By I. M. Yaglom
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The item of this ebook is the quantum mechanism that enables the macroscopic quantum coherence of a superconducting condensate to withstand to the assaults of hot temperature. way to this basic challenge of contemporary physics is required for the layout of room temperature superconductors, for controlling the decoherence results within the quantum desktops and for the certainty of a potential function of quantum coherence in residing subject that's debated this day in quantum biophysics.
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Extra resources for Felix Klein and Sophus Lie
The cotangent bundle T ∗ M = x∈M Tx∗ M is also a 2n-dimensional manifold. A one-form ω : M → T ∗ M is a smooth map assigning to each x ∈ M a covector ω(x) ∈ T ∗ M . In local coordinates ω(x) = ωi (x)dxi |x or simply ω = ωi dxi . The diﬀerential of a map. If τ : M → N is a smooth map between two manifolds then its diﬀerential dτ : T M → T N is a linear map deﬁned pointwise for ﬁxed x as follows: let w ∈ Tx M be an arbitrary vector and h : N → R an arbitrary smooth function. Then a new vector (dτ w)|τ (x) ∈ Tτ (x)N is deﬁned by (dτ w)|τ (x) h := w(h ◦ τ )(x).
On Tx M the tensor g induces a scalar product v·w = g|x (v, w), and likewise for vector-ﬁelds. By forming the inverse matrix g ij (x) one deﬁnes a (2, 0)-tensor ﬁeld g ij ∂xi ⊗ ∂xj . Raising and lowering indices. Type conversion. g. if v = aij dxi ⊗ dxj is a (0, 2)-tensor-ﬁeld then cij = g il alj deﬁnes the coeﬃcients of a (1, 1)-tensor C = cij ∂xi ⊗ dxj . This operation is called raising an index. Similarly, indices can be lowered by multiplication with gij . Covariant diﬀerentiation, Christoﬀel symbols.
The graph of such a function u is a subset of M × Rk . The total space M × Rk is an (n + k)-dimensional smooth manifold. Each tangent space has the simple structure T(x,u) (M × Rk ) = Tx M × Rk . Vector ﬁelds w on M × Rk are written in local coordinates as w = ξ i (x, u)∂xi + φα (x, u)∂uα . We use the notation w = ξ(x, u) + φ(x, u) with ξ(x, u) ∈ Tx M and φ(x, u) = (φ1 (x, u), . . , φk (x, u)) ∈ Rk . Partial derivatives Consider a smooth function f : M × Rk → R. Partial derivatives of f are deﬁned as follows: for ﬁxed x partial derivatives of the function f (x, ·) : Rk → R with respect to uα , α = 1, .