Deposition of Palaeo–Mesoproterozoic sedimentary rocks on the São Francisco–Congo craton started during Statherian taphrogenesis (1.8–1.75 Ga), as verified by ages of c. 1.7 Ga determined for volcanic rocks of the lower part of the Espinhaço Supergroup in the states of Minas Gerais and Bahia (Brazil). These basins contain volcanic rocks and conglomerates alternating with sandstones, argillites and dolomites, deposited in continental, transitional and marine environments. The rocks in the westernmost sector of the Congo Craton (Central Africa) compose the Chela Group, comprising sandstones, argillites and dolomites. In the easternmost region of the Congo Craton the Kibaran, Akanyaru, Kagera and Muva supergroups occur: the first three in the Kibaran Belt and the last in the Irumide Belt and on the Bangweulu Block. They consist predominantly of pelites and schists, sandstones and, in lesser proportion, conglomerates, deposited in shallow marine, fluvial and lacustrine environments. Their sedimentation ages are constrained through ages on felsic tuff layers as follows: Chela Group 1790±17 Ma, Kagera Supergroup 1780 ± 9 Ma, and Muva Supergroup 1879±13 Ma. These data show that broadly coeval and sedimentologically similar epi-continental sedimentary basins occurred on the São Francisco and Congo cratons, suggesting the possible existence of a long-lived wide epi-continental sea covering large areas of these cratons during Statherian times.


text:
Flick Baseball or MLB Power Pros Touch 2010 on a big screen while you're waiting for the first pitch to be thrown at Yankee Stadium? You'd better think again. Yahoo! Sports is reporting that the Yankees' security department is sending ticketholders home who show up to the park with Apple's">
Yankee Stadium bans the iPad
Jim Squires on May 24, 2010 at 4:40pm
Thinking about squeezing in a few innings of Flick Baseball or MLB Power Pros Touch 2010 on a big screen while you're waiting for the first pitch to be thrown at Yankee Stadium? You'd better think again. Yahoo! Sports is reporting that the Yankees' security department is sending ticketholders home who show up to the park with Apple's latest must-have item.
After hearing reports that a fan was turned away at a recent Yankees/Red Sox game for trying to bring their iPad with them, Yahoo! Sports contacted Yankees security for comment. Much to everyone's surprise, the story turned out to be true. iPads fall under the "no laptops" rule in their existing security policy, thereby banning the device from the stadium.
While not being able to bring your iPad to a Yankees game really shouldn't be that big of a deal (you're there to watch the game, remember?) I can't see how it could possibly be a security threat. For that matter, I can't imagine how a laptop could pose a problem either. It's a baseball stadium, not an airplane.
Regardless, if you were thinking of bringing along a big screen device for your big league day – presumably to enjoy the exceptional radio coverage provided by MLB At Bat – you'd better opt for the iPhone instead.
[via Yahoo! Sports]

redpajama_set_name: RedPajamaCommonCrawl


text:
NEO is Making Education in Minnesota Stronger
This feature will focus on educators who are using Epicenter and are making a difference in their communities.
Located in the North Star State, Novation Education Opportunities (NEO) is a leading authorizer of innovative, diverse, and effective charter schools throughout Minnesota.
While the Mall of America, hockey, and snow might be the first thing that comes to mind when people think of Minnesota, it is also one of the founding states of the charter school movement. Early on Minnesotans realized the transformational power of charter schools, and they looked to charters to improve and strengthen education throughout Minnesota.
In 2009 NEO formed as a nonprofit corporation and began authorizing schools in 2010. NEO currently authorizes 23 charter schools.
NEO was approved by the Minnesota Department of Education as a single-purpose authorizer and, therefore, is not part of a larger organization or part of another non-profit. Its mission is "to authorize and oversee charter schools through consistent, ongoing and robust evaluation to achieve significant and measurable student growth."
Prior to 2010, there were more than 50 authorizers in Minnesota; now there are 14. In 2009, the laws governing charters changed in Minnesota; more requirements were placed on authorizers, making it difficult for some to continue. Therefore, many authorizers chose not to continue and NEO received a number of schools through transfers, in addition to starting 10 new schools since that time.
"One thing that makes us unique is that, in our focus on academic performance, we also facilitate sharing, which was a main impetus for charter schools in Minnesota," said Wendy Swanson Choi, NEO's Executive Director. "The idea was that charter schools would find more effective and efficient ways to meet student needs and share that information, so everyone could learn through sharing effective practices."
NEO has also been using Epicenter for over four years and has found that Epicenter helps it fulfill its mission.
"Epicenter has given us the ability to collect information, organize it, secure it, and review it," Swanson Choi said. "Thanks to Epicenter, we have been able to reduce time in collecting documents and free up our time to provide meaningful and useful feedback to schools and facilitate sharing to reach more students with a higher quality education."
One of the exciting things about NEO is that it has 10 new schools, eight of which are already in operation. These schools had strong first years in academic and financial performance. NEO was able to provide meaningful and useful feedback and facilitate the sharing of effective practices with these schools to support their work in meeting student needs.
NEO's new schools will face the upcoming school year with the same positive outlook as their more established counterparts with strong enrollment and with strong academic growth, proficiency, or both—thanks in part to the effective practices that schools have shared with them.
"I can't imagine overseeing 23 schools in a meaningful and useful way without Epicenter," Swanson Choi said. "We have a list of 20-30 documents we must collect and review from each school throughout the year. Twenty documents from 23 schools is a lot of paperwork! If it weren't for Epicenter we would be spending our time on the phone with the schools trying to figure out who submitted what and when, leaving little time for feedback or sharing."
"One of the useful services with Epicenter is that schools get a heads-up on what is due to the state and can then check a box to certify that they submitted a document on time. That way, they don't need to duplicate services by sending documents to NEO and to the state; yet we know the school has met the requirement."
Not only does NEO serve schools statewide, they serve a very diverse portfolio of schools that serve a very diverse student population, which is also part of their mission. One of the reasons that they have achieved this is because they reach out to many community leaders and encourage them to think about starting schools to meet student needs. And they have had a positive response. For example, they authorize one of the only Korean language immersion language programs in the state. They also authorize one of the only high school Montessori programs in Minnesota. They authorize Avalon Charter School that is completely flat-managed, which means the teachers, and to an extent the students, are responsible for the management and decision making. In addition, NEO authorizes Lionsgate Academy, a school with a high-quality program for meeting the needs of all students, including students on the Autism Spectrum, and high schools that focus on a successful school to college and career transition. This diversity strengthens the knowledge and skill base of the portfolio of schools. Whatever a school team needs, there is likely a NEO-authorized school that can help by sharing strategies.
As the Executive Director, Wendy Swanson Choi first got involved in charters by being an ESL teacher and starting the ESL Department in the South St. Paul school district. She then became an education consultant, which she successfully turned into a full-time career for many years. That gave her the opportunity to wear many hats in both traditional district schools and charter schools, and meet school leaders statewide.
Her focus was on collecting data, analyzing it, and identifying strengths and needs to develop strategies and monitor the effectiveness of implementing the strategies.
She also served as an interim director and board member at Nova Classical Academy while her son was a student there, overseeing the financial turnaround and expansion to a high school. She served on the founding team and board of a new charter school, Venture Academy. And she worked at Minneapolis Public Schools, serving as their School Improvement Specialist to improve student performance. From there, she took her vast experience in education and went to NEO.
But Swanson Choi's experience goes beyond Minnesota; she had her start in education teaching English in Korea.
When she is not working hard to help Minnesotan students, Swanson Choi likes to go to the Boundary Waters Wilderness Area, which straddles the United States-Canada border and is located just west of Lake Superior.
"And to balance the work, I really like being out in the environment, especially canoeing and hiking," said Swanson Choi. "I also like being with my family—my husband and three sons, my three sisters and brother, and my parents who live two doors down. And I'm fortunate because I love my job and it is my passion!"
Filed Under: Articles, Newsletter, Uncategorized Tagged With: authorizer, education, minnesota, Neo, swanson choi


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Civilisation in the Service of Humanity Inter-Civilisation Dialogue A response to the paper by MP Jawad Abdullah Bo Hussein.
We greatly welcome this unique opportunity to participate in such an important dialogue.
The theme that has been taken up in this conference is a very necessary one in the world today. We live at a time when we are seeing more rapid changes than ever before in the history of civilization. Each year sees an increase in instability and violence, not only between nations but also within nations, and we also see a great deal of sorrow and suffering in the world, as negative traits (vices) become more and more prevalent.
So much of this can be avoided when nations and the individuals who comprise that nation find it possible to cultivate within themselves the shared human values that are being discussed here: the values of peace, respect, compassion and tolerance to name a few.
It is clear from this diverse gathering that that there are many individuals, groups and nations around the world who are diligently working towards inter-cultural harmony and understanding, and the Kingdom of Bahrain is to be commended for taking this excellent initiative in bringing together such an esteemed and enlightened group of people from many backgrounds.
The research paper presented by my learned brother the MP Jawad Abdullah Bo Hussein enlightens us to the fact that whatever cultural or religious background we come from, our essential and innate human values are the same. We may differ in some minor details, but there is certainly a platform of moral, ethical and spiritual principles that we can all agree on.
The coming together of people from many nations, cultures and religious backgrounds, to discuss, share and understand each other and their respective cultures better, is surely sowing seeds of change and hope for a better and more harmonious future. As my distinguished brother states in his paper, 'People are either your brothers in religion or in humanity'. At times we may not fully agree with each other on certain theological or other principles yet we can still aspire to a mutual human understanding, whilst at the same time accepting that others will have differing points of view and opinions. We can engender feelings of respect for the rights of another, we can listen with interest to one another's opinions, and we can draw upon and learn from one another's experience and culture. This is the point at which we can come together rather than allow ourselves to be divided.
It is a mark of the greatness of a human being when he can accept, without discrimination or dislike, a fellow human being whose opinions differ from their own. Such a person will realise that they themselves, their beliefs and opinions are not diminished by such. Rather; they would regard that the variety of life on earth with its many various facets is further testimony to the greatness, love and mercy of the Supreme.
We also understand, and it is made clear in the document, that the Creator made us all so that we can come together, know each other, respect each other and be in community. To know, to be open to dialogue, and to be educated in an enlightened way about other peoples and religions is an essential part of engendering good relationships with our neighbours. And this getting to know each other can be greatly enhanced when we approach it by seeing what unites us rather than simply what divides us: if we can hold the vision of another as a brother united not by religion, country or culture, but instead united by the fact that we are all a creation of the One Creator, then our vision will be a benevolent one. We can only profit by this type of interaction: it will add to the richness of our experience and depth of wisdom.
Here, the example of the Tree of Life, which you have so aptly taken as your logo, takes on an even deeper significance. It stands alone and its beauty is created by the multitude of branches, twigs and leaves, which are all connected and yet separate. We are all leaves of the tree: we may belong to different branches or religions, but we are all connected, and the Seed, which nourishes us all, is One.
I also fully agree with my learned brother that our mission on earth must be to incite whatever is good in humanity and to connect with fellow humans in a positive way to motivate their abilities, talents, and potential, so that we can all be enriched by the experience. When we are living in accordance with our highest principles and taking sustenance from the Supreme, then we are able to maintain an elevated way of living.
Whether we are referring to the physical body, or to a community, a nation or indeed the world itself, all parts need to be present, sound and in good working order for the whole to be complete and functioning as it is intended. The fingers of one hand, though they are separate and individual, each make their own unique contribution to the perfectly functioning hand, and each one has its necessary part to play.
Likewise, all levels and aspects of a society need to be functioning in a healthy way for the whole to be healthy. Each one of us needs to play our part in the best way we can. Values need to be apparent in all areas: governance, corporations, commerce, education and social enterprises, right down to the individual. In all of these areas, ethical and values education can play a part.
A person is fortunate who lives in a civilised community which upholds the rights of individuals. However, to accept civil rights and liberties as a part of a culture, also means to assume the responsibilities which go alongside those rights. When individuals can be motivated and inspired to live up to their higher nature and potential, then each 'cell' of this hypothetical body will contribute to a harmonious whole.
Taking personal responsibility is an essential part of living in a peaceful cohesive society, and my respected brother mentions the aspects of loyalty and paying taxes as the mark of a good citizen. This requires honesty, trust and a sense of community. People are able to co-exist successfully and to thrive as a community when they are also inspired to give back to society and each other. When there is a sense of belonging, responsibility and moral obligation, then society is strengthened. Other qualities equally important in building community are co-operation, humility and understanding. When a society's leaders display qualities that others can respect and aspire to, then trust is built. However, we as individuals need to learn to live peacefully with ourselves before we can live peacefully and successfully with others. Even the family unit today is in disorder because it is as though we have forgotten how to have respect for ourselves, or for each other. It is only possible to have respectful or loving relationship with those around us when we have respect for ourselves. Our birth family is the microcosm that reflects the macrocosm of the world. When we can remain stable, peaceful, strong, compassionate and loving within ourselves and with those immediately around us, then we can live peacefully with others: as a close family unit, as a valuable community member, as a participator in a civil or religious society, or as a citizen of the world.
Good governance guided by high moral values and respect for all, is a worthy aim for those in positions of responsibility, and we applaud the Kingdom of Bahrain and the individuals responsible for this conference as promoters of the ethical standards that Bahrain is aspiring to, and wishes to exemplify to the rest of the world.
Let us just spend a few minutes in reflection, going into the depth of the original values that we all share as human beings.


