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Let σ = { σ i ∣ i ∈ I } be some partition of the set of all primes and G a finite group. Then G is said to be σ -full if G has a Hall σ i -subgroup for all i ; σ -primary if G is a σ i -group for some i ; σ -soluble if every chief factor of G is σ -primary; σ -nilpotent if G is the direct product of σ -primary groups; G N σ denotes the σ -nilpotent residual of G , that is, the intersection of all normal subgroups N of G with σ -nilpotent quotient G / N . A subgroup A of G is said to be: σ -permutable in G provided G is σ -full and A permutes with all Hall σ i -subgroups H of G (that is, A H = H A ) for all i ; σ -subnormal in G if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G such that either A i - 1 ⊴ A i or A i / (A i - 1) A i is σ -primary for all i = 1 , ... , n . Let A σ G be the subgroup of A generated by all σ -permutable subgroups of G contained in A and A σ G be the intersection of all σ -permutable subgroups of G containing A . We prove that if G is a finite σ -soluble group, then the σ -permutability is a transitive relation in G if and only if G N σ avoids the pair (A σ G , A σ G) , that is, G N σ ∩ A σ G = G N σ ∩ A σ G for every σ -subnormal subgroup A of G . [ABSTRACT FROM AUTHOR]

The P–T paths of the exhumation of Precambrian granulite complexes at craton boundaries usually include two stages: subisothermal decompression and a decompression–cooling stage with a more gently sloped P–T path. Our goal is to understand the possible causes of the change in the slope of the P–T exhumation path of the Central Zone (CZ) of the Limpopo granulite complex, South Africa, located between the Kaapvaal and Zimbabwe cratons. For this purpose, rocks (mainly, metapelites) were studied in various structural settings within the Central Zone, i.e., in dome structures, regional cross folds, and in local and regional shear zones. The metapelites are gneisses of similar bulk composition. The rocks contain various amounts of relics of leucosomes composed of quartz–feldspar aggregates with garnet and biotite, and melanocratic domains that are enriched in cordierite and usually mark shear microzones that envelope and/or break garnet porphyroblasts. Study of polymineralic (crystallized melt and fluid) inclusions in the garnet, its zoning with respect to the major (Mg, Fe, and Ca) and some trace (P, Cr, and Sc) elements, fluid inclusions in quartz, as well as phase equilibria modeling (PERPLE_X) showed that the rocks coexisted with granite melts and saline aqueous carbonic fluids ( = 0.74–0.58) at the peak of metamorphism at 800–850°C and 10–11 kbar. Partial melting of the rocks initiated their subisothermal exhumation to 7.5–8 kbar during diapirism of granitic magmas in the Neoarchean (2.65–2.62 Ga). This is reflected in the specific zoning of the garnet grains in terms of the grossular content. A change in the rheology of the rocks as a result of partial removal and crystallization of melt activated the shear zones during further exhumation to 6–5.5 kbar along a decompression–cooling P–T path at 95–100°/kbar, reflecting the slower uplift of the rocks in the middle crust. This process was resumed due to thermal effects and interaction of the rocks with aqueous fluids ( > 0.85) in the Paleoproterozoic (~2.01 Ga). Such a scenario of metamorphic evolution implies that the Limpopo granulite complex in general and its Central Zone in particular resulted from the evolution of an ultrahot orogen, in which vertical tectonic movements associated with diapirism were coupled to horizontal tectonic processes caused by the convergence of continental blocks. [ABSTRACT FROM AUTHOR]

Xenoliths in kimberlites are the most promising material for studying the composition and structure of the lower levels of the continental crust. This study is aimed at the estimation of P–T parameters and fluid regime of metamorphism for garnet–biotite–feldspar and orthopyroxene–garnet–biotite–feldspar rocks found as xenoliths in kimberlites of the Yubileynaya and Sytykanskaya pipes, Yakutian kimberlite province. The seven studied samples show inverse dependences of relative contents of garnet and orthopyroxene, orthopyroxene and biotite, garnet and plagioclase, plagioclase and potassium feldspar. This indicates a consistent series of transformations of the assemblage garnet + plagioclase + orthopyroxene ± quartz to the assemblage garnet + biotite + potassium feldspar. In this process, the replacement of plagioclase by potassium feldspar was the leading reaction. Evidence of this reaction is specific reaction textures in the rocks, negative correlations between the contents of the minerals, and petrochemical characteristics of the rocks. Modeling of the mineral assemblages of the xenoliths using the pseudosection approach (PERPLE_X) revealed two groups of rocks corresponding to different depth levels of the Siberian cratonic crust. For rocks without orthopyroxene or with this mineral as single relics, the pressure was estimated at 9.5–10 kbar, and it is 6–7 kbar for orthopyroxene-bearing samples. The xenolith rocks have close metamorphic peak temperatures of 750–800°C. They experienced 200–250°C cooling and 3–4 kbar decompression, regardless of the level of the crust at which they had initially occurred. This indicates that the metamorphic evolution of the rocks during their exhumation was probably associated with collisional processes during the amalgamation of individual terrains of the Siberian craton. Xenoliths enriched in K-feldspar might have been products of metamorphic reactions with participation of aqueous–(carbonic)–salt fluids, which were sourced from basaltic magmas in the lower crust. The most strongly metasomatized rocks were located closest to the place of accumulation of crystallizing magmas. [ABSTRACT FROM AUTHOR]

To study the formation of S-type ultra-potassic granitic melts under the conditions of the continental crust, we conducted experiments on partial melting of plagioclase-free garnet–two-mica metapelite (with accessory apatite and ilmenite) at 6, 10, and 15 kbar and 700–900°C. Rock melting is driven by a series of peritectic melting reactios as a function of pressure and starts at temperatures of 750–800, ~800, and ~850°C at 6, 10, and 15 kbar, respectively, reflecting the positive dP/dT slope of the solidus. These P–T-conditions of melting, compositions of mineral assemblages and granitic melts formed during melting of plagioclase-free metapelites are similar to those formed during melting of plagioclase-bearing two-mica assemblages. The high K2O content (7–8 wt %) at a K2O/Na2O ratio of >8–10 is the main feature of the composition of the melt, which forms upon melting of plagioclase-free metapelite. These features are similar to the compositions of S-type ultra-potassic rhyolites, which form in extension tectonic settings. [ABSTRACT FROM AUTHOR]

