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Kanishka

Kanishka I, also known as Kanishka the Great, was an emperor of the Kushan dynasty, under whose reign (c. 127–150 CE) the empire reached its zenith. He is famous for his military, political, and spiritual achievements. A descendant of Kujula Kadphises, founder of the Kushan empire, Kanishka came to rule an empire extending from Central Asia and Gandhara to Pataliputra on the Gangetic plain. The main capital of his empire was located at Puruṣapura (Peshawar) in Gandhara, with another major capital at Mathura. Coins of Kanishka were found in Tripuri (present-day Jabalpur).

Although he never converted to the religion, his conquests and patronage of Buddhism played an important role in the development of the Silk Road, and in the transmission of Mahayana Buddhism from Gandhara across the Karakoram range to China. Around 127 CE, he replaced Greek with Bactrian as the official language of administration in the empire.

Earlier scholars believed that Kanishka ascended the Kushan throne in 78 CE, and that this date was used as the beginning of the Saka calendar era. However, historians no longer regard this date as that of Kanishka's accession. Falk estimates that Kanishka came to the throne in 127 CE.

Genealogy
There are two theories about Kanisha's origins, both based on the supposition of a separate Kanishka dynasty. The earlier Sten Konow's is that Kanishka came from Khotan, having been summoned as an ethnic ally at the time of troubles after Vima's reign. Konow supports this theory by citing a Tibetan tradition that a Khotanese expedition to India of about A.D. 120 was commanded by a King Vijayakirti along with a King Kanika and the king of Guzan.

Roman Ghirshman's similar theory is that Kanishka was originally king of Kashmir before becoming suzerain of the dynasty as a whole. He cites the above-mentioned Khalatse epigraph, which may allude to Kushan power reaching the northeast corner of Kashmir, and also the Rajatarangini, in which the list of Turushka kings of Kashmir gave Kanishka's name as the third of a sequence, along with the names of Hushka and Jushka. Finally, testimony of Kanishka's activity in Kashmir in favor of the Buddhist faith suggests that he favored the region above others and that he founded Peshawar as his capital to be close to Kashmir.

He was of Yuezhi ethnicity and his native language was probably Tocharian. Kanishka was the successor of Vima Kadphises, as demonstrated by an impressive genealogy of the Kushan kings, known as the Rabatak inscription. The connection of Kanishka with other Kushan rulers is described in the Rabatak inscription as Kanishka makes the list of the kings who ruled up to his time: Kujula Kadphises as his great-grandfather, Vima Taktu as his grandfather, Vima Kadphises as his father, and himself Kanishka: "for King Kujula Kadphises (his) great grandfather, and for King Vima Taktu (his) grandfather, and for King Vima Kadphises (his) father, and *also for himself, King Kanishka".

Conquests in India and Central Asia
Kanishka's empire was certainly vast. It extended from southern Uzbekistan and Tajikistan, north of the Amu Darya (Oxus) in the north west to Northern India, as far as Mathura in the south east (the Rabatak inscription even claims he held Pataliputra and Sri Champa), and his territory also included Kashmir, where there was a town Kanishkapur (modern day Kanispora), named after him not far from the Baramulla Pass and which still contains the base of a large stupa. The Buddhist text Śrīdharmapiṭakanidānasūtra—known via a Chinese translation made in AD 472—refers to the conquest of Pataliputra by Kanishka.

Knowledge of his hold over Central Asia is less well established. The Hou Hanshu, states that general Ban Chao fought battles near Khotan with a Kushan army of 70,000 men led by an otherwise unknown Kushan viceroy named Xie (謝) in 90 AD. Ban Chao claimed to be victorious, forcing the Kushans to retreat by use of a scorched-earth policy. The territories of Kashgar, Khotan and Yarkand were Chinese dependencies in the Tarim Basin, modern Xinjiang. Several coins of Kanishka have been found in the Tarim Basin.

Kanishka have fought off an invasion by the Parthian Empire late in his reign. This war is attested in a single source, a Chinese translation of a lost Sanskrit original, A History of the Buddha's Successors.

Kanishka's coins
Kanishka's coins portray images of Indian, Greek, Iranian and even Sumero-Elamite divinities, demonstrating the religious syncretism in his beliefs. Kanishka's coins from the beginning of his reign bear legends in Greek language and script and depict Greek divinities. Later coins bear legends in Bactrian, the Iranian language that the Kushans evidently spoke, and Greek divinities were replaced by corresponding Iranian ones. All of Kanishka's coins – even ones with a legend in the Bactrian language – were written in a modified Greek script that had one additional glyph (Ϸ) to represent /š/ (sh), as in the word 'Kushan' and 'Kanishka'.

On his coins, the king is typically depicted as a bearded man in a long coat and trousers gathered at the ankle, with flames emanating from his shoulders. He wears large rounded boots, and is armed with a long sword as well as a lance. He is frequently seen to be making a sacrifice on a small altar. The lower half of a lifesize limestone relief of Kanishka similarly attired, with a stiff embroidered surplice beneath his coat and spurs attached to his boots under the light gathered folds of his trousers, survived in the Kabul Museum until it was destroyed by the Taliban.

Hellenistic phase
A few coins at the beginning of his reign have a legend in the Greek language and script: ΒΑΣΙΛΕΥΣ ΒΑΣΙΛΕΩΝ ΚΑΝΗϷΚΟΥ, "[coin] of Kanishka, king of kings."

Greek deities, with Greek names are represented on these early coins:
 * ΗΛΙΟΣ (, Hēlios), ΗΦΑΗΣΤΟΣ (, Hephaistos), ΣΑΛΗΝΗ (, Selene), ΑΝΗΜΟΣ (, Anemos)

The inscriptions in Greek are full of spelling and syntactical errors.

Iranian / Indic phase
Following the transition to the Bactrian language on coins, Iranian and Indic divinities replace the Greek ones:

 * ΑΡΔΟΧϷΟ (ardoxsho, Ashi Vanghuhi)
 * ΛΡΟΟΑΣΠΟ (lrooaspo, Drvaspa)
 * ΑΘϷΟ (adsho, Atar)
 * ΦΑΡΡΟ (pharro, personified khwarenah)
 * ΜΑΟ (mao, Mah)
 * ΜΙΘΡΟ, ΜΙΙΡΟ, ΜΙΟΡΟ, ΜΙΥΡΟ (mithro, miiro, mioro, miuro, variants of Mithra)
 * ΜΟΖΔΟΟΑΝΟ (mozdaooano, "Mazda the victorious?")
 * ΝΑΝΑ, ΝΑΝΑΙΑ, ΝΑΝΑϷΑΟ (variants of pan-Asiatic Nana, Sogdian nny, in a Zoroastrian context Aredvi Sura Anahita)
 * ΜΑΝΑΟΒΑΓΟ (manaobago, Vohu Manah )
 * ΟΑΔΟ (oado, Vata)
 * ΟΡΑΛΑΓΝΟ (orlagno, Verethragna)

Only a few Buddhist divinities were used as well:
 * ΒΟΔΔΟ (boddo, Buddha),
 * ϷΑΚΑΜΑΝΟ ΒΟΔΔΟ (shakamano boddho, Shakyamuni Buddha)
 * ΜΕΤΡΑΓΟ ΒΟΔΔΟ (metrago boddo, the bodhisattava Maitreya)

Only a few Hindu divinities were used as well:
 * ΟΗϷΟ (oesho, Shiva). A recent study indicate that oesho may be Avestan Vayu conflated with Shiva.

Kanishka and Buddhism

In Buddhist tradition Kanishka is regarded as of utmost importance. Although he never converted to Buddhism, he encouraged its teachings and provided royal sponsorship. Notably, he administered the 4th Buddhist Council in Kashmir as the head of the council. It was presided by Vasumitra and Ashwaghosha. Images of the Buddha based on 32 physical signs were made during his time.

He encouraged both the Gandhara school of Greco-Buddhist Art and the Mathura school of art (an inescapable religious syncretism pervades Kushana rule). Kanishka personally seems to have embraced both Buddhism and the Persian attributes but he favored Buddhism, proven by his devotion to Buddhist teachings and prayer styles depicted in various books related to kushan empire.

His greatest contribution to Buddhist architecture was the Kanishka stupa at Purushapura, modern day Peshawar. Archaeologists who rediscovered the base of it in 1908–1909 estimated that this stupa had a diameter of 286 feet (87 metres). Reports of Chinese pilgrims such as Xuanzang indicate that its height was 600 to 700 (Chinese) "feet" (= roughly 180–210 metres or 591–689 ft.) and it was covered with jewels.

Kanishka is said to have been particularly close to the Buddhist scholar Ashvaghosha, who became his religious advisor in his later years.

Buddhist coinage
The Buddhist coins of Kanishka are comparatively rare (well under one percent of all known coins of Kanishka). Several show Kanishka on the obverse and the Buddha standing on the reverse. A few also show the Shakyamuni Buddha and Maitreya. Like all coins of Kanishka, the design is rather rough and proportions tend to be imprecise; the image of the Buddha is often slightly overdone, with oversize ears and feet spread apart in the same fashion as the Kushan king.