text:
Q: Rails 6 How to have multiple fields of a model as references to another table I'm currently creating a Rails app using rails 6. I have model named user_teams and a model named players. What I want to do is have multiple fields in the user_teams model such as player_one, player_two...player_six that are each an individual reference to a player.
I have implemented this but in a slightly janky way. I have the six fields in my user_team model and each is an integer of the id associated with the chosen player. I feel like there is a much better way to do this but I can't figure out how. Any help would be greatly appreciated!
Also once I have the references set up properly, how can I validate each player reference so that duplicate players can't be in the same user_team?
Example
UserTeam
  player_one: 1
  player_two: 1
Would not be allowed but
UserTeam
  player_one: 1
  player_two: 2
Would be allowed?

A: This is a standard many to many association and what you want is a join table that joins players and teams:
class Player < ApplicationRecord
  has_many :positions
  has_many :user_teams, through: :positions
end

class UserTeam < ApplicationRecord
  has_many :positions
  has_many :players, through: :positions
end

# generate the model and migration with:
# rails g model position player:belongs_to user_team:belongs_to
class Position < ApplicationRecord
  belongs_to :player
  belongs_to :user_team
end

Another way to set this up would be with has_and_belongs_to_many but as there is no join model there is no way to access any additional columns on the join table and you cannot join it directly.
You can validate the uniqueness of a player on a team by using validates_uniqueness with the scope option:
class Position < ApplicationRecord
  belongs_to :player
  belongs_to :user_team
  validates :player_id, uniqueness: { scope: :user_team_id }
end

This will prevent most duplicates but is prone to race conditions which could lead to duplicate data being inserted so it should be complimented with a unique index that enforces the constraint on the database level:
class CreatePositions < ActiveRecord::Migration[6.0]
  def change
    create_table :positions do |t|
      t.belongs_to :user_team, null: false, foreign_key: true
      t.belongs_to :player, null: false, foreign_key: true
      t.timestamps
    end
    add_index :positions, [:user_team_id, :player_id], unique: true
  end
end

A: UserTeam model should have field name &
Player model should have field user_team_id
model/UserTeam.rb
class UserTeam < ApplicationRecord
  has_many :players
end

model/Player.rb
class Player < ApplicationRecord
  belongs_to :user_team
end

redpajama_set_name: RedPajamaStackExchange


text:
Юрій Петрович Іванов ( 30 січня 1960, Сталіно, УРСР) — радянський український футболіст, нападник.

Життєпис
Вихованець футбольної школи СІ (Луганськ), перший тренер — В. Д. Добіжа. 1 липня 1978 року дебютував за «Зорю» (Ворошиловград) у чемпіонаті СРСР в домашній грі проти «Арарату» (1:0) — на 87-й хвилині замінив Юрія Колесникова. Влітку 1981 року перейшов у ЦСКА, за який зіграв два матчі в чемпіонаті — в поєдинках проти «Дніпра» і «Кубані» виходив за 15 хвилин до завершення матчу. У 1982 році грав у першій лізі за СКА Київ, у 1982-1984 роках — у другій лізі за «Колос» (Павлоград). У 1990 році виступав у чемпіонаті Сумської області за тростянецький «Автомобіліст» (4 матчі).

Примітки

Посилання

 Профіль гравця  на сайті «Український футбол»
 Профіль гравця на сайті footbook.ru
 Профіль гравця  на сайті teams.by
Уродженці Донецька
Радянські футболісти
Футболісти «Зорі» (Луганськ)
Футболісти ЦСКА (Москва)
Футболісти «Космоса» (Павлоград)

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}

A recent paper in the Monthly~\cite{Cha}
\emph{When are the roots of a polynomial real and distinct?} provided
the following elegant criterion:
\begin{theorem}\label{thm:inequality}
Let $f$ be a polynomial of degree $N\ge 1$ with real coefficients.
Then the zeros of $f$ are real and distinct if and only if
\begin{equation}\label{eqn:maininequality}
(f^{(j)}(x))^2 - f^{(j-1)}(x) f^{(j+1)}(x) > 0
\end{equation}
for all $x\in \mathbb R$ and all $1\le j \le n-1$.
\end{theorem}
The notation $f^{(j)}$ refers to the $j$th derivative of~$f$.

Suppose one had a polynomial and wanted to know if its roots are
real and distinct.
How could one actually check the condition in Theorem~\ref{thm:inequality}?
One possibility is to graph \eqref{eqn:maininequality} for each~$j$,
and observe whether the graphs always stay above the $x$-axis.
That idea was the motivation for seeking a condition that involved graphing the
polynomial itself.

To help build intuition,
consider the plots in Figure~\ref{fig:motivatingplots},
Each is a graph of part of a different high-degree polynomial which does \emph{not} have
only real zeros.  Try to decide if the information in each graph is sufficient
to conclude that the polynomial has a non-real zero.
Note that the graphs show the $x$-axis, making it possible to see
the zeros of the polynomials, but both the horizontal and vertical
scales are omitted because those are irrelevant to the discussion.

\begin{figure}[htp]
\scalebox{0.6}[0.6]{\includegraphics{polynonreal1.eps}}
\hskip 0.4in
\scalebox{0.6}[0.6]{\includegraphics{polynonreal2.eps}}
\vskip 0.4in
\scalebox{0.6}[0.6]{\includegraphics{polynonreal3.eps}}
\hskip 0.4in
\scalebox{0.6}[0.6]{\includegraphics{polynonreal4.eps}}
\caption{\sf
Graphs of portions of four different high-degree polynomials,
each of which has a pair of non-real zeros in the given range.
} \label{fig:motivatingplots}
\end{figure}

We review some basic facts about polynomials, and then discuss the
clues which can be seen in the graphs in Figure~\ref{fig:motivatingplots}.

\section{Algebraic and geometric properties of polynomials}

\term{Polynomials} are expressions of the form $f(x) = a_0 + a_1 x + \cdots + a_N x^N$.
The $a_j$, which are (real or complex) numbers, are the \term{coefficients}
of the polynomial.  If $a_N \not= 0$, then it is called the
\term{leading coefficient}, and we call $N$ the \term{degree}
of the polynomial. We write $\deg(f) = N$.
A \term{root}, also called a \term{zero}, of the polynomial
$f$ is a (real or complex) number $z$ such that $f(z)=0$.

If the degree of $f$ is at least one, then we say $f$ is a \term{nonconstant}
polynomial.  If $f$ is a nonconstant polynomial of degree~$N$ then
its derivative $f'$ is a polynomial of degree~$N-1$.  That is the first of
several statements we will encounter where a natural-looking assertion
is actually incorrect when applied to a constant polynomial.  Another
edge case is the \term{zero polynomial} $f(x) = 0$.  It is a polynomial,
but its degree is not defined.

The Fundamental Theorem of Algebra states that a degree $N$ polynomial
can be factored into a product of $N$ linear factors:
\begin{equation}\label{eqn:polyfactored}
f(x)=a_N(x-z_1)\cdots (x-z_N).
\end{equation}
The numbers $z_1$, \ldots, $z_N$ are the
roots of the polynomial.  If a root $z_j$ appears $k$ times then we say
that the root $z_j$ has \term{multiplicity}~$k$.  It follows from the
product rule that if a root of $f$ has multiplicity greater than~1,
then it is also a root of the derivative~$f'$, with multiplicity
decreased by one.
We observe that it takes $N+1$ numbers to specify a polynomial.
Those numbers could be the coefficients, or they could be the
leading coefficient and the zeros.

For the remainder of this paper, we will assume that all polynomials
have real coefficients.  We will refer to these as \term{real polynomials}.
A real polynomial defines a
differentiable function on the real numbers, so we can graph it,
as illustrated in Figure~\ref{fig:motivatingplots}.
The zeros of a real polynomial are either real, or they occur in
complex conjugate pairs:  if $z = x + i y$ is a root (here $x$ and $y$
are real, and $i$ is the imaginary unit, which satisfies $i^2 = -1$),
then its complex conjugate $\conj{z} = x - i y$ is also a root,
and those roots have the same multiplicity.  We say that the zeros of
a polynomial are \term{distinct} if every root has multiplicity one.
If we refer to ``the distinct real zeros'' of $f$, we mean the
list of roots where each root of higher multiplicity only appears once.
If we wish to refer to all zeros and want to be particularly clear that
we really mean \emph{all} zeros, we may say ``zeros counted with multiplicity.''

Since real polynomials are continuous and have continuous derivatives,
Rolle's theorem implies that between every consecutive pair of
real zeros, $(z_j, z_{j+1})$, with $z_j \not= z_{j+1}$,
there is a zero of the derivative.
Note that if $f$ has a multiple zero, meaning $z_j = z_{j+1}$,
the conclusion of Rolle's theorem still holds,
because of our previous observation on differentiation and multiple zeros.
However, we will not consider the case of multiple zeros in this paper.