One of the mechanisms explaining relationships between CO2-rich fluids and granitoid magmas at high-temperature crustal metamorphism is the melting of protoliths that originally contained carbonate minerals. In order to study the coupled processes of dehydration/decarbonation and melting, experiments were conducted with carbonate–biotite gneiss from an Archean greenstone belt at pressures of 6, 10, and 15 kbar in the temperature range of 800–950°C, and phase relations in this rock were modeled using the pseudo-section method. The experiments and modeling revealed a subvertical positive dP/dT slope of the solidus of the rock. In comparison to the calculated solidus temperatures, the experiments showed higher melting temperatures (~800°C at 6 kbar and ~850°C at 10 and 15 kbar). The products of the experiments at pressures of 6 and 10 kbar and temperatures >850°C were found out to contain assemblages of clinopyroxene, orthopyroxene, and ilmenite. The products of the experiments at 15 kbar did not contain either orthopyroxene or ilmenite, but calcium garnet and rutile were stable. The first portions of the near-solidus melts at 6 and 10 kbar were poor in SiO2 (44–50 wt %) and were formed because carbonate phases were involved in the melting reactions. With a temperature increase, the melts acquired a granite composition that was close to the composition of melts formed during the melting of the carbonate-free plagioclase + biotite + quartz assemblage. An aqueous–carbonic fluid containing Ca–Mg–Fe carbonate components coexisted with the melts. The phase assemblages and compositions of the granite melts obtained in the experiments are consistent with the modeling results. Comparison of the experimental results with published data on the partial melting of the carbonate-free plagioclase + biotite + quartz assemblages led us to the preliminary conclusion that Ca–Mg–Fe carbonates are able to decrease the melting temperature. The experiments have demonstrated that granite magmas can be derived together with aqueous–carbonic fluids from a carbonate-bearing protolith during high-grade metamorphism in the middle and lower crust. The occurrence of clinopyroxene or two-pyroxene assemblages in granitoids can be considered as a mineralogical indicator of this process. [ABSTRACT FROM AUTHOR]

The results of experimental study of crystallization of chromium-bearing Ba-titanates (redlegeite, lindsleyite, and hawthorneite) in the chromite–rutile/ilmenite system in the presence of the H2O–CO2–BaCO3 fluid at pressures of 1.8, 3.5, and 5.0 GPa and temperatures of 1000–1200°C, simulating the formation of these phases through metasomatism of peridotite of the upper mantle, are reported. Experiments showed the formation of Ba–Cr-titanates in the entire pressure range studied and the possibility of joint crystallization of titanates. However, redledgeite is formed only in the Fe-poor chromite–rutile–H2O–CO2–BaCO3 system, whereas minerals of the magnetoplumbite group are preferably crystallized in the system with ilmenite. Lindsleyite was not detected at a pressure of 1.8 GPa. A positive dependence of the Cr content in titanates on pressure was shown. The Raman spectra of redlegeite, lindsleyite, and hawthorneite are presented. [ABSTRACT FROM AUTHOR]

Gneiss domes cored by migmatites and granites represent the principal role of anatectic melts during the exhumation of high-grade metamorphic complexes. This study explores the exhumation history of a metapelitic granulite within the Ha-Tshanzi structure from the Central Zone of the Neoarchean–Paleoproterozoic Limpopo high-grade Complex, South Africa. Quartzofeldspathic garnet-bearing coarse-grained leucosomes in the rock alternate with attenuated shear bands consisting of biotite, cordierite, sillimanite and quartz that prominently modified the earlier garnet porphyroblasts. Cores of garnet porphyroblasts contain various polyphase inclusions that are interpreted as crystallized inclusions of melt. The phase equilibria modeling and regular zoning of garnet with respect to major (Mg, Fe, Ca) and some trace (P, Cr, Sc) elements reveals that a garnet + biotite + plagioclase + quartz ± sillimanite assemblage in the rock coexisted with the melt during the sub-isothermal (810–830 °C) exhumation from pressure of 10·0–10·2 kbar to 7·5–7·0 kbar during the Neoarchean event (2·68–2·62 Ga). The exhumation mediated by anatectic melt supports interpretation of the Ha-Tshanzi structure as a diapir-related granite–gneiss dome. During upwelling of the dome, the melt segregated into leucosome, while growth of cordierite sequestered water from the melt, assisting its crystallization at the end of the sub-isothermal decompression stage. As the rheology of the rock changed, melt-dominated deformation was transformed to solid-dominated ductile shear deformation. In contrast to the earlier sub-isothermal decompression P – T path, the gentler slope of ~75 °C kbar–1 of the decompression-cooling path marks the exhumation from pressures of ~7 kbar to pressures of 5–4·5 kbar and cooling to 600–550 °C. Dating of zircon, monazite and rutile shows that the Neoarchean evolution of the metapelite was strongly overprinted by the Paleoproterozoic event at c. 2·01 Ga. The results of the study highlight the significance of domed structures related to granitic diapirs in the exhumation of the Central Zone of the Limpopo Complex. [ABSTRACT FROM AUTHOR]

Let be a finite group and let be a subgroup of . Let be the subgroup of generated by all subnormal subgroups of contained in , and let be the intersection of all subnormal subgroups of containing . Let . Then we say that is -subnormal in if avoids every composition factor of between and , i.e., . In this paper, we give applications of -subnormality to the theory of groups with given -critical subgroups. In particular, using this notion, we give new characterizations of finite solvable groups, metanilpotent groups, and groups with nilpotent derived subgroup . [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G , that is, $AP=PA$. Let $A_{sG}$ be the subgroup of A generated by all S -permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S -permutable subgroups of G containing A. We prove that if G is a soluble group, then S -permutability is a transitive relation in G if and only if the nilpotent residual $G^{\mathfrak {N}}$ of G avoids the pair $(A^{s G}, A_{sG})$ , that is, $G^{\mathfrak {N}}\cap A^{sG}= G^{\mathfrak {N}}\cap A_{sG}$ for every subnormal subgroup A of G. [ABSTRACT FROM AUTHOR]

