When the ordinary citizen of Ancient Greece read inscriptions and the educated Greek read literature, what they saw was an all upper case Ionic alphabet: Α, Β, Γ, Δ, etc.

When Γ is the cocompact subgroup Z of the real numbers G = R, the Selberg trace formula is essentially the Poisson summation formula.

When Γ is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectrum of differential operators such as the Laplacian in terms of geometric data involving the lengths of geodesics on the Riemann surface.

When Heidegger and Sartre speak of a contrast between being and existence, they may be right, I don't know, but their language is too philosophical for me.

When they express themselves it is incandescent hatred that shines forth, the rage of repudiation, the ecstasy of negation.

When I try to work out my reasons for feeling that this passage is of critical significance, I come up with the following ideas, which I shall express very briefly here and revert to in a later essay.

When the proprietor dies, the establishment should become a corporation until it is either acquired by another proprietor or the government decides to drop it.

When the reactionary response is thus bolstered by an intellectual defense, the characteristics of that defense are explicable only in terms of the basic attitudes of unanalyzed reaction.

When I take over Taliesin, the first thing I'll do is fire you ''.

When someone in the audience rose and asked how does it feel to be a celebrity, Carl said, `` A celebrity is a fellow who eats celery with celerity ''.

When decision makers act within this frame they determine whether a claim put forward in the name of religion is to be accepted by the larger community as appropriate to religion.

When we consider the disorganized state of the world community, and the legacy of predispositions adversely directed against all who are identified as Jews, it is obvious that the struggle for the minds and muscles of men needs to be prosecuted with increasing vigor and skill.

When we turn to Aristotle's ideas on the moral measure of literature, it is at once apparent that he is at times equally concerned about the influence of the art.

When their faith in civil liberties is tested against strong pressures of social expediency in specific issues, e.g., suppression of `` dangerous ideas '', many waver and give in.

There is, of course, nothing new about dystopias, for they belong to a literary tradition which, including also the closely related satiric utopias, stretches from at least as far back as the eighteenth century and Swift's Gulliver's Travels to the twentieth century and Zamiatin's We, Capek's War With The Newts, Huxley's Brave New World, E. M. Forster's `` The Machine Stops '', C. S. Lewis's That Hideous Strength, and Orwell's Nineteen Eighty-Four, and which in science fiction is represented before the present deluge as early as Wells's trilogy, The Time Machine, `` A Story Of The Days To Come '', and When The Sleeper Wakes, and as recently as Jack Williamson's `` With Folded Hands '' ( 1947 ), the classic story of men replaced by their own robots.

When I first came across Samuel Johnson's pronouncement, `` the remedy for the ills of life is palliative rather than radical '', it seemed to me to sum up the profoundest of political and social truths.

When all else fails, it is argued that open sessions slow down governmental operations.

When a person has thoughtlessly or deliberately caused us pain or hardship it is not always easy to say, `` Just forget it ''.

When a person meets God's requirements for the experience of forgiveness he is forgiven.

When I hold my son he stiffens his whole body in my arms until he is as straight and stiff as a board.

When different colors are used, she is just as likely to color trees purple, hair green, etc..

When we become firm enough to stand for those ideals which we know to be right, when we become hard enough to refuse to aid nations which do not permit self-determination, when we become strong enough to resist any more drifts towards socialism in our own Nation, when we recognize that our enemy is Communism not war, and when we realize that concessions to Communists do not insure peace or freedom, then, and only then will we no longer be `` soft ''.

When one powerful nation strives to emulate the success of another, it is only natural.

When a man is laid to rest, he is entitled to stay put.

When I talked to Ching about it, he said, Everyone can learn, if he is not a Reactionary or lazy.

When molten antimony is slowly cooled, metallic antimony crystallizes in a trigonal cell, isomorphic with that of the gray allotrope of arsenic.

When n = 0, does not have zero elements ; rather, it is isomorphic to, since.

* When the subring is isomorphic to the dual number plane, then g acts as a shear mapping and preserves the dual angle.

* When the subring is isomorphic to the split-complex number plane, then g acts as a squeeze mapping and preserves the hyperbolic angle.

* When the subring is isomorphic to the ordinary complex number plane, then g acts as a rotation and preserves the Euclidean angle.

When n = 2, the dicyclic group is isomorphic to the quaternion group Q.

When the dimension of H is infinite, then S < sub > 1 </ sub >( H ) ( the trace class ) is not reflexive, because it contains a subspace isomorphic to ℓ < sup > 1 </ sup >, and S < sub >∞</ sub >( H ) = L ( H ) ( the bounded operators ) is not reflexive, because it contains a subspace isomorphic to ℓ < sup >∞</ sup > ( in both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of H ).

When the base field is an algebraically closed field of characteristic p, the n-torsion is still isomorphic to ( Z / nZ )< sup > 2g </ sup > when n and p are coprime.

When spoken about graph labeling with unique labels, commonly taken from the integer range 1 ,..., n, where n is the number of the vertices of the graph, two labeled graphs are said to be isomorphic if the corresponding underlying unlabeled graphs are isomorphic.

