Open Disks Research Articles

On the v-Picard Group of Stein Spaces

Abstract We study the image of the Hodge–Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $\mathbf{Z}_{p}$-module that maps, via the Bloch–Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit disks of dimension strictly greater than $1$.

A Class of Bi-Univalent Functions in a Leaf-Like Domain Defined through Subordination via q̧-Calculus

Bi-univalent functions associated with the leaf-like domain within open unit disks are investigated, and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with q-calculus. The class is proved to not be empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on |α2| and |α3| coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation.

On the convergence of series of fractional Hasse derivative bases in Fréchet spaces

This paper introduces a generalization of the Hasse derivative in the sense of the complex conformable derivative. The definition coincides with the classical version of the Hasse derivative of order . Accordingly, a new base, named complex conformable Hasse derivative bases (CCHDBs), is defined. We investigate the existence of expansions of analytic functions in a series of CCHDBs in Fréchet space on closed and open disks, open regions surrounding closed disks, for all entire functions and at the origin. Moreover, an upper bound for the order and type of the CCHDBs is obtained and proved to be attainable. The ‐property of CCHDBs is also discussed. Our results improve and extend the analog results in the complex analysis related to the classical complex derivative of a base of polynomials (BPs). The obtained results clarify several implications for the CCHDBs of special functions such as Euler, Bernoulli, Chebyshev, Bessel, and Gontcharaff polynomials.

Identification and cellular localization in Xenopus laevis photoreceptors of three Peripherin-2 family members, Prph2, Rom1 and Gp2l, which arose from gene duplication events in the common ancestors of jawed vertebrates

Rod and cone photoreceptors are named for the distinct morphologies of their outer segment organelles, which are either cylindrical or conical, respectively. The morphologies of the stacked disks that comprise the rod and cone outer segments also differ: rod disks are completely sealed and are discontinuous from the plasma membrane, while cone disks remain partially open to the extracellular space. These morphological differences between photoreceptor types are more prominent in non-mammalian vertebrates, whose cones typically possess a greater proportion of open disks and are more tapered in shape. In mammals, the tetraspanin prph2 generates and maintains the highly curved disk rim regions by forming extended oligomeric structures with itself and a structurally similar paralog, rom1. Here we determined that in addition to these two proteins, there is a third Prph2 family paralog in most non-mammalian vertebrate species, including X. laevis: Glycoprotein 2-like protein or “Gp2l”. A survey of multiple genome databases revealed a single invertebrate Prph2 ‘pro-ortholog’ in Amphioxus, several echinoderms and in a diversity of protostomes indicating an ancient divergence from other tetraspanins. Based on phylogenetic analysis, duplication of the vertebrate predecessor likely gave rise to the Gp2l and Prph2/Rom1 clades, with a further duplication distinguishing the Prph2 and Rom1 clades. Mammals have lost Gp2l and their Rom1 has undergone a period of accelerated evolution such that it has lost several features that are retained in non-mammalian vertebrate Rom1. Specifically, Prph2, Gp2l and non-mammalian Rom1 encode proteins with consensus N-linked glycosylation and outer segment localization signals; mammalian rom1 lacks these motifs. We determined that X. laevis gp2l is expressed exclusively in cones and green rods, while X. laevis rom1 is expressed exclusively in rods, and prph2 is present in both rods and cones. The presence of three Prph2-related genes with distinct expression patterns as well as the rapid evolution of mammalian Rom1, may contribute to the more pronounced differences in morphology between rod and cone outer segments and rod and cone disks observed in non-mammalian versus mammalian vertebrates.

Certain Results on Subclasses of Analytic and Bi-Univalent Functions Associated with Coefficient Estimates and Quasi-Subordination

The purpose of the present paper is to introduce and investigate new subclasses of analytic function class of bi-univalent functions defined in open unit disks connected with a linear q-convolution operator, which are associated with quasi-subordination. We find coefficient estimates of h2, h3 for functions in these subclasses. Several known and new consequences of these results are also pointed out. There is symmetry between the results of the subclass fq,∑ μ(ζ,n,ρ,σ,ϑ,γ,δ,φ) and the results of the subclass ℵ∑q,δλ,ζ,n,ρ,σ,ϑ,φ.

