in H. From this perspective the proof isn't too difficult. As H is a Hilbert space, it's a complete metric space, furthermore, as S is a closed subspace of a complete metric space, it must then also be complete in regards to it's metric (whether or not you've shown this in the past, I'm not sure, but the proof itself isn't too difficult), thus making it a Hilbert space.


TITLE: what are real and imaginary part of this expression
QUESTION [1 upvotes]: I have $M:=\sqrt{\frac{a\cdot(b+ic)}{de}}$ and all variables $a,b,c,d,e$ are real. Now I am looking for the real and imaginary part of this, but this square root makes it kind of hard.

REPLY [3 votes]: $$\sqrt{\frac{a(b+ic)}{de}}=\sqrt{\frac{a}{de}}\cdot\sqrt{b+ic}$$
Let $$\sqrt{b+ic}=x+iy$$
$$\implies b+ic=(x+iy)^2=x^2-y^2+2xyi$$
Equating the real & the imaginary parts, $b=x^2-y^2, c=2xy$
So, $b^2+c^2=(x^2-y^2)^2+(2xy)^2=(x^2+y^2)^2\implies x^2+y^2=\sqrt{b^2+c^2}$
We have $$x^2-y^2=b$$
$$\implies 2x^2=\sqrt{b^2+c^2}+b\implies x^2=\frac{\sqrt{b^2+c^2}+b}2$$
$$\implies x=\pm\frac{\sqrt{\sqrt{b^2+c^2}+b}}{\sqrt2}$$
and  $$\implies y^2=x^2-b=\frac{\sqrt{b^2+c^2}-b}2$$
$$\implies y=\pm\frac{\sqrt{\sqrt{b^2+c^2}-b}}{\sqrt2}$$
Now, the sign of $y=$ sign of $x\cdot$ sign of $c$


text:
\begin{document}

\begin{center}
{\LARGE Isochronous Spacetimes}

\bigskip

$^{\ast \heartsuit }$\textbf{Fabio Briscese}$^{1}$ and $^{+\lozenge }$
\textbf{Francesco Calogero}$^{2}\bigskip $

$^{\ast }$Istituto Nazionale di Alta Matematica Francesco Severi, Gruppo
Nazionale di Fisica Matematica, Citt\`{a} Universitaria, P.le A. Moro 5,
00185 Rome, Italy

$^{\heartsuit }$Dipartimento SBAI, Sezione di Matematica, University of Rome
\textquotedblleft La Sapienza", via Antonio Scarpa 16, 00161 Roma, Italy

$^{+}$Physics Department, University of Rome \textquotedblleft La Sapienza",
p. Aldo Moro, I-00185 ROMA, Italy

$^{\lozenge }$Istituto Nazionale di Fisica Nucleare, Sezione di Roma

$^{1}$fabio.briscese@sbai.uniroma1.it, briscese.phys@gmail.com

$^{2}$francesco.calogero@roma1.infn.it, francesco.calogero@uniroma1.it

\bigskip

\textit{Abstract}
\end{center}

The possibility has been recently demonstrated to manufacture
(nonrelativistic, Hamiltonian) many-body problems which feature an \textit{
isochronous} time evolution with an arbitrarily assigned period $T$ yet
mimic with good approximation, or even exactly, any given many-body problem
(within a quite large class, encompassing most of nonrelativistic physics)
over times $\tilde{T}$ which may also be arbitrarily large (but of course
such that $\tilde{T}<T$). In this paper we review and further explore the
possibility to extend this finding to a general relativity context, so that
it becomes relevant for cosmology.

\section{Introduction}

In this paper we revisit the findings reported in \cite{isochronous
cosmologies} and report some additional considerations relevant for a better
understanding of the validity---in the context of theoretical and
mathematical physics---of those results.

It has been recently shown \cite{CL2007,calogero} that---given a general
autonomous dynamical system $D$, other autonomous dynamical systems $\tilde{D
}$ can be manufactured, featuring two additional \textit{arbitrary} positive
parameters $T$ and $\tilde{T}$ with $T>\tilde{T}$ and possibly also two
additional dynamical variables---which are characterized by the following
two properties:

(i) For the same variables of the original dynamical system $D$ the new
dynamical system $\tilde{D}$ yields, over the time interval $\tilde{T}$,
hence over an \textit{arbitrarily long} time, a dynamical evolution which
mimics \textit{arbitrarily closely} that yielded by the original system $D$;
up to corrections of order $t/\tilde{T}$, or possibly even \textit{
identically}.

(ii) The system $\tilde{D}$ is \textit{isochronous}: for arbitrary initial
data all its solutions are completely periodic with a period $T$, which can
also be arbitrarily assigned, except for the\ (obviously necessary)
condition $T>\tilde{T}$.

Moreover it has been shown \cite{CL2007,calogero} that, if the dynamical
system $D$ is a many-body problem characterized by a (standard, autonomous)
Hamiltonian $H$ which is translation-invariant (i. e., it features no
external forces), other (also autonomous) Hamiltonians $\tilde{H}$
characterizing \textit{modified} many-body problems can be manufactured
which feature the \textit{same} dynamical variables as $H$ (i. e., in this
case there is no need to introduce two additional dynamical variables) and
which yield a time evolution \textit{quite close}, or even \textit{identical}
, to that yielded by the original Hamiltonian $H$ over the \textit{
arbitrarily assigned} time $\tilde{T}$, while being \textit{isochronous}
with the \textit{arbitrarily assigned} period $T$, of course with $T>\tilde{T
}$.

Let us emphasize that the class of Hamiltonians $H$ for which this result is
valid is quite general. In particular it includes the standard Hamiltonian
system describing an \textit{arbitrary} number $N$ of point particles with
\textit{arbitrary} masses moving in a space of \textit{arbitrary} dimensions
$d$ and interacting among themselves via potentials depending \textit{
arbitrarily} from the interparticle distances (including the possibility of
multiparticle forces), being therefore generally valid for any realistic
many-body problem, hence encompassing most of nonrelativistic physics. This
result is moreover true, \textit{mutatis mutandis}, in a quantal context.

For instance let us tersely review here the case of the standard Hamiltonian
describing the many-body problem, but focusing---merely for notational
simplicity---on the case with equal particles (and setting their mass to
unity) and a one-dimensional setting, which reads (in self-evident standard
notation)
\begin{subequations}
\begin{equation}
H\left( \underline{p},\underline{q}\right) =\frac{1}{2}\sum_{n=1}^{N}\left(
p_{n}^{2}\right) +V\left( \underline{q}\right)  \label{H}
\end{equation}
with the potential being translation-invariant, $V\left( \underline{q}
+a\right) =V\left( \underline{q}\right) $ (i. e., no external forces) but
otherwise unrestricted (except for the condition---again, for
simplicity---that the time-evolution yielded by this Hamiltonian be
nonsingular). The standard approach to treat this problem is to introduce
the center-of-mass coordinate $Q\left( \underline{q}\right) =\frac{1}{N}
\sum_{n=1}^{N}q_{n}$ and the total momentum $P\left( \underline{p}\right)
=\sum_{n=1}^{N}p_{n},$ and to then focus on the coordinate and momenta in
the center of mass, $x_{n}=q_{n}-Q\left( \underline{q}\right) ,$ $
y_{n}=p_{n}-P\left( \underline{p}\right) /N$, which characterize the physics
of the problem and whose evolution is determined by the Hamiltonian $h\left(
\underline{y},\underline{x}\right) $ defined as follows:
\begin{equation}
H\left( \underline{p},\underline{q}\right) =\frac{\left[ P\left( \underline{p
}\right) \right] ^{2}}{2N}+h\left( \underline{y},\underline{x}\right) ~,
\label{Hh}
\end{equation}
\begin{equation}
h\left( \underline{y},\underline{x}\right) =\frac{1}{2}\sum_{n=1}^{N}\left(
y_{n}^{2}\right) +V\left( \underline{x}\right) ~,  \label{h}
\end{equation}
where $V\left( \underline{x}\right) =V\left( \underline{q}\right) $ thanks
to the assumed translation-invariance of $V\left( \underline{q}\right) $. An
\textit{isochronous} Hamiltonian $\tilde{H}\left( \underline{p},\underline{q}
;T\right) $ reads then as follows,
\end{subequations}
\begin{subequations}
\begin{equation}
\tilde{H}\left( \underline{p},\underline{q};T\right) =\frac{1}{2}
\sum_{n=1}^{N}\left\{ \left[ P\left( \underline{p}\right) +h\left(
\underline{y},\underline{x}\right) \right] ^{2}+\left( \frac{2\pi }{T}
\right) \left[ Q\left( \underline{q}\right) \right] ^{2}\right\} ~.
\label{Htilde}
\end{equation}
It can indeed be shown \cite{CL2007,calogero} that it entails an \textit{
isochronous} evolution (with period $T$) of the center of mass coordinate $Q$
, of the total momentum $P$, and---most importantly---of all the particle
coordinates, whose evolution then reads
\begin{equation}
\underline{\tilde{x}}\left( t\right) =\underline{x}\left( \tau \left(
t\right) \right) ~,~~~\underline{\tilde{y}}\left( t\right) =\underline{y}
\left( \tau \left( t\right) \right) ~,  \label{xytildet}
\end{equation}
where we indicate as $\underline{\tilde{x}}\left( t\right) ,$ $\underline{
\tilde{y}}\left( t\right) $ the time evolution yielded by the tilded
Hamiltonian $\tilde{H}\left( \underline{p},\underline{q};T\right) $ and as $
\underline{x}\left( t\right) ,$ $\underline{y}\left( t\right) $ the time
evolution yielded by the original Hamiltonian $H\left( \underline{p},
\underline{q}\right) ,$ and
\begin{equation}
\tau \left( t\right) =A~\sin \left( \frac{2\pi t}{T}+\Phi \right)
\label{taut}
\end{equation}
where $A$ and $\Phi $ are \textit{constant} parameters given by simple
expressions in terms of the initial values, $Q\left( 0\right) $ and $P\left(
0\right) $, of the position of the center of mass of the system and of its
total momentum, and of the Hamiltonian $\tilde{H}\left( \underline{p},
\underline{q};T\right) $ (which is of course a constant of motion for this
time evolution). Note that, for $\left\vert t\right\vert <<T,$
\begin{equation}
\tau \left( t\right) =\alpha +\beta t+O\left[ \left( \frac{t}{T}\right) ^{2}
\right] ~~~\text{with \hspace{0in} }\alpha =A~\sin \left( \Phi \right)
~,~~~\beta =\frac{2\pi A}{T}\cos \left( \Phi \right) ~;
\end{equation}
hence up to small corrections (\textit{arbitrarily small} for an \textit{
arbitrarily large} assignment of the period $T$ as long as $\left\vert
t\right\vert $ is in an assigned interval $\tilde{T}$, of course such that $
\tilde{T}<<T$) the time evolution of the tilded coordinates approximates
that of the untilded coordinates, up to a constant shift and rescaling of
time.

