For the rational functions defined on the complex numbers, see Möbius transformation. The classical Möbius function μ is an important multiplicative function in number theory and combinatorics; the German mathematician August Ferdinand Möbius introduced it in 1832. It is a special case of a more general object in combinatorics. For any positive integer n, define μ as the sum of the primitive nth roots of unity, it has values in depending on the factorization of n into prime factors: μ = −1 if n is a square-free positive integer with an number of prime factors. Μ = − 1. Μ = − 0. The Möbius function can alternatively be represented as μ = δ ω Ω λ where δ is the Kronecker delta, λ is the Liouville function, ω is the number of distinct prime divisors of n, Ω is the number of prime factors of n, counted with multiplicity; the values of μ for the first 30 positive numbers are The first 50 values of the function are plotted below: The Möbius function is multiplicative. The sum of the Möbius function over all positive divisors of n is zero except when n = 1: ∑ d ∣ n μ = { 1 if n = 1, 0 if n > 1.

The equality above leads to the important Möbius inversion formula and is the main reason why μ is of relevance in the theory of multiplicative and arithmetic functions. Other applications of μ in combinatorics are connected with the use of the Pólya enumeration theorem in combinatorial groups and combinatorial enumerations. There is a formula for calculating the Möbius function without directly knowing the factorization of its argument: μ = ∑ gcd = 1 1 ≤ k ≤ n e 2 π i k n, i.e. μ is the sum of the primitive n-th roots of unity. Using μ = ∑ gcd = 1 1 ≤ k ≤ n e 2 π i k n, the formula ∑ d ∣ n μ = { 1 if n = 1, 0 if n > 1 can be seen as a consequence of the fact that the nth roots of unity sum to 0, since each nth root of unity is a primitive dth root of unity for one divisor d of n. However it is possible to prove this identity from first principles. First note that it is trivially true when n = 1. Suppose that n > 1. There is a bijection between the factors d of n for which μ ≠ 0 and the subsets of the set of all prime factors of n.

The asserted result follows from the fact that every non-empty finite set has an equal number of odd- and even-cardinality subsets. This last fact can be shown by induction on the cardinality |S| of a non-empty finite set S. First, if |S| = 1, there is one odd-cardinality subset of S, namely S itself, one even-cardinality subset, namely ∅. Next, if |S| > 1 divide the subsets of S into two subclasses depending on whether they contain or not some fixed element x in S. There is an obvious bijection between these two subclasses, pairing those subsets that have the same complement relative to the subset. One of these two subclasses consists of all the subsets of the set S \, therefore, by the induction hypothesis, has an equal number of odd- and even-cardinality subsets; these subsets in turn correspond bijectively to the even- and odd-cardinality -containing subsets of S. The inductive step follows directly from these two bijections. A related result is that the binomial coefficients exhibit alternating entries of odd and power which sum symmetrically.

In number theory another arithmetic function related to the Möbius function is the Mertens function, defined by M =

Apatin Brewery, a member of the Molson Coors Europe, is a Serbian brewery based in Apatin. It is majority owned by the American company Molson Coors, it was founded in 1756 as Imperial Brewery owned by the Imperial Chamber. It is one of the oldest breweries in Serbia; the production of 12,000 hectoliters of beer was recorded in the 18th century. The brewery was transferred to private ownership for the first time at the end of the 19th century. Major investments were characteristic of that phase, with a primary goal of modernizing the brewing process and enlarging production capacities. In the period prior to World War I, the yearly production of beer was 16,000 hectoliters; the brewery went through hard times in 1930s, which led to the disruption of production in 1935. Nine years after the liberation of Belgrade in World War II and reparation of the old equipment, a new phase of the company began. By 1950, more than 40,000 hectoliters of beer were produced. Continuous investments resulted in the reconstruction and modernization of the production facilities and the application of new technologies in the production process and the increase in the beer production capacities.

