text:
Question 1: Evaluate:

i) $3 \begin{bmatrix} 5 \\ -2 \end{bmatrix} = \begin{bmatrix} 15 \\ -6 \end{bmatrix}$

ii) $7 \begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} -7 & 14 \\ 0 & 7 \end{bmatrix}$

iii) $2 \begin{bmatrix} -1 & 0 \\ 2 & -3 \end{bmatrix} + \begin{bmatrix} 3 & 3 \\ 5 & 0 \end{bmatrix} = \begin{bmatrix} -2 & 0 \\ 4 & -6 \end{bmatrix} + \begin{bmatrix} 3 & 3 \\ 5 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 9 & -6 \end{bmatrix}$

iv) $6 \begin{bmatrix} 3 \\ 2 \end{bmatrix} - 2 \begin{bmatrix} -8 \\ 1 \end{bmatrix} = \begin{bmatrix} 18 \\ 12 \end{bmatrix} - \begin{bmatrix} -16 \\ 2 \end{bmatrix} = \begin{bmatrix} 34 \\ -14 \end{bmatrix}$

$\\$

Question 2: Find $x \ and \ y$ if

i) $3 \begin{bmatrix} 4 & x \end{bmatrix} +2 \begin{bmatrix} y & -3 \end{bmatrix} = \begin{bmatrix} 10 & 0 \end{bmatrix}$

ii) $x \begin{bmatrix} -1 \\ 2 \end{bmatrix} -4 \begin{bmatrix} -2 \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ -8 \end{bmatrix}$

i) $3 \begin{bmatrix} 4 & x \end{bmatrix} +2 \begin{bmatrix} y & -3 \end{bmatrix} = \begin{bmatrix} 10 & 0 \end{bmatrix}$

$\Rightarrow \begin{bmatrix} 12 & 3x \end{bmatrix} + \begin{bmatrix} 2y & -6 \end{bmatrix} = \begin{bmatrix} 10 & 0 \end{bmatrix}$

Therefore

$12+2y = \ 10 \ or y = -1$

$3x-6=0 \ or \ x = 2$

ii) $x \begin{bmatrix} -1 \\ 2 \end{bmatrix} -4 \begin{bmatrix} -2 \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ -8 \end{bmatrix}$

$\begin{bmatrix} -x \\ 2x \end{bmatrix} - \begin{bmatrix} -8 \\ 4y \end{bmatrix} = \begin{bmatrix} 7 \\ -8 \end{bmatrix}$

Therefore

$-x+8 = 7 \ or \ x = 1$

$2x-4y = -8 \ or \ y = 10$

$\\$

Question 3: Given $A = \begin{bmatrix} 2 & 1 \\ 3 & 0 \end{bmatrix}$$B = \begin{bmatrix} 1 & 1 \\ 5 & 2 \end{bmatrix}$$C = \begin{bmatrix} -3 & -1 \\ 0 & 0 \end{bmatrix}$, Find:

i)  $2A-3B+C$

ii)  $A+2C-B$

i)  $2A-3B+C$

$2 \begin{bmatrix} 2 & 1 \\ 3 & 0 \end{bmatrix} - 3 \begin{bmatrix} 1 & 1 \\ 5 & 2 \end{bmatrix} + \begin{bmatrix} -3 & -1 \\ 0 & 0 \end{bmatrix}$

$= \begin{bmatrix} 4 & 2 \\ 6 & 0 \end{bmatrix} - \begin{bmatrix} 3 & 3 \\ 15 & 6 \end{bmatrix} + \begin{bmatrix} -3 & -1 \\ 0 & 0 \end{bmatrix}$

$= \begin{bmatrix} -2 & -2 \\ -9 & -6 \end{bmatrix}$

ii)  $A+2C-B$

$\begin{bmatrix} 2 & 1 \\ 3 & 0 \end{bmatrix} + 2 \begin{bmatrix} -3 & -1 \\ 0 & 0 \end{bmatrix} - \begin{bmatrix} 1 & 1 \\ 5 & 2 \end{bmatrix}$

$= \begin{bmatrix} 2 & 1 \\ 3 & 0 \end{bmatrix} + \begin{bmatrix} -6 & -2 \\ 0 & 0 \end{bmatrix} - \begin{bmatrix} 1 & 1 \\ 5 & 2 \end{bmatrix}$

$= \begin{bmatrix} -5 & -2 \\ -2 & -2 \end{bmatrix}$

$\\$

Question 4: If  $\begin{bmatrix} 4 & -2 \\ 4 & 0 \end{bmatrix} +3A = \begin{bmatrix} -2 & -2 \\ 1 & -3 \end{bmatrix}$, Find $A$

$\begin{bmatrix} 4 & -2 \\ 4 & 0 \end{bmatrix} +3A = \begin{bmatrix} -2 & -2 \\ 1 & -3 \end{bmatrix}$

$\Rightarrow 3A = \begin{bmatrix} -2 & -2 \\ 1 & -3 \end{bmatrix} - \begin{bmatrix} 4 & -2 \\ 4 & 0 \end{bmatrix}$

$\Rightarrow 3A = \begin{bmatrix} -6 & 0 \\ -3 & -3 \end{bmatrix}$

$\Rightarrow A = \begin{bmatrix} -2 & 0 \\ -1 & -1 \end{bmatrix}$

$\\$

Question 5: Given $A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix}$,  $B = \begin{bmatrix} -4 & -1 \\ -3 & -2 \end{bmatrix}$, Find

i) $2A+B$

ii) Matrix $C$ such that $C+B = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

i) $2A+B$

$\Rightarrow 2 \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} + \begin{bmatrix} -4 & -1 \\ -3 & -2 \end{bmatrix} = \begin{bmatrix} -2 & 7 \\ 1 & 4 \end{bmatrix}$

ii) $C+B = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

$C = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} - \begin{bmatrix} -4 & -1 \\ -3 & -2 \end{bmatrix}$

$C = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$

$\\$

Question 6: If $2 \begin{bmatrix} 3 & x \\ 0 & 1 \end{bmatrix} + 3 \begin{bmatrix} 1 & 3 \\ y & 2 \end{bmatrix} = \begin{bmatrix} z & -7 \\ 15 & 8 \end{bmatrix}$, Find the value of $x, \ y, \ and \ z$.

$2 \begin{bmatrix} 3 & x \\ 0 & 1 \end{bmatrix} + 3 \begin{bmatrix} 1 & 3 \\ y & 2 \end{bmatrix} = \begin{bmatrix} z & -7 \\ 15 & 8 \end{bmatrix}$

$\Rightarrow \begin{bmatrix} 6 & 2x \\ 0 & 2 \end{bmatrix} + \begin{bmatrix} 3 & 9 \\ 3y & 6 \end{bmatrix} = \begin{bmatrix} z & -7 \\ 15 & 8 \end{bmatrix}$

$\Rightarrow \begin{bmatrix} 9 & 2x+9 \\ 3y & 8 \end{bmatrix}= \begin{bmatrix} z & -7 \\ 15 & 8 \end{bmatrix}$

Therefore

$z=9$

$2x+9 = -7 \ or \ x = -8$

$3y = 15 \ or \ y = 5$

$\\$

Question 7: Given $A = \begin{bmatrix} -3 & 6 \\ 0 & -9 \end{bmatrix}$ and $A^t$ is the transpose matrix. Find:

i) $2A + 3A^t$

ii) $2A^t-3A$

iii) $\frac{1}{2} A - \frac{1}{3}A^t$

iv) $A^t - \frac{1}{3}A$

If $A = \begin{bmatrix} -3 & 6 \\ 0 & -9 \end{bmatrix}$

Then $A^t = \begin{bmatrix} -3 & 0 \\ 6 & -9 \end{bmatrix}$

i) $2A + 3A^t$

$2 \begin{bmatrix} -3 & 6 \\ 0 & -9 \end{bmatrix} +3 \begin{bmatrix} -3 & 0 \\ 6 & -9 \end{bmatrix} = \begin{bmatrix} -15 & 12 \\ 18 & -45 \end{bmatrix}$

ii) $2A^t-3A$

$2 \begin{bmatrix} -3 & 0 \\ 6 & -9 \end{bmatrix} - 3 \begin{bmatrix} -3 & 6 \\ 0 & -9 \end{bmatrix} = \begin{bmatrix} 3 & -18 \\ 12 & 9 \end{bmatrix}$

iii) $\frac{1}{2} A - \frac{1}{3}A^t$

$\frac{1}{2}\begin{bmatrix} -3 & 6 \\ 0 & -9 \end{bmatrix} - \frac{1}{3}\begin{bmatrix} -3 & 0 \\ 6 & -9 \end{bmatrix} =\begin{bmatrix} -\frac{1}{2} & 3 \\ -2 & -\frac{3}{2} \end{bmatrix}$

iv) $A^t - \frac{1}{3}A$

$\begin{bmatrix} -3 & 0 \\ 6 & -9 \end{bmatrix}-\frac{1}{3} \begin{bmatrix} -3 & 6 \\ 0 & -9 \end{bmatrix}= \begin{bmatrix} -2 & -2 \\ 6 & -6 \end{bmatrix}$

$\\$

Question 8: Given $A = \begin{bmatrix} 1 & 1 \\ -2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}$. Solve for

i) $X+2A=B$

ii) $3X+B+2A=0$

iii) $3A-2X=X-2B$

i)     $X+2A=B$

$X = B-2A$

$X= \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}-2 \begin{bmatrix} 1 & 1 \\ -2 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -3 \\ 5 & 1 \end{bmatrix}$

ii)    $3X+B+2A=0$

$X = \frac{-1}{3}(B+2A)$

$X = \frac{-1}{3}(\begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}+2\begin{bmatrix} 1 & 1 \\ -2 & 0 \end{bmatrix}) = \begin{bmatrix} \frac{4}{3} & \frac{1}{3} \\ -1 & \frac{1}{3} \end{bmatrix}$

iii)   $3A-2X=X-2B$

$X = \frac{1}{3} (3A+2B)$

$X = \frac{1}{3} (3\begin{bmatrix} 1 & 1 \\ -2 & 0 \end{bmatrix} +2\begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}) = \begin{bmatrix} \frac{-7}{3} & \frac{1}{3} \\ \frac{-4}{3} & \frac{2}{3} \end{bmatrix}$

$\\$

Question 9: If $M = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$  and  $N = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$, show that $3M+5N=\begin{bmatrix} 5 \\ 3 \end{bmatrix}$

$3M+5N=3\begin{bmatrix} 0 \\ 1 \end{bmatrix} +5\begin{bmatrix} 0 \\ 1 \end{bmatrix} =\begin{bmatrix} 5 \\ 3 \end{bmatrix}$

Hence proved.