Since the polynomial does not
change sign on the interval $(z_j, z_{j+1})$, on that interval either
the polynomial is positive and it has a local maximum,
or it is negative and it has a local minimum.
(It may also have other critical points on that interval.)
In particular, if the
polynomial $f$ has $n$ distinct real zeros, then its derivative $f'$ has at least
$n-1$ distinct real zeros.  It will have exactly $n-1$ distinct real zeros
if and only if there is
exactly one critical point between each adjacent pair of real zeros,
and no critical points for $x < z_0$ or $x > z_N$.
In that situation $f'$ could have $n-1$ distinct real zeros but more than $n-1$ real zeros
counted with multiplicity.
That would happen, for example, if there was a local maximum where the 2nd and 3rd derivatives were also zero.

Since $f'$ has one fewer
zero than~$f$, the above discussion proves:

\begin{theorem}\label{thm:easydirection} If a real nonconstant polynomial has only distinct real zeros,
then the same is true of its derivative.  Furthermore, for each nonconstant
derivative, every local maximum is positive, and every local minimum is negative.
\end{theorem}

The plots in Figure~\ref{fig:motivatingplots} are meant to suggest
the converse of Theorem~\ref{thm:easydirection}:

\begin{theorem}\label{thm:harddirection} Suppose $f$ is a real nonconstant polynomial
such that it and all of its
nonconstant derivatives have  every local maximum is positive, and every local minimum is negative.
Then $f$ has only real zeros, and all zeros are distinct.
\end{theorem}

Theorem~\ref{thm:harddirection} goes back at least to
Joseph Fourier (1768-1830).  We give Fourier's proof in Section~\ref{sec:fourier}.

\section{Intuition for non-real roots}
We build some intuition for Theorem~\ref{thm:harddirection} before proving it
by induction in the next section,
and then giving a more conceptual proof in  Section~\ref{sec:fourier}.

The main idea is that if $f$ does not have all real zeros, the non-real
zeros cause \term{funny business} in the graph of~$f$.  A good eye for
detecting funny business requires looking at plots of many polynomials.
The examples in Figure~\ref{fig:motivatingplots} were chosen to
illustrate the main ideas.

Let's start with the upper-left plot.  One can count that there are 10 real
zeros in the given domain.  The graph goes up and down in a pleasant
way, constrained by the fact that it must cross the axis at the real zeros.
Polynomials don't like to change direction too quickly, so when there
is a big gap between zeros, the polynomial moves further from the $x$-axis
before turning around.  The gap between adjacent real zeros is the most
important factor in determining how far the graph wanders from the axis,
but it is not the only factor.  For example, the gap between the 1st and 2nd
zeros is the same as the gap between the 3rd and the 4th, yet the graph
becomes more negative between that second pair.  This is due to the
influence of the neighboring zeros.  The minimum being less negative between
the 1st and 2nd zero is a clue
about the zeros which are slightly to the left of the displayed range.

But what is going on between the 6th and 7th real zeros?
Why does the graph start wiggling for no good reason?  There is a good
reason for that wiggling:  it is caused by a pair of conjugate non-real zeros of the polynomial.
That behavior is the most blatant example of funny business:  a positive local
minimum.  From the graph we can deduce some information about those
non-real zeros:  its real part is slightly closer to the 7th zero than
to the 6th zero, and its imaginary part is not too large (on the scale
of the gap between the adjacent real zeros).

Let's move on to the upper-right plot.  Again we see funny business
between the 6th and 7th real zeros, although it is not as extreme
as in the previous example.  It is also caused by a pair of conjugate
zeros, but those zeros have slightly larger imaginary part
than in the previous graph
(in fact, 50\% larger).  There is only one critical point in that
interval, in particular no positive local minimum.  But what will happen
when we take the derivative of that function?  Looking carefully at
the graph, we can see that the derivative will have a negative local
maximum.

Now consider the lower-left plot.  The only thing which has changed is
the imaginary part of the complex roots, increasing by another 30\%.
Again we see funny business, in the form of an odd bulge to the graph
as it moves from the local maximum down to the 7th real zero.
That type of funny business might not have an official name,
but if you are alert to look out for it, you can immediately
identify it as coming from non-real zeros.  If you differentiate,
the funny business becomes more pronounced.  The second derivative
has a positive local minimum.

The reason the funny business becomes more extreme with each derivative
(until all the zeros end up on the real axis and the funny business
stops) is that the imaginary
part of the non-real zeros decreases with each derivative.  Zeros
with large imaginary part cause a slight disturbance, while zeros
with small imaginary part have a greater effect.
The size of the imaginary part causing funny business
should be measured relative to the
spacing between the nearby zeros.

Finally, let's look at the lower-right graph. The only change is that
the imaginary part of the complex roots increased by another 30\%.
Can we see any funny
business there?  There might be some asymmetry between the 6th and 7th
zeros, but maybe not enough to draw a definite conclusion.
There is even more asymmetry between the 1st and 2nd zeros,
and that has nothing to do with non-real zeros.
Looking more closely there are other clues.  The height of the
maximum between the 6th and 7th zeros is about the same as
between the 2nd and 3rd, but the gap between the 6th and 7th is
much larger.  That contradicts our previous observation about
the gap being the dominant factor in determining the size of the
maximum.  So, something (i.e., a conjugate pair of non-real zeros)
is causing the maximum to be smaller than expected.

There is another clue.  Look closely at the shape of the graph near the
two largest local maxima.  The one on the left is more sharply curved, and more
quickly becomes approximately straight.  The one on the left is what
the graph of a polynomial with only real zeros should look like.
The one on the right is an example of funny business.
The 3rd derivative of that polynomial has a positive local minimum.

\section{Induction proof of Theorem~\ref{thm:harddirection}}

The proof will proceed by induction on the degree of the polynomial.
A different and more interesting proof,
using ideas of Fourier,
is in Section~\ref{sec:fourier}

Suppose $f$ is a degree 1 polynomial.  It has no non-constant derivatives,
so the conditions in the statement are vacuously true, so we want to conclude
that $f$ has only real zeros.  That is true because the graph of $f$ is a line
of nonzero slope (nonzero because of the definition of `degree') so the
graph must cross the $x$-axis.  Thus, $f$ has at least one real root,
which is all roots by the Fundamental Theorem of Algebra,
and the root is distinct because there is only one of them.

For the induction step we
will prove the contrapositive, which is that if $f$ does not have
all distinct real roots, then either $f$ or some derivative of $f$ must have
a local maximum that is non-positive, or a local minimum
that is non-negative.
So, suppose the theorem is true for all real polynomials of degree~$N$,
and suppose $f$ is a real polynomial of degree~$N+1$ which either has
a non-real root, or it has a real root that is non-distinct.

In the case of a non-distinct real root,  if the multiplicity is even,
then the root is either a local maximum or local minimum.  Since the
function is zero at that root, the maximum or minimum is non-positive
and non-negative, which is what we wanted to show.
(This situation will appear repeatedly, so we will abbreviate it
as ``has a zero maximum or minimum''.) If the multiplicity
is odd, the multiplicity must be at least 3, so the derivative
has a multiple root of even multiplicity, which again is what we wanted
to show.

Now we handle the case that the real roots of $f$ are distinct,
but not all roots are real.  If $z_1 \le z_2 \le \cdots \le z_M$ are the real roots of~$f$,
then $M\le N-1$ because the non-real roots of $f$ come in conjugate
pairs, so there must be at least two of them.

Suppose $f'$ has $M-1$ distinct real roots, which it the minimum
allowed by Rolle's theorem.  If those roots all have multiplicity one,
then $f'$ has non-real roots because the degree of $f'$ is at least~$M+1$.
Since the degree of $f'$ is $N$, this case is covered by the induction hypothesis.

Now suppose $f'$ has $M-1$ distinct real roots, at least one of which
has multiplicity greater than one.  Then as we argued previously,
either $f'$ or $f''$ has a zero maximum or minimum.

This leaves the final case, where $f'$ has at least $M$ distinct real zeros.
Either there must be a zero of $f'(x)$ outside the interval $(z_1, z_M)$,
or by the pigeon hole principle
at least two zeros in an interval $(z_{j}, z_{j+1})$.

In the case of a zero outside $(z_1, z_M)$, by replacing $f(x)$ by
$f(-x)$ and/or by $-f(x)$ we may assume the zero of $f'(x)$ lies in $x > z_M$,
and furthermore $f(x) > 0$ for $x > z_M$.
If that zero of $f'$ was a local minimum of $f$, we are done.  If it
is a local maximum, then since $f$ is increasing for $x$ sufficiently large,
there must be a local minimum to the right of that local maximum,
so again we are done.  The final possibility is that it is neither,
in which case it corresponds to a multiple zero of $f'$,  so either
$f'$ or $f''$ has a zero maximum or minimum,
as in previous cases.

The final possibility is that $f'$ has at least two zeros in an interval $(z_{j}, z_{j+1})$.
If the two zeros of $f'$ are not distinct, then either $f'$ or~$f''$ has a
zero minimum or maximum, and we are done.  Thus,  we may assume those two zeros of $f'$
in that interval are distinct.
As in the previous case, we may assume $f$ is positive on that interval,
and so it has a local maximum, which corresponds to one of those
distinct zeros of $f'$.  If the other zero of $f'$ corresponds to a minimum of $f$,
we are done.  If it corresponds to a maximum, we are done because there must be a
minimum between two maxima.  That leaves the third possibility that
the other zero of $f'$ corresponds to neither a maximum nor a minimum of~$f$,
but that means it is a multiple zero of $f'$, so again we are done.

This covers all cases in the induction step, so the proof is complete.

\section{Fourier's proof}\label{sec:fourier}
We now give a different proof, using ideas due to Joseph Fourier
of Fourier series fame.  The proof is logically the same, in the sense
that there is a direct correspondence between the ideas of Fourier
and the cases in the induction proof.  But
Fourier's proof has better explanatory power, and it also illustrates
the value of introducing useful terminology.
This section is written as if we were reading Fourier's mind; the author
was introduced to these ideas by reading a paper of Y.-O.~Kim~\cite{Kim}.

