When X is an algebraic curve with field of definition the complex numbers, and if X has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of X ( its manifold of complex points ).

When D is the entire Riemann sphere, the field of meromorphic functions is simply the field of rational functions in one variable over the complex field, since one can prove that any meromorphic function on the sphere is rational.

When applying the Riemann constant in sigma mathematics, or when using integrals, it is a common practice to apply the quadratic function when ascertaining the equation's alpha variable.

When Yau was a graduate student, he started to generalize the uniformization theorem of Riemann surfaces to higher-dimensional complex Kähler manifolds.

When the topological spaces considered are Riemann surfaces or more generally analytic varieties, germs of holomorphic functions on them can be viewed as power series, and thus the set of germs can be considered to be the analytic continuation of an analytic function.

When, the complex projective space CP < sup > 1 </ sup > is the Riemann sphere, and when, CP < sup > 2 </ sup > is the complex projective plane ( see there for a more elementary discussion ).

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 the Riemann surfaces have empty boundary, they represent the worldsheets of closed strings.

When X = R < sup > n </ sup >, then, by the Riemann series theorem, the series is unconditionally convergent if and only if it is absolutely convergent.

When comparing two hypotheses and using some information, frequency methods would typically result in the rejection or non-rejection of the original hypothesis at a particular significance level, and frequentists would all agree that the hypothesis should be rejected or not at that level of significance.

When said biases are accounted, the original hypothesis — an ordering in skull volume ranging from Blacks to Mongols to Whites — is unsupported by the data.

When the hypothesis " All ravens are black " is accepted,

When Jupiter's South Equatorial Belt disappeared, one hypothesis put forward by Glenn Orten was that a large quantity of ammonia cirrus clouds had formed above it, hiding it from view.

When they hypothesized that character's guilt, they rated statements supporting that hypothesis as more important than conflicting statements.

When realization of paternity occurred, according to the hypothesis, men acted to claim a power to monopolize women and claim their offspring as possessions and patriarchy began.

When first introduced, the hypothesis boosting problem simply referred to the process of turning a weak learner into a strong learner.

When proximity during this critical period does not occur — for example, where a brother and sister are brought up separately, never meeting one another — they may find one another highly sexually attractive when they meet as adults, according to the hypothesis of genetic sexual attraction.

When Alfred Wegener first presented a hypothesis of continental drift in 1912, conservative geologists, especially in North America, demanded to know where the motive force could possibly lie.

When Streeter's two additional sources, M and L, are taken into account, this hypothesis is sometimes referred to as the Four Document Hypothesis.

When the complex relationships between pathogenic microorganisms and their hosts were finally identified — refuting the idea of holistic organisms — Schwendener's hypothesis began to gain popularity.

When an implication is translated by a hypothetical ( or conditional ) judgment the antecedent is called the hypothesis ( or the condition ) and the consequent is called the thesis.

When a theorem and its reciprocal are true we say that its hypothesis is the necessary and sufficient condition of the thesis ; that is to say, that it is at the same time both cause and consequence.

When the de Broglie wavelength was inserted into the Bragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.

When William Gilbert published de Magnete in 1600, he concluded that the Earth is magnetic and proposed the first hypothesis for the origin of this magnetism: permanent magnetism such as that found in lodestone.

When Hermogenes asks if he can provide another hypothesis on how signs come into being ( his own is simply ' convention '), Socrates initially suggests that they fit their referents in virtue of the sounds they are made of:

Form a conjecture ( hypothesis ): When nothing else is yet known, try to state an explanation, to someone else, or to your notebook.

Commenting on this hypothesis in an interview with WebMD, ophthalmologist Richard E. Bensinger stated " When we put drops in the eye to dilate the pupil, they paralyze the focusing muscles.

When the null hypothesis is rejected, the result is said to be statistically significant.

When he tests his hypothesis by attempting to kick West, the Twonky paralyzes his leg.

When he returned to England, he translated Michaelis's Introduction to the New Testament and added his own " hypothesis " on the inter-reliance of the Gospels ( that they might have derived from each other, see synoptic problem ).

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.