Delta IV is a group of five expendable launch systems in the Delta rocket family introduced in the early 2000s. Designed by Boeing's Defense, Space & Security division for the Evolved Expendable Launch Vehicle program, the Delta IV became a United Launch Alliance product in 2006; the Delta IV was and is a launch vehicle for United States Air Force military payloads, but has been used to launch a number of U. S. government non-military payloads and a single commercial satellite. The Delta IV had two main versions which allowed the family to cover a range of payload sizes and masses: the retired Medium and Heavy; as of 2019, only the Heavy remains active, with payloads that would fly on Medium moving to either the existing Atlas V or the forthcoming Vulcan. Retirement of the Delta IV is anticipated in 2024. Delta IV vehicles are built in the ULA facility in Alabama. Final assembly is completed at the launch site by ULA: at the Horizontal Integration Facility for launches from SLC-37B pad at Cape Canaveral and in a similar facility for launches from SLC-6 pad at Vandenberg Air Force Base.

The latest evolutionary development of the Delta rocket family, Delta IV was introduced to meet the requirements of the United States Air Force's Evolved Expendable Launch Vehicle program. While the Delta IV retains the name of the Delta family of rockets, major changes were incorporated; the most significant change was the switch from kerosene to liquid hydrogen fuel, with new tankage and a new engine required. During the Delta IV's development, a Small variant was considered; this would have featured the Delta II second stage, an optional Thiokol Star 48B third stage, the Delta II payload fairing, all atop a single Common Booster Core. The Small variant was dropped by 1999. In 2002, the Delta IV was first launched, with the RS-68 becoming the first large liquid-propellant rocket engine designed in the U. S. since the Space Shuttle main engine in the 1970s. The primary goal for the RS-68 was to reduce cost versus the SSME; some sacrifice in chamber pressure and specific impulse was made. The L3 Technologies Redundant Inertial Flight Control Assembly guidance system used on the Delta IV was common to that carried on the Delta II, although the software was different because of the differences between the Delta II and Delta IV.

The RIFCA featured six ring laser gyroscopes and accelerometers each, to provide a higher degree of reliability. Boeing intended to market Delta IV commercial launch services. However, the Delta IV entered the space launch market when global capacity was much higher than demand. Furthermore, as an unproven design it had difficulty finding a market in commercial launches, Delta IV launch costs are higher than comparable vehicles of the same era. In 2003, Boeing pulled the Delta IV from the commercial market, citing high costs. In 2005, Boeing stated; as of 2009, the USAF funded Delta IV EELV engineering and infrastructure work through contracts with Boeing Launch Services. On August 8, 2008, the USAF Space and Missile Systems Center increased the "cost plus award fee" contract with BLS for $1.656 billion to extend the period of performance through the end of FY09. In addition a $557.1 million option was added to cover FY10. However, the Delta IV series was at that time launched by the United Launch Alliance, a joint venture between Boeing and Lockheed Martin.

In February 2010, naturalized citizen Dongfan Chung, an engineer working with Boeing, was the first person convicted under the Economic Espionage Act of 1996. Chung passed on classified information on designs including the Delta IV rocket and was sentenced to 15 years. In March 2015, ULA announced plans to phase out the Delta IV Medium by 2018; the Delta IV will be replaced by Vulcan in the far term. The Delta IV Medium was retired on 22 August 2019. With the exception of the first launch, which carried the Eutelsat W5 commercial communications satellite, all Delta IV launches have been paid for by the US government. In 2015, ULA stated; the possibility of a higher performance Delta IV was first proposed in a 2006 RAND Corporation study of national security launch requirements out to 2020. A single National Reconnaissance Office payload required an increase in the lift capability of the Delta IV Heavy. Lift capacity was increased by developing the higher-performance RS-68A engine, which first flew on June 29, 2012.

