Shannon Entropy Formula

The Dirac Delta function, denoted as $\delta(x)$, is a mathematical construct that is not a function in the traditional sense but rather a distribution. It is defined to have the property that it is zero everywhere except at $x = 0$, where it is infinitely high, such that the integral over the entire real line equals one:

This unique property makes the Dirac Delta function extremely useful in physics and engineering, particularly in fields like signal processing and quantum mechanics. It can be thought of as representing an idealized point mass or point charge, allowing for the modeling of concentrated sources. In practical applications, it is often used to simplify the analysis of systems by replacing continuous functions with discrete spikes at specific points.

A Wavelet Matrix is a data structure that efficiently represents a sequence of elements while allowing for fast query operations, particularly for range queries and frequency counting. It is constructed using wavelet transforms, which decompose a dataset into multiple levels of detail, capturing both global and local features of the data. The structure is typically represented as a binary tree, where each level corresponds to a wavelet transform of the original data, enabling efficient storage and retrieval.

The key operations supported by a Wavelet Matrix include:

• Rank Query: Counting the number of occurrences of a specific value up to a given position.
• Select Query: Finding the position of the $k$-th occurrence of a specific value.

These operations can be performed in logarithmic time relative to the size of the input, making Wavelet Matrices particularly useful in applications such as string processing, data compression, and bioinformatics, where efficient data handling is crucial.

The Lamb Shift refers to a small difference in energy levels of hydrogen atoms that cannot be explained by the Dirac equation alone. This shift arises due to the interactions between the electron and the vacuum fluctuations of the electromagnetic field, a phenomenon explained by quantum electrodynamics (QED). The derivation involves calculating the energy levels of the hydrogen atom while accounting for the effects of these vacuum fluctuations, leading to a correction in the energy levels of the 2S and 2P states.

The energy correction can be expressed as:

where $\alpha$ is the fine-structure constant, $m_e$ is the electron mass, $c$ is the speed of light, and $n$ is the principal quantum number. The Lamb Shift is significant not only for its implications in atomic physics but also as an experimental verification of QED, illustrating the profound effects of quantum mechanics on atomic structure.

Endogenous Money Theory (EMT) within the Post-Keynesian framework posits that the supply of money is determined by the demand for loans rather than being fixed by the central bank. This theory challenges the traditional view of money supply as exogenous, emphasizing that banks create money through lending when they extend credit to borrowers. As firms and households seek financing for investment and consumption, banks respond by generating deposits, effectively increasing the money supply.

In this context, the relationship can be summarized as follows:

• Demand for loans drives money creation: When businesses want to invest, they approach banks for loans, prompting banks to create money.
• Interest rates are influenced by the supply and demand for credit, rather than being solely controlled by central bank policies.
• The role of the central bank is to ensure liquidity in the system and manage interest rates, but it does not directly control the total amount of money in circulation.

This understanding of money emphasizes the dynamic interplay between financial institutions and the economy, showcasing how monetary phenomena are deeply rooted in real economic activities.

Recurrent Networks, oder rekurrente neuronale Netze (RNNs), sind eine spezielle Art von neuronalen Netzen, die besonders gut für die Verarbeitung von sequenziellen Daten geeignet sind. Im Gegensatz zu traditionellen Feedforward-Netzen, die nur Informationen in eine Richtung fließen lassen, ermöglichen RNNs Feedback-Schleifen, sodass sie Informationen aus vorherigen Schritten speichern und nutzen können. Diese Eigenschaft macht RNNs ideal für Aufgaben wie Textverarbeitung, Sprachverarbeitung und zeitliche Vorhersagen, wo der Kontext aus vorherigen Eingaben entscheidend ist.

Die Funktionsweise eines RNNs kann mathematisch durch die Gleichung

beschrieben werden, wobei $h_t$ der versteckte Zustand zum Zeitpunkt $t$, $x_t$ der Eingabewert und $f$ eine Aktivierungsfunktion ist. Ein häufiges Problem, das bei RNNs auftritt, ist das Vanishing Gradient Problem, das die Fähigkeit des Netzwerks beeinträchtigen kann, langfristige Abhängigkeiten zu lernen. Um dieses Problem zu mildern, wurden Varianten wie Long Short-Term Memory (LSTM) und Gated Recurrent Units (GRUs) entwickelt, die spezielle Mechanismen enthalten, um Informationen über längere Zeiträume zu speichern.

Ergodic Theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. It primarily focuses on the long-term average behavior of systems evolving over time, providing insights into how these systems explore their state space. In particular, it investigates whether time averages are equal to space averages for almost all initial conditions. This concept is encapsulated in the Ergodic Hypothesis, which suggests that, under certain conditions, the time spent in a particular region of the state space will be proportional to the volume of that region. Key applications of Ergodic Theory can be found in statistical mechanics, information theory, and even economics, where it helps to model complex systems and predict their behavior over time.