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A conjecture of Erdős which has attracted considerable interest concerns the maximum length of a polynomial lemniscate ƒ(x, y) = 1 of degree 2n when p is monic, which Erdős conjectured was attained when p(z) = z n − 1. This is still not proved but Fryntov and Nazarov proved that p gives a local maximum. [1]
The inverse lemniscate sine also describes the arc length s relative to the x coordinate of the rectangular elastica. [36] This curve has y coordinate and arc length: y = ∫ x 1 t 2 d t 1 − t 4 , s = arcsl x = ∫ 0 x d t 1 − t 4 {\displaystyle y=\int _{x}^{1}{\frac {t^{2}\mathop {\mathrm {d} t} }{\sqrt {1-t^{4}}}},\quad s ...
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
[1] [2] Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. For the sake of simplicity, we confine our attention to the case of a function of two independent variables x and y for the moment. Consider a quasilinear PDE of the form [3]
In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2. Generalizations to more variables yield ...
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
Pell's equation for n = 2 and six of its integer solutions. Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form =, where n is a given positive nonsquare integer, and integer solutions are sought for x and y.
All non-isomorphic graphs on 3 vertices and their chromatic polynomials, clockwise from the top. The independent 3-set: k 3. An edge and a single vertex: k 2 (k – 1). The 3-path: k(k – 1) 2. The 3-clique: k(k – 1)(k – 2). The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics.