The early historical roots of Lee-Yang theorem

A deep and detailed historiographical analysis of a particular case study concerning the so-called Lee-Yang theorem of theoretical statistical mechanics of phase transitions, has emphasized what real historical roots underlie such a case study. To be precise, it turned out that some well-determined aspects of entire function theory have been at the primeval origins of this important formal result of statistical physics.Comment: History of Physics case study. arXiv admin note: substantial text overlap with arXiv:1106.4348, arXiv:math/0601653, arXiv:0809.3087, arXiv:1311.0596 by other author

Statistical convergence of double-complex Picard integral operators

AbstractIn this work, we study the statistical approximation properties of the double-complex Picard integral operators. We also show that our statistical approach is more applicable than the classical one

Q-periodicity, self-similarity and weierstrass-mandelbrot function

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2012Includes bibliographical references (leaves: 92-94)Text in English; Abstract: Turkish and Englishviii, 98 leavesIn the present thesis we study self-similar objects by method's of the q-calculus. This calculus is based on q-rescaled finite differences and introduces the q-numbers, the qderivative and the q-integral. Main object of consideration is the Weierstrass-Mandelbrot functions, continuous but nowhere differentiable functions. We consider these functions in connection with the q-periodic functions. We show that any q-periodic function is connected with standard periodic functions by the logarithmic scale, so that q-periodicity becomes the standard periodicity. We introduce self-similarity in terms of homogeneous functions and study properties of these functions with some applications. Then we introduce the dimension of self-similar objects as fractals in terms of scaling transformation. We show that q-calculus is proper mathematical tools to study the self-similarity. By using asymptotic formulas and expansions we apply our method to Weierstrass-Mandelbrot function, convergency of this function and relation with chirp decomposition

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

Robust optimization of control parameters for WEC arrays using stochastic methods

This work presents a new computational optimization framework for the robust control of parks of Wave Energy Converters (WEC) in irregular waves. The power of WEC parks is maximized with respect to the individual control damping and stiffness coefficients of each device. The results are robust with respect to the incident wave direction, which is treated as a random variable. Hydrodynamic properties are computed using the linear potential model, and the dynamics of the system is computed in the frequency domain. A slamming constraint is enforced to ensure that the results are physically realistic. We show that the stochastic optimization problem is well posed. Two optimization approaches for dealing with stochasticity are then considered: stochastic approximation and sample average approximation. The outcomes of the above mentioned methods in terms of accuracy and computational time are presented. The results of the optimization for complex and realistic array configurations of possible engineering interest are then discussed. Results of extensive numerical experiments demonstrate the efficiency of the proposed computational framework