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The concentration of carbon monoxide in an urban apartment is 48 µg/m^3. What mass of carbon monoxide in grams is present in a room measuring 11.0 ft by 11.5 ft by 20.5 ft? | Socratic
The concentration of carbon monoxide in an urban apartment is #48# #µg##/##m^3#. What mass of carbon monoxide in grams is present in a room measuring 11.0 ft by 11.5 ft by 20.5 ft?
1 Answer
The idea here is that you need to use the information provided by the problem to find the volume of the room.
Since you are told that the concentration of carbon monoxide in a typical apartment is equal to $48 \mu {\text{g/m}}^{3}$, finding the volume of the room in cubic meters will allow you to determine
how much carbon monoxide it contains.
Now, you can treat the room as a rectangular prism of length $l$, width $w$, and height $h$.
The volume of a rectangular prism is given by the formula
$\textcolor{b l u e}{V = l \times w \times h}$
You can use the conversion factor that exists between feet and meters to find the dimensions of the room in meters, then calculate the volume in cubic meters.
$\text{1 ft " = " 0.3048 m}$
The volume of the room will thus be
#V = (20.5color(red)(cancel(color(black)("ft"))) * "0.3048 m"/(1color(red)(cancel(color(black)("ft"))))) xx (11.5color(red)(cancel(color(black)("ft"))) * "0.3048 m"/(1color(red)(cancel(color
(black)("ft"))))) xx (11.0color(red)(cancel(color(black)("ft"))) * "0.3048 m"/(1color(red)(cancel(color(black)("ft")))))#
$V = {\text{73.43 m}}^{3}$
This means that the room will contain
#73.43color(red)(cancel(color(black)("m"^3))) * overbrace( (48color(white)(a)mu"g")/(1color(red)(cancel(color(black)("m"^3)))))^(color(purple)("given concentration")) = 3524.64mu"g"#
Rounded to two sig figs, the number of sig figs you have for the concentration of carbon monoxide, the answer will be
${M}_{C O} = \textcolor{g r e e n}{3500 \textcolor{w h i t e}{a} \mu \text{g}}$
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A New Approach for Assessing Heat Balance State along a Water Transfer Channel during Winter Periods
School of Civil Engineering, Hefei University of Technology, Hefei 230009, China
School of Engineering, University of Northern British Columbia, Prince George, BC V2N 4Z9, Canada
Power China Beijing Engineering Corporation Limited, Beijing 100024, China
China South-to-North Water Transfer Group Middle Line Co., Ltd., Beijing 100038, China
Yangtze River Scientific Research Institute, Wuhan 430010, China
Authors to whom correspondence should be addressed.
Submission received: 5 September 2022 / Revised: 6 October 2022 / Accepted: 11 October 2022 / Published: 17 October 2022
Ice problems in channels for water transfer in cold regions seriously affect the capacity and efficiency of water conveyance. Sometimes, ice problems such as ice jams in water transfer channels
create risk during winter periods. Recently, water temperature and environmental factors at various cross-sections along the main channel of the middle route of the South-to-North Water Transfer
Project in China have been measured. Based on these temperature data, the heat balance state of this water transfer channel has been investigated. A principal component analysis (PCA) method has been
used to analyze the complex factors influencing the observed variations of the water temperature, by reducing eigenvector dimension and then extracting the principal component as the input feature.
Based on the support vector machine (SVM) theory, a new approach for judging the heat loss or heat gain of flowing water in a channel during winter periods has been developed. The Gaussian radial
basis is used as the kernel function in this new approach. Then, parameters have been optimized by means of various methods. Through the supervised machine learning process toward the observed water
temperature data, it is found that the air–water temperature difference and thermal conditions are the key factors affecting the heat loss or heat absorption of water body. Results using the proposed
method agree well with those of measurements. The changes of water temperature are well predicted using the proposed method together with the state of water heat balance.
1. Introduction
Long-distance water transfer project is an important approach to effectively solve the water shortage problem in large- and medium-sized cities [
]. The total length of the main canal of the Middle Route of the South-to-North Water Transfer Project is 1267 km, as shown in
Figure 1
. Water is transported from the Danjiangkou Reservoir in the Hanjiang River to the North China Plain [
]. The purpose of this water transfer project is to transfer water to Henan Province, Hebei Province, and the cities of Beijing and Tianjin. The average annual amount of transferred water is 9.5
billion m
]. Since the start of the operation of this water transfer project in 2014, given the strong coupling of cascaded channels, it has been difficult to keep water levels at all gates along the route
being stable. The cross-section of the main canal has a trapezoidal shape, with a bottom width range from 7.0 to 26.5 m. Along the water transfer route, there are no lakes or reservoirs for water
detention, indicating the importance of safety operation of this project [
]. The main canal of this water transfer line is spanned from the place located at the north latitude of 33° to the north latitude of 40°. During winter periods, the temperature in northern China
such as in Beijing is generally low. As a consequence, the formation of ice cover in the water transfer cannel creates a big ice problem [
During an ice-covered flow condition, the characteristics of river hydraulics have been dramatically changed compared to that under an open flow condition. An ice cover adds an extra boundary to the
flow, which leads to considerable changes in the velocity profile, flow rate, bed shear stress distribution, and water temperature [
]. Water temperature is greatly affected by thermal factors and environmental variables [
]. Under the influence of air temperature, flow and boundary conditions during a winter period [
], water temperature is affected by the hydraulic response to the channel or the complex process of water transport during an ice-covered period. In the meantime, water temperature is an important
physical factor affecting the change of river ice regime [
]. Therefore, water temperature is often used as one of important indicators for forecasting river ice regime [
]. To avoid the development and formation of an ice jam during winter periods, the water temperature during the water transfer process should be carefully controlled [
Boyd et al. [
] calculated and analyzed the peak temperature and duration of each supercooling event by observing the water temperature of three controlled rivers, Kananaskis, North Saskatchewan and Peace Rivers
in Alberta, Canada. They pointed out that the water supercooling process is the key condition for the formation of suspended ice and anchor ice in rivers in cold regions. Osterkamp et al. [
] measured the temperature of water in the Goldstream Creek and Chatanika River in Alaska during the formation of frazil ice particles that forms in turbulent water. Frazil ice particles in flowing
water are essential for the development of ice pans which may lead to the development of ice jams and cause disaster [
]. The correlation between the cooling rate of a river and the heat loss of the water surface has been calculated for the periods before and during the formation of frazil ice. It was found that the
core of frazil ice in the river was cold organic matter, soil particles or a combination of these substances, which could enter the river through the mass exchange process of the air–water interface.
The degree of supercooling of the observed river was given. McFarlane et al. [
] compared five different methods for calculating sensible heat flux and latent heat flux based on air temperature, relative humidity, air pressure, wind speed and wind direction, short-wave and
long-wave radiation for the Dauphin River in Manitoba. By calculating the heat balance of the river, the supercooling phenomenon of the river was predicted. They claimed that the solar radiation is
an important factor in the heat budget of water bodies. Wang et al. [
] pointed out that both temperature field and velocity field of flowing water during an ice-covered period are essential for studying the formation and evolution of frazil ice. By applying the
Navier–Stokes equations and energy equation for flow under an ice cover, the velocity field and temperature field of flowing water have been analyzed. It is found that the temperature field of water
body is mainly affected by heat conduction on the surface of ice cover and convective heat transfer on the riverbed. Yang [
] developed both linear and non-linear models for calculating heat exchange between water and atmosphere based on meteorological data including solar radiation, long-wave radiation, evaporation and
convective heat exchange. The one-dimensional model for simulating temperature of flowing water in an open channel is developed. The model is verified and applied along some typical channel reaches
of the Middle Route of the South-to-North Water Transfer Project (MR-SNWTP).
Field observations along the main canal of the MR-SNWTP indicate that, with the decrease in air temperature and water temperature in winter, frazil ice appears and ice pans form. An ice cover will be
gradually developed on water surface of the MR-SNWTP. After the MR-SNWTP is covered by ice, both the growth of ice cover thickness and the decrease in water temperature slowed down. During the stable
ice-covered period, the water temperature was low with some minor fluctuations. With the increase in air temperature in spring, river ice starts to melt, and the water temperature begins to rise as
well [
]. Since the change of water temperature is also an indicator of the complex nonlinear evolution process of an ice cover during winter periods, the machine learning method with advantages in dealing
with nonlinear problems can be considered to predict the trend of water temperature [
]. Sun et al. [
] used the machine learning method to predict the river breakup time and peak flow of the Athabasca River in Fort McMurray, Canada. Their results show that the inclusion of certain inputs in the
optimal models can reveal some hints for the potential mechanism on river ice from the data-driven perspective. Seidou et al. [
] applied the artificial neural network and the lake ice thermodynamic model to the study ice cover thickness of several lakes and reservoirs in Canada. It is found that, in case of scarce or less
data available for use, result using the artificial neural network has higher prediction accuracy. Guo et al. [
] used the improved back propagation (BP) neural network method to predict the river-breakups in the Heilongjiang and improved the prediction accuracy by integrating multiple factors affecting water
temperature and development of ice conditions. The commonly used data-driven prediction models include multiple regression, projection pursuit, and cloud model. These prediction models have achieved
a series of results in different engineering cases [
]. The traditional multiple regression model is difficult to accurately describe the complex relationship between independent variables and dependent variables. When the process is highly random, the
prediction accuracy is often low. The machine learning models such as projection pursuit and cloud model possess the nonlinear mapping and adaptive ability, and their computational efficiency and
prediction accuracy are improved compared to the traditional multiple regression models. However, because the goal of train samples for these models is to minimize the fitting error, it is easy to
create problems such as overfitting or local optimality. The support vector machine (SVM) is an algorithm based on the statistical learning theory of small samples, which can be used to solve
nonlinear and high dimension problems [
]. Wang et al. [
] predicted the water level during an ice-jammed period by means of the BP neural network and the support vector machine. Their results show that the prediction accuracy using the support vector
machine method is higher than that using the BP neural network in case that the sample is small. The principal component analysis (PCA) method can be used to extract independent and effective
information from features, which is a statistical analysis method to reduce the input dimension [
]. Ren et al. [
] used the PCA method to eliminate redundant information and effectively analyzed the main factors affecting water temperature in the upper reach of the Yellow River. The identification of the heat
balance state of water body in the channel during a winter period has the characteristics of nonlinear process and high dimension, which is in line with the characteristics of the machine learning
method. However, no research work has been reported regarding the heat balance of a water body during an ice-covered period.
In summary, the seasonal change of water temperature in rivers affects the formation of river ice. Results of daily field observations at channel cross-sections in winters may be affected by ice
conditions in the MR-SNWTP. The heat balance of water body in rivers during a winter period is more complicated. The sluice gate for flow control located in Beijuma of the MR-SNWTP is the junction
point of the water transfer channel from an open channel to the culvert, and its location is shown in
Figure 1
. This channel cross-section of the MR-SNWTP was chosen in this study since relatively more field observation data are available and the temperature in this region is relatively low. Additionally,
this cross-section is vulnerable to the ice problem during winter periods. In this study, the machine learning method is used to study the heat balance and water temperature at this cross-section of
the MR-SNWTP.
2. Methodology
The change of temperature follows the principle of energy conservation [
], and the main parameters used in the water temperature model are shown in
Table 1
The general equation for describing temperature of water in an open channel is as follows:
$∂ ∂ t ( ρ w c p A w T w ) + ∂ ∂ x ( ρ w c p Q w T w ) − ∂ ∂ x ( E x ρ w c p A w ∂ T w ∂ x ) = B w φ w a$
Equation (1) is a mathematical model for describing water temperature field [
]. Symbols in Equation are explained in
Table 1
. From left to right of Equation (1), it is the time term, the convection term, the diffusion term, and the source term, respectively.
For a channel with a regular cross-section similar to ghat of the MR-SNWTP, the temperature gradient along the flow direction is small, and the diffusion term can often be ignored. Combined with the
continuity equation of the flow, by assuming the flow is incompressible with the fixed specific heat capacity, Equation (1) can be simplified. In the presence of an ice cover on water surface (either
partially or fully covered), both the boundary conditions and flow conditions for the model have been changed, resulting in a significant impact on the water temperature [
]. The exchange of heat occurs between the atmosphere, ice cover and water, which is related to the coverage of ice cover on water surface. Therefore, the equation for describing temperature of water
can be rewritten as follows:
$∂ ∂ t ( ρ w c p A w T w ) + ∂ ∂ x ( ρ w c p Q w T w ) − ∂ ∂ x ( E x ρ w c p A w ∂ T w ∂ x ) = B w [ N i φ w i + ( 1 − N i ) φ w a ]$
represents the extent that the cross-section is covered by ice. When an ice cover is present in a channel, the heat transfer of the water body is closely related to the heat loss of the ice body. The
above equations characterize the physical mechanism of the heat transfer of the water body during a winter period [
The heat exchange between atmosphere and water is affected by weather conditions, solar radiation and wind speed. In late fall or early winter, with the decrease in air temperature, cooling intensity
will be increased. Therefore, heat loss from water body results in the decrease in water temperature until the water surface is covered by an ice cover which is normally formed by frazil ice particle
and ice pans in rivers. After the entire water surface is covered by ice, the water temperature continues to fluctuate and drop, and the water temperature gradient at the front of the ice sheet is
large. During the stable ice-covered period, with the decrease in air temperature, the thickness of the ice cover gradually increases, and the heat exchange between water and air decreases. During
this stable ice-covered period, the water temperature can reach a dynamic equilibrium condition with some minor fluctuations. In the spring, with the increase in the air temperature and solar
radiation, an ice-covered river undergoes the thermodynamic breakup process (ice cover melting process). The water body gradually absorbs heat and results in the increase in temperature of water
body. Since the MR-SNWTP is 1267 km long and spans a latitude difference of 7°, there will be some changes in the temperature of water body along the MR-SNWTP. The change of air temperature and that
of water temperature has an obvious nonlinear relationship. The continuous change of meteorological elements in winter and spring affects the relevant parameters in Equation (2).
Equations (1) and (2) show the relationship between the change of water temperature and other factors. Since the parameters in these equations vary with various factors including the environmental
variables, the analytical solution of these equations will be very complicated. The hydrodynamic process also affects the heat balance of the water body. When the flow velocity is high enough, shore
ice along channel banks is normally difficult to be developed. Additionally, the relatively larger flow velocity during a freeze-up period is likely to cause the incoming ice pans/floes to be
entrained and submerged under the sheet ice cover to form an ice jam. During the break-up period, the surface of the entire channel (together with pools) is grouped into the open channel section,
ice-cover melting section and the fully ice-covered section. Along the melting section of the channel, ice cover becomes soft and loose with a lot of pores. Under the influence of hydrodynamic force,
this loose ice cover is gradually eroded, and the wavy-shaped surface appears on the bottom of an ice cover, which eventually causes the ice cover to be melted, broken, and transported downstream.
The hydrodynamics of the flowing water and the ice dynamics affect the heat transfer of water and changes of water temperature.
By taking the thermal and dynamic factors that affect the change of temperature of water body as the input features, an SVM-based discriminant model for water heat balance is established to assess
the heat loss and heat gain of water body. In this study, data were obtained through on-site observations. Among them, the observed data for water flow conditions at some cross-sections near the
control gate have been acquired from the management department of the MR-SNWTP. The temperature sensor and flow velocity meter have been used to collect data for describing water flow conditions,
which were measured 4 times every day. The meteorological data were acquired from the meteorological station which was setup by considering the locations for observations of the water flow. The
number of daily measurements of meteorological data is consistent with that of for water flow. The daily average approach is adopted to form a dataset for subsequent analysis. These environmental
variables have different effects on the latent heat flux and the sensible heat flux, which are correlated to the heat balance. The correlation analysis of each characteristic data has been carried
out, and the correlation matrix of the measured dataset is established. The calculated correlation coefficient is presented in
Figure 2
. The corresponding thermal factors and dynamic factors are abbreviated to better express the figure. These factors include: the temperature difference between air and water (Air and Water TEMP
diff), the previous water temperature (Pre-water TEMP), the cumulated seven-day average daily air temperature (7d Air TEMP CUM), the time effect (TM EFF), the maximum air temperature (Max Air TEMP),
the minimum air temperature (Min Air TEMP), the difference between wind speed and water velocity (Wind S and VEL diff), the flow Froude number of water flow (Fr), the solar radiation (SR), the air
pressure (Air PRESS), the cloudiness (Cloudiness), the weather conditions (WEA COND), the cloud height (Cloud H), and the precipitation (Precipitation).
Figure 2
, the correlation coefficient of 1 indicates that the two features are completely linear positive correlated; the correlation coefficient of −1 indicates that the two features are completely linear
negative correlation; and the correlation coefficient of 0 indicates that the two features are independent. It can be seen from
Figure 2
that there is a certain degree of correlation among the influencing features, that is, there is information overlap between the input features. The commonly used methods for data dimension reduction
include single-factor analysis, grey system analysis, set pair analysis, analytic hierarchy process and principal component analysis. Results of single-factor analysis have polarization problem. The
dimension reduction methods based on grey system theory, set pair analysis theory and analytic hierarchy process are controversial in the selection of index weight, and the application of the model
automatic calculation is limited. The core idea of the PCA method is to reduce the high-dimensional associated features to a few unrelated features, and to reflect the original information as much as
possible. The PCA method can be used to process the standardized sample data, extract the main information of input variables, and improve the accuracy of classification based on comprehensive
consideration of various influencing factors [
]. To ensure that the data have the same importance as the input scale, the daily average dataset of the selected measured parameters and measurement records are pre-processed by the standardized
method, and the covariance matrix
is calculated according to Equation (3):
is the number of sample data, and
is the normalized sample matrix. By solving the characteristic equation,
non-negative features
= 1, 2, ⋯,
) of the covariance matrix
are obtained and arranged in the order of
> ⋯ >
> 0. The corresponding orthogonal unit eigenvector
is solved, and the formula of principal component calculation is as follows:
In Equation (4),
represents the
th principal component (
), and the contribution rate
of the
th principal component is calculated using Equation (5):
$v i = λ i / ∑ k = 1 m λ k$
The upper limit of machine learning effect is controlled by the available data. The standardized input data are divided into training and verification sets. To reduce the difference introduced by
different sample division, the model is trained by the cross-validation method. For the part of training set, it is divided equally and one of them is selected as the test set, and the rest is used
as the training set to complete the cross-validation process.
The SVM algorithm is based on the principle of minimization of structural risk. Regarding the classification of samples, the sample dataset is set as
= [
is the corresponding label of
. The heat gain of water body is labeled as “+1”, and the heat loss of water body is labeled as “−1”. The hyperplane formula for the sample classification is expressed in Equation (6), and the
classification decision function is described as Equation (7) [
$f ( Z ) = sign ( w T ⋅ η ( Z ) + b )$
= (
; ⋯;
) is the weight vector corresponding to
is the displacement term which determines the distance between the hyperplane and the origin. The sign ( ) represents the sign function. The Lagrangian function for solving this problem can be
described as Equation (8):
$min L ( w , b , η , α ) = 1 2 ‖ w ‖ 2 − ∑ i = 1 p α i [ y i ( w T ⋅ η ( Z ) + b ) − 1 ]$
= (
; ⋯;
) is a Lagrangian operator,
> 0. By solving Equation (8) and substituting it into Equations (7) and (9) is obtained:
${ f ( Z ) = ∑ i = 1 p α i y i η ( Z i ) T η ( Z h ) + b s . t . { α i ≥ 0 ; i = 1 , 2 , ⋯ , p ∑ i = 1 p α i y i = 0$
The standardized sample data are inseparable in the original space. To solve this problem, the standardized sample data are mapped and transformed in the original space, and the value of the
transformed space inner product function can be transformed into the value directly calculated using the kernel function. The calculation formula is as follows:
$k ( Z i , Z h ) = η ( Z i ) T η ( Z h )$
) and
) are the mapping transformation functions of the original space; and
) is the kernel function. When meteorological data are relatively complete, based on the observed thermal-hydrodynamic data about winter ice-water regime of the MR-SNWTP, the heat budget process of
water body and the mutual feedback relationship of the related parameters are identified.
3. Analysis of Heat Budget of Water Body
The MR-SNWTP delivers water from the low latitude to high latitude. In winter, it is necessary to identify the influence of both thermal and dynamic factors of flowing water on the heat balance of
flowing water in the main canal of the MR-SNWTP. The key to the study is how to consider the influence of temperature, solar radiation, cloud cover, precipitation, wind speed and other factors on the
heat balance process of flowing water.
The supercooling process caused by heat loss of water body is a critical process for ice formation in channels. Hence, it is proposed to use environmental variables to predict the correct direction
of water temperature change. Then, by considering the difference between calculation accuracy and complexity, the parameter adjustment algorithm was selected to assess the heat balance state of water
body. The cumulative temperature during a certain time period is often used for analyzing river ice hydrology [
]. The difference between air temperature and water temperature, the cumulative air temperature, and wind speed and velocity gradient were selected for analyzing the impact these factors on the heat
balance of water body, as shown in
Figure 3
It can be seen from
Figure 3
that, under condition of the heat balance of water body, both thermal factors and dynamic factors contribute to the heat transfer of water body, and it is difficult to use a linear segmentation to
directly separate the heat transfer caused by thermal factors from that by dynamic factors. This is the characteristics of the heat budget of water body in river during a winter period. It has also
become difficult to accurately calculate the heat exchange of flowing water in winter by using both theoretical equations and numerical simulation. Due to the mixing process of flowing water and
latent heat from the phase transition from liquid water to solid ice, the heat balance state of water body cannot be simply expressed as a good linear function of either dynamic or thermal condition.
In the case of being linearly inseparable, the samples in the low-dimensional space are mapped to a high-dimensional feature space by using a nonlinear mapping function to make it to be linear. Then,
it is possible to use the linear algorithm to analyze the nonlinearity of samples in the high-dimensional feature space, and the optimal classification hyperplane can be found in this feature space.
The so-called optimal classification surface not only requires to correctly separate the categories, but also needs to maximize the classification interval. The purpose for correctly separation of
categories is to ensure that the empirical risk is minimized. Actually, the maximum classification interval is to minimize the confidence range in the generalization bound.
Considering the complexity and coupling of the water heat budget process, the Gaussian radial basis function (RBF) is used as the kernel function of the support vector machine:
$k ( Z i , Z h ) = exp ( − γ ⋅ ‖ Z i − Z h ‖ 2 )$
In Equation (11),
represent different input characteristics, and
is the Gaussian kernel bandwidth parameter. By introducing the multiplier
$α i *$
, and the penalty parameter
, a slack variable
needs to be added to the threshold to reduce the error of the dual problem. The slack variable represents that the error term whose deviation is less than
is not penalized, we can get:
$min L ( w , b , η , α , ε i , α * ) = 1 2 ‖ w ‖ 2 + C ∑ i = 1 p ε i − ∑ i = 1 p α i [ y i ( w T ⋅ k ( Z i , Z h ) + b ) − 1 − ε i ] − ∑ i = 1 p α i * ε i$
The parameter C represents the degree of punishment for prediction errors. The larger the C value, the more the model does not allow prediction errors. If the C value is too large, the model will
make fewer errors in the training data process and easily cause overfitting. On the contrary, if the C value is too small, the model will easily ignore the prediction error, which will lead to a poor
performance of the model. The value of γ will influence the range of the Gaussian function corresponding to each supporting vector. The weight of high-order features decays very fast, so that, the
larger the γ value, the less the support vectors; the smaller the γ value, the more the support vectors. The number of support vectors will affect the speed of the training and prediction process.
The parameters C and γ jointly determine the prediction results of the model. Appropriate selection of their optimal combination can lead to the good generalization ability and fitting effect of the
The selection of values of parameters in the kernel function is the key to the calculation accuracy. Thus, the optimization of parameters is needed to determine the values of
. The commonly used methods for parameter adjustment include the grid search (GS) method, the particle swarm optimization (PSO) method, and the genetic algorithm (GA) method. The grid search method
is to use the search data to form a grid space, and then obtain the value for each grid point using the exhaustive method. The concept about the PSO method is that each particle contains two
attributes of speed and position, each particle contains a fitness value determined by the objective function and knows the best position and the current position found by itself until now. In the
whole group, the best target position found by all individuals is also known by each particle. How each individual particle proceeds to the next position is determined by the parallel search of the
population. The GA method is a random search method by means of the evolutionary process of the biological world. Its main characteristics is to directly operate the data to be optimized. It can
automatically obtain and guide the optimized search space, adaptively adjust the search direction, and has strong robustness. As mentioned above, the heat exchange of water body during an ice-covered
period is a complex process and affected by many factors, which can be expressed as:
$d T w d t = g ( T w − h , T a − s , T t , δ v , F r , P , R a , R e l , R p , N , H c )$
represents the historical conditions of water temperature;
represents air temperature conditions including daily maximum, minimum, average temperature and cumulative temperature during a period;
represents the time-effect factor. Since the variation of water temperature approximately shows a periodic function and it fluctuates around the average annual water temperature, both sin (2π
) and cos (2π
) are selected to reflect the time-effect
; where
is the date count of the year (note: 1 January is 1, etc.); and
is the number of annual dates;
represents the component gradient of wind speed and flow velocity in the flow direction, which has an important influence on the heat transfer process;
is the flow Froude number;
is the average daily pressure;
is the solar radiation;
is the daily average relative humidity;
is the average daily precipitation;
is cloud amount;
is the cloud height. These parameters have either direct or indirect impact on the change of water temperature. Results of field observation showed that the changes of water temperature are
compatible with the simulations of the collected data. If
> 0, it indicates that the water body absorbs heat and the water temperature rises; if
< 0, water body loses heat and water temperature drops. After standardizing parameters in Equation (10) based on data measured at the typical survey stations, the PCA was used to extract the first
principal component and the second principal component as the input of the SVM. The cumulative contribution rate of each principal component output by using the PCA is shown in
Table 2
According to the cumulative contribution rate of each principal component in
Table 2
, results showed that the result for the first two principal components can explain more than 50% of the information for original data; the information for the first five principal components can
explain more than 80% of the information for original data; the information for the first seven principal components can explain more than 90% of the information for original data. With respect to
the prediction of water temperature and heat balance state, based on the data measured at the typical survey section during six winters from 2015 to 2021, the measured data from 2015 to 2020 are used
as the training samples to establish the model and optimize the model parameters, but the measured data from 2020 to 2021 are used as the verification samples to test the capacity for generalization
of the model. The flow chart for calculation for the assessment of heat balance state is shown in
Figure 4
Since the annual accumulated negative temperature of the training sample varies in a certain range, it can be used to describe different winter characteristics. During the winter period of 2019~2020,
the Huinanzhuang pumping station (on the Beijuma River) was under maintenance. The measured data for water transport in the Beiyishui River during an ice-covered period are used to act as the
observation data of the typical survey section, so that the training set also has a certain coverage on the hydraulic characteristics. PCA-1 is selected as the first axis and PCA-2 as the second
axis. By applying the GS, PSO and GA methods, results of prediction of classification are generated (
Figure 5
), and the corresponding parameter values are shown in
Table 3
. When the direction of water temperature change (heat gain vs. heat loss) is correctly identified, it is classified as a successful prediction.
Figure 5
, log
and log
are the logarithms of parameters
, respectively. According to the calculation process of Equations (3) and (4), the composition of the principal components includes all the influencing factors, and the contribution of each factor is
different. Due to the different features of various algorithms, the combination of parameters
obtained by the optimization also fluctuates within a certain range. It can be seen from
Figure 5
that the generated nonlinear partition hyperplane can better distinguish the heat budget of water body. In the prediction model, 63 out of the 89 classified data were predicted correctly, and the
accuracy reached at 70.79%. From the number of iterations, the GS method has fewer iterations with a higher optimization efficiency. In view of the cross-validation rate, the GA algorithm performs
well in the model training, and the number of the support vectors is less. By observing the execution time of the algorithm, with respect to the operation management of water transfer process, the
calculation time consumed using these three algorithms is within the acceptable range, and the time need using the GA algorithm is slightly higher. Furthermore, by selecting the first five principal
components in order to adequately reflect the influence of the original correlation factors, the SVM is used to predict the heat balance state of water body. The prediction results are summarized in
Table 4
With the increases in the number of principal components extracted, the contribution of each component in the model also increases accordingly. It can be seen from
Table 4
that the comprehensive impact factor formed by more principal components can reflect more information about the original data, and the prediction accuracy of the model can be further improved. It has
been found that the prediction results by means of two bionic algorithms, namely the PSO and GA methods, are better than those using the GS method. Results showed that the number of support vectors
generated by the PSO algorithm is less. From
Table 3
Table 4
, once can see that there are differences in computational performance using different approaches. When the number of input data are increased, the time of iterations also increases. This study is
concerned with the exploration of the heat balance state of the MR-SNWTP. In practical application, different parameters for describing the performance need to be trade-off according to the actual
situation of the project. Among these three methods, the times of iterations using the GA algorithm is relatively small, and the cross-validation rate of this method is convenient and outstanding.
This means that the GA algorithm method is clearly suitable for the parameter optimization process of the SVM model in this study. The changes of water temperature have been well predicted by means
of the proposed method together with the state of water heat balance.
According to the classification results of the model, the prediction results showed that water body began to absorb heat from 13 January 2021. At this time, the air temperature began to maintain
above 0 °C with the highest air temperature of about 5 °C when the weather became sunny with sufficient solar radiation and relative low humidity. On 17 January 2021, the main channel of the Beijuma
River survey section eventually broke up. After the breakup of the channel along this survey section, due to the release of latent heat from the upstream inflow and ice, the water body experienced a
repeatedly heat gain -heat loss process. By 21 January 2021, the main channel was eventually opened, and the water body continued to absorb heat, causing the water temperature to rise. Additionally,
then, the remaining shore ice vanished. As a machine learning method, the SVM model can mine the potential knowledge information of the data and complete the classification and prediction of the data
according to the features of datasets. In this study, the feasibility of the proposed approach has been verified by using results of the actual water transfer project. Result of this study can
provide reference for engineers in water resources engineering. To apply it, by replacing the input data and the value of the kernel parameters, the trend of variation of water temperature can be
identified, and the state of water heat budget can be classified. The proposed method in this study can be promising for controlling ice problems in channels for water transfer in cold regions and
can provide the operation guidelines for a river system under an ice-covered flow condition in winter.
4. Conclusions
In this study, starting with the thermodynamic characteristics of water body during winter periods, the heat loss and heat gain process of water body along a typical section of the MR-SNWTP has been
studied. The following conclusions have been drawn from this study:
By analyzing the factors that influence water heat balance during winter periods, the correlation coefficient matrix of thermal and hydrodynamic characteristic data is studied. By using the PCA
method to extract principal components as model input, the correlation between multiple variables was described by a few variables. By inputting data using the machine learning model, it can
effectively reduce data dimension and eliminate redundancy, and thus improve the computational efficiency. With the increases in the number of principal components extracted, the prediction
accuracy of the model increases accordingly.
Regarding the selection of the SVM parameters, the GS algorithm, the PSO algorithm and the GA algorithm are selected to optimize the parameters, which are applied to identify heat loss or heat
gain of water body in channels during winter periods. After the optimization of parameters, the recognition rate of model prediction algorithm is better. By comparing the differences between the
algorithms, the GA algorithm is more suitable for the SVM method used for the assessment of water heat balance state.
Aiming at the problem of heat budget of water body in the study channel in winter, considering water temperature change by means of environmental variables, a new approach is proposed to assess
the heat balance state by analyzing the observed data in the field. In view of the nonlinear change of water temperature, the RBF kernel function with better performance in classification is
selected. Regarding the sample learning, the SVM can quickly and accurately conduct classification. Thus, the SVM can used to effectively solve the identification problem of water heat exchange.
The changes of water temperature are crucial for the study of ice formation and evolution. The study of the heat balance state associated with flow water body and ice cover involves hydraulics,
thermodynamics, meteorology, engineering mechanics and other disciplines. When the prediction is about other water-ice problems, it is necessary to analyze the physical laws and causality of the
research content, and use machine learning to assist these prior knowledge to generate an appropriate solution. Based on the theory of this study, the simulation and prediction of water temperature
in winter can be carried out, or the modified multi-classification study regarding the ice-water mechanics can be conducted by considering different ice conditions, so as to provide the operation
guidelines for a river system under an ice-covered flow condition in winter. This will also be our future research topic. The method proposed in this study can be promising for controlling ice
problems in channels for water transfer in cold regions.
Author Contributions
Conceptualization, T.C., J.W. and J.S.; methodology, T.C.; validation, H.Z., Z.H. and M.H.; formal analysis, T.C. and Z.L.; investigation, J.W. and J.S.; writing—original draft preparation, T.C.;
writing—review and editing, J.S.; funding acquisition, J.W. All authors have read and agreed to the published version of the manuscript.
This research was funded by the National Natural Science Foundation of China, grant number 51879065; the National Key Research and Development Program of China, grant number 2018YFC1508401.
We thank the National Natural Science Foundation of China, grant number 51879065; the National Key Research and Development Program of China, grant number 2018YFC1508401. The authors are grateful for
the financial support.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 5. (a) the PCA-GS-SVM classification; (b) parameter optimization using the GS method; (c) the PCA-PSO-SVM classification; (d) the fitness curve of the PSO algorithm; (e) the PCA-GA-SVM
classification; (f) the fitness curve of the GA algorithm.
Parameter Interpretation Parameter Interpretation
ρ[w] mass density of water c[p] specific heat capacity of water
T[w] water temperature Q[w] flow discharge of water
A[w] flow cross-sectional area of the channel E[x] diffusion coefficient
B[w] width of water surface channel cross-section φ[wa] water—air heat flux
N[i] the ratio of the length of the water surface of the cross-section covered by ice cover to the width of the water surface of the same cross-section φ[wi] water—ice cover heat flux
Component 1 2 3 4 5 6 7
cumulative contribution rate 33% 52% 68% 77% 83% 89% 93%
Parameter PCA-GS-SVM PCA-PSO-SVM PCA-GA-SVM
Iteration times 276 286 291
C 0.11 0.10 0.13
γ 48.50 54.34 53.20
Execution time (ms) 83,139 83,085 86,598
Cross-validation rate 71.83% 72.02% 71.46%
Number of boundary support vectors 363 380 349
Number of support vectors 385 398 374
Number of classification predictions 89 89 89
Correct classification number 63 63 63
Classification accuracy 70.79% 70.79% 70.79%
Parameter PCA-GS-SVM PCA-PSO-SVM PCA-GA-SVM
Iteration times 3893 2146 950
C 48.50 19.64 7.82
γ 0.57 1.61 1.71
Execution time (ms) 87,288 96,072 87,401
Cross-validation rate 72.02% 71.08% 72.02%
Number of boundary support vectors 260 244 257
Number of support vectors 293 286 291
Number of classification predictions 89 89 89
Correct classification number 64 65 65
Classification accuracy 71.91% 73.03% 73.03%
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© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:
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MDPI and ACS Style
Cheng, T.; Wang, J.; Sui, J.; Zhao, H.; Hao, Z.; Huang, M.; Li, Z. A New Approach for Assessing Heat Balance State along a Water Transfer Channel during Winter Periods. Water 2022, 14, 3269. https://
AMA Style
Cheng T, Wang J, Sui J, Zhao H, Hao Z, Huang M, Li Z. A New Approach for Assessing Heat Balance State along a Water Transfer Channel during Winter Periods. Water. 2022; 14(20):3269. https://doi.org/
Chicago/Turabian Style
Cheng, Tiejie, Jun Wang, Jueyi Sui, Haijing Zhao, Zejia Hao, Minghai Huang, and Zhicong Li. 2022. "A New Approach for Assessing Heat Balance State along a Water Transfer Channel during Winter
Periods" Water 14, no. 20: 3269. https://doi.org/10.3390/w14203269
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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Metric System
Slide 1
Metric System
Scientific Measurements
Slide 2
Metric System
Developed by the French in the late 1700’s.
Based on powers of ten, so it is very easy to use.
Used by almost every country in the world, with the notable exception of the USA.
Especially used by scientists.
Abbreviated SI, which is French for Systeme International.
Slide 3
Metric Prefixes
Regardless of the unit, the entire metric system uses the same prefixes.
Common prefixes are:
kilo = 1000
centi = 1/100th
milli = 1/1000th
1 meter = 100 centimeters= 1000 millimeters
Slide 4
Length is the distance between two points.
The SI base unit for length is the meter.
We use rulers or meter sticks to find the length of objects.
Slide 5
Mass is the amount of matter that makes up an object.
A golf ball and a ping pong ball are the same size, but the golf ball has a lot more matter in it. So the golf ball will have more mass.
The SI unit for mass is the gram.
A paper clip has a mass of about one gram.
The mass of an object will not change unless we add or subtract matter from it.
Slide 6
Measuring Mass
We will use a triple beam balance scale to measure mass.
Gravity pulls equally on both sides of a balance scale, so you will get the same mass no matter what planet you are on.
Slide 7
Weight is a measure of the force of gravity on an object.
Your weight can change depending on the force of gravity. The gravity will change depending on the planet you are on.
The SI unit for weight is the Newton (N).
The English unit for weight is the pound.
Slide 8
Gravity is the force of attraction between any two objects with mass.
The force depends on two things:
more distance = less gravity = less weight
less distance = more gravity = more weight
more mass = more gravity = more weight
less mass = less gravity = less weight
Slide 9
Weight and Mass
Notice that Jill’s mass never changes. Her mother will not allow us to take parts off her, or add parts to her, so her mass stays the same. Jill is 30kg of little girl no matter where she goes!
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According to Wikipedia, “[t]he decibel (dB) is a logarithmic unit used to express the ratio between two values of a physical quantity, often power or intensity.” We usually hear it applied to sound.
It is also applied to radio communications. For instance, your phone usually has a setting that gives the signal strength in decibels.
In practice, this means there needs to be two measured values in order to have a ratio. The numerator of the ratio is the measured value. The denominator is a reference value, a standard number we
already know. For measuring sound this [latex]p_0[/latex], or the pressure at 20 micropascals in air. So when measuring sound,
[latex]L_p = 20\log_{10} \frac{p}{p_0}\,\text{dB.} [/latex]
But this definition implies I could apply it to any other physical quantities provided the base units were the same. So, being ridiculous, and using Earth as a reference value, we know the mass of
Jupiter is,
[latex]M_j = \log_{10} \frac{M_J}{M_\oplus} = \log_{10} \frac{1.898\times 10^{27}\,\text{kg}}{5.972\times 10^{24}\,\text{kg}} \approx 2.5\,\text{dB.}[/latex]
Image by Kevin Gill / Flickr.
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ABCD is a parallelogram. The position vectors of A and C are respectively, $3\
Hint: First, we need to find the midpoint of the diagonal DB, by considering the fact that both diagonals will have the same midpoint. OM and OC vectors will be with reference to the origin. Then we
will obtain the projection of the vector by using the dot product of the vectors, further divided by the magnitude of OC vector.
Complete step-by-step answer:
In the question, position vectors of A and C are given as:
$3\hat i + 3\hat j + 5\hat k$ and $\hat i - 5\hat j - 5\hat k$.
As we know that, in any parallelogram the midpoints of both the diagonals are the same. Thus, given M as the midpoint of DB will imply that M will also be the midpoint of AC. Midpoint is the mean
value of two vectors.
O\vec M = \dfrac{{O\vec A + O\vec C}}{2} \\
= \dfrac{{(3\hat i + 3\hat j + 5\hat k) + \hat i - 5\hat j - 5\hat k}}{2} \\
= 2\hat i - \hat j \\
So, the position vector of M is $2\hat i - \hat j$.
Now, we will obtain the magnitude of projection of vector OM on vector C, by dividing the magnitude of the dot product of vector Om and vector OC by magnitude of vector OC.
Thus, Magnitude of the projection is,
\dfrac{{\left| {O\vec M.O\vec C} \right|}}{{\left| {O\vec C} \right|}} \\
= \dfrac{{\left| {2 + 5} \right|}}{{\left| {\sqrt {1 + 25 + 25} } \right|}} \\
= \dfrac{7}{{\sqrt {51} }} \\
In the above expression we found the magnitude of OC as $\sqrt {51} $(=\[\sqrt {1 + 25 + 25} \]). Also, for the projection, the angle between the vectors will be zero.
The magnitude of the projection will be $\dfrac{7}{{\sqrt {51} }}$.
Note: Dot product of the vectors is also termed as the inner product or scalar product. The vector projection of some vector b onto another vector a is in the same direction or in the opposite
direction if the scalar projection is negative as of a. In another way, it is also termed as the component of b in the direction of a.
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Mathematical modelling, programming & computer software
Periodical includes the following sections:
• Mathematical Modelling,
• Programming and Computer Software,
• Survey Articles,
• Short Notes,
• Personalia,
• Mathematical Activity.
The section «Mathematical Modelling» contains articles considering classical and non-classical models of mathematical physics; differential and integral equations of mathematical models of natural
science; methods of solution of ill-posed problems; variational calculus; mathematical theory of optimal control, numerical mathematics, calculus and arithmetic models.
The section «Programming and Computer Software» contains articles considering theoretical questions of programming, software programming and design languages; testing, verification and validation of
programs; e-technologies and software systems; means of development and support of e-libraries and electronic edition; visualization system and system of virtual external environment; architecture,
system and applied control programming of computers and supercomputers; parallel computational technologies; distributed computing and grid-technologies; technologies of resource protection of
distributed information computation systems.
The section «Survey Articles» contains review articles in the field of publication.
The section «Short Notes» contains reports by postgraduate students and masters in the field of publication.
The section «Personalia» contains biographical articles about outstanding scientists.
The section «Mathematical Activity» announces and highlights the work of the mathematical community (conferences, seminars, workshops, etc.).
The size of the original articles shouldn’t exceed 12 pages in the sections «Mathematical Modelling» and «Programming and Computer Software», as for the «Survey Articles» they shouldn’t exceed 24
pages. The size of the reports shouldn’t exceed 5 pages.
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How to measure the width of curtains ?
Grommet Top
This is one of our most popular headings. Grommet top curtains feature 1.9 inch eyelet holes spaced evenly through out the top of the curtain. When closed, Grommet eyelet curtains give a beautiful
wave look. They are suitable for curtain poles only and should only be paired with poles with at least 1.5 inch in diameter.
Calculate the Recommended Curtain Width
If you already have a curtain pole, measure its full length. We recommend a fullness ratio of 1.5x-2xto create the waves and folds when the curtains are closed.
Fullness Ratio= Curtain width / Curtain pole length
If you do not have a pole yet, follow the steps below to measure the desired pole length and then calculate the recommended curtain width before you order curtains from us.
Referring to the diagram
1. Obtain Measurement A
Measure the full width of your window, if it is in a recess, measure the full width of the recess.
2. Obtain Measurement B
If you have a recessed window, you have the option of fitting the curtains inside the recess or above. If you want your curtains above the window, you need to decide how far your curtains will
over-hang the wall on each side of the window. This would normally be around 8 inches each side and enables the curtains to open wider. More daylight can enter the room and gives room for fixture on
the end of the curtain pole.
3. Obtain Measurement C
Simply add up Measurements A+B+B from steps 1 and 2, this will give you the desired curtain pole length. Please do not include curtain pole finials for measurement C.
4. Obtain the recommended curtain width
Finally, multiply C by 1.5-2, depending on how full you would like your curtains to look when closed, you will have the recommended curtain width.
Recommended Curtain Width = 1.5 to 2 x Measurement C
Choose Your Desired Curtain Length
Sill Length
Measure the distance between the top of the pole to the bottom of the window and then add 1.2", this should be the ideal length for the curtain if you are looking to achieve sill length.
Below Sill
Measure the distance between the top of the pole to the bottom of the window and then add 7.2”, this should be the ideal length for the curtain if you are looking to achieve below sill length (about
6“ below your window sill).
Floor Length
Measure the distance between the top of the pole to the floor and then add 0.2", this should be the ideal length for the curtain if you are looking to achieve floor length (about 1” above the floor).
Tab Top
Tab top curtains feature fabric loops at the top which the curtain pole passes through. Tab top can be used to showcase the pattern of the curtain as the curtain fabric is flat when the curtains are
shut. Please note they are suitable for curtain poles only.
Calculate the Recommended Curtain Width
If you already have a curtain pole, measure its full length. We recommend a fullness ratio of 1.5x-2x because we think most of the time curtains with tab top heading do not need too much stacks and
gathering at the top.
For example, if the pole’s full length is 80 inches, the single panel curtain’s width needs to be 40-60 inches if you are looking to have 2 panels of curtains.
Fullness Ratio= Curtain width / Curtain pole length
If you do not have a pole yet, follow the steps below to measure the desired pole length and then calculate the recommended curtain width before you order curtains from us.
Referring to the diagram
1. Obtain Measurement A
Measure the full width of your window, if it is in a recess, measure the full width of the recess.
2. Obtain Measurement B
If you have a recessed window, you have the option of fitting the curtains inside the recess or above. If you want your curtains above the window, you need to decide how far your curtains will
over-hang the wall on each side of the window. This would normally be around 20cm (8 inches) each side and enables the curtains to open wider. More daylight can enter the room and gives room for
fixture on the end of the curtain pole.
3. Obtain Measurement C
Simply add up Measurements A+B+B from steps 1 and 2, this will give you the desired curtain pole length. Please do not include curtain pole finials for measurement C.
4. Obtain the recommended curtain width
Finally, multiply C by 1.5-2, depending on how full you would like your curtains to look when closed, you will have the recommended curtain width.
Recommended Curtain Width = 1.5 to 2 x Measurement C
Choose Your Desired Curtain Length
Sill Length
Measure the distance between the top of the pole to the bottom of the window and then add 1.2", this should be the ideal length for the curtain if you are looking to achieve sill length.
Below Sill
Measure the distance between the top of the pole to the bottom of the window and then add 7.2”, this should be the ideal length for the curtain if you are looking to achieve below sill length (about
6“ below your window sill).
Floor Length
Measure the distance between the top of the pole to the floor and then add 0.2", this should be the ideal length for the curtain if you are looking to achieve floor length (about 1” above the floor).
Grommet Top & Bottom
This is our most windproof curtain work and highly recommended for outdoor use. They also have 1.9-inch eyelets like our grommet top curtains. They are suitable for curtain poles only and should only
be paired with poles of at least a 1.5 inch diameter. Grommet top and bottom curtains can block wind and rain real good as they are supported by two poles at the same time. Whatsmore, privacy is also
guaranteed when these curtains are closed!
Calculate the Recommended Curtain Width
If you already have a curtain pole, measure its full length. We recommend a fullness ratio of 1.5x-2xto create the waves and folds when the curtains are closed.
Fullness Ratio= Curtain width / Curtain pole length
If you do not have a pole yet, follow the steps below to measure the desired pole length and then calculate the recommended curtain width before you order curtains from us.
Referring to the diagram
1. Obtain Measurement A
Measure the full width of your window, if it is in a recess, measure the full width of the recess.
2. Obtain Measurement B
If you have a recessed window, you have the option of fitting the curtains inside the recess or above. If you want your curtains above the window, you need to decide how far your curtains will
over-hang the wall on each side of the window. This would normally be around 8 inches each side and enables the curtains to open wider. More daylight can enter the room and gives room for fixture on
the end of the curtain pole.
3. Obtain Measurement C
Simply add up Measurements A+B+B from steps 1 and 2, this will give you the desired curtain pole length. Please do not include curtain pole finials for measurement C.
4. Obtain the recommended curtain width
Finally, multiply C by 1.5-2, depending on how full you would like your curtains to look when closed, you will have the recommended curtain width.
Recommended Curtain Width = 1.5 to 2 x Measurement C
Calculate Curtain Length
Measure the distance between the top of the upper pole and the bottom of the lower pole and then add 4 inches, this should be the ideal length for your double grommet curtains.
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How to Calculate the Aggregate Adjustment for the Taxes in an Escrow Account | Sapling
The aggregate adjustment affects the amount of funds that are held in a mortgage borrower's escrow account at closing. This amount appears on Line 1007 of a HUD-1 standard real estate settlement
statement. It is typically a credit, or negative amount, offsetting the individual amounts that are shown for escrows for taxes, insurance and other payments on Lines 1001 through 1006 of the HUD-1.
Escrows usually apply to both taxes and insurance; there is unlikely to be an aggregate adjustment if only taxes are escrowed.
Step 1
Make rows on a spreadsheet for each of the first 12 months, starting with the month the first payment is due.
Step 2
In a separate column, list all payment amounts that will be paid out of the escrow account in the months they are due.
Step 3
Divide the total amount of payments from Step 2 by 12.
Step 4
Enter the figure from Step 3 in another column for each month as the tentative amount that will be paid into escrow monthly.
Step 5
Calculate the initial balance for each month. The initial balance is the initial balance from the preceding month (zero for the first month) plus the escrow deposit minus the month's payments.
Step 6
Find the amount of additional funds that would be needed to bring the lowest initial balance up to zero.
Step 7
Determine the "cushion," or lowest positive balance, that the lender requires for the account. Under federal rules, this cannot be more than one sixth of the total payments in Step 2, except that
mortgage insurance paid monthly cannot be included in the total payments for this purpose.
Step 8
Compute the initial escrow payment under aggregate adjustment rules by adding together the amounts from Steps 6 and 7.
Step 9
Subtract the amount from Step 8 from the sum of the escrow amounts using single-item accounting on Lines 1001 through 1006 of the HUD-1.
Step 10
Enter the amount from Step 9 as a credit, or negative amount, on Line 1007, Aggregate Adjustment. However, if the calculation from Step 9 results in a negative number or zero, enter zero on Line
You are not allowed to round dollar amounts.
If there is no aggregate escrow adjustment, make sure you put zero on Line 1007, since you are required by law to disclose the aggregate escrow adjustment.
The Federal Reserve Bank of Philadelphia reports that over-reliance on automated systems often causes errors in escrow calculations.
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Estimation in Errors-In-Variables Models.
Date of Submission
Institute Name (Publisher)
Indian Statistical Institute
Document Type
Doctoral Thesis
Degree Name
Doctor of Philosophy
Economic Research Unit (ERU-Kolkata)
Bhattacharya, Nikhilesh (ERU-Kolkata; ISI)
Abstract (Summary of the Work)
In many econometric investigations, the 'errors-in variables' (EIVs) are not negligible (Morgenstern, 1963). Examination of 25 series relating to national accounts by Langaskens and Rijekeghan (1974)
showed that the standard deviations of the errors ranged from 5 to 77 per cent of the average value of the corresponding variable. Such errors may vitiate least-squares (LS) estimation of regression
coefficients (Johnaton, 1972). The well-known methods (ML; IV, including grouping method) proposed for handling classical EIV model (EMM) in regression analysis muffer from serious linitations. Same
of them make strong distributional assumptions about the errors (and the regressors) and/or assume prier knowledge about the values of the error variancen; othera need auxiliary variables called
inatrumental-variables (IVs) which are supposed to be uncorrelated with the error tems, but strongly correlated with the true regressors. The IVs are thus not always handy and, in any case, one can
never check the assumptions.y1 = a + 8 X i- 1, 2, .., n .(0.1) where a and 8 are paraneters to be estimated; e, is the disturbance term distributed normally wi th mean zero and variance o for all and
X1, and Yi, are norobservable true values of the regressor and the coy of CV 2 regres and respectively. The e's are assumed to be independent of X'e where X is stochastic. The observed values x, and
y, are written as (0.2) where u, and vị are the EIVe which are independent of each other and of the true values Xi, and Yi,. u, s and v,s have neans zero and variances o and o respectively for all
i. Fori- 1, 2, ..., n, we assume that (X, Y, u, v, ) are i.i.d. random variables.one finds that 'ordinary least aquares' (OLS) regression of y on x gives an inconsistent estimator of 8 essentially
because eov(xi, wi)0 (Johns ton, 1972, p.262). Extension of this model to more than one and to regressor is obvious. Other important extensions allow u, be correlated or the distribution of ui, to
depend on the valIhe of Xi,. Various alternative methods of estimation have been suggested by previous researchers. These are based on different sets of assump- tions. Thas, some assume X to be
stochastic while others do not. Chapter 1 makes a critical survey of the different assumptions made in the literature on the distribution of errors and of the regressor X and reviews the different
methods of estimation śuggested so far. There are, of course, some models which oan not be fully identified at all (vide Section 1.6 of Chapter 1 see also Appendix 4.2 of Chapter 4). It may be
mentioned here that some good review articles on EVM already exist in the literature (Durbin, 1954; Madansky, 1959; Cochran, 1968; Moran, 1971; Pal, 1980a). Among other things, this chapter diecusses
how one can ob tain consistent estimators of 6 if (i) one has prior knowledge about the value of the error variances or of their ratio or if (ii) IVare available. Introduction of lagged values of
regressors/ regressand may also be helpful in finding consistent estimates of the parameters. Sometimes in the laboratory experiments repeated meagure- ments are available for the same value of the
variable. This may help in finding oónsistent estimates. The problem becomes more difficult if instead of one relation we have many relations in the model, but the variables are affected by EIV
apart from economists, sociologists have long beon applying such simul taneous equations models in path analysis and multiple indicator analysis.
ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843331
Control Number
Recommended Citation
Pal, Manoranjan Dr., "Estimation in Errors-In-Variables Models." (1982). Doctoral Theses. 295.
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How To Interpret CogAT Scores
BrianAJackson/iStock/Getty Images
The Cognitive Abilities Test, also known as the CogAT or CAT, is an exam administered to K-12 students to assess their abilities in three areas considered important in determining future academic
success: verbal, nonverbal and quantitative reasoning. This test is most commonly used by schools to determine placement for gifted and talented programs. CogAT scores are reported in terms of
percentiles and stanines rather than IQ, which is a better way of assessing where a student stands in relation to his peers. The score report lists four percentiles — one for each section, and one
for all three combined — ranging from 1 to 100, as well as four stanines, which are normalized standard score scales, ranging from 1 to 9, with 5 being the average.
Review the Percentiles
Step 1
Locate the number indicating the percentile in which your child was placed for verbal reasoning. For example, if your score report says that he was placed in the 98th percentile for verbal reasoning,
it means your child outperformed 98 percent of his peers and is in the top 2 percent for his age group.
Step 2
Locate the number indicating the percentile in which your child was placed for the nonverbal reasoning.
Step 3
Locate the number indicating the percentile in which your child was placed for the quantitative reasoning.
Step 4
Locate the number indicating the composite percentile for all three sections. This number combines all three scores and indicates where your child stands in comparison to other students who took the
test. Thus, a composite percentile score of 98 indicates that, overall, your child did better on all three sections combined than 98 percent of other students in her age group.
Review the Stanines
Step 1
Locate the number indicating your child's stanine for verbal reasoning. For example, a stanine of 9 corresponds to a percentile range of 96 to 99; a stanine of 8 corresponds to a percentile range of
89 to 95, and so on. A stanine above 5 means that your child scored above average on that section.
Step 2
Locate the number indicating your child's stanine for nonverbal reasoning.
Step 3
Locate the number indicating your child's stanine for quantitative reasoning.
TL;DR (Too Long; Didn't Read)
Generally, percentiles are a more descriptive way of understanding how your child did on the exam because they show how he ranked against his entire group of peers. Stanines are more confusing but
correspond directly to the percentile score.
A bar graph of the student's scores also appears on the score report and is a good way to visualize the numbers.
Additional information about your child's profile can be found at the Riverside Publishing website by typing in your child's profile code.
It is important to understand that the CogAT, like many IQ and cognitive tests administered to children, is an imperfect assessment measure that can vary greatly depending on a variety of external
factors. Thus, while these scores can be important in deciding placement, they should not be taken as the sole measurement of your child's abilities and skills.
Cite This Article
Paley, Irina. "How To Interpret CogAT Scores" sciencing.com, https://www.sciencing.com/interpret-cogat-scores-5931022/. 24 April 2017.
Paley, Irina. (2017, April 24). How To Interpret CogAT Scores. sciencing.com. Retrieved from https://www.sciencing.com/interpret-cogat-scores-5931022/
Paley, Irina. How To Interpret CogAT Scores last modified March 24, 2022. https://www.sciencing.com/interpret-cogat-scores-5931022/
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GNU Octave: Diagonal Matrix Functions
21.3.1 Diagonal Matrix Functions
inv and pinv can be applied to a diagonal matrix, yielding again a diagonal matrix. det will use an efficient straightforward calculation when given a diagonal matrix, as well as cond. The following
mapper functions can be applied to a diagonal matrix without converting it to a full one: abs, real, imag, conj, sqrt. A diagonal matrix can also be returned from the balance and svd functions. The
sparse function will convert a diagonal matrix efficiently to a sparse matrix.
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Systems of Equations | Algebra | Achievable CLT
Systems Questions
Systems is a term used to describe a combination of two or more equations where the task is typically to solve for one or more of the variables. On the CLT, systems will typically involve two
equations; in rare cases, on hard problems located near the end of the Quantitative Reasoning section, the test will present three equations and ask the test taker to solve for one of the variables.
Systems questions often present the student with multiple ways of solving them, including substitution, elimination, and graphing. We’ll explore these below.
Approach Question
How many points of intersection do the following equations have?
A. 0
B. 1
C. 2
D. 3
This question is a gold mine of possibility; there are at least three ways to approach it, each of which touches on key concepts and skills for the CLT.
Exploring the systems of equations aspect of this problem first, we can observe the important detail that is isolated in both equations; by the transitive property, which tells us that two quantities
equal to a third quantity must also be equal to each other, we knows we can set the right sides of the two equations equal to each other. Another way to say this is that we can substitute either
right side for in the other equation. Since y is equal to both quantities on the right, either of those quantities can always be exchanged for . This is the magic of the equals sign!
Setting the two right sides equal yields . As we note in the quadratics lesson, we encounter here the important principle of always setting quadratic expressions equal to zero. We want to retain the
positive coefficient in front of the terms, so we’ll subtract 2x from both sides and add 4 to both sides. This gives us .
At this point, we have two choices: factoring or quadratic formula. If you are comfortable factoring a quadratic with a leading coefficient (there are multiple methods), try it now. Here’s the
answer: . It is clear from this that there are two distinct solutions: and . The answer is C.
Because this sort of factoring can be challenging and does not occur very frequently on the CLT, we recommend honing your skills in using the quadratic formula so you always have the option to solve
this way. Using the quadratic formula, we create a numerator of and a denominator of .
That’s kind of a mess, but there’s good news: we don’t need to solve all the way! To identify the number of solutions when a quadratic is involved, we’ll use the discriminant: the portion of the
quadratic formula under the radical, or . If that solution is positive, there are two solutions to the equation; if it’s zero, then there is one solution; if negative, there are no real solutions
(the solutions are imaginary or, put another way, the parabola does not intersect with the x-axis on the real number xy-coordinate plane). In this case, we know we have a positive number when is
squared; we also know that we’ll be adding to that number because the two negatives will cancel out. The value is , but again, we don’t need the actual value; we just need to know that the
discriminant is positive. We have confirmed answer C, 2 solutions.
There is a takeaway shortcut from this process: if a quadratic has a negative c term, that quadratic will always have two positive roots. Why is that? Think about the nature of : we know that is
greater than or equal to zero and that will be positive if the is negative. So we’re adding a positive number to a nonnegative number. The result must always be positive! Note: this inference only
holds true with a positive leading coefficient, so make sure that when you set the quadratic equal to zero, you create a positive term. If that term is negative, you simply need to multiply the
entire equation by .
We have encountered two of three plausible ways to solve this problem: after substitution, we had factoring or the quadratic formula. If you are confident in your graphing skills, you may want to try
a graphing approach to this problem. Graphing on the CLT can be of limited value because it must be done by hand, without a calculator. But this problem doesn’t necessarily require precise graphing;
it only requires visualizing the line and the parabola and considering their intersection. If you remember how the basic “parent” form of the parabola, , is graphed, you’ll know that it intersects
the origin and rises on both sides of the origin. Our parabola here is translated down seven units because of the . The on the front stretches the parabola on the y-axis, making it “skinnier”.
Meanwhile, the line has a positive y-intercept at (0,1). If you can draw this well or even visualize it in your mind, you can see that we don’t even need to consider the line’s slope because it must
intersect the parabola twice, once on the right of the y-intercept and once on the left. We have graphing-based confirmation: two points of intersection.
One final note: graph paper is included among the permitted options for scratch paper on the CLT, so we recommend practicing with graph paper to get used to using it in cases like this question.
Topics for Cross-Reference
Though most CLT systems involve two equations, you may encounter a system of three equations late in the CLT Quantitative Reasoning section. Typically, one of two pathways will be available. 1) There
may be an opportunity to investigate the possibilities using “guess and check.” For example, if one of the equations is and you know the variables represent integers, there are only three
possibilities: either is and is , or is and is , or is and is . You can plug these possibilities into the other equations to see what works. 2) One of the equations will only employ two of the three
variables unknown - say, and , but not . In this case, you should use elimination with the other two equations to reduce them to another equation involving and . Then you can treat what remains as a
normal system of two equations.
The other variation you may encounter is a system of inequalities rather than equations. Some of the same tools used for systems of equations will apply with inequalities as long you keep the
following caveats in mind:
1. Elimination will work with systems of inequalities, but substitution will not.
2. You can only perform elimination if the two inequality symbols are pointing in the same direction (don’t try it with one “less than” and one “greater than”).
3. Eliminate only by adding, not by subtracting. (Subtracting affects the second inequality as if you’re multiplying by , which would change the direction of the inequality. That’s some quicksand
you want to avoid!)
Flashcard Fodder
With elimination, you may want to note the following on flashcards:
1. If the same variable in each equation has the same sign, you should subtract the two equations (Same = Subtract).
2. If the variables have coefficients that are equal except they have opposite signs, you should add the equations to eliminate a variable.
Sample Questions
Difficulty 1
Which of the following coordinate points is a solution to the system of equations below?
A. (0,-1)
B. (1,-1)
C. (2,1)
D. (3,4)
The answer is B. This is an excellent chance to practice elimination, because the coefficients are equal but opposite. By simply adding the equations as they stand, we will eliminate the variables
and be left with . So . We could plug 1 back into either equation and find that , but use the UnCLES and be smart: if , there’s only one possible answer already!
Difficulty 2
Which of the following is a solution to the system of inequalities below?
A. (0,5)
B. (1,2)
C. (2,1)
D. (0,2)
The answer is C. Systems of inequalities are a little different than systems of equations (see discussion under Variations in this lesson). We could work to switch the signs of inequality, which
would involve multiplying through that inequality by . But even then, the coefficients would not be the same either for or for . Much better to simply plug in the values.
Choice A doesn’t work for the first inequality (10 is not less than 5), so we can eliminate it without checking the second inequality. Choice B also doesn’t work for the first one (tricky, but is not
less than ). Choice C gives us for the first inequality, so we proceed to the second. That yields 5>4, so both are true; this is the answer. Choice D works for the first inequality (), but not for
the second ( is not greater than ).
Difficulty 3
Which of the following coordinate points is a solution to both the equations below?
A. (-8,-6)
B. (-2,-3/2)
C. (0,0)
D. (4,4)
The answer is A. We certainly have the option of backsolving (plugging in the answers) on this question, and that will be the fastest approach for many students. But let’s explore the textbook
approach here. Since y is isolated in the first equation, let’s substitute its equivalent value, , into the second equation. Then we’ll multiply through by to get rid of the negative fractional
coefficient before . We now have . Bringing all the terms to one side leaves , and that quadratic factors so that we end up with . So and . But only the first of these is present in the answer
choices. If we want to make sure, we can plug answer choice A back into both equations, using for , and in both cases .
If we do choose to backsolve, it would be sense to start with answer choice , since is a simple number to plug in. works for the first equation but not the second. We could then move to choice
because it has two positive integers, but that choice works for neither equation. We would likely try next to avoid dealing with the fraction in choice , and careful plugging in of the numbers in
choice confirms our answer.
Difficulty 4
How many solutions does the following system of equations have in the real number system?
A. 0
B. 1
C. 2
D. 3
The answer is A. This problem calls for finding the quadratic’s discriminant once again, but with a different result from our approach question. We start as usual by setting the right sides equal to
each other; however, we should eliminate the negative in front of the at the earliest opportunity, so let’s multiply the resulting equation by . We now have . Moving the terms to one side, we get .
That doesn’t look like it will factor, so let’s employ the quadratic formula or, more specifically, the discriminant. here leaves , which is . A negative result for the discriminant means zero
solutions in the real number system.
Difficulty 5
Which of the following is the x-coordinate of a point (x,y,z) that would satisfy all three equations below?
A. -8
B. -4
C. -1
D. 0
The answer is C. This question looks intimidating, but it helps to start with the most accessible equation. The last equation might look extremely simple, but remember that taking the square root of
both sides requires adding a “plus/minus” symbol to show that, in this case, could equal either or . The best approach is to plug in both of these options and see which works. If , then the first
equation is now , which simplifies to . We can combine that with the other equation containing and ; if you follow that process, you’ll find that x does not come out as a whole number and therefore
doesn’t match the answer choices.
By elimination, then, we now try . That makes the first equation , or . Combined with the equation , we find that .
Although it might be time-consuming, you can also do this problem in reverse. To abbreviate the process, we’ll go right to plugging in the right answer. If , then by the second equation . Using those
two values in the first equation yields , which makes the last equation true.
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Binary search¶
Binary search is a method that allows for quicker search of something by splitting the search interval into two. Its most common application is searching values in sorted arrays, however the
splitting idea is crucial in many other typical tasks.
Search in sorted arrays¶
The most typical problem that leads to the binary search is as follows. You're given a sorted array $A_0 \leq A_1 \leq \dots \leq A_{n-1}$, check if $k$ is present within the sequence. The simplest
solution would be to check every element one by one and compare it with $k$ (a so-called linear search). This approach works in $O(n)$, but doesn't utilize the fact that the array is sorted.
Binary search of the value $7$ in an array.
The image by AlwaysAngry is distributed under CC BY-SA 4.0 license.
Now assume that we know two indices $L < R$ such that $A_L \leq k \leq A_R$. Because the array is sorted, we can deduce that $k$ either occurs among $A_L, A_{L+1}, \dots, A_R$ or doesn't occur in the
array at all. If we pick an arbitrary index $M$ such that $L < M < R$ and check whether $k$ is less or greater than $A_M$. We have two possible cases:
1. $A_L \leq k \leq A_M$. In this case, we reduce the problem from $[L, R]$ to $[L, M]$;
2. $A_M \leq k \leq A_R$. In this case, we reduce the problem from $[L, R]$ to $[M, R]$.
When it is impossible to pick $M$, that is, when $R = L + 1$, we directly compare $k$ with $A_L$ and $A_R$. Otherwise we would want to pick $M$ in such manner that it reduces the active segment to a
single element as quickly as possible in the worst case.
Since in the worst case we will always reduce to larger segment of $[L, M]$ and $[M, R]$. Thus, in the worst case scenario the reduction would be from $R-L$ to $\max(M-L, R-M)$. To minimize this
value, we should pick $M \approx \frac{L+R}{2}$, then
$$ M-L \approx \frac{R-L}{2} \approx R-M. $$
In other words, from the worst-case scenario perspective it is optimal to always pick $M$ in the middle of $[L, R]$ and split it in half. Thus, the active segment halves on each step until it becomes
of size $1$. So, if the process needs $h$ steps, in the end it reduces the difference between $R$ and $L$ from $R-L$ to $\frac{R-L}{2^h} \approx 1$, giving us the equation $2^h \approx R-L$.
Taking $\log_2$ on both sides, we get $h \approx \log_2(R-L) \in O(\log n)$.
Logarithmic number of steps is drastically better than that of linear search. For example, for $n \approx 2^{20} \approx 10^6$ you'd need to make approximately a million operations for linear search,
but only around $20$ operations with the binary search.
Lower bound and upper bound¶
It is often convenient to find the position of the first element that is greater or equal than $k$ (called the lower bound of $k$ in the array) or the position of the first element that is greater
than $k$ (called the upper bound of $k$) rather than the exact position of the element.
Together, lower and upper bounds produce a possibly empty half-interval of the array elements that are equal to $k$. To check whether $k$ is present in the array it's enough to find its lower bound
and check if the corresponding element equates to $k$.
The explanation above provides a rough description of the algorithm. For the implementation details, we'd need to be more precise.
We will maintain a pair $L < R$ such that $A_L \leq k < A_R$. Meaning that the active search interval is $[L, R)$. We use half-interval here instead of a segment $[L, R]$ as it turns out to require
less corner case work.
When $R = L+1$, we can deduce from definitions above that $R$ is the upper bound of $k$. It is convenient to initialize $R$ with past-the-end index, that is $R=n$ and $L$ with before-the-beginning
index, that is $L=-1$. It is fine as long as we never evaluate $A_L$ and $A_R$ in our algorithm directly, formally treating it as $A_L = -\infty$ and $A_R = +\infty$.
Finally, to be specific about the value of $M$ we pick, we will stick with $M = \lfloor \frac{L+R}{2} \rfloor$.
Then the implementation could look like this:
... // a sorted array is stored as a[0], a[1], ..., a[n-1]
int l = -1, r = n;
while (r - l > 1) {
int m = (l + r) / 2;
if (k < a[m]) {
r = m; // a[l] <= k < a[m] <= a[r]
} else {
l = m; // a[l] <= a[m] <= k < a[r]
During the execution of the algorithm, we never evaluate neither $A_L$ nor $A_R$, as $L < M < R$. In the end, $L$ will be the index of the last element that is not greater than $k$ (or $-1$ if there
is no such element) and $R$ will be the index of the first element larger than $k$ (or $n$ if there is no such element).
Note. Calculating m as m = (r + l) / 2 can lead to overflow if l and r are two positive integers, and this error lived about 9 years in JDK as described in the blogpost. Some alternative approaches
include e.g. writing m = l + (r - l) / 2 which always works for positive integer l and r, but might still overflow if l is a negative number. If you use C++20, it offers an alternative solution in
the form of m = std::midpoint(l, r) which always works correctly.
Search on arbitrary predicate¶
Let $f : \{0,1,\dots, n-1\} \to \{0, 1\}$ be a boolean function defined on $0,1,\dots,n-1$ such that it is monotonously increasing, that is
$$ f(0) \leq f(1) \leq \dots \leq f(n-1). $$
The binary search, the way it is described above, finds the partition of the array by the predicate $f(M)$, holding the boolean value of $k < A_M$ expression. It is possible to use arbitrary
monotonous predicate instead of $k < A_M$. It is particularly useful when the computation of $f(k)$ is requires too much time to actually compute it for every possible value. In other words, binary
search finds the unique index $L$ such that $f(L) = 0$ and $f(R)=f(L+1)=1$ if such a transition point exists, or gives us $L = n-1$ if $f(0) = \dots = f(n-1) = 0$ or $L = -1$ if $f(0) = \dots = f
(n-1) = 1$.
Proof of correctness supposing a transition point exists, that is $f(0)=0$ and $f(n-1)=1$: The implementation maintaints the loop invariant $f(l)=0, f(r)=1$. When $r - l > 1$, the choice of $m$ means
$r-l$ will always decrease. The loop terminates when $r - l = 1$, giving us our desired transition point.
... // f(i) is a boolean function such that f(0) <= ... <= f(n-1)
int l = -1, r = n;
while (r - l > 1) {
int m = (l + r) / 2;
if (f(m)) {
r = m; // 0 = f(l) < f(m) = 1
} else {
l = m; // 0 = f(m) < f(r) = 1
Binary search on the answer¶
Such situation often occurs when we're asked to compute some value, but we're only capable of checking whether this value is at least $i$. For example, you're given an array $a_1,\dots,a_n$ and
you're asked to find the maximum floored average sum
$$ \left \lfloor \frac{a_l + a_{l+1} + \dots + a_r}{r-l+1} \right\rfloor $$
among all possible pairs of $l,r$ such that $r-l \geq x$. One of simple ways to solve this problem is to check whether the answer is at least $\lambda$, that is if there is a pair $l, r$ such that
the following is true:
$$ \frac{a_l + a_{l+1} + \dots + a_r}{r-l+1} \geq \lambda. $$
Equivalently, it rewrites as
$$ (a_l - \lambda) + (a_{l+1} - \lambda) + \dots + (a_r - \lambda) \geq 0, $$
so now we need to check whether there is a subarray of a new array $a_i - \lambda$ of length at least $x+1$ with non-negative sum, which is doable with some prefix sums.
Continuous search¶
Let $f : \mathbb R \to \mathbb R$ be a real-valued function that is continuous on a segment $[L, R]$.
Without loss of generality assume that $f(L) \leq f(R)$. From intermediate value theorem it follows that for any $y \in [f(L), f(R)]$ there is $x \in [L, R]$ such that $f(x) = y$. Note that, unlike
previous paragraphs, the function is not required to be monotonous.
The value $x$ could be approximated up to $\pm\delta$ in $O\left(\log \frac{R-L}{\delta}\right)$ time for any specific value of $\delta$. The idea is essentially the same, if we take $M \in (L, R)$
then we would be able to reduce the search interval to either $[L, M]$ or $[M, R]$ depending on whether $f(M)$ is larger than $y$. One common example here would be finding roots of odd-degree
For example, let $f(x)=x^3 + ax^2 + bx + c$. Then $f(L) \to -\infty$ and $f(R) \to +\infty$ with $L \to -\infty$ and $R \to +\infty$. Which means that it is always possible to find sufficiently small
$L$ and sufficiently large $R$ such that $f(L) < 0$ and $f(R) > 0$. Then, it is possible to find with binary search arbitrarily small interval containing $x$ such that $f(x)=0$.
Search with powers of 2¶
Another noteworthy way to do binary search is, instead of maintaining an active segment, to maintain the current pointer $i$ and the current power $k$. The pointer starts at $i=L$ and then on each
iteration one tests the predicate at point $i+2^k$. If the predicate is still $0$, the pointer is advanced from $i$ to $i+2^k$, otherwise it stays the same, then the power $k$ is decreased by $1$.
This paradigm is widely used in tasks around trees, such as finding lowest common ancestor of two vertices or finding an ancestor of a specific vertex that has a certain height. It could also be
adapted to e.g. find the $k$-th non-zero element in a Fenwick tree.
Practice Problems¶
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How do you find the roots for f(x) = 17x^15 + 41x^12 + 13x^3 - 10 using the fundamental theorem of algebra? | Socratic
How do you find the roots for #f(x) = 17x^15 + 41x^12 + 13x^3 - 10# using the fundamental theorem of algebra?
1 Answer
The FTOA does not help you find the zeros - it only tells you that this polynomial of degree $15$ has exactly $15$ Complex zeros counting multiplicity.
By roots, I will assume you mean zeros, i.e. values of $x$ for which $f \left(x\right) = 0$.
The so called fundamental theorem of algebra (FTOA) is neither fundamental nor a theorem of algebra, but what it does tell you is that any non-zero polynomial in one variable with Complex
coefficients has a zero in $\mathbb{C}$. That is, there is some Complex number ${x}_{1}$ such that $f \left({x}_{1}\right) = 0$.
A simple corollary of this - often stated as part of the FTOA - is that a polynomial in one variable of degree $n > 0$ with Complex coefficients has $n$ zeros counting multiplicity, all in $\mathbb
In our example, $f \left(x\right)$ is of degree $15$, so has $15$ zeros counting multiplicity, all in $\mathbb{C}$.
The FTOA does not help you actually find the zeros.
$\textcolor{w h i t e}{}$
What else can we find out about the zeros of this $f \left(x\right)$?
Note that the coefficients of $f \left(x\right)$ have only one change of sign, so only one positive Real zero.
Note $f \left(0\right) = - 10 < 0$ and $f \left(1\right) = 17 + 41 + 13 - 10 = 61 > 0$
So the positive Real zero is in $\left(0 , 1\right)$
$f \left(- x\right) = - 17 {x}^{5} + 41 {x}^{12} - 13 {x}^{3} - 10$
has two changes of sign, so $f \left(x\right)$ may have $0$ or $2$ negative Real zeros.
We find:
$f \left(- 1\right) = - 17 + 41 - 13 - 10 = 1 > 0$
So there is a Real zero in $\left(- 1 , 0\right)$ and another Real zero in $\left(- \infty , - 1\right)$.
Any other zeros will occur in Complex conjugate pairs, since all of the coefficients of $f \left(x\right)$ are Real.
Note that all of the degrees are multiples of $3$ so the zeros are all cube roots of the zeros of:
$g \left(t\right) = 17 {t}^{5} + 41 {t}^{4} + 13 t - 10$
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Abstract: The objectives of this research were to find out: (1) which learning model of the GI with CTL approach, TAI with CTL approach or conventional learning results in a better learning
achievement in mathematics; (2) which students attitudes toward mathematics of the positive, neutral, or negative types results in a better learning achievement in mathematics; (3) in each students
attitudes toward mathematics, which learning model of the GI with CTL approach, TAI with CTL approach or conventional learning results in a better learning achievement in mathematics; and (4) in each
learning model, which students attitudes toward mathematics of the positive, neutral, or negative types results in a better learning achievement in mathematics. This research used the quasi
experimental research method with the factorial design of 3x3. Its population was all of the students in Grade VII of State Junior High Schools in Ngawi regency. The samples of the research were
taken by using the stratified cluster random sampling technique. The data of the research were analyzed by using the unbalanced two-way analysis of variance at the significance level of 5%. The
results of this study showed that: (1) the GI and TAI with CTL approach learning models result in the same good learning achievement in mathematics, but both result in a better learning achievement
in mathematics than the conventional learning model; (2) the mathematics learning achievement with positive attitudes toward mathematics was better than that with neutral and negative attitudes
towards matematics, that with neutral attitude towards matematics was better than that with negative attitudes toward mathematics; (3) in each students attitudes toward mathematics type, the GI and
TAI with CTL approach learning models result in the same good learning achievement in Mathematics, but both result in a better learning achievement in mathematics than the conventional learning
model; (4) in each learning model, the mathematics learning achievement with positive attitudes toward mathematics was better than that with neutral and negative attitudes towards matematics, that
with neutral attitude towards matematics was better than that with negative attitudes toward mathematics.
Key words: learning model, GI, TAI, conventional, CTL approach, students attitudes toward mathematics.
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Closed Curve -- from Wolfram MathWorld
Closed Curve
In the plane, a closed curve is a curve with no endpoints and which completely encloses an area.
See also
Jordan Curve
Simple Curve
Explore with Wolfram|Alpha
Krantz, S. G. "Closed Curves." §2.1.2 in
Handbook of Complex Variables.
Boston, MA: Birkhäuser, pp. 19-20, 1999.
Referenced on Wolfram|Alpha
Closed Curve
Cite this as:
Weisstein, Eric W. "Closed Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClosedCurve.html
Subject classifications
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218 lbs to kg
To convert pounds (lbs) to kilograms (kg), you can use the following step-by-step instructions:
Step 1: Understand the conversion factor
1 lb is equal to approximately 0.45359237 kg. This is the conversion factor we will use to convert pounds to kilograms.
Step 2: Set up the conversion equation
To convert 218 lbs to kg, we can set up the equation as follows:
218 lbs * (0.45359237 kg/1 lb)
Step 3: Cancel out the units
In the equation, the unit “lbs” appears in both the numerator and denominator. This allows us to cancel out the unit and only keep the unit “kg” in the numerator.
Step 4: Perform the calculation
Now, we can multiply the numerical values:
218 * 0.45359237 = 98.88357466
Step 5: Round the answer (if necessary)
Since weight is typically rounded to a certain number of decimal places, you can round the answer to an appropriate number of decimal places. In this case, let’s round to two decimal places.
The final answer is approximately 98.88 kg.
Therefore, 218 lbs is approximately equal to 98.88 kg.
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Fixed-Structure Autopilot for a Passenger Jet
This example shows how to use slTuner and systune to tune the standard configuration of a longitudinal autopilot. We thank Professor D. Alazard from Institut Superieur de l'Aeronautique et de
l'Espace for providing the aircraft model and Professor Pierre Apkarian from ONERA for developing the example.
Aircraft Model and Autopilot Configuration
The longitudinal autopilot for a supersonic passenger jet flying at Mach 0.7 and 5000 ft is depicted in Figure 1. The autopilot main purpose is to follow vertical acceleration commands issued by the
pilot. The feedback structure consists of an inner loop controlling the pitch rate and an outer loop controlling the vertical acceleration . The autopilot also includes a feedforward component and a
reference model that specifies the desired response to a step command . Finally, the second-order roll-off filter
is used to attenuate noise and limit the control bandwidth as a safeguard against unmodeled dynamics. The tunable components are highlighted in orange.
Figure 1: Longitudinal Autopilot Configuration.
The aircraft model is a 5-state model, the state variables being the aerodynamic speed (m/s), the climb angle (rad), the angle of attack (rad), the pitch rate (rad/s), and the altitude (m). The
elevator deflection (rad) is used to control the vertical load factor . The open-loop dynamics include the oscillation with frequency and damping ratio = 1.7 (rad/s) and = 0.33, the phugoid mode =
0.64 (rad/s) and = 0.06, and the slow altitude mode = -0.0026.
load ConcordeData G
bode(G,{1e-3,1e2}), grid
title('Aircraft Model')
Note the zero at the origin in . Because of this zero, we cannot achieve zero steady-state error and must instead focus on the transient response to acceleration commands. Note that acceleration
commands are transient in nature so steady-state behavior is not a concern. This zero at the origin also precludes pure integral action so we use a pseudo-integrator with = 0.001.
Tuning Setup
When the control system is modeled in Simulink®, you can use the slTuner interface to quickly set up the tuning task. Open the Simulink model of the autopilot.
Configure the slTuner interface by listing the tuned blocks in the Simulink model (highlighted in orange). This automatically picks all Linear Analysis points in the model as points of interest for
analysis and tuning.
ST0 = slTuner('rct_concorde',{'Ki','Kp','Kq','Kf','RollOff'});
This also parameterizes each tuned block and initializes the block parameters based on their values in the Simulink model. Note that the four gains Ki,Kp,Kq,Kf are initialized to zero in this
example. By default the roll-off filter is parameterized as a generic second-order transfer function. To parameterize it as
create real parameters , build the transfer function shown above, and associate it with the RollOff block.
wn = realp('wn', 3); % natural frequency
zeta = realp('zeta',0.8); % damping
Fro = tf(wn^2,[1 2*zeta*wn wn^2]); % parametric transfer function
setBlockParam(ST0,'RollOff',Fro) % use Fro to parameterize "RollOff" block
Design Requirements
The autopilot must be tuned to satisfy three main design requirements:
1. Setpoint tracking: The response to the command should closely match the response of the reference model:
This reference model specifies a well-damped response with a 2 second settling time.
2. High-frequency roll-off: The closed-loop response from the noise signals to should roll off past 8 rad/s with a slope of at least -40 dB/decade.
3. Stability margins: The stability margins at the plant input should be at least 7 dB and 45 degrees.
For setpoint tracking, we require that the gain of the closed-loop transfer from the command to the tracking error be small in the frequency band [0.05,5] rad/s (recall that we cannot drive the
steady-state error to zero because of the plant zero at s=0). Using a few frequency points, sketch the maximum tracking error as a function of frequency and use it to limit the gain from to .
Freqs = [0.005 0.05 5 50];
Gains = [5 0.05 0.05 5];
Req1 = TuningGoal.Gain('Nzc','e',frd(Gains,Freqs));
Req1.Name = 'Maximum tracking error';
The TuningGoal.Gain constructor automatically turns the maximum error sketch into a smooth weighting function. Use viewGoal to graphically verify the desired error profile.
Repeat the same process to limit the high-frequency gain from the noise inputs to and enforce a -40 dB/decade slope in the frequency band from 8 to 800 rad/s
Freqs = [0.8 8 800];
Gains = [10 1 1e-4];
Req2 = TuningGoal.Gain('n','delta_m',frd(Gains,Freqs));
Req2.Name = 'Roll-off requirement';
Finally, register the plant input as a site for open-loop analysis and use TuningGoal.Margins to capture the stability margin requirement.
Req3 = TuningGoal.Margins('delta_m',7,45);
Autopilot Tuning
We are now ready to tune the autopilot parameters with systune. This command takes the untuned configuration ST0 and the three design requirements and returns the tuned version ST of ST0. All
requirements are satisfied when the final value is less than one.
[ST,fSoft] = systune(ST0,[Req1 Req2 Req3]);
Final: Soft = 0.966, Hard = -Inf, Iterations = 108
Use showTunable to see the tuned block values.
Block 1: rct_concorde/Ki =
D =
y1 -0.03033
Name: Ki
Static gain.
Block 2: rct_concorde/Kp =
D =
y1 -0.009176
Name: Kp
Static gain.
Block 3: rct_concorde/Kq =
D =
y1 -0.2884
Name: Kq
Static gain.
Block 4: rct_concorde/Kf =
D =
y1 -0.02324
Name: Kf
Static gain.
wn = 4.82
zeta = 0.514
To get the tuned value of , use getBlockValue to evaluate Fro for the tuned parameter values in ST:
Fro = getBlockValue(ST,'RollOff');
ans =
s^2 + 4.961 s + 23.27
Continuous-time transfer function.
Finally, use viewGoal to graphically verify that all requirements are satisfied.
viewGoal([Req1 Req2 Req3],ST)
Closed-Loop Simulations
We now verify that the tuned autopilot satisfies the design requirements. First compare the step response of with the step response of the reference model . Again use getIOTransfer to compute the
tuned closed-loop transfer from Nzc to Nz:
Gref = tf(1.7^2,[1 2*0.7*1.7 1.7^2]); % reference model
T = getIOTransfer(ST,'Nzc','Nz'); % transfer Nzc -> Nz
figure, step(T,'b',Gref,'b--',6), grid,
ylabel('N_z'), legend('Actual response','Reference model')
ans =
AxesLabel with properties:
String: "N_z"
FontSize: 11
FontWeight: "normal"
FontAngle: "normal"
Color: [0.1500 0.1500 0.1500]
Interpreter: "tex"
Visible: on
ans =
Legend (Actual response, Reference model) with properties:
String: {'Actual response' 'Reference model'}
Location: 'northeast'
Orientation: 'vertical'
FontSize: 9
Position: [0.6121 0.7695 0.2739 0.0789]
Units: 'normalized'
Use GET to show all properties
Also plot the deflection and the respective contributions of the feedforward and feedback paths:
T = getIOTransfer(ST,'Nzc','delta_m'); % transfer Nzc -> delta_m
Kf = getBlockValue(ST,'Kf'); % tuned value of Kf
Tff = Fro*Kf; % feedforward contribution to delta_m
step(T,'b',Tff,'g--',T-Tff,'r-.',6), grid
ylabel('\delta_m'), legend('Total','Feedforward','Feedback')
ans =
AxesLabel with properties:
String: "\delta_m"
FontSize: 11
FontWeight: "normal"
FontAngle: "normal"
Color: [0.1500 0.1500 0.1500]
Interpreter: "tex"
Visible: on
ans =
Legend (Total, Feedforward, Feedback) with properties:
String: {'Total' 'Feedforward' 'Feedback'}
Location: 'northeast'
Orientation: 'vertical'
FontSize: 9
Position: [0.6601 0.7614 0.2258 0.1144]
Units: 'normalized'
Use GET to show all properties
Finally, check the roll-off and stability margin requirements by computing the open-loop response at .
OL = getLoopTransfer(ST,'delta_m',-1); % negative-feedback loop transfer
The Bode plot confirms a roll-off of -40 dB/decade past 8 rad/s and indicates gain and phase margins in excess of 10 dB and 70 degrees.
See Also
systune (slTuner) (Simulink Control Design) | slTuner (Simulink Control Design) | TuningGoal.Gain | TuningGoal.Margins
Related Topics
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Introduction to SimEngine
Getting started
The goal of many statistical simulations is to compare the behavior of two or more statistical methods; we use this framework to demonstrate the SimEngine workflow. Most statistical simulations of
this type include three basic phases: (1) generate data, (2) run one or more methods using the generated data, and (3) compare the performance of the methods.
To briefly illustrate how these phases are implemented using , we use a simple example of estimating the rate parameter \(\lambda\) of a \(\text{Poisson}(\lambda)\) distribution. To anchor the
simulation in a real-world situation, one can imagine that a sample of size \(n\) from this Poisson distribution models the number of patients admitted daily to a hospital over the course of \(n\)
consecutive days. Suppose that the data consist of \(n\) independent and identically distributed observations \(X_1, X_2, \ldots, X_n\) drawn from a Poisson(\(\lambda\)) distribution. Since the \(\
lambda\) parameter of the Poisson distribution is equal to both the mean and the variance, one may ask whether the sample mean (denoted \(\hat{\lambda}_{M,n}\)) or the sample variance (denoted \(\hat
{\lambda}_{V,n}\)) is a better estimator of \(\lambda\).
1) Load the package and create a simulation object
After loading the package, the first step is to create a simulation object (an R object of class sim_obj) using the new_sim() function. The simulation object contains all data, functions, and results
related to the simulation.
2) Code a function to generate data
Many simulations involve a function that creates a dataset designed to mimic a real-world data-generating mechanism. Here, we write and test a simple function to generate a sample of n observations
from a Poisson distribution with \(\lambda = 20\).
3) Code the methods (or other functions)
With SimEngine, any functions declared (or loaded via source()) are automatically stored in the simulation object when the simulation runs. In this example, we test the sample mean and sample
variance estimators of the \(\lambda\) parameter. For simplicity, we write this as a single function and use the type argument to specify which estimator to use.
4) Set the simulation levels
Often, we wish to run the same simulation multiple times. We refer to each run as a simulation replicate. We may wish to vary certain features of the simulation between replicates. In this example,
perhaps we choose to vary the sample size and the estimator used to estimate \(\lambda\). We refer to the features that vary as simulation levels; in the example below, the simulation levels are the
sample size (n) and the estimator (estimator). We refer to the values that each simulation level can take on as level values; in the example below, the n level values are 10, 100, and 1000, and the
estimator level values are "M" (for “sample mean”) and "V" (for “sample variance”). By default, SimEngine runs one simulation replicate for each combination of level values — in this case, six
combinations — although the user will typically want to increase this; 1,000 or 10,000 replicates per combination is typical.
Note that we make extensive use of the pipe operators (%>% and %<>%) from the magrittr package; if you have never used pipes, see the magrittr documentation.
5) Create a simulation script
The simulation script is a user-written function that assembles the pieces above (generating data, analyzing the data, and returning results) to code the flow of a single simulation replicate. Within
a script, the current simulation level values can be referenced using the special variable L. For instance, in the running example, when the first simulation replicate is running, L$estimator will
equal "M" and L$n will equal 10. In the next replicate, L$estimator will equal "M" and L$n will equal 100, and so on, until all level value combinations are run. The simulation script will
automatically have access to any functions or objects that have been declared in the global environment.
sim %<>% set_script(function() {
dat <- create_data(n=L$n)
lambda_hat <- est_lambda(dat=dat, type=L$estimator)
return (list("lambda_hat"=lambda_hat))
The simulation script should always return a list containing one or more key-value pairs, where the keys are syntactically valid names. The values may be simple data types (numbers, character
strings, or boolean values) or more complex data types (lists, dataframes, model objects, etc.); see the Advanced Usage documentation for how to handle complex data types. Note that in this example,
the estimators could have been coded instead as two different functions and then called from within the script using the use_method() function.
6) Set the simulation configuration
The set_config() function controls options related to the entire simulation, such as the number of simulation replicates to run for each level value combination and the parallelization type, if
desired (see the Parallelization documentation). Packages needed for the simulation should be specified using the packages argument of set_config() (rather than using library() or require()). We set
num_sim to 100, and so SimEngine will run a total of 600 simulation replicates (100 for each of the six level value combinations).
7) Run the simulation
All 600 replicates are run at once and results are stored in the simulation object.
8) View and summarize results
Once the simulation replicates have finished running, the summarize() function can be used to calculate common summary statistics, such as bias, variance, mean squared error (MSE), and confidence
interval coverage.
sim %>% summarize(
list(stat="bias", name="bias_lambda", estimate="lambda_hat", truth=20),
list(stat="mse", name="mse_lambda", estimate="lambda_hat", truth=20)
#> level_id estimator n n_reps bias_lambda mse_lambda
#> 1 1 M 10 100 0.1510000 1.94630000
#> 2 2 V 10 100 -0.4021111 74.12680617
#> 3 3 M 100 100 0.1160000 0.17006800
#> 4 4 V 100 100 -0.1113414 9.69723645
#> 5 5 M 1000 100 0.0160700 0.01579209
#> 6 6 V 1000 100 0.1373756 0.85837283
In this example, we see that the MSE of the sample variance is much higher than that of the sample mean and that MSE decreases with increasing sample size for both estimators, as expected. From the
n_reps column, we see that 100 replicates were successfully run for each level value combination. Results for individual simulation replicates can also be directly accessed via the sim$results
#> sim_uid level_id rep_id estimator n runtime lambda_hat
#> 1 1 1 1 M 10 0.0003800392 20.1
#> 2 7 1 2 M 10 0.0001640320 18.3
#> 3 8 1 3 M 10 0.0001699924 20.5
#> 4 9 1 4 M 10 0.0001580715 21.4
#> 5 10 1 5 M 10 0.0001571178 18.6
#> 6 11 1 6 M 10 0.0001580715 19.5
Above, the sim_uid uniquely identifies a single simulation replicate and the level_id uniquely identifies a level value combination. The rep_id is unique within a given level value combination and
identifies the index of that replicate within the level value combination. The runtime column shows the runtime of each replicate (in seconds).
9) Update a simulation
After running a simulation, a user may want to update it by adding additional level values or replicates; this can be done with the update_sim() function. Prior to running update_sim(), the functions
set_levels() and/or set_config() are used to declare the updates that should be performed. For example, the following code sets the total number of replicates to 200 (i.e., adding 100 replicates to
those that have already been run) for each level value combination, and adds one additional level value for n.
sim %<>% set_config(num_sim = 200)
sim %<>% set_levels(
estimator = c("M", "V"),
n = c(10, 100, 1000, 10000)
After the levels and/or configuration are updated, update_sim() is called.
Another call to summarize() shows that the additional replicates were successfully:
sim %>% summarize(
list(stat="bias", name="bias_lambda", estimate="lambda_hat", truth=20),
list(stat="mse", name="mse_lambda", estimate="lambda_hat", truth=20)
#> level_id estimator n n_reps bias_lambda mse_lambda
#> 1 1 M 10 200 0.163000000 2.147800000
#> 2 2 V 10 200 0.146555556 77.451554321
#> 3 3 M 100 200 0.040450000 0.190544500
#> 4 4 V 100 200 -0.003796970 9.689442951
#> 5 5 M 1000 200 0.012795000 0.016100085
#> 6 6 V 1000 200 0.083129349 0.728864992
#> 7 7 M 10000 200 0.004467500 0.002133542
#> 8 8 V 10000 200 -0.007833964 0.078401278
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Gamma Function Using Spouge's Method
08-25-2015, 06:48 PM
(This post was last modified: 08-25-2015 09:43 PM by Dieter.)
Post: #28
Dieter Posts: 2,397
Senior Member Joined: Dec 2013
RE: Gamma Function Using Spouge's Methjod
(08-25-2015 01:56 AM)lcwright1964 Wrote: With these Lanczos approximations, the series is in 1/z, 1/z+1, 1/z+2, etc., rather than just constant, z, z^2, z^3, etc. This is where Viktor Toth's
rearrangement comes into play, to give
(z!/stuff in front)*Product(z+i, i=0..n) ~= polynomial in z of degree n
as the original approximation to be maximized.
All I know is that Mathematica and Maple (and most probably also others) offer some tools for rational and other approximations. But I do not have access to such software so I cannot say more about
(08-25-2015 01:56 AM)lcwright1964 Wrote: Or I could just stay with the original form, as you have, and fiddle with things in a spread sheet
...this approach has its special advantages. You can taylor an approximation exactly to meet your specific needs. For instance you may want to have better accuracy over a certain interval while for
the rest a somewhat less accurate result will do. Or you may define a different error measure, for instance a max. number of ULPs the approximation can be off. That's what I like about the manual
Les, you wanted an approxmation for n=5. I did some calculations on Free42 with 34-digit BCD precision. The exact coefficients are not fixed yet, but with c=5.081 it looks like the relative error for
x=0...70 is about ±4 E–13.
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A gentle introduction to OmicsPLS III
In the previous post, we discussed an alternative to cross-validation to select the number of components to retain. The scree plot can be a good visual alternative if data exploration is more
important than prediction. In this post, we will go through the interpretation of the scree plot and how we can derive the number of components to choose.
General characteristics of a scree plot
In short, cross-validation repeatedly evaluates the model performance in “independent” sets of samples that are left out when fitting the model. Usually, the prediction error ||y−ŷ||^2 is taken as
metric for how good a model is. For OmicsPLS, the numbers that minimize this prediction error is then taken as “the best”.
It’s this definition of “the best” where several approach differ. With the above steps, we select the model that can best predict y in samples that we haven’t observed yet. Sometimes, however, this
is not what we want. For example, in a descriptive study, the prediction error is of secundary interest, especially when we miss out on information about the samples. Or, when interpretation of the
model parameters is the main aim. A model that is based on the prediction error might be difficult to interpret (too few components retained for example). Therefore, we consider an elegant (at least,
in my opinion) alternative to the cross-validation.
The scree plot
The scree plot is a visual way to select the number of components in PCA, PLS, Factor analysis, and also OmicsPLS. The basic idea is that we look at the amount of variance that is explained by each
component. If the data are of “low rank type”, meaning that few components explain most of the variation, then this can be seen clearly in a scree plot. This is often the case with highly correlated
variables. If the data are very noisy, or the variables are not correlated, then it’s difficult to derive conclusions from a scree plot.
If we take PCA as example, we know that it summarizes the data into principal components. Each of these components explain part of the total variation in the data. The first component will explain
the most, then the second, etc. From a certain number of components on, we will have that these components explain very little variation of the data relative to the first few. Or relative to the
total variation. In that case, it makes sense to only retain the first few components, since the rest doesn’t seem to explain a lot.
The same idea can be used in OmicsPLS, although it’s slightly more complicated. Remember that we have three numbers of components: the number of joint, the number of $x$-specific, and the number of
$y$-specific. I developed a heuristic to get an estimate of numbers of components to retain, summarized in this workflow:
1. Load your input data matrices X and Y in R. We take simulated X and Y:
# First simulate X from 3 components
# Note that these components are not orthogonal!
X <- tcrossprod(
matrix(rnorm(100 * 3), nrow = 100),
matrix(runif(10 * 3), nrow = 10)
# Now define Y as X plus noise
Y <- X + matrix(rnorm(100 * 10), nrow = 100)
# Finally, add noise to X
X <- X + matrix(rnorm(100 * 10), nrow = 100)
1. Plot the eigenvalues of X, by running
scree_X <- svd(X, nu=0, nv=0)$d^2
scree_X <- scree_X / sum(scree_X)
plot(scree_X, type = "b")
comp_X <- 3 #or 2, or 4, whatever you like
choose the total number of components for X 3. Plot the eigenvalues of Y, by running
scree_Y <- svd(Y, nu=0, nv=0)$d^2
scree_Y <- scree_Y / sum(scree_Y)
plot(scree_Y, type = "b")
choose the total number of components for Y 4. Plot the singular values of the covariance between X and Y (not squared)
scree_XY <- svd(crossprod(X,Y), nu=0, nv=0)$d
scree_XY <- scree_XY / sum(scree_XY)
plot(scree_XY, type = "b")
Choose the number of joint components based on the plot 5. The number of joint components r is given by step 4, comp_XY. The number of X-specific components is given by step 2 minus step 4, comp_X -
comp_XY The number of y-specific components is given by step 3 minus step 4, comp_Y - comp_XY
The heuristic assumes that the number of eigenvalues in $x$ and $y$ are the sum of the number of joint eigenvalues, and the specific eigenvalues. Therefore, to get the number of specific components,
you subtract the two as described. Since no repeated fitting is involved, it’s much faster than cross-validation.
A final word of caution: calculating crossprod(X,Y) when both X and Y are high dimensional is a bad idea. The output is a matrix of size ncol(X)*ncol(Y) Instead, calculate tcrossprod(X,X) %*%
tcrossprod(Y,Y) as an approximation if the sample size is smaller than the number of variables. You then get
scree_XY <- sqrt(svd(tcrossprod(X,X) %*% tcrossprod(Y,Y), nu=0, nv=0)$d)
This matrix is of size nrow(X)*nrow(Y).
As an alternative to cross-validation, we went through the scree plot. From the scree plot, and after a subtraction step, the number of joint and specific components can be estimated. These numbers
can then be used to fit OmicsPLS to the data.
Questions? Comments? Let me know!
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An efficient Montgomery exponentiation algorithm by using signed-digit-recoding and folding techniques
The motivation for designing fast modular exponentiation algorithms comes from their applications in computer science. In this paper, a new CSD-EF Montgomery binary exponentiation algorithm is
proposed. It is based on the Montgomery algorithm using the canonical-signed-digit (CSD) technique and the exponent-folding (EF) binary exponentiation technique. By using the exponent-folding
technique of computing the common parts in the folded substrings, the same common part in the folding substrings can be simply computed once. We can thus improve the efficiency of the binary
exponentiation algorithm by decreasing the number of modular multiplications. Moreover, the "signed-digit representation" has less occurrence probability of the nonzero digit than binary number
representation. Taking this advantage, we can further effectively decrease the amount of modular multiplications and we can therefore decrease the computational complexity of modular exponentiation.
As compared with the Ha-Moon's algorithm 1.261718m multiplications and the Lou-Chang's algorithm 1.375m multiplications, the proposed CSD-EF Montgomery algorithm on average only takes 0.5m
multiplications to evaluate modular exponentiation, where m is the bit-length of the exponent.
• Algorithm analysis
• Canonical-signed-digit recoding
• Exponent-folding technique
• Modular exponentiation
• Montgomery algorithm
Dive into the research topics of 'An efficient Montgomery exponentiation algorithm by using signed-digit-recoding and folding techniques'. Together they form a unique fingerprint.
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Keras documentation: Adagrad
Adagrad class
Optimizer that implements the Adagrad algorithm.
Adagrad is an optimizer with parameter-specific learning rates, which are adapted relative to how frequently a parameter gets updated during training. The more updates a parameter receives, the
smaller the updates.
• learning_rate: Initial value for the learning rate: either a floating point value, or a tf.keras.optimizers.schedules.LearningRateSchedule instance. Defaults to 0.001. Note that Adagrad tends to
benefit from higher initial learning rate values compared to other optimizers. To match the exact form in the original paper, use 1.0. initial_accumulator_value: Floating point value. Starting
value for the accumulators (per-parameter momentum values). Must be non-negative. epsilon: Small floating point value used to maintain numerical stability. name: String. The name to use for
momentum accumulator weights created by the optimizer.
• weight_decay: Float, defaults to None. If set, weight decay is applied.
• clipnorm: Float. If set, the gradient of each weight is individually clipped so that its norm is no higher than this value.
• clipvalue: Float. If set, the gradient of each weight is clipped to be no higher than this value.
• global_clipnorm: Float. If set, the gradient of all weights is clipped so that their global norm is no higher than this value.
• use_ema: Boolean, defaults to False. If True, exponential moving average (EMA) is applied. EMA consists of computing an exponential moving average of the weights of the model (as the weight
values change after each training batch), and periodically overwriting the weights with their moving average.
• ema_momentum: Float, defaults to 0.99. Only used if use_ema=True. This is the momentum to use when computing the EMA of the model's weights: new_average = ema_momentum * old_average + (1 -
ema_momentum) * current_variable_value.
• ema_overwrite_frequency: Int or None, defaults to None. Only used if use_ema=True. Every ema_overwrite_frequency steps of iterations, we overwrite the model variable by its moving average. If
None, the optimizer does not overwrite model variables in the middle of training, and you need to explicitly overwrite the variables at the end of training by calling
optimizer.finalize_variable_values() (which updates the model variables in-place). When using the built-in fit() training loop, this happens automatically after the last epoch, and you don't need
to do anything.
• jit_compile: Boolean, defaults to True. If True, the optimizer will use XLA compilation. If no GPU device is found, this flag will be ignored.
• mesh: optional tf.experimental.dtensor.Mesh instance. When provided, the optimizer will be run in DTensor mode, e.g. state tracking variable will be a DVariable, and aggregation/reduction will
happen in the global DTensor context.
• **kwargs: keyword arguments only used for backward compatibility.
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t M
Potentially my ideal (or maybe worst) feature is that I'm totally dissatisfied with my very own expertise.
This might be why I've revealed a certain talent for mathematics and physics. Merely understanding is never ever sufficient for me; I need to minimum aim to carefully understand the thinking behind
the mathematics I do, then, take my understanding to its restrictions. Asking excessively about the reason something is the way it is, probably to the nuisance of my lecturers, is something I'm
compelled to do. I, in addition to numerous others, consider this Socratic technique of discovering as well as mentor to be unbelievably useful in developing a basic understanding of mathematics and
physics from fundamental principles, as well as I endeavour to inform in exactly this manner.
I wish I could encourage trainees with my intense love of mathematics and physics or, at the very least, reveal the topics as far much less overwhelming compared to they show up. Naturally not
everybody is a mathematician, as well as various minds find out at different paces, however I will intend to leave an enduring and valuable perception.
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What is the relationship between corresponding sides, altitudes, and medians in similar triangles? | HIX Tutor
What is the relationship between corresponding sides, altitudes, and medians in similar triangles?
Answer 1
The ratio of their lengths is the same.
Similarity can be defined through a concept of scaling (see Unizor - "Geometry - Similarity").
Accordingly, all linear elements (sides, altitudes, medians, radiuses of inscribed and circumscribed circles etc.) of one triangle are scaled by the same scaling factor to be congruent to
corresponding elements of another triangle. This scaling factor is the ratio between the lengths of all corresponding elements and is the same for all elements.
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Answer 2
In similar triangles:
1. Corresponding sides are proportional. This means that the ratio of corresponding sides in similar triangles is constant.
2. Altitudes are also proportional. The ratio of the lengths of corresponding altitudes in similar triangles is the same as the ratio of the lengths of corresponding sides.
3. Medians are also proportional. The ratio of the lengths of corresponding medians in similar triangles is the same as the ratio of the lengths of corresponding sides.
In summary, corresponding sides, altitudes, and medians of similar triangles are all proportional to each other.
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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
Not the question you need?
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2 3 4 5 Multiplication Worksheets
Mathematics, especially multiplication, creates the keystone of countless academic self-controls and real-world applications. Yet, for many learners, grasping multiplication can posture an obstacle.
To resolve this obstacle, teachers and parents have embraced a powerful device: 2 3 4 5 Multiplication Worksheets.
Introduction to 2 3 4 5 Multiplication Worksheets
2 3 4 5 Multiplication Worksheets
2 3 4 5 Multiplication Worksheets -
This page includes Long Multiplication worksheets for students who have mastered the basic multiplication facts and are learning to multiply 2 3 4 and more digit numbers Sometimes referred to as long
multiplication or multi digit multiplication the questions on these worksheets require students to have mastered the multiplication facts from 0 to 9
This basic Multiplication worksheet is designed to help kids practice multiplying by 2 3 4 or 5 with multiplication questions that change each time you visit This math worksheet is printable and
displays a full page math sheet with Horizontal Multiplication questions
Importance of Multiplication Technique Recognizing multiplication is essential, laying a strong foundation for sophisticated mathematical concepts. 2 3 4 5 Multiplication Worksheets supply structured
and targeted method, cultivating a deeper understanding of this basic arithmetic procedure.
Evolution of 2 3 4 5 Multiplication Worksheets
Commutative Property Of Multiplication Worksheets 2nd Grade Free Printable
Commutative Property Of Multiplication Worksheets 2nd Grade Free Printable
Our multiplication worksheets are free to download easy to use and very flexible These multiplication worksheets are a great resource for children in Kindergarten 1st Grade 2nd Grade 3rd Grade 4th
Grade and 5th Grade Click here for a Detailed Description of all the Multiplication Worksheets Quick Link for All Multiplication Worksheets
Math explained in easy language plus puzzles games quizzes videos and worksheets For K 12 kids teachers and parents Multiplication Worksheets Worksheets Multiplication Mixed Tables Worksheets
Worksheet Number Range Online Primer 1 to 4 Primer Plus 2 to 6 Up To Ten 2 to 10 Getting Tougher 2 to 12 Intermediate 3
From traditional pen-and-paper workouts to digitized interactive styles, 2 3 4 5 Multiplication Worksheets have evolved, dealing with varied learning styles and choices.
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Multiplication Worksheets 2 3 4 5 Multiplication Worksheets
Multiplication Worksheets 2 3 4 5 Multiplication Worksheets
Learn the multiplication tables in an interactive way with the free math multiplication learning games for 2rd 3th 4th and 5th grade The game element in the times tables games make it even more fun
learn Practice your multiplication tables Here you can find additional information about practicing multiplication tables at primary school
Liveworksheets transforms your traditional printable worksheets into self correcting interactive exercises that the students can do online and send to the teacher Skip to main content 2 3 4 and 5
Times tables YII MING HIE Member for 2 years 7 months Age 8 Level 2 Language English en ID 1105646 21 06 2021 Country code MY
Improved Mathematical Abilities
Constant practice sharpens multiplication efficiency, improving overall math abilities.
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Just How to Develop Engaging 2 3 4 5 Multiplication Worksheets
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Connecting multiplication to everyday scenarios includes significance and functionality to workouts.
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Multiplication Equipment and Games Technology-based resources use interactive learning experiences, making multiplication appealing and delightful. Interactive Sites and Applications On the internet
systems offer diverse and obtainable multiplication technique, supplementing standard worksheets. Personalizing Worksheets for Numerous Understanding Styles Visual Learners Aesthetic aids and
representations aid understanding for learners inclined toward aesthetic knowing. Auditory Learners Spoken multiplication troubles or mnemonics cater to learners that understand ideas through
auditory methods. Kinesthetic Learners Hands-on activities and manipulatives support kinesthetic students in recognizing multiplication. Tips for Effective Execution in Understanding Consistency in
Practice Routine technique enhances multiplication skills, promoting retention and fluency. Stabilizing Rep and Variety A mix of repeated workouts and diverse trouble layouts preserves interest and
understanding. Offering Positive Feedback Feedback help in identifying locations of renovation, urging continued progress. Challenges in Multiplication Technique and Solutions Motivation and
Involvement Hurdles Boring drills can cause uninterest; ingenious approaches can reignite inspiration. Conquering Worry of Math Negative perceptions around math can impede development; developing a
positive understanding setting is vital. Effect of 2 3 4 5 Multiplication Worksheets on Academic Efficiency Studies and Study Searchings For Research study indicates a positive connection in between
consistent worksheet use and enhanced math efficiency.
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sources, these worksheets not only enhance multiplication skills yet likewise promote important reasoning and analytical abilities.
4th Grade Multiplication Worksheets Best Coloring Pages For Kids 4th Grade Multiplication
Fill In Multiplication Worksheets 10 Multiplication Wo Printable multiplication worksheets
Check more of 2 3 4 5 Multiplication Worksheets below
Multiplication Strategies 3rd Grade Worksheets Times Tables Worksheets
Multiplication Grade 2 Math Worksheets
Multiplication Worksheets 1 2 3 4 5 Times Tables Worksheets
Multiplication Table Worksheets Grade 3
Multiplication Tables From 1 To 20 Printable Pdf Table Design Ideas
Printable Multiplication Worksheets X3 PrintableMultiplication
Multiply by 2 3 4 5 Horizontal Questions Full Page BigActivities
This basic Multiplication worksheet is designed to help kids practice multiplying by 2 3 4 or 5 with multiplication questions that change each time you visit This math worksheet is printable and
displays a full page math sheet with Horizontal Multiplication questions
Multiply by 2 3 or 4 worksheets K5 Learning
Multiplication facts with 2 3 or 4 as a factor Worksheet 1 provides the complete 2 3 and 4 times tables with and without answers 2 3 4 tables Worksheet 1 49 questions Worksheet 2 Worksheet 3 100
questions Worksheet 4 Worksheet 5 3 More Similar Multiply by 2 5 or 10 Multiply by 3 4 or 6 What is K5
This basic Multiplication worksheet is designed to help kids practice multiplying by 2 3 4 or 5 with multiplication questions that change each time you visit This math worksheet is printable and
displays a full page math sheet with Horizontal Multiplication questions
Multiplication facts with 2 3 or 4 as a factor Worksheet 1 provides the complete 2 3 and 4 times tables with and without answers 2 3 4 tables Worksheet 1 49 questions Worksheet 2 Worksheet 3 100
questions Worksheet 4 Worksheet 5 3 More Similar Multiply by 2 5 or 10 Multiply by 3 4 or 6 What is K5
Multiplication Table Worksheets Grade 3
Multiplication Grade 2 Math Worksheets
Multiplication Tables From 1 To 20 Printable Pdf Table Design Ideas
Printable Multiplication Worksheets X3 PrintableMultiplication
4 Digit Multiplication Worksheets Times Tables Worksheets
Multiplication Worksheets 2 3 4 5 Times Tables Worksheets
Multiplication Worksheets 2 3 4 5 Times Tables Worksheets
Multiplication Worksheets 2 And 3 PrintableMultiplication
FAQs (Frequently Asked Questions).
Are 2 3 4 5 Multiplication Worksheets ideal for every age teams?
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Consistent practice is essential. Normal sessions, ideally a few times a week, can generate considerable renovation.
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Worksheets are a beneficial device yet should be supplemented with different understanding techniques for detailed skill advancement.
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Exactly how can moms and dads sustain their youngsters's multiplication practice at home?
Motivating constant practice, providing aid, and developing a positive discovering environment are helpful actions.
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Optimising pointer subtraction with 2-adic integers.
Optimising pointer subtraction with 2-adic integers.
Here is a simple C type and a function definition:
struct A
char x[7];
int diff(struct A *a, struct A *b)
return a-b;
It doesn't seem like there could be much to say about that. The
structure is 7 bytes long so the subtraction implicitly divides by 7. That's about it. But take a look at the assembly language generated when it's compiled with gcc:
movl 4(%esp), %eax
subl 8(%esp), %eax
imull $-1227133513, %eax, %eax
Where is the division by 7? Instead we see multiplication by -1227133513. A good first guess is that maybe this strange constant is an approximate fixed point representation of 1/7. But this is a
single multiplication with no shifting or bit field selection tricks. So how does this work? And what is -1227133513? Answering that question will lead us on a trip through some suprising and
abstract mathematics. Among other things, we'll see how not only can you represent negative numbers as positive numbers in binary using twos complements, but that we can also represent fractions
similarly in binary too.
But first, some history.
Some n-bit CPUs
That's an Intel 4004 microprocessor, the first microprocessor completely contained on a single chip. It was a 4 bit processor equipped with 4 bit registers. With 4 bits we can represent unsigned
integers from 0 to 15 in a single register. But what happens if we want to represent larger integers?
Let's restrict ourselves to arithmetic operations using only addition, subtraction and multiplication and using one register per number. Then a curious thing happens. Take some numbers outside of the
range 0 to 15 and store only the last 4 bits of each number in our registers. Now perform a sequence of additions, subtractions and multiplications. Obviously we usually expect to get the wrong
result because if the final result is outside of our range we can't represent it in a single register. But the result we do get will have the last 4 bits of the correct result. This happens because
in the three operations I listed, the value of a bit in a result doesn't depend on higher bits in the inputs. Information only propagates from low bit to high bit. We can think of a 4004 as allowing
us to correctly resolve the last 4 bits of a result. From the perspective of a 4004, 1 and 17 and 33 all look like the same number. It doesn't have the power to distinguish them. But if we had a more
powerful 8 bit processor like the 6502 in this machine, we could distinguish them:
This is analogous to the situation we have with distances in physical world. With our eyes we can resolve details maybe down to 0.5mm. If we want to distinguish anything smaller we need more powerful
equipment, like a magnifying class. When that fails we can get a microscope, or an electron microscope, or these days even an atomic force microscope. The more we pay, the smaller we can resolve. We
can think of the cost of the equipment required to resolve two points as being a kind of measure of how close they are.
We have the same with computers. To resolve 1 and 17 we need an 8-bit machine. To resolve 1 and 65537 we need a 32-bit machine. And so on. So if we adopt a measure based on cost like in the previous
paragraph, there is a sense in which 1 is close to 17, but 1 is even closer to 257, and it's closer still to 65537. We have this inverted notion of closeness where numbers separated by large (in the
usual sense) powers of two are close in this new sense.
We have an interesting relationship between computing machines with different 'resolving' power. If we take an arithmetical computation on an N-bit machine, and then take the last M bits of the
inputs and result, we get exactly what the M-bit machine would have computed. So an M-bit machine can be thought of as a kind of window onto the last M-bits onto an N-bit machine. Here's a sequence
of machines:
Each machine provides a window onto the low bits of the previous machine in the sequence. But what happens at the "..." on the left? That suggests the bizarre idea that maybe all of these finite
machines could be thought of as window to some infinite bit machine. Does that idea make any kind of sense?
I'll try to convince you that's a sensible idea by pointing out that it's something familiar to anyone who's taken a rigorous analysis course. (And I'll mention in passing that the above diagram
illustrates a
in an appropriate
Mathematicians (often) build the real numbers from the rational numbers by a process known as completion. Consider a sequence like 1, 14/10, 141/100, 1414/1000, ... . The nth term is the largest
fraction, with 10
in the denominator, such that its square is less than 2. It's well known that there is no rational number whose square is 2. And yet it feels like this sequence ought to be converging to something.
It feels this way because successive terms in the sequence get as close to each other as you like. If you pick any ε there will be a term in the series, say x, with the property that later terms
never deviate from x by more than ε. Such a sequence is called a Cauchy sequence. But these sequences don't all converge to rational numbers. A number like √2 is a gap. What are we to do?
Mathematicians fill the gap by defining a new type of number, the real number. These are
by definition
Cauchy sequences. Now every Cauchy sequence converges to a real number because, by definition, the real number it converges to is the sequence. For this to be anything more than sleight of hand we
need to prove that we can do arithmetic with these sequences. But that's just a technical detail that can be found in any analysis book. So, for example, we can think of the sequence I gave above as
actually being the square root of two. In fact, the decimal notation we use to write √2, 1.414213..., can be thought of as shorthand for the sequence (1, 14/10, 141/100, ...).
The notion of completeness depends on an an idea of closeness. I've described an alternative to the usual notion of closeness and so we can define an alternative notion of Cauchy sequence. We'll say
that the sequence x
, x
, ... is a Cauchy sequence in the new sense if all the numbers from x
onwards agree on their last n bits. (This isn't quite the usual definition but it'll do for here.) For example, 1, 3, 7, 15, 31, ... define a Cauchy sequence. We consider a Cauchy sequence equal to
zero if x
always has zero for its n lowest bits. So 2, 4, 8, 16, 32, ... is a representation of zero. We can add, subtract and multiply Cauchy sequences pointwise, so, for example, the product and sum of x
and y
has terms x
. Two Cauchy sequences are considered equal if their difference is zero. These numbers are called 2-adic integers.
Exercise: prove that if x is a 2-adic integer then x+0=x and that 0x=0.
There's another way of looking at 2-adic integers. They are infinite strings of binary digits, extending to the left. The last n digits are simply given by the last n digits of x
. For example we can write 1, 3, 7, 31, ... as ...1111111. Amazingly we can add subtract and multiply these numbers using the obvious extensions of the usaul algorithms. Let's add ...1111111 to 1:
We get a carry of 1 that ripples off to infinity and gives us zeroes all the way.
We can try doing long multiplication of ...111111 with itself. We get:
It's important to notice that even though there are an infinite number of rows and columns in that multiplication you only need to multiply and add a finite number of numbers to get any digit of the
result. If you don't like that infinite arrangement you can instead compute the last n digits of the product by multiplying 11...n digits...111 by itself and taking the last n digits. The infinite
long multiplication is really the same as doing this for all n and organising it in one big table.
So ...1111111 has many of the properties we expect of -1. Added to 1 we get zero and squaring it gives 1. It is -1 in the 2-adic integers. This gives us a new insight into twos complement arithmetic.
The negative
are the truncated last n digits of the 2-adic representations of the negative integers. We should properly be thinking of twos-complement numbers as extending out to infinity on the left.
The field of analysis makes essential use of the notion of closeness with its δ and ε proofs. Many theorems from analysis carry over to the 2-adic integers. We find ourselves in a strange alternative
number universe which is a sort of mix of analysis and number theory. In fact, people have even tried studying physics in p-adic universes. (p-adics are what you get when you repeat the above for
base p numbers, but I don't want to talk about that now.) One consequence of analysis carrying over is that some of our intuitions about real numbers carry over to the 2-adics, even though some of
our intuitive geometric pictures seem like they don't really apply. I'm going to concentrate on one example.
The Newton-Raphson Method
I hope everyone is familiar with the Newton-Raphson method for solving equations. If we wish to solve f(x)=0 we start with an estimate x
. We find the tangent to y=f(x) at x=x
. The tangent line is an approximation to the curve y=f(x) so we solve the easy problem of finding where the tangent line crosses the x-axis to get a new estimate x
. This gives the formula
= x
With luck the new estimate will be closer than the old one. We can do some
to get some sufficient conditions for convergence.
The surprise is this: the Newton-Raphson method often works very well for the 2-adic integers even though the geometric picture of lines crossing axes doesn't quite make sense. In fact, it often
works much better than with real numbers allowing us to state very precise and easy to satisfy conditions for convergence.
