843 research outputs found
The sharp A(p) constant for weights in a reverse-Holder class
No full textCoifman and Fefferman established that the class of Muckenhoupt weights is equivalent to the class of weights satisfying the "reverse Holder inequality". In a recent paper V. Vasyunin [17] presented a proof of the reverse Holder inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function technique to find the sharp A(p) constants for weights in a reverse-Holder class on an interval; we also find the sharp constants for the higher-integrability result of Gehring [7].Additionally, we find sharp bounds for the A(p) constants of reverse-Holder-class weights defined on rectangles in R-n, as well as bounds on the A(p) constants for reverse-Holder weights defined on cubes in R-n, without claiming the sharpness.</p
[[alternative]]An Analogy of a Theorem of Gehring and Pommerenke under Nehari's
No full text[[abstract]]我們定義一個局部單葉的亞純函數 f 其 Schwarz 導數在 f'(z)≠0
的點為S_f=(f'')(f')'-1/2(f''/f')^2, 而在單極處則訂為 S_f=S_(1/
f). 我們早就知道S_f=0 若且唯若 f 是一個 Mobius 變換. 直接從定義
可得知 S_f 是一個可析函數.相反地, 如果 φ 是一個可析函數, 則存在
一個亞純函數 f 使得 S_f=φ. 早在1949年, Z. Nehari 就證明了 ∣
S_f(z)∣≦ 2/(1-∣z∣^2)^2) 及∣S_f(z)∣≦(π^2)/2 可導出 f 在單
位圓盤上的單葉性.F. W. Gehring 及 C.Pommerenke 更針對第一種條件
作深入的研究, 得到可同胚及可擬保角延拓的充分條件.而我們將注意力
集中在滿足Nehari第二種條件的函數和這些函數的延拓性.而這篇論文主
要是在單位圓盤D中就∣S_f(z)∣≦(π^2)/2 的條件作一檢視. 整個論文
的架構平行於Gehring 及 Pommerenke 的架構. 其中部份的技巧則模仿自
M. Chuaqui 及 B. Osgood 的研究工作.首先, 我們討論了 f 的解析性,
並且對 ∣f'∣及 ∣f∣ 的範圍作一估計.其次, 我們討論了 f 的單葉
性. 接著證明了 f 可連續延拓到 D 的邊界上. 再就此延拓是否一對一分
別討論. 如果是一對一, 則 f 可以擬保角延拓到整個複數平面; 若不是
一對一, f 則與 2/π tan πz/2 Mobius 共軛. 最後, 我們知道如果 f
不是與 2/ π tan πz/2 Mobius 共軛, 則 f 在 D 上滿足 Lipcshitz
條件.
We define the Schwarzian derivative of a function f which is
meromorphic andlocally univalent as S_f=(f''/f')'-(1/2)(f''/f')
^2 at the points where f'(0) is not equal to 0. And we define
S_f(z)=S_(1/f)(z) at the points which are simple poles. It's
shown in early age that S_f = 0 if and only if f is a Mobius
transformation. Directly from the definition, we know that S_f
is analytic. Conversely, suppose that φ is analytic
function, there exists a meromorphic function f such that S_f=
φ. Early in 1949, Z.
Nehari showed that conditions∣S_f(z)∣≦2/(1-∣z∣^2 )^2 and ∣
S_f(z)∣≦(π^2)/2 both imply that f is univalent in the unit
disk D. F. W. Gehring and C. Pommerenke focused their research
on the first condition that Nehari's deduced, and they had the
sufficient conditions that thefunction f can be extended to a
homeomorphic function and a quasiconformalfunction on the plane.
This paper mainly makes an investgation on which properties f
possessesunder the condition ∣S_f∣≦(π^2)/2 in the unit disc
D. The structure of this paper is parallel with the one of
Gehring and Pommerenke. And there are some skills imitated from
the research of M. Chuaqui and B. Osgood. First, we discuss the
analytic property of f under∣S_f∣≦(π^2)/2. We also estimate
the bound of ∣f'∣and ∣f∣. Next, we discuss the univalence of
fand prove that f has a continuous extension to the boundary of
D. Whether this extension is univalent or not give us two
directions. If the extension isone-to-one, f possess a
quasiconformal extension to the whole complex plane.If not, f is
a Mobius conjugate to 2/π tan πz/2. Finally, we prove that if
f is not a Mobius conjugate to f, then f satisfies Lipschitz
condition,which is the special case of Holder's continuity.
