11institutetext: Institute of Mathematics and Computer Science (ICMC),
University of São Paulo (USP), Sao Carlos, SP, Brazil

# The Importance of Open Data Policy to Tackle Pandemic in Latin America

Josimar E. Chire Saire 11

###### Abstract

Open Data Policies can provide transparency, impulse innovation and
citizenship participation. Access to the right data in right time can produce
huge benefits to population. But, in Latin America there is not enough
interest from governments to promote and use properly. By the other hand,
global pandemic has caused many damages in different levels, i.e. Economy,
Public Health, Education, etc. The paper opens a discussion about the
importance of Open Data Policy to mitigate the impact of Covid-19 and overpass
this problem.

###### Keywords:

Open Data Pandemic Coronavirus Covid-19 Latin America

##  1 Introduction

Governments constantly collect data through different institutions or agencies
about Health, Economy, Social and other areas. During the last decades, the
access to this data was restricted to specific entities. However, governments
noticed the open access to this data can produce benefits [4]: transparency,
promote innovation, participation of the citizenship.

There are initiatives to purpose a metric to analyze the previous concept
explained, i.e. Open Data Index (GODI). GODI [13] was purposed by Open
Knowledge Network, to evaluate openness: government budget, national
statistics, national laws, elections, living conditions (air, water, weather),
maps and others. It is important to notice that Public Health is not included.

Policymakers can face many challenges when they make a proposal, for many
reasons: complexity of problem, information access, constraints, etc. There
are many methodologies to make a proposal [2], the main concern in health
policies are the outcomes or effects on the population. For example, the steps
purposed by Dunn [17]: definition, prediction, prescription, description and
evaluation.

Global pandemic originated by Covid-19 started at the end of year 2019,
declared a pandemic and a global threat by the World Health Organization (WHO)
on 11-03-2020 [14]. WHO[15] defined some measures in the absence of an
approved cure, to control the spread of the virus. Many governments[16] soon
adopted and implemented these measures in their own countries, they include
physical distancing between persons, wearing face masks, avoiding physical
interactions like handshaking, and hugging, and lockdowns, etc.

Latina America [1] had the first reported cases in Brazil and Mexico, on 26
and 27 February 2020 respectively. A continent consisting of mostly developing
countries, Latin America faces a unique challenge in dealing with the
pandemic. A lack of strong Public Health sector, produced limited capability
of testing, slow and inefficient reaction to control the spread of the virus.
The primary measure was to set a partial or fully lockdown, closing schools,
universities to open online teaching. A lack of good administration of
approved budget to improve infrastructure of Public Health produced a not well
prepared Health Care System to deal with a increasing number of cases.
Finally, it is important to highlight every country take it’s own decisions
and set policies to combat Covid-19 consequences.

This paper wants to open the discussion about the importance of Open Data
Policies in Latin America. The strong relationship between Open Data and
Public Health, by consequence, the benefits of having access to relevant to
create appropriate policies to benefit the population and in this specific
context, reduce the impact of Covid-19 and tackle the pandemic

##  2 Open Data Availability

This sections discusses the Open Data Availability considering the existence
of national Statistics, Open Data Policy foundation and the global results of
GODI.

####  2.0.1 National Statistics

GODI indicator considers, statistics about demography, economy, i.e. Gross
Domestic Product, unemployment and population statistics. The next features
were evaluated: Open License, open and machine readable format, download at
once, up to date, publicly available, free of charge. After a search to find
National Statistics Institute for each country in South America, the table
Tab. 2.0.1 is presented. Names are similar, but the year of foundation is
different, in some countries the creation time is less than 50 years. And this
data is not easy to find, Venezuela and Paraguay do not have this data in
their websites.

Country | Institute/Organization | Foundation | Link
---|---|---|---
Argentina |  | Instituto Nacional de Estadística y Censos
---
de la República Argentina
1968 [6] | https://www.indec.gob.ar/
Bolivia | Instituto Nacional de Estadística | 1976 [8] | https://www.ine.gob.bo/
Brazil | Instituto Brasileiro de Geografia e Estatística | 1934 [5] | https://www.ibge.gov.br/
Chile | Instituto Nacional de Estadísticas | 1843 [9] | https://www.ine.cl/
Colombia | Departamento Administrativo Nacional de Estadística | 1953 [3] | https://www.dane.gov.co/
Ecuador | Instituto Nacional de Estadística y Censos | 1976 [11] | https://www.ecuadorencifras.gob.ec
Paraguay | Instituto Nacional de Estadística |  $\sim$1943 [7] | https://www.ine.gov.py/
Peru | Instituto Nacional de Estadística e Informática | 1975 [12] | https://www.inei.gob.pe/
Uruguay | Instituto Nacional de Estadística | 1994 [10] | https://www.ine.gub.uy
Venezuela | Instituto Nacional de Estadística | NA | http://www.ine.gov.ve/

Then, considering the previous table is possible to affirm that some countries
noticed the relevance of having a Statistics Center, the benefits and need.
Bolivia, Ecuador, Uruguay are the newest, this could imply maturing process to
consolidate organization and objectives. Brazil, Chile, Colombia are the
oldest, this should mean the organization could be more mature.

###  2.1 Open Data Policy

A second point to consider: the existence of open data website to share data
and when this started per country. Table 1 presents the year of starting the
Open Data Policy and the available link to access it.

Table 1: Open Data Foundation Country | Foundation | Link
---|---|---
Argentina | 2016 | http://datos.gob.ar/
Bolivia | NA | https://datos.gob.bo/
Brazil | 2011 | https://dados.gov.br/
Chile | 2008 | https://datos.gob.cl/
Colombia | 2014 | https://www.datos.gov.co/
Ecuador | 2020 | https://sni.gob.ec/datosabiertos
Paraguay | 2014 | https://www.datos.gov.py/
Peru | 2017 | https://www.datosabiertos.gob.pe/
Uruguay | 2015 | https://www.gub.uy/datos-abiertos
Venezuela | NA | NA

Considering the previous table, is possible to notice countries like Chile,
Brazil, Colombia and Paraguay are the oldest and the newest are: Peru,
Argentina, Uruguay. All this data were results of searching using: ”politica
datos abiertos”(open data policy) + country, and there was not data available
for Venezuela.

###  2.2 GODI Results

In this subsection are presented the results of GODI index. First, the figure
Fig. 2 presents the results of criterion: National Statistics, focusing in
Latin America countries. It is possible to notice, the top 5 countries are:
Dominican Republic, Mexico, Brazil, Venezuela, Argentina. And the last 5
countries: Bolivia, Puerto Rico, Peru, Guyana, Paraguay.

Figure 1: GODI: National Statistics 222The graphic was created using the
available table, Source: https://index.okfn.org/dataset/statistics/

It is important to highlight that six criterion were analyzed: Open licensed,
in an open and machine-readable format, downloadable at once, up to date,
publicly available, available free of charge.

Next figure 4 presents the available countries considered in the study. The
top 5 are: Brazil, Mexico, Colombia, Argentina and Uruguay. At the bottom are:
Dominican Republic, Panama, Costa Rica, Guyana, Venezuela.

Figure 2: GODI Indicators 444The graphic was created using the available
table, Source: https://index.okfn.org/place/

##  3 Covid-19 Statistics

The next figure Fig. 3 presents the statistics about Covid-19 of Latin
American countries. The top 5 countries with more cases: Brazil, Argentina,
Colombia, Mexico, Peru and the last 5 are: El Salvador, French Guyana,
Suriname, Guyana, Nicaragua.

Figure 3: Covid-19 Statistics of Latin American Countries

Next figure Fig. 4 presents the data considering total cases per country.

Figure 4: Total Cases

Again the next countries are in the top: Brazil, Argentina, Colombia, Mexico,
Peru.

###  3.1 Discussion

The discussion starts with the next question: the existence of Open Data
Policies can mitigate, help to combat Covid-19 pandemic? The discussion starts
with South America focus, but this can be extended for Latin American. The
start point is if there is information at the right time, then policymakers
can work properly and reduce/mitigate the impact of Covid-19 in each country
in Latin America. Summarizing all the tables and data presented previously,
the next figure Fig. 5 presents the National Statistics foundation year, Open
Data Policy foundation and ratio (total cases/population).