The paper summarizes authors' earlier and newly obtained experimental data on the formation of phlogopite and chromium-bearing potassium titanates of the crichtonite, magnetoplumbite, and hollandite groups, which are indicator minerals that characterize various stages of modal metasomatism in the upper mantle. Phlogopite-forming reactions were studied in the garnet–orthopyroxene system in the presence of H2O–KCl fluid at 3 and 5 GPa and 900–1000°C as simulating phlogopite formation in garnet peridotites and pyroxenites. Experiments have demonstrated regular variations in Ca and Cr contents in the garnet and Al in the pyroxenes, as well as the composition of the newly formed phlogopite, depending on the concentration of the potassium component (KCl or K2CO3) of the fluid. Experiments on the formation of potassium titanates (yimengite, mathiasite, and priderite) in the chromite–rutile/ilmenite–K2CO3–H2O–CO2 system at 3.5 and 5 GPa and 1200°C have proved that these minerals can be formed when chromite reacts with potassic aqueous–carbonic fluid, but these reactions require additional sources of titanium. These sources may be rutile and/or ilmenite, which are usually also produced by modal metasomatism of peridotites. The experiments thus confirm that the formation of titanates marks the most advanced stages of metasomatism in mantle peridotites. The experiments have shown that the formation of potassium titanates follows the formation of phlogopite, whereas assemblages of the titanates with phlogopite are produced at higher concentrations of the potassium component of the fluid than phlogopite alone. The relationships between the titanates are also a function of the activity of the potassium component in the fluid and, probably, pressure. The relationships and trends reproduced in the experiments clearly illustrate features of mineral assemblages and variations in compositions of minerals from metasomatized peridotites in the lithospheric mantle. [ABSTRACT FROM AUTHOR]

Throughout this article, all groups are finite and G is a group. Let σ = { σ i | i ∈ I } be some partition of the set of all primes P. Then σ (G) = { σ i | σ i ∩ π (G) ≠ ∅ } ; σ + (G) = { σ i | G has a chief factor H/K such that σ (H / K) = { σ i } }. The group G is said to be: σ-primary if G is a σi-group for some i; σ-soluble if every chief factor of G is σ-primary. The symbol R σ (G) denotes the product of all normal σ-soluble subgroups of G. The chief factor H/K of G is said to be: σ-central (in G) if (H / K) ⋊ (G / C G (H / K)) is σ-primary; a σi-factor if H/K is a σi-group. We say that G is: σ-nilpotent if every chief factor of G is σ-central; generalized { σ i } -nilpotent if every chief σi-factor of G is σ-central. The symbol F { g σ i } (G) denotes the product of all normal generalized { σ i } -nilpotent subgroups of G. We call any function f of the form f : σ ∪ { ∅ } → { formations of groups } , where f (∅) ≠ ∅ , a generalized formation σ-function and we put B L F σ (f) = (G | G / R σ (G) ∈ f (∅) and G / F { g σ i } (G) ∈ f (σ i) for all σ i ∈ σ + (G)). If for some generalized formation σ-function f we have F = B L F σ (f) , then we say that the class F is Baer-σ-local and f is a generalized σ-local definitionof F. In this article, we describe basic properties, examples, and some applications of Baer-σ-local formations. In particular, we prove that the set of all Baer-σ-local formations containing all nilpotent groups forms a subsemigroup of the semigroup of all formations G G (Theorem 1.13) and the set of all Baer-σ-local formations containing all σ-nilpotent groups is a right ideal in G G (Theorem 1.16). [ABSTRACT FROM AUTHOR]

Data on the carbon and oxygen isotopic composition of carbonates from the rocks of the Giyani greenstone belt, Kaapvaal craton, South Africa are presented. This belt is immediately adjacent to the Hout River Shear Zone, along which the Southern Marginal Zone (SMZ) of the Limpopo granulite complex is overthrusted onto the Kaapvaal craton. The δ13C values are –2.1 and –2.2‰ for carbonates from two metapelite samples, –5.1‰ for a meta ultrabasite sample, and –7.1 and –7.7‰ for two metabasite samples, respectively. The δ18О values are 17.1 and 17.5‰ for carbonates from the metapelites, 14.3‰ for carbonates from the metaultrabasite, and 12.9 and 13.0‰ for carbonates from the metabasites. The comparison of the data obtained to the published data on the isotopic composition of carbonates from the rocks of the different greenstone belts indicates both hydrothermal (in the metabasites and the metaultrabasite) and sedimentary (in the metapelites) origins of the carbonate material in the rocks studied. Based on the comparison of these data to δ13C of magnesite that formed in ultrabasic granulites during the interaction of the Limpopo complex with the craton, the graphite and fluid inclusions in leucocratic garnet-bearing granitoids that transported fluids, and the carbon isotopic composition of graphite from metapelites of the SMZ, it is concluded that the carbon source was located outside the SMZ and represented a heterogeneous carbonate-bearing lithologies. It could be carbonate-bearing volcanogenic–sedimentary sequences of the greenstone belts of the Kaapvaal craton, which underwent prograde metamorphism during the interaction with the SMZ granulites. [ABSTRACT FROM AUTHOR]

Let σ = { σ i | i ∈ I } be some partition of the set of all primes. A subgroup A of a finite group G is said to be: (i) σ-subnormal in G if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G such that either A i − 1 ⊴ A i or A i / (A i − 1) A i is a σ j -group, j = j (i) , for all i = 1 , ... , n ; (ii) modular in G if the following conditions are held: (1) 〈 X , A ∩ Z 〉 = 〈 X , A 〉 ∩ Z for all X ≤ G , Z ≤ G such that X ≤ Z , and (2) 〈 A , Y ∩ Z 〉 = 〈 A , Y 〉 ∩ Z for all Y ≤ G , Z ≤ G such that A ≤ Z ; (iii) σ-quasinormal in G if A is σ -subnormal and modular in G. We obtain a description of finite groups in which σ -quasinormality (respectively, modularity) is a transitive relation. Some known results are extended. [ABSTRACT FROM AUTHOR]

