**Integration** is a core concept of advanced

mathematics, specifically, in the fields of

calculus and

mathematical analysis. Given a

function *f*(

*x*) of a real

variable *x* and an

interval [

*a*,

*b*] of the

real line, the

**integral** represents the

area of a region in the

*xy*-plane bounded by the

graph of

*f*, the

*x*-axis, and the vertical lines

*x*=

*a* and

*x*=

*b*.

The term "integral" may also refer to the notion of

antiderivative, a function

*F* whose derivative is the given function

*f*. In this case it is called an

**indefinite integral**, while the integrals discussed in this article are termed

**definite integrals**. Some authors maintain a distinction between antiderivatives and indefinite integrals.

The principles of integration were formulated by

Isaac Newton and

Gottfried Wilhelm Leibniz in the late seventeenth century. Through the

fundamental theorem of calculus, that they independently developed, integration is connected with

differentiation, and the definite integral of a function can be easily computed once an antiderivative is known. Integrals and derivatives became the basic tools of

calculus, with numerous applications in science and engineering.

A rigorous mathematical definition of the integral was given by

Bernhard Riemann. It is based on a

limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integral began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A

line integral is defined for functions of two or three variables, and the interval of integration [

*a*,

*b*] is replaced by a certain

curve connecting two points on the plane or in the space. In a

surface integral, the curve is replaced by a piece of a

surface in the three-dimensional space. Integrals of

differential forms play a fundamental role in modern

differential geometry. These generalizations of integral first arose from the needs of

physics, and they play an important role in the formulation of many physical laws, notably those of

electrodynamics. Modern concepts of integration are based on the abstract mathematical theory known as

Lebesgue integration, developed by

Henri Lebesgue.

** History** Integration can be traced as far back as ancient Egypt,

*circa* 1800 BC, with the

Moscow Mathematical Papyrus demonstrating knowledge of a formula for the volume of a

pyramidal frustum. The first documented systematic technique capable of determining integrals is the

method of exhaustion of

Eudoxus (

*circa* 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. This method was further developed and employed by

Archimedes and used to calculate areas for parabolas and an approximation to the area of a circle. Similar methods were independently developed in China around the 3rd Century AD by

Liu Hui, who used it to find the area of the circle. This method was later used by

Zu Chongzhi to find the volume of a sphere.

Significant advances on techniques such as the method of exhaustion did not begin to appear until the 16th Century AD. At this time the work of

Cavalieri with his

*method of indivisibles*, and work by

Fermat, began to lay the foundations of modern calculus. Further steps were made in the early 17th Century by

Barrow and

Torricelli, who provided the first hints of a connection between integration and

differentiation.

** Pre-calculus integration** The major advance in integration came in the 17th Century with the independent discovery of the

fundamental theorem of calculus by

Newton and

Leibniz. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern

Calculus, whose notation for integrals is drawn directly from the work of Leibniz.

** Newton and Leibniz** While Newton and Leibniz provided systematic approach to integration, their work lacked a degree of rigour.

Bishop Berkeley memorably attacked

infinitesimals as "the ghosts of departed quantity". Calculus acquired a firmer footing with the development of

limits and was given a suitable foundation by

Cauchy in the first half of the 19th century. Integration was first rigorously formalised, using limits, by

Riemann. Although all piecewise continuous and bounded functions are Riemann integrable on a bounded interval, subsequently more general functions were considered, to which Riemann's definition does not apply, and

Lebesgue formulated a different definition of integral, founded in

measure theory. Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed.

** Formalising integrals** Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with

or

, which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.

The modern notation for the indefinite integral was introduced by

Gottfried Leibniz in 1675 (

Burton 1988, p. 359;

Leibniz 1899, p. 154). He derived the integral symbol, "∫", from an

elongated letter S, standing for

*summa* (Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by

Joseph Fourier in

*Mémoires* of the French Academy around 1819–20, reprinted in his book of 1822 (

Cajori 1929, pp. 249–250;

Fourier 1822, §231). In

Arabic mathematical notation which is written from right to left, an inverted integral symbol

is used

(W3C 2006).

** Notation** If a function has an integral, it is said to be

**integrable**. The function for which the integral is calculated is called the

**integrand**. The region over which a function is being integrated is called the

**domain of integration**. In general, the integrand may be a function of more than one variable, and the domain of integration may be an area, volume, a higher dimensional region, or even an abstract space that does not have a geometric structure in any usual sense.

The simplest case, the integral of a real-valued function

*f* of one real variable

*x* on the interval [

*a*,

*b*], is denoted by

The ∫ sign, an elongated "S", represents integration;

*a* and

*b* are the

**lower limit** and

**upper limit** of integration, defining the domain of integration;

*f* is the integrand, to be evaluated as

*x* varies over the interval [

*a*,

*b*]; and

*dx* can have different interpretations depending on the theory being used. For example, it can be seen as merely a notation indicating that

*x* is the 'dummy variable' of integration, as a reflection of the weights in the Riemann sum, a measure (in Lebesgue integration and its extensions), an infinitesimal (in non-standard analysis) or as an independent mathematical quantity: a

differential form. More complicated cases may vary the notation slightly.

** Terminology and notation** Integrals appear in many practical situations. Consider a swimming pool. If it is rectangular, then from its length, width, and depth we can easily determine the volume of water it can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope it). But if it is oval with a rounded bottom, all of these quantities call for integrals. Practical approximations may suffice at first, but eventually we demand exact and rigorous answers to such problems.

