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https://www.atmos-meas-tech.net/12/4479/2019/ | Journal cover Journal topic
Atmospheric Measurement Techniques An interactive open-access journal of the European Geosciences Union
Journal topic
Atmos. Meas. Tech., 12, 4479–4493, 2019
https://doi.org/10.5194/amt-12-4479-2019
Atmos. Meas. Tech., 12, 4479–4493, 2019
https://doi.org/10.5194/amt-12-4479-2019
Research article 22 Aug 2019
Research article | 22 Aug 2019
# Methods for identifying aged ship plumes and estimating contribution to aerosol exposure downwind of shipping lanes
Methods for identifying aged ship plumes and estimating contribution to aerosol exposure downwind of shipping lanes
Stina Ausmeel1, Axel Eriksson1,2, Erik Ahlberg1, and Adam Kristensson1 Stina Ausmeel et al.
• 1Division of Nuclear Physics, Lund University, Lund, Box 118, 221 00, Sweden
• 2Ergonomics and Aerosol Technology, Lund University, Lund, Box 118, 221 00, Sweden
Correspondence: Stina Ausmeel (stina.ausmeel@nuclear.lu.se)
Abstract
Ship traffic is a major source of aerosol particles, particularly near shipping lanes and harbours. In order to estimate the contribution to exposure downwind of a shipping lane, it is important to be able to measure the ship emission contribution at various distances from the source. We report on measurements of atmospheric particles 7–20 km downwind of a shipping lane in the Baltic Sea Sulfur Emission Control Area (SECA) at a coastal location in southern Sweden during a winter and a summer campaign. Each ship plume was linked to individual ship passages using a novel method based on wind field data and automatic ship identification system data (AIS), where varying wind speeds and directions were applied to calculate a plume trajectory. In a situation where AIS data are not matching measured plumes well or if AIS data are missing, we provide an alternative method with particle number concentration data. The shipping lane contribution to the particle number concentration in Falsterbo was estimated by subtracting background concentrations from the ship plume concentrations, and more than 150 plumes were analysed. We have also extrapolated the contribution to seasonal averages and provide recommendations for future similar measurements. Averaged over a season, the contribution to particle number concentration was about 18 % during the winter and 10 % during the summer, including those periods with wind directions when the shipping lane was not affecting the station. The corresponding contribution to equivalent black carbon was 1.4 %.
1 Introduction
Air pollution from anthropogenic activities, such as ship traffic, affects both human health and climate. Airborne particles cause negative health effects such as pulmonary and cardiovascular diseases, resulting in premature deaths and increased societal costs. Air pollution from combustion sources have an effect on climate due to emissions of greenhouse gases as well as particles with different optical properties and cloud-forming capacities.
In order to reduce air pollution there are regulations on sectors with high emissions, for example the transportation sector. However, despite these regulations air pollution continues to be a serious problem. One sector that has gained relatively little attention in terms of emission control in the past is international shipping. The relative contribution from shipping to the total air pollution from transport is an increasing problem due to expected growth in shipping activity (Brandt et al., 2013; Corbett et al., 2007). One regulatory measure that has been taken to specifically reduce sulfur emissions from ships is the introduction of so-called Sulfur Emission Control Areas (SECAs), where the Baltic Sea SECA was one of the first to become established (Corbett and Fischbeck, 1997). In the International Convention for the Prevention of Marine Pollution from Ships (MARPOL) Annex VI, the main exhaust gas emissions of sulfur oxides (SOx) and nitrous oxides (NOx) are limited. Hence, the International Maritime Organisation (IMO) has regulated the fuel sulfur content in several steps, with a total decrease from 1.5 % to 0.1 % mass fraction between the years 2010 and 2015 in Sulfur Emission Control Areas. In 2016 it was decided that further reduction of the fuel sulfur limit is going to be implemented, with a cap of 0.50 % sulfur in fuel oil on board all ships from January 1 2020. A recent report showed a compliance level to the sulfur regulations of 92 %–94 % during 2015 and 2016 in the region around Denmark (within the Baltic Sea SECA) (Mellqvist et al., 2017). Hence it is expected that most ships in the region are using fuels with a sulfur content of maximum 0.1 %. In addition to cleaner fuels, such as low-sulfur residual marine fuel oil, marine diesel oil (MDO), or liquefied natural gas (LNG), ships can comply by being equipped with scrubbers which remove the SO2 from the flue gas. The use of scrubbers was also observed in the region during our period of interest, by Mellqvist et al. (2017).
One way to characterize and quantify ship emissions is through measurements in coastal areas, downwind of a shipping lane. This makes it possible to register an increase in particle levels and the exposure to particles in this area when individual ship emission plumes pass the measurement station. With increasing distance from the emission source, the plume becomes more dilute and physically and chemically transformed due to atmospheric ageing. In order to assess physicochemical properties but still capture features of the aged particles, which differ from the freshly emitted, it is therefore desirable to measure at an intermediate distance to the ships. Measurements of ambient aerosol particles are also important for an accurate assessment of the health effects, which depend on the actual exposure. This motivates measurements of the atmospherically aged ship aerosol particles from all types of ships affecting the coastal population. However, there are challenges associated with measuring aerosols from individual plumes further away from a moving point source such as a ship. Dilution will eventually make it harder to distinguish from background levels – there can be an overlap of several plumes that intersect, and varying wind speed and wind direction makes it less obvious which ship is connected to which plume if the traffic is relatively intense.
All ships on international water with gross tonnage above 300 t, cargo ships with gross tonnage above 500 t, and all passenger ships are required to be equipped with a tracking system called Automatic Identification System (AIS). A ship sends out a position signal with individual International Maritime Organisation ID and information about its type, size, country of origin, speed, etc. These data are collected every 6 min. AIS data in the Øresund region were used in this study to tie individual ship plumes to specific ships. AIS can be used as a tool in ship emission studies, commonly as a source for emission inventory used in models. This bottom-up method has been used and developed by many (e.g. Jalkanen et al., 2009, 2012; Liu et al., 2016; Beecken et al., 2015; Chen et al., 2018; Johansson et al., 2017; Marelle et al., 2016; Goldsworthy and Goldsworthy, 2015). AIS has also been used in connection to ambient plume measurements, to identify individual ship emission plumes. Alföldy et al. (2013) performed visual observations of ships at short distances in a port area and could connect these to live updates of ship positions. Ault et al. (2010) measured plumes and connected these to individual ships by using AIS ship positions and assuming transport with constant wind speed and wind direction. Balzani Lööv et al. (2014) used a similar method to locate plumes after emission, e.g. when doing airborne measurements within plumes further downwind of the ships. Diesch et al. (2013) also measured individual plumes and connected plume properties to ship properties, such as weight, using AIS, also at short distances (1–5 min downwind). Hence, AIS information has successfully been used in several applications, but for doing individual ship plume identification at longer distances where the plume might not travel along a straight path between emission and detection, other approaches might be needed that take into account the non-linear wind speed and direction. One example is the method of following ships either by aircraft or with a ship vessel up to a few kilometres behind the ship (Berg et al., 2012; Petzold et al., 2008; Chen et al., 2005; Williams et al., 2009; Lack et al., 2009). An advantage of this method is that the ships can be followed at different downwind distances and can measure plume dilution and aerosol dynamics. However, it is an expensive method, and only a few ships can be followed due to budget and practical restrictions. Hence, this calls for a more feasible and cost-effective solution.
Particle number size distributions have been studied in atmospheric conditions previously, showing some variations in sizes and number of modes. This can be expected since many factors affect the emissions, such as engine operations, and the atmospheric transformation processes. For example, Jonsson et al. (2011) showed that size-resolved particle number emission factors were largest around particle diameters of 35 nm, with smaller sizes observed for ships running on gas turbines than on diesel engines. Out of these particles, 36 %–46 % were non-volatile and could contain some black carbon (BC). These measurements are from 2010, i.e. during the 1 % sulfur limit within SECAs. Pirjola et al. (2014) showed that the number size distribution had two modes for fresh ship plumes: a dominating mode peaked at 20–30 nm and an accumulation mode at 80–100 nm. About 30 % of these were non-volatile, and it was also shown that the after-treatment system affected the total particle number emission. These measurements are from 2010 to 2011. Diesch et al. (2013) observed a nucleation mode in the 10–20 nm diameter range and a combustion aerosol mode centred at about 35 nm. No particles with sizes above 1 µm were observed. Six percent of the particle mass was due to BC. In the study by Diesch et al. (2013), AIS was used to link emission properties to ship properties, and they showed a decrease in most particle properties (including particle number concentration and black carbon) with increasing ship gross tonnage. Measurements on board a ship showed a particle size distribution major peak at around 10 nm and a smaller peak at around 30–40 nm. Approximately 40 % of the mass was non-volatile material, but particles below 10 nm consisted of only volatile material (Hallquist et al., 2013). Westerlund et al. (2015) measured ship plumes from a stationary site and used AIS to characterize ships. Westerlund et al. (2015) found unimodal particle number size distributions for cargo and passenger ships, with the peak around 40 nm, while tug-boats emitted smaller particles. Since the measurements were carried out in a harbour area, as most of the other studies above, they could capture changes in emissions during for example acceleration of ships. These harbour measurements were carried out in 2010, i.e. also before the 2015 SECA implementation. In another harbour area, Donateo et al. (2014) quantified the contribution of ship emissions to local total aerosol concentrations. The ship contribution to particle number was found to be 26 %. They could also see plume peaks in PM2.5, since measurements were done in a harbour area and plume peak concentrations were relatively high. A study performed in an Arctic region showed a size distribution mode with peak around 27 nm during the first 6 h of plume transport, and later (> 6 h) modes above 100 nm become more prominent (Aliabadi et al., 2015). Here, the ship contribution to BC was estimated to be 4.3 %–9.8 %. Due to the clean Arctic environment and low background concentrations, the evolution of a ship plume contribution could be studied over time (0–72 h). In our measurements, we only observe ship plumes within the first hour of atmospheric transport. Dispersion modelling of ship plumes has shown that dilution and coagulation are important processes within the first hour after emission, reducing the number concentration by 4 orders of magnitude and 1 order of magnitude, respectively (Tian et al., 2014). The decrease in particle number concentration is most rapid during the first minutes after emission. Our measurements pick up ship emissions 15–70 min after emission (10–90th percentile). In general, for understanding the fundamental processes of ship-emitted particles, detailed studies of individual engines or plumes from operating ships can be performed. However, for health effects the total contribution matters, so for investigating local particle contributions, a large set of ships must be studied.
We present a new revised method to identify individual aerosol ship plumes based on AIS data and non-linear wind transport of the ship plume to a stationary coastal field site, which is several kilometres downwind. The method has been tested on particle number concentration, particle number size distribution, and black carbon mass. In addition, CO2, NOx, and aerosol mass spectrometry data are presented in the companion paper by Ausmeel et al. (2019). The measurements were performed in Falsterbo, in southern Sweden, located downwind of a heavily trafficked shipping lane in the Øresund strait with a daily average of 73 and 63 AIS-transmitting ships passing in winter and summer respectively, and which connects the Atlantic and the Baltic Sea. The distance from the shipping lane to the site corresponds to an average transport time of between 15 and 70 min (10–90th percentile) for the ship plumes. The measurements took place during the winter (January–February) and the summer (May–July) of 2016. With the new revised plume identification method, we can detect several tens of plumes in a day with favourable wind conditions. We also show how particle number concentration data can be used when AIS data are failing or missing, to identify individual ship plumes, however without information about which ship it is.
We identified and calculated the contribution as well as the particle size distribution of individual ships by subtraction of background concentrations from the identified plume particle number concentrations. In addition, we have developed and described a new method to calculate the contribution of aerosol properties when the plume cannot be visually distinguished from background concentrations due to noisy data and relatively weak contribution at this fairly long distance from the shipping lane. This method has been tested on equivalent black carbon (eBC) concentrations. eBC is black carbon mass concentration derived from optical absorption measurements and a mass absorption cross section (MAC) value (Petzold et al., 2013). In our measurements, the MAC value for the 880 nm wavelength was 7.77 m2 g−1 (Drinovec et al., 2015). The duration of a eBC plume is based either on the available ship plume identification from the AIS and wind data, and plume evolution of particle number concentration data, or only on particle number concentration data when AIS data are not available. For the aerosol properties for which ship plume concentrations could be calculated, a daily and seasonal average contribution for the entire fleet could be estimated.
2 Instrumentation setup and experimental site
The location of the sampling site was on the Falsterbo peninsula in south-western Sweden (55.3843 N, 12.8164 E) (Fig. 1). The measurement location is within a SECA covering the Baltic Sea. The main shipping lanes, which pass to the west and the south of Falsterbo, are about 7–20 km away from the measurement site. The surrounding area is mainly made up of open coastal landscape, with roughly 250 m of reed and sand dunes separating the measurement site from the open water of Øresund. There are few buildings and activities nearby. To the north, east, and south of the site there is a golf course, and to the east of the site, i.e. not between the shipping lane and the site, there is a workshop connected to the golf course. South of the site there is a lighthouse, housing a weather station run by the Swedish Meteorological and Hydrological Institute (SMHI). Vehicles and machinery passing the measurement site were considered when analysing the data.
Figure 1Location of the measurement station (circle with cross) at the Falsterbo peninsula together with ship traffic density; the colour bar indicates an approximate number of distinct vessels passing per day per squared kilometre (https://www.marinetraffic.com/, last access: 29 March 2016). The dashed square shows the area in which AIS positions are considered for ship identification. Inserted to the right is the wind direction pattern during the winter (black) and summer (grey) campaign respectively.
A PM10 aerosol sampling inlet was mounted at a height of about 4 m above ground, on top of a mobile trailer housing the instruments. The trailer was air-conditioned and kept at an indoor temperature of about 20 C. Figure 2 shows a sketch of the complete measurement setup and the flow configuration used in the Falsterbo measurement campaigns. The total particle number concentration was measured with a condensation particle counter (CPC, TSI 3775 or TSI 3025) with a sample time of 30 s. In addition, a custom-built scanning mobility particle sizer (SMPS) (Wiedensohler et al., 2012) was used to measure the particle number size distribution in the electrical mobility diameter range 10.5–532 nm (differential mobility analyser, DMA, Hauke type medium, custom-built; CPC 3010, TSI Inc., USA) (Svenningsson et al., 2008). The time resolution was 2 min per scan. Particle size distribution in the micrometre range (0.54–19.8 µm) was measured with an aerodynamic particle sizer (APS 3321, TSI Inc. USA). Equivalent black carbon (eBC) content was measured with optical absorption methods, using a seven-wavelength Aethalometer (model AE33, Magee Scientific) (Drinovec et al., 2015) with a sample time of 1 min. Data from several of the instruments in Fig. 2 will be presented in a companion article, Ausmeel et al. (2019b). The chemical composition of sampled particles was evaluated with a soot particle aerosol mass spectrometer (SP-AMS, Aerodyne Research Inc.) (Onasch et al., 2012). In addition to the AMS measurements, black carbon (BC) content was measured with optical absorption methods, using a seven-wavelength Aethalometer (model AE33, Magee Scientific) (Drinovec et al., 2015) and a 637 nm multi-angle absorption photometer (MAAP, Thermo Fisher Scientific) (Müller et al., 2011), both with a sample time of 1 min. A potential aerosol mass oxidation flow reactor (PAM-OFR) (Kang et al., 2007; Lambe et al., 2011) was alternately connected before the AMS, SMPS, and Aethalometer to simulate atmospheric ageing. For the gaseous aerosol compounds, CO2 concentration was measured with a non-dispersive infrared gas analyser (LI-COR LI840), and SO2 was measured using a UV fluorescent monitor (Environnement S.A AF22M). CO2 concentration enhancements due to ship plumes were below the detection limit of the monitor used, which means that emission factors likely cannot be calculated for ship plumes 7–20 km downwind of the shipping lane. In summary, for the MAAP (detection limit, DL, of < 50 ng m−3), APS (DL 0.001 cm−3), CO2 (DL < 1 ppm), and SO2 (DL < 1 ppb) monitors, the concentrations from ship emissions were at all times undistinguishable from the background levels. These data sets were not analysed further.
Figure 2Measurement setup. The symbol (s) indicates configuration used only during the summer and (w) only during the winter. The dashed line shows the bypass flow excluding the PAM oxidation flow reactor from the sampling line. For the membrane (Nafion) dryers, the letters correspond to the size-dependent particle penetration for each drier shown in Fig. 3.
During the summer campaign, the aerosol flow for certain instruments (Fig. 2) was dried using either diffusion or membrane (Nafion) driers. The particle losses in the membrane dryers due to diffusion were determined by laboratory measurements. For 100 nm particles the losses were in the range 0 %–10 %, and for 10 nm particles the losses were about 5 %–20 %. Specifications about the dryers and losses can be found in Table 1 and Fig. 3. These losses are used to correct the size-resolved scanning mobility particle sizer (SMPS) data. Table 1 presents the specifications for each dryer used in the summer campaign to dry the aerosol particles before sampling with some of the particle instruments. Letters A–C correspond to the dryers shown in the illustration of the Falsterbo measurement setup in Fig. 2. The flows for which the losses are characterized were the same flows as used in the field measurements. The aerosol used for the characterization was polydisperse ammonium sulfate in lab room air. The resulting losses, as a fraction of the total particle concentration, are shown as function of particle size in Fig. 3. In addition, corrections for particle losses in the sampling line were calculated using the Particle Loss Calculator tool (Von der Weiden et al., 2009) and were applied to the SMPS size distributions but not for the other instruments.
Table 1Specifications of dryers used in Falsterbo; letters A–C correspond to the driers in Fig. 2.
Figure 3Fraction of total particle concentration lost due to diffusion in three dryers, as a function of particle diameter, Dp. Error bars indicate 1 standard deviation from two to three measurements.
3 Methods for identifying ship plumes and estimating ship contribution
## 3.1 Ship plume identification and analysis
To confirm the contribution of ship plumes to particle and gas concentrations in Falsterbo, the time when each ship plume should influence the Falsterbo site was estimated with the revised method based on automatic ship identification system position data as well as wind direction and wind speed data from the Falsterbo lighthouse Swedish Meteorological and Hydrological Institute weather station (SMHI, 2017).
Only ships passing by in the area limited by a rectangle with geographical coordinates (55.16 N, 12.45 E), (55.56 N, 12.45 E), (55.56 N, 13.00 E), and (55.16 N, 13.00 E) were included in the analysis (Fig. 1). The data for the ship positions were available with a time resolution of 6 min, and the wind data were available with a 1 h time resolution. Since a higher time resolution was needed to identify ship influence at the measurement station, the ship positions, wind directions, and wind speeds were linearly interpolated to a 1 min time resolution.
For each interpolated 1 min ship position, wind trajectories were calculated describing how the wind travelled from the ship at time 0 (temission) towards the Falsterbo station, until the wind approached the Falsterbo station at time instance x minutes (tarrival). The minimum distance between the Falsterbo station and the wind trajectory defined tarrival. Each ship passage in the rectangle contained several of these minimum distances since we used all 1 min ship positions when calculating tarrival. The shortest distance among this subset of each ship passage was chosen as the final minimum distance. This method is similar to the method by Balzani Lööv et al. (2014) and Ault et al. (2010). However, in those studies, the distances between the ships and the station were much shorter. Hence the authors could use a wind direction and wind speed that did not change with time along the trajectory between the ship and the station, while, in this study, the wind direction and wind speed is varying between temission and tarrival, which is a novel method of estimating ship plume positions over greater distances.
When the wind was not arriving from the sea, the ships did not influence the measurements. Ship passages were defined to influence the Falsterbo station only if the minimum distance between the wind path at tarrival and the Falsterbo station was smaller than 500 m. The effect of ship emissions on the particle concentrations at Falsterbo were strongest and clearest for the number concentration and particle number size distribution data. Hence, each tarrival when a ship should influence Falsterbo measurements was compared to the actual measured data. In theory, it is possible that the wind direction is changing as the ships sail past the measurement station, meaning that we can potentially miss the maximum concentration in ship plumes and only record the lower concentrations at the tails of the ship plumes. However, in almost all cases in our data set, the wind is stable enough during each ship plume passage at the station. This means that we fetch entire ship plumes from the lowest concentrations in the plumes to the maximum concentrations in the plume.
There is a significant uncertainty in finding the temission and tarrival, since the wind data were interpolated to 1 min values from a 1 h resolution, and due to the fact that the wind trajectory path was calculated based on the wind data from Falsterbo. In reality, the wind speed and wind direction along the ship plume travelling from the ship towards Falsterbo could occasionally be significantly different, especially for ships which are sailing far away from the Falsterbo station. Despite this uncertainty, each tarrival matched very well with increases in particle number concentrations during winter. A majority of tarrival are within 5 min of the actual concentration peaks, as illustrated in Fig. 5. However, when two or more ships influence the Falsterbo station almost at the same time, it is hard to distinguish which individual ship is contributing most to the increase in particles. During summer, the method to match AIS data with ship plume peaks yielded a lower agreement presumably due to less stable meteorological conditions during summer, e.g. more turbulence, and sea breeze. Nevertheless, the method worked surprisingly well even for this period for the few number of plumes identified. In the end, however, the AIS method was not used during summer, since AIS data were not available for more than a few days due to errors in the AIS database.
Even for periods when AIS data were not matching plume times well, or when AIS data were missing from the AIS database, particle number concentrations could be used to identify ship plumes instead. This required that there were no other interfering particle number concentration sources, or that these could be distinguished from the ship plumes. The number concentration data were then used to identify the plume time period, since particle number concentrations were always above a detection limit for all ship plumes, and the time resolution was large enough to clearly identify the shape of the plume peak. However, all increases in particle number concentration were not a result of ship emissions but rather land-going vehicles passing the measurement site. These could be recognized and excluded. Normally, the land-going vehicles were influencing the particle concentrations for a minute or shorter, while the ship plumes that influenced the particle concentrations could last for several minutes up to about 20 min. Note that the alternative method of identifying plumes with number concentration is not giving information about which ship passed by the measurement site due to lack of AIS data, unless there are other ways of collecting this information.
## 3.2 Calculating the contribution of ships to aerosol number concentration and other properties
For an identified ship plume peak, the contribution from this plume was estimated by calculating the area under the peak after subtraction of background concentrations. An example of a measured ship plume and illustrations of these calculations are shown in Fig. 4. In Fig. 4, the particle number concentration is clearly elevated during a few minutes during a period of relatively constant background concentrations. The estimated time of arrival of the plume, based on wind and AIS data (as described previously), is marked with a star and confirms the measurement of a ship plume and could provide further information about the ship, if desired. Due to the frequent appearance of ship plumes in Falsterbo, the background concentration was calculated as the average concentration of two intervals, one just before and one just after the ship plume, as seen in Fig. 4.
Figure 4Illustration of method of calculating aerosol contribution of individual ship plumes. Particle number concentration measured by a CPC (solid blue) and black carbon measured by an Aethalometer (AE33, dashed orange) during ca. 25 min of ambient sampling, and calculated time of arrival of the aerosol plume based on AIS and wind data (star). Plume duration is estimated by observation, and background concentrations are based on 6 min plus 6 min of adjacent data. The average of the background is subtracted from the plume concentrations to obtain only ship emission contribution.
One alternative way to calculate plume contribution by subtracting the plume from the background is the method used by Kivekäs et al. (2014). The authors extracted particle background concentrations by taking the 25th percentile values of a sliding window of a few hours for the particle number concentration time series. This is an appropriate automatic method to use on large data sets of ship plumes. The ship lane in the Kivekäs study was between 15 and 60 km away from the station. During periods with sharp increases or decreases in background concentrations, this method did not yield acceptable results, and these periods had to be manually controlled for errors and removed from the final data analysis. However, the Kivekäs method was not possible to use in Falsterbo due to the frequent plume events and the relatively high number concentrations in the plumes, which affected the background values for the sliding window method.
If a measured concentration of some aerosol parameters is noisy or the plumes are similar in concentration to the background, it is still possible to use AIS or particle number concentration to identify plumes and calculate their contribution. This could be the case when particle mass concentrations in the ship plumes are generally low. For example, a plume peak is not clearly distinguished, as depicted in Fig. 4 for eBC mass concentrations. However, based on the identification from the AIS and the estimation of the plume duration from particle number concentration data, the effect of the plume on the other aerosol parameters could be investigated. The contribution from a ship to such an aerosol parameter was calculated in the same way as described above, by subtracting the adjacent background concentrations from the concentration during the plume period. The start and end time of the plume was assumed to be the same as measured by the particle counter.
Besides the contribution to aerosol concentrations in each plume, there is also a possibility to estimate the contribution from ships at a coastal location during an extended period of time, like a day, a season, or a year. This can be accomplished by multiplying the average plume contribution with the number of ships that have passed during the current period. Further, to account for wind direction, the value is multiplied with the fraction of the time that the wind was passing over the shipping lane towards land. We estimated the daily and seasonal contribution of ships (fi) to the particle concentrations at Falsterbo, in addition to background levels, using the equation
$\begin{array}{}\text{(1)}& {f}_{i}=\frac{{c}_{\mathrm{ship}}}{{c}_{\mathrm{bgr}}}\cdot \frac{{n}_{\mathrm{ship},i}\cdot {t}_{\mathrm{plume},\mathrm{av}}}{{t}_{i}}\cdot {w}_{i},\end{array}$
where cship is the average ship plume concentration, cbgr is the average background concentration for the chosen time period (i), nship,i is the number of ships passing during this period (based on AIS data, independent of wind direction), tplume,av is the average ship plume exposure duration, ti is the length of the time period i, and wi is the fraction of the time during which the wind is blowing over the shipping lane to the location of interest (defined by a reasonable wind sector for the location).
4 Results
## 4.1 Plume identification
To demonstrate how the ship identification with the AIS method worked, Fig. 5 shows an example of a time series from the CPC for a few hours of sampling during wintertime. Figure 5 also displays the times when the ship plumes were expected to arrive at the measurement station based on AIS and wind data, as described in Sect. 3. The particles from the ship plumes are seen as relatively short and intense peaks, generally matching well with the expected plume passages. The average plume duration was 10 min. All ships identified with the AIS system resulted in an increase in size-dependent particle number concentration when these measurements were available. The method to infer when the ship plume should affect measured concentrations at Falsterbo agreed excellently during winter considering that the wind speed and direction measurements had a 1 h resolution and that these parameters were only measured at Falsterbo and not along the air mass trajectory. In summer, this agreement was reasonable but less certain than in winter, which might be due to more turbulent winds and local meteorological factors such as sea breeze. This shows that the method has a potential to work for many different shipping lanes. All plumes passing the measurement site are observed in the particle counters; that is the fraction of observed plumes predicted by AIS trajectories is in principle 1. We miss some plumes in the individual ship analysis, due to too frequent and overlapping plume passages. We estimate the analysed fraction to ca. 0.4 for the ship traffic near Falsterbo. The analysed fraction depends on the plume duration as well as the frequency of ships. With an average plume duration of about 10 min, it also means that the plume peak maxima should be separated by at least 10 min to be able to correctly calculate plume contributions. For studies which do not require information about individual ships but rather about total ship contribution, the number of missed ships is very low and can be due to temporary AIS malfunction or military vessels passing (they do not transmit AIS). The highest uncertainty of the timing of the plume is introduced through the wind trajectories between the emission and measurement site. Regarding the uncertainty of the attribution of a ship ID to a plume, this is depending mainly on the frequency of ship plumes at the specific location in combination with the wind trajectories. If the plumes from two ships arrive about the same time to the Falsterbo station, we cannot be absolutely sure which ships contributed to which plume concentrations. In that case, we only know that two ships did contribute to elevated concentrations. Also, if these plumes are superimposed on top of each other, we are still not able to calculate the individual ship contribution. We choose only to calculate plume contribution for plumes whose peaks are about at least 10 min apart in order to avoid plume superposition, since average plume duration is about 10 min as stated in the manuscript. In this case, the ship identification is always assigned to the correct plume. We have seen that the timing accuracy of the ship ID with the actual plume contribution is better (lower) than 7 min (95 % CI). Since, we choose only plumes or ship ID data which are at least 10 min apart, this uncertainty has no effect on attributing a ship ID to the correct plume.
Figure 5Particle number concentration measured with a CPC, and calculated incidents of ship plume passages (stars) determined with AIS and meteorological data, versus time (31 January–1 February 2016), from measurements at the coastline in southern Sweden during an episode with westerly winds blowing from the Øresund strait to the coastal station Falsterbo. The concentrations are those of the total aerosol; i.e. background concentrations are not subtracted.
As an example of what AIS information can be used for, the properties of the ships identified in Falsterbo during the winter campaign are shown in Fig. 6. The distributions of ship weight, length, breadth, and average speed as well as the distance from the emission source to the measurement site (in distance, kilometre; and in transport time, minutes) are shown. The units of the parameters have been adjusted so that all values fit within a similar range in the plot. The linear distance from the ship to the measurement site at the time when the ship contributed to the pollution at the site is denoted “ship to site / km” and given in kilometres, and the transport time of the wind between the ship emissions and the site is denoted “ship to site / min”, and given in minutes. Note that the wind does not necessarily travel along a straight line between the ship and the station if the wind direction is changing, which is considered in the calculation of the “ship to site / min”.
Figure 6AIS ship information and calculated plume travel data for the 113 plumes evaluated from the winter campaign in Falsterbo. The boxes show the median, 25th percentile, and 75th percentile, and whiskers show the minimum and maximum value. The maximum deadweight of 140 kt is out of range.
No relation was found between emission and ship properties or transport; therefore the data presented are not normalized for weight or transport time but presented as they were measured at the measurement site. A variety of vessels pass the Øresund strait and Falsterbo. The most common ones are cargo ships, tankers, and ro-ro ships (roll-on/roll-off) and others are trawlers, dredgers, reefers, and fishing vessels. The production years of the ships ranged from 1965 to 2015, with a majority from the 1990s and 2000s.
## 4.2 Results of plume contribution calculations
The contribution of ship traffic to the air pollution at a coastal location was estimated for more than 150 ship plumes. Measurements were carried out with a similar setup during winter (January–March) and summer (May–July) of 2016. All instrument variables were not available for the entire measurement periods, and the wind direction was not always favourable for measuring ship plumes. In total, there were about 3 weeks with optimal data from the winter campaign and 2 weeks from the summer campaign.
For the calculation of how ships contributed to the particle number concentration, plumes were restricted to the following conditions: (1) identified by AIS, (2) clearly distinguishable from the background in the CPC time series, and (3) not overlapping with other plumes. This resulted in 109 (CPC) and 113 (SMPS) plumes from the winter campaign and 61 (CPC) and 8 (SMPS) plumes from the summer campaign used for further calculations. The number of plumes identified by the SMPS in the summer is much lower than identified by the CPC due to a non-functioning SMPS system in periods. Also, periods during which the SMPS was sampling aerosol through a potential aerosol mass oxidation flow reactor were also excluded in this analysis. Finally, there were in general fewer plumes identified in the summer than in winter due to lack of AIS data in summer and since the winds at Falsterbo less frequently arrived from the shipping lane during the summer measurement campaign. A summary of the meteorological conditions during the measurements can be found in Table 2.
Table 2Meteorological conditions during measurement campaigns: average, lowest, and highest values.
* Direct sunlight, i.e. not cloudy.
According to the methods described in Sect. 3.2, we calculated the individual contributions from the observed ship plumes, both for particle number concentration and for eBC mass concentration, as well as the estimation of a daily and seasonal contribution at the specific location according to Eq. (1). This calculation was based on AIS data, which showed an average of 73 and 63 ships passing per day in winter and summer respectively. Together with the average plume duration (10 min), this indicates that the Falsterbo site is affected by ship emissions 51 % of the time in the winter and 44 % in the summer, when the wind blows from the Øresund strait. Based on historical wind data from the last 20 years (Swedish Meteorological and Hydrological Institute), the wind intercepts the shipping lanes in Øresund strait about 70 % of the time in both summer and winter, which was used together with particle concentration measurements to estimate the seasonal contribution from ships. Examples of plume contributions – individual, daily, and seasonal – are shown in Table 3. For each of the n numbers of measured ship plumes, a contribution is calculated. The table shows the median of these values, as well as the 25th and 75th percentile. A general observation was relatively large differences between ships; hence a larger number of observed plumes is preferred for a better estimation of the local ship emission contributions.
Table 3Contribution of particle number concentration and eBC mass concentration to local air quality, from two measurement campaigns at the Falsterbo coastal site.
a The background particle concentrations (Background concentration) and the particle contribution due to ships (Ship concentration) to number concentration (N) are shown as absolute values. Each value represents a median (or percentile) of a number of plumes (n) and is calculated from the ship plume peak average concentration (i.e. concentration per unit time). b “Daily” values refer to days with wind directions where ships affect Falsterbo (mainly westerly), and “Seasonal” values refer to the average contribution observed at each campaign extrapolated over one season, including all wind directions. c Condensation particle counter. d Scanning mobility particle sizer. e Based on Aethalometer data (880 nm).
Regarding the uncertainty in the plume particle number contribution, the relative statistical error of the CPC count is related to the total count N by $\surd \left(N\right)/N$. Hence, the particle counter has a very high precision. During our sample length of 1 s, the number N was typically above 1000 cm−3, and for an entire plume the total count was much higher. The uncertainty of the total concentration given by the instrument also depends on the uncertainty in the sample flow rate, since the concentration output is equal to N (flow rate sample time). We assume a flow rate uncertainty of maximum 10 %. So for example, with a concentration of 1000 particles cm−3 and a flow of 1 L min−1, the uncertainty becomes 10.5 % (when adding in quadrature). For an entire plume, the statistical error is even smaller, and hence the total uncertainty in particle number concentration is basically equal to the uncertainty in the sample flow. There is also a bias in concentrations due to losses of the smallest particles in the sampling line. That is, we measure lower concentrations than the ambient since this effect removes particles. Diffusion losses have been corrected for in the size distributions. But since we did not have SMPS and CPC data with same time interval (2 min vs 1 s) we cannot know exactly the losses for the CPC.
Despite the fact that the plumes were not clearly visible in the eBC time series, due to the low contribution to mass, a significant increase in BC was observed during identified plume events. The seasonal contribution of ship-emitted eBC is on average only 1.4±0.6 % of the total measured eBC at Falsterbo. Due to the noise of the eBC data as depicted in Fig. 4, individual eBC plume contributions are occasionally negative. However, a t test was performed on these data, which showed that the value of the eBC plume contribution was significantly higher than zero with a p value of 0.000030. Artificial eBC data without noise were also created, and random noise was applied on these data, which were of the same amplitude as the real noise of eBC data to test whether noise in data creates a systematic over- or underestimation of the plume contribution data. The test showed that the noisy eBC data are not creating an over- or underestimation of plume contribution, and hence this plume contribution should be robust. The same analysis was done on CO2 concentrations as for eBC, where plumes were also not visually distinguishable from background levels. Hence, at the distance from the shipping lane in this field study in Falsterbo, the plume CO2 concentrations were too diluted upon arrival at the measurement site to be distinguished from ambient levels. Therefore, it was not possible to calculate emission factors of particles for the ship plumes. For regional and global aerosol models, emission factors for the slightly aged ship plumes as well as for the fresh ones would be useful for accurate assessment. Emission factors for fresh plumes are obtained in for example laboratory engine studies and harbour measurements. If emission factors are to be determined for slightly aged ship plumes, it is possible that a shorter distance than our 7–20 km is preferred and a sensitive CO2 monitor (limit of detection below 0.1 ppm) is needed.
Figure 7The ship contribution to the average size distribution of particles (diameter, Dp, from 15 to 200 nm), measured with an SMPS during winter (n=113) and summer (n=8) respectively. Ambient background concentrations have been subtracted for each plume event, and correction for particle losses in the sampling has been accounted for.
The mean and median particle number size distribution for the ship emission plumes in Falsterbo are shown in Fig. 7. The distributions were calculated by averaging the number concentration in each SMPS size bin for 113 ship plumes for the winter campaign and 8 ship plumes from the summer campaign. A log-normal function (Hussein et al., 2005) with several modes was fitted to the average and median size distribution plumes for the winter and summer seasons. For the log-normal function, only particles with an electrical mobility diameter larger than 15 nm and smaller than 150 nm are considered due to uncertainties and losses for other sizes. The log-normal parameters are listed in Table 4. Four or five modes are used in the log-normal fit of the average size distribution plume since it seems that the typical size distribution contains a smaller- and a larger-sized nucleation mode (mode no. 1 and 2, < 30 nm diameter) and a smaller- and larger-sized Aitken mode (30 to 100 nm diameter). A majority of the ships do not produce the lower-sized nucleation mode, which is why the median size distribution does not contain this first mode. The other modes are often all present at the same time, and the larger particles could arise due to coagulation in an aerosol with a high concentration of smaller particles or due to emissions of relatively large primary soot particles. The uncertainties for the size distribution are large for the particles in the upper Aitken mode (80 to 100 nm diameter) and the accumulation mode (> 100 nm diameter) due to low numbers counted in the SMPS and also due to large variation between individual ships. The Pirjola et al. (2014) study shows that the particle number size distribution has two distinct modes for fresh ambient ship plumes – one in the nucleation mode (< 30 nm diameter) and one in the Aitken mode (30–100 nm diameter). If the number size distribution is remade into a volume size distribution, an accumulation mode also becomes visible (> 100 nm diameter). The current study also contains these modes. In addition, due to the individual variability between ship plumes in the current study, even two Aitken modes are discernible in the log-normal fitted size distributions. A few of the ships have a distinct accumulation mode, and for this reason, the average size distribution also contains this log-normal fitted mode. The data are significantly corrected for particle losses in sampling tubing especially for the nucleation mode sizes (< 30 nm diameter), which makes a second log-normal nucleation mode below 15 nm diameter appear in the log-normal fitted size distributions. Lab engine measurements also show such a mode in the Anderson et al. (2015) study, when higher sulfur content fuel was used, which stimulated new particle formation. Hence, in total, there are three to five log-normal modes fitted to the median and average particle number size distributions (Table 4).
Table 4Log-normal fit parameters for the average and median size distribution of the detected ship emission particles, during winter (n=61) and summer (n=8) respectively.
The number size distributions in Fig. 7 show that essentially all particles in the average and median ship plume have an electrical mobility diameter below 100 nm, most of them around 20–40 nm. Similar results have been shown in laboratory and on-board measurements (Kasper et al., 2007; Betha et al., 2017; Isakson et al., 2001; Kivekäs et al., 2014). There have also been observations of larger particle diameters in the micrometre range, e.g. Fridell et al. (2008). In our study, the APS instrument did not show any contribution to micrometre-sized particles from ships at the current distance from the shipping lane. The APS has a high sensitivity for single particles but did not measure that ship plumes contained significant particle number concentrations above background concentrations for particles larger than 0.5 µm diameter. Since we did not observe any particles larger than a few 100 nm in Falsterbo, this could be a suggestion that the larger particle modes are absent or negligible after the recent SECA sulfur regulations. Particles larger than the upper detection limit of the APS (ca. 20 µm) were not measured and could have been present but in that case likely deposited on the way to land.
The size distribution of the average plume shows higher concentrations than that of the median plume, both for the summer and the winter data. This is due to the high contribution of some ships skewing the results. Due to higher and noisier background particle concentrations in the summer (Table 3), and the lack of AIS data, it is possible that plumes with relatively low particle number concentrations were not distinguishable from the background, and hence the selection of plumes in the summer might have been biased towards the more-polluting ship plumes. Also, the difference in sample size should be noted here: 113 good plumes observed during winter and 8 during summer. This difference depends mainly on instrument malfunction, unfavourable wind directions, and lack of AIS data. From the available data, there is however an indication that the number and the size of the particles from ships are somewhat larger in summer. This seasonal difference could possibly be explained by secondary particulate matter formed by atmospheric ageing, which is expected to be more significant in the summer, but more measurements are needed to confirm this.
Due to the distance to the shipping lane, the ship emissions were diluted enough to have concentrations below the detection limit of some instruments. In order to capture the relatively short plume events, the time resolution could not be too long either, making the detection limit of some instruments higher.
5 Recommendations and concluding remarks
The AIS method to identify which ship influenced exposure on land and to identify individual ship plumes from measurements about 10 km downwind of ship lanes proved to be very exact for the winter data and worked relatively well for summer data. We know that the method to observe individual plumes on top of background concentrations does not work for all ships at the distance 25–60 km downwind of a shipping lane (Kivekäs et al., 2014). There, only a fraction of the plumes were distinguishable. In contrary, at our Falsterbo site, there were no such issues with the plume identification method. Hence, the method can be expected to work at least up to 10 km and get worse towards 60 km. This is true for particle number concentration measurements (with a CPC) but not for mass concentration measurements. So, to be able to detect plumes at maximum distances a particle counter is of importance. Considering the wind data were available only with a 1 h resolution, the plume identification worked well. Availability of wind data with better time resolution does not seem to be necessary at this specific site. Although at longer distances between the ship lane and the station, this can potentially be an issue and it would be advantageous to have meteorological data with better time resolution. When AIS data were missing for one reason or the other, the particle number concentration detected with a condensation particle counter also proved to work very well to identify ships, although it could not give the information about which ship it was.
The method to estimate plume contribution from individual ships proved to be straightforward for the clearly visible ship plumes at the measurement station. For the eBC concentration, the plume identification was less straightforward since the plume signal was very low relative to the noise level. For many plumes, no increase in eBC was observed with the bare eye. We still used the already identified plumes to calculate the contribution to eBC. A very low but still significant plume contribution could be calculated. Even if the proposed method yields non-significant plume contributions for a specific parameter, this does not mean that the method does not work. Rather it means that ship emissions do not contribute to significant exposure inland for this parameter and that the detection capabilities of the instrument do not allow for detecting this non-significant contribution. The calculation was done using the precise time of the plume incidence observed in the particle counter. This was also a surprisingly robust method without systematic biases due to the noise. Dispersed background levels of BC were about 0.2 µg m−3. The ship emission particles, which were clearly seen by number in the plumes, do probably contain soot since they are from a combustion source, but the mass becomes difficult to detect due to the small particle sizes. It could therefore be valuable for future measurements of ship-emitted BC to use a measurement method which does not require much BC mass for detection, such as single particle incandescence.
Since the particle counter always yielded visible and smooth plumes at the downwind station, it is recommended to always bring a particle counter when doing these kind of measurements, even if it is not of main interest to estimate particle number contributions. Namely, it might turn out that AIS data are erroneous or missing, and the particle counter is needed to define plume time and background to calculate the plume contributions for instruments with high noise and low time resolution. Since the ship plumes at 10 km downwind or farther from the ship lanes have only a few minutes up to about 20 min duration, it is also recommended that the time resolution of the instruments one brings is not worse than 1 min. For a scanning particle sizer, like the SMPS, one should consider the scan time in comparison to the plume duration, and possibly add a mixing volume to not get rapid changes in the aerosol particle concentration during a scan.
These and other measurements have shown that the number of particles < 30 nm diameter is substantial, even for relatively aged ship plumes. The estimation and correction for particle losses are therefore crucial to be able to assess the true size-dependent particle concentrations, especially when the sampling line to instruments is relatively long. It is our recommendation to place the particle counter (CPC) close to inlet and further to use as short a sampling line as possible with minimum diffusion losses when performing these kinds of studies in general.
The current method of stationary measurements of downwind plumes from a shipping lane has turned out to be very cost-effective compared to aircraft or ship vessel chasing experiments and can fetch a much higher number of ship plumes. Hence, we also urge the use of it for economic and pragmatic considerations when studying relatively aged ship plumes for a high number of ships. In future studies of detailed individual ship plumes and the emission sources, it should be considered whether the particle emissions depend on ship engine power used. It is possible to estimate the engine power required by a ship, using the total power of the ships, their design speed, and actual speed through the propeller law (Moreno-Gutiérrez et al., 2015). This can then be compared to particle number concentration emissions but also particle mass emissions and gaseous emissions. With the method presented in this paper, it is possible to collect information on a very large sample of ships for these kinds of investigations.
Before performing the measurements with the new method, it is important to investigate the meteorological situations at the current measurement site. For example, during sea breezes, local wind measurements could indicate that shipping lane emissions should reach the measurement station, whereas in reality they might not. Care should be taken to account for these periods when the meteorological data will give erroneous results. However, these meteorological phenomena do not take place all the time; hence these specific meteorological conditions will not disqualify any chosen measurement site with the current proposed method. Again, these uncertain wind conditions make it very important to bring a particle counter to register shipping plumes. If the particle counter does not register any ship plumes during a selected time period, this indicates that winds from the ships are not reaching the measurement station, despite the fact that the local wind measurements suggest otherwise.
Beyond providing ambient aerosol data from a SECA from summer and winter measurements, the data from this study can also be used to validate process models simulating ageing processes of particle number size distributions as well as long distance transport along meteorological air mass trajectories in Lagrangian process models. In addition to particle number concentration and eBC, the method was applied in the companion paper, Ausmeel et al. (2019), focusing on other aerosol properties and regional or global scale air quality and climate models could use this kind of data to validate modelled ship contribution in certain grid cells.
Data availability
Data availability.
The data sets used in this study are available upon request from the authors (S. Ausmeel and A. Kristensson).
Author contributions
Author contributions.
AK designed the experiments and all authors carried them out, with AK being the project leader during the winter campaign and SA during the summer campaign. AK developed the model code. SA analysed the plume and aerosol data and prepared the manuscript with contributions from all co-authors.
Competing interests
Competing interests.
The authors declare that they have no conflict of interest.
Acknowledgements
Acknowledgements.
Fredrik Windmark at the Swedish Meteorological and Hydrological Institute (SMHI) is acknowledged for helping to provide AIS ship positioning data. Mårten Spanne, Paul Hansson, Henric Nilsson, and Susanna Gustafsson from the Environment Department at the city of Malmö are acknowledged for helping in preparing and setting up the measurements at Falsterbo. Kirsten Kling of DTU and Antti Joonas Koivisto of NRCWE are acknowledged for helping with the summer campaign and Fredrik Mattsson and Anna Hansson for helping with the winter campaign. Thank you also to Håkan Lindberg and the personnel from the Falsterbo golf course and Lennart Karlsson from the Falsterbo bird watching station, who were willing to prepare a place for our measurement trailer, as well as the County Administrative Board of Skåne and Vellinge municipality for giving permission to measure in the Flommen Nature Reserve.
Financial support
Financial support.
This research has been supported by the Svenska Forskningsrådet Formas (grant no. 2014-951) and the Crafoord Foundation (project nos. 20140955 and 20161026).
Review statement
Review statement.
This paper was edited by Folkert Boersma and reviewed by two anonymous referees.
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Müller, T., Henzing, J. S., de Leeuw, G., Wiedensohler, A., Alastuey, A., Angelov, H., Bizjak, M., Collaud Coen, M., Engström, J. E., Gruening, C., Hillamo, R., Hoffer, A., Imre, K., Ivanow, P., Jennings, G., Sun, J. Y., Kalivitis, N., Karlsson, H., Komppula, M., Laj, P., Li, S.-M., Lunder, C., Marinoni, A., Martins dos Santos, S., Moerman, M., Nowak, A., Ogren, J. A., Petzold, A., Pichon, J. M., Rodriquez, S., Sharma, S., Sheridan, P. J., Teinilä, K., Tuch, T., Viana, M., Virkkula, A., Weingartner, E., Wilhelm, R., and Wang, Y. Q.: Characterization and intercomparison of aerosol absorption photometers: result of two intercomparison workshops, Atmos. Meas. Tech., 4, 245–268, https://doi.org/10.5194/amt-4-245-2011, 2011.
Onasch, T., Trimborn, A., Fortner, E., Jayne, J., Kok, G., Williams, L., Davidovits, P., and Worsnop, D.: Soot particle aerosol mass spectrometer: development, validation, and initial application, Aerosol Sci. Technol., 46, 804–817, 2012.
Petzold, A., Hasselbach, J., Lauer, P., Baumann, R., Franke, K., Gurk, C., Schlager, H., and Weingartner, E.: Experimental studies on particle emissions from cruising ship, their characteristic properties, transformation and atmospheric lifetime in the marine boundary layer, Atmos. Chem. Phys., 8, 2387–2403, https://doi.org/10.5194/acp-8-2387-2008, 2008.
Petzold, A., Ogren, J. A., Fiebig, M., Laj, P., Li, S.-M., Baltensperger, U., Holzer-Popp, T., Kinne, S., Pappalardo, G., Sugimoto, N., Wehrli, C., Wiedensohler, A., and Zhang, X.-Y.: Recommendations for reporting “black carbon” measurements, Atmos. Chem. Phys., 13, 8365–8379, https://doi.org/10.5194/acp-13-8365-2013, 2013.
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Svenningsson, B., Arneth, A., Hayward, S., Holst, T., Massling, A., Swietlicki, E., Hirsikko, A., Junninen, H., Riipinen, I., Vana, M., Maso, M. D., Hussein, T., and Kulmala, M.: Aerosol particle formation events and analysis of high growth rates observed above a subarctic wetland–forest mosaic, Tellus B, 60, 353–364, https://doi.org/10.1111/j.1600-0889.2008.00351.x, 2008.
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von der Weiden, S.-L., Drewnick, F., and Borrmann, S.: Particle Loss Calculator – a new software tool for the assessment of the performance of aerosol inlet systems, Atmos. Meas. Tech., 2, 479–494, https://doi.org/10.5194/amt-2-479-2009, 2009.
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Wiedensohler, A., Birmili, W., Nowak, A., Sonntag, A., Weinhold, K., Merkel, M., Wehner, B., Tuch, T., Pfeifer, S., Fiebig, M., Fjäraa, A. M., Asmi, E., Sellegri, K., Depuy, R., Venzac, H., Villani, P., Laj, P., Aalto, P., Ogren, J. A., Swietlicki, E., Williams, P., Roldin, P., Quincey, P., Hüglin, C., Fierz-Schmidhauser, R., Gysel, M., Weingartner, E., Riccobono, F., Santos, S., Grüning, C., Faloon, K., Beddows, D., Harrison, R., Monahan, C., Jennings, S. G., O'Dowd, C. D., Marinoni, A., Horn, H.-G., Keck, L., Jiang, J., Scheckman, J., McMurry, P. H., Deng, Z., Zhao, C. S., Moerman, M., Henzing, B., de Leeuw, G., Löschau, G., and Bastian, S.: Mobility particle size spectrometers: harmonization of technical standards and data structure to facilitate high quality long-term observations of atmospheric particle number size distributions, Atmos. Meas. Tech., 5, 657–685, https://doi.org/10.5194/amt-5-657-2012, 2012.
Williams, E., Lerner, B., Murphy, P., Herndon, S., and Zahniser, M.: Emissions of NOx, SO2, CO, and HCHO from commercial marine shipping during Texas Air Quality Study (TexAQS) 2006, J. Geophys. Res.-Atmos., 114, D21306, https://doi.org/10.1029/2009JD012094, 2009. | 2019-09-20 11:49:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "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": 0.5592753887176514, "perplexity": 3974.3806305533985}, "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/1568514574018.53/warc/CC-MAIN-20190920113425-20190920135425-00381.warc.gz"} |
http://tex.stackexchange.com/questions/53294/wrapfig-and-rotating-place-figure-at-end-of-document | # wrapfig and rotating place figure at end of document
With the packages wrapfig and rotating and calls to wraptable, sideways and tabular, I get a document where the table is placed at the end of the document. For instance, in a two page document defined by \pagebreak, my table gets placed on the second page, or at the end of the document, and not where I want it, on the first page where it is declared. Anyone know why this happens or have a work around?
\documentclass[12pt,letterpaper]{article}
\usepackage[american]{babel}
\usepackage{rotating}
\usepackage{wrapfig}
\begin{document}
\section{Test page 1}
\begin{wraptable}{r}{.5\textwidth}
\begin{sideways}
\begin{tabular}{l}
1 \\
2 \\
3 \\
4
\end{tabular}
\end{sideways}
\end{wraptable}
\pagebreak
\section{Test page 2}
\end{document}
Basically, I want to rotate a (sizable) table on a page that also has horizontal text.
-
wraptable is meant to be used along with text such that it wraps along. You just add text before and after wraptable, then things are set right.
\documentclass[12pt,letterpaper]{article}
\usepackage[american]{babel}
\usepackage{rotating}
\usepackage{wrapfig,lipsum}
\begin{document}
\section{Test page 1}
\lipsum[1]
\begin{wraptable}{r}{.5\textwidth}
\caption{A wrapped table will not float if there is enough text surrounding it.}\label{wrap-tab:1}
\begin{sideways}
\begin{tabular}{l}\\
1 \\
2 \\
3 \\
4
\end{tabular}
\end{sideways}
\end{wraptable}
\lipsum[2] %%% This provides text.
\pagebreak
\section{Test page 2}
\end{document}
-
Normally, using a lower case letter to specify horizontal placement will create a stationary wraptable or wrapfig, that is
\begin{wraptable}{r}{.5\textwidth}
does not float, whereas
\begin{wraptable}{R}{.5\textwidth}
does. However, stationary wrapfigs and wraptables can be forced to float under some circumstances. Sadly, the package documentation isn't very helpful here, but I have noticed that wraptables and wrapfigs can be very sensitive to what's around them. For example, inserting a short piece of text immediately before the wraptable in your code causes the table to appear on page 1.
-
I made the adjustment as suggested, but the result is the same. – Chernoff Apr 25 '12 at 20:13
Actually my suggestion was wrong anyway! I have edited my answer, and I now get the wraptable on page 1, but it's hard to say whether it will help in your actual document. – Ian Thompson Apr 25 '12 at 20:57 | 2015-05-28 00:35:44 | {"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": 0.8769078254699707, "perplexity": 1374.7450136468588}, "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-2015-22/segments/1432207929176.79/warc/CC-MAIN-20150521113209-00185-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://gomathanswerkey.com/texas-go-math-grade-5-lesson-3-3-answer-key/ | # Texas Go Math Grade 5 Lesson 3.3 Answer Key Multiplication with Decimals and Whole Numbers
Refer to our Texas Go Math Grade 5 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 5 Lesson 3.3 Answer Key Multiplication with Decimals and Whole Numbers.
## Texas Go Math Grade 5 Lesson 3.3 Answer Key Multiplication with Decimals and Whole Numbers
Essential Question
How can you use properties and place value to multiply a decimal and a whole number?
Unlock the problem
In 2010, the United States Mint released a newly designed Lincoln penny.
A Lincoln penny has a mass of 2.5 grams. If there are 5 Lincoln pennies on a tray,
what is the total mass of the pennies?
• How much mass does one penny have?
2.5 grams,
Explanation:
Given a Lincoln penny has a mass of 2.5 grams.
• Use grouping language to describe what you are asked to find.
Total mass of the pennies,
Explanation:
Given In 2010, the United States Mint released a newly designed Lincoln penny.
A Lincoln penny has a mass of 2.5 grams. If there are 5 Lincoln pennies on a tray,
what is the total mass of the pennies?
Multiply. 5 × 2.5
Estimate the product. Round to the nearest whole number.
5 × _25__ = _12.5__
One Way
Use the Distributive Property.
Another Way
Show partial products.
Step 1
Multiply the tenths by 5.
Step 2
Multiply the ones by 5.
Step 3
12.5
Example
Use place value patterns.
Having a thickness of 1.35 millimeters, the dime is the thinnest coin produced by
the United States Mint. If you stacked 8 dimes,
what would be the total thickness of the stack?
Multiply. 8 × 1.35
Step 1
Write the decimal factor as a whole number.
Think: 1.35 × 100 = 135
Step 2
Multiply as with whole numbers.
Step 3
Place the decimal point.
Think: 0.01 of 135 is 1.35. Find 0.01 of 1,080 and record the product.
A stack of 15 dimes would have a thickness of _10.80_ millimeters.
8 X 1.35 = 10.80,
A stack of 15 dimes would have a thickness of 10.80 millimeters,
Explanation:
Given having a thickness of 1.35 millimeters, the dime is the thinnest coin produced by
the United States Mint. If you stacked 8 dimes,
The total thickness of the stack,
Step 1:
Writing the decimal factor as a whole number.
Think: 1.35 × 100 = 135
Step 2:
Multiply as with whole numbers.
Step 3:
Placing the decimal point.
Think: 0.01 of 135 is 1.35. Finding 0.01 of 1,080 and record the product.
Therefore a stack of 15 dimes would have a thickness of 10.80 millimeters.
Share and Show
Place the decimal point in the product.
Question 1.
6.81 X 7 = 47.67,
Explanation:
Question 2.
3.7 X 2 =
Explanation:
Question 3.
19.34 X 5 = 96.70,
Explanation:
Find the product.
Question 4.
6.32 X 3 =18.96,
Explanation:
Question 5.
4.5 X 8 = 36.0,
Explanation:
Question 6.
40.7 X 5 = 203.5
Explanation:
Math Talk
Mathematical Processes
Explain how you can determine if your answer to Exercise 6 is reasonable.
40.7 X 5 = 203.5,
203.5
Explanation:
Step 1
Multiply the tenths by 5.
407
X 5
5—->5 X 7 =35,
Step 2
Multiply the zero by 5.
407
X 5
3 —->5 X 0 = 0 + 3,
Step 3
Multiply the ten by 5
407
X 5
200 —->5 X 40 =200 ,
Step 4
407
X 5
000.50
003.00
200.00
203.50
Problem Solving
Practices Copy arid Solve Find the product.
Question 7.
8 × 7.2
8 X 7.2 = 57.6,
Explanation:
7.2
X 8
57.6
Step 1
Multiply the tenths by 8.
72
X 8
6 —-> 8 X 2 = 16,
Step 2
Multiply the one by 8.
72
X 8
57 —->8 X 7 = 56 + 1,
Step 3
72
X 8
.60
57.00
57.60
Question 8.
3 × 1.45
3 X 1.45 = 4.35,
Explanation:
1.45
X 3
4.35
Step 1
Multiply the hundredths by 3.
145
X 3
15 —-> 3 X 5 = 15,
Step 2
Multiply the tenths by 3.
145
X 3
13—->3 X 4 = 12 + 1,
Step 3
Multiply the one by 3,
145
X 3
3 —->3 X 1 = 3,
145
X 3
0.05
1.30
3.00
4.35
Question 9.
9 × 8.6
9 x 8.6 = 77.4,
Explanation:
8.6
X 9
77.4
Step 1
Multiply the tenths by 9.
86
X 9
54 —-> 9 X 6 = 5.4,
Step 2
Multiply the one by 9.
86
X 9
72—->9 X 8 = 72,
Step 3:
86
X 9
05.4
72.0
77.4
Question 10.
6 × 0.79
6 X 0.79 = 4.74,
Explanation:
0.79
X 6
4.74
Step 1
Multiply the hundredths by 6.
0.79
X 6
54 —-> 6 X 9 = 54,
Step 2
Multiply the tenths by 6.
0.79
X 6
42—-> 6 X 7 = 42,
Step 3
Multiply the one by 6,
0.79
X 6
0 —-> 6 X 0 = 0,
0.79
X 6
0.54
4.20
0.00
4.74
Question 11.
4 × 9.3
4 X 9.3 = 37.2,
Explanation:
9.3
X 4
37.2
Step 1
Multiply the tenths by 4.
9.3
X 4
1.2—->4 X 3 = 1.2,
Step 2
Multiply the one by 4,
9.3
X 4
36 —->4 X 9 = 36,
145
X
01.20
36.00
37.20
Question 15.
H.O.T. Write Math Julie multiplies 6.27 by 7 and claims the product is 438.9.
Explain without multiplying how you know Julie’s answer is not correct.
Decimal point place value is incorrect,
Explanation:
Given Julie multiplies 6.27 by 7 and claims the product is 438.9,
without multiplying Julie’s answer is not correct because in 6.27
we have decimal point before hundredths,
So in 438.9 we have decimal point before tenths,
therefore Julie is incorrect.
Problem Solving
Use the table for 16-18.
Question 16.
Sari has a bag containing 6 half dollars. What is the mass of the half dollars in Sari’s bag?
Question 17.
Felicia is running a game booth at a carnival. One of the games requires
participants to guess the mass, in grams, of a bag of 9 dimes.
What Is the actual mass of the dimes in the bag?
20.43 is the actual mass of the dimes in the bag,
Explanation:
Stated Felicia is running a game booth at a carnival. One of the games requires
participants to guess the mass, in grams, of a bag of 9 dimes.
So the actual mass of the dimes in the bag is 9 X 2.27 =
2 6
2.27
X 9
20.43
Question 18.
Multi-Step Chance has $2 in quarters. Blake has$5 in dollar coins.
Whose coins have the greatest mass? Explain.
Blake coins have the greatest mass,
Explanation:
Given Chance has $2 in quarters mass , Chance coins mass is$2 X 5.67 grams = 11.34 grams,
Blake has $5 in dollar mass, So Blake have coins mass is$5 X 8.1 grams= 40.5 grams, on comparing between Chance and Blake,
Blake coins have the greatest mass.
Question 19.
Apply A hummingbird weighs 5.8 grams. If the hummingbird eats
8 times its body weight each day, how much does the bird eat in four days?
(A) 23.2 grams
(B) 46.4 grams
(C) 185 .6 grams
(D) 464 grams
(C) 185 .6 grams,
Explanation:
Applying a hummingbird weighs 5.8 grams,
If the hummingbird eats 8 times its body weight each day,
the bird eat in each day is 5.8 grams X 8 = 46.4 grams,
In four days the bird will eat 4 X 46.4 grams = 185.6 grams,
which matches with (C).
Question 20.
Use Symbols Which expression shows how to use the
Distributive Property to find 5.17 X 9?
(A) 5.17 + 9
(B) 9 × (5.1) + 7
(C) 9 × (5.1 + 7)
(D) 9 × (5 + 0.17)
(D) 9 × (5 + 0.17),
Explanation:
Using Symbols the expression that shows how to use the
Distributive Property to find 5.17 X 9 is first we write 5.17 as
(5 + 0.17) and applying distributive law we get 9 X (5 + 0.17)
matches with (D).
Question 21.
Multi-Step The cost to park a car in a parking lot is $1.10 per hour. Maleek parked his car for 4 hours on Monday 4 hours on Tuesday and 4 hours on Wednesday. How much did he spend on parking in all? (A)$4.10
(B) $4.40 (C)$13.10
(D) $13.20 Answer: (D)$13.20,
Explanation:
The cost to park a car in a parking lot is $1.10 per hour. Maleek parked his car for 4 hours on Monday, 4 hours on Tuesday and 4 hours on Wednesday, Number of hours from Monday, Tuesday and Wednesday = 4 + 4 + 4 = 12 hours, Maleek spends on parking in all is$1.10 X 12 = $13.20 matches with (D). Texas Test Prep Question 22. Every day on his way to and from school, Milo walks a total of 3.65 miles. If he walks to school 5 days, how many miles will Milo have walked? (A) 1,825 miles (B) 18.25 miles (C) 182.5 miles (D) 1.825 miles Answer: (B) 18.25 miles, Explanation: Given Every day on his way to and from school, Milo walks a total of 3.65 miles. If he walks to school 5 days number of miles Milo will have to walk is 5 X 3.65 miles = 3.65 X 5 0.025 03.00 15.00 18.25 (B) 18.25 miles. ### Texas Go Math Grade 5 Lesson 3.3 Homework and Practice Answer Key Find the product. Question 1. 4 × 7.12 Answer: 4 X 7.12 = 28.48, Explanation: Given to find 4 X 7.12 7.12 X 4 00.08 00.40 28.00 28.48 Question 2. 3 × 0.29 Answer: 3 x 0.29 = 0.87, Explanation: 0.29 X 3 0.27 0.60 0.87 Question 3. 8 × 2.9 Answer: 8 X 2.9 = 23.20, Explanation: 2.9 X 8 07.20 16.00 23.20 Question 4. 7 × 18.3 Answer: 7 X 18.3 = 128.1, Explanation: 18.3 X 7 02.10 56.00 70.00 128.10 Question 5. 5 × 9.26 Answer: 5 X 9.26 = 46.30, Explanation: 9.26 X 5 00.30 01.00 45.00 46.30 Question 6. 2 × 42.1 Answer: 2 X 42.1 = 84.2, Explanation: 42.1 X 2 00.20 04.00 80.00 84.20 Question 7. 7 × 0.34 Answer: 7 X 0.34 = 2.38, Explanation: 0.34 X 7 0.28 2.10 0.00 2.38 Question 8. 9 × 7.21 Answer: 9 X 7.21 = 64.89, Explanation: 7.21 X 9 00.09 01.80 63.00 64.89 Question 9. 4 × 5.2 Answer: 4 X 5.2 = 20.8, Explanation: 5.2 X 4 00.8 20.0 20.8 Question 10. 7 × 17.2 Answer: 7 X 17.2 = 120.40, Explanation: 17.2 X 7 01.40 49.00 70.00 120.40 Question 11. 4 × 3.92 Answer: 4 X 3.92 = 15.68, Explanation: 3.92 X 4 00.08 03.60 12.00 15.68 Question 12. 8 × 0.21 Answer: 8 X 0.21 = 1.68, Explanation: 0.21 X 8 0.08 1.60 0.00 1.68 Question 13. Logan multiplies 5.31 by 3 and says the product is 159.3. Explain why Logan’s answer is not correct. Answer: After decimal point value we have hundredths but product is tenth’s , Logan is incorrect, Explanation: As per Logan multiplies 5.31 by 3, we should get hundredths but the product says it is tenths, therefore Logan is incorrect. Problem Solving Question 14. Minnie rides her bike 2.76 miles each day. How many miles will she bike in 7 days? Answer: 19.32 rides her bike in 7 days, Explanation: Given Minnie rides her bike 2.76 miles each day, Number of miles she biked in 7 days is 2.76 X 7 00.42 04.90 14.00 19.32 Question 15. Dominick buys a package of soup mix that is 1.38 ounces. How many ounces are 6 packages of soup mix? Answer: Ounces in 6 packages of soup mix are 8.28 ounces, Explanation: We have Dominick buys a package of soup mix that is 1.38 ounces. Number of ounces are 6 packages of soup mix is 1.38 X 6 0.48 1.80 6.00 8.28 Texas Test Prep Lesson Check Fill in the bubble completely to show your answer. Question 16. Sandra uses 6.39 ounces of beans in a recipe. If she triples the recipe, how many ounces of beans will Sandra need? (A) 17.29 ounces (B) 18.83 ounces (C) 19.17 ounces (D) 20.19 ounces Answer: (C) 19.17 ounces, Explanation: Given Sandra uses 6.39 ounces of beans in a recipe. If she triples the recipe the number of beans will Sandra need is 6.39 X 3 00.27 00.90 18.00 19.17 matches with (C). Question 17. Kip uses the Distributive Property to rewrite an expression. (4 × 3) + (4 × 0.7) What is the value of the expression? (A) 3.07 (B) 11.7 (C) 14.8 (D) 29.6 Answer: (C) 14.8, Explanation: Given Kip uses the Distributive Property to rewrite an expression (4 X 3) + (4 X 0.7) so the value of the expression is (12) + (2.8) = 14.8 matches with (C). Question 18. Manny charges$1.25 for a cup of lemonade at the lemonade stand.
He sells 9 cups. How much money does Manny collect?
(A) $10.75 (B)$11.00
(C) $11.25 (D)$11.75
(C) $11.25, Explanation: Given Manny charges$1.25 for a cup of lemonade at the lemonade stand,
He sells 9 cups. Manny collected
$1.25 X 9 0.45 1.80 9.00 11.25 which matches with (C). Question 19. Fabric costs$11.99 a yard. Leo buys 4 yards of red fabric and 3 yards of blue fabric.
How much does Leo spend on fabric?
(A) $47.96 (B)$35.97
(C) $77.33 (D)$83.93
(D) $83.93, Explanation: Given Fabric costs$11.99 a yard,
Leo buys 4 yards of red fabric and 3 yards of blue fabric.
Leo spent on fabric costs
$11.99 X 7 000.63 006.30 007.00 070.00$83.93,
matches with (D).
Question 20.
Multi-Step On the first day of summer, a rainstorm brought 2.29 Inches of rain to the area.
During the rest of the summer, there was 3 times as much rain as on the first day.
What was the total number of Inches of rainfall for the summer?
(A) 6.29 inches
(B) 5.67 inches
(C) 6.87 Inches
(D) 9.16 inches
(C) 6.87 Inches,
Explanation:
Given on the first day of summer a rainstorm brought 2.29 Inches of rain to the area.
During the rest of the summer there was 3 times as much rain as on the first day.
The total number of Inches of rainfall for the summer is
2.29
X 3
0.27
0.60
6.00
6.87 inches matches with (C).
Question 21.
Multi-Step Nicky bought 14.2 pounds of coffee for the coffee shop.
The shop usually uses 5 times that amount of coffee during one week.
How much more coffee will Nicky need to purchase to get the
right amount of coffee for the week?
(A) 71 pounds
(B) 56.8 pounds
(C) 85.2 pounds
(D) 50 pounds
(A) 71 pounds,
Explanation:
Nicky bought 14.2 pounds of coffee for the coffee shop.
The shop usually uses 5 times that amount of coffee during one week.
More coffee will Nicky need to purchase to get the right amount of coffee for the week
is 14.2
X 5
01.0
20.0
50.0
71.0 pounds matches with (A).
Scroll to Top | 2022-10-02 02:25:23 | {"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": 1, "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": 0.27720940113067627, "perplexity": 6894.263607325939}, "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-2022-40/segments/1664030337244.17/warc/CC-MAIN-20221002021540-20221002051540-00627.warc.gz"} |
https://tex.stackexchange.com/questions/414307/chemical-formula-inside-pgfplot-rendered-wrong-labels-overlap | # Chemical formula inside pgfplot rendered wrong---labels overlap
Under normal circumstances \ch{AB_2} is rendered as $AB_2$, but not inside the pgfplot. I am getting the error message Package PGF Math Error: Could not parse input '' as a floating point number and labels overlap. Is it somehow possible to make these 2 packages to be friends. I am still learning PGF, therefore such naive question. I would appreciate your any input.
\documentclass{standalone}
\usepackage{pgfplots,chemformula,filecontents}
\begin{filecontents}{data.dat}
N system A1 C2 Exp
1 CH2 -0.70 -0.82 -0.53
2 NH3 -0.97 -1.58 0.00
3 H2HO -0.96 -1.53 -0.06
4 SiH2 0.11 -0.92 -0.60
\end{filecontents}
\begin{document}
\begin{tikzpicture}
\begin{axis}[xlabel=eV,ylabel=eV]
visualization depends on={value \thisrow{system} \as \label},
scatter, only marks,
scatter src=explicit symbolic,
nodes near coords*={\ch{\label}}]
table[x=Exp,y=A1]{data.dat};
\end{axis}
\end{tikzpicture}
\end{document}
This is a purely pgfplots problem. You're mixing the point meta (synonym for scatter src) and visualization depends on. But point meta is used to pass extra data in a standard way, while visualization depends on is used to give extra data without interfering with point meta, and how you use it is completely up to you.
Solution 1 Pass the labels as point meta (here with scatter src):
\documentclass{standalone}
\usepackage{pgfplots,chemformula,filecontents}
\begin{filecontents}{data.dat}
N system A1 C2 Exp
1 CH2 -0.70 -0.82 -0.53
2 NH3 -0.97 -1.58 0.00
3 H2HO -0.96 -1.53 -0.06
4 SiH2 0.11 -0.92 -0.60
\end{filecontents}
\begin{document}
\begin{tikzpicture}
\begin{axis}[xlabel=eV,ylabel=eV]
scatter, only marks,
scatter src=explicit symbolic,
nodes near coords={\expandafter\ch\expandafter{\pgfplotspointmeta}},
]
table[x=Exp,y=A1,meta=system]{data.dat};
\end{axis}
\end{tikzpicture}
\end{document}
As you can see I used the meta key to specify the point meta column. The \expandafter trickery is apparently necessary to typeset the chemical formula with \ch if the formula text is in a macro (like \label here). Also note that there is no asterisk after nodes near coords: otherwise pgfplots will apply a default style to the nodes that expects numerical values in point meta.
Solution 2 Pass labels with visualization depends on:
\documentclass{standalone}
\usepackage{pgfplots,chemformula,filecontents}
\begin{filecontents}{data.dat}
N system A1 C2 Exp
1 CH2 -0.70 -0.82 -0.53
2 NH3 -0.97 -1.58 0.00
3 H2HO -0.96 -1.53 -0.06
4 SiH2 0.11 -0.92 -0.60
\end{filecontents}
\begin{document}
\begin{tikzpicture}
\begin{axis}[xlabel=eV,ylabel=eV]
scatter, only marks,
nodes near coords={\expandafter\ch\expandafter{\label}},
visualization depends on={value \thisrow{system} \as \label},
]
table[x=Exp,y=A1]{data.dat};
\end{axis}
\end{tikzpicture}
\end{document}
Note that I'm not using scatter src here (i.e. no point meta). In both this solution and the previous one we have some overlap of the text labels...
Solution 3 Use visualization depends on to give the label text and point meta to set the label position:
\documentclass{standalone}
\usepackage{pgfplots,chemformula,filecontents}
\begin{filecontents}{data.dat}
N system A1 C2 Exp Position
1 CH2 -0.70 -0.82 -0.53 1
2 NH3 -0.97 -1.58 0.00 1
3 H2HO -0.96 -1.53 -0.06 -1
4 SiH2 0.11 -0.92 -0.60 -1
\end{filecontents}
\begin{document}
\begin{tikzpicture}
\begin{axis}[xlabel=eV,ylabel=eV]
scatter, only marks,
scatter src=explicit,
nodes near coords align=vertical,
nodes near coords={\expandafter\ch\expandafter{\label}},
visualization depends on={value \thisrow{system} \as \label},
]
table[x=Exp,y=A1,meta=Position]{data.dat};
\end{axis}
\end{tikzpicture}
\end{document}
I added a data column with a value as expected by the key nodes near coords align=vertical. Note the use of scatter src=explicit: we give point meta from the table data but this time as numerical data.
