Springer Science and Business Media LLC

Observables are defined as homomorphisms from the Borelσ-algebra into a family of fuzzy sets considered with respect to the Giles connectives. Algebraic operations with observables are introduced and their relation to the corresponding operations with fuzzy random variables is explained.

In this paper, we study the formation of Shannon information entropy in a two axes rotating precessing Bose-Einstein condensates subject to dipole-dipole interactions. Our results show that how the Shannon information entropy depends on the second rotational frequency Ωx and dipole strength gd. With the increasing of gd, the position component Sr increases and momentum component Sk decreases at the same Ωx. Furthermore, we find that Landsberg’s order parameter η monotonically decreases with increasing Ωx and η increases with increasing gd at the same Ωx. Finally, we find that the increasing of dipole strength has no effect on the dynamic formation process of information entropy.

Kantowski-Sachs model is studied with source cosmic cloud strings coupled with electromagnetic field in Rosen’s (Gen. Relativ. Gravit. 4:435, 1973) bimetric theory of relativity. It is shown that there is no contribution from Maxwell fields in this theory. Hence geometric string and vacuum cosmological models are established.

We investigate the usefulness of a recently introduced seven-qubit maximally entangled state by Zha et al. (J. Phys. A: Math. Theor. 45:255–302, [10]) for quantum information splitting. It is shown that such a seven-qubit entangled state can be utilized for quantum information splitting of an arbitrary two-qubit state by performing only the Bell-state measurements.

An evolving secret sharing scheme is supposed to accommodate unbounded number (potentially infinite) of new participants over time and the dealer should not know the set of participants in advance. In this paper we initiate the study of quantum evolving schemes to share a quantum secret and show how to construct such a scheme where the qualified subsets (the sets which can reconstruct the secret) of participants are of increasing size as the number of participants increases with time. Share is given to new participants without modifying shares of old participants. While quantum secret sharing schemes which can accommodate new participants over time have been constructed before, a quantum scheme which can handle an unbounded and unspecified number of participants has not been considered so far. We also discuss the resilience of our scheme against a bounded number of quantum queries by an adversary/eavesdropper. This is achieved by using special functions for which lower bounds on their quantum query complexity is known.

In the special Conte truncated expansion approach one obtains different solutions of the Prigogine–Lefever equation by use of various solutions of a type of Riccati equation, including the periodic soliton solutions and singular soliton solutions. In order to acquire conveniently the soliton solutions of the Boussinesq equation, a proper transformation is applied. Using the special Conte truncated expansion approach yields the known bell-shape solutions and some new soliton solutions like cot2 × sec2, tan2 × c sec2, tanh2 × sech2, etc. We also study the soliton solutions of the modified Burgers equation (MBE). Using leading term analysis, we find the exponent is a fraction, i.e., − $$ \frac{1}{2} $$ . Therefore, the special Conte truncated expansion approach cannot be used directly. A transformation is first made to them another form of the MBE. Various soliton solutions of MBE are then presented, including the periodic solutions and singular soliton solutions.

We study the production of pairs in non-analytic potentials. While, when the potential is analytic, the average number of produced pairs is exponentially small in ħ, when the potential is non-analytic, using the W.K.B. method, we prove that the average number of produced pairs is $$\mathcal{O}$$ (αħ 2N ), where N is the regularity of the potential and α is the fine structure constant. We give a rigorous proof of the Schwinger's formula.