All this was conducted in the period 1958–1966 and it represented a basis for further development of the brewery and transformation into large company. In the 1970s, the process of automated production lines was introduced, the result of, production over 630,000 hectoliters of beer in 1977, as well as the introduction of new non-alcoholic beverages tasting of orange, lemon and Alpine herbs; the result of the economic sanctions in the 1990s was a brief period of decrease in the brewery production, so that 475,000 hectoliters of beer was produced yearly. The company began transferring into the private ownership in 1991, in 2003 it became a part of the international InterBrew network which merged in 2005 in InBev corporation. Over three years, InBew invested around 100 million euros in technology of the company. In 2008, the company became a part of Anheuser-Busch InBev. In December 2009, Apatin Brewery became a part of the StarBev regional brewery group based in Prague, founded by a leading investment company CVC Capital Partners.

Except Serbia, the group operated on the markets of Bosnia and Herzegovina, Croatia, Czech Republic, Montenegro and Slovakia. On 3 April 2012, Molson Coors brewing company has completed its announced acquisition of StarBev for €2.65 billion. This way, Molson Coors took over the regional brewing group Starbev, as part of which Apatin Brewery in Serbia has been operating. Today, it is the leading brewery on the Serbian market. Brands produced in Apatin Brewery are: Jelen, Jelen Cool, Jelen Fresh, Beck's, Stella Artois, Nikšićko pivo and Nikšićko tamno. Jelen is the leading beer brand on the Serbian market, has its share in Bosnia and Herzegovina and Macedonia. Apatin Brewery is known for its slogans for each of its brands: Jelen Pivo – "Men know why". According to the results of a 2009 study of TNS Medium Gallup, Apatin Brewery ranked third in Serbia in advertising and overall business results. Apatin Brewery is the sponsor of the Guča Trumpet Festival. Official website History of Apatin Brewery Molson Coors acquired StarBev Molson Coors Europe Official

Grant Hardy is professor of history and religious studies and director of the humanities program at the University of North Carolina at Asheville. He earned his B. A. in ancient Greek in 1984 from Brigham Young University and his Ph. D. in Chinese language and literature from Yale University in 1988 Having written, cowritten, or edited several books in the fields of history and religious texts as literature, Hardy is known for literary studies of the Book of Mormon. Hardy's contributions in Mormon studies are The Book of Mormon: A Reader's Edition, Understanding the Book of Mormon: A Readers' Guide and The Book of Mormon: Another Testament of Jesus Christ: Maxwell Institute Study Edition. Individual worksAccording to a review by Michael Austin of The Book of Mormon: Another Testament of Jesus Christ, Maxwell Institute Study Edition, "By combining with a serious and thoughtful scholar like Grant Hardy, the Church has produced and authorized a version of its signature scripture, orders of magnitude more helpful, more scholarly, than anything it has produced before."Hardy's Understanding the Book of Mormon has been received favorably for what its publisher, Oxford University Press, describes as "comprehensive analysis of the work's narrative structure."

In generalGrant Shreve says the Book of Mormon's text, "once derided as'a fiction of hob-goblins and bugbears,'" now is being examined by non-Mormon academics and university students, its inclusion on syllabi facilitated by "attractive reader’s editions of the Book of Mormon armed with immaculate scholarly introductions framing it for non-Mormon audiences" by Hardy and by Laurie Maffly-Kipp. Neal Rappleye, in the Mormon apologetics journal Interpreter, lauds Hardy's demonstrations of the Book of Mormon's "depth and complexity, multiple voices, insightful readings," that Rappleye believes bolsters its truth claims. In April 2016, the Journal of Book of Mormon Studies devoted an issue to Hardy's Book of Mormon studies, its editor saying, "We see his work as crucially transitional, bringing the scripture to the attention of the broader academy." Hardy, a Latter-day Saint, has joined proponents advocating tolerance within the faith for struggles with doubt. Providing context for his work, in his August 5, 2016 address at the annual FairMormon conference, he said, "Academics have little interest in debates about whether Mormonism is true or false, but they are interested in Mormonism as a religious and social movement."

When asked during the question-and-answer session concerning believers who harbor questions about the Book of Mormon's historicity, he said, "Can faith in the Book of Mormon as inspired fiction be a saving faith? And I think the answer is, absolutely."A two-part 2017 article by Duane Boyce in the LDS apologetics journal Interpreter questions the didactic effectiveness of Hardy's study's ascriptions of possible psychological motives to individuals categorized as prophets within the Book of Mormon. Boyce believes literary analysis of this type retrograde to the book's divine purpose. Ralph C. Hancock published in Interpreter his argument that Hardy’s reading of the Book of Mormon is "in a way more religious than any other because it is more rational—that is, by allowing natural questions to arise and to resonate, he reveals characters to us (especially the three authors" "that are more miraculous because they are more human." Studies of the Book of Mormon–the foundational scripture of the Latter-day Saints–usually were apologetic or counter-apologetic prior to about 2010.

