Here are some statements that require the axiom of choice in the sense that they are not provable from ZF but are provable from ZFC ( ZF plus AC ).

Here each vertex ( or node ) represents a point of choice for a player.

Here, according to an allegorical parable, " The Choice of Heracles ", invented by the sophist Prodicus ( c. 400 BC ) and reported in Xenophon's Memorabilia 2. 1. 21-34, he was visited by two nymphs — Pleasure and Virtue — who offered him a choice between a pleasant and easy life or a severe but glorious life: he chose the latter.

He wrote in his autobiography that it was Here, in this courageous New Englander's refusal to pay his taxes and his choice of jail rather than support a war that would spread slavery's territory into Mexico, I made my first contact with the theory of nonviolent resistance.

Here lay the strength and advantage of Saxe's choice: the descent in many places along the position formed a natural glacis, and throughout the whole length, from Antoing to the wood, the prolonged slope offered an even and deadly field for cannon and musketry fire.

Here ’ s an important distinction: punishment was inflicted on people who had been shown to break the law ; at least you had to have proof the law was broken. But, Foucault argued, why have proof? This was a ploy Foucault considered, proof was far more rigorous and exact in its approach because some form of finality had to be reached, a consensus, of return or recycling of punishment inflicted on those who had very little choice which was a transition from torture and straightforward execution by contrast, panopticism was a form of social control ( and power ) that is inflicted on everyone.

When we specify the internal energy to be U, what we really mean is that the total energy of the gas lies somewhere in an interval of length around U. Here is taken to be very small, it turns out that the entropy doesn't depend strongly on the choice of for large N. This means that the above " area " must be extended to a shell of a thickness equal to an uncertainty in momentum, so the entropy is given by:

Here, second, the plot builds to a point of no return, from where the protagonist – who in a case study need not be a person but may be an organization, a project, or a community – has no choice but to deal with matters, and thus is tested.

Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or interstitial packing.

for any torsion sheaf F. Here j is any open immersion of X into a scheme Y with a proper morphism g to S ( with f = gj ), and as before the definition does not depend on the choice of j and Y. Cohomology with compact support is the special case of this with S a point.

Here we have c < sup > a </ sup > for the ghost field while α fixes the gauge's choice for the quantization.

Here, PTFE or a high grade stainless steel may be a more appropriate choice.

James Monaco said that " Here, for the first time the focus is clearly set on the ethical and existential question of choice.

Here, second, the plot builds to a point of no return, from where the protagonist – who in a case study need not be a person but may be a company, an organization, a project, or a community – has no choice but to deal with matters, and thus is tested.

Here, the choice is mostly between Latin letters with diacritics, as used in many Latin-based orthographies of other Slavic languages, and digraph combinations, as used in English: ж → ž / zh, ч → č / ch, ш → š / sh, щ → št / ŝ / sht.

Here, in periods of crisis, " the capitalist class appears to have a choice between devaluing money or commodities, between inflation or depression.

Here indicates the best choice of action i for each state j.

Here, on the hottest day, it is cool beneath the stone and fresh from the water flowing in the sluices at the bottom of the vaults.

Here in these little rooms -- or stages arched open to the sky and river -- they choose a few lines out of the hundreds they may know and sing them according to one of the modes into which Persian music is divided.

Here, on a desk, is a stack of pamphlets representing the efforts of some of the best men of the day to penetrate these questions.

Here, if anywhere, it is not wholly incontrovertible.

Here we may observe that at least one modern philosophy of history is built on the assumption that ideas are the primary objectives of the historian's research.

Here an important caveat is in order.

Here, then, is what Swift would have called a modest proposal by way of a beginning.

But this we know: Here is a great life that in every area of American politics gives the American people occasion for pride and that has invested the democratic process with the most decent qualities of honor, decency, and self-respect.

Here is a word of advice when you go shopping for your pansy seeds.

Here then is our problem: aircraft are vital to winning a war today because they can perform those missions which a missile is totally incapable of performing ; ;

Here is truly a `` Great Recording of the Century '', and its greatness is by no means diminished by the fact that it is not quite perfect.

Here is an original kedgeree recipe from the Family Club's kitchen:

Here is the promise of a vacation trip they can afford.

Here is where things stand today:

Here the Af distance is 2.44 Aj.

Here the pulmonary vein, as in type 2,, is noted to draw away from the bronchus, and to follow a more direct, independent course to the hilum ( figs. 23, 24 ).

Here the number of trials is a random variable, not a fixed number.

Here there is a specific preventive component which applies in a more generalized sense to any casework situation.

( Here an entry is a form plus the information that pertains to it.

Here again, in the written language it is possible to help the reader get his stresses right by using underlining or italics, but much of the time there is simply reliance on his understanding in the light of context.

Here is the best short explanation of the origins of the Cold War that has been written.

Here, in the old days -- when they had come to see the moon or displays of fireworks -- sat the king and his court while priests, soldiers, and other members of the party lounged in the smaller alcoves between.

Here Keys and others, such as Dr. A. E. Ahrens of the Rockefeller Institute, took over to demonstrate the chemical difference between vegetable and animal fats -- and even between different varieties of each.

Here the left-hand side represents the true difference between the value of g at a and at, whereas the right-hand side represents the approximation determined by the derivative plus an error term.

Here, the difference between the two notions of program mentioned in the last paragraph becomes clear ; one is easily recognized by some grammar, while the other requires arbitrary computation to recognize.

Here we see Hersilia between her father and husband as she adjures the warriors on both sides not to take wives away from their husbands or mothers away from their children.

Here is the center of the ellipse, and is the angle between the < math > X </ Math >- axis and the major axis of the ellipse.

Here the relation between genotype and phenotype is illustrated, using a Punnett square, for the character of petal colour in pea.

Here English may serve as a useful lingua franca between ethnic and language groups.

Here Marx is drawing a distinction between exchange value ( which is the subject of the LTV ) and use value.

Here, Mies explored the relationship between people, shelter, and nature.

Here constant α quantifies the strength of the interaction between adjacent molecules.

Here, the term " parallax " is the angle or semi-angle of inclination between two sight-lines to the star.

Here Z is atomic number and n is a number which varies between 4 and 5.

Here Steiner suggests that there is an inconsistency between Kant's philosophy, which posits that knowledge is representation and that the essential verity of the world is inaccessible to human consciousness, and modern science, which assumes that all influences can be found in what Steiner termed the “ sinnlichen und geistlichen ” ( sensory and mental / spiritual ) world to which we have access.

Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.

Here, symptoms usually begin between the ages of 25 and 40 and may worsen with straining or any activity that causes cerebrospinal fluid pressure to fluctuate suddenly.

Here the terminology is different between the UK and the USA:

Here the authors make an explicit connection between urban legends and popular folklore, such as Grimm's Fairy Tales where similar themes and motifs arise.

Here ( RV 10. 75. 5 ), the sequence " Ganges, Yamuna, Sarasvati, Shutudri " places the Sarasvati between the Yamuna and the Sutlej, which is consistent with the Ghaggar identification.

Here d is the spacing between diffracting planes, is the incident angle, n is any integer, and λ is the wavelength of the beam.

Here the filament absorbs soft X-rays between 100 and 300 eV, indicating that the hot gas is located behind the filament.