Maxwell’s Equations

Fermat's Theorem, auch bekannt als Fermats letzter Satz, besagt, dass es keine drei positiven ganzen Zahlen $a$, $b$ und $c$ gibt, die die Gleichung

für einen ganzzahligen Exponenten $n > 2$ erfüllen. Pierre de Fermat formulierte diesen Satz im Jahr 1637 und hinterließ einen kurzen Hinweis, dass er einen "wunderbaren Beweis" für diese Aussage gefunden hatte, den er jedoch nicht aufschrieb. Der Satz blieb über 350 Jahre lang unbewiesen und wurde erst 1994 von dem Mathematiker Andrew Wiles bewiesen. Der Beweis nutzt komplexe Konzepte der modernen Zahlentheorie und elliptischen Kurven. Fermats letzter Satz ist nicht nur ein Meilenstein in der Mathematik, sondern hat auch bedeutende Auswirkungen auf das Verständnis von Zahlen und deren Beziehungen.

Fiber Bragg Grating (FBG) sensors are advanced optical devices that utilize the principles of light reflection and wavelength filtering. They consist of a periodic variation in the refractive index of an optical fiber, which reflects specific wavelengths of light while allowing others to pass through. When external factors such as temperature or pressure change, the grating period alters, leading to a shift in the reflected wavelength. This shift can be quantitatively measured to monitor various physical parameters, making FBG sensors valuable in applications such as structural health monitoring and medical diagnostics. Their high sensitivity, small size, and resistance to electromagnetic interference make them ideal for use in harsh environments. Overall, FBG sensors provide an effective and reliable means of measuring changes in physical conditions through optical means.

The Knuth-Morris-Pratt (KMP) algorithm is an efficient method for substring searching that improves upon naive approaches by utilizing preprocessing. The preprocessing phase involves creating a prefix table (also known as the "partial match" table) which helps to skip unnecessary comparisons during the actual search phase. This table records the lengths of the longest proper prefix of the substring that is also a suffix for every position in the substring.

To construct this table, we initialize an array $\text{lps}$ of the same length as the pattern, where $\text{lps}[i]$ represents the length of the longest proper prefix which is also a suffix for the substring ending at index $i$. The preprocessing runs in $O(m)$ time, where $m$ is the length of the pattern, ensuring that the subsequent search phase operates in linear time, $O(n)$, with respect to the text length $n$. This efficiency makes the KMP algorithm particularly useful for large-scale string matching tasks.

Spin-valve structures are a type of magnetic sensor that exploit the phenomenon of spin-dependent scattering of electrons. These devices typically consist of two ferromagnetic layers separated by a non-magnetic metallic layer, often referred to as the spacer. When a magnetic field is applied, the relative orientation of the magnetizations of the ferromagnetic layers changes, leading to variations in electrical resistance due to the Giant Magnetoresistance (GMR) effect.

The key principle behind spin-valve structures is that electrons with spins aligned with the magnetization of the ferromagnetic layers experience lower scattering, resulting in higher conductivity. In contrast, electrons with opposite spins face increased scattering, leading to higher resistance. This change in resistance can be expressed mathematically as:

where $R(H)$ is the resistance as a function of magnetic field $H$, $R_{AP}$ is the resistance in the antiparallel state, $R_{P}$ is the resistance in the parallel state, and $H_{C}$ is the critical field. Spin-valve structures are widely used in applications such as hard disk drives and magnetic random access memory (MRAM) due to their sensitivity and efficiency.

The Ricardian Model of international trade, developed by economist David Ricardo, emphasizes the concept of comparative advantage. This model posits that countries should specialize in producing goods for which they have the lowest opportunity cost, leading to more efficient resource allocation on a global scale. For instance, if Country A can produce wine more efficiently than cloth, and Country B can produce cloth more efficiently than wine, both countries benefit by specializing and trading with each other.

Mathematically, if we denote the opportunity costs of producing goods as $OC_{wine}$ and $OC_{cloth}$, countries will gain from trade if:

This principle allows for increased overall production and consumption, demonstrating that trade not only maximizes individual country's outputs but also enhances global economic welfare.

Eigenvalue Perturbation Theory is a mathematical framework used to study how the eigenvalues and eigenvectors of a linear operator change when the operator is subject to small perturbations. Given an operator $A$ with known eigenvalues $\lambda_n$ and eigenvectors $v_n$, if we consider a perturbed operator $A + \epsilon B$ (where $\epsilon$ is a small parameter and $B$ represents the perturbation), the theory provides a systematic way to approximate the new eigenvalues and eigenvectors.

The first-order perturbation theory states that the change in the eigenvalue can be expressed as:

where $\langle \cdot, \cdot \rangle$ denotes the inner product. For the eigenvectors, the first-order correction can be represented as:

This theory is particularly useful in quantum mechanics, structural analysis, and various applied fields, where systems are often subjected to small changes.