Abstract: Abstract There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean metric. Solutions are constructed in all dimensions and solutions are classified in dimension at most 4. Techniques are given for determining when two solutions are linearly conformally inequivalent. PubDate: 2021-01-16
Abstract: Abstract In this paper, we consider the conical Radon transform on all one-sided circular cones in \(\mathbf{R}^3\) with horizontal central axis whose vertices are on a vertical line. We derive an explicit inversion formula for such transform. The inversion makes use of the vertical slice transform on a sphere and V-line transform on a plane. PubDate: 2021-01-09
Abstract: Abstract We assume that \(\mathfrak {M}^{n}\) is an n-dimensional cigar metric measure space (CMMS for short) endowed with cigar metric and certain smooth potential on \(\mathbb {R}^{n}\) . When the dimension is two, it is a cigar soliton, which can be viewed as Euclidean-Witten black hole under the first-order Ricci flow of the world-sheet sigma model in general relative theory. Thus, the CMMS is of great significant in both geometry and physics. In this paper, we investigate the eigenvalue problem with Dirichlet boundary condition for the Witten-Laplacian on CMMS \(\mathfrak {M}^{n}\) and establish some intrinsic formulas by applying some auxiliary lemmas to replace the corresponding extrinsic formulas due to Chen and Cheng. Combining with a general formula given by the first author and Sun, we establish an inequality for the eigenvalues with lower order. As further interesting applications, we obtain several eigenvalue inequalities of Payne–Pólya–Weinberger type in low-dimensional topology. PubDate: 2021-01-09
Abstract: Abstract In this paper, a generalized nonlinear Camassa–Holm equation with time- and space-dependent coefficients is considered. We show that the control of the higher order dispersive term is possible by using an adequate weight function to define the energy. The existence and uniqueness of solutions are obtained via a standard Picard iterative method, so that there is no loss of regularity of the solution with respect to the initial condition in some appropriate Sobolev space. PubDate: 2021-01-08
Abstract: Abstract In this paper, we study the steady-state distribution of heat on long pipes in \({\mathbb {R}}^3\) heated along some regions of their surfaces. In particular, we prove that, if the pipe \(P=\{(x,y,z):\,x^2+y^2<1\}\) is heated along its surface belt \(B(a)=\{(x,y,z):\,x^2+y^2=1,-a<z<a\}\) , \(a>0\) , then the temperature in its cross-sections \(D_c=\{(x,y,z)\in P:\, z=c\}\) is increasing in the radial direction for all c in the interval \([-a, a]\) . PubDate: 2021-01-08
Abstract: Abstract In this paper we present briefly basics of the generation theory for one-parameter semigropus of holomorphic mappings in the one dimensional case and point out some new trends and problems related to this issue. A special attention we pay to the geometric properties of the nonlinear resolvent of holomorphic generators in the spirit of classical geometric function theory. PubDate: 2021-01-07
Abstract: Abstract We describe the set of meromorphic univalent functions in the class \(\Sigma \) , for which the sequence of the Faber polynomials \(\{F_j\}_{j=1}^\infty \) have the roots with following properties \( F_n (z_0) >0=\sum _{\begin{array}{c} j=1 \\ j\not =n \end{array}} F_j (z_0 ) \) . For such functions we found an explicit form of the Faber polynomials as well as we discussed some properties. PubDate: 2021-01-07
Abstract: Abstract Let W be the wedge \(\{z: \mathrm {arg}\,z\in (-\pi /4,\pi /4)\}\) in the complex plane \({\mathbb {C}}\) and assume that \({\mathbb {D}}\) is the unit disk. Assume further that \(\Sigma \subset {\mathbb {C}}\times {\mathbb {R}}\) is a minimal surface over W. We obtain some sharp point-wise estimates of Gram–Schmidt norm of the first derivative of a harmonic diffeomorpism of the unit disk onto W. Further we apply this result to obtain some upper estimates of the Gaussian curvature of the minimal surface \(\Sigma \) at a given point Q in term of underlaying point \(w\in W\) . This improves some result by Abu-Muhanna and Schober (Can J Math 39(6):1489–1530, 1987). PubDate: 2021-01-07
Abstract: Abstract The Dunkl–Coulomb and the Dunkl oscillator models in arbitrary space-dimensions are introduced. These models are shown to be maximally superintegrable and exactly solvable. The energy spectrum and the wave functions of both systems are obtained using different realizations of the Lie algebra \(\text{ so }(1,2).\) The N-dimensional Dunkl oscillator admits separation of variables in both Cartesian and polar coordinates and the corresponding separated solutions are expressed in terms of the generalized Hermite, Laguerre and Gegenbauer polynomials. The N-dimensional Dunkl–Coulomb Hamiltonian admits separation of variables in polar coordinates and the separated wave functions are expressed in terms of the generalized Laguerre and Gegenbauer polynomials. The symmetry operators generalizing the Runge–Lenz vector operator are given. Together with the Dunkl angular momentum operators and reflection operators they generate the symmetry algebra of the N-dimensional Dunkl–Coulomb Hamiltonian which is a deformation of \(\text{ so }(N+1)\) by reflections for bound states and is a deformation of \(\text{ so }(N,1)\) by reflections for positive energy states. The symmetry operators of the N-dimensional Dunkl oscillator are obtained by the Schwinger construction and generate its invariance algebra which is a deformation of \({\textit{su}}(N)\) by reflections. PubDate: 2021-01-06
