For a brief period each year, the rays of the sun are warm enough to melt some of the snows piled a mile deep at the base of the headwalls, and then the pinnacles glisten in the daytime at high noon, and billions of gallons of water begin their slow seepage under the glaciers and across the rockstrewn hanging valleys on their long, meandering journey to the sea -- running east past the sky-carving massifs of Gurla Mandhata and Kemchenjunga, then turning south and curling down through the jungles of Assam, past the Khasi Hills, and into Bengal, past Sirinjani and Madaripur, until the hard water of the melting snows mingles with the soft drainage of fields and at length fans out to meld with the teeming salt depths of the Bay of Bengal.

For centuries the North Atlantic nations dominated the world and as long as they did they could afford the luxury of fighting each other.

For instance -- what about all those people Harold Rhodes went toward unhesitatingly, as if this were the one moment they would ever have together, their one chance of knowing each other??

For each State ( except Puerto Rico, Guam, the Virgin Islands, and, prior to 1962, Alaska and Hawaii ) determine average per capita income based on the last three years.

( For each State, make all computations set forth in items 1 to 8 above, and then add the results obtained for each State in item 8.

For each State ( except the Virgin Islands, Guam, Puerto Rico, and, prior to 1962, Alaska and Hawaii ), determine the average per capita income for the last three years.

For the year 2000 this means servicing ( A ) the protection requirements of a watershed producing at least 200 million acre-feet of water each year, ( B ) recreation and wildlife resources used each year by 635 million visitors, ( C ) a timber resource supporting an annual cut of 21 billion board feet, and ( D ) 60 million acres of rangelands.

For added strength, I also fastened a small block on each side of every frame and batten joint.

Moreland fixed us each another drink, and said, `` For God's sake, tell me something truly amusing ''.

For each mass flow rate the arc voltage was measured.

For each i, let Af.

For an experiment to qualify as a binomial experiment, it must have four properties: ( 1 ) there must be a fixed number of trials, ( 2 ) each trial must result in a `` success '' or a `` failure '' ( a binomial trial ), ( 3 ) all trials must have identical probabilities of success, ( 4 ) the trials must be independent of each other.

For each of these lines meets Q in three points, namely two points on **zg and one point on one of the multiple secants.

For purposes of sample selection only ( individual tests were given later ) we obtained group test scores of reading achievement and intelligence from school records of the entire third-grade population in each school system.

The Wechsler Intelligence Scale For Children was administered to each sample third-grade child by a clinical worker.

For example, one hebephrenic man used to annoy me, month after month, by saying, whenever I got up to leave and made my fairly steoreotyped comment that I would be seeing him on the following day, or whenever, `` You're welcome '', in a notably condescending fashion -- as though it were his due for me to thank him for the privilege of spending the hour with him, and he were thus pointing up my failure to utter a humbly grateful, `` thank you '' to him at the end of each session.

For deserts, puddings and pies are each served once a week.

For easy viewing, the middle 2 beads on each wire ( the 5th and 6th bead ) usually are of a different colour from the other eight beads.

For each picture frame, the 6502 writes graphics commands for the DVG into a defined area of RAM ( the vector RAM ), and then asks the DVG to draw the corresponding vector image on the screen.

For example for any ( even infinite ) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks ( assumed to have no distinguishing features ), such a selection can be obtained only by invoking the axiom of choice.

For example, suppose that each member of the collection X is a nonempty subset of the natural numbers.

For any real number the absolute value or modulus of is denoted by ( a vertical bar on each side of the quantity ) and is defined as

For ƒ ∈ C ( X ) ( with a compact Hausdorff space X ), one sees that:

For every a ∈ A, there is λ ∈ C such that

For any x ∈ A,

For example, if u < sub > 1 </ sub > is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u < sub > 1 </ sub >, with α ∈ R, is an invariant curve of the map.

For example, the function for x ∈ Q satisfies f ( 0 )

For this rule to be applicable, each element y ∈ Y must correspond to no more than one x ∈ X ; a function ƒ with this property is called one-to-one, or information-preserving, or an injection.

For instance, for each n ∈ N, let

For instance, if Y is a normed space, then this topology is defined by the seminorms indexed by x ∈ X:

For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.

for all g < sub > 1 </ sub >, g < sub > 2 </ sub >, h < sub > 1 </ sub >, h < sub > 2 </ sub > ∈ G. For a congruence on a group, the equivalence class containing the identity element is always a normal subgroup, and the other equivalence classes are the cosets of this subgroup.

For every dyadic fraction r ∈ ( 0, 1 ), we are going to construct an open subset U ( r ) of X such that:

: 3 ) For every compact set K ⊂ D there exists a constant C such that for every x ∈ K and every non-negative integer k the following bound holds:

For example, a smooth curve α ( t ): 1 → M has tangent vector α ′( t < sub > 0 </ sub >) in the tangent space TM ( α ( t < sub > 0 </ sub >)) at any point t < sub > 0 </ sub > ∈ ( 0, 1 ), and each such vector has length ‖ α ′( t < sub > 0 </ sub >)‖, where ‖·‖ denotes the norm induced by the inner product on TM ( α ( t < sub > 0 </ sub >)).

** For any i ∈

:: For any open neighborhood N of A, there is a positive constant T such that f ( t, b ) ∈ N for all real t > T.

For 1 < p, q < ∞ and f ∈ L < sup > p </ sup >( μ ) and g ∈ L < sup > q </ sup >( μ ), Hölder's inequality becomes an equality if and only if | f |< sup > p </ sup > and | g |< sup > q </ sup > are linearly dependent in L < sup > 1 </ sup >( μ ), meaning that there exist real numbers α, β ≥ 0, not both of them zero, such that α | f |< sup > p </ sup > = β | g |< sup > q </ sup > μ-almost everywhere.

# For every root α ∈ Φ, the set Φ is closed under reflection through the hyperplane perpendicular to α.

For each weight λ ∈ h *, there exists a unique ( up to isomorphism ) simple highest-weight g-module with highest weight λ, which is denoted L ( λ ).

For example, if there is no inner model for a measurable cardinal, then the Dodd-Jensen core model, K < sup > DJ </ sup > is the core model and satisfies the Covering Property, that is for every uncountable set x of ordinals, there is y such that y ⊃ x, y has the same cardinality as x, and y ∈ K < sup > DJ </ sup >.

For every element a of a linearly ordered group G either a ∈ G < sub >+</ sub >, or − a ∈ G < sub >+</ sub >, or a = 0.