The easiest way to describe this release is to say that it reproduces an interesting and effective Steinberg performance with minimal alteration of its musical values.

The characteristic polynomial for A is Af and this is plainly also the minimal polynomial for A ( or for T ).

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.

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

( C ) if Af is the operator induced on Af by T, then the minimal polynomial for Af is Af.

It is clear that each vector in the range of Af is in Af for if **ya is in the range of Af, then Af and so Af because Af is divisible by the minimal polynomial P.

Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

Hence the minimal polynomial for Af is Af.

In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.

for since it is nilpotent, the minimal polynomial for this operator is of the form Af for some Af ; ;

but then since the operator is diagonalizable, the minimal polynomial cannot have a repeated root ; ;

hence Af and the minimal polynomial is simply x, which says the operator is 0.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

Acceptance of radiopasteurization is likely to be delayed, however, for two reasons: ( 1 ) the storage life of fresh chicken under refrigeration is becoming a minimal problem because of constantly improved sanitation and distributing practices, and ( 2 ) treatment by antibiotics, a measure already approved by the Federal Food and Drug Administration, serves to extend the storage life of chicken at a low cost of about 0.5 cents per pound.

Let me pass over the trip to Sante Fe with something of the same speed which made Mrs. Roebuck `` wonduh if the wahtahm speed limit '' ( 35 m.p.h. ) `` is still in ee-faket ''.

Let me quote him even more fully, for his analysis is important to my theme.

Let us not confuse the issue by labeling the objective or the method `` psychoanalytic '', for this is a well established term of art for the specific ideas and procedures initiated by Sigmund Freud and his followers for the study and treatment of disordered personalities.

Let us quote once more from R. G. Collingwood: `` History is properly concerned with the actions of human beings Regarded from the outside, an action is an event or series of events occurring in the physical world ; ;

Let us see just how typical Krim is.

Let us assume that Af is identical to the form of an occurrence Af which preceded Af in the text.

Let us also assume the existence of only one class or type of service, all of which is supplied at the same voltage, phase, etc. to residential, commercial, and industrial customers.

Let us therefore consider each of the three types of cost in turn, recognizing that this simplified classification is used only for illustrative purposes ; ;

Let Af where the maximization is over all admissible policies Af.

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

Let us suppose ( as is the practical case ) that the star is sufficiently distant that all light from the star travels in parallel paths to the Earth observer, regardless of where the Earth is in its orbit.

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

Let us call this act a ' to be ,' in contradistinction to what is commonly called ' a being.

Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function

Let be a Hilbert space and is a vector in.

Let A be a complex unital Banach algebra in which every non-zero element x is invertible ( a division algebra ).

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.

:: Let no-one deceive you by any means, for that day will not come unless the falling away comes first, and the man of sin is revealed, the son of perdition, who opposes and exalts himself above all that is called God or that is worshiped, so that he sits as God in the temple of God, showing himself that he is God. Thess 2: 3-4

`` That is, if we can be sure this is Colcord's money '' --

`` Dandy is to be our house guest, Louis.

It is hard to see how the situation could be otherwise.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

Ratified in the Republican Party victory in 1952, the Positive State is now evidenced by political campaigns being waged not on whether but on how much social legislation there should be.

Recognizing that the Rule of Law is `` a dynamic concept which should be employed not only to safeguard the civil and political rights of the individual in a free society '', the Congress asserted that it also included the responsibility `` to establish social, economic, educational and cultural conditions under which his legitimate aspirations and dignity may be realized ''.

To him, law is the command of the sovereign ( the English monarch ) who personifies the power of the nation, while sovereignty is the power to make law -- i.e., to prevail over internal groups and to be free from the commands of other sovereigns in other nations.

Work is under way to see whether new restraining devices should be installed on all nuclear weapons.

The box is internally wired so the door can never be opened without setting off a screeching klaxon ( `` It's real obnoxious '' ).

Others are confined to vast reservations, and not only does the Australian government justifiably not wish them to be viewed as exhibits in a zoo, but on their reservations they are extremely fugitive, shunning camps, coming together only for corroborees at which their strange culture comes to its highest pitch -- which is very low indeed.

Unfortunately, it was Muzak, which automatically is piped into the public rooms, and which nolens volens had to be endured.

I consider it to be my job to expose the public to what is being written today ''.

Since attack serves to stimulate interest in broadcasts, I added to my opening statement a sentence in which I claimed that German youth seemed to lack the enthusiasm which is a necessary ingredient of anger, and might be classified as uninterested and bored rather than angry.

William Styron, while facing the changing economy with a certain uneasy reluctance, insists he is not to be classified as a Southern writer and yet includes traditional Southern concepts in everything he publishes.

Lacking the pioneer spirit necessary to write of a new economy, these writers seem to be contenting themselves with an old one that is now as defunct as Confederate money.

As his disciples boast, even though his emphasis is elsewhere, Faulkner does show his awareness of the changing order of the South quite keenly, as can be proven by a quick recalling of his Sartoris and Snopes families.

The approach to the depiction of the experience of creation may be analytic, as it is for Miss Litz, or spontaneous, as it is for Merle Marsicano.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

An order can be chanced rather than chosen, and this approach produces an experience that is `` free and discovered rather than bound and remembered ''.

The sequence of movements in a Cunningham dance is unlike any sequence to be seen in life.

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.