Now let's consider the computation of reciprocals of real numbers. To find 1/a we wish to solve f(x)=0 where f(x)=1/x-a. Newton's method gives the iteration x
= x
). This is a well know iteration that is used internally by CPUs to compute reciprocals. But for it to work we need to start with a good estimate. The famous
Pentium divide bug
was a result of it using an incorrect lookup table to provide the first estimate. So let's say we want to find 1/7. We might start with an estimate like 0.1 and quickly get estimates 0.13, 0.1417,
0.142848, ... . It's converging to the familiar 0.142857...
But what happens if we start with a bad estimate like 1. We get the sequence:
It's diverging badly. But now let's look at the binary:
Our series may be diverging rapidly in the usual sense, but amazingly it's converging rapidly in our new 2-adic sense!
If it's really converging to a meaningful reciprocal we'd expect that if we multiplied the last n digits of these numbers by 7 then we'd get something that agreed with the number 1 in the last 7
digits. Let's take the last 32 digits:
and multiply by 7:
The last 32 bits are
So if we're using 32 bit registers, and we're performing multiplication, addition and subtraction, then this number is, to all intents and purposes, a representation of 1/7. If we interpret as a twos
complements number, then in decimal it's -1227133513. And that's the mysterious number gcc generated.
There are many things to follow up with. I'll try to be brief.
Try compiling C code with a struct of size 14. You'll notice some extra bit shifting going on. So far I've only defined the 2-adic integers. But to get he reciprocal of every non-zero number we need
numbers whose digits don't just extend leftwards to infinity but also extend a finite number of steps to the right of the "binary point". These are the full 2-adic numbers as opposed to merely the
2-adic integers. That's how the extra shifts can be interpreted. Or more simply, if you need to divide by 14 you can divide by 2 first and then use the above method to divide by 7.
I don't know how gcc generates its approximate 2-adic reciprocals. Possibly it uses something based on the Euclidean GCD algorithm. I wasn't able to find the precise line of source in a reasonable
An example of a precise version of the Newton-Raphson method for the p-adics is the
Hensel lemma
The last thing I want to say is that all of the above is intended purely to whet your appetite and point out that a curious abstraction from number theory has an application to compiler writing. It's
all non-rigourous and hand-wavey. Recommend reading further at
. I learnt most of what I know on the subject from the first few chapters of Koblitz's book
p-adic Numbers, p-adic Analysis, and Zeta functions
. The proof of the
von Staudt–Clausen theorem
in that book is mindblowing. It reveals that the real numbers and the p-adic numbers are equally valid ways to approximately get a handle on rational numbers and that there are whole alternative
p-adic universes out there inhabited by weird versions of familiar things like the Riemann zeta function.
(Oh, and please don't take the talk of CPUs too literally. I'm fully aware that you can represent big numbers even on a 4 bit CPU. But what what I say about a model of computation restricted to
multiplication, addition and subtraction in single n-bit registers holds true.)
Some Challenges
1. Prove from first principles that the iteration for 1/7 converges. Can you prove how many digits it generates at a time?
2. Can you find a 32 bit square root of 7? Using the Newton-Raphson method? Any other number? Any problems?
Update: I have replaced the images with images that are, to the best of my knowledge, public domain, or composited from public domain images. (Thanks jisakujien. My bad.)
If you want to play with some p-adics yourself, there is some code to be found
. That also has code for transcendental functions applied to p-adics.
Here's some C code to compute inverses of odd numbers modulo 2
(assuming 32 bit ints). Like the real valued Newton method, it doubles the number of correct digits at each step so we only need 5 iterations to get 32 bits. (C code as I think it's traditional to
twiddle one's bits in C rather than Haskell.)
#include <stdio.h>
#include <assert.h>
typedef unsigned int uint;
uint inverse(uint x)
uint y = 2-x;
y = y*(2-x*y);
y = y*(2-x*y);
y = y*(2-x*y);
y = y*(2-x*y);
return y;
int main()
uint i;
for (i = 1; i<0xfffffffe; i += 2)
assert (i*inverse(i) == 1);
24 Comments:
Unknown said...
Great post Dan, and very interesting.
First of all, it is important to note that the GCC code is not a general answer to the division by 7, the formula only works for numbers exactly divisible by 7.
You can find this without the need of Newton. I actually was asked that question in my first interview in the U.S. and below is how I quickly derived it during the interview. Funnily enough, the
interviewers did not expect that such a solution could exist :)
x = 7y
x = 8y - y
y = 8y - x = 2^3*y - x (1)
Multiplying (1) by 2^3 gives
2^3*y = 2^6*y - 2^3*x (2)
Using (2) in (1) gives
y = 2^6*y - x - 2^3*x
Repeating the same procedure 10 times gives
y = 2^33*y - x - 2^3*x - 2^6*x - 2^9*x - 2^12*x - 2^15*x - 2^18*x - 2^21*x - 2^24*x - 2^27*x - 2^30*x
Since we are working with 32 bits numbers we are working modulo 2^32 and therefore 2^33*y = 0 [2^32]
Therefore y = - x - 2^3*x - 2^6*x - 2^9*x - 2^12*x - 2^15*x - 2^18*x - 2^21*x - 2^24*x - 2^27*x - 2^30*x
And then y = x * -( 1 + 2^3 + 2^6 + 2^9 + 2^12 + 2^15 + 2^18 + 2^21 + 2^24 + 2^27 + 2^30 )
y = x * -1227133513
You can use the same method to derive a method that works for all integers. And I leave the challenge to the readers that works for all integers without the use of the multiplication :)
I thought you were going to say that -1227133513 is the inverse of 7 in the ring Z/(2^32 Z) of integers modulo 2^32, i.e., since -1227133513 * 7 == 1 mod 2^32 dividing by 7 is the same as
multiplying by -1227133513. The number -1227133513 may be computed with Euclid's algorithm.
P.S. I think that - in the source code should be /.
Andrej, yes you could use Euclid algorithm but I think my calculation is easier and less error prone.
Since Dan mentioned first principles, it led me to try to find the simplest and easiest method to find the positive number n so that n * 7 = 1 [2^32] and I found that the easiest way could be
done by kids in 4th or 5th grade without knowledge of Euclid or Bezout.
Looking for the number in hexadecimal is the easiest way due to the modulo. Write the products of 1 to F with 7 you get 7, e, 15, 1c, 23, 2a, 31, 38, 3f, 46, 4d, 54, 5b, 62 and 69.
Since the last digit of the product number with 7 is 1 only 7 works, so you have the last digit.
Now ?7 x 7 = 01, given that 7x7 was 31, you need the last digit of the multiplication of ? and 7 to be 10 - 3 = D which comes from Bx7.
You repeat the same procedure with ?B7 x 7 = 001, and you finally reach B6DB6DB7.
I am interested by any simpler method.
Dan, not sure if I understood the challenge about the square root since with the method above I can prove that there is no square root for 7.
This comment has been removed by a blog administrator.
I did drop a big hint when I said "any problems?" :-) The situation roughly mimics the situation with the real numbers. Some of the real numbers have a square root, some have two, and zero has
just one.
Anyway, I like your method as it's very easy to understand. But the Newton method here is very easy to code up (one short line of code), and runs fast.
Just linking to a page about the GFDL (which isn't even the text of the GFDL--not that that would be sufficient either) is not enough. There are quite a lot of hoops that you have to jump through
that not even Wikipedia itself gets right.
You also have the problem that the images used in Wikipedia are not necessarily available under the GFDL in the first place.
Thankfully, Wikipedia has transitioned to also using CC-BY-SA-3.0 which is much simpler to understand and meet the requirements of (with the caveat that the transition itself complicates things
to some extent). Because of this, you really should take the small amount of effort to at least read the non-legal text of the license and attempt to follow it.
Just saying 'oh, it is too hard and confusing' while making no real effort is quite dishonest. Saying that you took various images from Wikipedia and they might be available under some license
isn't even a nominal attempt. Is it really that hard to have a link to the original image pages with their licensing info? That would at least be a good faith effort.
> Is it really that hard to have a link to the original image pages with their licensing info?
Yes it is. Not because linking is hard, but because I don't have a lawyer to tell me whether that is an acceptable solution either. I'll replace these images with public domain images within 12
hours and I appreciate that with your prompting I will then be 100% legal. Thanks for setting me straight.
Great post, thank you. I had never studied p-adic numbers and had only encountered them a few times; I thought they were just a curiosity and didn't realise they had applications. Hensel's lemma
is really cool.
[BTW, I'm not a lawyer, but as I understand it, in general, the only requirements of images under free licenses (like most images on Wikipedia, except a few that are used as "fair use") are that
1. You credit the author. GFDL or Creative Commons aren't very clear on how to do this, but Wikipedia simply arranges so that clicking on the image will take you to a page that has author
information, and if you link to the same page, it should be ok. (Or better, also add a line under the image crediting the author credited on Wikipedia.)
2. Copyleft. "If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one." (The strictest and possibly incorrect
interpretation would be that including the image makes your blog post freely licensed as well, but I don't think so.)]
I received a post from a user called Gery but it was attached to the wrong blog post so I'm posting it here:
The colimit of your diagram is simply the Intel 4004 while the limit of the same diagram is an hypothetical infinite machine which projects onto every finite machines.
So, I guess you meant to talk about limits, instead of colimits.
@Gery I'll fix that. When first I learnt about p-adics I was using direct/inverse limit language but these days I use limit/colimit language. It drives me crazy that *co*limit corresponds to
*inverse* limit.
There is also a rather more 'hacky' introduction to this in the book and site
'Hacker's Delight'
See the section on 'Magic Numbers'
@ardencaple You can see the Newton method if you view the source to this page.
I'll repeat the "Great post, Dan" comment. You continue to amaze me with the examples and situations you use to motivate abstract concepts.
One problem, though: you say the rationals are constructed using Cauchy sequences. Do you, perhaps, mean the reals?
Which exact sentence do you think I wrote incorrectly? I'm a bit blind when it comes to spotting my own typos.
@sigfpe, in your reply to Cedrick, dated 17 May 2010, where you said, "Some of the real numbers have a square root, some have two, and zero has just one." should the "a" above be a "no", perhaps?
@BlackMeph Yes, well spotted. Substitute 'no' for 'a'.
A simpler way to find the number, based on Cedrick's method, is as follows:
x = 7y
y = (a+1)y - ay = (a+1)y - (a/7)x
We want to choose a such that (1) a+1=0 [mod 2^32], and (2) a is divisible by 7. It turns out that a=2^33-1 is such a number, and a/7=1227133513. This means that mod 2^32,
y = -(a/7)x = -1227133513x
"a+1=0 [mod 2^32], and (2) a is divisible by 7"
is essentially a restatement of the original problem: we're trying to find a such that 7a=1 mod 2^32. 2^33-1 is a solution, so that explains why gcc chose the constant for this particular
example. But what method are you proposing to find this solution?
You are correct, of course, I now realize that what I wrote is trivial. I used trial and error, which didn't take too long, obviously - 2^33-1 is the second choice, right after 2^32-1 which
didn't work. Note that I'm trying to describe how a human being might reach this particular number, not how a computer could solve the general case.
This did get me thinking, however, about long division: Divide 11111... by b in binary (in this case, b=7) and stop where the remainder is zero (after at least 32 bits). I don't know if this will
always work when b is odd (it will never work when b is even, of course).
Apologies if I'm being trivial again, or just plain wrong.
Quick comment: In this case, you can invert -7 pretty easily using the geometric series.
-1/7 = 1/(1-8) = 1 + 8 + 8^2 + 8^3 + ...
which converges 2-adically ....1001001001001001. Then get 1/7 by taking the "2's complement" negative.
Nice one! BTW It works for any odd integer, but it goes faster for one less than a power of two. I should have picked a harder example than 7 :-)
tlawson this is the same method as me except that I did not feel comfortable using the second = sign in 1/(1-8) = 1 + 8 + 8^2 + 8^3 + ...
hence my manual development.
What makes you feel comfortable using that equality? You can justify it a posteriori but seems not obvious to me that 1/(1-x) =series(x^n) [2^32] when 1<x .
Dan about R's method: 2^3 = 1 [7]
And therefore 2^3k = 1 [7] so the result appears magically by itself.
If x is even x^32=0 mod 2^32, and so 1/(1-x) = 1+x+x^2+...+x^31.
From the 2-adic perspective, if x is even, the terms of the series are getting 'smaller' and 'smaller', so the infinite series converges. This is really just another way of saying (1), but in
Sure for even numbers. I should have slept more, I did not even spot your original comment :)
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"Noise traders only" order book model by Bak et al. - Physics of Risk
"Noise traders only" order book model by Bak et al.
Last time we have discussed what order book is, and now we will present a simple model for the order book [1], which was inspired by reaction-diffusion model from physics.
Note that the full model considered in [1] is more complex than we discuss in this post. Here we only reproduce the results of the model with "noise traders" only (as discussed in Section IV B of the
The "noise traders" only model
The model itself is rather simple. It is assumed that there are \( N \) agents (here \( N \) must be even). Half of the agents will always submit bid orders, while the other half will always submit
ask orders.
Initially all agents submit their orders. Log-prices for the bid orders are uniformly distributed between \( -(\Delta P -1) /2 \) and \( 0 \). While prices for the ask orders are uniformly
distributed between \( 0 \) and \( (\Delta P-1)/2 \). Note that, unlike in this interpretation, in the original formulation of the model linear prices were used.
After initialization, during each time tick, randomly selected agent revises his order price. He moves the order one unit toward the spread, with probability \( \frac{1+D}{2} \), or one unit away
from the spread, with probability \( \frac{1-D}{2} \). If ask order meets bid order (or vice versa), both orders are anihilated and new current price is set. Afterwards the agents resubmit their
orders - bid order price is picked randomly between \( - (\Delta P-1) /2 \) and \( P_t \), ask order price is picked randomly between \( P_t \) and \( (\Delta P -1) /2 \).
In real markets such behavior is possible, but not reasonable as some stock exchanges may apply charges. It would be too costly to adjust the order's price often. Hence the model appears to be not
plausible. But the model has its redeemable feature - it reminds of certain physical system.
Imagine a long tube. You inject two different types of particles from the sides of the tube. Inside the tube particles move randomly (diffuse) until they collide with particles of other type and
anihilate (react). The same model describes a physical and financial system.
Interactive applets
To undestand the model better study the interactive applets below. The first applet shows us the structure of the order book. A similar applet was published in the previous text, but it used another
model (which will be discussed in the near future). As in this model order book often has alot of orders, we have chosen lines instead of points to show the amount of order per price level (though
the price levels are still discrete as in the previous example).
The second applet has a somewhat more traditional look. By using the applet below you can see the log-price and absolute return time series, as well as the main statistical features of absolute
return (PDF and spectral density). Note that this model does not have diserable statistical features, we show them simply out of tradition and our own curiosity.
Acknowledgment. This post was written while reviewing literature relevant to the planned activities in postdoctoral fellowship ''Physical modeling of order-book and opinion dynamics''
(09.3.3-LMT-K-712-02-0026) project. The fellowship is funded by the European Social Fund under the No 09.3.3-LMT-K-712 ''Development of Competences of Scientists, other Researchers and Students
through Practical Research Activities'' measure.
• P. Bak, M. Paczuski, M. Shubik. Price variations in a stock market with many agents. Physica A 246: 430-453 (1997). doi: 10.1016/S0378-4371(00)00067-4.
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File [_cd8_]<altodocs>float.tty!2
FLOAT is a floating-point package for the Alto, intended for use with
BCPL. (It uses standard Alto microcode -- no special instructions are
needed.) A microcoded version is also available, and is documented in
the last section. There are 32 floating-point accumulators, numbered
0-31. These accumulators may be loaded, stored, operated on, and
tested with the operations provided in this package. 'Storing' an
accumulator means converting it to a 2-word packed format (described
below) and storing the packed form.
In the discussion below, 'ARG' means: if the 16-bit value is less than
the number of accumulators, then use the contents of the accumulator of
that number. Otherwise, the 16-bit value is assumed to be a pointer to
a packed floating-point number.
All of the functions listed below that do not have "==>" after them
return their first argument as their value.
1. Floating point routines
FLD (acnum,arg) Load the specified accumulator from
source specified by arg. See above for a
definition of 'arg'.
FST (acnum, ptr-to-num) Store the contents of the accumulator
into a 2-word packed floating point
format. Error if exponent is too large
or small to fit into the packed
FTR (acnum) ==> integer Truncate the floating point number in
the accumulator and return the integer
value. FTR applied to an accumulator
containing 1.5 is 1; to one containing
-1.5 is -1. Error if number in ac cannot
fit in an integer representation.
FLDI (acnum,integer) Load-immediate of an accumulator with
the integer contents (signed 2's
FNEG (acnum) Negate the contents of the accumulator.
FAD (acnum,arg) Add the number in the accumulator to the
number specified by arg and leave the
result in the accumulator. See above for
a definition of 'arg'.
FSB (acnum,arg) Subtract the number specified by 'arg'
Copyright Xerox Corporation 1979
FLOAT December 26, 1977 2
from the number in the accumulator, and
leave the result in the accumulator.
FML (acnum,arg) [ also FMP ] Multiply the number specified by
'arg' by the number in the accumulator,
and leave the result in the ac.
FDV (acnum,arg) Divide the contents of the accumulator
by the number specified by arg, and
leave the result in the ac. Error if
attempt to divide by zero.
FCM (acnum,arg) ==> integer Compare the number in the ac with the
number specified by 'arg'. Return
-1 IF ARG1 < ARG2
0 IF ARG1 = ARG2
1 IF ARG1 > ARG2
FSN (acnum) ==> integer Return the sign of the floating point
-1 if sign negative
0 if value is exactly 0 (quick test!)
1 if sign positive and number non-zero
FEXP(acnum,increment) Adds 'increment' to the exponent of the
specified accumulator. The exponent is
a binary power. Thus
FLDV (acnum,ptr-to-vec) Read the 4-element vector into the
internal representation of a floating
point number.
FSTV (acnum,ptr-to-vector)Write the accumulator into the 4-element
vector in internal representation.
2. Double precision fixed point
There are also some functions for dealing with 2-word fixed point
numbers. The functions are chosen to be helpful to DDA scan-converters
and the like.
FSTDP(ac,ptr-to-num) Truncates the contents of the floating
point ac and stores it into the
specified double-precision number.
First word of the number is the integer
part, second is fraction. Two's
complement. Error if exponent too
FLDDP(ac,ptr-to-num) Loads floating point ac from dp number.
Same conventions for integer and
fractional part as FSTDP.
DPAD(a,b) => ip a and b are both pointers to dp numbers.
The dp sum is formed, and stored in a.
FLOAT December 26, 1977 3
Result is the integer part of the
DPSB(a,b) => ip Same as DPAD, but subtraction.
DPSHR(a) => ip Shift a double-precision number right
one bit, and return the integer part.
3. Format of a packed floating point number
structure FP: [
sign bit 1 //1 if negative.
expon bit 8 //excess 128 format (complemented if number <0)
mantissa1 bit 7 //High order 7 bits of mantissa
mantissa2 bit 16 //Low order 16 bits of mantissa
Note this format permits packed numbers to be tested for sign, to be
compared (by comparing first words first), to be tested for zero (first
word zero is sufficient), and (with some care) to be complemented.
4. Saving and Restoring Work Area
FLOAT has a compiled-in work area for storing contents of floating
accumulators, etc. The static FPwork points to this area. The first
word of the area (i.e. FPwork!0) is its length and the second word is
the number of floating point accumulators provided in the area. The
routines use whatever pointer is currently in FPwork for the storage
area. Thus, the accumulators may be "saved" and "restored" simply by:
let old=FPwork
let new=vec enough; new!1=old!1 //Copy AC count
...routines use "new" work area; will not affect "old"
This mechanism also lets you set up your own area, with any number of
accumulators. The length of work area required is 4*(number of
accumulators)+constant. (The constant may change when bugs are fixed
in the floating point routines. As a result, you should calculate it
from the compiled-in work area as follows: constant←FPwork!0-
4*FPwork!1.) It is not essential that the length word (FPwork!0) be
exact for the routines to work.
5. Errors
If you wish to capture errors, put the address of a BCPL subroutine in
the static FPerrprint. The routine will be called with one parameter:
0 Exponent too large -- FTR
1 Exponent too large -- FST
2 Dividing by zero -- FDV
3 Ac number out of range (any routine)
4 Exponent too large -- FSTDP
FLOAT December 26, 1977 4
The result of the error routine is returned as the result of the
offending call to the floating point package.
6. Floating point microcode
A microcoded version of the FLOAT package is also available. The
microcode is from four to six times faster than the assembly code.
Execution times are about 80 microseconds for multiply and divide, and
40 microseconds for addition and subtraction. The file MicroFloat.DM
is a dump-format file containing MicroFloat.BR and MicroFloatMC.BR.
These modules should be loaded with your program, along with the
LoadRam procedure, available separately as LoadRam.BR. The microcode
RAM must be loaded with the appropriate microcode. This is
accomplished by calling LoadRam(MicroFloatRamImage) After this call,
the memory space used for MicroFloatMC.BR and LoadRam.BR can be
released. Microfloat.BR must remain resident, but it only takes up
about 60 words. The floating point routines can also be invoked as
single assembly code instructrions, with op codes 70001 through 70021.
The correspondence between op codes and floating point operations is
documented in MicroFloat.ASM.
In contrast to the assembly coded version, the microcode does not
allocate any memory work space, and any number of accumulators may be
used. Four words of memory are needed for each accumulator, and this
memory space MUST be provided by the user by calling FPSetup(workArea),
where workArea is the block of memory to be used for mainintaining the
ACS, and workArea!0 is the number of accumulators to be used. The
length of workArea must be at least (4*numACs)+1 words long. The
contents of workArea are not re-iitialized, so that reusing a
previously used work area will have the effect of restoring the values
of the ACs to their previous state. The static FPwork will be set to
the current workArea. So, "save" and "restore" the accumulators by:
let old=FPwork
let new=vec (4*numACs)+1; new!0=numACs
...routines use "new" work area; will not affect "old"
Loading the RAM, calling FPSetup, and the (shorter) work area format
are the only changes from the assembly coded routines.
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[CP2K-user] [CP2K:19923] Re: Problem with the calculation of the CUBE file and Wannier Centres using the ROKS method?
Ilya Fedorov ilyafedorov19 at gmail.com
Fri Feb 16 21:21:28 UTC 2024
Dear colleagues,
I found a bug and I have not corrected all issues with the implementation
of orbitals printing with ROKS. I have now added make_mo_eig(), which
returns orbitals from OT space. I have limited it to nspin=1 and everything
now works consistently with CPMD. (https://github.com/cp2k/cp2k/pull/3274)
If you do not mind, may I replace the regtest with oxygen with
formaldehyde, since it is a simpler example and has a larger dE in ROKS?
Best regards,
понедельник, 12 февраля 2024г. в 19:09:44 UTC+3, Ilya Fedorov:
> Dear Matthias,
> I apologise for such a long reply. The problem turned out to be not so
> simple.
> At this moment it seems that ROKS does not work properly in CP2K.
> An oxygen molecule turned out to be a poor example to use with ROKS. I
> just copied it from an existing regtest for ROKS, but CPMD and CP2K gave
> the S1 energies lower than the S0 energies.
> A better example would be a formaldehyde molecule or a hydrogen molecule.The
> advantages of formaldehyde are that it has SOMO-1 and SOMO-2 separated in
> space, which, as will be discussed below, allows for better manifestation of the
> differences between CP2K and CPMD could be traced. I used a formaldehyde
> example from the original ROKS paper [Frank, I., Hutter, J., Marx, D., &
> Parrinello, M. (1998). JPC, 108, 4060].
> *Here are the main observations:*
> 1. The excitation energy E(S1) - E(S0) *in both CPMD and CP2K packages
> is about 3.5 eV*, which is in agreement with the paper of Frank et al.
> 2. The *Wannier centres* of all orbitals are almost the *same* between
> CPMD and CP2K (see Picture 1), but their *order is different*. This is
> important for SOMO-1 and SOMO-2, since we have the rotations turned off for
> these orbitals and their centres should have coincided.
> 3. *The most important difference* is in the visualisation of the
> orbitals. In the attached pdf-file (S0_S1_CP2K_CPMD.pdf) I visualise the
> orbitals for the ground state S0 and of the excited state S1. The
> isosurfaces are shown in red (+0.07) and in blue (-0.07). You can see that
> for the ground state the orbitals from CPMD and from CP2K coincide. For the
> excited state CPMD shows some change of order (with respect to S0) and the
> general picture remains close (*the resemblance HOMO ~ SOMO-1 takes
> place*). *But for the S1 state in CP2K all orbitals look different,
> except last two (SOMO-1 and SOMO-2)*, and the last two orbitals seem
> to have switched places (*the resemblance HOMO ~ SOMO-2 takes place*)
> (the same pattern as with Wannier centres.).
> I attach the codes for calculating the S0 and S1 states in CPMD and CP2K
> for formaldehyde.
> - The standard cpmd2cube.x script (
> https://github.com/CPMD-code/Addons/tree/main/cpmd2cube) should be
> used to convert the CPMD orbitals to .cube files (cmd: ./cpmd2cube.x
> WAVEFUNCTION.*)
> - To calculate Wannier centres in CPMD, a separate code
> (cpmd_s1_wannier.inp) should be used which takes 1 MD step, this is because
> a separate case of calculating Wannier centres for ROKS is only implemented
> in MD. At the same time, in CPMD the calculation of Wannier centres itself
> affects the final output of orbitals
> Best regards,
> Ilya
> P.S.
> Although the energies of SOMO-1 and SOMO-2 judging by dE ~3.5 eV should be
> significantly different and could not simply get mixed up. In addition, it
> is worth noting that despite the complete difference in orbitals 1-5 in
> CP2K and CPMD, the final Wannier centres are the same, so we can talk about
> the influence of the difference of the basis or something else?
> Could it be that the OT method does not handle single occupied orbitals
> correctly or the ROKS method in CP2K does not handle double occupied
> orbitals correctly? Because in the case of CPMD we see that the shape of
> the double occupied orbitals has not changed, the order has changed and one
> new SOMO-2 orbital has been added. In CP2K, the new single occupied
> orbitals look correct, but the double occupied ones have coordinately
> changed in ROKS.
> вторник, 21 ноября 2023г. в 13:27:51 UTC+3, Ilya Fedorov:
>> Dear Matthias,
>> I am sorry for the delay in answering your question.
>> Indeed, sometimes I observed poor convergence of OT with ROKS, but
>> sometimes the convergence is much better than diagonalization in СPMD. But
>> at the moment I do not have a systematic understanding of the roots of this
>> problem.
>> I made various comparisons between CPMD and CP2K in terms of calculations
>> with ROKS: comparing forces, trajectories, orbitals and Wannier centers, in
>> general I got a fairly close agreement.
>> In the near future I plan to collect these results into one report and
>> post it here.
>> Best
>> Ilya
>> четверг, 16 ноября 2023г. в 15:31:03 UTC+3, Krack Matthias:
>>> Hi Ilya
>>> Thanks for contributing and fixing the print issue.
>>> Indeed, ROKS produces only one set of orbitals, since there is also only
>>> one Hamiltonian, i.e. Kohn-Sham matrix, but different orbital occupations
>>> are applied for spin-up (alpha) and spin-down (beta) electrons which
>>> results in an alpha and beta electronic density matrix.
>>> The LOW-SPIN ROKS implementation in CP2K based on OT plus ROTATION is
>>> experimental as indicated by a warning printed in the output. That
>>> implementation has not been maintained since a long time. Therefore, I am
>>> wondering, if the implementation works properly, e.g. for the new test case
>>> O2_mo_cubes.inp. I tried tighter thresholds and a larger cutoff and box
>>> size, but the SCF did not converge. What’s your experience and how do the
>>> results compare to ROKS in CPMD?
>>> Best
>>> Matthias
>>> *From: *cp... at googlegroups.com <cp... at googlegroups.com> on behalf of
>>> Ilya Fedorov <ilyafe... at gmail.com>
>>> *Date: *Thursday, 16 November 2023 at 10:59
>>> *To: *cp2k <cp... at googlegroups.com>
>>> *Subject: *[CP2K:19518] Re: Problem with the calculation of the CUBE
>>> file and Wannier Centres using the ROKS method?
>>> I solved the problem and it was accepted into the code.
>>> 1. As I understand it, for ROKS the UKS base is used, and
>>> different spins of the same orbital have the same density. This allows to
>>> rely on the UKS codes, while you can use MO_CUBES only for spin-1.
>>> (regtest: QS/regtest-lsroks/O2_mo_cubes.inp)
>>> 2. I added such an implementation in the code. Only two method
>>> supports:
>>> o NONE: No rotation, just print the orbital centers.
>>> o JACOBI: Only doubly occupied orbitals rotate; for singly occupied
>>> orbitals, no rotation occurs (like the NONE method). (regtest:
>>> QS/regtest-lsroks/O2_loc_wan_jac.inp)
>>> It is possible to use only NONE and JACOBI. Other methods can be added
>>> in the future, but ROKS calculates quite slowly, and speeding up the
>>> Wannier calculation does not make a big contribution here.
>>> Commits: https://github.com/cp2k/cp2k/pull/3108
>>> четверг, 24 августа 2023г. в 15:19:45 UTC+3, Ilya Fedorov:
>>> Dear colleagues,
>>> Could you please help me to understand the CP2K implementation of ROKS
>>> and the Wannier Centers.
>>> I am facing the following two problems. I can't find an answer in the
>>> documentation. And, unfortunately, I don't really understand in the code.
>>> 1) The ROKS implementation in CP2K uses a spin-restricted calculation
>>> with multiple density matrices. As I understand it, this leads to some
>>> problems with printing CUBE files (for example, for each orbital it prints
>>> two cube files for spin_1 and spin_2). Which one is correct? Maybe the sum
>>> of them gives a correct density?
>>> (in log it print “Unclear how we define MOs in the restricted case ...
>>> skipping”)
>>> 2) I’m trying to use Wannier Centres with ROKS, but it doesn’t work in
>>> CP2K. As I understand this is also a multiple density matrix problem. I
>>> need Wannier Centres only (Many-body Position operator, no Jacobi
>>> rotation). Right now I use Wannier + ROKS in CPMD, and in CPMD the rotation
>>> is just switched off for the last two orbitals (diag(1,1)).
>>> Best regards
>>> Ilya
>>> --
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Carl Pomerance
Factorization of large numbers amounts to trial and retrial. The quadratic and algebraic number sieves produce sophisticated trials which are efficient for numbers up to 150 digits.
(pp. 1473)
Constance Reid
Reminiscences about the development of the mathematical career of Julia Robinson.
(pp. 1486)
John D. Fulton
(pp. 1493)
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Delta-Stepping Algorithm - My Online Vidhya
Read Time:4 Minute, 45 Second
Delta-Stepping Algorithm
Delta-stepping is a parallel algorithm for finding the shortest path in a weighted graph. It was first introduced by David Johnson and Brian Sandberg in 1989.
The algorithm is similar to Dijkstra’s algorithm, but it works by processing the graph in “buckets” of nodes, where each bucket contains nodes whose shortest distance from the source node is within a
certain range (known as the “delta”). The algorithm starts with a delta of 0 and gradually increases the delta until all nodes have been processed.
At each iteration of the algorithm, the nodes in the current bucket are processed in parallel. For each node, the algorithm updates the distance to all of its neighbors, and if the new distance is
within the range of the next bucket, the neighbor is added to the next bucket.
Delta-stepping has several advantages over Dijkstra’s algorithm, particularly for large-scale graphs. Because it processes nodes in parallel, it can be more efficient on multi-core processors and
distributed systems. Additionally, by processing nodes in buckets, it can reduce the number of nodes that need to be processed at each iteration, leading to faster convergence.
However, delta-stepping has some limitations as well. In particular, it is more complex to implement than Dijkstra’s algorithm, and it may not be as effective on small graphs or graphs with highly
variable edge weights.
1. Initialize the distance of all nodes to infinity, except for the source node, which has distance 0.
2. Set the delta to a small value, such as 1 or the square root of the number of nodes in the graph.
3. Create a list of “buckets”, where each bucket contains nodes whose distance from the source node is within the range [deltak, delta(k+1)] for some integer k.
4. Add the source node to the first bucket (i.e., the bucket for k=0).
5. While there are non-empty buckets:
a. Take the first non-empty bucket.
b. Process the nodes in the bucket in parallel:
i. For each node, consider all its neighbors and compute the distance to each neighbor as the sum of the current distance to the node and the weight of the edge to the neighbor.
ii. For each neighbor, if its new distance is less than the current distance, update its distance and add it to the corresponding bucket.
c. If the current bucket is empty and there are remaining non-empty buckets, increase the delta and reassign the nodes to new buckets based on their updated distances.
6. When all buckets are empty, the algorithm terminates and the distances to all nodes from the source node are computed.
The choice of delta can affect the performance of the algorithm. A smaller delta can lead to more buckets and finer-grained parallelism, but may increase the overhead of managing the buckets. A
larger delta can reduce the number of buckets and simplify the algorithm, but may reduce the parallelism and increase the convergence time. Therefore, the delta value should be chosen based on the
characteristics of the graph and the computing environment.
Implementation of the delta-stepping algorithm in Python:
import sys
import heapq
# Function to read in the graph from a file
def read_graph(file):
graph = {}
with open(file, 'r') as f:
for line in f:
# Ignore comment lines starting with #
if line.startswith('#'):
# Parse the line to extract the source node, destination node, and weight of the edge
parts = line.split()
if len(parts) != 3:
u, v, w = map(int, parts)
# Add the nodes and the edge to the graph
if u not in graph:
graph[u] = []
if v not in graph:
graph[v] = []
graph[u].append((v, w))
graph[v].append((u, w))
return graph
# Function to perform delta-stepping algorithm
def delta_stepping(graph, start, delta):
# Initialize the distance of all nodes to infinity, except for the start node
dist = {v: sys.maxsize for v in graph}
dist[start] = 0
# Initialize the heap with the start node
heap = [(0, start)]
# Loop until the heap is empty
while heap:
# Extract the node with the minimum distance from the heap
d, u = heapq.heappop(heap)
# If the distance to this node has already been updated, skip it
if d > dist[u]:
# Update the distances of all neighbors of the node
for v, w in graph[u]:
new_dist = dist[u] + w
# If the new distance is smaller than the current distance, update the distance and add the node to the heap
if new_dist < dist[v]:
dist[v] = new_dist
# If the new distance is a multiple of delta, add the node to the heap with its exact distance
if new_dist % delta == 0:
heapq.heappush(heap, (new_dist, v))
# Otherwise, add the node to the heap with a distance rounded up to the nearest multiple of delta
heapq.heappush(heap, (new_dist + delta - (new_dist % delta), v))
# Return the distances to all nodes from the start node
return dist
if __name__ == '__main__':
# Read in the graph from the file
graph = read_graph('graph.txt')
# Set the start node and the delta value
start = 1
delta = 2
# Run the delta-stepping algorithm and print the distances to all nodes
dist = delta_stepping(graph, start, delta)
The sample “graph.txt” file that corresponds to the input provided :
# Sample graph file
# Each line contains three integers representing an edge (u, v) with weight w
# Nodes are numbered starting from 1
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|
Interest Calculators
The simple & compound interest calculators are the collection of web based tools used to calculate, analyze and determine how much extra money a lender or borrower will receive or pay for a specific
period of time. These are all some of the fundamental tools to find the time value of money.
|
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|
parallel to the axes, and the values of x and y at their intersections completely define the point. In honor of Descartes, this way of labeling points is known as a cartesian system and the two
numbers (x,y) that define the position of any point are its cartesian coordinates.
Graphs use this system, as do some maps.
Very simple and clear, once a decision is made on which side of the sheet z is positive. By common agreement the positive branches of the (x,y, z) axes, in that order, follow the thumb and the first
two fingers of the right hand when extended in a way that they make the largest angles with each other.
What follows uses the trigonometric functions sine and cosine; if these are not familiar to you, either skip the rest of the section, or go learn about them.
Polar Coordinates
x,y) are not the only way of labeling a point P on a flat plane by a pair of numbers. Other ways exist, and they can be more useful in special situations.
One system ("polar coordinates") uses the length r of the line OP from the origin to P (i. e. the distance from P to the origin) and the angle that line makes with the x-axis. Angles are often
denoted by Greek letters, and here we follow conventions by marking it with φ (Greek f). Note that while in the cartesian system x and y play very similar roles, here roles are divided: r gives
distance and φ direction.
The two representations are closely related. From the definitions of the sine and cosine
x = r cos φ
y = r sin φ
That allows (x,y) to be derived from polar coordinates. To go in the opposite direction and derive (r,φ) from (x,y), note that from the above equations (or from the theorem of Pythagoras) one can
derive r:
r^2 = x^2 + y^2
Once r is known, the rest is easy
cos φ = x/r
sin φ = y/r
These relations fail only at the origin, where x = y = r = 0. At that point, φis undefined and one can choose for it whatever one pleases.
In three dimensional space, the cartesian labeling (x,y,z) is nicely symmetric, but sometimes it is convenient to follow the style of polar coordinates and label distance and direction separately.
Distance is easy: you take the line OP from the origin to the point and measure its length r. You can even show from the theorem of Pythagoras that in this case
r^2 = x^2 + y^2 + z^2
All the points with the same value of r form a sphere of radius r around the origin O. On a sphere we can label each point by latitude λ (lambda, small Greek L) and longitude φ (phi, small Greek F),
so that the position of any point in space could defined by the 3 numbers (r, λ, φ).
Azimuth and Elevation
│An old surveyor's tele-│
│scope (theodolite). │
The surveyor's telescope is designed to measure two such angles. The angle φ is measured counterclockwise in a horizontal plane, but surveyors (and soldiers) work with azimuth, a similar angle
measured clockwise from north. Thus the directions (north, east, south, west) have azimuth (0°, 90°, 180°, 270°). A rotating table allows the telescope to be pointed in any azimuth.
The angle λ is called elevation and is the angle by which the telescope is lifted above the horizontal (if it looks down, λ is negative). The two angles together can in principle specify any
direction: φ ranges from 0 to 360°, and λ from –90° (straight down or "nadir") to +90° (straight up or "zenith").
Again, one needs to decide from what direction is the azimuth measured--that is, where is azimuth zero? The rotation of the heavens (and the fact most humanity lives north of the equator) suggests
(for surveyor-type measurements) the northward direction, and this is indeed the usual zero point. The azimuth angle (viewed from the north) is measured counterclockwise.
Mathematicians however prefer their own notation and replace "latitude" (or elevation) λ with co-latitude θ= 90 – λ deg., the angle not to the horizon but to the vertical. The angle θ (theta, one of
two t-s in Greek) goes from 0 to 180°, not from –90° to + 90°. This actually may make more sense, because it is easier to measure an angle between two lines (OP and the vertical) rather than between
a line and a flat plane (OP and the horizontal).]
[And in case you have to know: In 3 dimensions, when deriving the spherical coordinates (r, θ, φ) that correspond to cartesian (x,y,z) from the same origin, θ is measured from the +z-axis and φ
is measured in the (x,y) plane, counterclockwise from the x-axis as seen from the +z side. This needs to be said, because viewed from –z ("from below") clockwise becomes counterclockwise, and
vice versa.]
Teachers using this web page will find a related lesson plan at Lcelcoor.htm
It belongs to a set of lesson plans whose home page is at Lintro.htm.
Questions from Users:
*** Drawing a Perpendicular Line in Rectangular Coordinates
Next Stop: #6 The Calendar
Timeline Glossary Back to the Master List
Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .
Last updated: 10 October 2016
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|
rams to Pounds
Grams to Pounds Converter
⇅ Switch toPounds to Grams Converter
How to use this Grams to Pounds Converter 🤔
Follow these steps to convert given weight from the units of Grams to the units of Pounds.
1. Enter the input Grams value in the text field.
2. The calculator converts the given Grams into Pounds in realtime ⌚ using the conversion formula, and displays under the Pounds label. You do not need to click any button. If the input changes,
Pounds value is re-calculated, just like that.
3. You may copy the resulting Pounds value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Grams to Pounds?
The formula to convert given weight from Grams to Pounds is:
Weight[(Pounds)] = Weight[(Grams)] / 453.59237
Substitute the given value of weight in grams, i.e., Weight[(Grams)] in the above formula and simplify the right-hand side value. The resulting value is the weight in pounds, i.e., Weight[(Pounds)].
Calculation will be done after you enter a valid input.
Consider that a high-end smartwatch weighs 50 grams.
Convert this weight from grams to Pounds.
The weight of smartwatch in grams is:
Weight[(Grams)] = 50
The formula to convert weight from grams to pounds is:
Weight[(Pounds)] = Weight[(Grams)] / 453.59237
Substitute given weight of smartwatch, Weight[(Grams)] = 50 in the above formula.
Weight[(Pounds)] = 50 / 453.59237
Weight[(Pounds)] = 0.1102
Final Answer:
Therefore, 50 g is equal to 0.1102 lbs.
The weight of smartwatch is 0.1102 lbs, in pounds.
Consider that a gold investment coin weighs 31.1 grams.
Convert this weight from grams to Pounds.
The weight of gold investment coin in grams is:
Weight[(Grams)] = 31.1
The formula to convert weight from grams to pounds is:
Weight[(Pounds)] = Weight[(Grams)] / 453.59237
Substitute given weight of gold investment coin, Weight[(Grams)] = 31.1 in the above formula.
Weight[(Pounds)] = 31.1 / 453.59237
Weight[(Pounds)] = 0.06856376354
Final Answer:
Therefore, 31.1 g is equal to 0.06856376354 lbs.
The weight of gold investment coin is 0.06856376354 lbs, in pounds.
Grams to Pounds Conversion Table
The following table gives some of the most used conversions from Grams to Pounds.
Grams (g) Pounds (lbs)
0.01 g 0.00002204623 lbs
0.1 g 0.00022046226 lbs
1 g 0.00220462262 lbs
2 g 0.00440924524 lbs
3 g 0.00661386787 lbs
4 g 0.00881849049 lbs
5 g 0.01102311311 lbs
6 g 0.01322773573 lbs
7 g 0.01543235835 lbs
8 g 0.01763698097 lbs
9 g 0.0198416036 lbs
10 g 0.02204622622 lbs
20 g 0.04409245244 lbs
50 g 0.1102 lbs
100 g 0.2205 lbs
1000 g 2.2046 lbs
The gram is a metric unit of mass. It is equal to one thousandth of a kilogram. Grams are commonly used for small measurements of mass, especially in scientific and everyday contexts.
The pound is a unit of mass used in the imperial system and the United States customary system. One pound is equivalent to 0.45359237 kilograms. The pound is commonly used for measuring the weight of
objects in everyday contexts, and it is a common unit in the United States and some other countries.
Frequently Asked Questions (FAQs)
1. How do I convert grams to pounds?
Divide the number of grams by 453.59237 to get the equivalent in pounds. For example, 1,000 grams ÷ 453.59237 ≈ 2.20462 pounds.
2. What is the formula for converting grams to pounds?
The formula is: pounds = grams ÷ 453.59237.
3. How many pounds are in a gram?
There are approximately 0.00220462 pounds in 1 gram.
4. Is 1,000 grams equal to 2.20462 pounds?
Yes, 1,000 grams is approximately equal to 2.20462 pounds.
5. How do I convert pounds to grams?
Multiply the number of pounds by 453.59237 to get the equivalent in grams. For example, 2 lbs × 453.59237 = 907.18474 grams.
6. What is the difference between grams and pounds?
Grams and pounds are both units of mass, but grams are part of the metric system while pounds are part of the imperial system. One pound is equal to 453.59237 grams.
7. How many pounds are there in 500 grams?
500 grams ÷ 453.59237 ≈ 1.10231 pounds.
8. How many pounds are in 250 grams?
250 grams ÷ 453.59237 ≈ 0.55116 pounds.
9. How do I use this grams to pounds converter?
Enter the value in grams that you want to convert, and the converter will automatically display the equivalent in pounds.
10. Why do we divide by 453.59237 to convert grams to pounds?
Because there are 453.59237 grams in 1 pound, so dividing by this number converts grams to pounds.
11. What is the SI unit of mass?
The SI unit of mass is the kilogram, but grams are commonly used as a smaller unit.
12. Are grams lighter than pounds?
Yes, grams are lighter than pounds. One gram equals approximately 0.00220462 pounds.
13. How many pounds are in 1,500 grams?
1,500 grams ÷ 453.59237 ≈ 3.30693 pounds.
14. How to convert 3,000 grams to pounds?
3,000 grams ÷ 453.59237 ≈ 6.61387 pounds.
15. Is 1 gram equal to 0.00220462 pounds?
Yes, 1 gram is approximately equal to 0.00220462 pounds.
Weight Converter Android Application
We have developed an Android application that converts weight between kilograms, grams, pounds, ounces, metric tons, and stones.
Click on the following button to see the application listing in Google Play Store, please install it, and it may be helpful in your Android mobile for conversions offline.
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[50 Word Friday] MathFour to Appear on Create Chatter TV
MathFour is the guest on Create Chatter TV this Thursday at 8:00pm CST (find your time here). In the spirit of promotion I’ve written this 50-Word-Friday article.
Math goes to the show!
Math is in the spotlight. It’s not just for engineers and academics. Everyone does math. Everyday.
Math isn’t your dirty little secret. You do math. Come learn how to recognize it and be proud of what you already know and do. Children should see this and it begins with you!
Will you be there?
Learn more about 50 Word Friday here.
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News around the ICM 2018
The ICM 2018 is currently taking place in Rio de Janeiro. Here are the prize winners of the IMU prizes and of the K-theory foundation, and the new IMU Executive Committee.
The Fields medallists 2018 are
There are also many more prizes and medals that are awarded at the ICM:
Preceding the ICM the K-theory foundation awards at a satellite conference of the ICM its prize.
Starting 2019, the IMU Executive Committee will consist of
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CIMA F3 Capital structure (theories)
Reader Interactions
1. During the explanation of M&M proposition with taxes (1963) you stated that, the cost of bankruptcy has been completely ignored and therefore with the inclusion of bankruptcy cost WACC would
increase, which is similar to the traditional theory. If it so, then isn’t it an assumption that Financial distress does not carry any cost, is correct in the Example of M&M assumptions. Please
2. Well presented, I now understand it much better after watching the lecture video
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The monkeys that beat the market
The craziest investment strategy that actually worked
“A blindfolded monkey throwing darts at a newspaper’s financial pages could select a portfolio that would do just as well as one carefully selected by experts.”
Burton Malkiel wrote this in 1973, in his book “A Random Walk on Wall Street” and it stirred up a lot of controversy. Of course! The entire basis for active fund management was under attack, and a
statement like this made them look not just incompetent, but comically absurd as well.
What took the cake was the fact that someone actually tested out Malkiel and showed that he might be right! In an experiment tracked by the Wall Street Journal, a monkey picked a limited number of
stocks that consistently beat the market. Famous TV presenter John Stossel did his own version of the experiment taking the monkey's place and reported higher returns as well.
Monkeys were not the only animals that proved to be genius traders.
• In 2012, a cat named Orlando made headlines for beating a team of investors narrowly over the span of a year. It made its “trading decisions” by dropping a toy mouse on a grid of numbers
allocated to different companies. The result? The cat turned £5,000 into £5,542 while the investment professionals made £5,176!
• Another absurd case was that of Michael Marcovici, an Austrian concept artist breeding and training “rat traders” with names like Morgan Kleinsworth and Mr. Lehmann whom he claimed had a 57%
accuracy rate!
Crazy, right? But there's an unspoken truth here: The vast majority of animal prophets who pick stocks would not make the news, because they fail. Even a broken clock is correct twice a day, so is
there more to this than just entertaining anecdotes? Can the “monkey index fund” stand the test of an experiment?
That's exactly what Robert Arnott and his team at Research Affiliates LLC set out to find.
The great monkey experiment
The idea was simple: Instead of actually managing a monkey, the team simulated a monkey’s picks by randomly selecting 30 stocks from the top 1000 stocks by market capitalization and making an equally
weighted index from it. The same process was repeated 100 times and the average returns were compared from 1964 to 2012.
Burton Malkiel had said that a monkey would do as well as mutual funds. He was wrong. The monkey actually beat the market 96 times out of 100!
The graph above shows the distribution of the returns. In 75% of the cases, the monkey beat the market cap benchmark by more than 1%, and 30% of the time, it got a relative profit of greater than 2%!
The returns of the monkey index fund were not only higher, but they were also better in terms of risk-adjusted return based on the Sharpe ratio. In terms of standard deviation though, the risk was
slightly higher.
Round 2
This was not the only experiment. Researchers from Cass university leveraged Rob Arnott's experiment to design their own: They picked stocks from a pool of 1000 without the restriction of equal
weighting and compared the returns against the market - They did this for 10 million different portfolios for each year between 1968 and 2011. The results were mindblowing:
• An investment of $100 in the US market in 1968 would have made just under $5000 by the end of 2011.
• Half the monkeys generated more than $8,700.
• A quarter returned more than $9,100.
• 10 percent made more than $9,500 - A 940% profit or more!
But the study went further. The 3-year rolling average of the monkeys’ performance was taken and compared against the market cap fund year by year from 1972-2012 to check: What proportion of monkeys
beat the market every year? These were the results:
These numbers show how all-or-nothing the whole venture is. All the monkeys beat the market about 57% of the time, but they all underperformed the market about 31% of the time. The timing is also
revealing: The monkeys win during bull runs but underperform for long stretches during bear markets. This is where the psychological element comes into play. It’s all fun and games when the going is
good, but would you have the confidence to perform worse than the market for 5 to 6 years in a row betting on a monkey’s predictions?
Having said that, it’s still no joke that the monkeys beat the market cap funds about 60% of the time. The figures from earlier reveal that the average performance over the entire period is also
better than market cap funds.
So how did they do it? Were the monkeys stock-picking geniuses?
Indexing is the key
Before you run to the pet store looking for a dart-throwing monkey, let's try to figure out why this works in the first place. The reason for the monkeys’ success is hidden in where they came up
short. If you look again at the data collected by the Rob Arnott team, you can see two things:
1. Volatility (beta) is more in the case of the monkey index fund
2. Equally weighted market index funds beat even the monkey index funds
How market index funds are created might play an even bigger role than the individual stocks that are picked for the fund. In market cap based index funds like the S&P500, the weightage given to the
companies isn't equal - It's based on their market capitalization. In times of turmoil, these funds are supposed to give more stability to the portfolio because they don't fluctuate as much. But that
also means that your scope for growth is limited because there's only so much that big companies can grow.
In an equally-weighted index fund, on the other hand, the returns from the growth of small and value stocks are captured as well, but the price you pay is (supposedly) higher volatility. In the case
of the monkeys picking random stocks, this is what happened - Equal exposure to a few small stocks that saw massive growth balanced out the losses from other stocks. It seemed like a gamble though,
because of the risk involved.
From 2000 onwards, equally-weighted index funds have outperformed SPY by more than 100%. The catch is that though Sharpe ratio looks similar, the volatility was more in the case of equally-weighted
funds as shown by the standard deviation, with higher drawdowns as well during times of crashes. The higher returns provided by these stocks were a trade-off for the exposure to this volatility.
If one puts an infinite number of monkeys in front of (strongly built) typewriters and lets them clap away (without destroying the machinery), there is a certainty that one of them will come out
with an exact version of the 'Iliad.' Once that hero among monkeys is found, would any reader invest their life's savings on a bet that the monkey would write the 'Odyssey' next?
- Nicholas Nassim Taleb
The problem with fascinating strategies is that they might not be repeatable.
The extraordinary performance of randomly picked stocks does not mean that any pick you make will work. It just means that where there are outsized rewards, there are outsized risks as well. Most
unknown stocks that make the headlines for exponential growth fall into the category of either small stocks or value stocks, and if you invest in them, you are rewarded for the risk you are taking
on. Market beta, Value, and Size - Exposure to these three decide the nature of your portfolio.
What you should take away from this article is that your exposure to these three factors need not be constant! A market cap based Index fund might not be the only solution for safe investing. There
are other possibilities out there with a slightly different risk-reward ratio.
It might be worth it to take a look at what the alternatives are.
See you next time with another quick read!
This is a quick read. I write more in-depth articles like my strategy for consistent returns from the crypto market and investing strategies for every risk level on my weekly newsletter Market
Sentiment. You can subscribe here:
Disclaimer: I am not a financial advisor. Do not consider this as financial advice.
If you enjoyed this piece, please do us the HUGE favor of simply liking and sharing it with one other person who you think would enjoy this article! Thank you.
Always interesting, well researched and useful information. I'm so glad I signed up for your blog. Thanks again for your hard work!
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I just saw a video of michael reeves that made a fish pick stocks and compared it to listening to wall street bets. I think that the fish also outperformed the market.
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Section: New Results
A fractional Brownian field indexed by ${L}^{2}$ and a varying Hurst parameter
Participant : Alexandre Richard.
Using structures of Abstract Wiener Spaces and their reproducing kernel Hilbert spaces, we define a fractional Brownian field indexed by a product space $\left(0,1/2\right]×{L}^{2}\left(T,m\right)$,
where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Lévy fractional
Brownian motion and multiparameter fractional Brownian motion, and provides a setup for new ones. We prove that it has good incremental variance in both coordinates and derive certain continuity and
Hölder regularity properties. Then, we apply these general results to multiparameter and set-indexed processes, which proves the existence of processes with prescribed local Hölder regularity on
general indexing collections.
The family of fBm can be considered for the different Hurst parameters as a single Gaussian process indexed by $\left(h,t\right)\in \left(0,1\right)×{ℝ}_{+}$, which is the position we adopt. Besides,
the “time” indexing is replaced by any separable ${L}^{2}$ space. We prove that there exists a Gaussian process indexed by $\left(0,1/2\right]×{L}^{2}\left(T,m\right)$, with the additional constraint
that the variance of its increments is as well behaved as it was on $\left(0,1\right)×{ℝ}_{+}$, that is, for any compact of ${L}^{2}$, there is a constant $C>0$ such that for any $f$ in this compact,
and any $h,{h}^{"}\in \left(0,1/2\right)$,
$𝔼{\left({B}_{f}^{h}-{B}_{f}^{{h}^{"}}\right)}^{2}\le C\phantom{\rule{4pt}{0ex}}{\left(h-{h}^{"}\right)}^{2}.$ (22)
When looking at the ${L}^{2}$–fBf with a fixed $h$, we have the following covariance: for each $h\in \left(0,1/2\right]$,
$\begin{array}{c}\hfill {k}_{h}:\left(f,g\right)\in {L}^{2}×{L}^{2}↦\frac{1}{2}\left(m{\left({f}^{2}\right)}^{2h}+m{\left({g}^{2}\right)}^{2h}{-m\left(|f-g|}^{2}{\right)}^{2h}\right)\phantom{\ (23)
An important subclass of these processes is formed by processes restricted to indicator functions of subsets of $T$. In particular, multiparameter when $\left(T,m\right)=\left({ℝ}_{+}^{d},\mathrm
{Leb}.\right)$, and more largely set-indexed processes [62] ,[20] naturally appear and thus motivate generalization b), besides the inherent interest of studying processes over an abstract space.
To define this field, we used fractional operators on the Wiener space $W$ introduced in [56] , and first expressed the fractional Brownian field (indexed by $\left(0,1/2\right]×{ℝ}_{+}$) as a white
noise integral over $W$:
$\begin{array}{c}\hfill \phantom{\rule{1.em}{0ex}}\left\{{\int }_{W}〈{𝒦}_{h}{R}_{h}\left(·,t\right),w〉\phantom{\rule{4pt}{0ex}}\mathrm{d}{𝔹}_{w},\phantom{\rule{4pt}{0ex}}\left(h,t\right)\in \left
The advantage of this approach is to allow the transfer of techniques of calculus on the Wiener space to any other linearly isometric space with the same structure (those spaces are called Abstract
Wiener Spaces). Using the separability and reproducing kernel property of the Cameron-Martin spaces built from the kernels ${k}_{h},h\in \left(0,1/2\right]$, we prove the existence of a Brownian
field $\left\{{𝐁}_{h,f},\phantom{\rule{4pt}{0ex}}h\in \left(0,1/2\right],f\in {L}^{2}\left(T,m\right)\right\}$ over some probability space $\left(\Omega ,ℱ,ℙ\right)$. Some Hilbert space analysis then
provides the desired bound (22 ). Then, we used this to derive a sufficient condition for almost sure continuity of the fractional Brownian field, in terms of metric entropy.
For fixed $h$, we proved that the $h$-fractional Brownian motion has the strong local nondeterminism property, which allowed to compute a sharp estimate of its small deviations, that is, for a
compact $K$ of ${L}^{2}$:
$exp\left(-C\phantom{\rule{4pt}{0ex}}N\left(K,{d}_{h},\epsilon \right)\right)\le ℙ\left(\underset{f\in K}{sup}|{𝐁}_{f}^{h}|\le \epsilon \right)\le exp\left(-{C}^{-1}\phantom{\rule{4pt}{0ex}}N\left(K,
{d}_{h},\epsilon \right)\right)\phantom{\rule{4pt}{0ex}},$
where $N\left(K,{d}_{h},\epsilon \right)$ is the metric entropy of $K$, i.e., the minimal number of balls necessary to cover $K$ with ${d}_{h}$-balls (the metric induced by the $h$-fBm) of radius at
most $\epsilon$.
Finally, we looked at the Hölder regularity of the fBf, when the ${L}^{2}$ indexing collection is restricted to the indicator functions of the rectangles of ${ℝ}^{d}$ (multiparameter processes) or to
some indexing collection (in the sense of [62] ). This restriction permits to use local Hölder regularity exponents, in the flavour of what was done in [24] . When a regular path $𝐡:{L}^{2}\to \left
(0,1/2\right]$ is specified, this defines a multifractional Brownian field as ${𝐁}_{f}^{𝐡}={𝐁}_{𝐡\left(f\right),f}$, whose Hölder regularity at each point is proved to equal $𝐡\left(f\right)$ almost
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chaotic attractor reconstruction.
A system can be described by a vector of real numbers, called its state, that aims to provide a complete description of the system at some point in time. The set of all possible states is the
system’s phase space or state space. This space and a rule specifying its evolution over time defines a dynamical system. These rules often take the form of differential equations.
An ordered set of state values over time is called a trajectory. Depending on the system, different trajectories can evolve to a common subset of phase space called an attractor. The presence and
behavior of attractors gives intuition about the underlying dynamical system. We can visualize the system and its attractors by plotting the trajectory of many different initial state values and
numerically integrating them to approximate their continuous time evolution on discrete computers. For dynamical systems with more than 3 coordinates, the state space can be partially visualized by
mapping a subset of its coordinates to the x, y, and z axes.
Consider the Lorenz attractor, defined by the following differential equations:
\[ \begin{eqnarray} \frac{dx}{dt} &=& \sigma(y - x) \
\frac{dy}{dt} &=& x(\rho - z) - y \
\frac{dz}{dt} &=& xy - \beta z \
\end{eqnarray} \]
A common set of initial conditions is \(x = -8.0, y = 8.0, z = 27.0\) with parameters \(\sigma = 10, \rho = \frac{8}{3}, \beta = 28\). When integrated using these values, a butterfly-like attractor
is revealed. This is an example of a chaotic attractor, defined by aperiodic trajectories that diverge exponentially fast.
Of course, dynamical systems and chaos theory are entire fields of study that cannot be adequately summarized here. Our purpose is to demonstrate the reconstruction of chaotic attractors from
incomplete system measurements: for example, a time series from only the first of the Lorenz equations. Takens’ Embedding Theorem explains how the phase space of an attractor can be reconstructed
using time-delayed measurements of a single variable. These ideas will further be explored using the information theory functions found in the Computational Mechanics in Python (CMPy) package.
generating a time series.
We first need a framework for generating trajectories from dynamical systems with different numbers of equations and parameters. Fortunately, Python makes this relatively straightforward. Consider
the following code to generate a trajectory given a set of ODEs:
def generate(data_length, odes, state, parameters):
data = numpy.zeros([state.shape[0], data_length])
for i in xrange(5000):
state = rk4(odes, state, parameters)
for i in xrange(data_length):
state = rk4(odes, state, parameters)
data[:, i] = state
return data
This function allocates and fills a Numpy array with data_length measurements of a system’s state over time, resulting from the numerical integration of the ODEs. The first 5000 iterates are regarded
as transient and discarded: this number is a guess and may be increased, decreased, or removed altogether if desired.