Rolle's theorem for a polynomial $f$ implies that there is a zero of $f'$
between each consecutive pair of real zeros, but sometimes you get ``extra''
zeros of $f'$.  Figure~\ref{fig:fourier} illustrates some of the possibilities.

\begin{figure}[htp]
\scalebox{1.3}[1.3]{\includegraphics{polyfourierplot.eps}}
\vskip 0.4in
\scalebox{1.3}[1.3]{\includegraphics{dpolyfourierplot.eps}}
\caption{\sf
Part of the graph of a high degree polynomial (top),
and its derivative (bottom).
The labeled points $A$, $B$, \ldots , $L$ are critical points in the top graph, equivalently, zeros of the derivative in the bottom graph.
In the bottom graph the subscript indicates the multiplicity of the zero.
} \label{fig:fourier}
\end{figure}

Fourier did not think all critical point were equally interesting.  For example,
critical point $B$ is not particularly interesting:  by Rolle's theorem
there has to be a critical point between the zeros on either side of $B$,
so there is nothing special about the point $B$.  Similarly,
none of $A$, $C$, $F$, $H$, $I$, $K$, or $L$ is interesting to Fourier.
The remaining critical points are interesting:  $D$, $E$, and $J$ because in
addition to $f'$ vanishing, so do some additional derivatives, so those
points contribute ``extra'' zeros to $f$; and point $G$ is interesting because
it is a negative local maximum, which is not predicted by
Rolle's theorem.  Table~\ref{tab:fourier} compiles information about the
behavior at each critical point.  The entries $+$, $0$, and $-$ refer to
the function being positive, negative, or zero; a $*$ means a nonzero
value (which could be positive or negative; and a blank entry means
the value is irrelevant in terms of Fourier's classification of critical points.

\begin{table}[]
\begin{tabular}{cccccccc}
$x$ &  $f(x)$  & $f'(x)$  & $f''(x)$ & $f'''(x)$ & $f^{(4)}(x)$ & $f^{(5)}(x)$ & $k_{f'}(x)$ \\
\hline
$A$   & $-$   & 0  &  $+$  &  & &&0 \\
$B$   & $+$   & 0  &  $-$  &  & &&0 \\
$C$   & $-$   & 0  &  $+$  &  & &&0 \\
$D$   & $*$   & 0  &  0  & $*$  & &&1  \\
$E$   & $+$   & 0  &  0  & 0  & $-$ &&1 \\
$F$   & $-$   & 0  &  $+$  &  &&&0  \\
$G$   & $-$   & 0  &  $-$  &  & &&1 \\
$H$   & $-$   & 0  &  $+$  &  &&&0  \\
$I$   & $+$   & 0  &  $-$  &  &&&0  \\
$J$   & $*$   & 0  &  0  & 0  & 0 & $*$ & 2\\
$K$   & $-$   & 0  &  $+$  &  &&&0  \\
$L$   & $+$   & 0  &  $-$  &  & &&0 \\
\end{tabular}
\caption{Signs of the function in Figure~\ref{fig:fourier} and
its derivatives.  The final column indicates the number of pairs of
non-real zeros which cause the behavior of the other entries in the row.
}\label{tab:fourier}
\end{table}

The final column of Table~\ref{tab:fourier} is a
number that expresses how ``interesting'' the critical point $x$ is,
in Fourier's view.
More specifically, $2k_{f'}(x)$ is the number of ``extra'' zeros that $f'(x)$ has
arising from the point~$x$,
where by \emph{extra} we mean ``more than one would expect only
from Rolle's theorem.''
The extra zeros might be at~$x$, which happens at $D$, $E$, and $J$.
Or they might be nearby: the second extra zero from $G$ is either $F$ or $H$,
although it is not meaningful to specify which one it is.

 Looking at the plots in Figure~\ref{fig:fourier},
we see that $f$ has 9 real zeros in that interval, so by Rolle's
theorem $f'$ should have at least 8 real zeros in that interval.
We can count that $f'$ actually has 18 real zeros counting multiplicity
(because $D$ has multiplicity~2,
$E$ has multiplicity~3, and $J$ has multiplicity~4).  There are 10
extra zeros, which is twice the sum of the values of $k_{f'}$ in
Table~\ref{tab:fourier}.

Another interpretation of $k_{f'}(\alpha)$ is that it is the number of pairs of complex
conjugate zeros of $f$ which are needed to explain the funny business at
the critical point~$\alpha$.

To relate to our earlier discussions, if $k_{f'}(\alpha) > 0$,
then either $f$ or $f'$ or $f''$
has a non-negative minimum or non-positive maximum at~$\alpha$.

Here is a formula for~$k_{f'}$.  Suppose $\alpha$ is a critical point
which is not a zero of $f$, and furthermore suppose
\begin{equation}
f'(\alpha) = f''(\alpha) = \cdots = f^{(m)}(\alpha) = 0
\end{equation}
and $f^{(m+1)}(\alpha) \not = 0$.  That is, exactly the first $m$
derivatives of $f$ vanish at~$\alpha$.  Then
\begin{equation}
k_{f'}(\alpha) = \begin{cases}
\frac{m}{2} & \text{if $m$ is even} \cr
\frac{m-1}{2} & \text{if $m$ is odd and $f(\alpha)f^{(m+1)}(\alpha) < 0$} \cr
\frac{m+1}{2} & \text{if $m$ is odd and $f(\alpha)f^{(m+1)}(\alpha) > 0$}.
\end{cases}
\end{equation}
That formula agrees with the values in Table~\ref{tab:fourier},
and we leave it as an exercise to check the other cases.
We set $k_{f'}(\alpha) = 0$ if $f'$ is identically zero,
or if $\alpha$ is not a zero of $f'$,
so $k_{f'}(\alpha)$ is defined for all $\alpha \in \mathbb C$,
but it is $0$ for almost all~$\alpha$.
We will call $k_{f'}(\alpha)$ the \emph{Fourier multiplicity of the
critical zero $\alpha$ of $f'$}.

Now we assemble the above ideas into Fourier's proof of
Theorem~\ref{thm:harddirection}, introducing two new functions
which help us keep track of the quantities of interest.  First is the
function $K(g)$ which adds up all of the Fourier multiplicities of $g$:
\begin{equation}
K(g) =  \sum_{\alpha \in \mathbb C} k_{g} (\alpha).
\end{equation}
The sum is over all complex numbers~$\alpha$, but recall that
$k_{g} (\alpha) = 0$ for all but finitely many~$\alpha$,
so really it is a finite sum.

We use $K(g)$ to express an observation we made earlier.
Since $k_{f'}(\alpha)$ counts the number of pairs of non-real
zeros of $f'$ which are causing the funny business near $x=\alpha$
in the graph of $f(x)$, we have:
\begin{equation}\label{eqn:countfunny}
\text{(number of non-real zeros of $f$)}
- \text{(number of non-real zeros of $f'$)}
= 2 K(f').
\end{equation}
The factor of $2$ of the right of~\eqref{eqn:countfunny} comes from the
fact that $k_{f'}(x)$ counts \emph{pairs} of non-real zeros.
As a formula, $Z_C(f) - Z_C(f') = 2 K(f')$, where
$Z_C(g)$ counts the number of non-real zeros of the nonzero polynomial~$g$
(if $g$ is identically zero, we set $Z_C(g)=0$).
That was Fourier's key insight into the proof of Theorem~\ref{thm:harddirection}.
Rearranging and then using induction, we have
\begin{align}
Z_C(f) =\mathstrut& Z_C(f') + 2 K(f') \cr
Z_C(f) =\mathstrut& Z_C(f'') + 2 K(f') + 2 K(f'')\cr
\vdots\ \,& \cr
Z_C(f) =\mathstrut& Z_C(f^{(n)}) + 2 \sum_{j=1}^n K(f^{(j)})
\end{align}
for any $n \ge 1$.

If $n$ is larger than the degree of $f$ then $f^{(n)}$ is identically zero
so $Z_C(f^{(n)})=0$ and we have $Z_C(f) = 2 \sum_{j=1}^n K(f^{(j)})$.
If $f$ does not have all real zeros, then $Z_C(f) > 0$, so
$ K(f^{(j)}) > 0$ for some $j$.  But that implies
at least one of $f^{(j-1)}$, $f^{(j)}$, or $f^{(j+1)}$ has a non-negative
minimum or non-positive local maximum.  That completes Fourier's
proof of Theorem~\ref{thm:harddirection}.

The reader is invited to locate where in Fourier's proof we used
the fact that the roots of $f$ are distinct.

\section{Infinitely many zeros}

A question on MathOverflow~\cite{MO} asked
about an equivalence to the Riemann Hypothesis stated in the
official description~\cite{Cl} of the Riemann Hypothesis Millennium Problem:
\begin{quote}
\emph{
The Riemann hypothesis is equivalent to the statement that all local maxima
of $\Xi(t)$ are positive and all local minima are negative, \ldots.
}
\end{quote}
Here $\Xi(t)$, called the Riemann $\Xi$-function,
is closely related to the Riemann zeta-function.
See Section~\ref{sec:zeta} for definitions and background.
For this immediate discussion, it is sufficient to know that
$\Xi(t)$ is real for real~$t$, and the Riemann Hypothesis
is the assertion that all zeros of $\Xi(t)$ are real.
So, it is similar to the real polynomials we have been discussing,
except it has infinitely many zeros.

The condition for a polynomial to have only real distinct zeros
involved maxima and minima of the original function, and also
of all derivatives; however, the quote above does not
mention derivatives.  Is our understanding of polynomials giving
us good intuition for functions with infinitely many zeros,
and so we should be skeptical of that claimed equivalence to
the Riemann Hypothesis?

As we will explain,
our intuition for polynomials is very likely guiding us in the
right direction.  However, establishing an analogue of Theorem~\ref{thm:harddirection}
for a class of functions which includes the Riemann $\Xi$-function,
is an unsolved problem!

Understanding the similarities between polynomials and certain
(but not all)
functions with infinitely many zeros,
and the relevance to the Riemann Hypothesis,
is the topic of the remainder of this paper.

\subsection{The Riemann zeta-function and the prime numbers}
The Riemann $\zeta$-function is defined by the Dirichlet series
\begin{equation}\label{eqn:zetaseries}
\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} .
\end{equation}

That series was first studied by Euler, who used it to prove that there are
infinitely many prime numbers.  The infinitude of primes had already been proven by Euclid,
but Euler's result is
better:  he showed that the series $\sum_{n=1}^\infty 1/p_n$ diverges.
Here $p_1, p_2, p_3,\ldots = 2, 3, 5, \ldots$ is the sequence of prime numbers.
If the sequence of primes increased very quickly (a possibility that is
allowed by Euclid's proof), then Euler's series $\sum_{n=1}^\infty 1/p_n$
might converge.  Euler's result puts limitations on how quickly the
sequence of primes can grow.

Euler's proof uses the series~\eqref{eqn:zetaseries} for $s>1$ a real number.
He exploits the fact that the series grows without bound as $s\to 1^+$,
which follows from the fact that the harmonic series diverges.  But it is Riemann's name, not Euler's,
which is attached to the $\zeta$-function, and rightly so.
Riemann had the brilliant insight to consider $\zeta(s)$ for $s$ a complex number.

