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In music, if a note has frequency f, integer multiples of that frequency, 2f,3f,4f and so on, are known as harmonics. As a result, the mathematical study of overlapping waves ...

Given collinear points W, X, Y, and Z, Y and Z are harmonic conjugates with respect to W and X if (|WY|)/(|YX|)=(|WZ|)/(|XZ|). (1) W and X are also harmonic conjugates with ...

The harmonic conjugate to a given function u(x,y) is a function v(x,y) such that f(x,y)=u(x,y)+iv(x,y) is complex differentiable (i.e., satisfies the Cauchy-Riemann ...

For all integers n and nonnegative integers t, the harmonic logarithms lambda_n^((t))(x) of order t and degree n are defined as the unique functions satisfying 1. ...

A device consisting of two coupled pendula, usually oscillating at right angles to each other, which are attached to a pen. The resulting motion can produce beautiful, ...

Let s_i be the orders of singular points on a curve (Coolidge 1959, p. 56). Harnack's first theorem states that a real irreducible curve of order n cannot have more than ...

A linkage which draws the inverse of a given curve. It can also convert circular to linear motion. The rods satisfy AB=CD and BC=DA, and O, P, and P^' remain collinear while ...

Any one of the eight Apollonius circles of three given circles is tangent to a circle H known as a Hart circle, as are the other three Apollonius circles having (1) like ...

The hat is a caret-shaped symbol commonly placed on top of variables to give them special meaning. The symbol x^^ is voiced "x-hat" (or sometimes as "x-roof") in mathematics, ...

The hat polykite is an aperiodic monotile discovered by Smith et al. (2023). It is illustrated above in an aperiodic tiling (Smith et al. 2023).