Functional Analysis and Its Applications最新文献

We study the problem of describing the triples ((Omega,g,mu)), (mu=rho,dx), where (g= (g^{ij}(x))) is the (co)metric associated with a symmetric second-order differential operator (mathbf{L}(f) = frac{1}{rho}sum_{ij} partial_i (g^{ij} rho,partial_j f)) defined on a domain (Omega) of (mathbb{R}^d) and such that there exists an orthonormal basis of (mathcal{L}^2(mu)) consisting of polynomials which are eigenvectors of (mathbf{L}) and this basis is compatible with the filtration of the space of polynomials by some weighted degree.

In a joint paper of D. Bakry, M. Zani, and the author this problem was solved in dimension 2 for the usual degree. In the present paper we solve it still in dimension 2 but for a weighted degree with arbitrary positive weights.

We consider partial theta series associated with periodic sequences of coefficients, namely, (Theta(tau):= sum_{n>0} n^nu f(n) e^{ipi n^2tau/M}), where (nuinmathbb{Z}_{ge0}) and

(fcolonmathbb{Z} to mathbb{C}) is an (M)-periodic function. Such a function (Theta) is analytic in the half-plane ({ operatorname {Im}tau>0}) and in the asymptotics of (Theta(tau)) as (tau) tends nontangentially to any (alphainmathbb{Q}) a formal power series appears, which depends on the parity of (nu) and (f). We discuss the summability and resurgence properties of these series; namely, we present explicit formulas for their formal Borel transforms and their consequences for the modularity properties of (Theta), or its “quantum modularity” properties in the sense of Zagier’s recent theory. The discrete Fourier transform of (f) plays an unexpected role and leads to a number-theoretic analogue of Écalle’s “bridge equations.” The main thesis is: (quantum) modularity (=) Stokes phenomenon (+) discrete Fourier transform.

We construct a Springer-type resolution of singularities of the odd nilpotent cone of the orthosymplectic Lie superalgebras (mathfrak{osp}(m|2n)).

We give an estimate of the rate of convergence to zero of the norms of remotest projections on three subspaces of a Hilbert space with zero intersection for starting vectors in the sum of orthogonal complements to these subspaces.

The notion of the limit spectral measure of a metric triple (i.e., a metric measure space) is defined. If the metric is square integrable, then the limit spectral measure is deterministic and coincides with the spectrum of the integral operator on (L^2(mu)) with kernel (rho). An example in which there is no deterministic spectral measure is constructed.

New results on the strong solvability in Sobolev spaces of the quasilinear Venttsel’ problem for parabolic equations with discontinuous leading coefficients are obtained.

The work is devoted to solving a problem of L. N. Shevrin and M. V. Sapir (Question 3.81b of the Sverdlovsk Notebook), namely, to constructing a finitely presented infinite nil-semigroup satisfying the identity (x^9 = 0). This problem is solved with the help of geometric methods of the theory of tilings and aperiodic tessellations. A semigroup of paths on a tiling, under certain conditions, inherits some properties of the tiling itself. Moreover, the defining relations in the semigroup correspond to a set of equivalent paths on the tiling.

The relationship between the geometric and the automaton approaches previously used in the construction of finitely presented objects is discussed. As noted by S. P. Novikov, the property of determinacy in the coloring of partition nodes and its extension inward is very similar to properties of a solution of a partial differential equation with a given boundary condition. The author believes that understanding this relationship between the theories of aperiodic mosaics and their arrangements and the theory of numerical methods and grids is very promising.

In this paper, we generalize the notions of quermassintegrals, harmonic quermassintegrals, and affine quermassintegrals to (p)-quermassintegrals so that the cases (p=1, -1, -n) of (p)-quermassintegrals are quermassintegrals, harmonic quermassintegrals, and affine quermassintegrals, respectively. Further, we obtain some inequalities associated with (p)-quermassintegrals, including (L_q) Brunn–Minkowski-type inequalities, a monotonic inequality, and a Bourgain–Milman-type inequality.

Large classes of nonnegative Schrödinger operators on (Bbb R^2) and (Bbb R^3) with the following properties are described:

1. The restriction of each of these operators to an appropriate unbounded set of measure zero in (Bbb R^2) (in (Bbb R^3)) is a nonnegative symmetric operator (the operator of a Dirichlet problem) with compact preresolvent;

2. Under certain additional assumptions on the potential, the Friedrichs extension of such a restriction has continuous (sometimes absolutely continuous) spectrum filling the positive semiaxis.

The obtained results give a solution of a problem by M. S. Birman.

On the basis of a new approach to the Calderón construction (X_0^{theta} X_1^{1-theta}) for ideal spaces (X_0) and (X_1) and a parameter (theta in [0,1]), final results concerning a description of multipliers spaces are obtained. In particular, it is shown that if ideal spaces (X_0) and (X_1) have the Fatou property, then (M(X_0^{theta_0} X_1^{1-theta_0},{to},X_0^{theta_1} X_1^{1-theta_1}) = M(X_1^{theta_1 - theta_0} to X_0^{theta_1 -theta_0})) for (0 <theta_0 <theta_1 <1). Due to the absence of constraints on the ideal spaces (X_0) and (X_1), the obtained results apply to a large class of ideal spaces.