Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music.This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music.
The opposite of symmetry is asymmetry, which refers to the absence or a violation of symmetry.
Assume that ##P## is a polynomial over a commutative ring ##R##. Then there exists a ring ##\tilde R## extending ##R## where ##P## splits into linear factor (not necessarily uniquely). This theorem, whose proof is given below, is difficult to find in the literature (if someone know a source, it...
Is calculating the Euler angles analitically possible?
I am trying to obtain the angles to transform the body-fixed reference frame to the inertial reference frame. I can get them without problems with numerical methods. But I would to validate them analitically, if possible.
I followed the...
Case 1)
Two rockets (no Earth involved) have an exactly the same acceleration profile/flight-plan during round trip but they dispatched to opposite directions. At the start both rockets are docked to the same space station...both rockets have an identical engine operation plan during the round...
The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials.
I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do...
Hi Guys,
at the moment I got a bit confused about the notation in some QM textbooks. Some say the operators should be symmetric, some say they should be self-adjoint (or in many cases hermitian what maybe means symmetric or maybe self-adjoint). Which condition do we need for our observables...
Homework Statement
Let K and L be symmetric PSD matrices of size N*N, with all entries in [0,1]. Let i be any number in 1...N and K’, L’ be two new symmetric PSD matrices, each with only row i and column i different from K and L. I would like to obtain an upper bound of the equation below...
I'm hoping someone can clarify for me, I have seen the following used:
\frac{\partial}{\partial g^{ab}}\left( g^{cd} \right) = \frac{1}{2} \left( \delta_a^c \delta_b^d + \delta_b^c \delta_a^d\right)
I understand the two half terms are used to account for the symmetry of the metric tensor...
Homework Statement
Let A=RxR=the set of all ordered pairs (x,y), where x and y are real numbers. Define relation P on A as follows: For all (x,y) and (z,w) in A, (x,y)P(z,w) iff x-y=z-w
Homework Equations
R is reflexive if, and only if, for all x ∈ A,x R x.
R is symmetric if, and only if, for...