To start off, consider the curve

*y* =

*f*(

*x*) between

*x* = 0 and

*x* = 1, with

*f*(

*x*) = √

*x*. We ask:

What is the area under the function

*f*, in the interval from 0 to 1?

and call this (yet unknown) area the

**integral** of

*f*. The notation for this integral will be

.

As a first approximation, look at the unit square given by the sides

*x*=0 to

*x*=1 and

*y*=

*f*(0)=0 and

*y*=

*f*(1)=1. Its area is exactly 1. As it is, the true value of the integral must be somewhat less. Decreasing the width of the approximation rectangles shall give a better result; so cross the interval in five steps, using the approximation points 0, )/(

*q*+1).)

Historically, after the failure of early efforts to rigorously define infinitesimals, Riemann formally defined integrals as a

limit of ordinary weighted sums, so that the

*dx* suggested the limit of a difference (namely, the interval width). Shortcomings of Riemann's dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral, which is founded on an ability to extend the idea of "measure" in much more flexible ways. Thus the notation

refers to a weighted sum in which the function values are partitioned, with μ measuring the weight to be assigned to each value. (Here

*A* denotes the region of integration.)

Differential geometry, with its "calculus on

manifolds", gives the familiar notation yet another interpretation. Now

*f*(

*x*) and

*dx* become a

differential form, ω =

*f*(

*x*)

*dx*, a new

differential operator **d**, known as the

exterior derivative appears, and the fundamental theorem becomes the more general

Stokes' theorem,

from which

Green's Theorem, the

divergence theorem, and the

fundamental theorem of calculus follow.

More recently, infinitesimals have reappeared with rigor, through modern innovations such as

non-standard analysis. Not only do these methods vindicate the intuitions of the pioneers, they also lead to new mathematics.

Although there are differences between these conceptions of integral, there is considerable overlap. Thus the area of the surface of the oval swimming pool can be handled as a geometric ellipse, as a sum of infinitesimals, as a Riemann integral, as a Lebesgue integral, or as a manifold with a differential form. The calculated result will be the same for all.

** Introduction** There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals.

** Formal definitions** *Main article: Riemann integral* ** Riemann integral** *Main article: Lebesgue integral* ** Lebesgue integral** Although the Riemann and Lebesgue integrals are the most important definitions of the integral, a number of others exist, including:

The

Riemann-Stieltjes integral, an extension of the Riemann integral.

The

Lebesgue-Stieltjes integral, further developed by

Johann Radon, which generalizes the

Riemann-Stieltjes and

Lebesgue integrals.

The

Daniell integral, which subsumes the

Lebesgue integral and

Lebesgue-Stieltjes integral without the dependence on

measures.

The

Henstock-Kurzweil integral, variously defined by

Arnaud Denjoy,

Oskar Perron, and (most elegantly, as the gauge integral)

Jaroslav Kurzweil, and developed by

Ralph Henstock.

** Other integrals** ** Linearity** A number of general inequalities hold for Riemann-integrable

functions defined on a

closed and

bounded interval [

*a*,

*b*] and can be generalized to other notions of integral (Lebesgue and Daniell).

This is a generalization of the above inequalities, as *M*(*b* − *a*) is the integral of the constant function with value *M* over [*a*, *b*].

If *f* is Riemann-integrable on [*a*, *b*] then the same is true for |*f*|, and

Moreover, if *f* and *g* are both Riemann-integrable then *f* are also Riemann integrable and the following Minkowski inequality holds: ** Inequalities for integrals**

In this section *f* is a real-valued Riemann-integrable function. The integral

over an interval [*a*, *b*] is defined if *a* < *b*. This means that the upper and lower sums of the function *f* are evaluated on a partition *a* = *x*_{0} ≤ *x*_{1} ≤ . . . ≤ *x*_{n} = *b* whose values *x*_{i} are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating *f* within intervals [*x*_{ i} , *x*_{ i +1}] where an interval with a higher index lies to the right of one with a lower index. The values *a* and *b*, the end-points of the interval, are called the limits of integration of *f*. Integrals can also be defined if *a* > *b*:

This, with *a* = *b*, implies:

The first convention is necessary in consideration of taking integrals over subintervals of [*a*, *b*]; the second says that an integral taken over a degenerate interval, or a point, should be zero. One reason for the first convention is that the integrability of *f* on an interval [*a*, *b*] implies that *f* is integrable on any subinterval [*c*, *d*], but in particular integrals have the property that:

With the first convention the resulting relation

is then well-defined for any cyclic permutation of *a*, *b*, and *c*.

