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Page "Einstein–Cartan theory" ¶ 10
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We and denote
We will denote the values of f{t} on different components by Af.
We will use the notation to denote the multiplicative inverse of, it is defined exactly when and are coprime ; the following construction explains why the coprimality condition is needed.
We call the set C the pair of A and B, and denote it
We usually denote this set using set-builder notation as
We write hom ( a, b ) ( or hom < sub > C </ sub >( a, b ) when there may be confusion about to which category hom ( a, b ) refers ) to denote the hom-class of all morphisms from a to b. ( Some authors write Mor ( a, b ) or simply C ( a, b ) instead.
We denote ( and would continue to denote ) noise from as "".
We can see this from the Bayesian update rule: letting U denote the unlikely outcome of the random process and M the proposition that the process has occurred many times before, we have
We use the shorthand notation to denote the joint probability of by.
We denote the subgroup of principal fractional ideals by Prin ( R ).
We are now left with the word tawdry to denote, essentially, the inferior garb worn by the poor and not a general put-down for St Ives itself, which remains a charming and attractive small river town.
We know from Plutarch that Xenocrates, if he did not explain the Platonic construction of the world-soul as Crantor after him did, nevertheless drew heavily on the Timaeus ; and further that he was at the head of those who, regarding the universe as unoriginated and imperishable, looked upon the chronological succession in the Platonic theory as a form in which to denote the relations of conceptual succession.
Suppose X is a normed vector space over R or C. We denote by its continuous dual, i. e. the space of all continuous linear maps from X to the base field.
We denote the canonical probability distributions for the Hamiltonian and the trial Hamiltonian by and, respectively.
We denote the eigenstates of by.
We denote the diagonal components of the density matrices for the canonical distributions for and in this basis as:
We denote the unitary operators we get by U ( a, A ), and these give us a continuous, unitary and true representation in that the collection of U ( a, A ) obey the group law of the inhomogeneous SL ( 2, C ).
We denote this map by Ad < sub > g </ sub >:
We will denote this space by C. We define
We can rephrase the above proof, using partitions, which we denote as:
We denote the spectrum of a commutative ring A by Spec ( A ).
" is reporting the sentence " We shall all sin from time to time " ( assuming the archbishop is including himself in the proposition ), where shall is used to denote simple futurity.
We end up with a sequence of numbers which denote the height of each throw to be made.
Precisely, let T be a triangulation of an n-manifold M. Let S be a simplex of T. We denote the dual cell ( to be defined precisely ) corresponding to S by DS.

We and tensor
We can introduce here the angular velocity tensor associated to the angular speed:
We assume for the moment the system in question consists of a subsystem being studied, A and the " environment ", and the total Hilbert space is the tensor product of a Hilbert space describing A, H < sub > A </ sub > and a Hilbert space describing E,: that is,
We therefore can give a definition of tensor field, namely as a section of some tensor bundle.
We can now give it a central extension into the Lie algebra spanned by H ', P '< sub > i </ sub >, C '< sub > i </ sub >, L '< sub > ij </ sub > ( antisymmetric tensor ), M such that M commutes with everything ( i. e. lies in the center, that's why it's called a central extension ) and
We can give the tensor product the structure of an algebra by defining
We may define the Hessian tensor
We may alternatively rewrite the geodesic equation in terms of the contortion tensor defined above:
We can now cleverly observe that the expression for the variation of Riemann curvature tensor above is equal to the difference of two such terms,
We may now obtain the variation of the Ricci curvature tensor simply by contracting two indices of the variation of the Riemann tensor,
We can describe geometries in general relativity in terms of a tetrad field instead of the usual metric tensor field.
We have to obtain a suitable tensor to describe the geometry of space in the presence of an energy field.
We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation-reversing coordinate transformation.
We can think of a fourth rank tensor such as the Weyl tensor, evaluated at some event, as acting on the space of bivectors at that event like a linear operator acting on a vector space:
We also need to specify an electromagnetic field by defining an electromagnetic field tensor on our Lorentzian manifold.
We can still use tensor math to use the maxima to select groups of gradients to package into several different tensor ellipsoids in the same voxel, or use more complex higher rank tensors analyses, or we can do a true " model free " analysis that just picks the maxima and goes on about doing the tractography.
We say that a natural transformation is tensor-preserving if it is the identity map on the trivial representation of G, and if it preserves tensor products in the sense that.
We can turn this vector space tensor product into an inner product space by defining

