Starlike Domains Research Articles

Potential regulation of dietary lipophilic antioxidants by mitochondrial contact sites

BackgroundRecent evidence has brought the mitochondrial contact sites, in communication with other organelles, to the centre of interest in cellular metabolism because of their essential role in many events surrounding the homeostasis of the human organism, including mitohormesis. Scope and approachThese mitochondrial dynamic structures are particularly active in response to feeding, after which the complex metabolic processes that produce energy from macronutrients in the mitochondria are also the largest producers of reactive oxygen species. However, little is known about the potential role of the rich lipid-binding protein coat of the outer mitochondrial membrane and the membrane of other organelles at mitochondrial contact sites as a place for transient storage and mobilisation of lipophilic antioxidant micronutrients, where they could function to promote mitohormesis. This commentary aims to assess the extent to which some of these lipid-binding proteins at mitochondrial contact sites are able to regulate the disposition of some lipophilic dietary antioxidants, such as β-carotene. Key findingsIt is found that the mitochondrial lipid-binding protein steroidogenic acute regulatory (StAR)-related lipid transfer (START) domain 1 (STARD1), the endoplasmic reticulum lipid-binding protein ASTER (with a StAR-like domain), and the endosomal lipid-binding proteins stabilins (with multiple fasciclin-1 adhesion domains) could work together at mitochondrial contact sites to allow the compartmentalisation and distribution of β-carotene and other dietary lipophilic antioxidants, which could help maintain the state of mitochondrial resistance to oxidative distress in the context of a healthy diet. ConclusionsThe surface of the outer mitochondrial membrane at contact sites during mitochondrial dynamics may have the potential to act as a functional platform for lipophilic antioxidant micronutrients that can eventually be mobilised to cope with supraphysiological levels of reactive oxygen species.

On starlikeness of regular Coulomb wave functions

In this paper, we study some geometric properties of a class of analytic functions which is defined from the J J -fraction expansion of the ratio z f ′ ( z ) / f ( z ) {zf’(z)}/{f(z)} . We find the disk domain which is mapped into a starlike domain by these functions. Moreover, we study similar results for two different normalized forms of regular Coulomb wave functions and a normalized Bessel function of the first kind by using continued fractions expansions.

A Collection of Results Relating the Geometry of Plane Domains and the Exit Time of Planar Brownian Motion

We prove a number of results relating exit times of planar Brownian motion with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; similarly probabilistic proofs of some recent results of Karafyllia on harmonic measure on starlike domains; examples of domains and their complements which are simultaneously large when measured by the moments of exit time of Brownian motion, and examples of domains and their complements which are simultaneously small; and proofs of several identities involving the Cauchy distribution using the optional stopping theorem.

On the convergence of Broyden’s method and some accelerated schemes for singular problems

Abstract We consider Broyden’s method and some accelerated schemes for nonlinear equations having a strongly regular singularity of first order with a one-dimensional nullspace. Our two main results are as follows. First, we show that the use of a preceding Newton-like step ensures convergence for starting points in a starlike domain with density 1. This extends the domain of convergence of these methods significantly. Second, we establish that the matrix updates of Broyden’s method converge q-linearly with the same asymptotic factor as the iterates. This contributes to the long-standing question of whether the Broyden matrices converge by showing that this is indeed the case for the setting at hand. Furthermore, we prove that the Broyden directions violate uniform linear independence, which implies that existing results for convergence of the Broyden matrices cannot be applied. Numerical experiments of high precision confirm the enlarged domain of convergence, the q-linear convergence of the matrix updates and the lack of uniform linear independence. In addition, they suggest that these results can be extended to singularities of higher order and that Broyden’s method can converge r-linearly without converging q-linearly. The underlying code is freely available.

A linear barycentric rational interpolant on starlike domains

When an approximant is accurate on an interval, it is only natural to try to extend it to multi-dimensional domains. In the present article we make use of the fact that linear rational barycentric interpolants converge rapidly toward analytic and several-times differentiable functions to interpolate them on two-dimensional starlike domains parametrized in polar coordinates. In the radial direction, we engage interpolants at conformally shifted Chebyshev nodes, which converge exponentially for analytic functions. In the circular direction, we deploy linear rational trigonometric barycentric interpolants, which converge similarly rapidly for periodic functions but now for conformally shifted equispaced nodes. We introduce a variant of a tensor-product interpolant of the above two schemes and prove that it converges exponentially for two-dimensional analytic functions–up to a logarithmic factor–and with an order limited only by the order of differentiability for real functions (provided that the boundary enjoys the same order of differentiability). Numerical examples confirm that the shifts permit one to reach a much higher accuracy with significantly fewer nodes, a property which is especially important in several dimensions.