Question 10: If $I$ is the unit matrix of order $2 \times 2$, find the matrix $M$ such that

i) $M-2I= 3\begin{bmatrix} -1 & 0 \\ 4 & 1 \end{bmatrix}$

ii) $5M+3I= 4\begin{bmatrix} 2 & -5 \\ 0 & -3 \end{bmatrix}$

Given $I$ is a unit matrix of order $2 \times 2$, we have

$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

i)  $M = 2\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + 3\begin{bmatrix} -1 & 0 \\ 4 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 12 & 5 \end{bmatrix}$

ii) $M= \frac{1}{5} (-3\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + 4\begin{bmatrix} 2 & -5 \\ 0 & -3 \end{bmatrix}) =\begin{bmatrix} 1 & -4 \\ 0 & -3 \end{bmatrix}$

$\\$

Question 11. If $\begin{bmatrix} 1 & 4 \\ -2 & 3 \end{bmatrix} +2M = 3 \begin{bmatrix} 3 & 2 \\ 0 & -3 \end{bmatrix}$, find the matrix $M$

$2M =3 \begin{bmatrix} 3 & 2 \\ 0 & -3 \end{bmatrix} - \begin{bmatrix} 1 & 4 \\ -2 & 3 \end{bmatrix}$
$2M =\begin{bmatrix} 9 & 6 \\ 0 & -9 \end{bmatrix} - \begin{bmatrix} 1 & 4 \\ -2 & 3 \end{bmatrix} = \begin{bmatrix} 8 & 2 \\ 2 & -12 \end{bmatrix}$
Therefore $M =\begin{bmatrix} 4 & 1 \\ 1 & -6 \end{bmatrix}$
$\\$


