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Asymptotic analysis

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

As an illustration, suppose that we are interested in the properties of a function $f (n)$ as $n$ becomes very large. If $f(n) = n^{2} + 3n$, then as $n$ becomes very large, the term $3n$ becomes insignificant compared to $n^{2}$. The function $f(n)$ is said to be "asymptotically equivalent to $n^{2}$, as $n → ∞$". This is often written symbolically as $f (n) ~ n^{2}$, which is read as "$f(n)$ is asymptotic to $n^{2}$".

An example of an important asymptotic result is the prime number theorem. Let $π(x)$ denote the prime-counting function (which is not directly related to the constant pi), i.e. $π(x)$ is the number of prime numbers that are less than or equal to $x$. Then the theorem states that $$\pi(x)\sim\frac{x}{\ln x}.$$

Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation.

Definition
Formally, given functions $f (x)$ and $g(x)$, we define a binary relation $$f(x) \sim g(x) \quad (\text{as } x\to\infty)$$ if and only if $$\lim_{x \to \infty} \frac{f(x)}{g(x)} = 1.$$

The symbol $~$ is the tilde. The relation is an equivalence relation on the set of functions of $x$; the functions $f$ and $g$ are said to be asymptotically equivalent. The domain of $f$ and $g$ can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.

The same notation is also used for other ways of passing to a limit: e.g. $x → 0$, $x ↓ 0$, $|x| → 0$. The way of passing to the limit is often not stated explicitly, if it is clear from the context.

Although the above definition is common in the literature, it is problematic if $g(x)$ is zero infinitely often as $x$ goes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, in little-o notation, is that $f ~ g$ if and only if $$f(x)=g(x)(1+o(1)).$$

This definition is equivalent to the prior definition if $g(x)$ is not zero in some neighbourhood of the limiting value.

Properties
If $$f \sim g$$ and $$a \sim b$$, then, under some mild conditions, the following hold:

 * $$f^r \sim g^r$$, for every real $r$
 * $$\log(f) \sim \log(g)$$ if $$\lim g \neq 1 $$
 * $$f\times a \sim g\times b$$
 * $$f / a \sim g / b$$

Such properties allow asymptotically equivalent functions to be freely exchanged in many algebraic expressions.

Examples of asymptotic formulas
H_\alpha^{(1)}(z) &\sim \sqrt{\frac{2}{\pi z}} e^{ i\left(z - \frac{2\pi\alpha - \pi}{4}\right)} \\ H_\alpha^{(2)}(z) &\sim \sqrt{\frac{2}{\pi z}} e^{-i\left(z - \frac{2\pi\alpha - \pi}{4}\right)} \end{align}$$
 * Factorial $$n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$$ —this is Stirling's approximation
 * Partition function For a positive integer n, the partition function, p(n), gives the number of ways of writing the integer n as a sum of positive integers, where the order of addends is not considered. $$p(n)\sim \frac{1}{4n\sqrt{3}} e^{\pi\sqrt{\frac{2n}{3}}}$$
 * Airy function The Airy function, Ai(x), is a solution of the differential equation $y″ − xy = 0$; it has many applications in physics. $$\operatorname{Ai}(x) \sim \frac{e^{-\frac{2}{3} x^\frac{3}{2}}}{2\sqrt{\pi} x^{1/4}}$$
 * Hankel functions $$\begin{align}

Asymptotic expansion
An asymptotic expansion of a function $f(x)$ is in practice an expression of that function in terms of a series, the partial sums of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for $f$. The idea is that successive terms provide an increasingly accurate description of the order of growth of $f$.

In symbols, it means we have $$f \sim g_1,$$ but also $$f - g_1 \sim g_2$$ and $$f - g_1 - \cdots - g_{k-1} \sim g_{k}$$ for each fixed k. In view of the definition of the $$\sim$$ symbol, the last equation means $$f - (g_1 + \cdots + g_k) = o(g_k)$$ in the little o notation, i.e., $$f - (g_1 + \cdots + g_k)$$ is much smaller than $$g_k.$$

The relation $$f - g_1 - \cdots - g_{k-1} \sim g_{k}$$ takes its full meaning if $$g_{k+1} = o(g_k)$$ for all k, which means the $$g_k$$ form an asymptotic scale. In that case, some authors may abusively write $$f \sim g_1 + \cdots + g_k$$ to denote the statement $$f - (g_1 + \cdots + g_k) = o(g_k).$$ One should however be careful that this is not a standard use of the $$\sim$$ symbol, and that it does not correspond to the definition given in.

In the present situation, this relation $$g_{k} = o(g_{k-1})$$ actually follows from combining steps k and k−1; by subtracting $$f - g_1 - \cdots - g_{k-2} = g_{k-1} + o(g_{k-1})$$ from $$f - g_1 - \cdots - g_{k-2} - g_{k-1} = g_{k} + o(g_{k}),$$ one gets $$g_{k} + o(g_{k})=o(g_{k-1}),$$ i.e. $$g_{k} = o(g_{k-1}).$$

In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. This optimal partial sum will usually have more terms as the argument approaches the limit value.

Examples of asymptotic expansions
\ (x \to \infty)$$
 * Gamma function $$\frac{e^x}{x^x \sqrt{2\pi x}} \Gamma(x+1) \sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots
 * Exponential integral $$xe^xE_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \to \infty) $$
 * Error function $$ \sqrt{\pi}x e^{x^2}\operatorname{erfc}(x) \sim 1+\sum_{n=1}^\infty (-1)^n \frac{(2n-1)!!}{n!(2x^2)^n} \ (x \to \infty)$$ where $m!!$ is the double factorial.

Worked example
Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. For example, we might start with the ordinary series $$\frac{1}{1-w}=\sum_{n=0}^\infty w^n$$

The expression on the left is valid on the entire complex plane $$w \ne 1$$, while the right hand side converges only for $$|w|< 1$$. Multiplying by $$e^{-w/t}$$ and integrating both sides yields $$ \int_0^\infty \frac{e^{-\frac{w}{t}}}{1 - w} \, dw = \sum_{n=0}^\infty t^{n+1} \int_0^\infty e^{-u} u^n \, du$$

The integral on the left hand side can be expressed in terms of the exponential integral. The integral on the right hand side, after the substitution $$u=w/t$$, may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion $$e^{-\frac{1}{t}} \operatorname{Ei}\left(\frac{1}{t}\right) = \sum _{n=0}^\infty n! \; t^{n+1} $$

Here, the right hand side is clearly not convergent for any non-zero value of t. However, by keeping t small, and truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of $$\operatorname{Ei}(1/t)$$. Substituting $$x = -1/t$$ and noting that $$\operatorname{Ei}(x) = -E_1(-x)$$ results in the asymptotic expansion given earlier in this article.

Asymptotic distribution
In mathematical statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables $Z_{i}$ for $i = 1, …, n$, for some positive integer $n$. An asymptotic distribution allows $i$ to range without bound, that is, $n$ is infinite.

A special case of an asymptotic distribution is when the late entries go to zero—that is, the $Z_{i}$ go to 0 as $i$ goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.

This is based on the notion of an asymptotic function which cleanly approaches a constant value (the asymptote) as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon.

An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation $$y = \frac{1}{x},$$ y becomes arbitrarily small in magnitude as x increases.

Applications
Asymptotic analysis is used in several mathematical sciences. In statistics, asymptotic theory provides limiting approximations of the probability distribution of sample statistics, such as the likelihood ratio statistic and the expected value of the deviance. Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics, however. Non-asymptotic bounds are provided by methods of approximation theory.

Examples of applications are the following.
 * In applied mathematics, asymptotic analysis is used to build numerical methods to approximate equation solutions.
 * In mathematical statistics and probability theory, asymptotics are used in analysis of long-run or large-sample behaviour of random variables and estimators.
 * In computer science in the analysis of algorithms, considering the performance of algorithms.
 * The behavior of physical systems, an example being statistical mechanics.
 * In accident analysis when identifying the causation of crash through count modeling with large number of crash counts in a given time and space.

Asymptotic analysis is a key tool for exploring the ordinary and partial differential equations which arise in the mathematical modelling of real-world phenomena. An illustrative example is the derivation of the boundary layer equations from the full Navier-Stokes equations governing fluid flow. In many cases, the asymptotic expansion is in power of a small parameter, $ε$: in the boundary layer case, this is the nondimensional ratio of the boundary layer thickness to a typical length scale of the problem. Indeed, applications of asymptotic analysis in mathematical modelling often center around a nondimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at hand.

Asymptotic expansions typically arise in the approximation of certain integrals (Laplace's method, saddle-point method, method of steepest descent) or in the approximation of probability distributions (Edgeworth series). The Feynman graphs in quantum field theory are another example of asymptotic expansions which often do not converge.

Asymptotic versus Numerical Analysis
Debruijn illustrates the use of asymptotics in the following dialog between Dr. N.A., a Numerical Analyst, and Dr. A.A., an Asymptotic Analyst: N.A.: I want to evaluate my function $$f(x)$$ for large values of $$x$$, with a relative error of at most 1%.

A.A.: $$f(x)=x^{-1}+\mathrm O(x^{-2}) \qquad (x\to\infty)$$.

N.A.: I am sorry, I don't understand.

A.A.: $$|f(x)-x^{-1}|<8x^{-2} \qquad (x>10^4).$$

N.A.: But my value of $$x$$ is only 100.

A.A.: Why did you not say so? My evaluations give"f(x)-x^{-1}"

N.A.: This is no news to me. I know already that $$0<f(100)<1$$.

A.A.: I can gain a little on some of my estimates. Now I find that"f(x)-x^{-1}"

N.A.: I asked for 1%, not for 20%.

A.A.: It is almost the best thing I possibly can get. Why don't you take larger values of $$x$$?

N.A.: !!! I think it's better to ask my electronic computing machine.

Machine: f(100) = 0.01137 42259 34008 67153

A.A.: Haven't I told you so? My estimate of 20% was not far off from the 14% of the real error.

N.A.: !!! . . . !

Some days later, Miss N.A. wants to know the value of f(1000), but her machine would take a month of computation to give the answer. She returns to her Asymptotic Colleague, and gets a fully satisfactory reply.

title: Asymptotic analysis


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Staten Island Yankees

The Staten Island Yankees were a minor league baseball team located in the New York City borough of Staten Island from 1999 to 2020. Nicknamed the "Baby Bombers", the Yankees were the Class A Short Season affiliate of the New York Yankees and played in the New York–Penn League at Richmond County Bank Ballpark along the waterfront in St. George. The Yankees won six New York–Penn League championships (2000, 2002, 2005, 2006, 2009, and 2011).

History
The franchise originated in 1983 as the Watertown Pirates (affiliated with the Pittsburgh Pirates), and became the Watertown Indians (affiliated with the Cleveland Indians) in 1989.

In 1999, in a deal brokered by New York City Mayor Rudy Giuliani, the Yankees affiliation was transferred from the Oneonta Yankees to the Watertown franchise, which was relocated and renamed the Staten Island Yankees. The Oneonta Yankees, in turn, became a Detroit Tigers affiliate and were renamed the Oneonta Tigers.

The Staten Island Yankees played their first two seasons at College of Staten Island Baseball Complex before moving into the Richmond County Bank Ballpark for the 2001 season.

The first SI Yankee to reach the major leagues as a New York Yankee was in 2003; pitcher Jason Anderson, pitching in relief in an 8–4 Yankee win over the Toronto Blue Jays at the Rogers Centre. (The first Staten Island Yankee to reach the majors for any team was Wily Mo Peña, who broke in with the Cincinnati Reds, in 2002.)

