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Fractal Measure

Open set condition

In fractal geometry, the open set condition (OSC) is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction. Specifically, given an iterated function system of contractive mappings ψ_i, the open set condition requires that there exists a nonempty, open set S satisfying two conditions:

1. $\bigcup _{i=1}^{m}\psi _{i}(V)\subseteq V,$
2. Each $\psi _{i}(V)$ is pairwise disjoint.

Introduced in 1946 by P.A.P Moran, the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.

An equivalent statement of the open set condition is to require that the s-dimensional Hausdorff measure of the set is greater than zero.

Computing Hausdorff measure

When the open set condition holds and each ψ_i is a similitude (that is, a composition of an isometry and a dilation around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for s of the following:

$\sum _{i=1}^{m}r_{i}^{s}=1.$

where r_i is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three non-collinear points a₁, a₂, a₃ in the plane R² and let ψ_i be the dilation of ratio 1/2 around a_i. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension s is the unique solution of

$\left({\frac {1}{2}}\right)^{s}+\left({\frac {1}{2}}\right)^{s}+\left({\frac {1}{2}}\right)^{s}=3\left({\frac {1}{2}}\right)^{s}=1.$

Taking natural logarithms of both sides of the above equation, we can solve for s, that is: s = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

Hand-eye calibration problem

Hand-eye calibration problem

In robotics, the hand-eye calibration problem, or the robot-sensor calibration problem, is the problem of determining the transformation between a robot end-effector and a camera or the transformation between a robot base and the world coordinate system. It takes the form of AX=ZB, where A and B are two systems, usually a robot base and a camera, and X and Z are unknown transformation matrices. A highly studied special case of the problem occurs where X=Z, taking the form of the problem AX=XB. Solutions to the problem take the forms of several types of methods, including "separable closed-form solutions, simultaneous closed-form solutions, and iterative solutions". The covariance of X in the equation can be calculated for any randomly perturbed matrices A and B.

Methods

Many different methods and solutions developed to solve the problem, broadly defined as either Separable, simultaneous solutions. Each type of solution has specific advantages and disadvantages as well as formulations and applications to the problem. A common theme throughout all of the methods is the common use of quaternions to represent rotation matrices.

Separable solutions

Given the equation AX=ZB, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where R_A represents a 3×3 rotation matrix and t_A a 3×1 translation vector, the equation can be broken into two parts:

R_AR_X=R_ZR_B

R_At_X+t_A=R_Zt_B+t_Z

Equation 2 becomes linear if R_Z is known. As such, the most frequent approach is to R_x and R_z using the first equation then using it to solve for the second two variables in the second equation. Rotation is represented using quaternions, allowing for a linear solution to be found. While separable methods are useful, any error in the estimation for the rotation matrices is compounded when being applied to the translation vector. Other solutions avoid this problem.

Simultaneous solutions

Simultaneous solutions are based on solving for both X and Z at the same time (rather than basing the solution of one part off of the other as in seperable solutions), propogation of error is significantly reduced. By formulating the matrices as dual quaternions, it is possible to get a linear equation by which X is solvable in a linear format. An alternative way applies the least squares method to the Kronecker product of the matrices A⊗B. As confirmed by experimental results, simultaneous solutions have less error than seperable quaternion solutions.

Iterative solutions

Iterative solutions are another method used to solve the problem of error propagation. One example of an iterative solution is a program based on minimizing ||AX−XB||. As the program iterates, it will converge on a solution to X independent to the initial robot orientation of R_B. Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into dual quaternions. However, while iterative solutions to the problem are generally simultaneous and accurate, they can be computationally taxing to carry out and may not always converge on the optimal solution.

• [1] - Octonion solution

What is algebra?

What is Algebra

Algebra is a complex branch of mathematics in which many subjects are vastly different from others. Essentially, algebra is manipulation of symbols and operations based on given properties about them. For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.

The most simple parts of algebra begin with computations similar to those of arithmetic but with variables that take on the properties of numbers. This allows proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

$ax^{2}+bx+c=0,$

where $a,b,c$ are any given numbers (except that $a$ cannot be $0$), the quadratic formula can be used to find the two unique values of the unknown quantity $x$ which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the quadratic equation above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry.

Algebra also considers entities that do not stand for just one number; using sets of numbers as algebras results in the ability to define relations between objects such as vectors, matrices, and polynomials. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra.

Because an entity can be anything with well defined properties, it is possible to define entities that are unlike any set of real or complex numbers. These entities are created using only their properties, and involve strict definitions to create a set. The entities, along with defined operations, form algebraic structures such as groups, rings, and fields. Abstract algebra is the study of these entities and more.

In geometry, algebra can be used in the manipulation of geometric properties; the interplay between geometry and algebra allows for studies of geometric structures such as constructible numbers and singularities. Reducing properties of geometric structures into algebraic structures has created subjects such as algebraic geometry, geometric algebra, and algebraic topology.

Today, the study of algebra includes many branches of mathematics, as can be seen in the Mathematics Subject Classification where none of the first level areas (two digit entries) is called algebra. Algebra instead includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used in 14-Algebraic geometry and 11-Number theory via algebraic number theory.

Antiassociative algebra

Antiassociative algebra

An algebra antiassociative if (xy)z = -x(yz) for every case of x,y, and z.

Ugandan Knuckles

Ugandan knuckles

Ugandan Knuckles is an internet meme from January 2018 depicting a deformed version of Knuckles the Echidna. Players would go in hords to the virtual reality video game VRChat to troll other players. The people would say quotes such as "Do you know the way?", which originate from the 2010 Ugandan action film Who Killed Captain Alex?, as well as "spitting" on other users whom they felt did not know "de way". The meme was a significant trend followed by several news organisations, including USA Today.

History

On February 20 2017, YouTuber Gregzilla uploaded a video on Sonic Lost World featuring a parody picture of Knuckles the Echidna. On December 22 2017, a 3D model of the Knuckles sketch was released on DeviantArt. That day, YouTuber Stahlsby uploaded a video in which several VRChat players wearing the parody costume trolled others by making clicking noises and saying "You do not know the way". After that, more and more people flooded VRChat to troll others as Ugandan Knuckles, leading to controversy, as the mock Ugandan accent and quotations used were widely regarded as racist. However, The meme continued to gain popularity until about mid-January 2018, but had mostly subsided by February.

Controversy

Because of its use of a fake Ugandan accent as well as the quotations from Who Killed Captain Alex?, the meme was widely criticized for being racially insensitive; Polygon described it as problematic. On January 27 2018, the company Razer was brought under fire for posting a Ugandan Knuckles meme that was widely criticised as a racist misstep.

The original creator of the 3D avatar, DeviantArt user "tidiestflyer", showed regret over the character, saying that he hoped it would not be used to annoy players of VRChat and that he enjoys the game and does not want to see anyone's rights get taken away because of the avatar. In response to the trolling in the game, the developers of VRChat published an open letter on Medium, stating that they were developing "new systems to allow the community to better self moderate" and asking users to use the built-in muting features.