title: Stochastic process

text: A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration formula_117, on a probability space formula_30 is a family of sigma-algebras such that formula_119 for all formula_120, where formula_121 and formula_122 denotes the total order of the index set formula_35. With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process formula_15 at formula_42, which can be interpreted as time formula_16. The intuition behind a filtration formula_127 is that as time formula_16 passes, more and more information on formula_15 is known or available, which is captured in formula_127, resulting in finer and finer partitions of formula_31.


title: Stochastic process

text: A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process formula_11 that has the same index set formula_35, state space formula_36, and probability space formula_135 as another stochastic process formula_136 is said to be a modification of formula_136 if for all formula_42 the following


title: Stochastic process

text: holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law and they are said to be stochastically equivalent or equivalent.


title: Stochastic process

text: Instead of modification, the term version is also used, however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.


title: Stochastic process

text: If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version. The theorem can also be generalized to random fields so the index set is formula_1-dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces.


title: Stochastic process

text: Two stochastic processes formula_11 and formula_136 defined on the same probability space formula_142 with the same index set formula_35 and set space formula_36 are said be indistinguishable if the following


title: Stochastic process

text: holds. If two formula_11 and formula_136 are modifications of each other and are almost surely continuous, then formula_11 and formula_136 are indistinguishable.


title: Stochastic process

text: Separability is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space, which means that the index set has a dense countable subset.


title: Stochastic process

text: More precisely, a real-valued continuous-time stochastic process formula_11 with a probability space formula_135 is separable if its index set formula_35 has a dense countable subset formula_152 and there is a set formula_153 of probability zero, so formula_154, such that for every open set formula_155 and every closed set formula_156, the two events formula_157 and formula_158 differ from each other at most on a subset of formula_159.


title: Stochastic process

text: The definition of separability can also be stated for other index sets and state spaces, such as in the case of random fields, where the index set as well as the state space can be formula_1-dimensional Euclidean space.


title: Stochastic process

text: The concept of separability of a stochastic process was introduced by Joseph Doob. The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process. Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.


title: Stochastic process

text: Two stochastic processes formula_11 and formula_136 defined on the same probability space formula_142 with the same index set formula_35 are said be independent if for all formula_165 and for every choice of epochs formula_166, the random vectors formula_167 and formula_168 are independent.


title: Stochastic process

text: Two stochastic processes formula_169 and formula_170 are called uncorrelated if their cross-covariance formula_171 is zero for all times. Formally:


title: Stochastic process

text: If two stochastic processes formula_11 and formula_136 are independent, then they are also uncorrelated.


title: Stochastic process

text: Two stochastic processes formula_169 and formula_170 are called orthogonal if their cross-correlation formula_177 is zero for all times. Formally:


title: Stochastic process

text: A Skorokhod space, also written as Skorohod space, is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as formula_179 or formula_180, and take values on the real line or on some metric space. Such functions are known as càdlàg or cadlag functions, based on the acronym of the French phrase "continue à droite, limite à gauche". A Skorokhod function space, introduced by Anatoliy Skorokhod, is often denoted with the letter formula_181, so the function space is also referred to as space formula_181. The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, formula_183 denotes the space of càdlàg functions defined on the unit interval formula_179.


title: Stochastic process

text: Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space. Such spaces contain continuous functions, which correspond to sample functions of the Wiener process. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process (on the real line), are also members of this space.


title: Stochastic process

text: In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues. For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.


title: Stochastic process

text: Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.


title: Stochastic process

text: The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.


title: Stochastic process

text: A Markov chain is a type of Markov process that has either discrete state space or discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space). It has been argued that the first definition of a Markov chain, where it has discrete time, now tends to be used, despite the second definition having been used by researchers like Joseph Doob and Kai Lai Chung.


title: Stochastic process

text: Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.


title: Stochastic process

text: The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as formula_1-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.


title: Stochastic process

text: A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, but they can also be complex-valued or even more general.


title: Stochastic process

text: A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. For a sequence of independent and identically distributed random variables formula_186 with zero mean, the stochastic process formed from the successive partial sums formula_187 is a discrete-time martingale. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.


title: Stochastic process

text: Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the "compensated Poisson process". Martingales can also be built from other martingales. For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.


title: Stochastic process

text: Martingales mathematically formalize the idea of a fair game, and they were originally developed to show that it is not possible to win a fair game. But now they are used in many areas of probability, which is one of the main reasons for studying them. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.


title: Stochastic process

text: Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance.


title: Stochastic process

text: Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their stationarity and independence properties, so they were known as "processes with stationary and independent increments". In other words, a stochastic process formula_11 is a Lévy process if for formula_1 non-negatives numbers, formula_190, the corresponding formula_191 increments


title: Stochastic process

text: are all independent of each other, and the distribution of each increment only depends on the difference in time.


title: Stochastic process

text: A Lévy process can be defined such that its state space is some abstract mathematical space, such as a Banach space, but the processes are often defined so that they take values in Euclidean space. The index set is the non-negative numbers, so formula_192, which gives the interpretation of time. Important stochastic processes such as the Wiener process, the homogeneous Poisson process (in one dimension), and subordinators are all Lévy processes.


title: Stochastic process

text: A random field is a collection of random variables indexed by a formula_1-dimensional Euclidean space or some manifold. In general, a random field can be considered an example of a stochastic or random process, where the index set is not necessarily a subset of the real line. But there is a convention that an indexed collection of random variables is called a random field when the index has two or more dimensions. If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.


title: Stochastic process

text: A point process is a collection of points randomly located on some mathematical space such as the real line, formula_1-dimensional Euclidean space, or more abstract spaces. Sometimes the term "point process" is not preferred, as historically the word "process" denoted an evolution of some system in time, so a point process is also called a random point field. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear.


title: Stochastic process

text: Other authors consider a point process as a stochastic process, where the process is indexed by sets of the underlying space on which it is defined, such as the real line or formula_1-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.


title: Stochastic process

text: Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago, but very little analysis on them was done in terms of probability. The year 1654 is often considered the birth of probability theory when French mathematicians Pierre Fermat and Blaise Pascal had a written correspondence on probability, motivated by a gambling problem. But there was earlier mathematical work done on the probability of gambling games such as "Liber de Ludo Aleae" by Gerolamo Cardano, written in the 16th century but posthumously published later in 1663.


title: Stochastic process

text: After Cardano, Jakob Bernoulli wrote Ars Conjectandi, which is considered a significant event in the history of probability theory. Bernoulli's book was published, also posthumously, in 1713 and inspired many mathematicians to study probability. But despite some renowned mathematicians contributing to probability theory, such as Pierre-Simon Laplace, Abraham de Moivre, Carl Gauss, Siméon Poisson and Pafnuty Chebyshev, most of the mathematical community did not consider probability theory to be part of mathematics until the 20th century.


title: Stochastic process

text: In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics, where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as Rudolf Clausius, most of the work had little or no randomness.


title: Stochastic process

text: This changed in 1859 when James Clerk Maxwell contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities. The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, Ludwig Boltzmann and Josiah Gibbs, which would later have an influence on Albert Einstein's mathematical model for Brownian movement.


title: Stochastic process

text: At the International Congress of Mathematicians in Paris in 1900, David Hilbert presented a list of mathematical problems, where his sixth problem asked for a mathematical treatment of physics and probability involving axioms. Around the start of the 20th century, mathematicians developed measure theory, a branch of mathematics for studying integrals of mathematical functions, where two of the founders were French mathematicians, Henri Lebesgue and Émile Borel. In 1925 another French mathematician Paul Lévy published the first probability book that used ideas from measure theory.


title: Stochastic process

text: In 1920s fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as Sergei Bernstein, Aleksandr Khinchin, and Andrei Kolmogorov. Kolmogorov published in 1929 his first attempt at presenting a mathematical foundation, based on measure theory, for probability theory. In the early 1930s Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as Eugene Slutsky and Nikolai Smirnov, and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.


title: Stochastic process

text: In 1933 Andrei Kolmogorov published in German, his book on the foundations of probability theory titled "Grundbegriffe der Wahrscheinlichkeitsrechnung", where Kolmogorov used measure theory to develop an axiomatic framework for probability theory. The publication of this book is now widely considered to be the birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics.


title: Stochastic process

text: After the publication of Kolmogorov's book, further fundamental work on probability theory and stochastic processes was done by Khinchin and Kolmogorov as well as other mathematicians such as Joseph Doob, William Feller, Maurice Fréchet, Paul Lévy, Wolfgang Doeblin, and Harald Cramér.


title: Stochastic process

text: Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory". World War II greatly interrupted the development of probability theory, causing, for example, the migration of Feller from Sweden to the United States of America and the death of Doeblin, considered now a pioneer in stochastic processes.


title: Stochastic process

text: After World War II the study of probability theory and stochastic processes gained more attention from mathematicians, with significant contributions made in many areas of probability and mathematics as well as the creation of new areas. Starting in the 1940s, Kiyosi Itô published papers developing the field of stochastic calculus, which involves stochastic integrals and stochastic differential equations based on the Wiener or Brownian motion process.


