Exponential Fields: Lack of Generic Derivations

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP1-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.

Motivated by the decidability question for the theory of real exponentiation and by the Transfer Conjecture for o‐minimal exponential fields, we show that, under the assumption of Schanuel's Conjecture, the prime model of real exponentiation is embeddable into any o‐minimal exponential field, where the embedding is not necessarily elementary. This is a consequence of an unconditional model theoretic embeddability result that we obtain by applying Kőnig's Lemma.

In Ax (Ann. Math. 93(2):252–268, 1971), J. Ax proved a transcendency theorem for certain differential fields of characteristic zero : the differential counterpart of the still open Schanuel conjecture about the exponential function over \({\mathbb{C}}\) (Lang, Introduction to transcendental numbers, 1966). In this article, we derive from Ax’s theorem transcendency results in the context of differential valued exponential fields. In particular, we obtain results for exponential Hardy fields, Logarithmic-Exponential power series fields, and Exponential-Logarithmic power series fields.

An exponential $\exp $ on an ordered field $(K,+,-,\cdot ,0,1,<)$ is an order-preserving isomorphism from the ordered additive group $(K,+,0,<)$ to the ordered multiplicative group of positive elements $(K^{>0},\cdot ,1,<)$ . The structure $(K,+,-,\cdot ,0,1,<,\exp )$ is then called an ordered exponential field (cf. [6]). A linearly ordered structure $(M,<,\ldots )$ is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M.The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields $(K,+,-,\cdot ,0,1,<,\exp )$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp (0) = 1$ . This study is mainly motivated by the Transfer Conjecture, which states as follows:Any o-minimal exponential field $(K,+,-,\cdot ,0,1,<,\exp )$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp (0)=1$ is elementarily equivalent to $\mathbb {R}_{\exp }$ .Here, $\mathbb {R}_{\exp }$ denotes the real exponential field $(\mathbb {R},+,-,\cdot ,0,1,<,\exp )$ , where $\exp $ denotes the standard exponential $x \mapsto \mathrm {e}^x$ on $\mathbb {R}$ . Moreover, elementary equivalence means that any first-order sentence in the language $\mathcal {L}_{\exp } = \{+,-,\cdot ,0,1, <,\exp \}$ holds for $(K,+,-,\cdot ,0,1,<,\exp )$ if and only if it holds for $\mathbb {R}_{\exp }$ .The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of $\mathbb {R}_{\exp }$ . To the date, it is not known if $\mathbb {R}_{\exp }$ is decidable, i.e., whether there exists a procedure determining for a given first-order $\mathcal {L}_{\exp }$ -sentence whether it is true or false in $\mathbb {R}_{\exp }$ . However, under the assumption of Schanuel’s Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for $\mathbb {R}_{\exp }$ exists (cf. [7]). Also a positive answer to the Transfer Conjecture would result in the decidability of $\mathbb {R}_{\exp }$ (cf. [1]). Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture, and the decidability question of $\mathbb {R}_{\exp }$ .Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields—the residue field and the value group—with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends—the smallest elementary substructures being prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable Henselian valuations, and strongly NIP ordered fields.Parts of this thesis were published in [2–5].Abstract prepared by Lothar Sebastian KrappE-mail: sebastian.krapp@uni-konstanz.deURL: https://d-nb.info/1202012558/34

I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property. A corollary is that there are at most countably many essential counterexamples to Schanuel's conjecture.

We introduce normalized exponential Yang-Mills energy functional ym0e , stress-energy tensor Se,ym0 associated with the normalized exponential Yang-Mills energy functional yM0e , e-conservation law. We also introduce the notion of the e-degree de which connects two separate parts in the associated normalized exponential stress-energy tensor Se,YM0 (cf. (3.10) and (4.15)), derive monotonicity formula for exponential Yang-Mills fields, and prove a vanishing theorem for exponential Yang-Mills fields. These monotonicity formula and vanishing theorem for exponential Yang-Mills fields augment and extend the monotonicity formula and vanishing theorem for F-Yang-Mills fields in [18] and [68, 9.2]. We also discuss an average principle (cf. Proposition 8.1), isoperimetric and Sobolev inequalities, convexity and Jensen’s inequality, p-Yang-Mills fields, an extrinsic average variational method in the calculus of variation and Φ(3)-harmonic maps, from varied, coupled, generalized viewpoints and perspectives (cf. Theorems 6.1, 7.1, 9.1, 9.2, 10.1, 10.2, 11.13, 11.14, and 11.15).