text: References of "Scarpetta, G"      in Complete repository Arts & humanities   Archaeology   Art & art history   Classical & oriental studies   History   Languages & linguistics   Literature   Performing arts   Philosophy & ethics   Religion & theology   Multidisciplinary, general & others Business & economic sciences   Accounting & auditing   Production, distribution & supply chain management   Finance   General management & organizational theory   Human resources management   Management information systems   Marketing   Strategy & innovation   Quantitative methods in economics & management   General economics & history of economic thought   International economics   Macroeconomics & monetary economics   Microeconomics   Economic systems & public economics   Social economics   Special economic topics (health, labor, transportation…)   Multidisciplinary, general & others Engineering, computing & technology   Aerospace & aeronautics engineering   Architecture   Chemical engineering   Civil engineering   Computer science   Electrical & electronics engineering   Energy   Geological, petroleum & mining engineering   Materials science & engineering   Mechanical engineering   Multidisciplinary, general & others Human health sciences   Alternative medicine   Anesthesia & intensive care   Cardiovascular & respiratory systems   Dentistry & oral medicine   Dermatology   Endocrinology, metabolism & nutrition   Forensic medicine   Gastroenterology & hepatology   General & internal medicine   Geriatrics   Hematology   Immunology & infectious disease   Laboratory medicine & medical technology   Neurology   Oncology   Ophthalmology   Orthopedics, rehabilitation & sports medicine   Otolaryngology   Pediatrics   Pharmacy, pharmacology & toxicology   Psychiatry   Public health, health care sciences & services   Radiology, nuclear medicine & imaging   Reproductive medicine (gynecology, andrology, obstetrics)   Rheumatology   Surgery   Urology & nephrology   Multidisciplinary, general & others Law, criminology & political science   Civil law   Criminal law & procedure   Criminology   Economic & commercial law   European & international law   Judicial law   Metalaw, Roman law, history of law & comparative law   Political science, public administration & international relations   Public law   Social law   Tax law   Multidisciplinary, general & others Life sciences   Agriculture & agronomy   Anatomy (cytology, histology, embryology...) & physiology   Animal production & animal husbandry   Aquatic sciences & oceanology   Biochemistry, biophysics & molecular biology   Biotechnology   Entomology & pest control   Environmental sciences & ecology   Food science   Genetics & genetic processes   Microbiology   Phytobiology (plant sciences, forestry, mycology...)   Veterinary medicine & animal health   Zoology   Multidisciplinary, general & others Physical, chemical, mathematical & earth Sciences   Chemistry   Earth sciences & physical geography   Mathematics   Physics   Space science, astronomy & astrophysics   Multidisciplinary, general & others Social & behavioral sciences, psychology   Animal psychology, ethology & psychobiology   Anthropology   Communication & mass media   Education & instruction   Human geography & demography   Library & information sciences   Neurosciences & behavior   Regional & inter-regional studies   Social work & social policy   Sociology & social sciences   Social, industrial & organizational psychology   Theoretical & cognitive psychology   Treatment & clinical psychology   Multidisciplinary, general & others     Showing results 1 to 20 of 64 1 2 3 4     OGLE-2014-BLG-1186: gravitational microlensing providing evidence for a planet orbiting the foreground star or for a close binary source?Dominik, M.; Bachelet, E.; Bozza, V. et alin Monthly Notices of the Royal Astronomical Society (2019), 484Discussing the particularly long gravitational microlensing event OGLE-2014-BLG-1186 with a time-scale t[SUB]E[/SUB] ˜ 300 d, we present a methodology for identifying the nature of localised deviations ... [more ▼]Discussing the particularly long gravitational microlensing event OGLE-2014-BLG-1186 with a time-scale t[SUB]E[/SUB] ˜ 300 d, we present a methodology for identifying the nature of localised deviations from single-lens point-source light curves, which ensures that (1) the claimed signal is substantially above the noise floor, (2) the inferred properties are robustly determined and their estimation is not subject to confusion with systematic noise in the photometry, (3) alternative viable solutions within the model framework are not missed. Annual parallax and binarity could be separated and robustly measured from the wing and the peak data, respectively. We find matching model light curves that involve either a binary lens or a binary source, and discover hitherto unknown model ambiguities. Our binary-lens models indicate a planet of mass M[SUB]2[/SUB] = (45 ± 9) M[SUB]⊕[/SUB], orbiting a star of mass M[SUB]1[/SUB] = (0.35 ± 0.06) M[SUB]⊙[/SUB], located at a distance D[SUB]L[/SUB] = (1.7 ± 0.3) kpc from Earth, whereas our binary-source models suggest a brown-dwarf lens of M = (0.046 ± 0.007) M[SUB]⊙[/SUB], located at a distance D[SUB]L[/SUB] = (5.7 ± 0.9) kpc, with the source potentially being a (partially) eclipsing binary involving stars predicted to be of similar colour given the ratios between the luminosities and radii. Further observations might resolve the ambiguity in the interpretation in favour of either a lens or a source binary. We experienced that close binary source stars pose a challenge for claiming the detection of planets by microlensing in events where the source passes very close to the lens star hosting the planet. [less ▲]Detailed reference viewed: 27 (4 ULiège) OGLE-2014-BLG-1186: gravitational microlensing providing evidence for a planet orbiting the foreground star or for a close binary source?Dominik, M.; Bachelet, E.; Bozza, V. et alin arXiv e-prints (2018), 1808(abridged) Using the particularly long gravitational microlensing event OGLE-2014-BLG-1186 with a time-scale $t_\mathrm{E}$ ~ 300 d, we present a methodology for identifying the nature of localised ... [more ▼](abridged) Using the particularly long gravitational microlensing event OGLE-2014-BLG-1186 with a time-scale $t_\mathrm{E}$ ~ 300 d, we present a methodology for identifying the nature of localised deviations from single-lens point-source light curves, which ensures that 1) the claimed signal is substantially above the noise floor, 2) the inferred properties are robustly determined and their estimation not subject to confusion with systematic noise in the photometry, 3) there are no alternative viable solutions within the model framework that might have been missed. Annual parallax and binarity could be separated and robustly measured from the wing and the peak data, respectively. We find matching model light curves that involve either a binary lens or a binary source. Our binary-lens models indicate a planet of mass $M_2$ = (45 $\pm$ 9) $M_\oplus$, orbiting a star of mass $M_1$ = (0.35 $\pm$ 0.06) $M_\odot$, located at a distance $D_\mathrm{L}$ = (1.7 $\pm$ 0.3) kpc from Earth, whereas our binary-source models suggest a brown-dwarf lens of $M$ = (0.046 $\pm$ 0.007) $M_\odot$, located at a distance $D_\mathrm{L}$ = (5.7 $\pm$ 0.9) kpc, with the source potentially being a (partially) eclipsing binary involving stars predicted to be of similar colour given the ratios between the luminosities and radii. The ambiguity in the interpretation would be resolved in favour of a lens binary by observing the luminous lens star separating from the source at the predicted proper motion of $\mu$ = (1.6 $\pm$ 0.3) mas yr$^{-1}$, whereas it would be resolved in favour of a source binary if the source could be shown to be a (partially) eclipsing binary matching the obtained model parameters. We experienced that close binary source stars pose a challenge for claiming the detection of planets by microlensing in events where the source passes very close to the lens star hosting the planet. [less ▲]Detailed reference viewed: 16 (3 ULiège) VizieR Online Data Catalog: WASP-22, WASP-41, WASP-42, WASP-55 (Southworth+, 2016)Southworth, J.; Tregloan-Reed, J.; Andersen, M. I. et alin Monthly Notices of the Royal Astronomical Society (2018), 74517 light curves of transits of the extrasolar planetary systems WASP-22, WASP-41, WASP-42 and WASP-55 are presented. 13 of the light curves were obtained using the Danish 1.54m telescope at ESO La Silla ... [more ▼]17 light curves of transits of the extrasolar planetary systems WASP-22, WASP-41, WASP-42 and WASP-55 are presented. 13 of the light curves were obtained using the Danish 1.54m telescope at ESO La Silla, Chile, in the Bessell R or Bessell I passbands. The other 4 light curves were obtained using the 84cm telescope at Observatorio Cerro Armazones, Chile, using either an R filter or no filter. The errorbars for each transit have been scaled so the best-fitting model (obtained using the JKTEBOP code) has a reduced chi-squared value of 1.0. (4 data files). [less ▲]Detailed reference viewed: 17 (2 ULiège) MiNDSTEp differential photometry of the gravitationally lensed quasars WFI 2033-4723 and HE 0047-1756: Microlensing and a new time delayGiannini, E.; Schmidt, R. W.; Wambsganss, J. et alin Astronomy and Astrophysics (2017), 597Aims. We present V and R photometry of the gravitationally lensed quasars WFI 2033-4723 and HE 0047-1756. The data were taken by the MiNDSTEp collaboration with the 1.54 m Danish telescope at the ESO La ... [more ▼]Aims. We present V and R photometry of the gravitationally lensed quasars WFI 2033-4723 and HE 0047-1756. The data were taken by the MiNDSTEp collaboration with the 1.54 m Danish telescope at the ESO La Silla observatory from 2008 to 2012. Methods. Differential photometry has been carried out using the image subtraction method as implemented in the HOTPAnTS package, additionally using GALFIT for quasar photometry. Results. The quasar WFI 2033-4723 showed brightness variations of order 0.5 mag in V and R during the campaign. The two lensed components of quasar HE 0047-1756 varied by 0.2-0.3 mag within five years. We provide, for the first time, an estimate of the time delay of component B with respect to A of Δt = (7.6 ± 1.8) days for this object. We also find evidence for a secular evolution of the magnitude difference between components A and B in both filters, which we explain as due to a long-duration microlensing event. Finally we find that both quasars WFI 2033-4723 and HE 0047-1756 become bluer when brighter, which is consistent with previous studies. © ESO, 2016. [less ▲]Detailed reference viewed: 37 (4 ULiège) Faint-source-star planetary microlensing: The discovery of the cold gas-giant planet OGLE-2014-BLG-0676LbRattenbury, N. J.; Bennett, D. P.; Sumi, T. et alin Monthly Notices of the Royal Astronomical Society (2017), 466(3), 2710-2717We report the discovery of a planet - OGLE-2014-BLG-0676Lb- via gravitational microlensing. Observations for the lensing event were made by the following groups: Microlensing Observations in Astrophysics ... [more ▼]We report the discovery of a planet - OGLE-2014-BLG-0676Lb- via gravitational microlensing. Observations for the lensing event were made by the following groups: Microlensing Observations in Astrophysics; Optical Gravitational Lensing Experiment; Wise Observatory; RoboNET/Las Cumbres Observatory Global Telescope; Microlensing Network for the Detection of Small Terrestrial Exoplanets; and μ-FUN. All analyses of the light-curve data favour a lens system comprising a planetary mass orbiting a host star. The most-favoured binary lens model has a mass ratio between the two lens masses of (4.78 ± 0.13) × 10-3. Subject to some important assumptions, a Bayesian probability density analysis suggests the lens system comprises a 3.09-1.12+1.02 MJ planet orbiting a 0.62-0.22+0.20 M⊙ host star at a deprojected orbital separation of 4.40-1.46+2.16 au. The distance to the lens system is 2.22-0.83+0.96 kpc. Planet OGLE- 2014-BLG-0676Lb provides additional data to the growing number of cool planets discovered using gravitational microlensing against which planetary formation theories may be tested. Most of the light in the baseline of this event is expected to come from the lens and thus high-resolution imaging observations could confirm our planetary model interpretation. © 2016 The Authors. [less ▲]Detailed reference viewed: 32 (3 ULiège) High-precision photometry by telescope defocussing - VIII.WASP-22, WASP-41,WASP-42 andWASP-55Southworth, J.; Tregloan-Reed, J.; Andersen, M. I. et alin Monthly Notices of the Royal Astronomical Society (2016), 457(4), 4205-4217We present 13 high-precision and four additional light curves of four bright southernhemisphere transiting planetary systems: WASP-22, WASP-41, WASP-42 and WASP-55. In the cases of WASP-42 and WASP-55 ... [more ▼]We present 13 high-precision and four additional light curves of four bright southernhemisphere transiting planetary systems: WASP-22, WASP-41, WASP-42 and WASP-55. In the cases of WASP-42 and WASP-55, these are the first follow-up observations since their discovery papers. We present refined measurements of the physical properties and orbital ephemerides of all four systems. No indications of transit timing variations were seen. All four planets have radii inflated above those expected from theoretical models of gas-giant planets; WASP-55 b is the most discrepant with a mass of 0.63MJup and a radius of 1.34 RJup. WASP-41 shows brightness anomalies during transit due to the planet occulting spots on the stellar surface. Two anomalies observed 3.1 d apart are very likely due to the same spot. We measure its change in position and determine a rotation period for the host star of 18.6 ± 1.5 d, in good agreement with a published measurement from spot-induced brightness modulation, and a sky-projected orbital obliquity of λ = 6 ± 11°. We conclude with a compilation of obliquity measurements from spot-tracking analyses and a discussion of this technique in the study of the orbital configurations of hot Jupiters. © 2016 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. [less ▲]Detailed reference viewed: 29 (1 ULiège) SPITZER PARALLAX of OGLE-2015-BLG-0966: A COLD NEPTUNE in the GALACTIC DISKStreet, R. A.; Udalski, A.; Novati, S. C. et alin Astrophysical Journal (2016), 819(2), We report the detection of a cold Neptune mplanet = 21 ± 2 M⊕ orbiting a 0.38 Mo M dwarf lying 2.5-3.3 kpc toward the Galactic center as part of a campaign combining ground-based and Spitzer observations ... [more ▼]We report the detection of a cold Neptune mplanet = 21 ± 2 M⊕ orbiting a 0.38 Mo M dwarf lying 2.5-3.3 kpc toward the Galactic center as part of a campaign combining ground-based and Spitzer observations to measure the Galactic distribution of planets. This is the first time that the complex real-time protocols described by Yee et al., which aim to maximize planet sensitivity while maintaining sample integrity, have been carried out in practice. Multiple survey and follow up teams successfully combined their efforts within the framework of these protocols to detect this planet. This is the second planet in the Spitzer Galactic distribution sample. Both are in the near to mid-disk and are clearly not in the Galactic bulge. © 2016. The American Astronomical Society. All rights reserved. [less ▲]Detailed reference viewed: 39 (1 ULiège) SPITZER OBSERVATIONS of OGLE-2015-BLG-1212 REVEAL A NEW PATH TOWARD BREAKING STRONG MICROLENS DEGENERACIESBozza, V.; Shvartzvald, Y.; Udalski, A. et alin Astrophysical Journal (2016), 820(1), Spitzer microlensing parallax observations of OGLE-2015-BLG-1212 decisively break a degeneracy between planetary and binary solutions that is somewhat ambiguous when only ground-based data are considered ... [more ▼]Spitzer microlensing parallax observations of OGLE-2015-BLG-1212 decisively break a degeneracy between planetary and binary solutions that is somewhat ambiguous when only ground-based data are considered. Only eight viable models survive out of an initial set of 32 local minima in the parameter space. These models clearly indicate that the lens is a stellar binary system possibly located within the bulge of our Galaxy, ruling out the planetary alternative. We argue that several types of discrete degeneracies can be broken via such space-based parallax observations. © 2016. The American Astronomical Society. All rights reserved.. [less ▲]Detailed reference viewed: 42 (1 ULiège) Erratum to A detailed census of variable stars in the globular cluster NGC 6333 (M9) from CCD differential photometryArellano Ferro, A.; Bramich, D. M.; Figuera Jaimes, R. et alin Monthly Notices of the Royal Astronomical Society (2016), 458(2), 1188-1189Detailed reference viewed: 29 (2 ULiège) Erratum: Estimating the parameters of globular cluster M 30 (NGC 7099) from time-series photometry (Astronomy and Astrophysics (2013) 555 (A36))Kains, N.; Bramich, D. M.; Arellano Ferro, A. et alin Astronomy and Astrophysics (2016), 588[No abstract available]Detailed reference viewed: 24 (1 ULiège) Physical properties of the planetary systems WASP-45 and WASP-46 from simultaneous multiband photometryCiceri, S.; Mancini, L.; Southworth, J. et alin Monthly Notices of the Royal Astronomical Society (2016), 456(1), 990-1002Accurate measurements of the physical characteristics of a large number of exoplanets are useful to strongly constrain theoretical models of planet formation and evolution, which lead to the large variety ... [more ▼]Accurate measurements of the physical characteristics of a large number of exoplanets are useful to strongly constrain theoretical models of planet formation and evolution, which lead to the large variety of exoplanets and planetary-system configurations that have been observed. We present a study of the planetary systemsWASP-45 andWASP-46, both composed of a mainsequence star and a close-in hot Jupiter, based on 29 new high-quality light curves of transits events. In particular, one transit of WASP-45 b and four of WASP-46 b were simultaneously observed in four optical filters, while one transit of WASP-46 b was observed with the NTT obtaining a precision of 0.30 mmag with a cadence of roughly 3 min. We also obtained five new spectra of WASP-45 with the FEROS spectrograph. We improved by a factor of 4 the measurement of the radius of the planet WASP-45 b, and found that WASP-46 b is slightly less massive and smaller than previously reported. Both planets now have a more accurate measurement of the density (0.959 ± 0.077 ρJup instead of 0.64 ± 0.30 ρJup for WASP-45 b, and 1.103 ± 0.052 ρJup instead of 0.94 ± 0.11 ρJup for WASP-46 b). We tentatively detected radius variations with wavelength for both planets, in particular in the case of WASP-45 b we found a slightly larger absorption in the redder bands than in the bluer ones. No hints for the presence of an additional planetary companion in the two systems were found either from the photometric or radial velocity measurements. © 2015 The Authors. [less ▲]Detailed reference viewed: 16 (1 ULiège) Exploring the crowded central region of ten Galactic globular clusters using EMCCDs: Variable star searches and new discoveriesFiguera Jaimes, R.; Bramich, D. M.; Skottfelt, J. et alin Astronomy and Astrophysics (2016), 588Aims. We aim to obtain time-series photometry of the very crowded central regions of Galactic globular clusters; to obtain better angular resolution thanhas been previously achieved with conventional CCDs ... [more ▼]Aims. We aim to obtain time-series photometry of the very crowded central regions of Galactic globular clusters; to obtain better angular resolution thanhas been previously achieved with conventional CCDs on ground-based telescopes; and to complete, or improve, the census of the variable star population in those stellar systems. Methods. Images were taken using the Danish 1.54-m Telescope at the ESO observatory at La Silla in Chile. The telescope was equipped with an electron-multiplying CCD, and the short-exposure-time images obtained (ten images per second) were stacked using the shift-and-add technique to produce the normal-exposure-time images (minutes). Photometry was performed via difference image analysis. Automatic detection of variable stars in the field was attempted. Results. The light curves of 12 541 stars in the cores of ten globular clusters were statistically analysed to automatically extract the variable stars. We obtained light curves for 31 previously known variable stars (3 long-period irregular, 2 semi-regular, 20 RR Lyrae, 1 SX Phoenicis, 3 cataclysmic variables, 1 W Ursae Majoris-type and 1 unclassified) and we discovered 30 new variables (16 long-period irregular, 7 semi-regular, 4 RR Lyrae, 1 SX Phoenicis and 2 unclassified). Fluxes and photometric measurements for these stars are available in electronic form through the Strasbourg astronomical Data Center. © ESO, 2016. [less ▲]Detailed reference viewed: 19 (0 ULiège) THE SPITZER MICROLENSING PROGRAM AS A PROBE for GLOBULAR CLUSTER PLANETS: ANALYSIS of OGLE-2015-BLG-0448Poleski, R.; Zhu, W.; Christie, G. W. et alin Astrophysical Journal (2016), 823(1), The microlensing event OGLE-2015-BLG-0448 was observed by Spitzer and lay within the tidal radius of the globular cluster NGC 6558. The event had moderate magnification and was intensively observed, hence ... [more ▼]The microlensing event OGLE-2015-BLG-0448 was observed by Spitzer and lay within the tidal radius of the globular cluster NGC 6558. The event had moderate magnification and was intensively observed, hence it had the potential to probe the distribution of planets in globular clusters. We measure the proper motion of NGC 6558 ((μcl (N, E) = +0.36 ± 0.10, +1.42 ± 0.10 mas yr-1) as well as the source and show that the lens is not a cluster member. Even though this particular event does not probe the distribution of planets in globular clusters, other potential cluster lens events can be verified using our methodology. Additionally, we find that microlens parallax measured using Optical Gravitational Lens Experiment (OGLE) photometry is consistent with the value found based on the light curve displacement between the Earth and Spitzer. © 2016. The American Astronomical Society. All rights reserved. [less ▲]Detailed reference viewed: 20 (0 ULiège) MASS MEASUREMENTS of ISOLATED OBJECTS from SPACE-BASED MICROLENSINGZhu, W.; Calchi Novati, S.; Gould, A. et alin Astrophysical Journal (2016), 825(1), We report on the mass and distance measurements of two single-lens events from the 2015 Spitzer microlensing campaign. With both finite-source effect and microlens parallax measurements, we find that the ... [more ▼]We report on the mass and distance measurements of two single-lens events from the 2015 Spitzer microlensing campaign. With both finite-source effect and microlens parallax measurements, we find that the lens of OGLE-2015-BLG-1268 is very likely a brown dwarf (BD). Assuming that the source star lies behind the same amount of dust as the Bulge red clump, we find the lens is a 45 ±7 BD at 5.9 ±1.0 kpc. The lens of of the second event, OGLE-2015-BLG-0763, is a 0.50 ±0.04 star at 6.9 ±1.0 kpc. We show that the probability to definitively measure the mass of isolated microlenses is dramatically increased once simultaneous ground- and space-based observations are conducted. © 2016. The American Astronomical Society. All rights reserved. [less ▲]Detailed reference viewed: 9 (1 ULiège) Many new variable stars discovered in the core of the globular cluster NGC 6715 (M 54) with EMCCD observationsFiguera Jaimes, R.; Bramich, D. M.; Kains, N. et alin Astronomy and Astrophysics (2016), 592Context. We show the benefits of using electron-multiplying CCDs and the shift-and-add technique as a tool to minimise the effects of atmospheric turbulence, such as blending between stars in crowded ... [more ▼]Context. We show the benefits of using electron-multiplying CCDs and the shift-and-add technique as a tool to minimise the effects of atmospheric turbulence, such as blending between stars in crowded fields, and to avoid saturated stars in the fields observed. We intend to complete, or improve on, the census of the variable star population in globular cluster NGC 6715. Aims. Our aim is to obtain high-precision time-series photometry of the very crowded central region of this stellar system via the collection of better angular resolution images than has been previously achieved with conventional CCDs on ground-based telescopes. Methods. Observations were carried out using the Danish 1.54-m telescope at the ESO La Silla observatory in Chile. The telescope is equipped with an electron-multiplying CCD that enables short-exposure-time images to be obtained (ten images per second) that were stacked using the shift-and-add technique to produce the normal-exposure-time images (minutes). The high precision photometry was performed via difference image analysis employing the DanDIA pipeline. We attempted automatic detection of variable stars in the field. Results. We statistically analysed the light curves of 1405 stars in the crowded central region of NGC 6715 to automatically identify the variable stars present in this cluster. We found light curves for 17 previously known variable stars near the edges of our reference image (16 RR Lyrae and 1 semi-regular) and we discovered 67 new variables (30 RR Lyrae, 21 irregular (long-period type), 3 semi-regular, 1 W Virginis, 1 eclipsing binary, and 11 unclassified). Photometric measurements for these stars are available in electronic form through the Strasbourg Astronomical Data Centre. © 2016 ESO. [less ▲]Detailed reference viewed: 15 (1 ULiège) OGLE-2015-BLG-0479LA,B: BINARY GRAVITATIONAL MICROLENS CHARACTERIZED by SIMULTANEOUS GROUND-BASED and SPACE-BASED OBSERVATIONSHan, C.; Udalski, A.; Gould, A. et alin Astrophysical Journal (2016), 828(1), We present a combined analysis of the observations of the gravitational microlensing event OGLE-2015-BLG-0479 taken both from the ground and by the Spitzer Space Telescope. The light curves seen from the ... [more ▼]We present a combined analysis of the observations of the gravitational microlensing event OGLE-2015-BLG-0479 taken both from the ground and by the Spitzer Space Telescope. The light curves seen from the ground and from space exhibit a time offset of ∼13 days between the caustic spikes, indicating that the relative lens-source positions seen from the two places are displaced by parallax effects. From modeling the light curves, we measure the space-based microlens parallax. Combined with the angular Einstein radius measured by analyzing the caustic crossings, we determine the mass and distance of the lens. We find that the lens is a binary composed of two G-type stars with masses of ∼1.0 M⊙ and ∼0.9 M⊙ located at a distance of ∼3 kpc. In addition, we are able to constrain the complete orbital parameters of the lens thanks to the precise measurement of the microlens parallax derived from the joint analysis. In contrast to the binary event OGLE-2014-BLG-1050, which was also observed by Spitzer, we find that the interpretation of OGLE-2015-BLG-0479 does not suffer from the degeneracy between (±, ±) and (±, ∓) solutions, confirming that the four-fold parallax degeneracy in single-lens events collapses into the two-fold degeneracy for the general case of binary-lens events. The location of the blend in the color-magnitude diagram is consistent with the lens properties, suggesting that the blend is the lens itself. The blend is bright enough for spectroscopy and thus this possibility can be checked from future follow-up observations. © 2016. The American Astronomical Society. All rights reserved. [less ▲]Detailed reference viewed: 16 (1 ULiège) Searching for variable stars in the cores of five metal-rich globular clusters using EMCCD observationsSkottfelt, J.; Bramich, D. M.; Figuera Jaimes, R. et alin Astronomy and Astrophysics (2015), 573Aims. In this paper, we present the analysis of time-series observations from 2013 and 2014 of five metal-rich ([Fe/H] > -1) globular clusters: NGC 6388, NGC 6441, NGC 6528, NGC 6638, and NGC 6652. The ... [more ▼]Aims. In this paper, we present the analysis of time-series observations from 2013 and 2014 of five metal-rich ([Fe/H] > -1) globular clusters: NGC 6388, NGC 6441, NGC 6528, NGC 6638, and NGC 6652. The data have been used to perform a census of the variable stars in the central parts of these clusters. Methods. The observations were made with the electron-multiplying charge-couple device (EMCCD) camera at the Danish 1.54m Telescope at La Silla, Chile, and they were analysed using difference image analysis to obtain high-precision light curves of the variable stars. Results. It was possible to identify and classify all of the previously known or suspected variable stars in the central regions of the five clusters. Furthermore, we were able to identify and, in most cases, classify 48, 49, 7, 8, and 2 previously unknown variables in NGC 6388, NGC 6441, NGC 6528, NGC 6638, and NGC 6652, respectively. Especially interesting is the case of NGC 6441, for which the variable star population of about 150 stars has been thoroughly examined by previous studies, including a Hubble Space Telescope study. In this paper we are able to present 49 new variable stars for this cluster, of which one (possibly two) are RR Lyrae stars, two are W Virginis stars, and the rest are long-period semi-regular or irregular variables on the red giant branch. We have also detected the first double-mode RR Lyrae in the cluster. © ESO 2015. [less ▲]Detailed reference viewed: 28 (0 ULiège) Transits and starspots in the WASP-6 planetary systemTregloan-Reed, J.; Southworth, J.; Burgdorf, M. et alin Monthly Notices of the Royal Astronomical Society (2015), 450(2), 1760-1769We present updates to PRISM, a photometric transit-starspot model, and GEMC, a hybrid optimization code combining MCMC and a genetic algorithm. We then present high-precision photometry of four transits ... [more ▼]We present updates to PRISM, a photometric transit-starspot model, and GEMC, a hybrid optimization code combining MCMC and a genetic algorithm. We then present high-precision photometry of four transits in the WASP-6 planetary system, two of which contain a starspot anomaly. All four transits were modelled using PRISM and GEMC, and the physical properties of the system calculated. We find the mass and radius of the host star to be 0.836 ± 0.063 M and 0.864 ± 0.024 R, respectively. For the planet, we find a mass of 0.485 ± 0.027 MJup, a radius of 1.230 ± 0.035 RJup and a density of 0.244 ± 0.014 ρJup. These values are consistent with those found in the literature. In the likely hypothesis that the two spot anomalies are caused by the same starspot or starspot complex, we measure the stars rotation period and velocity to be 23.80 ± 0.15 d and 1.78 ± 0.20 km s-1, respectively, at a colatitude of 75.8°. We find that the sky-projected angle between the stellar spin axis and the planetary orbital axis is λ = 7.2° ± 3.7°, indicating axial alignment. Our results are consistent with and more precise than published spectroscopic measurements of the Rossiter-McLaughlin effect. These results suggest that WASP-6 b formed at amuch greater distance from its host star and suffered orbital decay through tidal interactions with the protoplanetary disc. © 2015 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society. [less ▲]Detailed reference viewed: 18 (0 ULiège) OGLE-2011-BLG-0265Lb: A jovian microlensing planet orbiting an m dwarfSkowron, J.; Shin, I.-G.; Udalski, A. et alin Astrophysical Journal (2015), 804(1), We report the discovery of a Jupiter-mass planet orbiting an M-dwarf star that gave rise to the microlensing event OGLE-2011-BLG-0265. Such a system is very rare among known planetary systems and thus the ... [more ▼]We report the discovery of a Jupiter-mass planet orbiting an M-dwarf star that gave rise to the microlensing event OGLE-2011-BLG-0265. Such a system is very rare among known planetary systems and thus the discovery is important for theoretical studies of planetary formation and evolution. High-cadence temporal coverage of the planetary signal, combined with extended observations throughout the event, allows us to accurately model the observed light curve. However, the final microlensing solution remains degenerate, yielding two possible configurations of the planet and the host star. In the case of the preferred solution, the mass of the planet is Mp = 0.9 ± 0.3 MJ, and the planet is orbiting a star with a mass M = 0.22 ± 0.06 M. The second possible configuration (2? away) consists of a planet with Mp = 0.6 ± 0.3 MJ and host star with M = 0.14 ± 0.06 M. The system is located in the Galactic disk 34 kpc toward the Galactic bulge. In both cases, with an orbit size of 1.52.0 AU, the planet is a cold Jupiterlocated well beyond the snow line of the host star. Currently available data make the secure selection of the correct solution difficult, but there are prospects for lifting the degeneracy with additional follow-up observations in the future, when the lens and source star separate. © 2015. The American Astronomical Society. All rights reserved. [less ▲]Detailed reference viewed: 16 (0 ULiège) Pathway to the galactic distribution of planets: Combined Spitzer and ground-based Microlens parallax measurements of 21 single-lens eventsNovati, S. C.; Gould, A.; Udalski, A. et alin Astrophysical Journal (2015), 804(1), We present microlens parallax measurements for 21 (apparently) isolated lenses observed toward the Galactic bulge that were imaged simultaneously from Earth and Spitzer, which was ∼1 AU west of Earth in ... [more ▼]We present microlens parallax measurements for 21 (apparently) isolated lenses observed toward the Galactic bulge that were imaged simultaneously from Earth and Spitzer, which was ∼1 AU west of Earth in projection. We combine these measurements with a kinematic model of the Galaxy to derive distance estimates for each lens, with error bars that are small compared to the Sun's galactocentric distance. The ensemble therefore yields a well-defined cumulative distribution of lens distances. In principle, it is possible to compare this distribution against a set of planets detected in the same experiment in order to measure the Galactic distribution of planets. Since these Spitzer observations yielded only one planet, this is not yet possible in practice. However, it will become possible as larger samples are accumulated. © 2015. The American Astronomical Society. All rights reserved. [less ▲]Detailed reference viewed: 20 (0 ULiège) 1 2 3 4

