Gaussoids are two-antecedental approximations of Gaussian conditional independence structures

The gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely describes regular Gaussian CI as the ground set grows. In this article we show that the gaussoid axioms logically imply every inference rule of at most two antecedents which is valid for regular Gaussians over any ground set. The proof is accomplished by exhibiting for each inclusion-minimal gaussoid extension of at most two CI statements a regular Gaussian realization. Moreover we prove that all those gaussoids have rational positive-definite realizations inside every ε-ball around the identity matrix. For the proof we introduce the concept of algebraic Gaussians over arbitrary fields and of positive Gaussians over ordered fields and obtain the same two-antecedental completeness of the gaussoid axioms for algebraic and positive Gaussians over all fields of characteristic zero as a byproduct.

Algebraic Solution of Tropical Polynomial Optimization Problems

We consider constrained optimization problems defined in the tropical algebra setting on a linearly ordered, algebraically complete (radicable) idempotent semifield (a semiring with idempotent addition and invertible multiplication). The problems are to minimize the objective functions given by tropical analogues of multivariate Puiseux polynomials, subject to box constraints on the variables. A technique for variable elimination is presented that converts the original optimization problem to a new one in which one variable is removed and the box constraint for this variable is modified. The novel approach may be thought of as an extension of the Fourier–Motzkin elimination method for systems of linear inequalities in ordered fields to the issue of polynomial optimization in ordered tropical semifields. We use this technique to develop a procedure to solve the problem in a finite number of iterations. The procedure includes two phases: backward elimination and forward substitution of variables. We describe the main steps of the procedure, discuss its computational complexity and present numerical examples.

Ordered fields dense in their real closure and definable convex valuations

In this paper, we undertake a systematic model- and valuation-theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah–Hasson Conjecture (specialized to ordered fields) and provide an example limiting its valuation-theoretic conclusions.

Generic derivations on o-minimal structures

Let [Formula: see text] be a complete, model complete o-minimal theory extending the theory [Formula: see text] of real closed ordered fields in some appropriate language [Formula: see text]. We study derivations [Formula: see text] on models [Formula: see text]. We introduce the notion of a [Formula: see text]-derivation: a derivation which is compatible with the [Formula: see text]-definable [Formula: see text]-functions on [Formula: see text]. We show that the theory of [Formula: see text]-models with a [Formula: see text]-derivation has a model completion [Formula: see text]. The derivation in models [Formula: see text] behaves “generically”, it is wildly discontinuous and its kernel is a dense elementary [Formula: see text]-substructure of [Formula: see text]. If [Formula: see text], then [Formula: see text] is the theory of closed ordered differential fields (CODFs) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that [Formula: see text] has [Formula: see text] as its open core, that [Formula: see text] is distal, and that [Formula: see text] eliminates imaginaries. We also show that the theory of [Formula: see text]-models with finitely many commuting [Formula: see text]-derivations has a model completion.

Internality, transfer, and infinitesimal modeling of infinite processes†

ABSTRACT A probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson’s transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields (surreals, Levi-Civita field, Laurent series) may have advantages over hyperreals in probabilistic modeling. We show that probabilities developed over such fields are less expressive than hyperreal probabilities.

Relativistic AGN jets – III. Synthesis of synchrotron emission from double-double radio galaxies

ABSTRACT The class of double-double radio galaxies (DDRGs) relates to episodic jet outbursts. How various regions and components add to the total intensity in radio images is less well known. In this paper, we synthesize synchrotron images for DDRGs based on special relativistic hydrodynamic simulations, making advanced approximations for the magnetic fields. We study the synchrotron images for three different radial jet profiles; ordered, entangled, or mixed magnetic fields; spectral ageing from synchrotron cooling; the contribution from different jet components; the viewing angle and Doppler (de-)boosting; and the various epochs of the evolution of the DDRG. To link our results to observational data, we adopt to J1835+6204 as a reference source. In all cases, the synthesized synchrotron images show two clear pairs of hotspots, in the inner and outer lobes. The best resemblance is obtained for the piecewise isochoric jet model, for a viewing angle of approximately ϑ ∼ −71°, i.e. inclined with the lower jet towards the observer, with predominantly entangled (≳70 per cent of the magnetic pressure) in turbulent, rather than ordered fields. The effects of spectral ageing become significant when the ratio of observation frequencies and cut-off frequency νobs/ν∞, 0 ≳ 10−3, corresponding to ∼3 × 102 MHz. For viewing angles ϑ ≲ |−30°|, a DDRG morphology can no longer be recognized. The second jets must be injected within ≲ 4 per cent of the lifetime of the first jets for a DDRG structure to emerge, which is relevant for active galactic nuclei feedback constraints.