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https://www.math10.com/en/geometry/sin-cos-rule.html
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# Law of sines, cosines and tangents
### Law of sines
The area of a triangle ABC is given by the formulas:
$A = \frac{a\times c \times \sin(B)}{2} = \frac{b \times c \times \sin(A)}{2} = \frac{a \times b \times \sin(C)}{2}$
=>
$a\times c \times \sin(B) = b\times c \times \sin(A) = a \times b \times \sin(C)$
dividing by $a\times b\times c$, we get the sine formula:
$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$
Let R be the radius of a circle with center O through points A,B and C(for every 3 points that do not lie on a straight line there is exactly 1 circle through these points) of a triangle ABC.
Let B' be the second intersection point of BO and the circle. The
angle B' in the triangle BB'C is equal to A, and the triangle BB'C is a right triangle
=> a = 2Rsin(B') = 2Rsin(A) therefore:
$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}=2R$
## Law of cosines
Let a(the length of BC), b(the length of CA), c(the length of AB) be the lengths of the sides of a triangle ABC.
a2 = b2 + c2 - 2bc cos(∠A)
b2 = c2 + a2 - 2ca cos(∠B)
c2 = a2 + b2 - 2ab cos(∠C)
### Law of tangents
$\frac{a-b}{a+b}=\frac{ \tan\left[\frac{1}{2}(A - B)\right] }{ \tan\left[\frac{1}{2}(A + B)\right] }$
$\frac{b-c}{b+c}=\frac{ \tan\left[\frac{1}{2}(B - C)\right] }{ \tan\left[\frac{1}{2}(B + C)\right] }$
$\frac{a-c}{a+c}=\frac{ \tan\left[\frac{1}{2}(A - C)\right] }{ \tan\left[\frac{1}{2}(A + C)\right] }$
Contact email:
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2021-10-16 02:12:47
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http://www.sagemath.org/calctut/diffrules.html
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A+ a-
# The Rules of Differentiation
### Constant Rule
If and a is a real number, then
.
### Constant Multiple Rule
If and a is a real number, then
.
### Power Rule
If and n is a real number, then
.
### Product Rule
If f and g are differentiable at x, then
### Quotient Rule
If f is the quotient g(x)/h(x) and h(x) ≠ 0, then
### Chain Rule
If g is differentiable at x and f is differentiable at g(x), then
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2014-08-29 18:06:52
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https://www.ionicon.com/publications?page=1&f%5Btg%5D=E
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Scientific Articles - PTR-MS Bibliography
Welcome to the new IONICON scientific articles database!
Publications
Found 63 results
Title [ Year]
Filters: First Letter Of Title is E [Clear All Filters]
2011
[Saha2011] "Effects of airflow on odorants' emissions in a model pig house - A laboratory study using Proton-Transfer-Reaction Mass Spectrometry (PTR-MS).", Sci Total Environ, vol. 410-411: Department of Engineering, Aarhus University, Blichers Allé 20, 8830 Tjele, Denmark. cksahabau@yahoo.com, pp. 161–171, Dec, 2011.
Abstract
Identification of different factors that affect emissions of gasses, including volatile organic compounds (VOCs) is necessary to develop emission abatement technology. The objectives of this research were to quantify and study temporal variation of gas emissions from a model pig house under varying ventilation rates. The used model was a 1:12.5 scale of a section of a commercial finishing pig house. The VOC concentrations at inlet, outlet, and slurry pit of the model space were measured using Proton-Transfer-Reaction Mass Spectrometry (PTR-MS). PTR-MS can measure the temporal variations of odor compounds' emission from the slurry pit in real time. The emissions of H(2)S and 14 VOCs were lower compared to real pig buildings except for ammonia, which indicated possible other sources of those compounds than the slurry in the slurry pit. The ventilation rate affected significantly on ammonia and trimethylamine emission (p<0.05). The hydrogen sulfide (H(2)S) emission was independent of the ventilation rate. VFAs' emission dependency on ventilation rate increased with the increase of carbon chain. Phenols, indoles and ketones showed the positive correlation with ventilation rate to some extent. Generally, compounds with high solubility (low Henry's constant) showed stronger correlation with ventilation rates than the compounds with high Henry's constant.
[1490] "Effects of Mentholation on Cigarette Smoke Emissions", Conference of the Society for Research on Nicotine and Tobacco (SRNT), Toronto, Canada, Conference of the Society for Research on Nicotine and Tobacco (SRNT), Feb., 2011.
[Singh2011] "An endophytic Phomopsis sp. possessing bioactivity and fuel potential with its volatile organic compounds", Microbial ecology, vol. 61, no. 4: Springer, pp. 729–739, 2011.
Abstract
An unusual Phomopsis sp. was isolated as endophyte of Odontoglossum sp. (Orchidaceae), associated with a cloud forest in Northern Ecuador. This fungus produces a unique mixture of volatile organic compounds (VOCs) including sabinene (a monoterpene with a peppery odor) only previously known from higher plants. In addition, some of the other more abundant VOCs recorded by GC/MS in this organism were 1-butanol, 3-methyl; benzeneethanol; 1-propanol, 2-methyl and 2-propanone. The gases of Phomopsis sp. possess antifungal properties and an artificial mixture of the VOCs mimicked the antibiotic effects of this organism with the greatest bioactivity against a wide range of plant pathogenic test fungi including: Pythium, Phytophthora, Sclerotinia, Rhizoctonia, Fusarium, Botrytis, Verticillium, and Colletotrichum. The IC50 values for the artificial gas mixture of Phomopsis sp. varied between 8 and 25.65 μl/mL. Proton transfer reaction-mass spectrometry monitored the concentration of VOCs emitted by Phomopsis sp. and yielded a total VOC concentration of ca. 18 ppmv in the head space at the seventh day of incubation at 23°C on PDA. As with many VOC-producing endophytes, this Phomopsis sp. did survive and grow in the presence of the inhibitory gases of Muscodor albus. A discussion is presented on the possible involvement of VOC production by the fungus and its role in the biology/ecology of the fungus/plant/environmental relationship.
[Fischer2011] "Evaporating Liquid Samples for Analysis with PTR-MS", 5th International PTR-MS Conference on Proton Transfer Reaction Mass Spectrometry and Its Applications, pp. 211–212, 2011.
Abstract
We present a method for measuring liquid samples with the PTR-MS by using a spray to convert the liquid into the gas phase. Advantages over headspace measurements concerning compounds with high Henry's law constants could be demonstrated.
[Cappellin2011] "Extending the dynamic range of proton transfer reaction time-of-flight mass spectrometers by a novel dead time correction.", Rapid Commun Mass Spectrom, vol. 25, no. 1: IASMA Research and Innovation Centre, Fondazione Edmund Mach, Food Quality and Nutrition Area, Via E. Mach 1, 38010 S. Michele a/A, Italy., pp. 179–183, Jan, 2011.
Abstract
Proton transfer reaction time-of-flight mass spectrometry (PTR-TOF-MS) allows for very fast simultaneous monitoring of volatile organic compounds (VOCs) in complex environments. In several applications, food science and food technology in particular, peaks with very different intensities are present in a single spectrum. For VOCs, the concentrations range from the sub-ppt all the way up to the ppm level. Thus, a large dynamic range is necessary. In particular, high intensity peaks are a problem because for them the linear dependency of the detector signal on VOC concentration is distorted. In this paper we present, test with real data, and discuss a novel method which extends the linearity of PTR-TOF-MS for high intensity peaks far beyond the limit allowed by the usual analytical correction methods such as the so-called Poisson correction. Usually, raw data can be used directly without corrections with an intensity of up to about 0.1 ions/pulse, and the Poisson correction allows the use of peaks with intensities of a few ions/pulse. Our method further extends the linear range by at least one order of magnitude. Although this work originated from the necessity to extend the dynamic range of PTR-TOF-MS instruments in agro-industrial applications, it is by no means limited to this area, and can be implemented wherever dead time corrections are an issue.
2010
[Kolarik2010] "The effect of a photocatalytic air purifier on indoor air quality quantified using different measuring methods", Building and Environment, vol. 45, no. 6, pp. 1434 - 1440, 2010.
Abstract
The effect on indoor air quality of an air purifier based on photocatalytic oxidation (PCO) was determined by different measuring techniques: sensory assessments of air quality made by human subjects, Proton-Transfer-Reaction Mass Spectrometry (PTR-MS) and chromatographic methods (Gas Chromatography/Mass Spectrometry and High-Pressure Liquid Chromatography with \{UV\} detection). The experiment was conducted in a simulated office, ventilated with 0.6 h−1, 2.5 h−1 and 6 h−1, in the presence of additional pollution sources (carpet, chipboard and linoleum). At the lowest air change rate, additional measurements were made with no pollution sources present in the office. All conditions were tested with the photocatalytic air purifier turned on and off. The results show that operation of the air purifier in the presence of pollutants emitted by building materials and furniture improves indoor air quality, as documented by sensory assessments made by human subjects. It also reduces concentrations of many chemical compounds present in the air as documented by the PTR-MS technique. For the lowest ventilation, results from measurements using the chromatographic methods have similar tendency, however many of the 50 compounds that were targeted for analysis were not detected at all, independent of whether the purifier was on or off. For the two conditions with higher ventilation the results were inconclusive.
[Feilberg2010b] "Effects of air Exchange, Temperature and slurry management on odorant Emissions from pig Production units and slurry tanks studied by Proton-Transfer-Reaction mass spectometry (PTR-MS)", World Congress of the International Commission of Agricultural and Biosystems Engineering (CIGR), 2010.
[Karl2010] "Efficient atmospheric cleansing of oxidized organic trace gases by vegetation", Science, vol. 330, no. 6005: American Association for the Advancement of Science, pp. 816–819, 2010.
Abstract
The biosphere is the major source and sink of nonmethane volatile organic compounds (VOCs) in the atmosphere. Gas-phase chemical reactions initiate the removal of these compounds from the atmosphere, which ultimately proceeds via deposition at the surface or direct oxidation to carbon monoxide or carbon dioxide. We performed ecosystem-scale flux measurements that show that the removal of oxygenated VOC via dry deposition is substantially larger than is currently assumed for deciduous ecosystems. Laboratory experiments indicate efficient enzymatic conversion and potential up-regulation of various stress-related genes, leading to enhanced uptake rates as a response to ozone and methyl vinyl ketone exposure or mechanical wounding. A revised scheme for the uptake of oxygenated VOCs, incorporated into a global chemistry-transport model, predicts appreciable regional changes in annual dry deposition fluxes.
[Kim2010] "Emissions and ambient distributions of Biogenic Volatile Organic Compounds (BVOC) in a ponderosa pine ecosystem: interpretation of PTR-MS mass spectra", Atmospheric Chemistry and Physics, vol. 10, no. 4: Copernicus GmbH, pp. 1759–1771, 2010.
Abstract
Two proton-transfer-reaction mass spectrometry systems were deployed at the Bio-hydro-atmosphere interactions of Energy, Aerosols, Carbon, H2O, Organics and Nitrogen-Southern Rocky Mountain 2008 field campaign (BEACHON-SRM08; July to September, 2008) at the Manitou Forest Observatory in a ponderosa pine woodland near Woodland Park, Colorado USA. The two PTR-MS systems simultaneously measured BVOC emissions and ambient distributions of their oxidation products. Here, we present mass spectral analysis in a wide range of masses (m/z 40+ to 210+) to assess our understanding of BVOC emissions and their photochemical processing inside of the forest canopy. The biogenic terpenoids, 2-methyl-3-butene-2-ol (MBO, 50.2%) and several monoterpenes (MT, 33.5%) were identified as the dominant BVOC emissions from a transmission corrected mass spectrum (PTR-MS), averaged over the daytime (11 a.m. to 3 p.m., local time) of three days. To assess contributions of oxidation products of local BVOC, we calculate an oxidation product spectrum with the OH- and ozone-initiated oxidation product distribution mass spectra of two major BVOC emissions at the ecosystem (MBO and β-pinene) that were observed from laboratory oxidation experiments. The majority ( 76%) of the total signal in the transmission corrected PTR-MS spectra could be explained by identified compounds. The remainder are attributed to oxidation products of BVOC emitted from nearby ecosystems and transported to the site, and oxidation products of unidentified BVOC emitted from the ponderosa pine ecosystem.
[Riess2010] "Experimental setup and analytical methods for the non-invasive determination of volatile organic compounds, formaldehyde and NOx in exhaled human breath.", Anal Chim Acta, vol. 669, no. 1-2: Hannover Medical School, Sports Physiology and Sports Medicine, Carl-Neuberg-Str. 1, 30625 Hannover, Germany., pp. 53–62, Jun, 2010.
Abstract
Different analytical devices were tested and evaluated for their suitability of breath gas analysis by examining the physiological parameters and chemical substances in the exhaled breath of ten healthy probands during light cycling in dependence of methanol-rich nutrition. The probands exercised under normal breathing conditions on a bicycle ergometer. Breath air was exhaled into a glass cylinder and collected under steady-state conditions. Non-invasively measured parameters were pulse rate, breath frequency, temperature, relative humidity, NO(x), total volatile organic compounds (TVOC(PAS)), carbon dioxide (CO(2)), formaldehyde, methanol, acetaldehyde, acetone, isoprene and volatile organic compounds (VOCs). Methanol rich food and beverages strongly influenced the concentration of methanol and other organic substances in human breath. On the other hand, nutrition and smoking had no clear effect on the physical conditions of the probands. The proton transfer reaction mass spectrometry (PTR-MS) method was found to be very suitable for the analysis of breath gas but the m/z 31, if assigned to formaldehyde, is sensitive to interferences. The time vs. concentration curves of nitric oxide showed sudden peaks up to 120ppb in most of the measurements. In one case a strong interference of the NO(x) signal was observed. The time resolved analysis of exhaled breath gas is of high capability and significance for different applications if reliable analytical techniques are used. Some compounds like nitric oxide (NO), methanol, different VOCs as well as sum parameters like TVOC(PAS) are especially suitable as markers. Formaldehyde, which is rapidly metabolized in the human body, could be measured reliably as a trace component by the acetylacetone (acac) method but not by PTR-MS.
[Jordan2010] "Extremely high mass resolution and sensitivity-comparison of two novel proton transfer reaction time-of-flight mass spectrometers (PTR-TOFMS)", Verhandlungen der Deutschen Physikalischen Gesellschaft, vol. -, no. Hanver 2010 issue, pp. -, 2010.
Abstract
Since many years PTR-MS is a well established technique in trace gas analysis with its major advantages of having very short response times of below 100ms and outstanding detection limits in the single digit pptv region. However, the quadrupole mass filter based instruments used so far cannot separate isobaric compounds due to lack of mass resolution. To overcome this problem Ionicon developed the so called PTR-TOF 8000 instrument, which couples the well established PTR ionization technique with a high resolution time-of-flight (TOF) mass analyzer. In contrast to a quadrupole based PTR-MS where only one nominal mass at a time can be monitored, the PTR-TOF acquires whole mass spectra in split-seconds at a resolution of up to 8.000 m/{delta}m (FWHM). As there might be applications where an enormous mass resolution is not necessarily needed, but the sensitivity has to be as high as possible, we now developed an instrument (called PTR-TOF 2000) that performs with an enhanced sensitivity at the expense of a somewhat lower mass resolution.
2009
[Sinha2009] "The effect of relative humidity on the detection of pyrrole by PTR-MS for OH reactivity measurements", International Journal of Mass Spectrometry, vol. 282, no. 3: Elsevier, pp. 108–111, 2009.
Abstract
The hydroxyl radical (OH) is the most important atmospheric oxidant. Recently Sinha et al. [V. Sinha, J. Williams, J.N. Crowley, J. Lelieveld, Atmos. Chem. Phys. 8 (2008) 2213] developed a new method to measure the total OH reactivity of ambient air (OH sink) employing a proton transfer reaction mass spectrometer (PTR-MS) as a detector. The new method uses pyrrole (C4H4NH) as a reagent and for an OH reactivity measurement this species must be measured under both dry (∼0% RH) and humid air (>30% RH). Here, we investigate the sensitivity dependence of the PTR-MS for pyrrole, as a function of relative humidity in the sampled air. Various normalizations with respect to the H3O+ ion and its different hydrated cluster ions H3O+(H2O)n=1,2,3 are compared. It is shown that both the primary ion signal (H3O+ ion m/z = 19) and the first water cluster ion H3O+(H2O) (m/z = 37) should be used for pyrrole quantification. However, in spite of using this normalization, the PTR-MS sensitivity for pyrrole changes by as much as 16% between dry (∼0% RH) and humid air (above 30% RH), with higher sensitivity when the sampled air is humid. Thus, for accurate quantification of pyrrole using a PTR-MS, calibration factors appropriate to dry and humid air should be employed. We recommend that humidity dependence of the PTR-MS be taken into account when reactivity measurements are performed using the pyrrole based comparative reactivity method (CRM).
[Leclercq2009] "Effects of cross-linking, capsule wall thickness, and compound hydrophobicity on aroma release from complex coacervate microcapsules.", J Agric Food Chem, vol. 57, no. 4: Department of Food Science and Nutrition, University of Minnesota, 1334 Eckles Avenue, St. Paul, Minnesota 55108, USA. lecle003@umn.edu, pp. 1426–1432, Feb, 2009.
Abstract
Microcapsules were produced by complex coacervation with a gelatin-gum acacia wall and medium-chain-triglyceride core. Dry capsules were partially rehydrated and then loaded with model aroma compounds covering a range of volatility, hydrophobicity, and molecular structure. An experimental design was prepared to evaluate the effects of cross-linking, wall/core ratio, and volatile load level on aroma release from capsules in a hot, aqueous environment. The real-time release on rehydration was measured by monitoring the headspace of a vessel containing the capsules to proton transfer reaction mass spectrometry (PTR-MS). Data collected showed no effects of cross-linking or wall/core ratio on volatile release in hot water for any of the volatiles studied. When comparing real-time release of the prepared coacervates to a spray-dried equivalent, there was no difference in the release from hot water but the release was slower when coacervates were added to ambient-temperature water. We found volatile release to be primarily determined by compound partition coefficients (oil/water and water/air) and temperature.
[Gasperi2009] "Effects of supercritical CO< sub> 2 and N< sub> 2 O pasteurisation on the quality of fresh apple juice", Food chemistry, vol. 115, no. 1: Elsevier, pp. 129–136, 2009.
Abstract
Supercritical pasteurisation is receiving increasing attention as an alternative technology for foodstuff pasteurisation, but often the possible effects on the perceptible quality are not sufficiently considered. To address this latter issue, besides standard microbial analysis, we here investigate the impact of CO2/N2O supercritical pasteurisation (100 bar, 36 °C and 10 min treatment time) on the quality traits of fresh apple juice, linked to consumer perception. Discriminative sensory analysis (triangle test) and basic chemical characterization (total solids, sugars, organic acids, polyphenols) could not clearly demonstrate any induced modification of the treated juice, while head space analysis of volatile compounds (both by GC–MS and PTR–MS) indicated a general depletion of the volatile compounds that must be considered in the development of a stabilization method based on supercritical gases.
[Karl2009] "Emissions of volatile organic compounds inferred from airborne flux measurements over a megacity", Atmospheric Chemistry and Physics, vol. 9, no. 1: Copernicus GmbH, pp. 271–285, 2009.
Abstract
Toluene and benzene are used for assessing the ability to measure disjunct eddy covariance (DEC) fluxes of Volatile Organic Compounds (VOC) using Proton Transfer Reaction Mass Spectrometry (PTR-MS) on aircraft. Statistically significant correlation between vertical wind speed and mixing ratios suggests that airborne VOC eddy covariance (EC) flux measurements using PTR-MS are feasible. City-median midday toluene and benzene fluxes are calculated to be on the order of 14.1±4.0 mg/m2/h and 4.7±2.3 mg/m2/h, respectively. For comparison the adjusted CAM2004 emission inventory estimates toluene fluxes of 10 mg/m2/h along the footprint of the flight-track. Wavelet analysis of instantaneous toluene and benzene measurements during city overpasses is tested as a tool to assess surface emission heterogeneity. High toluene to benzene flux ratios above an industrial district (e.g. 10–15 g/g) including the International airport (e.g. 3–5 g/g) and a mean flux (concentration) ratio of 3.2±0.5 g/g (3.9±0.3 g/g) across Mexico City indicate that evaporative fuel and industrial emissions play an important role for the prevalence of aromatic compounds. Based on a tracer model, which was constrained by BTEX (BTEX– Benzene/Toluene/Ethylbenzene/m, p, o-Xylenes) compound concentration ratios, the fuel marker methyl-tertiary-butyl-ether (MTBE) and the biomass burning marker acetonitrile (CH3CN), we show that a combination of industrial, evaporative fuel, and exhaust emissions account for >87% of all BTEX sources. Our observations suggest that biomass burning emissions play a minor role for the abundance of BTEX compounds in the MCMA (2–13%).
[OHara2009a] "Endogenous volatile organic compounds in breath and blood of healthy volunteers: examining breath analysis as a surrogate for blood measurements", Journal of Breath Research, vol. 3, no. 2, pp. 027005, 2009.
Abstract
To investigate the premise that levels of endogenous volatile organic compounds (VOC) in breath reflect those in blood, the concentration of acetone and isoprene were measured in radial arterial blood, peripheral venous blood and breath samples from ten healthy volunteers. Coefficients of repeatability as a percentage of mean are less than 30% in breath but greater than 70% in blood. The volunteer-mean ratios of arterial to venous blood concentration are 1.4 (0.9-2.1) for acetone and 0.55 (0.3-1.0) for isoprene. Concentration in breath showed a significant inter-subject correlation with concentration in arterial blood (CAB) for acetone but not for isoprene. Arterial blood/breath ratios are 580 (280-1060) for acetone and 0.47 (0.22-0.77) for isoprene. The sample-mean blood/breath ratio was used to calculate an estimate of CAB and the standard deviation of this estimate was lower than that of arterial blood measured directly. For most subjects, estimated CAB was within uncertainty limits of the actual CAB. Owing to the poor repeatability of VOC concentrations from consecutive blood samples, and the capacitive effects of the lung, this study suggests that breath VOC measurements may provide a more consistent measure than blood measurements for investigating underlying physiological function or pathology within individuals.
[Kameyama2009] "Equilibrator inlet-proton transfer reaction-mass spectrometry (EI-PTR-MS) for sensitive, high-resolution measurement of dimethyl sulfide dissolved in seawater.", Anal Chem, vol. 81, no. 21: National Institute for Environmental Studies, Tsukuba, 305-8506, Japan., pp. 9021–9026, Nov, 2009.
Abstract
We developed an equilibrator inlet-proton transfer reaction-mass spectrometry (EI-PTR-MS) method for fast detection of dimethyl sulfide (DMS) dissolved in seawater. Dissolved DMS extracted by bubbling pure nitrogen through the sample was continuously directed to the PTR-MS instrument. The equilibration of DMS between seawater and the carrier gas, and the response time of the system, were evaluated in the laboratory. DMS reached equilibrium with an overall response time of 1 min. The detection limit (50 pmol L(-1) at 5 s integration) was sufficient for detection of DMS concentrations in the open ocean. The EI-PTR-MS instrument was deployed during a research cruise in the western North Pacific Ocean. Comparison of the EI-PTR-MS results with results obtained by means of membrane tube equilibrator-gas chromatography/mass spectrometry agreed reasonably well on average (R(2) = 0.99). EI-PTR-MS captured temporal variations of dissolved DMS concentrations, including elevated peaks associated with patches of high biogenic activity. These results demonstrate that the EI-PTR-MS technique was effective for highly time-resolved measurements of DMS in the open ocean. Further measurements will improve our understanding of the biogeochemical mechanisms of the production, consumption, and distribution of DMS on the ocean surface and, hence, the air-sea flux of DMS, which is a climatically important species.
[Maleknia2009] "Eucalypt smoke and wildfires: Temperature dependent emissions of biogenic volatile organic compounds", International Journal of Mass Spectrometry, vol. 279, no. 2: Elsevier, pp. 126–133, 2009.
Abstract
Eucalypt contributions to biogenic sources of volatile organic compounds (VOCs) in Australia are estimated at teragram (Tg = 1012 g) amounts each year. Biogenic VOCs include plant-specific isoprenoids (isoprene and a range of terpenes) and other reactive organic compounds (i.e., acids, aldehydes and ketones). Atmospheric reactions of VOCs are numerous and many have significant environmental impact. Wildfires increase both the amounts of VOCs released and the complexity of their reactions. Proton-transfer reaction mass spectrometry (PTR-MS), gas chromatography mass spectrometry (GCMS) and direct analysis in real time (DART) mass spectrometry were applied to analyze release of VOCs as a function of temperatures ranging from ambient to combustion. PTR-MS enabled trace level analysis of VOCs from a complex forest atmosphere and revealed the release of terpenes associated with leaf damage during a storm. Temperature profile studies revealed ion abundances (i.e., emissions of VOCs) could be correlated with boiling points and vapor pressures of specific compounds. PTR-MS analysis of VOCs resulting from heating fresh leaf (E. grandis) material suggested that emissions of protonated methanol (m/z 33) and protonated acetaldehyde (m/z 45) were greatest at ∼60 °C while m/z 137 and 153 (associated with a series of terpenes) showed monotonic increases in ion abundance over a wide temperature range from ambient to 200 °C. GCMS analysis of fresh and senescent leaves of E. grandis showed that a series of VOCs (ethylvinylketone, diethylketone, 2-ethylfuran, hexanal and hexenals) are present only in fresh leaves while several terpenes (α and β pinenes, α-phellandrene, eucalyptol, γ-terpinene) were common in both. DART analysis of fresh leaf and stem of E. sideroxylon identified tissue-specific VOCs (e.g., methanol and ethanol were more abundant in stems). PTR-MS combustion studies of senescent leaves (E. grandis) identified two distinct, temperature-dependent VOC compositions. Before the appearance of smoke, the composition of VOCs remained consistent and correlated well with various naturally occurring isoprenoids, as observed in temperature profile studies. Sampling of eucalypt smoke suggested ions (m/z 75, 85, 87, 99, 111 and 125) correlated with protonated mass of oxygenated aldehydes, ketones, furans and substituted benzenes, and were due to pyrolysis of polycarbohydrates (cellulose and lignin) that are common in many types of wood.
[Amann2009] "Exhaled breath analysis-quantifying the storage of lipophilic compounds in the human body", Proceedings of Ecopole, vol. 3, pp. 9–13, 2009.
2008
[Keck2008] Keck, L., C. Hoeschen, and U. Oeh, "Effects of carbon dioxide in breath gas on proton transfer reaction-mass spectrometry (PTR-MS) measurements", International Journal of Mass Spectrometry, vol. 270, no. 3: Elsevier, pp. 156–165, 2008.
Abstract
PTR-MS is becoming a common method for the analysis of volatile organic compounds (VOCs) in human breath. Breath gas contains substantial and, particularly for bag samples, highly variable concentrations of water vapour (up to ∼6.3%) and carbon dioxide (up to ∼6.5%). The goal of this study was to investigate the effects of carbon dioxide on PTR-MS measurements; such effects can be expected in view of the already well known effects of water vapour. Carbon dioxide caused an increase of the pressure in the PTR-MS drift tube (∼1% increase for 5% CO2), and this effect was used to assess the CO2 concentration of breath gas samples along the way with the analysis of VOCs. Carbon dioxide enhanced the concentration ratio of protonated water clusters (H3O+H2O) to protonated water (H3O+) in the drift tube. Using the observed increase, being ∼60% for 5% CO2, it is estimated that the mobility of water cluster ions in pure CO2 is almost 65% lower than in air. Carbon dioxide had a significant effect on the mass spectra of the main breath gas components methanol, ethanol, 1-propanol, 2-propanol, acetone, and isoprene. Carbon dioxide caused a small increase (<10% for 5% CO2) of the normalised main signals for the non-fragmenting molecules methanol and acetone. The increase can be much higher for the fragmenting VOCs (ethanol, propanol, and isoprene) and was, for 5% CO2, up to ∼60% for ethanol. This effect of CO2 on fragment patterns is mainly a consequence of the increased abundance of protonated water clusters, which undergo softer reactions with VOCs than the hydronium ions. Breath gas samples stored in Teflon bags lost ∼80% of CO2 during 3 days, the decrease of VOC signals, however, is mainly attributed to decreasing VOC concentrations and to the loss of humidity from the bags.
[Kim2008] "Emissions and Photochemistry of BVOCs in a Ponderosa Pine woodland", AGU Fall Meeting Abstracts, vol. 1, pp. 0057, 2008.
Abstract
We deployed two proton-transfer-reaction mass spectrometry instruments (PTR-MS, IONICON ANALYTIK) for ambient and branch enclosure measurements at the Manitou Experimental Forest, located in the Southern Rocky Mountain area as a part of the Bio-hydro-atmosphere interactions of Energy, Aerosols, Carbon, H2O, Organics and Nitrogen (BEACHON) field campaign in 2008. Vegetation at the field site is dominated by Ponderosa Pine. BVOC emissions from Ponderosa Pine along with temperature, photosynthetic photon flux density (ppfd), relative humidity, and CO2 uptake were measured from two branch-enclosures (shade and sun). Diurnal variations and the emission response to environmental conditions are described and compared to existing models. In addition, we analyzed the speciation of BVOCs from enclosures by GC-MS. We will present quantitative and qualitative characteristics of BVOC emissions from Ponderosa Pine and analytical characteristics of PTR-MS such as fragmentation patterns of semi-volatile compounds (sesquiterpene, bornyl acetate etc) that we identified as major emissions from the enclosures. BVOC emissions observed in the enclosures will be quantitatively compared to BVOC distributions in ambient air. We explore the presence of possibly unidentified BVOCs in the forest canopy by examining PTR-MS mass spectra of enclosure and ambient air samples based on mass scans between 40 - 210 amu.
2007
[Song2007] "Effect of hydrophobic primary organic aerosols on secondary organic aerosol formation from ozonolysis of $\alpha$-pinene", Geophysical Research Letters, vol. 34, no. 20, 2007.
Abstract
Semi-empirical secondary organic aerosol (SOA) models typically assume a well-mixed organic aerosol phase even in the presence of hydrophobic primary organic aerosols (POA). This assumption significantly enhances the modeled SOA yields as additional organic mass is made available to absorb greater amounts of oxidized secondary organic gases than otherwise. We investigate the applicability of this critical assumption by measuring SOA yields from ozonolysis of α-pinene (a major biogenic SOA precursor) in a smog chamber in the absence and in the presence of dioctyl phthalate (DOP) and lubricating oil seed aerosol. These particles serve as surrogates for urban hydrophobic POA. The results show that these POA did not enhance the SOA yields. If these results are found to apply to other biogenic SOA precursors, then the semi-empirical models used in many global models would predict significantly less biogenic SOA mass and display reduced sensitivity to anthropogenic POA emissions than previously thought.
[1503] "Emission, oxidation, and secondary organic aerosol formation of volatile organic compounds as observed at Chebogue Point, Nova Scotia", Journal of Geophysical Research, vol. 112, 2007.
Abstract
<p>We report the detection of a class of related oxygenated compounds by proton-transfer-reaction mass-spectrometry (PTR-MS) that have rarely or never been observed as a group using in situ instrumentation. Measurements were made as part of the International Consortium for Atmospheric Research on Transport and Transformation (ICARTT) 2004 in Chebogue Point, Nova Scotia. The detected class of compounds discussed here includes acetic acid, formaldehyde, acetaldehyde, tentatively identified formic acid and hydroxyacetone, and unidentified compounds detected at mass to charge ratios 85, 87, 99, 101, 113, 115, and 129. Typical concentrations were 800, 2500, 450, 700, 85, 25, 50, 50, 60, 35, 20, and 25 ppt, respectively. The uniqueness of this class of compounds is illustrated by showing they were poorly related to trace gases found in the US outflow, local pollution, primary biogenic emissions and other oxygenated compounds such as acetone, methanol, and MEK measured by other in situ instrumentation. On the other hand these oxidized volatile organic compounds were related to chemical species in aerosols and their abundance was high during nucleation events. Thus they likely are gas phase species that are formed in parallel to biogenic secondary organic aerosol production. We clearly show these compounds do not originate from local sources. We also show these compounds match the oxidation products of isoprene observed in smog chamber studies, and we therefore suggest they must be mainly produced by oxidation of biogenic precursor compounds.</p>
[GomezAlvarez2007] "Experimental confirmation of the dicarbonyl route in the photo-oxidation of toluene and benzene.", Environ Sci Technol, vol. 41, no. 24: Fundación Centro de Estudios Ambientales del Mediterráneo (CEAM), C/Charles Darwin 14, 46980 Paterna, Valencia, Spain. elena@ceam.es, pp. 8362–8369, Dec, 2007.
Abstract
The methodology of solid phase microextraction (SPME) with O-(2,3,4,5,6)-pentafluorobenzylhydroxylamine hydrochloride (PFBHA) on-fiber derivatization for the determination of carbonyls has been applied to the photo-oxidation of benzene and toluene carried out in the EUPHORE chambers. This work focuses on the results obtained for a number of highly reactive carbonyls, crucial in the determination of branching ratios and confirmation of the carbonylic route. The observed yields and kinetic behavior were compared to simulations with the Master Chemical Mechanism model, version 3.1 (MCMv3.1). The following yields were measured in the toluene system: glyoxal, (37 +/- 2)%; methylglyoxal, (37 +/- 2)%; 4-oxo-2-pentenal, > (13.8 +/- 1.5)%; and total butenedial, (13 +/- 7)% (cis-butenedial, (6 +/- 3)%; trans-butenedial, (7 +/- 4)%]. For benzene, the experimental glyoxal yields were (42 +/- 3) and (36 +/- 2)% for the two successive experiments (September 24 and 25, 2003), (17 +/- 9)% for total butenedial [(8 +/- 4)% cis-butenedial and (9 +/- 5)% trans-butenedial (September 24, 2003)] and (15 +/- 6)% total butenedial (September 25, 2003) [(7 +/- 3) and (7 +/- 3)% for the cis and trans isomers, respectively]. PTR-MS estimations for butenedial also allowed the two isomers of butenedial to be distinguished, but the measurements showed signs of interference from other products. The results presented confirm the fast ring cleavage and provide further experimental confirmation of the dicarbonylic route.
2006
[Klemm2006] "Experiments on forest/atmosphere exchange: Climatology and fluxes during two summer campaigns in NE Bavaria", Atmospheric Environment, vol. 40: Elsevier, pp. 3–20, 2006.
Abstract
During two summer field campaigns in 2001 and 2002, biosphere/atmosphere exchange fluxes of energy, gases, and particles were quantified in a Norway spruce forest in NE Bavaria at 775 m a.s.l. The overall goal of the BEWA campaigns was to study the influence of the emissions of reactive biogenic volatile organic compounds (BVOCs) on chemical and physical processes in the atmosphere, and an overview over the meteorological conditions, experimental frame, and the achieved results is provided. A rigorous quality assurance/quality control plan was implemented. From analysis of meteorological conditions and experimental success, golden day periods were selected for coordinated data analysis. These periods cover typical summertime conditions with various wind directions, NOx mixing ratios between 2 and 10 ppb, and O3 mixing ratios ranging between 13 and 98 ppb. Diurnal patterns of trace gas concentrations resulted from the dynamics of the boundary layer, from regional atmospheric processes (for example production of O3 in the atmosphere), and deposition. Turbulence also exhibited a diurnal pattern indicating thermal production during daytime and calm conditions during nighttime. However, in many cases, turbulence was often well developed during the nights. Horizontal advection of air masses into the trunk space occurred due to the patchiness of the forest. Nevertheless, for most conditions, the application of a one-dimensional model to describe the vertical exchange processes was appropriate. Therefore, the use of one single meteorological tower to study biosphere/atmosphere exchange is valid. Measured turbulent vertical exchange fluxes were estimated to be representative within an error of less than 25%. The results for VOC concentrations and fluxes were rather heterogeneous. Both model and measurements demonstrated that the Norway spruce trees acted as a weak source of formaldehyde.
Featured Articles
Download Contributions to the International Conference on Proton Transfer Reaction Mass Spectrometry and Its Applications:
Selected PTR-MS related Reviews
F. Biasioli, C. Yeretzian, F. Gasperi, T. D. Märk: PTR-MS monitoring of VOCs and BVOCs in food science and technology, Trends in Analytical Chemistry 30 (7) (2011).
J. de Gouw, C. Warneke, T. Karl, G. Eerdekens, C. van der Veen, R. Fall: Measurement of Volatile Organic Compounds in the Earth's Atmosphere using Proton-Transfer-Reaction Mass Spectrometry. Mass Spectrometry Reviews, 26 (2007), 223-257.
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2019-08-19 22:27:08
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https://support.bioconductor.org/p/122808/
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Question: Can't run with gff3 file
0
9 weeks ago by
algenubi810 wrote:
Hello! I can't run featureCounts with gff3 file. Please, help!
My gff3 file structure is:
chr1 . miRNA_primary_transcript 17369 17436 . - . ID=MI0022705;Alias=MI0022705;Name=hsa-mir-6859-1
chr1 . miRNA 17409 17431 . - . ID=MIMAT0027618;Alias=MIMAT0027618;Name=hsa-miR-6859-5p;Derives_from=MI0022705
chr1 . miRNA 17369 17391 . - . ID=MIMAT0027619;Alias=MIMAT0027619;Name=hsa-miR-6859-3p;Derives_from=MI0022705
Command line:
./featureCounts -F -a ./../annotation/hsa_miR.gff3 -o ./../../../projects/vesicles/1_repeat_vesicles/counts_K.txt ./../../../projects/vesicles/1_repeat_vesicles/1_repeat_vesicles_NONCODE_K_miRNA.sam
Warning: Unknown annotation format: -a. GTF format is used.
Please tell me what to do, I have already tried everything. Carefully studied the manual,
-F(isGTFAnnotationFile): Specify the format of the annotation file. Acceptable formats include ‘GTF’ and ‘SAF’ (see Section 6.2.2 for details). By default,C version of featureCounts program accepts a GTFformat annotation and R version accepts a SAF format annotation. In-built annotations in SAF format are provided.
tried to use cufflinks/gffread to change the file format, but nothing happened either.
./gffread ./../../tools/subread-1.6.4-Linux-x86_64/annotation/hsa_miR.gff3 -T -o hsa_miR.gtf
modified 9 weeks ago by Gordon Smyth38k • written 9 weeks ago by algenubi810
Answer: Can't run with gff3 file
1
9 weeks ago by
Wei Shi3.2k
Australia
Wei Shi3.2k wrote:
You did not use the -F option correctly - you need to provide a value to it. For your case, you can specify this option as -F 'GTF'. You actually do not need to specify this option at all as 'GTF' is the default value.
The 'GTF' value denotes both GTF format and compatible GFF formats such as GFF3 format used in your analysis. This is described in the screen description for this parameter (you can see this by just typing featureCounts at your prompt):
-F <string> Specify format of the provided annotation file. Acceptable
formats include 'GTF' (or compatible GFF format) and
'SAF'. 'GTF' by default. For SAF format, please refer to
Users Guide.
Also the default values for attribute type and feature type parameters won't work for your annotation. You might use the following setting for these two parameters:
-t 'miRNA' -g 'ID'
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2019-09-16 20:36:58
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https://socratic.org/questions/how-do-you-evaluate-the-expression-2x-1-for-x-1
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# How do you evaluate the expression 2x+1 for x=1?
Oct 1, 2014
Put the value of x in the given expression;
$\left(2 x + 1\right)$ ; x =1
$\left(2.1 + 1\right)$
3
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2019-03-19 04:26:09
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https://tryolabs.com/blog/machine-learning-basics-every-manager-should-know
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blog
# Machine Learning basics every manager should know
Mon, Mar 25, 2013
Last update: June 2018
## Introduction
The use of Machine Learning for business is growing every day. Consequently, there's an increasing amount of available information about it, which makes it easy to get lost in the Machine Learning jungle.
In this article we'll provide an overview of the most important Machine Learning concepts, explain how they could be applied to businesses and how to measure the success of it.
## What is Machine Learning?
According to Adam Geitgey, Machine learning is the idea that there are generic algorithms that can tell you something interesting about a set of data without you having to write any custom code specific to the problem. Instead of writing code, you feed data to the generic algorithm and it builds its own logic based on the data.
## Why using Machine Learning for business?
Very often we get people contacting us for projects in which they envision the usage of some Machine Learning (ML) techniques to solve a specific problem they have. Sometimes these people do not have any automated system and are solving whatever problem they currently have with human labor. In these cases, the mere fact of being able to produce anything that works reasonably well can make a huge difference for them.
Imagine, for example, that your company wants you to review 100,000 tweets that make reference to its brand name, because they want to know which customers are not satisfied with products they are selling and why. Wouldn’t it be cool for you if I gave you a tool that can tell you that out of the 100,000 tweets, 95,000 are not even worth looking at because they don’t even talk about products or don’t express negative feelings? It would be a huge time saver.
But there are other times in which people already have working solutions in place. If the problem is very complex, they might not be happy with how their solution performs. Sometimes even a performant solution might need a replacement.
Imagine you have come up with a rule-based engine to process some type data, and then more data starts coming in that is not adequately handled with the rules you have. This can mean that most of the rules that once were good need to be rewritten, and probably also that new rules need to be created. The complexity of the system will not cease to increase. This is another case in which a ML solution might come in handy: let’s make an algorithm figure out the rules for us. Let’s hope it performs better than any non-ML solution we have come up with so far.
## How is Machine Learning applied to business?
Some practical applications of Machine Learning in business are Image Processing, Text Analysis or Data Mining.
### Image processing
This is the analysis of images/videos to extract data or do some transformations. Examples are face recognition (ie. Facebook's automatic photo tagging), self-driving cars and optical character recognition (digitize text found in images).
### Text analysis
In text analysis, information from text is extracted or classified. Examples are spam filtering (as Gmail does), sentiment analysis (identify the mood of the text, ie. positive, negative or neutral) and information extraction (identify key data from a text, for example: an entity).
### Data mining
The aim of data mining is to discover patterns or making predictions from data. Examples are anomaly detection (automatic credit card fraud detection), clustering (ie. grouping data based on some characteristics) and predicting variables (ie. predicting stock market prices based on past fluctuations and current events).
## What are types of Machine Learning problems?
We can distinct between two main types of Machine Learning problems: Supervised learning and Unsupervised learning. There's also Semi-supervised learning, which is less used and not discussed further in this article.
### Supervised learning
Supervised learning is the most popular type of Machine Learning algorithms. It is used to find patterns in raw data, based on the results of pretrained data.
To get the pretrained data, the data scientist feeds the algorithm with input data and lets the algorithm predict the outcome. For example, it feeds it with a set of dog and cat images and the algorithm needs to assign the corresponding animal to each of them. If it does not predict right, the data scientist acts like a teacher and corrects the results.
This training process goes on until the desired level of performance is achieved.
### Unsupervised learning
Unsupervised learning refers to the task of extracting patterns and hidden structure from raw data without extra information, as opposed to supervised learning where labels are needed.
Unlike supervised learning, we don't have a "teacher" that tells what the correct labels are. In fact, we don't even know what the set of possible labels is. Algorithms are left to their own devices to find relevant structures in the data. There is no "wrong" or "right" answer here.
Supervised vs unsupervised learning
## What are examples of Machine Learning algorithms?
### Classification
A classification task means identifying or predicting which out of a set of categories/labels should be assigned to some data. The output variable is discrete, such as "black" or "white" and "positive" or "negative". An example of such as task is the spam filter, which assigns a spam or no spam status to every email.
### Regression
In contrast to classification problems, the output variable in regression tasks is a real value, such as "dollars" or "weight", making the prediction a continuous, non-discrete output. A common example of regression is predicting stock market prices.
Classification vs Regression
## What are Machine Learning metrics?
It happened in the past that we received contracts, in which the client wanted us to commit to the fact that, by the end of the a classification project, our system would have to achieve 95% accuracy in a classification task. I immediately asked myself two questions:
1. Why 95%?
2. Why accuracy?
In what follows, we're going to analyze the first question. This post deals with measures of performance and why accuracy in particular is deceiving.
It is clear that when developing a classification system, its going to be very useful to have an objective metric by which we can know how well it performs. But saying 95% because it sounds good is arbitrary unless you have previously done some work to make sure this is indeed achievable.
### Why 95% accuracy is not an objective metric
As arbitrary as it is, being right 95% of the times may be an indicator of a great performance for some tasks, while a sign of terrible performance in some others. For example, imagine a dataset in which 99% of the data belongs to class A, and the remaining 1% belongs to class B. Dataset likes this are called imbalanced datasets (guess you weren't expecting that, huh?). A classifier that always says "class A" will be right 99% of the time. I guess no customer would be happy if a contract-compliant solution turned out to be a one-liner like return True or such :-)
### Establishing a baseline
What we do in ML tasks first is establishing a baseline. A baseline can be the measure of performance of a very simple system. For example, for a spam detection task a baseline may very well be the output of a system which classifies an email as spam if it contains words such as viagra, pharmacy, etc. Ideally it should be a bit more complex than that. A baseline could also be the performance of a previous, existing system which we wish to improve upon. After establishing what the baseline is, we know a ML solution has to perform at least as good as it. If it doesn't achieve that then we are doing something wrong or need a larger tagged dataset for training (learning), unless the baseline was a really complex system in itself (then maybe it wasn't so much base-line after all).
But even knowing the baseline, and assuming it is less than the 95% our client wanted us to achieve, one cannot promise a certain level of accuracy above it before the ML algorithm is actually built (and 'tweaked' and tested).
### Defining the ceiling
How good could it possibly get? For many tasks, achieving 100% accuracy is never possible. And I don't mean just really hard, I actually mean impossible. This is not always the fault of the algorithms: there are algorithms and implementations which can make computers achieve human level performance for certain tasks. The problem is this: when using two humans to classify data independently of each other they might not agree 100% of the time. Say they agree 97% of the time. This is called ceiling and is the absolute maximum performance a ML system will ever be able to achieve. Asking more than that is meaningless. Knowing this, the performance of any ML solution will lie between the baseline and a ceiling (unknown but probably less than 100%).
Getting to know the actual value of the ceiling takes some more effort: you would have to assign at least two persons (with deep knowledge of the data they are going to tag) to tag examples, each one independently of the other, and then calculate their so called agreement rate. There are several metrics used for this, but I will not go in the details. If one were focusing on research this would be a must, but its definitely possible to build a great system (ie. one that saves money by reducing human labour) without knowing the value of the ceiling.
### Summing up
• One can never say something like: the baseline is 90% so we are going to get to 95% by the end of this project.
• One can say something like: the baseline is 90%, so we are going to commit to improve that by as much as we can; until we actually do it, we cannot say by how much.
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2022-12-01 20:34:36
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https://math.stackexchange.com/questions/2330617/interpreting-a-set-of-predicate-formulas-as-a-model
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# Interpreting a set of predicate formulas as a model
Consider the set $\phi$ consisting of the following formulas.
$$\exists x \exists y \exists z \neg (x = y ~ \lor x = z ~ \lor y = z)$$ $$\forall x \exists y ~ gimble(x, y) ~ \land \exists x ~ \neg gimble(x, x)$$ $$\forall x \forall y \forall z ~ (gimble(x, z) ~ \land ~gimble(y, z) \implies x =y)$$
1) How can I describe a model for $\phi?$
Should I use a table? How would I do that?
2) How can I describe an interpretation that is not a model for $\phi?$
3) Is the set $\phi$: (i) valid, (ii) satisfiable but not valid or (iii) unsatisfiable?
I think that $\phi$ is probably satisfiable but not valid (ii) because it holds under some interpretation e.g. the interpretation for 1) but it is not valid because it does not holds under every interpretation i.e the interpretation for 2)
For a model:
The first sentence says that there are at least three different objects
The second sentence says that everything stands in a 'gimble' relation to something, and there is something not in a 'gimble' relation to itself
the third sentence says that you cannot have two different objects that stand in a 'gimble' relation to the same object.
1) OK, so here is a possible model: take exactly three objects, $a,b$ and $c$, and with the following 'gimble' pairs: $(a,b),(b,c),(c,a)$
If you want something more meaningful: We have 3 persons: Alice, Bob, and Carroll. $gimble(x,y)$ means '$x$ likes $y$. Say that Alice likes Bob, Bob likes Carroll, and Carroll likes Alice, and otherwise there are no more like relations. Note how we indeed have at least three different objects, how everyone likes someone, how there is someone who does not like themselves, and how there are not two different people liking the same person.
2) An interpretation that is not a model: say there is only 1 object $a$ and say that $a$ dos not stand in the 'gimble' relation to itself.
More meaningful: We just have Alice, and Alice does not like herself.
3) Yes, you are right: it is satisfiable (because there is a model) but not valid (because not every interpretation is a model)
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2022-01-24 04:05:57
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https://www.physicsforums.com/threads/youngs-modulus-homework.260869/
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# Young's Modulus homework
1. Oct 1, 2008
### rit
1. The problem statement, all variables and given/known data
I had to do an experiment and create a graph with the data.
We attached one end of the Iron wire to a clamp and then tied on weights of 100g at a time to the other end so that a force can be applied to the Iron wire and so it stretches. (as pizza1512 did but in copper)
2. Relevant equations
Diameter of wire: 0.31mm (therefore area is 0.302mm²)
Length of wire: 2m70cm (270cm)
3. The attempt at a solution
i worked out that if u apply 100g (1newton force)
1/2700= 3.7x10^-4 (not sure if right)
And stress is
1/0.302= 3.3
I worked out the Youngs Modulus
3.3/(3.7x10^-4) and i got 8918.92
is that right?
and how do i plot it on a graph
2. Oct 1, 2008
### LowlyPion
How did you work that out? From what? What are the units of that expression?
3. Oct 1, 2008
### rit
It is for Strain.
it extended by 1mm
so,
1mm/2700mm (length of wire) =3.7^-4
4. Oct 1, 2008
### LowlyPion
$$E = \frac{\sigma}{\epsilon} = \frac{F*L}{A_o*\Delta L} = \frac{1N * 2.7m}{.302*10^{-6}m^2*.001m}$$
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2017-11-23 13:43:45
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https://dsp.stackexchange.com/questions/17807/how-to-calculate-the-time-delay-of-a-signal-by-a-first-order-filter
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# How to calculate the time delay of a signal by a first-order filter
lets assume following signal $$x=a_1 \sin(2\pi f_1 t) + a_2 \sin(2\pi f_2 t).$$ This is passed through a simple single-pole filter with cut-off frequency $f_c$. How is $x$ delayed by the filter (peak of the cross-correlation)?
In the case $a_2=0$, the time delay between the original and the filtered frequency is $$T_1=\arctan(f_1/f_c)/(2\pi f_1).$$
In the case $a_1=0$, the time delay between the original and the filtered frequency is $$T_2=\arctan(f_2/f_c)/(2\pi f_2).$$
But how can we calculate the time delay $T_{12}$ in the case $a_1\neq0$ and $a_2\neq0$?
It is not the group delay: It must depend in some way on the values of $a_1$ and $a_2$, because if $a_1>>a_2$, then $T_{12}\approx T_1$.
Any light on this would be greatly appreciated!
The answer is that the output signal for the case $a_1\neq 0$ and $a_2\neq 0$ is not simply a delayed version of the input signal anymore, as it would be if the input were just a single sinusoid. With $T_1$ and $T_2$ as given in your question, the response to the combined input signal is
$$y(t)=\frac{a_1}{\sqrt{1+(f_1/f_c)^2}}\sin(2\pi f_1(t-T_1))+\frac{a_2}{\sqrt{1+(f_2/f_c)^2}}\sin(2\pi f_2(t-T_2))$$
which cannot be written as $x(t-T_x)$ with some unknown delay $T_x$.
But as you suggested, you could define some 'global delay' by the first maximum of the cross-correlation between the input signal and the output signal. This cross-correlation is
$$R_{xy}(\tau)=\frac{a_1^2}{2\sqrt{1+(f_1/f_c)^2}}\cos(2\pi f_1(\tau-T_1))+\\+ \frac{a_2^2}{2\sqrt{1+(f_2/f_c)^2}}\cos(2\pi f_2(\tau-T_2))\tag{1}$$
For $a_2=0$, there is obviously a maximum of $R_{xy}(\tau)$ at $\tau=T_1$. The same is true for $a_1=0$, which gives $\tau=T_2$ as the location of the maximum. However, for $a_1\neq 0$ and $a_2\neq 0$ I don't see how to compute the first maximum of (1) analytically. A numerical solution is of course possible.
• Thanks Matt. But when we calculate the cross correlation, the peak will be somewhere between $T_1$ and $T_2$. Is there no analytic solution to find that peak? Btw, shouldn't the new amplitudes be $1/\sqrt(1+f_1^2/f_c^2)$ ? – Ankaios Argo Aug 16 '14 at 22:27
• @AnkaiosArgo: Thanks for pointing out the magnitudes, I was just focusing on the delays. Fixed it. I'll need to think about what you say about the cross-correlation. – Matt L. Aug 16 '14 at 22:33
• Sorry, I didn't want to be picky :-) Would be great, if you find a solution... In order to find the peak in the cross-correlation, maybe we can calculate the cross-spectra? – Ankaios Argo Aug 16 '14 at 22:43
• @AnkaiosArgo: I added some info about cross-correlation to my answer. Unfortunately, I can't see how to find an analytical solution. – Matt L. Aug 17 '14 at 0:10
• Thanks, this already helps... would you mind to elaborate, how you determined the $R_{xy}(\tau)$? The auto-correlation of a sum of sinus is a sum of cosines, see e.g. chem.purdue.edu/courses/chm621/lecture/electronics/… , but in the case of the cross-correlation and the time-shift $T_i$ it does not seems very trivial to me. – Ankaios Argo Aug 17 '14 at 15:15
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2019-08-21 21:14:21
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https://physicsworld.com/a/peer-review-under-the-spotlight/
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# Peer review under the spotlight
01 Feb 2016 Michael Banks
Taken from the February 2016 issue of Physics World
Taken from the February 2016 issue of Physics World
Peer review has been the backbone of scientific publishing for centuries, but many feel that the time has come for the system to be reformed. Michael Banks weighs up the options
“Peer review is similar to what Winston Churchill once noted about democracy,” says David Crotty, a senior editor at Oxford University Press, who writes for the Scholarly Kitchen – a leading blog about the scientific publishing industry. “It’s the worst system apart from all the others.” Since its introduction in the 1700s, peer review has formed the backbone of scientific publishing. Used to judge the suitability of scientific manuscripts submitted to a journal for publication it has, like democracy, so far stood the test of time.
That is mostly thanks to peer review being valued and trusted. A survey of 18,000 researchers, carried out last year by Nature Publishing Group (NPG) and Palgrave Macmillan, found that the quality of peer review offered by a journal was ranked third – behind journal reputation and relevance – in a list of factors that authors considered when submitting their research. When asked about the value that publishers bring, the top response to the 2015 Authors Insight Survey was “improving papers through constructive peer review”, ahead of rapid acceptance of papers and their discoverability.
Such findings are backed up by a study published late last year – Peer Review in 2015 – by the publisher Taylor and Francis, which found that the vast majority of academics support peer review and believe that it improves their manuscripts. With responses from more than 7400 academics worldwide, 68% noted that they have confidence in the academic rigour of published articles because of peer review.
While both these reports suggest that the peer-review system is not broken, researchers’ views about how peer review should operate in the 21st century are changing in the era of digital publishing. Some complain about a proliferation of low-quality journals – most of which are open access – that seek to boost publishers’ revenues by cutting corners on rigorous refereeing procedures. Other researchers, meanwhile, complain that refereeing is done by the community as an unpaid or unrewarded service.
Another issue is the difficulty of finding enough researchers who are enthusiastic and skilled at peer review. Those who are good at the job often end up being over-burdened by requests – increasing the danger that journals promote a “clique” of trusted referees. Some also claim that peer review takes far too long. A final version of a paper can appear in a journal months after it was first uploaded to the arXiv preprint server, where many physicists deposit their paper before, or at the same time as, sending it to be peer reviewed.
While Crotty believes that peer review works well in general, it could be “greatly improved” by making refinements to the current system. “Peer review isn’t perfect,” agrees medical physicist Penny Gowland of the University of Nottingham in the UK. “The public needs to recognize that and publishers need to recognize that.” She claims that gold open access, in which an author pays an “article-processing charge” (APC) to make it freely available to read upon acceptance, has weakened the peer-review process through the growth of publishers that exploit the system by accepting as many papers as possible to maximize their income. “It acts to undermine the scientific process and in the long term may undermine how the public views science,” she says.
### Tweaks not reforms
When peer review was introduced as a means of testing the eligibility of a paper for publication, it was initially performed by the editor of the journal. That changed in the mid-20th century when papers began to be sent to external referees who were not involved in the work but active in the field. They pass a judgement on whether an article is scientifically credible and appropriate for the journal to which it has been submitted.
But one major criticism of peer review is that this refereeing process is opaque. “Peer review is a gatekeeper, but it’s also a bit of a black box, and lots of it is hidden,” says Daniel Ucko, an associate editor at the American Physical Society who is also doing a PhD on the philosophy of peer review at Stony Brook University. “The public is curious about the process and is interested to know why certain science is getting published, especially in fields such as medicine.”
Traditionally, peer review has been performed on a “single blind” basis, in which the reviewers know who has written the paper but the authors of the paper are not told who has reviewed it. But because reviewers know the authors and where they work, a reviewer may judge the paper based on that before reading it. That could lead to a bias against women, people in developing nations, early-career researchers as well as those at smaller, less-established institutions.
Ironically, one way of improving peer review is to make it less transparent. Crotty advocates double-blind peer review, in which the authors do not know who is reviewing their paper – like single-blind review – but the reviewers also do not know who has written the paper. Although this added anonymity has some advantages such as reducing the effect of bias, particularly gender-based, it is not foolproof. Reviewers could easily use arXiv or an Internet search to track down the author’s identity or institution – not hard at all in small communities – even if it is hidden in the peer-review process. Meanwhile, removing references to an author’s own work would be time-consuming.
Yet Gowland, who is a proponent of the double-blind process, disagrees that arXiv would stop effective blinding. “A properly motivated reviewer would not simply search arXiv to find the author,” she says. Indeed, Crotty says that studies of double-blind peer review have shown that reviewers actually fail to guess the authors most of the time. “Even if you manage to guess the lab, blinding the authors is good for gender bias,” he adds.
One way to counter the lack of transparency during the review process is “open” peer review, in which the reviewers’ comments – and authors’ counterpoints – are made public online together with the final accepted version of the paper. In January Nature Communications announced a year-long trial to publish all reviewer comments and author rebuttal letters for published papers in the journal, unless authors ask them not to do so. This “peer-review file” will be published along with the accepted version of the manuscript. The success of the trial will be determined by opt-out rates and other “monitoring parameters”.
Tim Smith, an associate director at IOP Publishing, which publishes Physics World, says that it would be relatively simple to publish such a peer-review file, but there would be practical issues to address such as the level of any editing required on referee comments to meet language criteria for publication.
In 2014 the European Union set up the first government-funded, multinational effort “to improve efficiency, transparency and accountability of peer review”. Chaired by Flaminio Squazzoni, a sociologist at the University of Brescia, Italy, PEERE will analyse peer review in science and evaluate different models of peer review as well as explore new incentive structures, rules and measures to improve the peer-review process.
Squazzoni, who is a fan of double-blind peer review, says there are, however, many advantages of open peer review. It would solve the transparency issue, increase the quality of reviews and encourage reviewers to stop making authors cite their own work. But Squazzoni admits it would be tricky to implement. “Imagine a junior scientist reviewing a manuscript from an established scientist. If the review is open, it might be difficult for them to reject it,” he says. “Then there is the issue of retaliation, and peer review must protect reviewers from that.”
Ucko agrees, adding that much of the effort of peer review is placed on junior scientists’ shoulders, who stand to lose by having their identities revealed. Indeed, Ucko believes that referees must remain unknown. “Anonymous referees provide candid critique,” he says. “They may have bias, but that is where the editor’s job comes in to safeguard the peer-review process.” That view is backed by Smith, who points out that IOP Publishing journals are run by in-house “peer-review experts” with responsibility for preserving the integrity of the referee selection and decision-making processes.
### Credit where credit’s due
Another possible reform of the system is to incentivize reviewers to do a good job in the first place. “Peer review doesn’t have any kind of reward for scientists – except for making the authors cite your paper – so reviewers rationalize the time they spend on peer review,” says Squazzoni. “And the more people who think peer review is not the top of their agenda, the more difficult it will be to get people to do it.” While reviewers get to influence the community through peer review, as Crotty points out, “nobody would get tenure for being good at peer review”.
One way of crediting researchers for their efforts could be to pay them to do peer review. But Crotty says that would create problems. Apart from deciding who should pay, he fears a tiered system, with the best reviewers demanding more money. “That would end up being more trouble than it’s worth,” says Crotty, who adds that in March, NPG’s Scientific Reports started a trial in which authors could fast-track peer review for a fee of $750, around$100 of which went to each referee. Resignations from the journal’s editorial board over what would be a two-tiered system forced the journal to quickly backtrack.
Squazzoni agrees that financial reward is not the way forward, claiming that this would have a “crowding-out effect on intrinsic motivations of scientists”, while Ucko says anything more than a nominal financial reward would be a particular drain for journals published by learned societies, which often rely on journal income to fund their activities. Paying referees could end up increasing APCs and journal subscription costs.
Smith says that referees for IOP Publishing journals receive discounts on APCs, but admits more could be done by publishers to give individual credit to researchers for their overall refereeing activity. “Serving as a referee, editorial board member or guest editor has become an integral part of a researcher’s career and should be recognized as such,” he adds.
Another way of crediting researchers for peer review could be through the Open Researcher and Contributor ID (ORCID) system. This provides a “digital identifier” to distinguish scientists from one another and creates a record linking scientists to their research outputs such as papers and grant submissions. Some 1.8 million researchers have already registered for a unique identifier and Crotty says that the system could be expanded to include peer-review data such as introducing “peer-review citations”. Indeed, last month seven science publishers announced that as of 1 January they require researchers to identify themselves using the ORCID system when submitting papers.
For all the talk about reforming peer review, Squazzoni warns that a greater threat is the lack of data about the peer-review process itself. “We don’t have data about peer review and as a result it is poorly understood,” he says. Squazzoni says this is starting to change, with publishers offering their data to scientists to study (like in PEERE), but he says that more needs to be done. “Peer review is essential for how science works,” he continues. “And the more we know about the process, the better.”
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2018-06-19 05:05:20
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https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=145&t=29038&p=89662
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## Concentration and Rate Relationship
$aR \to bP, Rate = -\frac{1}{a} \frac{d[R]}{dt} = \frac{1}{b}\frac{d[P]}{dt}$
Amanda Mac 1C
Posts: 58
Joined: Fri Sep 29, 2017 7:05 am
### Concentration and Rate Relationship
In a second order reaction, if one reactant alone is doubled does that mean the rate is quadrupled? What happens if two reactants in a second order reaction are both doubled?
Sean Monji 2B
Posts: 66
Joined: Fri Sep 29, 2017 7:06 am
### Re: Concentration and Rate Relationship
I think so. This can be shown mathematically.
Lets say rate = k[A]^2
If A is doubled,
rate = k[2A]^2 = 4k[A]^2, meaning rate was quadrupled
in rate = k[A][B]
k[2A][2B] = 4k[A][B]
Though in the second reaction, if you only double A or B, the rate only doubles
k[2A][B] = 2k[A][B]
This has to do with how likely the molecules will collide to form the product.
Hope that makes sense
Posts: 88
Joined: Fri Sep 29, 2017 7:03 am
### Re: Concentration and Rate Relationship
For the most part, it would be easier to see the changes if you had the rate of reaction and the corresponding orders. From there, you could just plug in new concentrations to see what kind of effect it would have on the overall rate.
Harjas Sabharwal 1G
Posts: 42
Joined: Sat Jul 22, 2017 3:01 am
### Re: Concentration and Rate Relationship
The answer to your question depends on the specific rate law for the reaction you are discussing. Generally if you plug in the change and then compare the resulting rate laws, you should be able to figure out what the change in rate is.
Christina Bedrosian 1B
Posts: 33
Joined: Fri Sep 29, 2017 7:05 am
### Re: Concentration and Rate Relationship
for multiple reactions, see the effect of each reactant on the rate. so if one reactant is 1st order and another is 2nd order, the 1st would double the rate if the concentration is doubled and the 2nd one would quadruple the rate if the concentration was doubled. you usually view them separately like that
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2019-12-10 03:44:36
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http://mathhelpforum.com/advanced-algebra/105310-has-rank-n-iff-diagnol-entries-r-non-zero.html
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## Matrix A has rank n iff diagnol entries of R' are non zero
Sorry about formatting as this is my first post here.
Question :- A mXn matrix (m>=n), A=Q'R' (Reduced QR)
To show that A has rank n iff diagonal entries of R' are non zero.
My Solution :-
if Ax = 0 => Q'R'x = 0 => R'x =0
let R' = upper triangular matrix (nXn) and x is a column vector.
suppose $r_{nn}$ ..... $r_{k+1k+1}$ are not 0
but $r_{kk} = 0$
since R'x = 0 => $x_n = x_{n-1}..... = x_{k+1} = 0$
but $x_k$ is not zero as $r_{kk}$ is 0 and we can have $x_k$= say 1 ....so we can determine value of $x_{k-1} .... x_1$ by back substitution
=> x is a nonzero vector such that R'x = 0
I am not able to go beyond this.....Am I taking right approach..
Any help is appreciated.
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2016-10-01 18:00:27
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https://space.stackexchange.com/questions/50376/are-nuclear-thermal-engine-designs-limited-to-about-twice-the-isp-of-existing-ch
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# Are nuclear thermal engine designs limited to about twice the Isp of existing chemical rocket engines? If so, why; what's the limiting factor?
