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Computer algebra system

A computer algebra system is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials. Computer algebra systems may be divided into two classes: general-purpose; the specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics. General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as: a user interface allowing a user to enter and display mathematical formulas from a keyboard, menu selections, mouse or stylus.

A programming language and an interpreter, a simplifier, a rewrite system for simplifying mathematics formulas, a memory manager, including a garbage collector, needed by the huge size of the intermediate data, which may appear during a computation, an arbitrary-precision arithmetic, needed by the huge size of the integers that may occur, a large library of mathematical algorithms and special functions. The library must not only provide for the needs of the users, but the needs of the simplifier. For example, the computation of polynomial greatest common divisors is systematically used for the simplification of expressions involving fractions; this large amount of required computer capabilities explains the small number of general-purpose computer algebra systems. The main ones are Axiom, Magma, Maple and SageMath. Computer algebra systems began to appear in the 1960s and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence.

A prime example for the first development was the pioneering work conducted by the Nobel Prize laureate in physics Martinus Veltman, who designed a program for symbolic mathematics high-energy physics, called Schoonschip in 1963. Another early system was FORMAC. Using Lisp as the programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial-intelligence research environment. MATHLAB was made available to users on PDP-6 and PDP-10 systems running TOPS-10 or TENEX in universities. Today it can still be used on SIMH emulations of the PDP-10. MATHLAB should not be confused with MATLAB, a system for numerical computation built 15 years at the University of New Mexico; the first popular computer algebra systems were muMATH, Reduce and Macsyma. Reduce became free software in 2008; as of today, the most popular commercial systems are Mathematica and Maple, which are used by research mathematicians and engineers. Available alternatives include SageMath. In 1987, Hewlett-Packard introduced the first hand-held calculator CAS with the HP-28 series, it was possible, for the first time in a calculator, to arrange algebraic expressions, limited symbolic integration, Taylor series construction and a solver for algebraic equations.

In 1999, the independently developed CAS Erable for the HP 48 series became an integrated part of the firmware of the emerging HP 49/50 series, a year into the HP 40 series as well, whereas the HP Prime adopted the Xcas system in 2013. The Texas Instruments company in 1995 released the TI-92 calculator with a CAS based on the software Derive; the TI-89 series, first released in 1998 contains a CAS. Casio released their first CAS calculator with the CFX-9970G and succeeded it with the Algebra FX Series in 1999-2003 and the current ClassPad Series; the symbolic manipulations supported include: simplification to a smaller expression or some standard form, including automatic simplification with assumptions and simplification with constraints substitution of symbols or numeric values for certain expressions change of form of expressions: expanding products and powers and full factorization, rewriting as partial fractions, constraint satisfaction, rewriting trigonometric functions as exponentials, transforming logic expressions, etc. partial and total differentiation some indefinite and definite integration, including multidimensional integrals symbolic constrained and unconstrained global optimization solution of linear and some non-linear equations over various domains solution of some differential and difference equations taking some limits integral transforms series operations such as expansion and products matrix operations including products, etc. statistical computation theorem proving and verification, useful in the area of experimental mathematics optimized code generationIn the above, the word some indicates that the operation cannot always be performed.

Many include: a programming language, allowing users to implement their own algorithms arbitrary-precision numeric operations exact integer arithmetic and number theory functionality Editing of mathematical expressions in two-dimensional form plotting graphs and parametric plots of fun

William Ashbury Buchanan

William Ashbury Buchanan, was a journalist and politician based in Alberta. Following a career as a journalist in Ontario, Buchanan moved to Alberta and purchased the Lethbridge Weekly Herald in 1905, he attempted, unsuccessfully, to turn the paper into a daily in 1907 but it soon resumed a weekly publishing schedule. Buchanan entered provincial politics when he contested the riding of Lethbridge City on behalf of the Alberta Liberal Party in the 1909 provincial election and won a seat in the Alberta Legislature, he switched to federal politics in the 1911 federal election and was elected for the seat of Medicine Hat as a federal Liberal. As a result of the Conscription Crisis of 1917, Buchanan crossed the floor to support the government of Sir Robert Laird Borden, he was re-elected to the House of Commons of Canada as a Unionist for the new riding of Lethbridge in the 1917 general election. He did not run in the 1921 election. Buchanan was named to the Senate of Canada at the age of 49 in 1925 and served until his death in 1954 for the Liberal Party of Canada.