Figure 5: Comparison

In spite of an oldest foundation in Chile, the ratio is a medium level around
8.24% population got infected by Covid-19 and Brazil, got around 8.92%. And,
Colombia has 8.8% ratio. Finally, considering figure 4, Brazil, Colombia and
Chile are in the top 10. Then, Open Data Policies is enough to face this kind
of challenges as a pandemic, this a question. In spite of, the previous
results is important to mention the importance of Open Data Policy to support
work of policymakers, and other question is open: what other factors are
important to combat efficiently this pandemic? We have not explored the
citizenship, industry participation yet. And, the existence of National
Statistics and Open Data Policy guaranteed the proper collect of data about
social, economical conditions of the population. Besides, in the specific
situation of this pandemic the access to data can open new opportunities to
create, develop tools can monitor, track population, have a global sight of
the situation to have access to vaccine, i.e. Covax and get help from other
countries.

##  4 Conclusions

  * •

Most of the countries of Latin American have National Statistics, but this
paper does not explore the frequency of the data collection and other factors
that can make useful this data.

  * •

Many countries in South America have create a Open Data policy, Venezuela is
an exception, because the url is not available.

  * •

Countries with oldest foundation data of National Statistics and Open Data
policy got high rates of Covid-19, i.e. Chile, Brazil.

  * •

The public availability of data about one country can contribute to
create/produce innovation and facilitate vaccine initiatives as Covax.

  * •

Other tools can be used to get real time data about population, i.e.
Infodemilogy based in Social Networks analysis.

## References

  * [1] Chire-Saire, J., Mahmood, K.: Hope amid of a pandemic: Is psychological distress alleviating in south america while coronavirus is still on surge? (2020)
  * [2] Collins, T.: Health policy analysis: a simple tool for policy makers. Public Health 119(3), 192–196 (2005). https://doi.org/https://doi.org/10.1016/j.puhe.2004.03.006, https://www.sciencedirect.com/science/article/pii/S003335060400099X
  * [3] Dane: Generalidades. Accessed on 11 July 2021 in URL:https://www.dane.gov.co/index.php/acerca-del-dane/informacion-institucional/generalidades
  * [4] Foundation, O.K.: Welcome to open government data. Accessed on 11 July 2021 in URL:http://opengovernmentdata.org/
  * [5] Ibge: O ibge. Accessed on 11 July 2021 in URL:https://www.ibge.gov.br/acesso-informacao/institucional/o-ibge.html
  * [6] Indec: Ley 17622. Accessed on 11 July 2021 in URL: https://www.indec.gob.ar/tp/documentos/Ley_17622.htm
  * [7] Ine: Accessed on 11 July 2021 in URL:https://www.ine.gov.py
  * [8] Ine: Dl 14100 ley del sistema nacional de información estadística. Accessed on 11 July 2021 in URL:https://www.ine.gob.bo/index.php/wpfd_file/dl-14100-ley-del-sistema-nacional-de-informacion-estadistica/
  * [9] Ine: Historia. Accessed on 11 July 2021 in URL:https://www.ine.cl/institucional/nuestra-institucion/historia
  * [10] Ine: Marco jurídico. Accessed on 11 July 2021 in URL:https://www.ine.gub.uy/web/guest/ley16616
  * [11] Inec: Historia del instituto nacional de estadística y censos (inec). Accessed on 11 July 2021 in URL:https://www.ecuadorencifras.gob.ec/historia-del-instituto-nacional-de-estadistica-y-censos-inec/
  * [12] Inei: Historia. Accessed on 11 July 2021 in URL:https://www.inei.gob.pe/historia/
  * [13] International, O.K.: Global open data index. Accessed on 11 July 2021 in URL:http://2015.index.okfn.org/place/
  * [14] Organisation, T.W.H.: World health organisation (who) coronavirus disease (covid‐19) outbreak. Accessed on 11 July 2021 in URL: https://www.euro.who.int/en/health-topics/health-emergencies/coronavirus-covid-19/news/news/2020/3/who-announces-covid-19-outbreak-a-pandemic
  * [15] Organisation, W.H.: Covid-19 strategy update. Accessed on 11 July 2021 in URL: https://www.who.int/docs/default-source/coronaviruse/covid-strategy-update-14april2020.pdf
  * [16] Saire, J.C., Panford-Quainoo, K.: Twitter interaction to analyze covid-19 impact in ghana, africa from march to july (2020)
  * [17] W, D.: Public policy. an introduction p. 14 (1981)


# An Entropic Approach to Classical Density Functional Theory

Ahmad Yousefi
Department of Physics
University at Albany-SUNY
Albany, NY 12222
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
& Ariel Caticha
Department of Physics
University at Albany-SUNY
Albany, NY 12222
<EMAIL_ADDRESS>

###### Abstract

The classical Density Functional Theory (DFT) is introduced as an application
of entropic inference for inhomogeneous fluids in thermal equilibrium. It is
shown that entropic inference reproduces the variational principle of DFT when
information about expected density of particles is imposed. This process
introduces a family of trial density-parametrized probability distributions,
and consequently a trial entropy, from which the preferred one is found using
the method of Maximum Entropy (MaxEnt). As an application, the DFT model for
slowly varying density is provided, and its approximation scheme is discussed.

_Keywords_ entropic inference; relative entropy; density functional theory;
contact geometry; optimal approximations

##  1 Introduction

The Density Functional Theory was first developed in the context of quantum
mechanics and only later extended to the classical regime. The theory was
first introduced by Kohn and Hohenberg (1964)[1] as a computational tool to
calculate the spatial density of an electron gas in the presence of an
external potential at zero temperature. Soon afterwards, Mermin provided the
extension to finite temperatures [2]. Ebner, Saam, and Stroud (1976)[3]
applied the idea to simple classical fluids, and Evans (1979) provided a
systematic formulation in his classic paper [4] "The nature of the liquid-
vapour interface and other topics in the statistical mechanics of non-uniform,
classical fluids.".
The majority of physicists and chemists today are aware of the quantum DFT and
the Kohn-Sham model [5], while fewer are familiar with the classical DFT, a
historical review of quantum DFT and its vast variety of applications is found
in [6, 7]. The classical DFT, similarly, is a "formalism designed to tackle
the statistical mechanics of inhomogeneous fluids"[8], which has been used to
investigate a wide variety of equilibrium phenomena, including surface
tension, adsorption, wetting, fluids in porous materials, and the chemical
physics of solvation.
Just like the Thomas-Fermi-Dirac theory is usually regarded as a precursor of
quantum DFT, the van der Waals’ thermodynamic theory of capillarity under the
hypothesis of a continuous variation of density [9] can be regarded as the
earliest work on classical DFT, without a fundamental proof of existence for
such a variational principle.
"The long-term legacy of DFT depends largely on the continued value of the DFT
computer programs that practitioners use daily."[6] The algorithms behind the
computer programs, all starting from an original Hartree-Fock method to solve
the N-particle Schrödinger equation, have evolved by many approximations and
extensions implemented over time by a series of individuals, although the
algorithms produce accurate results, they do not mention the HK variational
principle. Without the variational principle the computer codes are suspect of
being ad hoc or intuitively motivated without a solid theoretical foundation,
therefore, the DFT variational principle not only scientifically justifies the
DFT algorithms, but also provides us with a basis to understand the repeatedly
modified algorithms behind the codes.
In this work we derive the classical DFT as an application of the method of
maximum entropy [10, 11, 12, 13, 14]. This integrates the classical DFT with
other formalisms of classical statistical mechanics (canonical, grand
canonical, etc.) as an application of information theory. Our approach not
only enables one to understand the theory from the Bayesian point of view, but
also provides a framework to construct equilibrium theories on the foundation
of MaxEnt. We emphasize that our goal is not derive an alternative to DFT. Our
goal is purely conceptual. We wish to find a new justification or derivation
of DFT that makes it explicit how DFT fits within the MaxEnt approach to
statistical mechanics. The advantage of such an understanding is the potential
for future applications that are outside the reach of the current version of
DFT.
In section 2, we briefly review entropic inference as an inference tool which
updates probabilities as degrees of rational belief in response to new
information. Then we show that any entropy maximization, produces a contact
structure which is invariant under the Legendre Transformations; this enables
us to take advantage of these transformations for maximized entropy functions
(functionals here) found from constraints other than those of thermal
equilibrium as well as thermodynamic potentials.
In section 3 we briefly review the method of relative entropy for optimal
approximation of probabilities, which allows us to derive and then generalize
the Bogolyubov variational principle. Then we apply it for the especial case
wherein the trial family of probabilities are parametrized by the density
function $n(x)$.
In section 4, the Density Functional formalism is introduced as an extension
of the existing ensemble formalisms of statistical mechanics (canonical, grand
canonical, etc.) and we show that the core DFT theorem is an immediate
consequence of MaxEnt: we prove that in the presence of an external potential
$v(x)$, there exists a trial density functional entropy $S_{v}(E;n]$ maximized
at the equilibrium density. We also prove that this entropy maximization is
equivalent to minimization of a density functional potential $\Omega(\beta;n]$
given by