Note that in this case---again, for the sake of simplicity---the modified,
\textit{isochronous} Hamiltonian $\tilde{H}\left( \underline{p},\underline{q}
;T\right) $ features only one additional constant parameter, $T$; but it has
been shown \cite{CL2007,calogero} that an analogous treatment---involving a
somewhat more complicated definition of a modified, \textit{isochronous}
Hamiltonian, featuring then two arbitrarily assigned parameters, $T$ and $
\tilde{T}<T$---allows to manufacture a modified Hamiltonian that reproduces
\textit{exactly} (up to a constant rescaling and shift of the time variable)
the same time evolution yielded by the original Hamiltonian over the
interval $\tilde{T}$ but is \textit{isochronous} with period $T$---and let
us again emphasize that this can be done for a many-body problem involving
an \textit{arbitrary} number of (possibly different) particles, moving in a
space of \textit{arbitrary} dimension $d$ and interacting via \textit{
arbitrary} interparticle forces. Which therefore includes most of
(nonrelativistic) physics.

Since it is impossible to distinguish experimentally dynamical systems that
behave \textit{arbitrarily closely}, or even \textit{identically}, over an
\textit{arbitrarily long} period of time, this finding has various
remarkable implications in the context of theoretical and mathematical
physics. In particular it raises \cite{CL2007,calogero} interesting
questions about the distinction between integrable and nonintegrable
evolutions, about the definition of chaotic behavior, and---for macroscopic
systems featuring, say, a number of particles of the order of Avogadro's
number---about statistical mechanics and the second principle of
thermodynamics. And for, say, $10^{85}$ particles it seems to have some
cosmological relevance.

But the proper setting of cosmological theories is general relativity. For
this reason in our previous paper \cite{isochronous cosmologies} we
investigated whether and how this result could be extended to that context.
We concluded that this is possible provided one extends general relativity
by allowing \textit{degenerate} (i.e. non invertible) metrics. In particular
we have shown \cite{isochronous cosmologies} that for any homogeneous,
isotropic and spatially flat metric $g_{\mu \nu }$ satisfying Einstein's
equations and providing a model of the universe, it is possible to find a
different (also homogeneous, isotropic and spatially flat) metric solution $
\tilde{g}_{\mu \nu }$ which is \textit{locally} (in time) diffeomorphic to $
g_{\mu \nu }$---hence yields the same cosmology as $g_{\mu \nu }$ for any
observation over an \textit{arbitrary} time interval $\tilde{T}$---and is
cyclic\footnote{
We prefer to talk of cyclic instead of periodic solutions due to the fact
that periodicity is not an invariant concept, since it is not invariant
under a redefinition of time. For a review of cyclic models see \cite{cyclic}
.} (in fact periodic with an \textit{arbitrary} period $T>\tilde{T}$ in the
time coordinate $t\,$); but it is \textit{degenerate} at an infinite,
discrete sequence of times $t_{n}=t_{0}\pm nT/2,$ $n=0,1,2,...$ . We
interpreted these metrics as corresponding to \textit{isochronous
cosmologies }\cite{isochronous cosmologies}.

Let us also recall \cite{isochronous cosmologies} that this result is not
restricted to homogeneous, isotropic and spatially flat metrics: it can be
easily extended to any synchronous metric, therefore it is quite general
since most metrics can be written in synchronous form by a diffeomorphic
change of coordinates.

Due to the diffeomorphic correspondence between $g_{\mu \nu }$ and $\tilde{g}
_{\mu \nu }$ \textit{locally} in time---for time intervals of order $\tilde{T
}$---these two metrics give the same physics locally in time, and therefore
there is no way to distinguish them using observations local in time; which
is essentially the same finding valid in the context of the Hamiltonian
systems considered in \cite{CL2007,calogero} and tersely recalled above. In
particular, the metric $\tilde{g}_{\mu \nu }$---while being cyclic on time
scales larger than $\tilde{T}$---may yield over the time interval $\tilde{T}$
just the standard sequence of domination firstly by radiation, then by
matter and dark energy, as well as the cosmological perturbations consistent
with all observational tests \cite
{cmb,leansing,bao,lss,supernovae,ref0,ref1,ref2,ref3,ref4,bicep2}, which
characterize the metric $g_{\mu \nu }$ of the standard $\Lambda $-CDM
cosmological model (see for instance \cite{mukhanov} for a review).
Furthermore, it has been shown that such isochronous metrics $\tilde{g}_{\mu
\nu }$ can be manufactured to be geodesically complete \cite{isochronous
cosmologies} and therefore singularity-free\footnote{
A spacetime is singularity-free if it is geodesically complete, i.e. if its
geodesics can be always past- and future-extended \cite{poisson,MTW}.}, so
that the geodesic motion as well as all physical quantities described by
scalar invariants are always well defined \cite{isochronous cosmologies} and
the Big Bang singularity may be avoided---even when some of the
phenomenological observations can nevertheless be interpreted as remnants of
a past Big Bang, which however may never be actually attained by the metric $
\tilde{g}_{\mu \nu }$, neither in the past nor in the future. This implies
that these isochronous metrics $\tilde{g}_{\mu \nu }$ may describe a
singularity-free universe which---while featuring a time evolution which
reproduces identically (up to diffeomorphic time reparameterization; \textit{
locally} in time, except at a discrete set of instants $t_{n}$) the standard
cosmological model characterized by the metric $g_{\mu \nu }$---features an
expansion which stops at some instant, to be followed by a period of
contraction, until this phase of evolution stops and is again followed by an
expanding phase, this pattern being repeated \textit{ad infinitum}. Let us
re-emphasize that the universe characterized by the metric $\tilde{g}_{\mu
\nu }$ may thereby avoid to experience the Big Bang singularity, even if the
universe which it mimics locally in time, characterized by the metric $
g_{\mu \nu },$ does encounter that singularity. Let us also note that this
class of metrics realize \textit{de facto} the reversal of time's arrow, as
discussed for instance in \cite{sakharov}.

But in order to do so, these isochronous metrics must be \textit{degenerate}
at a discrete set of instants $t_{n}$ when the time reversals occur, say
from expansion to contraction and viceversa; a phenomenology whose inclusion
in general relativity might be considered problematic, because at these
instants $t_{n}$---when the metric is \textit{degenerate}---the "equivalence
principle" corresponding to the requirement that the metric tensor have a
Minkowskian signature is violated. On the other hand a \textit{physically
unobservable} violation of a "principle" can be hardly considered \textit{
physically relevant}.

