Algebraic properties of Manin matrices II: q-analogues and integrable systems

We study a natural q-analogue of a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory, (called Manin Matrices in [5]) . These matrices we shall call q-Manin matrices(qMMs). They are defined, in the 2x2 case, by the relations M_21 M_12 = q M_12 M_21; M_22 M_12 = q M_12 M_22; [M_11;M_22] = 1/q M_21 M_12 - q M_12 M_21: They were already considered in the literature, especially in connection with the q-Mac Mahon master theorem [16], and the q-Sylvester identities [25]. The main aim of the present paper is to give a full list and detailed proofs of algebraic properties of qMMs known up to the moment and, in particular, to show that most of the basic theorems of linear algebras (e.g., Jacobi ratio theorems, Schhur complement, the Cayley-Hamilton theorem and so on and so forth) have a straightforward counterpart for q-Manin matrices. We also show how this classs of matrices ?ts within the theory of quasi-determninants of Gel'fand-Retakh and collaborators (see, e.g., [17]). In the last sections of the paper, we frame our definitions within the tensorial approach to non-commutative matrices of the Leningrad school, and we show how the notion of q-Manin matrix is related to theory of Quantum Integrable Systems.Comment: 62 pages, v.2 cosmetic changes, typos fixe

Algebraic properties of Manin matrices II: q-analogues and integrable systems

We study a natural q-analogue of a class of matrices with non-commutative entries, which were first considered by Yu.I. Manin in 1988 in relation with quantum group theory, (called Manin matrices in [5]). We call these q-analogues q-Manin matrices  . These matrices are defined, in the 2×22×2 case by the following relations among their matrix entries: M21M12=qM12M21, M22M12 = qM12M22 [M11,M22]=q-1M21M12-qM12M21 They were already considered in the literature, especially in connection with the q-MacMahon master theorem [10], and the q-Sylvester identities [22]. The main aim of the present paper is to give a full list and detailed proofs of the algebraic properties of q-Manin matrices known up to the moment and, in particular, to show that most of the basic theorems of linear algebras (e.g., Jacobi ratio theorems, Schur complement, the Cayley–Hamilton theorem and so on and so forth) have a straightforward counterpart for such a class of matrices. We also show how q-Manin matrices fit within the theory of quasideterminants of Gelfand–Retakh and collaborators (see, e.g., [11]). We frame our definitions within the tensorial approach to non-commutative matrices of the Leningrad school in the last sections. We finally discuss how the notion of q-Manin matrix is related to theory of Quantum Integrable Systems

SOS model partition function and the elliptic weight functions

We generalize a recent observation [arXiv:math/0610433] that the partition function of the 6-vertex model with domain-wall boundary conditions can be obtained by computing the projections of the product of the total currents in the quantum affine algebra $U_{q}(\hat{\mathfrak{sl}}_{2})$ in its current realization. A generalization is proved for the the elliptic current algebra [arXiv:q-alg/9703018,arXiv:q-alg/9601022]. The projections of the product of total currents are calculated explicitly and are represented as integral transforms of the product of the total currents. We prove that the kernel of this transform is proportional to the partition function of the SOS model with domain-wall boundary conditions.Comment: 21 pages, 5 figures, requires iopart packag

Morphofunctional reorganization of plantar aponeurosis in experimental modeling of fasciopathy by synthetic analogue of prostaglandin E1

Foundation. Chronic plantar fasciopathic pain syndrome is a pathology that significantly affects the quality of life of patients of all age categories. Insufficient knowledge of the etiological and pathogenetic factors in the development of fasciopathies explains the multiplicity, and sometimes inconsistency, of conservative and surgical treatment regimens. The choice of the optimal variant of therapeutic or surgical intervention may be associated with experimental modeling of fasciopathies and the study of the dynamics of the pathological process.The aim. To study the morphological changes in structures identical to the human plantar aponeurosis in experimental modeling of fasciopathy in animals.Research methods. The material for the study was fragments of the tendonaponeurotic complex of the foot of laboratory animals (control group: animals with the introduction of physiological sodium chloride solution (n = 12); main group: animals with the introduction of alprostadil (n = 12)). The methods of light microscopy (staining with alcian and toluidine blue, according to Van Gieson, Weigert – Van Gieson and Picro-Mallory) and morphometry were used.Results and discussion. As a result of the study, it was found that the four-fold administration of alprostadil had a significant effect on the structure of the dense fibrous connective tissue of the plantar foot of laboratory animals. The mechanisms of damage (edema, microhemorrhages, infiltration by lymphocytes, plasmocytes and leukocytes, dystrophy by the type of mucoid and fibrinoid swelling, delamination and rupture of collagen fibers), adaptation and regeneration (the appearance of a large number of activated fibrocytes, fibroblasts, microvessels, neoplasm of collagen fibers) were activated. All this together led to spatial focal histotopographic changes, consisting in an increase in the cellular composition of connective tissue structures against the background of a noticeable violation of their spatial orientation.Conclusion. Modeling of fasciopathy using alprostadil was accompanied by the appearance of mosaic reversible and irreversible heteromorphic and heterochronous changes in all connective tissue aponeurotic structures. Such histotopographic changes should be considered as one of the reasons for the clinical manifestations of plantar fasciopathies, causing functional insufficiency and explaining the clinical recurrent nature of the pathological process