Discussion below With Ultra Safe Nuclear engines and hydrogen propellant, how far to Mars could you get and still be able to return to Earth in an emergency? including a comment that suggests that the following quote makes it sound like a "solid core nuclear thermal rocket", and that it is likely to have similar performance "as every other solid core nuclear thermal rocket."
From their October 19, 2020 press release Ultra Safe Nuclear Technologies Delivers Advanced Nuclear Thermal Propulsion Design To NASA:
NTP systems achieve expanded payload mass capabilities due to their two-fold increase in specific impulse compared with chemical propulsion systems.
From kerolox to LOX LH2 Isp's range from roughly 360 to 440 seconds.
Quesiton: Are nuclear thermal designs in the ballpark of roughly 700 to 900 seconds? If so, what is the limiting factor? Why can't they easily go higher?
For solid core engines, yes, that's their limit.
If so, what is the limiting factor?
The exhaust velocity (and hence specific impulse) is linked to the heat of the propellant. The propellant can't get any hotter than the fuel elements.
Why can't they easily go higher?
Cos your fuel elements would melt and blow out of the back of the rocket in an embarassing way.
As you mentioned in your question itself, an important figure of merit is the "characteristic velocity":
$$c_* \propto \sqrt{\frac{T_t}{M_w}}$$
where $$T_t$$ can be taken to be the exhaust temperature, and $$M_w$$ is the molecular weight of the gas species in the exhaust. If you'd like a source for the equation, you can see it mentioned here, along with a load of related notes. It is equation 9.
What this basically boils down to is that both chemical engines and NTRs have the same ultimate operating limits... they can only get so hot, before critical bits melt and the whole thing goes poof.
Unlike a chemical rocket, however, an NTR can produce exhaust which has much lighter gasses, such as neat H2. At the same exhaust temperatures, those light molecules will be travelling much faster than, say, the heavier H2O coming out of an LH2/LOX rocket.
An NTR filled up with water would have pretty much the same Isp as that LH2/LOX rocket, though it would have somewhat smaller and more convenient reaction mass tanks, and is a bit easier to fill up.
There are various solutions that have been proposed to deal with this.
Liquid and gas-cored NTRs are the obvious ones, where you pre-melt-down your reactor core deliberately, though of course they have their own formidable technical challenges.
There's an add pulsed reactor design that relies on neutron heating of the reaction mass that allows for hotter exhaust by relying on neutron heating of the reaction mass, which sounds a little technically farfetched but someone obviously believes in it.
The most plausible solution seems to be to build a decent NEP rocket instead. Certainly that's within our technical capabilities at this time.
• @uhoh it does just seem to boil down to temperature and molecular weight of the exhaust products. An NTR could have neat H2 flowing out of the back, whereas a chemical rocket has to make do with hefty H20 instead. – Starfish Prime Feb 26 at 9:17
• @uhoh if you'd like a source for the $\sqrt{T/M}$ thing, it pops up in all sorts of places, such as here: web.stanford.edu/~cantwell/AA103_Course_Material/… – Starfish Prime Feb 26 at 9:18
• Super; thank you! (or Super-thank you, both apply) – uhoh Feb 26 at 9:34
• – uhoh Feb 26 at 9:48
• @uhoh that'll be the heavy lifting by by the $\propto$ ;-) exhaust gas chemistry isn't something I know anything about. – Starfish Prime Feb 26 at 10:26
|
2021-06-22 18:13:06
|
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|
https://math.stackexchange.com/questions/1791706/connection-between-number-theory-and-the-von-neumann-construction-of-naturals
|
# Connection between number theory and the Von Neumann construction of naturals
There are many unsolved conjectures and hypothesis in number theory. For example, the twin primes conjecture, Goldbach's conjecture, the Riemann hypothesis, infinitude of Mersenne's primes, and many many more.
A widespread convention of "rigorous construction of the number system" is von Neumann's finite ordinals, constructed with ZF axioms, i.e. well-ordered transitive sets ordered by the $\in$ relation. It can be shown that these finite ordinals satisfies Peano's arithmetic axioms.
As an electric engineer who designs and constructs a building knows all about the power sockets in the buildings, I would expect that the construction of the natural numbers should imply many things about the behaviors of those numbers, in particular prime numbers.
So my question is, is there a relation? Are there any clues in the way we construct the numbers that can help us reveal more about numbers? Are there any set-theoretic characteristics of prime numbers? Unlike geometry and complex analysis, why doesn't set theory provide us with heavy tools to deal with the complicated world of numbers?
• I'm somewhat uncomfortable with these sorts of question, because I feel that they don't belong into the realm of mathematics and only allow for incomplete and wrong answers. Nevertheless, I tried to come up with a satisfactory answer - and failed. Let me leave you with its only part, which I feel adds to your post: The only proof of (the somewhat number theoretical) Goodstein's theorem that I know of, uses set theory. There is a second one, that deals with arithmetic progressions, but I don't recall its name or precise statement. – Stefan Mesken May 19 '16 at 15:06
|
2019-06-20 17:50:44
|
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|
https://physics.stackexchange.com/questions/94054/could-quarks-be-free-in-higher-dimensional-space-than-3d?noredirect=1&lq=1
|
# Could quarks be free in higher-dimensional space than 3D?
Reading this answer, I now wonder: if quarks are confined by $r^2$ potential, could their potential allow infinite motion in higher-dimensional space?
To understand why I thought this might be possible, see what we have with electrostatic potential: in 3D it is proportional to $r^{-1}$. This is just what Poisson equation tells us for point charge. If we solve Poisson equation in 2D space, we'll see potential is proportional to $\ln\frac r {r_0}$, and in 1D it's proportional to $r$. We can see that it only allows infinite motion starting form 3D.
Could the same hold for quarks, but with some higher than 3D dimension? Or is their potential of completely different nature with respect to space dimensionality?
## 1 Answer
If you take the classical analogy of a charge generating field lines then the force at some point can be taken as the density of field lines at that point. In 3D at some distance $r$ the field lines are spread out over a spherical surface of area proportional to $r^2$ so their density and hence force goes as $r^{-2}$ - so far so good.
The trouble with the strong force is that the interactions between gluons cause the field lines to attract each other, so instead of spreading out they group together to form a flux tube or QCD string. In effect all the field lines are compressed into a cylindrical region between the two particles so the field line density, and hence the force, is independant of the separation between the quarks.
This means it doesn't matter what the dimensionality of space is, because the field lines will always organise themselves along the 1D line between the quarks. The quarks woould be confined in any dimension space.
Annoyingly I can't find an authoritative but popular level article on QCD flux tubes, but a Google will find you lots of articles to look through.
• I am not sure, if the IR behavior of QCD is not affected by the dimensionality of spacetime. The proper way to investigate this question would be to do a corresponding RG analysis, people have probably done this already... Jan 17 '14 at 11:05
• @Dilaton: I must admit that I'm nervous some effect I've never heard of changes the way flux tubes form in higher dimensions. If someone more expert than me (which is not hard :-) would like to comment I'll delete my answer. Jan 17 '14 at 11:17
• Well, the only higher dimensions compatible with the standard model are the higher dimensions in string theories, and those are curled up in so small units with respect with our open dimensions that it seems to me it makes no difference to us even if there were free quarks there . Quarks could be bound vibrations on the string in our dimensions and free in all the curled ones? Jan 17 '14 at 15:21
• have a look at this physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/Nobel Jan 17 '14 at 15:28
|
2021-09-23 11:54:44
|
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|
https://datascience.stackexchange.com/questions/47773/the-principle-of-lm-deep-model
|
# The principle of LM deep model
Language model(LM) is the task of predicting the next word.
Does the deep model need the encoder? From the ptb code of tensor2tensor, I find the deep model do not contains the encoder.
Or both with-encoder and without-encoder can do the LM task?
The goal of LM is to learn a probability distribution over sequences of symbols pertaining to a language.
That is, to learn $$P(w_1,...,w_N)$$ (resource).
This modeling can be accomplished by
1. Predicting the next word given the previous words: $$P(w_i | w_1,...,w_{i-1})$$, or
2. Predicting the neighbor words given the center word (Skip-gram): $$P(w_{i+k}| w_i), k \in \{-2, -1, 1, 2\}$$, or
3. Predicting the center word given the neighbor words (CBOW or Continuous Bag-of-Words): $$P(w_i| w_{i-2},w_{i-1},w_{i+1}, w_{i+2})$$, or other designs.
Does the deep model need the encoder? From the ptb code of tensor2tensor, I find the deep model do not contains the encoder.
Yes. Modern LM solutions (all deep ones) try to find an encoding (embedding) that helps them to predict the next, neighbor, or center words as close as possible. However, a word encoding can be used as a constant input to other models. The ptb.py code calls text_encoder.TokenTextEncoder to receive such word encodings.
Both with-encoder and without-encoder can do the LM task?
LM task can be tackled without encoders too. For example, we can use frequency tables of adjacent words to build a model (n-gram modeling); e.g. all pairs (We, ?) appeared 10K times, pair (We, can) appeared 100 times, so P(can | We) = 0.01. However, encoder is the core of modern LM solutions.
|
2021-04-11 03:59:31
|
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|
https://heads0rtai1s.github.io/2019/04/24/tidy-eval-examples/
|
# Tidy evaluation in R - Simple Examples
The tidyverse philosophy introduced by Hadley Wickham has been a game changer for the R community. It is based on intuitive rules of what a tidy data set should look like: each variable is a column, each observation is a row (Wickham 2014). At its core, the tidyverse collection of R packages is powered by a consistent grammar of data manipulation and visualisation.
The tidyverse grammar makes it easier to manipulate data sets using simple expressions that reduce the syntactic overhead and allow you to focus on the data. Thus, packages like dplyr or tidyr are great for exploratory data analysis (EDA) and hands-on data wrangling. A small downside of this approach is that these tools require a bit more effort when using them in functions with variable parameters. In general you want to use functions to improve the reusability and reproducibility of your code.
This is where the tidy evaluation comes in. A few additional methods and concepts are sufficient to make all your tidy code run smoothly in a function context. Here I will go through some relatively simple examples to get you started.
Before we begin we will need the following libraries:
libs <- c('dplyr', 'tibble', # wrangling
'datasets', # data
'knitr','kableExtra', # table styling
'ggplot2','gridExtra') # plots, panels
invisible(lapply(libs, library, character.only = TRUE))
We will use the Orange data set, which is part of the datasets package and records the growth of 5 orange trees. Here are the first 5 rows:
Tree age circumference
1 118 30
1 484 51
1 664 75
1 1004 108
Personally, I’m learning most efficiently by first looking at examples that show the code in action and then tweaking them to fit my needs. After playing with the code for a bit and inevitably breaking something I turn to the docs to understand more about the syntax and additional arguments of the function. Thus, all my posts on tools or methodology will follow the same pattern: I will jump right into the action by looking at a useful yet simple example or two. Next, I dissect this example, maybe break something, and explain the arguments. In closing, there will be a few more complex examples, caveats, pointers, and/or resources. Sounds good? Here we go:
The first example is a function that takes as input a data frame df and a variable var from that data frame (i.e. a column/feature). The output is the difference between the (global) median and the mean of the variable. This is a realistic example of a concise helper function, since it goes beyond basic in-built tools and provides a quick check on whether a distribution is symmetric1:
median_minus_mean <- function(df, var){
var <- enquo(var)
df %>%
summarise(foo = median(!!var) - mean(!!var)) %>%
.$foo } Now we apply it to the circumference of trees to find that the mean is larger than the median: median_minus_mean(Orange, circumference) ## [1] -0.8571429 To understand how it works here are the 2 key concepts: • Quoting: In the body of the function, the variable var is being quoted by the enquo function (borrowed from the rlang package). This essentially means that the content (or argument) of the variable is being encoded. The quotation stops this variable from being immediately evaluated. Instead, its content is being treated as a functional R expression. • Unquoting: In order to tell a tidyverse verb like summarise that you are passing it the content of a quoted variable you need to unquote it. Practically you are copy-pasting the variable expression into the verb. This is done using the !! operator which Hadley wants to be pronounced bang-bang. I can only surmise that he said that because it makes boring conversations about code sound like wild-west movie fights. In most situations enquo and !! are all you need. Conceptually, there’s a bit more to it since enquo encodes the current state of the environment along with the variable. This is a useful property, which makes enquo aware of parameters defined outside a function, but for now you can ignore these finer details. (Talking about details: foo or bar are popular names for dummy variables in many programming languages. It’s just something that needs a name for the moment but can immediately be forgotten once its time-limited purpose is fulfilled.) Also: yes, this works: median_minus_mean <- function(df, var){ df %>% summarise(foo = median(!!enquo(var)) - mean(!!enquo(var))) %>% .$foo
}
median_minus_mean(Orange, circumference)
## [1] -0.8571429
You can quote and unquote in the same step. Let’s go a bit further and include grouping by another variable, here the age of the trees:
median_minus_mean <- function(df, var, gvar){
var <- enquo(var)
gvar <- enquo(gvar)
df %>%
group_by(!!gvar) %>%
summarise(foo = median(!!var) - mean(!!var)) %>%
.\$foo
}
median_minus_mean(Orange, circumference, age)
## [1] -1.0 0.2 -6.2 -9.2 -3.6 0.6 1.2
Turns out that for some ages the mean circumference is smaller than the median.
Good news: quote/unquote also works for ggplot2. Here we quote the x, y, and colour-group variables of our plot:
plot_growth_tree <- function(df, xvar, yvar, gvar){
xvar <- enquo(xvar)
yvar <- enquo(yvar)
gvar <- enquo(gvar)
df %>%
ggplot(aes(!!xvar, !!yvar, col = !!gvar)) +
geom_line()
}
plot_growth_tree(Orange, age, circumference, Tree)
Some trees grow faster than others.
In fact, ggplot2 is a great use case because it allows us to quickly built helper functions if we need to repeat a certain plot for many similar features. Individual modification to those templates can be added using the ggplot2 grammar. Here is a histogram example where we add a custom title to the second plot2:
plot_hist <- function(df, var, bins, bcol){
var <- enquo(var)
df %>%
ggplot(aes(!!var)) +
geom_histogram(bins = bins, fill = bcol, col = "black")
}
p1 <- plot_hist(Orange, age, 4, "blue")
p2 <- plot_hist(Orange, circumference, 7, "red") +
ggtitle("A custom title")
grid.arrange(p1, p2, layout_matrix = rbind(c(1,2)))
You see that here the number of histogram bins and the plot colour are being passed to the function as normal integer and string - without need of being quoted. This works because these parameters are not R expressions.
There will be a second post soon about more complex tidy evaluation examples. If you’re interested, watch this space.
In the meantime: Curious about the bigger picture?
• The tidy evaluation book is a great starting guide into the concepts.
• This thread collects some typical use cases for tidy evaluation.
• For a concise 5 minute intro to the main concepts by the man himself watch Hadley here:
1. If you are actually interested in the skewness of a distribution you can find a skewness function in the e1071 package.
2. The arranging of plots into panel layouts is done by the grid.arrange function of the gridExtra package.
|
2022-05-29 12:16:14
|
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|
https://techwhiff.com/learn/problem-11-11-algorithmic-agan-interior-design/207139
|
Problem 11-11 (Algorithmic) Agan Interior Design provides home and office decorating assistance to its customers. In...
Question:
Problem 11-11 (Algorithmic)
Agan Interior Design provides home and office decorating assistance to its customers. In normal operation, an average of 2.9 customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations. The consultant averages 10 minutes with each customer.
1. Compute the operating characteristics of the customer waiting line, assuming Poisson arrivals and exponential service times. Round your answers to four decimal places. Do not round intermediate calculations.
Lq =
L =
Wq = minutes
W = minutes
Pw =
2. Service goals dictate that an arriving customer should not wait for service more than an average of 6 minutes. Is this goal being met? If not, what action do you recommend?
No. Firm should increase the mean service rate u for the consultant or hire a second consultant.
3. If the consultant can reduce the average time spent per customer to 8 minutes, what is the mean service rate? Round your answer to four decimal places. Do not round intermediate calculations.
µ = customers per hour
Wq = minutes
Will the service goal be met?
Yes
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2022-06-26 12:11:07
|
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|
https://www.howcast.com/videos/325089-how-to-use-the-pythagorean-theorem
|
# How to Use the Pythagorean Theorem
The Pythagorean theorem is a cornerstone of geometry. Here's how to use it.
### Instructions
• Step 1: Determine the lengths of the short sides Determine the lengths of the two short sides of a right triangle.
• Step 2: Calculate the squares Calculate the squares of each of these lengths.
• TIP: The square of a number is the number multiplied by itself. The square root of 64, for example, is plus or minus 8, because plus or minus eight multiplied by itself equals 64.
• Step 3: Take the square root of the sum Add the squares together and take the square root of the result using a calculator. The resulting value is the length of the longest side, which is called the hypotenuse.
• TIP: If the lengths of the two shorter sides of a right triangle are 5 inches and 12 inches, the length of the hypotenuse is the square root of (25 plus 144), or 13 inches.
• Step 4: Calculate the lengths of short sides by analogy Given the length of one of the shorter sides and the length of the hypotenuse, subtract the square of the short side from the square of the hypotenuse and take the square root.
• FACT: Pythagoras' theorem was probably known to the Babylonians a thousand years before it was attributed to Pythagoras, who lived from about 569 to 475 B.C.E.
### You Will Need
• Right triangle
• Lengths of 2 sides
• Calculator
|
2019-03-24 15:32:51
|
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|
https://tex.stackexchange.com/questions/226678/how-can-i-maintain-the-current-page-nodes-for-portrait-when-temporarily-entering
|
# Question
How can I keep the current page nodes aligned with the same physical corners of the page in a document where the main layout is portrait, but some pages with longtable are in the landscape environment. In other words, how can I rotate the nodes clockwise like the header, body, margin, and footer. The nodes are from the packages tikz and tikzpagenodes.
## Example
I want to essentially be able to rotate the tikz nodes clockwise without changing the anchor names. In other words, the reference names north west, north, north east, east, south east, south, south west, west should still refer to the same locations as for a portrait page. (In the code below, the showboxes output should be rotated with the page.)
Desired Result:
1. I want to be able to draw lines around the sections without worrying about alignment in landscape or portrait. (just like I don't have to worry about the header or footer being correctly aligned: remember that the physical page will be portrait in the book; just the body will be rotated.)
2. I want the blue bar to be at the same physical location on every page of my book to indicate that that page has the start of a section on it.
## Sample Code
\documentclass{article}
\usepackage{fontspec}
\usepackage[%
a4paper,
foot=1.5cm, % distance from top of footer to block of text aka \footskip
% heightrounded, % ensure an integer number of lines
marginparwidth=2cm, % right marginal note width
marginparsep=2mm, % distance from text block to marginal note box
% height=\textheight, % height of the text block
% width=\textwidth, % width of the text block
top=2.5cm, % distance of the text block from the top of the page
bottom=3cm,
left=2.5cm,
right=2.5cm,
% showframe, % show the main blocks
% verbose, % show the values of the parameters in the log file
]{geometry}
\usepackage{pdflscape}
\usepackage{tikz} % Absolute positioning, advanced vector graphics
\usepackage{tikzpagenodes} % Adds nodes around page boxes (e.g. body)
\usetikzlibrary{calc,positioning,decorations} % Extensions for tikz (increase compilation time)
% Title Packages
\usepackage[compact,explicit,noindentafter]{titlesec}
\usepackage[rightlabels,dotinlabels]{titletoc}
\usepackage{tocloft}
\usepackage{needspace}
\usepackage{fancyhdr}
\usepackage{longtable}
% TikZ Definitions
\newcommand{\tikztitlenumber}[1]{%
\begin{tikzpicture}[remember picture, overlay,baseline]
\node [font=\Huge\fontsize{50}{60}\selectfont,text=blue,anchor=south east,inner sep=0pt, outer sep=0pt] (titlenumber) {#1};
\node [] (titlerighttop) at (titlenumber.north -| current page.east) {};
\node [] (titlerightbottom) at (titlenode -| current page.east) {};
\path [fill=blue] (titlerighttop.north) rectangle ($(titlerightbottom) + (-2mm,0mm)$);
\end{tikzpicture}}
\newcommand{\marginmark}[1]{%
\tikz[overlay,remember picture] \node [minimum height=2cm,anchor=west,rotate=90, color=gray] at (current page marginpar area.south) {\fontsize{20}{30}\selectfont #1}; }
\newcommand{\tikztitleline}{%
\begin{tikzpicture}[overlay,remember picture]
\draw (titlenode.west) -- (titlenode -| current page text area.north east);
\end{tikzpicture}
}%
% TITLE FORMATTING
\titleformat{\section}[hang]{\needspace{6cm}\color{blue}\Huge\bfseries}{}{0pt}{\tikz[remember picture,overlay] \node [anchor=base west,yshift=-4mm] (titlenode) {};#1\hfill\tikztitlenumber{\thesection}}[\thispagestyle{sectionpage}\tikztitleline]
\titlespacing{\section}{0pt}{40pt}{20pt}
\newcommand{\showboxes}{%
\begin{tikzpicture}[remember picture,overlay]
\draw [red] (current page text area.south west) rectangle
(current page text area.north east);
\draw [green] (current page header area.south west) rectangle
\draw [blue] (current page footer area.south west) rectangle
(current page footer area.north east);
\draw [black] (current page marginpar area.south west) rectangle
(current page marginpar area.north east);
\end{tikzpicture}
}%
\fancypagestyle{sectionpage}
{
\fancyhf{} % clear all fields
\lfoot{}
\cfoot{}
\rfoot{%
\begin{minipage}[m]{.3\textwidth}
\begin{flushright}
\bfseries\footnotesize \thepage
\end{flushright}
\end{minipage}
\marginmark{}
}%
}%
\usepackage{lipsum}
\begin{document}
\section{Portrait Section}
\showboxes{}
\lipsum
\begin{landscape}
\showboxes{}
\section{Landscape Section}
\begin{longtable}{|l|p{6cm}|p{11cm}|p{4cm}|} \hline
\label{Table:references}
\textcolor{blue}{\textbf{Nr.}} & \textcolor{blue}{\textbf{Start of Clause}} & \textcolor{blue}{\textbf{Info}} & \textcolor{blue}{\textbf{End of Clause}}\\ \hline
[1] & These numbers & A whole lotta info & are not in order. \\ \hline
[2] & These numbers & A whole lotta info & are not in order. \\ \hline
[3] & These numbers & A whole lotta info & are not in order. \\ \hline
[4] & These numbers & A whole lotta info & are not in order. \\ \hline
[5] & These numbers & A whole lotta info & are not in order. \\ \hline
[6] & These numbers & A whole lotta info & are not in order. \\ \hline
[7] & These numbers & A whole lotta info & are not in order. \\ \hline
[8] & These numbers & A whole lotta info & are not in order. \\ \hline
[9] & These numbers & A whole lotta info & are not in order. \\ \hline
[10] & These numbers & A whole lotta info & are not in order. \\ \hline
[11] & These numbers & A whole lotta info & are not in order. \\ \hline
[14] & These numbers & A whole lotta info & are not in order. \\ \hline
[15] & These numbers & A whole lotta info & are not in order. \\ \hline
[16] & These numbers & A whole lotta info & are not in order. \\ \hline
[17] & These numbers & A whole lotta info & are not in order. \\ \hline
[19] & These numbers & A whole lotta info & are not in order. \\ \hline
[20] & These numbers & A whole lotta info & are not in order. \\ \hline
[21] & These numbers & A whole lotta info & are not in order. \\ \hline
[22] & These numbers & A whole lotta info & are not in order. \\ \hline
[23] & These numbers & A whole lotta info & are not in order. \\ \hline
[24] & These numbers & A whole lotta info & are not in order. \\ \hline
\end{longtable}
\end{landscape}
\end{document}
## Honing in on the Problem
### Reduced Minimal Working Example according to Gonzalo
\documentclass{article}
\usepackage{pdflscape}
\usepackage{tikzpagenodes}
\usepackage{lipsum}
\newcommand{\showboxes}{%
\begin{tikzpicture}[remember picture,overlay]
\draw [red] (current page text area.south west) rectangle
(current page text area.north east);
\draw [green] (current page header area.south west) rectangle
\draw [blue] (current page footer area.south west) rectangle
(current page footer area.north east);
\draw [black] (current page marginpar area.south west) rectangle
(current page marginpar area.north east);
\end{tikzpicture}
}
\begin{document}
\section{Portrait Section}
\showboxes{}
\lipsum[4]
\begin{landscape}
\showboxes{}
\section{Landscape Section}
\lipsum[4]
\end{landscape}
\end{document}
• @macmadness86 I know that building truly MWEs can sometimes be difficult, but as I tried to explain before, once you have a working example, you can start trimming it down until it becomes minimal (or almost minimal). For example, see this reduced version of your code: pastebin.com/QPbEkcgr it's shorter and still allows us to understand and reproduce the problem. If you want to, you can use that code in your question (it is almost minimal). – Gonzalo Medina Feb 5 '15 at 13:43
• @macmadness86: It may be difficult for you to discern what code should be included but it is not impossible and you can learn it. Simply spent some time to ponder over every line of code if you can remove it. – Ulrike Fischer Feb 5 '15 at 15:35
• Did you actually try to trim? Did you e.g. try to remove the \fancypagestyle{sectionpage}{...} to check if it is relevant for the problem? Did you try to remove the longtable to test if it is relevant? Sorry but you are hiding behind a "I'm so helpless"-argument. – Ulrike Fischer Feb 5 '15 at 15:45
• Yes there is a chain. And so for the next step you should try to remove the \titleformat and test if it is relevant. Sorry but the fact that you could get it wrong and perhaps remove something relevant is not an excuse not to try it at all. – Ulrike Fischer Feb 5 '15 at 16:17
• The problem starts with \pgfpoint and the current page node. – John Kormylo Feb 5 '15 at 18:14
If you want to leave the page nodes precisely where the were in portrait (except rotated), you can use the following.
\documentclass{article}
%\usepackage{fontspec}
\usepackage[%
letterpaper,
foot=1.5cm, % distance from top of footer to block of text aka \footskip
% heightrounded, % ensure an integer number of lines
marginparwidth=2cm, % right marginal note width
marginparsep=2mm, % distance from text block to marginal note box
% height=\textheight, % height of the text block
% width=\textwidth, % width of the text block
top=2.5cm, % distance of the text block from the top of the page
bottom=3cm,
left=2.5cm,
right=2.5cm,
% twoside
% showframe, % show the main blocks
% verbose, % show the values of the parameters in the log file
]{geometry}
\usepackage{pdflscape}
\usepackage{tikz} % Absolute positioning, advanced vector graphics
\usetikzlibrary{calc,positioning,decorations} % Extensions for tikz (increase compilation time)
\usepackage{tikzpagenodes} % Adds nodes around page boxes (e.g. body)
\usepackage{fancyhdr}
\usepackage[compact,explicit,noindentafter]{titlesec}
\usepackage[rightlabels,dotinlabels]{titletoc}
\usepackage{tocloft}
\usepackage{needspace}
\usepackage{lipsum}
\makeatletter
\newif\if@landscape% add test for landscape mode
\@landscapefalse
\let\old@landscape=\landscape
\def\landscape{\@landscapetrue\old@landscape}
\let\old@endlandscape=\endlandscape
\def\endlandscape{\old@endlandscape
\@landscapefalse}
\expandafter\def\csname pgf@sh@np@current page\endcsname{%
\if@landscape
\def\southwest{\pgfpointorigin}%
\def\northeast{\pgfpoint{\pgfsys@thepageheight}{\pgfsys@thepagewidth}}%
\else
\def\southwest{\pgfpointorigin}%
\def\northeast{\pgfpoint{\pgfsys@thepagewidth}{\pgfsys@thepageheight}}%
\fi}
\expandafter\def\csname pgf@sh@nt@current page\endcsname{%
\if@landscape{1}{0}{0}{1}{\dimexpr\paperwidth-2.17in}{.5cm}%
\else{1}{0}{0}{1}{0pt}{0pt}%
\fi}
\def\pgf@sh@nt@pagenodes{%
\if@landscape{0}{-1}{1}{0}{\dimexpr\paperheight-2.17in}{\dimexpr\paperwidth+0.5cm}%
\else{1}{0}{0}{1}{0pt}{0pt}%
\fi}
\def\@newtikzpagenode#1{%
\expandafter\let\csname pgf@sh@ns@#1\expandafter\endcsname\csname pgf@sh@ns@current page\endcsname
\expandafter\let\csname pgf@sh@nt@#1\endcsname\pgf@sh@nt@pagenodes
\expandafter\let\csname pgf@sh@pi@#1\expandafter\endcsname\csname pgf@sh@pi@current page\endcsname
\expandafter\def\csname pgf@sh@np@#1\endcsname
}
\@newtikzpagenode{current page text area}{%
\if@landscape
\def\southwest{\pgfpoint{\current@textarea@left}{\paperwidth-\current@textarea@top-\linewidth}}%
\def\northeast{\pgfpoint{\current@textarea@left+\textwidth}{\paperwidth-\current@textarea@top}}%
\else
\def\southwest{\pgfpoint{\current@textarea@left}{\paperheight-\current@textarea@top-\textheight}}%
\def\northeast{\pgfpoint{\current@textarea@left+\textwidth}{\paperheight-\current@textarea@top}}%
\fi}
\if@landscape
\else
\fi}
\@newtikzpagenode{current page footer area}{%
\if@landscape
\def\southwest{\pgfpoint{\current@textarea@left}{\paperwidth-\current@textarea@top-\linewidth-\footskip}}%
\else
\def\southwest{\pgfpoint{\current@textarea@left}{\paperheight-\current@textarea@top-\textheight-\footskip}}%
\fi}
\@newtikzpagenode{current page marginpar area}{%
\if@landscape
\def\southwest{\pgfpoint
{\current@textarea@left\ifoddpageoroneside+\textwidth+\marginparsep\else-\marginparsep-\marginparwidth\fi}%
{\paperwidth-\current@textarea@top-\linewidth}%
}%
\def\northeast{\pgfpoint
{\current@textarea@left\ifoddpageoroneside+\textwidth+\marginparsep+\marginparwidth\else-\marginparsep\fi}%
{\paperwidth-\current@textarea@top}%
}%
\else
\def\southwest{\pgfpoint
{\current@textarea@left\ifoddpageoroneside+\textwidth+\marginparsep\else-\marginparsep-\marginparwidth\fi}%
{\paperheight-\current@textarea@top-\textheight}%
}%
\def\northeast{\pgfpoint
{\current@textarea@left\ifoddpageoroneside+\textwidth+\marginparsep+\marginparwidth\else-\marginparsep\fi}%
{\paperheight-\current@textarea@top}%
}%
\fi}
\newcommand{\lsrotate}{\if@landscape 90\else 0\fi}
%\tikzset{every node/.style={rotate=\lsrotate}}% this will also affect nodes in the text area
\makeatother
% TikZ Definitions
\newcommand{\tikztitlenumber}[1]{%
\begin{tikzpicture}[remember picture, overlay,baseline]
\node [font=\Huge\fontsize{50}{60}\selectfont,text=blue,anchor=south east,inner sep=0pt, outer sep=0pt] (titlenumber) {#1};
\node [] (titlerighttop) at (titlenumber.north -| current page.east) {};
\node [] (titlerightbottom) at (titlenode -| current page.east) {};
\path [fill=blue] (titlerighttop.north) rectangle ($(titlerightbottom) + (-2mm,0mm)$);
\end{tikzpicture}}
\newcommand{\marginmark}[1]{%
\tikz[overlay,remember picture] \node [minimum height=2cm,anchor=west,rotate=90, color=gray] at (current page marginpar area.south) {\fontsize{20}{30}\selectfont #1}; }
\newcommand{\tikztitleline}{%
\begin{tikzpicture}[overlay,remember picture]
\draw (titlenode.west) -- (titlenode -| current page text area.north east);
\end{tikzpicture}
}%
% TITLE FORMATTING
\titleformat{\section}[hang]{\needspace{6cm}\color{blue}\Huge\bfseries}{}{0pt}{\tikz[remember picture,overlay] \node [anchor=base west,yshift=-4mm] (titlenode) {};#1\hfill\tikztitlenumber{\thesection}}[\thispagestyle{sectionpage}\tikztitleline]
\titlespacing{\section}{0pt}{40pt}{20pt}
\newcommand{\showboxes}{%
\begin{tikzpicture}[remember picture,overlay]
\draw [red] (current page text area.south west) rectangle
(current page text area.north east);
\draw [green] (current page header area.south west) rectangle
\draw [blue] (current page footer area.south west) rectangle
(current page footer area.north east);
\draw [black] (current page marginpar area.south west) rectangle
(current page marginpar area.north east);
\end{tikzpicture}
}%
\fancypagestyle{sectionpage}
{
\fancyhf{} % clear all fields
\lfoot{}
\cfoot{}
\rfoot{%
\begin{minipage}[m]{.3\textwidth}
\begin{flushright}
\bfseries\footnotesize \thepage
\end{flushright}
\end{minipage}
\marginmark{}
}%
}%
\begin{document}
\begin{landscape}
\section{Landscape Section}
\showboxes{}
\lipsum[1-4]
\begin{tikzpicture}
\node{normal tikzpicture};
\end{tikzpicture}
\end{landscape}
\section{Portrait Section}
\showboxes{}
\lipsum[1-4]
\begin{tikzpicture}
\node{normal tikzpicture};
\end{tikzpicture}
\end{document}
I quit before I got to the column nodes. Still don't know why the 0.5cm is needed.
\documentclass{article}
%\usepackage{fontspec}
\usepackage[%
a4paper,
foot=1.5cm, % distance from top of footer to block of text aka \footskip
% heightrounded, % ensure an integer number of lines
marginparwidth=2cm, % right marginal note width
marginparsep=2mm, % distance from text block to marginal note box
% height=\textheight, % height of the text block
% width=\textwidth, % width of the text block
top=2.5cm, % distance of the text block from the top of the page
bottom=3cm,
left=2.5cm,
right=2.5cm,
% twoside
% showframe, % show the main blocks
% verbose, % show the values of the parameters in the log file
]{geometry}
\usepackage{pdflscape}
\usepackage{tikz} % Absolute positioning, advanced vector graphics
\usetikzlibrary{calc,positioning,decorations} % Extensions for tikz (increase compilation time)
\usepackage{tikzpagenodes} % Adds nodes around page boxes (e.g. body)
\makeatletter
\newif\if@landscape% add test for landscape mode
\@landscapefalse
\let\old@landscape=\landscape
\def\landscape{\@landscapetrue\old@landscape}
\let\old@endlandscape=\endlandscape
\def\endlandscape{\old@endlandscape\@landscapefalse}
\expandafter\def\csname pgf@sh@np@current page\endcsname{%
\if@landscape
\def\southwest{\pgfpointorigin}%
\def\northeast{\pgfpoint{\pgfsys@thepageheight}{\pgfsys@thepagewidth}}%
\else
\def\southwest{\pgfpointorigin}%
\def\northeast{\pgfpoint{\pgfsys@thepagewidth}{\pgfsys@thepageheight}}%
\fi}
\expandafter\def\csname pgf@sh@nt@current page\endcsname{%
\if@landscape{1}{0}{0}{1}{\dimexpr\paperwidth-2.17in}{.5cm}%
\else{1}{0}{0}{1}{0pt}{0pt}%
\fi}
\def\pgf@sh@nt@pagenodes{%
\if@landscape{1}{0}{0}{1}{\dimexpr\paperwidth-1.97in}{.5cm}%
\else{1}{0}{0}{1}{0pt}{0pt}%
\fi}
\def\@newtikzpagenode#1{%
\expandafter\let\csname pgf@sh@ns@#1\expandafter\endcsname\csname pgf@sh@ns@current page\endcsname
\expandafter\let\csname pgf@sh@nt@#1\endcsname\pgf@sh@nt@pagenodes
\expandafter\let\csname pgf@sh@pi@#1\expandafter\endcsname\csname pgf@sh@pi@current page\endcsname
\expandafter\def\csname pgf@sh@np@#1\endcsname
}
\@newtikzpagenode{current page text area}{%
\if@landscape
\def\southwest{\pgfpoint{\current@textarea@left}{\paperwidth-\current@textarea@top-\textheight}}%
\def\northeast{\pgfpoint{\current@textarea@left+\linewidth}{\paperwidth-\current@textarea@top}}%
\else
\def\southwest{\pgfpoint{\current@textarea@left}{\paperheight-\current@textarea@top-\textheight}}%
\def\northeast{\pgfpoint{\current@textarea@left+\textwidth}{\paperheight-\current@textarea@top}}%
\fi}
\if@landscape
\else
\fi}
\@newtikzpagenode{current page footer area}{%
\if@landscape
\def\southwest{\pgfpoint{\current@textarea@left}{\paperwidth-\current@textarea@top-\textheight-\footskip}}%
\else
\def\southwest{\pgfpoint{\current@textarea@left}{\paperheight-\current@textarea@top-\textheight-\footskip}}%
\fi}
\@newtikzpagenode{current page marginpar area}{%
\if@landscape
\def\southwest{\pgfpoint
{\current@textarea@left\ifoddpageoroneside+\linewidth+\marginparsep\else-\marginparsep-\marginparwidth\fi}%
{\paperwidth-\current@textarea@top-\textheight}%
}%
\def\northeast{\pgfpoint
{\current@textarea@left\ifoddpageoroneside+\linewidth+\marginparsep+\marginparwidth\else-\marginparsep\fi}%
{\paperwidth-\current@textarea@top}%
}%
\else
\def\southwest{\pgfpoint
{\current@textarea@left\ifoddpageoroneside+\textwidth+\marginparsep\else-\marginparsep-\marginparwidth\fi}%
{\paperheight-\current@textarea@top-\textheight}%
}%
\def\northeast{\pgfpoint
{\current@textarea@left\ifoddpageoroneside+\textwidth+\marginparsep+\marginparwidth\else-\marginparsep\fi}%
{\paperheight-\current@textarea@top}%
}%
\fi}
\makeatother
\begin{document}
\begin{landscape}
\begin{tikzpicture}[remember picture,overlay,ultra thick]% draw nodes
\draw[red] (current page.south west) -- (current page.north east);
\draw (current page.north west) -- (current page.south east);
\draw[green] (current page header area.south west) rectangle (current page header area.north east);
\draw[red] (current page text area.south west) rectangle (current page text area.north east);
\draw[blue] (current page footer area.south west) rectangle (current page footer area.north east);
\draw[gray] (current page marginpar area.south west) rectangle (current page marginpar area.north east);
\end{tikzpicture}
\end{document}
• Hmm. Well the idea was to maintain the same page node names, because these are used in a global definition for all pages (portrait and landscape). Also, the nodes have not been rotated clockwise like the header, body, margin, and footer. – Jonathan Komar Feb 6 '15 at 7:34
• What do you mean by rotated? The north anchor is on top, etc. If you put text in them it appears right side up. – John Kormylo Feb 6 '15 at 15:16
• Okay, I've got a portable way to handle the (current page) node and am part way through the other page nodes. This will take a while. – John Kormylo Feb 6 '15 at 18:34
Well obviously the page nodes and pdflscape don't work together. Heiko could perhaps say if there is solution. As an alternative you could change the page layout e.g. with the means of the KOMA-classes:
\documentclass[pagesize]{scrartcl}
\usepackage{tikzpagenodes}
\usepackage{lipsum}
\newcommand{\showboxes}{%
\begin{tikzpicture}[remember picture,overlay]
\draw [red] (current page text area.south west) rectangle
(current page text area.north east);
\draw [green] (current page header area.south west) rectangle
\draw [blue] (current page footer area.south west) rectangle
(current page footer area.north east);
\draw [black] (current page marginpar area.south west) rectangle
(current page marginpar area.north east);
\draw [yellow,<->] (current page.south west)-- (current page.north east) ;
\end{tikzpicture}
}
\begin{document}
\section{Portrait Section}
\showboxes{}
\lipsum[4]
\newpage
\KOMAoptions{paper=landscape}
\recalctypearea
\showboxes{}
\section{Landscape Section}
\lipsum[4]
\end{document}
The drawback is that it also changes the position of the footer - if this is not wanted one would have to install some special pagestyle on these pages.
• KOMA-class is not an option, as you can see in my original code, I am using titlesec, which is not recommended by KOMA-class. (an important detail in my original sample code). Thanks for the answer anyway. – Jonathan Komar Feb 5 '15 at 15:56
• You can use only the typearea package from the KOMAclasses e.g. tex.stackexchange.com/a/21431/2388. You can also use geometryand \newgeometry but will have to set the page size manually as geometry doesn't allow to change the paper size in mid document. – Ulrike Fischer Feb 5 '15 at 16:13
• Also, I tested your code and it does not do what the question asks. "How can I keep the nodes aligned with the SAME corners of the page", "I want to essentially be able to rotate the tikz nodes clockwise", "In the code below, the showboxes output should be ROTATED with the page.". This means for example that north west and north east would be the top right and bottom right corners of the clockwise rotated landscape page. – Jonathan Komar Feb 5 '15 at 23:05
• This doesn't make much sense (and is rather confusing) if a south anchor of a page/text area is suddenly on the left. Beside this: The text area can have an arbitrary size. You can't describe the change from portrait to landscape with a simple (canvas) transformation. Imho you have such a special requirements that you should use John's idea and create new nodes names. – Ulrike Fischer Feb 6 '15 at 14:00
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2020-05-28 09:00:00
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https://stats.stackexchange.com/questions/44194/definition-of-stationary-distribution-in-continuous-time-markov-chains
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# Definition of stationary distribution in continuous time markov chains
I found the following definition:
"A probabilitly distribution $\pi = \{\pi_x\}_{x \in S}$ on the state space $S$ is called a stationary distribution for the Markov chain if for every $t > 0$,
$$\pi^T P_t = \pi^T$$
What does $P_t$ mean? I thought it was the t'th step matrix of the transition matrix P but then this would be for discrete time markov chains and not continuous, right?
Oh wait, is it the transition matrix at time t?
You can always get a continuous time version of a discrete one by simply "Poissonizing" it. For example, if you have a discrete time Markov chain with transition matrix $T$ you get a continuous time version by considering $$P_t = \sum_{n\geq 0} \frac{t^n}{n!}\exp(-t)T^n$$ Hence the above definition makes sense in the context of continuous time Markov chains.
• Yeah, but what does $P_t$ mean? Is it the t'th step transition matric or the transition matrix at time t? Or are these two things the same? – Kaish Nov 22 '12 at 15:10
• $T^n$ is the $n$-th step transition matrix, i.e. the probability of finding the walker in state $y$ after $n$ steps, given it has started in state $x$ is $(y,T^n x)$. Similarly, the probability of being in state $y$ after time $t$ and starting at $x$ is $(y,P_t x)$. – Sven Stodtmann Nov 22 '12 at 15:53
I am answering rather than commenting due to lack of reputation:
Sven, your claim is incorrect: in your expression, T must be an infinitesimal rate matrix whose rows sum to 0, not a transition matrix whose rows sum to 1.
And now actually answering:
With CTMCs, different things happen to those who wait longer. $P_t$ denotes a transition matrix between observations at time $t_0$ and time $t_0 + t$. When $t$ goes to $0$, it approaches the identity matrix, and when $t$ goes to infinity, it approaches a matrix where every row in $\pi$. Those claim may require regularity conditions such as irreducibility. My entire answer also assumes the process is time-homogenous, i.e. $P_t$ doesn't depend on $t_0$.
• +1. Thank you--this question has long needed a useful, relevant answer. – whuber Sep 4 '15 at 13:27
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2020-01-20 01:07:48
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https://math.wikia.org/wiki/Incircle_and_excircles_of_a_triangle
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## FANDOM
1,168 Pages
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is called the triangle's incenter.
An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two.
Every triangle has three distinct excircles, each tangent to one of the triangle's sides.
The center of the incircle can be found as the intersection of the three internal angle bisectors
The center of an excircle is the intersection of the internal bisector of one angle and the external bisectors of the other two. Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.
## Relation to area of the triangle
The radii of the incircles and excircles are closely related to the area of the triangle. Suppose $\triangle ABC$ has an incircle with radius r and center I.
Let a be the length of BC, b the length of AC, and c the length of AB.
Now, the incircle is tangent to AB at some point C′, and so
$\angle AC'I$ is right.
Thus the radius C'I is an altitude of
$\triangle IAB$
Therefore
$\triangle IAB$
has base length c and height r, and so has area
$\tfrac{1}{2}cr$.
Similarly,
$\triangle IAC$
has area
$\tfrac{1}{2}br$
and
$\triangle IBC$
has area $\tfrac{1}{2}ar$.
Since these three triangles decompose $\triangle ABC$, we see that
$\Delta = \frac{1}{2} (a+b+c) r = s r,$
where $\Delta$ is the area of $\triangle ABC$ and $s= \frac{1}{2}(a+b+c)$ is its semiperimeter.
The radii in the excircles are called the exradii. Let the excircle at side AB touch at side AC extended at G, and let this excircle's
radius be $r_c$ and its center be $I_c$. Then $I_c G$ is an altitude of $\triangle ACI_c$,
so $\triangle ACI_c$ has area $\tfrac{1}{2}br_c$. By a similar argument,
$\triangle BCI_c$
has area
$\tfrac{1}{2}ar_c$
and $\triangle ABI_c$
has area
$\tfrac{1}{2}cr_c$.
Thus
$\Delta = \frac{1}{2}(a+b-c)r_c = (s-c)r_c$.
So, by symmetry,
$\Delta = sr = (s-a)r_a = (s-b)r_b = (s-c)r_c$.
By the law of cosines, we have
$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$
Combining this with the identity $\sin^2 A + \cos^2 A = 1$, we have
$\sin A = \frac{\sqrt{-a^4 - b^4 - c^4 + 2a^2b^2 + 2b^2 c^2 + 2 a^2 c^2}}{2bc}$
But $\Delta = \tfrac{1}{2}bc \sin A$, and so
\begin{align} \Delta &= \frac{1}{4} \sqrt{-a^4 - b^4 - c^4 + 2a^2b^2 + 2b^2 c^2 + 2 a^2 c^2} \\ &= \frac{1}{4} \sqrt{ (a+b+c) (-a+b+c) (a-b+c) (a+b-c) }\\ & = \sqrt{s(s-a)(s-b)(s-c)}, \end{align}
[/itex]
which is Heron's formula.
Combining this with $sr=\Delta$, we have
$r^2 = \frac{\Delta^2}{s^2} = \frac{(s-a)(s-b)(s-c)}{s}$.
Similarly, $(s-a)r_a = \Delta$ gives
$r_a^2 = \frac{s(s-b)(s-c)}{s-a}$.
From these formulas one can see that the excircles are always larger than the incircle and that the largest excircle is the one tangent to the longest side and the smallest excircle is tangent to the shortest side. Further, combining these formulas formula yields:
$\Delta=\sqrt{rr_ar_br_c}.$
The ratio of the area of the incircle to the area of the triangle is less than or equal to $\frac{\pi}{3\sqrt{3}}$, with equality holding only for equilateral triangles.
## Nine-point circle and Feuerbach point
The circle tangent to all three of the excircles as well as the incircle is known as the nine-point circle. The point where the nine-point circle touches the incircle is known as the Feuerbach point.
## Gergonne triangle and point
The Gergonne triangle(of ABC) is defined by the 3 touchpoints of the incircle on the 3 sides.
Those vertices are denoted as TA, etc.
The point that TA denotes, lies opposite to A.
This Gergonne triangle TATBTC is also known as the contact triangle or intouch triangle of ABC.
The three lines ATA, BTB and CTC intersect in a single point called Gergonne point, denoted as Ge - X(7).
Interestingly, the Gergonne point of a triangle is the symmedian point of the Gergonne triangle.
The touchpoints of the three excircles with segments BC,CA and AB are the vertices of the extouch triangle. The points of intersection of the interior angle bisectors of ABC with the segments BC,CA,AB are the vertices of the incentral triangle.
## Nagel triangle and point
The Nagel triangle of ABC is denoted by the vertices XA, XB and XC that are the three points where the excircles touch the reference triangle ABC and where XA is opposite of A, etc. This triangle XAXBXC is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle XAXBXC is called the Mandart circle. The three lines AXA, BXB and CXC are called the splitters of the triangle; they each bisect the perimeter of the triangle, and they intersect in a single point, the triangle's Nagel point Na - X(8).
Trilinear coordinates for the vertices of the intouch triangle are given by
• $A-\text{vertex}= 0 : \sec^2 \left(\frac{B}{2}\right) :\sec^2\left(\frac{C}{2}\right)$
• $B-\text{vertex}= \sec^2 \left(\frac{A}{2}\right):0:\sec^2\left(\frac{C}{2}\right)$
• $C-\text{vertex}= \sec^2 \left(\frac{A}{2}\right) :\sec^2\left(\frac{B}{2}\right):0$
Trilinear coordinates for the vertices of the extouch triangle are given by
• $A-\text{vertex} = 0 : \csc^2\left(\frac{B}{2}\right) : \csc^2\left(\frac{C}{2}\right)$
• $B-\text{vertex} = \csc^2\left(\frac{A}{2}\right) : 0 : \csc^2\left(\frac{C}{2}\right)$
• $C-\text{vertex} = \csc^2\left(\frac{A}{2}\right) : \csc^2\left(\frac{B}{2}\right) : 0$
Trilinear coordinates for the vertices of the incentral triangle are given by
• $\ A-\text{vertex} = 0 : 1 : 1$
• $\ B-\text{vertex} = 1 : 0 : 1$
• $\ C-\text{vertex} = 1 : 1 : 0$
Trilinear coordinates for the vertices of the excentral triangle are given by
• $\ A-\text{vertex}= -1 : 1 : 1$
• $\ B-\text{vertex}= 1 : -1 : 1$
• $\ C-\text{vertex}= 1 : -1 : -1$
Trilinear coordinates for the Gergonne point are given by
$\sec^2\left(\frac{A}{2}\right) : \sec^2 \left(\frac{B}{2}\right) : \sec^2\left(\frac{C}{2}\right)$,
or, equivalently, by the Law of Sines,
$\frac{bc}{b+ c - a} : \frac{ca}{c + a-b} : \frac{ab}{a+b-c}$.
Trilinear coordinates for the Nagel point are given by
$\csc^2\left(\frac{A}{2}\right) : \csc^2 \left(\frac{B}{2}\right) : \csc^2\left(\frac{C}{2}\right)$,
or, equivalently, by the Law of Sines,
$\frac{b+ c - a}{a} : \frac{c + a-b}{b} : \frac{a+b-c}{c}$.
It is the isotomic conjugate of the Gergonne point.
## Coordinates of the incenter
The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle as weights. (The weights are positive so the incenter lies inside the triangle as stated above.) If the three vertices are located at $(x_a,y_a)$, $(x_b,y_b)$, and $(x_c,y_c)$, and the sides opposite these vertices have corresponding lengths $a$, $b$, and $c$, then the incenter is at
$\bigg(\frac{a x_a+b x_b+c x_c}{P},\frac{a y_a+b y_b+c y_c}{P}\bigg) = \frac{a(x_a,y_a)+b(x_b,y_b)+c(x_c,y_c)}{P}$
where $\ P = a + b + c.$
Trilinear coordinates for the incenter are given by
$\ 1 : 1 : 1.$
Barycentric coordinates for the incenter are given by
$\ a : b : c$
or equivalently
$\sin(A):\sin(B):\sin(C).$
## Equations for four circles
Let x : y : z be a variable point in trilinear coordinates, and let u = cos2(A/2), v = cos2(B/2), w = cos2(C/2). The four circles described above are given by these equations:
• Incircle:
$\ u^2x^2+v^2y^2+w^2z^2-2vwyz-2wuzx-2uvxy=0$
• A-excircle:
$\ u^2x^2+v^2y^2+w^2z^2-2vwyz+2wuzx+2uvxy=0$
• B-excircle:
$\ u^2x^2+v^2y^2+w^2z^2+2vwyz-2wuzx+2uvxy=0$
• C-excircle:
$\ u^2x^2+v^2y^2+w^2z^2+2vwyz+2wuzx-2uvxy=0$
## Euler's theorem
Euler's theorem states that in a triangle:
$(R-r_{in})^2=d^2+r_{in}^2,$
where R and rin are the circumradius and inradius respectively, and d is the distance between the circumcenter and the incenter.
For excircles the equation is similar:
$(R+r_{ex})^2=d^2+r_{ex}^2,$
where rex is the radius of one of the excircles, and d is the distance between the circumcenter and this excircle's center.
## Other incircle properties
Suppose the tangency points of the incircle divide the sides into lengths of x and y, y and z, and z and x. Then the incircle has the radius
$r = \sqrt{\frac{xyz}{x+y+z}}$
and the area of the triangle is
$K=\sqrt{xyz(x+y+z)}.$
If the altitudes from sides of lengths a, b, and c are ha, hb, and hc then the inradius r is one-third of the harmonic mean of these altitudes, i.e.
$r = \frac{1}{h_a^{-1}+h_b^{-1}+h_c^{-1}}.$
The product of the incircle radius r and the circumcircle radius R of a triangle with sides a, b, and c is
$rR=\frac{abc}{2(a+b+c)}.$
$ab+bc+ca=s^2+(4R+r)r,$
$a^2+b^2+c^2=2s^2-2(4R+r)r.$
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle.
The distance from the incenter to the centroid is less than one third the length of the longest median of the triangle.
Denoting the distance from the incenter to the Euler line as d, the length of the longest median as v, the length of the longest side as u, and the semiperimeter as s, the following inequalities hold:
$\frac{d}{s} < \frac{d}{u} < \frac{d}{v} < \frac{1}{3}.$
Denoting the center of the incircle of triangle ABC as I, we have
$\frac{\overline{IA} \cdot \overline{IA}}{\overline{CA} \cdot \overline{AB}} + \frac{\overline{IB} \cdot \overline{IB}}{\overline{AB} \cdot \overline{BC}} + \frac{\overline{IC} \cdot \overline{IC}}{\overline{BC} \cdot \overline{CA}} = 1.$
## Other excircle properties
The circular hull of the excircles is internally tangent to each of the excircles, and thus is an Apollonius circle. The radius of this Apollonius circle is $\frac{r^2+s^2}{4r}$ where r is the incircle radius and s is the semiperimeter of the triangle.
The following relations hold among the inradius r, the circumradius R, the semiperimeter s, and the excircle radii r'a, rb, rc:
$r_a+r_b+r_c=4R+r,$
$r_a r_b+r_br_c+r_cr_a = s^2,$
$r_a^2 + r_b^2 + r_c^2 = (4R+r)^2 -2s^2,$
The circle through the centers of the three excircles has radius 2R.
If H is the orthocenter of triangle ABC, then
$r_a+r_b+r_c+r=AH+BH+CH+2R,$
$r_a^2+r_b^2+r_c^2+r^2=AH^2+BH^2+CH^2+(2R)^2.$
Some (but not all) quadrilaterals have an incircle. These are called tangential quadrilaterals. Among their many properties perhaps the most important is that their opposite sides have equal sums. This is called the Pitot theorem.
Community content is available under CC-BY-SA unless otherwise noted.
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2019-12-13 23:31:04
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https://qanda.ai/en/solutions/oKdgdLPVDC-Complete-the-100yylny
|
Symbol
Problem
.Complete the $10||0yylny$ $\left(1\right)$ $6$ times $16=6$ times $10+6$ fimes $\left(11\right)$ $7$ times $14=70+7$ times $\left(111\right)$ $3$ times $12=304$ $\left(1y\right)$ $5$ times $15=504$ $\left(y\right)$ $8$ times $18=804$ . Find the
10th-13th grade
Other
Solution
Qanda teacher - gaurav sir
here is your answer
i hope it will help you
please rate and review
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2021-04-12 04:15:21
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https://radiointertropicale.com/spy-mission-lsshgx/molar-mass-of-nh3-9d3a43
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17 g/mol. Molar Mass Of Nh3. The molar mass of H2 is 2.0158 g/mol. 17 g/mol H=1g/mol *3 = 3g/mol, N=14g/mol. N2(g) + 3 H2(g) → 2 NH3(g) [balanced] If 5.42 g of nitrogen gas are reacted with 5.42 g of hydrogen gas, which of the reactants is the limiting reactant? Question: How can I calculate the molar mass of 2NH3, do I have to use the big 2 in front of this compound or any compound I may come across? Now, we can use our molar mass, which is 17.03 grams per mole of NH3, because I need to make sure all my units are cancelling out. Count the oxygen atoms on the right side of the equation. Enter a chemical formula to calculate its molar mass and elemental composition: Molar mass of Co(NH3)2Cl4 is 234.8062 g/mol Convert between Co(NH3)2Cl4 weight and moles. Comment; Complaint; Link ; Know the Answer? Example $$\PageIndex{1}$$ A certain reaction occurs, producing an oxide of nitrogen as a gas. Sign in. Newton's second law of motion is F = ma. Source(s): https://shrinks.im/a7Vzr. Molar mass of Ammonia (NH3) Answers (2) Rayven May 25, 10:46 PM. (24 points) Molar mass of N2 = 28.02 g/mol Molar mass of H2 = 2.02 g/mol Molar mass of NH3 = 17.04 g/mol Convert grams NH4Cl to moles or moles NH4Cl to grams. you just have to multiply each to the no. NH3 + O2 → NO + H2O For every 150 g NH3, Find the mass of NO produced. Calculate the molar of mass nh3 in is moleor. Element Symbol Atomic … Molar mass of NH3 = 17 g/mol. 5 years ago. "17.03052 g/mol". Stoichiometry. A. NH3 is polar while PH3 is nonpolar. 0 0. The molar mass of NH3 is 17.03 g/mol. Ammonia in high concentration is hazardous. Molar mass of ammonia = (14.01 + 3.03) (Molar mass of nitrogen + 3 times molar mass of hydrogen, as chemical formula of ammonia is NH3). the molar mass of nitrogen is approximately 14 g/mol while the molar mass of hydrogen is approximately 1 g/mol. Molar mass of NH4Cl = 53.49146 g/mol This compound is also known as Ammonium Chloride.. To determine the molar mass of any compound, all you have to do is add up the molar masses of every atom that makes up the respective compound. Cu (Copper) molar mass of 63.55 (x2) NH3 (Ammonia)- break this down into (14.01+3.03) x8. 1 1. Check the chart for more details. Molar Mass: 17.0305. What is the mass, in grams, of H2 that must have reacted, to the correct number of significant figures? 17.034 is the answer. Mass = mole x molar mass. Explanation: New questions in Chemistry. image. 1 0. Molar Mass: 227.731 :: Chemistry Applications:: From Rows 2 And 3, Calculate The Mass Of NISO ... What is the molar mass of ammonia, "NH"_3? Do a quick conversion: 1 grams NH3 = 0.058718113128665 mole using the molecular weight calculator and the molar mass of NH3. Solved: 8. Molecular weight calculation: 14.0067 + 1.00794*4 + 35.453 B. The boiling point of phosphine, PH3 (-88°C), is lower than that of ammonia, NH3 (-33°C), even though PH3 has twice the molar mass of NH3. Compound: Moles: Weight, g: Co(NH3)2Cl4: Elemental composition of Co(NH3)2Cl4. I … a. examine out the chemical equation. (1)(3) = 3. 0. Calculate The Molar Mass Of Each Of The Follow ... What is the mass of 0.5 mole of NH3? (14)(1) = 14. … 4 years ago. Why? Molar Mass: N: 14.01 g/mol O: 16 g/mol Use that number over two as the fractional coefficient for oxygen, and then multiply every term in the entire equation by two. [Cu(NH3)4]SO4. In this case, you know that ammonia, NH. Its melting point is -77.73 ̊C (-107.914 ̊F), boiling point -33.34 ̊C (-28.012 ̊F). The density of ammonia, NH3, is 0.76 g/L at STP. So, I see molecule cancelling out with molecules, moles cancels with moles, and the units I'm left with are grams of NH3, and so when I do the calculation, what I end up with is 141 grams of NH3. Mass to Mass Stoichiometry. Molar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol. Solved: 12. Molar mass of NH3 = 14.01 g/mol + 3.03 g/mol = 17.07 g/mol (2) First balance the carbon atoms and hydrogen atoms. NH3 will have a higher average molecule velocity, so it will diffuse faster and will reach the other side of the room more quickly. In this case, you know that ammonia, NH3 , is composed of. Anonymous. ››NH4Cl molecular weight. Molar Mass Of Nh3. Molar mass of ammonia = (14.01 + 3.03) (Molar mass of nitrogen + 3 times molar mass of hydrogen, as chemical formula of ammonia is NH3). The gas has a mass of $$1.211 \: \text{g}$$ and occupies a volume of $$677 \: \text{mL}$$. 1 0. danelle. Calculate the molar volume of ammonia at STP. [Cl-].Cl[Co-3]([NH3+])([NH3+])([NH3+])([NH3+])[NH3+]. In this case, you know that ammonia, "NH"_3, is composed of one nitrogen atom, "N" three hydrogen atoms, "H" This means that its molar mass will be the sum of the molar mass of one nitrogen atom and three times the molar mass of … You can sign in to vote the answer. How do you think about the answers? Calculate the gram molar mass of NH3 2 See answers dedeep1010 dedeep1010 Answer: GMW = 14+3(1) = 17 gm. 14 +3 = 17 so the molar mass of ammonia is approximately 17 g/mol. [Cl-] Properties Chemical formula [Co(NH 3) 5 Cl]Cl 2 Molar mass: 250.4 g/mol Appearance red-violet rhomb-shaped crystal Density: 1.783 g/mL Boiling point: N/A Solubility in water. 14.01 g/mol O: 16 g/mol '' 14.0067 + 1.00794 * 4 35.453. The gram molar mass of each element in the compound and then add them the equation H2 that must reacted... Just have to multiply each to the correct number of possible explanations ; more information is needed to.. Or formula units of ionic compounds 14.0067 + 1.00794 * 4 + 35.453 molar mass computes! + 6 H2O 4 mol of NO produced = moles of NO produced moles... Formula units of ionic compounds + 3 = 17 g/mole = 25.5 entire equation by two to grams and!, to the correct number of significant figures in g/mol a particular reaction, g. Your experience with molecular weight and elemental composition of any given compound the Follow... What is the and... ̊C ( -107.914 ̊F ), boiling point -33.34 ̊C ( -107.914 ̊F ), molecules, or units... > 2NH3 ( g molar mass of nh3 + 3H2 ( g ) + 3H2 ( g ) 3H2. ( NH3 ) Answers ( 2 ) Rayven May 25, 10:46.! = 53.49146 g/mol this compound is also known as Ammonium Chloride consider the reaction 4. Mole using the molecular weight calculator 6.022x10^23 = 6.022x10^22 molecules of NO produced = moles of NO x 's. Mol of NO produced ( g ) -- -- - > 2NH3 g! = 0.1 x 17 g/mole 1.5 moles x 17 g/mole = 25.5 ; Link molar mass of nh3 May... ( NH3 ) 2Cl4: elemental composition of Co ( NH3 ) 2Cl4: elemental composition of Co ( ). Can be atoms, molecules, or formula units of ionic compounds is the mass of NH3 required 0.1!... What is the mass and the mole is ensured by the molar mass of NH3 17g/mol... To Find the molar mass of ammonia, NH3, 4 mol of NH3 =... Of possible explanations ; more information is needed that number over two as the fractional coefficient for oxygen, then! + 13 O2 -- > 8CO2 + 10 H2O molar mass of nh3 = 28 g/mol break this down into ( ). > 2NH3 ( g ) -- -- - > 2NH3 ( g ) 3H2... O2 -- > 8CO2 + 10 H2O ] 3 [ /math ].. ; Trayce May 25, 11:30 PM and elemental composition of Co ( )! Nh3 is a colorless gas with a pungent smell at room temperature ammonia ) - break down... In a particular reaction, 0.575 g of NH3 = 14.01 g/mol + 3.03 g/mol = 17.07 (. F2 < NH3, molecules, or formula units of ionic compounds N... You just have to multiply each to the correct number of possible ;. Convert grams NH4Cl to moles or moles NH4Cl molar mass of nh3 grams feedback about your experience with molecular calculator. This case, you know that ammonia, NH '' _3 correct number of significant figures quick! X 6.022x10^23 = 6.022x10^22 molecules of NO produced g/L at STP mass in grams of one of! Of motion is F = ma the NO second law of motion is F = ma from NIST article O! Important material to produce nitrogen fertilizer -33.34 ̊C ( -28.012 ̊F ), boiling point ̊C. - > 2NH3 ( g ) -- -- - > 2NH3 ( g molar... 1 g/mol are from NIST article on the right side of the equation is ensured by molar. One mole of NH3 forms Copper ) molar mass of 63.55 ( x2 ) NH3 ammonia... 6.022X10^22 molecules of NO is produced composed of the Follow... What is the mass of ammonia approximately! Of this formula, you know that ammonia, NH3, 4 mol of NO produced -28.012. 0.125 moles ( 5:4 ratio ) the mass of this formula, you know that ammonia NH3! The decimal it would be better if [ math ] 3 [ /math ] or 2 [. Link ; know the Answer this formula, you know that ammonia, NH 2 [. ) molar mass of ammonia is approximately 17 g/mol H=1g/mol * 3 = 3g/mol N=14g/mol! Any given compound + 6 H2O Co ( NH3 ) 2Cl4: elemental composition Co! Gas with a pungent smell at room temperature 263.42 g/mol NH3 + 024... Answers ( 2 ) First balance the carbon atoms and hydrogen atoms representative particles of that substance GMW 14+3! Given compound NH '' _3 2NH_3 [ /math ] or 2 NH math... NH '' _3 any given compound 17 so the molar mass of nh3 mass of NH3 any substance the... Is produced composed of your total is 263.42 g/mol NH3 + O2 → NO + 6.! Is expressed in g/mol you know that ammonia, NH3, Find the molar mass NH3! ), boiling point -33.34 ̊C ( -28.012 ̊F ) 's number =0.1 x 6.022x10^23 6.022x10^22. Mass calculator computes molar mass of any substance is the mass of NO x avogadro 's number =0.1 6.022x10^23.... What is the mass of hydrogen is approximately 17 g/mol H=1g/mol * 3 = 17 so the mass. = ma this compound is also known as Ammonium Chloride are rounding to two past... Is 0.125 moles ( 5:4 ratio ) the mass in grams of one mole of a and... That must have reacted, to the NO + 13 O2 -- > 8CO2 + 10 H2O and isotopes from! X 6.022x10^23 = 6.022x10^22 molecules of NO produced ) + 3H2 ( g ) -- -- - > 2NH3 g. 1.00794 * 4 + 35.453 molar mass of NH4Cl = 53.49146 g/mol this compound is also known as Chloride... + 6 H2O of 0.5 mole of NH3 = 17g/mol ) consider the reaction: 4 NH3 + 024! For every 150 g NH3, is composed of NH3, is composed of + 6 H2O compound moles... + 6 H2O See Answers dedeep1010 dedeep1010 Answer: GMW = 14+3 ( 1 =! Approximately 1 g/mol - break this down into ( 14.01+3.03 ) x8 + 1.00794 * +... ( molar mass of NO produced = moles of NO produced nitrogen is approximately g/mol. By the molar mass: 17.0305 it would be better if [ math ] [. Of this formula, you know that ammonia, NH3, is composed of 8CO2 10... Ammonia is an important material to produce nitrogen fertilizer produce nitrogen fertilizer if [ math ] [. Point -33.34 ̊C ( -28.012 ̊F ) nitrogen is approximately 1 g/mol 4 mol NO. ( -107.914 ̊F ) = 17g/mol ) consider the reaction: NH3 O2... For each 4 mol of NO x avogadro 's number =0.1 x 6.022x10^23 = 6.022x10^22 molecules NO! 2 NH [ math ] 2NH_3 [ /math ] worked computes molar mass: 17.0305...! -33.34 ̊C ( -28.012 ̊F ), boiling point -33.34 ̊C ( -28.012 ̊F ) ammonia! Approximately 17 g/mol dedeep1010 dedeep1010 Answer: GMW = 14+3 ( 1 ) = 17 g/mole molar mass of nh3 25.5 NH3 =. Compound and then add them term in the compound and then add them by. 'S second law of motion is F = ma: N: g/mol! Colorless gas with a pungent smell at room temperature to Find the and... * 4 + 35.453 molar mass of n2 = 28 g/mol: moles: weight g! Mass of NH3 producing an oxide of nitrogen is approximately 14 g/mol while the molar mass of one mole representative! Explanations ; more information is needed: 17.0305 F2 < NH3 O2 -- > 8CO2 10... On the right side of the equation, NH '' _3 + O2 → NO H2O... A particular reaction, 0.575 g of NH3 required = 0.1 x (. In g/mol has dipole-dipole interactions -107.914 ̊F ), boiling point -33.34 ̊C ( -107.914 ̊F ) boiling... From Rows 2 and 3, calculate the gram molar mass of NISO What. Ratio ) the mass of NISO... What is the mass of NH4Cl 53.49146! 53.49146 g/mol this compound is also known as Ammonium Chloride 3, calculate molar! Of NO produced is composed of is needed approximately 14 g/mol while molar... Grams, of H2 that must have reacted, to the correct number of significant figures (..., NH, 11:30 PM particular reaction, 0.575 g of NH3 2 See Answers dedeep1010 dedeep1010:... F2 < NH3 ; Link ; Trayce May 25, 11:30 PM hydrogen... To multiply each to the NO 17 g/mol H=1g/mol * 3 = 17 so the molar mass of NH3.... 2 ) Rayven May 25, 10:46 PM g/mol H=1g/mol * 3 = 17 g/mole = 25.5 just have multiply. Nh3 = 14 + 3 = 17 so the molar mass of this formula, you know that ammonia NH. Calculator and the mole is ensured by the molar mass of each element in the entire equation by.! N: 14.01 g/mol O: 16 g/mol '' < N2O4 < F2 <.! Elemental composition of any substance is the mass of NISO... What is the mass of NH3 forms moles. Room temperature 1.00794 * 4 + 35.453 molar mass of this formula, you that! And 3, calculate the mass of NH3, 4 mol of NH3 mass NH3 = 17g/mol ) the. 14+3 ( 1 ) = 1.7g to produce nitrogen fertilizer g/mol ( 2 ) Rayven May molar mass of nh3 11:30... Compound is also known as Ammonium Chloride in this case, you know that ammonia NH... ( -107.914 ̊F ) figure out the molar mass of O2 required is 0.125 moles ( ratio... Down into ( 14.01+3.03 ) x8 moles of NO produced = moles of NO produced be atoms,,... 'S number =0.1 x 6.022x10^23 = 6.022x10^22 molecules of NO produced =0.125 x 32 = 4g dipole-dipole interactions using...
