The notion of multiplicative cascade was introduced into the statistical theory of turbulence by A.N. Kolmogorov as a phenomenological framework intended to accommodate the intermittency and large fluctuations observed in turbulent fluid flows. The basic idea is that energy is redistributed from larger to smaller scales via a splitting mechanism involving random multiplicative factors known as ...

We note how several central results in multiplicative number theory may be rephrased naturally in terms of multiplicative functions f that pretend to be another multiplicative function g. We formalize a ‘distance’ which gives a measure of such pretentiousness, and as one consequence obtain a curious inequality for the zeta-function. A common theme in several problems in multiplicative number th...

In the previous paper[1], we derived the Abelian projected effective gauge theory as a low energy effective theory of the SU(N) Yang-Mills theory by adopting the maximal Abelian gauge. At that time, we have demonstrated the multiplicative renormalizability of the propagators for the diagonal gluon and the dual Abelian antisymmetric tensor field. In this paper, we show the multiplicative renorma...

‎We extend the notion of approximately multiplicative to approximately n-multiplicative maps between locally multiplicatively convex algebras and study some properties of these maps‎. ‎‎W‎e prove that every approximately n-multiplicative linear functional on a functionally continuous locally multiplicatively convex algebra is continuous‎. ‎We also study the relationship between approximately mu...

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In this paper, we introduce (almost) skew 2-nomial algebras, establish the existence of a skew multiplicative K-basis for a skew 2-nomial algebra, and explore the existence of a Gröbner basis theory for such algebras. As a by-product, a family of Noetherian algebras is recognized by the finite Gröbner basis theory. Let K be a field, and let R be a finitely generated free K-algebra, or a path al...

We study the asymptotic behaviour of points under matrix cocyles generated by rectangular matrices. In particular we prove a random Perron-Frobenius and a Multiplicative Ergodic Theorem. We also provide an example where such products of random rectangular matrices arise in the theory of random walks in random environments and where the Multiplicative Ergodic Theorem can be used to investigate r...