George David Birkhoff
George David Birkhoff was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, during his time he was considered by many to be the preeminent American mathematician; the George D. Birkhoff House, his residence in Cambridge, has been designated a National Historic Landmark, he was born in Overisel Township, the son of David Birkhoff and Jane Gertrude Droppers. The mathematician Garrett Birkhoff was his son. Birkhoff obtained his A. B. and A. M. from Harvard University. He completed his Ph. D. in 1907, on differential equations, at the University of Chicago. While E. H. Moore was his supervisor, he was most influenced by the writings of Henri Poincaré. After teaching at the University of Wisconsin–Madison and Princeton University, he taught at Harvard from 1912 until his death. In 1923, he was awarded the inaugural Bôcher Memorial Prize by the American Mathematical Society for his paper in 1917 containing, among other things, what is now called the Birkhoff curve shortening process.
He was elected to the National Academy of Sciences, the American Philosophical Society, the American Academy of Arts and Sciences, the Académie des Sciences in Paris, the Pontifical Academy of Sciences, the London and Edinburgh Mathematical Societies. The George David Birkhoff Prize in applied mathematics is awarded jointly by the American Mathematical Society and the Society for Industrial and Applied Mathematics in his honor. Vice-president of the American Mathematical Society, 1919. President of the American Mathematical Society, 1925–1926. Editor of Transactions of the American Mathematical Society, 1920–1924. In 1912, attempting to solve the four color problem, Birkhoff introduced the chromatic polynomial. Though this line of attack did not prove fruitful, the polynomial itself became an important object of study in algebraic graph theory. In 1913, he proved Poincaré's "Last Geometric Theorem," a special case of the three-body problem, a result that made him world-famous. In 1927, he published his Dynamical Systems.
He wrote on the foundations of relativity and quantum mechanics, publishing the monograph Relativity and Modern Physics in 1923. In 1923, Birkhoff proved that the Schwarzschild geometry is the unique spherically symmetric solution of the Einstein field equations. A consequence is that black holes are not a mathematical curiosity, but could result from any spherical star having sufficient mass. Birkhoff's most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics; the ergodic theorem has had repercussions for dynamics, probability theory, group theory, functional analysis. He worked on number theory, the Riemann–Hilbert problem, the four colour problem, he proposed an axiomatization of Euclidean geometry different from Hilbert's. In his years, Birkhoff published two curious works, his 1933 Aesthetic Measure proposed a mathematical theory of aesthetics.
While writing this book, he spent a year studying the art and poetry of various cultures around the world. His 1938 Electricity as a Fluid combined his ideas on science, his 1943 theory of gravitation is puzzling since Birkhoff knew that his theory allows as sources only matter, a perfect fluid in which the speed of sound must equal the speed of light. Albert Einstein and Norbert Wiener, among others, accused Birkhoff of advocating anti-Semitic hiring practices. During the 1930s, when many Jewish mathematicians fled Europe and tried to obtain jobs in the USA, Birkhoff is alleged to have influenced the hiring process at American institutions to exclude Jews. Birkhoff's anti-Semitic views and remarks are well-documented, but Saunders Mac Lane has argued that Birkhoff's efforts were motivated less by animus towards Jews than by a desire to find jobs for home-grown American mathematicians. However, Birkhoff took a particular liking to certain Jewish mathematicians, including Stanislaw Ulam. Gian-Carlo Rota writes: "Like other persons rumored to be anti-Semitic, he would feel the urge to shower his protective instincts on some good-looking young Jew.
Ulam's sparkling manners were diametrically opposite to Birkhoff's hard-working, touchy personality. Birkhoff tried to keep Ulam at Harvard, but his colleagues balked at the idea." Birkhoff, George David. "A determinant formula for the number of ways of coloring a map". Ann. Math. 14: 42–46. Doi:10.2307/1967597. Birkhoff, George David. "Proof of Poincaré's geometric theorem". Trans. Amer. Math. Soc. 14: 14–22. Doi:10.1090/s0002-9947-1913-1500933-9. Birkhoff, George David. "Dynamical Systems with Two Degrees of Freedom". Trans. Amer. Math. Soc. 18: 199–300. Doi:10.1090/s0002-9947-1917-1501070-3. PMC 1091243. Birkhoff, George David and Ralph Beatley. 1959. Basic Geometry, 3rd ed. Chelsea Publishing Co. Birkhoff factorization Birkhoff–Grothendieck theorem Birkhoff's theorem Birkhoff's axioms Birkhoff interpolation Equidistribution theorem Aubin, David, 2005, "Dynamical systems" in Grattan-Guinness, I. ed. Landmark Writings in Western Mathematics. Elsevier: 871–81. Mac Lane, Saunders. "Jobs in the 1930s and the views of George D. Birkhoff".