Instead of viewing the above as conventions, one can also adopt the point of view that integration is performed on *oriented* manifolds only. If *M* is such an oriented *m*-dimensional manifold, and *M'* is the same manifold with opposed orientation and *ω* is an *m*-form, then one has (see below for integration of differential forms):

*Reversing limits of integration.* If *a* > *b* then define

*Integrals over intervals of length zero.* If *a* is a real number then

*Additivity of integration on intervals.* If *c* is any element of [*a*, *b*], then ** Conventions**

*Main article: Fundamental theorem of calculus* ** Fundamental theorem of calculus**

then *F* is continuous on [*a*, *b*]. If *f* is continuous at *x* in [*a*, *b*], then *F* is differentiable at *x*, and *F* ′(*x*) = *f*(*x*).

is an anti-derivative of *f* on [*a*, *b*]. Moreover,

*Fundamental theorem of calculus.* Let *f* be a real-valued integrable function defined on a closed interval [*a*, *b*]. If *F* is defined for *x* in [*a*, *b*] by

*Second fundamental theorem of calculus*. Let *f* be a real-valued integrable function defined on a closed interval [*a*, *b*]. If *F* is a function such that *F* ′(*x*) = *f*(*x*) for all *x* in [*a*, *b*] (that is, *F* is an antiderivative of *f*), then

*Corollary*. If *f* is a continuous function on [*a*, *b*], then *f* is integrable on [*a*, *b*], and *F*, defined by ** Statements of theorems**

** Extensions**

*Main article: Improper integral* ** Improper integrals** *Main article: Multiple integral* ** Multiple integration** *Main article: Line integral* ** Line integrals** *Main article: Surface integral* ** Surface integrals** *Main article: differential form* ** Integrals of differential forms** ** Methods and applications** The most basic technique for computing integrals of one real variable is based on the

fundamental theorem of calculus. It proceeds like this:

Note that the integral is not actually the antiderivative, but the fundamental theorem allows us to use antiderivatives to evaluate definite integrals.

The difficult step is often finding an antiderivative of

*f*. It is rarely possible to glance at a function and write down its antiderivative. More often, it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include:

Even if these techniques fail, it may still be possible to evaluate a given integral. The next most common technique is

residue calculus, whilst for

nonelementary integrals Taylor series can sometimes be used to find the antiderivative. There are also many less common ways of calculating definite integrals; for instance,

Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see

Gaussian integral.

Computations of volumes of

solids of revolution can usually be done with

disk integration or

shell integration.

Specific results which have been worked out by various techniques are collected in the

list of integrals.

Choose a function

*f*(

*x*) and an interval [

*a*,

*b*].

Find an antiderivative of

*f*, that is, a function

*F* such that

*F'* =

*f*.

By the fundamental theorem of calculus, provided the integrand and integral have no

singularities on the path of integration,

Therefore the value of the integral is

*F*(

*b*) −

*F*(

*a*).

Integration by substitution Integration by parts Integration by trigonometric substitution Integration by partial fractions ** Computing integrals** *Main article: Symbolic integration* ** Numerical quadrature** Table of integrals - integrals of the most common functions.

Lists of integrals Multiple integral Antiderivative Numerical integration Integral equation Riemann integral Riemann sum Differentiation under the integral sign Product integral
**Growth capital** is a very flexible type of

financing. The money borrowed under a growth

capital line of

credit can be used for any

corporate purposes. There are no requirements to provide

invoices or other backup material when borrowing under this type of facility, so administration is simplified as well.

Growth capital can be a beneficial way to extend a company's runway between rounds of financing. The extra time can be used to complete additional milestones that will raise the company's valuation, or as insurance to ensure that all intended milestones are successfully accomplished.

The

**Court system of Canada** is made up of many

courts differing in levels of legal superiority and separated by jurisdiction. Some of the courts are

federal in nature while others are provincial or territorial.

The Canadian constitution gives the federal government the exclusive right to legislate criminal law while the provinces have exclusive control over civil law. The provinces have jurisdiction over the administration of justice in their territory. Almost all cases, whether criminal or civil, start in provincial courts and may be eventually appealed to higher level courts. The quite small system of federal courts only hear cases concerned with matters which are under exclusive federal control, such as immigration. The federal government appoints and pays for both the judges of the federal courts and the judges of the superior-level court of each province. The provincial governments are responsible for appointing judges of the lower provincial ("inferior-level") courts.

This intricate interweaving of federal and provincial powers is typical of the Canadian constitution.

** Outline of the Court system** Although created by an Act of the

Parliament of Canada in 1875, its decisions could be reviewed by the

Judicial Committee of the Privy Council until 1949 when the

Supreme Court of Canada truly became the final and highest court in the country. The court currently consists of nine justices, which include the

Chief Justice of Canada, and its duties include hearing appeals of decisions from the

appellate courts (to be discussed next) and, on occasion, delivering

references (i.e., the court's opinion) on constitutional questions raised by the federal government. By law, three of the nine justices are appointed from Quebec; because of Quebec's use of

civil law.

** Supreme Court of Canada** These courts of appeal (as listed below by province and territory in alphabetical order) exist at the provincial and territorial levels and were separately constituted in the early decades of the 20th century, replacing the former Full Courts of the old Supreme Courts of the provinces, many of which were then re-named Courts of Queens Bench. Their function is to review decisions rendered by the superior-level courts and to do references (i.e., deliver a judicial opinion) when requested by a provincial or territorial government. These appellate courts do not normally conduct trials and hear witnesses.

These courts are Canada's equivalent of the Court of Appeal in

England and the various

State Supreme Courts and

US Courts of Appeals in the

United States. Each of the above-listed appellate courts is the highest court from its respective province or territory. A province's chief justice (i.e., highest ranking judge) sits in the appellate court of that province.

Alberta Court of Appeal British Columbia Court of Appeal Manitoba Court of Appeal New Brunswick Court of Appeal Supreme Court of Newfoundland (Court of Appeal) Court of Appeal for the Northwest Territories Nova Scotia Court of Appeal Nunavut Court of Appeal Ontario Court of Appeal Supreme Court of Prince Edward Island - Appeal Division Quebec Court of Appeal Saskatchewan Court of Appeal Court of Appeal of the Yukon Territory ** Appellate courts of the provinces and territories** These courts (as listed below by province and territory in alphabetical order) exist at the provincial and territorial levels. The superior courts are the

courts of first instance for divorce petitions, civil lawsuits involving claims greater than small claims, and criminal prosecutions for "

indictable offences" (i.e., "

felonies" in American legal terminology). They also perform a reviewing function for judgements from the local "inferior" courts and administrative decisions by provincial or territorial government entities such as labour boards, human rights tribunals and licensing authorities.

Furthermore, some of these superior courts (like the one in Ontario) have specialized branches that deal only with certain matters such as family law or small claims. To complicate things further, the Ontario Superior Court of Justice has a branch called the Divisional Court that hears only appeals and judicial reviews of administrative tribunals and whose decisions have greater binding authority than those from the "regular" branch of the Ontario Superior Court of Justice. Although a court, like the Supreme Court of British Columbia, may have the word "supreme" in its name, it is not necessarily the highest court from its respective province or territory.

Court of Queen's Bench of Alberta Supreme Court of British Columbia Court of Queen's Bench of Manitoba Court of Queen's Bench of New Brunswick Supreme Court of Newfoundland and Labrador (Trial Division)

Supreme Court of the Northwest Territories Supreme Court of Nova Scotia Nunavut Court of Justice Ontario Superior Court of Justice Supreme Court of Prince Edward Island - Trial Division

Quebec Superior Court Court of Queen's Bench for Saskatchewan Supreme Court of the Yukon Territory ** Superior-level courts of the provinces and territories** *Main article: Provincial Court* ** Provincial and territorial ("inferior") courts** The Federal Court and the more specialized Tax Court of Canada exists primarily to review administrative decisions by federal government bodies such as the

immigration board and hear lawsuits under the federal government's jurisdiction such as

intellectual property and

maritime law.

The Federal Court of Appeal hears appeals from decisions rendered by the Federal Court, the Tax Court of Canada and a certain group of federal administrative tribunals like the National Energy Board and the federal labour board. All judges of the Federal Court are

*ex officio* judges of the Federal Court of Appeal, and vice versa, although it is rare that a judge of one court will sit as a member of the other.

Before 2003, the Federal Court was known as the Federal Court of Canada - Trial Division while the Federal Court of Appeal was known as the Federal Court of Canada - Appeal Division. In turn, the Federal Court of Canada is descended from the old Exchequer Court of Canada created back in 1875.

Although the federal type courts can be said to have the same prestige as the superior courts from the provinces and territories, the federal ones lack the "inherent jurisdiction" (to be explained later) possessed by superior courts such as the Ontario Superior Court of Justice.

Federal Court

Tax Court of Canada

Federal Court of Appeal

** Courts of the federal level** The "courts martial" are conducted and presided over by military personnel and exist for the prosecution of military personnel, as well as civilian personnel who accompany military personnel, accused of violating the

*Code of Service Discipline*, which is found in the

*National Defence Act* (R.S.C. 1985, Chapter N-5) and constitutes a complete code of military law applicable to persons under military jurisdiction.

The decisions of the courts martial can be appealed to the Court Martial Appeal Court of Canada which, in contrast, exists outside the military and is made up of civilian judges. This appellate court is the successor of the Court Martial Appeal Board which was created in 1950, presided over by civilian judges and lawyers, and was the first ever civilian-based adjudicating body with authority to review decisions by a military court. The Court Martial Appeal Court is made up of civilian judges from the Federal Court, Federal Court of Appeal, and the superiour courts of the provinces. The current Chief Justice of the Court Martial Appeal Court (as of September 17, 2004) is Edmond P. Blanchard.

Court Martial Appeal Court of Canada

various military courts called "courts martial"

- General Court Martial

Disciplinary Court Martial

Standing Court Martial

Special General Court Martial ** Courts of military law**

Known in Canada as simply "tribunals", these are non-judicial adjudicative bodies, which means that they adjudicate (hear evidence and render decisions) like the courts do BUT are not presided over by judges. Instead, the adjudicators may be experts of the very specific legal field handled by the tribunal (e.g., labour law, human rights law, immigration law, energy law, liquor licensing law, etc.) who hear arguments and evidence provided by lawyers before making a written decision on record. Its decisions can be reviewed by a court through an appeal or a process called "judicial review". The reviewing court may be required to show some deference to the tribunal if the tribunal possesses some highly specialized expertise or knowledge that the court does not have. The degree of deference will also depend on such factors as the specific wording of the legislation creating the tribunal.

Tribunals may take into consideration the Canadian Charter of Rights and Freedoms, which is part of Canada's constitution. The extent to which tribunals may use the Charter in their decisions is a source of ongoing legal debate.

