text:
\begin{document}

	\title{The Burgers equations and the Born rule}

	\author{Dimiter Prodanov}

\begin{abstract}
The present work demonstrates the connections between the  Burgers, diffusion, and Schr\"odinger's equations.
The starting point is a formulation of the stochastic mechanics, which is modelled along the lines of the scale relativity theory.
The resulting statistical description obeys the  Fokker-Planck equation.

This paper further demonstrates the  connection between the two approaches,  embodied by the study of the Burgers equation, which from this perspective appears as a stochastic geodesic equation.
The main  result of the article is the transparent derivation of the Born rule from the starting point of a complex stochastic process, based on a complex Fokker-Planck formalism.

keywords: { Burgers equation; Schroedinger equation; diffusion; stochastic mechanics; scale relativity }

\end{abstract}

\address{
	 Environment, Health and Safety Department, IMEC, Leuven, Belgium \\
	 MMSIP, IICT, Bulgarian Acacdemy of Sciences, Bulgaria
}

\maketitle

\section{Introduction}
\label{sec:intro}

The present paper reveals deep connections between the Burgers equation and the Born's rule in quantum mechanics.
The Born rule assigns a probability to any possible outcome of a quantum measurement. It asserts that the probability, associated with an experimental outcome is equal to the squared modulus of the wave function \cite[Ch. 2]{Dirac1981}.
The Born rule completes the Copenhagen interpretation of quantum mechanics.
On the other hand, it leaves open the question on how these probabilities are to be interpreted.
Some authors managed to derive the rule using the machinery of Hilbert spaces (i.e. the Gleason theorem \cite{Gleason1957}), while others resorted to operational approaches \cite{Saunders2004, Masanes2019}.
However, using such abstractions, it may be difficult to distinguish epistemological from ontological aspects of the underlying physics.
The Born's rule was also implicitly demonstrated in the scope of scale relativity theory \cite{Nottale2007}, which was the inspiration of the present work.

The Burgers' equation was initially formulated by Bateman while modelling the weakly viscous liquid motion \cite{Bateman1915}.
The equation can be derived from the full Navier-Stocks equations of fluid dynamics under some simplifying assumptions.
The equation was later studied extensively by Burgers as a cartoon model of turbulence as an attempt for a simplified mean field theory of  turbulence \cite{Burgers1974}.
The Burgers' equation reads

\begin{equation}\label{eq:burgers}
 \partial_t a    + a \, \partial_x a   - \nu \partial_{xx} a = 0
\end{equation}

In the initial applications, it was assumed that the viscosity parameter is real, however the equation can be analytically continued for complex values of $\nu$.
The present paper uses all three manifestations of the viscosity coefficient -- positive, negative and imaginary -- hence the plural form in its title.

There is abundant mathematical literature about the Burgers equation.
The one-dimensional solutions are surveyed in \cite{Benton1972}, while the similarity solutions have been investigated in \cite{Ozis2017}.
At present, the number of applications of the Burgers' equation is very diverse.
The equation has numerous applications in modelling of a wide variety of physical processes.
It has been used to model physical systems, such as  surface perturbations, acoustic waves, electromagnetic waves, density waves, or traffic (see for example \cite{Gurbatov1992}).
The stochastic representation  of the Burgers' equation can be traced back to the seminal works of Busnello et al. \cite{Busnello1999, Busnello2005}.

On the other hand, the Burgers' equation can be introduced from the drastically different perspective.
It represents the equation of the drift function of a Brownian diffusion, considered as a stochastic process.
This can be rigorously demonstrated using the apparatus of It\^o calculus.
Recent literature employing this perspective  includes \cite{Constantin2007, Eyink2014}, however, the focus there was on the inviscid Burgers equation.
Einik and Drivas show that the entropy solutions of Burgers have Markov  processes of  Lagrangian trajectories backward in time for which the Burgers velocity is a backward martingale.
A more general approach for the viscous Navier-Stokes equations  was introduced by Constantin and Iyer who derived a probabilistic representation of the deterministic 3-dimensional Navier-Stokes equations \cite{Constantin2007, Constantin2011}.

The Burgers' equation also arises in the theory of the Langevin equation as the resulting equation for the velocity of a particle subject to random forces.
It is in this latter aspect that the equation can be linked to the theory of quantum mechanics and the Schr\"odinger equation.
The connection arises in two ways -- by Nelson's \textit{stochastic mechanics} and by Nottale's \textit{scale relativity} theory.
The two theories are complementary to each other.
In scale relativity any material particle is assumed to follow fractal geodesic path, while the stochastic mechanics assumes that particles follow a conservative Brownian motion \cite{Carlen1984}.
Such Brownian motion can be used the model fractal trajectories in terms of their stochastic Markov process embedding \cite{Prodanov2018b}.

It is noteworthy that the Burgers equation can be linearised exactly by means of the Cole-Hopf transformation, which brings the non-linear equation into one-to-one correspondence with the diffusion equation.
From this perspective, the findings of the above works are not surprising.
It is also remarkable that, if the viscosity coefficient becomes imaginary, the Cole-Hopf transformation produces the Schr\"odinger equation.
This is not only a formal similarity as the complex viscosity coefficient can be endowed with a precise meaning as will be demonstrated in the present work.

What is not widely recognized is that the Cole-Hopf transformation can be extended into multiple dimensions.
The objective of the present paper is to use such non-homogenous Cole-Hopf transform in order to investigate some aspects of the Burgers' equation in view of multidimensional linear equations, notably the Schr\"odinger's  equation.
This is achieved in the general manner using the techniques of the Geometric algebra and It\^o calculus.

Moreover, assuming reversibility of the diffusion, a complex structure can be imposed over the process, which naturally leads to the Schr\"odinger equation and its conjugate as exact linearizations of the corresponding Burgers equations with imaginary viscosity.
This may seem as an arbitrary choice, however it is not so, since introducing such complex structure leads naturally to the Born's rule for the interpretation of the squared modulus of the Schr\"odinger's $\psi$ function as the probability density, appearing in the Fokker-Planck equation of the initial pair of diffusion processes.

As applied to quantum  mechanics, it is demonstrated here that the Born's rule is not arbitrary but stems from the microscopic reversibility in a stochastic sense.
To the best of this author's knowledge, such derivation has not been demonstrated in literature.

Derivation of the results presented in this manuscript is facilitated by the language of the Geometric Algebra, which allows for a straightforward verification in computer algebra systems  \cite{Prodanov2016}.

\section{Stochastic mechanics}\label{sec:char}

The equations of stochastic mechanics were formulated initially  by F\'enyes \cite{Fenyes1952} and Weizel \cite{Weizel1953} and later developed by Nelson \cite{Nelson1966} towards a comprehensive interpretation of quantum mechanics.
The stochastic mechanics draws on the formal similarities between the classical statistical mechanics and the Schr\"odinger equation.
In the treatment of the stochastic mechanics, quantum phenomena are described in terms of Brownian motions instead of wave functions.

The main equation of motion is in fact the Langevin equation employing a Wiener driving process, which can be handled by the apparatus of It\^o calculus.

Consider the stochastic integral equation with continuous drift and diffusion coefficients
\[
X_t - x_0 = \int_{0}^{t} a(X, t) d t + \int_{0}^{t} b(X, t) d W_t
\]
where $a(X,t)$ and $b(X,t)$ are smooth functions of the variables  and $d W_t$ is an increment of a Wiener process $ d W_{  t}  \sim N (0, d t)$ adapted to the past filtration $\mathbb{F}_{t>0}$ -- i.e. starting from the initial state $t=0$ and $x_0$ is the  deterministic initial condition.
The requirement of a filtration is essential for the development of the stochastic calculus.
This means that, in a way, one is recording the response of the system at a pre-defined but infinite sequence of intervals in the past.
The differential form of this equation is the Langevin equation
  \[
 d X_t^{+}  = a(X, t) d t + b(X, t) d W_t
 \]
 called also stochastic differential equation in the mathematical literature.
 The superscript indicates adaptation to the past filtration.
 In most of the derivations in the subsequent sections the dependencies of the \textit{a} and \textit{b} parameters will be assumed but not denoted explicitly.

 The drift and the diffusion fields (e.g. coefficient) can be calculated in the following way.

 Following Nelson the forward and backward  \textit{drift}, respectively \textit{diffusion} coefficients, can be identified as the ensemble-averaged velocities \cite{Nelson1966, Guerra1995}.

 Therefore, one can define a pair of differential operators (e.g.  directional derivatives) in the mean sense \cite{Nelson2012}:

 \begin{definition}[Mean velocities]\label{def:nelsond}
 	\[
 	\mathcal{D}^{+}_t X:= \llim{dt}{0}{  \mathbb{E}  \left( \left. \frac{X_{t+ d t} - X_t }{ dt} \right| X_t=x \right) },
 	\quad \mathcal{D}_t^{-} X:= \llim{dt}{0}{  \mathbb{E}  \left( \left. \frac{ X_t - X_{t- d t} }{ dt} \right| X_t=x \right) }
 	\]
 \end{definition}

 Defined in this way, it is not necessary to resort to the techniques of non-standard analysis as initially explored by Nelson.
 In this way,
 \[
 a = \mathcal{D}^{+}_t X^{+}_t
 \]
 In a similar way, the diffusion coefficient can be rigorously interpreted as the expectation of the \textit{fractional velocity} \cite{Prodanov2018, Prodanov2018b}:
 \[
 |b|  = \llim{dt}{0}{  \mathbb{E}  \left( \left. \frac{\left| X_{t+ d t} - X_t \right|}{ \sqrt{dt} } \right| X_t=x \right)}
 \]
The evolution of the probability density of the stochastic process can be computed from the forward Fokker-Planck equation \cite{Oksendal2003}
\begin{flalign}
	\partial_t \rho  +   \partial_x  \left( a \rho  \right)  - \frac{1}{2} \partial_{xx} \left( b^2  \rho \right) = 0
\end{flalign}
which can be recognized as a conservation law
\[
{\partial_t} \rho  +  {\partial_x} j =0
\]
for the probability current

$ j:=  a \rho-  \frac{1}{2}\partial_{x} b^2 \rho $.

One can define also a backwards process in the sense of the integral equation
\[
x_T - X_t   = \int_{t}^{T }\hat{a}(X, t)  d t + \int_{t}^{T }\hat{b}(X, t) d \hat{W}_t
\]
which is adapted to the future filtration $\mathbb{F}_{t<T}$ -- i.e. starting from the final state -- and $x_T$ is the  deterministic final condition.
The backwards diffusion process leads to the anticipative (i.e. anti-It\^o) stochastic integrals.
It should be noted that the anticipative stochastic integrals are, in a sense, dual to the more common It\^o integrals.
The differential form can be written in a similar way as
\[
d X_{t}^{-}   = \hat{a}(X, t)  d t + \hat{b}(X, t) d \hat{W}_t
\]
Then, in a similar way
\[
\mathcal{D}_t^{-} X_t^{-} = \hat{a}
\]
Note that in general, $a$ and $\hat{a}$ are different velocity fields!

Another result will be important for the subsequent presentation.
According to F\"ollmer \cite{Follmer1986}:
\begin{proposition}
	Suppose that $b =\hat{b}$ and

	$$
	E \left( \int_{0}^{T} a (X,t)^2 dt \right) < \infty
	$$
	Then the backwards diffusion process has the same \textit{density} $\rho$.
\end{proposition}

Under this condition, the backwards process has the Fokker-Planck equation
\begin{flalign}
 {\partial_t} \rho  +    {\partial_x}  \left( \hat{a} \rho  \right)  + \frac{1}{2} \partial_{xx} \left( b^2  \rho \right) = 0
\end{flalign}

\subsection{Velocity fields}\label{sec:velnelson}
Given this background, the Nelson's \textit{osmotic velocity} can be defined from
\[
  a -  \hat{a}  = b^2  {\partial_x} \log{ b^2  \rho } + \phi (t)
\]
where $ \phi (t)$ is an arbitrary $\mathcal{C}^{1}$ function of time
as
\[
u := \frac{1}{2} \left(   a -  \hat{a} \right)  =  \frac{b^2}{2}  {\partial_x} \log{ b^2  \rho }
\]
and the \textit{current velocity} as
\[
 v := \frac{1}{2} \left( a +  \hat{a} \right)
\]
so that a continuity equation holds for the density
\[
 {\partial_t} \rho +  {\partial_x}  \left( v \rho\right)  =0
\]
In order to derive the Schr\"odinger equation, Nelson's theory posits a special form of the acceleration without further physical argumentation \cite{Nelson2012}.
This can be considered as a drawback of the original theory.

\section{Scale relativity}\label{sec:screl}

The nature of the random Wiener process described in the previous section could look mysterious and contrived.
This is not so. An intuitive rationale is given by the \textit{Scale relativity} theory of L. Nottale.
The main tenet of the scale relativity  theory is that there is no preferred scale of description of the physical reality.
Therefore, a physical phenomenon must be described simultaneously at all admissible scales.
This lead Nottale to postulating some kind of fractal character of the underlying mathematical variety (i.e. a pseudo-manifold) describing the observables.
The theory of such varieties is still underdeveloped, therefore Nottale's argument should be taken only as analogy.

Nottale further posits that the fractal driving process can be approximated in stochastic sense using a Markov process.
While Nottale presents a heuristic argument and claims that the prescription of a Wiener process may be generalized he does not proceed to rigorously develop  the argument.
A rigorous treatment supporting this claim was presented in \cite{Prodanov2018b}.

On the other hand, the stochastic mechanics fixes from the start the Wiener process as the driving noise.
The question of why the Wiener process takes central stage must be addressed further.
The answer to this question can be given more easily by an approach inspired by Nottale and is partially given by the argument presented by Gillepsie \cite{Gillespie1996}.
The original formulation in \cite{Gillespie1996} contains an explicit assumption of existence of the second moment of the distribution, which  amounts to assuming H\"older continuity of order $1/2$ as demonstrated  in  \cite{Prodanov2018b}.

The scale relativistic approach results in corrections of the Hamiltonian mechanics that arise due to the non-differentiability of paths.
Nottale introduces a complex operator differential operator, that he calls the scale derivative.
The velocity in scale relativity is not interpreted as a mathematical derivative but as finite difference.
Therefore, from mathematical point of view, the fundamental quantities should be treated as asymptotics.
The non-differentiability leads to introduction of two velocity fields:
$ v_{+}$ for the \textit{forward} and $v_{-}$ for the \textit{backward} velocity.
This double field can be embedded in a complex space.
Following Nottale \cite{Nottale1993}, the pair of velocity  fields is represented by a single complex-valued vector field  as
\[
\mathbf{v} = V  - i U
\]
with components given by
$
V := \frac{1}{2} \left( v_{+}+ v_{-}  \right) ,
U := \frac{1}{2} \left( v_{+}- v_{-}  \right)
$
where \textit{V} is interpreted as the  "classical" velocity and \textit{U} is a new quasi-velocity quantity (i.e. the \textit{osmotic} velocity in the terminology of Nelson).
Such representation will be called \textit{complex lifting}.
Under this lifting Nottale introduces a complex material derivative, which is a pseudo-differential operator acting on scalar functions as
\[
\diffc{F}  =  \partial_t F   + \mathbf{v} \cdot \nabla F  - i \sigma^2 \, \nabla^2 F
\]
where $\sigma$ is a constant, quantifying the effect of changing the resolution scale.
Using this tool, Nottale gives a heuristic derivation of the Schr\"odinger equation from the classical Newtonian equation of dynamics.

On the other hand, a different embedding choice is also possible
\[
\mathbf{u} = V  + i U
\]
resulting in the complex-conjugated differential operator
\[
\mathcal{D}^*F  =  \partial_t F   + \mathbf{u} \cdot \nabla  F  + i \sigma^2 \, \nabla^2 F
\]

\section{The Burgers equation as a stochastic  geodesic equation for the velocity field}\label{sec:geodesic}

The use of the Wiener process entails the application of the fundamental It\^o Lemma for the forward (i.e. adapted to the past, plus sign) or the backward processes (i.e. adapted to the future, minus sign), respectively.
In differential notation it reads
\begin{flalign}\label{eq:ito2}
dF (t, X_t) =\partial_{t} F dt + dX^{+}_t  \partial_{x} F    + \frac{1}{2} \left[ dX_t^{+},dX_t^{+}  \right]\partial_{xx} F   \\
dF (t, X_t) = \partial_{t} F dt + dX^{-}_t  \partial_{x} F    - \frac{1}{2} \left[ dX_t^{-},dX_t^{-}  \right]\partial_{xx} F
\end{flalign}
where, $\left[ dX^{\pm}_t, dX^{\pm}_t \right] = b^2 dt $ is the \textit{quadratic variation} of the process (see for example \cite[Ch. 4]{Oksendal2003}).
\begin{remark}
In its essence, the It\^o Lemma is just the generalized Taylor development in the \textit{t} variable using the algebraical substitution rules $ dt^2 \rightarrow 0$, $ dW_t^2 \rightarrow dt$, $ dW_t dt \rightarrow 0$.

\end{remark}
The term \textit{geodesic} will be interpreted as a solution of a stochastic variational problem \cite{Zambrini1986, Yasue1981a}.
A brief treatment is given in Appendix. \ref{sec:stochvar}.
The stochastic variational problem reads
\[
\delta \int_0^T \left( \left( \mathcal{D}_{\tau}^{+}  {X}_{\tau}^{+}\right) ^2 - b^2 \right) d\tau =0
\]
which implies
$
\mathbb{E} \ d a= 0
$.
By application of It\^o's Lemma the forward geodesic equation can be obtained as:
\begin{equation}\label{eq:geodes}
 {\partial_t} a  + a  \partial_{x} a  + \frac{b^2}{2}  \partial_{xx} a  =0
\end{equation}
This can be recognized as a Burgers equation with negative kinematic viscosity for the drift field \cite{Burgers1974}.

The backward geodesic equation follows from variational problem for the anticipative process
\[
\delta \int_0^T \left( \left( \mathcal{D}_{\tau}^{-}  {X}_{\tau}^{-}\right) ^2 + b^2\right)  d\tau =0 \rightarrow  \mathbb{E} \ d \hat{a}= 0
\]
By an application of the It\^o's lemma for the anticipative process one obtains
\begin{equation}\label{eq:geodes1}
 {\partial_t} \hat{a}  + \hat{a}  \partial_{x} \hat{a}  - \frac{b^2}{2}  \partial_{xx} \hat{a}  =0
\end{equation}

This can be recognized as a Burgers' equation with positive kinematic viscosity for the drift field.

\section{The real-valued Cole–Hopf transform}\label{sec:colehopftransf}

Normalization $b=1$ will be assumed further in the section to simplify presentation.
The Burgers equation can be linearized by the Cole–Hopf transformation \cite{Hopf1950, Cole1951}.
This mapping transforms the non-linear Burgers equation into the linear heat conduction equation in the following way.
Let
\[
u= {\partial_x} \log a
\]
Substitution into Eq. \ref{eq:geodes} leads to
\[
\frac{1}{2 \, u^2} \left( u\,  {\partial_{{xxx}} u}
+2 u\,   {\partial_{tx}} u
-  {\partial_x}u \,  { \partial_{xx} u}
-2 \, { \partial_t} u  \, {\partial_x}u
\right)  =0
\]
This can be recognized as a total spatial derivative
\[
{\partial_x} \frac{1 }{u} \left( {\partial_t}  u +  \frac{1 }{2}   \partial_{xx} u \right)  = 0
\]
Therefore, the transformed equation is equivalent to a solution of the diffusion equation in reversed time
\[
{\partial_t}  u +  \frac{1 }{2}   \partial_{xx} u = 0
\]
wherever $u\neq 0$.

It should be noted that if instead of the forward development (i.e prediction)  one takes the backward development (i.e. retrodiction) the usual form of the Burgers equation is recovered:
 \[
 {\partial_t} \hat{a}  + \hat{a} {\partial_x} \hat{a}   - \frac{1}{2} \partial_{xx} \hat{a}  = 0
 \]
This corresponds to the anticipative Wiener process, which is subject to the anticipative It\^o calculus \cite{Constantin2007, Dunkel2008}.

\section{The Complex Material Derivatives}
\label{sec:appdiff}

For simplicity of the discussion, the section focuses on the one-dimensional case.

\subsection{Complex embedding}\label{sec:emcomp}

Consider two real-valued Brownian motions
\begin{flalign*}
 dX_t := a dt + b dW_t \\
 d\hat{X}_{t} := \hat{a} dt +\hat{b} d\hat{W}_{t}
\end{flalign*}

The drift, resp. diffusion coefficients can be further embedded in a complex space as proposed by   Pavon \cite{Pavon1995}.
Such embedding is an isomorphism:
\begin{flalign*}
a \,\otimes  \hat{a}  & \mapsto \mathcal{V} := v  - i u \\
dX_t \, \otimes d\hat{X}_{t} & \mapsto d \mathcal{X} = \frac{1}{2} \left( dX_t \, + d\hat{X}_{t} \right)
-  \frac{i}{2} \left( dX_t - d\hat{X}_{t} \right)
\end{flalign*}

\begin{multline*}
d \mathcal{X} = \left( v- i u\right) dt+ \frac{b}{2} dW_t +\frac{\hat{b}}{2} d\hat{W}_t
- i \left( \frac{b}{2} dW_t +\frac{\hat{b}}{2} d\hat{W}_t\right) =\\
\mathcal{V}  dt + \frac{1 - i}{2} b \; dW_t + \frac{1 + i}{2} \hat{b} \; d \hat{W}_t
= \mathcal{V}  dt + \sqrt{-i}\sigma\left(  \frac{b   dW_t + i  \hat{b} \; d \hat{W}_t }{2 \sigma}\right)
\end{multline*}
where
\[
\sigma =\sqrt{\frac{b^2 + \hat{b}^2}{2}}
\]
Therefore, purely algebraically, we can designate a new complex stochastic variable
\[
 dZ_t: = \frac{ b dW_t+ i \hat{b} d \hat{W}_t }{\sqrt{2} \sigma}
\]
so that in differential form
\begin{equation}\label{eq:itonot}
d \mathcal{X} = \mathcal{V}  dt + \sqrt{-i}\sigma  dZ_t
\end{equation}
So far the complex variable $Z_t$ is not completely specified.
As an additional postulate, we assume independence of the processes.

We further form the double filtration

\begin{definition}[Double filtration]
	Consider the interval $[0, T]$ and define the double filtration
	\[
	\mathbb{F}_t^2: = \mathbb{F}_{t>0} \times \mathbb{F}_{t<T}
	\]
	where the future filtration is constrained as

	\[
	[t_1, t_2] \in \mathbb{F}_{t>0} \Longleftrightarrow [T-t_2, T- t_1] \in \mathbb{F}_{t<T}
	\]
\end{definition}

The variable $dZ_t$ is  adapted to the double filtration and retains the martingale properties according to the Levy Characterization of Brownian motion.
Notably,
$\mathbb{E} dZ_t =0$.
In addition,
\[
  dZ_t^2=\frac{b^2 -\hat{b}^2}{ b^2 +\hat{b}^2}    dt
\]

Moreover,
\[
dZ_t dZ_t^* = \frac{b^2 dW_t^2 + \hat{b}^2 d\hat{W}_t^2}{b^2 + \hat{b}^2} = dt
\]
Therefore, a complex quadratic variation process can be introduced as a lift
\[
[dZ_t, dZ_t] := \frac{1}{2} \left( dZ_t -i\, dZ_t^* \right) ^2
\]
and extended by linearity
so that
\[
[dZ_t, dZ_t] = -i dZ_t dZ_t^*
\]

We further specialize the argument by assuming that  $\hat{b}=b(T-t)$ for the stopping time $T$.
Then, since $b$ is constant, we immediately obtain
$
dZ_t^2=0
$.

\subsection{The complex It\^o-Nottale Lemma}\label{sec:compito}
The next derivations follow the technique introduced by Pavon \cite{Pavon1997}.
Adding and subtracting equations \ref{eq:ito2} and 5 gives
\begin{multline*}
2 dF   = 2 \partial_{t} F dt +  \left( dX^{+}_t + dX^{-}_t\right)  \partial_{x} F    + \frac{1}{2} \left[ dX_t^{+},dX_t^{+}  \right]\partial_{xx} F  - \frac{1}{2} \left[ dX_t^{-},dX_t^{-}  \right]\partial_{xx} F = \\
\left( dX^{+}_t + dX^{-}_t\right)  \partial_{x} F = 2 \partial_{t} F dt + 2 v dt \partial_{x} F
\end{multline*}
and
\begin{multline*}
0  = \left(  dX^{+}_t - dX^{-}_t\right)   \partial_{x} F    + \frac{1}{2} \left[ dX_t^{+},dX_t^{+}  \right]\partial_{xx} F + \frac{1}{2} \left[ dX_t^{-},dX_t^{-}  \right]\partial_{xx} F = \\
2 u dt \partial_{x} F + b^2 dt \, \partial_{xx} F \\
\end{multline*}
Therefore, in components one can write
\begin{equation}\label{eq:compito2}
d F = \left(\partial_t F + \mathcal{V}  \partial_{x} F   -\frac{ i b^2}{2}  \partial_{xx}  F \right)   dt + \sqrt{-i} b \, \partial_{x} F  \, d Z_t
\end{equation}
Therefore, a complex lifted It\^o-Nottale differential can be introduced in exactly the same way
\begin{equation}\label{eq:compito}
d F  :=  \partial_t F dt + d \mathcal{X} \partial_{x} F +\frac{1}{2}  \left[   d \mathcal{X}, d \mathcal{X} \right]   \partial_{xx} F
\end{equation}
with   quadratic variation  $ \left[  d \mathcal{X}, d \mathcal{X} \right] =  -i dZ_t dZ_t^*= -i b^2 dt$.

It should be noted that the complex embedding is not unique.
An alternative complex embedding is given by
\begin{flalign*}
a \,\otimes  \hat{a}  & \mapsto \mathcal{U} := v  + i u \\
dX_t \, \otimes d\hat{X}_{t} & \mapsto d \mathcal{X} = \frac{1}{2} \left(  X_{t+ d t} + X_{t - d t} \right)
+ i \frac{1}{2} \left(  X_{t+ d t} - X_{t - d t} \right)
\end{flalign*}
with the quadratic variation is
$ \left[  d \mathcal{X}, d \mathcal{X} \right] = -i b^2 dt$.
In this case, the quadratic variation can be defined as
$$ \left[  dZ_t, dZ_t \right]^* =   \frac{1}{2}(dZ_t^* +i dZ_t )^2$$
implying,  $ \left[  d \mathcal{X}, d \mathcal{X} \right] =  i dZ_t dZ_t^*= i b^2 dt$.