title: Stochastic process

text: Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of potential theory, with early ideas by Shizuo Kakutani and then later work by Joseph Doob. Further work, considered pioneering, was done by Gilbert Hunt in the 1950s, connecting Markov processes and potential theory, which had a significant effect on the theory of Lévy processes and led to more interest in studying Markov processes with methods developed by Itô.


title: Stochastic process

text: In 1953 Doob published his book "Stochastic processes", which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability.


title: Stochastic process

text: Other fields of probability were developed and used to study stochastic processes, with one main approach being the theory of large deviations. The theory has many applications in statistical physics, among other fields, and has core ideas going back to at least the 1930s. Later in the 1960s and 1970s fundamental work was done by Alexander Wentzell in the Soviet Union and Monroe D. Donsker and Srinivasa Varadhan in the United States of America, which would later result in Varadhan winning the 2007 Abel Prize. In the 1990s and 2000s the theories of Schramm–Loewner evolution and rough paths were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in Fields Medals being awarded to Wendelin Werner in 2008 and to Martin Hairer in 2014.


title: Stochastic process

text: The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.


title: Stochastic process

text: Although Khinchin gave mathematical definitions of stochastic processes in the 1930s, specific stochastic processes had already been discovered in different settings, such as the Brownian motion process and the Poisson process. Some families of stochastic processes such as point processes or renewal processes have long and complex histories, stretching back centuries.


title: Stochastic process

text: The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied. The process is a sequence of independent Bernoulli trials, which are named after Jackob Bernoulli who used them to study games of chance, including probability problems proposed and studied earlier by Christiaan Huygens. Bernoulli's work, including the Bernoulli process, were published in his book "Ars Conjectandi" in 1713.


title: Stochastic process

text: In 1905 Karl Pearson coined the term "random walk" while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields. Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks. For example, the problem known as the "Gambler's ruin" is based on a simple random walk, and is an example of a random walk with absorbing barriers. Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods, and then more detailed solutions were presented by Jakob Bernoulli and Abraham de Moivre.


title: Stochastic process

text: For random walks in formula_1-dimensional integer lattices, George Pólya published, in 1919 and 1921, work where he studied the probability of a symmetric random walk returning to a previous position in the lattice. Pólya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.


title: Stochastic process

text: The Wiener process or Brownian motion process has its origins in different fields including statistics, finance and physics. In 1880, Thorvald Thiele wrote a paper on the method of least squares, where he used the process to study the errors of a model in time-series analysis. The work is now considered as an early discovery of the statistical method known as Kalman filtering, but the work was largely overlooked. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time.


title: Stochastic process

text: The French mathematician Louis Bachelier used a Wiener process in his 1900 thesis in order to model price changes on the Paris Bourse, a stock exchange, without knowing the work of Thiele. It has been speculated that Bachelier drew ideas from the random walk model of Jules Regnault, but Bachelier did not cite him, and Bachelier's thesis is now considered pioneering in the field of financial mathematics.


title: Stochastic process

text: It is commonly thought that Bachelier's work gained little attention and was forgotten for decades until it was rediscovered in the 1950s by the Leonard Savage, and then become more popular after Bachelier's thesis was translated into English in 1964. But the work was never forgotten in the mathematical community, as Bachelier published a book in 1912 detailing his ideas, which was cited by mathematicians including Doob, Feller and Kolmogorov. The book continued to be cited, but then starting in the 1960s the original thesis by Bachelier began to be cited more than his book when economists started citing Bachelier's work.


title: Stochastic process

text: In 1905 Albert Einstein published a paper where he studied the physical observation of Brownian motion or movement to explain the seemingly random movements of particles in liquids by using ideas from the kinetic theory of gases. Einstein derived a differential equation, known as a diffusion equation, for describing the probability of finding a particle in a certain region of space. Shortly after Einstein's first paper on Brownian movement, Marian Smoluchowski published work where he cited Einstein, but wrote that he had independently derived the equivalent results by using a different method.


title: Stochastic process

text: Einstein's work, as well as experimental results obtained by Jean Perrin, later inspired Norbert Wiener in the 1920s to use a type of measure theory, developed by Percy Daniell, and Fourier analysis to prove the existence of the Wiener process as a mathematical object.


title: Stochastic process

text: The Poisson process is named after Siméon Poisson, due to its definition involving the Poisson distribution, but Poisson never studied the process. There are a number of claims for early uses or discoveries of the Poisson


title: Stochastic process

text: At the beginning of the 20th century the Poisson process would arise independently in different situations.


title: Stochastic process

text: In Sweden 1903, Filip Lundberg published a thesis containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.


title: Stochastic process

text: Another discovery occurred in Denmark in 1909 when A.K. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.


title: Stochastic process

text: In 1910 Ernest Rutherford and Hans Geiger published experimental results on counting alpha particles. Motivated by their work, Harry Bateman studied the counting problem and derived Poisson probabilities as a solution to a family of differential equations, resulting in the independent discovery of the Poisson process. After this time there were many studies and applications of the Poisson process, but its early history is complicated, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and various physical scientists.


title: Stochastic process

text: Markov processes and Markov chains are named after Andrey Markov who studied Markov chains in the early 20th century. Markov was interested in studying an extension of independent random sequences. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption, which had been commonly regarded as a requirement for such mathematical laws to hold. Markov later used Markov chains to study the distribution of vowels in Eugene Onegin, written by Alexander Pushkin, and proved a central limit theorem for such chains.


title: Stochastic process

text: In 1912 Poincaré studied Markov chains on finite groups with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest in 1907, and a branching process, introduced by Francis Galton and Henry William Watson in 1873, preceding the work of Markov. After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by Irénée-Jules Bienaymé. Starting in 1928, Maurice Fréchet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.


title: Stochastic process

text: Andrei Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as Norbert Wiener's work on Einstein's model of Brownian movement. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes. Independent of Kolmogorov's work, Sydney Chapman derived in a 1928 paper an equation, now called the Chapman–Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement. The differential equations are now called the Kolmogorov equations or the Kolmogorov–Chapman equations. Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in the 1930s, and then later Eugene Dynkin, starting in the 1950s.


title: Stochastic process

text: Lévy processes such as the Wiener process and the Poisson process (on the real line) are named after Paul Lévy who started studying them in the 1930s, but they have connections to infinitely divisible distributions going back to the 1920s. In a 1932 paper Kolmogorov derived a characteristic function for random variables associated with Lévy processes. This result was later derived under more general conditions by Lévy in 1934, and then Khinchin independently gave an alternative form for this characteristic function in 1937. In addition to Lévy, Khinchin and Kolomogrov, early fundamental contributions to the theory of Lévy processes were made by Bruno de Finetti and Kiyosi Itô.


title: Stochastic process

text: In mathematics, constructions of mathematical objects are needed, which is also the case for stochastic processes, to prove that they exist mathematically. There are two main approaches for constructing a stochastic process. One approach involves considering a measurable space of functions, defining a suitable measurable mapping from a probability space to this measurable space of functions, and then deriving the corresponding finite-dimensional distributions.


title: Stochastic process

text: Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem to prove a corresponding stochastic process exists. This theorem, which is an existence theorem for measures on infinite product spaces, says that if any finite-dimensional distributions satisfy two conditions, known as "consistency conditions", then there exists a stochastic process with those finite-dimensional distributions.


title: Stochastic process

text: When constructing continuous-time stochastic processes certain mathematical difficulties arise, due to the uncountable index sets, which do not occur with discrete-time processes. One problem is that is it possible to have more than one stochastic process with the same finite-dimensional distributions. For example, both the left-continuous modification and the right-continuous modification of a Poisson process have the same finite-dimensional distributions. This means that the distribution of the stochastic process does not, necessarily, specify uniquely the properties of the sample functions of the stochastic process.


title: Stochastic process

text: Another problem is that functionals of continuous-time process that rely upon an uncountable number of points of the index set may not be measurable, so the probabilities of certain events may not be well-defined. For example, the supremum of a stochastic process or random field is not necessarily a well-defined random variable. For a continuous-time stochastic process formula_11, other characteristics that depend on an uncountable number of points of the index set formula_35 include:


title: Stochastic process

text: One approach for avoiding mathematical construction issues of stochastic processes, proposed by Joseph Doob, is to assume that the stochastic process is separable. Separability ensures that infinite-dimensional distributions determine the properties of sample functions by requiring that sample functions are essentially determined by their values on a dense countable set of points in the index set. Furthermore, if a stochastic process is separable, then functionals of an uncountable number of points of the index set are measurable and their probabilities can be studied.


title: Stochastic process

text: Another approach is possible, originally developed by Anatoliy Skorokhod and Andrei Kolmogorov, for a continuous-time stochastic process with any metric space as its state space. For the construction of such a stochastic process, it is assumed that the sample functions of the stochastic process belong to some suitable function space, which is usually the Skorokhod space consisting of all right-continuous functions with left limits. This approach is now more used than the separability assumption, but such a stochastic process based on this approach will be automatically separable.


title: Stochastic process

text: Although less used, the separability assumption is considered more general because every stochastic process has a separable version. It is also used when it is not possible to construct a stochastic process in a Skorokhod space. For example, separability is assumed when constructing and studying random fields, where the collection of random variables is now indexed by sets other than the real line such as formula_1-dimensional Euclidean space.