The structures of two K-richterite crystals with the crystal chemical formulas (K0.44Na0.320.24)Σ=1(Ca1.18Na0.82)Σ=2Mg5Si8O22OH2 and (K0.83Na0.020.15)Σ=1(Ca1.11Na0.89)Σ=2Mg5Si8O22OH2 synthesized at a pressure of 3 GPa and a temperature of 1000°С in the MgSiO3 + CaMgSi2O6 + K2CO3 + Na2CO3 + CO2 + H2O system were studied by single-crystal X-ray analysis and Raman spectroscopy. The monoclinic cell parameters were obtained: a = 10.0256(5) and 9.9748(11) Å, b = 17.9874(7) and 17.9879(16) Å, c = 5.2687(3) and 5.2746(6) Å, Ve.c. = 916.17(18) and 918.52(8) Å3, β = 104.520(12)° and 104.821(5)°, sp. gr. С2/m (12), and Z = 2. The sites M(1), M(2), and M(3) are inhabited by Mg2+, while site M(4) is occupied simultaneously by Ca2+ and Na+. The Na+ cations not included in M(4) are located in position A, which also accommodates K+ cations. Raman spectroscopy made it possible to reveal vacancies in position A in both samples. The structure corresponds to the "ideal" structure of richterite group minerals. The unit cell volume of the measured crystals is directly proportional to the K content in position A. Based on the generalization of new and published data, an equation of the dependence of Vu.c. for amphiboles of the richterite Na(NaCa)Mg5Si8O22(OH)2–K-richterite K(NaCa)Mg5Si8O22(OH)2 series with a low tremolite component on the K content in position A is proposed. [ABSTRACT FROM AUTHOR]

Let G be a finite group and ℒ(G) the subgroup lattice of G. A subgroup M of G is called: (i) modular in G, if M is a modular element (in the sense of Kurosh) of the lattice ℒ(G); (ii) submodular in G if G has a chain of subgroups M = M0 ≤ M0 ≤⋯ ≤ Mt = G, where Mi is modular in Mi+1 for all i = 0, 1,…,t − 1. If A is a subgroup of G, then we denote by AmG the subgroup of A, generated by all of its subgroups that are modular in G.We say that a subgroup A is N-modular in G (N ≤ G), if for some modular subgroup T of G, containing A, N avoids the pair (T,AmG), i.e. N ∩ T = N ∩ AmG.We prove that if G is a soluble finite group and each of its submodular subgroups is N-modular in G, where N is the nilpotent residual of G, then the lattice ℒ(G) is modular. [ABSTRACT FROM AUTHOR]

Throughout this paper, all groups are finite and G always denotes a finite group; L(G) is the lattice of all subgroups of G; σ = {σι | i ∈ I} is some partition of the set of all primes. A subgroup A of G is said to be: (i) o-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤...≤ An = G such that either Ai-1... Ai or Ai/(Ai-1)A; is o-primary for all i = 1,..., n; (ii) o-permutable in G if G is σ-full, that is, G has a Hall σi-subgroup for every σi ∈ σ(G) and A permutes with all such Hall subgroups of G. In this review we discuss some new results on finite groups with given systems σ-subnormal and σ-permutable subgroups. In particular, we discuss groups in which σ-permutability is a transitive relation in the group. [ABSTRACT FROM AUTHOR]

The salt components of aqueous and aqueous-carbonic fluids are very important agents of metasomatism and partial melting of crustal and mantle rocks. The paper presents examples and synthesized data on mineral associations in granulite- and amphibolite-facies rocks of various composition in the middle and lower crust and in upper-mantle eclogites and peridotites that provide evidence of reactions involving salt components of fluids. These data are analyzed together with results of model experiments that reproduce some of these associations and make it possible to more accurately determine their crystallization parameters. [ABSTRACT FROM AUTHOR]

In this paper, we examine the application of the well-known hysteresis switching algorithm of adaptive control in the case of a continuously parameterised set of candidate controllers. Subject to existence of a feasible controller in the set, we prove that if the performance functions used at each time by the hysteresis algorithm are selected to be uniformly positive definite in the controller parameters, then one may safely set the hysteresis constant to zero without inducing chattering instability, and ensuring convergence to a performance-optimising controller. Expanding preliminary version of our work that we presented at the 2011 American Control Conference, the present paper adds new numerical simulation examples to demonstrate the application and benefit of the theory. [ABSTRACT FROM AUTHOR]

Throughout this paper, all groups are finite. Let be some partition of the set of all primes. If n is an integer, the symbol denotes the set ; and. We call any function f of the form a formation σ-function, and we put If for some formation σ-function f we have , then we say that the class is σ-local and f is a σ-local definition of. We suppose that every formation is 0-multiply σ-local; for n > 0, we say that the formation is n-multiply σ-local provided either is the class of all identity groups or where is multiply σ-local for all. In this paper, we describe some properties and examples of n-multiply σ-local formations. In particular, we prove that the Gaschütz product of any two n-multiply σ-local formations is also n-multiply σ-local. We also consider one application of such formations in the theory of finite factorizable groups. [ABSTRACT FROM AUTHOR]

Let σ = { σ i ∣ i ∈ I } \sigma=\{\sigma_{i}\mid i\in I\} be some partition of the set of all primes and 퐺 a finite group. Then 퐺 is said to be 휎-full if 퐺 has a Hall σ i \sigma_{i} -subgroup for all i ∈ I i\in I and 휎-primary if 퐺 is a σ i \sigma_{i} -group for some 푖. In addition, 퐺 is 휎-soluble if every chief factor of 퐺 is 휎-primary and 휎-nilpotent if 퐺 is a direct product of 휎-primary groups. We write G N σ G^{\mathfrak{N}_{\sigma}} for the 휎-nilpotent residual of 퐺, which is the intersection of all normal subgroups 푁 of 퐺 with 휎-nilpotent G / N G/N . A subgroup 퐴 of 퐺 is said to be 휎-permutable in 퐺 provided 퐺 is 휎-full and 퐴 permutes with all Hall σ i \sigma_{i} -subgroups 퐻 of 퐺 (that is, A ⁢ H = H ⁢ A AH=HA ) for all 푖. And 퐴 is 휎-subnormal in 퐺 if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G A=A_{0}\leq A_{1}\leq\cdots\leq A_{n}=G such that either A i − 1 ⁢ ⊴ ⁢ A i A_{i-1}\trianglelefteq A_{i} or A i / ( A i − 1 ) A i A_{i}/(A_{i-1})_{A_{i}} is 휎-primary for all i = 1 , … , n i=1,\ldots,n . We prove that if 퐺 is a 휎-soluble group, then 휎-permutability is a transitive relation in 퐺 if and only if G N σ ∩ A G = G N σ ∩ A G G^{\mathfrak{N}_{\sigma}}\cap A^{G}=G^{\mathfrak{N}_{\sigma}}\cap A_{G} for every 휎-subnormal subgroup 퐴 of 퐺. [ABSTRACT FROM AUTHOR]