We define the Schwarzian derivative of a function f which is
Molecular insights into the eye evolution of bivalvian molluscs
No full textThe intention of my PhD project was to gain more insights into eye evolution and to provide further evidence for the recently proposed idea that all eye-types found in eumetazoans derive from a common Pax6-dependent proto-type eye (Gehring and Ikeo, 1999). To do so, we decided to focus on eyes found in bivalves. Two main reasons prompted us to investigate the molecular basis of bivalvian eye formation. In the first place, all major eye-types, the compound eye, consisting of numerous ommatidia, the camera eye with a single lens and the mirror eye with a reflecting mirror in the back of the eye, are found in bivalves. Hence, the occurrence of different eye-types within the same phylogenetic class makes it very unlikely that these eyes arose as independent formations during evolution. A more elegant alternative is to assume that the compound-, camera-, and mirror eyes of clams evolved monophyletically from a common ancestral precursor. The second reason why we decided to investigate bivalvian eyes is their unusual anatomical position, the edge of the mantle. So far, molecular data and most prominently Pax6 expression were exclusively gathered from “cerebral eyes” of bilaterians, with the only exception of the non-cerebral Hesse eyecups of the lancelet, which by the way do not show any Pax6 expression (Glardon et al., 1998). In this study we focused on two bivalvian species, Arca noae and Pecten maximus, representing the compound eye-type and the mirror eye-type, respectively. We isolated two genes, Pax6 and Six1/2, known to be high up in the genetic regulatory cascade of eye development, from Arca and Pecten. Our expression studies of Pax6 and Six1/2 support the idea that these two genes are necessary for the formation of the olfactory system throughout the animal kingdom. In contrast, we could not assign Pax6 and Six1/2 expression to the visual system with absolute certainty. In a second project, we isolated three opsin genes, one from Arca and two opsin genes from Pecten. A Go-coupled opsin was isolated from Pecten which was shown to be exclusively expressed in the rhabdomeric photoreceptor cells of the proximal retina. The second opsin gene isolated from Pecten and the opsin gene from Arca were shown to be expressed in various tissues, suggesting a putative role in the photic regulation of peripheral circadian clocks. Moreover, phylogenetic analysis indicate that each of these two opsin genes may constitute a novel opsin subfamily
Univalence criteria for analytic functions.
No full textThis thesis is devoted to the study of univalence criteria for analytic functions, in particular a domain constant known as the inner radius of D. Let D be a simply connected plane domain and let B be the unit disk. We define the inner radius of D, by\sigma(D) = \sup\{a : a \geq 0,\ \Vert S\sb{f}\Vert\sb{D}\ \leq\ a\ {\rm implies}\ {\it f\/}\ {\rm is\ univalent\ in}\ D\}.Here S\sb{f} is the Schwarzian derivative of f, \rho\sb{D} the hyperbolic density on D and\Vert S\sb{f}\Vert\sb{D} = {\sup\limits\sb{z\in D}}\vert S\sb{f}(z)\vert\rho\sbsp{D}{-2}(z).Domains for which the value of is known include disks, angular sectors and regular polygons. All of the mentioned domains except non-convex angular sectors have an interesting property in common, namely that = 2-\Vert S\sb{h}\Vert\sb{B} where h maps B conformally onto D. Because of the importance of this property, we say that D is a regular domain if = 2 -\Vert S\sb{h}\Vert\sb{B} is satisfied. First we use regularity to give a simple new proof of the result on regular n-sided polygons P\sb{n}. Next we study rectangles and equiangular hexagons. We prove that if R is a rectangle whose ratio of longer over shorter side is bounded from above by a specific constant ( 1.52346 then R is regular and = = \sigma(P\sb4). In a similar fashion, we prove that if H is an equiangular hexagon whose sides form the sequence baabaa with 1.67117 then H is regular and = = \sigma(P\sb6).. An interesting problem is to characterize regular domains. For domains of bounded boundary rotation with convex corners we show that{\limsup\limits\sb{\vert z \vert\to1}}\vert S\sb{h}(z)\vert(1 - \vert z\vert\sp2)\sp{\sp2} = \Vert S\sb{h}\Vert\sb{B}is a sufficient condition for regularity, where h maps B conformally onto D. Some results previously known for B can now be extended for all regular domains, in particular theorems of Gehring-Pommerenke, Ahlfors and Minda. The last part of the thesis is devoted to investigating an alternative domain constant, The only domains for which is known are disks. We demonstrate some bounds on for convex angular sectors.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/104352/1/9513432.pdfDescription of 9513432.pdf : Restricted to UM users only
The geometry of discrete groups.
