Reports on Mathematical Physics最新文献

We classify the Lorentzian manifolds of dimension n ≥ 3 admitting some diagonalizable operators which satisfy the Codazzi equation. This classification is applied to characterize three-dimensional weakly-Einstein Lorentzian manifolds which fall in the conformal class of the flat metrics.

In this paper, we investigate a multidimensionally consistent coupled quadrilateral system of the discrete Boussinesq type, proposed by Fordy and Xenitidis recently. It is distinguished from the known discrete Boussinesq type equations by a special dispersion relation. A Bäcklund transformation is constructed and a one-soliton solution is derived using the Bäcklund transformation. We also give bilinear forms of the coupled equations and present formulae for multisoliton solutions. Both plane wave factor and phase factor in two-soliton solutions indicate the coupled systems belong to the discrete Boussinesq family, but there is no continuous correspondence in terms of the Miwa coordinates.

We discuss canonical rigged Hilbert space constructions. We focus on cyclic self-adjoint operators and use the one-dimensional free Hamiltonian on the half-line as a model. We propose a nonstandard construction that can be generalized to many quantum systems. Our construction is motivated by the Stone–von Neumann uniqueness theorem.

We consider four subsets of the complexified spacetime algebra, namely the real even part, the real odd part, the imaginary even part and the imaginary odd part. This naturally leads to the four connected components of the Lorentz group, supplemented each time by an additional symmetry. We then examine how these four parts impact the Dirac equation and show that four types of matter arise with positive and negative masses as well as positive and negative charges.

Generalized Ricci recurrent spacetimes (GR)_n are investigated in Gray's seven subspaces. It is proved that a (GR)_n spacetime in all subspaces but one is an Einstein spacetime. The subspace $ℐ$ cannot contain a (GR)_n spacetime. Further, the subspaces $ℐ\oplus A$ and $ℐ\oplus B$ reduce to $A$ and $B$, respectively. Next, we prove that a (GR)_n spacetime is Ricci semi-symmetric if and only if either the spacetime is Einstein or the vector field A^l is closed. Further, it is shown that the Ricci tensor of (GR)_n is Riemann compatible if A^l is closed. Finally, sufficient conditions are given on a (GR)_n warped product manifold to guarantee that the factor manifolds are Einstein. Moreover, it is shown that a generalized Ricci recurrent GRW spacetime is an Einstein spacetime.

In this note, we extend some results on positive and completely positive trace-preserving maps (called quantum channels in the CPTP case) from finite-dimensional to infinite-dimensional Hilbert space. Specifically, we mainly consider whether the fixed state of a quantum channel Φ on $T\left(\mathscr{H}\right)$ exists, where $T\left(\mathscr{H}\right)$ is the Banach algebra of all trace-class operators on the Hilbert space $\mathscr{H}$. We show that there exist the approximation states ρ_n for every quantum channel Φ. In particular, there is a quantum channel on $T\left(\mathscr{H}\right)$, which has not a fixed state. Also, we get the relationship between the fixed points of $\Phi \left(\left|A\right|\right)=\left|A\right|$ and Φ(A) = ωA, where ω is the complex number with |ω| = 1 and $A\in T\left(\mathscr{H}\right)$.

In this paper, we investigate the problem of stability with respect to a part of variables of nonlinear time-varying systems. We derive some sufficient conditions that guarantee exponential stability and practical exponential stability with respect to a part of the variables of perturbed systems based on Lyapunov techniques where converse theorems are stated. Furthermore, illustrative examples to show the usefulness and applicability of the theory of stability with respect to a part of variables are provided. In particular, we show that our approach can be applied to the spring-mass-damper model.

Every slice of a 3D Young diagram on the plane z = n in the coordinate system O - xyz is a 2D Young diagram. In this paper, we show how Jack polynomials of 2D Young diagrams constitute a Jack polynomial of 3D Young diagram, which is called 3-Jack polynomial. We give the specific expressions of 3-Jack polynomials of 3D Young diagrams of total box number less than 4, and we find a method to obtain 3-Jack polynomials for every 3D Young diagram. 3-Jack polynomials become Jack polynomials of 2D Young diagrams when 3D Young diagrams have only one layer in the z-axis direction.

In this study, we analyze almost pseudo-Ricci symmetric spacetimes endowed with Gray's decomposition, as well as generalized Robertson—Walker spacetimes. For almost pseudo-Ricci symmetric spacetimes, we determine the form of the Ricci tensor in all the O (n)-invariant subspaces provided by Gray's decomposition of the gradient of the Ricci tensor. In three cases we obtain that the Ricci tensor is in the form of perfect fluid and in one case the spacetime becomes a generalized Robertson—Walker spacetime. In other cases we obtain some algebraic results. Finally, it is shown that an almost pseudo-Ricci symmetric generalized Robertson—Walker spacetime is a perfect fluid spacetime.