Recent zbMATH articles in MSC 34Mhttps://zbmath.org/atom/cc/34M2021-06-15T18:09:00+00:00WerkzeugTwo-dimensional neighborhoods of elliptic curves: formal classification and foliations.https://zbmath.org/1460.320152021-06-15T18:09:00+00:00"Loray, Frank"https://zbmath.org/authors/?q=ai:loray.frank"Thom, Olivier"https://zbmath.org/authors/?q=ai:thom.olivier"Touzet, Frédéric"https://zbmath.org/authors/?q=ai:touzet.fredericSummary: We classify two-dimensional neighborhoods of an elliptic curve \(C\) with torsion normal bundle, up to formal equivalence. The proof makes use of the existence of a pair (indeed a pencil) of formal foliations having \(C\) as a common leaf, and the fact that neighborhoods are completely determined by the holonomy of such a pair. We also discuss analytic equivalence and for each formal model, we show that the corresponding moduli space is infinite dimensional.Stable pairs, flat connections and Gopakumar-Vafa invariants.https://zbmath.org/1460.141242021-06-15T18:09:00+00:00"Stoppa, Jacopo"https://zbmath.org/authors/?q=ai:stoppa.jacopo\textit{R. P. Thomas} [J. Differ. Geom. 54, No. 2, 367--438 (2000; Zbl 1034.14015)] has introduced an enumerative invariant ``counting'' stable coherent sheaves on a complex projective Calabi-Yau or Fano threefold \(X\), with respect to some Kähler class and in the absence of strictly semistables. His work is based on previous gauge-theoretic intuition, developed with \textit{S. K. Donaldson} [in: The geometric universe: science, geometry, and the work of Roger Penrose. Proceedings of the symposium on geometric issues in the foundations of science, Oxford, UK, June 1996 in honour of Roger Penrose in his 65th year. Oxford: Oxford University Press. 31--47 (1998; Zbl 0926.58003)]. The author focuses on an application of the Donaldson-Thomas (DT) theory, namely the correspondence between sheaf-theoretic curve counts and the enumerative theory of holomorphic maps from stable curves to \(X\), i.e. Gromov-Witten (GW) invariants. This GW/DT correspondence is governed by a famous conjecture of \textit{D. Maulik} et al. [Compos. Math. 142, No. 5, 1263--1285 (2006; Zbl 1108.14046)]. After an introduction to this circle of ideas the author presents a new computation which recasts the GW/DT correspondence in the language of flat connections, their monodromy, and their flat sections.
Reviewer: Vladimir P. Kostov (Nice)\(q\)-Racah ensemble and \(q\)-\(\mathrm{P} \left (E_7^{(1)}/A_1^{(1)}\right )\) discrete Painlevé equation.https://zbmath.org/1460.330182021-06-15T18:09:00+00:00"Dzhamay, Anton"https://zbmath.org/authors/?q=ai:dzhamay.anton"Knizel, Alisa"https://zbmath.org/authors/?q=ai:knizel.alisaSummary: The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlevé equations. Namely, we consider the \(q\)-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete \(q\)-\(\mathrm{P} \left (E_7^{(1)}/A_1^{(1)}\right )\) equation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.Discrete semiclassical orthogonal polynomials of class 2.https://zbmath.org/1460.330082021-06-15T18:09:00+00:00"Dominici, Diego"https://zbmath.org/authors/?q=ai:dominici.diego-ernesto"Marcellán, Francisco"https://zbmath.org/authors/?q=ai:marcellan-espanol.franciscoSummary: In this contribution, discrete semiclassical orthogonal polynomials of class \(s\leq 2\) are studied. By considering all possible solutions of the Pearson equation, we obtain the canonical families in each class. We also consider limit relations between these and other families of orthogonal polynomials.
For the entire collection see [Zbl 1459.33001].Special function solutions of Painlevé equations: theory, asymptotics and applications.https://zbmath.org/1460.330232021-06-15T18:09:00+00:00"Deaño, Alfredo"https://zbmath.org/authors/?q=ai:deano.alfredoSummary: In this paper we review the construction of special function solutions of the Painlevé differential equations. We motivate their study using the theory of orthogonal polynomials, in particular deformation of classical weight functions, as well as unitarily invariant ensembles in random matrix theory. The asymptotic behavior of these Painlevé functions can be studied in at least two different regimes, using the Riemann-Hilbert approach and the classical saddle point method for integrals.
