Contemporary mathematics 

About: Contemporary mathematics is an academic journal. The journal publishes majorly in the area(s): Eigenvalues and eigenvectors & Matrix (mathematics). It has an ISSN identifier of 0271-4132. Over the lifetime, 494 publications have been published receiving 8486 citations.

Quantum Amplitude Amplification and Estimation

Abstract: Consider a Boolean function $\chi: X \to \{0,1\}$ that partitions set $X$ between its good and bad elements, where $x$ is good if $\chi(x)=1$ and bad otherwise. Consider also a quantum algorithm $\mathcal A$ such that $A |0\rangle= \sum_{x\in X} \alpha_x |x\rangle$ is a quantum superposition of the elements of $X$, and let $a$ denote the probability that a good element is produced if $A |0\rangle$ is measured. If we repeat the process of running $A$, measuring the output, and using $\chi$ to check the validity of the result, we shall expect to repeat $1/a$ times on the average before a solution is found. *Amplitude amplification* is a process that allows to find a good $x$ after an expected number of applications of $A$ and its inverse which is proportional to $1/\sqrt{a}$, assuming algorithm $A$ makes no measurements. This is a generalization of Grover's searching algorithm in which $A$ was restricted to producing an equal superposition of all members of $X$ and we had a promise that a single $x$ existed such that $\chi(x)=1$. Our algorithm works whether or not the value of $a$ is known ahead of time. In case the value of $a$ is known, we can find a good $x$ after a number of applications of $A$ and its inverse which is proportional to $1/\sqrt{a}$ even in the worst case. We show that this quadratic speedup can also be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover's and Shor's quantum algorithms to perform amplitude estimation, a process that allows to estimate the value of $a$. We apply amplitude estimation to the problem of *approximate counting*, in which we wish to estimate the number of $x\in X$ such that $\chi(x)=1$. We obtain optimal quantum algorithms in a variety of settings.

The Application of Total Energy as a Lyapunov Function for Mechanical Control Systems

Abstract: Examination of total energy shows that the global limit behavior of a dissipative mechanical system is essentially equivalent to that of its constituent gradient vector field. The class of “navigation functions” is introduced and shown to result in “almost global” asymptotic stability for closed loop mechanical control systems upon which a navigation function has been imposed as an artifical potential energy. Two examples from the engineering literature satellite attitude tracking and robot obstacle avoidance are provided to demonstrate the utility of these observations. For more information: Kod*Lab Disciplines Electrical and Computer Engineering | Engineering | Systems Engineering Comments Preprint version. First published in Control Theory and Multibody Systems , in Volume 97, 1989, pages 131-158, published by the American Mathematical Society in the Contemporary Mathematics series. NOTE: At the time of publication, author Daniel Koditschek was affiliated with Yale University. Currently, he is a faculty member in the Department of Electrical and Systems Engineering at the University of Pennsylvania. This journal article is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/672

A Volterra Type Operator on Spaces of Analytic Functions

Abstract: The main results are conditions on g such that the Volterra type operator Jg(f)(z) = ∫ z

A Weil theorem for singular curves

Abstract: We generalize Weil's theorem on the number of rational points of smooth curves over a finite field to singular ones.