On March 26, 2006, the Staten Island Advance reported that the team's majority owners, the Getzler family, were considering selling their 51% share of the team, and were asking for between three and five million dollars. The New York Yankees purchased the Getzlers’ interest in the team and in return hired Mandalay Sports Properties to run the day-to-day operations of the team. Part of the agreement was that the New York Yankees and Mandalay become equal partners with Mandalay owning 50% of the Staten Island Yankees.

In 2006, the Yankees were managed by Gaylen Pitts, noted for frequently being ejected from games. In a game on August 25, 2006, Pitts was ejected and then returned to the field in sandals after a batter was hit by a pitch in the 9th inning of a 21–6 victory over the Brooklyn Cyclones.

In 2007, the Yankees were managed by Mike Gillespie, who had led the 1998 USC Trojans to a College World Series championship. Gillespie led the Baby Bombers to their third consecutive playoff appearance before losing to the Brooklyn Cyclones in 2007 NYPL Playoffs, 2 games to none.

Former Florida Gator coach Pat McMahon managed the Yankees for the 2008 season.

Former major league catcher Josh Paul managed the Yankees for the 2009 and 2010 seasons. in 2010, due to Dave Eiland taking a leave of absence, SI Yankees manager Paul was summoned to fill in as the New York Yankees bullpen coach, during which time former major leaguer Jody Reed filled in as the interim manager for the Staten Island Yankees.

On June 20, 2016, the team launched a campaign to rename the organization starting with the 2017 season. On September 8, 2016, the potential names were whittled down to five, with an online vote beginning on the team's website from that date; the candidate names were the Bridge Trolls, Heroes, Killer Bees, Pizza Rats, and Rock Pigeons. Ultimately, the team decided to retain the Yankees moniker but to call itself the Staten Island Pizza Rats for select games as an alternate identity.

Prior to the 2020 season, it was announced that the team would be shuttered when the MLB/MiLB agreement ended at the conclusion of the 2020 season under a proposed plan calling for the Short Season Class A designation to be eliminated. The start of the 2020 season was postponed due to the COVID-19 pandemic before ultimately being cancelled on June 30. On November 7, 2020, the New York Yankees announced that they were withdrawing from Staten Island as well as abandoning their affiliation with the Trenton Thunder as they were moving forward with their new Double-A affiliate: the Somerset Patriots, but they would seek to place a team from the independent Atlantic League at Richmond County Bank Ballpark in 2021. On December 3, 2020, the Staten Island Yankees announced that the club would cease operations. The Staten Island FerryHawks began play in the Atlantic League in 2022.

Notable alumni
Over a hundred Staten Island Yankees players reached the major leagues, not including MLB players who were with the team on a rehab assignment.

• Manny Acosta

• Jason Anderson

• John Axford

• TJ Beam

• Colter Bean

• Justin Berg

• Danny Burawa

• Melky Cabrera

• Andy Cannizaro

• Robinson Canó

• Matt Carson

• Bernie Castro

• Francisco Cervelli

• Justin Christian

• Brandon Claussen

• Phil Coke

• Matt DeSalvo

• Shelley Duncan

• Michael Dunn

• Anderson Garcia

• Brett Gardner

• Alex Graman

• John-Ford Griffin

• Eric Hacker

• Brad Halsey

• Sean Henn

• Mitch Jones

• Jeff Karstens

• Ian Kennedy

• Charlie Manning

• Daniel McCutchen

• Mark Melancon

• Wily Mo Pena

• David Phelps

• Andy Phillips

• Omir Santos

• Bronson Sardinha

• Matt Smith

• Chien-Ming Wang

• Chase Wright

The following MLB players made rehab appearances with the Staten Island Yankees:
 * Orlando Hernandez (2001)
 * José Contreras (2003)
 * Kevin Brown (2004)
 * Steve Karsay (2004)
 * Felix Rodríguez (2005)
 * Octavio Dotel (2006)
 * Jeff Karstens (2007) – the only former SI Yankee to return on an MLB rehab assignment
 * Darrell Rasner (2007)
 * Jonathan Albaladejo (2008)
 * Phil Hughes (2011)

Mascots
The original Staten Island Yankees Mascot was Scooter the "Holy Cow." A combination of the nickname of former New York Yankees shortstop and broadcaster Phil "Scooter" Rizzuto and his commentating catch phrase "Holy Cow!" Scooter debuted for the Staten Island Yankees when the franchise relocated from Watertown, New York, in 1999. In summer 2003, the Baby Bombers debuted Scooter's sisters Red and Huckleberry (or "Huck"). Scooter and his sisters were a staple at SI Yankee games, leading fan rallies and between-inning on-field games. The cows were known to have a The Three Stooges-like relationship in which Red and Huck often teamed up to trick and trap Scooter

Retired numbers

 * 6: Brett Gardner
 * 17: Robinson Canó
 * 19: Jason Anderson
 * 41: Chien-Ming Wang
 * 42: Jackie Robinson

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Staten Island Yankees


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Dave Catching

David Catching (born June 7, 1961) is an American musician from Memphis, Tennessee. He is a founding member of the California stoner rock bands earthlings? from 1993-present, and Queens of the Stone Age from 1996-2000, a former touring member of Eagles of Death Metal from 2003-2017 and the current operator of the Rancho De La Luna recording studio.

Career
Catching has played the guitar for well-known hard rock bands Eagles of Death Metal, Queens of the Stone Age, Tex and the Horseheads, as well as Mark Lanegan, The Ringling Sisters, earthlings?, Mondo Generator and the Masters of Reality, Yellow #5, and Gnarltones.

Catching has been associated with the desert country rock band, Smith & Pyle. He contributed to the debut album of country rock duo Smith & Pyle, It's OK to be Happy, which was recorded at his recording studio Rancho de la Luna and released in 2008. He currently resides in Joshua Tree, California.

Catching was on stage with Eagles of Death Metal during the November 2015 Paris attacks, and escaped the Bataclan with the rest of the band. Catching talked about this at length on the show Conan Neutron's Protonic Reversal, Rolling Stone and other media.

Musical equipment
With earthlings?, Mojave Lords and other live performances Catching plays a 1984 Gibson Les Paul, gifted to him by his brother in law, Frank Watts. He also uses for live shows and recording Echopark Guitars Albert 58, and Yamaha AES 1500. His pedalboard consists of Earthquaker Devices, Dispatch Master and Astral Destiny, a T.C. Electronics Polytune 2 tuner, and Spark, Jim Dunlop Wah pedal and Rotovibe pedal, Origin Me-Q Drive, Dr No Effects Colossus, Solid Gold FX Lysis and EM-III, Boss DD#3 Delay, Radial BigShot, into a combination of Bruce Zinky Tremoverb, and Mofo, or Lanham Amps Dragonslayer with Colby Amps Elpico. With the Eagles of Death Metal, Catching played his 1967 Gibson Flying V through a tuner and distortion pedal and Supro amplifier with a 2x12 cabinet and Orange Amplifiers. He also used an Ampeg Dan Armstrong guitar when playing with Eagles of Death Metal. Catching also uses HipStrap vintage styled guitar straps. He is endorsed by and uses Ernie Ball ultra slinky guitar strings, coiled red Bullet Cable.

Catching released a signature fuzz wah pedal in 2015, the Roadrunner, through Dr. No Effects.

In 2023 partnered with Grand Casino Factory to imagine and develop the F.R.E.D. or Frequency Re-arranging Experimentation Device, cited by Catching as a tribute to Fred Drake, his departed former partner in Rancho de la Luna as 'an unequalled sonic enhancement device befitting such a fantastic, uniquely wonderful, talented man as Fred Drake

From 1993-2015 with earthlings?, he used his 1958 Fender Stratocaster or his 1972 double cutaway Gibson Les Paul through a tuner and a distortion Rat pedal. His Les Paul is 1 of 6 ever made by Strings & Things in Memphis, Tennessee. Other guitars were bought by Ace Frehley, Jeff Beck, Pat Travers, and Michael Woods, guitarist for the group America. Photos have circulated of Dave's very guitar nearly being bought by Greg Lake of Emerson, Lake, and Palmer fame, but opting out at the last minute for an acoustic bass.

Catching co-founded Rancho de la Luna recording studio with Fred Drake in 1993. After Drake's death in 2002, Catching has since run the studio and continues to this day, hosting such musical acts as Foo Fighters, Queens of the Stone Age, Arctic Monkeys, Iggy Pop, Daniel Lanois and many others

Partial discography

 * The Modifiers (1980s)
 * 60 Watt Reality (The Ringling Sisters, 1990)
 *  earthlings? earthlings? (1996]]
 * Queens of the Stone Age, 18AD on Burn One Up! compilation album (1997)
 * The Desert Sessions 1 & 2 (1998)
 * Queens of the Stone Age (Queens of the Stone Age, 1998)
 * "Scraps at Midnight ((Mark Lanegan 1998)
 * The Desert Sessions 3 & 4 (1998)
 * The Desert Sessions 5 & 6 (1999)
 * Rated R (Queens of the Stone Age, 2000)
 * Human Beans Man's Ruin boxset (earthlings?, 2000)
 * Deep in the Hole (Masters of Reality, 2001)
 * Songs for the Deaf (Queens of the Stone Age, 2002)
 * A Drug Problem That Never Existed (Mondo Generator, 2003)
 * The Desert Sessions 9 & 10 (2003)
 * Bubblegum (Mark Lanegan Band, 2004)
 * III the EP (Mondo Generator, 2004)
 * Cowboy Coffee and Burnt Knives (Hulk 2004)
 * Lullabies to Paralyze (Queens of the Stone Age, 2005)
 * I Got a Brand New Egg Layin' Machine (Goon Moon, 2005)
 * Impeach my Bush Peaches
 * Death By Sexy... (Eagles of Death Metal, 2006)
 * Powder Burns (The Twilight Singers, 2006)
 * Dead Planet (Mondo Generator, 2006)
 * Licker's Last Leg (Goon Moon, 2007)
 * Neptune (The Duke Spirit, 2008)
 * Prime Motivator (Giraffes 2008)
 * Heart On (Eagles of Death Metal 2008)
 * Saturnalia (Gutter Twins 2008)
 * Humalien (earthlings?, 2009)
 * Pine Cross Dover (Masters of Reality, 2009)
 * Blues Funeral (Mark Lanegan Band, 2012)
 * Spine Hits (Sleepy Sun, 2012)
 * Tomorrowland Blues (Star & Dagger, 2013)
 * Do to the Beast (Afghan Whigs 2014)
 * Unfuckwithable (Mojave Lords, 2014)
 * ecaps (earthlings? 2014)
 * Wild Nights (Pins 2015)
 * Your Desert My Mind (The Mutants, 2016)
 * Mudda Fudda (earthlings? 2016)
 * Mojave Lords (Gloria In Absentia) (2017)
 * The Golden West (album) (Gayle Skidmore, 2017)
 * Shared Hallucinations Pt.1 (David Catching and Friends 2017)
 * Mojave Lords Expensive Feelings (2021)
 * Speedy Ortiz : Rabbit Rabbit (2023)

title: Dave Catching


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User talk:Suzyrice

Hi Suzy, and welcome to Wikipedia.

I've replied to your comments with an explanation at Talk:Suzy Rice. I hope I've not offended you; I was merely trying to fix a duplication error. -- Hadal 05:38, 8 May 2004 (UTC) - (PS: See the "welcome" link above for links helpful to newcomers.)