$\Omega_{v}(\beta;n]=\int d^{3}xv(x)n(x)+F(\beta;n]$ (1.1)

where $F(\beta;n]$ is independent of $v(x)$. This formulation achieves two
objectives. i) It shows that the density functional variational principle is
an application of MaxEnt for non-uniform fluids at equilibrium and therefore,
varying the density $n(x)$ in $S_{v}(E;n]$ does not imply that the functional
represents entropy of any non-equilibrium system, this trial entropy, although
very useful, is just a mathematical construct which allows us to incorporate
constraints which are related to one another by definition. ii) By this
approach we show that the Bayesian interpretation of probability liberates the
fundamental theorem of the DFT from an imaginary grand-canonical ensemble,
i.e. the thermodynamic chemical potential is appropriately defined without
need to define microstates for varying number of particles.
Finally, in section 5, as an illustration we discuss the already well-known
example of a slowly varying inhomogeneous fluid. We show that our entropic DFT
allows us to reproduce the gradient approximation results derived by Evans
[4]. There are two different approximations involved, i) rewriting the non-
uniform direct correlation function in terms of the uniform one; and ii) the
use of linear response theory to evaluate Fourier transform of direct
correlation functions. The former assumes that the density is uniform inside
each volume element, and the latter assumes that difference of densities for
neighboring volume elements is small compared to their average.

##  2 Entropic inference

A discussion of the method of maximum entropy as a tool for inference is found
in [14]. Given a prior Probability Distribution Function (PDF) $Q(X)$, we want
to find the posterior PDF $P(X)$ subject to constraints on expected values of
functions of $X$.
Formally, we need to maximize the relative entropy

$S_{r}[P|Q]\equiv-\sum_{X}(PlogP-PlogQ)\;,$ (2.1)

under constraints $A_{i}=\sum_{X}P(X)\hat{A}_{i}(X)$ and
$1=\sum_{X}P({\color[rgb]{0,0,0}X})$; where $A_{i}$’s are real numbers, and
$\hat{A}_{i}$’s are real-valued functions on the space of $X$, and $1\leq
i\leq m$ for $m$ number of constraints.
The maximization process yields the posterior probability

$P(X)=Q(X)\frac{1}{Z}e^{-\sum_{i}\alpha_{i}\hat{A}_{i}(X)}\;,\quad\quad\text{where
}Z=\sum_{X}Q(X)e^{-\sum_{i}\alpha_{i}\hat{A}_{i}(X)}\;,$ (2.2)

and $\alpha_{i}$’s are Lagrange multipliers associated with $A_{i}$’s.
Consequently the maximized entropy is

$S=\sum_{i}\alpha_{i}A_{i}+logZ\;.$ (2.3)

Now we can show that the above entropy readily produces a contact structure,
we can calculate the complete differential of equation (2.3) to define the
vanishing one-form $\omega_{cl}$ as

$\omega_{cl}\equiv dS-\sum_{i}\alpha_{i}dA_{i}=0\;.$ (2.4)

Therefore any classical entropy maximization with $m$ constraints produces a
contact structure $\\{{\color[rgb]{0,0,0}\mathbb{T}},\omega_{cl}\\}$ in which
manifold $\mathbb{T}$ has $2m+1$ coordinates $\\{q_{0},q_{1},\dots
q_{m},p_{1},\dots,p_{m}\\}$.
The physically relevant manifold $\mathbb{M}$ is an $m$-dimensional sub-
manifold of $\mathbb{T}$, on which $\omega_{cl}$ vanishes; i.e. $\mathbb{M}$
is determined by $1+m$ equations

$q_{0}\equiv S(\\{A_{i}\\})\;,\quad\quad p_{i}\equiv\alpha_{i}=\frac{\partial
S}{\partial A_{i}}\;.$ (2.5)

Legendre Transformations defined as,

$q_{0}\longrightarrow\;q_{0}-\sum_{j=1}^{l}p_{j}q_{j}\;,$ (2.6)
$q_{i}\longrightarrow\;p_{i}\;,\quad
p_{i}\longrightarrow\;-q_{i}\;,\quad\quad\text{for }1\leq i\leq l\;,$

are coordinate transformations on space $\mathbb{T}$ under which $\omega_{cl}$
is conserved. It has been shown [15, 16] that the laws of thermodynamics
produce a contact structure conforming to the above prescription, here we are
emphasizing that the contact structure is an immediate consequence of MaxEnt,
and therefore it can be utilized in applications of information theory, beyond
thermodynamics.

##  3 Maxent and Optimal Approximation of Probabilities

The posterior PDF found from entropic inference, is usually too complicated to
be used for practical purposes. A common solution is to approximate the
posterior PDF by a more tractable family of PDFs $\\{p_{\theta}\\}$ [17].
Given the exact probability $p_{0}$, the preferred member of tractable family
$p_{\theta^{*}}$ is found by maximizing the entropy of $p_{\theta}$ relative
to $p_{0}$:

$\frac{\delta
S_{r}[p_{\theta}|p_{0}]}{\delta\theta}\Big{|}_{\theta=\theta^{*}}=0\;.$ (3.1)

The density functional formalism is a systematic method in which the family of
trial probabilities is parametrized by the density of particles; in section 4
we shall use the method of maximum entropy to determine the family of trial
distributions parametrized by $n(x)$, $p_{\theta}\equiv p_{n}$. So that we can
rewrite equation 3.1 as

$\frac{\delta}{\delta
n(x^{\prime})}\Bigg{[}S_{r}[p_{n}|p_{0}]+\alpha_{eq}[N-\int
d^{3}xn(x)]\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$ (3.2)

We will see that the canonical distribution itself is a member of the trial
family, therefore in this case, the exact solution to equation (3.2) is
$p_{0}$ itself:

$p_{n}\Big{|}_{n(x)=n_{eq}(x)}=p_{0}\;.$ (3.3)

##  4 Density Functional Formalism

An equilibrium formalism of statistical mechanics is a relative entropy
maximization process consisting of three crucial elements: i) One must choose
the microstates that describe the system of inference. ii) The prior is chosen
to be uniform. iii) One must select the constraints that represent the
information that is relevant to the problem in hand.
In the density ensemble, microstates of the system are given as positions and
momenta of all $N$ particles of the same kind, given the uniform prior
probability distribution

$Q(\\{\vec{x}_{1},\dots,\vec{x}_{N};\vec{p}_{1},\dots,\vec{p}_{N}\\})=constant.$
(4.1)

Having in mind that we are looking for thermal properties of inhomogeneous
fluids, it is natural to choose the density of particles $n(x)$ as
computational constraint, and the expected energy $E$ as thermodynamic
constraint, in which $n(x)$ represents the inhomogeneity and $E$ defines the
thermal equilibrium.
Note that all constraints (computational, thermal, etc.) in the framework can
be incorporated as inferential constraints and can be imposed as prescribed in
section 2.
The density constraint holds for every point in space, therefore we have
$1+1+\mathbb{R}^{3}$ constraints, one for normalization, one for total energy,
and one for density of particles at each point in space; so we have to
maximize the relative entropy

$S_{r}[P|Q]=-\frac{1}{N!}\int d^{3N}xd^{3N}p(PlogP-PlogQ)\equiv-Tr_{c}(PlogP-
PlogQ),$ (4.2)

subject to constraints

$\displaystyle 1=$ $\displaystyle\langle 1\rangle,\quad
E=\langle\hat{H}_{v}\rangle,$ (4.3a) $\displaystyle n(x)=$
$\displaystyle\langle\hat{n}_{x}\rangle\quad\text{where}\int d^{3}xn(x)=N,$
(4.3b)

where $\langle.\rangle\equiv\frac{1}{N!}\int(.)Pd^{3N}xd^{3N}p$. The classical
Hamiltonian operator $\hat{H}$ and the particle density operator $\hat{n}_{x}$
are given as