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text:
## Craps Variants

Craps is a suprisingly fair game. I remember calculating the probability of winning craps for the first time in an undergraduate discrete math class: I went back through my calculations several times, certain there was a mistake somewhere. How could it be closer than $\frac{1}{36}$? (Spoiler Warning If you haven’t calculated these odds for yourself then you may want to do so before reading further. I’m about to spoil it for you rather thoroughly in the name of exploring a more general case.


text:
Next: Electromagnetic radiation Up: Electromagnetic energy and momentum Previous: Electromagnetic momentum

## Momentum conservation

It follows from Eqs. (1036) and (1050) that the momentum density of electromagnetic fields can be written
 (1058)

Now, a momentum conservation equation for electromagnetic fields should take the integral form
 (1059)

Here, and run from 1 to 3 (1 corresponds to the -direction, 2 to the -direction, and 3 to the -direction). Moreover, the Einstein summation convention is employed for repeated indices (e.g., ). Furthermore, the tensor represents the flux of the th component of electromagnetic momentum in the -direction. This tensor (a tensor is a direct generalization of a vector with two indices instead of one) is called the momentum flux density. Hence, the above equation states that the rate of loss of electromagnetic momentum in some volume is equal to the flux of electromagnetic momentum across the bounding surface plus the rate at which momentum is transferred to matter inside . The latter rate is, of course, just the net electromagnetic force acting on matter inside : i.e., the volume integral of the electromagnetic force density, . Now, a direct generalization of the divergence theorem states that
 (1060)

where , , etc. Hence, in differential form, our momentum conservation equation for electromagnetic fields is written
 (1061)

Let us now attempt to derive an equation of this form from Maxwell's equations.

Maxwell's equations are written:

 (1062) (1063) (1064) (1065)

We can cross Eq. (1065) divided by with , and rearrange, to give
 (1066)

Next, let us cross with Eq. (1064) times , rearrange, and add the result to the above equation. We obtain
 (1067)

Next, making use of Eqs. (1062) and (1063), we get
 (1068)

Now, from vector field theory,
 (1069)

with a similar equation for . Hence, Eq. (1068) takes the form
 (1070)

Finally, when written in terms of components, the above equation becomes
 (1071)

since , and . Here, is a Kronecker delta symbol (i.e., if , and otherwise). Comparing the above equation with Eq. (1061), we conclude that the momentum flux density tensor of electromagnetic fields takes the form
 (1072)

The momentum conservation equation (1061) is sometimes written

 (1073)

where
 (1074)

is called the Maxwell stress tensor.

Next: Electromagnetic radiation Up: Electromagnetic energy and momentum Previous: Electromagnetic momentum
Richard Fitzpatrick 2006-02-02


text:
# Re: [ddlm-group] Eliding in triple-quoted strings: Proposals C andD. .. .. .

Here is a possible compromise.  This thread began with
Ralf's concern about the lack of an elide mechanism
in treble quoted strings.  Simon's suggestion does
not really answer that question, but it is a reasonable
step in that direction.  So, how about ...

1.  Immediately adopt Simon's suggestion to allow the
\\n and \\ elides in treble quoted strings.  Except for
the confusion in the meaning of \"""" if a more general
elide is eventually adopted that should cause very little
stress for anybody.

2.  Add Ralf's proosed new section 7 to the CIF2
document as a proposal under discussion, with the
advice that people may wish to avoid creating treble-quoted
string that conflict with the full python elide
conventions.

3.  Provide a coherent discussion document for COMCIFS
and the community at large on the alternatives in
to the list prior to the Madrid meeting.  I would suggest
that Ralf be asked to contribute a page or 2 on the
merits of his proposal and that either John B. or James
contribute a page or 2 on their objections and alternatives.

4.  Discuss it face to face at the Msdrid meeting and
try to come to a resolution.

5.  Move forward with the rest of CIF2 as proposed in
the meantime so we will be ready to discuss all of CIF2
at the Madrid meeting, with a effort to have sample parsers
and data sets available on the web prior to the meeting.

Regards,
Herbert

=====================================================
Herbert J. Bernstein, Professor of Computer Science
Dowling College, Kramer Science Center, KSC 121
Idle Hour Blvd, Oakdale, NY, 11769

+1-631-244-3035
yaya@dowling.edu
=====================================================

On Sat, 8 Jan 2011, Herbert J. Bernstein wrote:

> Dear James,
>
>  You are clearly a much better programmer than I am. When I got down into
> the interactions among the treble quote, single quotes, text fields, elides,
> the bracketed constructs and comments in the lexical scan, I found the going
> tough.  If you have it done neatly, I would greatly appreciate seeing it.
>
>  I think we need a face to face meeting or Skype meeting to resolve not just
> this one issue, but the process of getting a workable CIF2.  Perhaps we can
> finally get to do that in Madrid.
>
>  Regards,
>    Herbert
>
>
> =====================================================
> Herbert J. Bernstein, Professor of Computer Science
>   Dowling College, Kramer Science Center, KSC 121
>        Idle Hour Blvd, Oakdale, NY, 11769
>
>                 +1-631-244-3035
>                 yaya@dowling.edu
> =====================================================
>
> On Sat, 8 Jan 2011, James Hester wrote:
>
>> I can't let these assertions go unchallenged:
>>
>> On Sat, Jan 8, 2011 at 12:04 PM, Herbert J. Bernstein
>> <yaya@bernstein-plus-sons.com> wrote:
>>> Dear Simon,
>>>
>>>   Adoption of Ralf's proposal will ...
>>>
>>>   1.  Make it much easier to create a CIF2 parser, because for one of
>>> the messiest parts of the code we will have a clear specification,
>>> sample code and a way to validate the tough cases.
>>
>> If we adopt a simpler spec than the Python in toto spec:
>> - there will be many fewer tough cases
>> - there will be a simpler and therefore clearer specification
>> - for many alternative schemes the lexer will be unchanged from the
>> current version, with the elide behaviour
>>  simply requiring a search and replace following lexing
>> Triple-quoted string handling is not currently a messy part of the
>> code, I don't understand why you think this.  It will become
>> significantly more complex under Ralf's proposal.
>>
>>>   2.  Make it easier for users to conform the the quoting rules, because
>>> at least that one part of CIF2 will be thoroughly documented with lots
>>> of examples.
>>
>> Quoting rules are not rocket science.  About 3 examples will be
>> enough, if we adopt a simple specification rather
>> than the unicode+raw+lots of escapes that the Python proposal entails.
>> Doing things the Python way would
>> imply more chance for user misunderstanding, especially bearing in
>> mind that CIF2 users are not necessarily
>> Python programmers or even programmers at all.  For these users, there
>> is absolutely no benefit in adopting Python or any other language's
>> approach - they are unfamiliar with them all.
>>
>>>   3.  Make is easier for the journals and archives to deal with "odd"
>>> CIF2 files containing complex treble quoted strings because at
>>> least  that one part of CIF2 will be throughly documented with lots
>>> of examples, and, with a utility (IDLE) all ready to allow them
>>> to pull out a troublesome treble-quoted string and figure out what
>>> it means or what it might mean if some intuitive change were made.
>>
>> The simpler the spec, the less likely mistakes will be made and the
>> less chance of ambiguity.
>>
>>>   Yes, if Ralf's proposal happens to be rejected, we will still have
>>> a problem in the lack of elide handling, and yes we will have to
>>> put in the time an effort to consider those alternatives, but, first,
>>> in order to have some chance of finishing the specification of CIF2
>>> before the summer meeting deadlines (at least one of which is in
>>> just a little more than 3 weeks), might it not be a good idea
>>> to discuss and consider what was actually proposed instead of
>>> chasing after lots of plausible alternatives that we already discussed
>>> and rejected, and so are not very likely to agree upon rapidly now.
>>
>> I have some hope that, by restricting our discussion to treble-quoted
>> strings, we can make progress compared to previous attempts.  I have
>> considered and discussed at length Ralf's proposal, and would be
>> interested in your responses to my particular objections.
>>
>>>   So, before I will delve into the many subtle variations of elide
>>> mechanisms, I would appreciate our finishing consideration of Ralf's
>>> actual proposal:
>>>
>>> =======================
>>>
>>> His revised wording (with one correction) is:
>>>
>>> ========================
>>>
>>> CHANGE 7 NEW
>>>
>>>
>>> Triple-quote delimited strings.
>>>
>>> The following ASCII sequences delimit the beginning of a string:
>>>
>>>     """
>>>     '''
>>>     r"""
>>>     r'''
>>>     u"""
>>>     u'''
>>>
>>> The characters following the delimiter sequence are interpreted
>>> with exactly the same algorithm as implemented for triple-quoted
>>> strings in the Python programming language version 2 series.
>>> In this algorithm, triple-quoted strings are terminated by matching
>>> """ or ''' delimiters.
>>>
>>> For example
>>>
>>>     """He said "His name is O'Hearly"."""
>>>     r'''In {\bf \TeX} the accents are \' and \".'''
>>>
>>> Triple-quoted strings provide a reliable mechanism for storing any
>>> arbitrary string in a CIF2 file.
>>>
>>> =========================
>>>
>>> This is cleaner and simpler than the original change 7 wording.
>>> It probably does not conflict with existing CIF1 documents and the
>>> _only_ CIF2 documents it can conflict with are the very few
>>> that happen to end in \""" or \''''.  The new leading delimiters
>>> r""", r''', u""" and u''' will have to be added to the list of forbidden
>>> starts to white-space delimited data values in change 5.  In exchange for
>>> this minor adjustments to valid CIF2 syntax we gain a fully documented,
>>> software supported way to include arbitrary strings in a CIF2 document
>>> that people are already used to working with.
>>>
>>> I have reviewed the discussion of the "use of elides in strings"
>>> thread in the ddlm-group discussion list, and, while I did not
>>> then and do not now understand the objections to the general use
>>> of elides in quoted strings, I particularly do not understand
>>> the logic of objecting to the use of elides in treble-quoted strings,
>>> which are a construct completely new to CIF and therefore in
>>> conflict with no existing data files.
>>>
>>> Would those who have an objection to Ralf's proposal please
>>> state their objections.  An objection that says we object because
>>> in past discussions another body could not manage to come to an
>>> agreement and just gave up does not speak to the merits of this
>>> specific proposal.
>>>
>>> I have no idea why we are considering other proposals before
>>> settling the status of Ralf's proposal.
>>
>> It is also useful to know what the alternatives might be when
>> considering a proposal.
>>
>>> I agree with Ralf's proposal.
>>>
>>> Regards,
>>>   Herbert
>>>
>>> At 12:37 AM +0000 1/8/11, SIMON WESTRIP wrote:
>>>> Dear Herbert
>>>>
>>>> I fail to see how the adoption of python string quoting rules is going to
>>>> make life easier for anyone other than a python programmer?
>>>> Even then, the mechanism is restricted to treble-quoted strings,
>>>> which are only
>>>> one part of CIF. Maybe I've missed something, but just because CIF might
>>>> share
>>>> common syntax with a programming language in one respect, does not
>>>> necessarily mean
>>>> that the tools of that medium are available to CIF?
>>>>
>>>> If you're looking to base CIF extensions on established mechanisms,
>>>> the minimal \(newline) and \\ escape sequences, which in essence are
>>>> the same as
>>>> the established CIF line-folding protocol (just dropping the initial
>>>> \ following the opening
>>>> delimiter and formalising the protocol as an inherent part of the
>>>> spec). Afterall, I beleive you
>>>> have already been using it, or at least interpreted it, as a means
>>>> to escape 'semicolon delimiters' within
>>>> semicolon-delimited values (I seem to recall discussions that
>>>> identified an issue with the published 'trip tests'
>>>> relating to line folding).
>>>>
>>>> Forgive me if I have missed something regarding the usefulness of
>>>> python in CIF; please enlighten me
>>>> as to its benefits if I were to write a CIF reader using anything
>>>> but python. As far as I can see, the only
>>>> advantages lie in the fact that the logic is established and thus
>>>> unquestionable; but that does not mean it is
>>>> necessarily entirely appropriate for CIF (which afterall isn't a
>>>> programming language).
>>>>
>>>> Cheers
>>>>
>>>> Simon
>>>>
>>>>
>>>>
>>>> From: Herbert J. Bernstein <yaya@bernstein-plus-sons.com>
>>>> To: Group finalising DDLm and associated dictionaries
>>>> <ddlm-group@iucr.org>
>>>> Sent: Friday, 7 January, 2011 23:07:40
>>>> Subject: Re: [ddlm-group] Eliding in triple-quoted strings:
>>>> Proposals C and D. .. .. .
>>>>
>>>> Dear Colleagues,
>>>>
>>>>   Ralf's proposal is what it is.  Before we go haring off in other
>>>> directions, we should respond constructively to what he has proposed.
>>>> I support it.  Ralf and John W. support it.  John B. and James H.
>>>> oppose it.  I think they are mistaken because ...
>>>>
>>>>   It is well and good to adopt a "Real Programmers Don't Each
>>>> Quiche" let's-start-from-scratch-and-roll-our-own approach when
>>>> you have the resources to accomplish our goals that way.  It
>>>> is a lot of fun, and has the potential to truly advance the
>>>> field, but it is also, in the current funding climate, unrealistic.
>>>>
>>>>   In the U.S., there is a serious prospect to science funding being
>>>> cut back so severely that the hit rates on grants next year may
>>>> be as low as 1 in 10.  I suspect an honest review of funding prospects
>>>> in other countries will uncover similarly dire warnings.
>>>>
>>>>   This does not mean we are all going out of buisness, but we do have
>>>> to be careful to conserve resources and focus our do-it-from-scratch
>>>> efforts on those areas that have the highest priority, and I fear,
>>>> for most of our community, CIF2, while important, is not likely to
>>>> be seen as worth that approach, and certainly filing the edges of
>>>> a brand-new treble quote spec is likely to be very far down
>>>> on anybody's priority list.
>>>>
>>>> Ralf has made a proposal that will save all of us a lot of effort
>>>> and allow us to devote more resources to higher priority problems.
>>>>
>>>> Not only is he right on this one point, but I urge us to look for
>>>> other areas where we can get to CIF2 by building on work that is
>>>>
>>>> This is not a good time for wheel-reinvention.
>>>>
>>>> I would appreciate knowing from those who wish to reinvent this
>>>> particular wheel, why they wish to do that and from where they
>>>> expect to get the resources to do it?
>>>>
>>>> Regards,
>>>>   Herbert
>>>>
>>>> =====================================================
>>>>   Herbert J. Bernstein, Professor of Computer Science
>>>>     Dowling College, Kramer Science Center, KSC 121
>>>>         Idle Hour Blvd, Oakdale, NY, 11769
>>>>
>>>>                   +1-631-244-3035
>>>>                   <mailto:yaya@dowling.edu>yaya@dowling.edu
>>>> =====================================================
>>>>
>>>> On Fri, 7 Jan 2011, Bollinger, John C wrote:
>>>>
>>>>>
>>>>>  On Friday, January 07, 2011 3:14 PM, Herbert J. Bernstein wrote:
>>>>>
>>>>>>  We seem not to be communicating effectively.
>>>>  >>
>>>>>>  What I am asking for is an _existing_, supported treble quote
>>>>>> specification
>>>>>>  from an _existing_ language with _existing_ documentation and
>>>>>>  _existing_ software as an alternative to the Python specification,
>>>>>>  documentation and software to which we all have access, that is being
>>>>>>  proposed as an alternative
>>>>>>  to what Ralf has proposed.
>>>>>
>>>>>  Thank you for that clarification.  You are right, I didn't understand
>>>>>  what you were asking for.
>>>>>
>>>>>  I hope this will likewise clarify my position: I reject the premise
>>>>> that
>>>>>  the system we choose must meet those criteria, and I oppose adopting
>>>>> the
>>>>>  full Python syntax and semantics.
>>>>>
>>>>>>  The Python specification is available at
>>>>>>
>>>>>>
>>>>>> <http://docs.python.org/reference/index.html>http://docs.python.org/reference/index.html
>>>>>>
>>>>>>  with the lexical analysis at
>>>>>>
>>>>>>
>>>>>> <http://docs.python.org/reference/lexical_analysis.html>http://docs.python.org/reference/lexical_analysis.html
>>>>>
>>>>>  Thanks, though that is exactly what I was looking at already.  It
>>>>> leaves
>>>>>  several details unclear, some of which I discussed in previous
>>>>> messages.
>>>>>  Hence, I consider it slightly short of a *full* specification.  It
>>>>> does,
>>>>>  however, provide my grounds for opposing adoption of that scheme for
>>>>>  CIF.
>>>>>
>>>>>>  The complete source code and binaries are available at:
>>>>>
>>>>>  Unless you propose to append a particular set of Python sources to the
>>>>>  CIF specification as a reference, I have no interest in perusing the
>>>>>  source code to seek answers to such questions of detail as I have.
>>>>>  Furthermore, I would oppose adding such an appendix on the grounds that
>>>>>  it would be exceedingly difficult to use to resolve questions such as
>>>>>  mine.
>>>>>
>>>>>  I am likewise unwilling to rely on the behavior the python binary that
>>>>>  happens to be installed on my computer to answer them.  If the correct
>>>>>  behavior is not documented independent of the program then there is no
>>>>>  particular reason to trust that it won't change in future versions, or
>>>>>  that any particular implementation is correct or bug-free.
>>>>>
>>>>>
>>>>>  Regards,
>>>>>
>>>>>  John
>>>>>
>>>>>  --
>>>>>  John C. Bollinger, Ph.D.
>>>>>  Department of Structural Biology
>>>>>  St. Jude Children's Research Hospital
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>  Email Disclaimer:
>>>>> <http://www.stjude.org/emaildisclaimer>www.stjude.org/emaildisclaimer
>>>>>
>>>>>  _______________________________________________
>>>>>  ddlm-group mailing list
>>>>>  <mailto:ddlm-group@iucr.org>ddlm-group@iucr.org
>>>>>
>>>>> <http://scripts.iucr.org/mailman/listinfo/ddlm-group>http://scripts.iucr.org/mailman/listinfo/ddlm-group
>>>>>
>>>> _______________________________________________
>>>> ddlm-group mailing list
>>>> <mailto:ddlm-group@iucr.org>ddlm-group@iucr.org
>>>> <http://scripts.iucr.org/mailman/listinfo/ddlm-group>http://scripts.iucr.org/mailman/listinfo/ddlm-group
>>>>
>>>>
>>>> _______________________________________________
>>>> ddlm-group mailing list
>>>> ddlm-group@iucr.org
>>>> http://scripts.iucr.org/mailman/listinfo/ddlm-group
>>>
>>>
>>> --
>>> =====================================================
>>>  Herbert J. Bernstein, Professor of Computer Science
>>>    Dowling College, Kramer Science Center, KSC 121
>>>         Idle Hour Blvd, Oakdale, NY, 11769
>>>
>>>                  +1-631-244-3035
>>>                  yaya@dowling.edu
>>> =====================================================
>>> _______________________________________________
>>> ddlm-group mailing list
>>> ddlm-group@iucr.org
>>> http://scripts.iucr.org/mailman/listinfo/ddlm-group
>>>
>>
>>
>>
>> --
>> T +61 (02) 9717 9907
>> F +61 (02) 9717 3145
>> M +61 (04) 0249 4148
>> _______________________________________________
>> ddlm-group mailing list
>> ddlm-group@iucr.org
>> http://scripts.iucr.org/mailman/listinfo/ddlm-group
>
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text:
# Math Help - Proof by contradiction.

I'm trying to write a contradiction to the statement:
If three positive real numbers x,y,z are chose between 0 and 1 with 0<x<y<z<1, then the distance from x to y is less than .5 or the distance from y to z is less than .5.

I belive it is:

If three positive real numbers x,y,z are chose between 0 and 1 with0<x<y<z<1, then the distance from x to y is not less than .5 and thedistance from y to z is not less than .5.

But my teacher tells me this is wrong.

2. ## Re: Proof by contradiction.

I'm trying to write a contradiction to the statement:
If three positive real numbers x,y,z are chose between 0 and 1 with 0<x<y<z<1, then the distance from x to y is less than .5 or the distance from y to z is less than .5.

I belive it is:

If three positive real numbers x,y,z are chose between 0 and 1 with0<x<y<z<1, then the distance from x to y is not less than .5 and thedistance from y to z is not less than .5.