Purpose and scope of this paper is to better clarify the meaning and the
interpretation of the \textit{isochronous spacetimes} introduced in \cite
{isochronous cosmologies}, and to discuss some additional technical aspects.
As stated above, these isochronous metrics have the property to be \textit{
degenerate} at an infinite set of times $t_{n}$, i.e. over an infinite
number of $3$-dimensional hypersurfaces, where generally the scale factor
reaches its maximum and minimum values and the spacetime changes from an
expanding to a contracting phase and vice versa \cite{isochronous
cosmologies}.

The inclusion of degenerate metrics in general relativity is not a trivial
matter; indeed, it is forbidden if it were required---as an absolute (as it
were, "metaphysical") rule---that general relativity be consistent with the
"equivalence principle", i. e. that spacetime \textit{always} be a
Pseudo-Riemannian manifold \textit{with Minkowskian signature}. This would
indeed imply that the solutions of Einstein's general relativity, to be
acceptable, must be \textit{nondegenerate} metric tensors, since wherever in
spacetime a metric is \textit{degenerate} it is not possible to define its
inverse, hence the Christoffel symbols as well as the Riemann, Ricci and
Einstein tensors are not defined there. Hence---so the argument of some
critics goes---our isochronous metrics, which are not Minkowskian (locally,
on the hypersurfaces $t=t_{n}$) should be discarded in general relativity
because they violate the equivalence principle at those degeneracy surfaces.
Our rejoinder is that the metrics we introduced \cite{isochronous
cosmologies} are solutions of a version of general relativity whose
dynamics, while still governed by Einstein's equations\footnote{
Einstein's equations by itself do not determine the signature of spacetime,
see the discussion in \cite{teitelboim,ellis1}.}, does allow the violation
of the equivalence principle at a discrete set of hypersurfaces. It can be
argued that such theories are therefore \textit{different} from standard
general relativity, if validity of the equivalence principle---enforced as
an absolute, universal rule---is considered an essential element of that
theory; although the difference is in fact \textit{physically unobservable}.
This situation is indeed quite analogous with the findings described above,
valid in the nonrelativistic context of Hamiltonian many-body problems
(classical or quantal), where the solutions of a physical theory described
by a Hamiltonian $H$ are arbitrarily well approximated or even identically
reproduced over an arbitrary time interval by periodic solutions of a
\textit{different} Hamiltonian $\tilde{H}$, i.e. by a \textit{different}
physical theory. The main difference is that in the general relativistic
context in\textbf{\ }the two \textit{different} physical theories the
dynamic is given by the\textit{\ same} Einstein's equations, only the class
of acceptable solutions is \textit{different}.

The consideration in the context of general relativity of degenerate metrics
is in any case not a novelty. For instance they were already introduced \cite
{ellis1,ellis2}, and subsequently investigated in a series of papers \cite
{elliscarfora,elliscomment,ellis3,ellis4,ellis5,ellis6,ellis8,ellis9,ellis10,ellis11,ellis12,ellis13,ellis14,ellis15,ellis16,ellis17,ellis18,ellis19}
, in order to investigate signature-changing spacetimes. This class of
signature-changing metrics corresponds to a classical realization of the
change of signature in quantum cosmology conjectured by Hartle and Hawking
\cite{hartlehawking3} (see also \cite
{hartlehawking1,hartlehawking2,hartlehawking4}), which has philosophical
implications on the origin of the universe \cite{hawkinghistory}.

Below we will also consider a different realization of isochronous
cosmologies via non-degenerate metrics featuring a jump in their first
derivatives at the inversion times $t_{n}$, which then implies a
distributional contribution in the stress-energy tensor at $t_{n}$. These
metrics are diffeomorphic to the continuous and degenerate isochronous
metrics everywhere except on the inversion hypersurfaces $t=t_{n}$, hence
they describe the same physics except at the infinite set of discrete times $
t_{n};$ therefore they may also be manufactured so as to agree with all
cosmological observations (\textit{locally} in time). Since these two
realizations of isochronous cosmologies are \textit{locally} but \textit{not
globally} diffeomorphic, they correspond to different spacetimes and may be
considered to emerge from \textit{different} theories: the nondegenerate
ones are generalized (in the sense of distributions) solutions of
Einsteinian general relativity (including universal validity of the
equivalence principle, but allowing jumps---whose physical significance and
justification is moot---at the discrete times $t_{n}$); the degenerate ones
feature no mathematical or physical pathologies but fail to satisfy the
"equivalence principle" at the discrete times $t_{n}$. They are essentially
indistinguishable---among themselves and from non-isochronous
cosmologies---by feasible experiments (unless one considers feasible an
experiment lasting an arbitrarily long time...).

\section{Isochronous Cosmologies}

The procedure used to construct, for an autonomous dynamical system $D$,
another autonomous dynamical system $\tilde{D}$ whose solutions \textit{
approximate} arbitrarily closely, or even reproduce exactly, those of the
system $D$ over an \textit{arbitrary} time interval $\tilde{T}$ but are
\textit{isochronous} with an \textit{arbitrary} period $T>\tilde{T}$, is
based on the introduction of an auxiliary variable $\tau $ in place of the
physical time variable $t$ \cite{CL2007,calogero}. This procedure formally
entails a change in the time dependence of any physical quantity $f(t)$, so
that $f(t)$ gets replaced by $f(\tau \left( t\right) )$ with $\tau \left(
t\right) $ a periodic function of $t$ (with period $T$: $\tau \left( t\pm
T\right) =\tau \left( t\right) $), hence the new dynamics entails that all
physical quantities evolve periodically with period $T$. This change, in the
case of a nonrelativistic dynamical system, implies a change in the
dynamical equations: for instance, in the case of the (quite general)
many-body problem described above, the theory is characterized by a new
(autonomous) Hamiltonian $\tilde{H}$---different from the original
(autonomous) Hamiltonian $H$---which causes any physical variable originally
evolving as $f\left( t\right) $ under the dynamics yielded by $H$, to evolve
instead as $f(\tau (t))$ under the dynamics yielded by $\tilde{H}$. This
change of the physical laws determining the time evolution of the system
produces the \textit{isochronous} evolution \cite{calogero,CL2007}. But let
us stress---as we already did in \cite{isochronous cosmologies}---that any
attempt to attribute to the new variable $\tau $ the significance of "time"
would be improper and confusing: the variable playing the role of "time" is
the same, $t$, for both the original dynamical system $D$ and the modified
dynamical system $\tilde{D}$; in particular, for that characterized by the
standard many-body Hamiltonian $H,$ see (\ref{H}), as well as for that
characterized by the modified many-body Hamiltonian $\tilde{H}$, see (\ref
{Htilde}).

The dynamical systems $D$ and $\tilde{D}$ mentioned above feature the same
trajectories in phase space, but while the time evolution of the dynamical
system $D$ corresponds to a \textit{uniform} forward motion along those
trajectories, the time evolutions of the modified dynamical systems $\tilde{D
}$---although produced by time-independent equations of motion---correspond
to a \textit{periodic} (with assigned period $T$), forward and backward,
time evolution along those same trajectories, exploring therefore only a
portion of them; with the possibility to manufacture $\tilde{D}$ in such a
way that, for an \textit{arbitrary} subinterval $\tilde{T}<T,$ the dynamics
of $D$ and $\tilde{D}$ be \textit{very close} or even \textit{identical}.

In the context of general relativity, an analogous phenomenology exists \cite
{isochronous cosmologies}. However, in this case the situation is a bit
different, due to the physical invariance of Einstein's equations under
diffeomorphisms, which implies that the procedure used to construct the
isochronous solutions, while again formally entailing the introduction of a
new auxiliary variable $\tau \left( t\right) $, does not imply any change of
the fundamental Einstein equations, corresponding instead to the
identification of an enlarged class of solutions of these equations. But
again any attempt to attribute \textit{globally} to this auxiliary variable $
\tau $ the significance of \textit{time} would be improper and confusing;
while by attributing to it \textit{locally }(in time) the significance of
\textit{reparameterized time}, it is immediately seen that the enlarged
class of metrics entails dynamical evolutions which, \textit{locally in time}
, are \textit{physically equivalent} to that yielded by the original,
unmodified metric.

Our point of departure is to assume a homogeneous, isotropic and spatially
flat metric $\ \tilde{g}_{\mu \nu }$ which, in a given reference frame $(t,
\vec{x})$, is defined by the line element
\end{subequations}
\begin{equation}
ds^{2}=b(t)^{2}~dt^{2}-a(t)^{2}~d\vec{x}^{2}~,  \label{newmetric}
\end{equation}
where $b\left( t\right) ^{2}$ is commonly termed the "lapse" function
(although we did not use this term in \cite{isochronous cosmologies}).

