Differential Monomials Research Articles

Normality and uniqueness of homogeneous differential polynomials

The primary goal of this work is to determine whether the results from [19, 20] still hold true when a differential polynomial is considered in place of a differential monomial. In this perspective, we continue our study to establish the uniqueness theorem for homogeneous differential polynomial of an entire and its higher order derivative sharing two polynomials using normal family theory as well as to obtain normality criteria for a family of analytic functions in a domain concerning homogeneous differential polynomial of a transcendental meromorphic function satisfying certain conditions. Meanwhile, as a result of this investigation, we proved three theorems that provide affirmative responses for the purpose of this study. Several examples are offered to demonstrate that the conditions of the theorem are necessary.

Meromorphic functions sharing a set with their differential monomials

Meromorphic functions sharing a set with their differential monomials

Characteristic estimation of differential polynomials

In this paper, we give the characteristic estimation of a meromorphic function f with the differential polynomials f^{l}(f^{(k)})^{n} and obtain that T(r,f)≤MN‾(r,1fl(f(k))n−a)+S(r,f)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} T(r,f)\\leq M\\overline{N} \\biggl(r,\\frac{1}{f^{l}(f^{(k)})^{n}-a} \\biggr)+S(r,f) \\end{aligned}$$ \\end{document} holds for M=min {frac{1}{l-2},6}, integers l(geq 2), n(geq 1), k(geq 1), and a non-zero constant a. This quantitative estimate is an interesting and complete extension of earlier results. The value distribution of a differential monomial of meromorphic functions is also investigated.

Further study on the Brück conjecture and some non-linear complex differential equations

PurposeThe purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramaniket al.Design/methodology/approach39B32, 30D35.FindingsIn the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex numberain the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Letfbe a non-constant entire function such thatσ2(f)<∞,σ2(f)is not a positive integer andδ(0, f)>0. LetM[f]be a differential monomial offof degreeγMandα(z), β(z)∈S(f)be such thatmax{σ(α), σ(β)} <σ(f). IfM[f]+βandfγM−αshare the value 0 CM, thenM[f]+βfγM−α=c,wherec≠0is a constant.Originality/valueThis is an original work of the authors.

Normal Forms for Rigid $\mathfrak{C}_{2,1}$ Hypersurfaces $M^5 \subset \mathbb{C}^3$

Consider a $2$-nondegenerate constant Levi rank $1$ rigid $\mathscr{C}^{\omega}$ hypersurface $M^5 \subset \mathbb{C}^3$ in coordinates $(z, \zeta, w = u+iv)$: \[ u = F(z,\zeta,\overline{z},\overline{\zeta}). \] The Gaussier-Merker model $u = \frac{z \overline{z} + \frac{1}{2} z^2 \overline{\zeta} + \frac{1}{2} \overline{z}^2 \zeta}{1 - \zeta \overline{\zeta}}$ was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone $\{ x_1^2 + x_2^2 - x_3^2 = 0 \}$. Another representation is the tube $u = \frac{(\operatorname{Re}z)^2}{1 - \operatorname{Re} \zeta}$. The Gaussier-Merker model has $7$-dimensional rigid automorphisms group. Inspired by Alexander Isaev, we study rigid biholomorphisms: \[ (z,\zeta,w) \longmapsto (f(z,\zeta), g(z,\zeta), \rho w + h(z,\zeta)) =: (z',\zeta',w'). \] The goal is to establish the Poincaré-Moser complete normal form: \[ u = \frac{z\overline{z} + \frac{1}{2} z^2 \overline{\zeta} + \frac{1}{2} \overline{z}^2 \zeta}{1 - \zeta \overline{\zeta}} + \sum_{\substack{a,b,c,d \in \mathbb{N} \\ a+c \geq 3}} G_{a,b,c,d} z^a \zeta^b \overline{z}^c \overline{\zeta}^d \] with $0 = G_{a,b,0,0} = G_{a,b,1,0} = G_{a,b,2,0}$ and $0 = G_{3,0,0,1} = \operatorname{Im} G_{3,0,1,1}$.

Value distribution of some differential monomials

Let f be a transcendental meromorphic function defined in the complex plane C and k ? N. We consider the value distribution of the differential polynomial fq0(f(k))qk, where q0(?2), qk(?1) are integers. We obtain a quantitative estimation of the characteristic function T(r,f) in terms of N?(r, 1/fq0(f(k))qk-1). Our result generalizes the results obtained by Xu et al. (Math. Inequal. Appl., Vol. 14, PP. 93-100, 2011); Karmakar and Sahoo (Results Math., Vol. 73, 2018) for a particular class of transcendental meromorphic functions.

Slowly changing function oriented growth measurement of differential polynomial and differential monomial

In the paper we establish some new results depending on the comparative growth properties of composite entire and meromorphic functions using relative $_{p}L^{\ast }$-order, relative $_{p}L^{\ast }$-lower order and differential monomials, differential polynomials generated by one of the factors.

Advancement on the study of growth analysis of differential polynomial and differential monomial in the light of slowly increasing functions

Study of the growth analysis of entire or meromorphic functions has generally been done through their Nevanlinna's characteristic function in comparison with those of exponential function. But if one is interested to compare the growth rates of any entire or meromorphic function with respect to another, the concepts of relative growth indicators will come. The field of study in this area may be more significant through the intensive applications of the theories of slowly increasing functions which actually means that $L(ar)\sim L(r)$ as $ r\rightarrow \infty $ for every positive constant $a$, i.e. $\underset{ r\rightarrow \infty }{\lim }\frac{L\left( ar\right) }{L\left( r\right) }=1$, where $L\equiv L\left( r\right) $ is a positive continuous function increasing slowly. Actually in the present paper, we establish some results depending on the comparative growth properties of composite entire and meromorphic functions using the idea of relative $_{p}L^{\ast }$-order, relative $_{p}L^{\ast }$- type, relative $_{p}L^{\ast }$-weak type and differential monomials, differential polynomials generated by one of the factors which extend some earlier results where $_{p}L^{\ast }$ is nothing but a weaker assumption of $L.$

Uniqueness of polynomial and differential monomial

Abstract In this paper, we discuss the problem of meromorphic functions sharing small function and present one theorem which extend a result of K. S. Charak and Banarasi Lal [16].