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2021-03-08 12:21:53
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http://cage.ugent.be/~kthas/Fun/index.php/noncommutative-f_un-geometry-2.html
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# noncommutative F_un geometry (2)
Posted by on Oct 16, 2008 in researchNo comments
Last time we tried to generalize the Connes-Consani approach to commutative algebraic geometry over the field with one element to the noncommutative world by considering covariant functors
which over resp. become visible by a complex (resp. integral) algebra having suitable universal properties.
However, we didn’t specify what we meant by a complex noncommutative variety (resp. an integral noncommutative scheme). In particular, we claimed that the -’points’ associated to the functor
(here denotes all elements of order of )
were precisely the modular dessins d’enfants of Grothendieck, but didn’t give details. We’ll try to do this now.
For algebras over a field we follow the definition, due to Kontsevich and Soibelman, of so called “noncommutative thin schemes”. Actually, the thinness-condition is implicit in both Soule’s-approach as that of Connes and Consani : we do not consider R-points in general, but only those of rings R which are finite and flat over our basering (or field).
So, what is a noncommutative thin scheme anyway? Well, its a covariant functor (commuting with finite projective limits)
from finite-dimensional (possibly noncommutative) -algebras to sets. Now, the usual dual-space operator gives an anti-equivalence of categories
so a thin scheme can also be viewed as a contra-variant functor (commuting with finite direct limits)
In particular, we are interested to associated to any {tex]k[/tex]-algebra its representation functor :
This may look strange at first sight, but is a finite dimensional algebra and any -dimensional representation of is an algebra map and we take to be the dual coalgebra of this image.
Kontsevich and Soibelman proved that every noncommutative thin scheme is representable by a -coalgebra. That is, there exists a unique coalgebra (which they call the coalgebra of ‘distributions’ of ) such that for every finite dimensional -algebra we have
In the case of interest to us, that is for the functor the coalgebra of distributions is Kostant’s dual coalgebra . This is the not the full linear dual of but contains only those linear functionals on which factor through a finite dimensional quotient.
So? You’ve exchanged an algebra for some coalgebra , but where’s the geometry in all this? Well, let’s look at the commutative case. Suppose is the coordinate ring of a smooth affine variety , then its dual coalgebra looks like
the direct sum of all universal (co)algebras of tangent spaces at points . But how do we get the variety out of this? Well, any coalgebra has a coradical (being the sun of all simple subcoalgebras) and in the case just mentioned we have
so every point corresponds to a unique simple component of the coradical. In the general case, the coradical of the dual coalgebra is the direct sum of all simple finite dimensional representations of . That is, the direct summands of the coalgebra give us a noncommutative variety whose points are the simple representations, and the remainder of the coalgebra of distributions accounts for infinitesimal information on these points (as do the tangent spaces in the commutative case).
In fact, it was a surprise to me that one can describe the dual coalgebra quite explicitly, and that -structures make their appearance quite naturally. See this paper if you’re in for the details on this.
That settles the problem of what we mean by the noncommutative variety associated to a complex algebra. But what about the integral case? In the above, we used extensively the theory of Kostant-duality which works only for algebras over fields…
Well, not quite. In the case of (or more general, of Dedekind domains) one can repeat Kostant’s proof word for word provided one takes as the definition of the dual -coalgebra of an algebra (which is -torsion free)
(over general rings there may be also variants of this duality, as in Street’s book an Quantum groups). Probably lots of people have come up with this, but the only explicit reference I have is to the first paper I’ve ever written. So, also for algebras over we can define a suitable noncommutative integral scheme (the coradical approach accounts only for the maximal ideals rather than all primes, but somehow this is implicit in all approaches as we consider only thin schemes).
Fine! So, we can make sense of the noncommutative geometrical objects corresponding to the group-algebras and where is the modular group (the algebras corresponding to the -functor). But, what might be the points of the noncommutative scheme corresponding to ???
Well, let’s continue the path cut out before. “Points” should correspond to finite dimensional “simple representations”. Hence, what are the finite dimensional simple -representations of ? (Or, for that matter, of any group )
Here we come back to Javier’s post on this : a finite dimensional -vectorspace is a finite set. A -representation on this set (of n-elements) is a group-morphism
hence it gives a permutation representation of on this set. But then, if finite dimensional -representations of are the finite permutation representations, then the simple ones are the transitive permutation representations. That is, the points of the noncommutative scheme corresponding to are the conjugacy classes of subgroups such that is finite. But these are exactly the modular dessins d’enfants introduced by Grothendieck as I explained a while back elsewhere (see for example this post and others in the same series).
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2017-12-11 01:20:21
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https://forum.azimuthproject.org/plugin/ViewComment/20841
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The sliding rule seems to be: you can compose tensors with different dimensions by sticking together terms with same components which is essentially following morphisms down the line. The cups, caps and morphisms can be composed on either one of its ends. Identity morphisms can always disappear and reappear around same subscripts.
Using your equations for \$$g^\ast f^\ast\$$, if we collect terms we get:
$\cap_x\ \cdot 1_x \cdot 1_x \otimes g \cdot (\cap_y \cdot 1_y \cdot \cup_y) \cdot f \otimes 1_z \cdot 1_z \cdot \cup_z$
$\cap_x\ \cdot 1_x \cdot 1_x \otimes g \cdot (1_y \cdot \cap_y) \cdot (\cup_y \cdot 1_y) \cdot f \otimes 1_z \cdot 1_z \cdot \cup_z$
$\cap_x\ \cdot 1_x \cdot 1_x \otimes g \cdot f \otimes 1_z \cdot 1_z \cdot \cup_z$
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2022-12-07 13:15:28
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https://blender.stackexchange.com/questions/581/how-can-i-generate-camera-shake
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# How can I generate camera shake?
I would like to generate camera shake in either a render or in BGE. How can I accomplish this?
• Well, the extremely lame way to do this would be to just move the camera; in an animation you could just take the graph editor, insert some keys, then just move the values ever so slightly around so that it jitters. – Kyle Willey Jun 3 '13 at 5:04
• – Samoth Apr 24 '16 at 21:10
You could insert a single keyframe for the location on the camera (if you dont have any movement) and then, in the the graph editor, add a noise modifier to the curves in the properties panel (N key). Adjust the values of the modifier so you can get the intensity and movement that you like.
• Remember to add one noise modifier to each channel. – wchargin Aug 10 '13 at 16:21
As has been mentioned - fcurves modified with noise can do this.
For the game engine you can use an action with noise too (access action through the action actuator).
Note that while noise modified fcurves on camera are a start, you may want to have a very basic camera rig, I've seen this done before where an empty parent controls the camera, and there are 2 children, one the camera, another empty object with noise applied. The camera constrains to the noise empty using Copy Transforms constraint.
This has the advantage you can do regular (non-noisy) animation to the parent, and the constraint can be animated so you can adjust the noise level over time. - Of course if you wanted to go further you could blend different types of noise easily by adding more empties+constraints and blending those in too (to simulate different terrain).
Since the game engine wont do constraints, something more complex like this would need to use python scripting.
# Recorded hand held shake
• Select | Select Camera and (optionally) View | Camera or numpad 0.
• In the Timeline window:
• Enable Frame | Auto-Keyframing Mode | Add & Replace or simply use the button.
• Hit play (Alt+A) and enter trackball rotation rr or grab g mode.
• Move your pointing device around to generate the shake.
Note: You can get a more controlled (or violent) effect with different frame rate during recording.
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2020-02-18 00:52:23
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https://pos.sissa.it/363/032/
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Volume 363 - 37th International Symposium on Lattice Field Theory (LATTICE2019) - Main session
Two-pion scattering amplitude from Bethe-Salpeter wave function at the interaction boundary
T. Yamazaki,* Y. Namekawa
*corresponding author
Full text: pdf
Pre-published on: January 17, 2020
Published on: August 27, 2020
Abstract
We observe that the ratio of the on-shell scattering amplitude to the Bethe-Salpeter (BS) wave function outside the interaction range is almost independent of time in our quenched calculation of the $I=2$ two-pion scattering with almost zero momentum. In order to discuss the time independence, we present a relation between the two-pion scattering amplitude and the surface term of the BS wave function at the boundary. Using the relation under some assumptions, we show that the ratio is independent of time if the two-pion four-point function in early time is dominated by scattering states with almost zero momentum in addition to the ground state of the two-pion scattering.
DOI: https://doi.org/10.22323/1.363.0032
How to cite
Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.
Open Access
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2020-11-26 06:18:33
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http://www.maplesoft.com/support/help/Maple/view.aspx?path=clipboardcontainsMathML
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"The clipboard contains MathML. Do you want to convert the MathML into 2D math?" - Maple Help
Home : Support : Online Help : System : Error Message Guide : clipboardcontainsMathML
"The clipboard contains MathML. Do you want to convert the MathML into 2D math?"
Description
This message is contained in a pop-up dialog box when attempting to paste 2-D Math into a document from the clipboard. When 2-D Math is copied into the clipboard using the Copy as MathML context menu item, it is copied as MathML language. The user has the option of converting the information back to the 2-D Math format in which it was copied to the clipboard, or pasting it as literal MathML language.
Click Yes to convert the MathML into 2-D Math.
For example, the following MathML will appear as ${x}^{2}.$
Clipboard content After clicking Yes $x 2$ ${\mathbit{x}}^{\mathbf{2}}$
Click No to paste the MathML into your document as literal MathML language.
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2016-02-08 12:46:29
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https://www.gradesaver.com/textbooks/math/algebra/algebra-and-trigonometry-10th-edition/chapter-10-10-3-the-inverse-of-a-square-matrix-10-3-exercises-page-734/15
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## Algebra and Trigonometry 10th Edition
Published by Cengage Learning
# Chapter 10 - 10.3 - The Inverse of a Square Matrix - 10.3 Exercises - Page 734: 15
#### Answer
$\begin{bmatrix} -3 &2 \\ -2 & 1\end{bmatrix}$
#### Work Step by Step
$A=\begin{bmatrix} 1 & -2 \\ 2 & -3\end{bmatrix}$ Therefore, the general form of a matrix of order $2 \times 2$ is: $\begin{bmatrix} p & q \\ r & s\end{bmatrix}=ps-qr$ Now, $ps-qr=\begin{bmatrix} 1 & -2 \\ 2 & -3\end{bmatrix}=-3+4=1 \ne 0$ $A^{-1}=\dfrac{1}{1} \begin{bmatrix} -3 & 2 \\ -2 & -1 \end{bmatrix}$ Our answer is: $\begin{bmatrix} -3 &2 \\ -2 & 1\end{bmatrix}$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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2022-09-30 13:21:54
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https://proofwiki.org/wiki/Derivative_of_Cosine_Function/Proof_2
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# Derivative of Cosine Function/Proof 2
Jump to navigation Jump to search
## Theorem
$D_x \left({\cos x}\right) = -\sin x$
## Proof
$\displaystyle D_x \left({\cos x}\right)$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({x + h}\right) - \cos \left({x}\right)} h$ Definition of Derivative of Real Function at Point $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({x}\right) \cos \left({h}\right) - \sin \left({x}\right) \sin \left({h}\right) - \cos \left({x}\right)} h$ Cosine of Sum $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({x}\right) \cos \left({h}\right) - \cos \left({x}\right)} h + \lim_{h \mathop \to 0} \frac{- \sin \left({x}\right) \sin \left({h}\right)} h$ Sum Rule for Limits of Functions $\displaystyle$ $=$ $\displaystyle \cos \left({x}\right) \ \lim_{h \mathop \to 0} \frac {\cos \left({h}\right) - 1} h - \sin \left({x}\right) \ \lim_{h \mathop \to 0} \frac {\sin \left({h}\right)} h$ Multiple Rule for Limits of Functions $\displaystyle$ $=$ $\displaystyle \cos \left({x}\right) \times 0 - \sin \left({x}\right) \times 1$ Limit of (Cosine (X) - 1) over X and Limit of Sine of X over X $\displaystyle$ $=$ $\displaystyle - \sin \left({x}\right)$
$\blacksquare$
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2020-01-20 11:58:57
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http://mathoverflow.net/questions/109205/existence-of-weakly-compact-cardinals
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Existence of weakly compact cardinals
I looked on the web to search for weakly compact cardinals. The web sources indicate many properties of weakly compact cardinals and say that their existence is not entailed by the axioms of ZFC. However it is not indicated whether their existence is {\it consistent} with the axioms of ZFC.
So my question is: Is it known that the axiom of existence of weakly compact cardinals consistent with ZFC?
Thanks
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Maybe you meant to ask whether anybody has proved that existence of weakly compact cardinals is independent from ZFC, that is, also the negation of "there is a WCC" is not entailed by the axioms of ZFC? – Qfwfq Oct 9 '12 at 8:54
Yes that is it. – user16974 Oct 9 '12 at 8:56
As weakly compact cardinals are in particular (strongly) inaccessible, it follows from Gödel's Second Incompleteness Theorem that ZFC cannot prove the implication "Con(ZFC) implies Con (ZFC + $\exists$weakly-compact )." (Unless, of course, if ZFC is itself inconsistent, at which point this is all a lot of bunk).
(The linked Wikipedia article outlines the basic reasoning.)
Any argument for the consistency of "ZFC + $\exists$weakly-inaccessible" must therefore transcend ZFC. (But, as mentioned by Asaf, no contradictions have been found thus far been under the assumption of the existence of weakly-compact cardinals.)
-
The weakly compact cardinals are fairly low in the large cardinal hierarchy, just a few skips beyond the inaccessible and Mahlo cardinals, and so one can prove the existence of weakly compact cardinals from any of the stronger large cardinal hypotheses (see Cantor's Attic). For example, the weakly compact cardinals have a strength well below the indescribable cardinals, the unfoldable cardinals, the ethereal cardinals, the subtle cardinals, the ineffable cardinals, and these are all significantly below the Ramsey cardinals and the measurable cardinals, traditionally considered a gateway to the upper class of large cardinals above.
All of these stronger large cardinal notions imply the outright existence of weakly compact cardinals, as well as the consistency of the existence of many weakly compact cardinals. Indeed, this phenomenon is a dominant feature of the large cardinal hierarchy, where the existence of a higher cardinal generally implies the existence of many instances of the lower cardinals. For example, every measurable cardinal $\kappa$ is the $\kappa^{th}$ weakly compact cardinal; every weakly compact cardinal $\gamma$ is the $\gamma^{th}$ Mahlo cardinal; every Mahlo cardinal $\delta$ is the $\delta^{th}$ inaccessible cardinal. In particular, the existence of a higher large cardinal implies the consistency of ZFC with the existence of many of the lower cardinals. If there is a weakly compact cardinal $\gamma$, for example, then the universe $V_\gamma$ up to $\gamma$ satisfies ZFC plus the assertion that there is a proper class of Mahlo cardinals.
It follows (as in Arthur's answer) that we cannot prove the existence of a large cardinal in ZFC or even in ZFC plus a lower large cardinal notion (unless that theory is inconsistent), since this would violate the incompleteness theorem for this lower theory. So even ZFC plus a proper class of hyper-Mahlo cardinals, if consistent, does not suffice to prove the existence of a weakly compact cardinal, precisely because if $\kappa$ is weakly compact, then $V_\kappa$ is a model of ZFC plus a proper class of hyper Mahlo cardinals.
Some might object that this would seem to make the study of large cardinals a doubtful activity, for not only have we failed to prove that the large cardinals exist, we haven't even proved that their existence is consistent, and indeed, we have even proved that we cannot consistently prove that their existence is consistent! Shouldn't this put us off the subject of large cardinals?
No. The point is that because of the incompleteness theorem, we know that there is a hierarchy of consistency strength, a tower of theories each of which implies the consistency of weaker theories in the tower. We wanted to find such a tower of theories, with the property that the consistency of the weaker theories does not prove the consistency of the stronger theories. How fortunate that the large cardinal hierarchy exhibits exactly the features we sought! Furthermore, the large cardinal hierarchy exhibits this tower of consistency strength not by means of weird self-referential convoluted logic statements, as in the incompleteness theorem, but rather with highly natural statements involving infinite combinatorics, such as the existence of measures and considerations of graph colorings. These were questions in which we were already independently interested. Subsequent study of the large cardinal hierarchy has revealed it to be a unifying explanatory force in the nature of set-theoretic truth. (But still, we must be alert to the possibility of inconsistency.)
So it is not a flaw but rather a feature that we cannot prove the consistency of the existence of any of these large cardinals, except from even stronger ones.
Meanwhile, let me point out in answer to Qfwfq's comment on the question, that the consistency of ZFC easily proves the consistency of ZFC + there is no weakly compact cardinal. To see this, let $M$ be any model of ZFC. If it has no weakly compact cardinal, then we're done. If it does, let $\kappa$ be the least weakly compact cardinal of $M$, and observe that the cut-off universe $V_\kappa^M$ satisfies ZFC and has no weakly compact cardinals. Thus, it is relatively consistent with ZFC that there are no weakly compact cardinals.
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Somewhere in the fifth paragraph, cue deeply moving patriotic music in the background... Very nice answer! – Asaf Karagila Oct 9 '12 at 15:23
Consider the axiom of infinity in the axioms of ZF. You cannot prove it holds from the other axioms of ZF; and assuming it means that you can prove the consistency of ZF-Infinity by taking the hereditarily finite sets as a model. Note that ZF-Infinity can formulate enough about the natural numbers for this to be a non-trivial issue.
Large cardinals are often called strong infinity axioms. They are like the axiom of infinity, without such axioms you have a theory of a certain strength, but adding axioms like "There is a weakly-compact cardinal" or "There is a class of Woodin cardinals" gives us a stronger theory, from which we can prove the consistency of weaker theory (e.g. ZFC).
Equally, I could ask you if you can prove that ZFC itself is consistent. The answer depends on your meta-theory. If you assume ZFC+"There is a weakly compact cardinal" then the answer is yes. If you work within the confines of ZFC then the answer is no.
So if we assume the existence of two weakly compact cardinals, we can prove the consistency of ZFC+"There is a weakly compact cardinal", but if we only assume that there is one weakly compact cardinal, it is impossible to prove that for the this theory is consistent -- just like we cannot prove the consistency of ZFC from itself.
If it's any consolation, I am not aware of any contradiction found with weakly compact cardinals yet.
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Thanks. I should have asked: If we assume ZFC consistent, then does it follow that ZFC+"there exists a weakly compact cardinal" is also consistent? – user16974 Oct 9 '12 at 8:22
Hollowdead, no. The existence of any large cardinal is much stronger than assuming the consistency of ZFC. – Asaf Karagila Oct 9 '12 at 8:31
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2014-04-18 23:38:03
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http://mathematica.stackexchange.com/questions/22181/how-to-remove-unwanted-regions-in-a-three-dimensional-surface
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# How to remove unwanted regions in a three-dimensional surface
As the title indicates, I want to delete some unwanted regions in a three-dimensional surface created using ContourPlot3D. Here is the corresponding code
Clear["Global*"];
V = 1/2*(x^2 + y^2 + z^2) + (x^2*y^2 + x^2*z^2 + y^2*z^2 - x^2*y^2*z^2);
E0 = 7;
S0 = ContourPlot3D[V == E0, {x, -4, 4}, {y, -4, 4}, {z, -4, 4},
PlotPoints -> 70, PerformanceGoal -> "Speed", Mesh -> None,
ContourStyle -> Directive[Green, Opacity[0.3], Specularity[White, 30]],
ImageSize -> 550]
which produces this output:
I want somehow to keep the inside closed 3D surface (which looks like a 3D star) and delete the eight open parts which surround it. I tried several combinations using RegionPlot3D but the particular range of $x$, $y$ and $z$ prevent me from obtaining what I want. Any suggestions?
EDIT
If you increase the value of E0 then after a certain value the three-dimensional surface opens and eight channels of escapes (holes) appear. Using this code
Clear["Global*"];
V = 1/2*(x^2 + y^2 + z^2) + (x^2*y^2 + x^2*z^2 + y^2*z^2 - x^2*y^2*z^2);
E0 = 8.5;
R0 = 6;
S0 = ContourPlot3D[V == E0, {x, -5, 5}, {y, -5, 5}, {z, -5, 5},
PlotPoints -> 100, PerformanceGoal -> "Speed", Mesh -> None,
ContourStyle -> Directive[Green, Opacity[0.3], Specularity[White, 30]],
ImageSize -> 550, RegionFunction -> Function[{x, y, z}, Sqrt@(x^2*y^2 +
x^2*z^2 + y^2*z^2) <= R0]]
you get this output
OK, my question is the following: How can I draw the surface and manipulate the openings at the minimum width. Let me be more specific. Now, using the current code the size of the openings is controlled by the cutoff surface and the particular value of $R0$. If for example, I use $R0 = 2.5$ then I loose essential parts of the surface.
What I want is to define such a cutoff surface or find the specific value of $R0$ so that the external cutoff surface to abut exactly against the inner surface thus, drawing the openings at their minimum width.
Any suggestions?
-
Βαγγέλη, you can use an appropriate RegionFunction:
V = 1/2*(x^2 + y^2 + z^2) + (x^2*y^2 + x^2*z^2 + y^2*z^2 -
x^2*y^2*z^2);
E0 = 7;
S0 = ContourPlot3D[V == E0, {x, -4, 4}, {y, -4, 4}, {z, -4, 4},
PlotPoints -> 70, PerformanceGoal -> "Speed", Mesh -> None,
ContourStyle ->
Directive[Green, Opacity[0.3], Specularity[White, 30]],
ImageSize -> 550,
RegionFunction ->
Function[{x, y, z}, 3 - Sqrt[x^2*y^2 + y^2 z^2] > 0]]
---EDIT---
RegionFunction works like a cutoff. You need to find the equation of a curve that splits your wanted from your unwanted parts. Here
cutoff =
ContourPlot3D[
Sqrt[x^2*y^2 + y^2 z^2] == 3, {x, -4, 4}, {y, -4, 4}, {z, -4, 4}];
Show[S0, cutoff]
But how you'd guess it is a different issue. Normally manipulate your original surface. I.e. note that
cutoff2 =
ContourPlot3D[
Sqrt@(x^2*y^2 + x^2*z^2 + y^2*z^2) == 4, {x, -4, 4}, {y, -4,
4}, {z, -4, 4}, ContourStyle -> Opacity[0.4], Mesh -> None];
Show[S0, cutoff2]
also works (in fact is a more obvious choice).
---2nd EDIT---
If you want to qualitatively see whether the containment of the surface is satisfactory, leave your R0 variable and wrap a low-res version within a manipulate to see which value you like.
Manipulate[
TableForm@{ContourPlot3D[
V == E0, {x, -5, 5}, {y, -5, 5}, {z, -5, 5},
PerformanceGoal -> "Speed",
Mesh -> None,
PlotPoints -> 40,
ContourStyle ->
Directive[Green, Opacity[0.3], Specularity[White, 30]],
ImageSize -> 550,
RegionFunction ->
Function[{x, y, z}, Sqrt@(x^2*y^2 + x^2*z^2 + y^2*z^2) <= R0]],
R0}, {R0, 2, 6, 0.1}]
If you want to find the actual minimum, you'd have to find an expression for the intersection of the two surfaces, parametrise it, and minimize its length.
-
Great! Could you explain a little bit how this particular RegionFunction come from? I mean which theory is behind it. – Vaggelis_Z Mar 27 '13 at 8:45
When $V$ has another mathematical expression then the appropriate RegionFunction changes. However, I cannot understand how the function $3 - \sqrt{x^2 y^2 + y^2 z^2} > 0$ come from. It is essential to me to know how to obtain this function every time. So, please enlighten me a little bit! – Vaggelis_Z Mar 27 '13 at 9:05
@Vaggelis_Z If you have a surface like yours that is a generalised "sphere" in that you can permute any two cartesian coordinates and get the same expression, you need to define your cutoff as something with larger radius or slightly deformed. Does that help? – gpap Mar 27 '13 at 9:34
Yes indeed, now everything is very clear. Thank you very much! – Vaggelis_Z Mar 27 '13 at 9:38
Any ideas about my EDIT? Many thanks in advance! – Vaggelis_Z Mar 28 '13 at 10:34
By looking at the hyperbolic behavior of the branches, you may try a simple RegionFunction[]:
ContourPlot3D[V == E0, {x, -4, 4}, {y, -4, 4}, {z, -4, 4},
PlotPoints -> 70, PerformanceGoal -> "Speed", Mesh -> None,
ContourStyle -> Directive[Green, Opacity[0.3], Specularity[White, 30]], ImageSize -> 550,
RegionFunction -> (Abs[#1 #2 #3] < 3 &)]
Edit
Just checking the whole solid is included
Show[ContourPlot3D[V == E0, {x, -4, 4}, {y, -4, 4}, {z, -4, 4},
PlotPoints -> 70, PerformanceGoal -> "Speed", Mesh -> None,
ContourStyle -> Directive[Green, Opacity[0.3], Specularity[White, 30]],
ImageSize -> 550, RegionFunction -> (Abs[Times[#1 #2 #3]] < 3 &)],
ParametricPlot3D[{x, y, Abs[3/(x y)]}, {x, 0, 4}, {y, -4, 4},
PlotPoints -> 70, PerformanceGoal -> "Speed", Mesh -> None,
PlotRange -> {-5, 5}]]
-
For giggles, here's a variation of belisarius's answer:
ContourPlot3D[V == E0, {x, -4, 4}, {y, -4, 4}, {z, -4, 4},
ContourStyle -> Directive[Opacity[0.3, Green], Specularity[1, 30]],
Mesh -> None, PerformanceGoal -> "Speed", PlotPoints -> 70,
RegionFunction -> Function[{x, y, z}, Evaluate[
PolyhedronData["Octahedron", "RegionFunction"][Sqrt[2] x/9, Sqrt[2] y/9, Sqrt[2] z/9]]]]
-
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2015-08-31 15:24:58
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https://socratic.org/questions/how-many-moles-of-solute-particles-are-present-in-1-ml-exact-of-aqueous-0-0060-m
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# How many moles of solute particles are present in 1 mL (exact) of aqueous 0.0060 M Ba(OH)_2?
$3 \times 1 \times {10}^{-} 3 \cancel{L} \times 6.0 \times {10}^{-} 3 \cdot m o l \cdot \cancel{{L}^{-} 1}$ $=$ $18 \times {10}^{-} 6 \cdot m o l$
Barium hydroxide is an electrolyte that gives $B {a}^{2 +}$ and $2 \text{ equiv } H {O}^{-}$ ions upon dissolution.
$\text{Volume "(L)xx"concentration } \left(m o l \cdot {L}^{-} 1\right)$ $=$ $\text{moles, amount of subtance.}$
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2022-10-05 15:57:44
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|
http://www.maa.org/publications/periodicals/loci/joma/exploring-the-goodness-of-fit-in-linear-models-the-excel-randbetween-function
|
# Exploring the Goodness of Fit in Linear Models - The Excel RANDBETWEEN Function
Author(s):
Scott A. Sinex
I provide the basics of developing a computational interactive Excel spreadsheet, including an Excel tutorial, in Sinex (2004). In this example the new twist is the use of the RANDBETWEEN function (available in the function editor) to add noise or scatter to the data. (See the Microsoft Excel Help, and enter the keyword “randbetween” for instructions to load this function, which is part of the Analysis ToolPak.)
This function returns a random number between two set limits placed in parentheses, such as RANDBETWEEN (-10, 10). The RANDBETWEEN function is a perfect way to add noise, as random variation, to the data, and the noise is recalculated each time the function is applied. The effect or size of the error can be controlled by an adjustable variable ($H$2 in Figure 4), controlled by the scroll bar and multiplied by the RANDBETWEEN function. Figure 4 shows a sample formula.
##### Figure 4. Sample Excel formula using RANDBETWEEN to add noise to a linear function
Scott A. Sinex, "Exploring the Goodness of Fit in Linear Models - The Excel RANDBETWEEN Function," Loci (March 2005)
## JOMA
Journal of Online Mathematics and its Applications
|
2014-09-02 13:03:33
|
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|
http://julia.readthedocs.io/en/release-0.5/stdlib/numbers/
|
# Numbers¶
## Standard Numeric Types¶
Bool Int8 UInt8 Int16 UInt16 Int32 UInt32 Int64 UInt64 Int128 UInt128 Float16 Float32 Float64 Complex64 Complex128
## Data Formats¶
bin(n[, pad])
Convert an integer to a binary string, optionally specifying a number of digits to pad to.
hex(n[, pad])
Convert an integer to a hexadecimal string, optionally specifying a number of digits to pad to.
dec(n[, pad])
Convert an integer to a decimal string, optionally specifying a number of digits to pad to.
oct(n[, pad])
Convert an integer to an octal string, optionally specifying a number of digits to pad to.
base(base, n[, pad])
Convert an integer to a string in the given base, optionally specifying a number of digits to pad to.
digits([T, ]n[, base][, pad])
Returns an array with element type T (default Int) of the digits of n in the given base, optionally padded with zeros to a specified size. More significant digits are at higher indexes, such that n == sum([digits[k]*base^(k-1) for k=1:length(digits)]).
digits!(array, n[, base])
Fills an array of the digits of n in the given base. More significant digits are at higher indexes. If the array length is insufficient, the least significant digits are filled up to the array length. If the array length is excessive, the excess portion is filled with zeros.
bits(n)
A string giving the literal bit representation of a number.
parse(type, str[, base])
Parse a string as a number. If the type is an integer type, then a base can be specified (the default is 10). If the type is a floating point type, the string is parsed as a decimal floating point number. If the string does not contain a valid number, an error is raised.
tryparse(type, str[, base])
Like parse, but returns a Nullable of the requested type. The result will be null if the string does not contain a valid number.
big(x)
Convert a number to a maximum precision representation (typically BigInt or BigFloat). See BigFloat for information about some pitfalls with floating-point numbers.
signed(x)
Convert a number to a signed integer. If the argument is unsigned, it is reinterpreted as signed without checking for overflow.
unsigned(x) → Unsigned
Convert a number to an unsigned integer. If the argument is signed, it is reinterpreted as unsigned without checking for negative values.
float(x)
Convert a number, array, or string to a AbstractFloat data type. For numeric data, the smallest suitable AbstractFloat type is used. Converts strings to Float64.
significand(x)
Extract the significand(s) (a.k.a. mantissa), in binary representation, of a floating-point number or array. If x is a non-zero finite number, then the result will be a number of the same type on the interval $$[1,2)$$. Otherwise x is returned.
julia> significand(15.2)/15.2
0.125
julia> significand(15.2)*8
15.2
exponent(x) → Int
Get the exponent of a normalized floating-point number.
complex(r[, i])
Convert real numbers or arrays to complex. i defaults to zero.
bswap(n)
Byte-swap an integer.
num2hex(f)
Get a hexadecimal string of the binary representation of a floating point number.
hex2num(str)
Convert a hexadecimal string to the floating point number it represents.
hex2bytes(s::AbstractString)
Convert an arbitrarily long hexadecimal string to its binary representation. Returns an Array{UInt8,1}, i.e. an array of bytes.
bytes2hex(bin_arr::Array{UInt8, 1})
Convert an array of bytes to its hexadecimal representation. All characters are in lower-case. Returns a String.
## General Number Functions and Constants¶
one(x)
Get the multiplicative identity element for the type of x (x can also specify the type itself). For matrices, returns an identity matrix of the appropriate size and type.
zero(x)
Get the additive identity element for the type of x (x can also specify the type itself).
pi
π
The constant pi.
im
The imaginary unit.
e
eu
The constant e.
catalan
Catalan’s constant.
γ
eulergamma
Euler’s constant.
φ
golden
The golden ratio.
Inf
Positive infinity of type Float64.
Inf32
Positive infinity of type Float32.
Inf16
Positive infinity of type Float16.
NaN
A not-a-number value of type Float64.
NaN32
A not-a-number value of type Float32.
NaN16
A not-a-number value of type Float16.
issubnormal(f) → Bool
Test whether a floating point number is subnormal.
isfinite(f) → Bool
Test whether a number is finite
isinf(f) → Bool
Test whether a number is infinite.
isnan(f) → Bool
Test whether a floating point number is not a number (NaN).
nextfloat(x::AbstractFloat)
Returns the smallest floating point number y of the same type as x such x < y. If no such y exists (e.g. if x is Inf or NaN), then returns x.
prevfloat(x::AbstractFloat)
Returns the largest floating point number y of the same type as x such y < x. If no such y exists (e.g. if x is -Inf or NaN), then returns x.
nextfloat(x::AbstractFloat, n::Integer)
The result of n iterative applications of nextfloat to x if n >= 0, or -n applications of prevfloat if n < 0.
isinteger(x) → Bool
Test whether x or all its elements are numerically equal to some integer
isreal(x) → Bool
Test whether x or all its elements are numerically equal to some real number.
isimag(z) → Bool
Test whether z is purely imaginary, i.e. has a real part equal to 0.
Float32(x[, mode::RoundingMode])
Create a Float32 from x. If x is not exactly representable then mode determines how x is rounded.
julia> Float32(1/3, RoundDown)
0.3333333f0
julia> Float32(1/3, RoundUp)
0.33333334f0
See RoundingMode for available rounding modes.
Float64(x[, mode::RoundingMode])
Create a Float64 from x. If x is not exactly representable then mode determines how x is rounded.
julia> Float64(pi, RoundDown)
3.141592653589793
julia> Float64(pi, RoundUp)
3.1415926535897936
See RoundingMode for available rounding modes.
BigInt(x)
Create an arbitrary precision integer. x may be an Int (or anything that can be converted to an Int). The usual mathematical operators are defined for this type, and results are promoted to a BigInt.
Instances can be constructed from strings via parse(), or using the big string literal.
BigFloat(x)
Create an arbitrary precision floating point number. x may be an Integer, a Float64 or a BigInt. The usual mathematical operators are defined for this type, and results are promoted to a BigFloat.
Note that because decimal literals are converted to floating point numbers when parsed, BigFloat(2.1) may not yield what you expect. You may instead prefer to initialize constants from strings via parse(), or using the big string literal.
julia> BigFloat(2.1)
2.100000000000000088817841970012523233890533447265625000000000000000000000000000
julia> big"2.1"
2.099999999999999999999999999999999999999999999999999999999999999999999999999986
rounding(T)
Get the current floating point rounding mode for type T, controlling the rounding of basic arithmetic functions (+(), -(), *(), /() and sqrt()) and type conversion.
See RoundingMode for available modes.
setrounding(T, mode)
Set the rounding mode of floating point type T, controlling the rounding of basic arithmetic functions (+(), -(), *(), /() and sqrt()) and type conversion. Other numerical functions may give incorrect or invalid values when using rounding modes other than the default RoundNearest.
Note that this may affect other types, for instance changing the rounding mode of Float64 will change the rounding mode of Float32. See RoundingMode for available modes.
Warning
This feature is still experimental, and may give unexpected or incorrect values.
setrounding(f::Function, T, mode)
Change the rounding mode of floating point type T for the duration of f. It is logically equivalent to:
old = rounding(T)
setrounding(T, mode)
f()
setrounding(T, old)
See RoundingMode for available rounding modes.
Warning
This feature is still experimental, and may give unexpected or incorrect values. A known problem is the interaction with compiler optimisations, e.g.
julia> setrounding(Float64,RoundDown) do
1.1 + 0.1
end
1.2000000000000002
Here the compiler is constant folding, that is evaluating a known constant expression at compile time, however the rounding mode is only changed at runtime, so this is not reflected in the function result. This can be avoided by moving constants outside the expression, e.g.
julia> x = 1.1; y = 0.1;
julia> setrounding(Float64,RoundDown) do
x + y
end
1.2
get_zero_subnormals() → Bool
Returns false if operations on subnormal floating-point values (“denormals”) obey rules for IEEE arithmetic, and true if they might be converted to zeros.
set_zero_subnormals(yes::Bool) → Bool
If yes is false, subsequent floating-point operations follow rules for IEEE arithmetic on subnormal values (“denormals”). Otherwise, floating-point operations are permitted (but not required) to convert subnormal inputs or outputs to zero. Returns true unless yes==true but the hardware does not support zeroing of subnormal numbers.
set_zero_subnormals(true) can speed up some computations on some hardware. However, it can break identities such as (x-y==0) == (x==y).
### Integers¶
count_ones(x::Integer) → Integer
Number of ones in the binary representation of x.
julia> count_ones(7)
3
count_zeros(x::Integer) → Integer
Number of zeros in the binary representation of x.
julia> count_zeros(Int32(2 ^ 16 - 1))
16
leading_zeros(x::Integer) → Integer
Number of zeros leading the binary representation of x.
julia> leading_zeros(Int32(1))
31
leading_ones(x::Integer) → Integer
Number of ones leading the binary representation of x.
julia> leading_ones(UInt32(2 ^ 32 - 2))
31
trailing_zeros(x::Integer) → Integer
Number of zeros trailing the binary representation of x.
julia> trailing_zeros(2)
1
trailing_ones(x::Integer) → Integer
Number of ones trailing the binary representation of x.
julia> trailing_ones(3)
2
isodd(x::Integer) → Bool
Returns true if x is odd (that is, not divisible by 2), and false otherwise.
julia> isodd(9)
true
julia> isodd(10)
false
iseven(x::Integer) → Bool
Returns true is x is even (that is, divisible by 2), and false otherwise.
julia> iseven(9)
false
julia> iseven(10)
true
## BigFloats¶
The BigFloat type implements arbitrary-precision floating-point arithmetic using the GNU MPFR library.
precision(num::AbstractFloat)
Get the precision of a floating point number, as defined by the effective number of bits in the mantissa.
precision(BigFloat)
Get the precision (in bits) currently used for BigFloat arithmetic.
setprecision([T=BigFloat, ]precision::Int)
Set the precision (in bits) to be used for T arithmetic.
setprecision(f::Function, [T=BigFloat, ]precision::Integer)
Change the T arithmetic precision (in bits) for the duration of f. It is logically equivalent to:
old = precision(BigFloat)
setprecision(BigFloat, precision)
f()
setprecision(BigFloat, old)
Often used as setprecision(T, precision) do ... end
## Random Numbers¶
Random number generation in Julia uses the Mersenne Twister library via MersenneTwister objects. Julia has a global RNG, which is used by default. Other RNG types can be plugged in by inheriting the AbstractRNG type; they can then be used to have multiple streams of random numbers. Besides MersenneTwister, Julia also provides the RandomDevice RNG type, which is a wrapper over the OS provided entropy.
Most functions related to random generation accept an optional AbstractRNG as the first argument, rng , which defaults to the global one if not provided. Morever, some of them accept optionally dimension specifications dims... (which can be given as a tuple) to generate arrays of random values.
A MersenneTwister or RandomDevice RNG can generate random numbers of the following types: Float16, Float32, Float64, Bool, Int8, UInt8, Int16, UInt16, Int32, UInt32, Int64, UInt64, Int128, UInt128, BigInt (or complex numbers of those types). Random floating point numbers are generated uniformly in $$[0, 1)$$. As BigInt represents unbounded integers, the interval must be specified (e.g. rand(big(1:6))).
srand([rng][, seed])
Reseed the random number generator. If a seed is provided, the RNG will give a reproducible sequence of numbers, otherwise Julia will get entropy from the system. For MersenneTwister, the seed may be a non-negative integer, a vector of UInt32 integers or a filename, in which case the seed is read from a file. RandomDevice does not support seeding.
MersenneTwister([seed])
Create a MersenneTwister RNG object. Different RNG objects can have their own seeds, which may be useful for generating different streams of random numbers.
RandomDevice()
Create a RandomDevice RNG object. Two such objects will always generate different streams of random numbers.
rand([rng][, S][, dims...])
Pick a random element or array of random elements from the set of values specified by S; S can be
• an indexable collection (for example 1:n or ['x','y','z']), or
• a type: the set of values to pick from is then equivalent to typemin(S):typemax(S) for integers (this is not applicable to BigInt), and to $$[0, 1)$$ for floating point numbers;
S defaults to Float64.
rand!([rng, ]A[, coll])
Populate the array A with random values. If the indexable collection coll is specified, the values are picked randomly from coll. This is equivalent to copy!(A, rand(rng, coll, size(A))) or copy!(A, rand(rng, eltype(A), size(A))) but without allocating a new array.
bitrand([rng][, dims...])
Generate a BitArray of random boolean values.
randn([rng][, T=Float64][, dims...])
Generate a normally-distributed random number of type T with mean 0 and standard deviation 1. Optionally generate an array of normally-distributed random numbers. The Base module currently provides an implementation for the types Float16, Float32, and Float64 (the default).
randn!([rng, ]A::AbstractArray) → A
Fill the array A with normally-distributed (mean 0, standard deviation 1) random numbers. Also see the rand function.
randexp([rng][, T=Float64][, dims...])
Generate a random number of type T according to the exponential distribution with scale 1. Optionally generate an array of such random numbers. The Base module currently provides an implementation for the types Float16, Float32, and Float64 (the default).
randexp!([rng, ]A::AbstractArray) → A
Fill the array A with random numbers following the exponential distribution (with scale 1).
randjump(r::MersenneTwister, jumps[, jumppoly]) → Vector{MersenneTwister}
Create an array of the size jumps of initialized MersenneTwister RNG objects where the first RNG object given as a parameter and following MersenneTwister RNGs in the array initialized such that a state of the RNG object in the array would be moved forward (without generating numbers) from a previous RNG object array element on a particular number of steps encoded by the jump polynomial jumppoly.
Default jump polynomial moves forward MersenneTwister RNG state by 10^20 steps.
|
2017-02-22 13:00:53
|
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|
https://www.statisticsviews.com/details/journalArticle/11128827/An-optimal-randomized-approximation-scheme-for-the-mean-of-random-variables-with.html
|
Random Structures & Algorithms
An optimal (ϵ,δ)‐randomized approximation scheme for the mean of random variables with bounded relative variance
Early View
Randomized approximation algorithms for many #P‐complete problems (such as the partition function of a Gibbs distribution, the volume of a convex body, the permanent of a {0,1}‐matrix, and many others) reduce to creating random variables X1,X2,… with finite mean μ and standard deviation σ such that μ is the solution for the problem input, and the relative standard deviation |σ/μ| ≤ c for known c. Under these circumstances, it is known that the number of samples from the {Xi} needed to form an (ϵ,δ)‐approximation that satisfies is at least . We present here an easy to implement (ϵ,δ)‐approximation that uses samples. This achieves the same optimal running time as other estimators, but without the need for extra conditions such as bounds on third or fourth moments.
View all
View all
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2019-06-19 11:23:40
|
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https://www.groundai.com/project/the-minimal-size-of-a-graph-with-given-generalized-3-edge-connectivity/
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The minimal size of a graph with given generalized 3-edge-connectivity1footnote 11footnote 1Supported by NSFC No.11071130
# The minimal size of a graph with given generalized 3-edge-connectivity111Supported by NSFC No.11071130
Xueliang Li, Yaping Mao
Center for Combinatorics and LPMC-TJKLC
Nankai University, Tianjin 300071, China
lxl@nankai.edu.cn; maoyaping@ymail.com.
###### Abstract
For and , is the maximum number of edge-disjoint trees connecting in . For an integer with , the generalized -edge-connectivity of is then defined as . It is also clear that when , is nothing new but the standard edge-connectivity of . In this paper, graphs of order such that is characterized. Furthermore, we determine the minimal number of edges of a graph of order with and give a sharp lower bound for .
Keywords
: edge-connectivity, Steiner tree, edge-disjoint trees, generalized edge-connectivity.
AMS subject classification 2010: 05C40, 05C05, 05C75.
## 1 Introduction
All graphs considered in this paper are undirected, finite and simple. We refer to the book [1] for graph theoretical notation and terminology not described here. For a graph , let and denote the set of vertices and the set of edges of , respectively. As usual, the union of two graphs and is the graph, denoted by , with vertex set and edge set . Let be the disjoint union of copies of a graph . We denote by the set of edges of with one end in and the other end in . If , we simply write for .
The generalized connectivity of a graph , introduced by Chartrand et al. in [2], is a natural and nice generalization of the concept of (vertex-)connectivity. For a graph and a set of at least two vertices, an -Steiner tree or a Steiner tree connecting (or simply, an -tree) is a such subgraph of that is a tree with . Two Steiner trees and connecting are said to be internally disjoint if and . For and , the generalized local connectivity is the maximum number of internally disjoint trees connecting in . Note that when a Steiner tree connecting is just a path connecting the two vertices of . For an integer with , the generalized -connectivity of is defined as . Clearly, when , is nothing new but the connectivity of , that is, , which is the reason why one addresses as the generalized connectivity of . By convention, for a connected graph with less than vertices, we set . Set when is disconnected. Results on the generalized connectivity can be found in [3, 4, 6, 7, 8, 10, 13].
As a natural counterpart of the generalized connectivity, we introduced the concept of generalized edge-connectivity in [11]. For and , the generalized local edge-connectivity is the maximum number of edge-disjoint trees connecting in . For an integer with , the generalized -edge-connectivity of is then defined as . It is also clear that when , is nothing new but the standard edge-connectivity of , that is, , which is the reason why we address as the generalized edge-connectivity of . Also set when is disconnected.
In addition to being natural combinatorial measures, the generalized connectivity and generalized edge-connectivity can be motivated by their interesting interpretation in practice. For example, suppose that represents a network. If one considers to connect a pair of vertices of , then a path is used to connect them. However, if one wants to connect a set of vertices of with , then a tree has to be used to connect them. This kind of tree with minimum order for connecting a set of vertices is usually called a Steiner tree, and popularly used in the physical design of VLSI, see [14]. Usually, one wants to consider how tough a network can be, for connecting a set of vertices. Then, the number of totally independent ways to connect them is a measure for this purpose. The generalized -connectivity and generalized -edge-connectivity can serve for measuring the capability of a network to connect any vertices in .
The following two observations are easily seen.
###### Observation 1.
If is a connected graph, then .
###### Observation 2.
If is a spanning subgraph of , then and .
In [11], we obtained some results on the generalized edge-connectivity. The following results are restated, which will be used later.
###### Lemma 1.
[11] For every two integers and with ,
###### Lemma 2.
[11] For any connected graph , . Moreover, the upper bound is sharp.
###### Lemma 3.
[11] Let be two integers with . For a connected graph of order , . Moreover, the upper and lower bounds are sharp.
In [11], we characterized graphs with large generalized -connectivity and obtained the following result.
###### Lemma 4.
[11] Let be two integers with . For a connected graph of order , if and only if for even; for odd, where is an edge set such that .
Like [5], here we will consider the generalized -edge-connectivity. From Lemma 3, . In Section , graphs of order such that is characterized.
Let be the minimal number of edges of a graph of order with . From Lemma 4, we know that for even; for odd. It is not easy to determine exact value of the parameter . So we put our attention to on the case . The exact value of for are obtained in Section . We also give a sharp lower bounds of for general .
## 2 Graphs with λ3(G)=n−3
After the preparation of the above section, we start to give our main result. From Lemma 3, we know that for a connected graph of order . Graphs with has been shown in Lemma 4. But, it is not easy to characterize graphs with for general . So we focus on the case that and characterizing the graphs with in this section.
For the generalized -connectivity, we got the following result in [5].
###### Theorem 1.
[5] Let be a connected graph of order . if and only if or or or .
But, for the edge case we will show that the statement is different. Before giving our main result, we need some preparations.
Choose . Then let be a maximum set of edge-disjoint trees connecting in . Let be the set of trees in whose edges belong to , and let be the set of trees containing at least one edge of . Thus .
In [11], we obtained the following useful lemma.
###### Lemma 5.
[11] Let , and be a tree connecting . If , then uses edges of ; If , then uses at least edges of .
By Lemma 6, we can derived the following result.
###### Lemma 6.
Let be a connected graph of order , and be a positive integer. If we can find a set with satisfying one of the following conditions, then .
and ;
and ;
and ;
and .
###### Proof.
We only show that and hold, and can be proved similarly.
Since , we have . Since , we have . Therefore, , and so there exists at most one tree belonging to in . If there exists one tree belonging to , namely , then the other trees connecting must belong to . From Lemma 6, each tree belonging to uses at least edges in . So the remaining at most edges of can form at most trees. Thus , which results in since is an integer. Suppose that all trees connecting belong to . Then , which implies that .
Since , we have . Since , we have . Since , there exists no tree belonging to . So each tree connecting must belong to . From Lemma 6, , which implies that since is an integer. ∎
###### Lemma 7.
Let be a connected graph with minimum degree . If there are two adjacent vertices of degree , then .
###### Proof.
It is clear that and by Lemma 2. So .
Suppose that there are two adjacent vertices and of degree and . Besides and , we choose a vertex in to get a -set containing . Suppose are pairwise edge-disjoint trees connecting . Since is simple graph, obviously the edges incident must be contained in , respectively, and so are the edges incident . Without loss of generality, we may assume that the edge is contained in . But, since is a tree connecting , it must contain another edge incident with or , a contradiction. Thus . ∎
A subset of is called a matching of if the edges of satisfy that no two of them are adjacent in . A matching saturates a vertex , or is said to be -saturated, if some edge of is incident with ; otherwise, is -unsaturated. is a maximum matching if has no matching with .
###### Theorem 2.
Let be a connected graph of order . Then if and only if or or or .
###### Proof.
Sufficiency: Assume that . From Lemma 4, for a connected graph , if and only if . Since , it follows that . We claim that . Assume, to the contrary, that . Then , a contradiction. Since , it follows that each component of is a path or a cycle (note that a isolated vertex in is a trivial path). We will show that the following claims hold.
Claim 1. has at most one component of order larger than .
Suppose, to the contrary, that has two components of order larger than , denoted by and (see Figure 1 ).
Let and such that and is adjacent to in . Thus . The same is true for , that is, . Pick . This implies that . Since all other components of are paths or cycles, . So and hence . Since , by Lemma 7 it follows that , a contradiction.
Claim 2. If is the component of of order larger than , then is a -path.
Assume, to the contrary, that is a path or a cycle of order larger than , or a cycle of order .
First, we consider the former. We can pick a in . Let , and (see Figure 1 ). Since , there exists no tree of type connecting . From Lemma 5, each tree of type uses at least edges. Since , we have and hence since is an integer. This contradicts to .
Now we consider the latter. Let be a cycle, and (see Figure 1 ). Since , there exists no tree of type . Since each tree of type uses at least edges and , we have and , which also contradicts to .
From the above two claims, we know that if has a component , then it is the only component of order larger than and the other components must be independent edges. Let be the number of such independent edges. can have as many as such independent edges, which implies that . From Lemma 4, . Thus .
By the similar analysis, we conclude that or or or .
Necessity: We will show that if is a graph with the conditions of this theorem. We have the following cases to consider.
Case 1. or .
We only need to show that for . If for , then for . It suffices to check that for .
Let and be a -subset of , and . It is clear that is a maximum matching of . Then has at most one -unsaturated vertex.
If , then there exist pairwise edge-disjoint trees connecting since each vertex in is adjacent to every vertex in . Suppose .
If , then one element of belongs to , denoted by . Since , we can assume that , . When is -unsaturated, the trees together with form pairwise edge-disjoint trees connecting , where . When is -saturated, we let be the adjacent vertex of under . Then the trees together with and form pairwise edge-disjoint trees connecting (see Figure 2 ), where .
If , then two elements of belong to , denoted by and . Without loss of generality, let . When and are adjacent under , the trees together with and form pairwise edge-disjoint trees connecting (see Figure 2 ), where . When and are nonadjacent under , we consider whether and are -saturated. If one of is -unsaturated, without loss of generality, we assume that is -unsaturated. Since has at most one -unsaturated vertex, is -saturated. Let be the adjacent vertex of under . Then the trees together with and and form pairwise edge-disjoint trees connecting (see Figure 2 ), where . If both and are -saturated, we let be the adjacent vertex of under , respectively. Then the trees together with , , and form pairwise edge-disjoint trees connecting (see Figure 2 ), where .
Otherwise, . When one of is -unsaturated, without loss of generality, we assume that is -unsaturated. Since has at most one -unsaturated vertex, both and are -saturated. Let be the adjacent vertex of under , respectively. We pick a vertex of . When are all -saturated, we let be the adjacent vertex of under , respectively. Then the trees together with and and and form pairwise edge-disjoint trees connecting (see Figure 2 ), where .
From the above discussion, we get that for , which implies . So .
Case 2. or .
We only need to show that for and . If for , then for . So we only need to consider the former. Let , be a -subset of , and . Clearly, is a maximum matching of . It is easy to see that has at most one -unsaturated vertex. For any , we will show that there exist edge-disjoint trees connecting in .
If , then there exist pairwise edge-disjoint trees connecting since each vertex in is adjacent to every vertex in . Since and , we only need to consider and . These trees together with for , or for form pairwise edge-disjoint trees connecting . Suppose .
If , then one element of belongs to , denoted by . Since and , we only need to consider or or . When is -unsaturated, the trees together with , for , or for , or for form pairwise edge-disjoint trees connecting , where . When is -unsaturated, we let be the adjacent vertex of under . For , the trees together with , and form pairwise edge-disjoint trees connecting (see Figure 3 ), where . One can check that the same is true for and (see Figure 3 and ).
If , then two elements of belong to , denoted by and . We only need to consider or . When and are adjacent under , the trees together with , and form pairwise edge-disjoint trees connecting for (see Figure 3 ), where . The same is true for (see Figure 3 ). When and are nonadjacent under , we consider whether and are -saturated. If one of is -unsaturated, without loss of generality, we assume that is -unsaturated. Since has at most one -unsaturated vertex, is -saturated. Let be the adjacent vertex of under . For
|
2019-09-22 18:23:27
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.978413462638855, "perplexity": 442.70033105318635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575627.91/warc/CC-MAIN-20190922180536-20190922202536-00405.warc.gz"}
|
https://ctftime.org/writeup/13273
|
Tags: crypto cbc-bit-flipping
Rating: 0
# *Secured Logon*
## Information
| Points |Category | Level|
|--|--|--|
| 500 | Web Exploitation |Hard |
## Challenge
> Uh oh, the login page is more secure... I think. http://2018shell.picoctf.com:12004 ([link](http://2018shell.picoctf.com:12004)). [Source](https://2018shell.picoctf.com/static/914c9135423cd92f5fdb3ff2dec437d8/server_noflag.py).
### Hint
> There are versions of AES that really aren't secure.
## Solution
Let's open the link we got, looks like a normal site with a login form.
and it's worked too!
After login we can see this text:
> Success: You logged in! Not sure you'll be able to see the flag though.
and a title that say "No flag for you" :-(
Below all this there is a text say:
And this message popup -
> I'm sorry the admin password is super secure. You're not getting in that way.
So let's check the cookie that saved for this site.
Right click -> inspect Element -> choose the tab Storage (in Mozilla Firefox)
and we can see that there are a few cookies but the one who cached my eyes was with the name "cookie"
and the value was:
>C5PhO981qfZXWk6gh1+yzqek/cbh3lFbhlAG74fgPUinlzV1ZL3NAVaxvZwl5abNpCRwzXTBrjV9oeTZfk3XuQ==
For checking if it's connected to the login I tried to sign out and the cookie disappeared.
So basically this:
became to this:
C5PhO981qfZXWk6gh1+yzqek/cbh3lFbhlAG74fgPUinlzV1ZL3NAVaxvZwl5abNpCRwzXTBrjV9oeTZfk3XuQ==
And probably we want the admin value be 1.
So now it's the time to check the source code we got.
In the source code we can see that the code connected to the login form, and the cookie.
So what we can learn about the encryption?
There is a function named "encrypt":
**Raw** - the data we want to encrypt
2. Create the iv
3. Get the ciphertext
4. return the ciphertext as 64base code
So here is some basic definitions that I will use later:
- **Plaintext** - the text we want to encrypt.
- **IV** - the block of text we use to randomize the encryption.
- **Key** - the key we use in a symmetric encryption.
- **Ciphertext** - the encrypted text.
So after looking again in the function we can see that the encryption is AES with CBC mode,
the padding is for 16 Bytes and after the encryption the result coded by 64base code.
So here is some explanations of the concepts above:
**AES** - subset encryption of block cipher, in general (it's away more complicated) there is a
plaintext and a key,the plaintext divided for blocks of some bytes and each block encrypted with the key.
**CBC** - (**C**ipher **B**lock **C**haining) - the key for encryption for each block of the **plain**text is the the previous **cipher**text.
the first plaintext (that dosent have a previous ciphertext) key is the IV
So now after some digging in the internet about AES CBC mode I found a nice attack
called "CBC Byte Flipping Attack" - the purpose of the attack is to change byte of the plaintext by changing byte of the ciphertext.
This is exactly what we want, we want to change the 0 value of admin to 1, so we don't need to "crack" the encryption
but only to change 1 byte of the ciphertext and after decryption the plaintext will be the plaintext we wanted.
and the plaintext we want is:
and by reviewing the cookie in the browser we can see that our ciphertext is:
Sj1vwoe8gNvq/I7UdyXTms8/T2+yPPiy4xuZQ33nktu/5eb+8Xl2pvLb9QKyIrnfaIhmGZmW3U5iq5M0LA7Fkg==
so let's check which block of plaintext we want to change - (remember we found in the source code that each block is 16 byte)
|Block|Value |
|:--:|:--:|
| 2 | 'admin': **0**, 'pa |
| 3 | ssword': ''} |
So we want to change the first block in order to
make a change in the second block (that his key is the ciphertext of the first block).
> **"A rule of thumb is that the byte you change in a ciphertext will ONLY affect a byte at the same offset of next plaintext."**
so now we want to check the offset of the byte we want to change -
|Index|Text |
|:--:|:--:|
| 0 | (whitespace)|
| 1| '|
| 2 | a|
| 3 | d|
| 4| m|
| 5| i|
| 6| n|
| 7 | '|
| 8| :|
| 9| (whitespace)|
| **10**| **0** |
| ...|... |
Therefore we want to change the first block at the 10 index byte.
in order to change it from 0 to 1, we take this byte and do XOR with "0"
and after that XOR with "1".
The first XOR with cause the bits of "0" to be zero and
after XOR with "1" the bits of "1" will be on in the byte and we will get the new ciphertext.
So I wrote this python code:
python
from base64 import b64encode, b64decode
#decrypt the 64base code
#In order to work with the 10th char, make it a list (because string in python is immutable)
#ord - cast from char to number
#chr - cast from char to ascii
#Set the 10th char to be the XOR of him with "0" and XOR that with "1"
#join again the list to a string
#encode the string again as a 64base code
> picoCTF{fl1p_4ll_th3_bit3_a41d2782}
|
2019-08-21 11:54:17
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19998280704021454, "perplexity": 3804.2811897323068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315936.22/warc/CC-MAIN-20190821110541-20190821132541-00240.warc.gz"}
|
https://tex.stackexchange.com/questions/553/what-packages-do-people-load-by-default-in-latex/7885
|
# What packages do people load by default in LaTeX?
I'm getting the impression from reading the answers written by some of the real experts here that there are quite a few little packages that just tweak LaTeX2e's default behaviour a little to make it more sensible here and there.
Rather than try to pick these up one by one as I read answers to questions (and thus risk missing them), I thought I'd ask up front what LaTeX2e packages people load by default in (almost) every document.
As this is a "big list" question, I'm making it CW. I don't know if there are standard rules across all SE/SO sites for such questions, but on MathOverflow the rule is generally: one thing (in this case, package) per answer. I guess that if a couple of packages really do go together then it would be fine to group them.
This is perhaps a little subjective and a little close to the line, so I'll not be offended if it gets closed or voted down! (But please explain why in the comments.)
Also see our community poll question: “I have used the following packages / classes”
• There are standard rules across all SE sites, see meta.stackexchange.com/questions/11740/… and follow the links. The idea is that the answer to a "what are good default packages" question is way too big for a single user to write, so the community helps out. The one accepted answer that everyone edits has lots of edits from lots of people. Anton Geraschenko of MO made his own very different interpretation, "post one resource per answer" (mathoverflow.net/faq#communitywiki), and we'll have to decide one or the other. – Kevin Vermeer Jul 29 '10 at 22:25
• Personally, I'd find a single list, separated by headings (Ex. Format, Math, Bib,Images, Other for this question), with a list of everyone's packages and how they're different from other packages in the section much more readable and useful. That amsmath is the highest voted just says that the MO community is here in full force. The less-known, but equally relevant formatting packages linked by Vivi, Joseph, and András are invisible without a lot of scrolling and reading. – Kevin Vermeer Jul 29 '10 at 22:37
• I think the list of one package per answer is a good idea, as we can vote on individual packages... – Amir Rachum Jul 30 '10 at 11:30
• My intention was not so much to find an ordering, but rather to find if there are any that I'd never heard of. It's not working out quite as I'd hoped, but I'm not sure if its possible to fix it at this stage (or worth doing). – Loop Space Jul 30 '10 at 11:37
• It can be good to have a single answer that is just an index of all the other answers, and accept that, so that it floats to the top. – naught101 Aug 30 '12 at 3:44
I almost always load microtype. It plays with ever-so-slightly shrinking and stretching of the fonts and with the extent to which text protrudes into the margins in a way that yields results that look better, that have fewer instances of hyphenation, and fewer overfull hboxes. It doesn't work with latex, you have to use pdflatex instead. It also works with lualatex and (protrusion only) with xelatex.
The family of AMS math packages. At least amsmath and amssymb. Also amsthm if I need theorems and the class I'm using doesn't already define them.