Abstract: Abstract The aim of this paper is to establish some necessary and sufficient conditions for the boundedness of the Hausdorff operators on two-weighted central Morrey, Herz, and Morrey–Herz spaces on the p-adic fields. Moreover, the sufficient conditions of boundedness of commutators of p-adic Hausdorff operators with symbols in the Lipschitz spaces on two-weighted central Morrey and Morrey–Herz spaces are also given. PubDate: 2021-01-05
Abstract: Abstract In this paper, we consider the bifurcation method of dynamical systems for solving time fractional nonlinear evolution equations. We adapt and modify the methodology, incorporating new ideas from the conformable fractional derivative, to investigate exact travelling wave solutions and bifurcations of phase transitions for nonlinear evolution equations. In this study, we show the existence of periodic wave solutions, kink and anti-kink wave solutions, a bright and dark solitary wave solution and parabolic solutions. Moreover, numerical simulations method is applied to show the richer dynamical behavior of the spatial and temporal fractional order of nonlinear evolutions systems and verify the theoretical results. PubDate: 2021-01-05
Abstract: Abstract We consider massless Dirac operators on the half-line with compactly supported potentials. We solve the inverse problems in terms of Jost function and scattering matrix (including characterization). We study resonances as zeros of Jost function and prove that a potential is uniquely determined by its resonances. Moreover, we prove the following: (1) resonances are free parameters and a potential continuously depends on a resonance, (2) the forbidden domain for resonances is estimated, (3) asymptotics of resonance counting function is determined, (4) these results are applied to canonical systems. PubDate: 2021-01-05
Abstract: Abstract In this note we study the differentiability with respect to the time-parameter of semigroups consisting of Lipschitzian or smooth self-mappings of a domain in a Banach space. PubDate: 2021-01-05
Abstract: Abstract The purpose of the present work is to investigate exact solutions of the fractional order multi Kaup–Boussinesq system with \(l=2\) by using the group invariance approach and power series expansion method. Due to the significance of conserved vectors in terms of integrability and behaviour of nonlinear systems, the conservation laws are also derived by testing the nonlinear self-adjointness. PubDate: 2021-01-05
Abstract: Abstract In this paper we introduce a new family of the KP tau-functions. This family can be described by a deformation of the generalized Kontsevich matrix model. We prove that the simplest representative of this family describes a generating function of the cubic Hodge integrals satisfying the Calabi–Yau condition, and claim that the whole family describes its generalization for the higher spin cases. To investigate this family we construct a new description of the Sato Grassmannian in terms of a canonical pair of the Kac–Schwarz operators. PubDate: 2021-01-04
Abstract: Abstract In this paper we obtain lower bounds on the radius of spatial analyticity of solutions to the Kawahara equation \(u_t + uu_x + \alpha u_{xxx} + \beta u_{xxxxx} = 0\) , \(\beta \ne 0\) , given initial data which is analytic with a fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time t can decay no faster than 1/ t as \( t \rightarrow \infty \) . PubDate: 2021-01-04
Abstract: Abstract We use the method of pseudoanalytic continuation to obtain a characterization of spaces of holomorphic functions with boundary values in Besov spaces in terms of polynomial approximations. PubDate: 2021-01-04
Abstract: Abstract The problem of Liouvillian integrability for the classical force-free generalized Duffing oscillators is solved completely. All the cases when the generalized Duffing oscillators possess Liouvillian first integrals are classified. It is shown that the general solutions in integrable cases are expressible via elliptic and hyperelliptic functions. The relationship between the generalized Duffing systems and the Newell–Whitehead–Segel equation is used to characterize algebraically invariant traveling waves of the latter. PubDate: 2021-01-04
Abstract: Abstract We consider perturbations of dynamical semigroups on the algebra of all bounded operators in a Hilbert space generated by covariant completely positive measures on the semi-axis. The construction is based upon unbounded linear perturbations of generators of the preadjoint semigroups on the space of nuclear operators. As an application we construct a perturbation of the semigroup of non-unital *-endomorphisms on the algebra of canonical anticommutation relations resulting in the flow of shifts. PubDate: 2021-01-04
Abstract: Abstract Let \(p\in (1,\infty )\) , \(q\in [1,\infty )\) , \(s\in {\mathbb Z}_{+}\) , \(\alpha \in [0,\infty )\) , and \(\mathcal {X}\) be \(\mathbb R^n\) or a cube \(Q_0\subsetneqq \mathbb R^n\) . In this article, the authors first introduce the local John–Nirenberg–Campanato space \(jn_{(p,q,s)_{\alpha }}(\mathcal {X})\) and show that the localized Campanato space is the limit case of \(jn_{(p,q,s)_{\alpha }}(\mathcal {X})\) as \(p\rightarrow \infty \) . By means of local atoms and the weak- \(*\) topology, the authors then introduce the local Hardy-kind space \(hk_{(p',q',s)_{\alpha }}(\mathcal {X})\) which proves to be the predual space of \(jn_{(p,q,s)_{\alpha }}(\mathcal {X})\) . Moreover, the authors prove the invariance of \(hk_{(p',q',s)_{\alpha }}(\mathcal {X})\) with respect to \(q\in (1,p)\) , where \(p'\) or \(q'\) denotes the conjugate number of p or q, respectively. All these results are new even for the localized John–Nirenberg space. PubDate: 2021-01-04
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