Parallel the burgeoning of Mormon studies as a field of more neutral scholarship in early 21st century, university courses began including literary studies of this book. In 2016, Nicholas J. Frederick said, "With a few notable exceptions, such as Philip Barlow’s Mormons and the Bible and Grant Hardy’s Understanding the Book of Mormon, full-length monographs devoted to have been lacking." In 2017, organizers of a Book of Mormon studies symposium classed Hardy's scholarship, that brings "the content and the depth of the Book of Mormon into the larger academic world". In addition to chapters and journal articles, Hardy has published the following books: Andrew Feldherr, eds.. Beginnings to AD 600. Oxford History of Historical Writing. 1. Oxford University Press. CS1 maint: uses editors parameter Understanding the Book of Mormon: A Readers' Guide. Oxford University Press. 2010. ISBN 978-0-19-974544-9. Grant Hardy, ed.. The Book of Mormon: A Reader's Edition. University of Illinois Press. Anne Behnke Kinney.

The Establishment of the Han Empire and Imperial China. Greenwood guides to historic events of the ancient world. Greenwood Publishing Group. ISBN 978-0-313-32588-5. Worlds of Bronze and Bamboo: Sima Qian's Conquest of History. Columbia University Press. 1999. ISBN 978-0-231-50451-5. Mark Ashurst-McGee, Robin Scott Jensen & Sharalyn D. Howcroft, eds.. "Textual Criticism and the Book of Mormon". Foundational Texts of Mormonism. Oxford University Press. Pp. 37–73. ISBN 9780190274405. CS1 maint: uses editors parameter Sacred Texts of the World; the Great Courses. Teaching Company. 2013. ISBN 978-1-62997-044-8. Great Minds of the Eastern Intellectual Tradition; the Great Courses. Teaching Company. 2011. ISBN 978-1-59803-742-5. Origin of the Book of Mormon Wor

The Dampier to Bunbury Natural Gas Pipeline is the longest natural gas pipeline in Australia. It is 660mm in diameter, which makes it one of Australia's largest in terms of transmission capacity. At the time of its commissioning in 1984, it was one of the longest gas pipelines in the world; the pipeline runs within Western Australia from a point near Withnell Bay, on the Burrup Peninsula near Dampier, to Bunbury in the south-west of the state. It carries natural gas, most of which enters the pipeline at the domestic gas plant associated with the North West Shelf Venture project; the other main inlet point is 135 km south of Dampier, where one of the sales gas pipelines from Varanus Island Processing Hub connects with the DBNGP. In June 2008, following a pipeline rupture and explosion at the Varanus Island facility, the DBNGP carried additional volumes of gas from the North West Shelf plant to the south-west of the state, for a period of several months. See: 2008 Western Australian gas crisis As a single trunkline it is 1530 km long, extending from the Burrup Peninsula in the Pilbara region, to Bunbury in the south west of Western Australia.

It supplies gas to industrial and residential customers in Perth and major regional centres along the pipeline route. It is covered by Western Australian pipeline licence PL-40. A number of lateral pipelines are connected to this pipeline, most of which are covered by separate licences, although PL-40 itself covers the main trunkline and some laterals totalling a length of 1789 km; the idea of a pipeline to transport gas from the North West Shelf to the south-west of Western Australia had its origins in 1975, following the discovery of large offshore reserves by WAPET and Woodside Petroleum. Around this time, the State Energy Commission of WA reviewed the state's future gas requirements in conjunction with the partners in the North West Shelf consortium; the developers of the North West Shelf were in the preliminary stages of planning a system of production facilities based on the Rankin and Goodwyn fields, linked to an LNG processing plant and a domestic gas plant situated at Withnell Bay.