The same application as above gives the It\^o equation for the drift
\begin{equation}\label{eq:compito3}
d G = \left(\partial_t G + \mathcal{U}  \partial_{x} G   +\frac{ i b^2}{2}  \partial_{xx}  G \right)   dt + \sqrt{i} b \, \partial_{x} G    \; d Z_t
\end{equation}

Moreover,
\begin{equation}\label{eq:compito34}
d G^* = \left(\partial_t G^* + \mathcal{U}^*  \partial_{x} G^*   -\frac{ i b^2}{2}  \partial_{xx}  G^* \right)   dt + \sqrt{-i} b \, \partial_{x} G^*    \; d Z_t^*
\end{equation}
which, can be recognized as the forward It\^o equation.
Therefore, the equations are dual by complex conjugation.

\section{The Complex Cole-Hopf Transform}\label{sec:colehpfcompl}

The stochastic geodesic equation can be introduced in the complex setting as well.
In this case, the geodesic equation reads
\[
\mathbb{E} \ d \mathcal{V} = 0
\]
and can be derived from the variational problem
\[
\delta \int_0^T \left( \mathcal{D}_\tau \mathcal{X}\right) ^2 d\tau =0
\]
Note that in this case no regularization of the drift is necessary. This is so because for a constant diffusion coefficient the quadratic variation vanishes: $dZ_t^2=0$.

In the complex case, starting from the generalized It\^o differential, the complex velocity field becomes
\[
d \mathcal{V} = \left({\partial_t} \mathcal{V} + \mathcal{V} {\partial_x} \mathcal{V} -\frac{ i b^2}{2} \partial_{xx}  \mathcal{V} \right)   dt + \sqrt{-i} b\, {\partial_x} \mathcal{V} \; d Z_t
\]
Therefore, the geodesic equation reads
\[
{\partial_t} \mathcal{V} + \mathcal{V} {\partial_x} \mathcal{V} -\frac{ i b^2}{2}\, \partial_{xx} \mathcal{V} = 0
\]
which can be recognized as a generalized Burgers' equation with imaginary kinematic viscosity coefficient.
Applying the complex  Cole–Hopf transformation as \cite{Lage2002}
\[
\mathcal{V} = -i { \partial_x} \log{U}, \quad  -\pi <\arg{U} <\pi
\]
and specializing to $b=1$ leads to the equation

\[
-  {\partial_x} \frac{ 1 }{U} \left( i \,  {\partial_t} U + \frac{ 1}{2}\, \partial_{{xx}} U \right) = 0
\]
which can be recognized as a gradient.
The last equation is equivalent to the solution of the free Schr\"odinger equation.
On the other hand, the diffusion part is simply
\[
- \sqrt{i} \left(  \partial_{ xx} \log U \right)  dZ_t = 	- \sqrt{i} \left( {\partial_x} \frac{1}{U}  {\partial_x}  U \right)
\]
since $  -i \sqrt{-i}=- \sqrt{i} $.

\begin{remark}
The above calculations can be reproduced using the computer algebra system Maxima \cite{Prodanov2018a}.
\end{remark}

Having demonstrated the solution technique, it is instructive to investigate multidimensional generalizations of the Burgers equation and the Cole-Hopf transform.

\section{Geometric algebra}
In the following section we use the convection of denoting the scalars with Greek letters, the vectors with lowercase Latin letters and multivectors or blades with capital Latin letters.
The Euclidean geometric algebra $\fclass{G}{3} \left( \fclass{R}{} \right)  $ is generated by the set of 3 orthonormal basis vectors $ E= \{e_{1}, e_{2}, e_{3} \}$
for which the so-called \textit{geometric} product is defined with properties
\begin{flalign}
e_{1} e_{1}  & = e_{2} e_{2} = e_{3}  e_{3} = 1 \\
e_{i} e_{j}  & = -  e_{j} e_{i}, \quad i \neq j
\end{flalign}

An overview of the topic can be found, for example in the book \cite{Macdonald2011}.
The geometric product of two vectors can be decomposed into a symmetrical \textbf{scalar product} and an antisymmetrical \textbf{wedge product}:
\[
a \, b = a \ast b + a \wedge b \,  ,  \quad a \ast b = b \ast a \, , \quad  a \wedge b = -  b \wedge a
\]
The scalar product is defined simply as the scalar part of the geometric product between multivectors:
\[
A \ast B = \left\langle A B \right\rangle_{0}
\]
where the notation $\left\langle  \right\rangle_{k}$ the part of the multivector sum of grade $k$.
Furthermore, the wedge product is extended for blades (products of basis vectors) as
\[
a \wedge A_k = \frac{1}{2} \left(a A_ k + (-1)^k A_k a \right)
\]
The Hestenes contraction is a symmetrical operation defined for multivectors of grades \textit{r} and  \textit{l}, respectively, as
\[
A_r \cdot B_l := \sum_{|r-s|>0} \left\langle AB \right\rangle_ {|r-s|}
\]
while for scalars $ \alpha \cdot A =0$.
Therefore, for vectors
\[
a \cdot b = a \ast b
\]
It is noteworthy also that for a vector and a blade \cite[Ch. 1, p.3]{Hestenes2015}
\[
a \cdot A_k= \frac{1}{2} \left(a A_k - (-1)^k A_k a \right)
\]
which extends the geometric product decomposition also to the products of vectors and blades:
\[
a A = a \cdot A + a \wedge A
\]
It should be noted that unlike the scalar product the contraction operation is not associative in the general case.

In the most general setting the geometric algebra is a subset of the Clifford algebra \Cl{p,q}.
What is remarkable is the Clifford algebra embedding theorem, which states that the Euclidean geometric algebra is isomorphic to the even part of the Space-Time Algebra $\Cl{1,3}^{+} $:
\[
\fclass{G}{3}  \cong \Cl{1,3}^{+}
\]
Therefore, all statements concerning Euclidean vectors can be translated into statements about space-time bivectors and vice versa.
This allows for an immediate generalization of the theorems of vector analysis using similar notation.

\subsection{Geometric calculus}\label{sec:gcalc}
The geometric derivative subsumes divergence, curl and gradient operations of the vector calculus.
Introduction on the topic can be found in  \cite{Macdonald2012}.
It is defined in the simplest way as
\[
\nabla f= e^j \partial_{x_j} f
\]
where $e^j $ are the components of the dual or reciprocal basis, such that
\[
x= x_i e^j=x^i e_i
\]
for an arbitrary vector $x$.
For the dual basis $e^i \ast x = x_i $ since $e^i e_j = e^i \ast e_j = \delta_{i,j}$, where the last symbol is the Kronecker's symbol.

The geometric derivative is co-ordinate independent.
Moreover, it splits into a grade-lowering and grade-increasing parts
\[
\nabla f= \nabla \cdot f + \nabla \wedge f
\]
The dot represents the Hestenes contraction operation (see discussion in \cite{Dorst2002}).

\subsection{The stretched gradient operator}\label{sec:strgrad}

\begin{definition}\label{def:strg}
The \textit{stretched gradient} operator
$\mathcal{C}(\nabla)$ is the linear operator acting on the gradient by anisotropically scaling the reciprocal basis vectors along the vector \textit{c}
\[
\mathcal{C}: e^k \mapsto c_k e^k
\]

So that
\begin{equation}\label{eq:coletr}
\mathcal{C}(\nabla) = \left(  c \cdot e^k \right)  e^k \partial_{x_k}
\end{equation}
\end{definition}

Then it is clear that the stretched gradient commutes with the gradient and the time derivative
\[
\partial_t \mathcal{C}(\nabla) = \mathcal{C}(\nabla) \partial_t
\]
and
\[
\nabla^2 \mathcal{C}(\nabla) =  \mathcal{C}(\nabla) \nabla^2
\]
under the assumption that $c$ is spatially constant.
Also, in components
\[
\left( \mathcal{C}(\nabla) F \right) \cdot \nabla = c_i\partial_{x_i} F \partial_{x_i}
\]
for a scalar function \textit{F}.
Furthermore, for a spatially constant scaling $c$
\[
\mathcal{C}(\nabla) \mathcal{C}(\nabla) = c_i^2 \partial^2_{x_i}
\]

\begin{proposition}\label{prop:comm1}
\[
\mathcal{C}(\nabla) F \cdot \nabla
= \left(  c \cdot e^i \right)  \partial_{x_i} F \partial_{x_i}
=  \partial_{x_i} F \left(  c \cdot e^i \right) \partial_{x_i} =
\nabla F \cdot \mathcal{C}(\nabla)
\]
for a scalar function \textit{F} and spatially constant \textit{c}.
\end{proposition}
On the other hand, the following identity holds true

\begin{proposition}\label{prop:comm2}
\[
\left( \mathcal{C}(\nabla) F \cdot \nabla\right)  \mathcal{C}(\nabla) F =
\mathcal{C}(\nabla) F \cdot \left(\nabla\,  \mathcal{C}(\nabla) F\right) =\frac{1}{2}\nabla \left(  \mathcal{C}(\nabla) F \right) ^2
\]
for a scalar function \textit{F}.
\end{proposition}

\section{The vectorized Cole-Hopf transform}\label{sec:colehpfinh}

Consider the complex, stochastic It\^o-Nottale process
\begin{equation}\label{eq:itonottale}
d \mathcal{X} = \mathcal{V} dt + d \mathcal{Z}_t
\end{equation}
where now $d \mathcal{Z}_t= e_i dZ^i_{t}$ is also Clifford vector-valued.
Using the apparatus of Geometric algebra, the complex It\^o differential of Eq. \ref{eq:compito2} generalizes to
\begin{equation}\label{eq:itocomplex}
d F = \left(  \partial_t F + \left( \mathcal{V} \cdot  \nabla \right)  F   - \frac{ i b^2}{2}  \nabla^2    F \right)   dt + \sqrt{-i} b  \left( d \mathcal{Z}_t \cdot \nabla   \right)  F
\end{equation}
in \Cl{p,q} over the complex numbers \fclass{C}{} for a smooth function $F$.

A sketch of the proof is provided in the remark below.

\begin{remark}
The  restrictions on the validity of the formula above are the assumptions that the diffusion coefficient must be a scalar (i.e. homogeneity of space) and
the co-ordinate processes $d{Z}_t^i$ are uncorrelated.

In the multidimensional case, the Taylor development of $F$ in the direction of the process $dX$ is
\[
dF=\partial_t F dt +  (dX \cdot  \nabla) F + \frac{1}{2} (dX \cdot  \nabla) ^2 F +\bigoh{dX^2+dt + dX dt}
\]
On the other hand, in matrix notation
\[
(dX \cdot  \nabla) ^2 F=  \mathbf{dX} \cdot \mathbf{H}(F) \cdot \mathbf{dX}^T
\]
where $\mathbf{H}(F) = \{ h_{ij}:= \partial_{x_i}\partial_{x_j} F \}$ is the Hessian matrix,
which is the usual statement of  the Multidimensional It\^o lemma \cite{Oksendal2003}.
Evaluating for the Wiener process  $dx^i= dW^i_t$ and
using the algebraical rules $dW^i_t dW^j_t \rightarrow 0$ (independence),
 $dW^i_t dt \rightarrow 0$  and $dW^i_t \,dW^i_t    \longrightarrow b^2 dt$ we obtain
\[
\mathbf{dX} \cdot \mathbf{H}(F) \cdot \mathbf{dX}^T = b^2 dt \nabla^2 F
\]
in Geometric Algebra language.
\end{remark}

\begin{theorem}[Cole-Hopf linearization]\label{th:colhopf1}
	The complex godesic equation
	\[
{\partial_t} \mathcal{V} + \left( \mathcal{V} \cdot \nabla \right)  \mathcal{V} -\frac{ i b^2}{2} \nabla^2  \mathcal{V} =- \nabla U
	\]
	is linearised by the Cole-Hopf transform
	\[
	\mathcal{V}= -i b^2 \nabla \log{F}
	\]
	into the Schr\"odinger-type equation
	\begin{equation}\label{eq:schro1}
	i b^2 \partial_t F       =     -\frac{b^4}{2} \nabla^2 F +U F
	\end{equation}
\end{theorem}

