Denote the free group on 2 letters by F2 and the SL(2,C)-representation variety of F2 by R = Hom(F2, SL(2,C)). The group SL(2,C) acts on R by conjugation. We construct an isomorphism between the coordinate ring C[SL(2,C)] and the ring of matrix coefficients, providing an additive basis of C[R]SL(2,C) in terms of spin networks. Using a graphical calculus, we determine the symmetries and multipli...

Let K be a p-adic local field and E an elliptic curve defined over K. The component group of E is the group E(K)/E0(K), where E0(K) denotes the subgroup of points of good reduction; this is known to be finite, cyclic if E has multiplicative reduction, and of order at most 4 if E has additive reduction. We show how to compute explicitly an isomorphism E(K)/E0(K) ∼= Z/NZ or E(K)/E0(K) ∼= Z/2Z× Z/2Z.

A linear process is a system of events and states related by an inner product, on which are defined the behaviorally motivated operations of tensor product or orthocurrence, sum or concurrence, sequence, and choice. Linear process algebra or LPA is the theory of this framework. LPA resembles Girard’s linear logic with the differences attributable to its focus on behavior instead of proof. As wi...

Interaction nets are graph rewriting systems which are a generalisation of proof nets for classical linear logic. The linear chemical abstract machine (CHAM) is a term rewriting system which corresponds to classical linear logic, via the Curry-Howard isomorphism. We can obtain a textual calculus for interaction nets which is surprisingly similar to linear CHAM based on the multiplicative fragme...

This work deals with semantics of programming languages (or equivalently, thanks to Curry-Howard isomorphism, with semantics of proofs). We introduce a framework in which both static semantics (Ehrhard’s hypercoherences) and dynamic semantics (Hyland-Ong’s games), can be presented. The work is carried in a multiplicative subsystem of Laurent’s polarized linear logic with weakening. Like Böhm tr...

Beta is found to be a function of several leading economic indicators and government policy variables within the context of the Variable Beta Model which incorporates economic characteristics in the single index model in a multiplicative manner. When contemporaneous macroeconomic descriptors are replaced with reporting-period-lagged macroeconomic descriptors, in the Lagged Variable Beta Model, ...

We prove that for no nontrivial ordered abelian group G, the ordered power series field R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a nonsurjective logarithm. For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through t...

Nonnegative matrix factorization (NMF) is the problem of approximating an input nonnegative matrix, $V$, as product two smaller matrices, $W$ and $H$. In this paper, we introduce a general framework to design multiplicative updates (MU) for NMF based on $\beta$-divergences ($\beta$-NMF) with disjoint equality constraints, penalty terms in objective function. By disjoint, mean that each variable...

We considers how a particular kind of graph corresponds to multiplicative intuitionistic linear logic formula. The main feature of the graphical notation is that it absorbs certain symmetries between conjunction and implication. We look at the basic definitions and present details of an implementation in the functional programming language Standard ML. This provides a functional approach to gra...