The lapse function is not determined by Einstein's equations; its
introduction in (\ref{newmetric}) corresponds to the freedom to
reparametrize time, which in general relativity is permissible, via a
diffeomorphic transformation, with no physical effect:\ indeed, in general
relativity it is the \textit{geometry} of spacetime that characterizes the
physics, and this geometry is not affected by the coordinates used to
describe it. An additional requirement which is generally considered
essential---amounting to the "equivalence principle"---is that the metric
have a \textit{Lorentzian signature} (everywhere in spacetime). This implies
that, in the case of the metric (\ref{newmetric}), the lapse function $b(t)$
must never vanish, so that one can always put this metric in its synchronous
form via the reparameterization of the time variable from $t$ to $\tau $
such that
\begin{equation}
d\tau =b(t)~dt~,~\hspace{0in}~\tau \left( t\right) =B\left( t\right) \equiv
\int_{0}^{t}b\left( t^{\prime }\right) dt^{\prime }~,  \label{repar}
\end{equation}
which maps the metric (\ref{newmetric}) into the Friedmann-Robertson-Walker
metric $g_{\mu \nu }$ with line element
\begin{subequations}
\begin{equation}
ds^{2}=d\tau ^{2}-\alpha (\tau )^{2}~d\vec{x}^{2}~,  \label{FRWmetric1}
\end{equation}
where
\begin{equation}
\alpha (\tau (t))\equiv a(t)~.  \label{alphadefinition}
\end{equation}

As stated above, our trick to construct \textit{isochronous} dynamical
systems is \textit{formally} based on the eventual replacement of the
physical time variable $t$ with another variable $\tau $. In a general
relativistic context this corresponds to have a \textit{periodic} lapse
function (with an \textit{\ arbitrarily assigned} period $T$) having
moreover a vanishing mean value, so that its integral $B\left( t\right) ,$
see (\ref{repar}), is also periodic with period $T$:
\end{subequations}
\begin{equation}
b(t+T)=b(t)~;~~~\tau \left( t+T\right) =\tau \left( t\right) ~.
\label{bBperiodic}
\end{equation}

The choice (\ref{bBperiodic})---when inserted in (\ref{newmetric})---implies
the following consequences \cite{isochronous cosmologies}.

(i) For any FRW metric (\ref{FRWmetric1}), the corresponding---via the
relation (\ref{bBperiodic})---metric (\ref{newmetric}) is periodic in the
time variable $t$, and therefore it is cyclic, defining thereby an \textit{
isochronous spacetime}.

(ii) Since the lapse function is periodic with period $T$ and has zero mean
value, it must vanish at an infinite set of times $t=t_{n}\equiv t_{0}\pm
nT/2$ with $n=0,1,2,3,...$, implying that the metric (\ref{newmetric}) is
degenerate at these times.

(iii) If the metric (\ref{FRWmetric1}) is a solution of Einstein's equations
(namely, of the FRW equations) corresponding to the stress energy tensor of
a perfect fluid (or a mixture of different perfect fluids) $T_{\mu \nu }=
\left[ r(\tau )+p(\tau )\right] u_{\mu }u_{\nu }-g_{\mu \nu }p(\tau )$,
where $r(\tau )$ and $p(\tau )$ are the energy density and pressure of the
fluid and $u_{\mu }=\delta _{\mu 0}$ is its 4-velocity, then (\ref{newmetric}
) is also a solution of Einstein's equations corresponding to a stress
energy tensor $T_{\mu \nu }=\left[ \rho (t)+P(t)\right] U_{\mu }U_{\nu
}-g_{\mu \nu }P(\tau )$ of a perfect fluid with energy density $\rho
(t)=r(\tau (t))$, pressure $P(t)=p(\tau (t))$ and 4-velocity $U_{\mu
}=|b(t)|\delta _{\mu 0}$. Therefore the metrics (\ref{newmetric}) and (\ref
{FRWmetric1}) describe two universes filled with the same perfect fluid(s).

(iv) Since for $t\neq t_{n}$ the two metrics (\ref{newmetric}) and (\ref
{FRWmetric1}) are connected by a diffeomorphic transformation, they give the
same physics in any time interval within which $b(t)$ does \textit{not}
vanish, so that they are in fact experimentally indistinguishable there.

(v) One can conveniently choose the lapse function in such a way that the
scale factor is always positive, i.e. $a(t)=\alpha (\tau (t))>0$ for any $t$
. This choice implies that all the scalar invariants as well as the pressure
and energy density of the fluid remain finite at any $t$ and the spacetime
is geodesically complete, thus singularity free.

(vi) Geodesics have, in spacetime, a helical structure.

These solutions of course violate the "equivalence principle" at the
infinite set of discrete times $t_{n}.$ It is thus seen that, with the
choice (\ref{bBperiodic}), the class of spacetime solutions of the equations
of general relativity is enlarged by allowing unobservable violations of the
"equivalence principle" at a discrete, numerable set of times $t_{n}$. This
more general class of spacetimes do not seem to entail any \textit{
experimentally observable} differences; although they might entail a quite
different structure of spacetime, for instance absence versus presence of
Bing Bang singularities.

The problem of finding a generalization of general relativity which includes
degenerate metrics has been studied in the past, motivated by the
consideration of a class of metrics, the so called signature-changing
metrics \cite
{ellis1,ellis2,elliscarfora,elliscomment,ellis3,ellis4,ellis5,ellis6,ellis8,ellis9,ellis10,ellis11,ellis12,ellis13,ellis14,ellis15,ellis16,ellis17,ellis18,ellis19}
, which give a classical realization of the change of signature in quantum
cosmology conjectured by Hartle and Hawking \cite
{hartlehawking3,hartlehawking1,hartlehawking2,hartlehawking4,hawkinghistory}
. The classical change of signature for a homogeneous, isotropic and
spatially flat universe is realized by a metric tensor defined by the
following line element
\begin{equation}
ds^{2}=N(t)~dt^{2}-R(t)^{2}~d\vec{x}^{2}~,
\label{changeofsignaturedegenerate}
\end{equation}
where $N(t)$ is a continuous function changing sign at some time $t_{0}$,
for instance $N(t)$ is positive for $t>t_{0}$, negative for $t<t_{0}$ and
zero for $t=t_{0}$ in correspondence to the hypersurface of change of
signature where the metric (\ref{changeofsignaturedegenerate}) is degenerate
(note the difference from (\ref{newmetric}), where the coefficient $
b^{2}\left( t\right) $ of $dt^{2}$ might vanish at some points but never
becomes \textit{negative, }$b^{2}\left( t\right) \geq 0$).

These generalizations of standard general relativity including degenerate
metrics as (\ref{newmetric}) and (\ref{changeofsignaturedegenerate}) are not
unique since they depend on their defining prescriptions: \textit{different}
prescriptions yield \textit{different} physical theories \cite{elliscomment}
. In particular, it has been pointed out that different generalizations
based on different prescriptions give different junction conditions on the
degeneracy hypersurfaces, see for instance the discussion in \cite
{elliscarfora,elliscomment}. In fact "one can also consider discontinuities
in the extrinsic curvature" \cite{elliscarfora} on the degeneracy
hypersurface of a degenerate solution of Einstein's equations, "but attempts
to relate these to a distributional matter source at the boundary require
some form of field equations valid on the surface", see discussion in \cite
{elliscomment}, and this would require some special prescription on what
Einstein's equations are on the degeneracy hypersurfaces. Thus, if one
accepts degenerate solutions of Einstein's equations as physically
acceptable spacetimes, there is no reason to infer that a discontinuity of
the extrinsic curvature on a degeneracy surface implies that the stress
energy tensor associated to such a metric tensor must have a distributional
form there. What is more, delta functions with support on degeneracy
hypersurfaces have no significant effect in covariant integrals \cite
{ellis15,ellis19}, due to the fact that the covariant volume element $dx^{4}
\sqrt{-g}$ associated to a degenerate metric is null on its degeneracy
hypersurface, because the determinant $g$ vanishes there. For instance, for
the metric (\ref{newmetric}) one has, for any integrable function $f\left(
x\right) $,
\begin{equation}
\int dx^{4}\sqrt{-g}\delta (t-t_{n})f\left( x\right) =\int
dtdx^{3}|b(t)||a(t)|^{3}\delta (t-t_{n})f\left( x\right)
=|b(t_{n})|~|a(t_{n})|^{3}\int \,dx^{3}f\left( x\right) =0
\end{equation}
(because $b\left( t_{n}\right) =0$). Hence delta-like distributions centered
on the degeneracy hypersurfaces $t=t_{n}$ disappear from integrals (see also
the discussion in \cite{ellis15}).

So, the question of how to include degenerate metrics in general relativity
is moot.