As the state government had access to more attractive interest rates than the commercial venturers, the state agreed to fund and build a 1540 km gas pipeline to transport the output from the domestic gas plant. In addition, SECWA entered into long term'take-or-pay' contracts with the North West Shelf partners, in which SECWA agreed to pay for fixed volumes of gas which exceeded the market demand for gas in the south west. Engineering design commenced in 1979. and the pipeline was constructed between 1983 and 1984, with the extension south to Bunbury commissioned in 1985. Construction involved the welding of 127,000 sections of 12-metre pipe. Gas first flowed into the pipe on 16 August 1984. Following the disaggregation of SECWA in 1995, the pipeline came under the ownership and control of the government's gas utility, AlintaGas; as part of a government policy of privatisation, Alinta sold the pipeline in 1998 to Epic Energy, a consortium of two U. S. pipeline companies at a price of A$2.407 billion.

Epic Energy owned and operated the pipeline for six years selling the pipeline in October 2004 to Dampier Bunbury Pipeline Ltd, the trading name of the DBNGP group of companies. DBP had two institutional shareholders: D. U. E. T. and Alcoa up until 2017. The DBNGP is owned and operated by Dampier Bunbury Pipeline, part of the Australian Gas Infrastructure Group, Australian Gas Infrastructure Group is owned by various consortia of Hong Kong-based entities listed on the Hong Kong Stock Exchange; the consortia include CK Asset Holdings Ltd, CK Infrastructure Holdings Ltd, Power Assets Holdings Ltd and CK Hutchison Holdings Ltd. When first constructed, the capacity of the system was 360 TJ/day. A$150m enhancement carried out in 1991 increased the capacity to 450 TJ/day; the current capacity of the pipeline is 785 TJ/day. The pipeline is undergoing a series of significant expansions in its capacity which are intended to increase the maximum throughput of the pipeline to 895 TJ/day; the expansion project consists of two types of upgrade.

Several of the pipeline's compressor stations will be upgraded with higher capacity gas turbines, thus increasing the actual pressure of the gas inside the line. In addition, the expansion involves a process of'looping' the pipeline; this is the installation of additional lengths of pipe alongside, connected to, the existing pipeline. The effect of looping is to provide additional capacity at critical sections of the main trunkline, thus increasing the total possible throughput; the DBNGP underwent a range of expansion projects under previous ownership structures. However, the most advanced have been undertaken by the current owners, who have invested in three major expansion projects since acquiring the asset, these projects are known as Stage 4, Stage 5A and the most recent, Stage 5B; these three projects were carried out in direct response to increasing demand for gas in the south-west of WA. The owners worked with major gas customers to deliver the capacity required within the timeframes needed to support those customers' projects.

Increasing the gas haulage capacity of the pipeline was crucial in supporting economic growth in WA. The expansions were built to meet DBP's contractual obligations; as a result of the expansions, 83% of the pipeline has now been duplicated – creating a seco

The A class were four minesweepers of the Royal Netherlands Navy. They were the first purpose-built minesweepers of the Dutch Navy, as earlier minesweepers were converted tugboats; the A class was based on a German design used during the First World War. All ships of the A class were built between 1930 at Willemsoord, Den Helder. All four ships were commissioned on 4 August 1930, two days on 6 August 1930, all four sailed to the Dutch East Indies, arriving at Surabaya on 30 October 1930. During the Japanese invasion of the Dutch East Indies in March 1942 all four ships were scuttled by their crews. Three of the ships, A, B and C, were salvaged and put into service with the Imperial Japanese Navy as auxiliary submarine chasers no. 113, 112 and 116, being rearmed with a single 47 mm gun, three 25 mm cannons, a 13.2 mm machine gun and eight depth charges. All three were sunk by US forces in 1944–1945. HNLMS A - Laid down 9/1928. Launched 19 April 1929 and completed on 4 August 1930. Scuttled at Surabaya, Java 1 March 1942, raised by Japan and commissioned as Cha.113 on 17 January 1943.

