Distinct Real Roots Research Articles

Expansion formulae for flexural-gravity wave propagation during wave blocking

Flexural-gravity wave blocking occurs in the presence of lateral compressive force. The existing expansion formulae for the velocity potential were limited in the case of distinct real roots of the dispersion relation with the multiplicity one. This limitation is overcome, and we obtain the complete solution for the flexural-gravity wave-maker problem at blocking and saddle points by deriving the expansion formula for velocity potential using the Fourier transform corresponding to the roots of the dispersion relation with multiplicity two and three, respectively. Further, we check the convergence of the newly derived expansion formulae by using the spectral representation of the associated eigenfunction. It has been proved that the eigenfunctions related to the expansion formulae are linearly dependent and satisfy the orthogonal mode-coupling relation. Next, we solve a model problem of flexural-gravity wave scattering by a crack using the derived expansion formulae and establish the energy balance relation by applying Green’s integral theorem at blocking points for infinite water depth. In addition, we have presented the validation of energy balance relation involving the generalized scattering coefficients.

Nilpotent Bicenters in Continuous Piecewise ℤ2-Equivariant Cubic Polynomial Hamiltonian Vector Fields: Cusp–Cusp Type

In this paper, we study the global dynamics for a class of continuous piecewise [Formula: see text]-equivariant cubic Hamiltonian vector fields with nilpotent bicenters at [Formula: see text]. We consider these polynomial vector fields with a challenging case where the bicenters [Formula: see text] come from the combination of two nilpotent cusps separated by [Formula: see text]. We call it a cusp–cusp type. We use the Poincaré compactification, the blow-up theory, the index theory and the theory of discriminant sequence for determining the number of distinct or negative real roots of a polynomial, to classify the global phase portraits of these vector fields in the Poincaré disc.

Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate

This paper uses Müntz orthogonal functions to numerically solve the fractional Bagley–Torvik equation with initial and boundary conditions. Müntz orthogonal functions are defined on the interval 0 , 1 and have simple and distinct real roots on this interval. For the function f ∈ L 2 0 , 1 , we obtain the best unique approximation using Müntz orthogonal functions. We obtain the Riemann–Liouville fractional integral operator for Müntz orthogonal functions so that we can reduce the complexity of calculations and increase the speed of solving the problem, which can be seen in the process of running the Maple program. To solve the fractional Bagley–Torvik equation with initial and boundary conditions, we use Müntz orthogonal functions and consider simple and distinct real roots of Müntz orthogonal functions as collocation points. By using the Riemann–Liouville fractional integral operator that we define for the Müntz orthogonal functions, the process of numerically solving the fractional Bagley–Torvik equation that is solved using Müntz orthogonal functions is reduced, and finally, we reach a system of algebraic equations. By solving algebraic equations and obtaining the vector of unknowns, the fractional Bagley–Torvik equation is solved using Müntz orthogonal functions, and the error value of the method can be calculated. The low error value of this numerical solution method shows the high accuracy of this method. With the help of the Müntz functions, we obtain the error bound for the approximation of the function. We have obtained the error bounds for the numerical method using which we solved the fractional Bagley–Torvik equation with initial and boundary conditions. Finally, we have given a numerical example to show the accuracy of the solution of the method presented in this paper. The results of solving this example using Müntz orthogonal functions and comparing the results with other methods that have been used the solve this example show the higher accuracy of the method proposed in this paper.

Real Morse Polynomials of Degree 5 and 6

A real polynomial p of degree n is called a Morse polynomial if its derivative has n−1 pairwise distinct real roots and values of p at these roots (critical values) are also pairwise distinct. The plot of such a polynomial is called a “snake.” By enumerating critical points and critical values in increasing order, we construct a permutation a1, . . . , an−1, where ai is the number of the polynomial’s value at the ith critical point. This permutation is called the passport of the snake (polynomial). In this work, for Morse polynomials of degree 5 and 6, we describe the partition of the coefficient space into domains of constant passport.

