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The equivalence of the two definitions follows from the fact that an algorithm on such a non-deterministic machine consists of two phases, the first of which consists of a guess about the solution which is generated in a non-deterministic way, while the second consists of a deterministic algorithm which verifies or rejects the guess as a valid solution to the problem.
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equivalence and two
It was possible, however, to decompose the compliance into a sum of a frequency-independent component and two viscoelastic mechanisms, each compatible with the Boltzmann superposition principle and with a consistent set of time-temperature equivalence factors.
The major tool one employs to describe such a situation is called equivalence of categories, which is given by appropriate functors between two categories.
Such a definition can be formulated in terms of equivalence classes of smooth functions on M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x.
One can show that this map is an isomorphism, establishing the equivalence of the two definitions.
Next, it was necessary to identify and prove the equivalence of two notions of effective calculability.
Kleene's Church – Turing Thesis: A few years later ( 1952 ) Kleene would overtly name, defend, and express the two " theses " and then " identify " them ( show equivalence ) by use of his Theorem XXX:
The completion of M can be constructed as a set of equivalence classes of Cauchy sequences in M. For any two Cauchy sequences ( x < sub > n </ sub >)< sub > n </ sub > and ( y < sub > n </ sub >)< sub > n </ sub > in M, we may define their distance as
Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are " a ~ b " and " a ≡ b ", which are used when R is the obvious relation being referenced, and variations of " a ~< sub > R </ sub > b ", " a ≡< sub > R </ sub > b ", or " aRb ".
This is an equivalence relation, which partitions the integers into two equivalence classes, the even and odd integers.
However, if the approximation is defined asymptotically, for example by saying that two functions f and g are approximately equal near some point if the limit of f-g is 0 at that point, then this defines an equivalence relation.
If ~ and ≈ are two equivalence relations on the same set S, and a ~ b implies a ≈ b for all a, b ∈ S, then ≈ is said to be a coarser relation than ~, and ~ is a finer relation than ≈.
For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.
This relation gives rise to exactly two equivalence classes: one class consisting of all even numbers, and the other consisting of all odd numbers.
Every two equivalence classes and are either equal or disjoint.
In other words, if is an equivalence relation on a set, and and are two elements of, then these statements are equivalent:
The morphisms from the point p to the point q are equivalence classes of continuous paths from p to q, with two paths being equivalent if they are homotopic.
In the following, we state two equivalent forms of the theorem, and show their equivalence.
For such data, one must use a hash function that is compatible with the data equivalence criterion being used: that is, any two inputs that are considered equivalent must yield the same hash value.
Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic Principles of Quantum Mechanics.
In other words, physical states can be identified with equivalence classes of vectors of length 1 in H, where two vectors represent the same state if they differ only by a phase factor.
Moral equivalence is a term used in political debate, usually to criticize any denial that a moral hierarchy can be assessed of two sides in a conflict, or in the actions or tactics of two sides.
" Hence an argument which claimed that the two parties could be viewed as " equally " culpable in a struggle for supremacy, would be advocating " moral equivalence.
equivalence and definitions
The equivalence of these definitions can be proved using the Dandelin spheres.
The equivalence of this GCD definition with the other definitions is described below.
As a result, general theorems about left / right adjoint functors, such as the equivalence of their various definitions or the fact that they respectively preserve colimits / limits ( which are also found in every area of mathematics ), can encode the details of many useful and otherwise non-trivial results.
See mass – energy equivalence for a discussion of definitions of mass.
In this paper he unified Gallai's result with several similar results by defining perfect graphs, and he conjectured the equivalence of the perfect graph and Berge graph definitions ; Berge's conjecture was later proven as the strong perfect graph theorem.
" The International Electrotechnical Commission began work on terminology in 1909 and established Technical Committee 1 in 1911, its oldest established committee, " to sanction the terms and definitions used in the different electrotechnical fields and to determine the equivalence of the terms used in the different languages.
In general, there is a wide range of possible definitions of functional equivalence covering comparisons between different levels of abstraction and varying granularity of timing details.
The equivalence between the two definitions is credited to Jean-Pierre Serre in Faisceaux algébriques cohérents.
) The equivalence of the definitions follows from the closedness of the symplectic form and Cartan's magic formula for the Lie derivative in terms of the exterior derivative.
equivalence and follows
Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.
It follows from the properties of an equivalence relation that
If X is a set with an equivalence relation denoted by infix, then a groupoid " representing " this equivalence relation can be formed as follows:
The Klein bottle can be seen as a fiber bundle over S < sup > 1 </ sup > as follows: one takes the square ( modulo the edge identifying equivalence relation ) from above to be E, the total space, while the base space B is given by the unit interval in y, modulo 1 ~ 0.
It follows from this that equality is the smallest equivalence relation on any set S, in the sense that it is a subset of any other equivalence relation on S. An equation is simply an assertion that two expressions are related by equality ( are equal ).
Consider the polynomial ring R, and the irreducible polynomial The quotient space is given by the congruence As a result, the elements ( or equivalence classes ) of are of the form where a and b belong to R. To see this, note that since it follows that,,, etc.
Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:
As follows from the correspondence between equivalence classes of connected coverings and conjugacy classes of subgroups of the fundamental group of the base space discussed below, a connected covering is determined by a subgroup of, where is the induced homomorphism.
Under an equivalence T, the symmetric matrix A of φ and the symmetric matrix B of ψ are related as follows:
From the monotonicity and continuity of relation composition, it follows immediately that the set of the bisimulations is closed under unions ( joins in the poset of relations ), and a simple algebraic calculation shows that the relation of bisimilarity — the join of all bisimulations — is an equivalence relation.
However, Einstein was the first scientist to propose the E = mc < sup > 2 </ sup > formula and the first to interpret mass – energy equivalence as a fundamental principle that follows from the relativistic symmetries of space and time.
From this follows that if f is a homotopy equivalence, then f < sub >*</ sub > is an isomorphism.
Note that one need not assume that the other inclusion is also a simple homotopy equivalence — that follows from the theorem.
If we assume that the binary relationship is also reflexive, then it follows that thermal equilibrium is an equivalence relation.
Classifications up to homeomorphism and homotopy equivalence are known, as follows.
The holding of the equivalence property also follows.
for, and the equivalence follows, since
Moreover, classifications up to homeomorphism and homotopy equivalence are known, as follows.
The above equivalence follows as the relative change in operating income with one more unit dX equals the contribution margin divided by operating income while the relative change in sales with one more unit dX equals price divided by revenue ( or, in other words, 1 / X with X being the quantity ).
Define an equivalence relation as follows.
Let p be a point of M. Consider the space consisting of smooth maps defined in some neighborhood of p. We define an equivalence relation on as follows.
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