In 1956, he applied the same thinking to the Riemann – Roch theorem, which had already recently been generalized to any dimension by Hirzebruch.

In general, one can specify a point in a space of any dimension n by use of n Cartesian coordinates, the signed distances from n mutually perpendicular hyperplanes.

In the same way one defines a Cartesian space of any dimension n, whose points can be identified with the tuples ( lists ) of n real numbers, that is, with.

They are not extended in space in any dimension.

The divergence can be defined on any manifold of dimension n with a volume form ( or density ) e. g. a Riemannian or Lorentzian manifold.

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it.

* Member: an ( element or object ) in a ( system or dimension ) at any ( level or rank )

In mathematics, the Hausdorff dimension ( also known as the Hausdorff – Besicovitch dimension ) is an extended non-negative real number associated with any metric space.

Inner product spaces generalize Euclidean spaces ( in which the inner product is the dot product, also known as the scalar product ) to vector spaces of any ( possibly infinite ) dimension, and are studied in functional analysis.

Their work led to the development of geometric measure theory which studies various notions of surface area for irregular objects of any dimension.

Spheres can be generalized to spaces of any dimension.

This is an example of the following fact: any subset of R of Hausdorff dimension strictly greater than zero must be uncountable.

Cull and de Curtins proved that on any rectangular board whose smaller dimension is at least 5, there is a ( possibly open ) knight's tour.

More generally, for points in any dimension in general position, each facet of the convex hull is either oriented upwards ( separating the hull from points directly above it ) or downwards ; the union of the upward-facing facets forms a topological disk, the upper hull, and similarly the union of the downward-facing facets forms the lower hull.

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

Those members who are not members of the cabinet are not entitled to make any suggestions or to take any decisions of political dimension.

In the broadest sense, any social science analysis that has a spatial dimension is embraced by regional scientists.

Note that neither of these statements implies the other, since there are complete metric spaces which are not locally compact ( the irrational numbers with the metric defined below ; also, any Banach space of infinite dimension ), and there are locally compact Hausdorff space which are not metrizable ( for instance, any uncountable product of non-trivial compact Hausdorff spaces is such ; also, several function spaces used in Functional Analysis ; the uncountable Fort space ).

Using a third dimension or sending any of the connections through another company or cottage is disallowed.

Likewise De Broglie's hypothesis ( 1924 ) states that any particle can be associated with a wave, and that the momentum p of the particle is related to the wavelength λ of such a wave, in one dimension, by:

This last formula is also valid for the curvature of curves in a Euclidean space of any dimension.

The formula is valid in any dimension.

In defiance, a chinless reprobate, Jake Camaret, marched down the aisle in St. Peter's one Sunday morning, followed by one of the women from the Bordel, whose dress and walk plainly showed the lack of any shame.

The specific staining by both direct and indirect methods showed that WTV antigen was concentrated in the pseudophloem tissue and in a few thick-walled cells in the xylem region, but was not detectable in any other tissues of the root and stem tumors.

Yet to determine precisely to what extent and exactly in what ways any individual showed the effects of Christianity would be impossible.

Nevertheless, a 2009 systematic review and meta-analysis of trials in people diagnosed with schizophrenia found that less than half ( 41 %) showed any therapeutic response to an antipsychotic, compared to 24 % on placebo, and that there was a decline in treatment response over time, and possibly a bias in which trial results were published.

The data showed noncompliance climbed to, or almost 15 % of all Bt corn grown, suggesting in some areas ample acreage does not exist to support pests without resistance to mate with any resistant pests that survived the Bt corn.

While at university, Gauss independently rediscovered several important theorems ; his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime ( and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2 ) can be constructed by compass and straightedge.

After 1080, he never showed any indication of repentance, and therefore remained excommunicate for twenty-six years ( 1080 – 1106 ).

He never wrote or showed any ill will towards those who wronged him.

This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type.

The production of Dolly showed that genes in the nucleus of such a mature differentiated somatic cell are still capable of reverting back to an embryonic totipotent state, creating a cell that can then go on to develop into any part of an animal.

Arthur Fine subsequently showed that any theory satisfying the inequalities can be modeled by a local hidden variable theory.

Fine's work showed that, taking locality as a given, there exist scenarios in which two statistical variables are correlated in a manner inconsistent with counter-factual definiteness, and that such scenarios are no more mysterious than any other, despite the inconsistency with counter-factual definiteness seeming ' counter-intuitive '.

More precisely, he showed that a random graph on vertices, formed by choosing independently whether to include each edge with probability has, with probability tending to 1 as goes to infinity, at most cycles of length or less, but has no independent set of size Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than in which each color class of a coloring must be small and which therefore requires at least colors in any coloring.

But having abandoned abstract philosophical speculation in his youth, Marx himself showed great reluctance during the rest of his life about offering any generalities or universal truths about human existence or human history.

Hershey and Chase showed that the introduction of deoxyribonuclease ( referred to as DNase ), an enzyme that breaks down DNA, into a solution containing the labeled bacteriophages did not introduce any < sup > 32 </ sup > P into the solution which demonstrated that the phage, while intact, is resistant to the enzyme.

Originally his work focused on technique-some argue that it had that focus at the cost of relevance, especially in Rain ( Regen, 1929 ), a 10-minute short filmed over 2 years which features impressive cinematography and a number of ' characters ' but no information about them aside from what was visible, and in The Bridge ( De Brug, 1928 ), which showed a frank admiration of engineering and also featured a number of " characters " but again did not give any information about them.

The onset of World War II showed that the League had failed its primary purpose, which was to prevent any future world war.

Among the laws enforced by Banda, it was illegal for women to wear see-through clothes, pants of any kind or skirts which showed any part of the knee.

Katherine Henderson and Barbara McManus consider he " showed more empathy for women than any other ancient writer ", citing " relatively modern critics " to support their claim.

The even-lower bitrates were introduced because tests showed that MPEG-1 Audio could provide higher quality than any existing ( circa 1994 ) very low bitrate ( i. e. speech ) audio codecs.

Maria Eleonore showed little affection for her daughter and was not allowed any influence in Christina's upbringing.

At age 19, he showed there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations.

Experiments conducted at the end of June and early July at Schilling, near Wilhelmshaven, showed that the submersible tanks functioned best when they were kept moving along the seabed as, if halted for any reason, they tended to sink into the sand.