On the symmetric group action on rigid disks in a strip

In this paper we decompose the rational homology of the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 as a direct sum of induced \(S_{n}\)-representations. In (Alpert in Generalized representation stability for disks in a strip and no-k-equal spaces, arXiv:2006.01240, 2020), Alpert proves that the \(k^{\text {th}}\)-integral homology of the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 is an FI\(_{k+1}\)-module by studying certain operations on homology called “high-insertion maps.” The integral homology groups \(H_{k}\big (\text {Conf}(n,2)\big )\) are free abelian, and Alpert computes a basis for \(H_{k}\big (\text {Conf}(n,2)\big )\) as an abelian group. In this paper, we study the rational homology groups as \(S_{n}\)-representations. We find a new basis for \(H_{1}\big (\text {Conf}(n,2);\mathbb {Q}\big ),\) and use this, along with results of Ramos (Proc Am Math Soc 145(11):4647–4660, 2017), to give an explicit description of \(H_{k}\big (\text {Conf}(n,2);\mathbb {Q}\big )\) as a direct sum of induced \(S_{n}\)-representations arising from free FI\(_{d}\)-modules. We use this decomposition to calculate the rank of the rational homology of the unordered configuration space of n open unit-diameter disks in the infinite strip of width 2.

What Can You Draw?

We address the problem of which planar sets can be drawn with a pencil and eraser. The pencil draws any union of black open unit disks in the plane . The eraser produces any union of white open unit disks. You may switch tools as many times as desired. Our main result is that drawability cannot be characterized by local obstructions: a bounded set can be locally drawable, while not being (globally) drawable. We also show that if drawable sets are defined using closed unit disks, then the cardinality of the collection of drawable sets is strictly larger compared with the definition involving open unit disks.

Approximation of functions by complex conformable derivative bases in Fréchet spaces

In the present paper, the representation, in different domains, of analytic functions by complex conformable fractional derivative bases (CCFDB) and complex conformable fractional integral bases (CCFIB) in Fréchet space are investigated. Results are proved to show that such representation is possible in closed disks, open disks, open regions surrounding closed disks, at the origin, and for all entire functions. Also, some results concerning the growth order and type of CCFDB and CCFIB are determined. Moreover, the ‐property of CCFDB and CCFIB is discussed. The obtained results recover some known results when . Finally, some applications to the CCFDB and CCFIB of Bernoulli, Euler, Bessel, and Chebyshev polynomials have been studied.

An over view of geometric function theory (GFT) and analytic properties of meromorphic functions

This article presents an over view of Geometric Function Theory (GFT) and utilized the analytic properties of meromorphic functions. Geometric Function Theory is a branch of mathematics that focuses on the geometric interpretations and implications of analytic functions defined in the complex plane. Our exploration begins with an itemized discussion of key concepts within GFT, emphasizing their relevance and theoretical underpinnings. Central to our study is the investigation of meromorphic functions, which are functions that are analytic except for isolated singularities where they may have poles. We examined various classes of meromorphic functions and elucidate their properties, including their behavior near singularities and their broader geometric implications. A significant portion of our inquiry involves the Hadamard product of functions. This operation allows us to explore the combined effect of two analytic functions, considering their series expansions and how their product transforms under this operation. By studying the Hadamard transformation, we uncover analogues and interesting results that shed light on the interplay between analytic functions and their geometric representations. We also provide detailed diagrammatic descriptions of fundamental geometric shapes such as circles, open unit disks, and closed unit disks. These diagrams serve to visually illustrate key concepts and relationships within GFT, aiding in the understanding of how analytic functions behave in different spatial configurations. Our article offers a comprehensive exploration of Geometric Function Theory, emphasizing its foundational concepts and their applications in analyzing analytic and meromorphic functions.

Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres

Let $\Sigma$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(\Sigma,G)$ is a quantization of the ring of functions of $X_G(\Sigma)$, the variety of $G$-characters of $\pi_1(\Sigma)$, introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $\Sigma$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the "daisy" graph on $\Sigma$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=\epsilon$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring $\mathcal{O}(G^n)$. We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(\Sigma)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ with this quantum moduli algebra specialized at $q=\epsilon$.

Structure of the maximal ideal space of 𝐇^{∞} on the countable disjoint union of open disks

The maximal ideal space of the algebra of bounded holomorphic functions on the countable disjoint union of open unit disks D ⊂ C \mathbb {D}\subset \mathbb {C} is studied from a topological point of view. The results are similar to those for the maximal ideal space of the algebra H ∞ ( D ) H^\infty (\mathbb {D}) .

Metric uniformization of morphisms of Berkovich curves via p-adic differential equations

We consider a finite rig-etale morphism f: Y → X of quasi-smooth Berkovich curves over a complete algebraically closed valued field extension k of ℚp and a skeleton Γf = (ΓY, ΓX) of the morphism f. We prove that Γf radializes f if and only if ΓX controls the pushforward of the constant p-adic differential equation $${f_*}\left( {{{\cal O}_Y},{d_Y}} \right)$$ . Furthermore, when f is a finite morphism of open unit discs and k is of arbitrary characteristic, we prove that f is radial if and only if the number of preimages of a point x ∈ X, counted without multiplicity, only depends on the radius of the point x.

Investigation of the enhanced ability of bile salt surfactants to solubilize phospholipid bilayers and form mixed micelles.

The self-assembly in mixtures of the anionic bile salt surfactant sodium deoxycholate (NaDC) and the zwitterionic phospholipid 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) in physiological saline solution has been investigated using light scattering, small-angle X-ray scattering and cryo-transmission electron microscopy. Rather small tri-axial ellipsoidal NaDC-DMPC mixed micelles form at a high content of bile salt in the mixture, which increase in size as an increasing amount of DMPC is incorporated into the micelles. Eventually, the micelles begin to grow substantially in length to form long wormlike micelles. At higher mole fractions of DMPC, the samples become turbid and cryo-TEM measurements reveal the existence of large perforated vesicles (stomatosomes), coexisting with geometrically open disks. To our knowledge, stomatosomes have not been observed before for any bile salt-phospholipid system. Mixed micelles are found to be the sole aggregate structure in a very wide regime of bile salt-phospholipid compositions, i.e. up to about 77 mol% phospholipid in the micelles. This is much higher than the corresponding value of 25 mol% observed for the conventional surfactant hexadecyltrimethylammonium bromide (CTAB) mixed with DMPC in the same solvent. The enhanced ability of bile salt surfactants to solubilize phospholipid bilayers and form mixed micelles is rationalized using bending elasticity theory. From our theoretical analysis, we are able to conclude that amphiphilic molecules rank in the following order of increasing spontaneous curvature: phospholipids < conventional surfactants < bile salts. The bending rigidity of the different amphiphilic molecules increases according to the following sequence: bile salts < conventional surfactants < phospholipids.

On the Ekeland–Hofer symplectic capacities of the real bidisc

In $\mathbb{C}^2$ with the standard symplectic structure we consider the bidisc $D^2\times D^2$ constructed as the product of two open real discs of radius $1$. We compute explicit values for the first, second and third Ekeland-Hofer symplectic capacity of $D^2\times D^2$. We discuss some applications to questions of symplectic rigidity.

Spectrum of a linear differential equation with constant coefficients

In this paper we compute the spectrum, in the sense of Berkovich, of an ultrametric linear differential equation with constant coefficients, defined over an affinoid domain of the analytic affine line $A_k^{1,an}$. We show that it is a finite union of either closed disks or topological closures of open disks and that it satisfies a continuity property.

Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras

Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and is a combinatorial quantization of the moduli space of flat connections on $\Sigma_{g,n}$. Here we focus on the two building blocks $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$ under the assumption that the gauge Hopf algebra $H$ is finite-dimensional, factorizable and ribbon, but not necessarily semisimple. We construct a projective representation of $\mathrm{SL}_2(\mathbb{Z})$, the mapping class group of the torus, based on $\mathcal{L}_{1,0}(H)$ and we study it explicitly for $H = \overline{U}_q(\mathfrak{sl}(2))$. We also show that it is equivalent to the representation constructed by Lyubashenko and Majid.

