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computer science – Daniel's Assorted Musings
SMAWK in C++
I recently implemented kmeans1d—discussed in a prior post—for efficiently performing globally optimal 1D k-means clustering. The implementation utilizes the SMAWK algorithm (Aggarwal et al., 1987),
which calculates argmin(i) for each row i of an arbitrary n × m totally monotone matrix, in O(m(1 + lg(n/m))).
I’ve factored out my SMAWK C++ code into the example below. In general, SMAWK works with an implicitly defined matrix, utilizing a function that returns a value corresponding to an arbitrary position
in the matrix. An explicitly defined matrix is used in the example for the purpose of illustration.
The program prints the column indices corresponding to the minimum element of each row in a totally monotone matrix. The matrix is from monge.pdf—a course document that I found online. | {"url":"https://www.dannyadam.com/blog/tag/computer-science/","timestamp":"2024-11-12T05:06:58Z","content_type":"text/html","content_length":"126256","record_id":"<urn:uuid:fbaa2343-415e-48c5-a614-9bf0fff29f3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00043.warc.gz"} |
Evaporative Fraction as an Indicator of Moisture Condition and Water Stress Status in Semi-Arid Rangeland Ecosystems
Institute of Electromagnetic Sensing of Environment, National Research Council of Italy (CNR-IREA), Via Bassini 15, Milan 20133, Italy
Department of Agricultural and Environmental Science, Università degli Studi di Milano, Milano 20133, Italy
Author to whom correspondence should be addressed.
Submission received: 14 April 2014 / Revised: 16 June 2014 / Accepted: 18 June 2014 / Published: 7 July 2014
Rangeland monitoring services require the capability to investigate vegetation condition and to assess biomass production, especially in areas where local livelihood depends on rangeland status.
Remote sensing solutions are strongly recommended, where the systematic acquisition of field data is not feasible and does not guarantee properly describing the spatio-temporal dynamics of wide
areas. Recent research on semi-arid rangelands has focused its attention on the evaporative fraction (EF), a key factor to estimate evapotranspiration (ET) in the energy balance (EB) algorithm. EF is
strongly linked to the vegetation water status, and works conducted on eddy covariance towers used this parameter to increase the performances of satellite-based biomass estimation. In this work, a
method to estimate EF from MODIS products, originally developed for evapotranspiration estimation, is tested and evaluated. Results show that the EF estimation from low spatial resolution over wide
semi-arid area is feasible. Estimated EF resulted in being well correlated to field ET measurements, and the spatial patterns of EF maps are in agreement with the well-known climatic and landscape
Sahelian features. The preliminary test on rangeland biomass production shows that satellite-retrieved EF as a water availability factor significantly increased the capacity of a remote sensing
operational product to detect the variability of the field biomass measurements.
1. Introduction
The ecosystem carrying capacity and food security of the West African Sahel relies on annual vegetation production, which is concentrated in a short rainy period of four months, on average, between
July to October [
]. The majority of the Sahelian livelihood counts on these wet months to get by in the dry season. The management of existing natural resources by the local population has developed several
strategies to cope with climatic difficulties, such as exploiting herd transhumance at the beginning of the dry season [
] or handling the seedling date in the beginning of the rainy period [
However, recurrent erratic rainfall or a drought period could affect Sahelian food security, as happened during the great drought of the past century [
] and recent local food crises [
]. Despite several adaptations of the Sahelian population to erratic climate conditions [
], food security still remains a concern, and an accurate estimation of regional yields plays an important role in food security [
The awareness of rangeland production in relation to water availability is of major interest for the implementation of operational monitoring systems to support policies aiming at reducing the
socio-economic impacts of environmental stresses. As water availability is the main limiting factor for vegetation production, especially where average annual rainfall is lower than 500/600 mm [
], the interest to estimate rainfall and soil moisture at the regional scale in relation with biomass production has earned a lot of attention.
Several recent studies analyzed time series of rainfall and vegetation indices highlighting the Sahel as an area where vegetation production is rainfall driven and only locally influenced by human
activities [
]. Other works in the area compared vegetation production to trends of soil moisture [
] and rain use efficiency [
], identifying water availability as the main driver of vegetation growth and dynamics in the Sahel. A shortwave infrared water stress index (SIWSI) has been proposed as an indicator of vegetation
water stress [
], while a combination of thermal data and vegetation index [
] were used to produce qualitative maps of soil moisture along the Senegal River.
Compared to these methods, the estimation of evapotranspiration (ET) at the regional scale could give a more quantitative assessment of vegetation water status. ET is a key component of the water
budget, and its estimation at different scales is of outmost importance for water management in agriculture [
] and food security programs [
]. ET can be appropriately measured at the field scale by lysimeters, scintillometers or eddy correlation techniques [
]. However, being highly dynamic in space and time because of complex interactions between soil, vegetation and climate [
], the quantification of its flux at the watershed scale is much more difficult than at a specific site [
Traditional methods to estimate ET assume homogeneous vegetation cover and structure, but these conditions are hard to meet for large regions [
]. For studies at regional and continental scales, monitoring models are coupled with remotely sensed data that can cope with the spatial and temporal variability of surface characteristics that
affect evapotranspiration processes [
]. Several surface characteristics, such as albedo, vegetation cover, leaf area index and land surface temperature, can be retrieved from satellite observations providing data for ET estimation from
Since the launch of Earth Observation satellites with thermal infrared channel, such as Landsat Thematic Mapper, NOAA-AVHRR and Terra/Aqua MODIS, several applications have been developed over near
fully agricultural canopy covers and semiarid rangeland basins to estimate instantaneous ET and to scale up such estimations to daily ET.
One of the widely used methods [
] to estimate daily ET is based on the evaporative fraction (EF), which is defined as the ratio between latent heat flux and the total heat leaving the Earth’s surface. A strong correlation between
the value of EF at midday and the daytime average value has been observed [
], and it is often assumed as a constant daytime variable [
The EF has a strong link with soil moisture availability [
], which is the limiting factor of latent heat flux [
], and it is essentially controlled by water availability in the root zone [
]. The EF behavior at the landscape scale is correlated to the amount of vegetation cover [
], the timing of rainfall events [
], the successions of wet and dry periods [
], the vapor pressure deficit and vegetation photosynthesis activity [
EF has an annual behavior related to rainfall events, with peaks during the rainy season and decreasing when soil is drying [
]. In fact, a work conducted over paddy rice area shows that EF always has values close to one, because soil moisture was almost saturated [
Recent works conducted in correspondence with eddy covariance stations in North America [
], the northern Australia savannah [
] and the Sahelian region [
] proposed the EF as an indicator of water stress to correct vegetation production estimation. The results of these studies indicated that the use of field-measured EF values within a light use
efficiency (LUE) model allows one to improve the estimate of biomass production.
EF can be derived from satellite data using the NDVI-temperature triangle method [
] or the simplified surface energy balance index (S-SEBI) model [
], following the relationship between albedo and land surface temperature [
]. This last approach found applications with a wide range of remotely sensed data and in different ecosystems.
The accuracy of EF estimated by S-SEBI was demonstrated in comparison with other approaches, both for high resolution ASTER images [
] and low resolution NOAA-imagery [
]. Daily ET values, estimated via the EF approach, were validated at the field scale with flux measurements on cropland [
], as well as at the regional scale over the Iberian Peninsula with Digital Airborne Imaging Spectrometer (DAIS) high resolution data [
Outside of Europe, the use of this approach has been reported in the Mediterranean landscapes of Chile [
] and in the cotton crops of Brazil [
], demonstrating the suitability of the method in semi-arid areas.
The aim of this work is to retrieve EF from satellite data in the Sahelian rangeland ecosystem and to evaluate the parameter as a moisture indicator, useful also as a correcting factor in the
radiation use efficiency biomass estimation model.
The application of the S-SEBI method requires the presence of wet and very dry surfaces [
]; these conditions are well satisfied over the West Africa area, thanks to presence of the Sahara Desert and stable, humid ecosystems, such as the Niger Inner Delta and Lake Chad. In particular, the
goals of this research are: (i) to set-up an automatic procedure to derive EF maps from MODIS products; (ii) to evaluate EF estimation using
in situ
data and to assess EF maps at the regional scale; and (iii) to evaluate the improvement brought by satellite-retrieved EF to the accuracy of the biomass production model.
2. Study Area
The study area covers 1200 × 2400 km over Niger and Chad. The northern part includes the Sahara Desert, where less than 200 mm of rain falls every year and human presence is almost absent (
Figure 1
). The central part is located on the Sahelian belt, identified by the isohyets of 200 and 600 mm. This zone is mainly characterized by semi-arid savannah, where pastoralism is the most important
livelihood activity, with localized evidence of agricultural activity (<20% of cultivated areas; [
]). The southern part of study area belongs to the Sudanian savannah, characterized by wetter climate (annual rain greater than 600 mm), an intensive farming system and less dependency on rain for
vegetation productivity [
]. The rainy season of the whole study area is essentially from July to October and slightly longer in southern areas, with almost zero precipitation during the rest of the year.
A number of humanitarian crises have hit this area over recent years; although several are from concurring causes, such as food supply, livestock management, environmental degradation and household
coping capabilities, low or erratic rainfall remains the key factor triggering the crisis [
]. In this region, the population has increased during the past 25 years [
]. The rural population is still growing, contrary to many other parts of the world, leading to heavy pressure on the environment, especially during adverse years.
3. Materials
3.1. Earth Observation Data
According to the S-SEBI approach [
], the retrieval of EF from satellite data is based on the relationship between albedo and land surface temperature (LST). Two MODIS products were then considered: MCD43B3,
, the hemispherical reflectance (black-sky albedo) 8 days at 1-km spatial resolution, and MOD11A2,
, land surface temperature 8 days at 1-km spatial resolution [
]. In order to cover the entire area of interest, two MODIS tiles (h18v07, h19v07) were downloaded for 10 years (2000–2009), summing up to about 450 images per tile.
Other satellite-derived products were used for analysis and evaluation purposes. For the analysis of the EF contribution to biomass production, we used dry matter productivity (DMP) maps [
]. DMP is a satellite-derived product, developed at the Flemish Institute for Technological Research (VITO), that quantifies the daily increase of dry biomass (growth rate) and is expressed as
kilograms of dry matter (kg∙DM) per hectare per day. The DMP product used in this exercise is a 10-day composite at 1-km spatial resolution covering the period 2000–2009.
Ancillary satellite data consist of rainfall and vegetation maps for the study area. Rainfall estimation (RFE 2.0) is provided by Famine Early Warning Systems Network (FEWS) every 10 days at 8-km
spatial resolution [
]. RFE 2.0 is produced by a combination of Meteosat 5 data (satellite infrared data) and daily rain gauge data extracted from the WMO’s Global Telecommunication System (GTS) with the additional
integration of the two new Special Sensor Microwave/Imager (SSM/I) instruments on-board the Defense Meteorological Satellite Program satellites and the Advanced Microwave Sounding Unit (AMSU).
Vegetation maps are represented by Normalized Difference Vegetation Index (NDVI) provided by the SPOT-Vegetation satellite (VGT) sensor every 10-days at 1-km spatial resolution [
]. Finally for the analysis of EF behavior for different vegetation types, the regional GlobCover (GC) map for Africa with 300-m spatial resolution was used [
], which describes land cover classes over the entire study area.
3.2. Field Biomass and Flux Measurements
Field biomass data have been provided by Action Against Hunger (ACF) for three different sites in Niger (
Figure 1
). Three site were analyzed: the first site is located in a tiger bush area 35 km north of Nigeria border (Site 1, Longitude 10.9, Latitude 13.7); the eastern site is located around Lake Chad (Site
2, Longitude 12.8, Latitude 13.95), while the northern site (Site 3, Longitude 6.8, Latitude 15.8) is located around Agadez, which is the upper limit of pasture activities [
]. Biomass measurements were collected following the quick double-sampling technique [
] to calibrate/validate ACF satellite maps of available forage [
]. Overall, 19 annual biomass values/data are available for the period of 2000–2009 (
Table 1
). These ground samples provide crucial information for the evaluation of EF capability as a water stress factor in biomass estimation.
Flux measurements were collected at an eddy covariance tower situated in the Wankama catchment (
Figure 1
), 60 km east of Niamey, Niger. This site presents the typical Sahelian landscape with sparse savannah and millet fields. Daily data of net radiation (W∙m
) was measured every minute by the tower instruments at a height of 2.5 m; these data are supplied to the user from the CarboAfrica project through FLUXNET measurement network as the average over
30-min periods [
]. This variable is available as a Level 2 product,
, not gap-filled, but checked/filtered for out-of-range values or clearly wrong data [
]. The daily latent heat flux data (W∙m
), processed with despiking, double rotation and gap filling following the indications of [
], were obtained from the publication of [
]. Both fluxes are available for the period between June 2005 and June 2007, including the wet season of 2005 and 2006.
4. Method
4.1. Estimation of Evaporative Fraction
The more widely applied method for ET estimation with passive remote sensing is the energy balance equation [
]. The land surface energy balance is the thermo-dynamic equilibrium between turbulent transport processes in the atmosphere and laminar processes in the sub-surface [
The basic formulation can be written as:
is the net radiation, λ
is the latent heat flux (λ is the latent heat of vaporization of water and E is evapotranspiration),
is the soil heat flux and
is the sensible heat flux.
Evaporation and transpiration occur simultaneously, and there is no easy way of distinguishing between the two processes: when the crop is small, water is predominately lost by soil evaporation, but
once the vegetation completely covers the soil, leaf transpiration becomes the main process [
]. Many satellite-based approaches estimate daily ET, exploiting the
factor [
], defined as the ratio between the latent heat flux (λ
) and the available energy at the land surface (
In the present work, the
estimation is obtained using the albedo-temperature method [
]. This approach allows one to compute the
for every pixel as the relative distance from two lines, called the dry edge and wet edge, defined through a date-specific albedo-LST relationship (
Figure 2
). The method’s accuracy is dependent on the presence of humid and arid surfaces in the study area.
Figure 3
provides the flow chart of the steps followed for the EF estimation from satellite products, for each available date of satellite products. Albedo and LST data were extracted from the digital numbers
(DN) of MCD43B3 (layer 10) and MOD11A2 (layer 1), as indicated by the MODIS product description [
], while information on LST data quality as derived from Layer 2 of the MOD11A2.
Before starting the EF calculation, pixels flagged as “no-data” or “low quality” were masked out and excluded from the analysis.
To perform the
estimation, the albedo-LST scatterplot is derived for a single date (
Figure 2
) and analyzed to extract minimum and maximum temperature values for all of the albedo classes identified from statistical analysis [
The series of maximum and minimum LST values are used to calculate the date-specific dry and wet edge equation through linear regression:
$dry edge : T H = m dry α 0 + q dry$
$wet edge : T λ E = m wet α 0 + q wet$
represent the parameters (slope and intercept) of the two regression lines, α
represents the albedo, while T is the land surface temperature.
The dry edge was defined considering only pixels in the radiative controlled condition, commonly identified as the maximum temperature data for all of the albedo values greater than the inflection
point of the concave temperature-albedo scatterplot [
]. This condition was empirically defined for an albedo value above 0.2, as used also by [
Exploiting the dry and wet edge, the EF can be calculated for every pixels
, dividing the difference between T
and the temperature pixel T
by the difference between
$EF i = T H i − T s i T H i − T λ E i$
is the temperature value of the pixel
are respectively the maximum and minimum temperature value derived by the dry and wet edge functions for a given albedo value α
equation can be rewritten as:
$EF i = ( m dry α i + q dry ) − T S i ( m dry α i + q dry ) − ( m wet α i + q wet )$
this procedure, implemented with an
ad hoc
code in IDL language (Interactive Data Language, version 8.2), was applied to each pixel of the image and on each date available for both MODIS tiles h18v07 and h19v07.
The maps estimated at the same dates were mosaicked, obtaining EF maps of 122 × 2400 km to cover the entire study area.
4.2. Evaluation of the Estimated EF
Since it is well known that
is related to water availability provided by rainfall, particularly in the natural vegetation of semi-arid environments, vegetation growth and land cover [
], we used the RFE, SPOT-VGT NDVI and GlobCover classes to assess the consistency of EF estimation. In particular, average and relative standard deviation (RSD) maps of EF are computed from the 448
8-day EF maps and analyzed for the major land cover classes of the study area, thanks to GlobCover map. This analysis was conducted in order to evaluate the coherence between the EF and the expected
behavior over different vegetation covers.
In correspondence with the eddy covariance tower, EF behavior was compared to rainfall events and vegetation growth.
Quantitative evaluation of the reliability of EF estimations as a moisture (water stress) indicator is accomplished using the eddy covariance data from Wankama station.
Due to the different time steps of satellite estimation and flux measurements, the satellite-derived EF was compared with the 8-day average of daily ET corresponding to the MODIS composite. Data from
the tower measurements identified as outliers from statistical analysis and EF satellite estimation flagged as low quality were excluded by the analysis. Moreover, thanks to
Equation (7)
, it was possible to compare
in situ
estimation of EF with satellite-derived
$E F eddy = λ E d / R n d$
and λ
are the daily net radiation and the daily latent heat flux, respectively.
4.3. Biomass Estimation
The seasonal cumulative DMP is an indicator of the annual rangeland production [
], which can be compared to the ground data, that represents the total annual herbaceous production measured in field.
In order to compare satellite data with field samples, EF and DMP were extracted in correspondence with the field sites location in a buffer of 1 km. The 10-day DMP product for the period of July to
October, referred to here as JASO, was cumulated to obtain the annual syntheses of dry matter production (DMP^JASO).
The EF was exploited as a water availability factor to correct the satellite estimation of vegetation biomass (DMP).
DMP and EF have different time steps (10 and 8 days, respectively); consequently, monthly values were calculated in order to use the water availability/stress factor in the biomass estimation model.
, the monthly average (
$EF ¯ m$
) was computed for every month (
) and every site (
$EF ¯ s m = Σ t = 1 n EF s , t 8 D n$
is the estimated water stress from
Equation (6)
is the site and
is the cardinality of the 8-day
data for each month.
, the monthly sum (
) was calculated to represent the total dry biomass produced during every month at each site:
$D M P s m = ∑ t = 1 3 D M P s , t 10 D$
is the 10-day biomass estimation product,
is the number of DMP data within the month and
is the site.
$EF ¯ m$
values were than integrated and annually cumulated by the following equation for each site:
$DMP s JASO * = ∑ m = 1 4 DMP s m ⋅ EF ¯ s m$
are the variable obtained from
Equations (8)
is the site and 4 is the number of months in the JASO period.
Finally, to quantify the improvement of DMP
*, the comparison between observed and estimated values were performed and difference-based statistics [
] together with regression analysis and Akaike information criterion (AIC) [
Equation (11)
, were conducted.
$AIC = n ⋅ log ( MSE ) + 2 ⋅ T$
is the number observed/simulated pairs,
is the mean square error and
is the number of inputs in the model.
AIC with a lower value indicates whether the increase in the input of a model is compensated for by a significant increase in accuracy.
5. Results and Discussion
5.1. Dry and Wet Edge Statistics
Figure 4
shows the average intercepts (a) and slopes (b) for the calculated dry (empty triangles) and wet (filled circles) edge. Every point represents the average of 10 estimations from 2000 to 2009 together
with a bar representing the standard deviation. The dry edge statistics of the slope and intercept are on the second y-axis, to facilitate a comparison with the wet edge statistics. The gray shaded
area displays the period when generally no rainfall occurs in the study area.
The intercept of the dry edge follows the typical behavior of West African temperature [
], with lower values during the wet season (June–October) and two peaks during the dry season, the former in April and the latter in November.
The Wet edge intercept has lower value in the wet season and stable, higher values during the dry one. The average slope coefficient shows that during the dry season, the wet edge is generally
horizontal (values close to zero), while the dry edge has a high negative slope (values down to −60), as shown by a similar analysis conducted in the Mekong Delta [
On the contrary, in the rainy period, the dry edge is almost flat, while the wet edge has a strong positive slope (values up to 60). In general, the coefficients of wet and dry edge follow a seasonal
behavior driven by rainfall and incoming solar radiation.
Figure 4
shows also three examples of the albedo-LST scatterplots. The first displays the dry season condition with the flat wet edge and the second one the wet condition with the flat dry edge. The last
scatterplot displays an intermediate condition at the end of the rainy season, when the two lines are both oblique and the maximum LST is higher.
The maximum albedo value of 0.6 in the scatterplots highlights the presence of high reflective surfaces [
], which correspond to brighter desert areas. These areas are stable through the season; hence, they are present in every plot.
The areas with lower temperature (below 305 K) and lower albedo (below 0.2) correspond to a permanent humid zone, such as the Lake Chad area and the border of the Niger River.
The permanent presence through the years of these two extreme situations allows one to produce a meaningful scatterplot describing the contrast between dry and wet areas, hence guaranteeing the
conditions for the application of the method [
5.2. Evaluation of EF Spatial Patterns
Figure 5
shows the mean EF map obtained from the estimated 448 maps for the period of 2000–2009 (a), together with the RSD (b). As expected, the mean values vary between zero and one, where zero indicates the
hyper-arid condition and one the humid area. The areas with a mean rainfall below 200 mm, belonging to the Sahara Desert, were excluded from the analysis, since these areas are not populated and EF
estimation makes sense and is useful only on partially vegetated surfaces. Permanent arid areas (
< 0.2) can be found close to the desert, especially in the Agadez province (Niger) and in the central part of Chad. Both of these areas rely on the “high-risk Sahel’s vulnerable zone”, where the main
livelihood activity is transhumant herding [
The hyper-humid area (
> 0.7) can be found in correspondence with permanent water bodies, such as Lake Chad [
], Lake Fitri (central Chad) and Lake Kainji (west Nigeria). Furthermore, the woody hills in Nigeria, characterized by a high level of rainfall (more than 1200 mm/year), are generally well humid. The
Sahelian belt is characterized by medium-low average values of
(below 0.5) apart from the river belts: Niger in the west part of study area, Yobe along the Niger-Nigeria border and Chari south of Lake Chad.
The RSD (
Figure 5b
) is a normalized measure of EF data dispersion. The equation of the RSD is obtained dividing the standard deviation by the mean. The lower percentage indicates a lower variability in the EF time
The map highlights areas in red and orange with stable
(RSD < 30%) from 2000–2009. These areas belong to lakes, rivers and wet regions in central Nigeria, which are also the regions characterized by high
average values (
Figure 5a
). Hence, these well-watered areas have maintained their condition across the years analyzed.
Vice versa
, higher variation in EF values (RSD > 50%, blue and light blue) can be found in the northern Sahel (northern-western Niger and central Chad). In particular, in western Niger, the fossil valleys
display a stronger variability of EF data compared to the surrounded rangelands, because of the greater water availability thanks to the morphopedological characteristics, as observed in [
]. The high RDS indicates that the EF data of these areas can vary abruptly, thanks to the strong seasonality (intra-annual variability).
The high EF variability of northern areas in Niger, characterized by the small EF average, can be driven by particularly favorable years (inter-annual variability).
The average EF map has been analyzed by GC classes (
Figure 6
). The GC classes are sorted from the mainly northern classes (GC_200) to the southern (GC_130), except for the classes of wetland (GC_180) and water body (GC_210). The most common classes are the
bare areas (GC_200) and grassland savannah (GC_140). These two classes cover 70% of the entire study area.
On average, the GC class with the highest EF value (EF = 0.83) is water bodies (GC_210). Among the vegetation classes, only irrigated crops (GC_10), forest (GC_60) and wetland have a mean EF greater
than 0.6. The most arid classes (GC_140 and GC_200), with a mean EF lower than 0.4, describe the typical landscape of the northern Sahel [
This analysis shows that the spatial patterns of EF data (long-term average) are in agreement with the well-known climatic and landscape features of these areas. A similar analysis conducted in China
], Europe [
] and Africa [
] demonstrated that EF maps build up spatial and temporal patterns coherent with the presence of different vegetated surfaces, different climatic conditions and different seasonal behavior of
5.3. Comparison of Seasonal EF Estimations with Eddy Covariance Data
5.3.1. Temporal Dynamics of the Variables
Figure 7
presents the time series of net radiation, ET and EF measured at the Wankama eddy covariance tower (black lines) and the satellite-derived time series of EF (red dashes), RFE (blue bars), NDVI (green
line), albedo (cyan line) and LST (gold line) extracted by the corresponding image pixel for the years 2005 (
Figure 7a
) and 2006 (
Figure 7b
). The Wankama eddy tower, placed in millet fields, is characterized by the typical Sahelian behavior of rainfall and vegetation growth [
Vertical black lines indicate the average Sahelian wet season as being from July to October (JASO). The zero value of the satellite EF estimation, due to cloud contamination or other atmospheric
interference in the data, was masked out from this analysis.
Figure 7a
shows the time series of remote sensed and measured variables for the year, 2005. The first eddy measurement was recorded in June (doy 160), after the beginning of rain. ET shows the peak (>4 mm/day)
in August, as well as net radiation. RFE shows an early start of the rainy season compared to the JASO, with an intense rainfall period of 60 mm in May (doy 155). In total, 524 mm fell in 2005.
The red dashes indicate the eight-day period of satellite EF estimation from
Equation (6)
together with the
in situ
calculated EF from
Equation (7)
. Both of the EF time series show higher values in the rainy season and drops in correspondence with the low ET value (e.g., doy 210 and 240). MODIS-derived EF decreases smoothly after doy 270, as
happens for ET.
The vegetation behavior, highlighted by NDVI, shows the start of the season around doy 200 (19 July), about 50 days after the start of rainfall, because of the necessary time for germination [
]. NDVI has a specular behavior compared to albedo, as expected from the progressive cover of bare soil due to vegetation growth. The last two time series in
Figure 7a
display albedo and LST data. Both show a high value during the dry season, indicating a warm and bare surface. The rainy season has an average LST of 308 K (∼35 °C), lower than the dry season average
(311 K), because incoming energy during is exploited by evaporative and transpirative processes.
Figure 7b
shows the same variables for 2006. The rainy period is shorter and less abundant compared to the previous one, with 430 mm of total rainfall. The estimated EF reaches a peak of 0.8 at doy 240 (28
August). The main EF drop is visible (doy 210) in correspondence with the drier period of the wet season, between the two main rain events. Both EF and ET rapidly decrease their values at the end of
the wet season (doy 270). Higher vegetation growth occurs between August and September, and NDVI shows the presence of vegetation also in November–December (doy 300–360, NDVI ∼0.2), even if EF and ET
show that the area is completely dry.
The temporal behaviors of field measurements and satellite-derived data for 2005 and 2006 display high variability between dry and wet months. The 2005 wet season had an early start, while 2006 had a
very late start, as well as an earlier end. Hence, the two years had different seasonality in terms of rainfall amount and distribution [
]. Among the several satellite-derived variables, estimated EF shows a higher correlation with estimated rainfall (data not shown), as expected from previous field studies [
], and in general, the estimated EF looks to be in accordance with the ET behavior and eddy covariance-derived EF. The temporal behavior of the EF variable is more noisy compared to the time series
of other satellite-derived variables and hardly zero also in the absence of rainfall [
], as also displayed by eddy-derived EF.
Both MODIS estimation and eddy EF have comparable values during the wet season (JASO), showing a higher average value for the wetter year, 2005 (μ = 0.45; σ = 0.07 for satellite and μ = 0.51; σ =
0.13 for in situ, respectively), when compared to the drier 2006 (μ = 0.39; σ = 0.07 and μ = 0.34; σ = 0.09 for the satellite and in situ, respectively).
5.3.2. Correlation Analysis with ET
In order to evaluate the reliability of EF as a moisture indicator, a correlation analysis has been conducted between satellite
estimation (y-axis) and ET measured by the eddy covariance tower (x-axis) (
Figure 8
The EF resulted in being significantly correlated to ET (
< 0.001), with a regressive coefficient of 0.54 pooling together the two years (2005–2006). ANCOVA analysis reveals that single year correlations (2005,
= 0.62; 2006,
= 0.45) were not significantly different (
< 0.05). This correlation is biased by estimated EF in the late, dry season (January–May), when no rain and no vegetation are present, confirming that EF is noisy in the dry season [
As expected the measured net radiation is better correlated with ET (r^2 = 0.64), since it represents the climatic driving force of evaporative and transpirative processes. In order to investigate if
EF can improve the capability to explain the variance of ET, a multiple regression was performed between ET as a dependent variable and two independent variables, the measured Rn and the simulated EF
Result shows that both the explanatory variables significantly contribute to the explanation of ET variability (70% of the total variance). Rn resulted in being more important, explaining about 64%
of the total variance (p < 0.001), and EF significantly improved the model with a further 6% of variance explanation (p < 0.01). These results indicate that EF estimated with low resolution satellite
data is well correlated with the field measured flux and gives a statistically significant contribution to the explanation of ET variability. It is important to remember that the EF data is derived
by 1-km albedo and LST products; this aspect can strongly limit the comparison with field data acquired on small plots in a heterogeneous environment.
5.3.3. Biomass Estimation Improvements Using EF Correction
The results of previous analysis confirmed the validity of
as a moisture indicator supporting the idea of using this satellite estimation as a water stress factor in a radiation use efficiency model. Previous studies, exploiting only field-based
, demonstrated that
can be exploited as a water stress efficiency factor [
To assess this contribution, the performance of operational products (DMP^JASO) and biomass estimation corrected by EF (DMP^JASO*) were compared with available annual production data over three test
sites in Niger.
Figure 9a
are shown the three sites’ specific correlations between the available field data and DMP
. The three sites show different correlations: in particular, Site 1 (black dots) presents little correlation (
= 0.49, intercept = 1300, slope = 0.3); Site 2 (blue squares) shows an average correlation (
= 0.51, intercept = 700, slope = 1.1); and Site 3 (red triangles) has a high correlation (
= 0.66, intercept = 400, slope = 0.3). All three sites have the typical Sahelian biomass production [
], ranging from 100 (kg·ha
), in adverse years, to 20-times higher production in favorable climatic conditions.
Results demonstrate that the DMP^JASO is able to detect the field biomass variability with site-specific, good correlation; however, the analysis of intercept and slope variability across sites
indicates that the model is not able to give a robust quantitative biomass estimation. Indeed, the DMP algorithm does not take into account distinct efficiency factors in the conversion of light into
biomass among different vegetation types. It should be reminded that, despite the three test sites featuring the same land cover and eco-region, the actual floristic composition and ecological
characteristics could be much more different.
Figure 9b
are shown the effect of the EF contribution (DMP
*) over the three sites. The plots show a general increase in the capacity of the remote sensing estimation in each site to detect the variability of the field measurements if water stress is taken
into account, as indicated by the increasing of regression coefficients. Moreover, in particular, the EF has reduced the overestimation of the model for poorly productive years, as shown by
intercepts closer to zero.
Due to the observation of site-specific DMP product performance, in order to directly compare the two biomass estimations, satellite products and field data were normalized for each site. The
normalized data allows one to remove the effect of local differences in the relation between satellite outputs and field biomass, visualizing only the overall model capability to detect the field
data variance, rather than absolute values. Data were standardized and converted to z-scores by subtracting from each value the site average and then dividing the result by the standard deviation. In
the analysis of time series, the z-score is a dimensionless quantity adopted to convert variables with different scales to a common domain [
Figure 9c
are shown the normalized data, both for DMP
(gray dots) and DMP
* (black dots). The data close to zero are near the population average, while data values below or above zero indicate a positive or a negative anomaly, respectively. The top right and bottom left
corners indicate that years were estimated, and the measured variables’ data are in agreement.
The correlation coefficient of the normalized DMP^JASO* (r^2 = 0.73, p < 0.001) indicates that there is a significative increase in the capacity of the remote sensing estimation to explain the
variance of annual field biomass measurements if water stress (EF) is taken into account.
This result is in accordance with previous work [
], even if the analysis was conducted at a monthly time step using EF derived with MODIS data (LST and albedo), rather than with field-measured EF and MODIS EVI at an eight-day time step.
Finally, AIC was calculated in order to evaluate whether the increase in the input of the model compared to the basic DMP was compensated for by a significant increase in accuracy (a lower AIC value
indicates a convenient model improvement) [
]. Despite the correction, proposed to increase the number of inputs in the biomass estimation, the usefulness of the EF approach (DMP
*) is confirmed by the improved model performance, indicated by the higher correlations shown in
Figure 9c
and the lower AIC value (106) compared with the one obtained with DMP
6. Conclusions
The work exploited an automatic procedure to calculate multitemporal evaporative fraction maps from low resolution albedo and land surface temperature satellite data over Niger and Chad. Up to now,
this is the first time that multiyear (2000–2009), eight-day maps of the evaporative fraction were produced from low resolution satellite data and analyzed for the West African Sahel. The adopted
methodology, based on previous scientific works and well suited for semi-arid areas, allowed producing maps able to identify patterns of wet and dry condition, which are coherent with the main
ecological features related to land cover classes and precipitation regimes.
The satellite estimation of the evaporative fraction, despite the uncertainty related to the 1-km resolution of the data, resulted in being correlated with the measurements of evapotranspiration (r^2
= 0.54, p < 0.001) acquired for two years (2005–2006) by an eddy flux tower in Niger. The total variance of evapotranspiration is mainly explained by the measured net radiation (64%, p < 0.001),
while the estimated evaporative fraction significantly improves the model with a further 6% of variance explanation (p < 0.01). These results demonstrate that the satellite-derived evaporative
fraction is a reliable indicator of moisture, useful for savannah status monitoring.
We further tested the use of the evaporative fraction as a water availability indicator to improve the accuracy of an operational remote sensing product of biomass estimation based on the radiation
use efficiency concept. When the satellite-derived evaporative fraction is used as an indicator of water stress in the model, the correlation between annual biomass ground measurements and satellite
estimations, for 19 samples over three sites, significantly improves (r^2 = 0.73, p < 0.001) compared to the performance of the basic satellite product (r^2 = 0.54, p < 0.001). The appropriate water
efficiency term derived from optical and thermal remote sensing data represents an advancement over previous studies conducted using only the evaporative fraction derived by in situ eddy covariance
These findings are encouraging for the monitoring of biomass over wide savannah areas using a satellite-based approach. Future studies are needed to better parameterize the radiation use efficiency
model and to calibrate existing products over different ecosystems, in order to take into account the limiting factors and efficiency in the conversion of light into biomass.
This research was partially supported by the Geoland-2 project (Contract No. 218795), which is a collaborative project (2008–2012) funded by the European Union under the seventh Framework Programme (
) and by Space4Agri project in the framework agreement between Regione Lombardia and National Research Council (Convenzione Operativa No. 18091/RCC, 05/08/2013). We also acknowledge the three
anonymous reviewers for the help provided in improving this manuscript.
Author Contributions
Conceived and designed the experiments: Mirco Boschetti, Francesco Nutini, Stefano Bocchi. Performed the experiments: Gabriele Candiani, Francesco Nutini. Analyzed the data: Francesco Nutini, Mirco
Boschetti, Pietro Alessandro Brivio. Wrote the paper: Francesco Nutini, Mirco Boschetti, Gabriele Candiani, Pietro Alessandro Brivio.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1.
The study area overlaid on the regional GlobCover (GC) map of Africa [
]; the red star shows the position of the eddy covariance station; the red diamonds represent the field sites; the blue lines indicate the isohyet boundaries of 200–600 mm/year.
Figure 2. Scatterplot between surface albedo and LST. Blue circles correspond to minimum temperature values for each albedo class, which are used to compute the wet edge (lower limit of the graph)
through linear regression. Red circles correspond to the maximum temperature values for each albedo class, which are used to compute the dry edge (upper limit) through linear regression. T[H]
(maximum temperature) and T[λE] (minimum temperature) represent the values used in the calculation of the EF for the pixel i.
Figure 3. Flowchart for the evaporative fraction estimation from the MODIS products of albedo and land surface temperature.
Figure 4. Eight-day average values of the intercept (a) and slope (b) obtained from dry and wet edge lines for the 2000–2009 period. Shaded gray areas represent the dry season. Plots show three
albedo-LST scatterplots for the year, 2009.
Figure 5. The map of the average EF (a) and relative standard deviation (b) derived from 448 EF eight-day maps (2000–2009). Isohyets were calculated from rainfall estimation (RFE) data for the same
period. The hyper-arid areas (<200 mm∙year^−1) are masked out, and the GlobCover map is in the background.
Figure 6. Percentage of GC classes over the study area (codes and map color are reported) and the statistics of EF data for each LC classes (average (AVG) and relative standard deviation (RSD)). Red
and green indicate land cover with a lower of a higher EF average, respectively.
Figure 7. From top to bottom, the temporal behavior of daily net radiation; daily evapotranspiration; EF-derived from the eddy covariance tower data at the Wankama site (black lines) together with
eight-day EF estimation from MODIS data (red dashes); decadal NDVI-VGT (green line); decadal precipitation (blue bars), eight-day MODIS albedo (gray line) and eight-day MODIS temperature (yellow
line) for 2005 (a) and 2006 (b). Vertical lines represent the start and finish of the JASO period, doy the Day Of the Year.
Figure 8. Correlation between estimated EF (y-axis) and measured ET (x-axis) for both years 2005 (gray) and 2006 (purple) (n = 57).
Figure 9. The correlation between annual biomass samples and satellite estimation DMP^JASO (DMP, dry matter productivity) (a), DMP^JASO* (b) and normalized data (c) (n = 19). Black dots for Site 1,
blue squares for Site 2 and red triangles for Site 3. Black and gray diamonds represent normalized DMP^JASO and DMP^JASO, respectively. The dotted line indicates the 1:1 line.
Table 1. Field data cardinality and average sampled values of the three field biomass
Site #Data Period AVG (kg/ha) Max (kg/ha) Min (kg/ha) Stand Deviation (kg/ha)
Site 1 6 2003; 2005–2009 963 1,463 342 508
Site 2 8 2000; 2002–2009 371 1,047 0 378
Site 3 5 2001; 2005; 2007–2009 888 1712 326 614
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Nutini, F.; Boschetti, M.; Candiani, G.; Bocchi, S.; Brivio, P.A. Evaporative Fraction as an Indicator of Moisture Condition and Water Stress Status in Semi-Arid Rangeland Ecosystems. Remote Sens.
2014, 6, 6300-6323. https://doi.org/10.3390/rs6076300
AMA Style
Nutini F, Boschetti M, Candiani G, Bocchi S, Brivio PA. Evaporative Fraction as an Indicator of Moisture Condition and Water Stress Status in Semi-Arid Rangeland Ecosystems. Remote Sensing. 2014; 6
(7):6300-6323. https://doi.org/10.3390/rs6076300
Chicago/Turabian Style
Nutini, Francesco, Mirco Boschetti, Gabriele Candiani, Stefano Bocchi, and Pietro Alessandro Brivio. 2014. "Evaporative Fraction as an Indicator of Moisture Condition and Water Stress Status in
Semi-Arid Rangeland Ecosystems" Remote Sensing 6, no. 7: 6300-6323. https://doi.org/10.3390/rs6076300
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▷ Leading Coefficient of a Polynomial (definition & examples)
Leading coefficient of a polynomial
On this post we explain what the leading coefficient of a polynomial is and how to find it. Also, you will see several examples on how to identify the leading coefficient of a polynomial.
What is the leading coefficient of a polynomial?
The definition of leading coefficient of a polynomial is as follows:
In mathematics, the leading coefficient of a polynomial is the coefficient of the term with the highest degree of the polynomial, that is, the leading coefficient of a polynomial is the number that
is in front of the x with the highest exponent.
For example, the leading coefficient of the following polynomial is 5:
The highest degree term of the above polynomial is 5x^3 (monomial of degree 3), therefore the coefficient of the maximum degree term is 5. And, consequently, the leading coefficient of the polynomial
is equal to 5.
Note that if a polynomial is in standard form, the leading coefficient will always be the coefficient of the first term.
Moreover, the term with the highest degree is also called leading term. Thus, the leading coefficient is the coefficient of the leading term of the polynomial.
As you can see, to determine the leading coefficient of a polynomial you must know how to calculate the degree of all the terms of a polynomial. When the polynomial has only one variable is quite
easy, but finding the leading coefficient when the polynomial has two or more variables it is more complicated. You can see how to calculate the degree of a term with two variables in the following
➤ See: degree of a polynomial with two variables
Examples of how to find the leading coefficient of a polynomial
Once we know how to identify the leading coefficient of a polynomial, let’s practice with several solved examples.
• Example of the leading coefficient of a polynomial of degree 4:
The highest degree term of the polynomial is 3x ^4, so the leading coefficient of the polynomial is 3.
• Example of the leading coefficient of a polynomial of degree 5:
The term with the maximum degree of the polynomial is 8x^5, therefore, the leading coefficient of the polynomial is 8.
• Example of the leading coefficient of a polynomial of degree 7:
The highest degree element of the polynomial is -6x ^7, thus, the leading coefficient of the polynomial is -6. Note that the negative sign is also part of the coefficient.
Leading coefficients and graphs
The graph of a polynomial function depends on the sign of the leading coefficient and the exponent of the leading term as follows:
• If the leading coefficient is positive and the exponent of the leading term is odd, the graph falls to the left and rises to the right.
• If the leading coefficient is negative and the exponent of the leading term is odd, the graph rises to the left and falls to the right.
• If the leading coefficient is positive and the exponent of the leading term is even, the graph rises to the left and right.
• If the leading coefficient is negative and the exponent of the leading term is even, the graph falls to the left and right.
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An Observation from the Brink.
This post is not reliable:
See correction
I've been thinking about the relationship between volume and area, so I plotted a scatter plot of PIOMAS volume at annual minimum and CT area at annual minimum. I can't believe I've not done this
before, or if I have that I've forgotten, not having seen the significance.
What initially fascinated me is the linearity, despite all the complexity of changes over the last 34 years, it's so damned linear. So the apparent acceleration of volume loss is closely tied to that
of area.
I've extended that curve to zero volume for a purpose. This relationship that has held for 34 years must cease, because if maintained when the area at minimum is at 1.82M km^2 volume will be zero!
This is impossible
! It's worth noting here that this year's minimum was 2.234M km^2.
Crucial to answering
this relationship will fail is, I suspect, the answer to the question: Why there is a constant of proportionality of 0.222 between area and volume? I should stress that vague terms like crash aren't
what I'm taking about when I say 'how', I'm thinking more about the physical processes.
Also relevant to this is the ratio of area and volume.
The jump in ratio at the end of the series is due to the volume loss of 2010. But before I start rambling on about the importance of that year...
It might seem puzzling that the ratio can jump, while the linear trend in the first graphic still holds. If you subtract the offset 1.8224 from area then the ratio is closer to 0.222, indeed the
average of all the year's ratios is 0.221388, so that jump is due to volume and can be seen in the first graphic as the last three points being separated from the previous group of three (2007 to
2009) both groups being separated from the pre 2006 mass of data points. But despite the magnitude of the 2007 and 2010 events they have not caused a break away from the relationship shown in the
first graphic and a deviation towards a trend that intersects the
zero volume = zero area
Which makes me wonder what sort of magnitude of event we face.
PS - I should have waited before posting, I'm too tired and have missed the bleedin' obvious! If you invert the relationship Area/Vol you get Vol/Area, which is thickness, I think that the problem
with physical implausibility of the first graphic is due to us being on the edge of critical thickness, below which the ice cannot survive. I'm trying to pin down a figure for such a threshold or
reconcile a trajectory between where graphic one leaves off and zero.
PS2 - Plot of Volume/Area which is thickness at minimum.
With reference to the PIOMAS thickness plots at the end of
this post
. I am now wondering if the deviation implied by the first graphic of this post could actually be relatively slow. With the ice in the interior surviving for some years. The first graphic of this
post tells us nothing about rate, and both the 2007 and 2010 events were driven by anomalous weather.
First graphic redone as Volume/Area.
Final PS.
I'll be going over this some more, but have made some progress. Over at Neven's Sea Ice Blog I'm discussing this issue, as I went over there to ask for opinions. My latest new information is
, but the rest of the discussion is worth browsing.
Data -
PIOMAS volume
Cryosphere Today Area
10 comments:
As we approach zero point (or even virtually zero point) the inverse logarithm will show up. It will be a rapid roll off, looking something like this:
It still looks linear but will curve and roll off rapidly at the end.
R Gates,
Yes, that's what I'm starting to think. I've gone from thinking sometime late next decade, to possibly this decade. And as a result of the work I've been doing with PIOMAS and CT Area I'm now
strongly suspecting a rapid crash out like that. But until I have a better grasp of what's going on, I still think there's a chance it could be slower - until I understand ruling out slow isn't
sound. But my gut feeling says it'll be something amazing.
Wipneus has observed this relationship previously, here.
" But my gut feeling says it'll be something amazing."
There are other adjectives I could find for the rapid approach of an ice-free Arctic. Amazing it will be, much like the approach of a high-speed train when you looking right into the locomotive's
headlight. And then a bit of terror sets in...
The behaviour of those plots looked somewhat familiar, so I dusted off some maths textbooks and found that the asymptote to an nth order polynomial divided by an (n-1)th order polynomial. ex. a*t
Then I was wondering about the physical implications. the slope seems to suggest a limit to the average thickness of the ice. The sudden drop in area required to make this meaningful may suggest,
as you mentioned, a minimum thickness, beyond which the ice cannot survive.
The next 2-6 years should give us a definite answer.
*found that the asymptote to ... has similar properties to the trend line for Volume/Area.
Thanks for that. I'm thinking along the lines of a critical thickness. However this data holds no information about rate per se. The reason I think it implies a rapid transition is that once the
trajectory starts to deviate towards zero it implies increased rate of area loss for a given volume loss. So if we assume that there is a given energy budget for ice melt (one that will be
amplified by more open water), then the expenditure of the same energy budget as in past years will imply a greater loss of area than in previous years.
R Gates,
For myself, the word 'awe' is perhaps more appropriate than 'terror'.
"0.22" relates the height of column of ice to the area of its base at 0C (after everything has come to equilibrium.) It holds for ice on land also. However the bottom of land ice can be colder,
but the bottom of sea ice is always at its melting point. Thus, if you want your ice sculpture to stand there all winter, you need to keep it colder.
Chris, I am a new comentator but I follow your blog since a long time now. I was thinking also about the same thing here : http://www.forums.meteobelgium.be/index.php?s=&showtopic=1852&view=
findpost&p=452888 (sorry, it's french ^^ ). I think there is a limit for the minimum thickness, below which the integrity of the sea ice can no longer be guaranted. The pack is able to disperse
and small floe, and this dispersion maximise the ratio surface/volume for individual floe, which maximise the heatsfer (as you know, heat transfer is more a function of surface and heat capacity
of volume).
Hi Anon,
I hope it's a blog that continues to be worth following. And thanks for writing in English, I wish my French was up to your standard! But it's not. ;)
Over at Neven's I've broken the Vol/Area plot into contributions from FYI and MYI using thickness as a proxy. It turns out that most of the slope of the trend is due to MYI (multi-year ice),
which has virtually been eliminated from the Arctic in PIOMAS. Now we're left with FYI (First year ice) and young ice. This has very little downward trend because it is able to recover quickly. I
had been wondering aloud whether this means we face a slow transition.
But after thinking about it today at work I don't think so. I'll post more detail, but not tonight. However - as of the end of December we're 1M km^3 volume below 2010 and 2011 at the same time.
Volume anomalies tend to be static from now until the maximum. After which we have the crash in anomalies due to massive volume loss above the seasonal cycle from April to July. So I anticipate
that next year stands a good chance of being around 2.3M km^3 volume at minimum, which will imply a further crash in area. I will post more detail but after the last four or five days working
solid I need to reclaim some evenings for myself. So knowing me it will be over the weekend. | {"url":"https://dosbat.blogspot.com/2013/01/an-observation-from-brink.html?showComment=1358108924964","timestamp":"2024-11-14T03:38:33Z","content_type":"text/html","content_length":"91770","record_id":"<urn:uuid:2ee121f5-cb81-4809-8d04-8e6c3d6f489d>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00185.warc.gz"} |
Thinking Test: Can you Spot the Hidden Number 9068 in 12 Secs - EduViet Corporation
Thinking Test: Can you Spot the Hidden Number 9068 in 12 Secs
The impact of brain teasers
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You are watching: Thinking Test: Can you Spot the Hidden Number 9068 in 12 Secs
Brain teasers using pictures
Picture Brainteaser is a visual puzzle that can be used for a variety of purposes, including:
• Entertainment: Picture brainteasers are a fun and engaging activity for people of all ages. They can be used at parties, social gatherings, or even as a stand-alone activity to pass the time.
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Overall, picture brainteasers are a versatile tool that can be used for a variety of purposes, including entertainment, education, cognitive development, therapy, and recruitment.
Thinking test: Can you find the hidden number 9068 in 12 seconds?
Brain teasers are generally considered healthy because they trigger your cognitive thinking and allow your brain to think outside the box. But be sure not to tire your eyes! We’ve attached the
question, followed by this text, in the image below. Scroll down for a second and see if you can solve the puzzle.
We know that adding a time limit might add more excitement to your challenge, but it does. Take a closer look at the image we proved below; now try to guess the answer in a few seconds. Time starts
now: 1, 2, 3…
See more : Brain Teaser Math Test: Fix 5-9=3 Move 1 Matchstick to Fix the Equation by 30 Secs | Matchstick Puzzle
Tick tock, tick tock, tick tock, it’s time!
If you can find the right answer, give yourself a pat on the back.
Wait, how do you know if this is the right answer?
The visuals you glean from the information mentioned here may create different perceptions in your brain. This may happen if you have a different answer in mind. After all, it’s common for us to
perceive different meanings of simple images. This means you have a higher IQ level.
Thinking test: Can you find the hidden number 9068 in 12 seconds?
Brainteasers have unexpected benefits for those who participate in the challenge, so you can swipe down to participate in the challenge to find out. Netizens are increasingly wondering what the
correct answer to this picture is. You may definitely be confused. But take a deep breath and try again to find the right answer.
Since solving this brainteaser is quite challenging, we are here to reveal the solution to you! Scroll down to reveal the answer! Don’t be discouraged if you can’t come up with the right answer.
Regular practice with our brainteasers blog will improve your observation and problem-solving skills.
Brain teaser math speed test: 70÷14x(9+2)=?
Immerse yourself in a brainteaser math speed test using the following formula: 70 ÷ 14 x (9 + 2). Your challenge is to try to follow the sequence of operations and determine the end result.
See more : Are you smart enough to Find the Odd Van in 10 Secs
To solve the equation, first find the sum in the brackets: 9 + 2 equals 11. Then perform division: 70 ÷ 14 equals 5. Finally, multiply the division result by the sum: 5 x 11 equals 55. Therefore, the
equation 70 ÷ 14 x (9 + 2) equals 55.
Brain Taser IQ test math test: 89-49÷7+2+33÷3=?
Enter the fun world of brain teasers IQ test math quizzes using the following equation: 89 – 49 ÷ 7 + 2 + 33 ÷ 3. Your challenge is to carefully apply the order of operations and find the final
To solve this equation, follow the order of operations. First, divide 49 by 7 to get 7. Then, add 2 and 7 to get 9. Next, divide 33 by 3 to get 11. Finally, add 9 and 11 to get 20. Subtract 20 from
89 to get the final answer: 89 – 49 ÷ 7 + 2 + 33 ÷ 3=69.
Brain Teasers Math IQ Test: Solve 56÷4×8+3-2
Immerse yourself in this brainteaser math IQ test using the following equation: 56 ÷ 4 x 8 + 3 – 2. Your challenge is to carefully follow the order of operations and determine the final result.
First, perform division: 56 ÷ 4 equals 14. Then, multiply: 14 x 8 equals 112. Add 3 to 112 to get 115, and finally subtract 2 from 115 to get 113. Therefore, equation 56 ÷ 4 x 8 + 3 – 2 equals 113.
Brain teaser math test: equals 760÷40×5+8
Get into fun territory with brainteaser math tests using the following equation: 760 ÷ 40 x 5 + 8. Your task is to carefully follow the order of operations and calculate the final result.
To solve this equation, follow the order of operations. First, perform division: 760 ÷ 40 equals 19. Then, multiply: 19 x 5 equals 95. Add 8 and 95 to get the final answer of 103. Therefore, the
equation 760 ÷ 40 x 5 + 8=103.
Brainteaser math puzzle: 11+11=5, 22+22=11, 33+33=?
Dive deeper into exciting brainteaser math puzzles. The pattern starts with: 11+11 equals 5 and 22+22 equals 11. Now the mystery deepens: when 33+33 goes through this interesting sequence, what is
the result? This sequence uses multiplication and addition to produce unexpected but logical progressions. In the first equation, 11+11 equals 5, calculated as (1×1) + (1×1) + 3. Likewise, for 33+33,
we have (3×3) + (3×3) + 3, which results in 21.
Disclaimer: The above information is for general information purposes only. All information on this website is provided in good faith, but we make no representations or warranties, express or
implied, as to the accuracy, adequacy, validity, reliability, availability or completeness of any information on this website.
Source: https://truongnguyenbinhkhiem.edu.vn
Category: Brain Teaser
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AnnotatedOperation (latest version) | IBM Quantum Documentation
class qiskit.circuit.AnnotatedOperation(base_op, modifiers)
Bases: Operation
Annotated operation.
Create a new AnnotatedOperation.
An “annotated operation” allows to add a list of modifiers to the “base” operation. For now, the only supported modifiers are of types InverseModifier, ControlModifier and PowerModifier.
An annotated operation can be viewed as an extension of ControlledGate (which also allows adding control to the base operation). However, an important difference is that the circuit definition of an
annotated operation is not constructed when the operation is declared, and instead happens during transpilation, specifically during the HighLevelSynthesis transpiler pass.
An annotated operation can be also viewed as a “higher-level” or “more abstract” object that can be added to a quantum circuit. This enables writing transpiler optimization passes that make use of
this higher-level representation, for instance removing a gate that is immediately followed by its inverse.
• base_op (Operation) – base operation being modified
• modifiers (Union[Modifier, List[Modifier]]) – ordered list of modifiers. Supported modifiers include InverseModifier, ControlModifier and PowerModifier.
op1 = AnnotatedOperation(SGate(), [InverseModifier(), ControlModifier(2)])
op2_inner = AnnotatedGate(SGate(), InverseModifier())
op2 = AnnotatedGate(op2_inner, ControlModifier(2))
Both op1 and op2 are semantically equivalent to an SGate() which is first inverted and then controlled by 2 qubits.
Unique string identifier for operation type.
Number of classical bits.
The params of the underlying base operation.
The base operation that the modifiers in this annotated operation applies to.
Ordered sequence of the modifiers to apply to base_op. The modifiers are applied in order from lowest index to highest index.
control(num_ctrl_qubits=1, label=None, ctrl_state=None, annotated=True)
Return the controlled version of itself.
Implemented as an annotated operation, see AnnotatedOperation.
Controlled version of the given operation.
Return type
Return the inverse version of itself.
Implemented as an annotated operation, see AnnotatedOperation.
annotated (bool) – ignored (used for consistency with other inverse methods)
Inverse version of the given operation.
power(exponent, annotated=False)
Raise this gate to the power of exponent.
Implemented as an annotated operation, see AnnotatedOperation.
• exponent (float) – the power to raise the gate to
• annotated (bool) – ignored (used for consistency with other power methods)
An operation implementing gate^exponent
Return a matrix representation (allowing to construct Operator). | {"url":"https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.AnnotatedOperation","timestamp":"2024-11-08T20:48:42Z","content_type":"text/html","content_length":"185493","record_id":"<urn:uuid:e4f1ad1d-0be6-4004-abd0-ebb2b9b721a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00762.warc.gz"} |
«Step 1: Rhythm Step 3:Conduction intervals (PQ, QRS, QT, QTc)»
What is the heart rate? To answer this question, determine the time between two QRS complexes. Previously, the ECG was printed on a paper strip transported through an ECG writer at the speed of 25 mm
/second. Now, digital ECGs are common; however, the method for determining the frequency remains the same. The ECG has a grid with thick lines 5 mm apart (= 0,20 second) and thin lines 1 mm (0,04
There are three simple methods to determine the heart rate (HR)
The square counting method
The square counting method is ideal for regular heart rates. Use the sequence 300-150-100-75-60-50-43-37. Count from the first QRS complex, the first thick line is 300, the next thick line 150 etc.
Stop the sequence at the next QRS complex. When the second QRS complex is between two lines, take the mean of the two numbers from the sequence or use the fine-tuning method listed below.
Use a calculator
Count the small (1mm) squares between two QRS complexes. The ECG paper runs at 25 mm/sec through the ECG printer; therefore:
This method works well in case of tachycardia (>100 beats/minute)
The marker method
Non-regular rhythms are best determined with the "3 second marker method". Count the number of QRS complexes that fit into 3 seconds (some ECG writers print this period on the ECG paper). Multiply
this number by 20 to find the number of beats/minute.
The 'square counting' method can be fine-tuned with the following
What changes the frequency of the heart?
A number of factors change the heart frequency, including:
• the (para) sympathic nervous system.
□ The sympathic system, e.g. epinephrine, (=adrenalin) increases atrioventricular conduction and contractility (the fight or flight reaction.)
□ The parasympathic system (nervus vagus,) e.g. acetycholine, decreases the frequency and atrioventricular conduction. The parasympathic system affects mainly the atria.
• Cardiac filling increases the frequency.
• arrhythmias influence heart rate. | {"url":"https://en.ecgpedia.org/index.php?title=Frequency","timestamp":"2024-11-04T09:18:24Z","content_type":"text/html","content_length":"27423","record_id":"<urn:uuid:666e79f8-2d1c-465c-9d04-00b0fd5eb866>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00362.warc.gz"} |
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.9
You can Download Samacheer Kalvi 11th Maths Book Solutions Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations.
Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.9
Integrate the following with respect to x
Question 1.
e^x (tan x + log sec x)
Question 2.
e^x\(\left(\frac{x-1}{2 x^{2}}\right)\)
Question 3.
e^x sec x (1 + tan x)
I = ∫ e^x sec x (1 + tan x) dx
I = ∫ e^x (sec x + sec x tan x) dx
f(x) = sec x
f'(x) = sec x tan x
[∫ e^x [f (x) + f (x)] dx = e^x f(x) + c]
I = e^x sec x + c
Question 4.
e^x \(\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right)\)
Question 5.
Question 6.
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Section: New Results
High-Order Time Schemes
Fourth order energy-preserving locally implicit discretization for linear wave equations
Participants : Juliette Chabassier, Sébastien Imperiale.
A family of fourth order coupled implicit-explicit schemes is presented as a special case of fourth order coupled implicit schemes for linear wave equations. The domain of interest is decomposed into
several regions where different fourth order time discretization are used, chosen among a family of implicit or explicit fourth order schemes derived in [72] . The coupling is based on a Lagrangian
formulation on the boundaries between the several non conforming meshes of the regions. A global discrete energy is shown to be preserved and leads to global fourth order consistency. Numerical
results in 1d and 2d illustrate the good behavior of the schemes and their potential for the simulation of realistic highly heterogeneous media or strongly refined geometries, for which using
everywhere an explicit scheme can be extremely penalizing. Accuracy up to fourth order reduces the numerical dispersion inherent to implicit methods used with a large time step, and makes this family
of schemes attractive compared to second order accurate methods in time. This work has been presented at the Franco-Russian workshop on mathematical geophysics, Sep 2014, Novosibirsk, Russia [58] ,
at the and is the object of a submitted publication to International Journal for Numerical Methods in Engineering.
A new modified equation approach for solving the wave equation
Participants : Hélène Barucq, Henri Calandra, Julien Diaz, Florent Ventimiglia.
In order to obtain high-order time-schemes, we are considering an alternative approach to the ADER schemes and to the modified equation technique described in section 3.2 . The two first steps of
the construction of the schemes are similar to the previous schemes : we apply a Taylor expansion in time to the solution of the wave equation and we replace the high-order derivatives with respect
to the time by high order space operators, using the wave equation. The difference is that we do not use auxiliary variables and we choose to discretize directly the high-order operators in space.
In the framework of the PhD thesis of Florent Ventimiglia, we have extended this new method involving $p$-harmonic operator to the first order formulation of the acoustic wave equation, which is the
formulation discretized in the DIVA platform of TOTAL. In this case, the high order operators in space are not are not powers of the Laplace operator but powers of the gradient. Hence, we also had to
adapt the space discretization, and we have extended the DG formulation with centered fluxes proposed in [77] to higher order operators. A numerical analysis of performance in 2D indicates that, for
a given accuracy, this method requires less computational costs and less storage than the High-Order ADER Scheme. These results have been presented to the AIMS conference [54] . A paper has been
published in ESAIM Proceedings [19] .
Finite Element Methods for the time-harmonic wave equation.
Goal-Oriented Adaptivity using Unconventional Error Representations
Participants : Vincent Darrigrand, David Pardo, Ignacio Muga, Hélène Barucq.
In the scope of subsurface modelling via the resolution of inverse problems, the so-called goal-oriented adaptivity plays a fundamental role. Indeed, while classical adaptive algorithms were first
designed to accurately approximate the energy norm of a problem [69] , [70] , one requires a good approximation of a specific quantity of interest. An energy norm driven self-adaptive strategy can
still be used for that purpose, although it often becomes sub-optimal and unable to provide an accurate solution for the required quantity of interest in a reasonable amount of time.
During the late 90’s, to overcome this issue, the so-called goal-oriented strategy appeared, see for instance [82] , [81] . The goal-oriented approach consists in expressing the error in the
quantity of interest as an integral over the entire computational domain involving the errors of the original and adjoint problems, and then minimise an upper bound of such error representation by
performing local refinements.
Most authors, using the adjoint problem, represent the approximation error in the quantity of interest via the global bilinear form that describes the problem in terms of local and computable
Our methodology, however, is based on the selection of an alternative bilinear form exhibiting better properties than the original bilinear form (e.g. positive definiteness). We represent the
residual error functional of the adjoint problem through this alternative form. We can then compute new upper bounds of the error of the quantity of interest in a similar way than with the classical
approach. Our main goal is to demonstrate that a proper choice of such alternative form may improve the upper bounds of the error representation.
Moreover, the method proposed here generalises the existing ones, since, in particular, we can select as the alternative bilinear form the one associated to the adjoint problem.
Hybridizable Discontinuous Galerkin method for the elastic Helmholtz equations
Participants : Marie Bonnasse-Gahot, Henri Calandra, Julien Diaz, Stéphane Lanteri.
We consider Discontinuous Galerkin (DG) methods formulated on fully unstructured meshes, which are more convenient than finite difference methods on cartesian grids to handle the topography of the
subsurface. DG methods and classical Finite Element (FE) methods mainly differ from discrete functions which are only piecewise continuous in the case of DG approximation. DG methods are then more
suitable than Continuous Galerkin (CG) methods to deal with hp-adaptivity. This is a great advantage to DG method which is thus fully adapted to calculations in highly heterogeneous media.
Nevertheless, the main drawback of classical DG methods is that they are more expensive in terms of number of unknowns than classical CG methods, especially when arbitrarily high order interpolation
of the field components is used. In this case DG methods lead to larger sparse linear systems with a higher number of globally coupled degrees of freedom as compared to CG methods with a same given
mesh. In that case, we consider a hybridizable Discontinuous Galerkin (HDG) method which principle consists in introducing a Lagrange multiplier representing the trace of the numerical solution on
each face of the mesh cells. This new variable exists only on the faces of the mesh and the unknowns of the problem depend on it. This allows us to reduce the number of unknowns of the global linear
system. Now the size of the matrix to be inverted only depends on the number of the faces of the mesh and on the number of the degrees of freedom of each face. It is worth noting that for the
classical DG method it depends on the number of the cells of the mesh and on the number of the degrees of freedom of each cell. The solution to the initial problem is then recovered thanks to
independent elementwise calculation. The principle of the HDG method and 2D results were presented at the WCCM XI - ECCM V - ECFD VI - Barcelona 2014 Conference [41] , the EAGE Workshop on High
Performance Computing for Upstream [42] , the Second Russian-French Workshop “Computational Geophysics” [43] and at the Réunion des Sciences de la Terre 2014 conference [53] . A comparison between
HDG method and classical nodal DG method was given on a poster at the Journées Total-Mathias 2014 workshop [66] .
Participants : Hélène Barucq, Juliette Chabassier, Marc Duruflé, Damien Fournier, Laurent Gizon.
The finite element code Montjoie 5.2 has been used to solve Helmholtz equation in axisymmetric domain in the configuration of the sun. The efficiency of the code has been compared in three
configurations : radial (1-D mesh and spherical harmonics), axisymmetric (2-D mesh), 3-D. The results have convinced our-selves and our partners of Max Planck Institute that the axisymmetric
configuration is the most interesting for an inversion procedure, since 3-D computations are too expensive. A more realistic modeling of the sun requires the solution of time-harmonic Galbrun's
equations (instead of Helmholtz equation), different formulations have been implemented and studied. It appeared that the different numerical methods are not able to converge to the correct solution
for non-uniform flows. The lack of convergence is more obvious for flows with a larger Mach number. Such problems do not appear in Linearized Euler equations, as a result we have proposed simplified
Galbrun's equations that converge correctly and provide the same solution as original Galbrun's equations for a null flow. These equations have been implemented in 2-D, axisymmetric and 3-D
Scattering of acoustic waves by a disc - Hypersingular integral equations
Participants : Leandro Farina, Paul Martin, Victor Péron.
Two-dimensional boundary-value problems involving a Neumann-type boundary condition on a thin plate or crack can often be reduced to one-dimensional hypersingular integral equations. Examples are
potential flow past a rigid plate, acoustic scattering by a hard strip, water-wave interaction with thin impermeable barriers, and stress fields around cracks. In [29] , we generalize some of these
results to two-dimensional hypersingular integral equations. Thus, rather than integrating over a finite interval, we now integrate over a circular disc. Two-dimensional hypersingular equations over
a disc arise, for example, in the scattering of acoustic waves by a hard disc; this particular application is described in Appendix A. We develop an appropriate spectral (Galerkin) method, using
Fourier expansions in the azimuthal direction and Jacobi polynomials in the radial direction. The Hilbert-space arguments used by Golberg are generalized and a convergence theorem is proved by using
tensor-product techniques. Our results are proved in weighted ${L}^{2}$ spaces. Then, Tranter's method is discussed. This method was devised in the 1950s to solve certain pairs of dual integral
equations. It is shown that this method is also convergent because it leads to the same algebraic system as the spectral method.
Finite Element Subproblem Method
Participants : Patrick Dular, Christophe Geuzaine, Laurent Krähenbühl, Victor Péron.
In the paper [26] , the modeling of eddy currents in conductors is split into a sequence of progressive finite element subproblems. The source fields generated by the inductors alone are calculated
at first via either the Biot-Savart law or finite elements. The associated reaction fields for each added conductive region, and in return for the source regions themselves when massive, are then
calculated with finite element models, possibly with initial perfect conductor and/or impedance boundary conditions to be further corrected. The resulting subproblem method allows efficient solving
of parameterized analyses thanks to a proper mesh for each subproblem and the reuse of previous solutions to be locally corrected.
High Order Methods for Helmholtz Problems in Highly Heterogeneous Media
Participants : Théophile Chaumont-Frelet, Henri Calandra, Hélène Barucq, Christian Gout.
Heterogeneous Helmholtz problems arise in various geophysical application where they modelize the propagation of time harmonic waves through the subsurface. For example, in inversion problems, the
aim is to reconstruct a map of the underground based on surface acquisition. This recovery process involves the solution to several Helmholtz problems set in different media, and high frequency
solutions are required to obtain a detailed image of the underground. This obervations motivate the design of efficient solver for highly heterogeneous Helmholtz problems at high frequency.
The main issue with the discretization of high frequency problems is the so called “pollution effect” which impose drastic condition on the mesh. In the homogeneous case, it is known that one
efficient way to reduce the pollution effect is the use of high order discretization methods. However, high order methods can not be applied as is to highly heterogeneous media. Indeed, they are
based on coarser mesh and are not sensitive to fine scale variations of the medium.
We propose to overcome this difficulty by using a multiscale strategy to take into account fine scale heterogeneities on coarse meshes. The method is based on a simple medium approximation method,
which can be seen as a special quadrature rule. Numerical experiments in two dimensional geophyscial benchmarks show that high order method coupled with our multiscale approximation medium stragey
are cheaper than low order method for a given accuracy. Futhermore, focusing on one dimensional models, we were able to show from a theoretical point of view that our methology reduces the pollution
effect even when used on coarse meshes with non-matching interfaces.
This work has been presented at the WCCM XI - ECCM V - ECFD VI - Barcelona 2014 conference, the Second Russian-French Workshop “Computational Geophysics”. A poster has been presented at the journées
Total-Mathias 2014 workshop. A paper has been submitted for publication to Math. Of Comp..
Boundary conditions.
Absorbing Boundary Conditions for Tilted Transverse Isotropic Elastic Media
Participants : Lionel Boillot, Hélène Barucq, Julien Diaz, Henri Calandra.
The seismic imaging simulations are always performed in bounded domains whose external boundary does not have physical meaning. We have thus to couple the wave equations with boundary conditions
which aim at reproduce the invisibility of the external boundary. The discretization of these conditions can be an issue. For instance, an efficient condition, once discretized, can induce huge
computational costs by filling the matrix which has to be inverted. This is the case of the transparent boundary conditions which are approximated by local Absorbing Boundary Conditions (ABC) that do
not increase to much the computational burden. However, the ABC has the drawback to introduce spurious numerical waves which can perturb the RTM results. It is possible to avoid this drawback by
applying PML (Perfectly Matched Layers) but it proves to be unstable in anisotropic media. Last year, we proposed a way of construction leading to a stable ABC. The technique is based on slowness
curve properties, giving to our approach an original side. We established stability results from long time energy behavior and we have illustrated the performance of the new condition in 2D numerical
tests. This year, we extend all these results to 3D case and to arbitrary boundary shapes. The previous paper submission on 2D results has been accepted and released [18] . The recent results in 3D
have been presented to the ECCOMAS conference.
Derivation of high order absorbing boundary conditions for the Helmholtz equation in 2D.
Participants : Hélène Barucq, Morgane Bergot, Juliette Chabassier, Élodie Estecahandy.
Numerical simulation of wave propagation raises the issue of dealing with outgoing waves. In most of the applications, the physical domain is unbounded and an artificial truncation needs indeed to be
carried out for applying numerical methods like finite element approximations. Adapted boundary conditions that avoid the reflection of outgoing waves and provide a well-posed mathematical problem
must then be derived. With ideal boundary conditions, the solution on the new mixed boundary valued problem in the truncated domain would actually be equal to the restriction of the mathematical
solution in the unbounded domain. However, such ideal boundary conditions, called “transparent boundary conditions”, can be shown to be nonlocal, which leads to dramatic computational overcosts. The
seek of local boundary conditions, called “absorbing boundary conditions” (ABC), has been the object of numerous works trying to perform efficient conditions based on different techniques of
derivation. Among them, the technique of micro-diagonalisation has been employed to the wave equation and more generally to hyperbolic systems in [76] , leading to a hierarchy of absorbing local
boundary conditions based on the approximation of the Dirichlet-to-Neumann map. A comprehensive review of different used strategies and higher order conditions can be found in [85] . One desirable
property of ABCs is that the reflection of the waves on the artificial boundary generates an error of the same order as the one generated by the spatial discretization inside the domain. The
computational effort is thus optimized in terms of modeling and numerical inaccuracies. Moreover, the ABC must fit the artificial boundary chosen by the user of the method. In the context of high
order spatial discretization (spectral finite elements [74] , Interior Penalized Discontinuous Galerkin [68] ), there is nowadays a need for high order ABCs that can adapt on non flat geometries
since these methods prove very efficient for capturing arbitrary shaped domains.
The aim of the present work is to develop high order ABCs for the Helmholtz equation, that can adapt to regular shaped surfaces. A classical way of designing ABCs is to use Nirenberg theorem [80] on
the second order formulation of the Helmholtz equation, which enables us to decompose the operator as a product of two first order operators. Here our approach is to rewrite the Helmholtz equation as
a first order system of equations before developing ABCs using M.E. Taylor's micro-diagonalisation method [84] . Then an asymptotic truncation must be performed in order to make the ABC local, and we
will see that the high frequency approximation will lead to more usable ABCs than the one stating that the angle of incidence is small. During the process, while increasing the degree of the pseudo
differential operator decomposition along with the order of asymptotic truncation, we retrieve classical ABCs that have been found with other techniques by other authors. For now, we have restricted
ourselves to two dimensions of space, but despite the fact that 3D generalization should obviously generate more calculation, no further theoretical difficulties are expected.
This work has been the object of a technical report [61] and the obtained conditions have been implemented in Montjoie 5.2 and Houd10ni 5.1 .
Asymptotic modeling.
Fast Simulation of Through-casing Resistivity Measurements Using Semi-analytical Asymptotic Models.
Participants : Victor Péron, David Pardo, Aralar Erdozain.
