In this paper I propose the non-Archimedean multiple-validity. Further, I build an infinite-order predicate logical language in that predicates of various order are considered as fuzzy relations. Such a language can have non-Archimedean valued semantics. For instance, infinite-order predicates can have an interpretation in the set [0, 1] of hyperreal (hyperrational) numbers. Notice that there e...

A Dedekind cut of an ordered abelian group G is a pair (X, Y) of nonempty subsets of G where Y=G−X and every member of X precedes every member of Y. A Dedekind cut (X, Y) is said to be continuous if X has a greatest member or Y has a least member, but not both; if every Dedekind cut of G is a continuous cut, G is said to be (Dedekind) continuous. The ordered abelian group R of real numbers is, ...

Let $L$ be a line bundle on proper, geometrically reduced scheme $X$ over non-trivially valued non-Archimedean field $K$. Roughly speaking, the volume of continuous metric Berkovich analytification measures asymptotic growth space small sections tensor powers $L$. For semipositive in sense Zhang, we show first that agrees with energy. The existence such yields is nef. A second result differenti...

We study perfect matchings, or close-packed dimer coverings, of finite sections the eleven Archimedean lattices and give a constructive proof showing that any two matchings can be transformed into each other using small sets local ring-exchange moves. This result has direct consequences for formulating quantum models with resonating valence bond ground state, i.e., superposition all coverings c...

We study model theoretic properties of valued fields (equipped with a real-valued multiplicative valuation), viewed as metric structures in continuous first order logic. For technical reasons we prefer to consider not the valued field (K, |·|) directly, but rather the associated projective spaces KP, as bounded metric structures. We show that the class of (projective spaces over) metric valued ...

For a point T and a circle δ, if two congruent circles of radius r touching at T also touch δ at points different from T , we say T generates circles of radius r with δ, and the two circles are said to be generated by T with δ. If the generated circles are Archimedean, we say T generates Archimedean circles with δ. Frank Power seems to be the earliest discoverer of this kind Archimedean circles...

A non-Archimedean antiderivational line analog of the Cauchy-type line integration is defined and investigated over local fields. Classes of non-Archimedean holomorphic functions are defined and studied. Residues of functions are studied; Laurent series representations are described. Moreover, non-Archimedean antiderivational analogs of integral representations of functions and differential for...

We explore the distinction between convergence and absolute convergence of series in both Archimedean and non-Archimedean ordered fields and find that the relationship between them is closely connected to sequential (Cauchy) completeness.

The Archimedean components of triangular norms (which turn the closed unit interval into an abelian, totally ordered semigroup with neutral element 1) are studied, in particular their extension to triangular norms, and some construction methods for Archimedean components are given. The triangular norms which are uniquely determined by their Archimedean components are characterized. Using ordina...

Let V be an algebraic subvariety of a torus Gm ↪→ P and denote by V ∗ the complement in V of the Zariski closure of the set of torsion points of V . By a theorem of Zhang, V ∗ is discrete for the metric induced by the normalized height ĥ. We describe some quantitative versions of this result, close to the conjectural bounds, and we discuss some applications to study of the class group of some n...