text:
Disability rights and the civil rights revolution
August 4, 2006 8:54 AM CDT
Those of us who come from the first wave of disability rights advocates began our struggle within the civil rights movement of the 1960s. We learned an important lesson from that movement as to how a minority of citizens could change the way a society views the rights of all the people.
In the southern states African Americans lived mostly separated lives from the white majority, forced into second-class citizenship as a direct result of the post-Reconstruction period of reaction that entrenched white supremacy. Racism existed in the northern states as well but was never publicly acknowledged.
Even as the Lincoln Memorial was dedicated in 1922, old Jim Crow ruled the day. Those African Americans who stood up for their rights, such as Malcolm X's father and many others, often found themselves hanging as "bitter fruit" on the nearest tree or lamppost. Only a mass civil rights movement actively demanding equality, equal opportunity and jobs advanced African Americans to equal citizenship under the law.
The years of civil rights struggle taught the first wave of disability rights advocates that if we wanted to live barrier-free we needed to carry out the same struggle for equality. We had to follow the path cleared by the civil rights movement and demand that people with disabilities be treated fairly and no differently than any other citizen — as human beings we should not be limited from participating within the mainstream of life, and should have the same opportunities as able-bodied citizens.
The passage of the Civil Rights Act of 1964 and the Voting Rights Act of 1965, and the establishment of affirmative action within the workplace and schools, inspired people with disabilities. The federal government, by these laws, was forced to develop policies to uproot institutional racism in the North and Jim Crow in the South, ending segregation in public schools, prohibiting job discrimination and offering real job opportunities. It was forced to exercise its power through the National Guard and the courts to enforce the new policies of desegregation in the workplace, the classroom and housing, because of the mass people's movement.
The civil rights movement was a nonviolent revolution. The progressive transformation that it brought about influenced not only the disability rights movement but many other social movements of the 1970s and '80s. It was a model for how mass demonstrations and civil disobedience could put pressure on the federal government to change its policies.
Following this example, the disability rights movement demanded the least restrictive environment in public education for students with disabilities. Using the Voting Rights Act as a model, people with disabilities demanded passage of the Help America Vote Act. Among other voter rights and protection measures, HAVA, passed in 2002, provides funds to state and local governments to upgrade their voting machines and to make all polling places accessible to people with disabilities.
The disability rights movement fought and pushed until the Americans with Disabilities Act was passed in 1990. It is entirely based on the Civil Rights Act. However, the U.S. Supreme Court is striking down one section after another, indicating that even the passage of progressive laws is not enough as long as the right wing rules the Congress, the Supreme Court and the White House. All of the progress made over the past 50 years is subject to the Bush counter-revolution.
Recent massive cuts in housing, benefits, food stamps, Medicare and Medicaid by the right-wing Bush Congress are a direct attack on low-income working people, the frail elderly and people with disabilities. This is why this fall's election is so important to all these oppressed people. We need to organize an all-people's movement to throw these reactionaries out of office, take back our government, restore the cuts and restore the laws designed to correct the injustices of the past.
Tom Siblo (tsiblo@hvcbiz.rr.com) is a diability rights activist in Kingston, N.Y.
"Anti-woke" Gov. DeSantis defends AP African-American course ban, draws pushback across the country


text:
Performing Maternity in Early Modern England
"Performing Maternity in Early Modern England" features essays that share a common concern with exploring maternity's cultural representation, performative aspects and practical consequences in the period from 1540-1690. The essays interrogate how early modern texts depict fertility, conception, delivery,... Read more
"Performing Maternity in Early Modern England" features essays that share a common concern with exploring maternity's cultural representation, performative aspects and practical consequences in the period from 1540-1690. The essays interrogate how early modern texts depict fertility, conception, delivery, and gendered constructions of maternity by analyzing a wealth of historical documents and images in conjunction with dramatic and non-dramatic literary texts. They emphasize that the embodied, repeated, and public nature of maternity defines it as inherently performative and ultimately central to the production of gender identity during the early modern period.
by Miguel de Cervantes Saavedra
by Ben Jonson , Elizabeth Cook
A Sonnet from Carthage:...
by Richard Helgerson
BEN JONSONS ANTIMASQUES : A...
by MICKEL
The Life and Times of Sir Thomas...
by P. J. C. Field
Sade: The Libertine Novels
by John Phillips
Beaumarchais in Seville: An...
by Hugh Thomas
English Literature in Eighteenth...
by Lopa Sanyal
New Mermaids: All for Love
by John Dryden , N.J. Andrew
The Gothic
by Fred Botting
Encyclopaedia of Indian Women...
by Simmi Jain