$\hat{H}_{v}\equiv\sum_{i=1}^{N}v(x_{i})+\hat{K}(p_{1},\dots,p_{N})+\hat{U}(x_{1},\dots,x_{N}),$
(4.4) $\hat{n}_{x}{\color[rgb]{0,0,0}\equiv}\sum_{i}^{N}\delta(x-x_{i}).$
(4.5)

The density $n(x)$ is not an arbitrary function; it is constrained by a fixed
total number of particles,

$\int d^{3}xn(x)=N.$ (4.6)

Maximizing (4.2) subject to (4.3) gives the posterior probability
$P(x_{1},\dots,x_{N};p_{1},\dots,p_{N})$ as

$P=\frac{1}{Z_{v}}e^{-\beta\hat{H}_{v}-\int
d^{3}x\alpha(x)\hat{n}_{x}}\quad\text{under condition}\int d^{3}xn(x)=N.$
(4.7)

where $\alpha(x)$ and $\beta$ are Lagrange multipliers.
The Lagrange multiplier function $\alpha(x)$ is implicitely determined by

$\frac{\delta logZ_{v}}{\delta\alpha(x)}=-n(x)$ (4.8)

and by equation (4.6)

$-\int d^{3}x\frac{\delta logZ_{v}}{\delta\alpha(x)}=N\;.$ (4.9)

Substituting the trial probabilities from (4.7) into (4.2) gives the trial
entropy $S_{v}(E{\color[rgb]{0,0,0};}n]$ as

$S_{v}(E;n]=\beta E+\int d^{3}x\alpha(x)n(x)+logZ_{v},$ (4.10)

where $Z_{v}(\beta;\alpha]$ is the trial partition function defined as

$Z_{v}(\beta;\alpha]=Tr_{c}e^{-\beta\hat{H}_{v}-\int
d^{3}x\alpha(x)\hat{n}_{x}}\;.$ (4.11)

The equilibrium density $n_{eq}(x)$ is that which maximizes
$S_{v}(E{\color[rgb]{0,0,0};}n]$ subject to $\int d^{3}xn(x)=N$:

$\frac{\delta}{\delta n(x^{\prime})}\Bigg{[}S_{v}(E;n]+\alpha_{eq}[N-\int
d^{3}xn(x)]\Bigg{]}=0\quad\textit{for fixed E}.$ (4.12)

Next, perform a Legendre transformation and define the Massieu functional
$\tilde{S}_{v}(\beta,n]$ as

$\tilde{S}_{v}\equiv S_{{\color[rgb]{0,0,0}v}}-\beta E,$ (4.13)

so that we can rewrite equation (4.12) as

$\frac{\delta}{\delta
n(x^{\prime})}\Bigg{[}\tilde{S}_{v}(\beta;n]-\alpha_{eq}\int
d^{3}xn(x)\Bigg{]}=0\quad\textit{for fixed $\beta$}\;.$ (4.14)

Combine (4.10), (4.13), and (4.14), and use the variational derivative
identity $\frac{\delta n(x)}{\delta n(x^{\prime})}=\delta(x-x^{\prime})$ to
find

$\int d^{3}x\Bigg{[}\frac{\delta
logZ_{v}(\beta;\alpha]}{\delta\alpha(x)}+n(x)\Bigg{]}\frac{\delta\alpha(x)}{\delta
n(x^{\prime})}=\alpha_{eq}-\alpha(x^{\prime}).$ (4.15)

The LHS of equation (4.15) vanishes by (4.6) and therefore the RHS must vanish
for an arbitrary $n(x)$ which implies that

$\alpha(x)=\alpha_{eq}\;,\quad\text{and}\quad\frac{\delta
logZ_{v}}{\delta\alpha(x)}{\color[rgb]{0,0,0}\Big{|}_{\alpha_{eq}}}=-n_{{\color[rgb]{0,0,0}eq}}(x)\;.$
(4.16)

Substituting (4.16) into (4.7) yields the equilibrium probability distribution

$P^{*}(x_{1},\dots,x_{N};p_{1},\dots,p_{N})=\frac{1}{Z_{v}^{*}}e^{-\beta\hat{H}_{v}-\alpha_{eq}\int
d^{3}x\hat{n}_{x}}=\frac{1}{Z_{v}^{*}}e^{-\beta\hat{H}_{v}-\alpha_{eq}N}$
(4.17)

where $Z_{v}^{*}(\beta,\alpha_{eq})=Tr_{c}e^{-\beta\hat{H}_{v}-\alpha_{eq}N}.$
From the inferential point of view, the variational principle for the grand
potential and the equilibrium density[4] is proved at this point; we showed
that for an arbitrary classical Hamiltonian $\hat{H}_{v}$, there exists a
trial entropy $S_{v}(E;n(x)]$ defined by equation (4.10), which assumes its
maximum at fixed energy and varying $n(x)$ under the condition $\int
d^{3}xn(x)=N$ at the equilibrium density, and gives the posterior PDF
(equation 4.17) equal to that of the canonical distribution.
The massieu function $\tilde{S}_{v}(\beta;n(x)]$ from equation (4.13) defines
the density functional potential $\Omega_{v}(\beta;n(x)]$ by

$\Omega_{v}(\beta;n]\equiv\frac{-\tilde{S}_{v}(\beta;n]}{\beta}=-\int
d^{3}x\frac{\alpha(x)}{\beta}n(x)-\frac{1}{\beta}logZ_{v}(\beta;\alpha]\;,$
(4.18)

so that the maximization of $S_{v}(E;n]$ (4.10) in the vicinity of
equilibrium, is equivalent to the minimization of $\Omega_{v}(\beta;n(x)]$
(4.18) around the same equilibrium point

$\frac{\delta}{\delta
n(x^{\prime})}\Bigg{[}\Omega_{v}(\beta;n]+\frac{\alpha_{eq}}{\beta}\int
d^{3}xn(x)\Bigg{]}=0\;.$ (4.19)

After we find $\Omega_{v}$, we just need to recall that
$\alpha(x)=-\beta\frac{\delta\Omega_{v}}{\delta n(x)}$ and substitute in
equation (4.16) to recover the "core integro-differential equation"[4] of DFT
as

$\nabla\Big{(}\frac{\delta\Omega(\beta;n]}{\delta n(x)}\Big{)}_{eq}=0$ (4.20)

which implies that

$\Omega_{v;eq}\leq\Omega_{v}(\beta;n],$ (4.21)

where

$\Omega_{v,eq}(\beta;n]=-\frac{\alpha_{eq}}{\beta}\int
d^{3}xn(x)-\frac{1}{\beta}logZ_{v}^{*}(\beta,\alpha_{eq}).$ (4.22)

From equation (4.18) it is clear that

$\Omega_{v}(\beta;n]=\int
d^{3}xv(x)n(x)+\langle\hat{K}+\hat{U}\rangle-\frac{S_{v}(E;n]}{\beta}\;.$
(4.23)

It is convenient to define the intrinsic density functional potential $F_{v}$
as

$F_{v}(\beta;n]\equiv\langle\hat{K}+\hat{U}\rangle-\frac{S_{v}}{\beta}$ (4.24)

to have

$\Omega_{v}(\beta;n]=\int v(x)n(x)+F_{v}(\beta;n]\;.$ (4.25)

Now we are ready to restate the fundamental theorem of the classical DFT:

######  Theorem 4.1.

The intrinsic functional potential $F_{v}$ is a functional of density $n(x)$
and is independent of the external potential:

$\frac{\delta F_{v}(\beta;n]}{\delta v(x^{\prime})}=0\quad\text{for fixed
$\beta$ and $n(x)$.}$ (4.26)

###### Proof.

The crucial observation behind the DFT formalism is that $P$ and $Z_{v}$
depend on the external potential $v(x)$ and the Lagrange multipliers
$\alpha(x)$ only through the particular combination
$\bar{\alpha}(x)\equiv\beta v(x)+\alpha(x)$. Substitute equation (4.10) in
(4.24) to get

