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"A study, by two of the major contributors to the theory, of the inverse scattering transform and its application to problems of nonlinear dispersive waves that arise in fluid dynamics, plasma physics, nonlinear optics, particle physics, crystal lattice theory, nonlinear circuit theory and other areas. A soliton is a localized pulse-like nonlinear wave that possesses remarkable stability properties. Typically, problems that admit soliton solutions are in the form of evolution equations that describe how some variable or set of variables evolve in time from a given state. The equations may take a variety of forms, for example, PDEs, differential difference equations, partial difference equations, and integrodifferential equations, as well as coupled ODEs of finite order. What is surprising is that, although these problems are nonlinear, the general solution that evolves from almost arbitrary initial data may be obtained without approximation. For such exactly solvable problems, the inverse scattering transform provides the general solution of their initial value problems. It is equally surprising that some of these exactly solvable problems arise naturally as models of physical phenomena. Simply put, the inverse scattering transform is a nonlinear analog of the Fourier transform used for linear problems. Its value lies in the fact that it allows certain nonlinear problems to be treated by what are essentially linear methods."

"Pencitraan gelombang mikro sudah banyak digunakan di dunia dalam berbagai bidang yang berbeda, salah satunya digunakan pada bidang kesehatan. Penerapan dari gelombang mikro dapat digunakan untuk mendiagnosis tumor/kanker payudara dengan menggunakan sensitivitas yang tinggi untuk mendeteksi jaringan abnormal payudara yang memiliki kontras dielektrik yang rendah, dibandingkan dengan jaringan normal lainnya. Skripsi ini merancang sistem validasi algoritma sensitivity-maps pencitraan gelombang mikro yang telah dilakukan dipenelitian sebelumnya. Metode ini memanfaatkan pengukuran dua jenis objek sebagai kalibrasi sistem, yaitu: objek referensi sebagai latar yang tidak terdapat penghambur dan objek kalibrasi berupa objek kecil sebagai penghambur (scattering). Objek yang diuji terdiri dari objek dengan kontras dielektrik rendah yang terbuat dengan menggunakan material Polyurethane Foam, Balsa Wood, dan Expanded Polystyrene. Rekonstruksi dilakukan pada tiga jenis data pengukuran S-Parameter yaitu S11, S21, dan gabungan keduanya. S-parameter diukur pada dua frekuensi, yaitu 3 dan 10 GHz. Hasil pengukuran akan direkonstruksi menggunakan MATLAB untuk dijadi sebuah citra. Selian itu, parameter relative root mean squared error (RRMSE) dan structural similarity index (SSIM) digunakan untuk menganalisis citra secara kuantitatif. Hasil rekonstruksi menunjukkan pengukuran gabungan ( S11 dan S21) dengan kualitas citra terbaik dengan nilai RRMSE 0.082  dan SSIM 0.477

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Microwave imaging has been widely used in the world in a variety of different fields, one of which is used in the health sector. The application of microwaves can be used to diagnose tumors/breast cancer by using a high sensitivity to detect abnormal breast tissue that has a low dielectric contrast, compared to other normal tissues. This thesis designs a validation system for the sensitivity-maps algorithm for microwave imaging that has been carried out in previous studies. This method utilizes the measurement of two types of objects as system calibration, namely: a reference object as a background where there are no scatterers and a calibration object in the form of small objects as scattering. The objects tested consist of objects with low dielectric contrast made using Polyurethane Foam, Balsa Wood, and Expanded Polystyrene materials. Reconstruction was carried out on three types of S-Parameter measurement data, namely S11, S21, and a combination of both. S-parameters are measured at two frequencies, namely 3 and 10 GHz. The measurement results will be reconstructed using MATLAB to become an image. In addition, the relative root mean squared error (RRMSE) and structural similarity index (SSIM) parameters are used to analyze the image quantitatively. The reconstruction results show the combined measurements (S11 and S21) with the best image quality with an RRMSE value of 0.082 and SSIM 0.477."

"Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems."