Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filter and Cauchy net.

The circular Cauchy distribution expressed in complex form has finite moments of all orders

First, the Cauchy – Riemann equations may be written in complex form

The finite-dimensional case of this inequality for real vectors was proved by Cauchy in 1821, and in 1859 Cauchy's student Bunyakovsky noted that by taking limits one can obtain an integral form of Cauchy's inequality.

When n = 3 the Cauchy – Schwarz inequality can also be deduced from Lagrange's identity, which takes the form

Cauchy gave a form of the ( ε, δ )- definition of limit, in the context of formally defining the derivative, in the 1820s ,< ref >

Augustin-Louis Cauchy exploited infinitesimals in defining continuity and an early form of a Dirac delta function.

The special case p = q = 2 gives a form of the Cauchy – Schwarz inequality.

The determinant can be expressed in closed form, as a special case of the Cauchy determinant.

By the Cauchy – Schwarz inequality, the degenerate elements, x in A satisfying ρ ( x * x )= 0, form a vector subspace I of A.

Note also that a factor consisting of a sum where both types of variables are involved ( e. g. a factor of the form ) cannot be factorized in this fashion ( except in some cases where occurring directly in an exponent ); this is why, for example, the Cauchy distribution and Student's t distribution are not exponential families.

If the boundary has the form of a curve or surface that gives a value to the normal derivative and the problem itself then it is a Cauchy boundary condition.

Then a Cauchy – Euler equation of order n has the form

This theorem by Cauchy was only published many years later in 1974 in a hand-written form and so is quite difficult to read.

Omega Cauchy ( ω ) and Tau Cauchy ( τ ) form a pair of domes near the crater Cauchy.

These form thirteen established families ( plus perhaps Shompen, which is poorly attested, as a fourteenth ), which have traditionally been grouped into two, as Mon – Khmer and Munda.

In sum, the idea was that Turkic, Mongolic, and Tungusic form a Sprachbund – the result of convergence through intensive borrowing and long contact among speakers of languages that are not necessarily closely related.

" This was thought to favour the doctrine of absolute non-resistance, and accordingly the convention parliament enacted the form that has been in use since that time – " I do sincerely promise and swear that I will be faithful and bear true allegiance to His Majesty ..."

As a compromise, the MCC commissioned a trophy in the form of a larger replica of the urn in Waterford Crystal to award to the winning team of each series from 1998 – 99.

* 1957 – The United States and Canada form the North American Air Defense Command ( NORAD ).

* 1964 – Tanganyika and Zanzibar merge to form Tanzania.

* 1967 – Vietnam War: China agrees to give North Vietnam an undisclosed amount of aid in the form of a grant.

Some early scientists such as Georg Ernst Stahl ( 1659-1734 ) and Francisque Bouillier ( 1813 – 1899 ) had supported a form of animism which life and mind, the directive principle in evolution and growth, holding that all cannot be traced back to chemical and mechanical processes, but that there is a directive force which guides energy without altering its amount.

* 1939 – The United Kingdom and Poland form a military alliance in which the UK promises to defend Poland in case of invasion by a foreign power.

* Beijing beef – This dish exists in native Chinese form, but using gai-lan ( Chinese broccoli ) rather than American broccoli.

As president, he implemented his own form of Presidential Reconstruction – a series of proclamations directing the seceded states to hold conventions and elections to re-form their civil governments.

As president, he implemented his own form of Presidential Reconstruction – a series of proclamations directing the seceded states to hold conventions and elections to re-form their civil governments.

Conversely, some elements do not maintain distinct allotropes in different phases – for example phosphorus has numerous solid allotropes, which all revert to the same P < sub > 4 </ sub > form when melted to the liquid state.

In English, which has mostly lost the case system, the definite article and noun – " the car " – remain in the same form regardless of the grammatical role played by the words.

Roma returned to form after being inconsistent for much of the late 1930s ; Roma recorded an unexpected title triumph in the 1941 – 42 season by winning their first ever scudetto title.

This became the dominant form in the Restoration, when composers such as Henry Purcell ( 1659 – 1695 ) and John Blow ( 1649 – 1708 ) wrote elaborate examples for the Chapel Royal with orchestral accompaniment.

Chemically, Agent Orange is an approximately 1: 1 mixture of two phenoxyl herbicides – 2, 4-dichlorophenoxyacetic acid ( 2, 4-D ) and 2, 4, 5-trichlorophenoxyacetic acid ( 2, 4, 5-T ) – in iso-octyl ester form.

As defined by Arrhenius, acid – base reactions are characterized by Arrhenius acids, which dissociate in aqueous solution to form hydrogen ions (), and Arrhenius bases, which form hydroxide () ions.

These axons transmit signals in the form of electrochemical pulses called action potentials, which last less than a thousandth of a second and travel along the axon at speeds of 1 – 100 meters per second.

Moreover, this disposition – concave in relation to the Allied army – gave Marlborough the opportunity to form a more compact line, drawn up in a shorter front between the ‘ horns ’ of the French crescent ; when the Allied blow came it would be more concentrated and carry more weight.

Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ ( s ) is non-zero for all complex values of the variable s that have the form s = 1 + it with t > 0.

The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

A Riemannian manifold is a space form if its sectional curvature is equal to a constant K. The Riemann tensor of a space form is given by

* Prime form of a Riemann surface

One of his results was a 1901 theorem proving that the Riemann hypothesis is equivalent to a stronger form of the prime number theorem.

* Quadratic differential, a form on a Riemann surface that locally looks like the square of an abelian differential

Such a form on X is usually called a ( non-degenerate ) Riemann form.

Initially proved as Riemann's inequality by, the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student.

On a general Riemann surface of genus g, K has degree 2g − 2, independently of the meromorphic form chosen to represent the divisor.

Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces ( the complex plane and the hyperbolic plane ).

The form of the Riemann – Roch theorem on a surface was also worked out.

Note that this latter form is the functional equation for the Riemann zeta function, as originally given by Riemann.

An L-function L ( E, s ) can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p. This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form.

Titchmarsh ( 1960 ) provided a Trace formula involving a sum over the Möbius function and zeros of Riemann Zeta in the form

The functional equation in question for the Riemann zeta function takes the simple form

This is simply the Riemann curvature tensor in a different form.

Hasse's theorem is equivalent to the determination of the absolute value of the roots of the local zeta-function of E. In this form it can be seen to be the analog of the Riemann hypothesis for the function field associated with the elliptic curve.

Here, ζ ( s ) is the usual Riemann zeta function and L ( s, E ) is called the L-function of E / Q, which takes the form

Examples of Lambert series in this form, with, occur in expressions for the Riemann zeta function for odd integer values ; see Zeta constants for details.

The complex numbers, together with a special point designated as ∞, form a topological space known as the Riemann sphere.