title: Serdar Berdimuhamedow

text: Serdar Gurbangulyýewiç Berdimuhamedow (born 22 September 1981) is the third and current president of Turkmenistan, serving since 19 March 2022. Berdimuhamedow had previously served in several other positions within the government of his father, Gurbanguly Berdimuhamedow.


title: Serdar Berdimuhamedow

text: In 2021, he became one of several deputy chairmen in the Cabinet of Ministers of Turkmenistan. Having won 73 percent of the vote in the 2022 Turkmenistan presidential election, considered to be neither free nor fair, he succeeded his father's 15-year-long authoritarian tenure as president.


title: Serdar Berdimuhamedow

text: Serdar Berdimuhamedow was born on 22 September 1981 in Ashgabat. He studied at Secondary School No. 43 in Ashgabat from 1987 to 1997. Between 1997 and 2001, he studied at the Turkmen Agricultural University, graduating as an engineer-technologist.


title: Serdar Berdimuhamedow

text: From July to November 2001, he worked in the Directorate for Foreign Economic Relations of the Food Processing Association, followed by two years of mandatory military service. From 2003 to 2008, he worked again at the Food Processing Association in the fruit and vegetable department and in the non-alcoholic beer and wine department.


title: Serdar Berdimuhamedow

text: Between 2008 and 2011, Berdimuhamedow studied at the Diplomatic Academy of the Russian Ministry of Foreign Affairs, earning a diploma in international relations. During this period he was also assigned to the Turkmen Embassy in Moscow as counselor of embassy. From 2011 to 2013, he studied at the Geneva Centre for Security Policy, earning a post-graduate degree in European and international security affairs. He was concurrently assigned to the Turkmen Mission to the United Nations in Geneva as a counselor of embassy.


title: Serdar Berdimuhamedow

text: From August to December 2013, he served as head of the European Department of the Turkmen Ministry of Foreign Affairs. From 2013 to 2016 he was deputy director of the State Agency for Management and Use of Hydrocarbon Resources. From 2016 to 2017 he was chairman of the Department of International Information of the Turkmen Ministry of Foreign Affairs.


title: Serdar Berdimuhamedow

text: On 18 August 2014, Serdar Berdimuhamedow defended a dissertation at the Turkmen Academy of Sciences to earn the degree of candidate of science. In July 2015, he was awarded a doctorate in technical sciences.


title: Serdar Berdimuhamedow

text: Berdimuhamedow served two years of mandatory military service from 2001 to 2003. On 27 October 2017, Berdimuhamedow was promoted from the rank of major to lieutenant colonel. His promotion ceremony was broadcast on national television. His branch of service is unknown, but on 7 March 2021, he was shown on national television in uniform wearing the shoulder patch of the Armed Forces of Turkmenistan. The same video showed him wearing shoulder boards indicating the rank of colonel, and the decree awarding him the honorary title of ""Defender of Homeland Turkmenistan"" referred to him as a colonel.


title: Serdar Berdimuhamedow

text: In November 2016, Berdimuhamedow won a seat in the Mejlis representing the 25th parliamentary district centered on Dushak, a town in Ahal Province. In March 2018, he was re-elected to the Assembly of Turkmenistan with over 90% of the votes cast in his district. He chaired the committee on legislation and norms as of March 2017.


title: Serdar Berdimuhamedow

text: Berdimuhamedow rose to the political spotlight following his appointment as deputy minister of foreign affairs in March 2018. In January 2019, Berdimuhamedow was shifted to the position of deputy governor of Ahal Province, the native region of most of the Turkmen elite. In June 2019, Berdimuhamedow was elevated to governor ("häkim") of Ahal Province. In February 2020, Berdimuhamedow was appointed minister of industry and construction materials of Turkmenistan.


title: Serdar Berdimuhamedow

text: On 11 February 2021, Berdimuhamedow was promoted to deputy chairman of the Cabinet of Ministers for innovation and digitization, a new position. He was simultaneously appointed to the State Security Council and chairman of the Supreme Control Chamber of Turkmenistan. On 9 July 2021, Berdimuhamedow was relieved of his positions on the State Security Council and Supreme Control Chamber, and as deputy chairman was assigned the economics and finance portfolio with specific responsibility for "economic and banking issues and international financial organizations".


title: Serdar Berdimuhamedow

text: He is chairperson of the Turkmen Alabay Dog Association, and president of the International Ahal Teke Horse Breeding Association.


title: Serdar Berdimuhamedow

text: It was confirmed in February 2022 that Berdimuhamedow was running for an early presidential election on 12 March, fuelling speculation that he would be the successor to his father as President of Turkmenistan. He won the election with 72.97% of the vote, and became his father's successor as President, establishing a political dynasty. The election was seen by many international observers as neither free nor fair.


title: Serdar Berdimuhamedow

text: Serdar was inaugurated on 19 March at the Ruhyýet Palace. He took the oath to the people on the Constitution of Turkmenistan and the Quran. He was then presented the presidential badge of office by Turkmen elders. From the palace, Berdimuhamedow, accompanied by a mounted escort, drove to Galkynysh Square and then to the Oguzhan Presidential Palace Complex where the Armed Forces of Turkmenistan performed a military parade. Deputy Prime Minister and Secretary of the State Security Council of Turkmenistan Çarymyrat Amanow reported to Berdimuhamedow. Later that day an official reception was given at the Audience Center of Turkmenistan. A festive Sadaqah was also given at the Hazret Omar Mosque.


title: Serdar Berdimuhamedow

text: On 25 March, he announced the new composition of his cabinet. The only new government member he appointed was Muhammetguly Muhammedow, who replaced him as Deputy Chairman of the Cabinet of Ministers for Economics and Finance. According to some, he intends to replace Foreign Minister and "de facto" Vice President of Turkmenistan Raşit Meredow with Esen Aydogdyev, rector of the International University of Humanities and Development.


title: Serdar Berdimuhamedow

text: On the first day of his presidency, Berdimuhamedow accepted the credentials of Fazal Muhammad Sabir, making Turkmenistan the first country in Central Asia to accept the diplomats of the Taliban government of Afghanistan. Former Indian President Ram Nath Kovind made a state visit to Turkmenistan on 1–4 April, marking the first visit of the President of India to Turkmenistan, and the first foreign head of state to be received by Berdimuhamedow. The two leaders discussed expanding trade and energy cooperation. President Berdimuhamedow made his first foreign visit, to Saudi Arabia, on 1–2 June. Later that month, he made visits to Russia and Iran, on the visit to the latter he used the first Aurus Senat limousine purchased by the Turkmen government.


title: Serdar Berdimuhamedow

text: During a State Security Council meeting in early April, he also removed Colonel General Çarymyrat Amanow from the post of Secretary of the State Security Council of Turkmenistan in favor of the head of the Defense Ministry, Begench Gundogdyev, due to Amanow's "transfer to another job." He also abolished Amanow's concurrent office of Deputy Chairman of the Cabinet responsible for Security, Military, and Justice. He also replaced the Minister of Internal Affairs, Colonel Öwezdurdy Hojanyýazow, with Colonel Muhammet Hydyrow, due to "shortcomings" and "improper control over public order, traffic safety as well as activity of local representatives of the police."


title: Serdar Berdimuhamedow

text: Berdimuhamedow is married and has four children. A classmate at the Russian Diplomatic Academy described him as even-tempered and unassuming. Instructors and other classmates reportedly described him as "modest, responsive, polite, and calm."


title: Serdar Berdimuhamedow

text: On 30 May 2022, Berdimuhamedow's mother Ogulgerek Berdimuhamedova was awarded the title of Honored Carpet Weaver of Turkmenistan for her many years at the carpet factory in the city of Geoktepe. On 1–2 June 2022, he undertook his first foreign visit to Saudi Arabia, where he and his son performed Umrah.


title: Tim Curry performances

text: This article presents the theatre, film, television, video games, and audiobook credits of English actor Tim Curry, who has a diverse range of work in these media formats. His most frequent roles are villainous roles or character parts.


title: Carlo Acutis

text: Carlo Acutis (3 May 1991 – 12 October 2006) was an English-born Italian Catholic youth and amateur computer programmer, who is best known for documenting Eucharistic miracles around the world and cataloguing them onto a website which he created before his death from leukemia. He was noted for his cheerfulness, computer skills, and deep devotion to the Eucharist, which became a core theme of his life. He was beatified on 10 October 2020, two days before the 14th anniversary of his death.


title: Carlo Acutis

text: Carlo Acutis was born in London on 3 May 1991 to a wealthy Italian family. His baptism took place on 18 May 1991 in the church of Our Lady of Dolours, Chelsea. His parents, Andrea Acutis and Antonia Salzano, who were not especially religious, had worked in London and Germany, finally settled in Milan in September 1991, not long after their first son's birth. In 1995, when Acutis was four years old, his maternal grandfather died and was said to have appeared to him in a dream asking to be prayed for. When the child evinced a precocious interest in religious practice, his questions were answered by the family's Polish babysitter. Three years later he requested to receive his First Communion at the age of seven. After consulting a prelate and providing instruction, the family arranged this at the convent of Sant'Ambrogio ad Nemus. After that he made the effort, either before or after Mass, to reflect before the tabernacle. Acutis became a frequent communicant and would make a weekly confession. He is said to have had several models as guides for his life, especially Francis of Assisi, as well as Francisco and Jacinta Marto, Dominic Savio, Tarcisius, and Bernadette Soubirous.