No full textThis dissertation is concerned with discrete groups of Mobius transformations. A Mobius transformation is a conformal self mapping of the extended complex plane and a Mobius group is discrete if it does not contain any convergent sequence of distinct elements. We are mainly interested in two generator groups since for essentially all Mobius groups, G is discrete if and only if every two generator subgroup of G is discrete. We obtain discreteness conditions for Mobius groups by considering various iterative commutator sequences. For example, we show that if is discrete and f and g are conjugate, then 2 2 cos(/7) is the sharp lower bound for the distance from the trace of the commutator (f,g\rbrack = fgf\sp{-1}g\sp{-1} to 2. This result gives rise to many geometric constraints for discrete Mobius groups. In the study of the iterative commutator sequences, one makes use of the fact that three trace parameters tr\sp2(f),\ {\rm tr}\sp2(g), and tr (f,g) determine the two generator group up to conjugacy whenever tr (f,g) is not equal to 2. In particular, one can replace g by an elliptic h of order two so that is discrete and tr (f,h) equals tr (f,g). In contrast to the two generator case, we show that the corresponding six trace parameters determine two different conjugacy classes of three generator Mobius groups and that the groups in one conjugacy class can be discrete while the groups in the other conjugacy class are not. These two conjugacy classes coincide if the generators are not parabolic and the axes of two generators intersect orthogonally. We also give an example to show that using an order two element in a three generator discrete group may change the discreteness. The chordal norm d(f) = d(f, id) is the maximum of the chordal distance d(fz,z) over all points z in \bar\IR\sp{n}. It measures the maximum chordal derivation of f from the identity, and d(f) = 2 if and only if f maps one point of a pair of antipodal points of \bar\IR\sp{n} onto the other. By means of the trace inequalities we obtain lower bounds of max for a discrete group in dimension 2 as well as in dimension n.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/104924/1/9624583.pdfDescription of 9624583.pdf : Restricted to UM users only
Domain constants of injectivity.
No full textLet f be a locally injective mapping of a simply connected hyperbolic domain D into \bar\doubc. Under what circumstances is f injective? We assign norms to f in such a way that if f is sufficiently small, then f is injective. Using these norms we associate domain constants of injectivity to D. We consider the cases where f is meromorphic and locally K-quasiconformal. For f meromorphic let S\sb{\rm f} denote the Schwarzian derivative and let \rho\sb{\rm D} denote the density of the Poincare metric in D. The inner radius of univalence (D) is defined as the supremum of the numbers a 0 such that \vert\rm S\sb{f}(z)\vert a\rho\rm\sb{D}(z)\sp2 for all z D is a sufficient condition for f to be injective. We define normal circular triangles and show that (D) = 2k\sp2 if D is a normal circular triangle whose smallest angle is k. Using this result we show that if D is a regular n-sided polygon, then (D) = 2 . For locally K-quasiconformal mappings we define (D,K) as the supremum of the numbers b with the property that if log J\sb{\rm f}\Vert\sb\* \leq b, then f is injective; when no such constants b exist, we set (D,K) = 0. Here \Vert\cdot\Vert\sb\* denotes the BMO norm in D. Then we define K(D) as the supremum of the numbers K for which (D,K) 0. It is known that K(D) 1 if and only if D is a quasidisk, and that K(D) 2 for all D, with equality if D is a disk. We prove that K(D) = 2 also when D = z: z 1, Re (z) cos(k) for k 1/7. For this we show that D has a length-area property which we call the crosscut property. We prove also that domains with this property are convex and are K-quasidisks where K is bounded by an absolute constant.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128207/2/8821554.pd
Conservation of the retinal determination gene cascade in the jellyfish "Cladonema radiatum"