Math. Intelligencer. 16: 9–10. Doi:10.1007/bf03024350. Kip Thorne, 19nn. Black Holes and Time Warps. W. W. Norton. ISBN 0-393-31276-3. Vandiver
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms; these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics; the mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, by Pierre de Fermat and Blaise Pascal in the seventeenth century. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, Pierre Laplace completed what is today considered the classic interpretation. Probability theory considered discrete events, its methods were combinatorial. Analytical considerations compelled the incorporation of continuous variables into the theory; this culminated on foundations laid by Andrey Nikolaevich Kolmogorov.
Kolmogorov combined the notion of sample space, introduced by Richard von Mises, measure theory and presented his axiom system for probability theory in 1933. This became the undisputed axiomatic basis for modern probability theory. Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately; the measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, more. Consider an experiment that can produce a number of outcomes; the set of all outcomes is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the power set of the sample space of die rolls; these collections are called events. In this case, is the event that the die falls on some odd number.
If the results that occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events, the probability that any of these events occurs is given by the sum of the probabilities of the events; the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6; this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, the event has a probability of 1, that is, absolute certainty; when doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable.
A random variable is a function that assigns to each elementary event in the sample space a real number. This function is denoted by a capital letter. In the case of a die, the assignment of a number to a certain elementary events can be done using the identity function; this does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable X could assign to the outcome "heads" the number "0" and to the outcome "tails" the number "1". Discrete probability theory deals with events. Examples: Throwing dice, experiments with decks of cards, random walk, tossing coins Classical definition: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability. For example, if the event is "occurrence of an number when a die is
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1, not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes, unique up to their order; the property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality test, fast but has a small chance of error, the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
Fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled; the first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved; these include Goldbach's conjecture, that every integer greater than 2 can be expressed as the sum of two primes, the twin prime conjecture, that there are infinitely many pairs of primes having just one number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.
Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural number is called a prime number if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it; the numbers greater than 1 that are not prime are called composite numbers. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid, more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, 5 are the prime numbers, as there are no other numbers that divide them evenly. 1 is not prime, as it is excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of a natural number n are the numbers.
Every natural number has both itself as a divisor. If it has any other divisor, it cannot be prime; this idea leads to a different but equivalent definition of the primes: they are the numbers with two positive divisors, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 divides n evenly; the first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. No number n greater than 2 is prime because any such number can be expressed as the product 2 × n / 2. Therefore, every prime number other than 2 is an odd number, is called an odd prime; when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are and decimal numbers that end in 0 or 5 are divisible by 5; the set of all primes is sometimes denoted by P or by P.
The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements proves the infinitude of primes and the fundamental theorem of arithmetic, shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide
Thermodynamics is the branch of physics that deals with heat and temperature, their relation to energy, work and properties of bodies of matter. The behavior of these quantities is governed by the four laws of thermodynamics, irrespective of the specific composition of the material or system in question; the laws of thermodynamics are explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to a wide variety of topics in science and engineering physical chemistry, chemical engineering and mechanical engineering. Thermodynamics developed out of a desire to increase the efficiency of early steam engines through the work of French physicist Nicolas Léonard Sadi Carnot who believed that engine efficiency was the key that could help France win the Napoleonic Wars. Scots-Irish physicist Lord Kelvin was the first to formulate a concise definition of thermodynamics in 1854 which stated, "Thermo-dynamics is the subject of the relation of heat to forces acting between contiguous parts of bodies, the relation of heat to electrical agency."
The initial application of thermodynamics to mechanical heat engines was extended early on to the study of chemical compounds and chemical reactions. Chemical thermodynamics studies the nature of the role of entropy in the process of chemical reactions and has provided the bulk of expansion and knowledge of the field. Other formulations of thermodynamics emerged in the following decades. Statistical thermodynamics, or statistical mechanics, concerned itself with statistical predictions of the collective motion of particles from their microscopic behavior. In 1909, Constantin Carathéodory presented a purely mathematical approach to the field in his axiomatic formulation of thermodynamics, a description referred to as geometrical thermodynamics. A description of any thermodynamic system employs the four laws of thermodynamics that form an axiomatic basis; the first law specifies that energy can be exchanged between physical systems as work. The second law defines the existence of a quantity called entropy, that describes the direction, thermodynamically, that a system can evolve and quantifies the state of order of a system and that can be used to quantify the useful work that can be extracted from the system.