Three types of Kanishka's Buddhist coins are known:

Standing Buddha
Only six Kushan coins of the Buddha are known in gold (the sixth one is the centerpiece of an ancient piece of jewellery, consisting of a Kanishka Buddha coin decorated with a ring of heart-shaped ruby stones). All these coins were minted in gold under Kanishka I, and are in two different denominations: a dinar of about 8 gm, roughly similar to a Roman aureus, and a quarter dinar of about 2 gm. (about the size of an obol).

The Buddha is represented wearing the monastic robe, the antaravasaka, the uttarasanga, and the overcoat sanghati.

The ears are extremely large and long, a symbolic exaggeration possibly rendered necessary by the small size of the coins, but otherwise visible in some later Gandharan statues of the Buddha typically dated to the 3rd–4th century CE (illustration, left). He has an abundant topknot covering the usnisha, often highly stylised in a curly or often globular manner, also visible on later Buddha statues of Gandhara.

In general, the representation of the Buddha on these coins is already highly symbolic, and quite distinct from the more naturalistic and Hellenistic images seen in early Gandhara sculptures. On several designs a mustache is apparent. The palm of his right hand bears the Chakra mark, and his brow bear the urna. An aureola, formed by one, two or three lines, surrounds him.

The full gown worn by the Buddha on the coins, covering both shoulders, suggests a Gandharan model rather than a Mathuran one.

"Shakyamuni Buddha"
The Shakyamuni Buddha (with the legend "Sakamano Boudo", i.e. Shakamuni Buddha, another name for the historic Buddha Siddharta Gautama), standing to front, with left hand on hip and forming the abhaya mudra with the right hand. All these coins are in copper only, and usually rather worn.

The gown of the Shakyamuni Buddha is quite light compared to that on the coins in the name of Buddha, clearly showing the outline of the body, in a nearly transparent way. These are probably the first two layers of monastic clothing the antaravasaka and the uttarasanga. Also, his gown is folded over the left arm (rather than being held in the left hand as above), a feature only otherwise known in the Bimaran casket and suggestive of a scarf-like uttariya. He has an abundant topknot covering the ushnisha, and a simple or double halo, sometimes radiating, surrounds his head.

"Maitreya Buddha"
The Bodhisattva Maitreya (with the legend "Metrago Boudo") cross-legged on a throne, holding a water pot, and also forming the Abhaya mudra. These coins are only known in copper and are quite worn out. On the clearest coins, Maitreya seems to be wearing the armbands of an Indian prince, a feature often seen on the statuary of Maitreya. The throne is decorated with small columns, suggesting that the coin representation of Maitreya was directly copied from pre-existing statuary with such well-known features.

The qualification of "Buddha" for Maitreya is inaccurate, as he is instead a Bodhisattva (he is the Buddha of the future).

The iconography of these three types is very different from that of the other deities depicted in Kanishka's coinage. Whether Kanishka's deities are all shown from the side, the Buddhas only are shown frontally, indicating that they were copied from contemporary frontal representations of the standing and seated Buddhas in statuary. Both representations of the Buddha and Shakyamuni have both shoulders covered by their monastic gown, indicating that the statues used as models were from the Gandhara school of art, rather than Mathura.

Buddhist statuary under Kanishka
Several Buddhist statues are directly connected to the reign of Kanishka, such as several Bodhisattva statues from the Art of Mathura, while a few other from Gandhara are inscribed with a date in an era which is now thought to be the Yavana era, starting in 186 to 175 BCE.

Kanishka stupa
The "Kanishka casket" or "Kanishka reliquary", dated to the first year of Kanishka's reign in 127 CE, was discovered in a deposit chamber under Kanishka stupa, during the archaeological excavations in 1908–1909 in Shah-Ji-Ki-Dheri, just outside the present-day Ganj Gate of the old city of Peshawar. It is today at the Peshawar Museum, and a copy is in the British Museum. It is said to have contained three bone fragments of the Buddha, which are now housed in Mandalay, Burma.

The casket is dedicated in Kharoshthi. The inscription reads:

 * "(*mahara)jasa kanishkasa kanishka-pure nagare aya gadha-karae deya-dharme sarva-satvana hita-suhartha bhavatu mahasenasa sagharaki dasa agisala nava-karmi ana*kanishkasa vihare mahasenasa sangharame"

The text is signed by the maker, a Greek artist named Agesilas, who oversaw work at Kanishka's stupas (caitya), confirming the direct involvement of Greeks with Buddhist realisations at such a late date: "The servant Agisalaos, the superintendent of works at the vihara of Kanishka in the monastery of Mahasena" ("dasa agisala nava-karmi ana*kaniskasa vihara mahasenasa sangharame").

The lid of the casket shows the Buddha on a lotus pedestal, and worshipped by Brahma and Indra. The edge of the lid is decorated by a frieze of flying geese. The body of the casket represents a Kushan monarch, probably Kanishka in person, with the Iranian sun and moon gods on his side. On the sides are two images of a seated Buddha, worshiped by royal figures, can be assumed as Kanishka. A garland, supported by cherubs goes around the scene in typical Hellenistic style.

The attribution of the casket to Kanishka has been recently disputed, essentially on stylistic ground (for example the ruler shown on the casket is not bearded, to the contrary of Kanishka). Instead, the casket is often attributed to Kanishka's successor Huvishka.

Kanishka in Buddhist tradition
In Buddhist tradition, Kanishka is often described as an aggressive, hot tempered, rigid, strict, and a bit harsh kind of King before he got converted to Buddhism of which he was very fond, and after his conversion to Buddhism, he became an openhearted, benevolent, and faithful ruler. As in the Sri-dharma-pitaka-nidana sutra:
 * "At this time the King of Ngan-si (Pahlava) was very aggressive and of a violent nature....There was a bhikshu (monk) arhat who seeing the harsh deeds done by the king wished to make him repent. So by his supernatural force he caused the king to see the torments of hell. The king was terrified and repented and cried terribly and hence dissolved all his negatives within him and got self realised for the first time in life ."

Additionally, the arrival of Kanishka was reportedly foretold or was predicted by the Buddha, as well as the construction of his stupa:
 * ". . . the Buddha, pointing to a small boy making a mud tope....[said] that on that spot would erect a tope by his name." Vinaya sutra

The same story is repeated in a Khotanese scroll found at Dunhuang, which first described how Kanishka would arrive 400 years after the death of the Buddha. The account also describes how Kanishka came to raise his stupa:
 * "A desire thus arose in [Kanishka to build a vast stupa]....at that time the four world-regents learnt the mind of the king. So for his sake they took the form of young boys....[and] began a stūpa of mud....the boys said to [Kanishka] 'We are making the .'....At that time the boys changed their form....[and] said to him, 'Great king, by you according to the Buddha's prophecy is a to be built wholly (?) with a large stūpa and hither relics must be invited which the meritorious good beings...will bring."

Chinese pilgrims to India, such as Xuanzang, who travelled there around 630 CE also relays the story:
 * " became sovereign of all Jambudvīpa (Indian subcontinent) but he did not believe in Karma, but he treated Buddhism with honor and respect as he himself converted to Buddhism intrigued by the teachings and scriptures of it. When he was hunting in the wild country a white hare appeared; the king gave a chase and the hare suddenly disappeared at [the site of the future stupa]....[when the construction of the stūpa was not going as planned] the king lost his patience and took the matter in his own hands and started resurrecting the plans precisely, thus completing the stupas with utmost perfection and perseverance. These two stupas are still in existence and were resorted to for cures by people afflicted with diseases."

King Kanishka because of his deeds was highly respected, regarded, honored by all the people he ruled and governed and was regarded the greatest king who ever lived because of his kindness, humbleness and sense of equality and self-righteousness among all aspects. Thus such great deeds and character of the king Kanishka made his name immortal and thus he was regarded "THE KING OF KINGS"

Transmission of Buddhism to China
Buddhist monks from the region of Gandhara played a key role in the development and the transmission of Buddhist ideas in the direction of northern Asia from the middle of the 2nd century CE. The Kushan monk, Lokaksema (c. 178 CE), became the first translator of Mahayana Buddhist scriptures into Chinese and established a translation bureau at the Chinese capital Loyang. Central Asian and East Asian Buddhist monks appear to have maintained strong exchanges for the following centuries.

Kanishka was probably succeeded by Huvishka. How and when this came about is still uncertain. The inscription on The Sacred Rock of Hunza also shows the signs of Kanishka.

title: Kanishka


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Talk:Red Spider Nebula

Name
Shouldn't this nebula be at Red Spider Nebula instead of its NGC number? See Naming conventions (common names). --mav 00:43 22 Jun 2003 (UTC)

 * Sure, why not? :) I only put it here in the first place because I was de-orphaning NGC images. Bryan

The spelling of Sagittarius is inconsistent in the piped link. Also, can't it just go to Sagittarius?