I've replied again at Talk:Suzy Rice. And again, I apologise for any offense. -- Hadal 10:28, 8 May 2004 (UTC)

title: User talk:Suzyrice


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Common hill myna

The common hill myna (Gracula religiosa), sometimes spelled "mynah" and formerly simply known as the hill myna or myna bird, is the myna most commonly sighted in aviculture, where it is often simply referred to by the latter two names. It is a member of the starling family (Sturnidae), resident in hill regions of South Asia and Southeast Asia. The Sri Lanka hill myna, a former subspecies of G. religiosa, is now generally accepted as a separate species G. ptilogenys. The Enggano hill myna (G. enganensis) and Nias hill myna (G. robusta) are also widely accepted as specifically distinct, and many authors favor treating the southern hill myna (G. indica) from the Nilgiris and elsewhere in the Western Ghats of India as a separate species.

The common hill myna is a popular talking bird. Its specific name religiosa may allude to the practice of teaching mynas to repeat prayers.

Taxonomy
The common hill myna was formally described in 1758 by the Swedish naturalist Carl Linnaeus in the tenth edition of his Systema Naturae under the current binomial name Gracula religiosa. The type location is the Indonesian island of Java. The genus name is from Latin graculus, an unknown bird sometimes identified as the western jackdaw. The specific epithet religiosa is from Latin religiosus meaning "sacred".

Seven subspecies are recognised:

 * G. r. peninsularis Whistler & Kinnear, 1933 – central east India
 * G. r. intermedia Hay, 1845 – north India to south China, Indochina and Thailand
 * G. r. andamanensis (Beavan, 1867) – Coco, Andaman and Nicobar Islands
 * G. r. religiosa Linnaeus, 1758 – Malay Peninsula, Sumatra, Java, Borneo and nearby islands
 * G. r. miotera Oberholser, 1917 – Simeulue (west of north Sumatra) (sometimes synonymised with the nominate G. r. religiosa)
 * G. r. batuensis Finsch, 1899 – Batu Islands and Mentawai Islands (off west Sumatra)
 * G. r. palawanensis (Sharpe, 1890) – Palawan (southwest Philippines)

The southern hill myna (Gracula indica), the Nias hill myna (Gracula robusta), the Enggano hill myna (Gracula enganensis) and the Tenggara hill myna (Gracula venerata) have all been classified as subspecies.

Description
This is a stocky jet-black myna, with bright orange-yellow patches of naked skin and fleshy wattles on the side of its head and nape. At about 29 cm length, it is somewhat larger than the common myna (Acridotheres tristis).

It is overall green-glossed black plumage, purple-tinged on the head and neck. Its large, white wing patches are obvious in flight, but mostly covered when the bird is sitting. The bill and strong legs are bright yellow, and there are yellow wattles on the nape and under the eye. These differ conspicuously in shape from the naked eye-patch of the common myna and bank myna (A. ginginianus), and more subtly vary between the different hill mynas from South Asia: in the common hill myna, they extend from the eye to the nape, where they join, while the Sri Lanka hill myna has a single wattle across the nape and extending a bit towards the eyes. In the southern hill myna, the wattles are separate and curve towards the top of the head. The Nias and Enggano hill mynas differ in details of the facial wattles, and size, particularly that of the bill.

Sexes are similar; juveniles have a duller bill. The subspecies differ in size, in the pattern of wattles on the head and in the glossiness of the plumage.

A 2020 study found that the subspecies G. religiosa miotera likely represents a distinct species and was likely driven to extinction in the wild in the late 2010s due unsustainable collecting for the wildlife trade. The paper recommends rescuing the last genetically pure captive individuals for the purpose of captive breeding. The International Ornithological Congress tentatively recognises it as a subspecies.

Vocalisations
The common hill myna is often detected by its loud, shrill, descending whistles followed by other calls. It is most vocal at dawn and dusk, when it is found in small groups in forest clearings high in the canopy.

Both sexes can produce an extraordinarily wide range of loud calls – whistles, wails, screeches, and gurgles, sometimes melodious and often very human-like in quality. Each individual has a repertoire of three to 13 such call types, which may be shared with some near neighbours of the same sex, being learned when young. Dialects change rapidly with distance, such that birds living more than 15 km apart have no call-types in common with one another.

Unlike some other birds, such as the greater racket-tailed drongo (Dicrurus paradiseus), the common hill myna does not imitate other birds in the wild, although it is a widely held misconception that they do. On the other hand, in captivity, they are among the most renowned mimics, the only bird, perhaps, on par with the grey parrot (Psittacus erithacus). They can learn to reproduce many everyday sounds, particularly the human voice, and even whistled tunes, with astonishing accuracy and clarity.

Distribution and ecology
This myna is a resident breeder from Kumaon division in India (80° E longitude) east through Nepal, Sikkim, Bhutan and Arunachal Pradesh, the lower Himalayas, terai and foothills up to 2,000 m ASL. Its range continues east through Southeast Asia northeastwards to southern China, and via Thailand southeastwards across northern Indonesia to Palawan in the Philippines. It is virtually extinct in Bangladesh due to habitat destruction and overexploitation for the pet trade. A feral population on Christmas Island has likewise disappeared. Introduced populations exist in Saint Helena, Puerto Rico and perhaps in the mainland United States and possibly elsewhere; feral birds require at least a warm subtropical climate to persist.

This myna is almost entirely arboreal, moving in large, noisy groups of half a dozen or so, in tree-tops at the edge of the forest. It hops sideways along the branch, unlike the characteristic jaunty walk of other mynas. Like most starlings, the hill myna is fairly omnivorous, eating fruit, nectar and insects.

They build a nest in a hole in a tree. The usual clutch is two or three eggs. There is no sexual dimorphism in these birds, which results in a limited possibility of choosing the sex to work with for mating.

Pet trade and conservation
The hill mynas are popular cage birds, renowned for their ability to imitate speech. The widely distributed common hill myna is the one most frequently seen in aviculture. Demand outstrips captive breeding capacity, so they are rarely found in pet stores and usually purchased directly from breeders or importers who can certify the birds are traded legally.

This species is widely distributed and locally common, and if adult stocks are safeguarded, it is able to multiply quickly. On a worldwide scale, the IUCN thus considers the common hill myna a Species of Least Concern. But in the 1990s, nearly 20,000 wild-caught birds, mostly adults and juveniles, were brought into trade each year. In the central part of its range, G. r. intermedia populations have declined markedly, especially in Thailand, which supplied much of the thriving Western market. Its neighbor countries, from where exports were often limited due to political or military reasons, nevertheless supplied a burgeoning domestic demand, and demand in the entire region continues to be very high. In 1992, Thailand had the common hill myna put on CITES Appendix III, to safeguard its stocks against collapsing. In 1997, at the request of the Netherlands and the Philippines, the species was uplisted to CITES Appendix II. The Andaman and Nicobar Islands subspecies G. r. andamanensis and (if valid) G. r. halibrecta, described as "exceedingly common" in 1874, qualified as Near Threatened in 1991. The former is not at all common anymore in the Nicobar Islands and the latter—if distinct—has a very limited range.

Elsewhere, such as on the Philippines and in Laos, the decline has been more localized. It is also becoming increasingly rare in the regions of northeastern India due to capture of fledged birds for the illegal pet trade. In the Garo Hills region, however, the locals make artificial nests of a split-bamboo framework covered with grass, and put them up in accessible positions in tall trees in a forest clearing or at the edge of a small village to entice the mynas to breed there. The villagers are thus able to extract the young at the proper time for easy hand-rearing, making common hill myna farming a profitable, small-scale cottage industry. It helps to preserve the environment, because the breeding birds are not removed from the population, while habitat destruction is curtailed because the mynas will desert areas of extensive logging and prefer more natural forest to plantations. As the mynas can be somewhat of a pest of fruit trees when too numerous, an additional benefit to the locals is the inexpensive means of controlling the myna population: failing stocks can be bolstered by putting out more nests than can be harvested, while the maximum proportion of nestlings are taken when the population becomes too large.

title: Common hill myna


text:
Southern hill myna

The southern hill myna (Gracula indica) is a member of the starling family. It is a resident of the forested regions of south India and Sri Lanka. This species moves around in flocks in the forest, and members of a flock keep in contact with a wide range of vocalizations.

Etymology
The French word for the bird mainate is probably derived from the Indo-Portuguese mainato, from the name of a songbird living near water, and is probably a metaphor since it refers to a man whose job is to wash the clothes of others.

Description
The southern hill myna has a shiny black plumage. It generally measures about twenty centimeters. It can live between 15 and 30 years. The legs are yellow, the beak is bright orange at the base and yellow at the tip. It has white spots on the primary remiges. Like other species of hill myna, the southern hill myna has yellow wattles on the head.

Habitats and behavior
The southern hill myna is present in southwestern India and Sri Lanka. It inhabits tropical and subtropical dry broadleaf forests, mountains, mangrove forests and plantation areas. It is usually found high in the canopy, moving in large noisy groups of about half a dozen, in treetops at the edge of the forest. It hops sideways along the branch. Like most starlings, the hill myna is omnivorous, eating fruit, nectar and insects.

Its voice is melodious, it is able to emit various croaks and calls. The hill myna is considered to be one of the best talking birds, often imitating human sounds and voice, better than parrots. It is also bred in captivity for this particularity. The southern hill myna is claimed to be less vocally gifted than the common hill myna, a very similar species whose range does not overlap.

Reproduction
It builds a nest in a hole in a tree. They are usually two or three eggs in the nest. Male and female birds look alike.

Conservation status
Although the population is declining it is considered "Least Concern" by the IUCN.

title: Southern hill myna


text:
Laplace's method

In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form

 * $$\int_a^b e^{Mf(x)} \, dx,$$

where $$f(x)$$ is a twice-differentiable function, $$M$$ is a large number, and the endpoints $$a$$ and $$b$$ could be infinite. This technique was originally presented in the book.

In Bayesian statistics, Laplace's approximation can refer to either approximating the posterior normalizing constant with Laplace's method or approximating the posterior distribution with a Gaussian centered at the maximum a posteriori estimate. Laplace approximations are used in the integrated nested Laplace approximations method for fast approximations of Bayesian inference.

Concept

Suppose the function $$f(x)$$ has a global maximum at $$x_0$$. Let $$M>0$$ be a constant and consider the following two functions:

 * $$\begin{align}

g(x) &= Mf(x), \\ h(x) &= e^{Mf(x)}. \end{align}$$

Note that $$x_0$$ will be the global maximum of $$g$$ and $$h$$ as well. Now observe:

 * $$\begin{align}

\frac{g(x_0)}{g(x)} &= \frac{M f(x_0)}{M f(x)} = \frac{f(x_0)}{f(x)}, \\[4pt] \frac{h(x_0)}{h(x)} &= \frac{e^{M f(x_0)}}{e^{M f(x)}} = e^{M(f(x_0) - f(x))}. \end{align}$$

As M increases, the ratio for $$h$$ will grow exponentially, while the ratio for $$g$$ does not change. Thus, significant contributions to the integral of this function will come only from points $$x$$ in a neighbourhood of $$x_0$$, which can then be estimated.

General theory
To state and motivate the method, one must make several assumptions. It is assumed that $$x_0$$ is not an endpoint of the interval of integration and that the values $$f(x)$$ cannot be very close to $$f(x_0)$$ unless $$x$$ is close to $$x_0$$.

$$f(x)$$ can be expanded around x0 by Taylor's theorem,

 * $$f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{1}{2} f''(x_0)(x-x_0)^2 + R$$

where $$R = O\left((x-x_0)^3\right)$$ (see: big O notation).

Since $$f$$ has a global maximum at $$x_0$$, and $$x_0$$ is not an endpoint, it is a stationary point, i.e. $$f'(x_0)=0$$. Therefore, the second-order Taylor polynomial approximating $$f(x)$$ is

 * $$f(x) \approx f(x_0) + \frac{1}{2} f''(x_0) (x-x_0)^2.$$

Then, just one more step is needed to get a Gaussian distribution. Since $$x_0$$ is a global maximum of the function $$f$$ it can be stated, by definition of the second derivative, that $$f''(x_0) < 0$$, thus giving the relation

$$f(x) \approx f(x_0) - \frac{1}{2} |f''(x_0)| (x-x_0)^2$$

for $$x$$ close to $$x_0$$. The integral can then be approximated with:

 * $$\int_a^b e^{M f(x)}\, dx\approx e^{M f(x_0)}\int_a^b e^{-\frac{1}{2} M|f''(x_0)| (x-x_0)^2} \, dx$$

This latter integral is a Gaussian integral if the limits of integration go from $$-\infty$$ to $$+\infty$$ (which can be assumed because the exponential decays very fast away from $$x_0$$), and thus it can be calculated. This gives

 * $$\int_a^b e^{M f(x)}\, dx\approx \sqrt{\frac{2\pi}{M|f''(x_0)|}}e^{M f(x_0)} \text { as } M\to\infty.$$

A generalization of this method and extension to arbitrary precision is provided by the book.

Formal statement and proof
Suppose $$f(x)$$ is a twice continuously differentiable function on $$[a,b],$$ and there exists a unique point $$x_0 \in (a,b)$$ such that:

 * $$f(x_0) = \max_{x \in [a,b]} f(x) \quad \text{and} \quad f''(x_0)<0.$$

Then:

 * $$\lim_{n\to\infty} \frac{\int_a^b e^{nf(x)} \, dx}{e^{nf(x_0)} \sqrt{\frac{2\pi}{n\left(-f''(x_0)\right)}}}= 1. $$

Lower bound: Let $$\varepsilon > 0$$. Since $$f$$ is continuous there exists $$\delta >0$$ such that if $$|x_0-c|< \delta$$ then $$f(c) \ge f''(x_0) - \varepsilon.$$ By Taylor's Theorem, for any $$x \in (x_0 - \delta, x_0 + \delta),$$

 * $$f(x) \ge f(x_0) + \frac{1}{2}(f''(x_0) - \varepsilon)(x-x_0)^2.$$

Then we have the following lower bound:

 * $$\begin{align}

\int_a^b e^{nf(x)} \, dx &\ge \int_{x_0 - \delta}^{x_0 + \delta} e^{nf(x)} \, dx \\ &\ge e^{nf(x_0)} \int_{x_0 - \delta}^{x_0 + \delta} e^{\frac{n}{2}(f''(x_0) - \varepsilon)(x-x_0)^2} \, dx \\ &=  e^{nf(x_0)} \sqrt{\frac{1}{n(-f(x_0) + \varepsilon)}} \int_{-\delta \sqrt{n(-f(x_0) + \varepsilon)} }^{\delta \sqrt{n(-f''(x_0) + \varepsilon)} } e^{-\frac{1}{2}y^2} \, dy \end{align}$$

where the last equality was obtained by a change of variables

 * $$y= \sqrt{n(-f''(x_0) + \varepsilon)} (x-x_0).$$

Remember $$f''(x_0)<0$$ so we can take the square root of its negation.

If we divide both sides of the above inequality by

 * $$e^{nf(x_0)}\sqrt{\frac{2 \pi}{n(-f''(x_0))}}$$

and take the limit we get:

 * $$\lim_{n \to \infty} \frac{\int_a^b e^{nf(x)} \,dx}{e^{nf(x_0)}\sqrt{\frac{2\pi}{n(-f(x_0))}}} \ge \lim_{n \to \infty} \frac{1}{\sqrt{2\pi}} \int_{-\delta\sqrt{n(-f(x_0) + \varepsilon)} }^{\delta \sqrt{n(-f(x_0) + \varepsilon)}} e^{-\frac{1}{2}y^2} \, dy \, \cdot \sqrt{\frac{-f(x_0)}{-f(x_0) + \varepsilon}} = \sqrt{\frac{-f(x_0)}{-f''(x_0) + \varepsilon}}$$

since this is true for arbitrary $$\varepsilon$$ we get the lower bound:

 * $$\lim_{n \to \infty} \frac{\int_a^b e^{nf(x)} \, dx}{ e^{nf(x_0)}\sqrt{\frac{2\pi}{n(-f''(x_0))}}} \ge 1$$

Note that this proof works also when $$a = -\infty$$ or $$b= \infty$$ (or both).

Upper bound: The proof is similar to that of the lower bound but there are a few inconveniences. Again we start by picking an $$\varepsilon >0$$ but in order for the proof to work we need $$\varepsilon$$ small enough so that $$f(x_0) + \varepsilon < 0.$$ Then, as above, by continuity of $$f$$ and Taylor's Theorem we can find $$\delta>0$$ so that if $$|x-x_0| < \delta$$, then

 * $$f(x) \le f(x_0) + \frac{1}{2} (f''(x_0) + \varepsilon)(x-x_0)^2.$$

Lastly, by our assumptions (assuming $$a,b$$ are finite) there exists an $$\eta >0$$ such that if $$|x-x_0|\ge \delta$$, then $$f(x) \le f(x_0) - \eta$$.

Then we can calculate the following upper bound:

 * $$\begin{align}

\int_a^b e^{nf(x)} \, dx &\le \int_a^{x_0-\delta} e^{nf(x)} \, dx + \int_{x_0-\delta}^{x_0 + \delta} e^{nf(x)} \, dx + \int_{x_0 + \delta}^b e^{nf(x)} \, dx \\ &\le (b-a)e^{n(f(x_0)-\eta)} + \int_{x_0-\delta}^{x_0 + \delta} e^{n f(x) } \, dx \\ &\le (b-a)e^{n(f(x_0)-\eta)} + e^{nf(x_0)} \int_{x_0-\delta}^{x_0 + \delta} e^{\frac{n}{2}(f''(x_0)+\varepsilon)(x-x_0)^2} \, dx\\ &\le (b-a)e^{n(f(x_0)-\eta)} + e^{nf(x_0)} \int_{-\infty}^{+\infty} e^{\frac{n}{2}(f''(x_0)+\varepsilon)(x-x_0)^2} \, dx \\ &\le (b-a)e^{n(f(x_0)-\eta)} + e^{nf(x_0)} \sqrt{\frac{2 \pi}{n (-f''(x_0) - \varepsilon)}} \end{align}$$

If we divide both sides of the above inequality by

 * $$e^{nf(x_0)}\sqrt{\frac{2 \pi}{n (-f''(x_0))}}$$

and take the limit we get:

 * $$\lim_{n \to \infty} \frac{\int_a^b e^{nf(x)} \, dx}{e^{nf(x_0)}\sqrt{\frac{2\pi}{n(-f(x_0))}}} \le \lim_{n \to \infty} (b-a) e^{-\eta n} \sqrt{\frac{n(-f(x_0))}{2\pi}} + \sqrt{\frac{-f(x_0)}{-f(x_0) - \varepsilon}} = \sqrt{\frac{-f(x_0)}{-f(x_0) - \varepsilon}}$$

Since $$\varepsilon$$ is arbitrary we get the upper bound:

 * $$\lim_{n \to \infty} \frac{\int_a^b e^{nf(x)} \, dx}{e^{nf(x_0)}\sqrt{\frac{2 \pi}{n (-f''(x_0))}}} \le 1$$

And combining this with the lower bound gives the result.

Note that the above proof obviously fails when $$a = -\infty$$ or $$b = \infty$$ (or both). To deal with these cases, we need some extra assumptions. A sufficient (not necessary) assumption is that for $$n = 1,$$

 * $$\int_a^b e^{nf(x)} \, dx < \infty,$$

and that the number $$\eta$$ as above exists (note that this must be an assumption in the case when the interval $$[a,b]$$ is infinite). The proof proceeds otherwise as above, but with a slightly different approximation of integrals:

 * $$\int_a^{x_0-\delta} e^{nf(x)} \, dx + \int_{x_0 + \delta}^b e^{nf(x)} \, dx \le \int_a^b e^{f(x)}e^{(n-1)(f(x_0) - \eta)} \, dx = e^{(n-1)(f(x_0) - \eta)} \int_a^b e^{f(x)} \, dx.$$

When we divide by

 * $$e^{nf(x_0)}\sqrt{\frac{2\pi}{n(-f''(x_0))}},$$

we get for this term

 * $$\frac{e^{(n-1)(f(x_0) - \eta)} \int_a^b e^{f(x)} \, dx}{e^{nf(x_0)}\sqrt{\frac{2\pi}{n(-f(x_0))}}} = e^{-(n-1)\eta} \sqrt{n} e^{-f(x_0)} \int_a^b e^{f(x)} \, dx \sqrt{\frac{-f(x_0)}{2\pi}}$$

whose limit as $$n \to \infty$$ is $$0$$. The rest of the proof (the analysis of the interesting term) proceeds as above.

The given condition in the infinite interval case is, as said above, sufficient but not necessary. However, the condition is fulfilled in many, if not in most, applications: the condition simply says that the integral we are studying must be well-defined (not infinite) and that the maximum of the function at $$x_0$$ must be a "true" maximum (the number $$\eta > 0$$ must exist). There is no need to demand that the integral is finite for $$n=1$$ but it is enough to demand that the integral is finite for some $$n=N.$$

This method relies on 4 basic concepts such as

 * 1. Relative error

The “approximation” in this method is related to the relative error and not the absolute error. Therefore, if we set

 * $$s = \sqrt{\frac{2\pi}{M\left|f''(x_0)\right|}},$$

the integral can be written as

 * $$\begin{align}

\int_a^b e^{M f(x)} \, dx &= se^{Mf(x_0)} \frac{1}{s}\int_a^b e^{M(f(x)-f(x_0))}\, dx \\ & = se^{Mf(x_0)} \int_{\frac{a-x_0}{s}}^{\frac{b-x_0}{s}} e^{M(f(sy+x_0)-f(x_0))}\,dy \end{align}$$

where $$s$$ is a small number when $$M$$ is a large number obviously and the relative error will be
 * $$\left| \int_{\frac{a-x_0}{s}}^{\frac{b-x_0}{s}} e^{M(f(sy+x_0)-f(x_0))} dy-1 \right|.$$

Now, let us separate this integral into two parts: $$y\in[-D_y,D_y]$$ region and the rest.

 * 2. $$e^{M (f(sy+x_0)-f(x_0))} \to e^{-\pi y^2}$$ around the stationary point when $$M$$ is large enough

Let’s look at the Taylor expansion of $$M(f(x)-f(x_0))$$ around x0 and translate x to y because we do the comparison in y-space, we will get

 * $$M(f(x)-f(x_0)) = \frac{Mf(x_0)}{2}s^2y^2 +\frac{Mf'(x_0)}{6}s^3y^3+ \cdots = -\pi y^2 +O\left(\frac{1}{\sqrt{M}}\right).$$

Note that $$f'(x_0)=0$$ because $$x_0$$ is a stationary point. From this equation you will find that the terms higher than second derivative in this Taylor expansion is suppressed as the order of $$\tfrac{1}{\sqrt{M}}$$ so that $$\exp(M(f(x)-f(x_0)))$$ will get closer to the Gaussian function as shown in figure. Besides,

 * $$\int_{-\infty}^{\infty}e^{-\pi y^2} dy =1.$$

 * 3. The larger $$M$$ is, the smaller range of $$x$$ is related

Because we do the comparison in y-space, $$y$$ is fixed in $$y\in[-D_y,D_y]$$ which will cause $$x\in[-sD_y, sD_y]$$; however, $$s$$ is inversely proportional to $$\sqrt{M}$$, the chosen region of $$x$$ will be smaller when $$M$$ is increased.

 * 4. If the integral in Laplace's method converges, the contribution of the region which is not around the stationary point of the integration of its relative error will tend to zero as $$M$$ grows.

Relying on the 3rd concept, even if we choose a very large Dy, sDy will finally be a very small number when $$M$$ is increased to a huge number. Then, how can we guarantee the integral of the rest will tend to 0 when $$M$$ is large enough?

The basic idea is to find a function $$m(x)$$ such that $$m(x)\ge f(x)$$ and the integral of $$e^{Mm(x)}$$ will tend to zero when $$M$$ grows. Because the exponential function of $$Mm(x)$$ will be always larger than zero as long as $$m(x)$$ is a real number, and this exponential function is proportional to $$m(x),$$ the integral of $$e^{Mf(x)}$$ will tend to zero. For simplicity, choose $$m(x)$$ as a tangent through the point $$x=sD_y$$ as shown in the figure:

If the interval of the integration of this method is finite, we will find that no matter $$f(x)$$ is continue in the rest region, it will be always smaller than $$m(x)$$ shown above when $$M$$ is large enough. By the way, it will be proved later that the integral of $$e^{Mm(x)}$$ will tend to zero when $$M$$ is large enough.

If the interval of the integration of this method is infinite, $$m(x)$$ and $$f(x)$$ might always cross to each other. If so, we cannot guarantee that the integral of $$e^{Mf(x)}$$ will tend to zero finally. For example, in the case of $$f(x)=\tfrac{\sin(x)}{x},$$ $$\int^{\infty}_{0}e^{Mf(x)} dx$$ will always diverge. Therefore, we need to require that $$\int^{\infty}_{d}e^{Mf(x)} dx$$ can converge for the infinite interval case. If so, this integral will tend to zero when $$d$$ is large enough and we can choose this $$d$$ as the cross of $$m(x)$$ and $$f(x).$$

You might ask why not choose $$\int^{\infty}_{d}e^{f(x)} dx$$ as a convergent integral? Let me use an example to show you the reason. Suppose the rest part of $$f(x)$$ is $$-\ln x,$$ then $$e^{f(x)}=\tfrac{1}{x}$$ and its integral will diverge; however, when $$M=2,$$ the integral of $$e^{Mf(x)}=\tfrac{1}{x^2}$$ converges. So, the integral of some functions will diverge when $$M$$ is not a large number, but they will converge when $$M$$ is large enough.

Based on these four concepts, we can derive the relative error of this method.

Other formulations
Laplace's approximation is sometimes written as

 * $$\int_a^b h(x) e^{M g(x)}\, dx \approx \sqrt{\frac{2\pi}{M|g''(x_0)|}} h(x_0) e^{M g(x_0)} \ \text { as } M\to\infty$$

where $$h$$ is positive.

Importantly, the accuracy of the approximation depends on the variable of integration, that is, on what stays in $$g(x)$$ and what goes into $$h(x).$$

First, use $$x_0=0$$ to denote the global maximum, which will simplify this derivation. We are interested in the relative error, written as $$|R|$$,

 * $$\int_a^b h(x) e^{M g(x)}\, dx = h(0)e^{Mg(0)}s \underbrace{\int_{a/s}^{b/s}\frac{h(x)}{h(0)}e^{M\left[ g(sy)-g(0) \right]} dy}_{1+R},$$

where

 * $$s\equiv\sqrt{\frac{2\pi}{M\left| g'' (0) \right|}}.$$

So, if we let

 * $$A\equiv \frac{h(sy)}{h(0)}e^{M\left[g(sy)-g(0) \right]}$$

and $$A_0\equiv e^{-\pi y^2}$$, we can get

 * $$\left| R\right| = \left| \int_{a/s}^{b/s}A\,dy -\int_{-\infty}^{\infty}A_0\,dy \right|$$

since $$\int_{-\infty}^{\infty}A_0\,dy =1$$.

For the upper bound, note that $$|A+B| \le |A|+|B|,$$ thus we can separate this integration into 5 parts with 3 different types (a), (b) and (c), respectively. Therefore,

 * $$|R| < \underbrace{\left| \int_{D_y}^{\infty}A_0 dy \right|}_{(a_1)} + \underbrace{\left| \int_{D_y}^{b/s}A dy \right|}_{(b_1)}+ \underbrace{\left| \int_{-D_y}^{D_y}\left(A-A_0\right) dy \right|}_{(c)} + \underbrace{\left| \int_{a/s}^{-D_y}A dy \right|}_{(b_2)} + \underbrace{\left| \int_{-\infty}^{-D_y}A_0 dy \right|}_{(a_2)}$$

where $$(a_1)$$ and $$(a_2)$$ are similar, let us just calculate $$(a_1)$$ and $$(b_1)$$ and $$(b_2)$$ are similar, too, I’ll just calculate $$(b_1)$$.

For $$(a_1)$$, after the translation of $$z\equiv\pi y^2$$, we can get

 * $$(a_1) = \left| \frac{1}{2\sqrt{\pi}}\int_{\pi D_y^2}^{\infty} e^{-z}z^{-1/2} dz\right| <\frac{e^{-\pi D_y^2}}{2\pi D_y}.$$

This means that as long as $$D_y$$ is large enough, it will tend to zero.

For $$(b_1)$$, we can get

 * $$(b_1)\le\left| \int_{D_y}^{b/s}\left[\frac{h(sy)}{h(0)}\right]_{\text{max}} e^{Mm(sy)}dy \right|$$

where

 * $$m(x) \ge g(x)-g(0) \text{as} x\in [sD_y,b]$$

and $$h(x)$$ should have the same sign of $$h(0)$$ during this region. Let us choose $$m(x)$$ as the tangent across the point at $$x=sD_y$$, i.e. $$m(sy)= g(sD_y)-g(0) +g'(sD_y)\left( sy-sD_y \right)$$ which is shown in the figure

From this figure you can find that when $$s$$ or $$D_y$$ gets smaller, the region satisfies the above inequality will get larger. Therefore, if we want to find a suitable $$m(x)$$ to cover the whole $$f(x)$$ during the interval of $$(b_1)$$, $$D_y$$ will have an upper limit. Besides, because the integration of $$e^{-\alpha x}$$ is simple, let me use it to estimate the relative error contributed by this $$(b_1)$$.

Based on Taylor expansion, we can get

 * $$\begin{align}

M\left[g(sD_y)-g(0)\right] &= M\left[ \frac{g(0)}{2}s^2D_y^2 +\frac{g'(\xi)}{6}s^3D_y^3 \right] && \text{as } \xi\in[0,sD_y] \\ & = -\pi D_y^2 +\frac{(2\pi)^{3/2}g'(\xi)D_y^3}{6\sqrt{M}|g(0)|^{\frac{3}{2}}}, \end{align}$$

and

 * $$\begin{align}

Msg'(sD_y) &= Ms\left(g(0)sD_y +\frac{g'(\zeta)}{2}s^2D_y^2\right) && \text{as } \zeta\in[0,sD_y] \\ &= -2\pi D_y +\sqrt{\frac{2}{M}}\left( \frac{\pi}{|g(0)|} \right)^{\frac{3}{2}}g'(\zeta)D_y^2, \end{align}$$

and then substitute them back into the calculation of $$(b_1)$$; however, you can find that the remainders of these two expansions are both inversely proportional to the square root of $$M$$, let me drop them out to beautify the calculation. Keeping them is better, but it will make the formula uglier.

 * $$\begin{align}

(b_1) &\le \left|\left[ \frac{h(sy)}{h(0)} \right]_{\max} e^{-\pi D_y^2}\int_0^{b/s-D_y}e^{-2\pi D_y y} dy \right| \\ &\le \left|\left[ \frac{h(sy)}{h(0)} \right]_{\max} e^{-\pi D_y^2}\frac{1}{2\pi D_y} \right|. \end{align}$$

Therefore, it will tend to zero when $$D_y$$ gets larger, but don't forget that the upper bound of $$D_y$$ should be considered during this calculation.

About the integration near $$x=0$$, we can also use Taylor's Theorem to calculate it. When $$h'(0) \ne 0$$

 * $$\begin{align}

(c) &\le \int_{-D_y}^{D_y} e^{-\pi y^2} \left| \frac{sh'(\xi)}{h(0)}y \right|\, dy \\ &< \sqrt{\frac{2}{\pi M |g''(0)|}} \left| \frac{h'(\xi)}{h(0)} \right|_\max \left( 1-e^{-\pi D_y^2} \right) \end{align}$$

and you can find that it is inversely proportional to the square root of $$M$$. In fact, $$(c)$$ will have the same behave when $$h(x)$$ is a constant.

Conclusively, the integral near the stationary point will get smaller as $$\sqrt{M}$$ gets larger, and the rest parts will tend to zero as long as $$D_y$$ is large enough; however, we need to remember that $$D_y$$ has an upper limit which is decided by whether the function $$m(x)$$ is always larger than $$g(x)-g(0)$$ in the rest region. However, as long as we can find one $$m(x)$$ satisfying this condition, the upper bound of $$D_y$$ can be chosen as directly proportional to $$\sqrt{M}$$ since $$m(x)$$ is a tangent across the point of $$g(x)-g(0)$$ at $$x=sD_y$$. So, the bigger $$M$$ is, the bigger $$D_y$$ can be.

In the multivariate case, where $$\mathbf{x}$$ is a $$d$$-dimensional vector and $$f(\mathbf{x})$$ is a scalar function of $$\mathbf{x}$$, Laplace's approximation is usually written as:

 * $$\int h(\mathbf{x})e^{M f(\mathbf{x})}\, d\mathbf{x} \approx \left(\frac{2\pi}{M}\right)^{d/2} \frac{h(\mathbf{x}_0)e^{M f(\mathbf{x}_0)}}{\left|-H(f)(\mathbf{x}_0)\right|^{1/2}} \text { as } M\to\infty$$

where $$H(f)(\mathbf{x}_0)$$ is the Hessian matrix of $$f$$ evaluated at $$\mathbf{x}_0$$ and where $$|\cdot|$$ denotes matrix determinant. Analogously to the univariate case, the Hessian is required to be negative-definite.

By the way, although $$\mathbf{x}$$ denotes a $$d$$-dimensional vector, the term $$d\mathbf{x}$$ denotes an infinitesimal volume here, i.e. $$d\mathbf{x} := dx_1dx_2\cdots dx_d$$.

Steepest descent extension
In extensions of Laplace's method, complex analysis, and in particular Cauchy's integral formula, is used to find a contour of steepest descent for an (asymptotically with large M) equivalent integral, expressed as a line integral. In particular, if no point x0 where the derivative of $$f$$ vanishes exists on the real line, it may be necessary to deform the integration contour to an optimal one, where the above analysis will be possible. Again, the main idea is to reduce, at least asymptotically, the calculation of the given integral to that of a simpler integral that can be explicitly evaluated. See the book of Erdelyi (1956) for a simple discussion (where the method is termed steepest descents).

The appropriate formulation for the complex z-plane is

 * $$\int_a^b e^{M f(z)}\, dz \approx \sqrt{\frac{2\pi}{-Mf''(z_0)}}e^{M f(z_0)} \text{ as } M\to\infty.$$

for a path passing through the saddle point at z0. Note the explicit appearance of a minus sign to indicate the direction of the second derivative: one must not take the modulus. Also note that if the integrand is meromorphic, one may have to add residues corresponding to poles traversed while deforming the contour (see for example section 3 of Okounkov's paper Symmetric functions and random partitions).

Further generalizations
An extension of the steepest descent method is the so-called nonlinear stationary phase/steepest descent method. Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann–Hilbert factorization problems.

Given a contour C in the complex sphere, a function $$f$$ defined on that contour and a special point, such as infinity, a holomorphic function M is sought away from C, with prescribed jump across C, and with a given normalization at infinity. If $$f$$ and hence M are matrices rather than scalars this is a problem that in general does not admit an explicit solution.

An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann–Hilbert problem to that of a simpler, explicitly solvable, Riemann–Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.

The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou. As in the linear case, "steepest descent contours" solve a min-max problem. In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov).

The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.

Median-point approximation generalization
In the generalization, evaluation of the integral is considered equivalent to finding the norm of the distribution with density

 * $$e^{M f(x)}.$$

Denoting the cumulative distribution $$F(x)$$, if there is a diffeomorphic Gaussian distribution with density

 * $$e^{-g -\frac{\gamma}{2}y^2}$$

the norm is given by

 * $$\sqrt{2\pi\gamma^{-1}}e^{-g}$$

and the corresponding diffeomorphism is

 * $$y(x)=\frac{1}{\sqrt{\gamma}}\Phi^{-1}{\left(\frac{F(x)}{F(\infty)}\right)},$$

where $$\Phi$$ denotes cumulative standard normal distribution function.

In general, any distribution diffeomorphic to the Gaussian distribution has density

 * $$e^{-g -\frac{\gamma}{2}y^2(x)}y'(x)$$

and the median-point is mapped to the median of the Gaussian distribution. Matching the logarithm of the density functions and their derivatives at the median point up to a given order yields a system of equations that determine the approximate values of $$\gamma$$ and $$g$$.

The approximation was introduced in 2019 by D. Makogon and C. Morais Smith primarily in the context of partition function evaluation for a system of interacting fermions.

Complex integrals
For complex integrals in the form:

 * $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} g(s)e^{st} \,ds$$

with $$t \gg 1,$$ we make the substitution t = iu and the change of variable $$s=c+ix$$ to get the bilateral Laplace transform:

 * $$\frac{1}{2 \pi}\int_{-\infty}^\infty g(c+ix)e^{-ux}e^{icu} \, dx.$$

We then split g(c + ix) in its real and complex part, after which we recover u = t/i. This is useful for inverse Laplace transforms, the Perron formula and complex integration.

Example: Stirling's approximation
Laplace's method can be used to derive Stirling's approximation

 * $$N!\approx \sqrt{2\pi N} N^N e^{-N}\,$$

for a large integer N.

From the definition of the Gamma function, we have

 * $$N! = \Gamma(N+1)=\int_0^\infty e^{-x} x^N \, dx.$$

Now we change variables, letting $$x=Nz$$ so that $$dx = Ndz.$$ Plug these values back in to obtain

 * $$\begin{align}

N! &= \int_0^\infty e^{-Nz} (Nz)^N N \, dz \\ &= N^{N+1} \int_0^\infty e^{-Nz} z^N \, dz \\ &= N^{N+1} \int_0^\infty e^{-Nz} e^{N\ln z} \, dz \\ &= N^{N+1} \int_0^\infty e^{N(\ln z-z)} \, dz. \end{align}$$

This integral has the form necessary for Laplace's method with

 * $$f(z) = \ln{z}-z$$

which is twice-differentiable:

 * $$f'(z) = \frac{1}{z}-1,$$
 * $$f''(z) = -\frac{1}{z^2}.$$

The maximum of $$f(z)$$ lies at z0 = 1, and the second derivative of $$f(z)$$ has the value −1 at this point. Therefore, we obtain

 * $$N! \approx N^{N+1}\sqrt{\frac{2\pi}{N}} e^{-N}=\sqrt{2\pi N} N^N e^{-N}.$$

title: Laplace's method


text:
Fyodorov

Fyodorov or Fedorov (Фёдоров, masculine) and Fyodorova or Fedorova (Фёдорова, feminine) is a common East Slavic last name that is derived from the given name Fyodor and literally means Fyodor's. It is transliterated in Polish as Fiodorow (masculine) and Fiodorowa (feminine), in Belarusian as Fiodaraŭ, and in Estonian Fjodorov. In Ukrainian it is always spelled as "Fedorov" (Федоров), because the Ukrainian alphabet does not have "ё". Another Ukrainian variant is Fedoriv.

Notable people with the name include:
 * Aleksandr Fyodorov (disambiguation), multiple people
 * Aleksey Fyodorov (disambiguation), multiple people
 * Alfred Fyodorov (1935–2001), Soviet football player and coach
 * Alicja Fiodorow (born 1985), Polish Paralympic athlete
 * Andrei Fyodorov (disambiguation), multiple people
 * Anna Fedorova (born 1990), Ukrainian pianist
 * Anthony Fedorov (born 1985), American Idol Season Four finalist
 * Boris Fyodorov (1958–2008), Russian economist and politician
 * Evgraf Fedorov (1853–1919), Russian mathematician, crystallographer and mineralogist
 * Evgraf Fedorov Jr. (1880–1965), Russian climatologist
 * Fedor Fedorov (ice hockey) (born 1981), Russian ice hockey player
 * Fiodar Fiodaraŭ (1911–1994), Belarusian physicist
 * Igor Fedorov (born 2000), Russian boxer
 * Irina Feodorova, Soviet historian and ethnographer
 * Ivan Fyodorov (disambiguation), multiple people
 * Joanna Fiodorow (born 1989), Polish hammer thrower
 * Leonid Feodorov (1879–1935), Russian ecclesiastical figure
 * Marina Fedorova (born 1997), Russian football forward
 * Miron Fyodorov (born 1985), Russian rapper
 * Nikolai Fyodorovich Fyodorov (1829–1903), Russian cosmist and philosopher
 * Nikolay Fyodorov (disambiguation), multiple people
 * Oleksiy Fedorov (1901–1989), leader of the Soviet Ukrainian partisan movement
 * Oxana Fedorova (born 1977), Russian Miss Universe 2002
 * Sergei Fyodorov (disambiguation), multiple people
 * Svyatoslav Fyodorov (1927–2000), Russian ophthalmologist and eye microsurgeon
 * Victor Fyodorov (1885–1922), Russian World War I flying ace
 * Viktoriya Fyodorova (born 1973), Russian high jumper
 * Vladimir Fyodorov (disambiguation), multiple people
 * Yevgeny Fyodorov (disambiguation), multiple people
 * Yuri Fedorov (born 1949), Soviet ice hockey player
 * Zoya Fyodorova (1907–1981), Russian actress

title: Fyodorov


text:
Canada Health and Social Transfer

The Canada Health and Social Transfer (CHST) was a system of block transfer payments from the Canadian government to provincial governments to pay for health care, post-secondary education and welfare, in place from the 1996–97 fiscal year until the 2004–05 fiscal year. It was split into the Canada Health Transfer (CHT) and Canada Social Transfer (CST) effective April 1, 2004, to provide greater accountability and transparency for federal health funding.

Background
The CHST was announced in the 1995 Canadian federal budget as an amalgamation of two federal programs prior to 1996:
 * The Established Programs Financing program, which paid for health care and post-secondary education and was established in 1977 ;
 * And the Canada Assistance Plan, which supported social assistance and was established in 1966.

Under the Constitution of Canada, health, education and social assistance are all areas of provincial responsibility and authority. The federal government does not directly participate in the administration of government services in these areas, though federal money through the CHST and its successors is used to fund them. But unlike equalization payments, which provinces can spend on anything, money distributed through the CHST is conditional and must be spent on health, post-secondary education or welfare. Legislation such as the Canada Health Act specify standards that the provinces must maintain in order to receive funding.

The rate of increase to the amounts in the transfer are driven by economic growth, due to the underlying program being designed on the basis of the "transfer" of tax points, however, occasional supplements to the program can be made either through increases in the base rate, such $11 billion more in funding per year through bilateral amendment agreements in 2016, or through special time-limited funding, such as the 2006 Wait Time Reductions Fund, which provided funding over a decade. The CHST and its successors consist of both cash transfers and tax transfers. Cash transfers are direct transfers of money from the federal government to the provinces. Tax transfers work because both federal and provincial governments collect personal and corporate income tax. A tax transfer involves the federal government reducing its income tax rates, leaving the provinces room ("tax points") by which they can increase their own taxes (and thus their revenues) without increasing the total tax burden on their citizens.

Criticism
The amount of transfer payments from the federal to the provincial governments is always controversial, particularly in the area of health care, which is of particular importance to Canadians. The premiers of Canadian provinces allege that federal funding has decreased markedly since the beginning of publicly funded health care, from fifty to sixteen cents of every dollar. Some argue against the principle of the Canada Health Transfer cash contributions being at 50%, arguing that reforms in 1977 led to the removal of that cost sharing principle due to incorporating the transfer of tax points.

Critics of the CHST, CHT and CST note that the programs have allowed the federal government to interfere in areas of provincial jurisdiction by giving Ottawa a power through the threat of withdrawing the transfers to any province that administers programs contrary to federal requirements.

title: Canada Health and Social Transfer


text:
Gullah

The Gullah are a subgroup of the African American ethnic group, who predominantly live in the Lowcountry region of the U.S. states of South Carolina, North Carolina, Georgia, and Florida within the coastal plain and the Sea Islands. Their language and culture have preserved a significant influence of Africanisms as a result of their historical geographic isolation and the community's relation to their shared history and identity.

Historically, the Gullah region extended from the Cape Fear area on North Carolina's coast south to the vicinity of Jacksonville on Florida's coast. The Gullah people and their language are also called Geechee, which may be derived from the name of the Ogeechee River near Savannah, Georgia. Gullah is a term that was originally used to designate the creole dialect of English spoken by Gullah and Geechee people. Over time, its speakers have used this term to formally refer to their creole language and distinctive ethnic identity as a people. The Georgia communities are distinguished by identifying as either "Freshwater Geechee" or "Saltwater Geechee", depending on whether they live on the mainland or the Sea Islands.

Because of a period of relative isolation from whites while working on large plantations in rural areas, the Africans, enslaved from a variety of Central and West African ethnic groups, developed a creole culture that has preserved much of their African linguistic and cultural heritage from various peoples; in addition, they absorbed new influences from the region. The Gullah people speak an English-based creole language containing many African loanwords and influenced by African languages in grammar and sentence structure. Sometimes referred to as "Sea Island Creole" by linguists and scholars, the Gullah language is sometimes considered as being similar to Bahamian Creole, Barbadian Creole, Guyanese Creole, Belizean Creole, Jamaican Patois and the Sierra Leone Krio language of West Africa. Gullah crafts, farming and fishing traditions, folk beliefs, music, rice-based cuisine and story-telling traditions all exhibit strong influences from Central and West African cultures.

Etymology
The origin of the word "Gullah" can be traced to the Kikongo language, spoken around the Congo River's mouth, from which the Gullah language dialects spoken by black Americans today come. Some scholars suggest that it may be cognate with the name Kongo, where the ancestors of some of the Gullah people likely originated. They created a new culture synthesized from that of the various African peoples brought into Charleston and other parts of South Carolina. Some scholars have suggested that it may come from the name of the Gola, an ethnic group living in the border area between present-day Sierra Leone and Liberia in West Africa, another area of enslaved ancestors of the Gullah people. British planters in the Caribbean and the Southern colonies of North America referred to this area as the "Grain Coast" or "Rice Coast"; many of the tribes are of Mandé or Manding origins. The name "Geechee", another common name for the Gullah people, may derive from the name of the Kissi people, an ethnic group living in the border area between Sierra Leone, Guinea, and Liberia.

