Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces

In this paper we study the Pettis integral of fuzzy mappings in arbitrary Banach spaces. We present some properties of the Pettis integral of fuzzy mappings and we give conditions under which a scalarly integrable fuzzy mapping is Pettis integrable

Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

Chaotic multi-objective optimization based design of fractional order PI{\lambda}D{\mu} controller in AVR system

In this paper, a fractional order (FO) PI{\lambda}D\mu controller is designed to take care of various contradictory objective functions for an Automatic Voltage Regulator (AVR) system. An improved evolutionary Non-dominated Sorting Genetic Algorithm II (NSGA II), which is augmented with a chaotic map for greater effectiveness, is used for the multi-objective optimization problem. The Pareto fronts showing the trade-off between different design criteria are obtained for the PI{\lambda}D\mu and PID controller. A comparative analysis is done with respect to the standard PID controller to demonstrate the merits and demerits of the fractional order PI{\lambda}D\mu controller.Comment: 30 pages, 14 figure

Dynamical aspects of the fuzzy CP2^{2} in the large NN reduced model with a cubic term

``Fuzzy CP^2'', which is a four-dimensional fuzzy manifold extension of the well-known fuzzy analogous to the fuzzy 2-sphere (S^2), appears as a classical solution in the dimensionally reduced 8d Yang-Mills model with a cubic term involving the structure constant of the SU(3) Lie algebra. Although the fuzzy S^2, which is also a classical solution of the same model, has actually smaller free energy than the fuzzy CP^2, Monte Carlo simulation shows that the fuzzy CP^2 is stable even nonperturbatively due to the suppression of tunneling effects at large N as far as the coefficient of the cubic term ($\alpha$) is sufficiently large. As \alpha is decreased, both the fuzzy CP$^2$ and the fuzzy S^2 collapse to a solid ball and the system is essentially described by the pure Yang-Mills model (\alpha = 0). The corresponding transitions are of first order and the critical points can be understood analytically. The gauge group generated dynamically above the critical point turns out to be of rank one for both CP^2 and S^2 cases. Above the critical point, we also perform perturbative calculations for various quantities to all orders, taking advantage of the one-loop saturation of the effective action in the large-N limit. By extrapolating our Monte Carlo results to N=\infty, we find excellent agreement with the all order results.Comment: 27 pages, 7 figures, (v2) References added (v3) all order analyses added, some typos correcte

Nonperturbative studies of fuzzy spheres in a matrix model with the Chern-Simons term

Fuzzy spheres appear as classical solutions in a matrix model obtained via dimensional reduction of 3-dimensional Yang-Mills theory with the Chern-Simons term. Well-defined perturbative expansion around these solutions can be formulated even for finite matrix size, and in the case of $k$ coincident fuzzy spheres it gives rise to a regularized U($k$) gauge theory on a noncommutative geometry. Here we study the matrix model nonperturbatively by Monte Carlo simulation. The system undergoes a first order phase transition as we change the coefficient ($\alpha$) of the Chern-Simons term. In the small $\alpha$ phase, the large $N$ properties of the system are qualitatively the same as in the pure Yang-Mills model ($\alpha =0$), whereas in the large $\alpha$ phase a single fuzzy sphere emerges dynamically. Various `multi fuzzy spheres' are observed as meta-stable states, and we argue in particular that the $k$ coincident fuzzy spheres cannot be realized as the true vacuum in this model even in the large $N$ limit. We also perform one-loop calculations of various observables for arbitrary $k$ including $k=1$. Comparison with our Monte Carlo data suggests that higher order corrections are suppressed in the large $N$ limit.Comment: Latex 37 pages, 13 figures, discussion on instabilities refined, references added, typo corrected, the final version to appear in JHE

Gaussian expansion analysis of a matrix model with the spontaneous breakdown of rotational symmetry

Recently the gaussian expansion method has been applied to investigate the dynamical generation of 4d space-time in the IIB matrix model, which is a conjectured nonperturbative definition of type IIB superstring theory in 10 dimensions. Evidence for such a phenomenon, which is associated with the spontaneous breaking of the SO(10) symmetry down to SO(4), has been obtained up to the 7-th order calculations. Here we apply the same method to a simplified model, which is expected to exhibit an analogous spontaneous symmetry breaking via the same mechanism as conjectured for the IIB matrix model. The results up to the 9-th order demonstrate a clear convergence, which allows us to unambiguously identify the actual symmetry breaking pattern by comparing the free energy of possible vacua and to calculate the extent of ``space-time'' in each direction.Comment: 23 pages, 20 figures, LaTe

Robust Estimation of Reliability in the Presence of Multiple Failure Modes

In structural design, every component or system needs to be tested to ascertain that it satisfies the desired safety levels. Due to the uncertainties associated with the operating conditions, design parameters, and material systems, this task becomes complex and expensive. Typically these uncertainties are defined using random, interval or fuzzy variables, depending on the information available. Analyzing components or systems in the presence of these different forms of uncertainty increases the computational cost considerably due to the iterative nature of these algorithms. Therefore, one of the objectives of this research was to develop methodologies that can efficiently handle multiple forms of uncertainty. Most of the work available in the literature about uncertainty analysis deals with the estimation of the safety of a structural component based on a particular performance criterion. Often an engineering system has multiple failure criteria, all of which are to be taken into consideration for estimating its safety. These failure criteria are often correlated, because they depend on the same uncertain variables and the accuracy of the estimations highly depend on the ability to model the joint failure surface. The evaluation of the failure criteria often requires computationally expensive finite element analysis or computational fluid dynamics simulations. Therefore, this work also focuses on using high fidelity models to efficiently estimate the safety levels based on multiple failure criteria. The use of high fidelity models to represent the limit-state functions (failure criteria) and the joint failure surface facilitates reduction in the computational cost involved, without significant loss of accuracy. The methodologies developed in this work can be used to propagate various types of uncertainties through systems with multiple nonlinear failure modes and can be used to reduce prototype testing during the early design process. In this research, fast Fourier transforms-based reliability estimation technique has been developed to estimate system reliability. The algorithm developed solves the convolution integral in parts over several disjoint regions spanning the entire design space to estimate the system reliability accurately. Moreover, transformation techniques for non-probabilistic variables are introduced and used to efficiently deal with mixed variable problems. The methodologies, developed in this research, to estimate the bounds of reliability are the first of their kind for a system subject to multiple forms of uncertainty