Proofs of operator monotonicity of some functions by using Lowner\u27s integral representation

We give somewhat simple proofs of operator monotonicity of some functions by using Lowner’s integral representation of an operator monotone function

A remark on rational operator monotone functions

Recently, M. Nagisa presented the general form of rational operator monotone functions for any open interval in ℝ= (−∞,∞), directly deduced from the property of Pick function. Using his results, we, in particular, determine the form of positive rational operator monotone functions on (0,∞)

Weakly selfadjoint operators II

In this note, we show some basic results related to the spectrum of a weakly selfadjoint operator

Order relations among some interpolating families of means

Some interpolating families of means of two positive numbers with a parameter are studied. They are concerned with some of the most familiar means, i.e., the arithmetic, geometric, harmonic, logarithmic means and so on. Their monotonicity with respect to the parameter and the order relations among them are discussed

Weakly selfadjoint operators

Extending the notion of a weakly positive operator introduced by Wigner, we define a weakly selfadjoint operator as the product of two selfadjoint operators, one of whose factors is positive, and show some facts on the operator

REE characteristics of Yamato-82162 and -86720 meteorites and their inference to classification

Abundances of REE, Ba, Sr, Rb, K, Mg, Ca and Fe were determined for Yamato-82162 and -86720 carbonaceous chondrites by mass spectrometric isotope dilution. The CI-normalized REE abundance patterns for Y-82162 are nearly flat (0.9-1.5×CI), but show small negative Ce anomalies. A small fraction of the chondrite shows a clear light/heavy REE fractionation. The REE abundance patterns of Y-86720 are also almost flat (1.4-1.9×CI) and show positive Eu anomalies. The degree of Eu anomaly seems to correlate with the absolute abundance of the REE. For the abundances of elements other than REE, Y-82162 shows enrichments of Fe and Mg and Y-86720 displays depletions of the relatively volatile elements K and Rb. From these results, the following conclusions are obtained : (1) Y-82162 may contain a high temperature component formed in the nebula. (2) Y-82162 is classified as a CI chondrite which has experienced thermal metamorphism. (3) Y-86720 is composed of a refractory-rich component and a refractory-poor component. (4) Y-86720 cannot be strictly classified into any group but is intermediate between CM and CO chondrites

"Dehydrated" chondrules from the Murchison (CM) chondrite

Two "dehydrated" chondrules (MC-5 and MC-27) were obtained from close to the fusion crust of the Murchison (CM) meteorite. They have a porphyritic texture with angular or rounded (relict) olivines containing abundant voids, recrystallized olivines, and numerous interstitial Fe oxide grains embedded in groundmass glass. This peculiar texture is similar to that of the fusion crust from the Orgueil (CI) chondrite, indicating that the texture formed during the reheating at the atmospheric entry of the meteorite. Isotope dilution analyses indicate that these chondrules have light-REE depleted pattern with a large negative Eu anomaly (CI normalized Sm/La ratio=1.3-2.4,Eu/Eu^*=0.40-0.57). They also show relatively low Ba and Sr (0.4-1.0×CI) concentrations and depletion of alkalies (0.2-0.4×CI). The lithophile element abundances of the two chondrules are in a similar range of those for other altered CM chondrules in the Murchison and the Yamato-793321 chondrites. It is suggested that the chemical compositions of chondrules, specifically REE and alkali abundances, remained unchanged but significant changes of petrographic texture occurred during atmospheric heating

A preliminary study of REE abundances in chondrules, an inclusion and mineral fragments from Yamato-793321 (CM2) chondrite

In order to examine REE distributions in components of CM chondrites, eight spherical silicate materials (four chondrules [PO, POP and GO], one CAI, two olivine fragments and one unknown inclusion [chondrule?]) were separated from the Yamato (Y)-793321 meteorite, one of the least altered CM2 chondrites, and have been analyzed for REE, Ba, Sr, Rb, K, Ca, Mg and Fe by isotope dilution, together with petrographic examination. The olivine fragments (YO-1,YO-2) (Fo>99%) with rounded metal inclusions show depletion of alkalis (~10^×CI), low refractory element abundances (Sr, Ba and REE=0.2-0.5×CI) and a fractionated (V-shaped) REE pattern, indicative of solid/liquid partitioning of REE. It is suggested that YO-1 and YO-2 formed from melt. The CAI (YI-5) consisting of olivine, fassaite and euhedral spinel shows no sign of aqueous alteration. It has low alkali (<0.1×CI) and high refractory element abundances (2-20×CI) and indicates a light REE depletion and generally smooth pattern with a light/heavy REE discontinuity. The unaltered PO chondrule (YC-7) shows alkali depletion and unfractionated abundances of REE, Ba, Sr and Ca. The altered PO and GO chondrules (YC-8,YC-38) and unknown spherule (Y-9) indicate a light-REE depleted pattern with a negative Eu anomaly and low Ba, Sr and alkalis (&acd;0.1×CI). This REE fractionation seems to be a new type for a chondrule, indicating that a unique REE fractionation occurred during the formation and/or evolution of the Y-793321 CM meteorite

Compositions of REE, K, Rb, Sr, Ba, Mg, Ca, Fe, and Sr isotopes in Antarctic"unique" meteorites

The abundances of Mg, Ca, Fe, K, Rb, Sr, Ba, and REE in five Antarctic "unique" meteorites, acapulcoite-lodranite type, Yamato (Y)-74063,Y-74357,Y-8002,and Allan Hills (ALH)-78230,and winonaite type, Y-75300 were determined by isotope dilution mass spectrometry. The Sr isotopic compositions of Y-74063,Y-75300,Y-8002,and ALH-78230 were also measured to investigate the whole-rock Rb-Sr isotopic systematics. In terms of the REE abundance pattern, these meteorites are grouped as follows; (1) Y-74063 and ALH-78230 with chondritic REE patterns, (2) Y-74357 with light REE depletion plus alkali and alkaline earth element depletion, and (3) Y-75300 and Y-8002 with middle REE depletion (V-shaped REE pattern) plus a positive Eu anomaly). The mineralogy of these meteorites is closely related to the above REE classification. The ^Rb-^Sr age determined from \u27whole-rock samples\u27 for Y-74063,Y-75300,Y-8002,and ALH-78230 is consistent with their formation of 4.5Ga, albeit with a large uncertainty due to the blank correction. Using the mineral/liquid partition coefficients of trace elements, the petrogenetic model calculations were performed. The results give indication on the origin of these "unique" meteorites. Y-74063 and ALH-78230 could be formed through a small degree of partial melting (less than a few%) from a chondritic starting material. Y-74357 could be formed by a larger degree (12%) of partial melting. On the other hand, the V-shaped REE pattern of the third group (Y-75300 and Y-8002) could not be explained by a simple partial melting process. The V-shaped REE pattern may be explained only by assuming the solid state equilibration within a reservoir with chondritic composition. It is suggested that Y-75300 and Y-8002 might be derived from parental materials depleted in phosphate and clinopyroxene