A Riemann–Hurwitz formula for skeleta in non-Archimedean geometry

Let k be an algebraically closed non-Archimedean non-trivially real valued field which is complete with respect to its valuation. Let \(\phi :C' \rightarrow C\) be a finite morphism between smooth projective irreducible k-curves. The morphism \(\phi \) induces a morphism \(\phi ^{\mathrm {an}} :C'^{\mathrm {an}} \rightarrow C^{\mathrm {an}}\) between the Berkovich analytifications of the curves. We construct a pair of deformation retractions of \(C'^{\mathrm {an}}\) and \(C^{\mathrm {an}}\) which are compatible with the morphism \(\phi ^{\mathrm {an}}\) and whose images \(\Sigma _{C'^{\mathrm {\mathrm {an}}}},\Sigma _{C^{\mathrm {\mathrm {an}}}}\) are closed subspaces of that are homeomorphic to finite metric graphs. We refer to such closed subspaces as skeleta. In addition, the subspaces \(\Sigma _{C'^{\mathrm {\mathrm {an}}}}\) and \(\Sigma _{C^{\mathrm {\mathrm {an}}}}\) are such that their complements in their respective analytifications decompose into the disjoint union of isomorphic copies of Berkovich open disks. The pair of compatible deformation retractions forces the morphism \(\phi ^{\mathrm {an}}\) to restrict to a map \(\Sigma _{C'^{\mathrm {\mathrm {an}}}} \rightarrow \Sigma _{C^{\mathrm {\mathrm {an}}}}\). The first Betti number of the skeleton C is well defined and an invariant of the curve which we call \(g^{\mathrm {an}}(C)\). We study how the genus of \(\Sigma _{C'^{\mathrm {\mathrm {an}}}}\) can be calculated using the morphism \(\phi ^{\mathrm {an}}_{|\Sigma _{C'^{\mathrm {\mathrm {an}}}}}\) and invariants defined on \(\Sigma _{C^{\mathrm {an}}}\) via the deformation retractions.

On compact packings of the plane with circles of three radii

A compact circle-packing P of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle S∈P, there exists a maximal indexed set {A0,…,An−1}⊆P so that, for every i∈{0,…,n−1}, the circle Ai is tangent to both circles S and Ai+1modn.We show that there exist at most 13617 pairs (r,s) with 0<s<r<1 for which there exist a compact circle-packing of the plane consisting of circles with radii s, r and 1.We discuss computing the exact values of such 0<s<r<1 as roots of polynomials and exhibit a selection of compact circle-packings consisting of circles of three radii. We also discuss the apparent infeasibility of computing all these values on contemporary consumer hardware.

Projective Representations of Mapping Class Groups in Combinatorial Quantization

Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The graph algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev--Grosse--Schomerus and Buffenoir--Roche and is a combinatorial quantization of the moduli space of flat connections on $\Sigma_{g,n}$. We construct a projective representation of the mapping class group of $\Sigma_{g,n}$ using $\mathcal{L}_{g,n}(H)$ and its subalgebra of invariant elements. Here we assume that the gauge Hopf algebra $H$ is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We also give explicit formulas for the representation of the Dehn twists generating the mapping class group; in particular, we show that it is equivalent to a representation constructed by V. Lyubashenko using categorical methods.

A triple boundary lemma for surface homeomorphisms

Given an orientation-preserving and area-preserving homeomorphism $f$ of the sphere, we prove that every point which is in the common boundary of three pairwise disjoint invariant open topological disks must be a fixed point. As an application, if $K$ is an invariant Wada type continuum, then $f^n|_K$ is the identity for some $n>0$. Another application is an elementary proof of the fact that invariant disks for a nonwandering homeomorphisms homotopic to the identity in an arbitrary surface are homotopically bounded if the fixed point set is inessential. The main results in this article are self-contained.