Infimum and supremum
In mathematics, the infimum of a subset S of a ordered set T is the greatest element in T, less than or equal to all elements of S, if such an element exists. The term greatest lower bound is commonly used; the supremum of a subset S of a ordered set T is the least element in T, greater than or equal to all elements of S, if such an element exists. The supremum is referred to as the least upper bound; the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary ordered sets are considered; the concepts of infimum and supremum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the positive real numbers ℝ+ does not have a minimum, because any given element of ℝ+ could be divided in half resulting in a smaller number, still in ℝ+.
There is, however one infimum of the positive real numbers: 0, smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. A lower bound of a subset S of a ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S. An upper bound of a subset S of a ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b. Infima and suprema do not exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique. Ordered sets for which certain infima are known to exist become interesting. For instance, a lattice is a ordered set in which all nonempty finite subsets have both a supremum and an infimum, a complete lattice is a ordered set in which all subsets have both a supremum and an infimum.
More information on the various classes of ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique. If S contains a greatest element that element is the supremum. If S contains a least element that element is the infimum; the infimum of a subset S of a ordered set P, assuming it exists, does not belong to S. If it does, it is a minimum or least element of S. Similarly, if the supremum of S belongs to S, it is a maximum or greatest element of S. For example, consider the set of negative real numbers; this set has no greatest element, since for every element of the set, there is another, element. For instance, for any negative real number x, there is another negative real number x 2, greater. On the other hand, every real number greater than or equal to zero is an upper bound on this set. Hence, 0 is the least upper bound of the negative reals, so the supremum is 0; this set has a supremum but no greatest element.
However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique. Whereas maxima and minima must be members of the subset, under consideration, the infimum and supremum of a subset need not be members of that subset themselves. A ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no smaller element, an upper bound; this does not say that each minimal upper bound is smaller than all other upper bounds, it is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a ordered set, like the real numbers, the concepts are the same; as an example, let S be the set of all finite subsets of natural numbers and consider the ordered set obtained by taking all sets from S together with the set of integers ℤ and the set of positive real numbers ℝ+, ordered by subset inclusion as above.
Both ℤ and ℝ+ are greater than all finite sets of natural numbers. Yet, neither is ℝ+ smaller than ℤ nor is the converse true: both sets are minimal upper bounds but none is a supremum; the least-upper-bound property is an example of the aforementioned completeness properties, typical for the set of real numbers. This property is sometimes called Dedekind completeness. If an ordered set S has the property that every nonempty subset of S having an upper bound has a least upper bound S is said to have the least-upper-bound property; as noted above, the set ℝ of all real numbers has the least-upper-bound property. The set ℤ of integers has the least-upper-bound property.
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer, it states that for any continuous function f mapping a compact convex set to itself there is a point x 0 such that f = x 0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself. Among hundreds of fixed-point theorems, Brouwer's is well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem; this gives it a place among the fundamental theorems of topology. The theorem is used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.
It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu; the theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods; this work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Luitzen Egbertus Jan Brouwer; the theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows: In the plane Every continuous function from a closed disk to itself has at least one fixed point; this can be generalized to an arbitrary finite dimension: In Euclidean space Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.
A more general version is as follows: Convex compact set Every continuous function from a convex compact subset K of a Euclidean space to K itself has a fixed point. An more general form is better known under a different name: Schauder fixed point theorem Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point; the theorem holds only for sets that are convex. The following examples show. Consider the function f = x + 1, a continuous function from R to itself; as it shifts every point to the right, it cannot have a fixed point. Note that R is convex and closed, but not bounded. Consider the function f = x + 1 2, a continuous function from the open interval to itself. In this interval, it shifts every point to the right, so it cannot have a fixed point. Note, convex and bounded, but not closed; the function f does have a fixed point for the closed interval, namely f = 1. Note that convexity is not necessary for BFPT; because the properties involved are invariant under homeomorphisms, BFPT is equivalent to forms in which the domain is required to be a closed unit ball D n.
For the same reason it holds for every set, homeomorphic to a closed ball. The following example shows. Consider the following function, defined in polar coordinates: f =, a continuous function from the unit circle to itself, it rotates every point on the unit circle 45 degrees counterclockwise, so it cannot have a fixed point. Note that the unit circle is closed and bounded, but it has a hole; the function f does have a fixed point for the unit disc. A formal generalization of BFPT for "hole-free" domains can be derived from the Lefschetz fixed-point theorem; the continuous function in this theorem is not required to be bijective or surjective. The theorem has several "real world" illustrations. Here are some examples. 1. Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will be at least one point of the crumpled sheet that lies directly above its corresponding point of the flat sheet.
