Reports on Mathematical Physics最新文献

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.

We propose a generalised Fisher information or a one-parameter extended class of the Fisher information for the case of one random variable. This new form of the Fisher information is obtained from the intriguing connection between the standard Fisher information and the variational principle together with the nonuniqueness property of the Lagrangian. A generalised Cramér--Rao inequality is also derived and a Fisher information hierarchy is also obtained from the two-parameter Kullback-Leibler divergence. An interesting point is that the whole Fisher information hierarchy, except for the standard Fisher information, does not follow the additive rule. Furthermore, the idea can be directly extended to obtain the one-parameter generalised Fisher information matrix for the case of one random variable with multi-estimated parameters. The hierarchy of the Fisher information matrices is obtained. The geometrical meaning of the first two matrices in the hierarchy is studied through the normal distribution. What we find is that these first two Fisher matrices give different nature of curvature on the same statistical manifold for the normal distribution.

We study the reduction by symmetries for optimality conditions in optimal control problems of left-invariant affine control systems with partial symmetry breaking cost functions. We recast the optimal control problem as a constrained problem with a partial symmetry breaking Hamiltonian and we obtain the reduced optimality conditions for normal extrema from Pontryagin's Maximum Principle and a Lie--Poisson bracket on the reduced state space. We apply the results to an energy-minimum obstacle avoidance problems.

We present the standard and nonstandard one-parameter deformations of two supergroups, one of which is the lower and the other upper triangular matrices, and using these we show that the two-parameter nonstandard deformation of the supergroup GL(1|1) can be achieved.