K-Theory for Triangulated Categories 3 $\frac{3}{4}$\frac{3}{4}: A Direct Proof of the Theorem of the Heart

Let be a triangulated category, and assume it admits at least one model. In this article, we define a K-theory for . The main theorem is that, given any bounded i-structure on , the K-theory of the heart agrees with the K-theory of . An immediate consequence tells us that, if two Abelian categories occur as hearts of a triangulated category for two different t-structures, then their K-theories must be isomorphic.The proof was also sketched in previous articles in this series. The virtue of this article is in the careful detail in which it is written down.     

Temperature-Dependence of Equilibrium States of a Two-Phase Elastic Medium with the Zero Surface Tension Coefficient

We consider the energy functional of a two-phase elastic medium with quadratic energy densities defined for such that ,where is a measurable characteristic function. Under some natural conditions on the data of the problem, we prove the existence of an interval (t ^-,t ⁺) of the change of temperature such that the energy functional has only a minimizer such that for or such that t^ + $$ " align="middle" border="0"> . The energy functional has no minimizers such that or if . We derive two-sided estimates for the numbers in terms of the characteristics of the two-phase elastic medium and the boundary condition. Bibliography: 3 titles.     

Extremal Decomposition Problems in the Space of Riemann Surfaces

An extension of a theorem on extremal decomposition of a Riemann surface is obtained. The problem of extremal decomposition is extended from the case of a Riemann surface with a prescribed set of distinguished points to the case of the Teichmüller space corresponding to under quasiconformal homeomorphisms f. For the functional of our problem on extremal decomposition of a surface , we consider a function expressing the dependence of the extremal value of on a point . Differentiation formulas for the function are derived. These formulas are different and depend on the genus g of the surface . The case where the function is pluriharmonic is considered. Bibliography: 8 titles.     

On the Growth of Entire Functions of Exponential Type near a Straight Line

Let be a sequence of complex numbers such that as . For close to the imaginary axis, upper bounds of the indicator of a nonzero entire function of exponential type with minimal growth vanishing on is obtained. These bounds give sufficient conditions for the system of exponents to be incomplete in an unbounded domain in .     

The Spectrum of the C*-Algebra of Pseudodifferential Boundary Value Problems on a Manifold with Smooth Edges

The C ^*-algebra generated by the operators of pseudodifferential boundary value problems on a manifold with smooth closed disjoint edges and boundary is studied. The operators act in the space L ₂( ) L ₂( ). The goal of this paper is to describe all (up to an equivalence) irreducible representations of the algebra Bibliography: 12 titles.     

A Uniqueness Theorem for the Dual Problem Associated to a Variational Problem with Linear Growth

Uniqueness is proved for solutions of the dual problem that is associated with the minimum problem among the mappings with prescribed Dirichlet boundary data and for smooth strictly convex integrands f of linear growth. No further assumptions on f or its conjugate function are imposed, in particular, is not assumed to be strictly convex. A special solution of the dual problem is seen to be a mapping into the image of , which immediately implies uniqueness. Bibliography: 13 titles.     

On the Uniqueness of the Recovery of Parameters of the Maxwell System from Dynamical Boundary Data

The paper deals with the problem of recovering the parameters (functions) and of the Maxwell dynamical system

Sharp Constant in a Jackson-Type Inequality for Approximation by Positive Linear Operators

In what follows, C is the space of -periodic continuous real-valued functions with uniform norm, is the first continuity modulus of a function with step h, H _n is the set of trigonometric polynomials of order at most n, is the set of linear positive operators (i.e., of operators such that for every ), is the space of square-integrable functions on ,

Direct Theorem about Approximation on a Family of Two Segments

Introduce the notation: , is the union of two segments [-1,1] and [-1 + ,1+ ], is a noninteger number, is the Hölder class with exponent on The following result announced by the authors in [J. Math. Sci. 117 (2003), No. 3] is proved. There exist numbers a ₁ ( ) , b ₁ ( ) 0 depending only on such that for any there exists a polynomial , such that . Bibliography: 11 titles.     

Effects Connected with the Coincidence of Velocities in the Two-Velocity Dynamical System

The paper deals with the system

A Higher-Order Analog of the Linking Number for a Pair of Divergence-Free Vector Fields

Pairs B, of divergence-free vector fields with compact support in are considered higher-order analog M(B, c (of order 3) of the Gauss helicity number H(B, )= , curl(A)=B; (of order 1) is constructed, which is invariant under volume-preserving diffeomorphisms. An integral expression for M is given. A degree-four polynomial m(B(x₁), B(x₂), ( ₁), ( ₂)), x₁, x₂, _{1 2} , is defined, which is symmetric in the first and second pairs of variables separately. M is the average value of m over arbitrary configurations of points. Several conjectures clarifying the geometric meaning of the invariant and relating it to invariants of knots and links are stated. Bibliography: 11 titles.     

Extremal Decompositions of Riemann Surfaces and Quasiconformal Mappings of Special Form. II

Let be a finite Riemann surface. For a quadratic differential on associated with a certain problem on extremal decomposition of into n domains, a parametric family of quasiconformal mappings , is defined. These mappings map the domains of the extremal decomposition of onto the domains of the extremal decomposition of ._K This allows one to study the functional dependence of the problem on the parameters. Bibliography: 6 titles.     

Sequence Spaces l p,q in Probabilistic Characterizations of Weak Type Operators

We study operators (not necessarily linear) defined on a quasi-Bahach space X and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space are proved with the help of the Lorentz sequence spaces . Sequences of functions belonging to fixed bounded sets in the spaces are characterized for and . The possibility of distinguishing weak type operators (bounded in the space ) from operators factorizable through is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in , is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.     

Orbits of Positive Operators from a Differentiable Viewpoint

Let be a unital C^*-algebra and G the group of units of . A geometrical study of the action of G over the set ⁺ of all positive elements of is presented. The orbits of elements with closed range by this action are provided with a structure of differentiable homogeneous space with a natural connection. The orbits are partitioned in 'components' which also have a rich geometrical structure.     

Properties of the Conformal Radius of a Domain

We consider the function , where is the conformal radius of a simply connected domain D at a point . We study relations between the values of the function at various points of the domain D. In Sec. 1, we establish exact inequalities relating the values of the function in an arbitrary simply connected domain with the position of the conformal center and with the maximal conformal radius of the domain D. The same values are related to the values of at another two points of the domain D. In Sec. 2, similar results are established for convex domains. This work supplements some recent results of Emel'yanov and Kovalev. Bibliography: 9 titles.     

Transversals of Families of Translations of a Planar Convex Compact

Let be a family of translations of a convex compact set such that every two elements of have a common point. Then there exist three points such that each element of contains one of these points. This answers in the affirmative an old question by Grünbaum. Bibliography: 13 titles.     

On Polynomial Bases for the Space C[-1,1]

For an arbitrary , we construct an orthogonal basis in the space consisting of algebraic polynomials such that their degrees do not exceed . It is not possible to decrease the order of growth of the degrees. Bibliography: 9 titles.