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https://en.wikipedia.org/wiki/Toll_(fee)
The Empire had a "passage system" whereby a number of toll stations would be established on a route where small tolls were collected
https://en.wikipedia.org/wiki/Noether's_second_theorem
Physics theorem for symmetries of action
Portrait of Emmy Noether, around 1900.
In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations.[1] The
https://en.wikipedia.org/wiki/Consistent_estimator
In this way one would obtain a sequence of estimates indexed by n, and consistency is a property of what occurs as the sample size “grows to infinity”
https://en.wikipedia.org/wiki/Lexicographer
Theoretical lexicography concerns the same aspects as lexicography, but aims to develop principles that can improve the quality of future dictionaries, for instance in terms of access to data and lexicographic information costs
https://en.wikipedia.org/wiki/Commutator_(ring_theory)
In linear algebra, if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis
https://en.wikipedia.org/wiki/Module_homomorphism
Schur's lemma says that a homomorphism between simple modules (modules with no non-trivial submodules) must be either zero or an isomorphism
https://en.wikipedia.org/wiki/Operator_algebra
The results obtained in the study of operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic
https://en.wikipedia.org/wiki/Ideal_(order_theory)
In Boolean algebras, the terms prime ideal and maximal ideal coincide, as do the terms prime filter and maximal filter
https://en.wikipedia.org/wiki/Korean_verbs
If speaking both to and of one's elders, one would use both the formal and the honorific suffixes
https://en.wikipedia.org/wiki/Euclidean_division
If 9 slices were divided among 3 people instead of 4, then each would receive 3 and no slice would be left over, which means that the remainder would be zero,
https://en.wikipedia.org/wiki/Embedding
The interest here is in how large n {\displaystyle n} must be for an embedding, in terms of the dimension m {\displaystyle m} of M {\displaystyle M}
https://en.wikipedia.org/wiki/Unmixed_ideal
In geometric terms, every regular scheme, for example a smooth variety over a field, is Cohen–Macaulay
https://en.wikipedia.org/wiki/Monotonic
To avoid ambiguity, the terms weakly monotone, weakly increasing and weakly decreasing are often used to refer to non-strict monotonicity
https://en.wikipedia.org/wiki/Projective_dimension
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https://en.wikipedia.org/wiki/Monotonic_function
To avoid ambiguity, the terms weakly monotone, weakly increasing and weakly decreasing are often used to refer to non-strict monotonicity
https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral
We specifically do not require g {\displaystyle g} to be continuous, which allows for integrals that have point mass terms
https://en.wikipedia.org/wiki/Metalanguage
Metalanguages of formal systems all resolve ultimately to natural language, the 'common parlance' in which mathematicians and logicians converse to define their terms and operations and 'read out' their formulae
https://en.wikipedia.org/wiki/Signal_recovery
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https://en.wikipedia.org/wiki/Injective_module
Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules
https://en.wikipedia.org/wiki/Hawker_Siddeley_Trident
[4]
In April 1956, Anthony Milward, chief executive of BEA, stated that he "would rather do without [jet airliners]
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/Variable_(mathematics)
[14] So for example, what would be written in modern notation as: x 3 − 2 x 2 + 10 x − 1 , {\displaystyle x^{3}-2x^{2}+10x-1,} would be written in Diophantus's syncopated notation as:
K υ α ¯ ζ ι ¯ ⋔ Δ υ β ¯ M α ¯ {\displaystyle \mathrm {K} ^{\upsilon }{\overline {\al
https://en.wikipedia.org/wiki/Geometrical_optics
of optics that describes light propagation in terms of rays
https://en.wikipedia.org/wiki/Mathematical_proof
In most mathematical literature, proofs are written in terms of rigorous informal logic
https://en.wikipedia.org/wiki/Apocalypse
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https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity
of the current error term or innovation as a function of the actual sizes of the previous time periods' error terms;[1] often the variance is related to the squares of the previous innovations
https://en.wikipedia.org/wiki/Russell's_paradox
[6][7] At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction (to Cantor's theorem)[8], as he told Hilbert and Richard Dedekind by letter
https://en.wikipedia.org/wiki/Common_hippo
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https://en.wikipedia.org/wiki/Hodge_star_operator
n}V} is defined in terms of an oriented orthonormal basis { e 1 , … , e n } {\displaystyle \{e_{1},\ldots ,e_{n}\}} of V as:
ω := ± e 1 ∧ ⋯ ∧ e n {\displaystyle \omega :=\pm e_{1}\wedge \cdots \wedge e_{n}} ,
where the sign is free to be chosen and fixed as plus or minus
https://en.wikipedia.org/wiki/Axiomatic_system
It is not, however, evident whether the existence of inverse elements was axiomatic: it would follow from the semantic assumption that groups were permutation groups (permutations being invertible by definition) or geometric transformations with the same property