In order to formulate a generalized gravitational theory which includes such
metrics, it is convenient to focus directly on the Einstein's equations
characterizing the geometry of spacetime, which themselves admit degenerate
metrics as their solutions. If, however, one considers desirable to derive
this theory from a variational principle, a possibility is to assume the
standard Einstein-Hilbert gravitational action of general relativity, such
that the total action of gravitation plus the matter (nongravitational)
fields is
\begin{equation}
S_{Tot}=\int_{M}\sqrt{-g}\left( -\frac{1}{2\mathit{k}}R+\mathit{L}
_{Mat}\right) d^{4}x~,  \label{actiontotal}
\end{equation}
where $\mathit{k}=8\pi G/c^{4}$, $G$ is the gravitational constant, $c$ the
speed of light, $M$ is the manifold defining the spacetime and $\mathit{L}
_{Mat}$ is the Lagrangian density of matter fields; and add the requirement
that, for a metric tensor which is degenerate on some hypersurface $\Sigma $
, the variation of the metric $g_{\mu \nu }$ vanish on $\Sigma $, i.e. $
\delta g_{\mu \nu }|_{\Sigma }=0$. With such an assumption the variation of
the action (\ref{actiontotal}) is
\begin{equation}
\delta S_{Tot}=\frac{1}{2}\int_{M/\Sigma }\sqrt{-g}\left( -\frac{1}{\mathit{k
}}G^{\mu \nu }+T^{\mu \nu }\right) d^{4}x~,  \label{actiontotalvariation}
\end{equation}
where $G^{\mu \nu }$ is the Einstein tensor and $T^{\mu \nu }$ is the
stress-energy tensor of matter fields, which gives the standard equations of
general relativity in the region where the metric tensor is nondegenerate.

To bypass the problems related to degenerate signature-changing metrics, it
has been proposed \cite{ellis1,ellis2} to consider a different,
nondegenerate but discontinuous, realization of the classical change of
signature, as given by the metric
\begin{equation}
ds^{2}=f(\tau )~d\tau ^{2}-R(\tau )^{2}~d\vec{x}^{2},
\label{changeofsignaturediscontinuous}
\end{equation}
with a discontinuous lapse function $f(\tau )=1$ for $\tau >\tau _{0}$ and $
f(\tau )=-1$ for $\tau <\tau _{0}$. It has been shown \cite{ellis1,ellis2}
that in this case it is possible to introduce a smooth (generalized)
orthonormal reference frame which allows a variational derivation of
Einstein's equations and an extension of the Darmois formalism \cite{ellis16}
to discontinuous metrics, in such a way that, also in the case of a
discontinuous metric tensor, the discontinuity of the extrinsic curvature on
a hypersurface is related to a distributional stress energy tensor \cite
{ellis15}. This is achieved by adding a surface term to the Einstein-Hilbert
action, in such a way that the gravitational action becomes
\begin{equation}
S_{g}=\int_{M/\Sigma }\sqrt{-g}Rd^{4}x+\oint_{\Sigma }\sqrt{-g}Kd\Sigma
\label{actionboundary}
\end{equation}
where $M$ is the manifold defining the spacetime, $\Sigma $ is the boundary
of $M$ where the metric tensor is discontinuous, $R$ is the Ricci scalar
curvature and $K$ the extrinsic curvature of $\Sigma $. The addition of such
a boundary term is always necessary when one considers manifolds with
boundaries \cite{HawkingHorowitz}. Incidentally we note that this action
cannot be used in the case of degenerate metrics, because in this case the
degeneracy surface $\Sigma $ has no unitary normal vector hence the
extrinsic curvature $K$ does not exist.

The two realizations of the classical change of signature, the continuous
and degenerate one, see (\ref{changeofsignaturedegenerate}), and the
discontinuous one, see (\ref{changeofsignaturediscontinuous}), are \textit{
locally but not globally} diffeomorphic, since they are related by a change
of the time variable $dt=d\tau /\sqrt{|N(t)|}$ which is not defined at the
time of signature change $t_{0}$ when $N(t)=0$. Hence they represent two
different spacetimes. However, except for the instants when $N(t)$ vanishes,
they are \textit{locally } diffeomorphic and thus they describe---excepts at
those instants---the same physics .

As in the case of signature-changing metrics, it is also possible to
consider a realization of \textit{isochronous} cosmologies different from
that we introduced in \cite{isochronous cosmologies} and reviewed above (see
(\ref{newmetric}) and the subsequent treatment), via \textit{nondegenerate}
metrics featuring a finite jump of their first derivatives at an infinite,
discrete set of equispaced times. Such a realization is given by the
following metric:
\begin{equation}
ds^{2}=d\eta ^{2}-\tilde{a}(\eta )^{2}d\vec{x}^{2}
\label{newmetricnondegenerate}
\end{equation}
with
\begin{equation}
\alpha (\tau (\eta ))\equiv \tilde{a}(\eta )
\label{newscalarfactornondegenerate}
\end{equation}
and
\begin{equation}
d\tau =C(\eta )~d\eta ,  \label{newCndegenerate}
\end{equation}
with a periodic function $C(\eta )$ of period $T$ given for instance by $
C(\eta )=1$ for $nT<\eta <(2n+1)T/2$ and $C(\eta )=-1$ for $(2n+1)T/2<\eta
<(n+1)T$ and $n$ integer, hence such that
\begin{subequations}
\begin{equation}
\tau \left( \eta \right) \equiv \int_{0}^{\eta }C\left( \eta ^{\prime
}\right) d\eta ^{\prime }=\eta -nT~~~\text{for}~~~nT<t<\left( 2n+1\right)
T/2~,
\end{equation}
\begin{equation}
\tau \left( \eta \right) \equiv \int_{0}^{\eta }C\left( \eta ^{\prime
}\right) d\eta ^{\prime }=\left( n+1\right) T-\eta ~~~\text{for}~~~\left(
2n+1\right) T/2<t<\left( n+1\right) T~.
\end{equation}

The metric (\ref{newmetricnondegenerate}) is periodic with period $T$ and is
not degenerate at the inversion times $\eta _{n}=nT/2$, where its first
derivatives feature a finite jump. Since (\ref{newmetricnondegenerate}) is
nondegenerate, now the Darmois formalism \cite{ellis16} applies and one can
relate the discontinuity of the first derivatives of the metric, hence the
discontinuity of the extrinsic curvature of the hypersurfaces $\Sigma _{\eta
_{n}}$ defined by $\eta =\eta _{n}$ (which is now well defined), with a
distributional stress-energy tensor. In fact defining the unitary normal
vector $n_{\alpha }=\delta _{\alpha 0}$, the tangent vectors $e_{\alpha
}^{a}=\delta _{\alpha }^{a}$ and the induced metric $h_{ab}=-\tilde{a}(\eta
)^{2}\delta _{ab}$ of a hypersurface $\Sigma _{\eta }$ of equation $\eta
=constant$, which also implies $n^{\alpha }=\delta ^{\alpha 0}$, $
e_{a}^{\alpha }=\delta _{a}^{\alpha }$ and $h^{ab}=-\delta _{ab}/\tilde{a}
(\eta )^{2}$, the extrinsic curvature of $\Sigma _{\eta }$ is given by \cite
{poisson}
\end{subequations}
\begin{equation}
K_{ab}\equiv n_{\alpha ;\beta }e_{a}^{\alpha }e_{b}^{\beta }= \alpha (\tau
(\eta )) \, \alpha^{\prime }(\tau (\eta )) \, C(\eta ) \,\delta _{ab}~,
\label{extrinsiccurvaturenondegenerate}
\end{equation}
where $\alpha^{\prime }(z)\equiv d\alpha(z)/dz$ \footnote{
Note that in \cite{isochronous cosmologies} we have calculated the extrinsic
curvature of the hypersurfaces $t=const$ of the metric (\ref{newmetric})
defining the normal vector of such hypersurfaces as $n^{\alpha }=\delta
^{\alpha }/b(t)$, the tangent vectors as $e_{\alpha }^{a}=a(t)\delta
_{\alpha }^{a}$ and $h_{ab}=-\delta _{ab}$. Instead, (\ref
{extrinsiccurvatureJUMPnondegenerate}) corresponds to the choice $n^{\alpha
}=\delta ^{\alpha 0}/|b(t)|$, $e_{\alpha }^{a}=\delta _{\alpha }^{a}$ and $
h_{ab}=-a(t)^{2}\delta _{ab}$.}.

Therefore the discontinuity of the extrinsic curvature on the hypersurface $
\Sigma _{\eta _{n}}$ is
\begin{equation}
\lbrack K_{ab}]|_{\Sigma _{\eta _{n}}}=2\,(-1)^{n}\,\alpha (\tau (\eta
))\,\alpha ^{\prime }(\tau (\eta ))\,\delta _{ab}~,
\label{extrinsiccurvatureJUMPnondegenerate}
\end{equation}
and this implies that the stress-energy tensor has, on the numerable set of
hypersurfaces $\Sigma _{\eta _{n}}$, a distributional contribution given by
\begin{equation}
T_{\alpha \beta }^{distr}=\sum_{n}\delta (\eta -\eta _{n})\,S_{ab}|_{\Sigma
_{\eta _{n}}}e_{\alpha }^{a}e_{\beta }^{b}~,
\label{StressEnergyTensorNondegenerate}
\end{equation}
where
\begin{equation}
S_{ab}|_{\Sigma _{\eta _{n}}}\equiv \frac{1}{8\pi }\left( [K_{ab}]|_{\Sigma
_{\eta _{n}}}-[K]|_{\Sigma _{\eta _{n}}}h_{ab}\right) =\frac{(-1)^{n+1}}{
2\pi }\,\alpha (\tau (\eta ))\,\alpha ^{\prime }(\tau (\eta ))\,\delta _{ab}
\end{equation}
and $[K]|_{\Sigma _{\eta _{n}}}\equiv \lbrack K_{ab}]|_{\Sigma _{\eta
_{n}}}h^{ab}$. We note that this distributional stress-energy tensor is
absent in the case of a vacuum Minkowskian universe, since in this case $
\alpha (\tau )$ is constant and its derivative vanishes.

