Pierre-Simon, marquis de Laplace was a French scholar whose work was important to the development of engineering, statistics and astronomy. He extended the work of his predecessors in his five-volume Mécanique Céleste; this work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed by Laplace. Laplace formulated Laplace's equation, pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming; the Laplacian differential operator used in mathematics, is named after him. He restated and developed the nebular hypothesis of the origin of the Solar System and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse. Laplace is remembered as one of the greatest scientists of all time. Sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries.
He was Napoleon's examiner when Napoleon attended the École Militaire in Paris in 1784. Laplace became a count of the Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration; some records of Laplace's life were burned in 1925 with the family château in Saint Julien de Mailloc, near Lisieux, the home of his great-great-grandson the Comte de Colbert-Laplace. Others had been destroyed earlier, when his house at Arcueil near Paris was looted by house breakers in 1871. Laplace was born in Beaumont-en-Auge, Normandy on 23 March 1749, a village four miles west of Pont l'Eveque. According to W. W. Rouse Ball, his father, Pierre de Laplace and farmed the small estates of Maarquis, his great-uncle, Maitre Oliver de Laplace, had held the title of Chirurgien Royal. It would seem. However, Karl Pearson is scathing about the inaccuracies in Rouse Ball's account and states: Indeed Caen was in Laplace's day the most intellectually active of all the towns of Normandy, it was here that Laplace was provisionally a professor.
It was here he wrote his first paper published in the Mélanges of the Royal Society of Turin, Tome iv. 1766–1769, at least two years before he went at 22 or 23 to Paris in 1771. Thus before he was 20 he was in touch with Lagrange in Turin, he did not go to Paris a raw self-taught country lad with only a peasant background! In 1765 at the age of sixteen Laplace left the "School of the Duke of Orleans" in Beaumont and went to the University of Caen, where he appears to have studied for five years and was a member of the Sphinx. The'École Militaire' of Beaumont did not replace the old school until 1776, his parents were from comfortable families. His father was Pierre Laplace, his mother was Marie-Anne Sochon; the Laplace family was involved in agriculture until at least 1750, but Pierre Laplace senior was a cider merchant and syndic of the town of Beaumont. Pierre Simon Laplace attended a school in the village run at a Benedictine priory, his father intending that he be ordained in the Roman Catholic Church.
At sixteen, to further his father's intention, he was sent to the University of Caen to read theology. At the university, he was mentored by two enthusiastic teachers of mathematics, Christophe Gadbled and Pierre Le Canu, who awoke his zeal for the subject. Here Laplace's brilliance as a mathematician was recognised and while still at Caen he wrote a memoir Sur le Calcul integral aux differences infiniment petites et aux differences finies; this provided the first intercourse between Lagrange. Lagrange was the senior by thirteen years, had founded in his native city Turin a journal named Miscellanea Taurinensia, in which many of his early works were printed and it was in the fourth volume of this series that Laplace's paper appeared. About this time, recognising that he had no vocation for the priesthood, he resolved to become a professional mathematician; some sources state that he broke with the church and became an atheist. Laplace did not graduate in theology but left for Paris with a letter of introduction from Le Canu to Jean le Rond d'Alembert who at that time was supreme in scientific circles.
According to his great-great-grandson, d'Alembert received him rather poorly, to get rid of him gave him a thick mathematics book, saying to come back when he had read it. When Laplace came back a few days d'Alembert was less friendly and did not hide his opinion that it was impossible that Laplace could have read and understood the book, but upon questioning him, he realised that it was true, from that time he took Laplace under his care. Another account is that Laplace solved overnight a problem that d'Alembert set him for submission the following week solved a harder problem the following night. D'Alembert was recommended him for a teaching place in the École Militaire. With a secure income and undemanding teaching, Laplace now threw himself into original research and for the next seventeen years, 1771–1787, he produced much of his original work in astronomy. From 1780–1784, Laplace and French chemist Antoine Lavoisier collaborated on several experimental investigations, designing their own equipment for the task.
In 1783 they published their joint paper, Memoir on Heat, in which they discussed the kinetic theory of molecular motion. In their experiments they measured the specific heat of various bodies, an
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise and in mathematics and physics as infinite-dimensional function spaces; the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis, ergodic theory. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications.
The success of Hilbert space methods ushered in a fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes, in analogy with Cartesian coordinates in the plane; when that set of axes is countably infinite, the Hilbert space can be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is in the older literature referred to as the Hilbert space.