In thermodynamics, interactions between large ensembles of objects are categorized. Central to this are the concepts of its surroundings. A system is composed of particles, whose average motions define its properties, those properties are in turn related to one another through equations of state. Properties can be combined to express internal energy and thermodynamic potentials, which are useful for determining conditions for equilibrium and spontaneous processes. With these tools, thermodynamics can be used to describe how systems respond to changes in their environment; this can be applied to a wide variety of topics in science and engineering, such as engines, phase transitions, chemical reactions, transport phenomena, black holes. The results of thermodynamics are essential for other fields of physics and for chemistry, chemical engineering, corrosion engineering, aerospace engineering, mechanical engineering, cell biology, biomedical engineering, materials science, economics, to name a few.
This article is focused on classical thermodynamics which studies systems in thermodynamic equilibrium. Non-equilibrium thermodynamics is treated as an extension of the classical treatment, but statistical mechanics has brought many advances to that field; the history of thermodynamics as a scientific discipline begins with Otto von Guericke who, in 1650, built and designed the world's first vacuum pump and demonstrated a vacuum using his Magdeburg hemispheres. Guericke was driven to make a vacuum in order to disprove Aristotle's long-held supposition that'nature abhors a vacuum'. Shortly after Guericke, the English physicist and chemist Robert Boyle had learned of Guericke's designs and, in 1656, in coordination with English scientist Robert Hooke, built an air pump. Using this pump and Hooke noticed a correlation between pressure and volume. In time, Boyle's Law was formulated, which states that pressure and volume are inversely proportional. In 1679, based on these concepts, an associate of Boyle's named Denis Papin built a steam digester, a closed vessel with a fitting lid that confined steam until a high pressure was generated.
Designs implemented a steam release valve that kept the machine from exploding. By watching the valve rhythmically move up and down, Papin conceived of the idea of a piston and a cylinder engine, he did not, follow through with his design. In 1697, based on Papin's designs, engineer Thomas Savery built the first engine, followed by Thomas Newcomen in 1712. Although these early engines were crude and inefficient, they attracted the attention of the leading scientists of the time; the fundamental concepts of heat capacity and latent heat, which were necessary for the development of thermodynamics, were developed by Professor Joseph Black at the University of Glasgow, where James Watt was employed as an instrument maker. Black and Watt performed experiments together, but it was Watt who conceived the idea of the external condenser which resulted in a large increase in steam engine efficiency. Drawing on all the previous work led Sadi Carnot, the "father of thermodynamics", to publish Reflections on the Motive Power of Fire, a discourse on heat, power and engine efficiency.
The book outlined the basic energetic relations between the Carnot engine, the Carnot cycle, motive power. It marked the start of thermodynamics as a modern scien
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous including engineering and physics; the notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set; the idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, the Anosov flow. Flows may be defined for systems of random variables and stochastic processes, occur in the study of ergodic dynamical systems; the most celebrated of these is the Bernoulli flow. A flow on a set X is a group action of the additive group of real numbers on X. More explicitly, a flow is a mapping φ: X × R → X such that, for all x ∈ X and all real numbers s and t, φ = x.
It is customary to write φt instead of φ, so that the equations above can be expressed as φ0 = Id and φs ∘ φt = φs+t. For all t ∈ ℝ, the mapping φt: X → X is a bijection with inverse φ−t: X → X; this follows from the above definition, the real parameter t may be taken as a generalized functional power, as in function iteration. Flows are required to be compatible with structures furnished on the set X. In particular, if X is equipped with a topology φ is required to be continuous. If X is equipped with a differentiable structure φ is required to be differentiable. In these cases the flow forms a one parameter subgroup of homeomorphisms and diffeomorphisms, respectively. In certain situations one might consider local flows, which are defined only in some subset d o m = ⊂ X × R called the flow domain of φ; this is the case with the flows of vector fields. It is common in many fields, including engineering and the study of differential equations, to use a notation that makes the flow implicit. Thus, x is written for φt, one might say that the "variable x depends on the time t and the initial condition x = x0".