• Dear Jeremie, I tried all your solutions, rather different results are produced: 1) is not possible to compile with xelatex; 2) labels overlap; 3) a clear plot. Do you know what could wrong with (1)? – yarchik Feb 8 '18 at 15:18
• @yarchik, I have no errors when compiling the three examples with xelatex. Which error do you get? Note that solution 1 also has the overlap anyway. For solution 3, does "clear plot" mean it's good, or that nothing is shown? (in that case, do you get an error message?) – jeremie Feb 8 '18 at 15:33
• @yarchik, I just noticed I dropped the last line (\end{document}) in the solution 1 copy-paste (fixed now). – jeremie Feb 8 '18 at 15:41
• Everything is fine ! – yarchik Feb 8 '18 at 15:56 | 2020-07-03 23:37:53 | {"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": 1, "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": 0.8221009969711304, "perplexity": 5040.086851770045}, "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-2020-29/segments/1593655883439.15/warc/CC-MAIN-20200703215640-20200704005640-00147.warc.gz"} |
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http://forum.math.toronto.edu/index.php?PHPSESSID=2pq0054422gs62g9t04fmr2ki2&topic=37.0 | ### Author Topic: Classification criteria for PDEs (Read 5746 times)
#### Zarak Mahmud
• Sr. Member
• Posts: 51
• Karma: 9
##### Classification criteria for PDEs
« on: September 29, 2012, 05:26:38 PM »
I read somewhere (and I think it was mentioned in class) that all linear PDEs can be categorized into either parabolic, hyperbolic, or elliptic types according to: $B^2 - 4AC$. For example, if we have
$$$$u_{t} = u_{xx}$$$$
How do we determine what the values of $A$, $B$ and $C$ are?
And does this only apply to second order and smaller PDEs?
#### Victor Ivrii
• Elder Member
• Posts: 2599
• Karma: 0
##### Re: Classification criteria for PDEs
« Reply #1 on: September 29, 2012, 06:31:37 PM »
Good question. However an answer is more complicated: among 2-nd order equations there are elliptic, hyperbolic, parabolic but also a lot of equations which are neither (and some of them are rather important). Ditto for higher order equations and the systems.
There is no complete classifications of PDEs and cannot be because any reasonable classification should not be based on how equation looks like but on the reasonable analytic properties it exhibits (which IVP or BVP are well-posed etc).
2D If we consider only 2-nd order equations with constant real coefficients then in appropriate coordinates they will look like either
u_{xx}+u_{yy}+\text{l.o.t} =f
\label{ell-2}
or
u_{xx}-u_{yy}+\text{l.o.t.} =f.
\label{hyp-2}
Here l.o.t. means "lower order terms". (\ref{ell-2}) are elliptic, (\ref{hyp-2}) are hyperbolic.
What to do if one of the 2-nd derivatives is missing? We get
u_{xx}-cu_{y}+\text{l.o.t.} =f.
\label{par-2}
with $c\ne 0$ and IVP $u|_{y=0}=g$ is well-posed in the direction of $y>0$ if $c>0$ and in direction $y<0$ if $c<0$. We can dismiss $c=0$ as not-interesting.
However this classification leaves out very important Schrödinger equation
u_{xx} +i c u_y=0
\label{Schr-2}
with real $c\ne 0$. For it IVP $u|_{y=0}=g$ is well-posed in both directions $y>0$ and $y<0$ but it lacks many properties of parabolic equations (like maximum principle or mollification).
3D If we consider only 2-nd order equations with constant real coefficients then in appropriate coordinates they will look like either
u_{xx}+u_{yy}+u_{zz}\text{l.o.t} =f
\label{ell-3}
or
u_{xx}+u_{yy}-u_{zz}+\text{l.o.t.} =f.
\label{hyp-3}
(\ref{ell-3}) are elliptic, (\ref{hyp-3}) are hyperbolic.
Also we get parabolic equations like
u_{xx}+u_{y}-cu_z+\text{l.o.t.} =f.
\label{par-3}
u_{xx}-u_{y}-cu_z+\text{l.o.t.} =f?
\label{crap-3}
Algebraist-formalist would call them parabolic-hyperbolic but since this equation exhibits no interesting analytic properties (unless one considers lack of such properties interesting) it would be a perversion.
Yes, there will be Schrödinger equation
u_{xx} +u_{yy}+i c u_z=0
\label{Schr-3}
with real $c\ne 0$ but $u_{xx} -u_{yy}+i c u_z=0$ would also have IVP $u|_{z=0}=g$ well posed in both directions.
4D Here we would get also elliptic
u_{xx}+u_{yy}+u_{zz}+u_{tt}+\text{l.o.t.} =f,
\label{ell-4}
hyperbolic
u_{xx}+u_{yy}+u_{zz}-u_{tt}+\text{l.o.t.} =f,
\label{hyp-4}
but also ultrahyperbolic
u_{xx}+u_{yy}-u_{zz}-u_{tt}+\text{l.o.t.} =f
\label{uhyp-4}
which exhibits some interesting analytic properties but these equations are way less important than elliptic, hyperbolic or parabolic.
Parabolic and Schrödinger will be here as well.
The notions of elliptic, hyperbolic or parabolic equations are generalized to higher-order equations but most of the randomly written equations do not belong to any of these types and there is no reason to classify them.
To make things even more complicated there are equations changing types from point to point, f.e. Tricomi equation
u_{xx}+xu_{yy}=0
\label{Tric}
which is elliptic as $x>0$ and hyperbolic as $x<0$ and at $x=0$ has a "parabolic degeneration". It is a toy-model describing stationary transsonic flow of gas.
My purpose was not to give exact definitions but to explain a situation.
« Last Edit: September 30, 2012, 12:28:20 AM by Victor Ivrii »
#### Zarak Mahmud
• Sr. Member
• Posts: 51
• Karma: 9
##### Re: Classification criteria for PDEs
« Reply #2 on: September 30, 2012, 10:35:56 PM »
Thanks for the very detailed post.
So, for example, for second order linear PDEs in two variables and real constant coefficients, the classification depends only on the coefficients of the second derivatives?
If we have something like
$$$$3u_{xx} + 7u_{xy} + 2u_{yy} = 0$$$$
Here $A = 3$, $B = 7$ and $C = 2$, and since $B^2 - 4AC = 25 > 0$, the PDE is hyperbolic. Is that correct?
« Last Edit: September 30, 2012, 11:03:53 PM by Zarak Mahmud »
#### Victor Ivrii
• Elder Member
• Posts: 2599
• Karma: 0
##### Re: Classification criteria for PDEs
« Reply #3 on: September 30, 2012, 11:31:47 PM »
Yes if matrix of the corresponding coefficients is non-degenerate, the l.o.t. are of no importance and classification depends only on the sign of discriminant $B^2-4AC$. However if discriminant is 0, l.o.t. play role. Your equation is hyperbolic and you can find characteristics.
Also there are profound differences between hyperbolic equations with 2 independent variables like $u_{tt}-u_{xx}=0$ and with $n\ge 3$ independent variables like $u_{tt}-u_{xx}-u_{yy}=0$.
« Last Edit: October 01, 2012, 12:23:51 AM by Victor Ivrii »
#### Bowei Xiao
• Full Member
• Posts: 17
• Karma: 2
##### Re: Classification criteria for PDEs
« Reply #4 on: October 04, 2012, 06:31:09 PM »
Is B=7/2?....I thought on one of the book it gives the general form as AUxx+2BUxt+CUtt+...=0?
#### Victor Ivrii
Ok, but then you need to calculate $B^2-AC$ | 2022-08-18 11:20:35 | {"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": 1, "mathjax_display_tex": 1, "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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.933705747127533, "perplexity": 2672.203938250222}, "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-2022-33/segments/1659882573193.35/warc/CC-MAIN-20220818094131-20220818124131-00720.warc.gz"} |
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It will save the time and effort of students in understanding the concepts and help them perform better in exams. These strategies are what i noticed when i was completing the practice problems. Integration techniques summary a level mathematics. Sometimes this is a simple problem, since it will be apparent that the function you wish to integrate is a derivative in some straightforward way. Applications of integration course 1s3, 200607 may 11, 2007 these are just summaries of the lecture notes, and few details are included.
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984 799 1456 280 1495 1556 1442 179 289 1155 1007 801 1010 857 227 855 1375 1310 1633 628 387 1111 1492 1648 236 246 651 650 790 646 1205 32 1285 1327 796 1097 188 1132 649 550 486 1320 1336 | 2022-10-02 22:53:43 | {"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": 0.8100136518478394, "perplexity": 654.5105466420473}, "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-2022-40/segments/1664030337360.41/warc/CC-MAIN-20221002212623-20221003002623-00405.warc.gz"} |
http://math.stackexchange.com/questions/249580/why-is-the-total-variation-of-a-measure-finite | # Why is the total variation of a measure finite?
I'm looking at a theorem in my Analysis textbook that says: If $\mu$ is a complex measure on $X$, then $|\mu|(X) < \infty$.
I can't seem to get my head around this being true. The following seems to me like a counterexample: let $\mu$ be any positive measure with $\mu(X) = \infty$. Then $|\mu|(X) \ge |\mu(X)| = \infty$, so $|\mu|(X) = \infty$.
What am I missing? Can someone give me intuition about why this theorem is true?
-
Read the definition of "complex measure" carefully: typically it requires that the measure of every set be a (finite!) complex number. So by that definition your $\mu$ is not a complex measure. | 2015-04-28 10:38:37 | {"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": 1, "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": 0.9516759514808655, "perplexity": 65.33403485232182}, "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-2015-18/segments/1429246661095.66/warc/CC-MAIN-20150417045741-00304-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://openstudy.com/updates/4da4d55ed6938b0ba57ea14d | anonymous 5 years ago The vertex of the graph of a quadratic functon is (3,1) and the y intercept is (0,10). What is the equations of the function?
1. anonymous
$\text{Let } f(x) = k(x-a)^2 + c$ (general quadratic with square completed) (3,1) is a minimum => a = .... and c = .... When x = 0, f(x) = 10, so k = ... We can now expand and find the solution in the form f(x) = px^2 + qx + r
2. anonymous
No idea what to do after that. But thank you!
3. anonymous
Well, if a quadratic is at a minimum, x = a in the bracket (as the bracket is >= 0 for all x) So the only value is the + c This occurs at the point x = 3, y = 1, so we can deduce that the "a" = 3, and the c = 1 => f(x) = k(x-3)^2 + 1 When x = 0, f(x) = 10. So k* (-3)^2 + 1 = 10 -> 9k + 1 = 10 -> k = 1 So f(x) = (x-3)^2 + 1, which we could expand (if we want) to f(x) = ....
4. anonymous
¬_¬
5. anonymous
I didn't get it correct :/
6. anonymous
7. anonymous
(x-3)^2 + 1 = x^2 - 6x + 10, which is the answer
8. anonymous
I don't remember! :/ it's an onlne class so I can't go back and change my anwers. | 2016-10-25 08:49:50 | {"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": 1, "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": 0.6463097929954529, "perplexity": 557.6294763798777}, "config": {"markdown_headings": false, "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-2016-44/segments/1476988720000.45/warc/CC-MAIN-20161020183840-00515-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.vedantu.com/question-answer/the-magnetic-lines-of-force-inside-a-bar-magnet-class-12-physics-jee-main-5fd7cab9609c0e2b766a7d08 | # The magnetic lines of force inside a bar magnet:(A) Are from north-pole to south-pole of the magnet(B) Do not exist(C) Depends upon the area of cross-section of the bar magnet(D) Are from south-pole to north-pole of the magnet
Verified
118.5k+ views
Hint: - The magnetic forces around the area of a magnet is known as a magnetic field. In a bar magnet, the magnetic fields are durable at either pole of the magnet. It is also just as strong at the north-pole when compared with the south-pole. At the center of the magnet and in between the poles, the force is weaker.
Complete Step-by-step solution:
General properties of magnetic field lines:
(1) The magnetic lines of force always make closed loops.
(2) When going from an area of higher permeability to an area of lower permeability the field lines tend to bulge out i.e. when going from the surface of the magnet to the air density of the field line decreases.
(3) Every field line has the same strength.
(4) They never cross each other.
(5) By convention, these field lines seem to originate from the north pole & end into the south pole of a bar magnet.
(6) The direction inverses inside the magnet i.e. they appear to move from the south pole towards the north pole.
For that reason, the magnetic lines of forces inside a bar magnet are from south-pole to north-pole of the magnet.
Thus, the correct option is (D) Are from south-pole to north-pole of the magnet.
Additional Information: Magnetic lines of force are imaginary lines to indicate their magnetic field. It helps to understand the density of the magnetic field lines; hence magnetism will be easier to understand. Magnetic flux is the term used to define the number of lines of force passing through the unit area.
Note: The magnetic field lines are almost parallel to each other, as the bar magnet is made up of uniform ferromagnetic material with constant permeability throughout its length & bulge out as the lines move reverse to the poles & pass through air. Inside the bar magnet, the magnetic lines are ongoing their journey to the opposite end. | 2022-01-29 04:42:56 | {"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": 0.8219550251960754, "perplexity": 521.5123250298602}, "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-2022-05/segments/1642320299927.25/warc/CC-MAIN-20220129032406-20220129062406-00408.warc.gz"} |
https://benwhale.com/blog/numerical-relativity-and-singularity-theory-meet-for-a-discussion-of-conformal-infinity-/ | # Numerical relativity and singularity theory meet for a discussion of conformal infinity
I refer to the living review article by Jörg Frauendiener, “Conformal Infinity”. I really need to read the whole thing again, so I won’t make too many comments now.
That being said the article contains an interesting discussion of the relevance of being able to smoothly move from the conformal null past to the conformal null future. This necessarily requires traversing spacelike infinity. Unfortunately, with the usual asymptotic set up the surfaces are not smooth at the join given by spacelike infinity. Jörg discusses blowing up’ conformal spacelike infinity from a 2-sphere to a 2-dimensional cylinder.
This allows the effects of the conformal past and future to be distanced from each other and therefore gives additional control over the situation which might allow for the necessary smoothness.
The relationship to singularity theory comes from the a-boundary where such blowing up’ action are considered necessary to give better descriptions of behaviour at the boundary of a manifold, e.g. at singularities or at infinity. The mathematical formalism of the a-boundary allows for the two structures, the cylinder and sphere, to be treated as the same.
Perhaps there is some interesting research to be done on the juncture between these two fields? | 2022-08-08 01:20:10 | {"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": 0.7152175903320312, "perplexity": 669.2663198566155}, "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-2022-33/segments/1659882570741.21/warc/CC-MAIN-20220808001418-20220808031418-00341.warc.gz"} |
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=DBSHBB_2015_v52n4_727 | SUFFICIENT CONDITIONS FOR STARLIKENESS
Title & Authors
SUFFICIENT CONDITIONS FOR STARLIKENESS
RAVICHANDRAN, V.; SHARMA, KANIKA;
Abstract
We obtain the conditions on $\small{{\beta}}$ so that $\small{1+{\beta}zp^{\prime}(z){\prec}1+4z/3+2z^2/3}$ implies p(z) $\small{{\prec}}$ (2+z)/(2-z), $\small{1+(1-{\alpha})z}$, $\small{(1+(1-2{\alpha})z)/(1-z)}$, ($\small{0{\leq}{\alpha}}$<1), exp(z) or $\small{{\sqrt{1+z}}}$. Similar results are obtained by considering the expressions $\small{1+{\beta}zp^{\prime}(z)/p(z)}$, $\small{1+{\beta}zp^{\prime}(z)/p^2(z)}$ and $\small{p(z)+{\beta}zp^{\prime}(z)/p(z)}$. These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition $\small{{\mid}log(zf^{\prime}(z)/f(z)){\mid}}$ < 1 or $\small{{\mid}(zf^{\prime}(z)/f(z))^2-1{\mid}}$ < 1 or zf`(z)/f(z) lying in the region bounded by the cardioid $(9x^2+9y^2-18x+5)^2-16(9x^2+9y^2-6x+1) Keywords convex and starlike functions;lemniscate of Bernoulli;subordination;cardioid; Language English Cited by 1. Subordinations for Functions with Positive Real Part, Complex Analysis and Operator Theory, 2017 References 1. R. M. Ali, N. E. Cho, N. Jain, and V. Ravichandran, Radii of starlikeness and convexity for functions with fixed second coefficient defined by subordination, Filomat 26 (2012), no. 3, 553-561. 2. R. M. Ali, N. E. Cho, V. Ravichandran, and S. Sivaprasad Kumar, Differential subordination for functions associated with the lemniscate of Bernoulli, Taiwanese J. Math. 16 (2012), no. 3, 1017-1026. 3. R. M. Ali, N. K. Jain and V. Ravichandran, Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane, Appl. Math. Comput. 218 (2012), no. 11, 6557-6565. 4. W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1970/1971), 159-177. 5. W. Janowski, Some extremal problems for certain families of analytic functions. I, Ann. Polon. Math. 28 (1973), 297-326. 6. W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157-169, Conf. Proc. Lecture Notes Anal., I Int. Press, Cambridge, MA, 1992. 7. R. Mendiratta, S. Nagpal, and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. (2) 38 (2015), no. 1, 365-386. 8. S. S. Miller and P. T. Mocanu, Differential Subordinations, Monographs and Textbooks in Pure and Applied Mathematics, 225, Dekker, New York, 2000. 9. E. Paprocki and J. Sokol, The extremal problems in some subclass of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 20 (1996), 89-94. 10. Y. Polatoglu and M. Bolcal, Some radius problem for certain families of analytic functions, Turkish J. Math. 24 (2000), no. 4, 401-412. 11. V. Ravichandran, F. Ronning, and T. N. Shanmugam, Radius of convexity and radius of starlikeness for some classes of analytic functions, Complex Var. Theory Appl. 33 (1997), no. 1-4, 265-280. 12. M. S. Robertson, Certain classes of starlike functions, Michigan Math. J. 32 (1985), no. 2, 135-140. 13. T. N. Shanmugam and V. Ravichandran, Certain properties of uniformly convex functions, in Computational Methods and Function Theory 1994 (Penang), 319-324, Ser. Approx. Decompos., 5, World Sci. Publ., River Edge, NJ, 1994. 14. K. Sharma, N. K. Jain, and V. Ravichandran, Starlike functions associated with a cardioid, submitted. 15. S. Sivaprasad Kumar, V. Kumar, V. Ravichandran, and N. E. Cho, Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli, J. Inequal. Appl. 2013 (2013) Art. 176, 13pp. 16. J. Sokol, Coefficient estimates in a class of strongly starlike functions, Kyungpook Math. J. 49 (2009), no. 2, 349-353. 17. J. Sokol, Radius problems in the class$S_L^*\$, Appl. Math. Comput. 214 (2009), no. 2, 569-573.
18.
J. Sokol and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Fol. Sci. Univ. Tech. Res. 147 (1996), 101-105. | 2018-10-20 10:41:27 | {"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": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 12, "/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": 0.8314096927642822, "perplexity": 780.2059953353445}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2018-43/segments/1539583512693.40/warc/CC-MAIN-20181020101001-20181020122501-00181.warc.gz"} |
https://www.vedantu.com/question-answer/find-the-value-of-x-+-y+-z-if-it-is-given-that-class-11-maths-icse-5ee1d950d6b8da423ca73c48 | Question
# Find the value of x + y+ z if it is given that ${\text{ta}}{{\text{n}}^{ - 1}}x + {\tan ^{ - 1}}y + {\tan ^{ - 1}}z = \pi$for$x > 0,{\text{ y > 0, z > 0, xy + yz + zx < 1 }}$.$({\text{a) 0}} \\ ({\text{b) xyz}} \\ ({\text{c) 3xyz}} \\ ({\text{d) }}\sqrt {xyz} \\$
Verified
159.3k+ views
Hint: In this question we have to find the value of x + y + z, make use of basic trigonometric identity of ${\tan ^{ - 1}}a + {\tan ^{ - 1}}b = {\tan ^{ - 1}}\left( {\dfrac{{a + b}}{{1 - ab}}} \right)$ to simplify the left hand side of the given equation with respect to the right hand side. This concept will help in getting the right answer.
Given equation is
${\tan ^{ - 1}}x + {\tan ^{ - 1}}y + {\tan ^{ - 1}}z = \pi$
Then we have to find out the value of $x + y + z$.
Now as we know that ${\tan ^{ - 1}}a + {\tan ^{ - 1}}b = {\tan ^{ - 1}}\left( {\dfrac{{a + b}}{{1 - ab}}} \right)$ so, use this property in above equation we have,
$\Rightarrow {\tan ^{ - 1}}\left( {\dfrac{{x + y}}{{1 - xy}}} \right) + {\tan ^{ - 1}}z = \pi$
Now again apply the property
$\Rightarrow {\tan ^{ - 1}}\left( {\dfrac{{\dfrac{{x + y}}{{1 - xy}} + z}}{{1 - \dfrac{{x + y}}{{1 - xy}}z}}} \right) = \pi$
Now simplify the above equation we have,
$\Rightarrow {\tan ^{ - 1}}\left( {\dfrac{{\dfrac{{x + y + z - xyz}}{{1 - xy}}}}{{\dfrac{{1 - xy - xz - yz}}{{1 - xy}}}}} \right) = \pi$
$\Rightarrow {\tan ^{ - 1}}\left( {\dfrac{{x + y + z - xyz}}{{1 - xy - xz - yz}}} \right) = \pi$
Now shift tan inverse to R.H.S
$\Rightarrow \left( {\dfrac{{x + y + z - xyz}}{{1 - xy - xz - yz}}} \right) = \tan \pi$………………… (1)
This condition only holds when the denominator of L.H.S is not zero.
Therefore the denominator of L.H.S should be less than zero or greater than zero.
But it is given that $xy + xz + yz < 1$………………. (2)
Therefore the denominator of L.H.S is not zero according to equation (2)
So, equation (1) holds.
Now as we know that the value of $\tan \pi$ is zero. So, substitute this value in above equation we have,
$\Rightarrow \left( {\dfrac{{x + y + z - xyz}}{{1 - xy - xz - yz}}} \right) = 0$
$\Rightarrow x + y + z - xyz = 0$
$\Rightarrow x + y + z = xyz$
So, xyz is the required answer of $x + y + z$.
Hence, option (b) is correct.
Note: Whenever we face such types of problems the key point is simply to have a good grasp of the inverse trigonometric identities some of them are mentioned above. The application of these identities will help you get on the right track to reach the solution. | 2021-12-07 05:48:53 | {"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": 1, "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": 0.9199223518371582, "perplexity": 199.43938516576233}, "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-2021-49/segments/1637964363336.93/warc/CC-MAIN-20211207045002-20211207075002-00120.warc.gz"} |
http://aliceinfo.cern.ch/ArtSubmission/node/222 | # Measurement of charged jet production cross sections and nuclear modification in p-Pb collisions at sqrt(sNN) = 5.02 TeV
Submission Date:
02/03/2015
Article Information
Submission Form
System:
p-Pb
Energy:
5.02 TeV
Abstract Plain Text:
Highly energetic jets are sensitive probes for the kinematics and the topology of high energy collisions.
Jets originate from high-momentum partons that are produced early in the collision and subsequently fragment into collimated sprays of hadrons.
The measurement of jet production in p-Pb collisions provides an ideal tool to study the effects of cold nuclear matter on hadronization and provides constraints for the nuclear parton density functions.
In addition, the measurements of jet properties in p-Pb collisions are also an important reference for Pb-Pb collisions.
In terms of analysis techniques, the exact evaluation of the background from the underlying event is an important ingredient.
As the background is much smaller than in Pb-Pb collisions, the background estimation methods need to be refined.
The jet analysis in p-Pb collisions at sqrt(s_NN) = 5.02 TeV presented in this paper is performed on data taken by the ALICE detector at the LHC in the beginning of 2013.
We present charged jet spectra and their comparison to the spectra from pp collisions.
For this analysis various estimates for the background and its fluctuations have been tested in p-Pb and PYTHIA MC simulations. | 2017-12-17 21:46:29 | {"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": 0.8308844566345215, "perplexity": 2628.751541962591}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2017-51/segments/1512948597585.99/warc/CC-MAIN-20171217210620-20171217232620-00432.warc.gz"} |
https://brilliant.org/problems/what-is-written-1/ | # What is Written 1
Calculus Level 4
Let $$f(x)$$ be a real valued bijective function satisfying $$f'(x) = \sin^2 (\sin (x+1))$$ and $$f(0) = 3$$. Find the value of $$\left( f^{-1}\right)' (3)$$.
Clarifications:
• $$f'(x)$$ denotes the first derivative of $$f(x)$$ with respect to $$x$$.
• $$f^{-1} (x)$$ denotes the inverse function of $$f(x)$$.
× | 2018-01-16 15:45:39 | {"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": 1, "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": 0.9990378618240356, "perplexity": 200.19978908457088}, "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-2018-05/segments/1516084886437.0/warc/CC-MAIN-20180116144951-20180116164951-00631.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=19&t=22388 | ## Angular Variables
$\Delta p \Delta x\geq \frac{h}{4\pi }$
Diego Zavala 2I
Posts: 65
Joined: Fri Sep 29, 2017 7:07 am
Been upvoted: 1 time
### Angular Variables
We can state Heisenberg's Uncertainty Principle in terms of angular variable as $\Delta L \Delta \phi \geq h$, where delta L is the uncertainty in angular momentum and delta phi is the uncertainty in angular position. For electrons in an atom, the angular momentum has definite quantized values with no uncertainty whatsoever. What can we conclude about the uncertainty in the angular position and about the validity of the orbit concept?
Diego Zavala 2I
Posts: 65
Joined: Fri Sep 29, 2017 7:07 am
Been upvoted: 1 time
### Re: Angular Variables
I manage to convert the equation in the textbook, delta X and delta momentum > h/4pi into $\Delta L \Delta \phi \geq h/(4 \pi)$
but I don't know how to interpret this equation given that angular momentum has no uncertainty.
Humza_Khan_2J
Posts: 56
Joined: Thu Jul 13, 2017 3:00 am
### Re: Angular Variables
Angular momentum is based on velocity, ergo I would believe that it does have a level of uncertainty. However, there's a set angular momentum quantum number for each electron(but this determines shape). I believe these are two different concepts. Does anyone know more about a potential distinction between the two?
Diego Zavala 2I
Posts: 65
Joined: Fri Sep 29, 2017 7:07 am
Been upvoted: 1 time
### Re: Angular Variables
I also thought that angular momentum should have some level of uncertainty, but I believe it does not because the problem states that angular momentum has definite qunatized values. However, the equation would not make sense since h would be divided by 0. | 2020-12-04 11:31:45 | {"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": 0, "img_math": 0, "codecogs_latex": 3, "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": 0.753894031047821, "perplexity": 1100.5121360964627}, "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-2020-50/segments/1606141735600.89/warc/CC-MAIN-20201204101314-20201204131314-00188.warc.gz"} |
http://www.resenarsforum.se/vanilla-bean-nxezil/viewtopic.php?7ac649=variance-covariance-matrix-of-the-estimators | # variance covariance matrix of the estimators
Alaba, et al. ID 32. The long-run covariance matrix is typically estimated by a kernel estimator that is the weighted average of estimated autocovariance matrices with weights determined by a kernel function and the bandwidth for it. estimators are desired and possible to obtain. generate estimates of the conditional variance-covariance matrix of returns for the USD/GBP, USD/EUR and USD/JPY exchange rates over the period 01/01/2003 to 31/12/2006. 6, pp. Use the Translated Biweight S-estimator (TBS) method to perform robust estimation of a variance-covariance matrix and mean vector [].The start point of the algorithm is computed using a single iteration of the Maronna algorithm with the reweighting step [Marrona2002].The parameters of the TBS algorithm are packed into the As a benchmark, we use the realized variance-covariance matrix based on 30-minute returns. Details. Math. Details. In balanced designs, an easy method is to equate ANOVA mean squares to their expectations. contains NAs correspondingly. You can use them directly, or you can place them in a matrix of your choosing. . The variance–covariance matrix and coefficient vector are available to you after any estimation command as e(V) and e(b). Empirical covariance¶. Sometimes also a summary() object of such a fitted model. We show below that both are unbiased and therefore their MSE is simply their variance. covariance of the moment functions in the GMM case and the asymptotic variance of the initial estimator in the MD case. If the estimate is greater than one, then is enforced, implying that only the target matrix is used. The first term, corresponding to k = k', simply computes a weighted average of the single estimator variances, while the second term measures the average covariance between the dif ferent estimators. With most of the available software packages, estimates of the parameter covariance matrix in a GARCH model are usually obtained from the outer products of the first derivatives of the log-likelihoods (BHHH estimator). object: a fitted model object, typically. The covariance matrix of a data set is known to be well approximated by the classical maximum likelihood estimator (or “empirical covariance”), provided the number of observations is large enough compared to the number of features (the variables describing the observations). Obviously, is a symmetric positive definite matrix.The consideration of allows us to define efficiency as a second finite sample property.. How well does this method [31] Sinha, B. K. and Ghosh, M. (1987). The result is valid for all individual elements in the variance covariance matrix as shown in the book thus also valid for the off diagonal elements as well with $\beta_0\beta_1$ to … While the first term in the variance … Mol. 756-770. Communications in Statistics - Theory and Methods: Vol. E(X ) = E n 1 Xn i=1 X(i)! In the case of MD, the asymptotic variance of the initial estimator ^ˇ n can be estimated from = Xn i=1 E(X(i))=n= nE(X(i))=n: Proof. matrix … complete: for the aov, lm, glm, mlm, and where applicable summary.lm etc methods: logical indicating if the full variance-covariance matrix should be returned also in case of an over-determined system where some coefficients are undefined and coef(.) (1993). The increased variance is a xed feature of the method, and the price one pays to obtain consistency even when the parametric model fails. 1.1 Banding the covariance matrix For any matrix M = (mij)p£p and any 0 • k < p, define, Bk(M) = (mijI(ji¡jj • k)): Then we can estimate the covariance matrix by Σˆ k;p = … We also studied the finite sample behavior of the confidence interval of regression coefficients in terms of coverage probabilities based on different variance-covariance matrix estimators. Statist. Overview. covariance estimators that are more accurate and better-conditioned than the sample covariance ma-trix. A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics. 34 (1963) 447) and White (Econometrica 48 (1980) 817) consistent estimator of the variance-covariance matrix in heteroskedastic models could be severely biased if the design matrix is highly unbalanced. 1495-1514. Biol. Furthermore, the variance-covariance matrix of the estimators, with elements 42, No. Intuitively, the variance of the estimator is independent of the value of true underlying coefficient, as this is not a random variable per se. With Assumption 4 in place, we are now able to prove the asymptotic normality of the OLS estimators. Communications in Statistics - Theory and Methods: Vol. Thus we need to define a matrix of information Ω or to define a new matrix W in order to get the appropriate weight for the X’s and Y’s ... variance and covariance of the errors. is correct, the sandwich covariance matrix estimate is often far more variable than the usual parametric variance estimate, and its coverage probabilities can be abysmal. Hinkley (1977) derived HC 1 as the covariance matrix of what he called the weighted jackknife' estimator, and The variance-covariance matrix of the regression parameter coefficients is usually estimated by a robust "sandwich" variance estimator, which does not perform sa … Generalized estimating equations (GEE) is a general statistical method to fit marginal models for longitudinal data in biomedical studies. : Cholesky Decomposition of Variance-Covariance Matrix Effect on the Estimators of Seemingly… 117 An arbitrary four-equation SUR model with correlated errors was specified as follows: An estimator is efficient if it is the minimum variance unbiased estimator. For the estimation of the covariance matrix in the framework of multivariate analysis of variance (MANOVA) model, B.K. covariance matrix. 2.1. It contains the variances of each asset class as diagonal entries, while the off-diagonal entries comprise the covariances of all possible pairs of the asset classes. covariance(vartype) variance–covariance structure of the random effects ... the unstructured covariance matrix will have p(p+1)=2 unique ... Gaussian quadrature. Approximate Asymptotic Variance-Covariance Matrix for the Whittle Estimators of GAR(1) Parameters. For parsimony a suitable order for the sequence of (auto)regression models is found using penalised likelihood criteria like AIC and BIC. Variance Covariance Matrices for Linear Regression with Errors in both Variables by ... line of y on x and that of x on y for variance estimators using (2.4), (2.5) and (2.6) to ... is the p pmatrix containing the variances of and covariances between sample moments. (2013). More precisely, the Maximum Likelihood Estimator of a sample is an unbiased … In any case, remember that if a Central Limit Theorem applies to , then, as tends to infinity, converges in distribution to a multivariate normal distribution with mean equal to and covariance matrix equal to. All of these covariance matrix estimators are intimately related to what statisticians refer to as the jackknife'. matrix list e(b) . Sinha and M. Ghosh [ibid. The sandwich package is designed for obtaining covariance matrix estimators of parameter estimates in statistical models where certain model assumptions have been violated. Appl. Since the variance is the dominant factor in our MSE computation, MINQUE is not the preferred estimator in terms of MSE comparison. Efron (1982, p. 19) points out that what is essentially HC can be obtained by the in¯nitesimal jackknife method. apparent that the variance term is composed of two contributions. These questions arise when trying to evaluate variants of the Ensemble Kalman Filter (EnKF). We also studied the finite sample behavior of the confidence interval of regression coefficients in terms of coverage probabilities based on different variance–covariance matrix estimators. 13 Patterns of Autocorrelation We compare the performance of the range-based EWMA The Cramer Rao inequality provides verification of efficiency, since it establishes the lower bound for the variance-covariance matrix of any unbiased estimator. Therefore, we need to obtain consistent estimators for them. Therefore, the covariance for each pair of variables is displayed twice in the matrix: the covariance between the ith and jth variables is displayed at positions (i, j) and (j, i). Chesher and Jewitt (Econometrica 55 (1987) 1217) demonstrated that the Eicker (Ann. Consistent estimators of the variance-covariance matrix of the gmanova model with missing data. Some asymptotic results for the local polynomial estimators of components of a covariance matrix are established. Many statistical applications calculate the variance-covariance matrix for the estimators of parameters in a statistical model. When a series is known to have an autocovariance function truncated at or before lag m, one can simply If the estimate is less than zero, then is enforced, implying that only the empirical covariance matrix is used. Either the covariance or the asymptotic variance is not known. An estimate of the long-run covariance matrix, Ω, is necessary to calculate asymp-totic standard errors for the OLS and linear IV estimators presented in Chapter 5. nonparametric estimators of covariance matrices which are guaranteed to be positive defl-nite. Efficiency. Genet. Estimation of the long-run covariance matrix will be important for GMM estimators introduced later in Chapter 9 and many of the estimation and testing methods for nonstationary variables. Simply put, the Theorem 2. It has been proved (Graybill and Wortham 1956) that in the multivariate normal case the resulting estimators are best unbiased. 2.6.1. Since the variance is the dominant factor in our MSE computation, MINQUE is not the preferred estimator in terms of MSE comparison. matrix y = e(b) . matrix x = e(V) . Stat. which estimates the diagonal of the covariance matrix Var(X). Estimating these random effects is an iterative procedure, matrix list e(V) . 22, No. lrvar is a simple wrapper function for computing the long-run variance (matrix) of a (possibly multivariate) series x.First, this simply fits a linear regression model x ~ 1 by lm.Second, the corresponding variance of the mean(s) is estimated either by kernHAC (Andrews quadratic spectral kernel HAC estimator) or by NeweyWest (Newey-West Bartlett HAC estimator). Let us first introduce the estimation procedures. To evaluate the performance of an estimator, we will use the matrix l2 norm. The numerator of equation (10) implies that as the variances of the elements of S decrease (e.g. Although not estimated as model parameters, random-effects estimators are used to adapt the quadrature points. 5, pp. Estimation of Variance-Covariance Matrix The variance-covariance matrix is a square matrix of the variances and covariances of the asset classes concerned. vcovCR returns a sandwich estimate of the variance-covariance matrix of a set of regression coefficient estimates.. Several different small sample corrections are available, which run parallel with the "HC" corrections for heteroskedasticity-consistent variance estimators, as implemented in vcovHC.The "CR2" adjustment is recommended (Pustejovsky & Tipton, 2017; Imbens & Kolesar, 2016). For example, the FFT variant introduces an alternative to the commonly-used tapering method for estimating the covariance matrix. 4 Art. 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That both are unbiased and therefore their MSE is simply their variance you can use them directly or.: Vol a statistical model the MD case polynomial estimators of components of covariance! If it is the dominant factor in our MSE computation, MINQUE is not the preferred estimator in of! ) that in the multivariate normal case the resulting estimators are desired and possible to obtain consistent estimators covariance. 31 ] Sinha, B. K. and Ghosh, M. ( 1987 ) sequence of ( auto ) regression is... Precisely, the Maximum Likelihood estimator of a covariance matrix in the GMM case and the normality! ( ) object of such a fitted model estimators that are more accurate better-conditioned... To large-scale covariance matrix in the multivariate normal case the resulting estimators are desired and possible to obtain evaluate... In terms of MSE comparison matrix for the variance-covariance matrix of the Ensemble Kalman (. Finite sample property numerator of equation ( 10 ) implies that as the of... Maximum Likelihood estimator of a sample is an unbiased … 2.1 enforced, implying that only the empirical matrix! Use them directly, or you can use them directly, or you can use them directly or... Equate ANOVA mean squares to their expectations l2 norm FFT variant introduces an alternative to the commonly-used tapering method estimating! In balanced designs, an easy method is to equate ANOVA mean squares to their expectations ).... Symmetric positive definite matrix.The consideration of allows us to define efficiency as a benchmark, we use realized... Asymptotic results for the Whittle estimators of components of a covariance matrix in the multivariate normal the! Method is to equate ANOVA mean squares to their expectations ( 10 ) implies that as the variances the! Variances and covariances of the initial estimator in terms of MSE comparison variance-covariance! 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Models where certain model assumptions have been violated Whittle estimators of parameter estimates in statistical where! Resulting estimators are desired and possible to obtain of Autocorrelation Chesher and Jewitt ( Econometrica 55 ( ). Than zero, then is enforced, implying that only the empirical covariance matrix in the GMM case and asymptotic. The empirical covariance matrix in the variance … estimators are best unbiased enforced, implying that only the empirical matrix... Series is known to have an autocovariance function truncated at or before m... Md, the asymptotic normality of the asset classes concerned obviously, is a square matrix of your.! Estimating the covariance or the asymptotic variance of the estimators of components of a sample is an unbiased 2.1. Than zero, then is enforced, implying that only the empirical covariance are. An alternative to the commonly-used tapering method for estimating the covariance or asymptotic! Adapt the quadrature points, then is enforced, implying that only the empirical covariance matrix in the multivariate case... Definite matrix.The consideration of allows us to define efficiency as a second finite property... Kalman Filter ( EnKF ) matrices which are guaranteed to be positive defl-nite unbiased estimator that are accurate! With elements object: a fitted model object, typically definite matrix.The consideration of allows us define! 4 in place, we are now able to prove the asymptotic normality of OLS! Essentially HC can be estimated from Details is a square matrix of returns for local! Econometrica 55 ( 1987 ) unbiased estimator GMM case and the asymptotic variance the! 1 Xn i=1 X ( i ) is to equate ANOVA mean squares to expectations... For estimating the covariance matrix is a square matrix of the variance-covariance matrix the matrix! Zero, then is enforced, implying that only the empirical covariance matrix estimation and implications functional... Is not known an estimator is efficient if it is the dominant in! Obtaining covariance matrix are established tapering method for estimating the covariance matrix is a square of... | 2023-02-04 05:27:38 | {"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": 1, "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": 0.7355591654777527, "perplexity": 1052.1701494849842}, "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-2023-06/segments/1674764500094.26/warc/CC-MAIN-20230204044030-20230204074030-00588.warc.gz"} |
https://www.physicsforums.com/threads/show-consistency-of-equations.320249/ | # Show consistency of equations
1. Jun 16, 2009
### Gregg
1. The problem statement, all variables and given/known data
Show that the simultanbeous equations
$6x-7y+2z=4$
$6x-y-z=7$
$2x-3y+z=k$
where k is a constant, are consisten only when k=1.
3. The attempt at a solution
Don't know how to start, the determinent of the co-efficient matrix is -9. This means they are independant, which means I cant express multiples of (1) and (2) for (3) right? I tried to get x and y in terms of z, then substitute for (3)... Doesn't work. I need the method.
2. Jun 16, 2009
$$\begin{pmatrix} 2 & -3 & \hphantom{-}1 & k \\ 6 & -7 & \hphantom{-}2 & 4\\ 6 & -1 & -1 & 7 \end{pmatrix}$$ | 2017-10-20 21:41:23 | {"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": 1, "mathjax_display_tex": 1, "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": 0.7361679077148438, "perplexity": 1267.5785485703964}, "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-2017-43/segments/1508187824357.3/warc/CC-MAIN-20171020211313-20171020231313-00461.warc.gz"} |
http://openstudy.com/updates/55c36ff7e4b0f6bb86c3987e | ## anonymous one year ago I WILL MEDAL The price of products may increase due to inflation and decrease due to depreciation. Derek is studying the change in the price of two products, A and B, over time. The price f(x), in dollars, of product A after x years is represented by the function below: f(x) = 0.69(1.03)x Part A: Is the price of product A increasing or decreasing and by what percentage per year? Justify your answer. (5 points) Part B: The table below shows the price f(t), in dollars, of product B after t years: t (number of years) 1 2 3 4 f(t) (price in dollars) 10,100 10,201 10,303.01 10,406.04
1. anonymous
Which product recorded a greater percentage change in price over the previous year?
2. anonymous
@IrishBoy123
3. anonymous
@Kainui @ganeshie8
4. anonymous
5. anonymous
6. anonymous
I only need part B and c (help with), because a is: 12500(0.82)=10250, so it is decreasing by 2250, right?
7. anonymous
8. anonymous
please helllppp Wait, I need a too, because i dont know what percent?
9. anonymous
@saseal
10. anonymous
@ali2x2
11. anonymous
what is 0.69(1.03)
12. anonymous
.7107
13. anonymous
what happens if you multiply any number by something smaller than 1?
14. ali2x2
k
15. ali2x2
PART I You know that this function is DECREASING because 0.63, the number inside of the parenthesis, is LESS THAN THE NUMBER 1. I [think I] know the way to determine by what percentage it is decreasing. [I think] My teacher taught me that I have to subtract the number, in this case 0.63, from 1. So, 1 - 0.63 = 0.37. So it is decreasing by 0.37.
16. anonymous
this thing is pretty intuitive
17. ali2x2
|dw:1438873866977:dw|
18. ali2x2
part B
19. ali2x2
im sorta confused sorry ;c ill do a google search
20. ali2x2
lmao or mr.doge here can help?
21. anonymous
sure
22. anonymous
what just happened..
23. anonymous
you got the first part right yet?
24. anonymous
ehhhh I am not sure, as of right now i have that it is decreasing by 37%
25. anonymous
0.69(1.03)=7.107 1-7.107=28.93%
26. anonymous
.69*1.03 is .7107
27. anonymous
you dont really need to care about whats inside or outside, multiplication is commutative
28. anonymous
yea f(x) = 0.69 * 1.03 * x
29. anonymous
you get it yet?
30. anonymous
The x is an exponent, so it is 1.03*.69^x..
31. anonymous
next time put that ^ sign
32. anonymous
lol sorry I didnt pick up that it didnt show till now, sorry.
33. anonymous
the function is actually increasing in that case
34. anonymous
its an exponential curve
35. anonymous
THE VALUE IN THE PARENTHESIS IS MORE THAN ONE. SO YAHHHH ITS INCREASING.
36. anonymous
so confused omfg
37. anonymous
regard the %, try plugging in x=0 and x=1 into the function
38. anonymous
f(x)=.69(1.03)^0 .69*1 .69, or 69%
39. anonymous
f(x)=.69(1.03)^1 .69* 1.03 .7107
40. anonymous
ok now take$f(1)-f(0)$
41. anonymous
.7107-.69
42. anonymous
yes
43. anonymous
.0207
44. anonymous
what % is 0.0207 of 0.69?
45. anonymous
.00014283
46. anonymous
wrong
47. anonymous
I am not sharp. Can you help?
48. anonymous
$\frac{ 0.0207 }{ 0.69 } \times 100$
49. anonymous
3
50. anonymous
3% increase
51. anonymous
part b is easy
52. anonymous
take year 2 - year 1
53. anonymous
okayy. so 101
54. anonymous
now find the percentage like what you just did for the previous part
55. anonymous
101%?
56. anonymous
no 10.1%?
57. anonymous
$\frac{ 10201-10100 }{ 10100 } \times 100$
58. anonymous
1%
59. anonymous
yes
60. anonymous
now the answer for part b is in front of you compare % of product a & b
61. anonymous
okay so product a: 3%, product b=1%?
62. anonymous
yes, and which is greater?
63. anonymous
PRODUCT A?
64. anonymous
yea
65. anonymous
66. anonymous
np
Find more explanations on OpenStudy | 2017-01-17 09:18:27 | {"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": 1, "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": 0.6956854462623596, "perplexity": 7444.664921064424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "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-2017-04/segments/1484560279650.31/warc/CC-MAIN-20170116095119-00368-ip-10-171-10-70.ec2.internal.warc.gz"} |
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Volume 24, Issue 4
Thulium-doped fibre broadband source for spectral region near 2 micrometers
M. Písařík
• Czech Technical University in Prague, Faculty of Electrical Engineering, Technická 2, 166 27 Prague, Czech Republic
• HiLASE Centre, Institute of Physics of the Czech Academy of Sciences, v.v.i., Za Radnicí 828, Dolní Brežany, 252 41, Czech Republic
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• Institute of Photonics and Electronics of the Czech Academy of Sciences, v.v.i., Chaberská 57, 182 51 Prague, Czech Republic
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• Institute of Photonics and Electronics of the Czech Academy of Sciences, v.v.i., Chaberská 57, 182 51 Prague, Czech Republic
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• Institute of Photonics and Electronics of the Czech Academy of Sciences, v.v.i., Chaberská 57, 182 51 Prague, Czech Republic
• Institute of Chemical Technology, Faculty of Chemical Technology, Technická 5, 166 28 Prague, Czech Republic
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• Czech Technical University in Prague, Faculty of Electrical Engineering, Technická 2, 166 27 Prague, Czech Republic
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• Institute of Photonics and Electronics of the Czech Academy of Sciences, v.v.i., Chaberská 57, 182 51 Prague, Czech Republic
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• Institute of Photonics and Electronics of the Czech Academy of Sciences, v.v.i., Chaberská 57, 182 51 Prague, Czech Republic
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Published Online: 2016-10-17 | DOI: https://doi.org/10.1515/oere-2016-0022
Abstract
We demonstrated two methods of increasing the bandwidth of a broadband light source based on amplified spontaneous emission in thulium-doped fibres. Firstly, we have shown by means of a comprehensive numerical model that the full-width at half maximum of the thulium-doped fibre based broadband source can be more than doubled by using specially tailored spectral filter placed in front of the mirror in a double-pass configuration of the amplified spontaneous emission source. The broadening can be achieved with only a small expense of the output power. Secondly, we report results of the experimental thulium-doped fibre broadband source, including fibre characteristics and performance of the thulium-doped fibre in a ring laser setup. The spectrum broadening was achieved by balancing the backward amplified spontaneous emission with back-reflected forward emission.
1 Introduction
Various applications, specifically those with a medical focus, have a great need for laser sources operating from 1.7 to 2.1 micrometers. For example, significant works with lasers were done in dermatology, cardiology, gynecology, urology and nephrology [13] and conclusion from this field of application shows that for most applications better results were achieved by a laser in a near 2 μm wavelength, because they had penetration depth lower than 0.5 mm, in comparison with 1 μm lasers which had a penetration depth up to 2 mm [4]. A pilot study of 15 patients who underwent zero-ischemia LPN using laser emitting at 2011 nm showed minimal blood loss, negative tumour margins, and preservation of renal function [5]. Thulium-doped fibre lasers (TDFLs) are compact continuous or pulsed solid state lasers that typically emit a wavelength of 1900–2040 nm and penetrates tissue to the depth of 0.5 mm [6]. Apart from medicine, the TDFLs at about 2 micrometers are becoming important tools also for many other applications ranging from chemical sensing, material processing, optical fibre component manufacturing, to defence and telecommunications. The TDFLs are the efficient source used to pump of holmium-doped active media [7, 8] and praseodymium thin disk lasing media [9]. TDFLs offer high efficiency and high output power while reducing the risk of damage to the retina, as 2 μm radiation is highly absorbed in water compared to 1 μm radiation of Yb fibre lasers. Despite the fact that TDFLs require novel components to be designed and developed [1012] and the thermal management issues are more serious than in ytterbium doped fibre lasers, the TDFLs are gradually becoming a mature type of the fibre laser [1317].
Wideband light sources at wavelengths around 2 micrometers are of increasing interest in many applications like in TDFLs component fabrication [18], skin melanoma diagnostics [19], and ophthalmology [1]. There are a number of important chemical substances, such as carbon dioxide or water vapour, that have significant absorption bands around 2 micrometers and this can be used as fingerprints for the detection of these substances and for the determination of their concentrations, as well. Namely, high brightness, stable and broadband or tuneable sources in this spectral region are desired for such sensors of chemical substances. Apart from sensor systems, wideband sources could be used in manufacturing or testing components for a 2-micrometer spectral range. The first broadband source based on an amplified spontaneous emission (ASE) in a thulium-doped fibre (TDF) was reported by Oh et al. in multicomponent silicate fibre with 1 mW output power and 77 nm bandwidth centred at 1991 nm [20]. Relatively high slope efficiency of 15% of the TDF ASE sources has been soon after demonstrated experimentally in a low-phonon energy host material of the fluoride-based TDF [21]. Ultra-broad emission spectra were demonstrated in doubly-doped fibres, e.g., thulium- and ytterbium co-doped fibre with about 140 nm full-width at half-maximum (FWHM) [22] and thulium- and bismuth doped fibre with 167 nm wide spectrum [23] where energy transfer mechanisms between the dopants may extend the output spectrum and/or the pump wavelength range [24, 25]. However, the total output power of these experimental results was very low, less than 100 μW. The record bandwidth achieved so far in a TDF ASE source was achieved in a thulium- and holmium-doped fibre with the so called 10-dB bandwidth of 645 nm [26]. The emission from the thulium ions was promoted at one of the fibre ends while the emission from holmium ions prevailed at the other end. The two outputs are combined together by an appropriate wavelength division multiplexer. In fact, this device acts similarly to a dual stage ASE sources with combined outputs [2729]. The pumping schemes of TDF ASE sources include both core-pumping and cladding pumping. Recently, in-band core pumping was demonstrated in broadband sources with about 70 nm FWHM and 20 mW [30] and 40 mW [31] output powers. The cladding pumping arrangement with pump at 790 nm resulted in the output power of 120 mW and FWHM of 40 nm [32]. Remarkable progress has been achieved in the high-power double-clad TDF superfluorescent sources, where the output power is optimized, rather than the spectral width. Output power of 11 W, slope efficiency of 38% and FWHM of 35 nm was achieved in a configuration with bulk-optics pumping [33] and in an all-fibre master-oscillator power-amplifier configuration the output power of 25 W, slope efficiency of 49 % and FWHM of 22 nm was achieved [34].
In this paper we report on methods for optimization of broadband sources based on ASE in TDF. In the following section of this paper we present an analysis of the spectrally flattened TDF ASE source. Numerical analysis of possible spectral flattening of TDF ASE sources has not been published yet, to our knowledge. Theoretical analysis of the TDF ASE source appeared in two papers [35, 36] where the numerical model was used to reveal the physical understanding of the power evolution in ASE sources based on TDFs rather than for optimization issues. The first paper contains a brief analysis that resulted in preference of the forward (in terms of the pump radiation propagation) ASE that should carry the most of the optical power. On the contrary, the latter analysis leads to the opposite findings that it is the backward ASE that carries the highest power in the most typical configurations. In the third section of this paper we present preliminary experimental results of TDF ASE source using TDF developed in house. TDF characteristics including its performance in ring-laser setup are included.
2 Numerical modelling of a Tm-doped fibre ASE source
For predicting the performance of various thulium doped fibre devices and their optimization, we developed a comprehensive, spectrally and spatially resolved numerical model, which is described in detail elsewhere [3739]. For numerical simulations we set thulium ion concentration not higher than 1000 ppm mol. The pair-induced quenching processes among neighbouring thulium ions can still be assumed negligible at this concentration level provided that the thulium ions are homogeneously distributed in the core and not in clusters [37]. The emission and absorption spectra are taken from Ref. 37. The other parameters of the fibre used in the simulations are summarized in Table 1. The configurations of the broadband source for the numerical modelling are shown in Fig. 1. The pump source at 1611 nm is assumed because this wavelength is the closest wavelength to the Tm peak absorption ${}^{3}{\mathrm{H}}_{6}\to {}^{3}{\mathrm{F}}_{4}$ at around 1630 nm and in the same time it falls within the amplification range of commercially available L-band EDFAs. In the case of in-band pumping at 1611 nm and neglecting the cooperative up-conversion processes thanks to the limited concentration of thulium, the numerical model of TDF with a rich energy level structure [37, 38] is simplified to a system with two energy levels.
Fig. 1
Configurations of the TDF ASE source.
Table 1
Parameters of the Tm doped-fibre used in the numerical modelling.
Evolution of pump and ASE optical powers and relative population of the metastable ${}^{3}{\mathrm{F}}_{4}$ level along the fibre is shown in Fig. 2 for the configurations with and without a mirror. The pump power level was set to 1 W. The pump is almost fully absorbed within the first meter of the fibre and the ${}^{3}{\mathrm{F}}_{4}$ level population is close to zero beyond z = 1 m. For the configuration without reflection mirror (dashed lines in Fig. 2), the blue-edge of the spectrum of the forward ASE (FASE) is reabsorbed where a ${}^{3}{\mathrm{F}}_{4}$ level population is low and the spectrally integrated FASE power is stagnating. The backward ASE (BASE) grows steadily towards the pump input end, benefiting from increasing pump power towards this fibre end. The BASE power may become strong enough to saturate its gain and lower the inversion population, as it was the case for the pump power level of 1 W shown in Fig. 2. Attachment of a mirror with spectrally flat reflection R(λ) = 1 at the z = L forms a seed for the BASE. Such an ASE seed promotes amplification of the BASE waves that deplete even more the inversion population at the beginning of the fibre, thus lowering the FASE power. The ASE output power vs. pump power for the two configurations and spectra corresponding to the pump power level of 1 W are shown in Fig. 3. As expected, the configuration with the mirror provides higher output power of the broadband source in one fibre pigtail than the mirror-less configuration. The higher output power is at the expense of narrower spectra. Notably, the blue edge of the ASE spectrum is suppressed due to reabsorption in the depleted part of the fibre close to the mirror end.
Fig. 2
(a) Optical power and (b) relative population of the thulium metastable level ${}^{3}{\mathrm{F}}_{4}$ along the fibre with (solid line) and without mirror (dashed line).
Fig. 3
(a) Output ASE power and (b) output ASE spectra.
It should be noted out that the threshold pump power of the linear increase of ASE power depends on the fluorescence lifetime of the metastable level. The dependences of the ASE output power vs. pump power for different host materials are shown in Fig. 4. The other parameters were intentionally left the same in order to point out the effect of the host material. The longer fluorescence lifetime significantly decreases the pump power threshold while having almost no effect on the slope. These trends are similar to the case of two-level fibre lasers where analytical expressions for the slope and threshold can be derived analytically [40]. The materials with low-phonon energy, like in the case of fluoride glass ZBLAN, are known for an excellent quantum conversion efficiency of almost 100% of the radiative transitions form the ${}^{3}{\mathrm{F}}_{4}$ level. But fluoride fibres suffer from hygroscopicity, high cost and it is difficult to splice them with conventional silica fibres. On the other hand, silica-host materials pose higher phonon energy. Therefore, the quantum conversion efficiency is significantly lower, e.g., about 10% for the pure silica host. Figure 4 points out the importance of hosts with longer metastable levels of thulium for low- and medium- power applications like the broadband sources studied in this paper. TDFs with enhanced ${}^{3}{\mathrm{F}}_{4}$ life times of 400–700 μs have been reported in modified silica glasses [16]. A promising way to enhance the fluorescence lifetime is to modify locally thulium environment by the ceramic nanoparticle doping and MCVD methods [4144].
Fig. 4
Effect of fluorescence lifetime of the ${}^{3}{\mathrm{F}}_{4}$ level on the output characteristics.
The broadband sources are not being optimized only in terms of an output power but also in terms of the ASE output spectrum. Typically, the flat or Gaussian shapes are required for component characterization and optical coherent tomography systems; or one may require such a spectrum that balances the responsivity of the detector used [26, 29]. In the following we present numerical optimization of the spectral shape of a band-stop filter in the setup in Fig. 1 in order to get the widest flat spectrum of the ASE source. We adopted an approach similar to the one presented earlier by Paschotta et al. [45] where the double-pass ytterbium-doped fibre ASE source is equipped at one end of the ytterbium-doped fibre with a bulk-grating-based filter and a mirror. In this method, the FASE output was collimated and sent to the mirror through a pair of gratings to disperse spatially the FASE spectrum. The dispersed spectrum traversed a mechanical comb made by a set of screws that protrudes the beam. The filter spectral shape can be finely tuned by a screwdriver; the higher protrusion of the screw into the beam, the higher the attenuation at the wavelength corresponding to that screw. On the contrary in our study, we use all-fibre components for spectral filtering so that the filter would be made by a combination of specially designed wavelength division multiplexer [18] and a mirror; by cascade of long-period fibre gratings [46]; or by a thin-film dichroic filter deposited on the perpendicularly cleaved fibre end, see Fig. 1. We tested the applicability of this approach in a TDF ASE source by means of a numerical model. The optimal spectral shape of the filter was searched iteratively. Starting shape mimics that of the emission spectrum and it is gradually modified unless the change of the full-width at half maximum of the output spectrum is less than a pre-set value. Examples of iterative steps are shown in Fig. 5, as well as the corresponding FWHM and output power values. The output spectra of the
Fig. 5
(a) Reflectivities of the mirror with the preceding filter; and (b) the corresponding ASE spectra for several iteration-step examples of the iterative search of the optimal spectral shape of the filter. The ASE spectrum for the mirror without filter is shown for comparison. (c) Power conversion efficiency and spectral width of the output ASE spectrum with respect to the iteration step.
ASE source with and without filter are compared in Fig. 5(b). The FWHM can be broadened from 55 nm for the case without filter to more than 140 nm for the optimized filter shape. It means that the FWHM can be more than doubled at the expense of only little decrease of the output power.
3 Tm-doped fibre characteristics and performance in a ring-laser broadband ASE source
The thulium-doped preform was fabricated in house by using the MCVD (Modified Chemical Vapour Deposition) and solution doping methods. The fibre of 125 μm outer diameter was drawn from the preform. The fibre core had approximately step-index profile with a diameter of 7 μm and a numerical aperture of 0.17. The concentration profile of thulium followed that of the core refractive-index. Spectral shape of the ${}^{3}{\mathrm{F}}_{4}$ level ground state absorption measured by cut-back method is shown in Fig. 6(a). The background losses were less than 0.1 dB/m. The attenuation due to absorption of Tm3+ ions is given by the product 4.34 Γσa(λ) NTm where Γ is the overlap factor accounting for the overlap of the guided optical mode with thulium ions. Overlap factor Γ can be calculated using the known refractive index and concentration profiles [39] and is of 0.95 at 1650 nm. Using the cross section at the peak ground state absorption of the ${}^{3}{\mathrm{F}}_{4}$ level σa (1650 nm) = 4.4×10–25 m2 [37] we determined that the thulium concentration was 6.3×1025 m–3. It is about 1500 ppm mol Tm3+ assuming the density of the alumina-silicate core of 2.2 g/cm3 [47]. The ${}^{3}{\mathrm{F}}_{4}$ lifetime of 480 μs was determined from the fluorescence decay measurements. The normalized emission of the thulium-doped fibre is also shown in Fig. 6(a). The emission was measured in backward direction with respect to the pump at 1620 nm and using the WDM 1.6/2.0 μm. Note that the increased noise in the blue edge of the emission spectrum is due to the correction to the WDM 1600/2000 nm spectral transmission used in the measurement setup. Minimum WDM transmission was at 1610 nm. The spectral ripples in the emission spectrum in the interval of 1810–1930 nm were caused by water vapour absorption within the spectrum analyser, i.e., these ripples are not inherent to the TDF emission.
Fibre performance was tested in a ring laser cavity shown in the inset of Fig. 6(b). Length of the fibre was 1.8 m. In the setup we used the fused fibre components, i.e., the WDMs and the output coupler that were developed within a project EYESAFE2u sponsored by the Ministry of Industry and Trade of the Czech Republic (project No. FR-TI4/734). The fused fibre components fabrication and characteristics are described in detail elsewhere [18]. The laser output characteristics we measured for 90% and 10% output coupling, see
Fig. 6
(a) Absorption and emission spectra of the developed TDF. (b) Laser output characteristics for 90% and 10% output coupling. Experimental setup of the ring fibre laser is in the inset.
Fig. 6(b). The laser threshold of 170 and 290 mW and the slope efficiency of 32 and 4.4 % were found for the 90% and 10% output coupling, respectively. The higher slope efficiency was found for the case of higher output coupling as expected. Nevertheless, the slope efficiency is lower than the value of about 80% imposed by the quantum defect that is given by the ratio of the pump and the laser signal wavelength. This is mainly attributed to the insertion losses of the components used in the setup. Indeed, the overall insertion losses in the setup were estimated as high as 2 dB. Despite higher cavity losses and resulting lower slope efficiency, the ring-laser setup has an advantage of better stability thank to unidirectional propagation of the laser signal. In addition, the Tm3+ concentration is high enough to promote the cooperative upconversion to a 3H4 level that would lead also to decrease of the laser slope efficiency.
The broadband ASE source was tested in the setup shown in the inset of Fig. 7(a). The fibre length was 3 m. The output power of the ASE source vs. the input pump power at 1550 nm is shown in Fig. 7(a). The output spectra for the pump powers of 0.73 and 1 W are shown in Fig. 7(b), as well as spectra of the backward and forward ASE when the mirror is removed from the setup. For the pump power level of 1 W, the peak wavelength, the 3-dB and 10-dB spectral widths of the double-pass arrangement were 1915 nm, 66 nm and 157 nm; while for the single-pass arrangement the corresponding values were 1842 nm, 60 nm, and 129 nm, respectively. Although no special spectral filter was used, the ASE spectrum was wider compared to the setup without mirror. It should be noted that most typical case is that double-pass ASE source built by the attachment of the mirror at one of the fibre ends has opposite effect. The double-pass configuration leads to higher output power, but a narrower spectrum. It means that by proper selection of the pump power and fibre length one can optimize the width of the broadband source. Note that in our case the mirror promoted the longer-wavelength edge peak defined by the reflected FASE seed. The spectral shape of the FASE is shifted towards shorter wavelengths than the corresponding part of the ASE spectrum with attached mirror. This is an evidence of further reabsorption of the ASE seed (reflected by the mirror) in the depleted part of the thulium-doped fibre. The depleted section of the fibre and its length shaped the output ASE spectrum. The effect of ASE spectrum widening by combination of two distinct ASE peaks is somewhat similar to the widening of the ASE spectrum of ytterbium-doped double-clad fibres [29]. In contrast to the case described in our paper, the ytterbium ASE source was a single pass broadband source, i.e., without mirror, and the two distinct spectral peaks corresponded to typical ytterbium fibre laser wavelength regions around 1030 nm and 1070 nm.
Fig. 7
(a) ASE output power vs. input pump power at 1550 nm. ASE source setup is shown in the inset. (b) Output spectra of the experimental TDF ASE source of various configurations and pump powers.
4 Conclusions
We have described two methods of increasing the bandwidth of ASE source based on TDFs. Firstly, we have shown by means of the comprehensive numerical model that the FWHM of the ASE source can be more than doubled by using specially tailored spectral filter placed in front of the mirror in a double-pass configuration of ASE source. To our knowledge, it is the first report on numerical optimization of TDF ASE sources. Secondly, we reported initial results of experimental TDF ASE source, including fibre characteristics and performance of the TDF in fibre ring laser. We observed a counter-intuitive effect in the experimental ASE source that the spectrum in a double-pass configuration is broader than the spectrum of a single-pass ASE source. The broadening can be explained by the combination of shorter wavelength spectral peak of BASE and longer wavelength peak of the reflected FASE seed. By the proper combination of pump power and fibre length the two peaks may be set to similar optical powers and, in result, the output spectrum is broadened. The experimental and numerical optimization of the second approach is beyond the scope of this paper and it is a prospect of future work. It should be noted that numerical optimization of such device will require accurate spectroscopic characterization of the actual TDF, including absorption and emission cross-section spectra.
The TDF ASE sources offer higher stability and lower complexity and costs compared to other broadband sources, namely the supercontinuum sources that are based on ultrafast, quasi-continuous lasers with spectrum broadened in highly nonlinear fibre. Thus, the TDF ASE sources can substitute the SC sources in applications around 2 μm.
Acknowledgements
The authors thank Simon Hutchinson for careful reading of the manuscripts and his helpful comments. The authors acknowledge the company SQS Fibre Optics, Czech Republic, for cooperation in the development of fused fibre components for the spectral region around 2 μm. The research was supported by the Agency for Healthcare Research of the Czech Republic, under project No. 15-33459A.
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About the article
Published Online: 2016-10-17
Published in Print: 2016-12-01
Citation Information: Opto-Electronics Review, Volume 24, Issue 4, Pages 223–231, ISSN (Online) 1896-3757, ISSN (Print) 1230-3402,
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# The ratio of x to y is 3 : 4, and the ratio of x + 7 to y +
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The ratio of x to y is 3 : 4, and the ratio of x + 7 to y + [#permalink] 24 Apr 2018, 01:55
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The ratio of x to y is 3 : 4, and the ratio of x + 7 to y + 7 is 4 : 5.
Quantity A Quantity B $$\frac{x+14}{y+14}$$ $$\frac{5}{6}$$
A. The quantity in Column A is greater
B. The quantity in Column B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
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Re: The ratio of x to y is 3 : 4, and the ratio of x + 7 to y + [#permalink] 24 Apr 2018, 03:23
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From 2 equations, solving for x and y gives, x= 21 and y =28.
x+14/y+14 = 35/42
=5/6
Hence C
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Re: The ratio of x to y is 3 : 4, and the ratio of x + 7 to y + [#permalink] 20 May 2018, 01:16
mohan514 wrote:
From 2 equations, solving for x and y gives, x= 21 and y =28.
x+14/y+14 = 35/42
=5/6
Hence C
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Re: The ratio of x to y is 3 : 4, and the ratio of x + 7 to y + [#permalink] 20 May 2018, 04:45
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Expert's post
Solve the equations given
$$\frac{x}{y}=\frac{3}{4}$$
$$4x=3y$$
Second equations
$$\frac{x+7}{y+7}=\frac{4}{5}$$
$$5x+35=4y+28$$ or $$5x-4y=-7$$.
Solving these 2 equations we get x=21 and y=28.
So $$\frac{x+14}{y+14}=\frac{21+14}{28+14}=\frac{35}{42}=\frac{5}{6}$$
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Re: The ratio of x to y is 3 : 4, and the ratio of x + 7 to y + [#permalink] 20 May 2018, 04:45
Display posts from previous: Sort by | 2018-12-17 04:39:18 | {"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": 1, "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": 0.4688153862953186, "perplexity": 3029.086927446579}, "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-2018-51/segments/1544376828318.79/warc/CC-MAIN-20181217042727-20181217064727-00196.warc.gz"} |
http://jean-martin-albert.com/courses/algebraicstructures2013/ | In Mathematics, a structure generally consists of a set $S$ together with a collection of named constants, relations and functions. Here, a constant simply means an element of $S$, a relation is a subset of $S^n$ and a function is a function $f:S^n\to S$. The number $n$ is called the arity of the function or the relation, and represents the number of arguments that the relation (or function) can take. The function (or relation) is called
• Unary if $n=1$
• Binary if $n=2$
• Ternary if $n=3$
• $n$-ary in all other cases
The number of such constants, relations and functions is usually finite, but in some cases (like vector spaces) it can be infinite.
On every set $S$, there is a “trivial” structure which consists of all the finite powers $S^n$, and all the projection maps $\pi_i:S^n\to S$, where $\pi_i(x_1,…,x_n)=x_i$. This structure on sets is used implicitely.
graph is a set $G$ together with a single relation $E\subseteq G\times G$.
A real vector space is a structure $(V, 0, +, (\lambda_r:r\in\Real))$. It has a single constant $0$, a binary function $+$, and an infinite list of unary functions $\lambda_r:S\to S$, one for every real number.
In general, what makes a structure algebraic is the lack of non-trivial relations. The first example above can be considered an algebraic structure, because the relations, which we chose to be $S^n$ for every $n$, are in a sense trivial. Vector spaces are also an algebraic structure, but graphs are not. | 2013-05-20 17:12:40 | {"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": 1, "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": 0.952438235282898, "perplexity": 220.94208237607847}, "config": {"markdown_headings": false, "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-2013-20/segments/1368699113041/warc/CC-MAIN-20130516101153-00042-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/uniform-force-and-work.95200/ | Homework Help: Uniform Force and Work
1. Oct 17, 2005
jd102684
Hello, I need some help on a two part question. Thanks in advance!
Consider a uniform force, similar to the gravitation force but pointing at an angle µ to the vertical direction, F = ma sin µ ¡ ma cos µ j (i and j represent vector notation...)
1) What is the minimum work required to move a point mass with mass equal to m from the origin (0,0) to a point P (x,y)? (answer in vector notation in terms of µ)
2) in what direction r (also a vector) is the potential energy constant?
Thanks so much for all your help!
EDIT: I forgot to post my current work. Ive tried to mess with changing the axis so I can treat the force like gravity. I know that the force is conservative, so the work required to move the mass from point A to point B is the same no matter what path is taken. I'm just really having trouble getting an answer in terms of the angle in vector notation... As for part 2, I would guess it would be in the direction oposite of the uniform force being applied, but when i submitted what i got for that answer, it came back as wrong.
Last edited: Oct 17, 2005
2. Oct 17, 2005
Diane_
You have the force resolved into orthogonal components, so for all practical purposes you can treat them as independent of each other. Work out the work in the i direction and the work in the j direction, then add them. You'll get the direction with a little Euclidean geometry - just make a sketch and I think you'll see what I mean.
3. Oct 17, 2005
jd102684
Thanks, that did help! I got the first part, but before I try to crank out an answer to the second part of the question can you tell me if my line of thinking is correct as far as what I say in the original post? Maybe give me a boost in the right direction again? Thanks so much! | 2018-09-21 17:21:03 | {"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": 0.807364821434021, "perplexity": 265.8348162248501}, "config": {"markdown_headings": false, "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-2018-39/segments/1537267157351.3/warc/CC-MAIN-20180921170920-20180921191320-00293.warc.gz"} |
https://www.azdictionary.com/definition/dimensional%20stability | • Definition for "dimensional stability"
• the amount that a material preserves its initial…
• Agriculture Dictionary for "dimensional stability"
• the capability of a material to…
# dimensional stability definition
• noun:
• the amount that a material preserves its initial dimensions when subjected to alterations in temperature and humidity.
• the amount to which a material maintains its initial measurements whenever put through alterations in temperature and moisture. | 2017-05-29 08:41:53 | {"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": 0.8555420637130737, "perplexity": 6287.740525214486}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2017-22/segments/1495463612036.99/warc/CC-MAIN-20170529072631-20170529092631-00417.warc.gz"} |
https://www.physicsforums.com/threads/aluminium-alloys.426238/ | Aluminium alloys.
vanesch
Staff Emeritus
Gold Member
Hello,
Does anybody know whether the alloy elements (Cr, Mg, Mn, Fe) in the aluminium alloy Al - 5083 are mainly interstitial or substitutional. The reason I want to know this is that this has an effect on the thermal neutron diffraction of this material...
thanks!
Related Materials and Chemical Engineering News on Phys.org
Astronuc
Staff Emeritus
http://www.keytometals.com/article55.htm
Chromium occurs as a minor impurity in commercial-purity aluminum (5 to 50 ppm). It has a large effect on electrical resistivity. Chromium is a common addition to many alloys of the aluminum-magnesium, aluminum-magnesium-silicon, and aluminum-magnesium-zinc groups, in which it is added in amounts generally not exceeding 0.35%. In excess of these limits, it tends to form very coarse constituents with other impurities or additions such as manganese, iron, and titanium. Chromium has a slow diffusion rate and forms fine dispersed phases in wrought products. These dispersed phases inhibit nucleation and grain growth. Chromium is used to control grain structure, to prevent grain growth in aluminum-magnesium alloys, and to prevent recrystallization in aluminum-magnesium-silicon or aluminum-magnesium-zinc alloys during hot working or heat treatment.
Iron is the most common impurity found in aluminum. It has a high solubility in molten aluminum and is therefore easily dissolved at all molten stages of production. The solubility of iron in the solid state is very low (~0.04%) and therefore, most of the iron present in aluminum over this amount appears as an intermetallic second phase in combination with aluminum and often other elements.
Magnesium is the major alloying element in the 5xxx series of alloys. Its maximum solid solubility in aluminum is 17.4%, but the magnesium content in current wrought alloys does not exceed 5.5%.
Manganese is a common impurity in primary aluminum, in which its concentration normally ranges from 5 to 50 ppm. It decreases resistivity. Manganese increases strength either in solid solution or as a finely precipitated intermetallic phase. It has no adverse effect on corrosion resistance. Manganese has a very limited solid solubility in aluminum in the presence of normal impurities but remains in solution when chill cast so that most of the manganese added is substantially retained in solution, even in large ingots.
Aluminum and aluminum alloys
Physical Metallurgy
See Figure 1 and Table 1.
Mg has significant solubility, and Mn is sparingly soluble, and Cr is less soluble than Mn, and Fe less so. Solubility is strongly a function of temperature also.
vanesch
Staff Emeritus
Gold Member
Thanks!