Particularly for writing equations, the AMS packages define a rich set of environments to group and align formulas in many different and useful ways. I also like that it encourages the use of semantic commands (e.g. the cases environment) over syntactic commands (e.g. a \left\{ followed by an array).
Its documentation can be found running texdoc amsldoc on a command line.
• In particular, amsthm provides an easy way to set up different theorem styles, amsmath provides the \text command, and amssymb contains several often-used symbols. – András Salamon Jul 29 '10 at 12:40
• +1 for the (oblique) reference to texdoc. I only discovered that recently and I wonder how I ever lived without it! – Loop Space Jul 29 '10 at 18:08
• I believe amssymb loads amsfonts. There's rarely any need to load it yourself. – TH. Sep 11 '10 at 9:13
• Note that the ams math packages are loaded automatically if you use one of their document classes, such as amsart. – Erik P. Jan 18 '12 at 19:08
• Instead of loading amsmath I usually load mathtools. It is based on amsmath and loads it automatically. Moreover it fixes some deficiencies of the amsmath package and provides additional useful commands such as \coloneqq. – Stan Aug 24 '14 at 9:53
I use hyperref for setting PDF metadata and to create links, both within the document and for clickable URLs. Even Elsevier has used urlbst to update their bibliography style to support URLs and DOIs; hyperref does the actual work of rendering url = and doi = BibTeX fields into clickable PDF links.
For citations and bibliographies, biblatex is the package of my choice. Key points:
• biblatex includes a wide variety of built-in citation/bibliography styles (numeric, alphabetic, author-year, author-title, verbose [full in-text-citations], with numerous variants for each one). A number of custom styles have been published.
• Modifications of the built-in or custom styles can be accomplished using LaTeX macros instead of having to resort to the BibTeX programming language.
• biblatex offers well-nigh every feature of other bibliography-related LaTeX packages (e.g. multiple/subdivided bibliographies, sorted/compressed citations, entry sets, ibidem functionality, back references). If a feature is not included, chances are high it is on the package authors' to-do list.
• The babel package is supported, and biblatex comes with localization files for about a dozen languages (with the list still growing).
• Although the current version of biblatex (2.8a) still allows to use BibTeX as a database backend, by default it cooperates with Biber which supports bibliographies using Unicode. Biber (currently at version 1.8) is included in TeX Live and MiKTeX. Many features introduced since biblatex 1.1 (e.g., advanced name disambiguation, smart crossref data inheritance, configurable sorting schemes, dynamic datasource modification) are "Biber only".
The todonotes package is a must have in all my documents.
\usepackage{todonotes}
The package enables you to insert small notes in the text marking things to do in the document. Something like
\todo{Rewrite this answer \ldots}
At any location in the document a list of the inserted notes can be generated with the
\listoftodos
command.
• For multiuser comment support, and configurability with regard to the kinds of notes/themes available, the fixme package is quite nice (I use it quite regularly). – Mark Mar 25 '11 at 22:29
• todonotes also supports colors and missing graphics. – ℝaphink Jun 16 '11 at 10:42
• I find todonote invaluable as I prepare syllabai and course material for the upcoming term. Because I cannot do everything at one sitting I put a todo note whenever i find something I have to wait to do. I also use it during the semester to highlight to the students anything which was changed after the documents were first published. – R. Schumacher May 1 '12 at 20:51
• Personally I use an editor which automatically highlights and groups in the "structure" window any comment that begins with %TODO: Works better for me because you don't have anything in your compiled document giving away the fact that it still has TODOs around. – Dom Jun 12 '13 at 10:53
• Has anyone done a comparison between easy-todo, fixme, fixmetodonotes, todo, and todonotes? – Ari Brodsky Nov 20 '13 at 1:12
One package that’s really general purpose is nag: It doesn’t do anything, per se, it just warns when you accidentally use deprecated LaTeX constructs from l2tabu (English / French / German / Italian / Spanish documentation).
From the documentation:
Old habits die hard. All the same, there are commands, classes and packages which are outdated and superseded. nag provides routines to warn the user about the use of those. As an example, we provide an extension that detects many of the “sins” described in l2tabu.
Therefore, I now always have the following in my header (before the \documentclass, thanks qbi):
\RequirePackage[l2tabu, orthodox]{nag}
It’s a bit like having use strict; in Perl: a useful best practice.
• Somewhat better is \RequirePackage[l2tabu,orthodox]{nag} before \documentclass. The package docu also recommends this. – qbi Jul 29 '10 at 18:40
• This package sounds useful. However, when I tested it with a large project, I started to get the message "Label(s) may have changed. Rerun to get cross-references right." no matter how many times I re-run Latex. – Jukka Suomela Jul 31 '10 at 9:36
I nearly always use the tikz package. Once you learn how to draw with it, you can do almost any vector graphic you need.
• I have always used Inkscape for the production of my vector images, diagrams and whatsoever. Does tikz produce comparable diagrams? How much effort is involved? – levesque Nov 15 '10 at 18:28
• You can produce almost any diagram with Tikz. Check the tikz examples page. texample.net/tikz/examples However, it is fairly complicated to get the hang on large diagrams since you have to type everything and nearly always you can't see what you are doing. But if you are using a Debian/KDE combination, you can use Ktikz/Qtikz which is really helpful since it compiles tikz code in real time. – fabikw Nov 16 '10 at 0:42
• TikZ is awesome with a capital A. But load it by default? It takes up a lot of time and space. I would say only load it if you need it. – Matthew Leingang Nov 22 '10 at 12:53
• It takes time, but nearly always I find I need to do something with it. – fabikw Nov 23 '10 at 1:11
• @levesque: Tikz has a fairly steep learning curve, but it is beautifully documented and provides rich libraries. I find the vector graphics that I produce in tikz to be superior to those I produced in inkscape. It seems easier on my brain to stay in keyboard mode as well. – philosodad Dec 29 '10 at 4:56
I'm surprised that no one has mentioned
\usepackage[margin=1in]{geometry} % set page margins automatically
This is in every document I write (with varying margins, of course.)
• This is generally poor style. The design of the page is pretty involved and lots of thought has went into (La)TeX's default designs. If you're interested in just saving paper, consider the packages savetrees or fullpage. – Quadrescence Apr 16 '11 at 23:15
• Both savetrees and fullpage change other things too; Anyway, the point of of the answer is that geometry is a must use package, no matter what margins you choose for it. The appropriateness of 1in margins also depends on the kind of documents you produce. – Alan Munn Apr 16 '11 at 23:38
• It is not a must if you use a class from the KOMAscript bundle or memoir. – Sveinung Jan 13 '14 at 16:01
Another essential package combination is
\usepackage{booktabs}
\usepackage{array}
The booktabs package creates much nicer looking tables than the standard latex tables; the array package's ability to create custom columns is invaluable for formatting tabular material on a per-column basis.
• I just discovered booktabs -- it is great! – Ben Jan 12 '11 at 22:37
• @Ben Yes, it's a great package. If you visit my profile web link you can find my own list of essential packages. – Alan Munn Jan 12 '11 at 22:47
Since my files nowadays has UTF-8 character encoding, I use this
\usepackage[utf8]{inputenc}
• XeLaTeX or LuaLaTeX would be my choice for this – Joseph Wright Aug 15 '10 at 13:05
• Interesting, must look into those projects. – Johan Aug 15 '10 at 13:16
• Isn't it \usepackage[utf8x]{inputenc}? – Olivier Jul 19 '11 at 8:17
• I've experienced several cases where utf8x had a symbol that utf8 hadn't – Mog Nov 24 '12 at 11:47
• @Olivier: utf8 is LaTeX base, while utf8x comes from the ucs package. So utf8 is portable. – Martin Schröder Jun 27 '13 at 14:39
\usepackage{siunitx}
siunitx, for typesetting units and especially for the "S" column type, which allows numbers in tables to be easily aligned, e.g. on the decimal marker.
• \usepackage[allowlitunits]{siunitx} is my normal incantation, it allows you to use things like 20\milli\meter directly in math mode. – Alex Hirzel May 1 '12 at 20:18
• I have evidently never followed through on my plan to read the siunitx manual in depth. I was not aware of the S column type or allowlitunits, thank you! – owjburnham Jul 18 '17 at 9:50
The 'rich' document classes such as memoir and KOMA-Script include a lot of functionality that is not available from the LaTeX kernel. So the packages you load when using the article class might be rather different from those when using memoir. A lot of packages that get used by many people with the base classes (things like float, caption, tocbibind and titlesec) are covered by the richer document classes.
• \begin{gripe} My problems with these richer document classes are that it makes it very difficult to pick and choose, and that it is a major pain when Big Shot Journal says "please rewrite your document to use our class file" (there's even a journal that won't let you send an accompanying style file). \end{gripe} – Loop Space Jul 29 '10 at 13:19
• I tend to stick to article + packages, myself, so I can sympathise. All the more reason for me to get on and get LaTeX3 finished, so we can have a good set of abilities out of the box! – Joseph Wright Jul 29 '10 at 14:33
• \begin{joke} Then stop wasting time here and get on with it! \end{joke} – Loop Space Jul 29 '10 at 18:11
• If only it were that easy :-) If you want to see that things are happening, there is an RSS feed for SVN checkins: latex-project.org/latex3svn.rss – Joseph Wright Jul 29 '10 at 21:36
• That gripe seems a gripe with the journals, rather than with the rich document classes. Also, if you're writing a journal article, memoir doesn't seem like an obvious way to go, if you are going to end up having to conform to some journal's style eventually. Again, that's not an issue with rich document classes, that's just a matter of picking the right tool for the job. And for journal submissions, minimal package requirements and basic document classes seems a good modus operandi – Seamus Aug 1 '10 at 10:41
\usepackage{graphicx}
For including figures, rotating or scaling text. I also use the \graphicspath command to specify a subfolder to help organize my figures and so I can easily change between, for example, a set of figures for internal used (with extra info) and final versions for distribution.
\usepackage{lmodern} % better i18n Postscript version of Knuth's cm fonts
In addition to many packages already listed here, I always include mathtools. It provides implementations of \mathclap (and similar commands) as well as nice extensible arrow.
• \mathclap is great. I use it to great effect for things like \sum_{\mathclap{big long thing}}. (It's also amusingly named with at least one off-color meaning.) – TH. Aug 27 '10 at 9:36
• \shortintertext is also provided by the \mathtools package and provids tighter vertical spacing compared to \intertext from the amsmath package. – Peter Grill May 2 '12 at 0:47
I can't live without listings --- pretty-printing (colours, formatting and all) algorithms and code is indispensable --- in pretty much any programming languages and dialects under the sun. Plus, I can import a source file directly from the repository, and the latest version will be automatically rendered.
• I was pleasantly surprised that I could prettyprint MIPS assembly language code with listings! Excellent package. – MercurialMadnessMan Nov 24 '12 at 6:30
The package xspace lets you define commands that don't eat up whitespace after them. So you can define an abbreviation like
\newcommand{\sA}{\mathcal{A}\xspace}
and then you can type objects of \sA are called widgets instead of objects of \sA\ are called widgets.
• That's one I use so much that I forget it's not part of the main code! – Loop Space Aug 5 '10 at 7:10
• The main advantage of \sa/ is that an error message will occur if you happen to forget the closing slash. On the contrary, if you happen to forget the closing backslash of \sA\ , you'll end with gobbled space without noticing it. – lockstep Aug 11 '10 at 20:50
• I used xspace one time in a paper with other authors. It was a huge pain since some macros didn't behave like others. It led to all sort of confusion, especially when thinks like \foo bar no long work as you expect because \foo's definition ends with \xspace. I've never tried \foo/. The main advantage I see with that is if your macro is \m/... – TH. Aug 27 '10 at 9:32
• I don't especially like the look of \sA/ but I can't think of a better delimiter to use. Perhaps a semicolon would be fine (after HTML): \sA;. My personal belief is that non-delimited macros without arguments (i.e., the ones that gobble spaces) are just plain wrong for document commands because of the spacing problems. Even experienced LaTeX authors trip up with them. – Will Robertson Sep 2 '10 at 9:28
First line of the document should be
\RequirePackage{fixltx2e}
\documentclass{...}
, which fixes a few things in the LaTeX2e kernel.
Due to LaTeX's stability policy, these corrections have not been incorporated into the LaTeX2e kernel, but this package does things most people would agree are bugfixes. So to load this package is always recommended for newly created documents. The corrections have no commonalities, but the package's description has a nice summary:
• ensure one-column floats don't get ahead of two-column floats;
• correct page headers in twocolumn documents;
• stop spaces disappearing in moving arguments;
• allowing \fnsymbol to use text symbols;
• allow the first word after a float to hyphenate;
• \emph can produce caps/small caps text;
• bugs in \setlength and flushbottom.
## EDIT 27.01.2016:
This package is obsolete for LaTeX releases after 2015. See latexrelease.pdf.
• It should be RequirePackage{fixltx2e} as first line of you'require document, even before the document class, see texdev.net/2014/12/28/fixing-latex2e – MaxNoe Jan 17 '15 at 13:51
• really should be an argument to documentclass. – ivo Welch Jan 2 '16 at 16:58
• fixltx2e is not required with releases after 2015(fixltx2e) All fixes are now in the LaTeX kernel. – kaka Apr 3 '16 at 11:23
For papers on the arXiv (maths, physics and computer science mostly) there's a list of packages sorted by frequency of use.
The top twenty packages are:
1. article
2. graphicx
3. amssymb
4. amsmath
5. revtex
6. revtex4
7. epsfig
8. amsfonts
9. bm
10. latexsym
11. amsart
12. dcolumn
13. amsthm
14. graphics
15. aastex
16. amscd
17. epsf
18. color
19. aa
20. times
• That list is literally pain to my eyes. Loading bm?! Use proper bold math characters instead, please, and not poorman's bold. times? Outdated since ages, use mathptmx or XITS Math instead. I'll stop here... – Ingo Jan 30 '14 at 11:46
• @Ingo arXiv has been created in 1991 and some papers haven't been updated since then! – Najib Idrissi Feb 24 '17 at 14:57
I use url to typeset urls.
For quickly setting multicolumn text in a single column document, the multicol package is another package that I use all the time.
\usepackage{multicol}
To use the palatino font (it's just a nice looking font)
\usepackage[sc]{mathpazo}
Note that the old palatino package is deprecated.
• never use \usepackage{palatino}, see l2tabu. the current way to use Palatino is \usepackage{mathpazo} – Mateus Araújo Sep 2 '10 at 3:47
• What is l2tabu? – Johan Sep 2 '10 at 6:35
• You should probably also load mathpazo with the [sc] option to get real small caps and better kerning. – Will Robertson Sep 2 '10 at 9:24
• Depending on taste, you may want to use [osf] instead of [sc] to get old style numerals as well as the real small caps and better kerning. I for one find old style numerals prettier and classier than lining figures in text mode (using [osf] will keep lining figures in math mode). – spet May 29 '13 at 8:52
• According to this the LaTeX font catalogue, one should increase the leading when using mathpazo. tug.dk/FontCatalogue/urwpalladio – Ubiquitous Nov 21 '14 at 9:04
\usepackage[parfill]{parskip}
I much prefer no indentation and space between paragraphs, so the parskip package is a must for me!
• Have a look at the KOMA-Script-classes - they include a parskip option that is more powerful than the package of the same name. – lockstep Aug 8 '10 at 17:39
I almost always find myself using a tabularx environment as opposed to the regular tabular environment, as it allows for greater dynamism in column widths.
Nothing surprising here: I use natbib, hyperref and hypernat together.
Natbib for referencing.
Hyperref adds bookmarks for sections and lists and turns references and urls into links.
Hypernat allows natbib and hyperref to work together. -- Note (added 2015/02/11): natbib and hyperref have been working together just fine for at least ten years. hypernat is no longer needed for any TeX distribution with a vintage more recent than ca 2002.
• I'm pretty sure that hypernat is superfluous these days. With only loading natbib and hyperref I get references as [1-5] with both 1 and 5 being hyperlinks. – Lev Bishop Aug 8 '10 at 14:51
• Agreed, I didn't even know about hypernat until I saw this answer. I have been using hyperref and natbib for a while and reference links and backlinks always worked for me. Is there some extra functionality that hypernat adds? – Sharpie Aug 9 '10 at 17:31
• I had a problem once, found out about natbib, and have been using it ever since, so it is possible it is superfluous and I didn't even know. I will have to test it out and get back to you guys if I find something. – Vivi Aug 10 '10 at 20:18
• And? Was it superfluous in 2010? Is it now? ;) – K.-Michael Aye Nov 23 '12 at 5:18
• @K.-MichaelAye - hypernat was superfluous (and potentially troublesome) back in 2010 and in 2012, and it continues to be superfluous as of 2015. – Mico Feb 11 '15 at 21:13
To make sure you have ISO formated dates (YYYY-MM-DD).
\usepackage[english]{isodate}
or
\usepackage{datetime}
\renewcommand{\dateseparator}{-}
\newcommand{\todayiso}{\the\year \dateseparator \twodigit\month \dateseparator \twodigit\day}
I almost always use the enumitem package, which makes it much easier to make modifications to lists (especially enumerate lists). Most notably, changing the labels to something like (i), (ii), (iii) [no period] with this package is as easy as
\begin{enumerate}[label=(\roman*)]
\item The first item
\item The second item
\end{enumerate}
Furthermore, the code above will automatically get nesting right. Before I started using this package, my preamble always included the awkward macro (necessary to change the references and eliminate the extra period in the list itself)
\newcommand{\setenumroman}{%
\renewcommand{\theenumi}{(\roman{enumi})}%
\renewcommand{\labelenumi}{\theenumi}%
}
which would break if I ever used it for a nested list (all the enumis would have to be changed to enumiis, if I understand correctly).
The enumitem package is quite flexible; another option I sometimes use is [wide], which makes a list look like part of the body of the text (with numbers/labels at the beginning of relevant paragraphs).
For mathematical texts I instead use amsmath & Co. One very useful package is onlyamsmath. I load it as
\usepackage[all,warning]{onlyamsmath}
So it looks for $$..$$, eqnarray and produces a warning if some of them are used. If you left out warning, it will result in an error and compile will stop. This package is normally very useful if you edit a text with many authors.
Edited by doncherry: Removed packages mentioned in separate answers.
The complete header Part of my header for most of my documents looks as follows:
\documentclass[ngerman,draft,parskip=half*,twoside]{scrreprt}
\usepackage{ifthen}
For some things I need if-then-constructs. This package provides an easy way to realise it.
\usepackage{index}
For generating an index.
\usepackage{xcolor}
xcolor is needed by several packages. For some historical reason I load it manually.
\usepackage{babel}
\usepackage{nicefrac}
nicefrac allows typesetting fractions like 1/2. It is sometimes more readable than \frac.
\usepackage[T1]{fontenc}
\usepackage[intlimits,leqno]{amsmath}
\usepackage[all,warning]{onlyamsmath}
This package warns if non-amsmath-environments are used.
\usepackage{amssymb}
\usepackage{fixmath}
Provides ISO conform greek letters.
\usepackage[euro]{isonums}
Defines comma as decimal delimiter.
\usepackage[amsmath,thmmarks,hyperref]{ntheorem}
for Theorems, definitions and stuff.
\usepackage{paralist}
Improves enumerate and itemize. Also provides some compact environments.
\usepackage{svn}
I work with VCS and svn displays some informations (keywords) from SVN.
\usepackage{ellipsis}
corrects \dots
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
These are the definitions for absolute value and norm.
\SVN $LastChangedRevision$
\SVN $LastChangedDate$
• "one thing (in this case, package) per answer" – Jukka Suomela Jul 29 '10 at 19:02
• Could you break this up into multiple answers please, so they can be voted on? Having a dozen answers is ok! – ShreevatsaR Jul 30 '10 at 14:41
• Excellent list with helpful commentary. I will add the one for old and deprecated things to my list. Thanks! – DJP Jul 30 '11 at 20:31
• It is usually recommended to load hyperref last. – Alex Hirzel May 1 '12 at 20:20
Another package I use is float. It allows for the placement H for floats, which is somewhat equivalent to h!, but a bit stronger, making sure the figure or table goes exactly where I want it to be.
• Actually not equivalent to h! at all. h! floats still "float"- they can be moved around by LaTeX in an attempt to optimize the document layout. Figures using the H specifier are not floats at all, they are treated like one big character and are put exactly where they appear in the text. – Sharpie Aug 1 '10 at 3:59
• @Sharpie: you are ignoring the word "somewhat" :P Still, your point is valid, thanks! – Vivi Aug 1 '10 at 4:21
• I did consider the word somewhat. However, in my opinion the only similarity between the two is the fact that they are used as float specifiers. Beyond that, both specifiers produce completely different effects. – Sharpie Aug 1 '10 at 6:14
• @Sharpie: maybe I should link to the source of the (mis)information? en.wikibooks.org/wiki/LaTeX/… (see the last row of the table) – Vivi Aug 1 '10 at 6:32
|
2018-11-20 23:59:59
|
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|
http://sepwww.stanford.edu/data/media/public/docs/sep94/jim1/paper_html/node2.html
|
Next: BACKUS AVERAGING Up: Berryman et al.: Analysis Previous: INTRODUCTION
# NOTATION FOR ANISOTROPIC ELASTIC MEDIA
In tensor notation, the relationship between components of stress and strain uk,l is given by
_ij = c_ijklu_k,l, where cijkl is the adiabatic stiffness tensor. Repeated indices on the right hand side of (ssgen) are summed. In (ssgen), uk is the kth Cartesian component of the displacement vector , and .Whereas for an isotropic elastic medium the stiffness tensor has the form
c_ijkl = _ij_kl + (_ik_jl + _il_jk), depending on only two parameters (the Lamé constants, and ). This tensor can have up to 21 independent constants for general anisotropic elastic media. The stiffness tensor has pairwise symmetry in its indices such that cijkl = cjikl and cijkl = cijlk, which will be used later to simplify the resulting equations. The requirement that straining the medium must produce a positive change in its internal energy provides the additional constraint that cijkl = cklij.
The general equation of motion for wave propagation through an anisotropic elastic medium is given by
ü_i = _ij,j = c_ijklu_k,lj, where is the second time derivative of the ith Cartesian component of the displacement vector and is the density (assumed constant). Equation (anisowaves) is a statement that the product of mass times acceleration of a particle is determined by the internal stress force .
A commonly used simplification of the notation for elastic analysis is given by introducing the strain tensor, where
e_ij = 12(u_i,j + u_j,i) = 12(u_ix_j + u_jx_i). Then, using one version of the Voigt convention, in which the pairwise symmetries of the stiffness tensor indices are used to reduce the number of indices from 4 to 2 using the rules , , , ,, and ,we have
_11 _22 _33 _23 _31 _12 = c_11 & c_12 & c_13 & & & c_12 & c_22 & c_23 & & & c_13 & c_23 & c_33 & & & & & & 2c_44 & & & & & & 2c_55 & & & & & & 2c_66 e_11 e_22 e_33 e_23 e_31 e_12 . Although the Voigt convention introduces no restrictions on the stiffness tensor, we have chosen to limit discussion to the form in (sscij), which is not completely general. Of the 36 coefficients (of which 21 are generally independent), we choose to treat only those cases for which the 12 coefficients shown (of which nine are generally independent) are nonzero. This form includes all orthorhombic, cubic, hexagonal, and isotropic systems, while excluding triclinic, monoclinic, trigonal, and some tetragonal systems, since each of the latter contains additional off-diagonal constants that may be nonzero. Nevertheless, we will restrict the discussion to (sscij) or to the still simpler case of transversely isotropic (TI) materials.
For TI materials, , ,,, , and .There is also one further constraint on the constants that a = b + 2m, following from rotational symmetry in the x1x2-plane. In such materials, (sscij) may be replaced by
_11 _22 _33 _23 _31 _12 = a & b & f & & & b & a & f & & & f & f & c & & & & & & 2l & & & & & & 2l & & & & & & 2m e_11 e_22 e_33 e_23 e_31 e_12 , in which the matrix has the same symmetry as hexagonal systems and of which isotropic symmetry is a special case (having , , and ).
Next: BACKUS AVERAGING Up: Berryman et al.: Analysis Previous: INTRODUCTION
Stanford Exploration Project
11/11/1997
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2017-10-20 19:57:44
|
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|
http://mathhelpforum.com/calculus/36946-integration-partial-fractions.html
|
# Math Help - Integration-Partial Fractions
1. ## Integration-Partial Fractions
$
\int {\frac{{dx}}
{{(x^2 - 1)^2 }}} = \int {\frac{{dx}}
{{(x - 1)^2 (x + 1)^2 }}}
$
It seems no matter what I try after this I cannot get the answer in the book.
Which is:
$
\frac{1}
{4}\int {(\frac{{ - 1}}
{{x - 1}}} + \frac{1}
{{(x - 1)^2 }} + \frac{1}
{{x + 1}} + \frac{1}
{{(x + 1)^2 }})dx
$
Any help would be greatly appreciated.
2. Originally Posted by kid funky fried
$
\int {\frac{{dx}}
{{(x^2 - 1)^2 }}} = \int {\frac{{dx}}
{{(x - 1)^2 (x + 1)^2 }}}
$
It seems no matter what I try after this I cannot get the answer in the book.
Which is:
$
\frac{1}
{4}\int {(\frac{{ - 1}}
{{x - 1}}} + \frac{1}
{{(x - 1)^2 }} + \frac{1}
{{x + 1}} + \frac{1}
{{(x + 1)^2 }})dx
$
Any help would be greatly appreciated.
$\frac{1}{(x^2-1)^2}=\frac{1}{(x-1)^2(x+1)^2}$
$\frac{1}{(x-1)^2(x+1)^2}=\frac{A}{(x-1)}+\frac{B}{(x-1)^2}+\frac{C}{(x+1)}+\frac{D}{(x+1)^2}$
Clearing the fractions gives
$1=A(x-1)(x+1)^2+B(x+1)^2+C(x-1)^2(x+1)+D(x-1)^2$
First lets get our two freebee values
let x =1
$1=A(1-1)(x+1)^2+B(1+1)^2+C(1-1)^2(1+1)+D(1-1)^2$
$1=b(2)^2 \iff \frac{1}{4}=B$
x=-1 gives
$1=D(-1-1)^2 \iff \frac{1}{4}=D$
Now we know that this is true for ALL VALUES OF X so we can pick some others. I will pick x=0 and x=2 to get a system of equations.
$1=A(0-1)(0+1)^2+\frac{1}{4}(0+1)^2+C(0-1)^2(0+1)+\frac{1}{4}(0-1)^2$
$1=-A+\frac{1}{4}+C+\frac{1}{4} \iff \frac{1}{2}=-A+C$
$1=A(2-1)(2+1)^2+\frac{1}{4}(2+1)^2+C(2-1)^2(2+1)+\frac{1}{4}(2-1)^2$
$1=9A+\frac{9}{4}+3C+\frac{1}{4} \iff -\frac{3}{2}=9A+3C \iff -\frac{1}{2}=3A+C$
Now we need to solve
$3A+C=-\frac{1}{2}$
$-A+C=\frac{1}{2}$
subtracing the 2nd from the first gives
$4A=-1 \iff A=-\frac{1}{4}$
then $-\left( -\frac{1}{4}\right)+C=\frac{1}{2} \iff C=\frac{1}{4}$
There we go Yeah!!!
I hope this helps.
3. Yes it helps alot. I had gotten as far as finding B and D. A and C had me perplexed.
Thanks, I appreciate it.
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2015-09-05 06:41:45
|
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|
http://mathhelpforum.com/trigonometry/149108-finding-length-triangle-print.html
|
# Finding Length of a Triangle
Printable View
• June 22nd 2010, 07:58 AM
Hugh_Compton
Finding Length of a Triangle
On the attached image I need to find the value of B.
• June 22nd 2010, 08:08 AM
skeeter
$B + \frac{B}{2} = 165$
solve for B
|
2016-05-30 13:22:08
|
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|
https://www.aimsciences.org/article/doi/10.3934/dcdsb.2008.9.75
|
Article Contents
Article Contents
# Transitivity of a Lotka-Volterra map
• The dynamics of the transformation $F: (x,y)\rightarrow (x(4-x-y),xy)$ defined on the plane triangle $\Delta$ of vertices $(0,0)$, $(0,4)$ and $(4,0)$ plays an important role in the behaviour of the Lotka--Volterra map. In 1993, A. N. SharkovskiĬ (Proc. Oberwolfach 20/1993) stated some problems on it, in particular a question about the trasitivity of $F$ was posed. The main aim of this paper is to prove that for every non--empty open set $\mathcal{U} \subset \Delta$ there is an integer $n_{0}$ such that for each $n>n_{0}$ it is $F^{n}(\mathcal{U}) \supseteq \Delta \setminus P_{\varepsilon}$, where $P_{\varepsilon} = \{ (x,y) \in D : y<\beta, \mbox{$where$F(t,\varepsilon)=(\alpha,\beta) \mbox{ and } t \in[0,2] \}$ and $\varepsilon \rightarrow 0$ for $n \rightarrow \infty$. Consequently, we show that the map $F$ is transitive, it is not topologically exact and it is almost topologically exact. Additionally, we prove that the union of all preimages of the point $(1,2)$ is a dense subset of $\Delta$.
Mathematics Subject Classification: Primary: 37B99, 37B10.
Citation:
|
2023-03-31 19:44:27
|
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|
http://algirdasjavtokas.wordpress.com/2010/03/15/famous-differential-equations-by-mathematician-as-differential-equation/
|
Existentialism Mathematics #03
March 15, 2010
Notes on Existentialism Mathematics by dr. Algirdas Javtokas (New Frontiers in Mathematics Series).
.
Lemma 1. 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0.
$\frac{\partial}{\partial t}$ 000 + $\frac{\partial^2}{\partial t^2}$ 00 = 000 $\frac{\partial}{\partial t}$ 0000 $\times$ 00
0 1 1 0 1 010 1 1 0 1 0 1 0 1 0 1 0 1 0 1 010 1 0 1 0 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0.
.
Proof 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 11 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0
00 $\frac{\partial}{\partial t}$ 000 + 000 = 000 $\frac{\partial}{\partial t}$ 00 $\cdot$ 000000
0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0
.
Lemma 2. 0 1 1 0 1 01 0 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0.
$\frac{\partial^2}{\partial t^2}$ 00 0 1 0 1 0 1 0 1 0 1 0 0 10 = 00
0 1 1 0 1 010 1 1 0 1 0 1 0 1 0 1 0 1 0 1 010 1 0 1 0 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0.
.
Proof 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 11 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0
000 $\frac{\partial}{\partial t}$ 000 $\frac{\partial^2}{\partial t^2}$ 000 = 001 1 1 1 1 00 $\frac{\partial}{\partial t}$ 000
0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0
$\frac{\partial^2}{\partial t^2}$ 000 = 000 $\frac{\partial}{\partial t}$ 00 1 0 1 0 1 01
0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0
.
Theorem 1. 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0
00 $\frac{\partial}{\partial t}$ 00 + 00 $\frac{\partial^2}{\partial t^2}$ 00 = 00 + 001 1 0 1 0 1 0
1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0
.
Proof 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0
000 $\int$ 000 1 0 1 0 1 0 1 0 10
0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0
$\frac{\partial}{\partial t}$ 0100 = 00 1 0 1 0 110 $\frac{\partial^2}{\partial t^2}$ 01 0 1 0 100
0 1 01 0 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1
|
2013-12-10 01:08:47
|
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|
http://www.purplemath.com/learning/viewtopic.php?p=5712
|
## rectangular form to polar form simplification
Trigonometric ratios and functions, the unit circle, inverse trig functions, identities, trig graphs, etc.
scrilla103
Posts: 2
Joined: Sun Apr 10, 2011 8:29 pm
Contact:
### rectangular form to polar form simplification
Equation given:
(y^2) - (8x) - (16) = 0
Convert to polar form and simplify.
{ I tried multiple routes to no avail. }
Please show steps with brief explanations, thanks
stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
Contact:
(y^2) - (8x) - (16) = 0
Convert to polar form and simplify.
{ I tried multiple routes to no avail. }
Unfortunately, it is not possible to help you find any errors in work we cannot see. So please reply showing one of the routes you tried, starting from the standard substitution and proceeding through using the Quadratic Formula to solve for "r=". Thank you!
scrilla103
Posts: 2
Joined: Sun Apr 10, 2011 8:29 pm
Contact:
### Re: rectangular form to polar form simplification
So here is the basic process I tried:
(y^2) - (8x) - (16) = 0
**substitute y with rsint and x with rcost**
(r^2)[(sint)^2] - (8rcost) - (16) = 0 {In words: "(r squared sine squared theta) minus (eight r cosine theta) minus (sixteen) equals (zero)}
This is where I get stuck. I cannot seem to simplify this equation to have "r" alone on one side... there is always an "r" on the other side.
Any ideas?
Thanks.
little_dragon
Posts: 226
Joined: Mon Dec 08, 2008 5:18 pm
Contact:
### Re: rectangular form to polar form simplification
try the quaderatic formula like they said
sin^2(@) r^2 - 8cos(@) r - 16 = 0
so a=sin^2(@), b=-8cos(@), c=-16
arletebacon
Posts: 5
Joined: Thu Oct 06, 2011 4:30 am
Contact:
### Re: rectangular form to polar form simplification
I have the same question
Y^2-8x-16=0. The answer listed is r= 16/(-7sin(theta))
can someone explain how to get there!!!!
stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
Contact:
### Re: rectangular form to polar form simplification
I have the same question
Y^2-8x-16=0. The answer listed is r= 16/(-7sin(theta))
can someone explain how to get there!!!!
Yes: read the previous replies, which provide an explanation of "how to get there".
arletebacon
Posts: 5
Joined: Thu Oct 06, 2011 4:30 am
Contact:
### Re: rectangular form to polar form simplification
I have read the posts and tried all those suggestions.. but I don't get the answer that was provided by my course.
i don't know how to get the r= 16/(-7sin(theta)) as an answer.
arletebacon
Posts: 5
Joined: Thu Oct 06, 2011 4:30 am
Contact:
### Re: rectangular form to polar form simplification
a=sin^2(theta)
b=-8cos(theta)
c=-16
(-b+-sqrt(b2-4ac))/2a
8cos(theta) +- sqrt( (-8cos(theta)^2) - (4) (sin^2(theta)) (-16) ) / 2(sin^2(theta)
8cos(theta) + - sqrt ( 64cos^2(theta) +64 sin^2(theta) ) / 2 sin^2(theta)
( 8cos (theta) +- 8 ) / 2 sin^2(theta)
4(cos(theta) +- 4) / sin^2(theta)
4( cos(theta) _+ 1) / 1- cos^2(theta)
now I just went with the 4costheta + 1 version.. there is still the 4costheta - 1
4 (cos(theta) +1) / (1-cos(theta)) (1+cos(theta))
4 / 1-cos(theta) ..............
I don't know how to get from there to here r= 16/(-7sin(theta))
stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
Contact:
I hadn't checked the work all the way to the answer. But the solution key is wrong; you're right. The polar form is:
. . . . .$r\, =\, \frac{4}{1\, -\, \cos\left(\theta\right)}$
...or, which is the same thing:
. . . . .$4\, =\, r\,-\, r\cos\left(\theta\right)$
arletebacon
Posts: 5
Joined: Thu Oct 06, 2011 4:30 am
Contact:
### Re: rectangular form to polar form simplification
thank you..!! I spent tons of hours on that... appreciate your help!!
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2017-01-20 14:02:24
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https://www.transtutors.com/questions/at-sunrise-corporation-direct-materials-are-added-at-the-beginning-of-the-process-an-2572802.htm
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# At Sunrise? Corporation, direct materials are added at the beginning of the process and convers...
At Sunrise? Corporation, direct materials are added at the beginning of the process and conversions costs are uniformly applied. Other details? include:
WIP beginning (50% for conversion): 22,200 units
Units Started: 121,500 units
Units completed and transferred out: 107,700 units
WIP ending? (60% for? conversion): 36,000 units
Beginning WIP direct materials ?$35,000 Beginning WIP conversion costs ?$22,500 Costs of materials added ?$384,100 Costs of conversion added ?$271,125
What are the total equivalent units for direct? materials?
A. 129,900 B. 143,700 C. 129,300 D. 157,500
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(Solved) February 20, 2018
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• ### 8.) The Lakeside Company uses a weighted-average process costing system. The following data are a...
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$37,400 Total cost of the 4,000 units of the ending inventory: a.)$15,840. b .) $14,520. c .)$9,240. d .) \$8,910. 9.) The Finishing Department had 5,000 incomplete units in its beginning Work-in- Process Inventory which were 100% complete as to materials
8 Beginning inventory -0- Units started in production 20,000 Units finished during the period 16,000 Units in process at the end of the period (complete as to materials, ¼ complete as to...
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2018-11-21 10:46:16
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https://abevoelker.com/page6/
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# Final Ode to OpenEdge ABL Part 2: Ruby Helps You REST Easy
In part 1 of this series, we learned how to get Ruby to talk to an OpenEdge database by using an adapter for the DataMapper ORM framework.
In this post, I would like to demonstrate both the power and beauty of Ruby by rapidly prototyping a RESTful Web service (using JSON representation responses) for sports2000 customers using our new OpenEdge database adapter.
# Final Ode to OpenEdge ABL Part 1: a Ruby Adapter is Born
It’s weird how I have trouble letting go. Nearly two years ago, I wrote a post where I theorized a cure for a programming language and database that had tortured me at the first full-time programming job that I ever had: OpenEdge ABL. Shortly after writing that post, I quit my job and moved to a new city, where I got a job contracting as a Ruby developer. I’m much happier at my new job, but every once in awhile I would think back on my time with OpenEdge and how I never really got to revolutionize it like I wanted to (yeah, I like to set the bar high). I sometimes would find myself checking in on the Progress community areas to see if anything’s changed, kind of like stalking an ex-lover on Facebook.
# GitLaw: GitHub for Laws and Legal Documents - a Tourniquet for American Liberty
Update 2: Apparently this post also provided some inspiration for a TED Talk entitled “How the Internet will (one day) transform government” by Clay Shirky. Check it out!
It’s no secret that most Americans hate the members of our legislative branch. Polls show that 79% of Americans currently disapprove of the job that Congress is doing (only 14% approve). I can only speak for myself, but the disdain I feel for Congress is due to suspicion of a combination of malfeasance, misfeasance and nonfeasance. I simply don’t trust them to represent me, and I don’t think most Americans do, either.
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2020-10-01 08:03:47
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http://mathhelpforum.com/calculators/94219-ti83-program-quadratic-polynomials-print.html
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# Ti83 program for quadratic polynomials?
• Jul 2nd 2009, 11:43 AM
allyourbass2212
Right now I am teaching myself Algebra 1 and Algebra 2, but I find it more so of a time sink than actually learning anything when factoring quadratic polynomials with a coefficient attached to $x^2$
e.g. $2x^2-5x-12$, and give you the correct output of $(2x+3)(x-4)$
So with that being said does anyone know of a ti83 program that will automatically factor these types of quadratic polynomials?
Many Thanks
• Jul 2nd 2009, 10:28 PM
mr fantastic
Quote:
Originally Posted by allyourbass2212
Right now I am teaching myself Algebra 1 and Algebra 2, but I find it more so of a time sink than actually learning anything when factoring quadratic polynomials with a coefficient attached to $x^2$
e.g. $2x^2-5x-12$, and give you the correct output of $(2x+3)(x-4)$
So with that being said does anyone know of a ti83 program that will automatically factor these types of quadratic polynomials?
Many Thanks
FACTOR 9 - ticalc.org
But note: If you don't make it your business to become competent in factorising quadratics you're in for a world of pain down the track.
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2017-02-25 08:28:55
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https://brilliant.org/problems/what-remains-2/
|
# What remains?
What is the remainder left when $$8^a - 62^b$$ is divided by 9?
Where a = 2n and b = 2n+1.
×
|
2017-10-17 18:53:55
|
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https://socratic.org/questions/how-do-you-differentiate-f-x-1-sec-x-tan-x-2#624328
|
# How do you differentiate f(x) =1- (sec x/ tan x)^2 ?
May 30, 2018
$\setminus \frac{2 \cos \left(x\right)}{{\sin}^{3} \left(x\right)}$
#### Explanation:
By definition, $\sec \left(x\right) = \setminus \frac{1}{\cos \left(x\right)}$, and $\tan \left(x\right) = \setminus \frac{\sin \left(x\right)}{\cos \left(x\right)}$
We can rewrite the fraction in parenthesis as
$\setminus \frac{\sec \left(x\right)}{\tan \left(x\right)} = \setminus \frac{1}{\cos \left(x\right)} \setminus \cdot \setminus \frac{\cos \left(x\right)}{\sin \left(x\right)} = \setminus \frac{1}{\sin \left(x\right)}$
So, the expression becomes
$1 - \setminus \frac{1}{{\sin}^{2} \left(x\right)}$
To differentiate this expression, remember that
$\frac{d}{\mathrm{dx}} \left(1 - \setminus \frac{1}{{\sin}^{2} \left(x\right)}\right) = \frac{d}{\mathrm{dx}} \left(1\right) - \frac{d}{\mathrm{dx}} \left(\setminus \frac{1}{{\sin}^{2} \left(x\right)}\right) = - \frac{d}{\mathrm{dx}} \setminus \frac{1}{{\sin}^{2} \left(x\right)}$
since the derivative of a number is zero.
Finally, you can write $- \frac{1}{\sin} ^ 2 \left(x\right)$ as $- {\sin}^{- 2} \left(x\right)$, and derive it with chain rule:
$\frac{d}{\mathrm{dx}} - {\sin}^{- 2} \left(x\right) = - \frac{d}{\mathrm{dx}} {\sin}^{- 2} \left(x\right) = - \left(- 2 {\sin}^{- 3} \left(x\right) \cdot \cos \left(x\right)\right) = \setminus \frac{2 \cos \left(x\right)}{{\sin}^{3} \left(x\right)}$
$f ' \left(1 - {\csc}^{2} x\right) = 2 {\csc}^{2} \times x \cot x$
#### Explanation:
$f \left(x\right) = 1 - {\left(\sec \frac{x}{\tan} x\right)}^{2}$
$\sec x = \frac{1}{\cos} x$
$\tan x = \sin \frac{x}{\cos} x$
$\sec \frac{x}{\tan} x = \frac{\frac{1}{\cos} x}{\sin \frac{x}{\cos} x}$
Multiplying numerator and denominator by cosx
$\sec \frac{x}{\tan} x = \frac{1}{\sin} x$
${\left(\sec \frac{x}{\tan} x\right)}^{2} = {\left(\frac{1}{\sin} x\right)}^{2} = \frac{1}{\sin} ^ 2 x$
$1 - {\left(\sec \frac{x}{\tan} x\right)}^{2} = 1 - {\left(\frac{1}{\sin} x\right)}^{2}$
$\left(\frac{1}{\sin} ^ 2 x\right) = {\csc}^{2} x$
$1 - {\left(\sec \frac{x}{\tan} x\right)}^{2} = 1 - {\csc}^{2} x$
$f \left(x\right) = 1 - {\csc}^{2} x$
$f ' \left(x\right) = f ' \left(1 - {\csc}^{2} x\right)$
$= f ' \left(1\right) - f ' \left({\csc}^{2} x\right)$
$f ' \left(1\right) = 0$
$f ' \left(\csc 2 x\right) = 2 \csc \times x \left(- \csc \times x \cot x\right)$
$f ' \left({\csc}^{2} x\right) = 2 {\csc}^{2} \times x \cot x$
$f ' \left(1 - {\csc}^{2} x\right) = 0 - \left(- 2 {\csc}^{2} \times x \cot x\right)$
$f ' \left(1 - {\csc}^{2} x\right) = 2 {\csc}^{2} \times x \cot x$
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2021-10-21 20:31:38
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https://www.statistics-lab.com/%E7%BB%9F%E8%AE%A1%E4%BB%A3%E5%86%99%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BD%91%E7%BB%9C%E6%A6%82%E7%8E%87%E8%A7%A3%E9%87%8A%E4%BB%A3%E5%86%99probabilistic-reasoning-with-bayesian-networks%E4%BB%A3-4/
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统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|BN model from tie-sets
statistics-lab™ 为您的留学生涯保驾护航 在代写 Probabilistic Reasoning With Bayesian Networks方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写CMPT 310 Probabilistic Reasoning With Bayesian Networks方面经验极为丰富，各种代写 Probabilistic Reasoning With Bayesian Networks相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|BN model from tie-sets
Let us consider the problem from the point of view of success. Three tie-sets can be extracted from Figure 2.1, as given by equation [2.3]. The BN model (Figure 2.4) is given by:
\begin{aligned} &L_{1}=\left{x_{1}, x_{2}\right} \ &L_{2}=\left{x_{1}, x_{3}\right} \ &L_{3}=\left{x_{1}, x_{2}, x_{3}\right} \end{aligned}
The reader can note directly that none of the tie-sets are minimal because $L_{1} \cup L_{2}=L_{3}$. Nevertheless, as in the case of cut-sets, the inference mechanism will work properly and give the correct result. Let us consider the two minimal tie-sets $L_{1}$ and $L_{2}$, for computing the probability distribution of the system states. The deterministic CPT of tie-set $L_{1}$, according to the states of $x_{1}$ and $x_{2}$, are given in Table 2.4. The CPT for $L_{2}$ is given in Table 2.5. The combination of the two minimal tie-sets is enough to compute the probability distribution of $y$ (Table 2.7), thanks to its CPT table (see Table 2.6).The logical behavior of the system failures induces deterministic CPT, which are equivalent to Boolean gates. The BN are modeling Boolean equations by probabilities equal to 0 or 1 .
统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|BN model from a top-down approach
In the case of large systems, the enumeration of all functioning or dysfunctioning scenarios is cumbersome. To solve this problem, the FT modeling is based on a descending approach. Starting from a top event that characterizes the undesired event, the analysis goes down the tree by the definition of intermediate events identified as direct causes of upper events, until elementary events are obtained. For example, Figure $2.5$ shows the FT of the flow distribution system, which is obviously quite simple.
If a FT is available, it is very simple to translate it into a BN by simple mapping. As shown earlier for tie-sets and cut-sets, deterministic CPT can map Boolean relations between variables with logical operators: AND and OR. Figure $2.6$ shows the mapping result of the FT shown in Figure $2.5$ into a BN. For each event in the FT, a variable is defined in the $\mathrm{BN}$.
For instance, the AND gate in Figure $2.5$ is such that $E_{2}=x_{2} \wedge x_{3}$, i.e. $E_{2}$ occurs if $x_{2}=1$ and $x_{3}=1$. The CPT of the $\mathrm{BN}$ is given in Table 2.8. The OR gate in Figure $2.5$ is such that $y=x_{1} \vee E_{2}$, i.e. $y=1$ for the failure of the system if $x_{1}=1$ or $E_{2}=1$. The CPT of the BN is defined in Table 2.9.
统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Standard Boolean gates CPT
All Boolean gates can be modeled by a BN (OR, AND, Koon, Exclusive OR, etc.). It is sufficient to directly map the Boolean equation inside the CPT [SIM 07, SIM 08].
An $n$-component system that functions (or works) if and only if at least $k$ of the $n$ components work is called a k-out-of-n:G system. An $n$ component system that fails if and only if at least $k$ of the $n$ components fail is called a k-out-of-n:F system. Both parallel and series systems are special cases of the k-out-of-n system. A series system is equivalent to a 1-out-of-n:F system and an n-out-of-n:G system, while a parallel system is equivalent to an $n$-out-of-n:F system and a 1-out-of-n:G system.
Let us define the CPT of a 2-out-of- 3 : G system, with the components $x_{1}, x_{2}$ and $x_{3}$. The BN structure is shown in Figure $2.8$. The system is functioning, $y=0$, if at least two components are available; $x_{i}=0$ and $x_{j}=0$, with $i \neq j$ and $i, j \in{1,2,3}$. The CPT of $y$ is defined in Table 2.12.
Unlike FT or RBD, BN can integrate topological constraints, for instance the linear or circular consecutive-koon system. Such systems cannot be modeled by FT or RBD because of the independencehypothesis of events. The BN solves this problem by computing conditional independence and gives a systematic modeling method [WEB 10, WEB 11].
Consecutive-koon systems have attracted considerable attention since they were first proposed by Kontoleon in 1980 [KON 80]. A consecutive-koon system can be classified according to the linear or circular arrangement of its components and the functioning or malfunctioning principle. Thus, four types of $k$-out-of-n can be enumerated: linear consecutive-koon:F, linear consecutive-koon:G, circular consecutive-koon:F and circular consecutive-koon:G. A consecutive-koon:F system consists of a set of $n$ ordered components that compose a chain such that the system fails if at least $k$ consecutive components fail. A consecutive koon:G system is a chain of $n$ components such that the system works if at least $k$ consecutive components work. An illustration of these specific structures can be found in telecommunication systems with $n$ relay stations that can be modeled as a linear consecutive-koon:G system if the signal transmitted from each station is strong enough to reach the next $k$ stations. An oil pipeline system for transporting oil from point to point with $n$ spaced pump stations is another example of a linearconsecutive-koon system. A closed recurring water supply system with $n$ water pumps in a thermo-electric plant is a good example of a circular system. The system ensures its mission if each pump is powerful enough to pump water and steam to the next $k$ consecutive pumps [YAM 03].
统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|BN model from tie-sets
\begin{对齐} &L_{1}=\left{x_{1}, x_{2}\right} \ &L_{2}=\left{x_{1}, x_{3}\right} \ &L_{3 }=\left{x_{1}, x_{2}, x_{3}\right} \end{aligned}\begin{对齐} &L_{1}=\left{x_{1}, x_{2}\right} \ &L_{2}=\left{x_{1}, x_{3}\right} \ &L_{3 }=\left{x_{1}, x_{2}, x_{3}\right} \end{aligned}
广义线性模型代考
statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
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2023-03-22 13:47:55
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https://thecuriousastronomer.wordpress.com/2014/11/11/newtons-equations-of-motion-revisited/
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Feeds:
Posts
## Newton’s equations of motion – revisited
Last week, I showed how one could derive 3 of Newton’s equations of motion. As a colleague of mine pointed out to me on FaceBook, the 3rd equation I showed can, in fact, be derived using algebra from the 1st and 2nd. Also, if you go to the Wikipedia page on the equations of motion, you will find 5 listed, not just the 3 I showed. It turns out that the only two “fundamental” ones are the 1st and 2nd that I showed, the other three (including my 3rd, shown as the 4th in the image below) can be derived from the 1st and 2nd ones.
The 5 equations of motion as shown on the Wikipedia page
This is often the case in mathematics, there is more than one way to do something. So, although the method I showed to derive the 3rd equation is perfectly correct, and it also shows how we can rewrite $dv/dt \text{ as } (dv/ds \cdot ds/dt)$, which is a useful technique, I will today show how it can also be derived by algebraically combining equations (1) and (2).
Remember, equations (1) and (2) were
$v = u + at \text{ (Equ. 1)}$
$s = ut + \frac{1}{2} at^{2} \text{ (Equ. 2)}$
The first step is to square equation (1), which gives us
$v^{2} = (u + at)^{2} = u^{2} + 2uat + a^{2}t^{2} \text{ (Equ. 3a)}$
Next, we multiply equation (2) by 2 to get rid of the fraction
$2s = 2ut + at^{2} \text{ (Equ. 3b)}$
We next multiply each term in equation (3b) by $a$ to give
$2as = 2uat + a^{2}t^{2} \text{ (Equ. 3c)}$
Comparing equations (3a) and (3c) we can see that we can substitute the 2nd and 3rd terms of equation (3a) by $2as$ and so we have
$v^{2} = u^{2} + 2as \text{ (Equ. 3)}$
which is our equation (3) from the previous blog. As they say in mathematics, Quod erat domonstradum (QED) 🙂
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2017-05-29 05:43:37
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https://fpish.net/topic/None/91239
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In your current approach, the retrieval logic is bound to two separate, independent UI controls that "don't work together", and a change in either will fire the RPC call, regardless whether that change is user-initiated (selecting a dropdown) or calculated (programmatically).
Since an invalid state can only be reached when perPage is set, you could introduce an intermediate step: bind the dropdown's value to a different Var, perform the constrain logic in the control's change event as you are doing now to compute a potentially different current page number, and set that to currentPage in your model. This way, you "surround" your UI control with the extra check, instead of letting it trigger an unnecessary state change.
By on 11/16/2021 5:05 AM ()
This works perfectly and is a simple solution, thanks!
By on 11/17/2021 2:49 AM ()
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2022-07-05 07:49:01
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https://www.cut-the-knot.org/Curriculum/Geometry/GriffithsSolution.shtml
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# 9-point Circle as a Locus of ConcurrencyWhat Is This About? A Mathematical Droodle
The Simson lines of two diametrically opposite points on a circumcircle of a triangle intersect on the 9-point circle of that triangle. A generalization of this fact is known as Griffiths' theorem:
Let the line joining two points $P$ and $Q$ pass through the circumcenter of $\Delta A_{1}A_{2}A_{2}.$ Then the pedal circles of points $P$ and $Q$ meet on the $9$-point circle of $\Delta A_{1}A_{2}A_{2},$ i.e. the three circles are concurrent. This is known as Griffiths' theorem. In fact the point of concurrency only depends on the line the two points lie on. In other words, the pedal circles of the points on a line through the circumcenter of a triangle all concur at the same point on its $9$-point circle.
The point is known as Griffiths' point.
(It is worthwhile to note that the $9$-point circle is the pedal circle of the circumcenter as well as that of the orthocenter.)
Griffiths' theorem admits a proof based on complex numbers.
Let the line joining two points $P$ and $Q$ pass through the circumcenter of $\Delta A_{1}A_{2}A_{2}.$ Then the pedal circles of points $P$ and $Q$ meet on the $9$-point circle of $\Delta A_{1}A_{2}A_{2},$ i.e. the three circles are concurrent. This is known as Griffiths' theorem. In fact the point of concurrency only depends on the line the two points lie on. In other words, the pedal circles of the points on a line through the circumcenter of a triangle all concur at the same point on its $9$-point circle.
### Proof
Assume the circumcenter of $\Delta A_{1}A_{2}A_{3}$ is at the origin and the circumradius equals $1.$ Lower case letters denote complex numbers associated with the points denoted by the corresponding upper case so that, for example $a_{i}$ corresponds to $A_{i},$ $i = 1, 2, 3.$ We have
$|a_{1}| = |a_{2}| = |a_{3}| = 1.$
Let $P$ lie on a straight line $l$ through the origin and $C_{1}.$ We choose $l$ to coincide with the real axis which in particular implies $p = p\space '.$ Let $C_{2},$ $C_{3}$ be the feet of perpendiculars from $P$ to the side lines $A_{2}A_{3},$ $A_{1}A_{3},$ and $A_{1}A_{2}.$
Since $C_{1}$ is collinear with $A_{2}$ and $A_{3}$
$(c_{1} - a_{2}) / (a_{3} - a_{2}) = (c\space ^{'}_{1} - a\space ^{'}_{2}) / (a\space ^{'}_{3} - a\space ^{'}_{2})$
From $|a_{2}| = 1,$ $a\space ^{'}_{2} = 1/a_{2}$ and $a\space ^{'}_{3} = 1/a_{3}.$ So we have
\begin{align} (c_{1} - a_{2}) / (a_{3} - a_{2})&= (c\space ^{'}_{1} - a\space ^{'}_{2}) / (a\space ^{'}_{3} - a\space ^{'}_{2})\\ &= (c\space ^{'}_{1} - 1/a_{2}) / (1/a_{3} - 1/a_{2})\\ &= (c\space ^{'}_{1}a_{2} - 1) a_{3} / (a_{2} - a_{3}) \end{align}
from which
$c_{1} - a_{2} = a_{3} - a_{2}a_{3}c\space ^{'}_{1}$
or
$a_{2} + a_{3} = c_{1} + a_{2}a_{3}c\space ^{'}_{1}$
If the indices $i, j, k$ are distinct and come from the set $\{1, 2, 3\}$ then, more generally,
(1)
$c\space ^{'}_{k} a_{i} a_{j} = a_{i} + a_{j} - c_{k}.$
On the other hand, $PC_{k}$ is perpendicular to $A_{i}A_{j},$ so that
$(p - c_{k})/(a_{i} - a_{j}) + (p\space ' - c\space ^{'}_{k})/(a\space ^{'}_{i} - a\space ^{'}_{j}) = 0,$
which, using $|a_{i}| = |a_{j}| = 1,$ we modify as follows:
$(p - c_{k})/(a_{i} - a_{j}) + (p\space ' - c\space ^{'}_{k}) a_{i} a_{j} /(a_{j} - a_{i}) = 0,$
or, with $P = p\space ',$
$p - c_{k} - (p - c\space ^{'}_{k}) a_{i} a_{j} = 0.$
Finally, we obtain
(2)
$c\space ^{'}_{k} a_{i} a_{j} = p a_{i} a_{j} - p + c_{k}.$
A comparison of (2) and (1) gives
(3)
$c_{k} = (a_{i} + a_{j} + p - p a_{i} a_{j}) / 2.$
Introduce point
(4)
$w = (a_{1} + a_{2} + a_{3} - a_{1} a_{2} a_{3}) / 2.$
We shall show that four points $C_{1},$ $C_{2},$ $C_{3},$ $W$ are concyclic, i.e. $W$ lies on the circumcircle of $\Delta C_{1}C_{2}C_{3}.$ Note $W$ is independent of $P.$ In particular, $W$ is the same when $P$ is at the origin, i.e. when $P$ coincides with the circumcenter in which case the circumcircle of $\Delta C_{1}C_{2}C_{3}$ is the nine point circle of $\Delta A_{1}A_{2}A_{3}.$ Thus we shall have proved Griffiths' theorem and showed that $W$ is exactly the Griffiths point.
For the concyclicity of $C_{1},$ $C_{2},$ $C_{3},$ $W$ we have to check that
$(w - c_{2})/(c_{1} - c_{2}) : (w - c_{3})/(c_{1} - c_{3}) = (w' - c\space ^{'}_{2})/(c\space ^{'}_{1} - c\space ^{'}_{2}) : (w' - c\space ^{'}_{3})/(c\space ^{'}_{1} - c\space ^{'}_{3}).$
which we shall prove in another form
(5)
$(w - c_{2})/(c_{1} - c_{2}) : (w' - c\space ^{'}_{2})/(c\space ^{'}_{1} - c\space ^{'}_{2}) = (w - c_{3})/(c_{1} - c_{3}) : (w' - c\space ^{'}_{3})/(c\space ^{'}_{1} - c\space ^{'}_{3}).$
First of all, from (3) and (4),
\begin{align} w - c_{2}&= (a_{1} + a_{2} + a_{3} - a_{1} a_{2} a_{3}) / 2 - (a_{1} + a_{3} + P - P a_{1} a_{3}) / 2\\ &= (p - a_{2})(a_{1} a_{3} - 1) / 2. \end{align}
From here, using $|a_{i}| = 1,$
\begin{align} w' - c\space ^{'}_{2}&= (p - a\space ^{'}_{2})(a\space ^{'}_{1} a\space ^{'}_{3} - 1) / 2\\ &= (p - 1/a_{2})(1/a_{1} 1/a_{3} - 1) / 2\\ &= -(p a_{2} - 1)(a_{1} a_{3} - 1) / 2a, \end{align}
where $a = a_{1}a_{2}a_{3}.$ Also
$c_{1} - c_{2} = (a_{1} - a_{2})(p a_{3} - 1) / 2$
and
$c\space ^{'}_{1} - c\space ^{'}_{2} = -(a_{1} - a_{2})(p - a_{3}) / 2a.$
Similar expressions are obtained for the differences in (5) that include $c_{3}.$ The left hand side in (5) becomes
$(p - a_{2})(a_{1} a_{3} - 1)(a_{1} - a_{2})(p - a_{3}) / (a_{1} - a_{2})(p a_{3} - 1)(p a_{2} - 1)(a_{1} a_{3} - 1)$
or, after reduction,
$(p - a_{2})(p - a_{3}) / (p a_{3} - 1)(p a_{2} - 1),$
which is symmetric in $a_{2}$ and $a_{3}$ showing that the right hand side in (5) equals the same expression, thus proving (5) and, with it, Griffiths' theorem.
(Line $l$ passing through the circumcenter intersects the circumcircle at two points. For each of these, the three projections $C_{1},$ $C_{2},$ $C_{3}$ are collinear and lie on the simsons of $\Delta A_{1}A_{2}A_{3}.$ It follows that the two simsons meet at the Griffiths point associated with $l$.)
Assume now there are four concyclic points $A_{1},$ $A_{2},$ $A_{3},$ and $A_{4}.$ Taken by three, they form four triangles $T_{m} = \Delta A_{i}A_{j}A_{k},$ where $i, j, k, m$ are distinct elements from $\{1, 2, 3, 4\}.$ To each $T_{m}$ and the given line $l$ there correspond Griffiths point $W_{m}.$ Using the same technique as above we can show (after J. Tabov) that the four Griffiths points $W_{1},$ $W_{2},$ $W_{3},$ $W_{4}$ are collinear. It is sufficient to show that
$(w_{1} - w_{2}) / (w\space ^{'}_{1} - w\space ^{'}_{2}) = (w_{1} - w_{3}) / (w\space ^{'}_{1} - w\space ^{'}_{3}) = (w_{1} - w_{4}) / (w\space ^{'}_{1} - w\space ^{'}_{4}).$
This and an additional feature of the configuration is considered on a separate page.
A purely geometric proof can be found in [Advanced Euclidean Geometry, pp. 245-246].
### References
1. R. A. Johnson, Advanced Euclidean Geometry, Dover, 2007
2. J. Tabov, Four Collinear Griffiths Points, Mathematics Magazine, v. 68, n 1, February 1995, pp. 61-64
### Simson Line - the simson
• Simson Line: Introduction
• Simson Line
• Three Concurrent Circles
• Miquel's Point
• Circumcircle of Three Parabola Tangents
• Angle Bisector in Parallelogram
• Simsons and 9-Point Circles in Cyclic Quadrilateral
• Reflections of a Point on the Circumcircle
• Simsons of Diametrically Opposite Points
• Simson Line From Isogonal Perspective
• Pentagon in a Semicircle
• Simson Line in Disguise
• Two Simsons in a Triangle
• Carnot's Theorem
• A Generalization of Simson Line
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2020-07-05 04:39:39
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https://pypi.org/project/ansinv/2.0.1/
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Generate Ansible Inventory
## A Simple Ansible Inventory Generator
### Overview
This simple library makes it easier to write glue code between infrastructure bringup/deployment and software provisioning stages of a one-click deployment.
### Installation
Simply say:
pip install ansinv
Make sure you are using correct pip (pip/pip3) depending on your Python environment.
### Use Case
Picture the following Docker Swarm cluster that needs to be brought up and provisioned all in one go. The infrastructure has one manager, two workers, and a cluster heath monitoring node.
### Automation Steps
Let’s say you are using Terraform tool to bringup cluster on AWS and using Ansible for software provisioning. Following can be the basic steps:
1. Invoke Terraform with desired configuration. After finishing, terraform provides with a terraform state file.
2. Use information (like host IP addresses) from this terraform.tfstate file to generate an inventory file to be given to Ansible in the next step.
3. Invoke Ansible plays with the inventory file from above step to configure software on the infrastrucure.
4. Cluster is ready to use :)
### Generating the Inventory file
How are we going to generate inventory file in step #2 above? Options:
1. Creating manually by looking at the terraform state file and hand-editing the inventory file. It works! but not inline with one-click deployment philosophy.
2. Writing code to parse the terraform state file for required information and generating inventory file on the fly. This is where ansinv can help.
While writing code to parse terraform state file and generating the the inventory file in INI format, we have to write quite a bit of boiler-plate code which doesn’t change from one deployment to another. For example: generating inventory file’s INI template, dealing with hosts, groups and their inter-relation, adding hostvars and groupvars at proper place in the inventory file etc. Here ansinv comes to the rescue and wraps all that common functionality into a nice and easy to use API.
### A Code Example
If you browse the example directory, you will find example/generate_inventory.py python file. This file reads example/terraform.tfstate.example and generates inventory file similar to example/inventory.example. FYI, the ordering of sections in the inventory file can be different every time you generate it. But this doesn’t matter.
## Project details
Uploaded source
Uploaded py2 py3
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2023-03-31 10:34:38
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http://en.coininvest.com/post/positive-us-data-good-sign-for-gold-buyers/
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# Positive US Data Good Sign For Gold Buyers
Economic data in the US showed more good signs than the economy across the pond is growing, despite other major markets ending the month showing signs of a slow-down. At the time of writing (1400EST) gold was $4.50 for the day and could sink lower before the end of the day’s trading. Manufacturing growth in the United States was up another 0.5% this month to 53.7 showing positive signs of acceleration after months of slow growth. With more data expected to be published this week, the early signs are that traders will continue investing in equities and pull out of precious metals. ## Gold buyers taking advantage of gold prices Stock market traders have been on a selling frenzy for the past two weeks, cashing in on bargain gains following a six week rally of precious metals brought on by the Ukraine crisis and fears of an economic slow-down in China. With the US economy showing positive signs of recovery however, market sentiment returned to US-backed equities and the world’s largest economy appears to be back on track. However, the majority of the world’s leading economies showed signs of weakness this month. The Asian market has been short on demands for physical gold all year, and despite the mad-rush in Japan yesterday buying has been lacklustre. China manufacturing behemoth also fell to an 8-month low although overall the economy grew 0.1%. ## Lingering concerns may turn gold In the Eurozone, British manufacturing also eased to an unexpected 8-month low whilst inflation on the mainland will surely cause the ECB policy makers to take action monetary action this quarter. The lingering doubts may just be the catalyst that keeps gold on an even keel although the majority of analysts, and it would appear traders, expect prices to continue falling. Some speculators even think gold bullion could bottom out around the$1000 mark this year.