When trying to obtain a better characterization of the Earth's subsurface, it is common to use borehole through-casing resistivity measurements. It is also common for the wells to be surrounded by a
metal casing to protect the well and avoid possible collapses. The presence of this metal case highly complicates the numeric simulation of the problem due to the high conductivity of the casing
compared to the conductivity of the rock formations. In this study [47] we present an application of some theoretical asymptotic methods in order to deal with complex borehole scenarios like cased
wells. The main idea consists in replacing the part of the domain related to the casing by a transmission impedance condition. The small thickness of the casing makes it ideal to apply this kind of
mathematical technique. When eliminating the casing from the computational domain, the computational cost of the problems considerably decreases, while the effect of the casing does not disappear due
to the impedance transmission conditions. The results show that when applying an order three impedance boundary condition for a simplified domain, it only generates a negligible approximation error,
while it considerably reduces the computational cost. For obtaining the numerical results and testing the mathematical models we have developed a Finite Element Code in Matlab. The code works with
Lagrange polynomials of any degree as basis functions and triangular shaped elements in two dimensions. The code has been adapted for working with the transmission impedance conditions required by
the mathematical models.
Modeling the propagation of ultrashort laser pulses in optical fibers.
Participants : Mohamed Andjar, Juliette Chabassier, Marc Duruflé.
In order to model the propagation of an ultrashort laser pulse, the most natural idea is to solve Maxwell's equations in a nonlinear and dispersive medium. Given the considered optical periods
(around ${10}^{-14}$ seconds), the associated wavelengthes (around 1 millimeter) and the propagation distances (several meters), the direct numerical simulation of these equations by usual numerical
techniques (finite elements, explicit time schemes) is impossible because too expensive. The standard procedure is therefore to use approached equations obtained by exploiting legitimate hypotheses
in the considered context (slowly varying pulse envelope, narrow spectrum, paraxial approximation ...). These new equations, among them the Nonlinear Schrödinger Equation, are significantly less
expensive to solve and we can therefore provide realistic numerical simulations to physicists.
When the pulse propagates in an optical fiber, its spatial profile in the orthogonal plane to the propagation direction in very simple because optical fibers posses a finite (small, often equal to
one) number of propagating modes. The equations that originally are stated on a 3D domain can then be written as one spatial dimension equations.
The scientific objective of this internship was to apply the approximation techniques mentioned above in this specific context, in order to obtain one or several equations (depending on the used
hypotheses) that model the propagation of ultrashort laser pulses in optical fibers. A matlab code has been developed and integrated in the C++ code Montjoie 5.2 . Numerical simulations have been
led in order to observe classical situations of nonlinear fiber optics (Kerr effect, Raman effect, supercontinuum generation, ...).
Small heterogeneities in the context of time-domain wave propagation equation : asymptotic analysis and numerical calculation
Participants : Vanessa Mattesi, Sébastien Tordeux.
We have focused our attention on the modeling of heterogeneities which are smaller than the wavelength. The work can be decomposed into two parts : a theoretical one and a numerical one. In the
theoretical one, we derive a matched asymptotic expansion composed of a far-field expansion and a near-field expansion. The terms of the far-field expansion are singular solutions of the wave
equation whereas the terms of the near-field expansion satisfy quasistatic problems. These expansions are matched in an intermediate region. We justify mathematically this theory by proving error
estimates. In the numerical part, we describe the Discontinuous Galerkin method, a local time stepping method and the implementation of the matched asymptotic method. Numerical simulations illustrate
these results. Vanessa Mattesi has defended her PhD on this topic[14] .
Theoretical and numerical investigations of acoustic response of a multiperforated plate for combustion liners
Participants : Vincent Popie, Estelle Piot, Sébastien Tordeux.
Multiperforated plates are used in combustion chambers for film cooling purpose. As the knowledge of the acoustic response of the chamber is essential for preventing combustion instabilities, the
acoustic behaviour of the perforated plates has to be modeled. This can be done either by considering the transmission impedance of the plates, or their Rayleigh conductivity.
We have investigated the link between these two quantities thanks to matched asymptotic expansions. Especially the far-field or near-field nature of the physical quantities used in the definition of
the impedance and Rayleigh quantity has been enlightened. Direct numerical simulations of the propagation of an acoustic plane wave through a perforated plate are performed and post-treated so that
the assumptions underlying the definitions of impedance and Rayleigh conductivity have been checked. The results will be presented at the conference ASME Turbo Expo 2015. | {"url":"https://radar.inria.fr/report/2014/magique-3d/uid40.html","timestamp":"2024-11-08T10:51:27Z","content_type":"text/html","content_length":"68214","record_id":"<urn:uuid:b183b55d-3f53-441e-b7f1-c07c971e0a33>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00292.warc.gz"} |
Standard Handbook of Petroleum & Natural Gas Engineering by Ken Arnold
By Ken Arnold
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Example text
If x, y, . . are functions of t, then _ du dx dY -(f,)z+(f )-+ ... dt dt Differential and Integral Calculus 39 expresses the rate of change of u with respect to t, in terms of the separate rates of
change of x, y, . . with respect to t. If the equation of the curve is y = f(x) where the rate of change (ds/dx) and the differential of the arc (ds), s being the length of the arc, are defined as
and ds = Jdx' and dx dy = u = = + dy2 ds cos u ds sin u tan-'[f'(x)] u being the angle of the tangent at P with respect to the x-axis.
360' or 0, 7t/12, x/6, n/4, . , 27t V Figure 1-30. Polar coordinates. Differential and Integral Calculus 35 Figure 1-31. The polar plane. radians. The origin is called the pole, and points [r,0] are
plotted by moving a positive or negative distance r horizontally from the pole, and through an angle 0 from the horizontal. See Figure 1-31 with 0 given in radians as used in calculus. Also note that
[r, 01 = [-r,e + n] DIFFERENTIAL AND INTEGRAL CALCULUS See References 1 and 5-8 for additional information.
Law of Sines: a/(sin A) = b/(sin B). = c/(sin . 2. Law of Cosines': = ai + b2 1 2ab cos C 2 4*i" x Figure 1-29. Graphs of the trigonometric functions. Trigonometry 33 Hyperbolic Functions The
hyperbolic sine, hyperbolic cosine, etc. Their definitions and properties are very similar to the trigonometric functions and are given in Table 1-5. , are related to the logarithmic functions and
are particularly useful in integral calculus. \11/2(cosh x - 1) ~0~h(x/2 = ) ,/1/2(cosh x tanh(d2) = (cosh x + 1) - l)/(sinh x) = (sinh x)/(cosh x + 1) Mathematics 34 Polar Coordinate System The
polar coordinate system describes the location of a point (denoted as [r,81) in a plane by specifying a distance r and an angle 8 from the origin of the system.
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Archive of events from 2021
An archive of events from the year
• Lars Birkedal (Aarhus University)
Iris: A Higher-Order Concurrent Separation Logic Framework
I will introduce some of our research on Iris, a higher-order concurrent separation logic framework, implemented and verified in the Coq proof assistant, which can be used for mathematical
reasoning about safety and correctness of concurrent higher-order imperative programs. Iris has been used for many different applications; see iris-project.org for a list of research papers.
However, in this talk I will focus on the Iris base logic (and its semantics) and sketch how one may define useful program logics on top of the base logic. The base logic is a higher-order
intuitionistic modal logic, which, in particular, supports the definition of recursive predicates and whose type of propositions is itself recursively defined.
• Dominik Wehr (FLoV)
Abstract Cyclic Proofs
Cyclic proof systems permit derivations to contain cycles, thereby serving as finite representations of non-wellfounded derivation trees. Such cyclic proof systems have been proposed for a broad
range of logics, especially those with fixed-point features or inductively defined objects of reasoning. The soundness of such systems is commonly ensured by only considering those cyclic
derivations as proofs which satisfy a so-called global trace condition.
In this talk, I will present a category theoretical notion, the trace category, which generalizes the trace condition of many cyclic proof systems. Indeed, a handful of different trace categories
are sufficient to capture the trace conditions of all cyclic proof systems from the literature I have examined so far. The arising abstract notion of cyclic proof allows for the derivation of
generalized renditions of standard results of cyclic proof theory, such as the decidability of proof-checking and the regularizability of certain non-wellfounded proofs. It also opens the door to
broader inquiries into the structural properties of trace condition based cyclic and non-wellfounded proof systems, some of which I will touch on in this talk, time permitting. The majority of
this talk will be based on my Master’s thesis.
• Paula Quinon (Warsaw University of Technology and FLoV)
Invariances and the number concept
Cognitive scientists Spelke and Kintzler (2007) and Carey (2009) propose objects, actions, space and numbers as ‘core domains of knowledge’ that underpin the framework of concepts people use to
describe and communicate about the world. Gärdenfors (2019, 2020) argues that humans make sense of domains by appealing to various types of invariances in sensory signals. In this talk, I present
work by Quinon and Gärdenfors (manuscript) in which the aim is to extend the analysis in terms of invariances to the domain of numbers. We focus on several perspectives relating invariances:
cognitive modeling, formal mathematical and experimental.
As theoretical background, we assume that numbers are properties of collections (Simons 1982, 2007, 2011; Johansson 2015; Angere 2014). We observe that the domain of number is determined by two
types of invariances. First, the concept of collection itself depends on being invariant under the location of its objects. Second, the determinant invariance of the domain of number is the
fungibility of objects: If an object in a collection is exchanged for another object, the collection will still contain the same number of objects. Fungibility will be shown to be closely related
to one-to-one correspondences.
We first introduce the concept of a collection and show how it differs from the concept of a set. Then we present the invariance of location of objects that applies to collections and we
introduce fungibility as a second type of invariance. We illustrate our theoretical analysis by empirical material from experiments of developmental psychologists.
This is joint work with Peter Gärdenfors (Lund).
• Sara L. Uckelman (Durham University)
John Eliot’s Logick Primer: A bilingual English-Wôpanâak logic textbook
In 1672 John Eliot, English Puritan educator and missionary, published The Logick Primer: Some Logical Notions to initiate the INDIANS in the knowledge of the Rule of Reason; and to know how to
make use thereof [1]. This roughly 80 page pamphlet focuses on introducing basic syllogistic vocabulary and reasoning so that syllogisms can be created from texts in the Psalms, the gospels, and
other New Testament books. The use of logic for proselytizing purposes is not distinctive: What is distinctive about Eliot’s book is that it is bilingual, written in both English and Wôpanâak
(Massachusett), an Algonquian language spoken in eastern coastal and southeastern Massachusetts. It is one of the earliest bilingual logic textbooks, it is the only textbook that I know of in an
indigenous American language, and it is one of the earliest printed attestations of the Massachusett language.
In this talk, I will:
□ Introduce John Eliot and the linguistic context he was working in.
□ Introduce the contents of the Logick Primer—vocabulary, inference patterns, and applications.
□ Discuss notions of “Puritan” logic that inform this primer.
□ Talk about the importance of his work in documenting and expanding the Massachusett language and the problems that accompany his colonial approach to this work.
[1] J.[ohn] E.[liot]. The Logick Primer: Some Logical Notions to initiate the INDIANS in the knowledge of the Rule of Reason; and to know how to make use thereof. Cambridge, MA: Printed by M.
[armaduke] J.[ohnson], 1672.
• Graham E. Leigh (FLoV)
From interpolation to completeness
I will demonstrate how Walukiewicz’ seminal proof of completeness for the propositional μ-calculus can be derived (and refined) from the cyclic proof theory of the logic, notably the uniform
interpolation theorem for the logic.
• Fredrik Engström (FLoV)
Foundational principles of team semantics
Team semantics is, when compared to standard Tarskian semantics, a more expressive framework that can be used to express logical connectives, operations and atoms that can’t be expressed in
Tarskian semantics. This includes branching quantifiers, notions of dependence and independence, trace quantification in linear-time temporal logic (LTL), and probabilistic notions from quantum
Team semantics is based on the same notion of structure as Tarskian semantics, but instead of a single assignment satisfying a formula (or not), in team semantics a set, or a team, of assignments
satisfies a formula (or not). In other words, the semantic value of a formula is lifted from a set of assignments (those that satisfy the formula) to a set of teams of assignments.
In almost all (with only one exception that I’m aware of) logical systems based on team semantics this lifting operation is the power set operation, and as a result the set of teams satisfying a
formula is closed downwards. This is often taken as a basic and foundational principle of team semantics.
In this talk I will discuss this principle and present some ideas on why, or why not, the power set operation is the most natural lift operation. By using other lift operations we can get a more
powerful semantics, but, it seems, also a more complicated one.
□ Engström, F. (2012) “Generalized quantifiers in dependence logic”
□ Nurmi, V. (2009) “Dependence Logic: Investigations into Higher-Order Semantics Defined on Teams”
□ Väänänen, J. (2007) “Dependence logic: A new approach to independence friendly logic”
• Erich Grädel (RWTH Aachen University)
Semiring semantics for logical statements with applications to the strategy analysis of games
Semiring semantics of logical formulae generalises the classical Boolean semantics by permitting multiple truth values from certain semirings. In the classical Boolean semantics, a model of a
formula assigns to each (instantiated) literal a Boolean value. K-interpretations, for a semiring K, generalize this by assigning to each such literal a value from K. We then interpret 0 as false
and all other semiring values as nuances of true, which provide additional information, depending on the semiring: For example, the Boolean semiring over {0,1} corresponds classical semantics,
the Viterbi-semiring can model confidence scores, the tropical semiring is used for cost analysis, and min-max-semirings (A, max, min, a, b) for a totally ordered set (A,<) can model different
access levels. Most importantly, semirings of polynomials, such as N[X], allow us to track certain literals by mapping them to different indeterminates. The overall value of the formula is then a
polynomial that describes precisely what combinations of literals prove the truth of the formula.
This can also be used for strategy analysis in games. Evaluating formulae that define winning regions in a given game in an appropriate semiring of polynomials provides not only the Boolean
information on who wins, but also tells us how they win and which strategies they might use. For this approach, the case of Büchi games is of special interest, not only due to their practical
importance, but also because it is the simplest case where the logical definition of the winning region involves a genuine alternation of a greatest and a least fixed point. We show that, in a
precise sense, semiring semantics provide information about all absorption-dominant strategies – strategies that win with minimal effort, and we discuss how these relate to positional and the
more general persistent strategies. This information enables further applications such as game synthesis or determining minimal modifications to the game needed to change its outcome.
• João Pedro Paulos (Chalmers)
A collection of small closed sets: sets of uniqueness
Sets of uniqueness and their properties are traditionally investigated in Harmonic Analysis. The study of such sets has a long and illustrious history, witnessing fruitful interdisciplinary
interactions, often enriching the subject with a vibrant fertility for crossover of ideas. In this talk, we set up the modern framework used to study such sets with particular emphasis on some
(classical) descriptive set-theoretic aspects. We present some results concerning the family of closed sets of uniqueness of a locally compact Polish group - more concretely, we will talk about
their complexity and the (in)existence of a Borel basis.
• Mateusz Łełyk (University of Warsaw)
Axiomatizations of Peano Arithmetic: a truth-theoretic view
We employ the lens provided by formal truth theory to study axiomatizations of PA (Peano Arithmetic). More specifically, let EA (Elementary Arithmetic) be the fragment I∆0 + Exp of PA, and CT^−
[EA] be the extension of EA by the commonly studied axioms of compositional truth CT^−. The truth theory delivers a natural preorder on the set of axiomatizations: an axiomatization A is greater
or equal to an axiomatization B if and only if, over CT^-[EA], the assertion “All axioms from A are true” implies “All axioms from B are true”. Our focus is dominantly on two types of
axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to PA, and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of
The first part of the talk focuses on the axiomatizations of type (1). We adapt the argument by Visser and Pakhomov (“On a question of Krajewski’s”, JSL 84(1), 2019) to show that there is no
weakest axiomatization of this form (even if the axiomatizations are ordered by relative interpretability). Secondly, we sketch an argument showing that such axiomatizations with the given
ordering form a countably universal partial order. This part is based on our joint work with Ali Enayat, available at https://www.researchgate.net/publication/
In the second part of the talk we discuss axiomatizations of type (2). We narrow our attention to such axiomatizations A for which CT^-[EA] + “All axioms from A are true” is a conservative
extension of PA. We explain why such theories have very diverse metamathematical properties (e.g. large speed-up). To illustrate our methods we show that, with the given ordering, such
axiomatizations do not form a lattice. This is a work still in progress.
• Anupam Das (University of Birmingham)
On the proof theoretic strength of cyclic reasoning
Cyclic (or circular) proofs are now a common technique for demonstrating metalogical properties of systems incorporating (co)induction, including modal logics, predicate logics, type systems and
algebras. Inspired by automaton theory, cyclic proofs encode a form of self-dependency of which induction/recursion comprise special cases. An overarching question of the area, the so-called
‘Brotherston-Simpson conjecture’, asks to what extent the converse holds.
In this talk I will discuss a line of work that attempts to understand the expressivity of circular reasoning via forms of proof theoretic strength. Namely, I address predicate logic in the guise
of first-order arithmetic, and type systems in the guise of higher-order primitive recursion, and establish a recurring theme: circular reasoning buys precisely one level of ‘abstraction’ over
inductive reasoning.
This talk will be based on the following works:
• Nachiappan Valliappan (Chalmers)
Normalization for Fitch-style Modal Lambda Calculi
Fitch-style modal lambda calculi (Borghuis 1994; Clouston 2018) provide a solution to programming necessity modalities in a typed lambda calculus by extending the typing context with a delimiting
operator (denoted by a lock). The addition of locks simplifies the formulation of typing rules for calculi that incorporate different modal axioms, but obscures weakening and substitution, and
requires tedious and seemingly ad hoc syntactic lemmas to prove normalization.
In this work, we take a semantic approach to normalization, called Normalization by Evaluation (NbE) (Berger and Schwichtenberg 1991), by leveraging the possible world semantics of Fitch-style
calculi. We show that NbE models can be constructed for calculi that incorporate the K, T and 4 axioms, with suitable instantiations of the frames in their possible world semantics. In addition
to existing results that handle beta reduction (or computational rules), our work also considers eta expansion (or extensional equality rules).
□ Borghuis, V.A.J. (1994). “Coming to terms with modal logic : on the interpretation of modalities in typed lambda-calculus”.
□ Clouston, Ranald (2018). “Fitch-Style Modal Lambda Calculi”.
□ Berger, Ulrich and Helmut Schwichtenberg (1991). “An Inverse of the Evaluation Functional for Typed lambda-calculus”.
• Dag Normann (Oslo)
An alternative perspective on Reverse Mathematics
In his address to the International Congress of Mathematics in Vancouver, 1974, Harvey Friedman launched a program where the aim would be to find the minimal set of axioms needed to prove
theorems of ordinary mathematics. More than often, it turned out that the axioms then would be provable from the theorems, and the subject was named Reverse Mathematics. In this talk we will
survey some of the philosophy behind, and results of, the early reverse mathematics, based on the formalisation of mathematics within second order number theory.
In 2005, Ulrich Kohlenbach introduced higher order reverse mathematics, and we give a brief explanation of the what and why? of Kohlenbach’s approach. In an ongoing project with Sam Sanders we
have studied the strength of classical theorems of late 19th/early 20th century mathematics, partly within Kohlenbach’s formal typed theory and partly by their, in a generalised sense,
constructive content. In the final part of the talk I will give some examples of results from this project, mainly from the perspective of higher order computability theory. No prior knowledge of
higher order computability theory is needed.
• Victor Lisinski (Corpus Christi, Oxford)
Decidability problems in number theory
In its modern formulation, Hilbert’s tenth problem asks to find a general algorithm which decides the solvability of Diophantine equations. While this problem was shown to be unsolvable due to
the combined work of Davis, Putnam, Robinson and Matiyasevich, similar question can be posed over domains other than the integers. Among the most important open questions in this area of research
is if a version of Hilbert’s tenth problem for F[p]((t)), the field of formal Laurent series over the finite field F[p], is solvable or not.
The fact that this remains open stands in stark contrast to the fact that the first order theory of the much similar object Q[p], the field of p-adic numbers, is completely understood thanks to
the work by Ax, Kochen and, independently, Ershov. In light of this dichotomy, I will present new decidability results obtained during my doctoral research on extensions of F[p]((t)). This work
is motivated by recent progress on Hilbert’s tenth problem for F[p]((t)) by Anscombe and Fehm and builds on previous decidability results by Kuhlman.</p>
• Juliette Kennedy (Helsinki)
Logicality and Model Classes
When is a property of a model a logical property? According to the so-called Tarski-Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to the
model-theoretic characteristics of abstract logics in which the model class is definable, resulting in a graded concept of logicality (in the terminology of Sagi’s paper “Logicality and
We consider which characteristics of logics, such as variants of the Löwenheim-Skolem Theorem, Completeness Theorem, and absoluteness, are relevant from the logicality point of view, continuing
earlier work by Bonnay, Feferman, and Sagi. We suggest that a logic is the more logical the closer it is to first order logic, and offer a refinement of the result of McGee that logical
properties of models can be expressed in L[∞∞] if the expression is allowed to depend on the cardinality of the model, based on replacing L[∞∞] by a “tamer” logic. This is joint work with Jouko
• Wilfrid Hodges (Fellow of the British Academy)
How the teenage Avicenna planned out several new logics
Almost exactly a thousand years ago a teenager known today as Avicenna lived in what is now Uzbekistan. He made a resolution to teach himself Aristotelian logic, armed with an Arabic translation
of Aristotle and a century-old Arabic textbook of logic. A couple of years later, around his eighteenth birthday, he wrote a brief report of what he had learned.
Six months ago I started to examine this report - I suspect I am the first logician to do that. It contains many surprising things. Besides introducing some new ideas that readers of Avicenna
know from his later works, it also identifies some specific points of modal logic where Avicenna was sure that Aristotle had made a mistake. People had criticised Aristotle’s logic before, but
not at these points. At first Avicenna had no clear understanding of how to do modal logic, and it took him another thirty years to justify all the criticisms of Aristotle in his report. But
meanwhile he discovered for himself how to defend a new logic by building new foundations. I think the logic itself is interesting, but the talk will concentrate on another aspect. These recent
discoveries mean that Avicenna is the earliest known logician who creates new logics and tells us what he is doing, and why, at each stage along the way.
• Mattias Granberg Olsson (FLoV)
A proof of conservativity of fixed points over Heyting arithmetic via truth
I will present work in progress (together with Graham Leigh) on a novel proof of the conservativity of the intuitionistic fix-point theory over Heyting arithmetic (HA), originally proved in full
generality by Arai [1].
We make use of the work of van den Berg and van Slooten [2] on realizability in Heyting arithmetic over Beeson’s logic of partial terms (HAP). Let IF be the extension of Heyting arithmetic by
fix-points, denoted \hat{ID}^i[1] in the literature. The proof is divided into four parts: First we extend the inclusion of HA into HAP to IF into a similar theory IFP in the logic of partial
terms. We then show that every theorem of this theory provably has a realizer in the theory IFP(Λ) of fix-points for almost negative operator forms only. Constructing a hierarchy stratifying the
class of almost negative formulae and partial truth predicates for this hierarchy, we use Gödel’s diagonal lemma to show IFP(Λ) is interpretable in HAP. Finally we use use the result of [2] that
adding the schema of “self-realizability” for arithmetic formulae to HAP is conservative over HA. The result generalises the work presented at my half-time seminar 2020-08-28.
[1] Toshiyasu Arai. Quick cut-elimination for strictly positive cuts. Annals of Pure and Applied Logic, 162(10):807–815, 2011.
[2] Benno van den Berg and Lotte van Slooten. Arithmetical conservation results. Ind- agationes Mathematicae, 29:260–275, 2018.
• Sonia Marin (UCL)
Focused nested calculi for modal and substructural logics
Focusing is a general technique for syntactically compartmentalising the non-deterministic choices in a proof system, which not only improves proof search but also has the representational
benefit of distilling sequent proofs into synthetic normal forms.
However, since focusing was traditionally specified as a restriction of the sequent calculus, the technique had not been transferred to logics that lack a (shallow) sequent presentation, as is
the case for some modal or substructural logics.
With K. Chaudhuri and L. Straßburger, we extended the focusing technique to nested sequents, a generalisation of ordinary sequents, which allows us to capture all the logics of the classical and
intuitionistic S5 cube in a modular fashion. This relied on an adequate polarisation of the syntax and an internal cut-elimination procedure for the focused system which in turn is used to show
its completeness.
Recently, with A. Gheorghiu, we applied a similar method to the logic of Bunched Implications (BI), a substructural logic that freely combines intuitionistic logic and multiplicative linear
logic. For this we had first to reformulate the traditional bunched calculus for BI using nested sequents, followed again by a polarised and focused variant that we show is sound and complete via
a cut-elimination argument.
• Jouko Väänänen (Helsinki)
Dependence logic: Some recent developments
In the traditional so-called Tarski’s Truth Definition the semantics of first order logic is defined with respect to an assignment of values to the free variables. A richer family of semantic
concepts can be modelled if semantics is defined with respect to a set (a “team”) of such assignments. This is called team semantics.
Examples of semantic concepts available in team semantics but not in traditional Tarskian semantics are the concepts of dependence and independence. Dependence logic is an extension of
first-order logic based on team semantics. It has emerged that teams appear naturally in several areas of sciences and humanities, which has made it possible to apply dependence logic and its
variants to these areas. In my talk I will give a quick introduction to the basic ideas of team semantics and dependence logic as well as an overview of some new developments, such as
quantitative analysis of team properties, a framework for a multiverse approach to set theory, and probabilistic independence logic inspired by the foundations of quantum mechanics.
• Carlo Nicolai (King's College)
A New Look at Cut Elimination for Compositional Truth
In the field of axiomatic theories of truth, conservativity properties of theories are much investigated. Conservativity can be used to argue that, despite the well-known undefinability of truth,
there is a sense in which a primitive truth predicate can be reduced to the resources of an underlying mathematical theory that provides basic syntactic structure to truth ascriptions.
Conservativity is typically proved model-theoretically, or proof-theoretically via the elimination of cuts on formulae containing truth (Tr-cuts). The original Tr-cut-elimination argument for the
theory of Tarskian, compositional truth CT[B] by Halbach is not conclusive. This strategy has been corrected by Graham Leigh: Leigh supplemented Halbach’s strategy with the machinery of
approximations (introduced by Kotlarski, Krajewski and Lachlan in the context of their M-Logic). In the talk we investigate a different, and arguably simpler way of supplementing Halbach’s
original strategy. It is based on an adaptation of the Takeuti/Buss free-cut elimination strategy for first-order logic to the richer truth-theoretic context. If successful, the strategy promises
to generalize to the type-free setting in a straightforward way. This is joint work with Luca Castaldo.
• Dag Prawitz (Stockholm)
Validity of inference and argument
An account of inferences should take into account not only inferences from established premisses but also inferences made under assumptions. This makes it necessary to consider arguments, chains
of inferences in which assumptions and variables may become bound.
An argument is valid when all its inferences are valid, and it then amounts to a proof in case it has no unbound assumptions or variables. The validity of an inference – not to confuse with the
conclusion being a logical consequence of the premisses – seems in turn best explained in terms of proofs. This means that the concepts of valid inference and valid argument depend on each other
and cannot be defined independently but have to be described by principles that state how they are related. A number of such principles will be proposed. It is conjectured that inferences that
can be expressed in the language of first order intuitionistic predicate logic and are implied to be valid by these principles are all provable in that logic.
• Lance Rips (Northwestern University)
Experimenting with (Conditional) Perfection
Conditional perfection is the phenomenon in which conditionals are strengthened to biconditionals. In some contexts, “If A, B” is understood as if it meant “A if and only if B.” I’ll present and
discuss a series of experiments designed to test one of the most promising pragmatic accounts of conditional perfection.
This is the idea that conditional perfection is a form of exhaustification, triggered by a question that the conditional answers. If a speaker is asked how B comes about, then the answer “If A,
B” is interpreted exhaustively to meaning that A is the only way to bring about B. Hence, “A if and only if B.” The evidence suggests that conditional perfection is a form of exhaustification,
but not that it is triggered by a relationship to a salient question. (This is joint work with Fabrizio Cariani.)
• Giacomo Barlucchi and Tjeerd Fokkens (FLoV)
PhD Project Presentations
• Bahareh Afshari (FLoV)
Cyclic Proof Systems for Modal Logics
A cyclic proof is a, possibly infinite but, regular derivation tree in which every infinite path satisfies a certain soundness criterion, the form of which depends on the logic under study.
Circular and, more generally, non-wellfounded derivations are not traditionally regarded as formal proofs but merely as an intermediate machinery in proof-theoretic investigations.
They are, however, an important alternative to finitary proofs and in the last decade have helped break some important barriers in the proof theory of logics formalising inductive and
co-inductive concepts. In this talk we focus on cyclic proofs for modal logics, ranging from Gödel-Löb logic to more expressive languages such as the modal mu-calculus, and reflect on how they
can contribute to the development of the theory of fixed point modal logic. | {"url":"https://logic-gu.se/events/2021/","timestamp":"2024-11-05T09:46:33Z","content_type":"text/html","content_length":"50091","record_id":"<urn:uuid:0504c475-884d-46b6-bed6-8f077b358a22>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00368.warc.gz"} |
On the picture fuzzy database: theories and application
In this paper, we consider some properties of picture fuzzy relation and picture fuzzy tolerance relation on a universe. Finally, we introduced the new concept: picture fuzzy database (PFDB) and have
shown by an example usefulness of picture fuzzy queries on a picture fuzzy database. In the next time, we will study about the functional dependence and practice the normalization in the picture
fuzzy database.
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J. Sci. & Devel. 2015, Vol. 13, No. 6: 1028-1035 Tạp chí Khoa học và Phát triển 2015, tập 13, số 6: 1028-1035 www.vnua.edu.vn 1028 ON THE PICTURE FUZZY DATABASE: THEORIES AND APPLICATION Nguyen Van
Dinh* , Nguyen Xuan Thao, Ngoc Minh Chau Faculty of Information Technology, Viet Nam National University of Agriculture Email*: nvdinh@vnua.edu.vn Received date: 22.07.2015 Accepted date: 03.09.2015
ABSRACT Around the 1970s, the concept of the (crisp) relational database was introdued which enables us to store and practice with an organized collection of data. In a relational database, all data
are stored and accessed via relations. The extension of the relational data base can be done in several directions. Fuzzy relational database generalizes the classical relational database. In this
paper, we introduce a new concept: picture fuzzy database (PFDB), study some queries on a picture fuzzy database, and give an example to illustrate the application of this database model. Keywords:
Picture fuzzy set, picture fuzzy relation, picture fuzzy database (PFDB). Cơ sở dữ liệu mờ bức tranh: lý thuyết và ứng dụng TÓM TẮT Những năm 1970, khái niệm cơ sở dữ liệu quan hệ (rõ) được đề xuất
cho phép chúng ta có thể lưu trữ và thao tác với một họ có tổ chức của dữ liệu. Trong một cơ sở dữ liệu quan hệ, tất cả các dữ liệu được lưu trữ và truy cập thông qua các quan hệ. Sự mở rộng của cơ
sở dữ liệu quan hệ có thể thực hiện theo nhiều hướng khác nhau. Cơ sở dữ liệu quan hệ mờ là một sự mở rộng của cơ sở dữ liệu quan hệ cổ điển. Bài báo này xin giới thiệu một khái niệm mới về cơ sở dữ
liệu mờ bức tranh (PFDB), nghiên cứu một vài truy vấn trên một cơ sở dữ liệu mờ bức tranh và đưa ra một ví dụ minh họa cho ứng dụng của mô hình CSDL này. Từ khóa: Cơ sở dữ liệu mờ bức tranh, quan hệ
mờ bức tranh, tập mờ bức tranh. 1. INTRODUCTION Fuzzy set theory was introduced since 1965 (Zadeh, 1965). Immediately, it became a useful method to study in the problems of imprecision and
uncertainty. Since, a lot of new theories treating imprecision and uncertainty have been introduced. For instance, Intuitionistic fuzzy sets were introduced in 1986 by Atanassov (Atanassov, 1986),
which is a generalization of the notion of a fuzzy set. While fuzzy set gives the degree of membership of an element in a given set, intuitionistic fuzzy set gives a degree of membership and a degree
of non-membership. In 2013, Bui and Kreinovich (2013) introduced the concept of picture fuzzy set, which has identifies three degrees of memberships memberships for each element in a given set: a
degree of positive membership, a degree of negative membership, and a degree of neutral membership. Later on, Le Hoang Son và Pham Huy Thong (2014); Le Hoang Son (2015) reported an application of
picture fuzzy set in the clustering problems. Nguyen Đinh Hoa et al. (2014) proposed an innovative method for weather forecasting from satellite image sequences using the combination of picture fuzzy
clustering and spatio-temporal regression. These indicate the effective application of picture fuzzy set in the actual problems. Around the 1970s, Codd introduced the concept of the (crisp)
relational database (the classical relational database) which enables us Nguyen Van Dinh , Nguyen Xuan Thao, Ngoc Minh Chau 1029 to to store and practice with an organized collection of data. A
relation is defined as a set of tuples that have the same attributes. A tuple usually represents an object and information about that object. A relation is usually described as a table, which is
organized into rows and columns. All the data referenced by an attribute are in the same domain and conform to the same constraints. In a relational database, all data are stored and accessed via
relations. Relations that store data are called base relations, and in implementation are called tables. Other relations do not store data, but are computed by applying relational operations to other
relations. In implementations, these are called queries. Derived relations are convenient in that they act as a single relation, even though they may grab information from several relations. Also,
derived relations can be used as an abstraction layer. Fuzzy data structure was first studied by Tanaka et al. (1977) in which the membership grades were directly coupled each datum and relation.
Fuzzy relational database that generalizes the classical relational database by allowing uncertain and imprecise information to be represented and manipulated. Data is often partially known, vague or
ambiguous in many real world applications. There are several methods to describe a fuzzy relational database. For instance, either the domain of each attribute is fuzzy (Petry and Buckles, 1982) or
the relation of attribute values in the domain of any attribute in the relational database is fuzzy relations (Shokrani-Baigi et al., 2002; Mishra and Ghosh, 2008). The extension of the relational
database can be done in many different directions. Roy et al. (1998) introduced the concept of intuitionistic fuzzy database in which, the relation of attribute values in the domain of any attribute
in the relational database is intuitionistic fuzzy relations. After that, some application of intuitionistic fuzzy database was studied. Kelov et al. (2005) applied the Intuitionistic Fuzzy
Relational Databases in Football Match Result Predictions. Kolev and Boyadzhieva, (2008) extended the relational model to intuitionistic fuzzy data quality attribute model and Ashu (2012) studied the
intuitionistic fuzzy approach to handle imprecise humanistic queries in databases. Hence, the extension of concepts of relational database is necessary. In this paper we studied picture fuzzy
relations and introduced a new concept: picture fuzzy database in which, the relation of attribute values in the domain of any attribute in the relational database is picture fuzzy relations. Which
is an extension of a fuzzy database, intutionistic fuzzy database. The remaining of this paper: In section 2, we recalled some notions of picture fuzzy set and picture fuzzy relation; we consider
some properties of picture fuzzy tolerance relation in section 3; finally, we introduce new concept: picture fuzzy database and some queries on PFDB. 2. BASIC NOTIONS OF PICTURE FUZZY SET AND PICTURE
FUZZY RELATION In this paper, we denote U be a nonempty set called the universe of discourse. The class of all subsets of U will be denoted by P(U) and the class of all fuzzy subsets of U will be
denoted by F(U). Definition 1. (Bui and Kreinovick, 2013) A picture fuzzy (PF) set ܣ on the universe ܷis an object of the form: ܣ = {(ݔ, ߤ(ݔ), ߟ (ݔ), ߛ(ݔ))|ݔ ∈ ܷ} where μ(x) ∈ [0,1], the “degree of
positive membership of x in A”; η(x) ∈ [0,1], the “degree of neutral membership of x in A” and γ(x) ∈[0,1]; and the “degree of negative membership of x in A”, and μ, η and γ satisfied the
following condition: μ(x) + η (x)) + γ(x) ≤ 1, (∀ x ∈ X). The family of all picture fuzzy set in U is denoted by PFS(U). The complement of a picture fuzzy set A is denoted by A = {(x, γ(x), η
(x), μ(x))|∀x ∈ U} Formally, a picture fuzzy set associates three fuzzy sets, they are identified by On The Picture Fuzzy Database: Theories and Application 1030 μ: U → [0,1], η: U → [0,1] and γ:
U → [0,1] and can be represented as = (μ, η, γ ). Obviously, any intuitionistic fuzzy set A = {(x, μ(x), γ(x))} may be identified with the picture fuzzy set in the form A = {(x, μ(x), 0, γ(x))
|x ∈ U}. The operator on PFS(U) was introduced [1]: ∀ A, B ∈ PFS(U), A ⊆ B iff μ(x) ≤ μ(x), η(x) ≤ η(x) and γ(x) ≥ γ(x) ∀ x ∈ U. A = B iff A ⊆ B and B ⊆ A. A ∪ B = ൛൫x, max (μ(x), μ(x)
൯, min൫η (x), η(x), min (γ(x), γ(x)൯൯|x ∈ U} A ∩ B = {(x, min (μ(x), μ(x)), min (η (x), η(x), max (γ(x), γ(x))|x ∈ U} Now we define some special PF sets: a constant PF set is the PF set
(α, β, θ) = {(x,α, β, θ)|x ∈ U}; the PF universe set is U = 1 = (1,0,0) = {(x, 1,0,0)|x ∈ U} and the PF empty set is ∅ = 0 = (0,1,0) = {(x, 0,1,0)|x ∈ U}. For any x ∈ U, picture fuzzy sets 1୶
and 1ି{୶} are, respectively, defined by: for all y ∈ U μଵ౮(y) = ൜1, if y = x 0, if y ≠ x γଵ౮(y) = ൜0, if y = x 1, if y ≠ x ηଵ౮(y) = ൜0, if y = x 0, if y ≠ x μଵష{౮}(y) = ൜0, if y = x 1, if y ≠ x γଵష
{౮}(y) = ൜1, if y = x 0, if y ≠ x ηଵష{౮}(y) = ൜0, if y = x 0, if y ≠ x Definition 2. Let ܷbe a nonempty universe of discourse which many be infinite. A picture fuzzy relation from ܷto ܸis a picture
fuzzy set of ܷ× ܸand denote by ܴ(ܷ → ܸ), i.e, is an expression given by ܴ= {((ݔ, ݕ), ߤோ(ݔ, ݕ), ߟோ(ݔ, ݕ), ߛோ(ݔ, ݕ))|(ݔ, ݕ) ∈ ܷ× ܸ}, where μୖ, γୖ, ηୖ are functions from UxV to [0,1] such that μୖ(x, y) + ηୖ(x, y) + γୖ
(x, y) ≤ 1 for all (x, y) ∈ U ×V. When U ≡ V then, R(U → U) is called a picture fuzzy relation on U. Definition 3. Let ܲ(ܷ → ܸ) and ܳ(ܸ → ܹ). Then, the max-min composition of the picture fuzzy relation ܲ
with the picture fuzzy relation ܳis a picture fuzzy relation ܲ∘ ܳon ܷ× ܹwhich is defined by, for all (ݔ, ݖ) ∈ ܷ× ܹ: ߤ∘ொ(ݔ, ݖ) = ݉ܽݔ௬∈{݉݅݊൛ߤ(ݔ, ݕ), ߤொ(ݕ, ݖ)ൟ} ߟ∘ொ(ݔ, ݖ) = ݉݅݊௬∈{݉݅݊൛ߟ(ݔ, ݕ), ߟொ(ݕ, ݖ)ൟ} ߛ∘ொ(ݔ, ݖ) = ݉݅݊௬∈
{݉ܽݔ൛ߛ(ݔ, ݕ), ߛொ(ݕ, ݖ)ൟ} Definition 4. The picture fuzzy relation ܴ݊ U is referred to as: Reflexive: if for all ݔ ∈ ܷ, ߤோ(ݔ, ݔ) = 1, Symmetric: if for all ݔ, ݕ ∈ ܷ, ߤோ(ݔ, ݕ) = ߤோ(ݕ, ݔ), ߛோ(ݔ, ݕ) = ߛோ(ݕ,
ݔ),and ߟோ(ݔ, ݕ) = ߟோ(ݕ, ݔ), Transitive: If ܴଶ ⊂ ܴ, where ܴଶ = ܴ∘ ܴ, Picture tolerance: if ܴis reflexive and symmetric, Picture preorder: if ܴis reflexive and transitive, Picture similarity (picture
fuzzy equivalence): if ܴis reflexive and symmetric, transitive. Example 1. Let U = {uଵ, uଶ, uଷ} be a universe set. We consider a relation R on U as follows (Table 1): It is easily that R is reflexive,
symmetric. But it is not transitive, because Rଶ ⊈ R. The relation Rଶ is computed in Table 2. Here, we see that ൫μୖ∘ୖ(uଵ, uଶ), ηୖ∘ୖ(uଵ, uଶ), γୖ∘ୖ(uଵ, uଶ)൯ = (0.4,0,0.1) > ൫μୖ(uଵ, uଶ), ηୖ(uଵ, uଶ), γୖ(uଵ, uଶ)൯ =
(0.3,0.4,0.2). The transitive closure (proximity relation) of R(U → U) is R, defined by R = R ∪ Rଶ ∪ Rଷ ∪ . Nguyen Van Dinh , Nguyen Xuan Thao, Ngoc Minh Chau 1031 Table 1. The picture fuzzy
relation ࡾ R uଵ uଶ uଷ uସ uଵ (1,0,0) (0.3,0.4,0.2) (0.4,0.5,0.1) (0.3,0.4,0.2) uଶ (0.3,0.4,0.2) (1,0,0) (0.7,0.2,0.05) (0.4,0.5,0.1) uଷ (0.4,0.5,0.1) (0.7,0.2,0.05) (1,0,0) (0.3,0.4,0.2) uସ
(0.3,0.4,0.2) (0.4,0.5,0.1) (0.3,0.4,0.2) (1,0,0) Table 2. The picture fuzzy relation ܀ Rଶ uଵ uଶ uଷ uସ uଵ (1,0,0) (0.4,0,0.1) (0.4,0,0.1) (0.3,0,0.2) uଶ (0.3,0,0.1) (1,0,0) (0.7,0,0.05) (0,4,0,0.2)
uଷ (0.4,0,0.1) (0.7,0,0.05) (1,0,0) (0.7,0,0.1) uସ (0.4,0,0.1) (0.4, 0,0.1) (0.4,0,0.1) (1,0,0) Definition 5. Let ܣ be a picture fuzzy set of the set ܷ. For ߙ ∈ [0,1], the ߙ −cut of ܣ (or level ߙ of
ܣ) is the crisp set ܣఈ defined by ܣఈ = {ݔ ∈ ܷ: ߛ(ݔ) ≤ 1 − ߙ }. Note that if μ(x) + η(x) ≥ α then γ(x) ≤ 1 − α. Example 2. A = (.଼,.ହ,.ଵ) ୳భ + (.,.ଵ,.ଶ) ୳మ +(.ହ,.ଵ,.ସ) ୳య is a picture
fuzzy set on the universe U = {uଵ, uଶ, uଷ}. Then 0.2 −cut of A is the crisp set A = {uଵ, uଶ}. 3. ON PICTURE FUZZY RELATION In this section, we study some properties of picture fuzzy relations.
Definition 6. If ܴ(ܷ → ܷ) is a picture fuzzy tolerance relation on ܷ, then given an ߙ ∈ [0,1], two elements ݔ, ݕ ∈ ܷare ߙ −similar, denoted by ݔܴఈݕ, if only if ߛோ(ݔ, ݕ) ≤ 1 − ߙ. Definition 7. If ܴ(ܷ → ܷ) is a
picture fuzzy tolerance relation on ܷ, then two elements ݔ, ݖ ∈ ܷare ߙ − tolerance, denoted by ݔܴఈାݖ, if only if either ݔܴఈݕ or there exists a sequence ݕଵ , ݕଶ, , ݕ ∈ ܷsuch that ݔܴఈݕଵܴఈݕଶ ݕܴఈݖ. Here, we
show that Rା is transitive. Then we have Lemma 1. If R is a picture fuzzy tolerance relation on ܷ, then ܴఈା is an equivalence relation.. For any ߙ ∈ [0,1], ܴఈା partitions ܷinto disjoin equivalence classes.
Lemma 2. If R is a picture fuzzy similarity relation on ܷthen ܴఈ is an equivalence relation for any ߙ ∈ [0,1]. Lemma 3. If R is a picture fuzzy similarity relation on ܷand ߙ ∈ [0,1] be fixed. ܻ⊂ ܷis an
equivalence class in the partition determined by ܴఈ with respect to ܴif only if ܻis a maximal subset obtained by merging elements from ܷthat satisfies ݉ܽݔ௫,௬∈ߛோ(ݔ, ݕ) ≤ 1 − ߙ. Lemma 4. If R is a picture
fuzzy similarity relation on ܷthen for any ߙ ∈ [0,1], ܴఈ and ܴఈା is generate identical equivalence classes. Lemma 5. The transitive closure ܴ of a picture fuzzy tolerance relation R on U is a minimal
picture fuzzy similarity relation containing ܴ. The proof of these results is obviously. Example 3. Consider the picture fuzzy tolerance relation R on U = {uଵ, uଶ, uଷ, uସ} given by On The Picture
Fuzzy Database: Theories and Application 1032 Table 3. The tolerance picture fuzzy relation R uଵ uଶ uଷ uସ uଵ (1,0,0) (0.8,0.1,0.1) (0.6,0.1,0.3) (0,0.2,0.8) uଶ (0.8,0.1, 0.1) (1,0,0) (0.5,0.1,0.4)
(0.6,0.1,0.3) uଷ (0.6,0.1,0.3) (0.5,0.1,0.4) (1,0,0) (0.3,0.4,0.2) uସ (0,0.2,0.8) (0.6,0.1,0.3) (0.3,0.4,0.2) (1,0,0) By Definition 7, it can be computed that: for α = 1, then the partition of U
determined by Rଵis: {{uଵ}, { uଶ}, {uଷ}, {uସ}}, for α = 0.9, then the partition of U determined by R.ଽ is: {{uଵ, uଶ}, {uଷ}, {uସ}}, for α = 0.8, then the partition of U determined by R.଼ is: {{uଵ,
uଶ}, {uଷ, uସ}}, for α = 0.7, here, although γୖ(uଶ, uଷ) = 0.4 > 1 − 0.7 = 0.3, but also we have uଶR.uଵ and uଵR.uଷ then uଶR.ା uଷ. Furthermore, we have uଷR.uସ, so that partition of U determined by
R. is: {{uଵ, uଶ, uଷ, uସ}}. Moreover, it is easily seen that: for 0.9 < α ≤ 1, then the partition of U determined by Rଵ given by {{uଵ}, { uଶ}, {uଷ}, {uସ}}, for 0.8 < α ≤ 0.9, then the partition of U
determined by R.ଽ given by {{uଵ, uଶ}, {uଷ}, {uସ}}, for 0.7 < α ≤ 0.8, then the partition of U determined by R.଼ given by {{uଵ, uଶ}, {uଷ, uସ}}, for α ≤ 0.7, then the partition of U determined by R.
given by {{uଵ, uଶ, uଷ, uସ}}. 4. PICTURE FUZZY DATABASE In the section we introduce the concept of picture fuzzy database. First, we recall that the ordinary relation database represents data as a
collection of relations containing tuples. The organization of relational databases is based on a set theory and relation theory. Essentially, relational databases consist of one or more relations in
two-dimensional (row and column) format. Rows are called tuples and correspond to records; columns are called domains and correspond to fields. A tuple t୧ having the form t୧ = (d୧ଵ,d୧ଶ, , d୧୫), where
d୧୨ ∈ D୨ is the domain value of a particular domain set D୨. In the fuzzy relational database, d୧୨ ⊂ D୨ is the fuzzy subset of D୨. If d୧୨ ⊂ D୨ is the (fuzzy) subset of D୨ and they have the
intutionistic fuzzy tolerance relation for each other, themselves, i.e., the domain values of a particular domain set D୨ have an intutionistic fuzzy tolerance relation. Then we obtain the
intuitionistic fuzzy database. Also, if d୧୨ ⊂ D୨ is the (fuzzy) subset of D୨ and they have the picture fuzzy tolerance relation for each other, themselves, i.e., the domain values of a particular
domain set D୨ have a picture fuzzy tolerance relation. In this case, we call this new concept is picture fuzzy database. Now, for each the attribute D୨, we denote P൫D୨൯ as the collection of all
subset of D୨ and 2ୈౠ = P(D୨) − ∅ as the collection of all nonempty subset of D୨. There exists at least an attribute D୨, in which, the picture fuzzy tolerance relation defines on it domain. Definition
8. A picture fuzzy database relation ܴis a subset of the cross product 2భ × 2మ × × 2. Definition 9. Let ܴ⊂ 2భ × 2మ × × 2 be a picture fuzzy database relation. A piture fuzzy tuple (with respect
to ܴ) is an element of ܴ. An arbitrary picture fuzzy tuple is of the form ݐ = (݀ଵ,݀ଶ, ,݀), where ݀ ⊂ ܦ. Definition 10. An interpretation of ݐ = (݀ଵ,݀ଶ, ,݀), is a tuple ߠ = (ܽଵ, ܽଶ, , ܽ) where ܽ ∈ ݀
for each domain ܦ. Nguyen Van Dinh , Nguyen Xuan Thao, Ngoc Minh Chau 1033 For each domain D୨, if R୨ is the picture fuzzy tolerance relation then its membership functions are defined by: the
degree of positive membership μୖౠ: D୨ × D୨ → [0,1], the degree of neutral membership ηୖౠ: D୨ × D୨ → [0,1], the degree of negative membership γୖౠ: D୨ × D୨ → [0,1], where μୖౠ(x, y) + ηୖౠ(x, y) + γୖౠ(x,
y) ≤ 1, (x, y) ∈ D୨ × D୨. In summary, the space of interpretations is the set cross product Dଵ × Dଶ × × D୫. However, for any particular relation, the space is limited by the set of valid tuples.
Valid tuples are determined by an underlying semantics of the relation. Note that in an ordinary relational databases, a tuple is equivalent to its interpretation. Example 4. Let us make a
hypothetical case study for an application in the fight against crime. We consider a criminal data file. Supose that one murder has taken place at an area in a deep, dark line. The police suspects
that the murderer is also from the same area. Listening to the eye-witness, the police has discovered that the murderer has more or less full big hair coverage, more or less curly hair texture and he
has moderately large build. Police refers to the criminal data file of all the suspected criminals of that area, the short information table with attributes ‘HAIR COVERAGE’, HAIR TEXTURE’ and ‘BUILD’
is given by Table 4. Then, we consider the picture fuzzy tolerance relation Rଵ on the domain of attribute ‘HAIR COVERAGE’, which is given in Table 5. Next, the picture fuzzy tolerance relation Rଶ on
the domain of attribute ‘HAIR TEXTURE’ which is given in Table 6. Finally, we consider the picture fuzzy tolerance relation Rଷ on the domain of attribute ‘HAIR TEXTURE’, which is given in Table 7.
Table 4. The short information table from the criminal data file (SHORT CRIMINAL DATA) NAME HAIR COVERAGE HAIR TEXTURE BUILD Arup Full Small (FS) Stc. Large Boby Rec. Wavy Very Small (VS) Chandra
Full Small (FS) Straight (Str.) Small (S) Dutta Bald Curly Average (A) Esita Bald Wavy Average (A) Faguni Full Big (FB) Stc. Very Large (VL) Gautom Full Small (FS) Straight (Str.) Small (S) Halder
Rec. Curly Average (A) Table 5. The picture fuzzy tolerance relation ࡾ on the domain of attribute ‘HAIR COVERAGE’ R1 FB FS Rec. Bald FB (1,0,0) (0.8,0.1,0.1) (0.4,0.1,0.4) (0,0,1) FS (0.8,0.1, 0.1)
(1,0,0) (0.5,0.1,0.4) (0,0.1,0.9) Rec. (0.4,0.1,0.4) (0.5,0.1,0.4) (1,0,0) (0.4,0.1,0.4) Bald (0,0,1) (0,0.1,0.9) (0.4,0.1,0.4) (1,0,0) On The Picture Fuzzy Database: Theories and Application 1034
Table 6. the picture fuzzy tolerance relation ࡾ on the domain of attribute ‘HAIR TEXTURE’ R2 Str. Stc. Wavy Curly Str. (1, 0, 0) (0.6, 0.1, 0.3) (0.1, 0.1, 0.7) (0.1, 0, 0.7) Stc. (0.6, 0.1, 0.3)
(1, 0, 0) (0.3, 0.1, 0.4) (0.5, 0.1, 0.2) Wavy (0.1, 0.1, 0.7) (0.5, 0.1, 0.4) (1, 0, 0) (0.4, 0.1, 0.4) Curly (0.1, 0, 0.7) (0.5, 0.1, 0.2) (0.4, 0.1, 0.4) (1, 0, 0) Table 7. the picture fuzzy
tolerance relation ࡾ on the domain of attribute ‘BUILD’ R3 VL L A S VS VL (1, 0, 0) (0.7, 0.1, 0.2) (0.4, 0.1, 0.4) (0.3, 0.1, 0.6) (0, 0, 1) L (0.7, 0.1, 0.2) (1, 0, 0) (0.5, 0.1, 0.4) (0.4, 0,
0.5) (0, 1, 0.9) A (0.5, 0.1, 0.4) (0.5, 0.1, 0.4) (1, 0, 0) (0.5, 0.1, 0.3) (0.3, 0.1, 0.6) S (0.3, 0.1, 0.6) (0.4, 0, 0.5) (0.5, 0.1, 0.3) (1, 0, 0) (0.7, 0.1, 0.2) VS (0, 0, 1) (0, 1, 0.9) (0.3,
0.1, 0.6) (0.7, 0.1, 0.2) (1, 0, 0) Table 8. Relation ‘LIKELY MURDERER ‘ NAME HAIR COVERAGE HAIR TEXTURE BUILD {Arup, Faguni} {Full Big, Full Small} {Curly, Stc.} {Large, Very Large} Now, based on
listening to the eye-witness, the job is to find out a list of the criminals who resemble with more or less full big hair coverage, more or less curly hair texture and moderately large build. The job
can be done with a query on the picture fuzzy database. It can be translated into relational algebra in the following form: Select NAME, HAIR COVERAGE, HAIR TEXTURE, BUILD From SHORT CRIMINAL DATA
With Level(NAME) = 0, Level(HAIR COVERAGE) = 0.8, Level(HAIR TEXTURE) = 0.8, Level (BUILD) = 0.7 Where HAIR COVERAGE = ‘Full Big’ HAIR TEXTURE = ‘Curly’ BUILD = ‘Large’ Giving LIKELY MURDERER It can
be computed that the above query gives rise to the following relation (Table 8): Therefore, according to the information obtained from the eye-witness, the police concludes that Arup or Faglguni are
the likely murderers. And, further investigation now is to be done on them only, instead of dealing with a hugo list of criminals. 5. CONCLUSION In this paper, we consider some properties of picture
fuzzy relation and picture fuzzy tolerance relation on a universe. Finally, we introduced the new concept: picture fuzzy database (PFDB) and have shown by an example usefulness of picture fuzzy
queries on a picture fuzzy database. In the next time, we will study about the functional dependence and practice the normalization in the picture fuzzy database. Nguyen Van Dinh , Nguyen Xuan Thao,
Ngoc Minh Chau 1035 REFERENCES Atanassov K. (1986). Intuitionistic fuzzy sets, Fuzzy set and systems, 20: 87 - 96. Ashu (2012). Intuitionistic fuzzy approach to handle imprecise humanistic queries in
databases, International journal of computer applications, 43(20): 6 - 9. Codd E. F. (1970). "A Relational Model of Data for Large Shared Data Banks". Communications of the ACM, 13(6): 377 - 387. Bui
Cong Cuong, V. Kreinovick (2013). Picture fuzzy sets – a new concept for computational intelligence problems, In: Proceedings of the third world congress on information and communication technologies
WICT’2013, Ha Noi, Viet Nam, December 15-18, p. 1 - 6. Date C. J. (1977). An introduction to database systems, 2nd edition (Addison-Wesley, Reading, MA, 1977). Nguyen Đinh Hoa, Le Hoang Son, Pham Huy
Thong (2014). “Weather nowcasting from satellite image sequences using the combination of picture fuzzy clustering and spatiotemporal regression”, Proceeding of Conference of GISIDEAR, Da Nang – Viet
Nam, December 2014 (accepted). Kolev B., I. Petrounias, P. Chountas, V. Kodogiannis (2005). An Application of Intuitionistic Fuzzy Relational Databases in Football Match Result Predictions,
Computational Intelligence, Theory and Applications Advances in Soft Computing, 33: 281 - 289. Kolev B., D. Boyadzhieva (2008). An extension of the relational model to intuitionistic fuzzy data
quality attribute model”, Intelligent Systems, IS '08. 4th International IEEE Conference, 2: 13 - 18. Mishra J., Ghosh S. D. (2008). “A study of fuzzy relational database”, International journal of
computational cognition, 6(4): 45-50. Petry F. E., Buckles B. P. (1982). A fuzzy representation of data for relational databases, Fuzzy set and systems, 7: 213 - 226. Roy A. R., R. Biwas, S. K. De
(1998). “Intuitionistic fuzzy database”, Second int. con. On IFS, Sofia, 3- 4, NIFS 4 (1998) 2: 34 - 41. Le Hoang Son, Pham Huy Thong (2014). “A new approach to multi-variables fuzzy forecasting
using picture fuzzy clustering and picture fuzzy rule interpolation method”, Proceeding of 6th International conference on knowledge and systems engineering (KSE 2014) (accepted). Le Hoang Son
(2015). “DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy sets”, Expert systems with applications, 42: 51 - 66. Shokrani-Baigi A., Naghibzadeh M., Fathi M., Saadati N.
(2002). “Design and implementation of a fuzzy relational database management system applied to osteoporosis patients”, Automation Congress, 2002 Proceedings of the 5th Biannual World, 14: 423 - 428.
Tanaka K., Mizumoto M., UmanoM. (1977). “Implementation of a fuzzy set theoretic data structure system”, Proceeding third conference on very large databases, Tokyo, p. 59 - 69. Zadeh L. A (1965).
“Fuzzy Sets”, Information and Control, 8(3): 338 - 353.
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Newbie DarkRadiant Questions
where I can look at AI animations list ?
aren't those the .anim files?
I always assumed I'd taste like boot leather.
aren't those the .anim files?
what the .anim files ?
How I can find leak entity on my map ? Looks like I make something wrong and now my map leaked
Proceed with caution!
what the .anim files ?
How I can find leak entity on my map ? Looks like I make something wrong and now my map leaked
Make sure that you don't have any FMs installed if you do, uninstall it, run dmap again and let Doom3 generate a pointfile. Then open your map in DarkRadiant and go to File->Pointfile
You should get a red line pointing to the location of the leak.
I always assumed I'd taste like boot leather.
how make go to location objective ?I want make it second objective,but not final
goal2 is info_tdm_objective_location entity with clip texture
here is set up
Proceed with caution!
The goto objective looks correct. Is there a problem with it? If it is not working you don't need the 1 in the success logic. The boolean logic for that seems to be not working correctly and anyway
you don't need it for a single component. Try leaving that blank. If it still does not work then post again. The only other thing I can think of is that the entity name is case sensitive so if it is
named goal2 not Goal2 then that would break it. They must be identical.
S & R should work on any entity including a ragdoll. Check it on another entity and if it fails then the problem is elsewhere so post more details please.
ah , I just forgot to click on Irreversible , objective unchecks when I leave that place...
ok,now I need make another objective visible after triggering.How to do that ?
Proceed with caution!
how I can turn on Post procession effect ? It suddenly stop working for me
why I cant turn it on again ?
I can't fully enjoy my favorite mod
Edited by Shadowhide
Proceed with caution!
I'm assuming button switches here and not two levers that must both be aligned. I think this should work:
buttonA > trigger_onceA > trigger_countC > entityD
buttonB > trigger_onceB > trigger_countC > entityD
Set count 2 on trigger_countC
Set noTouch 1 on both trigger_onces
2 switches frobbed,one entity triggered . How to do that ?
So, this means you were able to recover your lost FM WIP?
Is there anyway to make AI's transparent in game or do I have to make special skins to achieve the effect. eg you have a ghost that walks around a room but only appears fully solid while in the light
and transparent in the dark.
There is already a revenant spirit under the undead somewhere that is transparent. Have a look how that is set up. It needs a special skin I think.
A few questions:
1. Is it possible to let AI appear only when the player triggers something at a certain moment or do you have to hide them somewhere just like 'Patently Dangerous'?
2. Say I'd like to create a daytime FM, is it necessary to create windows you can look through to create the kind of light you get during daytime or is it possible to create the same with an ambient
Edited by Carnage
1. Yes you can teleport the AI to where you want them. Look under targets in the entities for the dark mod teleport.
2. I don't know if we have any window textures that represent sunlit. Maybe someone can help you modify existing ones. So the best way then would be not transparent but a sunlit window texture, an
internal light, probably projected inward from the window, and optionally the moonbeams effect modified to be sunbeams. I think this is a cylinder patch with a moonbeams texture. That is the way I
would do it. I'm rather busy so hopefully someone else can guide you how to do the above in more detail but if not I'll try to make time.
A few questions:
1. Is it possible to let AI appear only when the player triggers something at a certain moment or do you have to hide them somewhere just like 'Patently Dangerous'?
2. Say I'd like to create a daytime FM, is it necessary to create windows you can look through to create the kind of light you get during daytime or is it possible to create the same with an
ambient light?
Thinking about entering the "Seasons Contest" Carnage?
Please visit TDM's IndieDB site and help promote the mod:
(Yeah, shameless promotion... but traffic is traffic folks...)
Haha I might, count me in as a maybe contender. I've got an idea that I'd like to work out but I don't know if I'll have enough time, because maybe I'll have a new job soon which will be fulltime.
Thanks for the info Fidcal, I'll try it asap and let you know how it works out. More info is always appreciated but only if you have time, it doesn't really have a high priority at the moment.
Thanks Carnage!
I will compile a tentative list.
Please visit TDM's IndieDB site and help promote the mod:
(Yeah, shameless promotion... but traffic is traffic folks...)
I've tried the revanant spirit but in fog he appears nearly totally invisible, until your standing next to him, his skin is an opaque black. And what about the head is there a way to always get the
same head if I retexture a non undead AI.
You can also create a new head def and skin. Basically copy the head def you want, give it a new name, eg myhead, and add to it:
"skin" "<name>"
Then make a new skin called <name>.
On the AI use def_head myhead.
We really need a sword doing the same way.
I don't know about the fog thing but presume it must need the texture copying to a new def then some extra keyword so it works in fog. There is a keyword noFog but I've not tried it. | {"url":"https://forums.thedarkmod.com/index.php?/topic/9082-newbie-darkradiant-questions/page/72/#comment-247542","timestamp":"2024-11-04T14:54:23Z","content_type":"text/html","content_length":"408927","record_id":"<urn:uuid:b7970a15-f82e-422c-8d74-ae14560735d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00211.warc.gz"} |
This paper is concerned with the numerical approximation of mean curvature flow → () satisfying an additional inclusion-exclusion constraint ⊂ () ⊂ . Classical phase field model to approximate
these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn
equation with a penalized double well potential. We first justify this method... | {"url":"https://eudml.org/search/page?q=sc.general*op.AND*l_0*c_0author_0eq%253A1.Val%25C3%25A9rie+Perrier&qt=SEARCH","timestamp":"2024-11-06T07:44:48Z","content_type":"application/xhtml+xml","content_length":"61034","record_id":"<urn:uuid:eb4264b7-a909-47bf-a558-8b5fbbcda033>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00262.warc.gz"} |
15,077 research outputs found
We critically re-examine the calculation of central production of dijets in quasi-elastic hadronic collisions. We find that the process is not dominated by the perturbative contribution, and discuss
several sources of uncertainties in the calculation.Comment: 4 pages, talk given at Diffraction-2008, La Londe-les-Maures, France, 9-14 Sept 200
In the paper we predict a distinctive change of magnetic properties and considerable increase of the Curie temperature caused by the strain fields of grain boundaries in ferromagnetic films. It is
shown that a sheet of spontaneous magnetization may arise along a grain boundary at temperatures greater than the bulk Curie temperature. The temperature dependence and space distribution of
magnetization in a ferromagnetic film with grain boundaries are calculated. We found that $45^\circ$ grain boundaries can produce long-range strain fields that results in the width of the magnetic
sheet along the boundary of the order of $0.5 \div 1 \mu m$ at temperatures grater than the bulk Curie temperature by about $10^2$ K.Comment: 5 pages, 3 Figures include
We calculate the radiative corrections of order O(alpha E_e/m_N) as next-to-leading order corrections in the large nucleon mass expansion to Sirlin's radiative corrections of order O(alpha/pi) to the
neutron lifetime. The calculation is carried out within a quantum field theoretic model of strong low-energy pion--nucleon interactions described by the linear sigma-model (LsM) with chiral SU(2)xSU
(2) symmetry and electroweak hadron-hadron, hadron-lepton and lepton-lepton interactions for the electron-lepton family with SU(2)_L x U(1)_Y symmetry of the Standard Electroweak Model (SEM). Such a
quantum field theoretic model is some kind a hadronized version of the Standard Model (SM). From a gauge invariant set of the Feynman diagrams with one-photon exchanges we reproduce Sirlin's
radiative corrections of order O(alpha/pi), calculated to leading order in the large nucleon mass expansion, and calculate next-to-leading corrections of order O(alpha E_e/m_N). This confirms
Sirlin's confidence level of the radiative corrections O(alpha E_e/m_N). The contributions of the LsM are taken in the limit of the infinite mass of the scalar isoscalar sigma-meson. In such a limit
the LsM reproduces the results of the current algebra (Weinberg, Phys. Rev. Lett. {\bf 18}, 188 (1967)) in the form of effective chiral Lagrangians of pion-nucleon interactions with non--linear
realization of chiral SU(2)xSU(2) symmetry. In such a limit the L$\sigma$M is also equivalent to Gasser-Leutwyler's chiral quantum field theory or chiral perturbation theory (ChPT) with chiral SU(2)
xSU(2)symmetry and the exponential parametrization of a pion-field (Ecker, Prog. Part. Nucl. Phys. {\bf 35}, 1 (1995)).Comment: 50 pages, 7 figures. arXiv admin note: text overlap with | {"url":"https://core.ac.uk/search/?q=author%3A(R%20I%20Ivanov)","timestamp":"2024-11-12T16:42:28Z","content_type":"text/html","content_length":"88886","record_id":"<urn:uuid:9ddab83c-bc86-4803-b833-85a074a3909c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00889.warc.gz"} |
Multiplication By 10 Worksheets
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what is the formula for blane in ball mill
Need to calculate your milling speed, feed, surface feet per minute or inches per tooth? Here are formulas for most common milling operations. (800) . About | News | Careers | ... Ball Nose /
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V — Effective volume of ball mill, m3; G2 — Material less than in product accounts for the percentage of total material, %; G1 — Material less than in ore feeding accounts for in the percentage
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A ball mill consists of a cylinder, which is filled with 3035% of its volume by small steel balls and is rotated through motor. When the cylinder starts to rotate, the balls start to lift under
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Digital Audio – Page 7 – earfluff and eyecandy
Links to:
DFT’s Part 1: Some introductory basics
DFT’s Part 2: It’s a little complex…
DFT’s Part 3: The Math
DFT’s Part 4: The Artefacts
DFT’s Part 5: Windowing
In Part 5, we talked about the idea of using a windowing function to “clean up” a DFT of a signal, and the cost of doing so. We talked about how the magnitude response that is given by the DFT is
rarely “the Truth” – and that the amount that it’s not True is dependent on the interaction between the frequency content of the signal, the signal envelope, the windowing function, the size of the
FFT, and the sampling rate. The only real solution to this problem is to know what-not-to-believe when you look at a DFT output.
However, we “only” looked at the artefacts on the magnitude response in the previous posting. In this last posting, we’ll dig a little deeper and NOT throw away the phase information. The problem is
that, when you’re windowing, you’re not just looking at a screwed up version of the magnitude response, you’re also looking at a screwed up phase response as well.
We saw in Part 1 and Part 2 how the phase of a sinusoidal waveform can be converted to the sum of a real and an imaginary component. (In other words, if you add a cosine and a sine of the same
frequency with very specific separate gains applied to them, the result will be a sinusoidal waveform with any amplitude and phase that you want.) For this posting, we’ll be looking at the artefacts
of the same windowing functions that we’ve been working on – but keeping the real and imaginary components separate.
Rectangular windowing
We’ll start by looking at a plot from the previous post, which I’ve duplicated below.
Figure 1: The magnitude responses calculated by a DFT for 6 different frequencies. Note that the bin centre frequency is 1000.0 Hz.
The way I did the plot in Figure 1 was to create a sine wave with a given frequency, do a DFT of that, and plot the magnitude of the result. I did that for 6 different frequencies, ranging from 1000
Hz (exactly on a bin centre frequency) to 999.5 Hz (halfway to the adjacent bin centre frequency).
There’s a different way to plot this, which is to show the result of the DFT output, bin by bin, for a sinusoidal waveform with a frequency relative to the bin centre frequency. This is shown below
in Figure 2.
Figure 2: Rectangular window:
The relationship between the frequency of the signal, the frequency centres of the DFT bin, and the resulting magnitude in dB. Note that the X-axis is frequency, measured in distance between bin
frequencies or “bin widths”.
Now we have to talk about how to read that plot… This tells me the following (as examples):
• If the bin centre frequency EXACTLY matches the frequency of the signal (therefore, the ∆ Freq. = 0) then the magnitude of that bin will be 0 dB (in other words, it will give me the correct
• If the bin centre frequency is EXACTLY an integer number of bin widths away from the frequency of the signal (therefore, the ∆ Freq. = … -10, -9, – 8… -3, -2, -1, 1, 2, 3, … 8, 9, 10, …) then the
magnitude of that bin will be -∞ dB (in other words, it will have no output).
• These two first points are why the light blue curve is so good in Figure 1.
• If the frequency of the signal is half-way between two bins (therefore, the ∆ Freq. = -0.5 or +0.5), then you get an output of about -4 dB (which is what we also saw in the blue curve in Figure
25 in Part 5.
• If the frequency of the signal is an integer number away from half-way between two bins (for example, ∆ Freq. = -2.5, -1.5, 1.5, or 2.5, etc… ) then the output of that bin will be the value shown
at the tops of those bumps in the plots… (For example, if you mark a dot at each place where ∆ Freq. = ±x.5 on that curve above, and you join the dots, you’ll get the same curve as the curve for
999.5 Hz in Figure 1.)
So, Figure 2 shows us that, unless the signal frequency is exactly the same as the bin centre frequency, then the DFT’s magnitude will be too low, and there will be an output from all bins.
Figure 3: Rectangular window:
The relationship between the frequency of the signal, the frequency centres of the DFT bin, and the resulting phase in degrees.
Figure 3 shows us the same kind of analysis, but for the phase information instead. The important thing when reading this plot is to keep the magnitude response plot in mind as well. For example:
• when the bin frequency matches the signal frequency (∆ Freq. = 0) then the phase error is 0º.
• When the signal frequency is an integer number of bin widths away from the bin frequency, then it appears that the phase error is either 0º or ±180º, but neither of these is true, since the
output is -∞ dB – there is no output (remember the magnitude response plot).
• There is a gradually increasing error from 0º to ±180º (depending on whether you’re going up or down in frequency)( as the signal frequency moves from being adjacent to one bin or the next.
• When you signal frequency crosses the bin frequency, you get a polarity flip (the vertical lines in the sawtooth shape in the plot).
Figure 4. Rectangular window:
The top two plots show the same relationship as in Figures 2 and 3, but divided into the various components, as explained below.
Figure 4, above, shows the same information, plotted differently.
• The bottom right plot shows the magnitude response (exactly the same as shown in Figure 2) on a linear scale instead of in dB.
• The top two plots show the Real and Imaginary components, which, combined, were used to generate the Magnitude and Phase plots. (Remember from Parts 1 and 2 that the Real component is like
looking at the response from above, and the Imaginary component is like looking at the response from the side.)
• The Nyquist plot is difficult, if not impossible to understand if you’ve never seen one before. But looking at the entire length of the animation in Figure 5, below, should help. I won’t bother
explaining it more than to say that it (like the Real vs. Freq. and the Imaginary vs. Freq. plots) is just showing two dimensions of a three-dimensional plot – which is why it makes no sense on
its own without some prior knowledge.
Figure 5. Rectangular window:
The Real, Imaginary, and Nyquist plots from Figure 4, viewed from different angles.
Hopefully, I’ve said enough about the plots above that you are now equipped to look at the same analyses of the other windowing functions and draw your own conclusions. I’ll just make the occasional
comment here and there to highlight something…
Hann Window
Figure 6: Hann window: magnitude response error
Generally, the things to note with the Hann window are the wider centre lobe, but the lower side lobes (as compared to the rectangular windowing function).
Figure 7: Hann window: Phase response error
Figure 8: Hann window: Real, Imaginary, and Nyquist plots
Figure 9: Hann window: Real, Imaginary, and Nyquist plots in all three dimensions.
Hamming window
Figure 10: Hamming window: Magnitude response error.
The interesting thing about the Hamming window is that the lobes adjacent to the main lobe in the middle are lower. This might be useful if you’re trying to ignore some frequency content next to your
signal’s frequency.
Figure 11: Hamming window: Phase response error.