title: Carlo Acutis

text: He was educated in Milan at the Jesuit Instituto Leone XIII high school. On the social side, Acutis would worry about friends of his whose parents were divorcing and would invite them to his home to support them. He defended disabled peers at school when bullies mocked them. Outside school, he did volunteer work with the homeless and destitute. He also liked films, comic editing and playing PlayStation video games. Although he greatly enjoyed travel, the town of Assisi remained a particular favorite.


title: Carlo Acutis

text: Those around him considered him a "computer geek" on account of his passion and skill with computers and the internet. Acutis applied himself to creating a website dedicated to cataloguing each reported Eucharistic miracle in the world. He completed this in 2005, having started compiling the catalogue at the age of eleven. He admired Giacomo Alberione's initiatives to use the media to evangelize and proclaim the Gospel and aimed to do likewise with the website he had created.


title: Carlo Acutis

text: When he developed leukemia, he offered his suffering both for Pope Benedict XVI and for the Catholic Church, saying: "I offer to the Lord the sufferings that I will have to undergo for the Pope and for the Church." He had asked his parents to take him on pilgrimages to the sites of all the known Eucharistic miracles in the world, but his declining health prevented this from happening.


title: Carlo Acutis

text: The doctors treating his final illness had asked him if he was in great pain, a question to which he responded that "there are people who suffer much more than me". He died on 12 October 2006 at 06:45 from the M3 subtype of fulminating acute promyelocytic leukemia. He was 15 years old at the time of his death. He was buried in Assisi in accordance with his wishes.


title: Carlo Acutis

text: Acutis's mother, Antonia, is said to attribute to his intercession the fact that, at the age of 44, she gave birth to twins, born exactly four years to the day after his death. Following the Catholic Church's recognition of a miracle in 2020, attributed to Acutis, Antonia told the press that her son had appeared to her in dreams saying that he will not only be beatified but also canonised a saint in the future.


title: Carlo Acutis

text: In memory of Acutis, bishops Raffaello Martinelli and Angelo Comastri have helped to organize a travelling photo exhibition of all the Eucharistic miracle sites. It has since travelled to dozens of different countries across five continents.


title: Carlo Acutis

text: On 15 December 2020, under the provisions of Canon Law, Archbishop Bernard established a new parish in the Archdiocese of Birmingham under the patronage of the Blessed Carlo Acutis. The parish incorporates the three churches of St Peter & Paul, St Michael and St Bernadette in Wolverhampton.


title: Carlo Acutis

text: In July 2021, the parishes of St. John Berchmans in Logan Square, Chicago, and St. Hedwig in Bucktown combined under the patronage of Blessed Carlo Acutis.


title: Carlo Acutis

text: In April 2022, the first life-sized statue of Acutis in the United Kingdom was erected at Carfin Grotto, North Lanarkshire, Scotland.


title: Carlo Acutis

text: The call for him to be beatified began not long after Acutis's death. The campaign gained momentum in 2013 after he was named a Servant of God, the first stage on the path towards sainthood. The Lombardy Episcopal Conference approved the petition for the official canonization cause to proceed at a meeting in 2013. The opening of the diocesan investigation was held on 15 February 2013, with Cardinal Angelo Scola inaugurating the process, and concluding it on 24 November 2016. The formal introduction to the cause took place on 13 May 2013, and Acutis became titled a "Servant of God". Pope Francis next confirmed his life as one of heroic virtue on 5 July 2018, and declared him Venerable.


title: Carlo Acutis

text: On 14 November 2019, the Vatican's Medical Council of the Congregation for the Causes of Saints expressed a positive opinion about a miracle in Brazil attributed to Acutis's intercession. Luciana Vianna had taken her son, Mattheus, who was born with a pancreatic defect that made eating difficult, to a prayer service. Beforehand, Vianna had already prayed a novena asking for the teenager Acutis's intercession. During the service her son had simply asked that he should not "throw up as much". Immediately following the service, Mattheus told his mother that he felt healed and asked for solid food when he came home. Until then he had been on an all-liquid diet. After a detailed investigation, Pope Francis confirmed the miracle's authenticity in a decree on 21 February 2020, leading to Acutis's beatification.


title: Carlo Acutis

text: Within a month of the decree, the beatification ceremony was postponed due to the COVID-19 pandemic in Italy, during which the country was placed on lockdown. It was rescheduled for 10 October 2020 and was held in the Upper Church of the Basilica of Saint Francis of Assisi in Assisi, Italy, with Cardinal Agostino Vallini presiding on the Pope's behalf. , the postulator for Acutis's cause is Nicola Gori.


title: Carlo Acutis

text: Since the beatification ceremony on 10 October 2020, silent crowds have been filing past the exposed relics of the blessed youth in the one-time cathedral of Assisi, the church of Santa Maria Maggiore.


title: Kangxi Emperor

text: The Kangxi Emperor (4 May 1654– 20 December 1722), also known by his temple name Emperor Shengzu of Qing, born Xuanye, was the third emperor of the Qing dynasty, and the second Qing emperor to rule over China proper, reigning from 1661 to 1722.


title: Kangxi Emperor

text: The Kangxi Emperor's reign of 61 years makes him the longest-reigning emperor in Chinese history (although his grandson, the Qianlong Emperor, had the longest period of "de facto" power, ascending as an adult and maintaining effective power until his death) and one of the longest-reigning rulers in history. However, since he ascended the throne at the age of seven, actual power was held for six years by four regents and his grandmother, the Grand Empress Dowager Xiaozhuang.


title: Kangxi Emperor

text: The Kangxi Emperor is considered one of China's greatest emperors. He suppressed the Revolt of the Three Feudatories, forced the Kingdom of Tungning in Taiwan and assorted Mongol rebels in the North and Northwest to submit to Qing rule, and blocked Tsarist Russia on the Amur River, retaining Outer Manchuria.


title: Kangxi Emperor

text: The Kangxi Emperor's reign brought about long-term stability and relative wealth after years of war and chaos. He initiated the period known as the "Prosperous Era of Kangxi and Qianlong" or "High Qing", which lasted for several generations after his death. His court also accomplished such literary feats as the compilation of the "Kangxi Dictionary" and "Quan Tangshi" poetry anthology.


title: Kangxi Emperor

text: Born on 4 May 1654 to the Shunzhi Emperor and Empress Xiaokangzhang in Jingren Palace, the Forbidden City, Beijing, the Kangxi Emperor was originally given the personal name Xuanye (; Manchu transliteration: "hiowan yei"). He was enthroned at the age of seven (or eight by East Asian age reckoning), on 7 February 1661. However, his era name "Kangxi", only started to be used on 18February 1662, the first day of the following lunar year.


title: Kangxi Emperor

text: Sinologist Herbert Giles, drawing on contemporary sources, described the Kangxi Emperor as "fairly tall and well proportioned, he loved all manly exercises, and devoted three months annually to hunting. Large bright eyes lighted up his face, which was pitted with smallpox."


title: Kangxi Emperor

text: Before the Kangxi Emperor came to the throne, Grand Empress Dowager Xiaozhuang (in the name of Shunzhi Emperor) had appointed the powerful men Sonin, Suksaha, Ebilun, and Oboi as regents. Sonin died after his granddaughter became Empress Xiaochengren, leaving Suksaha at odds with Oboi in politics. In a fierce power struggle, Oboi had Suksaha put to death and seized absolute power as sole regent. The Kangxi Emperor and the rest of the imperial court acquiesced to this arrangement.


title: Kangxi Emperor

text: In the spring of 1662, the regents ordered a Great Clearance in southern China that evacuated the entire population from the seacoast to counter a resistance movement started by Ming loyalists under the leadership of Taiwan-based Ming general Zheng Chenggong, also titled Koxinga.


title: Kangxi Emperor

text: In 1669, the Kangxi Emperor had Oboi arrested with the help of his grandmother Grand Dowager Empress Xiaozhuang, who had raised him. and began taking personal control of the empire. He listed three issues of concern: flood control of the Yellow River; repair of the Grand Canal; the Revolt of the Three Feudatories in south China. The Grand Empress Dowager influenced him greatly and he took care of her himself in the months leading up to her death in 1688.


title: Kangxi Emperor

text: Kangxi's relatives from the Han Chinese Banner Tong 佟 clan of Fushun in Liaoning falsely claimed to be related to the Jurchen Manchu Tunggiya 佟佳 clan of Jilin, using this false claim to get themselves transferred to a Manchu banner in the reign of Kangxi emperor.


title: Kangxi Emperor

text: The main army of the Qing Empire, the Eight Banners Army, was in decline under the Kangxi Emperor. It was smaller than it had been at its peak under Hong Taiji and in the early reign of the Shunzhi Emperor; however, it was larger than in the Yongzheng and Qianlong emperors' reigns. In addition, the Green Standard Army was still powerful with generals such as Tuhai, Fei Yanggu, Zhang Yong, Zhou Peigong, Shi Lang, Mu Zhan, Shun Shike and Wang Jingbao.


