is the length of the coordinate vector corresponding to u < sub > i </ sub >.

# If n = 2k + 1, then the action of the unit vector u on the left ideal decomposes the space into a pair of isomorphic irreducible eigenspaces ( both denoted by Δ ), corresponding to the respective eigenvalues + 1 and − 1.

For example, given a graph G =( V, E ), the shortest path p from a vertex u to a vertex v exhibits optimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then the path p < sub > 1 </ sub > from u to w and p < sub > 2 </ sub > from w to v are indeed the shortest paths between the corresponding vertices ( by the simple cut-and-paste argument described in CLRS ).

Furthermore, if u < sub > 1 </ sub > and u < sub > 2 </ sub > are two left-singular vectors which both correspond to the singular value σ, then any normalized linear combination of the two vectors is also a left-singular vector corresponding to the singular value σ.

The Hebrew letters מלך ( mlk ) usually stand for mele < u > k </ u > ' king ' ( Proto-Northwest Semitic malku ) but when vocalized as mōle < u > k </ u > in Masoretic Hebrew text, they have been traditionally understood as a proper name Μολοχ ( molokh ) ( Proto-Northwest Semitic Mulku ) in the corresponding Greek renderings in the Septuagint translation, in Aquila, and in the Middle Eastern Targum.

For homogeneous turbulence ( i. e., statistically invariant under translations of the reference frame ) this is usually done by means of the energy spectrum function, where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field u ( x ):

One chooses a relation between x and u, determines the corresponding relation between dx and du by differentiating, and performs the substitutions.

Thus, a and b are the eigenvalues corresponding to u and v, respectively.

The next step is to insert a variable y into the perfect square on the left side of equation ( 2 ), and a corresponding 2y into the coefficient of u < sup > 2 </ sup > in the right side.

Here < u > n </ u > is the numeral corresponding to the natural number n, which is defined to be the closed term 1 + ··· + 1 ( n ones ), and < u > f ( n )</ u > is the numeral corresponding to f ( n ).

* A function problem consists of a function f from a set U to a set V. An instance of the problem is to compute, given an element u in U, the corresponding element f ( u ) in V. For example, U and V may be the set of all finite binary strings, and f may take a string and return the string obtained by reversing the digits of the input ( so f ( 0101 ) = 1010 ).

Since broadcast frequencies are very limited in number, these objectives are to some extent inconsistent in that not all of them can be fully realized, and to the extent that each is realized, there is a corresponding reduction of the possibilities for fullest achievement of the others.

The Y.M.C.A. and Y.W.C.A. at Carleton are connected with the corresponding national organizations and carry out their general purposes.

The video signal is amplified and then switched, in synchronism with the three ultraviolet light sources which are sequenced by the rotating mirror so that during one-twentieth of a second only one wavelength, corresponding to red, green, or blue, is seen.

In the f-plane the coordinates of the corresponding point are Af.

We must now show that on some component of the graph there exist two points for which the corresponding diagonal points in the C-plane are on opposite sides of C.

The values Af are the ordinary values at Af of a multi-valued function g{t} which has components corresponding to those of f{t}.

There are also corresponding members and outside advisory groups.

The coldest zones are in high latitudes, with the coldest regions corresponding to the areas covered by sea ice.

Vowels not preceded by a consonant may be represented with a zero consonant letter, modified to indicate the vowel, or separate letters for each vowel, that are distinct from the corresponding dependent vowel signs.

Except for AcPO < sub > 4 </ sub >, they are all similar to the corresponding lanthanum compounds and contain actinium in the oxidation state + 3.

Members of the southeast Asian planthopper genus Ancyra are well known for having a pair of prolonged filaments at the tips of the forewings that arise near a pair of small glossy spots ; this creates the impression of a pair of antennae, with corresponding " eyes " ( a remarkable case of automimicry ).

Hindu astrology relies on the sidereal zodiac in which the signs of the zodiac are aligned to the position of the corresponding constellations in the sky.

He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves.

All but one of the actinides are f-block elements, corresponding to the filling of the 5f electron shell ; lawrencium, a d-block element, is also generally considered an actinide.

But if the Frisian names Akkrum, Akkeringa and Dutch name Akkerghem, are derived from the personal name Akker there may be a corresponding Old English name from which Accrington may be derived.

But these are physical representations of the corresponding mathematical entities ; the line and the curve are idealized concepts whose width is 0 ( see Line ).

When the price of a stock on the NYSE and its corresponding futures contract on OCX are out of sync, one can buy the less expensive one and sell it to the more expensive market.

They are of two kinds: the " parallel texts ", which are parallel developments of the corresponding passages in the base text, and the speeches of Elihu ( Chapters 32-37 ), which consist of a polemic against the ideas expressed elsewhere in the poem, and so are claimed to be interpretive interpolations.

Due to their larger size, and corresponding difficulty with surviving the stomach, colon and liver, biopharmaceuticals are typically injected.

respectively, where e < sub > x </ sub >, e < sub > y </ sub >, e < sub > z </ sub > denotes the cartesian basis vectors ( all are orthogonal unit vectors ) and A < sub > x </ sub >, A < sub > y </ sub >, A < sub > z </ sub > are the corresponding coordinates, in the x, y, z directions.

It is a common misconception that in a liquid mixture at a given pressure, each component boils at the boiling point corresponding to the given pressure and the vapors of each component will collect separately and purely.

The maximum absolute value of g < sub > em </ sub >, corresponding to purely left-or right-handed circular polarization, is therefore 2.

It is a compromise between a purely abstract diagram ( e. g. the schematic of the Washington Metro ) and an exclusively realistic representation ( e. g. the corresponding aerial view of Washington ).

" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions ; for instance, in the case corresponding to the extension being separable, one known proof involves the use of the primitive element theorem in the context of Galois extensions.

It is quickly found that memorization purely of letters corresponding to moves on the cube is extremely difficult.

Later model televisions and radios were tuned by a rack of momentary push buttons ; some of the earlier types were purely mechanical and adjusted the capacitance or inductance of the tuned circuit to a preset number of positions corresponding to the frequencies of popular local stations.

The area of mathematics known as Stone duality is founded on the fact that the Stone representation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a duality exists between Boolean algebras and Boolean spaces.

For Libet, these subjective referrals would appear to be purely a mental function with no corresponding neural basis in the brain.

* The incoming radiation from the Sun is mostly in the form of visible light and nearby wavelengths, largely in the range 0. 2 – 4 μm, corresponding to the Sun's radiative temperature of 6, 000 K. Almost half the radiation is in the form of " visible " light, which our eyes are adapted to use.

For a nearby object emitting a given amount of light, radiative flux decreases with the square of the distance to the object, but the physical area corresponding to a given solid angle ( e. g. 1 square arcsecond ) increases in the same fashion, resulting in the same surface brightness.

Types of reactions frequently encountered are s: scattering,: radiative capture, a: absorption ( radiative capture belongs to this type ), f: fission, the corresponding notation for cross-sections being:,,, etc.

During radiative recombination, a form of spontaneous emission, a photon is emitted with the wavelength corresponding to the energy released.