\begin{proof}
Under so-identified assumptions, the drift equation becomes
\[
d \mathcal{V} = \left({\partial_t} \mathcal{V} + \left( \mathcal{V} \cdot \nabla \right)  \mathcal{V} -\frac{ i b^2}{2} \nabla^2  \mathcal{V} \right)   dt + \sqrt{-i} b\, \left(  d Z_t \cdot \nabla \right) \mathcal{V}
\]
Therefore, the geodesic equation reads
\[
{\partial_t} \mathcal{V} + \left( \mathcal{V} \cdot \nabla \right)  \mathcal{V} -\frac{ i b^2}{2} \nabla^2  \mathcal{V} =0
\]
Under the separate assumption of irrotational flow $\nabla \wedge \mathcal{V} =0 $,
a generalized, inhomogeneous Cole-Hopf transform can be introduced by analogy with the scalar case as
\[
\mathcal{V} =
 \mathcal{C}(\nabla) \log{F}   = \frac{\mathcal{C}(\nabla) F}{F}
\]
using the stretched gradient.
If $F=0$ then trivially $dF=0$.
Without loss of generality, assume $F>0$.
Under the above substitution using Props. \ref{prop:comm1} and \ref{prop:comm2}
\begin{multline*}
{\partial_t}  \mathcal{C}(\nabla) \log{F}   + \left(  \mathcal{C}(\nabla) \log{F}   \cdot \nabla \right)   \mathcal{C}(\nabla) \log{F}   -\frac{ i b^2}{2} \nabla^2   \mathcal{C}(\nabla) \log{F}   = \\
\mathcal{C}(\nabla) {\partial_t} \log{F}
+ \frac{1}{2} \nabla \left( \mathcal{C}(\nabla) \log{F} \right) ^2
 -\frac{ i b^2}{2}   \mathcal{C}(\nabla) \nabla^2 \log{F}
=0
\end{multline*}
On the other hand,
\[
\nabla^2 \log{F} = \frac{\nabla^2 F}{F} - \frac{\left( \nabla F \right) ^2}{F^2}
\]
Therefore,
\[
 \mathcal{C}(\nabla) \nabla^2 \log{F} =  \mathcal{C}(\nabla) \frac{\nabla^2 F}{F} -  \mathcal{C}(\nabla) \frac{\left( \nabla F \right) ^2}{F^2}
 = \mathcal{C}(\nabla) \frac{\nabla^2 F}{F} -  \mathcal{C}(\nabla)
 \left( \nabla \log{F} \right) ^2
\]
Therefore,
\begin{multline*}
\frac{1}{2} \nabla \left( \mathcal{C}(\nabla) \log{F} \right) ^2  -\frac{ i b^2}{2}   \mathcal{C}(\nabla)\nabla^2 \log{F}= \\
\frac{1}{2} \nabla \left( \mathcal{C}(\nabla) \log{F} \right) ^2
-\frac{ i b^2}{2} \mathcal{C}(\nabla) \frac{\nabla^2 F}{F} + \frac{ i b^2}{2}  \mathcal{C}(\nabla)
\left( \nabla \log{F} \right) ^2
\end{multline*}
Finally, we obtain
\[
  \mathcal{C}(\nabla) \left(  \partial_t \log{F} -\frac{ i b^2}{2}     \frac{\nabla^2 F}{F} \right) +\frac{1}{2}\left(
   \nabla \left( \mathcal{C}(\nabla) \log{F} \right) ^2
  +  { i b^2}  \mathcal{C}(\nabla)
  \left( \nabla \log{F} \right) ^2 \right)   = 0
\]
Therefore, for an exact linearisation, the following equation must hold
\[
\frac{1}{2}\left(
\nabla \left( \mathcal{C}(\nabla) \log{F} \right) ^2
+  { i b^2}  \mathcal{C}(\nabla)
\left( \nabla \log{F} \right) ^2 \right)=0
\]
Therefore, one obtains an algebraic system of equations for the coefficients of the stretched gradient in function of the diffusion constant.
Let $\log F =u$, so that in components the equation reads
 \[ u= \log{F}, \quad
e^i \left( \partial_{x_i} c_i^2 \left( \partial_{x_i} u \right) ^2 + i b^2 c_i \partial_{x_i} \left( \partial_{x_i} u \right) ^2 \right) =
e^i \left(  c_i^2   + i b^2 c_i  \right)\partial_{x_i} \left( \partial_{x_i} u \right) ^2=0
\]
Therefore, the scaling is homogenous so the coefficient can be relabeled as  $c_i \equiv \lambda$ and
\[
\lambda^2 +i b^2 \lambda =0
\]
Therefore, $ \lambda =-i b^2$.
Direct calculation verifies the identity:
\begin{multline*}
\nabla \left( \mathcal{C}(\nabla) \log{F} \right)^2 + i b^2 \mathcal{C}(\nabla)  \left( \nabla \log{F} \right) ^2
= (-i b^2)^2 \nabla \left( \nabla \log{F} \right)^2+ i b^2 (-i b^2) \nabla \left( \nabla \log{F} \right)^2 \\
=\left(  (-i b^2)^2 +  i b^2 (-i b^2)\right) \nabla \left( \nabla \log{F} \right)^2=b^4\left(\left( -i\right) ^2 - i ^2\right) \nabla \left( \nabla \log{F} \right)^2=0
\end{multline*}
for a real constant scalar \textit{b}.
Therefore,  exact linearisation is possible and
\begin{multline*}
 \mathcal{C}(\nabla) \left(  \partial_t \log{F} -\frac{ i b^2}{2} \frac{\nabla^2 F}{F} \right) =
-i b^2 \nabla \left(  \partial_t \log{F} -\frac{ i b^2}{2}  \frac{\nabla^2 F}{F} \right)= \\ \nabla \left(  \frac{ 1 }{F}\left( -i b^2 \partial_t F -\frac{b^4}{2} \nabla^2F \right)  \right) =0
\end{multline*}
Nothing in the present derivation depends on  the properties of the right-hand side (RHS) of the equation.
The left-hand side can be equated to a potential gradient from the RHS $-\nabla U $, representing physically a central force.
Therefore,
\[
\nabla \left(  \frac{ 1 }{F}\left( -i b^2 \partial_t F -\frac{b^4}{2} \nabla^2F \right)  \right) = -\nabla U
\]
So that

$$
   i b^2 \partial_t F       =     -\frac{b^4}{2} \nabla^2 F +U F
$$

and we recognize the form of the Schr\"odinger equation.
\end{proof}

\begin{corollary}
	In the geodesic setting, the drift equation reads
	\[
	d \mathcal{V} = \sqrt{-i} b \left( dZ_t \cdot \nabla \right) \mathcal{V}
	\]
	Under so-identified Cole-Hopf transform the diffusion term transforms as
	\[
\sqrt{-i} b\; dZ_t \cdot \nabla  \mathcal{C}(\nabla) \log{F} =	-i \sqrt{-i} b^3 \; \left( dZ_t \cdot \nabla\right) \nabla \log{F}
= -i \sqrt{-i} b^3 \; \nabla \left( dZ_t \cdot \left(  \nabla  \log{F}\right) \right)
	\]
\end{corollary}

\subsection{The Conjugated Shr\"odinger  Equation}\label{sec:conjug}
The conjugated Shr\"odinger equation can be derived in the same way.

\begin{theorem}
	The complex geodesic equation
	\[
 {\partial_t} \mathcal{U} + \left( \mathcal{U} \cdot \nabla \right)  \mathcal{V} +\frac{ i b^2}{2} \nabla^2  \mathcal{U} = - \nabla U
	\]
	can be linearised into
	\begin{equation}\label{eq:schro2}
	i b^2 \partial_t G -\frac{b^4}{2} \nabla^2 G   +  U G =0
	\end{equation}
	where
	\[
	G = F^*, \quad \mathcal{U}= i b^2 \nabla \log{G}
	\]
\end{theorem}

\begin{proof}

Starting from the It\^o formula
	\[
	d F = \left(  \partial_t F + \left( \mathcal{U} \cdot  \nabla \right)  F   + \frac{ i b^2}{2}  \nabla^2    F \right)   dt + \sqrt{i} b  \left( d Z_t \cdot \nabla   \right)  F
	\]
The drift equation  becomes
\[
d \mathcal{U} = \left({\partial_t} \mathcal{U} + \left( \mathcal{U} \cdot \nabla \right)  \mathcal{V} +\frac{ i b^2}{2} \nabla^2  \mathcal{U} \right)   dt + \sqrt{i} b\, \left(  d Z_t \cdot \nabla \right) \mathcal{U}
\]
Therefore, the geodesic equation reads
\[
{\partial_t} \mathcal{U} + \left( \mathcal{U} \cdot \nabla \right)  \mathcal{U} +\frac{ i b^2}{2} \nabla^2  \mathcal{U} =0
\]
Therefore, using the procedure as above we obtain the linearisation condition
\[
\lambda^2 - ib^2 \lambda =0, \quad c_i=\lambda
\]
for the transform
\[
\mathcal{U}= i b^2 \nabla \log{G}
\]
Finally,
\[
\nabla \left(  \frac{ 1 }{G}\left( i b^2 \partial_t G -\frac{b^4}{2} \nabla^2 G \right)  \right) = -\nabla U
\]
and
\[
  i b^2 \partial_t G -\frac{b^4}{2} \nabla^2 G   +  U G =0
\]
Therefore, we can identify $G= F^*$.
\end{proof}

\section{The Complex  Fokker-Planck equation implies the Born rule}\label{sec:born}
The key to the subsequent derivation is the use the Schr\"odinger equation and its conjugate on equal grounds.

A complex Fokker-Planck equation can be introduced based on the reversibility of the process.

\begin{theorem}[Complex Fokker-Planck equation]
	The pair of Fokker-Planck equations for the real-valued processes
	\begin{flalign}
	{\partial_t} \rho  +   \nabla \left(a \rho \right) -\frac{1}{2} \nabla^2 \left( b^2 \rho \right) =0 \\
	{\partial_t} \rho  +   \nabla \left(\hat{a} \rho \right) +\frac{1}{2} \nabla^2 \left( b^2 \rho \right) =0
	\end{flalign}
	implies the complex Fokker-Planck equation and its conjugate
	\begin{align}\label{eq:compfp}
	\partial_t \rho + \nabla \cdot \left( \mathcal{V} \rho \right) +\frac{i }{2} \nabla^2 \left( b^2 \rho \right) &=0 \\
	\partial_t \rho + \nabla \cdot \left( \mathcal{U} \rho \right) -\frac{i }{2} \nabla^2 \left( b^2 \rho \right) &=0, \quad \mathcal{U}= \mathcal{V}^*
	\end{align}
	for the It\^o-Nottale process.
\end{theorem}
\begin{proof}

Starting from the  Fokker-Planck equations for the forward and backward processes
\begin{flalign*}
{\partial_t} \rho  +   \nabla \left(a \rho \right) -\frac{1}{2} \nabla^2 \left( b^2 \rho \right) =0 \\
{\partial_t} \rho  +   \nabla \left(\hat{a} \rho \right) +\frac{1}{2} \nabla^2 \left( b^2 \rho \right) =0
\end{flalign*}
and taking sums and differences we obtain
\begin{flalign*}
2 {\partial_t} \rho  +   \nabla \left( \left( a +\hat{a}  \right) \rho     \right)  = 2 {\partial_t} \rho  +   2 \nabla \left(  v \rho     \right)= 0 \\
     \nabla \left( \left( a -\hat{a}  \right) \rho \right)  -   \nabla^2 \left( b^2  \rho \right) =2\nabla  \left( u \rho \right) -  \nabla^2 \left( b^2  \rho \right) =  0 \\
\end{flalign*}
Therefore, we can formulate a pair of Fokker-Planck equations for the complex velocity and its conjugate as
\begin{align*}
\partial_t \rho + \nabla \cdot \left( \mathcal{V} \rho \right) +\frac{i }{2} \nabla^2 \left( b^2 \rho \right) &=0 \\
\partial_t \rho + \nabla \cdot \left( \mathcal{U} \rho \right) -\frac{i }{2} \nabla^2 \left( b^2 \rho \right) &=0, \quad \mathcal{U}= \mathcal{V}^*
\end{align*}

\end{proof}

Finally, the Born rule can be derived as simple consequence of the complex Fokker-Planck equations.

\begin{theorem}[Born's rule]\label{th:born}
	Suppose that the above complex Fokker-Planck equations \ref{eq:compfp} hold.
	Then
	\[
	\rho= F F^{\star}
	\]
	where $F$ and $F^{\star}$ are solutions of the Schr\"odinger-type equations \ref{eq:schro1} and \ref{eq:schro2}.
 	Moreover,
 	\[
 	F = \sqrt{\rho} e^{-i S}
 	\]
 	for an analytic phase function $S(r, t)$.
\end{theorem}

\begin{proof}
	We use the same notation as in the proof above.
Subtracting the two equation leads to
\[
\nabla \left( \mathcal{V} \rho - \mathcal{U} \rho \right)  + i \nabla^2 b^2 \rho =
\nabla \cdot \left( \left( \mathcal{V}- \mathcal{U}  \right)    \rho  + i \nabla b^2 \rho \right)  =0
\]

As shown in the previous section, the Cole-Hopf transform can be specialized to
\[
\mathcal{V} =-i  b^2 \nabla \log{F h(t)}, \quad \mathcal{U}=\mathcal{V}^* =i  b^2 \nabla \log{F^* h^*(t)},
\]
up to an arbitrary analytic function of time $h(t)$.
Then it follows that
\[
\mathcal{V}- \mathcal{U}  = -i b^2\nabla \log{F} - i b^2\nabla \log{F^*}= -i b^2 \nabla \log{F F^*}
\]
Therefore,
\[
\nabla \cdot \left(-i b^2 \left(\nabla \log{ F F^* h h^*  }\right)    \rho  + i \nabla b^2 \rho\right)
= i \nabla \cdot \left( \rho \, b^2 \nabla \log{\frac{F F^* h h^*}{\rho \, b^2}} \right) =0
\]
Since, in general, $\rho$ is a function of the position it follows that
\[
 \nabla \log{\frac{F F^* h (t) h^*(t)}{\rho}}=0
\]
must hold.
Therefore, in general,
\[
\rho= F F^* f(t)
\]
where, $f(t)$ is a smooth function of time.
Since, $h$ is arbitrary but analytic function, we can choose
\[
h (t) = e^{i  g(t)}
\]
where $g(t)$ is another smooth function.

Therefore,
\[
F F^* =\rho
\]
which is the statement of the Born rule!

Therefore, one can write $F$ in the form
\[
F = \sqrt{\rho} e^{-i S} \equiv \psi
\]
for an analytic phase function $S$.

