Abdou Youssef

Technical Aspects of the Digital Library of Mathematical Functions

The NIST Digital Library of Mathematical Functions (DLMF) Project, begun in 1997, is preparing a handbook and Web site intended for wide communities of users. The contents are primarily mathematical formulas, graphs, methods of computation, references, and links to software. The task of developing a Web handbook of this nature presents several technical challenges. We describe the goals of the Digital Library of Mathematical Functions Project and the realities that constrain those goals. We propose practical initial solutions, in order to ease the authoring of adaptable content: a LaTeX class which encourages a modestly semantic markup style; and a mathematical search engine that adapts a text search engine to the task.

The banyan-hypercube networks

The authors introduce a family of networks that are a synthesis of banyans and hypercubes and are called the banyan-hypercubes (BH). They combine the advantageous features of banyans and hypercubes and thus have better communication capabilities. The networks can be viewed as consisting of interconnecting hypercubes. It is shown that many hypercube features can be incorporated into BHs with regard to routing, embedding of rings and meshes, and partitioning, and that improvements over the hypercube result are made. In particular, it is shown that BHs have better diameters and average distances than hypercubes, and they embed pyramids and multiple pyramids with dilation cost 1. An optimal routing algorithm for BHs and an efficient partitioning strategy are presented. >

Transform-based indexing of audio data for multimedia databases

Since the relative proportion of multimedia (video, image and audio) data within databases is expected to increase substantially in the future, keyword-based indexing would be inadequate and efficient content-based query and retrieval are required. The problem of devising content based query, indexing, and retrieval for these newer data types remains an open and challenging problem. While considerable attention has recently been given to image (and, to some extent, video) indexing, much less has been devoted to the problem of indexing its unidimensional counterpart-audio data. The paper proposes content-based indexing schemes for audio data in multimedia databases. The methods are based on transform techniques used in signal processing which transform data from time (or spatial) domain to frequency domain. This offers many advantages such as easy removal of noise, efficient compression and different types of processing. Two algorithms for indexing are presented along with experimental results.

Roles of math search in mathematics

Math-aware fine-grain search is expected to be widely available. A key question is what roles it can play in mathematics. It will be argued that, besides finding information, math search can help advance and manage mathematical knowledge. This paper will present the short-term goals and state of the art of math-aware fine-grain search. Afterwards, it will focus on how math search can help advance and manage mathematical knowledge, and discuss what needs to be done to fulfill those roles, emphasizing two key components. The first is similarity search, and how it applies to (1) discovering and drawing upon connections between different fields, and (2) proof development. The second is math metadata, which math search will surely encourage and benefit from, and which will be pivotal to mathematical knowledge management.

Design and analysis of product networks

In this paper a unified theory of Cartesian product networks is developed. Product networks (PN) include meshes, tori, and hypercubes among others. This paper studies the fundamental issues of topological properties, cost-performance ratio optimization, scalability, routing, embedding, and fault tolerance properties of PNs. In particular, the degree, diameter, average distance, connectivity, and node-symmetry of PNs are related to those of their constituent factor networks. Cost/performance analysis and comparison between different PNs, especially n-dimensional meshes/tori and n-dimensional r-ary hypercubes, are conducted, and the optimal trade-off between the number of dimensions and the size along each dimension are identified. Fast generic algorithms for point-to-point routing, broadcasting and permuting on PNs are designed, making use of the corresponding algorithms of the factor networks. Finally, efficient embeddings on PNs are constructed for linear arrays, rings, meshes, tori and trees.<<ETX>>

Methods of Relevance Ranking and Hit-content Generation in Math Search

To be effective and useful, math search systems must not only maximize precision and recall, but also present the query hits in a form that makes it easy for the user to identify quickly the truly relevant hits. To meet that requirement, the search system must sort the hits according to domain-appropriate relevance criteria, and provide with each hit a query-relevant summary of the hit target. The standard relevance measures in text search, which rely mostly on keyword frequencies and document sizes, turned out to be inadequate in math search. Therefore, alternative relevance measures must be defined, which give more weight to certain types of information than to others and take into account cross-reference statistics. In this paper, new, multidimensional relevance metrics are defined for math search, methods for computing and implementing them are discussed, and comparative performance evaluation results are presented. Query-relevant hit-summary generation is another factor that enables users to quickly determine the relevance of the presented hits. Although the hit title accompanied by a few leading sentences from the target document is simple to produce, this often fails to convey to the user the documents relevant excerpts. This shifts the burden onto the user to pursue many of the hits, and read significant portions of their target documents, to finally locate the wanted documents. Clearly, this task is too time-consuming and should be largely automated. This paper presents query-relevant hit-summary generation methods, outlines implementation strategies, and presents performance evaluation results.

Wavelet-based indexing of audio data in audio/multimedia databases

The phenomenal increase in the amounts of audio (and multimedia) data being generated, processed, and used in several computer applications have necessitated the development of audio (and multimedia) database systems with newer features such as content-based queries and similarity searches to manage and use such data. Example applications of audio databases are in digital libraries, entertainment industry, forensic laboratories and virtual reality, among others. Fast and accurate retrieval for content-based queries are crucial for such systems to be useful. Efficient content-based indexing and similarity searching schemes are keys to providing fast and relevant data retrieval. This paper presents a scheme for indexing of audio data based on wavelets. The performance of this scheme has been experimentally evaluated and is seen to be more resilient to noise than the indexing schemes using signal-level statistics, and give better retrieval precision than DCT-based indexing.

Anomaly Detection in Time Series of Graphs using Fusion of Graph Invariants

Given a time series of graphs <i>G</i>(<i>t</i>)=(<i>V</i>,<i>E</i>(<i>t</i>)) , <i>t</i>=1,2,... , where the fixed vertex set <i>V</i> represents “actors” and an edge between vertex <i>u</i> and vertex <i>v</i> at time <i>t</i>(<i>uv</i> ∈ <i>E</i>(<i>t</i>)) represents the existence of a communications event between actors <i>u</i> and <i>v</i> during the <i>t</i>^th time period, we wish to detect anomalies and/or change points. We consider a collection of graph features, or invariants, and demonstrate that adaptive fusion provides superior inferential efficacy compared to naive equal weighting for a certain class of anomaly detection problems. Simulation results using a latent process model for time series of graphs, as well as illustrative experimental results for a time series of graphs derived from the Enron email data, show that a fusion statistic can provide superior inference compared to individual invariants alone. These results also demonstrate that an adaptive weighting scheme for fusion of invariants performs better than naive equal weighting.

Equivalence detection using parse-tree normalization for math search

In recent years, efforts have begun to put math contents on the Web. As for other types of Web information, search capabilities should be provided to enable users to find what they need because without the ability to search the data for specific items, the data is useless. Conventional (i.e. textbased or even multimedia-based) search engines fall short of providing math-search capabilities. Preliminary efforts to create math-search systems have started, and many of the issues and the challenges for building such systems have been identified. One of the more difficult challenges is the detection of mathematical equivalence between expression in users’ queries and expressions in math contents. The purpose of this research is to develop techniques and algorithm for equivalence-detection based math search. In particular, this research aims to explore some proposed normalization rules, then to develop a general way that can be utilized to transform both the repository contents and users’ input expressions into a unified normalized form.

A general framework for developing adaptive fault-tolerant routing algorithms

It is shown that Cartesian product (CP) graph-based network methods provide a useful framework for the design of reliable parallel computer systems. Given component networks with prespecified connectivity, more complex networks with known connectivity and terminal reliability can be developed. CP networks provide systematic techniques for developing reliable fault-tolerant routing schemes, even for very complex topological structures. The authors establish the theoretical foundations that relate the connectivity of a CP network, the connectivity of the component networks, and the number of faulty components: present an adaptive generic algorithm that can perform successful point-to-point routing in the presence of faults: synthesize, using the theoretical results, this adaptive fault-tolerant algorithm from algorithms written for the component networks: prove the correctness of the algorithm: and show that the algorithm ensures following an optimal path, in the presence of many faults, with high probability. >