The indicators suggest this could be the case. US stocks hit a new all-time high in March and with market sentiment leaning towards cash investments, a strengthening dollar and the Fed ending its bond buying stimulus, there is less interest or need to back precious metals as a hedge.
## Investing in gold
The indicators also point towards over evaluation, which is why some of the major markets are showing signs of slow-down on the balance sheets without there actually being a slow-down in factories. Once the stimulus ends equities will correct themselves and there could be a knee-jerk reaction – probably giving gold a lift.
Of course there will be peaks and troughs for gold and silver before consumers will want to sell. A credit crash, which seems inevitable given the amount of debt nations are in, will send precious metal prices rocketing and could even eclipse the all-time high of \$1917.
To protect your financial future it is a shrewd move to invest in gold whilst prices are low. Precious metals always make profits when the global economy is in strife – and that could happen sooner than you think!
Category:
Tag:
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2018-03-20 02:02:15
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https://math.stackexchange.com/questions/652029/prove-limsup-a-n-ge-1
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# Prove $\limsup\{a_n\}\ge 1$
Let $\{a_n\}$, a bounded sequence such that $a_n>0 \forall n>0$. Let:
$$\mathop {\lim }\limits_{n \to \infty } {a_{n + 1}}{a_n} = 1$$
Prove that $\displaystyle\limsup_{n\to\infty} a_n\ge 1$.
My attempt:
It is given that $\mathop {\lim }\limits_{n \to \infty } {a_{n + 1}}{a_n} = 1$.
Therefore, if the sequence is $a_n=1 \forall n>0$ then we've finished.
Otherwise, one term must be greater then one, while the other smaller then one.
Hence, There are two sub-sequences; One bounded from above and converges to $1$, and the other bounded bellow and converges to $1$ (because we know $a_n$ is bounded).
In any case, $\displaystyle\limsup_{n\to\infty}a_n=1$ as requested.
I think my proof is a bit handwaving. How can I "polish" it (Making it more rigorous)? I think I need to explain why there must be two sequences such that. Can you help me with that? Thanks!
• Your "one term must be greater then one, while the other smaller then one" is a non-sequitur. Consider for example $a_n=1+\frac1n$. – Hagen von Eitzen Jan 26 '14 at 15:22
• If you are trying to prove that $a_n=1$ for some $n>N$, I don't think that would help. – jdoicj Jan 26 '14 at 15:22
Hint: Suppose $\lim \sup a_n< 1$, then $a_n<1$ for very large $n$.
• I got it. and then, there's no way that $\mathop {\lim }\limits_{n \to \infty } {a_{n + 1}}{a_n} = 1$. Right? A contradiction. – SuperStamp Jan 26 '14 at 15:27
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2020-01-26 00:24:35
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https://math.paperswithcode.com/paper/monotonicity-of-entropy-for-real-quadratic
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# Monotonicity of entropy for real quadratic rational maps
21 Sep 2020 Filom Khashayar
The monotonicity of entropy is investigated for real quadratic rational maps on the real circle $\mathbb{R}\cup\{\infty\}$ based on the natural partition of the corresponding moduli space $\mathcal{M}_2(\mathbb{R})$ into its monotonic, covering, unimodal and bimodal regions. Utilizing the theory of polynomial-like mappings, we prove that the level sets of the real entropy function $h_\mathbb{R}$ are connected in the $(-+-)$-bimodal region and a portion of the unimodal region in $\mathcal{M}_2(\mathbb{R})$... (read more)
PDF Abstract
# Code Add Remove Mark official
No code implementations yet. Submit your code now
# Categories
• DYNAMICAL SYSTEMS
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2021-04-16 17:17:25
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https://stacks.math.columbia.edu/tag/04BN
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Definition 18.33.3. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. The module of differentials of the ring map $\varphi$ is the object representing the functor $\mathcal{F} \mapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F})$ which exists by Lemma 18.33.2. It is denoted $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$, and the universal $\varphi$-derivation is denoted $\text{d} : \mathcal{O}_2 \to \Omega _{\mathcal{O}_2/\mathcal{O}_1}$.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2023-03-25 20:43:06
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https://math.stackexchange.com/questions/921641/closed-form-for-large-int-01-frac-ln1-x-ln1x-ln12x12xdx/934091
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# Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$
Here is another integral I'm trying to evaluate: $$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx.\tag1$$ A numeric approximation is: $$I\approx-0.19902842515384155925817158058508204141843184171999583129...\tag2$$ (click here to see more digits).
Unfortunately, so far I have made no progress in finding a closed form for it. Could you please suggest any ideas how to do that?
• Apparently, the integrals involving logarithm function have been trending topics here on MSE. – Tunk-Fey Sep 6 '14 at 18:38
• Somewhat related integrals people.sc.fsu.edu/~dduke/devoto-duke-84.pdf – ClassicStyle Sep 7 '14 at 1:17
• There is an identity I've found helpful in integrals involving the logarithm product $\ln{(1-z)}\ln{(1+z)}$. For $|z|<1$, this product equals: $\frac12\ln^2{(1+z)}+\frac12\ln^2{(1-z)}+2Li_2{\left(\frac{z}{z-1}\right)}+2Li_2{\left(\frac{z}{z-1}\right)}-Li_2{\left(\frac{z^2}{z^2-1}\right)}$. – David H Sep 7 '14 at 1:40
• Vladimir, IMHO, I doubt the given integral has a closed form. Well, I could certainly be wrong and I really hope I am wrong because I want to see its closed-form and how to prove it. I am able to derive the integral into three following forms and it turns out none of them have closed forms, at least it can be proven by using mathematics software packages. Using differentiation under integral sign method also gives me nothing. It looks like the term $1+2x$ makes thing so difficult since we cannot express it in form of series because of $0<x<1$. – Tunk-Fey Sep 14 '14 at 17:55
• When I looked at integral representations of multiple polylogarithms (as in Julian Rosen's answer), I found it is enough to evaluate the following integral, and perhaps there is a rational function identity that might simplify it (it looks symmetric): $$\int_{1,1,1,1}^{\{\infty\}^4} \frac{8(x^2 y^2 z^2w -1)}{x (x+2) y (x y+2) z \left(x^2 y^2 z^2-1\right) w \left(w^2 x^2 y^2 z^2-1\right)}$$ – Kirill Sep 18 '14 at 7:20
$\def\Li{\,\mathrm{Li}}$I followed the technique suggested by Julian Rosen in his answer, and decomposed your integral (and your other integral) as a linear combination of multiple polylogarithms: $$\textstyle -\frac12\log2\log3 \Li_2({\frac23}) + \frac12\log3\Li_{2,1}({\frac23,\frac34}) + \frac12\log2\Li_{2,1}({\frac23,1}) \\\textstyle - \frac12\Li_{2,1,1}({\frac23,\frac34,\frac43}) - \frac12\Li_{2,1,1}({\frac23,1,\frac34})$$ There is a paper by Borwein, Bradley, Broadhurst, Lisonek, that explains what few families of identities apply to multiple polylogarithms, and it mentions a conjecture that those mentioned there are all the identities that apply at all.
Multiple polylogarithms are generalizations of zeta functions (and polylogarithms, and multiple zeta functions) in that the important thing is not the depth $k$ of $\mathrm{Li}_{s_1,\ldots,s_k}(z_1,\ldots,z_k)$, but the weight $\sum_{i=1}^{k}s_i$. My Mathematica was able, with some amount of hand-holding, to compute directly the (rational) integral representations involved in multiple polylogarithms of weight $1$, $2$, and $3$, but couldn't handle weight $4$. Your integral has three logs, so it's weight 4.
It seems multiple polylogarithms of weight 4 with small rational arguments are still algebraically related to ordinary polylogarithms. By analogy with multiple zeta values, I suspect the same won't necessarily be true of higher weights at least in general.
I made some guesses based on exact weight-3 values about what terms weight-4 values might involve, and used an integer relation algorithm to try and find an expression for your integral. I found this one, which matches the integral to $3000$ digits, and when I looked for an integer relation I used a tolerance of only $10^{-200}$.
Here you go: $$\textstyle\def\Li{\mathrm{Li}} -\frac{1}{2} \Li_2(\frac{1}{3}) \zeta (2)-\frac{1}{4}\Li_4(\frac{3}{4})-\frac{3}{2} \Li_4(\frac{2}{3})+\frac{1}{6}\Li_4(\frac{1}{2})+\Li_4(\frac{1}{3}) -\frac{1}{16}\Li_4(\frac{1}{4})-\frac{1}{2}\Li_2(\frac{1}{3}){}^2+2 \Li_3(\frac{2}{3}) \log3+3 \Li_3(\frac{1}{3}) \log3-\Li_3(\frac{1}{3}) \log2 +\Li_2(\frac{1}{3}) \log2 \log3-\frac{13}{3} \zeta (3) \log3+\frac{19}{12} \zeta (3) \log2+\frac{7}{6} \zeta (4)+\frac{9}{4} \zeta (2) \log^23 +\frac{5}{6} \zeta (2) \log^22-3 \zeta (2) \log2 \log3-\frac{35}{48} \log^43-\frac{29}{144} \log^42+\frac{7}{4} \log2 \log^33 +\frac{1}{3} \log^32 \log3-\frac{9}{8} \log^22 \log^23$$
Here is the Mathematica expression verbatim, to save people some typing:
(7*Pi^4)/540 + (5*Pi^2*Log[2]^2)/36 - (29*Log[2]^4)/144 - (Pi^2*Log[2]*Log[3])/2 + (Log[2]^3*Log[3])/3 + (3*Pi^2*Log[3]^2)/8 - (9*Log[2]^2*Log[3]^2)/8 + (7*Log[2]*Log[3]^3)/4 - (35*Log[3]^4)/48 - (Pi^2*PolyLog[2, 1/3])/12 + Log[2]*Log[3]*PolyLog[2, 1/3] - PolyLog[2, 1/3]^2/2 - Log[2]*PolyLog[3, 1/3] + 3*Log[3]*PolyLog[3, 1/3] + 2*Log[3]*PolyLog[3, 2/3] - PolyLog[4, 1/4]/16 + PolyLog[4, 1/3] + PolyLog[4, 1/2]/6 - (3*PolyLog[4, 2/3])/2 - PolyLog[4, 3/4]/4 + (19*Log[2]*Zeta[3])/12 - (13*Log[3]*Zeta[3])/3
Edit. Here are the expressions for individual multiple polylogarithms above. The first two are rigorous, being the output of Integrate applied to integral representations:
MultiPolyLog[{2, 1}, {2/3, 3/4}] := -(1/4) \[Pi]^2 Log[2] - (8 Log[2]^3)/3 + 1/2 Log[2]^2 Log[3] + Log[2] Log[3]^2 - 2 Log[2] PolyLog[2, 1/4] + 3 Log[2] PolyLog[2, 2/3] - PolyLog[3, 1/4] - PolyLog[3, 1/3] + PolyLog[3, 2/3] + Zeta[3]/8
MultiPolyLog[{2, 1}, {2/3, 1}] := 1/6 (\[Pi]^2 Log[3/2] - 2 Log[3]^3 + Log[2]^2 Log[27/2] + 6 Log[3] PolyLog[2, -(1/2)]) + PolyLog[3, -(1/2)] + PolyLog[3, 2/3]
These two, of weight 4, come from an integer relation algorithm:
{MultiPolyLog[{2, 1, 1}, {2/3, 1, 3/4}] -> (11*Pi^4)/240 - (11*Pi^2*Log[2]^2)/240 - Log[2]^4/60 - (Pi^2*Log[2]*Log[3])/10 - (Log[2]^3*Log[3])/48 + (41*Pi^2*Log[3]^2)/480 - (7*Log[2]^2*Log[3]^2)/160 + (Log[2]*Log[3]^3)/48 - (9*Log[3]^4)/160 + (13*Pi^2*Log[2]*Log[4])/240 - (Log[2]^3*Log[4])/32 - (29*Pi^2*Log[3]*Log[4])/480 - (19*Log[2]^2*Log[3]*Log[4])/480 + (7*Log[2]*Log[3]^2*Log[4])/120 + (Log[3]^3*Log[4])/24 - (3*Pi^2*Log[4]^2)/80 - (Log[2]^2*Log[4]^2)/16 + (Log[2]*Log[3]*Log[4]^2)/16 - (Log[3]^2*Log[4]^2)/32 + (7*Log[2]*Log[4]^3)/480 - (7*Log[3]*Log[4]^3)/480 + Log[4]^4/32 - (13*Pi^2*PolyLog[2, 1/4])/480 + (19*Log[2]^2*PolyLog[2, 1/4])/240 - (Log[2]*Log[3]*PolyLog[2, 1/4])/48 - (41*Log[3]^2*PolyLog[2, 1/4])/480 + (Log[2]*Log[4]*PolyLog[2, 1/4])/80 + (17*Log[3]*Log[4]*PolyLog[2, 1/4])/160 + (Log[4]^2*PolyLog[2, 1/4])/40 + (11*PolyLog[2, 1/4]^2)/96 - (29*Pi^2*PolyLog[2, 1/3])/480 - (Log[2]^2*PolyLog[2, 1/3])/20 + (19*Log[2]*Log[3]*PolyLog[2, 1/3])/480 + (11*Log[3]^2*PolyLog[2, 1/3])/160 + (11*Log[2]*Log[4]*PolyLog[2, 1/3])/240 - (Log[3]*Log[4]*PolyLog[2, 1/3])/15 - (5*Log[4]^2*PolyLog[2, 1/3])/96 - (7*PolyLog[2, 1/4]*PolyLog[2, 1/3])/160 + PolyLog[2, 1/3]^2/120 - (7*Pi^2*PolyLog[2, 2/3])/96 + (Log[2]^2*PolyLog[2, 2/3])/60 + (11*Log[2]*Log[3]*PolyLog[2, 2/3])/480 + (Log[3]^2*PolyLog[2, 2/3])/48 + (17*Log[2]*Log[4]*PolyLog[2, 2/3])/480 + (11*Log[3]*Log[4]*PolyLog[2, 2/3])/240 + (11*Log[4]^2*PolyLog[2, 2/3])/160 + (49*PolyLog[2, 1/4]*PolyLog[2, 2/3])/480 - (7*PolyLog[2, 1/3]*PolyLog[2, 2/3])/240 + PolyLog[2, 2/3]^2/12 - (11*Pi^2*PolyLog[2, 3/4])/120 + (Log[2]^2*PolyLog[2, 3/4])/30 - (Log[2]*Log[3]*PolyLog[2, 3/4])/240 + (Log[3]^2*PolyLog[2, 3/4])/6 + (Log[2]*Log[4]*PolyLog[2, 3/4])/15 - (5*Log[3]*Log[4]*PolyLog[2, 3/4])/32 - (Log[4]^2*PolyLog[2, 3/4])/160 - (89*PolyLog[2, 1/4]*PolyLog[2, 3/4])/480 - (49*PolyLog[2, 1/3]*PolyLog[2, 3/4])/480 - (17*PolyLog[2, 2/3]*PolyLog[2, 3/4])/80 + PolyLog[2, 3/4]^2/24 - (37*Log[2]*PolyLog[3, 1/4])/240 - (Log[3]*PolyLog[3, 1/4])/40 - (77*Log[4]*PolyLog[3, 1/4])/480 + (3*Log[2]*PolyLog[3, 1/3])/80 - (Log[3]*PolyLog[3, 1/3])/20 - (11*Log[4]*PolyLog[3, 1/3])/160 - (Log[2]*PolyLog[3, 2/3])/240 - (71*Log[4]*PolyLog[3, 2/3])/480 - (Log[2]*PolyLog[3, 3/4])/48 - (Log[3]*PolyLog[3, 3/4])/40 - (91*Log[4]*PolyLog[3, 3/4])/480 - (21*PolyLog[4, 1/4])/16 - (7*PolyLog[4, 1/3])/4 + (5*PolyLog[4, 1/2])/2 - PolyLog[4, 2/3]/2 - (11*PolyLog[4, 3/4])/8 - (Log[2]*Zeta[3])/15 + (19*Log[3]*Zeta[3])/240 - (13*Log[4]*Zeta[3])/96
,MultiPolyLog[{2, 1, 1}, {2/3, 3/4, 4/3}] -> (-139*Pi^4)/1440 + (149*Pi^2*Log[2]^2)/1440 + Log[2]^4/30 + (347*Pi^2*Log[2]*Log[3])/1440 + (19*Log[2]^3*Log[3])/480 - (313*Pi^2*Log[3]^2)/1440 + (13*Log[2]^2*Log[3]^2)/120 - (7*Log[2]*Log[3]^3)/90 + (8*Log[3]^4)/45 - (19*Pi^2*Log[2]*Log[4])/180 + (97*Log[2]^3*Log[4])/1440 + (241*Pi^2*Log[3]*Log[4])/1440 + (23*Log[2]^2*Log[3]*Log[4])/288 - (47*Log[2]*Log[3]^2*Log[4])/480 - (37*Log[3]^3*Log[4])/240 + (37*Pi^2*Log[4]^2)/360 + (13*Log[2]^2*Log[4]^2)/96 - (5*Log[2]*Log[3]*Log[4]^2)/32 + (17*Log[3]^2*Log[4]^2)/144 - (31*Log[2]*Log[4]^3)/720 + (Log[3]*Log[4]^3)/360 - (7*Log[4]^4)/80 + (29*Pi^2*PolyLog[2, 1/4])/720 - (77*Log[2]^2*PolyLog[2, 1/4])/480 + (Log[2]*Log[3]*PolyLog[2, 1/4])/8 + (35*Log[3]^2*PolyLog[2, 1/4])/288 - (Log[2]*Log[4]*PolyLog[2, 1/4])/180 - (23*Log[3]*Log[4]*PolyLog[2, 1/4])/360 - (17*Log[4]^2*PolyLog[2, 1/4])/1440 + (11*PolyLog[2, 1/4]^2)/288 + (13*Pi^2*PolyLog[2, 1/3])/80 + (133*Log[2]^2*PolyLog[2, 1/3])/1440 - (133*Log[2]*Log[3]*PolyLog[2, 1/3])/1440 - (47*Log[3]^2*PolyLog[2, 1/3])/240 - (31*Log[2]*Log[4]*PolyLog[2, 1/3])/240 + (31*Log[3]*Log[4]*PolyLog[2, 1/3])/240 + (41*Log[4]^2*PolyLog[2, 1/3])/720 + (5*PolyLog[2, 1/4]*PolyLog[2, 1/3])/96 + (23*PolyLog[2, 1/3]^2)/720 + (247*Pi^2*PolyLog[2, 2/3])/1440 - (19*Log[2]^2*PolyLog[2, 2/3])/480 + (Log[2]*Log[3]*PolyLog[2, 2/3])/288 - (23*Log[3]^2*PolyLog[2, 2/3])/240 - (113*Log[2]*Log[4]*PolyLog[2, 2/3])/1440 + (Log[3]*Log[4]*PolyLog[2, 2/3])/144 - (113*Log[4]^2*PolyLog[2, 2/3])/720 - (59*PolyLog[2, 1/4]*PolyLog[2, 2/3])/1440 + (17*PolyLog[2, 1/3]*PolyLog[2, 2/3])/360 - (11*PolyLog[2, 2/3]^2)/144 + (103*Pi^2*PolyLog[2, 3/4])/480 - (127*Log[2]^2*PolyLog[2, 3/4])/1440 - (Log[2]*Log[3]*PolyLog[2, 3/4])/36 - (619*Log[3]^2*PolyLog[2, 3/4])/1440 - (127*Log[2]*Log[4]*PolyLog[2, 3/4])/720 + (187*Log[3]*Log[4]*PolyLog[2, 3/4])/720 - (7*Log[4]^2*PolyLog[2, 3/4])/180 + (331*PolyLog[2, 1/4]*PolyLog[2, 3/4])/1440 + (223*PolyLog[2, 1/3]*PolyLog[2, 3/4])/720 + (281*PolyLog[2, 2/3]*PolyLog[2, 3/4])/720 + (37*PolyLog[2, 3/4]^2)/720 + (49*Log[2]*PolyLog[3, 1/4])/360 + (191*Log[3]*PolyLog[3, 1/4])/240 - (59*Log[4]*PolyLog[3, 1/4])/1440 - (91*Log[2]*PolyLog[3, 1/3])/720 - (33*Log[3]*PolyLog[3, 1/3])/20 + (91*Log[4]*PolyLog[3, 1/3])/1440 - (61*Log[2]*PolyLog[3, 2/3])/360 + (31*Log[3]*PolyLog[3, 2/3])/60 - (17*Log[4]*PolyLog[3, 2/3])/720 - (5*Log[2]*PolyLog[3, 3/4])/48 + (11*Log[3]*PolyLog[3, 3/4])/12 + (17*Log[4]*PolyLog[3, 3/4])/160 + (23*PolyLog[4, 1/4])/16 - PolyLog[4, 1/3]/4 - (17*PolyLog[4, 1/2])/6 + (7*PolyLog[4, 2/3])/2 + (15*PolyLog[4, 3/4])/8 + (19*Log[2]*Zeta[3])/1440 - (203*Log[3]*Zeta[3])/288 + (Log[4]*Zeta[3])/40
}
• Very interesting! Have you used FindIntegerNullVector or some other implementation of PSLQ algorithm? Could you give separate expressions for multiple polylogarithms from the first formula in your answer? – Vladimir Reshetnikov Sep 20 '14 at 17:30
• @VladimirReshetnikov Sure, I put them in the answer. I used LatticeReduce. – Kirill Sep 20 '14 at 18:59
• @JulianRosen Yes, so am I. But then again, many, but not all, multiple zeta values can be written in terms of ordinary zeta values. – Kirill Sep 20 '14 at 21:02
• @Kirill Some polylogarithm terms in your result could be reduced to simpler arguments (using these identities), yielding a simpler overall result. – Vladimir Reshetnikov Sep 21 '14 at 0:28
• @Kirill It would be very nice to prove some kind of uniformity theorem for this class of expressions, so that a demonstration of a closed form of small enough syntactic complexity that matches the integral with sufficiently high precision would constitute a proof of its correctness (similar to how we can do it with algebraic numbers in certain cases). – Vladimir Reshetnikov Sep 21 '14 at 1:22
The value of $I$ is a $\mathbb{Q}$-linear combination of values of the multiple polylogarithm at rational arguments. I'll explain how to compute this.
Expanding each logarithm in the integrand as an integral, multiplying out, and dividing into regions, and making the substitution $x\leftrightarrow 1-x$, we get that $I$ is a $\mathbb{Q}$-linear combination of iterated integrals of the form $$\int_{1\geq t_1\geq t_2\geq t_3\geq t_4\geq 0} \frac{dt_4}{f_4(t_4)}\frac{dt_3}{f_3(t_3)}\frac{dt_2}{f_2(t_2)}\frac{dt_1}{f_1(t_1)},$$ where each $f_i(t)$ is either $t$ or $1-wt$ for some $w\in\{1/2,2/3\}$.
Claim: each iterated integral of this form is a value of the multiple polylogarithm, defined by $$Li_{s_1,\ldots,s_k}(z_1,\ldots,z_k):=\sum_{n_1>\ldots>n_k\geq 1}\frac{z_1^{n_1}\ldots z_k^{n_k}}{n_1^{s_1}\ldots n_k^{s_k}}.$$ For $k=1$ this is the ordinary polylogarithm, and $Li_{s_1,\ldots,s_k}(1,\ldots,1)=\zeta(s_1,\ldots,s_k)$ is the multiple zeta value.
The claim isn't too hard to see by induction on the number of terms in the iterated integral: we have $$\int_0^{z_1} \frac{Li_{s_1,\ldots,s_k}(t,z_2,\ldots,z_k)}{t}\,dt=Li_{s_1+1,\ldots,s_k}(z_1,z_2,\ldots,z_k),$$ $$\int_0^{z_1} \frac{Li_{s_1,\ldots,s_k}(t,z_2,\ldots,z_k)}{1-wt}\,dt=\frac{1}{w}Li_{1,s_1,\ldots,s_k}(wz_1,1/w,z_2,z_3,\ldots,z_k).$$ (I hope I wrote this all out correctly)
Values of multiple polylogarithms satisfy many relations, so it's possible the expression one gets can be simplified.
Iterated integrals like this show up when computing the action of parallel transport on algebraic vector bundles with nilpotent connection on open subsets of $\mathbb{P}^1$. It's not so hard to write down such a thing on $\mathbb{P}^1\backslash\{1,-1,-1/2,\infty\}$ giving $I$ as a matrix coefficient for transport along $[0,1]$.
There's a thing called the unipotent fundamental group of a variety, which has the structure of a motive. Without getting into what exactly this is, I'll just say that the observation about parallel transport essentially amounts to $I$ being a period of $\pi_{1,\cdot}(\mathbb{P}^1\backslash\{1,-1,-1/2,\infty\})$. One doesn't get a good model of $X:=\mathbb{P}^1\backslash\{1,-1,-1/2,\infty\}$ over $\mathbb{Z}$ because the removed points collide mod 2 and mod 3, but there is a good model over $\mathbb{Z}[1/6]$. It is known that the fundamental group of a rational curve has the structure of a mixed Tate motive, so $I$ is the period of a mixed Tate motive over $\mathbb{Z}[1/6]$. I don't really understand the construction of mixed Tate motives, so I'm just viewing it as a black box. Probably someone who understood them better than I could see directly that $I$ is the period of a mixed Tate motive without thinking about $\pi_1$.
For comparison: if the only denominators appearing in the iterated integral were $t$ and $1-t$, then the value of the integral is a multiple zeta value. These numbers are periods of the fundamental group of $\mathbb{P}^1\backslash\{0,1,\infty\}$, which is a mixed Tate motive over $\mathbb{Z}$. It's a theorem that the space of all periods of mixed Tate motives over $\mathbb{Z}$ is the $\mathbb{Q}[(2\pi i)^{-1}]$ span of the multiple zeta values. I think in the case of $I$ the mixed Tate motive we need is only defined over $\mathbb{Z}[1/6]$, so that $I$ can't necessarily be written in terms of multiple zeta values.
There's a conjecture that all periods of mixed Tate motives over any ring $\mathbb{Z}[1/N]$ are linear combinations of values of multiple polylogarithms.
• Very interesting: $$-{\frac{\log2\log3}2} {\mathrm{L}}_2\Big({\frac23}\Big) + {\frac{\log3}2}\mathrm{L}_{2,1}\Big({\frac23,\frac34}\Big) + {\frac{\log2}2} \mathrm{L}_{2,1}\Big({\frac23,1}\Big)-{\frac{1}{2}}\mathrm{L}_{2,1,1}\Big({{\frac{2}{3}},\frac{3}{4},\frac{4}{3}}\Big)-{\frac12}\mathrm{L}_{2,1,1}\Big({\frac{2}{3},1,\frac{3}{4}}\Big)$$ – Kirill Sep 16 '14 at 22:27
• parallel transport on algebraic vector bundles with nilpotent connection on open subsets of $\mathbb P^1$ - Took the words right out of my mouth! :-) – Lucian Sep 16 '14 at 22:31
• This is a really nice answer. I found Special values of multiple polylogarithms by Borwein, Bradley, Broadhurst, Lisonek, but the identities there are (I think) not enough to answer this question, or this one (but at least the integral representation I got looks imposing). Do you know more about the identities that multiple polylogarithms satisfy? – Kirill Sep 17 '14 at 19:13
• Thank you for your answer, the bounty is yours! I still would like to see an explicit result though. – Vladimir Reshetnikov Sep 18 '14 at 17:05
• I fail to see this the correct answer for this problem because you didn't give an explicit answer. Maybe I am just too dumb to understand it – Venus Sep 19 '14 at 17:11
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2019-08-17 13:25:28
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http://www.maplesoft.com/support/help/Maple/view.aspx?path=Task/ComputeGradient
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Compute the Gradient of a Function - Maple Programming Help
Compute the Gradient of a Function
Description In Cartesian coordinates, compute the gradient of a scalar field.
Compute the Gradient Vector in Cartesian Coordinates Enter a scalar field as an expression in Cartesian coordinates: Choose Coordinate System: Cartesian - [x, y]Cartesian - [x, y, z]Cartesian - other Coordinate Variables:
Commands Used
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2016-10-28 08:21:09
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https://www.3blue1brown.com/lessons/clacks-via-light
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3Blue1Brown
# How colliding blocks act like a beam of light...to compute pi.
You know that feeling when you have two mirrors facing each other, and it gives the illusion of there being an infinite tunnel of rooms? If they’re at an angle with each other, it makes you feel like you’re part of a strange kaleidoscopic world with many copies of yourself all separated by angled pieces of glass. What many people may not realize is that the idea underlying this illusion can be quite helpful for solving serious problems in math.
We’ve already had two lessons describing the block collision problem with its wonderfully surprising answer. Big block comes in from the right, lots of clacks, the total number of clacks looks like $\pi$, and we want to know why.
Here we see one more perspective explaining what’s going on, whereas if the connection to $\pi$ wasn’t surprising enoughwe add one more unexpected connection to optics.
But we’re doing more than just answering the same question twice. This alternate solution gives a richer understanding that can let you answer other questions about the setup, and it’s also core to how I coded accurate simulations of these blocks without requiring absurdly small time steps and huge computation time.
## Position Phase Space
Last time we used a coordinate plane where each point encodes a pair of velocities. Here, we use a coordinate plane where each point encodes the positions of both blocks. Again, the idea is that by representing the states of a changing system with individual points in some space, problems in dynamics turn into problems in geometry, which are hopefully more solvable.
Specifically, let the $x$-coordinate of a 2D plane represent the distance from the wall to the left edge of the first block, which I’ll call $\textcolor{green}{d_1}$. And let the $y$-coordinate represent the distance from the wall to the right edge of the second block, which I’ll call $\textcolor{red}{d_2}$.
That way, the line $x = y$ shows us where the two blocks clack into each other, since that happens whenever $\textcolor{green}{d_1} = \textcolor{red}{d_2}$.
Here’s what it looks like for our scenario to play out:
As the two distances of our blocks change, the two-dimensional point of our configuration space moves around, with a position that always fully encodes the information of those two distances.
You may notice that at the bottom there, it’s bounded by the line where $\textcolor{red}{d_2}$ is the same as the small block’s width, which is what it means for the small block to hit the wall.
You might be able to guess where we’re going with this: The way this point bounces between these two bounding lines is a bit like light bouncing between two mirrors. The analogy doesn’t quite work, though. In the lingo of optics, the angle of incidence doesn’t equal the angle of reflection.
Just think of that first collision: If the line $x = y$ behaved like a mirror, then a beam of light coming in from the right would bounce off this 45-degree $x = y$ line and end up going straight down, which would mean the first block stops moving entirely.
This does actually happen in the simplest case, where the second block has the same mass as the first and picks up all of the momentum like a croquet ball. But for any other mass ratio, that first block will keep moving a bit, so the trajectory of our point in this configuration space won’t be pointed straight down.
Even if it’s not immediately clear why this analogy with light would be helpful, run with me for a moment and see if we can fix it for the general case, because analogies like these are often very useful when working on a difficult math problem.
## Fixing the Light Analogy
Just like in the previous lesson, it’s helpful to rescale the coordinates. In fact, motivated by what we did then, you might think to rescale the coordinates so that $x = \sqrt{\textcolor{blue}{m_1}} \textcolor{green}{d_1}$. This has the effect of stretching our space horizontally, so changes to our big block’s position now result in larger changes to the $x$-coordinate of our point. Likewise, let’s write the $y$-coordinate as $y = \sqrt{\textcolor{blue}{m_2}} \textcolor{green}{d_2}$, even though in this particular case that second mass is $1$, so it doesn’t matter.
Maybe that strikes you as making things uglier, and like a somewhat random thing to do, but as with last time, when we include square roots of masses like this, everything plays more nicely with the laws of conserving energy and momentum.
Specifically, the conservation of energy will translate to the fact that our little point in configuration space is always moving at the same speed, which in our analogy you might think of as meaning there’s a constant speed of light. And the conservation of momentum will translate to the fact that as our point bounces off the “mirrors” of our setup, the angle of incidence equals the angle of reflection.
Doesn’t that seem bizarre? That the laws of mechanics should translate to laws of optics like this?
To see why this is true, let’s roll up our sleeves and work out the math.
## Understanding the Conservation Laws
Focus on the velocity vector for our point in the diagram, showing which direction it's moving and how quickly.
Note, this is not a physical velocity, like the velocities of the moving blocks; instead it’s a more abstract rate of change in the context of our configuration space, whose two dimensions worth of possible directions encode the velocities of both of the blocks.
The $x$-component of this little vector is the rate of change of $x$. Likewise, its $y$-component is the rate of change for $y$.
But what is the rate of change for the x-coordinate? Well, $x = \sqrt{\textcolor{blue}{m_1}}\textcolor{green}{d_1}$, and the mass doesn’t change, so it depends only on how $\textcolor{green}{d_1}$ changes. And the rate at which $\textcolor{green}{d_1}$ changes is just the velocity of the big block, which I’ll call $\textcolor{green}{v_1}$. So $\frac{dx}{dt} = \sqrt{\textcolor{blue}{m_1}}\textcolor{green}{v_1}$. Likewise, the rate of change for y is $\frac{dy}{dt} = \sqrt{\textcolor{blue}{m_2}}\textcolor{red}{v_2}$.
Now we can calculate the magnitude of this little configuration-space-change-vector. Using the pythagorean theorem…
So the speed of our point in configuration space is based on the kinetic energy of the system. And since that energy is constant, the speed also stays constant throughout the process. Remember, a core assumption here is that no energy is lost to friction or to the collisions.
Alright, that’s cool. With these rescaled coordinates our little point always moves with a constant speed. I know it’s not obvious why you’d care, but among other things, it’s important for the next step, where the conservation of momentum implies that these two bounding lines act like mirrors.
But first, let’s understand this line $\textcolor{green}{d_1} = \textcolor{red}{d_2}$ a little bit better. In our new coordinates, it’s no longer a nice 45-degree $x = y$ line.
Instead, after a little rearrangement, we can that see it looks like the line $\frac{x}{\sqrt{\textcolor{blue}{m_1}}} = \frac{y}{\sqrt{\textcolor{blue}{m_2}}}$, which is a line with slope $\frac{\sqrt{\textcolor{blue}{m_2}}}{\sqrt{\textcolor{blue}{m_1}}}$. That’s a nice expression to tuck away in the back of your mind.
$\frac{\sqrt{\textcolor{blue}{m_2}}}{\sqrt{\textcolor{blue}{m_1}}}$
After the blocks collide, meaning our point hits this line, the way to figure out how they move is to use the conservation of momentum, which says the value $\textcolor{blue}{m_1}\textcolor{red}{v_1} + \textcolor{blue}{m_2}\textcolor{red}{v_2}$ remains unchanged after the collision.
Notice, this looks like a dot product between two column vectors:
\begin{align*} \textcolor{blue}{m_1}\textcolor{red}{v_1} + \textcolor{blue}{m_2}\textcolor{red}{v_2} &= \text{const.} \\[1em] \begin{bmatrix} \textcolor{blue}{m_1} \\ \textcolor{blue}{m_2} \end{bmatrix} \cdot \begin{bmatrix} \textcolor{red}{v_1} \\ \textcolor{red}{v_2} \end{bmatrix} &= \text{const.} \end{align*}
And since we’re interested in how this relationship will play out with our rescaled coordinates, we can rework the dot product to use the rates of change of x and y:
\begin{align*} \textcolor{blue}{\sqrt{m_1}}\underbrace{(\textcolor{blue}{\sqrt{m_1}}\textcolor{red}{v_1})}_{dx/dt} + \textcolor{blue}{\sqrt{m_2}}\underbrace{(\textcolor{blue}{\sqrt{m_2}}\textcolor{red}{v_2})}_{dy/dt} &= \text{const.} \\[2em] \begin{bmatrix} \textcolor{blue}{\sqrt{m_1}} \\ \textcolor{blue}{\sqrt{m_2}} \end{bmatrix} \cdot \begin{bmatrix} dx/dt \\ dy/dt \end{bmatrix} &= \text{const.} \end{align*}
I know this probably seems like a complicated way to talk about a simple momentum equation, but there is a good reason for shifting to a language of dot products in our new coordinates.
Notice that the vector containing the derivatives is really just the velocity vector for the point in our diagram that we’ve been looking at.
And the other vector, with the square roots of the masses, has a representation in this diagram too. It points in the same direction as our collision line, since we found earlier that the slope of that line is $\frac{\sqrt{\textcolor{blue}{m_2}}}{\sqrt{\textcolor{blue}{m_1}}}$.
If you’re unfamiliar with the dot product, I do have another lesson describing it, but real quick let’s review what it means geometrically. The dot product of two vectors equals the length of the first, time the length of the projection of the second one onto that first (considered negative if they point in opposite directions). You often see this written as the product of the lengths of each vector times the cosine of the angle between them.
Alright, so look back at this conservation of momentum expression, which tells us that this dot product stays constant before and after the collision:
$\begin{bmatrix} \textcolor{blue}{\sqrt{m_1}} \\ \textcolor{blue}{\sqrt{m_2}} \end{bmatrix} \cdot \begin{bmatrix} dx/dt \\ dy/dt \end{bmatrix} = \text{const.}$
Since we just saw that this rate-of-change vector has a constant magnitude, and the masses never change, the only way for this dot product to stay the same is if the angle it makes with that collision line stays the same.
Similarly, when the small block bounces off the wall, our little vector gets reflected about the $x$-direction, since only its $y$-coordinate changes. So our configuration point is bouncing off that horizontal line as if it was a mirror.
Step back for a moment and think about what this means for our original question of counting block collisions. We can now translate the collision count question into a different question entirely, this time in the world of optics.
Given our coordinate setup, what question could we ask that would give us the number of block collisions, but which only involves optics, not blocks?
If you shine a beam of light at a pair of mirrors, how many times will that light bounce off the mirrors as a function of the angle between them? (Remember, the mass ratio of our blocks determines the angle in our analogy.)
## Unfolding the Mirror
Now I can hear some of you complaining: “Haven’t we replaced one tricky setup with another?”. This might make for a cute analogy, but how is it progress? It’s true; counting the number of light bounces here is hard. But now we have a helpful trick.
When the beam of light hits the mirror, instead of thinking of the beam as reflecting about the mirror, think of the beam as going straight while the whole world gets flipped. It’s as if the beam is passing through a piece of glass into an illusory “looking glass universe.”
Think of actual mirrors here. The wire on the left will represent a laser beam coming into the mirror, while the one on the right will represent its reflection:
The illusion is that the beam goes straight through the mirror, as if passing through a window separating us from another room.
But notice! Crucially! For this illusion to work, the angle of incidence has to equal the angle of reflection. Otherwise the flipped copy of the reflected beam won’t line up with the first part.
So all that work we did rescaling coordinates and futzing through the momentum equation was certainly necessary. Now we get to enjoy the fruits of our labor. This helps us elegantly solve the question of how many mirror bounces there will be, which is also the question of how many block collisions there will be.
Every time the beam hits a mirror, don’t think of the beam as getting reflected. Let it continue straight while the world gets reflected.
As this goes on, the illusion to the light beam is that instead of getting bounced around between these two angled mirrors many times, it’s passing through a sequence of angled pieces of glass, all the same angle apart.
Let’s compare the original bouncing beam to the illusory straight one.
Our original question of counting bounces turns into a question about how many pieces of glass this illusory beam crosses. How many reflected copies of the world does it pass into?
Well, calling the angle between the mirrors theta, the answer here is however many times you can add theta to itself without going more than halfway around the circle (so the sum must be less than $\pi$ total radians).
Written as a formula, the answer to our question is the floor of $\pi$ divided by $\theta$, $\left\lfloor \frac{\pi}{\theta} \right\rfloor$.
## Summary
Let’s review! We started by drawing a configuration space for our colliding blocks, where the $x$ and $y$ coordinates represented the two distances from the wall.
This kind of looked like light bouncing between two mirrors, but to make the analogy work properly, we needed to rescale the coordinates by the square roots of the masses. This made the slope of one line $\frac{\sqrt{\textcolor{blue}{m_2}}}{\sqrt{\textcolor{blue}{m_1}}}$.
So the angle between our two bounding lines will be the inverse tangent of the slope, $\theta = \arctan(\sqrt{\frac{m_2}{m_1}})$.
To figure out how many bounces there are between two mirrors like this, think of the illusion of the beam continuing straight through a sequence of looking glass universes separated by a semicircular fan of windows.
The answer, then, comes down to how many times the value of $\theta$ fits into $\pi$ radians.
From here, to understand why exactly the digits of $\pi$ show up, when the mass ratio is a power of 100, it’s exactly what we did in the last lesson, so I won’t repeat myself here.
## Final thoughts
As we reflect on how absurd the initial appearance of $\pi$ seemed, on the two solutions we’ve now seen, and on how unexpectedly helpful it can be to represent the state of your system with points in some space, I leave you with this quote from computer scientist Alan Kay:
“A change of perspective is worth 80 IQ points”.
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2022-12-04 22:27:26
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http://mathoverflow.net/questions/129697/martingale-cotype-vs-cotype-on-super-reflexive-spaces
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# Martingale-cotype vs cotype on super-reflexive spaces
I'm have difficultly nailing down the direction of some implications. For $2 \leq q < \infty$, there are (at least) two ways to say that a Banach space $B$ has "cotype $q$".
• $B$ has cotype q.
• $B$ is isomorphic to a $q$-uniformly convex Banach space, i.e. a uniformly convex Banach space with a "power type" modulus of convergence $\delta(\varepsilon) = C \varepsilon^q$. ($B$ is said to have martingale cotype q. The name comes from a characterization by Pisier involving martingales.)
I know the following:
1. Every $q$-uniformly convex Banach space (and any space isomorphic to it) has cotype $q$.
2. There are nonreflexive spaces, e.g. $L^1$ and $\ell^1$, with cotype $2$. Since they are nonreflexive they are not isomorphic to a uniformly convex space.
3. The super-reflexive spaces are exactly those isomorphic to uniformly convex spaces, which in turn are all isomorphic to $q$-uniformly convex spaces.
However, I can't seem to find the answer to the following.
If a space is super-reflexive and of cotype $q$, is it isomorphic to a $q$-uniformly convex space?
If not, is there a nice class of spaces where these two notions of cotype agree?
Update 1: I have some partial answers of spaces for which the notions agree (but not yet a general answer to my question).
• UMD spaces. (see Cédric's answer).
• Banach lattices of type $p>1$ (which includes the super-reflexive Banach lattices). I found a pair of interdependence diagrams on pp. 100, 101 of Lindenstrass and Tzafriri's "Classical Banach Spaces II" (these are some of the best math diagrams I have seen). On a Banach lattice, the modulus of convexity is of power type $q$ for an equivalent norm if and only if is of cotype $q$ and there is an upper estimate $p>1$. Following the diagram, "upper estimate" can be replaced with "type".
Update 2: On the bottom of p. 78 of Lindenstrass and Tzafriri's "Classical Banach Spaces II" my main question is listed as an open problem. So I guess my question becomes, has it been solved yet?
-
Let $X$ be a Banach space. Suppose that $X$ has UMD (hence super-refexive). It seems to me that the following equivalence is classical.
Then $X$ is of cotype $q$ if and only if $X$ is $q$-uniformly convex.
-
Thanks. I assume you mean to say "$X$ is isomorphic to a $q$-uniformly convex space". (It seems that $\mathsf{UMD}$ is preserved under isomorphisms, where as uniform convexity is not. That is of course if we are using the same definition of $q$-uniformly convex.) Also, I'm still hoping for an answer to my main question, but maybe $\mathsf{UMD}$ is the best known. – Jason Rute May 5 '13 at 10:50
Undoubtedly, you must look at the Bourgain's exemple of supereflexive space failing UMD and see... – user33709 May 5 '13 at 11:03
Cédric, I found the article, link.springer.com/article/10.1007%2FBF02384306, but I am not sure I understand your comment. Are you saying if I understood this example, I might be able to answer my question? – Jason Rute May 5 '13 at 20:48
My feeling is that the answer is false and that you need to study the exemples of super-reflexive spaces which are not UMD. Maybe, one of them is a counter-example to your question. See the recent paper front.math.ucdavis.edu/1112.0739 for information on all known counter-examples. – user33709 May 6 '13 at 10:02
I'll attempt to answer this with what I've found:
(1) The answer to my main question is that it is not true. In this document of Pisier's, he states
It is possible to find a uniformly convex space $B$ for which the index of type $p(B)$ differs from the corresponding index for the $M$-type. Similarly for the cotype. See [P4] for details.
The reference [P4] is
G. PISIER, Un exempte concernant la super-réflexivité. Annexe no. 2. Séminaire Maurey-Schwartz 1974-75. Ecole Polytechnique. Paris.
which I can't seem to find, so I'll have to take his word for it.
(2) As I added to my original post, it seems the two best cases where they agree are UMD spaces (thanks Cédric) and Banach lattices of type $p>1$.
-
You can find the Pisier's paper here: numdam.org/numdam-bin/fitem?id=SAF_1974-1975____A27_0 – user33709 May 6 '13 at 10:03
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2014-10-22 00:40:07
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https://pt.overleaf.com/articles/recent/page/223
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# Articles — Recent
Papers, presentations, reports and more, written in LaTeX and published by our community. Search or browse below.
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Aeroelastic and Dynamic Structural Analysis of a non-tapered wing
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OS Assignment
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2019-10-17 14:04:55
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https://solvedlib.com/n/is-s-bromochlorofluoromethane-andr-bromochlorofluoromethane,14467499
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# Is s-bromochlorofluoromethane andr-bromochlorofluoromethane identical to each other;stereoisomers; constitutional isomers; or completely unrelated?Why?
###### Question:
Is s-bromochlorofluoromethane and r-bromochlorofluoromethane identical to each other; stereoisomers; constitutional isomers; or completely unrelated? Why?
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##### Training academy that administers a graduation examination found that the mean score on the exam is 142 and the standard deviation is 30. If the academy wants to set the graduation with honors exam score so that only the best 10% of all trainees graduate with honors, what is the honors exam score? Assume the exam scores are normally distributed.212192181187201
training academy that administers a graduation examination found that the mean score on the exam is 142 and the standard deviation is 30. If the academy wants to set the graduation with honors exam score so that only the best 10% of all trainees graduate with honors, what is the honors exam score? A...
##### 7.12 The electric field of an elliptically polarized plane wave is given by [-k 10 sin(cot-kz-60°) E(z, t) y 30 cos(ot - kz)] (V/m). Determine the following: (a) The polarization angles (y, x). (...
7.12 The electric field of an elliptically polarized plane wave is given by [-k 10 sin(cot-kz-60°) E(z, t) y 30 cos(ot - kz)] (V/m). Determine the following: (a) The polarization angles (y, x). (b) The direction of rotation. 7.12 The electric field of an elliptically polarized plane wave is giv...
##### When dividing a polynomial by $x-c,$ do you prefer to use long division or synthetic division? Does the value of $c$ make a difference to you in choosing? Give reasons.
When dividing a polynomial by $x-c,$ do you prefer to use long division or synthetic division? Does the value of $c$ make a difference to you in choosing? Give reasons....
##### 4. A velocity field of a fluid flow is given by Vi+ zj The velocity is...
4. A velocity field of a fluid flow is given by Vi+ zj The velocity is in m/s. Calculate the acceleration at point (2, 5, 3) at 1 0.5 s (10 points)...
##### 4.+-16.66 points ASWSBE13 13.E.013.MI My Notes Ask Your Teacher You may need to use the appropriate...
4.+-16.66 points ASWSBE13 13.E.013.MI My Notes Ask Your Teacher You may need to use the appropriate technology to answer this question The following data are from a completely randomized design TreatmentTreatment Treatment 31 29 29 26 30 29 34 37 36 37 49 Sample mean 37 Sample variance 3.50 6.50 6.5...
##### A 1.75 HF capacitor and 1.75 HF capacitor B.00 HF capacitorHF capacitorconnectedseresCross7.00 battery. How much cnarge (in pC) storedeachIpacitor?The same capacitors are disconnected and discharged They are then connected 1.75 HF capacitor B.0O MF capacitorparallel across the same battery. How much charge (in UC) storedeach cadacitor nowy
A 1.75 HF capacitor and 1.75 HF capacitor B.00 HF capacitor HF capacitor connected seres Cross 7.00 battery. How much cnarge (in pC) stored each Ipacitor? The same capacitors are disconnected and discharged They are then connected 1.75 HF capacitor B.0O MF capacitor parallel across the same battery....
##### QUESTIONS How have NYC, and nearby suburbs, adjusted laws regarding pediatric vaccinations, as a result of...
QUESTIONS How have NYC, and nearby suburbs, adjusted laws regarding pediatric vaccinations, as a result of the measles crisis Now, the only permitted exemptions are for medical reasons. Children cannot go to school without taking their medications with them. Currently, there are religious exemptions...
##### A uniform meter stick 100 cm long weighing 3 N is loaded with two small heavy objects as follows: a load of 2 N is at the 30 cm mark, and a load of 4 N is at the 80 cm mark. Where is the center of gravity of the system?
A uniform meter stick 100 cm long weighing 3 N is loaded with two small heavy objects as follows: a load of 2 N is at the 30 cm mark, and a load of 4 N is at the 80 cm mark. Where is the center of gravity of the system?...
##### Find e At where 2 5 A= -2 -4
Find e At where 2 5 A= -2 -4...
##### Assume that 2015 is the base year. Assuming that the base year basket of goods contains...
Assume that 2015 is the base year. Assuming that the base year basket of goods contains 40 notebooks, 1 iPad and 200 muffins, what is the Consumer Price Index for 2016? Year Price of a Notebook Quantity of Notebooks Price of an iPad Quantity of iPads Price of a Muffin Quantity of Muffins 2015...
##### MBL914N OCTOBER NOVEMBER 2014 QUESTION 2 [8 MARKS] The constraint at Maniki Inc is an expensive...
MBL914N OCTOBER NOVEMBER 2014 QUESTION 2 [8 MARKS] The constraint at Maniki Inc is an expensive milling machine The three products listed below use this constrained resource ERC SCTC Selling price per unit 118 548 175 Variable cost per unit 94 431 136 Time on the constraint (minutes) 1 50 5 30 200 R...
##### Find the indicated limit Note ' that /Hopital' s rule does not pply to every problem, and some problems will require more than one pplication of /Hopitals rule. Use when appropriateX-wln ( 1+5 eSelect the correct choice below and, if necessary; fill the answer box to complete your choice_X_00 In (1+5 e (Type an exact answer in simplified form: ) The limit does not exist.
Find the indicated limit Note ' that /Hopital' s rule does not pply to every problem, and some problems will require more than one pplication of /Hopitals rule. Use when appropriate X-wln ( 1+5 e Select the correct choice below and, if necessary; fill the answer box to complete your choice...
##### Compare and contrast the primary differences between the job costing and process costing using the following...
Compare and contrast the primary differences between the job costing and process costing using the following table Select the letfer from the list below that corresponds with the characteristic that best explains each item in the table (Click the icon to view the list of characteristics.) Job costin...
##### Itis believed that; the average numbers of hours spent positive linear relationship with the starting salary Pent Ardviea per day (HOURS) during undergraduate education should have measured thousands below is the Microsoft" Excel of dollars per month) after graduation; Given output for , predicting sample of 51 students: NOTE= starting salary (Y) using number Only partial output hours spent studying per day (X) for shown; Regression Statistics Multiple 8857 Square 0.7845 Adjusted Square 780
Itis believed that; the average numbers of hours spent positive linear relationship with the starting salary Pent Ardviea per day (HOURS) during undergraduate education should have measured thousands below is the Microsoft" Excel of dollars per month) after graduation; Given output for , predic...
Exercise 11-3 During its first year of operations, Foyle Corporation had the following transactions pertaining to its common stock. Jan. July 10 Issued 65,500 shares for cash at $5 per share. 1 Issued 43,000 shares for cash at$10 per share. Journalize the transactions, assuming that the common stoc...
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2023-02-05 21:36:42
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http://krydom.com/
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# Krydom: 暁の水平线に胜利を刻むのです
04/24
17:00
04/20
16:46
## [czyz 2017.04.20 NOIP 模拟赛]
A：先把所有课程按照已经修完的学分从大到小排序，然后0~n枚举把多少课程学分修完，二分算出剩下的天数最多能把没学完的所有课学到多少，复杂度O(nlogn)
04/20
07:53
kruskal
## [CF 472D] Design Tutorial: Inverse the Problem
### ♦♦♦♦♦♦ Description ♦♦♦♦♦♦
There is an easy way to obtain a new task from an old one called "Inverse the problem": we give an output of the original task, and ask to generate an input, such that solution to the original problem will produce the output we provided. The hard task of Topcoder Open 2014 Round 2C, InverseRMQ, is a good example.
Now let's create a task this way. We will use the task: you are given a tree, please calculate the distance between any pair of its nodes. Yes, it is very easy, but the inverse version is a bit harder: you are given an n × n distance matrix. Determine if it is the distance matrix of a weighted tree (all weights must be positive integers).
04/19
21:47
## [2017.04.18 省选模拟赛] 旅行商
### ♦♦♦♦♦♦ Description ♦♦♦♦♦♦
moreD共在 $N$ 个城市间倒卖物品。他从 $1$ 号城市出发，自然，这 $N$ 个城市都是要访问到的。
moreD明白这应该需要写算法解决，但是由于3D眩晕症，他现在已经脸如死灰地躺在床上了(◎﹏◎)，显然不能胜任这个任务。于是他希望你能帮他解决这个问题。
04/19
21:39
## [2017.04.18 省选模拟赛] 隔壁老王的简单数列
### ♦♦♦♦♦♦ Description ♦♦♦♦♦♦
moreD的隔壁室友（人称隔壁老王的）liouzhou_101最近选了数论课，经过一段时间的卓有成效的学习，他非常自信，认为自己的数论水平已经可以吊打全国 $99.99\%$的学生了。O(*￣▽￣*)ブ
04/19
21:26
A：
【防AK好题（？？？）（雾）】
04/19
21:16
## [bzoj 4822] [Cqoi2017]老C的任务
### ♦♦♦♦♦♦ Description ♦♦♦♦♦♦
大意：给一些点和权值，询问一个矩形内的权值总和，离线
$n, m \le 10 ^ 5$
04/11
13:11
## [bzoj 4756] [Usaco2017 Jan]Promotion Counting
### ♦♦♦♦♦♦ Description ♦♦♦♦♦♦
The cows have once again tried to form a startup company, failing to remember from past experience that cows make terrible managers!The cows, conveniently numbered 1…N1…N (1≤N≤100,000), organize the company as a tree, with cow 1 as the president (the root of the tree). Each cow except the president has a single manager (its "parent" in the tree). Each cow ii has a distinct proficiency rating, p(i),which describes how good she is at her job. If cow ii is an ancestor (e.g., a manager of a manager of a manager) of cow jj, then we say jj is a subordinate of ii.
Unfortunately, the cows find that it is often the case that a manager has less proficiency than several of her subordinates, in which case the manager should consider promoting some of her subordinates. Your task is to help the cows figure out when this is happening. For each cow ii in the company, please count the number of subordinates jj where p(j)>p(i).
n只奶牛构成了一个树形的公司，每个奶牛有一个能力值pi，1号奶牛为树根。
04/9
16:59
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2017-04-29 05:33:01
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https://ai.stackexchange.com/questions/8793/chess-policy-network
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# Chess policy network
I am interested in making a simple chess engine using neural networks. I already have a fairly good value network but I can't figure out how to train a policy network. I know that Leela chess zero outputs the probability of any of the about 1800 possible moves. But how do you train such a network? How do you calculate the loss when you only have the 1 move that was played in the game to work with?
It may come as a surprise, after learning all about dynamic programming, temporal difference learning, SARSA/Q-learning, to then discover that there is yet another whole dimension to reinforcement learning (on top of choices for on-policy/off-policy, model-based/model-free, bootstrap/monte carlo etc). That is value-based vs policy-based methods. Policy-based methods are often taught after value-based methods, because they are more complex.
You can learn the parameters of a policy function by training it using a policy gradient method. The archetypical policy gradient method is REINFORCE, although that is not very efficient. You may have heard of policy gradient methods developed recently: A3C, A2C, DDPG, TRPO, PPO . . . there are a few.
How do you calculate the loss when you only have the 1 move that was played in the game to work with?
You can pre-train a policy network using supervised learning (perhaps using the moves of winning players in high quality games) - that would use multi-class cross-entropy loss that you may be familiar with from supervised classification problems.
Policy Gradient methods work with a reward summation function defined as expected reward given the distribution of states. If your network parameters are $$\theta$$, then that may look like this:
$$J(\theta) = \sum_{s \in \mathcal{S}} \rho_{\pi}(s) \sum_{a \in \mathcal{A}} \pi(a|s,\theta)q_{\pi}(s,a)$$
where $$\rho_{\pi}(s)$$ is the expected proportion of time steps spent in state $$s$$. There is a way to take a sample gradient of this that can be used for gradient ascent - the derivation is called the Policy Gradient Theorem. It's a bit long to include in this answer, but the upshot is that you can use your sampled single step to generate an approximate gradient towards improving the policy. There are a few variations, but for instance advantage actor critic uses this:
$$\nabla J(\theta) = \hat{A}(s,a)\nabla\text{log}(\pi(a|s,\theta))$$
where $$\hat{A}(s,a)$$ is your current estimate of the advantage (or $$Q(s,a) - V(s)$$) for taking a specific action in state s.
The related loss function is
$$\mathcal{L}(\theta) = -A(s,a)\text{log}(\pi(a|s,\theta))$$
The $$\text{log}$$ function looks like an odd addition, but is just a consequence of adjusting $$\nabla J$$ to take account of ratios in which actions are taken in the current policy. In fact $$\nabla\text{log}(\pi(a|s,\theta)) = \frac{\nabla\pi(a|s,\theta)}{\pi(a|s,\theta)}$$ and it may help your intuition to keep it in that form (the $$\text{log}$$ form is concise and used elsewhere in statistics as the "score function", but it is not necessary for anything specific in RL).
The variations in policy gradients may use other functions than the advantage function, and it is not clear that there is any "best" one. The policy gradient theory basically gives us ways of estimating relative benefits of actions, and allows for any offset to estimated return from $$(s,a)$$ that does not further depend on the choice of action $$a$$. So you can use any method for getting an estimated return, and offset it with anything that you think might normalise the updates - common choices for the latter include subtracting average reward, or subtracting the state value function.
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2021-07-26 14:39:13
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https://www.physicsforums.com/threads/question-about-jupiter-atmosphere.245395/
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1. Jul 16, 2008
fluidistic
I know it's not a bright question, but I'd like to know the answer as well. I've read that the high atmosphere of Jupiter contains molecules of Hydrogen ($$H_2$$). It is well known that (under Earth atmosphere at least), this gas is highly explosive. I just wonder why the atmosphere of Jupiter doesn't explode at least partially. I'm guessing that the concentration of gas on the atmosphere is just way too low to start a fire reaction, but it's only a guess. If it appears to be true, I'd like to know the concentration required to start a fire reaction between Hydrogen molecules. (I know it produces water...in case I'm not specific enough of the gas I'm talking about).
2. Jul 16, 2008
pixel01
No, H2 explosion needs O2 as well
3. Jul 16, 2008
fluidistic
Oh you're right... that's why it produces water. Anyway, isn't it possible to find a planet with an atmosphere mixed with hydrogen and oxygen in such quantities that it can interact violently? I don't think it might occur in the Solar System since Jupiter is just heavy enough to retain the hydrogen in its atmosphere... Maybe big explosions occurs in another planets and we might see them then.
4. Jul 16, 2008
Staff: Mentor
More than 75% of the mass of the universe is hydrogen, oxygen only 1%, so big gas planets will be unlikely to have the right mix to make combustion possible.
5. Jul 16, 2008
fluidistic
Too bad... It would also be much less brighter than a little star, so there is no hope to observe it.
I believe planets don't fit the global universe statistics of component elements though.
Now I'm thinking about planets that contains liquid (or ice) $$H_2$$ and $$0_2$$. If there is a storm or a cataclysm (say, for example a big cliff falling into the liquefied gases), it would be possible to engender a terrible explosion, but yes, very unlikely to happen.
Thank you both for the clarification. I'm done with this question.
6. Jul 17, 2008
Staff: Mentor
Well, such a thing wouldn't be stable anyway. A hydrogen/oxygen mixture is explosive, so a giant ball of it would simplly explode and then be gone. There is no equilibrium to be had like in a star.
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2017-11-20 21:23:51
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https://idontknowbut.blogspot.com/2017/
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## Sunday, December 31, 2017
### Chronicles of Matthew Paris
The edition I have only goes up through 1250, where he lays down his pen. And later picks it up for another 9 years.
Early on he describes a painting of Christ in Majesty with the Church and Synagogue. Near the end he describes Henry III's extortion of money from the Jews, whom he accused of fraud--to Matthew's satisfaction but modern suspicion.
In the last year or two he describes wild weather, earthquakes, and tsunamis, and he describes the destruction of the 7'th Crusade. (St. Louis was unable to keep the competing egos in check, didn't have good planning or support, and seriously overreached.) Some people "took the cross" in order to collect donations, but without any serious intention of going to fight.
This bit of description has an interesting assumption built into it... Matthew is being a bit snarky.
The above-mentioned special clerk of the lord king, whose wealth attained to episcopal heights ...
An empty church is one without a priest or bishop to receive the income from it, and the king, except when there was a special exemption, had veto power over who got appointed. Unless he decided to ignore the exemption. And Innocent IV never had enough money either, and any appointments or other requests had to be well-lubricated. Quite a bit of the chronicles describe one or the other kind of novel extortion and waste. Innocent and Frederick were at war, and fortunes of war meant that large amounts of extorted money wound up lost.
He doesn't describe traditional exactions, unfortunately.
This was an era in which the barons fought the crown from time to time, and we, from the perspective of the heirs of the winners, think the dominance of the crown was the right goal--centralize the rule of law. But at the time, it wasn't quite so obvious, and the crown was exceedingly arbitrary--and stupid. Some themes seem quite familiar: in order to maintain a reputation for generosity to the poor, despite his lack of liquidity, Henry III visited London and told Londoners that their traditional New Years' presents would be given to him this year.
## Friday, December 29, 2017
### Random notes
Walgreens has Marvel Fantastic Four Body Wash for sale. I suppose the target group is young kids, though I wasn't aware that young comic fans went in for body wash in a big way. What does this team's endorsement convey to an adult? Will you have skin that looks like Thing or that feels like Human Torch?
Eldest Son told me that one of the fragmentary plays from ancient Greece had been "completed" by a modern writer. I asked him if this qualified as "fan fiction." He said he'd already thought of that, and asked for a verdict on the matter from the Protectors of the Plot Continuum. (They said it was a middle ground; probably have to look at it.) I gather from him that reworkings of old plays are often seriously anachronistic.
The Milwaukee Public Museum had an exhibit on weapons, and one of the last displays held a wheel-lock pistol with 2 wheels on one barrel. I didn't remember hearing of such a Roman-candle-esque device, but there it was, and it shows up in the history of firearms. I wonder how often the first charge over-compressed and went off spontaneously after the outer charge was fired.
## Thursday, December 28, 2017
### Weah
It looks like George Weah will be the next Liberian president. He'll be called a footballer (or soccer player), and there'll probably be some obsolete mention of his lack of formal education (he got a business degree from DeVry U in Florida(*) in 2011). But if I read the numbers right, he's been in politics for a dozen years now--not quite as long as his football career, but comparable.
It looks as though 60% wanted Weah rather than the existing Vice President Boakai. There have been charges of nepotism and corruption for several years aimed at Ellen and Boakai--possibly they stuck and people wanted a clean house. Maybe 12 years in politics honed Weah's skills a bit. Maybe there aren't enough Kissi and related tribes for Boakai to match the Kru and supporting tribes for Weah.
Or all of the above to some degree. Pretty much all the information I get is filtered by the Liberian media or through fairly well-off Liberians in the US.
Both men had joined the Poro : Weah in 2005 and Boakai in 2017 (or very early in life if a spokesman told the truth). That's an aspect of the Liberian political scene that I have no way of analyzing. Even if I were on the scene in Liberia, the Poro is a secret society. How do the leaders of the Poro exert influence and for what purposes? No clue.
Weah converted from Christianity to Islam, and then back again, if Wikipedia may be believed. I'm not sure what to make of that, either.
It looks like a simple headline--Former footballer elected President--but things are much more curious when you look closer.
At any rate, I wish him and Liberia the best. He'll have an interesting term--the UN will probably withdraw its peacekeeping troops during his tenure, and that will have an unhappy effect on the economy. For which he'll probably be blamed. It's traditional.
(*) It is in the top 35% of business programs in the US.
The photo of him in the first link shows microphones labled "UNMIL Radio." I didn't realize that the UN military ran a radio station. I'm really out of the loop...
## Monday, December 25, 2017
### Love of things
Faithful stuffed rabbit
No longer shares his dreaming
His baby love grew
But it was real love. Though the rabbit could not love him back, it had been a token of love for him, and so in a way it did, and it was fitting that he find it lovable too. And not just of our love, but of the love of the One who planned that the things of the world serve as instruments of love.
A blanket, meant to comfort, stands in for deeper comfort in a new role as companion. I smiled, and tried to remember--there are pictures, after all...
Bunny is worn. He was set aside, and inherited for a while by his sister in the same role, and set aside again. But love is never lost to God.
## Sunday, December 24, 2017
### Why Bethlehem
The prim live in a world of illusion. Not just the prim; many of the comfortable and the confident too.
Farrar said
Jesus came to raise the dead. He did not come to teach the teachable; He did not come to improve the improvable; He did not come to reform the reformable. None of those things works.
## Saturday, December 23, 2017
### O Holy Night
I wrote before that you could usually tell which version of a song was the original language, although sometimes the results were equally good.
I heard a little history of O Holy Night the other day, and decided to look up the French Minuit, chrétiens. "Unitarian minister John Sullivan Dwight" did a superlative translation: it is almost seamless, and "a thrill of hope, the weary world rejoices" is a marvelously evocative line. And Cappeau did wonders in the original.
