non-Archimedean Model Research Articles

On the Notion of Emotional Truth

In this paper, we propose an explication of the notion of emotional (pragmatic) truth due to defining performative propositions on non-Archimedean models [Formula: see text]. In these models, it is possible to fix an appropriate behavioral scenario for each atomic performative proposition. Meanwhile, each performative proposition [Formula: see text] has an appropriate form of cognitive bias [Formula: see text] changing the meaning of [Formula: see text], but this bias is the result or foundation for interaction with the outside world. Therefore, instead of the usual domain [Formula: see text] of realization of atomic informative propositions [Formula: see text], which can be finite, we always have an infinite domain [Formula: see text] of realization of performative propositions [Formula: see text], where [Formula: see text] is an informative proposition, distorted by some kind of cognitive biases [Formula: see text]. This semantics allows us to shift the focus from artificial intelligence (AI) to emotional artificial intelligence (EAI).

Non-Archimedean quantum mechanics via quantum groups

We present a new non-Archimedean realization of the Fock representation of the q-oscillator algebras where the creation and annihilation operators act on complex-valued functions, which are defined on a non-Archimedean local field of arbitrary characteristic, for instance, the field of p-adic numbers. This new realization implies that many quantum models constructed using q-oscillator algebras are non-Archimedean models, in particular, p-adic quantum models. In this framework, we select a q-deformation of the Heisenberg uncertainty relation and construct the corresponding q-deformed Schrödinger equations. In this way we construct a p-adic quantum mechanics which is a p-deformed quantum mechanics. We also solve the time-independent Schrödinger equations for the free particle, and a particle in a non-Archimedean box. In the last case, we show the existence of a discrete sequence of energy levels. We determine the eigenvalues of Schrödinger operator for a general radial potential. By choosing the potential in a suitable form we recover the energy levels of the q-hydrogen atom.

Logarithms, constructible functions and integration on non-archimedean models of the theory of the real field with restricted analytic functions with value group of finite archimedean rank

Given a model of the theory of the real field with restricted analytic functions such that its value group has finite archimedean rank we show how one can extend the restricted logarithm to a global logarithm with values in the polynomial ring over the model with dimension the archimedean rank. The logarithms are determined by algebraic data from the model, namely by a section of the model and by an embedding of the value group into its Hahn group. If the archimedean rank of the value group coincides with the rational rank the logarithms are equivalent. We illustrate how one can embed such a logarithm into a model of the real field with restricted analytic functions and exponentiation. This allows us to define constructible functions with good lifting properties. As an application we establish a Lebesgue measure and integration theory with values in the polynomial ring, extending and strengthening the construction in [T. Kaiser: Lebesgue measure and integration theory on non-archimedean real closed fields with archimedean value group. Proc. Lond. Math. Soc. 116 (2018), no. 2, 209-247.].

Non-Archimedean replicator dynamics and Eigen’s paradox

We present a new non-Archimedean model of evolutionary dynamics, in which the genomes are represented by p-adic numbers. In this model the genomes have a variable length, not necessarily bounded, in contrast with the classical models where the length is fixed. The time evolution of the concentration of a given genome is controlled by a p-adic evolution equation. This equation depends on a fitness function f and on mutation measure Q. By choosing a mutation measure of Gibbs type, and by using a p-adic version of the Maynard Smith ansatz, we show the existence of threshold function Mc(f,Q), such that the long term survival of a genome requires that its length grows faster than Mc(f,Q). This implies that Eigen’s paradox does not occur if the complexity of genomes grows at the right pace. About twenty years ago, Scheuring and Poole, Jeffares and Penny proposed a hypothesis to explain Eigen’s paradox. Our mathematical model shows that this biological hypothesis is feasible, but it requires p-adic analysis instead of real analysis. More exactly, the Darwin–Eigen cycle proposed by Poole et al takes place if the length of the genomes exceeds Mc(f,Q).

Lipschitz stratifications in power-bounded $o$-minimal fields

We propose to grok Lipschitz stratifications from a non-archimedean point of view and thereby show that they exist for closed definable sets in any power-bounded o -minimal structure on a real closed field. Unlike the previous approaches in the literature, our method bypasses resolution of singularities andWeierstraß preparation altogether; it transfers the situation to a non-archimedean model, where the quantitative estimates appearing in Lipschitz stratifications are sharpened into valuation-theoretic inequalities. Applied to a uniform family of sets, this approach automatically yields a family of stratifications which satisfy the Lipschitz conditions in a uniform way.