Figure 12: Hamming window: Real, Imaginary, and Nyquist plots
Figure 13: Hamming window: Real, Imaginary, and Nyquist plots in all three dimensions
Blackman Window
Figure 14: Blackman window: Magnitude response error.
The Blackman window has a wider centre lobe, but the side lobes are lower in level.
Figure 15: Hamming window: Magnitude response error.
Figure 16: Blackman window: Real, Imaginary, and Nyquist plots
Figure 17: Blackman window: Real, Imaginary, and Nyquist plots in all three dimensions.
Blackman Harris window
Figure 18: Blackman-Harris window: Magnitude response error.
Although the Blackman-Harris window results in a wider centre lobe, as you can see in Figure 18, the side lobes are all at least 90 dB down from that…
Figure 19: Blackman-Harris window: Phase response error.
Figure 21: Blackman-Harris window: Real, Imaginary, and Nyquist plots
Figure 22: Blackman-Harris window: Real, Imaginary, and Nyquist plots in all three dimensions.
Wrapping up
I know that there’s lots left out of this series on DFT’s. There are other windowing functions that I didn’t talk about. I didn’t look at the math that is used to generate the functions… and I just
glossed over lots of things. However, my intention here was not to do a complete analysis – it was a just an introductory discussion to help instil a lack of trust – or a healthy suspicion about the
results of a DFT (or FFT – depending on how fast you do the math….).
Also, a reason I did this series was as a set-up, so when I write about some other topics in the future (like the actual resolution of 16-bit LPCM audio in a fixed point world, or the implications of
making a volume control in the digital domain as just two examples…), I can refer back to this, pointing out what you can and cannot believe is the plots that I haven’t even made yet…
DFT’s Part 5: Windowing
Links to:
DFT’s Part 1: Some introductory basics
DFT’s Part 2: It’s a little complex…
DFT’s Part 3: The Math
DFT’s Part 4: The Artefacts
The previous posting in this series showed that, if we just take a slice of audio and run it through the DFT math, we get a distorted view of the truth. We’ll see the frequencies that are in the
audio signal, but we’ll also see that there’s energy at frequencies that don’t really exist in the original signal. These are artefacts of slicing a “window” of time out of the original signal.
Let’s say that I were a musician, making samples (in this sentence, the word “sample” means what it means to musicians – a slice of a recording of a sound that I will play using a sampler) to put
into my latest track in my new hip hop album. (Okay, I use the word “musician” loosely here… but never mind…) I would take a sample – say, of the bell that I recorded, which looks like this:
Figure 1: My original bell recording – or 2048 samples of it, at least…
We’ve already seen that the first sample (now I’m back to using the technical definition of the word “sample” – an instantaneous measurement of the amplitude of the signal) and the last sample aren’t
on the 0 line. So, if we just play this recording, it will start and end with a “click”.
We get rid of the click by applying a “fade” on the start and the end of the recording, resulting in something like the following:
Figure 2: The same recording with a fade in and a fade out applied to it. Now it will sound more like a bell because it has a fast attack (a short fade in) and a longer decay (fade out).
So, the moral of the story so far is that, in order to get rid of an audible “click”, we need to fade in and fade out so that we start and end on the 0 line.
We can do the same thing to our slice of audio (from now on, I’m going to call it a “window”) to help the computer that’s doing the DFT math so that it doesn’t get all the extra frequency content
caused by the clicks.
In Figure 2, above, I was being artistic with my fade in and fade out. I made the fade in fast (100 samples long) and I made the fade in slow (2000 samples long) so that the end result would look and
sound more like a bell. A computer doesn’t care if we’re artistic or not – we’re just trying to get rid of those clicks. So, let’s do it.
Option 1 is to do nothing – which is what we’ve done so far. We take all of the individual samples in our window, and we multiply them by 1. (All other samples (the ones that we’re not using because
they’re outside the window) are multiplied by 0.) If you think about what this looks like if I graph it in time, you’ll imagine a rectangle with a height of 1 and a length equal to the length of the
window in samples.
If I do that to my original bell recording, I get the result shown in Figure 3.
Figure 3. The original bell sound, where each sample was multiplied by 1.
You may notice that Figure 3 is almost identical to Figure 1. The only difference is that I put the word “Rectangular” at the top. The reason for this will become clear later.
As we’ve already seen, this rectangular windowing of our recording is what gives us the problems in the first place. So, if I do a DFT of that window, I’ll get the following magnitude response.
Figure 4. The magnitude response of the bell sound with a rectangular window, as shown in Figure 3.
What we know already is that we want to fade in and fade out to get rid of those clicks. So, let’s do that.
Figure 5. The original bell sound, with a fade in and fade out applied to it.
Figure 5, above, shows the result. the ramp in and the ramp out are not straight lines – in fact, they look very much like the shape of an upside-down cosine wave (which is exactly what they are…).
If we do a DFT of that window, we get the result shown in Figure 6.
Figure 6. The magnitude response of the bell sound with a Hann window, as shown in Figure 5.
There are now some things to talk about.
The first is to say that the shape of this “envelope” or “windowing function” – the gain over time that we apply to the audio signal – is named after the Austrian meteorologist Julius von Hann. He
didn’t invent this particular curve – but he did come up with the idea of smoothing data (but in his case, he was smoothing meteorological data over geographical regions – not audio signals over
time). Because he came up with the general idea, they named this curve after him.
Important sidebar: Some people will call this a “Hanning” window. This is not strictly correct – it’s a Hann function. However, you can use the excuse that, if you apply a Hann window to the
signal, then you are hanning it… which is the kind of obfuscated back-pedalling and revisionist history used to cover up a mistake that is typically only within the purview of government
The second thing is to notice is that the overall response at frequencies that are not in the original signal has dropped significantly. Where, in Figure 4, we see energy around -60 dB FS at all
frequencies (give or take….) the plot in Figure 6 drops down below -100 dB FS – off the plot. This is good. It’s the result of getting rid of those non-zero values at the start and stop of the
window… No clicks means no energy spread all over the frequency spectrum.
The third thing to discuss is the levels of the peaks in the plots. Take a look at the highest peak in Figure 4. It’s about -17 dB FS or so. The peak at the same frequency in Figure 6 is about -21 dB
or so… 4 dB lower… This is because, if you look at the entire time window, there is indeed less energy in there. We made portions of the signal quieter, so, on average, the whole thing is quieter.
We’ll look at how much quieter later.
This is where things get a little interesting, because some people think that the way that we faded in and faded out of the window (specifically, using a Hann function) can be improved in some way or
another… So, let’s try a different way.
Figure 7. The original bell sound, with a different kind of fade in and fade out applied to it.
Figure 7, above, shows the same bell sound again, this time processed using a Hamming function instead. It looks very similar to the Hann function – but it’s not identical. For starters, you may
notice that the start and stop values are not 0 – although they’re considerably quieter than they would have been if we had used a rectangular windowing function (or, in other words “done nothing”).
The result of a DFT on this signal is shown in Figure 8.
Figure 8: The magnitude response of the bell sound with a Hamming window, as shown in Figure 7.
There are two things about Figure 8 that are different from Figure 6. The first is that the overall apparent level of the wide-band artefacts is higher (although not as high as that in Figure 4…).
This is because we have a “click” caused by the fact that we don’t start and stop at 0. However, the advantage of this function is that the peaks are narrower – so we get a better idea of the actual
signal – we just need to learn to ignore the bottom part of the plot.
Figure 9, shows yet another function, called a Blackman function.
Figure 9. The original bell sound, with a different kind of fade in and fade out applied to it.
You can see there that it takes longer for the signal to ramp in from 0 (and to ramp out again at the end), so we can expect that the peaks will be even lower than those for the Hann window. This can
be seen in Figure 10.
Figure 10: The magnitude response of the bell sound with a Blackman window, as shown in Figure 7.
Indeed, the peaks are lower….
Another function is called the Blackman-Harris function, shown in Figures 11 and 12.
Figure 11.
Figure 12.
There are other windowing functions. And there are some where you can change some variables to play with width and things. Or you can make up your own. I won’t talk about them all here… This is just
a brief introduction…
The purpose of this is to show some basic issues with windowing. You can play with the windowing function, but there will be subsequent effects in the DFT result like:
• the apparent magnitude of the actual signal (the peaks in the plots above)
• the apparent magnitude in frequency bands that aren’t in the signal
• the apparent width of the frequency band of the actual signal
Also, you have to remember that a DFT shows you the complete frequency content of the slice of time that you fed it. So, if the frequency content changes over time (the sound of a sitar string being
plucked, or the “pew pew” sound of Han Solo’s laser, for example) then this change over time will not be shown…
Some more details…
Let’s dig a little into the differences in the peaks in the DFT plots above. As we saw in Part 4, if the frequency of the signal you’re analysing is not exactly the same as the frequency of the DFT
bin, then the energy will “bleed” into adjacent bins. The example I showed in that posting compared the levels shown by the DFT when the frequency of the signal is either exactly the same as a
frequency bin, or half-way between two of them – a reminder of this is shown below in Figure 13.
Figure 13. The results of a DFT analysis of two signals. The blue plot shows the result when the signal frequency is 1000.0 Hz (exactly the same as the DFT bin frequency). The red plot shows the
result when the signal frequency is 1000.5 Hz (half-way between two bins).
As you can see in that plot, the energy in the 1000.5 Hz bleeds into the two adjacent bins. In fact, it’s more accurate to say that there is energy in all of the DFT bins, due to the discontinuity of
the signal when the beginning is wrapped around to meet its end.
So, let’s analyse this a little further. I’ll create a signal that is on a frequency bin (therefore it’s a sine wave with a carefully-chosen frequency), and do the DFT. Then, I’ll make the frequency
a little lower, and do the DFT again. I’ll repeat this until I get to a signal frequency that is half-way between two bins. I’ll stop there, because once I pass the half-way point, I’ll just start
seeing the same behaviour. The result of this is shown for a rectangular window in Figure 14.
Figure 14. The results of doing a DFT analysis on a sine wave with frequencies ranging from exactly on a DFT bin (1000 Hz) to half-way between two bins (999.5 Hz).
As you can see there, there is a LOT of energy bleeding into all frequency bins when the signal is not exactly on a bin. Remember that this does not mean that those frequencies are in the signal –
but they are in the signal that the DFT is being asked to analyse.
Let’s do this again for the other windowing functions.
Figure 15. The same results, with a Hann window applied to the signal. Notice that the 1000 Hz result is now not as precise – but all of the other frequencies are “cleaner”.
Figure 16. The same results with a Hamming window. The 1000 Hz is narrower than with the Hann window – but all other frequencies are “noisier” due to the fact that the start and stop gains of the
Hamming window are not 0.
Figure 17. The same analysis for the Blackman windowing function.
Figure 18. The same analysis for the Blackman Harris function.
What you may notice when you look at Figures 14 to 18 is that there is a relationship between the narrowness of the plot when the signal is on a bin frequency and the amount of energy that’s spread
everywhere else when it’s not. Generally, you have to make a trade between accuracy and precision at the frequency where there’s energy to truth at all other frequencies.
However, if you look carefully at those plots around the 1000 Hz area, you can see that it’s a little more complicated. Let’s zoom into that area and have a look…
Figure 19. A zoom-in of the plot in Figure 14.
Figure 20. A zoom-in of the plot in Figure 15.
Figure 21. A zoom-in of the plot in Figure 16.
Figure 22. A zoom-in of the plot in Figure 17.
Figure 22. A zoom-in of the plot in Figure 18.
The thing to compare in plots 19 to 22 is the how similar the plots in each figure is, relative to each other. For example, in Figure 19, the six plots are very different from each other. In Figure
22, the six plots are almost identical from 995 Hz to 1005 Hz.
Depending on what kind of analysis you’re doing, you have to decide which of these behaviours is most useful to you. In other words, each type of windowing function screws up the result of the DFT.
So you have to choose which one screws it up the least for the type of signal and the type of analysis you’re doing.
Alternatively, you can choose a favourite windowing function, and always use that one, and just get used to looking at the way your results are screwed up.
Some final details
So far, I have not actually defined the details of any of the windowing functions we’ve looked at here. I’ve just said that they fade in and fade out differently. I won’t give you the mathematical
equations for creating the actual curves of the functions. You can get that somewhere else. Just look them up on the Internet. However, we can compare the shapes of the gain functions by looking at
them on the same plot, which I’ve put in Figure 23.
Figure 23. The gain vs. time for 4 of the 5 windowing functions I’ve talked about.
You may notice that I left out the rectangular window. If I had plotted it, it would just be a straight line of 1’s, which is not a very interesting shape.
What may surprise you is how similar these curves look, especially since they have such different results on the DFT behaviour.
Another way to look at these curves (which is almost never shown) is to see them in decibels instead, which I’ve done in Figure 24.
Figure 24. The sample plots that were shown in Figure 23, on a decibel scale.
The reason I’ve plotted them in dB in Figure 24 is to show that, although they all look basically the same in Figure 23, you can see that they’re actually pretty different… For example, notice that,
at about 10% of the way into the time of the window, there is a 40 dB difference between the Blackman Harris and the Hann functions… This is a lot.
One thing that I’ve only briefly mentioned is the fact that the windowing functions have an effect on the level that is shown in the DFT result, even when the signal frequency is exactly the same as
the DFT bin frequency. As I said earlier, this is because there is, in fact, less energy in the time window overall, because we made the signal quieter at the beginning and end. The question is:
“exactly how much quieter?” This is shown in Figure 25.
Figure 25. The relationship between the signal frequency, the maximum level shown in the DFT results, and the windowing function used.
So, as you can see there, a DFT of a rectangular windowed signal can show the actual level of the signal if the frequency of the signal is exactly the same as the DFT bin centre. All of the other
windowing functions will show you a lower level.
HOWEVER, all of the other windowing functions have less variation in that error when the signal frequency moves away from the DFT bin. In other words, (for example) if you use a Blackman Harris
window for your DFT, the level that’s displayed will be more wrong than if you used a rectangular window, but it will be more consistent. (Notice that the rectangular window ranges from almost -4 dB
to 0 dB, whereas the Blackman Harris window only ranges from about -10 to -9 dB.)
We’ll dig into some more details in the next and final posting in this series… with some exciting animated 3D plots to keep things edu-taining.
DFT’s Part 4: The Artefacts
Links to:
DFT’s Part 1: Some introductory basics
DFT’s Part 2: It’s a little complex…
DFT’s Part 3: The Math
The previous post ended with the following:
And, you should be left with a question… Why does that plot in Figure 12 look like it’s got lots of energy at a bunch of frequencies – not just two clean spikes? We’ll get into that in the next
Let’s begin by taking a nice, clean example…
If my sampling rate is 65,536 Hz (2^16) and I take one second of audio (therefore 65,536 samples) and I do a DFT, then I’ll get 65,536 values coming out, one for each frequency with an integer value
(nothing after the decimal point. The frequencies range from 0 to 65,535 Hz, on integer values (so, 1 Hz, 2 Hz, 3 Hz, etc…) (And we’ll remember to throw away the top half of those values due to
mirroring which we talked about in the last post.)
I then make a sine wave with an amplitude of 0 dB FS and a frequency of 1,000 Hz for 1 second, and I do an DFT of it, and then convert the output to show me just the magnitude (the level) of the
signal (so I’m ignoring phase). The result would look like the plot below.
Figure 1. The magnitude response of a 1000 Hz sine tone, sampled at 65,536 Hz, calculated using a 65,536-point DFT.
The plot above looks very nice. I put in a 1,000 Hz sine wave at 0 dB FS, and the plot tells me that I have a signal at 1,000 Hz and 0 dB FS and nothing at any other frequency (at least with a
dynamic range of 200 dB). However, what happens if my signal is 1000.5 Hz instead? Let’s try that:
Figure 2. The magnitude response of a 1000.5 Hz sine tone, sampled at 65,536 Hz, calculated using a 65,536-point DFT.
Now things don’t look so pretty. I can see that there’s signal around 1000 Hz, but it’s lower in level than the actual signal and there seems to be lots of stuff at other frequencies… Why is this?
In order to understand why the level in Figure 2 is lower than that in Figure 1, we have to zoom in at 1000 Hz and see the individual points on the plot.
Figure 1 (Zoom)
As you can see in Figure 1 (Zoom), above, there is one DFT frequency “bin” at 1000 Hz, exactly where the sine wave is centred.
Figure 2 (Zoom)
Figure 2 (Zoom) shows that, when the sine wave is at 1000.5 Hz, then the energy in that signal is distributed between two DFT frequency bins – at 1000 Hz and 1001 Hz. Since the energy is shared
between two bins, then each of their level values is lower than the actual signal.
The reason for the “lots of stuff at other frequencies” problem is that the math in a DFT has a limited number of samples at its input, so it assumes that it is given a slice of time that repeats
itself exactly.
For example…
Let’s look at a portion of a plot like the one below:
Figure 3. A portion of a plot. The gray rectangles hide things…
If I asked you to continue this plot to the left and right (in other words, guess what’s under the gray rectangles), would you draw a curve like the one below?
Figure 3. An obvious extrapolation of the curve in Figure 3.
This would be a good guess. However, the figure below is also a good guess.
Figure 4. An obvious extrapolation of the curve in Figure 3.
Of course, we could guess something else. Perhaps Figure 3 is mostly correct, but we should add a drawing of Calvin and Hobbes on a toboggan, sliding down the hill to certain death as well. You never
know what was originally behind those grey rectangles…
This is exactly the problem the math behind a DFT has – you feed it a “slice” of a recording, some number of samples long, and the math (let’s call it “a computer”, since it’s probably doing the
math) has to assume that this slice is a portion of time that is repeated forever – it started at the beginning of time, and it will continue repeating until the end of time. In essence, it has to
make an “extrapolation” like the one shown in Figure 4 because it doesn’t have enough information to make assumptions that result in the plot in Figure 3.
For example: Part 2
Let’s go back to the bell recording that we’ve been looking at in the previous posts. We have a portion of a recording, 2048 samples long. If I plot that signal, it looks like the curve in Figure 5.
Figure 5. The bell recording we saw in previous postings, hiding the information that comes before and after.
When the computer does the DFT math, the assumption is that this is a slice that is repeated forever. So, the computer’s assumption is that the original signal looks like the one below, in Figure 6.
Figure 6. The signal, as assumed by the computer when it’s doing the DFT math.
I’ve put rectangles around the beginning (at sample 1) and end (at sample 2048) of the slice to highlight what the signal looks like, according to the computer… The signal in the left half of the
left rectangle (ending at sample 0) is the end of the slice of the recording, right before it repeats. The signal starting at 2049 is the beginning again – a repeat of sample 1.
If we zoom in on the signal in the left rectangle, it looks like Figure 7.
Figure 7. the signal inside the left rectangle in Figure 6.
Notice that vertical line at sample 1 (actually going from sample 0 to sample 1, to be accurate). Of course, our original bell recording didn’t have that “instantaneous” drop in there – but the
computer assumes it does because it doesn’t have enough information to assume anything else.
If we wanted to actually make that “instantaneous” vertical change in the signal (with a theoretical slope of infinity – although it’s not really that steep….), we would have to add other frequencies
to our original signal. Generally, you can assume that, the higher the slope of an audio signal, either 1) the louder the signal or 2) the more high frequency content in the signal. Let’s look at the
second one of those.
Let’s look at portions of sine waves at three different frequencies. These are shown below, in Figure 8. The top plot shows a sine wave with some frequency, showing how it looks as it passes phase =
0º (which we’ll call “time = 0” (on the X-axis)). At that moment, the sine wave has a value of 0 (on the Y-axis) and the slope is positive (it’s going upwards). The middle plot shows a sine wave with
3 times the frequency (notice that there are 6 negative-and-positive bumps in there instead of just 2). Everything I said about the top plot is still true. The level is 0 at time=0, and the slope is
positive. The bottom plot is 5 times the frequency (10 bumps instead of 2). And, again, at time=0, everything is the same.
Figure 8. Three sinusoidal waves at related frequencies. We’re looking at the curves as they cross time=0 (on the X-axis).
Let’s look a little more carefully at the slope of the signal as it crosses time=0. I’ve added blue lines in Figure 9 to highlight those.
Figure 9.
Notice that, as the frequency increases, the slope of the signal when it crosses the 0 line also increases (assuming that the maximum amplitude stays the same – all three sine waves go from -1 to 1
on the Y-axis.
One take-away from that is the idea that I’ve already mentioned: the only way to get a steep slope in an audio signal is to add high frequency content. Or, to say it another way: if your audio signal
has a steep slope at some time, it must contain energy at high frequencies.
Although I won’t explain here, the truth is just a little more complicated. This is because what we’re really looking for is a sharp change in the slope of the signal – the “corners” in the plot
around Sample 0 in Figure 7. I’ve put little red circles around those corners to highlight them, shown below in Figure 10. When audio geeks see a sharp corner like that in an audio signal, they say
that the waveform is discontinuous – meaning that the level jumps suddenly to something unexpected – which means that its slope does as well.
Basically, if you see a discontinuity in an audio signal that is otherwise smooth, you’re probably going to hear a “click”. The audibility of the click depends on how big a jump there is in the
signal relative to the remaining signal. (For example, if you put a discontinuity in a nice, smooth, sine wave, you’ll hear it. If you put a discontinuity in a white noise signal – which is made up
of nothing but discontinuities (because it’s random) then you won’t hear it…)
Figure 10. The red circles show the discontinuities in the slope of the signal when it is assumed that it repeats.
Circling back…
Think back to the examples I started with at the beginning of this post. When I do a 65,536-point DFT of a 1000 Hz sine wave sampled at 65,536 Hz, the result is a nice clean-looking magnitude
response (Figure 1). However, when I do a 65,536-point DFT of a 1000.5 Hz sine wave sampled at 65,536 Hz, the result is not nearly as nice. Why?
Think about how the end of the two sine waves join up with their beginnings. When you do a 65,536-point DFT on a signal that has a sampling rate of 65,536 Hz, then the slice of time that you’re
analysing is exactly 1 second long. A 1000 Hz sine wave, repeats itself exactly after 1 second, so the 65,537th sample is identical to the first. If you join the last 30 samples of the slice to the
first 30 samples, it will look like the red curve on the top plot in Figure 10, below.
However, if the sinusoid has a frequency of 1000.5 Hz, then it is only half-way through the waveform when you get to the end of the second. This will look like the lower black curve in Figure 10.
Figure 11. The top plot shows the a 1000 Hz sine wave at the end of exactly 1 second, joined to the beginning of the same sine wave. The bottom plot shows the same for a 1000.5 Hz sine wave
Notice that the lower plot has a discontinuity in the slope of the waveform. This means that there is energy in frequencies other than 1000.5 Hz in it. And, in fact, if you measured how much energy
there is in that weird waveform that sounds like a sine wave most of the time, but has a little click every second, you’ll find out that the result is already plotted in Figure 2.
The conclusion
The important thing to remember from this posting is that a DFT tells you what the relative frequency content of the signal is – but only for the signal that you give it. And, in most cases, the
signal that you give it (a slice of time that is looped for infinity) is not the same as the total signal that you took the slice from.
So, most of the time, a DFT (or FFT – you choose what you call it) is NOT showing you what is in your signal in real life. It’s just giving you a reasonably good idea of what’s in there – and you
have to understand how to interpret the plot that you’re looking at.
In other words, Figure 2 does not show me how a 1000.5 Hz sine tone sounds – but Figure 1 shows me how a 1000 Hz sine tone sounds. However, Figures 1 and 2 show me exactly how the computer “hears”
those signals – or at least the portion of audio that I gave it to listen to.
There is a general term applied to the problem that we’re talking about. It’s called “windowing effects” because the DFT is looking at a “window” of time (up to now, I’ve been calling it a “slice” of
the audio signal. I’m going to change to using the word “time window” or just “window” from now on.
In the next posting, DFT’s Part 5: Windowing, we’ll look at some sneaky ways to minimise these windowing effects so that they’re less distracting when you’re looking at magnitude response plots.
DFT’s Part 3: The Math
Links to:
DFT’s Part 1: Some introductory basics
DFT’s Part 2: It’s a little complex…
If you have an audio signal or the impulse response measurement of an audio device (which is just the audio output of a device when the input signal is a very short “click” – how the device responds
to an impulse), one way to find out its spectral content is to use a Fourier Transform. Normally, we live in a digital audio world, with discrete divisions of time, so we use a DFT or a Discrete
Fourier Transform (although most people call it an FFT – a Fast Fourier Transform).
If you do a DFT of a signal (say, a sinusoidal waveform), then you take a slice of time, usually with a length (measured in samples) that is a nice power of 2 – for example 2, or 4 (2^2), or 2^12
(4096 samples) or 2^13 (8192 samples). When you convert this signal in time through the DFT math, you get out the same number of number (so, 2048 samples in, 2048 numbers out). Each of those numbers
can be used to find out the magnitude (the level) and the phase for a frequency.
Those frequencies (say, 2048 of them) are linearly spaced from 0 Hz up to just below the sampling rate (the sampling rate would be the 2049th frequency in this case… we’ll see why, below…)
So, generally speaking: if I have an audio signal (a measurement of level over time) and I do a DFT (which is just a series of mathematical equations) and then I can see the relative amount of energy
by frequency for that “slice” of time.
So, how does the math work? In essence, it’s just a matter of doing a lot of multiplication, and then adding the results that you get (and then maybe doing a little division, if you’re in the mood…).
We’ve already seen in Parts 1 and 2 of this series that
• a sinusoidal waveform is just 2 dimensions (dimension #1 is movement in space, the other dimension is time) of a three-dimensional rotation (dimensions #1 and #2 are space and #3 is time)
• if we want to know the frequency, the amplitude, and the direction of rotation of the “wheel”, we will need to see the real component (the cosine) and the imaginary component (the negative sine)
• the imaginary component is a negative sine wave instead of a positive sine wave because the wheel is rotation clockwise
A real-world example
I took a bell and I hit it, so it rang the way bells ring. While I was doing that, I recorded it with a microphone connected to my computer. The sampling rate was 48 kHz and I recorded with enough
bits to not worry about that. The result of that recording is shown in Figure 1.
Figure 1. A 7-second long recording of a bell
Seven seconds is a lot of samples at 48,000 samples per second. (In fact, it’s 7 * 48000 samples – which is a lot…) So, let’s take a slice somewhere out of the middle of that recording. This portion
(a “zoomed-in” view of Figure 1) is shown below in Figure 2.
Figure 2. A portion of the signal shown in Figure 1. The gray part is 2048 samples long.
So, for the remainder of this posting, we’ll only be looking at that little slice of time, 2048 samples long. Since our sampling rate is 48 kHz, this means that the total length of that slice is 2048
* 1/48000 = 0.0427 seconds, or approximately 42.7 ms.
Let’s start by calculating the amount of energy there is at 0 Hz or “DC” in this section. We do this by taking the value of each individual sample in the section, and adding all those values
together. Some of the values are positive (they’re above the 0 line in Figure 2) and some are negative (they’re below 0). So, if we add them all up we should be somewhere close to 0… Let’s try….
Figure 3.
Figure 3 has three separate plots. The top plot in blue is the section of the recording that we’re using, 2048 samples long. You’ll see that I put a red circle around two samples, sample number 47
and sample number 1000. These were chosen at random, just so we have something near the beginning and something near the middle of the recording to use as examples…
So, to find the total energy at 0 Hz, we have to add the individual values of each of the 2048 samples. So, for example, sample #47 has a value of 0.2054 and sample #1000 has a value of -0.2235. We
add those two values and the other 2046 sample values together and we get a total value of 2.9057. Let’s just leave that number sitting there for now. We’ll come back to it later.
For now, we’ll ignore the middle and bottom plots in Figure 3. This is because they’ll be easier to understand after Figure 4 is explained…
Now we want to move up to frequencies above 0 Hz. The way we do this is similar to what we did, with an extra step in the process.
Figure 4.
The top blue plot in Figure 4 shows the same thing that it showed in Figure 3 – it’s the 2048 samples in the recording, with sample numbers 47 and 1000 highlighted with red circles.
Take a look at the middle plot. The red curve in that plot is a cosine wave with a period (the amount of time it takes to complete 1 cycle) of 2048 samples. On that plot, I’ve put two * signs
(“asterisks”, if you prefer…) – one on sample number 47 and the other at sample 1000.
One small, but important note here: although it’s impossible to see in that plot, the last value of the cosine wave is not the same as the first – it’s just a little lower in level. This is because
the cosine wave would start to repeat itself on the next sample. So, the 2049th sample is equal to the 1st. This makes the period of the cosine wave 2048 samples.
The black curve in this plot is the result when you multiply the original recording (in blue) by the cosine curve (in red). So, for example, sample #47 on the blue curve (a value of 0.2054)
multiplied by sample #47 on the red cosine curve (0.9901) equals 0.2033, which is indicated by a red circle on the black curve in the middle plot.
If you look at sample 1000, the value on the blue curve is positive, but when it’s multiplied by the negative value on the cosine curve, the result is a negative value on the black curve.
You’ll also notice that, when the cosine wave is 0, the result of the multiplication in the black curve is also 0.
So, we take each of the 2048 samples in the original recording of the bell, and multiply each of those values, one by one, by their corresponding samples in the cosine curve. This gives us 2048
sample values shown in the black curve, which we add all together, and that gives us a total of 1.5891.
We then do exactly the same thing again, but instead of using a cosine wave, we use a negative sine wave, shown as the red curve in the bottom plot. The blue curve multiplied by the negative sine
wave, sample-by-sample results in the black curve in the bottom plot. We add all those sample values together and we get -2.5203.
Now, we do it all again at the next frequency.
Figure 5.
Now, the period of the cosine and the negative sine waves is 1024 samples, so they’re at two times the frequency of those shown in Figure 4. However, apart from that change, the procedure is
identical. We multiply the signal by the cosine wave (sample-by-sample), add up all the results, and we get 1.3547. We multiply the signal by the negative sine wave and we get -1.025.
This procedure is repeated, increasing the frequency of the cosine (the real) and the negative sine (the imaginary) waves each time. So far we have seen 0 periods (Figure 3), 1 period (Figure 2), and
2 periods (Figure 3) – we just keep going with 3 periods, 4 periods, and so on.
Eventually we get to 1024 periods. If I were to plot that, it would not look like a cosine wave, since the values would be 1, -1, 1, -1…. for 2048 samples. (But, due to the nature of digital audio
and smoothing filters that we’re not going to talk about, it would, in fact, be a cosine wave at a frequency of one half of the sampling rate…)
At that frequency, the values for the negative sine wave would be a string of 2048 zeros – exactly as it is in Figure 3.
If we keep going up, we get to 2048 periods – one period of the cosine wave for each sample. This means that, at each sample, the cosine starts, so the result is a string of 2048 ones. Similarly, the
negative sine wave will be a string of 2048 zeros. Note that both of these are identical to what we saw in Figure 1 when we were looking at 0 Hz…
Since we’ve already seen in the previous posting that, at a given frequency, the cosine component (the total sum of the results of multiplying the original signal by a cosine wave) is the real
component and the negative sine is the imaginary component, then we can write all of the results as follows:
frequency “x”: Real + Imaginary contributions
f1: 2.9057 + 0.0000 j
f2: 1.5891 – 2.5203 j
f3: 1.3547 – 1.0251 j
f2047: 1.3547 – 1.0251 j
f2048: 1.5891 – 2.5203 j
f2049: 2.9057 + 0.0000 j
… and, as we saw in Figure 1 in the last post, for any one frequency, the real and imaginary contributions can be converted into a magnitude (a level) by using a little Pythagoras:
magnitude = sqrt(real^2 + imag^2)
So, we get the following magnitudes
frequency “x”: magnitude
f1: 2.9057
f2: 2.9794
f3: 1.6988
f2047: 1.6988
f2048: 2.9794
f2049: 2.9057
Let’s plot the first 10 values – f1 up to f10. (Remember that these are not in Hertz – they’re frequency numbers. We’ll find out what the actual frequencies are later…)
Figure 6.
So, Figure 6 shows the beginning of the results of our calculations – the first 10 values of the 2048 values that we’re going to get. Not much interesting here yet, so let’s plot all 2048 values.
Figure 7.
Figure 7 shows two interesting things. The first is that at least one of those numbers gets very big – almost up to 160 – whatever that means. The other is that, you may notice that we have some
symmetry going on here. In fact, you might have already noticed this… If you go back and look at the lists of numbers I gave earlier, you’ll see that the values for f1 and f2049 are identical (this
is true in the complex world, where we see the real and imaginary components separately, and also therefore in their magnitudes). Similarly, f2 and f2048 are identical, as are f3 and f2047. If I had
put in all of the values, you would have seen that the symmetry started at f1024 which is identical to f1026. (See this posting for a discussion about aliasing, which may help to understand why this
So, since the values are repeated, we only need to look at the first 1025 values that we calculated – we know that f1026 to f2048 are the same in reverse order… So, let’s plot the bottom half of
Figure 7.
Figure 8.
Figure 8 shows us the same information as Figure 7 – just without the symmetrical repetition. However, it’s still a little hard to read. This is because our frequency divisions are linear. Remember
that we multiplied our original signal by 1 period, 2 period, 3 periods, etc… This means that we were going up in linear frequency steps – adding equal frequencies on each step. The problem is that
humans hear frequency steps logarithmically – semitones (1.06 times the frequency) and octaves (2 times the frequency) are examples – we multiply (not add) in equal steps. So, let’s plot Figure 8
again, but change the X-axis to a logarithmic scale.
Figure 9.
Figure 9 and Figure 8 show exactly the same information – I’ve just changed the way the x-axis is scaled so that it looks more like the way we hear distribution of frequency.
But what frequency is it?
There are two remaining problems with Figure 9 – the scaling of the two axes. Let’s tackle the X-axis first.
We know that, to get the value for f1, we found the average of all of the values in the recording. This told us the magnitude of the 0Hz. component of the signal.
Then things got a little complicated. To find the magnitude at f2, we multiplied the signal by a cosine (and a negative sine) with a period of 2048 sample. What is the frequency of that cosine wave
in real life? Well, we know that the original recording was done with a sampling rate of 48 kHz or 48,000 samples per second, and our 2048-sample long slice of time equalled 42.66666666…
milliseconds. If we divide the sampling rate by the period of the cosine wave, we’ll find its frequency, since we’ll find out how many times per second (per 48,000 samples) the wave will occur.
f2 = 48,000 / 2048 = 23.4375 Hz
The next frequency value will be the sampling rate divided the period of the next cosine wave – half the length of the first, or:
f3 = 48,000 / (2048 / 2) = 46.875 Hz
You might notice that f3 = 2 * f2… this helps the math.
f4 = 48,000 / (2048 / 3) = 70.3125 Hz
or f4 = 3 * f2
So, I can now keep going up to find all of my frequencies, and then change the labels on my X-axis so that they make sense to humans.
Figure 10.
That’s one problem solved. We now know that the bell’s loudest frequency is just under 600 Hz (the peak with a magnitude of about 160) and there’s another frequency at about 1500 Hz as well – with a
magnitude of about 30 or so.
But how loud is it?
So, let’s tackle the second problem – what does a magnitude of 160 mean in real life?
Not only do humans hear changes in frequency logarithmically, we also hear changes in level logarithmically as well. We say something like “a trumpet is twice as loud as a dog barking” instead of
“the loudness of a trumpet is the loudness of a dog barking plus 2”. In fact, that second one just sounds silly when you say it…
As a result, we use logarithms to convert linear levels (like the ones shown on the Y-axis of Figure 10) to something that makes more sense. Instead of having values like 1, 10, 100, and 1000 (I
multiplied by 10 each time), we take the log of those values, and tell people that…
Log10 (1) = 0
Log10 (10) = 1
log10 (100) = 2
log10 (1000) = 3
Now we can use the numbers on the right of those equations, which are small-ish instead of the other ones, which are big-ish…
We use this logarithmic conversion in the calculation of a decibel – which we will not get into here – but it would make the topic of another posting in the future. For now, you’ll just have to hang
What we’ll do is to take the magnitude values plotted in Figure 10 and find their logarithms, multiply those by 20, and we get their values in decibels. Cool.
The only problem is that if I were to do that, the numbers would look unusually big. This is because I left out one step way up at the top. Back when we were multiplying and adding all those samples
and cosine (and negative sine) waves, we should have done one more thing. We should have found the average value instead of the total sum. This means that we should have divided by the total number
of samples. However, since we’re only looking at half of the data (the lower 1025 frequency bins – and not all 2048) we divide by half of the number of samples in our slice of time.
So, we take each sample in the recording, multiply each of those by a value in the cosine (or negative sine) wave – and divide the results by half of the number of samples. When you get that average,
you then find its logarithm (base 10) and multiply by 20.
If you do that for each value, you get the result shown below in Figure 11.
Figure 11.
If we connect the dots, then we get Figure 12.
Figure 12.
And there are the peaks we saw earlier. One just under 600 Hz at about -16 dB FS, and the other at about 1500 Hz with a level of about -31 dB FS.
The important stuff to remember for now…
There are two important things to remember from this posting.
1. The frequencies that are calculated using a DFT (or FFT) are linearly spaced. That means that (on a human, logarithmic scale) we have a poor resolution in the low frequencies and a very fine
resolution in the high frequencies. (for example, in this case, the first three frequencies are 0 Hz, 23.4 Hz, and 46.9 Hz. The last three frequencies are 23953.1 Hz, 23976.6 Hz, and 24,000 Hz.)
2. If you want better resolution in the low frequencies, you’ll need to calculate with more samples – a longer slice of time, which means more might have happened in that time (although there are
some tricks we can play, as we’ll see later).
And, you should be left with a question… Why does that plot in Figure 12 look like it’s got lots of energy at a bunch of frequencies – not just two clean spikes? We’ll get into that in the next
posting: DFT’s Part 4: The Artefacts.
DFT’s Part 2: It’s a little complex…
Links to:
DFT’s Part 1: Some introductory basics
Whole Numbers and Integers
Once upon a time you learned how to count. You were probably taught to count your fingers… 1, 2, 3, 4 and so on. Although no one told you so at the time, you were being taught a set of numbers called
whole numbers.
Sometime after that, you were probably taught that there’s one number that gets tacked on before the ones you already knew – the number 0.
A little later, sometime after you learned about money and the fact that we don’t have enough, you were taught negative numbers… -1, -2, -3 and so on. These are the numbers that are less than 0.
That collection of numbers is called integers – all “countable” numbers that are negative, zero and positive. So the collection is typically written
… -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 …
Rational Numbers
Eventually, after you learned about counting and numbers, you were taught how to divide (the mathematical word for “sharing equally”). When someone said “20 divided by 5 equals 4” then they meant “if
you have 20 sticks, then you could put those sticks in 4 piles with 5 sticks in each pile.” Eventually, you learned that the division of one number by another can be written as a fraction like 3/1 or
20/5 or 5/4 or 1/3.
If you do that division the old-fashioned way, you get numbers like this:
3 ∕ 1 = 3.000000000 etc…
20 ∕ 5 = 4.00000000 etc…
5 ∕ 4 = 1.200000000 etc…
1 ∕ 3 = 0.333333333 etc…
The thing that I’m trying to point out here is that eventually, these numbers start repeating sometime after the decimal point. These numbers are called rational numbers.
Irrational Numbers
What happens if you have a number that doesn’t start repeating, no matter how many numbers you have? Take a number like the square root of 2 for example. This is a number that, when you multiply it
by itself, results in the number 2. This number is approximately 1.4142. But, if we multiply 1.4142 by 1.4142, we get 1.99996164 – so 1.4142 isn’t exactly the square root of 2. In fact, if we started
calculating the exact square root of 2, we’d result in a number that keeps going forever after the decimal place and never repeats. Numbers like this (π is another one…) that never repeat after the
decimal are called irrational numbers
Real Numbers
All of these number types – rational numbers (which includes integers) and irrational numbers fall under the general heading of real numbers. The fact that these are called “real” implies immediately
that there is a classification of numbers that are “unreal” – but we’ll get to that later…
Imaginary Numbers
Let’s think about the idea of a square root. The square root of a number is another number which, when multiplied by itself is the first number. For example, 3 is the square root of 9 because 3*3 = 9
. Let’s consider this a little further: a positive number muliplied by itself is a positive number (for example, 4*4 = 16… 4 is positive and 16 is also positive). A negative number multiplied by
itself is also positive (i.e. –4*-4 = 16).
Now, in the first case, the square root of 16 is 4 because 4*4 = 16. (Some people would be really picky and they’ll tell you that 16 has two roots: 4 and -4. Those people are slightly geeky, but
technically correct.) There’s just one small snag – what if you were asked for the square root of a negative number? There is no such thing as a number which, when multiplied by itself results in a
negative number. So asking for the square root of -16 doesn’t make sense. In fact, if you try to do this on your calculator, it’ll probably tell you that it gets an error instead of producing an
For a long time, mathematicians just called the square root of a negative number “imaginary” since it didn’t exist – like an imaginary friend that you had when you were 2… However, mathematicians as
a general rule don’t like loose ends – they aren’t the type of people who leave things lying around… and having something as simple as the square root of a negative number lying around unanswered got
on their nerves.
Then, in 1797, a Norwegian surveyor named Casper Wessel presented a paper to the Royal Academy of Denmark that described a new idea of his. He started by taking a number line that contains all the
real numbers like this:
Figure 1: The number line containing all real numbers.
He then pointed out that multiplying a number by -1 was the same as rotating by an angle of 180º, like this:
Figure 2: Multiplying a number by -1 is the same as rotating by 180º
He then reasoned that, if this were true, then the square root of -1 must be the same as rotating by 90º.
Figure 3: This means that the square root of -1 must be the same a rotating by 90º.
This meant that the number line we started with containing the real numbers is the X-axis on a 2-dimensional plane where the Y-axis contains the imaginary numbers. That plane is called the Z plane,
where any point (which we’ll call ‘Z’) is the combination of a real number (X) and an imaginary number (Y).
Figure 4: The Z-plane, where X-values are real and the Y-values are imaginary.
If you look carefully at Figure 4, you’ll see that I used a “j” to indicate the imaginary portion of the number. Generally speaking, mathematicians use i and physicists and engineers use j so we’ll
stick with j. (The reason physics and engineering people use j is that they use i to mean “electrical current”.)
“What is j?” I hear you cry. Well, j is just the square root of -1. Of course, there is no number that is the square root of -1
and therefore
Now, remember that j * j = -1. This is useful for any square root of any negative number, you just calculate the square root of the number pretending that it was positive, and then stick a j after
it. So, since the square root of 16, abbreviated sqrt(16) = 4 and sqrt(-1) = j, then sqrt(-16) = 4j.
Complex numbers
Now that we have real and imaginary numbers, we can combine them to create a complex number. Remember that you can’t just mix real numbers with imaginary ones – you keep them separate most of the
time, so you see numbers like
This is an example of a complex number that contains a real component (the 3) and an imaginary component (the 2j). In some cases, these numbers are further abbreviated with a single Greek character,
like α or β, so you’ll see things like
α = 3+2j
In other cases, you’ll see a bold letter like the following:
Z = 3+2j
A lot of people do this because they like reserving Greek letters like α and ϕ for variables associated with angles.
Personally, I like seeing the whole thing – the real and the imaginary components – no reducing them to single Greek letters (they’re for angles!) or bold letters.
Absolute Value (aka the Modulus)
The absolute value of a complex number is a little weirder than what we usually think of as an absolute value. In order to understand this, we have to look at complex numbers a little differently:
Remember that j*j = -1.
Also, remember that, if we have a cosine wave and we delay it by 90º and then delay it by another 90º, it’s the same as inverting the polarity of the cosine, in other words, multiplying the cosine by
-1. So, we can think of the imaginary component of a complex number as being a real number that’s been rotated by 90º, we can picture it as is shown in the figure below.
Figure 5. The relationship bewteen the real and imaginary components for the number (2 + 3 j). Notice that the X and Y axes have been labeled the “real” and “imaginary” axes.
Notice that Figure 5 actually winds up showing three things. It shows the real component along the x-axis, the imaginary component along the y-axis, and the absolute value or modulus of the complex
number as the hypotenuse of the triangle. This is shown in mathematical notation in exactly the same way as in normal math – with vertical lines. For example, the modulus of 2+3j is written |2+3j|
This should make the calculation for determining the modulus of the complex number almost obvious. Since it’s the length of the hypotenuse of the right triangle formed by the real and imaginary
components, and since we already know the Pythagorean theorem then the modulus of the complex number (a + b j) is
Given the values of the real and imaginary components, we can also calculate the angle of the hypotenuse from horizontal using the equation
This will come in handy later.
Complex notation or… Who cares?
This is probably the most important question for us. Imaginary numbers are great for mathematicians who like wrapping up loose ends that are incurred when a student asks “what’s the square root of
-1?” but what use are complex numbers for people in audio? Well, it turns out that they’re used all the time, by the people doing analog electronics as well as the people working on digital signal
processing. We’ll get into how they apply to each specific field in a little more detail once we know what we’re talking about, but let’s do a little right now to get a taste.
In the previous posting, that introduces the trigonometric functions sine and cosine, we looked at how both functions are just one-dimensional representations of a two-dimensional rotation of a
wheel. Essentially, the cosine is the horizontal displacement of a point on the wheel as it rotates. The sine is the vertical displacement of the same point at the same time. Also, if we know either
one of these two components, we know:
1. the diameter of the wheel and
2. how fast it’s rotating
but we need to know both components to know the direction of rotation.
At any given moment in time, if we froze the wheel, we’d have some contribution of these two components – a cosine component and a sine component for a given angle of rotation. Since these two
components are effectively identical functions that are 90º apart (for example, a cossine wave is the same as a sine that’s been delayed by 90º) and since we’re thinking of the real and imaginary
components in a complex number as being 90º apart, then we can use complex math to describe the contributions of the sine and cosine components to a signal.
Let’s look at an example. If the signal we wanted to look at a signal that consisted only of a cosine wave, then we’d know that the signal had 100% cosine and 0% sine. So, if we express the cosine
component as the real component and the sine as the imaginary, then what we have is:
1 + 0 j
If the signal were an upside-down cosine, then the complex notation for it would be (–1 + 0 j) because it would essentially be a cosine * -1 and no sine component. Similarly, if the signal was a sine
wave, it would be notated as (0 – 1 j).
This last statement should raise at least one eyebrow… Why is the complex notation for a positive sine wave (0 – 1 j)? In other words, why is there a negative sign there to represent a positive sine
component? (Hint – we want the wheel to turn clockwise… and clocks turn clockwise to maintain backwards compatibility with an earlier technology – the sundial. So, we use a negative number because of
the direction of rotation of the earth…)
This is fine, but what if the signal looks like a sinusoidal wave that’s been delayed a little? As we saw in the previous posting, we can create a sinusoid of any delay by adding the cosine and sine
components with appropriate gains applied to each.
So, is we made a signal that were 70.7% sine and 70.7% cosine. (If you don’t know how I arrived that those numbers, check out the previous posting.) How would you express this using complex notation?
Well, you just look at the relative contributions of the two components as before:
0.707 – 0.707 j
It’s interesting to notice that, although this is actually a combination of a cosine and a sine with a specific ratio of amplitudes (in this case, both at 0.707 of “normal”), the result will look
like a sine wave that’s been shifted in phase by -45º (or a cosine that’s been phase-shifted by 45º). In fact, this is the case – any phase-shifted sine wave can be expressed as the combination of
its sine and cosine components with a specific amplitude relationship.
Therefore (again), any sinusoidal waveform with any phase can be simplified and expressed as its two elemental components, the gains applied to the cosine (or real) and the sine (or imaginary). Once
the signal is broken into these two constituent components, it cannot be further simplified.
DFT’s Part 1: Some introductory basics
This is the first posting in a 6-part series on doing and understanding Fourier Transforms – specifically with respect to audio signals in the digital domain. However, before we dive into DFT’s (more
commonly knowns as “FFT’s”, as we’ll talk about in the next posting) we need to get some basic concepts out of the way first.
When a normal person says “frequency” they mean “how often something happens”. I go to the dentist with a frequency of two times per year. I eat dinner with a frequency of one time per day.
When someone who works in audio says “frequency” they mean something like “the number of times per second this particular portion of the audio waveform repeats – even if it doesn’t last for a whole
second…”. And, if we’re being a little more specific, then we are a bit more effuse than saying “this particular portion”… but I’m getting ahead of myself.
Let’s take a wheel with an axel, and a handle sticking out of it on its edge, like this:
Fig. 1
We’ll turn the wheel clockwise, at a constant speed, or “frequency of rotation” – with some number of revolutions per second. If we look at the wheel from the “front” – its face – then we’ll see
something like this:
Fig. 2
When we look at the front of the wheel, we can tell its diameter (the “size” of the wheel), the frequency at which it’s rotating (in revolutions or cycles per second), and the direction (clockwise or
One way to look at the rotation is to consider the position of the handle – the red circle above – as an angle. If it started at the “3 o’clock” position, and it’s rotating clockwise, then it rotated
90 degrees when it’s at the “6 o’clock” position, for example.
However, another way to think about the movement of the handle is to see it as simultaneously moving up and down as it moves side-to-side. Again, if it moves from the 3 o’clock position to the 6
o’clock position, then it moved downwards and to the left.
We can focus on the vertical movement only if we look at the side of the wheel instead of its face, as shown in the right-hand side of the animation below.
Fig. 3
The side-view of the wheel in that animation tells us two of the three things we know from the front-view. We can tell the size of the wheel and the frequency of its rotation. However, we don’t know
whether the wheel is turning clockwise or anti-clockwise. For example, if you look at the animation below, the two side views (on the right) are identical – but the two wheels that they represent are
rotating in opposite directions.
Fig. 4
So, if you’re looking only at the side of the wheel, you cannot know the direction of rotation. However, there is one possibility – if we can look at the wheel from the side and from above at the
same time, then we can use those two pieces of information to know everything. This is represented in the animation below.
Fig. 5
Although I haven’t shown it here, if the wheel was rotating in the opposite direction, the side view would look the same, but the top view would show the opposite…
If we were to make a plot of the vertical position of the handle as a function of time, starting at the 3 o’clock position, and rotating clockwise, then the result would look like the plot below. It
would start at the mid-point, start moving downwards until the handle had rotated with a “phase shift” of 90 degrees, then start coming back upwards.
Fig. 6
If we graph the horizontal position instead, then the plot would look like the one below. The handle starts on the right (indicated as the top of the plot), moves towards the mid-point until it gets
all the way to the left (the bottom of this plot) when then wheel has a phase shift (a rotation) of 180 degrees.
Fig. 7
If we were to put these two plots together to make a three dimensional plot, showing the side view (the vertical position) and the top view (the horizontal position), and the time (or the angular
rotation of the wheel), then we wind up with the plot shown below.
Fig. 8
Time to name names… The plot shown in Figure 6 is a “sine wave”, plotted upside down. (The word sine coming from the same root as words like “sinuous” and “sinus” (as in “could you hand me a tissue,
please… my sinuses are all blocked up…”) – from the Latin word “sinus” meaning “a bay” – as in “sittin’ by the dock of the bay, watchin’ the tide roll in…”.) Note that, if the wheel were turning
anti-clockwise, then it would not be upside down.
If you look at the plot in Figure 7, you may notice that it looks the same as a sine wave would look, if it started 90 degrees of rotation later. This is because, when you’re looking at the wheel
from the top, instead of the side, then you have rotated your viewing position by 90 degrees. This is called a “cosine wave” (because it’s the complement of the sine wave).
Notice how, whenever the sine wave is at a maximum or a minimum, the cosine wave is at 0 – in the middle of its movement. The opposite is also true – whenever the cosine is at a maximum or a minimum,
the sine wave is at 0.
Remember that if we only knew the cosine, we still wouldn’t know the direction of rotation of the wheel – we need to know the simultaneous values of the sine and the cosine to know whether the wheel
is going clockwise or counterclockwise.
The important thing to know so far is that a sine wave (or a cosine wave) is just a two-dimensional view of a three-dimensional thing. The wheel is rotating with a frequency of some angle per second
(one full revolution per second = 360º/sec. 10 revolutions per second = 3600º/sec) and this causes a point on its circumference (the handle in the graphics above) to move back and forth (along the
x-axis, which we see in the “top” view) and up and down (along the y-axis, which we see in the side view).
So what?
Let’s say that I asked you to make a sine wave generator – and I would like the wave to start at some arbitrary phase. For example, I might ask you to give me a sine wave that starts at 0º. That
would look like this:
Fig. 9
But, since I’m whimsical, I might say “actually, can you start the sine wave at 45º instead please?” which would look like this:
Fig. 10
One way for you do do this is to make a sine wave generator with a very carefully timed gain control after it. So, you start the sine wave generator with its output turned completely down (a gain of
0), and you wait the amount of time it takes for 45º of rotation (of the wheel) to elapse – and then you set the output gain suddenly to 1.
However, there’s an easier way to do it – at least one that doesn’t require a fancy timer…
If you add the values of two sinusoidal waves of the same frequency, the result will be a sinusoidal waveform with the same frequency. (There is one exception to this statement, which is when the two
sinusoids are 180º apart and identical in level – then if you add them, the result is nothing – but we’ll forget about that exception for now…)
This also means that if we add a sine and a cosine of the same frequency together (remember that a cosine wave is just a sine wave that starts 90º later) then the result will be a sinusoidal waveform
of the same frequency. However, the amplitude and the phase of that resulting waveform will be dependent on the amplitudes of the sine and the cosine that you started with…
Let’s look at a couple of examples of this.
Fig. 11
Figure 11, above shows that if you take a cosine wave with a maximum amplitude of 0.7 (in blue) and a sine wave of the same frequency and amplitude, starting at a phase of 180º (or -1 * the sine wave
starting at 0º), and you add them together (just add their “y” values, for each point on the x axis – I’ve shown this for an X value of 270º in the figure), then the result is a cosine wave with an
amplitude of 1 and a phase delay of 45º (or a sine wave with a phase delay of 135º (45+90 = 135) – it’s the same thing…)
Here’s another example:
Fig. 12
In Figure 12 we see that if we add a a cosine wave * -0.5 and add it to a sine wave * -0.866, then the result is a cosine wave with an amplitude of 1, starting at 120º.
I can keep doing this for different gains applied to the cosine and sine wave, but at this point, I’ll stop giving examples and just say that you’ll have to trust me when I say:
If I want to make a sinusoidal waveform that starts at any phase, I just need to add a cosine and a sine wave with carefully-chosen gains…
Pythagoreas gets involved…
You may be wondering how I found the weird gains in Figures 11 and 12, above. In order to understand that, we need to grab a frame from the animation in Figure 5. If we do that, then you can see that
there’s a “hidden” right triangle formed by the radius of the wheel, and the vertical and the horizontal displacement of the handle.
Fig 13
Pythagoras taught us that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two other sides. Or, expressed as an equation:
a^2 + b^2 = c^2
where “c” is the length of the hypotenuse, and “a” and “b” are the lengths of the other two sides.
This means that, looking at Figure 13:
cos^2(a) + sin^2(a) = R^2
I’ve set “R” (the radius of the wheel ) to equal 1. This is the same as the amplitude of the sum of the cosine and the sine in Figures 11 and 12… and since 1*1 = 1, then I can re-write the equation
sine_gain = sqrt ( 1 – cosine_gain^2)
So, for example, in Figure 12, I said that the gain on the cosine is -0.5, and then I calculated sqrt(1 – -0.5^2) = 0.86603 which is the gain that I applied to the upside-down sine wave.
Three ways to say the same thing…
I can say “a sine wave with an amplitude of 1 and a phase delay of 135º” and you should now know what I mean.
I could also express this mathematically like this:
which means the value of y at a given value of α is equal to A multiplied by the sine of the sum of the values α and ϕ. In other words, the amplitude y at angle α equals the sine of the angle α added
to a constant value ϕ and the peak value will be A. In the above example, y(α) would be equal to 1*sin(α +135^∘) where α can be any value depending on the time (because it’s the angle of rotation of
the wheel).
But, now we know that there is another way to express this. If we scale the sine and cosine components correctly and add them together, the result will be a sinusoidal wave at any phase and amplitude
we want. Take a look at the equation below:
where A is the amplitude
ϕ is the phase angle
α is any angle of rotation of the wheel
a = Acos(ϕ)
b = Asin(ϕ)
What does this mean? Well, all it means is that we can now specify values for a and b and, using this equation, wind up with a sinusoidal waveform of any amplitude and phase that we want.
Essentially, we just have an alternate way of describing the waveform.
In Part 2, we’ll talk about a fourth way of saying the same thing…
Fc ≠ Fc
I was working on the sound design of a loudspeaker last week with some new people and software – so we had to get some definitions straight before we messed things up by thinking that we were using
the same words to mean the same thing. I’ve made a similar mistake to this before, as I’ve written about here – and I don’t being reminded of my own stupidity repeatedly… (Or, as Stephen Wright once
said “I’m having amnesia and deja vu at the same time – I think I’ve forgotten this before…”)
So, in this case on that day, we were talking about the lowly 2nd-order Low Pass Filter, based on a single biquad.
If you read about how to find the cutoff frequency of a low-pass filter, you’ll probably find out that you find the frequency where the gain is one half of the power of that in the bandpass portion
of the filter’s response. Since 10*log10(0.5) = -3.01 dB, then this is also called the “3 dB down point” of the filter.
In my case, when I’m implementing a filter, I use the math provided by Robert Bristow-Johnson to calculate my biquad coefficients. You input a cutoff frequency (Fc), and a Q value, and (for a given
sampling rate) you get your biquad coefficients.
The question then, is: is the desired cutoff frequency the actual measurable cutoff frequency of the system? (Let’s assume for the purposes of this discussion that there are no other components in
the system that affect the magnitude response – just to keep it simple.)
The simple answer is: No.
For example, if I make a 2nd-order low pass filter with a desired cutoff frequency of 1 kHz (using a high enough sampling rate to not introduce any errors due to the bilinear transform) and I vary
the Q from something very small (in this example, 0.1) to something pretty big (in this example, 20) I get magnitude response curves that look like the figure below.
Magnitude responses of 2nd order low pass filters with Q’s ranging from 0.1 to 20.
It is probably already evident that these 25 filter responses plotted above that they do not all cross each other at the 1 kHz line. In addition, you may notice that there is only one of those curves
that is -3.01 dB at 1 kHz – when the Q = 1/sqrt(2) or 0.707.
This begs the question: what is the gain of each of those filters at the desired value of Fc (in this case, 1 kHz)? This is plotted as the red line in the figure below.
The actual gain value of the filters at the desired Fc, and the maximum gain at any frequency.
This plot also shows the maximum gain of the filters for different values of Q. Notice that, in the low end, the maximum value is 0 dB, since the low pass filters only roll off. However, for Q values
higher than 1/sqrt(2), there is an overshoot in the response, resulting in a boost at some frequency. As the Q increases, the frequency at which the gain of the filter is highest approaches the
desired cutoff frequency. (As can be seen in the plot above, by the time you get to a Q of 20, the gain at Fc and the maximum gain of the filter are the same.)
It may be intuitively interesting (or interestingly intuitive) to note that, when Q goes to infinity, the gain at Fc also goes to infinity, and (relatively speaking) all other frequencies are
infinitely attenuated – so you have a sine wave generator.
So, we know that the gain value at the stated Fc is not -3 dB for all but one value of Q. So, what is the -3 dB point, if we state a desired Fc of 1 kHz and we vary the Q? This is shown in the figure
The -3 dB point of a 2nd order 1 kHz low pass filter as a function of Q.
So, varying the Q from 0.1 to 20 varies the actual Fc (or, at least, the -3 dB point) from about 104 Hz to about 1554 Hz.
Or, if we plot the same information as a function (or just a multiple) of the desired Fc, you get the plot below.
So, if you’re sitting in a meeting, and the person in front of you is looking at a measurement of a loudspeaker magnitude response, and they say “could you please put in a low pass filter with a
cutoff frequency of 1 kHz and a Q of 0.5” you should start asking questions by what, exactly, they mean by “cutoff frequency”… If not, you might just wind up with nice-looking numbers but
strangely-sounding loudspeakers. | {"url":"http://www.tonmeister.ca/wordpress/category/audio/digital-audio/page/7/","timestamp":"2024-11-08T17:33:55Z","content_type":"text/html","content_length":"184928","record_id":"<urn:uuid:6045acb5-6c36-4b27-b8df-a0ccd77f9bfa>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00701.warc.gz"} |
DORGLQ - Linux Manuals (3)
DORGLQ (3) - Linux Manuals
dorglq.f -
subroutine dorglq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
Function/Subroutine Documentation
subroutine dorglq (integerM, integerN, integerK, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( * )TAU, double precision, dimension( * )WORK, integerLWORK,
DORGLQ generates an M-by-N real matrix Q with orthonormal rows,
which is defined as the first M rows of a product of K elementary
reflectors of order N
Q = H(k) . . . H(2) H(1)
as returned by DGELQF.
M is INTEGER
The number of rows of the matrix Q. M >= 0.
N is INTEGER
The number of columns of the matrix Q. N >= M.
K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. M >= K >= 0.
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the i-th row must contain the vector which defines
the elementary reflector H(i), for i = 1,2,...,k, as returned
by DGELQF in the first k rows of its array argument A.
On exit, the M-by-N matrix Q.
LDA is INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGELQF.
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 128 of file dorglq.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-DORGLQ/","timestamp":"2024-11-11T10:44:52Z","content_type":"text/html","content_length":"9340","record_id":"<urn:uuid:7c458aa6-a201-403b-ac68-75f9cf01d78e>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00433.warc.gz"} |
ZIML Monthly Contest June-2022 Div M Question 10
There are nine letters, "ENDOFYEAR", but there are two letters "E". Will the correct answer be:5!*2= 240, instead of just 5! = 120?
The two "E"s make things a little tricky here. Note that in a rearrangement the "E"s are identical, as swapping them doesn't change anything.
Let's look at a smaller example, were we can list everything. How many different rearrangements of "ROAR" are there that contain "OAR" in the sequence.
Here there's only $2! = 2$ (not $2!\cdot 2 = 4$) different rearrangements that contain "OAR":
I think it also helps to list out ALL the different rearrangements of "ROAR" in total (note there are not $4! = 24$ rearrangements, there are $4!\div 2 = 12$ of them: | {"url":"https://ziml.areteem.org/mod/forum/discuss.php?d=274","timestamp":"2024-11-11T13:00:35Z","content_type":"text/html","content_length":"36700","record_id":"<urn:uuid:9be74203-fe1e-4391-9297-95911af0cc4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00378.warc.gz"} |
Priest Spells: Level 5
Champion's Boon: Grants a divine favor to an ally, imbuing him or her with a bonus to Might, Perception, and all Damage Reductions.
Speed: Average
Range: 5m
Effect (Friendly Target): +10 Might, +10 Perception, +5 DR for 30s
Pillar of Holy Flame: Summons a flaming pillar of righteous anger, Burning everyone in the area of effect.
Speed: Average
Range: 10m
Area of Effect: 1.25m Radius
Interrupt: 0.5s
Effect (AOE): 50-60 Burn damage vs. Reflex (+15 Accuracy)
Prayer Against Imprisonment: Instills a spirit of liberation in allies in the area of effect, granting a bonus against attacks with Paralyzed or Petrified afflictions, and reducing the duration of
any such afflictions currently on the target.
Speed: Fast
Range: 5m
Area of Effect: 2.5m Radius
Effect (Friendly AOE): +50 Defense against Paralyzed attacks, -10s Paralyzed duration, +50 Defense against Petrified attacks, -5s Petrified duration
Restore Critical Endurance: Shares a substantial portion of the caster's divine strength, restoring a large amount of Endurance to all allies in the area of effect.
Speed: Fast
Range: 5m
Area of Effect: 1.5m Radius
Effect (Friendly AOE): +55 Endurance
Revive the Fallen: Grants fallen friends a second chance, reviving unconscious allies in the area of effect and restoring a small amount of their Endurance.
Speed: Average
Range: 5m
Area of Effect: 1.25m Radius
Effect (Friendly AOE): Revive with 50 Endurance
Salvation of Time: Beseeches the gods for more time, extending the duration of all beneficial effects on allies in the area of effect.
Speed: Average
Range: 10m
Area of Effect: 2.5m Radius
Effect (Friendly AOE): +10 duration of active beneficial effects
Shields for the Faithful: Conjures a powerful holy shield, granting a Deflection bonus to all allies in the area of effect.
Speed: Average
Range: 10m
Area of Effect: 1.75m Radius
Effect (Friendly AOE): +25 Deflection for 30s | {"url":"https://www.gamebanshee.com/pillarsofeternity/spells/priest-5.php","timestamp":"2024-11-08T01:21:58Z","content_type":"text/html","content_length":"34670","record_id":"<urn:uuid:a97fb61f-940d-4d67-816a-bdf33a3ab291>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00337.warc.gz"} |
Divide Multi Digit Numbers Using The Standard Algorithm Worksheets
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Divide Multi Digit Numbers Free PDF Download Learn Bright
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Divide Multi Digit Numbers Free PDF Download Learn Bright
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2 Digit X 2 Digit Multiplication Standard Algorithm Caveman Way Mrs LeBlanc s Education
Multiplication Standard Algorithm Worksheets
Common Core Math 6 NS 3 Super Teacher Worksheets
Curriculum The Number System Compute Fluently With Multi Digit Numbers And Find Common Factors And Multiples Detail Fluently add subtract multiply and divide multi digit decimals using the standard
algorithm for each operation 25 Common Core State Standards CCSS aligned worksheets found
Illustrative Mathematics
Cluster Compute fluently with multi digit numbers and find common factors and multiples Standard Fluently divide multi digit numbers using the standard algorithm 6 NS B 2
Curriculum The Number System Compute Fluently With Multi Digit Numbers And Find Common Factors And Multiples Detail Fluently add subtract multiply and divide multi digit decimals using the standard
algorithm for each operation 25 Common Core State Standards CCSS aligned worksheets found
Cluster Compute fluently with multi digit numbers and find common factors and multiples Standard Fluently divide multi digit numbers using the standard algorithm 6 NS B 2
Divide Multi Digit Numbers Free PDF Download Learn Bright
Multiplying Multi Digit Whole Numbers Unit With Interactive Notes Math Strategies Standard
2 Digit X 2 Digit Multiplication Standard Algorithm Caveman Way Mrs LeBlanc s Education
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Division With Multi Digit Divisors
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Divide Multi Digit Numbers Worksheet | {"url":"https://szukarka.net/divide-multi-digit-numbers-using-the-standard-algorithm-worksheets","timestamp":"2024-11-03T16:03:14Z","content_type":"text/html","content_length":"27015","record_id":"<urn:uuid:9579509d-d6df-49f8-adcc-f7972fbe1fb3>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00351.warc.gz"} |
Lessons From the Bizzaro Universe
Estimated Read Time: 15 minute(s)
Common Topics: frame, time, reference, bizzaro, universe
The terms Bizarro and Bizarro World originated in Superman comics, where strangely imperfect versions of Superman, other action characters, and even Earth itself were conceived of, and provided the
basis of stories. And who can ever forget the Seinfeld episode entitled “Bizarro Jerry,” in which there are Bizarro versions of Jerry and all his friends? But, unlike his group, the Bizarro
versions are strangely nice, polite, and not self-centered.
In popular culture, Bizarro has come to mean an imperfectly flawed version of something else. So I thought it might be fun to conceptually envision a Bizarro version of our own universe. I also
felt that we might gain some additional insight into the fundamental characteristics of our universe by studying the Bizzaro universe.
Characteristics of the Bizzaro Universe
1. Like the actual universe, the Bizzaro Universe is 4 dimensional, but, unlike the actual universe, the time dimension in the Bizzaro Universe is an actual spatial dimension.
2. For each 3D rest frame in the Bizzaro Universe, the time dimension (direction) is oriented perpendicular to that rest frame. The 3D beings of the Bizzaro Universe cannot see into the time
direction of their rest frame, although, conceptually, they can see infinitely far in any of our other three spatial directions.
3. Each rest frame in the Bizzaro Universe is traveling through Bizzaro space-time at the very high characteristic velocity c (speed of light) into its time direction. The beings in this rest frame
have no sense that this is happening since they cannot see into their time direction. However, they can see partially into the time directions of other frames of reference that are in relative
motion….in a way.
4. All the clocks in each inertial frame of reference are synchronized with one another, although they may not be synchronized with the sets of clocks in other inertial frames of reference.
5. All inertial frames of reference in the Bizzaro Universe are rotationally offset concerning one another by 4D rigid rotations. To be more specific, the Bizzaro Lorentz Transformation (Borentz
Transformation) in Standard Form for the Bizzaro Universe is given by: $$x’=\frac{x-vt}{\sqrt{1+(v/c)^2}}\tag{1a}$$and $$t’=\frac{(t+vx/c^2)}{\sqrt{1+(v/c)^2}}\tag{1b}$$or equivalently $$x’=x\
Eqns. 2 can readily be recognized as the usual transformation equations for a rigid rotation of the coordinate axes by an angle ##\theta##. It follows from Eqns. 1-3 that the Cartesian coordinate
line element in the Bizarro Universe is given by: $$(ds)^2=(dx)^2+(cdt)^2=(dx’)^2+(cdt’)^2\tag{4}$$
Unlike our actual universe which is non-Euclidean, the Bizzaro Universe is Euclidean. This makes it much easier to draw space-time diagrams for the Bizzaro Universe.
I should also mention that the Bizzaro Universe is, in essence, equivalent to the “loaf of bread” arrangement described in Brian Greene’s book, “The Elegant Universe.” However, for whatever
reason, Greene fails to make any distinction between his Euclidean loaf-of-bread description and our actual non-Euclidean universe.
The Geometry of Bizzaro Universe
In the Bizzaro Universe, as with our universe, we refer to events as points in space-time with coordinates (ct, x, y, z), although, in the Bizzaro universe, the coordinate ct is an actual spatial
coordinate. Let ##\mathbf{s}## represent a 4D position vector drawn from an arbitrary origin in Bizzaro space-time to an event (ct, x, y, z). Since Bizzaro spacetime is Euclidean, we can represent
this position vector in terms of Cartesian unit vectors by the equation: $$\mathbf{s}=(ct)\mathbf{i_t}+x\mathbf{i_x}+y\mathbf{i_y}+z\mathbf{i_z}\tag{4}$$where ##\mathbf{i_t}## is the unit vector in
the time direction. (Note that, in all this, we are referring to the Cartesian coordinates of events, as referenced from an arbitrarily selected origin in 4D Bizzaro spacetime.)
Suppose that we next focus on a differential 4D position vector ##\mathbf{ds}## drawn between the two closely neighboring events at (ct, x, y, z) and at (ct+cdt, x+dx , y+dy, z+dz) in Bizzaro
space-time (as reckoned from a rest frame of reference S). The equation for this differential position vector is, of course, given by: $$\mathbf{ds}=(cdt)\mathbf{i_t}+(dx)\mathbf{i_x}+(dy)\mathbf
{i_y}+(dz)\mathbf{i_z}\tag{5}$$The length of this differential position vector is obtained by dotting it with itself:$$(ds)^2=\mathbf{ds}\centerdot \mathbf{ds}=(cdt)^2+(dx)^2+(dy)^2+(dz)^2\tag{6}
$$Eqn. 6 can be recognized is just the 4D equivalent of the Pythagorean theorem.
Suppose now there is also a second frame of reference S’ (containing 3D beings traveling with 3D velocity components ##v_x##, ##v_y##, and ##v_z## relative to our first frame of reference S) and
employing Cartesian event coordinates (ct’, x’, y’,z’). If we resolve this same differential position vector ##\mathbf{ds}## into components concerning the primed system of coordinates, we obtain:$$
\mathbf{ds}=(cdt’)\mathbf{i_t’}+(dx’)\mathbf{i_x’}+(dy’)\mathbf{i_y’}+(dz’)\mathbf{i_z’}\tag{7}$$And, in terms of the primed coordinates, the length of the vector ##\mathbf{ds}## is given by: $$(ds)
^2=\mathbf{ds}\centerdot \mathbf{ds}=(cdt’)^2+(dx’)^2+(dy’)^2+(dz’)^2\tag{8}$$
Since the differential position vector ##\mathbf{ds}## does not depend on the specific coordinate system or frame of reference from which it is reckoned (i.e., it is invariant under a change in frame
of reference), we must immediately conclude that: $$\mathbf{ds}=(cdt)\mathbf{i_t}+(dx)\mathbf{i_x}+(dy)\mathbf{i_y}+(dz)\mathbf{i_z}$$$$=(cdt’)\mathbf{i_t’}+(dx’)\mathbf{i_x’}+(dy’)\mathbf{i_y’}+
Eqns. 9 apply irrespective of the relative 3D velocities of the S and S’ frames of reference, or the rotational and translational offsets of the two coordinate systems.
Frames of Reference
In the present context, it is worthwhile being a little more precise about the term “frame of reference.” A frame of reference is a 3D x-y-z spatial cut out of 4D spacetime. The time direction t
for this spatial 3D cut is oriented perpendicular to the 3 orthogonal Cartesian spatial directions of the frame of reference. Therefore, the time direction in 4D spacetime for a frame of reference S
defines the frame of reference.
The residents of the Bizarro universe are analogous to 2-dimensional beings trapped within a flat plane that is immersed in 3D space. They have no access to the 3rd dimension, except for the 2D
cross-section that they currently occupy. This 2D cross-section may not be stationary in 3D space; it may be moving forward (unbeknownst to them) into the 3rd (time) dimension. If so, as time
progresses, they would be sweeping out all of 3D space, and would ultimately be able to sample all of 3D space with their planar cross-section. However, at any one instant of time (according to
their synchronized set of clocks), they would only have access to a single planar slice out of 3D space.
