We prove that every JB⁎-triple E (in particular, every C⁎-algebra) satisfying the Daugavet property also satisfies the stronger polynomial Daugavet property, that is, every weakly compact polynomial P:E⟶E satisfies the Daugavet equation ‖IdX+P‖=1+‖P‖. The analogous conclusion also holds for the alternative Daugavet property.

We prove that every commutative JB ^* -triple has numerical index one. We also revisit the notion of commutativity in JB ^* -triples to show that a JBW ^* -triple M has numerical index one precisely when it is commutative, while e^-1≤ n(M) ≤ 2^-1 otherwise. Consequently, a JB ^* -triple E is commutative if and only if n(E^*) =1 (equivalently, n(E^**) =1 ). In the general setting we prove that the numerical index of each JB ^* -triple E admitting a non-commutative element also satisfies e^-1≤ n(E) ≤ 2^-1 , and the same holds when the bidual of E contains a Cartan factor of rank ≥ 2 in its atomic part.

It is known, by Gelfand theory, that every commutative JB*-triple admits a representation as a space of continuous functions of the form C-0(T) (L) = {alpha epsilon C-0(L) : alpha(lambda t) =lambda alpha(t), A lambda epsilon T, t epsilon L}, where L is a principal T-bundle and T denotes the unit circle in C. We provide a full technical description of all orthogonality preserving (non-necessarily continuous nor bijective) linear maps between commutative JB*-triples. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB*-triples is automatically continuous and bi-orthogonality preserving.

In a previous paper, the authors implemented a machine learning k-nearest neighbors (KNN) algorithm and Python libraries to create two 3D interactive models of the stratigraphic architecture of the Quaternary onshore Llobregat River Delta (NE Spain) for groundwater exploration purposes. The main limitation of this previous paper was its lack of routines for evaluating the confidence of the 3D models. Building from the previous paper, this paper refines the programming code and introduces an additional algorithm to evaluate the confidence of the KNN predictions. A variant of the Similarity Ratio method was used to quantify the KNN prediction confidence. This variant used weights that were inversely proportional to the distance between each grain-size class and the inferred point to work out a value that played the role of similarity. While the KNN algorithm and Python libraries demonstrated their efficacy for obtaining 3D models of the stratigraphic arrangement of sedimentary porous media, the KNN prediction confidence verified the certainty of the 3D models. In the Llobregat River Delta, the KNN prediction confidence at each prospecting depth was a function of the available data density at that depth. As expected, the KNN prediction confidence decreased according to the decreasing data density at lower depths. The obtained average-weighted confidence was in the 0.44−0.53 range for gravel bodies at prospecting depths in the 12.7−72.4 m b.s.l. range and was in the 0.42−0.55 range for coarse sand bodies at prospecting depths in the 4.6−83.9 m b.s.l. range. In a couple of cases, spurious average-weighted confidences of 0.29 in one gravel body and 0.30 in one coarse sand body were obtained. These figures were interpreted as the result of the quite different weights of neighbors from different grain-size classes at short distances. The KNN algorithm confidence has proven its suitability for identifying these anomalous results in the supposedly well-depurated grain-size database used in this study. The introduced KNN algorithm confidence quantifies the reliability of the 3D interactive models, which is a necessary stage to make decisions in economic and environmental geology. In the Llobregat River Delta, this quantification clearly improves groundwater exploration predictability.

In 2022, Hatori gave a sufficient condition for complex Banach spaces to have the complex Mazur–Ulam property. In this paper, we introduce a class of complex Banach spaces B that do not satisfy the condition but enjoy the property that every surjective isometry on the unit sphere of such B admits an extension to a surjective real linear isometry on the whole space B . Typical examples of Banach spaces studied in this note are the spaces Lip ([0,1]) of all Lipschitz complex-valued functions on [0, 1] with norm ‖ a ‖ _B=|a(c)|+L(a) , where c is a fixed element in [0, 1], and the space C^1([0,1]) of all continuously differentiable complex-valued functions on [0, 1] equipped with the norm |f(0)|+‖ f' ‖ _∞ .