When we say that any two models of the above axioms are isomorphic, we mean that for any two models ( R, 0 < sub > R </ sub >, 1 < sub > R </ sub >, +< sub > R </ sub >, *< sub > R </ sub >, ≤< sub > R </ sub >) and ( S, 0 < sub > S </ sub >, 1 < sub > S </ sub >, +< sub > S </ sub >, *< sub > S </ sub >, ≤< sub > S </ sub >), there is a bijection f: R → S preserving both the field operations and the order.

When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is of particular interest: A simple module over k-algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k. This is in general stronger than being irreducible over the field k, and implies the module is irreducible even over the algebraic closure of k.

When n is even the algebra Cℓ < sub > n </ sub >( C ) is central simple and so by the Artin-Wedderburn theorem is isomorphic to a matrix algebra over C. When n is odd, the center includes not only the scalars but the pseudoscalars ( degree n elements ) as well.

When k is strictly less than n then the special orthogonal group SO ( n ) also acts transitively on V < sub > k </ sub >( R < sup > n </ sup >) with stabilizer subgroup isomorphic to SO ( n − k ) so that

When Q = P, however, the line can be any line through P. These lines correspond to the space of directions through P, which is isomorphic to P < sup > 1 </ sup >.

When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations.

When analyzing a stock, futures contract, or currency using fundamental analysis there are two basic approaches one can use ; bottom up analysis and top down analysis.

When one applies this process to nouns such as electrons, energy, contract, happiness, space, time, truth, causality, and God, ontology becomes fundamental to many branches of philosophy.

When applying link-state algorithms, each node uses as its fundamental data a map of the network in the form of a graph.

When other frequencies are present, the term " amplitude " refers to that of the fundamental only.

When the oscillator is long and thin, such as a guitar string, or the column of air in a trumpet, many of the partials are integer multiples of the fundamental frequency ; these are called harmonics.

When a resonant system such as a blown pipe or plucked string is excited, a number of overtones may be produced along with the fundamental tone.

When the input is a pure sine wave, the measurement is most commonly the ratio of the sum of the powers of all higher harmonic frequencies to the power at the first harmonic, or fundamental, frequency:

In the 1983 " Liber Hymnarius " from Solesmes, it is stated that, " When an ordinary syllable is set to one note, this represents the fundamental rhythmic value used in Gregorian chant ( i. e. valor syllabicus medius ).

When the bar and resonator are properly in tune with each other, the vibrating air beneath the bar travels down the resonator and is reflected off the closure at the bottom, then returns back to the top and is reflected back by the bar, over and over, creating a much stronger standing wave and amplifying the fundamental frequency.

When the fans are closed ( horizontal ) the resonators are partially occluded, reducing the resonance of the fundamental pitch.

When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus.

When it was reported that the Lambert W function provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges — a fundamental problem in physics — Corless and developers of the Maple Computer algebra system made a library search to find that this function was in fact ubiquitous to nature.

When asked what Messiaen's main influence had been on composers, George Benjamin said, " I think the sheer [...] colour has been so influential, [...] rather than being a decorative element, showed that colour could be a structural, a fundamental element, [...] the fundamental material of the music itself.

When the arm and hand move to pick up the rock (" E ") this is not caused by the preceding mental event M, nor by M and P together, but only by P. The physical causes are in principle reducible to fundamental physics, and therefore mental causes are eliminated using this reductionist explanation.

When the size of the observable universe becomes smaller than any particular structure, no interaction by any of the fundamental forces ( gravitational, electromagnetic, weak, or strong ) can occur between the most remote parts of the structure.

When one understands the fundamental reality that there is only one entity called the Universe, one is freed of the illusion of ownership.

When provincial legislation intrudes deeply into fundamental freedoms of speech, religion, association or assembly, the provincial legislature is creating criminal legislation, which under the distribution of powers is reserved exclusively to the Parliament of Canada by section 91 ( 27 ) of the Constitution Act, 1867.

When he revised the book in 1875, his modifications were very slight, and it is conceivable that, had he recast it, as he often expressed the desire to do in the last years of his life, he would not have abandoned any part of his fundamental thesis.

When a string is bowed or tone in a wind instrument initiated by vibrating reed or lips, a phenomenon called mode-locking counteracts the natural inharmonicity of the string or air column and causes the overtones to lock precisely onto integer multiples of the fundamental pitch, even though these are slightly different than the natural resonance points of the instrument.

When Yau was working on his thesis about manifolds with non-positive curvature and their fundamental groups, he realized that it is possible to use harmonic maps to give alternative proofs of some results there.

When the overtones are integer multiples ( e. g., 2 ×, 3 × ... 6 × ... ) of the fundamental frequency ( called harmonics ), then the oscillation is periodic, i. e., it vibrates in exactly the same way over and over again.

When the theory was put on an organised basis around 1950 ( with the definitions reduced to homotopy theory ) it became clear that the most fundamental characteristic classes known at that time ( the Stiefel-Whitney class, the Chern class, and the Pontryagin classes ) were reflections of the classical linear groups and their maximal torus structure.

When using the functor Ω to pass from Top to Frm, all set-theoretic elements of a space are lost, but-using a fundamental idea of category theory-one can as well work on the function spaces.