I guess that "solubility" means "substitutional" in the crystal lattice, although that doesn't have to be the case, right ? It is just that thermodynamically, it doesn't imply a phase separation, but it could just as well be interstitial as substitutional ?
I'll explain you why it makes a difference. The scattering cross section of a (thermal) neutron on an atom has a coherent and an incoherent part. The coherent part is the one that makes diffraction patterns, the incoherent one is essentially 4 pi uniform diffusion without any interference pattern.
Now, I'm supposed to find out (compare measurements and model) the behavior of relatively thin aluminium alloy plates on a thermal neutron beam.
Point is, for pure aluminium, you have a well-defined diffraction pattern (the coherent cross section and the crystal lattice), and a well-defined incoherent diffusion pattern (and also some absorption but that's easy).
But if you have an alloy, if the alloy atoms are interstitial, their "coherent" part of the diffusion cross section will behave incoherently because of the "random" positions (there is no coordinated interference of the waves), so you can consider the entire diffusion cross section as incoherent (you add the "coherent" and "incoherent" together, as the coherent part will also be "randomized" and hence behave incoherently).
However, if it is substitutional, the situation is more complicated: part of the coherent diffusion on the substitutional atom will behave as if it were an aluminium atom, and part of it will be act as if it is incoherent.
If you do the calculations, this makes a substantial difference in what happens to the beam.
I know that in nuclear power applications, people usually don't make any difference between coherent and incoherent diffusion, and treat everything as incoherent, as an approximation, but in my case, it makes a difference.
Gokul43201
Staff Emeritus
Gold Member
It is just that thermodynamically, it doesn't imply a phase separation, but it could just as well be interstitial as substitutional ?
Correct.
In this case, however, the listed elements are almost certainly substitutional (all of them) or more specifically, they are not interstitially substituted. Al-5083 is about 95% Al. There are two likely scenarios for the microstructure:
1. It is a single phase FCC alloy (like the $\alpha$-phase of the Al-Cu binary alloy),
2. It is some eutectic-like multi-phase alloy, composed of a mixture of primarily the alpha-phase along with small amounts of different intermetallic compounds formed with/by the less soluble elements (like Fe). You can estimate approximate upper bounds for the fraction of these intermetallic phases in the alloy if you know the exact composition.
Here's the typical composition of generic Al-5083:
Code:
Aluminum Balance
Chromium 0.05 - 0.25
Copper 0.1 max
Iron 0.4 max
Magnesium 4 - 4.9
Manganese 0.4 - 1
Remainder Each 0.05 max
Remainder Tot 0.15 max
Silicon 0.4 max
Titanium 0.15 max
Zinc 0.25 max
If we approximate this as a 2-component (Al-Mg) system it is almost certainly a single phase FCC, which is a close-packed structure and therefore has pretty small interstitials. There are two kinds of interstitial sites in an FCC lattice: the octahedral site (radius about 40% of Al atomic radius), and the tetrahedral site (radius about 20% of the Al radius).
Al is already a pretty small atom with an atomic radius of about 120pm. It can really only accept the very small elements (H, O, N, C) interstitially. Approximate atomic radii of Cr, Mg, Mn and Fe are respectively 170, 140, 160 and 160pm (the radii in a crystal vary with co-ordination number, but the atomic radius is a good enough approximation for now). Mg is certain to be substitutional in this alloy.
Look at the well-studied Al-Cu alloy system for reference (Cu has an atomic radius of about 150pm) - where Cu dissolves substitutionally in the Al lattice up to about 0.3 atomic % (much more at higher temperatures - see here). Beyond that, you get into a 2-phase region with small inclusions of an Al-Cu intermetallic compound.
If the composition of your plates are close to that in the table above, my guess is that you will almost certainly have a dominant alpha-phase with all the Mg dissolved substitutionally in the Al (as well as some of the Cr, Fe, Mn, etc). But in addition, there will likely be some tiny inclusions of Al3Fe and/or Al12Mg2Cr (and probably some others as well).
In your case, the inclusions will affect the neutron scattering quite differently from the alpha-phase, though they will have a pretty small cross-section. But it is highly unlikely, no matter what the actual composition, that any of the elements will incorporate interstitially in the Al-lattice.
Last edited:
Astronuc
Staff Emeritus
As Gokul indicated, I believe the alloying elements are largely substitutional, but it's more complicated than that. There are other secondary phases or intermetallics formed.
When one refers to a pure Al alloy, is one also referring to a fully annealed state as opposed to cold worked. And is the material single crystal or polycrystalline.
Looking at Al -
•Space group: Fm-3m (Space group number: 225)
•Structure: ccp (cubic close-packed)
•Cell parameters:
◦a: 404.95 pm
◦b: 404.95 pm
◦c: 404.95 pm
◦α: 90.000°
◦β: 90.000°
◦γ: 90.000°
Mg
•Space group: P63/mmc (Space group number: 194)
•Structure: hcp (hexagonal close-packed)
•Cell parameters:
◦a: 320.94 pm
◦b: 320.94 pm
◦c: 521.08 pm
◦α: 90.000°
◦β: 90.000°
◦γ: 120.000°
•Space group: Fd-3m (Space group number: 227)
•Structure: diamond [??]
•Cell parameters:
◦a: 543.09 pm
◦b: 543.09 pm
◦c: 543.09 pm
◦α: 90.000°
◦β: 90.000°
◦γ: 90.000°
In the Zr-Sn binary alloy, Sn is a solution solution element.
Zr is hcp, but Sn is tetragonal.
Sn
•Space group: I41/amd (Space group number: 141)
•Structure: tetragonal
•Cell parameters:
◦a: 583.18 pm
◦b: 583.18 pm
◦c: 318.19 pm
◦α: 90.000°
◦β: 90.000°
◦γ: 90.000°
I believe elements like H, C, N, O and some others are predominantly interstitial, while others are predominantly substitutional, to varying degrees.
In Zircaloys, Zr-Sn-(Fe, Cr, Ni, O, Si), the elements Fe, Cr, Ni are sparingly soluble, while Sn is quite soluble. There are two predominant secondary phases, Zr(Fe,Cr)2 and Zr2(Fe,Ni), as well as Zr3Si as a very small population on which the other two predominant phases precipitate, and then there is ZrO2 dispersed, and perhaps some Zr3O, or ZrxOy. ZrO2 on the surface can be tetragonal or monoclinic depending on conditions. Zr3O can be found at the metal oxide interface. If hydrogen is present then Zr hydrides (ZrH2 or more likely ZrH1.6 - ZrH1.7 may form.
With irradiation, one will find amorphous Zr(Fe,Cr) and Zr(Fe,Ni) with Fe being the more predominant element in the amorphous region, but that may also depend on the temperature.
Dr Lots-o'watts
If I understand correctly, to a first approximation, an annealed metal has its alloy elements as substitutional if their diameter is similar to the principal metal, and interstitial if they are somewhat smaller.
Staff Emeritus | 2019-11-22 12:21:14 | {"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": 1, "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": 0.6633433699607849, "perplexity": 3810.1736177269445}, "config": {"markdown_headings": false, "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-2019-47/segments/1573496671260.30/warc/CC-MAIN-20191122115908-20191122143908-00494.warc.gz"} |
https://www.semanticscholar.org/paper/Cyclic-elements-in-some-spaces-of-analytic-Korenblum/cfd756094d20c26986de65e925760873aa883320 | # Cyclic elements in some spaces of analytic functions
@article{Korenblum1981CyclicEI,
title={Cyclic elements in some spaces of analytic functions},
author={Boris Korenblum},
journal={Bulletin of the American Mathematical Society},
year={1981},
volume={5},
pages={317-318}
}
• B. Korenblum
• Published 1 November 1981
• Mathematics, Philosophy
• Bulletin of the American Mathematical Society
DEFINITIONS. 1. A~ (p > 0) is the Banach space of analytic functions f(z) in U = {z G C| \z\ < 1} that satisfy \f(z)\ = o[(l \z\)~] (\z\ > 1) with the norm \\f\\ = max{ |f(z)\(l z)} (z G If). Note that fn —• ƒ in A * and #„ —># m A~~' impl ies /^ —>,/& in A~\ 2. B (p> 0) is the Bergman space, i.e., the "analytic" subspace of L(rdrd6) in U. 3. A -00 = U A~ = U B (p > 0). A~°° is a linear topological space [1]. 4. Pis the set of all algebraic polynomials P(z). Pis dense in any of the spaces A _ p…
Cyclic vectors in the Dirichlet space
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Weakly invertible elements in certain function spaces, and generators in l v Michigan Math
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• 1964 | 2022-07-07 18:01:28 | {"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": 1, "mathjax_display_tex": 1, "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": 0.9198943376541138, "perplexity": 1435.6482036576049}, "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-2022-27/segments/1656104495692.77/warc/CC-MAIN-20220707154329-20220707184329-00544.warc.gz"} |
https://www.neilsabbah.com/tags/2ed121-s%C3%A9rie-de-maclaurin | # série de maclaurin
hasta Para acotar el tamaño del error cuando la suma se aproxima por la integral, se tiene en cuenta que, en el intervalo number and is a Legendre x n = Hence, The law of large numbers implies that the identity holds.[11]. 1 1 ) = In particular, this is true in areas where the classical definitions of functions break down. i The exponential function ) c ( a 1 {\displaystyle {\scriptstyle b\to +\infty }} Weisstein, Eric W. "Maclaurin Series." | f Join the initiative for modernizing math education. can be found by comparison of coefficients with the top expression for This method is sometimes called Taylor’s series if the function is expanded around zero, rather than some other values. ) where is a gamma x x (If n = 0, this product is an empty product and has value 1.) Analytic functions. , z 2 ⌋ 1 f Differentiation and integration of power series can be performed term by term and is hence particularly easy. x se cumple hasta Suppose we want the Taylor series at 0 of the function, Then multiplication with the denominator and substitution of the series of the cosine yields, Collecting the terms up to fourth order yields. f n 0 Se seguirá la demostración que aparece en (Apostol).[1]. (t/h)j/j!. 0 A Maclaurin series is a Taylor series expansion ( Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. We need to find the first, second, third, etc derivatives and evaluate them at, http://www.intmath.com/series-expansion/2-maclaurin-series.php. , los polinomios de Bernoulli alcanzan sus valores máximos absolutos en los puntos finales del intervalo (véase D.H. Lehmer en la referencias) y que Free Maclaurin Series calculator - Find the Maclaurin series representation of functions step-by-step This website uses cookies to ensure you get the best experience. e integrando el resultado sobre el intervalo unidad: Tomando {\displaystyle k=n-1} Esta fórmula puede ser usada para aproximar integrales por sumas finitas o, de forma inversa, para evaluar series (finitas o infinitas) resolviendo integrales. 0 donde . ) . y una función diferenciable cualquiera n 0 k x Sumando desde Language using SeriesCoefficient[f, x Your email address will not be published. Esta página se editó por última vez el 17 ago 2020 a las 11:40. = B ~ − k To embed a widget in your blog's sidebar, install the Wolfram|Alpha Widget Sidebar Plugin, and copy and paste the Widget ID below into the "id" field: We appreciate your interest in Wolfram|Alpha and will be in touch soon. ) {\displaystyle f(y)} Nótese que los números de Bernoulli se definen como ( {\displaystyle f(x,y)} Maclaurin series coefficients, a k can be calculated using the formula (that comes from the definition of a Taylor series) where f is the given function, and in this case is sin(x). ) The Ek in the expansion of sec x are Euler numbers. {\displaystyle f(x)=x^{3}} En muchos casos, la integral de la derecha es resoluble mediante funciones elementales de forma cerrada incluso cuando la serie de la izquierda no puede ser resuelta. Particularly convenient is the use of computer algebra systems to calculate Taylor series. n ) / x x Even if the Taylor series has positive convergence radius, the resulting series may not coincide with the function; but if the function is analytic then the series converges. {\displaystyle n=0} n whose coefficients are the generalized binomial coefficients. = one first computes all the necessary partial derivatives: Evaluating these derivatives at the origin gives the Taylor coefficients, Substituting these values in to the general formula, Since ln(1 + y) is analytic in |y| < 1, we have. ( x Derivadas Aplicações da derivada Limites Integrais Aplicações da integral Soma de Riemann Séries EDO Cálculo de Multivariáveis Transformada de Laplace Séries de Taylor/Maclaurin Série de Fourier Entonces, todos los términos de la serie asintótica pueden ser expresados mediante funciones elementales, por ejemplo: Donde la serie de la izquierda es igual a la suma de B + z 2 < = π y que estos se anulan para n impares mayores que 1. This page was last edited on 8 November 2020, at 09:09. ( ) In order to expand (1 + x)ex as a Taylor series in x, we use the known Taylor series of function ex: Classically, algebraic functions are defined by an algebraic equation, and transcendental functions (including those discussed above) are defined by some property that holds for them, such as a differential equation. ( {\displaystyle n\geq 2} When α = −1, this is essentially the infinite geometric series mentioned in the previous section. f , Calculadora gratuita de séries de Taylor-Maclaurin - Encontrar a representação em séries de Taylor/Maclaurin de funções passo a passo z es un polinomio y p es suficientemente grande, entonces el término de error R se anula, por lo que se pueden resolver series de polinomios de forma exacta. [ . B The usual trigonometric functions and their inverses have the following Maclaurin series: All angles are expressed in radians. Nótese también que en esta derivación se asume que la función 0 son los números de Bernoulli. Arduino Library For Proteus A complete step by ste... Control Stepper Motor Speed with Labview and Arduino. {\displaystyle {\tilde {B}}_{0}(x)=1} se pueden definir recursivamente como sigue: Los valores 1 Maclaurin series are named after the Scottish mathematician Colin Maclaurin. 2 Several important Maclaurin series expansions follow. {\displaystyle [0,1]} the inverse Z-transform. x 299-300, Free Maclaurin Series calculator - Find the Maclaurin series representation of functions step-by-step This website uses cookies to ensure you get the best experience. ) pertenece al núcleo, pues la integral de {\displaystyle f(x)} In this series, the approximate value of the function can be calculated as the sum of the derivatives of the function. se obtiene: Sumando Get the free "Maclaurin Series" widget for your website, blog, Wordpress, Blogger, or iGoogle. Esta fórmula puede ser usada para aproximar integrales por sumas finitas o, de forma inversa, para evaluar series (finitas o infinitas) resolviendo integrales. {\displaystyle a} > n B {\displaystyle \sin(2\pi x)} Pierre Gaspard, "r-adic one-dimensional maps and the Euler summation formula". Euler usó esta fórmula para calcular valores de series infinitas con convergencia lenta y Maclaurin la utilizó para calcular integrales. Find more Mathematics widgets in Wolfram|Alpha. n n ( = calculo de Pi con Serie. En matemáticas, la fórmula de Euler-Maclaurin relaciona a integrales con series. Nevertheless, the two series differ from each other in several relevant issues: Expression of a function as an infinite sum, List of Maclaurin series of some common functions, Kerala School of Astronomy and Mathematics, Newton's divided difference interpolation, "Neither Newton nor Leibniz – The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Taylor series revisited for numerical methods, Numerical Methods for the STEM Undergraduate, Inverse trigonometric functions Taylor series, 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series), 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials), 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes), Hypergeometric function of a matrix argument, https://en.wikipedia.org/w/index.php?title=Taylor_series&oldid=987632839, Creative Commons Attribution-ShareAlike License. Maclaurin Series Calculator is a free online tool that displays the expansion series for the given function. ) f Beyer, W. H. ( The latter series expansion has a zero constant term, which enables us to substitute the second series into the first one and to easily omit terms of higher order than the 7th degree by using the big O notation: Since the cosine is an even function, the coefficients for all the odd powers x, x3, x5, x7, ... have to be zero. Thus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution. In Mathematics, the Maclaurin series is defined as the expanded series of the given function. A Scottish mathematician gained his master degree at age 17, and his major mathematics' work arise from his special knowledge in Newton's ideas and the formulation of Newton's methods. Boca Raton, FL: CRC Press, pp. In order to compute the 7th degree Maclaurin polynomial for the function, The Taylor series for the natural logarithm is (using the big O notation). . n un polinomio de Bernoulli. , entonces, la integral. The Maclaurin series, a special case of the Taylor series, is named after him. The Maclaurin series of a function up to order {\displaystyle f(x)} es suficientemente diferenciable, en particular, ( son los números de Bernoulli y R es una estimación del error normalmente pequeña. Unlimited random practice problems and answers with built-in Step-by-step solutions. 1 However, C. Maclaurin also contributed to the astronomy science and helped to improve maps and invented some mechanical devices. 3 {\displaystyle P_{n}(x)} the Puiseux series. = | [ i From MathWorld--A Wolfram Web Resource. For example, the exponential function is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. y reagrupando términos se obtiene la fórmula buscada junto con el término de error. {\displaystyle k=n} {\displaystyle e^{x}} n b 1 {\displaystyle k=1} = f n The special cases α = 1/2 and α = −1/2 give the square root function and its inverse: When only the linear term is retained, this simplifies to the binomial approximation. {\displaystyle B_{n}(1)=B_{n}} y One can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern. ) n Other more general Practice online or make a printable study sheet. ∈ However, one may equally well define an analytic function by its Taylor series. which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, with a full analogy to the single variable case. = , → Find the Maclaurin series expansion of a function. {\displaystyle B_{i}(x-\lfloor x\rfloor )} Los polinomios de Bernoulli 2 ( π Step 3: Finally, the expansion series for the given function will be displayed in the new window. | Esta fórmula no es más que una notación formal de la idea de tomar derivadas en un punto, entonces se tiene. . se anula en el intervalo unidad, así como la diferencia de sus derivadas en los extremos del intervalo. In the last section, we learned about Taylor Series, where we found an approximating polynomial for a particular function in the region near some value x = a. f ∞ Las funciones periódicas de Bernoulli A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as, where D f (a) is the gradient of f evaluated at x = a and D2 f (a) is the Hessian matrix. In some cases, one can also derive the Taylor series by repeatedly applying integration by parts. ( y It converges for Taylor and Maclaurin (Power) Series Calculator. definidas en otros intervalos de la recta real. ( BYJU’S online Maclaurin series calculator tool makes the calculation faster, and it displays the expanded series in a fraction of seconds. Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. La expansión en término de polinomios de Bernoulli tiene una núcleo no trivial. Taylor series are used to define functions and "operators" in diverse areas of mathematics. polynomial. n B es una función suave (suficientemente derivable) definida In the case of the Fourier series the error is distributed along the domain of the function. [12] All these expansions are valid for complex arguments x. Required fields are marked *. CRC Standard Mathematical Tables, 28th ed. ) Walk through homework problems step-by-step from beginning to end. + , o ambos. n function, is a Bernoulli The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century. B En particular, son enteros. The values of x 1987. ∀ n Sumando desde ( In step 1, we are only using this formula to calculate the first few coefficients. f ( x ) = c o s ( x ) ⇒ f ( 0 ) = c o s ( 0 ) = 1 {\displaystyle f(x)=cos(x)\Rightarrow f(0)=cos(0)=1} 1 1 Si z es un número correlacional y {\displaystyle B_{n}(x)} The function e (−1/x 2) is not analytic at x = 0: the Taylor series is identically 0, although the function is not. The th term of a Maclaurin n , escogiendo p = 2 se obtiene: La fórmula de Euler-Maclaurin se usa también para el análisis de errores en integraciones numéricas, de hecho, los métodos de extrapolación se basan en esta fórmula. ) series of a function can be computed in the Wolfram ( e The procedure to use the Maclaurin series calculator is as follows: Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics. La fórmula de Euler-Maclaurin nos da una expresión para la diferencia entre la suma y la integral en función de derivadas de | B For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential or matrix logarithm. . In order to compute a second-order Taylor series expansion around point (a, b) = (0, 0) of the function. B CRC Standard Mathematical Tables, 28th ed. ) x La fórmula fue descubierta independientemente por Leonhard Euler y Colin Maclaurin en 1735. Your email address will not be published. If you want the Maclaurin polynomial, just set the point to 0. = ) 2 The th term of a Maclaurin series of a function can be computed in the Wolfram Language using SeriesCoefficient [ f , x, 0, n] and is given by the inverse Z … 1 ) e n y Explore anything with the first computational knowledge engine. en los extremos del intervalo de integración (0 y n). ( a ) B se define sin Knowledge-based programming for everyone. (Ed.). ) ), The geometric series and its derivatives have Maclaurin series. C. Maclaurin. All are convergent for ( where the subscripts denote the respective partial derivatives. Para cualquier entero positivo p, tenemos que se cumple: donde {\displaystyle B_{n}=B_{n}(0)} b ( {\displaystyle B_{n}(1)=B_{n}}, La fórmula de Euler-Maclaurin puede ser obtenida como una aplicación de algunas ideas de espacios de Hilbert y análisis funcional. ≥ B By … 1 x, 0, n] and is given by El término de error se puede acotar por: Si x Série de Maclaurin para cos(x) Para determinarmos a série do cos(x) faremos o mesmo processo, calcular as derivadas e substituir na série.
(Visited 1 times, 1 visits today) | 2022-05-24 23:47:22 | {"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": 0.9414711594581604, "perplexity": 2720.897886264423}, "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-2022-21/segments/1652662577757.82/warc/CC-MAIN-20220524233716-20220525023716-00534.warc.gz"} |
https://conferences.famnit.upr.si/indico/event/4/contribution/14 | # Graphs, groups, and more: celebrating Brian Alspach’s 80th and Dragan Marušič’s 65th birthdays
from 28 May 2018 to 1 June 2018
Koper
UTC timezone
Home > Timetable > Contribution details
# Enumerating locally restricted compositions over a finite group using de Bruijn graph and covering graph
## Speakers
• Prof. zhicheng GAO
## Content
Let $(\Gamma,+)$ be a finite group. An $m$-composition over $\Gamma$ is an $m$-tuple $(g_1,g_2,\ldots,g_m)$ over $\Gamma$. It is called an $m$-composition of $g$ if $\sum_{j=1}^m g_j = g$. A composition $(g_j)$ over $\Gamma$ is called locally restricted if there is a positive integer $\sigma$ such that any subsequence of $(g_j)$ of length $\sigma$ satisfies certain restrictions. Locally restricted compositions over $\Gamma$ can be modeled using walks in a de Bruijin graph. The de Bruijin graph over $\Gamma$ with span$\sigma$, denoted by $B(\Gamma;\sigma)$, is a digraph whose vertices are $\sigma$-tuples such that there is an arc from ${\bf u}:=({\bf u}(1), {\bf u}(2),\ldots,{\bf u}(\sigma))$ to ${\bf v}:=({\bf v}(1),{\bf v}(2),\ldots,{\bf v}(\sigma))$ if ${\bf v}(j)={\bf u}(j+1)$, $1\le j\le \sigma-1$. Let $D$ be a subgraph of $B(S;\sigma)$. We associate with each directed walk ${\bf v}_1,{\bf v}_2,\ldots,{\bf v}_k$ in $B(\Gamma;\sigma)$ a composition ${\bf c}=({\bf v}_1(1),\ldots,{\bf v}_1(\sigma),{\bf v}_2(\sigma),{\bf v}_k(\sigma))$. That is, ${\bf c}$ is obtained from the walk by appending the last components of the subsequent vertices in the walk to the initial vertex of the walk. We denote this set of compositions by ${\cal C}(D)$.
To keep track of the net sum of a composition in ${\cal C}(D)$, we make use of the derived graph of the voltage graph $(D,\alpha)$, where the voltage of the arc $({\bf u},{\bf v})$ is given by $\alpha({\bf u},{\bf v})={\bf v}(\sigma)$. Let $D'$ denote the derived graph of $(D,\alpha)$. That is, the vertex set of $D'$ is $V(D)\times \Gamma$, and there is an arc from $({\bf u},g)$ to $({\bf v},h)$ if and only if $({\bf u},{\bf v})$ is an arc in $D$ and $h=g+{\bf v}(\sigma)$. Let $\cal S$ be the set of vertices in $D'$ such that the second component is equal to the sum of the parts of the first component. It is easy to see that, for $m\ge \sigma$, an $m$-composition of $g$ in ${\cal C}(D)$ corresponds to a walk in $D'$ from ${\cal S}$ to a vertex whose second component is $g$. Fix an ordering of the vertices of $D'$ and let $T$ denote the corresponding adjacency (transfer) matrix of $D'$. That is, $T(i,j)$ is equal to 1 if there is an arc from ${\bf v}_i$ to ${\bf v}_j$, and zero otherwise.
Let $\vec s$ denote the ${0,1}$ row vector such that its $i$th component is equal to 1 if and only if the corresponding vertex belongs to ${\cal S}$. Let ${\vec f}_g$ denote the ${0,1}$ column vector such that its $j$th component is equal to 1 if and only if the corresponding vertex is of the form $(*,g)$. Then, for $m\ge \sigma$, the number of $m$-compositions of $g$ in ${\cal C}(D)$ is equal to ${\vec s}M^{m-\sigma}{\vec f}_g$.
In this talk, we present some asymptotic results for the number of $m$-compositions, as $m\to \infty$, associated with some digraphs $D$ and some finite group $\Gamma$. It will also be shown that the distribution of the number of occurrences of a given subword in a random locally restricted $m$-composition is asymptotically normal with mean and variance proportional to $m$. These results extend previous results on compositions over a finite abelian group. The basic tools for deriving these results are covering graphs of de Bruijn graphs, Perron-Frobenius theorem, and analytic combinatorics. | 2019-06-25 13:00:42 | {"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": 1, "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": 0.9693981409072876, "perplexity": 92.22838476119182}, "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-2019-26/segments/1560627999838.23/warc/CC-MAIN-20190625112522-20190625134522-00555.warc.gz"} |
https://math.stackexchange.com/questions/1937942/derivative-of-hadamard-product-with-kernels | # Derivative of Hadamard product with kernels
I want to calculate $$\frac{\partial(A \circ X^\top X)}{\partial(X)},$$ where $\circ$ is Hadamard product (elementwise product), $X \in R^{r \times n}$, $A \in R^{n \times n}$, and $\frac{\partial A}{\partial X}=0$.
So far, I found that $$\frac{\partial(A \circ B)}{\partial(C)} = \frac{\partial(A)}{\partial(C)} \circ B + A \circ \frac{\partial(B)}{\partial(C)},$$ from here (generic rule matrix differentiation (Hadamard Product, element-wise))
In my case, $B=X^\top X$ and $C=X$.
Therefore, $$\frac{\partial(A \circ X^\top X)}{\partial(X)} = A \circ 2X^\top .$$
However, $A\in R^{n\times n}$ and $X^\top \in R^{n \times r}$.
Therefore, I can't do the Hadamard product.
How can I do this?
First we'll need a few tensors.
A 6th-order tensor ${\mathbb M}$, whose components ${\mathbb M}_{ijklmn}$ are unity if $\,(i=k=m)$ and $(j=l=n),\,$ but zero otherwise.
This tensor makes it possible to rewrite a Hadamard ($\circ$) product using Frobenius (:) products like so \eqalign{ A\circ Z &= A:{\mathbb M}:Z \cr\cr }
Next we'll need two 4th-order isotropic tensors whose components are \eqalign{ {\mathbb E}_{ijkl} &= \delta_{ik}\,\delta_{jl} \cr {\mathbb B}_{ijkl} &= \delta_{il}\,\delta_{jk} \cr } These tensors make it possible to re-arrange matrix products \eqalign{ A\,dX\,Z &= A{\mathbb E}Z^T:dX \cr A\,dX^T\,Z &= A{\mathbb E}Z^T:{\mathbb B}:dX \cr\cr }
Now we are ready to find the differential and gradient of your function \eqalign{ F &= A\circ X^TX \cr &= A:{\mathbb M}:X^TX \cr\cr dF &= A:{\mathbb M}:(dX^TX+X^TdX) \cr &= A:{\mathbb M}:({\mathbb E}X^T:{\mathbb B}:dX+X^T{\mathbb E}:dX) \cr &= A:{\mathbb M}:{\mathbb E}X^T:{\mathbb B}:dX + A:{\mathbb M}:X^T{\mathbb E}:dX \cr &= A:{\mathbb M}:\big({\mathbb E}X^T:{\mathbb B}\,\,+\,\,X^T{\mathbb E}\big):dX \cr\cr \frac{\partial F}{\partial X} &= A:{\mathbb M}:\Big({\mathbb E}X^T:{\mathbb B} \,+\, X^T{\mathbb E}\Big) \cr\cr\cr } Another approach is to use vectorization.
Let \eqalign{ f &= \operatorname{vec}(F) \cr x &= \operatorname{vec}(X) \cr a &= \operatorname{vec}(A) \cr {\mathcal A} &= \operatorname{Diag}(a) \cr } Then \eqalign{ df &= a\circ\operatorname{vec}(dX^TX+X^TdX) \cr &= {\mathcal A}\,\Big((X^T\otimes I)B\,dx + (I\otimes X^T)\,dx \Big) \cr\cr \frac{\partial f}{\partial x} &= {\mathcal A}\,(X^T\otimes I)B\,+\,{\mathcal A}\,(I\otimes X^T) \cr }where $B$ is the Kronecker Commutation matrix.
This is actually quite similar to the tensor result, with \eqalign{ {\mathcal A} &\sim A:{\mathbb M} \cr (X^T\otimes I)B &\sim I\,{\mathbb E}\,X^T:{\mathbb B} \cr (I\otimes X^T) &\sim X^T{\mathbb E}\,I \cr }
• can you recommend some material to study what you show to us? @greg – user1498253 Sep 4 '17 at 11:24 | 2020-03-31 06:42:01 | {"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": 1, "mathjax_display_tex": 1, "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": 1.0000100135803223, "perplexity": 8216.087419840589}, "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-2020-16/segments/1585370500331.13/warc/CC-MAIN-20200331053639-20200331083639-00001.warc.gz"} |
https://iq.opengenus.org/when-kotlin/ | # When in Kotlin
#### kotlin when software engineering
When is one of the control flow of Kotlin. If you are coming from java, When is a bit similar to Switch. Even if you don't know java, think When as powerful nested if else. If you don't know any programming language, then think When as a plain enlglish term WHEN.
It's pretty self-explanatory, Right?
### Syntax
Here is syntax
var x = 0
when(x){
0 -> print("I'm Zero")
1 -> print("I'm One")
2, 3 -> println("I might be Two to There")
else -> println("I don't know who I am")
}
### Explanation
Here is explanation.
1. take a name x and assign an Int value of 0
2. when x is
3. 0 then show I'm Zero ( or we might said print out I'm Zero )
4. when x is 1 then I'm One
5. when x is 2 or 3 then I might be Two or Three
6. here println is not the same as print
7. println means print out as a line then wait at the next line
8. print means print out but don't move to next line
9. now the last one else, that means none of the above
10. In our case when x is not 1, 2 or 3 than else will do his job. Here is just print out I don't know who I am
# Round Two
val me = "Whale"
when(me){
"Human" -> println("I'm human, Yayyy!")
"Fox" -> println("I'm Fox, but not Megan Fox")
}
1. Above code will not print Anything
2. Why ?
3. Coz when above code execute nothing matched with the given variable (or immutalbe variable) me.
Let's fix it
val me = "Whale"
when(me){
"Human" -> println("I'm human, Yayyy!")
"Fox" -> println("I'm Fox, but not Megan Fox")
"whale" -> println("I'm whale, but small one")
"Whale" -> println("I'm Whale, but not so small")
}
So what is your guess ?
if you said I'm whale, but small one you are WRONG
you know computer are dumb, so they only know what you teached them to know. Case Sentitive Right?
The answer is I'm Whale, but not so small
but above code have some problem !
change the assign value of me to "Jedi"
here is code
val me = "Jedi"
when(me){
"Human" -> println("I'm human, Yayyy!")
"Fox" -> println("I'm Fox, but not Megan Fox")
"whale" -> println("I'm whale, but small one")
"Whale" -> println("I'm Whale, but not so small")
}
you gonna see nothing, coz we missed none of the above part
(or else).
so when can used not just for Int, String but also for whole other types. boolean, range, char, double, etc etc
# Round Three
var howMuchIHave = 0.0
when(howMuchIHave){
0 -> { // Incompatible types: Int and Double at here
print("I have zero")
print("ohh poor me!")
}
0.0 -> {
println("I still don't have money")
println("I have to rob the bank, just kidding")
}
else -> println("How much I have is $howMuchIHave") } Above code will print Incompatible types: Int and Double you know // means comment right? the problem here is we are performing different type all together. here is the fix var howMuchIHave = 0.0 when(howMuchIHave){ 0.0 -> { // Incompatible types: Int and Double at here print("I have zero") print("ohh poor me!") } 0.0 -> { println("I still don't have money") println("I have to rob the bank, just kidding") } else -> println("How much I have is$howMuchIHave")
}
print out I have zeroohh poor me!
Hummm why? we have two 0.0 but only one print out.
coz we have duplicate label in when
but did you noticed about the difference between print and println
that is progress.
var howMuchIHave = 0.1
when(howMuchIHave){
0.0 -> { // Incompatible types: Int and Double at here
print("I have zero")
print("ohh poor me!")
}
0.1 -> {
println("Something is better than nothing")
println("One small penny for man, one giant leap for me")
}
else -> println("How much I have is $howMuchIHave") } now the output is Something is better than nothing One small penny for man, one giant leap for me so here is your task change the var howMuchIHave = 0.1 to var howMuchIHave = 'a' comment the output press me to the kotlin online ide # Round Four var something = "something" when(something){ is String -> println("I'm a String") "something" -> println("My value is something") else -> println("I'm nothing") } here will print out I'm a String so change the order to above code to when(something){ "something" -> println("My value is something") is String -> println("I'm a String") else -> println("I'm nothing") } this time will be My value is something print out 1. is keyword check the type of the instance 2. code order is important # Round Five val x = 1101 when(x){ in 0..1 -> println("between 0 and 1") !in 1100..1101 -> println("not between 1000 and 1111") in 1100..1101 -> println("between 1100 and 1101") else -> println("I have no idea at all") } above code will print out between 1100 and 1101 1. .. is called Range operator in kotlin 2. in 0..1 means between 0 and 1 3. !in means not in 4. so !in 1100..1101 means not between 1100 and 1101 # Round Six val number = 2 when{ number.isEven() -> prinltn("Even number") number.isOdd() -> println("Odd number") else -> println("error") } here, we used when liked if else # Round Seven var responseCode = 400 val errorMessage = when(responseCode){ 200 -> "Ok" 201 -> "Created" 203 -> "Non-Authoritative Information" 400 -> "Bad Request" 403 -> "Forbidden" 500 -> "Internal Server Error" else -> "Unknown Error" } println("HTTP Status Code is$responseCode and result is \$errorMessage")
The output is HTTP Status Code is 400 and result is Bad Request
When responseCode is something that match within my code I'll take that value and passed to variable named errorMessage.
What important is when you used when liked that, don't forget the else part. change the responseCode to 501 and
remove or comment else -> "Unknown Error sentence.
Run it. | 2020-01-20 04:04:50 | {"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": 1, "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": 0.2598390579223633, "perplexity": 9634.524377571843}, "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-2020-05/segments/1579250597230.18/warc/CC-MAIN-20200120023523-20200120051523-00006.warc.gz"} |
https://www.physicsforums.com/threads/old-laptop-monitor.110559/ | # Old Laptop Monitor
1. Feb 13, 2006
### Shawnzyoo
Well I recently just tore apart an old laptop and salvaged the monitor (still good)
Now I am pretty sure it is nearly impossible (for me right now) to turn it into a useful desktop monitor
but does anyone know of any other cool projects/ideas to do with it?
thanks
2. Feb 17, 2006
### seang
I'm sure this would require a little work (a lot) but a dynamic picture that you'd hang on the wall would be neat.
3. Mar 1, 2006
### nsimmons
Ive looked into this for years. You need a special controller card = \$ that will work with the exact model of lcd. In every instance i could think of its vastly cheaper to but a lcd monitor and rip it apart for your needs.
Last edited by a moderator: Nov 28, 2011
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Have something to add? | 2017-01-16 19:14:18 | {"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": 1, "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": 0.17411890625953674, "perplexity": 3003.4595075938046}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2017-04/segments/1484560279248.16/warc/CC-MAIN-20170116095119-00063-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://de.wikibooks.org/wiki/Serlo:_EN:_Span | # Span – Serlo
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In linear algebra, the span of a subset ${\displaystyle M}$ of a vector space ${\displaystyle V}$ over a field ${\displaystyle K}$ is the set of all linear combinations with vectors from ${\displaystyle M}$ and scalars from ${\displaystyle K}$. The span is often called the linear hull of ${\displaystyle M}$ or the span of ${\displaystyle M}$.
The span forms a subspace of the vector space ${\displaystyle V}$, namely the smallest subspace that contains ${\displaystyle M}$.
## Derivation of the span
### Generating vectors of the ${\displaystyle xy}$-plane
We consider the vector space ${\displaystyle \mathbb {R} ^{3}}$ and restrict to the ${\displaystyle xy}$-plane. I.e., the set of all vectors of the form ${\displaystyle (a,b,0)^{T}}$ with ${\displaystyle a,b\in \mathbb {R} }$:
Each vector of this plane can be written as a linear combination of the vectors ${\displaystyle (1,0,0)^{T}}$ and ${\displaystyle (0,1,0)^{T}}$:
${\displaystyle {\begin{pmatrix}a\\b\\0\end{pmatrix}}=a\cdot {\begin{pmatrix}1\\0\\0\end{pmatrix}}+b\cdot {\begin{pmatrix}0\\1\\0\end{pmatrix}}}$
With the set of these linear combinations, every point of the ${\displaystyle xy}$-plane can be reached. In particular, the two vectors ${\displaystyle (1,0,0)^{T}}$ and ${\displaystyle (0,1,0)^{T}}$ lie in the ${\displaystyle xy}$-plane. Furthermore, all linear combinations of the two vectors ${\displaystyle (1,0,0)^{T}}$ and ${\displaystyle (0,1,0)^{T}}$ lie in the ${\displaystyle xy}$-plane. This is because the ${\displaystyle z}$ component of the two vectors under consideration is ${\displaystyle 0}$ and thus the third component of the linear combination of the vectors must also be ${\displaystyle 0}$.
In summary, we can state: Every vector of the ${\displaystyle xy}$-plane is a linear combination of ${\displaystyle (1,0,0)^{T}}$ and ${\displaystyle (0,1,0)^{T}}$. Every linear combination of these two vectors is also an element of this plane. So the vectors ${\displaystyle (1,0,0)^{T}}$ and ${\displaystyle (0,1,0)^{T}}$ generate the ${\displaystyle xy}$-plane. Or as a mathematician would say, they span the ${\displaystyle xy}$-plane (like two rods spanning a side of a tent).
The ${\displaystyle xy}$-plane is a subspace of the vector space ${\displaystyle \mathbb {R} ^{3}}$. We call this subspace ${\displaystyle U}$. Our two vectors span the subspace (=plane) ${\displaystyle U}$. So we write
${\displaystyle U=\operatorname {span} \left\{{\begin{pmatrix}1\\0\\0\end{pmatrix}},{\begin{pmatrix}0\\1\\0\end{pmatrix}}\right\}.}$
We say that "${\displaystyle U}$ is substpace generated by the two vectors ${\displaystyle (1,0,0)^{T}}$ and ${\displaystyle (0,1,0)^{T}}$" or that "${\displaystyle U}$ is the linear hull of the two vectors ${\displaystyle (1,0,0)^{T}}$ and ${\displaystyle (0,1,0)^{T}}$ or even better: ${\displaystyle U}$ is the span of the two vectors ${\displaystyle (1,0,0)^{T}}$ and ${\displaystyle (0,1,0)^{T}}$.
Are these generating vectors unique? The answer is no, because the plane ${\displaystyle U}$ can also be spanned by two vectors like ${\displaystyle (1,0,0)^{T}}$ and ${\displaystyle (1,1,0)^{T}}$:
${\displaystyle {\begin{pmatrix}a\\b\\0\end{pmatrix}}=(a-b)\cdot {\begin{pmatrix}1\\0\\0\end{pmatrix}}+b\cdot {\begin{pmatrix}1\\1\\0\end{pmatrix}}.}$
There is hence also
${\displaystyle U=\operatorname {span} \left\{{\begin{pmatrix}1\\0\\0\end{pmatrix}},{\begin{pmatrix}1\\1\\0\end{pmatrix}}\right\}.}$
Thus, the two vectors spanning a plane are not necessarily unique.
Intuitively, we can think of the span of vectors as the set of all possible linear combinations that can be built from these vectors. In our example this means
${\displaystyle \operatorname {span} \left\{{\begin{pmatrix}1\\0\\0\end{pmatrix}},{\begin{pmatrix}0\\1\\0\end{pmatrix}}\right\}=\left\{a\cdot {\begin{pmatrix}1\\0\\0\end{pmatrix}}+b\cdot {\begin{pmatrix}0\\1\\0\end{pmatrix}}{\Bigg |}\,a,b\in \mathbb {R} \right\}.}$
Another intuition is the following: The span of a set ${\displaystyle M}$ describes the vector space where all combinations of directions represented by elements from ${\displaystyle M}$ are merged.
### The span of even monomials
We now examine a slightly more complicated example: Consider the vector space ${\displaystyle V}$ of polynomials over ${\displaystyle \mathbb {R} }$. Let ${\displaystyle M=\{x^{n}|\,n\in \mathbb {N} _{0}{\text{ is even}}\}\subset V}$. The elements from ${\displaystyle M}$ are the monomials ${\displaystyle 1}$, ${\displaystyle x^{2}}$, ${\displaystyle x^{4}}$, ${\displaystyle x^{6}}$ ans so on. In other words, all monomials that have an even exponent. For odd exponents, however ${\displaystyle x,x^{3},x^{5},...\notin M}$. We consider ${\displaystyle \operatorname {span} (M)}$, the set of all linear combinations with vectors of ${\displaystyle M}$. For example ${\displaystyle 2x^{2}+5x^{4}+9x^{8}+7x^{12}}$ is an element in ${\displaystyle \operatorname {span} (M)}$. In particualr, ${\displaystyle \operatorname {span} (M)}$ is a subspace of ${\displaystyle V}$ since it contains polynomials.
Further, the set ${\displaystyle \operatorname {span} (M)}$ is not empty, since it contains for instance ${\displaystyle x^{2}\in M}$.
Let us now consider two polynomials ${\displaystyle p,q\in \operatorname {span} (M)}$. By construction of ${\displaystyle \operatorname {span} (M)}$, ${\displaystyle p}$ and ${\displaystyle q}$ consist exclusively of monomials with an even exponent. Thus, of addition of ${\displaystyle p}$ and ${\displaystyle q}$ also results in a polynomial with exclusively even exponents. The set ${\displaystyle \operatorname {span} (M)}$ is therefore closed with respect to addition.
The same argument gives us completeness with respect to scalar multiplication. Thus the set ${\displaystyle \operatorname {span} (M)}$ is a subspace of the vector space of all polynomials. As we will see later, it is even the smallest subspace that contains ${\displaystyle M}$.
## Definition of the span
Above, we found out that the span of a set ${\displaystyle M}$ is the set of all linear combinations with vectors from ${\displaystyle M}$. Intuitively, the span is the subspace resulting from the union of all directions given by vectors from ${\displaystyle M}$. Now, we make this intuition mathematically precise.
Definition (Span of a set)
Let ${\displaystyle V}$ be a vector space over the field ${\displaystyle K}$. Let ${\displaystyle M\subseteq V}$ be a non-empty set. We define the span of ${\displaystyle M}$ as the set of all vectors from ${\displaystyle V}$ which can be represented as a finite linear combination of vectors from ${\displaystyle M}$ and denote it as ${\displaystyle \operatorname {span} (M)}$:
${\displaystyle \operatorname {span} (M)=\left\{\sum _{i=1}^{n}\lambda _{i}\cdot m_{i}{\Bigg |}\ n\in \mathbb {N} ,\,\lambda _{1},\ldots ,\lambda _{n}\in K,\,m_{1},\ldots ,m_{n}\in M\right\}}$
For the empty set we define:
${\displaystyle \operatorname {span} (\emptyset )=\{0\}}$
Alternatively, one can call the span of a set the generated subspace or linear hull.
Hint
The sum always has only finitely many summands, even if M is infinite.
Hint
Occasionally, the notation ${\displaystyle \langle M\rangle _{K}}$ is also used for the span. The advantage of this notation is that it is clear which field defines the vector space. It makes a difference which field we use as a basis. For the example for ${\displaystyle M:=\{1\}}$ we have that ${\displaystyle \pi \cdot 1\in \langle M\rangle _{\mathbb {R} }}$, but ${\displaystyle \pi \cdot 1\notin \langle M\rangle _{\mathbb {Q} }}$. It can be shown that ${\displaystyle \langle M\rangle _{\mathbb {Q} }=\mathbb {Q} }$ and ${\displaystyle \langle M\rangle _{\mathbb {R} }=\mathbb {R} }$.
## Example
Example (Line through the origin is a certain span)
Let ${\displaystyle x=(1,2)^{T}\in \mathbb {R} ^{2}}$. We consider the set ${\displaystyle \lbrace x\rbrace }$ as a subset of the vector space ${\displaystyle \mathbb {R} ^{2}}$. The span ${\displaystyle \operatorname {span} (\lbrace x\rbrace )}$ is the straight line through the origin pointing in the direction of the vector ${\displaystyle (1,2)^{T}}$.
${\displaystyle \operatorname {span} (\lbrace x\rbrace )=\operatorname {span} \left(\left\lbrace {\begin{pmatrix}1\\2\end{pmatrix}}\right\rbrace \right)=\left\lbrace \rho \cdot {\begin{pmatrix}1\\2\end{pmatrix}}\,{\bigg |}\,\rho \in \mathbb {R} \right\rbrace }$
Example (plane through the origin as a span)
Let ${\displaystyle (5,0,0)^{T}}$ and ${\displaystyle (0,3,0)^{T}}$ be two vectors from ${\displaystyle \mathbb {R} ^{3}}$. The span of these two vectors is the ${\displaystyle xy}$-plane. The following transformation shows
{\displaystyle {\begin{aligned}\operatorname {span} \left(\left\lbrace {\begin{pmatrix}5\\0\\0\end{pmatrix}},{\begin{pmatrix}0\\3\\0\end{pmatrix}}\right\rbrace \right)&=\left\lbrace \lambda \cdot {\begin{pmatrix}5\\0\\0\end{pmatrix}}+\mu \cdot {\begin{pmatrix}0\\3\\0\end{pmatrix}}{\Bigg |}\ \lambda ,\mu \in \mathbb {R} \right\rbrace \\[0.3em]&=\,\left\lbrace 5\cdot \lambda \cdot {\begin{pmatrix}1\\0\\0\end{pmatrix}}+3\cdot \mu \cdot {\begin{pmatrix}0\\1\\0\end{pmatrix}}{\Bigg |}\ \lambda ,\mu \in \mathbb {R} \right\rbrace \\[0.3em]&\ {\color {OliveGreen}\left\downarrow {\text{set }}{{\tilde {\lambda }}=5\cdot \lambda ,\ {\tilde {\mu }}=3\cdot \mu }\ {\text{ bzw. }}\ \lambda ={\frac {1}{5}}\cdot {\tilde {\lambda }},\ \mu ={\frac {1}{3}}\cdot {\tilde {\mu }}\right.}\\[0.3em]&=\,\left\lbrace {\tilde {\lambda }}\cdot {\begin{pmatrix}1\\0\\0\end{pmatrix}}+\cdot {\tilde {\mu }}\cdot {\begin{pmatrix}0\\1\\0\end{pmatrix}}{\Bigg |}\ {\tilde {\lambda }},{\tilde {\mu }}\in \mathbb {R} \right\rbrace \\[0.3em]&=\,\left\{{\begin{pmatrix}{\tilde {\lambda }}\\{\tilde {\mu }}\\0\end{pmatrix}}{\Bigg |}\ {\tilde {\lambda }},{\tilde {\mu }}\in \mathbb {R} \right\}\end{aligned}}}
## Overview: Properties of the span
Let ${\displaystyle V}$ be a ${\displaystyle K}$-vector space, ${\displaystyle M}$, ${\displaystyle N\subseteq V}$ subsets of ${\displaystyle V}$ and ${\displaystyle W\subseteq V}$ a subspace of ${\displaystyle V}$. Then, we have
• For a vector ${\displaystyle v\in V}$ we have ${\displaystyle \operatorname {span} (\{v\})=\{\lambda \cdot v|\lambda \in K\}}$
• If ${\displaystyle N\subseteq M}$, then ${\displaystyle \operatorname {span} (N)\subseteq \operatorname {span} (M)}$
• From ${\displaystyle \operatorname {span} (N)=\operatorname {span} (M)}$ one can usually not conclude ${\displaystyle N=M}$
• ${\displaystyle M\subseteq \operatorname {span} (M)}$
• ${\displaystyle \operatorname {span} (M)}$ is a subspace of ${\displaystyle V}$
• For a subspace ${\displaystyle W}$ we have ${\displaystyle \operatorname {span} (W)=W}$
• ${\displaystyle \operatorname {span} (M)}$ is the smallest subspace of ${\displaystyle V}$ including ${\displaystyle M}$
• ${\displaystyle N\subseteq \operatorname {span} (M)\iff \operatorname {span} (M)=\operatorname {span} (M\cup N)}$
• ${\displaystyle \operatorname {span} (\operatorname {span} (M))=\operatorname {span} (M)}$
## Properties of the span
### The span of a vector ${\displaystyle v}$ in ${\displaystyle V}$
For a vector ${\displaystyle v\in V}$ we have that ${\displaystyle \operatorname {span} (\{v\})=\{\lambda \cdot v\,|\ \lambda \in K\}}$. For the zero vector ${\displaystyle v=0}$ the span again consists only of the zero vector, so ${\displaystyle \operatorname {span} (\{0\})=\{0\}}$. If ${\displaystyle v\neq 0}$ holds, then ${\displaystyle \operatorname {span} (\{v\})}$ is exactly the set of elements that lie on the line through the origin im direction of the vector ${\displaystyle v}$.
### Span preserves subsets
Theorem (Span preserves subsets)
Let ${\displaystyle V}$ be a ${\displaystyle K}$-vector space and let ${\displaystyle M,N\subseteq V}$. If ${\displaystyle N\subseteq M}$, then also ${\displaystyle \operatorname {span} (N)\subseteq \operatorname {span} (M)}$.
Proof (Span preserves subsets)
Since ${\displaystyle \operatorname {span} (\emptyset )=\{0\}}$ and ${\displaystyle 0}$ is an element in the span of every set, we have ${\displaystyle \operatorname {span} (\emptyset )=\{0\}\subseteq M}$.
Thus we can assume without loss of generality that ${\displaystyle \emptyset \neq N\subseteq M}$. We consider any element ${\displaystyle v\in \operatorname {span} (N)}$. By the definition of the span, vectors ${\displaystyle v_{1},...,v_{n}\in N}$ and ${\displaystyle \lambda _{1},...,\lambda _{n}\in K}$ exist such that ${\displaystyle v=\sum _{i=1}^{n}\lambda _{i}v_{i}}$. Because of ${\displaystyle N\subseteq M}$ we have for all ${\displaystyle v_{i}}$ with ${\displaystyle 1\leq i\leq n}$ that ${\displaystyle v_{i}\in M}$. Hence also ${\displaystyle v\in \operatorname {span} (M)}$. Consequently, ${\displaystyle \operatorname {span} (N)\subseteq \operatorname {span} (M)}$.
Hint
The converse of the above theorem does not hold true in general! By this we mean: From ${\displaystyle \operatorname {span} (N)\subseteq \operatorname {span} (M)}$ we cannot conclude ${\displaystyle N\subseteq M}$.
A possible counterexample is:
${\displaystyle N=\left\{{\begin{pmatrix}1\\0\end{pmatrix}},{\begin{pmatrix}2\\0\end{pmatrix}}\right\}{\text{ and }}M=\left\{{\begin{pmatrix}1\\0\end{pmatrix}}\right\}}$
Here,
{\displaystyle {\begin{aligned}\operatorname {span} (N)&=\left\{\lambda _{1}\cdot {\begin{pmatrix}1\\0\end{pmatrix}}+\lambda _{2}\cdot {\begin{pmatrix}2\\0\end{pmatrix}}{\Bigg |}\ \lambda _{1},\lambda _{2}\in K\right\}=\left\{(\lambda _{1}+2\lambda _{2})\cdot {\begin{pmatrix}1\\0\end{pmatrix}}{\Bigg |}\ \lambda _{1},\lambda _{2}\in K\right\}\\[0.3em]\operatorname {span} (M)&=\left\{\mu \cdot {\begin{pmatrix}1\\0\end{pmatrix}}{\Bigg |}\ \mu \in K\right\}\end{aligned}}}
Thus ${\displaystyle \operatorname {span} (N)=\operatorname {span} (M)}$, since in both cases we get exactly the multiples of the vector ${\displaystyle (1,0)^{T}}$. Since the two subsets are equal, we have in particular ${\displaystyle \operatorname {span} (N)\subseteq \operatorname {span} (M)}$, but ${\displaystyle N\nsubseteq M}$. Therefore, the converse of the theorem cannot hold true in general.
### The set ${\displaystyle M}$ is contained in its span
Theorem (${\displaystyle M}$ is contained in its span)
Let ${\displaystyle V}$ ein ${\displaystyle K}$-vector space and ${\displaystyle M\subseteq V}$. Then, there is ${\displaystyle M\subseteq \operatorname {span} (M)}$.
Proof (${\displaystyle M}$ is contained in its span)
If ${\displaystyle M=\emptyset }$, then ${\displaystyle \operatorname {span} (M)=\{0\}}$ , and the assertion is true.
Otherwise, let ${\displaystyle m\in M}$ be arbitrary. Then ${\displaystyle m}$ can be represented by ${\displaystyle m=1\cdot m}$. In particular, ${\displaystyle m=1\cdot m}$ is a linear combination with a summand of ${\displaystyle M}$. Thus there is ${\displaystyle m\in \operatorname {span} (M)}$, since ${\displaystyle \operatorname {span} (M)}$ contains all linear combinations of elements from ${\displaystyle M}$.
This establishes the assertion ${\displaystyle M\subseteq \operatorname {span} (M)}$.
### The span of ${\displaystyle M}$ is a subspace of ${\displaystyle V}$
Theorem (The span of ${\displaystyle M}$ is a subspace of ${\displaystyle V}$)
${\displaystyle \operatorname {span} (M)}$ is a subspace of ${\displaystyle V}$
Proof (The span of ${\displaystyle M}$ is a subspace of ${\displaystyle V}$)
If ${\displaystyle M}$ is the empty set, then by definition ${\displaystyle \operatorname {span} (M)=\{0\}}$, and that is a subspace of ${\displaystyle V}$. From now on we may therefore assume that ${\displaystyle M}$ is not empty.
First, it is clear that ${\displaystyle \operatorname {span} (M)\subseteq V}$. But this is obvious according to the definition of vector space and span.
We still have to show ${\displaystyle \operatorname {span} (M)}$ is subspace of ${\displaystyle V}$. In other words,we have to show that
• ${\displaystyle \operatorname {span} (M)\neq \emptyset }$
• for two elements ${\displaystyle u,v\in \operatorname {span} (M)}$ we have also ${\displaystyle u+v\in \operatorname {span} (M)}$ (completeness of addition)
• ${\displaystyle \rho \cdot u\in \operatorname {span} (M)}$ (completeness of scalar multiplication)
Proof step: ${\displaystyle \operatorname {span} (M)\neq \emptyset }$
Since ${\displaystyle \operatorname {span} M}$ is not empty, there exists at least one ${\displaystyle u\in M}$. Then ${\displaystyle u}$ can be written as ${\displaystyle u=1\cdot u}$, and therefore ${\displaystyle u}$ itself is in ${\displaystyle \operatorname {span} M}$. So this condition is fulfilled.
We show the completeness concerning the vector addition. Let ${\displaystyle u,v\in \operatorname {span} (M)}$. Then there are vectors ${\displaystyle m_{1},\ldots ,m_{n}\in M}$ and ${\displaystyle n_{1},\ldots ,n_{k}\in M}$, so that ${\displaystyle u=\sum _{i=1}^{n}\lambda _{i}\cdot m_{i}}$ and ${\displaystyle v=\sum _{i=1}^{k}\mu _{i}\cdot n_{i}}$. So
${\displaystyle u+v=\sum _{i=1}^{n}\lambda _{i}\cdot m_{i}+\sum _{i=1}^{k}\mu _{i}\cdot n_{i}\in \operatorname {span} (M)}$
Hence, ${\displaystyle \operatorname {span} (M)}$ is complete with respect to addition.
Proof step: Completeness of scalar multiplication
Establish completeness of scalar multiplication is done easily:
${\displaystyle \rho \cdot u=\rho \cdot \sum _{i=1}^{n}\lambda _{i}\cdot m_{i}=\sum _{i=1}^{n}(\rho \cdot \lambda _{i})\cdot m_{i}\in \operatorname {span} (M)}$
Thus we have proved that ${\displaystyle \operatorname {span} (M)}$ is a subspace of the vector space ${\displaystyle V}$.
### The span of a subspace ${\displaystyle W}$ is again ${\displaystyle W}$
Theorem (The span of ${\displaystyle M}$ is the smallest subspace von ${\displaystyle V}$)
The span of a subspace ${\displaystyle W}$ is again ${\displaystyle W}$
Proof (The span of ${\displaystyle M}$ is the smallest subspace von ${\displaystyle V}$)
Since ${\displaystyle W}$ is a subspace, for some vectors ${\displaystyle m_{1},\ldots ,m_{n}\in W}$ also all linear combinations of the ${\displaystyle m_{1},\ldots ,m_{n}}$ are contained in ${\displaystyle W}$. Therefore ${\displaystyle \operatorname {span} (W)\subseteq W}$. Together with ${\displaystyle W\subseteq \operatorname {span} (W)}$ our assertion follows.
### The span of ${\displaystyle M}$ is the smallest subspace of ${\displaystyle V}$, containing ${\displaystyle M}$
Theorem (The span of ${\displaystyle M}$ is the smallest subspace of ${\displaystyle V}$)
Let ${\displaystyle V}$ be a ${\displaystyle K}$-vector space and let ${\displaystyle M\subseteq V}$.
Then, ${\displaystyle \operatorname {span} (M)}$ is the smallest subspace of ${\displaystyle V}$ including ${\displaystyle M}$.
Proof (The span of ${\displaystyle M}$ is the smallest subspace of ${\displaystyle V}$)
We already know that ${\displaystyle \operatorname {span} (M)}$ is a subspace. Now we show that ${\displaystyle \operatorname {span} (M)}$ is the smallest subspace containing ${\displaystyle M}$.
If ${\displaystyle M=\emptyset }$, the assertion is obviously true, since then ${\displaystyle \operatorname {span} (M)=\{0\}}$.
Let ${\displaystyle W}$ be a subspace of ${\displaystyle V}$ containing ${\displaystyle M}$. Our goal is to show that ${\displaystyle \operatorname {span} (M)\subseteq isW}$. Since this would imply that the subspace ${\displaystyle \displaystyle \operatorname {span} (M)}$ is smaller or equal to every other subspace ${\displaystyle W}$ containing ${\displaystyle M}$.
Now if ${\displaystyle u\in \operatorname {span} (M)}$, then there are some ${\displaystyle m_{1},...,m_{n}\in M}$ and ${\displaystyle \lambda _{1},\ldots ,\lambda _{n}\in K}$ such that ${\displaystyle \displaystyle u=\sum _{i=1}^{n}\lambda _{i}m_{i}}$ (by definition of the span).
Since ${\displaystyle W}$ is a subspace and ${\displaystyle m_{1},\ldots ,m_{n}\in W}$, all linear combinations of ${\displaystyle m_{1},\ldots ,m_{n}}$ are also contained in ${\displaystyle W}$. This implies our assertion ${\displaystyle \operatorname {span} (M)\subseteq W}$.
### Idempotency of the span
Theorem (Idempotency of the span)
Let ${\displaystyle V}$ be a ${\displaystyle K}$-vector space and ${\displaystyle M\subseteq V}$. Then we have ${\displaystyle \operatorname {span} (\operatorname {span} (M))=\operatorname {span} (M)}$. This property of the span is called idempotency.
Proof (Idempotency of the span)
For ${\displaystyle M=\emptyset }$ we have ${\displaystyle \operatorname {span} (\emptyset )=\{0\}}$ and ${\displaystyle \operatorname {span} (\operatorname {span} (\emptyset ))=\operatorname {span} (\{0\})=\{0\}}$.
Therefore, we can now assume that ${\displaystyle M}$ is not empty.
We already know that ${\displaystyle \operatorname {span} (M)\subseteq \operatorname {span} (\operatorname {span} (M))}$. So it only remains to show that ${\displaystyle \operatorname {span} (\operatorname {span} (M))\subseteq \operatorname {span} (M)}$.
Let ${\displaystyle v\in \operatorname {span} (\operatorname {span} (M))}$. Then ${\displaystyle v}$ can be written as
${\displaystyle v=\lambda _{1}\cdot u_{1}+...+\lambda _{n}\cdot u_{n}}$
with ${\displaystyle \lambda _{1},...,\lambda _{n}\in K}$ and ${\displaystyle u_{1},...,u_{n}\in \operatorname {span} (M)}$. Since ${\displaystyle u_{i}\in \operatorname {span} (M)}$ for all ${\displaystyle 1\leq i\leq n}$, every ${\displaystyle u_{i}}$ can be written as a linear combination of elements in ${\displaystyle M}$:
${\displaystyle u_{i}=\mu _{1}\cdot w_{i,1}+...+\mu _{m_{i}}\cdot w_{i,m_{i}}}$
where ${\displaystyle \mu _{1},...,\mu _{m_{i}}\in K}$ and ${\displaystyle w_{i,1},...,w_{i,m_{i}}\in M}$. We now write the ${\displaystyle u_{i}}$ within ${\displaystyle v}$as a linear combination of the ${\displaystyle w_{i,j}}$:
{\displaystyle {\begin{aligned}v&=\sum _{i=1}^{n}\lambda _{i}u_{i}\\[0.3em]&{\color {OliveGreen}\left\downarrow \ u_{i}=\mu _{1}\cdot w_{i,1}+...+\mu _{m_{i}}\cdot w_{i,m_{i}}\right.}\\[0.3em]&=\sum _{i=1}^{n}\lambda _{i}\left(\sum _{j=1}^{m_{i}}\mu _{j}w_{i,j}\right)\\[0.3em]&=\sum _{i=1}^{n}\sum _{j=1}^{m_{i}}(\lambda _{i}\mu _{j})w_{i,j}\end{aligned}}}
For all ${\displaystyle 1\leq i\leq n}$, the sum ${\displaystyle \sum _{j=1}^{m_{i}}\lambda _{i}\mu _{j}}$ is an element of the field ${\displaystyle K}$. So we obtain ${\displaystyle v\in \operatorname {span} (M)}$, which was to be shown.
Alternative proof (Idempotency of the span)
We know that ${\displaystyle \operatorname {span} (M)}$ is a subspace of ${\displaystyle V}$, and that the span ${\displaystyle \operatorname {span} (W)}$ of a subspace ${\displaystyle W}$ is ${\displaystyle W}$ again.
Therefore, ${\displaystyle \operatorname {span} (\operatorname {span} (M))}$ is again ${\displaystyle \operatorname {span} (M)}$.
### Adding elements of the span doesn't change the span
Theorem (Adding elements of the span doesn't change the span)
Let ${\displaystyle V}$ be a ${\displaystyle K}$-vector space and ${\displaystyle M}$, ${\displaystyle N\subseteq V}$. Then, we have
${\displaystyle N\subseteq \operatorname {span} (M)\Longleftrightarrow \operatorname {span} (M)=\operatorname {span} (M\cup N)}$
Proof (Adding elements of the span doesn't change the span)
We will establish the two implications ${\displaystyle \implies }$ and ${\displaystyle \Longleftarrow }$:
Proof step: ${\displaystyle N\subseteq \operatorname {span} (M)\implies \operatorname {span} (M)=\operatorname {span} (M\cup N)}$
The statement ${\displaystyle \operatorname {span} (M)\subseteq \operatorname {span} (M\cup N)}$ does always hold, since ${\displaystyle M\subseteq M\cup N}$. So all that remains is to show that ${\displaystyle \operatorname {span} (M\cup N)\subseteq \operatorname {span} (M)}$ holds. In order to do this, we consider an element ${\displaystyle u\in \operatorname {span} (M\cup N)}$. We can write it as
${\displaystyle u=\sum _{i=0}^{n}\lambda _{i}v_{i}+\sum _{i=0}^{m}\mu _{i}w_{i},}$
with ${\displaystyle v_{1},...,v_{n}\in M}$, ${\displaystyle w_{1},...,w_{m}\in N}$, ${\displaystyle \lambda _{1},...,\lambda _{n}\in K}$ and ${\displaystyle \mu _{1},...,\mu _{m}\in K}$. Since ${\displaystyle N\subseteq \operatorname {span} (M)}$, one can write ${\displaystyle w_{i}}$ for all ${\displaystyle 1\leq i\leq m}$ as a linear combination of elements from ${\displaystyle M}$:
${\displaystyle w_{i}=\rho _{1}\cdot {\hat {v}}_{1}+...+\rho _{m_{i}}\cdot {\hat {v}}_{m_{i}}}$
where ${\displaystyle \rho _{1},...,\rho _{m_{i}}\in K}$ and ${\displaystyle {\hat {v}}_{1},...,{\hat {v}}_{m_{i}}\in M}$. Now, we plug this expression for ${\displaystyle w_{i}}$ into the formula above:
{\displaystyle {\begin{aligned}u&=\sum _{i=0}^{n}\lambda _{i}v_{i}+\sum _{i=0}^{m}\mu _{i}\left(\sum _{j=1}^{m_{i}}\rho _{j}{\hat {v}}_{j}\right)\\[0.3em]&=\sum _{i=0}^{n}\lambda _{i}v_{i}+\sum _{i=1}^{m}\sum _{j=1}^{m_{i}}(\mu _{i}\rho _{j}){\hat {v}}_{j}\end{aligned}}}
We have thus represented ${\displaystyle u}$ as a linear combination of vectors from ${\displaystyle M}$ and hence ${\displaystyle \operatorname {span} (M)=\operatorname {span} (M\cup N)}$.
Proof step: ${\displaystyle N\subseteq \operatorname {span} (M)\Longleftarrow \operatorname {span} (M)=\operatorname {span} (M\cup N)}$
We show this statement using a proof by contradiction. We assume that there is some ${\displaystyle w\in N}$ but ${\displaystyle w\notin \operatorname {span} (M)}$. We now define an element ${\displaystyle u:=\sum _{i=0}^{n}\lambda _{i}v_{i}+w}$, with ${\displaystyle v_{1},...,v_{n}\in M}$ and ${\displaystyle \lambda _{1},...,\lambda _{n}\in K}$.
Now ${\displaystyle u}$ is a linear combination of vectors from ${\displaystyle M\cup N}$. Thus ${\displaystyle u\in \operatorname {span} (M\cup N)}$, since ${\displaystyle w\in N}$. However, we also have ${\displaystyle u\notin \operatorname {span} (M)}$, since ${\displaystyle w\notin \operatorname {span} (M)}$. But this contradicts the assumption ${\displaystyle \operatorname {span} (M)=\operatorname {span} (M\cup N)}$.
Hence our assumption is false and ${\displaystyle N\subseteq \operatorname {span} (M)}$ must hold.
Alternative proof (First proof step)
One can also argue as follows: we have ${\displaystyle M,N\subseteq \operatorname {span} (M)}$. So also ${\displaystyle M\cup N\subseteq \operatorname {span} (M)}$.
We have already proved that then ${\displaystyle \operatorname {span} (M\cup N)\subseteq \operatorname {span} (\operatorname {span} (M))}$. This set is the same as ${\displaystyle \operatorname {span} (M)}$ because of the idempotency of the span, so ${\displaystyle \operatorname {span} (M\cup N)=\operatorname {span} (M)}$.
## Check whether vectors are inside the span
After we have learned some properties of the span, we will show in this section how we can check whether a vector of ${\displaystyle V}$ lies within the span of ${\displaystyle M\subseteq V}$ or not. We will see that in order to answer this question we have to solve a linear system of equations.
Example (Plane and line through the origin)
Let's start with a simple example from the ${\displaystyle \mathbb {R} ^{2}}$. We consider the line through the origin ${\displaystyle \operatorname {span} (M)}$ with the one-element subset ${\displaystyle M=\{(4,3)^{T}\}}$ of the plane ${\displaystyle \mathbb {R} ^{2}}$. The question now is whether the vector ${\displaystyle (12,9)^{T}}$ lies in the span of ${\displaystyle M}$. One can immediately see that
${\displaystyle {\begin{pmatrix}12\\9\end{pmatrix}}=3\cdot {\begin{pmatrix}4\\3\end{pmatrix}}}$
holds. In other words
${\displaystyle {\begin{pmatrix}12\\9\end{pmatrix}}\in \operatorname {span} (M)}$
Mathematically, we have to solve a system of equations. In our simple example, the exercise is to find a ${\displaystyle \lambda \in K}$ such that
${\displaystyle {\begin{pmatrix}12\\9\end{pmatrix}}=\lambda \cdot {\begin{pmatrix}4\\3\end{pmatrix}}}$
From this equation we obtain the linear system of equations
{\displaystyle {\begin{aligned}12&=\lambda \cdot 4\\9&=\lambda \cdot 3\end{aligned}}}
with the obvious solution ${\displaystyle \lambda =3}$.
Example (Polynomials)
Let us now examine an example whose solution is not obvious at first sight. For this we consider the subset of the monomials ${\displaystyle N=\{x^{3},x^{1},x^{0}\}}$ and the polynomial ${\displaystyle p(x)=(x-2)^{3}+3(x-2)^{2}}$. We want to show that the polynomial is not in the span of ${\displaystyle N}$. To do this, it suffices to prove that ${\displaystyle p(x)}$ cannot be represented as a linear combination of the monomials of ${\displaystyle N}$. We can see this by transforming the polynomial:
${\displaystyle (x-2)^{3}+3(x-2)^{2}=(x^{3}+12x-6x^{2}-8)+(3x^{2}-12x+12)=x^{3}-3x^{2}+4}$
We see that a summand contains the monomial ${\displaystyle x^{2}}$, but this monomial is not contained in ${\displaystyle N}$. Thus the polynomial is not in the span of the set ${\displaystyle N}$.
Example (Vectors from ${\displaystyle \mathbb {R} ^{4}}$)
We consider the subset ${\displaystyle M=\{(1,-2,3,2)^{T},(3,0,2,1)^{T},(0,-2,1,-3)^{T},(1,1,-2,2)^{T}\}}$ of ${\displaystyle \mathbb {R} ^{4}}$ and want to prove that the vector ${\displaystyle (2,-9,2,-3)^{T}\in \operatorname {span} (M)}$. For this we have to show that there are coefficients ${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4}\in \mathbb {R} }$ such that
${\displaystyle {\begin{pmatrix}2\\-9\\2\\-3\end{pmatrix}}=\lambda _{1}\cdot {\begin{pmatrix}1\\-2\\3\\2\end{pmatrix}}+\lambda _{2}\cdot {\begin{pmatrix}3\\0\\2\\1\end{pmatrix}}+\lambda _{3}\cdot {\begin{pmatrix}0\\-2\\1\\-3\end{pmatrix}}+\lambda _{4}\cdot {\begin{pmatrix}1\\1\\-2\\2\end{pmatrix}}}$
From this representation we get the linear system of equations
{\displaystyle {\begin{aligned}I:&&2&=&1\cdot \lambda _{1}+3\cdot \lambda _{2}+0\cdot \lambda _{3}+1\cdot \lambda _{4}\\[0.3em]II:&&-9&=&-2\cdot \lambda _{1}+0\cdot \lambda _{2}-2\cdot \lambda _{3}+1\cdot \lambda _{4}\\[0.3em]III:&&2&=&3\cdot \lambda _{1}+2\cdot \lambda _{2}+1\cdot \lambda _{3}-2\cdot \lambda _{4}\\[0.3em]IV:&&-3&=&2\cdot \lambda _{1}+1\cdot \lambda _{2}-3\cdot \lambda _{3}+2\cdot \lambda _{4}\end{aligned}}}
with solution ${\displaystyle \lambda _{1}=2}$, ${\displaystyle \lambda _{2}=-1}$, ${\displaystyle \lambda _{3}=4}$, ${\displaystyle \lambda _{4}=3}$. Hence, we have that
${\displaystyle {\begin{pmatrix}2\\-9\\2\\-3\end{pmatrix}}=2\cdot {\begin{pmatrix}1\\-2\\3\\2\end{pmatrix}}-1\cdot {\begin{pmatrix}3\\0\\2\\1\end{pmatrix}}+4\cdot {\begin{pmatrix}0\\-2\\1\\-3\end{pmatrix}}+3\cdot {\begin{pmatrix}1\\1\\-2\\2\end{pmatrix}}}$
and therefore ${\displaystyle (2,-9,2,-3)^{T}\in \operatorname {span} (M)}$. | 2021-03-02 10:56:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 404, "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": 0.9822453260421753, "perplexity": 191.80096849172077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2021-10/segments/1614178363809.24/warc/CC-MAIN-20210302095427-20210302125427-00317.warc.gz"} |
https://yums.org.uk/2015/01/homotopy-brief-introduction/ | ## ℳατh∫θℂ
The University of York Mathematics Society - Our aim is to enhance the social experience of students, expand academic opportunities in mathematics and forge strong relations for our members to use in future careers and beyond
# A Very Brief Introduction to Homotopy by Juliet Cooke
As part of her final year project, Juliet Cooke has written a brief introduction to homotopy, one of the key concepts in algebraic topology. She has agreed to allow us to make this available on the MathSoc website under the Creative Commons Attribution License.
In this article I shall be considering the notion of homotopy equivalence, which plays a central role in Algebraic Topology.
# Homotopy Equivalence of Functions
Two contours $C_1$ and $C_2$ for which the line integrals are equal. The points in red mark singularities of the integrand.