Let be a finite group and let be the lattice of all subnormal subgroups of . Let and be subgroups of and let be a sublattice in ; i.e., , for all . Then: is the -closure of in ; i.e., the intersection of all subgroups in which includes and is the -core of in , i.e., the subgroup in generated by all subgroups of belonging to . A subgroup is an - -subgroup in if either or and avoids each composition factor of between and ; i.e., . Using these notions, we give some new characterizations of soluble and supersoluble subgroups and generalize a few available results. [ABSTRACT FROM AUTHOR]

Abstract: Data on the carbon isotope composition of graphite and CO2 from inclusions in quartz of granitoids of the Southern Marginal Zone (SMZ) of the Limpopo granulite belt, South Africa, are presented. The average values of δ13C for graphite and СО2 from fluid inclusions in quartz are found within the range that is typical of deep (mantle) carbon sources. The data are evidence of an external source of granitoid magmas. They are likely to be hybrid rocks that appeared during assimilation of host metapelites by trondhjemite magmas, while graphite in trondhjemites and granodiorites is a product of high-temperature (~900-1000°С) evolution of the CO2-rich fluid phase of granitoid magmas when they intrude the rocks of the granulite complex and was formed during assimilation that led to reduction of CO2 when sulfide-bearing metapelite material was contaminated by the magma. [ABSTRACT FROM AUTHOR]

Abstract: The paper summarizes data on the petrology of granitoid intrusions in the Southern Marginal Zone (SMZ) of the Neoarchean Limpopo granulite complex, South Africa, and discusses the thermal and fluid effects of these intrusions on the granulites. The intrusions were emplaced in SMZ at 2680-2640 Ma, when the granulite complex was overthrust on the rocks of the adjacent Kaapvaal Craton. The mineral assemblages of the granitoids reflect temperatures above 900°C for the magmas that crystallized under pressures of 6-9 kbar. The granitoid magmas assimilated host rocks and were thereby enriched in MgO, FeO, and Al2O3, which was favorable for the crystallization of garnet, spinel, sillimanite, and corundum in the granitoids. Fluid inclusions in the granitoids and estimates of the fluid composition based on mineral equilibria indicate that the CO2/H2O ratio of the fluids broadly varied. Along with H2O-CO2 fluid, the magmas carried H2O-salt fluids, which penetrated the host rocks and triggered various metasomatic reactions in them. The thermal effects of the intrusions on the host granulites resulted in the development of partial melting zones that host orthopyroxene with >7 wt % Al2O3. Depending on the fluid regime and temperature, the orthopyroxene is found in equilibrium with either garnet and potassic feldspar or with biotite. The ages of the partial melting zones are comparable with those of the intrusions. [ABSTRACT FROM AUTHOR]

Summary: The possibility of mismatch between prior uncertainty modeling assumptions and reality is a problem both for robust control and for model‐based adaptive control algorithms that aim to use real‐time data to adaptively identify and correct such problems. Mismatches between prior model assumptions may fool adaptive algorithms intended to improve robustness into persistently preferring destabilizing controllers over stabilizing ones even when the instability is patently obvious to the eyes of the most casual observer. To eliminate all possibility of being thusly fooled, the assumption‐free unfalsified control concept was introduced in the early 1990s and has since been developed to the point where it now provides a unifying overarching theoretical framework for understanding the relationships, benefits, and weakness of various adaptive control methods. We review how the theory allows one to parsimoniously sift the individual elements of accumulating evidence and experimental data to determine precisely which of the elements falsify a given performance level and very briefly discuss recent research on cost‐detectable fading‐memory cost functions for time‐varying plants. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Evidence is presented for the localized melting of cordierite–orthopyroxene–biotite metapelites within the Petronella shear zone in the Southern Marginal Zone of the Limpopo Belt of Southern Africa. This process is expressed by the formation of centimetre-scale, K-feldspar-rich, garnet–orthopyroxene-bearing leucosome patches. Structural data indicate that the leucosomes formed after regional peak metamorphic and deformational events, but were syn- to late tectonic with respect to the shear fabric of the host metapelites. Phase equilibria modelling using the PERPLE_X software shows that the leucosomes were produced via fluid-deficient partial melting of the metapelites at temperatures above 900°C and pressures of 6·0–6·5 kbar. Melting reactions involving mainly biotite, plagioclase, quartz and accessory phases (including sulfides) produced peritectic Cr-enriched orthopyroxene, as well as Sc, Y, Cr, P-enriched garnet along with the potassium-rich melt. The melt was segregated around the peritectic minerals into the leucosomes. During near-isobaric cooling, the segregated melts crystallized to produce abundant K-feldspar and Sc, Y, Cr and P-poorer garnet inside the leucosome patches. Partial melt loss from the patches on cooling assisted with the preservation of anhydrous assemblages inside the patches and with re-hydration of the surrounding melanosome, with extensive formation of biotite + quartz±sillimanite assemblages after orthopyroxene and cordierite, as well as late Sc, Y, Cr, and P-free garnet (+sillimanite + quartz) at temperatures down to < 600°C. Field and P–T data suggest a link between the localized melting of metapelites and trondhjemite intrusions that were syn- to late-tectonically injected into the Petronella shear-zone at about 2·67 Ga after the peak metamorphic and deformational event in the Southern Marginal Zone. This link is proven by U–Pb ages for monazites (a weighted average 207Pb/235U age of 2666 ± 4 Ma) from the leucosome patches. The trondhjemites provided additional heat and, probably, also a small amount of fluids, for the leucosome formation. [ABSTRACT FROM AUTHOR]

