Formation of the binary pulsars J1141-6545 and B2023+46

The binaries PSR J1141-6545 and PSR B2303+46 each appear to contain a white dwarf which formed before the neutron star. We describe an evolutionary pathway to produce these two systems. In this scenario, the primary transfers its envelope onto the secondary which is then the more massive of the two stars, and indeed sufficiently massive later to produce a neutron star via a supernova. The core of the primary produces a massive white dwarf which enters into a common envelope with the core of the secondary when the latter evolves off the main sequence. During the common envelope phase, the white dwarf and the core of the secondary spiral together as the envelope is ejected. The evolutionary history of PSR J1141-6545 and PSR B2303+46 differ after this phase. In the case of PSR J1141--6545, the secondary (now a helium star) evolves into contact transferring its envelope onto the white dwarf. We propose that the vast majority of this material is in fact ejected from the system. The remains of the secondary then explode as a supernova producing a neutron star. Generally the white dwarf and neutron star will remain bound in tight, often eccentric, systems resembling PSR J1141-6545. These systems will spiral in and merge on a relatively short timescale and may make a significant contribution to the population of gamma ray burst progenitors. In PSR B2303+46, the helium-star secondary and white dwarf never come into contact. Rather the helium star loses its envelope via a wind, which increases the binary separation slightly. Only a small fraction of such systems will remain bound when the neutron star is formed (as the systems are wider). Those systems which are broken up will produce a population of high-velocity white dwarfs and neutron stars.Comment: 9 pages, 10 figures; MNRAS in pres

Random Bit Multilevel Algorithms for Stochastic Differential Equations

We study the approximation of expectations \E(f(X)) for solutions $X$ of SDEs and functionals $f \colon C([0,1],\R^r) \to \R$ by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals $f$ from the Lipschitz class w.r.t.\ the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid for all random bit Monte Carlo algorithms, and we show that, for the given quadrature problem, random bit Monte Carlo algorithms are at least almost as powerful as general randomized algorithms

Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces

We study the approximation of expectations \E(f(X)) for Gaussian random elements $X$ with values in a separable Hilbert space $H$ and Lipschitz continuous functionals $f \colon H \to \R$. We consider restricted Monte Carlo algorithms, which may only use random bits instead of random numbers. We determine the asymptotics (in some cases sharp up to multiplicative constants, in the other cases sharp up to logarithmic factors) of the corresponding $n$-th minimal error in terms of the decay of the eigenvalues of the covariance operator of $X$. It turns out that, within the margins from above, restricted Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms, and suitable random bit multilevel algorithms are optimal. The analysis of this problem leads to a variant of the quantization problem, namely, the optimal approximation of probability measures on $H$ by uniform distributions supported by a given, finite number of points. We determine the asymptotics (up to multiplicative constants) of the error of the best approximation for the one-dimensional standard normal distribution, for Gaussian measures as above, and for scalar autonomous SDEs

Richard Cosway as Teacher at Large

Thimblerigging, hanky-panky, and hugger-mugger : an unforgettable formula from Professor Cosway\u27s class in creditors\u27 rights. He used the formula to illustrate situations in which a court of equity or bankruptcy feels impelled to punish a miscreant somehow, regardless of the literal import of apparently applicable statutes. The same formula serves as a paradigm of Professor Cosway\u27s teaching technique. It is not that he practices the enumerated vices (heaven forbid), but that his use of the formula exemplifies certain qualities that have made his teaching so effective and popular, both in the classroom and as teacher-at-large to the profession

Sales—Automobile Warranties

A defective switch in a new automobile started a fire that destroyed the entire machine. Under the standard warranty given by all automobile manufacturers, the dealer\u27s only obligation was to supply a new switch. By so deciding, Norway v. Root illustrated the worthlessness of the standard warranty. But the importance of the case lies in the Washington court\u27s intimation that it might invalidate disclaimers of implied warranty in automobile sales were the issue squarely presented

Defining International Electronic Commerce

Electronic commerce has become a practical reality for thousands of businesses throughout the world. By combining the functional capabilities of computers and telecommunication systems, companies can now exchange information electronically rather than sending and receiving paper documents. In so doing, businesses are achieving remarkable and unparalleled improvements in the accuracy, speed and efficiency with which commercial transactions may be negotiated, confirmed and performed. By eliminating reliance upon paper as the medium through which commerce occurs, new and radically different approaches are emerging regarding how commercial relationships are defined and maintained. Used for international business transactions, the technologies of electronic commerce are confronting and overcoming traditional barriers to international trade presented by geographic, lingual and cultural disparities between possible trading partners. As a result, facilitating international electronic commerce throughout the global trading community is now fairly considered a project of the highest priority on the international agenda