Isomorphic spectrum and isomorphic length of a Banach space

We prove that, given any ordinal $\delta < \omega_2$, there exists a transfinite $\delta$-sequence of separable Banach spaces $(X_\alpha)_{\alpha < \delta}$ such that $X_\alpha$ embeds isomorphically into $X_\beta$ and contains no subspace isomorphic to $X_\beta$ for all $\alpha < \beta < \delta$. All these spaces are subspaces of the Banach space $E_p = \bigl( \bigoplus_{n=1}^\infty \ell_p \bigr)_2$, where $1 \leq p < 2$. Moreover, assuming Martin's axiom, we prove the same for all ordinals $\delta$ of continuum cardinality.

Closure of the Bangong–Nujiang Tethyan Ocean in the central Tibet: Results from the provenance of the Duoni Formation

ABSTRACT The closure of the Bangong–Nujiang Tethyan Ocean (BNTO) and consequent Lhasa–Qiangtang collision is vital to reasonably understanding the early tectonic history of the Tibetan Plateau before the India-Eurasia collision. The timing of the Lhasa–Qiangtang collision was mainly constrained by the ophiolite and magmatic rocks in previous studies, with only limited constraints from the sedimentary rocks within and adjacent to the Bangong–Nujiang suture zone. In the middle segment of the Bangong–Nujiang suture zone, the Duoni Formation, consisting of a fluvial delta sequence with minor andesite interlayers, was originally defined as the Late Cretaceous Jingzhushan Formation and interpreted as the products of the Lhasa–Qiangtang collision during the Late Cretaceous. Our new zircon U-Pb data from two samples of andesite interlayers demonstrate that it was deposited during the latest Early Cretaceous (ca. 113 Ma) rather than Late Cretaceous. Systemic studies on the sandstone detrital model, heavy-mineral assemblage, and clasts of conglomerate demonstrate a mixed source of both Lhasa and Qiangtang terranes and ophiolite complex. Clasts of conglomerate contain abundant angular peridotite, gabbro, basalt, chert, andesite, and granite, and minor quartzite and gneiss clasts also exist. Sandstones of the Duoni Formation are dominated by feldspathic–lithic graywacke (Qt25F14L61 and Qm13F14L73), indicative of a mixture of continental-arc and recycled-orogen source origin. Detrital minerals of chromite, clinopyroxene, epidote, and hornblende in sandstone also indicate an origin of ultramafic and mafic rocks, while garnets indicate a metamorphosed source. Paleocurrent data demonstrate bidirectional (southward and northward) source origins. Thus, we suggest that the deposition of the Duoni Formation took place in the processes of the Lhasa–Qiangtang collision during the latest Early Cretaceous (∼ 113 Ma), and the BNTO had been closed by this time.

Remarks on the Generalized Fractional Laplacian Operator

The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way of Lévy flights. The fractional Laplacian has many applications in the boundary behaviours of solutions to differential equations. The goal of this paper is to investigate the half-order Laplacian operator ( − Δ ) 1 2 in the distributional sense, based on the generalized convolution and Temple’s delta sequence. Several interesting examples related to the fractional Laplacian operator of order 1 / 2 are presented with applications to differential equations, some of which cannot be obtained in the classical sense by the standard definition of the fractional Laplacian via Fourier transform.

Quantifying the stratigraphic and spatial facies distribution in an ancient mixed-influence delta

Analysis of the sedimentology and stratigraphic architecture of tightly spaced three dimensional outcrops reveals that the Turonian (Upper Cretaceous) Wall Creek Member of the Frontier Formation in the western Powder River Basin, Wyoming, USA, is not composed of one continuous coarsening upward succession but of a complex stacked delta system containing three distinct sequences (S1-S3), each with a unique facies distribution and architectural heterogeneity. The basal sequence S1 consists of a fluvial dominated delta with two distinct lobes. These lobes are spatially constrained to the northeastern study area and show a rapid facies transition from trough crossbedded mouthbar deposits to lower delta front turbidites. Low angle clinoforms suggest a low accommodation setting with main sediment transport to the south. The middle S2 sequence is common throughout the study area and contains an abundance of storm-derived deposits, including hummocky cross stratification, suggesting the transition to a wave and storm-dominated delta setting. Sediment transport is largely to the south controlled by wind induced shear stresses. Lastly, heterolithic trough crossbedded sandstones with flaser bedding and abundant thin mudstones and rip-up clasts are characteristic for sequence S3. These deposits are interpreted as tidal bars in a tidal influenced delta. Quantitative evaluation of facies in the Wall Creek Member sequences shows that the dimensions and connectivity (baffle or barrier competence) of fine-grained thin beds varies systematically within the three delta types. The S1 fluvial delta is largely composed of laterally continuous delta front turbidites with continuous fine-grained thin beds (mean length 21.1 m or 69.2 ft, max length 83.9 m or 275.2 ft) separating individual sandstone beds. Conversely, abundant bioturbation and intense scouring by storms results in high amalgamation of sandy beds in sequence S2 and a limited length of fine-grained thin beds (mean 8.5 m or 27.9 ft) in the wave-dominated delta sequence. Tidally influenced deposits of sequence S3 are largely composed of heterolithic trough crossbedded sandstones and mudstones with low bioturbation, resulting in an intermediate fine-grained thin bed deposit (mean 12.1 m or 39.7 ft).