If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

In other words, if F satisfies the differential equation Af, then F is uniquely expressible in the form Af where Af satisfies the differential equation Af.

The expected value of F is ( where n is the treatment sample size )

where n < sub > d </ sub >, n < sub > F </ sub > and n < sub > C </ sub > are the refractive indices of the material at the wavelengths of the Fraunhofer d -, F-and C-spectral lines ( 587. 6 nm, 486. 1 nm and 656. 3 nm respectively ).

where F ' and C ' are the blue and red cadmium lines at 480. 0 nm and 643. 8 nm, respectively.

Thus, all the antiderivatives of x < sup > 2 </ sup > can be obtained by changing the value of C in F ( x ) = ( x < sup > 3 </ sup >/ 3 ) + C ; where C is an arbitrary constant known as the constant of integration.

Most modern assassinations have been committed either during a public performance or during transport, both because of weaker security and security lapses, such as with U. S. President John F. Kennedy and former Pakistani Prime Minister Benazir Bhutto, or as part of coups d ' état where security is either overwhelmed or completely removed, such as with Patrice Lumumba and likely Salvador Allende.

The Rathaus Schöneberg, where John F. Kennedy made his famous " Ich bin ein Berliner!

The case where X is F, and we have a bilinear form, is particularly useful ( see for example scalar product, inner product and quadratic form ).

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.

The Blue Angels ' six demonstration pilots fly the F / A-18 Hornet in more than 70 shows at 34 locations throughout the United States each year, where they still employ many of the same practices and techniques used in their aerial displays in 1946.

The temperature range from 41 ° F to 135 ° F ( 5 ° C to 57 ° C ) is the " Danger zone " where bacteria is likely to proliferate, food therefore should not be stored in this temperature range.

The marketplace where farmers brought in crops from surrounding towns to sell survives today as the small park at the corner of John F. Kennedy ( J. F. K.

This situation arises for all homonuclear diatomic molecules and is particularly a problem for F < sub > 2 </ sub >, where the minimum energy of the curve with molecular orbital theory is still higher in energy than the energy of two F atoms.

where ε denotes the Levi-Civita symbol, the metric tensor is used to lower the index on F, and the Einstein summation convention implies that repeated indices are summed over.

where F is the underlying field of the vector space being considered.

In the special case of a planet in circular orbit around its star, for example, where is zero, the centripetal acceleration alone is the entire acceleration of the planet, curving its path toward the sun under the force of gravity, the negative F ( r ).

The F-corona ( F for Fraunhofer ) is created by sunlight bouncing off dust particles, and is observable because its light contains the Fraunhofer absorption lines that are seen in raw sunlight ; the F-corona extends to very high elongation angles from the Sun, where it is called the zodiacal light.

The common notation for the divergence ∇· F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of ∇ ( see del ), apply them to the components of F, and sum the results.

* G — Games played: number of games where the player played, in whole or in part

A binary relation R is usually defined as an ordered triple ( X, Y, G ) where X and Y are arbitrary sets ( or classes ), and G is a subset of the Cartesian product X × Y.

Watterson was born in Washington, D. C., where his father, James G. Watterson ( born 1932 ), worked as a patent examiner while going to George Washington University Law School before becoming a patent attorney in 1960.

For example " GOOD DOG " can be encrypted as " PLLX XLP " where " L " substitutes for " O ", " P " for " G ", and " X " for " D " in the message.

" This anecdote is briefly discussed in G. Waldo Dunnington's Gauss, Titan of Science where it is suggested that it is an apocryphal story.

In view of the success of her novels, particularly Jane Eyre, Charlotte was persuaded by her publisher to visit London occasionally, where she revealed her true identity and began to move in a more exalted social circle, becoming friends with Harriet Martineau and Elizabeth Gaskell, and acquainted with William Makepeace Thackeray and G. H. Lewes.

for every automorphism φ of G ( where φ ( H ) denotes the image of H under φ ).

Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.

:# Mark the points E, F, G, H, where each tangent line meets the adjacent tangent.

It is now estimated that 1 % to 1. 5 % of sunlike stars possess such a planet, where " sunlike star " refers to any main-sequence star of spectral classes late-F, G, or early-K without a close stellar companion.

This leaves only the case where G has a vertex of degree 5 ; but Kempe's argument was flawed for this case.

One can compose functors, i. e. if F is a functor from A to B and G is a functor from B to C then one can form the composite functor G ∘ F from A to C. Composition of functors is associative where defined.

In addition, the entire field of synthetic biology began with researchers such as Erik T. Kool, where bases other than A, C, T and G are used in a synthetic DNA.

The resulting quotient is written, where G is the original group and N is the normal subgroup.

( This is pronounced " G mod N ," where " mod " is short for modulo.

The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism is isomorphic to where ker ( φ ) denotes the kernel of φ.

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

`` Clay '', he said, `` where are we goin ''??

`` Stay right here where you are, kid '', he called.

I asked the same questions inside the launch-control rooms of an Atlas missile base in Wyoming, where officers who wear sidearms are manning the `` commit buttons '' that could start a war -- accidentally or by design -- and in the command centers where other pistol-packing men could give orders to push such buttons.

they are the places where Persians live.

In Persia, where practically speaking there are no museums or libraries or, for that matter, hardly any books, the twins run free.

Even in domains where detailed and predictive understanding is still lacking, but where some explanations are possible, as with lightning and weather and earthquakes, the appropriate kind of human action has been more adequately indicated.

Not discussed here are some military problems of modern times such as undersea warfare, where the surveillance, sending, transmitting, and receiving are all so inadequate that networks and decision making are not the bottlenecks.

Ideally speaking, it should be allowed to operate only where the public has a great stake in the continuity of supply or services, and where the actions of a single proprietor are secondary to the needs of society.

Examples are in public utilities, making military aircraft and accessories, or where the investment and risk for a proprietorship would be too great for a much needed project impossible to achieve by any means other than the corporate form, e.g. constructing major airports or dams.

Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.

`` On each side of the Hoogli, where we are now sailing, are the Hindoo cottages, as thick together as the houses in our seaports.

but it is perennially restructured in situations where anchors are dragged or lost.

In this essay, we are, along with most historians, interested in the more general or more inclusive ideas, that are so to speak `` writ large '' in history of literature where they recur continually.

The novel, which is not merely dystopian but also brilliantly satiric, describes a future America where one-sixteenth of the population, the men who run advertising agencies and big corporations, control the rest of the people, the submerged fifteen-sixteenths who are the workers and consumers, with the government being no more than `` a clearing house for pressures ''.

Around that statue in the green park where children play and lovers walk in twos and there is a glowing view of the whole city, in that park are the rows of marble busts of Garibaldi's fallen men, the ones who one day rushed out of the Porta San Pancrazio and, under fire all the way, up the long, straight narrow lane to take, then lose the high ground of the Villa Doria Pamphili.

For instance, so-called `` conservative '' organizations, some of them secret, are sprouting in the garden of joining where `` liberal '' organizations once took root.

They enlisted the help of the New Jersey congressman, who has been able to trace the letters to the national archives, where they are available on microfilm.

The speed is controlled by pressing on the two brake buttons located where the index finger and thumb are placed when holding the motor.