Again, let us point out that the two metrics (\ref{newmetric}) and (\ref
{newmetricnondegenerate}) are two \textit{different} realizations of an
\textit{isochronous} cosmology, since they are \textit{locally}, but \textit{
not globally}, diffeomorphic via the change of variable $d\eta =b(t)dt$
which is singular at $t_{n}$. In particular the statements (i,iii,iv,v,vi),
see above, which are valid for the metric (\ref{newmetric}), also apply to
the metric (\ref{newmetricnondegenerate}). Moreover, both (\ref{newmetric})
and (\ref{newmetricnondegenerate}) are \textit{locally} (but \textit{not
globally}) diffeomorphic to the FRW metric (\ref{FRWmetric1}) and therefore
they gives an \textit{identical} dynamics in any time interval such that $
b(t)\neq 0$ or $C(\eta )\neq 0$. The main difference is that the isochronous
evolution given by (\ref{newmetric}) entails the introduction of degenerate
metrics and the nondegenerate realization of the isochronous dynamics given
by (\ref{newmetricnondegenerate}) implies the existence of a distributional
contribution to the stress-energy tensor. In both cases the evolution of the
universe is locally the same as that described by the FRW equations,
implying that it is not possible to discriminate by experiments local in
time between an isochronous and a noncyclic evolution of the universe.

\section{Conclusions}

In this paper we have reviewed and further discussed the idea introduced in
\cite{isochronous cosmologies}, itself being an extension to a general
relativity context---hence relevant to discuss cosmological models---of a
finding valid in the wide context of nonrelativistic Hamiltonian dynamics.
Let us emphasize that---as in that context---the essence of our observation
is to point out the disturbing possibility, given a physical theory that
describes reality, to manufacture variations of it---i. e., different
theories, characterized by the introduction of an arbitrary parameter $T$
---which describe essentially the \textit{same} reality over time intervals
smaller than $T$ but are \textit{cyclic} with period $T$.

And let us conclude this paper by emphasizing that we are \textit{not}
ourselves arguing that our Universe evolves \textit{cyclically} indeed
\textit{isochronously}, much less wish we to enter the related philosophical
issues (not our cup of tea!); we have merely pointed out the (disturbing!)
fact that, in the context of theoretical and mathematical physics, such a
possibility does not seem to be excluded; nor does it seem feasible to
exclude it by realizable experiments.

\bigskip

\section{Acknowledgements}

This paper was completed while FB was a Marie Curie Fellow of the Istituto
Nazionale di Alta Matematica "Francesco Severi".


TITLE: Equation involving absolute values
QUESTION [0 upvotes]: I am trying to solve this (knowing that $w$ is bigger than zero and real):
$$|1-\frac{7}{6}w-\sqrt{19}|=|1-\frac{7}{6}w+\sqrt{19}|$$
I put it into wolframalpha and it spat out $w = 6/7$, but how is that possible when the $w$-part is the same on each side??

REPLY [0 votes]: There are two possibilities
$$1-\frac{7}{6}w-\sqrt{19}=1-\frac{7}{6}w+\sqrt{19}\lor 1-\frac{7}{6}w-\sqrt{19}=-\left(1-\frac{7}{6}w+\sqrt{19}\right)$$
The first is impossible, the second gives
$$1-\frac{7}{6}w-\sqrt{19}=-1+\frac{7}{6}w-\sqrt{19}$$
that is
$$\frac{7}{3} w=2\rightarrow w=\frac{6}{7}$$
This is much simpler than squaring both sides...


TITLE: A solution without using Gram schmidt required.
QUESTION [1 upvotes]: The question is given below:

I found a solution online using Gram Schmidt procedure but I need a solution without it, could anyone help me in this, please?
EDIT:
I was given a hint that I can show that there is a dual basis and that the inner product establishes an isomorphism $\delta$ between the space and its dual and to let $w_{j} = \delta^{-1} (v_{j}^*)$ where $v_{j}^*$ are the dual basis of $v_{j}$ and that $<v_{i}, w_{j}> \delta_{w_{j} (v_{i})}$ but still I do not understand the intuition of the hint. could anyone explain for me what is this hint doing?

REPLY [3 votes]: Write
$$
w_j= \sum_k x_{jk} v_k.$$
Then the condition $\langle v_i, w_j\rangle = \delta_{ij}$ is equivalent to
$$
\sum_k x_{jk} \langle v_i, v_k\rangle = \delta_{ij}, $$
which is the system of $n^2$ linear equations given in matrix form as $AX=I$, where
$$
A=\begin{bmatrix} \langle v_i, v_j\rangle\end{bmatrix}, \quad X=\begin{bmatrix} x_{ij}\end{bmatrix}, $$
and $I$ is the $n\times n$ identity matrix.

Therefore, what really needs to be done is proving that $A$ is invertible.

To do that, try exploring the concept of Gramian matrix, for example on Wikipedia https://en.wikipedia.org/wiki/Gramian_matrix


TITLE: Determine whether the given map $\phi$ makes these two binary structures isomorphic
QUESTION [1 upvotes]: $\langle \mathbb{Q}, \mathbb{+} \rangle$ with $\langle \mathbb{Q}, \mathbb{+} \rangle$ where $\phi(x) = x + 1$ for $x \in \mathbb{Q}$
In order to show that two binary structures are isomorphic, I need to check whether $\phi$ is one-to-one, onto, and a homorphism.
a) one-to-one check (pass)
$$
\begin{align}
\phi(x) &= \phi(y) \\
x + 1 &= y + 1 \\
-1 & ~~~-1 \\
x &= y
\end{align}
$$
b) onto check (pass)
$$\phi(x+1) = (x+1) + 1 = x + 2 \textrm{ for } x \in \mathbb{Q}$$
c) homomorphism check (fail?)
$$ \phi(x + y) = (x + y) + 1 \neq \phi(x) + \phi(y) = (x+1) + (y+1) = x + y + 2$$
I think that the homomorphism check failed here and that means these two binary structures are not isomorphic for $\phi$. Did I come to the right conclusion? Am I missing any considerations in any of the checks?

REPLY [3 votes]: Your check for a) is fine, though you might want to start with "For all $x,y\in\Bbb{Q}$...".
Your check for b) is incomplete, and has a typo (you write $\Bbb{Z}$ in stead of $\Bbb{Q}$). It would suffice to say

For all $x\in\Bbb{Q}$ we have $x=\phi(x-1)$, so $\phi$ is surjective.

Your check for c) is fine, and show that $\phi$ is indeed not an isomorphism. However, this doesn't prove that the two binary structures aren't isomorphic. In fact they are the same, so of course they are isomorphic.
In hindsight it suffices to check that c) fails, and there is no need to bother checking a) and b).


TITLE: Given two form $\omega$, find $\mu$ such that $d\mu=\omega$?
QUESTION [0 upvotes]: I am reading a textbook on differential forms and there is a question of the form: given a $k$-form, $\omega$, on say $\mathbb{R^3}$, find $\mu$ such that $d\mu=\omega$. For example, if $\omega= (x_{1}^2 +x_{2}^2)dx_1 \wedge dx_2$, what is $\mu$?
I am looking for a systematic method to solve these types of questions.
Thanks

REPLY [1 votes]: $d\mu=\omega$ implies that $d^2\mu=d\omega=0$. The Poincare Lemma implies that $\omega$ is exact, if you look at a standard proof of this lemma, it constructs $\mu$ such that $d\mu=\omega$ given a closed form $\omega$.
https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms#Poincar%C3%A9_lemma


TITLE: Fundamental-ness of fundamental forces
QUESTION [1 upvotes]: Example :
Consider a black hole and it's event horizon of radius R surrounding it . Suppose in the direction of a diameter of the black hole ( or event horizon) , there are two charged particles. Distance between these particles is 4R . In order for each particle to experience an electromagnetic force from another, a photon must travel between them as an information carrier. But in the midway, the photon will be captured by black hole. I conclude from this that the gravitational force of black hole affected the electromagnetic force between those two particles.
Question
Then why do we even call these forces as fundamental as if they are able to affect other forces by affecting their carrier particles? Had I wrongly interpret the meaning of fundamental ?

