Connecting gravity field, moment of inertia, and core properties in Jupiter through empirical structure models

<p>Constraining Jupiter's internal structure is crucial for understanding its formation and evolution history. Recent interior models of Jupiter that fit Juno's measured gravitational field suggest an inhomogeneous interior and potentially the existence of a diluted core. These models, however, strongly depend on the model assumptions and the equations of state used. A complementary modelling approach is to use empirical structure models. <br>These can later be used to reveal new insights on the planetary interior and be compared to standard models. <br>Here we present empirical structure models of Jupiter where the density profile is constructed by piecewise-polytropic equations. With these models we investigate the relation between the normalized moment of inertia (MoI) and the gravitational moments J<sub>2</sub> and J<sub>4</sub>. <br>Given that only the first few gravitational moments of Jupiter are measured with high precision, we show that an accurate and independent measurement of the MoI value could be used to further constrain Jupiter's interior. An independent measurement of the MoI with an accuracy better than ~0.1% could constrain Jupiter's core region and density discontinuities in its envelope. <br>We find that models with a density discontinuity at ~1 Mbar, as would produce a presumed hydrogen-helium separation, correspond to a fuzzy core in Jupiter. <br>We next test the appropriateness of using polytropes, by comparing them with empirical models based on polynomials. <br>We conclude that both representations result in similar density profiles and ranges of values for quantities like core mass and MoI.</p>

On the Core, Bargaining Set and the Set of Competitive Allocations of Fuzzy Exchange Economy with a Continuum of Agents

In this paper, we present a new model closer to the real-life — called the fuzzy exchange economy with a continuum of agents (FXE-CA) — that combines fuzzy consumption and fuzzy initial endowment with the agent’s fuzzy preference in the fuzzy consumption set. To characterize the fuzzy competitive allocations of the FXE-CA, we define the indifference fuzzy core of a FXE-CA as the set of all fuzzy allocations that cannot be dominated by any coalition of agents. We also propose the Mas-Colell indifference fuzzy bargaining set, in which no coalition has a justified objection at a fuzzy allocation against any other coalition. Finally, we verify that the indifference fuzzy core and the indifference fuzzy bargaining set of a FXE-CA coincide with the set of all fuzzy competitive allocations under some conditions, respectively. This indicates that the agents unanimously distribute the fuzzy competitive allocations of a FXE-CA.

Clustering applications of IFDBSCAN algorithm with comparative analysis

Density Based Spatial Clustering of Application with Noise (DBSCAN) is one of the mostly preferred algorithm among density based clustering approaches in unsupervised machine learning, which uses epsilon neighborhood construction strategy in order to discover arbitrary shaped clusters. DBSCAN separates dense regions from low density regions and simultaneously assigns points that lie alone as outliers to unearth the hidden cluster patterns in the datasets. DBSCAN identifies dense regions by means of core point definition, detection of which are strictly dependent on input parameter definitions: ε is distance of the neighborhood or radius of hypersphere and MinPts is minimum density constraint inside ε radius hypersphere. Contrarily to classical DBSCAN’s crisp core point definition, intuitionistic fuzzy core point definition is proposed in our preliminary work to make DBSCAN algorithm capable of detecting different patterns of density by two different combinations of input parameters, particularly is a necessity for the density varying large datasets in multidimensional feature space. In this study, preliminarily proposed DBSCAN extension is studied: IFDBSCAN. The proposed extension is tested by computational experiments on several machine learning repository real-time datasets. Results show that, IFDBSCAN is superior to classical DBSCAN with respect to external & internal performance indices such as purity index, adjusted rand index, Fowlkes-Mallows score, silhouette coefficient, Calinski-Harabasz index and with respect to clustering structure results without increasing computational time so much, along with the possibility of trying two different density patterns on the same run and trying intermediary density values for the users by manipulating α margin.

The challenge of forming a fuzzy core in Jupiter

Recent structure models of Jupiter that match Juno gravity data suggest that the planet harbours an extended region in its deep interior that is enriched with heavy elements: a so-called dilute or fuzzy core. This finding raises the question of what possible formation pathways could have lead to such a structure. We modelled Jupiter’s formation and long-term evolution, starting at late-stage formation before runaway gas accretion. The formation scenarios we considered include both primordial composition gradients, as well as gradients that are built as proto-Jupiter rapidly acquires its gaseous envelope. We then followed Jupiter’s evolution as it cools down and contracts, with a particular focus on the energy and material transport in the interior. We find that none of the scenarios we consider lead to a fuzzy core that is compatible with interior structure models. In all the cases, most of Jupiter’s envelope becomes convective and fully mixed after a few million years at most. This is true even when we considered a case where the gas accretion leads to a cold planet, and large amounts of heavy elements are accreted. We therefore conclude that it is very challenging to explain Jupiter’s dilute core from standard formation models. We suggest that future works should consider more complex formation pathways as well as the modelling of additional physical processes that could lead to Jupiter’s current-state internal structure.

Core for Game with Fuzzy Generalized Triangular Payoff Value

In this paper, we investigate cooperative game with fuzzy payoff value in the generalized triangular fuzzy number directly. Based on the fuzzy max order, we define three kinds of fuzzy cores, i.e., fuzzy strong core, fuzzy non-dominated core and fuzzy weak core. All three kinds of fuzzy cores can be regarded as the generalization of crisp core. Convexity is one of the sufficient conditions for the existence of fuzzy core. By the balanced cooperative game, a necessary and sufficient existence condition of fuzzy strong core is also given. Further, the fuzzy strong core is represented by crisp core, and the relationship between fuzzy strong core and crisp one is shown. For the fuzzy non-dominated core and fuzzy weak core, we show their necessary and sufficient existence condition, and their properties to construct the fuzzy imputations. Hence, the verification of fuzzy non-dominated core and fuzzy weak core become easier. The above three fuzzy cores are all the extensions of crisp core, but their stable conditions are not different. The weak core is least restricted, but is least stable. Hence, we could choose the fuzzy core according the stable neediness of fuzzy cooperative game.

Cooperative Fuzzy Games with Convex Combination Form

In this paper, a new class of cooperative fuzzy games named fuzzy games with convex combination form is introduced. This kind of fuzzy games considers two aspects of information. One is the contribution of the players to the associated crisp coalitions; the other is their participation levels. The explicit expression of the Shapley function is given, which is equal to the production of the Shapley function on crisp games and the player participation levels. Meanwhile, the relationship between the fuzzy core and the Shapley function is studied. Surprisingly, the relationship between them does coincide as in crisp case. Furthermore, some desirable properties are researched. Finally, an example is provided to illustrate the difference in fuzzy coalition values and the player Shapley values for four types of fuzzy games.