Differential Polynomials Research Articles

Reconstructing the Extreme Ultraviolet Emission of Cool Dwarfs Using Differential Emission Measure Polynomials

Abstract Characterizing the atmospheres of planets orbiting M dwarfs requires understanding the spectral energy distributions of M dwarfs over planetary lifetimes. Surveys like MUSCLES, HAZMAT, and FUMES have collected multiwavelength spectra across the spectral type’s range of T eff and activity, but the extreme ultraviolet (EUV, 100–912 Å) flux of most of these stars remains unobserved because of obscuration by the interstellar medium compounded with limited detector sensitivity. While targets with observable EUV flux exist, there is no currently operational facility observing between 150 and 912 Å. Inferring the spectra of exoplanet hosts in this regime is critical to studying the evolution of planetary atmospheres because the EUV heats the top of the thermosphere and drives atmospheric escape. This paper presents our implementation of the differential emission measure technique to reconstruct the EUV spectra of cool dwarfs. We characterize our method’s accuracy and precision by applying it to the Sun and AU Mic. We then apply it to three fainter M dwarfs: GJ 832, Barnard’s star, and TRAPPIST-1. We demonstrate that with the strongest far-ultraviolet (FUV, 912–1700 Å) emission lines, observed with the Hubble Space Telescope and/or Far Ultraviolet Spectroscopic Explorer, and a coarse X-ray spectrum from either the Chandra X-ray Observatory or XMM-Newton, we can reconstruct the Sun’s EUV spectrum to within a factor of 1.8, with our model’s formal uncertainties encompassing the data. We report the integrated EUV flux of our M dwarf sample with uncertainties of a factor of 2–7 depending on available data quality.

Irradiance transport equation applied to propagation of wavefront obtained by the Bi-Ronchi test using point cloud

In this work, we show a Bi-Ronchi test (BRT) proposal using point cloud for sensing the whole surface and wavefront of spherical mirrors as a fast computational test with efficiency comparable with the Ronchi test. We also show an application of the Irradiance Transport equation based on the differential form of the circular Zernike polynomials, to obtain the Phase Transport equation, with capabilities to study the wavefront propagation. To this aim, we experimentally analyze the propagation of W with the BRT, to validate the experimental propagation performed with the Irradiance Transport Equation, giving place to a Phase Transport equation. For this purpose, we use square rulings to observe bi-Ronchigrams and their corresponding Transversal Aberrations as well as their W. We validated our results by comparing the BRT with the Ronchi Test for several measurements of a given wavefront W with Ronchi rulings in the same position. Three different bRR were placed in several locations before and beyond the mirror curvature Radius in order to characterize the experimental propagation of W and test the reflection relation of W before and beyond the mirror Curvature Radius. Finally, we use the Phase Transport equation to analyze the propagation of an efficient W in terms of the differential circular Zernike Polynomials in order to obtain a Riemann-integrable function to numerically find a solution for validating the experimental propagation, the symmetry of the wavefronts, the existence of a symmetry matrix, and the BRT.

Bifurcations of the Riccati Quadratic Polynomial Differential Systems

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of [Formula: see text] adding the circle [Formula: see text] of the infinity) modulo topological equivalence.

Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

This paper study the type of integrability of differential systems with separable variables \(\dot{x}=h\left (x\right )f\left (y\right )\), \(\dot{y}= g\left (y\right )\), where \(h\), \(f\) and \(g\) are polynomials. We provide a criterion for the existence of generalized analytic first integrals of such differential systems. Moreover we characterize the polynomial integrability of all such systems.In the particular case \(h\left (x\right )=\left (ax+b\right )^{m}\) we provide necessary and sufficient conditions in order that this subclass of systems has a generalized analytic first integral. These results extend known results from Giné et al. (Discrete Contin. Dyn. Syst. 33:4531–4547, 2013) and Llibre and Valls (Discrete Contin. Dyn. Syst., Ser. B 20:2657–2661, 2015). Such differential systems of separable variables are important due to the fact that after a blow-up change of variables any planar quasi-homogeneous polynomial differential system can be transformed into a special differential system of separable variables \(\dot{x}=xf\left (y\right )\), \(\dot{y}=g\left (y\right )\), with \(f\) and \(g\) polynomials.

Some results on uniqueness of meromorphic functions concerning differential polynomials

In this paper, we study the uniqueness problem of certain differential polynomials generated by two meromorphic functions. The results of the paper extend some recent results due to Meng and Li (J Anal 28:1–6, 2019).