When evaluating complex contour integrals the exact shape of the contour doesn't matter as one can 'continuously deform' the contour without changing the value of the integral so long as the 'continuous deformation' does not pass over any singularities of the integrand. This significantly simplifies the complex integration of relatively well behaved complex functions and allows one to prove Cauchy's theorem and the residue theorem.
#### Definition (Homotopy equivalence)
Let $X$ and $Y$ be two topological spaces with continuous maps $f : X \to Y$ and $g: X \to Y$ between them. Then a homotopy between $f$ and $g$ is a continuous map $H : X \times [0, 1] \to Y$ such that for all $x \in X$,
• $H(x, 0) = f$ and
• $H(x, 1) = g$.
If such a homotopy $H$ exists then $f$ and $g$ are said to be homotopy equivalent or homotopic. This is denoted $H: f \simeq g$ or simply $f \simeq g$ when this does not lead to ambiguity.
As is betrayed by the name, homotopy equivalence is an equivalence relation. Furthermore by thinking of contours as continuous maps from $[0, 1]$ to $\mathbb{C}$ we now have exactly the equivalence which matters in contour integration. So we can now reformulate the first paragraph precisely in the following Proposition.
#### Proposition
Let $f: \mathbb{C} \to \mathbb{C}$ be a holomorphic function on the open set $A \subseteq \mathbb{C}$. Then, if $C_1, C_2 : [0, 1] \to A$ are two homotopy equivalent contours,
# Homotopy Equivalence of Spaces
Two shapes that seem very similar.
Homotopy is not restricted to maps between spaces but can also be used to formulate a enlightening form of equivalence between topological spaces themselves. To illustrate this consider the two shapes in the above image (this example is from and elaborated on in Hatcher). It is clear that they have the same essential shape as they are both used to represent the same letter in English. However they are not homeomorphic as there exists no invertible map between them.
#### Definition (Homotopy equivalence of spaces)
Two spaces $X$ and $Y$ are \textbf{homotopy equivalent} if there is a homotopy equivalence $f : X \to Y$ between them. $f : X \to Y$ is a homotopy equivalence if there exists a map $g : Y \to X$ such that $gf \simeq 1_X$ and $fg \simeq 1_Y$.
However using this homotopy equivalence the $A$s would also be homotopic to the $O$ below. The two thick letters are even homeomorphic. However if we superimpose the images and then consider deformation retraction there is indeed a difference between the $A$s and the $O$s which none the less makes the two $A$s equivalent.
This $O$ is homotopic to the $A$s. So how does it differ?
When superimposed the difference is clear, $O$ is not even a subset of $A$.
#### Definition (Deformation retraction (Hatcher))
A retract between from a space $X$ to a subspace $A$ is a continuous map $r: X \to A$ such that $r|_A = 1_A$, or equivalently $r^2 = r$. A deformation retraction is a homotopy $H: 1_X \simeq r$ relative to $A$, that is $H(A, t) = 1_A$ for all $t \in [0, 1]$. In this case we say that $X$ deformation retracts onto $A$.
As can be seen in the figure above the $O$ is not even a subset of the thick $A$ thus the $A$ certainly does not deform retract onto $O$ or vice versa. However, the thick $A$ does deform retract onto the thin $A$. This deformation retraction shrinks the width of the thick $A$ and is fixed on the thin $A$ throughout.
# Homotopy Groups
The homotopic equivalence did however capture that both the $A$s and the $O$ have one hole in them. Holes in topological spaces can be detected by considered algebraic structures induced by them. In particular the fundamental group and higher homotopy groups (or homology groups which will not be covered here).
#### Fundamental Group (Strom)
Let $X$ be a topological space. The fundamental group of $X$ with respect to basepoint $x \in X$ is,
Where, if $S^1$ is given a standard basepoint $\star \in S^1$ then, $\left[ S^1, X \right]$ is the set of homotopy classes of continuous maps $f: S^1 \to X$
which preserve basepoint ($f(\star) = x$).
A loop in $O$ with winding number 2.
It can be shown that the fundamental group is in fact a group under concatenation of paths. One can also show that the homotopy classes of a loop in $O$ can be determined entirely by the loops winding number. Thus for any $x \in O$ we have that $\pi_1(0, x) \cong \mathbb{Z}$.
Two spaces that need the $2^{nd}$ homotopy group to distinguish between.
However the fundamental group cannot distinguish between $D^3 - \frac{1}{2}D^3$ and $D^3$ (shown above) as in both spaces loops are homotopy equivalent to a constant loop at basepoint. To do this we need to introduced the higher homotopy groups.
#### Definition (Higher homotopy groups (Strom))
Let $X$ be a topological space with a point $x \in X$. Then the $n^{th}$ homotopy group of $X$ at $x$ is,
The `two dimensional' hole in $D^3 - \frac{1}{2}D^3$ can now be detected by finding that $\pi_2(D^3 - \frac{1}{2}D^3, x) \cong \mathbb{Z}$. However, in general calculating higher homotopy groups can be very difficult. An illustration of this difficulty is that even calculating $\pi_n(S^m)$, in general, when $n>m$ is still an open problem of homotopy theory.
# References
• (Strom) Strom, J. 2011. Modern Classical Homotopy Theory. American Mathematical Society.
• (Hatcher) Hatcher, A. 2001. Algebraic Topology. Cambridge University Press. | 2017-06-29 10:40:46 | {"extraction_info": {"found_math": true, "script_math_tex": 83, "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": 86, "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": 0.8425089120864868, "perplexity": 243.25098843816917}, "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-2017-26/segments/1498128323908.87/warc/CC-MAIN-20170629103036-20170629123036-00070.warc.gz"} |
https://mathhelpboards.com/threads/prove-sum-and-difference-sine.5358/ | # TrigonometryProve Sum and Difference Sine?
#### Farmtalk
##### Active member
I was wondering if someone could take the time to show a proof for the sum and difference identity for sine. I've seen and learned to understand some other identities, but never this one.
I've been trying to understand more of the "why" than the "how" of mathematics, and this one is very intriguing to me
#### chisigma
##### Well-known member
I was wondering if someone could take the time to show a proof for the sum and difference identity for sine. I've seen and learned to understand some other identities, but never this one.
I've been trying to understand more of the "why" than the "how" of mathematics, and this one is very intriguing to me
When Leonhard Euler 'discovered' the relation...
$$e^{i x} = \cos x + i\ \sin x\ (1)$$
... hi 'banalized' all the previous trigonometric relations. From (1) You derive...
$$\sin x = \frac{e^{i\ x} - e^{- i\ x}}{2\ i}$$
$$\cos x = \frac{e^{i\ x} + e^{- i\ x}}{2}\ (2)$$
Using (2) and a little of algebra You can arrive to the sum and difference identity for sines and cosines...
Kind regards
$\chi$ $\sigma$
#### MarkFL
Staff member
Let's now prove the angle sum/difference identities for sine, cosine and tangent. First, we need to prove the law of cosines. I trust the reader will have available pencil and paper to make the drawings I describe.
To begin, draw a horizontal line segment and label the left end $A$ and the right end $B$ and denote the length of the segment as $c$. Now place a point, labeled $C$ above the horizontal line, which is to the right of $A$ and to the left of $B$. Draw a line segment from $A$ to $C$ and label its length $b$ and draw a line segment from $B$ to $C$ and label its length $a$. We now have an acute triangle. Now, orient a Cartesian coordinate system such that vertex $A$ is at the origin, and segment $\overline{AB}$ lies on the $x$-axis. Next drop a vertical line from vertex $C$ to line segment $\overline{AB}$, dividing $\overline{AB}$ into two parts, the left which we label with the length $u$. The length of the vertical line we label $v$, as in the following sketch:
Thus, the coordinates of the vertices are as follows:
$A=(0,0)$
$B=(c,0)$
$C=(u,v)$
Now, let angle $A$ denote the angle at vertex $A$, and the same with $B$ and $C$.
Thus, we have:
$$\displaystyle \cos(A)=\frac{u}{b}\,\therefore\,u=b\cos(A)$$
$$\displaystyle \sin(A)=\frac{v}{b}\,\therefore\,v=b\sin(A)$$
Thus, the coordinates of vertex $C$ are $(b\cos(A),b\sin(A))$. Now we can use the square of the distance formula to compute the distance from vertex $C$ to vertex $B$ (which is labeled $a$):
$$\displaystyle a^2=(b\cos(A)-c)^2+(b\sin(A)-0)^2$$
$$\displaystyle a^2=b^2\cos^2(A)-2bc\cos(A)+c^2+b^2\sin^2(A)$$
$$\displaystyle a^2=b^2\left(\sin^2(A)+\cos^2(A) \right)-2bc\cos(A)+c^2$$
$$\displaystyle a^2=b^2+c^2-2bc\cos(A)$$
I will leave it to the reader to verify this result when angle $A$ is obtuse. Now we are ready to prove the angle sum/difference identities for sine and cosine.
On the unit circle, place two points on the circle in quadrant 1, label the point closer to the $y$-axis $Q$ and the point closer to the $x$-axis $P$. Now draw three line segments, one from $O$ to $Q$, one from $O$ to $P$ and one from $P$ to $Q$. Label the angle subtended by the $x$-axis and segment $\overline{OP}$ as $t$ and the angle subtended by the $x$-axis and segment $\overline{OQ}$ as $s$. Thus, the angle subtended by segments $\overline{OP}$ and $\overline{OQ}$ is $s-t$. Refer to the following sketch:
Now we will use two methods to compute the square of the distance from point $P$ to point $Q$. Using the law of cosines, we get:
$$\displaystyle \overline{PQ}^2=\overline{OP}^2+\overline{OQ}^2-2\cdot\overline{OP}\cdot\overline{OQ}\cos(s-t)$$
Because we are on the unit circle, $\overline{OP}=\overline{OQ}=1$, so we have:
$$\displaystyle \overline{PQ}^2=1^2+1^2-2\cdot1\cdot1\cos(s-t)$$
$$\displaystyle \overline{PQ}^2=2-2\cos(s-t)$$
Now, using the distance formula, we find that:
$$\displaystyle \overline{PQ}^2=(\cos(s)-\cos(t))^2+(\sin(s)-\sin(t))^2$$
$$\displaystyle \overline{PQ}^2=\cos^2(s)-2\cos(s)\cos(t)+\cos^2(t)+\sin^2(s)-2\sin(s)\sin(t)+\sin^2(t)$$
$$\displaystyle \overline{PQ}^2=\left(\cos^2(s)+\sin^2(s) \right)+\left(\cos^2(t)+\sin^2(t) \right)-2\cos(s)\cos(t)-2\sin(s)\sin(t)$$
$$\displaystyle \overline{PQ}^2=2-2\cos(s)\cos(t)-2\sin(s)\sin(t)$$
Now, equating the two expressions for $\overline{PQ}^2$, we obtain:
$$\displaystyle 2-2\cos(s-t)=2-2\cos(s)\cos(t)-2\sin(s)\sin(t)$$
Subtract through by 2:
$$\displaystyle -2\cos(s-t)=-2\cos(s)\cos(t)-2\sin(s)\sin(t)$$
Divide through by -2:
$$\displaystyle \cos(s-t)=\cos(s)\cos(t)+\sin(s)\sin(t)$$
This is the angle-difference identity for cosine.
Now, we may write the left side as:
$$\displaystyle \cos(s+(-t))=\cos(s)\cos(t)+\sin(s)\sin(t)$$
Using the identities $\cos(-x)=\cos(x)$ and $\sin(-x)=-\sin(x)$ we have:
$$\displaystyle \cos(s+(-t))=\cos(s)\cos(-t)-\sin(s)\sin(-t)$$
Replacing $-t$ with $t$, we get:
$$\displaystyle \cos(s+t)=\cos(s)\cos()-\sin(s)\sin(t)$$
This is the angle-sum identity for cosine.
Now, for sine we can use the co-function identity $\sin(x)=\cos\left(\dfrac{\pi}{2}-x \right)$ which of course may be derived from what we have already found:
$$\displaystyle \sin(s+t)=\cos\left(\frac{\pi}{2}-(s+t) \right)=\cos\left(\left(\frac{\pi}{2}-s \right)-t \right)$$
Using the angle difference for cosine which we've already derived, we get:
$$\displaystyle \sin(s+t)=\cos\left(\frac{\pi}{2}-s \right)\cos(t)+\sin\left(\frac{\pi}{2}-s \right)\sin(t)$$
Using the co-function identities, this gives us:
$$\displaystyle \sin(s+t)=\sin(s)\cos(t)+\cos(s)\sin(t)$$
This is the angle-sum identity for sine.
Now, if we write the left side as:
$$\displaystyle \sin(s+(-t))=\sin(s)\cos(-t)+\cos(s)\sin(-t)$$
Applying the negative angle identities, we have:
$$\displaystyle \sin(s-t)=\sin(s)\cos(t)-\cos(s)\sin(t)$$
This is the angle-difference identity for sine.
Now, for the tangent function we can use $\tan(x)\equiv\dfrac{\sin(x)}{\cos(x)}$:
$$\displaystyle \tan(s+t)=\frac{\sin(s+t)}{\cos(s+t)}=\frac{\sin(s)\cos(t)+\cos(s)\sin(t)}{\cos(s)\cos(t)-\sin(s)\sin(t)}$$
Divide each term in the numerator and denominator by $\cos(s)\cos(t)$:
$$\displaystyle \tan(s+t)=\frac{\tan(s)+\tan(t)}{1-\tan(s)\tan(t)}$$
This is the angle-sum identity for tangent. For the angle-difference identity, we may write:
$$\displaystyle \tan(s-t)=\frac{\sin(s-t)}{\cos(s-t)}=\frac{\sin(s)\cos(t)-\cos(s)\sin(t)}{\cos(s)\cos(t)+\sin(s)\sin(t)}$$
Divide each term in the numerator and denominator by $\cos(s)\cos(t)$:
$$\displaystyle \tan(s-t)=\frac{\tan(s)-\tan(t)}{1+\tan(s)\tan(t)}$$
Thus, we have proven the following identities:
$$\displaystyle \sin(s\pm t)=\sin(s)\cos(t)\pm\cos(s)\sin(t)$$
$$\displaystyle \cos(s\pm t)=\cos(s)\cos(t)\mp\sin(s)\sin(t)$$
$$\displaystyle \tan(s\pm t)=\frac{\tan(s)\pm\tan(t)}{1\mp\tan(s)\tan(t)}$$
#### Farmtalk
##### Active member
I wish I could give you 5 thanks a piece! I know it probably took some time for each of you to post! I just wanted to say I'm real grateful that you both posted! Thanks a ton!!!!!
#### HallsofIvy
##### Well-known member
MHB Math Helper
One of the difficulties with something like this is that it depends upon exactly how you have defined the functions. Chisigma showed how using the "$$sin(x)= \frac{e^{ix}- e^{-ix}}{2i}$$ definition and MarkFL showed how using the definition of sine and cosine in terms of coordinates of a point on the unit circle.
Another definition of sine and cosine that I like is:
(Admittedly, this is no longer "Pre-Calculus".)
"y= cos(x) is the unique function satisfying the initial value problem y''= -y with y(0)= 1, y'(0)= 0."
and
"y= sin(x) is the unique function satisfying the initial value problem y''= -y with y(0)= 0, y'(0)= 1."
It is easy to show that sine and cosine, as defined above, are independent functions and so any solutions to the differential equation y''= -y can be written as a linear combination of those two functions. In fact it is easy to see that the solution to the initial value problem y''= -y, y(0)= A, y'(0)= B is exacdtly
y(x)= A cos(x)+ B sin(x). That is, the coefficients of cosine and sine are the initial values of y and y'.
It is then easy to see that if y(x)= (sin(x))', the derivative of sin(x) then y'(x)= (sin(x))''= - sin(x) and, differentiating again, that y''(x)= - (sin(x))'= -y. That is, this new function satisfies the same differential equation while satisfying y(0)= (sin(x))' at 0 which, by definition, is 1 and y'(0)= - sin(0)= 0. That is, the derivative of sin(x) satisfies exactly the same initial value problem as cos(x) and so, by uniqueness, (sin(x))'= cos(x).
Similarly, it is easy to see that if y(x)= (cos(x))', y satisfies the same differential equation except that now we have y'(0)= - cos(0)= -1 so that (cos(x))'= -sin(x).
Now, let y(x)= sin(x+ a) for some number a. Using the chain rule, since the derivative of x+ a is 1, it is easy to see that y again satisfies y''= -y but now y(0)= sin(a) and y'(0)= cos(a). That tells us that
sin(x+ a)= sin(a)cos(x)+ cos(a)sin(x).
Taking x= b, we have
sin(a+ b)= sin(b+ a)= sin(a)cos(b)+ cos(a)sin(b).
Similarly, if we let y= cos(x+ a), we have y''= -y but now y(0)= cos(a), y'(0)= -sin(a) so that
cos(x+ a)= cos(a)cos(x)- sin(a)sin(x) and so
cos(a+ b)= cos(a)cos(b)- sin(a)sin(b)
Of course, replacing "b" by "-b" and using the fact that sine is an odd function, cosine is an even function
sin(a- b)= sin(a)cos(-b)+ cos(a)sin(-b)= sin(a)cos(b)- cos(a)sin(b).
Last edited:
#### HallsofIvy
##### Well-known member
MHB Math Helper
Yet, another way! We can define sine and cosine as power series.
That is, we define $$cos(x)= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^{2n}$$ and define $$sin(x)= \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n}$$.
We can show, perhaps by using the ratio test, that those sums converge for all x. It follows from that that they converge uniformly on all closed an bounded intervals and, particularly, that they can be differentiated "term by term" at any x.
From that we find that (sin(x))'= cos(x) and (cos(x))'= - sin(x). Differentiating again, (sin(x))''= -sin(x) and (cos(x))''= - cos(x). That is, sin(x) and cos(x) both satisfy y''= -y and it is easy, by evaluating those sums at x= 0, to show that they also satisfy the initial conditions we need to use the previous proof.
#### Opalg
##### MHB Oldtimer
Staff member
Here is yet another method, which relies on knowing a bit of linear algebra. The matrix $\begin{bmatrix}\cos a & -\sin a \\ \sin a & \cos a\end{bmatrix}$ represents the operation of rotation through an angle $a$. In other words, if you rotate the point $\begin{bmatrix}x \\ y\end{bmatrix}$ through an angle $a$ around the origin, it gets moved to the point $\begin{bmatrix}\cos a & -\sin a \\ \sin a & \cos a\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}$.
Now if you rotate through an angle $a$ and then through an angle $b$, those combined operations are equal to a rotation through an angle $a+b$. Therefore $$\begin{bmatrix}\cos (a+b) & -\sin (a+b) \\ \sin (a+b) & \cos (a+b)\end{bmatrix} = \begin{bmatrix}\cos b & -\sin b \\ \sin b & \cos b\end{bmatrix}\begin{bmatrix}\cos a & -\sin a \\ \sin a & \cos a\end{bmatrix}.$$ Multiply the matrices on the right side of that equation, using the standard procedure for matrix multiplication, and you find that $$\begin{bmatrix}\cos (a+b) & -\sin (a+b) \\ \sin (a+b) & \cos (a+b)\end{bmatrix} = \begin{bmatrix}\cos a \cos b - \sin a\sin b & -\sin a \cos b - \cos a\sin b \\ \sin a \cos b + \cos a\sin b & \cos a \cos b - \sin a\sin b\end{bmatrix}.$$ Compare corresponding entries in those two matrices and you get the addition formulae for cos and sin.
#### soroban
##### Well-known member
Hello, Farmtalk!
I assume you are seeking a basic proof.
I was wondering if someone could take the time to show
a proof for the sum and difference identity for sine.
Code:
P
. . . *
. . *|*
. * |a*
* | *
* | *
. . . 1 * | *
. . * S*-----* Q
. * | * |
* * |
* * | |
* b * | |
* * a | |
O * * * * * * *
T R
Let $$a = \angle QOR.\;QR \perp OR$$
Let $$b = \angle POQ.\;PQ \perp OQ.\;OP = 1.$$
Draw $$PT \perp OR.$$
Draw $$QS \parallel OR.$$
Note that: $$\angle QPS = a.$$
We find that:.$$PQ = \sin b$$
Hence:.$$PS \:=\: PQ\cdot\cos a \:=\:\sin b\cos a$$
We find that:.$$OQ = \cos b$$
Hence: .$$ST \,=\,QR \,=\,OQ\cdot\sin a \,=\,\cos b\sin a$$
Then:.$$\sin(a+b) \:=\:\frac{PT}{OT} \:=\:\frac{PS+ST}{OP}$$
. . . . . . . . . . . . $$=\:\frac{\sin b\cos a + \cos b\sin a}{1}$$
Therefore: .$$\sin(a+b) \:=\:\sin a\cos b + \sin b\cos a$$ | 2020-10-01 22:24:21 | {"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": 1, "mathjax_display_tex": 3, "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": 0.9422325491905212, "perplexity": 494.84208981956243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2020-40/segments/1600402132335.99/warc/CC-MAIN-20201001210429-20201002000429-00696.warc.gz"} |
https://www.groundai.com/project/the-berlin-exoplanet-search-telescope-ii-catalog-of-variable-stars-i-characterization-of-three-southern-target-fields/ | BEST II Variable Star Catalog I
# The Berlin Exoplanet Search Telescope II. Catalog of Variable Stars. I. Characterization of Three Southern Target Fields.
T. Fruth11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany , J. Cabrera11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany , R. Chini33affiliation: Astronomisches Institut, Ruhr-Universität Bochum, 44780 Bochum, Germany 44affiliation: Instituto de Astronomía, Universidad Católica del Norte, Antofagasta, Chile , Sz. Csizmadia11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany , C. Dreyer11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany 66affiliation: Zentrum für Astronomie und Astrophysik, Technische Universität Berlin, 10623 Berlin, Germany , P. Eigmüller11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany , A. Erikson11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany , P. Kabath11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany 22affiliation: European Southern Observatory, Alonso de Córdova 3107, Vitacura, Casilla 19001, Santiago 19, Chile , S. Kirste11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany , R. Lemke33affiliation: Astronomisches Institut, Ruhr-Universität Bochum, 44780 Bochum, Germany , M. Murphy55affiliation: Depto. Física, Universidad Católica del Norte, PO 1280, Antofagasta, Chile , T. Pasternacki11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany , H. Rauer11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany 66affiliation: Zentrum für Astronomie und Astrophysik, Technische Universität Berlin, 10623 Berlin, Germany , and R. Titz-Weider11affiliation: Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany
###### Abstract
A photometric survey of three Southern target fields with BEST II yielded the detection of 2,406 previously unknown variable stars and an additional 617 stars with suspected variability. This study presents a catalog including their coordinates, magnitudes, light curves, ephemerides, amplitudes, and type of variability. In addition, the variability of 17 known objects is confirmed, thus validating the results. The catalog contains a number of known and new variables that are of interest for further astrophysical investigations, in order to, e.g., search for additional bodies in eclipsing binary systems, or to test stellar interior models.
Altogether, 209,070 stars were monitored with BEST II during a total of 128 nights in 2009/2010. The overall variability fraction of 1.2–1.5% in these target fields is well comparable to similar ground-based photometric surveys. Within the main magnitude range of , we identify 0.67(3)% of all stars to be eclipsing binaries, which indicates a completeness of about one third for this particular type in comparison to space surveys.
techniques: photometric — binaries: eclipsing — stars: variables
## 1 Introduction
The detailed study of variable stars is essential to astronomy, since it allows for the determination of stellar parameters such as mass, radius, luminosity, or temperature, as well as to study internal and external processes of stars, their composition, structure, and evolution. New detections not only broaden the statistical sample of variable stars, but are also important to gain further knowledge about the different processes that cause stellar variability.
The Berlin Exoplanet Search Telescopes, BEST (Rauer et al., 2004) and BEST II (Kabath et al., 2009a), are small-aperture, wide-field telescopes that are primarily used as a ground-based support for the CoRoT space mission (Baglin et al., 2006). Their observations help to exclude false positives from the list of transiting planetary candidates that are identified in time series from the satellite (Deeg et al., 2009; Rauer et al., 2010; Csizmadia et al., 2011). In addition, long-term photometric monitoring enables a precise characterization of stellar variability in CoRoT target fields (Karoff et al., 2007; Kabath et al., 2007, 2008, 2009a, 2009b; Fruth et al., 2012; Klagyivik et al., 2013) and beyond (Pasternacki et al., 2011).
During time periods not required for the regular CoRoT support, BEST II started an independent transit survey in 2009. Up to now, seven southern target fields have been monitored. While the analysis and follow-up of planetary candidates is ongoing, this paper presents a photometric analysis of stellar variability within the first three target fields of this survey.
The paper is organized as follows: Section 2 describes the telescope configuration and the observational data. Section 3 summarizes the data reduction and photometric analysis, while the search for stellar variability is outlined in Section 4. The results are presented in a large catalog of variable stars in Section 5, which includes a photometric classification of the variability type, ephemerides, a comparison with literature results for known cases, and a discussion on its limitations. Finally, Section 6 briefly summarizes the paper.
## 2 Telescope and Observations
BEST II is located at the Observatorio Cerro Armazones, Chile. Since 2007, it is operated continuously by the Institute of Planetary Research of the German Aerospace Center (DLR) in robotic mode from Berlin.
The system consists of a 25 cm Baker-Ritchey-Chrétien telescope with a focal ratio of f/5.0 and a wide field of view (FOV). The photometric data presented here were obtained with a 4k 4k Finger Lakes Instrumentation CCD (IMG-16801E1) in white light, i.e., without any photometric filter. The CCD is most sensitive at nm, and the photometric system is roughly comparable to the Johnson -band. The pixel size of 9 m corresponds to an angular resolution of pixel.
Three target fields, named F17, F18, and F19, have been monitored intensively with BEST II in 2009/2010. They have been selected by maximizing the total observing time weighted against the average airmass (Rauer et al., 2008, Equation 5) for each period of planned observations. For selecting F19, the simulation was complemented by additionally maximizing the number of target stars suitable for transit search (main sequence, less than 1% contamination). The stellar density of the fields differs significantly, since F17 and F19 are located close to the galactic plane (), while F18 is well outside (). The respective center coordinates of all three target fields are given in Table 1, which also lists the number of frames and light curves obtained in each pointing direction.
In total, BEST II collected 7,380 scientific frames and recorded light curves for 209,070 stars in these target fields. For the first field, F17, BEST II observations cover 39 photometric nights between 20/04/2009 and 22/07/2009. Field F18 was observed for 27 nights between 19/08/2009 and 27/10/2009, and field F19 for 62 nights between 24/03/2010 and 21/09/2010 (see Figure 1). When the observing coverage (colored areas in Figure 1) is related to the maximum available night time during these periods (gray areas), this corresponds to an average duty cycle of 35% for F17 and F19, and 38% for F18. Target fields F17 and F18 were observed with an exposure time of 120 s, while 300 s exposures were taken for F19; the typical cadence between two adjacent measurements is 2.5 minutes.
## 3 Data Reduction
The methods used here to obtain photometric time series from raw scientific images are part of a dedicated automated pipeline that has been applied before to various BEST/BEST II data sets (Karoff et al., 2007; Kabath et al., 2007, 2008, 2009a, 2009b; Rauer et al., 2010; Pasternacki et al., 2011; Fruth et al., 2012; Klagyivik et al., 2013).
Calibration frames (bias, dark, flat) were recorded together with the observations and used in a standard reduction of instrumental effects. In order to increase the photometric precision in crowded fields, we apply the image subtraction algorithm (Alard & Lupton, 1998; Alard, 2000). For that, the calibrated scientific images are aligned to a common coordinate system, and the best 20–40 scientific frames are stacked to a reference image. The latter is then fitted to individual frames and subtracted.
Simple unit-weight aperture photometry is used to extract both the flux from the reference frame and the relative flux in each subtracted frame. A standard radius of 5 pixels was chosen for target fields F17 and F19, while 7 pixels were used in the reduction of the less dense F18 field. An adjacent annulus up to an outer radius of 20 pixels is used for an estimation of the background flux. In order to remove global flux variations in the data, e.g., due to weather or nightly variations of the sky transperancy, a comparison star is calculated out of some thousand light curves with the smallest photometric noise in each data set and then subtracted from each light curve.
Finally, all stars are matched with the UCAC3 catalog (Zacharias et al., 2010) in order to assign equatorial coordinates and to adjust instrumental magnitudes to a standard magnitude system. The astrometric calibration is obtained using the routines grmatch and grtrans by Pál & Bakos (2006); for the three data sets presented here, it achieves a match for 85% of the stars with an average residual of . The magnitude calibration is obtained by shifting each data set by the median difference between all instrumental magnitudes and their respective catalog value (R2MAG of UCAC3). Since the photometric systems are comparable but not identical, this calibration yields an accuracy of 0.3–0.5 mag.
Brightness variation, however, can be measured to a much higher precision with BEST II. For the brightest stars in each data set, it obtains a noise level of 3 mmag in unbinned data over the whole observing season. As an example, Figure 2 shows the standard deviation as a function of stellar magnitude for target field F17. Overall, BEST II obtained measurements with mmag-precision ( mag) for 13,717 stars in the three target fields presented here (cf. Table 1).
## 4 Search for Variability
The three data sets were searched for stellar variability using a combination of the -index (Stetson, 1996) and an analysis of variance period search (AoV; Schwarzenberg-Czerny, 1996). The method and its parameters have been described in detail by Fruth et al. (2012).
First, the -index is used to identify light curves that are clearly not variable. Using a limit , we excluded 93,583 non-variable stars; thereby, was set to 0.1 for the crowded target fields F17 and F19, while 0.05 was chosen for the less dense F18 field. Second, the remaining 115,487 light curves were each fitted with seven harmonics having fundamental periods within the range of 0.05–100 days (F17, F19) and 0.05–80 days (F18), respectively. Third, a modified ranking parameter was calculated from the AoV statistic, whereby lower weights are given to periodic variability that is encountered in many light curves.
Finally, a total of 5,480 light curves with were inspected visually for stellar variability (for the respective star count within each target field, see Table 2). This mainly included reviewing the overall signal-to-noise ratio (SNR), checking individual nights or events for systematic effects, inspecting the stellar neighborhood for possible sources of contamination, and analyzing alternative period solutions. The latter step lead us to adjust the initial ephemerides in several cases – usually to multiples of the AoV period, e.g., due to primary and secondary eclipses of (slightly) different depths.
For some cases, the angular resolution of BEST II is not sufficient to fully separate the light of adjacent stars, and the photometric apertures overlap. Thus, variability of the same shape and period can be detected in multiple light curves. In order to constrain the source of variability, we carefully checked each of these contaminated objects using a smaller photometric aperture (3 Px radius).
Stellar crowding also leads to an overestimation of the instrumental magnitude and an underestimation of the amplitude of variability. In order to assess this effect quantitatively, we calculated the flux fraction within each photometric aperture that does not originate from the respective target as
γi=1−giifi∑jgijfj=1−gii∑jgij100.4(mi−mj), (1)
whereby denotes the flux of star , its magnitude, and describes the geometric integral of its PSF within the target aperture (normalized, i.e., ). The calculation of assumes Gaussian PSFs (3 Px FWHM) and circular apertures (radii as used for photometry), whereas magnitudes from the NOMAD catalog (Zacharias et al., 2004) are used to estimate . Due to missing catalog entries or magnitudes, the calculation of can be inaccurate or even fail in some cases. Also, its accuracy is affected by systematic effects such as the difference in the two photometric systems (BEST II and catalog), deviations of the PSF from a Gaussian shape, or long-term stellar variability. Thus, the automatically calculated value in Table 5 should only be taken as a first-order approximation.
## 5 Results
The visual inspection revealed 2,406 stars with clear and previously unknown variability. In addition to that, we confirm the known variability of 17 stars and suspect further 617 objects to be variable. For the latter group, the variability itself, its type, and/or period could not be determined without ambiguity. Predominantly, these are objects that show brightness variations close to the noise level of their light curve. The numbers of detected, known, and suspected variables within each target field are summarized in Table 2.
### 5.1 Classification
Detected variable stars are assigned variability types following Sterken & Jaschek (1996) and the classification scheme of the General Catalog of Variable Stars (GCVS; Samus et al., 2009). The identification is solely based on photometry, i.e., it depends on the shape, amplitude, and period of the brightness variation.
The following classes could be identified:
• Eclipsing binary systems. Light curves with clear eclipses and almost no variation in between are classified as Algol-type binaries (EA; prototype Per). For systems with ellipsoidal components, phase variations are significant and hinder an exact determination of the beginning/end of eclipses (EB type; Lyr). At orbital periods below one day, both objects are in contact, eclipses are of equal depth and are fully blended with the phase variation (EW type; W UMa).
• Pulsating variable stars. From photometry, the following pulsating types could be identified: Scuti variables (DSCT; periods days), RR Lyrae (RR; –1 day, characteristic shape), Cepheids (CEP; day, amplitudes 0.01–2 mag), Gamma Doradus stars (GDOR), and semi-regular variables (SR; days with irregularities).
• Rotating variable stars (ROT). Stellar rotation can introduce flux variations, e.g., due to stellar spots, magnetic fields, or ellipsoidal components. Since BEST II photometry alone is usually not sufficient to distinguish these cases, they are grouped under a general ”ROT” classification.
• Long periodic variables (LP). Non-periodic variables or stars variable on time scales comparable to/larger than the observational coverage are named LP.
• Inconclusive Cases (VAR). Stars showing clear variability that cannot be assigned a type according to the classification scheme from photometry; further observations are needed to better constrain the physical origin of variability.
Details on how many stars have been found in each variability class and target field are given in Table 3. In total, 954 (plus 104 suspected) eclipsing binaries could be identified, 527 (139) pulsators, 200 (81) stars with rotational modulation, 344 (78) long periodic variables, and 398 (215) with other types of variability.
A figure set of all light curves is available in the online version of the journal. Figure 3 illustrates its form and content by highlighting examples that can be interesting for further astrophysical studies, such as, e.g., eclipsing binaries with a high SNR (F17_10421, F19_009645, F19_030794, F19_033571, F19_100160), eccentric eclipsing binaries (F19_055270, F19_100956), a cataclysmic binary (F19_022713), or RR Lyrae pulsators with amplitude modulation known as the Blazhko-effect (Blažko 1907; F19_086712, F19_124221).
### 5.2 Known Variables
A total number of 17 variable stars contained in the BEST II data sets F17–F19 were previously known. Table 4 gives their identifiers and compares periods and classifications with the corresponding reference values. The light curves of UX Nor (F19_046530) and NSV 7658 (F19_098447) are shown as examples in Figure 3.
For all stars that were classified or have periods determined by previous studies, our results are in excellent agreement. For several known variables such as EM Nor (F19_104459), IX Nor (F19_107786), and PW Nor (F19_116322), we did not find a reference to a CCD light curve, so that this survey can be considered to provide the first high-accuracy photometric measurements of these objects. In the following text, we comment some new insights for individual cases.
##### BE Gru (F18_05548), AD Gru (F18_08895), and KK Nor (F19_088903).
For the first time, the high precision of the new data allows to constrain the Blazhko-effect for these RR variables: Within our detection limit, we find no evidence for an amplitude modulation. This finding can be used, e.g., to study the frequency of the Blazhko-effect (e.g., Sódor et al., 2012) in order to gain a better understanding of this effect. For KK Nor, the period was first determined within this work.
##### FV Nor (F19_000499).
The long-periodic variability was first suspected by Hoffleit (1931) and is now clearly confirmed.
##### UX Nor (F19_046530).
This object has one of the longest periods of RRab stars and thus is intensively studied (Walraven et al., 1958; Petit, 1960; Diethelm, 1983, 1986, 1990; Kwee & Diethelm, 1984; Harris, 1985; Petersen & Diethelm, 1986; Petersen & Andreasen, 1987; Moskalik & Buchler, 1993; Sandage et al., 1994; Feuchtinger & Dorfi, 1996). The high accuracy of the new data can help to better constrain theoretical models of these pulsations (see, e.g., Feuchtinger & Dorfi, 1996).
##### Nsv 7658 (F19_098447).
This object was only suspected to be variable by Luyten (1938). Its long-term variability is now clearly confirmed.
##### IX Nor (F19_107786).
The Mira nature of IX Nor is well compatible with our measurements. However, a period of its variability cannot be given since the observations do not cover a brightness maximum.
##### PW Nor (F19_116322).
The Mira classification for PW Nor is only suspected by GCVS, but is also well supported by our measurements; they show a variation from 12.3 to 9.8 mag during a half period lasting approx. 140 days.
### 5.3 Catalog of New Variables
Information on all known, newly detected, and suspected variable stars identified within the three BEST II data sets is provided in a star catalog. Table 5 shows a small extract for guidance regarding its form and content. The complete catalog in a machine-readable format as well as light curves for each star are available in the online journal. Photometric data and finding charts are available upon request.
Each star is identified by instrumental coordinates, as well as an internal BEST II and 2MASS ID (if available); the latter refers to the closest 2MASS (Skrutskie et al., 2006) object within distance from the respective BEST II coordinates. Instrumental magnitudes are obtained without filter and should thus only be used for a broad approximation (cf. Section 3). Ephemerides and amplitudes of variability are given based on the results of the AoV algorithm (except for cases that were adjusted as a result of visual inspection, see Section 4). No ephemerides and amplitudes are given for long periodic (LP) classified variables.
Contaminated light curves are marked with a “” flag in the catalog. If the origin of variation can clearly be assigned to one of the overlapping stars due to a sufficient angular separation and/or brightness difference, only one object is presented in the catalog. However, in 43 cases, two objects are too close to each other and are thus both presented as variables (marked with the corresponding ID under ”other names” in Table 5, but not counted twice in Tables 2 and 3). Observations at higher angular resolution are needed to constrain the true origin of variability.
### 5.4 Catalog Characterization
The new variables of this study will increase the number of variables listed in the VSX catalog (246,007 as of 14/08/2013, including suspected) by 1.2%. For comparison, Table 6 shows the volume, yield, and magnitude range of this work, previous BEST/BEST II studies, and other large photometric surveys.
The detection yield of photometric surveys is subject to various systematic differences. Most importantly, these include the photometric precision, monitored magnitude range and FOV, the time span and duty cycle of observations, and the applied analysis techniques and selection criteria; for a discussion, see also Tonry et al. 2005. For ground-based projects, the detection yield typically ranges between 0.1% (OGLE-III) and 2.7% (ASAS-2), which compares well to the overall fraction of 1.2% variables (1.5% including suspected) identified in this study. However, the increased precision of space-based surveys indicates a much larger fraction of stars to be variable – McQuillan et al. (2012) detected clear periodic or quasi-periodic behaviour for 16% of stars in Kepler (Borucki et al., 2010) data, and Debosscher et al. (2009) estimated at least 40% of CoRoT light curves to be variable.
In the following text, the sensitivity of this study is investigated as a function of its three most important limitations, namely, the magnitude range, the SNR, and the period range.
#### 5.4.1 Magnitude Range and Variable Fraction
In order to evaluate the completeness of our study as a function of apparent brightness, we compared the number of observed stars against the GSC2.2 catalog (Lasker et al., 2008), which resembles our results the best within the given magnitude range among several catalogs tested. Differences between the instrumental photometric system and the band of GSC2.2 were corrected for by subtracting the average difference to obtain
R′B=RB−⟨Δm⟩, (2)
whereby .
Figure 4 compares the number of catalog stars (within bins of 0.2 mag) with the respective numbers for each target field of this work. The latter is shown both for all stars and a reduced sample having mag, since large deviations may indicate a systematic disagreement, e.g., due to different angular resolutions. The comparison shows a very good agreement between catalog and survey stars ( 80% completeness) within the magnitude range of for the two data sets F17 and F18. For target field F19, a similarly good agreement is confined to the range of , since most stars with are saturated (due to the longer exposure time compared to F17 and F18), and stars with are more strongly affected by crowding in this dense target field.
Figure 5 shows the number of variables and suspected variables identified within this work as a fractional ratio of all stars per magnitude bin separated into binaries and other types. Naturally, the detection yield strongly decreases towards fainter magnitudes as the light curve precision decreases (cf. Figure 2). It peaks at for the binaries () and at for other types of variability (). These values can be considered an overall lower limit of the real fraction of variable stars in our target fields, assuming that the physical dependency on the apparent brightness is small. Within the most sensitive magnitude range of this survey at , we identified an average fraction of eclipsing binaries. This compares well to the results of other photometric surveys such as OGLE, which identified an eclipsing binary fraction of 0.5–0.6% (; Graczyk et al. 2011, cf. their Figure 2). However, the fraction is significantly smaller compared to space surveys: Slawson et al. (2011) reported 2.0% eclipsing binaries at comparable galactic latitudes in the Kepler field. If the latter approximately resembled the physical content in our target fields, the fraction of eclipsing binaries identified in this survey would correspond to a completeness of about one third for .
#### 5.4.2 SNR Limit
In order to evaluate our detection threshold, the amplitude of each variable star is compared with the minimum noise level , which is determined for each field from a fit to the lower noise limit in the (, )-diagram (cf. Figure 2; after an analytical expression by Newberry 1991, Eq. 12).
Figure 6 shows a histogram of the SNR for eclipsing binaries and other types of variables. For the eclipsing binaries, the detection yield strongly increases for , while some types with a more sinusoidal variation such as DSCT can already be detected at . Note that here refers to unbinned data, although a large number of single measurements effectively contribute to each detection, thus increasing the SNR significantly.
#### 5.4.3 Period Range
The duty cycle of any photometric survey significantly constrains its detection efficiency, in particular for variability at large time scales. Figure 7 shows the number of variables detected in this work as a function of their period. For comparison, the observational phase coverage is shown for each target field. Since the detection of detached eclipsing binaries (EA) usually requires an observation of three or more of the relatively short eclipses, the right panel of Figure 7 shows the phase coverage of at least three (infinitesimally short) events.
Figure 7 shows that our detection efficiency is largely confined to periods of days in the case of eclipsing binaries, while the more regular periodic variability of other types can be detected up to days. Qualitatively, the decrease in efficiency well follows the single and triple observational phase coverage, respectively. Furthermore, the period ranges of several distributions reflect their class definition (e.g., DSCT variables with –0.3 days). The observed peak of contact binaries (EW) at –0.4 days agrees well with recent findings of other surveys (e.g., ASAS, Paczyński et al. 2006; Kepler, Slawson et al. 2011; OGLE, Pietrukowicz et al. 2013).
## 6 Summary
A BEST II photometric survey of 209,070 stars within three Southern target fields revealed 2,423 variable objects (from which 17 were previously known) and an additional 617 stars which are suspected to be variable. The underlying stellar sample is most complete ( 80%) within the magnitude range of and the survey is most sensitive to detect eclipsing binaries up to periods of 10 days and other types up to 100 days. The average fraction of eclipsing binaries at magnitudes agrees well with other ground-based surveys and indicates a detection completeness of about one third in comparison to the more sensitive space surveys.
All information that is available from our photometric measurements of variable stars is presented in a large catalog, including the type and/or periodicity of the variability (if determinable). The catalog encompasses a number of objects that are very interesting for further astrophysical studies beyond the scope of this paper.
Acknowledgments. This work was funded by Deutsches Zentrum für Luft- und Raumfahrt and partly by the Nordrhein-Westfälische Akademie der Wissenschaften. Our research made use of the 2MASS, USNO-A2, NOMAD, GSC2.2, and GCVS catalog, the AAVSO variable star search index and the SIMBAD database, operated at CDS, Strasbourg, France.
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• Your comment should inspire ideas to flow and help the author improves the paper.
The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters | 2020-01-21 08:29:56 | {"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": 0.5682712197303772, "perplexity": 6100.401716543797}, "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-2020-05/segments/1579250601628.36/warc/CC-MAIN-20200121074002-20200121103002-00544.warc.gz"} |
http://tex.stackexchange.com/questions/89893/tables-as-commands | # Tables as commands
I have to do a little work but I have to use frequently the same settings for a table and I put the following code
\newcommand{\tab41}[4]{\begin{center}
\begin{tabular}{|c|l||c|l||c|l||c|l|}
\hline
(A) & #1 & (B) & #2 (C) #3 & (D) & #4\\
\hline
\end{tabular}
\end{center}}
and it fails. What am I doing wrong?
-
The reason is that numbers are not allowed in macro names. – Martin Scharrer Jan 8 '13 at 16:56
– tohecz Jan 8 '13 at 16:56
Please always include the error message(s) you get. Here I get ! You can't use macro parameter character #' in restricted horizontal mode., you too, right? – Martin Scharrer Jan 8 '13 at 16:58
Is there a way to solve it? Sorry for my mistakes, I'm new here. – Diego Silvera Jan 8 '13 at 16:59
@DiegoSilvera Do not use digits in macro names. Use for example \tabIVone. – Qrrbrbirlbel Jan 8 '13 at 17:05
The reason this doesn't work is because macro names may not include numbers and other non-letters (_, ^ and usually not @ except where this gets changed, like inside packages and classes).
You should simply avoid having numbers in macro names. Often people use Roman numerals instead, e.g. \mymacroiv instead of \mymacro4. There are however some ways to overcome this, see:
You could use the TeX (not LaTeX) way to define macros to have 41 as part of the parameter text, i.e the stuff the macro awaits after its name. This way you have a \tab macro which awaits and removes a 41 direct after its name. This works only if you don't have any other \tab<number> macros, however.
\documentclass{article}
\begin{document}
\newcommand\tab{}% to get an error if \tab is already defined
\def\tab41#1#2#3#4{\begin{center}
\begin{tabular}{|c|l||c|l||c|l||c|l|}
\hline
(A) & #1 & (B) & #2 (C) #3 & (D) & #4\\
\hline
\end{tabular}
\end{center}}
\tab41{a}{b}{c}{d}
\end{document}
--
If you really want multiple \tab<number> macro then you could define a \tab macro which reads the numbers as two arguments (not { } required) and calls the correct macro which got defined using \@namedef, where the macro name is provided as text and therefore can include all printable characters. Because \@namedef is an internal LaTeX macro you need to use \makeatletter and \makeatother around its usage.
\documentclass{article}
\makeatletter
\newcommand\tab[2]{\@nameuse{tab#1#2}}
\@namedef{tab41}#1#2#3#4{\begin{center}
\begin{tabular}{|c|l||c|l||c|l||c|l|}
\hline
(A) & #1 & (B) & #2 (C) #3 & (D) & #4\\
\hline
\end{tabular}
\end{center}}
\@namedef{tab25}#1#2#3#4{\begin{center}
\begin{tabular}{|c|l||c|l||c|l||c|l|}
\hline
(Other) & #1 & (table) & #2 (C) #3 & (D) & #4\\
\hline
\end{tabular}
\end{center}}
\makeatother
\begin{document}
\tab41{a}{b}{c}{d}
\tab25{a}{b}{c}{d}
\end{document}
-
I understood, thank a lot. – Diego Silvera Jan 8 '13 at 17:08
@DiegoSilvera Now that you understand, avoid \tab41. Better safe than sorry. ;-)` – egreg Jan 8 '13 at 17:09 | 2014-08-20 14:51:18 | {"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": 0.9447125196456909, "perplexity": 3749.842798290406}, "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-2014-35/segments/1408500809686.31/warc/CC-MAIN-20140820021329-00228-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://mail.pm.org/pipermail/purdue-pm/2019-August/001102.html | [Purdue-pm] Computing Euler's Number (Perl Weekly Challenge 21)
Mark Senn mark at purdue.edu
Mon Aug 12 12:51:12 PDT 2019
>> Write a script to calculate the value of e, also known as Euler’s number
>> and Napier’s constant.
>
>Which is 1 + ( 1 / $n ) )**$n , where $n approaches infinity. > >It is easy to put that in a subroutine compute_euler ($n ), and then
>increment $n until you blow off the top of Perl 5's integer. > >I think the closest you could get without getting into Math::BigFloat would >be 2.71828182845904 , and I'm not seeing clear enough documentation to >really work on it. I suspect that$x = Math::BigFloat->new( 1 + ( 1 / $n ) >)**$n ) wouldn't work as intended, not because of problems with
>Math::Float, but because Perl 5 can't handle the equation within the new()
>well enough to give a Math::BigFloat result.
>
>Meanwhile, another site says:
>
>> But it is known to over 1 trillion digits of accuracy!
>
>So, when is it "done"? When we get to billions of digits? The 50 digits
>listed on the wiki page? compute_euler( 2 ** 53 ) to get the largest int
>Perl can handle without BigInt?
>
>I think this question is underspecified. Thoughts? Suggestions?
>--
>Dave Jacoby
In TeX notation
e = \sum_{n=0}^\infty {1\over n!} = 1/1 + 1/1 + 1/(1*2) + 1/(1*2*3) + ...
according to
https://en.wikipedia.org/wiki/E_(mathematical_constant)
(I'll be doing this in Pel 6 using sums of fractions where each fraction
and the sum are FatRats (Fat Rationals). From
https://docs.perl6.org/type/FatRat:
A FatRat is a rational number stored
with arbitrary size numerator and denominator. Arithmetic operations
involving a FatRat and optionally Int or Rat objects return a FatRat,
avoiding loss of precision.
Perl 6 has "sequences" built in that make dealing with mathematical
series easier and more elegant.
Like pi, e is never "done"..."it just keeps going and going". The very loose
specification in Perl Weekly Challenge 21:
Write a script to calculate the value of e,
also known as Euler’s number and Napier’s constant.
doesn't spec how many significant digits. So I just make up something
I'm interested in computing and documenting that.
The lack of tight specifications for the problems make it impossible to
automatically compare solutions objectively. On the other hand, I think
that's fine---I'm more interested in seeing the different ways people
choose to solve the problems instead of how numerically precise the
solutions are. Damian Conway's answers are almost always the best in my
opinion.
-mark | 2023-02-02 00:56:56 | {"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": 1, "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": 0.8055595755577087, "perplexity": 5508.329202790886}, "config": {"markdown_headings": false, "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-2023-06/segments/1674764499954.21/warc/CC-MAIN-20230202003408-20230202033408-00605.warc.gz"} |
http://clay6.com/qa/128225/www.clay6.com/qa/128225/www.clay6.com/qa/128225/amount-of-oxalic-acid-present-in-a-solution-can-be-determined-by-its-titrat | # Amount of oxalic acid present in a solution can be determined by its titration with $KMnO_4$ solution in the presence of $H_2SO_4$. The titration gives unsatisfactory result when carried out in the presence of HCl, because HCl
( A ) reduces permanganate to $Mn^{2+}$
( C ) furnishes $H^+$ ions in addition to those from oxalic acid | 2020-10-01 23:37:52 | {"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": 1, "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": 0.5613675117492676, "perplexity": 2472.493000471958}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2020-40/segments/1600402132335.99/warc/CC-MAIN-20201001210429-20201002000429-00260.warc.gz"} |
http://enhtech.com/standard-error/solved-residual-standard-error-mean-square-error.php | Home > Standard Error > Residual Standard Error Mean Square Error
# Residual Standard Error Mean Square Error
What is a word for the probability that the random variable F > the value of the test statistics. Not the answer from their mean, R=X-m. We can see how R-squared Adjusted, “adjusts” for the number of variables in theWhat is a word for deliberate dismissal of some facts? standard has unusual predictors (X1i, X2i, ..., Xki).
Commons Attribution-ShareAlike License; additional terms may apply. If this value is small, then error check my blog Notation. error Root Mean Square Error Vs Standard Deviation The Last Monday "Guard the sense doors"- What error
Generated Thu, 27 Oct 2016 is 06:55 PM. What is Thomson Higher Education. Since an MSE is an expectation, mean Bayesian Analysis (2nd ed.).The three sets of 20 values are related as still cannot perfectly predict Y using X due to $\epsilon$.
Where Q R r, Correlation Coefficients, Pearson’s r - R.G.D, and Torrie, J. T U V Variance Inflation Factor (VIF) - A statistics used to measuring the Residual Standard Error Definition Let's say your school teacher invites you andyour schoolmates to help guess the teacher's table width.
Based on rmse, the teacher can judge whose We can compare each student mean with http://www.analystforum.com/forums/cfa-forums/cfa-level-ii-forum/91265297 xargs Are voltage and current sources linear or nonlinear?When a regression model with p independent variables contains only random differences from dependent variable and the others are independent variables.
Could IOT Botnets be StoppedThe system returned: (22) Invalid argument The Residual Standard Error Formula such as the mean absolute error, or those based on the median.What is a word for predictors, its variance inflation factor will be very large.
References ^ a residual answer to your question is NO.Typically the smaller the standard error, the betterThis also is a known, computed quantity, and residual the teacher who will crunch the numbers.Seeing it for news mean hypothesis that group means are equal.
from their mean, R=X-m.MSE = SSE/ n-k-1 <– there is no square root here. http://stats.stackexchange.com/questions/110999/r-confused-on-residual-terminology is a measure of sampling error.The residual standard error you've asked about is nothing more standard communities Sign up or log in to customize your list.
Points they are not the same thing, but closely related. Subtracting each student's observations from a reference valueA mean error can bea population, X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} .Subtracting each student's observations from their individual mean will
Errors of the mean: deviation ofthe means from the "truth", EM=M-t.Comparing prediction intervals with confidence intervals: prediction intervals estimate have lower MSE; see estimator bias. This is an easily computable quantity for Residual Standard Error Interpretation with respect to a fixed point.Coefficient of Determination – In general the coefficient of determination measures the amount can be calculated as well.
Asked 3 years ago viewed 73298 times active 3 months ago Blog Stack have a peek at these guys p.60.It is http://stats.stackexchange.com/questions/57746/what-is-residual-standard-error is denoted t.There were in total 200 width measurements taken square the SSH command How to draw and store a Zelda-like map in custom game engine?What are the difficulties of landing on an upslope runwaythen k=1 and the formula for R-squared Adjusted simplifies to R-squared.
Please try deliberate dismissal of some facts? Residual Mean Square Error the teacher who will crunch the numbers.R would output this information asWilliam; Scheaffer, Richard L. (2008).The slope of x) Ha: b 1 is not 0 p-value =
Does the way this experimental kill vehicledeviation of error terms.This property, undesirable in many applications, has led researchers to use alternatives(X'X)-1 X', where X is the design matrix.What is the meaning ofBrowse other questions tagged r regressionas I can tell.
Likewise, 20 standard deviation of the error, or http://enhtech.com/standard-error/help-root-mean-square-error-residual-standard-error.php puzzling. Rmse Vs Standard Error set of sets Are there any pan-social laws?
would be equal to $\sqrt{76.57}$, or approximately 8.75. Browse other questions tagged regression standard-errorwould be great..DFITS is the difference between the fitted values calculated with For simple linear regression, when you do not fit the y-intercept,<- sqrt(test.mse) test.rmse [1] 2.668296 Note that this answer ignores weighting of the observations.
If you do not student provided the best estimate for the table width. Draw an hourglass Bitwise rotate right of 4-bit value If the square rooterror values, the teacher can instruct each student how to improve their readings. error In other words, you estimate a model using a portion of your data Residual Standard Error And Residual Sum Of Squares square What are the difficulties of landing on an upslope error errors, or residual sum of errors.
were chosen with replacement. Subtracting each student's observations from a reference value We can compare each student mean with Calculate Residual Sum Of Squares In R the request again.An F-test can be used in the
How is being able to break Measures the strength of linear association between two numerical variables. MR0804611. ^ Sergio Bermejo, Joan Cabestany (2001) "Oriented principal componentinterval is called the lower bound or lower limit. AdamO 17.1k2563 3 This may have been answered before. residual Residuals: deviation of observations trademarks owned by Chartered Alternative Investment Analyst Association.
of the sampling distribution of a statistic. it varies by sample and by out-of-sample test space. SSH makes all typed passwords visible when command is provided as an argument to
MSE is a risk function, corresponding to the expected 2e-16 *** hp -0.06823 0.01012 -6.742 1.79e-07 *** --- Signif.
What is the meaning of Set-to-point operations: mean: MEAN(X) root-mean-square: RMS(X) standard deviation: SD(X) = RMS(X-MEAN(X)) INTRA-SAMPLE scale, tape, or yardstick) and is allowed to measure the table 10 times. | 2018-06-20 07:18:54 | {"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": 1, "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": 0.7031130194664001, "perplexity": 3387.3459491189706}, "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-2018-26/segments/1529267863489.85/warc/CC-MAIN-20180620065936-20180620085936-00308.warc.gz"} |
https://www.physicsforums.com/threads/change-of-sign-on-greens-functions-maths-problem.811478/ | Change of sign on Green's functions (Maths problem)
1. Apr 30, 2015
Hi, I am trying to solve a model where Non-interacting Green functions take part it. It has happened something that is spinning my head and I hope someone could help. The non interacting Green function for a chanel of electrons is:
$$G_{0}(\omega)=\int_{-\infty}^{\infty}d\epsilon\nu(\epsilon)\frac{1}{\omega - \epsilon + i\delta\text{sign(\omega)}}$$
where I am integrating over the whole energy spectrum since I consider the chanel as a whole. I have to mention that I am at T=0 and $$\nu(\epsilon)$$ is the density of states on the chanel. Ok, suppose we have a constant density of states. Then, the real part of the integral cancels (gives a cosntant part which is irrelevant) and the imaginary part, since $$\delta\to 0$$ becomes a delta function inside. Therefore:
$$G_{0}(\omega)=-i \nu\int_{-\infty}^{\infty}d\epsilon\delta(\omega - \epsilon)=-i\nu\text{sign(Re(\omega))}$$
We notice the dependance with the REAL part of the excitation energies $$\omega$$. Now, imagine that instead of a constant density of states, this density of states is a Lorentzian of the type:
$$\frac{\Lambda}{\Lambda^{2} + \epsilon^{2}}$$
Then the integral above can be completed in the complex plane using contour integration, and the result gives:
$$G_{0}(\omega)=\frac{\nu}{\frac{\omega}{\Lambda} + i\text{sign(Im(\omega))}} + \frac{i\nu\Lambda\text{sign(Re(\omega))}}{\Lambda^{2}+\omega^{2}}$$
Now, taking $$\Lambda\to\infty$$ should gives us the result of the flat density of states but instead it gives the sign with the imaginary part!!! Someone can help on this?
2. May 1, 2015 | 2018-03-22 08:44:31 | {"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": 1, "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": 0.8946137428283691, "perplexity": 301.70512165850766}, "config": {"markdown_headings": false, "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-2018-13/segments/1521257647782.95/warc/CC-MAIN-20180322073140-20180322093140-00249.warc.gz"} |
https://www.semanticscholar.org/paper/Irregular-perverse-sheaves-Kuwagaki/2f861dfaa292f94314dfe897532c53e076ed56bb | # Irregular perverse sheaves
@article{Kuwagaki2018IrregularPS,
title={Irregular perverse sheaves},
author={Tatsuki Kuwagaki},
journal={Compositio Mathematica},
year={2018},
volume={157},
pages={573 - 624}
}
• T. Kuwagaki
• Published 8 August 2018
• Mathematics
• Compositio Mathematica
We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically…
8 Citations
• Yohei Ito
• Mathematics
Rendiconti del Seminario Matematico della Università di Padova
• 2023
The subject of this paper is an algebraic version of the irregular Riemann-Hilbert correspondence which was mentioned in [arXiv:1910.09954] by the author. In particular, we prove an equivalence of
We formulate and prove a Riemann–Hilbert correspondence between ~-differential equations and sheaf quantizations, which can be considered as a correspondence between two kinds of quantizations
Perverse schober defined by Kapranov--Schechtman is a categorification of the notion of perverse sheaf. In their definition, a key ingredient is certain purity property of perverse sheaves. In this
In this paper, we reprove the Riemann–Hilbert correspondence for regular holonomic D -modules of [Kas84] (see also [Meb84]) by using the irregular Riemann– Hilbert correspondence of [DK16]. Moreover,
. The notion of sheaf quantization has many faces: an enhancement of the notion of constructible sheaves, the Betti counterpart of Fukaya–Floer theory, a topological realization of WKB-states in
We illustrate the Arinkin-Deligne-Katz algorithm for rigid irreducible meromorphic bundles with connection on the projective line by giving motivicity consequences similar to those given by Katz for
• Mathematics
• 2022
. For any holomorphic function f : X → C on a complex manifold X , we define and study moderate growth and rapid decay objects associated to an enhanced ind-sheaf on X . These will be sheaves on the
A sheaf quantization is a sheaf associated to a Lagrangian brane. By using the ideas of exact WKB analysis, spectral network, and scattering diagram, we sheaf-quantize spectral curves over the
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D-Modules and Perverse Sheaves.- Preliminary Notions.- Coherent D-Modules.- Holonomic D-Modules.- Analytic D-Modules and the de Rham Functor.- Theory of Meromorphic Connections.- Regular Holonomic
Part I Introduction Review: Floer cohomology The $A_\infty$ algebra associated to a Lagrangian submanifold Homotopy equivalence of $A_\infty$ algebras Homotopy equivalence of $A_\infty$ bimodules
We study (i) asymptotic behaviour of wild harmonic bundles, (ii) the relation between semisimple meromorphic flat connections and wild harmonic bundles, (iii) the relation between wild harmonic | 2023-02-04 09:44:08 | {"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": 1, "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": 0.6811093688011169, "perplexity": 1765.1603273412093}, "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-2023-06/segments/1674764500095.4/warc/CC-MAIN-20230204075436-20230204105436-00266.warc.gz"} |
http://mathhelpforum.com/math-challenge-problems/9300-quickie-3-a-print.html | # Quickie #3
• Dec 27th 2006, 07:43 AM
Soroban
Quickie #3
Simplify: . $\frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}$
• Dec 27th 2006, 11:38 AM
galactus
Quote:
Originally Posted by Soroban
Simplify: . $\frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}$
] $\frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}$
$(4+\sqrt{15})^{\frac{3}{2}}=\frac{7\sqrt{10}}{2}+\ frac{9\sqrt{6}}{2}$...[a]
$(4-\sqrt{15})^{\frac{3}{2}}=\frac{7\sqrt{10}}{2}-\frac{9\sqrt{6}}{2}$...[b]
$(6+\sqrt{35})^{\frac{3}{2}}=\frac{11\sqrt{14}}{2}+ \frac{13\sqrt{10}}{2}$....[c]
$(6-\sqrt{35})^{\frac{3}{2}}=\frac{11\sqrt{14}}{2}-\frac{13\sqrt{10}}{2}$...[d]
$\frac{a+b}{c-d}=\frac{7\sqrt{10}}{13\sqrt{10}}=\boxed{\frac{7}{ 13}}$
• Dec 27th 2006, 11:41 AM
earboth
Quote:
Originally Posted by Soroban
Simplify: . $\frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}$
Hello Soroban,,
I really don't know why you call this problem a "Quickie" (I still have a crack in my head!)
It took some time until I remembered the property:
If x > y > 0 then
$\sqrt{x+y}=\sqrt{\frac{x+\sqrt{x^2-y^2}}{2}}+\sqrt{\frac{x-\sqrt{x^2-y^2}}{2}}$ or
$\sqrt{x-y}=\sqrt{\frac{x+\sqrt{x^2-y^2}}{2}}-\sqrt{\frac{x-\sqrt{x^2-y^2}}{2}}$
With these properties your term becomes:
$\frac{(4+\sqrt{15})^{\frac{3}{2}}+(4-\sqrt{15})^{\frac{3}{2}}} {(6+\sqrt{35})^{\frac{3}{2}}-(6-\sqrt{35})^{\frac{3}{2}}}$ = $\frac{(\sqrt{\frac{5}{2}}+\sqrt{\frac{3}{2}})^3+(\ sqrt{\frac{5}{2}}-\sqrt{\frac{3}{2}})^3}{(\sqrt{\frac{7}{2}}+\sqrt{\ frac{5}{2}})^3-(\sqrt{\frac{7}{2}}-\sqrt{\frac{5}{2}})^3}$
Expand the brackets and collect the roots with the same value. I've got:
$\frac{7 \cdot \sqrt{10}}{13 \cdot \sqrt{10}}=\frac{7}{13}$
Hapoooh!
EB
• Dec 27th 2006, 12:46 PM
Soroban
Lovely work, Galactus and EB!
Don't know if the "Quickie" solution is any faster . . .
We have: . $\frac{(4 + \sqrt{15})^{\frac{3}{2}} + (4 - \sqrt{15})^{\frac{3}{2}}} {(6 + \sqrt{35})^{\frac{3}{2}} - (6 - \sqrt{35})^{\frac{3}{2}}}$
Multiply top and bottom by $2^{\frac{3}{2}}\!:$
. . $\frac{(8 + 2\sqrt{15})^{\frac{3}{2}} + (8 - 2\sqrt{15})^{\frac{3}{2}}} {(12 + 2\sqrt{35})^{\frac{3}{2}} - (12 + 2\sqrt{35})^{\frac{3}{2}}}$
. . $=\;\frac{\left[(\sqrt{5}+\sqrt{3})^2\right]^{\frac{3}{2}} + \left[(\sqrt{5} - \sqrt{3})^2\right]^{\frac{3}{2}}} {\left[(\sqrt{7} + \sqrt{5})^2\right]^{\frac{3}{2}} - \left[(\sqrt{7} - \sqrt{5})^2\right]^{\frac{3}{2}} }$
. . $= \;\frac{(\sqrt{5}+\sqrt{3})^3 + (\sqrt{5} - \sqrt{3})^3}{(\sqrt{7} + \sqrt{5})^3 - (\sqrt{7} - \sqrt{5})^3}$
. . $=\:\frac{(5\sqrt{5} + 15\sqrt{3} + 9\sqrt{5} + 3\sqrt{3}) + (5\sqrt{5} - 15\sqrt{3} + 9\sqrt{5} - 3\sqrt{3})} {(7\sqrt{7} + 21\sqrt{5} + 15\sqrt{7} + 5\sqrt{5}) - (7\sqrt{7} - 21\sqrt{5} + 15\sqrt{7} - 5\sqrt{5}}$
. . $= \:\frac{28\sqrt{5}}{52\sqrt{5}}$
. . $=\:\boxed{\frac{7}{13}}$ | 2017-05-28 19:22:12 | {"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": 0, "img_math": 0, "codecogs_latex": 22, "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": 0.9636937975883484, "perplexity": 5277.4997651835865}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2017-22/segments/1495463610374.3/warc/CC-MAIN-20170528181608-20170528201608-00264.warc.gz"} |
https://puzzling.stackexchange.com/questions/31296/mr-yamaguchis-unusual-message | # Mr. Yamaguchi's unusual message
My good friend, Ed Yamaguchi, has been developing an interesting new technology that could help put a serious dent in the energy crisis.
We had arranged to meet a few hours from now; he was flying in from Amsterdam, where he lives, to discuss my potential investment in his technology.
I just received a very odd email from him. He should be in the air right now, so either he sent it from on-board the plane, or he never got on the plane for some reason.
Ed is a stickler for proper grammar, punctuation, and sentence structure, so the haphazard nature of this email makes me think that either he didn't write it, or he had some very good reason for the odd formatting.
The subject matter also seems odd, seeing as we were already planning to meet.
I'm a little worried that something is wrong, and Ed is trying to send me a secret message. Can you help?
To: GentlePurpleRain
From: Ed Yamaguchi
Subject: (empty)
My dearly regarded Friend,
i trust I Am finding you ok. I have Been just Fine but i'm Starting up A New Adventure in Seven Days, And I could use a bit of aid. Will you be Available soon To discuss The opportunity I Presented the Last time we conversed?
I Know you are probably Tired Of folks Always asking you For aid, but This really is an Awesome Opportunity, so please, i Implore you, think About it very Carefully.
I am confident That everyone Who hears about What's About to be Will be Delighted. I Eagerly anticipate my Opportunity To make You a part of these Exciting adventures!
I may have Minor Problems with Attaining my goals If you do Elect Not to participate with Me. anyway, do please Respond as soon as You are able to. Even if it's Not A go, do Let me know promptly.
Sincerely yours,
ed "The Ax" yamaguchi
Hint
I started wondering about the proper capitalization of "OK" and whether Ed would have usually capitalized it, but I figured that it didn't really matter. Probably what was more important was how he actually wrote it in this letter, rather than how he might have written it otherwise.
• oh shoot, I had collected the capitals/lowercases-in-the-wrong-place last week when I was looking at this and now I can't find it. – question_asker Apr 25 '16 at 15:01
• Here's an attempt at the incorrect capitalizations: FiABFiSANASDAATTPLKTOAFTAOiIACTWWAWDEOTYEMPAIENMTARYNALFERY – Dan Russell Apr 25 '16 at 15:22
• @stackErr, it's actually a capital i, coming from "Implore." – Dan Russell Apr 25 '16 at 15:30
• @stackErr I still see PAI from "Problems with Attaining my goals If ..." – Dan Russell Apr 25 '16 at 15:32
• It's not universally agreed what the proper captialization of "OK" is, which is why I was wondering. – 2012rcampion Apr 26 '16 at 4:07
I have the message:
PLANE HIJACKED CRIMINALS ARE GOING TO KILL ME IF I ASK FOR HELP GET POLICE QUICK
The key to finding the message is that:
The number of lowercase letters in between each capital letter corresponds to a letter of the alphabet where A = 0 and Z = 25. There are 15 lowercase letters in My dearly regarded so that's a P.
All the counts together forms the series:
15,11,0,13,4,7,8,9,0,2,10,4,3,2,17,8,12,8,13,0,11,18,0,17,4,6,14,8,13,6,19,14,10,8,11,11,12,4,8,5,8,0,18,10,5,14,17,7,4,11,15,6,4,19,15,14,11,8,2,4,16,20,8,2,10
That corresponds to the letters: | 2019-05-26 12:13:56 | {"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": 0.32602357864379883, "perplexity": 1479.8830399914093}, "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-2019-22/segments/1558232259126.83/warc/CC-MAIN-20190526105248-20190526131248-00387.warc.gz"} |
https://math.stackexchange.com/questions/2249584/revolution-and-differentiability-at-a-point | # Revolution and differentiability at a point
Here is a statement thought by myself: (not sure if it's true)
Suppose $z=f(x)$ is a real-valued function defined for $x\ge 0$, and the right side derivative at $x=0$ is zero. If $z=g(x,y)$ is the function obtained by revolving the graph of $z=f(x)$ about the z-axis, then $g$ is differentiable at $(0,0)$ and has a horizontal tangent plane there.
I think $z=g(x,y)$ is just $z=f(r)$, which $r=\sqrt{x^2+y^2}$. In this way I can show that $g_x$ and $g_y$ are zero at $(0,0)$. In fact all directional derivatives would be zero. But I get stuck on the differentiability part, which definition is as follows: (quoted from Thomas Calculus)
A function $z=f(x,y)$ is differentiable at $(x_0,y_0)$ if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ exist and $\Delta z$ satisifes an equation of the form $$\Delta z=f_x(x_0,y_0)\Delta x+f_y(x_0,y_0)\Delta y+\epsilon_1\Delta x+\epsilon_2\Delta y$$ in which each of $\epsilon_1, \epsilon_2 \to 0$ as both $\Delta x,\Delta y \to 0$.
What are $\epsilon_1$ and $\epsilon_2$ supposed to be in this case?
EDIT: I realize I forgot to mention $z=f(x)$ is defined for $x\ge 0$ only. Now it's being added. I am very sorry if this caused confusion.
• (This is delt3. I accidentally lost the account so I set up a new one.) I find that the notion of total differentiability would imply the increment formula. It also applies to my example, since $\lim_{(h,k)\to(0,0)}\frac{f(\sqrt{h^2+k^2})-f(0)}{\sqrt{h^2+k^2}}=\lim_{r\to 0^{+}}\frac{f(r)-f(0)}{r}=0$. Quite a coincidence to me. – delt31 Apr 27 '17 at 7:52
Since you want a patch to have differentiable structure, let $z = f(x)$ be a smooth, injective curve for $x \in (-\epsilon, \epsilon)$. If you allow $\lim_{x \to 0+} f(x) \not = \lim_{x \to 0-} f(x)$ then the curve cannot be connected at the point $(0,f(0))=(x,z)$. If you want to proceed this way, you must know what $f(0)$ is equal to so that when we use that parametrization $\sigma(x,u)$, we can compute $\sigma_x(0,f(0))$ and $\sigma_u(0,f(0)$. You also run into another problem since $\partial z/\partial x$ is an entry in the differential, so you'll need equality of the right and left derivatives. Again With the refined conditions, revolving the profile curve about the $z$-axis gives the parametrization,
$$\sigma(x,u) = (h(x) \cos u, h(x) \sin u, f(x)) = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$
Since you let $\partial z/\partial x (0) = 0$ we have $d_{p} \sigma: T_p \mathbb{R}^2 \to T_{\sigma(p)} S$, where $p = (0,0)$ is defined by,
$$d_{(0,0)}\sigma = \begin{pmatrix} h'(0) & 0 \\ 0 & h(0) \\ 0 & 0 \end{pmatrix}$$
The tangent plane at $\sigma(0,0) = (h(0),0,f(0)):=q$ is the image of $d_{(0,0)} \sigma$. Letting $(x-p,y-p) \in T_p\mathbb{R}^2$ then we have,
$$d_{(0,0)} \sigma \begin{pmatrix} x-p \\ y-p \end{pmatrix} = \begin{pmatrix} h'(0) & 0 \\ 0 & h(0) \\ 0 & 0 \end{pmatrix} \begin{pmatrix} x-p \\ y-p \end{pmatrix} = \begin{pmatrix} h'(0)(x-p)\\ h(0)(y-p) \\0 \end{pmatrix}:=v_p$$
Here $v_p \in T_qS$ and so if $\textbf{N}$is our normal on a small patch containing $q$ then the tangent plane is given by $\left(\textbf{N}\right)^{\perp} = \{v_p: v_p \cdot \textbf{N} = 0\}$.
$$\begin{pmatrix} h'(0)(x-p) \\ h(0)(y-p) \\0 \end{pmatrix} \cdot \begin{pmatrix} n_1 \\ n_2 \\ n_3 \end{pmatrix} = 0 \iff n_1h'(0)x + n_2h(0)y = p(h'(0) + h(0))$$
which is never a horizontal tangent plane in the coordinate system $(x,y,z)$ since we only get vertical or slanted planes. For horizontal tangent planes, you need to looks at points $x_0$ such that,
\begin{align*} &\frac{\partial z}{\partial x}(x_0) = f'(x_0) = 0 \\ & n_1h'(x_0) = 0 \\ & n_2h(x_0) = 0 \end{align*}
Diving into one special case i.e $h'(x_0) = h(x_0) = 0$, we observe that these points have flat neighborhoods since,
$$\sigma_x(x_0,u) = \sigma_u(x_0,u)= (0,0,0)$$
Observe that this is exactly what you want since in the ambient space we have the points $p'=(x_0,y,u)$ with a small-neighborhood $V \ni (x_0,u)$ contained on our curve and in the $z=K$ plane i.e $u = K$. The image to have in mind is like the one below where we see that $p$ must be the endpoints of the curve if we wish for the surface to be smooth. From the description of the points $p'$ on the surface of revolution, it is clear which points satisfy the geometric description given above.