The paper presents data on the formation of K–Na richterite in the enstatite + diopside association with K2CO3–Na2CO3–CO2–H2O fluid at 3 GPa and 1000°C as a model for the formation of this mineral in peridotites of the upper mantle. Richterite formation depends on the (H2O + CO2)/(K2CO3 + Na2CO3) and K2CO3/Na2CO3 ratios in the starting material. A high concentration of alkaline components in the fluid leads to the decomposition of clinopyroxene, the formation of olivine, and a change in the component composition of the pyroxene and amphibole. Fluids with a high potassium concentration are favorable for the formation of K-richterite similar in composition to that formed in metasomatized peridotites of the upper mantle. In some cases, such a fluid leads to the decomposition of amphibole and stabilization of alkaline melt. An increase in the activity of the sodium component results in richterite similar in composition to richterite from lamproites. The clarified relations can be used to assess the activities of fluid components and conditions for the formation of K-richterite. To update the data bank of the Raman spectra of minerals, the largest and most homogeneous amphibole crystals of different compositions were studied. [ABSTRACT FROM AUTHOR]

To constrain effects of chloride-bearing H2O–CO2 fluids on complex natural assemblages during high-grade metamorphism and anatexis, we report the results of experiments on the interaction of biotite–hornblende tonalitic gneiss from the Sand River Formation (Limpopo Complex, South Africa) with H2O–CO2, H2O–CO2–KCl, H2O–CO2–NaCl, and H2O–CO2–(K, Na)Cl fluids at 550 MPa, 750 and 800°C, and varying chloride/(H2O + CO2) ratios with molar CO2/(CO2 + H2O) = 0·5. Heating of solid cylinders of gneiss at both temperatures in the absence of a free fluid phase produced no changes in the gneiss phase assemblage. The equimolar H2O–CO2 fluid at 750°С also did not significantly influence the phase assemblage. Addition of KCl to the fluid at 750°С resulted in formation of the clinopyroxene + K-feldspar (+ ilmenite/titanite) assemblage after biotite, hornblende and plagioclase. Orthopyroxene accompanied by amphibole appeared only at 800°C as a result of biotite breakdown in the presence of H2O–CO2 and low-salinity H2O–CO2–KCl fluids. Increase in the KCl content in the fluid at 800°С resulted in the production of a clinopyroxene-bearing assemblage. Increase of the NaCl content stabilized amphibole in an assemblage with either orthopyroxene (at low NaCl concentrations) or clinopyroxene. Nevertheless, clinopyroxene (+ albite) is stable only at high salt concentrations. Comparison of the experimental results with the results of thermodynamic modeling using the Gibbs free energy minimization method (PERPLE_X software) showed that mineral reactions and assemblages in the run products were governed by the activities of alkali components imposed by KCl and NaCl in the H2O–CO2 fluids, and decrease of the water activity served as an additional factor stabilizing anhydrous assemblages. No melts formed at 750°C in the presence of the H2O–CO2–KCl fluids. These fluids provoked melting only at 800°C with formation of rhyolitic melts. With increasing KCl content of the fluid, the melt composition changed to potassic rhyolitic with Al2O3 < 13·5 wt %, CaO < 2 wt %, K2O + Na2O > 7 wt %, FeO/(FeO + MgO) > 0·8, K2O/Na2O > 1, and moderate enrichment in Cl (0·2–0·6 wt %). Increasing NaCl content caused melting at 750°С and shifted the melt composition towards trachytic and trachyandesitic compositions at both 750 and 800°С. The experiments support a model for the formation of ferroan A-type granite–syenite complexes via crustal melting in the presence of H2O–CO2–salt fluids in extensional tectonic settings. They demonstrate a possible link between A-type granitoids and mid-crustal dehydration zones in amphibolite- to granulite-facies terrains and allow a new interpretation of mineral assemblages within these zones in terms of variations in fluid salinity. [ABSTRACT FROM PUBLISHER]

A tonalitic vein from a representative traverse section from the Paleoarchean Sand River biotite-amphibole gneiss consists of two domains - orthopyroxene-bearing and orthopyroxene-absent domains - which grade from center to the margin, along its length. The orthopyroxene-bearing domain is heavily altered with respect to the surrounding orthopyroxene-absent domain, which preserves a prominent magmatic texture typical of crystallized melt. Contrasting textures were observed in terms of monazite occurrence in fluorapatite in the two domains. While monazite occurs as inclusions in fluorapatite in the orthopyroxene-bearing domain, it prominently occurs as irregular rims along fluorapatite margins in the orthopyroxene-absent domain. The textural relation of monazite, occurring as both inclusions and irregular rims in the same fluorapatite grains, and different mineral chemical characteristics of both types of monazite, supports the operation of melting accompanying fluid-induced dehydration in the Sand River orthogneiss. T-aH2O pseudosection modeling reproduced the mineral assemblages of the domain representing melt petition formed during dehydration. The Paleoproterozoic timing of the events is characterized by a 40Ar/39Ar amphibole age of 2037 ± 10 Ma from the dehydrated gneiss. [ABSTRACT FROM AUTHOR]

The paper presents experimental data on the interaction of amphibole with NaCl–H2O and (K, Na)Cl–H2O solutions at varying salt content. When interacting with H2O–NaCl fluid, amphibole remains the predominant mineral in all experiments, and the newly formed minerals are Na-phlogopite, plagioclase, and nepheline/sodalite. At > 0.6, the amphibole melts. When amphibole interacts with H2O–NaCl–KCl fluid at < 0.40 and ХKCl/(ХKCl + ХNaCl) in the fluid, defined as ХNaCl = 0.506 – 0.84ХKCl, the amphibole is replaced by the association of nepheline with sodic plagioclase, sodalite, and biotite. At ХKCl/(ХKCl + ХNaCl) > 0.3, nepheline, sodalite, and plagioclase become unstable, K-feldspar is formed, and biotite, clinopyroxene, and amphibole remain stable. At ХKCl/(ХKCl + ХNaCl) > 0.5, the association Cpx + Bt + Kfs + Grt (grossular–andradite) is stable. Thus, grossular–andradite garnet is an indicator of a high potassium activity in the fluid, whereas nepheline testifies that the sodium activity was high. Na → K exchange is typical of the amphibole and biotite, and Ca → Na exchange occurs in the clinopyroxene, and all of these minerals (but neither nepheline nor garnet) remain generally stable within a wide range of the K/Na ratio in the fluid. Clinopyroxene in the experiments spans Ca–Fe–Mg compositions with a varying, sometimes high, Al content, and the amphiboles belong to the pargasite–hastingsite series. With an increase in ( > 0.57), i.e., a decrease in the gross salinity of the fluid, melts are generated, and their composition varies from trachyte to phonolite. An increase in the ХKCl/(ХKCl + ХNaCl) ratio in the fluids leads to a decrease in alumina content of the melts. An increase in the total salinity of the fluid leads to an increase in the content of potassium in the melt and a decrease in the content of chlorine in it. The experiments have shown that interaction between amphibole and fluids containing high NaCl and KCl concentrations results in mineral associations typically produced by alkaline metasomatism of amphibole-bearing rocks and concomitant HCl enrichment in the fluid phase. The substitution of highly saline fluids for highly acidic ones leads to the leaching of Ca, Mg, Fe from the metamorphic rocks, and the transport and redeposition of these components. It follows that significant removal of FeO, MgO, CaO from rocks is sometimes a consequence of the interaction of the host rocks with saline aqueous solutions. [ABSTRACT FROM AUTHOR]