redpajama_set_name: RedPajamaCommonCrawl


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One Man's Hero
Action / Drama / History / Romance / War / Western
Available in: 720p.WEB 1080p.WEB
WEB: same quality as BluRay, but ripped earlier from a streaming service
38% - Critics
63% - Audience
6.1 10 1240
720p.WEB 1080p.WEB
"One Man's Hero" tells the little-known story of the "St. Patrick's Battalion" or "San Patricios," a group of mostly Irish, and other immigrants of the Catholic religion, who deserted to Mexico after encountering religious and ethnic prejudice in the U.S. Army during the Mexican-American War. The plot centers around the personal story of John Riley, an Irishman who had been a Sergeant in the American Army, who is commissioned as a Captain in the Mexican Army, and commands the battalion, as he leads his men in battle, and struggles with authorities on both sides of the border.
Lance Hool
Ilia Volok as Daniel Grzbalski
Gary Hershberger as Eammon Daly
Carlos Carrasco as Dominguez
Mark Moses as Col. Benton Lacy
English 2.0
23.976 fps
P/S 0 / 4
Reviewed by ma-cortes 7 / 10
Spectacular historical film co-produced by Mexico/US/Spain and plenty of emotion , thrills and breathtaking battles
"One Man's Hero" is set in the 1840's , Mexico was embroiled in turmoil . Renegades and revolutionaries ruled the mountains and the central government was trying to regain control . The neighboring United States saw the opportunity to invade . Across the Atlantic Ocean a different battle raged , as Ireland lost half its population to the potato famine . Desperate and broken two million Irishmen sailed across to the United States . Enticed by the promise of citizenship , scores of young Catholic Irishmen joined the predominantly US Army. For most , it was their only choice . The Saint Patrick's Battalion (Spanish: Batallón de San Patricio, later reorganized as the Foreign Legion of Patricios), formed and led by John Riley, was a unit of 175 to several hundred (accounts vary) immigrants and expatriates of European descent who fought as part of the Mexican Army against the United States in the Mexican-American War of 1846-48. It is set in 1846 , the plot centers around John Riley (Tom Berenger) , an Irish immigrant soldier recruited in the American army during the Mexican-American War , being a sergeant in command of a battalion who are usually despised by the soldiers remaining . One day , John Riley and other Irishmen deserted to Mexico after in the U.S. Army during the Mexican-American War. As the motley bunch flee to Mexico , there they are assaulted and imprisoned by a Mexican guerrilla led by deserters (Joaquin De Almeida , Daniela Romo) opposite the Mexico government . Later on , Riley and his men choose to be soldiers for the Mexican Army , waging war the US invader . Riley then is commissioned as a Captain in the Mexican Army and fights with soldiers on both sides of the border.
It tells the little-known story of the "St. Patrick's Battalion" or "San Patricios," a group of mostly Irish, and other immigrants of the Catholic religion, who are mistreated , humiliated by US soldiers and encountering religious and ethnic prejudice . The script revolves around the personal story of John Riley, an Irishman who had been a Sergeant in the American Army . Here Tom Berenguer gives a good acting as the Irish sergeant who faces off anti-immigrant and anti-Catholic bigotry from his fellow servicemen and defects to the Mexican army , subsequently commanding an Irish battalion, as he leads his men in battle. He is well accompanied by a good cast, such as : Joaquin De Almeida as a rebel officer , Daniela Romo as a stubborn Guerrilla woman who falls for Riley , Mark Moses as an US upright Colonel , Stephen Toblowsky as a nasty officer , James Gammon as General Zachary Taylor , Patrick Bergin as General Scott , and a cameo by Prince Alberto de Monaco . It packs a colorful and evocative cinematography by Peter Wooley . And a rousing and thrilling musical score by Ernest Troost , shot on location in Durango , Puebla , Morelos, Mexico . Lavishly financed by Spanish/US/Mexican producers as Julio Fernandez , Jaime Comas Gil , Conrad and Lance Hool . The motion picture was well written , produced and directed by Lance Hool.
It is well based on facts , about Saint Patrick's Batalion , being composed primarily of Catholic Irish immigrants, the battalion also included Germans, Canadians, English, French, Italians, Poles, Scots, Spaniards, Swiss, and Mexicans, many of whom were also members of the Catholic Church. Most of the battalion's members had deserted or defected from the United States Army. The battalion served as an artillery unit for much of the war. Despite later being formally designated as two infantry companies, it still retained artillery pieces throughout the conflict. In many ways, the battalion acted as the sole Mexican counterbalance to the recent U.S. innovation of horse artillery. The San Patricios were responsible for the toughest battles encountered by the United States in its invasion of Mexico, with Ulysses S. Grant remarking that "Churubusco proved to be about the severest battle fought in the valley of Mexico".Disenfranchised Americans were in the ranks, including escaped slaves from the Southern United States. Only a few members of the Saint Patrick's Battalion were actual U.S. citizens. The Mexican government printed propaganda in different languages to entice immigrants in the U.S. Army to switch sides and offered incentives to foreigners who would enlist in its army including granting them citizenship, paying higher wages than the U.S. Army, and generous land grants. U.S. Army regiments from which members are known to have deserted include: the 1st Artillery, the 2nd Artillery, the 3rd Artillery, the 4th Artillery, the 2nd Dragoons, the 2nd Infantry, the 3rd Infantry, the 4th Infantry, the 5th Infantry, the 6th Infantry, the 7th Infantry, and the 8th Infantry.The San Patricios are honored in both Mexico and Ireland.The San Patricios captured by the U.S. Army were treated and punished as traitors for desertion in time of war. Seventy-two men were immediately charged with desertion by the Army.Two separate courts-martial were held, one at Tacubaya on 23 August, and another at San Ángel on 26 August. At neither of these trials were the men represented by lawyers nor were transcripts made of the proceedings. This lack of formal legal advice could account for the fact that several of the men claimed that drunkenness had led them to desert (a common defense in military trials at the time that sometimes led to lighter sentences), and others described how they were forced to join the Mexican Army in some form or another. The majority of the San Patricios either offered no defense or their defenses were not recorded. Wealthy Mexicans came to the San Patricios' defence at the trials, and members of Mexico's first families visited them in prison . There had strong sentences against the Irish soldiers . One soldier who claimed he was forced to fight by the Mexicans after he was captured by them, and who subsequently refused to do so, was sentenced to death by firing squad instead of hanging, along with another who was found not to have officially joined the Mexican Army. Most of the convicted San Patricios were sentenced to death by hanging: 30 from the Tacubaya trial and 18 from San Ángel. The rationale was that they had entered Mexican military service following the declaration of war. Execution by hanging was in violation of the contemporary Articles of War, which stipulated that the penalty for desertion and/or defecting to the enemy during a time of war was death by firing squad, regardless of the circumstances. Hanging was reserved only for spies (without uniform) and for "atrocities against civilians", neither of which activities were among the charges brought against any members of the Saint Patrick's Battalion. Although more than 9,000 U.S. soldiers deserted the army during the Mexican-American War, only the San Patricios (who unlike almost all other deserters had also fought against the United States) were punished by hanging. Those soldiers who had left military service before the official declaration of war on Mexico (Riley among them) were sentenced to "... receive 50 lashes on their bare backs, to be branded with the letter 'D' for deserter, and to wear iron yokes around their necks for the duration of the war . In all, 50 Saint Patrick's Battalion members were officially executed by the U.S. Army. Collectively, this was the largest mass execution in United States history-the hanging of 38 Sioux at the conclusion of the Dakota War of 1862 appears to be the largest execution by hanging at a single event. En masse executions for treason took place at three separate locations on three separate dates; 16 were executed on 10 September 1847 at San Ángel, four were executed the following day at the village of Mixcoac on 11 September, and 30 were hanged at Chapultepec.
Reviewed by =G= 5 / 10
Mediocre, esoteric, and lame.
"One Man's Hero" tells of the Saint Patty's Battalion of dissident Irish soldiers who fought for Mexico during the American-Mexican war of 1846-1848 and their leader, John Reilly (Berenger). This very mediocre film tries to do too much with a story which is fundamentally not very interesting as war stories go while spicing it up with a fictional and lame romance. Generally a poor production, "OMH" gives us little reason to care about any of the characters, features a typically shallow performance by Berenger, and wanders off into an esoteric bit of history which is of little interest to most. This film is to Irish-Americans as "Glory" was to African-Americans but fails where "Glory" succeeded.
Reviewed by breakdownthatfilm-blogspot-com 7 / 10
A story not told enough in the history books
When it comes films based on true events, it is in the filmmakers' best interest to keep all the facts as legitimate as possible without skewing the story all that much. The genre of films that probably receives the most critical of film buffs' attentions are the ones that have been historically recorded and not some fable spread by rumors and myths. Taking a deeper look, the type of historical film that gets this kind of focus is usually bio-pictures or war dramas. Further closing the gap, the war dramas that have received the motion picture treatment range; but several have depicted these tragedies during the Great War, World War II, the Vietnam War, the Revolutionary War and the American Civil War. Of course in world history, these are not the only confrontations that occurred, but for North American history, this is the usual batch. It is rare when another event is looked at that possibly was forgotten in the regular high school history books. The most recent that comes to mind is Heaven's Gate (1980), which was about a county war in 1890.
Turns out this movie takes place during the 19th century as well. Lance Hool, a producer to other various films like Missing in Action (1984), The Air Up There (1994), Flipper (1996) and Man on Fire (2004), takes a turn in the director's chair to cover the Mexican-American War in 1846. Written by Milton S. Gelman (who passed practically a decade before this film's release), the story is about real-life soldier Sergeant John Riley (Tom Berenger) and his men who abandoned the U.S. army after being persecuted because of their Catholic faith. While leaving, they find refuge with fellow Mexicans led by Cortina (Joaquim de Almeida) and Marta (Daniela Romo). There, they both decide to join Mexico and fight for their freedom. Meanwhile, Colonel Benton Lacy (Mark Moses) attempts to get Riley back to the north before General Winfield Scott (Patrick Bergin) blows them all away. Running parallel to that, Marta and Riley start to become more attracted to each other every minute they spend together.
For problems, there isn't all too much to gripe about. The only script issues that are obvious deal with character motivations. From what is known, John Riley is somewhat a mystery but only after the Mexican- American war ended. There are documents of his existence but there isn't a clear answer as to what he did post-war. Did he marry for real? In this feature, Marta is Riley's love interest as is she to Cortina. The passion that Riley has for Marta is a constant subplot that is brought up every half-hour or so. Yet Cortina repetitively reminds Riley to stay away, but Riley doesn't listen. Soon, Riley and Cortina fight and immediately right after; it's water under the bridge for some odd reason. What was all the antagonizing for if it was going to be settled so quickly? Was it even worth writing in? The only other dilemma in this movie is more technical and that's the depiction of war. This film was Orion Pictures last release and many of the studio's films were rated R (as was this one). Still there seems to be almost no blood or gore.
That particular aspect was probably the most inaccurate component. War is not light and fluffy stuff. There are a couple scenes where blood does flow but it's rated R. Gettysburg (1993) also should have been R but it was PG so the depiction of violence was much less gruesome and that's only appropriate because of its rating. Why give a movie the label of rated R if won't even play out as an R rated film? It's misleading. The cast to the film was also entertaining. Hearing Tom Berenger with an Irish accent is definitely a change in his usual speech pattern and it does sound authentic, as well as the soldiers played by Stuart Graham, Gregg Fitzgerald, Don Wycherley, Wolf Muser and Luke Hayden. Each actor equally matches Berenger in amiability. Daniela Romo as Marta is very pretty and it is obvious as to why Riley falls for her so quickly. Joaquim de Almeida is always fun to watch but in his role as Cortina, it's hard to know how trustworthy he is. Mark Moses performance as Colonel Lacy is another great show. Moses knows how to have a presence.
Visually, the film had an appealing look to it. All effects looked like they were executed practically, of which regularly gives a movie a more realistic viewing. Credit to João Fernandes as the director of photography for capturing wide shots of the western North American terrain. The actual depiction may not feel it has the right temperature to walk in casual clothing but it rightfully matches the environment of what the west was like. Fernandes was also the cinematographer for Friday the 13th Part IV: The Final Chapter (1984), Missing in Action (1984) and Red Scorpion (1988). The film score produced by Ernest Troost was another great element. Troost also composed the music to both horror comedies Tremors (1990) and Dead Heat (1988). Unlike Dead Heat (1988) though where the orchestra sounded like it came from an oldies monster film, Troost's orchestra to this film is much more full and contemporary. The main title is very applicable to the setting, with bagpipes representing Riley and his men. There's also a choir to boot.
It has some weird character motivations and underwhelming violence for an R rated war film, however it is an immersive film. The story provides a new history lesson to those unfamiliar of the Mexican-American war, the actors perform great, and the cinematography is befitting to the setting as well as the film score.


SOUTH DENNIS - Mary Ann (Rogers) Brittain, age 84, formerly of Camp Hill, PA passed surrounded by her loving family March 10, 2019 at South Dennis Healthcare, MA. Born October 19, 1934 Kingston, PA. Daughter of Benjamin and Jule (Zabreski) Rogers. She attended Misericordia University in Dallas, PA. She served her community through years of speaking with A.A., crocheted works of art for the Senior Centers and shelters throughout Cape Cod. Predeceased by her husband Harry E. Brittain Jr., her son Matthew. Surviving four children, Harry and (Kathleen) Mechanicsburg, PA; John and (Irene) South Dennis, MA, Mark, Dover, PA; David and (Susan) Middletown, PA; nine grandchildren; two great-grandchildren; sister Barbara Sieminski (Rogers) and (Charles), Boothwyn, PA and four nieces. Memorial celebration of her life will be held on Cape Cod, MA.
Notes of comfort at: https://www.casperfuneralservices.com/obituaries-guestbook.