Consider, on the other hand, the case where
\[
\log{\frac{F F^* h (t) h^*(t)}{\rho}} = \log{q(t)} \Longrightarrow
F F^* h (t) h^*(t)= q(t) \rho
\]
for a given positive and continuous function $q(t)$.
By integration over 3D space
\[
\int_{\Omega} F F^* h (t) h^*(t) d \omega= \int_{\Omega}  q(t) \rho \, d \omega \Longrightarrow
q(t) = h (t) h^*(t) \int_{\Omega}  F F^* d \omega
\]
Therefore, we can transform $F$ as
\[
F^\prime = F \frac{ h(t)}{\sqrt{q(t)}}
\]
so that the normalization
\[
\int_{\Omega}  F^\prime F^{\prime*} d \omega =1
\]
holds.
Therefore, also in this case $F^\prime$ can be interpreted in agreement with the Born's rule.
\end{proof}

\section{Concluding Remarks}\label{sec:disc}

This work was motivated in part by the premise that inherently non-linear phenomena need development of novel mathematical tools for their description.
The second motivation of the present work was to investigate the potential of stochastic methods to represent quantum-mechanical and convection-diffusive systems.

The augmented, in terms of white noise,  Newtonian dynamic leads to the stochastic geodesic equation for the drift velocity, which can be recognized as the Burgers equation.
If in addition one assumes also path-wise reversibility, this leads to a stochastic description in terms of a pair of Burgers equations.
This pair can be put into correspondence with a Schr\"odinger equation and its conjugate for a wave-function by means of the vectorized Cole-Hopf transform.
The use of the Fokker-Planck equation together with the Cole-Hopf transform leads to the Born rule for the wave function.

The complex structure, and hence, the Schr\"odinger equation can be considered as an ingenious and economical description of the studied phenomena, however such complex structure is not necessarily fundamental.
This line of reasoning strongly points out towards the universal but epistemological (!) character of the Schr\"odinger equation and its unitary evolution.
The Born's rule stems from the time-reversibility of the modelled diffusion processes and does not need to be postulated separately.
This corresponds with the time-reversibility of the classical physics kinematics.

Moreover, nowhere in these developments have we assumed anything particular about a "scale" of observations.
Therefore, one can reasonably argue that quantum-like phenomena are not confined only to the nanoscale, but in fact can be observed as emerging phenomena on any scale of study.

\appendix

\section{Appendices}

\subsection{The Stochastic Variation Problem}\label{sec:stochvar}

The study of stochastic Lagrangian variational principles has been motivated initially by quantum mechanics and  optimal control problems.
This section gives only sketch for the treatment of the problem.
The reader is directed to \cite{Zambrini1986,Yasue1981a, Pavon1995} for more details.
In the simplest form this is the minimization of the regularized functional assuming a constant diffusion coefficient $b$.

\begin{definition}\label{def:part}
	Consider the interval $I=[a,b]$.
	A partition of $I$ is a set of $n$ numbers $ \partd_n [I]: = (a < x_1 \ldots x_{n-1} < b ) $.
\end{definition}

\begin{definition}
Let $\alpha \in \{0,1\}$. Define
\[
S_\alpha (X | t_0, T) := \llim{N}{\infty}{} \mathbb{E}\left( \left. (\mathcal{P}_N) \sum\limits_{t=t_0}^{t=T} \frac{1}{2} \frac{ \left( \Delta X_k \right)^2  }{ \Delta t_k  }  - \sigma \left(  \alpha - \frac{1}{2} \right)    b^2  \right| X_k = x (\alpha t_k   + (1-\alpha )  t_{k+1}) \right)
\]
for the sequence of partitions $\mathcal{P}_N \subset \mathcal{P}_{N+1} \in \mathbb{F}_\alpha $ and $ \sigma $ denoting the sign of the argument,
where $\mathbb{F}_0 =\mathbb{F}_{t>0}$ and 	$\mathbb{F}_1 =\mathbb{F}_{t<T}$.
\end{definition}

On the first place, suppose that $\alpha =1$ and $N$ is finite.
The expectation operator and the finite summation commute so
\begin{multline*}
\mathbb{E}\left( \left. (\mathcal{P}_N) \sum\limits_{t=t_0}^{t=T} \frac{1}{2} \frac{ \left( \Delta X_k \right)^2  }{ \Delta t_k  }  - \sigma \left(  \alpha - \frac{1}{2} \right)    b^2  \right| X_k = x (\alpha t_k   + (1-\alpha )  t_{k+1}) \right)
= \\
\left. (\mathcal{P}_N) \sum\limits_{t=t_0}^{t=T} \frac{1}{2} \mathbb{E}\left( \frac{ \left( \Delta X_k \right)^2  }{ \Delta t_k  }  - \sigma \left(  \alpha - \frac{1}{2} \right)    b^2  \right| X_k = x (\alpha t_k   + (1-\alpha )  t_{k+1}) \right)
\end{multline*}
Then the increments can be interpreted as It\^o  integrals so that by the It\^o isometry since finite summation and integration commute
\begin{multline*}
\mathbb{E}\left( \left. \frac{1 }{ 2  \Delta t_k   } \left( \Delta X_k \right)^2  - \frac{1}{2}     b^2 \right| X_k =x(t_k) \right) = \\
\frac{1}{2 \Delta t_k }\left( \int_{t_k}^{t_{k+1}} a  ds \right) ^2 + \frac{1 }{    \Delta t_k   }\left( \int_{t_k}^{t_{k+1}} a  ds\right)  \mathbb{E}\left(   \int_{t_k}^{t_{k+1}} b dw \right)
+  \frac{1}{2 \Delta t_k} \mathbb{E} \left( \int_{t_k}^{t_{k+1}} b  dw \right) ^2 - \frac{1}{2}     b^2 = \\
\frac{1}{2 \Delta t_k  }\left( \int_{t_k}^{t_{k+1}} a  ds \right) ^2 + \frac{1}{2 \Delta t_k} \int_{t_k}^{t_{k+1}} b^2  ds - \frac{1}{2} b^2  =  a (t^*)  \int_{t_k }^{t_{k+1}} a  ds, \quad t^* \in (t_k, t_{+1})
\end{multline*}
by the Middle Value Theorem, where we use the It\^o isometry
\[
\mathbb{E} \left( \int_{t_k}^{t_{k+1}} b  dw \right) ^2 = \mathbb{E} \left(  \int_{t_k}^{t_{k+1}} b^2  dt\right)
\]
Therefore, $S_\alpha (t_0, T) $ is minimal if the drift vanishes on  $\mathcal{P}_N$.
Suppose that $X_t$ is varied by a small smooth function $\lambda \phi(t,x)$, where the smallness is controlled by $\lambda$, then the It\^o lemma should be applied so that
$ \mathbb{E} (d \delta X_t | \mathcal{F} ) =0$ on the difference process $ \delta X_t = \lambda \phi(t,x) dt + b dW_t $.
Therefore,
\begin{equation}
\mathbb{E} \left(  d\phi | \mathcal{F} \right)  =  \lambda dt \left( {\partial_t} \phi  + \phi \ {\partial_x} \phi  + \frac{b^2}{2}  {\partial_{xx}} \phi \right)  =0
\end{equation}
should hold.
The same calculation can be performed for  $\alpha = 0$ if the It\^o  integral is replaced by the anticipative  It\^o  integral.
In this case, $\sigma=-1$ and the integration is reversed
\begin{multline*}
\mathbb{E}\left( \left. \frac{1 }{ 2  \Delta t_k   } \left( \Delta X_k \right)^2  + \frac{1}{2}     b^2 \right| X_k =x(t_{k+1}) \right) = \\
\frac{1}{2 \Delta t_k }\left( \int_{t_{k+1}}^{t_{k}} a  ds \right) ^2 + \frac{1 }{    \Delta t_k   }\left( \int_{t_{k+1}}^{t_{k}} a  ds\right)  \mathbb{E}\left(   \int_{t_{k+1}}^{t_{k}} b dw \right)
+  \frac{1}{2 \Delta t_k} \mathbb{E} \left( \int_{t_{k+1}}^{t_{k}} b  dw \right) ^2 + \frac{1}{2}     b^2 = \\
\frac{1}{2 \Delta t_k  }\left( \int_{t_{k+1}}^{t_{k}} a  ds \right) ^2 + \frac{1}{2 \Delta t_k} \int_{t_{k+1}}^{t_{k}} b^2  ds + \frac{1}{2} b^2  =  a(t^*)  \int_{t_k }^{t_{k+1}} a  ds, \quad t^* \in (t_k, t_{+1})
\end{multline*}
In this case also the backward It\^o formula applies as
\begin{equation}
\mathbb{E} \left(  d\phi | \mathcal{F} \right)  =  \lambda dt \left( {\partial_t} \phi  + \phi  {\partial_x} \phi  - \frac{b^2}{2}  {\partial_{xx}} \phi \right)  =0
\end{equation}

\begin{remark}
The treatment of Pavon \cite{Pavon1995} uses the symmetrized functional
$S = S_0 + S_1$ together with a constraint on the anti-symmetrized functional  $S_0 - S_1$
in the present notation.
\end{remark}
The complex geodesic principle is achieved in a more parsimonious way by defining the quantity

\begin{definition}
	Define
\[
\mathcal{L} ( \mathcal{X}| t_0, T) := \llim{N}{\infty}{} \mathbb{E}\left( \left. (\mathcal{P}_N) \sum\limits_{t=t_0}^{t=T} \frac{1}{2} \frac{ \left( \Delta \mathcal{X}_k \right)^2  }{ \Delta t_k  }     \right| \mathcal{X}_k = \mathcal{X} (  t_k  ) \right)
\]
for the sequence of partitions $\mathcal{P}_N \subset \mathcal{P}_{N+1} \in \mathbb{F}^2 $ and the process
\[
\mathcal{X}_t =\int^t_0 \mathcal{V}  ds + \int^t_0 \sqrt{-i}\, b \, dZ_s
\]
\end{definition}

Then, in a simlar way, we take complex It\^o integrals
\begin{multline*}
\mathbb{E}\left( \left. \frac{1 }{ 2  \Delta t_k   } \left( \Delta \mathcal{X}_k \right)^2   \right| \mathcal{X}_k =x(t_{k+1}) \right) = \\
\frac{1}{2 \Delta t_k }\mathbb{E}\left( \int_{t_{k+1}}^{t_{k}} \mathcal{V}  ds \right) ^2 + \frac{ \sqrt{-i}\, b}{ \Delta t_k }\left( \int_{t_{k+1}}^{t_{k}} \mathcal{V}  ds\right) \mathbb{E}\left(   \int_{t_{k+1}}^{t_{k}}  dZ_s \right)
+  \frac{-  i\, b^2 }{2 \Delta t_k} \mathbb{E} \left( \int_{t_{k+1}}^{t_{k}} dZ_s \right) ^2  = \\
\frac{1}{2 \Delta t_k  }\left( \int_{t_{k+1}}^{t_{k}} \mathcal{V}  ds \right) ^2   =  \mathcal{V}  (\tau)  \int_{t_{k}}^{t_{k+1}} \mathcal{V} ds , \quad \tau \in (t_k, t_{+1})
\end{multline*}
where we use the result
\[
\mathbb{E} \left( \int_{t_{k+1}}^{t_{k}}   dZ \right) ^2 = \mathbb{E} \, \Delta Z_k^2=
 \mathbb{E} \,  Z_k^2 + \mathbb{E} \,  Z_{k+1}^2 + 2 \mathbb{E} \,  Z_{k} Z_{k+1}
 =  \mathbb{E} \,  Z_k^2 + \mathbb{E} \,  Z_{k+1}^2 =0
\]
since
\[
 \mathbb{E}  Z_t^2 = \mathbb{E} \left(  W_t^2 - \hat{W}^2_t +  i  W_t \hat{W}_t\right) =  \mathbb{E} \left(  W_t^2 - \hat{W}^2_t \right) + i \mathbb{E}   \left(  W_t \hat{W}_t \right) = 0
\]
by the independence of the Brownian motions.
Therefore, the expectation of complex drift variation $ \delta \mathcal{X}_t= \lambda \psi dt + \sqrt{-i} b dZ_t $ must vanish  as well so that
\[
\mathbb{E} \left(  d\psi | \mathcal{F} \right)  =  \lambda dt \left( {\partial_t} \psi  + \psi  {\partial_x} \psi  - i \frac{b^2}{2}  {\partial_{xx}} \psi \right)  =0
\]
Therefore, in 3 dimensions it is immediately generalized to
\[
\mathbb{E} \left(  d\psi | \mathcal{F} \right)  =  \lambda dt \left( {\partial_t} \psi  + \left( \psi \cdot \nabla \right) \psi  - i \frac{b^2}{2} \nabla^2 \psi \right)  =0
\]

Moreover, using the notation of mean derivatives
\[
\mathcal{L} ( \mathcal{X}| 0, T) = \int_{0}^{T} \ \left( \mathcal{D}_t  \mathcal{X}\right) ^2 dt
\]
and
\[
S_{0,1} (X |  0, T)  = \int_{0}^{T} \ \left( \mathcal{D}_t^{\pm}   {X}  ^2 \mp b^2 \right) dt
\]
by the Fubini's theorem.
Therefore, the complex variational problem is homeomorphic to the pair of real-valued variational problems, as expected.

