In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length and volume. A important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space Rn. For instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to subsets of a set X, it must further be countably additive: the measure of a'large' subset that can be decomposed into a finite number of'smaller' disjoint subsets is equal to the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure.
This problem was resolved by defining measure only on a sub-collection of all subsets. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined are complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, Maurice Fréchet, among others; the main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space. Probability theory considers measures that assign to the whole set the size 1, considers measurable subsets to be events whose probability is given by the measure.
Ergodic theory considers measures that are invariant under, or arise from, a dynamical system. Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties: Non-negativity: For all E in Σ: μ ≥ 0. Null empty set: μ = 0. Countable additivity: For all countable collections i = 1 ∞ of pairwise disjoint sets in Σ: μ = ∑ k = 1 ∞ μ One may require that at least one set E has finite measure; the empty set automatically has measure zero because of countable additivity, because μ = μ = μ + μ + μ + …, which implies that μ = 0. If only the second and third conditions of the definition of measure above are met, μ takes on at most one of the values ±∞ μ is called a signed measure; the pair is called a measurable space, the members of Σ are called measurable sets. If and are two measurable spaces a function f: X → Y is called measurable if for every Y-measurable set B ∈ Σ Y, the inverse image is X-measurable – i.e.: f ∈ Σ X.
In this setup, the composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows. See Measurable function#Term usage variations about another setup. A triple is called a measure space. A probability measure is a measure with total measure one – i.e. Μ = 1. A probability space is a measure space with a probability measure. For measure spaces that are topological spaces various compatibility conditions can be
Stefan Banach was a Polish mathematician, considered one of the world's most important and influential 20th-century mathematicians. He was the founder of modern functional analysis, an original member of the Lwów School of Mathematics, his major work was the 1932 book, Théorie des opérations linéaires, the first monograph on the general theory of functional analysis. Born in Kraków, Banach attended IV Gymnasium, a secondary school, worked on mathematics problems with his friend Witold Wilkosz. After graduating in 1910, Banach moved to Lwów. However, during World War I Banach returned to Kraków. After Banach solved some mathematics problems which Steinhaus considered difficult, they published their first joint work. In 1919, with several other mathematicians, Banach formed a mathematical society. In 1920 he received an assistantship at the Lwów Polytechnic, he soon became a professor at the Polytechnic, a member of the Polish Academy of Learning. He organized the "Lwów School of Mathematics". Around 1929 he began writing his Théorie des opérations linéaires.
After the outbreak of World War II, in September 1939, Lwów was taken over by the Soviet Union. Banach became a member of the Academy of Sciences of Ukraine and was dean of Lwów University's Department of Mathematics and Physics. In 1941, when the Germans took over Lwów, all institutions of higher education were closed to Poles; as a result, Banach was forced to earn a living as a feeder of lice at Rudolf Weigl's Institute for Study of Typhus and Virology. While the job carried the risk of infection with typhus, it protected him from being sent to slave labor in Germany and from other forms of repression; when the Soviets recaptured Lwów in 1944, Banach reestablished the University. However, because the Soviets were removing Poles from Soviet-annexed formerly-Polish territories, Banach prepared to return to Kraków. Before he could do so, he died in August 1945, having been diagnosed seven months earlier with lung cancer; some of the notable mathematical concepts that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach–Mazur game, the Banach–Alaoglu theorem, the Banach fixed-point theorem.
Stefan Banach was born on 30 March 1892 at St. Lazarus General Hospital in Kraków part of the Austro-Hungarian Empire, into a Góral Roman Catholic family and was subsequently baptised by his father, while his mother abandoned him upon this event and her identity is ambiguous. Banach's parents were natives of the Podhale region. Greczek was a soldier in the Austro-Hungarian Army stationed in Kraków. Little is known about Banach's mother. According to his baptismal certificate, she was worked as a domestic help. Unusually, Stefan's surname was his mother's instead of his father's, though he received his father's given name, Stefan. Since Stefan Greczek was a private and was prevented by military regulations from marrying, the mother was too poor to support the child, the couple decided that he should be reared by family and friends. Stefan spent the first few years of his life with his grandmother, but when she took ill Greczek arranged for his son to be raised by Franciszka Płowa and her niece Maria Puchalska in Kraków.
