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In this note, we study the inverse of the Abel transform for the symmetric space ${\bf SU}^{\star}(2n)/{\bf Sp}(n)$. We start by giving a recursive formula for the dual of the Abel transform on the root system $A_{n-1}$. This formula allows us to consider a transmutation property on the generalized Abel transform associated to $A_{n-1}$. / This paper has not been published. However, it has been cited in peer reviewed venues.

We consider the problem of counting <i>transitive factorizations</i> of permutations; that is, we study tuples (&sigma;<i>r</i>,. . . ,&sigma;1) of permutations on {1,. . . ,<i>n</i>} such that (1) the product &sigma;<i>r</i>. . . &sigma;1 is equal to a given target permutation &pi;, and (2) the group generated by the factors &sigma;<i>i</i> acts transitively on {1,. . . ,<i>n</i>}. This problem is widely known as the <i>Hurwitz Enumeration Problem</i>, since an encoding due to Hurwitz shows it to be equivalent to the enumeration of connected branched coverings of the sphere by a surface of given genus with specified branching. Much of our work concerns the enumeration of transitive factorizations of permutations into a minimal number of transposition factors. This problem has received considerable attention, and a formula for the number <i>c</i>(&pi;) of such factorizations of an arbitrary permutation &pi; has been derived through various means. The formula is remarkably simple, being a product of well-known combinatorial numbers, but no bijective proof of it is known except in the special case where &pi; is a full cycle. A major goal of this thesis is to provide further combinatorial rationale for this formula. We begin by introducing an encoding of factorizations (into transpositions) as edge-labelled maps. Our central result is a bijection that allows trees to be "pruned" from such maps. This is shown to explain the appearance of factors of the form <i>k^k</i> in the aforementioned formula for <i>c</i>(&pi;). It also has the effect of shifting focus to the combinatorics of smooth maps (<i>i. e. </i> maps without vertices of degree one). By providing decompositions for certain smooth planar maps, we are able to give combinatorial evaluations of <i>c</i>(&pi;) when &pi; is composed of up to three cycles. Many of these results are generalized to factorizations in which the factors are cycles of any length. We also investigate the <i>Double Hurwitz Problem</i>, which calls for the enumeration of factorizations whose leftmost factor is of specified cycle type, and whose remaining factors are transpositions. Finally, we extend our methods to the enumeration of factorizations up to an equivalence relation induced by possible commutations between adjacent factors.

We consider the problem of counting <i>transitive factorizations</i> of permutations; that is, we study tuples (&sigma;<i>r</i>,. . . ,&sigma;1) of permutations on {1,. . . ,<i>n</i>} such that (1) the product &sigma;<i>r</i>. . . &sigma;1 is equal to a given target permutation &pi;, and (2) the group generated by the factors &sigma;<i>i</i> acts transitively on {1,. . . ,<i>n</i>}. This problem is widely known as the <i>Hurwitz Enumeration Problem</i>, since an encoding due to Hurwitz shows it to be equivalent to the enumeration of connected branched coverings of the sphere by a surface of given genus with specified branching. Much of our work concerns the enumeration of transitive factorizations of permutations into a minimal number of transposition factors. This problem has received considerable attention, and a formula for the number <i>c</i>(&pi;) of such factorizations of an arbitrary permutation &pi; has been derived through various means. The formula is remarkably simple, being a product of well-known combinatorial numbers, but no bijective proof of it is known except in the special case where &pi; is a full cycle. A major goal of this thesis is to provide further combinatorial rationale for this formula. We begin by introducing an encoding of factorizations (into transpositions) as edge-labelled maps. Our central result is a bijection that allows trees to be "pruned" from such maps. This is shown to explain the appearance of factors of the form <i>k^k</i> in the aforementioned formula for <i>c</i>(&pi;). It also has the effect of shifting focus to the combinatorics of smooth maps (<i>i. e. </i> maps without vertices of degree one). By providing decompositions for certain smooth planar maps, we are able to give combinatorial evaluations of <i>c</i>(&pi;) when &pi; is composed of up to three cycles. Many of these results are generalized to factorizations in which the factors are cycles of any length. We also investigate the <i>Double Hurwitz Problem</i>, which calls for the enumeration of factorizations whose leftmost factor is of specified cycle type, and whose remaining factors are transpositions. Finally, we extend our methods to the enumeration of factorizations up to an equivalence relation induced by possible commutations between adjacent factors.

RC4 is one of the most widely used ciphers in practical software applications. In this thesis we examine security and design aspects of RC4. First we describe the functioning of RC4 and present previously published analyses. We then present a new cipher, Chameleon which uses a similar internal organization to RC4 but uses different methods. The remainder of the thesis uses ideas from both Chameleon and RC4 to develop design strategies for new ciphers. In particular, we develop a new cipher, RC4B, with the goal of greater security with an algorithm comparable in simplicity to RC4. We also present design strategies for ciphers and two new ciphers for 32-bit processors. Finally we present versions of Chameleon and RC4B that are implemented using playing-cards.

The Besov class $B_{pq}^s$ is defined by ${ f : { 2^{|n|s}||W_n*f||_p } _{ninmathbb{Z}}in ell^q(mathbb{Z}) }$. When $s=1$, $p=q $, we know if $f in B_p$ if and only if $int_mathbb{D} |f^{(n)}(z)|^p(1-|z|^2)^{2pn-2}dv(z) <+infty$. It is shown in [5] that $int_{mathbb{D}}|f^{'}(z)|^q K(z,z)^{1-q}dv(z)= O(L(b(e^{-(q-p)^{-1}})))$ if $f in B_{L,p}$. In this paper we will show that $f in B_{L,p}$ if and only if $sum_{n=0}^{infty}2^{nq}||W_n*f||_p^q = O(L(b(e^{-(q-p)^{-1}})))$.

A strategy was proposed to make machining rate stable and the machining precision achieved by properly tool dwelling time when surface still has form error after previously machining. Using computer simulation to plan tool dwelling time and to estimate practicability of this strategy. As a result of curvatures are different on the every points of the work piece surface. Normal vectors that between tool and work pieces surface are not stable in polishing process.HDP conditions and film thickness will be changed by curvature radius of work pieces.So HDP conditions must be controlled when the planning of tool motion. Analyzing all of different aspheric surfaces to make sure this strategy can be used. The different thing that between axially symmetric and axially non-symmetric is tool dwelling time should be a linear function the product of the depth function of profile and the radius for symmetric work pieces, but that of axially non-symmetric work pieces only should be linearly proportional to the depth function of profile.

Thesis (Ph. D.)--University of Hong Kong, 2000. / Includes bibliographical references (leaves 50-53).