\bibliographystyle{plain}
\bibliography{posterbib1}

\end{document}


TITLE: Probability a study rejects the null hypothesis
QUESTION [0 upvotes]: I have the following question but am not too sure how to approach it:
"A researcher is interested in knowing whether early childhood education has an impact on graduate earnings. She finds 9 published studies that have tested the null hypothesis that there is no impact. Since the studies were conducted at different times in different countries, it is reasonable to assume that the samples are statistically independent.
Assuming that there is in fact no relationship between early childhood education and graduate earnings, what is the probability that at least one of the 9 studies will reject the null hypothesis at a 5% significance level?"
My working out is as follows but I'm not sure if my approach is correct. Could someone please help me out?
"Significance level = 5%
P(She rejects the null | no relationship exists) = 0.05
P(She does not reject the null | no relationship exists) = 1-0.05 = 0.95
Number of studies: n = 9
P(At least one is rejected) = 1 - P(None are rejected)
P(None are rejected) = [P(She does not reject the null | no relationship exists)]^9
                 = 0.95^9 = 0.63024 (5d.p.)

P(At least one is rejected) = 1 - 0.63024
                        = 0.36975 (5d.p.) = 36.975%"

Any help is appreciated.

REPLY [1 votes]: Your work is correct.  The question is framed in terms of multiple hypothesis tests (familywise Type I error control), but in fact, it is nothing more than a question in elementary probability.
If any single test has a Type I error of $\alpha = 0.05$, then the chance that all $9$ independent tests will not reject the null when the null is true is simply $(1 - 0.05)^9$.  Consequently, the probability that at least one of the tests will incorrectly reject the null is $1 - (1 - 0.05)^9$.


text:
\begin{document}

\maketitle

\def\thefootnote{\fnsymbol{footnote}}
\footnotetext[1]{Subdirecci{\' o}n de Ingenier{\' i}a y Posgrado, Universidad Aeronáutica en Quer\'etaro, Parque Aeroespacial de Quer\'etaro, 76278, Quer\'etaro, M\'exico, {\tt adrian.vazquez@unaq.edu.mx}.}

\begin{abstract}
A linear system is a pair $(P,\mathcal{L})$ where
$\mathcal{L}$ is a family of subsets on a ground finite set $P$ such that $|l\cap l^\prime|\leq 1$, for every $l,l^\prime \in \mathcal{L}$. The elements of $P$ and $\mathcal{L}$ are called points and lines, respectively, and the linear system is called intersecting if any pair of lines intersect in exactly one point.
A subset $D$ of points of a linear system $(P,\mathcal{L})$ is a dominating set of $(P,\mathcal{L})$ if for every $u\in P\setminus D$ there exists $v\in D$ such that $u,v\in l$, for some $l\in\mathcal{L}$. The cardinality of a minimum dominating set of a linear system $(P,\mathcal{L})$ is called domination number of $(P,\mathcal{L})$, denoted by $\gamma(P,\mathcal{L})$. On the other hand, a subset $R$ of lines of a linear system $(P,\mathcal{L})$ is a 2-packing if any three elements of $R$ do not have a common point (are triplewise disjoint). The cardinality of a maximum 2-packing of a linear system $(P,\mathcal{L})$ is called 2-packing number of $(P,\mathcal{L})$, denoted by $\nu_2(P,\mathcal{L})$.

It is known that for intersecting linear systems $(P,\mathcal{L})$ of rank $r$ it satisfies $\gamma(P,\mathcal{L})\leq r-1$. In this note, we prove if $q$ is an even prime power and $(P,\mathcal{L})$ is an intersecting linear system of rank $q+2$ satisfying $\gamma(P,\mathcal{L})=q+1$, then this linear system can be constructed from a spanning $(q+1)$-uniform intersecting linear subsystem $(P^\prime,\mathcal{L}^\prime)$ of the projective plane of order $q$ satisfying $\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)-1=q+1$.

\end{abstract}

{\bf Keywords.} Linear systems, domination,transversal, 2-packing, projective plane.

{\bf Math. Subj. Class.} 05C65, 05C69

\section{Introduction}\label{sec:intro}

A \emph{set system} is a pair $(X,\mathcal{F})$ where $
\mathcal{F}$ is a finite family of subsets on a ground set $X$. A
set system can be also thought of as a hypergraph, where the elements of $X$ and $\mathcal{F}$ are called \emph{vertices} and \emph{hyperedges}, respectively. The set system $(X,\mathcal{F})$ is \emph{intersecting} if $E\cap F\neq\emptyset$, for for every pair of distinct subsets $E,F \in \mathcal{F}$. On the other hand, the set system $(X,\mathcal{F})$ is a \emph{linear system} if it satisfies $|E\cap F|\leq 1$, for every pair of distinct subsets $E,F \in \mathcal{F}$; and it is denoted by $(P,\mathcal{L})$. The elements of $P$ and $\mathcal{L}$ are called \emph{{\tiny }points} and \emph{lines} respectively. For the remainder of this work we will only consider linear systems, and most of the following definitions can be generalized for set systems.

The \emph{rank} of a linear system is the maximum size of a line. An \emph{$r$-uniform} li\-near system is a linear system such that all lines contains exactly $r$ points. In this context, a simple graph is a 2-uniform linear system. Throughout this paper, we will only consider linear systems of rank $r\geq2$.

Let $(P,\mathcal{L})$ be a linear system and $p\in P$ be a point. The \emph{degree} of $p$ is the number of lines containing $p$, demoted by $deg(p)$, the maximum degree overall points of the linear systems is denoted by $\Delta(P,\mathcal{L})$. A point of degrees $2$ and $3$ is called \emph{double point} and \emph{triple point} respectively. A point of degree zero is called an \emph{isolated point}. Two points $p,q\in P$ are \emph{adjacent} if there is a line $l\in\mathcal{L}$ such that $p,q\in l$.

A {\emph{linear subsystem}} $(P^{\prime },\mathcal{L}^{\prime })$ of
a linear system $(P,\mathcal{L})$ satisfies that for any line $l^\prime\in\mathcal{L}^\prime$ there exists a line $l\in\mathcal{L}$
such that $l^\prime=l\cap P^\prime$. The \emph{linear subsystem induced} by a set of lines $\mathcal{L}^{\prime}\subseteq \mathcal{L}$ is the linear subsystem
$(P^{\prime },\mathcal{L}^{\prime })$ where
$P^{\prime}=\bigcup_{l\in \mathcal{L}^{\prime }} l$. The linear subsystem   $(P^\prime,\mathcal{L}^\prime)$ of $(P,\mathcal{L})$ is called \emph{spanning linear subsystem} if $P^\prime=P$. Given a linear system $(P,\mathcal{L})$, and a point $p\in P$, the linear system obtained from $(P,\mathcal{L})$ by \emph{deleting point $p$} is the linear system $(P^{\prime },\mathcal{L}^{\prime })$ induced by $\mathcal{L}^{\prime }=\{l\setminus \{p\}: l\in \mathcal{L}\}$. Given a linear system $(P,\mathcal{L})$ and a line $l\in \mathcal{L}$, the linear system obtained from $(P,\mathcal{L})$ by \emph{deleting the line $l$} is the linear system $(P^{\prime },\mathcal{L}^{\prime })$ induced by $\mathcal{L}^{\prime }=
\mathcal{L}\setminus \{l\}$. Let $(P^{\prime},\mathcal{L}^{\prime })$ and $(P,\mathcal{L})$ be two linear systems. $(P^{\prime},\mathcal{L}^{\prime })$ and $(P,\mathcal{L})$ are isomorphic, denoted by $(P^{\prime },\mathcal{L}^{\prime })\simeq(P,\mathcal{L})$, if after deleting points of degree 1 or 0 from both, the systems $(P^{\prime},\mathcal{L}^{\prime })$ and $(P,\mathcal{L})$ are isomorphic as hypergraphs, see \cite{MR3727901}.

A subset $D$ of points of a linear system $(P,\mathcal{L})$ is a \emph{dominating set} of $(P,\mathcal{L})$ if for every $u\in P\setminus D$ there exists $v\in D$ such that $u$ and $v$ are adjacent. The minimum cardinality of a dominating set of a linear system $(P,\mathcal{L})$ is called \emph{domination number}, and it is denoted by $\gamma(P,\mathcal{L})$. Domination in set systems was introduced by Acharya \cite{Acharya} and studied further in \cite{Acharya2,Arumugam,Buj,Dong,Jose}. A subset  $T$ of points of a linear system $(P,\mathcal{L})$ is a \emph{transversal} of $(P,\mathcal{L})$ (also called \emph{vertex cover} or \emph{hitting set}) if $T\cap l\neq\emptyset$, for every line $l\in\mathcal{L}$. The minimum cardinality of a transversal of a linear system $(P,\mathcal{L})$ is called \emph{transversal number}, and it is denoted by $\tau(P,\mathcal{L})$. On the other hand, a subset $R$ of lines of a linear system $(P,\mathcal{L})$ is a \emph{$2$-packing} of $(P,\mathcal{L})$ if the elements of $R$ are triplewise disjoint, that is, if three elements are chosen in $R$ then they are not incidents in a common point. The \emph{2-packing number} of $(P,\mathcal{L})$ is the maximum cardinality of a 2-packing of $(P,\mathcal{L})$ and it is  denoted by $\nu_2(P,\mathcal{L})$. Transversals and 2-packings in linear systems was studied in \cite{CGCA,CCA,CA,MR3727901}, while domination and 2-packing in simple graphs was studied in \cite{Avila}.

In \cite{Kang} it was proved that, if $(P,\mathcal{L})$ is an intersecting linear system of rank $r\geq2$, then $\gamma(P,\mathcal{L})\leq r-1$. In \cite{Shan} a characterization of set systems $(X,\mathcal{F})$ holding the equality when $r=3$ was given. On the other hand, in \cite{Dong} it was shown that all intersecting linear systems $(P,\mathcal{L})$ of rank 4 satisfying $\gamma(P,\mathcal{L})=3$ can be constructed by the Fano plane. In this note, we prove if $q$ is an even prime power and $(P,\mathcal{L})$ is an intersecting linear system of rank $(q+2)$ satisfying $\gamma(P,\mathcal{L})=q+1$, then this linear system can be constructed from a spanning $(q+1)$-uniform intersecting linear subsystem $(P^\prime,\mathcal{L}^\prime)$ of the projective plane of order $q$ satisfying $\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)-1=q+1$. This result generalizes the main result given in \cite{Dong}.
\section{Previous results}

Let $\mathcal{I}_r$ be the family of linear systems $(P,\mathcal{L})$ of rank $r$ with $\gamma(P,\mathcal{L})=r-1$. To better understand the main result, we need the following:
\begin{lemma}\cite{Dong}\label{lemma:primero}
For every linear system $(P,\mathcal{L})\in\mathcal{I}_r$, there exists an $r$-uniform spanning linear subsystem $(P^*,\mathcal{L}^*)$ of $(P,\mathcal{L})$ such that every line in $\mathcal{L}^*$ contains one point of degree one.
\end{lemma}
It is denoted by $(P^*,\mathcal{L}^*)$ to be the $r$-uniform intersecting spanning linear subsystem of a linear system of $(P,\mathcal{L})\in\mathcal{I}_r$ obtained from the Lemma \ref{lemma:primero}. Further, let $(P^\prime,\mathcal{L}^\prime)$ be the $(r-1)$-uniform intersecting linear subsystem obtained from $(P^*,\mathcal{L}^*)$ by deleting the point of degree one of each line of $\mathcal{L}^*$, see \cite{Dong}.

\begin{lemma}\cite{Dong}\label{lema:igualdades}
For every linear system $(P,\mathcal{L})\in\mathcal{I}_r$ it satisfies $$\gamma(P,\mathcal{L})=\gamma(P^*,\mathcal{L}^*)=\tau(P^*,\mathcal{L}^*)=\tau(P^\prime,\mathcal{L}^\prime)=r-1.$$
\end{lemma}

\begin{lemma}\cite{Dong}\label{lemma:gradomaximo}
Let $(P,\mathcal{L})\in\mathcal{I}_r$ $(r\geq3)$ then every line of $(P^\prime,\mathcal{L}^\prime)$ has at most one point of degree 2 and $\Delta(P^\prime,\mathcal{L}^\prime)=r-1$.
\end{lemma}

\begin{lemma}\cite{Dong}\label{lemma:desigualdadesplanoproy}
Let $(P,\mathcal{L})\in\mathcal{I}_r$ $(r\geq3)$ then
\begin{center}
 $3(r-2)\leq|\mathcal{L}^\prime|\leq(r-1)^2-(r-1)+1$ and $|P^\prime|=(r-1)^2-(r-1)+1$,
\end{center}
and so $\gamma(P^\prime,\mathcal{L}^\prime)=1$.
\end{lemma}

\begin{theorem}\cite{CGCA}\label{teo:segundopaper}
Let $(P,\mathcal{L})$ be a linear system and $p,q\in P$ be two points such that $deg(p)=\Delta(P,\mathcal{L})$ and $deg(q)=\max\{deg(x): x\in
P\setminus\{p\}\}$. If $|\mathcal{L}|\leq deg(p)+deg(q)+\nu_2(P,\mathcal{L})-3$, then $\tau(P,\mathcal{L})\leq\nu_2(P,\mathcal{L})-1$.
\end{theorem}
\section{Main result}

In this section we prove, if $q$ is an even prime power and $(P,\mathcal{L})$ is an intersecting linear system of rank $(q+2)$ satisfying $\gamma(P,\mathcal{L})=q+1$, then this linear system can be constructed from a spanning $(q+1)$-uniform intersecting linear subsystem $(P^\prime,\mathcal{L}^\prime)$ of the projective plane of order $q$ satisfying $\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)-1=q+1$.

