Authors:Brian Simanek Abstract: Given a bounded function \(\varphi \) on the unit disk in the complex plane, we consider the operator \(T_{\varphi }\) , defined on the Bergman space of the disk and given by \(T_{\varphi }(f)=P(\varphi f)\) , where P denotes the orthogonal projection to the Bergman space in \(L^2({\mathbb {D}},dA)\) . For algebraic symbols \(\varphi \) , we provide new necessary conditions on \(\varphi \) for \(T_{\varphi }\) to be hyponormal, extending recent results of Fleeman and Liaw. Our approach is perturbative and aims to understand how small changes to a symbol preserve or destroy hyponormality of the corresponding operator. We consider both additive and multiplicative perturbations of a variety of algebraic symbols. One of our main results provides a necessary condition on the complex constant C for the operator \(T_{z^n+C z ^s}\) to be hyponormal. This condition is also sufficient if \(s\ge 2n\) . PubDate: 2019-01-08 DOI: 10.1007/s13324-018-00279-2

Authors:Thomas Bieske; Robert D. Freeman Abstract: In this short note, we remark on the main theorem of Theorem 5.1]. PubDate: 2019-01-03 DOI: 10.1007/s13324-018-00278-3

Authors:Maxim Ya Mazalov Abstract: Bianalytic capacities appear naturally in problems of uniform approximation of functions by bianalytic functions on compact sets in the complex plane. They play a crucial role in constructions of approximants in several such problems. It turns out, that bianalytic capacities obey several unusual properties in comparison with other capacities studied in the approximation theory. In particular, bianalytic capacities do not satisfy the semiadditivity property. In this paper, we study these capacities and consider their relations with Calderon commutators. PubDate: 2018-12-18 DOI: 10.1007/s13324-018-0276-y

Authors:Keshav Raj Acharya Abstract: We develop the Titchmarsh–Weyl theory for vector-valued discrete Schrödinger operators. We show that the Weyl m functions associated with these operators are matrix valued Herglotz functions that map complex upper half plane to the Siegel upper half space. We discuss about the Weyl disk and Weyl circle corresponding to these operators by defining these functions on a bounded interval. We also discuss the geometric properties of Weyl disk and find the center and radius of the Weyl disk explicitly in terms of matrices. PubDate: 2018-12-17 DOI: 10.1007/s13324-018-0277-x

Authors:Anna Duyunova; Valentin Lychagin; Sergey Tychkov Abstract: Symmetries and the corresponding algebras of differential invariants of inviscid fluids on a spherical layer are given. Their dependence on thermodynamical states of the medium is studied, and a classification of thermodynamical states is given. PubDate: 2018-12-13 DOI: 10.1007/s13324-018-0274-0

Authors:M. Bertola; A. Minakov Abstract: We consider the compressive wave for the modified Korteweg–de Vries equation with background constants \(c>0\) for \(x\rightarrow -\infty \) and 0 for \(x\rightarrow +\infty \) . We study the asymptotics of solutions in the transition zone \(4c^2t-\varepsilon t<x<4c^2t-\beta t^{\sigma }\ln t\) for \(\varepsilon >0,\) \(\sigma \in (0,1),\) \(\beta >0.\) In this region we have a bulk of nonvanishing oscillations, the number of which grows as \(\frac{\varepsilon t}{\ln t}.\) Also we show how to obtain Khruslov–Kotlyarov’s asymptotics in the domain \(4c^2t-\rho \ln t<x<4c^2t\) with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann–Hilbert problem. PubDate: 2018-12-12 DOI: 10.1007/s13324-018-0273-1

Authors:P. V. Paramonov; X. Tolsa Abstract: Criteria for approximability of functions by solutions of homogeneous second order elliptic equations (with constant complex coefficients) in the norms of the Whitney \(C^1\) -spaces on compact sets in \(\mathbb {R}^2\) are obtained in terms of the respective \(C^1\) -capacities. It is proved that the mentioned \(C^1\) -capacities are comparable to the classic C-analytic capacity, and so have a proper geometric measure characterization. PubDate: 2018-12-08 DOI: 10.1007/s13324-018-0275-z

Authors:Wen-Xiu Ma Abstract: A Darboux transformation is presented for the Volterra lattice equation, based on a pair of \(2\times 2\) matrix spectral problems. The resulting DT is applied to construction of solitary wave solutions from a constant seed solution. A particular phenomenon is that only one condition is required in determining the corresponding Darboux matrix, but not two as for most pairs of \(2\times 2 \) matrix spectral problems. PubDate: 2018-12-08 DOI: 10.1007/s13324-018-0267-z

Authors:Li Cheng; Yi Zhang; Mei-Juan Lin Abstract: We aim to explore exact solutions and integrable properties to the (2+1)-dimensional DJKM equation. Based on the bilinear Bäcklund transformation, we first furnish Lax pair and complex exponential wave function solutions, and then give complexitons or hyperbolic function solutions. Moreover, via the nonlinear superposition formula, the construction procedure for presenting rational solutions is improved. The key step is that all the involved parameters are extended to the complex field. In particular, we show that the (2+1)-dimensional DJKM equation possesses a general class of lump solutions when \({\sigma }^2=-1\) . PubDate: 2018-12-05 DOI: 10.1007/s13324-018-0271-3

Authors:Veli B. Shakhmurov; Rishad Shahmurov Abstract: In this paper, the existence and uniqueness of solution of the Cauchy problem for abstract Boussinesq equation is obtained. The leading part of the equation include general elliptic operator and abstract positive operator in a Banach space E. Since the Banach space E and linear operators are sufficiently large classes, by choosing their we obtain the existence and uniqueness of solution of numerous classes of generalized Boussinesq type equations which occur in a wide variety of physical systems. By applying this result, the Wentzell–Robin type mixed problem for Boussinesq equations and the Cauchy problem for finite or infinite systems of Boussinesq equations are studied. PubDate: 2018-12-01 DOI: 10.1007/s13324-018-0265-1

Authors:Yuan-Jen Chiang Abstract: We show the maximum principle for exponential energy minimizing maps. We then estimate the distance of two image points of an exponentially harmonic map between surfaces. We also study the existence of an exponentially harmonic map between surfaces if the image is contained in a convex disc. We finally investigate the existence of an exponentially harmonic map \(f:M_1\rightarrow M_2\) between surfaces in case \(\pi _2 (M_2) = \emptyset \) . PubDate: 2018-12-01 DOI: 10.1007/s13324-018-0270-4

Authors:Sorin Dragomir; Guilin Yang Abstract: We investigate the relationship between solitonic metrics \(g_u = - (\sin ^2 \frac{u}{2}) c^2 d t^2 + (\cos ^2 \frac{u}{2}) \sum _{i=1}^n (d x^i )^2\) with \(u \in C^\infty ({{\mathbb {R}}}^{n+1})\) and stationary points of the functional \(E_\Omega (\Phi ) = \frac{1}{2} \int _\Omega \Vert d \Phi \Vert ^2 d^{n+1} \mathbf{x}\) with \(\Phi \in C^\infty ({{\mathbb {R}}}^{n+1}, S^2 )\) . Building on work by F.L. Williams (cf. Williams, in: Milton (ed) Quantum field theory under the influence of external conditions, Rinton Press, Princeton, pp 370–372, 2004; in: 4th international winter conference on mathematical methods in physics, Rio de Janeiro, http://pos.sissa.it/archive/conferences/013/003/wc2004_003.pdf, 2004; in: Chen (ed) Trends in soliton research, Nova Science Publications, Hauppauge, pp 1–14, 2006; in: Maraver, Kevrekidis, Williams (eds) The sine-gordon modeland its applications, Springer, Berlin, pp 177–205, 2014) we show that a map \(\Phi = (\cos \beta \sin \frac{u}{2}, \sin \beta \sin \frac{u}{2}, \cos \frac{u}{2} )\) with \(\beta (\mathbf{x}, t) = m \big ( 1 + \mathbf{v} ^2 \big )^{-1/2} (t + \mathbf{v} \cdot \mathbf{x})\) is harmonic if and only if \(u_{tt} + \Delta u = m^2 \sin u\) (the sine-Gordon equation) and \(u_t + \mathbf{v} \cdot \nabla u = 0\) (the convection equation). In the spirit of work by B. Solomon (cf. Solomon in J Differ Geom 21:151–162, 1985) we build a 1-parameter variation of \(\Phi \) which singles out [from the full Euler–Lagrange system of the variational principle \(\delta E_\Omega (\Phi ) = 0\) ] the convection equation. Williams’ harmonic maps \(\Phi ^\pm = (1 + v^2 )^{-1/2} (\tau \cos \beta , \tau \sin \beta , \pm (1 + v^2 )^{1/2}\tanh \rho )\) are shown to be harmonic morphisms of dilation \(\lambda = m(1 + v^2 )^{-1/2}\tau \) , further explaining the relationship between Jackiw–Teitelboim 2-dimensional dilation-gravity theory (cf. Jackiw, in: Christensen (ed) Quantum theory of gravity, MIT, Cambridge, pp 403–420, 1982; Teitelboim, in: Christensen (ed) Quantum theory of gravity, Adam Hilger Ltd, Bristol, pp 403–420, 1984) and harmonic map theory (cf. Baird and Wood Harmonic morphisms between Riemannian manifolds, London mathematical society monographs, new series, 29, The Clarendon Press, Oxford University Press, Oxford, ISBN 0-19-850362-8, 2003). We show that geodesic motion in a weak solitonic gravitational field \(g_{u_\epsilon }\) [ \(u_\epsilon = u_0 + 2 \epsilon \rho \) with \(\rho \in C^\infty ({{\mathbb {R}}}^{n+1})\) bounded and \(\epsilon<< 1\) ] in the Newtonian velocity limit ( \(\Vert \mathbf{u}\Vert /c<< 1\) ) obeys to the law of motion \(d^2 \mathbf{r}/d t^2 = - \nabla \phi \) PubDate: 2018-11-27 DOI: 10.1007/s13324-018-0269-x

Authors:Thomas Bieske; Robert D. Freeman Abstract: We show the equivalence of weak and viscosity solutions to the \({\texttt {p}}(x)\) -Laplacian in Carnot groups, under certain reasonable restrictions on the function \({\texttt {p}}(x)\) and on the viscosity solution. We also obtain a comparison principle for viscosity solutions. This then implies that viscosity solutions to Dirichlet problems are unique. PubDate: 2018-11-17 DOI: 10.1007/s13324-018-0266-0

Authors:Carlos Daniel Tamayo Castro; Ricardo Abreu Blaya; Juan Bory Reyes Abstract: In this paper sufficient conditions for continuous and compact embeddings of generalized higher order Lipschitz classes on a compact subset of n-dimensional real Euclidean spaces are obtained. PubDate: 2018-11-16 DOI: 10.1007/s13324-018-0268-y

Authors:Oleg Safronov; Ari Laptev; Francesco Ferrulli Abstract: We study the spectrum of a system of second order differential operator \(D_m\) perturbed by a non-selfadjoint matrix valued potential V. We prove that eigenvalues of \(D_m+V\) are located near the edges of the spectrum of the unperturbed operator \(D_m\) . PubDate: 2018-11-07 DOI: 10.1007/s13324-018-0262-4

Authors:Pranav Haridas; Jaikrishnan Janardhanan Abstract: It is known that if \(f: D_1 \rightarrow D_2\) is a polynomial biholomorphism with polynomial inverse and constant Jacobian then \(D_1\) is a 1-point poly-quadrature domain (the Bergman span contains all holomorphic polynomials) of order 1 whenever \(D_2\) is a complete circular domain. Bell conjectured that all 1-point poly-quadrature domains arise in this manner. In this note, we construct a 1-point poly-quadrature domain of order 1 that is not biholomorphic to any complete circular domain. PubDate: 2018-11-02 DOI: 10.1007/s13324-018-0263-3

Authors:Bilel Selmi Abstract: We prove a decomposition theorem of Besicovitch’s type for the relative multifractal Hausdorff measure and packing measure in a probability space. By obtaining a new necessary condition for the strong regularity with the multifractal measures in a more general framework, we extend in this paper the density theorem of Dai and Li (A multifractal formalism in a probability space. Chaos Solitons Fractals 27:57–73, 2006). In particular, this result is more refined than those found in Dai and Taylor (Defining fractal in a probability space. Ill J Math 38:480–500, 1994). PubDate: 2018-11-01 DOI: 10.1007/s13324-018-0261-5

Authors:Melkana A. Brakalova Abstract: We prove symmetric properties of the p-integrable subspaces \(T_p\) of the universal Teichmüller space T, for \(p>0.\) The elements of \(T_p\) are quasisymmetric automorphisms of the real line or of the unit circle, which have a q.c. extension with p-integrable complex dilatation with respect to the Poincaré metric. We show that \(T_p\subset T_S,\) where \(T_S\) is the symmetric Teichmüller space and that, in the case of the real line, \(T_p\subset T_0,\) where \(T_0\) is the little Teichmüller space. The proofs use properties of the boundary behavior of q.c maps such as uniform weak conformality. PubDate: 2018-10-31 DOI: 10.1007/s13324-018-0259-z