For the entire collection see [Zbl 1459.33001].Triangle groups: automorphic forms and nonlinear differential equations.https://zbmath.org/1460.110482021-06-15T18:09:00+00:00"Ashok, Sujay K."https://zbmath.org/authors/?q=ai:ashok.sujay-k"Jatkar, Dileep P."https://zbmath.org/authors/?q=ai:jatkar.dileep-p"Raman, Madhusudhan"https://zbmath.org/authors/?q=ai:raman.madhusudhanSummary: We study the relations governing the ring of quasiautomorphic forms associated to triangle groups with a single cusp, thereby extending our earlier results on Hecke groups. The Eisenstein series associated to these triangle groups are shown to satisfy Ramanujan-like identities. These identities in turn allow us to associate a nonlinear differential equation to each triangle group. We show that they are solved by the quasiautomorphic weight-2 Eisenstein series associated to the triangle group and its orbit under the group action. We conclude by discussing the Painlevé property of these nonlinear differential equations.Inverse hybrid linear multistep methods for solving the second order initial value problems in ordinary differential equations.https://zbmath.org/1460.341092021-06-15T18:09:00+00:00"Ibrahim, Oluwasegun M."https://zbmath.org/authors/?q=ai:ibrahim.oluwasegun-m"Ikhile, Monday N. O."https://zbmath.org/authors/?q=ai:ikhile.monday-ndidi-oziegbeSummary: Inverse linear multistep methods (ILMMs) for first and second order differential equations have been proved to be suitable numerical methods for the solution of inverse initial value problems (IVPs). This paper presents the hybrid version of the ILMMs for the numerical solution of second order inverse IVPs. The stability of the proposed methods is represented in the boundary locus graph. The applicability of the schemes is demonstrated herein for the solution of linear and nonlinear problems. Computational results on the problems are compared with those from the existing method and ode45 (the explicit Runge-Kutta method).Growth of \(\phi \)-order solutions of linear differential equations with meromorphic coefficients on the complex plane.https://zbmath.org/1460.341082021-06-15T18:09:00+00:00"Kara, Mohamed Abdelhak"https://zbmath.org/authors/?q=ai:kara.mohamed-abdelhak"Belaïdi, Benharrat"https://zbmath.org/authors/?q=ai:belaidi.benharratIn this paper, the authors study the growth of higher order linear differential equations with meromorphic coefficients of \(\varphi\)-order on the comlpex plane. They prove many theorems by using the concepts of \(\varphi\)-order and \(\varphi\)-type. These theorems extend previous results due to Chyzhykov, Semochko, Belaïdi, Cao, Xu, Chen and Kinnunen. This work is interesting.
Reviewer: Karima Hamani (Mostaganem)Growth of solutions of complex differential equations in a sector of the unit disc.https://zbmath.org/1460.341072021-06-15T18:09:00+00:00"Belaïdi, Benharrat"https://zbmath.org/authors/?q=ai:belaidi.benharratIn the paper, the author discuss the growth of solutions of homogeneous linear complex differential equation by using the concept of lower $[p,q]$-order and lower $[p,q]$-type in a sector of the unit disc instead of the whole unit disc, and they obtain similar results as in the case of the unit disc.
Furthermore, they establish the concept of lower $[p,q]$-order and lower $[p,q]$-type of a meromorphic function in a sector $\Omega$ and extension of some earlier results in this area.
The work carried out here is no doubt a good piece of modern research work. Moreover, it contains a resourceful and current reference list at the end.
Reviewer: Nityagopal Biswas (Kalyani)Canonical decomposition of irreducible linear differential operators with symplectic or orthogonal differential Galois groups.https://zbmath.org/1460.120022021-06-15T18:09:00+00:00"Boukraa, S."https://zbmath.org/authors/?q=ai:boukraa.salah"Hassani, S."https://zbmath.org/authors/?q=ai:hassani.saoud"Maillard, J.-M."https://zbmath.org/authors/?q=ai:maillard.jean-marie"Weil, J.-A."https://zbmath.org/authors/?q=ai:weil.jacques-arthurPearcey system re-examined from the viewpoint of \(s\)-virtual turning points and non-hereditary turning points.https://zbmath.org/1460.341102021-06-15T18:09:00+00:00"Hirose, Sampei"https://zbmath.org/authors/?q=ai:hirose.sampei"Kawai, Takahiro"https://zbmath.org/authors/?q=ai:kawai.takahiro"Takei, Yoshitsugu"https://zbmath.org/authors/?q=ai:takei.yoshitsuguConsider the Pearcey integral
\[
\int \exp(\eta\varphi(x,t)) dt, \quad \varphi(x,t)=t^4+x_2t^2+x_1t
\]
satisfying the over-determined system \(M\):
\[
(4\eta^{-3} (\partial/\partial x_1)^3 + 2x_2\eta^{-1} \partial/\partial x_1
+x_1) \psi=0, \quad (\eta^{-1}\partial/\partial x_2 -\eta^{-2} (\partial/
\partial x_1)^2 )\psi =0,
\]
\(\eta\) being a large parameter. This system restricted to
\(Y_1=\{ (x_1, x_2)\in \mathbb{C}^2; \,\, x_2=A_2 \,\,(\not=0) \}\) is the
Berk-Nevins-Roberts equation [\textit{H. L. Berk} and \textit{K. V. Roberts}, J. Math. Phys. 23, 988--1002 (1982; Zbl 0488.34050)], and the description of its Stokes geometry requires a new Stokes
curve and a virtual turning point [\textit{T. Aoki} et al., in: Analyse algébrique des perturbations singulières. I. Méthodes résurgentes. Conférences du symposium franco-japonais sur l'analyse algébrique des perturbations singulières, CIRM, Marseille-Luminy, France, October 20-26, 1991. Paris: Hermann. 69--84 (1994; Zbl 0831.34058)].