What does the series \eqref{eqn:zetaseries} mean when $s$ is not real?
Let $s=\sigma + i t$ be a complex variable, where $\sigma$ and $t$ are
real variables.  (The seemingly unusual choice of variable names
has become standard when discussing the Riemann $\zeta$-function.)
The exponential of a complex number is defined by Euler's formula
$e^{i \theta} = \cos(\theta) + i \sin(\theta)$ for $\theta\in \mathbb R$.
Thus,
\begin{equation}
n^s = n^{\sigma + i t} = n^\sigma n^{it} = n^\sigma e^{i t \log(n)}
= n^\sigma (\cos(t \log(n)) + i \sin(t \log(n)) ).
\end{equation}

That formula explains what the terms in the series mean for complex~$s$, but
where does the series converge?
For any real number $\theta$ we have
\begin{equation}
| \cos(\theta) + i \sin(\theta) | = \sqrt{\cos^2(\theta) + \sin^2(\theta)} = \sqrt{1} = 1 .
\end{equation}
Thus,  $|n^s| = n^\sigma$.  Therefore, by the so-called\footnote{We write ``so-called'' because it was a terrible idea to name a result after the
variable in the expression, since the letter used for the variable is
an arbitrary choice.  The name ``zeta test'' is better.} ``$p$-series test'' from
calculus, which we will call the ``zeta test,''
\eqref{eqn:zetaseries} converges absolutely for $\sigma > 1$.

The Dirichlet series for $\zeta(s)$ only makes sense for $\sigma > 1$,
but there are other expressions for the $\zeta$-function, which  we will
not describe here, which allow us to define $\zeta(s)$ for all $s$ except
for $s=1$.  At $s=1$ the $\zeta$-function has a ``simple pole,'' which means
that $|\zeta(s)| \sim c/|s-1|$  as $s\to 1$.  In other words, the
$\zeta$-function ``blows up'' for $s$ near~$1$, but in the nicest possible way.

Riemann discovered that the zeros of the $\zeta$-function encode
secrets about the prime numbers.  The \emph{prime number theorem} is
the assertion that $p_n \sim n\log(n)$, meaning that
$\lim_{n\to\infty} \frac{p_n}{n\log(n)} = 1$.  Here ``$\log$'' means
``natural logarithm.''
Here is a silly proof of Euler's theorem that $\sum_{n=1}^\infty 1/p_n$ diverges:
combine the
prime number theorem, the integral test, and the limit
comparison test.
That proof is silly because the prime number theorem is a deep result,
and Euler's theorem can be proven with much more mundane input.

The prime number theorem was proven in 1896
by Hadamard and de la Vall\`ee Poussin, independently.
The key step in their proofs was
showing that $\zeta(s)$ has no zeros on the $\sigma=1$ line.
Riemann knew that the $\zeta$-function had no zeros for $\sigma > 1$.
The prime number theorem followed by turning that into a non-strict inequality.
That may seem like a minor improvement, but in fact it was one of the
major results of 19th century mathematics.

For the remainder of this paper we will set aside connections to the prime numbers and just
focus on analytic properties of the $\zeta$-function.

\subsection{Symmetries of the Riemann zeta-function}\label{sec:zeta}
The $\zeta$-function has a hidden symmetry which is revealed by the
Riemann $\xi$-function, defined by
\begin{equation}\label{eqn:xi}
\xi(s) = s (s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s) .
\end{equation}
Here $\Gamma(s)$ is Euler's Gamma-function, defined by
\begin{equation}\label{eqn:Gamma}
\Gamma(s) = \int_0^\infty e^x x^s \frac{dx}{x} .
\end{equation}
The integral~\eqref{eqn:Gamma} converges for~$\sigma > 0$, and one can also
check that~$\Gamma(1)=1$.  Just as happened with
the Dirichlet series for $\zeta(s)$, we have an expression for $\Gamma(s)$
which is valid for some $s$, but we want to make use of the $\Gamma$-function for other
values of~$s$.  Euler figured out how to do that.
Integrating by parts (with $u=x^{s-1}$ and $dv = e^x dx$) we
find
\begin{equation}
\Gamma(s) = (s - 1) \Gamma(s-1) .
\end{equation}
which us usually written in the form (which is equivalent, because we are just
renaming variables) $\Gamma(s+1) = s \Gamma(s)$.
If we rearrange that to get $\Gamma(s) = \Gamma(s+1)/s$, then we see that the
right side is known for $\sigma > -1$ except for $s=0$, and we can use that
as the definition for the left side.  Thus, we now know $\Gamma(s)$ for
$\sigma > -1$, except for $s=0$.  Repeating the same procedure we can define
$\Gamma(s)$ for $\sigma > -2$, except for $s=-1$ or $0$, then for
$\sigma > -3$, except for $s=-2$, $-1$, or $0$, and so on.

Euler's motivation was somewhat different:
he used the expression $\Gamma(s+1) = s \Gamma(s)$
to show that if $n$ is a positive integer, then $\Gamma(n) = (n-1)!$.
In other words, the $\Gamma$-function provides a way of extending the
factorial function to any positive real number.

Riemann showed that \eqref{eqn:xi} defines $\xi(s)$ for all $s\in\mathbb{C}$
and furthermore it satisfies the following surprising symmetry
which is called the \term{functional equation}:
\begin{equation}
\xi(s) = \xi(1-s) .
\end{equation}
In other words, the $\xi$-function is symmetric around the line $\sigma=\frac12$.
Combining the functional equation with the fact that $\xi(s)$ is real when
$s$ is real, we can conclude that
the zeros of the $\xi$-function either lie on the $\frac12$-line,
or occur in pairs symmetrically on either side of it.  The \term{Riemann hypothesis} is
the conjecture, made by Riemann in 1859, that all zeros of the $\xi$-function
actually lie on the $\sigma=\frac12$ line.  These days the line $\sigma=\frac12$ is known
as the \term{critical line}.

Note that the zeros of $\xi(s)$ are also
zeros of $\zeta(s)$, but $\zeta(s)$ actually has some additional zeros which
are considered unimportant and go by the name~\term{trivial zeros}.
The Riemann hypotheses is usually stated in the form ``the nontrivial zeros of
the Riemann zeta function lie on the critical line.''

We are one step away from connecting to our earlier discussion on real polynomials
with real zeros.  We can convert $\xi(s)$ to a function which is real on the
real axis by a simple change of variables, creating the Riemann $\Xi$-function:
\begin{equation}
\Xi(z) = \xi(\tfrac12 + i z) .
\end{equation}
Here $z=x + i y$ is a complex variable, and $x$ and $y$ are real.
We have: $\Xi(z)$ is real when $z$ is real, its zeros lie either on the real line
or in complex conjugate pairs, and the Riemann Hypotheses (RH) is the conjecture
that all zeros of~$\Xi(z)$ are real.  It is universally believed by all experts,
even those who are skeptical about RH, that all zeros of $\Xi(z)$ are simple.
Thus, we are in a situation parallel to our discussion about real polynomials
with distinct real zeros,
except that $\Xi(z)$ has infinitely many zeros.
Although we don't know that all the zeros of $\Xi$ are real,
we do know that all zeros either are real, or they lie close to the
real axis.  Combining the functional equation with the fact that $\zeta(s)$
is nonzero for $\sigma \ge 1$ shows that all zeros of $\Xi(z)$ lie
in the strip $-\frac12 < y < \frac12$.

One of the magical facts about polynomials is they can be written
as a product of linear factors,
each factor corresponds to a zero, and the zeros determine everything about
the polynomial except for a constant factor.  We will see that something
similar holds for certain functions that have infinitely many zeros.

\subsection{Another way to factor a polynomial}
Suppose $f(z) = a_0 + a_1 z + \cdots + a_n z^n$ is a polynomial
of degree~$n$, with roots $z_1$, \ldots, $z_n$.  If none of the
$z_j$ equal~$0$, then we can rearrange \eqref{eqn:polyfactored}  to get
\begin{align}\label{eqn:polyprod}
f(z) =\mathstrut & a_0 \left(1 - \frac{z}{z_1}\right)\cdots \left(1-\frac{z}{z_n}\right) \cr
= &\mathstrut a_0 \prod_{j=1}^{n} \left(1 - \frac{z}{z_j}\right),
\end{align}
where in the second line we used product notation, which is analogous to
summation notation.
If $0$ is a root with multiplicity~$k$, and $z_{k+1}$, \ldots, $z_{n}$ are the
nonzero roots, then we have
\begin{equation}\label{eqn:polyprod0}
f(z)
= \mathstrut a_k z^k \prod_{j=k+1}^{n} \left(1 - \frac{z}{z_j}\right).
\end{equation}

The product form~\eqref{eqn:polyprod} is useful because it can generalize
to the case of infinitely many zeros.  But before going into the details,
let's consider the concept of an infinite product.

\subsection{Infinite products}
Infinite sums
are familiar:  $\sum_{n=1}^\infty a_n$.  That sum might converge
or it might diverge.  It converges if the $a_n$ go to 0 fast enough.
Otherwise, it diverges.  An empty sum
is defined to equal~$0$.

Similarly, an infinite product, $\prod_{n=1}^\infty b_n$, might
converge, or it might diverge.  It converges if the $b_n$ go to 1
fast enough.  Otherwise, it diverges.
The empty product is defined to equal~$1$.

The definition of convergent infinite product is analogous to
convergent infinite sum, with one subtle difference:
\begin{definition}[Convergence of an infinite product]
The infinite product
\begin{equation}
\prod_{n=1}^\infty b_n
\end{equation}
\term{converges} if there exists $n_0$ such that
$b_n$ is nonzero for $n \ge n_0$,
and the limit
\begin{equation}\label{eqn:prodlim}
\lim_{N\to\infty} \prod_{n = n_0}^N b_n
\end{equation}
exists and is nonzero.
If $P$ is the limit in \eqref{eqn:prodlim}, then we say that the
infinite product converges to $b_1 \cdots b_{n_0 -1} P$.
\end{definition}

A possibly unexpected feature of the definition is that in a convergent
infinite product, only finitely many terms can be zero, and the product of
the nonzero terms also cannot equal zero.
That condition becomes natural
when relating infinite products to infinite sums.  Taking logarithms we have
\begin{equation}\label{eqn:logprod}
\log\left(\prod_{n = n_0}^N b_n \right) = \sum_{n = n_0}^N \log(b_n) .
\end{equation}
Since each $b_n$ in \eqref{eqn:logprod} is nonzero, and (for the infinite
product to have any hope of converging)  close to $1$ if
$n$ is sufficiently large, its logarithm is well-defined and close to~$0$.
If the limit as $N\to\infty$ exists on the right side of \eqref{eqn:logprod},
then the limit inside the parentheses on the left side of \eqref{eqn:logprod}
exists \emph{and is nonzero}.  Thus, \emph{the infinite product
$\prod_{n=1}^\infty b_n$ converges if and only if the infinite sum
$\sum_{n=n_0}^\infty \log(b_n)$ converges}.

Before looking at the convergence of some infinite products,
let's recall the geometric series
\begin{equation}
\sum_{n=0}^\infty x^n = \frac{1}{1-x}
\end{equation}
which we can integrate to obtain
\begin{equation}
\sum_{n=0}^\infty\frac{x^{n+1}}{n+1} = -\log(1-x) .
\end{equation}
A useful way to rearrange that expression is
\begin{equation}
\log(1+x) = x - \tfrac12 x^2 + \text{smaller terms}.
\end{equation}
We will use this to determine whether certain infinite products converge.

