The LYM property of a finite standard graded poset is one of the central notions in Sperner theory. It is known that the product of two finite standard graded posets satisfying the LYM properties may not have the LYM property again. In 1974, Harper proved that if two finite standard graded posets satisfying the LYM properties also satisfy rank logarithmic concavities, then their product also satisfies these two properties. However, Harper's proof is rather non-intuitive. Giving a natural proof of Harper's theorem is one of the goals of this thesis. The main new result of this thesis is a characterization of rank-finite standard graded LYM posets that satisfy rank logarithmic concavities. With this characterization theorem, we are able to give a new, natural proof of Harper's theorem. In fact, we prove a strengthened version of Harper's theorem by weakening the finiteness condition to the rank-finiteness condition. We present some interesting applications of the main characterization theorem. We also give a brief history of Sperner theory, and summarize all the ingredients we need for the main theorem and its applications, including a new equivalent condition for the LYM property that is a key for proving our main theorem.

The LYM property of a finite standard graded poset is one of the central notions in Sperner theory. It is known that the product of two finite standard graded posets satisfying the LYM properties may not have the LYM property again. In 1974, Harper proved that if two finite standard graded posets satisfying the LYM properties also satisfy rank logarithmic concavities, then their product also satisfies these two properties. However, Harper's proof is rather non-intuitive. Giving a natural proof of Harper's theorem is one of the goals of this thesis. The main new result of this thesis is a characterization of rank-finite standard graded LYM posets that satisfy rank logarithmic concavities. With this characterization theorem, we are able to give a new, natural proof of Harper's theorem. In fact, we prove a strengthened version of Harper's theorem by weakening the finiteness condition to the rank-finiteness condition. We present some interesting applications of the main characterization theorem. We also give a brief history of Sperner theory, and summarize all the ingredients we need for the main theorem and its applications, including a new equivalent condition for the LYM property that is a key for proving our main theorem.

Growing volume of spam mails have not only decreased the productivity of people but also become a security threat on the Internet. Mail servers should have abilities to filter out spam mails which change time by time precisely and manage increasing spam rules which generated by mail servers automatically and effectively. Most paper only focused on single aspect (especially for spam rule generation) to prevent spam mail. However, in real word, spam prevention is not just applying data mining algorithm for rule generation. To filter out spam mails correctly in a real world, there are still many issues should be considered in addition to spam rule generation. In this paper, we integrate three modules to form a complete anti-spam system, they are spam rule generation module, spam rule reinforcement module and spam rule exchange module. In this paper, rule-based data mining approach is used to generate exchangeable spam rules. The feedback of user¡¦s returns is reinforced spam rule. The distributing spam rules are exchanged through machine-readable XML format. The results of experiment draw the following conclusion: (1) The spam filter can filter out the Chinese mails by analyzing the header characteristics. (2) Rules exchanged among mail improve the spam recall and accuracy of mail servers. (3) Rules reinforced improve the effectiveness of spam rule.

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The main purpose of this thesis is to find the number of convex sets on a product of two totally ordered spaces. We will give formulas to find this number for specific cases and describe a process to obtain this number for all such spaces. In the first chapter we briefly discuss the motivation behind the work presented in this thesis. Also, the definitions and notation used throughout the paper are introduced here The second chapter starts with examining the product spaces of the form {1; 2; : : : ;n} × {1; 2}. That is, we begin by analyzing a two-row by n-column space for n > N. Three separate approaches are discussed, and verified, to find the total number of convex sets on the space. A general formula is presented to obtain this total for all n. In the third chapter we take the same {1; 2; : : : ;n} × {1; 2} spaces from Chapter 2 and consider all the scenarios for adding a second disjoint convex set to the space. Adding a second convex set gives a collection of two mutually disjoint sets. Again, a general formula is presented to obtain this total number of such collections for all n. The fourth chapter takes the idea from Chapter 2 and expands it to product spaces {1; 2; : : : ;n} × {1; 2; : : : ;m} consisting of more than two rows. Here the creation of convex sets having z rows from those having z − 1 rows is exploited to obtain a model that will give the total number of z-row convex sets on any n × m space, provided the set occupies z adjacent rows. Finally, the fifth chapter describes all possible scenarios for convex sets to be placed in the {1; 2; : : : ;n}×{1; 2; : : : ;m} space. This chapter then explains the process needed to acquire a count of all convex sets on any such space as well. Chapter 5 ends by walking through this process with a concrete example, breaking it down into each scenario. We conclude by briefly summarizing the results and specifying future work we would like to further investigate, in Chapter 6.