$\beta F_{v}(\beta;n]=logZ(\beta;\bar{\alpha}]+\int
d^{3}x\bar{\alpha}(x)n(x),$ (4.27)

where
$Z(\beta;\bar{\alpha}]=Z_{v}(\beta;\alpha]=Tr_{c}e^{-\beta(\hat{K}+\hat{U})-\int
d^{3}x\bar{\alpha}(x)\hat{n}_{x}}\;.$ The functional derivative of $\beta
F_{v}$ at fixed $n(x)$ and $\beta$ is

$\frac{\delta(\beta F_{v}(\beta;n])}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}=\int
d^{3}x^{\prime\prime}\frac{\delta}{\delta\bar{\alpha}(x^{\prime\prime})}\Big{[}logZ(\beta;\bar{\alpha}]+\int
d^{3}x\bar{\alpha}(x)n(x)\Big{]}\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}\;.$ (4.28)

Since $n(x^{\prime})=-\frac{\delta
logZ(\beta;\bar{\alpha}]}{\delta\bar{\alpha}(x^{\prime})}$, keeping $n(x)$
fixed is achieved by keeping $\bar{\alpha}(x)$ fixed:

$\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}=\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,\bar{\alpha}(x)}=0\;,$ (4.29)

so that

$\frac{\delta F_{v}(\beta,n]}{\delta v(x^{\prime})}\Big{|}_{\beta,n(x)}=0\;,$
(4.30)

which concludes the proof; thus, we can write down the intrinsic potential as

$F(\beta,n(x)]=F_{v}(\beta,n(x)]\;.$ (4.31)

∎

######  Remark 4.2.

Note that since a change of the external potential $v(x)$ can be compensated
by a suitable change of the multiplier $\alpha(x)$ in such a way as to keep
$\bar{\alpha}(x)$ fixed, such changes in $v(x)$ will have no effect on $n(x)$.
Therefore keeping $n(x)$ fixed on the left hand side of (4.27) means that
$\bar{\alpha}(x)$ on the right side is fixed too.

Now we can substitute equation (4.25) in (4.19), and define the chemical
potential

$\mu\equiv\frac{-\alpha_{eq}}{\beta}\;,$ (4.32)

to have

$\frac{\delta}{\delta n(x^{\prime})}\Bigg{[}\int
d^{3}xv(x)n(x)+F(\beta;n]-\mu\int d^{3}xn(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$
(4.33)

We can also substitute (4.25) in (4.20) to find

$v(x)+\frac{\delta F}{\delta n(x)}\Big{|}_{eq}=\mu,$ (4.34)

which allows us to define and interpret $\mu_{in}(x;n]\equiv\frac{\delta
F}{\delta n(x)}$ as the intrinsic chemical potential of the system. To proceed
further we also split $F$ into that of ideal gas plus the interaction part as

$F(\beta;n]=F_{id}(\beta;n]-\phi(\beta;n].$ (4.35)

Differentiating with $\frac{\delta}{\delta n(x)}$ gives

$\beta\mu_{in}(x;n]=log(\lambda^{3}n(x))-c(x;n]\;,$ (4.36)

where for a monatomic gas
$\lambda=\Big{(}\frac{2\pi\hbar^{2}}{m}\Big{)}^{1/2}$. The additional one body
potential $c(x;n]=\frac{\delta\phi}{\delta n(x)}$ is related to the Ornstein-
Zernike direct correlation function of non-uniform fluid [18, 19, 20] by

$c^{(2)}(x,x^{\prime};n]\equiv\frac{\delta c(x;n]}{\delta
n(x^{\prime})}=\frac{\delta^{2}\phi(\beta;n]}{\delta n(x)\delta
n(x^{\prime})}\;.$ (4.37)

##  5 Slowly Varying Density and Gradient Expansion

We have proved that the solution to equation (4.33) is the equilibrium
density. But the functional $F(\beta;n]$ needs to be approximated because the
direct calculation of $F$ involves calculating the canonical partition
function, the task which we have been avoiding to begin with. Therefore
different models of DFT may vary in their approach for guessing $F(\beta;n]$.
Now assume that we are interested in a monatomic fluid with a slowly varying
external potential. In our language, it means that we use the approximation
$\int d^{3}x\equiv\sum(\Delta x)^{3}$ where $\Delta x$ is much longer than the
density correlation length and the change in density in each volume element is
small compared to average density. This allows us to interpret each volume
element $(\Delta x)^{3}$ as a fluid at grand canonical equilibrium with the
rest of the fluid as its thermal and particle bath.
Similar to [4], we expand $F(\beta;n]$ as

$F(\beta;n]=\int d^{3}x\Big{[}f_{0}(n(x))+f_{2}(n(x))|\nabla
n(x)|^{2}+\mathcal{O}(\nabla^{4}n(x))\Big{]}\;.$ (5.1)

Differentiating with respect to $n(x)$ we have

$\mu_{in}(x;n]=\frac{\delta F}{\delta
n(x)}=f^{\prime}_{0}(n(x))-f^{\prime}_{2}(n(x))|\nabla
n(x)|^{2}-2f_{2}(n(x))\nabla^{2}n(x).$ (5.2)

In the absence of an external potential, $v(x)=0$, the second and the third
terms in the RHS of (5.2) vanish, and also from equation (4.34),
$\mu_{in}=\mu$ , therefore we have

$f_{0}^{\prime}(n)=\mu(n(x)),$ (5.3)

where $\mu(n(x))$ is chemical potential of a uniform fluid with density
$n=n(x)$. On the other hand, with the assumption that each volume element
behaves as if it is in grand canonical ensemble for itself under influence of
both external potential and additional one body interaction $c(x;n]$ we know
that the second derivative of $F$ is related to Ornstein-Zernike theory by

$\beta\frac{\delta^{2}F}{\delta n(x)\delta
n(x^{\prime})}=\frac{\delta(x-x^{\prime})}{n(x)}-c^{(2)}(x,x^{\prime};n]\;.$
(5.4)

Therefore we have a Taylor expansion of $F$ around the uniform density as

$\displaystyle F[n(x)]=$ $\displaystyle F[n]+\int d^{3}x\Big{[}\frac{\delta
F}{\delta n(x)}\Big{]}_{n_{eq}(x)}\tilde{n}(x)$ (5.5)
$\displaystyle+\frac{1}{2\beta}\int\int
d^{3}xd^{3}x^{\prime}\Big{[}\frac{\delta(x-x^{\prime})}{n(x)}-c^{(2)}(|x-x^{\prime}|;n]\Big{]}_{n_{eq}(x)}\tilde{n}(x)\tilde{n}(x^{\prime})+\dots,$

where $\tilde{n}(x)\equiv n(x)-n$, and $c^{(2)}(|x-x^{\prime}|;n]$ is the
direct correlation function of a uniform fluid with density $n=n(x)$. The
Fourier transform of the second integral in (5.5) gives

$\displaystyle\frac{1}{2\beta}\int\int
d^{3}xd^{3}x^{\prime}\Big{[}\frac{\delta(x-x^{\prime})}{n(x)}-c^{(2)}(|x-x^{\prime}|;n]\Big{]}_{n_{eq}(x)}\tilde{n}(x)\tilde{n}(x^{\prime})$
(5.6) $\displaystyle=\frac{-1}{2\beta
V}\sum_{q}\Big{(}c^{(2)}(q;n]-\frac{1}{n(q)}\Big{)}\tilde{n}(q)\tilde{n}(-q)\;,$

and comparing with (5.1) yields

$f_{0}^{\prime\prime}(n)=\frac{-1}{\beta}(a(n)-\frac{1}{n})\;,\quad\quad
f_{2}(n)=\frac{-b(n)}{2\beta},$ (5.7)

where the functions $a(n)$ and $b(n)$ are defined as coefficients of Fourier
transform of the Ornstein-Zernike direct correlation function by
$c^{2}(q;n]=a(n(q))+b(n(q))q^{2}+\dots$.
$b(n)$ is evaluated by linear response theory to find that

$f_{2}(n(x))=\frac{1}{12\beta}\int
d^{3}x^{\prime}|x-x^{\prime}|^{2}c^{(2)}(|x-x^{\prime}|;n].$ (5.8)

We can substitute equations (5.8) and (5.3) in (5.2) and use equilibrium
identity $\nabla\mu=0$ to find the integro-differential equation

$\nabla\Bigg{[}v(x)+\mu(n(x))-f_{2}^{\prime}(n(x))|\nabla
n(x)|^{2}-2f_{2}(n(x))\nabla^{2}n(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0,$ (5.9)

that determines the equilibrium density $\hat{n}_{eq}(x)$ in the presence of
external potential $v(x)$ given Ornstein-Zernike direct correlation function
of uniform fluid $c^{(2)}[n(x),|x-x^{\prime}|]$.

