[permalink] [id link]
That axiom stated that the internal energy of a phase in equilibrium is a function of state, that the sum of the internal energies of the phases is the total internal energy of the system, and that the value of the total internal energy of the system is changed by the amount of work done adiabatically on it, considering work as a form of energy.
from
Wikipedia
Some Related Sentences
axiom and stated
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:
With this alternate notion of choice function, the axiom of choice can be compactly stated as
Until the late 19th century, the axiom of choice was often used implicitly, although it had not yet been formally stated.
( Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference to the concept of subset.
In cryptography, Kerckhoffs's principle ( also called Kerckhoffs's Desiderata, Kerckhoffs's assumption, axiom, or law ) was stated by Auguste Kerckhoffs in the 19th century: A cryptosystem should be secure even if everything about the system, except the key, is public knowledge.
Carathéodory's version of the first law of thermodynamics was stated in an axiom which refrained from defining or mentioning temperature or quantity of heat transferred.
In order to establish his proof, Reinhold stated an axiom that could not possibly be doubted.
Sometimes the requirement that the operation be valued in a set is explicitly stated, in which case it is known as the axiom of closure.
As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ch is a homomorphism of abelian groups from the K-theory K ( X ) into the rational cohomology of X.
For each theorem that can be stated in the base system but is not provable in the base system, the goal is to determine the particular axiom system ( stronger than the base system ) that is necessary to prove that theorem.
The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X, there is a sequence ( x < sub > n </ sub >) in X such that x < sub > n </ sub > R < nowiki ></ nowiki > x < sub > n + 1 </ sub > for each n in N. ( Here an entire binary relation on X is one such that for each a in X there is a b in X such that aRb.
There is a third underlying axiom of Psychohistory, which is trivial and thus not stated by Seldon in his Plan:
This is often stated as a Mayer-Vietoris axiom: for any CW complex W covered by two subcomplexes U and V, and any elements u ∈ F ( U ), v ∈ F ( V ) such that u and v restrict to the same element of F ( U ∩ V ), there is an element w ∈ F ( W ) restricting to u and v, respectively.
The fundamental principle, which functions as an axiom, and can be stated in symbolic logic, is that a thing is good insofar as it exemplifies its concept.
axiom and internal
Countability of the set of all internal numbers ( in conjunction with the fact that those form a densely ordered set ) implies that that set does not satisfy the full least-upper-bound axiom.
* For any internal formula without free occurrence of z, the universal closure of the following formula is an axiom:
The approach for internal set theory is the same as that for any new axiomatic system-we construct a model for the new axioms using the elements of a simpler, more trusted, axiom scheme.
axiom and phase
Heating a body, such as a segment of protein alpha helix ( above ), tends to cause its atoms to vibrate more, and to expand or change Phase ( matter ) | phase, if heating is continued ; an axiom of nature noted by Herman Boerhaave in the in 1700s.
axiom and is
only seldom is it so simple as to be a matter of his obviously parroting some timeworn axiom, common to our culture, which he has evidently heard, over and over, from a parent until he experiences it as part of him.
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".
Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin.
In many cases such a selection can be made without invoking the axiom of choice ; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin.
Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in ZFC, the standard form of axiomatic set theory.
One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.
The axiom of choice asserts the existence of such elements ; it is therefore equivalent to:
The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function.
However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.
" In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo.
") This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.
If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.
axiom and function
Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.
In the even simpler case of a collection of one set, a choice function just corresponds to an element, so this instance of the axiom of choice says that every nonempty set has an element ; this holds trivially.
A still weaker example is the axiom of countable choice ( AC < sub > ω </ sub > or CC ), which states that a choice function exists for any countable set of nonempty sets.
For example, the axiom AC < sub > 11 </ sub > can be paraphrased to say that for any relation R on the set of real numbers, if you have proved that for each real number x there is a real number y such that R ( x, y ) holds, then there is actually a function F such that R ( x, F ( x )) holds for all real numbers.
Some examples include the first primitive recursive function that results in complexity, the smallest universal Turing Machine, and the shortest axiom for propositional calculus.
This version of the axiom schema of replacement is now suitable for use in a formal language that doesn't allow the introduction of new function symbols.
According to the Church-Turing thesis, any effectively calculable function is calculable by a Turing machine, and thus a set S is recursively enumerable if and only if there is some algorithm which yields an enumeration of S. This cannot be taken as a formal definition, however, because the Church-Turing thesis is an informal conjecture rather than a formal axiom.
For each first-order-formula the axiom of choice implies the existence of a function
Applying the axiom of choice again we get a function from the first order formulas to such functions
By the axiom of choice, there exists a function such that.
In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property ( also known as the right lifting property or the covering homotopy axiom ) is a technical condition on a continuous function from a topological space E to another one, B.
In his reviews for Mathematical Reviews of the da Costa / Doria papers on P = NP, logician Andreas Blass states that " the absence of rigor led to numerous errors ( and ambiguities )"; he also rejects da Costa's " naïvely plausible condition ", as this assumption is " based partly on the possible non-totality of certain function F and partly on an axiom equivalent to the totality of F ".
In decision theory, a capacity is defined as a function, from S, the set of subsets of some set, into, such that is set-wise monotone and is normalized ( i. e. Clearly this is a generalization of a probability measure, where the probability axiom of countability is weakened.
: The formal version of this axiom resembles the axiom schema of replacement, and embodies the class function F. The next section explains how Limitation of Size is stronger than the usual forms of the axiom of choice.
0.360 seconds.