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 This combinatorial operation was recently introduced in toric topology in an unpublished paper of Bahri, Bendersky, Cohen and Gitler on generalized moment-angle complexes (also known as K-powers).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.     

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) $ .     

Approximation complexity of tensor product-type random fields with heavy spectrum

We consider a sequence of Gaussian tensor product-type random fields , where and are all positive eigenvalues and eigenfunctions of the covariance operator of the process X ₁, are standard Gaussian random variables, and is a subset of positive integers. For each d ∈ ?, the sample paths of X _d almost surely belong to L ₂([0, 1] ^d ) with norm ∥·∥_2,d . The tuples , are the eigenpairs of the covariance operator of X _d . We approximate the random fields X _d , d ∈ , by the finite sums X _d ⁽ⁿ⁾ corresponding to the n maximal eigenvalues λ _k , . We investigate the logarithmic asymptotics of the average approximation complexity $n_d^{pr} (\varepsilon ,\delta ): = \min \left\{ {n \in \mathbb{N}:\mathbb{P}(\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 > \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 ) \leqslant \delta } \right\},$ and the probabilistic approximation complexity $n_d^{avg} (\varepsilon ): = \min \left\{ {n \in \mathbb{N}:\mathbb{E}\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 \leqslant \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 } \right\}$ , as the parametric dimension d → ∞ the error threshold ? ∈ (0, 1) is fixed, and the confidence level δ = δ(d, ?) is allowed to approach zero. Supplementing recent results of M.A. Lifshits and E.V. Tulyakova, we consider the case where the sequence decreases regularly and sufficiently slowly to zero, which has not been previously 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.     

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.     

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 ?? .     

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.     

Maximal function on the dual of Laguerre hypergroup

In this paper, we are interested in the Laguerre hypergroup $\mathbb{K} = [0,\infty ) \times \mathbb{R}$ which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup which can be topologically identified with the so-called Heisenberg fan, the subset of ?²: $$\bigcup\limits_{j \in \mathbb{N}} {\left\{ {(\lambda ,\mu ) \in \mathbb{R}^2 :\mu = \left| \lambda \right|(2j + \alpha + 1),\lambda \ne 0} \right\} \cup \left\{ {(0,\mu ) \in \mathbb{R}^2 :\mu \geqslant 0} \right\}} ,$$ by means of which the maximal function is investigated. For 1 < p ?? ??, the L _p ( )-boundedness and weak L ₁( )-boundedness result for the maximal function is obtained.     

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).     

Coefficient characterizations and sections for some univalent functions

Let (α) denote the class of locally univalent normalized analytic functions f in the unit disk |z| < 1 satisfying the condition $Re\left( {1 + \frac{{zf''(z)}} {{f'(z)}}} \right) < 1 + \frac{\alpha } {2}for|z| < 1 $ and for some 0 < α ≤ 1. We firstly prove sharp coefficient bounds for the moduli of the Taylor coefficients a _n of f ∈ (α). Secondly, we determine the sharp bound for the Fekete-Szegö functional for functions in (α) with complex parameter λ. Thirdly, we present a convolution characterization for functions f belonging to (α) and as a consequence we obtain a number of sufficient coefficient conditions for f to belong to (α). Finally, we discuss the close-to-convexity and starlikeness of partial sums of f ∈ (α). In particular, each partial sum s _n (z) of f ∈ (1) is starlike in the disk |z| ≤ 1/2 for n ≥ 11. Moreover, for f ∈ (1), we also have Re(s′ _n (z)) > 0 in |z| ≤ 1/2 for n ≥ 11.     

Convolution functions of several variables with generalized bounded variation

Let L ¹ be the class of all complex-valued functions, with period 2π in each variable, in the space , where $\mathbb{T} = [0,2\pi )$ is the one-dimensional torus. Here, it is observed that L ¹ * E ? E for E = Lip(p; α ₁, α ₂, ..., α _N ) over , for , for , and for in the sense of Vitali as well as Hardy.     

On the class of groups with pronormal hall π-subgroups

Given a set π of prime numbers, we define the class of all finite groups in which Hall π-subgroups exist and are pronormal by analogy with the Hall classes , , and . We study whether is closed under the main class-theoretic closure operations. In particular, we establish that is a saturated formation.     

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.     

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(?).     

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.     

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.     

Upper estimates for the approximation numbers of the generalized Laplace transform

A Laplace-transform-type operator acting in the Lebesgue spaces of real functions on the half-axis is considered. Sufficient conditions under which belongs to some Schattentype classes are found. Upper asymptotic estimates for the approximation numbers of are obtained.