##  6 Conclusion

We have shown that the variational principle of classical DFT is a special
case of applying the method of maximum entropy to construct optimal
approximations in terms of those variables that capture the relevant physical
information namely, the particle density $n(x)$. It is worth emphasizing once
again: In this paper we have pursued the purely conceptual goal of finding how
DFT fits within the MaxEnt approach to statistical mechanics. The advantage of
achieving such an insight is the potential for future applications that lie
outside the reach of the current versions of DFT. As an illustration we have
discussed the already well-known example of a slowly varying inhomogeneous
fluid. Future research can be pursued in three different directions: i) To
show that the method of maximum entropy can also be used to derive the quantum
version of DFT. ii) To approach the Dynamic DFT [21], generalizing the idea to
non-equilibrium systems following the theory of maximum caliber [22]. iii) To
revisit the objective of section 5 and construct weighted DFTs [23, 24] using
the method of maximum entropy.

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# Spin-structures on real Bott manifolds with Kähler structure

Anna Gąsior and Rafał Lutowski  Anna Gąsior: Maria Curie-Skłodowska
University, Institute of Mathematics, pl. Marii Curie-Skłodowskiej 1, 20-031
Lublin, Poland<EMAIL_ADDRESS>Rafał Lutowski: Institute of Mathematics,
University of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland
<EMAIL_ADDRESS>
###### Abstract.

In 2011 Ishida gave a criterion for existence of Kähler structures on real
Bott manifolds. For those manifolds we present necessary and sufficients
condition for admitting spin structures.

###### Key words and phrases:

real Bott manifolds, Spin structure, Kähler structure

###### 2010 Mathematics Subject Classification:

Primary 53C27; Secondary 53C29, 53B35, 20H15

##  1\. Introduction

Let $\Gamma$ be a fundamental group of an $n$-dimensional _real Bott manifold_
$M$. From [7] we know that $\Gamma$ is a _semi-diagonal_ Bieberbach group (and
$M$ is a _semi-diagonal_ flat manifold). This means that:

  1. (i)

$\Gamma$ fits into a short exact sequence

$0\longrightarrow\operatorname{\mathbb{Z}}^{n}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\Gamma\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}C_{2}^{d}\longrightarrow
1,$ (1)

for some $1\leq d\leq n$, where $C_{2}$ is the cyclic group of order $2$;

  2. (ii)

the image of $\varrho\colon
C_{2}^{d}\to\operatorname{GL}(n,\operatorname{\mathbb{Z}})$ is a group of
diagonal matrices, where

$\varrho_{g}(z)=\iota^{-1}(\gamma\iota(z)\gamma^{-1}),$

for $z\in\operatorname{\mathbb{Z}}^{n},g\in C_{2}^{n}$ and $\gamma\in\Gamma$
is such that $\pi(\gamma)=g$.

Note that in the particular case of real Bott manifolds we can take $d=n$.

Up to diffeomorphism, $M$ is determined by a certain strictly upper triangular
matrix $A$ with coefficients in $\operatorname{\mathbb{F}}_{2}$. We call $A$ a
_Bott matrix_ and we denote the manifold $M$ by $M(A)$. In [6, Theorem 3.1]
Ishida gave a necessary and sufficient condition for existence of the Kähler
structure on $M(A)$: $A$ is of even dimension, say $2n$, and one can split its
columns into $n$ pairs of equal ones.

Let $M$ be a real Bott and Kähler manifold, an _RBK-manifold_ for short. In
this note we examine the existence of spin structures on $M$. We would like to
mention that the general condition for existence of spin structures on semi-
diagonal flat manifolds is considered in [4] and [5]. However, in the specific
case of RBK-manifolds, this condition can be formulated in much simpler –
purely combinatorial – form. Namely, if $\tilde{A}$ is obtained from $A$ by a
removal of one column from each pair of equal ones and $\tilde{S}_{i}$ is the
sum of elements from the $i$-th row of $\tilde{A}$, then we get:

######  Main Theorem.

An RBK-manifold $M(A)$, of dimension $2n$, admits a spin structure if and only
if for every $1\leq i\leq 2n$

$\tilde{S}_{i}=1\Longrightarrow A^{(i)}=0,$

where $A^{(i)}$ denotes the $i$-th column of the matrix $A$.

We prove the above theorem in Section 3. We take advantage of the description
of spin semi-diagonal flat manifolds presented in [9]. For the convenience of
readers it is recalled in Section 2.

##  2\. New definition of real Bott manifold

In this section we recall methods introduced in [10] and developed in [9]. Let
$S^{1}$ be a unit circle in $\mathbb{C}$ and we consider automorphisms
$g_{i}:S^{1}\to S^{1}$ given by

$g_{0}(z)=z,\;\;g_{1}(z)=-z,\;\;g_{2}(z)=\bar{z},\;\;g_{3}(z)=-\bar{z},$ (2)

for all $z\in S^{1}$. If we identify $S^{1}$ with
$\operatorname{\mathbb{R}}/\operatorname{\mathbb{Z}}$, then for each
${[t]\in\operatorname{\mathbb{R}}/\operatorname{\mathbb{Z}}}$ we get

$g_{0}([t])=[t],\;\;g_{1}([t])=\left[t+\frac{1}{2}\right],\;\;g_{2}([t])=[-t],\;\;g_{3}([t])=\left[-t+\frac{1}{2}\right].$
(3)

Let ${D}=\langle g_{i}:i=0,1,2,3\rangle$. Then ${D}\cong{C_{2}\times C_{2}}$
and $g_{3}=g_{1}g_{2}$. We define an action ${D}^{n}$ on $T^{n}$ by

$(t_{1},\ldots,t_{n})(z_{1},\ldots,z_{n})=(t_{1}z_{1},\ldots,t_{n}z_{n})$ (4)

for $(t_{1},\ldots,t_{n})\in{D}^{n}$ and $(z_{1},\ldots,z_{n})\in
T^{n}=\underbrace{S^{1}\times\ldots\times S^{1}}_{n}$.

By taking any $d$ generators of the group $C_{2}^{d}\subseteq D^{n}$, we
define a $(d\times n)$-matrix with entries in $D$. This in turn defines a
matrix with entries in the set $P=\\{0,1,2,3\\}$, under the identification
$i\leftrightarrow g_{i}$ for $i=0,1,2,3$. We call it a $P$-matrix of
$C_{2}^{d}$. Note that under the above identification, $P$ has the natural
structure of a vector space over $\operatorname{\mathbb{F}}_{2}$.

Although a group $C_{2}^{d}\subseteq D^{n}$ can have many $P$-matrices in
general, every such a matrix $E$ encodes some important properties of its
action the torus $T^{n}$. Namely, $C_{2}^{d}$ acts freely on $T^{n}$ if and
only if there is $1$ in the sum of any distinct collection of rows of $E$. In
this case $C_{2}^{d}$ is the holonomy group of the flat manifold
$T^{n}/C_{2}^{d}$ if and only if there is either $2$ or $3$ in the sum of any
distinct collection of rows of $E$ (see [9, Lemma 2.4]).

