Examples of symmetrical in the following topics:
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- The Shell Theorem states that a spherically symmetric object affects other objects as if all of its mass were concentrated at its center.
- For highly symmetric shapes such as spheres or spherical shells, finding this point is simple.
- A spherically symmetric object affects other objects gravitationally as if all of its mass were concentrated at its center,
- If the object is a spherically symmetric shell (i.e., a hollow ball) then the net gravitational force on a body inside of it is zero.
- That is, a mass $m$ within a spherically symmetric shell of mass $M$, will feel no net force (Statement 2 of Shell Theorem).
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- In this way it can be shown that an object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its center.
- For points inside a spherically-symmetric distribution of matter, Newton's Shell theorem can be used to find the gravitational force.
- Thus, if a spherically symmetric body has a uniform core and a uniform mantle with a density that is less than $\frac{2}{3}$ of that of the core, then the gravity initially decreases outwardly beyond the boundary, and if the sphere is large enough, further outward the gravity increases again, and eventually it exceeds the gravity at the core/mantle boundary.
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- Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path.
- All projectile motion happens in a bilaterally symmetrical path, as long as the point of projection and return occur along the same horizontal surface.
- Bilateral symmetry means that the motion is symmetrical in the vertical plane.
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- A matrix which equals its transpose ( $A^T = A$ ) is said to be symmetric.
- If $A^T=-A$ the matrix is said to be skew-symmetric.
- We can split any square matrix $A$ into a sum of a symmetric and a skew-symmetric part via
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- $\langle \cos\theta_f'\rangle$ is also zero because the scatter photon is forward-backward symmetric in the rest-frame of the electron so we find that
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- An important class of matrices for inverse theory are the real symmetric matrices.
- And these two matrices are manifestly symmetric.
- In the case of real symmetric matrices, the eigenvector/ eigenvalue decomposition is especially nice, since in this case the diagonalizing matrix $S$ can be chosen to be an orthogonal matrix $Q$ .
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- Now returning to the Fourier transform, suppose the spectrum of our time series $f(t)$ is zero outside of some symmetric interval $[-2\pi f_s, 2\pi f_s]$ about the origin.
- (In fact the assumption that the interval is symmetric about the origin is made without loss of generality, since we can always introduce a change of variables which maps an arbitrary interval into a symmetric one centered on 0. ) In other words, the signal does not contain any frequencies higher than $f_s$ hertz.
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- A direct approach to the SVD, attributed to the physicist Lanczos, is to make a symmetric matrix out of the rectangular matrix $A$ as follows: Let
- And since $S$ is symmetric it has orthogonal eigenvectors $\mathbf{w}_i$$\lambda _ i$ with real eigenvalues $\lambda _ i$
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- If we put the symmetric $1/\sqrt{2 \pi}$ normalization in front of both transforms, we end up with a left-over factor of $1/\sqrt{2 \pi}$ because we started out with two Fourier transforms and we ended up with only one and a convolution.
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- (Technically such a matrix is called Hermitian or self-adjoint--the operation of taking the complex conjugate transpose being known at the adjoint--but we needn't bother with this distinction here. ) In our case, since $Q$ is obviously symmetric, we have: