Two Compact Incremental Prime Sieves

A prime sieve is an algorithm that finds the primes up to a bound $n$. We say that a prime sieve is incremental, if it can quickly determine if $n+1$ is prime after having found all primes up to $n$. We say a sieve is compact if it uses roughly $\sqrt{n}$ space or less. In this paper we present two new results: (1) We describe the rolling sieve, a practical, incremental prime sieve that takes $O(n\log\log n)$ time and $O(\sqrt{n}\log n)$ bits of space, and (2) We show how to modify the sieve of Atkin and Bernstein (2004) to obtain a sieve that is simultaneously sublinear, compact, and incremental. The second result solves an open problem given by Paul Pritchard in 1994

Boson sampling with displaced single-photon Fock states versus single-photon-added coherent states---The quantum-classical divide and computational-complexity transitions in linear optics

Boson sampling is a specific quantum computation, which is likely hard to implement efficiently on a classical computer. The task is to sample the output photon number distribution of a linear optical interferometric network, which is fed with single-photon Fock state inputs. A question that has been asked is if the sampling problems associated with any other input quantum states of light (other than the Fock states) to a linear optical network and suitable output detection strategies are also of similar computational complexity as boson sampling. We consider the states that differ from the Fock states by a displacement operation, namely the displaced Fock states and the photon-added coherent states. It is easy to show that the sampling problem associated with displaced single-photon Fock states and a displaced photon number detection scheme is in the same complexity class as boson sampling for all values of displacement. On the other hand, we show that the sampling problem associated with single-photon-added coherent states and the same displaced photon number detection scheme demonstrates a computational complexity transition. It transitions from being just as hard as boson sampling when the input coherent amplitudes are sufficiently small, to a classically simulatable problem in the limit of large coherent amplitudes.Comment: 7 pages, 3 figures; published versio

University Scholar Series: Jonathan Roth

Roman Warfare On April 13, 2011 Jonathan Roth spoke in the University Scholar Series hosted by Provost Gerry Selter at the Dr. Martin Luther King, Jr. Library. Jonathan Roth is a Professor in the History Department at SJSU. In this seminar, he examines the evolution of Roman war over its thousand-year history. He highlights the changing arms and equipment of the soldiers, unit organization and command structure, and the wars and battles of each era.https://scholarworks.sjsu.edu/uss/1008/thumbnail.jp

Fair pay and a Wagebill Argument for Wage Rigidity and Excessive Employment Variability

This paper considers a two-period optimal contracting model in which firms make new hires in the second period subject to the constraint that they cannot pay discriminate either against or in favour of the new hires. Under an assumption on the information available to workers, it is shown that wages are less flexible than needed for efficient employment levels, with the result that too few hires are made in bad states of the world. Unemployment is involuntary. In an extension to the model, there may also be involuntary and excessive layoffs in some states of the world.implicit contract theory, wage rigidity, involuntary unemployment