Sunk by US submarine USS Hardhead off Java on 23 June 1945. HNLMS B - Laid down 9/1928. Launched 24 September 1929 and completed on 4 August 1930. Scuttled at Surabaya, Java 2 March 1942, raised by Japan, commissioned as Cha.112 on 20 December 1942. Sunk by US aircraft in the Java Sea on 4 June 1945. HNLMS C - Laid down 4/1929. Launched late 1929. Completed 4 August 1930. Scuttled at Surabaya, Java 6 March 1942, raised and commissioned as Japanese Cha.116 on 8 April 1943. Sunk by US carrier-based aircraft 20 miles west of Cavite, Luzon on 13 November 1944. HNLMS D - Laid down 1928. Launched 1929. Completed 8/1930. Scuttled at Surabaya, Java 2 March 1942 to prevent capture by Japanese forces. Jentschura, Dieter Jung and Peter Mikel. Warships of the Imperial Japanese Navy 1869–1945. Annapolis, Maryland, USA: Naval Institute Press, 1986. ISBN 0-87021-893-X. Mark, C. Schepen van de Koninklijke Marine in W. O. II Alkmaar: De Alk bv, 1997: 109-110. Netherlandsnavy.nl:: A class minesweepers

Yakov Naumovich Feld — Soviet and Russian physicist who worked in the field of electrical engineering and antennas. S. Popov prize of the USSR Academy of Sciences, doctor of technical Sciences, Professor. After graduating from a vocational school in 1927, Feld went on to complete his studies at the radio faculty of the Kiev College of Communications in 1931. In 1932 - 1941 he worked as a consultant at the Central Radio Laboratory in Leningrad under the guidance of Professor V. V. Tatarinov. In 1939 Feld wrote a PhD thesis on "The General Theory of certain Types of Feeder Systems"; as Feld did not have Bachelor or Master diploma at the time, he required a special permission of the All-Union Committee on Higher Education of the Union of People's Commissioners of the USSR allowing him to obtain a PhD. In 1941 Feld along with other employees of the Central Radio Laboratory were evacuated to Krasnoyarsk. There he worked until 1946 in what is today a scientific-industrial enterprise "Radiosvyaz". In 1946 Feld joined the laboratory of the academician N. D. Papalexi at the Academy of Sciences at the Institute of Physics of the USSR.

Feld completed an internship with the academician M. A. Leontovich. In 1947 Feld wrote his doctoral thesis on "The Fundamentals of the Theory of Slit Antennas" and was awarded the title of Professor in 1948. In February 1946 Feld was invited by M. A. Leontovich to join the Central Science and Research Institute -108. In 1949 to 1976 Feld headed Antennas department at that institute. Throughout his life Feld combined research activities with teaching: 1934-1941 – Lecturer of Leningrad Electrotechnical Institute of Communication M. A. Bonch-Bruevich, from 1939 - Associate Professor at that Institute. Feld died on August 28, 1995 in Moscow and was buried at the Preobrazhenskoe Jewish cemetery in St. Petersburg. Feld was author of 3 monographs and over 200 scientific papers, the founder of a scientific school in the field of electrical engineering worked with over 40 PhD graduates and 10 doctorate PhD graduates. Feld was the founder and leader of the All-Moscow Seminar on "Diffraction and Wave Propagation".

In 2002 that Seminar was renamed into the Moscow Feld's Seminar of Electrodynamics. In 1956-1995 Feld was a member of the Editorial Board of a scientific journal "Radio-engineering and Electronics" of the USSR Academy of Sciences. Feld left an extensive scientific legacy: Modern formulation of electromagnetic boundary value problems. I. e. the used method of calculation of aperture antenna radiation patterns corresponds to a different model with a black flange. Feld led the development of original designs of antennas with elliptical polarization, the surface wave antennas, waveguide slot and lens antennas, antennas with electrical swing beam, the axially symmetric broadband phased arrays. Feld Ya. N. Foundations of Slot Antenna Theory. – Moscow: Soviet Radio, 1948. – 164 pp. Feld Ya. N. Benenson L. S. Centimeter and decimeter wave antennas. – Moscow: Zhukovsky Air Force Academy, 1955. – 208 pp. Feld Ya. N. Benenson L. S. Antenna and feeder devices. – Moscow: Zhukovsky Air Force Academy, 1959. – 552 pp. Handbook on antennas, vol.1.

– Moscow: IPRZhR, 1997. – 249 pp.. ISBN 5-88070-003-8 Feld Ya. N. Benenson L. S. Antenna theory foundations. – Moscow: Drofa, 2007. – 492 pp. ISBN 978-5-358-01772-6