Recall that, a \emph{finite projective plane} (or merely \emph{projective plane}) is an uniform linear system satisfying that any pair of points have a common line, any pair of lines have a common point and there exist four points in general position (there are not three collinear points). It is well known that if $(P,\mathcal{L})$ is a projective plane then there exists a number $q\in\mathbb{N}$, called \emph{order of projective plane}, such that every point (line, respectively) of $(P,\mathcal{L})$ is incident to exactly $q+1$ lines (points, respectively), and $(P,\mathcal{L})$ contains exactly $q^2+q+1$ points (lines, respectively). In addition to this, it is well known that projective planes of order $q$, denoted by $\Pi_q$, exist when $q$ is a prime power. For more information about the existence and the unicity of projective planes see, for instance, \cite{B86,B95}. In \cite{MR3727901} it was proved, if $q$ is an even prime power then $\tau(\Pi_q)=\nu_2(\Pi_q)-1=q+1$, however, if $q$ is an odd prime power then $\tau(\Pi_q)=\nu_2(\Pi_q)=q+1$.

To prove the following lemma (Lemma \ref{lemma:igualdad}), we need a result shown in \cite{Avila2}. Therefore, we present the proof given in \cite{Avila2}.

	\begin{lemma}\cite{Avila2}\label{lemma:impar}
		Let $(P,\mathcal{L})$ be an $r$-uniform intersecting linear system with $r\geq2$ be an even integer. If $\nu_2(P,\mathcal{L})=r+1$ then $\tau(P,\mathcal{L})=\frac{r+2}{2}$.
\end{lemma}
\begin{proof}[\cite{Avila2}]
	Let $(P,\mathcal{L})$ be a $r$-uniform intersecting linear system with $\nu_2(P,\mathcal{L})=r+1$, where $r\geq2$ is a even integer. Let $R=\{l_1,\ldots,l_{r+1}\}$ be a maximum 2-packing of $(P,\mathcal{L})$. Since $(P,\mathcal{L})$ is an intersecting linear system then $l_i\cap l_j\neq\emptyset$, for $1\leq i<j\leq r+1$, and hence $|l_i|=r$, for $i=1,\ldots,r+1$. Let $l\in\mathcal{L}\setminus R$. Since $(P,\mathcal{L})$ is an intersecting linear system it satisfies $l\cap l_i=l\cap l_i\cap l_{j_i}\neq\emptyset$, for $i=1,\ldots,r+1$ and for some $j_i\in\{1,\ldots,r+1\}\setminus\{i\}$, however, by the pigeonhole principle there are a line $l_s\in R$ such that $l\cap l_s=\emptyset$, since there are an odd number of lines in $R$ and by linearity of $(P,\mathcal{L})$. Therefore $\mathcal{L}=R$ and $\tau(P,\mathcal{L})=\lceil \nu_{2}/2\rceil=\frac{r+2}{2}$ (see \cite{CA}).
\end{proof}

\begin{lemma}\label{lemma:igualdad}
Let $r\geq2$ be an even integer . For every $(P,\mathcal{L})\in\mathcal{I}_{r+2}$ it satisfies $\nu_2(P^\prime,\mathcal{L}^\prime)= r+2$. Hence $\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)-1$.
\end{lemma}
\begin{proof}Since $(P^\prime,\mathcal{L}^\prime)$ is an intersecting $(r+1)$-uniform linear system, then $|l^\prime|\geq\nu_2(P^\prime,\mathcal{L}^\prime)-1$, for any line $l^\prime\in\mathcal{L}^\prime$. Hence, $\nu_2(P^\prime,\mathcal{L}^\prime)\leq r+2$. If $r+2$ is odd integer, then by Lemma \ref{lemma:impar} it satisfies $\tau(P^\prime,\mathcal{L}^\prime)=\frac{r+3}{2}$, which is a contradiction, since $\tau(P^\prime,\mathcal{L}^\prime)=r+1$.

Let $p\in P^\prime$ such that $deg(p)=\Delta(P^\prime,\mathcal{L}^\prime)$, and let $\Delta^\prime(P^\prime,\mathcal{L}^\prime)=\max\{deg(x): x\in P^\prime\setminus\{p\}\}$. By Theorem \ref{teo:segundopaper} if $|\mathcal{L}^\prime|\leq\Delta(P^\prime,\mathcal{L}^\prime)+\Delta^\prime(P^\prime,\mathcal{L}^\prime)+\nu_2(P,\mathcal{L})-3\leq3r+1$ (see Lemma \ref{lemma:gradomaximo}) then $\tau(P^\prime,\mathcal{L}^\prime)\leq\nu_2(P^\prime,\mathcal{L}^\prime)-1$, which implies $\nu_2(P^\prime,\mathcal{L}^\prime)\geq r+2$, and the equality $\nu_2(P^\prime,\mathcal{L}^\prime)=r+2$ holds. Hence, by Lemma \ref{lemma:desigualdadesplanoproy}, if $|\mathcal{L}^\prime|=3r$ then $\nu_2(P^\prime,\mathcal{L}^\prime)=r+2$, which implies if $(P,\mathcal{L})\in\mathcal{I}_{r+2}$ then $\nu_2(P^\prime,\mathcal{L}^\prime)=r+2$, since if $(\hat{P},\mathcal{\hat{L}})$ is a spanning linear subsystem of $(P^\prime,\mathcal{L}^\prime)$ then $\nu_2(\hat{P},\mathcal{\hat{L}})\leq\nu_2(P^\prime,\mathcal{L}^\prime)$.
\end{proof}

\begin{theorem}\label{thm:main}
Let $q$ be an even prime power. For every $(P,\mathcal{L})\in\mathcal{I}_{q+2}$ the linear system $(P^\prime,\mathcal{L}^\prime)$ is a spanning $(q+1)$-uniform linear subsystem of $\Pi_q$ such that $\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)-1=q+1$ with $|\mathcal{L}^\prime|\geq3q$.
\end{theorem}

\begin{proof}
	Let $(P,\mathcal{L})\in\mathcal{I}_{q+2}$. Then $(P^\prime,\mathcal{L}^\prime)$ is an $(q+1)$-uniform intersecting linear system with $|P^\prime|=q^2+q+1$ (by Lemma \ref{lemma:desigualdadesplanoproy}
	). Furthermore, if $|\mathcal{L}^\prime|=q^2+q+1$ then all points of $(P^\prime,\mathcal{L}^\prime)$ have degree $q+1$ (it is a consequence of Lemma \ref{lemma:desigualdadesplanoproy}, see \cite{Dong}). Since projective planes are dual systems, the 2-packing number coincides with the cardinality of an oval, which is the maximum number of points in general position (no three of them collinear), and it is equal to $q+2=\nu_2(P^\prime,\mathcal{L}^\prime)$ (Lemma \ref{lemma:igualdad}), when $q$ is even, see for example \cite{B95}. Hence, the linear system $(P^\prime,\mathcal{L}^\prime)$ is a projective plane of order $q$, $\Pi_q$. Therefore, if $(P,\mathcal{L})\in\mathcal{I}_{q+2}$ then $(P^\prime,\mathcal{L}^\prime)$ is a spanning linear subsystem of $\Pi_q$ satisfying $\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)-1=q+1$, with $|\mathcal{L}^\prime|\geq3q$ (see Lemma \ref{lemma:desigualdadesplanoproy}).
\end{proof}

The following is a straightforward consequence of Theorem \ref{thm:main} which is the main result given in \cite{Dong}.

\begin{corollary}
If $(P,\mathcal{L})\in\mathcal{I}_{4}$ then either $(P^\prime,\mathcal{L}^\prime)$ is the Fano plane, $\Pi_2$, or $(P^\prime,\mathcal{L}^\prime)$ is obtained from $\Pi_2$ by deleting any line.
\end{corollary}

{\bf Acknowledgment}

The author would like to thank the referees for many constructive suggestions to improve this paper.

Research was partially supported by SNI and CONACyT.

config: arxiv


TITLE: Second Derivative as an average displacement of neighbouring points?
QUESTION [1 upvotes]: Ian Stewart suggested in one of his books that 'the second derivative of a point is the average displacement of its surrounding points from that point.' Can someone interpret this and explain how this is true? I've tried integrating to +/- h and taking h to zero, tried epsilon delta definition but hardly anything worked. All I could show was that the sign of second derivative was related to the sign of the displacement.

REPLY [2 votes]: One possible interpretation involves inverting the roles of "average" and "displacement".  Namely, the second derivative can be viewed as the displacement of the average, in the following sense:
Consider the average of the nearby values $f(c+h)$ and $f(c-h)$ - clearly, it is $\frac{f(c+h)+f(c-h)}{2}$. This is in general different from $f(c)$. By what amount is the average displaced compared to $f(c)$? by $$\frac{f(c+h)+f(c-h)}{2}-f(c).$$ Now divide by $h^2$, and let $h$ tend to zero.  What you get (up to factor of 1/2) is the second derivative $$f''(c)=\lim_{h \to 0} \frac{f(c+h)+f(c-h)-2f(c)}{h^2}.$$


TITLE: Rescaling of effective hamiltonian coupling constants in the Wilsonain renormalization group
QUESTION [3 upvotes]: I am confused about an aspect of coupling constant rescaling in the Wilsonian renormalization group procedure. (I'm following Kardar's "Statistical Physics of Fields, Ch5). I think I understand the basic idea of the renormalization group, but I'm in undergrad and haven't taken field theory or an advanced stat mech course so if I have a conceptual error somewhere I'd really appreciate any corrections.
The partition function for the Landau Ginzburg hamiltonian is written as ($\tilde{\vec{m}}(\mathbf{q}) \ \text{and }\sigma(\mathbf{q})$ are the splitting of the original field into slow and fast components)
$$
\begin{align}
Z &= \int D\tilde{\vec{m}}(\mathbf{q})D\sigma(\mathbf{q}) \exp{\bigg\{- \int_{0}^{\Lambda} \frac{d^d \mathbf{q}}{(2\pi)^d} \bigg( \frac{t + K q^2}{2} \bigg) (|\tilde{m}(\mathbf{q})|^2} + |\sigma(\mathbf{q})|^2)-U[\tilde{\vec{m}}(\mathbf{q}),\sigma(\mathbf{q})] \bigg\}\\
&= \int D\tilde{\vec{m}}(\mathbf{q}) \exp{\bigg\{- \int_{0}^{\Lambda} \frac{d^d \mathbf{q}}{(2\pi)^d} \bigg( \frac{t + K q^2}{2} \bigg) (|\tilde{m}(\mathbf{q})|^2}\bigg\}  \exp{\bigg\{-\frac{nV}{2} \int_{\Lambda/b}^{\Lambda} \frac{d^d \mathbf{q}}{(2\pi)^d} \log(t + K q^2) \bigg\}} \bigg\langle e^{-U[\tilde{\vec{m}},\vec{\sigma}]}\bigg\rangle_{\sigma}
\end{align}
$$
I think I understand the overall procedure: integrate out the momenta above the cutoff; rescale the momenta $\mathbf{q} = b^{-1} \mathbf{q}'$ and the field $\tilde{\vec{m}} = z {\vec{m}\,}'$. Then you get the new hamiltonian:
$$
(\beta H)'[m'] = V(\delta f_b^0 + u \delta f_b^1) + \int_{0}^{\Lambda} \frac{d^d \mathbf{q'}}{(2\pi)^d} b^{-d}z^2\bigg( \frac{\tilde{t} + K b^{-2} q'^2}{2} \bigg) |m'(\mathbf{q'})|^2 +u b^{-3d} z^4 \int_{0}^{\Lambda} \frac{d^d \mathbf{q}'_1 d^d \mathbf{q}'_2 d^d \mathbf{q}'_3 d^d \mathbf{q}'_4}{(2\pi)^d} \vec{m}(\mathbf{q}'_1)\cdot \vec{m}(\mathbf{q}'_2)\vec{m}(\mathbf{q}'_3)\cdot\vec{m}(\mathbf{q}'_4) \ \delta^d(\mathbf{q}'_1+\mathbf{q}'_2+\mathbf{q}'_3+\mathbf{q}'_4)
$$
where the $t$ is $$\tilde{t} = t+4u(n-2) \int_{\Lambda/b}^{\Lambda} \frac{d^d \vec{k}}{(2\pi)^d} \frac{1}{t+K\ k^2}$$
Then you choose $z=b^{1+\frac{d}{2}}$ so that $K$ stays the same: $K'=K, \ u' = b^{-3d} \ z^4 \ u, \ \text{and} \ t'= b^{-d} \ z^2 \ \tilde{t}$.
My question is: why doesn't the $u$ inside $\tilde{t}$ become a $u'$ ? As I understand it, the couplings change with the cutoff, so shouldn't the $u$ be replaced with $u'$ wherever it appears? If not, why not, and what is the physical meaning of this? (Originally asked here but I decided to split into separate questions.)