Comparative growth measures of differential monomials and differential polynomials depending on their relative orders

Comparative growth measures of differential monomials and differential polynomials depending on their relative orders

Resolution of singularities of the cotangent sheaf of a singular variety

The main problem studied here is resolution of singularities of the cotangent sheaf of a complex- or real-analytic space X0 (or of an algebraic variety X0 over a field of characteristic zero). Given X0, we ask whether there is a global resolution of singularities σ:X→X0 such that the pulled-back cotangent sheaf of X0 is generated by differential monomials in suitable coordinates at every point of X (“Hsiang–Pati coordinates”). Desingularization of the cotangent sheaf is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of σ. We prove resolution of singularities of the cotangent sheaf in dimension up to three. It was previously known for surfaces with isolated singularities (Hsiang–Pati 1985, Pardon–Stern 2001). Consequences include monomialization of the induced Fubini–Study metric on the smooth part of a complex projective variety X0; there have been important applications of the latter to L2-cohomology.

Uniqueness of a Polynomial and a Differential Monomial Sharing a Small Function

Abstract In this paper taking a question in [1] into background we investigate the uniqueness of a non-constant polynomial with the differential monomial generated by a non-constant mermorphic function f. Our result will also extend a result of Banerjee-Majumder[2] given earlier. An open question is also posed, in the paper, for future investigation.

Some normality criteria

In this article we prove some normality criteria for a family of meromorphic functions which involves sharing of a non-zero value by certain differential monomials generated by the members of the family. These results generalizes some of the results of Schwick.

On the Estimation of Relative Type Based Growth Properties of Differential Monomials

In this paper an approach is made to investigate the study of the comparative growth properties of composite entire and meromorphic functions on the basis

Uniqueness of Meromorphic Functions Concerning Differential Monomials

Considering the uniqueness of meromorphic functions concerning differential monomials ,we obtain that, if two non-constant meromorphic functions f(z) and g(z) satisfy ,where k and n are tow positive integers satisfying k ≥ 3 and n ≥ 11 , then either where c1, c2, c, are three constants, satisfying (c1 c2)n+1c>n+1=- 1 or f = tg for a constant t such that tn+1 = 1

Enumeration and Generating Functions of Differential Rota–Baxter Words

In this paper, the methods and results in enumeration and generation of Rota–Baxter words in Guo and Sit (Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS), Math. Comp. Sci., vol. 4, Sp. Issue (2,3), 2011) are generalized and applied to a free, non-commutative, non-unitary, ordinary differential Rota–Baxter algebra with one generator. A differential Rota–Baxter algebra is an associative algebra with two operators modeled after the differential and integral operators, which are related by the First Fundamental Theorem of Calculus. Differential Rota–Baxter words are words formed by concatenating differential monomials in the generator with images of words under the Rota–Baxter operator. Their totality is a canonical basis of a free, non-commutative, non-unitary, ordinary differential Rota–Baxter algebra. A free differential Rota–Baxter algebra can be constructed from a free Rota–Baxter algebra on a countably infinite set of generators. The order of the derivation gives another dimension of grading on differential Rota–Baxter words, enabling us to generalize and refine results from Guo and Sit to enumerate the set of differential Rota–Baxter words and outline an algorithm for their generation according to a multi-graded structure. Enumeration of a basis is often a first step to choosing a data representation in implementation of algorithms involving free algebras, and in particular, free differential Rota–Baxter algebras and several related algebraic structures on forests and trees. The generating functions obtained can be used to provide links to other combinatorial structures.

Elliptic and weakly coercive systems of operators in Sobolev spaces

It is known that an elliptic system of order is weakly coercive in , that is, all differential monomials of order on -functions are subordinated to this system in the -norm. Conditions for the converse result are found and other properties of weakly coercive systems are investigated.An analogue of the de Leeuw-Mirkil theorem is obtained for operators with variable coefficients: it is shown that an operator of variables with constant principal part is weakly coercive in if and only if it is elliptic. A similar result is obtained for systems with constant coefficients under the condition and with several restrictions on the symbols .A complete description of differential polynomials of two variables which are weakly coercive in is given. Wide classes of systems with constant coefficients which are weakly coercive in , but non-elliptic are constructed.Bibliography: 32 titles.

Differential standard bases under composition

We generalize Hoon Hong’s theorem on Gröbner bases under composition to the case of differential standard bases in the ordinary ring of differential polynomials {ie4152-01}. In particular, we prove that some ideals have finite differential standard bases. We construct special orderings on differential monomials such that ideals generated by some power of a quasi-linear polynomial acquire finite differential standard bases.

On the value distribution of two differential monomials

On the value distribution of two differential monomials

On standard bases in rings of differential polynomials

We consider Ollivier’s standard bases (also known as differential Grobner bases) in an ordinary ring of differential polynomials in one indeterminate. We establish a link between these bases and Levi’s reduction process. We prove that the ideal [x p ] has a finite standard basis (w.r.t. the so-called β-orderings) that contains only one element. Various properties of admissible orderings on differential monomials are studied. We bring up the question of whether there is a finitely generated differential ideal that does not admit a finite standard basis w.r.t. any ordering.