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title: Talk:Red Spider Nebula


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Implicit function

In mathematics, an implicit equation is a relation of the form $$R(x_1, \dots, x_n) = 0,$$ where $R$ is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is $$x^2 + y^2 - 1 = 0.$$

An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. For example, the equation $$x^2 + y^2 - 1 = 0$$ of the unit circle defines $y$ as an implicit function of $x$ if $−1 ≤ x ≤ 1$, and $y$ is restricted to nonnegative values.

The implicit function theorem provides conditions under which some kinds of implicit equations define implicit functions, namely those that are obtained by equating to zero multivariable functions that are continuously differentiable.

Inverse functions
A common type of implicit function is an inverse function. Not all functions have a unique inverse function. If $g$ is a function of $x$ that has a unique inverse, then the inverse function of $g$, called $g^{−1}$, is the unique function giving a solution of the equation

 * $$ y=g(x) $$

for $x$ in terms of $y$. This solution can then be written as

 * $$ x = g^{-1}(y) \,.$$

Defining $g^{−1}$ as the inverse of $g$ is an implicit definition. For some functions $g$, $g^{−1}(y)$ can be written out explicitly as a closed-form expression — for instance, if $g(x) = 2x − 1$, then $g^{−1}(y) = 1⁄2(y + 1)$. However, this is often not possible, or only by introducing a new notation (as in the product log example below).

Intuitively, an inverse function is obtained from $g$ by interchanging the roles of the dependent and independent variables.

Example: The product log is an implicit function giving the solution for $x$ of the equation $y − xe^{x} = 0$.

Algebraic functions
An algebraic function is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable $x$ gives a solution for $y$ of an equation

 * $$a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0 \,,$$

where the coefficients $a_{i}(x)$ are polynomial functions of $x$. This algebraic function can be written as the right side of the solution equation $y = f(x)$. Written like this, $f$ is a multi-valued implicit function.

Algebraic functions play an important role in mathematical analysis and algebraic geometry. A simple example of an algebraic function is given by the left side of the unit circle equation:

 * $$x^2+y^2-1=0 \,. $$

Solving for $y$ gives an explicit solution:

 * $$y=\pm\sqrt{1-x^2} \,. $$

But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as $y = f(x)$, where $f$ is the multi-valued implicit function.

While explicit solutions can be found for equations that are quadratic, cubic, and quartic in $y$, the same is not in general true for quintic and higher degree equations, such as

 * $$ y^5 + 2y^4 -7y^3 + 3y^2 -6y - x = 0 \,. $$

Nevertheless, one can still refer to the implicit solution $y = f(x)$ involving the multi-valued implicit function $f$.

Caveats
Not every equation $R(x, y) = 0$ implies a graph of a single-valued function, the circle equation being one prominent example. Another example is an implicit function given by $x − C(y) = 0$ where $C$ is a cubic polynomial having a "hump" in its graph. Thus, for an implicit function to be a true (single-valued) function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a true function only after "zooming in" on some part of the $x$-axis and "cutting away" some unwanted function branches. Then an equation expressing $y$ as an implicit function of the other variables can be written.

The defining equation $R(x, y) = 0$ can also have other pathologies. For example, the equation $x = 0$ does not imply a function $f(x)$ giving solutions for $y$ at all; it is a vertical line. In order to avoid a problem like this, various constraints are frequently imposed on the allowable sorts of equations or on the domain. The implicit function theorem provides a uniform way of handling these sorts of pathologies.

Implicit differentiation
In calculus, a method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.

To differentiate an implicit function $y(x)$, defined by an equation $R(x, y) = 0$, it is not generally possible to solve it explicitly for $y$ and then differentiate. Instead, one can totally differentiate $R(x, y) = 0$ with respect to $x$ and $y$ and then solve the resulting linear equation for $dy⁄dx$ to explicitly get the derivative in terms of $x$ and $y$. Even when it is possible to explicitly solve the original equation, the formula resulting from total differentiation is, in general, much simpler and easier to use.

Example 1
Consider

 * $$y + x + 5 = 0 \,.$$

This equation is easy to solve for $y$, giving

 * $$y = -x - 5 \,,$$

where the right side is the explicit form of the function $y(x)$. Differentiation then gives $dy⁄dx = −1$.

Alternatively, one can totally differentiate the original equation:

 * $$\begin{align}

\frac{dy}{dx} + \frac{dx}{dx} + \frac{d}{dx}(5) &= 0 \, ; \\[6px] \frac{dy}{dx} + 1 + 0 &= 0 \,. \end{align}$$

Solving for $dy⁄dx$ gives

 * $$\frac{dy}{dx} = -1 \,,$$

the same answer as obtained previously.

Example 2
An example of an implicit function for which implicit differentiation is easier than using explicit differentiation is the function $y(x)$ defined by the equation

 * $$ x^4 + 2y^2 = 8 \,.$$

To differentiate this explicitly with respect to $x$, one has first to get

 * $$y(x) = \pm\sqrt{\frac{8 - x^4}{2}} \,,$$

and then differentiate this function. This creates two derivatives: one for $y ≥ 0$ and another for $y < 0$.

It is substantially easier to implicitly differentiate the original equation:

 * $$4x^3 + 4y\frac{dy}{dx} = 0 \,,$$

giving

 * $$\frac{dy}{dx} = \frac{-4x^3}{4y} = -\frac{x^3}{y} \,.$$

Example 3
Often, it is difficult or impossible to solve explicitly for $y$, and implicit differentiation is the only feasible method of differentiation. An example is the equation

 * $$y^5-y=x \,.$$

It is impossible to algebraically express $y$ explicitly as a function of $x$, and therefore one cannot find $dy⁄dx$ by explicit differentiation. Using the implicit method, $dy⁄dx$ can be obtained by differentiating the equation to obtain

 * $$5y^4\frac{dy}{dx} - \frac{dy}{dx} = \frac{dx}{dx} \,,$$

where $dx⁄dx = 1$. Factoring out $dy⁄dx$ shows that

 * $$\left(5y^4 - 1\right)\frac{dy}{dx} = 1 \,,$$

which yields the result

 * $$\frac{dy}{dx}=\frac{1}{5y^4-1} \,,$$

which is defined for

 * $$y \ne \pm\frac{1}{\sqrt[4]{5}} \quad \text{and} \quad y \ne \pm \frac{i}{\sqrt[4]{5}} \,.$$

General formula for derivative of implicit function
If $R(x, y) = 0$, the derivative of the implicit function $y(x)$ is given by

 * $$\frac{dy}{dx} = -\frac{\,\frac{\partial R}{\partial x}\,}{\frac{\partial R}{\partial y}} = -\frac {R_x}{R_y} \,,$$

where $R_{x}$ and $R_{y}$ indicate the partial derivatives of $R$ with respect to $x$ and $y$.

The above formula comes from using the generalized chain rule to obtain the total derivative — with respect to $x$ — of both sides of $R(x, y) = 0$:

 * $$\frac{\partial R}{\partial x} \frac{dx}{dx} + \frac{\partial R}{\partial y} \frac{dy}{dx} = 0 \,,$$

hence

 * $$\frac{\partial R}{\partial x} + \frac{\partial R}{\partial y} \frac{dy}{dx} =0 \,,$$

which, when solved for $dy⁄dx$, gives the expression above.

Implicit function theorem

Let $(x, y)$ be a differentiable function of two variables, and $x^{2} + y^{2} = 1$ be a pair of real numbers such that $y(x)$. If $g_{1}(x) = √1 − x^{2}$, then $R(x, y)$ defines an implicit function that is differentiable in some small enough neighbourhood of $A$; in other words, there is a differentiable function $y$ that is defined and differentiable in some neighbourhood of $B$, such that $(a, b)$ for $(a, b)$ in this neighbourhood.

The condition $R(a, b) = 0$ means that $∂R⁄∂y ≠ 0$ is a regular point of the implicit curve of implicit equation $R(x, y) = 0$ where the tangent is not vertical.

In a less technical language, implicit functions exist and can be differentiated, if the curve has a non-vertical tangent.

In algebraic geometry
Consider a relation of the form $R(x, f(x)) = 0$, where $f$ is a multivariable polynomial. The set of the values of the variables that satisfy this relation is called an implicit curve if $∂R⁄∂y ≠ 0$ and an implicit surface if $(a, b)$. The implicit equations are the basis of algebraic geometry, whose basic subjects of study are the simultaneous solutions of several implicit equations whose left-hand sides are polynomials. These sets of simultaneous solutions are called affine algebraic sets.

In differential equations
The solutions of differential equations generally appear expressed by an implicit function.

Marginal rate of substitution
In economics, when the level set $R(x, y) = 0$ is an indifference curve for the quantities $a$ and $x$ consumed of two goods, the absolute value of the implicit derivative $R(x_{1}, …, x_{n}) = 0$ is interpreted as the marginal rate of substitution of the two goods: how much more of $R$ one must receive in order to be indifferent to a loss of one unit of $x$.

Marginal rate of technical substitution
Similarly, sometimes the level set $n = 2$ is an isoquant showing various combinations of utilized quantities $y$ of labor and $y$ of physical capital each of which would result in the production of the same given quantity of output of some good. In this case the absolute value of the implicit derivative $n = 3$ is interpreted as the marginal rate of technical substitution between the two factors of production: how much more capital the firm must use to produce the same amount of output with one less unit of labor.

