Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.

Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.

Then in 1986, William H. Shea, Ph. D. in Archeology, expanded on Lenglet's foundation to include the entire book of Daniel.

Then the Cartesian product set D < sub > 1 </ sub > D < sub > 2 </ sub > can be made into a directed set by defining ( n < sub > 1 </ sub >, n < sub > 2 </ sub >) ≤ ( m < sub > 1 </ sub >, m < sub > 2 </ sub >) if and only if n < sub > 1 </ sub > ≤ m < sub > 1 </ sub > and n < sub > 2 </ sub > ≤ m < sub > 2 </ sub >.

Then in 1963 he was awarded a Ph. D. in Electrical Engineering and Computer Science ( EECS ) from the University of California, Berkeley, supervised by the computer designer pioneer Harry Huskey.

Then suppose definition B contains 32 instances of A ; C contains 32 instances of B ; D contains 32 instances of C ; and E contains 32 instances of D. The design now contains 5 definitions ( A through E ) and 128 instances.

Then the Zariski tangent space at a point p ∈ X is the collection of K-derivations D: O < sub > X, p </ sub >→ K, where K is the ground field and O < sub > X, p </ sub > is the stalk of O < sub > X </ sub > at p.

Then ( X × Y, ( π < sub > 1 </ sub >, π < sub > 2 </ sub >)) is a terminal morphism from Δ to the object ( X, Y ) of D × D. This is just a restatement of the above since the pair ( f, g ) represents an ( arbitrary ) morphism from Δ ( Z ) to ( X, Y ).

Let U be a functor from D to C, and let X be an object of C. Then the following statements are equivalent:

Then we have a universal morphism for each object in C and D:

Then the wave equation is to be satisfied if x is in D and.

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.

Then F and G can be restricted to D < sub > 1 </ sub > and C < sub > 1 </ sub > and yield inverse equivalences of these subcategories.

Then Disney consultant and children's book author Robert D. San Souci suggested making a movie of the Chinese poem, " The Song of Fa Mu Lan " and Disney combined the two separate projects.

Then in December 2009 Phil Plait left the JREF due to involvement in a television project, and D. J. Grothe assumed the position of President on January 1, 2010.

Then, the markedness of the ruler comes into play: it is " anchored " at point A, and slided and rotated until one mark is at point C, and one at point D, i. e., CD = AB.

West Freehold School ( 587 ); Then, in grades 6-8, most students in Applegate and Catena go to Dwight D. Eisenhower Middle School ( 834 ) and most students in Laura Donovan and West Freehold go to Clifton T. Barkalow School ( 819 ).

Plaques of " Then and Now " covering McFarland's 150 years of history were placed at the McFarland High School, the local Culver's restaurant, the E. D.

Then in 1943 he received orders transferring him to the Secret Intelligence Branch of the Office of Strategic Services ( OSS ) in Washington, D. C.

Then he lived in Washington, D. C., summering at Rose Cliff, Newport, Rhode Island.

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.

Then in 1952 the Barbados Advocate newspaper polled several prominent Barbadian politicians, lawyers, businessmen, the Speaker of the Barbados House of Assembly and later as first President of the Senate, Sir Theodore Branker, Q. C.

Then, we say Q is a generator of a one parameter symmetry Lie group.

Then the signal is either sampled or processed by the four-quadrant multiplier which produces the complex envelope components ( I, Q ) through multiplying it by the heterodyne frequency.

Then Q ( x ) has a zero α of multiplicity r, and in the partial fraction decomposition, r of the partial fractions will involve the powers of ( x − α ).

Then, for some quotient polynomial Q ( x ) and remainder polynomial R ( x ) with degree ( R ) < degree ( D ),

Then for each t the measure Q restricted to the unaugmented sigma fields is equivalent to P restricted to

Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces.

For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divides the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable ( which is the case for every p but a finite number ) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P.

Then Q (√ d ) is a finite extension of Q, called a quadratic extension.

Let V a representation of G, and form the vector bundle V = Q ×< sub > G </ sub > V over M. Then the principal G-connection α on Q induces a covariant derivative on V, which is a first order linear differential operator

Then the Radon – Nikodym theorem provides the function g, equal to the density of μ with respect to Q.

Then xR is a direct summand of Q considered as R-modules.

Then Q is not Artinian as a left Z module, but it is Artinian a right Q module.

* Top Cow: Then and Now 2: Q & A with Publisher Filip Sablik by Chris Arrant, Newsarama, June 25, 2008

* Top Cow: Then and Now 1: Q & A with President Matt Hawkins by Chris Arrant, Newsarama, June 24, 2008

Then the sum S < sub > t </ sub > of the Q < sub > a, t </ sub >, where a runs over all the roots of P, can be taken as a particular Q < sub > t </ sub >.

Then each point p of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains p.

Then is a totally ordered subset of A, hence there exists a maximal totally ordered subset containing, in particular A contains a maximal totally ordered subset.

is partially ordered by set inclusion, therefore it contains a maximal totally ordered subset P. Then the set satisfies the desired properties.

Then the image set f ( I ) is also an interval, and either it contains f ( b ), or it contains f ( a ); that is,

Then there is Parco Forlani, which, with a size of 235 hectares is the largest park in Milan, and contains a hill and a pond.

Then, between 27 and 30 August 1865, he took up Parsifal again and made a prose draft of the work ; this contains a fairly brief outline of the plot and a considerable amount of detailed commentary on the characters and themes of the drama.

Suppose a partially ordered set P has the property that every chain ( i. e. totally ordered subset ) has an upper bound in P. Then the set P contains at least one maximal element.

Suppose a non-empty partially ordered set P has the property that every non-empty chain has an upper bound in P. Then the set P contains at least one maximal element.

Then Disc 2 of the album contains songs that have not been released officially on an album, or were rejected demos on prior albums, as well as an acoustic version of " A Certain Shade of Green ," and a cover of the Prince song " Let's Go Crazy.

Then each tile is analysed independently to find the dominant patterns in the part of the image it contains.

Then the total number of intersection points of X and Y with coordinates in an algebraically closed field E which contains F, counted with their multiplicities, is equal to the product of the degrees of X and Y.

Life is Real Only Then, When ' I Am ' contains a fragment of an autobiographical description of later years, as well as transcripts of some lectures.

It contains a 8-bit Renesas ( Then a part of Hitachi ) H8 / 300 microcontroller as its internal CPU.

The poem also contains a reference to Canto XXVI in its line " Poi s ' ascose nel foco che gli affina " (" Then hid him in the fire that purifies them ") which appears in Eliot's closing section of The Waste Land as it does to end Dante's canto.

* The song Then She Appeared by rock group XTC contains the line " Dressed in tricolour and Phrygian cap ".

Then Nancy gets a call from Hannah, her housekeeper, that contains information that leads Nancy to believe something has happened to her father.

Suppose that R is an algebra over the field C of complex numbers and M = N is a finite-dimensional simple module over R. Then Schur's lemma says that the endomorphism ring of the module M is a division ring ; this division ring contains C in its center, is finite-dimensional over C and is therefore equal to C. Thus the endomorphism ring of the module M is " as small as possible ".

The choice of this song was considered ironic by many, considering the song contains lyrics such as " So if I can shoot rabbits / Then I can shoot fascists ".