A detailed algebraic-geometric background is presented for the tropical approach to enumeration of singular curves on toric surfaces, which consists of reducing the enumeration of algebraic curves to that of non-Archimedean amoebas, the images of algebraic curves by a real-valued non-Archimedean valuation. This idea was proposed by Kontsevich and recently realized by Mikhalkin, who enumerated t...

In the paper we study a method extending fuzzy measures on the set N = {1, . . . , n} to n-ary aggregation functions on the interval [0, 1]. The method is based on a fixed suitable n-ary aggregation function and the Möbius transform of the considered fuzzy measure. This approach generalizes the wellknown Lovász and Owen extensions of fuzzy measures. We focus our attention on the special class o...

In this paper we are concerned with the spectral analysis for classes of finte rank perturbations diagonal operators in form A = D+F where D is a operator and F u’1 ⊗v’1+...+u’m ⊗vm an finite non-archimedean Banach space countable type. We compute spectrum using theory Fredholm non archimedean setting concept essential linear operators.

One dimensional optimization on non-Archimedean fields is presented. We derive first and second order necessary and sufficient optimality conditions. For first order optimization, these conditions are similar to the corresponding real ones; but this is not the case for higher order optimization. This is due to the total disconnectedness of the given non-Archimedean field in the order topology, ...

Spiral-wave breakup in strongly chaotic media with nonphase-coherent chaotic attractors is investigated. Spiral-wave dynamics is studied for the Rössler reaction diffusion equation as the local attractor changes from a phase-coherent to a funnel form. Stable spiral waves with an Archimedean structure are observed to persist even when the local chaotic attractor has a funnel form. The destructio...

The discrete Lotka–Volterra equation over p-adic space was constructed since p-adic space is a prototype of spaces with non-Archimedean valuations and the space given by taking the ultra-discrete limit studied in soliton theory should be regarded as a space with the non-Archimedean valuations given in my previous paper (Matsutani S 2001 Int. J. Math. Math. Sci.). In this paper, using the natura...

‖b+ c‖ ≤ max (‖b‖, ‖c‖) then the value ‖b‖ is called non-archimedean. The thus delimited nonarchimedean values are of considerable arithmetic interest. They are useful in questions of divisibility and irreducibility and in fact often correspond exactly to the prime ideals of the given ring. This paper is devoted to the explicit construction of non-archimedean values. More specifically, given al...

Recently, the distributivity of fuzzy implications over t-norms, t-conorms and uninorms was studied in many articles. In this paper we characterize functions which satisfy one of the four generalized distributivity equations of classical implication in the case when t-conorms are continuous and Archimedean. Using the obtained characterizations we describe some solutions which are fuzzy implicat...

A set is called a unique range set for a certain class of functions if each inverse image of that set uniquely determines a function from the given class. We show that a finite set is a unique range set, counting multiplicity, for non-Archimedean entire functions if and only if there is no non-trivial affine transformation preserving the set. Our proof uses a theorem of Berkovich to extend, to ...

The paper gives complete proofs of the properties of the RankinSelberg integrals for the group GL(n,R) and GL(n,C).