inverse

Examples of inverse in the following topics:

• Matrix Inverses

  • A left inverse of a matrix $A\in \mathbf{R}^{n \times m}$ is defined to be a matrix $B$ such that
  • For example, given a right inverse $A$ , then since $AC=I$ , we have $ACy=y$ .
  • is a left inverse.
  • But there are infinitely many other left inverses.
  • is a right inverse.

• Inverse Compton Scattering

  • Inverse Compton scattering corresponds to the situation where the photon gains energy from the electron because the electron is in motion.

• Discrete Fourier Transform Examples

  • What we will do is construct an unknown time series' DFT by hand and inverse transform to see what the resulting time series looks like.
  • Figures 4.10 and 4.11 show the real (left) and imaginary (right) parts of six time series that resulted from inverse DFT'ing an array $H_n$ which was zero except at a single point (i.e., it's a Kronecker delta: $H_i = \delta _{i,j} =1$ and zero otherwise; here a different $j$ is chosen for each plot).
  • The real part of the inverse DFT of this convolved signal is shown in the lower right plot.
  • These time series are reconstructed from the spectra by inverse DFT.

• The Law of Universal Gravitation

  • Objects with mass feel an attractive force that is proportional to their masses and inversely proportional to the square of the distance.
  • Theorizing that this force must be proportional to the masses of the two objects involved, and using previous intuition about the inverse-square relationship of the force between the earth and the moon, Newton was able to formulate a general physical law by induction.
  • The Law of Universal Gravitation states that every point mass attracts every other point mass in the universe by a force pointing in a straight line between the centers-of-mass of both points, and this force is proportional to the masses of the objects and inversely proportional to their separation This attractive force always points inward, from one point to the other.
  • The force is proportional to the masses and inversely proportional to the square of the distance.

• Examples of Least Squares

  • Let's make sure we get the same thing using the generalized inverse approach.
  • So the generalized inverse solution (i.e., the least squares solution) is

• Current and Voltage Measurements in Circuits

  • The electrical current is directly proportional to the voltage applied and inversely related to the resistance in a circuit.
  • Resistance is inversely proportional to current.

• Gauss's Law

  • In fact, any "inverse-square law" can be formulated in a way similar to Gauss's law: For example, Gauss's law itself is essentially equivalent to the inverse-square Coulomb's law, and Gauss's law for gravity is essentially equivalent to the inverse-square Newton's law of gravity.

• Problems

  • What is the inverse Compton emission from a single electron passing through a gas of photons field in terms of the energy density of the photons and the Lorentz factor of the electron?
  • What is the total inverse Compton emission from the region if you assume that the synchrotron emission provides the seed photons for the inverse Compton emission?

• Introduction to The Four Fundamental Spaces

  • This subspace is called the nullspace or kernel and is extremely important from the point of view of inverse theory.
  • As we shall see, in an inverse calculation the right hand side of a matrix equations is usually associated with perturbations to the data.
  • Figuring out what features of a model are unresolved is a major goal of inversion.

• Electric Field and Changing Electric Potential

  • Electric field is the gradient of potential, which depends inversely upon distance of a given point of interest from a charge.
  • Electric field is the gradient of potential, which depends inversely upon distance of a given point of interest from a charge.
  • Moving towards and away from the charge results in change of potential; the relationship between distance and potential is inverse.