If the fourth dimension is a physical concept and not purely metaphysical, through what medium does it extend??

If V is finite-dimensional, then V * has the same dimension as V. Given a basis

If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

If a spatial extra dimension is of radius R, the invariant mass of such standing waves would be M < sub > n </ sub > = nh / Rc with n an integer, h being Planck's constant and c the speed of light.

If the input feature vectors have very high dimension, the learning problem can be difficult even if the true function only depends on a small number of those features.

If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two Majorana – Weyl spinor representations.

If this surface is folded along a third dimension, it allows one to picture a wormhole " bridge ".

If the dimension of V is n and

( If utility U = f ( x, y ), U, in the third dimension, does not have a local maximum for any x and y values.

If the supposed rock was out of space-time dimension, then the question would not make sense, because it would be impossible to move an object from one location in space to another if there is no space to begin with, meaning the faulting is with the logic of the question and not God's capabilities.

If the Noetherianity hypothesis is dropped, then R can have dimension anywhere between d + 1 and 2d + 1.

If R is a commutative ring, and M is an R-module, we define the Krull dimension of M to be the Krull dimension of the quotient of R making M a faithful module.

If V has finite dimension n, then GL ( V ) and GL ( n, F ) are isomorphic.

If we take the modules to be over a field, then the dimension of the n-th homology of X turns out to be the number of " holes " in X at dimension n.

If the polytope lies in n-dimensions, a face in the ( n − 1 )- dimension is called a facet.

If the primary key of the source data is required for reporting, the dimension already contains that piece of information for each row.

If A is torsion-free then it embeds into a vector space over the rational numbers of dimension rank A.

If there exist one or more eigenvalues, then if the corresponding b < sub > i </ sub > ≠ 0, it is a paraboloid ( either elliptic or hyperbolic ); if the corresponding, the dimension i degenerates and does not get into play, and the geometric meaning will be determined by other eigenvalues and other components of b. When it is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic ; otherwise, it is hyperbolic.

An extension of the tensor field idea incorporates an extra line bundle L on M. If W is the tensor product bundle of V with L, then W is a bundle of vector spaces of just the same dimension as V. This allows one to define the concept of tensor density, a ' twisted ' type of tensor field.

If the graph has n vertices and m edges then the dimension is m − n + 1.

If R is a field, the cycle space is a vector space over R with dimension m-n + c, where c is the number of connected components of G. If R is any commutative ring, the cycle space is a free R-module with rank m-n + c.

If the source has a maximum overall dimension or aperture width ( D ) that is large compared to the wavelength λ, the far-field region is commonly taken to exist at distances from the source, greater than Fresnel parameter S = D < sup > 2 </ sup >/( 4λ ), S > 1.

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

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.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If Robinson was a liar and a slanderer, he was also a very canny gentleman, for nothing that Pike could do would pry so much as a single word out of him.

If the world comes to this, wouldn't it be the very time when courage and American know-how would be needed to help survivors rebuild??

If we thus spent our very first day in the midst of a large number of your people honoring a new hero and a great national achievement, our last day, to us at least, was equally impressive and very moving, even though the crowds were absent and there was almost complete silence.

If all the operating variables were varied simultaneously, Af operations would be required to do the same job, and as R increases this increases very much more rapidly than the number of operations required by the dynamic program.

If the administration does not succeed in passing the sales tax bill, or any other tax bill, it could very well be faced this spring at the fiscal session of the Legislature with an interesting dilemma.

If the Southerners were sufficiently aroused, they could very well cut the Kennedy legislative program to ribbons from their vantage point of committee chairmanships, leaving Sam Rayburn leading a truncated, unworkable party.

If it does, ocean levels would rise by a few metres in a very short period of time.

If they find one they will monitor its progress very thoroughly.

If a poet writes down several lines, intending them as a poem, the very procedure by which it is written makes it a poem.

If the angle u1 is very small, O ' 1 is the Gaussian image ; and O ' 1 O ' 2 is termed the longitudinal aberration, and O ' 1R the lateral aberration of the pencils with aperture u2.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If this were the case, such an area forming around a ship could cause it to sink very rapidly and without warning.

If the color burst signal were visible on a television screen, it would appear as a vertical strip of a very dark olive color.

If he sometimes seems to approach the symbolic view, at other times he comes very close to a strong realistic doctrine.

If h is negative, then a + h is on the low part of the step, so the secant line from a to a + h is very steep, and as h tends to zero the slope tends to infinity.

If the impostor is himself, the clinical setting would be the same as the one described as depersonalisation, hence jamais vus of oneself or of the very " reality of reality ", are termed depersonalisation ( or surreality ) feelings.

If a second hanging tom is used, it is 10 " diameter and 8 " deep for fusion, or 13 " diameter and one inch deeper than the 12 " diameter tom otherwise, or very occasionally a 14 " diameter hanging tom is added to the 12 ", both being 8 " deep.

" If we cannot move on to point B until we have proved point A, and if in order to prove point A we must establish it with absolute certainty, then it looks as though we will have a very hard time proving any point at all.

If the surgery is performed on very young patients ( 2 – 5 years old ), the remaining hemisphere may acquire some rudimentary motor control of the ipsilateral body ; in older patients, paralysis results on the side of the body opposite to the part of the brain that was removed.

If a player is " easy to read " he very seldom will win large pots even when holding a very good hand.

If the number of counts is not very large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time.

If buried where the earth is very cold, the earth acts like a refrigerator.

If this happens rapidly before significant decay to the organic tissue, very fine three-dimensional morphological detail can be preserved.