Young Stefan would regard Franciszka as Maria as his older sister. In his early years Banach was tutored by Juliusz Mien, a French intellectual and friend of the Płowa family, who had emigrated to Poland and supported himself with photography and translations of Polish literature into French. Mien taught Banach French and most encouraged him in his early mathematical pursuits. In 1902 Banach, aged 10, enrolled in Kraków's IV Gymnasium. While the school specialized in the humanities and his best friend Witold Wiłkosz spent most of their time working on mathematics problems during breaks and after school. In life Banach would credit Dr. Kamil Kraft, the mathematics and physics teacher at the gymnasium with kindling his interests in mathematics. While Banach was a diligent student he did on occasion receive low grades and would speak critically of the school's math teachers. After obtaining his matura at age 18 in 1910, Banach moved to Lwów with the intention of studying at the Lwów Polytechnic.
He chose engineering as his field of study since at the time he was convinced that there was nothing new to discover in mathematics. At some point he attended Jagiellonian University in Kraków on a part-time basis; as Banach had to earn money to support his studies it was not until 1914 that he at age 22, passed his high school graduation exams. When World War I broke out, Banach was excused from military service due to his left-handedness and poor vision; when the Russian Army opened its offensive toward Lwów, Banach left for Kraków, where he spent the rest of the war. He made his living as a tutor at the local gymnasiums, worked in a bookstore and as a foreman of road building crew, he attended some lectures at the Jagiellonian University at that time, including those of the famous Polish mathematicians Stanisław Zaremba and Kazimierz Żorawski, but little is known of that period of his life. In 1916, in Kraków's Planty gardens, Banach encountered Professor Hugo Steinhaus, one of the renowned mathematicians of the time.
According to Steinhaus, while he was strolling through the gardens he was surprised to overhear the term "Lebesgue integral" (Lebesgue integration was
Edward C. Elliott
Edward Charles Elliott was an American educational researcher and administrator. He was the chancellor of the public university system of Montana from 1916 to 1922 and the president of Purdue University from 1922 to 1945. Born in Chicago, Elliott grew up in North Platte and studied chemistry at the University of Nebraska–Lincoln, where he received his Bachelor of Science and Master of Arts degrees, he was hired as a high school science teacher in Leadville and became that city's superintendent after one year. As superintendent, Elliott wrote formal rules for paying the teachers. Leadville opened its first four-year high school under Elliott's leadership. In 1903, Elliott accepted a fellowship at Columbia University, his doctoral dissertation was among the first works to apply statistics to the study of school administration. Elliott continued his research at the University of Wisconsin–Madison and devised a unique scale to rate teachers' merit and competency. In a series of studies with Daniel Starch, Elliott showed that a student's assignment can receive a wide variety of grades depending on the teacher and the school.
He participated in commissions that carried out early school surveys of Boise, New York City and Portland, Oregon. As director of Wisconsin's committees for accrediting schools and training teachers, he raised the requirements for obtaining a teacher's certification, although most of his initiatives were undone after he left the university. One longer-lasting program of Elliott's committees was the establishment of Wisconsin High School, where the university could observe new teachers. Elliott was a charter member of the American Association of University Professors, he served on its Committee on Academic Freedom for a few months before moving to Montana. From 1916 to 1922, Elliott served as the first "Chancellor of the University of Montana"; this university system combined four separate campuses throughout the state. He worked to bring efficiency to the system's procedures for budget requests and high school recruiting. Although no buildings had been built in the ten years before his chancellorship, Elliott pushed for a property tax and a bond issue that funded the construction of 13 new buildings.
