* Schwarz lemma, a mathematical result about holomorphic functions

** Schwarz – Ahlfors – Pick theorem, an extension of the Schwarz lemma for hyperbolic manifolds

** Schwarz lemma

In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself.

# REDIRECT Schwarz lemma

In mathematics, the Schwarz – Ahlfors – Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model.

The Schwarz – Pick lemma states that every holomorphic function on the unit disk U, or the upper half-plane H, with distances defined by the Poincaré metric, is a contraction mapping.

# REDIRECT Schwarz lemma

# REDIRECT Schwarz lemma

A firmly non-expansive mapping is always non-expansive, via the Cauchy – Schwarz inequality.

A common German folk-tale is of the German priest / monk named Berthold Schwarz who independently invented gunpowder, thus earning it the German name Schwarzpulver or in English Schwarz's powder.

Schwarz is also German for black so this folk-tale, while likely containing elements of truth, is considered problematic.

In mathematics, the Cauchy – Schwarz inequality ( also known as the Bunyakovsky inequality, the Schwarz inequality, or the Cauchy – Bunyakovsky – Schwarz inequality, or Cauchy – Bunyakovsky inequality ), is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas.

The Cauchy – Schwarz inequality states that for all vectors x and y of an inner product space it is true that

In Euclidean space R < sup > n </ sup > with the standard inner product, the Cauchy – Schwarz inequality is

The triangle inequality for the inner product is often shown as a consequence of the Cauchy – Schwarz inequality, as follows: given vectors x and y:

The Cauchy – Schwarz inequality proves that this definition is sensible, by showing that the right hand side lies in the interval, and justifies the notion that ( real ) Hilbert spaces are simply generalizations of the Euclidean space.

The Cauchy – Schwarz is used to prove that the inner product is a continuous function with respect to the topology induced by the inner product itself.

The Cauchy – Schwarz inequality is usually used to show Bessel's inequality.

One can see from the proof that the Cauchy – Schwarz inequality is a consequence of the positivity and anti-symmetry inner-product axioms.

This is precisely the Cauchy – Schwarz inequality.

When Φ is 2-positive, a stronger assumption than merely positive, one has something that looks very similar to the original Cauchy – Schwarz inequality:

The general formulation of the Heisenberg uncertainty principle is derived using the Cauchy – Schwarz inequality in the Hilbert space of quantum observables.

An English translation of part of Slavata's report of the incident is printed in Henry Frederick Schwarz, The Imperial Privy Council in the Seventeenth Century ( Cambridge, Mass.

German anatomist Ernst Schwarz is credited with having discovered the bonobo in 1928, based on his analysis of a skull in the Tervuren museum in Belgium that previously had been thought to have belonged to a juvenile chimpanzee.

Schwarz ( an adjective ) is German for " black ".

In the first episode of the animated series, the Dark Side is called " The Schwarz side of the Schwartz ".

Yet another explanation is that it is a reference to FAO Schwarz, which sells lots of movie merchandise.

It is then enough to minimize the functional E, owing to the Cauchy – Schwarz inequality

Jonathan Schwarz of Mother Jones criticized the exhibit and called it an example of corporate propaganda from Rupert Murdoch's News Corporation.

Teenburger also provided record sleeve designs for the bands Brinsley Schwarz and Red Dirt, as well as Vertigo artists such as Cressida, Gracious!

The word brane, derived from " membrane ", refers to a variety of interrelated objects, such as D-branes, black p-branes and Neveu – Schwarz 5-branes.

This helped him survive the hostility of other party officials such as the party treasurer, Franz Xaver Schwarz, who regarded him as a drunken incompetent.

They were soon joined by a handful of London acts such as Bees Make Honey, Max Merritt and the Meteors, Ducks Deluxe, The Amber Squad and Brinsley Schwarz who had been victims of the prevailing big-venue system.

In addition to the albums, Brinsley Schwarz also issued a series of singles under their own name, and various pseudonyms, such as " The Hitters ", " The Knees ", " Limelight " and " The Brinsleys " but these all failed.

