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Jørgen Pedersen Gram
Jørgen Pedersen Gram (27 June 1850 – 29 April 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark. Important papers of his include ''On series expansions determined by the methods of least squares'', and ''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. For number theorists his main fame is the series for the Riemann zeta function (the leading function in Riemann's exact prime-counting function). Instead of using a series of logarithmic integrals, Gram's function uses logarithm powers and the zeta function of positive integers. It has recently been supplanted by a formula of Ramanujan that uses the Bernoulli numbers directly instead of the zeta function. In control theory, the Gramian or Gram matrix is an important contribution named after him. The Contr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Observability Gramian
In control theory, we may need to find out whether or not a system such as \begin \dot(t)\boldsymbol(t)+\boldsymbol(t)\\ \boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t) \end is observable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are, respectively, n\times n, n\times p,q\times n and q\times p matrices. One of the many ways one can achieve such goal is by the use of the Observability Gramian. Observability in LTI Systems Linear Time Invariant (LTI) Systems are those systems in which the parameters \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are invariant with respect to time. One can determine if the LTI system is or is not observable simply by looking at the pair (\boldsymbol,\boldsymbol). Then, we can say that the following statements are equivalent: 1. The pair (\boldsymbol,\boldsymbol) is observable. 2. The n\times n matrix \boldsymbol(t)=\int_^e^\boldsymbol^\boldsymbole^d\tau is nonsingular for any t>0. 3. The nq\times n observability matrix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycling Road Incident Deaths
Cycling, also, when on a two-wheeled bicycle, called bicycling or biking, is the use of cycles for transport, recreation, exercise or sport. People engaged in cycling are referred to as "cyclists", "bicyclists", or "bikers". Apart from two-wheeled bicycles, "cycling" also includes the riding of unicycles, tricycles, quadricycles, recumbent and similar human-powered vehicles (HPVs). Bicycles were introduced in the 19th century and now number approximately one billion worldwide. They are the principal means of transportation in many parts of the world, especially in densely populated European cities. Cycling is widely regarded as an effective and efficient mode of transportation optimal for short to moderate distances. Bicycles provide numerous possible benefits in comparison with motor vehicles, including the sustained physical exercise involved in cycling, easier parking, increased maneuverability, and access to roads, bike paths and rural trails. Cycling also offers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1916 Deaths
Events Below, the events of the First World War have the "WWI" prefix. January * January 1 – The British Royal Army Medical Corps carries out the first successful blood transfusion, using blood that had been stored and cooled. * January 9 – WWI: Gallipoli Campaign: The last British troops are evacuated from Gallipoli, as the Ottoman Empire prevails over a joint British and French operation to capture Constantinople. * January 10 – WWI: Erzurum Offensive: Russia defeats the Ottoman Empire. * January 12 – The Gilbert and Ellice Islands Colony, part of the British Empire, is established in present-day Tuvalu and Kiribati. * January 13 – WWI: Battle of Wadi: Ottoman Empire forces defeat the British, during the Mesopotamian campaign in modern-day Iraq. * January 29 – WWI: Paris is bombed by German zeppelins. * January 31 – WWI: An attack is planned on Verdun, France. February * February 9 – 6.00 p.m. – Tristan Tz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1850 Births
Year 185 ( CLXXXV) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Lascivius and Atilius (or, less frequently, year 938 ''Ab urbe condita''). The denomination 185 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Roman Empire * Nobles of Britain demand that Emperor Commodus rescind all power given to Tigidius Perennis, who is eventually executed. * Publius Helvius Pertinax is made governor of Britain and quells a mutiny of the British Roman legions who wanted him to become emperor. The disgruntled usurpers go on to attempt to assassinate the governor. * Tigidius Perennis, his family and many others are executed for conspiring against Commodus. * Commodus drains Rome's treasury to put on gladiatorial spectacles and confiscates property to suppo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19th-century Danish Mathematicians
The 19th (nineteenth) century began on 1 January 1801 ( MDCCCI), and ended on 31 December 1900 ( MCM). The 19th century was the ninth century of the 2nd millennium. The 19th century was characterized by vast social upheaval. Slavery was abolished in much of Europe and the Americas. The First Industrial Revolution, though it began in the late 18th century, expanding beyond its British homeland for the first time during this century, particularly remaking the economies and societies of the Low Countries, the Rhineland, Northern Italy, and the Northeastern United States. A few decades later, the Second Industrial Revolution led to ever more massive urbanization and much higher levels of productivity, profit, and prosperity, a pattern that continued into the 20th century. The Islamic gunpowder empires fell into decline and European imperialism brought much of South Asia, Southeast Asia, and almost all of Africa under colonial rule. It was also marked by the collapse of the large S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Royal Danish Academy Of Sciences And Letters
{{Infobox organization , name = The Royal Danish Academy of Sciences and Letters , full_name = , native_name = Det Kongelige Danske Videnskabernes Selskab , native_name_lang = , logo = Royal Danish Academy of Sciences and Letters seal.svg , logo_size = 150 , logo_alt = , logo_caption = , image = Carlsbergfondet.JPG , image_size = , alt = , caption = The building on H.C. Andersens Boulevard. , map = , map_size = , map_alt = , map_caption = , map2 = , map2_size = , map2_alt = , map2_caption = , abbreviation = , nickname = , pronounce = , pronounce ref = , pronounce comment = , pronounce 2 = , named_after = , motto = , predecessor = , merged ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Siegel Theta Function
In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as :\theta(t) = \arg \left( \Gamma\left(\frac+\frac\right) \right) - \frac t for real values of ''t''. Here the argument is chosen in such a way that a continuous function is obtained and \theta(0)=0 holds, i.e., in the same way that the principal branch of the log-gamma function is defined. It has an asymptotic expansion :\theta(t) \sim \frac\log \frac - \frac - \frac+\frac+ \frac+\cdots which is not convergent, but whose first few terms give a good approximation for t \gg 1. Its Taylor-series at 0 which converges for , t, 6.29, and has local extrema at \pm 6.289835988\ldots, with value \mp 3.530972829\ldots. It has a single inflection point at t=0 with \theta^\prime(0)= -\frac = -2.6860917\ldots, which is the minimum of its derivative. Theta as a function of a complex variable We have an infinite series expression for the log-gamma function :\log \Gamma \left(z\right) = -\g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edgeworth Series
The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants. The series are the same; but, the arrangement of terms (and thus the accuracy of truncating the series) differ. The key idea of these expansions is to write the characteristic function of the distribution whose probability density function is to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover through the inverse Fourier transform. Gram–Charlier A series We examine a continuous random variable. Let \hat be the characteristic function of its distribution whose density function is , and \kappa_r its cumulants. We expand in terms of a known distribution with probability density function , characteristic function \hat, and cumulants \gamma_r. The density is generally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gram's Theorem
In mathematics, Gram's theorem states that an algebraic set in a finite-dimensional vector space invariant under some linear group can be defined by absolute invariant Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descript ...s. . It is named after J. P. Gram, who published it in 1874. References *. Reprinted by Academic Press (1971), . *. Invariant theory Theorems in algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Curve
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric " bell curve" shape. The parameter is the height of the curve's peak, is the position of the center of the peak, and (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value and variance . In this case, the Gaussian is of the form g(x) = \frac \exp\left( -\frac \frac \right). Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