Optimization
Often in economic theory, some function such as a utility function or a profit function is to be maximized with respect to a choice vector $x$ even though the objective function has not been restricted to any specific functional form. The implicit function theorem guarantees that the first-order conditions of the optimization define an implicit function for each element of the optimal vector $R(x, y) = 0$ of the choice vector $L$. When profit is being maximized, typically the resulting implicit functions are the labor demand function and the supply functions of various goods. When utility is being maximized, typically the resulting implicit functions are the labor supply function and the demand functions for various goods.

Moreover, the influence of the problem's parameters on $dy⁄dx$ — the partial derivatives of the implicit function — can be expressed as total derivatives of the system of first-order conditions found using total differentiation.

title: Implicit function


text:
Battle of Lundy's Lane

The Battle of Lundy's Lane, also known as the Battle of Niagara, was fought on 25 July 1814, during the War of 1812, between an invading American army and a British and Canadian army near present-day Niagara Falls, Ontario. It was one of the bloodiest battles of the war, and one of the deadliest battles fought in Canada, with approximately 1,720 casualties including 258 killed.

The engagement was marked by intense musketry at close range and instances of friendly fire on both sides amidst the smoke and confusion, which caused several units to break entirely. The two armies fought each other to a stalemate; neither side held firm control of the field following the engagement. However, the casualties suffered by the Americans precipitated their withdrawal, and the British held the strategic initiative.

Background
On 3 July 1814 an American army under Major General Jacob Brown launched an attack across the Niagara River near its source on Lake Erie. His force quickly captured the British position at Fort Erie and then advanced north. Two days later one of his two brigades of regular U.S. Infantry under Brigadier General Winfield Scott defeated a British force commanded by Major General Phineas Riall at the Battle of Chippawa.

A few days after the battle, Brown outflanked the British defences along the Chippawa Creek and the British fell back to Fort George near the mouth of the Niagara on Lake Ontario. Brown lacked the necessary troops and heavy artillery to attack this position and a British naval squadron controlled the lake. Commodore Isaac Chauncey, commander of the American ships based at Sackett's Harbor, New York, was waiting for new frigates and armed brigs to be completed before he challenged the British squadron. The American squadron was further delayed in port when Chauncey fell ill. As a result, no reinforcements or heavy guns were sent to Brown, and the British were able to move several units across the lake from York to reinforce Fort George.

For most of July, Brown's army occupied Queenston, a few miles south of Fort George. In this forward position they were harassed by Canadian militia and First Nation warriors. On 24 July Brown fell back to Chippawa Creek, intending to secure his supplies before advancing west to Burlington. As soon as Brown retired, British light infantry and militia under Major General Riall advanced to Lundy's Lane 4 mi north of the Chippawa to allow light troops to maintain contact with the American main force.

Preliminary movements
Early on 25 July, the British Lieutenant Governor of Upper Canada, Lieutenant General Gordon Drummond, arrived in Fort George to take personal command on the Niagara peninsula. He immediately ordered a force under Lieutenant Colonel John Tucker to advance south from Fort Niagara (which the British had captured in December 1813) along the east side of the Niagara River, hoping this would force Brown to evacuate the west bank. Instead, Brown ordered an advance north, intending in turn to force the British to recall Tucker's column to protect Fort George. The Americans apparently did not know that the British held Lundy's Lane in strength. As soon as Riall knew the Americans were advancing, he ordered his troops to fall back to Fort George and ordered another column under Colonel Hercules Scott to move from St. Davids to Queenston to cover his withdrawal, rather than advance to his support. These orders were countermanded by Drummond, who had force-marched a detachment of reinforcements to Lundy's Lane from Fort George. The British were still reoccupying their positions when the first American units came into view, at about 6:00 pm.

Scott's attack
Lundy's Lane was a spur from the main Portage Road alongside the Niagara River. It ran along the summit of some rising ground (about 25 feet higher than the surrounding area) and therefore commanded good views of the area. The British artillery (two 24-pounder and two 6-pounder guns, one 5.5-inch howitzer and a Congreve rocket detachment) were massed in a cemetery at the highest point of the battlefield.

The American 1st Brigade of regulars under Winfield Scott, with Towson's artillery company, emerged in the late afternoon from a forest into an open field and was badly mauled by the British artillery. Scott sent the 25th U.S. Infantry, commanded by Major Thomas Jesup, to outflank the British left. The 25th found a disused track leading to a landing stage on the river and used it to pass round the British flank. They caught the British and Canadian units there (the light company of the 1st Battalion of the 8th (King's) Regiment and the Upper Canada Incorporated Militia Battalion) while they were redeploying and unaware of the American presence, and drove them back in confusion. The British and Canadians rallied, but had been driven off the Portage Road. Jesup sent Captain Ketchum's light infantry company to secure the junction of Lundy's Lane and the Portage Road. Ketchum's company captured large numbers of wounded and messengers, including Major General Riall, who had been wounded in one arm and was riding to the rear. Most of the prisoners escaped when Ketchum, having briefly rejoined Jesup, ran into an enemy unit while trying to return to the main body of the American army, although Riall and militia cavalry leader Captain William Hamilton Merritt remained prisoners.

Jesup's action and the steadiness of Scott's brigade persuaded Drummond to withdraw his centre to maintain alignment with his left flank, and also pull back the Glengarry Light Infantry, who had been harassing Scott's own left flank. The withdrawal of Drummond's center left the artillery exposed in front of the infantry.

Brown's attack
By nightfall, Scott's brigade had suffered heavy casualties. Brown had arrived late in the day with the American main body (the 2nd Brigade of regulars under Brigadier General Eleazer Wheelock Ripley and a brigade of volunteers from the militia under Brigadier General Peter B. Porter). As Ripley and Porter relieved Scott's brigade, Brown ordered the 21st U.S. Infantry under Lieutenant Colonel James Miller to capture the British guns. Miller famously responded, "I'll try, Sir".

While the British were distracted by another attack by the 1st U.S. Infantry on their right, Miller's troops deployed within a few yards of the British artillery. They fired a volley of musketry that killed most of the gunners and followed up with a bayonet charge; this captured the guns and drove the British centre from the hill. The British infantry immediately behind the guns (the 2nd Battalion of the 89th Foot) tried to counter-attack, but were driven back by Miller and Ripley.

Meanwhile, the British column under Colonel Hercules Scott was arriving on the field, already tired from its unnecessary diversion via Queenston. Unaware of the changed situation, they blundered into Ripley's brigade and were also driven back in disorder, losing their own three 6-pounder guns. These were recovered by a charge by the light company of the 41st Foot, but were either abandoned again or remained in British hands but could not be brought into action as the drivers and gunners had been scattered.

Drummond's counter-attack
While the Americans tried to deploy their own artillery among the captured British guns, Drummond (who had been wounded in the neck) reorganized his troops and mounted a determined attempt to retake his own cannon. There was no subtlety; Drummond launched an attack in line, without attempting to use his many light infantry to harass or disorder the American line, or to locate any weak points in it. The Americans beat back the attack after a short-range musketry duel over the abandoned British guns, in which both sides suffered heavy casualties. The Glengarry Light Infantry, who had once again begun to harass the American left flank, were mistaken for Americans by other British units and forced to withdraw after suffering casualties from British fire. Undeterred by his first failure, Drummond launched a second attack, using the same methods and formation as in the first. Although some American units wavered, they were rallied by Ripley and stood their ground. While the combat was taking place, Winfield Scott led his depleted brigade (which had been reorganized into a single ad hoc battalion under Major Henry Leavenworth) in an unauthorized attack against Drummond's centre. Scott's brigade was engaged both by the British and by units of Ripley's brigade, who were not aware of the identity of the troops at which they were shooting. Drummond's line was driven back but Scott's men broke in disorder and retreated, before rallying on the American left. Scott rode to join Jesup's regiment, still out on the right flank, but was severely wounded shortly afterwards.

Shortly before midnight, Drummond launched a third counter-attack, using every man he could find, although by this time the British line consisted of mixed-up detachments and companies, rather than organized regiments and battalions. The fighting over the artillery was closer than before, with bayonets being used at one point, but again the exhausted British fell back.

End of the battle
By midnight both sides were spent. On the American side only 700 men were still standing in the line. Winfield Scott and Jacob Brown were both severely wounded. Brown would soon recover but Scott's injury removed him from the campaign. With supplies and water short, Brown ordered a retreat. Porter and Lieutenant Colonel Jacob Hindman (Brown's artillery commander) protested but complied. Ripley apparently did not learn of Brown's order until he realised that Hindman's artillery had been withdrawn. Although urged by Porter to maintain his position, he also withdrew. The British still had 1,400 men on the field but they were in no condition to interfere with the American withdrawal. Drummond had ordered some units to hold the Portage Road and left some light infantry outposts near the Americans, but had withdrawn the remainder a short distance west along Lundy's Lane.

