Abstract: Abstract In this paper, the N-fold Darboux transformation (DT) of the combined Toda lattice and relativistic Toda lattice equation is constructed in terms of determinants. Comparing with the usual 1-fold DT of equations, this kind of N-fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications of the N-fold DT, we derive two kinds of N-fold explicit exact solutions from two different seed solutions and plot the figures with properly parameters to illustrate the propagation of solitary waves. What’s more, we present the relationships between the structures of exact solutions parameters with \(N=1\) , from which we find the 1-fold solutions may be one soliton solutions or periodic solutions and the waves pass through without change of shapes, amplitudes, wavelengths and directions, etc. The results in this paper might be helpful for interpreting certain physical phenomena. PubDate: 2020-06-27

Abstract: Abstract Let E be a set on the unit circle T of \(\mathbb C\). We prove that there exists an \(f \in H^\infty \) which has no radial limits on E but has unrestricted limit at each point of \(T {\setminus } E\) if and only if E is an \(F_\sigma \) of measure zero. The necessity of the condition that E is an \(F_\sigma \) is almost obvious and the necessity of the condition that E is of measure zero follows from Fatou’s theorem. PubDate: 2020-06-12

Abstract: Abstract In this note, we reformulate Aharonov’s univalence criterion in terms of projective Schwarzian derivatives of higher order. We also mention some relations between quasiconformal extendibility and the Aharonov criterion. PubDate: 2020-06-12

Abstract: Abstract In this article, we study regularity criteria for the 3D micropolar fluid equations in terms of one partial derivative of the velocity. It is proved that if $$\begin{aligned} \int ^{T}_{0}\Vert \partial _{3}u\Vert ^{\frac{2}{1-r}}_{\dot{B}^{-r}_{\infty ,\infty }} dt<\infty \quad \text {with} \quad 0< r<1, \end{aligned}$$then, the solutions of the micropolar fluid equations actually are smooth on (0, T). This improves and extends many previous results. PubDate: 2020-06-12

Abstract: Abstract In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms \(\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])\) and \(\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])\). These operators arise when one studies the interior problem of tomography. The diagonalization of \(\mathcal {H}_R,\mathcal {H}_L\) has been previously obtained, but only asymptotically when \(b_L\ne -b_R\). We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes \(\mathcal {H}_R,\mathcal {H}_L\) explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates. PubDate: 2020-06-10

Abstract: Abstract In this paper, we proved a general theorem dealing with the absolute Cesàro summability factors by using an almost increasing sequence. This new theorem also contains as particular cases several known and new results on the absolute Cesàro summability factors of infinite series. PubDate: 2020-06-06

Abstract: Abstract Inner functions play a central role in function theory and operator theory on the Hardy space over the unit disk. Motivated by recent works of C. Bénéteau et al. and of D. Seco, we discuss inner functions on more general weighted Hardy spaces and investigate a method to construct analogues of finite Blaschke products. PubDate: 2020-06-05

Abstract: Abstract We study the isoperimetric problem for the axially symmetric sets in the Heisenberg group \(\mathbb {H}^n\) with density \( z ^p\). At first, we prove the existence of weighted isoperimetric sets. Then, we characterize weighted isoperimetric sets uniquely as bubble sets. Finally, we deduce an interesting result that, up to a constant multiplicator, \( z ^p\) is the only horizontal radial density for which bubble sets can be weighted isoperimetric sets. PubDate: 2020-05-17

Abstract: Abstract First we introduce the two tau-functions which appeared either as the \(\tau \)-function of the integrable hierarchy governing the Riemann mapping of Jordan curves or in conformal field theory and the universal Grassmannian. Then we discuss various aspects of their interrelation. Subsequently, we establish a novel connection between free probability, growth models and integrable systems, in particular for second order freeness, and summarise it in a dictionary. This extends the previous link between conformal maps and large N-matrix integrals to (higher) order free probability. Within this context of dynamically evolving contours, we determine a class of driving functions for controlled Loewner–Kufarev equations, which enables us to give a continuity estimate for the solution of such an equation when embedded into the Segal–Wilson Grassmannian. PubDate: 2020-05-08

Abstract: Abstract Conjugations in space \(L^2\) of the unit circle commuting with multiplication by z or intertwining multiplications by z and \({{\bar{z}}}\) are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces. PubDate: 2020-04-10

Abstract: Abstract The barotropic vorticity equation is a classical model in atmospheric sciences. In this paper, we study the symmetry invariance properties of multipliers (integrating factors) admitted by this equation. The results are classified according to the ratio of the characteristic length scale to the Rossby radius of deformation and the variation of earth’s angular rotation. A plethora of conservation laws can be obtained by studying the interaction between Lie point symmetry generators and multipliers. PubDate: 2020-04-09

Abstract: Abstract We deduce Paley–Wiener results in the Bargmann setting. At the same time we deduce characterisations of Pilipović spaces of low orders. In particular we improve the characterisation of the Gröchenig test function space \(\mathcal {H}_{\flat _1}=\mathcal {S}_C\), deduced in Toft (J Pseudo-Differ Oper Appl 8:83–139, 2017). PubDate: 2020-04-07