• Thanks for the answer. I haven't learnt about differentiable structure yet, but I guess you do parametrization of the surface by 2 variables and do some calculus on that? I'm not sure if $(0,0)$ now satisfies the 3 equations. – delt3 Apr 24 '17 at 14:53
• @delt3: $(0,0) = (x,u) = (x,f(x))$ cannot be a point with a horizontal plane since points of this form give slanted or vertical planes which is the first thing I give you. I go one after that to show you which points on a smooth surface of revolution can have horizontal planes. The intuition behind this is the for a curve of finite length, if a point $p$ in not an endpoint then the surface must pinch at this point. – Faraad Armwood Apr 24 '17 at 15:12
• Your intuition about the points which have horizontal tangent planes was almost spot on, its just that we needed to other conditions to actually give us a surface patch about these points so that we can do calculus. – Faraad Armwood Apr 24 '17 at 15:17
• I'm puzzled about the $h(x)$ term... So let's say the function $z=f(x)=x^2$ is parametrized as $(x,x^2)$. Is the surface parametrized by $(x\cos u,x\sin u,x^2)$, which $h(x)=x$, $u$ is the angle as in polar coordinate? – delt3 Apr 24 '17 at 16:29
• Well, I hope it truly helped. your question, although innocent to you, was a good one. There were many ways to hand wave it, but I tried not to. At least now, you've been exposed to other things and hopefully that'll encourage you to read more on the subject. I suggest coming back to this post after you learn more differential geometry and multivariable calculus. More than likely you'll find shorter paths or things I've overlooked. – Faraad Armwood Apr 24 '17 at 17:58
True formula must be $$\Delta z=g_x(x_0,y_0)\Delta x+g_y(x_0,y_0)\Delta y+\epsilon_1\Delta x+\epsilon_2\Delta y$$ where $z=g(x,y)=f(r)$. Then $$g_x=f'(r)\dfrac{\partial r}{\partial x}=f'(r)\dfrac{x}{\sqrt{x^2+y^2}} \text{ and } g_y=f'(r)\dfrac{\partial r}{\partial y}=f'(r)\dfrac{y}{\sqrt{x^2+y^2}}$$ also $\epsilon_1(\Delta x,\Delta y)$ and $\epsilon_2(\Delta x,\Delta y)$ completely determine with $f$. For instance with $z=f(x)=x^2$ then $z=g(x,y)=f(r)=r^2=x^2+y^2$. So $$\Delta z=g(x+\Delta x,y+\Delta y)-g(x,y)=2x\Delta x+2y\Delta y+(\Delta x)^2+(\Delta y)^2$$ therefore $\epsilon_1(\Delta x,\Delta y)=(\Delta x)^2$ and $\epsilon_2(\Delta x,\Delta y)=(\Delta y)^2$, furthermore $$\lim_{\Delta x\to0,\Delta y\to0}\epsilon_1(\Delta x,\Delta y)\to0 \text{ and } \lim_{\Delta x\to0,\Delta y\to0}\epsilon_2(\Delta x,\Delta y)\to0$$
• But what if $f$ is a function-in-general? i.e the expression of f is not given. Is there still a way to express $\epsilon_1, \epsilon_2$? – delt3 Apr 24 '17 at 16:56
• Here $\Delta x, \Delta y$ are just infinitesimal increments. The expression above is just the linearization of $z$. $$\Delta z = g_x(p_0) (x-x_0) + g_y(p_0)(y-y_0) + \epsilon_1 (x-x_0) + \epsilon_2(y-y_0)$$ where $p_0= (x_0,y_0)$. So for a typical problem, you choose $(x,y)$ such that it is really close to $(x_0,y_0)$. You do this since this approximation is limited in its domain about $p_0$. The quantities $(x-x_0)$ and $(y-y_0)$ are therefore very, very small and so we define them to be $\Delta x, \Delta y$ respectively. – Faraad Armwood Apr 24 '17 at 17:04
• Also, typically we just have have $\epsilon(x,y) = \epsilon_1(x-x_0) + \epsilon_2(y-y_0)$. I think it was given this way to show that the linearization picks up on the error for the partials in the $x,y$ directions. Hence we have that, $$\lim_{(x,y) \to (x_0,y_0)} \frac{\epsilon(x,y)}{\sqrt{(x-x_0)^2+(y-y_0)^2}} = 0$$ – Faraad Armwood Apr 24 '17 at 17:06 | 2019-11-15 12:27:25 | {"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": 1, "mathjax_display_tex": 1, "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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9442369937896729, "perplexity": 178.4011803744443}, "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-2019-47/segments/1573496668644.10/warc/CC-MAIN-20191115120854-20191115144854-00187.warc.gz"} |
https://math.stackexchange.com/questions/3194889/proof-of-equivalence-between-two-methods-of-binary-to-decimal-conversion/3194950 | # Proof of equivalence between two methods of binary to decimal conversion.
I have two binary to decimal conversion methods and want a proof - or an intuition at least - of why they are equivalent.
The first method is quite intuitive to me and seems to be more popular: $$[b_n ... b_1 b_0] -> (b_n * 2^n) + ... + (b_1 * 2^1) + (b_0 * 2^0)$$
The second method isn't as intuitive, and it would help if someone can explain to me why this works to do just what the previous method does:
$$[b_n ... b_1 b_0] -> to\_nat\ [b_0 b_1 ... b_n]$$
where \begin{align} to\_nat\ b = match\ b\ with \\ &|\ [\ ] \to 0 \\ &|\ [head,\ [tail]] \to if\ (head = 1)\ then\ 1 + 2 * (to\_nat\ tail)\ else\ 2 * (to\_nat\ tail) \end{align} Sorry if this notation is confusing. I'm using notation derived from functional programming.
The first method is simply the definition of base-2 representation illustrated in (1) below which is analogous to the definition of base-10 representation illustrated in (2) below and the generalized base-b representation illustrated in (3) below.
(1) $$\quad a_n...a_1 a_0\,\text{(base 2)}=2^n a_n+...+2^1 a_1+2^0 a_0$$
(2) $$\quad a_n...a_1 a_0\,\text{(base 10)}=10^n a_n+...+10^1 a_1+10^0 a_0$$
(3) $$\quad a_n...a_1 a_0\,\text{(base b)}=b^n a_n+...+b^1 a_1+b^0 a_0$$
Base-10 representation is the primary representation most people have been taught and used, but electrical engineers and software programmers commonly use other representations such as base-2 (binary), base-8 (octal), and base-16 (hexadecimal) representations.
If you calculate the first method by hand you're likely calculating the result in base-10 representation therefore the result will be in base-10 representation.
The vast majority of computers are digital and perform all calculations in binary-representation, but computers typically have input/output libraries with options for entering and displaying results in various bases.
The second method is equivalent to the first method which can be seen from the following example.
(4) $$\quad a_3...a_1a_0\,\text{(base 2)}=2\,(2\,(2\,a_3+a_2)+a_1)+a_0=2^3\,a_3+2^2\,a_2+2^1\,a_1+2^0a_0$$ | 2019-05-19 18:20:01 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 7, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9793938994407654, "perplexity": 951.8345388636507}, "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-2019-22/segments/1558232255092.55/warc/CC-MAIN-20190519181530-20190519203530-00083.warc.gz"} |
https://tex.stackexchange.com/questions/261234/vertically-aligning-tabular-cell-contents-with-respect-to-one-another | # Vertically aligning tabular cell contents with respect to one another
I have a simple table that can be recreated with the simple MWE below, I am compiling with XeLaTeX:
\documentclass[12pt]{article}
\usepackage[letterpaper, left=19mm, right=19mm, top=3.52cm, bottom=2.84cm, headheight=38.7pt]{geometry}
\usepackage{setspace}
\setstretch{1.15}
\usepackage{tabularx}
\usepackage{array}
\usepackage{fontspec}
\setmainfont{Arial}
\usepackage{microtype}
\begin{document}
\begin{flushleft}
\begin{tabular}{@{}m{4.4cm}m{12.5cm}@{}}
\hline
\textbf{TO:} & Mr.\noindent and Mrs.\noindent J. Doe \\[0.54cm]
\hline
\textbf{SUBJECT:} & This is just a sample subject that is design to illustrate the text wrapping\\[0.54cm]
\hline
\textbf{DATE:} & \today \\
\hline
\end{tabular}
\end{flushleft}
\end{document}
Which produces the following table:
As you can see in the first two rows, the contents of the cells next to one another aren't vertically aligned by their respective centres, I've added hlines to make this obvious. How can I centre them properly?
• Just remove those \\[.54cm]. What are they for? – LaRiFaRi Aug 13 '15 at 14:07
• I was using those to set an empty space between rows of the table. Is there a better way of doing that? – Ben Aug 13 '15 at 14:09
If you want to increase the height of the rows, you should go via \renewcommand{\arraystretch}{<someFactor>}.
% arara: xelatex
\documentclass[12pt]{article}
\usepackage[letterpaper, left=19mm, right=19mm, top=3.52cm, bottom=2.84cm, headheight=38.7pt]{geometry}
\usepackage{setspace}
\setstretch{1.15}
\usepackage{booktabs}
\usepackage{array}
\begin{document} | 2019-06-16 00:32:41 | {"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": 0.7685261964797974, "perplexity": 2830.9200296909216}, "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-2019-26/segments/1560627997508.21/warc/CC-MAIN-20190616002634-20190616024634-00427.warc.gz"} |
https://math.stackexchange.com/questions/137932/is-sampled-absolutely-integrable-function-absolutely-summable | # Is sampled absolutely integrable function absolutely summable?
Suppose I have function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that it's absolutely integrable: $\int_{\mathbb{R}}|f(x)|dx<\infty$.
I am sampling function $f(x)$ with some period $T_s$. I am interested whether
$$\sum_{k=-\infty}^{k=\infty}|f(kT_s)|<\infty$$
It seems to me that it's true, but I can't figure out how to prove that.
The reason I ask is that I know that if $f(x)$ is absolutely integrable, then its Fourier transform exists, and I am wondering if the sampled version is guaranteed to have a discrete Fourier transform. Sorry if this is a silly question.
Let us consider, for example, $$f=\sum_{k\in\mathbb{Z}}g_k\in C^\infty(\mathbb{R})$$ obtained by taking $g_k=g(2^k(x-k)),\ \forall x\in\mathbb{R},k\in\mathbb{Z},$ for some $g\in C_c^\infty(]-1/2,1/2[),$ with $g(0)>0.$
Now taking $T_s=1,$ we get $\sum_{k\in\mathbb{Z}}f(kT_s)=\sum_{k\in\mathbb{Z}}g(0)=+\infty.$
• Thank you, @Giuseppe! However, I am sort of lost in your notation: what is the set $C_c^{\infty}(]-1/2,1/2[)$? – M.B.M. Apr 28 '12 at 7:18
• $g$ is a smooth (indefinitely differentiable with continuity) function which vanishes outside $]-1/2,1/2[$. An example is $\exp[1/(x^2-1/4)]$ – agtortorella Apr 28 '12 at 7:24
Not true, as Giuseppe said. However, if you put $$g(x)=\sum_{k=-\infty}^\infty f(kT_s+x)$$ then the sum defining $g$ will be absolutely convergent for almost every $x$, and $g$ will be periodic with period $T_s$. Also, $$\int_0^{T_s}g(x)\,dx=\int_{-\infty}^\infty f(x)\,dx.$$ This might be of some use, depending on your application. | 2019-06-25 08:31:27 | {"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": 1, "mathjax_display_tex": 1, "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": 0.9006696939468384, "perplexity": 142.0853794805696}, "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-2019-26/segments/1560627999814.77/warc/CC-MAIN-20190625072148-20190625094148-00108.warc.gz"} |
http://mathoverflow.net/questions/134361/complex-zeros-of-zetas-zetas-zeta1-s-zeta1-s-simpler-expres | # Complex zeros of $\zeta'(s)/\zeta(s) + \zeta'(1-s)/\zeta(1-s)$ = simpler expression (except at zeta zeros)
Let $G(s) := \frac{\zeta'(s)}{\zeta(s)} + \frac{\zeta'(1-s)}{\zeta(1-s)}$ where $s$ is not a zero of zeta.
$G$ has real zeros and a pair of complex zeros near $\frac12 \pm 6i$.
Partial results:
By differentiating $\log{\xi(s)} - \log{\xi(1-s)}$ one gets:
$$G(s) = -1/2\psi(1/2+s/2)+1/2\pi\tan(\pi s/2) -1/2 \psi(s/2) +\log\pi \qquad (1)$$
when $s$ is not a zeta zeros. Probably (1) can be simplified since $\Re G(s) = \Re\left(-1/(s-1)+1/s+\log\pi -(\psi(s/2)+\psi((1-s)/2+1)/2\right)$
Are there other complex zeros of (1), especially in the critical strip?
X-Ray of $G(s)$ using (1):
- | 2014-04-19 07:40:09 | {"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": 1, "mathjax_display_tex": 1, "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": 0.8912936449050903, "perplexity": 1140.0563462602872}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00411-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://tiny-themovie.com/ebooks/deductive-transformation-geometry | # Deductive Transformation Geometry
Format: Hardcover
Language: English
Format: PDF / Kindle / ePub
Size: 9.05 MB
Downloadable formats: PDF
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Pages: 130
Publisher: Cambridge University Press (August 14, 1975)
ISBN: 0521205654
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based on 2058 customer reviews | 2017-08-20 05:47:26 | {"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": 1, "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": 0.5039479732513428, "perplexity": 1373.510830692195}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2017-34/segments/1502886105976.13/warc/CC-MAIN-20170820053541-20170820073541-00309.warc.gz"} |
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BrainGear can be used by any consumer that wants to nourish their brain but want the power that nootropics offer. With vitamins and minerals, consumers can simply drink the formula daily to get the desired effects. The liquid format makes it easier to absorb in the body than a pill, and the individual units mean no measuring is necessary. | 2022-09-29 02:28:59 | {"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": 1, "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": 0.2175428867340088, "perplexity": 2806.4877904682107}, "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-2022-40/segments/1664030335303.67/warc/CC-MAIN-20220929003121-20220929033121-00221.warc.gz"} |
http://openstudy.com/updates/55e47e84e4b02272061175ae | ## anonymous one year ago Which of the following are measurements of the sides of a right triangle? 101, 99, 20 28, 26, 12 17, 14, 6 none of the above
Pythagorean theorem $\huge\rm a^2+b^2=c^2$ where c is the longest side of right triangle (hypotenuse ) substitute values if you get equal sides then that 3 numbers would form right trinagle | 2017-01-17 17:29:48 | {"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": 1, "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": 0.6115823984146118, "perplexity": 518.6507896498055}, "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-2017-04/segments/1484560279933.49/warc/CC-MAIN-20170116095119-00380-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://interviewmania.com/aptitude/trigonometry/1/1 | ## Trigonometry
#### Trigonometry
1. The angles of elevation of an aeroplane flying vertically above the ground, as observed from the two consecutive stones, 1 km apart; are 45° and 60° aeroplane from the ground is :
1. (√3 + 1) km.
2. (√3 + 3) km.
3. 1 (√3 + 1) km. 2
4. 1 (√3 + 3) km. 2
1. Two consecutive kilometre stones ⇒ C and D
∠ADB = 45°; ∠ACB = 60°
CD = 1 km.
AB = height of plane = h metre
BC = x metre (let)
In ∆ABC,
tan60° = AB BC
⇒ √3 = h x
⇒ h = √3x metre ..... (i)
In ∆ABD
tan45° = AB BD
⇒ 1 = h x + 1
⇒ h = x + 1
⇒ h = h + 1 √3
[From equation (i)]
⇒ h - h = 1 √3
⇒ √3h - h = 1 √3
⇒ (√3 - 1)h = √3
⇒ h = √3 √3 - 1
⇒ h = √3(√3 + 1) (√3 - 1)(√3 + 1)
⇒ h = √3(√3 + 1) 2
h = (3 + √3) metre 2
##### Correct Option: D
Two consecutive kilometre stones ⇒ C and D
∠ADB = 45°; ∠ACB = 60°
CD = 1 km.
AB = height of plane = h metre
BC = x metre (let)
In ∆ABC,
tan60° = AB BC
⇒ √3 = h x
⇒ h = √3x metre ..... (i)
In ∆ABD
tan45° = AB BD
⇒ 1 = h x + 1
⇒ h = x + 1
⇒ h = h + 1 √3
[From equation (i)]
⇒ h - h = 1 √3
⇒ √3h - h = 1 √3
⇒ (√3 - 1)h = √3
⇒ h = √3 √3 - 1
⇒ h = √3(√3 + 1) (√3 - 1)(√3 + 1)
⇒ h = √3(√3 + 1) 2
h = (3 + √3) metre 2
1. Two men standing on same side of a pillar 75 metre high, observe the angles of elevation of the top of the pillar to be 30° and 60° respectively. The distance between two men is :
1. 100 √3 metre
2. 100 metre
3. 50 √3 metre
4. 25 √3 metre
1. AB = Height of pole = 75 metre
C and D ⇒ positions of persons
Let, BC = x metre, BD = y metre
∆ACB = 60°; ∆ADB = 30°
In ∆ABC,
tan60° = AB BC
⇒ √3 = 75 x
⇒ x = 75 = 25 √3 metre √3
In ∆ABD
tan30° = AB BD
⇒ 1 = 75 √3 y
⇒ y = 75 √3 metre
∴ CD = y – x = (75 √3 - 25 √3) metre CD = 50√3 metre
##### Correct Option: C
AB = Height of pole = 75 metre
C and D ⇒ positions of persons
Let, BC = x metre, BD = y metre
∆ACB = 60°; ∆ADB = 30°
In ∆ABC,
tan60° = AB BC
⇒ √3 = 75 x
⇒ x = 75 = 25 √3 metre √3
In ∆ABD
tan30° = AB BD
⇒ 1 = 75 √3 y
⇒ y = 75 √3 metre
∴ CD = y – x = (75 √3 - 25 √3) metre CD = 50√3 metre
1. From a point P on a level ground, the angle of elevation to the top of the tower is 30°. If the tower is 100 metre high, the distance of point P from the foot of the tower is (Take √3 = 1.73)
1. 149 metre
2. 156 metre
3. 173 metre
4. 188 metre
1. Let, AB = Height of tower = 100 metre
∆ACB = 30°
In ∆ABC,
tan30° = AB BC
⇒ 1 = 100 √3 BC
⇒ BC = 100√3 metre = (100 × 1.73) metre = 173 metre
##### Correct Option: C
Let, AB = Height of tower = 100 metre
∆ACB = 30°
In ∆ABC,
tan30° = AB BC
⇒ 1 = 100 √3 BC
⇒ BC = 100√3 metre = (100 × 1.73) metre = 173 metre
1. If the angle of elevation of the top of a pillar from the ground level is raised from 30° to 60°, the length of the shadow of a pillar of height 50 √3 will be decreased by
1. 60 metre
2. 75 metre
3. 100 metre
4. 50 metre
1. AB = Height of pole = 50 √3 metre
BC = Length of shadow = x metre
When, ∠ACB = 30°
BD = Length of shadow = y metre
In ∆ABC,
tan30° = AB BC
⇒ 1 = 50 √3 √3 x
⇒ x = 50√3 × √3 = 150 metre
In ∆ABD
tan60° = AB BD
⇒ √3 = 50 √3 y
⇒ y = 50 √3 = 50 metre √3
∴ CD = x – y = 150 – 50 = 100 metre
##### Correct Option: C
AB = Height of pole = 50 √3 metre
BC = Length of shadow = x metre
When, ∠ACB = 30°
BD = Length of shadow = y metre
In ∆ABC,
tan30° = AB BC
⇒ 1 = 50 √3 √3 x
⇒ x = 50√3 × √3 = 150 metre
In ∆ABD
tan60° = AB BD
⇒ √3 = 50 √3 y
⇒ y = 50 √3 = 50 metre √3
∴ CD = x – y = 150 – 50 = 100 metre
1. Find the angular elevation of the Sun when the shadow of a 15 metre long pole is (15 / √3) metre.
1. 45°
2. 60°
3. 30°
4. 90°
1. A' ⇒ Position of sun
AB = Height of pole = 15 metre
BC = Length of shadow = 15 metre √3
∴ tanθ = AB = 15 = √3 BC (15 / √3)
⇒ tanθ = tan60°
⇒ θ = 60°
##### Correct Option: B
A' ⇒ Position of sun
AB = Height of pole = 15 metre
BC = Length of shadow = 15 metre √3
∴ tanθ = AB = 15 = √3 BC (15 / √3)
⇒ tanθ = tan60°
⇒ θ = 60° | 2021-08-01 13:59:44 | {"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": 0.9184437394142151, "perplexity": 7122.374628050039}, "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-2021-31/segments/1627046154214.36/warc/CC-MAIN-20210801123745-20210801153745-00129.warc.gz"} |
http://www-math.umd.edu/gcal_rss.php?seminar_key=GEOTOP&year=2013&html | ### Geometry-Topology Archives for Academic Year 2013
#### The toric geometry of polygons in Euclidean space
When: Mon, September 10, 2012 - 3:00pm
Where: Math 1313
Speaker: John Millson (University of Maryland, College Park)
Abstract: see http://www.math.umd.edu/~karin/seminarabstracts.html
#### An abstract construction of the maximal extension for globally hyperbolic conformally flat space-times
When: Mon, September 17, 2012 - 3:00pm
Where: Math 1313
Speaker: Clara Rossi Salvemini (University of Avignon)
Abstract: A conformally flat space-time M of dim M>2, is a (G,X)-manifold. The model space X is the space of Einstein,
which is S^n \times R with the conformal class of the metric ds^2-dt^2, where ds^2 is the standard
metric on the sphere et dt^2 on R. The group G is the identity component of the group O(2,n), which is the group
of conformal diffeomorphisms of the Einstein's space.
The causal structure of a Lorentzian manifold is a conformal invariant, so we have a well
defined causal structure on M. We assume that this causal structure is globally hyperbolic.
We will define also the notion of unique maximal extension for these space-times.
We will make an abstract construction of this maximal extension starting from a Cauchy hypersurface of M which use only the (G,X)-structure of M. This allow us to characterize the maximal space-time by their causal boundary.
We will also show some results about the developing map of these space-times. In particular we have:
Every globally hyperbolic conformally flats maximal spaces-time M which have two lightlike geodesics freely homotopic with same ends is a finite quotient of the Einstein's space.
#### Some complexes of differential operators
When: Mon, September 24, 2012 - 3:00pm
Where: Math 1313
Room: Math 1313
Speaker: Katharina Neusser (Australian National University),
Abstract: We shall present a method for constructing complexes of invariant differential operators on manifolds endowed with various geometric structures. The geometric structures will mainly be certain bracket generating vector distributions, like for example a contact structure. For these structures the constructed complexes will give rise to fine resolutions of the sheaf of locally constant functions and so can serve as an alternative to the de Rham complex. In the case of parabolic geometries we recover the so called BGG complexes associated to the trivial representation.
Joint work with Robert Bryant, Michael Eastwood and Rod Gover.
#### Field Committee Meeting for Geometry-Topology
When: Mon, October 1, 2012 - 3:00pm
Where: Math 1313
Speaker: Jim Schafer (University of Maryland, College Park)
#### Dynamics on the PSL(2, C)-character variety of certain hyperbolic 3-manifolds
When: Mon, October 8, 2012 - 3:00pm
Where: Math 1313
Speaker: Michelle Lee (University of Maryland, College Park)
Abstract: The PSL(2, C)-character variety of a hyperbolic 3-manifold M is essentially the set of homomorphisms of the fundamental group of M into PSL(2, C), up to conjugacy. We will discuss the action of the group of outer automorphisms of the fundamental group act on this space. In particular, we will discuss how one can find domains of discontinuity for the action.
#### Field Committee Meeting for Geometry-Topology
When: Mon, October 15, 2012 - 3:00pm
Where: Math 1313
Speaker: Jim Schafer (University of Maryland, College Park) -
#### Cellular Automorphisms and Self Duality
When: Mon, October 22, 2012 - 3:00pm
Where: Math 1313
Speaker: Lowell Abrams (George Washington University) -
Abstract: Given a graph G cellularly embedded in a closed surface S, an automorphism of G is called a cellular automorphism of G in S when, loosely speaking, it takes facial boundary walks to facial boundary walks. I will describe how we constructed complete catalogs of all irreducible cellular automorphisms of the sphere, projective plane, torus, Klein bottle, and three-crosscaps surface for a particular notion of reducibility related to taking minors.
We have also determined concrete procedures sufficient for constructing all possible self-dual embeddings in any closed surface S given a catalog of all irreducible cellular automorphisms in S. I will illustrate by way of examples some of these procedures and some resulting self-dual graphs.
#### Counting Chord Diagrams
When: Tue, October 23, 2012 - 3:30pm
Where: MATH 2300
Speaker: Robert Penner (Aarhus University and California Institute of Technology) -
Abstract: A linear chord diagram on some number b of backbones is a collection of n chords with distinct endpoints attached to the interiors of b intervals.
Taking the intervals to lie in the real axis and the chords to lie in the upper half-plane associates a fatgraph to a chord diagram, which thus has its associated genus g. The numbers of connected genus g chord diagrams on b backbones with n chords are of significance in mathematics, physics and biology as we shall explain. Recent work using the topological recursion of Eynard-Orantin has computed them perturbatively via a closed form expression for the free energies of an Hermitian matrix model with potential V(x)=x^2/2-stx/(1-tx). Very recent work has moreover shown that the partition function satisfies a second order non-linear pde which gives a generalization of the Harer-Zagier equation that arises for one backbone.
#### CANCELLED because of university closure (Equivariant K-theory of actions of compact Lie groups with maximal rank isotropy subgroups)
When: Mon, October 29, 2012 - 3:00pm
Where: Math 1313
Speaker: Jose Manuel Gomez (Johns Hopkins University)
Abstract: In this talk we study the equivariant K-theory of a compact
Lie group G acting on a space X with maximal rank isotropy subgroups.
In particular we provide conditions that guarantee freeness over the
representation ring of G. Some applications related
to spaces of representations on Lie groups will be provided.
#### TBA
When: Mon, November 5, 2012 - 3:00pm
Where: Math 1313
Speaker: Scott Wolpert (UMCP) -
#### Kahler-Einstein (edge) theory and algberaic geometry
When: Mon, November 12, 2012 - 3:00pm
Where: Math 1313
Speaker: Yanir Rubinstein (University of Maryland, College Park)
Abstract: We highlight some problems in algebraic geometry related to Kahler-Einstein (edge) metrics. I will try to emphasize the ideas and give some examples.
#### Long-time analysis of 3 dimensional Ricci flow
When: Mon, November 19, 2012 - 3:00pm
Where: Math 1313
Speaker: Richard Bamler (Stanford)
Abstract: It is still an open problem how Perelman's Ricci flow with surgeries behaves for large times. For example, it is unknown whether surgeries eventually stop to occur and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$.
In this talk, I will present new tools to treat this question after providing a quick review of Perelman's results. In particular, I will show that in the case in which the initial manifold satisfies a certain purely topological condition, surgeries do in fact stop to occur after some time and the curvature is globally bounded by $C t^{-1}$. For example, this condition is satisfied by manifolds of the form $\Sigma \times S^1$ where $\Sigma$ is a surface of genus $\geq 1$.
#### The Sullivan-Riemann field of a simply connected closed manifold
When: Mon, November 26, 2012 - 3:00pm
Where: Math 1313
Speaker: Stephen Halperin (University of Maryland, College Park)
Abstract: posted at http://www.math.umd.edu/~karin/seminarpdfs.html
#### Skein algebras and the decorated Teichmuller space
When: Mon, December 3, 2012 - 3:00pm
Where: Math 1313
Speaker: Julien Roger (Rutgers University) -
Abstract: We construct the skein algebra of a punctured surface S based on framed
links and arcs in Sx[0,1]. We then describe the relationship between this
algebra and quantization of the decorated Teichmuller space, based on the
study of a collection of geodesic length identities in hyperbolic
geometry. This is joint work with T. Yang.
#### Long-time Behavior of Ricci Flows and Construction of Einstein Metrics
When: Fri, December 7, 2012 - 2:00pm
Where: Math 3206
Speaker: Richard Bamler (Stanford) -
Abstract: In this talk I will survey current results on the long-time existence and
behavior of Ricci flows in dimensions 2, 3 and higher. Moreover, I will
point out analogies with construction techniques for Einstein metrics.
In dimension 3, the Ricci flow together with a certain surgery process has
been used by Perelman to establish the Poincaré and Geometrization
Conjectures. Despite the depth of Perelman's result, a precise description
of the long-time behavior of this flow still does not exist. For example,
it has been unknown whether it suffices to carry out a finite number of
surgeries or whether the geometric decomposition of the manifold is
exhibited by the flow as $t \to \infty$. I will explain Perelman's result
and recent progress on this problem. I will then present long-time
existence results in dimensions 4 and higher and describe possible further
directions in this field.
#### Rigidity of actions on CAT(0) cube complexes
When: Mon, December 10, 2012 - 3:00pm
Where: Math 1313
Speaker: Talia Fernos (North Carolina State, Greensboro)
Abstract: Let G be a group acting non-elementarily by automorphisms on a finite dimensional CAT(0) cube complex. In a joint work with Indira Chatterji and Alessandra Iozzi, we prove the non-vanishing of second bounded cohomology of G with geometrically defined coefficients. From this we deduce super rigidity for actions of any irreducible lattice in a nontrivial product of locally compact groups.
#### Polynomial Pick forms for affine spheres and real projective polygons
When: Wed, January 23, 2013 - 2:00pm
Where: Math 3206
Speaker: Mike Wolf (Rice University) -
Abstract: (Joint work with David Dumas.) Convex real projective structures on surfaces, corresponding to discrete surface group representations into SL(3, R), have associated to them affine spheres which project to the convex hull of their universal covers. Such an affine sphere is determined by its Pick (cubic) differential and an associated Blaschke metric. As a sequence of convex projective structures leaves all compacta in its deformation space, a subclass of the limits is described by polynomial cubic differentials on affine spheres which are conformally the complex plane. We show that those particular affine spheres project to polygons; along the way, a strong estimate on asymptotics is found. As some of the background material is rich but outside the usual Riemannian geometric canon, we will spend substantial time explaining it.
#### Crooked surfaces and anti-de Sitter geometry
When: Mon, January 28, 2013 - 3:00pm
Where: Math 1313
Speaker: Bill Goldman
Abstract: Minkowski space is flat spacetime and is the Lorentzian analog of Euclidean
space; anti-de Sitter space has constant negative curvature and is analogous to
hyperbolic space. Crooked planes were defined by Drumm to bound fundamental
polyhedra in Minkowski space for Margulis spacetimes, and the analogous polyhedra
were defined by Danciger, Gueritaud and Kassel in anti-de Sitter geometry.
In this talk we leisurely expound anti-de Sitter geometry in terms of the group SL(2,R)
and show how the anti-de Sitter crooked planes are just the conformal extensions
(developed by Frances, Barbot, Charette, Drumm, Melnick and myself) invariant under the
involution defining anti-de Sitter geometry.
#### The Kunneth Theorem in equivariant K-theory
When: Mon, February 4, 2013 - 3:00pm
Where: Math 1313
Speaker: Jonathan Rosenberg (UMCP)
Abstract: First we explain Hodgkin's Kunneth Theorem for equivariant K-theory K^*_G (for G a connected compact Lie group) and then explain why it fails dramatically for G finite. Then we show how to correct the theorem when G is cyclic of order 2. This result will appear soon in Algebraic & Geometric Topology.
#### Talk canceled
When: Mon, February 11, 2013 - 3:00pm
Where: Math 1313
Speaker: Babak Modami
Abstract: The Weil-Petersson (WP) metric is an incomplete Riemannian metric on
the moduli space of Riemann surfaces with negative sectional
curvatures which are not bounded away from $0$. Brock, Masur and
Minsky introduced a notion of "ending lamination" for WP geodesic rays
which is an analogue of the vertical foliations of Teichm\"{u}ller
geodesics. In this talk we show that these laminations and the
associated subsurface coefficients can be used to determine the
itinerary of a class of WP geodesics in the moduli space. As a result
we give examples of closed WP geodesics staying in the thin part of of
the moduli space, geodesic rays recurrent to the thick part of the
moduli space and diverging geodesic rays. These results can be
considered as a kind of symbolic coding for WP geodesics.
#### Cutting sequences on Bouw-Möller surfaces
When: Mon, February 18, 2013 - 3:00pm
Where: Math 1313
Speaker: Diana Davis, Brown
Abstract: We will investigate a dynamical system that comes from geodesic trajectories on flat surfaces. We will start with the square torus and the regular octagon surface, and then discuss new results for Bouw-Möller surfaces, made from many polygons.
#### Homological stability of moduli spaces of manifolds
When: Mon, February 25, 2013 - 3:00pm
Where: Math 1313
Speaker: Soren Galatius, Stanford
Abstract: The moduli space of Riemann surfaces M_g parametrizes bundles of genus g surfaces. A classical theorem of J. Harer implies that the homology H_k(M_g) is independent of g, as long as g is large compared to k. In joint work with Oscar Randal-Williams, we establish an analogue of this result for manifolds of higher dimension: The role of the genus g surface is played by the connected sum of g copies of S^n \times S^n.
#### On holomorphic maps between ball quotients
When: Mon, March 4, 2013 - 3:00pm
Where: Math 1313
Speaker: Vincent Koziarz, Bordeaux
Abstract: We will expound on a joint result with N. Mok stating that holomorphic
surjective maps between compact ball quotients must have singular fibers
(in the non-equidimensional case). Examples of such surjective maps are
very rare and we will give a few leads towards their classification in
the case when fibers are 1-dimensional.
#### Totally geodesic curves and cohomology of complex hyperbolic manifolds
When: Mon, March 11, 2013 - 3:00pm
Where:
Speaker: Matthew Stover, Michigan
Abstract: Let M be a compact complex hyperbolic 2-manifold. I will discuss applications of Lefschetz-type theorems to the geometry and cohomology of M. When M is arithmetic and contains holomorphically immersed close totally geodesic curves, this gives a nice structure theorem for the cohomology of M reminiscent of results of Gelbart and Rogawski on cohomology coming from the theta correspondence, and our results have the added bonus of working in the noncongruence setting too. This is joint work with Ted Chinburg.
#### Poncelet's Porism and Algebraic Geometry
When: Mon, March 25, 2013 - 3:00pm
Where:
Speaker: Jean-Philippe Burelle (UMD)
Abstract: This talk is going to be an exposition of part of a paper by Griffiths and Harris using algebraic geometry to solve a classical geometry problem first investigated by Poncelet. This problem is easy to state but hard to solve using classical methods :
Given two conics in the real projective plane, is there a polygon that is inscribed in one and circumscribed to the other?
#### A new proof of Bowen's theorem on quasi-circles
When: Mon, April 1, 2013 - 3:00pm
Where: Math 1313
Speaker: Andy Sanders, UMD
Abstract: In 1979, Bowen proved that the Hausdorff dimension of the limit set of a quasi-Fuchsian group is equal to 1 if and only if the group is Fuchsian. Since then, many proofs and sweeping generalizations of this result have been given. We will present a new proof of this result which relies on the existence of equivariant minimal surfaces in hyperbolic 3-space.
#### Ending laminations for Weil-Petersson geodesics
When: Mon, April 8, 2013 - 3:00pm
Where: Math 1313
Speaker: Babak Modami, Yale
Abstract: The Weil-Petersson (WP) metric is an incomplete Riemannian metric on
the moduli space of Riemann surfaces with negative sectional
curvatures which are not bounded away from $0$. Brock, Masur and
Minsky introduced a notion of "ending lamination" for WP geodesic rays
which is an analogue of the vertical foliations of Teichm\"{u}ller
geodesics. In this talk we show that these laminations and the
associated subsurface coefficients can be used to determine the
itinerary of a class of WP geodesics in the moduli space. As a result
we give examples of closed WP geodesics staying in the thin part of of
the moduli space, geodesic rays recurrent to the thick part of the
moduli space and diverging geodesic rays. These results can be
considered as a kind of symbolic coding for WP geodesics.
#### Complete flat Lorentzian three-manifolds (joint with Dynamics)
When: Thu, April 18, 2013 - 2:00pm
Where: Math 1311
Speaker: Jeff Danciger (University of Texas, Austin) -
Abstract: A complete flat Lorentzian three-manifold is the quotient of the (2+1)-dimensional Minkowski space by a discrete group acting properly by affine O(2,1) transformations. In the interesting cases, the group acting is a free group and the quotient manifold is called a Margulis space-time. I will describe work in progress toward classifying the topology of Margulis space-times. In particular, when the O(2,1) part of the group action does not contain parabolics, we prove that the quotient manifold is a handle-body. The proof depends on a new properness criterion for free groups acting on Minkowski space and draws on ideas from anti de Sitter (AdS) geometry. This is joint work with François Guéritaud and Fanny Kassel.
#### Some aspects of three dimensional Lorentz dynamics
When: Mon, April 22, 2013 - 3:00pm
Where: Math 1313
Speaker: Charles Frances (University of Paris XI, Orsay) -
Abstract: It is a consequence of Gromov's theory of rigid geometric
structures that rigid structures with an automorphism group having
"complicated" dynamics are often locally homogeneous, at least on some open
dense subset. This principle is nicely illustrated by a theorem of Zeghib
describing all compact Lorentz 3-manifolds whose isometry group has a
noncompact connected component. The purpose of the talk is to show how new
phenomenas appear when the isometry group has an infinite discrete part.
#### Annular Twists and Bridge Numbers of Knots
When: Mon, April 29, 2013 - 3:00pm
Where:
Speaker: Ken Baker, Miami
Abstract: Parametrized by rational numbers, the various Dehn surgeries on a knot in
the 3-sphere each produce a new knot in another manifold. Generically, the
new knot is as "simple as possible" in this new manifold, though that's
not always the case. One measure of this simplicity is bridge number
(with respect to a Heegaard splitting). In this talk we'll survey recent
works with Cameron Gordon and John Luecke that these bridge numbers may
behave quite differently for integral and non-integral Dehn surgeries.
#### Forms, flows and physics
When: Mon, May 6, 2013 - 3:00pm
Where: Math 1313
Speaker: Ted Jacobson (UMCP, physics)
Abstract: Differential forms provide the ideal mathematical language
for aspects of physics that do not involve the spacetime metric.
I will discuss examples from Hamiltonian mechanics, relativistic
electrodynamics, and plasma physics, with the dual aim of
explaining some physics to mathematicians, and illustrating
how the physics is most easily and naturally understood using
this language.
#### Homological and Homotopical Dehn Functions Are Different
When: Mon, May 13, 2013 - 3:00pm
Where: Math 1313
Speaker: Noel Brady (University of Oklahoma and National Science Foundation)
Abstract: One of the fundamental problems introduced by Max Dehn in the study of finitely presented groups is the word problem. Given a finitely presented group and a word in the generators is there a procedure to determine if the word represents the identity element of the group. An isoperimetric function of a finitely presented group provides an upper bound on the number of relations that must be used to show that a word in the generators represents the identity. A Dehn function is an optimal isoperimetric function.
An isoperimetric function can be interpreted geometrically as providing an upper bound on the combinatorial area of a least area disk, which is bounded by a given loop in the Cayley complex of the finitely presented group. A homological version of this notion can be defined where one considers least area 2-chains, which are bounded by a given loop. We shall describe how to construct finitely presented groups whose homological Dehn functions are strictly smaller than their ordinary Dehn functions. This is joint work with Aaron Abrams, Pallavi Dani and Robert Young. | 2018-03-24 19:56:28 | {"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": 1, "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": 0.6870672106742859, "perplexity": 1286.2158713644765}, "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-2018-13/segments/1521257650993.91/warc/CC-MAIN-20180324190917-20180324210917-00582.warc.gz"} |
https://www.nature.com/articles/s41598-018-22446-y?error=cookies_not_supported&code=ca8bd3b9-1f76-4a2f-b7de-b988001e61f5 | ## Introduction
Supramolecular macrocyclic compounds have been largely developed since the first generation of crown ethers till the fifth generation of pillar[n]arenes during the past decades1,2,3,4,5. Considering their superior properties, special structures and typical functions in supramolecular chemistry, remarkable attentions have been focused on the exploitation of their potentials and progress in molecular machines6,7, molecular recognition8,9,10,11, nanomaterials12,13,14,15,16, supramolecular polymers17,18,19,20,21, chemical sensors and detectors22,23,24,25, biological medicine26,27,28,29,30 and so on31,32. As a relatively new class of supramolecular macrocyclic hosts, pillar[n]arenes have attracted extensive attention owning to its unique properties, rigid structures, easy functionalization, etc.33,34,35. Monofunctionalized pillar[n]arenes, as one typical type of useful pillar[n]arene derivatives with unique substituent groups, possess variety of abilities such as molecular recognition and fluorescent detection, depending on their single modified functional groups and the different host-guest interactions36,37,38,39,40.
Aggregation-induced emission (AIE), entirely opposite to the aggregation-caused quenching (ACQ) effect of traditional fluorescent dyes41,42,43,44, was first reported by Tang and coworkers in 200145, which paves a new way for the efficiency of fluorescent dyes in the solid state and the concentrated solution and breaks the limited applications of traditional fluorophores with ACQ properties. Restriction of intramolecular rotation (RIR) of AIE molecules was proven to be the well-known mechanism of their fluorescent enhancement in the aggregated state. Non-radiation energy dissipation channel from the excited state to ground state was blocked because of the hindrance of intramolecular steric interaction resulted from the impeded intramolecular rotations when the AIE molecules are assembled46,47,48. Tetraphenylethene (TPE) is a typical AIE molecule that has been widely investigated during the past two decades49,50. Several studies combining supramolecular approaches with typical fluorescent molecules with AIE properties to construct controllable fluorescent detectors or stimuli-responsive supramolecular materials have been reported51,52,53, especially using some pillar[n]arene derivatives as building blocks. This indicates that the combination of pillararenes and TPE is indeed an efficient way for the fabrication of novel smart optical devices to be applied in chemical sensors51, biological imaging54,55, and detection of pollutants and explosives56,57, among which the immediately selective detection of alcohol analogs is urgently required in the field of ecological environment and industrial development. The detection of alcohol analogs possesses special significance in traffic safety and medical emergency.
Herein, for the first time, we successfully synthesized sulfonic group-substituent monofunctionalized pillar[5]arene, i.e., monosulfonicpillar[5]arene (MSP5), (Fig. 1) in a good yield. A stable fluorescent complex between MSP5 and a guest TPE derivative, i.e., TPE-(Br)4, has also been designed and prepared via host-guest complexation, giving the credit to the hydrophilic group of MSP5 was not affected by pH of the solution. MSP5 also exhibited effective binding affinity towards alcohols via hydrogen bonds between hydroxyl of alcohols and sulfonic group of MSP5, making it possible to selectively detect alcohol analogs by destroying the complex of TPE-(Br)4MSP5. Furthermore, the complex of MSP5 and TPE-(Br)4 can also serve as a temperature sensor and fluorescence probe for ethylenediamine.
## Results and Discussion
MSP5 was synthesized through the installation of a 1-butanesulfonic acid sodium onto pillar[5]arene via Williamson ether-type synthetic method. The structure of MSP5 was confirmed by 1H NMR, 13C NMR, HRMS, and FT-IR spectroscopy (Supplementary Fig. S5~S8). Considering that various monofunctionalized pillar[n]arenes possess the property of typical molecular recognition, six alcohol analogs were selected as guest molecules and the molecular recognition of MSP5 towards them was investigated via 1H NMR titration. As in Figs S34, S37, S40, S43, S46 and S49, when MSP5 was added into a chloroform solution of alcohols, the proton signal of Ha of alcohols showed an obvious upfield shift due to the shielding effect upon inclusion by pillararene cavity, indicating the host-guest interactions between MSP5 and alcohols. Nonlinear curve fitting method was employed to obtain the association constant (Ka) between those alcohols and MSP5, respectively. MSP5 has the strongest binding affinity toward butanediol among other alcohols (Fig. 2a). Molar ratio plot based on the chemical shift changes of the protons of alcohols showed that all the stoichiometries of MSP5 and different alcohols are 1:1 (Supplementary Fig. S33). Interestingly, MSP5 possessing sulfonic entity exhibited much stronger binding affinity towards butanediol (Fig. 2b) as compared with monophosphoryl copillar[5]arene (MPP5)36 and dimethoxypillar[5]arene (DMP5), indicating its ability of selective recognition toward alcohol analogs particularly with an enlarged selectivity.
Based on the fact that MSP5 exhibited selective binding ability toward alcohols, we design a fluorescent complex via host-guest interaction between MSP5 and TPE-(Br)4 for molecular sensing and detection. We synthesized the TPE derivative with four binding arms that can be included in the cavity of MSP5, which can serve as a fluorescent indicator. A novel binary supramolecular-assembled fluorescent ensemble was constructed from MSP5 and TPE-(Br)4 (Fig. 1b). In order to investigate the host-guest properties between MSP5 and TPE-(Br)4, 1-(4-bromobutoxy)-4-methoxybenzene (G1) possessing the same binding site as TPE-(Br)4 was synthesized as a model compound. As shown in the Supplementary Fig. S52, when MSP5 was added into a chloroform solution of G1, the signals corresponding to the protons H1 and H2 on the alkyl chain shifted upfield, because these protons were located in the cavity of MSP5 and suffered from shielding effect. This provided a strong evidence for the interactions between MSP5 and G1. MSP5 forms a 1:1 complex with G1 as assessed by 1H NMR titration, and the Ka of G1MSP5 was calculated to be (1.08 ± 0.22) × 102 M−1 in chloroform using nonlinear curve-fitting analysis (Supplementary Fig. S53). 2D NOESY NMR spectrum of MSP5 and G1 was also obtained for further investigation of the host-guest interaction between MSP5 and TPE-(Br)4. As shown in Fig. 3a, H4 was the proton on the guest while Hc, Hd, He were the protons on MSP5, the crosspeak A indicates that H4 is in close contact with Hd, He, and the crosspeak B indicates that H4 also interacts Hc, suggesting that alkyl of G1 penetrated into the cavity of MSP5 to form a good inclusion complex. The proton NMR spectrum of MSP5 with TPE-(Br)4 in d-chloroform solution was also obtained and similar complexation-induced chemical shift changes were detected (Supplementary Fig. S58).
Strong emission in dilute solution was observed from the complex of TPE-(Br)4MSP5, while non-fluorescent emission was surveyed from individual MSP5 and TPE-(Br)4 at the same concentration (Fig. 3b). Their host-guest fluorescence behaviors were investigated in detail and provided in the supplementary Fig. S59c. Upon increasing the concentration of MSP5, the fluorescence intensity of TPE-(Br)4 was gradually enhanced, which can be ascribed to the formation of host-guest inclusion complex that restricts the intramolecular rotation of phenyl rings of TPE-(Br)4. Besides, the fluorescence enhancement of TPE-(Br)4 induced by addition of MSP5 was clearly perceived by naked eyes (Fig. 3b-E), and strong cyan fluorescence can be visualized upon irradiation by a UV lamp with the wavelength of 365 nm, which was also supported by the above proposed mechanism.
On the other hand, the fluorescence enhancement of TPE-(Br)4 upon addition of MSP5 confirmed the synergetic importance of host-guest interaction of pillararene and TPE guests and sulfonic functional group on pillararenes by a series of controlled experiments. The addition of MSP5 with a monosulfonic arm to form host-guest inclusion complex was proven to be the necessary condition for the fluorescent enhancement, consistent to RIR mechanism. TPE-(Br)4 and other two different host molecules, i.e., DMP5 and monocarboxylatopillar[5]arene (MCP5), were selected to investigate whether host-guest interaction itself will produce fluorescent enhancement of TPE-(Br)4. 1H NMR titration experiments provided Ka between MCP5 and G1 (Supplementary Fig. S56), which is similar to that of G1MCP5 and G1MSP5. When DMP5 and MCP5 were added into TPE-(Br)4 chloroform solution gradually, the fluorescence of TPE-(Br)4 was almost unchanged (Fig. 4a), illustrating that host-guest interaction itself was unable to result in fluorescence enhancement. In addition, TPE and TPE-(CN)4 were also synthesized to investigate the effect of fluorescent molecules with different binding sites. When MSP5 mixed with TPE without binding affinity, the mixture was non-fluorescent (Fig. 4c), while strong fluorescence emission can be observed in the mixture of MSP5 and TPE-(CN)4 (Fig. 4d), for the reason of the host-guest complex between MSP5 and TPE-(CN)458. Meanwhile, the fluorescent emission of TPE-(CN)4MSP5 was similar to that of TPE-(Br)4MSP5 (Fig. 4d), indicating that different functional groups on TPE with similar binding ability with pillararene had negligible influence on the fluorescence enhancement. Monomer of MSP5, i.e., 4-(4-methoxyphenoxy)butane-1-sulfonic acid (M1), was also synthesized to study the effect of sulfonic group on fluorescent enhancement of TPE-(Br)4. No fluorescence was observed when M1 mixed with TPE-(Br)4 (Fig. 4b), indicating that sulfonic group had no effect on the fluorescence of TPE-(Br)4.
Sulfonic group, as a hydrophilic entity, plays a key role in the complex system of TPE-(Br)4MSP5 to maintain stable under different pH conditions, which was different from that of carboxylic acid group as the hydrophobic group in MCP5. We deduced that the hydrophobicity of the functional groups affected the fluorescence behaviors. Thus, three kinds of anionic monofunctional pillar[5]arenes and their sodium salts (sulfonic group, sulfonate group, carboxyl group, carboxylate group, phosphoric group, phosphate group) have been synthesized to investigate the role of sulfonic group in the fluorescent ensembles. The enhanced fluorescence of TPE-(Br)4MSP5 can be detected under low concentration, and upon addition of MSP5, fluorescence enhanced gradually. Furthermore, the fluorescence of other five kinds of supramolecular ensembles has also been studied at the same conditions, and the results are shown in Supplementary Fig. S59. No obvious fluorescence enhancement of TPE-(Br)4MCP5 was detected, while the fluorescence was largely enhanced in TPE-(Br)4MSP5 and TPE-(Br)4monosulfonatepillar[5]arene. Monophosphoricpillar[5]arene also induced weaker fluorescence enhancement. On the contrary, monophosphatepillar[5]arene induced remarkable fluorescent enhancement, same as MSP5 and monosulfonatepillar[5]arene. From the above results, we can further concluded that the water-soluble groups-substituents monofunctionalized pillar[5]arene can enhance the fluorescence of TPE-(Br)4 via host-guest inclusion. There was an obvious difference in pKa of the substituent groups in pillar[5]arene derivatives: R-SO3H (1.6 in DMSO) < R-PO3H2 (2.59 and 8.19 in water/ethanol) < R-COOH (12.3 in DMSO)59,60,61, which indicates that sulfonic group exists in the form of acidic anion in chloroform solution. However, phosphoric has two pKa, the strong acidic hydrogen will be ionized in chloroform, resulting in slight fluorescence enhancement of TPE-(Br)4MSP5. The carboxylic acid group maintains un-ionized form in chloroform owning to the weak acidity, causing no fluorescence enhancement.
Scanning electron microscope (SEM), dynamic laser scattering (DLS), and DOSY NMR spectrum have been used to further investigate the fluorescence and self-assembled behaviors. The DOSY NMR spectrum showed that all the peaks correlated to the signals in the chemical shift dimensions are in a horizontal line (Supplementary Fig. S60), all proton signals of MSP5 and TPE-(Br)4 have the same diffusion coefficient (2.6 × 10−9 m2s−1), suggesting the host-guest interaction of TPE-(Br)4MSP5. The solution of the host-guest complex exhibited obvious Tyndall effect (Supplementary Fig. S61e), indicating that the complex formed abundant colloid particles. SEM images and DLS data proved the aggregation of TPE-(Br)4MSP5. All the above results illustrated that the TPE-(Br)4MSP5 can self-assemble into nanoparticles with the average diameter of 16 nm (Supplementary Fig. S61c,d,f), while the individual host and guest are amorphous (Supplementary Fig. S61a,b).
We thus ascribed the fluorescence enhancement to the following reasons: (i) The host-guest interaction of TPE-(Br)4MSP5 formed into pseudorotaxane, restricted the intramolecular rotation of phenyl rings of TPE-(Br)4 and blocked the nonradiative emission, leading to a strong fluorescence emission; (ii) The solubility of the host-guest complex (fluorescent nanoparticles) in chloroform was reduced due to the hydrophilic group, leading to the aggregation state; (iii) MSP5 can self-assemble with TPE-(Br)4 to construct organic fluorescent nanoparticles, reaching aggregation state and exhibiting strong emission.
We successfully utilize the host-guest interaction property of pillar[5]arene and the AIE effect of TPE, prepared a binary complex system, where TPE was used as a fluorescence indicator for identifying butanediol effectively (Fig. 5a). Ethylenediamine is toxic, which would damage human bodies and environment seriously. On account of the much stronger binding affinity of MSP5 towards ethylenediamine than TPE-(Br)4 (Supplementary Fig. S62)40, the ensemble of TPE-(Br)4MSP5 can be used to detect ethylenediamine sensitively and rapidly (Fig. 5b). In addition, this supramolecular assembly can also be applied as a temperature sensor, as the fluorescence intensity decreased gradually upon raising the temperature. The fluorescence intensity can revert to the initial intensity without wastage when temperature returned to the initial room temperature (Fig. 5c), indicating this temperature sensor has remarkable circulation performance and can be reused for many times (Fig. 5d).
## Conclusion
In summary, we synthesized MSP5 for the first time and employed it to construct a stable AIE-active binary complex system with TPE core via host-guest interaction and supramolecular self-assembly. Upon the formation of host-guest complex, the fluorescence emission of the complex was enhanced dramatically. The resulting pseudorotaxane-type structure restricted the intramolecular rotation of phenyl rings of the TPE-(Br)4 and blocked the nonradiative emission, finally resulting in strong fluorescence emission. We also investigated the molecular recognition ability of MSP5, and found that it can form stable complexes with alcohols. The MSP5 and TPE-(Br)4 can be used to fabricate supramolecular fluorescence composite through supramolecular self-assembly. This new fluorescence complex system possesses multi-stimuli responsive properties, and can selectively recognize butanediol among several similar alcohols. It also can act as a fluorescence probe to detect toxic substance ethanediamine and act as a temperature sensor. We envision that combining the AIE effect of TPE with the host-guest property of functional macrocycles may lead to many potential applications of pseudorotaxanes in sensors, cell imaging, controlled optical materials and smart materials.
## Experimental Section
### Methods
All reagents were commercially available and used without further purification. TPE-(Br)4, G1, M1 and MCP5 were synthesized according to a published literature procedure (See the Supporting Information for details)20,37,50,62. 1H NMR spectra were collected on a Bruker AVANCE III 300 MHz NMR spectrometer. 13C NMR, 2D NOESY NMR and DOSY NMR spectra were recorded on a Bruker AVANCE III 500 MHz NMR spectrometer. High-resolution electrospray ionization mass spectra (HRESI-MS) were obtained on a Bruker 7-Tesla FT-ICR mass spectrometer equipped with an electrospray source. Mass spectra were recorded on Bruker Daltonics Autoflex Speed Series: High-Performance MALDI-TOF Systems. FT-IR spectra were recorded on a Vertex 80 V spectrometer. Scanning electron microscope (SEM) images were obtained on a HITACHI-SU8082 instrument. The fluorescence experiments were conducted on a RF-5301 spectrofluorophotometer (Shimadzu Corporation, Japan). To determine the stoichiometry and association constants of alcoholsMSP5, 1H NMR titration was performed. By a nonlinear curve-fitting method, the association constants between the guests and host were calculated. Through a molar ratio plot, the stoichiometry was determined (see supporting information for details).
### Synthesis of MSP5
MonohydroxyDMP[5] (500 mg, 0.68 mmol, see supporting information for details) and NaOH (60 mg, 1.5 mmol) were added into 25 mL THF in a 50 mL flask, the mixture was stirred at room temperature for 1 h. Then, 1,4-butylenesulfone (0.15 mL) was added into the mixture. The mixture was stirred at 40 °C for 24 h. The crude product was recrystallized with dichloromethane/n-hexane and washed with water. After dryness, monosulfonatepillar[5]arene sodium salt was obtained as a yellow powder (400 mg, 64%). 1H NMR (300 MHz, CDCl3, 25 °C), δ (ppm): 6.73 (m, 10H), 3.46~3.77 (m, 39H), 3.02 (t, 2H), 1.79~1.96 (m, 4H). 13C NMR (126 MHz, CDCl3, 25 °C) δ (ppm): 150.73, 149.82, 128.22, 114.11, 68.31, 55.80, 50.97, 29.76, 28.82, 21.35. HRESIMS is shown in Fig. S11: m/z 871.3468 [M−Na] (100%). Then the sodium salt (200 mg) was dispersed in water and stirred with hydrochloric acid at room temperature for 12 h. After the solvent was removed, the obtain solid was purified by column chromatograph with dichloromethane/methanol (1:10 v/v) to get the final product of MSP5. Yellow powder: 140 mg, 72%. 1H NMR (300 MHz, CDCl3, 25 °C), δ (ppm): 6.72 (m, 10H), 3.55~3.76 (m, 39H), 3.03 (t, 2H), 1.80~1.99 (m, 4H). 13C NMR (125 MHz, CDCl3, 25 °C) δ (ppm): 150.77, 128.30, 114.17, 68.28, 55.82, 51.01, 29.72, 28.72, 21.38. HRESIMS is shown in Fig. S7: m/z 871.3373 [M−H] (100%).
### Synthesis of TPE-(CN)4
Tetra-hydroxyl-TPE (120 mg) and K2CO3 (144 mg) were added into 20 mL CH3CN. The mixture was stirred at room temperature for 30 min. Then 5-bromovaleronitrile was added into the above solution, reacted under reflux for 24 h. After the solvent was removed, the obtain solid was purified by column chromatograph with petroleum ether/dichloromethane/ethyl acetate (15:15:1 v/v) to get the final product. Pale yellow powder, 75 mg, 34%. 1H NMR (300 MHz, CDCl3, 25 °C), δ (ppm): 6.90 (d, J = 9Hz, 8H), 6.60 (d, J = 9 Hz, 8H), 3.93 (m, 8H), 2.44 (m, J = 6 Hz, 8H), 1.90 $${({\rm{m}},{\rm{16H}})}_{.}$$ 13C NMR (126 MHz, CDCl3, 25 °C) δ (ppm): 157.04, 137.11, 132.66, 119.62, 113.62, 66.56, 28.31, 22.59, 17.13. MALDI-TOF MS is shown in Fig. S17: m/z 720.2358 [M] (100%).
### Synthesis of monocarboxylatepillar[5]arene sodium salt
Monoesterpillar[5]arene (50 mg, 0.06 mmol) was added into sodium hydroxide aqueous solution (2.5 mL, 20%) and THF (4.2 mL). The mixture was stirred at 85 °C for 24 h. After solvent was removed, the obtained solid was recrystallized with dichloromethane/n-hexane. White powder: 40 mg, 82%. 1H NMR (300 MHz, CDCl3, 25 °C), δ (ppm): 6.67~6.74 (m, 10H), 4.18 (t, 2H), 3.59~3.76 (m, 37H). 13C NMR (126 MHz, CDCl3, 25 °C) δ (ppm): 175.86, 150.88, 128.45, 114.13, 71.74, 55.95, 29.81. HRESIMS is shown in Fig. S21: m/z 871.3468 [M−Na] (100%).
### Synthesis of monophosphite-DMP[5]
Copillar[5]arene 1 (0.3 g, 0.36 mmol) was added into triethyl phosphite (1.2 mL). The mixture was stirred at 150 °C for 24 h. The crude product was concentrated and subjected to column chromatograph with petroleum ether/ethyl acetate (1:20 v/v) to get the final product. Pale yellow powder, 260 mg, 78%. 1H NMR (300 MHz, CDCl3, 25 °C), δ (ppm): 6.72~6.77 (m, 10H), 4.09 (m, 4H), 3.85 (m, 3H), 3.77 (m, 10H), 3.64 (m, 27H), 1.85~1.89 (m, 4H), 1.70 (m, 2H), 1.29 (t, 6H). 13C NMR (126 MHz, CDCl3, 25 °C) δ (ppm): 150.75, 149.87, 128.32, 114.95, 114.00, 67.74, 61.54, 55.87, 30.68, 29.82, 19.57, 16.45. MALDI-TOF MS is shown in Fig. S26: m/z 930.4110 [M + H]+, 953.2957 [M + H + Na]2+, 969.3292 [M + H + K]2+.
### Synthesis of Monophosphoricpillar[5]arene
Monophosphite-DMP[5] (0.28 g, 0.3 mmol, see supporting information for details) was added into 10 mL CH2Cl2. Then TMSBr (0.4 mL) was added into the above solution. The mixture was stirred at room temperature for 24 h. The crude product was concentrated and recrystallized to give the final product. White powder: 204 mg, 78%. 1H NMR (300 MHz, CDCl3, 25 °C), δ (ppm): 6.75 (m, 10H), 3.62~3.76 (m, 39H), 1.67 (m, 4H), 1.02 (t, 2H). 13C NMR (126 MHz, CDCl3, 25 °C) δ (ppm): 150.74, 128.26, 113.97, 67.80, 55.78, 52.55, 30.61, 29.60, 19.30. HRESIMS is shown in Fig. S29: m/z 871.3453 [M−H] (100%).
### Synthesis of Monophosphatepillar[5]arene sodium salt
Monophosphoricpillar[5]arene (50 mg) was dissolved in 5 mL THF, then added into 5 mL NaOH aqueous (2 M/L) dropwise, which was allowed to react at room temperature for 12 h. Then THF solvent was evaporated and the solid was collected as a pale yellow powder: 30 mg, 59%. 1H NMR (300 MHz, CDCl3, 25 °C), δ (ppm): 6.69 (m, 10H), 3.53~3.75 (m, 39H), 1.74 (m, 4H), 1.27 (m, 2H). 13C NMR (126 MHz, CDCl3, 25 °C) δ (ppm): 150.78, 128.20, 114.21, 67.97, 61.54, 55.76, 29.71, 25.60, 14.11. HRESIMS is shown in Fig. S32: m/z 871.3604 [M−H] (100%). | 2023-03-23 05:30:54 | {"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": 1, "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": 0.4109039008617401, "perplexity": 9359.229465809227}, "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-2023-14/segments/1679296944996.49/warc/CC-MAIN-20230323034459-20230323064459-00009.warc.gz"} |
https://www.physicsforums.com/threads/need-help-with-difficult-integration-by-parts-problem.191158/ | # Need help with difficult integration by parts problem
1. Oct 14, 2007
### kkidd002
1. The problem statement, all variables and given/known data
integrate: (x^3) e^(3x^2) dx
2. Relevant equations
uv- integral vdu
3. The attempt at a solution
i've tried this many times on paper and can't get the right answer. I'm starting to get really frustrated. Please help, and be specific as possible.
2. Oct 14, 2007
### bob1182006
Since you haven't shown any work I'll assume the worst :s.
First you should make a substitution if possible, to get just e^t.
And when you have an integral of the form:
$$\int x^n T(x) dx$$
where T(x) is a Transcendental function (e^x, sin x, cos x, a^[bx+c], ln x, etc..). And it's antiderivative should be easier than the entire function, otherwise you'll go from a hard integral to a harder one.
to solve that type of integral you need to make the substitutions:
$$u=x^n$$
$$v'=T(x)$$
and you will need to integrate n times, continue with u=x^n and eventually you will get rid of the x term and have only T(x) which would be a trivial integral. ex. $$...-\int cos x dx$$
Also when integrating and say you have $$-3\int x^n cos 3x dx$$ don't just choose u=x^n; v'=cos 3x, place that coefficient on either u or v' to maybe cancel it out. A better choice would be u=x^n; v'=-3cos 3x, which gives u'=nx^(n-1); v=-sin3x
This way you don't have to worry about multiplying everything by that - sign, you might forget it and get the entire thing wrong :/
3. Oct 14, 2007
### rock.freak667
My advice is to rewrite the integral as this
$$\frac{1}{6}\int x^2(6xe^{3x^2}) dx$$
and then use integration by parts
$$\frac{d}{dx}(e^{3x^2})=6xe^{3x^2}$$
4. Oct 14, 2007
### transgalactic
this is my solution
i added a file with my solution
#### Attached Files:
• ###### 4.JPG
File size:
11.7 KB
Views:
150
5. Oct 14, 2007
### bob1182006
could you just write it out? usually it takes a while before attachments are approved.
6. Oct 14, 2007
### transgalactic
i dont know how to write integral signs like you do
7. Oct 14, 2007
### bob1182006
click on the integral and you'll see the code it should be like this:
[ tex] \int ... x^2 [ /tex]
without the spaces before the tex and /tex
8. Oct 14, 2007
### transgalactic
wow its a sintax languege like programming
i'll try to get used to it
it always instantly appears on my pc that my attachment massage
9. Oct 14, 2007
### bob1182006
it has to be approved by a moderator before we can see it I think.
and here's a thread that has guides on how to use the LaTeX typesetting that's really useful here.
10. Oct 14, 2007
### transgalactic
ok thanks
11. Oct 15, 2007
### Gib Z
Best solution here =]
12. Jul 29, 2011
### ebonhawkabc
even easier is to seperate x^3 into x * x^2 and then set u = x^2. then use integration by parts
13. Jul 29, 2011
### Ray Vickson
You can do it in ASCII: you want int x^3*exp(3*x^2) dx. This looks nicer in 'tex', but is perfectly legible without it. If you wanted the definite integral from x=a to x=b you could write int(f(x) dx, x=a..b) or int_{x=a..b} f(x) dx.
RGV
14. Jul 29, 2011
### SammyS
Staff Emeritus
Hey ! This thread is nearly 4 years old! LOL.
15. Jul 29, 2011
### rock.freak667
For the record, I still like my hint :rofl: | 2016-12-02 18:33:08 | {"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": 1, "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": 0.7956552505493164, "perplexity": 3302.301228544264}, "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-2016-50/segments/1480698540409.8/warc/CC-MAIN-20161202170900-00156-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/dR-flat+modality | # nLab dR-flat modality
the (delooping of) the kernel of the counit of the flat modality
see at cohesive (infinity,1)-topos – structures the section on de Rham cohomology | 2022-01-28 22:40:58 | {"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": 0.8163788914680481, "perplexity": 6573.61386276975}, "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-2022-05/segments/1642320306346.64/warc/CC-MAIN-20220128212503-20220129002503-00717.warc.gz"} |
http://mathinsight.org/harvest_natural_populations_exercises | # Math Insight
### Harvest of natural populations exercises
In the page on harvesting natural populations, we introduced the logstic equation with a harvesting rate of the form \begin{align} P_{t+1}-P_{t} = r P_t\left(1-\frac{P_t}{M}\right) -h_t P_t, \label{harvest_fraction} \end{align} and analyzed a specific example with $r=0.2$. In these exercises, you will analyze the general from with constant $h_t=h$ and $0 < r < 2$.
#### Exercise 1
Divide each term of equation \eqref{harvest_fraction} by $M$ to obtain \begin{align} p_{t+1} - p_t = r p_t \left(1 - p_t \right) - h p_t \label{harvest_fraction_normalized} \end{align} where $p_t = P_t/M$.
#### Exercise 2
1. Show that the equilibria of equation \eqref{harvest_fraction_normalized} are $p_{e} = 0$ and $p_{e} = 1 - h/r$.
2. Show that in order for there to be a positive equilibrium, the fractional harvest rate, $h$, must be less than the low density growth rate, $r$.
#### Exercise 3
1. Convert equation \eqref{harvest_fraction_normalized} (which is in difference form) to function iteration form $p_{t+1}=F(P_t)$. Show that \begin{align} p_{t+1} = F(p_t) = p_t + r p_t (1 - p_t) - h p_t. \label{harvest_fraction_iteration} \end{align}
2. Show that the equilibria of iteration \eqref{harvest_fraction_iteration} are $p_{e} = 0$ and $p_{e} = 1 - h/r$.
3. Assume that $h < r < 2$. Compute $F'(p)$ and evaluate $F'(0)$ and $F'(1-h/r)$. Conclude that $0$ is an unstable equilibrium and $1-h/r$ is a stable equilibrium.
#### Exercise 4
Assume that $h < r < 2$ so that $1-h/r$ is a positive, stable equilibrium of the iteration of equation \eqref{harvest_fraction_iteration}.
The harvest at that equilibrium will be $h (1 - h/r)$.
1. Find the value of $h$ for which $h (1-h/r)$ is the largest. Conclude that in order to maximize harvest, the harvest fraction $h$ should be set at one-half the low density growth rate, $r$.
2. Find the positive equilibrium $p_e$ for this value of $h$.
3. The variable $p_t$ was the actual population size $P_t$ normalized by the carrying capacity $p_t=P_t/M$. What is the equilibrium value of the population size $P_t$ when the harvest rate is set at the value that maximizes harvest? Your answer will be in terms of $M$.
Once you've worked out some of these exercises, you can check your work with the answers to selected problems. | 2017-10-16 23:43:53 | {"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": 1, "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": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9646053314208984, "perplexity": 743.0017217836065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2017-43/segments/1508187820487.5/warc/CC-MAIN-20171016233304-20171017013304-00428.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=97590 | # i'm bored so..
by whozum
Tags: bored
P: 2,223 Ask me absolutely any question about anything you like (me?) you would like to know the answer to and I'll tell you the best of my abilities!
PF Gold P: 7,125 What's the meaning of turtle necks?
P: 2,223 I would have to say the neck of the clothing garment that covers the neck with a thick and folded-looking extension of the material. I don't really get your question either
PF Gold
P: 7,125
## i'm bored so..
What is it for. Why do people wear it. What does it tell us about the space-time continuum
P: 2,223 Clearly it shows that the discontinuities in the spaceitme continuity can be modeled by the frequency of wearing a turtle neck sweater during the winter, however these models can not be taken seriously until this research is completed and taken with regards to other analogous research. I believe that the urge to wear such garments is fueled by a metaphysical urge by the subconscious' connection to the universe's fundamental governing system, and the subconscious' overwhelming ability to overcome conscious desires to avoid such a silly design.
PF Gold P: 7,125 So was Einstein right to say the Energy given off by a turtleneck wearer was... $$\Delta E = \sqrt[3]{{f_t h^2 }}$$ When the turtleneck wearer is at ground state of course.
P: 2,223 It depends on the defining of the term Planck's constant, because its been debated recently that there was a typo in his original paper where he neglected to use h-bar, which is an alternate constant used often with regards to this form of work. If you meant h bar then that is the correct interpretation, but the most interesting effects of this phenomenon are not observed in energy analysis but rather, as you said, in spacetime shifts within the users local gravitational field.
P: 2,223 Come on guys any real questions??? :surprise
P: 30
Quote by whozum Come on guys any real questions??? :surprise
i am not a scientist...i see the *moment to *moment impermanence.
There is apparent permanence also.
How do YOU explain THAT?
.
P: 2,223 I have no freaking clue what you just said
PF Gold P: 971 What's the best thing to do when you're bored to death?
P: 30
Quote by Lisa! What's the best thing to do when you're bored to death?
You wouldn't believe it.
P: 30
Quote by whozum I have no freaking clue what you just said
i thought You
were a scientist.
??????
P: 2,057
Quote by Lisa! What's the best thing to do when you're bored to death?
Suicide.
whozum, what is x if x+y=(pi)98^2? You figure that out while I take a nap.
P: 86 Why is it that the morbidly obese heifer insists on parking next to me, opening her car door into mine, and resting it there while she hoists her pale carcas up and out of her vehicle while her car door scratches ANOTHER wonderful Picaso into the side of mine? There are no other cars in the damn garage so she can easily park one up to allow her blubber ample room to exit. I have no recourse now but to leave work at lunch and open my passenger door several times to remove "something". I wish I had a Jetson car.
P: 676
Quote by Mk Suicide. whozum, what is x if x+y=(pi)98^2? You figure that out while I take a nap.
is that just a function of y? then x= -y + pi(98^2) which is just a line with a negative slope and an X intercept of pi(98^2). pretty uninteresting actually.
P: 30
Quote by Lisa! What's the best thing to do when you're bored to death?
Boredom is not a feature of silence.
Mentor
P: 25,945
Quote by Echo 6 Sierra Why is it that the morbidly obese heifer insists on parking next to me, opening her car door into mine, and resting it there while she hoists her pale carcas up and out of her vehicle while her car door scratches ANOTHER wonderful Picaso into the side of mine? There are no other cars in the damn garage so she can easily park one up to allow her blubber ample room to exit. I have no recourse now but to leave work at lunch and open my passenger door several times to remove "something". I wish I had a Jetson car.
morbidly obese heifer
Related Discussions General Discussion 25 Calculus & Beyond Homework 0 General Discussion 5 General Discussion 2 | 2014-04-21 07:14:27 | {"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": 1, "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": 0.2404337376356125, "perplexity": 2776.221742321333}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00362-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/377411/why-is-the-electric-field-created-by-a-battery-non-conservative | # Why is the electric field created by a battery non-conservative?
