We prove the most general reflection positivity results for both scalar fields and Dirac fields on a Riemannian manifold, and comment on applications to quantum field theory. As another application, we prove the inequality C D ≤ C N between Dirichlet and Neumann covariance operators on a manifold with a reflection.

We revise a monogenic calculus for several non-commuting operators, which is defined through group representations. Instead of an algebraic homomorphism we use group covariance. The related notion of joint spectrum and spectral mapping theorem are discussed. The construction is illustrated by a simple example of calculus and joint spectrum of two non-commuting selfadjoint n× n matrices.

Abstract. We first give an elementary proof of a result relating the eigenvalues of the Dirac operator to Branson’s Q-curvature on 4-dimensional spin compact manifolds. In the case of n-dimensional closed compact (spin) manifolds we then use the conformal covariance of the Dirac, Yamabe and Branson-Paneitz operators to compare appropriate powers of their first eigenvalues. Equality cases are al...

This work presents a parametrized family of divergences, namely Alpha-Beta LogDeterminant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infinite-dimensional setting. The family of Alpha-Beta Log-Det divergences is highl...

We prove general reflection positivity results for both scalar fields and Dirac fields on a Riemannian manifold, and comment on applications to quantum field theory. As another application, we prove the inequality CD ≤ CN between Dirichlet and Neumann covariance operators on a manifold with a reflection.

The subject is the overview of the use of quasi-entropy in finite dimensional spaces. Matrix monotone functions and relative modular operators are used. The origin is the relative entropy and the f -divergence, monotone metrics, covariance and the χ2 divergence are the most important particular cases. The extension of the monotone metric to two variables is a new concept.

We prove general reflection positivity results for both scalar fields and Dirac fields on a Riemannian manifold, and comment on applications to quantum field theory. As another application, we prove the inequality C D ≤ C N between Dirichlet and Neumann covariance operators on a manifold with a reflection.

The particle filter (PF) is a novel technique that has sufficiently good estimation results for the nonlinear/non-Gaussian systems. However, PF is inconsistent that caused mainly by loss of particle diversity in resampling step and unknown a priori knowledge of the noise statistics. This paper introduces a new modified particle filter called adaptive unscented particle filter (AUPF) to overcome th...

in this paper, we determine the structure of the space of multipliers of the range of a composition operator cφ that induces by the conditional expectation between two lp() spaces.