Another of his initiatives had the state refund the costs for all students to travel to one of the university campuses once a year. A nationwide controversy began in 1919 with Elliott's dismissal of an economics professor. Elliott and Edward O. Sisson, president of the State University, encouraged professor Louis Levine to conduct a study of Montana's tax system. A draft of Levine's report, The Taxation of Mines in Montana, concluded that state laws gave an unfair advantage to the mining industry, that these companies should be made to pay a higher amount of taxes; the mining industry had a significant influence on the Montana legislature, Elliott warned Levine that his study could harm state appropriations to the university. Not wanting the university to be involved in a political controversy, Elliott refused to have the university's name associated with Levine's report; when Levine published it independently in February 1919, Elliott suspended him from the faculty for insubordination and unprofessional conduct.
Magazines The New Republic and The Nation, Upton Sinclair's book The Goose-Step, many newspapers considered this an attack on academic freedom and an example of the dominance of the mining industry in Montana. A review committee at the university upheld Elliott's decision to fire Levine, but asked the State Board of Education to reinstate the professor and reduce the chancellor's power to dismiss faculty in the future. For the next 23 years, from 1922 until 1945, Elliott served as the sixth president of Purdue University. During his presidency—the second longest in the university's history—enrollment rose from 3,200 to 8,600 students. There was a doubling in staff, course offerings, major buildings and land acreage, the physical plant value increased from 3.7 billion to 18.7 billion dollars. Elliott supported a more individualized curriculum; the president's probation policy replaced Purdue's previous practice of promptly expelling the pupils who were not passing. The university organized an orientation program and a graduate school and built the first of its current system of residence halls.
In 1935 Elliott hired aviator Amelia Earhart and industrial engineer Lillian Gilbreth as visiting faculty members to find ways to improve the education of women. Early in his presidency, Elliott changed Purdue's budgeting procedures by hiring its first comptroller and business manager. In 1924, Purdue developed a plan for campus development that remained unchanged for sixty years. Although state funding was reduced during the Depression, Elliott continued to grow the university by obtaining funds from the federal New Deal agencies. Growth was supported by four corporations that Elliott helped to establish with close ties to the university; these were the Ross–Ade Foundation, the Purdue Research Foundation, a re-incorporated Better Homes in America, the Purdue Aeronautics Corporation. At first, Elliott was skeptical of spending university money on musical organizations. Once, when asked to fund a new choir, Elliott shouted, "Never, as long as I am president, will this university spend one damn penny on music on this campus, young man!
Get that through your head!" By 1934, Elliott was proposing the construction of a new auditorium. In 1938 he lobbied the state legislature to fund this project; the Hall of Music opened in 1940 and has been known sinc
Purdue University is a public research university in West Lafayette and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and money to establish a college of science and agriculture in his name; the first classes were held on September 1874, with six instructors and 39 students. The main campus in West Lafayette offers more than 200 majors for undergraduates, over 69 masters and doctoral programs, professional degrees in pharmacy and veterinary medicine. In addition, Purdue has more than 900 student organizations. Purdue is a member of the Big Ten Conference and enrolls the second largest student body of any university in Indiana, as well as the fourth largest foreign student population of any university in the United States. In 1865, the Indiana General Assembly voted to take advantage of the Morrill Land-Grant Colleges Act of 1862, began plans to establish an institution with a focus on agriculture and engineering.
Communities throughout the state offered their facilities and money to bid for the location of the new college. Popular proposals included the addition of an agriculture department at Indiana State University or at what is now Butler University. By 1869, Tippecanoe County’s offer included $150,000 from Lafayette business leader and philanthropist John Purdue, $50,000 from the county, 100 acres of land from local residents. On May 6, 1869, the General Assembly established the institution in Tippecanoe County as Purdue University, in the name of the principal benefactor. Classes began at Purdue on September 1874, with six instructors and 39 students. Professor John S. Hougham was Purdue’s first faculty member and served as acting president between the administrations of presidents Shortridge and White. A campus of five buildings was completed by the end of 1874. Purdue issued its first degree, a Bachelor of Science in chemistry, in 1875 and admitted its first female students that fall. Emerson E. White, the university’s president from 1876 to 1883, followed a strict interpretation of the Morrill Act.