Let us consider the linear forms $\alpha,\beta\colon
P\to\mathbb{\operatorname{\mathbb{F}}}_{2}$ given by the following table

| 0 | 1 | 2 | 3
---|---|---|---|---
$\alpha$ | 0 | 1 | 1 | 0
$\beta$ | 0 | 1 | 0 | 1

Let $C_{2}^{d}\subseteq{D}^{n}$ and $j\in\\{1,2,\ldots,n\\}$. We define
epimorphisms

$\alpha_{j}:C_{2}^{d}\subseteq{D}^{n}\xrightarrow{pr^{\prime}_{j}}{P}\xrightarrow{\alpha}\operatorname{\mathbb{F}}_{2},\;\;\beta_{j}:C_{2}^{d}\subseteq{D}^{n}\xrightarrow{pr^{\prime}_{j}}{P}\xrightarrow{\beta}\operatorname{\mathbb{F}}_{2},$
(5)

where

$pr^{\prime}_{j}(g_{i_{1}},g_{i_{2}},\ldots,g_{i_{n}})=i_{j}$

for $g_{i_{1}},g_{i_{2}},\ldots,g_{i_{n}}\in D$. Since
$H^{1}(C_{2}^{d},\operatorname{\mathbb{F}}_{2})=\operatorname{Hom}(C_{2}^{d},\mathbb{\operatorname{\mathbb{F}}}_{2})$,
we can view $\alpha_{j}$ and $\beta_{j}$ as cohomology classes and define

$\theta_{j}=\alpha_{j}\cup\beta_{j}\in
H^{2}(C_{2}^{d},\operatorname{\mathbb{F}}_{2}),$ (6)

where $\cup$ denotes the cup product. Fix generators $b_{1},\ldots,b_{d}$ of
$C_{2}^{d}$. It is well known that
$H^{*}(C_{2}^{d},\operatorname{\mathbb{F}}_{2})\cong\operatorname{\mathbb{F}}_{2}[x_{1},\ldots,x_{d}]$
where $\\{x_{1},\ldots,x_{d}\\}$ is a basis of
$H^{1}(C_{2}^{d},\operatorname{\mathbb{F}}_{2})$ such that
$x_{i}(b_{j})=\delta_{ij}$ is the Kronecker delta. Hence, elements
$\alpha_{j}$ and $\beta_{j}$ correspond to

$\alpha_{j}=\sum_{i=1}^{d}\alpha(pr_{j}(b_{i}))x_{i},\;\;\;\beta_{j}=\sum_{i=1}^{d}\beta(pr_{j}(b_{i}))x_{i}\in
C_{2}^{d}[x_{1},\ldots,x_{d}],$ (7)

(see [1, Theorem 1.2 and Proposition 1.3]). Moreover, if $[p_{ij}]$ is a
$P$-matrix of $C_{2}^{d}$, which corresponds to the chosen generators, we can
write equations (6) and (7) as follows

$\alpha_{j}=\sum_{i=1}^{d}\alpha(p_{ij})x_{i},\;\;\beta_{j}=\sum_{i=1}^{d}\beta(p_{ij})x_{i},\;\;\theta_{j}=\alpha_{j}\beta_{j}.$
(8)

There is an exact sequence

$0\to
H^{1}(C_{2}^{d},\operatorname{\mathbb{F}}_{2})\xrightarrow{\pi^{*}}H^{1}(\Gamma,\operatorname{\mathbb{F}}_{2})\xrightarrow{\iota^{*}}H^{1}(\operatorname{\mathbb{Z}}^{n},\operatorname{\mathbb{F}}_{2})\xrightarrow{d_{2}}H^{2}(C_{2}^{d},\operatorname{\mathbb{F}}_{2})\xrightarrow{\pi^{*}}H^{2}(\Gamma,\operatorname{\mathbb{F}}_{2})$

where $d_{2}$ is the transgression and $\pi^{*}$ is induced by the quotient
map $\pi:\Gamma\to C_{2}^{d}$ (see [2]).

######  Proposition 2.1 ([9, Proposition 3.2]).

Suppose that free and diagonal action of $C_{2}^{d}$ on $T^{n}$ corresponds to
a $P$-matrix $E$, which defines elements $\alpha_{j},\beta_{j}$ and
$\theta_{j}$ as in (8), for $1\leq j\leq n$. Let $M=T^{n}/C_{2}^{d}$ and
$\Gamma=\pi_{1}(M)$ be associated to the group extension (1). Then

  1. (i)

$\theta_{l}=d_{2}(\varepsilon_{l})$, where
$\\{\varepsilon_{1},\ldots,\varepsilon_{n}\\}$ is the basis of
$H^{1}(\operatorname{\mathbb{Z}}^{n},\operatorname{\mathbb{F}}_{2})$ dual to
the standard basis of
$\operatorname{\mathbb{Z}}^{n}\otimes_{\operatorname{\mathbb{Z}}}\operatorname{\mathbb{F}}_{2}$,
for $l=1,\ldots,n$;

  2. (ii)

the total Stiefel-Whitney class of $M$ is given by

$w(M)=\pi^{*}\left(w\right)\in
H^{*}(\Gamma,\operatorname{\mathbb{F}}_{2})=H^{*}(M,\operatorname{\mathbb{F}}_{2}),$

where

$w=\prod_{j=1}^{n}\left(1+\alpha_{j}+\beta_{j}\right)\in\operatorname{\mathbb{F}}_{2}[x_{1},\ldots,x_{d}].$
(9)

We call the ideal

$\operatorname{I}_{E}=\langle\operatorname{Im}(d_{2})\rangle=\langle\theta_{1},\ldots,\theta_{n}\rangle\subseteq\operatorname{\mathbb{F}}_{2}[x_{1},x_{2},\ldots,x_{n}]$

the _characteristic ideal_ of $E$ and the quotient
$C_{E}=\operatorname{\mathbb{F}}_{2}[x_{1},\ldots,x_{d}]/\operatorname{I}_{E}$
– the _characteristic algebra_ of $E$.

######  Corollary 2.1 ([9, Corollary 3.3]).

Suppose that free and diagonal action of $C_{2}^{d}$ on $T^{n}$ corresponds to
a $P$-matrix $E$. There is a canonical homomorphism of graded algebras
$\Phi:C_{E}\to H^{*}(T^{n}/C_{2}^{d},\operatorname{\mathbb{F}}_{2})$ such that
$\Phi([w])=w(T^{n}/C_{2}^{d})$. Moreover, $\Phi$ is a monomorphism in degree
less that or equal to two.

######  Definition 2.1.

Given a $P$-matrix $E\in P^{d\times n}$, we define the Stiefel-Whitney class
of $E$, to be the class $[w]\in C_{E}$ defined by (9).

######  Corollary 2.2.

Using notation of Corollary 2.1, let $w_{2}$ be the sum of degree $2$ summands
of $w$. Then, by [3, Proposition on page 40], $T^{n}/C_{2}^{d}$ admits a spin
structure if and only if $w_{2}\in\operatorname{I}_{E}$.

Now, we describe a real Bott manifold $M(A)$ for strictly upper triangular
matrix $A=[a_{ij}]$ with entries 0 or 1. For every $1\leq i\leq n$ we define
an element $s_{i}$ of the group
$\operatorname{Iso}(\operatorname{\mathbb{R}}^{n})=O(n)\ltimes\operatorname{\mathbb{R}}^{n}$
of isometries of the euclidean space $\operatorname{\mathbb{R}}^{n}$ in the
following way. If $i\neq n$ then

$s_{i}=\left(\operatorname{diag}\left[1,\ldots,1,(-1)^{a_{i,i+1}},\ldots,(-1)^{a_{i,n}}\right],\left(0,\ldots,0,\frac{1}{2},0\ldots,0\right)^{T}\right)$
(10)

where $(-1)^{a_{i,i+1}}$ is at the $(i+1,i+1)$ position and $\frac{1}{2}$ is
on the $i$-th coordinate of the translational part. In addition
$s_{n}=\left(I,\left(0,0,\ldots,0,\frac{1}{2}\right)\right)$. The group
$\Gamma(A)$ generated by $s_{1},\ldots,s_{n}$ is a crystallographic group. The
subgroup generated by $s_{1}^{2},s_{2}^{2},...,s_{n}^{2}$ consists of all
translations of $\operatorname{\mathbb{Z}}^{n}$. The action of $\Gamma(A)$ on
$\operatorname{\mathbb{R}}^{n}$ is free and the orbit space
$M(A)=\operatorname{\mathbb{R}}^{n}/\Gamma(A)$ is a flat manifold – a closed
Riemannian manifold with zero sectional curvature.