Appearing before some administrative tribunals may feel like appearing in a court, but the tribunal's procedure is relatively less formal than that of the court, and more importantly, the rules of evidence are not as strictly observed. In other words, some evidence that would be inadmissible in a court hearing could be allowed in a tribunal hearing. The presiding adjudicator is normally called "Mister/Madam Chair", and lawyers routinely appear in tribunals advocating a matter for their clients. A person does not require a lawyer to appear before an administrative tribunal. Indeed, many of these tribunals are specifically designed to be less formal than courts. Furthermore, some of these tribunals are part of a comprehensive dispute-resolution system, which may emphasize mediation rather than litigation. For example, provincial human rights commissions routinely use mediation to resolve many human rights complaints without the need for a hearing.

What tribunals all have in common is that they are created by statute, their adjudicators are appointed by government, and they focus on very particular and specialized areas of law. Because some subject matters (e.g., immigration) fall within federal jurisdiction while others (e.g., liquor licensing) in provincial jurisdiction, some tribunals are created by federal law while others are created by provincial law. Yet, there are both federal and provincial tribunals for some subject matters such as unionized labour and "human rights" (in American legal parlance, the "civil rights" of marginalized or/and disadvantaged social groups such as women, racial minorities, the disabled, homosexuals, certain religious groups, etc.).

Most importantly, from a lawyer's perspective, is the fact that the principle of stare decisis does not apply to tribunals. In other words, a tribunal adjudicative could legally make a decision that differs from a past decision, on the same subject and issues, delivered by the highest court in the land. Because a tribunal is not bound by legal precedent, established by itself or by a reviewing court, A tribunal is not court even though it performs an important adjudicative function and contributes to the development of law like a court would do. Although stare decisis does not apply to tribunals, their adjudicators will nonetheless find a prior court decision on a similar subject to be highly persuasive and will likely follow the courts in order to ensure consistency in the law and to prevent the embarrassment of having their decisions overturned by the courts.

Among the federal tribunals, there is a small group of tribunals whose decisions must be appealed directly to the Federal Court of Appeal rather than to the Federal Court Trial Division. These so-called "super tribunals" are listed in Subsection 28(1) of the *Federal Court Act* (R.S.C. 1985, Chapter F-7) and some examples include the National Energy Board, Canadian International Trade Tribunal, the Competition Tribunal, the Canada Industrial Relations Board (i.e. federal labour board), the Copyright Board, and the Canadian Radio-television and Telecommunications Commission ("CRTC").

** Federal and provincial administrative tribunals**

These are the superior courts from the provinces and territories as discussed above. The words "inherent jurisdiction" refers to the fact that the jurisdiction of the superior courts is more than just what is conferred by statute. Following the principles of English common law, because the superior courts derive their authority from the Constitution, they can hear any matter unless there is a federal or provincial statute that says otherwise or that gives exclusive jurisdiction to some other court or tribunal. The doctrine of "inherent jurisdiction" gives superior courts greater freedom than statutory courts (to be explained next) to be flexible and creative in the delivering of legal remedies and relief.

** Courts of inherent jurisdiction**

These courts include the Supreme Court of Canada, the different types of federal courts, the various appellate courts from the provinces and territories, and the numerous low level "provincial" courts. Their decision-making power is granted by either the federal parliament or a provincial legislature.

The word "statutory" refers to the fact that these courts' powers are derived from a type of legislation called a statute and is defined and limited by a statute. A statutory court cannot try cases in areas of law that are not mentioned or suggested in the statute. In this sense, statutory courts are similar to non-judicial adjudicative bodies such as administrative tribunals, boards, commissions, etc. which are created and given limited power by legislation. The practical implication of this is that a statutory court cannot provide a type of legal remedy or relief that is not expressly or implicitly referred to in its enabling or empowering statute.

** Statutory courts**

*Main article: Judicial appointments in Canada*

**Purity** is the state of being

pure; the opposite of purity is

impurity.

**Purity** may refer to:

Morality, the concept of human ethics

Black Oil virus, codenamed

*Purity* in

*The X-files* Purity Supermarkets, a brand formerly used for a supermarket chain by

Woolworths Limited in Tasmania, Australia

Purity Distilling Company, the company responsible for the Boston molasses disaster

"Purity", a song by the band Slipknot on their self-titled album

*Slipknot*
In

Hinduism,

**Durga** (

Sanskrit:

"the inaccessible"

Durga is depicted as a warrior woman riding a lion or a tiger with multiple hands carrying weapons and assuming mudras, or symbolic hand gestures. This form of the Goddess is the embodiment of feminine and creative energy (Shakti).

** Durga in the Hindu tradition**

The 4 day Durga Puja is the biggest annual festival in Bengal and other parts of Eastern India, but it is celebrated in various forms throughout the Hindu universe.

The day of Durga's victory is celebrated as *Vijaya Dashami* (East and South India), Dashain (Nepal) or Dussehra (North India) - these words literally mean "the tenth" (day), vijaya means "of-victory". In Kashmir she is worshipped as *shaarika* (the main temple is in Hari Parbat in Srinagar).

The actual period of the worship however may be on the preceding nine days followed by the last day called *Vijayadashami* in North India or five days in Bengal, (from the sixth to tenth day of the waxing-moon fortnight). Nine aspects of Durga known as Navadurga are meditated upon, one by one during the nine-day festival by devout shakti worshippers.

In North India, this tenth day, signifying Rama's victory in his battle against the demon Ravana, is celebrated as *Dussehra* - gigantic straw effigies of Ravana are burnt in designated open spaces (e.g. Delhi's Ram Lila grounds), watched by thousands of families and little children.

