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Positive And Negative Parts
In mathematics, the positive part of a real number, real or extended real number line, extended real-valued function (mathematics), function is defined by the formula f^+(x) = \max(f(x),0) = \begin f(x) & \text f(x) > 0 \\ 0 & \text \end Intuitively, the graph of a function, graph of f^+ is obtained by taking the graph of f, 'chopping off' the part under the -axis, and letting f^+ take the value zero there. Similarly, the negative part of is defined as f^-(x) = \max(-f(x),0) = -\min(f(x),0) = \begin -f(x) & \text f(x) 0]f \\ f^- &= -[f<0]f. \end One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function. Measure-theoretic properties Given a sigma-algebra, measurable space , an extended real-valued function is measurable function, measurable if and only if its positive and negative parts are. Therefore, if such a function is measurab ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive And Negative Parts Of F(x) = X^2 - 4
Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a positive number * Positive operator, a type of linear operator in mathematics * Positive result, a result that has been found significant in statistical hypothesis testing * Positive test, a diagnostic test result that indicates some parameter being evaluated was present * Positive charge, one of the two types of electrical charge * Positive (electrical polarity), in electrical circuits * Positive lens, in optics * Positive (photography), a positive image, in which the color and luminance correlates directly with that in the depicted scene * Positive sense, said of an RNA sequence that codes for a protein Philosophy and humanities * Affirmative (policy debate), the team which affirms the resolution * Negative and positive rights, concerning the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Function
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real And Imaginary Parts
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even And Odd Functions
In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = x^n is even if ''n'' is an even integer, and it is odd if ''n'' is an odd integer. Even functions are those real functions whose graph is self-symmetric with respect to the and odd functions are those whose graph is self-symmetric with respect to the origin. If the domain of a real function is self-symmetric with respect to the origin, then the function can be uniquely decomposed as the sum of an even function and an odd function. Early history The concept of even and odd functions appears to date back to the early 18th century, with Leonard Euler playing a significant role in their formalization. Euler introduced the concepts of even and odd functions (using La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectifier (neural Networks)
In the context of Neural network (machine learning), artificial neural networks, the rectifier or ReLU (rectified linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: :\operatorname(x) = x^+ = \max(0, x) = \frac = \begin x & \text x > 0, \\ 0 & x \le 0 \end where x is the input to a Artificial neuron, neuron. This is analogous to half-wave rectification in electrical engineering. ReLU is one of the most popular activation functions for artificial neural networks, and finds application in computer vision and speech recognitionAndrew L. Maas, Awni Y. Hannun, Andrew Y. Ng (2014)Rectifier Nonlinearities Improve Neural Network Acoustic Models using Deep learning, deep neural nets and computational neuroscience. History The ReLU was first used by Alston Scott Householder, Alston Householder in 1941 as a mathematical abstraction of biological neural networks. Kunihiko Fukushima in 1969 used R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hahn Decomposition Theorem
In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space (X,\Sigma) and any signed measure \mu defined on the \sigma -algebra \Sigma , there exist two \Sigma -measurable sets, P and N , of X such that: # P \cup N = X and P \cap N = \varnothing . # For every E \in \Sigma such that E \subseteq P , one has \mu(E) \geq 0 , i.e., P is a positive set for \mu . # For every E \in \Sigma such that E \subseteq N , one has \mu(E) \leq 0 , i.e., N is a negative set for \mu . Moreover, this decomposition is essentially unique, meaning that for any other pair (P',N') of \Sigma -measurable subsets of X fulfilling the three conditions above, the symmetric differences P \triangle P' and N \triangle N' are \mu -null sets in the strong sense that every \Sigma -measurable subset of them has zero measure. The pair (P,N) is then called a ''Hahn decomposition'' of the signed measure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed Measure
In mathematics, a signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign. Definition There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. Signed measures are usually only allowed to take finite real values, while some textbooks allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures". Given a measurable space (X, \Sigma) (that is, a set X with a σ-algebra \Sigma on it), an extended signed measure is a set function \mu : \Sigma \to \R \cup \ such that \mu(\varnothing) = 0 and \mu is σ-additive – that is, it satisfies the equality \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n) for any sequence A_1, A_2, \ldots, A_n, \ldots of disjoint sets in \Sigma. The series on the right must converge absolute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, named after france, French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the Riemann integral. The Lebesgue integral also has generally better analytical properties. For instance, under mild conditions, it is possible to exchange limits and Lebesgue integration, while the conditions for doing this with a Riemann integral are comparatively baroque. Furthermore, the Lebesgue integral can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vitali Set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measure, Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. Each Vitali set is Uncountable set, uncountable, and there are uncountably many Vitali sets. The proof of their existence depends on the axiom of choice. Measurable sets Certain sets have a definite 'length' or 'mass'. For instance, the interval (mathematics), interval [0, 1] is deemed to have length 1; more generally, an interval [''a'', ''b''], ''a'' ≤ ''b'', is deemed to have length ''b'' − ''a''. If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set [0, 1] ∪ [2, 3] is composed of two intervals of length one, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2. There is a natural question here: if ''E'' is an arbitr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