Linear operators on a Hilbert space are fairly concrete objects: in good cases, they are transformations that stretch the space by different factors in mutually perpendicular directions in a sense, made precise by the study of their spectrum. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ3, equipped with the dot product; the dot product takes two vectors x and y, produces a real number x · y. If x and y are represented in Cartesian coordinates the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3; the dot product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is linear in its first argument: · y = ax1 · y + bx2 · y for any scalars a, b, vectors x1, x2, y, it is positive definite: for all vectors x, x · x ≥ 0, with equality if and only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a inner product. A vector space equipped with such an inner product is known as a inner product space.
Every finite-dimensional inner product space is a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted ||x||, to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ. Multivariable calculus in Euclidean space relies on the ability to compute limits, to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞ x n consisting of vectors in ℝ3 is convergent provided that the sum of the lengths converges as an ordinary series of real numbers: ∑ k = 0 ∞ ‖ x k ‖ < ∞. Just as with a series of scalars, a series of vectors that converges also converges to some limit vector L in the Euclidean space, in the sense that ‖ L − ∑ k = 0 N x k ‖ → 0 as N → ∞; this property expresses the completeness of
Jørgen Pedersen Gram
Jørgen Pedersen Gram was a Danish actuary and mathematician, born in Nustrup, Duchy of Schleswig and died in Copenhagen, Denmark. Important papers of his include On series expansions determined by the methods of least squares, Investigations of the number of primes less than a given number; the mathematical method that bears his name, the Gram–Schmidt process, was first published in the former paper, in 1883. Gram's theorem and the Gramian matrix are named after him. For number theorists his main fame is the series for the Riemann zeta function. Instead of using a series of logarithmic integrals, Gram's function uses logarithm powers and the zeta function of positive integers, it has been supplanted by a formula of Ramanujan that uses the Bernoulli numbers directly instead of the zeta function. Gram was the first mathematician to provide a systematic theory of the development of skew frequency curves, showing that the normal symmetric Gaussian error curve was but one special case of a more general class of frequency curves.
He died after being struck by a bicycle. Logarithmic integral function Prime number Riemann–Siegel theta function which contain Gram points. Notes Bibliography Gram, J. P.. "Undersøgelser angaaende Maengden af Primtal under en given Graeense". Det K. Videnskabernes Selskab. 2: 183–308
Gene H. Golub
Gene Howard Golub, Fletcher Jones Professor of Computer Science at Stanford University, was one of the preeminent numerical analysts of his generation. Born in Chicago, he was educated at the University of Illinois at Urbana-Champaign, receiving his B. S. M. A. and Ph. D. all in mathematics. His M. A. degree was more in Mathematical Statistics. His PhD dissertation was entitled "The Use of Chebyshev Matrix Polynomials in the Iterative Solution of Linear Equations Compared to the Method of Successive Overrelaxation" and his thesis adviser was Abraham Taub. Gene Golub succumbed to acute myeloid leukemia on the morning of 16 November 2007 at the Stanford Hospital, he arrived at Stanford in 1962 and became a professor there in 1970. He advised more than thirty doctoral students. Gene Golub was an important figure in numerical analysis and pivotal to creating the NA-Net and the NA-Digest, as well as the International Congress on Industrial and Applied Mathematics. One of his best-known books is Matrix Computations, co-authored with Charles F. Van Loan.
He was a major contributor to algorithms for matrix decompositions. In particular he published an algorithm together with William Kahan in 1970 that made the computation of the singular value decomposition feasible and, still used today. A survey of his work was published in 2007 by Oxford University Press as "Milestones in Matrix Computation". Golub was awarded the B. Bolzano Gold Medal for Merits in the Field of Mathematical Sciences and was one of the few elected to three national academies: the National Academy of Sciences, the National Academy of Engineering, the American Academy of Arts and Sciences, he was a Foreign Member of the Royal Swedish Academy of Engineering Sciences. He is listed as an ISI cited researcher, he held 11 honorary doctorates and was scheduled to receive an honorary doctorate from ETH Zürich on November 17, 2007. He was a visiting professor at Princeton, MIT, ETH, Oxford. Gene Golub served as the president of the Society for Industrial and Applied Mathematics from 1985 to 1987 and was founding editor of both the SIAM Journal on Scientific Computing and the SIAM Journal on Matrix Analysis and Applications.