In Gujarat it is celebrated as the last day of Navaratri, during which the Garba dance is performed to celebrate the vigorous victory of **Mahishasura-mardini** Durga.

The Goddess Durga worshipped in her peaceful form as Shree Shantadurga also known as santeri , is the patron Goddess of Goa. She is worshipped by all Goan Hindus irrespective of caste and even by some Christians in Goa.

Goddess Durga is worshipped in many temples of Dakshina Kannada district of Karnataka.

** See also**

The

**European Patent Organisation** (

**EPO** or

**EPOrg** The European Patent Organisation is not legally bound to the

European Union and has several members which are not themselves EU states.

The evolution of the Organisation is inherently linked to the European Patent Convention. See

European Patent Convention for the history of the European Patent system as set up by the European Patent Convention and operated by the European Patent Office.

European Patent Convention Biotech directive Community patent Brussels Regime European Patent Litigation Agreement (EPLA)

London Agreement ** Organs** The

**European Patent Office** (

**EPO** or

**EPOff**.

** European Patent Office** The European Patent Office is directed by a president, who is responsible for its activities to the Administrative Council.

Presidents of the European Patent Office:

Johannes Bob van Benthem (

1 November 1977 -

30 April 1985),

Dutch Paul Brändli (

1 May 1985 -

31 December 1995),

Swiss Ingo Kober (

1 January 1996 -

30 June 2004),

German Alain Pompidou (

1 July 2004 -

30 June 2007),

French Alison Brimelow (

1 July 2007 -

30 June 2010),

British ** President** The official languages of the European Patent Office are

English,

French and

German. However, other languages than these three are not all considered on the same footing. Non-admissible languages, such as

Japanese or

Chinese, should be distinguished from the "

*admissible non-EPO languages*", such as

Spanish,

Italian,

Dutch and any language that is at least an official language in one Contracting State. European patent applications can be validly filed by some applicants in an admissible non-EPO language provided that a translation is filed thereafter, while they cannot be validly filed in

Chinese or

Japanese whether a translation is filed thereafter or not.

** Languages** The European Patent Office includes the following departments, pursuant to Art. 15 EPC: a

**Receiving Section**, responsible for the examination on filing and the examination as to formal requirements of European patent applications,

**Examining Divisions**, responsible for prior art searches and the examination of European patent applications,

**Opposition Divisions**, responsible for the examination of

oppositions against any European patent, a

**Legal Division**,

**Boards of Appeal**, responsible for the examination of

appeals, and an

**Enlarged Board of Appeal** (see also:

Appeal procedure before the European Patent Office). In practice, the above departments of European Patent Office are organized into five "Directorates-General" (DG), each being directed by a Vice-President:

**DG 1** Operations,

**DG 2** Operational Support,

**DG 3** Appeals,

**DG 4** Administration, and

**DG 5** Legal/International Affairs.

The European Patent Office does not include any

court which can take decisions on

infringement matter. National

jurisdictions are competent for infringement matter regarding European patents.

** Departments and Directorates-General** The European Patent Office acts as a Receiving Office, an International Searching Authority and an International Preliminary Examining Authority in the international procedure according to the

Patent Cooperation Treaty (PCT). The Patent Cooperation Treaty provides an international procedure for dealing with patent applications, called international applications, during the first 30 months after their first filing in any country. The European Patent Office does not grant international patents - which do not exist. After 30 months an international application must be converted into national or regional patent applications, and then are subject to national/regional grant procedures.

** Other activities** The Administrative Council is made up of members of the contracting states and is responsible for overseeing the work of the European Patent Office,

** Administrative Council** There are, as of

June 15,

2007, 32 Contracting States to the EPC, also called member states of the European Patent Organisation: Slovenia, Romania, Lithuania and Latvia were all extension states prior to joining the EPC.

** Statistics**
**Montevideo** (

IPA:

[monteβi'deo]) is the largest city, capital and chief port of

Uruguay. Montevideo is the

primate city in Uruguay, the only city in country with a population over 100,000. Montevideo has a privileged harbour, one of the most important in the

Americas. Also, it has beautiful beaches, like Pocitos, Buceo, Malvin, Playa de los Ingleses, Playa Verde, Punta Gorda and Carrasco. Many monuments and museums cover the city, as well as historic buildings and squares. The city's mayor is

Ricardo Ehrlich. According to the Mercer Human Resource Consulting, Montevideo is the

Latin American city with the highest quality of life (followed by

Buenos Aires and

Santiago de Chile).

** Population** Montevideo is situated in the south of the country, The geographic coordinates are 34.5° S, 56°W.

*18 de Julio* is the city's main avenue and extends from the

Plaza Independencia, which is the junction between the

*Ciudad Vieja* (the historical quarter) and the rest of the city, to the neighbourhood of

*Cordón*.

** Geography** ** History** There are at least two explanations for the name

*Montevideo*: The first states that it comes from the

Portuguese "Monte vide eu" which means "I see a mountain". The second is that the Spaniards recorded the location of a mountain in a map as "Monte VI De Este a Oeste" meaning "The sixth mountain from east to west". The city's full original name is

**San Felipe y Santiago de Montevideo**.

** Origin of Name** Montevideo was first found by Juan Diaz De Solis. He arrived in 1516. He encountered the natives living there, and was killed by them, along with the rest of his group of travelers. The

Portuguese founded

Colonia del Sacramento in the

17th century despite Spanish claims to the area due to the

Treaty of Tordesillas. The

Spanish chased the Portuguese out of a fort in the area in

1724. Then,

Bruno Mauricio de Zabala –

governor of Buenos Aires – founded the city on

December 24,

1726 to prevent further incursions.