Extrema of polynomials with real roots and Diophantine equations

There are many results in the literature concerning polynomial values and (shifted) power values of polynomials with consecutive integer roots, or more generally, with roots forming an arithmetic progression. It is an interesting question that how far one can ‘disturb’ the structure of the roots such that the finiteness results still remain valid. Also there are many results into this direction, with adding or removing one or more terms (roots).In this paper we study a case where (part of) the symmetric root structure is preserved, however, we allow (possibly large) increasing gaps between the roots. We prove that the finiteness of the solutions can also be guaranteed under these generalized circumstances. In our proofs we combine Baker's method and the Bilu-Tichy theorem with a new result providing an increasing property of the extremal values of polynomials with distinct real roots satisfying certain symmetry and increasing gap properties.

Determining a Particular Solution for n-th Order Linear Differential Equations with Constant Coefficients that Have the Free Term in the Following form P(x). ln x and the Characteristic Equation Has Distinct Real Roots

Abstract For n-th order linear differential equations with constant coefficients the problem of determining the general solution involves determining the general solution of the attached homogeneous equation and, on the other hand, determining a particular solution. This paper aims to analyze the conditions under which the general solution can be obtained for a n-th order linear differential equation with constant coefficients which have the free term in the following form P(x). ln x under the condition that the attached characteristic equation has real and distinct roots, meaning the order of multiplicity for each root is 1. For this we will use the method of variation of constants which leads us to a system whose solution we will gain by using the Upper incomplete Gamma function.

On the stability and behavior of solutions in mixed differential equations with delays and advances

In this study, some new results are obtained on asymptotic behavior and stability analysis of the mixed type differential equation where , and are real nonnegative continuous functions on , and and are real valued functions of bounded variation on . These results were obtained by using an appropriate real root of the characteristic equation. Moreover, by the use of two appropriate distinct real roots of the corresponding characteristic equation, a new result on the behavior of solutions is established. Five examples are also given to illustrate our results. We also presented the application of the obtained results in the special case of constant coefficients and have given three different cases in one example. The results obtained in this article show that real roots play an important role.

New Solutions for an Elliptic Equation Method and Its Applications in Nonlinear Evolution Equations

In this paper, we study an elliptic equation with four distinct real roots and obtain five new solutions to this type of elliptic equation. Using these obtained new elliptic function solutions we can construct a series of explicit exact solutions for many nonlinear evolution equations. As examples, we choose combined KdV-MKdV equation, a fourth-order integrable nonlinear Schrödinger equation and generalized Dullin-Gottwald-Holm equation to demonstrate the effectiveness of these new elliptic function solutions. These new elliptic function solutions can be applied to many other nonlinear evolution equations.

ON RANDOM CONVEX CHAINS, ORTHOGONAL POLYNOMIALS, PF SEQUENCES AND PROBABILISTIC LIMIT THEOREMS

Let $T$ be the triangle in the plane with vertices $(0,0)$, $(0,1)$ and $(0,1)$. The convex hull of $(0,1)$, $(1,0)$ and $n$ independent random points uniformly distributed in $T$ is the random convex chain $T_n$. A three-term recursion for the probability generating function $G_n$ of the number $f_0(T_n)$ of vertices of $T_n$ is proved. Via the link to orthogonal polynomials it is shown that $G_n$ has precisely $n$ distinct real roots in $(-\infty,0]$ and that the sequence $p_k^{(n)}:=\mathbb{P}(f_0(T_n)=k)$, $k=1,\ldots,n$, is a Polya frequency (PF) sequence. A selection of probabilistic consequences of this surprising and remarkable fact are discussed in detail.

Univariate polynomials and the contractability of certain sets

We consider the set $\Pi ^*_d$ of monic polynomials $Q_d=x^d+\sum _{j=0}^{d-1}a_jx^j$, $x\in \mathbb{R}$, $a_j\in \mathbb{R}^*$, having $d$ distinct real roots, and its subsets defined by fixing the signs of the coefficients $a_j$. We show that for every choice of these signs, the corresponding subset is non-empty and contractible. A similar result holds true in the cases of polynomials $Q_d$ of even degree $d$ and having no real roots or of odd degree and having exactly one real root. For even $d$ and when $Q_d$ has exactly two real roots which are of opposite signs, the subset is contractible. For even $d$ and when $Q_d$ has two positive (resp. two negative) roots, the subset is contractible or empty. It is empty exactly when the constant term is positive, among the other even coefficients there is at least one which is negative, and all odd coefficients are positive (resp. negative).

