Preservation of quasi-isomorphisms of complexes

We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes.Let X and Y be two classes of R-modules with Ext ≥1(X,Y) = 0 for each object X ∈X and each object Y ∈Y.We show that if A,B ∈C■(R) are X-complexes and U,V ∈ C■(R) are Y-complexes,then U■V■Hom(A,U)■Hom(A,V);A■B■Hom(B,U)■Hom(A,U).As an application,we give a sufficient condition for the Hom evaluation morphism being invertible.     

Special framed Morse functions on surfaces

Let M be a smooth closed orientable surface. Let F be the space of Morse functions on M and $\mathbb{F}^1$ be the space of framed Morse functions both endowed with the C ^??-topology. The space $\mathbb{F}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping is a homotopy equivalence. In the case when at least x(M) + 1 critical points of each function of F are marked, the homotopy equivalences and are proved, where is the complex of framed Morse functions, is the universal moduli space of framed Morse functions, is the group of self-diffeomorphisms of M homotopic to the identity.     

Continuous embeddings of Besov-Morrey function spaces

We study embeddings of spaces of Besov-Morrey type, M Bp1,q1s1,r1(Rd ) → M Bp2 ,q2s2 ,r2 (R d ), and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation Bp1 ,r1s1 (R d , w) → M Bp2 ,q2s2 ,r2 (Rd ) for a Muckenhoupt A ∞ weight w, with wα(x) = |x|α , α -d1, as a typical example.     

Filippov Lemma for certain second order differential inclusions

In the paper we give an analogue of the Filippov Lemma for the second order differential inclusions with the initial conditions y(0) = 0, y??(0) = 0, where the matrix A ?? ? ^d×d and multifunction is Lipschitz continuous in y with a t-independent constant l. The main result is the following: Assume that F is measurable in t and integrably bounded. Let y ₀ ?? W ^2,1 be an arbitrary function fulfilling the above initial conditions and such that where p ₀ ?? L ¹[0, 1]. Then there exists a solution y ?? W ^2,1 to the above differential inclusions such that a.e. in [0, 1], .     

Double Hilbert transform on

We introduce a space , where is the testing function space whose functions are infinitely differentiable and have bounded support, and is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from onto itself.     

The strong asymptotic equivalence and the generalized inverse

We discuss the relationship between the strong asymptotic equivalence relation and the generalized inverse in the class of all nondecreasing and unbounded functions, defined and positive on a half-axis [a, +∞) (a > 0). In the main theorem, we prove a proper characterization of the function class IRV∩ , where IRV is the class of all -regularly varying functions (in the sense of Karamata) having continuous index function.     

An additivity formula for the strict global dimension of C(Ω)

Let A be a unital strict Banach algebra, and let K ₊ be the one-point compactification of a discrete topological space K. Denote by the weak tensor product of the algebra A and C(K ₊), the algebra of continuous functions on K ₊. We prove that if K has sufficiently large cardinality (depending on A), then the strict global dimension is equal to .     

Zur Verteilung der Quadrate ganzer Zahlen in rationalen Quaternionenalgebren

For γ ∈ ?letQ 〈γ〉 = ?[i]+?[i]j. where j. is a hypercomplex number withj2 = γ, and define addition and multiplication formally with respect to $zj = j\overline z $ for all z ∈ ?[i], so thatQ〈γ〉 becomes a quaternion algebra over the rationals. Further fix γ s.t.Q 〈γ 〉 is a division algebra and define for real X ≥ 1 where |Re(α)|, |Im(α)|, |Re(β)|, |Im(ö)|≤ X and Generalizing former results concerning Hamilton’s quaternions (i.e. the case γ =- 1) we show that, as X → ∞, when γ < 0, when γ > 0, when γ < 0, wheny γ 0. Thereby δ(t) is any upper bound of the error term in Dirichlet’s divisor problem, e.g. δ(t) =t0.315, C_γ, D_γ > 0 are numerical constants, and c, d are given by c := π(1 + log 2 - 2η) and d := π2(1 + 4 log 4 - 4π)/8, where π = 0.577 … is Euler’s constant.     