In

1828, the town became the capital of Uruguay.

The city fell under heavy

British influence from the early

19th century until the early

20th century as a way to circumvent

Argentine and

Brazilian commercial control. It was repeatedly besieged by Argentinean dictator

Juan Manuel de Rosas between

1838 and

1851. Between

1860 and

1911, British Owned Railway companies built an extensive

railroad network linking the city to the surrounding countryside.

** Early History** During

World War II, a famous incident involving the

German pocket battleship Admiral Graf Spee took place in Punta del Este, 200 km from Montevideo. After the

Battle of the River Plate with the

British navy on

December 13,

1939, the Graf Spee retreated to Montevideo's port, which was considered neutral then. To avoid risking the crew in what he thought would be a losing battle, Captain

Hans Langsdorff scuttled the ship on

December 17. Langsdorff committed suicide two days later. On

10 February 2006, the eagle figurehead of the

*Admiral Graf Spee* was

recovered. To protect the feelings of those still sensitive to Nazi Germany, the swastika on the figurehead was covered as it was pulled from the water.

Since

2005 the

Mayor of Montevideo (styled

*Intendente Municipal* in Spanish) has been

Ricardo Ehrlich, of the

Frente Amplio (Broad Front), gaining 61% of the vote in the Mayoral elections, beating

Pedro Bordaberry of the

Partido Colorado, who scored 27%.

** 20th Century** Montevideo began as a minor settlement. In

1860, Montevideo had a population of 37,787. By

1884, the population had grown to 104,472, including many immigrants.

During the mid-20th century,

military dictatorship and

economic stagnation caused a decline whose residual effects are still seen today. Many rural poor flooded the city, with a large concentration in Ciudad Vieja.

Recently, economic recovery and stronger trade ties with Uruguay's neighbours have led to renewed agricultural development and hopes for greater future prosperity.

As of

2004, the city has a population of 1.35 million out of a total of 3.43 million in the country as a whole.

[1] The greater

metropolitan area has 1.8 million people.

Montevideo is served by

Carrasco International Airport.

Template:Panorama2 ** Growth/economy** The city shows some evidence of

world city formation. The past lives on in style, though. Back in 1870, the average living standards were similar to those in the United States.

** Neighbourhoods** Instituto Preuniversitario JUAN XXIII Stella Maris College (Montevideo) The British Schools of Montevideo University of the Republic, Uruguay ORT Uruguay Lycée Français Scuola Italiana Deutsche Schule Public Primary School Public Secondary School Catholic University, Uruguay Instituto Preuniversitario de Montevideo - PRE/U

** Education** Montevideo hosted all the matches of the

1 FIFA World Cup in

1930. Uruguay won the tournament by defeating Argentina 4-2, and later in

1950 defeated heavy favored hosts Brazil 2-1, achieving its second World Cup Championship.

Uruguay also won several olympic medals in soccer including 2 gold medals, and has won the most "

Copa America" tournaments, the world's second most prestigious tournament after the World Cup (tied with Argentina at 14 wins). Its

Estadio Centenario is considered a temple of world football. The city is home to two of the most important South American football clubs:

Nacional and

Peñarol. Consistently since its early successes, Uruguayan footballers have been among the worlds best, recently producing such soccer greats as

Diego Forlan,

Paulo Montero and

Alvaro Recoba, and currently boasts the highest number of world class exports to the European leagues out of any country in South America including fellow soccer powerhouses Argentina and Brazil, a surprising feat when considering the population of just over 3 million is a small fraction of most other South American countries.

There is strong world-wide sentiment that the centennial anniversary Fifa World Cup tournament to be held in 2030, be played at least in part in Uruguay, namely in the Centenario Stadium which was built specifically as the innagurational stadium for the tournament of 1930, and fittingly translates to "The Centennial Stadium".

** Sites of interest** Québec City,

Canada Barcelona,

Spain Montevideo,

USA Miami,

USA Curitiba,

Brazil
**Wenzhou** (

Simplified Chinese:

温州;

Traditional Chinese:

溫州;

pinyin:

Wēnzhōu) is a

prefecture-level city (or provincial subregion) in southeastern

Zhejiang province,

People's Republic of China. It has a population of 7,777,000 in 2006, with 1,336,000 residents in the 3

districts (区) of the city with the same name. It also contains 2 more cities and 6 counties. It borders

Lishui to the west,

Taizhou to the north, and looks out to the

East China Sea to the east.

Wenzhou was a prosperous foreign treaty port, which remains well-preserved today. It is also known for its emigrants who leave their native land for

Europe and the

United States, with a reputation for being enterprising folk who start restaurants, retail and wholesale businesses in their adopted countries.

