In this note we correct a gap in the proof of the complexity estimates appearing in our papers [1],[2]. The complexity estimates are correct, but the proof was incomplete at a certain point. The same gap appears in our paper [3], but there the estimate for the space complexity has to be multiplied with a power of log(g), where g is the genus of the curve. First we fill in the gap in [2]. In thi...

Embedding is the widely-observed phenomenon that a good is assigned a higher value if evaluated on its own rather than as part of a more inclusive set. Embedding is considered a serious problem affecting the quality of many environmental management and health risk policy judgments. This paper presents the results of an experiment involving of a structured, small-group approach for conducting en...

The aim of this paper is to report on recent work on liftings of groups of au-tomorphisms of a formal power series ring over a eld k of characteristic p to characteristic 0, where they are realised as groups of automorphisms of a formal power series ring over a suitable valuation ring R dominating the Witt vectors W(k): We show that the lifting requirement for a group of automorphisms places se...

Rigged Hilbert space of the free coherent states is investigated. We prove that this rigged Hilbert space is isomorphous to the space of generalized functions over p–adic disk. We discuss the relation of the described isomorphism of rigged Hilbert spaces and noncommutative geometry and show, that the considered example realises the isomorphism of the noncommutative line and p–adic disk. In the ...

This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we study notions of ~-adic nonlocal vertex algebra and ~-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan’s notion of quantum vertex operator algebra. For any topologically free C[[~]]-module W , we study ~-adic...

Let p be a prime, Cp the completion of an algebraic closure of the p-adicnumbers Qp and K a finite extension of Qp contained in Cp. Let v be the valuation on Cp such that v(p) = 1 and let | | be the absolute value on Cp such that |x| = p for x ∈ Cp. Suppose N is a positive integer prime to p. Let X1(Np) denote the modular curve over K which represents elliptic curves with Γ1(Np)-structure and l...

This paper is the first one in the series devoted to the calculation of particle mass spectrum in Topological GeometroDynamics. In this paper p-adic conformal field theory limit of TGD is formulated. TGD Universe is critical at quantum level and the idea is to realize criticality via conformal invariance. Ordinary real numbers do not allow this but if one assumes that in long length scales p-ad...

In the present paper, which is the third in a series of four papers, we study the theory surrounding the log-theta-lattice, a highly noncommutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data, that includes an elliptic curve EF over a num...

This paper confirms a conjecture of Amdeberhan and Moll that the power of 2 dividing the number of plane partitions in an n-cube is greater than the power of 2 dividing the number of totally symmetric plane partitions in the same cube when n is even, and less when n is odd.

In this article, we present an effective encoding of dendrograms by embedding them into the Bruhat-Tits trees associated to p-adic number fields. As an application , we show how strings over a finite alphabet can be encoded in cyclotomic extensions of Qp and discuss p-adic DNA encoding. The application leads to fast p-adic agglomerative hierarchic algorithms similar to the ones recently used e....