Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of ${\cal L}(H)$, where ${\cal L}(H)$ denotes the set of all adjointable operators on H. The C*-subalgebras of ${\cal L}(H)$ generated by elements in {P − P ∧ Q, Q − P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation $(\tilde \pi ,X)$ of i(P, Q, H) such that $\matrix{{\tilde \pi (P - P \wedge Q) = \pi (P) - \pi (P) \wedge \pi (Q),} \hfill \cr {\tilde \pi (Q - P \wedge Q) = \pi (Q) - \pi (P) \wedge \pi (Q).} \hfill \cr } $

When (P, Q) is semi-harmonious, that is, $\overline {{\cal R}(P + Q)} $ and $\overline {{\cal R}(2I - P - Q)} $ are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I − Q, I − P, H) are unitarily equivalent via a unitary operator in ${\cal L}(H)$. A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I − Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C*-modules are also provided.

The authors study the existence of standing wave solutions for the quasilinear Schrödinger equation with the critical exponent and singular coefficients. By applying the mountain pass theorem and the concentration compactness principle, they get a ground state solution. Moreover, the asymptotic behavior of the ground state solution is also obtained.

The nonlinear Schrödinger (NLS for short) equation plays an important role in describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet. In this paper, the authors study the NLS approximation by providing rigorous error estimates in Sobolev spaces for the electron Euler-Poisson equation, an important model to describe Langmuir waves in a plasma. They derive an approximate wave packet-like solution to the evolution equations by the multiscale analysis, then they construct the modified energy functional based on the quadratic terms and use the rotating coordinate transform to obtain uniform estimates of the error between the true and approximate solutions.

In this paper, the authors study the gradient estimates for positive weak solutions to the following p-Laplacian equation ${\Delta _p}u + a{u^\sigma } = 0$ on complete noncompact Riemannian manifold, where a, σ are two nonzero real constants with p ≠ 2. Using the gradient estimate, they can get the corresponding Liouville theorem. On the other hand, by virtue of the Poincaré inequality, they also obtain a Liouville theorem under some integral conditions with respect to positive weak solutions.

In this paper, the authors show that the maximal operators of the multilinear Calderón-Zygmund singular integrals are bounded from a product of weighted Hardy spaces into a weighted Lebesgue spaces, which essentially extend and improve the previous known results obtained by Grafakos and Kalton (2001) and Li, Xue and Yabuta (2011). The corresponding estimates on variable Hardy spaces are also established.

The note studies certain distance between unitary orbits. A result about Riesz interpolation property is proved in the first place. Weyl (1912) shows that dist(U(x), U(y)) =δ(x,y) for self-adjoint elements in matrixes. The author generalizes the result to C*-algebras of tracial rank one. It is proved that dist(U(x),U(y)) = D _c(x,y) in unital AT-algebras and in unital simple C*-algebras of tracial rank one, where x, y are self-adjoint elements and D _C (x, y) is a notion generalized from δ(x,y).

This paper is concerned with convergence of stochastic gradient algorithms with momentum terms in the nonconvex setting. A class of stochastic momentum methods, including stochastic gradient descent, heavy ball and Nesterov’s accelerated gradient, is analyzed in a general framework under mild assumptions. Based on the convergence result of expected gradients, the authors prove the almost sure convergence by a detailed discussion of the effects of momentum and the number of upcrossings. It is worth noting that there are not additional restrictions imposed on the objective function and stepsize. Another improvement over previous results is that the existing Lipschitz condition of the gradient is relaxed into the condition of Hölder continuity. As a byproduct, the authors apply a localization procedure to extend the results to stochastic stepsizes.

For a positive integer s, the projection body of an s-concave function $f:{\mathbb{R}^n} \to [0, + \infty )$, a convex body in the (n + s)-dimensional Euclidean space ${\mathbb{R}^{n + s}}$, is introduced. Associated inequalities for s-concave functions, such as, the functional isoperimetric inequality, the functional Petty projection inequality and the functional Loomis-Whitney inequality are obtained.