Normalized Berkovich spaces and surface singularities

We define normalized versions of Berkovich spaces over a trivially valued field $k$, obtained as quotients by the action of $\mathbb R_{>0}$ defined by rescaling semivaluations. We associate such a normalized space to any special formal $k$-scheme and prove an analogue of Raynaud's theorem, characterizing categorically the spaces obtained in this way. This construction yields a locally ringed $G$-topological space, which we prove to be $G$-locally isomorphic to a Berkovich space over the field $k((t))$ with a $t$-adic valuation. These spaces can be interpreted as non-archimedean models for the links of the singularities of $k$-varieties, and allow to study the birational geometry of $k$-varieties using techniques of non-archimedean geometry available only when working over a field with non-trivial valuation. In particular, we prove that the structure of the normalized non-archimedean links of surface singularities over an algebraically closed field $k$ is analogous to the structure of non-archimedean analytic curves over $k((t))$, and deduce characterizations of the essential and of the log essential valuations, i.e. those valuations whose center on every resolution (respectively log resolution) of the given surface is a divisor.

Some applications of numerosities in measure theory

We present some applications of the notion of numerosity to measure theory, including the construction of a non-Archimedean model for the probability of infinite sequences of coin tosses.

Lexicographic expected utility with a subjective state space

This paper provides a model that allows for a criterion of admissibility based on a subjective state space. For this purpose, we build a non-Archimedean model of preference with subjective states, generalizing Blume et al. (Econometrica 59:61–79, 1991), who present a non-Archimedean model with exogenous states; and Dekel et al. (Econometrica 69:891–934, 2001), who present an Archimedean model with an endogenous state space. We interpret the representation as modeling an agent who has several “hypotheses” about her state space, and who views some as “infinitely less relevant” than others.

Some remarks on Schanuel's conjecture

Schanuel's Conjecture is the statement: if x 1,…,x n ∈ C are linearly independent over Q , then the transcendence degree of Q(x 1,…,x n, exp(x 1),…, exp(x n)) over Q is at least n. Here we prove that this is true if instead we take infinitesimal elements from any ultrapower of C , and in fact from any nonarchimedean model of the theory of the expansion of the field of real numbers by restricted analytic functions.

Non-archimedean model for the relaxation process of the photoinduced darkening in amorphous chalcogenide films

The relaxation of the darkening induced by nanosecond laser pulses in amorphous chalcogenide thin films (AsSe and As2Se3) was modelled on the basis of a non-archimedean (ultrametric) approach. While for AsSe only a single time scale of evolution was evidenced (tentatively related to the simple, molecular structure of the glass) in As2Se3 three time scales were evidenced and discussed in the frame of a complex hierarchical structure of the glass with layer fragments, clusters of layers (amorphites) and, possibly, bigger domains.

NONARCHIMEDEAN CONFORMAL FIELD THEORIES

We present a general formalism for conformal field theories defined on a non-Archimedean field. Such theories are defined by complex-valued correlation functions of fields of a [Formula: see text]-adic variable. Conformal invariance is imposed by requiring the correlation functions to be unchanged under fractional linear transformations, the latter forming the full analogue of the conformal group in two-dimensional, euclidean space-time. All fields in the theory can be taken to be "primary", under the "non-Archimedean conformal group". The conformal symmetry fixes completely the form of all correlation functions, once we are given the weight-spectrum of the theory and the OPE coefficients (which must be the structure constants of certain commutative, associative algebras). We explicitly construct non-Archimedean CFT's having the same weight spectrum as that of Archimedean models of central charge c < 1. The OPE coefficients of these "local" Archimedean and non-Archimedean models are related by adelic formulae.

Non-Archimedean models of mixing—I. Residence time distribution for systems with imperfect mixing

Residence time distributions in flow systems which exhibit randomness on many scales of time are studied here. A non-Archimedean framework of the mixing process was proposed. An expansion where the first term correspond to perfect mixing while the following terms are corrections for imperfect mixing is obtained. The approach is tested by fitting the model to the experimental data of other authors.

Non-Archimedean models of mixing—II. A nearly dispersional model for packed bed thermal storage systems

A non-Archimedean mathematical model for simulating the transient characteristics of heat transfer in a packed bed thermal storage system is presented. The use of Hermite polynomials is proposed for the analysis of systems which exhibit many scales of heat transfer. The obtained solution is an expansion in which the first term corresponds to the axial dispersion. A comparison of experimental data reported from literature with the numerical prediction shows a good correlation.