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Gibbs Lemma
200px, Josiah Willard Gibbs In game theory and in particular the study of Blotto games and operational research, the Gibbs lemma is a result that is useful in maximization problems. It is named for Josiah Willard Gibbs. Consider \phi=\sum_^n f_i(x_i). Suppose \phi is maximized, subject to \sum x_i=X and x_i\geq 0, at x^0=(x_1^0,\ldots,x_n^0). If the f_i are differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point i ..., then the Gibbs lemma states that there exists a \lambda such that :\begin f'_i(x_i^0)&=\lambda \mbox x_i^0>0\\ &\leq\lambda\mbox x_i^0=0. \end Notes References Game theory {{gametheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josiah Willard Gibbs -from MMS-
Josiah ( or ) or Yoshiyahu; la, Iosias was the 16th king of Judah (–609 BCE) who, according to the Hebrew Bible, instituted major religious reforms by removing official worship of gods other than Yahweh. Josiah is credited by most biblical scholars with having established or compiled important Hebrew scriptures during the " Deuteronomic reform" which probably occurred during his rule. Josiah became king of the Kingdom of Judah at the age of eight, after the assassination of his father, King Amon. Josiah reigned for 31 years, from 641/640 to 610/609 BCE. Josiah is known only from biblical texts; no reference to him exists in other surviving texts of the period from Egypt or Babylon, and no clear archaeological evidence, such as inscriptions bearing his name, has ever been found. Nevertheless, most scholars believe that he existed historically and that the absence of documents is due to few documents of any sort surviving from this period, and to Jerusalem having been occupie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blotto Games
A Colonel Blotto game is a type of two-person constant-sum game in which the players (officers) are tasked to simultaneously distribute limited resources over several objects (battlefields). In the classic version of the game, the player devoting the most resources to a battlefield wins that battlefield, and the gain (or payoff) is equal to the total number of battlefields won. The game was first proposed by Émile Borel in 1921. In 1938 Borel and Ville published a particular optimal strategy (the "disk" solution). The game was studied after the Second World War by scholars in Operation Research, and became a classic in game theory. Gross and Wagner's 1950 research memorandum states Borel's optimal strategy, and coined the fictitious Colonel Blotto and Enemy names. For three battlefields or more, the space of pure strategies is multi-dimensional (two dimensions for three battlefields) and a mixed strategy is thus a probability distribution over a continuous set. The game is a rar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operational Research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve decision-making. It is considered to be a subfield of mathematical sciences. The term management science is occasionally used as a synonym. Employing techniques from other mathematical sciences, such as mathematical model, modeling, statistics, and mathematical optimization, optimization, operations research arrives at optimal or near-optimal solutions to decision-making problems. Because of its emphasis on practical applications, operations research has overlap with many other disciplines, notably industrial engineering. Operations research is often concerned with determining the extreme values of some real-world objective: the Maxima and minima, maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost). Originating in mil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous inductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of Statistical ensemble (mathematical physics), ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he invented modern vector calculus (independently of the British scientist Oliver Heaviside, who carried out similar work during the same period). In 1863, Yale University, Yale awarded Gibbs the first American Doctor of Philosophy, doctorate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
