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) $ .     

Zeta functions in triangulated categories

We prove the 2-out-of-3 property for the rationality of the motivic zeta function in distinguished triangles in Voevodsky’s category . As an application, we show the rationality of motivic zeta functions for all varieties whose motives are in the thick triangulated monoidal subcategory generated by motives of quasi-projective curves in . Together with a result due to P. O’sullivan, this also gives an example of a variety whose motive is not finite-dimensional while the motivic zeta function is rational.     

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).     

Toral rank conjecture for moment-angle complexes

We consider an operation K ? L(K) on the set of simplicial complexes, which we call the “doubling operation.” This combinatorial operation was recently introduced in toric topology in an unpublished paper of Bahri, Bendersky, Cohen and Gitler on generalized moment-angle complexes (also known as K-powers). The main property of the doubling operation is that the moment-angle complex can be identified with the real moment-angle complex for the double L(K). By way of application, we prove the toral rank conjecture for the spaces by providing a lower bound for the rank of the cohomology ring of the real moment-angle complexes . This paper can be viewed as a continuation of the author’s previous paper, where the doubling operation for polytopes was used to prove the toral rank conjecture for moment-angle manifolds.     

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.     

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 .     

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.     

Cofiniteness with respect to a Serre subcategory

Let Φ be a system of ideals in a commutative Noetherian ring R, and let be a Serre subcategory of R-modules. We set $$ H_\Phi ^i ( \cdot , \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Ext_R^i (R/\mathfrak{b}| \otimes R \cdot , \cdot ). $$ . Suppose that a is an ideal of R, and M and N are two R-modules such that M is finitely generated and N ∈ . It is shown that if the functor $ D_\Phi ( \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Hom_R (\mathfrak{b}, \cdot ) $ is exact, then, for any $ \mathfrak{b} \in \Phi ,Ext_R^j (R/\mathfrak{b},H_\Phi ^i (M,N)) $ ∈ for all i, j ≥ 0. It is also proved that if there is a nonnegative integer t such that $ H_\mathfrak{a}^i (M,N) $ ∈ for all i < t, then $ Hom_R (R/\mathfrak{a},H_\mathfrak{a}^t (M,N)) $ ∈ , provided that is contained in the class of weakly LaskerianR-modules. Finally, it is shown that if L is an R-module and t is the infimum of the integers i such that $ H_\mathfrak{a}^i (L) $ ∈ , then $ Ext_R^j (R/\mathfrak{a},H_\mathfrak{a}^t (M,L)) $ ∈ if and only if $ Ext_R^j (R/\mathfrak{a},Hom_R (M,H_\mathfrak{a}^t (L))) $ ∈ for all j ≥ 0.     

Associative n-Tuple algebras

In the paper,we study algebras having n bilinearmultiplication operations : A×A → A, s = 1, …, n, such that (a b) c = a (b c), s, r = 1,..., n, a, b, c ∈ A. The radical of such an algebra is defined as the intersection of the annihilators of irreducible A-modules, and it is proved that the radical coincides with the intersection of the maximal right ideals each of which is s-regular for some operation . This implies that the quotient algebra by the radical is semisimple. If an n-tuple algebra is Artinian, then the radical is nilpotent, and the semisimple Artinian n-tuple algebra is the direct sum of two-sided ideals each of which is a simple algebra. Moreover, in terms of sandwich algebras, we describe a finite-dimensional n-tuple algebra A, over an algebraically closed field, which is a simple A-module.     

Martindale rings and H-module algebras with invariant characteristic polynomials

Under study is the category of the possibly noncommutative H-module algebras that are mapped homomorphically onto commutative algebras. The H-equivariant Martindale ring of quotients Q _H (A) is shown to be a finite-dimensional Frobenius algebra over the subfield of invariant elements Q _H (A) ^H and also the classical ring of quotients for A. We introduce a full subcategory of such that the algebras in are integral over its subalgebras of invariants and construct a functor ?? , which is left adjoined to the inclusion ?? .     

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.     

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.     

Spectral theory for operator matrices related to models in mechanics

We derive various properties of the operator matrix where A ₀ is a uniformly positive operator and A ₀ ^?1/2 DA ₀ ^?1/2 is a bounded nonnegative operator in a Hilbert space H. Such operator matrices are associated with second-order problems of the form $ \ddot z(t) + A_0 z(t) + D\dot z(t) = 0 $ , which are used as models for transverse motions of thin beams in the presence of damping.     

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.     

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.     

On the relation between the darlington realizations of matrix functions from the Carathéodory class and their J p,r -inner SI-dilations

The paper is devoted to establishing a two-sided relation between the Darlington realizations of matrix functions from the Carathéodory class and their J _p,r -inner SI-dilations.     

Inverse limits in representations of a restricted Lie algebra

Let(g,[p]) be a restricted Lie algebra over an algebraically closed field of characteristic p 0.Then the inverse limits of "higher" reduced enveloping algebras {uχs(g)|s∈N} with χ running over g* make representations of g split into different "blocks".In this paper,we study such an infinitedimensional algebra Aχ(g):= ■Uχs(g) for a given χ∈g*.A module category equivalence is built between subcategories of U(g)-mod and Aχ(g)-mod.In the case of reductive Lie algebras,(quasi) generalized baby Verma modules and their properties are described.Furthermore,the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized χ-reduced module category are precisely determined,and a higher reciprocity in the case of regular nilpotent is obtained,generalizing the ordinary reciprocity.     

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.     

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.     

Characterizing Artin stacks

We study properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site . There is a natural method for extending a property P of morphisms of sheaves on to a property ${\mathcal{P}}$ of morphisms of presheaves of groupoids. We prove that the property ${\mathcal{P}}$ is homotopy invariant in the local model structure on when P is stable under pullback and local on the target. Using the homotopy invariance of the properties of being a representable morphism, representable in algebraic spaces, and of being a cover, we obtain homotopy theoretic characterizations of algebraic and Artin stacks as those which are equivalent to simplicial objects in satisfying certain analogues of the Kan conditions. The definition of Artin stack can naturally be placed within a hierarchy which roughly measures how far a stack is from being representable. We call the higher analogues of Artin stacks n-algebraic stacks, and provide a characterization of these in terms of simplicial objects. A consequence of this characterization is that, for presheaves of groupoids, n-algebraic is the same as 3-algebraic for all n ≥ 3. As an application of these results we show that a stack is n-algebraic if and only if the homotopy orbits of a group action on it is.