Specifically, let G be a simply-connected Lie group with Lie algebra.

Specifically, let χ be a character modulo k. Then we can write its Dirichlet L-function as

Specifically, let G be a group and let X and Y be two associated G-sets.

Specifically, let X be a separable normed space and B the closed unit ball in X < sup >∗</ sup >.

Specifically, the programs Front Row and Cover Flow, utilized in conjunction with the Apple Remote, let users easily browse through and view any multimedia content stored on their Macs.

The Dedekind zeta function satisfies a functional equation relating its values at s and 1 − s. Specifically, let Δ < sub > K </ sub > denote discriminant of K, let r < sub > 1 </ sub > ( resp.

Specifically, let X and Y be a topological spaces with A a subspace of Y.

Specifically, let L be a tame oriented knot or link in Euclidean 3-space ( or in the 3-sphere ).

Specifically, let X

Specifically, a topological space is compact if, whenever a collection of open sets covers the space, some subcollection consisting only of finitely many open sets also covers the space.

Specifically, the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v ^< sub > g </ sub > =

Specifically, spinors are objects associated to a vector space with a quadratic form ( like Euclidean space with the standard metric or Minkowski space with the Lorentz metric ), and are realized as elements of representation spaces of Clifford algebras.

Specifically, if F is a vector space of linear functionals on X which separates points of X, then the continuous dual of X with respect to the topology σ ( X, F ) is precisely equal to F.

Specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quadratic form Q.

( Specifically, a thin space is endorsed for typesetting ; a regular word space is sufficient wherever typographical nicety is impractical.

Specifically, f (' F ', ' F ', ..., ' F ') = ' F ', and these functions correspond to linear maps over the Boolean vector space.

Specifically, the linear span of the basis must be dense in H, but it may not be the entire space.

Specifically, a space velocity along the line-of-sight means that the transformation from tangential linear velocity to ( angular ) proper motion is a function of time.

Specifically, the similarity between the fiber bundle E and a product space B × F is defined using a continuous surjective map

Specifically, the scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space.

Specifically, if a system is in a state described by a vector in a Hilbert space, the measurement process affects the state in a non-deterministic, but statistically predictable way.

Specifically, if X is a 1-dimensional CW-complex, the attaching map for a 1-cell is a map from a two-point space to X,.

Specifically, it is focussed on military, medical, aerospace, space and automotive markets.

Specifically, it shows that if M is a unital, self-adjoint operator algebra in the C *- algebra B ( H ), for some Hilbert space H, then the weak closure, strong closure and bicommutant of M are equal.

Specifically, it must meet less than two of the five Hugo criteria for consideration as a semiprozine: that the magazine had an average press run of at least one thousand copies per issue, paid its contributors and / or staff in other than copies of the publication, provided at least half the income of any one person, had at least fifteen percent of its total space occupied by advertising, and announced itself to be a semiprozine.

Specifically, experiments will study the adaptation of the vestibular system and space adaptation syndrome, the adaptation of the central nervous system and the pathways which control the ability to sense location in the absence of gravity, and the effect of microgravity on a developing nervous system.

Specifically, Fock showed that the Schrödinger wavefunction in the momentum space for the Kepler problem was the stereographic projection of the spherical harmonics on the sphere.

Specifically, a topological space X is compactly generated if it satisfies the following condition:

Specifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the situation of continuous functions between topological spaces.

Specifically in terms of MSPC, multiway modeling of batch and continuous processes is increasingly common in industry and remains an active area of research in chemometrics and chemical engineering.

" Specifically on the subject of creation / evolution, some Anglicans view " Big Bang cosmology " as being " in tune with both the concepts of creation out of nothing and continuous creation.

Specifically, it is the density altitude at which flying in a clean configuration, at the best rate of climb airspeed for that altitude and with all engines operating and producing maximum continuous power, will produce a given rate of climb ( usually 100 feet per minute climb or 30 meters per minute, and 500 feet per minute climb for jet airplanes ).

Specifically, the input to a digital logic gate must convert a continuous voltage value into either a 0 or a 1 which is typically sampled and then processed.

Specifically, the continuous bandwidth usage can adversely affect network performance, especially in operations where file sizes are large, such as multimedia and CAD design environments.

) Specifically, one maps an element of H < sup > k </ sup >( M ) to its cap product with a fundamental class of M, which will exist for oriented M.

Specifically, 3D terrain using RAW height maps suffered from a lack of working examples, shadows were substandard ( consisting only of a spherical shadow texture projected on the terrain ), the ability to use skinned meshes for animated models was not working ( non-skinned meshes worked ), and the 3D rigid-body physics suffered from several issues.

Specifically, the neighborhood at time is used to choose a function from some larger set of possible functions, that maps the state of the cell at time to its state at time.

Never let anyone not in the know take a turn at the valves -- even if the little boys do want to play space ship.

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

Let M be a smooth manifold and let x be a point in M. Let T < sub > x </ sub > M be the tangent space at x.

In a historical or geopolitical sense the term usually refers collectively to Christian majority countries or countries in which Christianity dominates or was a territorial phenomenon .“ Christendom is originally a medieval concept steadily to have evolved since the fall of the Western Roman Empire and the gradual rise of the Papacy more in religio-temporal implication practically during and after the reign of Charlemagne ; and the concept let itself to be lulled in the minds of the staunch believers to the archetype of a holy religious space inhabited by Christians, blessed by God, the Heavenly Father, ruled by Christ through the Church and protected by the Spirit-body of Christ ; no wonder, this concept, as included the whole of Europe and then the expanding Christian territories on earth, strengthened the roots of Romance of the greatness of Christianity in the world .”

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

Let X be a topological space, and let x < sub > 0 </ sub > be a point of X.

If anyone shall dare act contrary to this and, recognizing his crime, does not within the space of thirty days make proper amends, let him be cut off from the Church and anathematized.

Given a topological space X, let G < sub > 0 </ sub > be the set X.

Let X be a measurable space, let μ be a measure on X, and let N be a measurable set in X.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by

But let every one of you keep the Sabbath after a spiritual manner, rejoicing in meditation on the law, not in relaxation of the body, admiring the workmanship of God, and not eating things prepared the day before, nor using lukewarm drinks, and walking within a prescribed space, nor finding delight in dancing and plaudits which have no sense in them.

According to interviews with Norm Prescott, Paramount offered Roddenberry a substantial sum of money to abandon creative control of the project and let Filmation proceed with their " kiddy space cadet " idea.

As above, we let ( V, g ) be an n-dimensional complex vector space equipped with a nondegenerate bilinear form.

If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.

For any topological space X, let C ( X ) denote the family of real-valued continuous functions on X and let C *( X ) be the subset of bounded real-valued continuous functions.

Let us therefore assume that this space without doubt belongs to the still undiscovered satellites of Mars, let us also add that perhaps Jupiter still has around itself some smaller ones which have not been sighted yet by any telescope.