Birinci mertebeden gecikmeli integro-diferansiyel denklemlerde kararlılık

In this study, some results are given concerning the behavior of the solutions for linear delay integro-differential equations. These results are obtained by the use of two distinct real roots of the corresponding characteristic equation.

Some remarks on the non-real roots of polynomials

Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of real roots of $g(x)=\sum_{k=0}^s t_{s-k} x^{s-k}$. For any $\xi > | \beta_m |$, if $n-s$ is odd then the number of real roots of $f(\xi, x)$ is $\gamma+1$, and if $n-s$ is even then the number of real roots of $f(\xi, x)$ is $\gamma$, $\gamma+2$ if $t_s>0$ or $t_s < 0$ respectively. A special case of the above result is constructing a family of totally complex polynomials which are reducible over $\mathbb Q$.

Asymptotic expansions and numerical simulations of I-V relations via a steady state Poisson-Nernst-Planck system

A steady state Poisson-Nernst-Planck (PNP) system is studied both analytically and numerically with particular attention on I-V relations of ion channels. Assuming the dielectric constant $\varepsilon$ is small, the PNP system can be viewed as a singularly perturbed system. Due to the special structures of the zeroth order inner and outer systems, one is able to derive more explicit expressions of higher order terms in asymptotic expansions. For the case of zero permanent charge, under the assumption of electro-neutrality at both ends of the channel, our result concerning the I-V relation for two oppositely charged ion species is that the third order correction is \textit{cubic} in $V$, and, furthermore (Theorem \ref{3rd order}), up to the third order, the cubic I-V relation has \textit{three distinct real roots} (except for a very degenerate case) which corresponds to the bi-stable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model. Three numerical experiments are conducted to check the cubic-like feature of the I-V curve, study the boundary value effect on the I-V relation and investigate the permanent charge effect on the I-V curve, respectively.

On the coefficients of the polynomial in the number field sieve

Polynomial selection is very important in the number field sieve. If the number of relations a pair of polynomials can generate is closely correlated with the coefficients of the polynomials, we can select polynomials by checking the coefficients first, which can speed up the selection of good polynomials. In this paper, we aim to study the correlation between polynomial coefficients and the number of relations the polynomials can generate. By analyzing the zero roots, it is found that a polynomial with the ending coefficient containing more small primes usually can generate more relations than the one whose ending coefficient contains less. As a polynomial with more real roots usually can generate more relations, using the complete discrimination system, the requirements on the coefficients of a polynomial to obtain more real roots are analyzed. For instance, a necessary condition for a polynomial of degree d to have d distinct real roots is that the coefficient of degree d−2 should be negative or small enough. The result in the case d = 3 can be used directly in selecting polynomials generated by the nonlinear method, where d = 3 is already enough for practical purpose.

Numerical Tribute to Achievements of Euler

This paper aims to make a tribute to one of the world's brightest personalities, the mathematical physicist Leonhard Euler (1707-1783). Our purpose is to expose, in a concise and entertaining way by utilizing modern computational techniques, a better understanding of the influence of Euler; also, to establish the remarkable significance in the history of science. A first analysis was done with the series that defines Euler and Bernoulli numbers and polynomials of Bernoulli and Euler; an additional result is the characterization of the functions that lead to the Euler-Mascheroni constant. In hydrodynamics it is also feasible to evaluate graphically the connection between sizes in diameter and the exit angle of the height of Euler for turbo-machines. In differential equations of Cauchy-Euler solutions in the cases of distinct real roots and complex roots are generated. Furthermore we report the generation of the Fourier series and the Fourier transform calculated by using direct Matlab commands. In calculus of variations, it is conceivable to obtain plots from a problem of the Euler Lagrange equations. Finally, Euler totient function is graphically analyzed. We seek the following benefits: The association between Euler numbers and generating functions is more effectively understood, also the nature of the Euler constant and the Euler function The Fourier transform is more comprehensible; also, it can be established that Matlab is an alternative way to study the conceptual development of achievements of physics and mathematics. We show that the abundance of Euler work and the history of mathematics, acquire better knowledge with the application of numerical tools