The details of numerical integration are outside the scope of this tutorial, but the basic idea is this: we need a way to obtain the next state of a dynamical system given its current state and its
ODEs. Since we are dealing with continuous time equations using a discrete computer, we can approximate the solution of these ODEs by integrating over a discrete time step. Smaller steps trade
increased computation for greater accuracy. In addition to the step size dt, the integration technique also affects accuracy. The simplest method for integration is the Euler method. In the above
code the rk4 function is a fourth-order Runge-Kutta integrator, balancing accuracy and computation by averaging over a set of predictors:
def rk4(odes, state, parameters, dt=0.01):
k1 = dt * odes(state, parameters)
k2 = dt * odes(state + 0.5 * k1, parameters)
k3 = dt * odes(state + 0.5 * k2, parameters)
k4 = dt * odes(state + k3, parameters)
return state + (k1 + 2 * k2 + 2 * k3 + k4) / 6
Finally, we define the Lorenz equations and tie everything together with a helper function lorenz_generate that passes a typical set of initial state and parameter values to generate:
def lorenz_odes((x, y, z), (sigma, beta, rho)):
return numpy.array([sigma * (y - x), x * (rho - z) - y, x * y - beta * z])
def lorenz_generate(data_length):
return generate(data_length, lorenz_odes, \
numpy.array([-8.0, 8.0, 27.0]), numpy.array([10.0, 8/3.0, 28.0]))
We also define the Rössler equations:
def rossler_odes((x, y, z), (a, b, c)):
return numpy.array([-y - z, x + a * y, b + z * (x - c)])
def rossler_generate(data_length):
return generate(data_length, rossler_odes, \
numpy.array([10.0, 0.0, 0.0]), numpy.array([0.15, 0.2, 10.0]))
A time series from the first Lorenz equation is simple to plot:
data = lorenz_generate(2**13)
The Lorenz attractor was shown earlier; the code is below and uses Matplotlib’s experimental 3D plotting.
from mpl_toolkits.mplot3d.axes3d import Axes3D
figure = pylab.figure()
axes = Axes3D(figure)
axes.plot3D(data[0], data[1], data[2])
time delay embedding.
Delaying the time series produced by a single ODE creates a higher dimensional embedding and, by Takens’ Embedding Theorem, allows the phase space of the attractor to be reconstructed. If the
measurement variable at time \(t\) is defined by \(x(t)\), an \((n+1)\)-dimensional embedding is defined by:
\[ [x(t), x(t + \tau), \dots, x(t + n \tau)] \]
The choice of \(\tau\) determines the accuracy of the reconstructed attractor. Too small a value will plot the attractor along a line and too large a value will not reveal the structure of the
attractor (see this page for examples). Fraser and Swinney suggest using the first local minimum of the mutual information between the delayed and non-delayed time series, effectively identifying a
value of \(\tau\) for which they share the least information. (In the general case we may also need to identify the attractor’s phase space dimension — we ignore this problem here.)
The code below generates data from a dynamical system, discretizes the values into equal frequency bins, and then measures the mutual information \(I\) for increasing values of \(\tau\) up to some
maximum value. The numpy function roll performs the delay by shifting all values over to the left by \(\tau\). The series must then be shortened by \(\tau\) values since our data is of fixed length
and there are no additional values to shift in. The loop terminates early if the current \(I\) is larger than the previous \(I\), indicating we’ve found the first local minimum. We then embed this
time series into 3 dimensions using the corresponding value of \(\tau\).
# create time series
data = lorenz_generate(2**14)[0]
data = preprocess(data, quantize_cols=[0], quantize_bins=1000)
# find usable time delay via mutual information
tau_max = 100
mis = []
for tau in range(1, tau_max):
unlagged = data[:-tau]
lagged = numpy.roll(data, -tau)[:-tau]
joint = numpy.hstack((unlagged, lagged))
mis.append(mutual_information(joint, normalized=True))
if len(mis) > 1 and mis[-2] < mis[-1]: # return first local minima
tau -= 1
print tau, mis
# plot time delay embedding
figure = pylab.figure()
axes = Axes3D(figure)
data_lag0 = data[:-2].flatten()
data_lag1 = numpy.roll(data, -tau)[:-2].flatten()
data_lag2 = numpy.roll(data, -2 * tau)[:-2].flatten()
axes.plot3D(data_lag0, data_lag1, data_lag2)
Varying the code above, we can plot the mutual information for all values of \(\tau\):
The value corresponding to the first minimum of \(I\) leads to the following reconstruction:
Finally, the same reconstruction for the Rössler attractor:
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80 Bloor Street West | 263.4m | 78s | Krugarand | Arcadis
Member Bio
Apr 24, 2007
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http://app.toronto.ca/DevelopmentAp...icationsList.do?action=init&folderRsn=3444295 80 BLOOR ST W
Ward 27 - Tor & E.York District
Rezoning application to develop the site into a 68 storey residential mixed use building for 68 floors (plus mechanical penthouse) and will include 39,810m2 of residential area (85 bachelor units,
300 one-bedrooms, 123 two-bedrooms, and 57 three-bedrooms) and 3,465m2 of retail space. There will be 181 parking spaces provided below grade for residential use.
Member Bio
Apr 24, 2007
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Redevelopment proposal for this building,
80 Bloor St W
Google Maps view
. This address is one property west of the building that fronts the NW corner of Bay & Bloor, though with a proposal this size it's a possibility it may be combined (?). Currently it's an office
tower, with a couple of retailers at grade and also a Good Life Fitness. The new proposal details does not indicate any replacement office space, which will certainly raise some points of discussion.
To its west is the Harry Rosen flagship.
Last edited:
Member Bio
May 16, 2007
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Yeah, there's no way the city is going to let that office space go (with that said, I have no idea what the zoning is here).
Member Bio
Oct 1, 2012
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Damn it. Why couldn't they just buy that dingy Scotiabank down the street?
Should be around 232m plus or minus 10m, if it is built to that number of floors.
I took all the purely residential towers in Toronto that are or will be at least 180m tall (22 residential buildings built, under construction and proposed) and divided their combined official
heights (4.3917 km) by their combined number of floors (1,445 floors), and multiplied that by 68 floors to give a 'typical' official height for a 68-floor residential building of just over 232m.
Normalizing each of these 22 purely residential buildings to 68 floors results in a range from 218.6m to 245.7m, with the middle 90% of them falling between 221m and 245m in official height.
Last edited:
Member Bio
May 19, 2007
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Sep 1, 2012
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from a year ago when i was ArchitectsAlliance, i believe they were working on the tower for 80 Bloor st. west
their inspiration for the design of the tower were mainly based upon Jean Nouvel's MoMa Tower
Prelim Renders from July 2012
Last edited:
TOBuilt database entry
Completed : 1973
Architect : Bregman + Hamann Architects
Developer : Courtot Investments Ltd.
from a year ago when i was ArchitectsAlliance, i believe they were working on the tower for 80 Bloor st. west
their inspiration for the design of the tower were mainly based upon Jean Nouvel's MoMa Tower
If this turns out to be the case, it might end up a lot more than 232m in height. The Jean Nouvel tower is designed at 320m and 72 floors, so 68 floors at the same ratio would be 302m. Plus of course
being a very definite "non-box".
-- although now that I see those July 2012 renders, 80 Bloor West is not nearly as elongated at the MoMO proposal. But it could still be in the 250m-260m range.
Last edited:
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Sep 1, 2012
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Prelim Render of Courtyard from July 2012
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Sep 20, 2013
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Is aA designing the tower, or are those renderings just an exercise?
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Mar 8, 2010
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what it would look like at 240m, it is the white one:
Staff member
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Apr 23, 2007
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If this turns out to be the case, it might end up a lot more than 232m in height. The Jean Nouvel tower is designed at 320m and 72 floors, so 68 floors at the same ratio would be 302m. Plus of
course being a very definite "non-box".
edit -- although now that I see those July 2012 renders, 80 Bloor West is not nearly as elongated at the MoMO proposal. But it could still be in the 250m-260m range.
I doubt the MoMA Tower inspiration has anything to do with floor heights.
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Apr 26, 2007
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Wow, that would be something!
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May 5, 2007
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Completed : 1973
Architect : Bregman + Hamann Architects
Developer : Courtot Investments Ltd.
Nothing special and no love loss
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In this comprehensive guide, we will explore the BINOM.DIST.RANGE formula in Excel. This formula is used to calculate the probability of a specific range of successes in a given number of trials,
following a binomial distribution. The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of trials, each with the same probability of
success. The BINOM.DIST.RANGE formula is particularly useful in various fields, such as finance, statistics, and quality control, among others.
BINOM.DIST.RANGE Syntax
The syntax for the BINOM.DIST.RANGE formula in Excel is as follows:
=BINOM.DIST.RANGE(number_of_trials, probability_of_success, number_of_successes1, [number_of_successes2])
• number_of_trials (required) – The total number of trials or experiments.
• probability_of_success (required) – The probability of success in each trial, expressed as a decimal between 0 and 1.
• number_of_successes1 (required) – The lower bound of the range of successes for which you want to calculate the probability.
• number_of_successes2 (optional) – The upper bound of the range of successes for which you want to calculate the probability. If omitted, the formula will calculate the probability of exactly
number_of_successes1 successes.
BINOM.DIST.RANGE Examples
Let’s look at some examples of how to use the BINOM.DIST.RANGE formula in Excel.
Example 1: Suppose you have a coin that you will flip 10 times, and you want to know the probability of getting exactly 5 heads. You can use the BINOM.DIST.RANGE formula as follows:
=BINOM.DIST.RANGE(10, 0.5, 5)
In this example, the number_of_trials is 10, the probability_of_success is 0.5 (since there is a 50% chance of getting heads), and the number_of_successes1 is 5. The result will be the probability of
getting exactly 5 heads in 10 coin flips.
Example 2: Suppose you have a quality control process where the probability of a defective item is 0.02. You want to know the probability of finding between 2 and 4 defective items in a sample of 100
items. You can use the BINOM.DIST.RANGE formula as follows:
=BINOM.DIST.RANGE(100, 0.02, 2, 4)
In this example, the number_of_trials is 100, the probability_of_success is 0.02, the number_of_successes1 is 2, and the number_of_successes2 is 4. The result will be the probability of finding
between 2 and 4 defective items in a sample of 100 items.
BINOM.DIST.RANGE Tips & Tricks
Here are some tips and tricks to help you get the most out of the BINOM.DIST.RANGE formula in Excel:
• Remember that the probability_of_success should be expressed as a decimal between 0 and 1. To convert a percentage to a decimal, divide the percentage by 100.
• If you want to calculate the probability of a single specific outcome (e.g., exactly 5 successes), you can omit the number_of_successes2 argument.
• Use the BINOM.DIST.RANGE formula to analyze various scenarios by changing the number_of_trials, probability_of_success, and range of successes to see how the probability changes.
Common Mistakes When Using BINOM.DIST.RANGE
Here are some common mistakes to avoid when using the BINOM.DIST.RANGE formula:
• Using a percentage instead of a decimal for the probability_of_success. Remember to divide the percentage by 100 to convert it to a decimal.
• Using a negative number or a number greater than 1 for the probability_of_success. The probability should always be between 0 and 1, inclusive.
• Using non-integer values for the number_of_trials or the number_of_successes arguments. These values should always be whole numbers.
Why Isn’t My BINOM.DIST.RANGE Working?
If you’re having trouble with the BINOM.DIST.RANGE formula, consider the following troubleshooting tips:
• Double-check your formula syntax and ensure that you have entered the correct arguments in the correct order.
• Ensure that the probability_of_success is expressed as a decimal between 0 and 1.
• Make sure that the number_of_trials and number_of_successes arguments are whole numbers.
• If you’re still having trouble, try breaking down the formula into smaller parts and checking each part individually to identify the source of the issue.
BINOM.DIST.RANGE: Related Formulae
Here are some related formulae that you might find useful when working with the BINOM.DIST.RANGE formula:
• BINOM.DIST: Calculates the individual probability of a specific number of successes in a given number of trials, following a binomial distribution.
• BINOM.INV: Calculates the smallest value for which the cumulative binomial distribution is greater than or equal to a specified criterion.
• POISSON.DIST: Calculates the Poisson probability distribution for a given number of events in a fixed interval.
• NORM.DIST: Calculates the normal (Gaussian) probability distribution for a given value, mean, and standard deviation.
• HYPGEOM.DIST: Calculates the hypergeometric probability distribution for a given number of successes, sample size, population size, and number of successes in the population.
By understanding the BINOM.DIST.RANGE formula and its related formulae, you can perform a wide range of probability calculations and analyses in Excel. This can be particularly useful in fields such
as finance, statistics, and quality control, among others.
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Research Projects Ideas Computational Geometry
Computational Geometry Research Projects Ideas, Topics, and Areas
Latest and New Computational Geometry Research Topics Areas and Ideas for writing the research paper.
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A Gas At Fixed Temperature Is Kept In A Closed Vessel. Some More Amount Of The Same Gas Is Added To The Vessel Without Altering The Temperature. What Will Be The Change In Pressure? - ConceptEra
A gas at fixed temperature is kept in a closed vessel. Some more amount of the same gas is added to the vessel without altering the temperature. What will be the change in pressure?
For n moles of a gas the equation of state is given by, PV = n RT. If m and M be the mass and molecular weight of the gas, then n = m/M and the equation of statereduces to PV =m/MRT M or, P =mRT/MV
At a fixed temperature (T = constant) if more amount of gas is added to a closed vessel (V = constant), then mass (m) of the gas increases. From the equation, it is evident that pressure of the gas
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All taken from Strategywiki.
Chrono Trigger[edit]
Note that an asterisk (*) indicates that the move is magical; in most cases these are elemental techniques that will benefit or suffer depending on the enemy's elemental properties and defensive
Attacks performed by executing the "Attack" function in battle are non-elemental, physical techniques that strike an enemy a single time and have the chance to deal double damage if a "crit"
(critical strike) occurs.
• Hit/HIT: this is the character's "Hit" stat value (an integer); it represents a character's accuracy. It only applies to Lucca and Marle's calculations. 1 Hit is equal to .66 Attack for Lucca and
• Power/PWR: this is the character's "Power" stat value (an integer); it represents a character's physical strength. It applies to Ayla's and all of the males' calculations. 1 PWR is equal to 1.33
(repeating) attack for males, and 1.75 attack for Ayla.
• Level: the character's level. This affects the statistics of a character and Ayla's attack value formula.
• Weapon: this is the weapon's power value (integer). It affects all characters except for Ayla.
• Attack: the value next to the small weapon icon, found on the pause menu next to a character's name; this is calculated from the Power and Weapon values. The formulas that produce the attack
value first produce a decimal value that is then rounded to the nearest whole number.
• Damage: the value seen when attacking an enemy.
Character Attack value formula
Ayla Attack = ((Power × 1.75) + (Level² ÷ 45.5))
Lucca and Marle Attack = ((Hit + Weapon) × 2/3)
Males Attack = ((Power × 4/3) + (Weapon × 5/9))
At the start of the game, Crono's weapon, Wood Sword, has a Weapon value of 3. At level 1, Crono has a PWR value of 5. His attack value is first calculated as 8.33333333 recurring, but it is then
rounded down to 8.
The base damage (theoretical minimum) for a character's "Attack" is their attack value multiplied by two. Additionally, a random number is generated to modify the damage and give it a more realistic,
dynamic appearance. The random number is affected by the character's level and/or attack value (it is difficult to differentiate between the two because as characters level, they gain the stats (PWR
or HIT) that increase their attack value). Also note that the resulting damage can be modified by an enemy's augmented defense.
Damage = ((Attack * 2) + (Random))
Crono at the start of the game (attack 8) deals a base of 16 damage, with a possible increase from the random value.
Unique formulae[edit]
Robo's Crisis Arm[edit]
Robo's Crisis Arm is a very dynamic weapon that can be either very strong (over 4,000) or very weak (0 damage). The damage dealt is based off of Robo's calculated attack value and the last digit of
his HP (the attack value is multiplied by this digit). This weapon is best used later in the game when Robo has 999 HP so that the last digit is usually 9. If you want to maximize his damage, all you
have to do is keep him healed.
Attack formula
First calculate his attack with "Attack = ((Power × 4/3) + (Weapon × 5/9))". Since Crisis Arm only has a power value of 1, when it is multiplied by 5/9, the resulting fraction (.55 repeating) does
not get rounded up in the SNES version. Therefore, a power of 89 results in an attack value of 119, and a power of 91 results in an attack value of 121.
Damage formula
• LDHP: this is the last digit of Robo's current HP.
Crisis Arm Damage = ((Attack/2 * LDHP) * 2)
• Example: if Robo's PWR is 99 (minimum level 59), then his Attack will be 132 (66 when divided by two). The following expected damages are calculated based on the last digit of his HP:
Expected damage
Each digit (LDHP value; except for zero) is worth as much as the attack value (in this case, it's 132).
LDHP Value Formula Theoretical minimum
0 ((66 × 0) × 2) 0
1 ((66 × 1) × 2) 132
2 ((66 × 2) × 2) 264
3 ((66 × 3) × 2) 396
4 ((66 × 4) × 2) 528
5 ((66 × 5) × 2) 660
6 ((66 × 6) × 2) 792
7 ((66 × 7) × 2) 924
8 ((66 × 8) × 2) 1056
9 ((66 × 9) × 2) 1188
Theoretical maximum
The theoretical maximum for each digit is very difficult to calculate, as it is based on a range of random numbers that scale with attack power (perhaps even character level). However, the general
rule of thumb for the random range is: Random number for Crisis Arm damage = (LDHP × 14). Therefore, in the table above, the damage you can deal with Crisis Arm without an accessory or a critical
strike is 1188-1314.
Absolute maximum
If you equip Robo with the Crisis Arm and Prismspecs, it will boost his attack value to about 185 (instead of the expected 255). At this point, he can crit for over 3,500 damage!
Ayla's Bronze Fist[edit]
• How to get: reach level 99 with Ayla.
On a critical strike, this will deal 9,999 damage. By the time you acquire this, Ayla's crit chance will be extremely high; from then on you can expect to deal major damage consistently.
Lucca's Wondershot[edit]
This weapon randomly changes attack power, it can do the following:
• 1/10 × base damage.
• 1/2 × base damage.
• 1 × base damage.
• 2 × base damage.
• 3 × base damage.
Magus's Doomsickle[edit]
This weapon grows in strength significantly as your allies die.
Number of fallen allies Base damage multiplier
• Defense: decreases the amount of physical damage a character takes. Is affected by the total equipped armor value (helm and body) and stamina.
• EV./Evasion: increases a character's chance to dodge regular, physical attacks. Magical attacks cannot be dodged.
• M DEF./Magic Defense: decreases the amount of damage taken from magical attacks (including non-elemental, but excluding physical).
• Stamina/Vigor: increases defense by one per point.
Stat Formula Maximum value
Defense = (Helm + Body + Stamina) • Males: 229 (SNES), 232 (DS; Magus can reach 234)
Defense • Females: 243 (SNES/DS)
Evasion Chance to evade = ?? ** (99)
Magic damage reduction Magic damage taken = ??
Physical damage reduction Physical damage taken = ??
Maximizing physical defense[edit]
• SNES: equip OzziePants, Moon Armor, and Power Seal.
• DS: equip Ozzie Pants, Saurian Leathers or Regal Plate or Shadowplume Robe (Magus only), and Power Seal.
Both versions: equip OzziePants, Prism Dress, and Power Seal.
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Conservation of Angular Momentum - Knowunity
Conservation of Angular Momentum: AP Physics 1 Study Guide
Hello, budding physicists! Get ready to dive into the wonderful whirl of Conservation of Angular Momentum. So fasten your seat belts (or should I say tie your capes, because we're about to spin into
action like superheroes), and let's unravel some physics magic! 🌟🔄
Angular Momentum: The Spin Doctor
Ah, angular momentum. You might think of it as the cooler, edgier sibling of linear momentum. Angular momentum (L) is the measure of an object's rotation and is given by the equation ( L = I \omega
), where ( I ) is the moment of inertia, and ( \omega ) is the angular velocity. The units for angular momentum are kilogram meters squared per second (kg m²/s). If no external torques are acting on
the system, angular momentum remains as constant as your playlist on repeat. 🎧
To illustrate, consider the graceful ice skater:
• Wide Arms, Slow Spin: Imagine a skater with arms extended. This position has a higher moment of inertia (I) and lower angular velocity (( \omega )), meaning more mass is spread out.
• Hug That Body, Fast Spin: Now, as the skater pulls their arms in, the moment of inertia decreases but the angular velocity increases. 🎯
This of course means the skater spins faster—with all the elegance only Sir Isaac Newton could conjure up. 🌪️
Cosmic Ballet: The Dance of Planets
Even the stars and planets are dancing to the tune of angular momentum! Let’s take a peek at planetary motion:
• Constant Companion: Planets move in elliptical orbits, returning to the same point after a full orbit. Considering Kepler’s Second Law, they sweep out equal areas in equal time.
• Closest Friends, Fastest Dances: When a planet is nearer to its star (think of it as getting cozy), its linear velocity increases due to a stronger gravitational pull. Yet, the angular velocity
remains unchanged - a cosmic waltz of conserved angular momentum. ✨
Example Exercise: Unstoppable Rods and Disks
Imagine a rod pivoting on a frictionless surface—the stuff hardcore physics dreams are made of. A disk slides towards it and collides. Let’s perform some angular magic:
Pre-Collision: The disk has a rotational inertia and initial position set for ultimate adventure and wackiness.
During Collision: All the valuable angular momentum held by the sliding disk (its L = mvr) is transferred, making the rod and disk rotate together.
Post-Collision: Now combine the moment of inertia for both objects. The new angular velocity (( \omega )) is found by solving ( L = I \omega ). Bonus points if you do it while humming "The Circle of
Life". 🌐
In this scenario, if the disk bounces off rather than sticking, the rod will spin faster. Why? Because bouncing means a larger change in momentum direction—think of it like going from a rock band to
a pop band overnight. 🎸➡🎤
Key Concepts to Spin Your Noggin
• Conservation of Angular Momentum: The total angular momentum in a system stays the same unless external torque barges in. It’s like a rule at a dance-off—no external interruptions!
• Angular Speed & Velocity: These are your measures for how fast something spins. Angular speed is in radians per second (rad/s) and angular momentum comes in hot at ( L = I \omega ).
• Moment of Inertia: This describes how mass is distributed regarding rotation. More distributed mass equals a higher moment of inertia. Think of a ballerina and her tutu versus a pair of weighted
boots. 🩰
• Kepler's 2nd Law: Things change but some rules are eternal. Planets sweep out equal areas in equal time, leading to conserved angular momentum.
Final Takeaway
The conservation of angular momentum is like a professional dancer swirling across the floor, always keeping balance and grace. Whether it's a skater pulling in their arms or planets orbiting in
endless dance, the principles remain solid and impactful.
Now, go forth and spin that knowledge into gold on your AP Physics 1 exam—because you've got the moves, and now, you've got the know-how too! ✨🔄🎉🏆
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The Ultimate ACCUPLACER Math Formula Cheat Sheet
If you’re taking the ACCUPLACER Math test in a few weeks or months, you might be anxious about how to remember ALL the different formulas and math concepts and recall them during the test.
ACCUPLACER Math covers a wide range of topics—from as early as elementary school to high school.
While you have probably learned many of these formulas at some point, it may have been a long time since you’ve used them. This is where most test takers have a hard time preparing for the test.
So, what formulas do you need to have memorized for the ACCUPLACER Math before the test day?
Following is a quick formula reference sheet that lists all important ACCUPLACER Math formulas you MUST know before you sit down for the test.
If you learn every formula in this ACCUPLACER Math Formula Cheat Sheet, you will save yourself valuable time on the test and probably get a few extra questions correct.
Looking for a comprehensive and complete list of all ACCUPLACER Math formulas? Please have a look at ACCUPLACER Math Formulas.
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ACCUPLACER Math Cheat Sheet
A number expressed in the form \(\frac{a}{b}\)
Adding and Subtracting with the same denominator:
Adding and Subtracting with the different denominator:
Multiplying and Dividing Fractions:
\(\frac{a}{b} × \frac{c}{d}=\frac{a×c}{b×d}\)
\(\frac{a}{b} ÷ \frac{c}{d}=\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}\)
Is a fraction written in a special form? For example, instead of writing \(\frac{1}{2}\) you can write \(0.5\).
Mixed Numbers
A number is composed of a whole number and a fraction. Example: \(2 \frac{2}{ 3}\) Converting between improper fractions and mixed numbers: \(a \frac{c}{b}=a+\frac{c}{b}= \frac{ab+ c}{b}\)
Factoring Numbers
Factor a number means breaking it up into numbers that can be multiplied together to get the original number. Example:\(12=2×2×3\)
\( \{…,-3,-2,-1,0,1,2,3,…\} \)
Includes: zero, counting numbers, and the negative of the counting numbers
Real Numbers
All numbers that are on a number line. Integers plus fractions, decimals, and irrationals, etc.) (\(\sqrt{2},\sqrt{3},π\), etc.)
Order of Operations
(parentheses/ exponents/ multiply/ divide/ add/ subtract)
Absolute Value
Refers to the distance of a number from, the distances are positive as the absolute value of a number cannot be negative. \(|-22|=22\)
A ratio is a comparison of two numbers by division. Example: \(3 : 5\), or \(\frac{3}{5}\)
Use the following formula to find the part, whole, or percent
part \(=\frac{percent}{100}×whole\)
Proportional Ratios
A proportion means that two ratios are equal. It can be written in two ways:
\(\frac{a}{b}=\frac{c}{d}\) , \(a: b = c: d \)
Percent of Change
\(\frac{New \ Value \ – \ Old \ Value}{Old Value}×100\%\)
Expressions and Variables
A variable is a letter that represents unspecified numbers. One may use a variable in the same manner as all other numbers: Addition: \(2+a\): \(2\) plus a
Subtraction: \(y-3\) : \(y\) minus \(3\)
Division: \(\frac{4}{x}\) : 4 divided by x
Multiplication: \(5a\) : \(5\) times a
The values of the two mathematical expressions are equal.
Distance from A to B:
\(\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2 }\)
Parallel and Perpendicular lines:
Parallel lines have equal slopes. Perpendicular lines (i.e., those that make a \(90^° \) angle where they intersect) have negative reciprocal slopes: \(m_{1}\) .\(m_{2}=-1\).
Parallel Lines (l \(\parallel\) m)
Mid-point of the segment AB:
M (\(\frac{x_{1}+x_{2}}{2} , \frac{y_{1}+y_{2}}{2}\))
Slope of the line:
\(\frac{y_{2}- y_{1}}{x_{2} – x_{1} }=\frac{rise}{run}\)
Point-slope form:
Given the slope m and a point \((x_{1},y_{1})\) on the line, the equation of the line is
\((y-y_{1})=m \ (x-x_{1})\).
Slope-intercept form:
given the slope m and the y-intercept b, then the equation of the line is:
\(=x^2+(b+a)x +ab\) “Difference of Squares”
\(a^2-b^2= (a+b)(a-b)\)
\(a^2+2ab+b^2=(a+b)(a+b) \)
\(a^2-2ab+b^2=(a-b)(a-b)\) “Reverse FOIL”
\(x^2+(b+a)x+ab=\) \((x+a)(x+b)\)
Refers to the number of times a number is multiplied by itself.
\(8 = 2 × 2 × 2 = 2^3\)
Scientific Notation:
It is a way of expressing numbers that are too big or too small to be conveniently written in decimal form.
In scientific notation all numbers are written in this form: \(m \times 10^n\)
Scientific notation:
The number we get after multiplying an integer (not a fraction) by itself. Example: \(2×2=4,2^2=4\)
Square Roots:
A square root of \(x\) is a number r whose square is \(x : r^2=x\)
\(r\) is a square root of \(x\)
All triangles:
Area \(=\frac{1}{2}\) b . h
Angles on the inside of any triangle add up to \(180^\circ\).
These triangles have three equal sides, and all three angles are \(60^\circ\).
An isosceles triangle has two equal sides. The “base” angles (the ones opposite the two sides) are equal (see the \(45^\circ\) triangle above).
Area \(=πr^2\)
Circumference \(=2πr\)
Full circle \(=360^\circ\)
(Rhombus if l=w)
Regular polygons are n-sided figures with all sides equal and all angles equal.
The sum of the inside angles of an n-sided regular polygon is
\((n-2) .180^\circ\).
Area of a trapezoid:
\(A =\frac{1}{2} h (b_{1}+b_{2})\)
Surface Area and Volume of a Rectangular/right prism:
Surface Area and Volume of a Cylinder:
\(SA =2πrh+2πr^2\)
\(V =πr^2 h \)
Surface Area and Volume of a Cone
\(SA =πrs+πr^2\)
\(V=\frac{1}{3} \ πr^2 \ h\)
Surface Area and Volume of a Sphere
\(SA =4πr^2\)
\(V =\frac{4}{3} \ πr^3\)
(p \(=\) perimeter of base B; \(π ~ 3.14 \))
Simple interest:
(I = interest, p = principal, r = rate, t = time)
mean: \(\frac{sum \ of \ the \ data}{of \ data \ entires}\)
value in the list that appears most often
largest value \(-\) smallest value
The middle value in the list (which must be sorted)
Example: median of
\( \{3,10,9,27,50\} = 10\)
Example: median of
\( \{3,9,10,27\}=\frac{(9+10)}{2}=9.5 \)
\( \frac{sum \ of \ terms}{number \ of \ terms}\)
Average speed
\(\frac{total \ distance}{total \ time}\)
\(\frac{number \ of \ desired \ outcomes}{number \ of \ total \ outcomes}\)
The probability of two different events A and B both happening are:
P(A and B)=p(A) .p(B)
as long as the events are independent (not mutually exclusive).
Powers, Exponents, Roots
\(x^a .x^b=x^{a+b}\)
\(\frac{x^a}{x^b} = x^{a-b}\)
\(\frac{1}{x^b }= x^{-b}\)
\((xy)^a= x^a .y^a\)
\(\sqrt{xy}=\sqrt{x} .\sqrt{y}\)
\((-1)^n=-1\), if n is odd.
\((-1)^n=+1\), if n is even.
If \(0<x<1\), then
Simple Interest
The charge for borrowing money or the return for lending it.
Interest = principal \(×\) rate \(×\) time
Positive Exponents
An exponent is simply shorthand for multiplying that number of identical factors. So \(4^3\) is the same as (4)(4)(4), three identical factors of 4. And \(x^3\) is just three factors of x, \((x)(x)
Negative Exponents
A negative exponent means to divide by that number of factors instead of multiplying.
So \(4^{-3}\) is the same as \( \frac{1}{4^3}\) and
Factorial- the product of a number and all counting numbers below it.
8 factorial \(=8!=\)
5 factorial \(=5!=\)
2 factorial \(=2!=2× 1=2\)
Multiplying Two Powers of the SAME Base
When the bases are the same, you find the new power by just adding the exponents
\(x^a .x^b=x^{a+b }\)
Powers of Powers
For the power of power: you multiply the exponents.
Dividing Powers
\(\frac{x^a}{x^b} =x^a x^{-b}= x^{a-b}\)
The Zero Exponent
Anything to the 0 power is 1.
\(x^0= 1\)
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A deep learning approach to the weak lensing analysis of galaxy clusters
Spinelli, Claudia
A deep learning approach to the weak lensing analysis of galaxy clusters.
[Laurea magistrale], Università di Bologna, Corso di Studio in
Astrofisica e cosmologia [LM-DM270]
, Documento full-text non disponibile
Il full-text non è disponibile per scelta dell'autore. (
Contatta l'autore
In the present work we propose an innovative method which can potentially allow the unbiased estimate of galaxy cluster structural parameters from weak gravitational lensing maps using deep learning
techniques. This method represents a viable alternative to more classical and time consuming approaches which are not ideal when dealing with large datasets. We readapted the Inception-v4
architecture to our purpose and implemented it in Pytorch. This model is trained with labeled data and then applied to unlabeled data in order to predict from the input maps, for each cluster, the
virial mass, the concentration, the number of substructures and the mass fraction in substructures. The determination of these quantities is particular important because of their possible
applications in several cosmological tests. The training and test sets consist of maps produced with the MOKA software, which generates semi-analytical mass distributions of galaxy clusters and
computes convergence and reduced shear maps. The simulated halos are placed at different redshifts, i.e. z = 0.25, 0.5, 0.75, 1. The complexity and the level of realism of the simulations has been
increased while performing a sequence of experiments. We train the model on noiseless convergence maps, which are then replaced with noiseless reduced shear maps. Finally, the same model is trained
using reduced shear maps which include shape noise for a given number density of lensed galaxies. We find that our model produces more accurate and precise measurements of the virial masses and
concentrations compared to the standard approach of fitting the convergence profile. It is able to learn information about the triaxial shape of the clusters during the training phase. Consequently,
well known biases due to projection effects and substructures are strongly mitigated. Even when a typical observational noise is added to the maps, the network is capable to measure the cluster
structural parameters well.
In the present work we propose an innovative method which can potentially allow the unbiased estimate of galaxy cluster structural parameters from weak gravitational lensing maps using deep learning
techniques. This method represents a viable alternative to more classical and time consuming approaches which are not ideal when dealing with large datasets. We readapted the Inception-v4
architecture to our purpose and implemented it in Pytorch. This model is trained with labeled data and then applied to unlabeled data in order to predict from the input maps, for each cluster, the
virial mass, the concentration, the number of substructures and the mass fraction in substructures. The determination of these quantities is particular important because of their possible
applications in several cosmological tests. The training and test sets consist of maps produced with the MOKA software, which generates semi-analytical mass distributions of galaxy clusters and
computes convergence and reduced shear maps. The simulated halos are placed at different redshifts, i.e. z = 0.25, 0.5, 0.75, 1. The complexity and the level of realism of the simulations has been
increased while performing a sequence of experiments. We train the model on noiseless convergence maps, which are then replaced with noiseless reduced shear maps. Finally, the same model is trained
using reduced shear maps which include shape noise for a given number density of lensed galaxies. We find that our model produces more accurate and precise measurements of the virial masses and
concentrations compared to the standard approach of fitting the convergence profile. It is able to learn information about the triaxial shape of the clusters during the training phase. Consequently,
well known biases due to projection effects and substructures are strongly mitigated. Even when a typical observational noise is added to the maps, the network is capable to measure the cluster
structural parameters well.
Altri metadati
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Fuzzy sliding mode controller design for semi-active seat suspension with neuro-inverse dynamics approximation for MR damper
To improve the ride comfort of car, this paper proposed a semi-active seat suspension with magneto-rheological (MR) damper and designed a new fuzzy sliding mode controller with expansion factor
(FSMCEF) based on the neuro-inverse dynamics approximation of the MR damper. This FSMCEF combines the advantages of both sliding mode controller (SMC) and fuzzy controller (FC) with expansion factor
(EF), and it takes an ideal skyhook model as the reference, and creates a sliding mode control law based on the errors dynamics between the seat suspension and its reference model. Further fuzzy
rules are used to suppress the chattering occurred in the above sliding mode control by fuzzifying the sliding mode surface and its derivative. Moreover, in order to compute the required control
current for MR damper after solving the desired control force using FSMCEF, this paper presented a BP algorithm based neural network inverse model, located between the FSMCEF and the MR damper,
taking the displacement, velocity of the MR damper and the desired control force output by FSMCEF as its input, and predicting the control current required to input MR damper. The predicting error
and stability of the neural network inverse model for MR is investigated by sample testing. In addition, the stability analysis of FSMCEF is also completed by under nominal system and non-nominal
system with parameter uncertainty and external disturbance. The results of numerical simulations show that the vibration reduction effect of the semi-active seat is obviously improved using FSMCEF
compared with using PID controller and SMC.
1. Introduction
Suspension is the main factor that affects the car smoothness and ride comfort, and its type and design have been a basic and important topic for the new car development [1]. In all the three prime
suspension types, semi-active suspension is recognized to be the compromise solution to reduce vibration and improve ride comfort because of its higher performance improvement at less cost and energy
consumption relative to active suspension. MR dampers are usually employed in the practical semi-active suspension for they have low control voltage and satisfactory response speed [2]. However, MR
damper also has high nonlinear features such as hysteresis and saturation which make its control much more difficult. Recently, much attention has been paid to the control techniques of the car
suspension systems with MR dampers. Some control methods have been used, such as fuzzy control [3], optimal control [4], preview control [5], LPV control [6, 7] and robust ${H}_{\infty }$ control
[8]. Literature [9] studied neural network semi-active vibration control of a quarter car suspensions with MR damper based on the Bouc-Wen model of MR damper. Literature [10] studied the switch
control of a quarter car suspension vibration. Due to the inherent highly nonlinear characteristics of MR damper, how to determine the input voltage corresponding to the control force worked out by
suspension controller is need to be solved when MR damper used in vibration control. The solutions are usually based on switching control law to adjust the input voltage and switch the optimal
control algorithm [11-13]. The input voltage of the MR damper switches between the minimum and maximum without being a continuous adjustable control signal, which limits the performance of the MR
damper. Neural network can approximate any nonlinear function, so in this paper the neural network technology is used to simulate the inverse dynamic characteristics of MR damper and to create a
continuous signal for MR damper as its nonlinear controller.
D’Amato and Viassolo demonstrated a fuzzy control strategy for active suspension systems to minimize vertical car body acceleration for improving the ride comfort and to avoid hitting suspension
limits for preserving the component lifetime [14]. Miao et al. developed an adaptive fuzzy controller for a quarter-car active suspension system to effectively suppress the vehicle’s vibration and
disturbance so as to improve ride comfort [15]. Sliding mode control (SMC) has been widely applied as a robust nonlinear control algorithm and its application in active suspension has recently
attracted the interest of many researchers [16-20]. Yoshimura et al constructed an active suspension system for a quarter car model with pneumatic actuator, and used SMC with sliding mode surface
created by LQ theory [21]. Yao et al. built a polynomial model for MR damper by using experimental data, and designed a model reference sliding mode controller with uniform reaching law for the
semi-active suspension [22]. Chen and Zhao designed a sliding mode controller for a semi-active seat suspension system, but they did not consider the type and dynamics of semi-actuator [23]. SMC has
better robustness and can be applied in the presence of model uncertainties and external disturbances, ensuring the system stability. However, using SMC to control a plant often requires high control
gains, and results easily in a chattering phenomenon because the control variable is changed drastically during the control process. As a consequence, to investigate the combined advantages of SMC
with the fuzzy logic controller has become an active field of research [24-28]. Lin et al proposed a fuzzy sliding mode controller (FSMC) to control an active suspension system and evaluated its
control performance [29]. In order to improve the control precision, the variable universe fuzzy controller is a kind of high precision fuzzy controller. A stable adaptive fuzzy control of a
nonlinear system is implemented based on the variable universe method proposed first in [30, 31].
In this paper, the model of quarter-car suspension with MR damper was first established in Section 2. Neural network technology is used to establish the nonlinear control of MR damper to simulate the
inverse dynamic characteristics, which is also discussed in this section. The FSMCEF control method for semi-active control of vehicle suspension system is studied in Section 3. The FSMCEF with
skyhook model as the reference is designed. Considering that the chattering of SMC can excite undesirable high-frequency dynamics, and fuzzy control with expansion factor rules are used to overcome
these drawbacks. After the controller design is completed, the simulation model is built in Section 4. The simulation test and results analysis are also completed in this section, and the conclusion
of FSMCEF performance is drawn finally.
2. Model of quarter-car semi-active seat suspension with MR damper
2.1. Overview of quarter-car semi-active seat suspension model
Considering the quarter-car model to be with high accuracy in analyzing the suspension dynamics, it is employed to model the semi-active suspension in this paper. Fig. 1 presented a three DOF model
of quarter-car suspension system containing the seat suspension with a MR damper. In this figure the car body, seat and human body are included as the sprung masses: ${m}_{v}$ and ${m}_{s}$, and the
vertical dynamics of tire and axle is often considered by introducing an unsprung mass ${m}_{t}$ and spring ${k}_{t}$. MR damper is placed between the seat and the car body to form the seat
suspension together with a spring and a damper.
Based on Newton second law, the dynamic equations of seat suspension system is:
$\left\{\begin{array}{l}{m}_{s}{\stackrel{¨}{z}}_{s}=-{c}_{s}\left({\stackrel{˙}{z}}_{s}-{\stackrel{˙}{z}}_{v}\right)-{k}_{s}\left({z}_{s}-{z}_{v}\right)-{F}_{d},\\ {m}_{v}{\stackrel{¨}{z}}_{v}={c}_
right)\\ {m}_{t}{\stackrel{¨}{z}}_{t}={c}_{v}\left({\stackrel{˙}{z}}_{v}-{\stackrel{˙}{z}}_{t}\right)+{k}_{v}\left({z}_{v}-{z}_{t}\right)-{k}_{t}\left({z}_{t}-{z}_{0}\right),\end{array}\right\,$
where ${m}_{t}$, ${m}_{v}$ and ${m}_{s}$ are unsprung mass, quarter car body mass and seat (plus human body) mass respectively. ${k}_{t}$, ${k}_{v}$ and ${k}_{s}$ are the stiffness coefficients of
the tire, quarter car suspension and seat suspension respectively. ${c}_{v}$ and ${c}_{s}$ are the damping coefficients of quarter car suspension and seat suspension respectively. ${F}_{d}$ is the
semi-active damping force created by the MR damper. ${z}_{0}$, ${z}_{t}$, ${z}_{v}$ and ${z}_{s}$ are the road excitation, vertical displacements of car axle, body and seat, respectively.
Based on Eq. (1), the state equation of the system is:
$\left\{\begin{array}{l}\stackrel{˙}{X}=AX+BU,\\ Y=CX+DU.\end{array}\right\$
$A=\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ \frac{-{k}_{s}}{{m}_{s}}& \frac{-{c}_{s}}{{m}_{s}}& \frac{{k}_{s}}{{m}_{s}}& \frac{{c}_{s}}{{m}_{s}}& 0& 0\\ 0& 0& 0& 1& 0& 0\\ \frac{{k}_{s}}{{m}_
{v}}& \frac{{c}_{s}}{{m}_{v}}& \frac{-\left({k}_{s}+{k}_{v}\right)}{{m}_{v}}& \frac{-\left({c}_{s}+{c}_{v}\right)}{{m}_{v}}& \frac{{k}_{v}}{{m}_{v}}& \frac{{c}_{v}}{{m}_{v}}\\ 0& 0& 0& 0& 0& 1\\ 0& 0
& \frac{{k}_{v}}{{m}_{t}}& \frac{{c}_{v}}{{m}_{t}}& \frac{-\left({k}_{t}+{k}_{v}\right)}{{m}_{t}}& \frac{-{c}_{v}}{{m}_{t}}\end{array}\right],B=\left[\begin{array}{cc}0& 0\\ \frac{-1}{{m}_{s}}& 0\\ 0
& 0\\ \frac{1}{{m}_{v}}& 0\\ 0& 0\\ 0& \frac{{k}_{t}}{{m}_{t}}\end{array}\right],$
$C=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ \frac{-{k}_{s}}{{m}_{s}}& \frac{-{c}_{s}}{{m}_{s}}& \frac{{k}_{s}}{{m}_{s}}& \frac{{c}_{s}}
{{m}_{s}}& 0& 0\\ 1& 0& -1& 0& 0& 0\end{array}\right],D=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ \frac{-1}{{m}_{s}}& 0\\ 0& 0\end{array}\right],U=\left[\begin{array}{l}{F}_{d}\\ {z}_{0}\end
Fig. 1Model of quarter-car suspension
2.2. MR damper modeling and analysis
Bouc-wen hysteresis model paralleled with dashpot and spring is originally used to formulate the MR damper. It can describe the hysteretic nonlinearity of the MR damper, but it is unable to describe
the nonlinear and saturation dependence of the magnetic field yielded by the direct drive current. A modified Bouc-wen hysteresis model, proposed by Spencer [32], effectively overcomes the above
drawback and precisely describes the nonlinear saturated characteristic of the MR damper. The modified Bouc-wen hysteresis model is presented in Fig. 2 and it is finally obtained by adding a serial
viscous damping with the Bouc-wen model and further paralleling a linear spring to the serialized structure.
Fig. 2Modified Bouc-Wen model of MR damper
In this paper, the modified Bouc-Wen model is used to describe the mechanical properties of the MR damper. The model introduces two internal variables, and it constructs a differential equation model
with 14 parameters to be determined. According to Fig. 2, the mathematical equations of the modified Bouc-wen model are presented as follows:
$\stackrel{˙}{y}=\frac{1}{\left({c}_{0}+{c}_{1}\right)}\left(\alpha z+{c}_{0}\stackrel{˙}{x}+{k}_{0}\left(x-y\right)\right),$
$\stackrel{˙}{z}=-\gamma {\left|\stackrel{˙}{x}-\stackrel{˙}{y}\left|z\right|z\right|}^{n-1}-\beta \left(\stackrel{˙}{x}-\stackrel{˙}{y}\right){\left|z\right|}^{n}+A\left(\stackrel{˙}{x}-\stackrel{˙}
where $\stackrel{˙}{u}=-\eta \left(u-v\right)$, ${k}_{1}$ is the stiffness of the damper accumulator. ${c}_{0}$ is the viscous damping observed when larger velocities is represented. ${c}_{1}$ is a
dashpot, included in the model to produce the roll-off that was observed in the experimental data at low velocities. ${k}_{0}$ is presented to control the stiffness at large velocities, and ${x}_{0}$
is the initial displacement of spring ${k}_{1}$ is associated with the nominal damper force due to the accumulator. $u$ is given as the output of a first-order filter. $v$ is the commanded voltage
sent to the current driver.
As for the RD-1005-3 damper produced by Lord Corporation, its parameters are chosen as follows. $\alpha =$ 963 N/cm, ${c}_{0}=$ 53N·S/cm, ${k}_{0}=$ 14 N/cm, ${c}_{1}=$ 930 N·s/cm, ${k}_{1}=$ 5.4 N/
cm, $\gamma =$ 200 cm^2, $\beta =$ 200 cm^-2, $n=$ 2, $A=$ 207, and ${x}_{0}=$ 18.9 cm, the response of the proposed model at 2.5 Hz is obtained as shown in Fig. 3. It can be seen from Fig. 3 that MR
damper can provide the damping effect in the plane of I and III quadrant of velocity-force plane, unlike the active actuator in all the four quadrants. Therefore, the output of MR damper has to track
the desired damping force only when the expected force and velocity have same sign, otherwise it should output the least damping force, so the formula described is:
$f\left(t\right)=\left\{\begin{array}{l}{f}_{e}\left(t\right),\begin{array}{ll}& {f}_{e}\left(t\right)\cdot \stackrel{˙}{x}>0,\end{array}\\ {f}_{e\mathrm{m}\mathrm{i}\mathrm{n}},\begin{array}{lll}&
{f}_{e}\left(t\right)\cdot \stackrel{˙}{x}\le 0,& \end{array}\end{array}\right\$
where $f\left(t\right)$ is the damping force of MR damper, ${f}_{e}\left(t\right)$ is the desired force which is obtained by using suspension controller, ${f}_{e\mathrm{m}\mathrm{i}\mathrm{n}}$ is
the minimal damping force corresponding to the zero input current, however it is not a constant value changing with the instant velocity.
If the MR damper controller employs the switching control method of Eq. (6), the input voltage of the MR damper switches between the minimum and maximum without being a continuous adjustable control
signal, which limits the performance of the MR damper. In this paper neural network is used to simulate the inverse model of MR damper, and it is further be as the nonlinear controller of MR damper
to create a continuous signal for MR damper, which will be discussed in next section.
Fig. 3Experimentally obtained response of the model
a) Force vs. displacement
2.3. Neuro inverse model approximating of MR damper
The reverse model of MR damper is defined as solving the voltage corresponding to input MR damper after the desired damping force is obtained by FSMCEF control algorithm. The aim of the reverse model
is to make MR damper track this desired force as possible. The inverse model can be described using the following nonlinear function:
$\stackrel{^}{v}\left(k\right)=h\left(\phi \left(k\right),\theta \right),$
where $\stackrel{^}{v}\left(t\right)$ is the input voltage, namely the output of MR damper reverse model. $\theta$ is the neural network weight vector, determined by training process. $\phi$ is the
input vector as $\varphi \left(k\right)=\left[\stackrel{^}{v}\left(k-1\right),...,\stackrel{^}{v}\left(k-{n}_{v}\right),x\left(k\right),...,x\left(k-{n}_{x}\right),F\left(k\right),...,F\left(k-{n}_
{f}\right)\right]$, in which $k$ is the $k$th^(current) time step, ${n}_{v}$, ${n}_{x}$ and ${n}_{f}$ are the previous time step numbers of input voltage, damping force and displacement respectively.
The BP neural network can approximate an arbitrary nonlinear continuous function with arbitrary precision, and it is used for the inverse-dynamics approximation of MR damper is shown in Fig. 4.
Fig. 4Block diagram of neuro inverse dynamics model for MR damper
The typical BP network is divided into three layers, which are input layer, hidden layer and output layer. For the inverse dynamics model of Eq. (7), the network input layer is set with 9 nodes and
the hidden layer with 20 nodes (see Fig. 5, $p=$ 20), the output layer has one node, standing for the input voltage of the MR damper.
The output of the hidden node is:
${S}_{j}^{}=\sum _{i=1}^{n}{w}_{ij}{a}_{i}-{\theta }_{j},{b}_{j}^{k}=f\left({S}_{j}^{k}\right)=\frac{1}{1+{e}^{-{S}_{j}^{k}}},j=1,2,\dots ,p,k=1,2,\dots ,m.$
The output of the output node is:
${L}_{j}^{k}=\sum _{j=1}^{p}{v}_{jt}{b}_{j}-{\gamma }_{t},{C}_{t}^{k}=f\left({L}_{t}^{k}\right)=\frac{1}{1+{e}^{-{L}_{t}^{k}}},t=1,2,\dots ,q.$
Fig. 5Detailed neural network for the inverse dynamics approximation
This neural network training includes namely mode suitable transmission, error back propagation, memory training and learning convergence. The detailed training procedures are as follows.
(1) Initialization:
The weights of $\left\{{w}_{ij}\right\}$, $\left\{{v}_{jt}\right\}$ and threshold of $\left\{{\theta }_{j}\right\}$, $\left\{{\gamma }_{t}\right\}$ are all set as random values in (–1, 1).
(2) Randomly pick up a pair of samples for network training.
(3) Calculate the output of the hidden layer using Eq. (8).
(4) Calculate the output layer using Eq. (9).
(5) Calculate the average error of the output layer: ${d}_{t}^{k}=\left({y}_{t}^{k}-{C}_{t}\right){C}_{t}\left(1-{C}_{t}^{}\right)$.
(6) Calculate the hidden layer general error: ${e}_{j}=\left(\sum _{t=1}^{q}{d}_{t}^{k}{v}_{jt}\right)\cdot {b}_{j}\left(1-{b}_{j}^{}\right)$.
(7) Modify the output layer weights and thresholds:
$\left\{\begin{array}{l}{v}_{jt}\left(N+1\right)={v}_{jt}\left(N\right)+\alpha {d}_{t}^{k}{b}_{j},\begin{array}{l}\end{array}\alpha \in \left(0,1\right),\begin{array}{l}\end{array}t=1,2,\dots ,q,\
begin{array}{l}\end{array}j=1,2,\dots ,p,\\ {\gamma }_{jt}\left(N+1\right)={\gamma }_{jt}\left(N\right)+\left(-\alpha {d}_{t}^{k}\right).\end{array}\right\$
(8) Modify hidden layer weights and thresholds:
$\left\{\begin{array}{l}{w}_{ij}\left(N+1\right)={w}_{ij}\left(N\right)+\beta {e}_{j}{a}_{i},\\ {\theta }_{j}\left(N+1\right)={\theta }_{j}\left(N\right)+\left(-\beta {e}_{j}\right).\end{array}\right
(9) Take out the next pair of sample and return Step (3) to repeat until completing all training samples.
(10) Determine whether a global error is less than the preset value, otherwise, to return to Step (2) to continue until meeting the requirements.
The displacement and the input voltage, as the network input, are generated by Gaussian white noise with frequency ranges of 0-3 Hz and 0-4 Hz respectively. A data set of 10000 points used for
training and validation are created by 500 Hz sampling frequency in 20 s sample time. These data are used for training the network and another data set including 1200 points is further created to
validate the training. The BP neural network training and validating process is shown in Fig. 6 and the control voltage between the predict output and desired output shown in Fig. 7. The BP network
prediction error compared with the desired output is shown in Fig. 8.
Fig. 6BP neural network training and validating process
Fig. 7Control voltage between the predict output and the desired output
Fig. 8BP network prediction error
Table 1BP predict output data, desired output data and their errors
Sample No. Predict output Desired output Errors
1 1.154196198 1.139969109 0.014227089
2 1.144247759 1.139969109 0.00427865
3 1.141656099 1.139969109 0.00168699
4 1.140767273 1.139969109 0.000798164
5 1.140462485 1.139969109 0.000493376
6 1.140385454 1.139969109 0.000416345
7 1.140396186 1.139969109 0.000427077
8 1.151196722 1.235470511 –0.084273789
9 1.219608674 1.235470511 –0.015861836
10 1.242757353 1.235470511 0.007286842
11 1.228802917 1.235470511 –0.006667594
12 1.228274434 1.235470511 –0.007196077
13 1.229949686 1.235470511 –0.005520825
14 1.231869009 1.235470511 –0.003601502
15 1.227968248 1.326840347 –0.098872098
…… …… …… ……
1.140153642 1.135359005 0.004794637
1195 1.140148274 1.135359005 0.00478927
1196 1.140113462 1.135359005 0.004754457
1197 1.140047649 1.135359005 0.004688644
1198 1.139997841 1.135359005 0.004638836
1199 1.139988131 1.135359005 0.004629127
1200 1.108848138 1.115573823 -0.006725686
Table 1 shows the BP predict output data, the desired output data and their error. The sum of the absolute value of 1200 errors is 15.9525. Table 1 presented the detailed result of some samples, and
the results show that the BP neural network predict output values substantially agree with the actual situation, and the errors are small, demonstrating that this model is highly with the project
reality and the neural network model is effective.
3. Fuzzy sliding mode controller design
The difference of sliding mode variable structure control from other conventional control strategies is its discontinuity of control. the system structure make switch characteristics change over
time, and that control features can force the system make small amplitude and high frequency movement up and down along the switching surface, which called “sliding mode” (see Fig. 9). The sliding
mode can be designed irrelative with parameter perturbation and external disturbance, and the system under sliding mode has good robustness.
Fig. 9Sliding mode motion in 2-D phase plot
But the sliding mode motion under parameters perturbation and external disturbance is easy to cause the high frequency chattering, because the high frequency chattering is infinitely fast in theory,
but in practice no actual actuators can realize it. The chattering phenomenon gives rise to the application difficulties of sliding mode control. In the next sections, the sliding mode controller for
semi-active suspension will be designed and its combination with fuzzy logic will further completed to depress the chattering. The sliding mode controller takes an ideal skyhook model as the
reference, and creates a sliding mode control law based on the errors dynamics between the seat suspension and its reference model. Thus, the skyhook model is first established and the next is
discussing the errors dynamics. Further fuzzy rules are used to suppress the chattering occurred in the above sliding mode control by fuzzifying the sliding mode surface and its derivative.
Considering that the chattering results in the errors changing in a large range, an expansion factor is used to change the universe of the fuzzy logic but without changing the fuzzy rules, which
forms a new variable universe fuzzy controller with adaptive characteristic. The combining of the sliding mode controller and fuzzy controller with expansion factor is further studied, the new formed
FSMCEF design is presented and its stability further completed.
3.1. Skyhook reference model
Herein the skyhook model is used as the reference to form the fuzzy sliding control algorithm, and it is presented in Fig. 10.
The dynamics equations are derived based on Newton second law as follows:
$\left\{\begin{array}{l}{m}_{s}{\stackrel{¨}{z}}_{rs}=-{c}_{s}\left({\stackrel{˙}{z}}_{rs}-{\stackrel{˙}{z}}_{rv}\right)-{k}_{s}\left({z}_{rs}-{z}_{rv}\right)-{c}_{sh}{\stackrel{˙}{z}}_{rs},\\ {m}_
_{v}\left({z}_{rv}-{z}_{rt}\right),\\ {m}_{t}{\stackrel{¨}{z}}_{rt}={c}_{v}\left({\stackrel{˙}{z}}_{rv}-{\stackrel{˙}{z}}_{rt}\right)+{k}_{v}\left({z}_{rv}-{z}_{rt}\right)-{k}_{t}\left({z}_{rt}-{z}_
where ${z}_{rt}$, ${z}_{rv}$ and ${z}_{rs}$ are the road excitation, vertical displacements of unsprung mass, car body and seat respectively, which are the corresponding variables in the reference
system compared with the plant system in Fig. 1. ${c}_{sh}$ is the damping coefficient of “sky-hook” damper.
Fig. 10Skyhook reference model
The state vector for the reference system is taken as ${Z}_{r}=\left[{z}_{rs},{\stackrel{˙}{z}}_{rs},{z}_{rv},{\stackrel{˙}{z}}_{rv},{z}_{rt},{\stackrel{˙}{z}}_{rt}{\right]}^{T}$, and the output
vector is taken as $Y=\left[{z}_{rs},{\stackrel{˙}{z}}_{rs},{z}_{rv},{\stackrel{˙}{z}}_{rv}{\right]}^{T}$. According to Eq. (12) the state equations are established as:
$\left\{\begin{array}{l}{\stackrel{˙}{Z}}_{r}={A}_{r}{Z}_{r}+{B}_{r}{u}_{r},\\ {Y}_{r}={C}_{r}{Z}_{r}+{D}_{r}{u}_{r},\end{array}\right\$
${A}_{r}=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ \frac{-{k}_{s}}{{m}_{s}}& \frac{-\left({c}_{s}+{c}_{sh}\right)}{{m}_{s}}& \frac{{k}_{s}}{{m}_{s}}& \frac{{c}_{s}}{{m}_{s}}& 0& 0\\ 0& 0& 0& 1& 0
& 0\\ \frac{{k}_{s}}{{m}_{v}}& \frac{{c}_{s}}{{m}_{v}}& \frac{-\left({k}_{s}+{k}_{v}\right)}{{m}_{v}}& \frac{-\left({c}_{s}+{c}_{v}\right)}{{m}_{v}}& \frac{{k}_{v}}{{m}_{v}}& \frac{{c}_{v}}{{m}_{v}}\
\ 0& 0& 0& 0& 0& 1\\ 0& 0& \frac{{k}_{v}}{{m}_{t}}& \frac{{c}_{v}}{{m}_{t}}& \frac{-\left({k}_{t}+{k}_{v}\right)}{{m}_{t}}& \frac{-{c}_{v}}{{m}_{t}}\end{array}\right],{u}_{r}=\left[{z}_{0}\right].$
${C}_{r}=\left[\begin{array}{llllll}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\end{array}\right],{B}_{r}={\left[\begin{array}{llllll}0& 0& 0& 0& 0& \frac{{k}_{t}}{{m}_
{t}}\end{array}\right]}^{T},{D}_{r}={\left[\begin{array}{llll}0& 0& 0& 0\end{array}\right]}^{T}.$
3.2. Error dynamics model used for SMC and FSMCEF
Both the sliding mode controller and the fuzzy sliding mode controller are designed to make the actual seat suspension motion to track the reference mode, and so they are based on the dynamic errors
between the seat suspension and the skyhook reference model. Based on the above seat model and the reference model, the seat suspension displacement error, its integral and its differential (velocity
error) are taken as the control variables, and they form the general tracking error vector$e$as $e=\left[\begin{array}{lll}{e}_{1}& {e}_{2}& {e}_{3}\end{array}{\right]}^{T}={\left[\begin{array}{lll}\
int \left({z}_{s}-{z}_{rs}\right)& {z}_{s}-{z}_{rs}& {\stackrel{˙}{z}}_{s}-{\stackrel{˙}{z}}_{rs}\end{array}\right]}^{T}$ and its differential is $\stackrel{˙}{e}={\left[\begin{array}{lll}{z}_{s}-{z}
_{rs}& {\stackrel{˙}{z}}_{s}-{\stackrel{˙}{z}}_{rs}& {\stackrel{¨}{z}}_{s}-{\stackrel{¨}{z}}_{rs}\end{array}\right]}^{T}$. so, the error dynamic equation is obtained as:
$E=\left[\begin{array}{lll}0& 1& 0\\ 0& 0& 1\\ 0& \frac{-{k}_{s}}{{m}_{s}}& \frac{-{c}_{s}}{{m}_{s}}\end{array}\right],G=\left[\begin{array}{llllll}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& \frac
{{k}_{s}}{{m}_{s}}& \frac{{c}_{s}}{{m}_{s}}& 0& 0\end{array}\right],H=\left[\begin{array}{llllll}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& \frac{{c}_{sh}}{{m}_{s}}& \frac{-{k}_{s}}{{m}_{s}}& \frac{-
{c}_{s}}{{m}_{s}}& 0& 0\end{array}\right],$
$F={\left[\begin{array}{lll}0& 0& \frac{-1}{{m}_{s}}\end{array}\right]}^{T},u=\left[{F}_{d}\right].$
3.3. Sliding mode control based on pole placement
The switching surface is taken as:
As for $s={c}_{1}{e}_{1}+{c}_{2}{e}_{2}+{e}_{3}=0$, Eq. (14) can be written to partitioned matrix form as:
$\left[\begin{array}{l}{\stackrel{˙}{e}}_{1}\\ {\stackrel{˙}{e}}_{2}\end{array}\right]=\left[\begin{array}{ll}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{l}{e}_{1}\\ {e}_{2}\end{array}\right]+\
left[\begin{array}{l}0\\ 1\end{array}\right]{e}_{3}=\left[\begin{array}{ll}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{l}{e}_{1}\\ {e}_{2}\end{array}\right]+\left[\begin{array}{l}0\\ 1\end
{array}\right]\left(s-{c}_{1}{e}_{1}-{c}_{2}{e}_{2}\right)=\left[\begin{array}{cc}0& 1\\ -{c}_{1}& -{c}_{2}\end{array}\right]\left[\begin{array}{l}{e}_{1}\\ {e}_{2}\end{array}\right]+\left[\begin
{array}{l}0\\ 1\end{array}\right]s,$
As for $\stackrel{˙}{s}={c}_{1}{e}_{2}+{c}_{2}{e}_{3}+{\stackrel{˙}{e}}_{3}=0$, Eq. (16) can be written as:
$\left[\begin{array}{l}{\stackrel{˙}{e}}_{1}\\ {\stackrel{˙}{e}}_{2}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -{c}_{1}& -{c}_{2}\end{array}\right]\left[\begin{array}{l}{e}_{1}\\ {e}_{2}\end
The characteristic polynomial for Eq. (18) is $D\left(\lambda \right)={\lambda }^{2}+{c}_{2}\lambda +{c}_{1}$. To obtain the value of ${c}_{1}$ and ${c}_{2}$, its characteristic roots are equal to
the given poles.
