"As research in the field of mobile robotics continues to advance, legged robots in different forms and shapes find a variety of applications on rough terrain where wheeled robots fail to operate in practice. For this reason, a modular legged robot platform is being developed at WPI. This research focuses on developing a mathematical model and then building a simulation to verify the model for a single leg for this platform. The robot platform is modular in the sense that leg modules can be removed and added to predetermined ports on the robot chassis. The modularity of a legged robot is a significant advancement in mobile robotics technology as it enables a single robot to take on different body configurations depending on circumstances and environment to achieve its goals. It also poses a challenge in terms of overall design as it requires autonomous operation of the leg. The goal for this research is to in part fulfill the need for a mathematical model for an autonomous leg. This research investigates the development of a kinematic and dynamic model for the leg, a step trajectory for walking, a simulation of the system to verify the dynamic model, and various functions and scripts to identify shortcomings within the model. This research uses Mathworks Matlab and Wolfram Mathematica to develop the mathematical model, and Matlab Simulink SimMechanics and Matlab functions to build a simulation. Both the mathematical model and simulation follow the classic design of other legged robots, utilizing Lagrangian dynamics, the Jacobian, and simulation tools. The result is a project that is unique in that it drives a robot leg almost independently with very limited communication to a central controller."

Estágio realizado na Empresa ADIRA, S.A. e orientado pelo Eng.º Tiago Faro / Tese de mestrado integrado. Engenharia Mecãnica. Faculdade de Engenharia. Universidade do Porto. 2010

Modular multiplication is a mathematical operation fundamental to the RSA cryptosystern, a public-key cryptosystem with many applications in privacy, security, and authenticity. However, cryptosecurity requires that the numbers involved be extremely large, typically ranging from 512-1024 bits in length. Calculations on numbers of this magnitude are cumbersome and lengthy; this limits the speed of RSA. This thesis examines the problem of speeding up modular multiplication of large numbers in hardware, using the classical (add-and-shift) multiplication algorithm. The problem is broken down, and it is shown that the primary computational bottleneck occurs in the modular reduction step performed on each cycle. This reduction consists of an integer division step, a broadcast step, and a multiplication step. Various methods of speeding up these steps are examined, both for the special case of radix-2 multipliers (those shifting a single bit at a time) and the general case of radix-2r multipliers (those shifting r bits on every cycle.) The impacts of these techniques, both on cycle time and on chip area, are discussed. The scalability of these systems is examined, and several implementations of modular multiplication found in the literature are analyzed. Most significantly, the technique of pipelining of modular multipliers is examined. It is shown that it is possible to pipeline the modular reduction sequence, effectively eliminating the cycle time's dependence on either the size of the modulus, or on the size of the radius. Furthermore, a technique for constructing such multipliers is given. It is demonstrated that this technique is scalable with respect to time, and that pipelining eliminates many of the disadvantages inherent in previous high-radix implementations. It is also demonstrated that such multipliers have an area requirement which is linear with respect to both radix and modulus size. / Graduation date: 1995

In general, to compute the modular inverse is very important in information security, many encrypt/decrypt and signature algorithms always need to use it. In 2007, Liu, Horng, and Liu proposed a variation on Euclidean algorithm, which can calculate the modular inverses as simple as calculate GCDs. This paper analyzes another type of left-shift binary GCD algorithm, which is suitable for the variation and that needs the fewer bit-operations than LSBGCD, which is analyzed by Shallit, and Sorenson.

Complex systems are naturally understood as combinations of their distinguishing characteristics or \definit{features}. Distinct features differentiate between variations of configurable systems and also identify the novelties of extensions. The implementation of a conceptual feature is often scattered throughout an artifact, forcing designers to understand the entire artifact in order to reason about the behavior of a single feature. It is particularly challenging to independently develop novel extensions to complex systems as a result. This dissertation shows how to modularly reason about the implementation of conceptual features in both the formalizations of programming languages and object-oriented software product lines. In both domains, modular verification of features can be leveraged to reason about the behavior of artifacts in which they are included: fully mechanized metatheory proofs for programming languages can be synthesized from independently developed proofs, and programs built from well-formed feature modules are guaranteed to be well-formed without needing to be typechecked. Modular reasoning about individual features can furthermore be used to efficiently reason about families of languages and programs which share a common set of features. / text

No description available.

As a generalization of the Riemann zeta function, L-function has become one of the central objects in Number Theory. The theory of L-functions, which produces a large family of consequences and conjectures in a unified way, concerns their zeros and poles, functional equations, special values and the connections between objects in different fields. Although most generalizations are largely conjectural, there are many existing results that provide us the evidence. In this thesis, we shall consider some L-functions and look into some problems mentioned above. More explicitly, for the L-functions associated to newforms of fixed square-free level, we will consider an average version of the fourth moments problem. The final bound is proven by considering definite rational quaternion algebras and divisor functions in them, generalizing Maass Correspondence Theorem and one of Duke's results and eventually applying the solution to Basis Problem. We then consider the problem of expressing the central value at 1/2 of the Rankin-Selberg L-function associated to two newforms in terms of the Pertersson inner product, where one of the newforms is twisted by the derivative of some Eisenstein series. Finally, we consider the Artin L-functions attached to irreducible $4$-dimensional $S_5$-Galois representations and deal with the modularity problem. One sufficient condition on the modularity is given, which may help to find an affirmative example for Strong Artin Conjecture in this case.

As a generalization of the Riemann zeta function, L-function has become one of the central objects in Number Theory. The theory of L-functions, which produces a large family of consequences and conjectures in a unified way, concerns their zeros and poles, functional equations, special values and the connections between objects in different fields. Although most generalizations are largely conjectural, there are many existing results that provide us the evidence. In this thesis, we shall consider some L-functions and look into some problems mentioned above. More explicitly, for the L-functions associated to newforms of fixed square-free level, we will consider an average version of the fourth moments problem. The final bound is proven by considering definite rational quaternion algebras and divisor functions in them, generalizing Maass Correspondence Theorem and one of Duke's results and eventually applying the solution to Basis Problem. We then consider the problem of expressing the central value at 1/2 of the Rankin-Selberg L-function associated to two newforms in terms of the Pertersson inner product, where one of the newforms is twisted by the derivative of some Eisenstein series. Finally, we consider the Artin L-functions attached to irreducible $4$-dimensional $S_5$-Galois representations and deal with the modularity problem. One sufficient condition on the modularity is given, which may help to find an affirmative example for Strong Artin Conjecture in this case.

Thesis (Ph.D.) -- University of Adelaide, Dept. of Electrical Engineering, 1979. / Typescript (photocopy).

Thesis (M.S.)--Rochester Institute of Technology, 1988. / Typescript. Includes bibliographical references (leaves 75-77).