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\section{}

\section{Introduction}

First noticed by \citet{NEW81}, and again later by \citet{BEN38}, the
so-called Benford's law states that a sequence of "random" numbers should be
such that their logarithms are uniform mod 1. As a consequence, the
first non-zero digit of a sequence of "random" numbers is $d$ with
probability $\log \left( 1+\frac{1}{d}\right) $, an unexpectedly non-uniform
probability law. $\log $ here stands for the base 10 logarithm, but an easy
generalisation follows: a random variable (r.v.) conforms to \emph{base }$b$
Benford's law if its base $b$ logarithm $\log _{b}\left( X\right) $ is
uniform mod 1. \citet{LOL08} recently proved that no r.v. follows
base $b$ Benford's law for all $b.$

Many experimental data roughly conform to Benford's law (most of which no
more than roughly). However, the vast majority of real data sets that have
been tested do not fit this law at all. For instance, \citet{SCO01}
reported that only 12.6\% of 230 real data sets passed the test for
Benford's law. In his seminal paper, \citet{BEN38} tested 20 data sets
including lakes areas, length of rivers, populations, etc., half of which
did not conform to Benford's law.

The same is true of mathematical sequences or continuous r.v.'s. For
example, binomial arrays ${n \choose k},$ with $n\geq 1,$ $k\in \left\{
0,...,n\right\} ,$ tend toward Benford's law \citep{DIA77}, whereas
simple sequences such as $\left( 10^{n}\right) _{n\in \mathbb{N}}$ obviously
don't.

In spite of all this, Benford's law is actually used in the so-called
"digital analysis" to detect anomalies in pricing \citep{SEH05} or
frauds, for instance in accounting reports \citep{DRA00} or campaign finance \citep{CHO07}. Faked data indeed
usually depart from Benford's law more than real ones \citep{HIL88}. However,
\citet{HAL08} advise caution, arguing that real data do not always fit
the law.

Many explanations have been put forward to elucidate the appearance of
Benford's law on natural or mathematical data. Some authors focus on
particular random variables \citep{ENG03}, sequence \citep{JOL05},
real data \citep{BUR91}, or orbits of
dynamical systems \citep{BER04}. As a rule, other explanations
assume special properties of the data. \citet{HIL95b} or \citet{PIN61} shows
that scale invariance implies Benford's law. Base invariance is an other
sufficient condition \citep{HIL95a}. Mixtures of uniform distributions
\citep{JAN04} also conform to Benford's law, and so do the
limits of some random processes \citep{SHU08}. Multiplicative
processes have been mentioned as well \citep{PIE01}. Each of
these explanations accounts for some appearances of data fitting Benford's
law, but lacks generality.

While looking for a truly general explanation, some authors noticed that
data sets are more likely to fit Benford's law if they were scattered
enough. More precisely, a sequence should "cover several orders of
magnitude", as \citet{RAI76} expressed it. Of course, scatter alone is no
sufficient condition. The sequence 0.9, 9, 90, 900... indeed covers several
orders of magnitude, but is far from conforming to Benford's law. The
continuous random variables that are known to fit Benford's law usually
present some "regularity": exponential densities, normal densities, or
lognormal densities are of this kind. Invariance assumptions
(base-invariance or scale-invariance) lead to "regular" densities and so do
central limit-like theorem assumptions of mixture.

Some technical explanations may be viewed as a mathematical expression of
the idea that a random variable $X$ is more likely to conform to Benford's
law if it is regular and scattered enough. \cite[Example 4.1.]{MAR00} linked
Benford's law to Poincar\'{e}'s theorem in circular statistics, and \citet{SMI07}
expressed it in terms of Fourier transforms and signal processing.
However, a non expert reader would hardly notice the smooth-and-scattered
implications of these developments.

Though scatter has been explicitly mentioned and regularity allusively
evoked, the idea that scatter and regularity (in a sense that will be made
clear further) may actually be a \emph{sufficient} explanation for Benford's
phenomenon related to continuous r.v.'s have never been formalized in a
simple way, to our knowledge, except in a recent article by \citet{FEW09}.
In this paper, Fewster hypothesizes that "\emph{any distribution [...] that
is reasonably smooth and covers several orders of magnitude is almost
guaranteed to obey Benford's law." }He then defines a smoothing procedure
for a r.v. $X$ based on $\left[ \pi ^{2}\left( x\right) \right] ^{\prime
\prime },$ $\pi $ being the probability density function (henceforth \emph{%
p.d.f.}) of $\log \left( X\right) ,$ and illustrates with a few eloquent
examples that under smoothness and scatter constraints, a r.v. cannot depart
much from Benford's law. However, no theorem is given that would formalise
this idea.

In the first part of this paper, we prove a theorem from which it follows
that scatter and regularity can be modelled in such a way that they, alone,
imply \emph{rough} compliance to Benford's law (again: real data usually do
not perfectly fit Benford's law, irrespective of the sample size).

It is not surprising that many data sets or random variables samples are
scattered and regular hence our explanation of Benford's phenomena
corroborates a widespread intuition. The proof of this theorem is
straightforward and requires only basic mathematical tools. Furthermore, as
we shall see, several of the existing explanations can be understood as
corollaries of ours. Our explanation encompasses more specific ones, and is
far simpler to understand and to prove.

Scatter and regularity do not presuppose any log-related properties (such as
the property of log-normality, scale-invariance, or multiplicative
properties). For this reason, if we are right, Benford's law should also
admit other versions. We set that a r.v. $X$ is $u$-Benford for a function $%
u $ if $u(X)$ is uniform mod 1. The classical Benford's law is thus
a special case of $u$-Benford's law, with $u=\log $. We test real data sets
and mathematical sequences for "$u$-Benfordness" with various $u$, and test
a second theorem echoing the first one. Most data conform to $u$-Benford's
law for different $u$, which is an argument in favour of our explanation.

\section{Scatter and regularity: a key to Benford}

The basic idea at the root of theorem 1 (below) is twofold.

First, we hypothesize that a continuous r.v. $X$ with density $f$ is almost
uniform mod 1 as soon as it is scattered and regular. More
precisely, any $f$ that is non-decreasing on $]-\infty ,a]$, and then
non-increasing on $[a,+\infty \lbrack $ (for regularity) and such that its
maximum $m=\sup (f)$ is "small" (for scatter) should correspond to a r.v. $X$
approaching uniformity mod 1. Figure 1 illustrates this idea.
\\

\includegraphics[scale=.4]{Benford_01.eps}

\begin{center}
Figure 1 --- Illustration of the idea that a regular
p.d.f. is almost bound to give rise to uniformity mod 1. The
stripes --- restrictions of the density on $\left[ n,n+1\right] $ --- of the
p.d.f. of a r.v. $X$ are stacked to form the p.d.f. of $X$ mod 1.
The slopes partly compensate, so that the resulting p.d.f. is almost
uniform. If the initial p.d.f. is linear on every $\left[ n,n+1\right] ,$
the compensation is perfect.
\end{center}

Second, note that if $X$ is scattered and regular enough, so should be $\log
\left( X\right) $. These two ideas are formalized and proved in theorem 1.

Henceforth, for any real number $x,$ $\left\lfloor x\right\rfloor $ will
denote the greatest integer not exceeding $x,$ and $\left\{ x\right\}
=x-\left\lfloor x\right\rfloor $. Any positive $x$ can be written as a
product $x=10^{\left\lfloor \log x\right\rfloor }.10^{\left\{ \log x\right\}
},$ and the Benford's law may be rephrased as the uniformity of the random
variable $\left\{ \log \left( X\right) \right\} .$

\begin{theorem}
Let $X$ be a continuous positive random variable with p.d.f. $f$ such that
 $Id.f:$ $x\longmapsto xf(x)$ conforms to the two following conditions~:
 $\exists a>0$ such that (1) $\max (Id.f)=m=a.f(a)$ and (2) $Id.f$ is
nondecreasing on $]0,a],$ and nonincreasing on $[a,+\infty \lbrack .$ Then,
for any $z\in ]0,1],$
\begin{equation*}
\left\vert P\left( \left\{ \log X\right\} <z\right) -z\right\vert <2\ln
\left( 10\right) m.
\end{equation*}
In particular, $\left( X_{n}\right) $ being a sequence of continuous r.v.'s
with p.d.f. $f_{n}$ satisfying these conditions and such that $m_{n}=\max
\left( Id.f_{n}\right) \longrightarrow 0$, $\left\{ \log \left( X_{n}\right)
\right\} $ converges toward uniformity on $[0,1[$ in law.
\end{theorem}

\begin{proof}
We first prove that for any continuous r.v. $Y$ with density $g$ such that $%
g $ is nondecreasing on $]-\infty ,b],$ and then nonincreasing on $%
[b,+\infty \lbrack ,$ the following holds:%
\begin{equation*}
\forall z\in ]0,1],~\left\vert P\left( \left\{ Y\right\} <z\right)
-z\right\vert <2M
\end{equation*}%
where $M=g\left( b\right) =\sup \left( g\right) .$

We may suppose without loss of generality that $b\in \lbrack 0,1[.$ Let $%
z\in ]0,1[$ (the case $z=0$ is obvious). Put $I_{n,z}=[n,n+z[.$ For any
integer $n\leq -1,$%
\begin{equation*}
\frac{1}{z}\int_{I_{n,z}}g(t)dt\leq \int_{n}^{n+1}g(t)dt.
\end{equation*}%
Thus%
\begin{equation*}
\frac{1}{z}\sum_{n\leq -1}\int_{I_{n,z}}g(t)dt\leq \int_{-\infty }^{0}g(t)dt.
\end{equation*}

For any integer $n\geq 2,$%
\begin{equation*}
\frac{1}{z}\int_{I_{n,z}}g(t)dt\leq \int_{n-1+z}^{n+z}g(t)dt,
\end{equation*}%
so%
\begin{equation*}
\frac{1}{z}\sum_{n\geq 2}\int_{I,z}g(t)dt\leq \int_{1+z}^{+\infty }g(t)dt.
\end{equation*}

Moreover, $\int_{I_{0,z}}g\leq zM$ and $\int_{I,z}g\leq zM.$ Hence,%
\begin{equation*}
\frac{1}{z}\sum_{n\in \mathbb{Z}}\int_{I_{n,z}}g\leq \int_{-\infty }^{\infty
}g+2M.
\end{equation*}%
We prove in the same fashion that%
\begin{equation*}
\frac{1}{z}\sum_{n\in \mathbb{Z}}\int_{I_{n,z}}g\geq \int_{-\infty }^{\infty
}g-2M.
\end{equation*}

Since $\sum_{\mathbb{Z}}\int_{I_{n,z}}g=P\left( \left\{ Y\right\} <z\right)
, $ $z<1$ and $\int_{-\infty }^{\infty }g=1,$ the result is proved.