title: Kangxi Emperor

text: The main reason for this decline was a change in system between the Kangxi and Qianlong emperors' reigns. The Kangxi Emperor continued using the traditional military system implemented by his predecessors, which was more efficient and stricter. According to the system, a commander who returned from a battle alone (with all his men dead) would be put to death, and likewise for a foot soldier. This was meant to motivate both commanders and soldiers alike to fight valiantly in war because there was no benefit for the sole survivor in a battle.


title: Kangxi Emperor

text: By the Qianlong Emperor's reign, military commanders had become lax and the training of the army was deemed less important as compared to during the previous emperors' reigns.


title: Kangxi Emperor

text: After the Qing takeover of China in 1644, large parts of the south and west were given as fiefs to three Ming generals who aided the Qing; in 1673 the three feudatories were controlled by Wu Sangui, Geng Jingzhong, and Shang Zhixin. Going against the advice of most of his advisors, Kangxi attempted to force the feudal princes to give up their lands and retire to Manchuria, sparking a rebellion that lasted eight years. For years afterwards Kangxi ruminated on his mistakes and blamed himself in part for the loss of life during the revolt.


title: Kangxi Emperor

text: Wu Sangui's forces overran most of southwest China and he tried to ally himself with local generals such as Wang Fuchen. The Kangxi Emperor employed generals including Zhou Peigong and Tuhai to suppress the rebellion, and also granted clemency to common people caught up in the war. He intended to personally lead the armies to crush the rebels but his subjects advised him against it. The Kangxi Emperor used mainly Han Chinese Green Standard Army soldiers to crush the rebels while the Manchu Banners took a backseat. The revolt ended with victory for Qing forces in 1681.


title: Kangxi Emperor

text: In 1683, the naval forces of the Ming loyalists on Taiwan—organized under the Zheng dynasty as the Kingdom of Tungning—were defeated off Penghu by 300-odd ships under the Qing admiral Shi Lang. Koxinga's grandson Zheng Keshuang surrendered Tungning a few days later and Taiwan became part of the Qing Empire. Zheng Keshuang moved to Beijing, joined the Qing nobility as the "Duke Haicheng" (), and was inducted into the Eight Banners as a member of the Han Plain Red Banner. His soldiers—including the rattan-shield troops (, "tengpaiying")—were similarly entered into the Eight Banners, notably serving against Russian Cossacks at Albazin.


title: Kangxi Emperor

text: A score of Ming princes had joined the Zheng dynasty on Taiwan, including Prince Zhu Shugui of Ningjing and Prince Honghuan (朱弘桓), the son of Zhu Yihai. The Qing sent most of the 17 Ming princes still living on Taiwan back to mainland China, where they spent the rest of their lives. The Prince of Ningjing and his five concubines, however, committed suicide rather than submit to capture. Their palace was used as Shi Lang's headquarters in 1683, but he memorialized the emperor to convert it into a Mazu temple as a propaganda measure in quieting remaining resistance on Taiwan. The emperor approved its dedication as the Grand Matsu Temple the next year and, honoring the goddess Mazu for her supposed assistance during the Qing invasion, promoted her to "Empress of Heaven" ( "Tianhou") from her previous status as a "heavenly consort" ( "Tianfei"). Belief in Mazu remains so widespread on Taiwan that her annual celebrations can gather hundreds of thousands of people; she is sometimes even syncretized with Guanyin and the Virgin Mary.


title: Kangxi Emperor

text: The end of the rebel stronghold and capture of the Ming princes allowed the Kangxi Emperor to relax the Sea Ban and permit resettlement of the Fujian and Guangdong coasts. The financial and other incentives to new settlers particularly drew the Hakka, who would have continuous low-level conflict with the returning Punti people for the next few centuries.


title: Kangxi Emperor

text: In the 1650s, the Qing Empire engaged the Tsardom of Russia in a series of border conflicts along the Amur River region, which concluded with the Qing gaining control of the area after the Siege of Albazin.


title: Kangxi Emperor

text: The Russians invaded the northern frontier again in the 1680s. A series of battles and negotiations culminated in the Treaty of Nerchinsk of 1689, by which a border was agreed between Russia and China.


title: Kangxi Emperor

text: The Inner Mongolian Chahar leader Ligdan Khan, a descendant of Genghis Khan, opposed and fought against the Qing until he died of smallpox in 1634. Thereafter, the Inner Mongols under his son Ejei Khan surrendered to the Qing and he was given the title of Prince (Qin Wang, 親王). The Inner Mongolian nobility now became closely tied to the Qing royal family and intermarried with them extensively. Ejei Khan died in 1661 and was succeeded by his brother Abunai. After Abunai showed disaffection with Manchu Qing rule, he was placed under house arrest in 1669 in Shenyang and the Kangxi Emperor gave his title to his son Borni.


title: Kangxi Emperor

text: Abunai bided his time then, with his brother Lubuzung, revolted against the Qing in 1675 during the Revolt of the Three Feudatories, with 3,000 Chahar Mongol followers joining in on the revolt. The revolt was put down within two months, the Qing defeating the rebels in battle on 20 April 1675, killing Abunai and all his followers. Their title was abolished, all Chahar Mongol royal males were executed even if they were born to Manchu Qing princesses, and all Chahar Mongol royal females were sold into slavery except the Manchu Qing princesses. The Chahar Mongols were then put under the direct control of the Qing Emperor unlike the other Inner Mongol leagues which maintained their autonomy.


title: Kangxi Emperor

text: The Outer Khalkha Mongols had preserved their independence, and only paid tribute to the Qing Empire. However, a conflict between the houses of Tümen Jasagtu Khan and Tösheetü Khan led to a dispute between the Khalkha and the Dzungars over the influence of Tibetan Buddhism. In 1688, the Dzungar chief, Galdan Boshugtu Khan, attacked the Khalkha from the west and invaded their territory. The Khalkha royal families and the first Jebtsundamba Khutuktu crossed the Gobi Desert and sought help from the Qing Empire in return for submission to Qing authority. In 1690, the Dzungars and Qing forces clashed at the Battle of Ulan Butung in Inner Mongolia, in which the Qing eventually emerged as the victor.


title: Kangxi Emperor

text: In 1696 and 1697 the Kangxi Emperor personally led campaigns against the Dzungars in the early Dzungar–Qing War. The western section of the Qing army defeated Galdan's forces at the Battle of Jao Modo and Galdan died in the following year.


title: Kangxi Emperor

text: In 1700, some 20,000 Qiqihar Xibe were resettled in Guisui, modern Inner Mongolia, and 36,000 Songyuan Xibe were resettled in Shenyang, Liaoning. The relocation of the Xibe from Qiqihar is believed by Liliya M. Gorelova to be linked to the Qing's annihilation of the Manchu clan Hoifan (Hoifa) in 1697 and the Manchu tribe Ula in 1703 after they rebelled against the Qing; both Hoifan and Ula were wiped out.


title: Kangxi Emperor

text: In 1701, the Kangxi Emperor ordered the reconquest of Kangding and other border towns in western Sichuan that had been taken by the Tibetans. The Manchu forces stormed Dartsedo and secured the border with Tibet and the lucrative tea-horse trade.


title: Kangxi Emperor

text: The Tibetan "desi" (regent) Sangye Gyatso concealed the death of the 5th Dalai Lama in 1682, and only informed the emperor in 1697. He moreover kept relations with Dzungar enemies of the Qing. All this evoked the great displeasure of the Kangxi Emperor. Eventually Sangye Gyatso was toppled and killed by the Khoshut ruler Lha-bzang Khan in 1705. As a reward for ridding him of his old enemy the Dalai Lama, the Kangxi Emperor appointed Lha-bzang Khan Regent of Tibet (). The Dzungar Khanate, a confederation of Oirat tribes based in parts of what is now Xinjiang, continued to threaten the Qing Empire and invaded Tibet in 1717. They took control of Lhasa with a 6,000 strong army and killed Lha-bzang Khan. The Dzungars held on to the city for three years and at the Battle of the Salween River defeated a Qing army sent to the region in 1718. The Qing did not take control of Lhasa until 1720, when the Kangxi Emperor sent a larger expedition force there to defeat the Dzungars.


title: Kangxi Emperor

text: Manchu emperor Kangxi incited anti-Muslim sentiment among the Mongols of Qinghai (Kokonor) in order to gain support against the Dzungar Oirat Mongol leader Galdan. Kangxi claimed that Chinese Muslims inside China such as Turkic Muslims in Qinghai (Kokonor) were plotting with Galdan, who he falsely claimed converted to Islam. Kangxi falsely claimed that Galdan had spurned and turned his back on Buddhism and the Dalai Lama and that he was plotting to install a Muslim as ruler of China after invading it in a conspiracy with Chinese Muslims. Kangxi also distrusted Muslims of Turfan and Hami.


title: Kangxi Emperor

text: The Kangxi Emperor granted the title of Wujing Boshi (五經博士; "Wǔjīng Bóshì") to the descendants of Shao Yong, Zhu Xi, Zhuansun Shi, Ran family (Ran Qiu, Ran Geng, Ran Yong), Bu Shang, Yan Yan (disciple of Confucius), and the Duke of Zhou's offspring.


