Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.

Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.

Every one of the infinitely many vertices of G can be reached from v < sub > 1 </ sub > with a simple path, and each such path must start with one of the finitely many vertices adjacent to v < sub > 1 </ sub >.

Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero.

Every real number x is surrounded by an infinitesimal " cloud " of hyperreal numbers infinitely close to it.

This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of ( finitely or infinitely many ) copies of this module K < sup > n </ sup >.

Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3.

* Every quotient of a divisible group is divisible.

* Every existentially closed group ( in the model theoretic sense ) is divisible.

* Every abelian group can be embedded in a divisible group.

Every codeword in a Type III code has Hamming weight divisible by 3.

Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.

*" Every denunciation of existing law tends in some measure to increase the probability that there will be violation of it.

Every determines one probability distribution for.

Every quantum mechanical observable corresponds to a unique function or distribution on phase space, and vice versa, as specified by Hermann Weyl ( 1927 ) and supplemented by John von Neumann ( 1931 ); Eugene Wigner ( 1932 ); and, in a grand synthesis, by H J Groenewold ( 1946 ).

Every cumulant is just n times the corresponding cumulant of the corresponding Bernoulli distribution.

Every cumulant is just n times the corresponding cumulant of the corresponding geometric distribution.

Every country has differing rules regarding distribution of electricity for portable appliances and lighting.

Every branch of the Central Statistical Administration and its successor the State Statistics Committee ( Goskomstat ) also had a First Department to control access, distribution, and publication of official economic, population, and social statistics.

Every day the online version receives tens of thousands of hits and the CD-ROM of the first edition now in its fourth updated distribution.

Low overhead and no video or promotional cost partnered with very little distribution costs allow for hearty royalties " Every six months I send those guys royalty checks ," Werckman says.

Every year on January 1 a distribution of free food ( annadanam ) is made at the temple.

Every polynomial P in x corresponds to a function, ƒ ( x )

Every closed point of Hilb ( X ) corresponds to a closed subscheme of a fixed scheme X, and every closed subscheme is represented by such a point.

Every class prepares a float which corresponds with the Homecoming theme or related theme of school spirit as assign by school administrators.

Every infinitesimal region of space time may have its own proper time that corresponds to the gravitational time dilation there, where electromagnetic radiation and matter may be equally affected, since they are made of the same essence ( as shown in many tests involving the famous equation E = mc < sup > 2 </ sup >).

Every finite Boolean algebra can be represented as a whole power set-the power set of its set of atoms ; each element of the Boolean algebra corresponds to the set of atoms below it ( the join of which is the element ).

Every cotree T defines a cograph G having the leaves of T as vertices, and in which the subtree rooted at each node of T corresponds to the induced subgraph in G defined by the set of leaves descending from that node:

Every word of length over an-ary alphabet corresponds to a node in this tree at depth.

Every element of naturally corresponds the subset of V which assigns a 1 to its vertices.

Every natural-born citizen of a foreign state who is also an American citizen and every natural-born American citizen who is a citizen of a foreign land owes a double allegiance, one to the United States, and one to his homeland ( in the event of an immigrant becoming a citizen of the US ), or to his adopted land ( in the event of an emigrant natural born citizen of the US becoming a citizen of another nation ).

* Every group G acts on G, i. e. in two natural but essentially different ways:, or.

Every man-made product is composed of natural resources ( at its fundamental level ).

* natural language — Every good boy deserves fruit ; and

Every smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:

Every twin prime pair except ( 3, 5 ) is of the form ( 6n − 1, 6n + 1 ) for some natural number n, and with the exception of < var > n </ var > = 1, < var > n </ var > must end in 0, 2, 3, 5, 7, or 8.

Every natural hollow is full of water, around the margin of which, long grasses, reeds and other aquatic plants grow in the greatest profusion, often making it difficult to say where the land ends and the water begins.

Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.

Every smooth manifold defined in this way has a natural diffeology, for which the plots correspond to the smooth maps from open subsets of R < sup > n </ sup > to the manifold.

* Every natural number a has a successor, denoted by Sa.

Every step of the process is natural and healthy, it is only when a person gets stuck in one step for a long period of time then the grieving can become unhealthy, destructive and even dangerous.

* Every number of the form 2 < sup > m </ sup > p for a natural number m and a prime number p such that p < 2 < sup > m + 1 </ sup > is also semiperfect.

* Every finite or cofinite subset of the natural numbers is computable.

Every minimal pair of natural numbers has and, for otherwise the pair would belong to and be at least as small in both coordinates, contradicting the assumption that is minimal.

Every finite ordinal ( natural number ) is initial, but most infinite ordinals are not initial.

Every year Plum Creek replants approximately 85 million seedlings and plans for the natural regeneration of millions of trees.

Every classical theorem that applies to the natural numbers applies to the non-standard natural numbers.

Every C *- algebra has an approximate identity of positive elements of norm ≤ 1 ; indeed, the net of all positive elements of norm ≤ 1 ; in A with its natural order always suffices.

* Every individual will have a responsibility to contribute back to the greater natural and social community from his / her Brown School experience.

These rights are guaranteed to " Every individual ," that is, every natural person.

Every aspect of the ecosystem from the different species of plants to the diversity of animal species contributes to natural services like water purification, provision of food, fuel, fiber, and biochemicals, nutrient cycling, pollination and seed dispersal, and climate regulation, just to name a few.

Every August it is held the " Strawberry Festival " where the producers exhibit their natural production or their by-products elaborated from the fruit.

Every year two or more junior Bachelors of Arts students who had made the greatest progress in mathematics and natural philosophy were to be awarded a prize from the fund.