He continued as owner of the Lethbridge Herald until his death. A malignant growth was discovered in Buchanan's body sometime in August 1953 and he died on July 12, 1954, ten days after celebrating his 78th birthday. William Ashbury Buchanan – Parliament of Canada biography

International School of Kraków

The International School of Kraków – a school in Lusina, Lesser Poland Voivodeship in southern Poland, belonging to the Polish Embassy of United States. The ISK was established in 1993 in the heart of Kraków; the school was founded as the American International School of Kraków later changed its name to the International School of Kraków. In 2006, the school moved to a new campus in Lusina with funding assistance from the United States Embassy in Poland; the school belongs to the Central and Eastern European Schools Association and competes with schools throughout Central and Eastern Europe. It is accredited by the Council of International Schools and the New England Association of Schools and Colleges. Since 2012, ISK provides the IB Diploma Programme for high school students. ISK is housed on an enclosed 1.25 hectare campus in the quiet suburb of Lusina 20 minutes from the city center. There are landscaped areas with basketball courts and a soccer field. ISK features a spacious library, large multi-purpose hall, covered sports facility, computer lab.

School website Director's welcome PTA page International Baccalaureate Organisation website Council of International Schools website Central & Eastern European Schools Association website New England Association of Schools and Colleges website

Israel national under-19 football team

Israel's national Under-19 team known as Israel Under-19s or Israel U-19s, is considered to be the feeder team for the Israel national under-21 football team. This team is for Israeli players aged 19 or under at the start of a two-year UEFA European Under-19 Championship campaign, so players can be, are, up to 20 years old. In existence are teams for Under-21s and Under-20s and Under 17s; as long as they are eligible, players can play at any level, making it possible to play for the U-19s, senior side and again for the U-19s. It is possible to play for one country at youth level and another at senior level. Israel U-19s do not have a permanent home, they play in stadia dotted all around Israel in an attempt to encourage fans in all areas of the country to get behind Israel. Because of the lack of appeal compared to the senior national team, smaller grounds are used; the idea to form a youth team first came about in 1957, as the IFA considered entering a team to the 1958 UEFA European Under-18 Championship.

The youth team, an under-19 squad, played its first match against its England equivalent on 20 May 1962, losing 1–2. Two days the team recorded its first victory, winning 2–1 in a rematch against England. In 1964, the youth team participated for the first time in the AFC Youth Championship, sharing the cup with Burma in its first tournament; the team went on to win the cup five more times in the next 8 years, before Israel was banned from participating in AFC tournaments. Until 1992, the youth team's only official tournaments were FIFA Youth Championship qualification tournaments, twice participating in the process in the OFC U-20 Championship and once in the South American Youth Football Championship. At the same period of time, to give the youth squad its share of international matches, the IFA established an annual tournament for under-18 teams, held in December and January between 1974 and 1989. In 1992, as Israel was admitted to UEFA, the youth started participating in the UEFA European Under-19 Championship, appearing in the final tournament in 1997 and 2014.

Champions*: Title shared DNP: Did Not Participate DNQ: Did Not Qualify AFC U-19 Championship: Winners: 1964, 1965, 1966, 1967, 1971, 1972 Third Place: 1968 Fourth Place: 1969 OFC Championship: Runners-up: 1985, 1986 This squad is the squad selection for the Exhibition game against Belgium and Hungary. UEFA European Under-19 Football Championship Israel national football team Israel national under-21 football team Israel national under-18 football team Israel national under-17 football team

IBM 1130

The IBM 1130 Computing System, introduced in 1965, was IBM's least expensive computer at that time. A binary 16-bit machine, it was aimed at price-sensitive, computing-intensive technical markets like education and engineering, succeeding the decimal IBM 1620 in that market segment. Typical installations included a 1 megabyte disk drive that stored the operating system and object programs, with program source generated and maintained on punched cards. Fortran was the most common programming language used, but several others, including APL, were available; the 1130 was used as an intelligent front-end for attaching an IBM 2250 Graphics Display Unit, or as remote job entry workstation, connected to a System/360 mainframe. The total production run of the 1130 has been estimated at 10,000; the 1130 holds a place in computing history because it gave many people their first direct interaction with a computer. Its price-performance ratio was good and it notably included inexpensive, removable disk storage, with reliable, easy-to-use software that supported several high-level languages.

