Grassmannian Frames with Applications to Coding and Communication

For a given class ${\cal F}$ of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation $|< f_k,f_l >|$ among all frames $\{f_k\}_{k \in {\cal I}} \in {\cal F}$. We first analyze finite-dimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with uniform tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on uniform tight frames. We then introduce infinite-dimensional Grassmannian frames and analyze their connection to uniform tight frames for frames which are generated by group-like unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.Comment: Submitted in June 2002 to Appl. Comp. Harm. Ana

Context as foundation for a semantic desktop

Adoption of semantic web technologies and principles presents an opportunity to change the conceptual model of desktop computing. Moving from a traditional position where the desktop is largely tied to a specific computational device, a semantic desktop could exist as a broad, networked space defined relative to the user. In this position paper we argue that personal, computing, and knowledge contexts are the appropriate means by which to define and shape the desktop space, and that collectively they provide the foundation for novel functionality in a semantic desktop

Optimal CDMA signature sequences, inverse eigenvalue problems and alternating minimization

This paper describes the matrix-theoretic ideas known as Welch-bound-equality sequences or unit-norm tight frames that are used to alternate minimizing the total squared correlation. This paper shows the construction of an optimal signature sequences for the synchronous code-division multiple-access (S-CDMA) channel in the presence of white noise and uniform received powers to solve inverse eigenvalue problems that maximize the sum capacity of the S-CDMA channel

DEVELOPMENT OF A LOW-DOSE RADIATION THERAPY DEVICE FOR ACUTE RESPIRATORY DISTRESS SYNDROME

This research focuses on developing a low-dose radiotherapy (LD-RT) device for treating acute respiratory distress syndrome (ARDS). This respiratory illness affects millions of people a year. Using orthovoltage (200 – 500 keV) X-ray energy provides many advantages over traditional radiotherapy delivery with linear accelerators, such as low cost and greater accessibility. In addition, X-ray tubes have been shown throughout history to provide good treatment outcomes for pneumonia, and research has shown LD-RT to be just as effective with ARDS. This proposal summarizes my efforts in determining the dosimetric properties of an LD-RT system to deliver treatment quickly and effectively. My research involves Monte Carlo transport simulations in MCNP to calculate a whole lung treatment\u27s dose delivery and coverage. In addition, different setup geometries and beam modifications, such as filtration, are explored to optimize the treatment delivery. The development of such a system would greatly expand the treatment options for ARDS-type illness at a lower cost and greater accessibility than current radiotherapy options

DEVELOPMENT OF A LOW-DOSE RADIATION THERAPY DEVICE FOR ACUTE RESPRITORY DISTRESS SYNDROME

This research focuses on the development of a low dose radiotherapy (LD-RT) device for treatment of acute respiratory distress syndrome (ARDS), a respiratory illness that affects millions of people a year. The use of orthovoltage (200 – 500 keV) X-ray energy provides many advantages over traditional radiotherapy delivery with linear accelerators, such as low cost and greater accessibility. In addition, X-ray tubes have been shown throughout history to provide good treatment outcomes for pneumonia, and research has shown LD-RT to be just as effective with ARDS. This proposal summarizes my efforts in determining the dosimetric properties of a LD-RT system to deliver treatment in a fast and effective manner. My research involves Monte Carlo transport simulations in MCNP to calculate the dose delivery and coverage of a whole lung treatment. In addition, different setup geometries and beam modifications will be explored to optimize the treatment delivery. The development of such a system would greatly expand the treatment options for ARDS type illness at a lower cost and greater accessibility than current radiotherapy options

Construction of equiangular signatures for synchronous CDMA systems

Welch bound equality (WBE) signature sequences maximize the uplink sum capacity in direct-spread synchronous code division multiple access (CDMA) systems. WBE sequences have a nice interference invariance property that typically holds only when the system is fully loaded, and, to maintain this property, the signature set must be redesigned and reassigned as the number of active users changes. An additional equiangular constraint on the signature set, however, maintains interference invariance. Finding such signatures requires equiangular side constraints to be imposed on an inverse eigenvalue problem. The paper presents an alternating projection algorithm that can design WBE sequences that satisfy equiangular side constraints. The proposed algorithm can be used to find Grassmannian frames as well as equiangular tight frames. Though one projection is onto a closed, but non-convex, set, it is shown that this algorithm converges to a fixed point, and these fixed points are partially characterized

Designing structured tight frames via an alternating projection method

Tight frames, also known as general Welch-bound- equality sequences, generalize orthonormal systems. Numerous applications - including communications, coding, and sparse approximation- require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm

CDMA signature sequences with low peak-to-average-power ratio via alternating projection

Several algorithms have been proposed to construct optimal signature sequences that maximize the sum capacity of the uplink in a direct-spread synchronous code division multiple access (CDMA) system. These algorithms produce signatures with real-valued or complex-valued entries that generally have a large peak-to-average power ratio (PAR). This paper presents an alternating projection algorithm that can design optimal signature sequences that satisfy PAR side constraints. This algorithm converges to a fixed point, and these fixed points are partially characterized

Poetry and Theology in the Book of Lamentations: An Investigation of Lamentations 1-3 using the Aesthetic Analysis of Umberto Eco

The present study assesses how Lamentations 1-3 synthetically presents its theology. It accomplishes this task by assessing the poetry through the aesthetic analysis of Italian semiotician Umberto Eco to discover how, in terms of genre, structure, and poetics, theology is presented for the model reader of Lamentations 1-3. Chapter I introduces the problem of theology in Lamentations and the difficulties and possibility of focussing the present research upon Lamentations 1-3. Within this discussion, these chapters are set in their historical context. Chapter I concludes by suggesting that interpretation of theology remains a complex task and employs the metaphors of horizons "behind," "within" and "in front of' the text as theoretical tools by which different approaches could be categorised. Chapter 2 surveys past research using the metaphors of "behind," "Within," and "in front of' the text as a heuristic framework. Each horizon is assessed in tum: historical paradigms of Gottwald, Albrektson, Westermann, and Brandscheidt (world "behind" the text); literary paradigm of Renkema (world "within" the text); and the feminist approaches of Seidman, Guest, and O'Connor (world "in fromt of' the text). Finally, Chapter 2 adopts an "integrated" approach, typified by Dobbs-Allsopp, that takes seriously all three horizons in interpreting Lamentations 1-3. Chapter 3 provides an entree into the theory of Umberto Eco. His theory is useful because it coheres with the integrated approach adopted in the study, it provides a helpful means to assess aesthetic texts, such as Lamentations, and it enables distinctions between kinds of texts-how texts are designed differently to elicit different responses from model readers (open and closed). In light of the theological ambiguity in Lamentations, the open/closed distinction is shown to be useful. Finally, Eco employs the concept of the cultural encyclopaedia, a theoretical device that describes the cumulative amount of cultural data available to the producer of a text at the time of its production. Chapter 4 frames the borders of encyclopaedic content for research into Lamentations 1-3. It presents the possible genres, structures, and poetics suggested for Lamentations research in the past. The analysis shows that Lamentations cannot be reduced to one genre but rather exploits different genres to advance its theology. As to structure, analysis reveals that the acrostic is the most evident structuring device in the book. And finally, a number of poetic devices activated in the encyclopaedic world of Lamentations 1-3 are explored, including repetition, wordplay and enjambment, imagery, speaking voices, and allusion. This discussion frames the exegesis of Lamentations 1-3, accomplished in Chapters 5-7. Chapters 5-7 assess Lamentations 1-3 using Eco's aesthetic theory. Each chapter presents an introduction to the structure, genre, speaking voices, and strophic divisions of Lamentations 1-3, follows with detailed exegesis of the chapters, and then concludes with a catalogue of the ways in which structure, genre, and poetics impinge upon theological portrayal in the poetry. Analysis shows Lamentations 1-3 tends towards "open" rather than "closed" textual strategies for their model readers. Recognition and cataloguing of the persistent poetic use of repetition proves to be an area that the present study adds to scholarly discussion, as well as how repetition impinges upon theological presentation in the book. There are two primary functions of repetition: intensification (upon suffering, sin, judgment) or combination (to recast previously held understandings or to provide interpretative depth). Repetition provides a variety of interpretative horizons for the reader in regards to the book's theology. Chapter 8 concludes with a summary of results, an initial discussion of Lamentations 4 and 5, and the possible purpose of poetry and theology in Lamentations 1-3. The study concludes that the theology varies, but this is part of the function of the poetry. The poetry is designed to bring the reader on an interpretative journey through its contents rather than to teach a particular perspective. Despite the various ways in which the relationships can be configured, the poetry persistently drives the reader to address YHWH in prayer: each of the poems includes, and concludes with, prayer to the deity concerning various sources of pain. That the poetry highlights prayer to YHWH-even when he is the cause of pain-reveals this interpretative journey has a destination. The poetry of Lamentations 1-3 is designed to enable the reader to address God in light of the perspectives adopted and sufferings endured through the reading process