The chief problem of pole assignment is rationally to determine the desired closed-loop poles set. The standard form of the second-order system transfer function is:
$\mathrm{\Phi }\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{{{\omega }_{n}}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{{\omega }_{n}}^{2}}.$
The two closed-loop poles are ${s}_{1,2}=-\zeta {\omega }_{n}±j{\omega }_{n}\sqrt{1-{\zeta }^{2}}$, and the system works in less damping state ($0<\zeta <1$) which makes these two poles to be
conjugate complex roots located in the left half plane of $s$ domain and to have the appropriate oscillation and short transition process. For the three orders system are in Eqs. (16) and (17), the
desired poles number are $n=$ 3. The conjugate pole pair of ${s}_{1}$ and ${s}_{2}$ are selected as the dominant poles, and the third is a non-dominant one. The poles placement of our closed-loop
system is completed to ensure the two dynamic performance indices: peak time ${t}_{p}$ and overshoot$\sigma$ %. These two indices are set as $\sigma \le$ 15 %, ${t}_{p}\le$ 0.7, which determines the
dominant poles be at –2.7326±4.4886i and the non-dominant pole at –20. The corresponding parameters are $\zeta =\text{0.52}$ and ${\omega }_{n}=\text{5.255}$. So, the switching function coefficient
vector is $c=\left[\text{68}\begin{array}{l}\end{array}\text{4}\begin{array}{l}\end{array}\text{1}\right]$. The system performance is mainly determined by these two dominant poles and the
non-dominant pole only produces minimal effect.
When system is in sliding mode motion, $s=0$, $ds/dt=0$ and:
The equivalent control for system into fuzzy sliding mode or sliding mode is ${u}^{*}$, and:
In order to improve the dynamic quality of the movement, the approaching mode employs the constant speed reaching law as:
$ds/dt=-\epsilon \mathrm{s}\mathrm{g}\mathrm{n}\left(s\right),$
where $\epsilon =$ 3.
The final sliding mode control law is taken as:
$u={u}_{eq}+{u}_{sw}={u}^{*}+\left[cB{\right]}^{-1}\stackrel{˙}{s}={u}^{*}+\epsilon {m}_{s}\mathrm{s}\mathrm{g}\mathrm{n}\left(s\right).$
So, the desired real-time variable damping force is as:
${F}_{d}=\left\{\begin{array}{lll}{F}_{{d}_{eq}}+{F}_{{d}_{sw}},& & \left[{F}_{{d}_{eq}}+{F}_{{d}_{sw}}\right]\left({\stackrel{˙}{x}}_{s}-{\stackrel{˙}{x}}_{v}\right)\ge 0,\\ 0,& & \begin{array}{l}\
where ${F}_{{d}_{sw}}={u}_{sw}=\epsilon {m}_{s}\mathrm{s}\mathrm{g}\mathrm{n}\left(s\right)$, $\epsilon =$ 3.
3.4. Fuzzy sliding mode control
The fuzzy logic control is further added to overcome sliding mode controller “chattering” problem. The connection of the sliding mode controller and the fuzzy logic controller is shown in Fig. 11,
which formed the final fuzzy sliding mode controller. The detailed content of the fuzzy control block in Fig. 11 is presented in Fig. 12. Its inputs are $s\left(e\right)$ and $\stackrel{˙}{s}\left(e\
right)$, with one output $\epsilon$ sent to the sliding mode controller.
Fig. 11Block diagram of FSMCEF system
The fuzzy controller first change the range of $s\left(e\right)$ and $\stackrel{˙}{s}\left(e\right)$, namely from their original ranges of [–0.04, 0.03] and [–6×10^-3, +8×10^-3] both to the new range
of [–6, +6] for further discretization and fuzzification. The corresponding conversion equation is:
Fig. 12The structure of fuzzy controller
The variables after conversion are $S$ and $SC$ (Fig. 12), and they are further discretized and fuzzified to form fuzzy sets $\underset{_}{S}$, $\underset{_}{S}C$. In this process $S$ and $SC$ are
classified into seven grades, forming seven fuzzy subsets, including: NL (Negative Large), PL (Positive Large), NM (Negative Medium), PM (Positive Medium), NS (Negative Small), PS (Positive Small),
NE (0). The $S$ and $SC$ of domain $X$ and $Y$ are belonging to 7 fuzzy subsets, respectively. Similarly, the output value $\underset{_}{\epsilon }$ are also ranked into seven fuzzy subsets: NL, PL,
NM, PM, NS, PS, NE.
For this double input and single output fuzzy controller, the control rules can be written as the following form: If $S={\underset{_}{S}}_{i}$ and $SC=\underset{_}{S}{C}_{j}$ then $U={\underset{_}
{U}}_{ij}$, ($i=$1, 2,…, 7, $j=i=$1, 2,.., 7), where ${\underset{_}{S}}_{i}$, $\underset{_}{S}{C}_{j}$ are input fuzzy sets, and ${\underset{_}{U}}_{ij}$ is output fuzzy sets.
These fuzzy sets conditional statements can be summed up in a fuzzy relation $\underset{_}{R}$, and $\underset{_}{R}=\underset{ij}{\cup }\left({\underset{_}{E}}_{i}×\underset{_}{E}{C}_{j}\right)×{\
underset{_}{U}}_{ij}$. According to each inference rules, the corresponding fuzzy relations, ${\underset{_}{R}}_{1},{\underset{_}{R}}_{2},...,{\underset{_}{R}}_{n}$ can be calculated. So, the total
of the whole system corresponding fuzzy control rule $\underset{_}{R}$ is:
$\underset{_}{R}={\underset{_}{R}}_{1}\vee {\underset{_}{R}}_{2}\vee \cdots \vee {\underset{_}{R}}_{49}=\stackrel{49}{\underset{i=1}{\vee }}{R}_{i}.$
The final fuzzy rules are shown in Table 2, and according to Table 2 the 3-D input-output relation diagram of the fuzzy controller is obtained as shown in Fig. 13.
Table 2Fuzzy rules
$\underset{_}{\epsilon }$
NL NM NS ZE PS PM PL
NL NL NL NM NM NS NS ZE
NM NL NL NM NS NS NS ZE
NS NM NM NS ZE ZE ZE ZE
$\underset{_}{S}$ ZO NM NM NS ZE ZE PS PS
PS NS NS ZE ZE PS PS PM
PM ZE PS PS PS PS PM PL
PL PS PS PS PM PL PL PL
Fig. 133-D diagram of Fuzzy control rules
When $\underset{_}{R}$ is determined, according to $\underset{_}{S}=\left\{-6,-5,\cdots +5,+6\right\}$, $\underset{_}{S}C=\left\{-6,-5\cdots +5,+6\right\}$ and synthetic fuzzy reasoning rules, the
corresponding fuzzy sets of controls is $\underset{_}{U}=\left(\underset{_}{E}×\underset{_}{E}C\right)\circ R$, and:
${\mu }_{\underset{_}{\epsilon }}\left(z\right)=\vee {\mu }_{\underset{_}{R}}\left(x,y,z\right)\wedge \left[{\mu }_{\underset{_}{S}}\left(x\right)\wedge {\mu }_{\underset{_}{S}C}\left(y\right)\
${\epsilon }_{fuzzy}=\frac{\sum _{i=1}^{49}\mu \left({\epsilon }_{i}\right)*{\epsilon }_{i}}{\sum _{i=1}^{49}\mu \left({\epsilon }_{i}\right)},\begin{array}{l}\end{array}{\epsilon }_{fuzzy}\in \left
The final fuzzy sliding or sliding mode control law is taken as:
$u={u}_{eq}+{u}_{sw}={u}^{*}+\left[cB{\right]}^{-1}s={u}^{*}+{\epsilon }_{fuzzy}{m}_{s}\mathrm{s}\mathrm{g}\mathrm{n}\left(s\right).$
3.5. Fuzzy sliding mode control with expansion factor
Equidistant domain partitioning method is used in fuzzy control generally. When the error is large, the system has sufficient error resolution, and it is shown as the “big error” dotted line in Fig.
14. When the error is small, the system response only changes around “ZO” corresponding to the original fuzzy partition, and other fuzzy subsets obviously do not work. Ideally, when the error is
reduced, the domain of the fuzzy controller should be able to make self-adaptation adjustment. The accuracy of the fuzzy controller is related to the number of output variables and fuzzy rules.
Supposing the input is $n$-dimensional, the fuzzy control rule of the universe is divided into $m$; the total rule number is ${m}^{n}$, if the fuzzy subset of the domain is divided into smaller, the
number of fuzzy control rules will increase exponentially, increasing the difficulty of making rules. Therefore, under the premise of not affecting the control effect, we should try to use less fuzzy
subset to reduce the number of fuzzy control rules. With the same form of rules, the universe shrinks as the error becomes smaller, and the universe expands as the error increases. The contraction of
the domain is equivalent to increasing the fuzzy control rules to improve the control accuracy. The function of the scaling factor $\alpha \left(x\right)$ transforms the domain into $\left[-\alpha \
left(x\right)E,\alpha \left(x\right)E\right]$, where $\alpha \left(x\right)$ is a continuous function of the error variable $x$. The appropriate domain expansion factor $\alpha \left(x\right)$ is
chosen so that the range of the universe changes with the error, which can realize the adaptive implementation of the expansion of the domain, without the need for other auxiliary algorithms and
increase the control rules.
Fig. 14Adjustment of the domain
Let ${X}_{i}=\left[-E,E\right]$ be the universe of input variable ${x}_{i}\left(i=1,2,\dots ,n\right)$, $Y=\left[-U,U\right]$ is the universe of output variable $y$; ${\psi }_{i}=\left\{{A}_{ij}\
right\}$ is the fuzzy partition on ${X}_{i}$, ${\varphi }_{j}=\left\{{B}_{j}\right\}$ is the fuzzy partition on $Y$, $1\le j\le m$. As ${\psi }_{i}$, ${\varphi }_{j}$ are the linguistic variables,
fuzzy inference rule $R$ can be formed:
$\text{if}{x}_{1}\text{is}{A}_{1j}\text{and}{x}_{2}\text{is}{A}_{2j}\text{and}\dots {x}_{n}\text{is}{A}_{nj}\text{then}y\text{is}{B}_{j},$
where ${x}_{i}$ is ${A}_{ij}$ peak, and ${y}_{j}$ is ${B}_{j}$ peak $\left(i=1,2,\dots ,n\right)$, $\left(j=1,2,\dots ,m\right)$, The fuzzy control system of Eq. (31) can be expressed as an $n$-piece
piecewise interpolation function:
$y\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)=F\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\triangleq \stackrel{m}{\sum _{j=1}}\prod _{i=1}^{n}{A}_{ij}\left({x}_{i}\right){y}_{j}.$
The variable universe is that domain ${X}_{i}$ and $Y$ can be adjusted independently with the change of input variable ${x}_{i}$ and output $y$, respectively:
${X}_{i}\left({x}_{i}\right)=\left[-{\alpha }_{i}\left({x}_{i}\right){E}_{i},{\alpha }_{i}\left({x}_{i}\right){E}_{i}\right],$
$Y\left(y\right)=\left[-\beta \left(y\right)U,\beta \left(y\right)U\right],$
where ${\alpha }_{i}\left({x}_{i}\right)$ and $\beta \left(y\right)$ is the domain expansion factor. In contrast to the variable universe, the original universe ${X}_{i}$ and $Y$ is called the
initial universe. Eq. (32) can be expressed as $n$-piece dynamic interpolation function:
$y\left(x\left(t+1\right)\right)=\beta \left(y\left(x\left(t\right)\right)\right)\sum _{j=1}^{m}\prod _{i=1}^{n}{A}_{ij}\left(\frac{{x}_{i}\left(t\right)}{{\alpha }_{i}\left({x}_{i}\left(t\right)\
where $x\left(t\right)\triangleq \left[{x}_{1}\left(t\right),$${x}_{2}\left(t\right),\dots ,{x}_{n}\left(t\right){\right]}^{T}$, and ${\alpha }_{i}\left({x}_{i}\right)$ is chosen as:
$\alpha \left(x\right)={\left(\frac{\left|x\right|}{E}\right)}^{\tau },0<\tau <1,$
$\alpha \left(x\right)=1-\lambda \mathrm{e}\mathrm{x}\mathrm{p}\left(-k{x}^{2}\right),\lambda \in \left(0,1\right),k>0.$
In this paper $\alpha \left(x\right)=1-1/\sqrt{1+k{x}^{2}}$, $k=$ 10^4. $\beta \left(y\right)$ is chosen as:
$\beta \left(t\right)={K}_{i}\sum _{i=1}^{n}{p}_{i}\underset{0}{\overset{t}{\int }}e\left(\tau \right)d\tau +\beta \left(0\right),$
where ${K}_{i}$ is the proportionality constant, and $\beta \left(0\right)$ is according to the actual situation, usually try $\beta \left(0\right)=$ 1. The control law of the variable universe fuzzy
controller is:
$u\left(t\right)=\left({K}_{i}\sum _{i=1}^{n}{p}_{i}\underset{0}{\overset{t}{\int }}e\left(\tau \right)d\tau +\beta \left(0\right)\right)U\sum _{j=1}^{m}\prod _{i=1}^{n}{A}_{ij}\left(\frac{{x}_{i}\
left(t\right)}{{\alpha }_{i}\left({x}_{i}\left(t\right)\right)}\right){y}_{j}.$
$X=\left[-E,E\right]$, $Y=\left[-D,D\right]$ is two-dimensional input domain, respectively, and $Z=\left[-U,U\right]$ is the output domain. When $X$ and $Y$ are relatively independent, we can get the
expansion factor $\alpha \left(x\right)$, $\beta \left(y\right)$ and $Z$ expansion factor $\gamma \left(z\right)$. But in most cases, $Y$ and $X$ are related. If $X$ is the error domain and $Y$ is
often the domain of error variation, $Y=\left(-\stackrel{˙}{E},\stackrel{˙}{E}\right)$ and $\beta \left(y\right)$ should be defined on $X×Y$ and $\beta \left(y\right)=\beta \left(x,y\right)$, then
the input and output of the domain expansion factor are:
$\alpha \left(x\right)={\left(\frac{\left|x\right|}{E}\right)}^{\tau },0<\tau <1,$
$\beta \left(x,y\right)=\left[{\left(\frac{\left|x\right|}{E}\right)}^{\tau }+{\left(\frac{\left|y\right|}{E}\right)}^{{\tau }_{1}}\right],\text{or}\beta \left(x,y\right)={\left(\frac{\left|x\right|}
{E}\right)}^{\tau }{\left(\frac{\left|y\right|}{E}\right)}^{{\tau }_{1}},0<\tau ,{\tau }_{1}<1.$
Since the error variation depends on the error, $\beta$ can be simply taken as $\beta \left(y\right)$, Eq. (41) can be rewritten as:
$\beta \left(y\right)=\frac{1}{2}\left[{\left(\frac{\left|y\right|}{\stackrel{˙}{E}}\right)}^{{\tau }_{1}}\right],\text{or}\beta \left(y\right)={\left(\frac{\left|y\right|}{\stackrel{˙}{E}}\right)}^
{{\tau }_{1}}.$
3.6. Stability analysis
3.6.1. Stability analysis of nominal system based on Lyapunov theorem
The poles placement of our closed-loop system is completed to ensure the two dynamic performance indices: peak time ${t}_{p}$ and overshoot $\sigma$ %. These two indices are set as $\sigma \le \text
{15}$ %, ${t}_{p}\le \text{0.7}$ which determine the dominant poles be at –2.7326±4.4886i and the non-dominant pole at –20. The corresponding parameters are $\zeta =\text{0.52}$ and ${\omega }_{n}=\
text{5.255}$. So, the switching function coefficient vector is $c=\left[\text{68}\begin{array}{l}\end{array}\text{4}\begin{array}{l}\end{array}\text{1}\right]$. The energy function is taken as $V\
left(x\right)={s}^{2}/\text{2}$ and it is positive definite, and its differential $\stackrel{˙}{V}\left(x\right)=s\cdot \stackrel{˙}{s}=s\cdot \left(-{\epsilon }_{fuzzy}\mathrm{s}\mathrm{g}\mathrm{n}
\left(s\right)\right)\le 0$, namely negative definite, so the entire system is asymptotically stable.
3.6.2. Robust stability analysis of system under parameter uncertainty and external disturbance
Considering the general form of the linear uncertainty system is:
$\stackrel{˙}{X}=\left(A+\mathrm{\Delta }A\right)X+\left(B+\mathrm{\Delta }B\right)U+D\omega$
where $X\in {R}^{n}\text{,}$$U\in {R}^{m}\text{;}$$A\in {R}^{n×n}\text{,}$$B\in {R}^{n×m}\text{,}$$D\in {R}^{n×l}\text{;}$$\mathrm{\Delta }A\in {R}^{n×n}\text{,}$$\mathrm{\Delta }B\in {R}^{n×m}$ are
uncertainty matrix of A and B respectively, which describes the differences between the nominal value of parameters and the actual true values; $\omega \in {R}^{l}$ is the uncertainty of disturbance.
Without loss of generality, the nominal model $\left(A,B\right)$ of the controlled object Eq. (43) is supposed to be completely controllable.
In order to study the impact of various uncertainties on the control system, $\mathrm{\Delta }A$, $\mathrm{\Delta }B$ and $\mathrm{\Delta }C$ can be decomposed into:
$\mathrm{\Delta }A=BH+\delta A,\mathrm{\Delta }B=BE+\delta B,D=BF+\delta D,$
where $H\in {R}^{m×n}$, $E\in {R}^{m×m}$; $F\in {R}^{m×l}$, $\delta A\in {R}^{n×n}$, $\delta B\in {R}^{n×m}$, $\delta D\in {R}^{n×l}$.
The first term on the right-hand side of Eq. (44) satisfies the matching condition and is the matching part of the uncertainty factor; the second term is the residual part, which is the mismatching
uncertainty factor. Generally, the information easy to obtain for the uncertainty factor is its lower and upper bounds.
Hypothesis 1: The uncertain factors of the controlled object Eq. (43) are bounded:
$‖\mathrm{\Delta }A‖\le {\rho }_{\sigma },‖\mathrm{\Delta }B‖\le {\rho }_{v},‖\omega ‖\le {\rho }_{\omega },‖\delta A‖\le {\stackrel{-}{\rho }}_{\sigma },‖\delta B‖\le {\stackrel{-}{\rho }}_{v},$
where ${\rho }_{\sigma }\ge {\stackrel{-}{\rho }}_{\sigma }\ge 0$, ${\rho }_{v}\ge {\stackrel{-}{\rho }}_{v}\ge 0$, ${\rho }_{\omega }\ge 0$ are known constants.
When $\delta A$, $\delta B$ and $\delta D$ are equal to zero respectively, Eq. (44) is equivalent to uncertainty factor matching conditions or invariance conditions:
$rank\left[B\right]=rank\left[\mathrm{\Delta }A\begin{array}{l}\end{array}B\right]=rank\left[\mathrm{\Delta }B\begin{array}{l}\end{array}B\right]=rank\left[DB\right].$
The controlled object Eq. (43) is the matching uncertainty system; when either $\delta Ae 0$, $\delta Be 0$ or $\delta De 0$ is established, the controlled object is a linear mismatch uncertainty
For the linear mismatch uncertainty system described by Eq. (43), the design of m-dimensional sliding mode domain in n-dimensional state space is as:
$S=GX=0,G\in {R}^{m×n}.$
In order to guarantee the non-singularity of the variable structure control system, the sliding mode requirement is $\left|GB\right|e 0$. Thus, by the equivalent control method, the state equation of
the variable structure closed-loop equivalent system of the mismatched uncertain Eq. (43) can be deduced:
$\stackrel{˙}{X}=\left[I-B\left(GB{\right)}^{-1}G\right]AX+\left[I-B\left(GB{\right)}^{-1}G]\left(\delta AX+\delta BU+\delta D\omega \right).$
Due to when $‖\omega ‖$ is bounded, mismatch perturbation uncertainty factor $\delta D\omega$ has nothing to do with stability, so lose the generality: $\omega =\text{0}$. And mismatch parameters and
input uncertainty factors are introduced the equivalent system and will cause the disturbance of its Eigen values, affecting the dynamic characteristics and stability of closed-loop system. The
stability and robustness of the variable-structure closed-loop control system Eq. (49) will be studied by estimating the perturbations of $\delta A$ and $\delta B$ to the eigenvalues.
1. When $\delta A$ ($\delta Ae 0$, $\delta B=$ 0).
For a variable structure equivalent system Eq. (48):
$\stackrel{˙}{X}=\left[I-B\left(GB{\right)}^{-1}G\right]AX+\left[I-B\left(GB{\right)}^{-1}G\right]\left(\delta AX+\delta BU+\delta D\omega \right)={A}_{eq}X+\delta {A}_{eq}X.$
When ${‖\delta A‖}_{\infty }\le {\stackrel{-}{\rho }}_{\sigma }$, ${\stackrel{-}{\rho }}_{\sigma }\ge 0$; ${A}_{eq}$ has $n-m$ nonzero single eigenvalues ${\lambda }_{1}$, ${\lambda }_{2}$,…, ${\
lambda }_{n-m}$.
(1) There is always a similarity transformation matrix $P\in {C}^{n×n}$, and:
$\begin{array}{l}P{A}_{eq}{P}^{-1}=\left[\begin{array}{cc}{D}_{11}& {D}_{12}\\ 0& 0\end{array}\right],P\delta {A}_{eq}{P}^{-1}=\left[\begin{array}{cc}\delta {A}_{11}& \delta {A}_{12}\\ 0& 0\end
{array}\right],\\ {D}_{11}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n-m}\right),\delta {A}_{11}\in {R}^{\left(n-m\right)×\left(n-m\right)}.\end
(2) Because the eigenvalue has invariance to matrix similarity transformation, all the disturbance of the characteristic value caused by $\delta {A}_{eq}\left(\delta A\right)$ to ${A}_{eq}$ is
transformed into the disturbance of $\delta {A}_{11}$ to ${D}_{11}$ in Eq. (50). Let $\delta {A}_{11}=\left\{\mathrm{\Delta }{a}_{ij}^{11}\right\}\text{,}$$P\delta {A}_{eq}{P}^{-1}=\left\{\mathrm{\
Delta }{a}_{ij}^{1}\right\}$, and $\lambda \left({D}_{11}+\delta {A}_{11}\right)=\left\{{\mu }_{1},{\mu }_{2},\dots ,{\mu }_{n-m}\right\}$. Then by Gerschgorin in the theorem, for each ${\mu }_{j}$
there is always ${\lambda }_{i}$:
$\left|{\mu }_{j}-\left({\lambda }_{i}+\mathrm{\Delta }{a}_{ii}^{11}\right)\right|\le \sum _{\begin{array}{l}k=l\\ ke i\end{array}}^{n-m}\left|\mathrm{\Delta }{a}_{ik}^{11}\right|,i,j=1,2,\dots
$\begin{array}{l}\left|{\mu }_{j}-{\lambda }_{i}\right|\le \left|\mathrm{\Delta }{a}_{ii}^{11}\right|+\sum _{\begin{array}{l}k=l\\ ke i\end{array}}^{n-m}\left|\mathrm{\Delta }{a}_{ik}^{11}\right|\le
\stackrel{n-m}{\underset{i=1}{\mathrm{m}\mathrm{a}\mathrm{x}}}\sum _{k=1}^{n}\left|\mathrm{\Delta }{a}_{ik}^{p}\right|=\stackrel{n}{\underset{i=1}{\mathrm{m}\mathrm{a}\mathrm{x}}}\sum _{k=1}^{n}\left
|\mathrm{\Delta }{a}_{ik}^{p}\right|\\ ={‖P\mathrm{\Delta }{A}_{eq}{P}^{-1}‖}_{\infty }\le {‖\delta A‖}_{\infty }\cdot cond\left(P\right)\cdot {‖I-B\left(GB{\right)}^{-1}G‖}_{\infty },\end{array}$
$\left|{\mu }_{j}-{\lambda }_{i}\right|\le {\stackrel{-}{\rho }}_{\sigma }\cdot {\epsilon }_{A}.$
The above equation shows that the existence of parameter mismatch uncertainty factor $\delta A$ makes the eigenvalue of the equivalent system change from ${\lambda }_{1}$, ${\lambda }_{2}$,…, ${\
lambda }_{n-m}$ to ${\mu }_{1}$, ${\mu }_{2}$,…, ${\mu }_{n-m}$, and all the $n-m$ values of the eigenvalue ${\mu }_{j}$ (single or multiple) are in the union of $n-m$ circles with ${\stackrel{-}{\
rho }}_{\sigma }\cdot {\epsilon }_{A}$ as the radius.
The sufficient condition for the asymptotic stability of the variable structure equivalent systems is given by Eq. (49). To substitute Eq. (53) into the equation, the sufficient condition becomes $\
underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}{R}_{e}\left({\lambda }_{i}\right)\le -{\stackrel{-}{\rho }}_{\sigma }\cdot {\epsilon }_{A}$.
A sufficient condition for the asymptotic stability of the variable structure equivalent system Eq. (49) is given as:
$\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{R}\mathrm{e}\left({\lambda }_{i}\right)<-{\stackrel{-}{\rho }}_{\sigma }\cdot cond\left(P\right)‖\left[I-B\left(GB{\right)}^{-1}G\right]‖=-{\stackrel{-}{\rho }}
_{\sigma }\cdot {\epsilon }_{A},$
where $‖\delta A‖\le {\stackrel{-}{\rho }}_{\sigma }$, $‖\cdot ‖$ is absolute.
2. When $\delta B$ ($\delta A=0$, $\delta Be 0$).
$\delta B$ is as a mismatching input uncertainty factor, and the effect of disturbance on the eigenvalue and stability of the variable structure equivalent system is more complicated than that of the
mismatch parameter uncertain factor $\delta A$, and with the control of the variable structure control system law intertwined. According to the characteristics of nonlinear discontinuous feedback in
variable structure control systems, the control law has the general form:
$U=KX+\rho S\left(‖S‖+\delta {\right)}^{-1},$
where $K\in {R}^{m×n}$, $\rho >0$ and $\delta >0$ are the quiver factors.
For a variable structure equivalent system:
$\stackrel{˙}{X}=\left[I-B\left(GB{\right)}^{-1}G\right]AX+\left[\left[I-B\left(GB{\right)}^{-1}G\right]\right]\delta BU.$
Given that $‖\delta B‖\le {\stackrel{-}{\rho }}_{v}$, ${\stackrel{-}{\rho }}_{v}$ and $\left[I-B\left(GB{\right)}^{-1}G\right]A$ have $n-m$ single nonzero eigenvalues, then the sufficient conditions
for the asymptotic stability of the equivalence system is:
$\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{R}\mathrm{e}\left({\lambda }_{i}\right)<-{\stackrel{-}{\rho }}_{v}\cdot cond\left(P\right)‖\left[I-B\left(GB{\right)}^{-1}G\right]K‖=-{\stackrel{-}{\rho }}_{v}\
cdot {\epsilon }_{B},\begin{array}{l}\end{array}i\in \left\{1,2,\dots ,n-m\right\},$
where $P\in {C}^{n×n}$.
3. When $\delta A$ and $\delta B$ ($\delta Ae 0$, $\delta Be 0$).
For linear mismatched uncertain system Eq. (43), the sliding mode is Eq. (47), and the variable structure control law is given by Eq. (55), if $\left[I-B\left(GB{\right)}^{-1}G\right]A$ has $n-m$
nonzero single eigenvalues ${\lambda }_{1}$, ${\lambda }_{2}$,…, ${\lambda }_{n-m}$; then the sufficient conditions for large-scale asymptotic stability of variable structure equivalent systems is:
$\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}{R}_{e}\left({\lambda }_{i}\right)<-{\stackrel{-}{\rho }}_{\sigma }\cdot {\epsilon }_{A}-{\stackrel{-}{\rho }}_{v}\cdot {\epsilon }_{B},$
where ${\epsilon }_{A}$ and ${\epsilon }_{B}$ are definitude respectively.
4. Numerical simulation and performance analysis
To evaluate the effectiveness of the proposed FSMCEF, A Simulink model is completed according to a certain model of car parameters which are shown in Table 3. For comparison, the FSMCEF, SMC, PID and
passive mode are established for the same model.
The road input is ${\stackrel{˙}{x}}_{r}\left(t\right)=-2\pi {f}_{0}{x}_{r}\left(t\right)+2\pi \sqrt{{G}_{0}{U}_{0}}w\left(t\right)\text{,}$ where ${x}_{r}\left(t\right)$ is the vertical displacement
for pavement input; ${f}_{0}$ is the cut off frequency for road input; ${G}_{0}$ is the road roughness coefficient; ${U}_{0}$ is the speed; $w\left(t\right)$ is the input white noise. Simulation
parameter settings are as follows: ${G}_{0}=$ 6.4×10^-3 m^3, ${U}_{0}=$ 20 m/s, ${f}_{0}=$ 0.01 Hz. Fig. 15 presented the FSMCEF simulink model without MR of quarter-car seat suspension system. Fig.
16 is the result of comparing the proposed FSMCEF with the sky-hook reference model. It displays that the FSMCEF can effective track the sky-hook reference mode.
Table 3parameters of a certain model of car
Parameter Value Unit
${m}_{s}$ 80 kg
${m}_{v}$ 400 kg
${m}_{t}$ 40 kg
${k}_{s}$ 8000 N/m
${c}_{s}$ 250 N/(m·s^-1)
${c}_{s1}$ 700 N/(m·s^-1)
${c}_{sh}$ 2000 N/(m·s^-1)
${k}_{v}$ 18500 N/m
${c}_{v}$ 1500 N/(m·s^-1)
${k}_{t}$ 185000 N/m
${\epsilon }_{fuzzy}$ 2-5 –
Fig. 15FSMCEF Simulink model with MR of quarter-car seat suspension system
Fig. 16Comparing the proposed FSMCEF with the sky-hook reference model
Fig. 17, Fig. 18 and Fig. 19 are the $e1$-$e2$-$e3$, $t$-$e1$-$e2$ and $t$-$e2$-$e3$ phase diagram of fuzzy sliding with expansion factor and sliding mode movement respectively, and every figure has
two different view angle parameters: azimuth (AZ) and elevation (EL). In the beginning with $e1$, $e2$ and $e3$ are in the system initial states, and the defaults are zeros, through the results can
be seen in the graph, each cycle system is able to achieve balance, and it can be seen that FSMCEF can effectively restrain the chattering.
Fig. 17The e1-e2-e3 phase diagram of FSMCEF and SMC
Fig. 18The t-e1-e2 phase diagram of FSMCEF and SMC
Fig. 19The t-e2-e3 phase diagram of FSMCEF and SMC
In order to verify the effectiveness of MR damper neural network-based inverse dynamics model, the Simulink model of seat suspension without neural network-based inverse dynamics model is further
built as shown in Fig. 20, and Fig. 21 and Fig. 22 demonstrated the acceleration and force of FSMCEF controller with and without the inverse dynamic model of MR damper. It can be seen from the
experimental results that the neural network model is used to simulate the inverse dynamic characteristics of the MR damper for the highly nonlinear characteristics of the MR damper. The neural
network model directly provides the desired control force for the generation of the fuzzy sliding mode with expansion factor to obtain a continuous input voltage, and can be seen that FSMCEF
controller with the inverse dynamic model of MR damper can effectively follow the ideal FSMCEF controller.
To testify the performance of FSMCEF, other control methods including SMC, PID and passive suspension (no control) are also simulated as the comparisons. Fig. 23 and Fig. 24 presented the simulation
results of all the above methods. It can be seen that FSMCEF is much better than SMC, PID and passive mode at both acceleration and deflection aspects.
Fig. 20FSMCEF Simulink model without MR of quarter-car seat suspension system
Fig. 21Force result of FSMCEF with and without the inverse dynamic model of MR damper
Fig. 22Acceleration result of FSMCEF with and without the inverse dynamic model of MR damper
Table 4 and 5 presented the standard deviation (STD), maximum (max), minimum (min), mean value (mean) and Root Mean Square (RMS) of the deflection and acceleration of seat suspension under different
controllers. It can be seen that using FSMCEF the STD, max, min, mean and RMS of seat deflection and acceleration are all the best, compared with using SMC, PID and passive mode. The simulation
results are analyzed statistically, Table 6 and Table 7 presented the performance improvement of FSMCEF compared with other methods when employed in seat suspension. It can be seen that FSMCEF is the
best controller, and improves the riding comfort and ride comfort.
Fig. 23Simulation result of seat dynamic deflection
Fig. 24Simulation results of seat acceleration
Table 4Statistics results of seat deflection
Controller type mean / m STD / m max / m min / m RMS / m
Passive mode 0.0084 0.0105 0.0228 –0.0243 0.0104
PID 0.0079 0.0098 0.0223 –0.0226 0.0097
SMC 0.0069 0.0085 0.0183 –0.0208 0.0084
FSMCEF 0.0068 0.0083 0.0182 –0.0202 0.0082
Table 5Statistics results of seat acceleration
Controller type mean / m·s^-2 STD / m·s^-2 max / m·s^-2 min / m·s^-2 RMS / m·s^-2
Passive mode 0.4746 0.6283 2.3419 –1.8309 0.6281
PID 0.4186 0.5574 1.9041 –1.5512 0.5573
SMC 0.3416 0.4179 1.3151 –0.9721 0.4179
FSMCEF 0.2506 0.3040 0.8860 –0.7641 0.3042
From Table 6 it can be concluded that FSMCEF outperforms the other their control methods, especially with 51.57 % improvement relative to the traditional seat suspension in passive mode. Further, the
frequency domain performance of FSMCEF is verified, and the seat suspension acceleration power spectrum density under the random road excitation is shown in Fig. 25. It is shown that FSMCEF improves
significantly the ride comfort of vehicle in lower frequency compared with the SMC, PID and passive seat acceleration, and it improves the vehicle ride comfort. In vehicle body resonant vibration
range (1-1.5 Hz) and the low-mid frequency range (4-12.5 Hz) which human body is sensitive to, the FSMCEF also can effectively reduce the seat acceleration. So, the FSMCEF effectively reduces the
vehicle vibration influence on the human body, significantly improves the dynamic comfort of vehicle systems.
Table 6The performance deflection improvement of FSMCEF compared with other methods
Controller type mean STD RMS
FSMCEF vs Passive mode 19.05 % 20.95 % 21.15 %
FSMCEF vs PID 13.924 % 15.31 % 15.46 %
FSMCEF vs SMC 1.45 % 2.35 % 2.38 %
Table 7The performance acceleration improvement of FSMCEF compared with other methods
Controller type mean STD RMS
FSMCEF vs Passive mode 47.19 % 51.62 % 51.57 %
FSMCEF vs PID 40.13 % 45.46 % 45.42 %
FSMCEF vs SMC 26.63 % 27.26 % 27.21 %
Fig. 25Acceleration power spectrum density of seat suspension under random road excitation
5. Conclusions
In this paper, a fuzzy sliding mode controller with expansion factor (FSMCEF) is designed for the MR damper-based semi-active seat suspension. This FSMCEF takes the sky-hook model as the reference,
and can guarantee the output of MR damper to be effective damping when the motion direction frequently changes. Aiming at the high nonlinearity of MR damper, the neural network model is used to
simulate the inverse dynamic characteristic of MR damper. The neural network model directly provides the expected control force to generate the fuzzy control sliding-mode expansion factor to obtain
continuous input voltage. The FSMCEF is derived based on the error dynamics of the skyhook and the controlled plant, and its fuzzy control term can attenuate the chattering. Considering the
hysteresis nonlinearity of MR damper, a three-layer BP neural network is trained to approximate the MR damper’s reverse dynamics and taken as the controller of the MR damper. Numerical simulations
verified the effectiveness of the FSMCEF compared with PID control, SMC and passive mode for seat suspensions with same model parameters, and the performance of the vehicle suspension system can be
effectively improved by the introduction of the MR damper in the control strategy, and the active control of the MR damper can be realized at the same time.
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About this article
Vibration generation and control
semi-active suspension
MR damper
fuzzy sliding mode controller
expansion factor
neuro-inverse dynamics model
The research work is supported by Heilongjiang Province Science Foundation (Grant No. LC2015019) and supported by the Fundamental Research Funds for the Central Universities (Grant No. 2572015AB18).
Copyright © 2017 JVE International Ltd.
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Optimal Investment Strategy under Stochastic Interest Rates
1. Introduction
The variation of the interest rates affects business decision about how to save and invest [1] [2] . Fundamentally, a rise in interest rates has a sizable negative effect on capital expenditures by
businesses. In economic theory, the cost of capital has an important influence on decisions to invest and, therefore, affects business cycles in general [3] . The market interest rate is considered
to be a key building block in the firm’s user cost of capital, which, combined with the resulting stream of expected cash flows, constitute the primary determinants of whether and how much to invest
[4] [5] [6] . Calderón and Fuentes [7] point out that an increase in the country’s interest rates raises their effective labour costs. Labour cost is usually one of the basic considerations for
business expansion. It is also argued by Chetty [8] that finding out how does an increase in interest rates affect capital investment by firms has important implications for monetary and fiscal
policies. Studies have revealed that the impacts of interest rates on capital investment are more substantial in emerging market countries (EMCs) as compared to developed economies [2] [7] [9] [10] .
This sets the benchmark as to why business planning in the EMCs needs to consider the fluctuations of the interest rates.
This study focuses on the problem of a company that wants to optimize the use of its revenue in the expansion of its investment which is highly sensitive to the fluctuations in the interest rate. We
show how the firm’s financial managers can ensure maximum growth in the investment while avoiding high capital expenditures caused by increase in interest rates. The maximum growth in investment is
calculated as the present value of all the future investment expenditures. Most models on firm’s investment found in the literature do not account for the effect of interest rate fluctuations as one
of the key determinants of the investment strategy. Our main contribution, therefore, is to consider the effect of interest rate in the investment side of the model proposed by Décamps and Villeneuve
[11] , albeit in a modified and simplified presentation.
The effect of uncertainty on firm investment attracts a lot of attention in the literature [12] [13] [14] . Bader and Malawi [15] investigated the impact of real interest rate on investment level in
Jordan and found that real interest rate has a negative impact on investment. Bo and Sterken [12] analyzed the joint impact of the interest rate volatility and debt on firm investment of Dutch listed
firms. One of their findings showed that the effect of the level of the interest rate on investment is significant and negative, and the level effect of the interest rate on investment turns to be
larger for less-indebted firms than for highly indebted firms. Since it is logical that a less-indebted firm or especially a non-indebted firm is the one in a better position to use generated profit
for growth in investment, then consideration of interest rate in investment is very important. Their study provided the cross-effect of the interest rate volatility and debt on investment. However,
there misses an actual strategy which should be adopted by firms when high interest rates have a significant impact on investment. Our study fills this gap by providing a strategy that accounts for
interest rates on firms’ investments explicitly.
Most studies consider investment growth decision of a firm in connection with dividend payments and/or consumption rates [11] [13] [16] [17] . Décamps and Villeneuve [11] , for example, study the
interactions between dividend policy and investment decision in a growth opportunity and under uncertainty. In particular, they consider a firm with a technology in place that has the opportunity to
invest in a new technology that increases its profitability. The firm self-finances the opportunity cost on its cash reserve rather than concentrating on the collected revenue as in our case.
However, they do not present explicitly the optimal level of investment and the actual potential investment stopping times. On the other hand the study by Chevalier, Vath and Scotti [13] considered
the problem of determining the optimal control on the dividend and investment policy of a firm. In addition, they considered the fact that the firm carries a debt obligation in its balance sheet.
They came up with a combined singular and multiple-regime switching control problem whereby each regime corresponds to a level of debt obligation held by the firm. While they considered debt as a
means to further investment, in our case we consider the collected revenue of a firm as main source of funding the investment growth. Hugonnier, Malamud and Morellec [18] developed a model of
investment, financing, and cash management decisions in which investment is lumpy and firms face capital supply uncertainty. They assumed that firms have to search for investors when in need of
capital for investment and therefore it faces uncertainty regarding its ability to raise funds in capital markets. They showed that firms with high investment costs differ in their behaviors from
firms with low investment costs and firms may rise outside funds before exhausting internal resources. Their analysis also revealed that investment and payout do not always increase with slack and
that the choice between internal and external funds does not follow a strict pecking order. In our study we focus on internal funds, profit in particular, in which decision can be made whether to
invest or consume. As we appreciate the impacts of interest rates on making investment decision, we build on such studies by providing a means to make more effective investment decisions. This is a
requirement for growing firms particularly in emerging markets.
Several facts collectively stand as the basis of our motivation to undertake this study. First we are motivated by the fact that there is a great discrepancy on the effect of interest rate between
the developed and the emerging economies as explained in [2] [7] [9] [10] . Since firms operating in the emerging economies suffer most from the fluctuations in the interest rates, their contribution
to the economy will also be limited and therefore there is a necessity to make them plan more effectively under the same circumstance. Secondly, in such emerging economies business is more affected
by macroeconomic shocks, therefore providing a means for firms to plan against interest rates will pave the way to combat other factors such as transaction costs and exchange rates. Also the role of
firms’ investments in transforming the economies of emerging market countries (EMCs) is enormously significant thus needs special attention. Moreover, it is argued that due to the availability of
natural resources, labour and consumer markets in such countries, there are always opportunities for the firms to grow. This is why having effective strategies for the growth of firms in such
contexts remains a necessity.
In our study we model investment as related to revenue rather than simply the cash holding of a firm because we assume that from the collected revenue the decision is immediately made on whether to
expand investment or consume (or pay dividend). Such a managerial decision is made depending on the level of the interest rate which is assumed to be a continuous stochastic process that can
generally be modeled as a term structure. Gibson, Lhabitant and Talay [19] , and Huang, Sun and Chen [20] provide appealing presentations of the term structure models of interest rates. Akyildirim et
al. [21] provide a means of applying stochastic interest rates in the optimization of dividend payouts. Apart from the fact that their stochastic optimal control was on dividends and ours is on
investment, their interest rate had only two states one for good economic state and the other for bad economic state. In our case we consider the interest rate that varies continuously and apply a
stochastic discount factor in the optimization of investment level. This is because naturally the economy cannot have only two states but varies randomly in its spectrum.
The outline of this paper is as follows. Section 2 gives model formulation, explanation on threshold interest rate value and investment cost, and definition of the objective function. In section 3,
we state the properties of the value function and provide proofs. We show that the value function for our objective function exists and it is unique and concave in both the profit and interest rate
variables, thus can be a solution of the dynamic programming equation. In section 4, we carry-out numerical experiment. We provide a plot that gives the general overview of the value function over
the interest rates and the profit levels. We present a numerical determination of the threshold interest rate value. We also give a numerical test for the sensitivity of the value function on the
drift for different interest rates and profit levels. In section 5, we summarize the results of our study and make a conclusion.
2. Model Formulation
Uncertainty is described by a probability space $\left(\Omega ,\mathcal{F},ℙ\right)$ and a filtration ${\left({\mathcal{F}}_{t}\right)}_{t\ge 0}$ satisfying the common assumptions. Let ${B}_{t}$ be
one dimensional $\left({\mathcal{F}}_{t}\right)$ ―standard Brownian motion. We consider a firm whose business generates profit ${Y}_{t}$ that follows a Stochastic Deferential Equation (SDE) with a
drift ${\alpha }_{0}$ and volatility $\beta$ presented as
$\text{d}{Y}_{t}={\alpha }_{0}\text{d}t+\beta \text{d}{B}_{t}.$ (1)
The firm operates in a floppy macroeconomic environment whereby the interest rate r varies randomly between the lowest possible interest rate and the highest possible interest rate. The dynamics of
interest rates is assumed to be a continuous stochastic process representing the state of economy. Depending on the state of economy, the firm may opt to channel a portion G of the profit Y to
further its investment which will increase the drift from ${\alpha }_{0}$ to ${\alpha }_{G}\ge {\alpha }_{0}$ . We assume that there is no alteration in the volatility $\beta$ . The drift ${\alpha }_
{G}$ is proportional to the investment cost G. The firm has to define the threshold interest rate ${r}_{\theta }$ from which the firm can further its investment only when $r\le {r}_{\theta }$
otherwise channels its profit into other uses such as dividend payments. The investment cost ${G}_{t}$ at time t which must be financed from the profit made ${Y}_{t}$ is positive non-decreasing right
continuous and adapted to the interest rate process ${r}_{t}$ . The set $\left\{t:{r}_{t}\le {r}_{\theta }\right\}$ consists of all possible stopping times $\tau$ at which investment can be done. We
therefore model the profit trend subject to investment as follows;
This is a simplified version derived from the cash reserve model given by [11] from which we concentrate in the side of investment and consider stochastic interest rate in the optimization later.
We consider the fact that at any time t and the interest rate ${r}_{t}$ , $0\le {G}_{t}\le {Y}_{t}$ and also the more the profit the higher is the upper limit level of optimal investment. This is
represented by the following relationship
${G}_{t}\le \left(1-{\text{e}}^{-\left({r}_{\theta }-{r}_{t}\right)}\right){Y}_{t}.$ (3)
The increment in the drift at time $\tau$ , from ${\alpha }_{0}$ to ${\alpha }_{G}$ depends on the ratio of the investment made ${G}_{\tau }$ to the profit ${Y}_{\tau }$ . In fact, the relationship
between ${\alpha }_{G}$ and ${\alpha }_{0}$ is given by
${\alpha }_{G}={\alpha }_{0}+\frac{{G}_{\tau }}{{Y}_{\tau }}\le {\alpha }_{0}+\frac{\left(1-{\text{e}}^{-\left({r}_{\theta }-{r}_{\tau }\right)}\right){Y}_{\tau }}{{Y}_{\tau }}={\alpha }_{0}+\left(1-
{\text{e}}^{-\left({r}_{\theta }-{r}_{\tau }\right)}\right).$ (4)
We make an assumption that the company should make positive profit for its survival, otherwise it has to undergo bankruptcy. We therefore define the bankruptcy time by
$⊺=inf\left\{t\ge 0:{Y}_{t}<0\right\}.$ (5)
Our aim is to study the optimal investment strategy of a firm as related to the variations in the interest rates. The firm wants to invest as more as it is feasible but is challenged by the
fluctuation of the interest rates. Given as initial conditions the value of the profit y and the corresponding interest rate r, we denote the set of all admissible investment costs and the stopping
times $\left(G,\tau \right)$ by $\mathcal{A}\left(y\right)$ . The investment costs and the stopping times constitute the control policy $\left({G}_{t},\tau ,t\ge 0\right)$ . Mathematically, the
optimal investment problem is to maximize the value function
$H\left(y,r;G,\tau \right)={\mathbb{E}}^{y,r}\left[{\int }_{0}^{⊺}\text{ }{\Lambda }_{t}\text{ }\text{d}{G}_{t}\right],\text{\hspace{0.17em}}\text{where}\text{\hspace{0.17em}}{\Lambda }_{t}=\text
{exp}\left(-{\int }_{0}^{t}\text{ }{r}_{s}\text{d}s\right).$ (6)
The corresponding optimal value function is then defined as
$v\left(y,r\right)=\underset{\left(G,\tau \right)\in \mathcal{A}\left(y\right)}{sup}H\left(y,r;G,\tau \right)$ (7)
and the optimal policy $\left({G}_{t}^{*},{\tau }^{*}\right)$ is such that
$H\left(y,r;{G}_{t}^{*},{\tau }^{*}\right)=v\left(y,r\right).$ (8)
3. The Value Function
We present analytically the characterization of the optimal value function. Our main goal is to maximize the expected investment fund discounted under a stochastic interest rate and finding the
optimal stopping times for investments to be done. In fact the stopping times are evaluated from the threshold interest rate ${r}_{\theta }$ as elaborated in the previous section.
We state the following theorem which summarizes the key features of the optimal value function. Important to mention these features are the uniqueness and concavity properties. Prior to stating the
theorem we define the differential operator for v, by considering the stochastic interest rate r, as
and use it in the theorem statement.
Theorem 1. The optimal value function $v=v\left(y,r\right)$ is the unique concave function satisfying the following conditions: [label=.]
1. $v\in {C}^{2}\left(\left[0,\infty \right)\right)$ and $v\left(0,r\right)=0$ ,
2. $\frac{\partial }{\partial y}v\left(y,r\right)\ge 1$ for all $y,r$ ,
3. For every $y>0$ , $r>0$ ,
$\mathcal{L}v\left(y,r\right)-rv\left(y,r\right)\ge 0.$ (10)
The formation of this theorem is analogous to the one given by [21] when dealing with optimal dividend policy. However, the statements in this theorem consider a continuous stochastic interest rate
as it also appears in the third term of the differential operator. This leads to a different approach in the analysis since the value function depends on two continuous independent variables y and r.
We take into account the argument that it is preferable to allocate fund from profit to further investment when the interest rate is below the threshold interest rate value ${r}_{\theta }$ over when
the interest rate is above this threshold value.
From the theorem stated above we find that the optimal value function is represented as the solution of the following dynamic programming equation
$\mathrm{max}\left\{\mathcal{L}v\left(y,r\right)-rv\left(y,r\right),-{v}_{y}\left(y,r\right)+1\right\}=0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }y>0,\text{ }\text{ }r>0$ (11)
having the boundary condition $v\left(0,r\right)=0$ and initial condition $v\left(y,0\right)=0$ . We have used the subscript to represent a partial derivative with respect to a variable. Also from
this point onwards, we shall be using $\stackrel{^}{\alpha }$ in the place of the expression which simplifies the analysis without destroying the intended meaning. Actually, the fact that $\stackrel
{^}{\alpha }$ takes the value of either ${\alpha }_{0}$ or ${\alpha }_{G}$ at a time is dealt with numerically by the sensitivity of the value function v to $\stackrel{^}{\alpha }$ in Section 4.
The representation in Equation (11) is actually a Hamilton-Jacobi-Bellman (HJB) equation characterized by the following system of equations
$\mathcal{L}v\left(y,r\right)-rv\left(y,r\right)=0,\text{ }\text{ }\text{ }\text{for}\text{\hspace{0.17em}}y>0,\text{ }\text{ }r>0,$ (12)
${v}_{y}\left(y,r\right)=1,\text{ }\text{ }\text{ }\text{for}\text{ }\text{\hspace{0.17em}}y>0,\text{ }\text{ }r={r}_{\theta },$ (13)
$v\left(0,r\right)=0.$ (14)
It can be seen clearly that Equation (12) is a second order linear partial differential equation (PDE) of parabolic type with initial and boundary conditions given by Equations (13) and (14)
Next, we prove Theorem 1 above by considering the PDE with the initial- boundary conditions above. To the best of our experience, we have never come across an attempt to prove the existence and
uniqueness of the value function by means of reducing the dynamic programming equation into a simple diffusion equation as it is done here.
Proof. Since a simple diffusion equation has a solution, we attempt the proof for existence of solution for v as given in Equation (12) by transformation of the equation into a simple diffusion
equation. This is done by the change in variables. We write Equation (12) in an expanded form with subscript notation as
$-{v}_{r}+\frac{1}{2}{\beta }^{2}{v}_{yy}+\stackrel{^}{\alpha }{v}_{y}-rv=0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }y>0,\text{ }\text{ }r>0.$ (15)
Changing v to u by $u=v{\text{e}}^{\frac{1}{2}{r}^{2}}$ implies
$v={\text{e}}^{-\frac{1}{2}{r}^{2}}u,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }{v}_{r}=-r{\text{e}}^{-\frac{1}{2}{r}^{2}}u+{\text{e}}^{-\frac{1}{2}{r}^{2}}{u}_{r},\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }{v}_{y}={\text{e}}^{-\frac{1}{2}{r}^{2}}{u}_{y},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }{v}_{yy}={\text{e}}^{-\frac{1}{2}{r}^{2}}{u}_{yy}.$
Which after substitution simplify Equation (15) to
$-{u}_{r}+\frac{1}{2}{\beta }^{2}{u}_{yy}+\stackrel{^}{\alpha }{u}_{y}=0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }y>0,\text{ }\text{ }r>0.$ (16)
Next we transform the independent variables y, r to $\eta =y-\stackrel{^}{\alpha }r$ , $\rho =r$ which leads to a simple diffusion equation under w,
$-{w}_{\rho }+\frac{1}{2}{\beta }^{2}{w}_{\eta \eta }=0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\eta >0,\text{ }\text{ }\rho >0.$ (17)
This can be written as
${w}_{r}-\frac{1}{2}{\beta }^{2}{w}_{yy}=0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }y>0,\text{ }\text{ }r>0,$ (18)
with a complete transformation given by
$v\left(y,r\right)=w\left(y,r\right){\text{e}}^{\frac{\stackrel{^}{\alpha }}{{\beta }^{2}}\left(y-\stackrel{^}{\alpha }r\right)-\frac{{r}^{2}}{2}}.$ (19)
Therefore, the solution for (15) and thus for (12) exists.
Uniqueness can now be shown from Equation (18).
Lemma 1. The diffusion equation presented by (18) with some given initial and boundary conditions deduced from (13) and (14) has a unique solution.
Proof. Suppose that ${w}_{1}$ and ${w}_{2}$ are solutions to the Equation (18), we show that $x={w}_{1}-{w}_{2}=0$ . The function x also satisfies (18) and the boundary conditions. We define the
function $\psi$ by
$\psi \left(r\right):={\int }_{0}^{\infty }{\left[x\left(y,r\right)\right]}^{2}\text{d}y.$ (20)
Differentiation under the integral sign leads to
$\begin{array}{c}\frac{\text{d}}{\text{d}r}\psi \left(r\right)=2{\int }_{0}^{\infty }x\left(y,r\right){x}_{r}\left(y,r\right)\text{d}y\\ =2{\int }_{0}^{\infty }x\left(y,r\right)\left(\frac{{\sigma }^
Integrating by parts in the variable y we find
$\frac{\text{d}}{\text{d}r}\psi \left(r\right)={{\beta }^{2}x\left(y,r\right)\cdot {x}_{y}\left(y,r\right)|}_{0}^{\infty }-2{\int }_{0}^{\infty }{\left[{x}_{y}\left(y,r\right)\right]}^{2}\text{d}y.$
From the boundary and initial conditions we deduce that
${x}_{y}\left(0,r\right)={x}_{y}\left(\infty ,r\right)=x\left(y,0\right)=0.$
$\frac{\text{d}}{\text{d}r}\psi \left(r\right)=-2{\int }_{0}^{\infty }{\left[{x}_{y}\left(y,r\right)\right]}^{2}\text{d}y\le 0.$ (22)
So, $\psi \left(r\right)$ is decreasing in r. We find that since $\psi \left(0\right)=0$ and since $\psi \left(r\right)\ge 0$ then $\psi \left(r\right)=0$ for all $r>0$ . Hence $x=0$ .
Now, we prove the concavity of v. Different from what is common in most of the literature [21] [22] , where the concavity of the value function is proved only w.r.t the variable y, we prove the
concavity of v in both y and r. This is because v in our case varies in both variables y and r. We fix r and show that v is concave in y and conversely.
Lemma 2. The solution of the PDE given in (15) is concave on both of the independent variables y and r.
Proof. We first arbitrarily fix r at $\stackrel{¯}{r}$ and thus Equation (15) becomes
$\frac{1}{2}{\beta }^{2}{v}_{yy}+\stackrel{^}{\alpha }{v}_{y}-\stackrel{¯}{r}v=0.$ (23)
This is an ODE w.r.t the variable y with the general solution
$v\left(y,\stackrel{¯}{r}\right)={c}_{1}{\text{e}}^{{k}_{1}y}+{c}_{2}{\text{e}}^{{k}_{2}y},$ (24)
where ${c}_{1}$ , ${c}_{2}$ are real constants and ${k}_{1}$ , ${k}_{2}$ are also real numbers given by
${k}_{1}=\frac{1}{{\beta }^{2}}\left(-\stackrel{^}{\alpha }+\sqrt{{\stackrel{^}{\alpha }}^{2}+2{\beta }^{2}\stackrel{¯}{r}}\right),$ (25)
${k}_{2}=\frac{1}{{\beta }^{2}}\left(-\stackrel{^}{\alpha }-\sqrt{{\stackrel{^}{\alpha }}^{2}+2{\beta }^{2}\stackrel{¯}{r}}\right).$ (26)
It is observed that ${k}_{2}<0<{k}_{1}$ and from the boundary condition (14) we have ${c}_{1}+{c}_{2}=0$ . We consider a fixed point $\stackrel{¯}{y}$ satisfying the condition given by (13) and find
every function satisfying the Equations (12) to (14) has the form
$v\left(y,\stackrel{¯}{r}\right)=\left\{\begin{array}{ll}{c}_{1}\left({\text{e}}^{{k}_{1}y}-{\text{e}}^{{k}_{2}y}\right),\hfill & \text{ }0<y<\stackrel{¯}{y},\hfill \\ y-\stackrel{¯}{y}+v\left(\
stackrel{¯}{y},\stackrel{¯}{r}\right),\hfill & \text{ }\text{ }\text{ }y\ge \stackrel{¯}{y}.\hfill \end{array}$ (27)
Assuming that v is ${C}^{2}$ at $\stackrel{¯}{y}$ , we use the smooth-pasting condition for singular control to determine ${c}_{1}$ and $\stackrel{¯}{y}$ . The condition leads to
${c}_{1}{k}_{1}{\text{e}}^{{k}_{1}\stackrel{¯}{y}}+{c}_{2}{k}_{2}{\text{e}}^{{k}_{2}\stackrel{¯}{y}}=1,$ (28)
${c}_{1}{k}_{1}^{2}{\text{e}}^{{k}_{1}\stackrel{¯}{y}}+{c}_{2}{k}_{2}^{2}{\text{e}}^{{k}_{2}\stackrel{¯}{y}}=0,$ (29)
from which we obtain
$\stackrel{¯}{y}=-\frac{\mathrm{ln}\left({k}_{1}^{2}\right)-\mathrm{ln}\left({k}_{2}^{2}\right)}{{k}_{1}-{k}_{2}},$ (30)
and from ${k}_{2}<0<{k}_{1}$ ,
${c}_{1}={\left[{k}_{1}{|\frac{{k}_{2}}{{k}_{1}}|}^{\frac{2{k}_{1}}{{k}_{1}-{k}_{2}}}-{k}_{2}{|\frac{{k}_{2}}{{k}_{1}}|}^{\frac{2{k}_{2}}{{k}_{1}-{k}_{2}}}\right]}^{-1}>0.$ (31)
From (25) and (26) with (30) it can be established that $\stackrel{¯}{y}>0$ is necessary and sufficient for $\stackrel{^}{\alpha }>0$ . We find that the function v as presented by (27) is well
defined if and only if $\stackrel{^}{\alpha }>0$ and since $\stackrel{¯}{y}>0$ we get that v is concave w.r.t the variable y.
Next we also arbitrarily fix y at ${y}^{*}$ and therefore Equation (15) becomes
$-{v}_{r}-rv=0.$ (32)
We simply solve for v and have
$v\left({y}^{*},r\right)={\text{e}}^{-\frac{{r}^{2}}{2}}$ (33)
which is concave up.
As the consequence of the proof of the theorem given above we have the following corollary about the value function.