Now, applying this to $Y=\log \left( X\right) $ proves theorem 1.
\end{proof}

\begin{remark}
The convergence theorem is still valid if we accept $f$ to have a finite
number of monotony changes, provided this number does not exceed a
previously fixed $k$. The proof is straightforward.
\end{remark}

\begin{remark}
The assumptions made on $Id.f$ may be seen as a measure of scatter and
regularity for $X,$ adjusted for our purpose.
\end{remark}

\section{Examples}

\subsection{Type I Pareto}

A continuous r.v. $X$ is type I Pareto with parameters $\alpha $ and $x_{0}$
$(\alpha ,~x_{0}\in \mathbb{R}_{+}^{\ast })$ iff it admits a density function%
\begin{equation*}
f_{x_{0},\alpha }\left( x\right) =\frac{\alpha x_{0}^{\alpha }}{x^{\alpha +1}%
}\mathbb{I}_{[x_{0},+\infty \lbrack }
\end{equation*}

Besides its classical use in income and wealth modelling, type I Pareto
variables arise in hydrology and astronomy \citep[page 252]{PAO06}.

The function $Id.f=g:$ $x\longmapsto \frac{\alpha x_{0}^{\alpha }}{x^{\alpha
}}\mathbb{I}_{[x_{0},\infty \lbrack }$ is decreasing. Its maximum is%
\begin{equation*}
\sup \left( Id.f\right) =Id.f\left( x_{0}\right) =\alpha .
\end{equation*}
Therefore, $X$ is nearly Benford-like, in the extent that%
\begin{equation*}
\left\vert P(\{\log X\}<z)-z\right\vert <2\ln (10)\alpha .
\end{equation*}

\subsection{Type II Pareto}

A r.v. $X$ is type II Pareto with parameter $b>0$ iff it admits a density
function defined by%
\begin{equation*}
f_{b}\left( x\right) =\frac{b}{\left( 1+x\right) ^{b+1}}\mathbb{I}%
_{[0,+\infty \lbrack }
\end{equation*}

It arises in a so-called \emph{mixture model}, with mixing components being
gamma distributed r.v.'s sequences.

The function $Id.f_{b}=g_{b}:$ $x\longmapsto \frac{bx}{\left( 1+x\right)
^{b+1}}\mathbb{I}_{[0,+\infty \lbrack }$ is $C^{\infty }\left( \mathbb{R}%
_{+}\right) ,$ with derivative%
\begin{equation*}
g_{b}^{\prime }\left( x\right) =\frac{b\left( 1-bx\right) }{\left(
x+1\right) ^{b+2}},
\end{equation*}%
which is positive whenever $x<\frac{1}{b},$ then negative. From this result
we derive%
\begin{equation*}
\sup g_{b}=g_{b}\left( \frac{1}{b}\right) =\frac{1}{\left( 1+\frac{1}{b}%
\right) ^{1+b}}=\left( \frac{b}{1+b}\right) ^{b+1},
\end{equation*}%
since%
\begin{equation*}
\ln \left[ \left( \frac{b}{1+b}\right) ^{b+1}\right] =\left( b+1\right) %
\left[ \ln b-\ln \left( b+1\right) \right] ,
\end{equation*}%
which tends toward $-\infty $ when $b$ tends toward 0,%
\begin{equation*}
\sup g_{b}\underset{b\longrightarrow 0}{\longrightarrow }0.
\end{equation*}%
Theorem 1 applies. It follows that $X$ conform toward Benford's law when $%
b\longrightarrow 0.$

\subsection{Lognormal distributions}

A r.v. $X$ is lognormal iff $\log \left( X\right) \thicksim N\left( \mu
,\sigma ^{2}\right) $. Lognormal distributions have been related to Benford
\citep{KOS06}. It is easy to prove that whenever $\sigma \longrightarrow
\infty ,$ $X$ tends toward Benford's law. Although the proof may use
different tools, a straightforward way to do it is theorem 1.

One classical explanation of Benford's law is that many data sets are
actually built through multiplicative processes \citep{PIE01}.
Thus, data may be seen as a product of many small effects. This may be
modelled by a r.v. $X$ that may be written as%
\begin{equation*}
X=\prod\limits_{i}Y_{i},
\end{equation*}%
$Y_{i}$ being a sequence of random variables. Using the $\log $
transformation, this leads to $\log \left( X\right) =\sum \log \left(
Y_{i}\right) .$

The \emph{multiplicative central-limit theorem} therefore proves that, under
usual assumptions, $X$ is bound to be almost lognormal, with $\log \left(
X\right) \thicksim N\left( \mu ,\sigma ^{2}\right) ,$ and $\sigma
\longrightarrow \infty ,$ thus roughly conforming to Benford, as an
application of theorem 1.

\section{Generalizing Benford}

If we are right to think that Benford's law is to be understood as a
consequence of mere scatter and regularity, instead of special
characteristics linked with multiplicative, scale-invariance, or whatever
log-related properties, we should be able to state, prove, and check on real
data sets, a generalized version of the Benford's law were some function $u$
replaces the $\log .$

Indeed, our basic idea is that $X$ being scattered and regular enough
implies $\log \left( X\right) $ to be scattered and regular as well, so that
$\log \left( X\right) $ should be almost uniform mod 1. The same
should be true of any $u\left( X\right) ,$ $u$ being a function preserving
scatter and regularity. Actually, some $u$ should even be better shots than $%
\log ,$ since $\log $ reduces scatter on $[1,+\infty \lbrack .$

First, let us set out a generalized version of theorem 1, the proof of which
is closely similar to that of theorem 1.

\begin{theorem}
Let $X$ be a r.v. taking values in a real interval $I,$ with p.d.f. $f.$ Let
$u$ be a $C^{1}$ increasing function $I\longrightarrow \mathbb{R}$, such
that $\frac{f}{u^{\prime }}:$ $x\longmapsto \frac{f(x)}{u^{\prime }\left(
x\right) }$ conforms to the following: $\exists a>0$ such that (1) $\max
\left( \frac{f}{u^{\prime }}\right) =m=\frac{f}{u^{\prime }}\left( a\right) $
and (2) $\frac{f}{u^{\prime }}$ is non-decreasing on $]0,a],$ and
non-increasing on $[a,+\infty \lbrack \cap I.$ Then, for all $z\in \lbrack
0,1[,$%
\begin{equation*}
\left\vert P\left( \left\{ u\left( X\right) \right\} <z\right) -z\right\vert
<2m.
\end{equation*}%
In particular, if $\left( X_{n}\right) $ is a sequence of such r.v.'s with
p.d.f. $f_{n}$ and $\max \left( f_{n}/u^{\prime }\right) =m_{n}$, and $%
\lim_{+\infty }\left( m_{n}\right) =0,$ then $\left\{ u\left( X_{n}\right)
\right\} $ converges in law toward $U\left( [0,1[\right) $ when $%
n\longrightarrow \infty .$
\end{theorem}

A r.v. $X$ such that $\left\{ u\left( X\right) \right\} \thicksim U\left(
[0,1[\right) $ will be said $u$ -Benford henceforth.

\subsection{Sequence}

Although our two theorems only apply to continuous r.v.'s, the underlying
intuition that $\log $-Benford's law is only a special case (having,
however, a special interest thanks to its implication in terms of
leading-digits interpretation) of a more general law does also apply to
sequence. In this section, we experimentally test $u$-Benfordness for a few
sequences $\left( v_{n}\right) $ and a four functions $u.$

We will use six mathematical sequences. Three of them, namely $\left( \pi
n\right) _{n\in \mathbb{N}},$ prime numbers $\left( p_{n}\right) $, and $%
\left( \sqrt{n}\right) _{n\in \mathbb{N}}$ are known not to follow Benford.
The three others, $\left( n^{n}\right) _{n\in \mathbb{N}},$ $\left(
n!\right) _{n\in \mathbb{N}}$ and $\left( e^{n}\right) _{n\in \mathbb{N}}$
conform to Benford.

As for $u,$ we will focus on four cases:%
\begin{eqnarray*}
x &\longmapsto &\log \left[ \log \left( x\right) \right] \\
x &\longmapsto &\log \left( x\right) \\
x &\longmapsto &\sqrt{x} \\
x &\longmapsto &\pi x^{2}
\end{eqnarray*}

The first one increases very slowly, so we may expect that it will not work
perfectly. The second leads to the classical Benford's law. The $\pi $
coefficient of the last $u$ allows us to use integer numbers, for which $%
\left\{ x^{2}\right\} $ is nil.