On the other hand, for the tangential system of \(M\) to
\(Y_2=\{ (x_1, x_2)\in \mathbb{C}^2; \,\, x_1=A_1 \,\,(\not=0) \}\), no virtual
turning point appears (Section 1). This paper discusses what happens in
the Stokes geometry for the tangential system of \(M\) to
\(Y(c)=\{ (x_1, x_2)\in \mathbb{C}^2; \,\, x_2=c(x_1-1), \,\, c\not=0 \}\)
from the viewpoint of \(s\)-virtual turning points defined by \textit{Definition
1.1.1}. It is shown that, for \(c\not=0,\) there exist the \(s\)-virtual turning
point \(x\) \((=x_1) =0\) and the non-hereditary turning point \(x=x_*=(2-4c^{-3})/3\),
which does not cause any Stokes phenomena. When \(c=1\), for simplicity,
a bicharacteristic strip of this tangential system is computed to show that
\(z\) \((=x)=0\) is a (traditional) virtual turning point given by the
self-intersection point \((z,y)=(0, \sqrt{3}/18 +2/9)\) formed by a
bicharacteristic curve emanating from the non-hereditary turning point
\((z,y)=(-2/3,0).\)
Reviewer: Shun Shimomura (Yokohama)Roots of generalised Hermite polynomials when both parameters are large.https://zbmath.org/1460.330122021-06-15T18:09:00+00:00"Masoero, Davide"https://zbmath.org/authors/?q=ai:masoero.davide"Roffelsen, Pieter"https://zbmath.org/authors/?q=ai:roffelsen.pieterExistence of transcendental meromorphic solutions on some types of nonlinear differential equations.https://zbmath.org/1460.341062021-06-15T18:09:00+00:00"Hu, Peichu"https://zbmath.org/authors/?q=ai:hu.peichu"Liu, Manli"https://zbmath.org/authors/?q=ai:liu.manliSummary: We show that when \(n>m\), the following delay differential equation \[f^n(z)f'(z)+p(z)(f(z+c)-f(z))^m=r(z)e^{q(z)}\] of rational coefficients \(p,r\) doesn't admit any transcendental entire solutions \(f(z)\) of finite order. Furthermore, we study the conditions of \(\alpha_1, \alpha_2\) that ensure existence of transcendental meromorphic solutions of the equation \[f^n(z) + f^{n-2}(z)f'(z)+ P_d(z,f)=p_1(z)e^{\alpha_1( z)}+p_2(z)e^{\alpha_2 (z)}.\] These results have improved some known theorems obtained most recently by other authors.Linear differential equations in the complex domain. From classical theory to forefront.https://zbmath.org/1460.340022021-06-15T18:09:00+00:00"Haraoka, Yoshishige"https://zbmath.org/authors/?q=ai:haraoka.yoshishigeThe main purpose of this interesting book is an introduction to recent developments in the theory of linear ordinary differential systems and linear completely integrable total differential systems. It consists of fourteen chapters and six references. The book is designed to be suitable for use as a primary textbook in an advanced graduate course or as a supplementary source for beginning graduate courses. The first chapter consists of an introduction. The second chapter examines the relationship between linear scalar differential equations and linear systems of differential equations. The third chapter examines the properties of linear scalar differential equations and linear systems of differential equations in the neighborhood of regular points. The fourth chapter deals with the properties of linear scalar differential equations and linear systems of differential equations in the neighborhood of regular singular points. The fifth chapter studies the properties of the monodromy group of linear scalar differential equations and linear systems of differential equations. The sixth chapter deals with the connection problem for linear differential equations. The seventh chapter discusses in detail the analytical properties of Fuchsian differential equations. The problem of the existence of Fuchsian differential equations corresponding to a given monodromy group is also considered. The eighth chapter presents the theory of isomonodromic deformations for Fuchsian systems of ordinary differential equations. Linear scalar differential equations whose solutions are represented by integrals of the Euler type in the ninth chapter are considered. The tenth chapter deals with the properties of linear scalar differential equations and linear systems of differential equations in the neighborhood of irregular singular points. The eleventh chapter deals with the basic properties of completely integrable linear Pfaffian systems. In the twelfth chapter the regular singularities of completely integrable linear Pfaffian systems are considered. The thirteenth chapter deals with the monodromy representations of completely integrable linear Pfaffian systems. The fourteenth chapter deals with middle convolusions and its application of of completely integrable linear Pfaffian systems. The book contains numerous examples, which can be useful as a reference for researchers.
Reviewer: Valentine Tyshchenko (Grodno)