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Kato's conjecture. Kato's conjecture is a mathematical problem named after mathematician Tosio Kato, of the University of California, Berkeley. Kato initially posed the problem in 1953. [1] Kato asked whether the square roots of certain elliptic operators, defined via functional calculus, are analytic. The full statement of the conjecture as ...
The norm residue isomorphism theorem (or Bloch–Kato conjecture) states that for a field k and an integer ℓ that is invertible in k, the norm residue map. from Milnor K-theory mod-ℓ to étale cohomology is an isomorphism. The case ℓ = 2 is the Milnor conjecture, and the case n = 2 is the Merkurjev–Suslin theorem. [6][7]
The Kato theorem, or Kato's cusp condition (after Japanese mathematician Tosio Kato), is used in computational quantum physics. [1][2] It states that for generalized Coulomb potentials, the electron density has a cusp at the position of the nuclei, where it satisfies. Here denotes the positions of the nuclei, their atomic number and is the Bohr ...
The conjectures for (b) are called the Bloch–Kato conjectures for special values (for Spencer Bloch and Kazuya Kato; this circle of ideas is distinct from the Bloch–Kato conjecture of K-theory, extending the Milnor conjecture, a proof of which was announced in 2009).
In this sense, the apparently ad hoc definition of Milnor K-theory becomes a theorem: certain motivic cohomology groups of a field can be explicitly computed by generators and relations. A much deeper result, the Bloch-Kato conjecture (also called the norm residue isomorphism theorem), relates Milnor K-theory to Galois cohomology or étale ...
The Bloch-Kato conjecture on values of L-functions predicts that the order of vanishing of an L-function of X at an integer point is equal to the rank of a suitable motivic cohomology group. This is one of the central problems of number theory, incorporating earlier conjectures by Deligne and Beilinson.
In mathematics, the Lie product formula, named for Sophus Lie (1875), but also widely called the Trotter product formula, [1] named after Hale Trotter, states that for arbitrary m × m real or complex matrices A and B, [2] where eA denotes the matrix exponential of A. The Lie–Trotter product formula[3] and the Trotter–Kato theorem[4] extend ...
The full statement of the norm residue isomorphism theorem (also known as the Bloch-Kato conjecture) was proven by Voevodsky. In the late 1990s Merkurjev gave the most general approach to the notion of essential dimension , introduced by Buhler and Reichstein , and made fundamental contributions to that field.