The low price and well-balanced feature set enabled interactive "open shop" program development. The IBM 1130 uses the same electronics packaging, called Solid Logic Technology, used in System/360, it has a 16-bit binary architecture, as do minicomputers like the PDP-11 and Data General Nova. The address space is 15 bits; the 1130 uses magnetic-core memory, which the processor addresses on word boundaries, using direct and indexed addressing modes. IBM implemented five models of the 1131 Central Processing Unit, the primary processing component of the IBM 1130; the Model 1 through Model 5 describe the core memory cycle time, as well as the model's ability to support disk storage. A letter A through D appended to the model number indicates the amount of core memory installed. IBM 1131 Central Processing Unit weighs about 760/1050 lb; the Model 4 was a lower-priced product with a 5.6 µs cycle time. Some purchasers of performance upgrades observed that the field adjustment to achieve the improvement was trivial.

The IBM 1132 printer relies on the 1130 processor rather than internal logic to determine when to fire the print wheels as they rotated. Printers for the Model 4 run more but the slower processor still can not keep up with it; the hardware manual discloses that when the Model 4 was servicing the two highest-level interrupts, it ran at the faster 3.6 µs cycle time. Some users of the Model 4 would write a phony printer driver that did not dismiss the printer interrupt, in order to benefit from the higher processor speed. However, lower-level interrupts are disabled during this interval the end-of-card interrupt from the 1442 card reader. Follow-on productsThe IBM 1800 is a variant of the IBM 1130 for process control applications, it uses hardware rather than core memory for the three index registers and features two extra instructions plus extra interrupt and I/O capabilities. It is a successor to the IBM 1710, as the IBM 1130 is a successor to the IBM 1620; the IBM 1500 is a multi-user educational system based around either an IBM 1130 or an IBM 1800.

It supports up to 32 student work stations, each with a variety of audio-visual capabilities. Other than these, IBM produced no compatible successor systems to the 1130; the IBM System/7 is a process control and real-time system, the IBM Series/1 is a general-purpose 16-bit minicomputer. ChronologyFebruary 11, 1965 – IBM introduces the 1130. Announced is the IBM 1132 printer, the lowest cost online computer printer announced by IBM at that time. Fourth quarter 1965 – First customer shipments begin from the San Jose plant. March 31, 1966 – IBM introduces the IBM 1500 educational system. April 1966 – IBM 1800 ships. August 9, 1966 – IBM rolls out the 1130 synchronous communications adapter, which permits the small 1130 system to be connected by regular leased telephone lines to, function as a communications terminal for, any model of the IBM System/360. April 17, 1967 – A four-way expansion of the 1130 is announced, involving: Five times the disk storage and four times the core memory. January 1968 – First shipments begin of the 1130 Models 2C, 2D, 3B, 3C, 3D.

July 1968 – The Boca Raton plant begins shipping the 1130. July 22, 1971 – 1130 Models 4A and 4B are introduced at new levels of economy. September 1971 – First customer shipments begin of the 1130 Model 4. May 31, 1972 – Models 1C, 1D, 5B, 5C and 5D are announced. To maximize speed and conserve space, the operating system and compilers are written in assembly language and employ techniques that are rare today, including intermixing code and data as well as self-modifying code. Much user programming was done in Fortran; the 1130 Fortran compiler can run on a machine with only 4,096 words of core—though the compiled program might not fit on such a machine. In this multi-pass compiler, each "phase" processes the entire source program and takes it another step toward machine code. For example, the first phase reads the source statements into memory, discards comment lines, removes spaces except in text literals, concatenates continuation lines and identifies labels; the compiler was available in a disk-resident version as well as on 8-channel punched paper tape or punched cards.