No full textThe species Cladonema radiatum belongs to the Cnidaria, a basal animal phylum which represents the closest sister group to bilateria. Despite their low position in the metazoan phylogenetic tree, Cnidaria are the only non-bilaterian animals with a defined body axis, a nervous system, sensory organs of great complexity such as photoreceptors and statocysts, and a remarkable regeneration capacity. Therefore cnidarians, for their basal position and with their surprising level of complexity, have become in the last years the organism of choice for evolutionary developmental studies, representing the appropriate outgroup necessary to understand the ancestral bilaterian condition. The jellyfish of C. radiatum bears eight to twelve lens eyes at the bell margin, on the tentacle bulb. Each eye displays a cornea, a lens, pigmented cells and a retina. This species appears therefore suitable for studying the conservation of an important gene network, the Retinal Determination Gene Cascade (RDGC), that has been demonstrated to be responsible for the eye development in species as diverse as Drosophila and mice. This network is made up of four gene families: Pax, Eya, Six and Dac. The full length sequence of a Pax gene from C. radiatum (CrPaxA) was already known. During my Ph.D. studies, I was able to isolate, by means of degenerate PCR, two more members of the Pax family (CrPaxB and CrPaxE) and one member of the Eya family (CrEya), described for the first time in Cnidaria. I then characterized the expression patterns of these genes by in situ hybridization, and analyzed by Real Time PCR their expression in the different tissues during the development of the jellyfish and at the different stages of the life cycle. CrPaxA is expressed in the retina and in nematocytes precursor cells in the tentacle bulb, whereas both CrPaxB and CrPaxE are expressed in the manubrium, the feeding and reproductive organ of the jellyfish where the gonads develop. In particular it was possible to detect the signal for CrPaxB in the maturing oocytes. CrEya is expressed at the same time in the retina and in the manubrium where it shows the same pattern at the level of the oocytes as CrPaxB. Taking advantage of the capability of the jellyfish to regenerate the eye once it has been micro-surgically removed, I was able to investigate the involvement of these genes in the development of the eye. Surprisingly none of them seems to be clearly up-regulated during the eye regeneration. This could indicate that CrPaxA and CrEya are involved in the
maintenance of the adult eye.
To gain further insights on the role of the isolated genes in the eye determination we used
targeted gene expression in Drosophila. Taking advantage of the UAS/GAL4 system, we misexpressed
the jellyfish genes in the imaginal discs of the fly and analysed the adults for
ectopic eyes induction. At the same time we examined the capability of these genes to rescue
Drosophila mutant phenotypes. Indeed UAS-CrPaxA was able to induce ectopic eyes, and
both UAS-CrPaxA and UAS-CrPaxB were able to rescue the Drosophila Pax2 mutant
sparkling.
The expression of CrPaxA and CrEya in the retina taken together with the functional assays
carried out in Drosophila argue for a conserved role of this gene network in the jellyfish eye.
This result is also supported by data from a previous report, showing the expression of two
members of the Six genes family in the eye of Cladonema. These results overall indicate a
high structural conservation of the members of the RDGC between Cnidaria and Bilateria,
and are in agreement with the theory of the monophyletic origin of the eye. The evidence for
conservation is further strengthened by the expression of CrPaxB, CrEya and a third Six gene
CrSix4/5 in the oocytes, suggesting a possible preservation of the interactions among the
members of the network and its redeployment to a different context. Changes in the temporal
and spatial pattern of genes expression are one of the main mechanisms by which the
phenotypic diversity arises, the redeployment of the RDGC in Cladonema radiatum might
offer an example of this process
Gehring (Birth, 1882-09-09)
No full text5132/Pg 152/1882/M W/Ger./Ger./F. Moller,Mid.Original record filed in drawer labeled 'GATES-GEMM'
Gehring (Birth, 1879-06-09)
No full textAddress: 72 McMicken Ave.3386/Pg. 33/1879/F W/Ger./Ger./Louisa Kurz, Mid.Original record filed in drawer labeled 'GATES-GEMM'
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