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Search Results: 1 - 10 of 212510 matches for " Karen L. Shuman "
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Harmonic analysis of iterated function systems with overlap
Palle E. T. Jorgensen,Keri A. Kornelson,Karen L. Shuman
Mathematics , 2007, DOI: 10.1063/1.2767004
Abstract: In this paper we extend previous work on IFSs without overlap. Our method involves systems of operators generalizing the more familiar Cuntz relations from operator algebra theory, and from subband filter operators in signal processing.
Affine systems: asymptotics at infinity for fractal measures
Palle E. T. Jorgensen,Keri A. Kornelson,Karen L. Shuman
Mathematics , 2007,
Abstract: We study measures on $\mathbb{R}^d$ which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions. The construction includes measures arising from affine and contractive iterated function systems with and without overlap (IFSs), i.e., limit measures $\mu$ induced by a finite family of affine mappings in $\mathbb{R}^d$ (the focus of our paper), as well as equilibrium measures in complex dynamics. By a systematic analysis of the Fourier transform of the measure $\mu$ at hand (frequency domain), we identify asymptotic laws, spectral types, dichotomy, and chaos laws. In particular we show that the cases when $\mu$ is singular carry a gradation, ranging from Cantor-like fractal measures to measures exhibiting chaos, i.e., a situation when small changes in the initial data produce large fluctuations in the outcome, or rather, the iteration limit (in this case the measures). Our method depends on asymptotic estimates on the Fourier transform of $\mu$ for paths at infinity in $\mathbb{R}^d$. We show how properties of $\mu$ depend on perturbations of the initial data, e.g., variations in a prescribed finite set of affine mappings in $\mathbb{R}^d$, in parameters of a rational function in one complex variable (Julia sets and equilibrium measures), or in the entries of a given infinite positive definite matrix.
Scaling by 5 on a 1/4-Cantor Measure
Palle E. T. Jorgensen,Keri A. Kornelson,Karen L. Shuman
Mathematics , 2011,
Abstract: Each Cantor measure (\mu) with scaling factor 1/(2n) has at least one associated orthonormal basis of exponential functions (ONB) for L^2(\mu). In the particular case where the scaling constant for the Cantor measure is 1/4 and two specific ONBs are selected for L^2(\mu), there is a unitary operator U defined by mapping one ONB to the other. This paper focuses on the case in which one ONB (\Gamma) is the original Jorgensen-Pedersen ONB for the Cantor measure (\mu) and the other ONB is is 5\Gamma. The main theorem of the paper states that the corresponding operator U is ergodic in the sense that only the constant functions are fixed by U.
Scalar spectral measures associated with an Operator-Fractal
Palle E. T. Jorgensen,Keri A. Kornelson,Karen L. Shuman
Mathematics , 2012,
Abstract: We examine the operator $U_5$ defined on $L^2(\mu_{\frac14})$ where $\mu_{\frac14}$ is the 1/4 Cantor measure. The operator $U_5$ scales the elements of the canonical exponential spectrum for $L^2(\mu_{\frac14})$ by 5 --- that is, $Ue_{\gamma} = e_{5\gamma}$ where $e_{\gamma}(t) = e^{2\pi i \gamma t}$. It is known that $U_5$ has a self-similar structure, which makes its spectrum, which is currently unknown, of particular interest. In order to better understand the spectrum of $U_5$, we demonstrate a decomposition of the projection valued measures and scalar spectral measures associated with $U_5$. We are also able to compute associated Radon-Nikodym derivatives between the scalar measures. Our decomposition utilizes a system of operators which form a representation of the Cuntz algebra $\mathcal{O}_2$.
Additive spectra of the 1/4 Cantor measure
Palle E. T. Jorgensen,Keri A. Kornelson,Karen L. Shuman
Mathematics , 2013,
Abstract: In this paper, we add to the characterization of the Fourier spectra for Bernoulli convolution measures. These measures are supported on Cantor subsets of the line. We prove that performing an odd additive translation to half the canonical spectrum for the 1/4 Cantor measure always yields an alternate spectrum. We call this set an additive spectrum. The proof works by connecting the additive set to a spectrum formed by odd multiplicative scaling.
