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The works of Yamabe, Trudinger, Aubin, and Schoen together comprise the solution of the Yamabe problem, which asserts that there is a metric of constant scalar curvature in every conformal class. In 1989, Schoen was also able to adapt Karen Uhlenbeck 's bubbling analysis, developed for other geometric-analytic problems, to the setting of ...
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications −a and 2a of a vector a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra [1] [2] [3] (or more generally, a module in abstract algebra [4] [5]).
In 1950, when Academic Press published G. Kuerti’s translation of the second edition of volume 2 of Lectures on Theoretical Physics by Sommerfeld, vector notation was the subject of a footnote: "In the original German text, vectors and their components are printed in the same Gothic types. The more usual way of making a typographical ...
Mesons named with the letter "f" are scalar mesons (as opposed to a pseudo-scalar meson), and mesons named with the letter "a" are axial-vector mesons (as opposed to an ordinary vector meson) a.k.a. an isoscalar vector meson, while the letters "b" and "h" refer to axial-vector mesons with positive parity, negative C-parity, and quantum numbers I G of 1 + and 0 − respectively.
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
The conservative vector fields correspond to the exact-forms, that is, to the forms which are the exterior derivative of a function (scalar field) on . The irrotational vector fields correspond to the closed 1 {\displaystyle 1} -forms , that is, to the 1 {\displaystyle 1} -forms ω {\displaystyle \omega } such that d ω = 0 {\displaystyle d ...
The Outer space, denoted X n or CV n, comes equipped with a natural action of the group of outer automorphisms Out(F n) of F n. The Outer space was introduced in a 1986 paper [ 1 ] of Marc Culler and Karen Vogtmann , and it serves as a free group analog of the Teichmüller space of a hyperbolic surface.