title: Kangxi Emperor

text: The reasons for the declining trend in the later years of the Kangxi Emperor's reign were a huge expenditure on military campaigns and an increase in corruption. To fix the problem, the Kangxi Emperor gave Prince Yong (the future Yongzheng Emperor) advice on how to make the economy more efficient.


title: Kangxi Emperor

text: During his reign, the Kangxi Emperor ordered the compilation of a dictionary of Chinese characters, which became known as the Kangxi Dictionary. This was seen as an attempt by the emperor to gain support from the Han Chinese scholar-bureaucrats, as many of them initially refused to serve him and remained loyal to the Ming dynasty. However, by persuading the scholars to work on the dictionary without asking them to formally serve the Qing imperial court, the Kangxi Emperor led them to gradually taking on greater responsibilities until they were assuming the duties of state officials.


title: Kangxi Emperor

text: In 1705, on the Kangxi Emperor's order, a compilation of Tang poetry, the Quan Tangshi, was produced.


title: Kangxi Emperor

text: The Kangxi Emperor also was interested in Western technology and wanted to import them to China. This was done through Jesuit missionaries, such as Ferdinand Verbiest, whom the Kangxi Emperor frequently summoned for meetings, or Karel Slavíček, who made the first precise map of Beijing on the emperor's order.


title: Kangxi Emperor

text: From 1711 to 1723, Matteo Ripa, an Italian priest sent to China by the Congregation for the Evangelization of Peoples, worked as a painter and copper-engraver at the Qing court. In 1723, he returned to Naples from China with four young Chinese Christians, in order to groom them to become priests and send them back to China as missionaries. This marked the beginning of the Collegio dei Cinesi, sanctioned by Pope Clement XII to help the evangelization of Christianity in China. This Chinese Institute was the first school of Sinology in Europe, which would later develop to become the Istituto Orientale and the present day Naples Eastern University.


title: Kangxi Emperor

text: The Kangxi Emperor was also the first Chinese emperor to play a western musical instrument. Thomas Pereira taught him how to play the harpsichord, and he employed Karel Slavíček as court musician. Slavíček was playing Spinet; later the emperor would play on it himself. China's famed blue and white porcelain probably reached its zenith during the Kangxi Emperor's reign.


title: Kangxi Emperor

text: In the early decades of the Kangxi Emperor's reign, Jesuits played a large role in the imperial court. With their knowledge of astronomy, they ran the imperial observatory. Jean-François Gerbillon and Thomas Pereira served as translators for the negotiations of the Treaty of Nerchinsk. The Kangxi Emperor was grateful to the Jesuits for their contributions, the many languages they could interpret, and the innovations they offered his military in gun manufacturing and artillery, the latter of which enabled the Qing Empire to conquer the Kingdom of Tungning.


title: Kangxi Emperor

text: The Kangxi Emperor was also fond of the Jesuits' respectful and unobtrusive manner; they spoke the Chinese language well, and wore the silk robes of the elite. In 1692, when Pereira requested tolerance for Christianity, the Kangxi Emperor was willing to oblige, and issued the Edict of Toleration, which recognized Catholicism, barred attacks on their churches, and legalized their missions and the practice of Christianity by the Chinese people.


title: Kangxi Emperor

text: However, controversy arose over whether Chinese Christians could still take part in traditional Confucian ceremonies and ancestor worship, with the Jesuits arguing for tolerance and the Dominicans taking a hard-line against foreign "idolatry". The Dominican position won the support of Pope Clement XI, who in 1705 sent Charles-Thomas Maillard de Tournon as his representative to the Kangxi Emperor, to communicate the ban on Chinese rites. Through de Tournon, the Pope insisted on sending his own representative to Beijing to oversee Jesuit missionaries in China. Kangxi refused, wanting to keep missionary activities in China under his final oversight, managed by one of the Jesuits who had been living in Beijing for years.


title: Kangxi Emperor

text: On 19 March 1715, Pope Clement XI issued the papal bull "Ex illa die", which officially condemned Chinese rites. In response, the Kangxi Emperor officially forbade Christian missions in China, as they were "causing trouble".


title: Kangxi Emperor

text: A prolonged struggle between various princes emerged during the Kangxi Emperor's reign over who should inherit the throne – the Nine Lords' War ().


title: Kangxi Emperor

text: In 1674 the Kangxi Emperor's first spouse, Empress Xiaochengren, died while giving birth to his second surviving son Yinreng, who at the age of two was named crown prince – a Han Chinese custom, to ensure stability during a time of chaos in the south. Although the Kangxi Emperor left the education of several of his sons to others, he personally oversaw the upbringing of Yinreng, grooming him to be a perfect successor. Yinreng was tutored by the mandarin Wang Shan, who remained devoted to him, and spent the later years of his life trying to persuade the Kangxi Emperor to restore Yinreng as the crown prince.


title: Kangxi Emperor

text: Yinreng proved to be unworthy of the succession despite his father showing favoritism towards him. He was said to have beaten and killed his subordinates, and was alleged to have had sexual relations with one of his father's concubines, which was deemed incest and a capital offence. Yinreng also purchased young children from Jiangsu to satisfy his pedophiliac pleasure. In addition, Yinreng's supporters, led by Songgotu, gradually formed a "Crown Prince Party" (太子黨), that aimed to help Yinreng get the throne as soon as possible, even if it meant using unlawful methods.


title: Kangxi Emperor

text: Over the years, the Kangxi Emperor kept constant watch over Yinreng and became aware of his son's many flaws, while their relationship gradually deteriorated. In 1707, the emperor decided that he could no longer tolerate Yinreng's behavior, which he partially mentioned in the imperial edict as "never obeying ancestors' virtues, never obliged to my order, only doing inhumanity and devilry, only showing maliciousness and lust", and decided to strip Yinreng of his position as crown prince. The Kangxi Emperor placed his oldest surviving son, Yinzhi, in charge of overseeing Yinreng's house arrest. Yinzhi, an unfavored Shu son, knowing he had no chance of being selected, recommended the eighth prince, Yinsi, and requested his father to order Yinreng's execution. The Kangxi Emperor was enraged and stripped Yinzhi of his titles. The emperor then commanded his subjects to cease debating the succession issue, but despite this and attempts to reduce rumours and speculation as to who the new crown prince might be, the imperial court's daily activities were disrupted. Yinzhi's actions caused the Kangxi Emperor to suspect that Yinreng might have been framed, so he restored Yinreng as crown prince in 1709, with the support of the 4th and 13th princes, and on the excuse that Yinreng had previously acted under the influence of mental illness.


title: Kangxi Emperor

text: In 1712, during the Kangxi Emperor's last inspection tour of the south, Yinreng, who was put in charge of state affairs during his father's absence, tried to vie for power again with his supporters. He allowed an attempt at forcing the Kangxi Emperor to abdicate when his father returned to Beijing. However, the emperor received news of the planned coup d'etat, and was so angry that he deposed Yinreng and placed him under house arrest again. After the incident, the emperor announced that he would not appoint any of his sons as crown prince for the remainder of his reign. He stated that he would place his Imperial Valedictory Will inside a box in the Palace of Heavenly Purity, which would only be opened after his death.


title: Kangxi Emperor

text: Seeing that Yinreng was completely disavowed, Yinsi and some other princes turned to support the 14th prince, Yinti, while the 13th prince supported Yinzhen. They formed the so-called "Eighth Lord Party" () and "Fourth Lord Party" ().


title: Kangxi Emperor

text: Following the deposition of the crown prince, the Kangxi Emperor implemented groundbreaking changes in the political landscape. The 13th prince, Yinxiang, was placed under house arrest as well for cooperating with Yinreng. The eighth prince Yinsi was stripped of all his titles and only had them restored years later. The 14th prince Yinti, whom many considered to be the most likely candidate to succeed the Kangxi Emperor, was sent on a military campaign during the political conflict. Yinsi, along with the ninth and tenth princes, Yintang and Yin'e, pledged their support to Yinti.


title: Kangxi Emperor

text: In the evening of 20 December 1722 before his death, the Kangxi Emperor called seven of his sons to assemble at his bedside. They were the third, fourth, eighth, ninth, tenth, sixteenth and seventeenth princes. After the Kangxi Emperor died, Longkodo announced that the emperor had selected the fourth prince, Yinzhen, as the new emperor. Yinzhen ascended to the throne and became known as the Yongzheng Emperor. The Kangxi Emperor was entombed at the Eastern Tombs in Zunhua, Hebei.


title: Kangxi Emperor

text: A legend concerning the Kangxi Emperor's will states that he chose Yinti as his heir, but Yinzhen forged the will in his own favour. It has, however, long been refuted by serious historians. Yinzhen, later the Yongzheng Emperor, has attracted many rumours, and some novel-like private books claim he did not die of illness but was assassinated by a swordswoman, Lü Siniang (), the granddaughter of Lü Liuliang, though this is never treated seriously by scholars.