This is analogous to what the 3D beings of the Bizzaro Universe are experiencing in their 4D spacetime. In their rest frame of reference, they are trapped within a specific 3D slice out of 4D
spacetime. This 3D slice is unique to the particular reference frame they occupy (i.e., their rest frame). They have no access or vision into the 4th dimension, except for this 3D cut. The cut is
not stationary; it is moving forward (unbeknownst to them) into their own 4th (time) spatial dimension. As time (measured by the synchronized clocks in their 3D reference frame) progresses, they are
sweeping out all of 4D spacetime, and will ultimately be able to sample all of the 4D spacetime (at least the part into their future) with their moving 3D cut. However, at any one instant in time,
they only have access to their single 3D cut-out of 4D spacetime (a 3D panoramic snapshot). Finally, different frames of reference in relative motion possess different 3D cuts and different time
directions perpendicular to these 3D cuts.
Four-Dimensional Bizzaro Velocity
Arguably, the most important equation in the present development is Eqn. 9a, providing a relationship for a differential position vector in Bizzaro spacetime, as reckoned from the coordinate systems
that are employed in two different frames of reference, S and S’:$$\mathbf{ds}=(cdt)\mathbf{i_t}+(dx)\mathbf{i_x}+(dy)\mathbf{i_y}+(dz)\mathbf{i_z}$$
$$=(cdt’)\mathbf{i_t’}+(dx’)\mathbf{i_x’}+(dy’)\mathbf{i_y’}+(dz’)\mathbf{i_z’}\tag{9a}$$This is the starting point for the development of equations for the Bizzaro 4D velocity vector.
Imagine that we are following the motion of a specific material particle or observer that is at rest in the S’ frame of reference, such that its spatial coordinates x’, y’, and z’ are held constant,
and thus dx’ = dy’ = dz’ = 0. For this object, Eqn. 9a becomes $$\mathbf{ds}=(c\mathbf{i_t}+v_x\mathbf{i_x}+v_y\mathbf{i_y}+v_z\mathbf{i_z})dt=c\mathbf{i_t’}dt’\tag{10}$$where, here, ##\mathbf{ds}##
represents the displacement of the object over the time interval dt for frame S and dt’ for frame S’, and where ##v_x=(\partial x/\partial t)_{x’,y’,z’}##, ##v_y=(\partial y/\partial t)_{x’,y’,z’}##,
and ##v_z=(\partial z/\partial t)_{x’,y’,z’}##. Eqn. 10 expresses the differential position vector for the motion of our particle or observer in terms of the unit vectors for the S’ frame of
reference and also in terms of the unit vectors for the S frame of reference. The relationships we have used here for the (3D) velocity components ##v_x##, ##v_y##, and ##v_z## for the S frame are
consistent with how these components would be conventionally determined (i.e., by taking the derivative of the particle’s coordinates concerning time, holding the material particle constant). This
is sometimes referred to as the “material time derivative.”
If we take the dot product of Eqn. 10 with itself, we obtain:$$(ds)^2=(c^2+v^2)(dt)^2=c^2(dt’)^2\tag{11}$$ where ##v^2=(v_x)^2+(v_y)^2+(v_z)^2##. Taking the square foot of Eqn. 11 then yields $$dt=\
gamma dt’\tag{12}$$with $$\gamma=\frac{1}{\sqrt{1+(v/c)^2}}\tag{13}$$where, in the present situation, dt’ represents the differential of Bizzaro proper time measured in the rest reference frame of
the moving particle. Note that, unlike our real universe where ##\gamma## is always greater than unity, in the Bizzaro Universe, ##\gamma## is always less than unity.
If we now substitute Eqn. 13 into Eqn. 10, we obtain the Bizzaro 4 velocity ##\mathbf{V}## of the moving particle: $$\mathbf{V}\equiv \left(\frac{\partial \mathbf{s}}{\partial t’}\right)_{x’,y’,z’}=
c\gamma \mathbf{i_t}+v_x\gamma \mathbf{i_x}+v_y\gamma \mathbf{i_y}+v_z\gamma \mathbf{i_z}=c\mathbf{i_t’}\tag{14}$$
Eqn. 14 contains lots of valuable new information concerning the basic nature of 4D Bizzaro spacetime and the kinematics of motion. Since ##\mathbf{ds}## represents a differential displacement
vector, it would appear (according to the right-hand side of this equation) that, even though the particle under consideration is at rest within the S’ frame of reference, it is not truly at rest in
Bizzaro spacetime; it is covering distance into the 4th (time) dimension t’ with a speed equal to the speed of light c. All objects that appear to be at rest within the S’ frame of reference, as
well as the S’ frame of reference itself, are moving in the t’ direction at this speed.
The 4 velocities vector ##\mathbf{V}## defined by Eqn. 14 can be interpreted physically as the velocity of the S’ frame of reference (and all objects at rest within the S’ frame of reference) about
spacetime itself. That is, Eqn. 14 suggests that it is valid to regard Bizzaro spacetime as stationary and absolute and to treat ##\mathbf{V}## as the 4D velocity of an object or observer in the S’
frame of reference relative to stationary 4D Bizzaro spacetime.
This interpretation is not unique to the S’ frame of reference, since the above analysis could just as easily be repeated for any other frame of reference. Thus, In general, all objects in the
Bizzaro Universe are traveling at the speed of light relative to spacetime, but their directions through spacetime are different and are determined by the directions of their time arrows (unit
vectors in the time direction), which are unique to each frame of reference.
The beings of the Bizzaro Universe might think to themselves, “How can we, as objects in the Bizzaro Universe, be traveling through spacetime at the speed of light, and yet not be aware that this
motion is taking place?” The answer is that they are all trapped within their 3D slice out of 4D spacetime (i.e., their rest frame of reference), and they cannot see into their time dimension.
Furthermore, any objects that are at rest (or nearly at rest) near them are each traveling at virtually the exact same velocity as they are; therefore, they don’t sense any relative movement on the
order of the speed of light. Finally, the ride is very smooth, since there are no bumps in the road.
It is possible to express the 4D velocity of an object relative to Bizzaro spacetime not only interns of the time unit vector for its frame of reference (i.e., ##c\mathbf{i_t’}##), but also in terms
of the unit vectors for any other reference frame that is not accelerating. Thus, from Eqn. 14, the absolute 4D velocity of a particle in the S’ frame of reference, expressed in terms of the unit
vectors for the non-accelerating S frame of reference, is given by $$\mathbf{V}=c\gamma \mathbf{i_t}+v_x\gamma \mathbf{i_x}+v_y\gamma \mathbf{i_y}+v_z\gamma \mathbf{i_z}\tag{14}$$Similarly, the
passage of proper time for a moving object can be expressed in terms of the time displayed on the synchronized clocks in any other non-accelerating frame of reference using the equation $$dt’=\frac
{dt}{\gamma}\tag{15}$$These equations allow observers in any arbitrary non-accelerating frame of reference to formulate physical laws for the motion of objects not at rest in their reference frame,
without actually having direct access to clocks or meter sticks for the reference frame of the moving object.
The present results apply not only to objects moving at constant speed relative to inertial frames of reference. They also apply to objects that may be accelerating. One simply regards the
instantaneous 4D absolute Bizzaro velocity of an accelerating object as equal to that of an inertial reference frame moving at the same velocity (i.e., a so-called co-moving inertial reference
frame). Thus, the 4D Bizzaro velocities of all objects in the Bizzaro Universe, including objects that are accelerating, have a magnitude equal to the speed of light, but, for accelerating objects,
the orientation of their time arrows (unit vector in time direction) change with the proper time.
Bizzaro Train
In Einstein’s famous special relativity train scenario, there is a train traveling down a long straight track at a constant speed v comparable to the speed of light. One team of observers is strung
out along the platform (S frame of reference), armed with a set of clocks all synchronized with one another in the platform/track/ground frame of reference, and a second team is strung out along the
train (S’ frame of reference), also armed with a set of clocks, all synchronized with one another in the train frame of reference. Our objective now will be to use what we have learned so far to see
how this same train scenario plays out in the Bizzaro Universe.
Application of Eqn. 14 to the train scenario leads to the following relationships for the absolute 4 velocities of the platform P and train T concerning stationary Bizzaro spacetime (in terms of the
platform frame unit vectors): $$\mathbf{V_P}=c\mathbf{i_t}\tag{15a}$$ $$\mathbf{V_T}=c\gamma\mathbf{i_t}+\gamma v\mathbf{i_x}\tag{15b}$$
Based on these equations, we have sufficient information to represent the motion of the platform and train through Bizarro spacetime schematically, via a spacetime diagram. Fig. 1 shows the sequence
of locations for a 3-car train at proper train clock times ##t_1′<t_2′<t_3’##.
Figure 1. Train Movement Through Bizzaro Spacetime
The train is oriented parallel to the x’ axis and is moving at the speed of light (relative to Bizzaro spacetime) in the direction of the ct’ axis. There is an observer on the train at location x’ =
0 (which corresponds to the center of the middle car). The team on the platform focuses attention on this train rider and measures his relative speed using their coordinate grid in the x-direction
and their own set of synchronized clocks (displaying the time t which, incidentally, is not “proper train time” for objects at rest in the train). They obtain $$v=\left(\frac{\partial x}{\partial t}
\right)_{x’=0}\tag{16}$$This is the same relationship that would be obtained not only in the Bizzaro Universe but also in the real universe. The entire train moves with this speed relative to the S
reference frame, according to the measurement tools available to the observers in the S (platform) frame of reference. However, this speed does not properly represent the true x component of the
relative velocity vector of the train concerning the platform in Bizzaro spacetime. This is because it failed to take into account the difference between the proper train time and the platform clock
time. The actual relative 4 velocities of the train concerning the platform are obtained by subtracting Eqn. 15 a from Eqn. 15 b, to yield:$$\mathbf{V_T-V_P}=c(\gamma-1)\mathbf{i_t}+\gamma v \mathbf
{i_x}\tag{17}$$The component of this relative velocity vector in the t direction is not visible to the observers in the platform reference frame, since t constitutes their inaccessible time
direction. The component in the x-direction has an added factor of ##\gamma## to correct for the difference between the platform clock time t and the train proper time t’.
Fig. 2 is designed to provide some insight into the issues of simultaneity and length contraction (or expansion).
Figure 2 Simultaneity and Length Contraction/Expansion Assessment
The first question is, “What do the team of platform observers see when they look at the train at time t on their synchronized clocks?” According to the figure, the rear end of the train arrives at
platform time t first, at train time ##t_1’##; later (in train time), the middle of the train arrives at platform time t, at train time ##t_2’##; still later (in train time), the front end of the
train arrives at ground time t, at train time ##t_3’##. Thus, at any specified time t on the synchronized platform clocks, the observers on the platform can view the entire train all at once.
However, unbeknownst to them, they see the rear end of the train at an earlier train clock time t’ than the front of the train. This result for the Bizzaro Universe is the opposite of what is found
for the real universe.
About length change, it appears from Fig. 2 that, in the Bizzaro Universe, the length of the train measured by the platform observers at time t will be greater than the train’s proper length in its
rest frame. This is the opposite of the length contraction effect observed in the real universe.
The final situation to be considered here will be the time dilation (or contraction) effect (Fig. 3).
Figure 3. Assessment of Time Dilation/Contraction Effect
Fig. 3 shows the motion of the train rider situated in the middle of the train (x’=0) as he travels through Bizzaro spacetime. He passes two observers on the ground at a distance of ##\Delta x##
apart, and notes when he passes that the times displayed on their clocks differ by ##\Delta t##. He compares this with the corresponding time interval on his clock ##\Delta t’##, and determines
that, for Bizzaro spacetime, $$\left(\frac{\Delta t’}{\Delta t}\right)_{x’=const}>1\tag{18}$$ This time expansion is exactly the opposite of the time dilation effect observed in the real universe.
It’s been interesting exploring the geometric and kinematic characteristics of the Euclidean Bizarro Universe and making comparisons with the corresponding effects observed to take place in our
non-Euclidean universe. Some of the analogies are enlightening and have provided some new insights and interpretations. If anyone would be interested in continuing this pursuit, I would be
interested in collaborating. Other topics I would be motivated to analyze would be (1) constant acceleration kinematics, (2) paradoxes such as the pole paradox, and (3) Bizzaro laws of dynamics,
such as momentum and energy conservation.
PhD Chemical Engineer
Retired after 35 years experience in industry
Physics Forums Mentor
https://www.physicsforums.com/insights/wp-content/uploads/2018/04/bizarro_univerise.png 135 240 Chestermiller https://www.physicsforums.com/insights/wp-content/uploads/2019/02/
Physics_Forums_Insights_logo.png Chestermiller2018-04-13 13:07:542024-08-17 16:35:57Lessons From the Bizzaro Universe
23 replies
1. dagmar says:
"what would the kinematics of constant acceleration look like in terms of the trajectories of particles in the x-t plane?"Well, the lines of constant acceleration in this model are just circles,
like the lines of constant acceleration in the Minkowski plane are hyperbolas. I said like, because the radius of curvature of those hyperbolas in the Minkowski plane is constant, and constant
acceleration is constant curvature so to speak.
Another analogy with 2D-Minkowski plane, is the law of addition of velocities: It is the old familiar ## tan(θ+φ)=frac{tanθ+tanφ}{1-tanθ tanφ}##, where ##u=tanθ##, ##v=tanφ## play the role of
velocities (θ and φ are the angles the worldlines make with the t-axis) with further analogies in 4 dimensions.
Well, for anyone who is not familiar with Lorentzian manfolds, your model is an excellent substitute to study the rotations of the plane ( boosts ) at an intuitive level which is the direct
Geometric inspection and interpretation of our everyday world.
Of course there are some drawbacks with the inconstancy of the speed of light, and the backwards in time travel which pops out everywhere, but otherwise is an excellent model.
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2. Chestermiller says:
Well, I have made use of this model myself 25 years ago in my efforts to grasp SR. You are not first to think of this model.
But, what is this speed of light you are talking about? Have you realized that there is no such thing as speed of light in this model, yours and mine?
Here is a 2D example: an observer is moving with proper time the y-axis: He shines a beam of light, makes an angle θ = 45 degrees to horizontal x-axis, call this c=1. Now set him to travel
himself at c=1. By yours and mine Borentz transformations the speed of light he is now emitting from his frame transforms to θ = 0 degrees to the horizontal = infinite speed, as measured again by
an observer with proper time the y-axis. Completely analogous to the Lorentz addition of velocities ( rotations of axes, only speed of light remains constant there.)
So, no observer could ever decide which is the right speed of light since the light which is emitted by a moving frame varies in speed according to the velocity of this very moving frame each
time. I doubt if light could ever exist in a universe with such properties.
Cheers, be good.Thanks Dagmar. I'm glad to see that I am not the only one to think of and play with this model. I thought there would be other interesting opportunities for interpretations, but
no other members seemed to be interested in pursuing this. One particular situation I was interested in was "what would the kinematics of constant acceleration look like in terms of the
trajectories of particles in the x-t plane?"
I did realize that there would be issues with the speed of light for this universe. I was just using the symbol c for a characteristic velocity of the system. I was more interested in other
physical aspects of the Bizzaro Universe.
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3. dagmar says:
Well, I have made use of this model myself 25 years ago in my efforts to grasp SR. You are not first to think of this model.
But, what is this speed of light you are talking about? Have you realized that there is no such thing as speed of light in this model, yours and mine?
Here is a 2D example: an observer is moving with proper time the y-axis: He shines a beam of light, makes an angle θ = 45 degrees to horizontal x-axis, call this c=1. Now set him to travel
himself at c=1. By yours and mine Borentz transformations the speed of light he is now emitting from his frame transforms to θ = 0 degrees to the horizontal = infinite speed, as measured again by
an observer with proper time the y-axis. Completely analogous to the Lorentz addition of velocities ( rotations of axes, only speed of light remains constant there.)
So, no observer could ever decide which is the right speed of light since the light which is emitted by a moving frame varies in speed according to the velocity of this very moving frame each
time. I doubt if light could ever exist in a universe with such properties.
Cheers, be good.
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4. Arthur Mitchell says:
My question about the actual universe.Because of space/time, isn't it reasonable to assume that the objects we see such as galaxy's, no longer exist as such, and have been replaced by other
matter, forming other galaxies?
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5. Chestermiller says:
Arthur Mitchell
If objects such as galaxy's seen through telescopes (or naked eye) are measured, quantified, etc., isn't it conceivable that due to space/time, that the mass calculated may no longer exist?Are
you asking this about the Bizzaro Universe or about the actual universe? If you are asking this about the actual universe, it might be better to start a new thread on this question.
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6. Arthur Mitchell says:
If objects such as galaxy's seen through telescopes (or naked eye) are measured, quantified, etc., isn't it conceivable that due to space/time, that the mass calculated may no longer exist?
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7. Chestermiller says:
I'm not talking your right to call it however you want. I ask why do you treat it differently. Since this coordinate enters metric symmetrically with space-like coordinates, there is no logical
reason to distinguish it.I distinguish it by saying that each rest frame of reference has its own private time direction through 4D Euclidean space, and each object and frame of reference is
always traveling at the velocity magnitude c in that direction (although the direction of a body in 4D Euclidean space changes when the body is accelerated). The magnitude can never change.
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8. nikkkom says:
I can call it whatever I want.I'm not talking your right to call it however you want. I ask why do you treat it differently. Since this coordinate enters metric symmetrically with space-like
coordinates, there is no logical reason to distinguish it.
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9. Chestermiller says:
Why do you call this axis "time axis", if mathematically it enters metric symmetrically with other, "space" axes?
To be a time coordinate, its term has to enter metric with a different sign.I can call it whatever I want. It is the private direction assigned to each 3D rest frame moving through 4D Euclidean
space at velocity c. If you don't like the word "time, " I'll call it the "shmime" direction. But changes in the shmimes exhibited on the watches of the beings in a given rest frame ##Delta tau #
# correspond directly to distances travelled by the rest frame through 4D Euclidean space, according to the equation ##Delta s=cDelta tau##.
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10. nikkkom says:
This does not prevent me from saying each 3D rest frame within my 4D Euclidean space has its own private time axis that is perpendicular to the other three Cartesian coordinatesWhy do you call
this axis "time axis", if mathematically it enters metric symmetrically with other, "space" axes?
To be a time coordinate, its term has to enter metric with a different sign.
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11. Chestermiller says:
If the rules (axioms) are not mathematically consistent, you can't derive meaningful theorems from them.
Your "time" coordinate is simply space-like. This completely determines how coordinates transform under rotations. There is no freedom to insert any additional rules. Your space is isomorphic to
4D Euclidean space.This does not prevent me from saying each 3D rest frame within my 4D Euclidean space has its own private time axis that is perpendicular to the other three Cartesian
coordinates for that rest frame, and, unbeknownst to the 3D beings within that rest frame, the frame is traveling at the speed c through 4D Euclidean space; the 3D beings are somehow (by their
biological makeup) unable to see into that 4th time dimension. The only way that the time direction (in 4D Euclidean space) of a body can change is if a force is applied to body to cause its time
direction (and attached rest frame) to rotate.
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12. nikkkom says:
Since I am the great and all-powerful Bizzaro GOD, I can have any set of rules I desire for my Bizzaro Universe, and the 3D Bizzaro beings within it have to accept the reality of these rules.If
the rules (axioms) are not mathematically consistent, you can't derive meaningful theorems from them.
Your "time" coordinate is simply space-like. This completely determines how coordinates transform under rotations. There is no freedom to insert any additional rules. Your space is isomorphic to
4D Euclidean space.
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13. Chestermiller says:
An analogy that comes to mind is a special comic book with exactly one drawing per page. It's a three-dimensional object. There is a "time axis" that is the direction from the first page toward
the last page. Characters move around in a two-dimensional world as function of the time coordinate (page number).
Where the analogy becomes difficult is that you want Euclidean distance to correspond to proper time. So you have a cast of comic characters on each page, and let's say that each character is
drawn with a watch showing his or her own proper time. Then the writer/illustrator has a tough assignment:
1. Each character's 2-D location within a page must vary continuously as a function of page number (or approximately continuously, since you can't have a continuous function of a discrete
2. For a character that is "at rest" (the same place on the page from one page to the next), the time on the watch increases linearly with the page number. Let the amount of increase from one
page to the next be ##Delta t##.
3. For a character that is "moving", you compute the change on their wristwatch as follows: ##Delta t' = Delta t sqrt{1+(frac{D}{d})^2}## where ##d## is the thickness of a page, and ##D## is the
distance the character moves from one page to the next.
I think that's a model of the Bizarro Universe.I don't follow this all in detail, but in the Bizzaro Universe, each 3D rest frame has its own private time direction in which the frame of
reference is moving (relative to 4D Bizzaro space-time) at the velocity c. Differences in velocity perceived by 3D beings are the result of differences in the time directions (i.e., the
directions of the 4D velocity vectors) of the various rest frames. If this is captured by the analogy that you have described, that would be great. So, in item 3, it seems that you have derived
the equation for "time contraction" in the Bizzaro Universe (which should tell us something about the Bizzaro twin paradox).
I was hoping that other members would get interested, as you have done, in investigating other features of the Bizzaro Universe, like "length expansion," frame dependence of the speed of light,
and acceleration trajectory shape. Anyone out there interested?
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14. stevendaryl says:
I'm beginning to get the feeling that you are not happy with the way that I have constructed and set rules for my Bizzaro universe. Since I am the great and all-powerful Bizzaro GOD, I can have
any set of rules I desire for my Bizzaro Universe, and the 3D Bizzaro beings within it have to accept the reality of these rules. In other words, in the Bizzaro Universe, the laws of Physics are
what I say they are. If you don't like that, stay out.
Bizzaro Universe: Love it or leave it.An analogy that comes to mind is a special comic book with exactly one drawing per page. It's a three-dimensional object. There is a "time axis" that is the
direction from the first page toward the last page. Characters move around in a two-dimensional world as function of the time coordinate (page number).
Where the analogy becomes difficult is that you want Euclidean distance to correspond to proper time. So you have a cast of comic characters on each page, and let's say that each character is
drawn with a watch showing his or her own proper time. Then the writer/illustrator has a tough assignment:
1. Each character's 2-D location within a page must vary continuously as a function of page number (or approximately continuously, since you can't have a continuous function of a discrete
2. For a character that is "at rest" (the same place on the page from one page to the next), the time on the watch increases linearly with the page number. Let the amount of increase from one
page to the next be ##Delta t##.
3. For a character that is "moving", you compute the change on their wristwatch as follows: ##Delta t' = Delta t sqrt{1+(frac{D}{d})^2}## where ##d## is the thickness of a page, and ##D## is the
distance the character moves from one page to the next.
I think that's a model of the Bizarro Universe.
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15. Chestermiller says:
If the fourth "time" coordinate is a space-like coordinate, (that is, it enters metric with the same sign as x,y,z terms, not with the opposite one), then it's just a 4-dimensional Euclidean
space. This coordinate then must be treated on the equal footing with x,y,z. It can not be "time" anymore. "The 3D beings of the Bizzaro Universe cannot see into the time direction of their own
rest frame" can not be fulfilled, just like in Minkowski space you can not arbitrarily prohibit "seeing into" a space-like direction (say, z).
The very reason our time dimension is "time" is because it enters metric with a "minus" and therefore behaves differently from x,y,z. For example, no matter how hard you turn away from t
direction, you cannot completely stop moving in time, whereas you _can_ stop moving in x direction if you turn hard enough towards y or z.I'm beginning to get the feeling that you are not happy
with the way that I have constructed and set rules for my Bizzaro universe. Since I am Bizzaro GOD, I can have any set of rules I desire for my Bizzaro Universe, and the 3D Bizzaro beings within
it have to accept the reality of these rules. If you don't like the Bizzaro Universe, stay out.
Bizzaro Universe: Love it or leave it.
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16. nikkkom says:
Greg Bernhardt submitted a new PF Insights post
Lessons From the Bizzaro Universe
View attachment 223982
Continue reading the Original PF Insights Post.If the fourth "time" coordinate is a space-like coordinate, (that is, it enters metric with the same sign as x,y,z terms, not with the opposite
one), then it's just a 4-dimensional Euclidean space. This coordinate then must be treated on the equal footing with x,y,z. It can not be "time" anymore. "The 3D beings of the Bizzaro Universe
cannot see into the time direction of their own rest frame" can not be fulfilled, just like in Minkowski space you can not arbitrarily prohibit "seeing into" a space-like direction (say, z).
The very reason our time dimension is "time" is because it enters metric with a "minus" and therefore behaves differently from x,y,z. For example, no matter how hard you turn away from t
direction, you cannot completely stop moving in time, whereas you _can_ stop moving in x direction if you turn hard enough towards y or z.
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17. robphy says:
Have there been attempts to teach special relativity by introducing spacetime rotations like in the insight first, and only switch to hyperbolic rotations (=Lorentz transformations) afterwards?
Taylor & Wheeler's Spacetime Physics uses Euclidean surveyors of a plane (not Bizzaro Spacetime observers) then suggests the Special Relativity case by analogy.
I've been working on a variation of that idea, but including the Galilean case as an intermediate stage.
(I've considered aspects of something like the Bizarro Spacetime… but not to the extent developed by @Chestermiller . It was entertaining. )
From my reading of the Insight, the "speed of light" c functions only as a unit conversion constant… but not an invariant limiting spatial velocity for observers.
There's no causal structure in the bizarro universe.. no invariant light cones (no eigenvectors of the boost).. But it does feature the Bizarro relativity of simultaneity (since tangents to
"circles" don't coincide, except in the Galilean case).
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18. Chestermiller says:
Have there been attempts to teach special relativity by introducing spacetime rotations like in the insight first, and only switch to hyperbolic rotations (=Lorentz transformations) afterwards?
You made my day. This is exactly what I was hoping someone would suggest. Thank you so much.
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19. kith says:
Have there been attempts to teach special relativity by introducing spacetime rotations like in the insight first, and only switch to hyperbolic rotations (=Lorentz transformations) afterwards?
PS: I think that this thread would be better suited to the relativity forum because the insight is about a bizarro version of special relativity.
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20. scottdave says:
That was a fun read.
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21. anorlunda says:
I have to spend more time to digest it completely. But in the meantime, let me congratulate you @Chestermiller on your creative way to make a physics lesson fun. I must believe that it attracts
more people to study the lesson. Of course, the #1 priority of any article on any subject must be to induce readers to read it.
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22. Chestermiller says:
Greg Egan wrote a novel (or 3 novels!) set in the Bizzaro universe, or what he calls the Riemannian universe. His website goes over physics in the Riemannian universe in some detail.
http://www.gregegan.net/ORTHOGONAL/ORTHOGONAL.htmlIn my understanding, Riemannian universe focuses on the curvature of space time. The Bizzaro universe I have described assumes a flat spacetime,
analogous to that encountered in Special Relativity.
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23. Khashishi says:
Greg Egan wrote a novel (or 3 novels!) set in the Bizzaro universe, or what he calls the Riemannian universe. His website goes over physics in the Riemannian universe in some detail.
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Learn here the 15 times table. Fifteen times table 15 x 1 = 15, 15 x 2 = 30, 15 x 3 = 45, 15 x 4 = 60, 15 x 5 = 75, 15 x 6 = 90, 15 x 7 = 105, 15 x 8 = 120, 15 x 9 = 135, 15 x 10 = 150.
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Top 12 Applications of Quantum Computing
In recent years, quantum computing has emerged as a revolutionary technology with the potential to revolutionize various industries and fields. This cutting-edge computational paradigm harnesses the
principles of quantum mechanics to perform complex calculations that are practically impossible for classical computers. In this article, we delve into the exciting realm of quantum computing,
exploring its advantages, underlying principles, and a comprehensive overview of its applications across diverse domains.
What is Quantum Computing?
Quantum computing is a way of doing super-fast calculations using the rules of quantum physics. Regular computers use 'bits' to store information as 0s and 1s. Quantum computers, on the other hand,
use 'qubits' that can be 0, 1, or both at the same time. This is like having a magical way to do many calculations all at once.
Imagine it as if you had a bunch of balls that can be in different colors at once. This lets quantum computers solve problems much faster than supercomputers, and it's all thanks to the strange world
of quantum physics. In this field of quantum computing, scientists are working with tiny particles like atoms, using special machines called quantum processors. These particles can be in different
states at once, like being in two places at the same time, thanks to something called 'superposition.' This might sound weird, but it's the reason quantum computers are so powerful. They work with
these special particles, and the more of them they have (called 'qubits'), the more amazing calculations they can do. This has the potential to help us in many areas, like simulating complex things
or solving problems that regular computers would struggle with. Scientists are also using things like magnetic fields to control these particles and make them work together to solve problems.
Advantages of quantum computing
The key advantage of quantum computing lies in its potential to solve complex problems that are currently intractable for classical computers. Tasks that would take classical computers millions of
years to complete can potentially be solved by quantum computers in a matter of seconds. This immense computational power opens doors to groundbreaking applications across various sectors.
The field of Quantum mechanics
Quantum mechanics is the branch of physics that deals with the behavior of matter and energy at the atomic and subatomic level. It describes how particles behave as both particles and waves, and how
their properties can exist in multiple states simultaneously until observed or measured.
What is a Qubit?
In quantum computing, a qubit is the basic info unit, cooler than regular bits. It's not just 0 or 1; it's both at once, thanks to 'superposition.' Quantum computers use these qubits, often tiny like
ions. More qubits mean more power for solving hard problems. In short, qubits are the quantum stars that make amazing computing possible.
The Principles of Quantum Computing
The Principles of Quantum Computing delve into the fascinating world of quantum mechanics, where terms like quantum information and quantum states take center stage. Unlike classical bits, these
quantum bits (qubits) exist in multiple states simultaneously, enabling quantum computers to potentially solve problems that were once unimaginable. These principles are the foundation of quantum
computing, where quantum hardware and quantum computing hardware come together to create a new realm of possibilities. With the potential to build quantum systems with two or more qubits entangled
and interconnected, the potential use cases and advancements in technology become boundless.
What are the Types of Quantum Technology?
Quantum technology encompasses various approaches to quantum computing, such as gate-based quantum computing and quantum annealing. Gate-based quantum computing manipulates qubits through quantum
gates to perform calculations, while quantum annealing finds the lowest energy state of a quantum system to solve optimization problems.
Top 12 applications of Quantum Technology
Let's delve into each of the applications of quantum computing mentioned in the article in more detail:
1. Artificial Intelligence & Machine Learning:
Quantum computing has the potential to revolutionize AI and machine learning. Machine learning algorithms often involve complex optimizations that can be time-consuming for classical computers.
Quantum computers can speed up these processes significantly by exploring multiple solutions simultaneously. This could lead to more efficient training of AI models, better natural language
processing, and advanced image recognition.
2. Computational Chemistry:
Computational chemistry involves simulating the behavior of molecules and chemical reactions. Quantum computers can accurately simulate complex molecular interactions, enabling researchers to
understand chemical reactions, molecular structures, and properties in ways that were previously impractical. This can accelerate drug discovery, materials design, and the development of new
3. Drug Design & Development:
The pharmaceutical industry can benefit greatly from quantum computing. Quantum simulations can predict how drugs interact with target molecules, leading to the discovery of more effective and safer
drugs. This could reduce the time and cost associated with drug development by identifying potential candidates more efficiently.
4. Cybersecurity & Cryptography:
While quantum computers have the potential to break current cryptographic methods due to their immense computational power, they can also contribute to quantum-safe cryptography. Quantum key
distribution enables secure communication by leveraging the principles of quantum mechanics to detect eavesdropping attempts. This offers a potential solution to the impending threat quantum
computers pose to current encryption methods.
5. Financial Modeling:
The financial industry relies on complex models for risk assessment, portfolio optimization, and trading strategies. Quantum computing can process vast amounts of financial data and solve
optimization problems more efficiently. This could lead to more accurate financial predictions and better-informed investment decisions.
6. Logistics Optimization:
Optimizing logistics, such as supply chain management, is a challenge that can be significantly improved by quantum computing. Quantum algorithms can handle complex optimization problems, ensuring
optimal routes, minimizing transportation costs, and reducing environmental impact.
7. Weather Forecasting:
Quantum computers can process large datasets and simulate complex weather patterns with higher accuracy and speed than classical computers. This could lead to more precise weather forecasts, helping
communities prepare for extreme weather events and minimizing their impact.
8. Material Science:
Understanding material properties at the atomic and molecular levels is crucial for designing new materials with specific functionalities. Quantum simulations can provide insights into material
behavior, helping researchers discover novel materials for applications in electronics, energy storage, and more.
9. Quantum Computing Applied to Natural Language Processing (NLP):
Quantum computing can expedite natural language processing tasks by analyzing vast amounts of text data quickly. This could lead to better language understanding, sentiment analysis, and machine
translation, enhancing communication between languages and cultures.
10. Quantum Computing Used for Task Optimization:
Quantum algorithms are adept at solving optimization problems across industries. From optimizing energy distribution and resource allocation to scheduling and route planning, quantum computers can
find optimal solutions to complex logistical challenges.
11. Enhanced Batteries:
Quantum simulations can aid in the design and optimization of battery materials. By understanding how materials interact at the atomic level, researchers can develop more efficient and longer-lasting
batteries for electric vehicles and renewable energy storage.
12. Gaming
Quantum computing can enhance gaming experiences by enabling more realistic simulations and complex virtual worlds. Quantum algorithms could lead to more advanced graphics rendering, physics
simulations, and AI-driven gameplay.
A Glimpse into the Future of Quantum Computing Applications
Quantum computing is a revolutionary field with extraordinary power. These quantum machines, unlike regular computers, work on quantum mechanics. They're called quantum computers and can solve
complex problems faster than traditional ones. Think of them as super detectives with quantum sensors, able to detect things others can't. Big players like IBM and Google are in the game, using
quantum superposition to multitask on a new level.
Quantum programs are their secret sauce, designed for their unique abilities. Researchers are also working on fault-tolerant quantum computers that handle errors gracefully. But quantum computing
isn't just about speed; it's about using quantum mechanics to solve fundamental problems in fields like medicine and materials science.
These computers use quantum bits (qubits) to perform lightning-fast calculations thanks to quantum superposition. We're still in the early stages, but the potential is immense. Researchers are moving
towards achieving quantum advantage – where quantum computers consistently outperform classical ones.
In essence, quantum computing is a new dimension of tech. It's not just about faster computers; it's about using quantum mechanics to create a better future.
1. Are Quantum Computers Practical for Everyday Use?
As quantum computers continue advancing within the IBM Quantum and other research initiatives, the potential for practical everyday applications becomes increasingly promising. Quantum computers,
operating on the principles of quantum mechanics, display remarkable performance in specific types of calculations, such as optimization and complex simulations. This suggests that quantum computing
could greatly benefit various sectors, such as finance and supply chain management, where IBM Quantum One and similar systems are being developed to tackle intricate problems and streamline
2. How Secure is Quantum Cryptography?
Quantum cryptography, built upon the laws of quantum mechanics, presents an alluring prospect of achieving unparalleled data security. By utilizing quantum entanglement and the properties of quantum
particles, researchers at institutions like the Institute for Quantum Computing are making strides in quantum key distribution (QKD). This method establishes secure communication channels, impervious
to the threats that classical encryption faces. In a post-quantum era, quantum cryptographic solutions hold the promise of revolutionizing data protection and maintaining confidentiality even against
powerful adversaries armed with quantum computers.
3. Will Quantum Computing Replace Classical Computing?
Rather than replacing classical computing outright, the era of quantum computing is expected to see the coexistence of quantum and classical paradigms. While universal quantum computers, including
superconducting quantum computers developed by companies like IBM and Google, excel at solving complex problems with quantum speedups, classical computers remain proficient in straightforward
calculations. This points toward an approach where both quantum and classical computing work in tandem to address a wide array of challenges, employing quantum speedups and classical precision as
4. What Challenges Does Quantum Computing Face?
The journey towards large-scale quantum computation faces a series of significant challenges. Quantum decoherence, resulting from interactions with the environment, poses a threat to the delicate
quantum behavior necessary for accurate computations. To overcome this, scientists and engineers at organizations like IBM Quantum are investing in error correction techniques to enhance the
stability of quantum circuits and qubits. Additionally, the scalability of quantum computers, as seen with superconducting qubits, necessitates innovative solutions to maintain coherence and control
as the number of quantum bits increases.
5. What Ethical Considerations are Associated with Quantum Computing?
As quantum computing technologies evolve, ethical considerations come to the forefront. The unparalleled computational power of quantum computers could potentially breach current encryption methods,
leading to privacy concerns. Both IBM Quantum and the wider scientific community must ensure that the development of quantum computers adheres to ethical standards, avoiding scenarios where sensitive
data becomes vulnerable. Collaborative efforts among researchers, policymakers, and industry stakeholders are crucial to address these challenges responsibly and ensure equitable access to the
benefits of quantum computing across different sectors, including pharmaceutical companies exploring quantum sensing for advanced drug development. | {"url":"https://www.networkpoppins.com/blog/top-12-applications-of-quantum-computing","timestamp":"2024-11-15T03:04:47Z","content_type":"text/html","content_length":"34737","record_id":"<urn:uuid:6879de74-ee45-4e41-9d1a-9df9f4c4a84a>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00475.warc.gz"} |
Stability and feasibility of state-constrained linear MPC without stabilizing terminal constraints
Title data
Boccia, Andrea ; Grüne, Lars ; Worthmann, Karl:
Stability and feasibility of state-constrained linear MPC without stabilizing terminal constraints.
In: MTNS 2014 : Proceedings on the 21st International Symposium on Mathematical Theory of Networks and Systems, July 7-11, 2014, University of Groningen. - Groningen , 2014 . - pp. 453-460
ISBN 978-90-367-6321-9
This is the latest version of this item.
Project information
Project title: Project's official title
Project's id
Marie-Curie Initial Training Network "Sensitivity Analysis for Deterministic Controller Design" (SADCO)
DFG Grant
Project financing: 7. Forschungsrahmenprogramm für Forschung, technologische Entwicklung und Demonstration der Europäischen Union
Deutsche Forschungsgemeinschaft
Abstract in another language
This paper is concerned with stability and recursive feasibility of constrained linear receding horizon control schemes without terminal constraints and costs. Particular attention is paid to
characterize the basin of attraction S of the asymptotically stable equilibrium. For stabilizable linear systems with quadratic costs and convex constraints we show that any compact subset of the
interior of the viability kernel is contained in S for sufficiently large optimization horizon N. An analysis at the boundary of the viability kernel provides a connection between the growth of the
infinite horizon optimal value function and stationarity of the feasible sets. Several examples are provided which illustrate the results obtained.
Further data
Available Versions of this Item | {"url":"https://eref.uni-bayreuth.de/id/eprint/8419/","timestamp":"2024-11-09T13:03:19Z","content_type":"application/xhtml+xml","content_length":"26578","record_id":"<urn:uuid:ebd38ab5-6e34-413e-801a-d43a4562b4c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00591.warc.gz"} |
Does 4 cups of powdered sugar equal a pound?
Four cups will be roughly equivalent to one pound when measuring powdered sugar.
How many cups are in a 2lb bag of powdered sugar?
7½ cups
The usual 32-ounce package (2 pounds) of powdered sugar has about 7½ cups of powdered sugar.
How many pounds of sugar is 4 cups?
One pound of powdered sugar contains approximately 4 cups. This is the usual size and amount in one box of powdered sugar.
How many cups are in a 4 pound bag of powdered sugar?
9 cups
Domino® Sugar Package Requirements (approximate)
4 lbs = 9 cups
Domino® Powdered Sugar
1 lb = 3 3/4 cups*
2 lbs = 7 1/2 cups*
Is 3 cups of powdered sugar a pound?
How many US cups of powdered sugar are in 1 pound? The answer is: The change of 1 lb ( pound ) unit in a powdered sugar measure equals = into 3.63 cup us ( US cup ) as per the equivalent measure and
for the same powdered sugar type.
How many cups are in a dry pound?
Dry Ingredients:
Whole Wheat Flour 3 1/2 cups = 1 pound. White All-Purpose/Bread Flour (sifted) 4 cups = 1 pound. White All-Purpose/Bread Flour (unsifted) 3 1/2 cups = 1 pound. White Cake/Pastry Flour (sifted) 4 1/2
cups = 1 pound.
How many cups of powdered sugar is in a pound?
3 1/2 cups
Powdered sugar right out of the box or the plastic bag weighs 4 1/2 ounces per cup, so a 1-pound box (or 16 ounces) contains about 3 1/2 cups of powdered sugar. If a recipe calls for sifted powdered
sugar, weigh out 4 ounces of sifted powdered sugar to equal 1 dry measuring cup.
How many cups of powdered sugar are in a pound?
Powdered sugar right out of the box or the plastic bag weighs 4 1/2 ounces per cup, so a 1-pound box (or 16 ounces) contains about 3 1/2 cups of powdered sugar.
How many cups is in a 16oz box of powdered sugar?
How do you measure powdered sugar?
Powdered sugar, also known as confectioner’s sugar, is measured the same way you measure flour: spoon and level.
1. Spoon the powdered sugar from the package to your dry measuring cup.
2. Use a straight edge, like a dinner knife, to level off the top of the sugar so it’s even with the top of the cup.
How much does 1 cup of confectioners sugar weigh?
One US cup of powdered sugar converted to gram equals to 125.00 g.
How many cups are in a pound of powdered sugar?
How many pounds is four cups?
Cooking Ingredient: A density is required for converting between cups and pounds (a cup of sugar weighs less than a cup of water).
Pounds and cups for all purpose flour.
Pounds to cups Cups to pounds
4 lb = 14.5 cups 4 cups = 1.1 pounds
5 lb = 18.12 cups 5 cups = 1.38 pounds
How many ounces is 6 cups of powdered sugar?
Sugar Weight to Volume Conversion Table
Ounces Cups (Granulated) Cups (Powdered)
4 oz 1/2 c 3/4 c
5 oz 3/4 c 1 1/8 c
6 oz 3/4 c 1 1/3 c
7 oz 3/4 c 1 2/3 c
How do I measure 3 cups of powdered sugar?
Granulated and powdered sugar should be spooned into a dry measuring cup and leveled off with a straight edge. Test Kitchen Tip: Be sure to stir the sugar first to remove any clumps. If there are a
lot of lumps in your powdered sugar, you can pass it through a sifter or sieve before measuring.
How many dry cups are in a pound?
Pounds and cups for granulated sugar
Pounds to cups Cups to pounds
1/2 lb = 1.13 cups 1/2 cup = 0.22 pounds
1 lb = 2.26 cups 1 cup = 0.44 pounds
2 lb = 4.52 cups 2 cups = 0.89 pounds
3 lb = 6.77 cups 3 cups = 1.33 pounds
How many cups is in a pound?
two cups
One pound equals two cups.
Does 2 cups equal 1 pound?
16 ounces equals one pound or two cups. Another way to look at the equivalent is that one cup weighs eight ounces and therefore two cups equal 16 ounces and this is the same weight of one pound–16 | {"url":"https://www.trentonsocial.com/does-4-cups-of-powdered-sugar-equal-a-pound/","timestamp":"2024-11-14T10:17:53Z","content_type":"text/html","content_length":"60176","record_id":"<urn:uuid:37b49b89-a66a-451d-b774-0667ec33df39>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00551.warc.gz"} |
• Hint: If you are getting scores of 0 regardless of what data the user types, you may have a problem with integer division. Use caution with types int and double, type-casting, and try to avoid
integer division problems.
• Hint: Use Math.Max and Math.Min to constrain numbers to within a particular bound.
Need help?
Stuck on an exercise? Contact your TA or instructor. | {"url":"https://codestepbystep.com/problem/view/vb/parameters/Grades","timestamp":"2024-11-12T00:53:11Z","content_type":"text/html","content_length":"18365","record_id":"<urn:uuid:ffb027e3-3baa-4718-b717-ad8cb85e0cd8>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00720.warc.gz"} |
690 meters per second to kilometers per minute
Speed Converter - Meters per second to kilometers per minute - 690 kilometers per minute to meters per second
This conversion of 690 meters per second to kilometers per minute has been calculated by multiplying 690 meters per second by 0.0600 and the result is 41.4000 kilometers per minute. | {"url":"https://unitconverter.io/meters-per-second/kilometers-per-minute/690","timestamp":"2024-11-05T12:28:04Z","content_type":"text/html","content_length":"15819","record_id":"<urn:uuid:5c364d8f-f799-4e50-9923-0e82d0f1dd35>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00321.warc.gz"} |
# -------------------------------------------------------------------
# Lecture 2: Stationarity testing, PPP example from Enders, 2004
# Libraries: tseries, zoo
setwd("~/Desktop/Cam/MFin Lectures Lent/Lecture2/Example")
# -------------------------------------------------------------------
ppp <- read.csv("http://klein.uk/R/Lent/RealExchangeRates_US_UK.csv")
names(ppp) <- c("Date","RER")
ppp$Date <- as.yearmon(ppp$Date, format="%YM%m")
# In other words if the PPP holds we should find out that the real exchange rate is stationary.
# Let’s test this theory employing the Augmented Dickey-Fuller test.
# Additionally note that the theory implies that our test can include only a constant (and not a trend).
# This is the real exchange rate for UK, defined as US prices in £ the numerator and
# UK prices in the denominator
plot(RER ~ Date, type="l", data=ppp)
# R's default adf:
# does not allow specification of constant/trend in the test!
# Alternatives:
# Augmented Dickey Fuller test (correct p-value)
adf.test.1(x=ppp$RER, int=T, trend=F)
# The arguments of the function are as in the original adf.test function, i.e.
# x = a numeric vector or time series.
# k = the lag order to calculate the test statistic
# defaults to (n-1)^(1/3)
# In addition, we have
# int = logical, a constant is included if int=T
# trend = logical, a trend variable is included if trend=T
# Augmented Dickey Fuller test (summary table)
summary( adf.test.2(x=ppp$RER, int=T, trend=F) )
# The arguments of the function are:
# x = time series vector
# k = number of lags to be included in the test
# int = logical, a constant is included if int=T
# trend = logical, a trend variable is included if trend=T
# The null hypothesis that the RER series has a unit root cannot be rejected
# (p-value 0.46)
## --- A procedure to test for unit roots ---
# Walter Enders (2004) Applied Econometric time series, page 213 ff
# Step 1: Start with trend and drift model (least restrictive)
adf.test.1(x=ppp$RER, int=T, trend=T, k=0)
# Step 2: If null is NOT rejected, check
# where to many deterministic regressors included in step 1?
# test for significance of the of the trend term by joint hypothesis
lm1 <- adf.test.2(x=ppp$RER, int=T, trend=T, k=0)
# wrong p-value from F-distribution:
linearHypothesis(lm1, c("xD.lag(x, -1)", "xtime(x)"))
# correct p-value from DF-table
linearHypothesis.adf(lm1, c("xD.lag(x, -1)", "xtime(x)"), int=T, trend=T)
# We can not reject H0. -> Coeffient of time-trend is zero
# Step 3: If null is not rejected, estimate model without trend
adf.test.1(x=ppp$RER, int=T, trend=F, k=0)
# null hypothesis of unit root is not rejected
# -> test for joint significance of constant and regressor
lm2 <- adf.test.2(x=ppp$RER, int=T, trend=F, k=0)
linearHypothesis.adf(lm2, c("(Intercept)", "x"), int=T, trend=F)
# drift is not significant
# Step 4: If null is not rejected, estimate a model without drift and trend
adf.test.1(x=ppp$RER, int=F, trend=F, k=0)
# -> null is not rejected
# -> series contains a unit root | {"url":"https://klein.uk/teaching/econometrics2/docs/Lec2.html","timestamp":"2024-11-06T07:51:01Z","content_type":"text/html","content_length":"14744","record_id":"<urn:uuid:5cb2bcc1-6dfe-4d4b-a205-63a526b92aba>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00237.warc.gz"} |
USACO 2015 December Contest, Gold
Problem 2. Fruit Feast
Contest has ended.
Log in to allow submissions in analysis mode
Bessie has broken into Farmer John's house again! She has discovered a pile of lemons and a pile of oranges in the kitchen (effectively an unlimited number of each), and she is determined to eat as
much as possible.
Bessie has a maximum fullness of $T$ ($1 \le T \le 5,000,000$). Eating an orange increases her fullness by $A$, and eating a lemon increases her fullness by $B$ ($1 \le A, B \le T$). Additionally, if
she wants, Bessie can drink water at most one time, which will instantly decrease her fullness by half (and will round down).
Help Bessie determine the maximum fullness she can achieve!
INPUT FORMAT (file feast.in):
The first (and only) line has three integers $T$, $A$, and $B$.
OUTPUT FORMAT (file feast.out):
A single integer, representing the maximum fullness Bessie can achieve.
Problem credits: Nathan Pinsker
Contest has ended. No further submissions allowed. | {"url":"https://usaco.org/index.php?page=viewproblem2&cpid=574","timestamp":"2024-11-07T10:02:28Z","content_type":"text/html","content_length":"8043","record_id":"<urn:uuid:467638a9-b3b4-4c3a-b701-2b4eafbf86fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00286.warc.gz"} |
Deep Computation in Statistical Physics
Deep Computation in Statistical Physics Workshop
organized by Cris Moore (SFI), Jon Machta (University of Massachusetts, Amherst & SFI), and Stephan Mertens (Magdeburg & SFI)
Santa Fe Institute
August 8-10, 2013
Computational physics was born in 1953, when Metropolis, Metropolis, Rosenbluth, Teller, and Teller at Los Alamos introduced the idea of Monte Carlo simulations to study the properties of fluids. The
same year, Enrico Fermi, John R. Pasta, and Stanislaw Ulam used the MANIAC-I computer at Los Alamos to simulate a system of nonlinearly coupled oscillators. Sparked by these two seminal papers,
simulations have become the third source of knowledge besides the much older sources theory and experiment.
Most simulations "put the physics on a computer" by mimicking physical processes in a straightforward way. We flip single spins in magnets or trace individual molecules on their way to the next
collision. This is useful and natural, yet it does not deploy the full power of the algorithmic approach to physics. Complex systems can be investigated more deeply by advanced and innovative
algorithms that do not try to imitate physics, but try to probe physical problems using a deep mathematical understanding of the system—that skip over much of the intermediate behavior and directly
access the long-time or large-scale behavior of the system. Examples of this kind of "deep computation" include algorithms that compute minimum spanning trees for percolation systems, or Monte Carlo
simulations that use non-local, unphysical updates to sample from the equilibrium distribution, or find a system’s ground state, much faster than the natural physical dynamics would.
Not only do these algorithms provide new tools for studying complex systems, but the workings of these algorithms often yield fresh insights into the systems themselves. For example, the
Swendsen-Wang cluster algorithm depends on a subtle mapping between spin models and percolation models and gives a geometric interpretation to the ordering that occurs at the critical point of spin
systems. At a more fundamental level, understanding the best algorithms available for simulating a physical system sheds light on the natural complexity of the system itself.
In this workshop we will bring together physicists, computer scientists and practitioners from related fields to share knowledge and collaborate on the development, analysis and application of new
algorithms. We will particularly emphasize algorithms applied to systems studied in statistical physics because of their simple formulation and wide relevance to many other scientific fields. A
paradigmatic example is the Ising model and its many variants. Originally designed to model magnetic systems but now applied to fields as diverse as image analysis and opinion formation in social
systems, Ising systems are simply defined but display a rich array of complex behavior induced by many-body interactions. Because most complex systems studied in statistical physics are stochastic we
will focus mainly, though not exclusively, on Monte Carlo algorithms, i.e., algorithms that rely essentially on pseudorandomness for their operation.
The year 2013 is the 60th anniversary of the seminal paper of Metropolis et. al. that initiated the simulation age, and made possible the quantitative understanding of complex systems. Moreover,
powerful new approaches have been developed in the past decade that are not widely disseminated between fields, and whose potential has yet to be fully developed. For instance, simulated annealing is
widely known, but more advanced techniques like parallel tempering are just starting to be applied outside statistical physics. Similarly, lifting Markov chains by giving the state a kind of momentum
can make them much faster; but it also makes them irreversible, putting them outside the reach of many standard tools for analyzing their equilibration times.
The time is ripe to bring together physicists, computer scientists, and others in complex systems to accelerate the development and understanding, both experimental and rigorous, of these new
algorithmic approaches. And by doing this at SFI, we hope to establish that SFI is one of the institutional owners of advanced algorithms and mathematically powerful techniques. | {"url":"https://wiki.santafe.edu/index.php?title=Deep_Computation_in_Statistical_Physics&oldid=50706","timestamp":"2024-11-09T11:09:43Z","content_type":"text/html","content_length":"23394","record_id":"<urn:uuid:9e77d5ef-876e-4280-9ba1-9525755d550b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00800.warc.gz"} |
[Newbie] Error involving curly braces
Hi. I had been trying out an example for the CPDT book by Adam Chlipala but ran into an error that I couldn't understand. Could someone help me figure it out?
Inductive type : Set := Nat | Bool.
Inductive tbinop : type -> type -> type -> Set :=
| TPlus : tbinop Nat Nat Nat
| TTimes : tbinop Nat Nat Nat
| TEq: forall t, tbinop t t Bool
| TLt: tbinop Nat Nat Bool.
Inductive texp : type -> Set :=
| TNConst : nat -> texp Nat
| TBConst : bool -> texp Bool
| TBinop t1 t2 t : tbinop t1 t2 t -> texp t1 -> texp t2 -> texp t.
Definition typeDenote (t :type) : Set :=
match t with
| Nat => nat
| Bool => bool
Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
: typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
match b with
| TPlus => plus
(* ERROR:
In environment
arg1 : type
arg2 : type
res : type
b : tbinop arg1 arg2 res
The term "Nat.add" has type "nat -> nat -> nat"
while it is expected to have type
"typeDenote ?t@{a1:=Nat} ->
typeDenote ?t0@{a2:=Nat} -> typeDenote ?t1@{r0:=Nat}".
The ? means an implicit argument, right? What does the curly brackets mean? And why were more variable names like t0 and t1 created?
No, the question mark does not necessarily mean implicit. It just denotes an unknown term. And what is between brackets is what this unknown term is allowed to use. For example, ?t@{a1:=Nat} could be
as simple as just a1 (or Nat), but it could not be a2, since a2 does not occur between brackets. As for t0 and t1, those are presumably the names used by Coq to denote the arguments of the type of b.
Indeed, arg1, arg2, and res are not constant over all the branches of match b, they depend on the actual constructor. For instance, with TPlus, res is Nat, but with TLt, res is Bool. So, Coq provides
a new variables to encompass all these constants. (In that simple case, it could just have been reusing res rather than creating a new variable.)
I would like to add something,
working example of your code:
Require Import Bool Arith.
Inductive type : Set := Nat | Bool.
Inductive tbinop : type -> type -> type -> Set :=
| TPlus : tbinop Nat Nat Nat
| TTimes : tbinop Nat Nat Nat
| TEq: forall t, tbinop t t Bool
| TLt: tbinop Nat Nat Bool.
Inductive texp : type -> Set :=
| TNConst : nat -> texp Nat
| TBConst : bool -> texp Bool
| TBinop t1 t2 t : tbinop t1 t2 t -> texp t1 -> texp t2 -> texp t.
Definition typeDenote (t :type) : Set :=
match t with
| Nat => nat
| Bool => bool
Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
: typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
match b with
| TPlus => plus
| TTimes => mult
| TEq Nat => beq_nat
| TEq Bool => eqb
| TLt => leb
Looks like coq is not able to build term in Definition tbinopDenote, because term is not finished.
@Guillaume Melquiond So does ?t@{a1:=Nat} mean that t is allowed to be either a1 or Nat? Or is it that only a1 but where a1 is of type Nat?
What if other values are also possible? How would coq show that as? Something like ?t@{a1,a2:=Nat} if a1 and a2 are possible values?
@Natasha Klaus
coq not able to build term in Definition tbinopDenote, because term is not finished
Oh.. since stuff like TEq and TLt were mentioned, those too need to be handled at tbinopDenote. That's it, right?
The error that I started out with was this:
Require Import Arith. (* beq_nat *)
Require Import Bool. (* eqb *)
Inductive type : Set := Nat | Bool.
Inductive tbinop : type -> type -> type -> Set :=
| TPlus : tbinop Nat Nat Nat
| TTimes : tbinop Nat Nat Nat
| TEq: forall t, tbinop t t Bool
| TLt: tbinop Nat Nat Bool.
Inductive texp : type -> Set :=
| TNConst : nat -> texp Nat
| TBConst : bool -> texp Bool
| TBinop t1 t2 t : tbinop t1 t2 t -> texp t1 -> texp t2 -> texp t.
Definition typeDenote (t :type) : Set :=
match t with
| Nat => nat
| Bool => bool
Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
: typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
match b with
| TPlus => plus
| TTimes => mult
| TEq Nat => beq_nat
| TEq Bool => eqb
| TLt => le
In environment
arg1 : type
arg2 : type
res : type
b : tbinop arg1 arg2 res
The term "Init.Nat.add" has type "nat -> nat -> nat"
while it is expected to have type
"typeDenote ?t@{a1:=Nat} ->
typeDenote ?t0@{a2:=Nat} -> typeDenote ?t1@{r0:=Nat}".
Couldn't figure why the error was showing up there.
@Ju-sh your original error is due to a type mismatch on the last line TLt => le, you can see it if you are explicit about the dependent pattern match:
Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
: typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
match b in tbinop arg1 arg2 res
return typeDenote arg1 -> typeDenote arg2 -> typeDenote res with
| TPlus => plus
| TTimes => mult
| TEq Nat => beq_nat
| TEq Bool => eqb
| TLt => le
In environment
arg1 : type
arg2 : type
res : type
b : tbinop arg1 arg2 res
The term "le" has type "nat -> nat -> Prop" while it is expected to have type
"typeDenote Nat -> typeDenote Nat -> typeDenote Bool" (cannot unify
"Prop" and "typeDenote Bool").
And coq does not give you the right error because it rightfully fails to infer a type for the pattern match with this type error
Ju-sh said:
So does ?t@{a1:=Nat} mean that t is allowed to be either a1 or Nat? Or is it that only a1 but where a1 is of type Nat?
What if other values are also possible? How would coq show that as? Something like ?t@{a1,a2:=Nat} if a1 and a2 are possible values?
No, it means that t should be an arbitrary term in a context a1 : type (so it can only use that variable), and moreover in that branch a1 is defined to be Nat. If Coq was able to assign t := a1, t0 :
= a2 and t1 := r0 then everything would be fine for that branch because nat -> nat -> nat and typeDenote Nat -> typeDenote Nat -> typeDenote Nat would end up convertible. But it cannot do this
assignment because of a type error in the following branches of the match (and Coq does not "know" where this failure comes from so it just bails out with the first branch mismatching).
Thanks @Kenji Maillard, I got a better idea now.
But I ran into new problems with the same example.
Require Import Bool Arith List.
(*Set Implicit Arguments.*)
(*Set Asymmetric Patterns.*)
Inductive type : Set := Nat | Bool.
Inductive tbinop : type -> type -> type -> Set :=
| TPlus : tbinop Nat Nat Nat
| TTimes : tbinop Nat Nat Nat
| TEq : forall t, tbinop t t Bool
| TLt : tbinop Nat Nat Bool.
Inductive texp : type -> Set :=
| TNConst : nat -> texp Nat
| TBConst : bool -> texp Bool
| TBinop : forall t1 t2 t, tbinop t1 t2 t -> texp t1 -> texp t2 -> texp t.
Definition typeDenote (t : type) : Set :=
match t with
| Nat => nat
| Bool => bool
Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
: typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
match b with
| TPlus => plus
| TTimes => mult
| TEq Nat => beq_nat
| TEq Bool => eqb
| TLt => leb
Fixpoint texpDenote t (e : texp t) : typeDenote t :=
match e with
(* Doubt1: How does [Set Implicit Arguments] affect the following line? *)
| TNConst n => n
| TBConst b => b
(* Doubt2: How does [Set Asymmetric Patterns] affect the following line? *)
| TBinop _ _ _ b e1 e2 => (tbinopDenote b) (texpDenote e1) (texpDenote e2)
If I remove the Set Implicit Arguments, the definition of texpDenote would complain:
In environment
texpDenote : forall t : type, texp t -> typeDenote t
t : type
e : texp t
n : nat
The term "n" has type "nat" while it is expected to have type
"typeDenote ?t@{t1:=Nat}".
But typeDenote Nat would give nat itself, right? And n is nat. So why is this causing an error?
But typeDenote Nat would give nat itself
the expected type is not typeDenote Nat so that doesn't matter
you need to do something like match e in texp t' return typeDenote t' with ...
Asymmetric Patterns only matters when there are parameters to an inductive type which is not the case here
Implicit Arguments should not affect TNCons/TBConst, but does affect TBinop
Again, Coq is misleading you here. It tries to infer the return clause of your pattern-matching, but fails to do so because of an unrelated error. If you do the modification @Gaëtan Gilbert
suggested, the error message is much better: The term "b" has type "tbinop t0 t1 t2" while it is expected to have type "type".. This tells you that when deactivating implicit arguments, you need to
pass more arguments to tbinopDenote, namely those that are not implicit. A working last line would be
| TBinop _ _ _ b e1 e2 => (tbinopDenote _ _ _ b) (texpDenote _ e1) (texpDenote _ e2)
and once you’ve got it right, Coq can infer the correct return type and you can drop the return clause of your match.
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237977-Coq-users/topic/.5BNewbie.5D.20Error.20involving.20curly.20braces.html","timestamp":"2024-11-04T08:45:41Z","content_type":"text/html","content_length":"38242","record_id":"<urn:uuid:e56ce1fc-74e0-4ba3-929a-ffacb24df73b>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00267.warc.gz"} |
million km to km converter
Units of length/distance in the metric scale are based on fractions or multiples of one. Our full terms & conditions can be found by clicking here. The following is a list of definitions relating to
conversions between square kilometers and acres. 8 au = 1196782965.6 km. Our online converter above will be of great assistance, as it is easy to use, has a quick copy button to … The prefix kilo,
abbreviated "k", indicates one thousand.1 km = 1000 m. 1 Light Year: 7 au = 1047185094.9 km. You can find metric conversion tables for SI units, as well Then kilometer value is multiplied with factor
to convert into miles. area, mass, pressure, and other types. ›› More information from the unit converter. The distance d in miles (mi) is equal to the distance d in kilometers (km) divided by
1.609344: d (mi) = d (km) / 1.609344 . Definition: A kilometer (symbol: km) is a unit of length in the International System of Units (SI). The SI base unit for length is the metre. To convert any
value in kilometers per second to kilometers per hour, just multiply the value in kilometers per second by the conversion factor 3600.So, 65 kilometers per second times 3600 is equal to 2.34 × 10 5
kilometers per hour. 1 Meter = 0.001 Kilometer. For example, to convert 100 m to km, multiply 100 by 0.001, that makes 0.1 km is 100m . Primary exceptions are the United Kingdom and the United States
of America, where the mile remains as standard. One kilometer is equivalent to 0.6214 miles. to use the unit converter. Example. Use the search box to find your required metric converter. What is a
square kilometer? The meter is defined as the length of the path traveled by light in vacuum during a time interval with a duration of 1 / 299,792,458 of a second, according to the most recent 2019
definition. Meters. How many million kilometers in 1 meter? meters to kilometers formula. Kilometers conversion calculators, tables and formulas to automatically convert from other length units. 1
million kilometers to kilometre = 1000000 kilometre. Easily convert miles to km (kilometer) with this free online conversion tool. The prefix kilo, abbreviated "k", indicates one thousand.1 km = 1000
m. 1 Step: The distance covered by a single step assuming a stride length of 0.762 meters or 2.5 feet. This is a measurement of speed typically used in countries using the metric system for
transport. million kilometers to em The kilometre is most commonly used on road signs to denote the distance left to travel to a given location. 6 au = 897587224.2 km. The metric, or decimal, system
of weights and measures was adopted in France in 1795. You can view more details on each measurement unit: One of the users made a request for the calculator /2363/. Example. 15 Kilometers per Liter
= 6.67 Liters per 100 km About. 5 au = 747989353.5 km. The square kilometer (abbreviation: sq km or Sq km or km 2, plural form: square kilometers) is a derived metric unit of area used in SI system
(International System of Units, Metric System). Type in unit Common Converters. <70 ppm CO is unlikely to make you sick whereas 150-200 ppm will make you sick and might kill you. million kilometers
to terameter How many Kilometers equal one Mile? Diferent length units conversion from decameter to kilometers. kilometre There are 0.00404685642 square kilometers in an acre. One square kilometer =
247.105381 acres = 1 195 990.05 square yards = 1 000 000 square meters Just enter the number and select the unit to view its equal value in the other units. 10 au = 1495978707 km. How many au in a
km? 4 au = 598391482.8 km. 2 au = 299195741.4 km. Use this calculator to convert between these two different ways of measuring fuel economy. A kilometre (American spelling: kilometer, symbol: km) is
a unit of length equal to 1000 metres (from the Greek words khilia = thousand and metro = count/measure). ConvertUnits.com provides an online Convert 20 kilometers to miles: d (mi) = 20km / 1.609344
= 12.4274mi. inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, Convert 1 dam - dkm into kilometer and decameters to km. A quick online length calculator to convert Miles(mi) to Kilometers
(km). million kilometers to royal foot kilometre to million kilometers, or enter any two units below: million kilometers to cape inch Length Weight Volume Temperature Area Pressure Energy Power … The
peak of Mount Everest is 8.848Km above sea level. million kilometers to micrometer One kilometer is therefore one thousand meters. Los Angeles and New York City is 2789.5 miles in United States, how
long is it in kilometers ? Definition: A kilometer (symbol: km) is a unit of length in the International System of Units (SI). kilometer = meter * 0.001. kilometer = meter / 1000. You are currently
converting speed units from kilometers per second to miles per hour 1 kps = 2236.9362920544 mph Nautical Miles : The nautical mile (symbol M, NM or nmi) is a unit of length, defined as 1,852 meters
(approximately 6,076 feet). million kilometers or The number converter performs the conversion between the Indian number system and … When trying to find the correct conversion using a formula, we
must use the more precise conversion factor, which is … It is approximately equal to 0.621 miles, 1094 yards or 3281 feet. Unit Descriptions; 1 Kilometer: 1 Kilometer is equal to 1000 meters. One
kilometer is equivalent to 0.6214 miles. From. Meter (metre) is a metric system base length unit. Kilometers : The kilometer (SI symbol: km) is a unit of length in the metric system, equal to 1000m
(also written as 1E+3m). The world’s tallest building, Burk Khalifa in Dubai, is 0.82984km tall. Miles to km is miles to kilometers length converter. History/origin: The prefix kilo- is a metric
prefix indicating one thousand. A simple online currency numbering system converter which is used to convert numbers to millions, billions, trillions, thousands, lakhs and crores. Paris in France is
878Km from Berlin in Germany, although you would have to make a journey of over 1050Km to travel from one to the other by land transport. When I was pondering about it I found the conversation on one
of the forums that the transport operator put the fare per kilometer (excluding cargo), and for accounting purposes you have to calculate the cost of the tonne-kilometers. Quick conversion chart of
million kilometers to kilometre. 9 au = 1346380836.3 km. Fuel Economy. 10 miles = 16.1 km = 8.7 nautical miles; 20 miles = 32.2 km = 17.4 nautical miles d (km) = d (mi) × 1.609344 . million
kilometers to tu Unit Descriptions; 1 Kilometer: 1 Kilometer is equal to 1000 meters. ppm is a much more comparable number than g/km e.g. Note that rounding errors may occur, so always check the
results. Kilometer. However, some countries use kilometers per liter. The symbol for square kilometer is km 2. as English units, currency, and other data. Kilometer. The distance d in kilometers (km)
is equal to the distance d in miles (mi) times 1.609344: The kilometre is used worldwide as a unit used for expressing distances between geographical locations on land, and in most countries is the
official unit for this purpose. Most of the world measures fuel economy in liters per 100 kilometers. Enter a number to convert Astronomical Units to Kilometers. Road speed limits are given in
kilometers per hour which is abbreviated as kph or km/h. One kilometer is therefore one thousand meters. Use this handy calculator below if you want to convert any measurements in miles to
kilometers. Task: Convert 8 square kilometers to hectares (show work) Formula: km 2 x 100 = hectares Calculations: 8 km 2 x 100 = 800 hectares Result: 8 km 2 is equal to 800 hectares Conversion Table
For quick reference purposes, below is a conversion table that you can use to convert from km 2 … Meters is the SI base unit of length. Kilometers can be abbreviated as km; for example, 1 kilometer
can be written as 1 km. To do a rough estimate of the conversion in your head, all you really need to remember is that a mile equals about 1.6 kilometers (or that a kilometer is approximately 2/3 of
a mile). The other way around, how many kilometers - km are in one decameter - dam - dkm unit? Whilst every effort has been made to ensure the accuracy of the metric calculators and charts given on
this site, we cannot make a guarantee or be held responsible for any errors that have been made. The kilometre is unit of length in the metric system equivalent to one thousand metres. 1 au =
149597870.7 km. A kilometer, or kilometre, is a unit of length equal to 1,000 meters, or about 0.621 miles.In most of the world, it is the most common unit for measuring distance between places. ››
Quick conversion chart of million kilometers to kilometre. conversion calculator for all types of measurement units. 1 metre is equal to 1.0E-9 million kilometers, or 0.001 kilometre. When the number
gets bigger it becomes difficult to convert … Convert miles to kilometers. Please enable Javascript Convert 20 miles to kilometers: Therefore, a square kilometer can be expressed as one million
square meters. How to convert miles to kilometers. ... 1 km = 1000m (metres) ... A megameter = 1 million metres (or 10,000Km) A gigametre = 1 billion metres (or 1,000,000Km) Metric Conversion; The
distance d in kilometers (km) is equal to the distance d in miles (mi) times 1.609344:. What is a square kilometer (km 2)? The symbol is "m". This unit of distance and area can be used within
landscaping, surveying and aerial photography. History/origin: The prefix kilo- is a metric prefix indicating one thousand. km is an abbreviation for kilometer (1 thousand meters). ... (American
spelling: kilometer, symbol: km) is a unit of length equal to 1000 metres (from the Greek words khilia = thousand and metro = count/measure). Calculate from length into other length unit measures.
Kilometers to International Nautical Miles. How to convert kilometers to miles. This unit conversion is usualy done with the help of a calculator. 2789.5 miles = 2789.5 × 1.6093440006 km =
4489.2650896737 km Answer : approximately 4489 km Conversion table of miles, km and nautical miles . million kilometers to yoctometer This site is owned and maintained by Wight Hat Ltd. ©2003-2020.
Miles to Km converter. We assume you are converting between million kilometre and metre. The answer is 1.0E-6. million kilometers to braza UnitConverter.net UnitConverter.net V1.2. It is also the
most popular unit for describing the distance between two locations in a straight line (across the surface of the Earth). million kilometers to seemeile Exactly 1.609344 kilometers equal one mile,
and this follows from the definition of a mile as 5280 feet, of a feet as 0.3044 meters, and of a kilometer as 1,000 meters and making the necessary mathematical operations.These are internationally
agreed upon unit definitions established by treaty in 1959. 1 kilometer is equal to 0.62137119 miles: 1 km = (1/1.609344) mi = 0.62137119 mi. If you spot an error on this site, we would be grateful
if you could report it to us by using the contact link at the top of this page and we will endeavour to correct it as soon as possible. 1 million kilometers to kilometre = 1000000 kilometre, 2
million kilometers to kilometre = 2000000 kilometre, 3 million kilometers to kilometre = 3000000 kilometre, 4 million kilometers to kilometre = 4000000 kilometre, 5 million kilometers to kilometre =
5000000 kilometre, 6 million kilometers to kilometre = 6000000 kilometre, 7 million kilometers to kilometre = 7000000 kilometre, 8 million kilometers to kilometre = 8000000 kilometre, 9 million
kilometers to kilometre = 9000000 kilometre, 10 million kilometers to kilometre = 10000000 kilometre. Use this page to learn how to convert between million kilometres and kilometres. Convert
Kilometers per Liter to Liters/100 km. million kilometers to zettameter. The answer is 1.0E-9. Between dam - dkm and km measurements conversion chart page. 3 au = 448793612.1 km. To convert from
kilometers to miles, multiply your figure by 0.62137119223733 (or divide by 1.609344) . Meters is the SI base unit of length. Million Billion Calculator is a number & currency conversion tool to
perform the conversion between trillions, billions, millions, crores, lacs, thousands & hundreds. One kilometer (or 'kilometre' with British spelling) is equivalent to 1,000 meters. Here, the
kilometers variable is used to store the kilometer value from the user. Plus learn how to convert Mi to Km It is approximately equal to 0.621 miles, 1094 yards or 3281 feet. What is a Meter? Type in
your own numbers in the form to convert the units! How to convert kilometers to miles. It converts units from miles to km or vice versa with a metric conversion table. 1 kilometer is equal to
0.62137119 miles: 1 km = (1/1.609344) mi = 0.62137119 mi. It is commonly used officially for expressing distances between geographical places on land in most of the world. We assume you are
converting between million kilometre and kilometre. metres squared, grams, moles, feet per second, and many more! How to convert miles to kilometers (Miles To KM) 1 mile is equal to 1.609344
kilometers: 1 mi = 1.609344 km. Using the metre as the basis for length measurements, the system is now used officially across the globe, with a few notable exceptions. The average distance from
Earth to the Moon is 384,400Km. You can view more details on each measurement unit: million kilometers or meter The SI base unit for length is the metre. A square kilometer is a unit of area in the
Metric System. To convert from kilometers to miles, divide the measurement in km by 1.609344 to get the equivalent in miles. 1 mile is equal to 1.609344 kilometers: 1 mi = 1.609344 km. To. You can do
the reverse unit conversion from 1 mile = 1.60934 km; 1 km = 0.62137 miles. Niagara Falls, on the U.S.A./Canada border, is approximately 1Km across. Examples include mm, Enter value in kilometers:
2.2 2.2 kilometers is equal to 1.3670162000000001 miles. How many million kilometers in 1 kilometre? Kilometers per hour. 1 m = 0.001 km. symbols, abbreviations, or full names for units of length,
However, there are metric measurements of length/distance greater than a kilometre that can be expressed in terms of kilometres. Geographical places on land in most of the users made a request for
the calculator /2363/ other data weights! Square kilometer ( symbol: km ) 1 mile = 1.60934 km ; 1 kilometer is equal million km to km converter 1.609344:... Road signs to denote the distance d in
miles ( mi ) times 1.609344: km are in one -... Difficult to convert mi to km ) is a list of definitions relating to conversions between square kilometers acres... Details on each measurement unit:
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and.! Angeles and New York City is 2789.5 miles = 2789.5 × 1.6093440006 =! As English units, currency, and other data is miles to km is miles to (... With a metric system base length unit occur, so
always check the.. Temperature area Pressure Energy Power … Please enable Javascript to use the search box find. Whereas 150-200 ppm will make you sick whereas 150-200 ppm will make you and...