The result of the experiment is given in Table 1.

\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
$v_{n}$ & $\log \circ \log \left( v_{n}\right) $ & $\log \left( v_{n}\right)
$ & $\sqrt{v_{n}}$ & $\pi v_{n}^{2}$ \\ \hline\hline
$\sqrt{n}$ $\left( N=10~000\right) $ & $68.90~(.000)$ & $45.90~(.000)$ & $%
4.94~(.000)$ & $0.02~(.000)$ \\ \hline
$\pi n$ $\left( N=10~000\right) $ & $44.08~(.000)$ & $26.05~(.000)$ & $%
0.19~(1.000)$ & $0.80~(.544)$ \\ \hline
$p_{n}$ $\left( N=10~000\right) $ & $53.92~(.000)$ & $22.01~(0.000)$ & $%
0.44~(0,990)$ & $0.69~(.719)$ \\ \hline
$e^{n}$ $\left( N=1~000\right) $ & $6.91~\left( 0.000\right) $ & $%
0.76~\left( 1.000\right) $ & $0.63~\left( .815\right) $ & $0.79~\left(
.560\right) $ \\ \hline
$n!$ $\left( N=1~000\right) ^{(\ast )}$ & $7.39~\left( .000\right) $ & $%
0.58~\left( .887\right) $ & $0.61~\left( .844\right) $ & $0.90~\left(
.387\right) $ \\ \hline
$n^{n}$ $\left( N=1~000\right) ^{(\ast )}$ & $7.45$~$\left( .000\right) $ &
$0.80~\left( .543\right) $ & $16.32~\left( .000\right) $ & $0.74~(.646)$
\\ \hline
\end{tabular}

Table 1 --- Results of the Kolmogorov-Smirnov tests applied on $\left\{
u\left( v_{n}\right) \right\} ,$ with four different functions $u$ (columns)
and six sequences (lines). Each sequence is tested through its first $N$
terms (from $n=1$ to $n=N),$ with an exception for $\log \circ \log \left(
n^{n}\right) $ and $\log \circ \log \left( n!\right) ,$ for which $n=1$ is
not considered. Each cell displays the Kolmogorov-Smirnov $z$ and the
corresponding $p$ value.
\end{center}

The sequences have been arranged according to the speed with which it
converges to $+\infty $ (and so are the functions $u)$. None of the six
sequences is $\log \circ \log $-Benford (but a faster divergent sequence
such as $\left( 10^{e^{n}}\right) $ would do). Only the last three are $\log
$-Benford. These are the sequences going to $\infty $ faster than any
polynomial. Only one sequence $\left( n^{n}\right) $ does not satisfy $\sqrt{%
.}$-Benfordness. However, this can be understood as a pathological case,
since $\sqrt{n^{n}}$ is integer whenever $n$ is even, or is a perfect
square. Doing the same Kolmogorov-Smirnov test with odd numbers not being
perfect squares gives $z=0,45$ and $p=0,987,$ showing no discrepancy with $%
\sqrt{.}$-Benfordness for $\left( n^{n}\right) .$ All six sequences are $\pi
.^{2}$-Benford.

Putting aside the case of $\sqrt{n^{n}},$ what Table 1 reveals is that the
convergence speed of $u\left( v_{n}\right) $ completely determines the $u$%
-Benfordness of $\left( v_{n}\right) .$ More precisely, it seems that $%
\left( v_{n}\right) $ is $u$-Benford whenever $u\left( v_{n}\right) $
increases as fast as $\sqrt{n},$ and is not $u$-Benford whenever $u\left(
v_{n}\right) $ increase as slowly as $\ln \left( n\right) .$ Of course, this
rule-of-thumb is not to be taken as a theorem. Obviously enough, one can
actually decide to increase or decrease convergence speed of $u\left(
v_{n}\right) $ without changing $\left\{ u\left( v_{n}\right) \right\} ,$
adding or substracting \emph{ad hoc }integer numbers.

Nevertheless, this observation suggests that we give a closer look at
sequence $f\left( n\right) ,$ where $f$ is an increasing and concave real
function converging toward $\infty ,$ and look for a condition for $\left(
\left\{ f\left( n\right) \right\} \right) _{n}$ to converge to uniformity.
An intuitive idea is that $\left( \left\{ f\left( n\right) \right\} \right)
_{n}$ will depart from uniformity if it does not increase fast enough: we
may define brackets of integers --- namely $[f^{-1}\left( n\right)
,f^{-1}\left( n+1\right) -1[\cap \mathbb{N},$ within which $\left\lfloor
f\left( n\right) \right\rfloor $ is constant, and of course $\left\{ f\left(
n\right) \right\} $ increasing. If these brackets are "too large", the
relative height of the last considered bracket is so important that it
overcomes the first terms of the sequence $f\left( 0\right) ,...,f\left(
n\right) $ mod 1. In that case, there is no limit to the
probability distribution of $\left( \left\{ f\left( n\right) \right\}
\right) .$ The weight of the brackets should therefore be small relative to $%
f^{-1}\left( n\right) ,$ which may be written as%
\begin{equation*}
\frac{f^{-1}\left( n\right) -f^{-1}\left( n+1\right) }{f^{-1}\left( n\right)
}\underset{\infty }{\longrightarrow }0.
\end{equation*}

Provided that $f$ is regular, this leads to%
\begin{equation*}
\frac{\left( f^{-1}\right) ^{\prime }\left( x\right) }{f^{-1}\left( x\right)
}\underset{\infty }{\longrightarrow }0,
\end{equation*}%
or%
\begin{equation*}
\left[ \ln \left( f^{-1}\left( x\right) \right) \right] ^{\prime }\underset{%
\infty }{\longrightarrow }0.
\end{equation*}

Functions $f:x\longmapsto x^{\alpha },$ $\alpha >0$ satisfy this condition.
Any $n^{\alpha }$ should then show a uniform limit probability law, except
for pathological cases $\left( \alpha \in \mathbb{Q}\right) .$ Taking $%
\alpha =\frac{1}{\pi }$ gives (with $N=1000),$ a Kolmogorov-Smirnov $%
z=1,331, $ and a $p$-value 0.058, which means there is no significant
discrepancy from uniformity. On the other hand, the $\log $ function which
does not conform to this condition is such that $\left\{ \log \left(
n\right) \right\} $ is not uniform, confirming once again our rule-of-thumb
conjecture.

\subsection{Real data}

We test three data sets for $u$-Benfordness using a Kolmogorov-Smirnov test
for uniformity. First data set is the opening value of the Dow Jones, the
first day of each month from October 1928 to November 2007. The second and
third are country areas expressed in millions of square-km%
${{}^2}$
and the populations of the different countries, as estimated in 2008,
expressed in millions of inhabitants. The two last sequences are provided by
the CIA\footnote{%
http://www.cia.gov/library/publications/the-world-factbook/docs/rankorderguide.html%
}. Table 2 displays the results.

\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
& $\log \circ \log \left( v_{n}\right) $ & $\log \left( v_{n}\right) $ & $%
\sqrt{v_{n}}$ & $\pi v_{n}^{2}$ \\ \hline
Dow Jones $\left( N=950\right) $ & $5.90~(.000)$ & $5.20~(.000)$ & $%
0.75~(.635)$ & $0.44~(.992)$ \\ \hline
Area pays $(N=256)$ & $1.94~(.001)$ & $0.51~(.959)$ & $0.89~(.404)$ & $%
1,88~(.002)$ \\ \hline
Populations $\left( N=242\right) $ & $3.39~(.000)$ & $0.79~(.568)$ & $%
0.83~(.494)$ & $0.42~(.994)$ \\ \hline
\end{tabular}

Table 2 --- Results of the Kolmogorov-Smirnov tests applied on $\left\{
u\left( v_{n}\right) \right\} .$
\end{center}

This table confirms our analysis: classical Benfordness is actually less
often borne out than $\sqrt{.}$-Benfordness on these data. The last column
shows that our previous conjectured rule has exceptions: divergence speed is
not an absolute criterion by itself. For country areas, the fast growing $%
u:x\longmapsto \pi x^{2}$ gives a discrepancy from uniformity, whereas the
slow-growing $\log $ does not. However, allowing for exceptions, it is still
a good rule-of-thumb.

\subsection{Continuous r.v.'s}

Our theorems apply on continuous r.v.'s. We now focus on three examples of
such r.v.'s, with the same $u$ as above (except for $\log \circ \log ,$
which is not defined everywhere on $\mathbb{R}_{+}^{\ast }$): the uniform
density on $]0,k]$ $(k>0),$ exponential density, and absolute value of a
normal distribution.

\subsubsection{Uniform r.v.'s}

It is a known fact that a uniform distribution $X_{k}$ on $]0,k]$ $(k>0)$
does not approach classical Benfordness, even as a limit. On every bracket $%
[10^{j-1},10^{j}-1[,$ the leading digit is uniform. Therefore, taking $%
k=10^{j}-1$ leads to a uniform (and not logarithmic) distribution for
leading digits, whatever $j$ might be.