This paper introduces a methodology based on Python libraries and machine learning k-Nearest Neighbors (KNN) algorithms to create an interactive 3D HTML model (3D_Vertical_Sections_Faults_LRD.html) that combines 2D grain-size KNN-prediction vertical maps (vertical sections) from which syn-sedimentary faults and other features in sedimentary porous media can be delineated. The model can be visualized and handled with conventional web browsers. The grain-size physical parameter is measurable, constant over instrumental time, handleable math-ematically, and its range can be associated to lithological classes. Grain-size input data comes from a public database of 433 boreholes in the Llobregat River Delta (LRD) in NE Spain. Four lithological classes were defined: Pre-Quaternary basement, and Quaternary gravel, sand, and clay-silt. Using a new KNN-prediction algorithm, seven NW-SE (transversal) and three SW-NE (longitudinal) vertical sections were created following the orien-tation of faults identified in surface and detected in reflection seismic geophysical surveys. For exploratory K values in the 1-75 range were used. K around 25 provides the general and smoothy shape of the basement top surface, whereas K =1 is a optimal value to represent the heterogeneity of the LRD at short distance. Using a new KNN-prediction confidence algorithm inspired in the Similarity Ratio algorithm for machine-learning KNN, the vertical sections overall confidence was evaluated as satisfactory. A general decreasing confidence trend ac-cording to the decreasing data density with depth and from inland to seaward was found. The vertical sections created with K = 1 show horizontal interruptions (displacements or vertical steps) in the basement continuity and in the Quaternary coarse bodies (gravel and sand) attributable to the action of Quaternary active faults. These faults have been linked or correlated with well-known active faults in the area related in much cases with the Valencia Trough opening. Moreover, several faults detected in surface and other identified in this paper by the first time have been revealed as fault zones made of fault branches with different steps in an echelon-like arrangement. Faulting seems to be more evident in the Pleistocene Lower Detrital Complex and much less active or inactive in the Holocene Upper Detrital Complex. Syn-tectonic gravel channels faulty controlled, progradation of gravel lobes, and lateral migration of channel bars were also observed. At its current develop-ment stage, this methodology could also be applied to other geological environments, making the due minor modifications of the code, and is especially suitable to reduce the high (usually unmeasurable) uncertainty associated to the qualitative geological data used in more complex numerical tools aimed at modelling a lot of geological resources (groundwater, minerals, geothermal, petroleum) or different Earth phenomena.

We prove that every commutative JB $$^*$$ -triple, represented as a space of continuous functions $$C_0^{\mathbb {T}}(L),$$ satisfies the complex Mazur–Ulam property, that is, every surjective isometry from the unit sphere of $$C_0^{\mathbb {T}}(L)$$ onto the unit sphere of any complex Banach space admits an extension to a surjective real linear isometry between the spaces.

This paper introduces a Python application for visualizing the 3D stratigraphic architecture of porous sedimentary media. The application uses the parameter granulometry deduced from borehole lithological records to create interactive 3D HTML models of essential stratigraphic elements. On the basis of the high density of boreholes and the subsequent geological knowledge gained during the last six decades, the Quaternary onshore Llobregat River Delta (LRD) in northeastern Spain was selected to show the application. The public granulometry dataset produced by the Water Authority of Catalonia from 433 boreholes in this strategic coastal groundwater body was clustered into the clay–silt, coarse sand, and gravel classes. Three interactive 3D HTML models were created. The first shows the location of the boreholes granulometry. The second includes the main gravel and coarse sand sedimentary bodies (lithosomes) associated with the identified three stratigraphic intervals, called lower (>50 m b.s.l.) in the distal LRD sector, middle (20–50 m b.s.l.) in the central LRD, and upper (<20 m b.s.l.) spread over the entire LRD. The third deals with the basement (Pliocene and older rocks) top surface, which shows an overall steeped shape deepening toward the marine platform and local horsts, probably due to faulting. The modeled stratigraphic elements match well with the sedimentary structures reported in recent scientific publications. This proves the good performance of this incipient Python application for visualizing the 3D stratigraphic architecture, which is a crucial stage for groundwater management and governance.

The k-nearest neighbors (KNN) algorithm is a non-parametric supervised machine learning classifier; which uses proximity and similarity to make classifications or predictions about the grouping of an individual data point. This ability makes the KNN algorithm ideal for classifying datasets of geological variables and parameters prior to 3D visualization. This paper introduces a machine learning KNN algorithm and Python libraries for visualizing the 3D stratigraphic architecture of sedimentary porous media in the Quaternary onshore Llobregat River Delta (LRD) in northeastern Spain. A first HTML model showed a consecutive 5 m-equispaced set of horizontal sections of the granulometry classes created with the KNN algorithm from 0 to 120 m below sea level in the onshore LRD. A second HTML model showed the 3D mapping of the main Quaternary gravel and coarse sand sedimentary bodies (lithosomes) and the basement (Pliocene and older rocks) top surface created with Python libraries. These results reproduce well the complex sedimentary structure of the LRD reported in recent scientific publications and proves the suitability of the KNN algorithm and Python libraries for visualizing the 3D stratigraphic structure of sedimentary porous media, which is a crucial stage in making decisions in different environmental and economic geology disciplines.