AN APPLICATION OF SCHUR’S ALGORITHM TO VARIABILITY REGIONS OF CERTAIN ANALYTIC FUNCTIONS II

AbstractWe extend our study of variability regions, Ali et al. [‘An application of Schur algorithm to variability regions of certain analytic functions–I’, Comput. Methods Funct. Theory, to appear] from convex domains to starlike domains. Let $\mathcal {CV}(\Omega )$ be the class of analytic functions f in ${\mathbb D}$ with $f(0)=f'(0)-1=0$ satisfying $1+zf''(z)/f'(z) \in {\Omega }$ . As an application of the main result, we determine the variability region of $\log f'(z_0)$ when f ranges over $\mathcal {CV}(\Omega )$ . By choosing a particular $\Omega $ , we obtain the precise variability regions of $\log f'(z_0)$ for some well-known subclasses of analytic and univalent functions.

On the finiteness of moments of the exit time of planar Brownian motion from comb domains

A comb domain is defined to be the entire complex plain with a collection of vertical slits, symmetric over the real axis, removed. In this paper, we consider the question of determining whether the exit time of planar Brownian motion from such a domain has finite \(p\)-th moment. This question has been addressed before in relation to starlike domains, but these previous results do not apply to comb domains. Our main result is a sufficient condition on the location of the slits which ensures that the \(p\)-th moment of the exit time is finite. Several auxiliary results are also presented, including a construction of a comb domain whose exit time has infinite \(p\)-th moment for all \(p \geq 1/2\).

Some Properties of Generalized Bessel Integral Operators

In the space of functions analytic in an arbitrary starlike domain of the complex plane m-symmetric with respect to the origin, we defined a Bessel generalized integral operator ℐm of the mth order. We establish the equivalence between this operator and the Volterra integral operator of the mth order. The convolution is constructed for the indicated generalized Bessel integral operator. Some commutative properties of the operator ℐm are investigated.

Universally starlike and Pick functions

Denote by \({{\cal P}_{\log}}\) the set of all non-constant Pick functions f whose logarithmic derivatives f′/f also belong to the Pick class. Let \({\cal U}({\rm{\Lambda}})\) be the family of functions z · f(z), where \(f \in {{\cal P}_{\log}}\) and f is holomorphic on Λ ≔ ℂ [1, + √). Important examples of functions in \({\cal U}({\rm{\Lambda}})\) are the classical polylogarithms \(L{i_\alpha}(z): = \sum\nolimits_{k = 1}^\infty {{z^k}} /{k^\alpha}\) for α ≥ 0; see [5](2015).In this note we prove that every \(\varphi \in {\cal U}({\rm{\Lambda}})\) is universally starlike, i.e., φ maps every circular domain in Λ containing the origin one-to-one onto a starlike domain. Furthermore, we show that every non-constant function \(f \in {{\cal P}_{\log}}\) belongs to the Hardy space Hp on the upper half-plane for some constant p = p(f) > 1, unless f is proportional to some function (a − z)−θ with a ∊ ℝ and 0 <θ ≤ 1. Finally we derive a necessary and sufficient condition on a real-valued function υ for which there exists \(f \in {{\cal P}_{\log}}\) such that υ (x) = limε↓0 lim f(x + iε) for almost all x ∊ ℝ.

Unit stepsize for the Newton method close to critical solutions

As is well known, when initialized close to a nonsingular solution of a smooth nonlinear equation, the Newton method converges to this solution superlinearly. Moreover, the common Armijo linesearch procedure used to globalize the process for convergence from arbitrary starting points, accepts the unit stepsize asymptotically and ensures fast local convergence. In the case of a singular and possibly even nonisolated solution, the situation is much more complicated. Local linear convergence (with asymptotic ratio of 1/2) of the Newton method can still be guaranteed under reasonable assumptions, from a starlike, asymptotically dense set around the solution. Moreover, convergence can be accelerated by extrapolation and overrelaxation techniques. However, nothing was previously known on how the Newton method can be coupled in these circumstances with a linesearch technique for globalization that locally accepts unit stepsize and guarantees linear convergence. It turns out that this is a rather nontrivial issue, requiring a delicate combination of the analyses on acceptance of the unit stepsize and on the iterates staying within the relevant starlike domain of convergence. In addition to these analyses, numerical illustrations and comparisons are presented for the Newton method and the use of extrapolation to accelerate convergence speed.