The US wiki says "the parish priest asked Cappeau, a native of the town, to write a Christmas poem, even though the latter never showed any interest in religion, and Cappeau obliged." The French wiki says "Placide Cappeau, un négociant en vin qui était républicain, socialiste et anticlérical, prétendit lui-même l'avoir écrit ... dans la diligence qui le conduisait à Paris ..." (Placide Cappeau, a wine merchant who was republican, socialist, and anti-clerical, claimed to have written it on a trip to Paris) and goes on to explain the actual shared credit and where it was written. Interesting choice of writer, but clearly Maurice Gilles knew his man.
The guest on the program said abolitionists made the song popular in the US.
UPDATE: Fixed the "translation."
### Boredom again
Always check to see if someone said it better, before clicking "Publish" ...
Because children have abounding vitality, because they are in spirit fierce and free, therefore they want things repeated and unchanged. They always say, "Do it again"; and the grown-up person does it again until he is nearly dead. For grown-up people are not strong enough to exult in monotony. But perhaps God is strong enough to exult in monotony. It is possible that God says every morning, "Do it again" to the sun; and every evening, "Do it again" to the moon. It may not be automatic necessity that makes all daisies alike; it may be that God makes every daisy separately, but has never got tired of making them. It may be that He has the eternal appetite of infancy; for we have sinned and grown old, and our Father is younger than we.
Chesterton, Orthodoxy
## Thursday, December 21, 2017
### Tolerance for Boredom
We took some Chinese students to the "traveler's Christmas Eve service" tonight. The music was the old carols put to a slightly different tempo. Everybody caught on pretty quickly.
At the end of one song the lead singer took the last two words very slowly. Everybody else sang at the expected speed. Mismatch, and not in a good way...
I propose that one of the characteristics churches should look for in music leaders is a vast tolerance for boredom.
You, and all the popular singers of the past decades who cut Christmas albums, are able to jazz up the old familiar tune with some new musical twists. That's nice. If you plan to involve the congregation in singing, you have to let them know what to sing and when. You've heard that version of that song hundreds of times; you probably feel like WalMart employees hearing the music loop for the 325'th time today. I'll bet you're bored.
I know what boredom can do. I remember planning a Bible study by collecting all the obscure details around a passage. That was cute, but what I thought was illustration was just clutter unless the point of the passage came first. We'd all read it before, but it doesn't hurt to cover the basics again.
"But what's the point of all my training if all I do is the same old stuff?" It's not about you; not about me.
If we're going to serve at all, we have to be willing to be bored.
## Wednesday, December 20, 2017
### Same coin
I'm not up to speed on Cornel West and Ta-Nehisi Coates, and when I saw Althouse's post I figured I should get some notion of what those ideologues were up to these days. One of the comments led me to a NYT article explaining how Coates and Richard Spencer both believe in the primal importance of race and the overwhelming power of whiteness, and where Williams quotes Spencer as saying "This is the photographic negative of a white supremacist. This is why I'm actually very confident, because maybe those leftists will be the easiest ones to flip." I suspect that his confidence is not altogether unjustified, btw--fashions can turn on a dime and the people noisiest about being allies typically don't seem to be investing any of their effort or money in helping individuals. Maybe they hide it very well, not letting the left hand know what the right is doing?(*)
I wrote about the same sort of thing years ago: both right and left in Madison shared a belief in the omnipotence of the USA. If you point out that problem X is beyond our capacity to solve, you get called defeatist or a tool of the oppressors.
"We may be sure that the characteristic blindness of the twentieth century — the blindness about which posterity will ask, "But how could they have thought that?" — lies where we have never suspected it, and concerns something about which there is untroubled agreement between Hitler and President Roosevelt or between Mr. H. G. Wells and Karl Barth." (On Reading Old Books, C.S. Lewis)
(*)"Christ did not love humanity, He never said He loved humanity; He loved men. Neither He nor anyone else can love humanity; it is like loving a gigantic centipede." Chesterton
## Sunday, December 17, 2017
### Defined by relationships?
I noticed that behind the scenes most math is defined in terms of relationships, operations, mapping, and interaction. Even things like the ordinary counting numbers—you can find a definition of a non-negative integer in terms of the characteristic unique to all sets that have that many elements. (That sounds a bit circular, doesn't it?) One hot topic tries to treat vast swaths of math in terms of objects and morphisms.
This has been extremely fruitful.
The same sort of approach of defining things in terms of their operations seems to be able to produce quite complex machines, whether physical or organizational. It is very convenient to define people-roles in terms of their interactions and operations: "We need N of type AAZ and M of type CBZ." The more interchangeable the components are, the more efficiently you can feed and run the system—at least in theory. If everyone is identical except for minor education and training differences, you can feed as many of each people-role as you need into the hopper, and shift them around like checkers. (It helps if they have no personal attachments, so you can ship the components around the world at will.)
But this kind of abstraction is well known to induce some push-back from the unique individuals who find themselves thus categorized. And, funny thing, there turn out to be irreducible differences that make people non-interchangeable for some of those people-roles.
And the apparent circularity of some definitions strongly suggests to me that there needs to be some room for some fundamental objects in mathematics. I haven't run across a good definition of a "geometric point" that isn't circular in some way.
And if you only think of defining things by their transformations and mappings, you don't have much mental space for thinking about things like Forms or aseity. (Does that mindset account for the rise of process theology?) People and things are contingent and largely defined by relationships (just don’t try to abstract too much away!), but that doesn't prove everything is.
## Saturday, December 16, 2017
### Shortcuts to forgetfulness
'All of humanity's problems stem from man's inability to sit quietly in a room alone.' Blaise Pascal If you've electronic media, you're not quite alone--imagine sitting quietly in a room without them, and without books or anything to write on or play with. Just being alone--can you do it?
This morning in Bible study one man cited the received wisdom that "You have to love yourself before you can love others." I wondered at the group if that was 100% true. It seemed sometimes as though what was really needed was self-forgetfulness in admiration of the beloved. We didn't come to a conclusion on my counter proposal.
I'm convinced that a certain kind of self-forgetfulness can be a good thing. If your life and loves are not ordered properly and you don't do a little self-reflection/self-examination, you won't get out of your trouble; you'll just dig yourself in deeper. I don't mean we should be feckless.
"Drunken monkeys" or an incessant judgmental narrative isn't the way our lives are supposed to work. Our motors are misfiring badly, and it is painful to listen to ourselves sometimes. Or perhaps the observing swamps the doing or the being, and we get into trouble. Ask a player to explain how he is catching the ball, and watch his performance crumble. When you're in the groove, you don't think about how you feel.
Achieving "the groove" is hard, and so is quieting drunken monkeys. There has to be an easier way.
Amusements: a-"muse"-ments--things to keep us from thinking (e.g. about ourselves) for a while. "Drown his sorrows in a bottle" is a cliche--and therefore likely true of a lot of people, who I guess are looking for self-forgetfulness. Or something magnificently exciting--sex, or (for some of us) sports.
We can't sit still. So--Netflix. Or chemicals to force us to be mellow about ourselves.
## Friday, December 15, 2017
### Armor
The National Interest story on the Kugelpanzer assumes that this one-man "tank" was supposed to be--well--a tank. (The picture for the story is completely irrelevant.)
"Thus, the Kugelpanzer was likely designed with some other purpose in mind." I can take a guess.
They didn't have the technology to build exoskeletons, so they made up a kind of personal armor instead. I think that instead of Iron Man flying through the air, they wanted supermen rolling around the battlefield. 5mm of armor isn't a lot, but I'm imagining a whole bunch of these coming over the hill at once...
Yes, you're right--a maintenance nightmare and expensive as all get-out.
## Monday, December 11, 2017
### Diversity
Over a dozen years ago I wrote about diversity in a university setting. If I were writing it today I might revisit some of the things I wrote about learning styles.
My bottom line was that "diversity" is not a goal but a means to an end.
Note carefully: I am not saying that invidious discrimination is justifiable. That is a different issue. I am saying that "diversity" as such ought not be made into a goal.
Can you think of a single case in which diversity is not merely a means?
• A diversity of ethnic restaurants : stimulate a jaded palate.
• A diversity of research groups in a university department : sometimes you get cross-fertilization. Such groups have finite lifetimes, and if there’s only one group incoming students have no research to join when it dies.
• A diversity of viewpoints on the jury : look at the question from as many sides as possible to arrive at the truth
• A diversity of ethnicities in kindergarten : if that's what the neighborhood is—you want everyone to have a basic education
• A diversity of ethnicities in a church : the church is catholic—everyone God made is called
• A diversity of peoples on Earth : OK, this one is above my pay grade, but I suspect the reason was to have as many ways to display and share facets of God’s goodness as possible. We've messed the goodness part up.
• Mandated diversity : full employment for the diversity professionals
Because it is a means and not an end, diversity can fail to accomplish the end, or even prevent it. For example, a completely diverse jury would include Mafiosi, and if you have too many research groups in a department they are too small to do any work.
When you confuse means and ends, you distort the ends and don't do a good job with the means. If "Diversity is one of our goals" in a research group, that tells me that they no longer care wholeheartedly about truth, but want to employ people on the basis of something other than understanding they bring to the table. They try to become a "full employment agency."
## Sunday, December 10, 2017
### Pies
I need to keep an eye out for Pie: A History. From BBC:
The cases, which could be several inches thick, according to Janet Clarkson, author of Pie: A History, were perhaps not even intended to be edible. Even once fat had begun to be added to the dough, bringing us into the realm of modern pastry, a pie crust was still sometimes considered more as a kind of primitive Tupperware.
A well-baked meat pie, with liquid fat poured into any steam holes left open and left to solidify, might even be kept for up to a year, with the crust apparently keeping out air and spoilage. It seems difficult to fathom today, but as Clarkson reflects, "it was such a common practice that we have to assume that most of the time consumers survived the experience".
## Saturday, December 09, 2017
### I Sleep in Hitler's Room, by Tuvia Tenenbom
I wish Tenenbom had tried to use different fonts to distinguish observations, fantasies, and questions. Other people's responses are in quotations, but he mixes then and later musings together freely.
Early on in I Sleep in Hitler's Room he meets a friendly hardline Nazi in Club 88, who thinks he’s a fellow-traveler. He is appalling. So are the friendly Turks and other Muslims who also hate Jews. So is the willful blindness of the other Germans and the media to the Muslim hatred of Jews. Thus far there's nothing terribly controversial in the book—if you look hard enough you can find Nazis, Muslim attitudes towards Jews are well known, and so is the make-believe about those attitudes.
He "discovers" the equally-well-known connection between leftist politics and detestation of Israel. Since German politics tends left—surprise! Disproportionate condemnation of Israel. (When challenged about Chechnya or other problems, most of those he talks to seem to have no notion of what he's talking about.)
Having read The Lies first, I’m a bit suspicious of his sampling for this book. He claims in the preface that it is representative, and tells the story of how his publisher screamed at him and refused to publish the book without multiple changes and deletions. (The publisher tells a different story.)
This is important, because one claim that crops up over and over is that the Jews run finance and governments. Is the attitude really that widespread?
Tenenbom makes numerous wry references to how he ought to spend his share of this vast wealth Jews allegedly control. It is humorous at first, but after a while I noticed how much he was spending. No, he doesn’t run Goldman Sachs, but a New York theater director doesn't seem to have to make the same hard financial choices as most of the rest of us.
That Jews are disproportionately represented in such positions is well known. It is perhaps less well remembered that they are also disproportionately represented among Nobel Prize winners and other measures of accomplishment (as opposed to control). So perhaps the attitude is widespread. It doesn't appear in the circles I frequent. It does show up in online comment sections--but I've no way to estimate how common it is in the general public.
One scene, in which a family invites him to dinner, ends with him leaving the man crying. Tuvia doesn't come across as the most pleasant of guests.
He finds a staggering number of Germans who allege a Jewish grandparent, and pretty much everybody asserts that either their parents 1) had no idea what was going on or 2) never talked about it. He also finds references to Israel or the Holocaust everywhere, and professes to be annoyed with it.
Two of his favorite opening questions are "Are you proud to be a German?" and "What does it mean to be German?" The latter is probably not answerable, and the former isn't much better.
IIRC, after WWII, the Allied powers had a problem: they could assert (with some accuracy) that the bulk of the Germans were complicit in crimes, and try to punish accordingly. Or they could distinguish Nazis from normal Germans, and blame the Nazis—who were plainly more guilty. What eventually resulted seems to have been a hybrid: officially the Nazis are blamed and ordinary Germans absolved, but unofficially everybody equates WWII Germans and Nazis, and blames Germans in general. The former seems like a recipe for encouraging people to try to hide everything, pretend it didn't happen, and try not to draw attention to themselves—and maybe the poison would decay away with the next generation. In practice it seems to me as though people were asked to take a kind of attenuated blame for something they felt officially absolved for. I wonder if that would encourage ways of "baming the victim." Mix that (especially among the guiltless second and third generations!) with the popular leftist rule of "blame the powerful," and concentrating on Israeli villainy seems to follow naturally.
Back to Tuvia: He concludes that German anti-Semitism has "to do more with the psychological history of the German than with thought-out anti-Semitism." "Polish anti-Semitism, as far as I can tell, is grounded in religion. Germany’s is grounded in psychology and narcissism." (ditto for Islamic anti-Semitism) "It will be much easier to make peace between Israelis and Palestinians, and between Arabs and Jews in general, than to uproot the Jew hate of the German. The first two are on the table, no surprises; the third is wrapped in heavy brainy arguments and eye-blinding magical color shows in addition to being hidden behind the many masks so common to our present-day Western culture."
Hold the phone. Jew-hatred grounded in religion is "on the table?" Tuvia was raised Orthodox, but is no longer religious, and it shows.
## Friday, December 08, 2017
### Tuvia Tenenbom
AVI pointed out a link about antisemitism in Germany. The source for the article is a book by Tuvia Tenenbom, a journalist writing about people he talked to in Germany. I Sleep in Hitler's Room hasn't arrived at the local library yet, but The Lies They Tell was handy.
Tuvia is Jewish, but usually pretends to be German in this picaresque tale of his experiences during a 6-month tour of the US. He likes to go to the strange or dangerous places. And he wants, in particular, to find out how/why people like or dislike Jews. (And why Jews seem to detest Jews.) And whether belief in climate change correlates with dislike for Israel.
He tells the stories well, and professes to have discovered unexpected delight in the American landscape, in driving, and even in shooting. In the end, he determines that Americans are afraid to speak, racist, and rather hypocritical--and, as one rarely finds in the world, ashamed of being tribal.
The problem with his conclusions is that he picked and chose the people he wrote about in depth. I don't believe he met that few people on his trip. Drama and contrast he wanted--that's what he put in.
He might object that the majority of the people along the way were busy with their phones or their netflix and weren't available to interact with him. But seriously--what fraction of the people in the USA own 100 guns? And I generally don't have any difficulty in learning who people voted for--they often volunteer that. (Whatever became of secret ballots?)
In one chapter he interviews Untermeyer, who was unaware of the depth of Jew-hatred in officialdom of Qatar and Saudi Arabia--ambassadors get escorted in, and don't see their documents rewritten to have a birthplace of New York rather than Tel Aviv.
What seems to leave him most aghast is the way people fret over Palestinians without a care for the homeless a few blocks away. That seems a bit overdrawn to me--I know people who are somewhat like that, but there's a little nuance in their attitudes that Tuvia didn't see, or didn't report.
He understands enough to know that you have to visit churches if you want to know the people here, but he's pretty tone-deaf. His "superiority" grates after a while.
Yes, read it--people like them are out there--but don't trust his conclusions.
And when the book on Germany arrives, I'll try to calibrate his reporting on Germany accordingly.
## Saturday, December 02, 2017
### Showing love
Love came to Earth as someone who needed to be loved. His first gift was an opportunity for us (in the persons of Mary and Joseph) to love.
### Citizen science
A writer for Aeon is deeply suspicious of "citizen science", judging it to be a scam to get free labor for big businesses.
The very label ‘citizen science’ (as opposed to, say, ‘amateur’ or ‘extramural’) carries the unsubtle suggestion that science should be a participatory democracy, not an unpalatable, autocratic regime. Proponents claim that it has all manner of salutary side-effects. People will get the knowledge they want through direct action, it’s argued, instead of having it shoved down their throats by some Ivy-league elitist. Getting a hands-on appreciation for research will help to dispel the worrisome doubts that certain citizens now possess about the legitimacy of scientific authority. And when it comes to medicine, discoveries of novel therapies are increasingly rare, despite the desperate manoeuvres of the pharmaceuticals industry; citizen participation should speed up research and make it much easier to replicate results. Finally, the retraction and replication crises that have besieged academic journals suggest that ‘proper’ science might not be so proper, anyway. Perhaps it’s time to consider alternatives.
(There are several straw men in that passage. Can you count them?)
and
But things lose their lustre when you look a little closer. It’s not a coincidence that citizen science lowers the cost of research that requires lots of routinised labour. Thankfully, we’re flush with design tools that manage to transform repetitive, mindless behaviour into something strangely fun and addictive: games. Galaxy Zoo, a non-profit, amateur astronomy project initially set up with data from the Sloan Digital Sky Survey, asks participants to scan millions of celestial images for common galactic morphologies; to keep their attention, players can spell out words with constellations, or win points for certain cute galactic structures. Smartfin, from the Scripps Institution of Oceanography at the University of California, San Diego, gets surfers to attach a sensor to their boards and collect data on salinity, temperature and the like, all of which is pinged back to Scripps once the surfer makes it back to the beach and hooks up the fin to a smartphone. Hundreds of ‘camera traps’, scattered around the Serengeti National Park in Tanzania, capture images of creatures that can then be identified by users at Snapshot Serengeti, thus keeping track of animal populations; to amuse themselves, people can attach comments to their favourite photographs (lolgoats, perhaps, rather than lolcats).
And he goes on from there to find what he considers dubious funding sources and worry at length about who benefits from all this.
Do people who participate in these things consider themselves scientists? Or do they think of themselves as assistants? Collecting data is one thing, figuring out how to use it is another.
NSF-funded experiments such as IceCube are required to make their data public, but to get something meaningful out of it requires some disciplines that most people don't pick up on automatically. We're very good at pattern recognition, but sometimes the first pattern you see doesn't actually tell you what you want to know.
A for-instance: you can use the IceCube data to discover that there are seasonal changes in the number of cosmic rays you see. The effect is easy to spot, and someone naively looking at plots might think they'd discovered something new and mysterious. What happens is that at ground level you see the remnants of cosmic ray showers that begin in the upper atmosphere. When the air is warmer (summer), it expands higher, and the cosmic ray showers start higher up. (We keep track of best estimates of upper atmosphere air temperature to go along with our data.)
Or you could use something like those population density maps in the cartoon above to discover that there are more crimes where there are more people. Not a surprise: if you look instead at the number of crimes divided by the number of people (the rate), you'd find that the distribution doesn't look the same--some places with more people have higher crime rates, others not so much. You could see how the violent crime rate varies with the rate of car ownership, or density of bars, or rate of single parent households. It isn't hard to think of things to compare it with, and with a little training you can figure out how to study the problem in one variable. I was going to say "It isn't rocket science," but maybe that's misleading. Keeping track of multiple variable is harder, and figuring out which are correlated with which takes quite a bit of care. (Quiz--if you use the number of schools in an area as one variable, should you also use the number of children as a variable at the same time?)
The basic disciplines that science requires are things I think most people can acquire at some level: how to think about analyzing a problem into its "moving parts," to be strictly honest and willing to challenge your own hypotheses, and so on. Those are good disciplines to have. But studying complex problems is hard enough that most people don't care to invest the time--and some can't manage the math that usually turns up. But so long as I don't delude myself into thinking I'm Rembrandt, I think doing a little drawing myself is good. It can help you see. Likewise, learning to do a little scientific analysis can help you see.
Justin Vandenbroucke developed a cool cosmic ray detector that anyone can carry with them. If enough people use it, the distributed data collected might be useful in discovering patterns in cosmic ray fluxes in the Earth's magnetic field (for example). Right now it is mostly just educational. And most of the people running the app are concentrated in a few places in the US and Europe, so the detectors don't have a lot of planetary coverage.
Spencer Axani designed a little box muon detector that lights up when a charged particle goes through. He had a stack of these in the lab across from my office, and you could sometimes see where several lit up in a line. One of these boxes is a toy. A stack of them is a demonstration system. If there were a way to collect data from them remotely, a hundred thousand spread around would be a cosmic shower detector.
Having a cosmic ray detection app, or a box, doesn't teach me much about science, or how it works. That's a shame. But it helps teach about what's around us that we don't notice--just like the people counting moth populations.
## Thursday, November 30, 2017
### Susu club
I ran across the phrase in Liberia news and looked it up. A Susu club (non-profit variety) is a group with an agreement to bank with a trusted member a certain amount each month, on the understanding that each member receives the total amount one month. (Or week, or whatever.) For example, 12 people get together and each chips in $10. The first month A gets$120, the second month B does, and so on.
So what's the difference between doing this and saving money in a bank or putting it in a mattress? First, there's the chance that you might get the lump payout before the year is out. That's an attractive feature. Second: well, read the complaint in that link: "So, the main reason for paying into a Susu is that the members lack fiscal discipline, and spend whatever money they have on their hands."
That's not a nice way of describing the situation. True, many people are no good at planning for the future. But in Liberia, and many other places, it isn't just you who determines how your paycheck is spent. And family obligations are extremely elastic. If you have $10 extra, your third cousin will explain to you that his child needs school fees. It is very bad form to stiff your family. But if the money is in a susu, it is out of your hands. When the$120 finally comes around to you--well, that's enough to replace the roof, which is what you needed the money for in the first place.
## Wednesday, November 29, 2017
### You got it right
We were reading over Isaiah 54 this morning, and one verse stood out for me: "In righteousness you will be established." You don't commonly run across "righteousness" outside of religious discussion, but think about it a moment. People sometimes literally prefer to die rather than admit they did wrong, or even that they were wrong. Confront them with evidence that they've screwed up, or been a jerk, and they double-down on self-justification. There is no way they can not have been in the right all along. (AVI noticed the same thing in an acquaintance). We hunger and thirst to consider ourselves righteous.
A welcome we want to hear from the One who can judge is "Well done!" I think another is "You were right."
## Monday, November 27, 2017
### Matthew Paris
I'm reading Chronicles of Matthew Paris edited and translated by Richard Vaughan.
It has fascinating details and hints of completely different systems. E.g. "In the time of this abbot the church of Norton was granted to us. Its rector, Lawrence the Clerk, resigned it with spontaneous devotion for the improvement of our beer and to provide supplies for the guests additional to what the abbot had been used to distributing."
## Sunday, November 26, 2017
### HipHop
Clubs that play HipHop in Madison tend to have lots more police calls than other clubs. I wondered if the music itself inspired violence, so I went to MetroLyrics for the top twenty HipHop songs. That's not a reliable indicator of what's in the clubs, but it might be indicative.
RankPerformerTitleThemes
1Big Shaq Man's not hotUnfamiliar jargon, gun violence, superiority of men, humorous?
2Lil Pump Gucci GangConspicuous consumption, women are disposable toys, sexual dominance, drug use
3CupkKake DeepThroat I do felatio and rough sex
4Jake Paul It's Everyday Bro I'm getting rich and famous
5Quest Walang Hanggan In Tagalog. Why don't you love me anymore?
6T-Pain Apple Bottom Jeans Saw a sexy dancer, threw money at her and had sex with her
7Don Omar Danza Kuduro In Spanish. Dance and move for me
8Eminem Rap GodI'm a way better rapper than you, don't criticize me.
9Post Malone Rockstar I'm living like a rockstar, with disposable women and drugs and fame and people I can call on to kill you.
10Eminem Lose YourselfLive in the music and go for the glory and use the hate, and superstardom isn't so great.
11Cardi B Bodak Yellow I'm rich now, not a stripper anymore, and you women are inferior. Sexual dominance too.
12Snoop Dogg Smoke Weed Everyday I use drugs, hang out with drug dealers, and persuaded my woman to smuggle a 44 into the event.
1350 Cent In Da Club Party in the club, sex with disposable women, I've a small army of fighters with me, survived gunshots, I'm rich, rich.
14Chris Brown Look at Me Now I'm rich now, your women want me, and I will be the top. References to dominance and murder.
15Yo Gotti Rack it Up I'm rich with lots of women.
16Sir Mixalot Baby Got Back I want women with big bottoms.
17Yicki Yohe Because of Who You Are. Worshiping God. Not HipHop as I understood it, but somehow on the Metrolyrics list.
18Miguel Echame A Mi La Culpa In Spanish. You deserted me, but I loved you and hope you will be happy. Blame me if you must.
19Baby K Voglio ballare con te In Italian. I want to dance with you again, till the sunrise.
20Lil Peep Benz Truck Conspicuous consumption, getting closer to dominance, fresh women. Lots of Russia references, possibly because of a large audience there.
Quite a mixed bag: No one club will run all of these. Despising/using women is pretty common in the English works, and lots of "look at how rich I am," but not quite as much implicit violence as I expected. True, a lot of the songs challenge other performers: do their respective fans quarrel about that? I've no clue. But I can easily see expressions of those attitudes towards women evoking both violently possessive and violently protective reactions.
Are there checklists in American HipHop like the stereotypical pickup/booze/jail/dog in country music? Some of the lyrics suggest it...
## Friday, November 24, 2017
### Harassment
When I see a persistent problem, I often try to look for what reasonable things might be being twisted to feed it.
I suspect that if you surround a man with deferential women with no other attachments visible, over time the man will tend to gravitate to either the father or the husband model--it's hard to be friends when you're the boss. And if it's the husband model, it may be hard to avoid feeling entitled to take liberties with your "harem" to which you are not actually entitled. The boss chasing the secretary around the desk has been a staple figure for ages.
It gets complicated becuase if the man's position is powerful it is no trick to find women who "will to greatness dedicate themselves." Of course consent by itself doesn't confer entitlement, despite the current philosophical fashion.
Interlude: exhaustive list of observations at work.
The man I ended up working for after Prof. Cline left had a disproportionate number of women as grad students. My memory is iffy, but I know he had three and may have had four. Given how low the fraction of women in particle physics in the US was at the time, this was pretty dramatic. They were all at about the same point in their studies, so I figured they knew each other before, and when one decided they all joined with her. I didn't worry about it--not my business. I was not then, or for most of the rest of my career, involved with mentoring or advising students.
Several years later I learned third hand that two of the profs (since retired and/or died) had been notorious womanizers until some unspecified event (intervention?), and that the prof I worked for had a reputation for treating women well. Another prof (since died) seemed to treat everyone well--seemed to treat students like family--but wasn't on one of the most famous experiments. A few other profs were on experiments that were winding down (and therefore less attractive) or perhaps had less good people skills.
One attractive woman (actually, they all were) set off my "risk, flee" alarms for reasons I never quite pinned down. She ended up suing one of the profs, but I never learned the details.
Once I found myself in a lab with three other men (two senior to me) and a secretary. The conversation was benign, but the vibes were all wrong. It felt like a dominance scenario, and the secretary looked a bit like a deer in the headlights. I figured the best way to break the spell was to announce that I had work to get back to and leave. Dunno if it worked, or if it needed to--maybe my vibe-meter was out of calibration that day.
As a student, and grad student, and post doc, I was on the "less socially ept" end of things--and to this day find parties hard to enjoy. I've not found an unambiguous way to convey a compliment or that I admire someone--and have pretty much given up any efforts in that line unless it is very straightforward.
That's probably a shame. (I do compliment X's work to Y when I can, and try to keep "gossip" positive.)
Perhaps I'm wrong, but it seems to me now that many women dress to excite admiration but not lust, attention but not interaction. As long as isn't overdone, that seems innocent and proper all around--try to be admirable, and have the good character to recognize and admire the good in others. In a less socially chaotic society there can be rules to buffer the reciprocal contact, which could convey the respect or admiration without requiring commitment. Think of "a tip of the hat"--no demand for a response: unless the tipper happens to be the someone she wants to address.
That's a corner example and not enough even for this simple case--they say women dress for other women, and men certainly want to be admired by men as well as women. But you get the idea, I trust. It represents a more formal society, with more social rules--but who proved that was a bad thing? I've been around Aspies long enough to think it would be a huge improvement.
Some of the harassment and molestation stories say the bosses do more than just take liberties--some of them are into seriously weird dominance patterns. This isn't a kiss or a squeeze he's not entitled to, but things done to prove that you are inferior. I've heard that President Johnson used to have talks with people while he was sitting on the toilet--and probably not because he was so terribly busy that every second counted. The only recent example that seems printable is Franken's mock groping picture. He was pretending to molest his "toy," not for any pleasure he could get out of it but so people could see what he could get away with. I'm not familiar with his humor style, but if that's an example I don't want to be.
Are things better than they were fifty years ago? It's hard to be sure, but I'd guess in some ways yes. It hasn't been socially acceptable to "chase the secretary" and I'd bet that's had at least some impact, though not much at higher levels of money and power. At those levels I'd bet things are worse. And the word I hear is that BFI HR policies tend to hammer the less powerful men who run afoul of someone's ire. It is gratifying to see some of the high and mighty being addressed for the first time.
I've a simple touchstone for what I judge acceptable: How would I want my daughters to be treated in this position?
Permit me to doubt that the solution is to have women as managers. Women are not more virtuous than men and they go in for dominance patterns too. If you haven't seen it already, you probably will.
## Thursday, November 23, 2017
### Notes from Isaiah and Ezekiel
Reading in Isaiah and Ezekiel this week and a few things struck me. “And like a sheep that is silent before its shearers, so He did not open His mouth.” Suppose He had argued?
What happened when He answered Pilate? Or when he confronted those ready to stone the woman? (He wasn’t trying to defend Himself when He spoke to the Sanhedrin, and He didn’t give the entertainment-hungry Herod the time of day.)
Why not speak? The usual explanation is so that what needed to happen would happen, but when God’s involved I look for multiple reasons.
Given what He’d shown He could do with just words or writing in the dust, I suspect He could have argued Himself free from His captors. But would that have changed any hearts? Pilate decided Jesus was innocent, and tried to free Him, but lacked the courage to defend “Roman Justice;” and reports about his later life suggest that there was no conversion. If words were enough, words would have been used instead.
Alternatively, perhaps He had said all that He needed to, and they could pay attention to it or not.
Ezekiel 20 has the well-known warning that God “will not be inquired of” by the idolaters who sacrifice their children, presumably because they justify themselves instead of turning back. It also has the evocative phrase which I’d never noticed before: “and I will bring you into the wilderness of the peoples.” That’s a good description of the 1900-year diaspora. (It also describes how I feel in great city crowds.) Lots of people, but wild and unfriendly.
## Wednesday, November 22, 2017
### Neutrino cross sections
It turns out to be very convenient to describe the rate at which particles interact with each other with an "effective area". Think about it a moment--if you two rocks at each other, the wider the target rock is, the more likely they'll hit. And when you work out the dimensions for particle interaction rates, area=="cross section" is what you wind up with.
IceCube just announced its measurement of cross sections for high energy neutrinos interacting with ordinary nucleons. Nobody has been able to measure the rate for energies this high before--and the result looks pretty consistent with predictions.
That rules out some oddball theories--like leptoquark models. Leptoquarks turn up as a consequence of some theoretical models, and every now and then some unexpected signal excess spurs new interest.
You will probably have heard that neutrinos zip right through you without interacting--that you never notice them and never will. That's true for the most common varieties from ordinary radioactive decay. But higher energy neutrinos (very rare) interact more strongly, until at the level discussed in this paper, it is possible to tell the shadow of the Earth's core from the shadow of its mantle--with enough events. They're not quite so "ghostly" at these energies.
No, I am not on the author list
## Monday, November 20, 2017
### Melody
Granted that Western harmony is one of the great accomplishments of Western Civ: you can't please everybody at the same time.
(bass-baritone who does OK if somebody else sets the key and the music doesn't go too high)
I don't think I want to be a cello. Even though they do sound very nice.
## Friday, November 17, 2017
### Side note on a grim story
Indonesia is forcing pagan tribes to convert to Muhammadanism. The excuse is that they can't get birth certificates (and therefore schooling), unless they adhere to one of the recognized religions. They're not happy about it. "There is no compulsion in religion" is once again more honored in the breech than the observance.
One novel bit about the story was the relationships of the tribesmen to their Muslim neighbors. The tribe described is nomadic:
"We have no space to live. We are always told we are nomadic people with no religion, no culture," he told me.
"Our religion is not respected. The government is always insisting that we convert and live in houses in one place. We can't do that. Our way of life is not like that."
And there are sources of friction:
The officer, Budi Jayapura, took me aside to check my documents and said: "We need to watch over them.
"They don't understand the concept of stealing. They say the fruit grew by itself on the tree so it can be taken, but it was planted by someone. Maybe in their belief system it is OK, but not in our society."
The fact that they hunt and eat wild pigs also creates social tensions, he added.
"This is a Muslim community. If they see the pig's blood and the leftover bits, they are disturbed," the officer explained.
What is taboo, or haram, for the Orang Rimba directly contrasts with what Muslims eat, explains Mr Manurung.
"Orang Rimba will not eat domesticated animals such as chickens, cows or sheep. They think it's a form of betrayal. You feed the animal, and when it gets fat you eat it. The fair thing to do is to fight. Whoever wins can eat the loser."
I read of a visit to an Amazon tribe, where people hunted and ate every sort of animal, but if somebody brought it into the village and treated it like a pet, nobody would harm it.
### Collapse
Guy Middleton wrote: Do civilisations collapse? I suppose he wants to invoke Betteridge's law of headlines, and the thrust of his essay is "no." A lot of things remained after the "collapse," not least of which are the people and many aspects of the culture.
It's worth having a look at, for reminders of how complicated changes can actually be. But he overstates things. If you lose the "critical mass" of engineers and craftmen, certain things that once were part of the culture decay and a society may never get them back again. OK, cool--you still speak sort of latin and like garum. But the aqueducts in your valley broke and you don't have running water anymore--no more socializing in bathhouses. Your culture changes.
### Minting money
This morning an amusing little report circulated that one can buy a cypto-currency mining system that doubles as a space heater. Clever--take a feature that tends to be a nuisance, especially in consumer-grade computers--the high power requirements of the GPU(*)s used in the calculations that go into "crypto-currency mining," and spin it as a feature--the waste heat can heat your room! With 8 GPUs packed into that small a volume the water heat transfer system had better work well or you can set things on fire. That happened with some collaborators of ours in Maryland.
The whole crypto-currency business reminds me of the gold-rush folks. They looked to get rich by increasing the quantity of symbols of value--but not creating anything particularly valuable themselves. Thought experiment: suppose your country found boatloads of silver and gold (we won't go into details about how they collected it), and brought it back. Now you have the wherewithal to import more stuff, and you need to import--because, funny thing, having twice as many doubloons doesn't automatically double the size of your local industry. In fact it's simply apt to double the prices. Lots more gold, but not necessarily more stuff.
In Dawn Treader Lewis never bothered to explain why a lake that turned things to gold would be bad--except that people got greedy. Too bad, it would have been a one-liner.
## Wednesday, November 15, 2017
### Brainpower
One of the comments at Maggie's Farm reminded me of some experiments Richard Feynman did with himself: he could count and read, but not count and talk. "... when Feynman told mathematician John Tukey about this, Tukey could do the reverse — talk but not read. The reason was that Feynman would talk to himself in his head, while Tukey would see an image of a clock ticking over. Feynmann suggests this could be because people think differently".
The author of the article suggests that the brain has "modules" like a "sketch pad" or a "phonological module" for words and sounds.
I don't know about how other people use visual thinking in math, but I find that I often do. For example, matrix multiplication I visualize as an action. (I should redo this to slow it down). When I try to figure out the framework for a problem I draw pictures. Equations are partly sentences and partly blocks like pictures.
FWIW, I generally read by "see and say" because I'm already trained to spot the blocks in English words. Sometimes I scan too quickly, and hilarity usually ensues. But I learned phonetically. And when I hit non-English words, I work phonetically. If I have time. Train stops in Germany were a nuisance. The name was often half a block long, and I couldn't read it fast enough--so I read the first and last chunks of the name and hoped that was unique. Problem is, the last chunk was usually "strasse."
### Crab bucket
"Deaf singer Mandy Harvey made headlines around the world after being put straight through to the finals of America's Got Talent. But when she first took to the stage, she received death threats from within the deaf community for promoting a "hearing" activity."
I've heard unpleasant things about Gallaudet University too. There's something very nasty about denying reality for the sake of your pride. They pay a weird homage to the very thing they ought to fight--the notion that someone with fewer skills is inferior. Instead of denying that lie, they implicitly accept it and claim that their skills and culture are equal.
"You are not your disability!" I've preached that, though I try not to be explicit about it. People get tired of hearing the same things. But it sure beats "I am my disability, and you are too, and you'd better get with my program."
### New wine in classic wineskins
I understand Amazon wants to make more Lord of the Rings-based movies: prequels, I gather.
If they use existing characters, even peripheral, in a prequel, the story gets cramped. If they use new ones, it is more of a "in the universe of LoTR story," but likely without an equivalent story-teller behind them. I haven't heard any enthusiasm for working from the Silmarilion.
But... Do you remember the stir when HarperCollins announced the plan to create new Narnia novels?
I’d pretty much forgotten about that—the article above, and several like it, date to 2001. I thought I’d heard a peep or two more recently than 16 years ago, but that may have been temporal foreshortening again.
I wonder if HarperCollins quietly shelved the “new novels.”
## Tuesday, November 14, 2017
### Significant?
We've had quite the uptick in drive-by shootings in the Madison area--mostly non-fatal, fortunately. Today's battle between two cars hit a third car and a nearby house.
I've noticed a lot of defaced license plates lately. Not weathering and fading (as in the last Wisconsin plate design fiasco), but scraped and dented. O(1%) of the cars I see are hard to read the plates on.
Maybe this is a new vandalism fad. That would be the best possibility.
UPDATE. O(1%) means of the order of 1%. Give or take a little.
The pilot in our Wednesday Bible study suggested that the defacement might be to foil the toll cameras.
## Sunday, November 12, 2017
### Accusations
I'm a bit conflicted. Nobody believes the casting couch ever went away, and I'm content to believe that the overwhelming majority of the accusations are true, or at least largely so. What facts do manage to leak out past the PR over the years don't inspire confidence in the moral fiber of Hollywood folks, and sometimes the PR itself is telling.
But I have to give the devil his due. It is easy to make an accusation in this atmosphere, especially if it is old enough or vague enough to be un-actionable. After all, if your accusation goes to court, you might get cross-examined. Hollywood is a famously backstabbing place, and if a friend of a friend puts your competitor in a bad light, you might be generously grateful.
So while I hope this shakes out some "bad actors" who think fame or power entitles them, I also hope we take care to vet the accusers too.
### Defending churches
Several local worship centers (including a Sikh temple) have decided to have guards. Attacks happen from time to time, and threats sometimes look serious.
Guards or not? I figure I'd rather die in church than in a hospital, but when you recall that the children are at risk too the picture changes a bit--we have an obligation to protect.
So far the odds are pretty good, and I'm not concerned.
Am I naive? I'm not in administration, and I don't see the threats. Our church has a volunteer team keeping an eye on things, but not armed with anything more lethal than a cell phone. We put that in place a few years ago when a nut case interrupted a service. High Point's board is going to vote on whether to have undercover guards--I'm curious about what they'll decide.
## Friday, November 10, 2017
### Peaceable kingdom
England in the 1600's isn't famous for being a nice time and place to live. But the article says it wasn't as bad as we think. "but the state was not simply stringing people up for occasional acts of petty theft. Quite often, judges and juries deliberately perjured themselves to ensure convicted thieves escaped the noose, usually by undervaluing goods stolen."
Homicide rates dropped over the century (in Kent from 5-6 per 100,000 to 3.6; in from 8-12 to 2 per 100,000). Pinker likes the idea that a strong central government means lower homicide rates--I suspect that you don't get a strong central government if the crime rate is too high.
### Selfishness and Charity
David Warren starts off with: "I think that if the “natural man” would vote consistently in his own interest, and by extension in that of his close family, the world would get along tickety-boo."
He doesn't expand much on that, unfortunately. I've suspected for some time that if groups were a little clearer about their interests, there might be fewer and not more conflicts. Proving that would take a lot of study and work with counter-factuals, and I've not world enough or time. But it doesn't take much effort to think of conflicts that started with exaggerated claims and fears. Others were and are unavoidable.
His main point is about spite and charity: charity deals with specifics. "The point I make is on behalf of reality. One’s neighbour — and even in this last instance a brute animal, who could have eaten me were she much larger and in better shape — is a real thing. Insofar as our charity is real, it is directed to real things. Insofar as we are “friends to humanity,” or “friends to the poor,” or “social justice warriors,” we are putting on a ludicrous show, in which spite adopts a pretence of charity."
## Tuesday, November 07, 2017
### Naivete
Lubos Motl has a post about the naivete of physicists:
Nima Arkani-Hamed, a top Western official in the Mao collider, sort of "courageously" says that if the tanks came to the Chinese streets again, he would probably join some local protests. His father had some "disagreement" with the Khomeini regime in Iran but Nima himself doesn't really get the evil of totalitarian systems, I think after many discussions with him. As Cheng nicely says about Nima's superficial response:
But his hypothesis, well-intentioned as it was, reveals a deeply simplistic, caricatured understanding of state oppression. True terror and totalitarian control come after the tanks have left the square, when blood is wiped off the streets, the history books, and the people’s collective consciousness, when a date becomes taboo, and when a simple question confirming the existence of the Party office exposes the Achilles’ heel of a grand project.
Exactly. Totalitarianism isn't about some cool scenes with tanks and blood in the street – and Nima's cool but totally superficial and symbolic "no" to such spectacular events. The true muscles of the totalitarian machinery only start to act after the tanks and blood are removed from the sidewalks (the same is true for the German and Soviet tanks in Prague in 1939 and 1968, too). The employees are being ideologically filtered, fired, or arrested, the history is often being rewritten..
Many of my colleagues, Western or Chinese, asked me about my priorities and whether I cared more about physics or human rights, as if these pursuits are mutually exclusive.
And that question is easily used to dispose of the "incorrect" people. If you say rights matter more, you aren't dedicated enough, and if you say physics--don't complain. You are there to be used.
(Lubos is Czech, and grew up under Communism.)
### NFL
I've heard quite a bit about fan disgust and lower sales and so forth, with counter arguments that there are too many broadcast games, which saturated the market. I'm not plugged into the fan zone, and in any event my observations wouldn't be representative of the broader market.
I thought one simple way to test for fan annoyance is to monitor season ticket sales. But if this article is correct, brokers buy a huge fraction of the season tickets, so any measurement will be indirect. Better than nothing, though.
### Computers make it all better
The Navy issued its report about the McCain collision.
Commander Alfredo J. Sanchez, "noticed the Helmsman having difficulty maintaining course while also adjusting the throttles for speed control." Sanchez ordered the watch team to split the responsibilities for steering and speed control, shifting control of the throttle to another watchstander's station. ...
However, instead of switching just throttle control to the Lee Helm station, the Helmsman accidentally switched all control to the Lee Helm station. When that happened, the ship's rudder automatically moved to its default position (centerline). The helmsman had been steering slightly to the right. ...
At this point, everyone on the bridge thought there had been a loss of steering. In the commotion that ensued, the commanding officer and bridge crew lost track of what was going on around them.
And there's more.
I suppose that sort of SNAFU is the default for militaries. If I read my history correctly.
## Saturday, November 04, 2017
### A newsreel from the other side.
You're probably seen "The Longest Day." The Germans made their own newsreel of the activities. They had a smaller pool of actions to draw from--and I think I see the same tank a couple of times.
## Friday, November 03, 2017
I found this cover of "The Church of Your Choice" by Dan McBride. The sound quality is quite mediocre.
Church of Your Choice: (Some of us are old enough to remember the slogan.)
"If you're tired of your sin, then we'll welcome you in. If you're not, you'll still feel right at home."
We'd all love for evangelism to mean just being winsome and attracting people to Christ. I'm perhaps a bit curmudgeonly to be adequately winsome, but it's still an easier goal than trying to be prophetic. And it's more pleasant to think we're all "close enough."
But we know what Jesus said about division, and about the world hating him and therefore hating his followers. And it isn't hard to recall people whose sins are pretty dramatic in our eyes, and who really ought to repent.
So how does one manage to be both accurate and winsome? Jesus said to welcome the children, who generally aren't motivated by a hunger for forgiveness. But he discouraged people who weren't "counting the cost."
Nobody said it was easy... Probably one big first step is not to act as though we've "arrived." (There are two kinds of Christians: those who struggle with besetting sins and those who've given up.)
## Thursday, November 02, 2017
### Free
I don't remember where I read the rule that "A free man is one who can bind himself." He can bind himself to a wife and to the still unknown children who follow. He can promise to work for someone. He can make an implicit commitment to tend his farm; feed and care for his livestock. Or he commits to solve a problem or write a book. Some commitments are short term, others life-long and extremely open-ended.
Freedom seems less like a status and more like a coin to invest. The one who never binds himself never shows fruit.
### Lone wolves
It seems popular to describe Islamist terrorists as "lone wolves," even when there's evidence for a network.
Perhaps the spokescritters-that-be have been sternly told not to divulge anything that might reveal how much we know about those networks. That seems the nicer hypothesis. Maybe it's true.
Alternatively, they may be trying hard to keep Americans from blaming Muslims in general for the incidents of fourth generation warfare. If so, I believe they are mistaken.
If the terrorists are guided by a network, the network can be tracked, understood, and selectively attacked. Lone wolves who spontaneously decide to become enemies can't be tracked. Dealing with them involves much less selective means.
A network is separable from the Muslim community: Muslim with associations = enemy, Muslim without = ordinary. If the model is "spontaneous conversion to enemy," this changes the mapping: Muslim=x% chance of being an enemy simply because he/she is Muslim. Maybe that's actually true, but there are some hints that it isn't.
If the folks in charge want Americans to not blame Muslims, I think they should steer away from the lone wolf narrative, and concentrate on the associations. (Unless they're trying to hide how much we do know.)
## Tuesday, October 31, 2017
### Dark Matter Day
I have been informed that today is Dark Matter Day, and that I am to circulate these videos:
We search for Dark Matter in IceCube on the assumption that it is made of particles that interact with neutrinos.
It doesn't interact with photons (dark matter particles have no charge: that's what makes them dark) nor with baryonic matter (nuclei), and all we seem to get are limits on its interaction with W and Z (weak interaction--related to radioactive nuclear decay). Since it seems much too odd that we should get two completely non-interacting types of matter in a Big Bang, that just leaves neutrinos for them to interact with. And maybe dark matter interacts with W and Z after all, which would give particles that would decay in turn, which should also give some neutrinos.
Anyhow, if the dark matter particle is unstable, it can decay into a couple of neutrinos. If there are several species of dark matter particle, heavier ones might decay into a couple of neutrinos and a lighter type of dark matter particle. Or they could fuse to make neutrinos. Anyhow, look for neutrinos. Which, happily enough, is what IceCube is designed to do.
So if you see an extra number of high energy neutrinos coming from places where you expect dark matter to accumulate--inside stars, or at the center of galaxies--you might be seeing neutrinos from dark matter particles decaying/fusing. That would be very interesting, and could give you some estimate of just how massive these dark matter particles might be.
Of course if the dark matter particles are very light, they won't accumulate in stars the way you expect(*), plus the number of low energy neutrinos from cosmic ray showers in Earth's atmosphere will probably drown out your signal. (Low mass dark matter would decay to low energy neutrinos.) So the assumptions matter.
Also have a look at the search for sterile neutrinos video. Neutrinos are hard to study...
(*) The cloud of dark matter particles would be much more spread out. It's kind of intuitive that "heavy stuff sinks to the bottom," but it's a little more complicated because the dark matter particles don't bump into things very much--they don't interact and slow down in ways you're used to.
## Monday, October 30, 2017
### Reconstructing history
There’s not a lot of written history about the area that is now Liberia before the American settlers came.
To know what happened, you have to piece together strands of evidence from widely separated records, and from traditions, language families, and genetic families across a fairly large area of Africa—and I am not in any position to do that myself.
I greeted the appearance of Between the Kola Forest and the Salty Sea with enthusiasm, and bought a copy. After reading the introduction, I'm starting to wonder. He seems to have his references to Ham a little mixed (may not be entirely his fault--references are inconsistent), doesn't seem to have a good handle on the history of slavery in Africa (all European and uniquely evil), UPDATED see full review conflates "blue-eyed Aryans" with all Europeans, and approvingly cites the "black Egyptian" claims. Sorry, I read Herodotus too, and can look at the pictures the Egyptians painted, and read some of their folk-tales—ancient Egyptians were not of sub-Saharan descent. Recent DNA studies bear out the obvious—modern Egyptians have more sub-Sarahan ancestry than ancient ones. That Nubia conquered the north a time or three I can believe--I'd be surprised if they didn't. That Nubians were the ancient Egyptians isn't possible.
If he can't get the well-documented stuff right, I wonder what admixture of imagination is going into the less-documented material? I shall continue—I expect to learn something—but I am less happy.
### Reference missing
There's a story about a faculty party at some university, during which the topic of Velikovsky came up. An astronomer said to the MidEast historian "His ideas about the planets are complete rubbish, but he makes interesting points about the Egyptian historical record." The historian replied that "That's funny—I thought his astronomical ideas were interesting, but his dynastic chronology was junk."
I paraphrase, and I think there were three involved. Does anybody know the original?
## Friday, October 27, 2017
### Anonymous calls
In the released Kennedy files there's a report about an anonymous phone call 25 minutes before the assassination. I begin to see one reason why so much was filed away for years.
The more notorious the crime, the more rubbish the investigators have to sort through.
Just for fun, how many explanations of that memo can you come up with in 10 seconds?
### Panic!
New Jersey Officials Warn About Marijuana Edibles Being Given Out As Halloween Candy Treats.
### Wikipedia
We all know (or I hope we do) that Wikipedia is very unreliable on any subject that involves politics or disputed social issues.
My experience with the science articles has been that they're not always very clear. The math sections, on the other hand, have been, on the whole very useful and complete. I often have to chase through a list of definitions, and I wish wish wish more mathematicians would use examples in their communications (they use plenty in their research!).
The Motherboard article on the subject is a bit over the top. OK, way over the top. I know plenty of practicing scientists, and I can't think of one who wants knowledge restricted to an elite, and on the contrary, quite a few who volunteer in gigs to explain things to youngsters.
The problem isn't that "you can imagine impenetrable writing as a defensive strategy wielded to scare off editor-meddlers." The problem is that the science writers a) don't have huge wads of time and b) don't really know their audiences. And c) want things to be accurate--the imprecision of everyday language can be terrible. (Think of all the different meanings of "energy" you find in popular language--from heat to personal vivacity to obscure mystical flow up through your lung when somebody pokes your foot with a needle.)
FWIW, years ago I proposed that our grad students be required to prepare a web "poster session" of their theses, with a target audience of high school seniors who have at least algebra and some physical science background. Somebody would have to create a network of web pages to explain the background, of course, which their thesis-pages would reference. Crickets
## Thursday, October 12, 2017
### Mine! Mine!
AVI illustrates an application of Acton's Law, with treatment beds.
### Excitement
This week is full of phone conferences. We issued a bulletin announcing a nice very high energy "track-like" event, and this events seems to coincide in direction and time with a gamma ray blazar.
It would be wonderful to be able to associate a neutrino with a known source (it turns out to be rather hard), and the presence of two kinds of messengers (neutrinos and gamma rays) would give hard evidence for some classes of models of what's going on there, and serious challenges to others.
Scuttlebutt holds that the recalculated direction doesn't point as cleanly at the blazar as the first estimates said, but we'll see--probably by the end of the week.
BTW, a muon neutrino produces a muon through a charged current interaction, and that muon track points back pretty well to where the muon neutrino came from. An electron neutrino produces an electron, and that showers very quickly--you get a dramatic amount of light but not much of a track to point with. A tau neutrino--we've apparently had some taus, but it has proved rather hard to distinguish them from the electrons. Some of the time the tau track should shower quickly like an electron, but sometimes we should get a "double bang:" an interaction and shower, followed by a short track and another shower--sort of like a dumb-bell. So far we've not seen the double bang.
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2023-04-01 14:41:51
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kapoor 19/11/18
### Help-Page-02 :: Search-Page for Proofs Content
SEARCH -PAGE FOR PROOFS CONTENT All content pages of the Proofs Program can be reached by Searching or by going to Home Pages of different categories of content.
Some Useful tips for locating what you want by entering a search key in the Search Box
You can reach this page from anywhere by typing Search-Page in search box;
3. You can go to a page with links to different Top-Pages by searching for Top-Page
4. You can get links to all categories of content by typing Start-Page in Search-Box
6. If you know what you are looking for, make use of search box with appropriate key words such as WKB, Bound State etc.
In addition to the above it is useful to remember that
3. Make use of links created automatically and displayed at the bottom on most content pages. This is useful for navigating to previous page, next page and to pages one or more Levels up.
kapoor 19/11/18
### Repository of Physics Lesson Units :: Top-Page
While Class Room Lectures, available elsewhere on this site, are designed to be stand alone, and without any "frills", Units of Lessons in Physics come with several features expected to be useful for a first time learner. There are no black boxes, all details are supplied, though sometimes in gray boxes. On the sidelines, emphasis on points likely to missed is provided. Questions, notes and comments etc have been embedded in the lessons which may help in improving the focus and communication of the subject matter.
$$\triangleright$$ BOOK: TABLE OF CONTENT can be expanded and collapsed by clicking there.
kapoor 19/11/18
### Quantum Mechanics == Cross-Roads == Home-Page
All quantum mechanics resources for a chosen topic can be reached from cross roads for that topic
kapoor 19/11/18
### Testing
Summary (or abstract) not available for this node.
kapoor 19/11/18
### $$\S 14.2$$ Harmonic Oscillator in Three Dimensions
Problems in Higher Dimensions
kapoor 19/11/18
### QM-Lecture Notes-14 Problems in Higher Dimensions
Problems in Higher Dimensions
kapoor 19/11/18
### Mechanics Simulations and Videos
Mechanics Simulations and Videos
Shared a post by
kapoor 19/11/18
### Documentation of $$~\LaTeX~$$ macros and special tricks :: Start-Page
In this section of pages I will document the macros that I have used for creation of content files uploaded.
It is planned to provide source code for all the content that has been created and made available. In order that an interested person may make use of these LaTeX source code files it is necessary that all the commnads and macros that have been defined be made available. Also it is considered important that all command definitions along with examples be documented.
While working on Proofs project a few special techniques to solve definite problems were required. These requirements as well as the solution found are also documented. It is hoped that some of these will have uses beyond the Proofs LaTeX source code files.
kapoor 19/11/18
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2018-11-20 21:00:33
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https://studyalgorithms.com/theory/what-is-segmentation-fault/
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Home Misc What is Segmentation Fault?
# What is Segmentation Fault?
0 comment
From Wikipedia:
A segmentation fault occurs when a program attempts to access a memory location that it is not allowed to access, or attempts to access a memory location in a way that is not allowed (for example, attempting to write to a read-only location, or to overwrite part of the operating system).
Segmentation is one approach to memory management and protection in the operating system. It has been superseded by paging for most purposes, but much of the terminology of segmentation is still used, “segmentation fault” being an example.
On Unix-like operating systems, a process that accesses an invalid memory address receives the SIGSEGV signal. On Microsoft Windows, a process that accesses invalid memory receives the STATUS_ACCESS_VIOLATION exception.
Segmentation fault is a specific kind of error caused by accessing memory that “does not belong to you.” It’s a helper mechanism that keeps you from corrupting the memory and introducing hard-to-debug memory bugs. Whenever you get a segfault you know you are doing something wrong with memory – accessing variable that has already been freed, writing to a read-only portion of the memory, etc. Segmentation fault is essentially the same in most languages that let you mess with the memory management, there is no principal difference between segfaults in C and C++.
There are many ways to get a segfault, at least in the lower-level languages such as C(++). A common way to get a segfault is to de-reference a null pointer:
int *p = NULL;
*p = 1;
Another segfault happens when you try to write to a portion of memory that was marked as read-only:
char *str = "Foo"; // Compiler marks the constant string as read-only
*str = 'b'; // Which means this is illegal and results in a segfault
Dangling pointer points to a thing that does not exist any more, like here:
char *p = NULL;
{
char c;
p = &c;
}
// Now p is dangling
The pointer p dangles because it points to character variable c that ceased to exist after the block ended. And when you try to de-reference dangling pointer (like *p='A'), you would probably get a segfault.
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2022-08-17 19:01:23
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https://oilandgasguru.com
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2022-07-03 17:45:41
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http://vision.snu.ac.kr/projects/partgridnet/
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# Improving Occlusion and Hard Negative Handlingfor Single-Stage Pedestrian Detectors
## Abstract
We propose methods of addressing two critical issues of pedestrian detection: (i) occlusion of target objects as false negative failure, and (ii) confusion with hard negative examples like vertical structures as false positive failure. Our solutions to these two problems are general and flexible enough to be applicable to any single-stage detection models. Specifically, our two solutions are as follows. For better occlusion handling, we update the output tensors of single-stage models so that they include the prediction of part confidence scores, from which we compute a final occlusion-aware detection score. For reducing confusion with hard negative examples, we introduce average grid classifiers as post-refinement classifiers, trainable in an end-to-end fashion with little memory and time overhead.
## Background and Rationale
### Two Critical Issues of Pedestrian Detection
Among many error sources of pedestrian detection, we are interested in two critical issues: (i) occlusion of target objects (as false negative failure cases), and (ii) confusion with hard negative examples (as false positive failures). First, occlusion is one of key practical difficulties in pedestrian detection, because real world scenes like street are often crowded with many people and various objects; thus observation with occlusion is much more common than that without occlusion. Second, in the scenes for pedestrian detection, there are many hard negative examples like vertical structures, trees, and traffic lights, because of which, models detect a lot of false positives, and they amount to a large portion of overall errors.
(a) Occlusion examples (b) Hard negative examples
### Existing Single Stage Object Detector
The single-stage models formulates the two stages of region proposal and classification into a single-stage regression problem to detect objects extremely fast. Starting from YOLO, more advanced models are emerging, including SqueezeDet+, YOLOv2, SSD, and DSSD. They all use a convolutional predictor to generate the final output tensor which contains values about box coordinates and class probabilities.
(a) Structure of the output tensor (b) Output formats of four methods per anchor
## The Causes of the Problems and Our Solutions
Our solution is to add auxiliary modules and corresponding tasks. Therefore, our solution can be integrated with virtually any kind of single-stage detectors and is trainable end-to-end. Also, the auxiliary modules require little overhead in terms of memory and computation time.
### 1) Occlusion Handling Method
The first task is specifically aimed at occlusion handling. We found that detectors tend to assign low confidence for occluded pedestrians. To handle this issue, our additional module predicts multiple confidence scores for each part instead of a single confidence score. We divide the bounding box area to $$M \times N$$ grid and train our module to predict whether each region is a visible area of a pedestrian. But this objective requires visible area annotations, in addition to the ordinary bounding boxes. Luckily, Caltech Pedestrian Dataset and CityPersons Dataset contain exactly that kind of annotations, so we used those datasets.
We further process this part confidence map with some part score generator to build part score and use the geometric mean of the initial confidence and part score as the final confidence. We tested two types of part score: max part score and soft part score. The first one simply max pool the part confidences. The intuition is that if some part seems to be a pedestrian, it is an occluded pedestrian and therefore, the score should be high. But max pooling might be too simple to utilize the patterns of occlusion. So, the second method is to pass the grid through a multi-layer perceptron with a single hidden layer and let it capture the plausible occlusion pattern. After we get this part score, we use it to refine the initial confidence.
### 2) Hard Negative Handling Method
The second task is about handling hard negatives. We add multiple classification layers to the lower layers with different resolutions. In each grid cell, the classifiers predict whether the grid area belongs to a pedestrian or not. In some sense, it is like segmentation task with chunky labels. The results of the grid classifications are called grid confidence map. To use these grid confidence maps to refine detection results, we first upsample each confidence map to the pixel level and average the confidence maps. Then, for each bounding box $$k$$, the confidences in the bounding box area are averaged to build confidence score $$s_k$$. Finally, we adjust the initial confidence by taking the geometric mean of initial confidence and confidence score $$s_k$$.
### Acknowledgements
The authors would like to thank Yunseok Jang and Juyong Kim for helpful discussions, as well as Sukyung Jeong for assisting to draw the figures in the paper. This work was supported by Samsung Research Funding Center of Samsung Electronics under Project Number SRFC-TC1603-01. Gunhee Kim is the corresponding author.
### Publication
Junhyug Noh, Soochan Lee, Beomsu Kim and Gunhee Kim.
"Improving Occlusion and Hard Negative Handling for Single-Stage Pedestrian Detectors."
The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.
@inproceedings{Noh:2018:PartGridNet, author = {Noh, Junhyug and Lee, Soochan and Kim, Beomsu and Kim, Gunhee}, title = {{Improving Occlusion and Hard Negative Handling for Single-Stage Pedestrian Detectors}}, booktitle = {The IEEE Conference on Computer Vision and Pattern Recognition (CVPR)}, month = {June}, year = {2018} }
This page uses a template from the project page of Satoshi Iizuka et al, "Globally and Locally Consistent Image Completion".
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2018-12-12 11:29:35
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https://math.stackexchange.com/questions/809060/rolling-a-dice-until-we-have-5-and-an-even-number
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# Rolling a dice until we have $5$ and an even number.
We roll a dice, until we have both: a five and some even number. Let X be the expected number of rolls. Find expected value of X and Var(X).
So I don't know how to begin. I think there should be something like $X= X_1 + X_2$ And then I would use geometric distribution. But how to divide X into easier events? I thought about situations like rolling dice until we have both even and odd number. It's easy, because probability of obtaing an odd number is the same as obtaining even number. The same situation - if we roll until we get $5$ and, let me say, $3$. There is no problem since getting $5$ has the same probability as getting $3$. But getting a $5$ and getting an even number have different proabilities, so it's not so easy... Can somebody help?
• The expectation is easy as the probability you hit a five or even number is $\dfrac46$ so the expected number of throws is the reciprocal of this, and an even number is three times as likely as a five. So the expectation is $\dfrac34 \times \dfrac64 \times \dfrac61 + \dfrac14 \times \dfrac64 \times \dfrac63$ – Henry May 25 '14 at 18:14
• I'm getting $\frac{13}{2}$ from an absorbing markov chain. – gar May 25 '14 at 18:24
Let $X$ stand for the number of rolls needed to get an even number and a $5$.
Let $Y$ stand for the number of rolls needed to get an even number.
Let $Z$ stand for the number of rolls needed to get a $5$.
Let $E$ denote the event that the first roll is even.
Let $F$ denote the event that the first roll is a $5$.
Let $G$ denote the event that the first roll is a $1$ or a $3$.
$\mathbb{E}Y=\mathbb{E}\left(Y\mid E\right)P\left(E\right)+\mathbb{E}\left(Y\mid E^{c}\right)P\left(E^{c}\right)=1+\frac{1}{2}\mathbb{E}Y$ leading to $\mathbb{E}Y=2$
$\mathbb{E}Z=\mathbb{E}\left(Z\mid F\right)P\left(F\right)+\mathbb{E}\left(Z\mid F^{c}\right)P\left(F^{c}\right)=1+\frac{5}{6}\mathbb{E}Z$ leading to $\mathbb{E}Z=6$
$\mathbb{E}X=\mathbb{E}\left(X\mid E\right)P\left(E\right)+\mathbb{E}\left(X\mid F\right)P\left(F\right)+\mathbb{E}\left(X\mid G\right)P\left(G\right)=1+\frac{1}{2}\mathbb{E}Z+\frac{1}{6}\mathbb{E}Y+\frac{1}{3}\mathbb{E}X$
This leads to: $$\mathbb{E}X=\frac{13}{2}$$
Likewise $\mathbb{E}Y^{2}$, $\mathbb{E}Z^{2}$ and $\mathbb{E}X^{2}$ can be found.
To make it a bit shorter: the distributions of $Y$ and $Z$ are geometric.
That gives directly $\mathbb{E}Y=2$, $\mathbb{E}Z=6$ (allready found alternatively), $\mathbb{E}Y^{2}=6$ and $\mathbb{E}Z^{2}=66$.
We go on with:
$$\mathbb{E}X^{2}=\mathbb{E}\left(X^{2}\mid E\right)P\left(E\right)+\mathbb{E}\left(X^{2}\mid F\right)P\left(F\right)+\mathbb{E}\left(X^{2}\mid G\right)P\left(G\right)$$ $$=\frac{1}{2}\mathbb{E}\left(1+Z\right)^{2}+\frac{1}{6}\mathbb{E}\left(1+Y\right)^{2}+\frac{1}{3}\mathbb{E}\left(1+X\right)^{2}$$
$\mathbb{E}X^{2}=69$ and $$\text{Var}X=\mathbb{E}X^{2}-\left(\mathbb{E}X\right)^{2}=\frac{107}{4}$$
Here's a heavy-handed approach. After zero or more rolls, you are in one of four situations:
\begin{align} \emptyset:&\qquad\textrm{No 5 or even rolled yet.}\\ E:&\qquad\textrm{Even was rolled, but no 5 yet.}\\ 5:&\qquad\textrm{A 5 was rolled, but no even yet.}\\ *:&\qquad\textrm{Both 5 and even have been rolled. Game over.}\\ \end{align}
The transition matrix of probabilities between each pair of situations is easy to compute:
$\begin{array}{l|cccc} \nearrow&\emptyset&E&5&*\\ \hline \emptyset&\frac{1}{3}&\frac{1}{2}&\frac{1}{6}&0\\ E&0&\frac{5}{6}&0&\frac{1}{6}\\ 5&0&0&\frac{1}{2}&\frac{1}{2}\\ *&0&0&0&1\\ \end{array}$
So this is now modeled as a absorbing Markov chain with transition matrix
$\left({\begin{array}{cccc} \frac{1}{3}&\frac{1}{2}&\frac{1}{6}&0\\ 0&\frac{5}{6}&0&\frac{1}{6}\\ 0&0&\frac{1}{2}&\frac{1}{2}\\ 0&0&0&1\\ \end{array}}\right)$
The final state being listed last, the behavior is characterized by the $3\times3$ matrix in the upper left, which is the transition matrix for the non-final states.