Growth of solutions of non-homogeneous linear differential equations and its applications

In this paper, we investigate the growth properties of solutions of the non-homogeneous linear complex differential equation $L\left(f\right)=b\left(z\right)f+c\left(z\right)$, where $L\left(f\right)$ is a linear differential polynomial and $b\left( z\right) $, $c\left( z\right) $ are entire functions and give some of its applications on sharing value problems.

A note on the value distribution of $\phi f^2 f^{(k)}-1$

In this paper, we study the value distribution of the differential polynomial $\varphi f^2f^{(k)}-1$, where $f(z)$ is a transcendental meromorphic function, $\varphi (z)\;(\not\equiv 0)$ is a small function of $f(z)$ and $k\;(\geq 2)$ is a positive integer. We obtain an inequality concerning the Nevanlinna Characteristic function $T(r,f)$ estimated by reduced counting function only. Our result extends the result due to J.F. Xu and H.X. Yi [J. Math. Inequal., 10 (2016), 971-976].

On certain non-linear differential monomial sharing non-zero polynomial

UDC 517.5 With the idea of normal family we study the uniqueness of meromorphic functions and when and share two values, where and is a polynomial. The obtained result significantly improves and generalizes the result in [A. Banerjee, S. Majumder, <em>On certain non-linear differential polynomial sharing a non-zero polynomial</em>, Bol. Soc. Mat. Mex. (2016),https://doi.org/10.1007/s40590-016-0156-0].

On meromorphic solutions of non‐linear differential equations of Tumura–Clunie type

AbstractMeromorphic solutions of non‐linear differential equations of the form are investigated, where is an integer, h is a meromorphic function, and is differential polynomial in f and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case when h has the particular form , where are small functions of f and are entire functions. In such a case the order of h is either a positive integer or equal to infinity. In this article it is assumed that h is a meromorphic solution of the linear differential equation with rational coefficients , and hence the order of h is a rational number. Recent results by Liao–Yang–Zhang (2013) and Liao (2015) follow as special cases of the main results.

Value sharing of non linear differential polynomials with the glimpse of normal family

Value sharing of non linear differential polynomials with the glimpse of normal family

Uniqueness of differential q-shift difference polynomials of entire functions

Uniqueness of differential q-shift difference polynomials of entire functions

Erratum: Radicals in differential polynomial rings

Erratum: Radicals in differential polynomial rings

Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra

We study the differential polynomial identities of the algebra UTm(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the TL-ideal they constitute. Then we determine the Sn-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence.

Complex meromorphic functions $$f'P'(f)$$ and $$g'P'(g)$$ sharing small function with finite weight

The aim of the paper is to study the uniqueness problems of meromorphic functions when certain differential polynomials generated by them share a small function. With the aid of weighted sharing we prove four theorems first two of which improve the results due to Boussaf et al. (Indagationes Mathematicae 24:15–41, 2013) and last two improve a result due to Bhoosnurmath and Pujari (Bull Korean Math Soc 52:13–33, 2015).

Configurations of the Topological Indices of the Planar Polynomial Differential Systems of Degree (2, m)

Using the Euler–Jacobi formula there is a relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the polynomial differential systems \(\dot{x}=P(x,y)\), \(\dot{y} =Q(x,y)\) with degree of P equal to 2 and degree of Q equal to m when these systems have 2m finite singular points.

Normality and Uniqueness Property of Meromorphic Function in Terms of Some Differential Polynomials

In this paper, we will consider normality and uniqueness property of a family $\mathcal {F}$ of meromorphic functions when [Q(f)](k) and [Q(g)](k) share α ignoring multiplicities, for any $f,g\in \mathcal {F}$ , where Q is a polynomial and α is a small function. Our results do not need all of zeros of Q have large order as other authors’ results.

Uniqueness of differential difference polynomials of L-functions and meromorphic functions sharing a polynomial

In this article, we study the uniqueness of Differential difference polynomials of L-function and Differential difference polynomials of a meromorphic function concerning weighted sharing of a polynomial. Our result improves and generalizes results of Abhijit Banerjee, Saikat Bhattacharyya [1], N. Mandal, N. K. Datta [5].

Normality of meromorphic functions and their differential polynomials

Normality of meromorphic functions and their differential polynomials

涉及微分多项式分担函数的正规定则

涉及微分多项式分担函数的正规定则

Uniqueness theorems concerning L-functions and weakly weighted sharing

In this paper, we establish some uniqueness theorems when p(L) and P(L) share “(R(z),l)” where p, P and R denote polynomial function, homogeneous differential polynomial function and rational function respectively and L denotes an L-function in the extended Selberg class. Our results improve and generalize some recent results due to Mandal, Datta [11].