But my teacher tells me this is wrong.
I suggest you review your terminology. What you have given is not (even close to) a proof. All you've done is write the negation of the original statement.

To prove a statement by contradiction, you need to write the negation of your original statement, then go through a series of logical steps to arrive at a contradiction. This shows that the negation is false, and therefore that the original statement is true.

3. ## Re: Proof by contradiction.

That is what i am trying to do, is just find the correct statement. i have not proved it yet.

4. ## Re: Proof by contradiction.

the way i want to prove it, is if the distance from x to y is not less than .5 and the distance from y to z is not less than .5.
then we assume z - y >= 0.5 and y - x > =0.5. Adding the two we get z - x >= 1, which is impossible for two numbers between 0 and 1. I guess im confused as to what kind of proof im actually using

5. ## Re: Proof by contradiction.

the way i want to prove it, is if the distance from x to y is not less than .5 and the distance from y to z is not less than .5.
then we assume z - y >= 0.5 and y - x > =0.5. Adding the two we get z - x >= 1, which is impossible for two numbers between 0 and 1. I guess im confused as to what kind of proof im actually using
I think you're on the right track.

I would say something like...

\displaystyle \begin{align*}y - x &\geq 0.5 \\ y - x &= 0.5 + m\textrm{ for some }m \geq 0\end{align*}

and

\displaystyle \begin{align*}z - y &\geq 0.5 \\ z - y &= 0.5 + n\textrm{ for some } n \geq 0 \end{align*}

So

\displaystyle \begin{align*} (z - y ) + (y - x) &= (0.5 + m) + (0.5 + n) \\ z - x &= 1 + m + n \\ z - x &\geq 1 \end{align*}

which is where our contradiction lies, since $\displaystyle 0 < x < y < z < 1 \implies z - x \leq 1$.

6. ## Re: Proof by contradiction.

all right, so what would be my statement? Since my teacher keeps telling my contradiction statement is wrong.

7. ## Re: Proof by contradiction.

Am i even using proof by contradiction?

8. ## Re: Proof by contradiction.

Am i even using proof by contradiction?
I don't see any problem with the statement of the negation.

9. ## Re: Proof by contradiction.

So its proof by negation?... Is that valid?

10. ## Re: Proof by contradiction.

So its proof by negation?... Is that valid?
I suggest you reread Post 2.

11. ## Re: Proof by contradiction.

Then why does my teacher tell me its wrong?!?!

12. ## Re: Proof by contradiction.

I'm trying to write a contradiction to the statement:
If three positive real numbers x,y,z are chose between 0 and 1 with 0<x<y<z<1, then the distance from x to y is less than .5 or the distance from y to z is less than .5.

I belive it is:

If three positive real numbers x,y,z are chose between 0 and 1 with0<x<y<z<1, then the distance from x to y is not less than .5 and thedistance from y to z is not less than .5.
You forgot to change the universal quantification to existential quantification.

13. ## Re: Proof by contradiction.

So my statement

If positive real numbers x,y,z are chosen between 0 and 1 with 0<x<y<z<1, then the distance from x to y is less than .5 or the distance from y to z is less than .5
.

Then i have the statement P then Q where Q=(q1 v q2)
P= Positive real numbers x,y,z between 0 and 1 with 0<x<y<z<1
Q = (q1=the distance from x to y is less than .5) v (q2=the distance from y to z is less than .5)

Okay so now i use contradiction to prove it
The statement then must turn into P and ~Q ( P and not Q)
So
Positive real numbers x,y,z are between 0 and 1 with 0<x<y<z<1, AND the distance from x to y is not less than .5 and the distance from y to z is not less than .5

Then the statement can be written as :
Positive real numbers x,y,z are chosen between 0 and 1 with 0<x<y<z<1, AND the distance from x to y is greater than or equal to .5 and the distance from y to z is greater than or equal to .5.

This doesn't sound right to me

14. ## Re: Proof by contradiction.

Should my statement be written as:
If Real numbers x,y,z are 0<x<y<z1, then the distance from x to y is less than .5 or the distance from y to z is less than .5.

Real numbers x,y,z are 0<x<y<z1, AND the distance from x to y is greater than or equal to .5 and the distance from y to z is greater than or equal to .5.

What do you think?

15. ## Re: Proof by contradiction.

That exist three numbers $\{a,b,c\}\subset (0,1)$ such $0.


text:
If condition question. C++

This topic is 2423 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

Recommended Posts

Suppose I have a struct called "Hum":

 struct Hum { int du; }

Now I have the following code:

 Hum *hum = NULL; //Condition. if(hum && hum->du == 0) { //Do stuff... } else { //Error! }

My doubt is this: If the first condition is not met, is the following condition discarded right away without having it checked? On the case where "hum == NULL", an error would obviously occur if the second condition is checked.

I've done tests with this and concluded the answer to be no. But would like to know if that's applied to all cases.

Thanks

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The code is safe: http://en.wikipedia....cuit_evaluation

Nota: En este contexto "duda" se debe traducir como "question", no "doubt". A mí me costó bastante aprenderlo. Edited by alvaro

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I appreciate the reply and link alvaro, though I didn't quite understand your note since I'm brazillian , but I get the point.

Thanks again.

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É a mesma coisa em Português: A tradução correta de "dúvida" neste contexto é "question", não "doubt".

I lived in Rio for three years when I was a kid. De nada.

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É a mesma coisa em Português: A tradução correta de "dúvida" neste contexto é "question", não "doubt".

I lived in Rio for three years when I was a kid. De nada.

Um compatriota entao, é bom saber que nao sou o unico por aqui... : )

I don't live in Rio but visit it every now and then, it's a great city!

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Just for completeness it should be pointed out that && and || will evaluate both sides, causing a dereference to the pointer and a crash, if operators && or || are overloaded.

It doesn’t mean you should go out of your way to write code that is safe under such a condition, it means never ever overload operators && or ||.

L. Spiro

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It doesn’t mean you should go out of your way to write code that is safe under such a condition, it means never ever overload operators && or ||.

... or operator , (for the same reason).

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[quote name='L. Spiro' timestamp='1338459103' post='4944924']
It doesn’t mean you should go out of your way to write code that is safe under such a condition, it means never ever overload operators && or ||.

... or operator , (for the same reason).
[/quote]
Won't the comma operator always evaluate both the left side and the right side though (i.e. there shouldn't be any short circuiting, right?)? I don't think (guessing) it guarantees the order in which it evaluates its arguments when overloaded though, which certainly could be problematic (and is what I think you're pointing out).

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[quote name='alvaro' timestamp='1338473232' post='4944982']
[quote name='L. Spiro' timestamp='1338459103' post='4944924']
It doesn’t mean you should go out of your way to write code that is safe under such a condition, it means never ever overload operators && or ||.

... or operator , (for the same reason).
[/quote]
Won't the comma operator always evaluate both the left side and the right side though (i.e. there shouldn't be any short circuiting, right?)? I don't think (guessing) it guarantees the order in which it evaluates its arguments when overloaded though, which certainly could be problematic (and is what I think you're pointing out).
[/quote]
I think he just meant it as a general comment that the comma operator changes behavior when overloaded, like && and ||, not whether the two sides are evaluated or not. The built in comma operator is a sequence point, but the overloaded operator is a function call and subject to the usual rules about order of evaluation.

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[quote name='Cornstalks' timestamp='1338475494' post='4944993']
[quote name='alvaro' timestamp='1338473232' post='4944982']
[quote name='L. Spiro' timestamp='1338459103' post='4944924']
It doesn’t mean you should go out of your way to write code that is safe under such a condition, it means never ever overload operators && or ||.

... or operator , (for the same reason).
[/quote]
Won't the comma operator always evaluate both the left side and the right side though (i.e. there shouldn't be any short circuiting, right?)? I don't think (guessing) it guarantees the order in which it evaluates its arguments when overloaded though, which certainly could be problematic (and is what I think you're pointing out).
[/quote]
I think he just meant it as a general comment that the comma operator changes behavior when overloaded, like && and ||, not whether the two sides are evaluated or not. The built in comma operator is a sequence point, but the overloaded operator is a function call and subject to the usual rules about order of evaluation.
[/quote]
Yes, that's what I think he meant. Maybe I just misunderstood what "for the same reason" implies, but it makes it sound like the reason is short-circuiting (or not short-circuiting), and I was just trying to clarify that the "same reason" here is that it will screw up programmers because the evaluation of the arguments is different than what most would expect, and not the short-circuiting reason. Edited by Cornstalks

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text:
Variational principle & lorentz force law

Homework Statement

Show that the Lorentz force law follows from the following variational principle:
$$S=\frac{m}{2}\int\eta_{\mu\nu}u^\mu u^\nu ds-q\int A_\mu u^\mu ds$$

Homework Equations

Definition of Field Strength Tensor
Integration by Parts
Chain Rule & Product Rule for Derivatives

The Attempt at a Solution

In order to find the force equation, we need to vary the action and find when $\delta S=0$.

So, we begin by varying the action integral,
$$\delta S=\frac{m}{2}\int\eta_{\mu\nu}\delta(u^\mu u^\nu)ds-q\int\delta(A_\mu u^\mu)ds$$

Then apply product rule to expand all the terms,
$$\delta S=m\int\eta_{\mu\nu}u^\nu\frac{d(\delta x^\mu)}{ds}ds-q\int\left(\partial_\nu A_\mu u^\mu\delta x^\nu+A_\mu\frac{d(\delta x^\mu)}{ds}\right)ds$$

Simplified and renamed indices in the second term,
$$\delta S=m\int u_\mu\frac{d(\delta x^\mu)}{ds}ds-q\int\left(\partial_\mu A_\nu u^\nu\delta x^\mu+A_\mu\frac{d(\delta x^\mu)}{ds}\right)ds$$

Reverse integration by parts to get the variation out of the derivatives,
$$\delta S=m\int\left(\frac{d}{ds}\left(u_\mu\delta x^\mu\right)-\frac{du_\mu}{ds}\delta x^\mu\right)ds-q\int\left(\partial_\mu A_\nu u^\nu\delta x^\mu+\frac{d}{ds}\left( A_\mu\delta x^\mu\right)-\frac{dA_\mu}{ds}\delta x^\mu\right)ds$$

Grouping like terms we have,
$$\delta S=\int\left(\frac{d}{ds}\left(mu_\mu\delta x^\mu-qA_\mu\delta x^\mu\right)\right)ds+\int \left( q\left(\frac{dA_\mu}{ds}\delta x^\mu-\partial_\mu A_\nu u^\nu\delta x^\mu\right)-m\frac{du_\mu}{ds}\delta x^\mu\right)ds$$

After integration, the first term should be zero since the variations must vanish at the endpoints. Also, using the chain rule on the dA/ds derivative gives me
$$\delta S=\int \left( q\left(\partial_\nu A_\mu u^\nu-\partial_\mu A_\nu u^\nu\right)-m\frac{du_\mu}{ds}\right)\delta x^\mu ds$$

Rearranged terms to simplify further,
$$\delta S=\int \left( -q\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)u^\nu -m\frac{du_\mu}{ds}\right)\delta x^\mu ds$$

Then I use the definition of the field strength tensor to get,
$$\delta S=\int \left( -qF_{\mu\nu}u^\nu -m\frac{du_\mu}{ds}\right)\delta x^\mu ds$$

Then $\delta S=0$ for any arbitrary variation $\delta x^\mu$ only if the integrand itself is zero, so
$$qF_{\mu\nu}u^\nu+m\frac{du_\mu}{ds}=0$$

Rearranging the expression gives
$$\frac{du_\mu}{ds}=-\frac{q}{m}F_{\mu\nu}u^\nu$$

This is the correct form for the Lorentz force law, however I have an additional minus sign. It's probably a trivial problem, but I've double checked all the work several times and I can't figure out where it's coming from.