title: Kangxi Emperor

text: The Kangxi Emperor was a great consolidator of the Qing dynasty. The transition from the Ming dynasty to the Qing was a cataclysm whose central event was the fall of the capital Beijing to the peasant rebels led by Li Zicheng, then to the Manchus in 1644, and the installation of the five-year-old Shunzhi Emperor on their throne. By 1661, when the Shunzhi Emperor died and was succeeded by the Kangxi Emperor, the Qing conquest of China proper was almost complete. Leading Manchus were already using Chinese institutions and mastering Confucian ideology, while maintaining Manchu culture among themselves. The Kangxi Emperor completed the conquest, suppressed all significant military threats and revived the central government system inherited from the Ming with important modifications.


title: Kangxi Emperor

text: The Kangxi Emperor was a workaholic, rising early and retiring late, reading and responding to numerous memorials every day, conferring with his councilors and giving audiences – and this was in normal times; in wartime, he might be reading memorials from the warfront until after midnight or even, as with the Dzungar conflict, away on campaign in person.


title: Kangxi Emperor

text: The Kangxi Emperor devised a system of communication that circumvented the scholar-bureaucrats, who had a tendency to usurp the power of the emperor. This Palace Memorial System involved the transfer of secret messages between him and trusted officials in the provinces, where the messages were contained in locked boxes that only he and the official had access to. This started as a system for receiving uncensored extreme-weather reports, which the emperor regarded as divine comments on his rule. However, it soon evolved into a general-purpose secret "news channel." Out of this emerged a Grand Council, which dealt with extraordinary, especially military, events. The council was chaired by the emperor and manned by his more elevated Han Chinese and Manchu household staff. From this council, the mandarin civil servants were excluded – they were left only with routine administration.


title: Kangxi Emperor

text: The Kangxi Emperor managed to woo the Confucian intelligentsia into co-operating with the Qing government, despite their deep reservations about Manchu rule and loyalty to the Ming. He appealed to this very sense of Confucian values, for instance, by issuing the Sacred Edict in 1670. He encouraged Confucian learning and made sure that the civil service examinations were held every three years even during times of stress. When some scholars, out of loyalty to the Ming, refused to take the exams, he hit upon the expedient of a special exam to be taken by nomination. He personally sponsored the writing of the Ming Official History, the Kangxi Dictionary, a phrase-dictionary, a vast encyclopedia and an even vaster compilation of Chinese literature. To promote his image as a "sage ruler," he appointed Manchu and Chinese tutors with whom he studied the Confucian classics and worked intensively on Chinese calligraphy.


title: Kangxi Emperor

text: In the one military campaign in which he actively participated, against the Dzungar Mongols, the Kangxi Emperor showed himself an effective military commander. According to Finer, the emperor's own written reflections allow one to experience "how intimate and caring was his communion with the rank-and-file, how discriminating and yet masterful his relationship with his generals".


title: Kangxi Emperor

text: As a result of the scaling down of hostilities as peace returned to China after the Manchu conquest, and also as a result of the ensuing rapid increase of population, land cultivation and therefore tax revenues based on agriculture, the Kangxi Emperor was able first to make tax remissions, then in 1712 to freeze the land tax and corvée altogether, without embarrassing the state treasury (although the dynasty eventually suffered from this fiscal policy).


title: Tuckman's stages of group development

text: The "forming–storming–norming–performing" model of group development was first proposed by Bruce Tuckman in 1965, who said that these phases are all necessary and inevitable in order for a team to grow, face up to challenges, tackle problems, find solutions, plan work, and deliver results. As Tuckman knew these inevitable phases were critical to team growth and development, he hypothesized that along with these factors that interpersonal relationships and task activity would enhance the four-stage model that is needed to successfully navigate and create an effective group function.


title: Tuckman's stages of group development

text: The team meets and learns about the opportunities and challenges, and then agrees on goals and begins to tackle the tasks. Team members tend to behave quite independently. They may be motivated but are usually relatively uninformed of the issues and objectives of the team. Team members are usually on their best behavior but very focused on themselves. Mature team members begin to model appropriate behavior even at this early phase. The meeting environment also plays an important role to model the initial behavior of each individual. The major task functions also concern orientation. Members attempt to become oriented to the tasks as well as to one another. This is also the stage in which group members test boundaries, create ground rules, and define organizational standards. Discussion centers on defining the scope of the task, how to approach it, and similar concerns. To grow from this stage to the next, each member must relinquish the comfort of non-threatening topics and risk the possibility of conflict.


title: Tuckman's stages of group development

text: This is the second stage of team development, where the group starts to sort itself out and gain each others' trust. This stage often starts when they voice their opinions; conflict may arise between team members as power and status are assigned. When group members start to work with each other they start to learn about individual working styles and what it is like to work with each other as a team; it also identifies the hierarchy of positions in the group. At this stage there is often a positive and polite atmosphere, people are pleasant to each other, and they may have feelings of excitement, eagerness and positiveness. Others may have feelings of suspicion, fear and anxiety. The leader of the team will then describe the tasks to the group, describe the different behaviours to the group and how to deal and handle complaints. In this stage "...participants form opinions about the character and integrity of the other participants and feel compelled to voice these opinions if they find someone shirking responsibility or attempting to dominate. Sometimes participants question the actions or decision of the leader as the expedition grows harder..." Disagreements and personality clashes must be resolved before the team can progress out of this stage, and so some teams may never emerge from "storming" or re-enter that phase if new challenges or disputes arise. In Tuckman's 1965 paper, only 50% of the studies identified a stage of intragroup conflict, and some of the remaining studies jumped directly from stage 1 to stage 3. Some groups may avoid the phase altogether, but for those who do not, the duration, intensity and destructiveness of the "storms" can be varied. Tolerance of each team member and their differences should be emphasized; without tolerance and patience the team will fail. This phase can become destructive to the team and will lower motivation if allowed to get out of control. Some teams will never develop past this stage; however, disagreements within the team can make members stronger, more versatile, and able to work more effectively together. Supervisors during this phase may be more accessible, but tend to remain directive in their guidance of decision-making and professional behaviour. The team members will therefore resolve their differences and members will be able to participate with one another more comfortably. The ideal is that they will not feel that they are being judged, and will therefore share their opinions and views. Normally tension, struggle and sometimes arguments occur. This stage can also be upsetting.


title: Tuckman's stages of group development

text: "Resolved disagreements and personality clashes result in greater intimacy, and a spirit of co-operation emerges." This happens when the team is aware of competition and they share a common goal. In this stage, all team members take responsibility and have the ambition to work for the success of the team's goals. They start tolerating the whims and fancies of the other team members. They accept others as they are and make an effort to move on. The danger here is that members may be so focused on preventing conflict that they are reluctant to share controversial ideas.


title: Tuckman's stages of group development

text: "With group norms and roles established, group members focus on achieving common goals, often reaching an unexpectedly high level of success." By this time, they are motivated and knowledgeable. The team members are now competent, autonomous and able to handle the decision-making process without supervision. Dissent is expected and allowed as long as it is channelled through means acceptable to the team.


title: Tuckman's stages of group development

text: Supervisors of the team during this phase are almost always participating. The team will make most of the necessary decisions. Even the most high-performing teams will revert to earlier stages in certain circumstances. Many long-standing teams go through these cycles many times as they react to changing circumstances. For example, a change in leadership may cause the team to revert to "storming" as the new people challenge the existing norms and dynamics of the team.


title: Tuckman's stages of group development

text: In 1977, Tuckman, jointly with Mary Ann Jensen, added a fifth stage to the four stages: "adjourning"," "that involves completing the task and breaking up the team (in some texts referred to as "mourning"). After being invited by Group and Organizational Studies to publish and update of the model, they revisited the original model and reviewed the literature. They concluded that an important step in the small group life cycle was the ultimate separation which occurred at the end of this cycle.""'


title: Tuckman's stages of group development

text: Timothy Biggs suggested that an additional stage be added of "norming" after "forming" and renaming the traditional norming stage "re-norming". This addition is designed to reflect that there is a period after forming where the performance of a team gradually improves, and the interference of a leader who is content with that level of performance will prevent a team from progressing through the storming stage to true performance. This puts the emphasis back on the team and leader, as the storming stage must be actively engaged in order to succeed – too many "diplomats" or "peacemakers", especially in a leadership role, may prevent the team from reaching their full potential.


title: Tuckman's stages of group development

text: Rickards and Moger proposed a similar extension to the Tuckman model when a group breaks out of its norms, through a process of creative problem-solving.


title: Tuckman's stages of group development

text: Alasdair A. K. White together with his colleague, John Fairhurst, examined Tuckman's development sequence when developing the White-Fairhurst TPR model. They simplify the sequence and group the forming-storming-norming stages together as the "transforming" phase, which they equate with the initial performance level. This is then followed by a "performing" phase that leads to a new performance level which they call the "reforming" phase. Their work was developed further by White in his essay "From Comfort Zone to Performance Management" in which he demonstrates the linkage between Tuckman's work with that of Colin Carnall's "coping cycle" and the "comfort zone theory".


