Sensitivity of Boolean formulas

The sensitivity set of a Boolean function at a particular input is the set of input positions where changing that one bit changes the output. Analogously we define the sensitivity set of a Boolean formula in a conjunctive normal form at a particular truth assignment, it is the set of positions where changing that one bit of the truth assignment changes the evaluation of at least one of the conjunct in the formula. We consider Boolean formulas in a generalized conjunctive normal form. Given a set ?$?$ of Boolean functions, an ?$?$-constraint is an application of a function from ?$?$ to a tuple of literals built upon distinct variables, an ?$?$-formula is then a conjunction of ?$?$-constraints. In this framework, given a truth assignment I$I$ and a set of positions S$S$, we are able to enumerate all ?$?$-formulas that are satisfied by I$I$ and that have S$S$ as the sensitivity set at I$I$. We prove that this number depends on the cardinality of S$S$ only, and can be expressed according to the sensitivity of the Boolean functions in ?$?$.     

Optimal average sample number of the SPRT chart for monitoring fraction nonconforming

The Sequential Probability Ratio Test (SPRT) control chart is a powerful tool for monitoring manufacturing processes. It is highly suitable for the applications where testing is destructive or very expensive, such as the automobile airbags test. This article studies the effect of the Average Sample Number (ASN) (i.e., the average sample size) on the chart’s performance. A design algorithm is proposed to develop the optimal SPRT chart for monitoring the fraction nonconforming p of Bernoulli processes. By optimizing the ASN and other charting parameters, the average detection speed of the SPRT chart is almost doubled. It is also found that the optimal SPRT chart significantly outperforms the optimal np and binomial CUSUM charts, in terms of Average Number of Defectives (AND), under different combinations of the design specifications. It is observed that the SPRT chart using a relatively smaller ASN and a shorter sampling interval (h) has a higher overall detection effectiveness.     

The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup $C{l}_1(\mathbb{Z}P)$ of $S{K}_1(\mathbb{Z}P)$, in terms of a genetic basis of P. We also introduce a deflation map $C{l}_1(\mathbb{Z}P)\to C{l}_1(\mathbb{Z}(P/N))$, for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of $S{K}_1(\mathbb{Z}P)$, when P is an elementary abelian p-group.     

On Oliver’s p-group conjecture: II

Let p be an odd prime and S a finite p-group. B. Oliver’s conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup \mathfrakX(S){\mathfrak{X}(S)} of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver’s conjecture as a statement about the representation theory of the factor group S/\mathfrakX(S){S/\mathfrak{X}(S)}. We now verify the conjecture for a wide variety of groups S/\mathfrakX(S){S/\mathfrak{X}(S)}.     

Convergence of a non-monotone scheme for Hamilton–Jacobi–Bellman equations with discontinous initial data

We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton–Jacobi–Bellman equation of the form v _t + max _α (f(x, α)v _x ) = 0, v(0, x) = v ₀(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1–33, 2007). We show for general discontinuous initial data a first-order convergence of the scheme, in L ¹-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.     

Quantum effects in the differential cross Ssctions for the insertion reaction N(2D) + H2

The quantum (QM) scattering theory has been difficult to apply to the family of insertion reactions and the approximate quasiclassical trajectory (QCT) method or statistical calculations were mostly applied. In this Letter, we compare the experimental differential cross sections for the title insertion reaction with the results of QM and QCT calculations on an ab initio potential energy surface. The QM results reproduce well the crossed beam experiment, while a small, but significant, difference in the QCT ones points to quantum effects, possibly the occurrence of tunneling through the combined potential and centrifugal barrier.     

Solar cells based on a fullerene-azothiophene dyad

A power conversion efficiency of 0.37%, under white light of 80 mW cm-2 intensity, was obtained when a fullerene-azothiophene dyad was used as the active layer of a photovoltaic cell.     

A Simultaneous Discovery: The Case of Johannes Stark and Antonino Lo Surdo

In 1913 the German physicist Johannes Stark (1874–1957) and the Italian physicist Antonino Lo Surdo (1880–1949)discovered virtually simultaneously and independently that hydrogen spectral lines are split into components by an external electric field. Both of their discoveries ensued from studies on the same phenomenon, the Doppler effect in canal rays, but they arose in different theoretical contexts. Stark had been working within the context of the emerging quantum theory, following a research program aimed at studying the effect of an electric field on spectral lines. Lo Surdo had been working within the context of the classical theory, and his was an accidental discovery. Both discoveries, however, played important roles in the history of physics: Starks discovery contributed to the establishment of both the old and the new quantum theories; Lo Surdos discovery led Antonio Garbasso (1871–1933)to introduce research on the quantum theory into Italian physics. Ironically, soon after their discoveries, both Stark and Lo Surdo rejected developments in modern physics and allied themselves with the political and racial programs of Hitler and Mussolini.Matteo Leone is a doctoral student in the history of science at the University of Bari, Italy;his main fields of research are the history of spectroscopy and atomic physics. Alessandro Paoletti is Curator of the Museum of the Department of Physics at the University of Genoa, Italy. Nadia Robotti is Professor of the History of Physics at the University of Genoa; her main fields of research are the history of atomic physics, the old quantum theory, and spectroscopy.     

Dispersion-free pulse propagation in a negative-index material

The possibility of controlling the spectral position of the zero group-velocity dispersion point of a negative-index material can be exploited by varying the ratio between the electric and the magnetic plasma frequency to obtain dispersion-free propagation in spectral regions otherwise inaccessible using conventional positive-index materials. Our predictions are confirmed by pulse propagation simulations where all the orders of the complex dispersion of the material are taken into account.