##  3\. Main results

We keep the notation of the previous section. Using the same methods as in
[10] and [8], for each strictly upper triangular matrix $A=[a_{ij}]$ which
generates the fundamental group of real Bott manifold $M(A)$ we get that the
corresponding $P$-matrix $P_{A}=[p_{ij}]$ is of the form

$P_{A}=\begin{bmatrix}1&2a_{12}&\ldots&2a_{1,n-1}&2a_{1n}\\\
0&1&\ldots&2a_{2,n-1}&2a_{2n}\\\ &&\ddots&\\\ 0&0&\ldots&1&2a_{n-1,n}\\\
0&0&\ldots&0&1\end{bmatrix}.$ (11)

Note that in the above notation $2a_{ij}$ is multiplication in integers. To be
more specific, we have

$p_{ij}=\left\\{\begin{array}[]{ll}1&\text{ if }i=j,\\\ 0&\text{ if }i\neq
j\text{ and }a_{ij}=0,\\\ 2&\text{ if }i\neq j\text{ and }a_{ij}=1,\\\
\end{array}\right.$

for $1\leq i,j\leq n$. Using the form (11) of $P_{A}$ and definition of forms
$\alpha$ and $\beta$ we get that

$\displaystyle\alpha_{j}$
$\displaystyle=\sum_{i=1}^{j}\alpha\left(p_{ij}\right)x_{i}=\sum_{\begin{subarray}{c}i=1\end{subarray}}^{j-1}\alpha(p_{ij})x_{i}+\alpha(p_{jj})x_{j}=\sum_{\begin{subarray}{c}i=1\end{subarray}}^{j-1}a_{ij}x_{i}+x_{j},$
$\displaystyle\beta_{j}$
$\displaystyle=\sum_{i=1}^{j}\beta\left(p_{ij}\right)x_{i}=\beta\left(p_{jj}\right)x_{j}=x_{j}$

and hence

$\displaystyle\alpha_{j}+\beta_{j}$
$\displaystyle=\sum_{i=1}^{j-1}a_{ij}x_{i},$ (12) $\displaystyle\theta_{j}$
$\displaystyle=\sum_{\begin{subarray}{c}i=1\end{subarray}}^{j-1}a_{ij}x_{i}x_{j}+x_{j}^{2}=\sum_{i\neq
j}a_{ij}x_{i}x_{j}+x_{j}^{2}$ (13)

for all $1\leq j\leq n$. Note that in the last formula we take advantage of
the definition of a Bott matrix.

We go back to the matrix $A$ of the real Bott manifold $M(A)$. From [6] we
have the following necessary and sufficient condition for the existence of a
Kähler structure on $M(A)$

######  Theorem 3.1 ([6, Theorem 3.1]).

Let $A$ be $2n$-dimensional matrix of real Bott manifold $M(A)$. Then the
following conditions are equivalent:

  1. (i)

there exist $n$ subsets $\\{j_{1},j_{n+1}\\}\ldots\\{j_{n},j_{2n}\\}$ of the
set $\\{1,2,\ldots,2n\\}$ such that

$\bigcup_{k=1}^{n}\\{j_{k},j_{k+n}\\}=\\{1,2,\ldots,2n\\}$

and $A^{(j_{k})}=A^{(j_{k+n})}$ for all $1\leq k\leq n$, where $A^{(i)}$ is
the $i$-th column of the matrix $A$;

  2. (ii)

there exist a Kähler structure on $M(A)$.

Let $A$ be a Bott matrix of an RBK-manifold. Using the notation from Theorem
3.1, let $\tilde{A}=[A^{(j_{1})}A^{(j_{2})}\ldots A^{({j_{n}})}]$ be a matrix
obtained from $A$ by removing duplicated columns. Then
$\tilde{A}\in\operatorname{\mathbb{F}}_{2}^{2n\times n}$. Let

$\tilde{S}_{i}=\sum_{k=1}^{n}a_{ij_{k}}\in\operatorname{\mathbb{F}}_{2}$

denote the sum of elements in the $i$-th row of the matrix $\tilde{A}$, for
$1\leq i\leq 2n$. Let us recall our main theorem.

######  Main Theorem.

Let $A$ be a matrix of of $2n$-dimensional RBK-manifold $M(A)$. Then $M(A)$
admits a spin structure if and only if for every $1\leq i\leq 2n$

$\tilde{S}_{i}=1\Longrightarrow A^{(i)}=0.$

Proof. $M(A)$ is $2n$-dimensional RBK-manifold, so from (12) we get

$\alpha_{j_{k+n}}+\beta_{j_{k+n}}=\alpha_{j_{k}}+\beta_{j_{k}}=\sum_{i=1}^{j_{k}-1}a_{ij_{k}}x_{i},$

and

$\displaystyle w(M(A))$
$\displaystyle=\prod_{k=1}^{2n}\left(1+\alpha_{j_{k}}+\beta_{j_{k}}\right)=\prod_{k=1}^{n}\left(1+\alpha_{j_{k}}+\beta_{j_{k}}\right)^{2}$
$\displaystyle=\prod_{k=1}^{n}\left(1+\sum_{i=1}^{j_{k}-1}a_{ij_{k}}x_{i}\right)^{2}=\prod_{k=1}^{n}\left(1+\sum_{i=1}^{j_{k}-1}a_{ij_{k}}x_{i}^{2}\right).$

From the above considerations and from the definition of a Bott matrix we have

$\displaystyle w_{2}=w_{2}(M(A))$
$\displaystyle=\sum_{k=1}^{n}\sum_{i=1}^{j_{k}-1}a_{ij_{k}}x_{i}^{2}=\sum_{k=1}^{n}\sum_{i=1}^{2n}a_{ij_{k}}x_{i}^{2}$
$\displaystyle=\sum_{i=1}^{2n}\sum_{k=1}^{n}a_{ij_{k}}x_{i}^{2}=\sum_{i=1}^{2n}\tilde{S}_{i}x_{i}^{2}.$

Let $J=\\{j:\tilde{S}_{j}=1\\}\subset\\{1,\ldots,2n\\}$. Then

$w_{2}=\sum_{j\in J}x_{j}^{2}.$ (14)

By Corollary 2.2, the existence of a spin structure on $M(A)$ is equivalent to
$w_{2}\in\operatorname{I}_{P_{A}}$, which – by (13) – occurs if and only if

$\sum_{j\in J}x_{j}^{2}=\sum_{j\in J}\theta_{j}.$ (15)

The above holds if $A^{(j)}=0$, since then, using formula (13) again, we get
$\theta_{j}=x_{j}^{2}$, for $j\in J$.

Now, let

$K=\\{(i,j):a_{ij}=1,j\in J,1\leq i\leq 2n\\}.$

Then

$\sum_{j\in J}\theta_{j}=\sum_{j\in J}\left(\sum_{i\neq
j}a_{ij}x_{i}x_{j}+x_{j}^{2}\right)=\sum_{j\in J}x_{j}^{2}+\sum_{(i,j)\in
K}x_{i}x_{j}.$ (16)

From (14), (15) and (16) we have

$\sum_{j\in J}x_{j}^{2}=\sum_{j\in J}x_{j}^{2}+\sum_{(i,j)\in K}x_{i}x_{j},$

hence $K=\emptyset$, which means that the $j$-th column of the Bott matrix $A$
has only zero entries.

$\square$

######  Example 3.1.

Let

$A=\left[\begin{matrix}0&0&1&1&1&1\\\ 0&0&1&1&1&1\\\ 0&0&0&0&1&1\\\
0&0&0&0&1&1\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\end{matrix}\right]$

be a matrix of a manifold $M(A)$. Then

$\tilde{A}=\left[\begin{matrix}0&1&1\\\ 0&1&1\\\ 0&0&1\\\ 0&0&1\\\ 0&0&0\\\
0&0&0\end{matrix}\right],$

$\tilde{S}_{1}=\tilde{S}_{2}=\tilde{S}_{5}=\tilde{S}_{6}=0$,
$\tilde{S}_{3}=\tilde{S}_{4}=1$ and there are entries equal to $1$ in columns
$A^{(3)}$ and $A^{(4)}$, so $M(A)$ has no spin-structure.

At the end let us note an easy corollary of our main theorem.

######  Corollary 3.1.

Let $A\in\operatorname{\mathbb{F}}_{2}^{2n\times 2n}$ be a Bott matrix such
that $M(A)$ is a RBK-manifold. Let $T=\\{j:A^{(j)}\neq 0\\}$. If

$T=T_{1}\sqcup\ldots\sqcup T_{l}$

such that for every $1\leq k\leq l$ $T_{k}$ is a four-elements set and

$\forall_{i,j\in T_{k}}A^{(i)}=A^{(j)},$

then $M(A)$ admits a spin structure.

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