Still another possible linguistic source for "Gullah" are the Dyula ethnic group of West Africa, from whom the American Gullah might be partially descended. The Dyula civilization had a large territory that stretched from Senegal through Mali to Burkina Faso and the rest of what was French West Africa. These were vast savanna lands with lower population densities. Slave raiding was easier and more common here than in forested areas with natural forms of physical defenses. The word "Dyula" is pronounced "Gwullah" among members of the Akan ethnic group in Ghana and Côte d'Ivoire. The primary land route through which captured Dyula people then came into contact with European slavers was the "Grain Coast" and "Rice Coast" (present-day Liberia, Sierra Leone, Senegambia, and Guinea). The story of Gullah Jack (an African slave trafficked from Angola to the United States) may indicate that the word Gullah originated in Angola, as some commentators believe the word is a shortened version of the country's name.

Some scholars have also suggested indigenous American origins for these words. The Spanish named the South Carolina and Georgia coastal region as Guale, after a Native American tribe. The name of the Ogeechee River, a prominent geographical feature in coastal Georgia and central to Guale territory, may have been derived from a Creek Indian (Muskogee language) word. Sapelo Island, the site of the last Gullah community of Hog Hammock, was also principal place of refuge for Guale people who also fled slavery on the mainland.

The Gullah people have several West African words in their language that survived despite hundreds of years of slavery when African Americans were forced to speak English.

African roots

According to Port of Charleston records, African slaves shipped to the port came from the following areas: Angola (39%), Senegambia (20%), the Windward Coast (17%), the Gold Coast (13%), Sierra Leone (6%), and Madagascar, Mozambique, and the two Bights (Benin and Biafra) (5% combined) (Pollitzer, 1999:43). The term "Windward Coast" often referred to Sierra Leone, so the total figure of slaves from that region is higher than 6%.

Particularly along the western coast, the local peoples had cultivated African rice for what is estimated to approach 3,000 years. African rice is a species related to, yet distinct from, Asian rice. It was originally domesticated in the inland delta of the Upper Niger River. Once Carolinian and Georgian planters in the American South discovered that African rice would grow in that region, they often sought enslaved Africans from rice-growing regions because they had the skills and knowledge needed to develop and build irrigation, dams and earthworks.

Two British trading companies based in England operated the slave castle at Bunce Island (formerly called Bance Island), located in the Sierra Leone River. Henry Laurens was their main contact in Charleston and was a planter and slave trader. His counterpart in Britain was the Scottish merchant and slave trader Richard Oswald. Many of the enslaved Africans taken in West Africa were processed through Bunce Island. It was a prime export site for slaves to South Carolina and Georgia. Slave castles in Ghana, by contrast, shipped many of the people they handled to ports and markets in the Caribbean islands.

After Freetown, Sierra Leone, was founded in the late 18th century by the British as a colony for poor black people from London and black Loyalists from Nova Scotia resettled after the American Revolutionary War, the British did not allow slaves to be taken from Sierra Leone, protecting the people from kidnappers. In 1808 both Great Britain and the United States prohibited the African slave trade. After that date, the British, whose navy patrolled to intercept slave ships off Africa, sometimes resettled Africans liberated from slave trader ships in Sierra Leone. Similarly, Americans sometimes settled freed slaves at Liberia, a similar colony established in the early 19th century by the American Colonization Society. As it was a place for freed slaves and free blacks from the United States, some free blacks emigrated there voluntarily, for the chance to create their own society.

Origin of Gullah culture

The Gullah people have been able to preserve much of their African cultural heritage because of climate, geography, cultural pride, and patterns of importation of enslaved Africans. While some were transported to some areas of Brazil (including Bahia), the enslaved Gullah-Gheechee people were traded in what was then Charlestowne, South Carolina. According to British historian P.E.H. Hair, Gullah culture developed as a creole culture in the colonies and United States from the peoples of many different African cultures who came together there. These included the Bakongo, Mbundu, Vili, from Igbo, Angola, Calabar, Congo and the Gold Coast.

By the middle of the 18th century, thousands of acres in the Georgia and South Carolina Lowcountry, and the Sea Islands were developed as African rice fields. African farmers from the "Rice Coast" brought the skills for cultivation and tidal irrigation that made rice farming one of the most successful industries in early America.

The subtropical climate encouraged the spread of malaria and yellow fever, which were both carried and transmitted by mosquitoes. These tropical diseases were endemic in Africa and might have been carried by enslaved Africans to the colonies. Mosquitoes in the swamps and inundated rice fields of the Lowcountry picked up and spread the diseases to European settlers, as well. Malaria and yellow fever soon became endemic in the region.

Because they had acquired some immunity in their homeland, Africans were more resistant to these tropical fevers than were the Europeans. As the rice industry was developed, planters continued to import enslaved Africans. By about 1708, South Carolina had a black majority. Coastal Georgia developed a black majority after rice cultivation expanded there in the mid-18th century. Malaria and yellow fever became endemic. Fearing these diseases, many white planters and their families left the Lowcountry during the rainy spring and summer months when fevers ran rampant. Others lived mostly in cities such as Charleston rather than on the isolated plantations, especially those on the Sea Islands.

The planters left their European or African "rice drivers", or overseers, in charge of the rice plantations. These had hundreds of laborers, with African traditions reinforced by new imports from the same regions. Over time, the Gullah people developed a creole culture in which elements of African languages, cultures, and community life were preserved to a high degree. Their culture developed in a distinct way, different from that of the enslaved African Americans in states such as North Carolina, Virginia, and Maryland, where the enslaved lived in smaller groups, and had more sustained and frequent interactions with whites and British American culture.

Civil War period
When the U.S. Civil War began, the Union rushed to blockade Confederate shipping. White planters on the Sea Islands, fearing an invasion by the US naval forces, abandoned their plantations and fled to the mainland. When Union forces arrived on the Sea Islands in 1861, they found the Gullah people eager for their freedom, and eager as well to defend it. Many Gullah served with distinction in the Union Army's First South Carolina Volunteers. The Sea Islands were the first place in the South where slaves were freed. Long before the War ended, Unitarian missionaries from Pennsylvania came to start schools on the islands for the newly freed slaves. Penn Center, now a Gullah community organization on Saint Helena Island, South Carolina, was founded as the first school for freed slaves. After the Civil War ended, the Gullahs' isolation from the outside world increased in some respects. The rice planters on the mainland gradually abandoned their plantations and moved away from the area because of labor issues and hurricane damage to crops. Free blacks were unwilling to work in the dangerous and disease-ridden rice fields. A series of hurricanes devastated the crops in the 1890s. Left alone in remote rural areas of the Lowcountry, the Gullah continued to practice their traditional culture with little influence from the outside world well into the 20th century.

Recent history

In the 20th century, some plantations were redeveloped as resort or hunting destinations by wealthy whites. Gradually more visitors went to the islands to enjoy their beaches and mild climate. Since the late 20th century, the Gullah people—led by Penn Center and other determined community groups—have been fighting to keep control of their traditional lands. Since the 1960s, resort development on the Sea Islands has greatly increased property values, threatening to push the Gullah off family lands which they have owned since emancipation. They have fought back against uncontrolled development on the islands through community action, the courts, and the political process.

The Gullah have also struggled to preserve their traditional culture in the face of much more contact with modern culture and media. In 1979, a translation of the New Testament into the Gullah language was begun. The American Bible Society published De Nyew Testament in 2005. In November 2011, Healin fa de Soul, a five-CD collection of readings from the Gullah Bible, was released. This collection includes Scipcha Wa De Bring Healing ("Scripture That Heals") and the Gospel of John (De Good Nyews Bout Jedus Christ Wa John Write). This was also the most extensive collection of Gullah recordings, surpassing those of Lorenzo Dow Turner. The recordings have helped people develop an interest in the culture, because they get to hear the language and learn how to pronounce some words.

The Gullah achieved another victory in 2006 when the U.S. Congress passed the "Gullah/Geechee Cultural Heritage Corridor Act"; it provided $10 million over 10 years for the preservation and interpretation of historic sites in the Low Country relating to Gullah culture. The Heritage Corridor will extend from southern North Carolina to northern Florida. The project will be administered by the US National Park Service, with extensive consultation with the Gullah community.

The Gullah have also reached out to West Africa. Gullah groups made three celebrated "homecomings" to Sierra Leone in 1989, 1997, and 2005. Sierra Leone is at the heart of the traditional rice-growing region of West Africa where many of the Gullahs' ancestors originated. Bunce Island, the British slave castle in Sierra Leone, sent many African captives to Charleston and Savannah during the mid- and late 18th century. These dramatic homecomings were the subject of three documentary films—Family Across the Sea (1990), The Language You Cry In (1998), and Priscilla's Homecoming (in production).

African influences

 * The Gullah word guba (or goober) for peanut derives from the Kikongo and Kimbundu word N'guba.
 * Gullah rice dishes called "red rice" and "okra soup" are similar to West African "jollof rice" and "okra soup" and hog maws. Jollof rice is a traditional style of rice preparation brought by the Wolof people of West Africa.
 * The Gullah version of "gumbo" has its roots in African cooking. "Gumbo" is derived from a word in the Umbundu language of Angola, meaning okra, one of the dish's main ingredients.
 * Gullah rice farmers once made and used mortar and pestles and winnowing fanners similar in style to tools used by West African rice farmers.
 * Gullah beliefs about "hags" and "haunts" are similar to African beliefs about malevolent ancestors, witches, and "devils" (forest spirits).
 * Gullah "root doctors" protect their clients against dangerous spiritual forces by using ritual objects similar to those employed by African traditional healers.
 * Gullah herbal medicines are similar to traditional African remedies.
 * The Gullah "seekin" ritual is similar to coming of age ceremonies in West African secret societies, such as the Poro and Sande.
 * The Gullah ring shout is similar to ecstatic religious rituals performed in West and Central Africa.
 * Gullah stories about "Br'er Rabbit" are similar to West and Central African trickster tales about the figures of the clever and conniving rabbit, spider, and tortoise.
 * Gullah spirituals, shouts, and other musical forms employ the "call and response" method commonly used in African music.
 * Gullah "sweetgrass baskets" are coil straw baskets made by the descendants of enslaved peoples in the South Carolina Lowcountry. They are nearly identical to traditional coil baskets made by the Wolof people in Senegal.
 * Gullah "strip quilts" mimic the design of cloth woven with the traditional strip loom used throughout West Africa. Kente cloth from the Ashanti and the Ewe peoples, as well as Akwete cloth from the Igbo people are woven on the strip loom.
 * An African song, preserved by a Gullah family in coastal Georgia, was identified in the 1940s by linguist Lorenzo Turner and found to be a Mende song from Sierra Leone. It is probably the longest text in an African language to survive the transatlantic crossing of enslaved Africans to the present-day United States. Later, in the 1990s, researchers Joseph Opala, Cynthia Schmidt, and Taziff Koroma located a remote village in Sierra Leone where the song is still sung today, and determined it is a funeral hymn. This research and the resulting reunion between a Gullah family and a Mende family that have both retained versions of the song is recounted in the documentary The Language You Cry In (1998).
 * Some words coming from other African languages such as Yoruba, Fon, Ewe, Twi, Ga, Mende, and Bini are still used by Gullah people.
 * The Gullahs’ English-based creole language is strikingly similar to Sierra Leone Krio of West Africa and contains such identical expressions as bigyai ("greedy"), pantap ("on top of"), ohltu ("both"), tif ("steal"), yeys ("ear"), and swit ("delicious").

