Diverse <Emphasis Type="Italic">N</Emphasis>=(4,4) Twisted Multiplets in the <Emphasis Type="Italic">N</Emphasis>=(2,2) Superspace

We describe four different types of N=(4, 4) twisted supermultiplets in the two-dimensional N=(2, 2) superspace ℝ^(1,1|2,2). All these multiplets are represented by a pair of chiral and twisted chiral superfields and differ in the transformation properties under an extra hidden N=(2, 2) supersymmetry. The sigma-model N=(2, 2) superfield Lagrangians for each type of the N=(4, 4) twisted supermultiplet are real functions subjected to some differential constraints implied by the hidden supersymmetry. We prove that the general sigma-model action including all types of N=(4, 4) twisted multiplets and invariant under the N=(4, 4) supersymmetry reduces to a sum of sigma-model actions for separate types. An interaction between the multiplets of different sorts is possible only through the appropriate mass terms and only for those multiplets that belong to the same “self-dual” pair. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 66–86, October, 2005.     

Integrable <Emphasis Type="Italic">sl</Emphasis>(<Emphasis Type="Italic">N</Emphasis>, ℂ) tops as Calogero-Moser systems

We show a relation between systems of integrable tops on the algebras sl(N, ℂ) and Calogero-Moser systems of N particles. We construct classical Lax operators corresponding to these systems. We show that these operators are related to certain new trigonometric and rational solutions of the Yang-Baxter equations for the algebras sl(N, ℂ) and give explicit formulas for N = 2, 3. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 355–369, March, 2009.     

Defining relations on the Hamiltonians of <Emphasis Type="Italic">XXX</Emphasis> and <Emphasis Type="Italic">XXZ R</Emphasis>-matrices and new integrable spin-orbital chains

We present several complete systems of integrability conditions on the density of the Hamiltonian of a spin chain matrix. The corresponding formulas for R-matrices are also given. The latter are expressed via the local Hamiltonian density in a form similar to spin one half XXX and XXZ models. The result is applied to the problem of integrability of SU(2) × SU(2)-and SU(2) × U(1)-invariant spin-orbital chains (the Kugel-Homskii-Inagaki model). Eight new integrable cases are found. One of these cases corresponds to the Temperley-Lieb algebra, three cases correspond to the algebra associated with the XXX model, one case corresponds to the algebra associated with the XXZ model, and one case corresponds to the algebra associated with the graded XXZ model. The remaining two R-matrices are also presented. Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 50–58.     

<Emphasis Type="Italic">Q</Emphasis>-measures on <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript>λ

 We give a characterization of strongly compact cardinals in terms of Q _κ λ. We also prove that weakly normal Q-measures on Q _κ λ are ⊂κ-normal. Received: 29 September 2000 / Revised version: March 2002 Published online: 5 November 2002 This project is supported by the New Zealand Marsden Fund. The author wishes to thank the referee for numerous comments and suggestions which have been incorporated into this version of the paper.     

<Emphasis Type="Italic">G</Emphasis>-compactness and groups

Lascar described E _KP as a composition of E _L and the topological closure of E _L (Casanovas et al. in J Math Log 1(2):305–319). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M′ consisting of M and X as two sorts, where X is an affine copy of G and in M′ we have the structure of M and the action of G on X. We prove that the Lascar group of M′ is a semi-direct product of the Lascar group of M and G/G _L . We discuss the relationship between G-compactness of M and M′. This example may yield new examples of non-G-compact theories. The first author is supported by the Polish Goverment grant N N201 384134. The second author is supported by the Polish Goverment grant N201 032 32/2231.     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

Static spin susceptibility in the <Emphasis Type="Italic">t</Emphasis>-<Emphasis Type="Italic">J</Emphasis> model

Based on the equation-of-motion method for the relaxation function expressed in terms of Hubbard operators, we evaluate the static spin susceptibility and correlation functions in the two-dimensional t-J model. We show that at zero temperature, a long-range antiferromagnetic order exists in the lattice for the hole concentration δ < δ_c ≃ 0.018. The calculated dependences of the spin susceptibility and correlation length on the temperature and hole concentration agree qualitatively with the experiment. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 538–550, September, 2007.     

Interpolation of Ideal Measures by Abstract K and J Spaces

We work with the abstract K and J interpolation method generated by a sequence lattice Г. We investigate the deviation of an interpolated operator from a given operator ideal by establishing formulae for the ideal measure of the interpolated operator in terms of the ideal measures of restrictions of the operator. Formulae are given in terms of the norms of the shift operators on Г.     

The <Emphasis Type="Italic">p</Emphasis>-rank of Ext<Subscript>ℤ</Subscript>(<Emphasis Type="Italic">G</Emphasis>, ℤ) in certain models of <Emphasis Type="Italic">ZFC</Emphasis>

We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext _ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable strong limit cardinal μ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π₀ and Π₁, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2^μ = μ⁺ such that the p-rank of Ext _ℤ(G, ℤ) equals 2^μ = μ⁺ for every p ∈ Π₀ and 0 otherwise, that is, for p ∈ Π₁. Number 874 in Shelah’s list of publications. Supported by the German-Israeli Foundation for Scientific Research & Development project No. I-706-54.6/2001. Supported by a grant from the German Research Foundation DFG. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 369–397, May–June, 2007.     