The American artillerymen had suffered severely during the fighting, and Hindman had difficulty finding sufficient draught horses to get all his guns away. One American 6-pounder gun had been lost earlier during the close-range fighting, when its drivers had been hit by musket fire and the horses drawing it had bolted into the British lines. Hindman also had to abandon a howitzer with a broken carriage. The Americans were able to drag away one captured 6-pounder gun that had earlier been pushed to the bottom of the high ground in the centre of the former British position. Hindman later found more horses and sent a team back to recover one of the prized British 24-pounder guns. The team was captured by British parties who were wandering around the battlefield.

Aftermath
In the early hours of the morning of 26 July, Brown ordered Ripley to recover the abandoned British guns the next day. Reinforcing his exhausted men with detachments that had been left at Chippawa, Ripley moved out at daybreak with 1,200 soldiers but found that Drummond had reoccupied the battlefield with 2,200 men. Ripley withdrew, unmolested.

The American army fell back to Fort Erie, first deliberately destroying the old British fortifications along Chippawa Creek and burning the bridges behind them. Because they were short of transport, they had to abandon or destroy much equipment and supplies to make room for the wounded on the available wagons. Drummond was later to claim that the Americans had retreated in disorder. After burying some of the British and Canadian dead on the battlefield and burning many American corpses in large funeral pyres, the British withdrew to Queenston until Drummond received reinforcements.

In respect to the effect of the battle on the War, the British won a strategic victory, as the Americans on the Niagara had suffered so many casualties that they were now badly outnumbered, and were forced to retire to Fort Erie. Richard V. Barbuto says, "On 26 July, Brown's plan to advance on Burlington Heights was irretrievably shattered... Drummond had secured the forts at the northern end of the Niagara, and he had blunted an American advance. Although there was still a lot of fight in both forces, the balance of combat power on the Niagara Peninsula had swung from the invaders to the defenders".

Casualties
The British official casualty return had 84 killed, 559 wounded, 42 captured, and 193 missing. The Americans captured 19 British officers and 150 other ranks, giving a revised British casualty total of 84 killed, 559 wounded, 169 captured, and 55 missing.

The original American official casualty return, dated 30 July 1814, gave 171 dead, 572 wounded, and 117 missing. A slightly later version gave the same number of killed and wounded but 110 missing. Donald Graves, a Canadian historian employed at the Directorate of History, Department of National Defence Canada, says "British records state that four officers and 75 Americans of other ranks captured at Lundy's Lane were imprisoned at Quebec in the autumn of 1814. The remainder of the missing were probably the wounded or dead left on the field after Ripley withdrew". Three officers of the 22nd U.S. Regiment of Infantry; Lieutenants William Sturgis, John D. Kehr and Robert M. Davidson, who appear on the casualty list as "missing" and "supposed to be killed" were later confirmed as having been killed on 25 July. This gives a revised American loss of 174 killed, 572 wounded, 79 captured, and 28 missing. Graves speculates that the discrepancies in the proportions of killed and wounded men between British and American casualties may be accounted for by the Americans not collating their casualty returns until five days after the battle, when some of those originally listed as missing were confirmed to have been killed, and some severely wounded men had died of their wounds.

Analysis
There had been much fighting at close quarters. Veteran British officers, who had fought against French armies in the Peninsular War, were horrified at the carnage they had witnessed at Lundy's Lane. Drummond reported, "Of so determined a Character were [the American] attacks directed against our guns that our Artillery Men were bayonetted by the enemy in the Act of loading, and the muzzles of the Enemy's Guns were advanced within a few Yards of ours". The battle confirmed that the American regular forces had developed into a highly professional army. Scott is widely credited for this progress, having modelled and trained his troops using French Revolutionary Army drills and exercises, although not all the American units present at Lundy's Lane had benefitted from this training.

Of the British and Canadian regulars at Lundy's Lane, the inexperienced 103rd regiment engaged other British troops. Many of the other British regulars had been serving for over two years in Upper Canada, had suffered heavy casualties in earlier actions and were consequently war-weary. Towards the end of the battle, the loss of several colonels and the fact that there were few complete regiments present rather than mixtures of detached companies handicapped their final attacks. The detachments of Lincoln militia with Riall's division fired wildly and even shot the cockade off their commander's hat. Drummond redistributed their ammunition among the regulars and sent the militia to the rear. By comparison, the Incorporated Upper Canada militia battalion rallied and held the British left flank despite being surprised by Jessop and losing their colonel and reserve ammunition. They suffered 142 casualties, while the Lincoln and York militia had one killed and 19 wounded.

Evidence compiled by Donald Graves argues that General Drummond failed to use skirmish pickets to protect his guns, which were consequently captured by the Americans. Drummond also showed little tactical finesse during his counter-attacks, not using his light infantry to their best advantage and mounting only straightforward frontal attacks. American historian John R. Elting suggests that if Drummond had instead concentrated on the vulnerable American left flank, he might have won a decisive victory. (Drummond had much administrative experience, but had previously seen action only in the abortive Flanders Campaign in 1794 as a comparatively junior officer, and in the Egyptian campaign in 1801 as commander of a battalion.)

Battlefield and memorials
The site was designated as a National Historic Site of Canada in 1937.

The site of the battle is now a residential and commercial area of Niagara Falls, Ontario. Part of the battlefield site was preserved next to the Drummond Hill Cemetery on Lundy's Lane east of Drummond Road. Other memorials added to the site include:
 * Soldier's Monument – created by the Canadian Parliament and unveiled by Lundy's Lane Historical Society in 1895
 * Commemorative Wall – added 2004
 * Laura Secord is buried at the Drummond Hill Cemetery

Legacy

 * James Miller's response, "I'll try, Sir", to Brown's order to capture the British guns is now the motto of the 5th U.S. Infantry. The 21st were merged into this regiment in 1815 when Congress reduced the United States Army to a smaller peacetime establishment.
 * Ten active regular infantry battalions of the United States Army (1–2 Inf, 2-2 Inf, 1–3 Inf, 2–3 Inf, 4-3 Inf, 1–5 Inf, 2–5 Inf, 1–6 Inf, 2–6 Inf and 4–6 Inf) perpetuate the lineages of a number of American infantry units (the old 1st, 9th, 19th, 21st, 22nd, 23rd and 25th Infantry Regiments) that took part in the battle.
 * Four Regiments of the British Army, the 1st (Royal Scots), 8th (Kings), 41st and the 89th Regiments of Foot were awarded the "Niagara" Battle Honour for their service during the battle of Lundy's Lane. Their. respective modern successor units are: The Royal Regiment of Scotland, the Duke of Lancaster's Regiment, the Royal Welsh Regiment and the Royal Irish Regiment.
 * As the result of a decision taken in 2012, four Ontario-based regiments of the Canadian Army perpetuate the history and heritage of the Battalion of Incorporated Militia and thus carry the NIAGARA battle honour earned by that battalion. These regiments are: The Lincoln and Welland Regiment, the Queen's York Rangers, the Brockville Rifles and the Princess of Wales' Own Regiment.
 * Seven other Canadian Army regiments carry the same Battle Honour in commemoration of other Canadian Fencible and Militia units that took part in the battle. They are: the Stormont, Dundas and Glengarry Highlanders, the North Shore (New Brunswick) Regiment, the 56th Field Artillery Regiment, RCA, the Essex and Kent Scottish Regiment, the Royal Canadian Regiment, the Royal Hamilton Light Infantry, and The Lorne Scots.
 * In 1818, American poet John Neal published the poem "Battle of Niagara" about the battle. It is considered the best poetic description of Niagara Falls up to that time.
 * In 1908, Canadian poet Duncan Campbell Scott wrote a poem, "The Battle of Lundy's Lane," which won a contest sponsored by the Toronto Globe. The poem was reprinted in his 1916 collection, Lundy's Lane and Other Poems.
 * Lundy's Lane is mentioned in the unofficial Canadian patriotic anthem, "The Maple Leaf Forever" :
 * At Queenston Heights and Lundy's Lane our brave fathers, side by side
 * for freedom, homes, and loved ones dear, firmly stood and nobly died.
 * And those dear rights which they maintained, we swear to yield them never.
 * Our watchword evermore shall be, the Maple Leaf forever!

Catherine Lundy
Catherine Lundy was born in the late 1700s to Elizabeth Garvey and Lieutenant Daniel Shannon of the British army. As a teen, she married Thomas Lundy, with whom she would eventually have eight children. They settled at Lundy Lane, just one mile west of Niagara Falls. On July 25, 1814, British soldiers passed her house on their way to meet the Americans. Rather than fleeing with everyone else, she provided them with water after their 14-mile march. Furthermore, as a battle broke out just one mile east of her property, she opened her home to tend to the wounded. The British army was so appreciative of her help that one officer gave her his sword as a thanks. Catherine Lundy died in the middle of the 19th century, going down in history as a "saint".

title: Battle of Lundy's Lane


text:
Luminous efficiency function

A luminous efficiency function or luminosity function represents the average spectral sensitivity of human visual perception of light. It is based on subjective judgements of which of a pair of different-colored lights is brighter, to describe relative sensitivity to light of different wavelengths. It is not an absolute reference to any particular individual, but is a standard observer representation of visual sensitivity of theoretical human eye. It is valuable as a baseline for experimental purposes, and in colorimetry. Different luminous efficiency functions apply under different lighting conditions, varying from photopic in brightly lit conditions through mesopic to scotopic under low lighting conditions. When not specified, the luminous efficiency function generally refers to the photopic luminous efficiency function.

