A. A. Tret’yakov

Late Paleozoic subductional and collisional igneous complexes in the Naryn segment of the Middle Tien Shan (Kyrgyzstan)

The Late Paleozoic Tien Shan fold belt was formed in the course of subduction of the crust underlying the past Turkestan ocean under the Kazakh continent and subsequent collision of the latter with the Alai and Tarim massifs. The onset of subduction is evidenced by development of flysch sequences and olistostromes in the accretionary complex of the South Tien Shan: in the terminal Visean‐Serpukhovian (~330‐325 Ma ago) in the west and in the second half of the Bashkirian Age (~315 Ma ago) in the east of the Kyrgyz Tien Shan [1]. 1 The onset of collision between the Kazakh continent and Tarim Massif is dated back to the terminal Late Carboniferous based on the initiation of a foredeep along the northern margin of the latter [6]. The mature collision stage began in the mid-Asselian, when the last sea basins disappeared in the Tien Shan and granitoids intruded in its southern segment [1, 9]. The position and age of the volcanic arc that was forming in the course of convergence between the Kazakh continent and the Tarim Massif remain unclear. Recently, this problem acquired particular significance, since the data available for the territory of China imply subduction under the Tarim Massif and an Early Carboniferous age of collision, i.e., substantially older as compared with that assumed for the Kyrgyz region [7]. 1 Ages are given after [13].

Stenian Granitoids of the West Kyrgyz Ridge (North Tien Shan): Position, Structure, and Age Determination

In the structure of west Kyrgyz Ridge (North Tien Shan), a great role is played by complexly dislocated Upper Precambrian-Cambrian terrigenous-carbonate and shale strata, as well as by granitoids that comprise several coupled WNW-striking synforms and antiforms, the largest of which is the Makbal antiform. Southeast of the core of this antiform, granitoids comprise the large Kara Dzhilga massif and several massifs that are of lesser size and have tectonic correlations with the hosting terrigenous-carbonate strata. In the Kara Dzhilga massif, the rocks of three penetration phases are distinguished; contacts between rocks are often of tectonic character. The early phase is presented by monzonite and monzodiorite; the main one, by large-porphyric biotitic granites; and the additional one, by aplitic granites and pegmatites. By the chemical composition, granites of Kara Dzhilga massif of the main phase correspond to subalkaline granites of high-potassium calc-alkali series. The age of their crystallization (zircon-based U-Pb method) is 1131 ± 4 Ma (Stenian). The formation of Stenian granitoids in the North Tien Shan may be related to development of Grenville fold belts, whose fragments were identified in the units of the Central Asian Belt. Tectonic correlations between these granitoids and hosting terrigenous-carbonate strata appeared as a result of immersion to significant depths and subsequent exhumation into the upper crustal horizons in the Early Ordovician.

Late Riphean age of the Karsakpai massif of alkaline syenites of South Ulutau (Central Kazakhstan)

The Ulutau massif is located in the western part of Central Kazakhstan and is one of largest Precambrian sialic massif of this region. Geologically, the southern part of the massif is composed of volcanogenic and volcanogenic–sedimentary rocks of acid and rare basic composition metamorphosed under the green schist metamorphism facies. The northern and eastern parts of the massif are composed mainly of amphibolite facies metamorphic rocks, represented by amphibolites, gneisses, and amphibole crystalline schists (Fig. 1a).

Age and sources of Precambrian zircon–rutile deposits in the Kokchetav sialic massif (northern Kazakhstan)

The U–Pb geochronological data on detrital zircons from placers confined to Neoproterozoic quartzite–schist sequences, which are widespread in the Kokchetav massif of northern Kazakhstan, are discussed. Detrital zircons (332 grains in total) originate from the ore occurrences in the central, northern, and western parts of the massif. The concordant ages of detrital zircons from all the examined occurrences largely correspond to intervals of 1017–1528, 1628–1946, and 2653–2739 Ma. The obtained data imply that material of quartzite–schist sequences of the Kokchetav massif was provided by Mesoproterozoic, Paleoproterozoic, and Neoarchean rock complexes. The lower age limit determined for these sequences is approximately 1.06 Ga. The dates obtained for detrital zircons are most consistent with events that took place in Laurentia. They correspond to the formation and breakup of the Columbia/Nuna supercontinent (approximately 1650–1580 and 1450–1380 Ma ago, respectively) and formation of the Rodinia supercontinent in the period of 1300–900 Ma ago.

