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Showing 1 to 8 of 8 (0.144 seconds)Spelling suggestions: "subject:"harmonic 2analysis anda depresentation"" "subject:"harmonic 2analysis anda prepresentation""
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Some applications of self-affine sets to wavelet theoryFu, Xiaoye 10 1900 (has links)
<p>In this thesis, we study several applications of self-affine sets to wavelet theory. Five major topics are considered here: wavelet sets (scaling sets), multiwavelet sets (generalized scaling sets), self-affine tiles, integral self-affine multi-tiles, self-affine sets. We divide the thesis into six chapters to discuss these topics. In Chapter 1, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK=(K+d_1)\bigcup(K+d_2)$, where $B=A^t$ and $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$. We show that it must be a constant in dimension $n=1$ or $2$ and it is bounded by $2\lvert K\rvert$ for any $n$. This result shows that all $A$-dilation self-affine scaling sets must be $A$-dilation MRA scaling sets in dimensions one and two. There exist results on the connection between the theory of wavelets and the theory of integral self-affine tiles and in particular, on the construction of wavelet bases using integral self-affine tiles. However, there are many non-integral self-affine tiles which can also yield wavelet basis. In Chapter 2 and Chapter 3, we give a complete characterization of all two dimensional $A$-dilation scaling sets $K$ such that $K$ is a self-affine tile satisfying $BK=(K+d_1)\bigcup (K+d_2)$ for some $d_1, d_2\in\mathbb{R}^2$, where $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$. In Chapter 2, we deal with a particular case where $0\in\{d_1,d_2\}$, i.e. a self-affine tile $K$ satisfies $BK=K\bigcup (K+d)$ for some $d\in\mathbb{R}^2$. Chapter 3 is devoted to the general case with $d_1, d_2\in\mathbb{R}^2$. Moreover, we give a sufficient condition for a self-affine tile, possibly non-integral, to be an MRA scaling set in Chapter 3. Gabardo and Yu first considered using integral self-affine tiles in the Fourier domain to construct wavelet sets and they produced a class of compact wavelet sets with certain self-similarity properties. In Chapter 4, we generalize their results to the integral self-affine multi-tiles setting. We characterize some analytic properties of integral self-affine multi-tiles under certain conditions. We also consider the problem of constructing (multi)wavelet sets using integral self-affine multi-tiles. Suppose that a measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$ is an integral self-affine multi-tile associated with an $n\times n$ integral expansive matrix $B$. To our knowledge, no one considered how to represent an integral self-affine $\mathbb{Z}^n$-tiling set as the disjoint union of prototiles. In Chapter 5, we provide an algorithm to decompose $K$ into disjoint pieces $K_j$ which satisfy $K=\displaystyle\bigcup K_j$ such that the collection of the sets $K_j$ is an integral self-affine collection associated with matrix $B$ and the number of pieces $K_j$ is minimal. Using this algorithm, we can determine whether a given measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$ is an integral self-affine multi-tile associated with any given $n\times n$ integral expansive matrix $B$. Furthermore, the minimal decomposition we provide is unique. Let $B$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$. The self-affine set $K=K(B,\mathcal{D})$ is the unique compact set satisfying the set equation $BK=\displaystyle\bigcup_{d\in\mathcal{D}}(K+d)$. In Chapter 6, we not only consider the problem how to compute the Lebesgue measure of self-affine sets $K(B,\mathcal{D})$, but also consider the Hausdorff measure for those with zero Lebesgue measure under the assumption that $K(B,\mathcal{D})$ is a self-similar set. In the case where $\text{card}(\mathcal{D})=\lvert\det B\rvert,$ we relate the Lebesgue measure of $K(B,\mathcal{D})$ to the upper Beurling density of the associated measure $\mu=\lim\limits_{s\to\infty}\sum\limits_{\ell_0,\dotsc,\ell_{s-1}\in\mathcal{D}}\delta_{\ell_0+B\ell_1+\dotsb+B^{s-1}\ell_{s-1}}.$ If, on the other hand, $\text{card}(\mathcal{D})<\lvert\det B\rvert$ and $B$ is a similarity matrix, we relate the Hausdorff measure $\mathcal{H}^s(K)$, where $s$ is the similarity dimension of $K$, to a corresponding notion of upper density for the measure $\mu$.</p> / Doctor of Science (PhD)
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Compactness of Isoresonant PotentialsWolf, Robert G. 01 January 2017 (has links)
Bruning considered sets of isospectral Schrodinger operators with smooth real potentials on a compact manifold of dimension three. He showed the set of potentials associated to an isospectral set is compact in the topology of smooth functions by relating the spectrum to the trace of the heat semi-group. Similarly, we can consider the resonances of Schrodinger operators with real valued potentials on Euclidean space of whose support lies inside a ball of fixed radius that generate the same resonances as some fixed Schrodinger operator, an ``isoresonant" set of potentials. This isoresonant set of potentials is also compact in the topology of smooth functions for dimensions one and three. The basis of the result stems from the relation of a regularized wave trace to the resonances via the Poisson formula (also known as the Melrose trace formula). The second link is the small-t asymptotic expansion of the regularized wave trace whose coefficients are integrals of the potential function and its derivatives. For an isoresonant set these coefficients are equal due to the Poisson formula. The equivalence of coefficients allows us to uniformly bound the potential functions and their derivatives with respect to the isoresonant set. Finally, taking a sequence of functions in the isoresonant set we use the uniform bounds to construct a convergent subsequence using the Arzela-Ascoli theorem.
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The Boundedness of the Hardy-Littlewood Maximal Function and the Strong Maximal Function on the Space BMOZhang, Wenhao 01 January 2018 (has links)
In this thesis, we present the space BMO, the one-parameter Hardy-Littlewood maximal function, and the two-parameter strong maximal function. We use the John-Nirenberg inequality, the relation between Muckenhoupt weights and BMO, and the Coifman-Rochberg proposition on constructing A1 weights with the Hardy- Littlewood maximal function to show the boundedness of the Hardy-Littlewood maximal function on BMO. The analogous statement for the strong maximal function is not yet understood. We begin our exploration of this problem by discussing an equivalence between the boundedness of the strong maximal function on rectangular BMO and the fact that the strong maximal function maps A∞ weights into the A1 class. We then extend a multiparameter counterexample to the Coifman-Rochberg proposition proposed by Soria (1987) and discuss the difficulties in modifying it into an A∞ counterexample that would disapprove the boundedness of the strong maximal function.
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A New Subgroup Chain for the Finite Affine GroupLingenbrink, David Alan, Jr. 01 January 2014 (has links)
The finite affine group is a matrix group whose entries come from a finite field. A natural subgroup consists of those matrices whose entries all come from a subfield instead. In this paper, I will introduce intermediate sub- groups with entries from both the field and a subfield. I will also examine the representations of these intermediate subgroups as well as the branch- ing diagram for the resulting subgroup chain. This will allow us to create a fast Fourier transform for the group that uses asymptotically fewer opera- tions than the brute force algorithm.
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On Representations of the Jacobi Group and Differential EquationsWebster, Benjamin 01 January 2018 (has links)
In PDEs with nontrivial Lie symmetry algebras, the Lie symmetry naturally yield Fourier and Laplace transforms of fundamental solutions. Applying this fact we discuss the semidirect product of the metaplectic group and the Heisenberg group, then induce a representation our group and use it to investigate the invariant solutions of a general differential equation of the form .
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Fast Algorithms for Analyzing Partially Ranked DataMcDermott, Matthew 01 January 2014 (has links)
Imagine your local creamery administers a survey asking their patrons to choose their five favorite ice cream flavors. Any data collected by this survey would be an example of partially ranked data, as the set of all possible flavors is only ranked into subsets of the chosen flavors and the non-chosen flavors. If the creamery asks you to help analyze this data, what approaches could you take? One approach is to use the natural symmetries of the underlying data space to decompose any data set into smaller parts that can be more easily understood. In this work, I describe how to use permutation representations of the symmetric group to create and study efficient algorithms that yield such decompositions.
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BBT Acoustic Alternative Top Bracing CADD Data Set-NoRev-2022Jun28Hemphill, Bill 22 July 2022 (has links)
This electronic document file set consists of an overview presentation (PDF-formatted) file and companion video (MP4) and CADD files (DWG & DXF) for laser cutting the ETSU-developed alternate top bracing designs and marking templates for the STEM Guitar Project’s BBT (OM-sized) standard acoustic guitar kit. The three (3) alternative BBT top bracing designs in this release are (a) a one-piece base for the standard kit's (Martin-style) bracing, (b) 277 Ladder-style bracing, and (c) an X-braced fan-style bracing similar to traditional European or so-called 'classical' acoustic guitars.
The CADD data set for each of the three (3) top bracing designs includes (a) a nominal 24" x 18" x 3mm (0.118") Baltic birch plywood laser layout of (1) the one-piece base with slots, (2) pre-radiused and pre-scalloped vertical braces with tabs to ensure proper orientation and alignment, and (3) various gages and jigs and (b) a nominal 15" x 20" marking template.
The 'provided as is" CADD data is formatted for use on a Universal Laser Systems (ULS) laser cutter digital (CNC) device. Each CADD drawing is also provided in two (2) formats: Autodesk AutoCAD 2007 .DWG and .DXF R12. Users should modify and adapt the CADD data as required to fit their equipment. This CADD data set is released and distributed under a Creative Commons license; users are also encouraged to make changes o the data and share (with attribution) their designs with the worldwide acoustic guitar building community.
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BBT Acoustic Alternative Top Bracing CADD Data Set-NoRev-2022Jun28Hemphill, Bill 22 July 2022 (has links)
This electronic document file set consists of an overview presentation (PDF-formatted) file and companion video (MP4) and CADD files (DWG & DXF) for laser cutting the ETSU-developed alternate top bracing designs and marking templates for the STEM Guitar Project’s BBT (OM-sized) standard acoustic guitar kit. The three (3) alternative BBT top bracing designs in this release are (a) a one-piece base for the standard kit's (Martin-style) bracing, (b) 277 Ladder-style bracing, and (c) an X-braced fan-style bracing similar to traditional European or so-called 'classical' acoustic guitars.
The CADD data set for each of the three (3) top bracing designs includes (a) a nominal 24" x 18" x 3mm (0.118") Baltic birch plywood laser layout of (1) the one-piece base with slots, (2) pre-radiused and pre-scalloped vertical braces with tabs to ensure proper orientation and alignment, and (3) various gages and jigs and (b) a nominal 15" x 20" marking template.
The 'provided as is" CADD data is formatted for use on a Universal Laser Systems (ULS) laser cutter digital (CNC) device. Each CADD drawing is also provided in two (2) formats: Autodesk AutoCAD 2007 .DWG and .DXF R12. Users should modify and adapt the CADD data as required to fit their equipment. This CADD data set is released and distributed under a Creative Commons license; users are also encouraged to make changes o the data and share (with attribution) their designs with the worldwide acoustic guitar building community.
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