Abstract. Let u(x, t) be the solution to the free time-dependent Schrödinger equation at the point (x, t) in space-time Rn+1 with initial data f . We characterize the size of u in terms Lp(Lq)-estimates with power weights. Our bounds are given by norms of f in homogeneous Sobolev spaces Ḣs (Rn). Our methods include use of spherical harmonics, uniformity properties of Bessel functions and interp...

We prove a dispersive estimate for the evolution of Schrödinger operators H = −∆ + V (x) in R3. The potential is allowed to be a complex-valued function belonging to Lp(R3) ∩ Lq(R3), p < 3 2 < q, so that H need not be self-adjoint or even symmetric. Some additional spectral conditions are imposed, namely that no resonances of H exist anywhere within the interval [0,∞) and that eigenfunctions at...

The replacement theorem for classical and normal modal logics is a fundamental tool. It says that if A and B have been proved equivalent, occurrences of A in a formula may be replaced with occurrences of B to produce a formula equivalent to the original one. This theorem does not hold for LP, Logic of Proofs. A replacement for replacement is not simple to formulate. In this note I have provided...

We consider the Cauchy problem for a coupled system generalizing the thermoelastic plate equations. First we prove resolvent estimates for the stationary operator and conclude the analyticity of the associated semigroup in Lp-spaces, 1 < p < ∞, for certain values of the parameters of the system; here the Newton polygon method is used. Then we prove decay rates of the Lq(Rn)-norms of solutions, ...

Abstract. Let X1, X2, . . . , Xn be a sequence of independent random variables, let M be a rearrangement invariant space on the underlying probability space, and let N be a symmetric sequence space. This paper gives an approximate formula for the quantity ‖‖(Xi)‖N‖M whenever Lq embeds into M for some 1 ≤ q < ∞. This extends work of Johnson and Schechtman who tackled the case when N = lp, and re...

We study the left-invertibility of infinite matrices indexed by metric spaces with polynomial growth. Under different conditions on the rows and the columns, we prove that being bounded-below in lp for some 1 ≤ p ≤ ∞, implies that there is a left-inverse which is bounded in lq, for all 1 ≤ q ≤ ∞. In particular, this applies to matrices with polynomial decay, indexed by discrete groups of polyno...

In this paper, we are concerned with the asymptotic behavior of the solution for the non-autonomous parabolic equation with nonlinear Laplacian principal part in R n. The existence of the (L2(Rn),L2(Rn)) -uniform attractor, the (L2(Rn),Lp(Rn)) -uniform attractor and the (L2(Rn),W 1,p(Rn)∩Lq(Rn)) -uniform attractor will be proved. Mathematics subject classification (2010): 35B40, 35B41, 35L70.

A semidefinite relaxation σ(Γ) for the problem of finding the maximum number κ(Γ) of edges in a complete bipartite subgraph of a bipartite graph Γ = (V1 ∪ V2, E) is considered. For a large class of graphs, the relaxation is better than the LP-relaxation described in [9]. It is shown that σ(Γ) is bounded from above by the Lovász theta function θ(LQ(Γ)) of the graph LQ(Γ) related to the line grap...

We solve the linearly constrained linear-quadratic LQ and linear-quadratic-Gaussian LQG optimal control problems. Closed form expressions of the optimal controls are derived, and the Separation Theorem is generalized.