REPLY [1 votes]: Had I wrongly interpret the meaning of fundamental ?

For physics theories, yes you interpret wrongly.
The word "fundamental" in the way used in physics means

a : serving as a basis supporting existence or determining essential structure or function : basic

In the present mainstream mathematical physics theories, the basic theory from which all others emerge is the standard model of particle physics. It is the quantum field theory used for particles and is defined in dimensions of units compatible with h_bar , the Planck constant, where the model's calculations are validated. That is where the fundamental forces are defined as such: basic forces in nature.
All other theories can be shown mathematically to emerge from this underlying level,  thermodynamics, maxwell classical electricity and magnetism, .... Though the forces are fundamental because everything, at present, is supposed to emerge from these, the forces measured in larger dimensions are composite, and do not invalidate the word fundamental for the quantum field theory of the standard model.
For simplicity, let us take a different example of a sphere than your black hole. Take an apple, and two charged particles across it. The fundamental forces are interactions of the charged particle with the surface of the apple, not with each other as there is a multitude of charges shielding one particle from the other.


TITLE: Prove that if ${x_1, x_2, x_3}$ are roots of ${x^3 + px + q = 0}$ then ${x_1^3+x_2^3 + x_3^3 = 3x_1x_2x_3}$
QUESTION [4 upvotes]: How to prove that ${x_1^3+x_2^3 + x_3^3 = 3x_1x_2x_3}$ holds in case ${x_1, x_2, x_3}$ are roots of the polynomial?
I've tried the following approach:
If $x_1$, $x_2$ and $x_3$ are roots then
$$(x-x_1)(x-x_2)(x-x_3) = x^3+px+q = 0$$
Now find the coefficient near the powers of $x$:
$$
x^3 - (x_1 + x_2 + x_3)x^2 + (x_1x_2 + x_1x_3 + x_2x_3)x - x_1x_2x_3 = x^3+px+q
$$
That means that I can write a system of equations:
$$
\begin{cases}
-(x_1 + x_2 + x_3) = 0 \\
x_1x_2 + x_1x_3 + x_2x_3 =  p \\
- x_1x_2x_3 = q
\end{cases}
$$
At this point I got stuck. I've tried to raise $x_1 + x_2 + x_3$ to 3 power and expand the terms, but that didn't give me any insights. It feels like I have to play with the system of equations in some way but not sure what exact.

REPLY [1 votes]: Every symmetric polynomial can be expressed in terms of the elementary symmetric polynomials, in this case $s_1=x_1+x_2+x_3$, $s_2=x_1x_2+x_2x_3+x_3x_1$ and $s_3=x_1x_2x_3$. Since $x_1^3+x_2^3+x^3$ is homogeneous, we can find $a$, $b$ and $c$ such that
$$
x_1^3+x_2^3+x_3^3=as_1^3+bs_1s_2+cs_3
$$

For $x_1=1$, $x_2=0$, $x_3=0$: $1=a$
For $x_1=1$, $x_2=1$, $x_3=0$: $2=8a+2b$
For $x_1=1$, $x_2=1$, $x_3=1$: $3=27a+9b+c$

Therefore $a=1$, $b=-3$, $c=3$ and finally
$$
x_1^3+x_2^3+x_3^3=(x_1+x_2+x_3)^3-3(x_1+x_2+x_3)(x_1x_2+x_2x_3+x_3x_1)+3x_1x_2x_3
$$
This is a general result.
In your case, by Viète’s formulas
$$
x_1+x_2+x_3=0,\qquad
$$
so in the end
$$
x_1^3+x_2^3+x_3^3=3x_1x_2x_3
$$


TITLE: Decision rule that minmize the probability of error
QUESTION [1 upvotes]: Given
We consider a real-valued, discrete-time communication system with a channel gain $h$ and additive white Laplacian noise of unit scale with two possible signals $s \in (-\mu,+\mu)$ that are sent equiprobably: $y = hs + w$. To be more precise, the noise $W$ has the pdf
\begin{align}
   f_W(w) = \frac{1}{2}e^{-|w|}
\end{align}
Further, we define $SNR = \mu$, as a signal-to-noise ratio for noise of unit scale.
Question
Suppose first $h = 1$. Specify the decision rule that minimizes the probability of error in deciding on the transmitted message $s$, and compute the average error probability $P_e$ in terms of our definition of SNR.
Attempted solution
For $h = 1$ we have that
\begin{align}
   y = s + w
\end{align}
We will have two cases, first for $-\mu$
\begin{align}
   y = -\mu + w \\
   P(y|-\mu) = \frac{1}{2}e^{-|w + \mu|}
\end{align}
and call this hypothesis $H_0$
Secondly, for $+\mu$ we will have
\begin{align}
   y = \mu + w \\
   P(y|+\mu) = \frac{1}{2}e^{-|w-\mu|}
\end{align}
and this hypothesis is called $H_1$
My question is, do I always have to calculate conditional probability when when calculating the probability of error? And I also wonder that is the equation for the probability of error in this case?

REPLY [0 votes]: First question: Yes, When dealing with error probabilities you always have to calculate the conditional probabilities.


TITLE: Inner product of functions as integration
QUESTION [1 upvotes]: I am trying to teach my self some linear algebra in preparation for a module in machine learning. I am using Gilbert Strang's text Introduction to Linear Algebra and am having some difficulties.
My specific question is: How is the last equation below an example of integration by parts? It seems to be missing $x(t)y(t)$ to me and I don't know where that has gone.
The book gives the following equations leading up to it:
$$x^Ty = (x,y) = \int_{-\infty}^\infty x(t)y(t)~dt$$
From what I can tell, the above is saying that the inner product of the vectors $x$ and $y$ is equivalent or at least approximate to taking the integral of those two vectors as functions. Does the $(x,y)$ just mean inner product? I don't really understand what a vector of a function even is though.
$$(Ax)^Ty = x^T(A^Ty)$$
This associative rule for matrix-vector multiplication was given as a more rigorous view of what the transpose of $A$ actually is. It then partnered it with this:
$$\int_{-\infty}^\infty\frac{dx}{dt}y(t)~dt = \int_{-\infty}^\infty x(t)\left(-\frac{dy}{dt}\right)~dt$$
I can see how the two equations relate to eachother and that this suggests that $A^T$ is anti-symmetric. Gil mentions integration by parts here which I can definitely recognise but it doesn't seem complete, there should be an extra $x(t)y(t)$ surely?

REPLY [1 votes]: It seems that you are somewhat confused about the use of "vectors" terminology.
In the context as mentioned, vectors are quite abstract things. In fact, they
are a generalization of the well known "geometric entities endowed with magnitude
and direction". See the Wikipedia page about
Vector spaces ,
especially the section about Function spaces .
So there is not a "vector of a function": the functions themselves are
abstract vectors. And yes, an inner product - abstract again - for functions
can be defined by the integral as mentioned. What must been shown, however, is
that the thus defined integral preserves some essential properties of the common "geometric" inner product. See the Wikipedia page about Inner product spaces .
The main part of your question ( i.e. why $\left[x(t)y(t)\right]_{-\infty}^{+\infty} = 0$ ) has been answered by 'user161825' .
Hope this helps.


TITLE: Is there any risk to transform to $(B^{T} \otimes A)\operatorname{vec}(X)=\operatorname{vec}(C) $ for solving $AXB=C$ for X
QUESTION [1 upvotes]: To solve the equation $AXB=C$ for X, we can use the property of vec operator and kronecker product to transform to $(B^{T}\otimes A)\operatorname{vec}(X)=\operatorname{vec}(C)$, where $\operatorname{vec}(\cdot)$ is the vectorization operator.
Is there any risk to do like this, since $B^{T}\otimes A$ is a much larger matrix than A or B?
Thank you.

REPLY [3 votes]: We assume that the unknown $X\in M_{m,n}$. When $A\in M_{m,m},B\in M_{n,n}$ are full matrices, $B^T\otimes A$ is a full $mn\times mn$ matrix; then inversion of such a matrix has complexity $O(m^3n^3)$. As we will see, using this method (when $A,B$ are large matrices), is a very bad idea.
There are special methods to solve equations in the form $AXB+CXD=E$ where $A,C\in M_{m,m},B,D\in M_{n,n},E\in M_{m,n}$ are known and $X\in M_{m,n}$ is unknown. cf.
i)   http://www.maths.lth.se/na/courses/NUM115/NUM115-09/sylvester.pdf
uses an algorithm that is an extension of the Bartels–Stewart method and the Hessenberg-Schur method. Originally the Bartels –Stewart algorithm was used to solve the Sylvester equation.
ii)    http://www.dm.unibo.it/~simoncin/matrixeq.pdf
That is important, is that the previous algorithms have complexity $O(n^3+m^3)$ that is much smaller than the complexity of the Kronecker product method. For example, solving a Lyapounov equation ($AX+XA^T=B$ with $n\times n$ matrices) -with Bartels–Stewart- has the same complexity ($\approx 40 n^3$) as finding eigenvalues and eigenvectors of a $n\times n$ matrix.