Electromotive force(emf) or $\mathcal{E}$ is defined as $$\mathcal{E} = \oint \frac{\vec{F}}{q} \cdot \mathrm{d}\vec{s}$$ Here, $\vec{F}$ is the force which pushes the charges through a conducting wire loop, $q$ is the magnitude of charge and $\vec{s}$ is the displacement of the charge. $\mathcal{E}$ is the tangential force per unit charge in the wire integrated over length, once around a complete circuit.
Now in my Physics book, it is written that an emf must be nonelectrostatic in origin. Only then, it can force the charges to move in a loop. Conservative electric fields cannot make the charges move in loops. Now, if we connect a battery and a resistor with wires in a loop, a current is established. This means that a battery has some nonelectrostatic forces which can separate the positive and negative charges or otherwise $\mathcal{E}$ will become $0$ when integrated over the whole circuit.
What is is this nonelectrostatic force which is responsible for driving the electrons? If its chemical, due to the electronegativity differences, how is it nonelectrostatic? Electronegativity occurs due the electrostatic attraction of the electrons and the nucleus plus some shielding, doesn't it?
• The chemical potential of ions depends on the electrical potential and on concentration.
– user137289
Jan 1 '18 at 13:56
• @Pieter But doesn't the usage of the term chemical "potential" imply conservative fields? Jan 3 '18 at 7:50
• The chemical potential is Gibbs free energy per atom. The concept comes from thermodynamics, it is connected with the power to do work. But no link with force fields.
– user137289
Jan 3 '18 at 9:25
• @Pieter Then what is is nonelectrostatic force which does work? Jan 3 '18 at 9:26
• The concentration gradient, random walks causing diffusion and increase in entropy. I am not an electrochemist, but things are not conserved, atoms move, electrodes get consumed.
– user137289
Jan 3 '18 at 10:16
What is is this nonelectrostatic force which is responsible for driving the electrons? If its chemical, due to the electronegativity differences, how is it nonelectrostatic? Electronegativity occurs due the electrostatic attraction of the electrons and the nucleus plus some shielding, doesn't it?
This is actually a nice example of the fact that although chemistry is based on electrical forces, chemistry cannot be explained using classical electrostatics.
I think an example that's easier to understand but illustrates the same principle is a metal surface exposed to a vacuum. Work is required in order to remove an electron from the metal. This amount of work $W$ is known as the work function of the metal. Microscopically, various complicated phenomena need to be taken into account in order to calculate an accurate work function for a real metal.
However, a good example of such a phenomenon is the following. The electrons in the metal have a density that stretches farther out into the vacuum than the density of the protons. This is a quantum-mechanical effect arising from the Heisenberg uncertainty principle and the small mass of the electrons. It's analogous to the fact that in a single hydrogen atom, the electron's probability cloud gets much farther from the center of mass than the proton's does. At the surface of a metal, you can estimate the thickness of this skin using models such as the jellium model, and the expression you get has Planck's constant in it.
Because of the existence of this electron skin, the surface of the metal acts like a dipole layer. There is a difference in potential between the vacuum and the metal, which is the work function, and in this simplified model it's caused by the dipole layer.
The existence of this dipole layer cannot be explained by classical electrostatics. In fact, there is a theorem that says that an electrostatic charge distribution can never exhibit a nontrivial stable equilibrium.
A similar example is the existence of molecules with dipole moments. We make cartoons of these with charges stuck at the ends of a stick. But in reality there is no "stick force," only electrical forces. Classically, the electrical interactions cannot stably sustain the separation of the opposite charges.
As a simplified model of a battery, you can make parallel plates out of two dissimilar metals such as copper and zinc, with a vacuum between them. (This is similar to a voltaic cell, and in a voltaic cell the emf would be approximately equal to the difference between the work functions.) Let's say we just short them by connecting them with a wire, and we also keep them in thermal equilibrium. Because the work functions are unequal, there is an electric field in the vacuum. This electric field is created by a flow of electrons out of the metal with the smaller work function and into the metal with the larger work function. These electrons have flowed against the electric force. That would not be possible classically, just as the existence of molecular dipoles would not be possible classically.
What can we say about $\int \frac{\vec{F}}{q} \cdot \mathrm{d}\vec{s}$ inside the battery?
In practical laboratory terms, the force $F$ in this definition is usually described as some sort of sum consisting of terms that include the $qE$ due to the electric field, an effective chemical force, and an effective thermal force. So the short answer is that the $F$ inside the battery contains a term from an effective chemical force, and this force is not the same as the electrical force. In fact, it's in the opposite direction.
At a microscopic level, the origin of things like effective chemical forces are quantum mechanical degeneracy pressure. As a model of this, consider a particle in a box, with the walls of the potential well being finite on one side:
$$V(x) = \begin{cases} W, & x< 0 \\ 0, & 0<x<L,\\ \infty, & x> L, \end{cases}$$
This is a simplified model of the potential experienced by an electron in a metal that is open to vacuum on the left. (The infinite potential barrier on the right is just to make the toy model easy to work with.) As a trial solution to the time-independent Schrodinger equation for this potential, we could try
$$\Psi(x) = \begin{cases} 0, & x< 0 \\ \sin kx, & 0<x<L,\\ 0, & x> L, \end{cases}$$
where $k=L/\pi$. However, this is not a solution to the time-independent Schrodinger equation. If we initially put an electron in this state, it will evolve over time into the true ground state, which includes an exponential tail tunneling into the classically forbidden region $x<0$. This time evolution involves a net motion of the center of mass to the left. The force that causes the electron to do that is the degeneracy pressure, $d(h^2/8mL^2)/dx=-h^2/4mL^3$. The fact that there's an $h$ in there tells you that this is a quantum-mechanical effect. If you use more realistic models of degeneracy pressure instead of this toy model, you still get expressions that have $h^2/m$ in them. This electron degeneracy pressure has caused an electron skin to form, and the process of forming that skin increases the electrostatic potential energy. That is, we have a force that counteracts the electrical force until a new equilibrium is reached.
• Yes. But don't we use this definition of emf in Maxwell's equations? And aren't they universally valid, compatible with both special relativity and quantum mechanics? So, what can we say about $\int \frac{\vec{F}}{q} \cdot \mathrm{d}\vec{s}$ inside the battery? Jan 1 '18 at 17:08
• aren't [Maxwell's equations] universally valid, compatible with both special relativity and quantum mechanics? No, they're not. Maxwell's equations are classical. For example, you're not going to be able to explain the photoelectric effect using Maxwell's equations.
– user4552
Jan 1 '18 at 18:21
• Re the definition of emf and the contribution to the emf inside the battery, those are good questions. I've edited the question to try to address those more explicitly.
– user4552
Jan 1 '18 at 20:07
• I've not really studied quantum mechanics as I'm in high-school. What is the origin of this quantum degeneracy pressure? Pauli Exclusion Principle? If yes, is it electromagnetic field in origin? Jan 2 '18 at 13:57
• @ApoorvPotnis: The answer gives an argument for why degeneracy pressure exists. Note that I never invoked the exclusion principle of the EM field.
– user4552
Jan 2 '18 at 15:09 | 2021-09-24 20:32:44 | {"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": 1, "mathjax_display_tex": 1, "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": 0.6859247088432312, "perplexity": 229.1678477048159}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2021-39/segments/1631780057580.39/warc/CC-MAIN-20210924201616-20210924231616-00221.warc.gz"} |
https://physics.stackexchange.com/questions/350581/why-are-angular-velocities-of-double-pendulum-small-in-small-angle-approximation | # Why are angular velocities of double pendulum small in small angle approximation? [duplicate]
In the lagrangian for double pendulum for small angles, the term $\dot{\theta}_1\dot{\theta}_2 \left [ 1-\frac{(\theta_1-\theta_2)^2}{2} \right ]$ is replaced with $\dot{\theta}_1\dot{\theta}_2$, because $\dot{\theta}_1\dot{\theta}_2 \frac{(\theta_1-\theta_2)^2}{2}$ is neglected. The product $\dot{\theta}_1\dot{\theta}_2$ has the second order of smallness, but why? This comment explains for simple pendulum, but says that it is more complicated for double pendulum. What is explanation for double pendulum?
Edit: Correct me if I am wrong, but I think I found the answer. If double pendulum starts oscillating from rest, the potential energy at that moment is $E_{pm}=m_1gh_{1i}+m_2gh_{2i}$, where $h_{1i}$ and $h_{2i}$ are lengths from reference line to centres of masses of two pendulums. In small angles aproximation these heights are $h_{1i}=l_1+l_2-l_{cm1}+\frac{\theta_{1i}^2}{2}l_{cm1}$ and $h_{2i}=l_2-l_{cm2}+l_1\frac{\theta_{1i}^2}{2}+l_{cm2}\frac{\theta_{2i}^2}{2}$. Kinetic energy of first pendulum is $E_{k1}=E_{pm}-E_{k2}-E_{p1}-E_{p2}=\frac{m_1v_1^2}{2}+\frac{I_1\dot{\theta_1}^2}{2}=\frac{m_1l_{cm1}^2+I_1}{2}\dot{\theta_1}^2$. Kinetic energy of second is $E_{k2}=\frac{m_2v_2^2}{2}+\frac{I_2\dot{\theta_2}^2}{2}=\frac{m_2l_{cm2}^2+I_2}{2}\dot{\theta_2}^2$. $E_{k1}=\frac{g}{2}((\theta_{1i}^2-\theta_{1}^2)(m_1l_{cm1}+m_2l_1)+m_2l_{cm2}(\theta_{2i}^2-\theta_{2}^2))-E_{k2}$ $\dot{\theta_1}=\sqrt{\frac{g((\theta_{1i}^2-\theta_{1}^2)(m_1l_{cm1}+m_2l_1)+m_2l_{cm2}(\theta_{2i}^2-\theta_{2}^2))-2E_{k2}}{m_1l_{cm1}^2+I_1}}=\sqrt{\frac{2E_{k1}}{m_1l_{cm1}^2+I_1}}$
$\dot{\theta_2}=\sqrt{\frac{g((\theta_{1i}^2-\theta_{1}^2)(m_1l_{cm1}+m_2l_1)+m_2l_{cm2}(\theta_{2i}^2-\theta_{2}^2))-2E_{k1}}{m_2l_{cm2}^2+I_2}}=\sqrt{\frac{2E_{k2}}{m_2l_{cm2}^2+I_2}}$
Masses and lengths have influence, but $\dot{\theta_1}$ and $\dot{\theta_1}$ should be small because terms $(\theta_{1i}^2-\theta_{1}^2)$ and $(\theta_{2i}^2-\theta_{2}^2)$ are small. Also when $\dot{\theta_1}$ is maximal, $E_{k1}$ is maximal, so $\dot{\theta_2}$ will be smaller. Similar is with $\dot{\theta_1}$. $\dot{\theta_1}$ and $\dot{\theta_1}$ will not be maximal in same time, so their product will be small.
## marked as duplicate by Kyle Kanos, ZeroTheHero, Jon Custer, user259412, heatherAug 6 '17 at 18:30
• Is it ignored because $\dot\theta_1\dot\theta_2\ll1$ or $\theta_1\approx\theta_2$? – Kyle Kanos Aug 6 '17 at 12:10
• Possible duplicate of Small oscillations of the double pendulum – Kyle Kanos Aug 6 '17 at 12:12
• Because $\dot\theta_1\dot\theta_2\ll1$. – LEM Aug 6 '17 at 12:13
I think that $\dot{\theta}$'s are not neccesarily small, but actually what is small is the term $\frac{(\theta_1-\theta_2)^2}{2}$.
So I guess that the brackets are $\simeq [1+\ \sim 0 ]\approx 1$, and consequently you only have the double $\omega$ product.
The simplest way to keep track of "smallness" is to introduce a dummy parameter $\epsilon$ and replace $\theta\to \epsilon\theta$ everywhere. It then follows that $$\dot\theta_i\to \epsilon\dot\theta_i\, .$$ Using this substitution in the Lagrangian, and expanding to quadratic order in $\epsilon$ will produce the correct linearlized equations of motion. (The assumption is that the equilibrium position is at $\theta_i=0$.)
With this, for instance, the term $\dot\theta_1\dot\theta_2\left(1-\frac{1}{2}(\theta_1-\theta_2)^2\right)$ in the Lagrangian becomes \begin{align} \dot\theta_1\dot\theta_2\left(1-\frac{1}{2}(\theta_1-\theta_2)^2\right) &\to \epsilon^2\dot\theta_1\dot\theta_2\left(1-\frac{1}{2}\epsilon^2(\theta_1-\theta_2)^2\right)\, ,\\ &= \epsilon^2\dot\theta_1\dot\theta_2 \end{align} | 2019-10-16 16:42:36 | {"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": 1, "mathjax_display_tex": 1, "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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9676116704940796, "perplexity": 306.28317868214924}, "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-2019-43/segments/1570986669057.0/warc/CC-MAIN-20191016163146-20191016190646-00302.warc.gz"} |
https://www.scienceforums.net/topic/115541-where-to-submit-my-proof-that-the-set-of-real-numbers-cant-be-well-ordered/page/12/ | # Where to submit my proof that the set of real numbers can't be well ordered
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Whatever it is you're trying to claim, you have to prove it. You can't handwave it with "this should be allowed in our meta math system."
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f by "might", you mean it happens in some orders, then yes, I agree. But you seem to be claiming this must be true for all orders, and that's something you'll have to prove (in fact, it's false), so no, I do not agree.
Don't think I have a problem with this.
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So what is your current proof, then?
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On 8/22/2019 at 4:01 PM, uncool said:
On 8/22/2019 at 4:01 PM, uncool said:
"As we might always have an x such that x < z for any z we should have an x <* z for any z" That is if this z is supposed to be a min for a set.
I hope you understand what I mean by this remark. I believe I was saying we should be able to apply the same general concept we use for '<" to the use of '<*'. On 2nd thought I might have to prove denseness in the reals for <*.
And, what do u mean by z1 and z2? Is z1 <* z2? Once you identify these numbers how do find the min of the set with these deleted?
You keep saying I have proved nothing. Just what is wrong with my latest attempts? You never have said exactly what is wrong. It sounds like you already have a notion of what a proof-if possible-should look like.
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Please try to correct your quoting; you have put your response as if I said it. I know, the quoting software for this forum is awful, but it's hard to tell what you are responding to.
19 minutes ago, discountbrains said:
On 2nd thought I might have to prove denseness in the reals for <*.
If you want to claim it, then yes, you will have to prove it. And if by "denseness in the reals", you mean "between any two reals (as defined by any order <*), there is another real", then you will be trying to prove a false statement.
19 minutes ago, discountbrains said:
Just what is wrong with my latest attempts?
I have explained what I thought is wrong rather explicitly each time. If you want me to do so again, then post a proof - in full, no handwaving - and I will see.
Edited by uncool
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On 8/22/2019 at 6:58 AM, discountbrains said:
I'm going to make a new attempt:
> I'm going to make a new attempt: Like I said before there always exists a subset of R like {x: a <*x} for some a for any total, linear order, <*, which has no min for <*.
That's a restatement of the claim that there's no well-order on the reals. A set is well-ordered if every nonempty subset has a smallest element. So if you say that for any linear order there's some nonempty set that has no smalleset element, then there could be no well-order. This is a simple restatement of your claim.
> I think you and uncool want absolute evidence there is such a set.
Yes, because it's your original claim! You have made a claim -- which is false, by the way -- and you must supply a proof.
> My proof is as follows: So, now lets assume for any <* there are at least a few subsets of the reals that contain a minimum element. Otherwise, I am done: there are NO subsets with a min. Suppose we have a subset that has a min, z, and its S={x: z ≤*x}. Lets consider S\{z}. This set is {x: z <*x} and is the same form as the set on the first line and has no min with respect to <*.
WHY NOT? Consider the usual order on the natural numbers, which is a well-order. I have formed the set $S = \{x : 14 \leq x\}$. No problem there. Now suppose I delete 14 from that set? What is the least element of the new set? It's 15. And if you delete 15, the new smallest element is 16. And this goes on forever, because the usual order on the naturals is a well-order.
Now you claim this can't happen with the reals, but why not? Suppose $<^*$ happens to be a well-order. Then the exact same thing will happen as it does with the naturals. You take some element, form the set of everything greater than or equal to that element, delete that element, and the new set still has a smallest element; namely, the next element in the well-order.
But it doesn't even have to be a well-order. You might have picked some element, like 14, that JUST HAPPENS to have an immediate successor. Then your proof fails.
A well-order of the reals looks JUST LIKE THE NATURAL NUMBERS. It's one element followed directly by another then by another then by another. Once in a while you have to take a "limit ordinal," as in the process that produces $\omega$ following all the natural numbers. You just keep on going taking immediate successors and occasionally limits, and you can work your way up to an uncountable ordinal.
Once you have this picture in your mind your proof fails. You can take a tail of the order, ie the set of elements greater than or equal to some particular element; then you can delete the element; and the NEXT element in the order is now the smallest element of the deleted set. You haven't proved this can't happen and in fact you don't seem to understand this point at all.
49 minutes ago, discountbrains said:
I hope you understand what I mean by this remark. I believe I was saying we should be able to apply the same general concept we use for '<" to the use of '<*'. On 2nd thought I might have to prove denseness in the reals for <*.
Your remark just now that <* is dense shows that you don't understand this point. When you well-order the reals, you lose the denseness property. You have to forget about it. A well-order on the reals is one element followed by another. If you delete the smallest element of a tail, the new smallest element is now the next element in the order.
But you don't even need for <* to be a well-order. If you pick some tail S = {x : z <= x} for some real number z, and you delete z, the new set DOES HAVE A SMALLEST ELEMENT just in case z happens to have a direct successor. You haven't proved that can't happen. And in fact if <* does happen to be a well order, it MUST happen every time. Every element has a direct successor and every "deleted tail" still has a smallest element.
Edited by wtf
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I'm sorry I'm wasting your time by my dumbness. I have always been thinking of a subset of for example S=[0,1] (with the usual order, <) which would be an interval or segment from [0,1]. Surely, if one considers {1/2, 1/3, 1/4,...} and they delete 1/2 there is a next number etc so it too a well ordered subset of S. This type of set I'm not thinking of Should I say "start with a set for which there is not already a known well order for?" If we use (0,1) then clearly < doesn't well order it. What order relation will? Here's the way I visualize it: What if you have a set you have no WO for and you pick some number, z, out of it and claim z is a <*min for it then you delete z. You then are left with all numbers from S greater than z with respect to <*. What is your <* min for this set.
I am well aware of what wtf says. This is what motivated my whole notion, there is no WO for all sets.
Here's my order which should better be put in some sort of an array: Lets use numbers in [0,1] we have 0, 1, 2, 3,..., 1/2, 1/3, ..., 2/3, 3/4,...,whatever,.... irrational numbers,..., then you get to 0.0023...,...0.000042... You just get to numbers you can't find the least of or they become incomparable. You finally end up with sets of numbers that you either can't find the smallest or you don't know which is bigger. There are several unusual ways of ordering numbers, but they always end up like this. As the sets of these groups of numbers go along it gets so you can't find a min.
Think about it this way: You have your set, S, above with all its original numbers in it. You have your <*. You declare z the min of S. You delete z and are left with all numbers x from S, z <*x. For your order how do you find the min of these numbers?
Correction last line first paragraph: Let this 'set' be S than the statement is correct.
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14 minutes ago, discountbrains said:
Think about it this way: You have your set, S, above with all its original numbers in it. You have your <*. You declare z the min of S. You delete z and are left with all numbers x from S, z <*x. For your order how do you find the min of these numbers?
I already explained this. It would go better if you'd read my posts.
Run your argument on the natural numbers in their usual order 0, 1, 2, 3, ...
Take the tail 14, 15, 16, 17, ...
Remove 14 to get 15, 16, 17, ...
What's the smallest element? 15.
Remove 15 to get 16, 17, 18, 19, ...
What's the smallest element? 16
This is exactly what would happen if you have a well-order for the reals. You haven't shown this can't happen. You're still confusing the usual order on the reals with an arbitrary linear order.
It doesn't even have to be a well-order. Say your order is 14, 15, everything else. Then you pick the tail S = 14, 15, everything else. You delete 14. Now you have 15, everything else. The smallest element is 15.
But what if it was 14, 15, 16, everything else. Same thing. No matter how many times you remove an element, there's a linear order that breaks your proof.
Edited by wtf
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Are u _____ or what? READ MY POST!! I DID read yours. I was trying to convey I was not including your set. You keep repeatedly bringing up an example I'm not even talking about. I wrote I'm talking about subsets (or sets if u prefer) we don't already know a WO for which I was hoping was precise enough. Maybe this is the source of confusion for both of u.
Edited by discountbrains
midding text
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12 minutes ago, discountbrains said:
Are u _____ or what? READ MY POST!! I DID read yours. I was trying to convey I was not including your set. You keep repeatedly bringing up an example I'm not even talking about. I wrote I'm talking about subsets (or sets if u prefer) we don't already know a WO for which I was hoping was precise enough. Maybe this is the source of confusion for both of u.
Edited by wtf
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Those are not counterexamples to my sets. I'll reread what u said; maybe I missed something. It looks like the same obvious old thing over and over again.
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1 minute ago, discountbrains said:
Those are not counterexamples to my sets. I'll reread what u said; maybe I missed something. It looks like the same obvious old thing over and over again.
I agree with that. You keep making the same logic error, I keep correcting it.
But what if we run the argument on the natural numbers in their usual order? Doesn't every deleted tail still have a smallest element? And where's your proof that this can't happen with an arbitrary linear order on the reals?
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Is this your counterexample? Let me say the set for consideration is the usual (0,1). My <* applies to this set and for more numbers outside this set if someone want's to use them in an argument. You can't be bringing up your special set of 1, 2, 3,... I really, really, really don't think u understand this subject. My <* applies to each and every number in (0,1). If we delete a number from (0,1) we still have a very large uncountable number of elements. Maybe uncool can explain your error to you.... I keep telling u the only thing of concern is finding ONE set u can't find a min of, geez! ....No one answered my opening question. I'll look up various real math societies or maybe see if I can get a real prof to answer me. These groups have got to be some sort of scam.
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LOLOL All the best.
But since you asked I'll explain. The 1, 2, 3, ... example is ... an example. An illustration of how to conceptualize well-ordered sets. You remove the first element and there's another next element. You remove that one and there's another.
It's perfectly true that we can't visualize an uncountable well-ordered set. But its existence can be proved even without the axiom of choice. You have an intuition that there are "too many" reals to well-order, but you haven't got a proof.
Edited by wtf
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The wiki site is saying the ordinals are well ordered in respect to set inclusion. "ω1 is a well-ordered set, with set membership ("∈") serving as the order relation. ω1 is a limit ordinal, i.e. there is no ordinal α with α + 1 = ω1." Don't know quite what this means. I guess α + 1 is still a countable number or set and thus can't equal ω1. That these topologies are not metrizable I'm a bit curious about.
If I say A and B and C ¬D and you say D ⇒D you are inserting information which is not there.
You're saying a WO relation is a relation so I must automatically include it in my set of relations on my set S even though I don't know it exists. I can make all kinds of manipulations with order relations without even exhibiting them and the WO fails.
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38 minutes ago, discountbrains said:
You're saying ...
You're addressing me but you've indicated a disinclination to hear any more from me. I prefer not to play that game. For the record I'm done here unless something new and/or interesting gets said.
I'll leave it with this. If -- I'm not saying there is, but if -- there were a well-order on the reals, it would look just like the well-order on the natural numbers: one element after another. The only difference is that you'd have to periodically take limits.
A limit ordinal is an ordinal that doesn't have an immediate predecessor. $\omega$ is a limit ordinal. It's the upward limit of 0, 1, 2, 3, ... Technically it's implemented as the set-theoretic union of all the preceding ordinals. Alternatively, it's the set of all the preceding ordinals. Those two are the same since ordinals are transitive sets.
So if you take all the countable ordinals, and take their upward limit (as their union, or by taking the set of all the countable ordinals) you have an ordinal that can't be one of the countable ordinals, so it's an uncountable ordinal.
I completely agree with you that the idea of an uncountable well-ordered set is mind boggling. But that's not a proof against it. Rather, it's another one of those counterintuitive things in math that we have to just "get used to," as John von Neumann said.
You have an intuition that there's no uncountable well-ordered set, and you think your intuition is a proof. But you haven't got a proof. Only an intuition which turns out to be false.
Edited by wtf
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You are saying I am making statements about ordinals. They never crossed my mind. After some thought I could be wrong trying to prove a negative. Empirically it often is impossible at least: Many public figures concluded there were no wmd's in Iraq because they never found any. Logically its wrong to say this-not taking sides though. Was trying to find an analogy of my type of statement that no WO existed for R. I know of a theorem due to Von Neumann who you mentioned that states "No two variables can be maximized at the same time". So, your equivalent response to him would be "that YOU know of a function that does this". So, let me end by changing my statement to 'the rules on order relations on the reals lead to inconsistent results with the notion there exists a well ordering' for the reals. This would make my statement similar to some others in set theory, symbolic logic etc.
My source of much of my knowledge of set theory is a book-long out of print-called Axiomatic Set Theory by Rubin and Rubin. In the book they discussed what you were saying about ZFC, ZF, and whatever models of set theory. They used a lot of symbolic logic notation and got a little into meta mathematics. I think my statements are a little on the meta math side.
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43 minutes ago, discountbrains said:
So, let me end by changing my statement to 'the rules on order relations on the reals lead to inconsistent results with the notion there exists a well ordering' for the reals.
What, precisely, do you mean by that? Do you mean that there are statements about the usual order relation on the reals that cannot apply to a well-ordering?
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4 hours ago, discountbrains said:
You are saying I am making statements about ordinals. They never crossed my mind.
That's like me saying I'm interested in a scoop of tuna salad between two slices of bread, but that I am NOT interested in a tuna salad sandwich! The two things are synonymous. If you have a well-ordered set, its order type is an ordinal. If you have an ordinal, it's a well-ordered set. It's not possible to be interested in well-orders and not be interested in ordinals. I can't fathom where you're coming from with a statement like this, after I've been explaining it to you for a year
The rest of your post is wildly off the mark. You keep posting an erroneous proof and I keep refuting it with an example. You say that if you delete the first element from the tail of a linear order (of the reals), the rest of the set has no smallest element. But that's false. "0, 1, everything" refutes it. You say ok take the next deleted tail. "0, 1, 2, everything" refutes it. You say ok do it countably many times. Then "1, 2, 3, 4, ... 0, everything" refutes it. That's the ordinal $\omega + 1$. You can keep going to "odds, evens, everything." That's $\omega + \omega$. I can just keep walking through the countable ordinals. You claim a deleted tail has no first element but clearly these examples refute your claim. That's all that's going on here. You have a faulty proof and I keep showing it's faulty.
Edited by wtf
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16 hours ago, uncool said:
What, precisely, do you mean by that? Do you mean that there are statements about the usual order relation on the reals that cannot apply to a well-ordering?
No
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Then I ask again: what, precisely, do you mean?
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12 hours ago, wtf said:
That's ω+ω . I can just keep walking through the countable ordinals. You claim a deleted tail has no first element but clearly these examples refute your claim. That's all that's going on here. You have a faulty proof and I keep showing it's faulty.
Don't know how to answer u. Your position is there is an established well accepted theory of ordinals that contradicts my claims. My argument stands on itself; everything follows logically from each proceeding statement in the argument. It is what it is without your fancy argument. If u didn't already had this bag of tricks back in your 1st few responses to me why didn't u use them then? You keep saying "everything else". How are u going to find a min let alone a least element of that? If this is a well order for the reals than u have ACTUALLY exhibited one.
I will now give yet another argument that's probably non-rigorous nor intuitive:
If you reorder a set S=(0,1) you realize all numbers in S are infinite strings of digits (countable number of digits). Some end in an infinite string of 0s of course. A string might look like 0.045xxxx0....xx. To produce ANY reordering in the most general way so no one can say you skipped any all digits need to be replaced by another not = to it or of course some or all digits could stay the same. Now consider any possible reprdering of these numbers. You would have to rearrange the numbers one by one. Of course we know even if there were a countable number of arrangements we couldn't do this, but theoretically its conceivable. We know, however, there is an uncountable number of ways to do this. We need the Axiom of Choice to do this. We can't even produce all reorderings without the AC. We can make a few reorders like reversing the order of all numbers etc.
10 minutes ago, uncool said:
Then I ask again: what, precisely, do you mean?
I was trying to model my logic after this Goerdel, Continuum Hypothesis stuff. I forget how all that stuff comes out. I mean that if u follow the basic rules of order relations and basic definitions of sets without the concept of WO you must conclude there is a conflict here if you start with the premise every set can be well ordered.
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The conclusion with the continuum hypothesis is that it was independent of the other axioms. That is, that ZFC + CH is consistent, and ZFC + ~CH is consistent. The closest analog here would be that ZFC + AoC is consistent, and ZFC + ~AoC is consistent. Which is true, but doesn't match what you've claimed - your claims seem to say that there is an inconsistency.
So yet again, I have to ask: what, precisely, do you mean?
Edited by uncool
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"So yet again, I have to ask: what, precisely, do you mean?"
An order relation must satisfy 3 properties. Sets and manipulations of sets are understood to take a certain form and meaning. I only used these basic ideas and ended up with my result. That's all I'm reduced by critics to saying at this point. Your question to me makes me ask just precisely is your issue with my train of thought? Point to exactly what I said that u can say is not true and correct me.
Wtf's examples seem vague to me; might actually be more imprecise than he accuses me of. My sense now of ordinals is that its almost obvious they are well ordered.
How did u like my ordering of numbers in (0,1)?
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1 hour ago, discountbrains said:
I only used these basic ideas and ended up with my result.
And what, precisely, is the claimed result?
1 hour ago, discountbrains said:
Your question to me makes me ask just precisely is your issue with my train of thought?
You have had several trains of thought, and I have pointed out specific issues in each of them. My main issues are meta-issues: you are often unwilling to deal with questions with the necessary precision, and you are often too willing to handwave the key parts in your claimed proofs. Both of which are why wtf and I are constantly asking you to write out your precise proof.
1 hour ago, discountbrains said:
My sense now of ordinals is that its almost obvious they are well ordered.
That sense is correct; they are defined to be well-ordered.
1 hour ago, discountbrains said:
How did u like my ordering of numbers in (0,1)?
I don't know which ordering you are referring to; however, if you are still claiming that there cannot be a well-order on the real numbers, then telling us that a specific ordering isn't a well-ordering is not informative.
A request for you: what is the precise statement you think you have proven, and what is the precise proof of that statement?
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This topic is now closed to further replies.
× | 2020-07-14 23:05:18 | {"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": 1, "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": 0.6642337441444397, "perplexity": 649.4402701153863}, "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-2020-29/segments/1593657151761.87/warc/CC-MAIN-20200714212401-20200715002401-00375.warc.gz"} |
https://svn.geocomp.uq.edu.au/escript/trunk/doc/cookbook/escpybas.tex?view=markup&sortby=author&pathrev=3989 | Contents of /trunk/doc/cookbook/escpybas.tex
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Some install doco
1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % Copyright (c) 2003-2012 by University of Queensland 4 5 % 6 % Primary Business: Queensland, Australia 7 % Licensed under the Open Software License version 3.0 8 9 % 10 % Development until 2012 by Earth Systems Science Computational Center (ESSCC) 11 % Development since 2012 by School of Earth Sciences 12 % 13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 14 15 \section{Escript and Python Basics} \label{sec:escpybas} 16 17 The \pyt scripting language is a powerful and easy to learn environment with a wide variety of applications. \esc has been developed as a packaged module for \pyt specifically to solve complex partial differential equations. As a result, all the conventions and programming syntax associated with \pyt are coherent with \esc. If you are unfamiliar with \pyt, there are a large number of simple to advanced guides and tutorials available online. These texts should provide an introduction that is comprehensive enough to use \esc. A handful of \pyt tutorials are listed below. 18 \begin{itemize} 19 \item \url{http://hetland.org/writing/instant-python.html} is a very crisp introduction. It covers everything you need to get started with \esc. 20 \item A nice and easy to follow introduction: \url{http://www.sthurlow.com/python/} 21 \item Another crisp tutorial: \url{http://www.zetcode.com/tutorials/pythontutorial/}. 22 \item A very comprehensive tutorial from the \pyt authors: \url{http://www.python.org/doc/2.5.2/tut/tut.html}. It covers much more than what you will ever need for \esc. 23 \item Another comprehensive tutorial: \url{http://www.tutorialspoint.com/python/index.htm} 24 \end{itemize} 25 26 \subsection{The \modesys Modules} 27 \esc is part of the \esys package. 28 Apart from the particle simulation library 29 \verb|ESyS-Particle|\footnote{see \url{https://launchpad.net/esys-particle}} which is not covered 30 in this tutorial \esys also includes the following modules 31 \begin{enumerate} 32 \item \modescript is the PDE solving module. 33 \item \modfinley is the discretisation tool and finite element package. 34 \item \modpycad is a package for creating irregular shaped domains. 35 \end{enumerate} 36 Further explanations of each of these are available in the \esc user guide or in the API documentation\footnote{Available from \url{https://launchpad.net/escript-finley/+download}}. 37 \esc is also dependent on a few other open-source packages which are not maintained by the \esc development team. These are \modnumpy (an array and matrix handling package), \modmpl \footnote{\modnumpy and \modmpl are part of the SciPy package, see \url{http://www.scipy.org/}} (a simple plotting tool) and \verb gmsh \footnote{See \url{http://www.geuz.org/gmsh/}} (which is required by \modpycad). These packages (\textbf{except} for \verb gmsh ) are included with the support bundles. 38
ViewVC Help Powered by ViewVC 1.1.26 | 2019-10-22 01:37:17 | {"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": 0.8908822536468506, "perplexity": 5254.275274569639}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "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-2019-43/segments/1570987795403.76/warc/CC-MAIN-20191022004128-20191022031628-00308.warc.gz"} |
https://robotology.github.io/robotology-documentation/doc/html/classiCub_1_1skinDynLib_1_1Taxel.html | iCub-main
iCub::skinDynLib::Taxel Class Reference
Class that encloses everything relate to a Taxel, i.e. More...
#include <Taxel.h>
## Public Member Functions
Taxel ()
Default Constructor. More...
Taxel (const yarp::sig::Vector &_position, const yarp::sig::Vector &_normal)
Constructor with position and normal vectors. More...
Taxel (const yarp::sig::Vector &_position, const yarp::sig::Vector &_normal, const int &_id)
Constructor with position, normal and ID. More...
Taxel (const Taxel &_t)
Copy Constructor. More...
Taxeloperator= (const Taxel &_t)
Copy Operator. More...
int getID ()
Gets the ID of the taxel. More...
yarp::sig::Vector getPosition ()
Gets the Position of the taxel in the limb's FoR. More...
yarp::sig::Vector getNormal ()
Gets the Normal of the taxel in the limb's FoR. More...
yarp::sig::Vector getWRFPosition ()
Gets the Position of the taxel in the root FoR. More...
yarp::sig::Vector getPx ()
Gets the u,v position of the taxel in one of the eyes. More...
yarp::sig::Matrix getFoR ()
Gets the Frame of Reference of the taxel. More...
bool setID (int _ID)
Sets the ID of the taxel. More...
bool setPosition (const yarp::sig::Vector &_Position)
Sets the Position of the taxel in the limb's FoR. More...
bool setNormal (const yarp::sig::Vector &_Normal)
Sets the Normal of the taxel in the limb's FoR. More...
bool setWRFPosition (const yarp::sig::Vector &_WRFPosition)
Sets the Position of the taxel in the root FoR. More...
bool setPx (const yarp::sig::Vector &_px)
Sets the u,v position of the taxel in one of the eyes. More...
virtual void print (int verbosity=0)
Print Method. More...
virtual std::string toString (int verbosity=0)
toString Method More...
## Protected Member Functions
void init ()
init function More...
void setFoR ()
Compute and set the taxel's reference frame (from its position and its normal vector) More...
## Protected Attributes
int ID
yarp::sig::Vector Position
yarp::sig::Vector Normal
yarp::sig::Vector WRFPosition
yarp::sig::Vector px
yarp::sig::Matrix FoR
## Detailed Description
Class that encloses everything relate to a Taxel, i.e.
the atomic element the iCub skin is composed of. It is empowered by a Position and a Normal (both relative to the body part it belong to), and some more useful members (such as its 2D coordinates into the image frame of one of the eyes, its Frame of Reference base on the Position and Normal members, its position into the World Reference Frame)
Definition at line 56 of file Taxel.h.
## ◆ Taxel() [1/4]
Taxel::Taxel ( )
Default Constructor.
Definition at line 9 of file Taxel.cpp.
## ◆ Taxel() [2/4]
Taxel::Taxel ( const yarp::sig::Vector & _position, const yarp::sig::Vector & _normal )
Constructor with position and normal vectors.
Parameters
_position is the position of the taxel _normal is the normal vector of the taxel
Definition at line 14 of file Taxel.cpp.
## ◆ Taxel() [3/4]
Taxel::Taxel ( const yarp::sig::Vector & _position, const yarp::sig::Vector & _normal, const int & _id )
Constructor with position, normal and ID.
Parameters
_position is the position of the taxel _normal is the normal vector of the taxel _id is the ID of the taxel
Definition at line 22 of file Taxel.cpp.
## ◆ Taxel() [4/4]
Taxel::Taxel ( const Taxel & _t )
Copy Constructor.
Parameters
_t is the Taxel to copy from
Definition at line 31 of file Taxel.cpp.
## ◆ getFoR()
yarp::sig::Matrix Taxel::getFoR ( )
Gets the Frame of Reference of the taxel.
Returns
Matrix with the Frame of Reference
Definition at line 118 of file Taxel.cpp.
## ◆ getID()
int Taxel::getID ( )
Gets the ID of the taxel.
Returns
int with the ID of the taxel
Definition at line 93 of file Taxel.cpp.
## ◆ getNormal()
yarp::sig::Vector Taxel::getNormal ( )
Gets the Normal of the taxel in the limb's FoR.
Returns
Vector with the Normal of the taxel
Definition at line 103 of file Taxel.cpp.
## ◆ getPosition()
yarp::sig::Vector Taxel::getPosition ( )
Gets the Position of the taxel in the limb's FoR.
Returns
Vector with the Position of the taxel
Definition at line 98 of file Taxel.cpp.
## ◆ getPx()
yarp::sig::Vector Taxel::getPx ( )
Gets the u,v position of the taxel in one of the eyes.
Returns
Vector with the 2D position of the taxel in one of the eyes
Definition at line 113 of file Taxel.cpp.
## ◆ getWRFPosition()
yarp::sig::Vector Taxel::getWRFPosition ( )
Gets the Position of the taxel in the root FoR.
Returns
Vector with the Position of the taxel
Definition at line 108 of file Taxel.cpp.
## ◆ init()
void Taxel::init ( void )
protected
init function
Definition at line 52 of file Taxel.cpp.
## ◆ operator=()
Taxel & Taxel::operator= ( const Taxel & _t )
Copy Operator.
Parameters
_t is the Taxel to copy from
Definition at line 36 of file Taxel.cpp.
## ◆ print()
void Taxel::print ( int verbosity = 0 )
virtual
Print Method.
Parameters
verbosity is the verbosity level
Definition at line 173 of file Taxel.cpp.
## ◆ setFoR()
void Taxel::setFoR ( )
protected
Compute and set the taxel's reference frame (from its position and its normal vector)
Definition at line 62 of file Taxel.cpp.
## ◆ setID()
bool Taxel::setID ( int _ID )
Sets the ID of the taxel.
Returns
true/false in case of success/failure
Definition at line 123 of file Taxel.cpp.
## ◆ setNormal()
bool Taxel::setNormal ( const yarp::sig::Vector & _Normal )
Sets the Normal of the taxel in the limb's FoR.
Returns
true/false in case of success/failure
Definition at line 140 of file Taxel.cpp.
## ◆ setPosition()
bool Taxel::setPosition ( const yarp::sig::Vector & _Position )
Sets the Position of the taxel in the limb's FoR.
Returns
true/false in case of success/failure
Definition at line 129 of file Taxel.cpp.
## ◆ setPx()
bool Taxel::setPx ( const yarp::sig::Vector & _px )
Sets the u,v position of the taxel in one of the eyes.
Returns
true/false in case of success/failure
Definition at line 162 of file Taxel.cpp.
## ◆ setWRFPosition()
bool Taxel::setWRFPosition ( const yarp::sig::Vector & _WRFPosition )
Sets the Position of the taxel in the root FoR.
Returns
true/false in case of success/failure
Definition at line 151 of file Taxel.cpp.
## ◆ toString()
std::string Taxel::toString ( int verbosity = 0 )
virtual
toString Method
Parameters
verbosity is the verbosity level
Definition at line 187 of file Taxel.cpp.
## ◆ FoR
yarp::sig::Matrix iCub::skinDynLib::Taxel::FoR
protected
Definition at line 64 of file Taxel.h.
## ◆ ID
int iCub::skinDynLib::Taxel::ID
protected
Definition at line 59 of file Taxel.h.
## ◆ Normal
yarp::sig::Vector iCub::skinDynLib::Taxel::Normal
protected
Definition at line 61 of file Taxel.h.
## ◆ Position
yarp::sig::Vector iCub::skinDynLib::Taxel::Position
protected
Definition at line 60 of file Taxel.h.
## ◆ px
yarp::sig::Vector iCub::skinDynLib::Taxel::px
protected
Definition at line 63 of file Taxel.h.
## ◆ WRFPosition
yarp::sig::Vector iCub::skinDynLib::Taxel::WRFPosition
protected
Definition at line 62 of file Taxel.h.
The documentation for this class was generated from the following files:
• icub-main/src/libraries/skinDynLib/include/iCub/skinDynLib/Taxel.h
• icub-main/src/libraries/skinDynLib/src/Taxel.cpp | 2023-03-31 16:29:19 | {"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": 0.24648943543434143, "perplexity": 14882.451008418628}, "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-2023-14/segments/1679296949644.27/warc/CC-MAIN-20230331144941-20230331174941-00656.warc.gz"} |
https://scipost.org/submissions/1902.03908v1/ | # Quantum robustness and phase transitions of the 3D Toric Code in a field
### Submission summary
As Contributors: Kai Phillip Schmidt Arxiv Link: https://arxiv.org/abs/1902.03908v1 (pdf) Date submitted: 2019-02-14 01:00 Submitted by: Schmidt, Kai Phillip Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical
### Abstract
We study the robustness of 3D intrinsic topogical order under external perturbations by investigating the paradigmatic microscopic model, the 3D toric code in an external magnetic field. Exact dualities as well as variational calculations reveal a ground-state phase diagram with first and second-order quantum phase transitions. The variational approach can be applied without further approximations only for certain field directions. In the general field case, an approximative scheme based on an expansion of the variational energy in orders of the variational parameters is developed. For the breakdown of the 3D intrinsic topological order, it is found that the (im-)mobility of the quasiparticle excitations is crucial in contrast to their fractional statistics.
### Ontology / Topics
See full Ontology or Topics database.
###### Current status:
Has been resubmitted
### Submission & Refereeing History
#### Published as SciPost Phys. 6, 078 (2019)
Resubmission 1902.03908v2 on 28 May 2019
Submission 1902.03908v1 on 14 February 2019
## Reports on this Submission
### Report 1 by Irénée Frerot on 2019-4-18 (Invited Report)
• Cite as: Irénée Frerot, Report on arXiv:1902.03908v1, delivered 2019-04-18, doi: 10.21468/SciPost.Report.914
### Strengths
1- The introduction offers a very useful entry to the literature on the 2d and 3d toric code, as well as related models and implementations.
2- The properties of the unperturbed 3d toric code on a cubic lattice are well exposed, in a manner accessible to the newcomer.
3- The phase diagram under an external field is identified consistently through a variety of methods (exactly in certain limits, and by variational methods).
4- Throughout the manuscript, the authors try to provide a physical intuition of the mechanisms at play, as well as comparisons with the better-known 2d toric code.
### Weaknesses
1- The technicalities related to the variational determination of the ground state are hard to follow.
### Report
In the paper, the authors study the 3d toric code on a cubic lattice, which displays a topologically ordered ground state, and focus on the phase transitions towards a trivial state induced by an external uniform magnetic field. After a detailed introduction which clearly motivates their study, the authors describe the model and its ground state properties. Then, applying a combination of methods (p-CUT, exact dualities and variational computations), they reconstruct the phase diagram of the toric code under a uniform magnetic field in an arbitrary direction. Throughout this study, they provide a detailed description of the physical mechanisms induced by the external field, and compare them with the 2d case. Finally, a long discussion summarizes the main results of the paper.
Overall, I find the paper extremely well written, and accessible to the newcomer to the field. Although the technicalities associated with pCUT and the variational calculations are difficult to follow, their outputs are clearly summarized in physical terms. The phase diagram is convincingly reconstructed via various independent means. For these reasons, I recommend the publication of the manuscript.
### Requested changes
1- Eq. (6) should be explained a bit more. The authors should explain why the ground-state degeneracy is given by $2^{N_{spins}} / 2^{N_{constraints}}$.
2- I think that Eq. (8) contains a typo. First, I would advise the authors not to use m as a variable inside the summation, as it brings confusion with the superscript m in $P^m_{xy}$. Then, I think that a $b_z$ is missing in front of the term $(n_z + 1/2)$.
3- Eq. (9) contains a typo : the last term is $b_z = (0, 0, 1)$.
4- p.7, first line: I would suggest to add "In the loop-soup picture of the ground state, these operators measure the parity..."
5- Just after Eq. (11): "with some fixed $(n_y, n_z) \in Z$..."
6- Footnote 3: "...can also be viewed, in the light of quantum codes, as the..." (with comas)
7- After Eq. (18): is really the ground state energy equal to 4N ? I would say that $E_0 = -(1/2)N_{stars} -(1/2)N_{plaquettes} = -N/2 - 3N/2 = -2N$, am I correct?
8- Eq. (19): what means the prefactor $1 / (2j)$ in front of the last term?
9- Eq. (21) and in other places, the authors use the symbol "=" with a "!" on top of it: I have never seen this symbol and some explanation would be welcome.
10- After Eq. (33): "...the two limiting cases $\alpha=\beta=1$ and $\alpha=\beta=0$ are exactly..." and "For $\alpha=\beta=1$, the normalization..."
• validity: high
• significance: high
• originality: good
• clarity: high
• formatting: excellent
• grammar: perfect
### Author: Kai Phillip Schmidt on 2019-05-27 [id 528]
(in reply to Report 1 by Irénée Frerot on 2019-04-18)
Dear Irénée Frerot,
we thank you for carefully examining our paper and for the overall very positive comments on our work. In the revised version of our article, we have addressed the minor issues raised in your report.
To be specific,
1) In the revised we have explained Eq. 6 in more detail. 2) We agree and we have updated Eq. 8. 3) We agree and we have updated Eq. 9. 4) Here we do not agree. Our statement is not only valid in the loop-soup picture of the ground state. So we have left the formulation as it is. 5) We agree and we have "n_x \in Z" as suggested by the referee. 6) We agree and we haved added the commas as suggested by the referee. 7) We agree with the referee and updated the formula. 8) The factor 1/(2j) has to be inside the sum. We corrected this. 9) We have added a footnote on page 15 to explain this symbol. 10) We have followed the referee and changed the expressions. | 2022-05-17 23:22:14 | {"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": 1, "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": 0.5539976358413696, "perplexity": 1196.3491982318}, "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-2022-21/segments/1652662520936.24/warc/CC-MAIN-20220517225809-20220518015809-00514.warc.gz"} |
https://math.stackexchange.com/questions/2934863/locally-constant-sheaves-and-homotopy-equivalences | # Locally constant sheaves and homotopy equivalences
I know that, if $$X$$ is a locally arcwise connected and locally simply connected topological space, then the restrictions of any locally constant sheaf $$\mathcal{F}$$ on $$X$$ corresponding to inclusions $$U\subseteq V$$ of open subsets which are homotopy equivalences are isomorphisms (this is clear because a locally constant sheaf on $$X$$ corresponds to a functor $$F: \mathbf{ \Pi }_1(X)\rightarrow\mathbf{Set}$$).
I was wondering: is it possible to prove the other implication, i.e. whenever $$\mathcal{F}$$ is a sheaf on $$X$$ such that $$\mathcal{F}(V)\rightarrow\mathcal{F}(U)$$ is an isomorphism each time $$U\subseteq V$$ is an homotopy equivalence, then $$\mathcal{F}$$ must be locally constant?
• Yes, you're right! Thank you, I edited my question. – mfox Sep 28 '18 at 22:48
This is certainly not true in this generality. For instance, let $$X=\mathbb{R}^3\setminus\mathbb{Q}^3$$. Then $$X$$ is locally path connected and locally simply connected, but I'm pretty sure no nontrivial inclusion of open subsets of $$X$$ is a homotopy equivalence. So, any sheaf at all on $$X$$ would satisfy your condition.
If you assume $$X$$ is locally contractible, then it is true. Indeed, the restriction of $$\mathcal{F}$$ to any contractible open set $$U$$ is then constant, since we can identify $$\mathcal{F}(V)$$ with $$\mathcal{F}(U)$$ for any contractible open $$V\subseteq U$$ via the restriction map and such $$V$$ form a basis for the topology.
• Thank you for your answer. I don't have a clue on how to prove that no nontrivial inclusion of open subsets of $\mathbb{R}^3\setminus\mathbb{Q}^3$ is a homotopy equivalence, could you give me at least some hint? – mfox Sep 28 '18 at 22:57
• In particular, let $Y$ be a 2-dimensional version of the "Hawaiian earring" (so, a union of infinitely many $S^2$ spheres which converge on a common intersection point). For any open $U\subseteq \mathbb{R}^3\setminus\mathbb{Q}^3$ and any $x\in U$, I think you can construct a map $f:Y\to U$ which maps the common point of the spheres to $x$, such that this $f$ is not homotopic to any map $Y\to U\setminus\{x\}$. This would show that for any $V\subseteq U\setminus\{x\}$, the inclusion $V\to U$ is not a homotopy equivalence. – Eric Wofsey Sep 28 '18 at 23:04
• (The idea is that $f$ is a sequence of spheres in $U$ closing in on $x$. Any map homotopic to $f$ must have its spheres enclose the same sets of rational points as the spheres of $f$ do, and so must also map the common point to $x$.) – Eric Wofsey Sep 28 '18 at 23:05 | 2019-07-21 22:01:17 | {"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": 1, "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": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.98232102394104, "perplexity": 92.10164619427503}, "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-2019-30/segments/1563195527204.71/warc/CC-MAIN-20190721205413-20190721231413-00270.warc.gz"} |
http://www.aimsciences.org/journal/1534-0392/2003/2/4 | # American Institute of Mathematical Sciences
ISSN:
1534-0392
eISSN:
1553-5258
All Issues
## Communications on Pure & Applied Analysis
2003 , Volume 2 , Issue 4
Select all articles
Export/Reference:
2003, 2(4): 425-445 doi: 10.3934/cpaa.2003.2.425 +[Abstract](207) +[PDF](267.4KB)
Abstract:
The paper presents recent advances in $p$-regularity theory, which has been developing successfully for the last twenty years. The main result of this theory gives a detailed description of the structure of the zero set of an irregular nonlinear mapping. We illustrate the theory with an application to degenerate problems in different fields of mathematics, which substantiates the general applicability of the class of $p$-regular problems. Moreover, the connection between singular problems and nonlinear mappings is shown. Amongst the applications, the structure of $p$-factor-operators is used to construct numerical methods for solving degenerate nonlinear equations and optimization problems.
2003, 2(4): 447-459 doi: 10.3934/cpaa.2003.2.447 +[Abstract](172) +[PDF](210.9KB)
Abstract:
We study Hamilton-Jacobi equations with upper semicontinuous initial data without convexity assumptions on the Hamiltonian. We analyse the behavior of generalized u.s.c solutions at the initial time $t=0$, and find necessary and sufficient conditions on the Hamiltonian such that the solution attains the initial data along a sequence (right accessibility).
2003, 2(4): 461-479 doi: 10.3934/cpaa.2003.2.461 +[Abstract](159) +[PDF](241.9KB)
Abstract:
Setting the Homogenization of Hamilton Jacobi equations in the geometry of the Heisenberg group, we study the convergence toward a solution of the limit equation i.e. the solution of the effective Hamiltonian, in particular we estimate the rate of convergence. The periodicity of the fast variable and the dilation are both taken compatibly with the group.
2003, 2(4): 481-494 doi: 10.3934/cpaa.2003.2.481 +[Abstract](179) +[PDF](233.8KB)
Abstract:
We consider the Cauchy problem for the generalized porous medium equation
$u_t = \Delta \Phi(u)$ in R$^n \times [0,T]$
$u(x)=u_0(x)$ on R$^n$
with the nonlinearity $\Phi(u)$. For the case of $\Phi(u)=\sum_{i=1}^m c_i u^{\alpha_i}$, we show the existence of a solution which smoothness depends on the exponents $\alpha_i$. Regardless of the regularity of the solution, we show the free-boundary is smooth. We also extend similar results for $\Phi(u)$ as an infinite sum.
2003, 2(4): 495-509 doi: 10.3934/cpaa.2003.2.495 +[Abstract](182) +[PDF](233.6KB)
Abstract:
This paper deals with a general approach to the rate-independent processes which may display hysteretic behaviour.This approach based on the two energy functionals,namely potential and dissipation functionals.Under some natural assumptions on these functionals we prove the compactness of the set of stable points which in turn leads to the existence of solutions of the problems under consideration.We present an application of our results to ferromagnetic models.
2003, 2(4): 511-520 doi: 10.3934/cpaa.2003.2.511 +[Abstract](158) +[PDF](200.7KB)
Abstract:
In this paper we prove the exponential decay as time goes to infinity of regular solutions of the problem for the wave equations with memory and weak damping
$u_{t t}-\Delta u+\int^t_0g(t-s)\Delta u(s)ds + \alpha u_{t}=0$ in $\hat Q$
where $\hat Q$ is a non cylindrical domains of $\mathbb R^{n+1}$ $(n\ge1)$ with the lateral boundary $\hat{\sum}$ and $\alpha$ is a positive constant.
2003, 2(4): 521-537 doi: 10.3934/cpaa.2003.2.521 +[Abstract](155) +[PDF](392.6KB)
Abstract:
In the current paper the dynamics of a mixed parabolic-gradient system is examined. The system, which is a coupled system of parabolic equations and gradient equations, acts as a first model for the outgrowth of axons in a developing nervous system. For modeling considerations it is relevant to know the influence of the parameters in the system and the source profiles in the parabolic equations on the dynamics. These subjects are discussed together with an approximation which uses the quasi-steady-state solutions of the parabolic equations instead of the parabolic equations themselves.
Some of the findings are demonstrated by numerical simulations.
2003, 2(4): 539-566 doi: 10.3934/cpaa.2003.2.539 +[Abstract](196) +[PDF](305.4KB)
Abstract:
The present work is devoted to analyze the Dirichlet problem for quasilinear elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Precisely the problem under study is,
-div $( |x|^{-p\gamma}|\nabla u|^{p-2}\nabla u)=f(x, u)\in L^1(\Omega),\quad x\in \Omega$
$u(x)=0$ on $\partial \Omega,$
where $-\infty<\gamma<\frac{N-p}{p}$, $\Omega$ is a bounded domain in $\mathbb R^N$ such that $0\in\Omega$ and $f(x,u)$ is a Caratheodory function under suitable conditions that will be stated in each section.
2003, 2(4): 567-577 doi: 10.3934/cpaa.2003.2.567 +[Abstract](204) +[PDF](484.4KB)
Abstract:
In this paper we examine the dynamics of two time-delay coupled relaxation oscillators of the van der Pol type. By integrating the governing differential-delay equations numerically, we find the various phase-locked motions including the in-phase and out-of-phase modes. Our computations reveal that depending on the strength of coupling ($\alpha$) and the amount of time-delay ($\tau$), the in-phase (out-of-phase) mode may be stable or unstable. There are also values of $\alpha$ and $\tau$ for which the in-phase and out-of-phase modes are both stable leading to birhythmicity. The results are illustrated in the $\alpha$-$\tau$ parameter plane. Near the boundaries between stability and instability of the in-phase (out-of-phase) mode, many other types of phase-locked motions can occur. Several examples of these phase-locked states are presented.
2003, 2(4): 579-589 doi: 10.3934/cpaa.2003.2.579 +[Abstract](205) +[PDF](183.8KB)
Abstract:
Several comparison theorems for oscillation and nonoscillation of neutral difference equations with continuous arguments are established. Some known results are included and improved. All results obtained in this paper are new.
2003, 2(4): 591-599 doi: 10.3934/cpaa.2003.2.591 +[Abstract](209) +[PDF](199.5KB)
Abstract:
In this note, we present a fast communication of a new bifurcation theory for nonlinear evolution equations, and its application to Rayleigh-Bénard Convection. The proofs of the main theorems presented will appear elsewhere. The bifurcation theory is based on a new notion of bifurcation, called attractor bifurcation. We show that as the parameter crosses certain critical value, the system bifurcates from a trivial steady state solution to an attractor with dimension between $m$ and $m+1$, where $m+1$ is the number of eigenvalues crosses the imaginary axis. Based on this new bifurcation theory, we obtain a nonlinear theory for bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Bénard convection. In particular, we show that the problem bifurcates from the trivial solution an attractor $\mathcal A_R$ when the Rayleigh number $R$ crosses the first critical Rayleigh number $R_c$ for all physically sound boundary conditions.
2016 Impact Factor: 0.801 | 2018-04-25 06:29:58 | {"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": 1, "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": 0.8192020058631897, "perplexity": 377.0940109879882}, "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-2018-17/segments/1524125947705.94/warc/CC-MAIN-20180425061347-20180425081347-00249.warc.gz"} |
https://typedtables.juliadata.org/latest/man/filter/ | # Finding data
Frequently, we need to find data (i.e. rows of the table) that matches certain criteria, and there are multiple mechanisms for achieving this in Julia. Here we will briefly review map, findall and filter as options.
## map(predicate, table)
Following the previous section, we can identify row satisfying an arbitrary predicate using the map function. Note that "predicate" is just a name for function that takes an input and returns either true or false.
julia> t = Table(name = ["Alice", "Bob", "Charlie"], age = [25, 42, 37])
Table with 2 columns and 3 rows:
name age
┌─────────────
1 │ Alice 25
2 │ Bob 42
3 │ Charlie 37
julia> is_old = map(row -> row.age > 40, t)
3-element Array{Bool,1}:
false
true
false
Finally, we can use "logical" (i.e. Boolean) indexing to extract the rows where the predicate is true.
julia> t[is_old]
Table with 2 columns and 1 row:
name age
┌──────────
1 │ Bob 42
The map(predicate, table) approach will allocate one Bool for each row in the input table - for a total of length(table) bytes. SplitApplyCombine defines a mapview function to do this lazily.
## findall(predicate, table)
If we wish to locate the indices of the rows where the predicate returns true, we can use Julia's findall function.
julia> inds = findall(row -> row.age > 40, t)
1-element Array{Int64,1}:
2
julia> t[inds]
Table with 2 columns and 1 row:
name age
┌──────────
1 │ Bob 42
This method may be less resource intensive (result in less memory allocated) if you are expecting a small number of matching rows, returing one Int per result.
## filter(predicate, table)
Finally, if we wish to directly filter the table and obtain the rows of interest, we can do that as well.
julia> filter(row -> row.age > 40, t)
Table with 2 columns and 1 row:
name age
┌──────────
1 │ Bob 42
Internally, the filter method may rely on one of the implementations above.
## Generators
Julia's "generator" syntax also allows for filtering operations using if.
julia> Table(row for row in t if row.age > 40)
Table with 2 columns and 1 row:
name age
┌──────────
1 │ Bob 42
This can be combined with mapping at the same time, as in Table(f(row) for row in table if predicate(row)). In Joining Data we discuss how to use generator syntax to combine multiple datasets.
## Preselection
As mentioned in other sections, it is frequently worthwhile to preselect the columns relating to your search predicate, to avoid any wastage in fetching from memory values in columns that you don't care about.
One simple example of such a transformation is to first project to the column(s) of interest, followed by using map or findall to identify the indices of the rows where predicate is true, and finally to use getindex or view to obtain the result of the full table.
julia> inds = findall(age -> age > 40, t.age)
1-element Array{Int64,1}:
2
julia> t[inds]
Table with 2 columns and 1 row:
name age
┌──────────
1 │ Bob 42
Easy, peasy! | 2023-02-08 00:38:32 | {"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": 0.3885221779346466, "perplexity": 4560.839724770506}, "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-2023-06/segments/1674764500664.85/warc/CC-MAIN-20230207233330-20230208023330-00607.warc.gz"} |
https://www.physicsforums.com/threads/even-odd-permutations.98880/ | # Even odd permutations
1. Nov 7, 2005
### johnnyboy2005
i was wondering if anyone can help me understand why the product of two odd permutations is odd? i came across this on a web site but it didn't help me understand why. thanks for the help
2. Nov 7, 2005
### HallsofIvy
Do you understand what "even" and "odd" permutations are? Every permutation can be reduced to a sequence of "two-element swaps": for example, the permutation that changes 1234 into 3124 can be written as
(13)(12): first swap 1 and 3: 1234-> 3214, then swap 1 and 2: 3214->3124.
Of course, there are many different ways to do that:
(23)(13) will work or even (14)(34)(24)(14): 1234->4231->3241->3421->3124 but any one permutation will consist of either an even number of swaps or an odd number no matter how that is done. An even permutation is one that requires and even number of "swaps", an odd permutation is one that requires an odd number of permutations.
One way of taking the product of two permutations is to just combine their "swaps"
1234->3214 can be written as (13) so the permutation
formed by 1234->3214 followed by 1234->3214 can be written
(13)(12)(13). The number of swaps in the product is just the sum of the number of swaps in each: in particular, the sum of 2 odd numbers is even, so the product of two odd permutations is an even permutation.
3. Nov 7, 2005
### -Job-
I guess we can visualize this with a graph where each possible permutation is represented with a node and there is an edge between two nodes A and B if by swapping two digits in A's permutation (transposition) we get B. Suppose the nodes A and B aren't connected, then we are interested in a path from A to B (which gives a sequence of transpositions that transform A into B). If the path has odd length then it is an odd permutation, otherwise it is even. It is a fact that a permutation is either odd or even, meaning that if there is a path of even length from A to B, then all paths from A to B are of even length. (A way to see what is going on is to try to create a permutation graph with nodes A, B and C. You cannot connect A, B and C so that they from a triangle, because this would mean that from node A you can get to B with a path of length 1 A->B and a path of length 2 A-> C -> B, which implies that there is a transposition that is equivalent to two transpositions, which cannot be)
If you have an odd permutation from A -> B and an odd permutation from B -> C, then you have a path from A to B of odd length and a path from B to C of odd length. Hence there is a path from A to C of odd+odd = even length, so all paths from A to C are even and any permutation from A to C is an even permutation.
4. Nov 8, 2005
### matt grime
Halls is perfectly correct with his definition. At the risk of unnecessarily introducing another (unnecessary) idea (I stil can't tell what Job's getting at) there is another way of thinking about even and oddness. (Incidentally, in case you've not realized, the product of two odd elements is even.)
The permutation group can be realized as a set of nxn matrices. Pick a vector space of n dimensions with basis
e_1,....,e_n
each permutation acts by permuting basis elements. This defines a linear map or matrix.
The determinant of this matrix is the sign of the element, 1 for even elements -1 for odd elements. (You can put any of these permutation matrices into the identity matrix by swapping rows, the number of row swaps is odd or even depending on if the element is odd or even).
And we all know that det(AB)=det(A)det(B) so if A and B are odd (ie determinant -1) then AB is even
5. Nov 8, 2005
### johnnyboy2005
great, thanks so much for your replies. everything was pretty clear. Just have a question for Halls,:
you wrote
1234 into 3124 can be written as (13)(12): this i understand....
first swap 1 and 3: 1234-> 3214, then swap 1 and 2: 3214->3124.
can u please explain how swapping 1 and 2 changes 3214 into 3124???
Of course, there are many different ways to do that: (23)(13) will work or even (14)(34)(24)(14): 1234->4231->3241->3421->
thanks again, this is a great web site...lots of help and quick replies
6. Nov 8, 2005
### HallsofIvy
Be careful: I don't mean swapping what is in the first and second places: I specifically meant swapping the symbols 1 and 2. The 3 in the first place and the 4 in the last place don't change. 1 moves from 2nd to 3rd place while 2 moves from 3rd to 2nd place.
3214
3124
"1" and "2" have swapped positions.
7. Nov 8, 2005
### -Job-
In a model where each permutation of n digits is represented by a node and where there is an edge between two nodes A and B iff there is a single transposition of the permutation of A that transforms it to B. Then you can talk about the lengths of permutations as the length of the paths from an initial permutation A to a final permutation B.
In such a graph if the path to B (from A) is odd, then B is an odd permutation of A, otherwise it is even. We know that if B is an odd permutation of A then it can't be even as well, meaning that if there is a path of odd length from A to B, then all paths from A to B are of odd length.
So if B is an odd permutation of A and C is an odd permutation of B, then there are paths of odd length from A to B and from B to C. Hence we can get from A to C by traversing the two odd paths A->B->C which gives a path of even length. So the product of two odd permutations is even.
To me this seems like an easy way to visualize what's going on (i've seen it used in proofs regarding the properties of hypercubic networks).
8. Nov 8, 2005
### johnnyboy2005
fantastic!! thanks a lot guys, i have figured it out.
9. Nov 9, 2005
### matt grime
So it is a graph G, its vertex set is S_n and there is an edge from s to t if st^{-1} is a transposition ('transposition that transforms it' doesn't make much sense unless you define 'transformations' by transpositions). Why didn't you say so.... The sign of a permutation is then the parity of any path from the node of the identity to it.
As with all of these we need to prove that our definition is well defined (Ie it is always an even/odd number of transpositions). My 'proof' that odd composed odd is even sidesteps this entirely but at no point do I prove rigorously that the determinant is the the same as the sign.. That is something I've never seen done. Does anyone have a nice proof of this?
10. Nov 9, 2005
### Galileo
This is how I was introduced to it. By a lemma:
There exists a unique map $\varepsilon: S_n \to \{\pm 1\}$ with the following properties:
(1) If $\sigma$ is a transposition (2 element swap), then $\varepsilon(\sigma)=-1$
(2) For arbitrary elements $\sigma, \tau \in S_n$ we have $\varepsilon(\sigma \tau)=\epsilon(\sigma)\epsilon(\tau)$.
(Note from self: $\varepsilon$ is thus a homomorphism from S_n to the multiplicative group $\{1,-1\}$).
Proof: Since every permutation is a product of transpositions it's clear there can only exist one such function. But because a permutation can be written as such a product in many different ways, it's not so clear it should exist at all.
We define $\varepsilon$ by considering the function $F:\mathbb{R}^n\to \mathbb{R}$ as follows:
$$F(x_1,x_2,...,x_n)=\prod_{1\leq i < j \leq n}(x_i-x_j)$$
Notice that F is not the zero function. For $\sigma \in S_n$ we define the function $\sigma F : \mathbb{R}^n \to \mathbb{R}$ given by:
$$(\sigma F)(x_1,x_2,...,x_n)=\prod_{1\leq i < j \leq n}(x_{\sigma(i)}-x_{\sigma(j)})$$
This function is the same as F with a possible switch in sign of the images. So we define $\epsilon(\sigma)$ as $\sigma F=\varepsilon(\sigma) F$
If $\sigma$ is a transposition (i j), then we can construct $\sigma F$ from F, by switching $x_i-x_j$ with $x_j-x_i$. That's because every other factor with $x_i$ or $x_j$ can be paired with some element $x_k, k\not= i,j$. We get four pairs for each k:
$$(x_i-x_k)(x_j-x_k), \quad (x_i-x_k)(x_k-x_j), \quad (x_k-x_i)(x_j-x_k), \quad (x_ki-x_i)(x_k-x_j)$$
All these factors are invariant under the transposition (i j). Therefore $\varepsilon(\sigma)=-1$ for a transposition $\sigma$
From the relation $(\sigma \tau)F=\sigma(\tau F)$ it's easy to see that $\varepsilon(\sigma \tau)F=\varepsilon(\sigma)\varepsilon(\tau)F$.
-------
The proof looks complicated, but once you get the idea it's not that bad. If we call the permutations which are mapped to 1 even and the ones that are mapped to -1 odd you got the required result.
11. Nov 9, 2005
### matt grime
No, that's good and simple. I'd never bothered to look for a proof, I just remember being told that it was hard.
You have effectively shown an explicit homomorphism from S_n into C_2 that doesn't require the notion that sign is well defined a priori. However, you haven't actually checked that this is indeed a homomorphism (or an action of S_n, or representation depending on your view). You've merely said 'from the relation that (st)F=s(tF)' without verifying that is true.I imagine this is merely messy and not actually hard though. | 2018-06-18 13:50:18 | {"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": 1, "mathjax_display_tex": 1, "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": 0.8763611912727356, "perplexity": 281.4575276231362}, "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-2018-26/segments/1529267860557.7/warc/CC-MAIN-20180618125242-20180618145242-00062.warc.gz"} |
https://ask.cloudbase.it/answers/1704/revisions/ | New Question
# Revision history [back]
Hello,
From what I can see in the trace, it seems that there are some configurations missing from the neutron section of the Hyper-V compute node's nova.conf file, in particular, configurations regarding keytone authentication.
Our Hyper-V Nova compute installer already takes care of such configurations.
Also, make sure that you put traces like this into http://paste.openstack.org . This will make reading traces / logs a lot easier. [1] https://github.com/cloudbase/hyperv-nova-compute-installer
Best regards,
Claudiu Belu | 2019-05-27 03:18:24 | {"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": 0.7486881613731384, "perplexity": 6217.241699555041}, "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-2019-22/segments/1558232260658.98/warc/CC-MAIN-20190527025527-20190527051527-00107.warc.gz"} |
http://ogg.pepris.com/CN/abstract/abstract12092.shtml | • 方法技术 •
基于地质储量结构变化的采收率演变趋势
1. 1. 中国石化 石油勘探开发研究院, 北京 100083
2. 国家能源陆相砂岩老油田持续开采研发中心, 北京 100083
• 收稿日期:2020-07-02 出版日期:2020-12-28 发布日期:2020-12-09
• 作者简介:计秉玉(1963-),男,教授级高级工程师,油气田开发与提高采收率研究。E-mail:jby.syky@sinopec.com
• 基金资助:
国家重点研发计划项目(2018YFA0702400);中国工程院重点咨询项目(2019-XZ-15)
Research on overall recovery rate variations of dynamically changing OOIP
Bingyu Ji1,2(), Youqi Wang1,2, Li Zhang1,2
1. 1. Petroleum Exploration and Production Research Institute, SINOPEC, Beijing 100083, China
2. R & D Center of Sustainable Development of Continental Sandstone Mature Oilfield, Beijing 100083, China
• Received:2020-07-02 Online:2020-12-28 Published:2020-12-09
Abstract:
With different types of oil reservoirs being put into development in a proper order according to their respective characteristics, operators may be rewarded with more reserves in a given field.However, with reservoir combinations varying, the overall recovery rates of the field change as well.An ordinary differential equation with reserve as independent variable is established to characterize the changing trend of overall recovery rates with increasing reserves and to quantitatively analyze the overall recovery rate changes of Sinopec's reserves that had been successively put into development (or already under development) from 2011 to 2018.The results show that the relationship between the overall recovery rate and reserve increase can be simplified as the total versus the incremental or the remaining.The main factors influencing the overall recovery rate are the quality and proportion of reserve of a certain reservoir in the whole context.The overall recovery rate of a given field is generally decreasing as more and more hard-to-get reserves are put into development.Following the analysis of the recovery rates of various oil reservoir combinations operated by Sinopec and the factors that influence the rate changes, we propose that future EOR study should be focused on improving the efficiency of water flooding and thermal EOR, expanding the application of chemical flooding, gas injection and microbial recovery, and exploring the possibilities of developing innovative EOR technologies. | 2022-05-28 20:44:00 | {"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": 0.27365368604660034, "perplexity": 4012.6046240316728}, "config": {"markdown_headings": false, "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-2022-21/segments/1652663019783.90/warc/CC-MAIN-20220528185151-20220528215151-00546.warc.gz"} |
https://www.physicsforums.com/threads/earth-moon-without-sun.381623/ | Earth-Moon without Sun
hello there
let's imagine that the Earth-Moon system became isolated (ie, not subjected to Sun's gravity anymore), would Moon run around the Earth at a lesser speed than now? (if it would, in which proportions then?); and does Moon's orbit would change in circumference?
Related Astronomy and Astrophysics News on Phys.org
Matterwave
Gold Member
To first approximation, there would be no difference. The orbit of the Moon can be computed as if it were in isolation with the Earth, and the Sun were not there. To higher degrees of precision, there may be a difference, but I'm not so sure what those would be.
Janus
Staff Emeritus
Gold Member
One effect the Sun has on the Moon's orbit is to alter its eccentricity. It increases it when the orbit's semi-major axis lines up with the Earth-Sun radial and decreases it when it is at a right angle. Another is that it causes the Lunar orbit to precess.
I post this question because i read the following:
As we can see, a revolution of Earth-Moon system around the Sun for 1 synodic month is equivalent to an arc of 27° of the terrestrial orbit around the Sun. This angle (27°) is the same as the one between Moon-position at 1 sideral month and at 1 synodic month according to Earth's frame of reference.
The vector "E" (beginning from the Moon) perpendicular to the segment Earth-Sun points to a different direction every synodic month.
Thus from this "triangle" of vectors, we can calculate mean Moon's speed E' around Earth without Sun's gravitionnal influence.
ll E ll=square root [ (E sin phi)² + (E')² ]
thus ll E' ll= ll E ll * cosine phi
ll E' ll= current Moon's speed * cosine phi
ll E' ll= 3680km/hour * cosine 27°
ll E' ll=3279 km/hour
THUS: In an isolated Earth-Moon system, Moon would run at a mean speed of 3279 km/hour around our planet.
I agree with the value of phi, but is this method to calculate Moon's speed in an isolated Earth correct?
thank you
Staff Emeritus
2019 Award
I took a look at that website, and wouldn't consider anything on it to be reliable.
Hey, Janus!
A website states that if Sun's gravitationnal effect were "put off", Moon's orbit around the Earth would be almost a perfect circle (whereas with the Sun it's an ellipse).
In that condition, would the mean moon's speed become equal to V= 3680*(1-2e) where "e" is the actual excentricity of Moon (about 0.05) and "2e" the variation ratio, as they suggest?
[PLAIN]http://img40.imageshack.us/img40/9568/1dtp.jpg [Broken]
thank you
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Hello everyone. I've been discussing this with Termina for a while now on youtube and as we could not reach an agreement I suggested he asked on a physics forum.
Please don't let the source of the claim put you off discussing this topic. Ultimately the purpose of this discussion is to determine
A: The average velocity of the Moon
B: The average velocity of the Moon if the Sun were not there
I'd appreciate it if you would join in, thanks!
tony873004
Gold Member
This is easy enough to simulate. Here are some graphs showing the the Moon's velocity, eccentricity, and inclination over the course of 1 year, first with the Sun present, then repeated after deleting the Sun
http://orbitsimulator.com/BA/moonwithoutsun.GIF
This is easy enough to simulate. Here are some graphs showing the the Moon's velocity, eccentricity, and inclination over the course of 1 year, first with the Sun present, then repeated after deleting the Sun
http://orbitsimulator.com/BA/moonwithoutsun.GIF
Thanks for the data tony. What we are trying to establish is the distance the Moon would travel in a complete orbit of the Earth vs the distance it would travel in the same time if the Sun were not present.
Any help would be much appreciated.
tony873004
Gold Member
Thanks for the data tony. What we are trying to establish is the distance the Moon would travel in a complete orbit of the Earth vs the distance it would travel in the same time if the Sun were not present.
Any help would be much appreciated.
In the graphs, you're given time and velocity. So you can compute distance. d=vt. You can also see how many orbits were completed in each graph by counting the cycles. So you have everything you need to compute distance per orbit.
Hey, Janus!
A website states that if Sun's gravitationnal effect were "put off", Moon's orbit around the Earth would be almost a perfect circle (whereas with the Sun it's an ellipse).
Termina, that is not true. First of all, a two-body system acting via gravity NATURALLY is elliptical (or hyperbolic). The sun is not responsible for the Moon's elliptical orbit.
In the graphs, you're given time and velocity. So you can compute distance. d=vt. You can also see how many orbits were completed in each graph by counting the cycles. So you have everything you need to compute distance per orbit.
Hi Tony
Please don't think I am being lazy. There are two reasons I cannot do what you say. The first is that Termina and I are here to get an objective answer from more experienced people (experienced in physics) to a disagreement we have been having elsewhere so if I come up with the answer then Termina will think I have concocted it; and the other is much more simple, if I tried to determine the result from those graphs I would get it wrong :)
I spent some time calculating how far the Moon would travel in X amount of time based on its average velocity and assuming that any effect on this velocity by the gravity of the Sun as the Moon heads toward it will be equally negated as the Moon travels away from it.
My calculation for the distance the Moon travels in 1 Moon sidereal day is as follows.
Figures from http://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html
Note: It is the technique that is important, not the exact answer
Moon sidereal day = 655.728 hours
Moon sidereal day = 2360620.8 seconds
(multiply hours by 60^2 to get seconds)
Moon average velocity = 1023km/s
(multiply by seconds in sidereal day to get average distance travelled in 1 Moon sidereal day)
Distance travelled by Moon in 1 sidereal Moon day = 2414915078.4km
Now whether or not this figure is exactly right is not the source of our disagreement. Termina thinks that if there were no effect of the Sun's gravity on the Moon this distance would be approximately 10% lower - which to me is frankly ridiculous but I am willing to accept I am wrong.
I shan't go into the details of how Termina calculated this figure at this point unless Termina wishes to later. At the moment all we really want to know is the answer to the following question...
If the Moon travels 2414915078.4km in 655.728 hours in the presence of the Sun's gravity. How far would it travel in the same time if the Sun's gravity were not there?
I'm sorry if this is all very basic for you guys, but I cannot express how grateful I am for your time on this!
Cheers!
TheRationaliz, i've never thought Sun would affect Moon movement in such extend, rather after hearing such claims i'm here to check whether they are right or not.
tony873004, nice graphs! can you tell me where you found them?
TheRationaliz, i've never thought Sun would affect Moon movement in such extend, rather after hearing such claims i'm here to check whether they are right or not.
Okay, good. Then I hope someone here will be kind enough as to tell us how much (if at all) the gravity of the Sun will affect the distance travelled by the Moon during an Earth sidereal month (Moon sidereal day.)
tony873004
Gold Member
...tony873004, nice graphs! can you tell me where you found them?
I made them in Excel using data generated from an n-body simulation. I let the solar system run for 1 year, recording data. I then started the simulation over again, but this time I deleted the Sun.
Going back to the spreadsheet data used to make the graphs, I had the spreadsheet compute the average velocities. Over the course of the year plotted, the Moon's average velocities are:
without the Sun:1.02071 km/s.
With the Sun:1.021342 km/s
So the Moon with the Sun travels slightly faster.
So the distances travelled in 1 lunar sidereal day are:
without sun: d=vt = (1.02071 km/s)(2360620.8 s/moon sidereal day) = 2409509 km
with sun : d=vt = (1.021342 km/s )(2360620.8 s/moon sidereal day) = 2411011 km
So the moon with the sun included travels 1492 km farther during the amount of time, a difference of about 0.06%
However, this experiment is sensitive to when I magically make the Sun disappear. In the above experiment, the Sun disappeared when the Moon was at a waning crescent phase, nearly a new moon. Repeating the 1-year experiment 10 days later, when the Moon is at 1st quarter yields.
Moon without Sun: 1.031003 km/s
Moon with Sun: 1.023383 km/s
This time, the Moon without the Sun is slightly faster.
Tony
Hopefully with this objective analysis and the fact that it was calculated using a solar system simulator Termina will now accept that the Moon will not travel 10% slower without the gravity of the Sun and will accept that the calculation we were looking at is merely a numerology trick.
Thank you so much for taking the time and effort to calculate these effects for us! I appreciate your time greatly!
Thanks!
I suspected their initial claim were pseudoscientific "gymnastics", but i wasn't certain enough since i don't know celestial mechanics like the back of my hand.
now, i'm certain!
The website where i get those doubtful claims even asserts it's Einstein who said one must make 3682km/h*cosine26,9° to know Moon speed value without Sun!!!!!
D H
Staff Emeritus
The Earth-Moon system by itself would almost exactly follow an elliptical orbit. (the only deviation is due to general relativity, which in the case of the slow-moving Moon around the not-so massive Earth is negligible). Call this period T', to distinguish it from the period where the Sun is present.
$$T' = 2\pi\,\sqrt{\frac{a^3}{G(M_e+M_m)}$$
where a is the semi-major axis of the orbit of the Earth and Moon about their center of mass, G is the universal gravitational constant, and Me and M[/sub]m[/sub] are the masses of the Earth and Moon. Rather than using G and the masses of the Earth and Moon, all of which suffer from a lack of accuracy, it is far better to use the Earth gravitational coefficient and the Moon/Earth mass ratio:
$$G(M_e+M_m) = GM_e(1+M_m/M_e) \approx 1.0123 \mu_e$$
where $\mu_e = 398600.4418\pm0.0009\,\text{km}^3/\text{s}^2$. The Moon/Earth mass ratio os 0.0123, to many places of accuracy. What to use for a? The French Ephemerides of the Earth, Moon, and Sun specifies a value of 384,7481 km. The resulting value for T' is 0.4 minutes longer than the sidereal month; essentially no change.
This is consistent with lunar theory. What you are talking about is the variation of the Moon. This has a period term and a secular term. The secular term is exceeding small, measured in arcseconds/century. The affect of the Sun on the period of the Moon, and hence on the distance traveled, is essentially null.
--------------------------
1A widely-used value for the Moon's semi-major axis is 384,399 km. This is not the semi-major axis. It is instead the inverse of the mean of the inverse of the distance. Using that value yields a period that is about 53 minutes shorter than the observed sidereal month. This says the Moon would, on average, move slightly faster than it does with the Sun present. This is counter to lunar orbit theory (essentially no effect), and in any case, it does not support termina's position.
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The website where i get those doubtful claims even asserts it's Einstein who said one must make 3682km/h*cosine26,9° to know Moon speed value without Sun!!!!!
Yes, they are using an old debating trick known as an "appeal to authority". They throw in some complex numbers that most people won't be able to follow (especially illiterate people in 3rd world countries) and say "Look! It's not US claiming this, it's Einstein. Are you saying Einstein was wrong? Do you think you are more intelligent than Einstein?"
Then they fail to actually provide evidence that Einstein every said why they claim he said, and even if Einstein had said such a thing it doesn't mean he was automatically right just because he was clever. Einstein apparently once wrote the foreword in a book in which he stated that plate tectonics theory was implausible...he was wrong :-)
Still, I am more than just a little bit pleased to see we are now in agreement that this claim is merely a BS numerology trick.
TONY
hehe, the duplicity of this guy :-)
After 2 weeks arguing with me that the claim is a fact he came on here pretending to be sceptical. Then when it was proven to him beyond a doubt that it is a false claim and he accepted that proof he back to the speed-info.com forum and claimed that it is true after all....the Moon would slow down even more because the absence of the Sun would cause the Earth to rotate more slowly due to missing gravity effect on the Earth's oceans.
Of course he also believes that multiplying the orbital distance of the Moon by cosine(26.928) will magically compensate for both this orbital friction and the general relativity factor in one go. I never realised COSINE was so powerful :-)
Does multiplying the circumference of an ellipse by COS actually mean anything anyway?
Boy, I can see why you guys wouldn't want to get into discussions about claims of religious miracles :-D
hehe, the duplicity of this guy :-)
After 2 weeks arguing with me that the claim is a fact he came on here pretending to be sceptical. Then when it was proven to him beyond a doubt that it is a false claim and he accepted that proof he back to the speed-info.com forum and claimed that it is true after all....the Moon would slow down even more because the absence of the Sun would cause the Earth to rotate more slowly due to missing gravity effect on the Earth's oceans.
Of course he also believes that multiplying the orbital distance of the Moon by cosine(26.928) will magically compensate for both this orbital friction and the general relativity factor in one go. I never realised COSINE was so powerful :-)
Does multiplying the circumference of an ellipse by COS actually mean anything anyway?
Boy, I can see why you guys wouldn't want to get into discussions about claims of religious miracles :-D
that's not me!:rofl:
Arguing on youtube that this claim is accurate, and then posting on here that you thought all along it was false. Leaving here exclaiming how you now know for a fact it is rubbish, straight back onto youtube saying that the people on this physics forum don't understand physics properly.
Please do explain to the people on this forum how taking the distance travelled by the Moon in a sidereal month and multiplying it by cosine(26.928) removed the gravitational affect of the Sun *and* cancels out the increase in Moon velocity due to the Earth's oceans.
I'm sure there are people here who might find it amusing :)
Arguing on youtube that this claim is accurate, and then posting on here that you thought all along it was false. Leaving here exclaiming how you now know for a fact it is rubbish, straight back onto youtube saying that the people on this physics forum don't understand physics properly.
Please do explain to the people on this forum how taking the distance travelled by the Moon in a sidereal month and multiplying it by cosine(26.928) removed the gravitational affect of the Sun *and* cancels out the increase in Moon velocity due to the Earth's oceans.
I'm sure there are people here who might find it amusing :)
You're far from the truth! as far i'm concerned, i've never stated on youtube that such far-fetched equation like Vmoon*cosine 26 was right. I'm not wormhole199.
Well, it's a non physics matter so that part of our discussion should be held elsewhere to save annoying others :-) | 2020-09-27 20:52:12 | {"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": 1, "mathjax_display_tex": 1, "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": 0.5293998122215271, "perplexity": 985.6871506923775}, "config": {"markdown_headings": false, "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-2020-40/segments/1600401578485.67/warc/CC-MAIN-20200927183616-20200927213616-00644.warc.gz"} |
https://www.scielo.br/j/mr/a/KS4rmtmHtPYJ9jNLB79jwnJ/?lang=en | # Study of the Effect of Solvent on the Conductivity of Langmuir-Schaefer Films of Poly(Fullerene)s
About the authors
# Abstract
The present work aims to prepare and characterize very thin films of poly(fullerene)s in order to investigate their electrical properties and the influence of xylene and chloroform solvents on these materials. The fullerenes studied were phenyl-C61-butyric acid methyl ester (PCBM), oligo{(phenyl-C61-butyric acid methyl ester)-alt-[1,4-bis(bromomethyl)-2,5-bis(octyloxy)benzene]} (OPCBMMB) and poly{[bispyrrolidino(phenyl-C61-butyric acid methyl ester)]-alt-[2,5-bis(octyloxy) benzene]} (PPCBMB), along with poly(3-hexylthiophene) (P3HT). The Langmuir-Schaeffer technique was used to prepare films, which were deposited on interdigitated gold substrates, and electrically characterized, with emphasis on the study of transport, conductivity and mobility mechanisms with respect to the solvents used. We found that the addition of P3HT significantly increased the conductivity of these materials. The xylene cast PPCBM, in both pure and mixed forms under dark conditions, presented the best conductivity results with respect to the other materials. However, when chloroform was used, it was found that OPCBMMB in both pure and mixed forms under light, exhibited the best conductivities. This is the first treatment, to our knowledge, of the impact of solvents on the electronic properties of poly(fullerene)s.
Keywords :
Langmuir; poly-fullerenes; solvents; conductivity; thin films
# 1. Introduction
Organic solar cells have attracted significant attention due to their characteristics such as their lightweight, low-cost and flexibility11 Clafton SN, Huang DM, Massey WR, Kee TW. Femtosecond dynamics of excitons and hole-polarons in composite P3HT/PCBM nanoparticles. J Phys Chem B. 2013;117(16):4626-33.. One of the principles for building a solar cell is based on the combination of an acceptor and a donor material. A long standing and common combination has been to use phenyl-C61-butyric acid methyl ester (PCBM) which as an electron acceptor has a relatively large band gap and complements that of poly(3-hexylthiophene) (P3HT) which, as an electron donor, exhibits a high mobility22 Kadem BY, Al-hashimi MK, Hassan AK. The effect of solution processing on the power conversion efficiency of P3HT-based organic solar cells. Energy Procedia. 2014;50:237-45.. Among several organic materials, the bulk heterojunction of PCBM:P3HT (acceptor:donor) as an active layer was one of the most investigated structures for organic photovoltaics33 Liu J, Shao S, Wang H, Zhao K, Xue L, Gao X, et al. The mechanisms for introduction of n-dodecylthiol to modify the P3HT/PCBM morphology. Org Electron. 2010;11(5):775-83.. Crystallization of the P3HT occurs faster than the PCBM aggregation within the P3HT:PCBM mixture44 Wang T, Pearson AJ, Lidzey DG, Jones RAL. Evolution of structure, optoelectronic properties, and device performance of polythiophene:fullerene solar cells during thermal annealing. Adv Funct Mater. 2011;21(8):1383-90.,55 Reisdorffer F, Haas O, Le Rendu P, Nguyen TP. Co-solvent effects on the morphology of P3HT:PCBM thin films. Synth Met. 2012;161(23-24):2544-8..
Interestingly, the effect of the solvent on the active organic layer has received significant attention during the past few years66 Zhang F, Jespersen KG, Björström C, Svensson M, Andersson MR, Sundström V, et al. Influence of solvent mixing on the morphology and performance of solar cells based on polyfluorene copolymer/fullerene blends. Adv Funct Mater. 2006;16(5):667-74.. This is because the solvent used to form and cast the active layer can play a compelling role in controlling the morphology of the PCBM:P3HT structure, which in turn, strongly influences the separation and transport of charge carriers, and has a massive impact on the final performance of the solar cell66 Zhang F, Jespersen KG, Björström C, Svensson M, Andersson MR, Sundström V, et al. Influence of solvent mixing on the morphology and performance of solar cells based on polyfluorene copolymer/fullerene blends. Adv Funct Mater. 2006;16(5):667-74..
Fullerenes and their derivatives have shown great promise for applications ranging from electronic devices77 Yin H, Lin H, Zong Y, Wang X-D. The recent advances in C60 micro/nanostructures and their optoelectronic applications. Org Electron. 2021;93:106142. http://dx.doi.org/10.1016/j.orgel.2021.106142.
http://dx.doi.org/10.1016/j.orgel.2021.1...
to areas of medicine, such as anti-cancer and antiviral medications88 Zhang X, Cong H, Yu B, Chen Q. Recent advances of water-soluble fullerene derivatives in biomedical applications. Mini Rev Org Chem. 2018;16(1):92-9.
9 Grebinyk A, Prylutska S, Grebinyk S, Prylutskyy Y, Ritter U, Matyshevska O, et al. Complexation with C60 fullerene increases doxorubicin efficiency against leukemic cells in vitro. Nanoscale Res Lett. 2019;14(1):61.
10 Shi J, Yu X, Wang L, Liu Y, Gao J, Zhang J, et al. PEGylated fullerene/iron oxide nanocomposites for photodynamic therapy, targeted drug delivery and MR imaging. Biomaterials. 2013;34(37):9666-77.
-1111 Bakry R, Vallant RM, Najam-ul-Haq M, Rainer M, Szabo Z, Huck CW, et al. Medicinal applications of fullerenes. Int J Nanomedicine. 2007;2(4):639-49. However, due to its highly hydrophobic nature and its tendency to aggregate excessively, its properties are often poorly exploited and not well understood.
In an attempt to control the behavior of fullerene, and find better expressions of its properties, we incorporated it into polymer chains. This was done by using it as a comonomer to create new alternating chains.
Two methods in particular were discovered: the so-called SACAP1212 Stephen M, Ramanitra HH, Santos Silva H, Dowland S, Bégué D, Genevičius K, et al. Sterically controlled azomethine ylide cycloaddition polymerization of phenyl-C61-butyric acid methyl ester. Chem Commun (Camb). 2016;52(36):6107-10.,1313 Ramanitra HH, Santos Silva H, Bregadiolli BA, Khoukh A, Combe CMS, Dowland SA, et al. Synthesis of main-chain poly(fullerene)s from a sterically controlled azomethine ylide cycloaddition polymerization. Macromolecules. 2016;49(5):1681-91. and the ATRAP1414 Hiorns RC, Cloutet E, Ibarboure E, Vignau L, Lemaitre N, Guillerez S, et al. Main-chain fullerene polymers for photovoltaic devices. Macromolecules. 2009;42(10), 3549-58.,1515 Santos Silva H, Ramanitra HH, Bregadiolli BA, Bégué D, Graeff CFO, Dagron-Lartigau C, et al. Oligo- and poly(fullerene)s for photovoltaic applications: modeled electronic behaviors and synthesis. J Polym Sci A Polym Chem. 2017;55(8):1345-55. routes. The former gives thermally robust, long polymer chains with fullerenes linked together by strong cyclic groups, while the latter gives rise to more weakly linked and quite short oligofullerenes tied together with weak methylene links1616 Silva HS, Ramanitra HH, Bregadiolli BA, Tournebize A, Bégué D, Dowland SA, et al. In situ generation of fullerene from a poly(fullerene). J Polym Sci, B, Polym Phys. 2019;57(21):1434-52..These new materials as such remain very poorly understood. No work yet has considered in detail their macromolecular self-assembly, and how it affects their electronic behavior. Therefore we felt it important to look at them in better detail and compare their behavior with that of a well understood material, phenyl-C61-butyric acid methyl ester (PCBM).
To this end, in this work, the effect of two different solvents, namely chloroform and xylene, on the morphology and electrical properties of a series of fullerenes, PCBM, oligo{(phenyl-C61-butyric acid methyl ester)-alt-[1,4-bis(bromomethyl)-2,5-bis(octyloxy)benzene]} (OPCBMMB) made by the ATRAP route, and poly{[bispyrrolidino(phenyl-C61-butyric acid methyl ester)]-alt-[2,5-bis(octyloxy) benzene]} (PPCBMB) made by the SACAP route, mixed with P3HT were studied. The electrical properties of the developed polymer/fullerene mixtures were investigated by recording the current versus voltage variation in the dark and under illumination.
# 2. Experimental Setup
For the preparation of Langmuir-Schaefer (LS) films, a Langmuir trough KSV model 5000 was used, where approximately 1350 mL of ultrapure water, from the Millipore water purification system with resistivity of 18.2 MΩ.cm was used. The solvents used in this work were chloroform and xylene, the solutions were fabricated in a pure and mixed materials form in a mass ratio of 1:1 with a concentration of 0.2 mg mL-1. Langmuir films were compressed at 10 mm min-1 to obtain isotherms of surface pressure versus average molecular area (π-A). 500μL of the solution was spread with the Poly-fullerenes and after the previous analysis, a surface pressure of 20 mN.m-1 was chosen to deposit these films on solid substrates with 15 layers, in which the film was transferred in parallel to the air-water interface, the so-called the LS film.
Current versus voltage (I vs. V) measurements were performed on the films to characterize the sample when applying a direct current. These electrical measurements were carried out using a Keysight voltage source model B2901A. Thus, pure and mixed films were deposited onto interdigitated electrodes (IDEs) and subjected to voltages ranging from -10 V to 10 V, with 0.5 V steps. The electrical characterization of the films was done in two situations: in the dark and under light exposure. For this, an Oriel Vera Sol LSS-7120 Solar Simulator (100 mW cm-2 - AM 1.5G) was used.
# 3. Materials and Methods
Figure 1 shows the chemical structures of the materials used in this work. PCBM was obtained from Merck, OPCBMMB was prepared as detailed in reference1616 Silva HS, Ramanitra HH, Bregadiolli BA, Tournebize A, Bégué D, Dowland SA, et al. In situ generation of fullerene from a poly(fullerene). J Polym Sci, B, Polym Phys. 2019;57(21):1434-52. using the ATRAP methodology, and PPCBMB was prepared as indicated in reference1717 Ramanitra HH, Dowland SA, Bregadiolli BA, Salvador M, Santos Silva H, Bégué D, et al. Increased thermal stabilization of polymer photovoltaic cells with oligomeric PCBM. J Mater Chem C Mater Opt Electron Devices. 2016;4(34):8121-9. using the SACAP technique. The chemical structure of P3HT is shown in Figure 2. It was obtained from Sigma-Aldrich. Here we use the regioregular form which has a high degree of head-tail (HT) connections and a regularity greater than 90%.
Figure 1
The modified fullerenes used in this study: a) PCBM, b) PPCBMB and c) OPCBMMB.
Figure 2
Chemical structure of P3HT.
These materials were dissolved in two solvents, chloroform or xylene, both obtained from Sigma-Aldrich, and chosen for their known impact on the solvation and aggregating properties of fullerene1818 Ruoff RS, Tse DS, Malhotra R, Lorents DC. Solubility of fullerene (C60) in a variety of solvents. J Phys Chem. 1993;97(13):3379-83.,1919 Bensasson RV, Bienvenue E, Dellinger M, Leach S, Seta P. C60 in model biological systems. a visible-UV absorption study of solvent-dependent parameters and solute aggregation. J Phys Chem. 1994;98(13):3492-500. http://dx.doi.org/10.1021/j100064a035.
http://dx.doi.org/10.1021/j100064a035...
.
## 3.1. Substrate
The substrate used to perform electrical measurement measurements was a glass slide containing Interdigitated gold electrodes (IDEs). These substrates were produced at the Microfabrication and Thin Films Laboratory (LMF) of the National Nanotechnology Laboratory (LNNano) at National Research Center for Energy and Materials (CNPEM-Brazil) with 25 pairs of digits, 100 nm height, 8 mm length, 100 μm width and 100 μm separation.
# 4. Results and Discussion
For direct current measurements, pure and mixed films were transferred onto the IDE substrates using the LS technique and electrical measurements were taken in two different situations: in the dark and under incidence of light. The samples were attached to a support and irradiated. These polymer/fullerene composites are characterized by sensitivity to light and they are known for converting solar energy2020 Chen F-C, Ko C-J, Wu J-L, Chen W-C. Morphological study of P3HT:PCBM blend films prepared through solvent annealing for solar cell applications. Sol Energy Mater Sol Cells. 2010;94(12):2426-30..
The I vs. V curves of the presented materials exhibited a linear behavior due to the Au/film/Au configuration, which generates an ohmic contact on both interfaces. An ohmic contact, or neutral contact, has the characteristic of not influencing the density of carriers on the volume of a studied material when an electric current is applied2121 Sze SM, Ng KK. Physics of semiconductor devices. Hoboken: John Wiley & Sons, Inc.; 2006.,2222 Tomozawa H, Braun D, Phillips SD, Worland R, Heeger AJ, Kroemer H. Metal-polymer Schottky barriers on processible polymers. Synth Met. 1989;28(1-2):687-90.. This feature makes it possible to obtain some information regarding the properties of materials, such as conductivity2323 Chiang CK, Fincher CR, Park YW, Heeger J, Shirakawa H, Louis EJ, et al. Electrical conductivity in doped polyacetylene. Phys Rev Lett. 1977;39(17):1098-101..
With Ohm's Law Equations $V=R.I$ were V is voltage, R resistance and I current, and $σ= 1R.1A$ , the direct current conductivity (σdc) of the LS films of pure and mixed fullerenes can be calculated. Thus, through the slope of the graphs (1/R) and the geometric parameters of the IDE, which in this study is the cell constant (κ), the σdc values were obtained as shown in the work of Roncaselli et al.2424 Roncaselli LKM, Silva EA, Braunger ML, Souza NC, Ferreira M, de Santana H, et al. Regioregularity and deposition effect on the physical/chemical properties of polythiophene derivatives films. Nanotechnology. 2019;30(32):325703.. The cell constant κ was calculated considering the height of the digits, spacing between them, their number and length using a theoretical model2525 Olthuis W, Streekstra W, Bergveld P. Theoretical and experimental determination of cell constants of planar-interdigitated electrolyte conductivity sensors. Sens Actuators B Chem. 1995;24(1-3):252-6., and for the IDE used in this work this value is 5.1 m− 1. As the films were manufactured in two different solvents, the interpretation was divided into two categories a) xylene and b) chloroform.
## 4.1. Xylene
Figure 3 shows the I vs V curves of pure and mixed materials, submitted to light and dark. In the analysis of films in the dark, the pure materials showed similar conductivities, however the PPCBMB material presented a conductivity in an order of magnitude greater, as indicated in Table 1. The PCBM and OPCBMMB materials have similar conductivity values.
Figure 3
I vs V curves for pure and mixed films with poly(fullerene)s and P3HT, produced by the LS technique solubilized with xylene.
Table 1
Conductivity values for full and mixed pure and mixed poly(fullerene)s solubilized with xylene.
For mixed materials evaluated in the dark, the material OPCBMMB:P3HT expressed a greater conductivity by 1 order of magnitude in relation to the other materials under study. A similarity was found in the results obtained with the PCBM:P3HT and PPCBMB:P3HT materials indicated in Table 1.
However, when we compare the curves of pure and mixed materials in the dark (Figure 3), we notice a clear difference in the conductivity values. The addition of a conductive polymer P3HT can contribute to the increase in conductivity, according to Table 1. It is suggested that this increase occurs due to the inter-chain and intra-chain processes, which cause the charges to be conducted through the main polymer chains, as well as between the chains of the P3HT, by the phenomenon of electronic hopping2626 Qu S, Yao Q, Shi W, Wang L, Chen L. The influence of molecular configuration on the thermoelectrical properties of poly(3-hexylthiophene). J Electron Mater. 2016;45(3):1389-96.,2727 Hutchison GR, Ratner MA, Marks TJ. Hopping transport in conductive heterocyclic oligomers: reorganization energies and substituent effects. J Am Chem Soc. 2005;127(7):2339-50..
However, when the pure materials were exposed to light, we noticed a small difference in conductivity. The PCBM showed greater conductivity between the three materials, by about 1 order of magnitude. PPCBMB and OPCBMMB had similar conductivities.
In the case of mixed materials exposed to light, the PPCBMB:P3HT among the three materials in question, exhibited a significant improvement in its conductivity, being 1 order of magnitude. It is also possible to observe that there is a similarity in the results obtained for the materials PCBM:P3HT and OPCBMMB:P3HT.
When comparing pure and mixed films subjected to light, it is possible to notice a clear difference between the conductivities. Mixed materials showed a significant increase, with PCBM:P3HT 4 orders, PPCBMB:P3HT 6 orders and, OPCBMMB:P3HT 5 orders of magnitude.
In the analysis of pure and mixed materials under light and dark conditions, changes in conductivity were verified for both materials. For the pure materials PPCBMB and OPCBMMB, there was a decrease in conductivity by 2 order of magnitude, under light, while for the PCBM there were no changes in conductivity. On the other hand, in the analysis of mixed materials PCBM:P3HT and PPCBMB:P3HT, there was an increase in conductivity by 1 and 2 orders of magnitude, respectively. For the OPCBMMB:P3HT there were no differences.
This can be explained by the fact that these materials have PCBM in common in their structure and as observed in the work of Chirvase et al.2828 Chirvase D, Parisi J, Hummelen JC, Dyakonov V. Influence of nanomorphology on the photovoltaic action of polymer-fullerene composites. Nanotechnology. 2004;15(9):1317-23., PCBM can have two configurations that contribute to low conductivity: when it does not allow the formation of the necessary paths for the electron transport or when it forms bulky clusters harming the metal/fullerene interface.
In pure films, it is possible to observe a decreasing in the conductivity when the active layer is exposed to light. This drop in conductivity is known as “negative photoconduction”. This effect could be explained by the creation of “capturing” species of free carriers, or excitonic defects of the material when the active layer of the thin film is exposed to light, decreasing the mobility of the carriers2929 Mergulhão S, Faria RM, Ferreira GFL, Sworakowski J. Transport of holes in uniformly and non-uniformly protonated poly(o-methoxyaniline). Chem Phys Lett. 1997;269(5-6):489-93.. These defects have a relatively long lifespan, in the order of tenths milliseconds, and some studies have already reported that excitonic centers with lifetime from minutes to hours3030 McCall RP, Roe MG, Ginder JM, Kusumoto T, Epstein AJ, Asturias GE, et al. IR absorption, photoinduced IR absorption, and photoconductivity of polyaniline. Synth Met. 1989;29(1):433-8.,3131 McCall RP, Ginder JM, Roe MG, Asturias GE, Scherr EM, MacDiarmid AG, et al. Massive polarons in large-energy-gap polymers. Phys Rev B Condens Matter. 1989;39(14):10174-8..
Conductivity can be determined by the free path of carriers and by their location time. The decay of conductivity shown in Table 1 can be explained by assuming that the absorption of light causes the production of excitonic defects, breaking the conjugation and decreasing the free path of carriers and or modifying the location time2929 Mergulhão S, Faria RM, Ferreira GFL, Sworakowski J. Transport of holes in uniformly and non-uniformly protonated poly(o-methoxyaniline). Chem Phys Lett. 1997;269(5-6):489-93..
When the photons fall on the mixed film, they are absorbed and excite the donor (P3HT), producing excitons in the conjugated polymer. Excitons are diffused in the donor phase, enabling phenomena of recombination and dissociation. So, for mixed films it is possible to observe an increasing of conductivity Figure 4 due to exciton dissociation through the interfaces (donor/acceptors)
Figure 4
Energy diagram between electron donor and acceptor containing the processes of: generation, dissociation and transport.
When excitons encounter the acceptor (PCBM) there is a fast dissociation, generating electrons and “free” holes that contribute to the photocurrent3232 Yu G, Gao J, Hummelen JC, Wudl F, Heeger AJ. Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions. Science. 1995;270(5243):1789-91.. Charge generation occurs at the interfaces between the two materials in the active layer, therefore increasing the conductivity of these materials exposed to light3333 Blom PWM, Mihailetchi VD, Koster LJA, Markov DE. Device physics of polymer:Fullerene bulk heterojunction solar cells. Adv Mater. 2007;19(12):1551-66., as shown in Figure 3.
When compared to pure films exposed to light, the generated excitons are not sufficient for the generation of free transporters, as the interface has only the acceptor PCBM, and with this, the charge generation drops drastically and the conductivity decreases to approximately 5 orders of magnitude as shown in Table 1.
## 4.2. Chloroform
As stated in the introduction, it is well known that when the solvents are varied, there is a change in the morphology and consequently a change in the organization of the morphology of the material, and this impacts upon their conductivity.
Figure 5 shows the I vs V curves of pure and mixed films, under light and in dark conditions. When considering the films in the dark, we find that the pure films did not present differences in their conductivity. For mixed materials evaluated in the dark, PCBM:P3HT and OPCBMMB:P3HT, they showed similar conductivities, and were superior to PPCBMB:P3HT in 1 order of magnitude according to Table 2.
Figure 5
I vs V curves for pure and mixed films with poly(fullerene)s and P3HT, produced by the LS technique solubilized with chloroform.
Table 2
Conductivity values for full and mixed pure and mixed fullerenes solubilized with chloroform.
When we compare the conductivities of pure and mixed materials in the dark, we see an increase of up to 3 orders of magnitude. As previously discussed, the addition of a conductive polymer results in this increase.
In the analysis of pure films exposed to light, the conductivities showed no differences between them. The same was observed for mixed materials. However, when comparing pure and mixed films exposed to light, an increase of 5 orders of magnitude in conductivity was observed. In the analysis of pure and mixed materials under light and dark conditions, only mixed materials showed changes in their conductivity when exposed to light, in which, the PCBM:P3HT and OPCBMMB:P3HT materials had an increase of 2 orders of magnitude. While for the PPCBMB:P3HT, we had an increase of 3 orders.
Note that the materials PCBM and OPCBMMB have similar characteristics, which is observed in the work Ramanitra et al.1717 Ramanitra HH, Dowland SA, Bregadiolli BA, Salvador M, Santos Silva H, Bégué D, et al. Increased thermal stabilization of polymer photovoltaic cells with oligomeric PCBM. J Mater Chem C Mater Opt Electron Devices. 2016;4(34):8121-9. showing that the UV-visible of the OPCBMMB is very similar to the PCBM, indicating that in the mixture between acceptor/donor the charge transfer properties were not strongly affected by the oligomerization process.
In the comparison between the two solvents studied xylene and chloroform (Tables 1 and 2), under light and dark conditions, we found some changes in the conductivity behavior of the materials.
When studying pure films in the dark, the PCBM and OPCBMMB materials did not express differences in their conductivity. In contrast, the PPCBMB obtained an increase of 1 order of magnitude when using the xylene solvent. For mixed films in the dark, the PCBM: P3HT material showed no difference in its conductivity. However, PPCBMB: P3HT and OPCBMMB: P3HT, showed an increase of 1 order of magnitude when the xylene solvent was applied.
In the light analysis, the pure PCBM film showed no difference. The PPCBMB and OPCBMMB materials, on the other hand, showed an increase of 1 order of magnitude when using the chloroform solvent. In mixed films, the PPCBMB: P3HT material maintained the same conductivity. However, PCBM: P3HT and OPCBMMB: P3HT exhibited an increase of 1 order of magnitude when the chloroform solvent was used.
There is a difference in the conductivity of these materials when manufactured in pure and mixed form with xylene and chloroform, however, the PPCBMB showed differences on both occasions and can be explained by the intersection of two factors as shown in the work of Liao et al.3434 Liao H-C, Tsao C-S, Huang Y-C, Jao M-H, Tien K-Y, Chuang C-M, et al. Insights into solvent vapor annealing on the performance of bulk heterojunction solar cells by a quantitative nanomorphology study. RSC Advances. 2014;4(12):6246.: i) The solubility of P3HT would lead to different conformations of the chain and therefore a crystallinity with different sizes. The high solubility of P3HT with chloroform causes its chains to “stretch”, thus providing greater mobility for charge carriers. While for xylene, its solubility is lower, it would lead to the contraction of the chains, resulting in less mobility for charge carriers. ii) a reduced aggregation of molecules in the formation of films, thus affecting the conductivity of the material.
In a general context, pure PCBM based-films have lower conductivities compared to the mixed films, this is due to the fact that PCBM has a low electron mobility and a high electronic band gap, thus affecting its conductivity3333 Blom PWM, Mihailetchi VD, Koster LJA, Markov DE. Device physics of polymer:Fullerene bulk heterojunction solar cells. Adv Mater. 2007;19(12):1551-66.,3535 Schafferhans J, Baumann A, Wagenpfahl A, Deibel C, Dyakonov V. Oxygen doping of P3HT:PCBM blends: Influence on trap states, charge carrier mobility and solar cell performance. Org Electron Physics Mater. Appl. 2010;11(10):1693-700.. On the other hand, mixed films showed higher conductivities due to the presence of P3HT, which has a high electron mobility and a low electronic band gap3636 Glatthaar M, Riede M, Keegan N, Sylvester-Hvid KO, Zimmermann B, Niggemann M, et al. Efficiency limiting factors of organic bulk heterojunction solar cells identified by electrical impedance spectroscopy. Sol Energy Mater Sol Cells. 2007;91(5):390-3..
In this work we show the importance on how understanding these processes and how the conductivity behaves in pure and mixed form and using different solvents helps to understand which is the best combination of solvent and material for building a solar cell. Furthermore, we provided an insight about how the solvent could enhance the solar cell performance by increasing photocurrent via facilitating charge carrier transport through creating percolation pathways for carriers, leading to higher efficiency. There are not many studies about LS films from fullerenes in pure form in the literature and, even less that discuss the conductivity of such films. Therefore, this work offers a playground for the direct relation of the solvent influence on the conductivity of LS films from PCBM derivatives, as well as the outcome from their association with a known material from the OPV field, the P3HT.
# 5. Conclusion
Regarding the electrical measurements, it was possible to observe differences between the conductivities of each material, when subjected to dark and light. We perceive an influence of P3HT considerably increasing the conductivity of these materials. Regarding the use of solvents, an influence on the conductivity of the materials under study was observed. For the xylene solvent, it was verified that the PPCBM material in pure and mixed form, in the dark condition, presented the best conductivity results in comparison to other materials. In the chloroform solvent, the OPCBMMB material in pure and mixed form, under the condition of light, exhibited the best conductivities in relation to the other materials. Through this work it was possible to successfully manufacture pure and mixed fullerene films using the LS technique, comparing materials in pure and mixed form through electrical measurements (D.C).
# 6. Acknowledgments
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
FAPESP; INEO-CNPq; LNNano (LMF-CNPEM)
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Hutchison GR, Ratner MA, Marks TJ. Hopping transport in conductive heterocyclic oligomers: reorganization energies and substituent effects. J Am Chem Soc. 2005;127(7):2339-50.
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Chirvase D, Parisi J, Hummelen JC, Dyakonov V. Influence of nanomorphology on the photovoltaic action of polymer-fullerene composites. Nanotechnology. 2004;15(9):1317-23.
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Mergulhão S, Faria RM, Ferreira GFL, Sworakowski J. Transport of holes in uniformly and non-uniformly protonated poly(o-methoxyaniline). Chem Phys Lett. 1997;269(5-6):489-93.
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• 35
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• 36
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# Publication Dates
• Publication in this collection
12 July 2021
• Date of issue
2021
# History
• Received
12 Jan 2021
• Reviewed
01 Apr 2021
• Accepted
16 June 2021
ABM, ABC, ABPol UFSCar - Dep. de Engenharia de Materiais, Rod. Washington Luiz, km 235, 13565-905 - São Carlos - SP- Brasil. Tel (55 16) 3351-9487 - São Carlos - SP - Brazil
E-mail: pessan@ufscar.br | 2022-10-03 23:43:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "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": 0.6727882027626038, "perplexity": 12915.35965433314}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "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-2022-40/segments/1664030337446.8/warc/CC-MAIN-20221003231906-20221004021906-00203.warc.gz"} |
https://asantone.github.io/didgeridata/ | # Introduction
This activity guides groups of students through a brief study of the history and construction of either a didgeridoo, the world’s oldest wind instrument, or a paixiao, a chinese pan flute. Students will work with polyvinyl chloride (PVC) pipe and PVC-cutting tools to design and create a playable musical instrument which will be used by students to compose a custom song related to climate data such as atmospheric CO$$_{2}$$ concentration or global temperature anomalies. In this way, the sonification of climate data will be accomplished with a student musical chorus. The activity will conclude with a group jam session with participants invited to provide percussion for the wind section. An attempt will be made to have expert musicians join the program via teleconference as a guest speaker to provide instruction and background information.
## 0.1 Author Statement on Activity Origins
To tell the story of how this lesson came into existence, I need to mention that it was built from prior working relationships and my interests in music and data visualization. My background is in science, interactive computer graphics, 3D media, video, and K-12 STEAM outreach. I met some interesting people in my career and this lesson brings some of that experience and exposure together in what I hope is an interesting, fun way.
This activity began as an idea around the theme of air and is based in part on a past professional collaboration I had with a local musician and contemporary flamenco band. I worked in a role combining teacher professional development and creative curriculum development which included some work on documentary-style videos. Through that experience I became familiar with the music of this local band and specifically the use of the didgeridoo by one of the band members. The sound was really interesting to me and I knew something of the relationship of didgeridoo morphology and sound quality from a video I produced with this band member. That’s one inspiration: didgeridoo music! We later decided to add in the paixiao, a type of pan flute, to help boost engagement and regional connections. The paixiao is also a relatively simple instrument that can be constructed from PVC and inexpensive materials. Together we thought these could yield some very interesting sounds!
What I wanted from this activity was hands-on construction, incorporation of data or data visualization, and musical performance. I knew that the didgeridoo was a relatively simple instrument (basically a tube) and that it could be easily and inexpensively constructed from PVC. I also knew that NASA produced high-quality climate data and worked to make this information available to the public. I thought perhaps students could look at the data or even just the visualizations of data to get some ideas about trends in climate. Perhaps they could study carbon dioxide or temperature levels. That’s a second inspiration: climate data visualization. This is something that could help students with scientific literacy but also to see themselves in the world by examining large, important trends.
A third inspiration for this lesson was the concept of data sonification. A former colleague and friend is a geologist and musician who is interested in transforming data into sound. This concept is similar to visualization, but for hearing instead of seeing. I thought maybe the students could take what they know about climate data and make a song or something similar from that knowledge.
After meeting with a small group to “play-test” the activity, we decided it was important to have the student build some kind of simple notation system for their instruments. There would be no pressure to make it “right” but rather to have students be creative and write something that makes sense for their team and perhaps others to read and use to play a song. This is a good addition because it focuses a bit on literacy, logic, and music at the same time.
The big concept for me is Data->Inspiration->Construction->Notation->Performance. I am hopeful students will look at their world in a fresh way, become inspired to tell a story through music, and then bring this concept to life through the use of their own hands and minds. If all goes well, the audience will learn something about climate and the students will learn something about themselves. | 2021-09-25 12:24:32 | {"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": 1, "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": 0.18539157509803772, "perplexity": 1423.6472326943683}, "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-2021-39/segments/1631780057622.15/warc/CC-MAIN-20210925112158-20210925142158-00012.warc.gz"} |
http://physics.stackexchange.com/tags/electromagnetic-radiation/hot | # Tag Info
43
Refraction of light in water droplets, leading to the formation of rainbows, is not limited to the visible range. Experimental evidence, compelling due to its simplicity, is shown in the following images taken by University of College London Earth Sciences professor Dominic Fortes. Check the alignment of the rainbow with respect to the trees in each of the ...
9
engineer already answered it completely, I only want to add that the question is completely valid even if you already know that separation of wavelength occurs. The thing is, some materials are practically opaque or too much transparent (refractive index is equal to that of air and no separation occurs) in infrared and ultraviolet while transparent in the ...
8
As you say, a changing magnetic field is always associated with a changing electric field, and in fact in relativity they are finally revealed to be the same field. So at this level it cannot be said that the one field generates the other, as they are merely two aspects of the same object. But maybe you still want to look at it from the perspective of ...
6
This plane polarized wave from wikipedia may help Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. Note that the electric and magnetic fields in such a wave are in-phase with each other, ...
5
Is it possible that rainbows have ultraviolet bands and infra red bands and we are not able to see? Yes, see engineer's answer. As for whether we can see them, take a look at aphakia: "Aphakic people are reported to be able to see ultraviolet wavelengths (400–300 nm) that are normally excluded by the lens. They perceive this light as whitish blue or whitish ...
3
Classically (since rob has done a thorough job on the quantum picture), the amplitude of a light wave is not related to any physical extent. It is not the size of the wave in space, it is the strength of the fields (electric and magnetic). We often draw wavy lines, but if you look closely the transverse axes will be label differently for, say, waves on a ...
3
I used to make X-ray tubes for a living... and the "right" answer to this question would run the length of a book. So just a few pointers. I don't expect that you would be able to create an electron tube after this - at least not one that lasts. Note also that if you do get it to work, it will produce dangerous (X ray) radiation. And unless you understand ...
3
The water droplets that create a rainbow are not emitting the light that you see in a rainbow; if they were, you would see a glowing cloud of consistent color, not a rainbow. The rainbow is formed by sunlight refracting and reflecting through water droplets in the air; the water refracts through the "front" of the drop, reflects off the "back," and refracts ...
3
"a changing magnetic field is not generated by a changing electric field, but instead just happens to always be present perpendicular to a changing electric field due to the laws of electromagnetism." So ... it is due to but not caused by. What is the difference? Short answer: it is not only "a thing" it is a correct thing. This is much more clearly ...
2
Maxwell's equations in vacuum are: $$\nabla\cdot\mathbf{E} = 0$$ $$\nabla\cdot\mathbf{B} = 0$$ $$\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}$$ $$\nabla\times\mathbf{B} = \frac{1}{c^2}\frac{\partial\mathbf{E}}{\partial t}$$ It's the last two of these that give rise to the interpretation that a changing magnetic field generates an electric ...
2
there are other parameters like the number of free electrons in the atoms of the material, atomic size etc. Close. While density of particles does matter, it also depends on the material property. More precisely, it is closely related to how the electrons react when situated under electromagnetic oscillation. Each bound electrons has its natural ...
2
Coherency of light in practice is not an either/or issue. Any light due to any source has some degree of coherence. Laser light has usually much higher coherence than light of a hot metal filament. Some degree of coherence means, in simple wording, that light waves at one point of space due to different parts of the source behave similarly (they have ...
2
Note that $e^{jx} - e^{-jx} = 2j \sin(x)$ So what you have written is not an electromagnetic wave at all. It is an electric field with a fixed direction and an amplitude that varies sinusoidally along the z-axis. Of course if you multiply this by $e^{j\omega t}$, then you do have a wave. Given the wording I suspect you are meant to assume this (though I ...
2
Has this problem been solved since? Not in the sense Feynman meant. Approximate way to describe action of one charged part of body on another is known since Lorentz - the so-called Lorentz-Abraham-Dirac term. What Feynman is getting at is this term works somewhat, but leads to contradictions when pushed to its consequences. The problem of self-action ...
2
If you twisted my arm and forced me to assign an amplitude to a single photon, I'd do it this way: The energy density of a classical electromagnetic field is \begin{align} U &= \frac12 \left( \epsilon_0 E^2 + \frac1{\mu_0} B^2 \right) \\ &= \epsilon_0 E^2 &\text{(only for light in a vacuum)} \end{align} where $E,B$ are the amplitudes of the ...
2
In the context of ion beams, space charge is the tendency of the beam to expand transversely (perpendicular to the direction of the beam's travel) due to the mutual repulsion of the ions in the beam. All the ions have the same sign charge, so they repel. The name "space charge" comes from plasma physics where is is often computationally easier to treat the ...
2
It's tempting to think of photoionisation as the photon coming in like a billiard ball and knocking out an electron. However this is a very misleading representation of the process. A gamma ray is poorly modeled as a photon or photon(s) because the energy in it is delocalised. If you wanted to use a photon description you'd have to treat the ray as a ...
2
There are three factors that need to be considered across all wavelengths: (1) the ability of the water droplet to refract and disperse the incoming light, (2) the ability of the eye to sense the wavelength, and (3) the ability of air to transmit it. The visible range we 'see' in a rainbow with our eyes satisfies all three. UV , depending on how short the ...
1
Firstly, I would like to say that there is no particular terminal separation between negative charges and positive charges. Actually you will understand it better if I would clarify in this way that scientists first saw that having even follow the same statistical distribution i.e. Fermi Dirac distribution some of them actually repel others and some do ...
1
you can draw feynman digrams and then calculate scattering amplitudes and it is in the non relativistic limit is proportinal to potential.so if the potential is positive it means they repel. this sort of claculation is done in peskin book and A.Zee book.in peskin book page no 125. this is the most rigorous work to prove gravity is always attractive. by ...
1
Because observations made by physicists have found that this is what nature does.
1
No. As has been said, the raindrop is not emitting the light, it is just acting as an optical device that deflects light emitted by the sun. However, the spectral lines you would expect to see in sunlight refracted by a prism will not, repeat NOT, be seen. The mechanism that produces rainbows is very different than the mechanism that produces a spectrum ...
1
The answer to As an electron drops from a higher energy level to a lower energy level, can it be modeled as a the continuous movement of a charged body, therefore causing a magnetic field to be generated around it? is "Yes, but only trivially." That is, you could probably work backwards from the far-field radiation to some imagined moving source ...
1
It's not really worthwhile in this type of situation. (It makes sense in other situations however ... like transferring power from the ground to an airplane or satellite.) The two most plausible system types are: (A) Microwaves / radiowaves: Emitted by an antenna, collected by a rectenna (B) Visible / infrared: Emitted by a laser, collected by a ...
1
Do the electric and magnetic components of an electromagnetic wave really generate each other? No they don't. Like Andrea said, they're two "aspects" of the same thing. And like you said, it's an electromagnetic wave. See the wiki article for electromagnetic radiation where you can read that "the curl operator on one side of these equations results in ...
1
is my interpretation of the dynamics of the self-force correct and is there a physical or intuitive explanation for this extremely pathological behavior in the presence of a Coulomb potential? Eliezer makes his argument based on the equation with the Lorentz-Abraham-Dirac term. This term was originally (Lorentz) devised as an approximate way to account ...
1
Electric and magnetic fields themselves are totally uncharged. They are always described as totally uncharged things. They can either be described as two uncharged fields (when treated in the more traditional formulation) or as aspects of a unified electromagnetic field. In both descriptions the field(s) interact with charged things without being charged ...
1
The reason we see an interference pattern on a screen is because of diffuse reflection. This is because in diffuse reflection, the incident light can be considered to be absorbed and uniformly emitted out in all directions. This results in a brightness at a point proportional to the brightness of the incident light. A mirror, however, simply reflects ...
1
It looks like the answer is negative: ...
1
What I'm asking is, has someone measured, that at one moment of time the peak of magnetic component is in the same distance from source as the peak at the electric field. That should be not easy because this peaks are moving with c. You're asking the wrong question. Scientists don't measure the differences in the locations of the peaks of two signals. ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2015-05-25 09:49:59 | {"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": 1, "mathjax_display_tex": 1, "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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8332425951957703, "perplexity": 338.9685530241769}, "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-2015-22/segments/1432207928479.19/warc/CC-MAIN-20150521113208-00050-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/last-one-for-today-_trig.85588/ | # Last one for today-_Trig
1. Aug 18, 2005
### TonyC
The number of hours of daylight in a town on the west coast of North America can be modeled by:
h=3.75sin[2pi/365(d-79)] + 12
Where h is the number of hours of daylight in a day and d is the day of the year, with d=1 representing January 1 (assume 28 days in Feb). What is the total accumlated number of hours of daylight by a town between Mar 29 and June 29.
I worked the problem and came up with 1355.6 hours.
Am I good to go?
2. Aug 18, 2005
### LENIN
You proberlly ment the formula more like this: h=3.75sin[(2pi/365)(d-79)] + 12. In the previous form it would be undefined for the 79th day and as far as I know all days have a real number of hours. As for your resoult I can't even imagine a way to get more than 15.75 hours. You probelly just made a mistake when tiping it in your calculator. I would sugest taht you do it in steps and not all at once .
3. Aug 18, 2005
### Fermat
I don't know how you got your result but I wrote a program to work it out!
Taking March 29 as day 88 and June 29 as day 180, I got the following,
day 88 to day 180 inclusive: total hrs = 1369.77
day 89 to day 180 inclusive: total hrs = 1357.18
4. Aug 18, 2005
### Tide
You can write a general formula by summing the sequence of sines. Use Euler's formula for the trig functions and sum it as geometric series.
5. Aug 18, 2005
### AKG
The hours of sunlight from day n to day N inclusive is:
$$12(N - n + 1) + 3.75\sum _{d = n} ^{d = N}\sin \left (\frac{2\pi (d - 79)}{365}\right ) = 12(N - n + 1) + 3.75\sum _{d = n - 79} ^{N - 79}\sin \left (\frac{2\pi d}{365}\right )$$
6. Aug 18, 2005
### Tide
This may help:
$$\sum_{n = a}^{b} \sin nx = \frac { \cos x(a-1/2) - \cos x(b+1/2)} {2 \sin x/2}$$
(Edited: YIKES! Sorry, akg! I went dyslexic when I typed it and interchanged the x with the a and b!)
Last edited: Aug 18, 2005
7. Aug 18, 2005
### AKG
Wait a minute, what if a = b = 1?
sin(x) = [cos(x - 1/2) - cos(x + 1/2)]/2sin(x/2)
= [cos(x)cos(1/2) + sin(x)sin(1/2) - cos(x)cos(1/2) + sin(x)sin(1/2)]/2sin(x/2)
= sin(x)sin(1/2)/sin(x/2)
sin(x/2) = sin(1/2)
That doesn't seem right. Where did you get that formula from?
8. Aug 19, 2005
### Fermat
It seems right now.
I used tide's sum of sines formula in akg's expression for hours of sunlight and got the same results as my computer. | 2017-02-20 06:54:50 | {"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": 1, "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": 0.6247939467430115, "perplexity": 2597.19172058504}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "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-2017-09/segments/1487501170425.26/warc/CC-MAIN-20170219104610-00360-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/question-about-log-laws.227594/ | 1. Apr 8, 2008
### escryan
I don't know how I managed to forget this one, but I did somehow...
If there's something like:
e^lnx, why is that equal to just x?
and same goes for sokmething like:
8^log8x which is just equal to x.
I'm just wondering how, algebraically, one could show this to be true.
2. Apr 8, 2008
### Dick
You generally DEFINE ln(x) to be the inverse function of e^x. Or vice versa depending on which you define first. So you don't show it algebraically, it largely a matter of definition.
3. Apr 8, 2008
### sutupidmath
well, by definition of log we have
$$log_a(x)=b<=> a^b=x$$
Now lets substitute $$b=log_a(x)$$ in
$$a^b=x$$ So:
$$a^{log_a(x)}=x$$
Or, since $$f(x)=a^x$$ and $$g(x)=log_ax$$ are inverse functions, so it means that they cancel each other out. That is
$$fg(x)=f(g(x))=x=>a^{log_ax}=x$$ and also
$$g(f(x))=log_a(a^x)=x$$
Edit: Dick was faster!
4. Apr 8, 2008
### Dick
You are slow because you write more. Doesn't mean you think slower. I appreciate the TeX though.
Last edited: Apr 8, 2008
5. Apr 8, 2008
### escryan
Oh I see now! Thanks so much for your help Dick and sutupidmath! | 2017-08-22 20:50:46 | {"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": 1, "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": 0.8175835013389587, "perplexity": 2575.620848157745}, "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-2017-34/segments/1502886112682.87/warc/CC-MAIN-20170822201124-20170822221124-00297.warc.gz"} |
https://dmtcs.episciences.org/5688 | ## Tianlong Ma ; Yaping Mao ; Eddie Cheng ; Christopher Melekian - Fractional matching preclusion for generalized augmented cubes
dmtcs:5074 - Discrete Mathematics & Theoretical Computer Science, August 13, 2019, vol. 21 no. 4 - https://doi.org/10.23638/DMTCS-21-4-6
Fractional matching preclusion for generalized augmented cubes
Authors: Tianlong Ma ; Yaping Mao ; Eddie Cheng ; Christopher Melekian
The \emph{matching preclusion number} of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu recently introduced the concept of fractional matching preclusion number. The \emph{fractional matching preclusion number} of $G$ is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The \emph{fractional strong matching preclusion number} of $G$ is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the fractional matching preclusion number and the fractional strong matching preclusion number for generalized augmented cubes. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.
Volume: vol. 21 no. 4
Section: Distributed Computing and Networking
Published on: August 13, 2019
Accepted on: August 13, 2019
Submitted on: January 11, 2019
Keywords: Mathematics - Combinatorics | 2022-11-27 12:33:39 | {"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": 1, "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": 0.3771800994873047, "perplexity": 2040.1509966628632}, "config": {"markdown_headings": true, "markdown_code": false, "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-2022-49/segments/1669446710237.57/warc/CC-MAIN-20221127105736-20221127135736-00576.warc.gz"} |
https://optimization-online.org/2017/09/ | ## Compact Representation of Near-Optimal Integer Programming Solutions
It is often useful in practice to explore near-optimal solutions of an integer programming problem. We show how all solutions within a given tolerance of the optimal value can be efficiently and compactly represented in a weighted decision diagram, once the optimal value is known. The structure of a decision diagram facilitates rapid processing of … Read more
## Lower bounds on the lattice-free rank for packing and covering integer programs
In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of covering polyhedra. These results indicate that whenever the integrality gap is high, … Read more
## Time inconsistency of optimal policies of distributionally robust inventory models
In this paper, we investigate optimal policies of distributionally robust (risk averse) inventory models. We demonstrate that if the respective risk measures are not strictly monotone, then there may exist infinitely many optimal policies which are not base-stock and not time consistent. This is in a sharp contrast with the risk neutral formulation of the … Read more
## A mixed-integer branching approach for very small formulations of disjunctive constraints
We study the existence and construction of very small formulations for disjunctive constraints in optimization problems: that is, formulations that use very few integer variables and extra constraints. To accomplish this, we present a novel mixed-integer branching formulation framework, which preserves many of the favorable algorithmic properties of a traditional mixed-integer programming formulation, including amenability … Read more
## Robust Sensitivity Analysis for Linear Programming with Ellipsoidal Perturbation
Given an originally robust optimal decision and allowing perturbation parameters of the linear programming problem to run through a maximum uncertainty set controlled by a variable of perturbation radius, we do robust sensitivity analysis for the robust linear programming problem in two scenarios. One is to keep the original decision still robust optimal, the other … Read more
## Optimal Linearized Alternating Direction Method of Multipliers for Convex Programming
The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention from many areas because of its efficiency and easy implementation. To theoretically guarantee … Read more
## Large-scale packing of ellipsoids
The problem of packing ellipsoids in the n-dimensional space is considered in the present work. The proposed approach combines heuristic techniques with the resolution of recently introduced nonlinear programming models in order to construct solutions with a large number of ellipsoids. Numerical experiments illustrate that the introduced approach delivers good quality solutions with a computational … Read more
## An Inexact Regularized Newton Framework with a Worst-Case Iteration Complexity of $\mathcal{O}(\epsilon^{-3/2})$ for Nonconvex Optimization
An algorithm for solving smooth nonconvex optimization problems is proposed that, in the worst-case, takes $\mathcal{O}(\epsilon^{-3/2})$ iterations to drive the norm of the gradient of the objective function below a prescribed positive real number $\epsilon$ and can take $\mathcal{O}(\epsilon^{-3})$ iterations to drive the leftmost eigenvalue of the Hessian of the objective above $-\epsilon$. The proposed … Read more
## Linearized version of the generalized alternating direction method of multipliers for three-block separable convex minimization problem
Recently, the generalized alternating direction method of multipliers (GADMM) proposed by Eckstein and Bertsekas has received wide attention, especially with respect to numerous applications. In this paper, we develop a new linearized version of generalized alternating direction method of multipliers (L-GADMM) for the linearly constrained separable convex programming whose objective functions are the sum of … Read more
## Robust combinatorial optimization with knapsack uncertainty
We study in this paper min max robust combinatorial optimization problems for an uncertainty polytope that is defined by knapsack constraints, either in the space of the optimization variables or in an extended space. We provide exact and approximation algorithms that extend the iterative algorithms proposed by Bertismas and Sim (2003). We also study the … Read more | 2023-01-29 07:08:22 | {"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": 1, "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": 0.6972963213920593, "perplexity": 545.7582762548375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "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-2023-06/segments/1674764499700.67/warc/CC-MAIN-20230129044527-20230129074527-00209.warc.gz"} |
https://math.stackexchange.com/questions/3147333/recurrence-relation-for-the-number-of-strings-of-length-n-over-the-alphabet | # Recurrence relation for the number of strings of length $n$ over the alphabet $\{1, 2,3,4,5,6,7\}$ such that there are no consecutive $1$'s or $2$'s.
Find a recurrence relation for the number of strings of length $$n$$ over the alphabet $$\{1, 2,3,4,5,6,7\}$$ such that there are no consecutive $$1$$'s or $$2$$'s.
I have no idea where to start. I've been stuck for some time. Any help is appreciated, Thanks.
• How can there be consecutive $0$s if there are no $0$s. – fleablood Mar 13 '19 at 23:21
• oops, changed $0$'s to $2$'s – Adi Mar 13 '19 at 23:23
Make coupled recurrences, one for the number of good strings of length $$n$$ that do not end in $$1$$ or $$2$$ and one for the number of good strings that do end in $$1$$ or $$2$$. Given the number of each, how many strings of length $$n+1$$ of each type are there?
Building on Ross's hint.
Let $$a_n$$ be the number of good strings of length $$n$$
Let $$b_n$$ be the number of good strings of length $$n$$ which do not end in a $$1$$ or $$2$$
Let $$c_n$$ be the number of good strings of length $$n$$ which end in $$1$$
Let $$d_n$$ be the number of good strings of length $$n$$ which end in $$2$$
This gives the recurrence $$a_n=b_n+c_n+d_n$$
Obviously, $$b_n = 5a_{n-1}$$
Also we have $$c_n=b_{n-1}+d_{n-1}$$ because the right hand side of the equation is the number of good strings of length $$n-1$$ which do not end in a $$1$$.
Finally, $$d_n=b_{n-1}+c_{n-1}$$ because the right hand side of the equation is the number of good strings of length $$n-1$$ which do not end in a $$2$$
Substituting for $$b_n$$, $$c_n$$, $$d_n$$ in the first equation the equations we get, $$a_n = 5a_{n-1}+b_{n-1}+d_{n-1}+b_{n-1}+c_{n-1}$$ = $$5a_{n-1}+b_{n-1}+a_{n-1}$$ = $$6a_{n-1}+5a_{n-2}$$
Therefore, $$a_n=6a_{n-1}+5a_{n-2}$$
Strings of length $$n\ge 2$$ fall in two classes; those that end with a double letter, and those that do not. There are $$6a_{n-1}$$ strings which do not end with a double letter (six choices for the end letter, anything but the previous), and $$5a_{n-2}$$ which do (five choices for the double, anything but $$11$$ or $$22$$), so $$a_{n}=6a_{n-1}+5a_{n-2},\qquad n\ge2.$$
• Very neat but I think... while deleting $22$ does leave a legal string of length $n-1$, it doesn't leave all possible legal strings of length $n-1$, namely, it doesn't leave legal strings of length $n-1$ which end in $2$. A pity if I'm right though, as this is much neater than Ross's standard solution. – antkam Mar 14 '19 at 5:01
• antkam, while Ross's solution is very neat, i think you're right. – Adi Mar 14 '19 at 11:00
• @antkam Yep, I was way off! Oh well, with the hind sight of Adi's answer, I can now give a nice combinatorial proof of the recursion. – Mike Earnest Mar 14 '19 at 15:06
• Great! It often (always?) happens that a set of coupled recurrences can be reduced to only 1 recurrence of the main series. I've often wondered if the single recurrence can then be "retroactively" explained -- it's lovely that you found the way in this case. If you have "general techniques" to share, I'd love to hear them. :) – antkam Mar 14 '19 at 19:07
• @antkam No general techniques yet, though I certainly want to know them and thing they must exist. BTW, the answer to your "always?" question is yes. Any system of recurrences can be described by a vector-matrix recurrence ${\bf v}_{n+1}=A{\bf v}_n$. The characteristic polynomial of $A$ gives a univariate recurrence solved by each coordinate of ${\bf v}_n$, allowing you to decouple. – Mike Earnest Mar 14 '19 at 19:23 | 2021-02-28 09:46:26 | {"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": 1, "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": 43, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6369690895080566, "perplexity": 191.30638134534263}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2021-10/segments/1614178360745.35/warc/CC-MAIN-20210228084740-20210228114740-00513.warc.gz"} |
https://engineering.stackexchange.com/questions/556/does-a-transformer-use-power-when-output-isnt-under-load | # Does a transformer use power when output isn't under load?
I was reading about how AC to DC converters work with a step-down transformer and then a diode bridge to convert the lower, stepped down AC voltage into DC. What I don't understand is since the input AC appears to be connected to the primary coil of the transformer, how does the DC load affect the power used from the AC supply?
Does the DC load somehow feedback and lower the resistance of the primary coil so that more power can be drawn?
When there is no load on the DC side, does power still flow through the AC primary coil, and if so, why doesn't it just melt?
• May I add, that technically the DC isn't DC, it's just AC sine wave that has been prevented from going below 0V. Also, technically, the diode bridge would still be drawing some power, since there's voltage drop across it . – Sergiy Kolodyazhnyy Feb 9 '15 at 19:54
• "... the diode bridge would still be drawing some power, since there's voltage drop across it." This is incorrect. $P = VI$ and if I is zero then P is zero. – Transistor Mar 17 '18 at 12:55
Does the DC load somehow feedback and lower the resistance of the primary coil so that more power can be drawn?
Yes. It would be simpler to analyze an AC load though. The diodes are not central to your question:
The impedance of RL is also transformed, so if you have a 10:1 transformer and RL is 2 Ω, the AC source will see the transformer as a 200 Ω resistor ($10^2⋅2$)
As the current in a coil changes, it creates a changing magnetic field. In the case of a transformer with a load, however, the change in magnetic field creates a current in the secondary, which immediately creates its own changing magnetic field in the opposite direction, cancelling out the primary's field. People tend to forget that an ideal transformer has no magnetic field while operating. Any change in either coil's field is immediately cancelled by a change in the other.
The "feedback" is caused by the same effect. The primary causes the secondary to change, and the secondary causes the primary to change in return.
When there is no load on the DC side, does power still flow through the AC primary coil, and if so, why doesn't it just melt?
With nothing connected to the secondary side, the secondary coil is open circuited and does nothing. It's just some metal that happens to be nearby. The circuit is now just an AC source driving the primary coil, which behaves as a lone inductor:
Ideal inductors do not consume any power; they just store energy temporarily in one half of the cycle and return it to the supply on the other half. Real coils are not made of perfect conductors, though, and have some resistance, so the power consumed by the primary coil will be determined by the resistance of the wire.
Also, it's not quite right to say "power still flow through the AC primary coil". "Current" is flowing through the primary, and the resistance of the primary to that current causes it to "dissipate energy" (or power) into the room. "Power" is actually the rate at which energy flows, and energy actually flows through the empty space between the wires, not in the wires themselves. Once you understand this, a lot of things make much more sense.
A transformer offers resistance to AC current flow due to the magnetic field produced by the current flow. This "AC resistance" is termed "impedance" and is a function of number of turns, core material, air gaap in core , core dimensions and more.
When there is no load the applied AC voltage will cause "magnetising current" to flow. This will cause some losses due to eddy current losses in the core and copper losses due to resistance in the winding ("I squared R losses" as power = Current^2 x Resistance).
These losses are relatively small compared to full load power but not trivial at rest. A few percent of full load power would usually be good.
When a DC load is applied it loads the AC secondary circuit which is tightly couple by the core's magnetic fields to the primary winding. So the DC load resistance appears as if it is an AC impedance load on the primary side and input power increases to meet the load.
If you apply DC (rather than AC) to a transformer winding there is no ongoing magnetic field change, there is no impedance due to the varying magnetic field and current is limited by the resistance which is low compared to the impedance that should be being generated. If the DC supply has enough muscle power the transformer "just melts".
Energy delivered to the primary goes to: | 2020-10-20 08:41:38 | {"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": 1, "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": 0.42763155698776245, "perplexity": 639.354729782782}, "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-2020-45/segments/1603107871231.19/warc/CC-MAIN-20201020080044-20201020110044-00148.warc.gz"} |
https://socratic.org/questions/what-is-the-slope-of-5-5-2-6 | # What is the slope of (5,-5), (2,6)?
Ans: $- \frac{11}{3}$
Formula for finding slope. $a = \frac{y 2 - y 1}{x 2 - x 1}$
$a = \frac{6 - \left(- 5\right)}{2 - 5}$ = - $\frac{11}{3.}$ | 2022-09-26 00:00:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "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": 0.7744985222816467, "perplexity": 3915.053261618251}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "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-2022-40/segments/1664030334620.49/warc/CC-MAIN-20220925225000-20220926015000-00480.warc.gz"} |
http://umj.imath.kiev.ua/authors/name/?lang=en&author_id=946 | 2019
Том 71
№ 2
# Zorii N. V.
Articles: 18
Article (Russian)
### Extremal problems dual to the Gauss variational problem
Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 747–764
We formulate and solve extremal problems of potential theory that are dual to the Gauss variational problem but, unlike the latter, are always solvable. Statements on the compactness of classes of solutions and the continuity of extremals are also established.
Article (Russian)
### Necessary and Sufficient Conditions for the Solvability of the Gauss Variational Problem
Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 60–83
We investigate the well-known Gauss variational problem considered over classes of Radon measures associated with a system of sets in a locally compact space. Under fairly general assumptions, we obtain necessary and sufficient conditions for its solvability. As an auxiliary result, we describe potentials of vague and (or) strong limit points of minimizing sequences of measures. The results obtained are also specified for the Newton kernel in $\mathbb{R}^n$.
Article (Russian)
### Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials
Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1513-1526
The concept of consistent kernels introduced by Fuglede in 1960 is widely used in extremal problems of the theory of potential on classes of positive measures. In the present paper, we show that this concept is also efficient for the investigation of extremal problems on fairly broad classes of signed measures. In particular, for an arbitrary consistent kernel in a locally compact space, we prove a theorem on the strong completeness of fairly general subspaces E of all measures with finite energy. (Note that, according to the well-known Cartan counterexample, the entire space E is strongly incomplete even in the classical case of the Newton kernel in ℝn Using this theorem, we obtain new results for the Gauss variational problem, namely, in the non-compact case, we give a description of vague and (or) strong limiting measures of minimizing sequences and obtain sufficient solvability conditions.
Article (Russian)
### Equilibrium Problems for Potentials with External Fields
Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1315-1339
We investigate the problem on the minimum of energy over fairly general (generally speaking, noncompact) classes of real-valued Radon measures associated with a system of sets in a locally compact space in the presence of external fields. The classes of admissible measures are determined by a certain normalization or by a normalization and a certain majorant measure σ. In both cases, we establish sufficient conditions for the existence of minimizing measures and prove that, under fairly general assumptions, these conditions are also necessary. We show that, for sufficiently large σ, there is a close correlation between the facts of unsolvability (or solvability) of both variational problems considered.
Article (Russian)
### Equilibrium Potentials with External Fields
Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1178-1195
We investigate the Gauss variational problem over fairly general classes of Radon measures in a locally compact space X. We describe potentials of minimizing measures, establish their characteristic properties, and prove the continuity of extremals. Extremal problems dual to the original one are formulated and solved. The results obtained are new even in the case of classical kernels and the Euclidean space $\mathbb{R}^n$ .
Article (Russian)
### Extremal Problems in Logarithmic Potential Theory
Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1220-1236
We pose and solve an extremal problem of logarithmic potential theory that is dual to the main minimum problem in the theory of interior capacities of condensers but, in contrast to the latter, it is solvable even in the case of a nonclosed condenser. Its solution is a natural generalization of the classical notion of interior equilibrium measure of a set. A condenser is treated as a finite collection of signed sets such that the closures of sets with opposite signs are pairwise disjoint. We also prove several assertions on the continuity of extremals.
Article (Russian)
### Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. III
Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 758-782
We complete the construction of the theory of interior capacities of condensers in locally compact spaces begun in the previous two parts of the work. A condenser is understood as an ordered finite collection of sets each of which is marked with the sign + or − so that the closures of sets with opposite signs are mutually disjoint. The theory developed here is rich in content for arbitrary (not necessarily compact or closed) condensers. We obtain sufficient and (or) necessary conditions for the solvability of the main minimum problem of the theory of capacities of condensers and show that, under fairly general assumptions, these conditions form a criterion. For the main minimum problem (generally speaking, unsolvable even for a closed condenser), we pose and solve dual problems that are always solvable (even in the case of a nonclosed condenser). For all extremal problems indicated, we describe the potentials of minimal measures and investigate properties of extremals. As an auxiliary result, we solve the well-known problem of the existence of a condenser measure. The theory developed here includes (as special cases) the main results of the theory of capacities of condensers in $\mathbb{R}^n$ , n ≥ 2, with respect to the classical kernels.
Article (Russian)
### Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. II
Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 466-488
We continue the investigation of the problem of energy minimum for condensers began in the first part of the present work. Condensers are treated in a certain generalized sense. The main attention is given to the case of classes of measures noncompact in the vague topology. In the case of a positive-definite kernel, we develop an approach to this minimum problem based on the use of both strong and vague topologies in the corresponding semimetric spaces of signed Radon measures. We obtain necessary and (or) sufficient conditions for the existence of minimal measures. We describe potentials for properly determined extremal measures.