On a Property of Harmonic Measure on Simply Connected Domains

AbstractLet $D\subset \mathbb{C}$ be a domain with $0\in D$. For $R&gt;0$, let $\widehat{\unicode[STIX]{x1D714}}_{D}(R)$ denote the harmonic measure of $D\cap \{|z|=R\}$ at $0$ with respect to the domain $D\cap \{|z|&lt;R\}$ and let $\unicode[STIX]{x1D714}_{D}(R)$ denote the harmonic measure of $\unicode[STIX]{x2202}D\cap \{|z|\geqslant R\}$ at $0$ with respect to $D$. The behavior of the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ near $\infty$ determines (in some sense) how large $D$ is. However, it is not known whether the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ always have the same behavior when $R$ tends to $\infty$. Obviously, $\unicode[STIX]{x1D714}_{D}(R)\leqslant \widehat{\unicode[STIX]{x1D714}}_{D}(R)$ for every $R&gt;0$. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant $C$ such that for all simply connected domains $D$ with $0\in D$ and all $R&gt;0$, $$\begin{eqnarray}\unicode[STIX]{x1D714}_{D}(R)\geqslant C\widehat{\unicode[STIX]{x1D714}}_{D}(R)?\end{eqnarray}$$ In general, we prove that the answer is negative by means of two different counter-examples. However, under additional assumptions involving the geometry of $D$, we prove that the answer is positive. We also find the value of the optimal constant for starlike domains.

Forelli type theorem in harmonic map forms

In the current paper, we give and prove a formulation of the Forelli Theorem on a starlike domain in C n \mathbb {C}^n . We also generalize the Forelli Theorem from a function to a map with a real Riemannian manifold as its target manifold.

Generalized Mean-Value Theorem for an Analytic Function and an Algorithm for the Evaluation of the Function of Mean Values

We prove the mean-value theorem for functions analytic in starlike domains, propose an algorithm for finding the function of mean values, and study its analytic continuation. We present a differential equation for the function of mean values and the interpretation of the Lagrange formula for analytic functions in terms of the theory of differential equations. The set of points of the initial values of the function of mean values is analyzed and the upper bound for the radius of expansion of the function of mean values in Taylor’s series is established.

Coefficient inequality for a subclass of biholomorphic mappings in several complex variables

Let be the familiar class of starlike functions of order in the unit disk. In Keogh and Merkes proved that for a function in the class , then In this paper, we investigate the corresponding problem for the subclass of starlike mappings of order defined on the unit ball in a complex Banach space, on the unit polydisk in and the bounded starlike circular domain in , respectively.

Critical solutions of nonlinear equations: local attraction for Newton-type methods

We show that if the equation mapping is 2-regular at a solution in some nonzero direction in the null space of its Jacobian (in which case this solution is critical; in particular, the local Lipschitzian error bound does not hold), then this direction defines a star-like domain with nonempty interior from which the iterates generated by a certain class of Newton-type methods necessarily converge to the solution in question. This is despite the solution being degenerate, and possibly non-isolated (so that there are other solutions nearby). In this sense, Newtonian iterates are attracted to the specific (critical) solution. Those results are related to the ones due to A. Griewank for the basic Newton method but are also applicable, for example, to some methods developed specially for tackling the case of potentially non-isolated solutions, including the Levenberg–Marquardt and the LP-Newton methods for equations, and the stabilized sequential quadratic programming for optimization.

Hyperbolic Type Distances in Starlike Domains

We study the growth of hyperbolic type distances in starlike domains. We derive estimates for various hyperbolic type distances and consider the asymptotic sharpness of the estimates.

The Fekete and Szegö problem on bounded starlike circular domain in $\mathbb{C}^n$

The Fekete and Szegö problem on bounded starlike circular domain in $\mathbb{C}^n$

О решениях волнового уравнения с младшим членом

A correspondence between solutions of the wave equation and the wave equation with a minor term in a star-like domain was obtained. This result is an analogue of the corresponding result obtained earlier for both the Helmholtz equation and the Laplace equation

The exit time of planar Brownian motion and the Phragmén–Lindelöf principle

In this paper a version of the Phragmén–Lindelöf principle is proved using probabilistic techniques. In particular, we will show that if the pth moment of the exit time of Brownian motion from a planar domain is finite, then an analytic function on that domain is either bounded by its supremum on the boundary or else goes to ∞ along some sequence more rapidly than e|z|2p. We also provide a method of constructing domains whose exit time has finite pth moment. This allows us to give a general Phragmén–Lindelöf principle for spiral-like and star-like domains, as well as a new proof of a theorem of Hansen. A number of auxiliary results are presented as well.

Suboptimal Polynomial Meshes on Planar Lipschitz Domains

We construct norming meshes with cardinality 𝒪(n s ), s = 3, for polynomials of total degree at most n on the closure of bounded planar Lipschitz domains. Such cardinality is intermediate between optimality (s = 2), recently obtained by Kroó on multidimensional C 2 star-like domains, and that arising from a general construction on Markov compact sets due to Calvi and Levenberg (s = 4).