Corollary 1. Consider the maximization problem of the value function $H\left(y,r;G,\tau \right)$ over all strategies $\left(G,\tau \right)$ in the admissible set $\mathcal{A}\left(y\right)$ . Then
the concave solution v to the HJB Equation (11) with the drift $\stackrel{^}{\alpha }>0$ given by (27) under fixed r where the constants $\stackrel{¯}{y}$ and ${c}_{1}$ are as found in (30) and (31),
and that appears as (33) under fixed y, is the optimal value function.
4. Numerical Results and Discussion
Under this section, we illustrate how the value function varies in relation to the interest rates and the profit levels. We mainly consider the PDE as it appears in (12) and (15). We begin by
providing a three dimensional plot that gives a general outlook of the dependence of the value function on the interest rates and the profit levels. The next two figures show how the value function
varies over the interest rates at different profit levels and how it varies over the profit values at different interest rates. The former will be helpful in estimating the threshold interest rate $
{r}_{\theta }$ which is in fact the point where the value function is the same for high and low profit levels. The last two illustrations are about sensitivity of the value function to the
generalized profitability $\alpha$ for different values of interest rate and profit levels. The parameter values for profitability and volatility have been adopted from the study by [23] and the
interest rates are the results of estimation.
In Figure 1 we generally observe that the value function increases as the interest rate decreases and also it increases as the profit level increases. This automatically suggests that the favourable
time to invest is when the interest rate is low and the profit level is high, a fact which is detailed in Figure 2.
In Figure 2, we find that the value function drops exponentially as the interest rate increases when the profit level is high whereas a moderate increase is
Figure 1. General overview of the value function over the interest rate and profit level with α = 1.5 and β = 1.5.
Figure 2. Variation of the value function over interest rate r for different levels of profit y.
experienced as to the increase in interest rate when the profit level is low. At the middle level of profit there is a moderate increase followed by a moderate drop. The curve for high profit and the
curve for low profit cross each other at the point with $r\approx 0.18$ , this gives the threshold interest rate ${r}_{\theta }=18%$ . Since it is a contradiction to further investment under low
profit, we consider the high profits which give higher value function when the interest rate $r<{r}_{\theta }$ .
By inspection, Figure 3 establishes the fact that the value function has significantly high values when the interest rate is low and the profit level is high. However, at extreme high profit values
the value function starts to drop which means theoretically that it is not appropriate to consider business growth when profit is at extreme.
Both Figure 4 and Figure 5 portray that the value function v is more sensitive to the parameter α when the interest rates are high and less sensitive when the interest rates are low, regardless of
the profit levels. This indicates that business is more stable when the interest rates are low than when the interest rates are high. In addition, we learn from Figure 4 that at some points having
high profitability combined with high levels of profit is not suitable to expand investment as compared to relatively low profitability.
5. Conclusion, Recommendation and Possible Extensions
In this work, we have set up a strategy for a company that wants to maximize its investment under the context of randomly fluctuating interest rates. We considered a company that operates in a floppy
economy, such as that of developing countries, and generates profit which can be modeled by a SDE. The company has to identify the threshold interest rate value, above which investment is not a
feasible decision.
Figure 3. Variation of the value function over profit level y for different interest rates r.
Figure 4. Sensitivity of the value function w.r.t profitability α under the interest rates r.
Figure 5. Sensitivity of the value function w.r.t profitability α under the profit levels y.
From this study, we find mainly four results which we explain in summary. First we find that the value function increases as the interest rate decreases and it also increases as the profit level
increases. This is to say that the interest rate and the profit level have opposing influence in the maximization policy on investment. The second and vital result of this study is the existence of a
threshold value for the interest rate ${r}_{\theta }$ for a given firm which can stand as the basis for making interest rate based investment decision. For the parameters we have, such a threshold
interest rate is about 18%. Monetary policy makers may make sure that the interest rates are well standardized to be below this value in order to promote investment by firms in emerging market
countries. Thirdly we find that it is not advisable for companies to plan for expansion of their business when are already making extremely high profits from the business they are undertaking.
Possibly a good advice for such companies is to plan for investing on other kinds of business as they enjoy higher profits from the existing business. Lastly we revealed that business is more stable
when the interest rates are low than when the interest rates are high. Though this can be obvious in economic terms, we explicitly presented here in relation to the investment decision of firms, so
that the decision making on investment considers the role of interest rate to be of major concern.
As a consequence of this study, a combined optimal dividend and investment policy under stochastic interest rate can be studied. In such a study, the attention can be in the maximization of dividend
payments and accumulating some fund for investment or conversely. As an extension to this study, it can also be possible to find the optimal time for accessing loan for investment under stochastic
interest rates.
The study by Adeline P. Mtunya is under the sponsorship of the Government of The United Republic of Tanzania through the Commission for Science and Technology (COSTECH). The authors thank the sponsor
for the financial grant that was necessary towards accomplishment of this work. We also appreciate the material and moral support from the management of The Nelson Mandela African Institution of
Science and Technology (NM-AIST) and Mkwawa University College of Education (MUCE).
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Cartesian product
Cartesian product of the sets {x,y,z} and {1,2,3}
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B.^[1] In terms of set-builder
notation, that is ${\displaystyle A\times B=\{(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\}.}$^[2]^[3]
A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the
form (row value, column value).^[4]
One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered
pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.
The Cartesian product is named after René Descartes,^[5] whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.
Set-theoretic definition
A rigorous definition of the Cartesian product requires a domain to be specified in the set-builder notation. In this case the domain would have to contain the Cartesian product itself. For defining
the Cartesian product of the sets ${\displaystyle A}$ and ${\displaystyle B}$, with the typical Kuratowski's definition of a pair ${\displaystyle (a,b)}$ as ${\displaystyle \{\{a\},\{a,b\}\}}$, an
appropriate domain is the set ${\displaystyle {\mathcal {P}}({\mathcal {P}}(A\cup B))}$ where ${\displaystyle {\mathcal {P}}}$ denotes the power set. Then the Cartesian product of the sets ${\
displaystyle A}$ and ${\displaystyle B}$ would be defined as^[6] ${\displaystyle A\times B=\{x\in {\mathcal {P}}({\mathcal {P}}(A\cup B))\mid \exists a\in A\ \exists b\in B:x=(a,b)\}.}$
A deck of cards
Standard 52-card deck
An illustrative example is the
standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits
{♠, ♥, ♦, ♣
} form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52
ordered pairs
, which correspond to all 52 possible playing cards.
Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}.
Suits × Ranks returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}.
These two sets are distinct, even disjoint, but there is a natural bijection between them, under which (3, ♣) corresponds to (♣, 3) and so on.
A two-dimensional coordinate system
Cartesian coordinates of example points
The main historical example is the
. Usually, such a pair's first and second components are called its
coordinates, respectively (see picture). The set of all such pairs (i.e., the Cartesian product
${\displaystyle \mathbb {R} \times \mathbb {R} }$
, with
${\displaystyle \mathbb {R} }$
denoting the real numbers) is thus assigned to the set of all points in the plane.
Most common implementation (set theory)
A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, Kuratowski's definition, is ${\
displaystyle (x,y)=\{\{x\},\{x,y\}\}}$. Under this definition, ${\displaystyle (x,y)}$ is an element of ${\displaystyle {\mathcal {P}}({\mathcal {P}}(X\cup Y))}$, and ${\displaystyle X\times Y}$ is a
subset of that set, where ${\displaystyle {\mathcal {P}}}$ represents the
, and relations are usually defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is necessarily prior to most other definitions.
Non-commutativity and non-associativity
Let A, B, C, and D be sets.
The Cartesian product A × B is not
${\displaystyle A\times Beq B\times A,}$^[4]
because the
ordered pairs
are reversed unless at least one of the following conditions is satisfied:
For example:
A = {1,2}; B = {3,4}
A × B = {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)}
B × A = {3,4} × {1,2} = {(3,1), (3,2), (4,1), (4,2)}
A = B = {1,2}
A × B = B × A = {1,2} × {1,2} = {(1,1), (1,2), (2,1), (2,2)}
A = {1,2}; B = ∅
A × B = {1,2} × ∅ = ∅
B × A = ∅ × {1,2} = ∅
Strictly speaking, the Cartesian product is not
(unless one of the involved sets is empty).
${\displaystyle (A\times B)\times Ceq A\times (B\times C)}$
If for example
A = {1}
, then
(A × A) × A = {((1, 1), 1)} ≠ {(1, (1, 1))} = A × (A × A)
Intersections, unions, and subsets
The Cartesian product satisfies the following property with respect to intersections (see middle picture). ${\displaystyle (A\cap B)\times (C\cap D)=(A\times C)\cap (B\times D)}$
In most cases, the above statement is not true if we replace intersection with union (see rightmost picture). ${\displaystyle (A\cup B)\times (C\cup D)eq (A\times C)\cup (B\times D)}$
In fact, we have that: ${\displaystyle (A\times C)\cup (B\times D)=[(A\setminus B)\times C]\cup [(A\cap B)\times (C\cup D)]\cup [(B\setminus A)\times D]}$
For the set difference, we also have the following identity: ${\displaystyle (A\times C)\setminus (B\times D)=[A\times (C\setminus D)]\cup [(A\setminus B)\times C]}$
Here are some rules demonstrating distributivity with other operators (see leftmost picture):^[8] {\displaystyle {\begin{aligned}A\times (B\cap C)&=(A\times B)\cap (A\times C),\\A\times (B\cup C)&=(A
\times B)\cup (A\times C),\\A\times (B\setminus C)&=(A\times B)\setminus (A\times C),\end{aligned}}} ${\displaystyle (A\times B)^{\complement }=\left(A^{\complement }\times B^{\complement }\right)\
cup \left(A^{\complement }\times B\right)\cup \left(A\times B^{\complement }\right)\!,}$ where ${\displaystyle A^{\complement }}$ denotes the
absolute complement
Other properties related with subsets are:
${\displaystyle {\text{if }}A\subseteq B{\text{, then }}A\times C\subseteq B\times C;}$
${\displaystyle {\text{if both }}A,Beq \emptyset {\text{, then }}A\times B\subseteq C\times D\!\iff \!A\subseteq C{\text{ and }}B\subseteq D.}$^[9]
The cardinality of a set is the number of elements of the set. For example, defining two sets: A = {a, b} and B = {5, 6}. Both set A and set B consist of two elements each. Their Cartesian product,
written as A × B, results in a new set which has the following elements:
A × B = {(a,5), (a,6), (b,5), (b,6)}.
where each element of A is paired with each element of B, and where each pair makes up one element of the output set. The number of values in each element of the resulting set is equal to the number
of sets whose Cartesian product is being taken; 2 in this case. The cardinality of the output set is equal to the product of the cardinalities of all the input sets. That is,
|A × B| = |A| · |B|.^[4]
In this case, |A × B| = 4
|A × B × C| = |A| · |B| · |C|
and so on.
The set A × B is infinite if either A or B is infinite, and the other set is not the empty set.^[10]
Cartesian products of several sets
n-ary Cartesian product
The Cartesian product can be generalized to the n-ary Cartesian product over n sets X[1], ..., X[n] as the set ${\displaystyle X_{1}\times \cdots \times X_{n}=\{(x_{1},\ldots ,x_{n})\mid x_{i}\in X_
{i}\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}}$
of n-tuples. If tuples are defined as nested ordered pairs, it can be identified with (X[1] × ... × X[n−1]) × X[n]. If a tuple is defined as a function on {1, 2, ..., n} that takes its value at i to
be the i-th element of the tuple, then the Cartesian product X[1] × ... × X[n] is the set of functions ${\displaystyle \{x:\{1,\ldots ,n\}\to X_{1}\cup \cdots \cup X_{n}\ |\ x(i)\in X_{i}\ {\text{for
every}}\ i\in \{1,\ldots ,n\}\}.}$
n-ary Cartesian power
The Cartesian square of a set X is the Cartesian product X^2 = X × X. An example is the 2-dimensional plane R^2 = R × R where R is the set of real numbers:^[1] R^2 is the set of all points (x,y)
where x and y are real numbers (see the Cartesian coordinate system).
The n-ary Cartesian power of a set X, denoted ${\displaystyle X^{n}}$, can be defined as ${\displaystyle X^{n}=\underbrace {X\times X\times \cdots \times X} _{n}=\{(x_{1},\ldots ,x_{n})\ |\ x_{i}\in
X\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}.}$
An example of this is R^3 = R × R × R, with R again the set of real numbers,^[1] and more generally R^n.
The n-ary Cartesian power of a set X is
Infinite Cartesian products
It is possible to define the Cartesian product of an arbitrary (possibly infinite) indexed family of sets. If I is any index set, and ${\displaystyle \{X_{i}\}_{i\in I}}$ is a family of sets indexed
by I, then the Cartesian product of the sets in ${\displaystyle \{X_{i}\}_{i\in I}}$ is defined to be ${\displaystyle \prod _{i\in I}X_{i}=\left\{\left.f:I\to \bigcup _{i\in I}X_{i}\ \right|\ \forall
i\in I.\ f(i)\in X_{i}\right\},}$ that is, the set of all functions defined on the index set I such that the value of the function at a particular index i is an element of X[i]. Even if each of the X
[i] is nonempty, the Cartesian product may be empty if the axiom of choice, which is equivalent to the statement that every such product is nonempty, is not assumed. ${\displaystyle \prod _{i\in I}X_
{i}}$ may also be denoted ${\displaystyle {\mathsf {X}}}$${\displaystyle {}_{i\in I}X_{i}}$.^[11]
For each j in I, the function ${\displaystyle \pi _{j}:\prod _{i\in I}X_{i}\to X_{j},}$ defined by ${\displaystyle \pi _{j}(f)=f(j)}$ is called the j-th projection map.
Cartesian power is a Cartesian product where all the factors X[i] are the same set X. In this case, ${\displaystyle \prod _{i\in I}X_{i}=\prod _{i\in I}X}$ is the set of all functions from I to X,
and is frequently denoted X^I. This case is important in the study of
cardinal exponentiation
. An important special case is when the index set is
${\displaystyle \mathbb {N} }$
, the
natural numbers
: this Cartesian product is the set of all infinite sequences with the
-th term in its corresponding set
. For example, each element of
${\displaystyle \prod _{n=1}^{\infty }\mathbb {R} =\mathbb {R} \times \mathbb {R} \times \cdots }$
can be visualized as a
with countably infinite real number components. This set is frequently denoted
${\displaystyle \mathbb {R} ^{\omega }}$
, or
${\displaystyle \mathbb {R} ^{\mathbb {N} }}$
Other forms
Abbreviated form
If several sets are being multiplied together (e.g., X[1], X[2], X[3], ...), then some authors^[12] choose to abbreviate the Cartesian product as simply ×X[i].
Cartesian product of functions
If f is a function from X to A and g is a function from Y to B, then their Cartesian product f × g is a function from X × Y to A × B with ${\displaystyle (f\times g)(x,y)=(f(x),g(y)).}$
This can be extended to tuples and infinite collections of functions. This is different from the standard Cartesian product of functions considered as sets.
Let ${\displaystyle A}$ be a set and ${\displaystyle B\subseteq A}$. Then the cylinder of ${\displaystyle B}$ with respect to ${\displaystyle A}$ is the Cartesian product ${\displaystyle B\times A}$
of ${\displaystyle B}$ and ${\displaystyle A}$.
Normally, ${\displaystyle A}$ is considered to be the universe of the context and is left away. For example, if ${\displaystyle B}$ is a subset of the natural numbers ${\displaystyle \mathbb {N} }$,
then the cylinder of ${\displaystyle B}$ is ${\displaystyle B\times \mathbb {N} }$.
Definitions outside set theory
Category theory
Although the Cartesian product is traditionally applied to sets,
fiber product
Graph theory
In graph theory, the Cartesian product of two graphs G and H is the graph denoted by G × H, whose vertex set is the (ordinary) Cartesian product V(G) × V(H) and such that two vertices (u,v) and (u′,v
′) are adjacent in G × H, if and only if u = u′ and v is adjacent with v′ in H, or v = v′ and u is adjacent with u′ in G. The Cartesian product of graphs is not a product in the sense of category
theory. Instead, the categorical product is known as the tensor product of graphs.
See also
External links
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Disqus Comments
Some more Julia sets...
Super fun! Here's a tweak I made for the Julia set.
from math import floor, ceil, sin,cos
def linear_interpolation(color1, color2, t):
return color1 * (1-sin(t)) + (color2 * cos(t))
# Image size (pixels)
WIDTH = 1000
HEIGHT = 1000
# Plot window
RE_START = -0.60
RE_END = -0.00
IM_START = -0.20
IM_END = 0.40
c = complex(-0.8, 0.156)
• 3 years ago
It creates an image file called output in the folder/directory your .py file is run from.
I hate garbage code
• 3 years ago
It got executed perfectly. Too bad it literally does nothing.
It works on the Mandelbrot set too... very Lovecraft meets Alice in Wonderland!
I love how a 3D structure of roots, reaching into infinity, seems to appear with this color scheme.
from math import floor, ceil, sin, cos
def linear_interpolation(color1, color2, t):
return color1 * (1-sin(t)) + (color2 * cos(t))
# Image size (pixels)
WIDTH = 1000
HEIGHT = 1000
# Plot window
RE_START = -0.925
RE_END = -0.800
IM_START = -0.325
IM_END = -0.200
Vianney Hervy
• 3 years ago
Hello, I have a question. At the beginning of the definition, you assume that the sequence is not bounded if the modulus of one of its terms is greater than 2. I can't figure out a demonstration for
that fact. Could you help me or give me advice ?
Thanks a lot
• 3 years ago
One suggestion for extending the tool: Add a constant called NODES = 1, and replace "x=x*x+c" with "x=x^(NODES+1)+c".
This is because the standard Mandlebrot set is based on "X^2+C", but incrementing that exponent to X^n gives you n-1 nodes as the result. I replicated this by just adding more x's to the code above,
but since most people don't know that property of the set, it would be an easy way for them to play with it.
• 3 years ago
Proper notation to get this to work in Python 3.7 is "x=x**(NODES+1)+c" ** is the notation for exponent in python.
• 3 years ago
Thanks! I was struggling to understand The Mandelbrot Set, but now it's all clear.
• 3 years ago
I am not a programmer, but I am trying to get this code to work, the interpreter returned this line a syntax error,
color = 255 - int(m * 255 / MAX_ITER)
Im running Python 3.8.2, thank you in advance to anyone for their assistance. Nick
Ps, I'm working my way through this. The above issue has been solved; now on to the next issue! :)
• 4 years ago
I got this error:
from mandelbrot import mandelbrot, MAX_ITER
ModuleNotFoundError: No module named 'mandelbrot'
I use Google Colab.
• 4 years ago
Please note, that there are sources for two .py files plot.py and mandelbrot.py (see the tabs in the source code window). You should store both into the same folder/project.
• 4 years ago
why does it not work in python 3.8
• 5 years ago
When I run your first 2 examples, the smoothed mandelbrot does not look any different.
• 5 years ago
In the max_iter program it seems like you're going from -1 to 1. In the bw plot.py it seems like the real part RE_START, RE_END goes from -2 to 1 but the imagineary part IM_START, IM_END goes from -1
to 1. I was under the impression that the Mandelbrot set lies in the coordinate space of -1 to 1. Is that true?, if so why does the real part go from -2 to 1?
• 5 years ago
EDIT* Never mind, It works great with python3.
When I copy the two files plot.py and mandelbrot.py to my local computer I get an all white image for the black and white code and and all red image for the color code. I have pil installed. Am I
missing something?
• 7 years ago
Is it ok if I don't understand this in the 8th grade?
• 7 years ago
That's perfectly ok, you need to learn what's a coordinate system, a sequence and what is a complex number. Feel free to ask questions!
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Differentiation of two variable function - mathXplain
Differentiation of two variable function
Content of the topic
Functions of two variables and the partial derivatives
Functions of two variables take two real numbers and assign a third real number to them. In other words, they assign a third number to a pair of numbers.
We could look at these number-pairs as coordinates in a plane.
Functions of two variables assign a third coordinate, the height, to the points of this plane.
By assigning this third (height) coordinate to all points of the domain, a surface is taking shape above the x,y plane. This is the graph of the function.
Some properties of single variable functions can be transmitted to two-variable functions, but some properties cannot.
There is no point for instance talking about monotonicity in case of two-variable functions, as it would be quite difficult to determine whether a surface happens to be increasing or decreasing.
On the other hand, the concept of minimum and maximum can be transmitted.
We should imagine the maximum of a two-variable function as a peak of a mountain,
and the minimum as a valley.
Let's see some two-variable functions.
Our task is to find out where the minimum,
the maximum or even the saddle point of a two-variable function happens to be.
Just like in the one-variable case, we will have to differentiate here, too, but now we have x as well as y, so we have to differentiate with respect to x and also with respect to y, which should be
twice as much fun.
These derivatives are called partial derivatives.
Let's see the partial derivatives.
Let’s differentiate this function, for instance.
we differentiate with respect to x, while y is held constant
differentiate with respect to x
y is treated as a constant,
if it stands by itself, its derivative is zero
if it is multiplied by some expression with x, than it stays as is
we differentiate with respect to y, while x is held constant
differentiate with respect to y
x is treated as a constant,
if it stands by itself, its derivative is zero
if it is multiplied by some expression with y, than it stays as is.
There is another notation for partial derivatives.
We will use both notations.
Here comes another function, let’s differentiate this one, too.
Both first order partial derivatives can be further differentiated with respect to x as well as y, too.
This way we get four second order derivatives.
The two outer ones are called pure second order derivatives,
and the two middle ones are the mixed second order derivatives.
The two mixed second order derivatives are usually equal.
Well, to be exact, they are equal if the function is twice totally differentiable.
But instead, we should remember that they are always equal, except in the section that is for professionals only, where the precise definitions of multivariate differentiation will be discussed.
Now, let’s see how we can find local minima and maxima using partial differentiation.
How we can find local minima and maxima using partial differentiation 1.0
Now, let’s see how we can find local minima and maxima using partial differentiation.
Solving the system of equations:
The resulting number pairs are points in the x,y plane.
These points are called stationary points, and at these points
the function can have a minimum, a maximum or a saddle point.
The solutions of the linear system are the stationary points
And now we can get to the second order derivatives.
We arrange them neatly in a matrix that is called a Hessian matrix.
And then we substitute the stationary points.
We have to take these matrices and look at their ... ahem ... determinants.
If somebody have not heard about the determinants of matrices yet, well, that is understandable, it is a very simple thing.
Here is a 2x2 matrix,
and its determinant is a number.
This number can be positive, negative or zero.
Let’s say for this matrix here,
the determinant is -14.
We calculate the determinant of the Hessian matrix, which can be positive, negative or zero.
If the determinant is positive, that means there is a minimum or a maximum.
If it is negative, then there is a saddle point.
If it is zero, then further investigation is necessary, but it doesn’t happen very often.
We will try summarizing it in this tiny space here.
Let's see what happens at the two stationary points.
Well, it seems is a saddle point.
And is a local minimum.
Let's see another one like this.
Let’s find the local extrema and saddle points of the following function.
Here are the stationary points:
And now come the second derivatives.
Next, let's see what happens at the stationary points.
Solving the system of equations:
, ,
, ,
Two stationary points: and
Here comes the Hessian matrix:
Now let's see the stationary points!
First let's check .
Substitute zero for x, y and z:
This is indefinite, so is a saddle point.
Next, let's see .
Substitute one for x and y, and zero for z:
This is positive definite, so it is a local minimum.
How we can find local minima and maxima using partial differentiation 2.0
Now, let’s see how we can find local minima and maxima using partial differentiation.
Solving the system of equations:
The resulting number pairs are points in the x,y plane.
These points are called stationary points, and at these points
the function can have a minimum, a maximum or a saddle point.
The solutions of the linear system are the stationary points
And now we can get to the second order derivatives.
We arrange them neatly in a matrix that is called a Hessian matrix.
And then we substitute the stationary points.
We have to take these matrices and look at their ... ahem ... determinants.
If somebody have not heard about the determinants of matrices yet, well, that is understandable, it is a very simple thing.
Here is a 2x2 matrix,
and its determinant is a number.
This number can be positive, negative or zero.
Let’s say for this matrix here,
the determinant is -14.
We calculate the determinant of the Hessian matrix, which can be positive, negative or zero.
If the determinant is positive, that means there is a minimum or a maximum.
If it is negative, then there is a saddle point.
If it is zero, then further investigation is necessary, but it doesn’t happen very often.
We will try summarizing it in this tiny space here.
Let's see what happens at the two stationary points.
Well, it seems is a saddle point.
And is a local minimum.
Let's see another one like this.
Let’s find the local extrema and saddle points of the following function.
Here are the stationary points:
And now come the second derivatives.
Next, let's see what happens at the stationary points.
Solving the system of equations:
, ,
, ,
Two stationary points: and
Here comes the Hessian matrix:
Now let's see the stationary points!
First let's check .
Substitute zero for x, y and z:
This is indefinite, so is a saddle point.
Next, let's see .
Substitute one for x and y, and zero for z:
This is positive definite, so it is a local minimum.
The tangent plane
If we remember, the geometric interpretation of the derivative in case of single variable functions was the slope of the tangent.
The equation of the tangent for function at point is:
The tangent of a single variable function is a line, and the tangent of a two-variable function is a plane.
The number of coordinates is increased by 1, so it is not x and y, but x, y and z.
The equation of the plane tangent to function at point is:
Well, this is the equation of the tangent plane.
Let's see an example.
Here is this function, for instance:
and we are looking for the tangent plane at point .
Here comes the equation of the tangent plane,
and we have to calculate these.
Well, this is the equation of the tangent plane:
If we expand the parentheses, and get all terms on one side,
then we can see the normal vector of the plane.
And here is the normal vector:
The first two coordinates are the derivatives with respect to x and y,
and the third coordinate is negative one.
What should parameter be, so that the tangent at point
to function would also pass through point ?
A plane passes through a point if the equation holds when substituting the point’s coordinates into the equation of the plane.
Here is :
Now, let's see the vector.
The vector in the formula must be of unit length.
Since now is not a unit long,
we turn this into a unit vector.
We divide the vector by its own length:
The equation of the plane that is tangent to the surface given by at point is:
The normal vector of the tangent plane is . This is easy to see, if we move z to the right side of the equation of the tangent plane.
Gradient and directional derivative
The vector made up of the of the function's partial derivatives with respect to x and y is called the gradient of the function.
Here is the gradient:
, shortly .
The gradient helps us calculate the directional derivative. The directional derivative describes how steeply the surface of the function slopes along a given arbitrary direction.
So, it means that there is a mountain climber standing at point P on the surface, and decides to move in direction. The directional derivative tells him how steep he would have to climb.
Calculating the directional derivative is very simple: it is the dot product of the gradient and the unit-length vector .
The directional derivative of the function at point is:
( is a unit vector here)
Let's see an example of this!
Let's calculate the directional derivative of for direction at point .
According to the formula, the directional derivative is:
Here this funny symbol is the symbol of differentiation, and it is pronounced as "d", but there is a bit more friendly notation for the directional derivative: .
We need the partial derivatives for calculating the gradient.
So, the gradient is:
To get the directional derivative, we should create the dot product of the gradient and the vector, but it is now not a unit vector, its length is:
To turn this into a unit vector we divide the vector by its own length:
Now, let's see the vector.
The vector in the formula must be of unit length.
Since now is not a unit long,
we turn this into a unit vector.
We divide the vector by its own length:
Therefore the directional derivative is:
If a mountain climber asked us which direction he should take from point P in order to climb the steepest route, well...
we could actually give him an answer.
The steepest rise on a surface is always in the direction of the gradient vector.
That means if the climber starts climbing
in the direction of the gradient, then he will be climbing the steepest route.
The function is an explicit function, its derivative, as expected, is .
The implicit differentiation rule
The function is an explicit function, its derivative, as expected, is .
A function is implicit if y is not expressed, not in the form of y=...
We get an implicit function if we mess up the function, like this:
and then we take the square root, too
So, this is an implicit function.
If now, we have to differentiate this newly created implicit function, we could do that by differentiating both sides of the equation, and treat y as a function.
actually, it is a function, since .
Well, the derivative of the x on the right side is most definitely 1.
The left side is much more exciting. Here we have a composite function:
And then it also has to be multiplied by the derivative of the inside function.
We need , in other words, the derivative of the function that was given in implicit form.
Let’s try to express
Here it is.
Since , if we substitute this to y...
And this is the same as the explicit derivative.
It is fair to ask why we bothered so much with this, if at the end, we got the same result, except it was a lot more complicated.
Well, the answer is that unfortunately there are some functions that have no explicit forms.
This function has an explicit form, so in this case, it was unnecessary to suffer through the implicit differentiation.
But take a look at this one, for instance.
In this case, y cannot be expressed in any way, so we are forced to use implicit differentiation.
So, we differentiate both sides, but let's not forget that y is a function here.
So, for example is a composite function.
Therefore, we differentiate it as a composite:
Take the derivative of the outside function,
multiplied by the derivative of the inside function.
Now let’s see the implicit differentiation.
We differentiate both sides of the equation:
We need the derivative of y, so we collect all terms on one side, and send all others to the other side:
Then we factor out .
and finally, we divide (and conquer!):
Well, this is the derivative of our function that was given in implicit form.
Now let’s see the differentiation rule for implicit functions.
The point of this method is to make our life easier.
It says that if is an implicit function, then its derivative is:
Well, so far, there is nothing encouraging about this... But let’s see how it works in practice.
Here is the implicit function:
where all terms should be collected on one side,
and it should be called F.
Before we fall victims of a fatal mistake, we must make it clear that this is not a two-variable function, but an implicit function.
The difference between and is huge.
Let's see what the difference is.
Function is a two-variable function indeed, and x and y can be given freely, but
is not a two-variable function. , Let's just try to substitute 0 for x and 1 for y.
We will get 2=0, which is not true, so here only one of x and y can vary freely, the other cannot. So that is why this function is a single variable function.
Now, that we clarified all this, let’s see what the formula says.
The formula says that we should differentiate this function and using the customary partial derivation with respect to x and y.
And here is the implicit derivative.
It is exactly the same result as earlier,
only this time it was much simpler.
Now that's what the implicit differentiation rule is good for.
The rule works for more variables, too.
It says that if is a single variable implicit function, then its derivative is:
If is an n-variable implicit function, then the derivative of as an implicit function with respect to variable is:
Let's see an example for this!
This is a two-variable implicit function.
Even though it has three letters: x, y and z, notice that only two of them can be given freely, due to the equation.
In two-variable functions, x and y are usually the variables, so we can treat this function as
Z=something x and y
Let’s differentiate this with respect to x, and with respect to y!
Problem I find the local extrema and saddle points
Problem I find the local extrema and saddle points
Problem I find the local extrema and saddle points
Problem I find the local extrema and saddle points
Problem I find the local extrema and saddle points
Problem I find the local extrema and saddle points
Problem I find the local extrema and saddle points
Problem I find the local extrema and saddle points
Problem I find the local extrema and saddle points
Problem I find the local extrema and saddle points
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Boole's inequality
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Boole's inequality
In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater
than the sum of the probabilities of the individual events. Boole's inequality is named after George Boole.
Formally, for a countable set of events A1, A2, A3, ..., we have
In measure-theoretic terms, Boole's inequality follows from the fact that a measure (and certainly any probability measure) is σ-sub-additive.
Boole's inequality may be proved for finite collections of events using the method of induction.
For thecase, it follows that
Sinceand because the union operation isassociative, we have
by the first axiom of probability, we have
Proof without using induction
For any events inin ourprobability spacewe have
One of the axioms of a probability space is that ifare disjoint subsets of the probability space then
this is called countable additivity.
Indeed, from the axioms of a probability distribution,
Note that both terms on the right are nonnegative.
Now we have to modify the sets, so they become disjoint.
Therefore, we can deduce the following equation
Boole's inequality may be generalized to find upper and lower bounds on the probability of finite unions of events.^[1] These bounds are known as Bonferroni inequalities, after Carlo Emilio
Bonferroni, see Bonferroni (1936).
for all integers k in {3, ..., n}.
Then, for odd k in {1, ..., n},
and for even k in {2, ..., n},
Boole's inequality is recovered by setting k = 1. When k = n, then equality holds and the resulting identity is the inclusion–exclusion principle.
• Diluted inclusion–exclusion principle
• Schuette–Nesbitt formula
• Boole–Fréchet inequalities
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Langmuir analysis
The Analysis and Diagnostic framework is in active development at the moment. For the foreseeable future, the API will be in continuous flux as functionality is added and modified.
Langmuir analysis
Defines the Langmuir analysis module as part of the diagnostics package.
Characteristic(bias, current) Class representing a single I-V probe characteristic for convenient experimental data access and computation.
extract_exponential_section(probe_characteristic) Extract the section of exponential electron current growth from the probe characteristic.
extract_ion_section(probe_characteristic) Extract the section dominated by ion collection from the probe characteristic.
extrapolate_electron_current(...[, ...]) Extrapolate the electron current from the Maxwellian electron temperature obtained in the exponential growth region.
extrapolate_ion_current_OML(...[, visualize]) Extrapolate the ion current from the ion density obtained with the OML method.
get_EEDF(probe_characteristic[, visualize]) Implement the Druyvesteyn method of obtaining the normalized Electron Energy Distribution Function (EEDF).
get_electron_density_LM(...) Implement the Langmuir-Mottley (LM) method of obtaining the electron density.
get_electron_saturation_current(...) Obtain an estimate of the electron saturation current corresponding to the obtained plasma potential.
get_electron_temperature(exponential_section) Obtain the Maxwellian or bi-Maxwellian electron temperature using the exponential fit method.
get_floating_potential(probe_characteristic) Implement the simplest but crudest method for obtaining an estimate of the floating potential from the probe characteristic.
get_ion_density_LM(ion_saturation_current, ...) Implement the Langmuir-Mottley (LM) method of obtaining the ion density.
get_ion_density_OML(probe_characteristic, ...) Implement the Orbital Motion Limit (OML) method of obtaining an estimate of the ion density.
get_ion_saturation_current(probe_characteristic) Implement the simplest but crudest method for obtaining an estimate of the ion saturation current from the probe characteristic.
get_plasma_potential(probe_characteristic[, ...]) Implement the simplest but crudest method for obtaining an estimate of the plasma potential from the probe characteristic.
reduce_bimaxwellian_temperature(T_e, ...) Reduce a bi-Maxwellian (dual) temperature to a single mean temperature for a given fraction.
swept_probe_analysis(probe_characteristic, ...) Attempt to perform a basic swept probe analysis based on the provided characteristic and probe data.
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Multi-GPU processing of unstructured data for machine learning
We introduce a method for processing unstructured data for machine learning based on an LZ-complexity string distance. Computing the LZ-complexity is inherently a serial data compression process;
hence, we introduce a string distance computed by a parallel algorithm that utilizes multiple GPU devices to process unstructured data, which typically exists in large quantities. We use this
algorithm to compute a distance matrix representation of the unstructured data that standard learning algorithms can use to learn. Our approach eliminates the need for human-based feature definition
or extraction. Except for some simple data reformatting done manually, our proposed approach operates on the original raw data and is fully automatic. The parallel computation of the distance matrix
is efficient. It obtains a speed-up factor of 528 in computing the distance matrix between every possible pair of 16 strings of length 1M bytes. We show that for learning time-series classification,
relative to the ubiquitous TFIDF data representation, the distance-matrix representation yields a higher learning accuracy for most of a broad set of learning algorithms. Thus, the parallel algorithm
can be helpful in efficiently and accurately learning from unstructured data.
Publication series
Name Research Paper Proceedings of the ISC High Performance 2024
Conference 39th International Conference on High Performance Computing, ISC High Performance 2024
Country/Territory Germany
City Hamburg
Period 12/05/24 → 16/05/24
• CUDA
• LZ-complexity
• multi-GPU
• string distance
ASJC Scopus subject areas
• Artificial Intelligence
• Computational Theory and Mathematics
• Hardware and Architecture
• Computer Networks and Communications
• Computational Mathematics
• Theoretical Computer Science
Dive into the research topics of 'Multi-GPU processing of unstructured data for machine learning'. Together they form a unique fingerprint.
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Path Analysis Made Easy – Dr Martin Lea
Path Analysis Made Easy.
Take my Statistics Course
An Introduction to Path Analysis: Theory and Practice.
In this introductory module I introduce the concept of mediation and differentiate it from moderation. I’ll describe and illustrate what a simple case of mediation looks like with some real world
data. When you're comfortable with the idea of mediation, I'll show you some different techniques to test for significant mediated effects in your data and discuss the best one to use.
In the second module I'll introduce the main concepts you need to understand Path Analysis (or Causal Modeling as it is otherwise known) and show you how to do various kinds of modelling using the
AMOS program, which is available online. I’ll be showing you how to construct and test some simple models using techniques you can apply to your own data.
This course is useful for research who want to model their own data. However, a second purpose of this course is to provide you with the knowledge you need to interpret descriptions of causal models
that you may read about. Causal modeling is becoming increasingly popular, especially in social and clinical fields, and it is important to be able to interpret and evaluate a model you may come
across in your own research or in a journal article which an editor has asked you to review.
Path analysis and structural equation modeling are techniques to assess the direct causal contribution of one variable to another in a non experimental situation. They are therefore particularly
useful in field studies, and have become increasingly popular as modern psychology draws from real problems and non-laboratory research methods. However, as…
This slide summarizes Barron & Kenny's (1986) causal steps for establishing mediation, which we have just discussed. However, do all of the steps have to be met for there to be mediation? Certainly,
Step 4 does not have to be met unless the expectation is for complete mediation. Moreover, Step…
Statistics Training: Introduction to Path Analysis
Statistics Training: Introduction to Path Analysis
Statistics Training: Introduction to Path Analysis
What is Simple Regression? What is Multiple Regression? In simple regression a single dependent or criterion variable is related to a single independent variable or predictor variable. Multiple
regression is an extension of simple regression in which the criterion is regressed against several potential predictors. For example, a simple…
Path models are built up from basic models of moderation and/or mediation. It is common in psychology for the terms moderator and mediator to be used interchangeably. However, they are conceptually
different. “In general terms, a moderator is a qualitative (e.g., sex, race class) or quantitative (e.g., level of reward)…
This example illustrates the importance of clearly specifying your theory in terms of moderators and mediators. It's taken from an advisory session with a PhD student who approached me to discuss how
to test her theory. Her project was looking at the link between language deficit and self-esteem in young adults. Her hypothesis…
The simplest mediation analysis involves a single independent variable, a dependent variable, and a hypothesized mediator. The unmediated model is represented by the direct effect of x on y,
quantified as c. However, the effect of X on Y may be mediated by a process, or mediating variable M. Complete…
So how do we go about doing a mediation analysis? In the next four posts I'll take you through the main approaches to testing for a significant mediation effect. We'll first look at the Causal Steps
approach, made famous by Baron & Kenny (1986). Then we'll look at several modern…
Let's start by decomposing mediation into a number of causal steps as described by Baron & Kenny (1986). We'll use our mediation model for the effects of Visual Anonymity on Group Attraction,
mediated by Self-Categorization. The fist step is to show that the initial variable affects the outcome. In our…
In the second step, we need to show that the initial, or predictor, variable affects the mediator. So we perform another simple linear regression using the mediator as if it were the outcome variable
and regressing it on the predictor, which gives us an estimate of path a. In our…
Steps 3 and 4 are conducted simultaneously using multiple regression. Step 3 consists of regressing the outcome variable y onto both the mediator, m, and the predictor, x to provide an estimate of
path b. Note: it is not sufficient just to correlate the mediator with the outcome; the mediator…
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The goal of BayesSurvival is to perform unadjusted Bayesian survival analysis for right censored time-to-event data. The main function (BayesSurv) computes the posterior mean and a credible band for
the survival function and for the cumulative hazard, as well as the posterior mean for the hazard, starting from a piecewise exponential (histogram) prior with Gamma distributed heights that are
either independent, or have a Markovian dependence structure. A function (PlotBayesSurv) is provided to easily create plots of the posterior means of the hazard, cumulative hazard and survival
function, with a credible band accompanying the latter two. The priors and samplers are described in more detail in the preprint ‘Multiscale Bayesian survival analysis’ by Castillo and Van der
Pas (2020+). In that paper it is also shown that the credible bands for the survival function and the cumulative hazard can be considered confidence bands (under mild conditions) and thus offer
reliable uncertainty quantification.
You can install the released version of BayesSurvival from CRAN with:
This is a basic example which shows you how to solve a common problem:
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Likelihood function
Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
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Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory
In statistics, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some
observed outcomes is equal to the probability of those observed outcomes given those parameter values. Likelihood functions play a key role in statistical inference, especially methods of estimating
a parameter from a set of statistics.
In non-technical parlance, "likelihood" is usually a synonym for "probability" but in statistical usage, a clear technical distinction is made. One may ask "If I were to flip a fair coin 100 times,
what is the probability of it landing heads-up every time?" or "Given that I have flipped a coin 100 times and it has landed heads-up 100 times, what is the likelihood that the coin is fair?" but it
would be improper to switch "likelihood" and "probability" in the two sentences.
If a probability distribution depends on a parameter, one may on one hand consider—for a given value of the parameter—the probability (density) of the different outcomes, and on the other hand
consider—for a given outcome—the probability (density) this outcome has occurred for different values of the parameter. The first approach interprets the probability distribution as a function of the
outcome, given a fixed parameter value, while the second interprets it as a function of the parameter, given a fixed outcome. In the latter case the function is called the "likelihood function" of
the parameter, and indicates how likely a parameter value is in light of the observed outcome.
Definition[ ]
For the definition of the likelihood function, one has to distinguish between discrete and continuous probability distributions.
Discrete probability distribution[ ]
Let X be a random variable with a discrete probability distribution p depending on a parameter θ. Then the function
${\displaystyle \mathcal{L}(\theta |x) = p_\theta (x) = P_\theta (X=x), \, }$
considered as a function of θ, is called the likelihood function (of θ, given the outcome x of X). Sometimes the probability on the value x of X for the parameter value θ is written as ${\
displaystyle P(X=x|\theta)}$, but should not be considered as a conditional probability.
Continuous probability distribution[ ]
Let X be a random variable with a continuous probability distribution with density function f depending on a parameter θ. Then the function
${\displaystyle \mathcal{L}(\theta |x) = f_{\theta} (x), \, }$
considered as a function of θ, is called the likelihood function (of θ, given the outcome x of X). Sometimes the density function for the value x of X for the parameter value θ is written as ${\
displaystyle f(x|\theta)}$, but should not be considered as a conditional probability density.
The actual value of a likelihood function bears no meaning. Its use lies in comparing one value with another. E.g., one value of the parameter may be more likely than another, given the outcome of
the sample. Or a specific value will be most likely: the maximum likelihood estimate. Comparison may also be performed in considering the quotient of two likelihood values. That's why generally, ${\
displaystyle \mathcal{L}(\theta |x)}$ is permitted to be any positive multiple of the above defined function ${\displaystyle \mathcal{L}}$. More precisely, then, a likelihood function is any
representative from an equivalence class of functions,
${\displaystyle \mathcal{L} \in \left\lbrace \alpha \; P_\theta: \alpha > 0 \right\rbrace, \, }$
where the constant of proportionality α > 0 is not permitted to depend upon θ, and is required to be the same for all likelihood functions used in any one comparison. In particular, the numerical
value ${\displaystyle \mathcal{L}}$(θ | x) alone is immaterial; all that matters are maximum values of ${\displaystyle \mathcal{L}}$, or likelihood ratios, such as those of the form
${\displaystyle \frac{\mathcal{L}(\theta_2 | x)}{\mathcal{L}(\theta_1 | x)} = \frac{\alpha P(X=x|\theta_2)}{\alpha P(X=x|\theta_1)} = \frac{P(X=x|\theta_2)}{P(X=x|\theta_1)}, }$
that are invariant with respect to the constant of proportionality α.
A. W. F. Edwards defined support to be the natural logarithm of the likelihood ratio, and the support function as the natural logarithm of the likelihood function (the same as the log-likelihood; see
below).^[1] However, there is potential for confusion with the mathematical meaning of 'support', and this terminology is not widely used outside Edwards' main applied field of phylogenetics.
For more about making inferences via likelihood functions, see also the method of maximum likelihood, and likelihood-ratio testing.
Log-likelihood[ ]
For many applications involving likelihood functions, it is more convenient to work in terms of the natural logarithm of the likelihood function, called the log-likelihood, than in terms of the
likelihood function itself. Because the logarithm is a monotonically increasing function, the logarithm of a function achieves its maximum value at the same points as the function itself, and hence
the log-likelihood can be used in place of the likelihood in maximum likelihood estimation and related techniques. Finding the maximum of a function often involves taking the derivative of a function
and solving for the parameter being maximized, and this is often easier when the function being maximized is a log-likelihood rather than the original likelihood function.
For example, some likelihood functions are for the parameters that explain a collection of statistically independent observations. In such a situation, the likelihood function factors into a product
of individual likelihood functions. The logarithm of this product is a sum of individual logarithms, and the derivative of a sum of terms is often easier to compute than the derivative of a product.
In addition, several common distributions have likelihood functions that contain products of factors involving exponentiation. The logarithm of such a function is a sum of products, again easier to
differentiate than the original function.
As an example, consider the gamma distribution, whose likelihood function is
${\displaystyle \mathcal{L} (\alpha, \beta|x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}}$
and suppose we wish to find the maximum likelihood estimate of β for a single observed value x. This function looks rather daunting. Its logarithm, however, is much simpler to work with:
${\displaystyle \log \mathcal{L}(\alpha,\beta|x) = \alpha \log \beta - \log \Gamma(\alpha) + (\alpha-1) \log x - \beta x. \, }$
The partial derivative with respect to β is simply
${\displaystyle \frac{\partial \log \mathcal{L}(\alpha,\beta|x)}{\partial \beta} = \frac{\alpha}{\beta} - x}$
If there are a number of independent random samples x[1],…,x[n], then the joint log-likelihood will be the sum of individual log-likelihoods, and the derivative of this sum will be the sum of
individual derivatives:
${\displaystyle \frac{n \alpha}{\beta} - \sum_{i=1}^n x_i}$
Setting that equal to zero and solving for β yields
${\displaystyle \hat\beta = \frac{\alpha}{\bar{x}}}$
where ${\displaystyle \hat\beta}$ denotes the maximum-likelihood estimate and ${\displaystyle \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i}$ is the sample mean of the observations.
Likelihood function of a parameterized model[ ]
Among many applications, we consider here one of broad theoretical and practical importance. Given a parameterized family of probability density functions (or probability mass functions in the case
of discrete distributions)
${\displaystyle x\mapsto f(x\mid\theta), \!}$
where θ is the parameter, the likelihood function is
${\displaystyle \theta\mapsto f(x\mid\theta), \!}$
${\displaystyle \mathcal{L}(\theta \mid x)=f(x\mid\theta), \!}$
where x is the observed outcome of an experiment. In other words, when f(x | θ) is viewed as a function of x with θ fixed, it is a probability density function, and when viewed as a function of θ
with x fixed, it is a likelihood function.
Note: This is not the same as the probability that those parameters are the right ones, given the observed sample. Attempting to interpret the likelihood of a hypothesis given observed evidence as
the probability of the hypothesis is a common error, with potentially disastrous real-world consequences in medicine, engineering or jurisprudence. See prosecutor's fallacy for an example of this.
From a geometric standpoint, if we consider f (x, θ) as a function of two variables then the family of probability distributions can be viewed as level curves parallel to the x-axis, while the family
of likelihood functions are the orthogonal level curves parallel to the θ-axis.
Likelihoods for continuous distributions[ ]
The use of the probability density instead of a probability in specifying the likelihood function above may be justified in a simple way. Suppose that, instead of an exact observation, x, the
observation is the value in a short interval (x[j−1], x[j]), with length Δ[j], where the subscripts refer to a predefined set of intervals. Then the probability of getting this observation (of being
in interval j) is approximately
${\displaystyle \mathcal{L}_\text{approx}(\theta \mid x \text{ in interval } j) = f(x_{*}\mid\theta) \Delta_j, \!}$
where x[*] can be any point in interval j. Then, recalling that the likelihood function is defined up to a multiplicative constant, it is just as valid to say that the likelihood function is
${\displaystyle \mathcal{L}_\text{approx}(\theta \mid x \text{ in interval } j)= f(x_{*}\mid\theta), \!}$
and then, on considering the lengths of the intervals to decrease to zero,
${\displaystyle \mathcal{L}(\theta \mid x )= f(x\mid\theta). \!}$
Likelihoods for mixed continuous–discrete distributions[ ]
The above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components. Suppose that the distribution consists of a number of discrete
probability masses p[k](θ) and a density f(x | θ), where the sum of all the p's added to the integral of f is always one. Assuming that it is possible to distinguish an observation corresponding to
one of the discrete probability masses from one which corresponds to the density component, the likelihood function for an observation from the continuous component can be dealt with as above by
setting the interval length short enough to exclude any of the discrete masses. For an observation from the discrete component, the probability can either be written down directly or treated within
the above context by saying that the probability of getting an observation in an interval that does contain a discrete component (of being in interval j which contains discrete component k) is
${\displaystyle \mathcal{L}_\text{approx}(\theta \mid x \text{ in interval } j \text{ containing discrete mass } k)=p_k(\theta) + f(x_{*}\mid\theta) \Delta_j, \!}$
where ${\displaystyle x_{*}\ }$ can be any point in interval j. Then, on considering the lengths of the intervals to decrease to zero, the likelihood function for a observation from the discrete
component is
${\displaystyle \mathcal{L}(\theta \mid x )= p_k(\theta), \!}$
where k is the index of the discrete probability mass corresponding to observation x.
The fact that the likelihood function can be defined in a way that includes contributions that are not commensurate (the density and the probability mass) arises from the way in which the likelihood
function is defined up to a constant of proportionality, where this "constant" can change with the observation x, but not with the parameter θ.
Example 1[ ]
Let ${\displaystyle p_\text{H}}$ be the probability that a certain coin lands heads up (H) when tossed. So, the probability of getting two heads in two tosses (HH) is ${\displaystyle p_\text{H}^2}$.
If ${\displaystyle p_\text{H} = 0.5}$, then the probability of seeing two heads is 0.25.
In symbols, we can say the above as:
${\displaystyle P(\text{HH} | p_\text{H}=0.5) = 0.25.}$
Another way of saying this is to reverse it and say that "the likelihood that ${\displaystyle p_\text{H} = 0.5}$, given the observation HH, is 0.25"; that is:
${\displaystyle \mathcal{L}(p_\text{H}=0.5 | \text{HH}) = P(\text{HH} | p_\text{H}=0.5) = 0.25.}$
But this is not the same as saying that the probability that ${\displaystyle p_\text{H} = 0.5}$, given the observation HH, is 0.25.
Notice that the likelihood that ${\displaystyle p_\text{H} = 1}$, given the observation HH, is 1. But it is clearly not true that the probability that ${\displaystyle p_\text{H} = 1}$, given the
observation HH, is 1. Two heads in a row hardly proves that the coin always comes up heads. In fact, two heads in a row is possible for any ${\displaystyle p_\text{H} > 0}$.
The likelihood function is not a probability density function. Notice that the integral of a likelihood function is not in general 1. In this example, the integral of the likelihood over the interval
[0, 1] in ${\displaystyle p_\text{H}}$ is 1/3, demonstrating that the likelihood function cannot be interpreted as a probability density function for ${\displaystyle p_\text{H}}$.
Example 2[ ]
Main article: German tank problem
Consider a jar containing N lottery tickets numbered from 1 through N. If you pick a ticket randomly then you get positive integer n, with probability 1/N if n ≤ N and with probability zero if n > N.
This can be written
${\displaystyle P(n|N)= \frac{[n \le N]}{N}}$
where the Iverson bracket [n ≤ N] is 1 when n ≤ N and 0 otherwise. When considered a function of n for fixed N this is the probability distribution, but when considered a function of N for fixed n
this is a likelihood function. The maximum likelihood estimate for N is N[0] = n (by contrast, the unbiased estimate is 2n − 1).
This likelihood function is not a probability distribution, because the total
${\displaystyle \sum_{N=1}^\infty P(n|N) = \sum_{N} \frac{[N \ge n]}{N} = \sum_{N=n}^\infty \frac{1}{N}}$
is a divergent series.
Suppose, however, that you pick two tickets rather than one.
The probability of the outcome {n[1], n[2]}, where n[1] < n[2], is
${\displaystyle P(\{n_1,n_2\}|N)= \frac{[n_2 \le N]}{\binom N 2} .}$
When considered a function of N for fixed n[2], this is a likelihood function. The maximum likelihood estimate for N is N[0] = n[2].
This time the total
${\displaystyle \sum_{N=1}^\infty P(\{n_1,n_2\}|N) = \sum_{N} \frac{[N\ge n_2]}{\binom N 2} =\frac 2 {n_2-1} }$
is a convergent series, and so this likelihood function can be normalized into a probability distribution.
If you pick 3 or more tickets, the likelihood function has a well defined mean value, which is larger than the maximum likelihood estimate. If you pick 4 or more tickets, the likelihood function has
a well defined standard deviation too.
Relative likelihood[ ]
Suppose that the maximum likelihood estimate for θ is ${\displaystyle \hat{\theta}}$. Relative plausibilities of other θ values may be found by comparing the likelihood of those other values with the
likelihood of ${\displaystyle \hat{\theta}}$. The relative likelihood of θ is defined^[2] as ${\displaystyle \mathcal{L}(\theta | x)/\mathcal{L}(\hat \theta | x)}$.
A 10% likelihood region for θ is
${\displaystyle \{\theta : \mathcal{L}(\theta | x)/\mathcal{L}(\hat \theta | x) \ge 0.10\},}$
and more generally, a p% likelihood region for θ is defined^[2] to be
${\displaystyle \{\theta : \mathcal{L}(\theta | x)/\mathcal{L}(\hat \theta | x) \ge p/100 \}.}$
If θ is a single real parameter, a p% likelihood region will typically comprise an interval of real values. In that case, the region is called a likelihood interval.^[2]^[3]
Likelihood intervals can be compared to confidence intervals. If θ is a single real parameter, then under certain conditions, a 14.7% likelihood interval for θ will be the same as a 95% confidence
interval.^[2] In a slightly different formulation suited to the use of log-likelihoods, the e^−2 likelihood interval is the same as the 0.954 confidence interval (under certain conditions).^[3]
The idea of basing an interval estimate on the relative likelihood goes back to Fisher in 1956 and has been by many authors since then.^[3] If a likelihood interval is specifically to be interpreted
as a confidence interval, then this idea is immediately related to the likelihood ratio test which can be used to define appropriate intervals for multivariate parameters. This approach can be used
to define the critical points for the likelihood ratio statistic to achieve the required coverage level for a confidence interval. However a likelihood interval can be used as such, having been
determined in a well-defined way, without claiming any particular coverage probability.
Likelihoods that eliminate nuisance parameters[ ]
In many cases, the likelihood is a function of more than one parameter but interest focuses on the estimation of only one, or at most a few of them, with the others being considered as nuisance
parameters. Several alternative approaches have been developed to eliminate such nuisance parameters so that a likelihood can be written as a function of only the parameter (or parameters) of
interest; the main approaches being marginal, conditional and profile likelihoods.^[4]^[5]
These approaches are useful because standard likelihood methods can become unreliable or fail entirely when there are many nuisance parameters or when the nuisance parameters are high-dimensional.
This is particularly true when the nuisance parameters can be considered to be "missing data"; they represent a non-negligible fraction of the number of observations and this fraction does not
decrease when the sample size increases. Often these approaches can be used to derive closed-form formulae for statistical tests when direct use of maximum likelihood requires iterative numerical
methods. These approaches find application in some specialized topics such as sequential analysis.
Conditional likelihood[ ]
Sometimes it is possible to find a sufficient statistic for the nuisance parameters, and conditioning on this statistic results in a likelihood which does not depend on the nuisance parameters.
One example occurs in 2×2 tables, where conditioning on all four marginal totals leads to a conditional likelihood based on the non-central hypergeometric distribution. This form of conditioning is
also the basis for Fisher's exact test.
Marginal likelihood[ ]
Main article: Marginal likelihood
Sometimes we can remove the nuisance parameters by considering a likelihood based on only part of the information in the data, for example by using the set of ranks rather than the numerical values.
Another example occurs in linear mixed models, where considering a likelihood for the residuals only after fitting the fixed effects leads to residual maximum likelihood estimation of the variance
Profile likelihood[ ]
It is often possible to write some parameters as functions of other parameters, thereby reducing the number of independent parameters. (The function is the parameter value which maximizes the
likelihood given the value of the other parameters.) This procedure is called concentration of the parameters and results in the concentrated likelihood function, also occasionally known as the
maximized likelihood function, but most often called the profile likelihood function.
For example, consider a regression analysis model with normally distributed errors. The most likely value of the error variance is the variance of the residuals. The residuals depend on all other
parameters. Hence the variance parameter can be written as a function of the other parameters.
Unlike conditional and marginal likelihoods, profile likelihood methods can always be used, even when the profile likelihood cannot be written down explicitly. However, the profile likelihood is not
a true likelihood, as it is not based directly on a probability distribution, and this leads to some less satisfactory properties. Attempts have been made to improve this, resulting in modified
profile likelihood.
The idea of profile likelihood can also be used to compute confidence intervals that often have better small-sample properties than those based on asymptotic standard errors calculated from the full
likelihood. In the case of parameter estimation in partially observed systems, the profile likelihood can be also used for identifiability analysis.^[6] An implementation is available in the MATLAB
Toolbox PottersWheel.
Partial likelihood[ ]
A partial likelihood is a factor component of the likelihood function that isolates the parameters of interest.^[7] It is a key component of the proportional hazards model.
Historical remarks[ ]
In English, "likelihood" has been distinguished as being related to but weaker than "probability" since its earliest uses. The comparison of hypotheses by evaluating likelihoods has been used for
centuries, for example by John Milton in Aeropagitica: "when greatest likelihoods are brought that such things are truly and really in those persons to whom they are ascribed".
In Danish, "likelihood" was used by Thorvald N. Thiele in 1889.^[8]^[9]^[10]
In English, "likelihood" appears in many writings by Charles Sanders Peirce, where model-based inference (usually abduction but sometimes including induction) is distinguished from statistical
procedures based on objective randomization. Peirce's preference for randomization-based inference is discussed in "Illustrations of the Logic of Science" (1877–1878) and "A Theory of Probable
Inference" (1883)".
"probabilities that are strictly objective and at the same time very great, although they can never be absolutely conclusive, ought nevertheless to influence our preference for one hypothesis
over another; but slight probabilities, even if objective, are not worth consideration; and merely subjective likelihoods should be disregarded altogether. For they are merely expressions of our
preconceived notions" (7.227 in his Collected Papers).
"But experience must be our chart in economical navigation; and experience shows that likelihoods are treacherous guides. Nothing has caused so much waste of time and means, in all sorts of
researchers, as inquirers' becoming so wedded to certain likelihoods as to forget all the other factors of the economy of research; so that, unless it be very solidly grounded, likelihood is far
better disregarded, or nearly so; and even when it seems solidly grounded, it should be proceeded upon with a cautious tread, with an eye to other considerations, and recollection of the
disasters caused." (Essential Peirce, volume 2, pages 108–109)
Like Thiele, Peirce considers the likelihood for a binomial distribution. Peirce uses the logarithm of the odds-ratio throughout his career. Peirce's propensity for using the log odds is discussed by
Stephen Stigler.^[citation needed]
In Great Britain, "likelihood" was popularized in mathematical statistics by R.A. Fisher in 1922^[11]: "On the mathematical foundations of theoretical statistics". In that paper, Fisher also uses the
term "method of maximum likelihood". Fisher argues against inverse probability as a basis for statistical inferences, and instead proposes inferences based on likelihood functions. Fisher's use of
"likelihood" fixed the terminology that is used by statisticians throughout the world.
See also[ ]
• Principle of maximum entropy
• Conditional entropy
Notes[ ]
References[ ]
• John W. Pratt (May 1976). F. Y. Edgeworth and R. A. Fisher on the Efficiency of Maximum Likelihood Estimation. The Annals of Statistics 4 (3): 501–514. | jstor = 2958222
• Stephen M. Stigler (1978). Francis Ysidro Edgeworth, Statistician. Journal of the Royal Statistical Society, Series A 141 (3): 287–322. | jstor = 2344804
• Stephen M. Stigler. The History of Statistics: The Measurement of Uncertainty before 1900, Harvard University Press.
• Stephen M. Stigler. Statistics on the Table: The History of Statistical Concepts and Methods, Harvard University Press.
• Anders Hald (1999). On the History of Maximum Likelihood in Relation to Inverse Probability and Least Squares. Statistical Science 14 (2): 214–222. | jstor = 2676741
• Hald, A. (1998). A History of Mathematical Statistics from 1750 to 1930, New York: Wiley.
External links[ ]
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isInfinite property
Returns true if either component is double.infinity, and false if both are finite (or negative infinity, or NaN).