The first of NASA's high-level strategic goals is to extend and sustain human activities across the solar system. As the United States moves into the post-Shuttle era, meeting this goal is more challenging than ever. There are several desired outcomes for this goal, including development of an integrated architecture and capabilities for safe crewed and cargo missions beyond low Earth orbit. NASA's Flexible Path for the future human exploration of space provides the guidelines to achieve this outcome. Designing space system architectures to satisfy the Flexible Path starts early in design, when a downselection process works to reduce the broad spectrum of feasible system architectures into a more refined set that contains a handful of alternatives that are to be considered and studied further in the detailed design phases. This downselection process is supported by what is referred to as architecture space exploration (ASE). ASE is a systems engineering process which generates the design knowledge necessary to enable informed decision-making. The broad spectrum of potential system architectures can be impractical to evaluate. As the system architecture becomes more complex in its structure and decomposition, its space encounters a factorial growth in the number of alternatives to be considered. This effect is known in the literature as combinatorial explosion. For the Flexible Path, the development of new space system architectures can occur over the period of a decade or more. During this time, a variety of changes can occur which lead to new requirements that necessitate the development of new technologies, or changes in budget and schedule. Developing comprehensive and quantitative design knowledge early during design helps to address these challenges. Current methods focus on a small number of system architecture alternatives. From these alternatives, a series of 'one off' -type of trade studies are performed to refine and generate more design knowledge. These small-scale studies are unable to adequately capture the broad spectrum of possible architectures and typically use qualitative knowledge. The focus of this research is to develop a systems engineering method for system-level ASE during pre-phase A design that is rapid, exhaustive, flexible, traceable, and quantitative. Review of literature found a gap in currents methods that were able to achieve this research objective. This led to the development of the Set Theory-Influenced Architecture Space Exploration (STASE) methodology. The downselection process is modeled as a decision-making process with STASE serving as a supporting systems engineering method. STASE is comprised of two main phases: system decomposition and system synthesis. During system decomposition, the problem is broken down into three system spaces. The architecture space consists of the categorical parameters and decisions that uniquely define an architecture, such as the physical and functional aspects. The design space contains the design parameters that uniquely define individual point designs for a given architecture. The objective space holds the objectives that are used in comparing alternatives. The application of set theory across the system spaces enables an alternative form of representing system alternatives. This novel application of set theory allows the STASE method to mitigate the problem of combinatorial explosion. The fundamental definitions and theorems of set theory are used to form the mathematical basis for the STASE method. A series of hypotheses were formed to develop STASE in a scientific way. These hypotheses are confirmed by experiments using a proof of concept over a subset of the Flexible Path. The STASE method results are compared against baseline results found using the traditional process of representing individual architectures as the system alternatives. The comparisons highlight many advantages of the STASE method. The greatest advantage is that STASE comprehensively explores the architecture space more rapidly than the baseline. This is because the set theory-influenced representation of alternatives has a summation growth with system complexity in the architecture space. The resultant option subsets provide additional design knowledge that enables new ways of visualizing results and comparing alternatives during early design. The option subsets can also account for changes in some requirements and constraints so that new analysis of system alternatives is not required. An example decision-making process was performed for the proof of concept. This notional example starts from the entire architecture space with the goal of minimizing the total cost and the number of launches. Several decisions are made for different architecture parameters using the developed data visualization and manipulation techniques until a complete architecture was determined. The example serves as a use-case example that walks through the implementation of the STASE method, the techniques for analyzing the results, and the steps towards making meaningful architecture decisions.