Abstract: A revolution in control theory thought happened in the early 1970s when the dominant focus of research shifted from optimality to robustness in response to unexpected failures of optimal control theory to produce feedback control designs capable of tolerating normal differences between design models and reality. The robustness concept has since become such an integral part of present day control theory that it is difficult to imagine that time long ago when the concept lacked a clear mathematical representation and the tools of multivariable robustness analysis were yet to be identified. We shall revisit that time to examine the events that facilitated, and necessitated, this remarkable paradigm shift. Next, looking to the future, we will consider how failures of robust control designs to cope with incorrect uncertainty estimates are beginning to spur control theorists to consider data-driven problem formulations for estimation and control that tacitly question the roles of basic concepts like true model and uncertainty bounds, stochastic noise models and even Bayesian probability. We will discuss how and why Karl Popper’s scientific logic of unfalsification seems to be emerging as a central concept in these data-driven problem formulations, and how the unfalsification concept might again shift the focus of mathematical research in the areas of estimation and control. [Copyright &y& Elsevier]

Recent data suggest an active role for chloride-bearing alkali carbonatitic melts in the formation and evolution of kimberlites, whereas experiments indicate that chlorides could be responsible for liquid immiscibility during melting of carbonated mantle rocks. This study considers melting trends in kimberlite-related chloride–carbonate–silicate systems at a pressure of 5·5 GPa. The direction of these trends largely depends on the chlorine concentration in the system. All trends start with a Cl-rich carbonatitic liquid coexisting with crystalline phases. Melting of a peridotite–carbonate–chloride system containing 4·4 wt % Cl results in a gradual transition from a carbonatitic melt (5–7 wt % SiO2, MgO/CaO = 0·5–0·6, ∼2 wt % Cl) at 1000–1100°C through a Cl-rich carbonate–silicate melt (12–15 wt % SiO2, MgO/CaO = 0·6, and up to 14 wt % Cl) at 1360–1400°C towards a Cl-bearing ultrabasic carbonate–silicate melt (UCSM), i.e. kimberlite-like (26–29 wt % SiO2, MgO/CaO = 1·5–2·8, and 6–4 wt % Cl), at 1500–1600°C. This trend results from the specific behavior of chlorides, which are found to be stable crystalline phases up to about 300°C above the solidus. In addition, the trend touches a miscibility gap at about 1450°C, although immiscibility does not significantly influence the melt evolution. In contrast, the melting trend of a peridotite system with about 1·3 wt % Cl does not intersect the miscibility gap and proceeds from a carbonatitic melt (∼5 wt % SiO2, MgO/CaO = 0·4–0·5, ∼2 wt % Cl) at 1100–1150°C toward the UCSM (∼25 wt % SiO2, MgO/CaO = 1·7–1·8, and 0·6–1·0 wt % Cl) at 1500–1600°C. Chloride–carbonate–silicate systems containing 17 wt % Cl show an abrupt transition from the chloride–carbonate liquid toward the UCSM because of the immiscibility gap between carbonate–silicate and chloride–carbonate melts. Melting relations in all studied chloride–carbonate–silicate systems are exclusively regulated by peritectic reactions between the silicate phases and carbonate constituents (mostly CaCO3) of the melts. Chlorides decrease the solidus temperatures of carbonated peridotites, but do not reduce the temperature interval for the transition from carbonate- to silicate-dominated melts. The experimental results suggest that Cl-rich carbonatite liquids preserved in diamonds and the Cl-rich kimberlites of the Udachnaya-East kimberlite pipe (Yakutia) could represent a linked system of chloride-rich liquids that evolved over a wide temperature interval during incipient melting and evolution of kimberlite magma from a carbonated mantle source containing 3–4 wt % Cl. [ABSTRACT FROM AUTHOR]

It is proved that the lattice of all τ -closed totally saturated formations of finite groups is $ \mathfrak{G} $-separable. [ABSTRACT FROM AUTHOR]

It is proved that the lattice of totally saturated formations of finite groups is distributive. Thus, we give an affirmative answer to the problem proposed by Shemetkov, Skiba and Guo. [ABSTRACT FROM AUTHOR]

Abstract: The experiments in the model system CaMgSi2O6–(Na2CO3, CaCO3)–KCl are performed at 5 GPa and 1400–1600 °C in order to study the phase relations, including liquid immiscibility, in the chloride–carbonate–silicate systems with application to alkali and chlorine-rich liquids preserved in kimberlitic diamonds. Experiments in the boundary joins of the system demonstrated that both the carbonate–silicate and chloride–carbonate melts are homogeneous; while high-temperature (above 1800 °C) liquid immiscibility is assumed for the chloride–silicate join of the above system. Addition of silicate component into the chloride–carbonate melts and chloride component into the carbonate–silicate melts results in splitting of the homogeneous liquids into the immiscible chloride–carbonate brine and carbonate–silicate melt. Carbonate–silicate and chloride–carbonate branches of the miscibility gap converge within the carbonate-rich region of the system. Regular temperature evolution of the shape and size of the miscibility gap is deduced. With decreasing temperature, the convergence point moves toward more Si-rich compositions, expanding fields of homogeneous chloride–carbonate silica-saturated melts. This effect is governed by the precipitation of the silicate phases even from silica-bearing chloride–carbonate melts. In addition, experiments revealed regular evolution of both Cl-bearing carbonate–silicate melt and Si-bearing chloride–carbonate brine toward the low-temperature chlorine–bearing carbonatitic liquid with decreasing temperature. These trends are similar to the evolution of the melt and brine inclusions in some diamonds from Botswana, Brazil, Canada, and Yakutia, indicating their growth during cooling. The model for interaction of the chloride–carbonate brine with the mantle rocks is developed on the basis of the present experimental data. This model is applied to the chlorine-enriched kimberlites of the Udachnaya–East pipe. [Copyright &y& Elsevier]