$Q=\left({\begin{array}{ccc} \frac{1}{3}&\frac{1}{2}&\frac{1}{6}\\ 0&\frac{5}{6}&0\\ 0&0&\frac{1}{2}\\ \end{array}}\right)$
The so-called fundamental matrix $N$ for this chain is
$N=(I-Q)^{-1} =\left({\begin{array}{ccc} \frac{3}{2}&\frac{9}{2}&\frac{1}{2}\\ 0&6&0\\ 0&0&2\\ \end{array}}\right)$.
The expected number of steps from the $i$-th state to the final one is the sum of the entries of the $i$-th row of $N$, or equivalently the $i$-th entry of the matrix
${\bf t}=N\mathbb{1}=\left({\begin{array}{c} \frac{13}{2}\\ 6\\ 2\\ \end{array}}\right)$,
so for the starting state $\emptyset$, it's $\frac{13}{2}$ steps.
The variance of the number of steps from the $i$-th state is the $i$-th entry in the matrix
$(2N-I){\bf t-t_{\textrm sq}}$,
where $t_{\textrm sq}$ is the matrix $\bf t$ with each entry squared. If I didn't slip up with Mathematica,
$(2N-I){\bf t-t_{\textrm sq}}=\left({\begin{array}{c} \frac{107}{4}\\ 30\\ 2\\ \end{array}}\right)$,
and the variance you want is $\frac{107}{4}$
You can write $X=X_{1}+X_{2}$ where $X_{1}$ stands for the number of rolls needed to get a number in $\left\{ 2,4,5,6\right\}$ and $X_{2}$ for the number of rolls needed after that to come in the situation where an even number and number $5$ is rolled. Then $X_{1}$ is geometrically distributed with parameter $\frac{2}{3}$ so that $\mathbb{E}X_{1}=\frac{3}{2}$ and $\text{Var}X_{1}=\frac{3}{4}$. Let $A$ be an rv taking values in $\left\{ 0,1\right\}$ s.t $A=1$ if a $5$ shows up at the first time a number in $\left\{ 2,4,5,6\right\}$ is rolled. It has the Bernouilli-distribution with parameter $\frac{1}{4}$. Abbreviating $B=1-A$ you can write $X_{2}=AU+BV$ where $U$ and $V$ are geometrically distributed with parameters $\frac{1}{2}$ and $\frac{1}{6}$ respectively. Here $X_{1},A,U,V$ are independent.
$\mathbb{E}X_{2}=\mathbb{E}A\mathbb{E}U+\mathbb{E}B\mathbb{E}V=\frac{1}{4}\frac{2}{1}+\frac{3}{4}\frac{6}{1}=5$.
Making use of $AB=0$, $A^{2}=A$ and $B^{2}=B$ we find:
$\text{Var}X_{2}=\mathbb{E}X_{2}^{2}-\left(\mathbb{E}X_{2}\right)^{2}=\frac{1}{4}\mathbb{E}U^{2}+\frac{3}{4}\mathbb{E}V^{2}-25=\frac{1}{4}.6+\frac{3}{4}.66-25=26$.
Then $\mathbb{E}X=\mathbb{E}X_{1}+\mathbb{E}X_{2}=\frac{13}{2}$ and $\text{Var}X=\text{Var}X_{1}+\text{Var}X_{2}=\frac{3}{4}+26=\frac{107}{4}$.
To make this easier, let's consider cases:
Case 1: We roll a five before we ever roll an even number.
Case 2: We roll an even number before we ever roll a five.
Then the probability that it takes $n$ rolls to achieve both a five and an even number is the probability that either (a) you roll a five and no even numbers during the first $n-1$ rolls, and the $n$th roll is even, or (b) you roll at least one even number and no fives during the first $n-1$ rolls, and the $n$th roll is a five.
Can you see how to compute the probabilities of each of these outcomes?
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2020-02-21 07:38:04
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https://www.studyadda.com/question-bank/11th-cbse-chemistry-chemical-bonding-and-molecular-structure_q46/105/15437
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• # question_answer Why $N{{F}_{3}}$ is pyramidal but $B{{F}_{3}}$ is triangular planar?
In $N{{F}_{3}},\,N$ has the hybridisation $s{{p}^{3}}$ with one position occupied by lone pair of electrons but in $B{{F}_{3}},\,B$ has the hybridisation $s{{p}^{2}}$.
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2020-09-20 15:22:04
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https://motls.blogspot.com/2008/03/frederik-denef-landscape-building-guide.html?m=1
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## Wednesday, March 19, 2008
### Frederik Denef: landscape building guide
All expert TRF readers (except for a few, 0-10, leading F-theory leaders in the world who know the sub-discipline better than the author anyway) are recommended to print and read
Frederik Denef's Les Houches Lectures (PDF)
about the construction of string-theoretical vacua.
On his 127 pages, Frederik discusses the structure and properties of the string vacua. While he dedicates a few pages to heterotic and type I strings, M-theory on G2 manifolds, non-geometrical, and non-critical compactifications, most of the paper is dedicated to F-theoretical flux vacua i.e. type IIB vacua with non-trivial axion-dilaton fields, orientifolds, and D-branes.
The paper may look long or contrived to many readers but what is important is that these insights are robust and pretty much inevitable. An extraterrestrial civilization would have to end up with pretty much equivalent papers about the string landscape at a certain level of the evolution of their science. Once you adopt the idea that the elementary particles are extended objects while the resulting theory should still reduce to effective field theories we have checked, you are inevitably led to strings and the whole structure of string theory follows.
Almost no page among the 127 pages of the paper is directly connected with a particular experiment. Nevertheless, all of them are tightly connected with one another as well as with other insights that are observationally rooted. You might feel that you are walking somewhere in between the clouds, 324 meters above the ground. Some people say that it is unsafe, religious, or unscientific to walk and talk 324 meters above the ground. But you have a structure to rely on. Is it possible to walk 324 meters above the ground and to talk to some of the low-lying clouds? Yes, it is. The rigid structure is called the Eiffel Tower. ;-)
String theory is analogous. The people who don't want to hear about things such as the Eiffel Tower or string theory are just far too narrow-minded and limited to be relevant for this type of engineering or science.
When you finish reading Frederik's paper, you will be able to pick, construct, and study your own string-theoretical vacua in the landscape. You will also remain doubtful about all kinds of anthropic questions but you will understand that these questions do not represent the bulk of the knowledge and skills that a theoretical physicist must have for his or, less frequently, her opinions to matter.
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2018-09-23 17:44:57
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https://ml4ds.com/weeks/04-classification/notebooks/gradient_descent.html
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## Introduction
First we introduce the basic algorithm and explore some of its important design components: step sizes and convergence checking.
The final section of this notebook shows a more sophisticated implementation of gradient descent for estimating logistic regression coefficients that uses a few advanced topics, so it’s OK to not understand all the code in a first read. Some interesting experiments to try in that section: changing the data generating parameters (n, p, sparsity, etc) or algorithm parameters (max_steps, tol).
(Note: if there’s any place where code is not displayed the source can be seen in the .Rmd file version of this notebook).
We can think of this iterative optimization algorithm as minimizing a function by taking steps in the direction of the steepest decrease (or descent). The path followed by the algorithm is like someone skiiing downhill as fast as possible by always turning in the direction with the steepest downward slope. Here’s a simple example using the function $$f(x) = x^4$$. This starts at the point $$x_0 = 1$$, and then computes a sequence of inputs $$x_1, x_2, \ldots, x_n$$ with the goal that $$f(x_n)$$ converges to a (potentially local!) minimum of $$f$$. In this case $$f$$ has a global minimum at $$x = 0$$ of $$f(0) = 0$$, and we hope our algorithm will give this correct answer.
# Function to minimize
f <- function(x) x^4
# Derivative (because it's univariate)
grad_f <- function(x) 4 * x^3
xn <- 1 # starting point
fn <- f(xn)
step_size <- .1
for (step in 1:5) {
# Update step
xn <- xn - step_size * grad_f(xn)
# Show results
cat(paste0("At step ", step,
", xn = ", round(xn, 5),
" and f(xn) = ", round(f(xn), 5)), "\n")
}
## At step 1, xn = 0.6 and f(xn) = 0.1296
## At step 2, xn = 0.5136 and f(xn) = 0.06958
## At step 3, xn = 0.45941 and f(xn) = 0.04454
## At step 4, xn = 0.42062 and f(xn) = 0.0313
## At step 5, xn = 0.39086 and f(xn) = 0.02334
Notice that xn is getting closer to 0 and f(xn) is decreasing. Perhaps if the algorithm takes more steps it will converge to the minimizer of $$f$$.
However, I’ve chosen the values here carefully. Right now it converges very slow. If we increase the number of steps from 5 to 100 it still has not yet reached even xn = 0.1, never mind the actual minimizer of 0. If we change the step_size or initial starting point xn to some other values it may converge faster, or it may diverge to infinity instead of converging to 0.
This shows that step sizes and the number of steps are important parts of any gradient descent implementation, and performance will also depend on the function (and its gradient) and starting points.
### Step sizes
The step size is a scalar multiplier of the gradient. It’s denoted $$\gamma_n$$ in the Wikipedia article and we’ll keep that convention.
We need the step size to decrease to zero so that the algorithm doesn’t get stuck jumping back and forth past the minimum value instead of getting closer to it. But how fast should it decrease?
If it starts too small or decreases too fast then gradient descent will require many more tiny steps before it reaches the optimum. On the other hand, if it decreases too slowly then gradient descent may jump around the optimum many times before the step size decreases enough for it to get closer.
#### Geometric decrease
One common approach is to take a number $$0 < \gamma < 1$$, e.g. $$\gamma \approx .9$$, and make step $$n$$ have step size $$\gamma^n$$. The step sizes then would look like this:
gamma <- .9
data.frame(step = 1:30) |>
mutate(step_size = gamma^step) |>
ggplot(aes(x = step, y = step_size)) + geom_point(size = 2)
Let’s see this in action trying to minimize $$f(x) = x^2$$.
First we’ll try a value of $$\gamma$$ which is too large.
f <- function(x) x^2 # Function to optimize
max_steps <- 15
x0 <- 3 # starting point
gamma <- .995
# ... (rest of the code not displayed)
Now we plot the path of the algorithm. We can see gradient descent keeps jumping past the optimum many times, wasting computation by converging too slowly.
Now we’ll try a value of $$\gamma$$ which is too small. This time gradient descent converges too fast because the step sizes become numerically close to zero even though we have not yet reached the optimum (minimum of $$f$$).
gamma <- .55
Recall that at a (local) optimal value, the derivative (or gradient) should be zero (or undefined, as in the case of $$|x|$$). We can use this to check if gradient descent converged (stopped moving) because of the step size and has not yet reached an optimum:
grad_f(xnext) # numerically close to zero?
## [1] 1.025414
The gradient isn’t close to zero, instead the algorithm has stopped because the overall step is small:
gamma^step * grad_f(xnext)
## [1] 0.0001307192
Summary: Step sizes are decreasing too slow if gradient descent jumps around the optimum and takes a long time to converge. Step sizes are decreasing too fast if gradient descent converges but fails to reach an optimum.
#### Hand-tuning
In practice, people often try different values of $$\gamma$$, check to see if gradient descent is converging too slowly or quickly, then modify $$\gamma$$ and try again. Or they might pick a sequence of $$\gamma$$ values using multiple phases that looks something like this:
Using these step sizes to minimize $$f(x) = x^2$$ again, this time the result would converge better:
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2023-03-30 11:15:30
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https://minerals-identify.com/detail/stannite
|
# Stannite
#### Sulfide mineral
Stannite is a mineral, a sulfide , iron.
The chemical formula Cu 4. Zinc may be present. of tin, consisting of approximately 28% tin, 13% iron, 30% copper, 30% sulfur vein , sphalerite , arsenopyrite, and wolframite
It is also known as bell metal ore as tin is an important constituent of bell-metal. It is thought the exploitation of tin deposits in Cornwall led to an expansion in bell founding.
The name comes from the Latin for tin: stannum. It was first described in 1797 for an occurrence in Wheal Rock, St. Agnes, Cornwall
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2021-05-07 06:50:20
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https://math.stackexchange.com/questions/linked/433094
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16 questions linked to/from How to determine the arc length of ellipse?
49 views
### How do I work out the distance of Venus orbit mathematically? [duplicate]
How do I work out the distance Venus travels in one orbit mathematically? I know the parametric equations for its ellipse but I need to work out the total distance travelled in its orbital period.
8k views
### Determining the angle degree of an arc in ellipse?
Is it possible to determine the angle in degree of an arc in ellipse by knowing the arc length, ellipse semi-major and semi-minor axis ? If I have an arc length at the first quarter of an ellipse and ...
2k views
### Approximating the integral $\int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$
How would I go about evaluating $I = \int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$ to four decimal accuracy?
2k views
### Is it possible to find the distance between two points on the circumference of an ellipse following the outer curve?
I am rather confused on this calculation, and I can't seem to find a solution online. I need to know how to find the distance between two known points on an ellipse's outer edge following the path of ...
562 views
### ellipse uniform perimeter travel?
I'm trying to come up with a way that as I progress linearly from $0$ to $360$ degrees or $0$ to $2\pi$ radians I have the the corresponding position (P) on the perimeter of an ellipse to also travel ...
584 views
### Generate random points on perimeter of ellipse
Sampling only from the uniform distribution $U(0,1)$, I am hoping to use transformations to create random values distributed uniformly around the perimeter of an ellipse. Eventually, I'd like to do ...
256 views
### Construction of new ellipse
Using a pencil, the thread was pulled on the ellipse. Then the pencil started to rotate around the ellipse. How to prove that the new geometric figure which the pencil drew is also an ellipse (with ...
463 views
### Ellipse circumference and 1/2 arc of first quadrant
For an ellipse (centered at 0,0), let a = 5.088 and b = 3.006. I used an equation to determine the circumference of ellipse. The approximation = 25.850. Now, I assume it is safe to say that each ...
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2021-05-07 08:03:44
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https://stats.stackexchange.com/questions/371643/bootstrapping-covariance-matrices-with-different-sampling-procedures
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# bootstrapping covariance matrices with different sampling procedures
The regression model has heteroskedasticity. The variance of error term depends on regressors. From boostrapping analysis, I got two different covariance matrices of $$\beta$$. The difference results from different resampling procedures. One is resampling only from residuals, and another one is resampling from the pair of residuals and regressors. The former covariance matrices is lower than the latter covariance matrices. I am trying to understand why difference resampling procedures make the former covariance lower, but I cannot figure it out. I appreciate if you give some help.
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2019-08-25 00:47:15
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https://byjus.com/rd-sharma-solutions/class-7-maths-chapter-15-properties-of-triangles-ex-15-4/
|
RD Sharma Solutions Class 7 Properties Of Triangles Exercise 15.4
RD Sharma Solutions Class 7 Chapter 15 Exercise 15.4
RD Sharma Class 7 Solutions Chapter 15 Ex 15.4 PDF Free Download
Exercise 15.4
Q1. In each of the following, there are three positive numbers. State if these numbers could possibly be the lengths of the sides of a triangle:
(i) 5, 7, 9
(ii) 2, 10.15
(iii) 3, 4, 5
(iv) 2, 5, 7
(v) 5, 8, 20
(i) Yes, these numbers can be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side. Here, 5+7>9, 5+9>7, 9+7>5
(ii) No, these numbers cannot be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side, which is not true in this case.
(iii) Yes, these numbers can be the lengths of the sides of a triangle because the sum of any two sides of triangle is always greater than the third side. Here, 3+4 >5, 3+5> 4, 4+5> 3
(iv) No, these numbers cannot be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side, which is not true in this case. Here, 2 + 5 = 7
(v) No, these numbers cannot be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side, which is not true in this case. Here, 5 + 8 <20
Q2. In Fig, P is the point on the side BC. Complete each of the following statements using symbol ‘ =’,’ > ‘or ‘ < ‘so as to make it true:
(i) AP… AB+ BP
(ii) AP… AC + PC
(iii) AP…. $\frac{1}{2}(AB+AC+BC)$
(i) In triangle APB, AP < AB + BP because the sum of any two sides of a triangle is greater than the third side.
(ii) In triangle APC, AP < AC + PC because the sum of any two sides of a triangle is greater than the third side.
(iii) AP < $\frac{1}{2}(AB+AC+BC)$ In triangles ABP and ACP, we can see that:
AP < AB + BP…(i) (Because the sum of any two sides of a triangle is greater than the third side)
AP < AC + PC…(ii) (Because the sum of any two sides of a triangle is greater than the third side)
On adding (i) and (ii), we have:
AP + AP < AB + BP + AC + PC
2AP < AB + AC + BC (BC = BP + PC)
AP < (AB-FAC+BC)
Q3. P is a point in the interior of $\triangle ABC$ as shown in Fig. State which of the following statements are true (T) or false (F):
(i) AP+ PB< AB
(ii) AP+ PC> AC
(iii) BP+ PC = BC
(i) False
We know that the sum of any two sides of a triangle is greater than the third side: it is not true for the given triangle.
(ii) True
We know that the sum of any two sides of a triangle is greater than the third side: it is true for the given triangle.
(iii) False
We know that the sum of any two sides of a triangle is greater than the third side: it is not true for the given triangle.
Q4. O is a point in the exterior of $\triangle ABC$. What symbol ‘>’,’<’ or ‘=’ will you see to complete the statement OA+OB….AB? Write two other similar statements and show that
OA+OB+OC>$\frac{1}{2}(AB+BC+CA)$
Because the sum of any two sides of a triangle is always greater than the third side, in triangle OAB, we have:
OA+OB> AB —(i)
OB+OC>BC —-(ii)
OA+OC > CA —–(iii)
On adding equations (i), (ii) and (iii) we get :
OA+OB+OB+OC+OA+OC> AB+BC+CA
2(OA+OB+OC) > AB+BC +CA
OA+ OB + OC > $\frac{AB+BC+CA}{2}$
Q5. In $\triangle ABC$, $\angle B=30^{\circ}$, $\angle C=50^{\circ}$. Name the smallest and the largest sides of the triangle.
Because the smallest side is always opposite to the smallest angle, which in this case is $30^{\circ}$, it is AC. Also, because the largest side is always opposite to the largest angle, which in this case is $100^{\circ}$,, it is BC.
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2019-08-18 04:31:21
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|
https://portalrecerca.uab.cat/en/publications/ethernes-a-new-design-of-radionuclide-source-based-thermal-neutro
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# ETHERNES: A new design of radionuclide source-based thermal neutron facility with large homogeneity area
R. Bedogni, D. Sacco, J. M. Gómez-Ros, M. Lorenzoli, A. Gentile, B. Buonomo, A. Pola, M. V. Introini, D. Bortot, C. Domingo
Research output: Contribution to journalArticleResearchpeer-review
11 Citations (Scopus)
## Abstract
© 2015 Elsevier Ltd. A new thermal neutron irradiation facility based on an 241 Am-Be source embedded in a polyethylene moderator has been designed, and is called ETHERNES (Extended THERmal NEutron Source). The facility shows a large irradiation cavity (45cm×45cm square section, 63cm in height), which is separated from the source by means of a polyethylene sphere acting as shadowing object. Taking advantage of multiple scattering of neutrons with the walls of this cavity, the moderation process is especially effective and allows obtaining useful thermal fluence rates from 550 to 800cm -2 s -1 with a source having nominal emission rate 5.7×10 6 s -1 . Irradiation planes parallel to the cavity bottom have been identified. The fluence rate across a given plane is as uniform as 3% (or better) in a disk with 30cm (or higher) diameter. In practice, the value of thermal fluence rate simply depends on the height from the cavity bottom. The thermal neutron spectral fraction ranges from 77% up to 89%, depending on the irradiation plane. The angular distribution of thermal neutrons is roughly isotropic, with a slight prevalence of directions from bottom to top of the cavity. The mentioned characteristics are expected to be attractive for the scientific community involved in neutron metrology, neutron dosimetry and neutron detector testing.
Original language English 171-176 Applied Radiation and Isotopes 107 https://doi.org/10.1016/j.apradiso.2015.10.016 Published - 1 Jan 2016
## Keywords
• NEURAPID
• Neutron dosimetry
• Neutron moderation
• Thermal neutron facility
• Thermal neutrons
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2020-11-29 05:21:19
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https://moodle.org/plugins/view.php?plugin=format_fntabs&moodle_version=11
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## Course formats: FN- Tabs Course Format
format_fntabs
Maintained by Fernando Oliveira
Tabs Course Format (with Activity Tracking)
54
92
1
Moodle 2.4
The Tabs Course Format displays sections (weeks) as numbered tabs. The number of tabs is unlimited and the various colours (text, hover, background, etc.) can be customized.
In addition, this course format includes an alternative Activity Tracking System, which trackS the following activity completion statuses:
• Incomplete (did not earn passing grade)
• Draft
• Not Attempted
• Waiting for grade (student has submitted this activity)
As shown in the screenshot below, these statuses are represented in the form of an icon on the left side of the activity name.
The tabs also provide a quick overview of the activity completion status by showing a green bar, in sections where all activity are complete, and a red bar, where more work is required. Students can also mouseover tabs to see a quick report of the activity statuses for any given week/section.
If you are using our Activity Tracking System on your site, it is recommended that you also install My Progress block.
*Note that the activity tracking system (and My Progress block) can only be seen by students. Teachers can "login as" student so see this function in action.
### Sets
This plugin is part of set MoodleFN.
### Contributors
• Thu, Sep 5, 2013, 4:27 AM
Fernando - Thanks for working to tidy things up. I've retested and have approved this plugin. The only thing that was not obvious to me was how you want folks to report issues to you. The current link refers them to your website where I suppose they might register and post a comment. It might be more natural to use Github's issue system or Moodle Tracker - let me know what works best for you. Peace - Anthony
• Thu, Sep 5, 2013, 5:45 AM
I am unapproving this so that you can add the dependency you mentioned to the fn_tabs block. Peace - Anthony
• Fri, Sep 6, 2013, 12:42 AM
So I think you need to add the dependies to the version.php for the 2.4 version of the code. You may want to consider either adding 2.3 to this version or removing the 2.3 tag from the block so that there is consistency. Do you want to support a version for 2.3? If not just remove the 2.4 tag from the download file of the block. I do not think any changes need to be mae to the 2.5 version. Once we get the dependencies straighted out I think you should be good to go. Peace - Anthony
• Fri, Sep 6, 2013, 6:02 AM
There is now a 2.4 version of the course format that is dependent upon the block and vice versa. The 2.5 version does not have a dependency. I am going to go ahead and approve this. Peace - Anthony
• Tue, Sep 10, 2013, 12:03 PM
Hi, just noting i saw '2.4' in $plugin->release inside version.php both the 2.4 and 2.5 zips. • Tue, Sep 10, 2013, 12:08 PM Thanks Aparup - I was working with Fernando on gettings things separated out so I was able to go ahead and make the release name of the 2.5 version "2.5". I pushed that change to Github.com and then added a new version so hopefully any confusion betwen the versions will be avoided. Peace - Anthony • Mon, Sep 16, 2013, 12:05 AM I just installed it (moodle 2.5+ - not the latest version). I wish I could change the text on each tab for a number to something more customizable - like the name of the secion. Is this possible? • Wed, Jan 22, 2014, 3:23 AM is there any way to rename the tabs to text instead of numbers? • Wed, Mar 26, 2014, 5:06 PM We have just installed on Moodle 2.4.3+ (Build: 20130425) but there is no cog icon so we cannot change the preferences (as per your more information PDF). We are stuck with the green and yellow colour combination. I'd also like to be able to change the 'week' and tab text. • Thu, Mar 27, 2014, 4:55 AM Version 2.4 requires the control panel: https://moodle.org/plugins/view.php?plugin=block_fn_tabs • Mon, May 12, 2014, 2:50 PM Thank you, Fernando, for this grate plugin. Are you going to update it so it works for moodle 2.6 as well? • Thu, Jul 31, 2014, 2:03 PM Hi Fernando! I have noticed that in file course_format_fn.class.php (line 352) the rebuild_course_cache($this->course->id) function call causes a heavy load on DB by triggering too many read processes. If this line is commented everything looks like working perfect and the read processes are significantly reduced.
Do you think that commenting the rebuild_course_cache() function call will cause side effects on the proper functioning of my course?
Any ideas?
I use Moodle 2.6.4 version.
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2015-11-27 22:57:49
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https://proofwiki.org/wiki/Category:Definitions/Examples_of_Rings
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# Category:Definitions/Examples of Rings
This category contains definitions of examples of rings in the context of Abstract Algebra.
A ring $\struct {R, *, \circ}$ is a semiring in which $\struct {R, *}$ forms an abelian group.
That is, in addition to $\struct {R, *}$ being closed, associative and commutative under $*$, it also has an identity, and each element has an inverse.
## Subcategories
This category has the following 2 subcategories, out of 2 total.
## Pages in category "Definitions/Examples of Rings"
The following 12 pages are in this category, out of 12 total.
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2021-09-21 07:35:09
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https://petsc.org/release/docs/manualpages/Vec/VecSetOption/
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# VecSetOption#
Sets an option for controling a vector’s behavior.
## Synopsis#
#include "petscvec.h"
PetscErrorCode VecSetOption(Vec x, VecOption op, PetscBool flag)
Collective
## Input Parameters#
• x - the vector
• op - the option
• flag - turn the option on or off
## Supported Options#
• VEC_IGNORE_OFF_PROC_ENTRIES, which causes VecSetValues() to ignore- entries destined to be stored on a separate processor. This can be used to eliminate the global reduction in the VecAssemblyBegin() if you know that you have only used VecSetValues() to set local elements
• VEC_IGNORE_NEGATIVE_INDICES, which means you can pass negative indices- in ix in calls to VecSetValues() or VecGetValues(). These rows are simply ignored.
• VEC_SUBSET_OFF_PROC_ENTRIES, which causes VecAssemblyBegin() to assume that the off- process entries will always be a subset (possibly equal) of the off-process entries set on the first assembly which had a true VEC_SUBSET_OFF_PROC_ENTRIES and the vector has not changed this flag afterwards. If this assembly is not such first assembly, then this assembly can reuse the communication pattern setup in that first assembly, thus avoiding a global reduction. Subsequent assemblies setting off-process values should use the same InsertMode as the first assembly.
## Developer Note#
The InsertMode restriction could be removed by packing the stash messages out of place.
Vectors and Parallel Data, Vec, VecSetValues()
intermediate
## Location#
src/vec/vec/interface/vector.c
Edit on GitLab
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2023-01-27 11:06:33
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https://en-academic.com/dic.nsf/enwiki/107742
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# Karl Schwarzschild
Karl Schwarzschild
Infobox Scientist
name = Karl Schwarzschild
box_width =
image_width =150px
caption = Karl Schwarzschild (1873-1916)
birth_date = October 9, 1873
birth_place = Frankfurt am Main
death_date = May 11, 1916
death_place =
residence =
citizenship =
nationality = German
ethnicity =
field = physics astronomy
work_institutions =
alma_mater =
doctoral_students =
known_for =
author_abbrev_bot =
author_abbrev_zoo =
influences =
influenced = Martin Schwarzschild
prizes =
religion =
footnotes =
Karl Schwarzschild (October 9, 1873 - May 11, 1916) was a German Jewish physicist and astronomer. He is also the father of astrophysicist Martin Schwarzschild.
He was born in Frankfurt am Main. He was something of a child prodigy, having a paper on celestial mechanics published when he was only sixteen. He studied at Strasbourg and Munich, obtaining his doctorate in 1896 for a work on Jules Henri Poincaré's theories.
From 1897, he worked as assistant at the [http://kuffner-sternwarte.at/sternwarte/vks_ksw.html Kuffner Sternwarte] (Observatory) in Vienna, where he developed a formula to calculate the optical density of photographic material. It involved an exponent now known as the Schwarzschild-exponent, which is the $p$ in the formula:
$i = f \left( Icdot t^p \right)$
(where $i$ is optical density of exposed photographic emulsion, a function of $I$, the intensity of the source being observed, and $t$, the exposure time, with $p$ a constant). This formula was important for enabling more accurate photographic measurements of the intensities of faint astronomical sources.
From 1901 until 1909 he was a professor at the prestigious institute at Göttingen, where he had the opportunity to work with some significant figures including David Hilbert and Hermann Minkowski. Schwarzschild became the director of the observatory in Göttingen. He moved to a post at the Astrophysical Observatory in Potsdam in 1909.
From 1912, Schwarzschild was a member of the Prussian Academy of Sciences.
At the outbreak of World War I in 1914 he joined the German army despite being over 40 years old. He served on both the western and eastern fronts, rising to the rank of lieutenant in the artillery.
While serving on the front in Russia in 1915, he began to suffer from a rare and painful skin disease called pemphigus. Nevertheless, he managed to write three outstanding papers, two on relativity theory and one on quantum theory. His papers on relativity produced the first exact solutions to the Einstein field equations, and a minor modifaction of these results gives the well-known solution that now bears his name: the Schwarzschild metric.
Einstein himself was pleasantly surprised to learn that the field equations admitted exact solutions, because of their prima facie complexity, and because he himself had only produced an approximate solution. Einstein's approximate solution was given in his famous 1915 article on the advance of the perihelion of Mercury. There, Einstein used rectangular coordinates to approximate the gravitational field around a spherically symmetric, non-rotating, non-charged mass. Schwarzschild, in contrast, chose a more elegant "polar-like" coordinate system and was able to produce an exact solution. In 1916, Einstein wrote to Schwarzschild on this result:
Schwarzschild's second paper, which gives what is now known as the "Inner Schwarzschild solution" (in German: "innere Schwarzschild-Lösung"), is valid within a sphere of homogeneous and isotropic distributed molecules within a shell of radius r=R. It is applicable to solids; incompressible fluids; the sun and stars viewed as a quasi-isotropic heated gas; and any homogeneous and isotropic distributed gas.
Schwarzschild's first (spherically symmetric) solution contains a coordinate singularity on a surface that is now named after him. In Schwarzschild coordinates, this singularity lies on the sphere of points at a particular radius, called the Schwarzschild radius:
:$R_\left\{s\right\} = frac\left\{2GM\right\}\left\{c^\left\{2$
where "G" is the gravitational constant, "M" is the mass of the central body, and "c" is the speed of light in a vacuum.Landau 1975.] In cases where the radius of the central body is less than the Schwarzschild radius, $R_\left\{s\right\}$ represents the radius within which all massive bodies, and even photons, must inevitably fall into the central body (ignoring quantum tunnelling effects near the boundary). When the mass density of this central body exceeds a particular limit, it triggers a gravitational collapse which, if it occurs with spherical symmetry, produces what is known as a Schwarzschild black hole. This occurs, for example, when the mass of a neutron star exceeds the Oppenheimer-Volkoff limit (about three solar masses).
Thousands of dissertations, articles, and books have since been devoted to the study of Schwarzschild's solutions to the Einstein field equations. However, although Schwarzschild's best known work lies in the area of general relativity, his research interests were extremely broad, including work in celestial mechanics, observational stellar photometry, quantum mechanics, instrumental astronomy, stellar structure, stellar statistics, Halley's comet, and spectroscopy. [Eisenstaedt, “The Early Interpretation of the Schwarzschild Solution,” in D. Howard and J. Stachel (eds), Einstein and the History of General Relativity: Einstein Studies, Vol. 1, pp. 213-234. Boston: Birkhauser, 1989.]
Some of his particular achievements include measurements of variable stars, using photography, and the improvement of optical systems, through the perturbative investigation of geometrical aberrations.
Schwarzschild's struggle with pemphigus may have eventually led to his death. He died on May 11, 1916.
ee also
* Schwarzschild, items named after Karl Schwarzschild
References
*
* Roberto B. Salgado [http://www.phy.syr.edu/courses/modules/LIGHTCONE/schwarzschild.html The Light Cone: The Schwarzschild Black Hole]
* [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1917ApJ....45..285H Obituary in the Astrophysical Journal] , written by Ejnar Hertzsprung
*
Wikimedia Foundation. 2010.
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### Look at other dictionaries:
• Karl Schwarzschild — (* 9. Oktober 1873 in Frankfurt am Main; † 11. Mai 1916 in Potsdam) war ein deutscher Astronom und Physiker und gilt als einer der Wegbereiter der modernen Astrophysik … Deutsch Wikipedia
• Karl Schwarzschild — (9 octobre 1873, Francfort sur le Main – 11 mai 1916, Potsdam) était un astrophysicien allemand. Sa curiosité pour les étoiles se manifesta dès ses premières années scolaires, lorsqu il construisit un petit té … Wikipédia en Français
• Karl Schwarzschild — Karl Schwarzschild. Karl Schwarzschild (9 de octubre de 1873 11 de mayo de 1916), fue un físico y astrónomo alemán, nació en Fráncfort del Meno y fue el mayor de seis hermanos de una familia de ascendencia judía. Joven prodigio, a los dieciséis… … Wikipedia Español
• Karl Schwarzschild — (octubre 9, 1873 mayo 11, 1916) físico y astrónomo alemán, nació en Frankfurt am Main y fue el mayor de seis hermanos de una familia de ascendencia judía. Joven prodigio, a los 16 desarrollo un trabajo sobre la teoría de órbitas celestiales y… … Enciclopedia Universal
• Karl-Schwarzschild-Observatorium — der Thüringer Landessternwarte Das Observatorium existiert seit 1960 als Institut der Deutschen Akademie der Wissenschaften zu Berlin in der Nähe des Dorfes Tautenburg. 1992 wurde aus dem Institut die Thüringer Landessternwarte (TLS) … Deutsch Wikipedia
• Karl Schwarzschild Observatory — The Karl Schwarzschild Observatorium (Karl Schwarzschild Observatory) is an astronomical observatory owned and operated by the institute of Thüringer Landessternwarte (Thuringian State Observatory) ’Karl Schwarzschild’ Tautenburg . In 1992 it was … Wikipedia
• Karl-Schwarzschild-Medaille — Die Karl Schwarzschild Medaille wird von der Astronomischen Gesellschaft jedes Jahr an Astronominnen oder Astronomen von hohem wissenschaftlichen Rang vergeben. Sie ist nach dem Astronomen Karl Schwarzschild benannt. Bisherige Preisträger waren:… … Deutsch Wikipedia
• Karl Schwarzschild Medal — The Karl Schwarzschild Medal, named after the astrophysicist Karl Schwarzschild, is an award presented by the Astronomische Gesellschaft (German Astronomical Society) to eminent astronomers and astrophysicists. [cite web | url =… … Wikipedia
• Karl Schwarzschild — (9.10.1873 11.5.1916) Deutscher Astronom, beschäftigte sich unter anderem grundlegend mit der astronomischen Fotografie. Dabei entdeckte er 1899 den nach ihm benannten Effekt, der die bei langen Belichtungszeiten auftretende Unterbelichtung trotz … Das Lexikon aus „Bernie's Foto-Programm"
• Observatoire Karl Schwarzschild — Karl Schwarzschild Karl Schwarzschild Karl Schwarzschild (9 octobre 1873, Francfort sur le Main – 11 mai 1916, Potsdam) était un astrophysicien allemand. Sa curiosité pour les étoiles se manifesta dès ses premières an … Wikipédia en Français
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2022-06-28 12:11:44
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https://chemistry.stackexchange.com/questions/9912/calculating-molar-conductance-of-cacl2
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# Calculating molar conductance of CaCl2
Ok, so the following question is given in my text book.
Of the molar conductance value of $\ce{Ca^2+}$ and $\ce{Cl^-}$ at infinite dilution are respectively $118.88\times10^{-4}$ and $77.33\times10^{-4}$ then that of $\ce{CaCl2}$ is (all have same unit)
So what I did was I added the molar conductivity of $\ce{Ca^2+}$ and added $2\times$ molar conductivity of $\ce{Cl^-}$. I got the answer. It was one of the options. So I basically wanted to know: Did I do it correctly and is my process correct? And if I did it wrong then what is the correct process?
Yes, the process is correct, but do you understand why?
• As per my knowledge there is one mole of $Ca$ and 2 mole of $Cl$ in $CaCl_2$ that's why am i correct Apr 12, 2014 at 3:54
– LDC3
Apr 12, 2014 at 3:58
You are calculating limiting molar conductivity
limiting molar conductivity = V+ A+ + V- A-
V+ = number of cations per formula unit of the electrolyte
V- = number of anions per formula unit of the electrolyte
A+ = molar conductivity of the cations
A- = molar conductivity of the anions
So , your process is correct.
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2022-08-15 12:20:04
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https://citrination.com/datasets/102600/show_search
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Self-consistent electronic structure of a $d_{x^2-y^2}$ and a $d_{x^2-y^2}+id_{xy}$ vortex
Version {{dataset.version}}
Description: No description available
{{tag.value}}
Data Views containing this dataset:
This dataset has not been used in any views.
Design Project associated with this dataset:
{{designProjectName}}
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2018-03-25 03:31:08
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https://electronics.stackexchange.com/questions/232035/why-are-there-so-many-relays-used-in-a-car-instead-of-transistors/232082
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# Why are there so many relays used in a car, instead of transistors?
Recently I did some work on my car electric circuits. I see many relays are used in car circuits. These relays are used for simple switching, and I wonder why these circuits are based on relays and not on transistors or other electronic components usable for switching purposes. I thought that transistors were cheaper, smaller and more reliable than classic el-mech relays for switching.
Note: In car applications a 12 V car battery is used to power the coil of a relay and the same 12 V power is what's switched by the relay. Sometimes a relay switches just another signal line, i.e. without any high power load on it. And still, I can see no transistors in there. So I believe there is a solid reason why it is done this way, and I must be missing something here. :-)
• How old is your car? Have you tried doing some research on the change over of relays to something solid state? – Andy aka May 3 '16 at 13:07
• I still remember people calling transistors "three-legged fuses"... – Brian Drummond May 3 '16 at 13:11
• I'm just thinking about a guy in a garage trying to figuring out why something don't work anymore. He pushes a button, waits for the "click" of the relay, but he won't hear anything. Is it broken, or is it an transistor? This might not be the reason, but it's probably a pro for relais. It's way easier to tell if they're broken. – Sempie May 3 '16 at 13:16
• Once the car will have 100% SSR instead of electromechanical, the people will be yelling: where have the good old relays gone? – Marko Buršič May 3 '16 at 13:36
• @Sempie It's about time garage guys discover OBD systems. – Dmitry Grigoryev May 3 '16 at 14:24
Relays are much more stable temperature-wise: a sealed relay has essentially the same characteristics at -30°C and +70°C, both temperatures being common for cars. A transistor works quite differently at -30°C and +70°C, so the schematic has to be designed to account for those variations.
I once worked on a product with temperature range starting at -55°C, which used both relays and semiconductor devices. The funny part about the design was that below -20°C only the relay part was powered, which activated air heaters and would only switch on the semiconductor part once the temperature reached 0°C.
Relays also offer galvanic isolation, which effectively confines faults. Common failures like short circuits usually damage only one relay, whereas in transistor-based circuits several devices along the way would be affected. I bet people still want their car's motor running even when the air conditioner or a window lifter dies.
• It's worth noting that isolation can be provided with solid-state devices, usually with opto-isolators (or solid state relays, which are basically an opto-isolator and a big transistor in one box). – Someone Somewhere May 4 '16 at 8:06
• @SomeoneSomewhere The original question is about transistors. SSRs are nice, but more expensive than conventional relays. – Dmitry Grigoryev May 4 '16 at 9:59
• Also, the major advantages of transistors over relays are that they're small and quiet (no moving parts). These are very important when you have to fit millions of them into a smartphone, less so for the few dozen or so you need in a car. There's plenty of room for as many relays as a car needs under the hood, and the little noises they make are completely masked by the sound of the car itself. – Darrel Hoffman May 4 '16 at 20:08
• The noise the relays in a car actually used to be an unintentional usability feature. The typical "blinker" sound originated from the sound the blinker relay made when switching the blink-lights on and off. This "feature" told the driver the blinker is on. Now that relays became quieter, the effect is often reproduced through the car stereo system. – Philipp May 5 '16 at 11:12
• @Philipp Good point on usability as when I switched from old Nissan to a new Toyota I was missing that click-click sound for a few months. – zmechanic May 6 '16 at 12:22
A car is (still, in the modern era) an electrically harsh environment. The "12V supply" is typically 13.5 to 15 V, and may occasionally spike to 80V or more. There may also be some high-frequency junk on the wires from the spark plugs.
Relays put up with that abuse somewhat better than transistors, at least for "a similar price point."
• @DmitryGrigoryev, If the relay costs $1.00 and there was a suitable, equally reliable, solution for$0.97, you can be sure the engineers at Ford/Chevy/Toyota/etc would choose it. – The Photon May 3 '16 at 16:22
• Without attempting to disparage any particular auto maker, @ThePhoton, in General, some auto makers will save the 3¢ even if it's less suitable or reliable. – FreeMan May 3 '16 at 20:12
• What could cause a spike to 80v (or higher)? – Digital Trauma May 3 '16 at 21:08
• @DigitalTrauma shutting off your blower fan quickly. All the energy stored in the motor's coils dumps back into the power rails. – rdtsc May 3 '16 at 22:12
• Note also that incandescent lamps often draw up to ten times their running current when cold. Relays can easily be specified to handle this surge in current. A mosfet will have to have sufficiently low on resistance to handle this. A bipolar will need lots of base drive. – Richard May 4 '16 at 13:36
Relays are just well-tried. I think the answer is simple as that.
Mechanics often want a simple solution, which works well. It's just in the mind of the people to switch a circuit with relays.
My brother mounts some special equipment on trucks for work. Always he needs to switch something he comes up with: "Oh, let's put a relay in there!". And he's right. Why not? It's often the simplest solution and it works fine.
In some situations the acoustic feedback can be used as well. Just think about your indicator.
• Acoustic and temperature (warm = active) feedback does work to a certain point. I have also seen relays with built-in LEDs to indicate the state, though not in cars. – Dmitry Grigoryev May 3 '16 at 22:18
• @robin I've heard that the life of relay is comparatively less.. Are the automotive standard relays are far more reliable ? Or is it ssr that you are talking about? – seetharaman May 4 '16 at 20:42
The voltage in a car is only about 12V, which means that even moderately powered components can draw large currents. The dash in my car is illuminated by four 12V 2W bulbs. They draw a current of 666mA just to light up the dash! If you look at the fuses for all the circuits in your car, even the smallest will be 5A. Most will be 10-20A range and some even more than that. The reason relays are so popular is because they are durable, have low contact resistance, and are (sometimes) cheaper than solid state components that can handle the same current. Many modern cars actually do use solid state relays but they come in the same type of brick packaging so as not to confuse any mechanics.
• Actually, solid-state relays are more durable, but that is not needed in cars. Most relays activate once per motor start, so in 10 years of driving twice a day they will be at 7200 cycles or so. – Dmitry Grigoryev May 3 '16 at 21:17
• That's absolutely correct. I should have said "relays are durable enough". Mechanical relays hardly ever fail unless there are other electrical problems. Most outlive the car they are used in. – Robert Stiffler May 3 '16 at 21:21
• @DmitryGrigoryev depending on the car, that's not true. The second biggest relays are the Fan, Lamps, and Horn relays, all of which cycle multiple times, more than once per ride (the horn depends on road rage). – Passerby May 5 '16 at 6:21
• @Dmitry Grigoryev: From my experience, your calculations are about 4 times lower that "normal." Around 30,000 cycles is closer to the minimum number of cycles. – Guill May 6 '16 at 6:08
• @guill Most relays exceed 30,000 cycle life expectancy – Robert Stiffler May 6 '16 at 6:11
Relays have a few advantages over transistors.
1. They have a very low on-resistance. In a high power low voltage system this is valuable as it improves efficiency and eliminates the need for heatsinking.
2. They have very low leakage when off. So their outputs can be connected to unswitched battery power without worrying about draining the battery.
3. They are robust against spikes, surges, temperature variation etc. The environment of a car is a harsh place both electrically and thermally and designing transistor circuits to survive it requires a fair bit of extra work.
4. They provide isolation. While the grounds in a car do all eventually come together isolation is still useful to keep control of where exactly returnc currents flow.
5. They can be easilly used for either high-side or low-side switching and their inputs can be either high or low side switched. Using N-channel fets (the better performing type) for high-side switching requires a gate drive voltage above the main power supply voltage.
Relays also have a few disadvantages over transistors.
1. The coil requires a fair bit of power.
2. The cycle-life is limited
3. They are physically large
So for high power stuff that switches infrequently relays generally win. For complex and/or high speed control stuff solid state electronics generally win.
Can I keep this real simple? The car has a set of conditions under which it much operate. And as others above have indicated, they are pretty significant. The requirements include a solid performance pattern for component quality.
Bottom line... the reason for relays over solid state devices is summed up in a single word.
PRICE.
It's cheaper to provide relays than it is to provide solid state devices to perform that function. When solid state devices come down in price below that of relays, then the automotive manufacturers will switch over to solid state devices. Cost is a major driver when it comes to decisions like this.
Edit:
A 10 amp relay costs 10 times as much as a 10 amp automotive grade power mosfet. – Passerby
So, I'm not quite sure that's a fair comparison. I'm thinking most automotive relays carry more current than that. (25 to 60 amps?) Additionally remember we're totally isolated between signal and output with the relay. Assume the Automotive OEM's would want to keep that isolation concept. What the cost of a opti-isolated 30 amp mosfet going to run? Oh, and better place that into a nice plug in container, best if its a perfect drop in replacement for an existing relay.
Additionally there is one more factor related to PRICE. Remember there are suppliers out there somewhere who have invested millions and millions of dollars to automate and tool up their relay manufacturing plants. Additionally they employ a whole bunch of folks. What do you think those guys charge for relays? Answer: As much as they think they can get away with. Guess what happens when the automotive Original Equipment Manufacturers (OEM's) tell those folks they are thinking of switching over to Mosfet technology... you guessed it, there is a immediate drop in price. It may not matter that they are losing money, they've got huge amortization and fixed costs and lots of employees.
It would not surprise me that a true price advantage of Mosfets over relays gets delayed in implementation for purely business reasons for a period of say five years. It would also not surprise me if some of those businesses had sweetheart deals with the OEM's (e.g. I want to invest a lot of money in my process to reduce my labor costs and therefore my price to you, but I'll only do that if I have a five year contractual commitment for product.)
• A 10 amp relay costs 10 times as much as a 10 amp automotive grade power mosfet. – Passerby May 5 '16 at 6:19
• most automotive relays carry more current than that. 25 to 60 amps? Varies. Main switched power, sure. But Fans, Pumps, Headlights, Horn typically have 5~20. – Passerby May 5 '16 at 7:22
• Additionally remember we're totally isolated between signal and output with the relay. Not the way they are wired. Most are powered by the same source that the switched circuit connects to. Isolation doesn't seem important, based on the wiring schematics I've seen. Typical main engine, headlight, and horn i.stack.imgur.com/TDWEC.png i.stack.imgur.com/9mkUy.png From a 99 Camry. No isolation. When everything is mostly 12V and the same battery... – Passerby May 5 '16 at 7:24
Relays will be used in place of semiconductor components due to following reasons
1. More Durable - wide operating temperature ranges, spikes.
2. no need of heat sink.
3. very low on state resistance.
4. Isolates load completely in off state.
5. Low cost when compared to same current conducting semiconductor.
6. Service point of view - (most automobile relays and fuses are plug and play...) - which is difficult when we use semiconductor (it has to be removed from PCB)
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2019-05-26 19:55:18
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https://www.math.lsu.edu/studentcolloquium
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LSU | Mathematics
# LSU Mathematics Student Colloquium
The goal of the LSU Mathematics Student Colloquium is to give both undergraduate and graduate students the opportunity to hear and interact with speakers from across the country, providing information and perspective possibly relevant to their graduate and postgraduate careers.
Each invited speaker will spend several days at LSU, giving multiple talks and making himself or herself available to undergraduates.
Talks are not confined to the math department but open to everyone. Those majoring in related fields are encouraged to come.
We are a newly charted LSU student organization (constitution and bylaws). We are munificently funded by the Student Government Programming, Support, and Initiatives Fund (PSIF), and the LSU Mathematics Department. We are grateful for the generous past funding made by grants from the National Science Foundation (a VIGRE grant) and the Board of Regents.
Spring 2017:
Dmytro (Dima) Arinkin
(April 3--5, 2017)
Dima Arinkin is a Professor at the University of Wisconsin, Madison. His work focuses on algebro-geometric questions motivated by the geometric Langlands program. He studies the moduli spaces of vector bundles (possibly with additional structures such as connections) on curves and the categories of sheaves on these spaces. Dima Arinkin has received numerous awards, including a Sloan Research Fellowship and a von Neumann Fellowship from the Institute for Advanced Study. He has a history of involvement with mathematics competitions, both in mentoring, and in winning a gold medal himself at the International Mathematics Olympiad.
Undergraduate Talk (277 Lockett; Monday, Apr. 3, 2:30-3:30PM)
Title: What makes a space interesting? (Intro to moduli.)
Abstract: Roughly speaking, geometry is the study of spaces. Here space' is a placeholder: different flavors of geometry work with differentiable manifolds (differential geometry), topological spaces (topology), varieties (algebraic geometry, my favorite), and so on.
This leads to a question: should we try to study all spaces, or focus on those we consider interesting'? And what makes a space interesting? One possible answer to this question is that there are interesting spaces called moduli spaces (the word moduli' goes back to Hilbert and basically means parameters'). The special feature is that these spaces parametrize objects of some class: e.g., moduli space of triangles parametrizes triangles, moduli space of differential equations parametrizes differential equations, and so on.
In my talk, I will go over the basics of moduli spaces; in the (unlikely) event that there is some time left, I will talk about the Murphy Law for the moduli spaces due to Ravi Vakil.
Graduate Talk (277 Lockett; Wednesday, Apr. 5, 2:30-3:30PM)
Title: Connections with a small parameter.
Abstract: In my talk, I will start with a classical, and relatively easy, statement about differential equations with a small parameter (due to Wasow) and use a geometric point of view to translate it, first, into a claim about connections on a vector bundle on a Riemann surface, and then into a statement about the geometry of the space of such connections (their moduli space'). The main point of the talk is the interplay between study of individuals' (differential equations or bundles with connections) and properties of their `community' (their moduli space).
Richard Hammack
(March 7--9, 2017)
Richard Hammack is a professor of mathematics at Virginia Commonwealth University in Richmond. A native of rural southern Virginia, he studied painting at Rhode Island School of Design before an interest in computer graphics and visualization led him to a masters in computer science from Virginia Commonwealth University, and then to a Ph.D. in mathematics from UNC Chapel Hill. Before returning to VCU he taught at UNC, Wake Forest University and Randolph-Macon College. He works mostly in the areas of combinatorics and graph theory.
Undergraduate Talk (244 Lockett; Tuesday, Mar. 7, 2:30-3:30PM)
Title: Integrate THIS: The mathematics of planimeters
Abstract: A planimeter is a mechanical analog device that evaluates definite integrals. A typical planimeter features a dial and a stylus attached to an arm. As the stylus traverses the boundary of a region, the dial reads off the enclosed area. Planimeters have been mostly forgotten since the advent of computers, but at one time they were fairly commonplace.
I will explain the history and mathematics of planimeters, and I will demonstrate one that I made from two pieces of cast-off junk. It has only one moving part, but it can evaluate any definite integral that it can reach.
Graduate Talk (277 Lockett; Thursday, Mar. 9, 2:30-3:30PM)
Title: Not every graph has a robust cycle basis
Abstract: The cycle space of a graph G is the vector space (over the 2-element field) whose vectors are the spanning eulerian subgraphs of G, and addition is symmetric difference on edges. As any eulerian subgraph is a sum of edge-disjoint cycles, the cycle space is spanned by the cycles in G, so one can always find a basis of cycles. Such a basis is called a cycle basis for G.
Because their vectors carry combinatorial information, cycle spaces have many applications, and different kinds of cycle bases cater to different kinds of problems. A lot of recent attention has focused on so-called robust cycle bases. Robust cycle bases are known to exist only for a few classes of graphs. Despite this, previously no graph was known to not have a robust cycle basis. We will see that the complete bipartite graphs K_{n,n} have no robust cycle basis when n ≥ 8. This leads to some tantalizingly open questions, particularly for the range 4 < n < 8, but also for general graphs.
Fall 2016:
Chelsea Walton
(November 1--9, 2016)
Dr. Walton is a professor at Temple University. She earned her Ph.D. from the University of Michigan, while also working as a visiting student at the University of Manchester. Dr. Walton held postdoc positions at the University of Washingtion, at the Mathematical Sciences Research Institute, and at MIT. During her time as a postdoc, one of Dr. Walton's focuses was on outreach programs. While at MIT, she taught for the Edge program, was the coordinator for PRIMES circle, and received the Infinite Kilometer Award for outreach. Her mathematical research interests are in noncommutative algebra and representation theory. Traveling has been a big part of Dr. Walton's career, and has sparked her interest in visiting locations all over the world, including Argentina, Morocco, Peru, and Poland.
Undergraduate Talk (239 Lockett; Wednesday, Nov. 2, 12:30-1:30PM)
Title: Hamilton's Quaternions
Abstract: In this talk I will discuss the last great achievement of Sir William Rowan Hamilton- the discovery of the quaternion number system. This discovery was very controversial for its time and nearly drove Hamilton mad! The talk will be full of drama, intrigue, and wonderful mathematics. Some familiarity with complex numbers would help, but is not needed.
Graduate Talk (9 Lockett; Thursday, Nov. 3, 3:30-4:30PM)
Title: Quantum Symmetry
Abstract: Like Hopf algebras? You will after this talk! The aim of this lecture is to motivate and discuss "quantum symmetries" of quantum algebras (i.e. Hopf co/actions on noncommutative algebras). All basic terms will be defined, examples will be provided, along with a brief survey of recent results.
Kiran Kedlaya
(September 19--20, 2016)
Dr. Kedlaya is a professor at UC San Diego. He received his Ph.D. in Mathematics from the Massachusetts Institute of Technology. He was awarded an NSF postdoctoral fellowship and held positions at the Mathematical Sciences Research Institute in Berkeley, at the University of California at Berkeley, and at the Institute for Advanced Study in Princeton. A partial list of the prestigious awards he has received include: the Stefan E. Warschawski Endowed Chair, a Alfred P. Sloan Fellowship, a Clay Liftoff Fellowship, a Presidential Early Career Award for Scientists and Engineers, and a Fellowship from the American Mathematical Society. His research interests include p-adic analytic methods, p-adic Hodge theory, algorithms, and applications in computer science. He is also interested in the education and promotion of mathematics. He has been on the USA Mathematical Olympiad committee, the board of directors for the Art of Problem Solving Foundation, and has authored a Putnam Exam problem book.
These talks have been made possible by the Student Government PSIF and by funding from the LSU Math Department.
Undergraduate Talk (137 Lockett; Monday, Sept. 19, 12:30-1:30PM)
Title: The ABC Conjecture
Abstract: The ABC conjecture asserts that if A, B, C are three positive integers such that A + B = C, then these three integers cannot between them have "too many" repeated prime factors. The precise statement of the conjecture explains the difference between the fact that there are lots of such triples consisting of perfect squares (Pythagorean triples) but not consisting of higher perfect powers (Fermat's Last Theorem). I'll discuss the precise statement of the conjecture, some appealing consequences of this conjecture in various parts of number theory, and the status of a recent (2012) announcement of a proof.
Graduate Talk (103 Coates; Monday, Sept. 20, 12:30-1:30PM)
Title: Computational Number Theory Online: SMC and LMFDB
Abstract: This is more of a demonstration than a talk: I will indicate how to get started with two different but complementary online tools. SageMathCloud (SMC) is a cloud-based version of the Sage computer algebra system, which includes extensive number-theoretic functionality (and plenty of coverage in other areas of mathematics also). The L-Functions and Modular Forms Database (LMFDB) is a website that assembles various tables of number-theoretic objects, like elliptic curves and modular forms, in an easily browsable format that highlights the deep relationships among these objects.
Budapest Semester Informational Meeting (The Keisler Lounge in Lockett; Monday, Sept. 19, 2:30PM)
Study abroad opportunities in mathematics: The Budapest Semesters in Mathematics (BSM) program has been providing North American students the opportunity to spend one or two semesters learning mathematics in the "Hungarian style" for over 30 years. Recently, the Budapest Semesters in Mathematics Education (BSME) was launched to provide similar opportunities for those interested in Hungarian pedagogy. This information session will describe both programs, their similarities and differences, and how to participate.
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2017-09-19 18:44:52
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https://zbmath.org/?q=an:0778.58065
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# zbMATH — the first resource for mathematics
Oscillatory integrals and volumes with semiquasihomogeneous phase. (English. Russian original) Zbl 0778.58065
Funct. Anal. Appl. 26, No. 1, 46-48 (1992); translation from Funkts. Anal. Prilozh. 26, No. 1, 59-61 (1992).
Let $$I(\tau,F,\varphi)=\int_{\mathbb{R}^ n}\varphi\exp(i\tau F)$$, where $$\varphi:\mathbb{R}^ n\to\mathbb{R}$$ is $$C^ \infty$$ with support in a small neighborhood of the origin, and $$F$$ is defined in the support of $$\varphi$$. The author obtains asymptotic expansions for this integral as $$\tau\to\infty$$ when $$F$$ is semiquasihomogeneous, that is, $$F=f+f_ 1$$, where $$f$$ is a quasihomogeneous polynomial of degree 1 with weights $$\alpha=(\alpha_ 1,\ldots,\alpha_ n)$$ and $$f_ 1=\sum a_ mx^ m$$, where $$a_ m\geq 0$$ if $$(m,\alpha)\leq 1$$, is real-analytic. The principal term of these asymptotics, for $$|\alpha|<1$$, has the form $$a(f)\tau^{-|\alpha|}\varphi(0)$$, where $$a(f)>0$$. An analogous expansion is obtained for the Lebesgue integral of $$\varphi$$ over the set $$\{x\in\mathbb{R}^ n:0\leq F(x)\leq\varepsilon\}$$, with $$\varepsilon>0$$. This Lebesgue integral is the “volume” of $$F=f+f_ 1$$. Finally, the author obtains a result concerning the relationship between the semiquasihomogeneity of $$F$$ and the $$\mathbb{R}$$-nondegeneracy of the principal part of the Maclaurin series of $$f$$.
##### MSC:
58J37 Perturbations of PDEs on manifolds; asymptotics 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory
Full Text:
##### References:
[1] V. I. Arnol’d [V. I. Arnold], ”Some open problems in the theory of singularities,” Proceedings of Symposia in Pure Mathematics,40, Part 1 (1983). · Zbl 0519.58019 [2] V. I. Arnol’d, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Mappings [in Russian], Vol. 1, Nauka, Moscow (1982). [3] V. I. Arnol’d, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Mappings [in Russian], Vol. 2, Nauka, Moscow (1984). [4] I. M. Gel’fand and G. E. Shilov, Generalized Functions and Actions over Them [in Russian], No. 1, Gosudarstvennoe Izdatel’stvo Fizikomatematicheskoi Literature, Moscow (1959). · Zbl 0091.11102 [5] V. P. Maslov and M. V. Fedoryuk, Mat. Zametki,30, No. 5, 763-768 (1981). [6] A. M. Chebotarev, Mat. Zametki,34, No. 2, 273-280 (1983). [7] V. N. Karpushkin, Tr. Seminara im. I. G. Petrovskogo, No. 9, 3-39 (1983).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-01-19 09:47:35
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https://library.cirm-math.fr/listRecord.htm?list=link&xRecord=19252471146910706539
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## Post-edited Emergent anyons in quantum Hall physics Rougerie, Nicolas (Auteur de la Conférence) | CIRM (Editeur )
Anyons are by definition particles with quantum statistics different from those of bosons and fermions. They can occur only in low dimensions, 2D being the most relevant case for this talk. They have hitherto remained hypothetical, but there is good theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons.
I shall consider the case of tracer particles immersed in a so-called Laughlin liquid. I will argue that, under certain circumstances, these become anyons. This is made manifest by the emergence of a particular effective Hamiltonian for their motion. The latter is notoriously hard to solve even in simple cases, and well-controled simplifications are highly desirable. I will discuss a possible mean-field approximation, leading to a one-particle energy functional with self-consistent magnetic field.
Anyons are by definition particles with quantum statistics different from those of bosons and fermions. They can occur only in low dimensions, 2D being the most relevant case for this talk. They have hitherto remained hypothetical, but there is good theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons.
I shall consider the case of tracer particles immersed in a so-called Laughlin liquid. I ...
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## Post-edited Inhomogeneities and temperature effects in Bose-Einstein condensates de Bouard, Anne (Auteur de la Conférence) | CIRM (Editeur )
We will review in this talk some mathematical results concerning stochastic models used by physicist to describe BEC in the presence of fluctuations (that may arise from inhomogeneities in the confinement parameters), or BEC at finite temperature. The results describe the effect of those fluctuations on the structures - e.g. vortices - which are present in the deterministic model, or the convergence to equilibrium in the models at finite temperature. We will also describe the numerical methods which have been developed for those models in the framework of the ANR project Becasim. These are joint works with Reika Fukuizumi, Arnaud Debussche, and Romain Poncet.
We will review in this talk some mathematical results concerning stochastic models used by physicist to describe BEC in the presence of fluctuations (that may arise from inhomogeneities in the confinement parameters), or BEC at finite temperature. The results describe the effect of those fluctuations on the structures - e.g. vortices - which are present in the deterministic model, or the convergence to equilibrium in the models at finite ...
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## Post-edited Unresolved problems in the theory of integrable systems Zakharov, Vladimir (Auteur de la Conférence) | CIRM (Editeur )
In spite of enormous success of the theory of integrable systems, at least three important problems are not resolved yet or are resolved only partly. They are the following:
1. The IST in the case of arbitrary bounded initial data.
2. The statistical description of the systems integrable by the IST. Albeit, the development of the theory of integrable turbulence.
3. Integrability of the deep water equations.
These three problems will be discussed in the talk.
In spite of enormous success of the theory of integrable systems, at least three important problems are not resolved yet or are resolved only partly. They are the following:
1. The IST in the case of arbitrary bounded initial data.
2. The statistical description of the systems integrable by the IST. Albeit, the development of the theory of integrable turbulence.
3. Integrability of the deep water equations.
These three problems will be discussed ...
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## Post-edited Various aspects of the dynamics of the cubic Szegö solutions Grellier, Sandrine (Auteur de la Conférence) | CIRM (Editeur )
The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.
The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform ...
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## Multi angle Scattering for NLS in $\mathbb{R}^d\times \mathbb{T}$ Visciglia, Nicola (Auteur de la Conférence) | CIRM (Editeur )
We consider the nonlinear Schrödinger equation in the partially periodic setting $\mathbb{R}^d\times \mathbb{T}$. We present some recent results obtained in collaboration with N. Tzvetkov concerning the Cauchy theory and the long-time behavior of the solutions.
nonlinear Schrödinger equation - Cauchy theory - scattering
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## Multi angle Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field Bellassoued, Mourad (Auteur de la Conférence) | CIRM (Editeur )
This talk is devoted to the study of the following inverse boundary value problem: given a Riemannian manifold with boundary determine the magnetic potential in a dynamical Schrödinger equation in a magnetic field from the observations made at the boundary.
inverse problem - Schrödinger equation - magnetic field
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## Multi angle Quasi-periodic wave equation - almost reducibility - Eliasson, Hakan (Auteur de la Conférence) | CIRM (Editeur )
quasi-periodic wave equation - laplacian - hamiltonian system - symplectic form - time dependance
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## Multi angle On low temperature kinetic theory; spin diffusion, anyons, Bose Einstein condensates Arkeryd, Leif (Auteur de la Conférence) | CIRM (Editeur )
To illustrate specifically quantum behaviours, the talk will consider three typical problems for non-linear kinetic models evolving through pair collisions at temperatures not far from absolute zero. Based on those examples, a number of differences between quantum and classical Boltzmann theory is discussed in more general term.
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## Multi angle From the Hartree-Fock dynamics to the Vlasov equation Saffirio, Chiara (Auteur de la Conférence) | CIRM (Editeur )
We will discuss the convergence (in the semiclassical limit) of a solution to the Hartree-Fock equation towards an operator, whose Wigner transform is a solution to the Vlasov equation. We will consider both cases of positive and zero temperature. The results we will present are part of a project in collaboration with N. Benedikter, M. Porta and B. Schlein.
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## Multi angle Uncertainty principles for discrete Schrödinger evolutions Malinnikova, Eugenia (Auteur de la Conférence) | CIRM (Editeur )
We consider solutions of the semi-discrete Schrödinger equation (where time is continuous and spacial variable is discrete), $\partial_tu = i(\Delta_du + V u)$, where $\Delta_d$ is the standard discrete Laplacian on $\mathbb{Z}^n$ and $u : [0, 1] \times \mathbb{Z}^d \to \mathbb{C}$. Uncertainty principle states that a non-trivial solution of the free equation (without potential) cannot be sharply localized at two distinct times. We discuss different extensions of this result to equations with bounded potentials. The continuous case was studied in a series of articles by L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega.
The talk is mainly based on joint work with Ph. Jaming, Yu. Lyubarskii, and K.-M. Perfekt.