The bulk of Gene Golub's research work was collaborative. He had at least 181 distinct co-authors and the number may still increase as co-authored papers keep appearing posthumously. Home page at Stanford University Archived May 13, 2007, at the Wayback Machine Gene H. Golub at the Mathematics Genealogy Project Gene H Golub Memorial page Oral history interviews with Gene H. Golub, Charles Babbage Institute, University of Minnesota. Interview by Pamela McCorduck, 16 May 1979 and 8 June 1979, California. Gene Golub, Oral history interview by Thomas Haigh, 22–23 October 2005, Stanford University. Society for Industrial and Applied Mathematics, Philadelphia, PA, six-hour interview covers full career - transcript online. Gene Golub in pictures around the world. Gene Golub Papers "Because of space limitations... Master bibliography of matrix computation is online" from 4th edition of "Matrix computations"
Inner product space
In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors, they provide the means of defining orthogonality between vectors. Inner product spaces generalize Euclidean spaces to vector spaces of any dimension, are studied in functional analysis; the first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product induces an associated norm, thus an inner product space is a normed vector space. A complete space with an inner product is called a Hilbert space. An space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space.
Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C. Formally, an inner product space is a vector space V over the field F together with an inner product, i.e. with a map ⟨ ⋅, ⋅ ⟩: V × V → F that satisfies the following three axioms for all vectors x, y, z ∈ V and all scalars a ∈ F: Conjugate symmetry: ⟨ x, y ⟩ = ⟨ y, x ⟩ ¯ Linearity in the first argument: ⟨ a x, y ⟩ = a ⟨ x, y ⟩ ⟨ x + y, z ⟩ = ⟨ x, z ⟩ + ⟨ y, z ⟩ Positive-definite: ⟨ x, x ⟩ > 0, x ∈ V ∖. Positive-definiteness and linearity ensure that: ⟨ x, x ⟩ = 0 ⇒ x = 0 ⟨ 0, 0 ⟩ = ⟨ 0 x, 0 x ⟩ = 0 ⟨ x, 0 x ⟩ = 0 Notice that conjugate symmetry implies that ⟨x, x⟩ is real for all x, since we have: ⟨ x, x ⟩ = ⟨ x, x ⟩ ¯. Conjugate symmetry and linearity in the first variable imply ⟨ x, a y ⟩ = ⟨ a y, x ⟩ ¯ = a ¯ ⟨ y, x ⟩ ¯ = a ¯ ⟨ x, y ⟩ ⟨ x, y + z ⟩ = ⟨ y + z, x ⟩ ¯ = ⟨ y, x ⟩ ¯ + ⟨ z, x ⟩ ¯ = ⟨ x, y ⟩ + ⟨ x, z ⟩.
Erhard Schmidt was a Baltic German mathematician whose work influenced the direction of mathematics in the twentieth century. Schmidt was born in the Governorate of Livonia, his advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905. His doctoral dissertation was entitled Entwickelung willkürlicher Funktionen nach Systemen vorgeschriebener and was a work on integral equations. Together with David Hilbert he made important contributions to functional analysis. Ernst Zermelo credited conversations with Schmidt for the idea and method for his classic 1904 proof of the Well-ordering theorem from an "Axiom of choice", which has become an integral part of modern set theory. After the war, in 1948, Schmidt founded and became the first editor-in-chief of the journal Mathematische Nachrichten. During World War II Schmidt held positions of authority at the University of Berlin and had to carry out various Nazi resolutions against the Jews—a job that he did not do well, since he was criticized at one point for not understanding the "Jewish question."
At the celebration of Schmidt's 75th birthday in 1951 a prominent Jewish mathematician, Hans Freudenthal, who had survived the Nazi years, spoke of the difficulties that Schmidt faced during that period without criticism. He was, however, a conservative and a nationalist, defended Hitler after Kristallnacht, telling Issai Schur that "Suppose we had to fight a war to rearm Germany, unite with Austria, liberate the Saar and the German part of Czechoslovakia; such a war would have cost us half a million young men. But everybody would have admired our victorious leader. Now, Hitler has achieved great things for Germany. I hope some day you will be recompensed but I am still grateful to Hitler". Gram–Schmidt process Hilbert–Schmidt operator Lyapunov–Schmidt reduction Schmidt decomposition Hilbert–Schmidt integral operator List of Baltic German scientists Zermelo, Ernst. "Beweis, daß jede Menge wohlgeordnet werden kann". Mathematische Annalen. 59: 514–516. Doi:10.1007/BF01445300. Reprinted in English translation as "Proof that every set can be well-ordered", van Heijenoort 1976, pp. 139–141