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In this paper, the new concepts of generalized Menger probabilistic metric spaces and tripled common fixed point for a pair of mappings T : X × X × X → X and A : X → X are introduced. Utilizing the properties of the pseudo-metric and the triangular norm, some tripled common fixed point problems of hybrid probabilistic contractions with a gauge function φ are studied. The obtained results generalize some coupled common fixed point theorems in the corresponding literature. Finally, an example is given to illustrate our main results. MSC:47H10, 46S50.

We propose an Otto heat engine that undergoes processes involving a special class of feedback and analyze theoretically its response. We use stochastic thermodynamics to determine the performance characteristics of the heat engine and indicate the possibility that its maximum efficiency can surpass the Carnot value. The analytical expression for efficiency at maximum power, including the effects resulting from feedback, reduces to that previously derived based on an engine without feedback.

This article is concerned with the coupled system of fractional differential equations with nonlocal integral boundary conditions. The existence results are obtained by applying some standard fixed point theorems. Finally, an example is also provided to illustrate the availability of our main results.

This paper proposes a revised counter-propagation network (CPN) model by integrating rough set in structural damage detection, applicable for processing redundant and uncertain information as well as assessing structural health states. Firstly, rough set is used in the model to deal with a large volume of data; secondly, a revised training algorithm is developed to improve the capabilities of the CPN model; and lastly, the least input vectors are input to the revised CPN (RCPN) model, hence the rough set-based RCPN is proposed in the paper. To validate the model proposed, numerical experiments are conducted, and, as a result, six damage patterns have been successfully identified in a steel frame. The influence of measurement noise, the network models adopted, and the data preprocessing methods on damage identification is also discussed in the paper. The results show that the proposed model not only has good damage detection capability and noise tolerance, but also significantly reduces the data storage requirement and saves computing time.

In this paper, some Banach spaces are introduced. Based on these spaces and the coincidence degree theory, a 2m-point boundary value problem for a coupled system of impulsive fractional differential equations at resonance is considered, and the new criterion on existence is obtained. Finally, an example is also given to illustrate the availability of our main results. MSC:34A08, 34B10, 34B37.

We study the efficiency at maximum power, $\eta_m$, of irreversible quantum Carnot engines (QCEs) that perform finite-time cycles between a hot and a cold reservoir at temperatures $T_h$ and $T_c$, respectively. For QCEs in the reversible limit (long cycle period, zero dissipation), $\eta_m$ becomes identical to Carnot efficiency $\eta_{_C}=1-\frac{T_c}{T_h}$. For QCE cycles in which nonadiabatic dissipation and time spent on two adiabats are included, the efficiency $\eta_m$ at maximum power output is bounded from above by $\frac{\eta_{_C}}{2-\eta_{_C}}$ and from below by $\frac{\eta_{_C}}2$. In the case of symmetric dissipation, the Curzon-Ahlborn efficiency $\eta_{_{CA}}=1-\sqrt{\frac{T_c}{T_h}}$ is recovered under the condition that the time allocation between the adiabats and the contact time with the reservoir satisfy a certain relation., Comment: to be published in Phys. Rev. E (2012)

We establish a quantum Otto engine (QOE) of a two-level atom, which is confined in a one-dimensional (1D) harmonic trap and is coupled to single-mode radiation fields. Besides two adiabatic processes, the QOE cycle consists of two isochoric processes, along one of which the two-level atom as the working substance interacts with a single-mode radiation field. Based on the semigroup approach, we derive the time for completing any adiabatic process and then present a performance analysis of the heat engine model. Furthermore, we generalize the results to the performance optimization for a QOE of a single two-level atom trapped in a 1D power-law potential. Our result shows that the efficiency at maximum power output is dependent on the trap exponent $\ensuremath{\theta}$ but is independent of the energy spectrum index $\ensuremath{\sigma}$.

In 2004, Kirkpatrick discussed three ways (I), (II) and (III) of describing non-disturbance between quantum measurements X and Y, and showed that they are all equivalent to the compatibility of X and Y if they are both sharp measurements. In 2005, based on a special sequential product on the standard effect algebra, Gudder showed that if X and Y are unsharp measurements, then (I) holds if and only if X and Y are compatible and Y is sharp measurement; compatibility of X and Y implies (II), but the converse does not hold, and only (III) is equivalent to the compatibility of X and Y. Liu and Wu (J. Phys. A, Math. Theor. 42:185206, 2009) and Shen and Wu (J. Phys. A, Math. Theor. 42:345203, 2009) showed that there are many sequential products on the standard effect algebra. In this paper, we obtain the same conclusions as Gudder’s for all these sequential products of the standard effect algebra.

In this paper, we introduce the new concepts of multidimensional Menger probabilistic metric spaces and related fixed point for a pair of mappings T: $\underbrace{X\times X\times\cdots\times X}_{n}\rightarrow X $ and A: $X\rightarrow X$ . Utilizing the properties of the related triangular norm and the compatibility of A with T, some multidimensional common fixed point problems of hybrid probabilistic contractions with a gauge function φ are studied. The obtained results generalize some coupled and triple common fixed point theorems in the corresponding literature. Finally, an example is given to illustrate our main results.

In this paper, we introduce the concepts of generalized probabilistically bounded set $\Omega^{*}$ and Menger-Hausdorff metric $\widetilde{G}^{*}$ in Menger probabilistic G-metric spaces, and prove that $(\Omega^{*},\widetilde{G}^{*},\Delta)$ is also a Menger probabilistic G-metric space. Utilizing these concepts, we establish some common fixed point theorems for three hybrid pairs of mappings satisfying the common property $(E.A)$ in Menger probabilistic G-metric spaces. Finally, an example is given to exemplify the theorems.

In this paper, a new concept of the common property (E.A) for two hybrid pairs of mappings is introduced in Menger PM-spaces. Utilizing this concept, some common fixed point theorems, which shed some new light on the study of fixed point results for hybrid pairs in Menger PM-spaces, are obtained under strict contractive conditions. The corresponding results in metric spaces which generalize many known results are also obtained. Finally, an example is also given to exemplify our main results.