hermitian form sentence in Hindi
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- Krein was at the time still a young mathematician, only two years older than Naimark, but had already built a research group in functional analysis, and they worked together on some Naimark's first works on symmetric and Hermitian forms.
- With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an almost Hermitian manifold and is the reason why every almost complex manifold ( in particular every symplectic manifold ) has a Spin c structure.
- Lefschetz proved that the line bundle " L ", associated to the Hermitian form " H " is ample if and only if " H " is positive definite, and in this case " L " 3 is very ample.
- However, Hermitian forms have basis-independent signature in both the complex and the quaternionic case . ( The real case reduces to the symmetric case . ) A skew-Hermitian form on a complex vector space is rendered Hermitian by multiplication by, so in this case, only is interesting.
- However, Hermitian forms have basis-independent signature in both the complex and the quaternionic case . ( The real case reduces to the symmetric case . ) A skew-Hermitian form on a complex vector space is rendered Hermitian by multiplication by, so in this case, only is interesting.
- Freedman's classification can be extended to some cases when the fundamental group is not too complicated; for example, when it is "'Z "'there is a classification similar to the one above using Hermitian forms over the group ring of "'Z " '.
- At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts : the real part is symmetric ( orthogonal ), and the imaginary part is skew-symmetric ( symplectic ) and these are related by the complex structure ( which is the compatibility ).
- The CFT is called unitary if the space of states has a positive definite Hermitian form such that L _ 0 and \ bar L _ 0 are self-adjoint, L _ 0 ^ \ dagger = L _ 0 and \ bar L _ 0 ^ \ dagger = \ bar L _ 0.
- The complex projective orthogonal group, PO ( " n ", "'C "') should not be confused with the projective unitary group, PU ( " n " ) : PO preserves a symmetric form, while PU preserves a hermitian form PU is the symmetries of complex projective space ( preserving the Fubini Study metric ).
- It is an axiom of the theory that possible measurement that we can possibly do on the system and get a real number " must " be represented by some Hermitian form Q in the following sense : Measurements in quantum physics are always probabilistic, but the expected value of the measured result when we start from state \ psi is \ tfrac { Q ( \ psi, \ psi ) } { \ langle \ psi | \ psi \ rangle }.
- Similar arguments about the Hilbert inner product ( which can be demonstrated to be a Hermitian form, therefore justifying the name " inner product " ) lead to the conclusion that its neutral space is precisely K _ { 00 } = ( K _ 0 \ cap K _ + ) \ oplus ( K _ 0 \ cap K _-), that elements of this neutral space have zero Hilbert inner product with any element of K, and that the Hilbert inner product is positive semi-definite.
- There is a division algebra " D " with center " l " and degree over " l " 3 or 1, with an involution of the second kind which restricts to the nontrivial automorphism of " l " over " k ", and a nontrivial Hermitian form on a module over " D " of dimension 1 or 3 such that " G " is the special unitary group of this Hermitian form . ( As a consequence of and the work of Cartwright and Steger, " D " has degree 3 over " l " and the module has dimension 1 over " D " . ) There is one real place of " k " such that the points of " G " form a copy of PU ( 2, 1 ), and over all other real places of " k " they form the compact group PU ( 3 ).
- There is a division algebra " D " with center " l " and degree over " l " 3 or 1, with an involution of the second kind which restricts to the nontrivial automorphism of " l " over " k ", and a nontrivial Hermitian form on a module over " D " of dimension 1 or 3 such that " G " is the special unitary group of this Hermitian form . ( As a consequence of and the work of Cartwright and Steger, " D " has degree 3 over " l " and the module has dimension 1 over " D " . ) There is one real place of " k " such that the points of " G " form a copy of PU ( 2, 1 ), and over all other real places of " k " they form the compact group PU ( 3 ).
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