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.

But California is where we're goin ''.

He thought of the jungles below him, and of the wild, strange, untracked beauty there and he promised himself that someday he would return, on foot perhaps, to hunt in this last corner of the world where man is sometimes himself the hunted, and animals the lords.

Now I wish to enter the American market, where the competition is very strong.

from downstream, where the water level is much lower, it is a high, elaborately facaded pavilion.

The answers derived by these means may determine not only the temporal organization of the dance but also its spatial design, special slips designating the location on the stage where the movement is to be performed.

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.

Robert Penn Warren puts it this way in `` Brother To Dragons '': `` The recognition of complicity is the beginning of innocence '', where innocence, I think, means about the same thing as redemption.

A dominant motive is the poet's longing for his homeland and its boyhood associations: `` Not men-folk, but the fields where I would stray, The stones where as a child I used to play ''.

In a society where everything is for sale, Marlowe is the only man who cannot be bought.

This is done for simplicity of commands and to bring the hidden redundancy up to where misunderstanding has almost zero possibility.

As to benefits to employees, it is notorious for its callous disregard except where it depends on them for services.

`` The Moral Creed '' and `` The Will To Risk '' live happily together, if we do not examine where the line is to be drawn.

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.

Ptolemy's problem is to forecast where, against the inverted bowl of night, some particular light will be found at future times.

She was exposing herself to temptation which it is best to avoid where it can consistently be done.

Such performance is a great tribute to American scientists and engineers, who in the past five years have had to telescope time and technology to develop these long-range ballistic missiles, where America had none before.

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

`` History has this in common with every other science: that the historian is not allowed to claim any single piece of knowledge, except where he can justify his claim by exhibiting to himself in the first place, and secondly to any one else who is both able and willing to follow his demonstration, the grounds upon which it is based.

Finally we may note that the idea appears in educational theory where its influence is at present widespread.

The tangent galvanometer was used to measure currents using this effect, where the restoring force returning the pointer to the zero position was provided by the Earth's magnetic field.

This carried on to the point where Claudius openly lamented his bad wives, and began to comment on Britannicus ' approaching manhood with an eye towards restoring his status within the imperial family.

* Khomeini talked not about restoring the Caliphate, but about establishing an Islamic state where the leading role was taken by Islamic jurists ( ulama ) as the successors of Shia Imams until the Mahdi returned from occultation.

Joachim took some part in the political complications of the Scandinavian kingdoms, but the early years of his reign were mainly spent in the administration of his electorate, where he succeeded in restoring some degree of order through stern measures.

Reversible elastic deformation in metals can be described by Hooke's Law for restoring forces, where the stress is linearly proportional to the strain.

Not only was the country's principal resource and employment generating activity almost entirely depleted by the rapid mining done by the three countries, the mining companies had also failed to follow the basic principles of restoring and regenerating the lands where mining had been completed.

Systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the spring-mass system, are described mathematically by the simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion.

In mechanics and physics, simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement.

Frederick in the meantime was focused on restoring peace in the Rhineland, where he organized a magnificent celebration of the canonization of Charles the Great ( Charlemagne ) at Aachen, done under the authority of the antipope Paschal III.

Folk poetry has been the source of affectionate kisses where they sometimes played an important part, as when they had the power to cast off spells or to break bonds of witchcraft and sorcery, often restoring a man to his original shape.

France therefore embarked on a program of constitutional reform in the dependent republics, first in the Helvetic Republic, where Napoleon as Mediator imposed the Constitution of Malmaison in 1801 ( followed by the Second Helvetic Constitution a year later ), restoring the old confederal order.

The Church of Jesus Christ of Latter-day Saints believes a significant portion of its members to be descended from Ephraim, arguing that they are charged with restoring the lost tribes in the latter days as prophesied by Isaiah, and that the tribes of both Ephraim and Judah will play important leadership roles for covenant Israel in the last days ; some believe that this would be the fulfilment of part of the Blessing of Jacob, where it states that Joseph is a fruitful bough, even a fruitful bough by a well ; whose branches run over the wall (, interpreting the " wall " as the ocean ).

The Brazilian government liquidated the Guaíra Falls National Park, and dynamited the submerged rock face where the falls had been, facilitating safer navigation, but eliminating the possibility of restoring the falls in the future.

The flag of the Patria Vieja would wave again after Carrera's return to power 23 July 1814 until the Battle of Rancagua ( 1 – 2 October ) where the royalist victory ended the patriot government and began the Reconquista ( or Reconquest ) from 1814 to 1817, restoring the imperial standard.

The fortifications were consolidated here and there, but the chief attention was paid to restoring the roofing of the towers and the ramparts, where Viollet-le-Duc ordered the destruction of structures that had encroached against the walls, some of them of considerable age.

Fresh from work in the north of France, he made the error of using slates and restoring the roofs as pointed cones, where local practice was traditionally of tile roofing and low slopes, in a snow-free environment.

Some of the money was spent on restoring and decorating the church of the Chora monastery in the northwest of Constantinople, where Metochites ’ donor portrait can still be seen in a famous mosaic in the narthex, above the entrance to the nave.

The restoration will involve restoring the six places where the canal is culverted ( including the three places where the M6 Motorway construction blocked the route ), restoring Hincaster Tunnel, restoring the 5 dry miles, and a new crossing of the A590 road near Kendal, as well as many more minor works including work on 52 listed structures.

The second day takes the participants through Stratford St Mary where the Trust is restoring the lock, and on through the heart of Constable Country via Dedham and Flatford before ending at Cattawade.