This is different than comparing for equality with an instance that has both components set to double.infinity.
See also:
• isFinite, which is true if both components are finite (and not NaN).
bool get isInfinite => _dx >= double.infinity || _dy >= double.infinity;
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We study Banach spaces X with subspaces Y whose unit ball is densely remotal in X. We show that for several classes of Banach spaces, the unit ball of the space of compact operators is densely
remotal in the space of bounded operators. We also show that for several classical Banach spaces, the unit ball is densely remotal in the duals of higher even order. We show that for a separable
remotal set E ⊆ X, the set of Bochner integrable functions with values in E is a remotal set in L¹(μ,X).
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ball mill capacity
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If a ball mill uses little or no water during grinding, it is a 'dry' mill. If a ball mill uses water during grinding, it is a 'wet' mill. A typical ball mill will have a drum length that is 1 or
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The capacity of a ball mill depends on several factors including the size of the ball mill, the type of material being ground, and the operational parameters of the ball mill. The...
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The grinding mill's production capacity is generally calculated based on the newly generated powder ore of less than mm (200 mesh). V — Effective volume of ball mill, m3; G2 — Material less than
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The lack of constraints in ball mill capacity in the published ball mill models may result in unrealistic predictions of mill throughput. This paper presents an overfilling indicator for wet
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The acceleration factor of the ball or rod mass is a function of the peripheral speed of the mill. Thus. n = c9np/√D, the above equation becomes P = f1 (D²)·f5 (πD c9 np/√D) = cs np As a first
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An early 20thcentury oilseed rollermill from the Olsztyn district, Poland A late 19th century double roller mill displayed at Cook's Mill in Greenville, West Virginia in 2022 Closeup of Barnard's
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bmlm 1.3.15
Minor update: Use new Stan array syntax thanks to Andrew Johnson.
bmlm 1.3.14
Minor fix: Let rstantools generate Makevars
bmlm 1.3.13
Minor housekeeping:
• Updated compiler flags thanks to @jgabry
• Updated contact info
bmlm 1.3.12
Updated compiler flags for new version of RStan thanks to Andrew Johnson.
bmlm 1.3.11
Fix package for staged installation
bmlm 1.3.9
• Fix NOTE about methods package
bmlm 1.3.8
• Update to C++14, thanks to Ben Goodrich.
bmlm 1.3.7
• Removed tab2doc(), package no longer needs archived ReporteRs package.
bmlm 1.3.6
• Deprecate tab2doc() because required package ReporteRs is archived.
bmlm 1.3.5
• Fix (harmless) constructor error message
bmlm 1.3.4
• Minor cleaning of Stan code
• Fix typos in documentation
bmlm 1.3.3
• Change the label ‘%me’ to ‘pme’ (for proportion mediated effect) in output of mlm_path_plot(…, text = TRUE).
bmlm 1.3.2
• Add options to mlm_spaghetti_plot() to allow jittering and adjusting size of the error bars.
bmlm 1.3.1
• mlm_spaghetti_plot() now has argument mx which can be set to mx = "data" to plot the spaghetti plot of the M - Y relationship (b path) such that the X values are from data, and not fitted values
from the X - M model (a path). The argument defaults to mx = "fitted", such that the X axis values of the M - Y spaghetti plot are fitted values.
bmlm 1.3.0
• New function mlm_spaghetti_plot() for visualizing model-fitted values for paths a (X->M regression) and b (M->Y regression)
bmlm 1.2.10
• Default priors are now \(Normal(0, 1000)\) for regression coefficients, and \(Cauchy(0, 50)\) for group-level SDs
• mlm_summary() now gives only population level parameters by default, and group-level parameters when pars = "random"
• Renamed the mediated effect parameter to me to distinguish it from the product of a and b (similarly for group-level u_me)
• mlm_path_plot() now draws a template if no model is entered (i.e. template argument is deprecated)
• mlm_path_plot() now by default also shows SDs of group-level effects. This behavior can be turned off by specifying random = FALSE
• The fitted model object doesn’t contain the whole covariance matrix anymore, but now contains the group-level intercepts
• New example data set included in package: MEC2010
• Posterior standard deviation is now referred to as SE in mlm_summary()
bmlm 1.2.9
Removed sigma_y from being modeled when binary_y = TRUE.
bmlm 1.2.1
Removed posterior probabilities from default outputs.
Added type = “violin” as option for plotting coefficients with mlm_pars_plot().
bmlm 1.2.0
Users may now change each individual regression parameter’s prior, instead of classes of priors.
Users may now change the shape parameter of the LKJ prior.
bmlm 1.1.1
Coefficient plots now reorder parameter estimates, if user has requested varying effects.
Path plot now by default does not scale the edges.
bmlm 1.1.0
Major update
bmlm now uses pre-compiled C++ code for the Stan models, which eliminates the need to compile a model each time mlm() is run. This significantly speeds up model estimation.
Minor update
The Stan code used by mlm() is now built from separate chunks, allowing more flexible and robust model development.
bmlm 1.0.0
Initial release to CRAN.
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Non-trivial operations in RPN mode
05-04-2015, 08:09 PM
(This post was last modified: 05-04-2015 08:11 PM by Marcio.)
Post: #1
Marcio Posts: 438
Senior Member Joined: Feb 2015
Non-trivial operations in RPN mode
I read the manual but it does not mention it'd be possible for commands to take more than 1 argument from the stack. Does anybody know how to do that? Or is RPN still limited to basic operations like
cos(a), sin(b) etc?
05-04-2015, 08:13 PM
Post: #2
Jonathan Cameron Posts: 205
Member Joined: Dec 2013
RE: Non-trivial operations in RPN mode
(05-04-2015 08:09 PM)Marcio Wrote: I read the manual but it does not mention it'd be possible for commands to take more than 1 argument from the stack. Does anybody know how to do that? Or is
RPN limited to basic operations like cos(a), sin(b) etc?
Include the number of arguments as an argument to the command in RPN mode:
1 [ENTER]
2 [ENTER]
executes the 'cmd' command instructing it to take 2 numbers from the stack.
05-04-2015, 08:15 PM
Post: #3
Jonathan Cameron Posts: 205
Member Joined: Dec 2013
RE: Non-trivial operations in RPN mode
(05-04-2015 08:13 PM)Jonathan Cameron Wrote: Include the number of arguments as an argument to the command in RPN mode:
1 [ENTER]
2 [ENTER]
executes the 'cmd' command instructing it to take 2 numbers from the stack.
What I would like to know is how to create programmed functions that take a pre-determined number of parameters that you define in the function definition, so you do not need to tell the function how
many arguments are needed.
05-04-2015, 08:23 PM
(This post was last modified: 05-04-2015 09:24 PM by Marcio.)
Post: #4
Marcio Posts: 438
Senior Member Joined: Feb 2015
RE: Non-trivial operations in RPN mode
(05-04-2015 08:13 PM)Jonathan Cameron Wrote:
(05-04-2015 08:09 PM)Marcio Wrote: I read the manual but it does not mention it'd be possible for commands to take more than 1 argument from the stack. Does anybody know how to do that? Or
is RPN limited to basic operations like cos(a), sin(b) etc?
Include the number of arguments as an argument to the command in RPN mode:
1 [ENTER]
2 [ENTER]
executes the 'cmd' command instructing it to take 2 numbers from the stack.
Brilliant! Thanks Jonathan. This way I can kinda 'emulate' the 50g inside the Prime, and at a small price of writing specific programs for the commands I use the most. Luckily, I can group them into
a single program.
Very much appreciated. Made my day!
User(s) browsing this thread: 1 Guest(s)
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Description[ ]
Stats are what we call statistical characteristics of any given object in the game. They refer to different things in regards to different objects. Stats in a game perspective, are what defines any
given unit, or AirMech. The two main objects in AirMech that have stats are AirMechs and Units. These stats can be modified via Pilots and Items. For a listing of stat modifiers, who has which
modifiers and explanations of what they do hit the link.
AirMech Stats[ ]
Stats[ ]
Price: 1800 Kudos Pow Pow Pow Spd Tme Cost Carry
270 Diamonds Ground Air Repair M/Sec Build Build Weight
M-11 Striker 160 262 9 955 32.2 4.3 1000 1650
The Striker is the most common AirMech design, used by many countries during the war. Good all around combat abilities, also has an energy
Air mode
• Primary: AirMech Cannons
• Secondary: Homing Missiles
Ground mode
• Primary: AirMech Cannons
• Secondary: Energy Shield and Beam Sword
Stat descriptions[ ]
• Pow Ground: How much damage the AirMech deals to ground units.
• Pow Air: How much damage the AirMech deals to air units (only AirMechs in Air mode are considered air units right now)
• Pow Repair: How quickly the AirMech repairs itself.
• Def Armor: How much armor (HP) a mech has.
• Spd M/sec: How quickly the AirMech moves when in Air mode.
• Time Build: How quickly your AirMech rebuilds itself when it's destroyed.
• Cost Build: How much it costs to rebuild the AirMech each time it's destroyed.
• Carry Weight: How much unit mass the AirMech can carry at once.
Stat growths[ ]
The stats given in the stat sheets for AirMechs are default stats they have at in-game level 1 with no pilot or item modifiers applied. For all modifiers, the stats are adjusted. Also, a unique trait
of AirMechs is that their stats grow when they level up in-game. Pow Air, Pow Ground, Def Armor, Time Build, and Carry Weight all grow with level ups.
Unit Stats[ ]
Longhorn N/A N/A
DPS vs Ground 69 Carry Weight 1100
DPS vs Air 0 Move Speed 3.0
Attack Range 18 Repair 0
Blast Damage Undefined Upkeep Undefined
Weapon Heavy Build Time 7.0
Hit Points 1250 Build Cost 4800
Armor Heavy Salvage Value Undefined
Vision Range 20 Unlock Level -
Longhorn. The backbone tank of any army, only weak against airborne threats.
Stat descriptions[ ]
• DPS Ground: How much damage the unit deals to ground units.
• DPS Air: How much damage the unit deals to air units (only AirMechs in Air mode are considered air units right now)
• Pow Repair: How much armor the unit repairs.
• Hit Points: How much armor (HP) a unit has.
• Move Speed: How quickly the unit moves on the ground.
• Time Build: How much time it takes to build the unit.
• Cost Build: How much it costs to build the unit.
• Weapon Class: Class of the weapon determines how much damage the unit will deal to other units with different armor classes.
• Armor Class: Class of armor determines how much damage the unit will take from other units with different weapon classes.
• Unit Weight: How much the unit weights when it's carried by a AirMech.
• Upkeep Cost: How much power it demotes from the increase
• Power: How much power it uses
• Salvage:How much it costs to salvage
Stat modifications[ ]
The stats given are fixed throughout the game, but can be modified by choice of Pilot or item. There are currently no other ways to change unit stats.
Weapon and Armor Class System[ ]
This system modifies how much damage a unit or AirMech take from weapons based on their armor class and incoming fire's weapon class. Heavier armors take less damage from lighter weapons is the basic
idea. There are 4 tiers of armor: Light, Medium, Heavy, Ultra Heavy. And 3 Tiers of weapons: Light, Medium, Heavy. When the armor and weapon classes are the same, damage is 100%. When the armor is
one tier higher, it's reduced to 50% damage, 10% is two tiers higher and 0% if it's three tiers higher. Damage is 100% if weapon class is the same or higher tier than armor, there is no benefit or
disadvantage to using higher tier weapons on lower tier armors. Here it is in a neat table:
EDIT (31/07/2014 ver.27720): this table is outdated, now the damage reduction is similar, and it is calculated with this FORMULA:
ARMOR - WEAPON = DPS DAMAGE REDUCTION % (if the number is 0 or negative there is no reduction)
Example1: Longhorn ATTACK 90 (dps 96) attacks a Joker ARMOR 40 [40-90= -50 that is a negative number so Longhorn deal the whole 96 dps to the Joker without any reduction].
Example2: Brute ATTACK 1 (dps 200) attacks a Longhorn ARMOR 85 [85-1= 84% damage reduction! So the 200 dps of Brute is only 52 dps against the Longhorn because brute uses a very light weapon VS a
quite high armor]
Light Armor Medium Armor Heavy Armor Ultra Heavy Armor
Light Weapon 100 50 10 0
Medium Weapon 100 100 50 10
Heavy Weapon 100 100 100 50
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Pop-Con Presentation
Here is my explanation of how total consumption changes with population; that is, the total ecological impact of the world's population: the ecological footprint times the number of people. This is
the essence of my "population-consumption model."
The light green rectangle represents the total of what each person consumes regardless of how big the population gets. I call that the "extraction mass" because it's the amount of resources extracted
from the Earth either by, or for, each person. Note that because the amounts are so large, I'm using a unit of one Earth per year, and that the rectangle has a thickness of one of these units. The
total volume of the rectangle is currently about 1.5 Earths per year. Keep in mind that this diagram is not to scale.
As the population grows, some resources are consumed purely in the interaction between people. The consumption per interaction is what I call "transaction mass," and the total amount consumed in this
way is represented by the blue triangle. Notice that although the triangle has a large area is very large, it is also very thin, so its total volume -- the amount consumed -- is currently only about
0.1 Earth per year, or 1/16th of the total world consumption.
Relating to the graph of population versus footprint, the slope of the graph is half the transaction mass (because we're talking about a triangle, which is half a square), and the consumption at zero
population (really, one person) is just the extraction mass.
© Copyright 2011 Bradley Jarvis. All rights reserved.
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n glam::f64
pub struct DAffine3 {
pub matrix3: DMat3,
pub translation: DVec3,
Expand description
A 3D affine transform, which can represent translation, rotation, scaling and shear.
§matrix3: DMat3§translation: DVec3
The degenerate zero transform.
This transforms any finite vector and point to zero. The zero transform is non-invertible.
The identity transform.
Multiplying a vector with this returns the same vector.
Creates an affine transform from three column vectors.
Creates an affine transform from a [f64; 12] array stored in column major order.
Creates a [f64; 12] array storing data in column major order.
Creates an affine transform from a [[f64; 3]; 4] 3D array stored in column major order. If your data is in row major order you will need to transpose the returned matrix.
Creates a [[f64; 3]; 4] 3D array storing data in column major order. If you require data in row major order transpose the matrix first.
Creates an affine transform from the first 12 values in slice.
Panics if slice is less than 12 elements long.
Writes the columns of self to the first 12 elements in slice.
Panics if slice is less than 12 elements long.
Creates an affine transform that changes scale. Note that if any scale is zero the transform will be non-invertible.
Creates an affine transform from the given rotation quaternion.
Creates an affine transform containing a 3D rotation around a normalized rotation axis of angle (in radians).
Creates an affine transform containing a 3D rotation around the x axis of angle (in radians).
Creates an affine transform containing a 3D rotation around the y axis of angle (in radians).
Creates an affine transform containing a 3D rotation around the z axis of angle (in radians).
Creates an affine transformation from the given 3D translation.
Creates an affine transform from a 3x3 matrix (expressing scale, shear and rotation)
Creates an affine transform from a 3x3 matrix (expressing scale, shear and rotation) and a translation vector.
Equivalent to DAffine3::from_translation(translation) * DAffine3::from_mat3(mat3)
Creates an affine transform from the given 3D scale, rotation and translation.
Equivalent to DAffine3::from_translation(translation) * DAffine3::from_quat(rotation) * DAffine3::from_scale(scale)
Creates an affine transform from the given 3D rotation and translation.
Equivalent to DAffine3::from_translation(translation) * DAffine3::from_quat(rotation)
The given DMat4 must be an affine transform, i.e. contain no perspective transform.
Extracts scale, rotation and translation from self.
The transform is expected to be non-degenerate and without shearing, or the output will be invalid.
Will panic if the determinant self.matrix3 is zero or if the resulting scale vector contains any zero elements when glam_assert is enabled.
Creates a left-handed view transform using a camera position, an up direction, and a facing direction.
For a view coordinate system with +X=right, +Y=up and +Z=forward.
Creates a right-handed view transform using a camera position, an up direction, and a facing direction.
For a view coordinate system with +X=right, +Y=up and +Z=back.
Creates a left-handed view transform using a camera position, an up direction, and a focal point. For a view coordinate system with +X=right, +Y=up and +Z=forward.
Will panic if up is not normalized when glam_assert is enabled.
Creates a right-handed view transform using a camera position, an up direction, and a focal point. For a view coordinate system with +X=right, +Y=up and +Z=back.
Will panic if up is not normalized when glam_assert is enabled.
Transforms the given 3D points, applying shear, scale, rotation and translation.
Transforms the given 3D vector, applying shear, scale and rotation (but NOT translation).
To also apply translation, use Self::transform_point3() instead.
Returns true if, and only if, all elements are finite.
If any element is either NaN, positive or negative infinity, this will return false.
Returns true if any elements are NaN.
Returns true if the absolute difference of all elements between self and rhs is less than or equal to max_abs_diff.
This can be used to compare if two 3x4 matrices contain similar elements. It works best when comparing with a known value. The max_abs_diff that should be used used depends on the values being
compared against.
For more see comparing floating point numbers.
Return the inverse of this transform.
Note that if the transform is not invertible the result will be invalid.
Trait Implementations§
The resulting type after dereferencing.
Dereferences the value.
Mutably dereferences the value.
Converts to this type from the input type.
The resulting type after applying the * operator.
The resulting type after applying the * operator.
The resulting type after applying the * operator.
This method tests for self and other values to be equal, and is used by ==.
This method tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.
Method which takes an iterator and generates Self from the elements by multiplying the items.
Auto Trait Implementations§
Blanket Implementations§
Returns the argument unchanged.
Calls U::from(self).
That is, this conversion is whatever the implementation of From<T> for U chooses to do.
The resulting type after obtaining ownership.
Creates owned data from borrowed data, usually by cloning.
Read more
Uses borrowed data to replace owned data, usually by cloning.
Read more
The type returned in the event of a conversion error.
Performs the conversion.
The type returned in the event of a conversion error.
Performs the conversion.
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NCERT Solutions Class 11 Physics Chapter 6 Work, Energy and Power
This article contains solution for all the NCERT Physics Class XI Chapter 6 questions. All solutions are written in detail for better understanding and in simple language. In order to help students
become well - prepared in the subjects presented in this chapter, the textbook provides answers to worksheets, supplementary questions, sample problems, and other questions. Additionally, they aid in
improving their capacity to respond to tricky, complicated questions in tests and other competitive assessments. Students can get a quick overview of the key terminology and concepts that are
utilised in this chapter by consulting the NCERT Solutions for Class 11 Physics, which have been updated in accordance with the most recent CBSE Syllabus 2022-23.
In our daily lives, words like "effort," "energy," and "power" are employed. A person is considered to be doing their work if they are lugging bricks, sowing seeds, studying for examinations, etc.
Work has a clear and defined definition in physics.
NCERT Solutions for Class 11 Physics Chapter 6
Question 1:
The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative.
1. Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.
2. Work done by the gravitational force in the above case.
3. Work done by friction on a body sliding down an inclined plane.
4. Work done by an applied force on a body moving on a rough horizontal plane with uniformvelocity.
5. Work done by the resistive force of air on a vibrating pendulum in bringing it to rest.
1. It is obvious that the force and displacement are moving in the same direction, meaning that the effort done on it is productive.
2. It should be noticed that the object is moving upward, whilst the force of gravity is pulling the object downward. As a result, the work is detrimental.
3. It is obvious that the object is moving in the opposite direction from where the frictional force is acting. As a result, the work is unfavourable.
4. The frictional force acting on an object travelling in an uneven horizontal plane is directed in the opposite direction from the direction of the motion. A uniform force is supplied to the item
in order to keep it moving at the same speed. As a result, the object is moving in the same direction as the force being applied. The task completed is therefore fruitful.
5. It should be noticed that the bob is moving in the opposite direction from the air's resistance to it. As a result, the work is unsuccessful.
Question 2:
A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with the coefficient of kinetic friction = 0.1. Compute the
1. work done by the applied force in 10 s.
2. work done by friction in 10 s.
3. work done by the net force on the body in 10 s.
4. change in kinetic energy of the body in 10 s.
Mass of the body = 2 Kg
Horizontal force applied = 7 N
Co - efficient of kinetic friction = 0.1
Acceleration produced by the applied force is,
Change in Kinetic Energy = 635 - 0 = 635 J
Thus, the work done by the net force is equal to the final kinetic energy.
Question 3:
Given in the figure are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions,
if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these
potential energy shapes are relevant.
The total energy is given by E = K.E + P.E
K.E = E - P.E.
Kinetic energy is always positive. The region where K.E. would turn negative prevents the particle from existing.
(a) Potential energy is 0 for the region between x = 0 and x = a. Kinetic energy is therefore positive. The potential energy is greater than E for x > a. As a result, kinetic energy is zero.
Therefore, the particle won't be present in the region where x > a.
In this situation, the particle can have no total energy at all.
(b) If P.E. > E along the entire x - axis, the object's kinetic energy would be negative. As a result, the particle won't be present here.
(c) Because the P.E. is bigger than the E in this case (x = 0 to x = a and x > b), the kinetic energy is negative. In this area, the object is unable to be there.
(d) For the ranges of x = a/2 to b/2 and x = a/2 to - b/2. Positive kinetic energy exists, and the P.E. This area contains the particle.
Question 4:
The potential energy function for a particle executing linear simple harmonic motion is given by V(x) = kx^2/2, where k is the force constant of the oscillator. For k = 0.5 N m^ - 1, the graph of V
(x) versus x is shown in Fig. 6.12. Show that a particle of total energy 1 J moving under this potential must 'turn back' when it reaches x = ± 2 m.
Particle energy E = 1 J
K = 0.5 N m^ - 1
Question 5:
Answer the following:
(a) The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained? The rocket or the atmosphere?
Friction causes the casing to burn up, reducing the mass of the rocket.
According to the principle of energy conservation:
The decreased mass of the rocket will result in a decrease in total energy. As a result, the rocket provides the energy required for the casing to burn.
(b) Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet's velocity in general. Yet the work done by the
gravitational force over every complete orbit of the comet is zero. Why?
Gravitational force is a conservative force. The conservative force exerts no effort on a closed path. As a result, the gravitational force does zero work for each whole circle of the comet.
(c) An artificial satellite orbiting the earth in a very thin atmosphere loses its energy
gradually due to dissipation against atmospheric resistance, however small. Why, then, does its speed increase progressively as it comes closer and closer to the earth?
Since the system's overall energy should remain constant, the kinetic energy increases as the potential energy of the satellite rotating around the Earth diminishes. The satellite's velocity rises as
a result. Despite this, atmospheric friction causes a slight reduction in the system's overall energy.
(d) In Fig. 6.13 (i), the man walks 2 m carrying a mass of 15 kg on his hands. In Fig., he walks the same distance pulling the rope behind him. The rope goes over a pulley, and a mass of 15 kg hangs
at its other end. In which case is the work done greater?
Scenario I:
mass = 20 kg
Displacement of the object, s = 4 m
Work = F × s × cos θ
θ = angle between the force and displacement
F[s] = m × g[s] × cos θ
W = m × g[s] × cos θ
= 20 × 4 × 9.8 × cos 90°
= 0 [Because cos 90° = 0]
Scenario II:
Mass. m = 20 kg
Distance, s = 4 m
The applied force direction is the same as the direction of the displacement.
Therefore the angle between the force and displacement is zero degrees.
Since,cos 0° = 1
∴ W = F × s × cos θ
= m × g[s] × cos θ
= 20 × 4 × 9.8 × cos 0°
= 784 J
Thus the amount of work done is greater in scenario II.
Question 6:
Underline the correct alternative :
(a) When a conservative force does positive work on a body, the potential energy of
the body increases/decreases/remains unaltered.
When a body is moved in the direction of the force, the conservative force exerts positive work on the body, which causes the body to travel to the centre of the force. As a result, the distance
between the two gets smaller, and the body's potential energy gets smaller.
(b) Work done by a body against friction always results in a loss of its kinetic/potential energy.
Kinetic energy
When work is done in the direction that is counter to the direction of friction, the body's velocity is reduced. As a result, the kinetic energy drops.
(c) The rate of change of total momentum of a many - particle system is proportional
to the external force/sum of the internal forces on the system
External force
Regardless of their directions, internal forces cannot result in a change in momentum. As a result, the change in overall momentum is proportional to the force acting on the system from outside.
(d) In an inelastic collision of two bodies, the quantities which do not change after
the collision are the total kinetic energy/total linear momentum/total energy of
the system of two bodies
Total linear momentum.
The total linear momentum is unaffected by whether the impact is elastic or inelastic.
Question 7:
State if each of the following statements is true or false. Give reasons for your answer.
(a) In an elastic collision of two bodies, the momentum and energy of each body is
Both bodies' energy and momentum are conserved and not separately.
(b) The total energy of a system is always conserved, no matter what internal and external forces on the body are present.
The external factors acting on the system have the power to influence the body and alter its energy.
(c) Work done in the motion of a body over a closed loop is zero for every force in
In a closed loop, the conservative force exerts no work on the moving body.
(d) In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.
Question 8:
Answer carefully, with reasons:
(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e., when they are in contact)?
In an elastic collision, the kinetic energy at the start and at the end is equal. There is no kinetic energy conservation when the two balls contact. It transforms into kinetic energy.
(b) Is the total linear momentum conserved during the short time of an elastic collision of two balls?
In an elastic collision, the system's entire linear momentum is conserved.
(c) What are the answers to (a) and (b) for an inelastic collision?
In an inelastic collision, kinetic energy will be lost. Always, the K.E. following a collision is lower than the K.E. initially.
In an inelastic collision, the system's overall linear momentum is also conserved.
(d) If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic? (Note, we are talking here of potential energy
corresponding to the force during a collision, not gravitational potential energy).
Because the forces involved are conservative forces, the collision is elastic. It is based on how far apart the billiard balls' centres are from one another.
Question 9:
A body is initially at rest. It undergoes one - dimensional motion with constant acceleration. The power delivered to it at time t is proportional to:
1. t^1/2
2. t^3/2
3. t^2
4. t
A body is initially at rest. It undergoes one - dimensional motion with constant accelerration.
P = Fv
Where,P = power,F = force,v = velocity
Using equation of motion,
v = u + at
where, v = final velocity, u = initial velocity, a = acceleration, t = time
The fact that the body is initially at rest and has zero initial velocity is given (the body is not moving at all). u = 0.
v = 0 + at
v = at
Force is the product of mass and acceleration
F = ma
Therefore Power becomes,
P = Fv
P = Fv
P = ma × at
P = ma^2 t
In this case, acceleration and mass are both constants. Power therefore has a direct relationship with time.
P is directly proportional to t
Question 10:
A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to:
1. t^1/2
2. t^3/2
3. t^2
4. t
The power of the body can be calculated as,
P = [F][v]
By substituting the dimension values of F and v in the above equation, we get
P = [MLT^ - 2][LT^ - 1] = [ML^2T^ - 3]
The values of power and mass will remain constant since the body is moving while being acted upon by a source of constant power.
Consequently, the above equation may be expressed as,
[L^2 T^ - 3] = Constant
L^2 is directly proportional to T^3
Therefore,L is directly proportional to T^3/2
Question 11:
A body constrained to move along the z - axis of a coordinate system is subject to a constant forceF given by
where i, j, k, are unit vectors along the x - y - and z - axis of the system, respectively. What is the work done by this force in moving the body at a distance of 4 m along the z - axis?
The constant force applied on a body is,
The body is at a distance of 4 m, away from the origin and is moving along the z - axis. Therefore, the displacement vector of the body at this particular instant would be,
The work done in moving the object by the force would be given by,
Substituting values in the above equation we get,
So the work done by the force in moving the object 4 m along the direction of z axis is 12 J.
Question 12:
An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy 10 keV, and the second with 100 keV. Which is faster, the electron or the proton? Obtain the ratio of
their speeds. (electron mass = 9.11 × 10^ - 31kg, proton mass = 1.67 × 10^-27kg, 1 eV = 1.60 × 10^-19J)
Mass of the electron,m[e] = 9.11 × 10^ - 31 Kg
Mass of the proton,m[p] = 1.67 × 10^ - 27 Kg
Question 13:
A raindrop of radius 2 mm falls from a height of 500 m above the ground. It falls with decreasing acceleration (due to the viscous resistance of the air) until, at half its original height, it
attains its maximum (terminal) speed and moves with uniform speed thereafter. What is the work done by the gravitational force on the drop in the first and second half of its journey? What is the
work done by the resistive force in the entire journey if its speed on reaching the ground is 10 ms^ - 1?
Radius of the rain drop,r = 2 mm = 2 × 10^ - 3 m
Height from which the rain drop is falling,s = 500 m
Density of the water,ρ = 10^3 Kg/m^3
Question 14:
A molecule in a gas container hits a horizontal wall with speed 200 m s^ - 1 and angle 30° with the normal and rebounds with the same speed. Is momentum conserved in the collision? Is the collision
elastic or inelastic?
We know that in an inelastic collision, the momentum is always conserved.
The molecule approaches and rebounds with the same speed i.e. 200 m/s.
∴ Initial Velocity = Final Velocity = 200 m/s
Therefore the initial kinetic energy is given by,
Question 15:
A pump on the ground floor of a building can pump up water to fill a tank of volume 30 m^3 in 15 min. If the tank is 40 m above the ground, and the efficiency of the pump is 30%, how much electric
power is consumed by the pump?
The volume of the tank = 30 m^3
Time taken to fill the tank = 15 minutes = 15 × 60 = 900 seconds
Height of the tank above the ground = 40 m
Efficiency of the pump,η = 30%
Density of the water,ρ = 10^3 Kgm^3
Question 16:
Two identical ball bearings in contact with each other and resting on a frictionless table are hit head - on by another ball bearing of the same mass moving initially with speed V. If the collision
is elastic, which of the following figure is a possible result after collision?
Mass of the ball bearing is m.
Total Kinetic Energy of the system before collision is,
Question 17:
A ball A which is at an angle 30° to the vertical is released, and it hits a ball B of the same mass, which is at rest. Does ball A rise after collision? The collision is an elastic collision.
Ball B acquires the velocity of ball A when it collides with ball B, which is stationary, in an elastic collision, while ball A immediately comes to rest following the contact. Momentum is
transferred from the stationary body to the moving body. Ball B moves with the velocity of ball A as a result, and ball A comes to rest after the contact.
Question 18:
The bob of a pendulum is released from a horizontal position. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lowermost point, given that it dissipated 5%
of its initial energy against air resistance?
Length of the pendulam,l = 1.5 m
Potential energy of the bob at the horizontal position = mgh = mgl
The initial energy lost as a result of air resistance when the bob descends from horizontal to its lowest point is 5%.
The bob's total kinetic energy is equal to 95% of its total potential energy in its horizontal position.
Question 19:
A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, the sand starts leaking out of a hole on the floor of the
trolley at the rate of 0.05 kg s^ - 1. What is the speed of the trolley after the entire sandbag is empty?
The sandbag is put onto the trolley, which is moving at a constant speed of 27 km/h. There is no system of external forces at work. There won't be any outside force acting on the system, even if the
sand begins to leak out of the bag. As a result, the trolley's speed won't change. It will be 27 kilometres per hour.
Question 20:
A body of mass 0.5 kg travels in a straight line with velocity v = ax^3/2 where a = 5 m^ - 1/2 s^ - 1. What is the work done by the net force during its displacement from x = 0 to x = 2 m?
Mass of the body,m = 0.5 Kg
Velocity of the body,v = ax^3/2
Here,acceleration a = 5 m^ - 1/2s^ - 1
Initial velocity at x = 0,
Question 21:
The windmill sweeps a circle of area A with its blades. If the velocity of the wind is perpendicular to the circle, find the air passing through it in time t and also the kinetic energy of the air.
25 % of the wind energy is converted into electrical energy, and v = 36 km/h, A = 30 m^2 and the density of the air is 1.2 kg m^ - 3. What is the electrical power produced?
Density of air,ρ kg/m^3
Time taken = t sec
Velocity of air = v m/s
Question 22:
A person trying to lose weight (dieter) lifts a 10 kg mass one thousand times to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated. (a)
How much work does she do against the gravitational force? (b) Fat supplies 3.8 × 10^7J of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will
the dieter use up?
Mass of the person,m = 10 Kg
Height to which the mass is lifted,h = 0.5 m
Number of times,n = 1000
(a) Work done against gravitational force.
Question 23:
A family uses 8 kW of power. (a) Direct solar energy is incident on the horizontal surface at an average rate of 200 W per square meter. If 20% of this energy can be converted to useful electrical
energy, how large an area is needed to supply 8 kW?
(b) Compare this area to that of the roof of a typical house.
(a) Power used by the family,
Percentage of energy converted to useful electrical energy = 20%
As solar energy is incident at a rate of 200 Wm^ - 2
The area required to generate the desired energy is A.
Useful electrical energy produced per second,
(b) The area needed is comparable to the roof of a large house of dimension 14m × 14m.
Question 24:
A bullet of mass 0.012 kg and horizontal speed 70 m s^ - 1 strikes a block of wood of mass 0.4 kg and instantly comes to rest with respect to the block. The block is suspended from the ceiling by
means of thin wires. Calculate the height to which the block rises. Also, estimate the amount of heat produced in the block.
Mass of bullet,m[1] = 0.012 Kg
Initial speed of the bullet,u[1] = 70 m/s
Mass of the wooden block,m[2] = 0.4 Kg
Initial speed of the wooden block,u[2] = 0
Final speed of the system of the bullet and the block = v m/s
Applying the law of conservation of momentum,
Let h be the height to which the block rises.
Applying the law of conservation of energy to this system,
Potential energy of the combination = Kinetic energy of the combination
The wooden block will rise to a height of 0.212m.
The heat produced = Initial kinetic energy of the bullet - final kinetic energy of the combination
Question 25:
Two inclined frictionless tracks, one gradual and the other steep, meet at A, from where two stones are allowed to slide down from rest, one on each track. Will the stones reach the bottom at the
same time? Will they reach there at the same speed? Explain. Given θ[1] = 300, θ[2] = 600, and h = 10 m, what are the speeds and times taken by the two stones?
In the figure, the sides AB and AC are inclined to the horizontal at ∠θ[1] and ∠θ[2], respectively.
According to the law of conservation of mechanical energy,
Potential Energy at the top = Kinetic Energy at the bottom
Since the height of both sides is the same, both stones will reach the bottom at the same speed.
From (1) and (2), we get v[1] = v[2]
Hence, both stones will reach the bottom at the same speed.
For the stone 1
Net force acting on the stone is given by,
Question 26:
A 1 kg block situated on a rough incline is connected to a spring of spring constant 100 N m^ - 1, as shown in Fig. The block is released from rest with the spring in the unstretched position. The
block moves 10 cm down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is
Mass of the block,m = 1 Kg
Spring constant,k = 100 N m^ - 1
Displacement in the block,x = 10 cm = 0.1 m
At equilibrium,
Question 27:
A bolt of mass 0.3 kg falls from the ceiling of an elevator moving down with a uniform speed of 7 ms^ - 1. It hits the floor of the elevator (length of elevator = 3 m) and does not rebound. What is
the heat produced by the impact? Would your answer be different if the elevator were stationary?
Mass of the bolt,m = 0.3 Kg
Potential Energy of the bolt = mgh
= 0.3 × 9.8 × 3 = 8.82 J
The bolt doesn't bounce back. Therefore, all of the potential energy is transformed into heat energy. Since the acceleration due to gravity is the same in all inertial systems, the heat generated
will not change even if the lift is motionless.
Question 28:
On a frictionless track, a trolley moves with a speed of 36 km/h with a mass of 200 Kg. A child whose mass is 20 kg runs on the trolley with a speed of 4 m/s from one end to another, which is 20 m.
The speed is relative to the trolley in the direction opposite to its motion. Find the final speed of the trolley and the distance the trolley moved from the time the child began to run.
Mass of the trolley,m = 200 Kg
Speed,v = 36 Km/h = 10 m/s
Mass of the boy = 20 Kg
Question 29:
Which of the following does not describe the elastic collision of two billiard balls? The distance between the centres of the balls is r.
(i), (ii), (iii), (iv) and (vi).
In a system, the distance between any two masses has an inverse relationship with their potential energy. As the two balls approach closer to one another, the potential energy of the system will
decrease. The potential energy zeroes out when the balls collide, or when r = 2R. These requirements are not met by the potential energy curve in (i), (ii), (iii), (iv), and (vi). There is therefore
no elastic collision.
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Data Analysis and Modeling in Astronomy
Random values, discrete and continuous probability distributions, probability density, statistical description of data, moments of the probability distribution.
Statistical tests, testing hypotheses, t-test, F-test, Chi^2 test, Kolmogorov-Smirnov test.
Linear correlation, correlation coefficient, principal component analysis.
Modeling of data and estimation of the parameters of the model, method of maximum likelihood, least square method, central limit theorem, robust methods, linear models, non-linear models, estimation
of errors of parameters, Monte Carlo methods, bootstrap, Markov Chain Monte Carlo.
Methods for determining minimum of a n-dimensional function: simplex, Powell method, conjugate gradient method, Levenberg-Marquardt method, genetic algorithms.
Analysis of the time series, methods for determining periods: power spectrum, autocorrelation, Nyquist frequency, phase dispersion minimization, sampling, false periods.
Bayesian analysis - Bayes theorem, posterior probability density, examples.
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Learn Image Processing with Keras – A Practical Guide to Machine Learning with TensorFlow 2.0 & Keras
Check out a free preview of the full A Practical Guide to Machine Learning with TensorFlow 2.0 & Keras course
The "Image Processing with Keras" Lesson is part of the full, A Practical Guide to Machine Learning with TensorFlow 2.0 & Keras course featured in this preview video. Here's what you'd learn in this
Vadim demonstrates how to build training data, use a dataset, and visualize its images using the Matplotlib library.
Transcript from the "Image Processing with Keras" Lesson
>> We'll be talking about next is the computer vision. So let's create new notebook. Because I want just to show you a couple of datasets you can start using to play around with the images. For
instance, once again, let's import TensorFlow. We also kinda import Keras directly. So let's say from tensorflow import keras.
All right, just to make sure, we can print out a TensorFlow version. I think Keras should also have the same format, new version of the Keras, at least. So, keras__version__ should give you, yes,
2.2.4-tf. Meaning that Keras, this version of Keras is using TensorFlow, or actually part of TensorFlow, to be precise.
And what is interesting in Keras, There is additional functionality available. And the one I'm looking for is keras.datasets. So keras.datasets, if you press Enter and just press top afterwards, it
will show you the available datasets already embedded or available for you right now as part of the Keras framework itself.
So boston_housing, I believe it's data from 1970 about prices of houses in Boston depending on multiple different input values. For instance, the data points use the age of the person living there,
how many rooms, how much floor, how far away it is from the downtown. Criminal activities in the region, kinda a lot of different information which affects the prices, so you can play with it.
For instance, if you want to try different regression algorithms. cifar10 and cifar100, it's images, but pretty small images. I don't remember the exact resolution. I think it's 20, maybe similar to
mnist. That's the one I want to use. It's 28 by 28 small images with 10 different classes or 100 different classes.
fashion_mnist or mnist, that's the one I want to use. mnist is your handwritten digits, and fashion_mnist is just ten different types of clothing and shoes. So to basically visualize what we have
here, we can simply say load_data, right? And if you kinda curious what this function will return, that's the only function available in the mnist module.
You can just run this command and it will tell you that it will load the mnist data. So it will return back x_train, y_train and x_test, y_test. Okay, so let's just copy those and say these values
should be created, right? And after we call this function, ta-dah, okay.
So now in my x_train, I will have something, as well as y_train. So let's go back and actually discuss what exactly we have here. Remember that the whole machine learning process can be split into
two parts. First, we're doing the training of the model itself, right? And for training parts, we can use x_training, that's the input data, and y_training, that's the labels we should get.
So remember we need to provide input and output for model to modify their weights, right? And x_test and y_test, it's another set of similar inputs and outputs, but the one our model haven't seen
yet. And those can be used to just test your model, to see what's the accuracy of the predictions you're gonna get.
So let's actually visualize it. So what's what is train? If I just simply print out x_train, you will see that it looks like arrays, tons of arrays. So what if I just grab train 0, kinda the first
array? Okay, a lot of zeros, and some other numbers, all right.
That's pretty awesome, although it doesn't really clear how it looked like. I think if I zoom out just for a second, it will be slightly better. No, we still cannot see it easily. I know one trick.
What I can do, I can, yes, import numpy as np. And in numpy, there is very good command called set_printoptions, yes, printoptions.
And to this command, you can provide the linewidth. So let's return quickly back to the zoom. So what we did, we just import numpy as np and setting up the printoptions. Because I want full line to
be printed, so I can put it to something really large, right, I don't care.
All right, so now if I zoom out, You can actually see that it looks like number 5, right? So it's just the values of the intensity. So 0 is our background, and those numbers between 0 and 255 is just
the intensity of ink. So instead of just printing out those numbers, I can even visualize them, it's gonna be easier.
All right, back to 200%. What I can do, I can use matplotlib inline, import matplotlib.pyplot as plt. I'm just using this library for visualization. And I can, for instance, plt.imshow. imshow
function is just showing images. And I can say I want to visualize x_train[0]. Let's see if it works.
Yeah, it does, so basically in x_train[0], so x_train is the list of images. And for instance, in second image, if we execute it, is our 0, right? And at the same time, if we print our ys, so
y_train, for instance, that was number 5. So it actually shows us what the number we should recognize by looking at this image, right?
And in index number 1, we should see 0, which is true. So that’s gonna be the data we will be using to train our model, right? But the thing is when we print it out, the numbers were between 0 and
255, right, integers. Our machine learning models, they usually work not with integers, but with floating point numbers, right?
So what we will need to do, we need to, first of all, convert all of those inputs to floating points. And probably, we want to rescale it. It is recommended in a stabilization of your model for data
and signal to be normalized and ideally kinda shrink to, well, as close to 0.
So what we can do here, we can simply convert. So just all our x_trains, we can say we want all those to be divided by 255.0. So what this operation will do, it will take all individual pixels as
numbers, divide them by 255. Which will automatically cast them into floating point numbers and write them back to the original location.
So for instance, if I just do that and do the same thing with, not ys, x_test. And execute this. Now, for instance, if I just print x_train[0], We'll see that all my numbers, which were distributed
between 0 and 255, now shrink down to the values between 0 and 1.
That's what I want, all right.
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GCSE / National 5 Maths Glossary
Acute angle
An angle less than a right angle (ie less than 90°). See also obtuse angle, reflex angle and right angle.
A number to add
eg in 4 + 3, both 4 and 3 are addends.
Next to.
Where letters are used to represent an unknown number.
An angle is the rotational difference between two lines, normally measured in degrees.
A value close to the actual value of a number
Part of a circle's circumference.
The amount of skin a shape has.
Not symmetrical
One number that represents all the numbers in a set. Average is often used to mean the mean but median and mode are also averages.
One of the scales used to represent a point in a coordinate system. For a 2D graph, there's an x-axis and a y-axis, for a 3D graph there's also a z-axis.
A mnemonic to help you remember which order to carry out operations: Brackets, Indices, Division, Multiplication, Addition and Subtraction.
Doing operations in a different order makes the answer wrong. eg 2+1x3=5 but (2+1)x3=9
A mnemonic to help you remember which order to carry out operations: Brackets, Order, Division, Multiplication, Addition and Subtraction
Doing operations in a different order makes the answer wrong. eg 2+1x3=5 but (2+1)x3=9
1. The bottom side of a polygon.
2. The plave value scale of a number system. We normally count in base 10 because we have 0-9, and then a 1 in the next column. This is convenient because we have 10 fingers but other number
systems exist:
□ Binary
□ Hexadecimal
□ Sexagesimal
A three digit angle measured clockwise from north, eg 030°
Base 2
This number system is used in computers because it makes them easy to make. Because you only need 1 and 0, 1 can be electricity, and 0 no electricity. Instead of the columns (from right to left)
being 1, 10, (10x10=) 100, (10x10x10=) 1,000 etc, they are 1, 2, 4, 8, etc
So the number 5 in binary would be 1x4, 0x 2, 1x1 = 101.
Now you are ready for my binary joke:
There are 10 types of people in the world, those who understand binary and those who don't
To cut exactly in half
The part of an expression that must be carried out first. eg 2+1x3=5 but (2+1)x3=9
If there is a bracket inside another bracket, start with the innermost bracket.
Work out a value. Calculate does not mean you need to use a calculator.
Centilitre cl
A measure of volume. Centi- means a hundredth, so there are a hundred centilitres in a litre. One litre is 1,000 cm^3. You can think of this as a shape measuring 10cm x 10cm x 10cm.
Centimetre cm
A measure of distance. Centi means a hundredth, so there are 100cm in one meter. Things that are 1cm include: the width of a staple, the thickness of a note pad and the diameter of a belly
Centre of enlargement
The point around which scaling is defined. If your pencil was scaled by a factor of 3, it would be 3 times as long. If the centre of scaling was 1m to the left, then the scaled pencil would move
2 metres to the right because it is now 3m from the centre of enlargement.
Centre of rotation
The point that did not move when a shape is rotated. For a wheel the centre of rotation would, of course, be the centre of the wheel.
A straight line between two points on the circumference of a circle.
The perimeter of a circle.
Class width
The range of values (ie upper boundary minus lower boundary) represented by a group.
Age range Class width Frequency
4-7 3 4
8-11 3 6
12-15 3 7
15-18 3 5
A number that a variable is multiplied by. eg the coefficient of 3x is 3.
In a table, columns go up and down, and rows go from side to side.
Common denominator
A denominator is common if it's shared between 2 or more fractions
These two fractions can not be added together because they do not share a common denominator:
12 + 13
So we make a common denominator by multiplying each fraction by the other fraction's denominator:,
36 + 26 = 56
You can place one shape exactly on top of a shape that's congruent with it. You may need to rotate it, translate it and/or reflect it.
See also: similar, directly congruent.
A number that is always the same, represented by a symbol or a letter. One example is π (the ratio of a circle's diameter to its perimeter). Another example is c, the speed of light, used in E=mc
The normal way of doing things
Add money to a bank account.
Cross section
The shape when you slice a 3d shape along its length
A 3D shape where every surface is a square.
Cube number ^3^
A number which is the result of multiplying a whole number by itself, twice
4 cubed = 4^3 = 4 x 4 x 4 = 16 x 4 = 64
Cubed ^3^
A number which is the result of multiplying a whole number by itself, twice
4 cubed = 4^3 = 4 x 4 x 4 = 16 x 4 = 64
Like a cube but some sides are rectangles rather than squares.
Cumulative frequency
A running total of frequencies. eg
Age range Frequency Cumulative Frequency
4-7 4 4
8-11 6 10
12-15 7 17
15-18 5 22
A time period of 24 hours, normally measured from midnight to midnight. There are 7 days in a week and 365 ¼ in a year (365 years in most years, but 366 in a leap year)
Take money out of a bank account
10 sided polygon
A number that is not a whole number, expressed with a '.' in it, rather than as a fraction. eg 3.14 , 2.7
Decimal places
The number of digits after the decimal point.
To make a quantity smaller
Literally a measure of something. It is commonly used to mean either:
1. A measure of temperature, either as degrees centigrade or degrees farenheit
2. A measure of an angle. A circle is divided into 360 degrees. Why 360 and not 100? Because 360 has a lot of factors.
The bottom part of a fraction. numeratordenominator or dividenddivisor.
See also: common denominator
To use a mathematical function to convey the properties of something.
For example:
The green line above is decribed by the function y = - x
It is also described by x = - y but by convention, we write the y to the left of the equals sign.
The straight line length passing through the centre of a circle from one point on a circumference to an opposite point on a circumference.
Subtract the smaller value from the larger value. ie the difference between 5 and 2 is 3.
Showing the time using displayed numbers rather than with hands or a pointer.
Directly Congruent
Directly congruent shapes have the same sizes and angles as one another. They can be superimposed on one another by translation and rotation, but unlike shapes that are congruent, they do not
require flipping or reflection to match.
How far away something is. eg at its closest, the distance to the moon is 360,000km .
You also see distances (in miles) on road signs.
How data is scattered. This can be represented by numbers (ie standard deviation) or as a plot.
A number to divide by
eg in 12 ÷ 3, 3 is the dividend.
The bottom part of a fraction. numeratordenominator or dividenddivisor.
See also: common denominator
The top part of a fraction. numeratordenominator or dividenddivisor
Used to show that one quantity has the same value as another
A statement showing one expression has the same value as another. eg 2+3=7-2
If it has an equals sign in it, it's an equation.
A shape where all sides have the same length.
This could be an equilateral triangle, which is also a regular triangle...
...or an irregular equilateral octagon, for example
Equilateral triangle
A regular triangle. Equilateral or equi(equal) lateral(side) means all sides have the same length as each other. In the case of a triangle, if the side lengths are the same, the angles will also
be the same as one another.
Mathematicians don't like to 'overspecify' things. They (ok, we) like to give a tiny bit of information from which you can imply the rest of what you might want to know.
To find an approximate answer to a problem. This often involves solving the problem with easier numbers. ie 10.2 x 4.8 ≈ 50
A number which is a multiple of 2. All even numbers end in 0, 2 , 4 , 6 or 8. Whole numbers that aren't even are odd. They're not strange; that's just what they're called.
Even number
A number which is a multiple of 2. All even numbers end in 0, 2 , 4 , 6 or 8. Whole numbers that aren't even are odd. They're not strange; that's just what they're called.
To multiply out brackets. eg 2(x+3) expanded becomes 2x+6.
Or (x+1)(2x+3) = 2x^2+5x+3
Give a reason supporting your answer.
A collection of terms which can contain variables (letters) and numbers, eg 2x+y=8
Extapolation is the act of inferring one or more new data points beyond the end of a data series, based on the assumption that trends present in the data will continue.
Compare: interpolation
A number that fits exactly in another number. eg 6 is a factor of 12
The opposte of expand. Take a common factor out of an expression to put it into brackets. eg 2x+6 factorised is s(x+3)
Or 2x^2+5x+3 = (x+1)(2x+3)
Numbers. Fifty two in figures (or numerals) is 52.
An equation that describes the relationship between two or more variables.
The number of times something occurs. It's a tool to make data more helpful.
For example a dodgeball club may have the following members:
Age range Frequency Cumulative Frequency
4-7 4 4
8-11 6 10
12-15 7 4 17
15-18 5 22
Frequency density
A mesure of how many values a class represents. It is calculated as frequency divided by class width.
One or more mathematical operations carried out on the input. For example, if the function is 2x +1, and the input (x) is 2, the output would be 5 (2 × 2 +1 )
A measure of how steep a line is found by dividing the distance up by the distance across.. You see this expressed as a percentage (ie multiplied by 100) on road signs:
Gram g
Gram is a measure of mass. 1g is about the weight of a small paperclip or a plastic pen cap. 1 litre of water weighs 1,000g (= 1kg)
Gram is a measure of mass. 1g is about the weight of a small paperclip or a plastic pen cap. 1 litre of water weighs 1,000g (= 1kg)
The Highest Common Factor (HCF) is the largest factor common to a set of numbers. eg the HCF of 12, 24 and 32 is 4
The vertical dimension of a shape or solid.
A heptagon (also known as a septagon) is a 7 sided polygon (shape).
Is a 50p coin a septagon (or heptagon)?
A six sided polygon (shape).
A hexagon
Hexagons with friends:
Highest Common Factor
The Highest Common Factor (HCF) is the largest factor common to a set of numbers. eg the HCF of 12, 24 and 32 is 4
Like a bar chart except each bar represents a range of data points. The width of each bar is proportional to the class interval and the area is proportional to the frequency.
A 'flat' orientation. The surface of water is always horizontal.
The longest side of a right angled triangle.
This is essential to know for pythagoras.
The Inter Quartile Range (IQR) is the difference between the upper and lower quartile in a data set.
It gives an indication of how spread out the middle chunk of data is.
Improper fraction
An improper fraction is one where the numerator is larger than the denominator.
the mixed fraction 1 13 as an improper fraction would be 43
To make a quantity larger.
Indices ^4
eg 2^4 - 4 is the indice or power
Whole number.
See also irrational number, negative number, real number.
Coming between two things.
Interpolation is literally inserting something into something else. In maths, we use it to mean the insertion of an intermediate value based on the existing values.
Interpolation is literally inserting something into something else. In maths, we use it to mean the insertion of an intermediate value based on the existing values.
Interquartile range
The Inter Quartile Range (IQR) is the difference between the upper and lower quartile in a data set.
It gives an indication of how spread out the middle chunk of data is.
Irrational numbers are never ending and not recurring decimals that can't be written as a fraction of integers.
17 is never ending and not recurring but can be written as a fraction so is not irrational.
√2 is irrational.
A polygon that is not regular. ie its sides are not all the same length and/ or its angles are unequal
An iscosceles has two sides the same shape. It is often said of triangles.
But can also be said of other shapes:
Give a reason supporting your answer.
Kilogram kg
The SI unit of mass. About the same mass as a bag of sugar. 1kg is 1,000g
Kilometre km
A measure of distance. 1Km=1,000m
In maths, a kite is a quadrilateral with one line of symmetry stretching between two opposite corners. If there is another line of symmetry between the other two corners, it's a rhombus.
The LCM is the smallest multiple common to a set of numbers. The LCM of 6 and 4 is 12. For prime numbers, the LCM will be the product of the numbers.
Leap year
In a leap year there's an extra day, so there are 366 days in a leap year, instead of 365. Every 4th year is a leap year.
The reason for it is to keep summer in the summer months and winter in the winter months.
The longest dimension of an object that is not its height..
Line of symmetry
A line (or axis) of symmetry is a lline such that what's on one side of the line is mirrored by what's on the other side of the line
A square has 4 lines of symmetry. You can imagine folding it in half along opposite edges, or folding it in half across its corners, and having both sides of the fold exactly matching up.
Lines of symmetry
A line (or axis) of symmetry is a lline such that what's on one side of the line is mirrored by what's on the other side of the line
A square has 4 lines of symmetry. You can imagine folding it in half along opposite edges, or folding it in half across its corners, and having both sides of the fold exactly matching up.
Litre l
A measure of volume. One litre is 1,000 cm^3, so the volume of 10cm x 10cm x 10cm
Loci is the plural of locus. Most formulas you encounter have only one locus. An example of a formula with many loci would be a tangent plot (ie y=tan(x)).
A locus is a set of points which satisfy a condition (such as a formula). Most commonly the points form a line.
eg, the green line represents the locus of y = - x.
Lower quartile
The lower quartile is the lowest ¼ of data. If asked to find the lower quartile, you are trying to find the data point ¼ of the way along the data.
Lower range
The smallest value in a data set.
Lowest Common Multiple
The LCM is the smallest multiple common to a set of numbers. The LCM of 6 and 4 is 12. For prime numbers, the LCM will be the product of the numbers.
MIxed fraction
A mix of a whole number and a fraction
The improper fraction would be 43 is the same as the mixed fraction 1 13
The most common type of average, formed by adding up the data set and dividing by the number of items.
A type of average: the middle value when the data set is sorted form smallest to largest.
Metre m
A measure of distance. 1km =1,000m, 1,000mm =1m. Some things that are about 1m: the length of a guitar, the width of a door frame, the height of a kitchen counter.
A measure of volume 1cm cubed, or one thousandth of a litre. A teaspoon is about 5ml.
A tiny measure of distance. There are 1,000mm in 1 metre. Things that are 1mm include: a pencil tip, a mustard seed, a sewing needle tip, a grain of sand.
Relating to the mode (average). You could say the mode in a set of human height data is 178cm, or the modal height is 178cm.
The most commonly occurring value in a data set. You can remember it as it's the same word used in fashion a la mode.
If there are two (or more) equally common values, there would be two (or more) modes.
A time period dividing the year into 12. There are between 28 and 31 days in a month. You can remember them by the rhyme:
30 days hath September, April June and November
All the rest have 31
Except February all alone with 28 or 29 in a leap year.
A number that's been multiplied by another number. eg 6 is a multiple of 2 (and also a multiple of 3)
A number to multiply by
eg in 4 × 3, both 4 and 3 are multiplicands.
Natural number
A positive integer.
If you were counting sheep and came up with half a sheep or a negative number of sheep, that would be unnatural. Zero is not a natural number either. A shepherd with zero sheep is not a shepherd.
A number less than zero.
The temperature in a normal household freezer is about -20°c. Negative numbers are also seen around money.
Not 1.
Non-unitary fraction
A non-unitary fraction is a fraction that has a numerator that is not 1.
14 is a unitary fraction.
34 is a non-unitary fraction.
24 is also a non-unitary fraction, even though it can be simplified to a unitary fraction.
A 9 sided polygon.
Normal to
Perpendicular to a tangent of a line.
Numbers. Fifty two in figures (or numerals) is 52.
The top part of a fraction. numeratordenominator or dividenddivisor
The division sign, ÷
Obtuse angle
An angle between 90° and 180 °.
Angle b is obtuse.
An 8 sided polygon.
Odd number
A number that is not even. Odd numbers always end in 1, 3, 5, 7, or 9.
A simple function, most commonly addition, subtraction, multiplication or division.
The direction something is facing.
When two or more lines are parallel they are always the same distance apart. In other words the lines are aligned in exactly the same direction.
What would happen if train tracks weren't parallel?
A quadrilateral where each side is parallel to its opposite side. A rectangle is a special parallelogram.
A 5 sided polygon.
The distance round the outside edge of a shape.
Lines which meet at right angles.
Pi π
The ratio of a circle's circumference to its diameter.
By definition: π=circumferencediameter
This ratio stays the same no matter how big or small a circle gets:
Pi is an irational constant. For most uses 3.14 is precise enough.
Pi has been calculated to over 1 trillion digits but such calculations are entirely useless - 39 digits of pi would be enough to calculate the width of the universe to the precision of one atom.
NASA use 15 digits.
A shape with straight sides.
Positive number
A number greater than zero.
All natural numbers are positive
eg 2^4 - 4 is the indice or power
The number of decimal places that an answer can be quoted to.
The number of decimal places that an answer can be quoted to.
A number that has 2 factors - 1 and itself. 1 is not prime because it has only 1 factor. 0 is not prime either.
A 3D shape with the same cross section all the way through it. If you could make it with a cookie cutter, it's a prism.
A measure of how likely an event is (eg rolling a 3 with dice)
The value you get when multiplying two values together. eg The product of 2 and 3 is 6 (=2x3)
Proportions are equal ratios. For example 2 rectangles that are twice as wide as they are high, both have a width:height ratio of 2:1. They have equal proportions.
A method for finding side lengths in a right angled triangle.
The square on the hypotenuse is equal to the sum of the squares on the other two sides.
Pythagorean triples
Pythagorean triples are sets of whole side lengths that define right angle triangles. These come up often and spotting one saves you having to calculate square roots eg:
20, 21, 29
12, 35, 37
9, 40, 41
They can be disguised by scaling. For example a 3,4,5 triangle may be scaled as 6,8,10, or 9,12,15, etc.
Quadratic equation
An equation where the highest indice is 2. All these equations are quadratic:
□ y = x^2 + 3x -5
□ y = 4x^2 + 3x -5
□ y = x^2 -5
□ y = x^2 + 3x
A polygon with 4 sides ('quad' is 4 - lateral is 'sides').
A quotient is what is produced by dividing two numbers. This is often encountered as two numbers (ie a fraction or 12)
Less commonly a quotient is defined as the greatest whole number of times a divisor may be subtracted from a dividend—before making the remainder negative. For example, 7 ÷ 2 has a quotient of 3.
Radial symmetry
A shape is rotationally symmetrical if it still looks the same after turning it less than 1 complete rotation.
An equilateral triangle has 3 degrees of rotational symmetry, one for each place where the shape looks the same (including the full 360°).
Radians are a unit for measuring angles. Instead of dividing a circle into 360 equal parts (like degrees), radians divide a circle into 2 π (6.28) equal parts.
Why? Because 1 radian is the same distance as the radius of the circle
Most people prefer degrees but calculators often default to radians.
Plural of radius
The distance from the centre of a circle to its circumference.
Random sampling
A method of reducing a data set's size by choosing random data points. The assumption is that because the sample is chosen at random, that the random sample represents the whole data set.
There are lots of examples of this in opinion polls (where people say what they think) and in advertising (8 of 10 women agree their hair is silkier when they use ___). It would not be feasible
to ask everyone.
The difference between the largest number and the smallest in a data set.
The factor by which one number varies with another.
A decimal number that could be formed by dividing one whole number by another. You can often spot rational numbers spot because they either end (ie 0.125) or are recurring(eg 0.333333).
One divided by a number. The reciprocal of 2 is ½. The reciprocal of ½ is 2.
A 4 sided shape where each angle is a right angle. A square is a special rectangle where all the sides are the same length.
A decimal number which doesn't end but repeats after the decimal point (eg 1.3333... or 0.090909). It can be shown to be recurring by putting a dot over the recurring number, or all the recurring
numbers. (eg. 1.3 or 0.09)
Reflex angle
An angle greater than 180°.