Application of Maslov’s formula to finding an asymptotic solution to an elastic deformation problem

We propose a scheme of bifurcation analysis of equilibrium configurations of a weakly inhomogeneous elastic beam on an elastic base under the assumption of two-mode degeneracy; this scheme generalizes the Darinskii-Sapronov scheme developed earlier for the case of a homogeneous beam. The consideration of an inhomogeneous beam requires replacing the condition that the pair of eigenvectors of the operator from the linear part of the equation (at zero) is constant by the condition of the existence of a pair of vectors smoothly depending on the parameters whose linear hull is invariant with respect to . It is shown that such a pair is sufficient for the construction of the principal part of the key function and for analyzing the branching of the equilibrium configurations of the beam. The construction of the required pair of vectors is based on a formula for the orthogonal projection onto the root subspace of (from the theory of perturbations of self-adjoint operators in the sense of Maslov). The effect of the type of inhomogeneity of the beam on the formof its deflection is studied.     

On the geometry of normal locally conformal almost cosymplectic manifolds

Normal locally conformal almost cosymplectic structures (or -structures) are considered. A full set of structure equations is obtained, and the components of the Riemannian curvature tensor and the Ricci tensor are calculated. Necessary and sufficient conditions for the constancy of the curvature of such manifolds are found. In particular, it is shown that a normal -manifold which is a spatial form has nonpositive curvature. The constancy of ΦHS-curvature is studied. Expressions for the components of the Weyl tensor on the space of the associated G-structure are obtained. Necessary and sufficient conditions for a normal -manifold to coincide with the conformal plane are found. Finally, locally symmetric normal -manifolds are considered.     

Singular strictly increasing functions and a problem on partitions of closed intervals

We establish that the problem of constructing a strictly increasing singular function is equivalent to the problem of constructing subsets and of a closed interval [a; b] ? ? such that (1) ∩ = ø; (2) ∪ = [a; b]; (3) the Lebesgue measures of the intersections of and with an arbitrary interval J ? [a; b] are positive.     

Peierls substitution and the Maslov operator method

We consider a periodic Schrödinger operator in a constant magnetic field with vector potential A(x). A version of adiabatic approximation for quantum mechanical equations with rapidly varying electric potentials and weak magnetic fields is the Peierls substitution which, in appropriate dimensionless variables, permits writing the pseudodifferential equation for the new auxiliary function: , where is the corresponding energy level of some auxiliary Schrödinger operator, assumed to be nondegenerate, and µ is a small parameter. In the present paper, we use V. P. Maslov’s operator method to show that, in the case of a constant magnetic field, such a reduction in any perturbation order leads to the equation with the operator represented as a function depending only on the operators of kinetic momenta $ \hat P_j = - i\mu \partial _{x_j } + A_j \left( x \right) $ .     

Two identities related to dirichlet character of polynomials

Let q be a positive integer, χ denote any Dirichlet character mod q. For any integer m with (m, q) = 1, we define a sum C(χ, k,m; q) analogous to high-dimensional Kloosterman sums as follows: , where a · ā ≡ 1 mod q. The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value |C(χ, k,m; q)|, and give two interesting identities for it.     

On pointwise estimate for Gamma operators

For linear combinations of Gamma operators, if 0＜a＜2r, 1/2-1/2r≤λ≤l, or 0≤λ＜1/2-1/2r(r≥2),0＜a＜r+10＜a<(r+1)/1-λ, we obtain an equivalent theorem with ωuλρ(f ,t) instead of ωrλφ(f,t), where ωuφ(f,t) is theDitzian-Totik moduli of smoothness.     