Rather than emulate the classical universities, White believed Purdue should be an "industrial college" and devote its resources toward providing a liberal education with an emphasis on science and agriculture. He intended not only to prepare students for industrial work, but to prepare them to be good citizens and family members. Part of White's plan to distinguish Purdue from classical universities included a controversial attempt to ban fraternities; this ban was overturned by the Indiana Supreme Court and led to White's resignation. The next president, James H. Smart, is remembered for his call in 1894 to rebuild the original Heavilon Hall "one brick higher" after it had been destroyed by a fire. By the end of the nineteenth century, the university was organized into schools of agriculture and pharmacy, former U. S. President Benjamin Harrison was serving on the board of trustees. Purdue's engineering laboratories included testing facilities for a locomotive and a Corliss steam engine, one of the most efficient engines of the time.
The School of Agriculture was sharing its research with farmers throughout the state with its cooperative extension services and would undergo a period of growth over the following two decades. Programs in education and home economics were soon established, as well as a short-lived school of medicine. By 1925 Purdue had the largest undergraduate engineering enrollment in the country, a status it would keep for half a century. President Edward C. Elliott oversaw a campus building program between the world wars. Inventor and trustee David E. Ross coordinated several fundraisers, donated lands to the university, was instrumental in establishing the Purdue Research Foundation. Ross's gifts and fundraisers supported such projects as Ross–Ade Stadium, the Memorial Union, a civil engineering surveying camp, Purdue University Airport. Purdue Airport was the country's first university-owned airport and the site of the country's first college-credit flight training courses. Amelia Earhart joined the Purdue faculty in 1935 as a consultant for these flight courses and as a counselor on women's careers.
In 1937, the Purdue Research Foundation provided the funds for the Lockheed Electra 10-E Earhart flew on her attempted round-the-world flight. Every school and department at the university was involved in some type of military research or training during World War II. During a project on radar receivers, Purdue physicists discovered properties of germanium that led to the making of the first transistor; the Army and the Navy conducted training programs at Purdue and more than 17,500 students and alumni served in the armed forces. Purdue set up about a hundred centers throughout Indiana to train skilled workers for defense industries; as veterans returned to the university under the G. I. Bill, first-year classes were taught at some of these sites to alleviate the demand for campus space. Four of these sites are now degree-granting regional campuses of the Purdue University system. Purdue's on-campus housing became racially desegregated in 1947, following pressure from Purdue President Frederick L. Hovde and Indiana Governor Ralph F. Gates.
After the war, Hovde worked to expand the academic opportunities at the university. A decade-long construction program emphasized research. In the late 1950s and early 1960s the university established programs in veterinary medicine, industrial management, nursing, as well as the first computer science department in the United States. Undergraduate humanities courses were strengthened
Peer review is the evaluation of work by one or more people with similar competences as the producers of the work. It functions as a form of self-regulation by qualified members of a profession within the relevant field. Peer review methods are used to maintain quality standards, improve performance, provide credibility. In academia, scholarly peer review is used to determine an academic paper's suitability for publication. Peer review can be categorized by the type of activity and by the field or profession in which the activity occurs, e.g. medical peer review. Professional peer review focuses on the performance of professionals, with a view to improving quality, upholding standards, or providing certification. In academia, peer review is used to inform in decisions related to faculty tenure. Henry Oldenburg was a British philosopher, seen as the'father' of modern scientific peer review. WA prototype is a professional peer-review process recommended in the Ethics of the Physician written by Ishāq ibn ʻAlī al-Ruhāwī.