Let $z$ be any number.  Does the product
\begin{equation}
\prod_{j=1}^\infty \left(1 - \frac{z}{j}\right)
\end{equation}
converge?  First: are the terms nonzero if $j$ is sufficiently large?
Yes, we just need $j > |z|$.  Second, taking logarithms we have
\begin{align}
\log\left( \prod_{j=j_0}^\infty \left(1 - \frac{z}{j}\right) \right)
=\mathstrut &
\sum_{j=j_0}^\infty \log \left(1 - \frac{z}{j}\right) \cr
=\mathstrut &
\sum_{j=j_0}^\infty \left( - \frac{z}{j} + \frac12 \frac{z^2}{j^2} + \text{smaller terms}\right) .
\end{align}
We recognize the harmonic series, which diverges, so the product diverges.

Does the product
\begin{equation}
\prod_{j=1}^\infty \left(1 - \frac{z}{j^2}\right)
\end{equation}
converge?  First: are the terms nonzero if $j$ is sufficiently large?
Yes, we just need $j > \sqrt{|z|}$.  Second, taking logarithms we have
\begin{align}
\log\left( \prod_{j=j_0}^\infty \left(1 - \frac{z}{j^2}\right) \right)
=\mathstrut &
\sum_{j=j_0}^\infty \log \left(1 - \frac{z}{j^2}\right) \cr
=\mathstrut &
\sum_{j=j_0}^\infty \left( - \frac{z}{j^2} + \tfrac12 \frac{z^2}{j^4} + \text{smaller terms}\right) .
\end{align}
By the zeta test, those series converge, therefore so does the infinite product.

Does the product
\begin{equation}\label{eqn:prodwithexp}
\prod_{j=1}^\infty \left(1 - \frac{z}{j}\right) e^{z/j}
\end{equation}
converge? As in the first example above, the terms are nonzero if $j > |z|$.
When we take logarithms, the summand is
\begin{align}\label{eqn:expfactor}
\log \left( \left(1 - \frac{z}{j}\right) e^{z/j} \right)
=\mathstrut &
\log  \left(1 - \frac{z}{j}\right) + \log(e^{z/j}) \cr
= \mathstrut &
- \frac{z}{j} + \tfrac12 \frac{z^2}{j^2} + \text{smaller terms} + \frac{z}{j} \cr
= \mathstrut &  \tfrac12 \frac{z^2}{j^2} + \text{smaller terms} ,
\end{align}
so by the zeta test, the sum, and hence the product,
converges.

In the above discussion we have been somewhat imprecise in our use
of the approximation $\log(1 + x) = x - \tfrac12 x^2 + \cdots$.  The issue
which can arise is that just because a term is smaller, does not mean that an
infinite sum involving that term will be smaller.  For example,
consider
\begin{equation}
\sum_{n=1}^\infty \log\left(1 + \frac{(-1)^n}{\sqrt{\mathstrut n}}\right)
=\sum_{n=1}^\infty  \frac{(-1)^n}{\sqrt{\mathstrut n}} - \tfrac12 \sum_{n=1}^\infty \frac{1}{n} + \cdots .
\end{equation}
The first sum on the right converges (by the alternating series test).
The second sum has smaller terms (because $\frac{1}{\sqrt{n}} \ge \frac{1}{n}$),
but the second sum diverges because it is the harmonic series.
Thus, just because terms are smaller doesn't mean that we can ignore them.
But there is a case when we can:  when all the terms are positive.
That motivates this definition:
\begin{definition}\label{def:absconv}
The product
\begin{equation}
\prod_{n=1}^\infty (1 + c_n)
\end{equation}
\term{converges absolutely} if
\begin{equation}
\sum_{n=1}^\infty  c_n
\end{equation}
converges absolutely.
\end{definition}
The ideas in the previous discussion prove that an absolutely convergent
product converges.

\section{Entire functions of finite order}
Polynomials are classified by their degree, or equivalently, the number of zeros
counting multiplicity.  There is another, also equivalent way, to classify polynomials:
how fast they grow.  If $f(z)=a_0 + \cdots + a_n z^n$ is a polynomial of degree~$n$,
and $|z|$ is very large, then $|f(z)|$ is approximately~$|a_n z^n|$.
In other words, if $\mathcal N > \deg(f)$ and  $|z|$ is sufficiently large, then
\begin{equation}\label{eqn:degreebound}
\log(|f(z)|) < \mathcal N \log(|z|).
\end{equation}
We can use \eqref{eqn:degreebound} as an alternate definition of the degree of
a polynomial:  it is the infimum (greatest lower bound) of the numbers~$\mathcal N$
for which \eqref{eqn:degreebound} holds as $|z| \to \infty$.

The value of having multiple equivalent definitions is that some generalize
to other contexts, and some do not.  In this case, it is the rate of growth
which can be used to classify functions that have infinitely many zeros.

We will consider \term{entire} functions,
which means that the function has a Taylor series which converges for
all~$z\in \mathbb C$.  Examples are $\sin(z)$, $e^z$, polynomials, as well as
sums, products, and compositions of those functions.  Non-examples
are $\log(z)$, $1/z$, $\tan(z)$, and $\sqrt{z}$.
A good reference for the material in this section is~\cite{Lev}.

For polynomials there is a direct relationship between the growth of the
function and the number of zeros, but for entire functions the relationship
is not exact.  An entire function with many zeros must grow quickly, but
the function $e^z$ grows quickly even though it has no zeros.  It turns out
that is the complete story:  the growth of an entire function is
determined by its zeros, and by factors of the form~$e^{g(z)}$ where
$g$ is an entire function.  That motivates the definition of the
\term{order} of an entire function.

\begin{definition}  Suppose $f$ is an entire function, and suppose there
exists a real number $\mathcal N > 0$ such that if $z$ is sufficiently large,
\begin{equation}\label{eqn:orderN}
\log(|f(z)|) < |z|^{\mathcal N} .
\end{equation}
Then we say that $f$ \term{has finite order}.
If $f$ has finite order, then the \term{order} of~$f$ is the
infimum of the set of $\mathcal N$ such that~\eqref{eqn:orderN} holds
as $|z| \to \infty$.
\end{definition}

Combining~\eqref{eqn:degreebound} and \eqref{eqn:orderN}, and using the fact that
$\log(|z|)$ grows slower than any power of~$z$, we see that
polynomials have order~0.

The exponential function~$e^z$ has order~$1$,
and more generally if $g$ is a polynomial of degree~$N$ then
$e^{g(z)}$ has order~$N$.  Finite sums (or products) of functions of finite order
have finite order, and the order of the sum (or product) is at most
the order of the largest term (or factor).  The Riemann $\Xi$-function has
order~1, although that is not easy to deduce only from the information
we have provided in this paper.  The Euler $\Gamma$-function is not entire
because it is not defined at the negative integers, but $1/\Gamma(z)$ is an
entire function of order~1.

The beauty of entire functions of finite order is that they
can be written as an infinite product with a nice form.  This was
discovered by Hadamard, as part of his proof of the Prime Number Theorem.
We will state a special case which is sufficient for our needs.

\begin{theorem}[Hadamard]  Suppose $f$ is an entire function,
suppose~$0$ is a zero of $f$ of order~$m$, and
$z_1, z_2, \ldots$ are the non-zero zeros of~$f$.  If the order of $f$
is less than~1
then there exists $A\in \mathbb C$ such that
\begin{equation}\label{eqn:hadamard1}
f(z) = A z^m \prod_{n \ge 1} \left(1 - \frac{z}{z_n}\right) .
\end{equation}
If the order of $f$ is less than 2, then
 there exist numbers
$A, b \in \mathbb C$ such that
\begin{equation}\label{eqn:hadamard2}
f(z) = A e^{b z} z^m \prod_{n \ge 1} \left(1 - \frac{z}{z_n}\right) e^{z/z_n} .
\end{equation}
The products in \eqref{eqn:hadamard1} and \eqref{eqn:hadamard2}
converge absolutely for all~$z\in \mathbb C$.
\end{theorem}

By Definition~\ref{def:absconv},
the sum $\sum 1/|z_n|$ in \eqref{eqn:hadamard1} converges,
and the same ideas we used when discussing \eqref{eqn:prodwithexp} show that
in \eqref{eqn:hadamard2} the sum
$\sum 1/|z_n|^2$ converges.
In general, for a function of order~$\rho$
with non-zero zeros $z_1, z_2, \ldots$, the series $\sum 1/|z_n|^p$ converges
if~$p > \rho$.  Thus there is a close relationship between the order of a function
and the distribution of its zeros, with low order functions having zeros
distributed more sparsely.

Now we have the ingredients to extend our results on polynomials to
functions with infinitely many zeros.

\subsection{Zeros of derivatives}
Entire functions of order less than 2 are nice because the analogue of
Theorem~\ref{thm:easydirection} holds for them:
\begin{theorem}\label{thm:easyorder1} Suppose $f(z)$ is an entire function of
order less than~2, which is real
on the real axis and has only distinct real zeros.  Then all zeros of $f'$ are real,
and furthermore all local maxima of $f'$ are positive and all local
minima of $f'$ are negative.
\end{theorem}

Since the derivative of an entire function of order $\rho$ is entire
of order~$\rho$, the conclusion of Theorem~\ref{thm:easyorder1}
applies to every derivative~$f^{(n)}$ and not only~$f'$.

The proof of Theorem~\ref{thm:easydirection} used Rolle's theorem and the fact
that the derivative of a polynomial has one fewer zero than the original
polynomial.
However, it makes
no sense to talk about ``one fewer'' zero of the derivative when there
are infinitely many zeros.  Instead,
we will show that $f'$ has exactly one real zero between each pair of
consecutive real zeros of $f$, and no other zeros.

\begin{proof}
Suppose $f$ satisfies the conditions in the Theorem~\ref{thm:easyorder1}.
Since $f$ is an entire function of order less than~$2$, it has the
form~\eqref{eqn:hadamard2}.  Furthermore, $A$, $b$, and all $z_j$ are real.
Since $f'(z)/f(z) = \frac{d}{dz}\log(f(z))$,
we have
\begin{equation}\label{eqn:logderiv}
\frac{f'}{f}(z) = B + \sum_{j=1}^\infty \frac{1}{z - z_j} + \frac{1}{z_j} .
\end{equation}
Note that the series in~\eqref{eqn:logderiv} is defined whenever $z$ is not
a zero of $f$, and the sum converges absolutely wherever it is defined.

Suppose $z_j$ and $z_{j*}$ are consecutive zeros of $f$, and
$z_j < z < z_{j*}$.  We wish to show that $f'(z)$ has only
one zero in that interval.  Since $f(z)$ has constant sign on that interval,
this is equivalent to $f'(z)/f(z)$ having exactly one zero on that
interval.  That function has at least one zero on that interval because by
\eqref{eqn:logderiv}, that function is large and positive if $z$ is slightly
larger than $z_j$, and it is large and negative if $z$ is slightly
smaller than $z_{j*}$.

To show that $f'(z)$, equivalently $f'(z)/f(z)$, has only one zero in
that interval, it is sufficient to show that $f'(z)/f(z)$ is decreasing
on that interval.  But that follows from differentiating~\eqref{eqn:logderiv}:
\begin{equation}
\left(\frac{f'}{f}(z)\right)' = \sum_{j=1}^\infty -\frac{1}{(z - z_j)^2},
\end{equation}
which is negative for real $z$ whenever it is defined.
In particular it is negative on the interval $(z_j,z_{j*})$.