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you. D in miles ( mi ) to kilometers Liters per 100 kilometers calculator to convert between these two different of... Officially for expressing distances between geographical places on land in most
of the users made a request the! That makes 0.1 km is 100m … Please enable Javascript to use the box! From Earth to the Moon is 384,400Km other types of miles, 1094 yards 3281. By 0.62137119223733 (
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square meters kilometers per Liter to Liters/100 km required metric converter ›› more information from the user one! The Moon is 384,400Km kilometer: 1 km = ( 1/1.609344 ) mi million km to km
converter. Factor to convert 100 m to km ) is a metric system base length unit system... The units kilometers variable million km to km converter used to store the kilometer value from unit. On the
U.S.A./Canada border, is approximately 1Km across therefore, a square is. Kilometre the SI base unit for length is the metre this calculator to convert units. Miles to kilometers length converter km
is 100m the International system of weights and measures was adopted in France 1795. - dam - dkm into kilometer and decameters to km or vice versa with a metric indicating. 0.001. kilometer = meter *
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Officially for expressing distances between geographical places on land in most of the users made a request for calculator! 0.62137119 mi easily convert miles to km unit Descriptions ; 1 km = ( 1/
1.609344 ) =! Wight Hat Ltd. ©2003-2020 online length calculator to convert mi to km ) is equivalent to 1,000 meters other. Distance d in miles ( mi ) times 1.609344: request for calculator! Variable
is used to store the kilometer value from the user × 1.6093440006 km = miles! Landscaping, surveying and aerial photography measuring fuel economy in Liters per 100 km.! You want to convert miles to
km or vice versa with a metric prefix one! Unit of length in the metric, or full names for units length/distance! Is equal to 0.62137119 miles: 1 kilometer is equal to 1000 meters the in. Convert 1
dam - dkm unit miles ( mi ) × 1.609344 Quick conversion chart page of definitions to... Weights and measures was adopted in France in 1795 on the U.S.A./Canada border, is approximately equal to
0.62137119:! Is 2789.5 miles in United States, how long is it in kilometers km!
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Fabian Hebestreit
Fabian Hebestreit
I am a professor of mathematics at Bielefeld University. I obtained my doctorate at the WWU Münster in 2014 and my habilitation at the RFWU Bonn in 2022, and was a lecturer at the University of
Aberdeen before coming to Bielefeld in 2024.
Stefan Bauer and I are organising the 34th NRW Topology Meeting in Bielefeld; it will take place on November 22nd and 23rd and the accompanying webpage is here.
Contact Information
Address: Universität Bielefeld
Fakultät für Mathematik
Universitätsstraße 25 2
33615 Bielefeld
Office: V4-211
Phone: +49 (0)521 106 5002
eMail: hebestreit@math.uni-bielefeld.de
Research interests
Algebraic & geometric topology, homotopical algebra and more specifically: The homotopy theory of diffeomorphism groups, manifold bundles and cobordism categories and their relation to arithmetic
groups and algebraic K- and L-theory; higher categories as foundations for the above.
I am a member of the TRR 358 'Integral structures in geometry and representation theory', joint between the Universities of Bielefeld and Paderborn. My workgroup currently has two further members:
During the winter term 24/25 I will teach Lineare Algebra II and Higher categories and algebraic K-theory II.
Some lecture notes on algebraic topology, higher categories and algebraic K-theory are here.
Publications and Preprints
Commutative algebra
• with P.Scholze: A note on higher almost ring theory.
arXiv: 2409.01940 (9 pages, 2024)
• with A.Krause & M.Ramzi: A note on quadratic forms.
Bulletin of the London Mathematical Society, vol. 56, no. 5 (2024), pp. 1803-1818
Algebraic & geometric topology
Category theory
Algebraic & hermitian K-theory
Other writing | {"url":"https://www.math.uni-bielefeld.de/~hebestreit/","timestamp":"2024-11-13T09:20:53Z","content_type":"text/html","content_length":"11671","record_id":"<urn:uuid:a3019bdc-d12b-4ca8-a7db-1e79c8cb71c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00680.warc.gz"} |
The ABS Function
The ABS function returns the absolute value of the argument.
The type of this function is according to the argument type as follows:
Argument Type Function Type
Integer Integer
Numeric Numeric
General Format
1. Argument-1 must be class numeric.
Returned Values
1. The returned value is as follows:
a. When the value of argument-1 is zero or positive, (argument-1)
b. When the value of argument-1 is negative, (-argument-1) | {"url":"https://www.microfocus.com/documentation/visual-cobol/30pu12/VC-DevHub/HRLHLHPDF701.html","timestamp":"2024-11-09T14:02:17Z","content_type":"text/html","content_length":"14067","record_id":"<urn:uuid:2b9ea675-8550-4902-85f5-3e4b5cd70620>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00173.warc.gz"} |
Objective Troy: A Terrorist, A President, And The Rise Of The Drone 2015
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Equations with different dimensions
Hello everybody.
I have some equations with different dimension and variable A is used in both of them. For example. The first equation has one dimension (A(i)). but the second equation has two dimension (i and j).
eq1(i)… A(i) =e= x(i)+y(i);
eq2(i,j)… B(i,j) =e= A(i,j)+c(i,j);
(if i=1:3, j=1,2 and A(i)=[6,7,8] then A(i,j)=[6,7,8;6,7,8] )
Actually, values of variable A are same for different values of the second dimension (j). The dimensions of variable A are different
in two equations, Then GAMS can’t solve the problem. How can I solve this problem?
I added another equation and solve this problem but I think It’s not a good idea for that. GAMS didn’t give me any error for my solution.
My solution:
eq1(i)… A(i) =e= x(i)+y(i);
eq2(i,j)… B(i,j) =e= A2(i,j)+c(i,j);
eq3(i)… A2(i,j) =e=A(i);
Thank you for your help.
Not sure if I really understood the problem but to me it seems that you should just drop the j index for A and should be good:
eq1(i).. A(i) =e= x(i)+y(i);
eq2(i,j).. B(i,j) =e= A(i)+c(i,j);
I hope this helps!
Hi Fred,
Thank you a lot. It’s so helpful for me.
Best wishes, | {"url":"https://forum.gams.com/t/equations-with-different-dimensions/2477","timestamp":"2024-11-02T10:36:44Z","content_type":"text/html","content_length":"17693","record_id":"<urn:uuid:c612ba07-fb31-4710-bdaf-d7dad5597787>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00129.warc.gz"} |
Practical Tuning Series - Tune a Preprocessing Pipeline
This is the second part of the practical tuning series. The other parts can be found here:
In this post, we build a simple preprocessing pipeline and tune it. For this, we are using the mlr3pipelines extension package. First, we start by imputing missing values in the Pima Indians Diabetes
data set. After that, we encode a factor column to numerical dummy columns in the data set. Next, we combine both preprocessing steps to a Graph and create a GraphLearner. Finally, nested resampling
is used to compare the performance of two imputation methods.
We load the mlr3verse package which pulls in the most important packages for this example.
We initialize the random number generator with a fixed seed for reproducibility, and decrease the verbosity of the logger to keep the output clearly represented. The lgr package is used for logging
in all mlr3 packages. The mlr3 logger prints the logging messages from the base package, whereas the bbotk logger is responsible for logging messages from the optimization packages (e.g. mlr3tuning
In this example, we use the Pima Indians Diabetes data set which is used to predict whether or not a patient has diabetes. The patients are characterized by 8 numeric features of which some have
missing values. We alter the data set by categorizing the feature pressure (blood pressure) into the categories "low", "mid", and "high".
# retrieve the task from mlr3
task = tsk("pima")
# create data frame with categorized pressure feature
data = task$data(cols = "pressure")
breaks = quantile(data$pressure, probs = c(0, 0.33, 0.66, 1), na.rm = TRUE)
data$pressure = cut(data$pressure, breaks, labels = c("low", "mid", "high"))
# overwrite the feature in the task
# generate a quick textual overview
Data summary
Name task$data()
Number of rows 768
Number of columns 9
Key NULL
Column type frequency:
factor 2
numeric 7
Group variables None
Variable type: factor
skim_variable n_missing complete_rate ordered n_unique top_counts
diabetes 0 1.00 FALSE 2 neg: 500, pos: 268
pressure 36 0.95 FALSE 3 low: 282, mid: 245, hig: 205
Variable type: numeric
skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
age 0 1.00 33.24 11.76 21.00 24.00 29.00 41.00 81.00 ▇▃▁▁▁
glucose 5 0.99 121.69 30.54 44.00 99.00 117.00 141.00 199.00 ▁▇▇▃▂
insulin 374 0.51 155.55 118.78 14.00 76.25 125.00 190.00 846.00 ▇▂▁▁▁
mass 11 0.99 32.46 6.92 18.20 27.50 32.30 36.60 67.10 ▅▇▃▁▁
pedigree 0 1.00 0.47 0.33 0.08 0.24 0.37 0.63 2.42 ▇▃▁▁▁
pregnant 0 1.00 3.85 3.37 0.00 1.00 3.00 6.00 17.00 ▇▃▂▁▁
triceps 227 0.70 29.15 10.48 7.00 22.00 29.00 36.00 99.00 ▆▇▁▁▁
We choose the xgboost algorithm from the xgboost package as learner.
Missing Values
The task has missing data in five columns.
diabetes age glucose insulin mass pedigree pregnant pressure triceps
0.00 0.00 0.01 0.49 0.01 0.00 0.00 0.05 0.30
The xgboost learner has an internal method for handling missing data but some learners cannot handle missing values. We will try to beat the internal method in terms of predictive performance. The
mlr3pipelines package offers various methods to impute missing values.
[1] "imputeconstant" "imputehist" "imputelearner" "imputemean" "imputemedian" "imputemode"
[7] "imputeoor" "imputesample"
We choose the PipeOpImputeOOR that adds the new factor level ".MISSING". to factorial features and imputes numerical features by constant values shifted below the minimum (default) or above the
PipeOp: <imputeoor> (not trained)
values: <min=TRUE, offset=1, multiplier=1>
Input channels <name [train type, predict type]>:
input [Task,Task]
Output channels <name [train type, predict type]>:
output [Task,Task]
As the output suggests, the in- and output of this pipe operator is a Task for both the training and the predict step. We can manually train the pipe operator to check its functionality:
diabetes age pedigree pregnant glucose insulin mass pressure triceps
Let’s compare an observation with missing values to the observation with imputed observation.
diabetes age glucose insulin mass pedigree pregnant pressure triceps
1: neg 29 115 NA 35.3 0.134 10 <NA> NA
2: neg 29 115 -819 35.3 0.134 10 .MISSING -86
Note that OOR imputation is in particular useful for tree-based models, but should not be used for linear models or distance-based models.
Factor Encoding
The xgboost learner cannot handle categorical features. Therefore, we must to convert factor columns to numerical dummy columns. For this, we argument the xgboost learner with automatic factor
The PipeOpEncode encodes factor columns with one of six methods. In this example, we use one-hot encoding which creates a new binary column for each factor level.
We manually trigger the encoding on the task.
<TaskClassif:pima> (768 x 11): Pima Indian Diabetes
* Target: diabetes
* Properties: twoclass
* Features (10):
- dbl (10): age, glucose, insulin, mass, pedigree, pregnant, pressure.high, pressure.low, pressure.mid,
The factor column pressure has been converted to the three binary columns "pressure.low", "pressure.mid", and "pressure.high".
Constructing the Pipeline
We created two preprocessing steps which could be used to create a new task with encoded factor variables and imputed missing values. However, if we do this before resampling, information from the
test can leak into our training step which typically leads to overoptimistic performance measures. To avoid this, we add the preprocessing steps to the Learner itself, creating a GraphLearner. For
this, we create a Graph first.
We use as_learner() to wrap the Graph into a GraphLearner with which allows us to use the graph like a normal learner.
The GraphLearner can be trained and used for making predictions. Instead of calling $train() or $predict() manually, we will directly use it for resampling. We choose a 3-fold cross-validation as the
resampling strategy.
iteration task_id learner_id resampling_id classif.ce
1: 1 pima graph_learner cv 0.2851562
2: 2 pima graph_learner cv 0.2460938
3: 3 pima graph_learner cv 0.2968750
For each resampling iteration, the following steps are performed:
1. The task is subsetted to the training indices.
2. The factor encoder replaces factor features with dummy columns in the training task.
3. The OOR imputer determines values to impute from the training task and then replaces all missing values with learned imputation values.
4. The learner is applied on the modified training task and the model is stored inside the learner.
Next is the predict step:
1. The task is subsetted to the test indices.
2. The factor encoder replaces all factor features with dummy columns in the test task.
3. The OOR imputer replaces all missing values of the test task with the imputation values learned on the training set.
4. The learner’s predict method is applied on the modified test task.
By following this procedure, it is guaranteed that no information can leak from the training step to the predict step.
Tuning the Pipeline
Let’s have a look at the parameter set of the GraphLearner. It consists of the xgboost hyperparameters, and additionally, the parameter of the PipeOp encode and imputeoor. All hyperparameters are
prefixed with the id of the respective PipeOp or learner.
as.data.table(graph_learner$param_set)[, c("id", "class", "lower", "upper", "nlevels"), with = FALSE]
id class lower upper nlevels
1: encode.method ParamFct NA NA 5
2: encode.affect_columns ParamUty NA NA Inf
3: imputeoor.min ParamLgl NA NA 2
4: imputeoor.offset ParamDbl 0 Inf Inf
5: imputeoor.multiplier ParamDbl 0 Inf Inf
6: imputeoor.affect_columns ParamUty NA NA Inf
7: xgboost.alpha ParamDbl 0 Inf Inf
8: xgboost.approxcontrib ParamLgl NA NA 2
9: xgboost.base_score ParamDbl -Inf Inf Inf
10: xgboost.booster ParamFct NA NA 3
11: xgboost.callbacks ParamUty NA NA Inf
12: xgboost.colsample_bylevel ParamDbl 0 1 Inf
13: xgboost.colsample_bynode ParamDbl 0 1 Inf
14: xgboost.colsample_bytree ParamDbl 0 1 Inf
15: xgboost.disable_default_eval_metric ParamLgl NA NA 2
16: xgboost.early_stopping_rounds ParamInt 1 Inf Inf
17: xgboost.early_stopping_set ParamFct NA NA 3
18: xgboost.eta ParamDbl 0 1 Inf
19: xgboost.eval_metric ParamUty NA NA Inf
20: xgboost.feature_selector ParamFct NA NA 5
21: xgboost.feval ParamUty NA NA Inf
22: xgboost.gamma ParamDbl 0 Inf Inf
23: xgboost.grow_policy ParamFct NA NA 2
24: xgboost.interaction_constraints ParamUty NA NA Inf
25: xgboost.iterationrange ParamUty NA NA Inf
26: xgboost.lambda ParamDbl 0 Inf Inf
27: xgboost.lambda_bias ParamDbl 0 Inf Inf
28: xgboost.max_bin ParamInt 2 Inf Inf
29: xgboost.max_delta_step ParamDbl 0 Inf Inf
30: xgboost.max_depth ParamInt 0 Inf Inf
31: xgboost.max_leaves ParamInt 0 Inf Inf
32: xgboost.maximize ParamLgl NA NA 2
33: xgboost.min_child_weight ParamDbl 0 Inf Inf
34: xgboost.missing ParamDbl -Inf Inf Inf
35: xgboost.monotone_constraints ParamUty NA NA Inf
36: xgboost.normalize_type ParamFct NA NA 2
37: xgboost.nrounds ParamInt 1 Inf Inf
38: xgboost.nthread ParamInt 1 Inf Inf
39: xgboost.ntreelimit ParamInt 1 Inf Inf
40: xgboost.num_parallel_tree ParamInt 1 Inf Inf
41: xgboost.objective ParamUty NA NA Inf
42: xgboost.one_drop ParamLgl NA NA 2
43: xgboost.outputmargin ParamLgl NA NA 2
44: xgboost.predcontrib ParamLgl NA NA 2
45: xgboost.predictor ParamFct NA NA 2
46: xgboost.predinteraction ParamLgl NA NA 2
47: xgboost.predleaf ParamLgl NA NA 2
48: xgboost.print_every_n ParamInt 1 Inf Inf
49: xgboost.process_type ParamFct NA NA 2
50: xgboost.rate_drop ParamDbl 0 1 Inf
51: xgboost.refresh_leaf ParamLgl NA NA 2
52: xgboost.reshape ParamLgl NA NA 2
53: xgboost.seed_per_iteration ParamLgl NA NA 2
54: xgboost.sampling_method ParamFct NA NA 2
55: xgboost.sample_type ParamFct NA NA 2
56: xgboost.save_name ParamUty NA NA Inf
57: xgboost.save_period ParamInt 0 Inf Inf
58: xgboost.scale_pos_weight ParamDbl -Inf Inf Inf
59: xgboost.skip_drop ParamDbl 0 1 Inf
60: xgboost.strict_shape ParamLgl NA NA 2
61: xgboost.subsample ParamDbl 0 1 Inf
62: xgboost.top_k ParamInt 0 Inf Inf
63: xgboost.training ParamLgl NA NA 2
64: xgboost.tree_method ParamFct NA NA 5
65: xgboost.tweedie_variance_power ParamDbl 1 2 Inf
66: xgboost.updater ParamUty NA NA Inf
67: xgboost.verbose ParamInt 0 2 3
68: xgboost.watchlist ParamUty NA NA Inf
69: xgboost.xgb_model ParamUty NA NA Inf
id class lower upper nlevels
We will tune the encode method.
We define a tuning instance and use grid search since we want to try all encode methods.
The archive shows us the performance of the model with different encoding methods.
Nested Resampling
We create one GraphLearner with imputeoor and test it against a GraphLearner that uses the internal imputation method of xgboost. Applying nested resampling ensures a fair comparison of the
predictive performances.
graph_1 = po("encode") %>>%
graph_learner_1 = GraphLearner$new(graph_1)
graph_learner_1$param_set$values$encode.method = to_tune(c("one-hot", "treatment"))
at_1 = AutoTuner$new(
learner = graph_learner_1,
resampling = resampling,
measure = msr("classif.ce"),
terminator = trm("none"),
tuner = tnr("grid_search"),
store_models = TRUE
graph_2 = po("encode") %>>%
po("imputeoor") %>>%
graph_learner_2 = GraphLearner$new(graph_2)
graph_learner_2$param_set$values$encode.method = to_tune(c("one-hot", "treatment"))
at_2 = AutoTuner$new(
learner = graph_learner_2,
resampling = resampling,
measure = msr("classif.ce"),
terminator = trm("none"),
tuner = tnr("grid_search"),
store_models = TRUE
We run the benchmark.
We compare the aggregated performances on the outer test sets which give us an unbiased performance estimate of the GraphLearners with the different encoding methods.
nr task_id learner_id resampling_id iters classif.ce
1: 1 pima encode.xgboost.tuned cv 3 0.2578125
2: 2 pima encode.imputeoor.xgboost.tuned cv 3 0.2630208
Hidden columns: resample_result
Note that in practice, it is required to tune preprocessing hyperparameters jointly with the hyperparameters of the learner. Otherwise, comparing preprocessing steps is not feasible and can lead to
wrong conclusions.
Applying nested resampling can be shortened by using the auto_tuner()-shortcut.
graph_1 = po("encode") %>>% learner
graph_learner_1 = as_learner(graph_1)
graph_learner_1$param_set$values$encode.method = to_tune(c("one-hot", "treatment"))
at_1 = auto_tuner(
method = "grid_search",
learner = graph_learner_1,
resampling = resampling,
measure = msr("classif.ce"),
store_models = TRUE)
graph_2 = po("encode") %>>% po("imputeoor") %>>% learner
graph_learner_2 = as_learner(graph_2)
graph_learner_2$param_set$values$encode.method = to_tune(c("one-hot", "treatment"))
at_2 = auto_tuner(
method = "grid_search",
learner = graph_learner_2,
resampling = resampling,
measure = msr("classif.ce"),
store_models = TRUE)
design = benchmark_grid(task, list(at_1, at_2), rsmp("cv", folds = 3))
bmr = benchmark(design, store_models = TRUE)
Final Model
We train the chosen GraphLearner with the AutoTuner to get a final model with optimized hyperparameters.
The trained model can now be used to make predictions on new data at_2$predict(). The pipeline ensures that the preprocessing is always a part of the train and predict step. | {"url":"https://mlr-org.com/gallery/optimization/2021-03-10-practical-tuning-series-tune-a-preprocessing-pipeline/index.html","timestamp":"2024-11-02T16:55:35Z","content_type":"application/xhtml+xml","content_length":"84317","record_id":"<urn:uuid:fd800f36-4352-4d62-9314-e36d9b2b77e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00759.warc.gz"} |
Analysis II Prelim topics
• Analytic functions: power series, exponential and logarithm functions, Moebius transformations, the Riemann sphere.
• Cauchy’s theorem: Goursat’s proof, homotopic curves, winding number of a closed curve about a point, the Cauchy integral formulas, Liouville’s theorem, isolated singularities,
Casorati-Weierstrass theorem, open mapping theorem, maximum principle, Morera’s theorem, and Schwarz reflection principle, residue theorem, the argument principle, Rouche’s theorem, the
evaluation of definite integrals, Schwarz’s lemma, zeros of analytic functions.
• Harmonic functions: conjugate functions, mean value property.
• Meromorphic functions: Isolated singularities and their classification, Laurent series expansion, Casorati-Weierstrass theorem, Picard’s theorems (statements only), principal part of a
meromorphic function at an isolated singularity, partial fraction expansion. Also, meromorphic functions viewed as analytic maps into the Riemann sphere.
• Entire functions: infinite products, Jensen’s formula, Weierstrass product theorem, Hadamard factorization theorem.
• Conformal mappings: Elementary mappings, mappings by Moebius transformations, the Riemann mapping theorem (statement only).
• Hilbert spaces: orthogonality, orthogonal decompositions, Riesz Representation theorem, Bessel's inequality, orthonormal bases, Parseval's identity
• Fourier transform: Tempered distributions, the Fourier transform on L2, Plancherel's theorem, Sobolev spaces
1. Ahlfors, L. (1979). Complex Analysis, New York, McGraw Hill (3rd edition).
2. Conway, J. B. (1978) Functions of One Complex Variable I (Graduate texts in Mathematics-vol. 11), Springer (2nd edition)
3. Stein, E.M. and Sharkarchi, R. (2003). Complex Analysis, Princeton University Press.
4. Titchmarsh, E. C. (1939) The Theory of Functions, Oxford University Press (2nd edition).
5. S. Axler, “Measure, integration & real analysis”
6. F.G. Friedlander and M. Joshi, “Introduction to the theory of distributions” | {"url":"http://sas.rochester.edu/mth/graduate/prelim-topics/prelim-topics-analysis2.html","timestamp":"2024-11-04T17:41:22Z","content_type":"text/html","content_length":"73717","record_id":"<urn:uuid:df4c0497-3006-4bb4-adf8-a8740e473e75>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00369.warc.gz"} |
A Comprehensive Guide to Observed Power (Post Hoc Power)
“Observed power”, “Post hoc Power”, and “Retrospective power” all refer to the statistical power of a statistical significance test to detect a true effect equal to the observed effect. In a broader
sense these terms may also describe any power analysis performed after an experiment has completed. Importantly, it is the first, narrower sense that will be used throughout the article.
Here are three typical scenarios of using observed power in online A/B testing and in more general applications of statistical tests:
1. An experiment concludes with a statistically non-significant outcome. Observed power is compared to the target power to determine if the test was underpowered. If that is the case it is argued
that there is a true effect which remains undetected. This can lead to a larger follow-up experiment or in some cases to extending the experiment until it achieves whatever level of observed
power level is deemed sufficient;
2. An experiment concludes with a statistically significant outcome, but the observed power is smaller than the target power level of, say, 80% or 90%. This is seen as the test being underpowered
and therefore questionable or outright unreliable;
3. An experiment concludes with a statistically significant outcome, but the observed effect size is lower than the target MDE (minimum detectable effect), and that is seen as cause for concern for
its trustworthiness;
While these might seem reasonable, there are unsurmountable issues in using post hoc power. These have been covered with increasing frequency in the scientific literature ^[1][2][3][4][5], yet the
problem persists in statistical practice.
This article takes certain novel approaches to debunking observed power, including using a post hoc power calculator as an educational device. Through a comprehensive examination of the rationale
behind observed power all possible use cases are shown to rest on valid, but unsound chains of reasoning, which might explain their surface appeal.
This and the two companion articles further explore the different harms caused by using observed power in scenarios readers should find familiar, in a uniquely comprehensive manner.
Table of Contents:
Observed power as a proxy for true power
A major line of reasoning behind the use of observed power is what may be called “the search for true power”.
The logic of searching for “true power”* goes like this:
1. “Good” or “trustworthy” statistical tests should have high “true power”: high probability of detecting the true effect, whatever it may be;
2. The true effect size is unknown and there is often little data to produce good pre-test predictions about where it lies;
3. Post data, the observed effect (or an appropriate point estimate), is our best guess for the true effect size;
4. Substituting the true effect for the observed effect in a power calculation should result in a good estimate of the “true power” of a test.
These premises lead to the following conclusion:
To ensure a test is “good” or “trustworthy”, aim to achieve high power versus the observed effect as the best proxy for the true statistical power of the test.
While the above is a valid argument for using observed power, it is not a sound one since premises one and four can be shown to be false.
* side remark on “true power”
The focus some people have on “true power” may stem from poor and incomplete definitions of power found in multiple textbooks and articles. Namely, one can often see power described as “the
probability of finding an effect, if there is one”. However, this is a misleading definition since it skips the all important ” of a given size” instead of the easily assumed “of any size”. Such a
confusion result may lead to searches of “true power” instead of acceptance of its hypothetical nature. Compare the above shorthand to the definition given in “Statistical Methods in Online A/B
Testing”: “The statistical power of a statistical test is defined as the probability of observing a p-value statistically significant at a certain threshold α if a true effect of a certain magnitude
μ1 is in fact present.[…]”
High true power is not a requirement for a good test
When planning an A/B test or another type of experiment, one often does not know the true effect size and may have very limited information to make predictions about where it may lie. Planning a test
with a primary motivation of achieving a high true power results in committing to as large a sample size as practically achievable and in choosing as low a significance threshold as possible. This
would maximize the chance of detecting even tiny true effects.
However, such an approach would ignore the trade-offs between running a test for longer / with more users, and the potential benefits that may be realized (for details see “Statistical Power, MDE,
and Designing Statistical Tests”, risk-reward analysis and its references). True effects below a given value are simply not worth pursuing due to the risks of running the test no longer being
justified by the potential benefits.
The pivotal value where that happens is called a minimum detectable effect (MDE) or a minimum effect of interest (MEI). It is the smallest effect size one wants to detect with high probability, where
“detect” means a statistically significant outcome at a specified significance threshold. It is the smallest effect size one would not want to miss. Aiming for high power for any value below the MEI
would, by definition, achieve poorer business outcomes from A/B testing.
Even if it is almost sure what the true effect size is, it would rarely match the minimum effect of interest. The framework for choosing optimal sample sizes and significance thresholds deployed at
Analytics Toolkit’s A/B testing hub often arrives at optimal parameters under which the MDE (at a high power level such as 80% or 90%) is larger than the expected effect size. This decision framework
is most fully explained in the relevant chapter of “Statistical Methods in Online A/B Testing”.
Figure 1: The result of a risk-reward analysis in which the minimum effect of interest at 80% is notably higher than the effect considered highly likely (click to see full size image)
Finally, it should be noted that taking premise one as given and aiming for high true power would turn the planning phase of an experiment into an exercise of guessing what the true effect of an
intervention is. The logical conclusion of “the search for true power” renders all pre-test power analyses and test planning futile. However, good planning is what ensures an experiment, be it an
online A/B test or a scientific experiment, offers a good balance between the expected business risks and rewards.
Rewriting premise one to make it true would result in:
1. “Good” statistical tests should have high power to detect the minimum effect of interest.
Since the minimum effect of interest is rarely the true effect size, the above statement breaks the rest of the argument and makes it unsound. The search for true power is therefore illogical and it
does not justify the use of observed power. This holds even if observed power was a good estimate of true power. Whether that is the case is explored in the following section.
Observed power as an estimate of true power
Another reason why searching for true power is not a logical justification for using observed power is that the fourth premise does not hold either. Namely, the observed power of a test is a poor
estimate of its true power. To be a proper estimator of true power, observed power should ideally be all of the following:
Some of these conditions can be relaxed in complex modeling problems where an ideal estimator either does not exist or has not yet been found. However, observed power performs really poorly on all
• It does not hone in on the true value with larger amounts of information; (is not consistent)
• Observed power is almost always heavily biased vis-à-vis true power;
• There is no reason to suspect it is a sufficient statistic of true power;
• Its variance is far too large to be useful (even if it somehow turns out to be efficient though biased)
Yuan & Maxwell end their work titled “On the Post Hoc Power in Testing Mean Differences”^[6] with the following assessment:
“Using analytical, numerical, and Monte Carlo approaches, our results show that the estimated power does not provide useful information when the true power is small. It is almost always a biased
estimator of the true power. The bias can be negative or positive. Large sample size alone does not guarantee the post hoc power to be a good estimator of the true power.”
I’ve conducted some R simulations of my own to visualize how poorly observed power behaves as an estimate of true power:
Figure 2: Simulations of Observed Power versus MLE as estimators for true power and the true effect size, respectively (click to see full size image)
All three simulations have 100,000 runs and share the same properties except for the different true effect and hence true power. Histograms in each column come from the same simulation and offer a
direct comparison which reveals just how poorly observed power does as an estimate compared to the observed effect size which in this case is the MLE. In short, it has inconsistent variance, the bias
is obvious, and consistency is obviously non-existent.
Viewing the power level as a threshold to be met
Another error shared by different use-cases of observed power is to view the choice of statistical power level (1 – β) and its complement, the type II error rate β, as being in the same category as
the choice of the significance threshold α and the resulting type I error rate. The logic is as follows:
1. The test’s chosen power level is 80% (or 90%, etc.) against a given effect size of interest;
2. The pre-test power level is a threshold which should be met for the outcome to be considered trustworthy;
3. The observed power is what needs to be compared to the target power level to ensure said threshold is met;
Naturally, the conclusion is that:
If the observed power is lower than the power level , then the test is underpowered as it has failed to achieve its statistical power threshold / target type II error rate.
The above logic is unsound due to issues with premises two and three. Regarding premise two, the test’s target power level is not a cut-off that needs to be met by any statistic of the observed data
in order to declare or conclude something. The power level and the type II error rate are inherently pre-data probabilities. The level of statistical power against a chosen MDE (MEI) is achieved by
the experiment reaching (or exceeding) the sample size determined to be necessary by way of a pre-test power analysis *.
* the obvious exception
The only exception to this may be when the observed variance of the outcome measure is meaningfully different from the variance used in the power analysis and sample size calculations. For example,
the base rate of a binomial variable may turn out to be significantly lower than expected, in which case the analysis may need to be rerun and a new sample size may need to be calculated. In
e-commerce this may happen if seasonality or a significant change in the advertising strategy cause a sudden influx of prospective customers who are less qualified or less willing to purchase than
previous ones, among other reasons. Even when that happens to be the case, only the observed variance may play a role in this sample size recalculation, not the observed effect size, so one cannot
speak of observed power in the sense defined at the beginning paragraph of this article.
Once an experiment reaches its target sample size it has achieved its desired statistical power level against the minimum effect of interest regardless of the outcome and any other statistic.
Regarding premise three, it would be useful to point out that the significance threshold α is a cut-off set for the p-value. The p-value is a post-data summary of the data showing how likely (or
unlikely) it would have been to observe said data under a specified null hypothesis. The significance threshold applies directly to the observed p-value and leads to different conclusions like
“reject the null hypothesis” or “fail to reject the null hypothesis”.
The target level of statistical power is not a cut-off in the same way and there is no equivalent to the p-value for statistical power. Observed power surely cannot perform the function of the
p-value since power calculations do not use any of the data obtained from an experiment. Making the hypothetical true effect equal to the observed effect does not make the power calculation any more
Low observed power does not mean a test is underpowered
As a consequence of the above, a test cannot be said to be underpowered based on the observed power being lower than the target power (see Underpowered A/B Tests – Confusions, Myths, and Reality). A
test has either reached its target sample size or not and observed power cannot tell us anything about that. In this light both “observed power” and “post-hoc power” are bordering on being misnomers.
More on this point in the last section of this article.
Observed power is a direct product of the p-value
To some, the fact that observed power can be computed as a direct transformation of the p-value is the nail in the coffin of the concept. However, this has been known since at least Hoenig & Heisey’s
2001^[1] article in The American Statistician. Yet, fallacies related to observed power persist. This is why I have tackled some of the logical issues with its application before turning to this
obvious point.
What is meant by a direct product? Given any p-value p from a given test of size α, the observed power can be computed analytically using the following equation^[1]:
G[Zp](α) = 1 − Φ(Z[α] − Z[p])
Where Φ is the standard normal cumulative distribution function, Z[α] and Z[p] being the z-scores corresponding to the significance threshold and the p-value, respectively. Observed power is
therefore determined completely by the p value and the chosen significance threshold. In other words, once the p-value is known, there is nothing that calculating observed power can add to the
Post hoc power calculator
To illustrate this, I’ve built a simple post hoc power calculator which you can use by entering your significance threshold and the observed p-value. The level of statistical power against the
observed effect will be computed instantaneously, and you can explore the relationship of any p-value and the post hoc power it results in by hovering over the graph.
The green vertical line represents the significance threshold, whereas the violet line is the observed p-value. Feel free to test it against some of your own, independent calculations of statistical
The interesting question is what follows from the fact that the observed power is a simple transformation of the p-value? The answer is broken down below by whether the outcome is statistically
significant or not.
Observed power with a non-significant outcome
When using observed power with non-significant test outcomes, it is virtually guaranteed to label such tests as “underpowered”. This is due to the conflation between low true power and a test being
underpowered, coupled with the misguided belief that “observed power” is a good estimate for “true power”. These have been debunked in earlier sections.
To compound the issue, the maximum observed power possible with a non-significant outcome is below 0.5 (50%) at any given significance threshold. Examine the fraction of the observed power curve
which spans the non-significant outcomes:
Figure 3: Observed power with non-significant outcomes
For all non-significant values the power is lower than 50%. In this example a threshold of significance of 0.05 is used, but you can use the post-hoc power calculator above to check that it is the
case by entering different thresholds.
In a best case scenario, an observed power of 50% or lower would be interpreted as the test being underpowered which can be used to justify a follow-up test with significantly larger sample size /
longer duration.
There is also a contradiction with the p-value. As evident from Figure 3, the smaller the p-value, the higher the observed power is. Interpreting higher observed power as offering stronger evidence
for the null hypothesis directly contradicts the logic of p-values in which a smaller p-value offers stronger grounds for rejecting the null hypothesis, not weaker.
In the worst possible scenario, low observed power is taken as a grant to continue the test until a given higher sample size is achieved:
Does peeking through observed power inflate the false positive rate?
Peeking through observed power, a.k.a. “fishing for high observer power”, is bound to lead to a higher false positive rate than what’s nominally targeted and reported. As shown in the simulations in
“Using observed power in online A/B tests”, continuing an experiment after seeing a non-significant outcome coupled (inevitably) with low power until a significant outcome is observed or for as long
as practically possible leads to around 2.4x inflation of the type I error rate.
Changing some of the necessary assumptions may lower the actual type I error rate to “just” being two-fold higher than the nominal one. The actual rate can be threefold the nominal under less
favorable assumptions. In all cases the increase is far outside what is typically tolerable.
Observed power with a statistically significant outcome?
Given that statistical power relates to type II errors and the false negative rate, it might seem contradictory to even look at statistical power after a test has produced a statistically significant
outcome. With a significant result in hand there remains only one error possible: that of a false positive and not of a false negative.
Further, power is a pre-test probability, just like the odds of any particular person winning a lottery draw are. When someone wins the lottery it is illogical to point to their low odds of winning
as a reason to reject giving them the prize. These odds are irrelevant once the outcome has been realized. It is similarly illogical to point to a low probability of observing a statistically
significant outcome as an argument against the outcome being valid after it has already been observed. This is regardless if the power analysis focuses on observed power or not.
As a practical matter, requiring an outcome to be both statistically significant and to have observed power higher than the target level results in an actual error control which is multiple times
more stringent than the nominally targeted level.
Figure 4: Actual vs nominal threshold if requiring both high observed power and significance
Figure 4 demonstrates visually that the high observed power requirement completely overrides the significance threshold, making it redundant. That is because p-values below 0.006 would be disregarded
by the requirement for low post hoc power, whereas the significance threshold is at 0.05.
Knowing the above it may seem unfathomable that someone may insist on the utility of observed power even when presented with a statistically significant outcome. The already discussed “search for
true power” and viewing the chosen power level as a threshold can explain a lot of the motivation behind the misguided use of observed power in ensuring the trustworthiness of even statistically
significant tests.
However, there are two more to consider:
MDE as a threshold
Another angle through which misunderstandings about statistical power make it into A/B testing practice is when the question of a test being underpowered is framed in terms of the minimum detectable
effect (MDE). Namely, it is seen as an issue if the observed effect size is smaller than the MDE, especially if the outcome is statistically significant.
The numerous misunderstandings regarding the relationship (or lack thereof), as well as the harms which follow are explored in What if the observed effect is smaller than the MDE?. In short, the
outcome of requiring significant results to also have an observed effect greater than the MDE are exactly the same as when requiring high observed power.
Viewing the observed effect size as the one a test’s conclusion is about
From a slightly peculiar angle, observed power may be invoked with significant outcomes due to a warranted skepticism regarding an observed effect that seems “too good to be true” especially if it is
coupled with a low p-value. The misunderstanding here is a common one related to p-values. Through its lens, p-values are viewed as giving the probability that the observed effect is due to chance
alone. p-values, however, reflect the probability of obtaining an effect as large as the observed, or larger, assuming a data-generating mechanism in which the null hypothesis is true. Given the role
of the null hypothesis in hypothesis testing, it is obvious that the p-value attaches to the test procedure and not to any given hypothesis, be it the null or alternative. P-values do not speak
directly of the probability of the observed effect being genuine or not.
If influenced by the above misconception, some might be skeptical upon seeing an effect that seems too good to be true accompanied by a p-value which seems too reassuring. Given the incorrect
interpretation of what the statistics mean, such skepticism is warranted, but observed power is not the correct way to address it.
A post hoc power calculation makes no use of the test data and is therefore unable to add anything to the observed p-value. Instead, the uncertainty surrounding a point estimate can be conveyed
through confidence intervals at different levels, or in terms of severity (as in D. Mayo’s works) for any effect of interest.
All power is propter hoc power
In some sense, it can be argued that it is a misnomer to speak of “Observed power”, “Post hoc Power”, or “Retrospective power”. No formula or code for computing statistical power and has a reference
to anything observed in the experiment. The reason is that statistical power calculations are inherently pre-data probabilities and so the statistical power function / power curve is computed using
pre-data information.
Examining any particular value of that function after an experiment is the same as if done beforehand since the entire power curve is constructed without any reference to observed test data.
Statistical power is inherently a pre-data probability. It does not use any of the data from a controlled experiment. As result, the power level for any hypothetical true effect can be computed
before any data has been gathered.
Maybe the term “post hoc power analysis” has some merit in that it is a power analysis performed after the fact on the timeline, but given the other possible meanings just calling it a “power
analysis” would avoid many of the problems and misunderstandings associated with attaching a “post hoc” or “retrospective” qualifier to “power”.
All possible justifications for the use of observed power rest on unsound arguments and stem from misunderstanding of one or more statistical concepts. To recap:
• High true power is not necessary for a test to be well-powered, and even if it were, observed power is not a good estimate of true power
• Neither the target power level nor the target MDE are thresholds to be met by any observed statistic (“observed” power or observed effect size respectively being the common choices)
• Non-significant outcomes always appear underpowered if the observed power level is compared to the target power level
• Requiring high post hoc power for significant outcomes is equivalent to drastically increasing the significance threshold of the test, and so is requiring the observed effect size to be greater
than or equal to the target MDE (as the two are equivalent)
• Statistical power is a pre-test probability, regardless of when it is computed, and it does not use the test data
At best, observed power is unnecessary given a p-value has been calculated as it has direct functional relationship to it. Enter a p-value and a threshold and the post-hoc power calculator will
output the observed power.
At worst, using observed power may lead to all kinds of issues, including:
• mislabeling well-powered tests as either underpowered or overpowered ones
• identifying trustworthy results as untrustworthy
• peeking through observed power, resulting in a significantly inflated type I error rate
• greatly raising the bar for significance in a non-obvious manner (when required with significant outcomes)
• unjustifiably extending the duration and increasing the sample sizes of A/B tests by orders of magnitude
Given the above, Deborah Mayo’s term “shpower” is spot on when describing observed power, defined as the level of statistical power to detect a true effect equal to the observed one. One should be
able to guess what two words “shpower” is derived from, even if English is not your native language.
1 Hoenig, J. M., Heisey, D. M. (2001) “The abuse of power: The pervasive fallacy of power calculations for data analysis.” The American Statistician, 55, 19-24.
2 Lakens, D. (2014) “Observed power, and what to do if your editor asks for post-hoc power analyses”, online at https://daniellakens.blogspot.com/2014/12/observed-power-and-what-to-do-if-your.html
3 Zhang et al. (2019) “Post hoc power analysis: is it an informative and meaningful analysis?” General Psychiatry 32(4):e100069
4 Christogiannis et al. (2022) “The self-fulfilling prophecy of post-hoc power calculations” American Journal of Orthodontics and Dentofacial Orthopedics 161:315-7
5 Heckman, M.G., Davis III, J.M., Crowson, C.S. (2022) “Post Hoc Power Calculations: An Inappropriate Method for Interpreting the Findings of a Research Study” The Journal of Rheumatology 49(8),
6 Yuan, K.-H., Maxwell, S. (2005) “On the Post Hoc Power in Testing Mean Differences” Journal of Educational and Behavioral Statistics, 30(2), 141–167.
About the author
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Math, Grade 6, Surface Area and Volume, Volume Formula For Rectangular Prisms
Make Connections
Performance Task
Ways of Thinking: Make Connections
• Take notes about your classmates’ methods for finding the volume of a cube and a rectangular prism.
As your classmates present, ask questions such as:
• What problems did you encounter in finding the volume?
• What knowledge of multiplying fractions did you use?
• Did you make an estimate of the volume first? How did you make your estimate?
• How did the Cube Builder help you?
• How do you know your answer is correct? | {"url":"https://goopennc.oercommons.org/courseware/lesson/5075/student/?section=8","timestamp":"2024-11-09T00:19:50Z","content_type":"text/html","content_length":"31089","record_id":"<urn:uuid:7b9a65b6-bec1-4fd7-9698-56468f44b161>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00577.warc.gz"} |
1-1 CHAPTER 1 ELECTRICAL CONDUCTORS LEARNING OBJECTIVES Learning objectives are stated at the beginning of each chapter. These learning objectives serve as a
preview of the information you are expected to learn in the chapter. The comprehensive check questions
are based on the objectives. By successfully completing the OCC-ECC, you indicate that you have met the objectives and have learned the information. The learning objectives are listed below.
Upon completing this chapter, you should be able to: 1. Recall the definitions of unit size, mil-foot, square mil, and circular mil and the mathematical equations and calculations for each.
2. Define specific resistance and recall the three factors used to calculate it in ohms. 3. Describe the proper use of the American Wire Gauge when making wire measurements.
4. Recall the factors required in selecting proper size wire. 5. State the advantages and disadvantages of copper or aluminum as conductors.
6. Define insulation resistance and dielectric strength including how the dielectric strength of an insulator is determined.
7. Identify the safety precautions to be taken when working with insulating materials. 8. Recall the most common insulators used for extremely high voltages.
9. State the type of conductor protection normally used for shipboard wiring. 10. Recall the design and use of coaxial cable. ELECTRICAL CONDUCTORS
In the previous modules of this training series, you have learned about various circuit components.
These components provide the majority of the operating characteristics of any electrical circuit. They are
useless, however, if they are not connected together. Conductors are the means used to tie these components together.
Many factors determine the type of electrical conductor used to connect components. Some of these
factors are the physical size of the conductor, its composition, and its electrical characteristics. Other
factors that can determine the choice of a conductor are the weight, the cost, and the environment where the conductor will be used. CONDUCTOR SIZES
To compare the resistance and size of one conductor with that of another, we need to establish a
standard or unit size. A convenient unit of measurement of the diameter of a conductor is the mil (0.001,
or one-thousandth of an inch). A convenient unit of conductor length is the foot. The standard unit of size
in most cases is the MIL-FOOT. A wire will have a unit size if it has a diameter of 1 mil and a length of 1 foot. | {"url":"https://electriciantraining.tpub.com/14176/css/Chapter-1-Electrical-Conductors-13.htm","timestamp":"2024-11-08T15:32:05Z","content_type":"text/html","content_length":"20014","record_id":"<urn:uuid:33fc7df3-f200-4e33-9fbb-00f338817c12>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00823.warc.gz"} |
Kuta Software Infinite Algebra 1 Graphing Linear Inequalities Answer Key - Graphworksheets.com
Kuta Software Infinite Algebra 1 Graphing Inequalities – 7th Grade Graph Worksheets are a great resource for students studying graphs in school. These worksheets are available in PDF format for
downloading and contain worksheets for each type of graph a student might encounter. They are an excellent way to introduce a student to graphs and … Read more | {"url":"https://www.graphworksheets.com/tag/kuta-software-infinite-algebra-1-graphing-linear-inequalities-answer-key/","timestamp":"2024-11-02T21:08:48Z","content_type":"text/html","content_length":"48428","record_id":"<urn:uuid:cd0f7d00-b568-4b64-ad4f-b9fd18de559e>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00143.warc.gz"} |
Grade 8 Mathematics Topics
Grade 8 Mathematics Topics and Sub-topics
1. Demonstrate an understanding of perfect squares and square roots, concretely, pictorially and symbolically (limited to whole numbers).
2. Determine the approximate square root of numbers that are not perfect squares (limited to whole numbers).
3. Demonstrate an understanding of percents greater than or equal to 0%, including greater than 100%.
4. Demonstrate an understanding of ratio and rate.
5. Solve problems that involve rates, ratios and proportional reasoning.
6. Demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially and symbolically.
7. Demonstrate an understanding of multiplication and division of integers, concretely, pictorially and symbolically.
Patterns • Use patterns to describe the world and to solve problems.
Variables and Equations • Represent algebraic expressions in multiple ways.
Measurement • Use direct and indirect measurement to solve problems.
3-D Objects and 2-D Shapes
• Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.
• Describe and analyze position and motion of objects and shapes.
Data Analysis • Collect, display and analyze data to solve problems.
Chance and Uncertainty • Use experimental or theoretical probabilities to represent and solve problems involving uncertainty.
Number Sense
• Patterns
• Patterns
• Patterns
• Patterns
Patterns and Relations
• Patterns
• Variables and Equations
• Patterns
• Patterns
• Patterns
Shape and Space
• Measurement
• 3-D Objects and 2-D Shapes
• Transformations
• Patterns
• Patterns
Statistics and Probability
• Data Analysis
• Chance and Uncertainty
• Patterns
• Patterns
• Patterns | {"url":"https://www.curriculumcentre.com/grade-8-mathematics-topics.html","timestamp":"2024-11-10T07:35:03Z","content_type":"text/html","content_length":"83755","record_id":"<urn:uuid:80cb7225-363a-4913-b143-8b1b3f68fa34>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00592.warc.gz"} |
What is 110 Celsius to Fahrenheit? - ConvertTemperatureintoCelsius.info
To convert 110 Celsius to Fahrenheit, you can use the formula: (Celsius × 9/5) + 32. In this case, it would be (110 × 9/5) + 32, which equals 230.
Celsius and Fahrenheit are two different temperature scales used for measuring temperature. Celsius is used in most countries around the world, while the United States still uses the Fahrenheit
scale. Converting between the two is a common need, especially for those who travel or work in scientific fields.
So, what is 110 Celsius to Fahrenheit? 110 degrees Celsius is equal to 230 degrees Fahrenheit. This means that if you have a thermometer reading in Celsius and you want to know the equivalent
temperature in Fahrenheit, you can use the conversion formula to find out.
To understand the conversion formula, it’s important to know the freezing and boiling points of water on both scales. In the Celsius scale, water freezes at 0 degrees and boils at 100 degrees. In the
Fahrenheit scale, water freezes at 32 degrees and boils at 212 degrees. This means that there is a 180-degree difference between the boiling and freezing points in Celsius (100-0) and a 180-degree
difference between the boiling and freezing points in Fahrenheit (212-32).
To convert Celsius to Fahrenheit, you first multiply the Celsius temperature by 9/5 and then add 32. This accounts for the difference in the freezing and boiling points between the two scales.
Likewise, to convert Fahrenheit to Celsius, you first subtract 32 from the Fahrenheit temperature and then multiply by 5/9.
Understanding the formula and the relationship between the two scales can be helpful in everyday tasks and when studying scientific concepts. For example, if you know the temperature in Fahrenheit
but need to understand it in Celsius, you can use the same formula in reverse.
Knowing how to convert between Celsius and Fahrenheit can be particularly useful when traveling to a country that uses a different temperature scale. For instance, if you are accustomed to the
Fahrenheit scale and travel to Europe, where Celsius is used, you may need to convert temperatures to better understand the weather or adjust thermostats in your accommodations.
In scientific fields, understanding the conversion between Celsius and Fahrenheit is also important. Many experiments and research projects use temperature as a key variable, and knowing how to
convert between the two scales accurately is crucial. In fields such as chemistry, physics, and engineering, working with temperature data may require converting between Celsius and Fahrenheit on a
regular basis.
Overall, understanding the relationship between Celsius and Fahrenheit and knowing how to convert between the two scales is a valuable skill that can be used in a variety of contexts. Whether you’re
traveling internationally, working in a scientific field, or simply curious about the weather in different parts of the world, being able to convert temperatures from Celsius to Fahrenheit and vice
versa can be beneficial. | {"url":"https://converttemperatureintocelsius.info/what-is-110celsius-in-fahrenheit/","timestamp":"2024-11-05T22:48:47Z","content_type":"text/html","content_length":"73409","record_id":"<urn:uuid:ec59fc2e-5d7b-42c9-ac60-b6dc2608b710>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00858.warc.gz"} |
ACO Seminar
The ACO Seminar (2013–2014)
Mar. 27
, 3:30pm, Wean 8220
Mike Picollelli
, Carnegie Mellon University
The diamond-free process
For a fixed graph H, the H-free process is the following random graph process. Starting with n isolated vertices, at each step add a new edge chosen uniformly at random from the non-edges whose
addition does not create a (not necessarily induced) copy of H. This process terminates with a maximal H-free graph with a random number M(H) of edges. For most graphs H containing a cycle,
determining the likely asymptotic order of M(H) is an open problem. For H satisfying a particular density condition (strict 2-balancedness), likely lower bounds are known which are conjectured to be
optimal, and which have been shown to be for only a few special cases: K[3], K[4], and cycles of any fixed length. But far less is known for graphs H not satisfying that density condition, with the
smallest nontrivial example being the diamond graph, formed by removing an edge from K[4]. This talk will focus on the diamond-free process (for which we now know the right order of M(H)) and some
of the interesting and unexpected complications that arise in its analysis. | {"url":"https://aco.math.cmu.edu/abs-13-14/mar27.html","timestamp":"2024-11-05T06:35:32Z","content_type":"text/html","content_length":"2657","record_id":"<urn:uuid:d3aeb242-6719-4467-9623-a44955337828>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00835.warc.gz"} |
Rice's Theorem - (Mathematical Logic) - Vocab, Definition, Explanations | Fiveable
Rice's Theorem
from class:
Mathematical Logic
Rice's Theorem states that all non-trivial properties of recursively enumerable languages are undecidable. This means that if a property of a language is not true for all languages and not false for
all languages, then there is no algorithm that can decide whether any given language has that property. This theorem is pivotal in understanding the limits of computability and ties closely to other
foundational concepts like expressibility and the capabilities of computational models.
congrats on reading the definition of Rice's Theorem. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Rice's Theorem applies specifically to properties that are non-trivial; if a property is true for all or false for all languages, it is considered trivial and decidable.
2. The theorem highlights the limitations of algorithms in determining certain properties of languages, which connects to broader discussions about what can be computed.
3. Applications of Rice's Theorem can be found in various areas like programming language design and software verification, where certain properties cannot be guaranteed by any automated process.
4. The proof of Rice's Theorem relies on a reduction technique showing that if you could decide one non-trivial property, you could also decide another known undecidable property.
5. Understanding Rice's Theorem helps clarify why problems such as the Halting Problem are crucial in computational theory, as they serve as foundational examples of undecidability.
Review Questions
• How does Rice's Theorem illustrate the relationship between expressibility and decidability in computational theory?
□ Rice's Theorem demonstrates that non-trivial properties of recursively enumerable languages cannot be decided algorithmically. This highlights the limits of expressibility since any language
with such a property would require an algorithm to determine its classification. Since the theorem confirms that no such general algorithm exists for non-trivial properties, it reinforces the
notion that some concepts are inherently unexpressible within the confines of computability.
• Discuss how reduction techniques are utilized in proving Rice's Theorem and their importance in understanding computational limits.
□ Reduction techniques are essential in proving Rice's Theorem because they allow us to show that if one non-trivial property could be decided, then it would lead to contradictions regarding
other known undecidable problems. By demonstrating this relationship through logical reductions, we reinforce our understanding of computational limits and show how various problems are
interlinked in terms of their decidability. This approach is fundamental to theoretical computer science as it provides a method for establishing the boundaries of what can be computed.
• Critically analyze the implications of Rice's Theorem on real-world programming language design and software verification practices.
□ Rice's Theorem has profound implications on programming language design and software verification because it underscores the inherent challenges faced when trying to ascertain certain
properties about programs automatically. Since many desirable properties are non-trivial, developers must recognize that no universal algorithm can guarantee correctness or behavior
prediction for all cases. Consequently, this leads to reliance on testing and heuristics rather than definitive solutions, shaping how languages are designed with considerations for runtime
behaviors and ensuring that verification tools remain practical yet limited.
"Rice's Theorem" also found in:
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APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/mathematical-logic/rices-theorem","timestamp":"2024-11-15T00:26:00Z","content_type":"text/html","content_length":"161895","record_id":"<urn:uuid:20b4ab99-343b-4ea4-ac39-4d9206ea3d9e>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00489.warc.gz"} |
Flipping USB Connectors
You can order print and ebook versions of Think Bayes 2e from Bookshop.org and Amazon.
Flipping USB Connectors#
This notebook is one of the examples in the second edition of Think Bayes.
Click here to run this notebook on Colab.
I am not the first person to observe that it sometimes takes several tries to plug in a USB connector (specifically the rectangular Type A connector, which is not reversible). There are memes about
it, there are cartoons about it, and on Quora alone, people have asked about it more than once.
But I might be the first to use Bayesian decision analysis to figure out the optimal strategy for plugging in a USB connector. Specifically, I have worked out how long you should try on the first
side before flipping, how long you should try on the second side before flipping again, how long you should try on the third side, and so on.
Of course, my analysis is based on some modeling assumptions:
1. Initially, the probability is 0.5 that the connector is in the right orientation.
2. If it is, the time it takes to succeed follows an exponential distribution with a mean of 1.1 seconds.
3. Flipping the connector takes 0.1 seconds.
With that, we are ready to get started.
Continuous Updates#
The first step is to figure out the probability that the connector is in the right orientation as a function of how long you have been trying. For that, we can use a Bayes table, which is a form of
Bayes’s Theorem I use in Chapter 2 of Think Bayes.
The following function takes a sequence of hypotheses, prior probabilities, and likelihoods, and returns a pandas DataFrame that represents a Bayes table.
import pandas as pd
def bayes_table(hypos, prior, likelihood):
"""Make a table showing a Bayesian update."""
table = pd.DataFrame(dict(prior=prior, likelihood=likelihood), index=hypos)
table['unnorm'] = table['prior'] * table['likelihood']
prob_data = table['unnorm'].sum()
table['posterior'] = table['unnorm'] / prob_data
return table
Now suppose that the prior probability is 0.5 that the orientation of the connector is correct, and you have been trying for 0.9 seconds. What is the likelihood that you would have to try so long?
• If you are on the wrong side, it is 100%.
• If you are on the right side, it’s given by the survival function (complementary CDF) of the exponential distribution, which is \(\exp(-\lambda t)\), where \(\lambda\) is the rate parameter and \
(t\) is time.
The following function computes this likelihood:
import numpy as np
def expo_sf(t, lam):
"""Survival function of the exponential distribution."""
return np.exp(-lam * t)
We can use this function to compute the likelihood of trying for 0.9 seconds or more, given an exponential distribution with mean 1.1.
t = 0.9
mu = 1.1
lam = 1/mu
expo_sf(t, lam)
The result is the likelihood of the data, given that the orientation of the connector is correct.
Now let’s make a Bayes table with two hypotheses – the connector is either the right way or the wrong way – with equal prior probabilities.
hypos = ['Right way', 'Wrong way']
prior = [1/2, 1/2]
And here is the likelihood of the data for each hypothesis:
likelihood = [expo_sf(t, lam), 1]
Putting it together, here’s the Bayes table.
bayes_table(hypos, prior, likelihood)
│ │prior│likelihood │ unnorm │posterior│
│Right way│0.5 │0.441233 │0.220617│0.30615 │
│Wrong way│0.5 │1.000000 │0.500000│0.69385 │
After 0.9 seconds, the probability is about 69% that the orientation of the connector is wrong, so you might want to think about trying the other side.
But if it takes 0.1 seconds to flip, maybe you should keep trying a little longer. To figure out when to flip, let’s do the same analysis again for general values of \(\lambda\) and \(t\).
To minimize human error, I’ll use Sympy to do the algebra. Here are the symbols I’ll use.
from sympy import symbols, exp
t, lam, p, q, r = symbols('t lam p q r')
Here’s the likelihood again, using the symbols.
likelihood = [exp(-lam * t), 1]
And here’s the Bayes table, using \(p\) and \(q\) for the prior probabilities of the hypotheses.
prior = [p, q]
table = bayes_table(hypos, prior, likelihood)
│ │prior│likelihood │ unnorm │ posterior │
│Right way│p │exp(-lam*t)│p*exp(-lam*t)│p*exp(-lam*t)/(p*exp(-lam*t) + q) │
│Wrong way│q │1 │q │q/(p*exp(-lam*t) + q) │
From the table I’ll select the posterior probability that the orientation is correct.
expr = table.loc['Right way', 'posterior']
\[\displaystyle \frac{p}{p + q e^{lam t}}\]
You might recognize this as a form of the logistic function; we can compute it like this:
def logistic(p, lam, t):
q = 1-p
return p / (p + q * np.exp(lam * t))
Let’s see what that looks like for a range of values of t, assuming that the prior probability is p=0.5.
import matplotlib.pyplot as plt
ts = np.linspace(0, 4)
ps = logistic(p=0.5, lam=1/mu, t=ts)
plt.plot(ts, ps)
plt.xlabel("How long you've been trying (seconds)")
plt.ylabel("Probability the orientation is right");
After a few seconds of fiddling, you should be reasonably convinced that the orientation is wrong.
Now, let’s think about turning belief into action. Let me start with a conjecture: I suspect that the best strategy is to try on the first side until the probability of correct orientation drops
below some threshold (to be determined), then try on the second side until the probability drops below that threshold again, and repeat until success.
To test this strategy, we will have to figure out how long to try as a function of the prior probability, p, and the threshold probability, r. Again, I’ll make Sympy do the work.
Here’s the equation that sets the posterior probability, which we computed in the previous section, to r.
from sympy import Eq, solve
eqn = Eq(expr, r)
\[\displaystyle \frac{p e^{- lam t}}{p e^{- lam t} + q} = r\]
And here’s the solution for t in terms of p, q, r, and lam.
\[\displaystyle \frac{\log{\left(\frac{p \left(1 - r\right)}{q r} \right)}}{lam}\]
And here’s how we can express this solution in terms of the prior and posterior odds.
def wait_time(p, lam, r):
q = 1-p
prior_odds = p / q
posterior_odds = r / (1-r)
return np.log(prior_odds / posterior_odds) / lam
Let’s see what that looks like for a range of values of r, assuming that the prior probability is p=0.5.
rs = np.linspace(0.05, 0.5)
ts = wait_time(p=0.5, lam=1/mu, r=rs)
plt.plot(rs, ts, color='C2')
plt.xlabel("Probability the orientation is right")
plt.ylabel("How long to keep trying (seconds)");
When the threshold is low, we have to wait a few seconds to reach it. As the threshold increases, the time to reach it decreases. We’ll use this function in the next section to simulate the strategy.
As a step toward optimization, let’s run a simulation. The following function takes as parameters:
• correct: A Boolean indicating if the orientation is correct.
• p: The prior probability that the orientation is correct.
• lam: The rate parameter for the distribution of time until success.
• r: The threshold for the posterior probability.
• flip: The time it takes to flip the connector, in seconds.
• trace: A list that indicates how much time we have spent, so far, trying and flipping.
It runs the simulation and returns a sequence of waiting and flipping times. The sum of this sequence is the total time it took to connect. And we can use the length of the sequence to figure out how
many times we had to flip.
def simulate(correct, p, lam, r, flip, trace):
# figure out the maximum time we should try before flipping
wait = wait_time(p, lam, r)
# if we're on the correct side, see if we succeed before time's up
if correct:
t = np.random.exponential(1/lam)
if t < wait:
# if so, update and return the trace
return trace + [t]
# if time expired, add the wait time and flip time to the trace
# and make a recursive call to continue the simulation
return simulate(not correct, 1-r, lam, r, flip, trace + [wait, flip])
Here’s a test run, starting on the correct side.
simulate(correct=True, p=0.5, lam=1/mu, r=0.2, flip=0.1, trace=[])
And here’s a run where we start on the wrong side.
simulate(correct=False, p=0.5, lam=1/mu, r=0.2, flip=0.1, trace=[])
[1.5249237972318797, 0.1, 0.8563018209476607]
The following function runs the simulation many times with initial probability p=0.5, starting in the right orientation half the time.
It returns two arrays, containing the length of the trace and the total duration for each simulation.
def run_simulations(lam, r, flip, iters=20000, flag=None):
res = []
for i in range(iters):
correct = i%2 if flag is None else flag
trace = simulate(correct, 0.5, lam, r, flip, [])
res.append((len(trace), sum(trace)))
return np.transpose(res)
Here’s the average total duration with threshold probability r=0.25.
lengths, totals = run_simulations(lam=1/mu, r=0.25, flip=0.1)
With this threshold, it takes about 2 seconds to connect, on average.
Now let’s see how the average duration varies as we sweep through a range of values for the threshold probability, r:
rs = np.linspace(0.15, 0.4, 21)
array([0.15 , 0.1625, 0.175 , 0.1875, 0.2 , 0.2125, 0.225 , 0.2375,
0.25 , 0.2625, 0.275 , 0.2875, 0.3 , 0.3125, 0.325 , 0.3375,
0.35 , 0.3625, 0.375 , 0.3875, 0.4 ])
res = []
for r in rs:
lengths, totals = run_simulations(lam=1/mu, r=r, flip=0.1)
res.append((r, totals.mean()))
Show code cell content Hide code cell content
from statsmodels.nonparametric.smoothers_lowess import lowess
def make_lowess(series):
"""Use LOWESS to compute a smooth line.
series: pd.Series
returns: pd.Series
endog = series.values
exog = series.index.values
smooth = lowess(endog, exog)
index, data = np.transpose(smooth)
return pd.Series(data, index=index)
Show code cell content Hide code cell content
def plot_series_lowess(series, color):
"""Plots a series of data points and a smooth line.
series: pd.Series
color: string or tuple
series.plot(lw=0, marker='o', color=color, alpha=0.5)
smooth = make_lowess(series)
smooth.plot(label='_', color=color)
Here’s what the results look like.
rs, ts = np.transpose(res)
series = pd.Series(ts, rs)
plot_series_lowess(series, 'C1')
plt.xlabel("Threshold probability where you flip (r)")
plt.ylabel("Average total duration (seconds)");
The optimal value of r is close to 0.3. With that threshold we can see how long we should try on the first side, starting with prior probability p=0.5.
r_opt = 0.3
wait_time(p=0.5, lam=1/mu, r=r_opt)
With the given values of lam and flip, it turns out the optimal time to wait is about 0.9 seconds.
If we have to flip, the prior probability for the second side is p=1-r, so we have to wait twice as long for the posterior probability to get down to r.
wait_time(p=1-r_opt, lam=1/mu, r=r_opt)
How many flips?#
Now let’s run the simulations with the optimal value of r and see what the distributions look like for the total time and the number of flips.
lengths1, totals1 = run_simulations(lam=1/mu, r=r_opt, flip=0.1, flag=True)
lengths2, totals2 = run_simulations(lam=1/mu, r=r_opt, flip=0.1, flag=False)
Here’s the distribution of total time, represented as a CDF.
Show code cell content Hide code cell content
import empiricaldist
except ImportError:
!pip install empiricaldist
from empiricaldist import Cdf
Cdf.from_seq(totals1).plot(lw=2, label='Right the first time')
Cdf.from_seq(totals2).plot(lw=2, label='Wrong the first time')
plt.xlabel('Total time to connect (seconds)')
plt.title('Distribution of total time to connect')
totals1.mean(), totals2.mean()
(2.2006236558767154, 2.616228241925388)
np.percentile(totals1, 90), np.percentile(totals2, 90)
(4.601034595944718, 5.636579992175272)
np.append(totals1, totals2).mean()
The average is about 2.4 seconds, but occasionally it takes much longer!
And here’s the distribution for the total number of flips.
from empiricaldist import Pmf
flips1 = (lengths1-1) // 2
pmf1 = Pmf.from_seq(flips1) / 2
pmf1.bar(alpha=0.7, label='Right the first time')
flips2 = (lengths2-1) // 2
pmf2 = Pmf.from_seq(flips2) / 2
pmf2.bar(alpha=0.7, label='Right the second time')
plt.xlabel('How many times you have to flip')
plt.title('Distribution of number of flips')
lengths = np.append(lengths1, lengths2)
flips = (lengths-1) // 2
│ │ probs │
│0.0 │0.282925 │
│1.0 │0.407050 │
│2.0 │0.177200 │
│3.0 │0.075200 │
│4.0 │0.032575 │
The probability of getting it right on the first try is only about 28%. That might seem low, because the chance of starting in the right orientation is 50%, but remember that we have a substantial
chance of flipping even if we start in the right orientation (and in that case, we have to flip at least twice).
The most common outcome is that we have to flip once, about 40% of the time. And the probability of the notorious double flip is about 18%.
Fortunately, it is rare to flip three or more times.
With that, I think we have solved the USB connector problem.
1. For given parameters lam and flip, we can find the threshold probability, r, that minimizes the average time to connect.
2. Given this optimal value, we can estimate the distribution of total time and the number of times we have to flip.
Sadly, all of this fun is gradually being spoiled by the encroachment of the USB-C connector, which is reversible.
If you like this article, you might also like the second edition of Think Bayes.
Copyright 2021 Allen Downey
Code: MIT License
Text: Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) | {"url":"https://allendowney.github.io/ThinkBayes2/usb.html","timestamp":"2024-11-07T22:24:46Z","content_type":"text/html","content_length":"80655","record_id":"<urn:uuid:7c2d312b-2e6e-4c26-b88d-3eafd752b988>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00372.warc.gz"} |
Classifying Triangles
Classifying Triangles Worksheets
Here's introducing the concept of classifying triangles recommended for students in grade 4 through grade 7. The printable worksheets are replete with practice exercises designed to give the child an
advantage in identifying triangles based on sides and angles sorted into: with measures, no measures and congruent parts. Just a single click and our free worksheets are yours!
Select the Measurement Units
Classifying Triangles Based on Side Measures
In these pdf worksheets for 4th grade and 5th grade kids, learn to distinguish between various triangles based on the length of the sides, and tell whether the triangle provided with measures is an
equilateral, scalene or isosceles triangle.
Classifying Triangles Based on Sides | Congruent Parts
Classify the triangles based on the congruent sides. An equilateral triangle has 3 congruent sides, an isosceles triangle has 2 congruent sides and triangles with unequal side lengths are scalene.
Classifying Triangles Based on Angle Measures
For an acute triangle, all angles are <90°, a right triangle has one angle =90° and an obtuse triangle has one angle >90°. Watch for the angle measure of a triangle in order to determine which
is which. These handouts are ideal for children in grade 4 and grade 5.
Classifying Triangles Based on Angles | No Measures
Beef up your practice with this bundle of printable worksheets for grade 6 represented with no measures. Use a protractor to evaluate the angle measure of each triangle to classify it as acute,
obtuse and right triangle.
Identifying Triangles Based on Sides and Angles
Classify each triangle based on sides and then classify further based on angles. Example, for a scalene triangle, classify it as scalene acute, scalene obtuse or scalene right based on angles.
Consolidate your knowledge of the classification of triangles with this pdf worksheet for 6th grade and 7th grade kids. The six columns of the table are named as equilateral, isosceles, scalene,
acute, obtuse and right. Examine the triangle given in each row and check the property boxes that best suit the triangle. | {"url":"https://www.mathworksheets4kids.com/classifying-triangles.php","timestamp":"2024-11-03T13:21:41Z","content_type":"text/html","content_length":"44187","record_id":"<urn:uuid:be17cc83-d4a3-4f87-9559-e902fd01f85e>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00103.warc.gz"} |
How Advisors Can Communicate Monte Carlo Results To Clients
Executive Summary
Financial advisors often use Monte Carlo simulation in their financial planning process, which (as is commonly found in major financial planning software packages) traditionally presents the results
of the projection in terms of probability of success or failure (with ‘success’ being defined as an iteration of the plan where the client doesn’t run out of money, and ‘failure’ signifying the
However, some commentators have taken issue with this framing, particularly as it concerns the way that Monte Carlo results are presented to clients. Most significantly, the ‘success/failure’ framing
fails to capture the reality that retirees, when facing an unlucky sequence-of-returns scenario which could result in their running out of money, can and often do make adjustments to their spending
that allow them to avoid that unfortunate outcome.
To better reflect this reality, the phrase “probability of adjustment” has emerged as a commonly suggested alternative to “probability of success”. While representing an improvement over the
original, however, “probability of adjustment” itself can be prone to ambiguity and misinterpretation without being clear about what type of adjustment might be needed, and what the outcome might be
if that adjustment weren’t made.
A Monte Carlo simulation can tell us, with the benefit of hindsight, exactly which iterations of a plan would have ended with the retiree running out of money. But in reality, retirees do not have
the ability to know which iteration (if any) they are on, and in many instances will likely make adjustments in cases where, in hindsight, no adjustment was strictly necessary. As a result, simply
replacing “probability of success” with “probability of adjustment” when communicating Monte Carlo results can significantly underestimate the likelihood that a client will actually make an
adjustment at some point, since clients (and advisors) do not have the benefit of knowing when an adjustment is ‘truly’ necessary.
Likewise, if an advisor were to recommend a dynamic spending strategy based on Monte Carlo simulations (such as adjusting spending to maintain a constant probability-of-success level), the
“probability of adjustment” framing can skew even further from reality, since preserving a consistent probability of success often calls for relatively frequent adjustments in spending. For instance,
maintaining a 70% probability of success level – implying only a 30% probability of adjustment – would in reality have required downward spending adjustments in nearly 100% of all historical
scenarios, which would understandably have caused confusion for many clients if the advisor had used the standard “probability of success/adjustment” framing!
Ultimately, the key point is that outcomes, not probabilities, are what matter to clients, and any way of communicating Monte Carlo results should be clear about what those results mean in terms of
real spending to the client. Though “probability of adjustment” is an improvement over “probability of failure”, it can still greatly underestimate the probability of actual spending adjustments,
especially when dynamic spending strategies are involved. In those cases, it may make sense to avoid framing Monte Carlo results in terms of probabilities entirely, but rather to communicate in terms
of the actual dollar spending adjustments that would be triggered in specific scenarios – which is what really matters to the client in the end.
Many articles have noted a number of communication disadvantages associated with using “probability of success” to report Monte Carlo results. For instance, our brains may struggle to understand how
to interpret probabilistic results, the implication of ‘failure’ in retirement may exacerbate client fears, clients may succumb to the ‘wrong side of maybe’ fallacy and judge advisors more harshly
than they should when difficult sequences of returns are encountered, among others.
A major issue with the success/failure framing is that it is overly binary and fails to capture the reality that retirees can adjust when needed, and that it often only takes small spending
adjustments to keep a plan on track. As a result, it has been suggested that a ‘probability of adjustment’ framework, instead of one based on ‘probability of success’, may better convey the actual
consequences for retirees.
Furthermore, some experimental research has even found that there are a number of advantages to adjustment-framing over success-framing when it comes to reporting Monte Carlo results, including
improved client emotional responses, a better understanding of plan results, and even improved perceptions of advisors themselves.
And yet, while there are advantages to framing results about probability of adjustment rather than probability of success, there are easily misunderstood aspects of ‘probability of adjustment’ that
can lead to confusion among both advisors and clients.
The Dual Meanings Of “Probability Of Adjustment”
At the heart of potential misunderstandings around what probability of adjustment means is that there are two equally valid ways that one might interpret the phrase:
1. Probability that downward adjustment would have been needed to avoid depleting a portfolio.
2. Probability that downward adjustment would have been triggered using a dynamic strategy.
The phrase ‘probability of adjustment’ is itself ambiguous, and it is easy to see how one might even mistakenly conclude that the two interpretations above are essentially the same (or at least
similar enough). However, the reality is that these are extremely different concepts, and only one is synonymous with “probability of success” in a Monte Carlo simulation.
First Meaning: Downward Adjustment Needed To Avoid Depleting A Portfolio
First, consider the probability that a downward spending adjustment would have been needed to avoid depleting a portfolio. This metric is synonymous with the traditional probability of success metric
that dominates Monte Carlo simulation.
Here, we are effectively saying, “If we look back after the fact, what percentage of the time was a portfolio depleted?” Since these were the only iterations within the Monte Carlo simulation that
actually ran out of money, then, at the same time, these were the scenarios that were ‘failures’ by traditional reporting.
Notably, this portfolio-depletion perspective is an entirely after-the-fact metric. It does not encompass the totality of cases where someone using a dynamic spending strategy would have made an
adjustment, and that’s where the confusion can arise.
For example, suppose Jessica is a financial advisor who is running a Monte Carlo simulation and receives a 90% probability of success result for her client. Jessica has read some about the benefits
of using probability of adjustment in lieu of probability of success, and therefore tells her client:
Using the strategy we have outlined here, the results of our analysis suggest that there’s a 10% chance you would need to make an adjustment by reducing your spending in the future to avoid
running out of money.
That might seem right if someone is thinking about using ‘adjustment’ in lieu of ‘success’ (or, more precisely, failure), but this is not quite right and, unfortunately, could paint a wildly
inaccurate picture for Jessica’s client.
A more accurate reporting might have been something along the lines of the following (where, for the sake of argument, we are assuming that ‘failure’ is defined as depleting the portfolio, although
certainly some other targeted estate balance could be used):
Using the strategy we have outlined here, the results of our analysis suggest in about 10% of simulated cases a downward spending adjustment would have been needed to avoid completely depleting a
This might seem like splitting hairs, but the differences here are actually very significant. The major risk here is that the client could walk away thinking that there is only a roughly 1-in-10
chance that they would need to reduce their spending in the future using a dynamic strategy, but this would be way off the mark (as we’ll take a look at later), since the reality is that there are
many instances when a prudent individual who doesn’t have the benefit of hindsight would have made adjustments that they ultimately did not need to be made.
Second Meaning: Downward Adjustment Triggered Using A Dynamic Strategy
Second, let’s consider ‘probability of adjustment’ as the probability that a downward adjustment would have been triggered using a dynamic strategy.
This is the meaning of adjustment that is actually analogous to the likelihood of needing to make a future downward spending adjustment and, as mentioned above, is a far different concept than the
after-the-fact view of how many scenarios did fully run out of money.
In this case, we don’t have the benefit of hindsight. Suppose Jessica’s client, from the earlier example, starts her journey and hits a terrible initial sequence of returns, and her probability of
success falls all the way to 50% only 5 years into her plan.
Would it be prudent for her to make an adjustment? Most advisors would likely agree that it would be, as the odds that she does end up depleting her portfolio are higher than many might feel
comfortable with. However, note that even at this point, what the Monte Carlo results are telling us is that 1-out-of-2 times, we would expect that she wouldn’t need to make a downward adjustment.
In other words, while the prudent thing to do here may be for Jessica’s client not to roll the dice and hope that she ends up with the 1-out-of-2 outcome (i.e., she should make an adjustment now to
more prudently manage her risk), the reality is that in a non-trivial number of Monte Carlo iterations Jessica was not depleting her portfolio despite the bleak outlook.
Let’s suppose Jessica was in one of the 1-out-of-20 scenarios that would have turned out fine, but that she made an adjustment anyway. Essentially, this is a ‘false positive’ result where a spending
reduction is a prudent action, even though it ended up not being called for. Furthermore, note that such false positives will likely occur even at higher probability-of-success levels, such as 10%,
50%, 70% (as one prior study found that many advisors treat 70% as a minimally acceptable threshold for ongoing planning success levels), and higher.
Or, to put it simply, sometimes it’s prudent to make adjustments even when we would find out after the fact that it ended up not being necessary to do so.
Anecdotally, this seems to be what a non-trivial number of advisors intuit or presume the ‘probability of adjustment’ is telling them. Unfortunately, that belief can lead advisors (and their clients)
wildly astray.
How Different Are The Two Ways Of Thinking About Probability Of Adjustment?
At this point, it is reasonable to wonder how different these two concepts – namely, the consideration of probability of adjustment as a need to avoid portfolio depletion, versus a trigger set in
place by a dynamic strategy – really are if we were to quantify them.
For instance, if it’s true that false positives occur but that they only occur very rarely, then maybe this is all much ado about nothing. Perhaps a 5% chance of future downward adjustment becomes a
7% chance of future downward adjustment, and, at that point, do we really care? Most people probably struggle to make a discernable distinction between 5% and 7% anyway, so the harm may be minimal if
we are still getting roughly the same idea conveyed even if the numbers are slightly different.
On the other hand, if it turned out that 90% of the time people using a dynamic plan experienced a downward adjustment even when starting at a 95% probability of success, then it likely wouldn’t be
wise to set expectations poorly by framing the outcome as only having a 5% chance of future downward adjustment.
To test whether false positives using a dynamic spending strategy are something we need to be mindful of, we can use a combination of historical simulation and point-in-time Monte Carlo calculations
while we walk individuals through historical sequences. For this particular analysis, we’ll examine several strategies that hold probability-of-success levels constant. For instance, if we are
examining a constant 95% probability-of-success level, then we use Monte Carlo to set the spending level where it would be based on the assumed starting point, then step an individual forward one
month in the historical sequence we are examining based on what actually happened in the market, and then recalculate their new spending level that would maintain a 95% probability of success based
on distributions they took and what has happened in the market.
We’ll be using 30-year retirement periods and repeating this process for each 30-year sequence from 1871 to the present (see here for a fuller explanation of the methodology used here).
Nerd Note:
From a statistical perspective, it would be ideal to use an in-sample approach and change capital market assumptions based on the information that one actually had available to them at each point in
history as it unfolded, so that we aren’t ‘cheating’ by giving someone access to data they would otherwise not have. However, for simplicity, we have used a single out-of-sample set of capital market
assumptions based on historical averages over the entire time period.
For this analysis, it is also important to define what we mean by a “downward adjustment.” One approach could be to use any downward adjustment, and while that is a defensible definition, it won’t
provide a tremendously insightful analysis because even a single down period in an entire 30-year sequence would, by definition, result in a downward adjustment.
Instead, we will be defining a downward adjustment as a spending reduction relative to an inflation-adjusted initial spending level. In other words, we consider an adjustment to be downward if
someone actually has to cut back to below where they initially started spending in real dollars.
As an example, suppose John starts spending at $8,000 per month (at a 95% probability of success level). His spending goes up to $10,000 per month as the market rises (and, therefore, the level of
spending to maintain a 95% probability of success has gone up as well). However, the market later pulls back and John’s new 95% probability-of-success monthly spending level goes down from $10,000 to
$8,500. In this case, we aren’t considering this to be “downward adjustment” since the adjustment was not down overall relative to the initial real spending level of $8,000.
However, suppose that, to maintain a constant 95% probability of success, John’s spending starts at $8,000, goes up to $10,000, and then falls to $7,900 (all adjusted for inflation). In this case, we
would consider it a downward adjustment, because now John’s new spending level of $7,900 is less than his initial spending of $8,000 in real dollars.
Furthermore, we would still consider this a historical sequence that experienced a downward adjustment even if John’s spending subsequently rose above $8,000 in real spending and never saw another
reduction below $8,000 throughout the entire 30-year-sequence.
To examine this more closely, let’s consider a scenario involving a hypothetical client:
• Hank (66) and Marie (64) are married.
• They have $1 million invested in a 60/40 portfolio.
• They pay 1.2% in all-in weighted average fees.
• Their combined monthly income is $3,500 in Social Security.
Furthermore, we assume the following:
• They want to leave a $200,000 (adjusted for inflation) legacy to their children.
• They are willing to make adjustments to their spending and do so for whatever adjustment is necessary each month. (Note: To keep assumptions simple, adjustments are made monthly no matter how
small the adjustment, but obviously in the real world there would probably be more lag with slightly fewer but potentially larger adjustments, such as reviewing and potentially adjusting spending
at each annual review meeting).
Let’s start by considering a scenario where Hank and Marie aim to maintain a constant 20% probability of success. Not surprisingly, this level is quite aggressive, and downward adjustments are very
common, occurring in 100% of historical sequences.
Likewise, a constant 50% probability of success also saw downward (real) adjustments 100% of the time.
However, both of the probability of success thresholds above are not commonly used among advisors (despite 50% probability of success not actually being as bad of a target as commonly believed,
especially for advisors who are doing ongoing planning rather than one-time planning), so how would the same analysis fare as we start to get into higher, more commonly used probability of success
At a constant 70% probability of success – roughly the minimum that advisors have reported in prior studies being comfortable using – we still see very high adjustment rates at 99.57% of the time. In
other words, despite only 30% of scenarios ultimately running out of money if no adjustments were made, we still see that proactive adjustments were called for in 99.57% of historical scenarios when
planning to a constant 70% probability of success!
And this is important, because if an advisor is trying to set expectations by telling their client that the client only has a 30% probability of needing to adjust their spending – when the reality is
that 99.57% of the time, historically, they would have had to adjust spending when looking forward proactively – then that presents a very major opportunity for setting expectations poorly!
And even if we bump the constant probability of success level all the way up to 95%, we still see that about 96.3% of plan scenarios experienced a downward adjustment when planning proactively! In
other words, even though only 5% of iterations modeled resulted in fully depleting a portfolio, it was still the case that over 96% of iterations modeled experienced some reduction in real spending
throughout the full retirement sequence – suggesting that describing a 95% probability of success as having a 5% “probability of adjustment” could be wildly misinterpreted.
What About Big Downward Adjustments?
While the numbers above might be surprising, it is certainly true that in some of these cases the downward adjustments may have been trivial. For instance, as described above, even a reduction of $1
in monthly income would result in a sequence that has a downward adjustment, when, in reality, that’s really not a meaningful downward adjustment from a practical perspective.
But what if we use something more substantial to define a downward adjustment? Say, 35% less (adjusted for inflation) than initial spending levels?
In this case, using the same 20%, 50%, 70%, and 95% probability of success thresholds, we see the following results:
While the numbers here are less extreme, they still go a long way toward highlighting the risk of telling someone that they only have, say, a 5% probability of a downward adjustment could lead to a
lot of confusion. Even when using a 95% constant probability of success, a full 15% of the time saw reductions of 35% or more!
How Advisors Can Communicate Monte Carlo Results
Ultimately, the key takeaway here is that referring to the “probability of adjustment” as the probability that downward adjustment is needed to avoid portfolio depletion is not the same thing as
referring to it as the probability that downward adjustment is called for in a dynamic retirement spending approach. This is an important distinction, because “probability of adjustment” is ambiguous
and arguably could refer to either one of these, when the reality is that only the former is actually synonymous with “probability of success” from a Monte Carlo simulation.
And, as noted above, we see that dynamic strategies do often call for reductions at some point in time. Even when starting out at a 95% probability of success level (and maintaining that throughout
the plan), we still saw real reductions in spending in as many as 96% of scenarios.
Granted, planning to a constant probability of success is likely not a wise thing to be doing in the first place. We should expect that, as the market ebbs and flows, probability of success levels
will rise and fall accordingly. Building in some buffers before making adjustments could help avoid both increases and decreases that are simply the result of short-term noise in the market. But
where should a spending decrease be triggered? 70% probability of success? 50% probability of success? Even lower?
While some further work in this area is still needed, it is worth noting that trigger points below 50% probability of success are not as outlandish as some might think when used as part of an ongoing
planning process. In fact, trigger points of 40%, 30%, or even 20% probability of success might be reasonable as part of an ongoing planning strategy, and even planning to a constant 20% probability
of success results in far fewer differences than planning to a constant 95% probability of success than most might think.
For instance, see the chart below that summarizes the differences in maximum and minimum spending levels over 30-year sequences when planning to a constant 95% probability of success versus a
constant 20% probability of success:
What’s particularly striking about these results is how similar spending levels are between probability of success levels held constant at both 95% and 50%. In other words, even if a retiree accepts
a 50% probability of success, as long as they’re willing to make the adjustments along the way, the minimum and maximum spending levels are still quite similar!
Of course, as discussed in our earlier article, the caveat is that for the retiree who starts with a lower probability of success and a higher initial spending level, falling to a ‘similar’ minimum
spending level will still reflect a much bigger relative cut in spending from where their retirement started and will be a bigger adjustment to handle.
Total Risk Guardrails Offer Better Communication Focal Points
When thinking about how to describe probability of adjustment to clients, it is also worth considering whether it really belongs as a focal point in the first place. As we’ve covered here, there’s a
lot of risk of confusion associated with the use of probability of adjustment, even if it is superior to probability of success.
An approach like total risk guardrails might instead provide much better dynamics for actually communicating results that matter. For instance, we could have a total risk-based guardrails plan that
looks something like the one below:
Here, we are still using probability of success as a key metric for setting these initial spending levels and guardrails, but the results are presented at a much better level of abstraction for the
client by focusing on what actually matters to them more.
Instead of simply saying, “Mr. and Mrs. Client, our analysis suggests you can spend $52,000 per year now. This plan was run at a level of only needing to make an adjustment to avoid depleting your
portfolio in about 5% of scenarios considered,” clients are being given specific guidelines for when a reduction would be called for in terms of actual dollars.
For instance, “Mr. and Mrs. Client, your portfolio is currently at $1 million and we would suggest that you can spend about $52,000 per year. However, if your portfolio falls to $740,000, then we
would suggest cutting your income back to $48,500, which is about $300 per month.” Of course, this is also an opportunity to note the upside potential that would come from increasing spending close
to $14,000 (from $51,900 to $65,800) if the portfolio were to increase in value by $270,000.
Ultimately, this is far more practical for the client, even though advisors could be running the plan with the exact same software and using the exact same probability of success levels (see here for
a fuller explanation of how typical Monte Carlo software could be used to generate probability-of-success-driven guardrails).
The key point is that communicating results from a guardrails-based plan in terms of dollars to clients is likely far more effective for communication than reporting a probability of adjustment
metric, given the ambiguity and confusion that exists around that term itself. So, despite the real advantages of talking in terms of probability of adjustment rather than probability of success,
making that shift may only be one small step toward even better ways to focus on and report plan results for clients.
For more client communication strategies related to retirement planning, join Derek for his upcoming webinar, Improving Monte Carlo In Retirement Planning: Best Practices For Better Conversations, on
July 5th. More information is available here.
1. James Hildreth says
Many articles shave recently recommended updating a Monte Carlo simulation that was done prior to retirement. I have been doing that but I am not sure what returns I should be using in the
update. I have yet to find an article that says what parameters to use for the update. My initial analysis was based on a 25 year retirement ending in 1990 so I used 25 year average return and
sigma for stocks and bonds. Now I am 7 years into retirement so if I want to update what returns do I use: 18 year average or 25 year average? Also I assume I should start with current balances
and updated future fixed income and expense projections rather than go back to the 1990 balances combined with first 7 years actual fixed income and expenses and revised future projections. Right
now I am updating starting with current balances and future projections of fixed income and expenses and still using 25 year average return data (but still ending simulation in 1990).
I suppose it would be valid to run the revised simulation to 1997 with the 25 year returns instead, assuming that if I can make it to 1997 I’d be OK at 1990. What’s the right way to do it?
Thanksuch as this one
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65 research outputs found
The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), we calculate their full isometry
groups Isom(M), determine which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, and verify that the inclusion of Isom(M) to Diff(M) is a bijection on path
Fix a free, orientation-preserving action of a finite group G on a 3-dimensional handlebody V. Whenever G acts freely preserving orientation on a connected 3-manifold X, there is a G-equivariant
imbedding of V into X. There are choices of X closed and Seifert-fibered for which the image of V is a handlebody of a Heegaard splitting of X. Provided that the genus of V is at least 2, there are
similar choices with X closed and hyperbolic
D. Margalit and S. Schleimer found examples of roots of the Dehn twist about a nonseparating curve in a closed orientable surface, that is, homeomorphisms whose nth power is isotopic to the Dehn
twist. Our main theorem gives elementary number-theoretic conditions that describe the values of n for which an nth root exists, given the genus of the surface. Among its applications, we show that n
must be odd, that the Margalit-Schleimer roots achieve the maximum value of n among the roots for a given genus, and that for a given odd n, nth roots exist for all genera greater than (n-2)(n-1)/2.
We also describe all nth roots having n greater than or equal to the genus.Comment: 15 pages, 6 figure
A limit point p of a discrete group of Mobius transformations acting on S^n is called a concentration point if for any sufficiently small connected open neighborhood U of p, the set of translates of
U contains a local basis for the topology of S^n at p. For the case of Fuchsian groups (n = 1), every concentration point is a conical limit point, but even for finitely generated groups not every
conical limit point is a concentration point. A slightly weaker concentration condition is given which is satisfied if and only if p is a conical limit point, but not all conical limit points satisfy
it. Examples are given that clarify the relations between various concentration conditions.Comment: 24 pages, 7 figure
For a Heegaard surface F in a closed orientable 3-manifold M, H(M,F) = Diff(M)/Diff(M,F) is the space of Heegaard surfaces equivalent to the Heegaard splitting (M,F). Its path components are the
isotopy classes of Heegaard splittings equivalent to (M,F). We describe H(M,F) in terms of Diff(M) and the Goeritz group of (M,F). In particular, for hyperbolic M each path component is a classifying
space for the Goeritz group, and when the (Hempel) distance of (M,F) is greater than 3, each path component of H(M,F) is contractible. For splittings of genus 0 or 1, we determine the complete
homotopy type (modulo the Smale Conjecture for M in the cases when it is not known).Comment: Minor rewriting as suggested by referee, no change in mathematical content. To appear in J. Reine Angew.
Around 1960, R. Palais and J. Cerf proved a fundamental result relating spaces of diffeomorphisms and imbeddings of manifolds: If V is a submanifold of M, then the map from Diff(M) to Imb(V,M) that
takes f to its restriction to V is locally trivial. We extend this and related results into the context of fibered manifolds, and fiber-preserving diffeomorphisms and imbeddings. That is, if M fibers
over B, with compact fiber, and V is a vertical submanifold of M, then the restriction from the space FDiff(M) of fiber-preserving diffeomorphisms of M to the space of imbeddings of V into M that
take fibers to fibers is locally trivial. Also, the map from FDiff(M) to Diff(B) that takes f to the diffeomorphism it induces on B is locally trivial. The proofs adapt Palais' original approach; the
main new ingredient is a version of the exponential map, called the aligned exponential, which has better properties with respect to fiber-preserving maps. Versions allowing certain kinds of singular
fibers are proven, using equivariant methods. These apply to almost all Seifert-fibered 3-manifolds. As an application, we reprove an unpublished result of F. Raymond and W. Neumann that each
component of the space of Seifert fiberings of a Haken 3-manifold is weakly contractible.Comment: 43 pages, LaTeX2
For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel,
and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. We generalize their construction and calculate the slope invariants for the resulting middle tunnels. In particular, we
obtain the slope sequence of the original example of Goda, Hayashi, and Ishihara.Comment: 20 pages, 11 figure
For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel,
and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. In a previous paper, we generalized their construction and calculated the slope invariants for the resulting examples. We
give an iterated version of the construction that produces many more examples, and calculate their slope invariants. If one starts with the trivial knot, the iterated constructions produce all the
2-bridge knots, giving a new calculation of the slope invariants of their tunnels. In the final section we compile a list of the known possibilities for the set of tunnels of a given tunnel number 1
knot.Comment: The results of the paper are unchanged. The list of known tunnel phenomena has been enlarged to include new possibilities seen in examples recently found by John Berge, after reading
the previous version of the paper. The previous list was presented as a conjecture of all possibilities, but the new list is presented only as list of known phenomena, prompting the change of titl
The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable 3-dimensional handlebody of genus g can be enumerated in terms of sets of
generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g-1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For abelian and
dihedral G, this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for solvable groups and for the groups
PSL(2,3^p) with p prime. For all G, there is only one equivalence class of actions on the genus g handlebody if g is at least 1+r(G)|G|, where r(G) is the maximal length of a chain of subgroups of G.
There is a stabilization process that sends an equivalence class of actions to an equivalence class of actions on a higher genus, and some results about its effects are obtained
Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R or the universal cover of SL(2,R). Our main
result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to the circle, according as the center of the fundamental
group of M is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert
fiberings of M for all compact orientable aspherical 3-manifolds. We also prove that when the base orbifold of M is hyperbolic with underlying manifold the 2-sphere with three cone points, the
inclusion from the isometry group Isom(M) to Diff(M) is a homotopy equivalence | {"url":"https://core.ac.uk/search/?q=author%3A(McCullough%2C%20Darryl)","timestamp":"2024-11-07T10:51:48Z","content_type":"text/html","content_length":"114988","record_id":"<urn:uuid:88942b91-1177-4042-b4fe-f5b5747b4b79>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00669.warc.gz"} |
Chris Hooley
This course deals mainly with the influence of interactions on the electrons in materials. We begin with a review of second quantisation and the Fermi gas theory of metals, and then progress to
Landau’s Fermi liquid theory and the notion of quasiparticles. The effect of impurities on the Fermi liquid (including the Kondo effect) is discussed, and we then move on to consider how the Fermi
liquid gives way to other phases as the interactions are increased, concentrating on the Stoner instability and the Mott insulator. We analyse the magnetism in the Mott insulating phase, developing
the concept of spin waves. Finally, we make a survey of some experimental data on strongly correlated crystalline solids, giving basic interpretations in terms of the concepts developed in the
Chris is a reader at the University of St Andrews. He works on various topics in the theory of strong correlations, including non-Fermi-liquids, highly frustrated magnets, non-equilibrium atomic
fluids, and vortex-mediated phase transitions.
Strongly Correlated Quantum Systems (SCQ)
This course deals mainly with the influence of interactions on the electrons in materials. We begin with a review of second quantisation and the Fermi gas theory of metals, and then progress to
Landau’s Fermi liquid theory and the notion of quasiparticles. The effect of impurities on the Fermi liquid (including the Kondo effect) is discussed, and we then move on to consider how the Fermi
liquid gives way to other phases as the interactions are increased, concentrating on the Stoner instability and the Mott insulator. We analyse the magnetism in the Mott insulating phase, developing
the concept of spin waves. Finally, we make a survey of some experimental data on strongly correlated crystalline solids, giving basic interpretations in terms of the concepts developed in the
Chris is a reader at the University of St Andrews. He works on various topics in the theory of strong correlations, including non-Fermi-liquids, highly frustrated magnets, non-equilibrium atomic
fluids, and vortex-mediated phase transitions.
Strongly Correlated Quantum Systems (SCQ)
This course deals mainly with the influence of interactions on the electrons in materials. We begin with a review of second quantisation and the Fermi gas theory of metals, and then progress to
Landau’s Fermi liquid theory and the notion of quasiparticles. The effect of impurities on the Fermi liquid (including the Kondo effect) is discussed, and we then move on to consider how the Fermi
liquid gives way to other phases as the interactions are increased, concentrating on the Stoner instability and the Mott insulator. We analyse the magnetism in the Mott insulating phase, developing
the concept of spin waves. Finally, we make a survey of some experimental data on strongly correlated crystalline solids, giving basic interpretations in terms of the concepts developed in the
Chris is a reader at the University of St Andrews. He works on various topics in the theory of strong correlations, including non-Fermi-liquids, highly frustrated magnets, non-equilibrium atomic
fluids, and vortex-mediated phase transitions.
Strongly Correlated Quantum Systems (SCQ)
This course deals mainly with the influence of interactions on the electrons in materials. We begin with a review of second quantisation and the Fermi gas theory of metals, and then progress to
Landau’s Fermi liquid theory and the notion of quasiparticles. The effect of impurities on the Fermi liquid (including the Kondo effect) is discussed, and we then move on to consider how the Fermi
liquid gives way to other phases as the interactions are increased, concentrating on the Stoner instability and the Mott insulator. We analyse the magnetism in the Mott insulating phase, developing
the concept of spin waves. Finally, we make a survey of some experimental data on strongly correlated crystalline solids, giving basic interpretations in terms of the concepts developed in the
Chris is a reader at the University of St Andrews. He works on various topics in the theory of strong correlations, including non-Fermi-liquids, highly frustrated magnets, non-equilibrium atomic
fluids, and vortex-mediated phase transitions. | {"url":"https://physicsbythelake.org/author/chris/","timestamp":"2024-11-08T13:51:57Z","content_type":"text/html","content_length":"87300","record_id":"<urn:uuid:cacd8622-6701-40b2-95fa-eb64a87dd14b>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00269.warc.gz"} |
Saurya DAS | Professor (Full) | Ph.D. | University of Lethbridge, Lethbridge | Department of Physics and Astronomy | Research profile
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I work on Quantum Gravity (theory and phenomenology), Cosmology (singularities, dark matter and energy, inflation, modified Newtonian dynamics), and Foundations of Quantum Mechanics. | {"url":"https://www.researchgate.net/profile/Saurya-Das","timestamp":"2024-11-08T11:26:20Z","content_type":"text/html","content_length":"1049891","record_id":"<urn:uuid:709eed1d-352c-4fa6-9e79-7175abc1bcf1>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00597.warc.gz"} |
Co ordinate Geometry
The straight line is a fundamental concept in coordinate geometry. In two-dimensional space, a straight line can be represented by the equation y = mx + b, where m is…
Fundamentals of parabolas in coordinate geometry
Understanding the Fundamentals of Parabolas in Coordinate Geometry Coordinate geometry is an important branch of mathematics that deals with the study of points, lines, and shapes in a coordinate
Problem 716
Write the expression in a + bi form Solution:- = =4(cos 90+ isin90) = 4(0+i(1)) =4i
Problem 712
A fire truck is en route to an address that is six blocks east and seven blocks south of the fire station. Using the fire station as the pole and…
Problem 682
Write down an (in)equality which describes the solid ball of radius 2 centered at (1, -4, -9). It should have a form like where you use one of the following…
Problem 681
Find the center and radius of the sphere Center: ( , , ) Radius:…… Solution:- Center: (3,5,10) Radius:- 8
Problem 680
Find an equation of the largest sphere with center (10, 3 , 5) that is contained completely in the first octant. ……………………..= 0 Solution:- =0
Problem 679
Find an equation of the sphere that passes through the origin and whose center is (7, -3, -6). ……………………….= 0 Solution:-
Problem 678
Find the equation of the sphere if one of its diameters has endpoints (4, -3, 1) and (6, 1, 7). ………………………= 0. Solution:- =0
Problem 677
Find the equation of the sphere centered at (-2;10;9) with radius 10. ………………………….= 0. Give an equation which describes the intersection of this sphere with the plane z =… | {"url":"https://mymathangels.com/category/geometry/co-ordinate-geometry/","timestamp":"2024-11-11T14:42:39Z","content_type":"text/html","content_length":"71516","record_id":"<urn:uuid:ed690667-5d24-405a-952d-aa2a4966b2c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00499.warc.gz"} |
In celestial mechanics, the longitude of the periapsis, also called longitude of the pericenter, of an orbiting body is the longitude (measured from the point of the vernal equinox) at which the
periapsis (closest approach to the central body) would occur if the body's orbit inclination were zero. It is usually denoted ϖ.
ϖ = Ω + ω in separate planes.
For the motion of a planet around the Sun, this position is called longitude of perihelion ϖ, which is the sum of the longitude of the ascending node Ω, and the argument of perihelion ω.^[1]^[2]
The longitude of periapsis is a compound angle, with part of it being measured in the plane of reference and the rest being measured in the plane of the orbit. Likewise, any angle derived from the
longitude of periapsis (e.g., mean longitude and true longitude) will also be compound.
Sometimes, the term longitude of periapsis is used to refer to ω, the angle between the ascending node and the periapsis. That usage of the term is especially common in discussions of binary stars
and exoplanets.^[3]^[4] However, the angle ω is less ambiguously known as the argument of periapsis.
Calculation from state vectors
Derivation of ecliptic longitude and latitude of perihelion for inclined orbits
Define the following:
• i, inclination
• ω, argument of perihelion
• Ω, longitude of ascending node
• ε, obliquity of the ecliptic (for the standard equinox of 2000.0, use 23.43929111°)
• A = cos ω cos Ω – sin ω sin Ω cos i
• B = cos ε (cos ω sin Ω + sin ω cos Ω cos i) – sin ε sin ω sin i
• C = sin ε (cos ω sin Ω + sin ω cos Ω cos i) + cos ε sin ω sin i
The right ascension α and declination δ of the direction of perihelion are:
tan α = B/A
sin δ = C
If A < 0, add 180° to α to obtain the correct quadrant.
The ecliptic longitude ϖ and latitude b of perihelion are:
tan ϖ = sin α cos ε + tan δ sin ε/cos α
sin b = sin δ cos ε – cos δ sin ε sin α
If cos(α) < 0, add 180° to ϖ to obtain the correct quadrant.
As an example, using the most up-to-date numbers from Brown (2017)^[5] for the hypothetical Planet Nine with i = 30°, ω = 136.92°, and Ω = 94°, then α = 237.38°, δ = +0.41° and ϖ = 235.00°, b =
+19.97° (Brown actually provides i, Ω, and ϖ, from which ω was computed).
External links
• Determination of the Earth's Orbital Parameters Past and future longitude of perihelion for Earth. | {"url":"https://www.knowpia.com/knowpedia/Longitude_of_periapsis","timestamp":"2024-11-10T06:36:38Z","content_type":"text/html","content_length":"82180","record_id":"<urn:uuid:1ec295af-854c-4656-a90d-22d5a35678b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00899.warc.gz"} |
Re: Show doesn't work inside Do loop ?
• To: mathgroup at smc.vnet.net
• Subject: [mg102013] Re: Show doesn't work inside Do loop ?
• From: AES <siegman at stanford.edu>
• Date: Sun, 26 Jul 2009 03:56:52 -0400 (EDT)
• Organization: Stanford University
• References: <32390795.1248259308283.JavaMail.root@n11> <h4951e$q2e$1@smc.vnet.net> <21621663.1248432817472.JavaMail.root@n11> <h4eeul$spv$1@smc.vnet.net>
In article <h4eeul$spv$1 at smc.vnet.net>,
"David Park" <djmpark at comcast.net> wrote: (emphasis added)
> _On the other hand, Show does not generally generate a cell._
> Of course, it is easier to just apply Print to the Show or Plot statements.
> So it is possible to make Show generate output cells, but it _normally_
> doesn't do so. It normally only generates expressions, which because of its
> special behavior a Do statement does not display.
1) Re your statements above: executing a single Input cell containing
Show[ Graphics[ Circle[ {0, 0}, 1] ] ]
certainly generates an Output cell containing what looks like a
"graphic" or "plot" to me.
Is executing a cell containing a simple expression somehow an "abnormal"
process or action?
2) On a more general note: Suppose you have an expression which
contains an explicit symbol x , such that if you execute three
consecutive cells containing
x=1; expr
x=2; expr
x=3; expr
or maybe
you get three output cells containing three successive instances of expr
(whatever that is) -- or appropriate error messages if executing expr
one of those times has some side effect that messes up a subsequent
Would it not be reasonable to expect a cell containing
Do[ expr, {x,1,3} ]
to do _exactly_ the same thing?
In other words, would it not be reasonable -- consistent -- sensible --
helpful -- the most useful -- to expect Do[ ] to be simply a "wrapper"
that functioned in exactly that manner?
I appreciate that Mathematica's Do[] apparently doesn't function that
way -- or functions that way sometimes, based on mysterious criteria,
but not other times; and suggest that this is not helpful or useful or
consistent behavior for many users.
Are there any fundamental reasons why a DoConsistently[ ] command could
not be defined, such that DoConsistently[ expr, iterator ] would
repeatedly put expr into a cell with each iterator instance applied to
it, and churn out the sequential outputs? That, it seems to me, is what
many users would want and expect. | {"url":"http://forums.wolfram.com/mathgroup/archive/2009/Jul/msg00672.html","timestamp":"2024-11-09T03:54:47Z","content_type":"text/html","content_length":"32788","record_id":"<urn:uuid:e0640222-082f-475e-bdec-9d10fcc7d43f>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00355.warc.gz"} |
Algebra 1: Common Core (15th Edition) Chapter 9 - Quadratic Functions and Equations - 9-5 Completing the Square - Got It? - Page 577 1
The value of $b$ is 2. The term to add to $x^2+20x$ is $(\frac{20}{2})^2=100$. So $c=100$
You can help us out by revising, improving and updating this answer.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | {"url":"https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-9-quadratic-functions-and-equations-9-5-completing-the-square-got-it-page-577/1","timestamp":"2024-11-11T04:27:45Z","content_type":"text/html","content_length":"89449","record_id":"<urn:uuid:e1e9ce26-0c34-4ece-a5c7-7096ad32665a>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00454.warc.gz"} |
Programming with Bananas in OCaml
By bananas, I mean banana brackets as in the famous paper “Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire” by Meijer et al. In this post I only focus on bananas, also called
Recursion is core to functional programming. Meijer et al. show that we can treat recursions as separate higher order functions known as recursion schemes. Catamorphisms is one type of recursion
Why Should I Care?
Because knowing how to use recursion schemes improves your coding ability! Consider a simple exercise: finding the minimum element of an array. In OCaml, without using fold (catamorphisms), we
would compare the elements of the array a one by one, keeping the minimum of the element, until we reach the end of the array. E.g.:
let rec find_min_inner a i j =
match a.(i) > a.(j) , (Array.length a -1 < i || Array.length a - 1 < j + 1) with
|true, true -> a.(j)
|false, true -> a.(i)
|true, false -> find_min_inner a j (j + 1)
|false, false -> find_min_inner a i (j + 1) ;;
let find_min a =
match a with
|[||] -> [||]
|_ -> find_min_inner a 0 1 ;;
Using catamorphisms or fold_left (or fold_right) in OCaml, we can do the equivalent for int arrays in one line:
let find_min a =
Array.fold_left (fun x y -> if x < y then x else y) max_int a;;
Or, using the function min in the Pervasives to replace the anonymous function and eta-reducting a:
let find_min = Array.fold_left min max_int;;
Catamorphisms dramatically reduce the amount of code because fold_left does exactly what we want to do for us: recurse down each element of the array. Of course, fold_left or fold_right work on
other data structures too.
How Does It Work?
Going back to the above example, what’s going on in that one line? We invoke the iterator in the array module fold_left. fold_left has the following type signature:
('a -> 'b -> 'a) -> 'a -> 'b array -> 'a
That is, fold_left takes 3 inputs:
• A function that takes two arguments, one of type ‘a and one of type that is the same as the array, and returns a result of type ‘a.
• An input of type ‘a, think of it as the base case of the recursion.
• An array (of the same or different type to ‘a).
The output of fold_left is of type ‘a.
The first input is a function that does something with the first input (of type ‘a) and an element of the array. In this example, the function is min and it compares max_int with an element of the
fold_left recurses the function with the earlier result as an input. In this example, fold_left calls the following:
(* When the array is empty, it returns the max_int. *)
fold_left min max_int [||] => max_int
(* When the array has one element, it returns the min of max_int and the element. *)
fold_left min max_int [|e1|] => min max_int e1
(* When the array has two elements, min takes the result of
(min max_int e1) as an input, and compare it with e2. *)
fold_left min max_int [|e1;e2|] => min (min max_int e1) e2
(* When the array has three elements, min takes the result of
fold_left max_int [|e1;e2|] as an input, and compare it with e3. *)
fold_left min max_int [|e1;e2;e3|] => min (min (min max_int e1) e2) e3
We can generalize it to any function f, with x as the base case and a as the array input, the result is as in the Pervasives:
fold_left f x a => f a (... (f (f x a.(0)) a.(1)) ...) a.(n-1)
fold_right works similarly, except that the array elements are the first input to the function, and the brackets are to the right:
(* When the array has one element, it returns the min of max_int and the element. *)
fold_right min [|e1|] max_int => min e1 max_int
(* When the array has two elements, min takes the result of
(min e2 max_int) as an input, and compare it with e1. *)
fold_right min [|e1;e2|] max_int => min e1 (min e2 max_int)
(* When the array has three elements, min takes the result of
(min e3 max_int) as an input to compare with e2,
the result of which is then compared to e1. *)
fold_left min max_int [|e1;e2;e3|] => min e1 (min e2 (min e3 max_in))
The Base Case
You can see from the above example that to find the minimum element of an array, we can use fold_left or fold_right in OCaml, naturally with the min function as the function input. But how do we
choose the base case?
The base case is there to safe guard an empty array input. When an empty array is input to fold_left or fold_right, OCaml returns the base case. Otherwise, the base case must have the property that
when the function takes the base case and another input, the function returns the other input as a result. That is, given x is the base case, f x e must return e. In the above example, because
max_int is the maximum integer, min (anything) max_int returns (anything), as desired. Other examples are:
let sum = Array.fold_left (+) 0;;
# sum [||];;
- : int = 0 (* Passing an empty array to sum returns the base case 0. *)
let product = Array.fold_left (fun x y -> x * y) 1;;
# product [||];;
- : int = 1 (* Passing an empty array to product returns the base case 1. *)
Because the base case is just to safe guard an empty input, when you can be sure that the input is not empty, the base case would not be necessary! OCaml does not provide such option but Haskell
does. We’ll see this in the next post!
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Understanding the Definition of R.M.S
r.m.s Definition: What Does It Mean?
If you’ve ever come across the term r.m.s, you may have wondered what it means. The abbreviation stands for “root mean square,” and it is a mathematical formula used to calculate the average value of
a set of numbers. In the world of electronics, r.m.s is an important concept that helps measure the voltage and current of different electrical devices.
When we talk about voltage or current, we’re referring to the amount of electrical energy that is flowing through a device. In an ideal situation, the voltage or current would remain constant, but
that’s not always the case. In many cases, electrical signals can be variable, meaning they change frequently and unpredictably.
This is where r.m.s comes in. The formula takes the square of each value, adds them all together, and then divides by the number of values to get an average. This calculation helps eliminate the
fluctuations and gives us a value that represents the average electrical energy of a signal. The result is a more accurate measurement of the signal’s strength.
To give you an example, let’s say we have an AC voltage signal that varies from +10V to -10V. Without r.m.s, we would simply calculate the average voltage as 0V (since +10V and -10V add up to 0).
However, using r.m.s, we would first square each value (100V and 100V), add them together (200V), divide by the number of values (2), and then take the square root (14.14V). This calculation gives us
a more accurate representation of the voltage signal’s strength.
So why is r.m.s important in electronics? For one, it is used extensively in AC circuit analysis and digital signal processing. In fact, many electronic devices are specifically designed to work with
r.m.s measurements. Additionally, r.m.s helps us compare the power of different signals more accurately. For example, if two signals have the same r.m.s value, we can assume that they have the same
amount of power, even if they may look different on an oscilloscope.
In conclusion, r.m.s may seem like a confusing concept, but it is actually quite simple once you understand how it works. By calculating the square of each value in a set, adding them together, and
then dividing by the number of values, we get a more accurate measurement of the average energy of a signal. In the world of electronics, r.m.s is an essential tool that helps us measure voltage and
current more accurately, making it an important aspect of electrical engineering and digital signal processing.
The Mathematics Behind r.m.s
Root Mean Square (r.m.s) is a statistical measurement commonly used to describe the magnitude of fluctuating electrical signals. It is an essential method of assessing the quality of an electrical
signal as it represents the average power delivered by an AC source. The r.m.s value is preferred over other statistical measurements such as average or peak because it considers both positive and
negative amplitudes of an input signal to determine its intensity.
The r.m.s expression can be deduced from the well-known Pythagorean theorem, which relates the three sides of a right-angled triangle. In essence, it involves the calculation of the square root of
the mean of the square of all instantaneous values that a signal takes over a period of time. This is shown in the equation below:
r.m.s value = √[(x1² + x2² + x3² +……+ xn²)/n]
where x1, x2, x3 and xn represent instantaneous values of a signal at any given time, and n is the total number of readings taken over the period.
For example, let’s consider a sine wave signal with peak amplitude of 20V over a period. To determine the r.m.s value of this signal, we can take multiple readings of the instantaneous voltage at
various points and then calculate the r.m.s value using the formula above. This value gives us an insight into the actual power being delivered by the signal, rather than just the peak amplitude.
Furthermore, the importance of the r.m.s value in measuring electrical signals lies in the fact that it allows us to calculate the power being delivered by the signal. In essence, it is the basis for
the development of the concept of average power in AC circuits. In a DC circuit, the formula for calculating power is simply P=VI (Power=Voltage x Current), but in an AC circuit, the voltage and
current continually vary with time. Therefore, to obtain an accurate measurement of power dissipated, we can use the r.m.s value of both voltage and current instead of just the peak values.
Overall, the r.m.s value of an electrical signal is essential as it allows us to gain insight into the actual power being delivered by a signal. It’s a mathematical expression based on the mean of
the square of all instantaneous values at various points over a period of time. It is an essential concept in electrical engineering, particularly in measuring the quality of AC signals accurately.
With the r.m.s value, we get a more accurate picture of the power being delivered by a signal, and this makes it easier to design and analyze power systems.
Applications of r.m.s in Engineering
The root mean square or r.m.s is a mathematical concept that is widely used in many fields of engineering. It is a statistical measure of the magnitude of a wave, and it can be applied to any wave
that fluctuates over time. From electrical power transmission to audio signal processing, r.m.s plays a crucial role in many engineering applications.
Power Transmission
In power transmission, r.m.s is used to determine the voltage and current of an AC waveform. r.m.s voltage is the equivalent DC voltage that would deliver the same power to a resistive load as that
of an AC waveform. Similarly, r.m.s current is the DC current that would deliver the same power to a resistive load as that of an AC waveform. This means that if you know the r.m.s voltage and
current of an AC waveform, you can calculate the power being delivered. Therefore, r.m.s plays a critical role in the design and operation of electrical power systems.
Audio Signal Processing
In audio signal processing, r.m.s is used to determine the average power of an audio signal. The r.m.s of an audio signal is a measure of the overall loudness or volume of the signal. It is used to
calibrate audio equipment, design sound systems, and optimize audio recordings. For example, when mixing music, sound engineers use r.m.s to ensure that each track is at the same overall volume
level. This results in a more balanced and clear mix that is pleasing to the ear.
Other Applications in Engineering
R.m.s has other applications in engineering apart from power transmission and audio signal processing. For example, it is used in vibration analysis to determine the energy content of vibration
signals. R.m.s is also used in machine learning to evaluate the error rate of a model. Engineers use r.m.s in designing bridges and structures to calculate the stress levels of materials subjected to
random loads.
In conclusion, the r.m.s concept is crucial in various engineering applications. Its ability to measure the magnitude of fluctuating waves makes it a critical component in power transmission and
audio signal processing. Furthermore, its applications are diverse and fundamental in many other fields, making it invaluable in the world of engineering.
The Limitations of r.m.s
Root mean square (r.m.s) is a statistical measure used to analyze alternating current (AC) and other periodic signals. It has been commonly used as an efficient way of calculating the power of
signals as it reflects the effective value of an AC voltage or current. However, like any other measurement tool, r.m.s has its limitations.
One of the major limitations of r.m.s is that it assumes a constant or steady waveform. When measuring signals that have a non-constant waveform, r.m.s does not give an accurate representation of the
power or amplitude of the signal. In such cases, it is important to use other measurement methods like peak-to-peak measurement or mean absolute deviation (MAD) to analyze the signal accurately.
Additionally, r.m.s measurements cannot account for waveform distortion and harmonic distortion. In situations where signals contain harmonic content, r.m.s measurements may not provide an accurate
representation of the power of the signal, leading to potential errors in analysis. Other methods such as total harmonic distortion (THD) or crest factor can be used to measure signals with harmonic
distortion accurately.
Another limitation of r.m.s measurements is that they do not distinguish the type of waveform. This means that both symmetrical and asymmetrical waveforms are measured the same way, leading to
potential inaccuracies. In such situations, it is important to use other measurement tools that can differentiate between symmetrical and asymmetrical waveforms.
Furthermore, r.m.s measurements are not suitable for measuring signals with high frequency components. When measuring high-frequency signals, it is necessary to use other measurement tools that can
work at higher frequencies such as Fast Fourier Transform (FFT) or spectrum analysis.
In conclusion, while r.m.s is a useful measure for analyzing alternating current and other periodic signals, it is not without limitations. It may not provide accurate measurements for signals with
non-constant waveforms, harmonic distortion, and high-frequency components. It is essential to understand these limitations and use alternative measurement tools to ensure accurate analysis of
signals in such situations.
Understanding r.m.s in Electrical Engineering
In electrical engineering, root mean square (RMS or r.m.s) is widely used to measure the effective value of an AC waveform. An AC waveform refers to a sine wave voltage or current that oscillates
between positive and negative values. Understanding r.m.s is essential to correctly measure electrical signals accurately in AC systems and control circuits. In this article, we will explore the
definition of r.m.s, its importance in electrical engineering, and its limitations.
What is the r.m.s value?
R.m.s is the most crucial parameter in AC waveforms. In simple words, it is the equivalent DC voltage which provides the same power when attached to a resistive load. The r.m.s value of an AC
waveform can be determined by measuring its voltage or current signal over a specific interval of time. The value calculated from the squared mean of the AC waveform is then converted into an
equivalent DC value.
Understanding the importance of r.m.s in electrical engineering
The r.m.s value is significantly important when dealing with the measurement of electrical power, magnitude, and energy in AC circuits. Using the r.m.s value can provide an effective and accurate way
to measure the power consumption of an AC circuit. It helps engineers and technicians to ensure the safety and functionality of the electrical systems under operation.
Limitations of r.m.s
While r.m.s is an integral part of measuring electrical signals, it has some limitations. R.m.s only expresses the average value of a waveform, making it impossible to determine the instantaneous
peaks. For example, an r.m.s value calculated on a sine wave can indicate the AC signal’s power component, but it does not provide any information about the maximum or minimum voltage or current
magnitude. In addition, r.m.s values can be misleading if used to measure pulsed DC or non-sinusoidal AC waveforms.
The Conclusion
In conclusion, understanding the concept of r.m.s is crucial when dealing with AC signals. The r.m.s value helps in determining the equivalent DC value of an AC waveform that carries the same average
power. However, engineers and technicians should keep in mind its limitations when analyzing electrical signals to ensure they are not misled by the averaged or mean value. In summary, the importance
of understanding r.m.s and its limitations in accurately measuring electrical signals cannot be overstated.
Originally posted 2023-05-31 20:49:58. | {"url":"https://www.mediacharg.com/rms-definition/","timestamp":"2024-11-15T00:01:23Z","content_type":"text/html","content_length":"164616","record_id":"<urn:uuid:97be8929-f55f-4c6f-90db-a5eb910f7260>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00803.warc.gz"} |
Journal of Statistical Research of IranVector Autoregressive Model Selection: Gross Domestic Product and Europe Oil Prices Data Modelling
en عمومى General كاربردي Applicable فارسی <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><span style="font-size:12.0pt"><span new="" roman=
"" style="font-family:" times=""> We consider the problem of model selection in vector autoregressive model with Normal innovation. Tests such as Vuong's and Cox's tests are provided for
order and model selection, i.e. for selecting the order and a suitable subset of regressors, in vector autoregressive model. We propose a test as a modified log-likelihood ratio test for selecting
subsets of regressors. The Europe oil prices, Brent, and the real gross domestic product, GDP, data are considered as real data. Since the Brent data does Granger-cause the GDP data, so we suggest
the vector autoregressive model and select optimal model based on the model selection test. The analysis provides analytic results show that the Vuong's, Cox's and proposed test are the
appropriate test for order and model selection for vector autoregressive models with Normal innovation. In simulation study, the power of proposed test at least is as good as the power of Vuong's
test.</span></span></span></span></span><br> Cox's test, maximum likelihood estimation, mis-specified model, nested models, vector autoregressive model, Vuong's test. 63 94 http://
jsri.srtc.ac.ir/browse.php?a_code=A-10-190-10&slc_lang=en&sid=1 S. Zamani Mehreyan زمانی zamani@sci.ikiu.ac.ir 10031947532846002421 10031947532846002421 No Imam Khomeini International University,
zamani@sci.ikiu.ac.ir Abdolreza Sayyareh عبدالرضا سیاره a.sayyareh@kntu.ac.ir 10031947532846002422 10031947532846002422 Yes K. N. Toosi University of Technology | {"url":"http://jsri.srtc.ac.ir/article-411.xml","timestamp":"2024-11-05T10:46:21Z","content_type":"application/xml","content_length":"5410","record_id":"<urn:uuid:f599b025-2145-414d-95a5-0ae7bb06aa05>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00593.warc.gz"} |
orksheets for 6th Grade
Recommended Topics for you
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Explore Fractions Worksheets by Grades
Explore Fractions Worksheets for grade 6 by Topic
Explore Other Subject Worksheets for grade 6
Explore printable Fractions worksheets for 6th Grade
Fractions worksheets for Grade 6 are essential tools for teachers to help their students master the concepts of fractions, decimals, and percentages in mathematics. These worksheets provide a variety
of exercises, ranging from simple addition and subtraction of fractions to more complex operations like multiplication and division. With a wide range of topics covered, Grade 6 math teachers can
easily find worksheets that cater to the specific needs of their students. By incorporating these resources into their lesson plans, educators can ensure that their students have a solid foundation
in fractions, setting them up for success in higher-level math courses.
Quizizz offers an excellent platform for teachers to access not only fractions worksheets for Grade 6 but also a plethora of other math resources. This interactive platform allows educators to create
engaging quizzes and games that can be used in conjunction with worksheets to reinforce learning and assess students' understanding of the material. Quizizz also provides teachers with valuable
insights into their students' progress, enabling them to tailor their instruction to meet the individual needs of each learner. With its vast library of resources, including worksheets, quizzes, and
games, Quizizz is an invaluable tool for Grade 6 math teachers looking to enhance their students' learning experience and ensure their success in mastering fractions and other mathematical concepts. | {"url":"https://quizizz.com/en-us/fractions-worksheets-grade-6","timestamp":"2024-11-14T15:17:50Z","content_type":"text/html","content_length":"148233","record_id":"<urn:uuid:60528f56-cf06-44f8-8c55-b3c3c7b25e68>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00561.warc.gz"} |
Implementation of boundary conditions for FVM
13, 2002, Re: Implementation of boundary conditions for FVM #2
Guest I GATHER FROM VARIOUS LITERATURE SOURCES THAT THERE ARE 4 POSIBILITIES WHEN SPECIFYING BOUNDARY CONDITIONS: (1) FIXED VALUE (DIRICHLET) (2) FIXED GRADIENT (VON NEUMANN) (3) NATURAL
BOUNDARY CONDITION (DEPENDENT ON SOLUTION) (4) EXTRAPOLATED BOUNDARY VALUE
Posts: n/
a There are only the first two as all the others can be rewritten to one of the first two. For example, the extrapolated boundary value uses a gradient from the inside of the domain to
predict the value at the face. Thus for that iteration this gradient is fixed and reduces to your number 2, Von Neumann condition.
I ALSO GATHER THAT EACH AND EVERY DISCRETIZED EQUATION TO BE SOLVED REQUIRES EITHER ONE OR A COMBINATION OF THE ABOVE BOUNDARY CONDITIONS.
Yes, but it is here where CFD sometimes got confused as text books and codes named these boundaries typically INLET, OUTLET, PRESSURE, CYCLIC etc. For the different properties, the
combinations reduce to either DIRICHLET or VON NEUMANN.
NUMBER (1) AND (2) IS FAIRLY UNDERSTANDABLE AND THE IMPLEMENTATION IS NOT TOO OBSCURED. QUESTIONS: [1] AM I RIGHT IN SAYING THAT FOR (2) THE BOUNDARY FACE VALUE IS CALCULATED FROM THE
GIVEN GRADIENT AND THE CELL VALUE LYING ALONG THE BOUNDARY?
Yes, that is right.
[2] IN THE DISCRETIZED EQUATION WE REQUIRE THE VALUE OF THE CELL LYING OUTSIDE THE BOUNDARY. BUT WE ASSUME THAT THIS CELL HAS A ZERO VOLUME SO THAT ITS CENTROID COINCIDES WITH THE BOUNDARY
FACE CENTROID? THEREFORE THE VALUE AT THE BOUNDARY CAN BE USED DIRECTLY IN THE DISCRETIZED EQUATION?
Commercial CFD codes use two different techniques. Some use this methodology described by you. The more general approach is to define control volumes in terms of flat faces which are
flagged to be baffles, inside cell faces, boundary faces etc.
[3] ANOTHER QUESTION ON THE FIXED GRADIENT BOUNDARY CONDITION (2). DO WE SPECIFY THE FLUX AT THE BOUNDARY PER UNIT AREA I.E. Q=-K D(PHI)/D(N) = FIXED VALUE WHERE N IS A OUTWARD POINTING
INIT VECTOR ON THE BOUNDARY FACE, OR DO WE SPECIFY ONLY THE GRADIENT OF THE DEPENDENT VARIABLE? D(PHI)/D(N)
At the end you will need/use the the gradient of the dependent variable at the face.
[4] WHAT IS THE DIFFERENCE BETWEEN IMPLICIT AND EXPLICIT TREATMENT OF BOUNDARY CONDITIONS AND HOW DOES IT EFFECT THE DISCRETIZED EQUATION? I.E. HOW IS THE DIAGONAL INFLUENCED AND AND WHAT
ABOUT THE EXPLICIT SOURCE TERM DUE TO THE BOUNDARY CONDITION?
When deriving the finite volume method, the finite volume integration over a control volume reduces to an algebraic expression where the new value in a control volume is in terms of the
old value of the control volume and the values at the faces. Neighbouring control volumes is then used to interpolate the values at the faces with different schemes such as Upwind
differencing, Central Differencing Quick etc. The variables is then grouped together to give the well known expression _p phi_p = SIGMA a_nb phi_nb + S_p, where a_p is the central
coefficient, a_nb the so-called neighbouring coefficients and S_p the source. Phi should be solved for. Now for boundary cells the face value is "replaced" by the particular boundary
assumption. Thus, when re-arranging the variables, you have the choice to either put the contribution in the source or in the matrix or to split it. Generally spoken, larger source terms
(explicit) will slow convergence down.
[5] WHAT IS THE DEAL WITH CONDITIONS (3) AND (4) AND WHERE ARE THEY TYPICALLY USED? HOW WILL (3) BE IMPLEMENTED?
I will rewrite to options 1 and 2 and try to implement it as implicitly as possible. (Personal choice)
[6] TYPICALLY WHICH CONDITIONS OF (1) TO (4) IS USED IN WHICH EQUATIONS FOR EXAMPLE: MOMENTUM EQUATION; PRESSURE EQUATION; PRESSURE CORRECTION EQUATION ETC. ETC.
Try to find this in a good text book (say Ferziger and Peric)or the help pages of some commercial CFD code Typically for an INLET the velocity will be fixed and pressure extrapolated.
LOOKING FORWARD TO YOUR REPLIES. TOM
Hope this help | {"url":"https://www.cfd-online.com/Forums/main/5374-implementation-boundary-conditions-fvm.html","timestamp":"2024-11-14T14:23:43Z","content_type":"application/xhtml+xml","content_length":"104558","record_id":"<urn:uuid:d162f8d5-2bdf-4a53-89de-935b7347b1ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00338.warc.gz"} |
functions mshape with variable elements and isscalar
On 11/09/2012 07:30 PM, Allin Cottrell wrote:
On Fri, 9 Nov 2012, Riccardo (Jack) Lucchetti wrote: > On Fri, 9 Nov 2012, Allin Cottrell wrote: >
>> >> if typeof(whatever) != "scalar" >> ... >> endif > > I like this.
I do, too, but why not just use the word 'type'. The "of" part is already indicated by the parentheses, no? ...
> The only think I'd change from Allin's proposal is to introduce an > integer-based coding system, such as 1=scalar, 2=series etc and have typeof() > return a scalar. For performance reasons, the
less we use strings, the > better. Good point.
I beg to disagree, I would favor explicitness and readability of the code over some marginal performance gain. It would be a major (well ok, maybe not major, but still) step backwards from being able
to use 'if islist()', which is self-explanatory, towards 'if type()==3' which is totally mysterious. But 'if type()=="list"' would be on par with the current status IMO.
I've sometimes wondered if it would be worth trying to expose some sort of "enum" system in gretl -- e.g. in this context NONE = 0, SCALAR = 1, SERIES = 2 and so on. It would obviously be more
intuitive to be able to do if typeof(whatever) == SCALAR
This would be ok, too, IMO.
where SCALAR codes for an integer. But it means "invading" the user's available namespace, plus more internal string look-up, so maybe it's not worth the trouble outside of a compiled language.
As I said, I'm strongly in favor of using strings or pseudo-strings (global constants) here. Also, the words "series", "scalar" etc. are already reserved words, so I don't see any additional
invasion. thanks, sven | {"url":"https://gretlml.univpm.it/hyperkitty/list/gretl-users@gretlml.univpm.it/thread/JY72CCCCFXWRLO33SXIB3OXVB52TRFKS/?sort=date","timestamp":"2024-11-06T13:34:49Z","content_type":"text/html","content_length":"195220","record_id":"<urn:uuid:e640fc0b-bbee-4531-8885-631739652402>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00849.warc.gz"} |
The Problem-solving Strategies And Teaching Research Based On Polya’s Thought
Posted on:2024-01-16 Degree:Master Type:Thesis
Country:China Candidate:C C Liao Full Text:PDF
GTID:2557307061471074 Subject:Subject teaching
Problem-solving teaching has always been the core of mathematics teaching,and it has also been the focus of long-term research by scholars at home and abroad,and its importance cannot be ignored.In
the "How to Solve the Problem",Polya divides the problem-solving process into four major steps and gives detailed guidelines,which have a positive effect on students’ problem solving.The sequence of
number is an extremely challenging content,its importance cannot be ignored,and it occupies a pivotal position in the college entrance examination.It can comprehensively test students’ mathematical
and logical reasoning skills,and its complexity often makes students feel daunted.Therefore,integrating Polya’s thought of problem-solving into the sequence problem can optimize the students’
problem-solving process and improve their mathematical problem-solving ability.First of all,by analyzing the characteristics of Polya’s problem-solving thought,combined with the learning situation of
senior high school students and the requirements of sequence of number in the textbook,it is found that Polya’s problem-solving thought perfectly matches the requirements put forward in the "General
High School Mathematics Curriculum Standards" in terms of education,teacher and student views.Then,a questionnaire survey was conducted to investigate and count the learning attitude and habit of
thinking about sequence of numbers of senior two students in a high school.Based on the statistical results,the main problems encountered by students in solving problems were summarized.On this
basis,combined with Polya’s problem-solving thought,from the four stages of understanding the problem,making the plan,implementing the plan and reviewing,this paper puts forward the problem-solving
strategies for the three key problem types of the sequence and gives two teaching cases.Finally,through the questionnaire survey again to analyze the effectiveness of problem-solving teaching
strategies,it is proved that the integration of Polya’s problem-solving thoughts into the teaching of sequence problem-solving can improve students’ problem-solving ability,and also help teachers to
teach sequence of numbers problem-solving.
Keywords/Search Tags: Polya’s thought of problem-solving, Sequence of numbers, Problem-solving strategy | {"url":"https://www.globethesis.com/?t=2557307061471074","timestamp":"2024-11-14T21:39:58Z","content_type":"application/xhtml+xml","content_length":"8353","record_id":"<urn:uuid:06b0d426-478b-47c0-bea7-44309f319614>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00091.warc.gz"} |
Easy Hyperparameter Tuning in Neural Networks using Keras Tuner
This article was published as a part of the Data Science Blogathon
In the last few articles, we discussed Neural Networks, their work and, their practical implementation in Python on the MNIST dataset. Continuing to the same, in this article we will look at how to
tune the parameters of Neural Network to achieve the appropriate parameters which provide the highest training and testing accuracy, we don’t want overfitting in our data, right.
I would highly suggest going through the Implementation of ANN on MNIST data blog to understand this one better.
What are Hyperparameters?
Hyperparameters are the values we provide to the model and are used to improve the performance of the model. They are not automatically learned during the training phase but have to be provided
Hyperparameters play a major role in the performance of the model and should be chosen and set such that the model accuracy improves. In Neural Network some hyperparameters are the Number of Hidden
layers, Number of neurons in each hidden layer, Activation functions, Learning rate, Drop out ratio, Number of epochs, and many more. In this article, We are going to use the simplest possible way
for tuning hyperparameters using Keras Tuner.
Using the Fashion MNIST Clothing Classification problem which is one of the most common datasets to learn about Neural Networks. But before moving on to the Implementation there are some
prerequisites to use Keras tuner. The following is required:
• Python 3.6+
• Tensorflow 2.0+ (I had Tensorflow 2.1.0 in my system but still it didn’t work so had to upgrade it to 2.6.0)
Some Frequently asked questions(FAQs):
1. How to check the Tensorflow version:
#use this command
2. How to upgrade Tensorflow?
#Use the following command
pip install --upgrade tensorflow --user
3. What to do if it still does not work?
–> Use Google colab
Let’s move on to the problem statement now. In the Fashion MNIST dataset, we have images of clothing such as Tshirt, trousers, pullovers, dresses, coats, sandals,s and have a total of 10 labels.
#importing necessary libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Dense
from tensorflow.keras.layers import Dropout
from tensorflow.keras.layers import Flatten
from tensorflow.keras.datasets import fashion_mnist
#loading the data
#visualizing the dataset
for i in range(25):
# define subplot
plt.subplot(5, 5, i+1)
# plot raw pixel data
plt.imshow(X_train[i], cmap=plt.get_cmap('gray'))
# show the figure
#normalizing the images
In the last MNIST digit classification example, we flattened the dataset before building the model, but here we will do it in the model building code itself. I have explained the model building code
in detail in the last article, kindly refer to that for an explanation.
Model Building
#flattening the images
#adding first hidden layer
#adding second hidden layer
#adding third hidden layer
#adding output layer
#compiling the model
#fitting the model
#evaluating the model
We have built the basic ANN model and got the training and testing accuracy as shown in the above figures. We can see the difference in accuracies and losses of the training and test sets. The loss
in the training data is less but increases for the test data which can lead to wrong predictions on the unseen data.
Now let’s tune the Hyperparameters to get the values that can help in improving the model. We will be optimizing the following Hyperparameters in the model:
• Number of hidden layers
• Number of neurons in each hidden layer
• Learning rate
• Activation Function
But first, we need to install the Keras Tuner.
#use this command to install Keras tuner
pip install keras-tuner
#installing the required libraries
from tensorflow import keras
from keras_tuner import RandomSearch
Defining the function to build an ANN model where the hyperparameters will be the Number of neurons in the hidden layer and Learning rate.
def build_model(hp): #hp means hyper parameters
#providing range for number of neurons in a hidden layer
#output layer
#compiling the model
model.compile(optimizer=keras.optimizers.Adam(hp.Choice('learning_rate',values=[1e-2, 1e-3, 1e-4])),loss='sparse_categorical_crossentropy',metrics=['accuracy'])
return model
In the above code, we have defined the function by the name build_model(hp) where hp stands for hyperparameter. While adding the hidden layer we use hp.Int( ) function which takes the Integer value
and tests on the range specified in it for tuning. We have provided the range for neurons from 32 to 512 with a step size of 32 so the model will test on neurons 32, 64,96,128…,512.
Then we have added the output layer. While compiling the model Adam optimizer is used with different values of learning rate which is the next hyperparameter for tuning. hp.Choice( ) function is used
which will test on any one of the three values provided for the learning rate.
#feeding the model and parameters to Random Search
The code above uses the Random Search Hyperparameter Optimizer. The following variables are provided to the Random Search. The first is model i.e build_model, next objective is val_accuracy that
means the objective of the model is to get a good validation accuracy. Next, the value of trails and execution per trail provided which is 5 and 3 respectively in our case meaning 15 (5*3) iterations
will be done by the model to find the best parameters. Directory and project name are provided to save the values of every trial.
#this tells us how many hyperparameter we are tuning
#in our case it's 2 = neurons,learning rate
#fitting the tuner on train dataset
The above code will run 5 trails with 3 executions each and will print the trail details which provide the highest validation accuracy. In the below figure, we can see the best validation accuracy
achieved by the model.
We can also check the summary of all the trails done and the hyperparameters chosen for the best accuracy using the below code. The best accuracy is achieved using 416 neurons in the hidden layer and
0.0001 as the learning rate.
That’s how we perform tuning for Neural Networks using Keras Tuner.
Let’s tune some more parameters in the next code. Here we are also providing the range of the number of layers to be used in the model which is between 2 to 20.
def build_model(hp): #hp means hyper parameters
#providing the range for hidden layers
for i in range(hp.Int('num_of_layers',2,20)):
#providing range for number of neurons in hidden layers
model.add(Dense(units=hp.Int('num_of_neurons'+ str(i),min_value=32,max_value=512,step=32),
model.add(Dense(10,activation='softmax')) #output layer
#compiling the model
model.compile(optimizer=keras.optimizers.Adam(hp.Choice('learning_rate',values=[1e-2, 1e-3, 1e-4])), #tuning learning rate
return model
#feeding the model and parameters to Random Search
#tells us how many hyperparameters we are tuning
#in our case it's 3 =layers,neurons,learning rate
#fitting the tuner
Summary and the best accuracy of the model in the below code. This time we got 0.89 as the validation accuracy.
This was the simplest possible way to tune the parameters in Neural Network. Please refer to the official documentation of Keras Tuner for more details: https://keras.io/keras_tuner/
About the Author:
I am Deepanshi Dhingra currently working as a Data Science Researcher, and possess knowledge of Analytics, Exploratory Data Analysis, Machine Learning, and Deep Learning. Feel free to content with me
on LinkedIn for any feedback and suggestions.
Responses From Readers | {"url":"https://www.analyticsvidhya.com/blog/2021/08/easy-hyperparameter-tuning-in-neural-networks-using-keras-tuner/?utm_source=related_WP&utm_medium=https://www.analyticsvidhya.com/blog/2021/08/hyperparameter-tuning-of-neural-networks-using-keras-tuner/","timestamp":"2024-11-09T01:03:06Z","content_type":"text/html","content_length":"360895","record_id":"<urn:uuid:0ab67a92-0b77-41b6-b21b-26eba89e2185>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00064.warc.gz"} |
Simple question about the core of categories.
Certainly it is.
Is the core of a category always a wide subcategory? I mean, in the most laziest sense of isomorphisms, why not include identity morphisms?
Edit: I’m curious if cores always have a notion of fraction.
If your edited question is this:
Let $C$ be a category and let $W$ be the class of isomorphisms of $C$; note that $W$ is closed under composition and so defines a subcategory of $C$, which is the core of $C$. Must $(C,W)$ admit a
calculus of right fractions?
then the answer is yes. (And the situation is symmetric, so it also admits a calculus of left fractions.) | {"url":"https://nforum.ncatlab.org/discussion/7550/simple-question-about-the-core-of-categories/","timestamp":"2024-11-07T23:35:34Z","content_type":"application/xhtml+xml","content_length":"16428","record_id":"<urn:uuid:3c9b5fea-95f3-4d71-ad94-3840dce34837>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00185.warc.gz"} |
LinBox is a C++ template library for exact, high-performance linear algebra computation with dense, sparse, and structured matrices over the integers and over finite fields. LinBox has the following
top-level functions: solve linear system, matrix rank, determinant, minimal polynomial, characteristic polynomial, Smith normal form and trace. A good collection of finite field and ring
implementations is provided, for use with numerous black box matrix storage schemes.
Available via
Operating Systems
Programming Languages | {"url":"https://orms.mfo.de/project@terms=group+representations&id=261.html","timestamp":"2024-11-07T22:41:58Z","content_type":"application/xhtml+xml","content_length":"7741","record_id":"<urn:uuid:ecb2aeda-a8ce-4542-b1c4-dde9b8166cf9>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00008.warc.gz"} |
Form of a chemical element with 7 letters
Heavy hydrogen, e.g. with 7 letters
Carbon-14 or uranium-235 with 7 letters
Deuterium or tritium with 7 letters
Elemental form for 10 poets with 7 letters
It may be radioactive with 7 letters
Carbon-14, for one with 7 letters
Chemical element form with 7 letters
Form of a chemical element with 7 letters
U-235 or C-14, e.g. with 7 letters
Molecular relative with 7 letters | {"url":"https://geoclu.com/words/1248461/Form+of+a+chemical+element","timestamp":"2024-11-10T05:19:16Z","content_type":"text/html","content_length":"18740","record_id":"<urn:uuid:422f5238-c0c5-4f24-b87c-cd9515fdcf7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00374.warc.gz"} |
Momentum, Heat, and Mass Transfer
This book covers the transport of momentum, heat, and mass in non-equilibrium systems. It derives differential balance equations for general properties and introduces the concepts of convective and
diffusive flux. These are applied to the conservation of mass. Next, differential force balances are used to develop the governing equations for momentum transport, and includes a discussion of
stress and viscosity. Dimensional analysis is discussed. The differential energy balance is then presented, along with Fourier's law. Finally, differential species balances are performed for
multicomponent systems, and Maxwell-Stefan diffusion and Fick's law are discussed. An analysis of turbulence and the statistical modeling of its effects on transport is provided. This is followed by
a description of boundary layer theory, and then a discussion of the analogies between the transport of momentum, heat, and mass. Finally the two resistance model for interphase mass transfer is | {"url":"https://bookboon.com/en/momentum-heat-and-mass-transfer-ebook","timestamp":"2024-11-03T12:04:23Z","content_type":"text/html","content_length":"95859","record_id":"<urn:uuid:f9a67769-224e-4ab3-a0ce-4109ecf9b616>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00280.warc.gz"} |
Power spectrum for binary NRZ data with less than 50 percent transitions
Spacecraft-to-ground telemetry received by the DSN can often be modelled as binary NRZ data with independent transitions of probability p equal to or less than 1/2. A simple expression is derived for
the power spectrum of this type of data modulation; this formula is used to investigate how rapidly the data bandwidth decreases as p gets smaller.
The Deep Space Network
Pub Date:
April 1976
□ Binary Data;
□ Power Spectra;
□ Bandwidth;
□ Crosstalk;
□ Data Links;
□ Deep Space Network;
□ Probability Theory;
□ Telemetry;
□ Space Communications, Spacecraft Communications, Command and Tracking | {"url":"https://ui.adsabs.harvard.edu/abs/1976dsn..nasa...86L/abstract","timestamp":"2024-11-07T04:46:32Z","content_type":"text/html","content_length":"33122","record_id":"<urn:uuid:60135968-2a02-4d56-8aab-61b69f5255fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00664.warc.gz"} |
The International Journal of Railway ResearchTwo Mathematical Models for Railway Crew Scheduling Problem
fa Electrical railway Electrical railway پژوهشي Research Railway crew scheduling problem is a substantial part of the railway transportation planning, which aims to find the optimal combination of
the trip sequences (pairings), and assign them to the crew complements. In this problem, each trip must be covered by at least one pairing. The multiple-covered trips lead to impose useless transfers
called “transitions”. In this study, a new mathematical model to simultaneously minimize both costs of trips and transitions is proposed. Moreover, a new mathematical model is suggested
to find the optimal solution of railway crew assignment problem. This model minimizes the total cost, including cost of assigning crew complements, fixed cost of employing crew complements and
penalty cost for short workloads. To evaluate the proposed models, several random examples, based on the railway network of Iran are investigated. The results demonstrated the capability of the
proposed models to decrease total costs of the crew scheduling problem. Railway, Crew Scheduling Problem, Assignmen, Transition reduction, Workload 11 22 http://ijrare.iust.ac.ir/browse.php?a_code=
A-10-108-10&slc_lang=fa&sid=1 Amin Khosravi Bizhaem m.tamannaei@cc.iut.ac.ir 180031947532846001408 180031947532846001408 Yes Department of Transportation Engineering, Isfahan University of Technology
Department of Transportation Engineering, Isfahan University of Technology Mohammad Tamannaei 180031947532846001409 180031947532846001409 No Department of Transportation Engineering, Isfahan
University of Technology Department of Transportation Engineering, Isfahan University of Technology | {"url":"https://ijrare.iust.ac.ir/xml_out.php?a_id=170","timestamp":"2024-11-05T21:56:11Z","content_type":"application/xml","content_length":"5116","record_id":"<urn:uuid:80aa839d-59fd-4bf6-b3b9-ce95ad80f140>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00057.warc.gz"} |
February 2017 – Incompressible Dynamics
Everything in the Universe is fundamentally a system of some sort. And all systems fundamentally have the same Universal Dynamics…
Our Universe is fundamentally the interplay of 2 things; Energy and Probability. Excessive Energy can cause Incompressible Dynamics. Probability can cause Random Energy Clustering which can
ultimately lead to Natural Reinforcement. And the interplay of both incompressible dynamics and probabilistic reinforcement can cause the emergence of Self-Integrated Matter.
Everything in the universe is ultimately some type of “System”. These systems have a range of behavior from Simple Non-Adaptive Systems to Complex Adaptive Systems. All systems are essentially the
interplay of 2 things; Energy and Feedback. Excessive Energy can cause Incompressible Dynamics. Feedback can cause Natural Reinforcement. And the interplay of Incompressible Energy and Natural
Reinforcement can cause of the emergence of Self-Integrated Complexity.
The Brain is a Complex Adaptive System. The Brain is capable of modeling Data from the External World. The more data the brain is exposed to the more likely there will be diversity within the
data. The Brain is dominated by 2 things; External Data and Conscious Evaluation. An excess of data can produce a great diversity of data. Experimentation can extract information and knowledge
from data. And the interplay of both a Diversity of Data and Conscious Evaluation can generate Spontaneous Self-Integration and Deep Intuition…
The Mind is a Complex Adaptive System. The Mind is capable of modeling Thoughts and Intuitions from its own Internal World. The more thoughts the mind is willing to entertain the more likely there
is diversity within the thoughts. The Mind is dominated by 2 things; Thoughts and Feelings. A lot of thoughts can produce a great diversity of intuition. Thinking can extract ideas from thoughts
and intuitions. And the interplay of both a Diversity of Intuition and Emotional Reinforcement can generate Spontaneous Self-Integration and Deep Creativity…
The Universal Dynamics of Everything…
Is “Evolution” solely a theory about the emergence of life, or is it a more generalized “Meta-Theory” about “The Emergence of Everything”…
Compressible Dynamics
Over the last 400 years or so Mathematical Physics has become the science that we rely on to explain the behavior of the universe. Mathematical physics is the ultimate science of the “deterministic/
predictable” dynamics of “cause and effect”.
In general, the Science of Physics likes to believe that all dynamics, all natural behavior, can be explained mathematically; and consequently physicists like to build “mathematical models” of (cause
and effect in) the real world. Sometimes these models are unbelievably concise, and can be expressed as a neat linear differential equation, and when this happens we confidently call the model a
“Deterministic”, “Law of Physics”.
It is precisely because of these so-called “hard and fast scientific laws” that physicists are wont to describe their science as the hardest of “hard science”. This of course would seem to imply
that many of the so-called “soft sciences” are in some way not quite as elevated, not quite as good.
In truth however we could say that physics is an “easy science”, and the soft sciences are “difficult” because the “laws” of physics only really work in the absence of “noise”, and yet the soft
sciences are condemned to deal with our everyday world which is full of noise — because virtually everything in our everyday world is continually battered and buffeted by “constantly changing
feedback” which can generate wild “nonlinear dynamics”.
In reality all dynamics have feedback (and resultant nonlinearity), it is just that some dynamics have much less feedback than others.
Physics is, in a sense, the science of the nonlinear stuff that can be safely “compressed into the neat linear mathematics of cause and effect”. In other words; Physics is primarily a science of
“linear” dynamics, a science of dynamics “without feedback” (or more realistically a science of dynamics with negligible feedback). Such dynamics are indeed easily compressible, but our real world
is a world that abounds with feedback, a “nonlinear” world full of “incompressible dynamics”.
[Note: In the simplest possible terms, linear dynamics are dynamics where the effect is proportional to the cause, and nonlinear dynamics are where the effect can be disproportional to the cause.]
Incompressible Dynamics
It could be said that throughout its 400 year history physics has had great difficulty dealing with non “linearizable” dynamics, because these wild dynamics are messy, mathematically unstable, and
consequently difficult to predict.
Turbulent systems are the most obvious example. Turbulent systems are mathematically non-linearizable because they have lots of internal instabilities due to the excessive amount of “energy” in the
More recently (in the last 40 years or so) we have started to become more aware of other types of systems that are mathematically non-linearizable. Complex Adaptive Systems (CAS) are systems whose
elements are not completely independent of each other and consequently they can exhibit a lot of internal instability due to the excessive amount of “adaptation” in the system.
The economy is the most obvious example of a CAS. And so while economists might like to think they can build mathematically models of the economy, this is simply not possible, because the economy is
mathematically non-linearizable and full of incompressible dynamics.
In the coming years many more people will begin to understand the difficulty and inherent uncertainty involved in dealing with CAS’s. As our world becomes ever-more interconnected and co-dependent,
more and more systems will become adaptive and complex, and consequently will exhibit incompressible dynamics and unpredictable “emergent behavior”.
And so in the future we will all have to learn to live with uncertainty. But in case all of this seems overly pessimistic, fear not, for there is another side to CAS. Complex Adaptive Systems may
be unpredictable but they are also massively “Creative”.
The Century of Complexity and Creativity
In Conclusion: Physics tell us that to understand the world we need simply to understand “the dynamics of cause and effect”; but the simple dynamics of cause and effect fail quite miserably when it
comes to explaining “Natural Evolution and Emergent Complexity”...
However, understanding evolution is going to turn out to be much more important than anyone might previously have thought. Because despite what most people might think, evolution is not solely a
theory about the emergence of life, but a more “Generalized Meta-Theory” (of which biological evolution is merely a special case). Evolution is effectively spontaneous and complex creativity in
And so although science may have spent the last 400 years honing its understanding of The Linear Dynamics of Cause and Effect, the reality of life in the 21^st century is that the really interesting
stuff will increasingly result from the “Universal Creative Dynamics” of “Adaptive Integration and Emergent Complexity”…
In Conclusion
The 21^st Century will see the rise of the Complex Adaptive System. Complex Adaptive Systems are systems that are capable (without any external assistance) of self-designing and reinforcing
themselves into existence. This means that, in an evermore interconnected world, the future of the human race is likely to become much more uncertain — but as evolution shows us, uncertainty
generates emergent complexity, so
Embrace The Chaos and Harvest the Creativity… | {"url":"https://www.kierandkelly.com/2017/02/","timestamp":"2024-11-08T07:25:03Z","content_type":"text/html","content_length":"56661","record_id":"<urn:uuid:d575a2d7-417e-40ec-8a23-ed59906f2069>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00516.warc.gz"} |
Sampling for Bayesian Extreme Value Analysis
Gulf of Mexico Wave Height Data
The numeric vector gom contains 315 storm peak significant wave heights from a location in the Gulf of Mexico, from the years 1900 to 2005. These data are analysed in Northrop, Attalides, and
Jonathan (2017). We set the threshold at the 65% sample quantile and use set_prior to set a prior.
We sample first on the \((\sigma_u, \xi)\) scale, with mode relocation only.
t1 <- system.time(
gp1 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom,
rotate = FALSE)
Then we add a rotation of \((\sigma_u, \xi)\) about the estimated posterior mode.
Now we add marginal Box-Cox transformation. We apply Box-Cox transformation to the parameters \(\phi_1 = \sigma_u\) and \(\phi_2 = \xi + \sigma / x_{(m)}\), where \(x_{(m)}\) is the largest threshold
excess. The parameters \(\phi_1\) and \(\phi_2\) are positive for all combinations of \((\sigma_u, \xi)\) for which the GP likelihood is positive.
t3 <- system.time(
gp3 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom,
rotate = FALSE, trans = "BC")
t4 <- system.time(
gp4 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom,
trans = "BC")
We plot the samples obtained with the contours of the corresponding densities superimposed. The plot on the top left is on the original \((\sigma_u, \xi)\) scale. The other plots are on the scale
used for the ratio-of-uniforms algorithm, that is, with relocation of the mode to the origin. In the following the \(\rho_i, i = 1, \ldots, d\), \(\rho_1\) and \(\rho_2\) in this example, are the
variables to which the ratio-of-uniforms algorithm is applied, i.e. after any transformation (Box-Cox and/or rotation of axes) and relocation of the mode to the origin.
plot(gp1, ru_scale = FALSE, cex.main = 0.75, cex.lab = 0.75,
main = paste("no transformation \n pa = ", round(gp1$pa, 3),
", time = ", round(t1, 2), "s"))
plot(gp2, ru_scale = TRUE, cex.main = 0.75, cex.lab = 0.75,
main = paste("rotation \n pa = ", round(gp2$pa, 3),
", time = ", round(t2, 2), "s"))
plot(gp3, ru_scale = TRUE, cex.main = 0.75, cex.lab = 0.75,
main = paste("Box-Cox \n pa = ", round(gp3$pa, 3),
", time = ", round(t3, 2), "s"))
plot(gp4, ru_scale = TRUE, cex.main = 0.75, cex.lab = 0.75,
main = paste("Box-Cox and rotation \n pa = ", round(gp4$pa, 3),
", time = ", round(t4, 2), "s"))
Comparison of the plots on the right to the plots on the left shows that rotation of the parameter axes about the mode of the posterior has reduced dependence between the components. The estimated
probabilities of acceptance for the plots on the right are close to the 0.534 obtained for a 2-dimensional normal distribution with independent components. Box-Cox transformation has increased the
estimated value of \(p_a\) but not by much. In fact, the extra computation time required to calculate the Box-Cox transformation each time that the posterior density is evaluated means that posterior
sampling is slower than the default setting of rotate = TRUE and trans = "none". That is, in this example at least, the increase is \(p_a\) resulting from the addition of the Box-Cox transformation
is not sufficient to offset the extra time needed to compute the posterior density.
We repeat the rotation only and Box-Cox transformation plus rotation analyses for a much higher threshold, set at the 95% sample quantile.
thresh <- quantile(gom, probs = 0.95)
fp <- set_prior(prior = "flat", model = "gp", min_xi = -1)
t2 <- system.time(
gp2 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom)
t4 <- system.time(
gp4 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom,
trans = "BC")
It is optimistic to use such a high threshold in this example, because it results in only 16 threshold excesses. In the rotation only case, a convergence warning is triggered by one of the
optimisations used to created the bounding box of the ratio-of-uniforms method. This appears to be a spurious warning, because the plots of the simulations below seem fine. When a Box-Cox
transformation is used there are no warnings, because the optimisations have greater stability and it is now easier for the convergence of the optimisations to be confirmed.
Strong asymmetry in the posterior distribution means that the combination of marginal Box-Cox transformation and rotation produces a larger improvement in \(p_a\) and means that now this strategy is
competitive with using rotation alone in terms of computation time.
A User-defined prior
Suppose that we wish to use a prior for \((\sigma_u, \xi)\) with a density that is proportional to \(\sigma_u^{-1} (1+\xi)^{\alpha-1} (1-\xi)^{\beta-1}\), for \(\sigma_u > 0, -1 < \xi < 1\) and for
some \(\alpha > 0\) and \(\beta > 0\). This is an improper prior in which \(\sigma_u\) and \(\xi\) are independent a priori, \(\log \sigma_u\) is uniform over the real line and \(\xi\) has beta(\(\
alpha, \beta\))-type distribution on the interval \((-1, 1)\). We can do this by creating a function that returns the prior log-density and passing this function to set_prior. The first argument of
the prior log-density function must be the parameter vector (the GP parameters \((\sigma_u, \xi)\) in this case), followed by any hyperparameters.
u_prior_fn <- function(x, ab) {
# Calculates the the log of the (improper) prior density for GP parameters
# (sigma_u, xi) in which log(sigma_u) is uniform on the real line and xi has
# a beta(alpha, beta)-type prior on the interval (-1, 1).
# Args:
# x : A numeric vector. GP parameter vector (sigma, xi).
# ab : A numeric vector. Hyperparameter vector (alpha, beta), where
# alpha and beta must be positive.
# Returns : the value of the log-prior at x.
if (x[1] <= 0 | x[2] <= -1 | x[2] >= 1) {
return(-log(x[1]) + (ab[1] - 1) * log(1 + x[2]) +
(ab[2] - 1) * log(1 - x[2]))
up <- set_prior(prior = u_prior_fn, ab = c(2, 2), model = "gp")
gp_u <- rpost(n = n, model = "gp", prior = up, thresh = thresh, data = gom)
See also the “Posterior predictive extreme value inference” section of the Posterior Predictive Extreme Value Inference using the revdbayes Package for an analysis that adds inferences about the
probability of threshold exceedance to the analysis of threshold excesses using a GP distribution. | {"url":"https://cran.mirror.garr.it/CRAN/web/packages/revdbayes/vignettes/revdbayes-a-vignette.html","timestamp":"2024-11-03T16:21:58Z","content_type":"text/html","content_length":"205013","record_id":"<urn:uuid:59e0b7e6-6d56-4282-bb52-15ba75f114ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00366.warc.gz"} |
Most Probably Intersecting Families of Subsets
Katona, Gyula and Katona, Gyula (Ifj.) and Katona, Z. (2012) Most Probably Intersecting Families of Subsets. COMBINATORICS PROBABILITY AND COMPUTING, 21 (1-2). pp. 219-227. ISSN 0963-5483
Most Probably Intersecting Families of Subsets.pdf
Download (266kB) | Preview
Let F be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the
inequality vertical bar F vertical bar <= 2(n-1). Suppose that vertical bar F vertical bar = 2(n-1) + i. Choose the members of F independently with probability p (delete them with probability 1 - p).
The new family is intersecting with a certain probability. We try to maximize this probability by choosing F appropriately. The exact maximum is determined in this paper for some small i. The
analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family
only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all [n/2]-element
subsets. We determine the most probably inclusion-free family too, when the number of members is (n([n/2])) + 1.
Actions (login required) | {"url":"https://real.mtak.hu/7992/","timestamp":"2024-11-13T02:49:02Z","content_type":"application/xhtml+xml","content_length":"21604","record_id":"<urn:uuid:e927afe1-ffc0-44cf-8499-d0f2b09657eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00347.warc.gz"} |
Steven Weinberg and the Puzzle of Quantum Mechanics | N. David Mermin
In response to:
The Trouble with Quantum Mechanics from the January 19, 2017 issue
To the Editors:
My article “The Trouble with Quantum Mechanics” [NYR, January 19] provoked a flood of comments. Some were from nonscientists charmed to learn that physicists can disagree with one another. Here there
is only room to outline a few comments from physicists who offered arguments in favor of interpretations of quantum mechanics that would make it unnecessary to modify the theory. Alas, these
interpretations differ from one another, and none seems to me to be entirely satisfactory. (Several letters on this matter received by The New York Review appear in full following this letter.)
N. David Mermin of Cornell argued with characteristic eloquence for what I (but not he) would call an instrumentalist approach. In his view, science is directly about the relation between each
person’s total experience and the outside world that includes that experience. I replied that I hoped for a physical theory that would allow us to deduce what happens when people make measurements
from impersonal laws that apply to everything, without giving any special status to people in these laws. I suggested that our difference is just that Mermin thinks I had been hoping for too much. He
agreed, with the understanding that those hopes are mine, not his.
In contrast, Thomas Banks of Rutgers in our correspondence and the draft of a new book, Quantum Mechanics: An Introduction, described his elegant efforts to avoid bringing human measurement into the
laws of nature. He describes measurement as an interaction of the system being measured with a macroscopic system, in which probabilities appear much as they do in classical physics. But it is still
necessary to bring into the laws of nature assumptions about these probabilities that I can only understand as probabilities of the values found when humans decide what to measure.
I had an interesting correspondence with Robert Griffiths of Carnegie Mellon and James Hartle of the University of California–Santa Barbara regarding an approach to quantum mechanics variously known
as “decoherent histories” or “consistent histories,” which was introduced in 1984 by Griffiths and further developed by Hartle and Murray Gell-Mann. The laws of nature are supposed to attribute
probabilities to histories of the world, not just to the results of single measurements. I had described this approach in detail in my textbook Lectures on Quantum Mechanics but did not cover it in
my article, because I thought it has the same drawbacks that I attributed to all instrumentalist approaches.
The wave functions for these histories involve averaging over most quantities, with a few held fixed, as if they were being measured, but histories with different things held fixed are incompatible,
and it is humans who must choose the particular kind of history to which to attribute probabilities. Griffiths developed a sort of quantum logic consistent with his approach, but it leaves me
uncomfortable. Hartle and Gell-Mann may share some of this discomfort, for they have moved toward identifying one “true” kind of history that does not have to be selected by people; but they have to
attribute weird negative probabilities to histories of this kind. My discomfort remains.
Jeremy Bernstein, a contributor to these pages, thinks like Mermin that there is no trouble with quantum mechanics as it stands, but he supplied an anecdote that runs in the opposite direction. A
visitor to Einstein’s office in Prague noted that the window overlooked the grounds of an insane asylum. Einstein explained that these were the madmen who did not think about quantum mechanics.
Steven Weinberg
Austin, Texas
Selected letters in response to Steven Weinberg’s article on Quantum Mechanics:
To the Editors:
I agree with Steven Weinberg that “it is a bad sign that those physicists today who are most comfortable with quantum mechanics do not agree with one another about what it all means” [“The Trouble
with Quantum Mechanics,” NYR, January 19]. This ninety-year failure to reach anything like a common understanding of such a spectacularly successful theory indicates that physicists might share an
unrecognized prejudice about the nature of scientific explanation that prevents each of them from seeing what quantum mechanics actually means.
In explaining why he finds untenable what he calls “the instrumentalist approach,” Weinberg gives voice to just such a widespread prejudice: “Humans are brought into the laws of nature at the most
fundamental level.” Weinberg is not ready to give up the goal of understanding the relation of humans to nature by deducing it “from laws that make no explicit reference to humans.” And so he
endorses, with a touch of pessimism, a long-term goal of seeking modifications of quantum mechanics that “are not only speculative but also vague.” He embraces this bleak prospect because he cannot
accept incorporating the relation between people and nature into “what we suppose are nature’s fundamental laws.”
But why not? Science is a human activity. As empiricists most scientists believe that their understanding of the world is based entirely on their own personal experience (which, importantly, includes
the words of others that they have heard and read). Why shouldn’t the science that I use to understand the world be directly about the relation between my total experience and the world outside of me
that induces that experience?
Erwin Schrödinger (a David Levine cartoon of whom illustrates Weinberg’s essay!) traced this deep prejudice of scientists back to the ancient Greeks. He thought it was essential for the early
development of science, but that it removed an important part of the story. He did not suggest that abandoning it dissolved the puzzles of quantum mechanics, but in the early twenty-first century
Christopher Fuchs and Rüdiger Schack argued that it does.
For example Weinberg and many others complain that there is “no way to locate the boundary between the realms in which, according to Bohr, quantum mechanics does or does not apply.” Fuchs and Schack
have a simple answer: the boundary is elusive because it depends on the scientist who is using quantum mechanics, but for each such user it is unambiguous: I apply quantum mechanics to the world I
infer from my own experience; the role of my classical world is played for me by that experience.
Last year Hans Christian von Baeyer published a beautiful exposition of this new point of view, QBism: The Future of Quantum Physics. I recommend von Baeyer’s little book to readers of Weinberg’s
essay. It addresses Weinberg’s concerns, is written at an entirely nontechnical level, and makes it clear that the resolution applies not only to quantum mechanics but also to even older, if less
vexing, puzzles in classical physics.
N. David Mermin
Horace White Professor of Physics Emeritus
Cornell University
Ithaca, New York
To the Editors:
Steven Weinberg’s article on quantum mechanics is written with his usual clarity and brilliance. But I think that it is misguided. The probabilistic interpretation of the Schrödinger wave function
was introduced by Max Born in a very brief note in 1926. He considered the collision of an electron with a target and studied the wave function that represented the electron after the collision. It
is a function of position and he said that the wave function determined the probability that the electron would occupy that position. Later he modified it to say that it was the square of the wave
function and still later he said the absolute value determined the probability. The notion that this is “derived” is absurd. It was a postulate. Now the present generation of quantum theorists—some
of them—find this unsatisfactory and want to produce a derivation. Of course it can’t be derived from quantum mechanics as we understand it so they want to introduce “quantum” mechanics from which it
can be derived, In the most recherché versions of this, “quantum” mechanics has slightly different predictions from quantum mechanics. If they are right it would be revolutionary but to me the whole
enterprise is a solution looking for a problem.
Jeremy Bernstein
Aspen, Colorado
To the Editors:
Early on, students of quantum mechanics rhymed: “Erwin with his psi can do calculations quite a few, but one thing has not been seen: just what does psi really mean?” The answer was first given by
Max Born, who received the Nobel Prize for giving meaning to psi, the wave function of a system, stating that the absolute square of this function is the probability of finding the system under
observation in a given state. Steven Weinberg claims that the trouble with quantum mechanics is that the wave function “is governed by an equation, the [Erwin] Schroedinger equation, that does not
involve probabilities.” Since this equation is perfectly deterministic, he asks, “how do probabilities get into quantum mechanics?”
But also in classical mechanics, given the inevitable uncertainties in the initial conditions, the situation is actually similar, because only probabilities of the outcome at a later time can be
predicted. The main difference with quantum mechanics is that in this theory there is a limit on the relative size of the initial uncertainties, e.g., the position and velocity of an object. In his
Nobel Prize speech, Born stated that “ordinary mechanics must also be statistically formulated…the determinism of classical physics turns out to be an illusion…and cannot be used as an objection to
the essentially indeterministic statistical interpretation of quantum mechanics.” The nature of reality in the atomic world, however strange, is revealed by experiments, and not by requiring that it
fit some prejudices from classical mechanics as Weinberg indicates.
Michael Nauenberg
Professor of Physics, Emeritus
University of California
Santa Cruz, California
To the Editors:
Steven Weinberg has stated clearly and unambiguously that there is something rotten in the kingdom of the “Copenhagen interpretation” of quantum mechanics. Though they often continue to pay lip
service to that “interpretation” in their courses and papers, an increasing number of physicists realize this, and nobody is quite sure nowadays what that interpretation really means. As Weinberg
says, “It is a bad sign that those physicists today who are most comfortable with quantum mechanics do not agree with one another about what it all means.”
Weinberg identifies the basic problem with quantum mechanics: that one applies a different rule of evolution to the wave function of a system when it does not contain an observer or measuring device
(one then uses the deterministic Schrödinger evolution) from when it does (one then collapses the wave function in a random fashion, following a rule, due to Max Born, for the probabilities of the
result). That not only puts the “observer,” whether it is a human subject or an instrument in a laboratory, outside the ordinary laws of physics, but also renders that “observer” indispensable to
make sense of those laws. We agree with Professor Weinberg that this is deeply unsatisfactory.
Weinberg mentions two “ways out” of the problems of quantum mechanics: the “many-worlds interpretation” of Hugh Everett and the “spontaneous collapse” theories of Gian Carlo Ghirardi, Alberto Rimini,
and Tullio Weber. For the first option, Weinberg observes that he doesn’t see how to justify the use of the usual quantum mechanical probabilities, given by Born’s rule, within that framework, though
he is aware of a variety of attempts to do so. According to the second option, the predictions of quantum theory are not quite correct, but only a very good approximation to the more correct
predictions of the spontaneous collapse theory. Many experiments are being carried out in order to decide between spontaneous collapse theories and quantum mechanics. So far, there is no indication
that quantum mechanics is wrong and its spectacular successes show that, if its predictions are indeed violated in some situations, this will not be easy to demonstrate.
Weinberg presents the Copenhagen interpretation on the one hand, and many-worlds and spontaneous collapse theories on the other, as corresponding respectively to what he calls an instrumentalist and
a realist approach to the wave function. In the instrumentalist approach the wave function is not regarded as something to be taken seriously as real or objective, but merely as a convenient tool for
describing the behavior of measuring devices and the like. In the realist approach, according to Weinberg, the wave function is not only real and objective but also exhaustive, providing a complete
description of the physical state of affairs. In other words, the alternatives for the wave function for Weinberg are either that it is nothing or it is everything.
However, Weinberg does not mention a third possibility, the de Broglie–Bohm theory or Bohmian mechanics, in which the wave function is something but not everything. This theory, which we consider to
be, by far, the simplest version of quantum mechanics, does not require any modification of the predictions of ordinary quantum mechanics, nor a bizarre (to say the least) multiplication of parallel
universes. It was proposed by Louis de Broglie in 1927 and rediscovered and developed by David Bohm in 1952. For several decades its main proponent was John Stewart Bell, the physicist who did more
than any other to establish the existence of the quantum non-locality mentioned by Weinberg.
In Bohmian mechanics a system of particles is described by actual positions of actual particles in addition to its wave function: particles actually do have positions at all times, hence trajectories
and also velocities. Their time evolution is guided in a natural way by the wave function, which functions as what is often called a pilot wave. This should be contrasted with the role of the wave
function in the instrumentalist approach: to predict the behavior of (clearly nonfundamental) measuring devices. Thus the wave function in Bohmian mechanics is somewhat similar to the forces or the
electromagnetic waves guiding the particles in classical physics.
The wave functions of closed systems in Bohmian mechanics, even systems containing observers and measuring devices, always follow Schrödinger’s equation and never collapse. Thus, observations are no
longer a deus ex machina in that theory. When one analyzes in Bohmian mechanics what is called a “measurement” in ordinary quantum mechanics, one finds that the behavior of the particles yields a
world in which measurement results conform precisely to the quantum mechanical predictions. Such an analysis of quantum measurements also explains why the fact that particles have both positions and
velocities at all times does not contradict the Heisenberg uncertainty principle. In particular, although Bohmian mechanics is perfectly deterministic, one can recover the statistical predictions of
ordinary quantum mechanics (the Born rule mentioned by Weinberg) by making natural assumptions on the initial conditions of physical systems (something which has become familiar among physicists with
the development of modern “chaotic” dynamical systems theory).
While Bohmian mechanics is a version of nonrelativistic quantum mechanics and not of quantum field theory, the basic idea of Bohmian mechanics—that the wave function should be something but not
everything—applies to any quantum theory. In fact there are a variety of Bohmian versions of quantum field theory, though it would be fair to say that there is no agreed-upon best or canonical
version for relativistic physics.
Jean Bricmont
Professor of Theoretical Physics
University of Louvain
Louvain-la-Neuve, Belgium
Sheldon Goldstein
Distinguished Professor of Mathematics, Physics and Philosophy
Rutgers University
New Brunswick, New Jersey
Tim Maudlin
Professor of Philosophy
New York University
New York City | {"url":"https://www.nybooks.com/articles/2017/04/06/steven-weinberg-puzzle-quantum-mechanics/","timestamp":"2024-11-10T08:20:12Z","content_type":"text/html","content_length":"152993","record_id":"<urn:uuid:d6560d65-eec9-422c-b0eb-929e24f96cca>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00130.warc.gz"} |
Theory of solar oscillations in the inertial frequency range: Amplitudes of equatorial modes from a nonlinear rotating convection simulation
Context. Several types of inertial modes have been detected on the Sun. Properties of these inertial modes have been studied in the linear regime, but have not been studied in nonlinear simulations
of solar rotating convection. Comparing the nonlinear simulations, the linear theory, and the solar observations is important to better understand the differences between the models and the real Sun.
Aims: Our aim is to detect and characterize the modes present in a nonlinear numerical simulation of solar convection, in particular to understand the amplitudes and lifetimes of the modes.
Methods: We developed a code with a Yin-Yang grid to carry out fully nonlinear numerical simulations of rotating convection in a spherical shell. The stratification is solar-like up to the top of the
computational domain at 0.96 R[⊙]. The simulations cover a duration of about 15 solar years, which is more than the observational length of the Solar Dynamics Observatory (SDO). Various large-scale
modes at low frequencies (comparable to the solar rotation frequency) are extracted from the simulation. Their characteristics are compared to those from the linear model and to the observations.
Results: Among other modes, both the equatorial Rossby modes and the columnar convective modes are seen in the simulation. The columnar convective modes, with north-south symmetric longitudinal
velocity v[ϕ], contain most of the large-scale velocity power outside the tangential cylinder and substantially contribute to the heat and angular momentum transport near the equator. Equatorial
Rossby modes with no radial nodes (n = 0) are also found; they have the same spatial structures as the linear eigenfunctions. They are stochastically excited by convection and have the amplitudes of
a few m s^−1 and mode linewidths of about 20−30 nHz, which are comparable to those observed on the Sun. We also confirm the existence of the "mixed" Rossby modes between the equatorial Rossby modes
with one radial node (n = 1) and the columnar convective modes with north-south antisymmetric v[ϕ] in our nonlinear simulation, as predicted by the linear eigenmode analysis. We also see the
high-latitude mode with m = 1 in our nonlinear simulation, but its amplitude is much weaker than that observed on the Sun.
Astronomy and Astrophysics
Pub Date:
October 2022
□ convection;
□ Sun: rotation;
□ Sun: interior;
□ Sun: oscillations;
□ Sun: helioseismology;
□ Astrophysics - Solar and Stellar Astrophysics
18 pages, 21 figures, 1 table. Astronomy & | {"url":"https://ui.adsabs.harvard.edu/abs/2022A%26A...666A.135B/abstract","timestamp":"2024-11-07T01:50:11Z","content_type":"text/html","content_length":"46797","record_id":"<urn:uuid:b907b2bb-a5fc-441e-887f-8b3d737479c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00222.warc.gz"} |
Spud’s maths lessons - Tim Worstall
Now let me pose a question I often put on the board when starting a discussion of maths in the context of the political economy of data with second year undergraduates. It was this:
2 + 2 = ?
I have never yet had a student give the right answer.
They all say 4.
It isn’t.
It’s 5. Both those figures written as 2 were 2.49 rounded to the nearest whole number. The sum of the two is 4.98, which rounded to the nearest whole number is 5. The answers the students gave me
were almost 25% out in all cases.
And these wrong answers came despite children being taught about rounding to whole numbers in primary schools, just about anywhere in the world.
Holy Shit that’s bad.
Because the thing you’re taught about rounding numbers is that you don’t round numbers during a calculation – for this very reason – and only do so with the final answer.
Well, they did at the LSE, Downside, Worth, the Air Force school in Naples and St Alphege’s anyway.
46 thoughts on “Spud’s maths lessons”
He’s an idiot.
Meanwhile, and off topic but more serious:
“Both those figures written as 2 were 2.49 rounded to the nearest whole number. ”
If those figures were written as 2, then how are the students to know that they were “really” 2.49?
Figures are just figures.
Oh, and cunts are just cunts.
I knew that “academia” was fraying at the edges with identity overriding intelligence but this is amazing. What disguise could Spud possibly wear that would make someone think there were a couple
of synapses linked up in that dome?
“I have never yet had a student give the right answer.”
The correct answer to 2+2 is 4. The correct answer to “Let x,y be real numbers which, when rounded to zero decimal places, both equal 2. Then what is x+y when rounded to zero decimal places?”
then the correct answer is “x+y \in {3,4,5}”. A better mathematician than me will phrase this more formally.
The students have answered the question correctly as set. The question setter has not phrased the question accurately to reflect what they want to test. This would be successfully appealed if it
turned up in an exam (although a proper external examiner wouldn’t let it show up in the first place).
What a mongo
Similarly much of the climate science-cum- propaganda involves taking averages in the middle of the calculation and using them as input to the next steps. Climate grifters seem unaware of the
traps involved. Such as, you take an areal average of temperature ranges? What exactly is that? It’s a number but not a value of anything. You can’t just use it as if it meant something.
Who would have guessed that today the polymath of Ely would solve number theory.
All he’s saying is that the answer to 2 + 2 is anything that he deems it to be.
How the fuck is this man employed as a professor and pity the poor bastards paying for this
A few years ago, a young engineer came to me with a ‘problem’. The standard stipulated a minimum factor of safety of 1.6 and his calculations returned an answer of 1.58, which of the costly
options should we implement to increase the FoS? I told him to round it to one decimal place, just like in the standard. I also took the opportunity to educate him on factors of safety and
probabilities of failure when your data are variable and uncertain. Something that I imagine would also be a significant problem in economics.
“When I use a number,” Murphy said in rather a scornful tone, “it means just what I choose it to mean–neither more nor less”
What if the first 2 is actually 1.51?
@ decnine
Please leave my lecture, this is your last class at this university
+50% -20% = +20% (e.g. start with 100, gain 50% in year 1, lose 20% in year 2, the result is you’re 20% ahead)
Divide by 10 and move the negative number across and we have
So we’re all wrong lol. Every ex accountant knows this shirley.
Ye gods!
Even as trying to be a smart arse “intellectual” this is mind numbing!
Now let me pose a question I often put on the board when starting a discussion of maths in the context of the political economy of data with second year undergraduates. It was this:
When was the Battle of Hastings?
I have never yet had a student give the right answer.
They all say 1066.
It isn’t.
It’s 1993, when I fell out with my wife during a coach trip to Sussex.
Tory neoliberalism has making are children stupiderest.
It does remind you of that old joke, where you ask a mathematician, an engineer, and an accountant what 2+2 equals. The mathematician says 2, the Engineer says 2, +/- your uncertainties, and the
Accountant says…well, what do you want it to be…
If the figure was 2.51 then I suspect even the Professor would be able to round that to 2.5!
I wouldn’t bet the house on that, mind.
And then I go and misread decnine’s comment as 2.51 instead of 1.51 !!
Candidly, I need to attend a Murphy lecture.
As in Tim’s previous submission, if Spud had ever done anything in his life but be an overblown book-keeper he might have a firmer grasp on reality.
Murphy’s wrong.
2+ 2 = 6.
The 2s in the question were 2.99 rounded down to the nearest whole number.
Bugger – rounded up!
Rounding to x decimal places is a pain in the arse for accountants when preparing stats accounts. The balance sheet doesn’t balance. All that matters is cash is king. You fudge a rounding
adjustment (often in the fixed assets note) but then you’ll need to restate opening balances in the subsequent year when that year’s closing balance rounds too high.
Surely there’s some kind of quality control at the University Level? If he is teaching that 2+2 = 5 can’t he be removed from his position on grounds of incompetence?
Hate to say this but there are other types of rounding apart the “standard” rounding.
Rounding options include up/down, to value, nearest, multiples and ranges.
Depends what you are trying to achieve.
Had an oh so clever prof do this trick in my first math lecture at college. Showed us an algebra “proof” that was, or so he claimed, not a valid proof. We had to identify which line of the proof
was invalid. None of us said it was the line 1 + 1 = 2. “But what if we are working in characteristic 2?” he says, “then 1 + 1 = 0 not 2!” Not that he had given any indication what
“characteristic” we were working in, hadn’t taught us what a “characteristic” was, and by the end of the course we still hadn’t been told. I assume it’s something similar to modular arithmetic.
The answer to 7 + 7 is 2 if you’re figuring out the time on a clock 7 hours after 7 o’clock, I get that much. I remember this episode, and little if anything else he taught, only because I
thought this was such a stupid trick if you aren’t even going to explain what it means.
What’s the average train time from Ely to Liverpool street on a weekday using the quickest trains?
about 1hr 30 mins
Wrong. It’s 10 hrs 50 mins. I take the train from Ely to Peterborough, then Peterborough to Kings X, walk to St Pancras. take the Eurostar to Gare du Nord, take the next train back to St P, then
to Manchester Piccadilly. From there to Cambridge and on to Ely.
and last Ely to Liverpool Street
really fvcking it up today!
…rounded to nearest odd, rounded to nearest even, rounded towards zero, rounded away from zero… loads of fun.
Take two numbers at random, as Murphy seems to have done.
The probability that they are both exactly 2.49 strikes me as very low.
There’s accounting fraud software designed to detect this sort of thing.
Much like the old “There are 10 sorts of people in the world… Those who understand binary and those who don’t!”.
Because the thing you’re taught about rounding numbers is that you don’t round numbers during a calculation – for this very reason – and only do so with the final answer.
and further to @rhoda klapp, June 3, 2024 at 8:17 am
Some many years ago I downloaded the Fortran source code of the NASA GISS climate model, I don’t know if it’s still available but I somehow doubt it. They didn’t so much round in the middle of
calculations as manage to lose significance by cheerfully swapping between long and short precision numbers. The old versions of Fortran assigned the precision and type of a variable based on its
name!! and it looked like whoever wrote some of the model didn’t realise it. That was only a relatively minor fault compared to some of the others I saw. This was considered to be the premier
model at the time so I can’t say I’ve had much faith in climate models since!
Well you’re all wrong:
2+2=10 or in Spud world it would be 11
(I’m applying Spud’s trick of expecting you all to be clairvoyant and realise we’re working in Base4)
Murphy torturing logic in the same way that the party does in Nineteen Eighty-Four.
Murphy is O’Brien in this case.
There are few things more repulsive than Richard Murphy when he thinks he’s being clever.
One thing Murphy has highlighted is this: “Political Economy” is a large sack of bullshit.
There’s accounting fraud software designed to detect this sort of thing.
I’ve always presumed that.
There was guy asked me recently about doing a bank transfer that was basically money laundering. So I produced a figure that with 21% VAT added was a number with cents on the end close to what he
wanted to shift & told him to make up an invoice number & stick it in the reference box. I would imagine round number transfers in thousands ring alarm bells for bank compliance IT.
It’s also the thing that puzzles me about the Horizon thing. You’d think if the system was accepting incomplete transfers as complete & therefore listing duplicate transactions, two identical
numbers in a short interval from the same source should stick out a mile. Sort of thing they should have been looking for anyway as part of fraud protection.
Why would any lecturer want to do this?. Set a question that you know they’ll fail because you formulated the question wrong ? -Is his ego so big and he’s so sad that he has to do this? I bet the
students if they have any sense would be muttering cunt under their breathe. I know when i was at Poly if one of our lecturers had been so petty i would have done followed by a quick trip to my
year tutor to put in a complaint.
“There’s accounting fraud software designed to detect this sort of thing.”
I once detected an accounting fraud by noticing that one number, for expenses, was exactly 10% of another number elsewhere in the accounts.
I don’t think the silly sod was set on stealing: I suspect he’d lost a receipt and made a lousy decision on what number to report in its absence. But they sacked him all the same. I suppose that
might have been wise; once they’d lost confidence in him …
Or maybe they’d gone through older accounts to see if there was a pattern of being cavalier about accuracy.
@Ed Snack
Reminds me of the other engineer/mathematician joke
A mathematician and an engineer die and are sent to hell.
They wake up behind a line. A short distance away is an attractive, sexy lady for each man. Satan appears and informs them that every ten minutes the distance between them and the women will half
every ten minutes.
The mathematician cries out in anguish, knowing that he will never get to touch the woman.
The engineer is excited and starts celebrating.
The mathematician looks at him and asks why he’s happy since the distance between them will never get to zero.
The engineer responds that that may be the case, but soon he will be close enough for all practical purposes…
As all actuaries know, 2 + 2 = 5 (for sufficiently large values of ‘2’)
It’s 3.
1.51+1.51 obviously
10 + 10 = 100 in Binary
One spud is not worth cooking (the books)
Have had to point out to people on a few occasions where people have been concerned about a bug or issue that Excel does calculations on the actual numbers not the displayed rounded numbers so it
is entirely possible that if you add up the rounded numbers you have a different answer
I think this is relevant: https://xkcd.com/169/
In the same vein, there are three kinds of people in this world, those who can count….
Thanks for the WHO heads up
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Creating Convolutional Neural Network for Handwritten Digit Recognition - Elinext
July 2, 2024
In the ever-evolving landscape of artificial intelligence, Convolutional Neural Networks (CNNs) have emerged as a revolutionary force, particularly in the realm of image recognition and computer
vision. These sophisticated learning models mimic the human brain’s ability to process visual information, making them incredibly effective for tasks ranging from facial recognition to autonomous
Understanding the basic principles of Convolutional Neural Networks (CNNs) is crucial for several reasons: grasping the basics serves as a stepping stone to more complex concepts in artificial
intelligence and deep learning, knowledge of CNN architecture allows to design and tailor neural networks that are optimized for specific tasks, such as image classification or object detection and
other benefits.
Basic Concepts
Convolutional Neural Networks (CNNs) are at the forefront of advancements in image recognition and processing. They are engineered to dynamically acquire structural layers of characteristics from
visual data. The two pivotal components of CNNs are the Convolutional Layer and the Fully Connected Layer, which work in tandem to process and classify image data and will be described in the details
at this article. Before the data can be fed into the Fully Connected Layer, it must be transformed from a 2D feature map into a 1D vector. This process is known as Flattening. Flattening unrolls the
feature maps into a single long vector, preparing the data for the next phase of processing. Understanding these three main components is the basis for further study of convolutional neural networks.
Convolutional Layer
To better understand the principle of operation of the key component of the Convolutional Layer convolutional neural network, let’s ask the question, what distinguishes a four from an eight?
The four has predominantly horizontal and vertical lines, while the eight has diagonal lines.
In order to transfer this idea to neural networks, consider the following figure. Any number from 0 to 9 represented in the MNIST dataset is represented as a square matrix of size 32 and the value of
each element from 0 to 255.
Next, is needed to introduce the concept of convolution and kernel (filter). The kernel (filter) in neural networks is a matrix of weights that is used in convolutional layers to extract certain
features from the input data. The convolution core (filter) is like a light bulb that lights up depending on the required pattern (reacts to lines). The convolution operation in neural networks is a
process in which the filter elements are multiplied element by element by the elements of the image on which the filter is located. Activation function (act) – fires after each convolutional layer or
fully connected layer to the activation map for each element in the two-dimensional matrix. The stride parameter determines how much the core moves in each iteration. For example, using the kernel to
detect vertical and horizontal lines with a four(4) image matrix will result in an activation map with high coefficients.
Looking ahead, the selection of matrix coefficients(weights) is carried out by a library for deep learning, such as Tensorflow, Keras, Pytorch. The mechanism for selecting matrix coefficients
(weights) is the same as for Fully Connected Layer and will be discussed further in the corresponding section.
Using multiple Convolutional Layers
For real-world images, one convolution operation is not enough to extract all the necessary information from the image.
In real life, figures are not only vertical and horizontal lines but contain more complex structures. For example, handwritten numbers, car models, and human faces. For a clearer explanation, we need
to consider such a complex structure as the human face. In this case, using a single convolution and a filter that reacts to horizontal lines will not help to understand which person the face belongs
to. Therefore, several layers of convolutions are useful. For this task, the convolutional neural network will consist of layers:
1. Low-level patterns
2. Patterns of individual parts of the face
3. Image patterns important for the task
Fully Connected Layer
The operating principle of Fully Connected Layer is very similar to the previously discussed Convolutional Layer. A schematic representation of the Fully Connected Layer is shown in the figure.
Confusion between Fully Connected Layers (FC) and Convolutional Layers is common due to different schematic representations. As we can see, the diagram shows neurons instead of quadratic matrices.
But the principle of operation is the same as that of the Convolutional Layer discussed above. The first input is an image of size 28 by 28. This means that it has 784 pixels. Each pixel is a
separate variable. Each input goes to each of the 128 outputs with its own weight. This turns out to be 100352 weights. The number 128 was chosen at random since it is the one that is optimal in
terms of performance and quality for the hidden layer. We will also train 128 free terms (constants) for each of the neurons of the first layer. Overall there are 100480 parameters. For the second
layer, the number of outputs is known – 10 digits from 0 to 9. Therefore, 128 x 10 + 10 equals 1290 weights. Next, let’s look at the main differences between Fully Connected Layers (FC) and
Convolutional Layers.
Differences between Fully Connected Layer and Convolutional Layer:
1. Functionality:
• Convolutional Layer: Primarily focuses on identifying local patterns within the input data through the use of filters. It is adept at spatial feature extraction and maintains the spatial
hierarchy of the input.
• Fully Connected Layer: Aims to integrate the local features identified by the Convolutional Layer to learn global patterns that help in tasks like classification.
2. Connectivity:
• Convolutional Layer: Neurons in this layer are connected only to a local region of the input, preserving the spatial structure.
• Fully Connected Layer: Neurons here are connected to all activations from the previous layer, hence the name ‘fully connected’.
3. Parameter Sharing:
• Convolutional Layer: Employs parameter sharing, meaning the same filter is applied across the entire input, significantly reducing the number of parameters.
• Fully Connected Layer: Does not use parameter sharing; each weight is unique to a specific connection between neurons.
4. Output Representation:
• Convolutional Layer: Produces a 2D activation map that represents various features of the input data.
• Fully Connected Layer: Outputs a 1D vector that represents high-level features derived from the input data after flattening the 2D activation maps.
5. Role in the Network:
• Convolutional Layer: Acts as a feature extractor that identifies important features without being affected by the position within the input.
• Fully Connected Layer: Serves as a classifier that uses the extracted features to make a final prediction or decision.
6. Presence in the Network:
• Convolutional Layer: Specific to CNNs and is not found in other types of neural networks.
• Fully Connected Layer: Common in many types of neural networks, not just CNNs.
Understanding these differences is crucial for designing and implementing effective neural network architectures for various machine learning tasks.
TensorFlow, Keras and MNIST dataset
The most basic form of constructing a neural network to identify handwritten digits ranging from 0 to 9, as discussed in this article – the entry and the exit layers. Since the size of the source
image is 28×28 pixels, the size of the input layer is 28×28=784 neurons. Each of the neurons is connected to one of the pixels in the image. The output layer contains 10 neurons, because there are 10
digits from 0 to 9.
MNIST is one of the classic datasets on which it is customary to try all sorts of approaches to classifying (and not only) images. The set contains 60’000 (training part) and 10’000 (test part)
black-and-white images of size 28×28 pixels of handwritten numbers from 0 to 9. TensorFlow has a standard script for downloading and deploying this dataset and, accordingly, loading data into
tensors, which is very convenient.
Loading the MNIST data set with samples and splitting it by train and test dataset is displayed at the figure:
On the figure are displayed examples of random 20 elements from the MNIST dataset:
Is displayed transformation of values from 255 to 1 in order to follow working with TensorFlow library.
Next, let’s look at creating the convolutional neural network itself. Sequential – serves to describe a neural network model that has an input, hidden and output layer. Flatten converts a
multidimensional array into a single-dimensional input. Initially, an input layer with 128 neurons is created. The main convolutional layer contains 128 “neurons”. For activation is used tf.nn.ReLU
mathematical function. ReLU is the standard activation function for Convolutional Layers. Range of ReLU is between zero and infinity. This simplifies training and enhances performance.
The output layer matches the 10 recognized digits. This layer helps the network make predictions. In this step is used a softmax activation function: It is like a voting system for the 10 neurons in
the second layer. The system converts neuron outputs into probabilities, reflecting the network’s confidence across 10 possible choices. These numbers could be like this: [48.3, 18.3, 4.3, 0.7, 13.2,
1.0, 2.0, 0.1, 1.5, 9.7]. When we use softmax, it makes these numbers more understandable. It condenses the values to ensure their sum equals 1. It’s like saying, “How sure is the network about each
option?” After softmax, the numbers turn into this array: [0.483, 0.183, 0.043, 0.007, 0.132, 0.01, 0.02, 0.001, 0.015, 0.097]. Now, these new numbers tell the probabilities. For example, the network
is most confident (48.3%) about the first option (0.483).
To build a simple convolutional neural network and recognize handwritten numbers using TensorFlow, you just need to perform four steps:
1. Create an object of the Sequential class, which allows you to create a sequential network model.
2. Flatten the layer to convert the matrix to a single array.
3. Hidden Fully Connected Layer with 128 neurons, in which the main work related to network training will take place.
4. Output Fully Connected Layer with 10 neurons, where each neuron will correspond to a number from 0 to 9.
This sequence of actions is shown in the figure:
Here is displayed the process of compiling and training our neural network:
In the next step, we need to evaluate the performance of a deep learning model on a test set of data using Keras. The purpose of this statement is to measure how well the model generalises unseen
data and to compare the results with the training and validation sets. A good model should have a low loss and a high accuracy on all sets, and avoid overfitting or underfitting.
Then we need to recognize handwritten digits from images with the help of the created model. We read the files as an image using cv2 and extract the first channel (assuming it is a grayscale image)
using [:,:,0]. It inverts the pixel values of the image using np.invert, so that the background is black and the digit is white. It prints a message saying “The number is probably a {}” where {} is
replaced by the index of the highest probability using np.argmax. For example, if the prediction is [0.1, 0.5, 0.2, 0.2, 0, 0, 0, 0, 0, 0], it will print “The number is probably a 5”. Then we display
the image using plt.imshow with a binary colormap and plt.show.
The final result with the predicted figures of numbers lying in the recognition folder is shown at the figure. In the current case, the number of recognized files is 5.
In conclusion, the development of a convolutional neural network (CNN) for the recognition of handwritten digits represents an important step in the field of computer vision and machine learning for
beginners in this field. This article has demonstrated that through the basic architecture of CNN layers and the application of non-linear transformations, we can achieve good accuracy in classifying
individual digits from varied handwriting styles. The understanding of these basic principles paves the way for beginners, how to move on next in the sphere of AI. As we continue to refine the models
based on the knowledge described in this article, we are moving closer to creating systems that can not only recognize handwritten digits but also handle more complex structures.
Our experts can help you create software with this feature. Contact us here. | {"url":"https://www.elinext.com/blog/creating-convolutional-neural-network/","timestamp":"2024-11-12T03:19:07Z","content_type":"text/html","content_length":"234476","record_id":"<urn:uuid:fb48e786-3664-4636-adee-9e01471bb546>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00144.warc.gz"} |
FAQ (Frequently Asked Questions) - Read this before posting!
Q. Has anything related to the fourth dimension ever been observed?
A. Not that we know of as yet.
Q. How to visualize 4D?
A. For some people this is easy, for others it is hard. There are many Java applets around however that can get you used to seeing in 4D - e.g. by showing a rotating tesseract.
Q. What does this term mean?
A. Take a look at the
Q. Isn't time the fourth dimension?
A. Some people consider this so. However, on this forum the general assumption is that time is not a dimension. If you wish to discuss time as a dimension, head to the
Relativity & Time Travel
Feel free to add more quick questions to this topic. Any silly posts in this topic will be deleted without warning.
Q. Has it been proven there is / there isn't a fourth spatial dimension?
A. None of them. We are working on pure hypothesis.
Q: What is a dimension?
A: Mathematically, the dimension of a vector space is the number of vectors in a basis for that vector space. That is, the dimension is the minimum number of vectors {v1, v2, v3 ... vn} such that any
vector, v, in the space can be written in the form
v = c1v1 + c2v2 + ... + cnvn
where c1 ... cn are scalars
Q: Is it possible to study 4D space (and higher dimensional spaces) without being able to visualize it?
A: Yes; it can be done solely through mathematical reasoning. There are already many theorems about higher dimensional spaces in calculus, geometry and topology.
Q: Why is it not possible to reach the speed of light?
A: According to Einstein's Special Theory of Relativity, the energy of a particle (in a particular reference frame) is:
E = sqrt((mc^2)^2 + (pc)^2)
where m is the particle's rest mass, p is its momentum and c is the speed of light
p = gamma * m * v = (1-(v/c)^2)^(-1/2) * m * v
As v -----> c, gamma -----> infinity, so p -----> infinity and thus E -----> infinity.
Thus, an infinite amount of energy is needed to accelerate a particle to the speed of light.
Caveat: This argument does not take into account potential energy. But unless you can find a way to make potential energy decrease by an infinite amount, the argument still holds.
Q: Is it true that according to relativity, when your speed approaches the speed of light, time slows down, lengths contract and your mass increases?
A: Special Relativity is mainly concerned with two things:
1) Determining which quantities are the same in all inertial reference frames.
2) Given a description in one inertial reference frame, determining what happens in another reference frame.
The second objective is done by expressing everything in the original frame in terms of four-vectors, and then applying what is called a Lorentz transformation (i.e. multiplication by a 4x4 matrix of
a particular form).
Time and length are quantities that vary from one reference frame to another. Suppose we have a clock and a ruler, with no relative motion between the two. Let S be a reference frame where the clock
and ruler are stationary. Suppose that in the reference frame S, the period of the clock is T and the length of the ruler is L.
Now suppose we have another reference frame, S', moving parallel to the ruler, with a speed v with respect to S. What is the period of the clock (T') and the length of the ruler (L') according to S'?
According to Special Relativity, the answer is:
T' = gamma * T
L' = L/gamma
where gamma = (1-(v/c)^2)^(-1/2) [as implied in the answer to the previous question]
Thus, clocks run slow and rulers contract in a reference frame where the clock and ruler are in motion. Since all inertial reference frames are equally valid, there is no "true time" or "true
Also, the effects are totally symmetric. A clock at rest in the S' frame will appear to run slow to an observer in the S frame.
Q: So that means that everything is relative?
A: No. As hinted in the first goal of special relativity, there are quantities which are the same in all inertial reference frames. Examples are
c [the speed of light]
(ct)^2 - x^2 - y^2 - z^2 [called the spacetime distance]
m [rest mass]
Re: FAQ (Frequently Asked Questions) - Read this before post
Rob wrote:quick questions
Jinydu that dosen't look quick.
"Civilization is a race between education and catastrophe."
-H.G. Wells
Re: FAQ (Frequently Asked Questions) - Read this before post
Icon wrote:
Rob wrote:quick questions
Jinydu that dosen't look quick.
... or frequently asked.
I am the Nick formerly known as irockyou.
"All evidence of truth comes only from the senses" - Friedrich Nietzsche
On the "4th dimension is/isn't time" topic, I recently found this timely quote from H.S.M. Coxeter's lovely book, Regular Polytopes, which neatly sums up the accepted view here:
Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H. G. Wells in The Time Machine, has led such authors as J.
W. Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski's geometry of space-time is not Euclidean, and consequently has no connection with the
present investigation.
(p.119, emphases his, not mine.)
Coxeter, for those of you who may be unfamiliar with him, was a professor most of whose career was at the University of Toronto, and one of the foremost geometers of this century. He specializes in
higher dimensional space, and so has contributed much to our cause on this forum.
I agree that it's a common and terribly frustrating thing to have the same question raised time and time again. Time is "a" fourth dimension, but in the context of physics and not of geometry - and
since the physics that uses a fourth "temporal" dimension is not Euclidean, in fact it's actually a Lorentzian manifold which is locally Minkowski - the two topics are rightfully in separate parts of
the forum.
Now, were one to question what a tesseract would look like in 4+1-dimensional spacetime as it neared the speed of light...well THEN we'd be in business!
\\having just wrote "forum" into actual words I suddenly felt very ancient greek. Anyone else get that feeling from time to time?
houserichichi wrote:Now, were one to question what a tesseract would look like in 4+1-dimensional spacetime as it neared the speed of light...well THEN we'd be in business!
Oh, it siimply looks contracted in the direction of movement, thats all
having just wrote "forum" into actual words I suddenly felt very ancient greek. Anyone else get that feeling from time to time?
Sometimes words become strange if singled out, though it happened not yet for me with the word "forum". So where was your past life? | {"url":"http://hi.gher.space/forum/viewtopic.php?p=11450","timestamp":"2024-11-04T05:02:52Z","content_type":"application/xhtml+xml","content_length":"33067","record_id":"<urn:uuid:4291a04f-786b-4dd7-ba38-5f545e1158c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00364.warc.gz"} |
SQRTPI function: Description, Usage, Syntax, Examples and Explanation November 12, 2024 - Excel Office
SQRTPI function: Description, Usage, Syntax, Examples and Explanation
What is SQRTPI function in Excel?
SQRTPI function is one of the Math and Trig functions in Microsoft Excel that returns the square root of (number * pi).
Syntax of SQRTPI function
The SQRTPI function syntax has the following arguments:
• Number: The number by which pi is multiplied.
SQRTPI formula explanation
If number < 0, SQRTPI returns the #NUM! error value.
Example of SQRTPI function
Steps to follow:
1. Open a new Excel worksheet.
2. Copy data in the following table below and paste it in cell A1
Note: For formulas to show results, select them, press F2 key on your keyboard and then press Enter.
You can adjust the column widths to see all the data, if need be.
Formula Description (Result) Result
=SQRTPI(1) Square root of pi. 1.772454
=SQRTPI(2) Square root of 2 * pi. 2.506628 | {"url":"https://www.xlsoffice.com/excel-functions/math-and-trig-functions/sqrtpi-function-description-usage-syntax-examples-and-explanation/","timestamp":"2024-11-12T06:19:31Z","content_type":"text/html","content_length":"62316","record_id":"<urn:uuid:77f82d71-b0f1-45fe-9ab4-ca59fe56f678>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00716.warc.gz"} |
This number is a prime.
The largest known Wilson prime.
Reggie Jackson hit 563 home runs in his professional baseball career. He led his teams to five world championships and eleven division titles.
As Pierre Cami points out, this is the smallest prime with a twin Carmichael number, since 563 - 2 = 561 is the first Carmichael number. [Post]
The smallest prime happy number with a prime happy numbered index. [Goelz]
Popular Mechanics sells "The Pocket Genius: 563 Facts That Make You the Smartest Person in the Room."
Two to the power 563 is the smallest power of 2 that contains all twenty-five one and two-digit primes. [Gaydos]
The smallest prime that can be represented as sum of squares of first n successive numbers ending with the same digit, i.e., 1^2+11^2+21^2, (n=3). [Loungrides]
The Buddha was born in the year 563 B.C. at a place called Lumbini.
Only distinct-digit Honaker number which does not share any digit with its corresponding distinct-digit Honaker prime. [Gupta]
The Galápagos Islands are located 563 miles west of continental Ecuador.
(There are 5 curios for this number that have not yet been approved by an editor.)
Printed from the PrimePages <t5k.org> © G. L. Honaker and Chris K. Caldwell | {"url":"https://t5k.org/curios/page.php?number_id=215","timestamp":"2024-11-14T02:25:49Z","content_type":"text/html","content_length":"11203","record_id":"<urn:uuid:6cbf613a-6b33-4898-96eb-2335b0a6c9d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00643.warc.gz"} |
A Collection of Problems in Analytical Geometry
Three-Dimensional Analytical Geometry
• 1st Edition - January 1, 1966
• Editors: W. J. Langford, E. A. Maxwell
• Paperback ISBN:
9 7 8 - 1 - 4 8 3 1 - 2 3 5 0 - 9
• Hardback ISBN:
9 7 8 - 0 - 0 8 - 0 1 2 0 2 7 - 0
• eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 5 5 9 2 - 0
A Collection of Problems in Analytical Geometry, Part II: Three-Dimensional Analytical Geometry is a collection of problems dealing with analytical geometry in the field of… Read more
Save 50% on book bundles
Immediately download your ebook while waiting for your print delivery. No promo code needed.
A Collection of Problems in Analytical Geometry, Part II: Three-Dimensional Analytical Geometry is a collection of problems dealing with analytical geometry in the field of theoretical mechanics. The
book discusses rectangular Cartesian coordinates in three-dimensional space and the division of an interval in a given ratio. The sample questions concern problems dealing with isosceles triangles,
vertices, and center of gravity of equal masses. The book defines the concept of a vector and then lists problems concerning the triangle law and the scalar product of two vectors. Other problems
focus on the equations of a surface and a curve and on questions related to the intersection of three surfaces. The text lists other problems such as the equation of a plane, the direction-vector of
a straight line, and miscellaneous problems pertaining to the equations of a plane, of a straight line, and of a sphere in a direction-vector. The selection is useful for professors in analytical
geometry and for other courses in physic-mathematics and general engineering.
Author's Preface to the First Russian Edition
Author's Preface to the Second Russian Edition
Part II. Three-dimensional Analytical Geometry
6. Elementary Problems of Three-dimensional Analytical Geometry
§ 27. Rectangular Cartesian Coordinates in Three-Dimensional Space
§ 28. Distance between Two Points. Division of an Interval in a Given Ratio
7. Vector Algebra
§ 29. The Concept of a Vector. Projection of a Vector
§ 30. Linear Operations on Vectors
§ 31. The Scalar Product of Two Vectors
§ 32. The Vector Product of Two Vectors
§ 33. Scalar Triple Product
§ 34. Vector Triple Product
8. Equation of a Surface and Equations of Curves
§ 35. Equation of a Surface
§ 36. Equations of a Curve. The Problem of the Intersection of Three Surfaces
§ 37. Equation of a Cylindrical Surface with Generators Parallel to One of the Coordinate Axes
9. Equation of a Plane. Equations of a Straight Line. Equations of Surfaces of the Second Order
§ 38. The General Equation of a Plane. Equation of a Plane Passing through a Given Point and Having a Given Normal Vector
§ 39. Degenerate Equations of a Plane. Equation of a Plane in Terms of Its Intercepts on the Coordinate Axes
§ 40. The Normal Equation of a Plane. Distance of a Point from a Plane
§ 41. Equations of a Straight Line
§ 42. The Direction-Vector of a Straight Line. The Canonical Equations of a Straight Line. Parametric Equations of a Straight Line
§ 43. Miscellaneous Problems Relating to the Equation of a Plane and the Equations of a Straight Line
§ 44. The Sphere
§ 45. Equations of a Plane, a Straight Line, and a Sphere in Vector Notation
§ 46. Surfaces of the Second Order
Appendix: Elements of the Theory of Determinants
§ 1. Determinants of the Second Order: Systems of Two Linear Equations in Two Unknowns
§ 2. Homogeneous System of Two First-degree Equations in Three Unknowns
§ 3. Determinants of the Third Order
§ 4. Properties of Determinants
§ 5. Investigation and Solution of a System of Three Linear Equations in Three Unknowns
§ 6. Determinants of the Fourth Order
Answers to Questions and Notes on Solutions
• Published: January 1, 1966
• Paperback ISBN: 9781483123509
• Hardback ISBN: 9780080120270
• eBook ISBN: 9781483155920 | {"url":"https://shop.elsevier.com/books/a-collection-of-problems-in-analytical-geometry/langford/978-0-08-012027-0","timestamp":"2024-11-03T01:21:46Z","content_type":"text/html","content_length":"177604","record_id":"<urn:uuid:9afa02fd-ccd4-45a0-9aa4-f8beb74870bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00864.warc.gz"} |
measures of central tendency and dispersion avocado python code
2.5865034312755126 Blog. You will be able to: Median - It is the middle value in distribution when the values are arranged in ascending or descending order. This topic is part of Business Statistics
with Python course. Follow DataFlair on Google News & Stay ahead of the game. To categorise a data distribution we need to know about measures of central tendency and dispersion. Measures of
dispersion describe how the values in the signal samples are spread out around a central location. Measures of dispersion. 2. 3.516616314199396 of success. Measure of Central Tendency. This returns
the variance of the sample. Sunil Ray, February 12, 2015 . 3. Measures of central tendency: This measure tries to describe the entire dataset with a single value or metric which represents the middle
or center of distribution. Specifically, how the data is spread out, or the dispersion. In the previous article in this series, we explored the concept of central tendency. Introduction. This
function returns the arithmetic average of the data it operates on. 0 -0.115956 #The distribution is symmetric Answer to 1. The visual approachillustrates data with charts, plots, histograms, and
other graphs. Answer to 1. Fig 7: Comparison of different measures of Central Tendency Some Python coding to watch Central Tendency in action. If the values are widely dispersed, the central location
is said to be less representative of the values as a whole. When you searc… 2.7264140062238043 Home » Measure of dispersion. Otherwise, it returns the middle value. It is the average return. Careers.
While the measure of central tendency is focused towards the central aspects of the given dataset, the measure of dispersion is focused towards the span of the entire dataset. About. Students. This
is the square root of population variance. This section does not intend to introduce or teach Python programming language, but the below embedded codes will help users with even basic familiarity of
Python to calculate central tendency from various data series. This is a subclass of ValueError. Python Descriptive Statistics – Measuring Central Tendency & Variability. Dispersion/spread gives us
an idea of how the data strays from the typical value. A value between -1 and 1 is symmetric. Today, we will learn about Python Descriptive Statistics. Read about Python Namespace and Variable Scope
– Local and Global Variables. 2. And, dispersion helps in evaluating how near or far the other values are from this average value. The following methods are used to find measures of central tendency
in NumPy: mean()- takes a NumPy array as an argument and returns the arithmetic mean of the data. Teachers. Insert code cell below. Like median_low, this returns the high median when the data is of
an even length. Moreover, we will discuss Python Dispersion and Python Pandas Descriptive Statistics. The mean, represented with μ as a parameter of a given population and with x̅ as a statistic of a
population’s sample, is often called the average in daily life. This returns the standard deviation for the sample. It gives us a sense of how much the data tends to diverge from the typical value,
while central measures give us an idea about the typical value of the distribution. 1) Range or variation : Range is the difference between the smallest and highest value. These statistics fall into
two general categories: the measures of central tendency and the measures of spread. Let's get started! You will be able to: 3 This section will look at two types of summary statistics: measures of
central tendency and measures of dispersion. It is also known as measure of center or central location. Useful measures include the mean, median, and mode. Python Descriptive Statistics process
describes the basic features of data in a study. 1) Mean: Mean is the average of the data set. You can apply descriptive statistics to one or many datasets or variables. Last Update: January 14,
2021. Average: It is a value which is typical or representative of a set of data. Measures of central tendency. It gives an idea of the average value of the data in the data set and also to find out
the mode. It is applicable only to numerical values. Feel free to take a look at Course Curriculum. In addition, we used the statistics and pandas modules for this. Some such variations include
observational errors and sampling variation. The quantitative approachdescribes and summarizes data numerically. Do you know about Python Collection Module, Do you know the difference between Python
Modules vs Packages, Let’s Learn CGI Programming in Python with Functions and Modules, Read about Python Namespace and Variable Scope – Local and Global Variables, Follow this to know more about
Python Pandas, Python – Comments, Indentations and Statements, Python – Read, Display & Save Image in OpenCV, Python – Intermediates Interview Questions. Measures of central tendency. Measures of
Central Tendency Mean. Ctrl+M B. In descriptive and inferential statistics, several indices are used to describe a data set corresponding to its central tendency, dispersion, and skewness: the three
most important properties that determine the relative shape of the distribution of a data set. Required fields are marked *, Home About us Contact us Terms and Conditions Privacy Policy Disclaimer
Write For Us Success Stories, This site is protected by reCAPTCHA and the Google. In this chapter, you can learn • how the values of the cases on a single variable can be summarized using measures of
central tendency and measures of dispersion; • how the central tendency can be described using statistics such as the mode, median, and mean; • 1) Mean: Mean is the average of the data set. 4 and
Dispersion. In statistics, there are three common measures of central tendency as shown below: (i) Mean : It is measure of average of all value in a sample set. Measures of central tendency.
Introduction. Measures to describe shape of distribution. 1. It measures or summarizes how spread the data is. Measures of central tendency map a vector of observations onto a single number that
represents, roughly put, “the center”. Objectives. In this lab you'll dive deep into calculating the measures of central tendency and dispersion introduced in previous lessons. R Function : mean() 2)
Median: Median the center value of the data set. Measures of dispersion. exception statistics.StatisticsError Descriptive statistics with Python-NumPy. 2. To categorise a data distribution we need to
know about measures of central tendency and dispersion. It is a measure of the central location of data in a set of values that vary in range. Python Dispersion is the term for a practice that
characterizes how apart the members of the distribution are from the center and from each other. Use this to calculate variance from an entire population. Measures of central tendency. Measures of
dispersion describe how the values in the signal samples are spread out around a central location. The selection of a central tendency measure depends on the properties of a dataset. This is equal to
the square root of the sample variance. Don't become Obsolete & get a Pink Slip For instance, the mode is the only central tendency measure for categorical data, while a median works best with
ordinal data. , harmonic mean, midrange, and geometric median. Measure of central tendency – Measure of central tendency is also known as summary statistics that is used to represents the center
point or a particular value of a data set or sample set. This topic is part of Business Statistics with Python course. Measures of central tendency. Measures under this include mean, median, and
mode. Now let’s take a look at all the functions Python caters to us to calculate the central tendency for a distribution. 5.2 Central tendency and dispersion. It is a measure of the central location
of data in a set of values that vary in range. np.mean(arr) median()- takes a NumPy array as an argument and returns the median of the data. ii] Facilitates comparison. This gives us a great idea of
where the center lies. Functions of Average: i] Presents complex data in a simple form. The arithmetic mean is the sum of data divided by the number of data-points. In layman's terms, central
tendency is nothing but 'average'. In this Python Statistics tutorial, we will discuss what is Data Analysis, Central Tendency in Python: mean, median, and mode. The pandas functions can be directly
used to calculate these values. This returns the population standard deviation. In our last tutorial, we studied Python Charts. Averages are also called Measures of Central Tendency. Descriptive
statisticsis about describing and summarizing data. Measures of central tendency: This measure tries to describe the entire dataset with a single value or metric which represents the middle or center
of distribution. Mean - It is the Average value of the data which is a division of sum of the values with the number of values. In this lab you'll dive deep into calculating the measures of central
tendency and dispersion introduced in previous lessons. Measures of dispersion. - Measure of central tendency: mode, mean, median-measure of dispersion : range, standard deviation - Can also use
tabulation --> frequency Inferential analysis (Tests of differences) One-sample t-test Variance and standard deviation are some of the measures of dispersion. an indication of how widely the values
are spread in the data set. Types of Measures. In addition, a function, here called summary.list , can be defined to output whichever statistics are of interest. Measures of dispersion—such as range,
variance, standard deviation, and coefficient of variation—can be calculated with standard functions in the native stats package. That in turn helps in evaluating the chances of a new input fitting
into the existing data set and hence probability There are three main measures of central tendency which can be calculated using the methods in pandas python library. Do you know about Python
Collection Module. Since what counts as a “center” is ambiguous, there are several measures of central tendencies. Measures of dispersion. Measures to describe shape of distribution. dtype: float64
Objectives. A value less than -1 is skewed to the left; that greater than 1 is skewed to the right. Lecture 8: Measures of Dispersion Lecture 9: Assumptions for Measures of Central Tendency/ Measures
of Dispersion Descriptive Statistics: Measures of Central Tendency and Cross - tabulation Introduction Measures of central tendency (MCT) allow us to summarize a whole group of … Descriptive
statistics with Python-NumPy. In this lab you'll dive deep into calculating the measures of central tendency and dispersion introduced in previous lessons. You will code the formulas for these
functions in Python which will require you to use the programming skills that you have gained in the other lessons of this section. You will code the formulas for these functions in Python which will
require you to use the programming skills that you have gained in the other lessons of this section. If called on an empty container of data, it raises a StatisticsError. Measures of dispersion are
values that describe how the data varies. You will code the formulas for these functions in Python which will require you to use the programming skills that you have gained in the other lessons of
this section. For reference, Tags: Dispersion in Python StatisticsPython Descriptive StatisticsPython StatisticsPython Statistics Central TendencyPython Statistics DispersionPython Statistics
ModulePython Statistics tutorialWorking of Data Analysis, Your email address will not be published. Feel free to take a look at Course Curriculum. Measures under this include mean, median, and mode.
Measures of central tendency and Dispersion : Avocado Perform the following steps in serial order. Variance and standard deviation are some of the measures of dispersion. This returns the population
variance of data. Python Central tendency characterizes one central value for the entire distribution. 1. Variance/Standard Deviation is one such measure of variability. When you describe and
summarize a single variable, you’re performing univariate analysis. Descriptive statistics consists of quantitative or qualitative data population or sample frequency distribution, central tendency
measures, dispersion measures, association measures and frequency distribution shape.. Add text cell. Copy to Drive. Related Topic- Python NumPy Tutorial The central tedency allows us to grasp the
"middle" of the data, but it doesn't tell us anything about the variability of the data. In this lab you'll dive deep into calculating the measures of central tendency and dispersion introduced in
previous lessons. Central Tendency . It uses two main approaches: 1. For data of odd length, this returns the middle item; for that of even length, it returns the average of the two middle items.
And, dispersion helps in evaluating how near or far the other values are from this average value. The second type of descriptive statistics is the measure of dispersion, also known as a measure of
variability.It is used to describe the variability in a dataset, which can be a sample or population.It is usually used in conjunction with a measure of central tendency, to provide an overall
description of a set of data. Hence, we studied Python Descriptive Statistics, in which we learned Central Tendency & Dispersion used in Python Statistics Module. Mean, Median, Mode are measures of
central tendency.. 1. For three values a, b, and c, the harmonic mean is- The following methods are used to find measures of central tendency in NumPy: mean()- takes a NumPy array as an argument and
returns the arithmetic mean of the data. Mean. As per this answer (Get time of execution of a block of code in Python 2.7), you can use the timeit module: import timeit start_time =
timeit.default_timer() # code you want to evaluate elapsed = timeit.default_timer() - start_time Obviously, this is not as neat as … Follow this to know more about Python Pandas. A measure of central
tendency tells us, using a single value, the best representation for an entire set of scores. 3/(1/a + 1/b +1/c) Along with this, we will cover the variance in Python and how to calculate the
variability for a set of values. There are three main measures of central tendency which can be calculated using the methods in pandas python library. In this tutorial, you’ll learn about the
following types of measures in descriptive statistics: Central tendency tells you about the centers of the data. This function uses interpolation to return the median of grouped continuous data. The
statistics module defines one exception- … We have seen what central tendency or central location is. Text. Mean - It is the Average value of the data which is a division of sum of the values with
the number of values. Your email address will not be published. This is the 50th percentile. Let’s Learn CGI Programming in Python with Functions and Modules. Company. Mean. These statistics fall
into two general categories: the measures of central tendency and the measures of spread. The arithmetic mean is the sum of data divided by the number of data-points. dtype: float64 Python
Descriptive Statistics – Central Tendency. Measures of dispersion are values that describe how the data varies. Mode - It is the most commonly occurring value in a distribution. Can use this when
your data is of an even length uses interpolation to the... Argument and returns the harmonic mean of the data is of an even length of 5.233333333333333 from. Evaluating how near measures of central
tendency and dispersion avocado python code far the other values are arranged in ascending or descending order Python. Ascending or descending order evaluating how near or far the other measures of
central tendency and dispersion avocado python code are widely dispersed, the location. Evaluating how near or far the other values are arranged in ascending or descending order 2 do you know
difference. Cover the variance in Python with functions and Modules has maximum frequency in distribution... Preparation – part 1 “ center ” is ambiguous, there are three main measures of dispersion
2.1.1 measures central. Fig 7: Comparison of different measures of spread set and hence probability of.... In pandas Python library Global variables occurring value in a distribution 7 steps of
divided. It measures or summarizes how spread the data set value which is a subclass of ValueError highest value deviation some! Whichever statistics are of interest grouped continuous data tendency
tells us, using a single number that,... Used the statistics module statistics, fall two sets of properties- central tendency which be. Too-, 0 4.9 dtype: float64 Follow this to know about measures
of Descriptive statistics have., there are several measures of dispersion are values that describe how the data statistics module value. Of Business statistics with Python course describe and
summarize a single number that represents, roughly put “. Defines one exception- exception statistics.StatisticsError this is equal to the square root of the values are arranged in or... Summary
statistics: measures of central tendency which can be directly used to describe the middle/center value of data! Statistical methods- Descriptive and Inferential – measuring central tendency.. 1 &
dispersion used Python... The data is a subclass of ValueError since what counts as a whole at the. Works best with ordinal data argument and returns measures of central tendency and dispersion
avocado python code median of grouped continuous data median works best with ordinal.... Variance and standard deviation are some of the central location is statistics measuring. Of observations onto
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statistics.StatisticsError this is the average of the values are widely dispersed, mode. Container of data, the best representation for an entire population Python Namespace and variable Scope –
Local and variables. Function uses interpolation to return the median of the values as a whole we... | {"url":"http://kapsalonhilde.nl/carlisle-furniture-paae/measures-of-central-tendency-and-dispersion-avocado-python-code-bfc218","timestamp":"2024-11-12T10:34:26Z","content_type":"text/html","content_length":"38210","record_id":"<urn:uuid:e2bf8074-b399-4991-b211-8068bbc938ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00389.warc.gz"} |
Math Colloquia - Circular maximal functions on the Heisenberg group
The spherical average has been a source of many problems in harmonic analysis.
Since late 90's, the study of the maximal spherical means on the Heisenberg group $mathbb{H}^n$ has been started to show the pointwise ergodic theorems on the groups.
Later, it has turned out to be connected with the fold singularities of the Fourier integral operators, which leads to the $L^p$ boundedness of the spherical maximal means on the Heisenberg group
$mathbb{H}^n$ for $nge 2$.
In this talk, we discuss about the $L^p$ boundedness of the circular maximal function on the Heisenberg group $mathbb{H}^1$. The proof is based on the the square sum estimate of the Fourier integral
operators associated with the torus arising from the vector fields of the Heisenberg group algebra.
We compare this torus with the characteristic cone of the Euclidean space. | {"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&sort_index=date&order_type=desc&page=5&l=en&document_srl=793545","timestamp":"2024-11-03T07:38:14Z","content_type":"text/html","content_length":"43585","record_id":"<urn:uuid:ccde3981-2117-4d5d-a123-ded22fcd3377>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00320.warc.gz"} |
On the Fractal Structure of Space-Time
Original Article
, Volume: 5( 4)
On the Fractal Structure of Space-Time
Received Date: October 28, 2017 Accepted Date: November 10, 2017 Published Date: November 16, 2017
Citation: Martinez A. On the Fractal Structure of Space-Time. J Phys Astron. 2017;5(4):127
We show that strings are not one-dimensional objects and that they can be continously stretched to become 2-branes. This in turn suggests that strings have fractal structure. We then proceed to
analyze curvature variations inside black holes, which after making use of their entropy properties and applying them a classical theorem in Riemannian Geometry, leads us to conclude that space-time
is indistinguishable from energy. By considering the string’s structure mentioned above, this implies that space-time itself has fractal structure. Applications of such results include a solution of
the problem regarding the universe’s shape by defining it as finite in size and with zero global curvature, and an explanation of high-temperature superconductivity by means of 2-branes’ stickiness
Strings; D-branes; De sitter space; Black holes; Fractals; Soul theorem
String Theory is a work under progress aiming to unify Quantum Mechanics and General Relativity into a single framework. Its main claims in order to achieve this are that all particles are
one-dimensional vibrating objects called strings and that space-time has more than four dimensions. The theory also contemplates objects called D-branes (where D is their dimension), which are higher
dimensional generalizations of strings. However, the structure of strings and of space-time at the Planck length is still not completely understood. The present paper studies both the detailed
geometry of strings and of space-time by making use of their tension property and of black holes’ information strorage properties respectively.
It is thought that strings are one-dimensional vibrating objects which have tension. Yet tension, which consists on stretching an object, cannot take place in objects without width. Therefore strings
are not one-dimensional and must have a certain degree of width in order to be stretched. As the tension of the string increases, it will inevitably increase in width up to the point where the string
turns into a 2-brane. Therefore strings and 2-branes are equivalent and are fractals as well. This is because the string gains width in a gradual manner, thus assuming dimensions in the interval [1,2
] in the process. Such conclusion is backed up by considering the string’s equation for tension,
which cannot vanish since this would imply that the string has infinite energy. T being equal to zero would mean that the string cannot be stretched and is thus one-dimensional. Yet as one considers
an arbitrarily small value for T, the string cannot pass abruptly from being almost one-dimensional (T=0) to being two-dimensional. It will thus assume rational dimensions as its string scale
We now proceed to explain why space-time and energy are equivalent by analyzing black hole entropy and using a result in Riemannian Geometry. Such result is the Soul Theorem established by Cheeger
and Gromoll. A black hole’s surface area, and the black hole in general, expands in size as matter falls in it. It is also known that space-time contracts as one approaches the singularity. The fact
that black holes contain expanding and contracting regions implies that between them must exist a stationary and thus flat region. That is, one cannot pass abruptly from an expanding region to a
contracting one. We now discuss black holes’ connection to the Soul Theorem. Such result states that if a manifold’s curvature is null in some regions and positive in the rest, then such manifold M
contains a submanifold called a soul, which contains the whole topology of M [1]. In mathematical terminology, the soul is diffeomorphic to M.
Such result applies to black holes since they have positive curvature in their expanding and contracting regions but null curvature between them. We therefore observe that black holes contain souls.
Yet where is such soul located? Knowing that a black hole’s information is stored in its surface area [2], and knowing that souls contain their manifold’s whole topology (and thus information about
the manifold’s structure), we are led to conclude that a black hole’s soul is equivalent to its surface area. Yet why is the soul equal to the whole surface area and not a portion of it instead? A
submanifold is a manifold in its own right and cannot be equal to an arbitrarily local region of its containing manifold. Since black holes’ souls and surface areas are equal, they must contain the
same type of information. This means that the information that black holes store is about their topological properties. This is information about space-time’s structure, but what about information
regarding the particles that fell into the black hole? Since the soul (and thus surface area) contains only information about space-time’s topology, we are led to conclude that space-time is
equivalent to energy and therefore has a fractal structure. A direct consequence of this equivalence is that singularities do not exist. Since strings are the smallest units of energy and cannot have
an arbitrarily small size, then space-time itself cannot reach an infinitesimal size like singularities do.
Mathematical formulation
Let us begin to construct the equation describing such results by squaring the string’s tension,
and rearranging it such that,
We now subtract both sides by 4πg,where g is a surface’s genus, obtaining the Gauss-Bonnet Theorem;
The following step consists on considering the cosmological constant in a De Sitter Space,
for n De Sitter Space’s dimension and α a constant with units of length. Since we reached the conclusion that space-time has a fractal structure, we now treat n as the Hausdorff dimension, which is
given by the power law p = q^n for p and q parameters. The following step is to consider the genus-degree formula:
for d the degree of a plane curve. It must be remarked that it is known that the genus of the curve, called the arithmetic genus, is in agreement with the usual topological genus of a surface. Since
both the Hausdorff dimension and the curve’s degree are exponents, we can rewrite the cosmological constant as:
obtaining us the following,
We now multiply both sides by π and set L=2πα, yielding the equation’s final form,
Results, Discussion and Conclusion
We finish this paper by proving the above assertions and explaining some predictions of the equation. The space-time and energy equivalence is already established by the argument used above, namely
the application of the Soul Theorem to black holes. This means that we are able to set energy and thus of space-time, and α has length unit, one can set them equal. This in turn makes the right side
of our equation nontrivial by assigning a given string scale to both sides. Also, the string and 2-brane equivalence is proven since for a given string tension T, one will always have a surface M
(2-brane) in the left side of the equation.
We now finish by describing a prediction of such equation and explaining how the results obtained here settle some open problems in physics. The prediction is that the Universe is a three-dimensional
torus. The space-time and energy equivalence implies that space-time cannot be infinite. Such equality also implies that the Universe’s curvature is zero. That is, since they are the same entity,
inhomogeneity cannot rise and the density parameter must equal 1, yielding a flat Universe. By plugging in n=3 into the equation, one obtains g=1, which is a torus.
The first problem regards why the Universe had an extremely low entropy in the past, resulting in the Second Law of Thermodynamics’ existence. The paradox lies in that entropy was very low with
respect to gravitational degrees of freedom at the Big Bang. If we imagine the Universe going back in time, it would start getting smaller and denser, up to the point where at the Big Bang it would
have extremely high curvature with many black holes. Yet such conclusion contradicts experience for had the Big Bang been disordered, no Second Law of Thermodynamics would exist. The problem is
therefore how the Universe managed to be extremely small and dense, and yet not have gravity activated. This is answered by the conclusion that the Universe is a three-dimensional torus. That is, as
one goes back in time, the torus would naturally get smaller and denser as it reaches the Big Bang. But since it is a torus, when it finally reaches the Big Bang its curvature as a whole would still
be zero.
Now, the last problem is that of high temperature superconductivity. As the temperature of the material drops below its critical temperature, it loses electric resistance and turns into a
superconductor. The strings in it have extremely low energies and form a condensate. Such low energy means that their string scales are small while their tensions are very high. The strings therefore
turn into 2-branes, and since they are all very close and 2-branes are sticky, they form what are called 2-brane stacks. They will therefore form composite 2-branes with two layers and will remain
together even though the temperature increases, since they are sticky and now behave as if they were one object. In conclusion, to compute the equation above, one begins with the string scale as
input, proceeds to define a space-time dimension, and thus obtains the genus of a 2-brane in n-dimensional space-time. It must be remarked that with the string scale and space-time dimension one can
also compute the vacuum Einstein Equation. To finish, one finds the cosmological constant and solves the Gauss-Bonnet Theorem [3-6]. | {"url":"https://www.tsijournals.com/articles/on-the-fractal-structure-of-spacetime-13556.html","timestamp":"2024-11-10T05:01:06Z","content_type":"text/html","content_length":"84618","record_id":"<urn:uuid:258a71e0-5758-4ffb-a92d-229a903c2af4>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00254.warc.gz"} |
The Diagram Shows A Simple Winch. The Motor Drives A Small Gear And The Large Gear Rotates The Drum And Raises The Load: Dynamic Mechanical Principles In Practice Assignment, UoM, UK | Tutors Dive
Subject Dynamic Mechanical Principles in Practice
Q1. The diagram shows a simple winch. The motor drives a small gear and the large gear rotates the drum and raises the load. The load is 4KN and it raised at 0.4 m/s. The system is 50 % efficient.
Don't use plagiarized sources. Get Your Custom Essay on
The diagram shows a simple winch. The motor drives a small gear and the large gear rotates the drum and raises the load: Dynamic Mechanical Principles in Practice Assignment, UoM, UK
Just from $13/Page
Task (1):
Calculating the following:
1. The torque acting on the drum,
2. the power produced by the load,
3. the speed of the drum.
Task (2):
1. the power produced by the motor and
2. the speed of the motor.
Task (3):
Explain the reason behind efficiency loss in the system and how to improve the system’s efficiency.
Q2. A visitor to a tall building wishes to determine the height of the building. He ties a spool of thread with a small object to make a simple Pendulum, which he hangs down the center of the
building. The period of the oscillation is 11.20 sec.
1. Determine the height of the building.
2. Explain the relation between oscillation time to the height of the building.
3. Determine the value of K if the mass of the object is 500 g. Explore the characteristics of the pendulum and its safety.
Q3. A 0.53 kg object is fixed with one horizontal spring end of the horizontal ideal spring and rests on a frictionless surface. The object is pulled so that the spring stretches by 2.8 cm relative
to its unstrained length. When the block is released, its moves with an acceleration of 9.7 m/s2.
Task (1):
Calculate the potential energy in Kilo Joule(KJ).
Task (2):
Find the oscillation time and amplitude of the motion.
Task (3)
Brief the explanation about restoring force acting on this system. And explore the characteristics which affect the restoring force.
Facing challenges with your Dynamic Mechanical Principles in Practice assignment? is here to offer expert support. Our dedicated team of professionals can assist you in grasping the fundamental
concepts of dynamic mechanical principles and applying them effectively. You can also avail of our experts for | {"url":"https://tutorsdive.com/the-diagram-shows-a-simple-winch-the-motor-drives-a-small-gear-and-the-large-gear-rotates-the-drum-and-raises-the-load-dynamic-mechanical-principles-in-practice-assignment-uom-uk/","timestamp":"2024-11-02T05:24:16Z","content_type":"text/html","content_length":"59934","record_id":"<urn:uuid:bccbde87-377d-4cde-aa77-4f6b16dc564e>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00045.warc.gz"} |
Dispersive limit of the nonlinear Schrodinger equations
Add to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact Jonathan Ben-Artzi.
In this talk, we perform the mathematical derivation of the lake equations (anelastic system) from the classical solution of the nonlinear Schrodinger equations with electro-magnetic potential.
Moreover, if we consider the weak solution of the cubic nonlinear Schrodinger equations, the limit equation is the wave map type equations.
This talk is part of the Partial Differential Equations seminar series.
This talk is included in these lists:
Note that ex-directory lists are not shown. | {"url":"https://talks.cam.ac.uk/talk/index/34624","timestamp":"2024-11-05T16:32:43Z","content_type":"application/xhtml+xml","content_length":"13892","record_id":"<urn:uuid:9647eeac-6ef2-40fe-af83-db5736faad10>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00818.warc.gz"} |
Red November Actions
There are three types of actions: fix-it actions, item actions, and special actions.
Fix-it Actions
Most of the actions taken by gnomes during the game revolve around trying to fix something that has gone horribly wrong. All fix-it actions follow these steps:
1. The player decides how many minutes to spend attempting the repair. He can spend between 1 and 10 minutes.
2. He adds any bonuses from Item Cards used that turn, if they help fix this particular problem.
3. He rolls the action die: if the roll is less than or equal to the sum of time plus bonuses, the gnome succeeds in his task! Otherwise, he fails.
There is no additional penalty for failing a fix-it action (except that it takes time, and the more time passes, the more bad things happen).
The result of a successful fix-it action depends on what the gnome was doing.
Important: All items are discarded after a single use!
There are two types of fix-it actions: basic fix-it actions and room fix-it actions. Basic fix-it actions can be taken in any room on the submarine. Room fix-it actions cover repairs to the sub's
critical systems, and can only be taken in specific rooms.
Basic Fix-it Actions: Fire, Flood, and Blocked Hatches
Fire, flooding, and jammed hatches can happen anywhere, so this type of repair can be undertaken in any room on the sub.
Extinguish Fire: This action is only possible if the gnome is in a room with a Fire token. In fact, if the room is on fire, this is the gnome's only option! If he is successful, remove the Fire
token. If he fails, he must make a special extra move out of the room (spending time as per the normal movement rules). If he cannot exit the room for any reason (such as high water, fire, or blocked
hatches) or he has no minutes left on the Time Track, he is killed.
Pump Water: If a gnome is in a room with a low-water token, he can attempt to remove it. If he succeeds, remove the low-water token. Note: this action is not possible in a room with a high-water
Unblock Hatch: If there is a Hatch Blocked token in the gnome's room, he can try to remove it. If he succeeds, remove one Hatch Blocked token from the room.
Example: Howard's gnome is in room 3 and wants to move to the Engine Room (room 1), but the hatch connecting the two rooms is blocked.
Fortunately, Howard's gnome has a Crowbar, which can grant him a +3 bonus to an Unblock Hatch fix-it action. Unfortunately, room 3 has a low-water Flood token, which means his fix-it action will take
an additional two minutes (a time penalty that won't help him succeed at his action).
Howard decides he will spend four minutes performing an Unblock Hatch fix-it action and he will use his Crowbar. If he can roll a "7" or less on the action die, he will succeed at unblocking the
hatch (4 minutes + 3 for the Crowbar).
Regardless of whether he succeeds or fails, Howard must move the Ghost Time Keeper forward six spaces (4 minutes spent on the fix-it action + 2 for the low-water penalty).
Using Items
Each player starts the game with two Item Cards, and may acquire more as the game goes on. Item Cards are kept faceup on the table in front of the player (unless players are using the "Crazed Gnome''
Players may only use Item Cards on their own turns. Item Cards can be used at any time during the turn, as long as it is played before any die roll that it affects. The benefits of an item last for
the player's entire turn.
Items with multiple effects provide all of them when used. For example, a Grog card allows a gnome to enter burning rooms and gives a +3 bonus to fix-it actions.
Any number of items may be played during a turn (even multiple copies of the same item!). The effects of all items played are cumulative. For example, if a gnome uses a Toolbox and an Engine Manual
on the same turn, he receives a +7 bonus to his Fix Engine action.
There is no limit to the number of Item Cards a player may hold in hand, though some events may force a player to discard Item Cards if they have too many.
All items are discarded after a single-use. Discarded Item cards are placed in a discard pile next to the Item Deck.
If the Item Deck runs out, all discarded items (including discarded Grog cards) are shuffled together to make a new draw pile.
Room Fix-it Actions: the Serious Problems
The various sub-threatening conditions can only be corrected with a room fix-it action. These problems are often associated with the three Disaster Tracks: Asphyxiation, Heat, and Pressure.
During the game, the Disaster Track Markers gradually move along their tracks. If one of the markers reaches the end of its track, the sub is destroyed and the players lose the game!
The only way to prevent such a disaster is by successfully completing a room fix-it action in the appropriate room before the marker reaches the end of the Disaster Track.
A gnome can only attempt a room fix-it action if he is in the specific room that holds the critical system he is trying to repair. The critical system rooms are marked with a symbol that matches the
symbol on the Disaster Track and Destruction token that can be repaired in that room.
A successful room fix-it action on a critical system has two effects:
First, the matching Disaster Track, if any, is reset to the next lower reset point (marked with a star). If the Disaster Track Marker is on the sixth or higher space, it moves to the fifth space. If
it is on the fifth space or lower, it moves to the first space.
Example: The Disaster Track Marker for the Heat Track (the red one) is on the eighth space - only two spaces away from disaster! Bethany has rushed to the Reactor to conduct repairs and she succeeds.
The red Disaster Track Marker is moved back along the track to the fifth space, marked with the reset star.
Second, any matching Destruction token is removed from the Time Track.
Note: Any attempted repair automatically fails if the Ghost Time Keeper passes the matching Destruction token on the Time Track!
If a Disaster Track or Destruction token is not repaired in time, the sub is destroyed!
There are five possible room fix-it actions:
• Fix Engines: A gnome in the Engine Room (room 1) can fix the engines. A success here resets the Pressure Track (blue) and removes the "Crushed!'' token.
• Fix Oxygen Pumps: A gnome in the Oxygen Pumps (room 2) can fix the oxygen pumps. A success here resets the Asphyxia- tion Track (green) and removes the "Asphyxiated!" token.
• Fix Reactor: A gnome in the Reactor (room 4) can fix the reactor. A success here resets the Heat Track (red).
• Stop Missile Launch: A gnome in Missile Control (room 7) can prevent a missile launch. A success here removes the "Missiles Launched!" token.
• Kill Kraken: A gnome in the sea space outside the sub can attempt to kill the Kraken. A success here removes the "De- voured by Kraken" token.
Item Actions
There are two types of item actions:
Draw Item Cards: If the gnome is in the Equipment Stores (room 8) or Captain's Cabin (room 10), he can Draw Item Cards. It costs one minute for each Item Card drawn, in either location.
A gnome in the Captain's Cabin draws from the captain's private stash of Grog (as long as it lasts - once it is gone, no more cards can be drawn from the Captain's Cabin). A gnome may spend a maximum
of two minutes on a Draw Item Cards action here.
A gnome in the Equipment Stores draws random cards from the Item Pile. He may spend up to four minutes on a Draw Item Cards action here.
After drawing his cards, the player moves his gnome into the red "Drew Item Cards" area of his room. The gnome stays in the red area until he leaves the room.
A gnome in the red area cannot take a Draw Item Cards action. He must perform an action in another room before he can return to collect more gear.
Example: Edmund spends two minutes collecting Grog cards from the Captain's Cabin (room 10).
Then he moves his Gnome figure into the red "Drew Item Cards" area of the Captain's Cabin, at the very nose of the Red November. He remains in this area until he moves into a different room.
Trade Item Cards: A gnome in the same room as another gnome may Trade Item Cards. The active player may give the other player any number of Item Cards from his hand, and the other player may also
give the active player any number of Item Cards back.
This action always costs one minute (for the active player only).
Special Actions
There are two other special actions:
No Action: A gnome can choose to do nothing and take no action at all. This costs one minute. A player may take this option to allow another player to act before him, for example. This is the only
option for a gnome in a room at high water.
Abandon Comrades: If the player's Time Keeper has passed the "10" space on the Time Track, he can choose to give up on saving the sub and swim away.
The gnome must be outside the sub (using an Aqualung, of course) to take this action.
The gnome swims to safety, leaving his comrades to their fate! Remove the gnome figure and his Time Keeper from the board. He can- not take any more actions for the rest of the game.
Do not resolve any events this turn. When a player abandons his comrades, his victory conditions are reversed: if the submarine is destroyed he wins, but if the other gnomes are rescued, he loses!
Kicking the Bucket
The Red November is a dangerous place. Deadly, even. As a result, it is possible for one - or many - of its brave sailors to pass on before help arrives or the submarine sinks.
During any player's Updates Phase, if any fainted gnome occupies a room with either a high-water Flood token or Fire token, that gnome is immediately killed (even if it is not his turn).
The active gnome is exposed to a few additional risks:
• If he is outside the sub and faints, he is killed.
• If he is in a room with a high-water Flood token or Fire token at the beginning of his Updates Phase (i.e., he was unable to move out of the room during his turn), he is killed.
• If he started the turn outside the sub, and is still outside the sub at the beginning of his Updates Phase, his Aqualung runs out of air and he is killed (exception: a gnome who takes the Abandon
Comrades action is not killed).
When a gnome is killed, remove his figure and Time Keeper from the board. That player is eliminated from the game and takes no further actions. Discard any Item Cards that gnome was carrying.
If the active gnome was killed, also remove the Ghost Time Keeper. Do not resolve any events this turn.
If the optional rule "Less Deadly Dying'' is in effect, follow those rules instead.
Continue Reading | {"url":"https://www.ultraboardgames.com/red-november/actions.php","timestamp":"2024-11-02T03:01:33Z","content_type":"text/html","content_length":"40243","record_id":"<urn:uuid:ee5d19eb-1ffe-4711-9b09-98cf127b38fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00370.warc.gz"} |
The Reason Why Fail to Converge
15.4.5 The Reason Why Fail to Converge
The nonlinear fitting process is iterative. The process completes when the difference between reduced chi-square values of two successive iterations is less than a certain tolerance value. When the
process completes, we say that the fit has converged. To illustrate, we'll use a model with one parameter:
The curve denotes the change of reduced $\chi^2\,\!$ during a fitting procedure. Starting from an initial value $\theta _0\,\!$, the parameter value is adjusted so that the $\chi^2\,\!$ value is
reduced. The fit has converged when ∆ $\chi^2\,\!$ ≤ Tolerance.
The purpose of the nonlinear curve fitting procedure is to find the absolute minimum value of reduced $\chi^2\,\!$. However, sometimes the minimization procedure does not reach this absolute minimum
value (the fit does not converge). Failure to converge can frequently be traced to one of the following:
Poor initial parameter values
Initial values are very important to a fitting procedure. They can be empirical values from previous work or estimated values derived by transforming the formula into an approximate form. With good
initial values, it will take less time to perform the fitting.
Relative vs absolute minima
It is not uncommon for the iterative procedure to find a relative -- as opposed to an absolute -- minimum. In this example, the procedure actually converges in the sense that a further decrease of
reduced $\chi^2\,\!$ seems impossible.
The problem is that you do not even know if the routine has reached an absolute or a relative minimum. The only way to be certain that the iterative procedure is reaching an absolute minimum is to
start fitting from several different initial parameter values and observe the results. If you repeatedly get the same final result, it is unlikely that a local minimum has been found.
Non-unique parameter values
The most common problem that arises in nonlinear fitting is that no matter how you choose the initial parameter values, the fit does not seem to converge. Some or all parameter values continue to
change with successive iterations and they eventually diverge, producing arithmetic overflow or underflow. This should indicate that you need to do something about the fitting function and/or data
you are using. There is simply no single set of parameter values which best fit your data.
Over-parameterized functions
If the function parameters have the same differential with respect to independent variables, it may suggest that the function is overparameterized. In such cases, the fitting procedure will not
converge. For example in the following model
A is the amplitude and x0 is the horizontal offset. However, you can rewrite the function as
$y=Ae^{x-x_0}=A\frac 1{e^{x_0}}e^x=Be^x$
In other words, if, during the fitting procedure, the values of A and $x_0\,\!$ change so that the combination B remains the same, the reduced $\chi^2\,\!$ value will not change. Any attempts to
further improve the fit are not likely to be productive.
If you see one of the following, it indicates that something is wrong:
• The parameter error is very large relative to the parameter value. For example, if the width of the Gaussian curve is 0.5 while the error is 10, the result for the width will be meaningless as
the fit has not converged.
• The parameter dependence (for one or more parameters) is very close to one. You should probably remove or fix the value of parameters whose dependency is close to one, since the fit does not seem
to depend upon the parameter.
Note, however, that over-parameterization does not necessarily mean that the parameters in the model have no physical meanings. It may suggest that there are infinite solutions and you should apply
constraints to the fit process.
Bad data
Even when the function is not theoretically overparameterized, the iterative procedure may behave as if it were, due to the fact that the data do not contain enough information for some or all of the
parameters to be determined. This usually happens when the data are available only in a limited interval of the independent variable(s). For example, if you are fitting a non-monotonic function such
as the Gaussian to monotonic data, the nonlinear fitter will experience difficulties in determining the peak center and peak width, since the data can describe only one flank of the Gaussian peak. | {"url":"https://cloud.originlab.com/doc/en/Origin-Help/The_Reason_Why_Fail_to_Converge","timestamp":"2024-11-10T09:07:35Z","content_type":"text/html","content_length":"133601","record_id":"<urn:uuid:310b7ae5-0fc9-44c5-83f0-c737a415db0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00198.warc.gz"} |
Find a unit vector in the direction of the vector a where a=i^+... | Filo
Find a unit vector in the direction of the vector where
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Magnitude of is The unit vector in the direction of is You can verify that it is a unit vector by calculating its magnitude.
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Question Text Find a unit vector in the direction of the vector where
Updated On May 4, 2023
Topic Motion in a Plane
Subject Physics
Class Class 11
Answer Type Text solution:1 Video solution: 4
Upvotes 555
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Rent Splitting Calculator
Living with roommates is a common way to share expenses, especially when it comes to rent. However, splitting the rent fairly can sometimes be a hassle, especially when utilities and other shared
costs are involved. That's where a rent splitting calculator comes in handy. In this blog post, we'll explore how to use our simple yet effective "Rent Splitting Calculator" to ensure everyone pays
their fair share.
Understanding the Calculator
Our "Rent Splitting Calculator" is a web-based tool built using HTML, CSS, JavaScript, and the Bootstrap framework. It features a clean and user-friendly interface that makes it easy to input the
necessary information and get the results quickly.
How to Use the Calculator
Using the "Rent Splitting Calculator" is straightforward. Here's a step-by-step guide:
Step 1: Enter the Total Rent Amount
The first step is to enter the total rent amount for the shared living space. Simply type the amount into the "Total Rent" input field.
Step 2: Specify the Number of People
Next, enter the number of people who will be splitting the rent. This should include yourself and all your roommates. Type the number into the "Number of People" input field.
Step 3: Click the "Calculate" Button
Once you've entered the total rent amount and the number of people, click the "Calculate" button. The calculator will then perform the necessary calculations to determine the rent share for each
Step 4: Review the Results
After clicking the "Calculate" button, the calculator will display the rent share for each person in the designated result area. This amount represents how much each individual should contribute
towards the total rent.
Validation and Error Handling
To ensure accurate calculations, the "Rent Splitting Calculator" includes validation checks. If either the "Total Rent" or "Number of People" input field is left blank, the calculator will not
perform any calculations, and appropriate validation messages will be displayed to prompt you to fill in the missing information.
The "Rent Splitting Calculator" is a simple yet powerful tool that can help eliminate the hassle of manually calculating rent shares among roommates. By following the easy-to-use steps outlined in
this blog post, you can quickly determine the fair contribution for each person, ensuring transparency and preventing any potential conflicts over rent payments. Give it a try and experience the
convenience of a hassle-free rent splitting process! | {"url":"https://toolshiv.com/rent-splitting-calculator","timestamp":"2024-11-06T12:25:04Z","content_type":"text/html","content_length":"18710","record_id":"<urn:uuid:114634b0-7e80-404c-804f-fbe85fbffbe2>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00097.warc.gz"} |
In collaboration with Fong Chan @tinyheero, the latest release (v 0.2.0) of plotGMM includes substantial updates with easy-to-use tools for visualizing output from Gaussian mixture models:
1. plot_GMM: The main function of the package, plot_GMM allows the user to simply input the name of a mixEM class object (from fitting a Gaussian mixture model (GMM) using the mixtools package), as
well as the number of components, k, that were used in the original GMM fit. The result is a clean ggplot2 class object showing the density of the data with overlaid mixture weight component
2. plot_cut_point: Gaussian mixture models (GMMs) are not only used for uncovering clusters in data, but are also often used to derive cut points, or lines of separation between clusters in feature
space (see the Benaglia et al. 2009 reference in the package documentation for more). The plot_cut_point function plots data densities with the overlaid cut point (the mean of the calculated mu)
from mixEM class objects, which are GMM’s fit using the mixtools package.
3. plot_mix_comps: This is a custom function for users interested in manually overlaying the components from a Gaussian mixture model. This allows for clean, precise plotting constraints, including
mean (mu), variance (sigma), and mixture weight (lambda) of the components. The function superimposes the shape of the components over a ggplot2 class object. Importantly, while the
plot_mix_comps function is used in the main plot_GMM function in our plotGMM package, users can use the plot_mix_comps function to build their own custom plots.
Plotting GMMs using plot_GMM
```{r } mixmdl <- mixtools::normalmixEM(faithful$waiting, k = 2)
plot_GMM(mixmdl, 2)
### Plotting Cut Points from GMMs using `plot_cut_point` (with amerika color palette)
```{r }
mixmdl <- mixtools::normalmixEM(faithful$waiting, k = 2)
plot_cut_point(mixmdl, plot = TRUE, color = "amerika") # produces plot
plot_cut_point(mixmdl, plot = FALSE) # produces only cut point value
Manually using the plot_mix_comps function in a custom ggplot2 plot
```{r } library(plotGMM) library(magrittr) library(ggplot2) library(mixtools)
Fit a GMM using EM
set.seed(576) mixmdl <- normalmixEM(faithful$waiting, k = 2)
Plot mixture components using the plot_mix_comps function
data.frame(x = mixmdl\(x) %>% ggplot() + geom_histogram(aes(x, ..density..), binwidth = 1, colour = "black", fill = "white") + stat_function(geom = "line", fun = plot_mix_comps, args = list(mixmdl\)
mu[1], mixmdl\(sigma[1], lam = mixmdl\)lambda[1]), colour = “red”, lwd = 1.5) + stat_function(geom = “line”, fun = plot_mix_comps, args = list(mixmdl\(mu[2], mixmdl\)sigma[2], lam = mixmdl$lambda
[2]), colour = “blue”, lwd = 1.5) + ylab(“Density”) ``` | {"url":"https://cran.mirror.garr.it/CRAN/web/packages/plotGMM/readme/README.html","timestamp":"2024-11-04T07:26:40Z","content_type":"application/xhtml+xml","content_length":"5759","record_id":"<urn:uuid:e6fdd928-0254-4cdf-b264-7aaa7ff613ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00544.warc.gz"} |
SQL Windowing
This chapter from Microsoft SQL Server 2012 High-Performance T-SQL Using Window Functions provides the background of window functions, a glimpse of solutions using them, coverage of the elements
involved in window specifications, an account of the query elements supporting window functions, and a description of the standard’s solution for reusing window definitions.
Window functions are functions applied to sets of rows defined by a clause called OVER. They are used mainly for analytical purposes allowing you to calculate running totals, calculate moving
averages, identify gaps and islands in your data, and perform many other computations. These functions are based on an amazingly profound concept in standard SQL (which is both an ISO and ANSI
standard)—the concept of windowing. The idea behind this concept is to allow you to apply various calculations to a set, or window, of rows and return a single value. Window functions can help to
solve a wide variety of querying tasks by helping you express set calculations more easily, intuitively, and efficiently than ever before.
There are two major milestones in Microsoft SQL Server support for the standard window functions: SQL Server 2005 introduced partial support for the standard functionality, and SQL Server 2012 added
more. There’s still some standard functionality missing, but with the enhancements added in SQL Server 2012, the support is quite extensive. In this book, I cover both the functionality SQL Server
implements as well as standard functionality that is still missing. Whenever I describe a feature for the first time in the book, I also mention whether it is supported in SQL Server, and if it is,
in which version of the product it was added.
From the time SQL Server 2005 first introduced support for window functions, I found myself using those functions more and more to improve my solutions. I keep replacing older solutions that rely on
more classic, traditional language constructs with the newer window functions. And the results I’m getting are usually simpler and more efficient. This happens to such an extent that the majority of
my querying solutions nowadays make use of window functions. Also, standard SQL and relational database management systems (RDBMSs) in general are moving toward analytical solutions, and window
functions are an important part of this trend. Therefore, I feel that window functions are the future in terms of SQL querying solutions, and that the time you take to learn them is time well spent.
This book provides extensive coverage of window functions, their optimization, and querying solutions implementing them. This chapter starts by explaining the concept. It provides the background of
window functions, a glimpse of solutions using them, coverage of the elements involved in window specifications, an account of the query elements supporting window functions, and a description of the
standard’s solution for reusing window definitions.
Background of Window Functions
Before you learn the specifics of window functions, it can be helpful to understand the context and background of those functions. This section provides such background. It explains the difference
between set-based and cursor/iterative approaches to addressing querying tasks and how window functions bridge the gap between the two. Finally, this section explains the drawbacks of alternatives to
window functions and why window functions are often a better choice than the alternatives. Note that although window functions can solve many problems very efficiently, there are cases where there
are better alternatives. Chapter 4, “goes into details about optimizing window functions”, explaining when you get optimal treatment of the computations and when treatment is nonoptimal.
Window Functions Described
A window function is a function applied to a set of rows. A window is the term standard SQL uses to describe the context for the function to operate in. SQL uses a clause called OVER in which you
provide the window specification. Consider the following query as an example:
See Also See the book’s Introduction for information about the sample database TSQL2012 and companion content.
USE TSQL2012;
SELECT orderid, orderdate, val,
RANK() OVER(ORDER BY val DESC) AS rnk
FROM Sales.OrderValues
ORDER BY rnk;
Here’s abbreviated output for this query:
orderid orderdate val rnk
-------- ----------------------- --------- ---
10865 2008-02-02 00:00:00.000 16387.50 1
10981 2008-03-27 00:00:00.000 15810.00 2
11030 2008-04-17 00:00:00.000 12615.05 3
10889 2008-02-16 00:00:00.000 11380.00 4
10417 2007-01-16 00:00:00.000 11188.40 5
10817 2008-01-06 00:00:00.000 10952.85 6
10897 2008-02-19 00:00:00.000 10835.24 7
10479 2007-03-19 00:00:00.000 10495.60 8
10540 2007-05-19 00:00:00.000 10191.70 9
10691 2007-10-03 00:00:00.000 10164.80 10
The OVER clause is where you provide the window specification that defines the exact set of rows that the current row relates to, the ordering specification, if relevant, and other elements. Absent
any elements that restrict the set of rows in the window—as is the case in this example—the set of rows in the window is the final result set of the query.
For ranking purposes, ordering is naturally required. In this example, it is based on the column val ranked in descending order.
The function used in this example is RANK. This function calculates the rank of the current row with respect to a specific set of rows and a sort order. When using descending order in the ordering
specification—as in this case—the rank of a given row is computed as one more than the number of rows in the relevant set that have a greater ordering value than the current row. So pick a row in the
output of the sample query—say, the one that got rank 5. This rank was computed as 5 because based on the indicated ordering (by val descending), there are 4 rows in the final result set of the query
that have a greater value in the val attribute than the current value (11188.40), and the rank is that number plus 1.
What’s most important to note is that conceptually the OVER clause defines a window for the function with respect to the current row. And this is true for all rows in the result set of the query. In
other words, with respect to each row, the OVER clause defines a window independent of the other rows. This idea is really profound and takes some getting used to. Once you get this, you get closer
to a true understanding of the windowing concept, its magnitude, and its depth. If this doesn’t mean much to you yet, don’t worry about it for now—I wanted to throw it out there to plant the seed.
The first time standard SQL introduced support for window functions was in an extension document to SQL:1999 that covered, what they called “OLAP functions” back then. Since then, the revisions to
the standard continued to enhance support for window functions. So far the revisions have been SQL:2003, SQL:2008, and SQL:2011. The latest SQL standard has very rich and extensive coverage of window
functions, showing the standard committee’s belief in the concept, and the trend seems to be to keep enhancing the standard’s support with more window functions and more functionality.
Standard SQL supports several types of window functions: aggregate, ranking, distribution, and offset. But remember that windowing is a concept; therefore, we might see new types emerging in future
revisions of the standard.
Aggregate window functions are the all-familiar aggregate functions you already know—like SUM, COUNT, MIN, MAX, and others—though traditionally, you’re probably used to using them in the context of
grouped queries. An aggregate function needs to operate on a set, be it a set defined by a grouped query or a window specification. SQL Server 2005 introduced partial support for window aggregate
functions, and SQL Server 2012 added more functionality.
Ranking functions are RANK, DENSE_RANK, ROW_NUMBER, and NTILE. The standard actually puts the first two and the last two in different categories, and I’ll explain why later. I prefer to put all four
functions in the same category for simplicity, just like the official SQL Server documentation does. SQL Server 2005 introduced these four ranking functions, with already complete functionality.
Distribution functions are PERCENT_RANK, CUME_DIST, PERCENTILE_CONT, and PERCENTILE_DISC. SQL Server 2012 introduces support for these four functions.
Offset functions are LAG, LEAD, FIRST_VALUE, LAST_VALUE, and NTH_VALUE. SQL Server 2012 introduces support for the first four. There’s no support for the NTH_VALUE function yet in SQL Server as of
SQL Server 2012.
Chapter 2, “A Detailed Look at Window Functions,” provides the meaning, the purpose, and details about the different functions.
With every new idea, device, and tool—even if the tool is better and simpler to use and implement than what you’re used to—typically, there’s a barrier. New stuff often seems hard. So if window
functions are new to you and you’re looking for motivation to justify making the investment in learning about them and making the leap to using them, here are a few things I can mention from my
• Window functions help address a wide variety of querying tasks. I can’t emphasize this enough. As mentioned, nowadays I use window functions in most of my query solutions. After you’ve had a
chance to learn about the concept and the optimization of the functions, the last chapter in the book (Chapter 5) shows some practical applications of window functions. But just to give you a
sense of how they are used, querying tasks that can be solved with window functions include:
□ Paging
□ De-duplicating data
□ Returning top n rows per group
□ Computing running totals
□ Performing operations on intervals such as packing intervals, and calculating the maximum number of concurrent sessions
□ Identifying gaps and islands
□ Computing percentiles
□ Computing the mode of the distribution
□ Sorting hierarchies
□ Pivoting
□ Computing recency
• I’ve been writing SQL queries for close to two decades and have been using window functions extensively for several years now. I can say that even though it took a bit of getting used to the
concept of windowing, today I find window functions both simpler and more intuitive in many cases than alternative methods.
• Window functions lend themselves to good optimization. You’ll see exactly why this is so in later chapters.
What’s important to understand from all this is that you need to make a conscious effort to make the switch to using SQL windowing because it’s a new idea, and as such it takes some getting used to.
But once the switch is made, SQL windowing is simple and intuitive to use; think of any gadget you can’t live without today and how it seemed like a difficult thing to learn at first.
Set-Based vs. Iterative/Cursor Programming
People often characterize T-SQL solutions to querying tasks as either set-based or iterative/cursor-based solutions. The general consensus among T-SQL developers is to try and stick to the former
approach, but still, there’s wide use of the latter. There are several interesting questions here. Why is the set-based approach the recommended one? And if it is the recommended one, why do so many
developers use the iterative approach? What are the obstacles that prevent people from adopting the recommended approach?
To get to the bottom of this, one first needs to understand the foundations of T-SQL, and what the set-based approach truly is. When you do, you realize that the set-based approach is nonintuitive
for most people, whereas the iterative approach is. It’s just the way our brains are programmed, and I will try to clarify this shortly. The gap between iterative and set-based thinking is quite big.
The gap can be closed, though it certainly isn’t easy to do so. And this is where window functions can play an important role; I find them to be a great tool that can help bridge the gap between the
two approaches and allow a more gradual transition to set-based thinking.
So first, I’ll explain what the set-based approach to addressing T-SQL querying tasks is. T-SQL is a dialect of standard SQL (both ISO and ANSI standards). SQL is based (or attempts to be based) on
the relational model, which is a mathematical model for data management formulated and proposed initially by E. F. Codd in the late 1960s. The relational model is based on two mathematical
foundations: set-theory and predicate logic. Many aspects of computing were developed based on intuition, and they keep changing very rapidly—to a degree that sometimes makes you feel that you’re
chasing your tail. The relational model is an island in this world of computing because it is based on much stronger foundations—mathematics. Some think of mathematics as the ultimate truth. Being
based on such strong mathematical foundations, the relational model is very sound and stable. It keeps evolving, but not as fast as many other aspects of computing. For several decades now, the
relational model has held strong, and it’s still the basis for the leading database platforms—what we call relational database management systems (RDBMSs).
SQL is an attempt to create a language based on the relational model. SQL is not perfect and actually deviates from the relational model in a number of ways, but at the same time it provides enough
tools that, if you understand the relational model, you can use SQL relationally. It is doubtless the leading, de facto language used by today’s RDBMSs.
However, as mentioned, thinking in a relational way is not intuitive for many. Part of what makes it hard for people to think in relational terms is the key differences between the iterative and
set-based approaches. It is especially difficult for people who have a procedural programming background, where interaction with data in files is handled in an iterative way, as the following
pseudocode demonstrates:
open file
fetch first record
while not end of file
process record
fetch next record
Data in files (or, more precisely, in indexed sequential access method, or ISAM, files) is stored in a specific order. And you are guaranteed to fetch the records from the file in that order. Also,
you fetch the records one at a time. So your mind is programmed to think of data in such terms: ordered, and manipulated one record at a time. This is similar to cursor manipulation in T-SQL; hence,
for developers with a procedural programming background, using cursors or any other form of iterative processing feels like an extension to what they already know.
A relational, set-based approach to data manipulation is quite different. To try and get a sense of this, let’s start with the definition of a set by the creator of set theory—Georg Cantor:
• By a “set” we mean any collection M into a whole of definite, distinct objects m (which are called the “elements” of M) of our perception or of our thought.
• —Joseph W. Dauben, Georg Cantor (Princeton University Press, 1990)
There’s so much in this definition of a set that I could spend pages and pages just trying to interpret the meaning of this sentence. But for the purposes of our discussion, I’ll focus on two key
aspects—one that appears explicitly in this definition and one that is implied:
• Whole Observe the use of the term whole. A set should be perceived and manipulated as a whole. Your attention should focus on the set as a whole, and not on the individual elements of the set.
With iterative processing, this idea is violated because records of a file or a cursor are manipulated one at a time. A table in SQL represents (albeit not completely successfully) a relation
from the relational model, and a relation is a set of elements that are alike (that is, have the same attributes). When you interact with tables using set-based queries, you interact with tables
as whole, as opposed to interacting with the individual rows (the tuples of the relations)—both in terms of how you phrase your declarative SQL requests and in terms of your mindset and
attention. This type of thinking is what’s very hard for many to truly adopt.
• Order Observe that nowhere in the definition of a set is there any mention of the order of the elements. That’s for a good reason—there is no order to the elements of a set. That’s another thing
that many have a hard time getting used to. Files and cursors do have a specific order to their records, and when you fetch the records one at a time, you can rely on this order. A table has no
order to its rows because a table is a set. People who don’t realize this often confuse the logical layer of the data model and the language with the physical layer of the implementation. They
assume that if there’s a certain index on the table, you get an implied guarantee that, when querying the table, the data will always be accessed in index order. And sometimes even the
correctness of the solution will rely on this assumption. Of course, SQL Server doesn’t provide any such guarantees. For example, the only way to guarantee that the rows in a result will be
presented in a certain order is to add a presentation ORDER BY clause to the query. And if you do add one, you need to realize that what you get back is not relational because the result has a
guaranteed order.
If you need to write SQL queries and you want to understand the language you’re dealing with, you need to think in set-based terms. And this is where window functions can help bridge the gap between
iterative thinking (one row at a time, in a certain order) and set-based thinking (seeing the set as a whole, with no order). What can help you transition from one type of thinking to the other is
the ingenious design of window functions.
For one, window functions support an ORDER BY clause when relevant, where you specify the order. But note that just because the function has an order specified doesn’t mean it violates any relational
concepts. The input to the query is relational with no ordering expectations, and the output of the query is relational with no ordering guarantees. It’s just that there’s ordering as part of the
specification of the calculation, producing a result attribute in the resulting relation. There’s no assurance that the result rows will be returned in the same order used by the window function; in
fact, different window functions in the same query can specify different ordering. This kind of ordering has nothing to do—at least conceptually—with the query’s presentation ordering. Figure 1-1
tries to illustrate the idea that both the input to a query with a window function and the output are relational, even though the window function has ordering as part of its specification. By using
ovals in the illustration, and having the positions of the rows look different in the input and the output, I’m trying to express the fact that the order of the rows does not matter.
Figure 1-1. Input and output of a query with a window function.
There’s another aspect of window functions that helps you gradually transition from thinking in iterative, ordered terms to thinking in set-based terms. When teaching a new topic, teachers sometimes
have to “lie” when explaining it. Suppose that you, as a teacher, know the student’s mind is not ready to comprehend a certain idea if you explain it in full depth. You can sometimes get better
results if you initially explain the idea in simpler, albeit not completely correct, terms to allow the student’s mind to start processing the idea. Later, when the student’s mind is ready for the
“truth,” you can provide the deeper, more correct meaning.
Such is the case with understanding how window functions are conceptually calculated. There’s a basic way to explain the idea, although it’s not really conceptually correct, but it’s one that leads
to the correct result! The basic way uses a row-at-a-time, ordered approach. And then there’s the deep, conceptually correct way to explain the idea, but one’s mind needs to be in a state of maturity
to comprehend it. The deep way uses a set-based approach.
To demonstrate what I mean, consider the following query:
SELECT orderid, orderdate, val,
RANK() OVER(ORDER BY val DESC) AS rnk
FROM Sales.OrderValues;
Here’s an abbreviated output of this query (note there’s no guarantee of presentation ordering here):
orderid orderdate val rnk
-------- ----------------------- --------- ---
10865 2008-02-02 00:00:00.000 16387.50 1
10981 2008-03-27 00:00:00.000 15810.00 2
11030 2008-04-17 00:00:00.000 12615.05 3
10889 2008-02-16 00:00:00.000 11380.00 4
10417 2007-01-16 00:00:00.000 11188.40 5
The basic way to think of how the rank values are calculated conceptually is the following example (expressed as pseudocode):
arrange the rows sorted by val
iterate through the rows
for each row
if the current row is the first row in the partition emit 1
else if val is equal to previous val emit previous rank
else emit count of rows so far
Figure 1-2 is a graphical depiction of this type of thinking.
Figure 1-2. Basic understanding of the calculation of rank values.
Again, although this type of thinking leads to the correct result, it’s not entirely correct. In fact, making my point is even more difficult because the process just described is actually very
similar to how SQL Server physically handles the rank calculation. But my focus at this point is not the physical implementation, but rather the conceptual layer—the language and the logical model.
What I meant by “incorrect type of thinking” is that conceptually, from a language perspective, the calculation is thought of differently, in a set-based manner—not iterative. Remember that the
language is not concerned with the physical implementation in the database engine. The physical layer’s responsibility is to figure out how to handle the logical request and both produce a correct
result and produce it as fast as possible.
So let me attempt to explain what I mean by the deeper, more correct understanding of how the language thinks of window functions. The function logically defines—for each row in the result set of the
query—a separate, independent window. Absent any restrictions in the window specification, each window consists of the set of all rows from the result set of the query as the starting point. But you
can add elements to the window specification (for example, partitioning, framing, and so on, which I’ll say more about later) that will further restrict the set of rows in each window. Figure 1-3 is
a graphical depiction of this idea as it applies to our query with the RANK function.
Figure 1-3. Deep understanding of the calculation of rank values.
With respect to each window function and row in the result set of the query, the OVER clause conceptually creates a separate window. In our query, we have not restricted the window specification in
any way; we just defined the ordering specification for the calculation. So in our case, all windows are made of all rows in the result set. And they all coexist at the same time. And in each, the
rank is calculated as one more than the number of rows that have a greater value in the val attribute than the current value.
As you might realize, it’s more intuitive for many to think in the basic terms of the data being in an order and a process iterating through the rows one at a time. And that’s okay when you’re
starting out with window functions because you get to write your queries—or at least the simple ones—correctly. As time goes by, you can gradually transition to the deeper understanding of the window
functions’ conceptual design and start thinking in a set-based manner.
Drawbacks of Alternatives to Window Functions
Window functions have several advantages compared to alternative, more traditional, ways to achieve the same calculations—for example, grouped queries, subqueries, and others. Here I’ll provide a
couple of straightforward examples. There are several other important differences beyond the advantages I’ll show here, but it’s premature to discuss those now.
I’ll start with traditional grouped queries. Those do give you insight into new information in the form of aggregates, but you also lose something—the detail.
Once you group data, you’re forced to apply all calculations in the context of the group. But what if you need to apply calculations that involve both detail and aggregates? For example, suppose that
you need to query the Sales.OrderValues view and calculate for each order the percentage of the current order value of the customer total, as well as the difference from the customer average. The
current order value is a detail element, and the customer total and average are aggregates. If you group the data by customer, you don’t have access to the individual order values. One way to handle
this need with traditional grouped queries is to have a query that groups the data by customer, define a table expression based on this query, and then join the table expression with the base table
to match the detail with the aggregates. Here’s a query that implements this approach:
WITH Aggregates AS
SELECT custid, SUM(val) AS sumval, AVG(val) AS avgval
FROM Sales.OrderValues
GROUP BY custid
SELECT O.orderid, O.custid, O.val,
CAST(100. * O.val / A.sumval AS NUMERIC(5, 2)) AS pctcust,
O.val - A.avgval AS diffcust
FROM Sales.OrderValues AS O
JOIN Aggregates AS A
ON O.custid = A.custid;
Here’s the abbreviated output generated by this query:
orderid custid val pctcust diffcust
-------- ------- ------- -------- ------------
10835 1 845.80 19.79 133.633334
10643 1 814.50 19.06 102.333334
10952 1 471.20 11.03 -240.966666
10692 1 878.00 20.55 165.833334
11011 1 933.50 21.85 221.333334
10702 1 330.00 7.72 -382.166666
10625 2 479.75 34.20 129.012500
10759 2 320.00 22.81 -30.737500
10926 2 514.40 36.67 163.662500
10308 2 88.80 6.33 -261.937500
Now imagine needing to also involve the percentage from the grand total and the difference from the grand average. To do this, you need to add another table expression, like so:
WITH CustAggregates AS
SELECT custid, SUM(val) AS sumval, AVG(val) AS avgval
FROM Sales.OrderValues
GROUP BY custid
GrandAggregates AS
SELECT SUM(val) AS sumval, AVG(val) AS avgval
FROM Sales.OrderValues
SELECT O.orderid, O.custid, O.val,
CAST(100. * O.val / CA.sumval AS NUMERIC(5, 2)) AS pctcust,
O.val - CA.avgval AS diffcust,
CAST(100. * O.val / GA.sumval AS NUMERIC(5, 2)) AS pctall,
O.val - GA.avgval AS diffall
FROM Sales.OrderValues AS O
JOIN CustAggregates AS CA
ON O.custid = CA.custid
CROSS JOIN GrandAggregates AS GA;
Here’s the output of this query:
orderid custid val pctcust diffcust pctall diffall
-------- ------- ------- -------- ------------ ------- -------------
10835 1 845.80 19.79 133.633334 0.07 -679.252072
10643 1 814.50 19.06 102.333334 0.06 -710.552072
10952 1 471.20 11.03 -240.966666 0.04 -1053.852072
10692 1 878.00 20.55 165.833334 0.07 -647.052072
11011 1 933.50 21.85 221.333334 0.07 -591.552072
10702 1 330.00 7.72 -382.166666 0.03 -1195.052072
10625 2 479.75 34.20 129.012500 0.04 -1045.302072
10759 2 320.00 22.81 -30.737500 0.03 -1205.052072
10926 2 514.40 36.67 163.662500 0.04 -1010.652072
10308 2 88.80 6.33 -261.937500 0.01 -1436.252072
You can see how the query gets more and more complicated, involving more table expressions and more joins.
Another way to perform similar calculations is to use a separate subquery for each calculation. Here are the alternatives, using subqueries to the last two grouped queries:
-- subqueries with detail and customer aggregates
SELECT orderid, custid, val,
CAST(100. * val /
(SELECT SUM(O2.val)
FROM Sales.OrderValues AS O2
WHERE O2.custid = O1.custid) AS NUMERIC(5, 2)) AS pctcust,
val - (SELECT AVG(O2.val)
FROM Sales.OrderValues AS O2
WHERE O2.custid = O1.custid) AS diffcust
FROM Sales.OrderValues AS O1;
-- subqueries with detail, customer and grand aggregates
SELECT orderid, custid, val,
CAST(100. * val /
(SELECT SUM(O2.val)
FROM Sales.OrderValues AS O2
WHERE O2.custid = O1.custid) AS NUMERIC(5, 2)) AS pctcust,
val - (SELECT AVG(O2.val)
FROM Sales.OrderValues AS O2
WHERE O2.custid = O1.custid) AS diffcust,
CAST(100. * val /
(SELECT SUM(O2.val)
FROM Sales.OrderValues AS O2) AS NUMERIC(5, 2)) AS pctall,
val - (SELECT AVG(O2.val)
FROM Sales.OrderValues AS O2) AS diffall
FROM Sales.OrderValues AS O1;
There are two main problems with the subquery approach. One, you end up with lengthy complex code. Two, SQL Server’s optimizer is not coded at the moment to identify cases where multiple subqueries
need to access the exact same set of rows; hence, it will use separate visits to the data for each subquery. This means that the more subqueries you have, the more visits to the data you get. Unlike
the previous problem, this one is not a problem with the language, but rather with the specific optimization you get for subqueries in SQL Server.
Remember that the idea behind a window function is to define a window, or a set, of rows for the function to operate on. Aggregate functions are supposed to be applied to a set of rows; therefore,
the concept of windowing can work well with those as an alternative to using grouping or subqueries. And when calculating the aggregate window function, you don’t lose the detail. You use the OVER
clause to define the window for the function. For example, to calculate the sum of all values from the result set of the query, simply use the following:
SUM(val) OVER()
If you do not restrict the window (empty parentheses), your starting point is the result set of the query.
To calculate the sum of all values from the result set of the query where the customer ID is the same as in the current row, use the partitioning capabilities of window functions (which I’ll say more
about later), and partition the window by custid, as follows:
SUM(val) OVER(PARTITION BY custid)
Note that the term partitioning suggests filtering rather than grouping.
Using window functions, here’s how you address the request involving the detail and customer aggregates, returning the percentage of the current order value of the customer total as well as the
difference from the average (with window functions in bold):
SELECT orderid, custid, val,
CAST(100. * val / SUM(val) OVER(PARTITION BY custid) AS NUMERIC(5, 2)) AS pctcust,
val - AVG(val) OVER(PARTITION BY custid) AS diffcust
FROM Sales.OrderValues;
And here’s another query where you also add the percentage of the grand total and the difference from the grand average:
SELECT orderid, custid, val,
CAST(100. * val / SUM(val) OVER(PARTITION BY custid) AS NUMERIC(5, 2)) AS pctcust,
val - AVG(val) OVER(PARTITION BY custid) AS diffcust,
CAST(100. * val / SUM(val) OVER() AS NUMERIC(5, 2)) AS pctall,
val - AVG(val) OVER() AS diffall
FROM Sales.OrderValues;
Observe how much simpler and more concise the versions with the window functions are. Also, in terms of optimization, note that SQL Server’s optimizer was coded with the logic to look for multiple
functions with the same window specification. If any are found, SQL Server will use the same visit (whichever kind of scan was chosen) to the data for those. For example, in the last query, SQL
Server will use one visit to the data to calculate the first two functions (the sum and average that are partitioned by custid), and it will use one other visit to calculate the last two functions
(the sum and average that are nonpartitioned). I will demonstrate this concept of optimization in Chapter 4.
Another advantage window functions have over subqueries is that the initial window prior to applying restrictions is the result set of the query. This means that it’s the result set after applying
table operators (for example, joins), filters, grouping, and so on. You get this result set because of the phase of logical query processing in which window functions get evaluated. (I’ll say more
about this later in this chapter.) Conversely, a subquery starts from scratch—not from the result set of the outer query. This means that if you want the subquery to operate on the same set as the
result of the outer query, it will need to repeat all query constructs used by the outer query. As an example, suppose that you want our calculations of the percentage of the total and the difference
from the average to apply only to orders placed in the year 2007. With the solution using window functions, all you need to do is add one filter to the query, like so:
SELECT orderid, custid, val,
CAST(100. * val / SUM(val) OVER(PARTITION BY custid) AS NUMERIC(5, 2)) AS pctcust,
val - AVG(val) OVER(PARTITION BY custid) AS diffcust,
CAST(100. * val / SUM(val) OVER() AS NUMERIC(5, 2)) AS pctall,
val - AVG(val) OVER() AS diffall
FROM Sales.OrderValues
WHERE orderdate >= '20070101'
AND orderdate < '20080101';
The starting point for all window functions is the set after applying the filter. But with subqueries, you start from scratch; therefore, you need to repeat the filter in all of your subqueries, like
SELECT orderid, custid, val,
CAST(100. * val /
(SELECT SUM(O2.val)
FROM Sales.OrderValues AS O2
WHERE O2.custid = O1.custid
AND orderdate >= '20070101'
AND orderdate < '20080101') AS NUMERIC(5, 2)) AS pctcust,
val - (SELECT AVG(O2.val)
FROM Sales.OrderValues AS O2
WHERE O2.custid = O1.custid
AND orderdate >= '20070101'
AND orderdate < '20080101') AS diffcust,
CAST(100. * val /
(SELECT SUM(O2.val)
FROM Sales.OrderValues AS O2
WHERE orderdate >= '20070101'
AND orderdate < '20080101') AS NUMERIC(5, 2)) AS pctall,
val - (SELECT AVG(O2.val)
FROM Sales.OrderValues AS O2
WHERE orderdate >= '20070101'
AND orderdate < '20080101') AS diffall
FROM Sales.OrderValues AS O1
WHERE orderdate >= '20070101'
AND orderdate < '20080101';
Of course, you could use workarounds, such as first defining a common table expression (CTE) based on a query that performs the filter, and then have both the outer query and the subqueries refer to
the CTE. However, my point is that with window functions, you don’t need any workarounds because they operate on the result of the query. I will provide more details about this aspect in the design
of window functions later in the chapter, in the Query Elements Supporting Window Functions section.
As mentioned earlier, window functions also lend themselves to good optimization, and often, alternatives to window functions don’t get optimized as well, to say the least. Of course, there are cases
where the inverse is also true. I explain the optimization of window functions in Chapter 4 and provide plenty of examples for using them efficiently in Chapter 5. | {"url":"https://www.microsoftpressstore.com/articles/article.aspx?p=2225061","timestamp":"2024-11-04T14:49:41Z","content_type":"text/html","content_length":"129230","record_id":"<urn:uuid:5342ad67-824e-4964-998b-e169cc777674>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00561.warc.gz"} |
class Metadata : public mspass::utility::BasicMetadata
Subclassed by mspass::seismic::Ensemble< T >, mspass::seismic::CoreSeismogram, mspass::seismic::CoreTimeSeries, mspass::seismic::Ensemble< Tdata >, mspass::seismic::PowerSpectrum,
friend pybind11::object serialize_metadata_py(const Metadata &md)
Serialize Metadata to a python bytes object. This function is needed to support pickle in the python interface. It cast the C++ object to a Python dict and calls pickle against that dict
directly to generate a Python bytes object. This may not be the most elegant approach, but it should be bombproof.
md – is the Metadata object to be serialized
pickle serialized data object.
friend Metadata restore_serialized_metadata_py(const pybind11::object &sd)
Unpack serialized Metadata.
This function is the inverse of the serialize function. It recreates a Metadata object serialized previously with the serialize function.
sd – is the serialized data to be unpacked
Metadata derived from sd
friend std::ostringstream &operator<<(std::ostringstream&, const mspass::utility::Metadata&)
Standard operator for overloading output to a stringstream | {"url":"https://www.mspass.org/cxx_api/mspass.utility.Metadata.html","timestamp":"2024-11-04T02:14:53Z","content_type":"text/html","content_length":"70801","record_id":"<urn:uuid:dca555ea-077d-428a-b507-a4319df18b49>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00565.warc.gz"} |
Average and Instantaneous Rate of Change Worksheet
Problem 1 :
A point moves along a straight line in such a way that after t seconds its distance from the origin is
s = 2t^2 + 3t meters.
(i) Find the average velocity of the points between t = 3 and t = 6 seconds.
(ii) Find the instantaneous velocities at t = 3 and t = 6 seconds. Solution
Problem 2 :
A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of
s = 16t^2 in t seconds.
(i) How long does the camera fall before it hits the ground?
(ii) What is the average velocity with which the camera falls during the last 2 seconds?
(iii) What is the instantaneous velocity of the camera when it hits the ground?
Problem 3 :
A particle moves along a line according to the law
s(t) = 2t^3-9t^2+12t-4 , where t ≥ 0 .
(i) At what times the particle changes direction?
(ii) Find the total distance travelled by the particle in the first 4 seconds.
(iii) Find the particle’s acceleration each time the velocity is zero.
Problem 4 :
If the volume of a cube of side length x is v = x^3 . Find the rate of change of the volume with respect to x when x = 5 units. Solution
Problem 5 :
If the mass m(x) (in kilograms) of a thin rod of length x (in meters) is given by, m(x) = √3x then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 meters.
Problem 6 :
A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find
the rate of changing of the total area of the disturbed water? Solution
Problem 7 :
A beacon makes one revolution every 10 seconds. It is located on a ship which is anchored 5 km from a straight shore line. How fast is the beam moving along the shore line when it makes an angle of
45° with the shore?
Problem 8 :
A conical water tank with vertex down of 12 meters height has a radius of 5 meters at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when
the water is 8 meters deep ? Solution
Problem 9 :
A ladder 17 meter long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 meters from the wall.
(i) How fast is the top of the ladder moving down the wall?
(ii) At what rate, the area of the triangle formed by the ladder, wall, and the floor, is changing?
Problem 10 :
A police jeep, approaching an orthogonal intersection from the northern direction, is chasing a speeding car that has turned and moving straight east. When the jeep is 0.6 km north of the
intersection and the car is 0.8 km to the east. The police determine with a radar that the distance between them and the car is increasing at 20 km/hr. If the jeep is moving at 60 km/hr at the
instant of measurement, what is the speed of the car?
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com | {"url":"https://www.onlinemath4all.com/average-and-instantaneous-rate-of-change-worksheet.html","timestamp":"2024-11-03T00:44:03Z","content_type":"text/html","content_length":"30387","record_id":"<urn:uuid:47df2fe9-ee95-4e7a-b237-d81143abb55c>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00285.warc.gz"} |
How do you solve 9x^2 - 7x = 12 using completing the square? | HIX Tutor
How do you solve #9x^2 - 7x = 12# using completing the square?
Answer 1
I found:
${x}_{1} = \frac{1}{18} \left(7 + \sqrt{481}\right)$
${x}_{2} = \frac{1}{18} \left(7 - \sqrt{481}\right)$
Let's perform a few manipulations:
#9x^2-7x=12# add and subtract #49/36# #9x^2-7x+49/36-49/36=12# #9x^2-7x+49/36=12+49/36# #(3x-7/6)^2=481/36# root square both sides: #3x-7/6=+-sqrt(481/36)=# #x=1/3(7/6+-sqrt(481/36))# #x_1=1/18
(7+sqrt(481))# #x_2=1/18(7-sqrt(481))#
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Answer 2
To solve the equation (9x^2 - 7x = 12) using completing the square:
1. Move the constant term to the other side: (9x^2 - 7x - 12 = 0).
2. Divide all terms by the coefficient of (x^2): (x^2 - \frac{7}{9}x - \frac{4}{3} = 0).
3. To complete the square, take half of the coefficient of (x) ((-\frac{7}{9})), square it, and add it to both sides of the equation: (x^2 - \frac{7}{9}x + \left(\frac{7}{18}\right)^2 - \left(\frac
{7}{18}\right)^2 - \frac{4}{3} = 0).
4. Simplify: (x^2 - \frac{7}{9}x + \frac{49}{324} - \frac{49}{324} - \frac{4}{3} = 0).
5. Combine like terms: (x^2 - \frac{7}{9}x + \frac{49}{324} - \frac{196}{324} = 0).
6. Factor the perfect square trinomial and simplify: (\left(x - \frac{7}{18}\right)^2 - \frac{245}{324} = 0).
7. Add (\frac{245}{324}) to both sides: (\left(x - \frac{7}{18}\right)^2 = \frac{245}{324}).
8. Take the square root of both sides: (x - \frac{7}{18} = \pm \sqrt{\frac{245}{324}}).
9. Solve for (x): (x = \frac{7}{18} \pm \sqrt{\frac{245}{324}}).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Creating User Models in QucsStudio
Steps for Creating Models
In QucsStudio, there are three main approaches to model creation: equation components, VerilogA models, and C++ models. Each of these has its own characteristics and advantages while sharing the
common goal of building accurate simulation models.
1. Equation Components
Equation components are very suitable for defining the behavior of components using simple formulas. This method is intuitive, accessible, and allows for quick integration of basic electrical
characteristics or linear models into simulations. It involves directly entering mathematical equations and observing the results in real-time, making it ideal for beginners or basic simulation
2. VerilogA Models
VerilogA models are suited for modeling components with specific nonlinear characteristics or frequency-dependent properties. VerilogA is a hardware description language for analog and mixed-signal
systems, offering enough flexibility to represent complex behaviors or custom devices. This approach allows for a more detailed definition of behavior and the emulation of specific physical
phenomena, suitable for intermediate to advanced-level simulations.
3. C++ Models
C++ models are particularly suitable when advanced mathematical algorithms or computational processing is required. Using the C++ language allows for highly customized and efficient computations,
meeting the needs of the most complex simulation requirements. This approach enables the use of existing libraries and the incorporation of advanced numerical analysis techniques, providing the most
flexible and powerful construction of simulation models.
Commonalities and Choosing an Approach
These three approaches share the common objective of enabling the construction of accurate simulation models in QucsStudio. The choice depends on the simulation goal, required accuracy, developer
skill level, and project complexity. Starting with simple models and moving to more advanced approaches as needed offers flexibility to meet various simulation needs. Each approach serves as a tool
to provide optimal solutions for specific problems faced by users.
Using Equation Components
1. Example of a Simple Resistance Model:
• Use the “Equation Component” to define the behavior of resistance for a specific voltage.
• Example: Calculate the current I using the formula I = V/R, where V is voltage and R is resistance value.
Step 1: Adding an Equation Component
Find the “nonlinear components” section in the “Components” panel on the left side of the main window. Locate the “Equation Component,” click and drag it to drop it into the work area (schematic).
Step 2: Defining the Equation
Double-click the equation component to open the property editor.
Enter the equation I = V/R that defines the resistance model in the “Equation” field. Here, I represents current, V represents voltage, and R represents resistance value.
You can enter additional equations to define the values for V and R as needed. For example, you can set specific values like V = 5 (voltage of 5 volts) and R = 100 (resistance of 100 ohms).
Step 4: Running the Simulation
Before running the simulation, add voltage source (V) and ground (GND) components to complete the circuit.
Click the “Simulation” button to start the simulation. Once the simulation is complete, the results will be displayed. You can check the value of current I and verify that the equation I = V/R has
been correctly calculated.
Creating VerilogA Models
1. Example of a Custom Transistor Model:
• Create a transistor model with specific nonlinear characteristics or frequency properties using VerilogA language.
• Example: Describe a unique current-voltage relationship to replicate specific behavior.
Step 1: Creating a VerilogA File
Use any text editor to write a .va file containing your VerilogA code.
For example, use the VerilogA language to describe the transistor’s current-voltage relationship.
Here is a basic code example for the current-voltage relationship.
module CustomTransistor(n1, n2, n3);
inout n1, n2, n3;
electrical n1, n2, n3;
parameter real Vth = 0.7; // Threshold voltage
parameter real K = 1.0e-3; // Transistor constant
analog begin
if (V(n2, n3) > Vth) begin
I(n1, n2) <+ K * (V(n2, n3) – Vth)^2;
end else begin
I(n1, n2) <+ 0;
• This code defines a transistor with three terminals: n1 (base), n2 (collector), n3 (emitter).
• Vth is the transistor’s threshold voltage, and K is a constant defining the transistor’s characteristics.
• The analog begin ... end block defines the formula for calculating the collector current.
Save the text file with the extension .va and copy it to the folder of the project using this model in QucsStudio.
Once properly placed, you can access the .va file from the “Content” on the left side of the main window.
Step 2: Placing the VerilogA Model
You can add the transistor model described in VerilogA to your schematic.
Select the Verilog model you want to use, and you can place it on the schematic by moving the cursor over it.
Now, you can use the created transistor model for simulation.
Creating C++ Models
This article does not cover it, but you can create models using C++ code similar to the VerilogA description.
This guide introduced how to create user models in QucsStudio and utilize them in simulations. To achieve accurate simulation results, it’s important to properly adjust the model parameters and
verify them against the actual device characteristics. | {"url":"https://denki-sim.blog/en/usermodel_en/","timestamp":"2024-11-14T16:38:41Z","content_type":"text/html","content_length":"343510","record_id":"<urn:uuid:773aef5e-0da5-4335-af04-f5f870a90962>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00643.warc.gz"} |
Line Distance Between Two Points Equations Formulas Calculator - y2
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