The Bohr phenomenon for analytic functions on shifted disks

In this paper, we investigate the Bohr phenomenon for the class of analytic functions defined on the simply connected domain \(\Omega_{\gamma}=\bigg\{z\in\mathbb{C} \colon \bigg|z+\frac{\gamma}{1-\gamma}\bigg|<\frac{1}{1-\gamma}\bigg\}\) for \(0\leq \gamma<1.\) We study improved Bohr radius, Bohr-Rogosinski radius and refined Bohr radius for the class of analytic functions defined in \(\Omega_{\gamma}\), and obtain several sharp results.

Multicoated elastic inhomogeneities of arbitrary shape neutral to multiple fields

We rigorously establish the interesting result that in anti-plane elasticity an elastic epitrochoidal inhomogeneity can be made neutral to multiple uniform fields applied in the matrix via the insertion of two intermediate coatings. Using a two-term conformal mapping function, the simply connected domain occupied by the epitrochoidal inhomogeneity and its surrounding inner and outer coatings is mapped onto the interior of the unit circle in the image plane. The mismatch parameters are determined in an analytical manner by solving a set of two non-linear equations. An elastic inhomogeneity of arbitrary shape can be made neutral to multiple fields through the insertion of N coatings when the proposed mapping function for the simply connected domain occupied by the multicoated inhomogeneity is described in terms of a polynomial of finite degree containing N non-constant terms. In this case, the mismatch parameters are determined by iteratively solving a set of N non-linear equations.

On the convergence of the barycentric method in solving internal Dirichlet and Neumann problems in $\mathbb{R}^2$ for the Helmholtz equation

The application of the barycentric method for the numerical solution of Dirichlet and Neumann problems for the Helmholtz equation in the bounded simply connected domain $\Omega\subset\mathbb{R}^2$ is considered. The main assumption in the solution is to set the $\Omega$ boundary in a piecewise linear representation. A distinctive feature of the barycentric method is the order of formation of a global system of vector basis functions for $\Omega$ via barycentric coordinates. The existence and uniqueness of the solution of Dirichlet and Neumann problems for the Helmholtz equation by the barycentric method are established and the convergence rate estimate is determined. The features of the algorithmic implementation of the method are clarified.

Derivatives, Products, and Pullbacks in Forman's Combinatorial Differential Forms

We study derivatives, closedness, and exactness of 0-forms and 1-forms in the theory of combinatorial differential forms constructed by Robin Forman. We give an example of a closed but not exact 1-form on a non-simply connected domain. We give a sufficient condition on the domain for a closed 1-form to be exact. We show that the product of forms proposed by Forman is not anti-commutative. We propose a definition of pullbacks of forms and show that this operation has several properties analogous to pullbacks on smooth forms.

Conformal Invariance of CLEΚ on the Riemann Sphere for Κ ∈(4,8)

The conformal loop ensemble (${\textrm{CLE}}$) is the canonical conformally invariant probability measure on non-crossing loops in a simply connected domain in $\mathbbm{C}$ and is indexed by a parameter $\kappa \in (8/3,8)$. We consider ${\textrm{CLE}}_\kappa $ on the whole-plane in the regime in which the loops are self-intersecting ($\kappa \in (4,8)$) and show that it is invariant under the inversion map $z \mapsto 1/z$. This shows that whole-plane ${\textrm{CLE}}_\kappa $ for $\kappa \in (4,8)$ defines a conformally invariant measure on loops on the Riemann sphere. The analogous statement in the regime in which the loops are simple ($\kappa \in (8/3,4]$) was proven by Kemppainen and Werner and together with the present work covers the entire range $\kappa \in (8/3,8)$ for which ${\textrm{CLE}}_\kappa $ is defined. As an intermediate step in the proof, we show that ${\textrm{CLE}}_\kappa $ for $\kappa \in (4,8)$ on an annulus, with any specified number of inner-boundary-surrounding loops, is well defined and conformally invariant.

Transcendental Semigroup that has Simply Connected Fatou Components

We prove that there exists non-trivial transcendental semigroup S such that the periodic or pre-periodic or wandering components of Fatou set F(S) have simply connected domain D.

Toeplitz operators on Bergman spaces of polygonal domains

We study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces Ap(Ω), 1 < p < ∞, where Ω ⊂ ℂ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of ‘averages’ of symbol over certain Cartesian squares. We use the Whitney decomposition of Ω in the proof. We also give examples of bounded Toeplitz operators on Ap(Ω) in the case where polygon Ω has such a large corner that the Bergman projection is unbounded.

Monogenic functions in two dimensional commutative algebras to equations of plane orthotropy

Among all two-dimensional commutative and assosiative algebras of the second rank with the unity \(e\) over the field of complex numbers \(\mathbb{C}\) we find a semi-simple algebra \(\mathbb{B}_{0} := \{c_1 e+c_2 \omega: c_k\in\mathbb{C}, k=1,2\}\), \(\omega^2=e\), containing a basis \((e_1,e_2)\), such that \( e_1^4 + 2p e_1^2 e_2^2 + e_2^4 = 0 \) for any fixed \( p \) such that \(-1 \lt p \gt 1 \). A domain \(\mathcal{B}_{1}=\{(e_1,e_2)\}\), \(e_1=e\), is discribed in an explicit form. We consider an approach of \(\mathbb{B}_{0}\)-valued ''analytic'' functions \(\Phi(xe_1+ye_2) = U_{1}(x,y)e_1 + U_{2}(x,y)ie_1+ U_{3}(x,y)e_2 + U_{4}(x,y)ie_2\) (\((e_1,e_2)\in \mathcal{B}\), \(x\) and \(y\) are real variables) such that their real-valued components \(U_{k}\), \(k=\overline{1,4}\), satisfy the equation on finding the stress function \(u\) in the case of orthotropic plane deformations (with absence of body forses): \( \left(\frac{\partial^4}{\partial x^4} +2p\frac{\partial^4 }{\partial x^2\partial y^2}+ \frac{\partial^4 }{\partial y^4} \right) u(x,y)=0\) for every \((x,y)\in D\), where \(D\) is a domain of the Cartesian plane \(xOy\). A characterization of solutions \(u\) for this equation in a bounded simply-connected domain via real components \(U_{k}\), \(k=\overline{1,4}\), of the function \(\Phi\) is done in the following sense: let \(D\) be a bounded and simply-connected domain, a solution \(u\) is fixed, then \(u\) is a first component of monogenic function \(\Phi_{u}\). The variety of such \(\Phi_{u}\) is found in a complete form. We consider a particular case of \((e,e_2)\in \mathcal{B}_{1}\) for which \(\Phi_{u}\) can be found in an explicit form. For this case a function \(\Phi_{u}\) is obtained in an explicit form. Note, that in case of orthotropic plane deformations, when Eqs. of the stress function is of the form: \( \left(\frac{\partial^4}{\partial x^4} +2p\frac{\partial^4}{\partial x^2\partial y^2}+\frac{\partial^4 } {\partial y^4} \right) u(x,y)=0\), here \(p\) is a fixed number such that \(p>1\), a similar research is done in [Gryshchuk S. V. Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. I. Ukr. Mat. Zh. 2018. 70, No. 8. pp. 1058-1071 (Ukrainian); Gryshchuk S. V. Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. II. Ukr. Mat. Zh. 2018. 70, No. 10. pp. 1382-1389 (Ukrainian)].