We have shown that $f'(z)$ has only one zero in each interval $(z_j, z_{j*})$,
which proves that all maxima of $f$ are positive  and all minima of
$f$ are negative.  Now we must prove that $f'$ has no non-real zeros.
In \eqref{eqn:logderiv} let $z=x + i y$, and then take the imaginary
part of that expression, making use of the fact that $B$ and $z_j$ are real:
\begin{equation}
\Im\left(\frac{f'}{f}(x + i y)\right) =
\sum_{j=1}^\infty - \frac{y}{(x-z_j)^2 + y^2} .
\end{equation}
That is nonzero if $y\not=0$, because every term is nonzero and has
the same sign, therefore $f'(z)/f(z)$ cannot be $0$ if
$z$ is not real, because its imaginary part is not zero.
\end{proof}

The above proof is not quite right.  The first two paragraphs implicitly
assumed that the zeros of $f$ range from $-\infty$ to $\infty$, but that
might not be the case:  there might be a largest, or a smallest, zero,
or there might only be finitely many zeros.  We leave it as an exercise to
work out the details for those cases.  Illustrative examples to consider
are $e^{z} p(z)$ where $p$ is a real polynomial.

\subsection{The converse}
Does the analogue of Theorem~\ref{thm:harddirection} hold for entire
functions of order less than~2?
Definitely not.  The function $e^z - 1$ real on the real line and is entire
of order~$1$.  It has zeros at $2\pi i n$ for all $n\in \mathbb Z$,
so it has infinitely many complex zeros.
But it has no real maxima or minima.  Its first derivative, and
every higher derivative, is $e^z$, which also has no real maxima or minima.

All is not lost:  our motivation was the Riemann $\Xi$-function,
which has order~1 and is real on the real axis.  It has one more
key property:  its complex zeros, if they exist, must have imaginary
part between $-\frac12$ and~$\frac12$, which follows from the fact that
$\zeta(s)$ has no zeros in~$\sigma \ge 1$.  That is, all zeros lie in a
strip around the real axis.   The counterexample $e^z - 1$ has zeros
arbitrarily far from the real axis.  Maybe the analogue of
Theorem~\ref{thm:harddirection} holds for functions with all zeros
near the real axis?

Y-.O.~Kim~\cite{Kim} proved a version with that extra assumption,
plus an additional extra assumption.

\begin{theorem}\label{thm:kimhalf}  Suppose $f(z)$ is an entire function of order~1
which is real on the real axis and has all zeros in the strip
$-A < y < A$ for some $A > 0$.  Let $w_1, w_2, \ldots$ be the
non-real zeros of $f$ and further suppose
\begin{equation}\label{eqn:sumthird}
\sum_j \frac{1}{|w_j|^p} \text{ converges for some $p < \frac12$. }
\end{equation}
If $f$ and all of its derivatives have every maximum positive and
every minimum negative, then $f$ has only real zeros, and all zeros
are distinct.
\end{theorem}

In other words, if the zeros are in a strip and there are not too many
complex zeros, then the analogue of Theorem~\ref{thm:harddirection}
is true.

Might Theorem~\ref{thm:kimhalf} be true without the extra condition on
the non-real zeros?  Or maybe if there are many non-real zeros, they can
somehow conspire to stay away from the real axis as the function is differentiated?
Those are research questions.

One could apply Theorem~\ref{thm:kimhalf} in its current form, of one could show
that the Riemann $\Xi$-function had sufficiently few non-real zeros.  This is
far beyond the realm of possibility in the foreseeable future.  At present
it has not even been shown that at least half of the zeros of the $\Xi$-function
are real.

At the start of this section we asked whether there might be an equivalence
to the Riemann Hypothesis that only involved the maxima and minima of
the $\Xi$-function and its derivatives.  Such an equivalence cannot follow
from a general statement about real entire functions of order~1,
as illustrated by the example
and $\cos(a z)(z^2 + \frac14)$.
That function has zeros at $\pm i/2$ and all other zeros are real.
As $a$ increases, the real zeros move closer together but the pair of
complex zeros do not move.
If $a$ is small,
the function has a positive local minimum, but if $a$ is large
enough, it does not.  As $a$ increases, it takes more and more derivatives
before a positive local minimum or negative local maximum appears.
That is a another version of a lesson from Figure~\ref{fig:motivatingplots}:
funny business comes from complex zeros, complex zeros close to the
real axis cause more funny business, and closeness to the real axis
should be measured relative to the gaps between nearby zeros.

redpajama_set_name: RedPajamaArXiv


text:
Home FAQ DeafBlind Culture & Touch
DeafBlind Culture & Touch
Touch & Positive Access for Learners who are DeafBlind
Touch & Positive Access for Learners who are DeafBlind Part 1 — 1 hour 38 minutes
Presentation slides 760 Kb
Image Description: Heather is standing facing the camera, smiling, with a light green background. Her hair is long and curly and she is wearing a dark green short-sleeved shirt.
Heather Withrow, Mother & DeafBlind Advocate
Certified Orientation & Mobility Instructor
Heather Withrow, also known as Hex, is a mother of three children, one of who is DeafBlind, one of who is Deaf, and one of who is a KODA (KODA: Hearing Kid of Deaf Adults). In addition to serving as an advocate for Deaf and DeafBlind Texans, she has received a Bachelor of Arts in graphic design from Gallaudet University and a Master of Education in Special Education from Texas Tech University. With a concentration in Orientation and Mobility and receiving a graduate certificate in Deafblindness, Hex is now a Certified Orientation and Mobility Specialist (COMS). She has also taught art and has coached volleyball and track and field.
Heather has served on various committees and boards that are to increase better learning and communication access for people who are DeafBlind. In one of her roles she serves on the development team for one of the modules in the Open Hands, Open Access modules on intervener training. Heather is also a member of the Texas Association for the Deaf (TAD), a lifetime member of National Family Association of DeafBlind (NFADB), a lifetime member of DeafBlind Multihandicapped Association of Texas (DBMAT) and a member of the Association for Education and Rehabilitation for the Blind and Visually Impaired (AER). As a mother who is Deaf and a native signer, Hex brings a very unique perspective to the learning and world of access to her son and to others who are DeafBlind.
Image Description: Orion "chillin'" on the swing. Young white boy with short white hair laying on his back on a turquoise colored mesh swing with a smile on his face.
Becoming @BlindishLatina: Intersectionality, Disability Pride and Advocating
Becoming @BlindishLatina: Intersectionality, Disability Pride and Advocating 1 hour 24 minutes
Presentation slides 6 MB
Catarina Rivera MSEd, MPH, CPACC
[Image: Light-skinned female smiling at the camera in a tan coat and black shirt with a tan zig zag pattern. She has long brown hair. The photo is taken outside on a street with a blurred background.]
Catarina is a disability public speaker and DEI* consultant who works with companies to improve disability awareness, inclusion, and accessibility. She has worked with over 35+ clients since launching her company and has been featured in Nasdaq, Authority Magazine, and as one of the top 21 DEI influencers of 2021. Past clients have included HarperCollins Publishers, LinkedIn, Hachette Book Group, Whole30, Grant Thornton, and BCG Digital Ventures.
In the webinar, disabled public speaker, DEI* consultant, and content creator Catarina Rivera, MSEd, MPH, CPACC discussed her personal disability journey with Usher Syndrome as well as intersectionality and why it's important. She will share her experience advocating for disability awareness and inclusion as a speaker and content creator.
*Diversity, Equity & Inclusion
Back to School: Increasing Environmental & Tactile Access
Back to School: Increasing Environmental & Tactile Access 1 hour 24 minutes
Photo of Rhonda Voight-Campbell, a white middle-aged female with short, wavy brown hair, smiling at the camera. She is layered with a dark plaid shirt and a navy blue vest and is seated indoors in a restaurant.
Rhonda Voight-Campbell (white; she/her/hers)
Rhonda is adjunct faculty at the Rochester Institute of Technology/National Technical Institute for the Deaf in the ASL and Interpreting Education Department. Specifically, she teaches undergraduate students the necessary skills to work effectively with DeafBlind people. Rhonda also has extensive training in protactile communication strategies and provides consultation on the topic throughout the DeafBlind community and other parties.
She is also the on-going contributing author to the "Tapping Thoughts" series sponsored by NYDBC.
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Hayley Broadway (white; she/her/hers/they)
Hayley is known as the Community Nudger as her love of community work and teaching brings her happiness. She enjoys spending time with her teen boys and spouse, exploring ideas with friends and colleagues and poetry. Creating opportunities for DeafBlind teachers and DeafBlind students and sharing that experience for more nurturing growth of Protactile language is one of her biggest joys.
With expertise in educational services, her continued work in the protactile-related field, she has founded a company, called Touch Seeds, in order to provide support and resources to DeafBlind children and their families.
Daily Home Routines
Feeding & Swallowing


text:
'Captain Fantastic' director Matt Ross signs up for sci-fi 'Tomorrow And Tomorrow'
By Jeremy Kay2017-02-23T22:30:00+00:00
Sci-fi crime thriller for TriStar based on 2014 novel by Tom Sweterlitsch.
Matt Ross, the director of Captain Fantastic with Viggo Mortensen, has signed up to direct sci-fi Tomorrow And Tomorrow for TriStar.
Mark Gordon is producing for The Mark Gordon Company alongside Entertainment 360 and Lynette Howell Taylor, with whom Ross collaborated on Captain Fantastic.
Tomorrow And Tomorrow will explore the interaction between reality and virtual reality.
Entertainment 360 found Sweterlitsch's book and brought it to Gordon. They pitched to TriStar where it sold in what the parties described as "a competitive auction."
"The triumph of Captain Fantastic is that it is at once funny, emotional and thought provoking," TriStar president Hannah Minghella said. "It's this ability to explore a thematic idea in a way that is both intelligent and entertaining that makes Matt such an exciting director for Tomorrow And Tomorrow.
"As a futuristic crime thriller, the book uses the near future to reflect on timely questions about our world today and how virtual reality impacts actual reality. It's Matt's interest in exploring the complex intersection of technology with morality and love that makes his vision for Tomorrow And Tomorrow especially compelling."
Ross added: "Tomorrow And Tomorrow is prescient. It posits a world not so dissimilar from today; a direction we are all clearly headed where technology has altered the ways in which we interact with each other and the world around us.
"I hope to examine, following the book's lead, the degree to which our lives are enhanced, and deeply compromised, by the technology that is already an inseparable part of our daily existence. Lynette and I couldn't be more excited to collaborate with Hannah to translate this book into the complex and relevant film we all believe it can be."
Captain Fantastic garnered a best actor Oscar nod for Viggo Mortensen and opened theatrically via Bleecker Street last July.
Mary Steenburgen among new cast on gay rom-com 'Happiest Season' for TriStar, eOne
Principal photography begins this week in Pittsburgh
Why Rotterdam's slimmed-down CineMart is good for the international film business
Less projects, more room to shine.
Netflix passes 100m international subscribers but new rivals dent US growth
Record growth in Europe, Latin America, Asia Pacific.
Only registered users or subscribers can comment on this article.
'Cat In The Wall' duo win top award at Trieste's When East Meets West
Mina Mileva and Vesela Kazakova are prepping a new project based on on true events.