Counting the number of distinct real roots of certain polynomials by Bezoutian and the Galois groups over the rational number field

In this article, we count the number of distinct real roots of certain polynomials in terms of Bezoutian form. As an application, we construct certain irreducible polynomials over the rational number field which have given number of real roots and by the result of Oz Ben-Shimol [On Galois groups of prime degree polynomials with complex roots, Algebra Disc. Math. 2 (2009), pp. 99–107], we obtain an algorithm to construct irreducible polynomials of prime degree p whose Galois groups are isomorphic to S p or A p .

Using Abel's Theorem to Explain Repeated Roots of the Characteristic Equation

This document describes how one can derive the solutions to a linear constant coefficient homogeneous differential equation with repeated roots in the characteristic equation with Abel's Theorem. y (n) (t) + p1(t)y (n−1) (t) + p2(t)y (n−2) (t) + · · · + pn−1(t)y ' (t) + pn(t)y(t) = 0 (1.1) is encountered by every differential equations student. We most often study second order equations, the case when n = 2, due to Newton's laws of motion and with ease one can generalize the theory to higher order. We consider the case when the coefficient functions pi(t) are constant. Under these conditions, one makes the ansatz y = e rt which transforms (1.1) into an algebraic equation, which is more easily solved. In particular, if pi = ai, with the above ansatz, we have the characteristic equation r n + a1r n−1 + a2r n−2 + · · · + an−1r + an = 0. (1.2) By factoring out the e rt , one finds that solving an algebraic equation yields solutions to the original differential equation. Most textbooks first handle distinct real roots and later come back to discuss when r = r1 is a repeated solution to (1.2). We show an approach that relies on Abel's Theorem and the idea of reduction of order can constructively establish that the repeated root r = r1 with multiplicity m gives rise to m linearly independent solutions of the form yi(t) = t i−1 e r1t for 1 ≤ i ≤ m.

On the maximum rank of a real binary form

We show that a real binary form f of degree n has n distinct real roots if and only if for any $${(\alpha,\beta)\in\mathbb{R}^2{\setminus}\{0\}}$$ all the forms αf x + βf y have n − 1 distinct real roots. This answers to a question of Comon and Ottaviani (On the typical rank of real binary forms, available at arXiv:math/0909.4865, 2009), and allows to complete their argument to show that f has symmetric rank n if and only if it has n distinct real roots.

Verified computation of a disc containing exactly &lt;I&gt;k&lt;/I&gt; roots of a univariate nonlinear function

It is well known that it is an ill-posed problem to decide whether a function has a multiple root. For example, an arbitrarily small perturbation of a real polynomial may change a double real root into two distinct real or complex roots. In this paper we describe a computational method for the verified computation of a complex disc to contain exactly k roots of a univariate nonlinear function. The function may be given by some program. Computational results using INTLAB, the Matlab toolbox for reliable computing, demonstrate properties and limits of the method.

Asymptotic Expansions of I-V Relations via a Poisson–Nernst–Planck System

We investigate higher order matched asymptotic expansions of a steady-state Poisson-Nernst-Planck (PNP) system with particular attention to the I-V relations of ion channels. Assuming that the Debye length is small relative to the diameter of the narrow channel, the PNP system can be viewed as a singularly perturbed system. Special structures of the zeroth order inner and outer systems make it possible to provide an explicit derivation of higher order terms in the asymptotic expansions. For the case of zero permanent charge, our results concerning the I-V relation for two oppositely charged ion species are (i) the first order correction to the zeroth order linear I-V relation is generally quadratic in V; (ii) when the electro-neutrality condition is enforced at both ends of the channel, there is NO first order correction, but the second order correction is cubic in V. Furthermore (Theorem 3.4), up to the second order, the cubic I-V relation has (except for a very degenerate case) three distinct real roots that correspond to the bistable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model.