The CIE photopic luminous efficiency function $\overline{y}(λ)$ or $V(λ)$ is a standard function established by the Commission Internationale de l'Éclairage (CIE) and standardized in collaboration with the ISO, and may be used to convert radiant energy into luminous (i.e., visible) energy. It also forms the central color matching function in the CIE 1931 color space.

Details
There are two luminous efficiency functions in common use. For everyday light levels, the photopic luminosity function best approximates the response of the human eye. For low light levels, the response of the human eye changes, and the scotopic curve applies. The photopic curve is the CIE standard curve used in the CIE 1931 color space.

The luminous flux (or visible power) in a light source is defined by the photopic luminosity function. The following equation calculates the total luminous flux in a source of light:
 * $$\Phi_\mathrm{v} = 683.002\ (\mathrm{lm/W}) \cdot \int^\infin_0 \overline{y}(\lambda) \Phi_{\mathrm{e},\lambda}(\lambda)\, \mathrm{d}\lambda,$$

where
 * Φv is the luminous flux, in lumens;
 * Φe,λ is the spectral radiant flux, in watts per nanometre;
 * $\overline{y}$(λ), also known as V(λ), is the luminosity function, dimensionless;
 * λ is the wavelength, in nanometres.

Formally, the integral is the inner product of the luminosity function with the spectral power distribution. In practice, the integral is replaced by a sum over discrete wavelengths for which tabulated values of the luminous efficiency function are available. The CIE distributes standard tables with luminosity function values at 5 nm intervals from 380 nm to 780 nm.

The standard luminous efficiency function is normalized to a peak value of unity at 555 nm (see luminous coefficient). The value of the constant in front of the integral is usually rounded off to $683 lm/W$. The small excess fractional value comes from the slight mismatch between the definition of the lumen and the peak of the luminosity function. The lumen is defined to be unity for a radiant energy of 1/683 W at a frequency of 540 THz, which corresponds to a standard air wavelength of 555.016 nm rather than $555 nm$, which is the peak of the luminosity curve. The value of $\overline{y}$(λ) is $1$ at $555.016 nm$, so that a value of 683/$1$ = 683.002 is the multiplicative constant.

The number 683 is connected to the modern (1979) definition of the candela, the unit of luminous intensity. This arbitrary number made the new definition give numbers equivalent to those from the old definition of the candela.

Improvements to the standard
The CIE 1924 photopic V(λ) luminosity function, which is included in the CIE 1931 color-matching functions as the $\overline{y}$(λ) function, has long been acknowledged to underestimate the contribution of the blue end of the spectrum to perceived luminance. There have been numerous attempts to improve the standard function, to make it more representative of human vision. Judd in 1951, improved by Vos in 1978, resulted in a function known as CIE VM(λ). More recently, Sharpe, Stockman, Jagla & Jägle (2005) developed a function consistent with the Stockman & Sharpe cone fundamentals; their curves are plotted in the figure above.

Stockman & Sharpe has subsequently produced an improved function in 2011, taking into account the effects of chromatic adaptation under daylight. Their work in 2008 has revealed that "luminous efficiency or V(l) functions change dramatically with chromatic adaptation".

ISO standard
The ISO standard is ISO/CIE FDIS 11664-1. The standard provides an incremental table by nm of each value in the visible range for the CIE 1924 function.

Scotopic luminosity
For very low levels of intensity (scotopic vision), the sensitivity of the eye is mediated by rods, not cones, and shifts toward the violet, peaking around 507 nm for young eyes; the sensitivity is equivalent to $1,699 lm/W$ or $1,700 lm/W$ at this peak. The standard scotopic luminous efficiency function or V(λ) was adopted by the CIE in 1951, based on measurements by Wald (1945) and by Crawford (1949).

Luminosity for mesopic vision, a wide transitioning band between scotopic and phototic vision, is more poorly standardized. The consensus is that this luminous efficiency can be written as a weighted average of scotopic and mesopic luminosities, but different organizations provide different weighting factors.

Color blindness
Color blindness changes the sensitivity of the eye as a function of wavelength. For people with protanopia, the peak of the eye's response is shifted toward the short-wave part of the spectrum (approximately 540 nm), while for people with deuteranopia, there is a slight shift in the peak of the spectrum, to about 560 nm. People with protanopia have essentially no sensitivity to light of wavelengths more than 670 nm.

Most non-primate mammals have the same luminous efficiency function as people with protanopia. Their insensitivity to long-wavelength red light makes it possible to use such illumination while studying the nocturnal life of animals.

For older people with normal color vision, the crystalline lens may become slightly yellow due to cataracts, which moves the maximum of sensitivity to the red part of the spectrum and narrows the range of perceived wavelengths.

title: Luminous efficiency function


text:
Borda

Borda may refer to:

 * Qaṣīda al-Burda, a famous Sufi poem.
 * Borda (building) or borde, traditional cattle farmers' buildings in the Pyrenees, a barn, sheepfold, or stable
 * Places in India
 * Borda, Goa, a town and suburb of the city of Margao in the state of Goa, India
 * Borda, Maharashtra, a village in Osmanabad district of Maharashtra State, India
 * Borda, Bhopal, a village in Madhya Pradesh, India
 * Borda da Mata, a municipality in Minas Gerais, Brazil
 * Borda, the Hungarian name for Burda village, Budureasa Commune, Bihor County, Romania
 * Borda (crater), a lunar crater
 * Borda (legendary creature), in the culture of the Emilia-Romagna of the Po Valley, Italy
 * Borda count, a single-winner election method
 * Borda–Carnot equation in fluid dynamics
 * BORDA, Bremen Overseas Research and Development Association
 * House of Borda, family name of a French-Spanish noble house
 * Hospital Borda, the largest and most notable psychiatric hospital in Argentina

People

 * Aritz Borda, a Spanish footballer
 * César Borda (1993–2019), an Argentine footballer
 * Deborah Borda, an American orchestra executive
 * Jean-Charles de Borda (1733–1799), a French mathematician, physicist, political scientist, and sailor
 * Lidia Borda, an Argentine tango singer
 * Soleil Borda, an American teen actress
 * Lino (Carmel) Gauci Borda, Maltese politician
 * Manuel Borda, Maltese politician
 * Arturo Borda (1883–1953), a Bolivian surrealist painter
 * Pepe Serret Borda ( 1941-1993), a Spanish businessman
 * Anapaola Borda Rodas (born 2004), Argentine chess master

Ships

 * French ship Borda, six different French naval ships named Borda

title: Borda


text:
User:Ed Cormany

About me
I'm a linguist by trade. I currently live and work in New York City. From 2007–2013 I lived in Ithaca, New York while working on my PhD in linguistics at Cornell University. At Cornell I designed and taught a course on constructed languages.

You can find out much more about me elsewhere on the internet:

my personal website

twitter

I enjoy following sports, especially my University of Michigan Wolverines. I also follow Italian football (soccer), and help maintain articles on those topics, particularly the Coppa Italia articles and, when I feel like it, S.S. Lazio season articles.

title: User:Ed Cormany


text:
Automatic number identification

Automatic number identification (ANI) is a feature of a telecommunications network for automatically determining the origination telephone number on toll calls for billing purposes. Automatic number identification was originally created by the American Telephone and Telegraph Company (AT&T) for long distance service in the Bell System, eliminating the need for telephone operators to manually record calls.

Modern ANI has two components: information digits, which identify the class of service, and the calling party billing telephone number.

The term is also used to describe the functions of two-way radio selective calling that identify the transmitting user.

ANI is distinct from newer caller ID services, such as call display, which are solely for informing a subscriber.

Toll-free telephone numbers
Modern toll-free telephone numbers, which generate itemized billing of all calls received instead of relying on the special fixed-rate trunks of the Bell System's original Inward WATS service, depend on ANI to track inbound calls to numbers in special area codes such as +1-800, 888, 877, 866, 855, 844, and 833 with 822 reserved for future toll free use (United States and Canada), 1800 (Australia) or 0800 and 0808 (United Kingdom).

Privacy
ANI is conceptually and technically different from caller ID service. A caller's telephone number and line type are captured by ANI service even if caller ID blocking is activated. The destination telephone company switching office can relay the originating telephone number to ANI delivery services subscribers. Toll-free subscribers and large companies normally have access to ANI, either instantly via installed equipment, or from a monthly billing statement. Residential subscribers can obtain access to ANI information through third party companies that charge for the service.

ANI is generally not transmitted when a call is operator assisted; only the area code of the last switch to route the call is sent.

Automatic number announcement
ANI is used to provide automatic number announcement, a test facility of a central office for telephone installation technicians. The service, which is not advertised to the public, allows an installer to identify a line by dialing a telephone number. Such numbers are typically assigned in a a range reserved for testing purposes (such as 958-xxxx in much of North America).