REPLY [5 votes]: What you are doing as part of this calculation is deriving the relationship between the coupling constants of the model at the coarse scale and the original scale. The results you get,
$$K' = K$$
$$u' = b^{-3d} z^4 u,$$
$$t' = b^{-d} z^2 \left(t + (n-2)\int_{\Lambda/b}^\Lambda \frac{d^d\vec{k}}{(2\pi)^d} \frac{1}{t+K k^2}\right)$$
are the so-called recursion relations between the parameters at the original fine scale ($K$, $u$, and $t$) and the parameters at the coarse scale ($K'$, $u'$, and $t'$). You do not replace $u$ with $u'$ in $\tilde{t}$ for the same reason that you do not replace the $t$ with $t'$. i.e., the right hand sides are the old parameters at the original scale, the left hand sides are the new parameters at the coarsened scale.
The primed quantities are really just a relabeling of terms in your coarsened Hamiltonian so that it matches the original Hamiltonian (up to approximations); they are not a change of variables. The only change of variables you are actually performing is the rescaling of the degrees of freedom.
In case it helps to clarify what you are actually doing during this calculation, I'll elaborate on that last sentence: the renormalization group procedure consists of two distinct steps: 1) averaging over (integrating out) statistical degrees of freedom and 2) making a change of variables on the remaining degrees of freedom to restore the system to its original scale.
I like to think of this in terms of probability distributions: if you have a multivariate distribution $p(m_1,m_2,\dots,\sigma_1,\sigma_2,\dots) \propto \exp(-H(m_1,m_2,\dots,\sigma_1,\sigma_2,\dots))$, then the first step of the renormalization group procedure is to marginalize over (average out) the variables $\sigma_1,\dots$ to obtain the distribution $p(m_1,m_2,\dots)$. Then, you define a new set of rescaled variables $m_1',m_2',\dots$, giving a new distribution
$$p'(m_1',m_2',\dots) = \left|\frac{\partial \vec{m}}{\partial \vec{m'}} \right| p(m_1(\vec{m}'),m_2(\vec{m}'),\dots) \propto e^{-H'(m_1',m_2',\dots)},$$
where $\left|\frac{\partial \vec{m}}{\partial \vec{m'}} \right| $ is the Jacobian of the change of variables (I've assumed continuous variables for simplicity here).
Often this change of variables is just a rescaling $m' \sim z m$ (and the Jacobian therefore does not add any important terms to the Hamiltonian). After this, defining the primed coupling constants is in some sense just a matter of notational cleanup. Since we expect that in many cases our coarse-grained Hamiltonian will have more or less the same form as our original Hamiltonian, it makes sense to define new coupling constants so that the form of the two Hamiltonians is superficially similar. e.g., if the pairwise interaction term in the original Hamiltonian were $J m_i m_j$ and the pairwise term in the coarsened Hamiltonian is $f(J,{\rm other~couplings}) m_i' m_j'$, we define $J' = f(J,{\rm other~couplings})$.
The conceptual leap that follows is that, if we are able to perform this procedure in the first place, mashing on the coarsened Hamiltonian until it looks like the original Hamiltonian, then there is nothing* stopping us from doing this again and again, obtaining a new set of coarsened coarsened couplings $J'' = f(J',{\rm other~couplings}')$, which allows us to interpret this relationship between the couplings as a recursion relation at different scales. (*important caveat: if there is only a finite number of degrees of freedom, then we can only perform this procedure a finite number of times).


TITLE: Gradient on sphere
QUESTION [0 upvotes]: I would to like to calculate the $\nabla f$, where $f$ is a restriction of the function $F: \mathbb{R}^{n+1}\to \mathbb{R} $ to $\mathbb{S^{n}}$.
I know that $\nabla f = \sum_{i,j=1}^{n}  g^{ij}\frac{\partial f }{\partial x_{i}} \frac{\partial }{\partial x_{j}}$, where $g^{ij}$ is a inverse matrix of metric coefficients  given by stereo graphic projection. By formation law of the function
$\nabla f = \sum_{i,j=1}^{n}  g^{ij}\frac{\partial f }{\partial x_{i}} \frac{\partial }{\partial x_{j}}= \sum_{j=1}^{n}  g^{1j}\frac{\partial }{\partial x_{j}}$. However, it is a hard to finish. There exists a short hut for calculating the gradient of function?

REPLY [2 votes]: Let $M$ be a submanifold of $\mathbb R^n$ and $f : \mathbb R^n \to \mathbb R$ is a smooth function. Then
$$\tag{1}\nabla (f|_M) = (\nabla f)^\top,$$
where $\top$ is the projection onto the tangent space of $M$. To see this, note that $\nabla (f|_M)$ is the unique vector fields on $M$ so that
$$ \langle \nabla f|_M , v\rangle = \nabla_v (f|_M), \ \forall v\in TM,$$
where $\nabla_v$ denotes directional derivatives. To check $(1)$, let $x\in M$ and $v\in T_xM$. Then there is a curve $\gamma :(-\epsilon, \epsilon) \to M$ so that $\gamma(0)  = x$ and $\gamma'(0) = v$. Thus
$$\nabla_v (f|_M)  =\frac{d}{dt} f(\gamma(t))\bigg|_{t=0}.$$
But the right hand side is also (think of $\gamma $ mapping to $\mathbb R^n$)
$$\langle \nabla f, v\rangle. $$
Thus
$$\langle \nabla (f|_M), v\rangle = \nabla_v f = \langle \nabla f, v\rangle = \langle (\nabla f)^\top, v\rangle \ \ \forall v \Rightarrow \nabla (f|_M) = (\nabla f)^\top.$$


TITLE: Evaluate $\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$
QUESTION [5 upvotes]: Evaluate by complex methods
$$\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$$
Sis.

REPLY [1 votes]: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\on}[1]{\operatorname{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
\begin{align}
&\bbox[5px,#ffd]{\left.\int_{0}^{\pi}{\sin^{2}\pars{\theta} \over \bracks{1 - 2a\cos\pars{\theta} + a^{2}}
\bracks{1 - 2b\cos\pars{\theta} + b^{2}}}\,\dd\theta
\,\right\vert_{\, 0\ <\ a\ <\ b\ <\ 1}}
\\[5mm] = &\
{1 \over 2}
\int_{-\pi}^{\pi}{\pars{\expo{\ic\theta} - \expo{-\ic\theta}}^{2}/\pars{-4} \over
\pars{a - \expo{\ic\theta}}\pars{a - \expo{-\ic\theta}}
\pars{b - \expo{\ic\theta}}
\pars{b - \expo{-\ic\theta}}}\,\dd\theta
\\[5mm] = &\
-{1 \over 8}\,\Re
\int_{-\pi}^{\pi}{\pars{z - z^{-1}}^{2} \over
\pars{a - z}\pars{a - z^{-1}}
\pars{b - z}
\pars{b - z^{-1}}}\,{\dd z \over \ic z}
\\[5mm] = &\
-{1 \over 8ab}\,\Im
\int_{-\pi}^{\pi}{\pars{z^{2} - 1}^{2} \over
z\pars{z - a}\pars{z - a^{-1}}
\pars{z - b}
\pars{z - b^{-1}}}\,\dd z
\end{align}
The integrand has simple poles at $\ds{z \in \braces{0,a,b}}$ with
$$
\left\{\begin{array}{rcl}
\ds{\on{Res}_{\, z\ =\ 0}} & \ds{=} & \ds{\phantom{-}1}
\\[1mm]
\ds{\on{Res}_{\, z\ =\ a}} & \ds{=} &
\ds{-{\pars{1 - a^{2}}b \over \pars{b - a}\pars{1 - ab}}}
\\[1mm]
\ds{\on{Res}_{\, z\ =\ b}} & \ds{=} &
\ds{\phantom{-}
{a\pars{1 - b^{2}} \over \pars{b - a}\pars{1 - ab}}}
\end{array}\right.
$$
Then,
\begin{align}
&\bbox[5px,#ffd]{\left.\int_{0}^{\pi}{\sin^{2}\pars{\theta} \over \bracks{1 - 2a\cos\pars{\theta} + a^{2}}
\bracks{1 - 2b\cos\pars{\theta} + b^{2}}}\,\dd\theta
\,\right\vert_{\, 0\ <\ a\ <\ b\ <\ 1}}
\\[5mm] = &\
-{1 \over 8ab}\Im\bracks{2\pi\ic\pars{\on{Res}_{\, z\ =\ 0}\ +\
\on{Res}_{\, z\ =\ a}\ +\ \on{Res}_{\, z\ =\ b}}}
\\[5mm] = &\
\bbx{\pi/2 \over 1 - ab} \\ &
\end{align}


TITLE: If a continuous function is nonzero at a point $a$, there is a ball around $a$ in which it has the same sign as $f(a)$
QUESTION [6 upvotes]: Let $f$ be a scalar field continuous at an interior point a of a set $S\in \mathbb{R}$.
  If $f(a)\ne 0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$.

The above statement is what I want to prove. I am not sure whether I have to use epsilon/delta to prove it. Please help me.

REPLY [8 votes]: If $f(a) \neq 0$, set $\varepsilon = \frac{|f(a)|}{2}$. Since $\varepsilon > 0$, by continuity of $f$ at $a$, there exists a $\delta > 0$ such that if $x \in B(a; \delta)$, then $|f(x) - f(a)| < \varepsilon$. Hence, for all $x\in B(a; \delta)$, $|f(a)| - \varepsilon < f(x) < |f(a)| + \varepsilon$, i.e., $f(a) - \frac{|f(a)|}{2} < |f(x)| < f(a) + \frac{|f(a)|}{2}$. If $f(a) > 0$, then
$$f(x) > f(a) - \frac{|f(a)|}{2} = f(a) - \frac{f(a)}{2} = \frac{f(a)}{2} > 0\quad \text{for all $x\in B(a;\delta)$}.$$
If $f(a) < 0$, then $$f(x) < f(a) + \frac{|f(a)|}{2} = f(a) - \frac{f(a)}{2} = \frac{f(a)}{2} < 0 \quad \text{for all $x\in B(a;\delta)$}.$$
In either case, the sign of $f$ equals the sign of $f(a)$ in the $n$-ball $B(a;\delta)$.


TITLE: Writing u as a linear combination of the vectors in S.
QUESTION [3 upvotes]: Write vector
u = $$\left[\begin{array}{ccc|c}2 \\10 \\1\end{array}\right]$$
as a linear combination of the vectors in S. Use elementary row operations on an augmented matrix to find the necessary coefficients.
S = {
$v1$$\left[\begin{matrix}1\\2\\2\end{matrix}\right] , v2\left[\begin{matrix}4\\2\\1\end{matrix}\right],
v2\left[\begin{matrix}5\\4\\1\end{matrix}\right]
$ }. If it is not possible, explain why?

This is what i have so far:
S = {
$v1$$\left[\begin{matrix}1\\2\\2\end{matrix}\right] , v2\left[\begin{matrix}4\\2\\1\end{matrix}\right],
v3\left[\begin{matrix}5\\4\\1\end{matrix}\right].
v4\left[\begin{matrix}2\\10\\1\end{matrix}\right]
$ }

$c1$$\left[\begin{matrix}1\\2\\2\end{matrix}\right] , c2\left[\begin{matrix}4\\2\\1\end{matrix}\right],
c3\left[\begin{matrix}5\\4\\1\end{matrix}\right].
c4\left[\begin{matrix}2\\10\\1\end{matrix}\right]
=\left[\begin{matrix}0\\0\\0\end{matrix}\right]$
$c1$$\left[\begin{matrix}1\\2\\2\end{matrix}\right] , c2\left[\begin{matrix}4\\2\\1\end{matrix}\right],
c3\left[\begin{matrix}5\\4\\1\end{matrix}\right].
c4\left[\begin{matrix}2\\10\\1\end{matrix}\right]$
$
\begin{bmatrix}
1 & 4 & 5 & 2\\
2 & 2 & 4 & 10\\
2 & 1 & 1 & 1\\
\end{bmatrix}
$
Now i don't know how to do this. Help will greatly be appreciated.
Thanks

REPLY [2 votes]: $a\left[\begin{matrix}1\\2\\2\end{matrix}\right] + b\left[\begin{matrix}4\\2\\1\end{matrix}\right]+
c\left[\begin{matrix}5\\4\\1\end{matrix}\right]=\left[\begin{matrix}2\\10\\1\end{matrix}\right]$.
Putting this into matrix form gives:
$\begin{bmatrix}1&4&5&2\\2&2&4&10\\2&1&1&1\end{bmatrix}\to\begin{bmatrix}1&4&5&2\\1&1&2&5\\2&1&1&1\end{bmatrix}\to\begin{bmatrix}1&4&5&2\\0&3&3&-3\\0&7&9&3\end{bmatrix}\\\to\begin{bmatrix}1&4&5&2\\0&1&1&-1\\
0&7&9&3\end{bmatrix}\to\begin{bmatrix}1&0&1&6\\0&1&1&-1\\0&0&1&5\end{bmatrix}\to\begin{bmatrix}1&0&0&1\\0&1&0&-6\\0&0&1&5\end{bmatrix}$.
So, $a=1,b=-6,c=5.$
Plug the values back in to check if this is indeed correct:
$1\left[\begin{matrix}1\\2\\2\end{matrix}\right]  -6\left[\begin{matrix}4\\2\\1\end{matrix}\right]+
5\left[\begin{matrix}5\\4\\1\end{matrix}\right]=\left[\begin{matrix}2\\10\\1\end{matrix}\right]$$\implies$$\left[\begin{matrix}1\\2\\2\end{matrix}\right] + \left[\begin{matrix}-24\\-12\\-6\end{matrix}\right]+
\left[\begin{matrix}25\\20\\5\end{matrix}\right]=\left[\begin{matrix}2\\10\\1\end{matrix}\right]$.
So, it must be correct.