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1. Computability in infinite Galois theory and algorithmically random algebraic fields

   https://doi.org/10.1112/jlms.70017  

   Calvert, Wesley; Harizanov, Valentina; Shlapentokh, Alexandra (November 2024, Journal of the London Mathematical Society)

   Abstract We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of that are random according to our definition. We show that there are noncomputable algebraic fields which are not random. We also partially characterize the index set, relative to an oracle, of the set of random algebraic fields computable relative to that oracle. In order to carry out this investigation of randomness for fields, we develop computability in the context of the infinite Galois theory (where the relevant Galois groups are uncountable), including definitions of computable and computably enumerable Galois groups and computability of Haar measure on the Galois groups. 

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2. On the computability of rotation sets and their entropies

   https://doi.org/10.1017/etds.2018.45  

   BURR, MICHAEL A.; SCHMOLL, MARTIN; WOLF, CHRISTIAN (February 2020, Ergodic Theory and Dynamical Systems)

   null (Ed.)

   Let $$f:X\rightarrow X$$ be a continuous dynamical system on a compact metric space $$X$$ and let $$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$$ be an $$m$$ -dimensional continuous potential. The (generalized) rotation set $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is defined as the set of all $$\unicode[STIX]{x1D707}$$ -integrals of $$\unicode[STIX]{x1D6F7}$$ , where $$\unicode[STIX]{x1D707}$$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy $$\unicode[STIX]{x210B}(w)$$ to each $$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$$ . In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where $$f$$ is a subshift of finite type. We prove that $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is computable and that $$\unicode[STIX]{x210B}(w)$$ is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, $$\unicode[STIX]{x210B}$$ is not continuous on the boundary of the rotation set when considered as a function of $$\unicode[STIX]{x1D6F7}$$ and $$w$$ . In particular, $$\unicode[STIX]{x210B}$$ is, in general, not computable at the boundary of $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ . 

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3. Eccentricity evolution in gaseous dynamical friction

   https://doi.org/10.1093/mnras/stac1294  

   Szölgyén, Ákos; MacLeod, Morgan; Loeb, Abraham (May 2022, Monthly Notices of the Royal Astronomical Society)

   ABSTRACT We analyse how drag forces modify the orbits of objects moving through extended gaseous distributions. We consider how hydrodynamic (surface area) drag forces and dynamical friction (gravitational) drag forces drive the evolution of orbital eccentricity. While hydrodynamic drag forces cause eccentric orbits to become more circular, dynamical friction drag can cause orbits to become more eccentric. We develop a semi-analytic model that accurately predicts these changes by comparing the total work and torque applied to the orbit at periapse and apoapse. We use a toy model of a radial power-law density profile, ρ ∝ r−γ, to determine that there is a critical γ = 3 power index, which separates the eccentricity evolution in dynamical friction: orbits become more eccentric for γ < 3 and circularize for γ > 3. We apply these findings to the infall of a Jupiter-like planet into the envelope of its host star. The hydrostatic envelopes of stars are defined by steep density gradients near the limb and shallower gradients in the interior. Under the influence of gaseous dynamical friction, an infalling object’s orbit will first decrease in eccentricity and then increase. The critical separation that delineates these regimes is predicted by the local density slope and is linearly dependent on polytropic index. More broadly, our findings indicate that binary systems may routinely emerge from common envelope phases with non-zero eccentricities that were excited by the dynamical friction forces that drove their orbital tightening. 

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4. White-box Machine learning approaches to identify governing equations for overall dynamics of manufacturing systems: A case study on distillation column

   Subramanian, Renganathan; Moar, Raghav Rajesh; Singh, Shweta (March 2021, Machine learning with applications)

   null (Ed.)

   Dynamical equations form the basis of design for manufacturing processes and control systems; however, identifying governing equations using a mechanistic approach is tedious. Recently, Machine learning (ML) has shown promise to identify the governing dynamical equations for physical systems faster. This possibility of rapid identification of governing equations provides an exciting opportunity for advancing dynamical systems modeling. However, applicability of the ML approach in identifying governing mechanisms for the dynamics of complex systems relevant to manufacturing has not been tested. We test and compare the efficacy of two white-box ML approaches (SINDy and SymReg) for predicting dynamics and structure of dynamical equations for overall dynamics in a distillation column. Results demonstrate that a combination of ML approaches should be used to identify a full range of equations. In terms of physical law, few terms were interpretable as related to Fick’s law of diffusion and Henry’s law in SINDy, whereas SymReg identified energy balance as driving dynamics. 

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5. Computability of topological pressure on compact shift spaces beyond finite type*

   https://doi.org/10.1088/1361-6544/ac7702  

   Burr, Michael; Das, Suddhasattwa; Wolf, Christian; Yang, Yun (June 2022, Nonlinearity)

   We investigate the computability (in the sense of computable analysis) of the topological pressure P_top(ϕ) on compact shift spaces X for continuous potentials ϕ:X→R. This question has recently been studied for subshifts of finite type (SFTs) and their factors (sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalised gap shifts, and particular beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalised pressure function (X,ϕ) ↦P_top(X,ϕ|_X) is not computable for a large set of shift spaces X and potentials ϕ . In particular, the entropy map X↦h_top(X) is computable at a shift spaceXif and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability. 

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