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Let f(x,y)=u(x,y)+iv(x,y), (1) where z=x+iy, (2) so dz=dx+idy. (3) The total derivative of f with respect to z is then (df)/(dz) = ...

A fundamental system of logic based on the concept of a generalized function whose argument is also a function (Schönfinkel 1924). This mathematical discipline was ...

Given an expression involving known constants, integration in finite terms, computation of limits, etc., the constant problem is the determination of if the expression is ...

Einstein summation is a notational convention for simplifying expressions including summations of vectors, matrices, and general tensors. There are essentially three rules of ...

Euler (1772ab) conjectured that there are no positive integer solutions to the quartic Diophantine equation A^4=B^4+C^4+D^4. This conjecture was disproved by Elkies (1988), ...

Pick any two relatively prime integers h and k, then the circle C(h,k) of radius 1/(2k^2) centered at (h/k,+/-1/(2k^2)) is known as a Ford circle. No matter what and how many ...

Guilloché patterns are spirograph-like curves that frame a curve within an inner and outer envelope curve. They are used on banknotes, securities, and passports worldwide for ...

A hexagon tiling is a tiling of the plane by identical hexagons. The regular hexagon forms a regular tessellation, also called a hexagonal grid, illustrated above. There are ...

In statistical mechanics, the two-dimensional Ising model is a popular tool used to study the dipole moments of magnetic spins. The Ising model in two dimensions is a type of ...

An L-algebraic number is a number theta in (0,1) which satisfies sum_(k=0)^nc_kL(theta^k)=0, (1) where L(x) is the Rogers L-function and c_k are integers not all equal to 0 ...