Authors:Aymen Ammar; Aref Jeribi; Bilel Saadaoui Pages: 325 - 350 Abstract: The main goal of this paper is to give a characterization of the essential pseudospectra of \(2\times 2\) matrix of linear relations on a Banach space. We start by giving the definition and we investigate the characterization and some properties of the essential pseudospectra. Furthermore, we apply the obtained result to determine the essential pseudospectra of two-group transport equation with general boundary conditions in the Banach space. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0170-z Issue No:Vol. 8, No. 3 (2018)

Authors:Palle Jorgensen; Erin Pearse; Feng Tian Pages: 351 - 382 Abstract: Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces whose intersection contains a fixed vector space \(\mathscr {D}\) . In the case when \(\mathscr {D}\) is dense in one of the Hilbert spaces (but not necessarily in the other), we make precise an operator-theoretic linking between the two Hilbert spaces. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and the operator theory of reflection positivity. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0173-9 Issue No:Vol. 8, No. 3 (2018)

Authors:Diganta Borah; Pranav Haridas; Kaushal Verma Pages: 383 - 414 Abstract: We study several quantities associated to the Green’s function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and \(L^2\) -cohomology of the capacity metric and critical points of the Green’s function. The principal idea used is an affine scaling of the domain that furnishes quantitative boundary behaviour of the Green’s function and related objects. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0177-5 Issue No:Vol. 8, No. 3 (2018)

Authors:Hui Nie; Junyi Zhu; Xianguo Geng Pages: 415 - 426 Abstract: The Gerdjikov–Ivanov equation is investigated by the Riemann–Hilbert approach and the technique of regularization. The trace formula and new form of N-soliton solution are given. The dynamics of the stationary solitons and non-stationary solitons are discussed. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0179-3 Issue No:Vol. 8, No. 3 (2018)

Authors:Jin-Yun Yang; Wen-Xiu Ma; Zhenyun Qin Pages: 427 - 436 Abstract: Based on the Hirota bilinear form of the \((2+1)\) -dimensional Ito equation, one class of lump solutions and two classes of interaction solutions between lumps and line solitons are generated through analysis and symbolic computations with Maple. Analyticity is naturally guaranteed for the presented lump and interaction solutions, and the interaction solutions reduce to lumps (or line solitons) while the hyperbolic-cosine (or the quadratic function) disappears. Three-dimensional plots and contour plots are made for two specific examples of the resulting interaction solutions. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0181-9 Issue No:Vol. 8, No. 3 (2018)

Authors:Yehonatan Salman Pages: 437 - 463 Abstract: The aim of the article is to recover a certain type of finite parametric distributions and functions using their spherical mean transform which is given on a certain family of spheres whose centers belong to a finite set \(\Gamma \) . For this, we show how the problem of reconstruction can be converted to a Prony’s type system of equations whose regularity is guaranteed by the assumption that the points in the set \(\Gamma \) are in general position. By solving the corresponding Prony’s system we can extract the set of parameters which define the corresponding function or distribution. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0171-y Issue No:Vol. 8, No. 3 (2018)

Authors:Nguyen Xuan Hong; Tran Van Thuy Pages: 465 - 484 Abstract: In this paper, we prove the Hölder continuity for solutions to the complex Monge–Ampère equations on non-smooth pseudoconvex domains of plurisubharmonic type m. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0175-7 Issue No:Vol. 8, No. 3 (2018)

Authors:Rubén A. Hidalgo Abstract: A \((\gamma ,n)\) -gonal pair is a pair (S, f), where S is a closed Riemann surface and \(f:S \rightarrow R\) is a degree n holomorphic map onto a closed Riemann surface R of genus \(\gamma \) . If the signature of (S, f) is of hyperbolic type, then it admits a uniformizing pair \((\varGamma ,G)\) , where G is a Fuchsian group acting on the unit disc \({{\mathbb {D}}}\) containing \(\varGamma \) as an index n subgroup, such that f is induced by the inclusion \(\varGamma \le G\) . The uniformizing pair is uniquely determined by (S, f), up to conjugation by holomorphic automorphisms of \({{\mathbb {D}}}\) , and it permits to provide a natural complex orbifold structure on the Hurwitz space parametrizing (twisted) isomorphic classes of pairs topologically equivalent to (S, f). In order to produce certain compactifications of these Hurwitz spaces, one needs to consider the so called stable \((\gamma ,n)\) -gonal pairs, which are natural geometrical deformations of \((\gamma ,n)\) -gonal pairs. Due to the above, it seems interesting to search for uniformizations of stable \((\gamma ,n)\) -gonal pairs, in terms of certain class of Kleinian groups. In this paper we review such uniformizations by using noded Fuchsian groups, obtained from the noded Beltrami differentials of Fuchsian groups that were previously studied by Alexander Vasil’ev and the author, and which provide uniformizations of stable Riemann orbifolds. These uniformizations permit to obtain a compactification of the Hurwitz spaces together a complex orbifold structure, these being quotients of the augmented Teichmüller space of G by a suitable finite index subgroup of its modular group. PubDate: 2018-10-15 DOI: 10.1007/s13324-018-0253-5