Structure, age substantiation and tectonic setting of the Lower-Middle Ordovician volcanic-sedimentary and plutonic complexes of the western part of the Kyrgyz Range (Northern Tien Shan)

Among the Caledonides exposed in the western part of the Kyrgyz Range the Lower Ordovician volcanogenic-sedimentary, plutonic, and tuffaceous-terrigenous complexes were distinguished. Volcanogenic-sedimentary sequences are the Kentash Formation, composed of volcanic rocks, tuffs and subvolcanic bodies of dacitic, andesitic and basaltic composition, sandstones and tuffites with interlayers and lenses of limestone. On the basis of conodonts and U-Pb dating of zircon grains the age of this Formation is in the age interval between Late Tremadocian Stage and Early Darriwilian Stage. Differentiated volcanites are associated with ultramafic-gabbro massifs of the Kokkiya Complex of the Late Darriwilian age (U-Pb zirconology). Features of the chemical composition of rocks of the Kentash Formation and the Kokkiya Complex indicate that they formed in suprasubduction settings within the island arc with a thick heterogeneous basement. Tuffaceous-terrigenous deposits are presented by the olistostrome formation, and coarse-grained deposits of the Taldybulak and Kyzylkainar Formations. The formation of olistostrome formation is associated with the over-thrusting of Cambrian melanocratic complexes on terrigenous-carbonate and shale strata of the Upper Precambrian-Cambrian age. Deposits of the Taldybulak and Kyzylkainar Formations accumulated in the back-arc basin and on the island arc slope, made of rocks of the Kentash Formation.

New numerical methods and some applied aspects of the p-regularity theory

The theory of p-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu.G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the p-factor method for solving ODEs with a small parameter.

pth-order approximation of the solution set of nonlinear equations

Given a system of nonlinear equations, a formula is derived for the family of its approximate solutions of up to the pth order of smallness. A formula approximating an implicit function up to the third order of smallness is presented. Iterative methods converging with the pth order are constructed for solving systems of nonlinear equations. These results are extended to the degenerate case. Examples of applying the results are given.

The chelkar peridotite-gabbronorite pluton (Kokchetav massif, Northern Kazakhstan): Formation type and geochronology

1162 The occurrence of large plutons composed of ultra mafic, mafic, and alkaline rocks is a peculiar feature of many Precambrian sialic massifs in Kazakhstan. They are known in the Kokchetav, Shatsky, Ulutau, and Erementau–Niyaz massifs [1, 3, 5, 6, 8]. Their maxi mal number is characteristic of the Kokchetav massif of northern Kazakhstan, where they are united into the Zlatogorsk peridotite–pyroxenite–norite and Krasnyi Mai alkaline ultramafic complexes [5, 8].

P-Factor-Approach to Degenerate Optimization Problems

The paper describes and analyzes an application of the p-regularity theory to nonregular, (irregular, degenerate) nonlinear optimization problems. The p-regularity theory, also known as the factor-analysis of nonlinear mappings, has been developing successfully for the last twenty years. The p-factor-approach is based on the construction of a p-factor-operator, which allows us to describe and analyze nonlinear problems in the degenerate case.

New approach to optimality conditions for degenerate nonlinear programming problems

In the work, a new approach to constructing optimality conditions for degenerate smooth optimization problems with inequality constraints is proposed. The approach is based on the theory of p-regularity. A special case of degeneracy, when the first derivatives of some function-constraints are equal to zero up to some order, is considered. Optimality conditions for the general case of degeneracy with p = 2 are presented. Proposed constructions and optimality conditions are illustrated by an example. A general case of degeneracy is considered and optimality conditions for the case of p ≥ 2 are proposed.