Related papers: A Simple Proof of the Quadratic Formula
An analysis of a fractional cubic differential equation is presented, which is a generalization of different versions of fractional logistic equations, in order to obtain simpler numerical methods that globalize and extend the results…
We provide a sufficient condition for solvability of a system of real quadratic equations $p_i(x)=y_i$, $i=1, \ldots, m$, where $p_i: {\mathbb R}^n \longrightarrow {\mathbb R}$ are quadratic forms. By solving a positive semidefinite…
The purpose of this article is to compute the normal form of a class of general quadratic Hamiltonian systems that generalizes naturally Euler's equations from the free rigid body dynamics.
In this paper we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables $Q(x_1,\,x_2,\,x_3,\,x_4)=0$ can be expressed in terms of bilinear forms in four parameters. We use this…
This manuscript reviews theoretical results and applications related to quadratic forms in Gaussian random variables. It summarizes definitions, canonical representations, exact and approximate distributional results, numerical inversion…
It is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations, members of which are compatible between one another on a higher dimensional lattice. There turn out to be two such families, a…
This note is purely expository. The statement of the Gauss theorem on the constructibility of regular polygons by means of compass and ruler is simple and well-known. However, its proofs given in most textbooks rely upon much unmotivated…
We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.
We give several descriptions of positive quadrature formulas which are exact for trigonometric -, respectively, Laurent polynomials of degree less or equal $n-1-m$, $0\leq m\leq n-1$. A complete and simple description is obtained with the…
This paper provides a new simple proof of Hesse's theorem in projective geometry for any dimension.
We present a new proof of the celebrated quadratic reciprocity law. Our proof is based on group theory.
A robust, fast and accurate method for solving the Colebrook-like equations is presented. The algorithm is efficient for the whole range of parameters involved in the Colebrook equation. The computations are not more demanding than…
We give a half-page proof of the Lagrange-Good formula, using the Fourier representation of Dirac delta function.
We give a pen and paper and (comparatively) much simpler proof to verify of the Four Colour Theorem.
We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete.
This paper proposes a totally constructive approach for the proof of Hilbert's theorem on ternary quartic forms. The main contribution is the ladder technique, with which the Hilbert's theorem is proved vividly.
A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to…
The Doppler effect has many applications in science and engineering fields. Although the format of the classical Doppler effect equation is simple, the derivation for the equation in physics textbooks is not intuitive to many students. This…
Using the circle method, we obtain asymptotic formulae for the number of integer solutions to certain quadratic polynomials that are uniform in the coefficients of the polynomial.
This article provides a simple trigonometric method for determining how many roots of a quartic equation are real and how many are complex, without solving the equation. The approach replaces the quartic's classical discriminant -- a…