We prove that the finite length condition for the lattice of τ-closed totally saturated subformations of a τ-closed totally saturated formation is equivalent to the finite length condition for this lattice, and it is also equivalent for the formation to be soluble and one-generated. [ABSTRACT FROM AUTHOR]

It is proved that the lattice of all τ-closed totally saturated formations of finite groups is algebraic. [ABSTRACT FROM AUTHOR]

We study τ-closed totally saturated formations of finite groups. [ABSTRACT FROM AUTHOR]

Abstract: A novel benzimidazole series of small-molecule thrombopoietin receptor agonists has been discovered. Herein, we discuss the preliminary exploration of structure–activity relationships within this chemotype. [Copyright &y& Elsevier]

The join CaMgSi2O6-KAlSi3O8 has been studied at 6 GPa (890-1,500°C) and 3.5 GPa (1,000-1,100°C). K-rich melts in the join produce assemblages Cpx + Grt, Cpx + Opx, Cpx + San, and Cpx + Grt + San at 1,100-1,300°C. AtN<~70 mol%, sanidine is unstable on the solidus and appears at the liquidus, ifN>90 mol%. This explains a scarcity of San in mantle Cpx-rich assemblages and its association with high-K aluminosilicate melt inclusions in diamonds. In absence of San, KCpx is the only host for potassium. The K-jadeite content in KCpx systematically increases with decreasing temperature and reaches 10-12 mol% near the solidus. However, KCpx coexists with San atN>70 mol% and<1,300°C, being formed via reaction San + L=KCpx. The KJd content in KCpx is controlled by the equilibrium San=KJd + SiOthat displaces to the right with increasing pressure and decreasing both the temperature andThis equilibrium is considered to be responsible for the formation of San lamellae in natural UHP Cpx. In our experiments at 3.5 GPa, garnet is absent whereas the KJd and Ca-Eskola contents in Cpx are low, and the join CaMgSi2O6-KAlSi3O8 is close to binary (with the eutectic Cpx + San + L). Different topologies of the join at 6 and 3.5 GPa define a sequence of mineral crystallization from K-rich aluminosilicate melts during cooling and decompression: from KCpx + Grt without San atP>4 GPa to Cpx + San atP<4 GPa. Similar sequence of assemblages is observed in some eclogitic xenoliths from kimberlites and Grt-Cpx rocks of the Kokchetav Complex (Northern Kazakhstan). [ABSTRACT FROM AUTHOR]

Abstract: For block structured monotone or incrementally positive n-dimensional nonlinearities, the largest class of convolution operators (stability multipliers) that preserve positivity is derived. These multipliers can be used in conjunction with positivity and IQC stability criteria to evaluate stability and robustness of MIMO feedback systems. [Copyright &y& Elsevier]

Develops a method of designing the Venturi scrubber with adjustable throat section. Gas speed; Gas flow rate; Hydraulic resistance of a Venturi scrubber with built-in droplet trap; Liquid flow rate in a scrubber with adjustable throat.

It is proved that all proper totally local subformations of a non one-generated totally local formation $\backslash mathfrak\; F$ of finite groups are one-generated iff $\backslash ,\{\backslash mathfrak\; F\}$ coincides with a formation $\{\backslash mathfrak\; S\}\_\{\backslash pi\}$ of all soluble $\backslash pi$-groups, where $|\backslash pi|=2$. [ABSTRACT FROM AUTHOR]

The pseudo-binary system CaMgSi2O6-KAlSi2O6, modeling the potassium-bearing clinopyroxene (KCpx) solid solution, has been studied at 7 GPa and 1,100–1,650 °C. The KCpx is a liquidus phase of the system up to 60 mol% of KAlSi2O6. At higher content of KAlSi2O6 in the system, grossular-rich garnet becomes a liquidus phase. Above 75 mol% of KAlSi2O6 in the system, KCpx is unstable at the solidus as well, and garnet coexists with kalsilite, Si-wadeite and kyanite. No coexistence of KCpx with kyanite was observed. Above the solidus, KAlSi2O6 content of the KCpx coexisting with melt increases with decreasing temperature. Near the solidus of the system (about 1,250 °C) KCpx contains up to 5.6 wt% of K2O, i.e. about 22–26 mol% of KAlSi2O6. Such high concentration of potassium in KCpx is presumably the maximal content of KAlSi2O6 in the Fe-free clinopyroxene at 7 GPa. In addition to the major substitution MgM1CаМ2⇔AlМ1KМ2, the KCpx solid solution contains Ca-Eskola and only minor Ca-Tschermack components. Our experimental results indicate that the natural assemblage KCpx+grossular-rich garnet might be a product of crystallization of the ultra-potassic SiO2-rich alumino-silicate mantle melts (>200 km). [ABSTRACT FROM AUTHOR]

Presents information on a study which discussed compositional and structural data on an aluminum-rich ultrapotassic clinopyroxene (CPX) crystal. Geological setting and petrography of the studied rock sample; Experimental methods; Chemical composition;Origin of the ultra-potassic CPX inclusions in garnet.

Examines the total syntheses of the tumor cell growth inhibitory macrolides. Key features of the unified synthetic scheme; Assignment of absolute stereochemistries for (+)-zampanolide and (+)-dactylolide; Stereochemical disposition of the hydroxyl group.

In the period of the building of a communist society in our country, the problem of eliminating significant differences between town and countryside acquires profound and practical significance. [ABSTRACT FROM AUTHOR]