TITLE: A relation between Lebesgue and Riemann integrable functions
QUESTION [1 upvotes]: Is it true that a bounded function $ f:[a,b]\to \mathbb{R} $ is Lebesgue integrable if and only if there exists a function $f_1:[a,b]\to \mathbb{R}$ that is Riemann integrable and
$$  m^*\lbrace x: (f-f_1)(x)\ne 0\rbrace =0 $$
One side of the above assumption is true and obvious.

REPLY [1 votes]: No. Take $f=1_A$, where $A$ is a measurable set with the property that both $A$ and $A^c$ have positive measure intersection with any interval (See for example, Construction of a Borel set with positive but not full measure in each interval). It is not too hard to see that modification on a null set $N$ cannot make such an $1_A$ continuous a.e. (because $A\setminus N$ and $A^c\setminus N$ must accumulate at every point).


TITLE: Bounded linear space (elementary question)
QUESTION [1 upvotes]: Does exist  (nonzero) bounded normed space over any field?
Fix normed linear space $L$ over field K.
We have $x+x+x+\cdots\in L$
So $||nx||=n||x||\rightarrow \infty$ when $n \rightarrow \infty \\$
Is this ok?
Is zero space bounded normed space?

REPLY [1 votes]: In  general, $\|nx\| = n \|x\|$ can't work.  In the case of a field of nonzero characteristic, there are only finitely many different values for $n x$.


TITLE: Properties of ideals preserved under extension of scalars
QUESTION [2 upvotes]: The motivation for this question comes from a question in a book by a certain R.H dealing with geometrically reduced and irreducible schemes.
Let $k \subset K$ be algebraically closed fields and let $I$ be a prime ideal in the polynomial ring $k [x_1 \ldots x_n]$, generated, say, by $f_1, \ldots f_m$. Suppose that $I$ is prime or that the radical of $I$ is prime. These properties, I think, should be preserved if we consider $I$ as an ideal in the larger polynomial ring of $K$.
I have at least two arguments for that:
1) The first follows by translating those ideal theoretic to a system of polynomial equations and inequations in $K$ with coefficients in the small field $k$. By Hilbert's Nullstellensatz,  if there's a solution in the big field, then there's already a solution in the small one.
2) Use quantifier elimination. Every extension of algebraically closed fields is an elementary extension (in the sense of model theory). Translate the fact that I is not prime into a first order formula with parameters in the small field $k$. If there's a witness in the larger field, there is a witness in the small one already (Here one should be a little bit careful: each formula only handles polynomials of bounded degree. But compactness ensures that if an ideal is not prime, then there is a bound to the degree of the polynomials witnessing this fact as a function of the degrees of its generators).
However, I am not sure how to make these notions precise and write them down elegantly. I will appreciate help

REPLY [3 votes]: This is indeed true. As Keenan's answer stated, the geometric version of the notion you're talking about is geometric irreducibility and reducedness.
Consider the scheme $X$ cut out by $I$. Then $X$ is irreducible if and only if $\sqrt{I}$ is prime. Then $X$ is geometrically irreducible if it stays irreducible after any base extension. Similarly, $X$ is reduced if and only if $\sqrt{I} = I$ and $X$ is geometrically reduced if it stays reduced after base extension. Your case of $I$ being prime is the case of being both geometrically reduced and geometrically irreducible.
Then under this setup, indeed it is true that a scheme that is reduced (irreducible) over an algebraically closed field is geometrically reduced (geometrically irreducible). Therefore, in fact you can weaken your assumption of $K$ being algebraically closed. You just need $k$ to be algebraically closed.
For proofs of these statements, see sections 4 and 6 of this chapter of the stacks project: http://stacks.math.columbia.edu/download/varieties.pdf. You'll have to go back to some earlier sections where they reference algebra results but you should be able to chase around a full proof.


TITLE: determinant of symmetric matrix with zeros in diagonal is even
QUESTION [2 upvotes]: Question is to prove that

Determinant of a symmetric matrix of odd degree with integer entries  and zeros in the diagonal is even..

For $A=(a_{ij})$ with $a_{ii}=0$ and $a_{ij}=a_{ji}$..
Determinant is given by $$\sum_{\sigma}sgn(\sigma)\prod a_{i\sigma(i)}$$
Now, we can neglect $\sigma$ that fixes at least one element in $\{1,2\dots,n\}$ as we would then have $a_{ii}$ in that product which is zero..
Let $S$ be the collection of $\sigma$ that is remaining...
Let $\sigma\in S$ then $sgn(\sigma)=sgn(\sigma^{-1})$
$$\prod a_{i\sigma(i)}=\prod a_{\sigma(i)i}=\prod a_{j\sigma^{-1}(j)}$$
For pair $\sigma,\sigma^{-1}$ the product turns out to be same..
As $n$ is odd, there can be no element of order $2$ in $S$ so, $\sigma\neq \sigma^{-1}$
Suppose $\sigma$ is an element of order $2$.. We assume that $\sigma$ does not fix any element..
Start with $1$.. $\sigma(1)\neq 1$ we can suppose $\sigma(1)=2$..
As $\sigma^2=Id$  we should have $\sigma(2)=1$.. So, $\sigma=(1~2)\tau$
Start with $3$  $\sigma(3)\neq 3$ we can suppose $\sigma(3)=4$..
As $\sigma^2=Id$  we should have $\sigma(4)=3$.. So, $\sigma=(1~2)(3~4)\tau'$
Repeating this, we end up with $\sigma=(1~2)\cdots(2m-1~2m)(2m+1)$ a contradiction as we have assumed that $\sigma$ does not fix any element but here we have no option..
So, if $n$ is odd and $\sigma$ does not fix any element then $\sigma^2\neq Id$.. So, $\sigma\neq \sigma^{-1}$
So, sum has same number twice.. Thus determinant is even...
Help me to clear this...

REPLY [1 votes]: If we change the sign of any member of matrix the deterninant doesnt change modulo 2. So replace all $a_{ij}, i<j$ by $-a_{ij}$ and receive skew-simmetric matrix of odd order which determinant is even of course - because it becomes zero.


TITLE: Immersions of surfaces in $\mathbb{R}^3$
QUESTION [15 upvotes]: Stephen Smale famously proved in [Trans. Amer. Math. Soc. 90 (1959), 281-290] that any two $C^2$ immersions $S^2\to\mathbb R^3$ are regularly homotopic. This is how we knew that one can do a sphere eversion before ever constructing such a thing, for example.
Later, Morris Hirsch wrote [Trans. Amer. Math. Soc. 93 (1959), 242-276], where he moves from the problem of immersing spheres to immersing general manifolds. His paper is based on Smale's, he says, in roughly the same way that obstruction theory is based on the theory of homotopy groups.
In his paper Hirsch uses his results to solve various problems in immersion theory, but I cannot find in it, nor after a google search, an answer to this:

What is the classification up to regular homotopy of immersions  $M\to\mathbb{R^3}$ from a closed oriented surface?

In particular, is there just a single regular homotopy class of immersions of a closed orientable surface of positive genus? Can one everse a sphere with handles?
Later. Looking at the explicit construction of representatives of regular homotopy classes of immersions given in the paper by Joel Hass and John Hughes that Danny Ruberman mentions in his answer, the following question asks itself. In some sense, it is closer to the eversion problem, as when turning spheres inside-out we start and end with embeddings.

What is the classification up to regular homotopy of embeddings  $M\to\mathbb{R^3}$ of a closed oriented surface?

Of course, along the homotopy the embedding may degrade into just an immersion. Corollary 3.3 in the paper states that if $M$ has genus $g$, then of the $4^g$ regular homotopy classes of immersions $M\to M\times[0,1]$ which are homotopic to the the obvious map $\iota:p\in M\mapsto (p,0)\in M\times [0,1]$ only one can be represented by an embedding, the one which contains $\iota$. So the second question above is what happens when we replace $M\times[0,1]$ by $\mathbb{R^3}$.

REPLY [3 votes]: Regarding the classification up to regular homotopy of embeddings, U. Pinkall proves in [Topology 24 (1984), 421–434] that

if $f$, $g:M\to\mathbb R^3$ are two embeddings of a compact orientable surface, then there is a diffeomorphism $h:M\to M$ such that $f$ and $g\circ h$ are regularly homotopic.

This is not quite as good as «one can everse a compact orientable surface», though: that the usual embedding $f:S^2\to\mathbb R^3$ and $g=-f$ are related as in Pinkall's theorem is obvious.
Pinkall notes that it is not true that two embeddings $f$, $g:M\to\mathbb R^3$ are necesarily regularly homotopic, but does not say anything how often they are.