ULA phased out the baseline RS-68 engine with the launch of Delta flight 371 on March 25, 2015. All following launches have used the RS-68A, the engine's higher thrust allowed the use of a single standardized CBC design for all Delta IV Medium and M+ versions; this upgrade reduced cost and increased flexibility, since any standardized CBC could be configured for zero, two, or four solid boosters. However, the new CBC led to a slight performance loss for most medium configurations; the Delta IV Heavy still requires non-standard CBCs for boosters. Payload capacities after RS-68A upgrade Payload capacities with original RS-68 *Masses include a Payload Attach Fitting. Possible future upgrades for the Delta IV included adding extra strap-on solid motors, higher-thrust main engines, lighter materials, higher-thrust second stages, more strap-on CBCs, a cryogenic prope

In mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand, it has an important application in class field theory. If G is a finite cyclic group acting on a G-module A the cohomology groups Hn have period 2 for n≥1. A Herbrand module is an A. In this case, the Herbrand quotient h is defined to be the quotient h = |H2|/|H1|of the order of the and odd cohomology groups; the quotient may be defined for a pair of endomorphisms of an Abelian group, f and g, which satisfy the condition fg = gf = 0. Their Herbrand quotient q is defined as q = | k e r f: i m g | | k e r g: i m f | if the two indices are finite. If G is a cyclic group with generator γ acting on an Abelian group A we recover the previous definition by taking f = 1 - γ and g = 1 + γ + γ2 +.... The Herbrand quotient is multiplicative on short exact sequences. In other words, if0 → A → B → C → 0is exact, any two of the quotients are defined so is the third and h = hhIf A is finite h = 1.

For A is a submodule of the G-module B of finite index, if either quotient is defined so is the other and they are equal: more if there is a G-morphism A → B with finite kernel and cokernel the same holds. If Z is the integers with G acting trivially h = |G| If A is a finitely generated G-module the Herbrand quotient h depends only on the complex G-module C⊗A; these properties mean that the Herbrand quotient is relatively easy to calculate, is much easier to calculate than the orders of either of the individual cohomology groups. Class formation Atiyah, M. F.. T. C.. "Cohomology of Groups". In Cassels, J. W. S.. Algebraic Number Theory. Academic Press. Zbl 0153.07403. See section 8. Artin, Emil. Class Field Theory. AMS Chelsea. P. 5. ISBN 0-8218-4426-1. Zbl 1179.11040. Cohen, Henri. Number Theory – Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics. 239. Springer-Verlag. Pp. 242–248. ISBN 978-0-387-49922-2. Zbl 1119.11001. Janusz, Gerald J.. Algebraic number fields. Pure and Applied Mathematics.

55. Academic Press. P. 142. Zbl 0307.12001. Koch, Helmut. Algebraic Number Theory. Encycl. Math. Sci. 62. Springer-Verlag. Pp. 120–121. ISBN 3-540-63003-1. Zbl 0819.11044. Serre, Jean-Pierre. Local Fields. Graduate Texts in Mathematics. 67. Translated by Greenberg, Marvin Jay. Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016

The Piano Concerto No. 5 in D major, Op. 120 "Concerto Pastorale", by Ferdinand Ries was composed between 1813-1816 and published in 1823 by the firm of Sauer & Leidesdorf with a dedication to Prince Oscar of Sweden. According to Allen Bradley in the liner notes to the Naxos recording, the composers manuscript is undated making it difficult to determine at what point the composer started working on this concerto; however the works dedication to Prince Oscar of Sweden, suggests that he started work on the concerto during his time in Sweden and worked intermittently on it, along with revisions to the Piano Concerto Op.55 until around 1815-16. This would make the concerto the sixth of the composers eight piano concertos to be written; the concertos publication in 1823 came at a time when Ries was retiring from active performing and so no longer needed to keep secret the works he relied on in performance. The concerto follows the traditional three-movement structure: Allegro Andantino Rondo: Allegro To date the concerto has only been recorded once, by Uwe Grodd with the Bournemouth Symphony Orchestra and soloist Christopher Hinterhuber, this was released by Naxos Records in conjunction with a publication of the score in a critical edition prepared by Allen Bradley.

Notes SourcesBadley, Allan. Ries, F.: Piano Concertos, Vol. 4 - Piano Concertos, Opp. 115 and 120. Naxos Records. 8.557844. Hill, Cecil. Ferdinand Ries: A Thematic Catalogue. Armidale, NSW: University of New England. ISBN 0-85834-156-5. Hill, Cecil. Ferdinand Ries. A Study and Addenda. Armidale, NSW: University of New England. ISSN 0314-5999. McGorray, Ian. Ferdinand Ries and the Piano Concerto: Beethoven's Shadow and the Early Romantic Concerto. University of Cincinnati. Piano Concerto in D major, Op. 120: Scores at the International Music Score Library Project