Renormalization group and the <Emphasis Type="Italic">ɛ</Emphasis>-expansion: Representation of the <Emphasis Type="Italic">β</Emphasis>-function and anomalous dimensions by nonsingular integrals

In the framework of the renormalization group and the ɛ-expansion, we propose expressions for the β-function and anomalous dimensions in terms of renormalized one-irreducible functions. These expressions are convenient for numerical calculations. We choose the renormalization scheme in which the quantities calculated using R operations are represented by integrals that do not contain singularities in ɛ. We develop a completely automated calculation system starting from constructing diagrams, determining relevant subgraphs, combinatorial coefficients, etc., up to determining critical exponents. As an example, we calculate the critical exponents of the φ ³ model in the order ɛ ⁴ .     

Composite Integers n for Which φ（n）｜n- 1

In this note, we show that the number of composite integers n ≤ x such that φ（n）｜n - 1 is at most O（x^1/2（loglog x）^1/2）, thus improving earlier results by Pomerance and by Shan.     

Division and <Emphasis Type="Italic">k</Emphasis>-th root theorems for <Emphasis Type="Italic">Q</Emphasis>-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z _∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f: K → X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X ² and X × [0, 1] are Q-manifolds as well. We construct a countable family χ of spaces with DDP and cd-AP such that no space X ∈ χ is homeomorphic to the Hilbert cube Q whereas the product X × Y of any different spaces X, Y ∈ χ is homeomorphic to Q. We also show that no uncountable family χ with such properties exists. This work was supported by the Slovenian-Ukrainian (Grant No. SLO-UKR 04-06/07)     

On the value distribution of <Emphasis Type="Italic">ƒ</Emphasis> + <Emphasis Type="Italic">a</Emphasis>(<Emphasis Type="Italic">ƒ</Emphasis>′)<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

The nonlinear <Emphasis Type="Italic">N</Emphasis>-membranes evolution problem

The parabolic N-membranes problem for the p-Laplacian and the complete order constraint on the components of the solution are studied in what concerns the approximation, regularity, and stability of variational solutions. To the evolutionary case is extended the characterization of the Lagrange multipliers associated with the ordering constraint in terms of characteristic functions of coincidence sets. Continuous dependence results are given, and the asymptotic behavior as t → ∞ of the solution and the coincidence sets, showing that they converge to their stationary counterparts, is studied. Bibliography: 22 titles. Dedication to V. A. Solonnikov on the occasion of his 75th birthday with admiration and friendship Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 303–324.     

On Baer-Suzuki <Emphasis Type="Italic">π</Emphasis>-theorems

Given a set π of primes, say that the Baer-Suzuki π-theorem holds for a finite group G if only an element of O _π(G) can, together with each conjugate element, generate a π-subgroup. We find a sufficient condition for the Baer-Suzuki π-theorem to hold for a finite group in terms of nonabelian composition factors. We show also that in case 2 ∉ π the Baer-Suzuki π-theorem holds for every finite group.     

A method for the resolution of the Jacobi equation <Emphasis Type="Italic">Y</Emphasis>″ + <Emphasis Type="Italic">RY</Emphasis> = 0 on the manifold <Emphasis Type="Italic">Sp</Emphasis>(2)/<Emphasis Type="Italic">SU</Emphasis>(2)

In this paper a method for the resolution of the differential equation of the Jacobi vector fields in the manifold V ₁ = Sp(2)/SU(2) is exposed. These results are applied to determine areas and volumes of geodesic spheres and balls. Work partially supported by DGI (Spain) and FEDER Projects MTM 2004-06015-C02-01 and MTM 2007-65852 (first author) and by Research Project PGIDIT05PXIB16601PR (second author). Authors’ addresses: A. M. Naveira, Departamento de Geometría y Topología. Facultad de Matemáticas, Avda. Andrés Estellés, N1, 46100 – Burjassot, Valencia, Spain; A. D. Tarrío Tobar, E. U. Arquitectura Técnica, Campus A Zapateira. Universidad de A Coru?a, 15192 – A Coru?a, Spain     

Generalization of the Kneser theorem on zeros of solutions of the equation <Emphasis Type="Italic">y</Emphasis>″ + <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)<Emphasis Type="Italic">y</Emphasis> = 0

We establish conditions for the oscillation of solutions of the equation y″ + p(t)Ay = 0 in a Banach space, where A is a bounded linear operator and p: ℝ+ → ℝ+ is a continuous function. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 571–576, April, 2007.     

Phases of the goldstone multitrace matrix model in the large-<Emphasis Type="Italic">N</Emphasis> limit

We consider the Goldstone Hermitian matrix model with a multitrace term. When defining the solution on two intervals, we introduce a special parameter ξ describing the phase. We discuss the phase existence conditions at ξ = 0 (or 1) and at ξ = 1/2. We calculate the propagator and the vacuum energy in the symmetric case ξ = 1/2. In the general case, we discuss the solution structure and calculate the magnetization and other parameters expressed in terms of the sum of all the intervals. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 457–465, September, 2007.     

<Emphasis Type="Italic">D</Emphasis><Subscript>∞</Subscript>-differential <Emphasis Type="Italic">E</Emphasis><Subscript>∞</Subscript>-algebras and Steenrod operations in spectral sequences

This paper is devoted to the introduction of a D _∞-differential analog of the notion of an E _∞-(co)algebra and to the construction of generalized Steenrod operations in terms of multiplicative spectral sequences. In this paper, we investigate basic homotopy properties of D _∞-differential E _∞-(co)algebras and construct a spectral sequence of a D _∞-differential E _∞-(co)algebra. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.