The Taikov functional in the space of algebraic polynomials on the multidimensional Euclidean sphere

We discuss three related extremal problems on the set of algebraic polynomials of given degree n on the unit sphere $ \mathbb{S}^{m - 1} $ of Euclidean space ? ^m of dimension m ≥ 2. (1) The norm of the functional F(h) = FhP _n = ∫_?(h) P _n (x)dx, which is equal to the integral over the spherical cap ?(h) of angular radius arccos h, ?1 < h < 1, on the set with the norm of the space L( $ \mathbb{S}^{m - 1} $ ) of summable functions on the sphere. (2) The best approximation in L _∞( $ \mathbb{S}^{m - 1} $ ) of the characteristic function χ _h of the cap ?(h) by the subspace of functions from L _∞( $ \mathbb{S}^{m - 1} $ ) that are orthogonal to the space of polynomials . (3) The best approximation in the space L( $ \mathbb{S}^{m - 1} $ ) of the function χ _h by the space of polynomials . We present the solution of all three problems for the value h = t(n,m) which is the largest root of the polynomial in a single variable of degree n + 1 least deviating from zero in the space L ₁ ^? on the interval (?1, 1) with ultraspheric weight ?(t) = (1 ? t ²) ^α , α = (m ? 3)/2.     

The Agnihotri—Woodward—Belkale polytope and Klyachko cones

The Agnihotri—Woodward—Belkale polytope Δ (resp., the Klyachko cone ) is the set of solutions of the multiplicative (resp., additive) Horn problem, i.e., the set of triples of spectra of special unitary (resp. traceless Hermitian) n × n matrices satisfying AB = C (resp. A + B = C). The set is the tangent cone of Δ at the origin. The group G = ? _n ⊕ ? _n acts naturally on Δ. In this note, we report on a computer calculation showing that Δ coincides with the intersection of , g ∈ G, for n ≤ 14 but does not coincide with it for n = 15. Our motivation was an attempt to understand how to solve the multiplicative Horn problem in practice for given conjugacy classes in SU(n).     

Notes on derivations on algebras of measurable operators

Derivations on algebras of (unbounded) operators affiliated with a von Neumann algebra ? are considered. Let be one of the algebras of measurable operators, of locally measurable operators, and of τ-measurable operators. The von Neumann algebras ? of type I for which any derivation on is inner are completely described in terms of properties of central projections. It is also shown that any derivation on the algebra LS(?) of all locally measurable operators affiliated with a properly infinite von Neumann algebra ? vanishes on the center LS(?).     

Local convergence in measure on semifinite von Neumann algebras

Suppose that ? is a von Neumann algebra of operators on a Hilbert space $\mathcal{H}$ and τ is a faithful normal semifinite trace on ?. The set of all τ-measurable operators with the topology t _τ of convergence in measure is a topological *-algebra. The topologies of τ-local and weakly τ-local convergence in measure are obtained by localizing t _τ and are denoted by t _τ1 and t _wτ1, respectively. The set with any of these topologies is a topological vector space. The continuity of certain operations and the closedness of certain classes of operators in with respect to the topologies t _τ1 and t _wτ1 are proved. S.M. Nikol’skii’s theorem (1943) is extended from the algebra $\mathcal{B}(\mathcal{H})$ to semifinite von Neumann algebras. The following theorem is proved: For a von Neumann algebra ? with a faithful normal semifinite trace τ, the following conditions are equivalent: (i) the algebra ? is finite; (ii) t _wτ1 = t _τ1; (iii) the multiplication is jointly t _τ1-continuous from to ; (iv) the multiplication is jointly t _τ1-continuous from to ; (v) the involution is t _τ1-continuous from to .     

On the best convergence of trigonometric integrals

ОсНОВНОИ РЕжУльтАт Ё тОИ стАтьИ жАклУЧАЕт сь В слЕДУУЩЕМ. ЕслИ с ФИксИРОВАННыМr=2, 3,..., тО Дль тОгО, ЧтОБы ИМЕлО М ЕстО (*) гДЕ ДОстАтОЧНО ВыпОлНЕН ИЕ слЕДУУЩИх УслОВИИ: 1) ЕслИ $$1< \frac{{n_\nu + 1}}{{n_\nu }} = g_\nu \in N, \nu = 0,1,$$ тО Дль кАжДОгОΝ=0,1,... ИМЕ Ет МЕстО (**) 2) ЕслИ тО (***) $$n_{\nu + 1} \geqslant (2r - 1)n_\nu , \nu = 0,1,...$$ . ЕслИ И тО УслОВИь (**), (***) ьВльУтсь тАкжЕ НЕОБхОДИМыМИ Д ль (*).