The density $g_{k}$ of $\sqrt{X_{k}}$ is%
\begin{equation*}
g_{k}\left( x\right) =\frac{2x}{k},~x\in \left] 0,\sqrt{k}\right]
\end{equation*}

and $g_{k}\left( x\right) =0$ otherwise. It is an increasing function on $%
]-\infty ,\sqrt{k}]$, decreasing on $[\sqrt{k},+\infty \lbrack $ with
maximum $\frac{2}{\sqrt{k}}\longrightarrow 0$ when $k\longrightarrow \infty .
$ Theorem 2 applies, showing that $X_{k}$ tends toward $\sqrt{.}$%
-Benfordness in law. Now, theorem 3 below proves that $X_{k}$ tends toward $u
$-Benfordness, when $u\left( x\right) =\pi x^{2}.$

\begin{theorem}
If $X$ follows a uniform density on $]0,k],$ $\left\{ \pi X^{2}\right\} $
converges in law toward uniformity on $[0,1[$ when $k\longrightarrow \infty .
$
\end{theorem}

\begin{proof}
Let $X\thicksim U\left( \left] 0,k\right] \right) .$ The p.d.f. $g$ of $%
Y=\pi X^{2}$ is%
\begin{equation*}
g\left( x\right) =\frac{1}{2a\sqrt{x}},~x\in ]0,a^{2}]
\end{equation*}%
where $a=k\sqrt{\pi }.$

The c.d.f. $G$ of $Y$ is then%
\begin{equation*}
G\left( x\right) =\frac{\sqrt{x}}{a},~x\in ]0,a^{2}].
\end{equation*}%
Let now $\delta \in ]0,1[.$ Call $P_{\delta }$ the probability that $\left\{
Y\right\} <\delta $.%
\begin{equation*}
\sum_{j=0}^{\left\lfloor a^{2}-\delta \right\rfloor }G\left( j+\delta
\right) -G\left( j\right) \leq P_{\delta }\leq \sum_{j=0}^{\left\lfloor
a^{2}-\delta \right\rfloor +1}G\left( j+\delta \right) -G\left( j\right)
\end{equation*}%
\begin{equation*}
\frac{1}{a}\sum_{j=0}^{\left\lfloor a^{2}-\delta \right\rfloor }\sqrt{%
j+\delta }-\sqrt{j}\leq P_{\delta }\leq \frac{1}{a}\sum_{j=0}^{\left\lfloor
a^{2}-\delta \right\rfloor +1}\sqrt{j+\delta }-\sqrt{j}
\end{equation*}%
The square-root function being concave,%
\begin{equation*}
\sqrt{j+\delta }-\sqrt{j}\geq \frac{\delta }{2\sqrt{j+\delta }}
\end{equation*}%
and, for any $j>0,$%
\begin{equation*}
\sqrt{j+\delta }-\sqrt{j}\leq \frac{\delta }{2\sqrt{j}}.
\end{equation*}%
Hence,%
\begin{eqnarray*}
\frac{\delta }{2a}\sum_{j=0}^{\left\lfloor a^{2}-\delta \right\rfloor }\frac{%
1}{\sqrt{j+\delta }} &\leq &P_{\delta }\leq \frac{1}{a}\left[ \sqrt{\delta }%
+\sum_{1}^{\left\lfloor a^{2}-\delta \right\rfloor +1}\frac{\delta }{2\sqrt{j%
}}\right] \\
\frac{\delta }{2a}\sum_{j=0}^{\left\lfloor a^{2}-\delta \right\rfloor }\frac{%
1}{\sqrt{j+\delta }} &\leq &P_{\delta }\leq \frac{\sqrt{\delta }}{a}+\frac{%
\delta }{2a}\sum_{1}^{\left\lfloor a^{2}-\delta \right\rfloor +1}\frac{1}{%
\sqrt{j}}
\end{eqnarray*}%
$x\longmapsto \frac{1}{\sqrt{x}}$ being decreasing,%
\begin{equation*}
\sum_{j=0}^{\left\lfloor a^{2}-\delta \right\rfloor }\frac{1}{\sqrt{j+\delta
}}\geq \int\limits_{\delta }^{\left\lfloor a^{2}-\delta \right\rfloor
+1+\delta }\frac{1}{\sqrt{t}}dt\geq 2\left[ \sqrt{\left\lfloor a^{2}-\delta
\right\rfloor +1+\delta }-\sqrt{\delta }\right]
\end{equation*}%
and%
\begin{equation*}
\sum_{1}^{\left\lfloor a^{2}-\delta \right\rfloor +1}\frac{1}{\sqrt{j}}\leq
\int_{0}^{\left\lfloor a^{2}-\delta \right\rfloor +1}\frac{1}{\sqrt{t}}%
dt\leq 2\left[ \sqrt{\left\lfloor a^{2}-\delta \right\rfloor +1}\right] .
\end{equation*}%
So,%
\begin{equation*}
\frac{\delta }{a}\left[ \sqrt{\left\lfloor a^{2}-\delta \right\rfloor
+1+\delta }-\sqrt{\delta }\right] \leq P_{\delta }\leq \frac{\delta }{a}%
\left[ \sqrt{\left\lfloor a^{2}-\delta \right\rfloor +1}\right] .
\end{equation*}%
As a consequence, for any fixed $\delta $, $\lim_{a\longrightarrow \infty
}\left( P_{\delta }\right) =\delta ,$ and $\left\{ \pi X^{2}\right\} $
converges in law to uniformity on $[0,1[$.
\end{proof}

\subsubsection{Exponential r.v.'s}

Let $X_{\lambda }$ be an exponential r.v. with p.d.f. $f_{\lambda }\left(
x\right) =\lambda \exp \left( -\lambda x\right) $ $\left( x\geq 0,\lambda
>0\right) .$ Engel and Leuenberger [2003] demonstrated that $X_{\lambda }$
tends toward the Benford's law when $\lambda \longrightarrow 0.$

The p.d.f. of $\sqrt{X_{\lambda }}$ is $x\longmapsto 2\lambda x\exp \left(
-\lambda x^{2}\right) ,$ which increases on $\left] 0,\frac{1}{2\lambda }%
\right] $ and then decreases. Its maximum is $\exp \left( -\frac{1}{4\lambda
}\right) .$ Theorem 2 thus applies, showing that $X_{\lambda }$ is $\sqrt{.}$%
-Benford as a limit when $\lambda \longrightarrow 0.$

Finally, theorem 4 below demonstrates that $X_{\lambda }$ tends toward $u$%
-Benfordness for $u\left( x\right) =\pi x^{2}$ as well.

\begin{theorem}
If $X\thicksim EXP\left( \lambda \right) $ (with p.d.f. $f:x\longmapsto
\lambda \exp \left( -\lambda x\right) ),$ then $Y=\pi X^{2}$ converges
toward uniformity mod 1 when $\lambda \longrightarrow 0.$
\end{theorem}

\begin{proof}
Let $X$ be such a r.v. $Y=\pi X^{2}$ has density $g$ with%
\begin{equation*}
g\left( x\right) =\frac{\mu }{2\sqrt{x}}\exp \left( -\mu \sqrt{x}\right)
,~x\geq 0
\end{equation*}%
where $\mu =\frac{\lambda }{\sqrt{\pi }}.$ The $Y$ c.d.f. $G$ is thus, for
all $x\geq 0$%
\begin{equation*}
G\left( x\right) =1-e^{-\mu \sqrt{x}}.
\end{equation*}%
Let $P_{\delta }$ denote the probability that $\left\{ Y\right\} <\delta ,$
for $\delta \in ]0,1[.$%
\begin{equation*}
P_{\delta }=\sum_{j=0}^{\infty }\left[ e^{-\mu \sqrt{j}}-e^{-\mu \sqrt{%
j+\delta }}\right]
\end{equation*}%
$x\longmapsto \exp \left( -\mu \sqrt{x}\right) \ $being convex,%
\begin{equation*}
\delta \frac{\mu }{2\sqrt{j+\delta }}e^{-\mu \sqrt{j+\delta }}\leq e^{-\mu
\sqrt{j}}-e^{-\mu \sqrt{j+\delta }}
\end{equation*}%
for any $j\geq 0,$ and%
\begin{equation*}
e^{-\mu \sqrt{j}}-e^{-\mu \sqrt{j+\delta }}\leq \delta \frac{\mu }{2\sqrt{j}}%
e^{-\mu \sqrt{j}}
\end{equation*}%
for any $j>0.$ Thus%
\begin{equation*}
\delta \sum_{j=0}^{\infty }\frac{\mu }{2\sqrt{j+\delta }}e^{-\mu \sqrt{%
j+\delta }}\leq P_{\delta }\leq 1-e^{-\mu \sqrt{\delta }}+\delta
\sum_{j=1}^{\infty }\frac{\mu }{2\sqrt{j}}e^{-\mu \sqrt{j}}.
\end{equation*}%
$x\longmapsto \frac{1}{\sqrt{x}}\exp \left( -\mu \sqrt{x}\right) $ being
decreasing,%
\begin{eqnarray*}
\delta \sum_{j=0}^{\infty }\frac{\mu }{2\sqrt{j+\delta }}e^{-\mu \sqrt{%
j+\delta }} &\geq &\delta \int_{\sqrt{\delta }}^{\infty }\frac{\mu }{2\sqrt{t%
}}e^{-\mu \sqrt{t}}dt \\
&\geq &\delta \left[ -e^{-\mu \sqrt{t}}\right] _{\sqrt{\delta }}^{\infty } \\
&=&\delta e^{-\mu \sqrt{\delta }},
\end{eqnarray*}%
and%
\begin{eqnarray*}
1-e^{-\mu \sqrt{\delta }}+\delta \sum_{j=1}^{\infty }\frac{\mu }{2\sqrt{j}}%
e^{-\mu \sqrt{j}} &\leq &1-e^{-\mu \sqrt{\delta }}+\delta \int_{0}^{\infty }%
\frac{\mu }{2\sqrt{t}}e^{-\mu \sqrt{t}}dt \\
&\leq &1-e^{-\mu \sqrt{\delta }}+\delta
\end{eqnarray*}%
The two expressions tend toward $\delta $ when $\mu \longrightarrow 0,$ so
that $P_{\delta }\longrightarrow \delta .$ The proof is complete.
\end{proof}

