A new approach for trapezoidal approximation of fuzzy numbers using WABL distance

In this paper, we present a new approach to obtain trapezoidal approximation of fuzzy numbers with respect to weighted distance proposed by Nasibov [5] which the main property of this metric is flexibility in the decision maker's choice. Also, we prove some properties of the proposed method such as translation invariance, scale invariance and identity. Finally, we illustrate the efficiency of proposed method by solving some numerical examples

Emergent general relativity in the tensor models possessing Gaussian classical solutions

This paper gives a summary of the author's works concerning the emergent general relativity in a particular class of tensor models, which possess Gaussian classical solutions. In general, a classical solution in a tensor model may be physically regarded as a background space, and small fluctuations about the solution as emergent fields on the space. The numerical analyses of the tensor models possessing Gaussian classical background solutions have shown that the low-lying long-wavelength fluctuations around the backgrounds are in one-to-one correspondence with the geometric fluctuations on flat spaces in the general relativity. It has also been shown that part of the orthogonal symmetry of the tensor model spontaneously broken by the backgrounds can be identified with the local translation symmetry of the general relativity. Thus the tensor model provides an interesting model of simultaneous emergence of space, the general relativity, and its local gauge symmetry of translation.Comment: 15pages, 5 figures, based on the proceedings of VIII International Workshop, "Lie Theory and its Applications in Physics", Varna, 15 - 21 June 2009, and of XXV Max Born Symposium, ``The Planck Scale'', Wroclaw, 29 June - 3 July 200

Invariance transformations for processing NDE signals

The ultimate objective in nondestructive evaluation (NDE) is the characterization of materials, on the basis of information in the response from energy/material interactions. This is commonly referred to as the inverse problem. Inverse problems are in general ill-posed and full analytical solutions to these problems are seldom tractable. Pragmatic approaches for solving them employ a constrained search technique by limiting the space of all possible solutions. A more modest goal is therefore to use the received signal for characterizing defects in objects in terms of the location, size and shape. However, the NDE signal received by the sensors is influenced not only by the defect, but also by the operational parameters associated with the experiment. This dissertation deals with the subject of invariant pattern recognition techniques that render NDE signals insensitive to operational variables, while at the same time, preserve or enhance defect related information. Such techniques are comprised of invariance transformations that operate on the raw signals prior to interpretation using subsequent defect characterization schemes. Invariance transformations are studied in the context of the magnetostatic flux leakage (MFL) inspection technique, which is the method of choice for inspecting natural gas transmission pipelines buried underground;The magnetic flux leakage signal received by the scanning device is very sensitive to a number of operational parameters. Factors that have a major impact on the signal include those caused by variations in the permeability of the pipe-wall material and the velocity of the inspection tool. This study describes novel approaches to compensate for the effects of these variables;Two types of invariance schemes, feature selection and signal compensation, are studied. In the feature selection approach, the invariance transformation is recast as a problem in interpolation of scattered, multi-dimensional data. A variety of interpolation techniques are explored, the most powerful among them being feed-forward neural networks. The second parametric variation is compensated by using restoration filters. The filter kernels are derived using a constrained, stochastic least square optimization technique or by adaptive methods. Both linear and non-linear filters are studied as tools for signal compensation;Results showing the successful application of these invariance transformations to real and simulated MFL data are presented

Stochastic representation of the Reynolds transport theorem: revisiting large-scale modeling

We explore the potential of a formulation of the Navier-Stokes equations incorporating a random description of the small-scale velocity component. This model, established from a version of the Reynolds transport theorem adapted to a stochastic representation of the flow, gives rise to a large-scale description of the flow dynamics in which emerges an anisotropic subgrid tensor, reminiscent to the Reynolds stress tensor, together with a drift correction due to an inhomogeneous turbulence. The corresponding subgrid model, which depends on the small scales velocity variance, generalizes the Boussinesq eddy viscosity assumption. However, it is not anymore obtained from an analogy with molecular dissipation but ensues rigorously from the random modeling of the flow. This principle allows us to propose several subgrid models defined directly on the resolved flow component. We assess and compare numerically those models on a standard Green-Taylor vortex flow at Reynolds 1600. The numerical simulations, carried out with an accurate divergence-free scheme, outperform classical large-eddies formulations and provides a simple demonstration of the pertinence of the proposed large-scale modeling

Scalar field theory in Snyder space-time: alternatives

We construct two types of scalar field theory on Snyder space-time. The first one is based on the natural momenta addition inherent to the coset momentum space. This construction uncovers a non-associative deformation of the Poincar\'e symmetries. The second one considers Snyder space-time as a subspace of a larger non-commutative space. We discuss different possibilities to restrict the extra-dimensional scalar field theory to a theory living only on Sndyer space-time and present the consequences of these restrictions on the Poincar\'e symmetries. We show moreover how the non-associative approach and the Doplicher-Fredenhagen-Roberts space can be seen as specific approximations of the extra-dimensional theory. These results are obtained for the 3d Euclidian Snyder space-time constructed from \SO(3,1)/\SO(3), but our results extend to any dimension and signature.Comment: 24 pages