title: Tuckman's stages of group development

text: A healthcare research study "Maximizing Team Performance: The Critical Role of the Nurse Leader" examined the role of nursing leaders in facilitating the development of high performing change teams using the Tuckman model of group development as a guiding framework. Using qualitative research techniques, these authors linked the team development stages to leadership strategies, as well as identified keys to leader success. Some examples from the article:


title: Tuckman's stages of group development

text: In agile software development, high-performance teams will exhibit a swarm behavior as they come together, collaborate, and focus on solving a single problem. Swarming is a sometime behavior, in contrast to mob programming, which can be thought of as swarming all the time.


title: Tuckman's stages of group development

text: In project management, the Tuckman Ladder is referenced and used extensively by project managers to help them assemble and guide teams toward success. In project management, the Tuckman Ladder's phases are not always realized in a linear fashion; it is common for teams to progress to the next phase and then wind up back at a previous phase as the project timeline progreses.


title: Lake

text: A lake is an area filled with water, localized in a basin, surrounded by land, and distinct from any river or other outlet that serves to feed or drain the lake. Lakes lie on land and are not part of the ocean, although, like the much larger oceans, they do form part of the Earth's water cycle. Lakes are distinct from lagoons, which are generally coastal parts of the ocean. Lakes are typically larger and deeper than ponds, which also lie on land, though there are no official or scientific definitions. Lakes can be contrasted with rivers or streams, which usually flow in a channel on land. Most lakes are fed and drained by rivers and streams.


title: Lake

text: Natural lakes are generally found in mountainous areas, rift zones, and areas with ongoing glaciation. Other lakes are found in endorheic basins or along the courses of mature rivers, where a river channel has widened into a basin. Some parts of the world have many lakes formed by the chaotic drainage patterns left over from the last ice age. All lakes are temporary over long periods of time, as they will slowly fill in with sediments or spill out of the basin containing them.


title: Lake

text: Many lakes are artificial and are constructed for industrial or agricultural use, for hydroelectric power generation or domestic water supply, for aesthetic or recreational purposes, or for other activities.


title: Lake

text: The word "lake" comes from Middle English ('lake, pond, waterway'), from Old English ('pond, pool, stream'), from Proto-Germanic ('pond, ditch, slow moving stream'), from the Proto-Indo-European root ('to leak, drain'). Cognates include Dutch ('lake, pond, ditch'), Middle Low German ('water pooled in a riverbed, puddle') as in: , , German ('pool, puddle'), and Icelandic ('slow flowing stream'). Also related are the English words "leak" and "leach".


title: Lake

text: There is considerable uncertainty about defining the difference between lakes and ponds, and neither term has an internationally accepted definition across scientific disciplines or political boundaries. For example, limnologists have defined lakes as water bodies that are simply a larger version of a pond, which can have wave action on the shoreline or where wind-induced turbulence plays a major role in mixing the water column. None of these definitions completely excludes ponds and all are difficult to measure. For this reason, simple size-based definitions are increasingly used to separate ponds and lakes. Definitions for "lake" range in minimum sizes for a body of water from to . Pioneering animal ecologist Charles Elton regarded lakes as waterbodies of or more. The term "lake" is also used to describe a feature such as Lake Eyre, which is a dry basin most of the time but may become filled under seasonal conditions of heavy rainfall. In common usage, many lakes bear names ending with the word "pond", and a lesser number of names ending with "lake" are, in quasi-technical fact, ponds. One textbook illustrates this point with the following: "In Newfoundland, for example, almost every lake is called a pond, whereas in Wisconsin, almost every pond is called a lake."


title: Lake

text: One hydrology book proposes to define the term "lake" as a body of water with the following five characteristics:


title: Lake

text: With the exception of criterion 3, the others have been accepted or elaborated upon by other hydrology publications.


title: Lake

text: The majority of lakes on Earth are freshwater, and most lie in the Northern Hemisphere at higher latitudes. Canada, with a deranged drainage system, has an estimated 31,752 lakes larger than in surface area. The total number of lakes in Canada is unknown but is estimated to be at least 2 million. Finland has 187,888 lakes of in area, or larger, of which 56,000 are large ( or larger).


title: Lake

text: Most lakes have at least one natural outflow in the form of a river or stream, which maintain a lake's average level by allowing the drainage of excess water. Some lakes do not have a natural outflow and lose water solely by evaporation or underground seepage, or both. These are termed endorheic lakes.


title: Lake

text: Many lakes are artificial and are constructed for hydroelectric power generation, aesthetic purposes, recreational purposes, industrial use, agricultural use, or domestic water supply.


title: Lake

text: The number of lakes on Earth is undetermined because most lakes and ponds are very small and do not appear on maps or satellite imagery. Despite this uncertainty, a large number of studies agree that small ponds are much more abundant than large lakes. For example, one widely cited study estimated that Earth has 304 million lakes and ponds, and that 91% of these are or less in area. Despite the overwhelming abundance of ponds, almost all of Earth's lake water is found in fewer than 100 large lakes; this is because lake volume scales superlinearly with lake area.


title: Lake

text: Extraterrestrial lakes exist on the moon Titan, which orbits the planet Saturn. The shape of lakes on Titan is very similar to those on Earth. Lakes were formerly present on the surface of Mars, but are now dry lake beds.


title: Lake

text: In 1957, G. Evelyn Hutchinson published a monograph titled "A Treatise on Limnology", which is regarded as a landmark discussion and classification of all major lake types, their origin, morphometric characteristics, and distribution. Hutchinson presented in his publication a comprehensive analysis of the origin of lakes and proposed what is a widely accepted classification of lakes according to their origin. This classification recognizes 11 major lake types that are divided into 76 subtypes. The 11 major lake types are:


title: Lake

text: Tectonic lakes are lakes formed by the deformation and resulting lateral and vertical movements of the Earth's crust. These movements include faulting, tilting, folding, and warping. Some of the largest lakes on Earth are rift lakes occupying rift valleys, e.g. Central African Rift lakes and Lake Baikal. Other well-known tectonic lakes, Caspian Sea, the Sea of Aral, and other lakes from the Pontocaspian occupy basins that have been separated from the sea by the tectonic uplift of the sea floor above the ocean level.


title: Lake

text: Often, the tectonic action of crustal extension has created an alternating series of parallel grabens and horsts that form elongate basins alternating with mountain ranges. Not only does this promote the creation of lakes by the disruption of preexisting drainage networks, it also creates within arid regions endorheic basins that contain salt lakes (also called saline lakes). They form where there is no natural outlet, a high evaporation rate and the drainage surface of the water table has a higher-than-normal salt content. Examples of these salt lakes include Great Salt Lake and the Dead Sea. Another type of tectonic lake caused by faulting is sag ponds.


title: Lake

text: Volcanic lakes are lakes that occupy either local depressions, e.g. craters and maars, or larger basins, e.g. calderas, created by volcanism. Crater lakes are formed in volcanic craters and calderas, which fill up with precipitation more rapidly than they empty via either evaporation, groundwater discharge, or a combination of both. Sometimes the latter are called caldera lakes, although often no distinction is made. An example is Crater Lake in Oregon, in the caldera of Mount Mazama. The caldera was created in a massive volcanic eruption that led to the subsidence of Mount Mazama around 4860 BCE. Other volcanic lakes are created when either rivers or streams are dammed by lava flows or volcanic lahars. The basin which is now Malheur Lake, Oregon was created when a lava flow dammed the Malheur River. Among all lake types, volcanic crater lakes most closely approximate a circular shape.


title: Lake

text: Glacial lakes are lakes created by the direct action of glaciers and continental ice sheets. A wide variety of glacial processes create enclosed basins. As a result, there are a wide variety of different types of glacial lakes and it is often difficult to define clear-cut distinctions between different types of glacial lakes and lakes influenced by other activities. The general types of glacial lakes that have been recognized are lakes in direct contact with ice, glacially carved rock basins and depressions, morainic and outwash lakes, and glacial drift basins. Glacial lakes are the most numerous lakes in the world. Most lakes in northern Europe and North America have been either influenced or created by the latest, but not last, glaciation, to have covered the region. Glacial lakes include proglacial lakes, subglacial lakes, finger lakes, and epishelf lakes. Epishelf lakes are highly stratified lakes in which a layer of freshwater, derived from ice and snow melt, is dammed behind an ice shelf that is attached to the coastline. They are mostly found in Antarctica.


title: Lake

text: Fluvial (or riverine) lakes are lakes produced by running water. These lakes include plunge pool lakes, fluviatile dams and meander lakes.


title: Lake

text: The most common type of fluvial lake is a crescent-shaped lake called an "oxbow lake" due to the distinctive curved shape. They can form in river valleys as a result of meandering. The slow-moving river forms a sinuous shape as the outer side of bends are eroded away more rapidly than the inner side. Eventually a horseshoe bend is formed and the river cuts through the narrow neck. This new passage then forms the main passage for the river and the ends of the bend become silted up, thus forming a bow-shaped lake. Their crescent shape gives oxbow lakes a higher perimeter to area ratio than other lake types.


title: Lake

text: Lakes formed by other processes responsible for floodplain basin creation. During high floods they are flushed with river water. There are four types: 1. Confluent floodplain lake, 2. Contrafluent-confluent floodplain lake, 3. Contrafluent floodplain lake, 4. Profundal floodplain lake.