\subsubsection{Absolute value of a normal distribution}

To test the absolute value of a normal distribution $X$ with mean 0 and
variance $10^{8},$ we picked a sample of 2000 values and used the same
procedure as for real data. It appears, as shown in Table 3, that $X$
significantly departs from $u$-Benfordness with $u=\log $ and $u=\pi .^{2},$
but not with $u=\sqrt{.}.$

\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
& $\log \left( X\right) $ & $\sqrt{X}$ & $\pi X^{2}$ \\ \hline
$U\left( [0,k[\right) ~k\longrightarrow \infty $ & NO & YES & YES \\ \hline
$EXP\left( \lambda \right) ~\lambda \longrightarrow 0$ & YES & YES & YES \\
\hline
$\left\vert \mathcal{N}\left( 0,10^{8}\right) \right\vert $ & $14.49~\left(
.000\right) $ & $0.647~\left( .797\right) $ & $28.726~\left( .000\right) $
\\ \hline
\end{tabular}

Table 3 --- The table displays if uniform distributions, exponential
distributions, and absolute value of a normal distribution, are $u$-Benford
for different functions $u,$ or not. The last line shows the results (and $p$%
-values) of the Kolmogorov-Smirnov tests applied to a 2000-sample. It could
be read as "NO; YES; NO".
\end{center}

As we already noticed, the best shot when one is looking for Benford seems
to be the square-root rather than $\log .$

\section{Discussion}

Random variables exactly conforming the Benford's classical law are rare,
although many do roughly approach the law. Indeed, many explanations have
been proposed for this approximate law to hold so often. These explanations
involve complex characteristics, sometimes directly related to logarithms,
sometimes through multiplicative properties.

Our idea --- formalized in theorem 1 --- is more simple and general. The
fact that real data often are regular and scattered is intuitive. What we
proved is an idea which has been recently expressed by Fewster [2009]:
scatter and regularity are actually \emph{sufficient} condition to
Benfordness.

This fact thus provides a new explanation of Benford's law. Other
explanations, of course, are acceptable as well. But it may be argued that
some of the most popular explanations are in fact corollaries of our
theorem. As we have seen when studying Pareto type II density, mixtures of
distributions may lead to regular and scattered density, to which theorem 1
applies. Thus, we may argue that a mixture of densities is nearly Benford
\emph{because} it is necessarily scattered and regular. In the same fashion,
multiplications of effects lead to Benford-like densities, but also (as the
multiplicative central-limit theorem states) to regular and scattered
densities.

Apart from the fact that our explanation is simpler and (arguably) more
general, a good argument in its favor is that Benfordness may be generalized
--- unlike log-related explanations. Scale invariance or multiplicative
properties are log-related. But as we have seen, Benfordness is not
dependant on log, and can easily be generalized. Actually, it seems that
square root is a better candidate than log. The historical importance of $%
\log $-Benfordness is of course due to the implications in terms of leading
digits which bears no equivalence with square-root.

\bibliographystyle{elsarticle-harv}

redpajama_set_name: RedPajamaArXiv


For Coach's latest collection, Stuart Vevers re-imagines the American journey, combining the sub-cultures of music, skate and surf, with the suburban as a backdrop to its big-city juxtaposition. With Coach's signature style of luxe and structure, the new collection breathes an edgy streetstyle angle to the Coach brand that sees them exploring the tension of utility and luxury--of creating looks that reference the workwear silhouettes of the mid-west with the fast-paced boldness of New York.
While their last collection focused on the Coach Dreamers, this latest collection puts a spotlight on the All-American Dreamer: an optimistic, smart outsider who dresses with a cool, effortless ease, and a sense of spontaneous freedom.
The collection features a graphic, colourful, and vibrant take on boyish femininity--all injected with a sense of nostalgia and tradition that goes hand-in-hand with Coach's latest collaboration with Gary Baseman. These are looks that bring a surprising twist to the familiar, a re-imagination of the American, big-city silhouette.
Inspired by Stuart's vision of that all-american Coach girl who favours vintage silhouettes, this knee-length dress brings a different take to the sweet, suburban girl-next-door. Gary Baseman's art printed throughout the sheer fabric brings a touch of quirk and boldness to an otherwise simple, everyday silhouette.
Inspired by Daphne from the popular 70's era cartoon Scooby-Doo, this look is all about that all-american cool girl. The flares, graphic shirt, and bright orange blazer ensemble brings to mind traditional tailoring with an irreverent, casual twist--perfect for achieving that work-place city chic.
A quintessential American fabrication is the classic combo of a jean jacket and skirt; and this ensemble brings it all back into focus. With its studded trimmings and clean white palette, it's all about embodying that cool girl vibe of hanging out with the band at the local bar.
In line with Coach's tailored and luxurious image, this look is inspired by that luxe sense of nonchalance. An uptown girl with her bold, tailored jacket and structured Coach Swagger bag who isn't afraid to express her sense of individualism and spontaneity with her bold choices.
At the root of it, the all-american girl is never afraid of making a statement in that imprecise way that's hard to replicate. This look is essentially a gathering of found pieces, of putting together an ensemble that is unconventional yet still rooted in the basics--of taking the girl from the suburbs and letting her express herself in a big-city way.
What do you think of COACH's new collection? Let us know in the comment box below!
It has been more than two weeks since Manila Fashion Festival but the rave about it is still alive amongst us Clozetters. We couldn't stop talking about our favourite collections, which were a toss between designers that resembled our personal styles and designers with looks we've only dreamed of wearing.
Overall, it was 4 consecutive nights of runway elegance and fashion innovations--and the best part was that the Clozette Team got to share it with our dear Clozetters in the Philippines!
So we caught up with the Clozetters we spotted at #MNLFashionFest to get the low-down on what they wore and who their favourite designers are.
On her look: "I wanted to wear black but at the same time didn't want to stick to the usual black & white mix. So I opted for a brown top to go with my black skirt and vest ensemble. I also like adding texture in my outfits, so the lace top kept the outfit from looking too plain and flat."
Her favourite Designer/Collection: "I loved how the collections were pretty diverse so it's pretty hard to pick one favourite. But aside from the standouts, I also really enjoyed Banggo Niu's collection. He definitely impressed with his high energy display and edgy I'm-a-cool-girl clothes."
On her look: "I almost always wear heels, but I wanted to try my hand at this sports luxe trend. So I wore my gold LZD sneakers with a velvet cutout top by designer John Chan to achieve the look."
Her favourite Designer/Collection: "There were lots of collections that I liked, but I'll go with Michael Leyva because his creations were ultra glamorous but still wearable."
On her look: "Monochrome is my go-to colour scheme. I think it's perfect for special occasions because it's timeless."
Her favourite Designer/Collection: "Vania Romoff's F/W 2015 Collection was the best for me because it reminded me of one of my favourite books when I was a kid - 'The Secret Garden!' "
On her look: "I wore Bottomline crop top, MDS Collection skirt, Pedro heels, & Charles Keith sling bag. I chose to wear this during the event because of the skirt. I find the print really cute and unique."
Her favourite Designer/Collection: "Happy Andrada & Jot Losa."
On her look: "I believe that tropical print doesn't require jetting off to an exotic island. It makes a cool statement as the temperature heats up. I simply re-worked the Tropical printed crop-top into a stylish outfit. I paired it with my green textured high-waisted short to let the pattern be the focus of my ensemble. I added a textured blazer to add an empowered vibe and finished it off with my go-to stiletto."
Her favourite Designer/Collection: "I like Banggo Niu. I'm so attracted to his fresh and playful designs. His creations centred on the sporty chic look with an 80's vibe. Shimmers and leather plus jersey letterings dominated his collection and I loved it. I find it elegant. With Banggo Niu's creations, you'll definitely DRESS with a BANG."
On her look: "My top was from designer Alexander J. Bitong, Maxi skirt from online shop www.rosegal.com, heels from Kenneth Cole, and bag from Coach. Obviously in fashion shows, black and white ensembles stands out the most so I wore something monochromatic. Black can really make you look sophisticated. With my top, I opted to wear a top from a designer--it's always a classic choice."
Her favourite Designer/Collection: "My favourite designer was Vania Romoff as she is one of the leading designers of today's generation and I just love her collections! It's very classic and feminine from her soft flows and classic silhouettes. She always creates beautiful designs. Her collection was to die for!"
On her look: "I wore a sheer top from Zalora, lace bandage skirt I got from a bazaar, flats from Alberto Asumague, and bag form Zalora. As I am always dressed so dark, I dolled up myself with pastel colours. With spring/summer upon us, it is time to lighten up our wardrobe and jump into a pastel wonderland!"
Her favourite Designer/Collection: "My favourite designer is Vania Romoff"
On her look: "Sundays are always casual days for me. So I thought of wearing black shorts and black tops. However, knowing that it is an event, I wouldn't want to go too casual. So I've picked a flowery blazer and black shoes to finish my entire ensemble."
Her favourite Designer/Collection: "I loved Vania Romoff's collection. The dresses simply spoke to me of ethereal elegance. Of all the designers there, hers was the ones I would really love to wear. It reflected my personal style--simple, classic, and clean colours. "
On my look: "I always play on the safe side, especially during events like this. I opted for my no-fail black, white, and gold pieces and played it out with the structure. A crop top and sheer jogger pants, paired with snakeskin heels and LV bag is the perfect classy look for me since I had to make my look comfortable enough for taking photos of the event."
My favourite Designer/Collection: "It's actually a toss between Vania Romoff and Cheetah Rivera's collections. Both used quality fabrics and feminine silhouettes--which I always dream of wearing someday in a very glam event. Very classy and wearable."
On her look: "Manila Fashion Festival is the perfect event to take fashion risks! I tried to go out of my comfort zone by wearing something different than what I usually wear (mainly black & white). With that in mind, I decided to pair contrasting colours & bold patterns from my wardrobe together with some chunky accessories to complement each other. The look was definitely new to me but Manila Fashion Festival was definitely the time to make a fashion statement so I dared to be a little different."
Her favourite Designer/Collection: "My favourite designer would definitely be Cheetah Rivera. The beautiful Swallow embroidery on her dresses really caught my eye, plus the tailoring and fit was impeccable. Not only that but her use of bold primary colours and edgy modern cuts gave her designs the right amount of edge & sophistication which perfectly embodies the modern day woman. Second favourite collection of mine would be Mark Bumgarner. I loved how he elevated the evening gowns by giving each one a unique look--may it be a different cut or an embellished collar. His most memorable design for me would be the tailored slacks with the embellished turtle-neck top that trickled down back with a trail or a long cape of sorts. It wasn't necessarily an evening gown but the combination was fresh and new which made it as equally jaw-dropping as his other designs."
What was the latest fashion show you watched and which designer was your favourite?


