Recent zbMATH articles in MSC 30Dhttps://zbmath.org/atom/cc/30D2024-05-13T19:39:47.825584ZWerkzeugRiemann-Hurwitz theorem and Riemann-Roch theorem for hypermapshttps://zbmath.org/1532.051772024-05-13T19:39:47.825584Z"Cheng, Mengnan"https://zbmath.org/authors/?q=ai:cheng.mengnan"Cao, Tingbin"https://zbmath.org/authors/?q=ai:cao.tingbinSummary: In this paper, we try to answer some questions raised by \textit{L. Cangelmi} [Eur. J. Comb. 33, No. 7, 1444--1448 (2012; Zbl 1244.05110)]. We reinterpret the Riemann-Hurwitz theorem of orientable algebraic hypermaps by introducing tripartite graph morphisms and obtain Riemann-Roch theorems for orientable hypermaps by defining the divisor of a function \(f\) on darts. In addition, we extend Riemann-Roch theorem to non-orientable hypermaps by suitably replacing the orientable genus with the non-orientable genus. Finally, as an application of the Riemann-Hurwitz theorem, we establish the second main theorem from the viewpoint of Nevanlinna theory.Uniqueness of Dirichlet series in the light of shared set and valueshttps://zbmath.org/1532.111222024-05-13T19:39:47.825584Z"Banerjee, Abhijit"https://zbmath.org/authors/?q=ai:banerjee.abhijit"Kundu, Arpita"https://zbmath.org/authors/?q=ai:kundu.arpitaSummary: In this article, we have studied the uniqueness problem of Dirichlet series, which is convergent in a right half-plane and having analytic continuation in the complex plane as a meromorphic function sharing some sets and values. Our first result partially improve a result of \textit{N. Oswald} and \textit{J. Steuding} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 48, 117--128 (2018; Zbl 1413.11107)] by relaxing the sharing conditions. Most importantly, we have pointed out a number of big gaps in a recent paper [\textit{S. Halder} and \textit{P. Sahoo}, J. Contemp. Math. Anal., Armen. Acad. Sci. 56, No. 2, 80--86 (2021; Zbl 1469.11352)], which makes the existence of the paper under question. Finally, under a different approach, we have provided the corrected form of the result of [Halder and Sahoo loc. cit.] as much as practicable.One proof of the fundamental theorem of algebra (of polynomials)https://zbmath.org/1532.300012024-05-13T19:39:47.825584Z"Khodos, Olga V."https://zbmath.org/authors/?q=ai:khodos.olga-vSummary: One proof of the fundamental theorem of algebra (of polynomials) is given, based on the theorem on the number of zeros of an entire function.On an analog of Descartes' rule of signs and the Budan-Fourier theorem for entire functionshttps://zbmath.org/1532.300052024-05-13T19:39:47.825584Z"Prenov, Barlikbay B."https://zbmath.org/authors/?q=ai:prenov.barlikbai-bSummary: It is proven the analog of Descartes' rule of signs and the Budan-Fourier theorem for entire functions.Coefficient multipliers for the Privalov class in a diskhttps://zbmath.org/1532.300062024-05-13T19:39:47.825584Z"Rodikova, Eugenia G."https://zbmath.org/authors/?q=ai:rodikova.eugenia-gennadevnaSummary: We obtain exact estimates of the growth and the Taylor coefficients of analytic functions from the Privalov classes in the unit disk. Also we describe coefficient multipliers from the Privalov classes into the Hardy classes.An approach to determine the resultant of two entire functionshttps://zbmath.org/1532.300072024-05-13T19:39:47.825584Z"Khodos, Olga V."https://zbmath.org/authors/?q=ai:khodos.olga-vSummary: An approach to determine the resultant of two entire functions is studied.Meromorphic functions with three radially distributed valueshttps://zbmath.org/1532.300082024-05-13T19:39:47.825584Z"Bergweiler, Walter"https://zbmath.org/authors/?q=ai:bergweiler.walter"Eremenko, Alexandre"https://zbmath.org/authors/?q=ai:eremenko.alexandre-eThe authors consider transcendental meromorphic functions for which the zeros, 1-points and poles are distributed on three distinct rays. They prove the following three results.
Theorem 1.1. Let \(L_0, L_1\) and \(L_{\infty}\) be three distinct rays emanating from the origin. If the rays are not equally spaced, then there is no transcendental meromorphic function for which all but finitely many zeros lie on \(L_0\), all but finitely many 1-points lie on \(L_1\) and all but finitely many poles lie on \(L_{\infty}\).
Theorem 1.2. Let \(L_0, L_1\) and \(L_{\infty}\) be three distinct rays emanating from the origin and let \(0\leq r<R \leq \infty\). Let \(\mathcal{F}\) be the family of all functions meromorphic in \(\{z \in \mathbb{C}: r<|z|<R\}\) for which all zeros are on \(L_0\), all 1 -points are on \(L_1\) and all poles are on \(L_{\infty}\). Then \(\mathcal{F}\) is normal if and only if the rays are not equally spaced.
Theorem 1.3. Let \(L^1, L^2\) and \(L^3\) be three equally spaced rays and let \(f\) be a transcendental meromorphic function. Then there exist distinct values \(a_1, a_2\) and \(a_3\) such that all but finitely many \(a_j\)-points are on \(L^j\) if and only if \[S(f)(z)=e^{3 \theta i} z R\left(z^3\right),\] where \(\theta\) is the argument of one of the rays \(L^j\) and \(R\) is a real rational function satisfying \(0<R(\infty)<\infty\).
Reviewer: Si Duc Quang (Hanoi)Meromorphic functions sharing three values with their derivatives in an angular domainhttps://zbmath.org/1532.300092024-05-13T19:39:47.825584Z"Pan, Biao"https://zbmath.org/authors/?q=ai:pan.biao(no abstract)On the existence of solutions of Fermat-type differential-difference equationshttps://zbmath.org/1532.340812024-05-13T19:39:47.825584Z"Chen, Jun-Fan"https://zbmath.org/authors/?q=ai:chen.junfan"Lin, Shu-Qing"https://zbmath.org/authors/?q=ai:lin.shuqingSummary: : We investigate the non-existence of finite order transcendental entire solutions of Fermat-type differential-difference equations
\[
\left[ f(z)f^{\prime} (z)\right]^n +P^2 (z)f^m (z+\eta)=Q(z)
\]
and
\[
\left[ f(z)f^{\prime} (z)\right]^n +P(z)[\Delta_{\eta}f(z)]^m =Q(z),
\]
where \(P(z)\) and \(Q(z)\) are non-zero polynomials, \(m\) and \(n\) are positive integers, and \(\eta\in\mathbb{C}\setminus\{ 0\}\). In addition, we discuss transcendental entire solutions of finite order of the following Fermat-type differential-difference equation
\[
P^2 (z) \left[ f^{(k)} (z)\right]^2 +\left[ \alpha f(z+\eta)-\beta f(z)\right]^2 =e^{r(z)},
\]
where \(P(z)\not\equiv 0\) is a polynomial, \(r(z)\) is a non-constant polynomial, \(\alpha\neq 0\) and \(\beta\) are constants, \(k\) is a positive integer, and \(\eta\in\mathbb{C}\setminus\{ 0\}\). Our results generalize some previous results.Three results on transcendental meromorphic solutions of certain nonlinear differential equationshttps://zbmath.org/1532.340882024-05-13T19:39:47.825584Z"Li, Nan"https://zbmath.org/authors/?q=ai:li.nan.4"Yang, Lianzhong"https://zbmath.org/authors/?q=ai:yang.lianzhongSummary: In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: \( f^n +P(f)=R(z) e^{\alpha(z)}\) and \(f^n +P_{\ast}(f)=p_1 (z)e^{\alpha_1 (z)}+p_2 (z)e^{\alpha_2 (z)}\) in the complex plane, where \(P(f)\) and \(P_{\ast} (f)\) are differential polynomials in \(f\) of degree \(n-1\) with coefficients being small functions and rational functions respectively, \(R\) is a non-vanishing small function of \(f, \alpha\) is a nonconstant entire function, \(p_1, p_2\) are non-vanishing rational functions, and \(\alpha_1, \alpha_2\) are nonconstant polynomials. Particularly, we consider the solutions of the second equation when \(p_1, p_2\) are nonzero constants, and \(\deg \alpha_1 =\deg \alpha_2 =1\). Our results are improvements and complements of \textit{L.-W. Liao} [Complex Var. Elliptic Equ. 60, No. 6, 748--756 (2015; Zbl 1317.30039)], and \textit{J. X. Rong} and \textit{J. F. Xu}, Mathematics 7, No. 6, Paper No. 539, 11 p. (2019; \url{doi:10.3390/math7060539})], etc., which partially answer a question proposed by Li \textit{P. Li} [J. Math. Anal. Appl. 375, No. 1, 310--319 (2011; Zbl 1206.30046)].Some inequalities on the convergent abscissas of Laplace-Stieltjes transformshttps://zbmath.org/1532.440022024-05-13T19:39:47.825584Z"Xu, Hong-Yan"https://zbmath.org/authors/?q=ai:xu.hongyan"Xuan, Zu Xing"https://zbmath.org/authors/?q=ai:xuan.zuxingThe authors establish some inequalities concerning the abscissa of convergence, the abscissa of absolute convergence, and the abscissa of uniform convergence of the Laplace-Stieltjes transform \(\int_{0} ^{\infty}e^{st}d\alpha(t)\).
In the first section, the authors introduce the Laplace-Stieltjes transform and present the motivation behind the paper. Further, four fundamental lemmas associated with continuous functions and functions of bounded variation are discussed.
Section 2 presents three definitions of the convergence abscissas of Laplace-Stieltjes transform including the convergence abscissas, absolute convergence abscissas, and uniform convergence abscissas. It is shown that the abscissa of convergence of the Laplace-Stieltjes transform can be different from the abscissa of absolute convergence of the Laplace-Stieltjes transform.
In Section 3 the relationship between the three convergence abscissas is analysed. It is shown that the formulas for the convergence abscissas are consistent with the Valiron-Knopp-Bohr formula given by \textit{J.-r.~Yu} [Acta Math. Sin. 13, 471--484 (1963; Zbl 0147.33004)]. The calculation formulas and properties of the three convergence abscissas are discussed in Section~4, Section~5, and Section~6 of the paper, respectively. Lastly, in Section~7 conclusions of the study are discussed.
Reviewer: Shared Chander Pandey (Jaipur)Origins of parameters in adimensional modelshttps://zbmath.org/1532.810732024-05-13T19:39:47.825584Z"Fowlie, Andrew"https://zbmath.org/authors/?q=ai:fowlie.andrewSummary: We explore the origins of parameters in adimensional theories -- fundamental theories with no classical massive scales. If the parameters originate as draws from a distribution, it should be possible to write a distribution for them that doesn't depend on or introduce any massive scales. These distributions are the invariant distributions for the renormalization group (RG). If there exist RG invariant combinations of parameters, the RG invariant distributions are specified up to arbitrary functions of the RG invariants. If such distributions can be constructed, adimensional theories could predict the values of their parameters through distributions that are constrained by the RG. If they can't be constructed, the parameters must originate in a different way. We demonstrate the RG invariant distributions in QCD, the Coleman-Weinberg model and a totally asymptotically free theory.