Eggs over Easy had a large repertoire of over 50 of their own songs, and 50 covers, and were regularly joined on stage by members of Brinsley Schwarz and other performers, such as Loudon Wainwright III and Frankie Miller.

One of the former Tally Ho jazz players, Barry Richardson, was so impressed he formed Bees Make Honey, who were also given a Tally Ho residency, alongside a mix of new and existing bands, such as Max Merritt and the Meteors and Brinsley Schwarz.

* On April 28, 2009, Pennsylvania Senator Arlen Specter specifically mentioned the Club for Growth's previous activity in supporting conservative primary opponents against Republicans such as Wayne Gilchrest, Joe Schwarz, Lincoln Chafee and Heather Wilson, in his decision to run for re-election as a Democrat in 2010.

For indefinite systems, preconditioners such as incomplete LU factorization, additive Schwarz, and multigrid perform poorly or fail entirely, so the problem structure must be used for effective preconditioning.

That same year, Lionel made a bigger push to sell its train sets outside of hobby shops, selling them in stores such as FAO Schwarz, Macy's, and Target.

He also continues to run his indie record label Midi: Nette, and has produced for artists such as Schwarz Stein and Kanon Wakeshima.

Goebbels ' attempts to fill the space between theatre and opera left blank due to traditional genre borderline drawing has led to projects such as Schwarz auf Weiss ( Black on White, 1996 ) or Die Wiederholung ( The Repetition, 1997 ).

Academic contacts in Germany: Carathéodory's contacts in Germany were many and included such famous names as: Minkowski, Hilbert, Klein, Einstein, Schwarz, Fejér.

During Malan's tenure in parliament as defense minister his greatest opposition came from MPs of the Progressive Federal Party such as Harry Schwarz and Philip Myburgh, who both served as shadow defense ministers at various points during the 1980s.

According to Schwarz, in theory this seems like a great plan, but with football season over for a large majority of the high schools, ( football is the number one sport of concussions ) experts are finding that athletes have learned a couple of ways to get around the standards such as denying any concussion symptoms they are having, learning how to answer questions to hide any signs of concussions, or not saying anything about the possible concussion to the athletic trainer or physician working the game.

In Japan there are several musicians in the visual kei scene, such as Mana ( Moi dix Mois and Malice Mizer ), Kaya ( Schwarz Stein ), Hizaki and Jasmine You ( both Versailles ), who always or usually appear in full or semi-drag.

He has worked under such great conductors as Fabio Luisi, Fabio Mechetti, Giancarlo Guerrero, Lucas Foss, Eri Klas and Pinchas Steinberg, Gerard Schwarz, Lucas Vis, Jorge Mester, Theo Alcantara, Robert Bernhardt, JoAnn Falletta, Maximiano Valdes, Andrew Sewell, Bruce Ferden, Giséle Ben-dor, Christopher Wilkins, Enrique García Ascensio, Andrea Quinn, Kirk Trevor, Juan José García Caffi, Peter Jaffe, Arthur Fagen, David Alan Miller and Marvin Hamlisch.

From around 1970, competition from newer suburban shopping malls led management to reposition the Plaza with luxury hotels such as the Ritz-Carlton Hotel, higher end restaurants, and upscale retailers including Gucci, FAO Schwarz, Saks Fifth Avenue, Bally and Halls.

Instead, modern surrogate iron gall formulas such as blue-black bottled inks by Lamy, Montblanc, Chesterfield's Archival Vaultor, Diamine Registrar's Ink, Ecclesiastical Stationery Supplies Registrars Ink, Gutenberg Urkundentinte G10 Schwarz ( certificate ink G10 black ) or Rohrer & Klingner " Salix " and " Scabiosa " inks are offered for fountain pens.

Their sound fitted well with both 1970s British pub rock bands ( such as Brinsley Schwarz ) and British New Wave ( such as Elvis Costello ).

However, even modern solid-state receivers can be very large and heavy, such as the Plessey PR2250, the Redifon R551 or the Rohde & Schwarz EK070.