Algebra

Viscous Time Theory & Birch—Swinnerton-Dyer

Authors: Raoul Bianchetti, Payam Danesh
Comments: 10 Pages.

We develop a precise arithmetic version of Viscous Time Theory by replacing the original Sobolev field space with a finite Hilbert model of the Selmer complex. The VTT passive Hessian creates an anchored space for arithmetic. When we use a Schur complement it gets rid of the memory coupling. This results in an operator. The Bloch-Kato Selmer group is a sector, for this operator. The VTT passive Hessian and the Bloch-Kato Selmer group are connected in this way. The free part of the Bloch-Kato Selmer group gives us the Mordell-Weil rank. The finite arithmetic memory is what captures the Tate-Shafarevich and the Tamagawa contributions. The height-energy layer is also important. It connects with the Néron-Tate regulator. This connection happens through Arakelov and nonarchimedean potential theory. The VTT passive Hessian and the Néron-Tate regulator are related in this way. The main result is a rigorous VTT—BSD reduction theorem: after the active determinant germ is compared with L(E,s) at s=1, the BSD rank formula and refined leading coefficient follow. In this paper, we establish the variational, cohomological, height-theoretic and determinant-line structure needed for that comparison and it identifies the remaining analytic theorem.
Category: Algebra

Quaternions, Octonions, and Now, 16-Ons and 2^n-Ons: New Kinds of Numbers

Authors: Warren D. Smith
Comments: 68 Pages. Placing this 2004 paper on vixra for archival purposes.

"Cayley-Dickson doubling," starting from the real numbers, successively yields the complex numbers (dimension 2), quaternions (4), and octonions (8). Each contains all the previous ones as subalgebras. Famous Theorems, previously thought to be the last word, state that these are the full set of division (or normed) algebras with 1 over the real numbers. Their properties keep degrading: the complex numbers lose the ordering and self-conjugacy (x̅=x) properties of the reals; at the quaternions we lose commutativity; and at the octonions we lose associativity. If one keeps Cayley-Dickson doubling to get the 16-dimensional "sedenions," zero-divisors appear.

We introduce a different doubling process which also produces the complexes, quaternions, and octonions, but keeps going to yield 2ⁿ-dimensional normed algebraic structures, for every n≥0. Each contains all the previous ones as subalgebras. We'll see how these evade the Famous Impossibility Theorems. They also lead to a rational "vector product" operation in 2^k-1 dimensions for each k≥2; this operation is impossible in other dimensions.

But properties continue to degrade. The 16-ons lose distributivity, right-cancellation yx·x^-1=y, flexibility a·ba=ab·a, and antiautomorphism (c̅=b̅a̅ where c=ab). The 32-ons lose the properties that the solutions of generic division problems necessarily exist and are unique, and lose the "Trotter product limit formula." We introduce an important new notion to topology we call "generalized smoothness." The 2ⁿ-ons are generalized smooth for n≤4.

All 2ⁿ-ons have 1 and obey numerous identities including weakenings of the distributive, associative, and antiautomorphism laws. In the case of 16-ons these weakened distributivity laws characterize them, i.e. our 16-ons are, in a sense, unique and best-possible. Our 2ⁿ-ons also are unique, albeit in a much weaker sense. The 2ⁿ-ons with n≤4 support a version of the fundamental theorem of algebra. Normed algebras (rational but not nec. distributive) over the reals are impossible in dimensions other than powers of 2.
Category: Algebra