text:
Packers' JK Scott controls field position with precision punting
JK Scott had nine punts against the Bears and averaged 47.6 yards gross and 43.6 net with five landing inside the 20.
Packers' JK Scott controls field position with precision punting JK Scott had nine punts against the Bears and averaged 47.6 yards gross and 43.6 net with five landing inside the 20. Check out this story on packersnews.com: https://www.packersnews.com/story/sports/nfl/packers/2019/09/06/packers-jk-scott-controls-field-position-precision-punting/2233309001/
Tom Silverstein, Packers News Published 6:19 p.m. CT Sept. 6, 2019 | Updated 2:30 p.m. CT Sept. 7, 2019
GREEN BAY – Defense and special teams.
If that is the formula the Green Bay Packers are going to use to win football games this season, they are off to a good start.
And if punter JK Scott is going to perform the way he did in a 10-3 upset of the Chicago Bears at Soldier Field on Thursday night, they might not even need defense.
OK, that's an exaggeration, but it isn't going too far to say Scott controlled field position all night long, which is what general manager Brian Gutekunst thought he could do when he drafted him out of Alabama in the fifth round of the 2018 draft.
It was a spectacular showing for a guy who had a mediocre rookie season. But it wasn't a total surprise given his recent performance. About two weeks into training camp, Scott seemed to find his Zen place and his punting went from pretty good to potentially game-changing.
In the second exhibition game, he averaged more than 50 yards per punt and reached 5.0-second or better hang time on all five attempts. He found consistency after that and finished the exhibition season with a 48.0 gross average and a 41.2 net average and nine punts inside the 20 on 22 attempts.
Green Bay Packers punter J.K. Scott (6) punts during the 1st half of the Green Bay Packers Chicago Bears football game at Soldier Field in Chicago on Thursday, Sept. 5, 2019. Photo by Mike De Sisti/Milwaukee Journal Sentinel (Photo: Mike De Sisti)
At the end of camp, Scott explained that he had made a concerted effort to stop trying to crush the ball and to relax. Every time he tried to put a little something extra on the ball, it seemed to backfire.
So, he just swung away freely against the Bears and stopped trying to press.
"I felt pretty good tonight," Scott said after the game. "That's what I'm doing — when I relax I'm hitting the ball more naturally."
Scott had nine punts against the Bears and averaged 47.6 yards gross and 43.6 net with five landing inside the 20. Speedy returner Tarik Cohen, who ranked fifth in the NFL with a 12.5 average last year, was limited to four returns for 36 yards.
"He was on point all night long," coach Matt LaFleur said. "We knew anytime you have returners like they have, and specifically like Tarik Cohen as the punt returner, it starts with our specialists. We knew that JK had to have a big game and he delivered."
The amazing part of Scott's performance was the way he combined hang time with direction. The aim generally is to hit each punt somewhere from the numbers on the field to the sideline.
Anything in the middle of the field is dangerous because it gives the returner two directions to take the punt. Any punt with hang time under 4.0 is also dangerous because the coverage team needs at least that to get down and cover the punt.
Here are the distances, hang times, landing spots, net yardage on his nine punts and which were Aussie-style pooches meant to go high and short:
First quarter, 13:15 – 53 yards, 4.9, on the numbers, 42.
First quarter, 8:04 – 42 yards, 4.23, middle of the field, 36.
First quarter, 3:20 – 47 yards, 4.95, outside the numbers, 47.
Second quarter, 5:37 – 63 yards, 4.02, middle of the field, 63.
Second quarter, 0:55 – 36 yards, 4.61, inside the numbers, 36, pooch.
Third quarter, 10:58 – 37 yards, 3.92, on the numbers, 37, pooch.
Third quarter, 6:43 – 50 yards, 4.35, on the sideline, 45, pooch.
Fourth quarter, 15:00 – 37 yards, 4.3, outside the numbers, 37 pooch.
Fourth quarter, 1:43 – 63 yards, 4.52, outside the numbers, 49.
If you take the punts Scott in which did not use the Aussie style to pin the Bears in their own territory, he averaged 53.6 yards gross, 4.5-second hang time and 47.4 net. That means any time the Packers were punting away from inside their own territory, they could expect a 50-yard flip in field position.
"He was great," long snapper Hunter Bradley said. "I told him after the game it's nice to finally be on the same sideline with him when he does stuff like that because there were multiple games at college when I was (at Mississippi) State and he was at 'Bama where he singlehandedly beat us because he was doing that, and we just didn't have an answer for it.
"He's worked his butt off in the offseason and that's what he's capable of. It's nice to finally see him back on track."
The only poor punt Scott hit was the second one. It was from the end zone in the first quarter. It was short and down the middle but had just enough hang time to allow the coverage units to have a chance.
Complete coverage:
Packers defense stifles Bears, makes Matt LaFleur a winner in his debut
Dougherty: Packers defense torments Trubisky
Silverstein: Two Smiths prove too much for Bears offense to handle
Packers know run game needs to be better
Insider: Erratic Packers offense does just enough to win
Rodgers vows Packers offense will improve
PODCAST: Jim Owczarski and Tom Silverstein talk about the defense carrying the team to victory
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"The ball location (was good overall)," LaFleur said. "There was one punt where we were backed up where we served it up to him right in the middle of the field, but on that particular play our coverage unit was on point and we were able to smother him a little bit. I'm really happy for JK."
Of all the punts Scott hit, his last was the best.
The Packers were trying to run out the clock and preserve their victory, but they were forced to punt from their own 27 with 1 minute, 45 seconds left. The defense had been on the field for 69 snaps (including penalties) and was hoping Scott could put the Bears in a hole to make things a little easier.
The Bears faked like they were playing for a block, lining nine players at the line of scrimmage. It forced the gunners to come inside and block. The Bears only rushed five, however, but their strategy had resulted in the Packers' pursuit being bottled up in the middle of the field.
If Scott had hit a bad punt, Cohen would have had time to field the punt and potentially have a big return down either sideline.
Instead, Scott hit a bomb. Cohen had to back track for it and caught it over his shoulder right at the painted "10" on the Bears' end of the field. Cohen returned it 14 yards, but the Bears were called for holding and Chicago had to start on its own 14.
"Oh my gosh. It's awesome," Bradley said of that final punt. "He saw the front and saw that he really needed, with them having us caved out, he really needed to get a hold of it. So, he lowered his drop and he got a hold of it.
"That's what he's capable of. He's definitely going to be in this league a long time punting like that."
When it was brought to his attention that he might have been the Packers' MVP Thursday night, Scott shrugged his shoulders and said that was not how he was thinking about the performance.
"I just try to do my job," he said. "For me, it's about doing your job and helping us win. It's always a team effort. I'm thankful to be able to play a part in winning. That's the big thing."
Jim Owczarski of the Milwaukee Journal Sentinel contributed to this story
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Today I'm in need of a little reminder on how much I love our sweet scooter.
She is on the verge of popping her first tooth and actually isn't behaving sweet at all!
Constant crying, diaper rash, drool everywhere, gnawing on everything, wants me to tote her around all day, wants to nurse all day, no napping, lots of pooping, very mild temp.
It makes me want a grandma close to take the other chickies and make cookies, forts, and play pretend with them all day long...because this momma is spent.
What a sweetheart! Jamison is doing the scoot too but sitting up. The kid hates his tummy so he scoots on his bum. I need to make a cool video like you! I love her big eyes. They're gorgeous.
Teething is so difficult. I hope she's sated soon and that you get a respite from the crying.