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Runge’S Approximation Theorem

Runge's Approximation Theorem ist ein bedeutendes Resultat in der Funktionalanalysis und der Approximationstheorie, das sich mit der Approximation von Funktionen durch rationale Funktionen beschäftigt. Der Kern des Theorems besagt, dass jede stetige Funktion auf einem kompakten Intervall durch rationale Funktionen beliebig genau approximiert werden kann, vorausgesetzt, dass die Approximation in einem kompakten Teilbereich des Intervalls erfolgt. Dies wird häufig durch die Verwendung von Runge-Polynomen erreicht, die eine spezielle Form von rationalen Funktionen sind.

Ein wichtiger Aspekt des Theorems ist die Identifikation von Rationalen Funktionen als eine geeignete Klasse von Funktionen, die eine breite Anwendbarkeit in der Approximationstheorie haben. Wenn beispielsweise fff eine stetige Funktion auf einem kompakten Intervall [a,b][a, b][a,b] ist, gibt es für jede positive Zahl ϵ\epsilonϵ eine rationale Funktion R(x)R(x)R(x), sodass:

∣f(x)−R(x)∣<ϵfu¨r alle x∈[a,b]|f(x) - R(x)| < \epsilon \quad \text{für alle } x \in [a, b]∣f(x)−R(x)∣<ϵfu¨r alle x∈[a,b]

Dies zeigt die Stärke von Runge's Theorem in der Approximationstheorie und seine Relevanz in verschiedenen Bereichen wie der Numerik und Signalverarbeitung.

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Bode Gain Margin

The Bode Gain Margin is a critical parameter in control theory that measures the stability of a feedback control system. It represents the amount of gain increase that can be tolerated before the system becomes unstable. Specifically, it is defined as the difference in decibels (dB) between the gain at the phase crossover frequency (where the phase shift is -180 degrees) and a gain of 1 (0 dB). If the gain margin is positive, the system is stable; if it is negative, the system is unstable.

To express this mathematically, if G(jω)G(j\omega)G(jω) is the open-loop transfer function evaluated at the frequency ω\omegaω where the phase is -180 degrees, the gain margin GMGMGM can be calculated as:

GM=20log⁡10(1∣G(jω)∣)GM = 20 \log_{10} \left( \frac{1}{|G(j\omega)|} \right)GM=20log10​(∣G(jω)∣1​)

where ∣G(jω)∣|G(j\omega)|∣G(jω)∣ is the magnitude of the transfer function at the phase crossover frequency. A higher gain margin indicates a more robust system, providing a greater buffer against variations in system parameters or external disturbances.

Neural Mass Modeling

Neural Mass Modeling (NMM) is a theoretical framework used to describe the collective behavior of large populations of neurons in the brain. It simplifies the complex dynamics of individual neurons into a set of differential equations that represent the average activity of a neural mass, allowing researchers to investigate the macroscopic properties of neural networks. Key features of NMM include the ability to model oscillatory behavior, synchronization phenomena, and the influence of external inputs on neural dynamics. The equations often take the form of coupled oscillators, where the state of the neural mass can be described using variables such as population firing rates and synaptic interactions. By using NMM, researchers can gain insights into various neurological phenomena, including epilepsy, sleep cycles, and the effects of pharmacological interventions on brain activity.

Muon Tomography

Muon Tomography is a non-invasive imaging technique that utilizes muons, which are elementary particles similar to electrons but with a much greater mass. These particles are created when cosmic rays collide with the Earth's atmosphere and are capable of penetrating dense materials like rock and metal. By detecting and analyzing the scattering and absorption of muons as they pass through an object, researchers can create detailed images of its internal structure.

The underlying principle is based on the fact that muons lose energy and are deflected when they interact with matter. The data collected from multiple muon detectors allows for the reconstruction of three-dimensional images using algorithms similar to those in traditional X-ray computed tomography. This technique has valuable applications in various fields, including archaeology for scanning ancient structures, nuclear security for detecting hidden materials, and geology for studying volcanic activity.

Solow Residual Productivity

The Solow Residual Productivity, named after economist Robert Solow, represents a measure of the portion of output in an economy that cannot be attributed to the accumulation of capital and labor. In essence, it captures the effects of technological progress and efficiency improvements that drive economic growth. The formula to calculate the Solow residual is derived from the Cobb-Douglas production function:

Y=A⋅Kα⋅L1−αY = A \cdot K^\alpha \cdot L^{1-\alpha}Y=A⋅Kα⋅L1−α

where YYY is total output, AAA is the total factor productivity (TFP), KKK is capital, LLL is labor, and α\alphaα is the output elasticity of capital. By rearranging this equation, the Solow residual AAA can be isolated, highlighting the contributions of technological advancements and other factors that increase productivity without requiring additional inputs. Therefore, the Solow Residual is crucial for understanding long-term economic growth, as it emphasizes the role of innovation and efficiency beyond mere input increases.

Nonlinear System Bifurcations

Nonlinear system bifurcations refer to qualitative changes in the behavior of a nonlinear dynamical system as a parameter is varied. These bifurcations can lead to the emergence of new equilibria, periodic orbits, or chaotic behavior. Typically, a system described by differential equations can undergo bifurcations when a parameter λ\lambdaλ crosses a critical value, resulting in a change in the number or stability of equilibrium points.

Common types of bifurcations include:

  • Saddle-Node Bifurcation: Two fixed points collide and annihilate each other.
  • Hopf Bifurcation: A fixed point loses stability and gives rise to a periodic orbit.
  • Transcritical Bifurcation: Two fixed points exchange stability.

Understanding these bifurcations is crucial in various fields, such as physics, biology, and economics, as they can explain phenomena ranging from population dynamics to market crashes.

Arrow'S Impossibility

Arrow's Impossibility Theorem, formulated by economist Kenneth Arrow in 1951, addresses the challenges of social choice theory, which deals with aggregating individual preferences into a collective decision. The theorem states that when there are three or more options, it is impossible to design a voting system that satisfies a specific set of reasonable criteria simultaneously. These criteria include unrestricted domain (any individual preference order can be considered), non-dictatorship (no single voter can dictate the group's preference), Pareto efficiency (if everyone prefers one option over another, the group's preference should reflect that), and independence of irrelevant alternatives (the ranking of options should not be affected by the presence of irrelevant alternatives).

The implications of Arrow's theorem highlight the inherent complexities and limitations in designing fair voting systems, suggesting that no system can perfectly translate individual preferences into a collective decision without violating at least one of these criteria.