DNIS
Dialed Number Identification Service (DNIS) is a related service feature available to private branch exchange subscribers. It transmits information about the destination number, which a service provider can use to have several toll-free numbers directed to the same call center and provide unique service. DNIS can also be used to identify other call routing information. For example, toll-free service can be configured to send a specific DNIS number that is assigned to callers from geographic regions based on city, area code, state, or country.

Similar services

 * Europe: Calling Line Identification (CLI)
 * United Kingdom: Caller Line Identification (CLID)
 * Australia: Automatic number identification (ANI) 1800801920 or 12722123 (Telstra line only, local call cost)

title: Automatic number identification


text:
User talk:Ed Cormany

compliments :-)
Thanks for cleaning up the Aornis Hades article(s). To say I'm a Wiki-newb is an understatement, but I -do- know when I'm right. Thanks for your help. I hope that'll stop people from deleting my work just so they can mindlessly retype what Fforde said in his novel about the rivers. Usoki 20:37, 22 June 2007 (UTC)

Good job at implicit differentiation! Pizza Puzzle
 * Thanks. I didn't take AP BC Calc for nothing :-) Ed Cormany

Ditto for Geothermal areas of Yellowstone. You expanded the entry nicely! --mav 07:35 24 Jul 2003 (UTC)
 * Thank you, considering I have a lot more planned for that and related articles and just haven't got around to it. Somehow paid work and sleep take precedence over free work.  Funny how that is...  Ed Cormany 19:31 24 Jul 2003 (UTC)

Good job on List of chess engines! Thanks! (I came across it accidentally the other day and couldn't face it--plus I don't have the summer off. ;-) ) Elf | Talk 04:10, 14 May 2004 (UTC)

Thanks for your help with the Russia template! -- ran 00:41, Jun 23, 2004 (UTC)

Thanks a lot for fixing the article I put on cleanup! (boa knot) Just for the future- please remove all entries after fixing them up to prevent the cleanup page from getting more bloated than it already is. Thanks for your time. -Frazzydee 23:41, 4 Jul 2004 (UTC)

other
A while ago, you asked why any article called "Wikipedia Help the Eager Beaver Foundation" was blanked and protected instead of being deleted. I don't know if anyone got back to you on that, or if you figured it out on your own, but the reason was that the article had been deleted four times. It was continually recreated by the vandal, so Angela created a blank version and protected it. Once the vandal lost interest, it was deleted. -- Cyan 19:16, 3 Nov 2003 (UTC)

 * Thanks Cyan, that makes good sense, and I had never thought of such a clever solution to that type of problem. Ed Cormany

On the section of this article about the Wikipedia Help the Eager Beaver Article Foundation there was something written: The vandal: It's coming back to save the Eager Beaver article. Why did you erase that? Tell me why at my talk page. Heegoop, 10 Jul 2004 (UTC)

Hi Ed, FYI, I created a help file on archiving Current Events: How to archive Current Events As you will see, it is little more than the text currently at the top of the Talk page. I hope this useful information is now more visible. All the best, -- Viajero 17:12, 10 Nov 2003 (UTC)

Hi, Ed, and thanks for the nice note abt listing Cleanup on Cleanup. I hope it didn't seem sarcastic; my personality makes me good at doing things like getting the new day's date on promptly and adding headings that i think are encouraging others to do the periodic review of entries as they age (which i agonize over too much to be effective at). I don't see it as "grown out of control", but as a little sub-optimum; i'm hoping that things like the Cleanup lisitng and the headings (which make it more obvious that we, collectively, are neglecting the sensible aging reviews) are part of our decentralized control system's means applying a course correction. --Jerzy(t) 13:34, 2004 May 13 (UTC)

Thank you,Ed, your advice is helpful.I will make some change to Softswitch as you advised me. I will learn fast-:)!--Bigben 04:03, 17 May 2004 (UTC)

Hello Ed. I notice that you have redirected Amount of substance to Mole (unit). Please may I ask why? I've set out some reasons why both articles should exist in Talk:Amount of substance. Thanks Ian Cairns 22:35, 7 Jul 2004 (UTC)

Archived talk

Article Licensing
Hi, I've started a drive to get users to multi-license all of their contributions that they've made to either (1) all U.S. state, county, and city articles or (2) all articles, using the Creative Commons Attribution-Share Alike (CC-by-sa) v1.0 and v2.0 Licenses or into the public domain if they prefer. The CC-by-sa license is a true free documentation license that is similar to Wikipedia's license, the GFDL, but it allows other projects, such as WikiTravel, to use our articles. Since you are among the top 2000 Wikipedians by edits, I was wondering if you would be willing to multi-license all of your contributions or at minimum those on the geographic articles. Over 90% of people asked have agreed. For More Information:
 * Multi-Licensing FAQ - Lots of questions answered
 * Multi-Licensing Guide
 * Free the Rambot Articles Project

To allow us to track those users who muli-license their contributions, many users copy and paste the " " template into their user page, but there are other options at Template messages/User namespace. The following examples could also copied and pasted into your user page:

 * Option 1
 * I agree to multi-license all my contributions, with the exception of my user pages, as described below:

OR
 * Option 2
 * I agree to multi-license all my contributions to any U.S. state, county, or city article as described below:

Or if you wanted to place your work into the public domain, you could replace " " with "  ". If you only prefer using the GFDL, I would like to know that too. Please let me know what you think at my talk page. It's important to know either way so no one keeps asking. -- Ram-Man (comment| talk)

Proposed Georgia Move
As a past participant in the discussion on how to handle the Georgia pages, I thought you might be interested to know that there's a new attempt to reach consensus on the matter being addressed at Talk:Georgia (country). Please come by and share your thoughts to help form a consensus. --Vengeful Cynic 03:49, 8 July 2006 (UTC)

Coppa Italia
Hey, no worries about wiping them out. I was just *extremely* bored this evening, and noticed that it hadn't been updated (I was assuming because of the bracket), and thought I'd just throw some time into it. I did the best I could, even though it's real convoluted and hard on the eyes, so surely the bracket will make things all more the clearer, but don't worry about it! --Snojoe 05:54, 7 August 2007 (UTC)
 * Again regarding the bracket, and just a suggestion; obviously, you and I know better as to which bracket champions faces another in the semifinals, but for clarification, maybe there should be some sort of symbol representing each bracket, and which bracket winner faces the other in the semis. Example, declare the Roma/Sampdoria bracket "Bracket A" and in the semifinal/finals bracket, list "Bracket A" v "Bracket B" in each slot? Showing the page to my cousin, he had questioned the progression from quarters to semis, and thought maybe something should be added to clarify? You did a marvelous job with them though, definitely much better than what I had done! --Snojoe 20:54, 9 August 2007 (UTC)
 * More Coppa... More like just a general question, is there a specific website you're checking for live scores? I keep Googling some, and none seem to do a justice as all are *really* slow in updating the scores. I'd love to use it not just for Coppa scores, but any other competitions that Soccernet doesn't update as actively as they should. --Snojoe 20:34, 29 August 2007 (UTC)

UEFA Champions League draw
Why don't you readd the talbles from last season too? There is more consensus about deleting it then there is for keeping it. If I don't do it again then someone else will and I ca nganrantee you that. Kingjeff 02:30, 6 September 2007 (UTC)

12-team bracket
I'm hoping you might be able to help. Please read and leave a comment at Category talk:Tournament bracket templates. Thanks. Flibirigit (talk) 21:31, 24 February 2008 (UTC)

Why "Internazionale"?
Well, it is a boring idea which changing Inter to Internazionale. However, plenty of press (television of course) refer the club as "Internazionale". Inter is easier to call, but Internazionale is more proper one. That's why I made this edit. But I have no idea if the decision is correct. Raymond Giggs 01:30, 15 May 2008 (UTC)

Semi-protection for UEFA CL 2008-09
I would definitely support it. I tried to reach that reverter in German since he uses a German IP address, but this obviously still failed. Hockey-holic (talk) 16:50, 26 July 2008 (UTC)

Recent moves re: Serie C
If you look at what is now Talk:Lega_Pro_Prima_Divisione, you will find there was a requested move discussion ongoing - it would have been nice if that had been concluded before the move. Given Wikipedia's guidelines on official names, it is far from obvious that your decision based on official renaming is the correct one. Knepflerle (talk) 21:47, 30 July 2008 (UTC)

 * No problem, it was easily missed! I have a slight preference for the old name; but taking another look back, nobody else has commented there, and you and the editor who nominated the move are both in favour so I'm perfectly happy to leave it.  Good luck with your editing on Italian football - in general I think the articles on Italian football are of a really decent standard, I just hope other countries will catch up eventually! Knepflerle (talk) 13:13, 31 July 2008 (UTC)

Coppa Italia 2008-09 changes
I added the teams' positions before any results were posted and thus every team was just listed once. Someone else added the results and copied the winning team's name (link and position) on to the next braket. For most people, many of those team names in the lower divisions are so obscure that few readers know what level those teams are playing in. Once all the resutls are in, if I were to ask you how many Serie D teams survived the first round, you would be able to answer me in 20 seconds, otherwise, it would take you many minutes, or at least it would take me many minutes. I personally don't think the bracket is hard to read. Maybe an index might be needed to tell the reader what the codes mean. We also need to know which team was playing at home (maybe something like the 2007-08 Coppa Italia). Someone put up the seeding chart, but I think that is wrong too, since the tournament has expanded from 42 to 78 teams.