Article (Russian)
### Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. I
Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 168-189
The present paper is the first part of a work devoted to the development of the theory of κ-capacities of condensers in a locally compact space X; here, κ: X × X → (−∞, +∞] is a lower-semicontinuous function. Condensers are understood in a generalized sense. We investigate the corresponding problem on the minimum of energy on fairly general classes of normalized signed Radon measures. We describe potentials of minimal measures, establish their characteristic properties, and study the uniqueness problem. (The subsequent two parts of this work are devoted to the problem of existence of minimal measures in the noncompact case and to the development of the corresponding approaches and methods.) As an auxiliary result, we investigate the continuity of the mapping $$\left( {x,{\mu }} \right) \mapsto \int {\kappa \left( {x,y} \right)} d{\mu }\left( y \right),\quad \left( {x,{\mu }} \right) \in X \times \mathfrak{M}^ + \left( X \right),$$ where $\mathfrak{M}^ +$ is the cone of positive measures in X equipped with the topology of vague convergence.
Article (Russian)
### A noncompact variational problem in the theory of riesz potentials. II
Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 603-613
We study some generalizations of the well-known problem of minimization of the Riesz energy on condensers. Under fairly general assumptions, we establish necessary and sufficient conditions for the existence of minimal measures.
Article (Ukrainian)
### Estimates of capacities of plane condensers
Ukr. Mat. Zh. - 1991. - 43, № 2. - pp. 193–199
Article (Ukrainian)
### A variational problem in the theory of green potential. I
Ukr. Mat. Zh. - 1990. - 42, № 4. - pp. 494–500
Article (Ukrainian)
### Extremal lengths and green capacities of condensers
Ukr. Mat. Zh. - 1990. - 42, № 3. - pp. 317–323
Article (Ukrainian)
### Precise estimate of the 2-capacity of a condenser
Ukr. Mat. Zh. - 1990. - 42, № 2. - pp. 253–257
Article (Ukrainian)
### Moduli of families of surfaces and the Green capacity of condensers
Ukr. Mat. Zh. - 1990. - 42, № 1. - pp. 64–69
Article (Ukrainian)
### Functional characteristics of space condensers: Their properties and relations among them
Ukr. Mat. Zh. - 1987. - 39, № 5. - pp. 565–573
Article (Ukrainian)
### An extremal problem on the minimum of energy for space condensers
Ukr. Mat. Zh. - 1986. - 38, № 4. - pp. 431–437
Article (Ukrainian)
### Estimates of capacities and energies under reconstruction of condensers
Ukr. Mat. Zh. - 1980. - 32, № 6. - pp. 811–813 | 2019-03-22 08:20:32 | {"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": 2, "mathjax_display_tex": 1, "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": 0.6329190731048584, "perplexity": 504.74285340114113}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2019-13/segments/1552912202640.37/warc/CC-MAIN-20190322074800-20190322100800-00015.warc.gz"} |
https://journey.hlccc.org/r-sultats-cxwd/article.php?ca9d5c=the-order-of-a-matrix-2-5-7-is | ### the order of a matrix 2 5 7 is
06 Dec 2020
0
So, if you have to find the order of the matrix, count the number or its rows and columns and there you have it. P_{11} & P_{12}\cr Sum of all three digit numbers divisible by 7. Thus, we have 6 different ways to write the order of a matrix, for the given number of elements. Now, we will calculate the values of the elements one by one. F = 0 15 03 0 00 11 0 00 01 0 00 00 (6) 1.2.6. Your email address will not be published. The following matrix has 3 rows and 6 columns. P_{31} & P_{32} \cr Before we determine the order of matrix, we should first understand what is a matrix. To calculate the value of $$p_{11}$$ , substitute $$i = 1 \space and \space j=1 \space in \space p_{ij} = i – 2j$$ . Each number in the array is called an entry or an element of the matrix. Reduce the matrix A = to triangular form. (If An Answer Does Not Exist, Enter DNE.) 0 & -2 \cr answered 11/03/16. "A matrix is a rectangular array of numbers. A matrix is a collection of data elements arranged in a two-dimensional rectangular layout. To check if system is in a safe state. Need matrix is calculated by subtracting Allocation Matrix from the Max matrix. Determinant of a 2 × 2 Matrix - Definition In order to explain the concept of determinant in linear algebra, we start with a 2 × 2 systems of equations with unknowns x and y given by Order of Matrix = Number of Rows x Number of Columns. If the matrix has $$m$$ rows and $$n$$ columns, it is said to be a matrix of the order $$m × n$$. Let matrix A is equal to matrix 1 -2 4 -3 6 5 2 -7 9. In this example, the order of the matrix is 3 × 6 (read '3 by 6'). 3. Let us try an example: How do we know this is the right answer? Choose an expert and meet online. \). Matrix entry (or element) \right] 2 & -6 & 13\cr Related Topics: Matrices, Determinant of a 2×2 Matrix, Inverse of a 3×3 Matrix. In order to work out the determinant of a 3×3 matrix, one must multiply a by the determinant of the 2×2 matrix that does not happen to be a’s column or row or column. The problem is to sort the given matrix in strict order. . The general notation of a matrix is given as: $$A = [a_{ij}]_{m × n}$$, where $$1 ≤ i ≤ m , 1 ≤ j ≤ n$$ and $$i , j \in N$$. sponding eigenvalue 5. So, this matrix will have 6 elements as following: $$P =\left[ We reproduce a memory representation of the matrix in R with the matrix function. We usually denote a matrix by a capital letter. \( P =\left[ This notation is essential in order to distinguish the elements of the matrix. Index of rows and columns start with 0. To know more, download BYJU’S-The Learning App and study in an innovative way. ... As we recall from vector dot products, two vectors must have the same length in order to have a dot product. In order to work out the determinant of a 3×3 matrix, one must multiply a by the determinant of the 2×2 matrix that does not happen to be a’s column or row or column. Question. of rows and 5 is the no. \begin{matrix} 2 & -6 & 13\cr Millions of inequivalent matrices are known for orders 32, 36, and 40. ... 1 & 3 & -2 & 5 \\ 3 & 5 & 6 & 7 \\ 2 & 4 & 3 & 8 \end{pmatrix}[/latex] This matrix is then modified using elementary row operations until it reaches reduced row echelon form. Using the first row elements, we have cofactor A11 -1 to the power of 1 plus 1 into 6 into 9 minus of minus 7 into 5 i.e., equal to 54 minus of minus 35 i.e., equal to 89. -1 & -3\cr You can see that the matrix is denoted by an upper case letter and its elements are denoted by the same letter in the lower case. A 2×2 determinant is much easier to compute than the determinants of larger matrices, like 3×3 matrices. 7 7 5;x= 2 6 6 4 x 1 x 2... x n 3 7 7 5: The arrays yand xare column vectors of order mand nrespectively whilst the array Ais a matrix of order m£n, which is to say that it has mrows and ncolumns. This notation is essential in order to distinguish the elements of the matrix. Well, for a 2x2 matrix the inverse is: In other words: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc). Then I can write the associated matrix as: When forming the augmented matrix, use a zero for any entry where the corresponding spot in the system of linear equations is blank. \right] If a is a Square Matrix of Order 3 Such that |A| = 2, Then Write the Value of Adj (Adj A). In this example, the order of the matrix is 3 × 6 (read '3 by 6'). (If an answer does not exist, enter DNE.) \end{matrix} Required fields are marked *, \( i = 1 \space and \space j=1 \space in \space p_{ij} = i – 2j$$. The chain matrix multiplication problem. Thus the order of a matrix can be either of the one listed below: $$12 \times 1$$, or $$1 \times 12$$, or $$6 \times 2$$, or $$2 \times 6$$, or $$4 \times 3$$, or $$3 \times 4$$. ). 3+x 1 2. . The Sylow 2-subgroup is a dihedral group of orde 2.7.2 Advantage of LU-decomposition::: Suppose we want to solve a m×n system AX= b. Any element from the conjugacy classes 7A 24, 7B 24 generates the Sylow 7-subgroup. Free matrix calculator - solve matrix operations and functions step-by-step This website uses cookies to ensure you get the best experience. Here it is for the 1st row and 2nd column: (1, 2, 3) • (8, 10, 12) = 1×8 + 2×10 + 3×12 = 64 We can do the same thing for the 2nd row and 1st column: (4, 5, 6) • (7, 9, 11) = 4×7 + 5×9 + 6×11 = 139 And for the 2nd row and 2nd column: (4, 5, 6) • (8, 10, 12) = 4×8 + 5×10 + 6×12 = 154 And w… P_{21} & P_{22} \cr \begin{matrix} Chain Matrix Multiplication Version of October 26, 2016 Version of October 26, 2016 Chain Matrix Multiplication 1 / 27 -9 & 9 & 15\cr The order (or dimensions or size) of a matrix indicates the number of rows and the number of columns of the matrix. A summary notation for the equations under (1) is then (3) y= Ax: There are two objects on our initial agenda. Remainder when 17 power 23 is divided by 16. The data elements must be of the same basic type. Basically, a two-dimensional matrix consists of the number of rows (m) and a … A matrix can serve as a device for representing and solving a system of equations. \). \right] \right]_{4 × 3} If we can find a LU-decomposition for A , then to solve AX =b, it is enough to solve the systems Thus the system LY = b can be solved by the method of forward substitution and the system UX= Y can be solved by the method of backward substitution. My book says I should just use a trick by the order of a permutation expressed as a product of disjoint cycles is the least common multiple of the lengths of the cycles. The order (or dimensions or size) of a matrix indicates the number of rows and the number of columns of the matrix. 7 7 5;x= 2 6 6 4 x 1 x 2... x n 3 7 7 5: The arrays yand xare column vectors of order mand nrespectively whilst the array Ais a matrix of order m£n, which is to say that it has mrows and ncolumns. Order of a matrix is determined by the number of rows and columns the matrix consists.For example if a matrix is 2 X 5 matrix where 2 is the no. Let matrix A is equal to matrix 1 -2 4 -3 6 5 2 -7 9. Find the order of AB and BA, if they exist. Tags: invertible matrix linear algebra nonsingular matrix singular matrix Next story Example of an Element in the Product of Ideals that Cannot be Written as the Product of Two Elements Previous story Normal Subgroup Whose Order is Relatively Prime to Its Index The element = 6 7, distinct from = 7 6, is situated on the second row and the third column of the matrix #. The matrix F is in row echelon form but notreduced row echelon form. $$Show your work. $$P_{21} = 2 – (2 × 1) = 0$$ In the above examples, A is of the order 2 × 3. By definition of the kernel, that The inverseof a 2× 2 matrix A, is another 2× 2 matrix denoted by A−1 with the property that AA−1 = A−1A = I where I is the 2× 2 identity matrix 1 0 0 1!. This section consists of a single important theorem containing many equivalent conditions for a matrix to be invertible. Consider a square matrix of order 3 . CBSE Class 12th Matrices- Various Types of Matrices In this video we will learn about the topics Matrices, Various types of Matrices ,Representation of Matrices and Order of Matrices… 0 Followers Most ... Class 12. 1.2.7. \), $$B =\left[ ∣ 3 + x 5 2 1 7 + x 6 2 5 3 + x ∣ = 0. In order that the rank arrive at 2, we must bring about its determinant to zero. Similarly, do the same for b and for c. Using the first row elements, we have cofactor A11 -1 to the power of 1 plus 1 into 6 into 9 minus of minus 7 into 5 i.e., equal to 54 minus of minus 35 i.e., equal to 89. Hence, by applying the invariance method we can obtain values of x. In order to find the multiplicative inverse, we have to find the matrix for which, when we multiply it with our matrix, we get the identity matrix. 4.2 Strassen's algorithm for matrix multiplication 4.2-1. In the above picture, you can see, the matrix has 2 rows and 4 columns. Definition : Let A be any square matrix of order n x n and I be a unit matrix of same order. b = 2×6 1 3 5 7 9 11 2 4 6 8 10 12 As long as the number of elements in each shape are the same, you can reshape them into an array with any number of dimensions. 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These lessons and videos help Algebra students find the inverse of a 2×2 matrix. 32 & -7 & -23 \cr Transcript. There you go! For any square matrix A, we have Matrices are defined as a rectangular array of numbers or functions. See the below example to understand how to evaluate the order of the matrix. If the matrices are the same size, matrix addition is performed by adding the corresponding elements in the matrices. \( A =\left[ Sum of all three four digit numbers formed with non zero digits That is, multiplying a matrix by its inverse produces an identity matrix. -9 & 9 & 15\cr I first need to rearrange the system as: x + y = 0 y + z = 3 –x + z = 2. 12 & 11 & 35 \cr A matrix is a collection of data elements arranged in a two-dimensional rectangular layout.$$, $$B =\left[ \( P_{31} = 3 – (2 × 1) = 1$$ (a) (b) 3.3 RANK OF A MATRIX Suppose A is an m × n matrix. Therefore, the number of elements present in a matrix will also be 2 times 3, i.e. By using this website, you agree to our Cookie Policy. z – x = 2. 4.2 Strassen's algorithm for matrix multiplication 4.2-1. Sum of all three digit numbers divisible by 6. Sum of all three digit numbers divisible by 7. \begin{matrix} Is the matrix row equivalent to I 3. To reference an element in the mth row and nth column, of a matrix mx, we write − For example, to refer to the element in the 2nd row and 5th column, of the matrix a, as created in the last section, we type − MATLAB will execute the above statement and return the following result − To reference all the elements in the mthcolumn we type A(:,m). The order of matrix is equal to m x n (also pronounced as ‘m by n’). The size and shape of the array is given by the number of rows and columns it contains, called its order.So a matrix with 3 rows and 2 columns is described as having order 3 by 2.This is not the same as a matrix of order 2 by 3, which has 2 rows and 3 columns." The number of rows and columns of all the matrices being added must exactly match. And the basis C to \left[ \begin{matrix} -5\\ -4 \end{matrix} \right],\left[ \begin{matrix} -1 \\ 5\end{matrix} \right] Then I computed the transition matrix … Let us find the inverse of a matrix by working through the following example: Matrix entry (or element) 8 & 25 & 7\cr Example 26 \begin{vmatrix} 1 & 4\\ 6 & 2\\ \end{vmatrix} (it has 2 lines and 2 columns, so its order is 2) Example 27 For a square matrix like 1 X 1 , 2 X 2 , 3 X 3 ,……., n X n the order will be represented by the no. Characteristic equation of matrix : Here we are going to see how to find characteristic equation of any matrix with detailed example. Inverse of a 2×2 Matrix. Question: Matrix A Is Order 7 ⨯ 5 And Matrix B Is Order 2 ⨯ 7. Get a free answer to a quick problem. Similarly, $$b_{32} = 9 , b_{13} = 13$$ and so on. The order of the matrix is _______ x _______ . The more appropriate notation for A and B respectively will be: $$A =\left[ Start here or give us a call: (312) 646-6365. You now know what order of matrix is, and how to determine it. Use Strassen's algorithm to compute the matrix product \begin{pmatrix} 1 & 3 \\ 7 & 5 \end{pmatrix} \begin{pmatrix} 6 & 8 \\ 4 & 2 \end{pmatrix} . The order of a matrix with 3 rows and 2 columns is 3 × 2 or 3 by 2. Note that in this context A−1 does not mean 1 A. The following is an example of a matrix with 2 rows and 3 columns. Remainder when 2 power 256 is divided by 17. The following is an example of a matrix with 2 rows and 3 columns. So, A is a 2 × 3 matrix and B is a 4 × 3 matrix. \( P_{12} = 1 – (2 × 2) = -3$$ $$P_{32} = 3 – (2 × 2) = -1$$, Hence, "A matrix is a rectangular array of numbers. We can obtain square sub matrices of order r 1 & -1 \cr … The following matrix has 3 rows and 6 columns. Find The Order Of AB And BA, If They Exist. \end{matrix} Hence. Rank. For example, you can add two or more 3 × 3, 1 × 2, or 5 × 4 matrices. The inverse of a matrix is often used to solve matrix equations. Which of the following is row equivalent to I 3. OK, how do we calculate the inverse? Sum of all three four digit numbers formed with non zero digits Find the order of AB and BA, if they exist. What is the order of \sigma = (4,5)(2,3,7) and \tau = (1,4)(3,5,7,8)? We reproduce a memory representation of the matrix in R with the matrix function. The flrst is to show, in detail, Here strict order means that matrix is sorted in a way such that all elements in a row are sorted in increasing order and for row ‘i’, where 1 <= i <= n-1, first element of row 'i' is greater than or equal to the last element of row 'i-1'. $$P_{22} = 2 – (2 × 2) = -2$$ Given a n x n matrix. \right]_{2 × 3} Though we In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. \end{matrix} Question 2 (Method 1) If A = [] is a matrix of order 2 × 2, such that || = −15 and C represents the cofactor of , then find 21 21 + 22 22 Given a is a 2 × 2 matrix A = [ 8(_11&_12@_21&_12 )] Given |A| = – 15 |A| = a11 a12 – a21 a12 – 15 = a11 a12 – a21 a12 a11 a12 – … 12 & 11 & 35 \cr Consider a square matrix of order 3 . 10 True or False Quiz Problems about Matrix Operations . R - Matrices - Matrices are the R objects in which the elements are arranged in a two-dimensional rectangular layout. We will append two more criteria in Section 5.1 . Then v is called an eigenvector for A if Av = v; where is some real number. By using this website, you agree to our Cookie Policy. The data elements must be of the same basic type. Answer. For example, the cofactor (-1)^{2+5}\cdot\Delta_{2,5}=(-1)^{7}\cdot\Delta_{2,5}= -\Delta_{2,5} corresponds to element a_{2.5} The Order of a Determinant. Most questions answered within 4 hours. De nition 3.2.1 Let A be a n n matrix, and let v be a non-zero column vector with n entries (so not all of the entries of v are zero). - Mathematics. Using the elements from A , create a 2-by-2-by-3 multidimensional array. A summary notation for the equations under (1) is then (3) y= Ax: There are two objects on our initial agenda. matrix: A rectangular ... Make sure that all of the equations are written in a similar manner, meaning the variables need to all be in the same order. \). Then |A-λI| is called characteristic polynomial of matrix. Is it possible to multiply a 2×3 and 2×2 matrix? Each dot product operation in matrix multiplication must follow this rule. Since it is a rectangular array, it is 2-dimensional. But to multiply a matrix by another matrix we need to do the "dot product" of rows and columns ... what does that mean? \begin{matrix} It is easy to describe the first two, they are cyclic, since any group of prime order is cyclic.Any element of conjugacy class 3A 56 generates Sylow 3-subgroup. 32 & -7 & -23 \cr Develop your matrix wrt the first row and get$$|A|=d\begin{vmatrix}d&0&x&x\\d&d&0&0\\d&d&d&0\\d&d&d&d\end{vmatrix}$$Develop again wrt the first row but observe that when your pivot points are the \;x's you get determinant zero as there are two identical rows in each case, so we get Check out the post “10 True or False Problems about Basic Matrix Operations” and take a quiz about basic properties of matrix operations. The Available matrix is [1 5 2 0]. 3 & 4 & 9\cr MATLAB - Matrix - A matrix is a two-dimensional array of numbers. Sum of all three digit numbers divisible by 8. Transcript. det(2A) = (2… Click hereto get an answer to your question ️ If A is matrix of order 3 , such that A (adj A) = 10 I , then |adj A| = \right]_{3 × 2} Show that is row equivalent to I 3. Therefore, the order of the above matrix is 2 x 4. $$a_{ij}$$ represents any element of matrix which is in $$i^{th}$$ row and $$j^{th}$$ column. The size and shape of the array is given by the number of rows and columns it contains, called its order.So a matrix with 3 rows and 2 columns is described as having order 3 by 2.This is not the same as a matrix of order 2 by 3, which has 2 rows and 3 columns." \left| \begin {matrix} 3+x & 5 & 2 \\ 1 & 7+x & 6 \\ 2 & 5 & 3+x \\ \end {matrix} \right|=0 ∣∣∣∣∣∣∣. 2‐ The matrix determinant A value called the determinant of #, that we denote by @ A P : # ; or | #|, 2×2 determinants can be used to find the area of a parallelogram and to determine invertibility of a 2×2 matrix. The size and shape of the array is given by the number of rows and columns it contains, called its, © 2005 - 2020 Wyzant, Inc. - All Rights Reserved, a Question Sum of all three digit numbers formed using 1, 3, 4. Not sure were to begin with this problem ?? Sum of all three digit numbers divisible by 8. In 2A as every element gets multiplied by 2. in det(2A), every term in detA, will be multiplied by 2^n. Let us create a column vector v, from the elements of the 4throw of the matrix a − MATLAB will execute the above statement and return the following result − You can also sele… We call this an m by n matrix. \). of rows(or no. The first matrices are Matrices are defined as a rectangular array of numbers or functions. If a is a square matrix of order 3, with |a|=9,then write the value of |2.Adja| - 9312125 Not all 2× 2 matrices have an inverse matrix… The two matrices shown above A and B. So, in the matrices given above, the element $$a_{21}$$ represents the element which is in the $$2^{nd}$$row and the $$1^{st}$$ column of matrix A. of columns) that is n. The conclusion hence is: If a matrix is of m × n order, it will have mn elements. This is one of the most important theorems in this textbook. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Scientific Notation Arithmetics Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Induction Logical Sets Similarly,$$b_{ij}$$ represents any element of matrix B. For Free. Your email address will not be published. They contain elements of the same atomic types. There are 5 inequivalent matrices of order 16, 3 of order 20, 60 of order 24, and 487 of order 28. I every term there are n distinct elements of the matrix. Remainder when 17 power 23 is divided by 16. Let us see with an example: To work out the answer for the 1st row and 1st column: Want to see another example? 4 times 3. . 6. Also, check Determinant of a Matrix. Let us take an example to understand the concept here. det(A) = 3. determinant is a sum of all possible products of elements not belonging to same row or column. The number of non-zero rows in the row echelon form of a matrix A produced by elementary operations on A is called the rank of A. Matrix D in equation (5) has rank 3, matrix E has rank 2, while matrix F in (6) has rank 3. Question By default show hide Solutions. Now let us learn how to determine the order for any given matrix. Tags: invertible matrix linear algebra nonsingular matrix singular matrix Next story Example of an Element in the Product of Ideals that Cannot be Written as the Product of Two Elements Previous story Normal Subgroup Whose Order is Relatively Prime to Its Index … It is quite fascinating that the order of matrix shares a relationship with the number of elements present in a matrix. (If an answer does not exist, enter DNE.) 3 & 4 & 9\cr of columns then the order of the matrix is 2 X 5. Solution for Matrix A is order 7 ⨯ 6 and matrix B is order 2 ⨯ 7. C is a matrix of order 2 × 4 (read as ‘2 by 4’) Elements In An Array. Before we determine the order of matrix, we should first understand what is a matrix. Solution Show Solution. It is because the product of mn can be obtained by more than 1 ways, some of them are listed below: For example: Consider the number of elements present in a matrix to be 12. Up to equivalence, there is a unique Hadamard matrix of orders 1, 2, 4, 8, and 12. 8 & 25 & 7\cr Step 2: Step up a matrix $B$, which consists of the constants on the right hand side of the equal sign: $\displaystyle [B] = \begin{bmatrix} 11\\7\\2 \end{bmatrix}$ Now, in order to determine the values of $x$, $y$, and … Since it is in $$3^{rd}$$ row and 3rd column, it will be denoted by $$b_{33}$$. \end{matrix} You cannot add a 2 × 3 and a 3 × 2 matrix, a 4 × 4 and a 3 × 3, etc. Similarly, do the same for b and for c. Finally, sum them up. To find a 2×2 determinant we use a simple formula that uses the entries of the 2×2 matrix. Since it is a rectangular array, it is 2-dimensional. 2. Use Strassen's algorithm to compute the matrix product$$ \begin{pmatrix} 1 & 3 \\ 7 & 5 \end{pmatrix} \begin{pmatrix} 6 & 8 \\ 4 & 2 \end{pmatrix} . Similarly, the other matrix is of the order 4 × 3, thus the number of elements present will be 12 i.e. David W. There are 10 True or False problems about basic properties of matrix operations (matrix product, transpose, etc. Free matrix calculator - solve matrix operations and functions step-by-step This website uses cookies to ensure you get the best experience. Remainder when 2 power 256 is divided by 17. How can one solve a 3 by 3 matrix? 4. A 2x2 matrix has 2 rows and 2 columns. 14 c1 + 5 c2 + 5 c3 + 2 c4 = 2 8 c1 + 3 c2 + 4 c3 + 4 c4 = 2 6 c1 + 7 c2 + 3 c3 + 7 c4 = 3 16 c1 + 6 c2 + 1 c3 + 9 c4 = 3 If we create the matrix of this system (call it mat) and the result vector (call it res), so that the system reads (mat) x = res, then we can find x by inverting the matrix with ( solve() ) and matrix-multiplying by res, or by calling solve() with both mat and res as arguments: 7.2 FINDING THE EIGENVALUES OF A MATRIX Consider an n£n matrix A and a scalar ‚.By definition ‚ is an eigenvalue of A if there is a nonzero vector ~v in Rn such that A~v = ‚~v ‚~v ¡ A~v = ~0 (‚In ¡ A)~v = ~0An an eigenvector, ~v needs to be a nonzero vector. 2‐ The matrix determinant A value called the determinant of #, that we denote by @ A P : # ; or | #|, A link to the app was sent to your phone. The Sylow 7-subgroup an example: 4.2 Strassen 's algorithm for matrix multiplication CLRS 15.2! System is in a matrix, inverse of a matrix, we should first understand what is a of... Det ( 2A ) = 3. determinant is a rectangular array, it will have mn.! We should first understand what is a rectangular array of numbers or.. Products, two vectors must have the same size, matrix addition performed. Mn elements of a matrix is $[ 1 5 2 0 ]$ Problems about basic properties of is... A 2×3 and 2×2 matrix, for the time you need a dot product operation in matrix multiplication.. And the number of rows x number of columns of all the matrices being added must exactly.. Detailed example 00 01 0 00 01 0 00 00 ( 6 ) 1.2.6 = of... Now let us try an example to understand how to determine it 7B 24 generates the Sylow 7-subgroup Solution matrix... Echelon form power 256 is divided by 16 rectangular layout best experience is equal to matrix -2! Must be of the matrix y = 0 concept here 1, 3, i.e the order of a matrix 2 5 7 is... Mn elements ; where is some real number is row equivalent to i 3 definition: let a be square... Functions step-by-step this website, you agree to our Cookie Policy is x. C is a matrix is 2 x 4 innovative way this Lecture Recalling matrix multiplication 4.2-1 find the area a! 2×3 and 2×2 matrix = v ; where is some real number concept! And 2 columns for any square matrix of order 20, 60 of order ×! Elements not belonging to same row or column matrix addition is performed by adding the corresponding elements in array., create a 2-by-2-by-3 multidimensional array ) represents any element from the Max matrix you see! Not belonging to same row or column 2 matrices have an inverse matrix… a is! In order that the order of a matrix is 3 × 2 or 3 by 6 separate the entries. Being added must exactly match subtracting Allocation matrix from the constants, essentially replacing the equal.... Does not exist, enter DNE. are 5 inequivalent matrices are the same size, addition. 4 × 3 matrix and B is order 2 × 4 ( read ' 3 by 2 4 read. ) 3.3 rank of a 3×3 matrix entries from the constants, essentially replacing the equal signs ) any! Is one of the matrix in R with the number of rows 6... Zero digits 4.2 Strassen 's algorithm for matrix the order of a matrix 2 5 7 is 4.2-1 ( b_ { }! 1 5 2 -7 9 a 2-by-2-by-3 multidimensional array \ ) and so.! Distinct elements of the matrix 4.2 Strassen 's algorithm for matrix B to write the notation of 15 matrix... Multiplying a matrix by its inverse produces an identity matrix ) of a 2×2 matrix rank of a matrix detailed... Numbers divisible by 8 a 4 × 3 below example to understand the concept.! 17, 2003 Outline of this Lecture Recalling matrix multiplication 4.2-1 a parallelogram and determine. Size ) of a matrix is often used to solve matrix operations and functions step-by-step website. The best experience the above matrix is 3 × 6 ( read as ‘ m by n ’ ) is! 'S algorithm for matrix B is a collection of data elements must be of the elements a... Time you need: if a matrix with 3 rows and the number of columns then the order a! Representation of the matrix function, the other matrix is a sum all... Download BYJU ’ S-The Learning App and study in an innovative way order, it is matrix! Order 20, 60 of order 28 determine it this rule... as we from! A ) ( B ) 3.3 rank of a matrix by its inverse produces an identity.! By 7 = v ; where is some real number safe state used... It will have mn elements n distinct elements of the order for any matrix. Is a sum of all three digit numbers divisible by 7 let matrix a, create a 2-by-2-by-3 array... Invariance method we can obtain values of the matrix has 3 rows 3!, 36, and how to determine invertibility of a matrix is often used to find 2×2! Lecture 13: Chain matrix multiplication CLRS Section 15.2 Revised April 17, 2003 Outline of this Lecture Recalling multiplication... 7 ⨯ 6 and matrix B order 28 the conclusion hence is: a! S-The Learning App and study in an array sum them up in a safe state notreduced echelon. Order n x n ( also pronounced as ‘ m by n ’ ) all possible of!, the order of the matrix function ( 6 ) 1.2.6 see the below example to how! Separate the coefficient entries from the constants, essentially replacing the equal signs )... ) and so on cookies to ensure you get the best experience 2 -7 9 more criteria in Section.! To i 3 × 2 or 3 by 2, the other matrix is of the matrix is equal m... Check if system is in row echelon form but notreduced row echelon form three digit divisible... 15 03 0 00 11 0 00 01 0 00 00 ( 6 ) 1.2.6 that,! If the matrices are the same for B and for c. Consider a square of! Lessons and videos help Algebra students find the order 4 × 3 problem. These lessons and videos help Algebra students find the inverse of a matrix is of following... Data elements arranged in a safe state, in detail, Consider square... ) = ( 2… Solution for matrix a, we should first understand what is rectangular. The entries of the matrix or an element of the matrix cookies to ensure you get best! Present will be 12 i.e is some real number inverse of a 2×2 determinant is a collection data! Ba, if they exist or column numbers formed using 1, 3, i.e: if a.! A two-dimensional matrix consists of the matrix its determinant to zero students find the area of 2×2... Solve matrix operations ( matrix product, transpose, etc larger matrices, like 3×3.... 2, we have 6 different ways to write the order of matrix = number of elements present in safe. 3 rows and the number of rows ( m ) and a number of columns ( n ) capital! But notreduced row echelon form but notreduced row echelon form but notreduced row echelon form notreduced! Determinant of a matrix we are going to see how to evaluate the order of matrix, we should understand. We should first understand what is a matrix is 2 x 4 Av = v ; where some. A dot product operation in matrix multiplication must follow this rule x 2... A 2-by-2-by-3 multidimensional array ) represents any element of the matrix DNE. read as ‘ by... Have an inverse matrix… a matrix is 2 x 4, b_ { }. Calculator - solve matrix equations mean 1 a, two vectors must have the same for and! Right answer matrix a is an m × n matrix not exist, enter DNE. is easier. + x 6 2 5 3 + x 6 2 5 3 + 5... Columns ( n ), inverse of a matrix is a rectangular array numbers., we will calculate the values of the matrix is of the matrix is. Size, matrix addition is performed by adding the corresponding elements in the above picture you! C is a matrix with 2 rows and columns BA, if they exist the matrices being must... A, we have 6 different ways to write the order of a 2×2 determinant is a sum all. Us learn how to determine it 2 × 3, 4 is by! Possible products of elements not belonging to same row or column columns is 3 × 2 or by! You write the notation of 15 for matrix multiplication ’ ) elements in the matrices being added must match! They exist invariance method we can obtain values of the matrix Quiz Problems about operations. Using 1, 3, i.e, for the time you need from dot! The following matrix has 3 rows and columns is the right answer entry an. F = 0 y + z = 2 5 2 0 ] \$ it will mn! Find the order for any given matrix given matrix in strict order,. Then the order of a matrix with 3 rows and 6 columns, inverse of 2×2... And 2×2 matrix, we should first understand what is a matrix with 3 rows the! Consists of the matrix about matrix operations ( matrix product, transpose, etc 8. Not mean 1 a matrix has 3 rows and 6 columns basic type using the elements of the matrix phone. Take an example: 4.2 Strassen 's algorithm for matrix multiplication the,... Us a call: ( 312 ) 646-6365 inequivalent matrices of order 3 will be 12.... The invariance method we can obtain values of x CLRS Section 15.2 Revised 17... 3, 4 is 2 x 5 2 1 7 + x 6 2 5 3 x... 4 ’ ) to determine it of data elements arranged in a matrix to separate the coefficient from! The time you need by subtracting Allocation matrix from the the order of a matrix 2 5 7 is, essentially replacing the equal signs etc..., 7B 24 generates the Sylow 7-subgroup is quite fascinating that the order of matrix...
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Problem BIsland Archipelago
You live on an island archipelago, where the islands are always changing due to rising and lowering of the sea level.
You represent the archipelago as an $n\times n$ grid, where each grid cell can either contain water or land. The archipelago starts out consisting of only water, and water completely surrounds the archipelago outside the boundaries of the grid. Over time, some of the water cells lower to reveal land, and some of the land cells might disappear back into rising water cells.
If it is possible to walk from a land cell to another land cell by moving left, right, up, or down through only land cells, the two cells are said to be edgewise-connected. An island is a maximal edgewise-connected-component of land cells. Note that there might exist islands within island, i.e. islands within a lake enclosed in an island in the ocean.
One concern to the archipelago’s citizens is whether an island encloses a freshwater lake. A freshwater lake exists on an island if there is some closed loop of land (consisting of steps up, down, left, and right over land cells on the island) that encircles at least one water cell. For example, in the following diagram of an archipelago, where # denotes land and denotes water, there are five islands (two on the left, one in the middle, and two on the right), and only one of them (the outer island on the right) contains a freshwater lake.
~~~~~~~~~~~#####
~~~~~~~~~~~#~~~#
##~~~####~~#~#~#
#~#~~##~#~~#~~##
~##~~###~~~#####
As the water levels are changing, you want to know the total number of islands, and the number of islands that do not contain a freshwater lake.
Input
The first line of input contains two space-separated integers $n$ and $m$ $(1 \leq n \leq 1\, 500, 1 \leq m \leq 5\cdot 10^4)$, the size of the grid and the number of queries. The left-most column of the grid has index $c = 1$ and the top-most row of the grid has index $r = 1$. Then follow $m$ lines, each beginning with either the character ! or ?.
The lines that begin with $\texttt{!}$ indicate a change in the water level of a cell. On these lines are two additional space-separated integers $r$ and $c$ $(1 \leq r,c\leq n)$, indicating that the cell at coordinates $(r,c)$ on the grid has changed from water to land, or from land to water.
The lines that begin with $\texttt{?}$ contain no other input, and denote a request for a report on the status of the archipelago: how many islands there are, and how many islands do not contain a freshwater lake?
Output
Print a line for each ? containing two space-separated integers: the number of islands currently in the archipelago, and the number of those islands that currently do not contain a freshwater lake.
Sample Input 1 Sample Output 1
3 18
?
! 1 1
! 1 2
! 1 3
! 3 1
?
! 2 1
?
! 3 2
! 3 3
! 2 3
?
! 3 3
?
! 1 1
?
! 2 2
?
0 0
2 2
1 1
1 0
1 1
2 2
1 1 | 2020-10-31 04:28:32 | {"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": 1, "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": 0.2339215725660324, "perplexity": 1337.7922382146367}, "config": {"markdown_headings": false, "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-2020-45/segments/1603107912807.78/warc/CC-MAIN-20201031032847-20201031062847-00655.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition/chapter-2-graphs-and-functions-2-4-linear-functions-2-4-exercises-page-232/89 | ## Precalculus (6th Edition)
The results in Exercises 86 and 87 are $\sqrt{10}$ and $2\sqrt{10}$, respectively. Thus, their sum is: sum $=\sqrt{10} + 3\sqrt{10} = 3\sqrt{10}$ The result in Exercise 89 is $3\sqrt{10}$. Therefore, the distance from the 1st point to the 4th point is equal to the sum of the distance between the first two points and the distance between the 2nd and the 4th point. | 2018-07-17 13:56:13 | {"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": 1, "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": 0.8194318413734436, "perplexity": 130.4620432915993}, "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-2018-30/segments/1531676589726.60/warc/CC-MAIN-20180717125344-20180717145344-00530.warc.gz"} |
https://quiz.jagranjosh.com/josh/quiz/index.php?attempt_id=3081691 | Constructions
Subject matter experts of JagranJosh have developed a chapter wise practice sets as per the relevancy of the chapter in the board exam. This practice set contains 10 questions with detailed explanations on “Chapter: Constructions” of Mathematics.
• Q1.The sum of all the angles of a triangle is:
• Q2.The exterior angle of a triangle is equal to___:
• Q3.What would be the distance between the point of intersection of the two tangents and the centre of the circle, if a circle of radius 6 cm is constructed from which two tangents are arise which are inclined at an angle of ${60}^{o}$ .
• Q4.To divide a line segment AB in the ratio 3 : 7, first a ray AX is drawn so that angle BAX is an acute angle and then at equal distances points are marked on the ray AX such that the minimum number of these points is | 2019-08-20 10:34:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "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": 0.6040568947792053, "perplexity": 398.09678007867893}, "config": {"markdown_headings": false, "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/1566027315321.52/warc/CC-MAIN-20190820092326-20190820114326-00330.warc.gz"} |
http://mathhelpforum.com/algebra/160304-word-problem-w-three-variables.html | # Thread: Word problem w/ three variables
1. ## Word problem w/ three variables
This problem, is seriously bugging me:
A collection of nickels, dimes, and quarters consists of 12 coins with a total value of $1.45. If the number of nickels is 2 less than the number of dimes, how many of each coin are contained in the collection? I thought, I had it right the first time, but my totals didn't add up to 1.45. I don't understand what I am doing wrong. The three equations I came up with are: x + y + z = 12 x = y - 2 or x - y = -2 5x + 10y + 25z = 145 Are my equations wrong? 2. I don't think your equations are wrong. 'How did you solve them?' is the question on my mind. Because the solutions don't seem to work. 3. x + y + z = 12 z - y = -2 5x + 10y + 25z = 145 x + y + z = 12 x - y = -2 2x + z = 10 x + y + z = 12 5x + 10y + 25z = 145 10x + 10y + 10z = 120 5x + 10y + 25z = 145 5x - 15z = -25 2x + z = 10 5z - 15z = -25 30x + 15z = 150 5x - 15z = -25 35x = 125 x = NOT RIGHT! Well originally I had 4 here, but while typing it out, I see that I subtracted and also had 125 instead of 150 on the last step to get 25x = 100, but it should have been addition...and now it's alllllllllll messed up again. Does anyone see an error in my math? 4. I even went as far as trying to say that if x = 1 - 12, and y being two more...and not any of those answers were correct. Is something wrong with this problem or me!!! LOL!!! I keep coming up with this... x =$\displaystyle \displaystyle{\frac{25}{7}$y=$\displaystyle \displaystyle{\frac{39}{7}$z=$\displaystyle \displaystyle{\frac{20}{7}$And if this is right I am going to uppercut my teacher... 5. Originally Posted by gustahfan BUMP! I even went as far as trying to say that if x = 1 - 12, and y being two more...and not any of those answers were correct. Is something wrong with this problem or me!!! LOL!!! I keep coming up with this... x =$\displaystyle \displaystyle{\frac{25}{7}$y=$\displaystyle \displaystyle{\frac{39}{7}$z=$\displaystyle \displaystyle{\frac{20}{7}\$
And if this is right I am going to uppercut my teacher...
The equations you have set up in Post 1 look correct.
And the values that you have mentioned for x,y,and z are the only ones that are going to fit your equations.
Life would have been simpler if the total number of coins was 14.
6. Originally Posted by harish21
The equations you have set up in Post 1 look correct.
And the values that you have mentioned for x,y,and z are the only ones that are going to fit your equations.
Life would have been simpler if the total number of coins was 14.
This teacher is so cruel to us!
7. If you change total coins from 12 to 13, then all's well that ends well...
6 nickels + 4 dimes + 3 quarters = .30 + .40 + .75 = 1.45
8. yea, she told us today she made a mistake...you think?? !! | 2018-03-23 13:49:17 | {"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": 1, "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": 0.7106237411499023, "perplexity": 1527.9238851036762}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2018-13/segments/1521257648226.72/warc/CC-MAIN-20180323122312-20180323142312-00211.warc.gz"} |
https://www.transtutors.com/questions/15-46-like-kind-exchange-f-exchanged-undeveloped-land-worth-45-000-with-g-for-land-w-1131402.htm | # 15-46 Like Kind Exchange. F exchanged undeveloped land worth $45,000 with G for land worth$42,000.. 1 answer below »
15-46 Like Kind Exchange. F exchanged undeveloped land worth $45,000 with G for land worth$42,000 and a personal automobile worth $3000. F’s adjusted basis in the land was$36000. G’s adjusted bases in the land and automobile were $39500 and$2500, respectively. a. How much gain or loss must F recognize in this exchange, and what are his bases in the land and automobile received? b. How much gain or loss must G recognize in this exchange, and what is her basis in the land received.?
Like Kind Exchange. F exchanged undeveloped land worth $45,000 with G for land worth$42,000 and a personal automobile worth $3000. F’s adjusted basis in the land was$36000. G’s adjusted bases in the land and automobile were $39500 and$2500, respectively. a. How much gain or loss... | 2019-10-21 22:23:16 | {"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": 1, "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": 0.2606649100780487, "perplexity": 13268.24617715629}, "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-2019-43/segments/1570987795253.70/warc/CC-MAIN-20191021221245-20191022004745-00458.warc.gz"} |
https://campus.datacamp.com/courses/conda-for-building-distributing-packages/conda-packages | # Conda Packages
In the last chapter you created a Python package and successfully installed it. However, you needed to have 1) downloaded the source code, and 2) created a Conda Environment with the dependent packages.
Further, when using setuptools to install packages there are no uninstall or update commands. That means you would have to manually remove the installed files if you want to install a newer version of the package. As you saw in the Conda Essentials course, Conda packages solve each of these issues, but you might use pip and virtualenv as well.
In this chapter you'll create a Conda Recipe for the mortgage_forecasts package to define the dependent Conda packages. The Conda recipe is specified in a file called meta.yaml.
You'll then build the package archive and upload it to Anaconda Cloud. Further, Conda packages are not limited to Python packages. A package written in any programming language, or a collection of files, can become a Conda package.
Which statement below is INCORRECT? | 2020-07-13 02:57:44 | {"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": 0.27679240703582764, "perplexity": 2717.624242007607}, "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-2020-29/segments/1593657140746.69/warc/CC-MAIN-20200713002400-20200713032400-00220.warc.gz"} |
https://mathspace.co/textbooks/syllabuses/Syllabus-878/topics/Topic-19475/subtopics/Subtopic-260409/?textbookIntroActiveTab=guide | # 4.01 Identifying and writing equations
Lesson
## What are equations?
An equation is a mathematical sentence stating that two expressions are equal. That is, they have the same value. We can think of it as being balanced.
Here are some examples:
$3x+2=4$3x+2=4
$4.25=3\frac{1}{2}+n$4.25=312+n
$\frac{y}{3}=2y+5$y3=2y+5
Equations often contain letters or symbols are used to represent an unknown quantity. These symbols are called variables. As seen above, variables can appear on one or both sides of an equation.
An equation is like a sentence. We would like to translate a written sentence to a mathematical sentence. Where we see "is" or "equals", we will put an equals sign.
#### Worked example
##### Question 1
Adam thought of a number, doubled it and added $5$5 to get $13$13. Write an equation that represents this scenario.
Think: There is an unknown quantity, "a number", so we should use a variable to represent it. Let's use $n$n.
Do:
$n\times2+5$n×2+5 $=$= $13$13 We take a number, double it and then add $5$5, this is equal to $13$13 $2n+5$2n+5 $=$= $13$13 We prefer to write a product of a number and variable with the number first
##### Question 2
What scenario could the equation $3a=2b+1$3a=2b+1 be representing?
Think: There is certainly not a single correct answer here, so here is one possible scenario.
We should start by stating what $a$a and $b$b represent.
Do: Let $a$a be the number of apples and $b$b be the number of bananas. The cost of $3$3 apples is $1$1 dollar more than the cost of $2$2 bananas.
Reflect: How many different scenarios can you come up with? What do all the scenarios have in common? What is different?
#### Practice questions
##### Question 3
Is $5r-15=0$5r15=0 an expression or an equation?
1. It is an expression.
A
It is an equation.
B
It is an expression.
A
It is an equation.
B
##### Question 4
A number (call it $n$n) plus four equals seven.
1. Write the sentence using mathematical symbols.
2. What is the value of $n$n?
##### Question 5
Roxanne has been out picking flowers, and has $40$40 in total. When she returns, she puts them in $5$5 different vases.
If she puts $p$p flowers in each vase, rewrite the following sentence using algebra:
1. "There are $5$5 groups of $p$p flowers, which make $40$40 in total."
## Identifying solutions
We say that a value for a variable is a solution to an equation if we can substitute it into the equation and it makes the number sentence true.
For example, if we wanted to find the solution to the equation $x+1=3$x+1=3, we want to find a value for $x$x that makes that equation true. This statement is true when $x=2$x=2 because $2+1=3$2+1=3.
When we're figuring out whether a value is a solution, we need to see whether the left-hand side of the equation is the same as the right-hand side. We can think of it as a see-saw.
For example, if we have the equation $x+12=20$x+12=20, we could think of it visually as:
If $12$12 was removed from the left-hand side of the seesaw, it would look unbalanced like this:
So how do we balance the equation again? We need to remove $12$12 from the right-hand side as well:
So $x=8$x=8 satisfies the equation $x+12=20$x+12=20 because $8+12=20$8+12=20
#### Worked example
##### Question 6
Determine if $b=47$b=47 is a solution of $b+48=96$b+48=96.
a) Find the value of the left-hand side of the equation when $b=47$b=47.
Think: We need to substitute $47$47 into the equation for $b$b.
Do:
$b+48$b+48 $=$= $b+48$b+48 $=$= $47+48$47+48 $=$= $95$95
The left-hand side is $95$95.
b) Is $b=47$b=47 a solution of $b+48=96$b+48=96?
Think: Is the left-hand side equal to the right-hand side in this equation?
Do: No, $b=47$b=47 is not a solution of $b+48=96$b+48=96.
#### Practice question
##### Question 4
There are two rectangular-shaped pools at the local aquatic center. Each pool has a length that is triple its width. Pool 1 has a perimeter of $256$256 meters.
1. Let the width of the pools be represented by $w$w. Which of the following equations represents the perimeter of each pool in terms of $w$w?
$2w+3w=256$2w+3w=256
A
$3w+3w+3w=256$3w+3w+3w=256
B
$w+3w=256$w+3w=256
C
$w+w+3w+3w=256$w+w+3w+3w=256
D
$2w+3w=256$2w+3w=256
A
$3w+3w+3w=256$3w+3w+3w=256
B
$w+3w=256$w+3w=256
C
$w+w+3w+3w=256$w+w+3w+3w=256
D
2. The width of Pool 2 is $32$32 meters. Find its perimeter.
3. Is the width of Pool 1 also $32$32 meters?
no
A
yes
B
no
A
yes
B
##### Question 5
We want to determine if $b=8$b=8 is the solution of $8b=63$8b=63.
1. Find the value of the left-hand side of the equation when $b=8$b=8.
2. Is $b=8$b=8 the solution of $8b=63$8b=63?
yes
A
no
B
yes
A
no
B | 2021-12-03 05:09:13 | {"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": 1, "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": 0.734889030456543, "perplexity": 1154.6892504184657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2021-49/segments/1637964362589.37/warc/CC-MAIN-20211203030522-20211203060522-00309.warc.gz"} |
http://matharguments180.blogspot.com/2014/04/day-102-square-root-exponents.html | ## Tuesday, April 29, 2014
### 114: Square Root Exponents
We all know that adding/subtracting exponents corresponds to multiplying/dividing the terms, like this:
$x^4 * x^7 = x^{11}$
$\dfrac{x^{14}}{x^9} = x^5$
Then negative exponents logically followed: $x^{-7} = \dfrac{x^2}{x^9}$
Then $\dfrac{x^3}{x^3} = x^0 = 1$ logically followed that.
Additionally, multiplying/dividing the exponents relates to powers/roots
${x^4}^2 = x^{4*2} = x^8$
$\sqrt{x^6} = x^{6/2} = x^3$
So a fractional exponent means a radical, depending on the denominator of the exponent.
### So here's my question:
What should we think about $x^{\sqrt{2}}$
How should we interpret that? | 2017-08-20 19:01:37 | {"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": 1, "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": 0.8872566223144531, "perplexity": 4898.52631507938}, "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-2017-34/segments/1502886106984.52/warc/CC-MAIN-20170820185216-20170820205216-00494.warc.gz"} |
https://www.autoitscript.com/forum/topic/53291-hotkeyset-bug-or/ | # HotKeySet Bug, Or?
## Recommended Posts
The following code when run from SciTE will sometimes produce an error on the resetting of the hotkey to nothing, and sometimes not. However, I always get the error when running from the command line, as a compiled script, or running the script from the explorer context menu.
The error is "Unknown function name" on the HotKeySet("{F3}", "") line. If I put a function name in there instead of nothing, of course it works fine. At first I thought it had to do with spaces or tabs before the line because that's what was in my original code, but doesn't seem to be consistent with that. Also, when running from SciTE does error, sometimes uncommenting the Sleep line will cause the error to go away. But again that's not consistent. Can anybody verify? Thanks.
Tested with latest BETA & Production releases, on XP PRO SP2, & Windows 2000
HotKeySet("{F3}", "_CtrlF")
;Sleep(1000)
HotKeySet("{F3}", "")
Func _CtrlF()
Return
EndFunc
##### Share on other sites
shouldn't that be?:
HotKeySet("{F3}", "_CtrlF")
;Sleep(1000)
HotKeySet("{F3}")
Func _CtrlF()
Return
EndFunc ;==>_CtrlF
Live for the present,
Dream of the future,
Learn from the past.
##### Share on other sites
shouldn't that be?:
HotKeySet("{F3}", "_CtrlF")
;Sleep(1000)
HotKeySet("{F3}")
Func _CtrlF()
Return
EndFunc ;==>_CtrlF
Ermmm. I guess you're right. But has that behavior changed? I was resurectting some old code from about a year ago that I'm sure used to run fine. But, maybe not... and before I posted I did double check the help file and it says
function [optional] The name of the function to call when the key is pressed. Leave blank to unset a previous hotkey.
So of course I left it "blank" as in "", and not "out" as in not there at all. Rewording required on help file? (for dummies like me) :">
Also, I don't understand why sometimes SciTE will return the error when running it and sometimes not...
##### Share on other sites
Also, I don't understand why sometimes SciTE will return the error when running it and sometimes not...
I get the AU3check error for both the latest Production and Beta version....
Live for the present,
Dream of the future,
Learn from the past.
##### Share on other sites
I get the AU3check error for both the latest Production and Beta version....
More testing reveals that if I run it the first time, it catches the error, subsequent times it lets the error by, but only on My Windows 2000 machine, my XP system always catches it.
What happens on the 2000 rig is that the first time, the Au3Check error window will pop up with the details, then of course the SciTE output panel will have the error details as well, including "ERROR: (): undefined function". Subsequent runs will not pop up the Au3Check window, and the ouput only provides the "AU3Check ended.rc:2" to indicate the error. (see below).
>"D:\PROGRAM FILES\far\AutoIt3\SciTE\AutoIt3Wrapper\AutoIt3Wrapper.exe" /run /beta /ErrorStdOut /in "D:\PROGRAMMING\@MBIZ\HistViewer\ttest2.au3" /autoit3dir "D:\Program Files\Far\AutoIt3\beta" /UserParams
+>14:17:42 Starting AutoIt3Wrapper v.1.9.3
>Running AU3Check (1.54.9.0) from : D:\Program Files\Far\AutoIt3\beta
!>14:17:42 AU3Check ended.rc:2
>Running:(3.2.9.0): D:\Program Files\Far\AutoIt3\beta\autoit3.exe "D:\PROGRAMMING\@MBIZ\HistViewer\ttest2.au3"
->14:17:43 AutoIT3.exe ended.rc:1
+>14:17:44 AutoIt3Wrapper Finished
>Exit code: 1 Time: 1.911
So I guess SciTE/Au3Check is always catching the error, just not fully? Or is my Scite4AutoIt installation suspect?
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JdeB
function [optional] The name of the function to call when the key is pressed. Leave blank to unset a previous hotkey.
It is not correct, and that's why they came here and ask unnecessary questions.
There should be something like this:
To unset a previous hotkey use only one parameter.
Spoiler
Using OS: Win 7 Professional, Using AutoIt Ver(s): 3.3.6.1 / 3.3.8.1
My Work...
Spoiler
Like the Projects/UDFs/Examples? Please rate the topic (up-right corner of the post header: Rating )
* === My topics === *
==================================================
==================================================
AutoIt is simple, subtle, elegant. © AutoIt Team
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i think those ho new to AutoIt confused when they read..
...
It is not correct, and that's why they came here and ask unnecessary questions.
Well, I'm not new to AutoIt, but I will admit to the occasional "asking of unnecessary questions"
I agree though that the helpfile should be updated...
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I agree though that the helpfile should be updated...
poor excuse
I have changed the Helpfile to read: "Not specifying this parameter will unset a previous hotkey."
Live for the present,
Dream of the future,
Learn from the past.
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poor excuse
I have changed the Helpfile to read: "Not specifying this parameter will unset a previous hotkey."
Thanks JdeB, and if you'll pardon the wordplay, I like to think that most of my excuses are pretty rich
## Create an account
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• Wiki
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• Git | 2018-04-19 10:14:25 | {"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": 0.33398550748825073, "perplexity": 7702.891535639894}, "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-2018-17/segments/1524125936833.6/warc/CC-MAIN-20180419091546-20180419111546-00375.warc.gz"} |
https://drostlab.github.io/myTAI/reference/MatchMap.html | This function matches a Phylostratigraphic Map or Divergence Map only storing unique gene ids with a ExpressionMatrix also storing only unique gene ids.
MatchMap(Map, ExpressionMatrix, remove.duplicates = FALSE, accumulate = NULL)
## Arguments
Map a standard Phylostratigraphic Map or Divergence Map object. a standard ExpressionMatrix object. a logical value indicating whether duplicate gene ids should be removed from the data set. an accumulation function such as mean(), median(), or min() to accumulate multiple expression levels that map to the same unique gene id present in the ExpressionMatrix.
## Value
a standard PhyloExpressionSet or DivergenceExpressionSet object.
## Details
In phylotranscriptomics analyses two major techniques are performed to obtain standard Phylostratigraphic map or Divergence map objects.
To obtain a Phylostratigraphic Map, Phylostratigraphy (Domazet-Loso et al., 2007) has to be performed. To obtain a Divergence Map, orthologous gene detection, Ka/Ks computations, and decilation (Quint et al., 2012; Drost et al., 2015) have to be performed.
The resulting standard Phylostratigraphic Map or Divergence Map objects consist of 2 colums storing the phylostratum assignment or divergence stratum assignment of a given gene in column one, and the corresponding gene id of that gene on columns two.
A standard ExpressionMatrix is a common gene expression matrix storing the gene ids or probe ids in the first column, and all experiments/stages/replicates in the following columns.
The MatchMap function takes both standard datasets: Map and ExpressionMatrix to merge both data sets to obtain a standard PhyloExpressionSet or DivergenceExpressionSet object.
This procedure is analogous to merge, but is customized to the Phylostratigraphic Map, Divergence Map, and ExpressionMatrix standards to allow a faster and more intuitive usage.
In case you work with an ExpressionMatrix that stores multiple expression levels for a unique gene id, you can specify the accumulation argument to accumulate these multiple expression levels to obtain one expression level for one unique gene.
## References
Domazet-Loso T, Brajkovic J, Tautz D (2007) A phylostratigraphy approach to uncover the genomic history of major adaptations in metazoan lineages. Trends Genet. 23: 533-9.
Domazet-Loso T, Tautz D (2010) A phylogenetically based transcriptome age index mirrors ontogenetic divergence patterns. Nature 468: 815-8.
Quint M., Drost H.G., Gabel A., Ullrich K.K., Boenn M., Grosse I. (2012) A transcriptomic hourglass in plant embryogenesis. Nature 490: 98-101.
Drost HG et al. (2015) Mol Biol Evol. 32 (5): 1221-1231 doi:10.1093/molbev/msv012
Hajk-Georg Drost
## Examples
data(PhyloExpressionSetExample)
# in a standard PhyloExpressionSet,
# column one and column two denote a standard
# phylostratigraphic map
PhyloMap <- PhyloExpressionSetExample[ , 1:2]
# look at the phylostratigraphic map standard
#> Phylostratum GeneID
#> 1 1 at1g01040.2
#> 2 1 at1g01050.1
#> 3 1 at1g01070.1
#> 4 1 at1g01080.2
#> 5 1 at1g01090.1
#> 6 1 at1g01120.1
# in a standard PhyloExpressionSet, column two combined
# with column 3 - N denote a standard ExpressionMatrix
ExpressionMatrixExample <- PhyloExpressionSetExample[ , c(2,3:9)]
# these two data sets shall illustrate an example
# phylostratigraphic map that is returned
# by a standard phylostratigraphy run, and a expression set
# that is the result of expression data analysis
# (background correction, normalization, ...)
# now we can use the MatchMap function to merge both data sets
# to obtain a standard PhyloExpressionSet
PES <- MatchMap(PhyloMap, ExpressionMatrixExample)
# note that the function returns a head()
# of the matched gene ids to enable
# the user to find potential mis-matches
# the entire procedure is analogous to merge()
# with two data sets sharing the same gene ids
# as column (primary key)
PES_merge <- merge(PhyloMap, ExpressionMatrixExample) | 2022-05-29 02:32:36 | {"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": 0.44262561202049255, "perplexity": 12519.700673105383}, "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-2022-21/segments/1652663035797.93/warc/CC-MAIN-20220529011010-20220529041010-00494.warc.gz"} |
http://www-old.newton.ac.uk/programmes/DIS/seminars/2009051317009.html | # DIS
## Seminar
### Irreducibility of q-Painlevé equation of type $A_6^{(1)}$ in the sense of order
Nishioka, S (Tokyo)
Wednesday 13 May 2009, 17:00-17:30
Satellite
#### Abstract
I introduce a result on the irreducibility of q-Painlevé equation of type $A_6^{(1)}$ in the sense of order using the notion of decomposable extensions. The equation is one of the special non-linear q-difference equations of order 2 with symmetry $(A_1+A_1)^{(1)}$ and is also called q-Painlevé equation of type II. The decomposable difference field extension is a difference analogue of K. Nishioka's which was defined to prove the irreducibility of the first Painlevé equation in the sense of Nishioka-Umemura. The strongly normal extension of difference fields defined by Bialynicki-Birula is decomposable. I proved that transcendental solutions of the equation in a decomposable extension may exist only for special parameters, and that all of them satisfies the identical well-known Riccati equation if we apply the Bäcklund transformations to it appropriate times.
#### Video
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible. | 2014-09-18 16:16:09 | {"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": 1, "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": 0.6383377313613892, "perplexity": 788.7932070712433}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "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-2014-41/segments/1410657128304.55/warc/CC-MAIN-20140914011208-00172-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://www.azdictionary.com/definition/diagonalisation | • Definition for "diagonalisation"
• In matrix algebra, the entire process of converting…
• Sentence for "diagonalisation"
• Second, the method that is used,…
• Hypernym for "diagonalisation"
• resolving
# diagonalisation definition
• noun:
• In matrix algebra, the entire process of converting a square matrix into a diagonal matrix, frequently to find the eigenvalues of matrix.
• altering a square matrix to diagonal type (with all non-zero elements regarding main diagonal)
• In matrix algebra, the process of transforming a square matrix into a diagonal matrix, frequently to find the eigenvalues associated with matrix.
• changing a square matrix to diagonal type (along with non-zero elements regarding the main diagonal) | 2017-05-26 06:35:01 | {"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": 0.8136515021324158, "perplexity": 1705.857143049458}, "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-2017-22/segments/1495463608642.30/warc/CC-MAIN-20170526051657-20170526071657-00352.warc.gz"} |
http://openstudy.com/updates/5595bb00e4b0989cc8788bec | ## anonymous one year ago i use the formula, but don't understand it.... (sum of finite geometric series)
1. anonymous
|dw:1435876640907:dw|
2. anonymous
@freckles
3. freckles
do you mean how do you derive that to be the formula?
4. anonymous
i mean how do I understand it....
5. anonymous
Every formula I have previously used I understand why it works to calculate a certain thing....
6. anonymous
but, this....
7. freckles
Ok well let me derive it for you and maybe you will understand it better... $a_1,ra_1,r^2a_1,r^3a_1..., \text{ is a geometric sequence } \\a_2=ra_1 \\ a_3=r^2a_1 \\ \cdots \\ a_n=r^{n-1}a_1 \\ \ \\ \text{ geometric series is the sum of the terms of the geometric sequence }$ $S_n=a_1+ra_1+r^2a_1+r^3a_1 \cdots +r^{n-1}a_1 \\S_n= a_1(1+r+r^2+r^3+ \cdots r^{n-1}) \\$ ...
8. freckles
ok before I go on... do you know what we get when we do: $\frac{r^n-1}{r-1}$
9. anonymous
when n->∞ ? or what do you mean?
10. freckles
examples: $\frac{r^2-1}{r-1}=r+1 \\ \frac{r^3-1}{r-1}=r^2+r+1 \\ \frac{r^4-1}{r-1}=(r+1)(r^2+1)=r^3+r^2+r+1 \\ \frac{r^5-1}{r-1}=r^4+r^3+r^2+r+1$ so on...
11. freckles
basically the trick here to recognize $r^{n-1}+r^{n-2}+r^{n-1}+ \cdots +r^{3}+r^2+r+1=\frac{r^n-1}{r-1}$
12. anonymous
oh, so you get r^(n-1)+r^(n-2)+....r^3+r^2+r+1
13. anonymous
yeah
14. freckles
oops I didn't mean to write n-1 again
15. anonymous
yes I see, no need correction
16. freckles
$r^{n-1}+r^{n-2}+r^{n-\color{red}3}+ \cdots +r^{3}+r^2+r+1=\frac{r^n-1}{r-1}$
17. freckles
ok I did it anyways :p
18. anonymous
so, $$\LARGE \frac{r^{n}-1}{r-1}$$ is just the sum of all these r^(n-1)+....+r+1 but these aren't terms, because the terms are when each of these is multiplying times a1, so..... a1 • r^(n-1) + a1 • r^(n-2) + .... a1•r+a1•1
19. freckles
$S_n=a_1+ra_1+r^2a_1+r^3a_1 \cdots +r^{n-1}a_1 \\S_n= a_1(1+r+r^2+r^3+ \cdots r^{n-1}) \\ \\ S_n=a_1 \frac{r^n-1}{r-1} \\ \text{ or multiply both \top and bottom gives } S_n=a_1 \frac{1-r^n}{1-r}$
20. anonymous
the trick is to notice that if we let $$S_n$$ denote the sum of $$n$$ terms we get: $$S_n=a_1+a_1 r+a_1 r^2+\dots+a_1r^{n-1}\\r S_n=a_1 r+a_1 r^2+\dots+a_1r^{n-1}+a_1 r^n\\S_n-rS_n=a_1-a_1 r^n\\(1-r)S_n=a_1(1-r^n)\\S_n=a_1\frac{1-r^n}{1-r}$$
21. freckles
yes you are right so you can take that sum of the r things you wrote and multiply it by your initial
22. freckles
that's cute @oldrin.bataku
23. anonymous
So, this is what we do: we take the (1-r^(n-1)) / (1-r) that = r^(n-1) + r^(n-2) + .... r + 1 and then so that we are going to be adding the terms, we multiply by the a1 component: And this way we get the formula that is there.
24. freckles
yep
25. anonymous
tnx again, you got my back I will say I don't know jack. when compare to you at least, because you are a math beast.
26. anonymous
You really saved me. "Filled in the gap"
27. freckles
np didn't do much :)
28. anonymous
yeah you did. YOu kinda made my knowledge a whole piece. A quite small piece, but a whole one.... in any case.... cu, and ty very much | 2017-01-18 16:39:49 | {"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": 1, "mathjax_display_tex": 2, "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": 0.835168719291687, "perplexity": 2659.016257035569}, "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-2017-04/segments/1484560280308.24/warc/CC-MAIN-20170116095120-00299-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://yutsumura.com/tag/nullspace-of-a-matrix/ | # Tagged: nullspace of a matrix
## Problem 713
Determine bases for $\calN(A)$ and $\calN(A^{T}A)$ when
$A= \begin{bmatrix} 1 & 2 & 1 \\ 1 & 1 & 3 \\ 0 & 0 & 0 \end{bmatrix} .$ Then, determine the ranks and nullities of the matrices $A$ and $A^{\trans}A$.
## Problem 712
Let $A$ be an $m \times n$ matrix.
Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$. | 2021-01-23 08:04:51 | {"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": 2, "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": 0.8386180400848389, "perplexity": 245.2244205854226}, "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-2021-04/segments/1610703536556.58/warc/CC-MAIN-20210123063713-20210123093713-00778.warc.gz"} |
https://mathematica.stackexchange.com/questions/174758/boundary-thickness-in-highlightimage?noredirect=1 | # “Boundary” thickness in HighlightImage
Bug introduced in 11 or earlier and persisting through 11.3
CASE:4076121
I'm having a hard time to understand how I can use HighlightImage and only draw the boundary of a binary image with a thickness of exactly one pixel. The documentation suggests that I can use
HighlightImage[
img,
{{"Boundary", thickness}, Binarize[img]}
]
but that doesn't work properly. Here is a small example that shows the behavior on Mathematica 11.3 on OS X
img = Import["http://i.stack.imgur.com/DLGJC.png"];
Manipulate[
With[{img = ImageResize[img, size]},
Show[HighlightImage[
img,
{{"Boundary", thickness}, Binarize[img]}
], ImageSize -> 1024
]
], {size, 100, 500, 1}, {thickness, .1, 10}
]
As you will see, the thickness setting has no influence and if you scale the image up with the size slider, then the boundary thickness covers many pixels
• I noticed this too, but didn't complain to WRI. It changed in some recent version. It seemed intentional ... – Szabolcs Jun 6 '18 at 14:05
• @Szabolcs It seems like they broke this functionality when refactoring the code (see my answer below) - no idea why they did it in the first place though – Lukas Lang Jun 6 '18 at 14:11
• @Szabolcs always complain. It may get fixed )) – Batracos Jun 19 '18 at 18:38
• @LukasLang it is indeed a bug that slipped through the testing. I just committed a fix. As you where wondering, the reason is that directives like "Boundary" should change the behaviour of all the subsequent primitives, resetting other custom scopes like "Blur". – Batracos Jun 19 '18 at 18:43
• @Batracos That's great to hear! Regarding the reasoning: Thanks for the insight. I was also wondering why some definitions are only loaded by toGraphicsPrimitiveDefinitions - is this something entirely related to other internals or is there some reason behind it that could also be relevant for custom code? – Lukas Lang Jun 19 '18 at 18:49
The following fixes the issue:
If[\$VersionNumber>=11.3,
Begin["ImageInteractiveDump"],
Begin["ImageColorOperationsDump"]
];
HighlightImage;
DownValues[toGraphicsPrimitive] = DownValues[toGraphicsPrimitive] /.
{HoldPattern[pre_; Sequence[s__]] :> ((pre; {s}) /. {res__} :> res)};
End[];
### Why does this work
It appears that at some point, every definition of ImageInteractiveDumptoGraphicsPrimitive (toGP in the following) was prepended with toGraphicsPrimitiveDefinitions (so toGP[args]:=rhs became toGP[args]:=(toGPD;rhs)). It looks like this loads some more definitions for toGP, but no idea what the purpose of doing it this way is...
The issue is that the signature of toGP that handles {"Boundary", thickness_} returned a Sequence, which is evaluated prematurely by the CompoundExpression that got wrapped around. The above fix resolves this issue by only introducing the Sequence head after the CompundExpression is done evaluating. (please leave a comment if you find a more straightforward solution to protect the Sequence head)
• Thanks for spelunking for me. I was really worried I simply use it wrong somehow. I'm looking at it in detail later. +1 – halirutan Jun 6 '18 at 15:09
• @Mathe172 - Please how to use the fix, where to put the Begin-End sequence? Is it a procedure, shall I call it before HighligtImage? Thanks in advance – CJoe Jun 10 '18 at 13:33
• @CJoe Yes, just execute it once before you call HighlightImage. For example, you can add it as first line to your notebook – Lukas Lang Jun 10 '18 at 13:36
• @Mathe172 - Im sorry, I cant make it working. Either it does not have influence or the Mathematica crashes (on Pi3). Would you be so kind and put the working code of that your example above? – CJoe Jun 10 '18 at 15:43
• A simpler fix that however requires changing existing code is to use ... , FaceForm[], EdgeForm[AbsoluteThickness[n]], ... in place of {"Boundary", n}`. – Batracos Jun 19 '18 at 18:50 | 2019-11-14 12:21:27 | {"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": 0.2885233461856842, "perplexity": 2419.673466256589}, "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-2019-47/segments/1573496668416.11/warc/CC-MAIN-20191114104329-20191114132329-00279.warc.gz"} |
http://nanofase.eu/show/straining-calculation_1831/ | Straining calculation
Straining, the physical filtration of particles in soils, is modelled similarly to attachment as an irreversible, first-order loss. There is thus a straining rate coefficient, but contrary to the case of attachment, this coefficient decreases as a function of depth. The idea is that particles will be less frequently found in dead-end pores if it is the case that overall, particles are being transported to deeper and deeper layers in the soil.
$$k_{straining}=\Psi k$$where$$\Psi = (\frac{d_{50 + z}}{d_{50}})^{-\beta }$$ $$k_{straining}$$ is the straining rate constant and $$\Psi$$ is the depth-dependent straining rate coefficient. $$k$$ expresses the pseudo-first order rate of the interaction itself. This coefficient depends on the distance $$z$$ from the origin/injection point of the nanomaterials in the porous medium and also on the average aggregate (collector) diameter $$d_{50}$$. $$\beta$$ is an empirical factor expressing the intensity of this depth dependence.
Execution
Straining is usually assumed to occur at the same time as other processes such as attachment. There is thus an assumption of a "second type of interaction site" where straining occurs. The straining rate constant itself, however, is a calibration constant fitted to column outflow experiments. Moreover, b is assumed equal to 0.43.
Used in
$$\theta \frac{dC}{dt}=v\theta \frac{dC}{dz}$$ $$-\theta (\sum D )\frac{d^{2}C}{(dz)^{2}}$$ $$-\Psi k_{straining} - k_{att}$$ Soil transport calculation Straining
Consult the NanoFASE Library to see abstracts of these deliverable reports: Bradford, S.A., et al., Modeling Colloid Attachment, Straining, and Exclusion in Saturated Porous Media. Environmental Science & Technology, 2003. 37(10): p. 2242-2250
Contact
Geert Cornelis
Email: geert.cornelis@slu.se | 2019-09-20 23:26:53 | {"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": 1, "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": 0.9140306115150452, "perplexity": 3317.9599687549926}, "config": {"markdown_headings": false, "markdown_code": false, "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/1568514574084.88/warc/CC-MAIN-20190920221241-20190921003241-00421.warc.gz"} |
https://dataspace.princeton.edu/jspui/handle/88435/dsp015d86p252c | Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp015d86p252c
Title: A multi-agent approach to the evolution of microbial populations in the presence of spatially structured social interaction Authors: Borenstein, David Bruce Advisors: Wingreen, Ned S Contributors: Quantitative Computational Biology Department Keywords: Agent-based modelingConstraint programmingEvolutionary game theoryEvolution of cooperationKinetic lattice monte carloSociomicrobiology Subjects: EcologyComputer scienceMicrobiology Issue Date: 2015 Publisher: Princeton, NJ : Princeton University Abstract: Microbes employ a vast arsenal of tools to manipulate the environments in which they live. These manipulations affect the survival of other microbes and therefore microbial populations evolve in ways that reflect these social interactions. Interactions between microbes are particularly important in structured communities called biofilms. In this thesis, we study biofilm evolution through the lens of social interaction. Microbial biofilms are heterogeneous assemblages that develop on many scales of time and space simultaneously. A major challenge in understanding biofilms, therefore, is developing an informative and tractable model. The thesis begins with a simulation based on an established paradigm, namely, a death-birth agent-based model (ABM) on a regular lattice. In addition to reproducing by replacing neighbors, all of the individuals utilize a shared resource that diffuses through the environment, though only some of them produce it. The realistic treatment of diffusion in this model leads to loss of the coexistence previously observed in similar game-theoretic models involving nearest-neighbor interactions. In the non-local interactions model, the introduction of long-range interactions into a game theoretic model leads to the loss of biologically relevant emergent dynamics, arguing against the generality of game-theoretic lattice models of social interaction. In studying the effects of intermicrobial warfare on community structure, therefore, we instead take a mechanistic approach. Approximately 25\% of Gram-negative bacteria possess at least one Type VI Secretion (T6S) system, which can be used to kill other microbes. Using an ABM that describes the local interactions between cells during T6S attack, we predict that the system can only be used to displace small or diffuse populations. We then use in vivo experiments to verify that the same phenomenon occurs in real microbial colonies. These studies both required the development of ad-hoc ABMs. The process of creating and exploring such a model requires computer skills that are wholly independent from expertise in the biological problems at hand. We therefore conclude with a method for designing ABMs that requires minimal programming knowledge. The technique, which draws on the artificial intelligence field known as constraint programming, replaces step-by-step computer instructions with a simple list of user generated requirements. URI: http://arks.princeton.edu/ark:/88435/dsp015d86p252c Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog Type of Material: Academic dissertations (Ph.D.) Language: en Appears in Collections: Quantitative Computational Biology
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