An Operator-Fractal
Palle E. T. Jorgensen,Keri A. Kornelson,Karen L. Shuman
Mathematics , 2011,
Abstract: Certain Bernoulli convolution measures (\mu) are known to be spectral. Recently, much work has concentrated on determining conditions under which orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure known to be spectral, the orthonormal basis need not be unique; indeed, there are often families of such spectral bases. Let \lambda = 1/(2n) for a natural number n and consider the Bernoulli measure (\mu) with scale factor \lambda. It is known that L^2(\mu) has a Fourier basis. We first show that there are Cuntz operators acting on this Hilbert space which create an orthogonal decomposition, thereby offering powerful algorithms for computations for Fourier expansions. When L^2(\mu) has more than one Fourier basis, there are natural unitary operators U, indexed by a subset of odd scaling factors p; each U is defined by mapping one ONB to another. We show that the unitary operator U can also be orthogonally decomposed according to the Cuntz relations. Moreover, this operator-fractal U exhibits its own self-similarity.
Iterated function systems, moments, and transformations of infinite matrices
Palle Jorgensen,Keri Kornelson,Karen Shuman
Mathematics , 2008,
Abstract: We study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Our main object of study is the infinite matrix which encodes all the moment data of a Borel measure on R^d or C. To encode the salient features of a given IFS into precise moment data, we establish an interdependence between IFS equilibrium measures, the encoding of the sequence of moments of these measures into operators, and a new correspondence between the IFS moments and this family of operators in Hilbert space. For a given IFS, our aim is to establish a functorial correspondence in such a way that the geometric transformations of the IFS turn into transformations of moment matrices, or rather transformations of the operators that are associated with them. We first examine the classical existence problem for moments, culminating in a new proof of the existence of a Borel measure on R or C with a specified list of moments. Next, we consider moment problems associated with affine and non-affine IFSs. Our main goal is to determine conditions under which an intertwining relation is satisfied by the moment matrix of an equilibrium measure of an IFS. Finally, using the famous Hilbert matrix as our prototypical example, we study boundedness and spectral properties of moment matrices viewed as Kato-Friedrichs operators on weighted l^2 spaces.
Orthogonal Exponentials for Bernoulli Iterated Function Systems
Palle Jorgensen,Keri Kornelson,Karen Shuman
Mathematics , 2007,
Abstract: We investigate certain spectral properties of the Bernoulli convolution measures on attractor sets arising from iterated function systems on the real line. In particular, we examine collections of orthogonal exponential functions in the Hilbert space of square integrable functions on the attractor. We carefully examine a test case where the parameter lambda is equal to 3/4 and therefore the IFS has overlap. We also determine rational values of lambda for which infinite sets of orthogonal exponentials exist.
Families of spectral sets for Bernoulli convolutions
Palle Jorgensen,Keri Kornelson,Karen Shuman
Mathematics , 2009,
Abstract: In this paper, we study the harmonic analysis of Bernoulli measures. We show a variety of orthonormal Fourier bases for the L^2 Hilbert spaces corresponding to certain Bernoulli measures, making use of contractive transfer operators. For other cases, we exhibit maximal Fourier families that are not orthonormal bases.
Rhomboid 4 (ROM4) Affects the Processing of Surface Adhesins and Facilitates Host Cell Invasion by Toxoplasma gondii
Jeffrey S. Buguliskis,Fabien Brossier,Joel Shuman,L. David Sibley
PLOS Pathogens , 2010, DOI: 10.1371/journal.ppat.1000858
Abstract: Host cell attachment by Toxoplasma gondii is dependent on polarized secretion of apical adhesins released from the micronemes. Subsequent translocation of these adhesive complexes by an actin-myosin motor powers motility and host cell invasion. Invasion and motility are also accompanied by shedding of surface adhesins by intramembrane proteolysis. Several previous studies have implicated rhomboid proteases in this step; however, their precise roles in vivo have not been elucidated. Using a conditional knockout strategy, we demonstrate that TgROM4 participates in processing of surface adhesins including MIC2, AMA1, and MIC3. Suppression of TgROM4 led to decreased release of the adhesin MIC2 into the supernatant and concomitantly increased the surface expression of this and a subset of other adhesins. Suppression of TgROM4 resulted in disruption of normal gliding, with the majority of parasites twirling on their posterior ends. Parasites lacking TgROM4 bound better to host cells, but lost the ability to apically orient and consequently most failed to generate a moving junction; hence, invasion was severely impaired. Our findings indicate that TgROM4 is involved in shedding of micronemal proteins from the cell surface. Down regulation of TgROM4 disrupts the normal apical-posterior gradient of adhesins that is important for efficient cell motility and invasion of host cells by T. gondii.
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