Authors:Filippo Bracci; Manuel D. Contreras; Santiago Díaz-Madrigal Abstract: Let \((\phi _t)\) be a semigroup of holomorphic self-maps of \(\mathbb {D}\) . In this note, we use an abstract approach to define the Koenigs function of \((\phi _t)\) and “holomorphic models” and show how to deduce the existence and properties of the infinitesimal generator of \((\phi _t)\) from this construction. PubDate: 2018-10-15 DOI: 10.1007/s13324-018-0254-4

Authors:Ivan Gonzalez; Igor Kondrashuk; Eduardo A. Notte-Cuello; Ivan Parra-Ferrada Abstract: We observe a property of orthogonality of the Mellin–Barnes transformation of triangle one-loop diagrams, which follows from our previous papers (Kondrashuk and Kotikov in JHEP 0808:106, 2008; Kondrashuk and Vergara in JHEP 1003:051, 2010; Allendes et al. in J Math Phys 51:052304, 2010). In those papers it has been established that Usyukina–Davydychev functions are invariant with respect to the Fourier transformation. This has been proved at the level of graphs and also via the Mellin–Barnes transformation. We partially apply to the one-loop massless scalar diagram the same trick in which the Mellin–Barnes transformation was involved and obtain the property of orthogonality of the corresponding MB transforms under integration over contours in two complex planes with certain weight. This property is valid in an arbitrary number of dimensions. PubDate: 2018-10-10 DOI: 10.1007/s13324-018-0252-6

Authors:K. Saoudi Abstract: In this work we consider the following fractional Kirchhoff equations with singular nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} M\Big ( \int _{\mathbb {R}^{2N}}\frac{ u(x)-u(y) ^{2}}{ x-y ^{N+2s}}dx dy\Big )(-\Delta )^s u = \lambda a(x) u ^{q-2}u +\frac{1-\alpha }{2-\alpha -\beta } c(x) u ^{-\alpha } v ^{1-\beta }, \quad \text {in }\Omega ,\\ M\Big ( \int _{\mathbb {R}^{2N}}\frac{ v(x)-v(y) ^{2}}{ x-y ^{N+2s}}dx dy\Big ) (-\Delta )^s v= \mu b(x) v ^{q-2}v +\frac{1-\beta }{2-\alpha -\beta } c(x) u ^{1-\alpha } v ^{-\beta }, \quad \text {in }\Omega ,\\ u=v = 0,\;\; \text{ in } \,\mathbb {R}^N\setminus \Omega , \end{array} \right. \end{aligned}$$ where \(\Omega \) is a bounded domain in \(\mathbb {R}^n\) with smooth boundary \(\partial \Omega \) , \(N> 2s\) , \(s \in (0,1)\) , \(0<\alpha<1,\;0<\beta <1,\) \(0<\alpha +\beta<2\theta<q<2^*_s,\) \(2^*_s=\frac{2N}{N-2s}\) is the fractional Sobolev exponent, \(\lambda , \mu \) are two parameters, \(a,\, b, \,c \in C({\overline{\Omega }})\) are non-negative weight functions, M is a continuous function, given by \(M(t)=k+lt^{\theta -1}\) \(k>0,\,l,\,\theta \ge 1,\) and \((-\Delta )^s\) is the fractional Laplacien operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter \(\lambda \) and \(\mu \) . PubDate: 2018-10-03 DOI: 10.1007/s13324-018-0251-7

Authors:Wolfram Bauer; Daisuke Tarama Abstract: Pseudo-H-type groups \(G_{r,s}\) form a class of step-two nilpotent Lie groups with a natural pseudo-Riemannian metric. In this paper the question of complete integrability in the sense of Liouville is studied for the corresponding (pseudo-)Riemannian geodesic flow. Via the isometry group of \(G_{r,s}\) families of first integrals are constructed. A modification of these functions gives a set of \(\dim G_{r,s}\) functionally independent smooth first integrals in involution. The existence of a lattice L in \(G_{r,s}\) is guaranteed by recent work of K. Furutani and I. Markina. The complete integrability of the pseudo-Riemannian geodesic flow of the compact nilmanifold \(L \backslash G_{r,s}\) is proved under additional assumptions on the group \(G_{r,s}\) . PubDate: 2018-10-01 DOI: 10.1007/s13324-018-0250-8

Authors:Liangchen Qian; Yan Xu Abstract: In this paper, we obtain some normality criteria for families of meromorphic functions, which improve and generalize related results of Gu, Yang–Chang, Schwick, and Datt–Kumar. PubDate: 2018-09-28 DOI: 10.1007/s13324-018-0249-1

