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https://en.wikipedia.org/wiki/Injective_module
Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q
https://en.wikipedia.org/wiki/Moment_generating_function
If the expectation does not exist in an open neighborhood of 0, we say that the moment generating function does not exist
https://en.wikipedia.org/wiki/Bijective
A function is bijective if and only if it is invertible; that is, a function f : X → Y {\displaystyle f:X\to Y} is bijective if and only if there is a function g : Y → X , {\displaystyle g:Y\to X,} the inverse of f, such that each of the two ways for composing the two functions produces an identity function: g ( f
https://en.wikipedia.org/wiki/Signal_recovery
Bias can be desirable if false alarms and misses lead to different costs
https://en.wikipedia.org/wiki/S._R._Srinivasa_Varadhan
[22] He was elected to Fellow of the American Academy of Arts and Sciences (1988),[23] the Third World Academy of Sciences (1988), the Institute of Mathematical Statistics (1991), the Royal Society (1998),[24] the Indian Academy of Sciences (2004), the Society for Industrial and Applied Mathematics (2009),[25] and the
https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem
[3][4]
In simple terms, it states that if a system has a finite volume (or total probability measure) and preserves that volume over time, then almost every initial state will eventually return arbitrarily close to its starting position, and it will do so infinitely many times
https://en.wikipedia.org/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem
This is where the Lehmann–Scheffé theorem comes in, if T {\displaystyle T} is also complete, that is, for any real-valued measurable function φ {\displaystyle \varphi } it holds
E θ [ φ ( T ( X ) ) ] = 0 ⟹ φ ( T ( X ) ) = 0 P θ -a
https://en.wikipedia.org/wiki/Variable_(mathematics)
The "Aha problems" involve finding unknown quantities (referred to as aha, "stack") if the sum of the quantity and part(s) of it are given (The Rhind Mathematical Papyrus also contains four of these types of problems)
https://en.wikipedia.org/wiki/Slavery_in_ancient_Egypt
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https://en.wikipedia.org/wiki/Diogenes_Laertius
[5] The modern form "Diogenes Laertius" is much rarer, used by Stephanus of Byzantium,[6] and in a lemma to the Greek Anthology
https://en.wikipedia.org/wiki/Truth_tables
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https://en.wikipedia.org/wiki/Apocalypse
For the general concept of a world-ending disaster, see global catastrophic risk
https://en.wikipedia.org/wiki/White_noise
[3] In particular, if each sample has a normal distribution with zero mean, the signal is said to be additive white Gaussian noise
https://en.wikipedia.org/wiki/Interval_(mathematics)
[1]
The endpoints of an interval are its supremum (least upper bound), and its infimum (greatest lower bound), if they exist as real numbers
https://en.wikipedia.org/wiki/Semigroup
If the semigroup operation is commutative, then the semigroup is called a commutative semigroup or (less often than in the analogous case of groups) it may be called an abelian semigroup
https://en.wikipedia.org/wiki/Mathematical_proof
A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work
https://en.wikipedia.org/wiki/Ideal_(ring_theory)
If the ring is commutative, the definitions of left, right, and two-sided ideal coincide, and one talks simply of an ideal
https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity
The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model; if an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model
https://en.wikipedia.org/wiki/Russell's_paradox
If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves
https://en.wikipedia.org/wiki/Projective_varieties
A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial
https://en.wikipedia.org/wiki/Timeline_of_astronomy
[12] This remains the most accurate estimate for the length of the sidereal year anywhere in the world for over a thousand years
https://en.wikipedia.org/wiki/Axiomatic_system
The use of an axiomatic approach is a move away from informal reasoning, in which nouns may carry real-world semantic values, and towards formal proof
https://en.wikipedia.org/wiki/Autoregressive_model
Continuing this process shows that the effect of ε 1 {\displaystyle \varepsilon _{1}} never ends, although if the process is stationary then the effect diminishes toward zero in the limit
https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field
If only the electric field (E) is non-zero, and is constant in time, the field is said to be an electrostatic field
https://en.wikipedia.org/wiki/Excess_kurtosis
In particular if all of the ''i''</sub>"}},"i":0}}]}'>Xi have the same variance, then this simplifies to Kurt [ Y ] − 3 = 1 n 2 ∑ i = 1 n ( Kurt [ X i ] − 3 )
https://en.wikipedia.org/wiki/Theory_of_knots
Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 {\displaystyle \mathbb {R} ^{3}} upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself
https://en.wikipedia.org/wiki/P-adic_number
}
In general the series that represents a p-adic number is not convergent in the usual sense, but it is convergent for the p-adic absolute value | s | p = p − k , {\displaystyle |s|_{p}=p^{-k},} where k is the least integer i such that a i ≠ 0 {\displaystyle a_{i}\neq 0} (if all a i {\displaystyle a_{i}} are zero
https://en.wikipedia.org/wiki/Affine_connection
In differential geometry, an affine connection[a] is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space
https://en.wikipedia.org/wiki/Formal_power_series_ring
} If we studied this as a power series, its properties would include, for example, that its radius of convergence is 1 by the Cauchy–Hadamard theorem
https://en.wikipedia.org/wiki/Continent
1 Early concepts of the Old World continents
5