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Related Concept Videos

Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
Network Function of a Circuit01:25

Network Function of a Circuit

Frequency response analysis in electrical circuits provides vital insights into a circuit's behavior as the frequency of the input signal changes. The transfer function, a mathematical tool, is instrumental in understanding this behavior. It defines the relationship between phasor output and input and comes in four types: voltage gain, current gain, transfer impedance, and transfer admittance. The critical components of the transfer function are the poles and zeros.
First Order Systems01:21

First Order Systems

First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...
Types of Functions III01:28

Types of Functions III

Logarithmic and piecewise functions play central roles in mathematical modeling, particularly when capturing nonlinear or segmented behaviors in real-world phenomena. Although these functions differ fundamentally in structure and application, both serve to represent complex relationships in simplified mathematical terms.A logarithmic function is defined as the inverse of an exponential function, expressed as These functions grow quickly for small values of x but slow down as x increases,...
Frequency Response of a Circuit01:20

Frequency Response of a Circuit

Inductive circuits present intriguing challenges in electrical engineering, particularly during the transition from the time domain to the frequency domain. This transformation involves converting inductors into impedances and utilizing phasor representation.
The transfer function is pivotal in characterizing how these circuits react to various frequencies, facilitating a profound understanding of their behavior. An essential parameter is the time constant, signifying the...
Second Order systems II01:18

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
If  ζ...

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Related Experiment Video

Updated: May 11, 2026

Data Acquisition Protocol for Determining Embedded Sensitivity Functions
07:46

Data Acquisition Protocol for Determining Embedded Sensitivity Functions

Published on: April 20, 2016

Fukui function and response function for nonlocal and fractional systems.

Degao Peng1, Weitao Yang

  • 1Department of Chemistry, Duke University, Durham, North Carolina 27708, USA.

The Journal of Chemical Physics
|May 17, 2013
PubMed
Summary
This summary is machine-generated.

This study extends Fukui functions and linear-response functions to fractional and integer systems, providing new analytical derivatives. These findings enhance understanding of energy functionals and density functional theory development.

Related Experiment Videos

Last Updated: May 11, 2026

Data Acquisition Protocol for Determining Embedded Sensitivity Functions
07:46

Data Acquisition Protocol for Determining Embedded Sensitivity Functions

Published on: April 20, 2016

Area of Science:

  • Quantum Chemistry
  • Theoretical Chemistry
  • Materials Science

Background:

  • Density-functional theory (DFT) relies on energy functionals.
  • Fukui functions and linear-response functions are key descriptors in DFT.
  • Previous work established foundational concepts for these functions.

Purpose of the Study:

  • To extend existing Fukui functions and linear-response functions to fractional and integer systems.
  • To derive and verify new analytical derivatives for energy functionals.
  • To establish real-space dependency for exact conditions in DFT.

Main Methods:

  • Analytical derivation of second- and third-order energy derivatives.
  • Extension of Fukui functions and linear-response functions to fractional systems.
  • Verification using finite difference numerical derivatives.
  • Introduction of nonlocal potentials to define nonlocal functions.

Main Results:

  • Successfully extended second-order derivatives (linear-response, Fukui function, chemical hardness) to fractional systems.
  • Obtained third-order derivatives (second-order response, Fukui response, dual descriptor, hyperhardness) for integer systems.
  • Demonstrated real-space dependency for exact conditions on energy functionals.
  • Defined nonlocal Fukui and linear-response functions using external nonlocal potentials.

Conclusions:

  • The derived analytical derivatives significantly enrich information on exact conditions for energy functionals.
  • Nonlocal linear-response functions provide precise meaning for time-dependent DFT with generalized Kohn-Sham functionals.
  • These extensions are valuable for conceptual DFT and the development of new density functionals.