Examples are given below. In the case of a flow of a vector field V on a smooth manifold X, the flow is denoted in such a way that its generator is made explicit. For example, Φ V: X × R → X. Given x in X, the set is called the orbit of x under φ. Informally, it may be regarded as the trajectory of a particle, positioned at x. If the flow is generated by a vector field its orbits are the images of its integral curves. Let F: Rn→Rn be a vector field and x: R→Rn the solution of the initial value problem x ˙ = F, x = x 0. Φ = x is the flow of the vector field F. It is a well-defined local flow provided. Φ: Rn×R → Rn is Lipschitz-continuous wherever defined. In general it may be hard to show that the flow φ is globally defined, but one simple criterion is that the vector field F is compactly supported. In the case of time-dependent vector fields F: Rn×R→Rn, one denotes φt,t0 = x, where x: R→Rn is the solution of x ˙ = F, x = x 0. Φt,t0 is the time-dependent flow of F. It is not a "flow" by the definition above, but it can be seen as one by rearranging its arguments.
Namely, the mapping φ: ( R n ×
A call centre or call center is a centralised office used for receiving or transmitting a large volume of requests by telephone. An inbound call centre is operated by a company to administer incoming product support or information enquiries from consumers. Outbound call centres are operated for telemarketing, solicitation of charitable or political donations, debt collection and market research. A contact centre is a location for centralised handling of individual communications, including letters, live support software, social media, instant message, e-mail. A call centre has an open workspace for call centre agents, with work stations that include a computer for each agent, a telephone set/headset connected to a telecom switch, one or more supervisor stations, it can be independently operated or networked with additional centres linked to a corporate computer network, including mainframes, microcomputers and LANs. The voice and data pathways into the centre are linked through a set of new technologies called computer telephony integration.
The contact centre is a central point. Through contact centres, valuable information about company are routed to appropriate people, contacts to be tracked and data to be gathered, it is a part of company's customer relationship management. The majority of large companies use contact centres as a means of managing their customer interaction; these centres can be operated by either an in house department responsible or outsourcing customer interaction to a third party agency. The origins of call centres dates back to the 1960s with the UK-based Birmingham Press and Mail, which installed Private Automated Business Exchanges to have rows of agents handling customer contacts. By 1973, call centres received mainstream attention after Rockwell International patented its Galaxy Automatic Call Distributor for a telephone booking system as well as the popularization of telephone headsets as seen on televised NASA Mission Control Center events. During the late 1970s, call centre technology expanded to include telephone sales, airline reservations and banking systems.
The term "call centre" was first published and recognized by the Oxford English Dictionary in 1983. The 1980s experienced the development of toll-free telephone numbers to increase the efficiency of agents and overall call volume. Call centres increased with the deregulation of long distance calling and growth in information dependent industries; as call centres expanded, unionisation occurred in North America to gain members including the Communications Workers of America and the United Steelworkers. In Australia, the National Union of Workers represents unionised workers. In Europe, Uni Global Union of Switzerland is involved in assisting unionisation in this realm and in Germany Vereinte Dienstleistungsgewerkschaft represents call centre workers. During the 1990s, call centres expanded internationally and developed into two additional subsets of communication, contact centres and outsourced bureau centres. A contact centre is defined as a coordinated system of people, processes and strategies that provides access to information and expertise, through appropriate channels of communication, enabling interactions that create value for the customer and organisation.
In contrast to in-house management, outsourced bureau contact centres are a model of contact centre that provide services on a "pay per use" model. The overheads of the contact centre are shared by many clients, thereby supporting a cost effective model for low volumes of calls; the modern contact center has developed more complex systems, which require skilled operational and management staff that can use multichannel online and offline tools to improve customer interaction. Call centre technologies include speech recognition software to allow computers to handle first level of customer support, text mining and natural language processing to allow better customer handling, agent training by automatic mining of best practices from past interactions, support automation and many other technologies to improve agent productivity and customer satisfaction. Automatic lead selection or lead steering is intended to improve efficiencies, both for inbound and outbound campaigns; this allows inbound calls to be directly routed to the appropriate agent for the task, whilst minimising wait times and long lists of irrelevant options for people calling in.
For outbound calls, lead selection allows management to designate what type of leads go to which agent based on factors including skill, socioeconomic factors and past performance and percentage likelihood of closing a sale per lead. The universal queue standardises the processing of communications across multiple technologies such as fax and email; the virtual queue provides callers with an alternative to waiting on hold when no agents are available to handle inbound call demand. Call centres have been built on Private branch exchange equipment, owned and maintained by the call centre operator themselves; the PBX can provide functions such as automatic call distribution, interactive voice response, skills-based routing. In virtual call centre model, the call centre operator pays a monthly or annual fee to a vendor that hosts the call centre telephony equipment in their own data centre. In this model, the operator does not own, operate or host the equipment that the call centre runs on. Agents connect to the vendor's equipment through traditional PSTN telephone lines, or over voice over IP.
Calls to and from prospects or contacts originate from or terminate at the vendor's data centre, rather than at