Two-stage evolution of mantle peridotites from the Stalemate Fracture Zone, northwestern Pacific

This paper reports the results of a mineralogical study of 14 mantle peridotite samples dredged in 2009 from the eastern slope of the northwestern segment of the Stalemate Ridge in the northwestern Pacific during cruise SO201-KALMAR Leg 1b of the R/V Sonne. The sample collection included four serpentinized and silicified dunites and ten variably serpentinized lherzolites. The compositions of primary minerals (clinopyroxene, orthopyroxene, and spinel) change systematically from the lherzolites to dunites. Spinel from the lherzolites shows higher Mg# and lower Cr# values (0.65-0.68 and 0.26-0.33, respectively) compared with spinel from the dunites (Mg# = 0.56-0.64 and Cr# = 0.38-0.43). Clinopyroxene from the lherzolites is less magnesian (Mg# = 91.7-92.4) than clinopyroxene from dunite sample DR37-3 (Mg# = 93.7). Based on the obtained data, it was concluded that the lherzolites of the Stalemate Fracture Zone were derived by 10-12% near-fractional melting of a DMM-type depleted mantle reservoir beneath the Kula-Pacific spreading center. The dunites were produced by interaction of residual lherzolites with sodium- and titaniumrich melt and are probably fragments of a network of dunite channels in the shallow mantle. The moderately depleted composition of minerals clearly distinguishes the lherzolites from the strongly depleted peridotites of the East Pacific Rise and indicates the existence of slow-spreading mid-ocean ridges in the Pacific Ocean during the Cretaceous-Paleogene

Nested Bethe ansatz for "all" closed spin chains

We present in an unified and detailed way the Nested Bethe Ansatz for closed spin chains based on Y(gl(n)), Y(gl(m|n)), U_q(gl(n)) or U_q(gl(m|n)) (super)algebras, with arbitrary representations (i.e. `spins') on each site of the chain. In particular, the case of indecomposable representations of superalgebras is studied. The construction extends and unifies the results already obtained for spin chains based on Y(gl(n)) or U_q(gl(n)) and for some particular super-spin chains. We give the Bethe equations and the form of the Bethe vectors. The case of gl(2|1), gl(2|2$ and gl(4|4) superalgebras (that are related to AdS/CFT correspondence) is also detailed.Comment: 30 pages; New section on indecomposable representations added and the case of gl(2|1), gl(2|2) and gl(4|4) superalgebras (that are related to AdS/CFT correspondence) is also detaile

Superconformal mechanics

We survey the salient features and problems of conformal and superconformal mechanics and portray some of its developments over the past decade. Both classical and quantum issues of single- and multiparticle systems are covered.Comment: 1+68 pages, invited review for Journal of Physics A; v2: revised text extended by 4 pages and 11 references, published versio

Geochemical nature and age of the plagiogranite-gabbronorite association of the oceanic core complex of the Mid-Atlantic ridge at 5 degrees 10 ' S

Our newly obtained data on the geochemistry and age of plagiogranite-gabbronorite association in the oceanic core complex of the Mid-Atlantic Ridge (MAR) at 5A degrees 10'S suggest close genetic relations between these rocks in this segment of the ridge. The U/Pb zircon age of an oceanic plagiogranite (OPG) sample is 1.059 +/- 0.055 Ma and is in good agreement with the zircon age of plutonic rocks in the oceanic core complex of northern MAR. A distinctive geochemical feature of the rocks is their unusually depleted Sr-87/Sr-86 and Nd-143/Nd-144 ratios, which suggest that the plutonic rocks of the gabbronorite-plagiogranite association in MAR at 5A degrees 10'S could be derived from the most strongly depleted mantle reservoir of all known to occur beneath the axial MAR zone. The COMAGMAT-5.2 numerical thermodynamic simulation of the possible crystallization links between the plagiogranite and gabbronorite from the MAR segment at 5A degrees 10'S led us to conclude that the leading role in the origin of the plagiogranite was played by a two-stage process: the partial melting of the gabbronorite and the subsequent fractionation of the newly generated melt. The regional differences between the isotopic-geochemical parameters of MAR plagiogranites can, perhaps, reflect local specifics of so-called hydrothermal anatexis, such as the geochemical features of the rocks involved in this process and the parameters of the hydrothermal process, for example, variations in the W/R ratio