Assuming p, one proves R and ¬ R, and concludes from this that p → ( R ∧ ¬ R ).

Assuming ¬ p, one proves R and ¬ R, and concludes from this that ¬ p → ( R ∧ ¬ R ).

→ Assuming non-zero rates of labor growth complicates matters somewhat, but the basic logic still applies – in the short-run the rate of growth slows as diminishing returns take effect and the economy converges to a constant " steady-state " rate of growth ( that is, no economic growth per-capita ).

Assuming f: A → B is a unital ring homomorphism, is an ideal in A, is an ideal in B, then:

The original Nernst heat theorem makes the weaker and less controversial claim that the entropy change for any isothermal process approaches zero as T → 0:

A homomorphism between two Boolean algebras A and B is a function f: A → B such that for all a, b in A:

The systematic name for sucrose, O-α-D-glucopyranosyl -( 1 → 2 )- D-fructofuranoside, indicates four things:

The systematic name for lactose is O-β-D-galactopyranosyl -( 1 → 4 )- D-glucopyranose.

This is because we arrange things such that for every integer, there is a distinct odd integer: ... − 2 → − 3, − 1 → − 1, 0 → 1, 1 → 3, 2 → 5, ...; or, more generally, n → 2n + 1.

* Consider the set K of all functions ƒ: → satisfying the Lipschitz condition | ƒ ( x ) − ƒ ( y )| ≤ | x − y | for all x, y ∈.

The free ions recombine to create new chemical compounds ( for example, breaking atmospheric oxygen into single oxygen → 2O, which then recombine creating ozone ).

Some definitions also add that for any production rule of the form u → v of a context-sensitive grammar, it shall be true that | u |≤| v |.

A metric space X is complete if and only if every decreasing sequence of non-empty closed subsets of X, with diameters tending to 0, has a non-empty intersection: if F < sub > n </ sub > is closed and non-empty, for every n, and diam ( F < sub > n </ sub >) → 0, then there is a point x ∈ X common to all sets F < sub > n </ sub >.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

for some natural number n. Moreover, since, the commutator subgroup is normal in G. For any homomorphism f: G → H,

Namely φ is universal for homomorphisms from G to an abelian group H: for any abelian group H and homomorphism of groups f: G → H there exists a unique homomorphism F: G < sup > ab </ sup > → H such that.

As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization G < sup > ab </ sup > up to canonical isomorphism, whereas the explicit construction G → G / shows existence.

The flow now defines a map, the Poincaré map F: S → S, for points starting in S and returning to S. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes x < sub > 0 </ sub >.

A dynamical system is the tuple, with a manifold ( locally a Banach space or Euclidean space ), the domain for time ( non-negative reals, the integers, ...) and f an evolution rule t → f < sup > t </ sup > ( with ) such that f < sup > t </ sup > is a diffeomorphism of the manifold to itself.

A map τ: X → X is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has.

For the purposes of simplicity, given that X and Y are sets of attributes in R, X → Y denotes that X functionally determines each of the members of Y — in this case Y is known as the dependent set.

The addition of the empty string allows the statement that the context sensitive languages are a proper superset of the context free languages, rather than having to make the weaker statement that all context free grammars with no → λ productions are also context sensitive grammars.

In fact another theoretical representation of an amine oxide uses a dative covalent bond ≡ N → O with no formal charges.

Since there is no heat exchange between the calorimeter and surroundings → Q

Therefore, if Y is an unprovable statement in a formal system, there is no statement X in that system such that X is equivalent to the implication ( X → Y ).

A residue class character θ on ( Z / k )< sup >*</ sup > is primitive if there is no proper divisor d of k such that θ factors as a map ( Z / k )< sup >*</ sup > → ( Z / d )< sup >*</ sup > → C < sup >*</ sup >.

In this case a complex number λ is said to be in the spectrum of such an operator T: D → X ( where D is dense in X ) if there is no bounded inverse ( λI − T )< sup >− 1 </ sup >: X → D.

In some cases the derivation is concealed enough that it is no longer morphemic: France → French ( or Frenchman / Frenchwoman ) or Flanders → Flemish or Wales → Welsh.

The area had no official name and was not militarily important to either the Japanese or the Chinese, but it was only eight hundred metres away from the Chinese garrison of Beidaying ( → běidàyíng ), where troops under the command of the " Young Marshal " Zhang Xueliang were stationed.

which has no limit as x → 0.

Then no matter what the choice of system of parameters or lifting, the last map from R = K < sub > d </ sub > → F < sub > d </ sub > is not 0.

*- ino ,-ina, the most used one along with-etto: paese → paesino ( village → little village ); also in baby talk and after other suffixes: bello → bellino ( pretty ), giovane → giovanotto → giovanottino ( but there are no limits to suffixation, which could continue );

* córka ( daughter ) → córeczka, córunia, córcia ( Originally córka was created as diminutive from córa, which is no longer in common use.

* An expected second 17, 20-lyase reaction ( 17-hydroxyprogesterone → androstenedione ) is mediated so inefficiently in humans as to be of no known significance.

Therefore these bundles may have no fibers at all, although for sufficiently well behaved categories they do ; for instance, for a category with pullbacks and a terminal object 1 the points of B can be identified with morphisms p: 1 → B and the fiber of p is obtained as the pullback of p and π.

Second, there are no real examples of this change working in Slavic, and it is also doubtful that only this change ( ruki ) and no other such change of sibilants ( e. g. s → h ) was borrowed into Slavic.

When T → ∞, the gain of the amplifier goes to infinity as well, and in such a case the differential voltage driving the amplifier ( the voltage across the input transistor r < sub > π1 </ sub >) is driven to zero and ( according to Ohm's law when there is no voltage ) it draws no input current.

→ Including non-zero technological progress is very similar to the assumption of non-zero workforce growth, in terms of " effective labor ": a new steady state is reached with constant output per worker-hour required for a unit of output.

Linehan and others combined a commitment to the core conditions of acceptance and change through the Hegelian principle of dialectical progress ( in which thesis + antithesis → synthesis ) and assembled an array of skills for emotional self-regulation drawn from Western psychological traditions, such as cognitive behavioral therapy and an interpersonal variant, " assertiveness training ", and Eastern meditative traditions, such as Buddhist mindfulness meditation.

→ Given a fixed stock of labor, the impact on output of the last unit of capital accumulated will always be less than the one before.

Capitalist society is epitomized by the so-called circuit of commodity production, M-C-M ' and by renting money for that purpose where the aggregate of market actors determine the money price M, of the input labor and commodities and M ' the struck price of C, the produced market commodity. It is centered on the process M → M ', " making money " and the exchange of value that occurs at that point.