Joint estimation of heterogeneous exponential Markov Random Fields through an approximate likelihood inference

In this paper, some inequalities of Simons type for exponential Yang-Mills fields over compact Riemannian manifolds are established, and the energy gaps are obtained.

We have studied alpha attractor cosmology with exponential harmonic fields and reheating phase after inflation. The results show that, for large limit of alpha, the inflationary parameters of the used model confirm the observational data. For exponential harmonic fields, we have obtained constraints on reheating parameters. It arises that for these fields, reheating can last longer than canonical and tachyon fields.

Static black hole with the Power Maxwell invariant (PMI), Born–Infeld (BI), logarithmic (LN), exponential (EN) electromagnetic fields in three-dimensional spacetime with cosmological constant was studied. It was shown that the LN and EN fields represent the Born–Infeld type of nonlinear electrodynamics. It the framework of General Relativity the exact solutions of the field equations were obtained, corresponding thermodynamic functions were calculated and the [Formula: see text] criticality of the black holes in the extended phase-space thermodynamics was investigated.

Given an ordered fieldK, we compute the natural valuation and skeleton of the ordered multiplicative group (K >0, ·, 1, <) in terms of those of the ordered additive group (K,+,0,<). We use this computation to provide necessary and sufficient conditions on the value groupv(K) and residue field\(\bar K\), for theL ∞ε-equivalence of the above mentioned groups. We then apply the results to exponential fields, and describev(K) in that case. Finally, ifK is countable or a power series field, we derive necessary and sufficient conditions onv(K) and\(\bar K\) forK to be exponential. In the countable case, we get a structure theorem forv(K).

Reflection amplitudes of ADE Toda theories and thermodynamic Bethe ansatz

Spatial overlap between the electromagnetic fields and the analytes is a key factor for strong light-matter interaction leading to high sensitivity for label-free optical biosensors. Usually, the exponential fields of cavity modes or surface plasmon resonances are applied to monitor the refractive index variation from bio-reactions. The sensitivity is therefore limited by the influence of local index variation to the weak exponential field. In this paper, by constructing a metallic microstructure array-dielectric-metal (MDM) structure, a novel metamaterial integrated microfluidic (MIM) sensor is demonstrated in terahertz (THz) range, where the dielectric layer of the MDM metamaterial is hollow and acts as the microfluidic channel. Tuning the electromagnetic parameters of metamaterial, greatly confined electromagnetic fields can be obtained in the channel resulting in significantly enhanced interaction between the analytes and the THz wave. A record high sensitivity of 3.5 THz/RIU is predicted by numerical simulation. Normalized the sensitivity to the working frequency, the calculated and measured normalized sensitivity is 0.55/RIU and 0.31/RIU, respectively. The proposed idea to integrate metamaterial and microfluid with a large light-matter interaction can be extended to other frequency regions and has promising applications in biosensing and matter detection.

Turing meets Schanuel

In an extended abstract Ressayre considered real closed exponential fields and integer parts that respect the exponential function. He outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre's construction, which becomes canonical once we fix the real closed exponential field, a residue field section, and a well ordering of the field. The procedure is constructible over these objects; each step looks effective, but may require many steps. We produce an example of an exponential field $R$ with a residue field $k$ and a well ordering $<$ such that $D^c(R)$ is low and $k$ and $<$ are $\Delta^0_3$, and Ressayre's construction cannot be completed in $L_{\omega_1^{CK}}$.