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https://en.wikipedia.org/wiki/Toll_(fee)
In Europe, the road toll goes back to the practice of the Germanic tribes, who charged fees to travellers if they wanted to cross over mountain passages
https://en.wikipedia.org/wiki/Consistent_estimator
If the sequence of estimates can be mathematically shown to converge in probability to the true value θ0, it is called a consistent estimator; otherwise the estimator is said to be inconsistent
https://en.wikipedia.org/wiki/Noether's_second_theorem
Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by k arbitrary functions and their derivatives up to order m, then the functional derivatives of L satisfy a system of k differential equations
https://en.wikipedia.org/wiki/Measurable_function
} A function f : X → Y {\displaystyle f:X\to Y} is said to be measurable if for every E ∈ T {\displaystyle E\in \mathrm {T} } the pre-image of E {\displaystyle E} under f {\displaystyle f} is in Σ {\displaystyle \Sigma } ; that is, for all E ∈ T {\displaystyle E\in \mathrm {T} } f − 1 ( E ) := { x ∈ X ∣ f
https://en.wikipedia.org/wiki/Module_homomorphism
Explicitly, if M and N are left modules over a ring R, then a function f : M → N {\displaystyle f:M\to N} is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R,
f ( x + y ) = f ( x ) + f ( y ) , {\displaystyle f(x+y)=f(x)+f(y),} f ( r x ) = r f ( x )
https://en.wikipedia.org/wiki/Local_ring
1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit
https://en.wikipedia.org/wiki/Faithful_module
For example, if R {\displaystyle R} is a commutative ring and I {\displaystyle I} is an ideal of R {\displaystyle R} , we can consider the quotient ring R / I {\displaystyle R/I} to be an R {\displaystyle R} -module
https://en.wikipedia.org/wiki/Commutator_(ring_theory)
[1]
This element is equal to the group's identity if and only if g and h commute (that is, if and only if gh = hg)
https://en.wikipedia.org/wiki/Ideal_(order_theory)
In this case, the following equivalent definition can be given: a subset I of a lattice ( P , ≤ ) {\displaystyle (P,\leq )} is an ideal if and only if it is a lower set that is closed under finite joins (suprema); that is, it is nonempty and for all x, y in I, the element x ∨ y {\displaystyle x\vee y} of P is also
https://en.wikipedia.org/wiki/Euclidean_division
This can be confirmed using multiplication, the inverse of division: if each of the 4 people received 2 slices, then 4 × 2 = 8 slices were given out in total
https://en.wikipedia.org/wiki/Korean_verbs
If the verbal root ends in a vowel, the two vowels may merge or contract
https://en.wikipedia.org/wiki/Discrete_space
X {\displaystyle X} is defined by letting every superset of the diagonal { ( x , x ) : x ∈ X } {\displaystyle \{(x,x):x\in X\}} in X × X {\displaystyle X\times X} be an entourage, and X {\displaystyle X} is a discrete uniform space if it is equipped with its discrete uniformity
https://en.wikipedia.org/wiki/Embedding
[3] More explicitly, an injective continuous map f : X → Y {\displaystyle f:X\to Y} between topological spaces X {\displaystyle X} and Y {\displaystyle Y} is a topological embedding if f {\displaystyle f} yields a homeomorphism between X {\displaystyle X} and f ( X ) {\displaystyle f(X)} (where f ( X ) {\displaystyle
https://en.wikipedia.org/wiki/Unmixed_ideal
finitely generated) R-module M ≠ 0 {\displaystyle M\neq 0} is a Cohen–Macaulay module if d e p t h ( M ) = d i m ( M ) {\displaystyle \mathrm {depth} (M)=\mathrm {dim} (M)} (in general we have: d e p t h ( M ) ≤ d i m ( M ) {\displaystyle \mathrm {depth} (M)\leq \mathrm {dim} (M)} , see Auslander–Buchsbaum
https://en.wikipedia.org/wiki/Projective_dimension
However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem)
https://en.wikipedia.org/wiki/Monotonic
A function is termed monotonically increasing (also increasing or non-decreasing)[3] if for all x {\displaystyle x} and y {\displaystyle y} such that x ≤ y {\displaystyle x\leq y} one has f ( x ) ≤ f ( y ) {\displaystyle f\
https://en.wikipedia.org/wiki/Monotonic_function
A function is termed monotonically increasing (also increasing or non-decreasing)[3] if for all x {\displaystyle x} and y {\displaystyle y} such that x ≤ y {\displaystyle x\leq y} one has f ( x ) ≤ f ( y ) {\displaystyle f\
https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral
[2]
On the other hand, a classical result[3] shows that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with 1"}},"i":0}}]}'>α + β > 1
https://en.wikipedia.org/wiki/Noncommutative_ring
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https://en.wikipedia.org/wiki/Complex_analysis
}
A complex function is continuous if and only if its associated vector-valued function of two variables is also continuous
https://en.wikipedia.org/wiki/Non-associative_ring
That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative
https://en.wikipedia.org/wiki/Canonical_isomorphism
Two mathematical structures are isomorphic if an isomorphism exists between them, and this is often denoted as A ≅ B {\displaystyle A\cong B}
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/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/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/Truth_tables
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