text:
☰ Revisor of Missouri
Constitution Committee Publications
Other Links Help / FAQ Appendices and Tables
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Title VII CITIES, TOWNS AND VILLAGES
< > • Effective - 28 Aug 2011
87.205. Accidental disability retirement allowance. — 1. Upon retirement for accidental disability before August 28, 2011, a member shall receive seventy-five percent of the pay then provided by law for the highest step in the range of salary for the title or rank held by such member at the time of such retirement unless the member is permanently and totally incapacitated from performing any work, occupation or vocation of any kind whatsoever and is continuously confined to the member's home except for visits to obtain medical treatment, in which event the member may receive, in the discretion of the board of trustees, a retirement allowance in an amount not exceeding the member's rate of compensation as a firefighter in effect as of the date the allowance begins.
2. Anyone who has retired pursuant to the provisions of section 87.170 and has been reinstated pursuant to subsection 2 of section 87.130 who subsequently becomes disabled, as provided in section 87.200, shall receive a total benefit which is the higher of either the disability pension or the service pension.
3. Upon retirement for accidental disability on or after August 28, 2011, based on conditions of the heart, lungs, or cancer or based on permanent and total disability which will prevent the member from obtaining employment elsewhere, as determined by the board of trustees based on medical evidence presented by the retirement system's physicians, a member shall receive, regardless of his or her number of years of creditable service, seventy-five percent of the earnable compensation then provided for the step in the range of salary for the title or rank held by such member at the time of such retirement.
4. Except as provided in subsection 3 of this section, upon retirement for accidental disability on or after August 28, 2011, a member shall receive a base pension equal to twenty-five percent of the member's earnable compensation then provided for the step in the range of salary for the title or rank held by such member at the time of such retirement.
5. Except as provided in subsection 3 of this section, upon retirement for accidental disability on or after August 28, 2011, the member may elect to receive an education allowance in an amount not to exceed the tuition for a state resident at the University of Missouri-St. Louis. The accidentally disabled member shall enroll in a college, university, community college, or vocational or technical school at the first opportunity after the accidentally disabled member was retired and shall receive such educational allowance in the form of reimbursement upon proof of payment to such institution. The education allowance described in this subsection shall cease when the accidentally disabled member ceases to be a full-time student, fails to provide proof of achievement of a grade point average of two on a four-point scale or the equivalent on another scale for each academic term, or if the accidentally disabled member is restored to active service as a firefighter, but in no event shall such education allowance be available for more than five years after the member is retired under section 87.200.
6. Except as provided in subsection 3 of this section, upon retirement for accidental disability on or after August 28, 2011, in addition to the base pension provided for in subsection 4 of this section and the education allowance provided for in subsection 5 of this section, members with twenty-five years or less of creditable service shall receive an additional accidental retirement pension equal to two and three-fourths percent of the member's earnable compensation then provided for the step in the range of salary for the title or rank held by such member at the time of retirement for each year of creditable service equal to or greater than ten years but not more than twenty-five years.
7. Except as provided in subsection 3 of this section, upon retirement for accidental disability on or after August 28, 2011, in addition to the base pension provided for in subsection 4 of this section and the additional accidental retirement pension provided for in subsection 6 of this section, for members with twenty-five years or less of creditable service, then during such time that the disabled member is a full-time student in a college, university, community college, or vocational or technical school and is receiving the educational allowance provided for in subsection 5 of this section, such member shall also receive a supplemental disability retirement pension in the amount necessary so that his or her total accidental disability retirement pension, excluding the education allowance, shall be equal to one hundred percent of the earnable compensation then provided for the step in the range of salary for the title or rank held by such member at the time of such retirement. In no event shall such supplemental accidental disability pension be paid for a period more than five years after the member is retired under section 87.200.
8. Except as provided in subsection 3 of this section, upon retirement for accidental disability on or after August 28, 2011, in addition to the base pension provided for in subsection 4 of this section and the education allowance provided for in subsection 5 of this section, for members with more than twenty-five years of creditable service, such member shall also receive an additional pension equal to fifty percent of the member's earnable compensation then provided for the step in the range of salary for the title or rank held by such member at the time of such retirement.
9. Notwithstanding any other provisions in this section, upon retirement for accidental disability, other than as provided in subsection 3 of this section, on or after August 28, 2011, a member with more than twenty years of creditable service but not more than twenty-five years of creditable service may waive the right to receive the education allowance provided for in subsection 5 of this section, the right to additional pension retirement allowance provided for in subsection 6 of this section, and the right to receive the supplemental disability retirement pension provided for in subsection 7 of this section and may elect to receive instead in addition to the accidental disability retirement base pension as provided for in subsection 4 of this section an additional pension from the date of such member's retirement equal to forty percent of the member's earnable compensation then provided for the step in the range of salary for the title or rank held by such member at the time of such retirement. Any such election shall be made prior to such member's receipt of his or her first accidental disability pension payment.
--------
(L. 1959 S.B. 314 § 86.531, A.L. 1967 p. 168, A.L. 1996 H.B. 1404, A.L. 2001 S.B. 290, A.L. 2011 H.B. 282 merged with H.B. 664)
---- end of effective 28 Aug 2011 ----
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@02:28:54.6 9 :)


text:
Q: Handle nil in loop from list Golang I'm looping through some data received from a backend and might contain a nil in the middle, the program crashes with panic: runtime error: invalid memory address or nil pointer dereference
How can i handle the nil? Would like to set an nil to show "EMPTY"
I've tried this but still crashes:
    for _, res := range resp.Autombiles {
        for _, trk := range res.Trucks {
                if trk == nil {
                    println("EMPTY")
                } else {
                    println(*trk.EngineType)
                }
            }
        }
    }

Here's my loop:
    for _, res := range resp.Autombiles {
        for _, trk := range res.Trucks {
                fmt.Println(*trk.EngineType)
            }
        }
    }

A: Looks like res.Trucks is nil. So your program crashes before trk variable is created, here. for _, res := range res.Trucks{}
for _, res := range resp.Autombiles {

    //       res.Trucks can be nil
    for _, trk := range res.Trucks {
            ...
    }
}

So, add a check for existance of res.Trucks
for _, res := range resp.Autombiles {
    if res.Trucks != nil {
        for _, trk := range res.Trucks {
            if trk == nil || trk.EngineType == nil {
                println("EMPTY")
            } else {
                println(*trk.EngineType)
            }
        }
    }
}


text:
Q: How to impurely modify a state associated with an object? In Haskell, I have a container like:
data Container a = Container { length :: Int, buffer :: Unboxed.Vector (Int,a) }

This container is a flattened tree. Its accessor (!) performs a binary (log(N)) search through the vector in order to find the right bucket where index is stored.
(!) :: Container a -> Int -> a
container ! index = ... binary search ...

Since consecutive accesses are likely to be in the same bucket, this could be optimized in the following way:
if `index` is on the the last accessed bucket, skip the search

The tricky point is the last accessed bucket part. In JavaScript, I'd just impurely modify a hidden variable on the container object.
function read(index,object){

    var lastBucket = object.__lastBucket;

    // if the last bucket contains index, no need to search
    if (contains(object, lastBucket, index))
        var bucket = lastBucket;

    // if it doesn't
    else {
        // then we search the bucket
        var bucket = searchBucket(index,object);

        // And impurely annotate it on the container, so the
        // next time we access it we could skip the search.
        container.__lastBucket = bucket;
    }

    return object.buffer[bucket].value;
}

Since this is just an optimization and the result is the same independent of the branch taken, I believe it doesn't break referential transparency. How is it possible, in Haskell, to impurely modify an state associated with a runtime value?
~
I have thought in 2 possible solutions.

*

*A global, mutable hashmap linking pointers to the lastBucket value, and use unsafePerformIO to write on it. But I'd need a way to get the runtime pointer of an object, or at least an unique id of some sort (how?).

*Add an extra field to Container, lastBucket :: Int, and somehow impurely modify it within (!), and consider that field internal (because it obviously break referential transparency).

A: Using solution (1), I managed to get the following design. First, I added a __lastAccessedBucket :: IORef Int field to my datatype, as suggested by @Xicò:
data Container a = Container {
    length :: Int,
    buffer :: V.Vector (Int,a),
    __lastAccessedBucket :: IORef Int }

Then, I had to update the functions that create a new Container in order to create a new IORef using unsafePerformIO:
fromList :: [a] -> Container a
fromList list = unsafePerformIO $ do
    ref <- newIORef 0
    return $ Container (L.length list) buffer ref
    where buffer = V.fromList (prepare list)

Finally, I created two new functions, findBucketWithHint, a pure function which searches the bucket of an index with guess (i.e., the bucket where you think it might be), and the unsafeFindBucket function, which replaces the pure findBucket when performance is needed, by always using the last accessed bucket as the hint:
unsafeFindBucket :: Int -> Container a -> Int
unsafeFindBucket findIdx container = unsafePerformIO $ do
    let lastBucketRef = __lastAccessedBucket contianer
    lastBucket       <- readIORef lastBucketRef
    let newBucket     = findBucketWithHint lastBucket findIdx container
    writeIORef lastBucketRef newBucket
    return $ newBucket

With this, unsafeFindBucket is technically a pure function with the same API of the original findBucket function, but is an order of magnitude faster in some benchmarks. I have no idea how safe this is and where it could cause bugs. Threads are certainly a concern.

A: (This is more an extended comment than an answer.)
First I'd suggest to check if this isn't a case of premature optimization. After all, O(log n) ins't that bad.
If this part is indeed performance-critical, your intention is definitely valid. The usual warning for unsafePerformIO is "use it only if you know what you're doing", which you obviously do, and it can help to make things pure and fast at the same time.
Be sure that you follow all the precautions in the docs, in particular setting the proper compiler flags (you might want to use the OPTIONS_GHC pragma).
Also make sure that the IO operation is thread safe. The easiest way to ensure that is to use IORef together with atomicModifyIORef.
The disadvantage of an internal mutable state is that the performance of the cache will deteriorate if it's accessed from multiple threads, if they lookup different elements.
One remedy would be to explicitly thread the updated state instead of using the internal mutable state. This is obviously what you want to avoid, but if your program is using monads, you could just add another monadic layer that'd internally keep the state for you and expose the lookup operation as a monadic action.
Finally, you could consider using splay trees instead of the array. You'd still have (amortized) O(log n) complexity, but their big advantage is that by design they move frequently accessed elements near the top. So if you'll be accessing a subset of elements of size k, they'll be soon moved to the top, so the lookup operations will be just O(log k) (constant for a single, repeatedly accessed element). Again, they update the structure on lookups, but you could use the same approach with unsafePerformIO and atomic updates of IORef to keep the outer interface pure.