** Administration** Wenzhou, also known as Yongjia (or Yung-chia) has a history which goes back to about

2000 BC, when it became known for its pottery production. In the

2nd century BC it was called the Kingdom of

Dong'ou. Under the

Tang Dynasty, it was promoted to prefecture status and given its current name in

675 AD.

Throughout its history, Wenzhou's traditional economic role has been as a port giving access to the mountainous interior of southern Zhejiang Province. In

1876 Wenzhou was opened to the foreign

tea trade, but no foreign settlement was ever made there. In

1937–

1942 during the war with

Japan, Wenzhou became an important port due to its being one of the few Chinese ports still under Chinese control. It declined in the later years of the war but began to recover after coastal trade along the Zhejiang coast was re-established in

1955.

Wenzhou derives its present name from its mild climate. With jurisdiction over three districts, two county-level cities and six counties, Wenzhou covers a land area of 11,784 square kilometers and sea area of 11,000 square kilometers. The population of the prefectural level city is 7.7 million including 2 million urban residents, divided among 2 "county level" cities and 3 wards.

As a coastal city, Wenzhou is rich in natural resources. The 339-kilometer long coastline gives the city abundant marine resources and many beautiful islands. Dongtou, one of the counties in Wenzhou, is also called the "County of one hundred islands". Wenzhou also boasts wonderful landscapes with rugged mountains and tranquil waters, including three state-level scenic spots, namely the Yandang Mountain, the Nanxi River and the Baizhangji Fall-Feiyun Lake, and two national nature reserves, namely the Wuyanling Ridge and the Nanji Islands, among which Yandang Mountain has been named as World Geopark, while Nanji Islands listed as UNESCO's Marine Nature Reserve of World Biosphere Reserves. Scenic area accounts for 25% of the city's land space, which is a perfect integration of exotic mountains, tender water and charming sea.

** History** Wenzhou exports food,

tea,

wine,

jute,

timber,

paper,

Alunite (a non-metallic mineral used to make alum and fertilizer).

Alunite is quite abundant here and sometimes Wenzhou claims to be the "Alunite Capital of the World". Its main industries are food processing, papermaking, and building materials, with some engineering works producing mostly farm machinery. From the 1990s, low-voltage electric appliances manufacturing became a major industry in Wenzhou, with some of the large private enterprises setting up joint ventures with GE and Schneider. Since 1994, exploration for

oil and

natural gas has commenced in the

East China Sea 100 km off the coast of Wenzhou. Companies such as Texaco, Chevron, Shell and Japex have started to drill for oil but the operations have been largely unsuccessful.

Since the new government term started in 2004, the local government has initiated a brand-new development strategy of inviting investment from international market, which is dubbed as "Number 1 Project" of the city. It is a particularly bold endeavour for Wenzhou, especially with the backdrop of declining FDI in the national level.

Wenzhou is a city full of vitality, with creativity as the source of its vitality. Since China adopted the opening up and reform policy, Wenzhou has been the first to set up individual and private enterprises as well as shareholding cooperative economy in China. It has also taken the lead in carrying out the financial system reform and the structural reform in townships. Being a pioneer in utilizing marketing mechanism to develop urban constructions, Wenzhou has won a number of firsts in China and set many national records.

Vitality comes from Wenzhou natives. Without much dependence in state investments, the development of the city really lies on the efforts of the natives. Vitality results from business culture, which is the top feature of Wenzhou's economy. Wenzhou businessmen have set their feet on the way of accumulating capital and also made themselves one of the important forces of the overseas Chinese businessmen. "Big market with small commodities, small money with high capital intensity" has become the prominent character of Wenzhou's economy. Vitality also originates from opening up of market. In recent years, Wenzhou has continuously deregulated to embrace foreign investment and opened more widely to the outside world, encouraging the aspiring spirit of the local people to start-up businesses. With enduring vitality and sustained innovation, the economy of Wenzhou has always enjoyed healthy development. From 1978 through 2005, the GDP of the city has increased from 1.32 billion RMB to 160 billion RMB with the gross fiscal revenue increasing from 0.135 billion RMB to 20.49 billion RMB, and the net per capita income for rural residents increasing from 113.5 RMB to 6,845 RMB. What's more, the per capita disposable income for urban residents increased from 422.6 RMB in 1981 to 19,805 RMB in 2005.

Wenzhou is the birthplace of the China private economy. In the early days of opening up and reform, the people of Wenzhou took the lead in developing commodity economy, household industries and specialized markets. Thousands upon thousands of people and families were engaged in household manufacturing to develop individual and private economy. Up till now, Wenzhou has a total of 240,000 individually-owned commercial and industrial units and 130,000 private enterprises of which 180 are group companies, 4 among China's top 500 enterprises and 36 among national 500 top private enterprises. The quantity, industrial output, tax, export and number of employees of the private enterprises account for 99%, 96%, 75%, 95% and 80% of the whole city respectively. There are 27 national production bases such as "China's Shoes Capital" and "China's Capital of Electrical Equipment", 40 China's famous trademarks and China's famous-brand products and 67 national inspection-exempt products in the city. The development of private economy in Wenzhou has created the "Wenzhou Economic Model", which gives great inspiration to the modernization drive in China.

**Habitat destruction** is a process of

land use change in which one

habitat-type is removed and replaced with another habitat-type. In the process of land-use change,

plants and

animals which previously used the site are displaced or destroyed, reducing

biodiversity.

Urban Sprawl is one cause of habitat destruction. Other important causes of habitat destruction include

mining,

trawling, and

agriculture. Habitat destruction is currently ranked as the most important cause of species extinction worldwide.