He stated that a visiting physician had to make duplicate notes of a patient's condition on every visit. When the patient was cured or had died, the notes of the physician were examined by a local medical council of other physicians, who would decide whether the treatment had met the required standards of medical care. Professional peer review is common in the field of health care, where it is called clinical peer review. Further, since peer review activity is segmented by clinical discipline, there is physician peer review, nursing peer review, dentistry peer review, etc. Many other professional fields have some level of peer review process: accounting, engineering and forest fire management. Peer review is used in education to achieve certain learning objectives as a tool to reach higher order processes in the affective and cognitive domains as defined by Bloom's taxonomy; this may take a variety of forms, including mimicking the scholarly peer review processes used in science and medicine.
Scholarly peer review is the process of subjecting an author's scholarly work, research, or ideas to the scrutiny of others who are experts in the same field, before a paper describing this work is published in a journal, conference proceedings or as a book. The peer review helps the publisher decide whether the work should be accepted, considered acceptable with revisions, or rejected. Peer review requires a community of experts in a given field, who are qualified and able to perform reasonably impartial review. Impartial review of work in less narrowly defined or inter-disciplinary fields, may be difficult to accomplish, the significance of an idea may never be appreciated among its contemporaries. Peer review is considered necessary to academic quality and is used in most major scholarly journals, but it by no means prevents publication of invalid research. Traditionally, peer reviewers have been anonymous, but there is a significant amount of open peer review, where the comments are visible to readers with the identities of the peer reviewers disclosed as well.
The European Union has been using peer review in the "Open Method of Co-ordination" of policies in the fields of active labour market policy since 1999. In 2004, a program of peer reviews started in social inclusion; each program sponsors about eight peer review meetings in each year, in which a "host country" lays a given policy or initiative open to examination by half a dozen other countries and the relevant European-level NGOs. These meet over two days and include visits to local sites where the policy can be seen in operation; the meeting is preceded by the compilation of an expert report on which participating "peer countries" submit comments. The results are published on the web; the United Nations Economic Commission for Europe, through UNECE Environmental Performance Reviews, uses peer review, referred to as "peer learning", to evaluate progress made by its member countries in improving their environmental policies. The State of California is the only U. S. state to mandate scientific peer review.
In 1997, the Governor of California signed into law Senate Bill 1320, Chapter 295, statutes of 1997, which mandates that, before any CalEPA Board, Department, or Office adopts a final version of a rule-making, the scientific findings and assumptions on which the proposed rule are based must be submitted for independent external scientific peer review. This requirement is incorporated into the California Health and Safety Code Section 57004. Medical peer review may be distinguished in 4 classifications: 1) clinical peer review. Additionally, "medical peer review" has been used by the American Medical Association to refer not only to the process of improving quality and safety in health care organizations, but to the process of rating clinical behavior or compliance with professional society membership standards. Thus, the terminology has poor standardization and specificity as a database search term. To an outsider, the anonymous, pre-publication peer review process is opaque. Certain journals are accused of not carrying out stringent peer review in order to more expand their customer base in journals where authors pay a fee before public
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, measure, infinite series, analytic functions. These theories are studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Analysis may be distinguished from geometry. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids; the explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.
In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would be called Cavalieri's principle to find the volume of a sphere in the 5th century; the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series, his followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. Descartes and Fermat independently developed analytic geometry, a few decades Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations and partial differential equations, Fourier analysis, generating functions.
During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra used in earlier work by Euler. Instead, Cauchy formulated calculus in terms of geometric infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y, he introduced the concept of the Cauchy sequence, started the formal theory of complex analysis. Poisson, Liouville and others studied partial differential equations and harmonic analysis; the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis.
In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, introduced the "epsilon-delta" definition of limit. Mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. "monsters" began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, Baire proved the Baire category theorem.
In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, Hilbert introduced Hilbert spaces to solve integral equations; the idea of normed vector space was in the air, in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Much of analysis happens in some metric space. Examples of analysis without a metric include functional analysis. Formally, a metric space is an ordered pair where M is a set