Authors:Dmitri Prokhorov Abstract: We consider functionals L which are the real parts of linear combinations of two Taylor coefficients on the class S of holomorphic univalent functions in the unit disk. The Bombieri conjecture can be interpreted in the form that L are locally maximized by the Koebe function simultaneously on S and on its subclass \(S_R\) consisting of typically real functions. We derive necessary criteria for the Bombieri conjecture in terms of inequalities for solutions to systems of differential equations in variations for the Loewner ODE. PubDate: 2018-09-07 DOI: 10.1007/s13324-018-0248-2

Authors:Pavel Gumenyuk; István Prause Abstract: Becker (J Reine Angew Math 255:23–43, 1972) discovered a sufficient condition for quasiconformal extendibility of Loewner chains. Many known conditions for quasiconformal extendibility of holomorphic functions in the unit disk can be deduced from his result. We give a new proof of (a generalization of) Becker’s result based on Slodkowski’s Extended \(\lambda \) -Lemma. Moreover, we characterize all quasiconformal extensions produced by Becker’s (classical) construction and use that to obtain examples in which Becker’s extension is extremal (i.e. optimal in the sense of maximal dilatation) or, on the contrary, fails to be extremal. PubDate: 2018-09-06 DOI: 10.1007/s13324-018-0247-3

Authors:Tarul Garg; Ana Maria Acu; P. N. Agrawal Abstract: The present paper deals with the approximation properties of the bivariate operators which are the combination of Bernstein–Chlodowsky operators and the Szász–Kantorovich type operators. We investigate the degree of approximation of the bivariate operators for continuous functions in the weighted space of polynomial growth. Further, we introduce the Generalized Boolean Sum (GBS) of these bivariate Chlodowsky–Szász–Kantorovich type operators and examine the order of approximation in the Bögel space of continuous functions by means of the Lipschitz class and mixed modulus of smoothness. Besides this, we compare the rate of convergence of the Chlodowsky–Szász–Kantorovich type operators and the associated GBS operators by numerical examples and tables using Maple algorithms. It turns out that the GBS operator converges faster to the function than the original operator. PubDate: 2018-08-18 DOI: 10.1007/s13324-018-0246-4

Authors:Pu Zhang Abstract: We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy–Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces when the symbols b belong to the Lipschitz spaces, by which some new characterizations of Lipschitz spaces and nonnegative Lipschitz functions are obtained. Some equivalent relations between the Lipschitz norm and the variable exponent Lebesgue norm are also given. PubDate: 2018-08-11 DOI: 10.1007/s13324-018-0245-5

Authors:Natalia P. Bondarenko Abstract: In this paper, a partial inverse problem for the quadratic Sturm–Liouville pencil on a geometrical graph of arbitrary structure is studied. We suppose that the coefficients of differential expressions are known a priori on all the graph edges except one, and recover the coefficients on the remaining edge, using a part of the spectrum. The results of the paper are uniqueness theorems and a constructive algorithm for solving the partial inverse problem. PubDate: 2018-08-03 DOI: 10.1007/s13324-018-0244-6

Authors:Jihoon Lee Abstract: Let \(\Omega \) be a bounded domain in \(\mathbb {R}^N=\mathbb {R}^{N_1} \times \mathbb {R}^{N_2}\) with \(N_1, N_2 \ge 1\) , and \(N(s) = N_1 + (1+s)N_2\) be the homogeneous dimension of \(\mathbb {R}^N\) for \(s \ge 0\) . In this paper, we prove the existence and uniqueness of boundary blow-up solutions to the following semilinear degenerate elliptic equation $$\begin{aligned} {\left\{ \begin{array}{ll} G_s u = { x ^{2s}} u^p_+ \; &{}\text { in } \Omega ,\\ u(z)\rightarrow +\infty \; &{}\text { as } {d}(z) \rightarrow 0, \end{array}\right. } \end{aligned}$$ where \(u_+ = \max \{u,0\}\) , \(1<p<{{N(s)+2s} \over {N(s)-2}}\) , and d(z) denotes the Grushin distance from z to the boundary of \(\Omega \) . Here \(G_s\) is the Grushin operator of the form $$\begin{aligned} G_s u= \Delta _x u + x ^{2s}\Delta _y u, \; s\ge 0. \end{aligned}$$ It is worth noticing that our results do not require any assumption on the smoothness of the domain \(\Omega \) , and when \(s=0\) , we cover the previous results for the Laplace operator \(\Delta \) . PubDate: 2018-07-23 DOI: 10.1007/s13324-018-0241-9

Authors:Deepak Bansal; Manoj Kumar Soni; Amit Soni Abstract: In the present investigation we first introduce modified Dini function \(R^k_{\nu }(z)\) and then find sufficient conditions so that the modified Dini function \(R^k_{\nu }(z)\) have certain geometric properties like close-to-convexity, starlikeness and strongly starlikeness in the open unit disk. Relevance with some known results are also pointed out. PubDate: 2018-07-16 DOI: 10.1007/s13324-018-0243-7