This is a consequence of the n = 2 case o
Bernard Bolzano was a Bohemian mathematician, philosopher and Catholic priest of Italian extraction known for his antimilitarist views. Bolzano wrote in his native language. For the most part, his work came to prominence posthumously. Bolzano was the son of two pious Catholics, his father, Bernard Pompeius Bolzano, was an Italian who had moved to Prague, where he married Maria Cecilia Maurer who came from Prague's German-speaking family Maurer. Only two of their twelve children lived to adulthood. Bolzano entered the University of Prague in 1796 and studied mathematics and physics. Starting in 1800, he began studying theology, becoming a Catholic priest in 1804, he was appointed to the new chair of philosophy of religion at Prague University in 1805. He proved to be a popular lecturer not only in religion but in philosophy, he was elected Dean of the Philosophical Faculty in 1818. Bolzano alienated many faculty and church leaders with his teachings of the social waste of militarism and the needlessness of war.
He urged a total reform of the educational and economic systems that would direct the nation's interests toward peace rather than toward armed conflict between nations. Upon his refusal to recant his beliefs, Bolzano was dismissed from the university in 1819, his political convictions, which he was inclined to share with others with some frequency proved to be too liberal for the Austrian authorities. He was exiled to the countryside and devoted his energies to his writings on social, religious and mathematical matters. Although forbidden to publish in mainstream journals as a condition of his exile, Bolzano continued to develop his ideas and publish them either on his own or in obscure Eastern European journals. In 1842 he moved back to Prague, where he died in 1848. Bolzano made several original contributions to mathematics, his overall philosophical stance was that, contrary to much of the prevailing mathematics of the era, it was better not to introduce intuitive ideas such as time and motion into mathematics.
To this end, he was one of the earliest mathematicians to begin instilling rigor into mathematical analysis with his three chief mathematical works Beyträge zu einer begründeteren Darstellung der Mathematik, Der binomische Lehrsatz and Rein analytischer Beweis. These works presented "...a sample of a new way of developing analysis", whose ultimate goal would not be realized until some fifty years when they came to the attention of Karl Weierstrass. To the foundations of mathematical analysis he contributed the introduction of a rigorous ε–δ definition of a mathematical limit. Bolzano was the first to recognize the greatest lower bound property of the real numbers. Like several others of his day, he was skeptical of the possibility of Gottfried Leibniz's infinitesimals, the earliest putative foundation for differential calculus. Bolzano's notion of a limit was similar to the modern one: that a limit, rather than being a relation among infinitesimals, must instead be cast in terms of how the dependent variable approaches a definite quantity as the independent variable approaches some other definite quantity.
Bolzano gave the first purely analytic proof of the fundamental theorem of algebra, proven by Gauss from geometrical considerations. He gave the first purely analytic proof of the intermediate value theorem. Today he is remembered for the Bolzano–Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano's first proof and, called the Weierstrass theorem until Bolzano's earlier work was rediscovered. Bolzano's posthumously published work Paradoxien des Unendlichen was admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor, Richard Dedekind. Bolzano's main claim to fame, however, is his 1837 Wissenschaftslehre, a work in four volumes that covered not only philosophy of science in the modern sense but logic and scientific pedagogy; the logical theory that Bolzano developed in this work has come to be acknowledged as ground-breaking. Other works are a four-volume Lehrbuch der Religionswissenschaft and the metaphysical work Athanasia, a defense of the immortality of the soul.
Bolzano did valuable work in mathematics, which remained unknown until Otto Stolz rediscovered many of his lost journal articles and republished them in 1881. In his 1837 Wissenschaftslehre Bolzano attempted to provide logical foundations for all sciences, building on abstractions like part-relation, abstract objects, sentence-shapes and propositions in themselves and sets, substances, subjective ideas and sentence-occurrences; these attempts were an extension of his earlier thoughts in the philosophy of mathematics, for example his 1810 Beiträge where he emphasized the distinction between the objective relationship between logical consequences and our subjective recognition of these connections. For Bolzano, it was not enough that we have confirmation of natural or mathematical truths, but rather it was the proper role of the sciences to seek out justification in terms of the fundamental truths that may or may not appear to be obvious to our intuitions. Bolzano begins his work by explainin
Joseph-Louis Lagrange was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, both classical and celestial mechanics. In 1766, on the recommendation of Leonhard Euler and d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics, written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century. In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences, he remained in France until the end of his life. He was involved in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, became Senator in 1799.
Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He extended the method to take into account possible constraints, arriving at the method of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations, he proved. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series, he studied the three-body problem for the Earth and Moon and the movement of Jupiter's satellites, in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points. But above all, he is best known for his work on mechanics, where he transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, presented the so-called mechanical "principles" as simple results of the variational calculus.
Born as Giuseppe Lodovico Lagrangia, Lagrange was of French descent. His paternal great-grandfather was a French army officer who had moved to Turin, the de facto capital of the kingdom of Piedmont-Sardinia at Lagrange's time, married an Italian, his mother was from the countryside of Turin. He was raised as a Roman Catholic, his father, who had charge of the king's military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained a good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father, Lagrange seems to have accepted this willingly, he studied at the University of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, it was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident.
Alone and unaided he threw himself into mathematical studies. Charles Emmanuel III appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of Benjamin Robins and Leonhard Euler. In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D'Antoni, the academy's military commander and famous artillery theorist, Lagrange proved to be a problematic professor with his oblivious teaching style, abstract reasoning, impatience with artillery and fortification-engineering applications. In this Academy one of his students was François Daviet de Foncenex. Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of tautochrone, discovering a method of maximising and minimising functionals in a way similar to finding extrema of functions.
Lagrange wrote several letters to Leonhard Euler between 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and simplifying Euler's earlier analysis. Lagrange applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis. Euler was impressed with Lagrange's results, it has been stated that "with characteristic courtesy he withheld a paper he had written, which covered some of the same ground, in order that the young Italian might have time to complete his work, claim the undisputed invention of the new calculus". Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773. In 1758, with the aid of his pupils, Lagrange established a society, subsequently incorporated as the Turin Aca
Louis François Antoine Arbogast
Louis François Antoine Arbogast was a French mathematician. He was died at Strasbourg, where he was professor, he wrote on series and the derivatives known by his name: he was the first writer to separate the symbols of operation from those of quantity, introducing systematically the operator notation DF for the derivative of the function F. In 1800, he published a calculus treatise where the first known statement of what is known as Faà di Bruno's formula appears, 55 years before the first published paper of Francesco Faà di Bruno on that topic, he was professor of mathematics at the Collège de Colmar and entered a mathematical competition run by the St Petersburg Academy. His entry was to bring him fame and an important place in the history of the development of the calculus. Arbogast submitted an essay to the St Petersburg Academy in which he came down on the side of Euler. In fact he went much further than Euler in the type of arbitrary functions introduced by integrating partial differential equations, claiming that the functions could be discontinuous not only in the limited sense claimed by Euler, but discontinuous in a more general sense that he defined that allowed the function to consist of portions of different curves.
Arbogast won the prize with his essay and his notion of discontinuous function became important in Cauchy's more rigorous approach to analysis. In 1789 he submitted in Strasbourg a major report on the differential and integral calculus to the Académie des Sciences in Paris, never published. In the Preface of a work he described the ideas that prompted him to write the major report of 1789, he realised that there was no rigorous methods to deal with the convergence of series, Arbogast's career reached new heights. In addition to his mathematics post, he was appointed as professor of physics at the Collège Royal in Strasbourg and from April 1791 he served as its rector until October 1791 when he was appointed rector of the University of Strasbourg, his contributions to mathematics show him as a philosophical thinker. As well as introducing discontinuous functions, as we discussed above, he conceived the calculus as operational symbols; the formal algebraic manipulation of series investigated by Lagrange and Laplace in the 1770s has been put in the form of operator equalities by Arbogast in 1800.
We owe him the general concept of factorial as a product of a finite number of terms in arithmetic progression. The original version of this article was taken from the public domain resource the Rouse History of Mathematics. Discontinuous function Operational calculus A Short Account of the History of Mathematics at Project Gutenberg Itard, Jean, "Arbogast, Louis François Antoine", Dictionary of Scientific Biography, 1, New York: Charles Scribner's Sons, pp. 206–208, ISBN 0-684-10114-9. O'Connor, John J..
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson, he wrote: the idea of infinitely small or infinitesimal quantities seems to appeal to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus; as for the objection that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latter Robinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle.
Robinson continued: However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals fell into disrepute and was replaced by the classical theory of limits. Robinson continues: It is shown in this book that Leibniz's ideas can be vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics; the key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory. In 1973, intuitionist Arend Heyting praised non-standard analysis as "a standard model of important mathematical research". A non-zero element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1 n, for n a standard natural number. Ordered fields that have infinitesimal elements are called non-Archimedean.
More non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the transfer principle for real numbers is a hyperreal field, non-standard real analysis uses these fields as non-standard models of the real numbers. Robinson's original approach was based on these non-standard models of the field of real numbers, his classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print. On page 88, Robinson writes: The existence of non-standard models of arithmetic was discovered by Thoralf Skolem. Skolem's method foreshadows the ultrapower construction Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it is not enough to construct an ordered field with infinitesimals. See the article on hyperreal numbers for a discussion of some of the relevant ideas. In this section we outline one of the simplest approaches to defining a hyperreal field ∗ R.
Let R be the field of real numbers, let N be the semiring of natural numbers. Denote by R N the set of sequences of real numbers. A field ∗ R is defined as a suitable quotient of R N. Take a nonprincipal ultrafilter F ⊂ P. In particular, F contains the Fréchet filter. Consider a pair of sequences u =, v = ∈ R N We say that u and v are equivalent if they coincide on a set of indices, a member of the ultrafilter, or in formulas: ∈ F The quotient of R N by the resulting equivalence relation is a hyperreal field ∗ R, a situation summarized by the formula ∗ R = R N / F. There are at least three reasons to consider non-standard analysis: historical and technical. Much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity; as noted in the article on hyperreal numbers, these formulations were criticized by George Berkeley and others. It was a challenge to develop a consistent theory of analysis using infinitesimals and the first person to do this in a satisfactory way was Abraham Robinson.
In 1958 Curt Schmieden and Detlef Laugwit