Here-a-smooth-atlas-for-a-topological-manifold-M

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Here and smooth

Here, in a supposedly smooth step from one room to another, the Tramp loses his hat in one room, but it is instantly back on his head as he enters the next room.

Here π: P → X is required to be a smooth map between smooth manifolds, G is required to be a Lie group, and the corresponding action on P should be smooth.

Here, are arbitrary smooth functions of.

Here, can be any smooth functions ; they control the waveform of the two possible polarization modes of gravitational radiation.

Here f and g are real-valued smooth functions.

Here the canyon narrows to a flat stony track, in places as little as wide, with sheer and smooth rock walls up to high on each side.

Here X ∈ T x M, therefore X is a derivation defined on M and f is a smooth real-valued function on N. By definition, the pushforward of X at a given x in M is in T φ ( x ) N and therefore itself is a derivation.

Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is the flow domain.

Here and for

Here Wright gave a slight sigh of weariness, and continued, `` It means more long years lived across the social grain of the life of our people, making shift to live in the face of popular disrespect and misunderstanding as I best can for myself and those dependent upon me ''.

Here in the cool darkness Mr. Podger could still feel the warmth of midday, could still see the yellow butterflies dancing over the road, could still see the friendly grin on the young, sun-browned face as the driver looked back over his shoulder for a moment before the car streaked out of sight.

Here, for the most part, they were well treated, as a `` reminder of the Old Testament heritage of Christianity '' ; ;

Here briefly in this humble tribute I have sought for some simple and succinct summation that would define the immense service of this patriot to his country.

But this we know: Here is a great life that in every area of American politics gives the American people occasion for pride and that has invested the democratic process with the most decent qualities of honor, decency, and self-respect.

Here the reasonable mastery of the elements of administration can do much to free a president for his primary role.

Here is a word of advice when you go shopping for your pansy seeds.

Here are suggestions for the frankfurter trimmings: 1.

Here, this happy, roving son of good fortune proved that he could accept the disciplines of a new social-economic order fighting for its very existence and ideals in a truculent world.

Here, Prokofieff became a workman in the vineyards of Socialism -- producing music for the masses.

Here we shall restrict discussion to those methods that appear sufficiently sensitive and precise for determining the concentration of TSH in blood.

Here, for the case of squares inscribed in plane curves, we remove the restriction to convexity and give certain other results.

Here would be a powerful force for raising business activity.

Here in 1815 the great nations assembled to legislate not merely for Europe, but for the world.

Here, perhaps, Fromm is vulnerable, for he does not always use the best and most recent evidence available, and he sometimes selects and interprets the evidence in rather special ways.

Here the absent sitter makes a `` date '' with a communicator ( someone close to him who is deceased ), asking him to `` come in '' at a certain hour, when a channel will be open for him.

Here again laboratory approaches are being evolved, for it is recognized how `` elastic '' these readings can be, how they can apply to many people, and are often stated in general terms all too easily applied to any individual's own case.

Here are hatched plans for getting a share of the American bounty, the secret of the anti-missile missile, or an invitation to dinner.

Here I was accompanied by Mrs. Okamoto ( Fumio's mother ), her son, Mr. Washizu ( a prospective student with whom I have been corresponding for more than a year ), and Mr. Nishima, one of the science teachers.

Here was a man with an enormous gift for living as well as thinking.

Here they would lie in suspended animation for many years.

Here is the BNF for Algol60 and the ICL2900 compiler source, library documentation, and a considerable test suite including Brian Wichmann's tests.

Here he met his first love, Mary Wood, for whom he made a quilt.

Here for nearly four years he labored with the greatest perseverance and industry.

Here and topological

Here, G × G is viewed as a topological space by using the product topology.

Here for an n-cell σ in T and for g in G the cell g σ is exactly the translate of σ by a covering transformation of T corresponding to g. Moreover, C n ( T ) is a free ZG-module with free ZG-basis given by representatives of G-orbits of n-cells in T. In this case the standard topological chain complex

Here the quotient is in the category of topological spaces, rather than groups or rings.

Here we take a clockwise ( left-handed ) twist of the elastic ribbon to be a kink with topological charge.

Here by an arc from x to y we mean the image in M of a topological embedding f from an interval to M such that f ( a )= x and f ( b )= y.

Here, then, is understood to be the singular chain functor, which maps topological spaces to the category of chain complexes Comp ( or Kom ).

Here A ′ and B ′ mean the topological duals of

Here and manifold

Here d H denotes Hausdorff distance between subsets in M and the isometric embedding is understood in the global sense, i. e. it must preserve all distances, not only infinitesimally small ones ; for example no compact Riemannian manifold of negative sectional curvature admits such an embedding into Euclidean space.

Here is a linear transformation of the tangent space of the manifold ; it is linear in each argument.

Here, is a vector in, the tangent space to the manifold Q at point q.

Here the problem of evaluation of the modes of a n component mixture in a D dimensional space is reduced to identification of critical points ( local minima, maxima and saddle points ) on a manifold referred to as the ridgeline surface

Here describe the indices of coordinates of the submanifold while the functions encode the embedding into the higher-dimensional manifold whose tangent indices are denoted.

Here, is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold.

Here, the ∗ is the Hodge dual, thus the last form, ∗( 1 ), emphasizes that the volume form is the Hodge dual of the constant map on the manifold.

Here there are six spacetime dimensions, which constitute a symplectic manifold, and it turns out that the worldsheets are necessarily parametrized by pseudoholomorphic curves, whose moduli spaces are only finite-dimensional.

where C n is a universal constant only depending on the dimension of M. Here the homotopy systole sysπ 1 is by definition the least length of a noncontractible loop in M. A manifold is called essential if its fundamental class represents a nontrivial class in the homology of its fundamental group.

Here the stable 2-systole of a Riemannian manifold M is defined by setting

Here the boundary corresponds to the intersection of with the-sphere ( of sufficiently small radius ) and therefore it is a real manifold of dimension.

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