Cuisine

The Gullah have preserved many of their west African food ways growing and eating crops such as Sea island red peas, Carolina Gold rice, Sea island Benne, Sea island Okra, sorghum, and watermelon all of which were brought with them from West Africa. Rice is a staple food in Gullah communities and continues to be cultivated in abundance in the coastal regions of Georgia and South Carolina. Rice is also an important food in West African cultures. As descendants of enslaved Africans, the Gullah continued the traditional food and food techniques of their ancestors, demonstrating another link to traditional African cultures.

Rice is a core commodity of the Gullah food system: a meal was not considered complete without rice. There are strict rituals surrounding the preparation of rice in the Gullah communities. First, individuals would remove the darker grains from the rice, and then hand wash the rice numerous times before it was ready for cooking. The Gullah people would add enough water for the rice to steam on its own, but not so much that one would have to stir or drain it. These traditional techniques were passed down during the period of slavery and are still an important part of rice preparation by Gullah people.

The first high-profile book on Gullah cooking was published in 2022 by Emily Meggett, an 89-year-old Gullah cook.

Celebrating Gullah culture
Over the years, the Gullah have attracted study by many historians, linguists, folklorists, and anthropologists interested in their rich cultural heritage. Many academic books on that subject have been published. The Gullah have also become a symbol of cultural pride for blacks throughout the United States and a subject of general interest in the media. Numerous newspaper and magazine articles, documentary films, and children's books on Gullah culture, have been produced, in addition to popular novels set in the Gullah region. In 1991 Julie Dash wrote and directed Daughters of the Dust, the first feature film about the Gullah, set at the turn of the 20th century on St. Helena Island. Born into a Gullah family, she was the first African-American woman director to produce a feature film.

Gullah people now organize cultural festivals every year in towns up and down the Lowcountry. Hilton Head Island, for instance, hosts a "Gullah Celebration" in February. It includes "De Aarts ob We People" show; the "Ol’ Fashioned Gullah Breakfast"; "National Freedom Day," the "Gullah Film Fest", "A Taste of Gullah" food and entertainment, a "Celebration of Lowcountry Authors and Books," an "Arts, Crafts & Food Expo," and "De Gullah Playhouse". Beaufort hosts the oldest and the largest celebration, "The Original Gullah Festival" in May. The nearby Penn Center on St. Helena Island holds "Heritage Days" in November. Other Gullah festivals are celebrated on James Island, South Carolina, and Sapelo Island, Georgia.

Gullah culture is also being celebrated elsewhere in the United States. The High Art Museum in Atlanta has presented exhibits about Gullah culture. The Black Cultural Center at Purdue University in West Lafayette, Indiana conducted a research tour, cultural arts festival, and other related events to showcase the Gullah culture. The Black Cultural Center Library maintains a bibliography of Gullah books and materials, as well. Metro State College in Denver, Colorado, hosted a conference on Gullah culture, called The Water Brought Us: Gullah History and Culture, which featured a panel of Gullah scholars and cultural activists. These events in Indiana and Colorado are typical of the attention Gullah culture regularly receives throughout the United States.

Cultural survival

Gullah culture has proven to be particularly resilient. Gullah traditions are strong in the rural areas of the Lowcountry mainland and on the Sea Islands, and among their people in urban areas such as Charleston and Savannah. Gullah people who have left the Lowcountry and moved far away have also preserved traditions; for instance, many Gullah in New York, who went North in the Great Migration of the first half of the 20th century, have established their own neighborhood churches in Harlem, Brooklyn, and Queens. Typically they send their children back to rural communities in South Carolina and Georgia during the summer months to live with grandparents, uncles, and aunts. Gullah people living in New York frequently return to the Lowcountry to retire. Second- and third-generation Gullah in New York often maintain many of their traditional customs and many still speak the Gullah language.

The Gullah custom of painting porch ceilings haint blue to deter haints, or ghosts, survives in the American South. Having also been adopted by White Southerners, it has lost some of its spiritual significance.

Representation in art, entertainment, and media
Gullah Gullah Island is an American musical children's television series that was produced by and aired on the Nick Jr. programming block on the Nickelodeon network from October 24, 1994, to April 7, 1998. The show was hosted by Ron Daise—now the former vice president for Creative Education at Brookgreen Gardens in Murrells Inlet, South Carolina—and his wife Natalie Daise, both of whom also served as cultural advisors, and were inspired by the Gullah culture of Ron Daise's home of St. Helena Island, South Carolina, part of the Sea Islands.

Notable Americans with Gullah roots

 * Robert Sengstacke Abbott
 * Cornelia Walker Bailey
 * Jim Brown
 * Kardea Brown
 * Kwame Brown
 * Marion Brown
 * Craig Anthony Bullock (DJ Homicide)
 * Emory Campbell
 * Septima Poinsette Clark
 * Julie Dash
 * Sam Doyle
 * William Jonathan Drayton Jr. (Flavor Flav)
 * Edda L. Fields-Black
 * Joe Frazier
 * Candice Glover
 * Marquetta Goodwine
 * Gullah Jack
 * Mary Jackson
 * James Jamerson
 * Bumpy Johnson
 * Griffin Lotson
 * Earl Manigault
 * Lenard Larry McKelvey (Charlamagne Tha God)
 * Khris Middleton
 * Eric Milligan (The Blixunami)
 * Michelle Obama
 * Joseph Rainey
 * Philip Reid
 * Sallie Ann Robinson
 * Chris Rock
 * Tony Rock
 * Eden Royce
 * Raven Saunders
 * Philip Simmons
 * Robert Smalls
 * Vertamae Smart-Grosvenor
 * Clarence Thomas
 * Denmark Vesey
 * Kemba Walker
 * Robert Lee Watt
 * Maurice Samuel Young (Trick Daddy)

title: Gullah


text:
Kanchenjungha (film)

Kanchenjungha (Kanchonjônggha) is a 1962 Indian film directed by Satyajit Ray.

The film is about an upper class Bengali family on vacation in Darjeeling, a popular hill station and resort, near Kanchenjunga.

Plot
A wealthy family from Calcutta is on the last day of their vacation in Darjeeling, a hill station at the foot of Mount Kanchenjungha, the second highest peak of the Himalayas. Until now, they have been unable to catch a glimpse of the peak Kanchenjungha. The family members are dominated by the father, Indranath (Chhabi Biswas), an industrialist. He wants his daughter to marry a man of his choice and hopes that the man will propose if they are left together alone for some time.

Several long walks and long conversations form the main body of the film. The real-time drama unfolds the daughter's feelings about her father's idea, and the negative reactions to this by her mother and others. By accident, she meets an outsider, Ashoke, a young student who has refused a job offer from Indranath. Though nothing develops between them, his presence coupled with the setting of mountains and the failure of her sister's marriage prompts her to reject the proposed suitor.

At the end of his walk, the industrialist arrives at a rendezvous point, expecting to meet his family and the successful suitor. None of them is present to greet him. As the mist clears, the peak of Kanchenjungha is revealed in its full glory. But Indranath is too pre-occupied to admire it.

Cast

 * Chhabi Biswas: Indranath Roy
 * Karuna Banerjee: Labanya, wife
 * Anil Chatterjee: Anil, son
 * Alokananda Roy: Monisha, unmarried daughter
 * Anubha Gupta: Anima, elder daughter
 * Arun Mukherjee: Ashoke, young man from Calcutta
 * Subrata Sen: Sankar
 * Sibani Singh: Tuklu
 * Vidya Singh: Anil's girlfriend
 * Pahari Sanyal: Jagadish
 * N. Viswanathan: Bannerjee

Production
Kanchenjungha was Ray's first original screenplay and for the first time, he was shooting in color. The film shows about 100 minutes (in real time) in the life of a group of rich Bengalis on vacation. Unlike the usual Ray films, it has a linear narrative with no central characters, and no straight narrative in the classical sense.

It is a very structured and composed film that uses color and nature to heighten the drama. Ray told his biographer Andrew Robinson: "The idea was to have the film starting with sunlight. Then clouds coming, then mist rising, and then mist disappearing, the cloud disappearing, and then the sun shining on the snow-peaks. There is an independent progression to Nature itself, and the story reflects this."

As the weather becomes misty - the young daughter and the suitor part at that point, Indranath meets Ashoke, and the elder daughter, Anima and her husband have a bitter moment between them. And then when the sun appears again - Anima's daughter comes back to her parents and they accept her, the misunderstanding is cleared up, and the younger daughter and Ashoke develop a tentative relationship with a hint of future prospects.

Narrative structure
The structure of the film contrasts a primary plot with a secondary plot. The secondary plot involves two characters, largely static, and is played out in scenes involving only them. The primary plot pulls in most of the characters, in various groups coming together and breaking apart, across many different locations, as they all walk the circular path rising and falling over the mountainside.

The film happens in linear realtime, i.e. the runtime of the film covers events happening over approximately the same period. However frequent inter-cutting between the different groups fractures this linearity, as does the thematically opposed trajectories of the two plots, one proceeding from family to the individual while the other proceeds from the individual to the family.

However there are elements cutting across this chopped up structure. The youngest child of the family, riding in circles and singing a counting rhyme, keeps moving across the background of the scenes like a metronome. The shifting states of weather are a constant backdrop, anchoring everything in time. In the end the climax of all trajectories is capped by the snowy peaks of Kanchenjungha finally appearing out of the clouds, suggesting an underlying unity of everybody's private struggles and their resolutions.

Characters
The characters and the Nature are interrelated in mood. In his later film Asani Sanket (Distant Thunder, 1973), Ray attempted quite the opposite. While the Nature is shown in its full glory and lushness, in contrast, the characters starve and do inhuman things.

As the characters do not change their clothes during the film (it being depiction of a real time event that happens in 100 odd minutes), the colors of clothes too add a dimension to the characters.

About Kanchenjungha, Ray told in an interview to Cineaste magazine, "(It was) a very personal film. It was a good ten to fifteen years ahead of its time... Kanchenjungha told the story of several groups of characters and it went back and forth. ... It's a very musical form, but it wasn't liked. The reaction was stupid. Even the reviews were not interesting. But, looking back now, I find that it is a very interesting film."

title: Kanchenjungha (film)


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Sri Lanka hill myna

The Sri Lanka hill myna, Ceylon myna or Sri Lanka myna (Gracula ptilogenys) is a myna, a member of the starling family. This bird is endemic to Sri Lanka.

This passerine is typically found in forest and cultivation. The Sri Lanka myna builds a nest in a hole. The normal clutch is two eggs.

These 25 cm long birds have green-glossed black plumage, purple-tinged on the head and neck. There are large white wing patches, which are obvious in flight. The strong legs are bright yellow, and there are yellow wattles on the nape.

The different shape and position of the wattles and the stouter orange-red bill distinguish this species from the southern hill myna, which also occurs in Sri Lankan forests. The sexes are similar in plumage, but can be distinguished by iris color, which is pale in females and dark in males. Juveniles have a duller bill and smaller wattles, and are less glossy overall.

Like most starlings, the Sri Lanka myna is fairly omnivorous, eating fruit, nectar and insects.

In culture
In Sri Lanka, the Myna bird is known in many names including Sela lihiniya, Mal kawadiya, Kampatiya in the Sinhala language. The name Sela Lihiniya is often mentioned in poems and other similar literature and is quoted for its melodious calls. This bird appears in a 10 rupee Sri Lankan postal stamp. It is also the media icon of Sri Lanka Rupavahini Corporation.

title: Sri Lanka hill myna