Introduction to Real Coefficients

The dispersion relation is explicitly established like a fifth degree algebraic equation with real coefficients and hence it always allows some real roots, thus predicting existence of some standing waves.

Presented is a very detailed two-tier analysis of the location of the real roots of the general quartic equation $$x^4 + a x^3 + b x^2 + c x + d = 0$$ x 4 + a x 3 + b x 2 + c x + d = 0 with real coefficients and the classification of the roots in terms of a, b, c, and d, without using any numerical approximations.

In this case, it refers to generic rational functions stating that a generic rational function $ R : \mathbb{C}\mathbb{P}^1 \rightarrow \mathbb{C}\mathbb{P}^1$ with only real critical points can be transformed by post-composition with an automorphism of $\mathbb{C}\mathbb{P}^1$ into a quotient of polynomials with real coefficients.

Let n≥ 1 and Pn be the vector space of all polynomials of degree n or less with real coefficients.

In this paper, we consider two classes of analytic functions with real coefficients.

Thus, $\xi(s)=0$ corresponds to an algebraic equation with real coefficients and infinite degree.

Phase domains are described by Ginzburg-Landau amplitude equations with real coefficients.

Lyubashevsky about an efficient algorithm for calculating the parameter θ(f) that characterizes the value of the sup-norm of the product of elements of the ring of truncated polynomials modulo a given unitary polynomial f (x) with real coefficients.

It is possible to transfer the results onto the class KR(i ) of functions convex in the direction of the imaginary axis with real coefficients as well as onto the class T of typically real functions.

Obtained expression is explicit and it does not require any iterative calculation as it is built reposing on the analytical solution of a cubic equation of real coefficients.

In previous studies, a system of linear differential equations with real coefficients, but with interval forcing terms and interval initial values was investigated.

We provide a Sage script to do so, and extend our result to compute Euler products $\prod_{p\in\mathcal{A}}F(1/p)/G(1/p)$ where $F$ and $G$ are polynomials with real coefficients, when this product converges absolutely.

The initial conditions for the implementation of the former are the real coefficients of the nth degree polynomial and the signal amplitude.

For four masses, the homology groups for this energy range are fully determined; for arbitrary N, the homology with real coefficients is determined.

We identify "critical" families of scalarized black hole solutions such that the expansion of the metric functions and of the scalar field at the horizon no longer allows for real coefficients.

The positive semidefinite Gram matrices of a form f with real coefficients parametrize the sum-of-squares representations of f.

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity.

A recent paper of Braverman, Cohen, and Garg (STOC 2018) introduced the concept of a weighted pseudorandom generator (WPRG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with real coefficients that scale the acceptance probabilities of any potential distinguisher.

In this paper, for polynomials with real coefficients P , Q P,Q satisfying ∣ P ( x ) ∣ ≤ ∣ Q ( x ) ∣ | P\left(x)| \le | Q\left(x)| for each x x in a real interval I I , we prove the bound L ( P ) ≤ c L ( Q ) L\left(P)\le cL\left(Q) between the lengths of P P and Q Q with a constant c c , which is exponential in the degree d d of P P.

Networks of genetic expression can be modelled by hypergraphs with the additional structure that real coefficients are given to each vertex-edge incidence.