Abstract: Abstract Given a nonexpansive mapping which maps a closed subset of a complete metric space into the space, we study the convergence of its inexact iterates to its fixed point set in the case where the errors are nonsummable. Previous results in this direction concerned nonexpansive self-mappings of the complete metric space and inexact iterates with summable errors. PubDate: 2020-04-06

Abstract: Abstract A radial weight \(\omega \) belongs to the class \(\widehat{\mathcal {D}}\) if there exists \(C=C(\omega )\ge 1\) such that \(\int _r^1 \omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds\) for all \(0\le r<1\). Write \(\omega \in \check{\mathcal {D}}\) if there exist constants \(K=K(\omega )>1\) and \(C=C(\omega )>1\) such that \({\widehat{\omega }}(r)\ge C{\widehat{\omega }}\left( 1-\frac{1-r}{K}\right) \) for all \(0\le r<1\). These classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights (Peláez and Rättyä in Bergman projection induced by radial weight, 2019. arXiv:1902.09837). Classical results by Hardy and Littlewood (J Reine Angew Math 167:405–423, 1932) and Shields and Williams (Mich Math J 29(1):3–25, 1982) show that the weighted Bergman space of harmonic functions is not closed by harmonic conjugation if \({\omega \in \widehat{\mathcal {D}}\setminus \check{\mathcal {D}}}\) and \(0<p\le 1\). In this paper we establish sharp estimates for the norm of the analytic Bergman space \(A^p_\omega \), with \({\omega \in \widehat{\mathcal {D}}\setminus \check{\mathcal {D}}}\) and \(0<p<\infty \), in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights. PubDate: 2020-03-29

Abstract: Abstract This article continues to study the linearized Chandrasekhar equation. We use the Hilbert-type inequalities to accurately calculate the norm of the Fredholm integral operator and obtain the exact range for the parameters of the linearized Chandrasekhar equation to ensure that there is a unique solution to the equation in \(L^p\) space. A series of examples that can accurately calculate the norm of Fredholm integral operator shows that the Chandrasekhar kernel functions do not need to meet harsh conditions. As the symbolic part of the Chandrasekhar kernel function and the non-homogeneous terms satisfy the exponential decay condition, we yield a normed convergence rate of the approximation solution in \(L^p\) sense, which adds new results to the theory of radiation transfer in astrophysics. PubDate: 2020-03-14

Abstract: Abstract We present a simple proof of Tan’s theorem on asymptotic similarity between the Mandelbrot set and Julia sets at Misiurewicz parameters. Then we give a new perspective on this phenomenon in terms of Zalcman functions, that is, entire functions generated by applying Zalcman’s lemma to complex dynamics. We also show asymptotic similarity between the tricorn and Julia sets at Misiurewicz parameters, which is an antiholomorphic counterpart of Tan’s theorem. PubDate: 2020-03-06

Abstract: Abstract In this paper, we shall give a characterization for the strong and weak type Spanne type boundedness of the fractional maximal operator \(M_{\alpha }\), \(0\le \alpha <Q\) on Carnot group \({{\mathbb {G}}}\) on generalized weighted Morrey spaces \(M_{p,\varphi }({{\mathbb {G}}},w)\), where Q is the homogeneous dimension of \({{\mathbb {G}}}\). Also we give a characterization for the Spanne type boundedness of the fractional maximal commutator operator \(M_{b,\alpha }\) on generalized weighted Morrey spaces. PubDate: 2020-03-05

Abstract: We study certain dynamical systems which leave invariant an indefinite quadratic form via semigroups or evolution families of complex symmetric Hilbert space operators. In the setting of bounded operators we show that a \(\mathcal {C}\)-selfadjoint operator generates a contraction \(C_0\)-semigroup if and only if it is dissipative. In addition, we examine the abstract Cauchy problem for nonautonomous linear differential equations possessing a complex symmetry. In the unbounded operator framework we isolate the class of complex symmetric, unbounded semigroups and investigate Stone-type theorems adapted to them. On Fock space realization, we characterize all \(\mathcal {C}\)-selfadjoint, unbounded weighted composition semigroups. As a byproduct we prove that the generator of a \(\mathcal {C}\)-selfadjoint, unbounded semigroup is not necessarily \(\mathcal {C}\)-selfadjoint. PubDate: 2020-02-21

Abstract: Abstract We study closed n-dimensional manifolds of which the metrics are critical for quadratic curvature functionals involving the Ricci curvature, the scalar curvature and the Riemannian curvature tensor on the space of Riemannian metrics with unit volume. Under some additional integral conditions, we classify such manifolds. Moreover, under some curvature conditions, the result that a critical metric must be Einstein is proved. PubDate: 2020-02-05

Abstract: Abstract In the first part of this work we study the set of bicomplex numbers from the point of view of a hyperbolic module. We make use of the partial order defined on the set of hyperbolic numbers. We recall some properties of the hyperbolic geometrical objects that were defined in previous papers. With the help of these notions some fractal-type sets in the four dimensional space are constructed. PubDate: 2020-02-05