 * I would keep that information for the first listing of the team, unless there is some sort of consenses that its no good. If you think its so important that it should brought up in the football project page, please do so.  How about a compromise?........somehow keep the information on last year's placement near the team name (at the front maybe?) within the bracket area and not part of the name as it is now.  Just a thought.  Juve2000 (talk) 02:50, 11 August 2008 (UTC)

 * I just noticed what you did, and I just want to say that I totally diagree with the changes you made. The seeding chart makes no sense at all.  Teams are grouped based on their classification the previous year and then drawn based on the round they are to enter.  There is no seeding from 1 to 78,  you just made that up based on the order of the draw.  What you have done makes no sense.  By the way, thanks for destroying the hard work I put in into making the brackets understandable.  If everyone did what you did, we would all be reverting everything and nothing would get done.  What am I supposed to do now? revert your work?  Don't worry, that garbage will not be reverted by me.  Lets see now, if I want to see where Castellarano came from, i have to look up seed 64 on your famous chart, and then see that it finished third in D/D last year.  I see your point, the way I had it was way too easy, lets make it complicated and hard for the reader.  Good Job, Juve2000 (talk) 00:09, 12 August 2008 (UTC)

Winner parameter
I am quite convinced it is actually necessary to evidence the winning team in the playoffs, it is not so automatic to recognize it only by bolding club names. If you think it is merely a matter of colour, just change it; my only concern is to make the name of the winning team clear undeniably evident. --Angelo (talk) 14:31, 12 June 2009 (UTC)

bracket templates
Hi. I just noticed that some brackets used semi-finals and some used semifinals, Semi Finals, Semi-Finals, Semifinals. (Same for quarter-finals) Basically all of the different forms. Sometimes different forms in the same bracket template. So I looked it up at the Oxford English Directory through my library and they listed quarter-finals and semi-finals using the hyphen. Dictionary.com uses the word without the hyphens as one word. And of course in the media, anything goes, it seems. So it was basically a tidy-up, not trying to be bold. I think the OED is the 'most correct' spelling, so I went with that. I think that semi-final is defendable as the best choice as it shows in the spelling what is meant, that it is a derivation of the final. I do see the 'semi-final' format used extensively here in North America (mostly in books). There will definitely be opinions. I don't know of a project to go and discuss it at, so I just did a tidy. We have a ton of bracket templates that are virtually identical. The bracket templates could be reduced as another tidy-up project but that's a bigger job. Alaney2k (talk) 23:08, 16 June 2009 (UTC)
 * I just hope it doesn't cause too much of a fuss. I was trying to be helpful to make a default, not impose. (I looked at a large sample of the various articles linked and over 80% do not use the default titles anyway, so I was not that concerned about messing up articles) I would like to add some docs to a lot of the templates. They are so useful, and yet we probably don't use them enough, as shown by the duplication. Oh, and not sure what you meant about standardizing the 'other way around'. Alaney2k (talk) 01:05, 17 June 2009 (UTC)
 * You're right. Coulda just removed them. Still would have changed a bunch of capitalization styles. You know, I could have just used AWB because they are all in one or two cats. Damn, why didn't think of that? LOL. Oh yeah, because there were so many versions... I could turn it around a lot easier cause I've standardized. Alaney2k (talk) 01:32, 17 June 2009 (UTC)

S.S. Lazio 2009-2010
Thank you! I'm doing my best, this year I have the card to see every match at the stadium, so I can help also with many photos! My problem is the english, because I'm not very good to write...maybe you could work on the page correcting my wrong sentences ;). It would be great (and very useful!).-- A ndrea 93  (msg) 13:43, 23 August 2009 (UTC)

formatting of league tables
Hello,

I have replied to your inquiry at WT:FOOTY, and also changed the 2009–10 Serie B table to what all tables should look like. Cheers, Soccer-holicI hear voices in my head... 17:50, 6 August 2010 (UTC)

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text:
Lemniscate of Bernoulli

In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points $F_{1}$ and $F_{2}$, known as foci, at distance $F_{1}$ from each other as the locus of points $F_{2}$ so that $2c$. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from, which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4.

This lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed focal points is a constant. A Cassini oval, by contrast, is the locus of points for which the product of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli.

This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisector of its two foci). It may also be drawn by a mechanical linkage in the form of Watt's linkage, with the lengths of the three bars of the linkage and the distance between its endpoints chosen to form a crossed parallelogram.

Equations
The equations can be stated in terms of the focal distance $c$ or the half-width $a$ of a lemniscate. These parameters are related as $P$.
 * Its Cartesian equation is (up to translation and rotation):
 * $$\begin{align} \left(x^2 + y^2\right)^2 &= a^2 \left(x^2 - y^2\right) \\ &= 2 c^2 \left(x^2 - y^2\right) \end{align}$$
 * As a parametric equation:
 * $$x = \frac{a\cos t}{1 + \sin^2 t}; \qquad y = \frac{a\sin t \cos t}{1 + \sin^2 t} $$
 * A rational parametrization:
 * $$x = a \frac{t+t^3}{1+t^4}; \qquad y = a\frac{t-t^3}{1 + t^4} $$
 * In polar coordinates:
 * $$r^2 = a^2 \cos 2\theta$$
 * Its equation in the complex plane is:
 * $$|z-c||z+c|=c^2$$
 * In two-center bipolar coordinates:
 * $$rr' = c^2$$
 * In rational polar coordinates:
 * $$Q = 2s-1$$

Arc length and elliptic functions
The determination of the arc length of arcs of the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae). The period lattices are of a very special form, being proportional to the Gaussian integers. For this reason the case of elliptic functions with complex multiplication by √−1 is called the lemniscatic case in some sources.

Using the elliptic integral
 * $$\operatorname{arcsl}x \stackrel{\text{def}}{{}={}} \int_0^x\frac{dt}{\sqrt{1-t^4}}$$

the formula of the arc length $L$ can be given as
 * $$\begin{align} L &= 4\sqrt{2}\,c\int_{0}^1\frac{dt}{\sqrt{1-t^4}} = 4\sqrt{2}\,c\,\operatorname{arcsl}1 \\[6pt] &= \frac{\Gamma (1/4)^2}{\sqrt\pi}\,c =\frac{2\pi}{\operatorname{M}(1,1/\sqrt{2})}c\approx 7{.}416 \cdot c \end{align}$$

where $$\Gamma$$ is the gamma function and $$\operatorname{M}$$ is the arithmetic–geometric mean.

Angles
Given two distinct points $$\rm A$$ and $$\rm B$$, let $$\rm M$$ be the midpoint of $$\rm AB$$. Then the lemniscate of diameter $$\rm AB$$ can also be defined as the set of points $$\rm A$$, $$\rm B$$, $$\rm M$$, together with the locus of the points $$\rm P$$ such that $$|\widehat{\rm APM}-\widehat{\rm BPM}|$$ is a right angle (cf. Thales' theorem and its converse).

The following theorem about angles occurring in the lemniscate is due to German mathematician Gerhard Christoph Hermann Vechtmann, who described it 1843 in his dissertation on lemniscates.

 * $PF_{1}·PF_{2} = c^{2}$ and $a = c√2$ are the foci of the lemniscate, $F_{1}$ is the midpoint of the line segment $F_{2}$ and $O$ is any point on the lemniscate outside the line connecting $F_{1}F_{2}$ and $P$. The normal $F_{1}$ of the lemniscate in $F_{2}$ intersects the line connecting $n$ and $P$ in $F_{1}$. Now the interior angle of the triangle $F_{2}$ at $R$ is one third of the triangle's exterior angle at $OPR$ (see also angle trisection). In addition the interior angle at $O$ is twice the interior angle at $R$.

Further properties

 * The lemniscate is symmetric to the line connecting its foci $P$ and $O$ and as well to the perpendicular bisector of the line segment $F_{1}$.
 * The lemniscate is symmetric to the midpoint of the line segment $F_{2}$.
 * The area enclosed by the lemniscate is $F_{1}F_{2}$.
 * The lemniscate is the circle inversion of a hyperbola and vice versa.
 * The two tangents at the midpoint $F_{1}F_{2}$ are perpendicular, and each of them forms an angle of $a^{2} = 2c^{2}$ with the line connecting $O$ and $\pi⁄4$.
 * The planar cross-section of a standard torus tangent to its inner equator is a lemniscate.
 * The curvature at $$(x,y)$$ is $${3\over a^2}\sqrt{x^2+y^2}$$. The maximum curvature, which occurs at $$(\pm a,0)$$, is therefore $$3/a$$.

Applications
Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models.

title: Lemniscate of Bernoulli