The most widely

Aventine Triad

The Aventine Triad is a modern term for the joint cult of the Roman deities Ceres and Libera. The cult was established ca. 493 BC within a sacred district on or near the Aventine Hill, traditionally associated with the Roman plebs. Accounts describe the temple building and rites as "Greek" in style; some modern historians describe the Aventine Triad as a plebeian parallel and self-conscious antithesis to the Archaic Triad of Jupiter and Quirinus and the Capitoline Triad of Jupiter and Juno. The Aventine Triad and associated ludi served as a focus of plebeian identity, sometimes in opposition to Rome's original ruling elite, the patricians; the Aventine relationship between Ceres and Libera was based first on their functions as agricultural and fertility deities of the plebs as a distinct social group. Liber had been companion to both Ceres and to Libera in separate and disparate fertility cults that were widespread throughout the Hellenised Italian peninsula, long before their official adoption by Rome – or rather, their partial assimilation, as Ceres' own cult appears to have been considered more tractable and obedient than Liber's.

Their Aventine cults, reported in Roman sources as distinctively Greek in character, may have been further reinforced and influenced by their perceived similarities to particular Greek deities: Ceres to Demeter, Liber to Dionysus and Libera to either Persephone or Ariadne. In keeping with Roman theology, the internal and external equivalence of the Aventine Triad remained speculative and flexible. Long after its establishment, Cicero rejects the equivalence of Liber and Dionysus and asserts that Ceres is mother to Liber and Libera; the Aventine Triad was established soon after the overthrow of the Roman monarchy and establishment of the Republic. Rome's majority of citizen commoners were ruled by the patricians, a small number of powerful, landed aristocrats who asserted a traditional, exclusive right to Rome's highest religious and military offices; the plebs not only served in Rome's legions: they were the backbone of its economy – smallholders, skilled specialists, managers of landed estates, vintners and exporters of grain and wine.

Against a background of famine in Rome, an imminent war against the Latins and a threatened plebeian secession, the dictator A. Postumius vowed a temple to the patron deities of the plebs, Ceres and Libera on or near the Aventine Hill; the famine ended and Rome's plebeian citizen-soldiery co-operated in the conquest of the Latins. In 493 BC, a new built temple on or near the Aventine hill was dedicated to the Triad and Rome's first recorded ludi scaenici were held in honour of Liber, for the benefit of the Roman people. Liber's festival, the Liberalia, may date from this time. Patrician dominance was manifest in the Capitoline Triad of Jupiter and Quirinus on the Capitoline Hill, at the heart of the city; the Capitoline temple lay within Rome's sacred boundary. The Aventine lay outside it. In most versions of the Roman founding myth, this was the hill on which the unfortunate Remus lost to his brother Romulus in a contest of augury to decide Rome's foundation and leadership. Postumius' vow has been interpreted as a pragmatic, timely recognition of the plebeian citizenry as a distinct social and political grouping with its own values and traditions.

Some aspects of their cults were still considered morally "un-Roman" by Rome's authorities. Thus the Aventine Triad gave the plebs what has been variously described by modern historians as a parallel to the official Capitoline Triad, its "copy and antithesis". Among other religious innovations based on his antiquarian interests, the emperor Claudius redrew the pomerium to encompass the Aventine; the plebs continued to establish and administer their own laws and held formal assemblies from which patricians were excluded. They elected their own magistrates and sought religious confirmation of their decisions through their own augury, which in plebeian religious tradition had been introduced by Marsyas, a satyr or silen in the entourage of Liber. Meanwhile, the plebeian tribunes, an emergent plebeian nobility and a small but growing number of popularist politicians of patrician ancestry gained increasing influence over Rome's religious life and government. Any person who offended against the sacred rights and person of a plebeian tribune was liable to declaration as homo sacer, who could be killed with impunity and whose property was certainly, forfeit to Ceres.

So, official Ludi Cereales were not established until as late as 202 BC. Liber's festival and the Bacchic or Dionysian aspects of his cult were suppressed under the ferocious Senatus consultum de Bacchanalibus of 186 BC; the Liberalia rites were transferred to Cerealia. Varro's complex, investigative Late Republican theology groups Ceres with Tellus and Venus, therefore with Victoria. Evidence is lacking for the earliest priesthoods of the Aventine Triad, whether in joint or individual cult to its deities; the plebeian aediles, named after their service of aedes may have acted as cult priests for their community and may have served Liber and Libera in this capacity. Ceres was served by a flamen Cere