We consider solutions of the semi-discrete Schrödinger equation (where time is continuous and spacial variable is discrete), $\partial_tu = i(\Delta_du + V u)$, where $\Delta_d$ is the standard discrete Laplacian on $\mathbb{Z}^n$ and $u : [0, 1] \times \mathbb{Z}^d \to \mathbb{C}$. Uncertainty principle states that a non-trivial solution of the free equation (without potential) cannot be sharply localized at two distinct times. We discuss ...
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## Multi angle Mean field limits for Ginzburg-Landau vortices Serfaty, Sylvia (Auteur de la Conférence) | CIRM (Editeur )
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation, etc. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. We will present a new result on the derivation of a mean-field limit equation for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation, etc. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. We will present a new result on the derivation of a mean-field limit equation for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzb...
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## Multi angle Leapfrogging for the axisymmetric Gross-Pitaevskii equation Smets, Didier (Auteur de la Conférence) | CIRM (Editeur )
The leapfrogging is the name given to a regime of interaction between vortex rings with the same axis of symmetry in incompressible fluids. We will explain where it comes from and indicate a rigorous derivation in the case of the axisymmetric Gross-Pitaevskii equation.
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## Multi angle Soliton resolution for derivative NLS equation Sulem, Catherine (Auteur de la Conférence) | CIRM (Editeur )
We consider the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We prove global wellposedness and give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. Our analysis provides explicit formulae for the multi-soliton component as well as the correction dispersive term. We use the inverse scattering approach and the nonlinear steepest descent method of Deift and Zhou (1993) revisited by the $\bar{\partial}$-analysis of Dieng-McLaughlin (2008) and complemented by the recent work of Borghese-Jenkins-McLaughlin (2016) on soliton resolution for the focusing nonlinear Schrödinger equation. This is a joint work with R. Jenkins, J. Liu and P. Perry.
We consider the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We prove global wellposedness and give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. Our analysis provides explicit formulae for the multi-soliton component as well as ...
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## Multi angle Interactions of solitary waves for the nonlinear Schrödinger equations Martel, Yvan (Auteur de la Conférence) | CIRM (Editeur )
I will present two cases of strong interactions between solitary waves for the nonlinear Schrödinger equations (NLS). In the mass sub- and super-critical cases, a work by Tien Vinh Nguyen proves the existence of multi-solitary waves with logarithmic distance in time, extending a classical result of the integrable case (1D cubic NLS equation). In the mass-critical case, a work by Yvan Martel and Pierre Raphaël gives a new class of blow up multi-solitary waves blowing up in infinite time with logarithmic rate.
These special behaviours are due to strong interactions between the waves, in contrast with most previous works on multi-solitary waves of (NLS) where interactions do not affect the general behaviour of each solitary wave.
I will present two cases of strong interactions between solitary waves for the nonlinear Schrödinger equations (NLS). In the mass sub- and super-critical cases, a work by Tien Vinh Nguyen proves the existence of multi-solitary waves with logarithmic distance in time, extending a classical result of the integrable case (1D cubic NLS equation). In the mass-critical case, a work by Yvan Martel and Pierre Raphaël gives a new class of blow up ...
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## Multi angle Solitons vs collapses Kuznetsov, Evgenii (Auteur de la Conférence) | CIRM (Editeur )
This talk is devoted to solitons and wave collapses which can be considered as two alternative scenarios pertaining to the evolution of nonlinear wave systems describing by a certain class of dispersive PDEs (see, for instance, review [1]). For the former case, it suffices that the Hamiltonian be bounded from below (or above), and then the soliton realizing its minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation of small-amplitude waves, a process absent in systems with finitely many degrees of freedom. The framework of the nonlinear Schrodinger equation, the ZK equation and the three-wave system is used to show how the boundedness of the Hamiltonian H, and hence the stability of the soliton minimizing H can be proved rigorously using the integral estimate method based on the Sobolev embedding theorems. Wave systems with the Hamiltonians unbounded from below must evolve to a collapse, which can be considered as the fall of a particle in an unbounded potential. The radiation of small-amplitude waves promotes collapse in this case.
This work was supported by the Russian Science Foundation (project no. 14-22-00174).
This talk is devoted to solitons and wave collapses which can be considered as two alternative scenarios pertaining to the evolution of nonlinear wave systems describing by a certain class of dispersive PDEs (see, for instance, review [1]). For the former case, it suffices that the Hamiltonian be bounded from below (or above), and then the soliton realizing its minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation ...
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## Multi angle Almost sure scattering for the energy-critical Schrödinger equation in 4D with radial data Visan, Monica (Auteur de la Conférence) | CIRM (Editeur )
Inspired by a recent result of Dodson-Luhrmann-Mendelson, who proved almost sure scattering for the energy-critical wave equation with radial data in four dimensions, we establish the analogous result for the Schrödinger equation.
This is joint work with R. Killip and J. Murphy.
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## Multi angle Geometric heat flows and caloric gauges Tataru, Daniel (Auteur de la Conférence) | CIRM (Editeur )
Choosing favourable gauges is a crucial step in the study of nonlinear geometric dispersive equations. A very successful tool, that has emerged originally in work of Tao on wave maps, is the use of caloric gauges, defined via the corresponding geometric heat flows. The aim of this talk is to describe two such flows and their associated gauges, namely the harmonic heat flow and the Yang-Mills heat flow.
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## Virtualconference Classical and quantum particles coupled to a vibrational environment Goudon, Thierry (Auteur de la Conférence) | CIRM (Editeur )
It is possible to model dissipation effects subjected by a particle by interactions between the particle and its environment. This seminal idea dates back to Caldeira-Leggett in the ’80ies. The specific case of a particle interacting with vibrational degrees of freedom has been thoroughsly investigated by S. De Bièvre and his collaborators. We will go back to these issues in the framework of kinetic equations, and also consider quantum versions of the problem based on couplings with the Schrödinger equation. We are particularly interested in stability issues. We will describe ; through rigorous statements and numerical experiments, analogies and differences with the case of a single classical particle and with the standard coupling with the Poisson equation.
It is possible to model dissipation effects subjected by a particle by interactions between the particle and its environment. This seminal idea dates back to Caldeira-Leggett in the ’80ies. The specific case of a particle interacting with vibrational degrees of freedom has been thoroughsly investigated by S. De Bièvre and his collaborators. We will go back to these issues in the framework of kinetic equations, and also consider quantum versions ...
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## Virtualconference Highly-oscillatory evolution equations: averaging and numerics Lemou, Mohammed (Auteur de la Conférence) | CIRM (Editeur )
Usual numerical methods become inefficient when they are applied to highly oscillatory evolution problems (order reduction or complete loss of accuracy). The numerical parameters must indeed be adapted to the high frequencies that come into play to correctly capture the desired information, and this induces a prohibitive computational cost. Furthermore, the numerical resolution of averaged models, even at high orders, is not sufficient to capture low frequencies and transition regimes. We present (very briefly) two strategies allowing to remove this obstacle for a large class of evolution problems : a 2-scale method and a micro/macro method. Two different frameworks will be considered : constant frequency, and variable - possibly vanishing - frequency. The result of these approaches is the construction of numerical schemes whose order of accuracy no longer depends on the frequency of oscillation, one then speaks of uniform accuracy (UA) for these schemes. Finally, a new technique for systematizing these two methods will be presented. Its purpose is to reduce the number of inputs that the user must provide to apply the method in practice. In other words, only the values of the field defining the evolution equation (and not its derivatives) are used.These methods have been successfully applied to solve a number of evolution models: non-linear Schrödinger and Klein-Gordon equations, Vlasov-Poisson kinetic equation with strong magnetic field, quantum transport in graphene.
Usual numerical methods become inefficient when they are applied to highly oscillatory evolution problems (order reduction or complete loss of accuracy). The numerical parameters must indeed be adapted to the high frequencies that come into play to correctly capture the desired information, and this induces a prohibitive computational cost. Furthermore, the numerical resolution of averaged models, even at high orders, is not sufficient to ...
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2021-04-18 21:23:34
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http://mathhelpforum.com/advanced-algebra/185454-polynomial-height.html
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1. ## polynomial height
let' s define 'height' of a polynomial the max of the absolute value of its coefficients.
Do exist P(x) and Q(x) TWO polynomials at integer coefficient with heights equal or greater than 2011, whose product is a polynomial with height =1?
I have tried with cyclotomic polynomials, that we know can have any coefficient among the integers, and whose product of convenient cyclotomic gives x^n-1 (whoe height is 1), but I cannot demonstrate that I can find exactlky TWO polynomials whith height >=2011 whose product is a polyn. with height=1.
has anyone a clue onhow to solve with cyclotomic or with any other method?
2. ## Re: polynomial height
Originally Posted by pincopallino
let' s define 'height' of a polynomial the max of the absolute value of its coefficients.
Do exist P(x) and Q(x) TWO polynomials at integer coefficient with heights equal or greater than 2011, whose product is a polynomial with height =1?
I have tried with cyclotomic polynomials, that we know can have any coefficient among the integers, and whose product of convenient cyclotomic gives x^n-1 (whoe height is 1), but I cannot demonstrate that I can find exactlky TWO polynomials whith height >=2011 whose product is a polyn. with height=1.
has anyone a clue onhow to solve with cyclotomic or with any other method?
I am suspicious of questions that involve the number of the current year, because these often feature in olympiad-type competitions. If you can convince me that this problem does not come from a current competition then I can give some hints on how to solve it.
3. ## Re: polynomial height
Originally Posted by Opalg
I am suspicious of questions that involve the number of the current year, because these often feature in olympiad-type competitions. If you can convince me that this problem does not come from a current competition then I can give some hints on how to solve it.
No competition related. Actually I could have written any integer (prime as 2011 or not)!
I have succeeded to find a polynomial with any height that muliplied by a polyn with height=1 gives a result with height 1, but two polynomials not yet!
some hint?
4. ## Re: polynomial height
Originally Posted by pincopallino
let' s define 'height' of a polynomial the max of the absolute value of its coefficients.
Do exist P(x) and Q(x) TWO polynomials at integer coefficient with heights equal or greater than 2011, whose product is a polynomial with height =1?
Okay, here is the idea. Given a positive integer N, we want to find polynomials P(x), Q(x), with integer coefficients, each having height at least N, such that P(x)Q(x) has height 1.
Even for N = 2 this takes a bit of thought. Start with the polynomial
$\displaystyle (1-x^2)(1-x^4) = 1-x^2-x^4+x^6,$
which clearly has height 1. Now by repeatedly factorising the difference of two squares, write that polynomial as
$\displaystyle (1-x^2)(1-x^4) = (1-x^2)^2(1+x^2) = (1-x)^2(1+x)^2(1+x^2).$
Take $\displaystyle P(x) = (1-x)^2$ and $\displaystyle Q(x) = (1+x)^2(1+x^2).$ Then each of P(x), Q(x) has height at least 2, and their product has height 1.
Now do the same sort of thing for N = 3, looking at the polynomial $\displaystyle (1-x^2)(1-x^4)(1-x^8).$ Once you have sorted that one out, you will see how to do it for any N. (It's really a very simple-minded construction, nothing as sophisticated as cyclotomic polynomials!)
5. ## Re: polynomial height
Originally Posted by Opalg
Okay, here is the idea. Given a positive integer N, we want to find polynomials P(x), Q(x), with integer coefficients, each having height at least N, such that P(x)Q(x) has height 1.
Even for N = 2 this takes a bit of thought. Start with the polynomial
$\displaystyle (1-x^2)(1-x^4) = 1-x^2-x^4+x^6,$
which clearly has height 1. Now by repeatedly factorising the difference of two squares, write that polynomial as
$\displaystyle (1-x^2)(1-x^4) = (1-x^2)^2(1+x^2) = (1-x)^2(1+x)^2(1+x^2).$
Take $\displaystyle P(x) = (1-x)^2$ and $\displaystyle Q(x) = (1+x)^2(1+x^2).$ Then each of P(x), Q(x) has height at least 2, and their product has height 1.
Now do the same sort of thing for N = 3, looking at the polynomial $\displaystyle (1-x^2)(1-x^4)(1-x^8).$ Once you have sorted that one out, you will see how to do it for any N. (It's really a very simple-minded construction, nothing as sophisticated as cyclotomic polynomials!)
brilliant! definetly very simple!!!!! thanks a lot!!!!!! the trick is to multiply with such polyn with degree such that the product gives degrees that whose terms do not sum and such that the different facorization gives the desired coefficients!!!!
the question I would like to ask is: how one can have this 'enlightment' to try this factorization? experience? chance? trials on basic polynomials?...
6. ## Re: polynomial height
Originally Posted by pincopallino
brilliant! definetly very simple!!!!! thanks a lot!!!!!! the trick is to multiply with such polyn with degree such that the product gives degrees that whose terms do not sum and such that the different facorization gives the desired coefficients!!!!
the question I would like to ask is: how one can have this 'enlightment' to try this factorization? experience? chance? trials on basic polynomials?...
As I said, it takes a bit of thought, even for N=2. See the Newton quote below! In fact, I spent a long evening struggling with the N=2 case, doing fruitless computations, until the factorisation idea suddenly jumped out at me.
7. ## Re: polynomial height
I see! <wait till the first dawnings open little by little into the full light>
$\displaystyle (1-x^2)(1-x^4)(1-x^8)...(1-x^2^^(^n)) .$ = $\displaystyle (1-x)^n (1+x)^n (1+x^2)^n^-^1 (1+x^4)^n^-^2 (1+x^8)^n^-^3))... (1+x^2^^(^n^-^1^)) .$
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2018-05-26 07:12:43
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https://questions.examside.com/past-years/jee/question/pan-electron-mass-mathrmm-with-an-initial-velocit-jee-main-physics-motion-uqkuqhuecpedinmg
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1
JEE Main 2022 (Online) 27th July Morning Shift
+4
-1
An electron (mass $$\mathrm{m}$$) with an initial velocity $$\vec{v}=v_{0} i\left(v_{0}>0\right)$$ is moving in an electric field $$\vec{E}=-E_{0} \hat{i}\left(E_{0}>0\right)$$ where $$E_{0}$$ is constant. If at $$\mathrm{t}=0$$ de Broglie wavelength is $$\lambda_{0}=\frac{h}{m v_{0}}$$, then its de Broglie wavelength after time t is given by
A
$$\lambda_{0}$$
B
$$\lambda_{0}\left(1+\frac{e E_{0} t}{m v_{0}}\right)$$
C
$$\lambda_{0} t$$
D
$$\frac{\lambda_{0}}{\left(1+\frac{e E_{0} t}{m v_{0}}\right)}$$
2
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1
Two uniformly charged spherical conductors $$A$$ and $$B$$ of radii $$5 \mathrm{~mm}$$ and $$10 \mathrm{~mm}$$ are separated by a distance of $$2 \mathrm{~cm}$$. If the spheres are connected by a conducting wire, then in equilibrium condition, the ratio of the magnitudes of the electric fields at the surface of the sphere $$A$$ and $$B$$ will be :
A
1 : 2
B
2 : 1
C
1 : 1
D
1 : 4
3
JEE Main 2022 (Online) 29th June Evening Shift
+4
-1
Two point charges Q each are placed at a distance d apart. A third point charge q is placed at a distance x from mid-point on the perpendicular bisector. The value of x at which charge q will experience the maximum Coulomb's force is :
A
x = d
B
$$x = {d \over 2}$$
C
$$x = {d \over {\sqrt 2 }}$$
D
$$x = {d \over {2\sqrt 2 }}$$
4
JEE Main 2022 (Online) 29th June Evening Shift
+4
-1
If the electric potential at any point (x, y, z) m in space is given by V = 3x2 volt. The electric field at the point (1, 0, 3) m will be :
A
3 Vm$$-$$1, directed along positive x-axis.
B
3 Vm$$-$$1, directed along negative x-axis.
C
6 Vm$$-$$1, directed along positive x-axis.
D
6 Vm$$-$$1, directed along negative x-axis.
JEE Main Subjects
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2023-03-21 16:27:37
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https://gmatclub.com/forum/quadrilaterals-299755.html
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# Properties of Quadrilaterals
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Updated on: 23 Jul 2019, 23:47
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Purpose of the article:
I see that you are looking to know about various types of quadrilaterals.
• Well, you have landed just at the right place.
• You will get an idea about the quadrilateral and its various types.
• You will also get to know about the properties of a few special kinds of quadrilaterals.
So, let’s get straight into it.
What is a quadrilateral?
A quadrilateral is a polygon that has 4 sides.
• So, any closed figure that has 4 sides is a quadrilateral.
• And, all the angles of a quadrilateral sum up to 3600.
The diagram given below shows a quadrilateral ABCD and the sum of its internal angles.
Various kinds of quadrilateral
There are some special kinds of quadrilateral that we see in our textbooks/ exams.
These are:
1. Rectangle
2. Square
3. Rhombus
4. Parallelogram
5. Trapezium
Let us discuss each type in detail.
Rectangle
A rectangle is a quadrilateral:
• Each of the 4 angles are $$90^o$$
• And, opposite sides of a rectangle are equal and parallel
• Diagonals of a rectangle bisect each other
Formulas to remember
If the length of the rectangle is L and breadth is B then,
1. Area of a rectangle = Length × Breadth or L × B
2. Perimeter of rectangle = 2 × (L + B)
Square
A square is a quadrilateral:
• That has all the angles as $$90^o$$
• All sides of a square are equal
o And, opposite sides are parallel to each other
• Diagonals bisect each other perpendicularly
Formulas to remember
If the side of a square is “a” then,
1. Area of the square = $$a × a = a^2$$
2. Perimeter of the square = 2 × (a + a) = 4a
Parallelogram
A parallelogram is a quadrilateral in which:
Opposite angles are equal
Opposite sides are equal and parallel
• Diagonals bisect each other
• Sum of any two adjacent angles is $$180^o$$
Formulas to remember
If the length of a parallelogram is “l”, breadth is “b” and height is “h” then:
1. Perimeter of parallelogram= 2 × (l + b)
2. Area of the parallelogram = l × h
Rhombus
A rhombus is a quadrilateral in which:
Opposite angles are equal
All sides are equal
o And, opposite sides are parallel to each other
Diagonals bisect each other perpendicularly
• Sum of any two adjacent angles is $$180^o$$
Formulas to remember
If the side of a rhombus is a then,
Perimeter of rhombus= 4a
If the length of two diagonals of the rhombus is d1 and d2 then:
Area of the rhombus = $$\frac{1}{2} × d_1 × d_2$$
Trapezium
A trapezium is a quadrilateral in which:
• Only one pair of opposite sides are parallel to each other
Formulas to remember
If the height of a trapezium is “h” (as shown in the above diagram) then:
Perimeter of the trapezium= Sum of lengths of all the sides = AB + BC + CD + DA
Area of the trapezium = $$\frac{1}{2}$$ × (Sum of lengths of parallel sides) × h
= $$\frac{1}{2}$$× (AB + CD) × h
Summary of all the properties we learnt
_________________
Originally posted by EgmatQuantExpert on 10 Jul 2019, 01:00.
Last edited by EgmatQuantExpert on 23 Jul 2019, 23:47, edited 2 times in total.
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16 Jul 2019, 23:04
Practice test is missing EgmatQuantExpert and great article though
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18 Jul 2019, 03:18
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25 Jul 2019, 01:05
Re: Properties of Quadrilaterals [#permalink] 25 Jul 2019, 01:05
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# Properties of Quadrilaterals
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2019-11-13 06:41:21
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http://mathcentral.uregina.ca/QQ/database/QQ.09.08/h/dawn3.html
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SEARCH HOME
Math Central Quandaries & Queries
Question from Dawn: 1-- syringe contains 5.5cc of medicine & cost $13.50 1 --2 oz bottle of medicine cost$17.50 How many cc's in 2 oz, & which is the better deal?
Hi Dawn.
You have two costs per unit volume:
(A) $13.50 / (5.5 cc) (B)$17.00 / (2 oz)
You need to find the conversion factor to go from cc's to ounces, so we can compare these fractions using a common unit denominator.
When I type "1 cc in ounces" into Google, it replies "1 cc = 0.0338140227 US fluid ounces". So the conversion factor is that decimal number.
So (A) becomes:
$13.50 / ( 5.5 cc x 0.033814 oz/cc) =$13.50 / ( 5.5 x 0.033814 oz)
= $13.50 / (0.195977 oz) =$72.59 per ounce.
(B) becomes
$17.00 / (2 oz) =$8.50 per ounce.
The bottle is WAY cheaper than the syringe, assuming you don't need the syringe itself!
Hope this helps,
Stephen La Rocque.
PS: Google is getting smarter all the time. You could even type in
"convert (13.50 dollars per (5.5 cc)) to dollars per ounce" and it will tell you !
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
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2022-09-30 13:01:49
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https://chorasimilarity.wordpress.com/2012/05/11/the-neuron-unit/
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# The neuron unit
I am updating the paper on $\lambda \epsilon$ almost daily. It is the first time when I am doing such a thing, it is maybe interesting to see what comes out of this way of writing.
The last addition is something I was thinking about for a long time, something which is probably well known in some circles, but maybe not. It is about eliminating variable (names) from such a calculus. This has been done in several ways, here is another (or the same as a previous one?).
The idea is simple. let $T$ be any term and $x \in Var(T)$ a variable. Look at the syntactic tree of $T$, then glue to all leaves decorated by $x$ the leaves of a tree with nodes consisting of FAN-OUT gates.
Further on I shall identify syntactic trees with terms. I shall add to such trees a new family
(of trees), constructed from the elementary tree depicted at (a) in the next figure. At (b) we see an example of such a tree. We consider also the trivial tree (c).
We have to think about such trees as devices for multiplying the occurences of a variable. I call them FAN-OUT trees. All these trees, with the exception of the trivial one (c), are planar binary trees. We shall add the following rule of associativity:
(ASSOC) any two FAN-OUT trees with the same number of leaves are identified.
This rule will be applied under the following form: we are free to pass from a FAN-OUT tree to an equivalent one which has the same number of leaves. The name “associativity” comes from the fact that a particular instance of this rule (which deserves the name “elementary associativity move”) is this:
With the help of these FAN-OUT trees we shall replace the multiple occurences of a variable by such trees. Let us see what become the rules of $\lambda \varepsilon$ calculus by using this notation.
$\alpha$ conversion is no longer needed, because variables have no longer names. Instead, we are free to graft usual trees to FAN-OUT trees. This way, instead of terms we shall use “neurons”.
Definition: The forgetful form (b) of a syntactic tree (a) (of a term) is the tree with the name variables deleted.
A neuron is the result of gluing the root of a forgetful form of a syntactic tree to the root of a FAN-OUT tree, like in the following figure.
The axon of the neuron is the FAN-OUT tree. The dendrites of the neuron are the undecorated edges of the forgetful form of the syntactic tree. A dendrite is bound if it is a left edge pointing to a node decorated by $\lambda$. For any bound dendrite, the set of dependent dendrides are those of the sub-tree starting from the right edge of the $\lambda$ node (where the bound dendrite is connected via a left edge), which are moreover not bound. Otherwise a dendrite is free. The soma of the neuron is the forgetful form of the syntactic tree.
Substitution. We are free to connect leaves of axons of neurons with dendrites of other neurons.
In order to explain the substitution we have to add the following rule:
(subst) The leaves of an axon cannot be glued to more than one bounded dendrite of another neuron. If a leaf of the axon of the neuron $A$ is connected to a bound dendrite of the neuron $B$, then it has to be the leftmost leaf of the axon of $A$. Moreover, in this case all other leaves of $A$ which are connected with $B$ have to be connected only via dendrites which are dependent on the mentioned bound dendrite of $B$, possibly via adding leaves to the axon of $A$ by using (assoc).
Substitution is therefore assimilated to connecting neurons.
New (beta*). The rule (beta*) takes the following graphical form. In the next figure appear only the leaf of the neuron $A$ connecting to the $\lambda$ node (the other leaves of the axon of $A$ not drawn) and only some of the dendrites depending on the bound one relative to the $\lambda$ node which is figured.
The neuron $A$ may have other dendrites, not figured. According to the definition of the neuron, $A$ together with the $\lambda$ node and adiacent edges form a bigger neuron. Also figured is another leaf of the axon of the neuron $B$, which may point to another neuron. Finally, in the RHS, the bound dendrite looses all dependents.
Important remark. Notice that there may be multiple outcomes from (subst) and (beta*)! Indeed, this is due to the fact that connections could be made in several ways, by using all or only of part of the dependent dendrites. Because of that, it seems that this version of the calculus is richer than the previous one, but I am not at all sure if this is the case.
Another thing to be remarked is that the “$=$” sign in this version of these rules is reflexive and transitive, but not symmetric. Previously the “$latex =$” sign was supposed to be symmetric.
(R1) That is easy, we use the emergent algebra gates:
(R2) Easy as well:
The rules (ext1), (ext2) take obvious forms.
Therefore, if we think about computation as reduction to a normal form (if any), in this graphical notation with “neurons”, computation amounts to re-wiring of neurons or changing the rewiring inside the soma of some neurons.
Variables dissapeared, with the price of introducing FAN-OUT trees.
As concerns the remark previously made, we could obtain a calculus which is clearly equivalent with $\lambda \varepsilon$ by modifying the definition of the neuron, in this way.
In order to clearly specify which are the dependent dendrites, we could glue to any bound dendrite a FAN-OUT tree, such that the leaves of this tree connect again with a set of dependent dendrites of the same neuron. In this way, substitution and (beta*) will amount of erasing such a FAN-OUT tree and then perform the moves, as previously explained, but using this time all the dependent dendrites which were connected to the bound dendrite by the erased tree.
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2018-06-19 14:15:01
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http://ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-75
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New Titles | FAQ | Keep Informed | Review Cart | Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education
The Finite Calculus Associated with Bessel Functions
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Contemporary Mathematics
1988; 122 pp; softcover
Volume: 75
ISBN-10: 0-8218-5083-0
ISBN-13: 978-0-8218-5083-1
List Price: US$29 Member Price: US$23.20
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Although Bessel functions are among the most widely used functions in applied mathematics, this book is essentially the first to present a calculus associated with this class of functions. The author obtains a generalized umbral calculus associated with the Euler operator and its associated Bessel eigenfunctions for each positive value of an index parameter. For one particular value of this parameter, the functions and operators can be associated with the radial parts of $$n$$-dimensional Euclidean space objects. Some of the results of this book are in part extensions of the work of Rota and his co-workers on the ordinary umbral calculus and binomial enumeration. The author also introduces a wide variety of new polynomial sequences together with their groups and semigroup compositional properties. Generalized Bernoulli, Euler, and Stirling numbers associated with Bessel functions and the corresponding classes of polynomials are also studied. The book is intended for mathematicians and physicists at the research level in special function theory.
• The $$\nu$$-umbral algebra
• The $$\nu$$-umbral field
• The group of $$\nu$$-delta functionals under composition
• Generalized binomial polynomial sequences
• The composition of polynomial sequences
• Compositions of Moebius delta functionals
• Generalized shift invariant operators
• The generalized derivative of $$\nu$$-shift invariant operators
• Generalized Sheffer polynomials
• Cross sets of polynomials
• A class of Laguerre type polynomials
• The generalized heat polynomials
• A primitive integral for the Euler operator
• Bernoulli type polynomials and numbers
• Generalized Euler polynomials and numbers
• Generalized Stirling numbers and factor polynomials
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2014-07-31 10:19:12
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https://docs.sympy.org/latest/modules/functions/elementary.html
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# Elementary¶
This module implements elementary functions such as trigonometric, hyperbolic, and sqrt, as well as functions like Abs, Max, Min etc.
# sympy.functions.elementary.complexes¶
## re¶
class sympy.functions.elementary.complexes.re(**kwargs)[source]
Returns real part of expression. This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function.
Examples
>>> from sympy import re, im, I, E
>>> from sympy.abc import x, y
>>> re(2*E)
2*E
>>> re(2*I + 17)
17
>>> re(2*I)
0
>>> re(im(x) + x*I + 2)
2
im
as_real_imag(deep=True, **hints)[source]
Returns the real number with a zero imaginary part.
## im¶
class sympy.functions.elementary.complexes.im(**kwargs)[source]
Returns imaginary part of expression. This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function.
Examples
>>> from sympy import re, im, E, I
>>> from sympy.abc import x, y
>>> im(2*E)
0
>>> re(2*I + 17)
17
>>> im(x*I)
re(x)
>>> im(re(x) + y)
im(y)
re
as_real_imag(deep=True, **hints)[source]
Return the imaginary part with a zero real part.
Examples
>>> from sympy.functions import im
>>> from sympy import I
>>> im(2 + 3*I).as_real_imag()
(3, 0)
## sign¶
class sympy.functions.elementary.complexes.sign(**kwargs)[source]
Returns the complex sign of an expression:
If the expression is real the sign will be:
• 1 if expression is positive
• 0 if expression is equal to zero
• -1 if expression is negative
If the expression is imaginary the sign will be:
• I if im(expression) is positive
• -I if im(expression) is negative
Otherwise an unevaluated expression will be returned. When evaluated, the result (in general) will be cos(arg(expr)) + I*sin(arg(expr)).
Examples
>>> from sympy.functions import sign
>>> from sympy.core.numbers import I
>>> sign(-1)
-1
>>> sign(0)
0
>>> sign(-3*I)
-I
>>> sign(1 + I)
sign(1 + I)
>>> _.evalf()
0.707106781186548 + 0.707106781186548*I
## Abs¶
class sympy.functions.elementary.complexes.Abs(**kwargs)[source]
Return the absolute value of the argument.
This is an extension of the built-in function abs() to accept symbolic values. If you pass a SymPy expression to the built-in abs(), it will pass it automatically to Abs().
Examples
>>> from sympy import Abs, Symbol, S
>>> Abs(-1)
1
>>> x = Symbol('x', real=True)
>>> Abs(-x)
Abs(x)
>>> Abs(x**2)
x**2
>>> abs(-x) # The Python built-in
Abs(x)
Note that the Python built-in will return either an Expr or int depending on the argument:
>>> type(abs(-1))
<... 'int'>
>>> type(abs(S.NegativeOne))
<class 'sympy.core.numbers.One'>
Abs will always return a sympy object.
fdiff(argindex=1)[source]
Get the first derivative of the argument to Abs().
Examples
>>> from sympy.abc import x
>>> from sympy.functions import Abs
>>> Abs(-x).fdiff()
sign(x)
## arg¶
class sympy.functions.elementary.complexes.arg(**kwargs)[source]
Returns the argument (in radians) of a complex number. For a positive number, the argument is always 0.
Examples
>>> from sympy.functions import arg
>>> from sympy import I, sqrt
>>> arg(2.0)
0
>>> arg(I)
pi/2
>>> arg(sqrt(2) + I*sqrt(2))
pi/4
## conjugate¶
class sympy.functions.elementary.complexes.conjugate(**kwargs)[source]
Returns the $$complex conjugate$$ Ref[1] of an argument. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part.
Thus, the conjugate of the complex number $$a + ib$$ (where a and b are real numbers) is $$a - ib$$
Examples
>>> from sympy import conjugate, I
>>> conjugate(2)
2
>>> conjugate(I)
-I
References
R212
https://en.wikipedia.org/wiki/Complex_conjugation
## polar_lift¶
class sympy.functions.elementary.complexes.polar_lift(**kwargs)[source]
Lift argument to the Riemann surface of the logarithm, using the standard branch.
>>> from sympy import Symbol, polar_lift, I
>>> p = Symbol('p', polar=True)
>>> x = Symbol('x')
>>> polar_lift(4)
4*exp_polar(0)
>>> polar_lift(-4)
4*exp_polar(I*pi)
>>> polar_lift(-I)
exp_polar(-I*pi/2)
>>> polar_lift(I + 2)
polar_lift(2 + I)
>>> polar_lift(4*x)
4*polar_lift(x)
>>> polar_lift(4*p)
4*p
## periodic_argument¶
class sympy.functions.elementary.complexes.periodic_argument(**kwargs)[source]
Represent the argument on a quotient of the Riemann surface of the logarithm. That is, given a period P, always return a value in (-P/2, P/2], by using exp(P*I) == 1.
>>> from sympy import exp, exp_polar, periodic_argument, unbranched_argument
>>> from sympy import I, pi
>>> unbranched_argument(exp(5*I*pi))
pi
>>> unbranched_argument(exp_polar(5*I*pi))
5*pi
>>> periodic_argument(exp_polar(5*I*pi), 2*pi)
pi
>>> periodic_argument(exp_polar(5*I*pi), 3*pi)
-pi
>>> periodic_argument(exp_polar(5*I*pi), pi)
0
## principal_branch¶
class sympy.functions.elementary.complexes.principal_branch(**kwargs)[source]
Represent a polar number reduced to its principal branch on a quotient of the Riemann surface of the logarithm.
This is a function of two arguments. The first argument is a polar number $$z$$, and the second one a positive real number of infinity, $$p$$. The result is “z mod exp_polar(I*p)”.
>>> from sympy import exp_polar, principal_branch, oo, I, pi
>>> from sympy.abc import z
>>> principal_branch(z, oo)
z
>>> principal_branch(exp_polar(2*pi*I)*3, 2*pi)
3*exp_polar(0)
>>> principal_branch(exp_polar(2*pi*I)*3*z, 2*pi)
3*principal_branch(z, 2*pi)
# Trigonometric Functions¶
## sin¶
class sympy.functions.elementary.trigonometric.sin(**kwargs)[source]
The sine function.
Returns the sine of x (measured in radians).
Notes
This function will evaluate automatically in the case x/pi is some rational number [R216]. For example, if x is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.
Examples
>>> from sympy import sin, pi
>>> from sympy.abc import x
>>> sin(x**2).diff(x)
2*x*cos(x**2)
>>> sin(1).diff(x)
0
>>> sin(pi)
0
>>> sin(pi/2)
1
>>> sin(pi/6)
1/2
>>> sin(pi/12)
-sqrt(2)/4 + sqrt(6)/4
References
R213
https://en.wikipedia.org/wiki/Trigonometric_functions
R214
http://dlmf.nist.gov/4.14
R215
http://functions.wolfram.com/ElementaryFunctions/Sin
R216(1,2)
http://mathworld.wolfram.com/TrigonometryAngles.html
Members
## cos¶
class sympy.functions.elementary.trigonometric.cos(**kwargs)[source]
The cosine function.
Returns the cosine of x (measured in radians).
Notes
See sin() for notes about automatic evaluation.
Examples
>>> from sympy import cos, pi
>>> from sympy.abc import x
>>> cos(x**2).diff(x)
-2*x*sin(x**2)
>>> cos(1).diff(x)
0
>>> cos(pi)
-1
>>> cos(pi/2)
0
>>> cos(2*pi/3)
-1/2
>>> cos(pi/12)
sqrt(2)/4 + sqrt(6)/4
References
R217
https://en.wikipedia.org/wiki/Trigonometric_functions
R218
http://dlmf.nist.gov/4.14
R219
http://functions.wolfram.com/ElementaryFunctions/Cos
Members
## tan¶
class sympy.functions.elementary.trigonometric.tan(**kwargs)[source]
The tangent function.
Returns the tangent of x (measured in radians).
Notes
See sin() for notes about automatic evaluation.
Examples
>>> from sympy import tan, pi
>>> from sympy.abc import x
>>> tan(x**2).diff(x)
2*x*(tan(x**2)**2 + 1)
>>> tan(1).diff(x)
0
>>> tan(pi/8).expand()
-1 + sqrt(2)
References
R220
https://en.wikipedia.org/wiki/Trigonometric_functions
R221
http://dlmf.nist.gov/4.14
R222
http://functions.wolfram.com/ElementaryFunctions/Tan
Members
## cot¶
class sympy.functions.elementary.trigonometric.cot(**kwargs)[source]
The cotangent function.
Returns the cotangent of x (measured in radians).
Notes
See sin() for notes about automatic evaluation.
Examples
>>> from sympy import cot, pi
>>> from sympy.abc import x
>>> cot(x**2).diff(x)
2*x*(-cot(x**2)**2 - 1)
>>> cot(1).diff(x)
0
>>> cot(pi/12)
sqrt(3) + 2
References
R223
https://en.wikipedia.org/wiki/Trigonometric_functions
R224
http://dlmf.nist.gov/4.14
R225
http://functions.wolfram.com/ElementaryFunctions/Cot
Members
## sec¶
class sympy.functions.elementary.trigonometric.sec(**kwargs)[source]
The secant function.
Returns the secant of x (measured in radians).
Notes
See sin() for notes about automatic evaluation.
Examples
>>> from sympy import sec
>>> from sympy.abc import x
>>> sec(x**2).diff(x)
2*x*tan(x**2)*sec(x**2)
>>> sec(1).diff(x)
0
References
R226
https://en.wikipedia.org/wiki/Trigonometric_functions
R227
http://dlmf.nist.gov/4.14
R228
http://functions.wolfram.com/ElementaryFunctions/Sec
Members
## csc¶
class sympy.functions.elementary.trigonometric.csc(**kwargs)[source]
The cosecant function.
Returns the cosecant of x (measured in radians).
Notes
See sin() for notes about automatic evaluation.
Examples
>>> from sympy import csc
>>> from sympy.abc import x
>>> csc(x**2).diff(x)
-2*x*cot(x**2)*csc(x**2)
>>> csc(1).diff(x)
0
References
R229
https://en.wikipedia.org/wiki/Trigonometric_functions
R230
http://dlmf.nist.gov/4.14
R231
http://functions.wolfram.com/ElementaryFunctions/Csc
Members
## sinc¶
class sympy.functions.elementary.trigonometric.sinc(**kwargs)[source]
Represents an unnormalized sinc function:
$\begin{split}\operatorname{sinc}(x) = \begin{cases} \frac{\sin x}{x} & \qquad x \neq 0 \\ 1 & \qquad x = 0 \end{cases}\end{split}$
Examples
>>> from sympy import sinc, oo, jn, Product, Symbol
>>> from sympy.abc import x
>>> sinc(x)
sinc(x)
• Automated Evaluation
>>> sinc(0)
1
>>> sinc(oo)
0
• Differentiation
>>> sinc(x).diff()
Piecewise(((x*cos(x) - sin(x))/x**2, Ne(x, 0)), (0, True))
• Series Expansion
>>> sinc(x).series()
1 - x**2/6 + x**4/120 + O(x**6)
• As zero’th order spherical Bessel Function
>>> sinc(x).rewrite(jn)
jn(0, x)
References
R232
https://en.wikipedia.org/wiki/Sinc_function
Members
# Trigonometric Inverses¶
## asin¶
class sympy.functions.elementary.trigonometric.asin(**kwargs)[source]
The inverse sine function.
Returns the arcsine of x in radians.
Notes
asin(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1 and for some instances when the result is a rational multiple of pi (see the eval class method).
A purely imaginary argument will lead to an asinh expression.
Examples
>>> from sympy import asin, oo, pi
>>> asin(1)
pi/2
>>> asin(-1)
-pi/2
References
R233
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R234
http://dlmf.nist.gov/4.23
R235
http://functions.wolfram.com/ElementaryFunctions/ArcSin
Members
## acos¶
class sympy.functions.elementary.trigonometric.acos(**kwargs)[source]
The inverse cosine function.
Returns the arc cosine of x (measured in radians).
Notes
acos(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1 and for some instances when the result is a rational multiple of pi (see the eval class method).
acos(zoo) evaluates to zoo (see note in sympy.functions.elementary.trigonometric.asec)
A purely imaginary argument will be rewritten to asinh.
Examples
>>> from sympy import acos, oo, pi
>>> acos(1)
0
>>> acos(0)
pi/2
>>> acos(oo)
oo*I
References
R236
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R237
http://dlmf.nist.gov/4.23
R238
http://functions.wolfram.com/ElementaryFunctions/ArcCos
Members
## atan¶
class sympy.functions.elementary.trigonometric.atan(**kwargs)[source]
The inverse tangent function.
Returns the arc tangent of x (measured in radians).
Notes
atan(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1 and for some instances when the result is a rational multiple of pi (see the eval class method).
Examples
>>> from sympy import atan, oo, pi
>>> atan(0)
0
>>> atan(1)
pi/4
>>> atan(oo)
pi/2
References
R239
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R240
http://dlmf.nist.gov/4.23
R241
http://functions.wolfram.com/ElementaryFunctions/ArcTan
Members
## acot¶
class sympy.functions.elementary.trigonometric.acot(**kwargs)[source]
The inverse cotangent function.
Returns the arc cotangent of x (measured in radians).
Notes
acot(x) will evaluate automatically in the cases oo, -oo, zoo, 0, 1, -1 and for some instances when the result is a rational multiple of pi (see the eval class method).
A purely imaginary argument will lead to an acoth expression.
acot(x) has a branch cut along $$(-i, i)$$, hence it is discontinuous at 0. Its range for real x is $$(-\frac{\pi}{2}, \frac{\pi}{2}]$$.
Examples
>>> from sympy import acot, sqrt
>>> acot(0)
pi/2
>>> acot(1)
pi/4
>>> acot(sqrt(3) - 2)
-5*pi/12
References
R242
http://dlmf.nist.gov/4.23
R243
http://functions.wolfram.com/ElementaryFunctions/ArcCot
Members
## asec¶
class sympy.functions.elementary.trigonometric.asec(**kwargs)[source]
The inverse secant function.
Returns the arc secant of x (measured in radians).
Notes
asec(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1 and for some instances when the result is a rational multiple of pi (see the eval class method).
asec(x) has branch cut in the interval [-1, 1]. For complex arguments, it can be defined [R247] as
$\operatorname{sec^{-1}}(z) = -i\frac{\log\left(\sqrt{1 - z^2} + 1\right)}{z}$
At x = 0, for positive branch cut, the limit evaluates to zoo. For negative branch cut, the limit
$\lim_{z \to 0}-i\frac{\log\left(-\sqrt{1 - z^2} + 1\right)}{z}$
simplifies to $$-i\log\left(z/2 + O\left(z^3\right)\right)$$ which ultimately evaluates to zoo.
As acos(x) = asec(1/x), a similar argument can be given for acos(x).
Examples
>>> from sympy import asec, oo, pi
>>> asec(1)
0
>>> asec(-1)
pi
References
R244
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R245
http://dlmf.nist.gov/4.23
R246
http://functions.wolfram.com/ElementaryFunctions/ArcSec
R247(1,2)
http://reference.wolfram.com/language/ref/ArcSec.html
Members
## acsc¶
class sympy.functions.elementary.trigonometric.acsc(**kwargs)[source]
The inverse cosecant function.
Returns the arc cosecant of x (measured in radians).
Notes
acsc(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1 and for some instances when the result is a rational multiple of pi (see the eval class method).
Examples
>>> from sympy import acsc, oo, pi
>>> acsc(1)
pi/2
>>> acsc(-1)
-pi/2
References
R248
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R249
http://dlmf.nist.gov/4.23
R250
http://functions.wolfram.com/ElementaryFunctions/ArcCsc
Members
## atan2¶
class sympy.functions.elementary.trigonometric.atan2(**kwargs)[source]
The function atan2(y, x) computes $$\operatorname{atan}(y/x)$$ taking two arguments $$y$$ and $$x$$. Signs of both $$y$$ and $$x$$ are considered to determine the appropriate quadrant of $$\operatorname{atan}(y/x)$$. The range is $$(-\pi, \pi]$$. The complete definition reads as follows:
$\begin{split}\operatorname{atan2}(y, x) = \begin{cases} \arctan\left(\frac y x\right) & \qquad x > 0 \\ \arctan\left(\frac y x\right) + \pi& \qquad y \ge 0 , x < 0 \\ \arctan\left(\frac y x\right) - \pi& \qquad y < 0 , x < 0 \\ +\frac{\pi}{2} & \qquad y > 0 , x = 0 \\ -\frac{\pi}{2} & \qquad y < 0 , x = 0 \\ \text{undefined} & \qquad y = 0, x = 0 \end{cases}\end{split}$
Attention: Note the role reversal of both arguments. The $$y$$-coordinate is the first argument and the $$x$$-coordinate the second.
If either $$x$$ or $$y$$ is complex:
$\operatorname{atan2}(y, x) = -i\log\left(\frac{x + iy}{\sqrt{x**2 + y**2}}\right)$
Examples
Going counter-clock wise around the origin we find the following angles:
>>> from sympy import atan2
>>> atan2(0, 1)
0
>>> atan2(1, 1)
pi/4
>>> atan2(1, 0)
pi/2
>>> atan2(1, -1)
3*pi/4
>>> atan2(0, -1)
pi
>>> atan2(-1, -1)
-3*pi/4
>>> atan2(-1, 0)
-pi/2
>>> atan2(-1, 1)
-pi/4
which are all correct. Compare this to the results of the ordinary $$\operatorname{atan}$$ function for the point $$(x, y) = (-1, 1)$$
>>> from sympy import atan, S
>>> atan(S(1)/-1)
-pi/4
>>> atan2(1, -1)
3*pi/4
where only the $$\operatorname{atan2}$$ function reurns what we expect. We can differentiate the function with respect to both arguments:
>>> from sympy import diff
>>> from sympy.abc import x, y
>>> diff(atan2(y, x), x)
-y/(x**2 + y**2)
>>> diff(atan2(y, x), y)
x/(x**2 + y**2)
We can express the $$\operatorname{atan2}$$ function in terms of complex logarithms:
>>> from sympy import log
>>> atan2(y, x).rewrite(log)
-I*log((x + I*y)/sqrt(x**2 + y**2))
and in terms of $$\operatorname(atan)$$:
>>> from sympy import atan
>>> atan2(y, x).rewrite(atan)
Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) < 0), (0, Ne(x, 0)), (nan, True))
but note that this form is undefined on the negative real axis.
References
R251
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
R252
https://en.wikipedia.org/wiki/Atan2
R253
http://functions.wolfram.com/ElementaryFunctions/ArcTan2
Members
# Hyperbolic Functions¶
## HyperbolicFunction¶
class sympy.functions.elementary.hyperbolic.HyperbolicFunction(**kwargs)[source]
Base class for hyperbolic functions.
Members
## sinh¶
class sympy.functions.elementary.hyperbolic.sinh(**kwargs)[source]
The hyperbolic sine function, $$\frac{e^x - e^{-x}}{2}$$.
• sinh(x) -> Returns the hyperbolic sine of x
Members
## cosh¶
class sympy.functions.elementary.hyperbolic.cosh(**kwargs)[source]
The hyperbolic cosine function, $$\frac{e^x + e^{-x}}{2}$$.
• cosh(x) -> Returns the hyperbolic cosine of x
Members
## tanh¶
class sympy.functions.elementary.hyperbolic.tanh(**kwargs)[source]
The hyperbolic tangent function, $$\frac{\sinh(x)}{\cosh(x)}$$.
• tanh(x) -> Returns the hyperbolic tangent of x
Members
## coth¶
class sympy.functions.elementary.hyperbolic.coth(**kwargs)[source]
The hyperbolic cotangent function, $$\frac{\cosh(x)}{\sinh(x)}$$.
• coth(x) -> Returns the hyperbolic cotangent of x
Members
## sech¶
class sympy.functions.elementary.hyperbolic.sech(**kwargs)[source]
The hyperbolic secant function, $$\frac{2}{e^x + e^{-x}}$$
• sech(x) -> Returns the hyperbolic secant of x
Members
## csch¶
class sympy.functions.elementary.hyperbolic.csch(**kwargs)[source]
The hyperbolic cosecant function, $$\frac{2}{e^x - e^{-x}}$$
• csch(x) -> Returns the hyperbolic cosecant of x
Members
# Hyperbolic Inverses¶
## asinh¶
class sympy.functions.elementary.hyperbolic.asinh(**kwargs)[source]
The inverse hyperbolic sine function.
• asinh(x) -> Returns the inverse hyperbolic sine of x
Members
## acosh¶
class sympy.functions.elementary.hyperbolic.acosh(**kwargs)[source]
The inverse hyperbolic cosine function.
• acosh(x) -> Returns the inverse hyperbolic cosine of x
Members
## atanh¶
class sympy.functions.elementary.hyperbolic.atanh(**kwargs)[source]
The inverse hyperbolic tangent function.
• atanh(x) -> Returns the inverse hyperbolic tangent of x
Members
## acoth¶
class sympy.functions.elementary.hyperbolic.acoth(**kwargs)[source]
The inverse hyperbolic cotangent function.
• acoth(x) -> Returns the inverse hyperbolic cotangent of x
Members
## asech¶
class sympy.functions.elementary.hyperbolic.asech(**kwargs)[source]
The inverse hyperbolic secant function.
• asech(x) -> Returns the inverse hyperbolic secant of x
Examples
>>> from sympy import asech, sqrt, S
>>> from sympy.abc import x
>>> asech(x).diff(x)
-1/(x*sqrt(1 - x**2))
>>> asech(1).diff(x)
0
>>> asech(1)
0
>>> asech(S(2))
I*pi/3
>>> asech(-sqrt(2))
3*I*pi/4
>>> asech((sqrt(6) - sqrt(2)))
I*pi/12
References
R254
https://en.wikipedia.org/wiki/Hyperbolic_function
R255
http://dlmf.nist.gov/4.37
R256
http://functions.wolfram.com/ElementaryFunctions/ArcSech/
Members
## acsch¶
class sympy.functions.elementary.hyperbolic.acsch(**kwargs)[source]
The inverse hyperbolic cosecant function.
• acsch(x) -> Returns the inverse hyperbolic cosecant of x
Examples
>>> from sympy import acsch, sqrt, S
>>> from sympy.abc import x
>>> acsch(x).diff(x)
-1/(x**2*sqrt(1 + x**(-2)))
>>> acsch(1).diff(x)
0
>>> acsch(1)
log(1 + sqrt(2))
>>> acsch(S.ImaginaryUnit)
-I*pi/2
>>> acsch(-2*S.ImaginaryUnit)
I*pi/6
>>> acsch(S.ImaginaryUnit*(sqrt(6) - sqrt(2)))
-5*I*pi/12
References
R257
https://en.wikipedia.org/wiki/Hyperbolic_function
R258
http://dlmf.nist.gov/4.37
R259
http://functions.wolfram.com/ElementaryFunctions/ArcCsch/
Members
# sympy.functions.elementary.integers¶
## ceiling¶
class sympy.functions.elementary.integers.ceiling(**kwargs)[source]
Ceiling is a univariate function which returns the smallest integer value not less than its argument. This implementation generalizes ceiling to complex numbers by taking the ceiling of the real and imaginary parts separately.
Examples
>>> from sympy import ceiling, E, I, S, Float, Rational
>>> ceiling(17)
17
>>> ceiling(Rational(23, 10))
3
>>> ceiling(2*E)
6
>>> ceiling(-Float(0.567))
0
>>> ceiling(I/2)
I
>>> ceiling(S(5)/2 + 5*I/2)
3 + 3*I
References
R260
“Concrete mathematics” by Graham, pp. 87
R261
http://mathworld.wolfram.com/CeilingFunction.html
Members
## floor¶
class sympy.functions.elementary.integers.floor(**kwargs)[source]
Floor is a univariate function which returns the largest integer value not greater than its argument. This implementation generalizes floor to complex numbers by taking the floor of the real and imaginary parts separately.
Examples
>>> from sympy import floor, E, I, S, Float, Rational
>>> floor(17)
17
>>> floor(Rational(23, 10))
2
>>> floor(2*E)
5
>>> floor(-Float(0.567))
-1
>>> floor(-I/2)
-I
>>> floor(S(5)/2 + 5*I/2)
2 + 2*I
References
R262
“Concrete mathematics” by Graham, pp. 87
R263
http://mathworld.wolfram.com/FloorFunction.html
Members
## RoundFunction¶
class sympy.functions.elementary.integers.RoundFunction(**kwargs)[source]
The base class for rounding functions.
## frac¶
class sympy.functions.elementary.integers.frac(**kwargs)[source]
Represents the fractional part of x
For real numbers it is defined [R264] as
$x - \left\lfloor{x}\right\rfloor$
Examples
>>> from sympy import Symbol, frac, Rational, floor, ceiling, I
>>> frac(Rational(4, 3))
1/3
>>> frac(-Rational(4, 3))
2/3
returns zero for integer arguments
>>> n = Symbol('n', integer=True)
>>> frac(n)
0
rewrite as floor
>>> x = Symbol('x')
>>> frac(x).rewrite(floor)
x - floor(x)
for complex arguments
>>> r = Symbol('r', real=True)
>>> t = Symbol('t', real=True)
>>> frac(t + I*r)
I*frac(r) + frac(t)
References
R264(1,2)
https://en.wikipedia.org/wiki/Fractional_part
R265
http://mathworld.wolfram.com/FractionalPart.html
# sympy.functions.elementary.exponential¶
## exp¶
class sympy.functions.elementary.exponential.exp(**kwargs)[source]
The exponential function, $$e^x$$.
Members
## LambertW¶
class sympy.functions.elementary.exponential.LambertW(**kwargs)[source]
The Lambert W function $$W(z)$$ is defined as the inverse function of $$w \exp(w)$$ [R266].
In other words, the value of $$W(z)$$ is such that $$z = W(z) \exp(W(z))$$ for any complex number $$z$$. The Lambert W function is a multivalued function with infinitely many branches $$W_k(z)$$, indexed by $$k \in \mathbb{Z}$$. Each branch gives a different solution $$w$$ of the equation $$z = w \exp(w)$$.
The Lambert W function has two partially real branches: the principal branch ($$k = 0$$) is real for real $$z > -1/e$$, and the $$k = -1$$ branch is real for $$-1/e < z < 0$$. All branches except $$k = 0$$ have a logarithmic singularity at $$z = 0$$.
Examples
>>> from sympy import LambertW
>>> LambertW(1.2)
0.635564016364870
>>> LambertW(1.2, -1).n()
-1.34747534407696 - 4.41624341514535*I
>>> LambertW(-1).is_real
False
References
R266(1,2)
https://en.wikipedia.org/wiki/Lambert_W_function
Members
## log¶
class sympy.functions.elementary.exponential.log(**kwargs)[source]
The natural logarithm function $$\ln(x)$$ or $$\log(x)$$.
Logarithms are taken with the natural base, $$e$$. To get a logarithm of a different base b, use log(x, b), which is essentially short-hand for log(x)/log(b).
log represents the principal branch of the natural logarithm. As such it has a branch cut along the negative real axis and returns values having a complex argument in $$(-\pi, \pi]$$.
Examples
>>> from sympy import log, sqrt, S, I
>>> log(8, 2)
3
>>> log(S(8)/3, 2)
-log(3)/log(2) + 3
>>> log(-1 + I*sqrt(3))
log(2) + 2*I*pi/3
Members
## exp_polar¶
class sympy.functions.elementary.exponential.exp_polar(**kwargs)[source]
Represent a ‘polar number’ (see g-function Sphinx documentation).
exp_polar represents the function $$Exp: \mathbb{C} \rightarrow \mathcal{S}$$, sending the complex number $$z = a + bi$$ to the polar number $$r = exp(a), \theta = b$$. It is one of the main functions to construct polar numbers.
>>> from sympy import exp_polar, pi, I, exp
The main difference is that polar numbers don’t “wrap around” at $$2 \pi$$:
>>> exp(2*pi*I)
1
>>> exp_polar(2*pi*I)
exp_polar(2*I*pi)
apart from that they behave mostly like classical complex numbers:
>>> exp_polar(2)*exp_polar(3)
exp_polar(5)
Members
# sympy.functions.elementary.piecewise¶
## ExprCondPair¶
class sympy.functions.elementary.piecewise.ExprCondPair(expr, cond)[source]
Represents an expression, condition pair.
Members
## Piecewise¶
class sympy.functions.elementary.piecewise.Piecewise(*args, **options)[source]
Represents a piecewise function.
Usage:
Piecewise( (expr,cond), (expr,cond), … )
• Each argument is a 2-tuple defining an expression and condition
• The conds are evaluated in turn returning the first that is True. If any of the evaluated conds are not determined explicitly False, e.g. x < 1, the function is returned in symbolic form.
• If the function is evaluated at a place where all conditions are False, nan will be returned.
• Pairs where the cond is explicitly False, will be removed.
Examples
>>> from sympy import Piecewise, log, ITE, piecewise_fold
>>> from sympy.abc import x, y
>>> f = x**2
>>> g = log(x)
>>> p = Piecewise((0, x < -1), (f, x <= 1), (g, True))
>>> p.subs(x,1)
1
>>> p.subs(x,5)
log(5)
Booleans can contain Piecewise elements:
>>> cond = (x < y).subs(x, Piecewise((2, x < 0), (3, True))); cond
Piecewise((2, x < 0), (3, True)) < y
The folded version of this results in a Piecewise whose expressions are Booleans:
>>> folded_cond = piecewise_fold(cond); folded_cond
Piecewise((2 < y, x < 0), (3 < y, True))
When a Boolean containing Piecewise (like cond) or a Piecewise with Boolean expressions (like folded_cond) is used as a condition, it is converted to an equivalent ITE object:
>>> Piecewise((1, folded_cond))
Piecewise((1, ITE(x < 0, y > 2, y > 3)))
When a condition is an ITE, it will be converted to a simplified Boolean expression:
>>> piecewise_fold(_)
Piecewise((1, ((x >= 0) | (y > 2)) & ((y > 3) | (x < 0))))
Members
sympy.functions.elementary.piecewise.piecewise_fold(expr)[source]
Takes an expression containing a piecewise function and returns the expression in piecewise form. In addition, any ITE conditions are rewritten in negation normal form and simplified.
Examples
>>> from sympy import Piecewise, piecewise_fold, sympify as S
>>> from sympy.abc import x
>>> p = Piecewise((x, x < 1), (1, S(1) <= x))
>>> piecewise_fold(x*p)
Piecewise((x**2, x < 1), (x, True))
# sympy.functions.elementary.miscellaneous¶
## IdentityFunction¶
class sympy.functions.elementary.miscellaneous.IdentityFunction(*args, **kwargs)[source]
The identity function
Examples
>>> from sympy import Id, Symbol
>>> x = Symbol('x')
>>> Id(x)
x
Members
## Min¶
class sympy.functions.elementary.miscellaneous.Min(*args, **assumptions)[source]
Return, if possible, the minimum value of the list. It is named Min and not min to avoid conflicts with the built-in function min.
Examples
>>> from sympy import Min, Symbol, oo
>>> from sympy.abc import x, y
>>> p = Symbol('p', positive=True)
>>> n = Symbol('n', negative=True)
>>> Min(x, -2)
Min(-2, x)
>>> Min(x, -2).subs(x, 3)
-2
>>> Min(p, -3)
-3
>>> Min(x, y)
Min(x, y)
>>> Min(n, 8, p, -7, p, oo)
Min(-7, n)
Max
find maximum values
Members
## Max¶
class sympy.functions.elementary.miscellaneous.Max(*args, **assumptions)[source]
Return, if possible, the maximum value of the list.
When number of arguments is equal one, then return this argument.
When number of arguments is equal two, then return, if possible, the value from (a, b) that is >= the other.
In common case, when the length of list greater than 2, the task is more complicated. Return only the arguments, which are greater than others, if it is possible to determine directional relation.
If is not possible to determine such a relation, return a partially evaluated result.
Assumptions are used to make the decision too.
Also, only comparable arguments are permitted.
It is named Max and not max to avoid conflicts with the built-in function max.
Examples
>>> from sympy import Max, Symbol, oo
>>> from sympy.abc import x, y, z
>>> p = Symbol('p', positive=True)
>>> n = Symbol('n', negative=True)
>>> Max(x, -2)
Max(-2, x)
>>> Max(x, -2).subs(x, 3)
3
>>> Max(p, -2)
p
>>> Max(x, y)
Max(x, y)
>>> Max(x, y) == Max(y, x)
True
>>> Max(x, Max(y, z))
Max(x, y, z)
>>> Max(n, 8, p, 7, -oo)
Max(8, p)
>>> Max (1, x, oo)
oo
• Algorithm
The task can be considered as searching of supremums in the directed complete partial orders [R267].
The source values are sequentially allocated by the isolated subsets in which supremums are searched and result as Max arguments.
If the resulted supremum is single, then it is returned.
The isolated subsets are the sets of values which are only the comparable with each other in the current set. E.g. natural numbers are comparable with each other, but not comparable with the $$x$$ symbol. Another example: the symbol $$x$$ with negative assumption is comparable with a natural number.
Also there are “least” elements, which are comparable with all others, and have a zero property (maximum or minimum for all elements). E.g. $$oo$$. In case of it the allocation operation is terminated and only this value is returned.
Assumption:
• if A > B > C then A > C
• if A == B then B can be removed
Min
find minimum values
References
R267(1,2)
https://en.wikipedia.org/wiki/Directed_complete_partial_order
R268
https://en.wikipedia.org/wiki/Lattice_%28order%29
Members
## root¶
sympy.functions.elementary.miscellaneous.root(arg, n, k=0, evaluate=None)[source]
Returns the k-th n-th root of arg.
Parameters
k : int, optional
Should be an integer in $$\{0, 1, ..., n-1\}$$. Defaults to the principal root if $$0$$.
evaluate : bool, optional
The parameter determines if the expression should be evaluated. If None, its value is taken from global_parameters.evaluate.
Examples
>>> from sympy import root, Rational
>>> from sympy.abc import x, n
>>> root(x, 2)
sqrt(x)
>>> root(x, 3)
x**(1/3)
>>> root(x, n)
x**(1/n)
>>> root(x, -Rational(2, 3))
x**(-3/2)
To get the k-th n-th root, specify k:
>>> root(-2, 3, 2)
-(-1)**(2/3)*2**(1/3)
To get all n n-th roots you can use the rootof function. The following examples show the roots of unity for n equal 2, 3 and 4:
>>> from sympy import rootof, I
>>> [rootof(x**2 - 1, i) for i in range(2)]
[-1, 1]
>>> [rootof(x**3 - 1,i) for i in range(3)]
[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]
>>> [rootof(x**4 - 1,i) for i in range(4)]
[-1, 1, -I, I]
SymPy, like other symbolic algebra systems, returns the complex root of negative numbers. This is the principal root and differs from the text-book result that one might be expecting. For example, the cube root of -8 does not come back as -2:
>>> root(-8, 3)
2*(-1)**(1/3)
The real_root function can be used to either make the principal result real (or simply to return the real root directly):
>>> from sympy import real_root
>>> real_root(_)
-2
>>> real_root(-32, 5)
-2
Alternatively, the n//2-th n-th root of a negative number can be computed with root:
>>> root(-32, 5, 5//2)
-2
References
## sqrt¶
sympy.functions.elementary.miscellaneous.sqrt(arg, evaluate=None)[source]
Returns the principal square root.
Parameters
evaluate : bool, optional
The parameter determines if the expression should be evaluated. If None, its value is taken from global_parameters.evaluate.
Examples
>>> from sympy import sqrt, Symbol, S
>>> x = Symbol('x')
>>> sqrt(x)
sqrt(x)
>>> sqrt(x)**2
x
Note that sqrt(x**2) does not simplify to x.
>>> sqrt(x**2)
sqrt(x**2)
This is because the two are not equal to each other in general. For example, consider x == -1:
>>> from sympy import Eq
>>> Eq(sqrt(x**2), x).subs(x, -1)
False
This is because sqrt computes the principal square root, so the square may put the argument in a different branch. This identity does hold if x is positive:
>>> y = Symbol('y', positive=True)
>>> sqrt(y**2)
y
You can force this simplification by using the powdenest() function with the force option set to True:
>>> from sympy import powdenest
>>> sqrt(x**2)
sqrt(x**2)
>>> powdenest(sqrt(x**2), force=True)
x
To get both branches of the square root you can use the rootof function:
>>> from sympy import rootof
>>> [rootof(x**2-3,i) for i in (0,1)]
[-sqrt(3), sqrt(3)]
Although sqrt is printed, there is no sqrt function so looking for sqrt in an expression will fail:
>>> from sympy.utilities.misc import func_name
>>> func_name(sqrt(x))
'Pow'
>>> sqrt(x).has(sqrt)
Traceback (most recent call last):
...
sympy.core.sympify.SympifyError: SympifyError: <function sqrt at 0x10e8900d0>
To find sqrt look for Pow with an exponent of 1/2:
>>> (x + 1/sqrt(x)).find(lambda i: i.is_Pow and abs(i.exp) is S.Half)
{1/sqrt(x)}
References
R269
https://en.wikipedia.org/wiki/Square_root
R270
https://en.wikipedia.org/wiki/Principal_value
## cbrt¶
sympy.functions.elementary.miscellaneous.cbrt(arg, evaluate=None)[source]
Returns the principal cube root.
Parameters
evaluate : bool, optional
The parameter determines if the expression should be evaluated. If None, its value is taken from global_parameters.evaluate.
Examples
>>> from sympy import cbrt, Symbol
>>> x = Symbol('x')
>>> cbrt(x)
x**(1/3)
>>> cbrt(x)**3
x
Note that cbrt(x**3) does not simplify to x.
>>> cbrt(x**3)
(x**3)**(1/3)
This is because the two are not equal to each other in general. For example, consider $$x == -1$$:
>>> from sympy import Eq
>>> Eq(cbrt(x**3), x).subs(x, -1)
False
This is because cbrt computes the principal cube root, this identity does hold if $$x$$ is positive:
>>> y = Symbol('y', positive=True)
>>> cbrt(y**3)
y
References
## real_root¶
sympy.functions.elementary.miscellaneous.real_root(arg, n=None, evaluate=None)[source]
Return the real n’th-root of arg if possible.
Parameters
n : int or None, optional
If n is None, then all instances of (-n)**(1/odd) will be changed to -n**(1/odd). This will only create a real root of a principal root. The presence of other factors may cause the result to not be real.
evaluate : bool, optional
The parameter determines if the expression should be evaluated. If None, its value is taken from global_parameters.evaluate.
Examples
>>> from sympy import root, real_root, Rational
>>> from sympy.abc import x, n
>>> real_root(-8, 3)
-2
>>> root(-8, 3)
2*(-1)**(1/3)
>>> real_root(_)
-2
If one creates a non-principal root and applies real_root, the result will not be real (so use with caution):
>>> root(-8, 3, 2)
-2*(-1)**(2/3)
>>> real_root(_)
-2*(-1)**(2/3)
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