We and by
We began by declaring that all men are created equal.
We consider a rural community as an assemblage of inhabited dwellings whose configuration is determined by the location and size of the arable land sites necessary for family subsistence.
We are tempted to blame others for our problems rather than look them straight in the face and realize they are of our own making and possible of solution only by ourselves with the help of desperately needed, enlightened, competent leaders.
We note the use of rhetoric as a weapon, the manipulation of the masses by propaganda, the `` mobilization '' of effort and resources.
We are also struck by the fact that this story of a boy's love for his mother does not offend, while the incestuous love of the man, Paul Morel, sometimes repels.
`` We were possessed by visions of a new civilization to come, very pure and elevated '', he has said, `` in fact some ideal form of socialism such as we had dreamed of since the war of 1914-1918 ''.
We feel uncomfortable at being bossed by a corporation or a union or a television set, but until we have some knowledge about these phenomena and what they are doing to us, we can hardly learn to control them.
We must meet this situation by promoting a rising volume of exports and world trade.
We were given a job and we carried it out, and later, his case was taken up by the Disciplinary Committee.
We attempted to conclude this, and did so by having the papers burned.
`` We were requested by the Secretary General, as I understand it, to discuss with you such matters as appear to us to be relevant, and we are not of course either a formal group or a committee in the sense of being guided by any rules or regulations of the Secretariat.
We must avoid the notion, suggested to some people by examples such as those just mentioned, that ideas are `` units '' in some way comparable to coins or counters that can be passed intact from one group of people to another or even, for that matter, from one individual to another.
`` We are menaced for the first time in the history of the Republic by the open and unblushing effort of a multi-millionaire to purchase the Presidential nomination.
We stopped first at the amphitheater that lies at the foot of the height crowned by the Parthenon.
We were struck by the notable absence of banana skins and beer cans, but just so that we wouldn't go overboard on Greek refinement, perfection was side-stepped by a couple of braying portable radios.
There is, of course, nothing new about dystopias, for they belong to a literary tradition which, including also the closely related satiric utopias, stretches from at least as far back as the eighteenth century and Swift's Gulliver's Travels to the twentieth century and Zamiatin's We, Capek's War With The Newts, Huxley's Brave New World, E. M. Forster's `` The Machine Stops '', C. S. Lewis's That Hideous Strength, and Orwell's Nineteen Eighty-Four, and which in science fiction is represented before the present deluge as early as Wells's trilogy, The Time Machine, `` A Story Of The Days To Come '', and When The Sleeper Wakes, and as recently as Jack Williamson's `` With Folded Hands '' ( 1947 ), the classic story of men replaced by their own robots.
We believe that the list of vital things left undone to date by the Eighty-seventh Congress should have included repeal by the Senate of the Connally amendment.
If it is not enough that all of our internationalist One Worlders are advocating that we join this market, I refer you to an article in the New York Times' magazine section ( Nov. 12, 1961 ), by Mr. Eric Johnston, entitled `` We Must Join The Common Market ''.
We would be crippled by reduced output, industrial decline, widespread unemployment ''.
Once, Andrus walked by it, hastily scanned the bold black headline and the five-column lead of the article ( by Duane Bosch, staff correspondent -- age not given ), and muttered: `` We a buncha national celebrities ''.
We also continued to run a series of ads featuring endorsement of Rhode Island by industrialists who had recently established new plants here.

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