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.

He once wrote, " If I should go to Poland or Germany, every stone, every tree would remind me of contempt, hatred, murder, of children killed, of mothers burned alive, of human beings asphyxiated.

If not " true " crannogs, small occupied islets ( often at least partially artificial in nature ) may be referred to as island duns, although rather confusingly, 22 islet-based sites are classified as ' proper ' crannogs due to the different interpretations of the inspectors or excavators who drew up field reports Canmore search for crannog in the Western Isles Hebridean island dwellings or crannogs were commonly built on both natural and artificial islets, usually reached by means of a stone causeway.

If any one of the sides makes the bung reach that end of the churchyard it is victorious .” The actual word hockey was mentioned centuries before, in 1363, when King Edward III of England issued a declaration banning a list of games: " moreover we ordain that you prohibit under penalty of imprisonment all and sundry from such stone, wood and iron throwing ; handball, football, or hockey ; coursing and cock-fighting, or other such idle games.

If a player ends his stone with a point move he gets a " free turn ".

If this were the case, it would advance the earliest known stone structure at the monument by some 500 years.

If you don't kill it yourself, then the Lord, hammer-blow after hammer-blow, shall send various misfortunes, so as to crush this stone.

As James Hutton wrote: " If the stone, for example, which fell today, were to rise again tomorrow, there would be an end of natural philosophy science, our principles would fail, and we would no longer investigate the rules of nature from our observations.

If the opponent places a stone at B or D, the remaining hex can be filled to join the original two stones into a single group.

If the color is too pink, the stone is a pink sapphire.

If the monks did hide the stone, they hid it well ; no other stone fitting its description has ever been found.

If the being can create a stone that it cannot lift, then it seems that it can cease to be omnipotent.

If the being cannot create a stone which it cannot lift, then it seems it is already not omnipotent.

This is justified by observing that in order for the omnipotent agent to create such a stone, the omnipotent agent must already be more powerful than itself: such a stone is too heavy for the omnipotent agent to lift, but the omnipotent agent already can create such a stone ; If an omnipotent agent already is more powerful than itself, then it already is just that powerful.

If a being is accidentally omnipotent, then it can resolve the paradox by creating a stone which it cannot lift and thereby becoming non-omnipotent.

If God can remove His own omnipotence, then God can create an enormous stone, remove His own omnipotence, then not be able to lift the stone.

If the stone is positioned near a bone ( usually a rib in the case of kidney stones ), this treatment may be more uncomfortable because the shock waves can cause a mild resonance in the bone which can be felt by the patient.

If allowed to view the ultrasound or x-ray monitor, they may be able to see their stones change from a distinct bright point ( or dark spot depending on whether the fluoro unit is set up in native or bones white ) to a fuzzy cloud as the stone is disintegrated into a fine powder.

If carved from natural crystalline stone ( such as quartz, beryl, calcite, obsidian, or amethyst ), it may display the natural coloring and structure of the mineral from which it was fashioned.

Danaher asks one of the men in the crowd if the IRA had a hand in this, to which the reply was " If it were, not a scorched stone of your fine house would be standing.

If the stone is correctly interpreted as depicting this myth, it demonstrates that the myth was in a stable form for a period of about 500 years to the recording of the myth in the Prose Edda around the year 1220.

If a patient has a stone lodged higher in the urinary tract, the doctor may use a much finer calibre scope called a ureteroscope through the bladder and up into the ureter.