Last edited:

Answers and Replies

Perhaps the problem is this:

On the right hand side you have $$u^\nu$$
On the left hand side $$u_\mu$$

The relation between the two is given by $$\eta_{\mu\nu}$$ and that depends on your signature convention. This signature is expressed in the sign of the first term of your action principle.

I checked my notes and it's supposed to come out to
$$\frac{du_\mu}{ds}=\frac{q}{m}F_{\mu\nu}u^\nu$$

We've been using the convention that the spatial terms have the minus signs in the Minkowski metric (+---). In an earlier step I used $u_\mu=\eta_{\mu\nu}u^\nu$ to get the lower mu index in the first term with the mass. I can't find anything that says how the signature affects the process of raising/lowering indices though.

Question is why do you have minus sign in front of the second (or plus in front of the first) term in:

$$S=\frac{m}{2}\int\eta_{\mu\nu}u^\mu u^\nu ds-q\int A_\mu u^\mu ds$$

Landau and Lifgarbagez has this action

The first term is $$\sqrt{\eta_{\mu\nu}u^{\mu}u^{\nu}}\,ds$$ but it should not matter.

Last edited:


text:
## The Leech lattice neighbour

Here’s the upper part of Kneser‘s neighbourhood graph of the Niemeier lattices:

The Leech lattice has a unique neighbour, that is, among the $23$ remaining Niemeier lattices there is a unique one, $(A_1^{24})^+$, sharing an index two sub-lattice with the Leech.

How would you try to construct $(A_1^{24})^+$, an even unimodular lattice having the same roots as $A_1^{24}$?

The root lattice $A_1$ is $\sqrt{2} \mathbb{Z}$. It has two roots $\pm \sqrt{2}$, determinant $2$, its dual lattice is $A_1^* = \tfrac{1}{\sqrt{2}} \mathbb{Z}$ and we have $A_1^*/A_1 \simeq C_2 \simeq \mathbb{F}_2$.

Thus, $A_1^{24}= \sqrt{2} \mathbb{Z}^{\oplus 24}$ has $48$ roots, determinant $2^{24}$, its dual lattice is $(A_1^{24})^* = \tfrac{1}{\sqrt{2}} \mathbb{Z}^{\oplus 24}$ and the quotient group $(A_1^{24})^*/A_1^{24}$ is $C_2^{24}$ isomorphic to the additive subgroup of $\mathbb{F}_2^{\oplus 24}$.

A larger lattice $A_1^{24} \subseteq L$ of index $k$ gives for the dual lattices an extension $L^* \subseteq (A_1^{24})^*$, also of index $k$. If $L$ were unimodular, then the index has to be $2^{12}$ because we have the situation
$A_1^{24} \subseteq L = L^* \subseteq (A_1^{24})^*$
So, Kneser’s glue vectors form a $12$-dimensional subspace $\mathcal{C}$ in $\mathbb{F}_2^{\oplus 24}$, that is,
$L = \mathcal{C} \underset{\mathbb{F}_2}{\times} (A_1^{24})^* = \{ \tfrac{1}{\sqrt{2}} \vec{v} ~|~\vec{v} \in \mathbb{Z}^{\oplus 24},~v=\vec{v}~mod~2 \in \mathcal{C} \}$
Because $L = L^*$, the linear code $\mathcal{C}$ must be self-dual meaning that $v.w = 0$ (in $\mathbb{F}_2$) for all $v,w \in \mathcal{C}$. Further, we want that the roots of $A_1^{24}$ and $L$ are the same, so the minimal number of non-zero coordinates in $v \in \mathcal{C}$ must be $8$.

That is, $\mathcal{C}$ must be a self-dual binary code of length $24$ with Hamming distance $8$.

Marcel Golay (1902-1989) – Photo Credit

We now know that there is a unique such code, the (extended) binary Golay code, $\mathcal{C}_{24}$, which has

• one vector of weight $0$
• $759$ vectors of weight $8$ (called ‘octads’)
• $2576$ vectors of weight $12$ (called ‘dodecads’)
• $759$ vectors of weight $16$
• one vector of weight $24$

The $759$ octads form a Steiner system $S(5,8,24)$ (that is, for any $5$-subset $S$ of the $24$-coordinates there is a unique octad having its non-zero coordinates containing $S$).

Witt constructed a Steiner system $S(5,8,24)$ in his 1938 paper “Die $5$-fach transitiven Gruppen von Mathieu”, so it is not unthinkable that he checked the subspace of $\mathbb{F}_2^{\oplus 24}$ spanned by his $759$ octads to be $12$-dimensional and self-dual, thereby constructing the Niemeier-lattice $(A_1^{24})^+$ on that sunday in 1940.

John Conway classified all nine self-dual codes of length $24$ in which the weight
of every codeword is a multiple of $4$. Each one of these codes $\mathcal{C}$ gives a Niemeier lattice $\mathcal{C} \underset{\mathbb{F}_2}{\times} (A_1^{24})^*$, all but one of them having more roots than $A_1^{24}$.

Vera Pless and Neil Sloan classified all $26$ binary self-dual codes of length $24$.

## Borcherds’ favourite numbers

Whenever I visit someone’s YouTube or Twitter profile page, I hope to see an interesting banner image. Here’s the one from Richard Borcherds’ YouTube Channel.

Not too surprisingly for Borcherds, almost all of these numbers are related to the monster group or its moonshine.

Let’s try to decode them, in no particular order.

196884

John McKay’s observation $196884 = 1 + 196883$ was the start of the whole ‘monstrous moonshine’ industry. Here, $1$ and $196883$ are the dimensions of the two smallest irreducible representations of the monster simple group, and $196884$ is the first non-trivial coefficient in Klein’s j-function in number theory.

$196884$ is also the dimension of the space in which Robert Griess constructed the Monster, following Simon Norton’s lead that there should be an algebra structure on the monster-representation of that dimension. This algebra is now known as the Griess algebra.

Here’s a recent talk by Griess “My life and times with the sporadic simple groups” in which he tells about his construction of the monster (relevant part starting at 1:15:53 into the movie).

1729

1729 is the second (and most famous) taxicab number. A long time ago I did write a post about the classic Ramanujan-Hardy story the taxicab curve (note to self: try to tidy up the layout of some old posts!).

Recently, connections between Ramanujan’s observation and K3-surfaces were discovered. Emory University has an enticing press release about this: Mathematicians find ‘magic key’ to drive Ramanujan’s taxi-cab number. The paper itself is here.

“We’ve found that Ramanujan actually discovered a K3 surface more than 30 years before others started studying K3 surfaces and they were even named. It turns out that Ramanujan’s work anticipated deep structures that have become fundamental objects in arithmetic geometry, number theory and physics.”

Ken Ono

24

There’s no other number like $24$ responsible for the existence of sporadic simple groups.

24 is the length of the binary Golay code, with isomorphism group the sporadic Mathieu group $M_24$ and hence all of the other Mathieu-groups as subgroups.

24 is the dimension of the Leech lattice, with isomorphism group the Conway group $Co_0 = .0$ (dotto), giving us modulo its center the sporadic group $Co_1=.1$ and the other Conway groups $Co_2=.2, Co_3=.3$, and all other sporadics of the second generation in the happy family as subquotients (McL,HS,Suz and $HJ=J_2$)

24 is the central charge of the Monster vertex algebra constructed by Frenkel, Lepowski and Meurman. Most experts believe that the Monster’s reason of existence is that it is the symmetry group of this vertex algebra. John Conway was one among few others hoping for a nicer explanation, as he said in this interview with Alex Ryba.

24 is also an important number in monstrous moonshine, see for example the post the defining property of 24. There’s a lot more to say on this, but I’ll save it for another day.

60

60 is, of course, the order of the smallest non-Abelian simple group, $A_5$, the rotation symmetry group of the icosahedron. $A_5$ is the symmetry group of choice for most viruses but not the Corona-virus.

3264

3264 is the correct solution to Steiner’s conic problem asking for the number of conics in $\mathbb{P}^2_{\mathbb{C}}$ tangent to five given conics in general position.

Steiner himself claimed that there were $7776=6^5$ such conics, but realised later that he was wrong. The correct number was first given by Ernest de Jonquières in 1859, but a rigorous proof had to await the advent of modern intersection theory.

Eisenbud and Harris wrote a book on intersection theory in algebraic geometry, freely available online: 3264 and all that.

248

248 is the dimension of the exceptional simple Lie group $E_8$. $E_8$ is also connected to the monster group.

If you take two Fischer involutions in the monster (elements of conjugacy class 2A) and multiply them, the resulting element surprisingly belongs to one of just 9 conjugacy classes:

1A,2A,2B,3A,3C,4A,4B,5A or 6A

The orders of these elements are exactly the dimensions of the fundamental root for the extended $E_8$ Dynkin diagram.

This is yet another moonshine observation by John McKay and I wrote a couple of posts about it and about Duncan’s solution: the monster graph and McKay’s observation, and $E_8$ from moonshine groups.

163

163 is a remarkable number because of the ‘modular miracle’
$e^{\pi \sqrt{163}} = 262537412640768743.99999999999925…$
This is somewhat related to moonshine, or at least to Klein’s j-function, which by a result of Kronecker’s detects the classnumber of imaginary quadratic fields $\mathbb{Q}(\sqrt{-D})$ and produces integers if the classnumber is one (as is the case for $\mathbb{Q}(\sqrt{-163})$).

The details are in the post the miracle of 163, or in the paper by John Stillwell, Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76.

His description of the $j$-function (at 4:13 in the movie) is simply hilarious!
Borcherds connects $163$ to the monster moonshine via the $j$-function, but there’s another one.
The monster group has $194$ conjugacy classes and monstrous moonshine assigns a ‘moonshine function’ to each conjugacy class (the $j$-function is assigned to the identity element). However, these $194$ functions are not linearly independent and the space spanned by them has dimension exactly $163$.