A polygon where all sides are equal and all angles are equal.
The way one variable varies with another. For example y=3x+1 or y= 1x
The amount left over when a number can not be divided exactly.
eg 7 ÷ 4 = 1 remainder 3
A quadrilateral with two lines of symetry, each from corner to corner.
Right angle
An angle of 90°
Right angle triangle
A triangle with a right angle.
Right angled triangles are easy to work with because you can calculate side lengths with pythagoras and angles (and sides too) with simple trigonometry
Right angled triangle
A triangle with a right angle.
Right angled triangles are easy to work with because you can calculate side lengths with pythagoras and angles (and sides too) with simple trigonometry
To turn a shape a given angle and a given direction. To be precise, a centre of rotation has to be given.
Rotational symmetry
A shape is rotationally symmetrical if it still looks the same after turning it less than 1 complete rotation.
An equilateral triangle has 3 degrees of rotational symmetry, one for each place where the shape looks the same (including the full 360°).
Round number
A number that is exactly divisible by 10.
The more zeros a number has at the end, the rounder it is, eg:
□ 872 is not round
□ 870 is round
□ 900 is rounder than 870
□ 1000 is rounder than 900
To make a number easier to deal with but less precise by reducing the amount of significant figures. eg 3.14159 could be rounded as 3.142 or 3.14
In a table, columns go up and down, and rows go from side to side.
Running total
A total that is added to with each new number in the data set..
Dice throw Running total
The international system of units. It is wrtten SI rather than IS because it was named by the French: Système International.
Scale factor
How many times greater or smaller the length and width or height) a scaled shape will be scaled by.
A Scalene triangle is one where no two sides are equal and no two angles are equal - although either one of these tests enough on its own because if the sides are all different lengths then no
two angles can be the same. A right angled triangle can be a scalene but an iscosceles can't.
To change the size of shape. To define it precisely, the Centre of enlargement needs to be deifined.
A line passing through a circle but not through its centre.
A part of the area of a circle formed of two radii and part of the circumference between them.
An area between the circumference of a circle and a chord.
A heptagon (also known as a septagon) is a 7 sided polygon (shape).
Is a 50p coin a septagon (or heptagon)?
A list of numbers which follow a pattern. eg
□ 2,4,6,8...
□ 50,45,40,35...
□ 2,4,8,16,32....
Sexagesimal is literally base 60.
It was invented nearly 7,000 years ago and we still use it: 60 seconds make a minute, and then 60 minutes make an hour.
Significant figures
Each of the digits of a number that are used to express it to the required degree of accuracy, starting from the first non-zero digit.
Shapes are similar if they have the same angles and proportions. Unlike congruent shapes, they can be different sizes.
To write an expression in its simplest (lowest) terms. eg:
□ 24 = 12
□ x^2 + 3 + 3x^2 + 2x - 7 -3x +2 = 4x^2 - x - 2
A closed 3D shape.
Find a missing value. Frequently written as solve for the value to find. eg solve for x.
How fast something is moving, most commonly in miles per hour (ie how many miles something would travel in 1 hour) in the UK.
If you can't remember the formula, think of the units. We say 30 miles per hour ie 30mileshour so speed is distance (miles)time (hours)
A special rectangle where all the sides are the same length.
See also: square number.
Square number
A number multiplied by itself. It's called square because if you draw what's happening, it's the area of a square with the given side length. Some square numbers include 1 (=1x1) 4 (=2x2), 9 (=
3x3), 16(=4x4)
A number to subtract
eg in 4 - 3, 3 is the subtrahend.
The answer when adding values together. eg The sum of 3 and 2 is 5.
Surface area
The total 'skin' area of a 3D shape.
A shape with 1 or more lines of symmetry.
If you can place a mirror on a shape and the reflected portion behind the mirror is the same as the other half, that shape has symmetry. Where you placed the mirror is called a line of symmetry.
A way of recording a running total, by marking a number as groups of 5 (4 vertical lines and a horizontal one). It is useful for recording a running total.
[DEL:IIII:DEL] II (=7)
There are two uses of the word tangent in mathematics.
1. A straight line that touches a point on a curve wihtout passing through it. A tangent to a circle is, like the circumference it touches, perpendicular to the radius at that point.
2. The function which maps a gradient on to an angle, shown here with the angle in radians.
Part of an expression.
The metric system
The international system of units. It is wrtten SI rather than IS because it was named by the French: Système International.
Top heavy fraction
An improper fraction is one where the numerator is larger than the denominator.
the mixed fraction 1 13 as an improper fraction would be 43
Reflections, rotations, translations and / or enlargements are all transformations.
To slide a shape along the x or y axis.
A quadrilateral where 2 sides are parallel to one another. The other 2 sides are not parallel to one another, or it would be a parallelogram.
A quadrilateral where 2 sides are parallel to one another. The other 2 sides are not parallel to one another, or it would be a parallelogram.
Tree diagram
A list of the outcomes for a series of probable events. To work out the probability of 2 heads in 3 flips of a coin. you would start on the left wiht branches to heads and tails, and then each of
those branches would branch again into heads and tails, and those branches would branch further.
A 3 sided polygon.
Triangular number
A series of numbers where the amount added increases by 1 with each number in the series.
A branch of maths concerned with angles and calculations involving angles.
Unit fraction
A unitary fraction is a fraction that has a numerator of 1.
14 is a unitary fraction.
34 is a non-unitary fraction.
24 is also a non-unitary fraction, even though it can be simplified to a unitary fraction.
Yes, it's that simple. Unitary, just means 1.
Unitary fraction
A unitary fraction is a fraction that has a numerator of 1.
14 is a unitary fraction.
34 is a non-unitary fraction.
24 is also a non-unitary fraction, even though it can be simplified to a unitary fraction.
The scale of the quantity that the number expresses. For example, grams, metres, miles per hour, degrees, etc
Upper quartile
The upper quartile is the highest ¼ of data. If asked to find the upper quartile, you are trying to find the data point ¾ of the way along the data.
Upper range
The largest value in a data set.
A numerical quantity.
A letter in an expression. It may represent a fixed number (eg 2x+1=5 can only be true if x =2), or a variable can be a point anywhere along a line (eg y=3x+2)
A point where two straight lines meet.
An up-down orientation..
Vertically opposite angles
Where two straight lines meet (forming an X), any pair of opposite angles will be equal.
eg A= B and C = D
The horizontal line separating the numerator from denominator in a fraction
The amount of space an object takes up - like a 3D area.
Time period. There are 7 days in a week and roughly 52 weeks in a year.
The side to side distance. eg a swimming pool may be 12m wide (or have a width of 12m).
The side to side distance. eg a swimming pool may be 12m wide (or have a width of 12m).
Write down
Give the answer. You may need to use logical reasoning but you would not need to actually add, subtract, multily, divide, etc.
X angles
Where two straight lines meet (forming an X), any pair of opposite angles will be equal.
eg A= B and C = D
The horizontal axis. It is sometimes called y=0 because that is the formula of the line that lies along the x-axis.
The x-coordinate where a graph crosses the x-axis (eg y=0)
eg, the x intercept of the purple line is -1.
The vertical axis. It is sometimes called x=0 because that is the formula of the line that lies along the y-axis.
The y - coordinate of where a graph crosses the y-axis (eg x=0).
eg, the y intercept of the purple line is -2
The amount of time it takes for the Earth to travel once around the sun. A year is 365 days (366 in a leap year), 12 months or roughly 52 weeks.
The third (normally depth) axis for a 3D space or graph.
Where a line crosses two parallel lines, (forming a Z), the internal opposite angles will be equal.
In the diagram, where h and h^| are parallel, α = β:
non-unit fraction
A non-unitary fraction is a fraction that has a numerator that is not 1.
14 is a unitary fraction.
34 is a non-unitary fraction.
24 is also a non-unitary fraction, even though it can be simplified to a unitary fraction.
The definitions contained here are copyright © 2023 Influenca Ltd, with some diagrams and pictures copyright © Wikimeda commons, used with permission under the Wikimedia commons license. You may
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Ncert Solutions for Class 7 Maths Latest 2023 Chapterwise PDF - Utopper
Ncert Solutions for Class 7 Maths
NCERT Solutions Class 7 Maths CBSE 2022-23 Edition
Here are the NCERT Solutions for Class 7 Maths available made by Utopper experts. Practicing NCERT Solutions is an absolute necessity for students who wish to perform well on their Maths examination.
Students having difficulty solving problems from the NCERT textbook for Class 7 can consult the NCERT Solutions listed below.
Students are advised to practice the Class 7 NCERT Solutions, which are available in PDF format, as this facilitates their comprehension of fundamental concepts. Utopper has prepared step-by-step
solutions with thorough explanations for students who experience anxiety while searching for the most comprehensive and thorough NCERT Solutions for Class 7 Maths. Our expert tutors have designed
these exercises to aid in your exam preparation and help you earn high marks in the subject.Â
Class 7 Maths Ncert Solutions – Download Free PDFs
To perform well on exams, students must master the concepts thoroughly. If they comprehend it properly, they can quickly learn. Understanding and practicing problems based on mathematical concepts is
the essence of mathematics. They may have difficulty answering the questions if they do not have access to NCERT solutions. Therefore, Utopper provides students with chapter-by-chapter PDFs of NCERT
Maths Book Class 7 Solutions. Thus, students can cover all topics and strengthen their understanding by answering questions. The solutions are presented in a manner that students can easily
comprehend and remember.
At Utopper, the NCERT Maths Book Class 7 Solutions PDF is compiled by subject matter experts who are highly qualified and experienced. The solutions are intended to address any doubts that students
may have while solving NCERT Maths textbook problems. In this manner, students can achieve higher exam scores. Students can find solutions for every topic in every chapter, written in language that
is simple enough for every student to comprehend.Â
In addition, students have access to additional online study materials and resources, such as CBSE Revision Notes, Ncert books, previous question papers, exemplar solutions, important questions,
etc., via the Utopper website. Students are also encouraged to practice Class 7 Sample Papers to familiarise themselves with the question format of the final exam.
NCERT Solutions for Class 7 Maths Chaptewise PDF
Ncert Solutions For Class 7 Maths
NCERT Solutions for Class 7 Maths are created and prepared by a team of experienced teachers and subject matter experts, whereas private solutions are written by a single author or possibly two.
Therefore, the quality of the content flow in NCERT books is superior to that of any other publisher. Utilize these solutions to attempt to answer all of the questions. Simply read the NCERT books
thoroughly, and you may quickly locate the answers to the questions.Â
If you click on the name of the chapter whose solutions you wish to view, you will be taken to the page for that chapter. Then, you can review the detailed, step-by-step CBSE 7th-grade mathematics
questions and answers for every single question in that chapter.Â
In class 7, you must learn 15 chapters of mathematics. The following are notable chapters and their respective subjects:
NCERT Solutions for Class 7 Maths Chapter-wise Links :
• Chapter 1: Integers
• Chapter 2: Fractions and Decimals
• Chapter 3: Data Handling
• Chapter 4: Simple Equations
• Chapter 5: Lines and Angles
• Chapter 6: The Triangles and Its Properties
• Chapter 7: Congruence of Triangles
• Chapter 8: Comparing Quantities
• Chapter 9: Rational Numbers
• Chapter 10: Practical Geometry
• Chapter 11: Perimeter and Area
• Chapter 12: Algebraic Expressions
• Chapter 13: Exponents and Powers
• Chapter 14: Symmetry
• Chapter 15: Visualising Solid Shapes
Some Important Topics Present in the NCERT Class 7 Maths Solutions
Class 7 Chapter 1 Integers – This chapter contains four exercises; NCERT Class 7 Maths Solutions can be used to solve all the exercise examples and problems. With the aid of NCERT Solutions, you must
study the properties of integer addition and subtraction, positive and negative integer multiplication, and their properties. This chapter concludes with a discussion of the division of integers; to
further comprehend this topic, review its properties and solve instances.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Integers Exercise 1.1
• Class 7 Maths Integers Exercise 1.2
• Class 7 Maths Integers Exercise 1.3
• Class 7 Maths Integers Exercise 1.4
Class 7 Chapter 2 Fractions and Decimals – There are seven exercises in this chapter. Using the NCERT solutions for Class 7, you can study topics such as fraction addition and subtraction, fraction
multiplication by fractions, and whole numbers. You must also master the division of fractions by fractions and whole numbers. The following section will cover decimals. Use the NCERT Solutions to
complete the exercises and gain knowledge of the division of a decimal number by a decimal number, by 10, 100, and 1000, and by whole numbers. Utilize the examples and NCERT Class 7 Solutions to
prepare thoroughly.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Fractions and Decimals Exercise 2.1
• Class 7 Maths Fractions and Decimals Exercise 2.2
• Class 7 Maths Fractions and Decimals Exercise 2.3
• Class 7 Maths Fractions and Decimals Exercise 2.4
• Class 7 Maths Fractions and Decimals Exercise 2.5
• Class 7 Maths Fractions and Decimals Exercise 2.6
• Class 7 Maths Fractions and Decimals Exercise 2.7
Class 7 Chapter 3 Data Handling – This chapter has seven exercises. Learn and answer questions on data collecting and data organization. The following topic is the mean mode, and median, which is an
important topic that you must thoroughly understand. Utilize the NCERT solutions to practice and prepare for the exercise questions. In this chapter, you must also study the applications of bar
graphs. This chapter concludes with a discussion of chance and probability. Master all the formulas and procedures. The mode of huge data is another key issue; solve a few examples before moving on
to the exercise questions.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Data Handling Exercise 3.1
• Class 7 Maths Data Handling Exercise 3.2
• Class 7 Maths Data Handling Exercise 3.3
• Class 7 Maths Data Handling Exercise 3.4
Class 7 Chapter 4 Simple Equations – There are seven exercises in this chapter, and you can practice and solve all of the questions using the NCERT solutions. Learn how to solve an equation and
perform as many practice problems as possible. You must also understand its applications. Learn the methods and apply them thoroughly.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Simple Equations Exercise 4.1
• Class 7 Maths Simple Equations Exercise 4.2
• Class 7 Maths Simple Equations Exercise 4.3
• Class 7 Maths Simple Equations Exercise 4.4
Class 7 Chapter 5 Lines and Angles – This chapter is significantly shorter than the others and only contains two tasks. Learn about the several types of angles, including complementary, supplemental,
and adjacent. Additionally, you must learn about pairs of lines. Before beginning the exercises, the examples should be reviewed. Utilize the NCERT solutions to comprehend and comprehend the topics.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Lines and Angles Exercise 5.1
• Class 7 Maths Lines and Angles Exercise 5.2
Class 7 Chapter 6 The Triangle and Its Properties – One of the most salient chapters in your Class 7 Maths. You must be extremely thorough with this chapter; it has five exercises, all of which must
be completed. Use the NCERT solutions to understand and solve issues such as the medians of a triangle, the altitudes of a triangle, and the angle sum property of a triangle, etc. Learn the theorems
and proofs with the help of the NCERT Maths answers, which contain thorough explanations and additional examples.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths The Triangle and Its Properties Exercise 6.1
• Class 7 Maths The Triangle and Its Properties Exercise 6.2
• Class 7 Maths The Triangle and Its Properties Exercise 6.3
• Class 7 Maths The Triangle and Its Properties Exercise 6.4
• Class 7 Maths The Triangle and Its Properties Exercise 6.5
Class 7 Chapter 7 Congruence of Triangles: This chapter discusses triangle congruence. Two figures are said to be congruent if their shape and size are identical. Congruence of plane figures,
congruence among line segments, congruence of angles, congruence of triangles, criteria for congruence of triangles, and congruence of right-angled triangles are additional related topics.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Congruence of Triangles Exercise 7.1
• Class 7 Maths Congruence of Triangles Exercise 7.2
Class 7 Chapter 8 Comparing Quantities: Comparing Quantities is discussed in Chapter 8 of the NCERT textbook. Related topics include equivalent ratios, percentages as an alternative method for
comparing quantities, the definition of percentages, converting fractions to percentages, converting decimals to percentages, and converting percentages to fractions or decimals. Fun with estimation,
interpreting percentages, converting percentages to how many, converting ratios to percentages, increasing or decreasing as a percentage, profit or loss as a percentage, interest on borrowed money,
and simple interest are additional interesting topics.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Comparing Quantities Exercise 8.1
• Class 7 Maths Comparing Quantities Exercise 8.2
• Class 7 Maths Comparing Quantities Exercise 8.3
Class 7 Chapter 9 Rational Numbers – This chapter also has two exercises, and may be mastered with relative ease. Learn about positive and negative rational numbers, the standard form of rational
numbers, etc. You must also practice addition, subtraction, multiplication, and division problems with rational numbers.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Rational Numbers Exercise 9.1
• Class 7 Maths Rational Numbers Exercise 9.2
Class 7 Chapter 10 Practical Geometry: Practical Geometry is covered in Chapter 10 of the NCERT textbook. You are acquainted with a variety of shapes. During earlier classes, you learned how to draw
some of them. You can, for example, draw a line segment of a specified length, a line perpendicular to a specified line segment, an angle, an angle bisector, a circle, etc. You will now learn to draw
parallel lines and a variety of triangles. This chapter covers the construction of a line parallel to a given line, through a point that is not on the line, as well as the construction of various
types of triangles.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Practical Geometry Exercise 10.1
• Class 7 Maths Practical Geometry Exercise 10.2
• Class 7 Maths Practical Geometry Exercise 10.3
• Class 7 Maths Practical Geometry Exercise 10.4
• Class 7 Maths Practical Geometry Exercise 10.5
Class 7 Chapter 11 Perimeter and Area – There are four exercises in this chapter. You must learn about the properties of squares and rectangles. Determine the area of a triangle, a parallelogram, and
a circle. The final two subjects of this chapter are simple, but you must practice questions relating to them in order to be proficient. Topics include unit conversion and applications.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Perimeter and Area Exercise 11.1
• Class 7 Maths Perimeter and Area Exercise 11.2
• Class 7 Maths Perimeter and Area Exercise 11.3
• Class 7 Maths Perimeter and Area Exercise 11.4
Class 7 Chapter 12 Algebraic Expressions: In Chapter 12, “Algebraic Expressions,” of the NCERT textbook, the terms of an expression, coefficients, like and unlike terms, monomials, binomials,
trinomials, polynomials, addition and subtraction of algebraic expressions, and finding the values of an expression are covered.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Algebraic Expressions Exercise 12.1
• Class 7 Maths Algebraic Expressions Exercise 12.2
• Class 7 Maths Algebraic Expressions Exercise 12.3
• Class 7 Maths Algebraic Expressions Exercise 12.4
Class 7 Chapter 13 Exponents and Powers: Exponents and Powers is the subject of Chapter 13 of the NCERT textbook. Exponents are the result of rational numbers multiplied by themselves multiple times.
Laws of exponents, multiplying powers with the same base, dividing powers with the same base, taking the power of a power, multiplying powers with the same exponents, dividing powers with the same
exponents, the decimal number system, and expressing large numbers in standard form are some of the topics covered in this section.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Exponents and Powers Exercise 13.1
• Class 7 Maths Exponents and Powers Exercise 13.2
• Class 7 Maths Exponents and Powers Exercise 13.3
Class 7 Chapter 14 Symmetry: This chapter focuses on Symmetry. The definition of symmetry is that when a shape is rotated, flipped, or slid, it becomes identical to another. For two objects to be
symmetrical, they must be the same size and shape, but one must be oriented differently than the other. Symmetry can also exist within a single object, such as a face. This chapter discusses regular
polygon symmetry lines, rotational symmetry, line symmetry, and rotational symmetry.
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Symmetry Exercise 14.1
• Class 7 Maths Symmetry Exercise 14.2
• Class 7 Maths Symmetry Exercise 14.3
Class 7 Chapter 15 Visualizing Solid Shapes – This is the final chapter of your Class 7 Maths curriculum for CBSE. This chapter has four exercises, for which you can consult the Class 7 NCERT
solutions. You must become familiar with faces, vertices, and edges. You must also practice drawing solid shapes on a flat surface and learn how to see an object from different perspectives. Answer
the exercise questions and complete the examples thoroughly.Â
Links to exercise solutions for the topics covered in this chapter can be found here.
• Class 7 Maths Visualising Solid Shapes Exercise 15.1
• Class 7 Maths Visualising Solid Shapes Exercise 15.2
• Class 7 Maths Visualising Solid Shapes Exercise 15.3
• Class 7 Maths Visualising Solid Shapes Exercise 15.4
The NCERT solutions provide an excellent explanation for each of the aforementioned chapters, and you can utilize them to prepare. Use the Class 7 CBSE guide for improved preparation.Â
Benefits of NCERT Solution of Class 7 Maths
NCERT Solutions for Class 7 Maths are the key to success for every student because they help them build on their core knowledge and become experts in every subject. Some of the good things about the
Class 7 Ncert solutions are as follows:
• The answers are made according to the rules and format set by the CBSE board. This gives the students an idea of how the questions will be set up on the final exams.
• Some expert teachers with years of experience in this field put together the solutions, so you can be sure that there are no mistakes in them. The students can be sure that the solutions are
• The answers are given to the questions that are most likely to show up on the tests. This helps the students prepare well.
Key features of NCERT Solutions for Class 7
Here are a few advantages of referring to Utopper’s NCERT Solutions:
• These solutions are grade-appropriate and use easily-understood language so that children can comprehend them without difficulty.
• These solutions have been produced by the nation’s leading specialists, allowing you to study from highly qualified instructors with minimal effort.
• All solutions are crafted with the most recent recommendations in mind.
• The PDF will serve as a one-stop shop for all of your educational needs.
• It will assist you in establishing a solid basis for higher study.
• You will receive practice tests to evaluate your knowledge and learning.
• These solutions have been created by a group of Utopper specialists who have taken into account the grade level of the pupils. Students should not worry about the accuracy of the solutions
because they will receive the most accurate responses to the problems.
• If students practice these solutions beforehand, they will be able to complete the curriculum well ahead of schedule. In addition, the Class 7 All Subject Question Answer PDF will assist students
to have a deeper understanding of questions that may be asked in final exams.
Class 7 NCERT Solutions – Marks Weightage
All topics in Class 7 are worth a total of 100 marks, 20 of which come from an internal evaluation and 80 from the summative assessment administered at the conclusion of the school year by schools in
accordance with the requirements of the CBSE. The internal evaluation consists of a weighted average of the grades earned on examinations, projects, and other assignments. The final examination for
each subject is worth 80 marks and is weighted differently for each chapter and topic. A student’s ability to intelligently judge their readiness for tests might be aided by adhering to suitable
Internal Assessment Summative Assessment Total
Other Related Study Material for Class 7 :
Importance of Class 7 NCERT Solutions for All Subjects
Class 7 students have a great deal of homework to complete. Their curriculum comprises of numerous subjects, including mathematics, physics, English grammar, Hindi composition, and social studies. In
order to outperform the rest of the class, pupils must have a solid understanding of these disciplines. After finishing the chapters, students might seek out extra study materials to further their
Choosing Utopper’s NCERT solutions will assist students to solve difficult questions and gain an understanding of the curriculum. In addition, students will be able to comprehend the essential
concepts taught in Class 7 NCERT Solutions. Thus, individuals can develop a study plan that will aid them throughout their final exams. These NCERT solutions are unquestionable of great assistance to
students who wish to complete the Class 7 curriculum and do well on their tests. These solutions were developed by subject matter specialists at Utopper in accordance with CBSE guidelines. Therefore,
students can utilize these solutions as references to enhance their own skills in order to perform well on exams.
Frequently Asked Questions on NCERT Solutions for Class 7
Q.1 Which Maths Textbook Is the Best for Class 7 CBSE?
NCERT is sufficient for a 7th student. Try to thoroughly cover the ideas in the NCERT textbooks and use them as a resource when preparing.
Q.2 Where can I download Class 7 Math Ncert Solutions?
Candidates may freely access Class 7 Maths NCERT Solutions from our website. Use them whenever you feel the need to prepare and enhance your understanding of the ideas.
Q.3 How much do the NCERT Solutions for Class 7 PDFs Cost from Utopper?
PDFs of subject-specific solutions are available for free at Utopper. Students are encouraged to provide their information in order to better understand their needs. By doing so, pupils will have
easy access to download the subject-specific study materials. Students can consult the PDF solutions while answering textbook questions and cross-checking their approach to handling hard problems.
Q.4 Why should I study the NCERT Solutions for Class 7 to prepare for my CBSE examinations?
The CBSE examination is a significant milestone in a student’s life, as the total score reflects their conceptual understanding and overall performance. In seventh grade, it is essential that
students have a firm grasp on the ideas that carry the most exam weight. The straightforward language chosen to convey the principles facilitates pupils’ comprehension. Through consistent
preparation, pupils will be able to answer the challenging questions on the final examination.
Q.5 How can I get good marks in 7th grade?
The 7 grade is not treated as seriously as the board classes, yet it is a key developmental stage. To earn good grades in Class 7, you must study all the chapters and clarify key concepts, give each
topic equal attention and importance, and take notes wherever necessary to make revision easier. Additionally, practise with mock exams to gauge your readiness. You will be prepared to answer any
exam question if you thoroughly review your material.
Q.6 How many chapters do the Class 7 Math NCERT Solutions cover?
Mathematics is a difficult subject for the majority of students. Therefore, the difficult issue must be resolved using reference materials such as NCERT Solutions. The NCERT Solutions for Class 7
Maths is divided into 15 chapters. The following are:Â
Chapter 1: Integers
Chapter 2: Fractions and Decimals
Chapter 3: Data Handling
Chapter 4: Simple Equations
Chapter 5: Lines and Angles
Chapter 6: The Triangles and Its Properties
Chapter 7: Congruence of Triangles
Chapter 8: Comparing Quantities
Chapter 9: Rational Numbers
Chapter 10: Practical Geometry
Chapter 11: Perimeter and Area
Chapter 12: Algebraic Expressions
Chapter 13: Exponents and Powers
Chapter 14: Symmetry
Chapter 15: Visualising Solid Shapes
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CHPTRD - Linux Manuals (3)
CHPTRD (3) - Linux Manuals
chptrd.f -
subroutine chptrd (UPLO, N, AP, D, E, TAU, INFO)
Function/Subroutine Documentation
subroutine chptrd (characterUPLO, integerN, complex, dimension( * )AP, real, dimension( * )D, real, dimension( * )E, complex, dimension( * )TAU, integerINFO)
CHPTRD reduces a complex Hermitian matrix A stored in packed form to
real symmetric tridiagonal form T by a unitary similarity
transformation: Q**H * A * Q = T.
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
AP is COMPLEX array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, if UPLO = 'U', the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the unitary
matrix Q as a product of elementary reflectors; if UPLO
= 'L', the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the unitary matrix Q as a product
of elementary reflectors. See Further Details.
D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).
E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
TAU is COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Further Details:
If UPLO = 'U', the matrix Q is represented as a product of elementary
Q = H(n-1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary
Q = H(1) H(2) . . . H(n-1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
overwriting A(i+2:n,i), and tau is stored in TAU(i).
Definition at line 152 of file chptrd.f.
Generated automatically by Doxygen for LAPACK from the source code.
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Fourier interpolation – The Dan MacKinlay stable of variably-well-consider’d enterprises
Fourier interpolation
June 19, 2019 — June 19, 2019
feature construction
functional analysis
linear algebra
signal processing
sparser than thou
\[\renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\mmm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\ff}[1]{\mathfrak{#1}} \renewcommand
{\oo}[1]{\operatorname{#1}} \renewcommand{\cc}[1]{\mathcal{#1}}\]
Figure 1:
a.k.a. spectral resampling/differentiation/integration.
Rick Lyons, How to Interpolate in the Time-Domain by Zero-Padding in the Frequency Domain. Also more classic Rick Lyons: FFT Interpolation Based on FFT Samples: A Detective Story With a Surprise
Steven Johnson’s Notes on FFT-based differentiation is all I need; it points out a couple of subtleties about DTFT-based differentiation of functions.
A common numerical technique is to differentiate some sampled function y(x) via fast Fourier transforms (FFTs). Equivalently, one differentiates an approximate Fourier series. Equivalently, one
differentiates a trigonometric interpolation. These are also known as spectral differentiation methods. However, the implementation of such methods is prey to common confusions due to the
aliasing phenomenon inherent in sampling, and the treatment of the maximum-frequency (Nyquist) component is especially tricky. In this note, we review the basic ideas of spectral differentiation
(for equally spaced samples)…
Specifically, if I have some function sampled on \([0,L]\) at discrete equally spaced points
\[y_n = y(nL/N)\] then I will take the DTFT
\[\begin{aligned} Y_k&=\cc{F}(\vv{y})_k\\ &=\frac{1}{N}\sum_{n=0}^{N-1}y_n\exp -\left(\frac{2\pi i}{N}nk\right) \end{aligned}\]
and I will invert it
\[\begin{aligned} y_n&=\cc{F}^{-1}(\vv{Y})_n\\ &=\frac{1}{N}\sum_{n=0}^{N-1}Y_k\exp \left(\frac{2\pi i}{N}nk\right). \end{aligned}\]
Note that I will also be assuming, informally put, that \(y\) is periodic with period \(L,\) or, equivalently, assuming continuity of function and (enough) derivatives between the boundaries \(0\)
and \(L\). In practice, I rarely have such a given period for a function that I wish to interpolate in this fashion, so I enforce boundary conditions by windowing the function with a von Hann window
or similar. (Failure to do so will still “work”; it’ll just have abysmal convergence properties due to the implicitly super-Nyquist frequencies introduced at the discontinuity that I won’t be
handling properly, which we call the Gibbs phenomenon.) Windowing will obviously in general change the function. If I don’t want to change it, I can always use a flattish window such as the Tukey
window over a longer signal than I intend to interpolate. If I could sample at arbitrary points I might use Chebychev interpolation to make it effectively periodic. But this is already wandering
off-topic. I ignore that for now.
1 Minimum curvature interpolant
So far so normal. When we wish to interpolate and/or differentiate, things get a little less obvious; we wish to preserve minimum curvature for the interpolant \(\hat{y}\), which is under-determined
by the DTFT components due to aliasing.
For a trivial example of the pathologies, all possible \(m_k\in\mathbb{Z}\) following give equally valid DTFTs for the same sample vector with regard to having the same sample points
\[Y_k=\cc{F}(\vv{y})_k=\frac{1}{N}\sum_{n=0}^{N-1}y_n\exp -\left(\frac{2\pi i}{N}n(k+m_kN)\right)\] but inspection reveals they have different interpretations.
Some calculation reveals that a minimum mean-square-derivative interpolant has a relatively simple form for even \(N\)
\[\hat{y}(t)=Y_0+\sum_{0< k < N/2} \left[ Y_k\exp\left(\frac{2\pi i}{L} k t\right)+ Y_{N-k}\exp\left(-\frac{2\pi i}{L} k t\right) \right]+ Y_{N/2}\cos\left(\frac{\pi }{L} t\right) \] This is also a
strictly real interpolant for real-valued input, since in that case \(Y_{N-k}=Y_{k}\) and in fact
\[\begin{aligned} \hat{y}(t)&=Y_0+\sum_{0< k < N/2} \left[ Y_k\left( \exp\left(\frac{2\pi i}{L} k t\right)+ \exp\left(-\frac{2\pi i}{L} k t\right) \right) \right]+ Y_{N/2}\cos\left(\frac{\pi }{L} t\
right)\\ &=Y_0+\sum_{0< k < N/2} \left( 2Y_k\cos\left(\frac{2\pi }{L} k t\right) \right)+ Y_{N/2}\cos\left(\frac{\pi }{L} t\right) \end{aligned}\]
2 Derivatives
Now, suppose at some equally-spaced points \(rm,m\in\mathbb{Z}\) we wish to interpolate this sampled \(y\) using its DTFT coefficients. (What follows will also assume \(r\leq 1\) so I can avoid
talking about the Nyquist component.) The implied interpolant is
\[\begin{aligned} \hat{y}(rm) &=Y_0+\sum_{0< k < N/2} \left( 2Y_k\cos\left(\frac{2\pi }{L} k rm\right) \right)+ Y_{N/2}\cos\left(\frac{\pi }{L} rm\right)\\ \end{aligned}\]
Unless we know something special about \(r\) I can’t see any shortcuts to evaluate these; but I can see how we can get the derivatives of this function cheaply if we are going to take the trouble to
evaluate these.
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What does 9 mean in baseball?
Each position conventionally has an associated number, for use in scorekeeping by the official scorer: 1 (pitcher), 2 (catcher), 3 (first baseman), 4 (second baseman), 5 (third baseman), 6
(shortstop), 7 (left fielder), 8 (center fielder), and 9 (right fielder).
Why is baseball 9 inning?
Sensing that an official ruling was necessary as more and more baseball teams were formed, the Knickerbockers decided to form a committee in 1856 to tackle the issue. The desire for more competitive
defense won out, and nine innings -- and nine men -- became the standard for good. Why 162 games?
Is baseball 9 or 10 innings?
Baseball is a game played between two teams of nine players each. The game is divided into nine innings, each divided into two halves.
What does 10 mean in baseball?
6 = shortstop. 7 = left field. 8 = center field. 9 = right field. If you play a 10-player lineup, a “10” would indicate a short fielder or fourth outfielder.
What does hits per 9 mean?
Definition. H/9 represents the average number of hits a pitcher allows per nine innings pitched. It is determined by dividing a pitcher's hits allowed by his innings pitched and multiplying that by
nine. It's a very useful tool for evaluating pitchers, whose goal is to prevent runs, which are usually scored by hits.
28 related questions found
What is a good K per 9?
For starting pitchers the top and bottom 20th percentile are a K/9 above 7.56 and below 4.89. Relievers top and bottom 20th percentiles are a K/9 above 8.94 and below 5.54. Variations: Some people
prefer to use strikeouts per batter faced (K% or K/G) to express a player's ability to strike batters out.
How do you count hits in baseball?
There are four types of hits in baseball: singles, doubles, triples and home runs. All four are counted equally when deciphering batting average. If a player is thrown out attempting to take an extra
base (e.g., turning a single into a double), that still counts as a hit. Hits come in all varieties.
What does 21 mean in baseball?
21 on the back of their baseball caps for the rest of their major league careers. The award is given to a player at the end of every season for "extraordinary character, community involvement,
philanthropy and positive contributions, both on and off the field."
What is a 9 pitch inning called?
An immaculate inning occurs when a pitcher strikes out all three batters he faces in one inning, using the minimum possible number of pitches - nine.
What does 9 innings mean?
A full inning consists of six outs, three for each team; and, in Major League Baseball and most other adult leagues, a regulation game consists of nine innings.
How long is a baseball game?
MLB average game length 2000-2022
During the 2022 season, an average nine-inning game in the MLB lasted three hours and three minutes.
How long is a 9 inning baseball game?
The average length of a nine-inning MLB regular season game over the last 10 seasons is just north of three hours. The last time the average length was less than three hours was in 2015.
How many hours is a 9 inning game?
Baseball games can vary widely in length and are typically scheduled to last nine innings at the Major League level, seven innings for high school baseball, and six innings for Little League. The
duration of an inning is around twenty minutes each; thus, a nine-inning game usually lasts about three hours.
Is baseball 9 or 8 innings?
In baseball, an official game (regulation game in the Major League Baseball rulebook) is a game where nine innings have been played, except when the game is scheduled with fewer innings, extra
innings are required to determine a winner, or the game must be stopped before nine innings have been played, e.g. due to ...
Is baseball still 9 innings?
If not terminated early, regulation games last until the trailing team has had the chance to make 27 outs (nine innings). If the home team is leading after the visiting team has made three outs in
the top of the ninth inning, the home team wins and does not have to come to bat in the bottom of the ninth.
Does baseball always go to 9 innings?
A standard MLB game will typically last a total of nine innings, unless the game goes to extra innings. However, there are also instances where games are cut short due to poor weather conditions,
like rain, fog, lightning, or even snow.
What is 9 strikes in a row called in baseball?
An immaculate inning in baseball is when a pitcher throws nine consecutive strikes to complete the inning.
What is the highest inning in baseball?
Here is a look at the top 10 single-inning performances in the Modern Era (since 1900), as provided by the Elias Sports Bureau. Note: Technically, the all-time Major League record is 18 runs by the
Chicago White Stockings (now the Cubs) against the long-defunct Detroit Wolverines on Sept. 6, 1883.
How long is an inning?
Instead of a time clock, baseball is regulated by “innings.” Each inning has a top half and a lower half in which the teams take turns in batting and fielding; in the end, the winner is decided by
who has the most runs at the end of nine innings on most occasions. One inning takes an average of 20 minutes to complete.
What does +7.5 mean in baseball?
Over 7.5 (+110) Under 7.5 (-110) In this prop bet, you are betting on how many strikeouts Clayton Kershaw will have. Whether you pick over or under 7.5 strikeouts, the odds are -110, meaning you must
bet $110 for every $100 you want to win. Understanding the different baseball bets that are available is pretty easy.
What does a 5 4 3 mean in baseball?
5-4-3 triple play
The third baseman (5) fields a batted ball and steps on third base to force out a runner advancing from second, then throws to the second baseman (4) to force out a runner advancing from first. The
second baseman then throws to the first baseman (3) to force out the batter.
What does a 6 4 3 mean in baseball?
So, as an example, a 6 4 3 double play means the shortstop fielded the ball and threw it to the second baseman, who turned the double play by throwing it to first base.
What does 3 1 mean in baseball?
Usage. The count is usually announced as a pair of numbers, for example, 3–1 (pronounced as "three and one"), with the first number being the number of balls and the second being the number of
How many foul balls is a strike?
A foul ball is also counted as a strike when a hitter has less than two strikes. When a batter accumulates three strikes, he is out. If the batter bunts a foul ball with two strikes then it is
counted as a strike and the batter is out.
What does batting 300 mean?
A . 300 average would indicate that a player collected a hit on three of every 10 at bats. Ted Williams, Hall of Famer, two-time Triple Crown winner, and generally considered to be one of the best
hitters in history (.
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Intangible Assets (Part-1)
Welcome to 2 part series of Intangible Assets tutorial. Here in this first part, you will learn:
• Definition of Intangible Assets
• Example of Intangible Assets in Accounting
• Patents in Accounting
• Copyright in Accounting
Intangible assets are considered as the rights, privileges, and competitive advantages which resulted from the ownership of long term assets having no physical substance. The intangibles assets prove
their existence in the form of contracts or licenses. The Intangible assets are basically arising from some sources. For example:
• (1) Government grants, such as Patent, Copyrights and trademark .
• (2) Acquisition of another business, in which the purchase price includes a payment for the company™s favorable attributes that is called as Goodwill and
• (3) Private monopolistic arrangements which arises from contractual agreements, such as Franchising and leases.
Companies record intangible assets at their cost price. Intangibles are categorized according to the period of their life because these intangible assets will have a limited life or an indefinite
life. If an intangible has a limited life, then the company will allocate its cost over the asset™s useful life using a process which is similar to the process depreciation. The process of allocating
the cost of intangible assets is referred to as amortization. But the cost of intangible assets with indefinite lives will not be amortized. A company increases (debits) Amortization Expense, and
decreases (credits) the specific intangible asset in order to record amortization of an intangible asset. The amortization process of Intangible assets are usually done or calculated on a
straight-line basis method. For example, a patent has a useful life of 10years. Then the company amortizes the cost of a patent over its 10-year life or its useful life, whichever is shorter between
the two. There is a difference between intangible assets and plant assets in determining the cost. For example in case of plant assets, cost consists of both the purchase price of the asset and the
costs that are incurred in designing and constructing the asset. But in case of intangible assets costs will only include the purchase price. (Companies expense any costs incurred in developing an
intangible asset.)
A ‘ patent basically an exclusive right which is issued by the U.S. Patent Office. This office provides the right to the recipient to manufacture sell or otherwise organize a development for a period
of 20 years from the date of the grant. A patent is nonrenewable but companies can lengthen the legal life of a patent if they acquire new patents for improvements or for making any changes in the
basic design of the patent. The initial cost of a patent is the cash or cash equivalent price that is paid to obtain the patent.’ Many patents are related with the legal action. An owner incurs legal
cost for effectively protecting a patent in a contravention suit is measured necessary to establish the patent™s validity. The owner will add those costs to the account of patent, and amortizes the
costs of patent over the remaining useful life of the patent. The patent holder amortizes the cost of a patent over its 10-year legal life or its useful life, whichever is shorter between the two.
Generally obsolescence and inadequacy are considered or measured by the companies for determining the useful life of the patent. These factors could lead a patent to become economically unproductive
before the end of its legal life.
Copyrights are approved by the federal government .copyrights means the owner has the exclusive right to reproduce and sell an artistic or published work. Copyrights extend for the life of the
creator plus 70 years. The cost of a copyright is consists of both the cost of acquiring it and the cost for defending it. This cost could be very minimal which is paid to the U.S. Copyright Office
or the costs could be higher if there is an infringement suit is involved. The useful life of a copyright generally is shorter than its legal life. Therefore, copyrights usually are amortized over
its useful life.
Don’t Miss the Second Part of Intangible Assets tutorial series.
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45 grams of Almond in Milliliters • Recipe equivalences
45 grams of Almond in Milliliters
How many Milliliters are 45 grams of Almond?
45 grams of Almond in Milliliters is approximately equal to 73 milliliters.
That is, if in a cooking recipe you need to know what the equivalent of 45 grams of Almond measure in Ml, the exact equivalence would be 73.409461, so in rounded form it is approximately 73
Is this equivalence of 45 grams to Milliliters the same for other ingredients?
It should be noted that depending on the ingredient to be measured, the equivalence of Grams to Milliliters will be different. That is, the rule of equivalence of Grams of Almond in Ml is applicable
only for this ingredient, for other cooking ingredients there are other rules of equivalence.
Please note that this website is merely informative and that its purpose is to try to inform about the approximate equivalent values to estimate the weight of the products that can be used in a
cooking recipe, such as Almond, for example. In order to have an exact measurement, it is recommended to use a scale.
In the case of not having an accessible weighing scale and we need to know the equivalence of 45 grams of Almond in Milliliters, a very approximate answer will be 73 milliliters.
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Decimal to Hexadecimal Converter | Convertopedia
Use the below conversion tool to convert a decimal number to hexadecimal number:
Also Try:
Decimal Number
Whole number, a decimal point and a fractional value combines to form a decimal number. The decimal point separates the whole number part from the fractional part of the number. Each digit of a
decimal number can be any number from 0 to 9. Any value less than 1 is written to the right of decimal point. Decimal numbers are also known as base-10 number or counting numbers. Place value of
decimal number varies as the whole number powers of 10 starting from the left of decimal point. Similarly, the place value of digits left to decimal point varies as the division of power of tens.
Hexadecimal Number
Hexadecimal number system uses 16 different symbols to represent a numeric value. It uses numbers 0 to 9 and alphabets A to F for representation. . The place value of each digits of an hexadecimal
number varies as the whole number powers of 16 starting from the right (Least Significant Digit). The first single digit number in hexadecimal system is 0 and the last is F. Similarly, the first two
digit hexadecimal number is 10 and the last is FF and so on. It is used as an alternative for binary numbers by developers and programmers.
Decimal to hexadecimal conversion example
Convert 582[10] to hex
582[10] = 246[16]
Convert 859[10] to hex
859[10] = 35B[16]
Hexadecimal to Decimal conversion table
Hexadecimal Number Decimal Number
A 10
B 11
C 12
D 13
E 14
F 15
Also Try:
Other Converters
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Schwarzschild and Kerr Solutions of Einstein's Field Equation: an introduction
by Christian Heinicke, Friedrich W. Hehl
Publisher: arXiv 2015
Number of pages: 96
Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(-Droste) solution, and into
one specific stationary axially symmetric solution, the Kerr solution.
Download or read it online for free here:
Download link
(2.7MB, PDF)
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Complex Geometry of Nature and General Relativity
Giampiero Esposito
arXivAn attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex
manifolds, spinor and twistor methods, heaven spaces.
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Trefftz methods with cracklets and their relation to BEM and MFS
Alves, Carlos J. S.; Martins, Nuno F. M.; Valtchev, Svilen S.
Engineering Analysis and Boundary Elements, 95 (2018), 93-104
In this paper we consider Trefftz methods which are based on functions defined by the single layer or double layer potentials, integrals of the fundamental solutions, or their normal derivative, on
cracks. These functions are called cracklets, and verify the partial differential equation, as long as the crack support is not placed inside the domain. A boundary element method (BEM)
interpretation is to consider these cracks as elements of the original boundary, in a direct BEM approach, or elements of an artificial boundary, in an indirect BEM approach. In this paper we
consider the cracklets just as basis functions in Trefftz methods, as the method of fundamental solutions (MFS). We focus on the 2D Laplace equation, and establish some comparisons and connections
between these cracklet methods and standard approaches for the BEM, the indirect BEM, and the MFS. Namely, we propose the enrichment of the MFS basis with these cracklets. Several numerical
simulations are presented to test the performance of the methods, in particular comparing the results with the MFS and the BEM.
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Erik Palmgren's homepage
The logic group at the department consists at the moment of
• Per Martin-Löf, professor (emeritus).
• Erik Palmgren, professor
• Henrik Forssell, PhD, affiliated researcher
• Peter LeFanu Lumsdaine, PhD, assistant professor in Mathematical Logic
• Anders Mörtberg, PhD, assistant professor in Computational Mathematics
• Guillaume Brunerie, Postdoc
• Jacopo Emmenegger, Postdoc Genova
• Daniel Alhsén, PhD student (main advisor : Valentin Goranko)
• Menno de Boer, PhD student (main advisor : Peter Lumsdaine)
• Johan Lindberg, PhD student
• Anna Giulia Montaruli, PhD student
• Associated researcher:
• Roussanka Loukanova, PhD, associated researcher
Former members of logic group (2012 - 2018) Supervised PhD-students
• Jonas Eliasson until "Filosofie Licentiat" 2001. He continued to PhD with Steve Awodey and Viggo Stoltenberg as advisors.
• Johan Granström, PhD 2009 (joint with Per Martin-Löf)
• Anton Hedin, PhD 2011.
• Olov Wilander, PhD 2011.
• Christian Espindola, PhD 2016. (Henrik Forssell was assistant PhD-advisor)
• Håkon Robbestad Gylterud, PhD 2017. (Henrik Forssell was assistant PhD-advisor)
• Jacopo Emmenegger, PhD 2019.
Slides of talks
Regular courses in mathematical logic at the department
(partly outdated - see department course pages)
• Logic (intermediate level) - a first course presenting the semantics and a deductive system of predicate logic, including the completeness theorem and its consequences.
• Logic II (advanced level) - a second tier logic course including basic model theory, theory of computation and Gödel's incompleteness theorems.
• Computability and Constructive Mathematics - gives a high level introduction to dedicability, computability and constructivity in mathematics.
• Set theory and Forcing (Advanced level)
• Metamathematics and Proof Theory, Spring 2015. (Advanced level/PhD-level.)
Irregular courses in mathematical logic and related subjects at the department
Some links
March 15, 2019, Erik Palmgren. Email: palmgren [at] math (dot) su {dot} se
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SQL Math Functions with Use Cases
In this context,
• ABS(x): Returns the absolute value of the input value 'x'. For example, ABS(-10) would return 10.
• ROUND(x, d): Rounds the input value 'x' to the nearest whole number or to the specified number of decimal places 'd'. For instance, ROUND(3.14159, 2) would return 3.14.
• CEILING(x): Returns the smallest integer value greater than or equal to the input value 'x'. For example, CEILING(4.25) would return 5.
• FLOOR(x): Returns the largest integer value less than or equal to the input value 'x'. For instance, FLOOR(4.75) would return 4.
• POWER(x, y): Raises the input value 'x' to the power 'y'. For example, POWER(2, 3) would return 8.
• SQRT(x): Returns the square root of the input value 'x'. For instance, SQRT(16) would return 4.
Here are five advanced SQL queries that utilize SQL math functions:
Calculate the average salary of employees, rounding the result to two decimal places.
SELECT ROUND(AVG(salary), 2) AS average_salary
FROM employees;
Find the square root of the total sales for each product category.
SELECT category, SQRT(SUM(sales)) AS square_root_sales
FROM products
GROUP BY category;
Calculate the total revenue, rounding it to the nearest thousand.
SELECT ROUND(SUM(price * quantity), -3) AS total_revenue
FROM orders;
Find the ceiling value of the average rating for each product.
SELECT product_id, CEILING(AVG(rating)) AS ceiling_rating
FROM reviews
GROUP BY product_id;
Calculate the power of the discount percentage for each product.
SELECT product_id, POWER(discount, 2) AS discount_power
FROM products;
Hopefully you find this article helpful. Share your suggestion in comment.
Follow me in Linkedin, Instagram, Twitter, Github.
Email : ashsajal@yahoo.com
Top comments (4)
Aaron Reese •
If you need to include more information than just the grouped filed and aggregate you can also use the OVER() syntax.
SELECT category, SQRT(SUM(sales)) AS square_root_sales
FROM products
GROUP BY category;
SELECT category, ProductName, SQRT(SUM(sales) OVER (PARTITION BY category)) AS square_root_sales_of_category
FROM products;
This will aggregate the sales by category and display the result for the product category against the product row,
Ashfiquzzaman Sajal •
Thanks for the information Reese.
Ashfiquzzaman Sajal •
Don't forget to share your valuable comments!
Ashfiquzzaman Sajal •
All SQL math functions list : postgresql.org/docs/9.5/functions-...
For further actions, you may consider blocking this person and/or reporting abuse
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LoRa: Performance analysis
Here the performance of LoRa Network has been explained. For Matlab code please visit LoRa: SF performance comparison
LoRa uses CSS (chirp spread spectrum) as modulation with a scalable bandwidth of 125kHz, 250kHz or 500kHz. In the system model I used AWGN channel model, BW of 125kHz and spreading factor 10. To
encode data in CSS symbol, bits are transmitted to a specific starting frequency of the chirp signal. To decode the CSS symbol, correlation with copy of a base CSS symbol is used to extract the bits
based on the phase shift of the signal.
To understand LoRa Symbol generation please read LoRa: Symbol Generation To understand LoRa decoding please read LoRa Decoding
Following is the simulation result (BER vs SNR curve) for SF 10:
The analytical BER of chirp spread spectrum is as follows:
Where Eb/No is energy per bit to noise power spectral density ratio and Q(x) is the Q-function.
For the Matlab code of performance comparison of all SF please read LoRa: SF performance comparison
If you are a research student and want to sell your work on my Blog here, please reach me on sakshama.ghosliya@gmail.com
5 comments:
1. Could you share the reference for analytical BER of chirp spread spectrum?
1. You can find it in a conference paper with title "Range and coexistence analysis of long range unlicensed communication" published by "Brecht Reynders, Wannes Meert, Sofie Pollin" in June
If you have direct access to IEEE papers, here is the link: http://ieeexplore.ieee.org/document/7500415/
2. Hello,
Sorry, I am posting the message again because of an error.
I run your Matlab code from LoRa: SF performance comparison and compared the results with new ones obtained using the analytical expression given in your post, initially published by Reynders et
al. 2016. I expected similar results, but they turned out to be different. As an illustrative example, please see the figure I posted here for a spreading factor of 10 and a bandwidth of 125kHz.
The curve in red was computed using the analytical expression while the curve in blue was computed using your Matlab code.
Would it be possible to know the reason for such difference?
Many thanks
1. I wanted to add that, because the analytical expression takes Eb/N0 as input, I calculate the signal to noise ratio using the following:
Rb = ( sf * bw / ( 2 ^ sf )); % LoRa bit rate
SNR = (Rb * EbNo) / bw; % SNR from Eb/N0
SNR_dB = 10 * log10(SNR); % SNR in dB
The above conversion to plot BER against SNR is the only reason I may think of. The conversion makes sense to me, but I may be wrong.
Thanks, and I am looking forward for your comments.
3. Hello Sakshama,
How did you convert the Eb/No to SNR? Please share the reference.
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L. Calibration of the halo linear bias in Λ(
Issue A&A
Volume 691, November 2024
Article Number A62
Number of page(s) 20
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/202451230
Published online 30 October 2024
A&A, 691, A62 (2024)
Euclid preparation
L. Calibration of the halo linear bias in Λ(v)CDM cosmologies
^1 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy
^2 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy
^3 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy
^4 ICSC – Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum Computing, Via Magnanelli 2, Bologna, Italy
^5 Ludwig-Maximilians-University, Schellingstrasse 4, 80799 Munich, Germany
^6 Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal, 4, 20018 Donostia-San Sebastián, Guipuzkoa, Spain
^7 IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain
^8 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany
^9 Dipartimento di Fisica – Sezione di Astronomia, Università di Trieste, Via Tiepolo 11, 34131 Trieste, Italy
^10 Department of Astrophysics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
^11 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405 Orsay, France
^12 Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
^13 INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy
^14 SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste TS, Italy
^15 Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93/2, 40129 Bologna, Italy
^16 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy
^17 INFN-Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
^18 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany
^19 INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese (TO), Italy
^20 Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy
^21 INFN-Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy
^22 Department of Physics “E. Pancini”, University Federico II, Via Cinthia 6, 80126, Napoli, Italy
^23 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy
^24 INFN section of Naples, Via Cinthia 6, 80126 Napoli, Italy
^25 Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France
^26 Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy
^27 INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
^28 INAF – IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy
^29 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain
^30 Port d’Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra (Barcelona), Spain
^31 Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany
^32 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy
^33 Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
^34 Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n, 38204 San Cristóbal de La Laguna, Tenerife, Spain
^35 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
^36 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK
^37 European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy
^38 ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain
^39 Université Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, Villeurbanne 69100, France
^40 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland
^41 UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France
^42 Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, 1749-016 Lisboa, Portugal
^43 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal
^44 Department of Astronomy, University of Geneva, ch. d’Ecogia 16, 1290 Versoix, Switzerland
^45 INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere, 100, 00100 Roma, Italy
^46 INFN-Padova, Via Marzolo 8, 35131 Padova, Italy
^47 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191 Gif-sur-Yvette, France
^48 Institut d’Estudis Espacials de Catalunya (IEEC), Edifici RDIT, Campus UPC, 08860 Castelldefels, Barcelona, Spain
^49 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain
^50 Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy
^51 FRACTAL S.L.N.E., calle Tulipán 2, Portal 13 1A, 28231 Las Rozas de Madrid, Spain
^52 INAF-Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, 35122 Padova, Italy
^53 Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy
^54 Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway
^55 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
^56 Felix Hormuth Engineering, Goethestr. 17, 69181 Leimen, Germany
^57 Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
^58 Cosmic Dawn Center (DAWN), Denmark
^59 Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
^60 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France
^61 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
^62 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
^63 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
^64 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2, 00014 University of Helsinki, Finland
^65 Aix-Marseille Université, CNRS/IN2P3, CPPM, Marseille, France
^66 Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland
^67 Department of Physics, PO Box 64, 00014 University of Helsinki, Finland
^68 Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland
^69 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991PD, Dwingeloo, The Netherlands
^70 Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany
^71 INFN-Sezione di Roma, Piazzale Aldo Moro 2, c/o Dipartimento di Fisica, Edificio G. Marconi, 00185 Roma, Italy
^72 Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93/2, 40129 Bologna, Italy
^73 Department of Physics, Institute for Computational Cosmology, Durham University, South Road DH1 3LE, UK
^74 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France
^75 University of Applied Sciences and Arts of Northwestern Switzerland, School of Engineering, 5210 Windisch, Switzerland
^76 Institut d’Astrophysique de Paris, 98bis Boulevard Arago, 75014 Paris, France
^77 Institut d’Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France
^78 European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
^79 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
^80 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 155, 2200 Copenhagen, Denmark
^81 Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
^82 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
^83 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
^84 Space Science Data Center, Italian Space Agency, via del Politecnico snc, 00133 Roma, Italy
^85 Centre National d’Etudes Spatiales – Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France
^86 Institute of Space Science, Str. Atomistilor, nr. 409 Măgurele, Ilfov 077125, Romania
^87 Dipartimento di Fisica e Astronomia “G. Galilei”, Università di Padova, Via Marzolo 8, 35131 Padova, Italy
^88 Université St Joseph; Faculty of Sciences, Beirut, Lebanon
^89 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile
^90 Satlantis, University Science Park, Sede Bld 48940, Leioa-Bilbao, Spain
^91 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal
^92 Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202 Cartagena, Spain
^93 INFN-Bologna, Via Irnerio 46, 40126 Bologna, Italy
^94 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands
^95 Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA
^96 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna, Italy
^97 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, 11020 Nus (Aosta Valley), Italy
^98 Junia, EPA department, 41 Bd Vauban, 59800 Lille, France
^99 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain
^100 CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA
^101 Laboratoire Univers et Théorie, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, 92190 Meudon, France
^102 INFN – Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
^103 Departamento de Física Fundamental. Universidad de Salamanca. Plaza de la Merced s/n., 37008 Salamanca, Spain
^104 Departamento de Astrofísica, Universidad de La Laguna, 38206, La Laguna, Tenerife, Spain
^105 Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy
^106 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy
^107 Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, 67000 Strasbourg, France
^108 Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
^109 Max-Planck-Institut für Physik, Boltzmannstr. 8, 85748 Garching, Germany
^110 Minnesota Institute for Astrophysics, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA
^111 Institute Lorentz, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands
^112 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
^113 Department of Physics & Astronomy, University of California Irvine, Irvine, CA 92697, USA
^114 Department of Astronomy & Physics and Institute for Computational Astrophysics, Saint Mary’s University, 923 Robie Street, Halifax, Nova Scotia B3H 3C3, Canada
^115 Departamento Física Aplicada, Universidad Politécnica de Cartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain
^116 Instituto de Astrofísica de Canarias (IAC); Departamento de Astrofísica, Universidad de La Laguna (ULL), 38200 La Laguna, Tenerife, Spain
^117 Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK
^118 Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France
^119 CEA Saclay, DFR/IRFU, Service d’Astrophysique, Bat. 709, 91191 Gif-sur-Yvette, France
^120 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
^121 Department of Computer Science, Aalto University, PO Box 15400, Espoo 00 076, Finland
^122 Instituto de Astrofísica de Canarias, c/ Via Lactea s/n, La Laguna 38200, Spain. Departamento de Astrofísica de la Universidad de La Laguna, Avda. Francisco Sanchez, La Laguna, 38200, Spain
^123 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780 Bochum, Germany
^124 Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53, Avenue des Martyrs, 38000 Grenoble, France
^125 Department of Physics and Astronomy, Vesilinnantie 5, 20014 University of Turku, Finland
^126 Serco for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
^127 ARC Centre of Excellence for Dark Matter Particle Physics, Melbourne, Australia
^128 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia
^129 School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK
^130 Department of Physics and Astronomy, University of the Western Cape, Bellville, Cape Town 7535, South Africa
^131 ICTP South American Institute for Fundamental Research, Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil
^132 IRFU, CEA, Université Paris-Saclay 91191 Gif-sur-Yvette Cedex, France
^133 Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Stockholm 106 91, Sweden
^134 Astrophysics Group, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK
^135 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy
^136 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy
^137 Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal
^138 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal
^139 HE Space for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
^140 Aurora Technology for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
^141 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
^142 Dipartimento di Fisica, Università degli studi di Genova, and INFN-Sezione di Genova, via Dodecaneso 33, 16146 Genova, Italy
^143 Theoretical astrophysics, Department of Physics and Astronomy, Uppsala University, Box 515, 751 20 Uppsala, Sweden
^144 Department of Physics, Royal Holloway, University of London, TW20 0EX, UK
^145 Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK
^146 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA
^147 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark
^148 Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY 10003, USA
^149 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA
^★ Corresponding author; tiago.batalha@inaf.it
Received: 24 June 2024
Accepted: 2 September 2024
The Euclid mission, designed to map the geometry of the dark Universe, presents an unprecedented opportunity for advancing our understanding of the cosmos through its photometric galaxy cluster
survey. Central to this endeavor is the accurate calibration of the mass- and redshift-dependent halo bias (HB), which is the focus of this paper. Our aim is to enhance the precision of HB
predictions, which is crucial for deriving cosmological constraints from the clustering of galaxy clusters. Our study is based on the peak-background split (PBS) model linked to the halo mass
function (HMF), and it extends it with a parametric correction to precisely align with results from an extended set of N-body simulations carried out with the OpenGADGET3 code. Employing simulations
with fixed and paired initial conditions, we meticulously analyzed the matter-halo cross-spectrum and modeled its covariance using a large number of mock catalogs generated with Lagrangian
perturbation theory simulations with the PINOCCHIO code. This ensures a comprehensive understanding of the uncertainties in our HB calibration. Our findings indicate that the calibrated HB model is
remarkably resilient against changes in cosmological parameters, including those involving massive neutrinos. The robustness and adaptability of our calibrated HB model provide an important
contribution to the cosmological exploitation of the cluster surveys to be provided by the Euclid mission. This study highlights the necessity of continuously refining the calibration of cosmological
tools such as the HB to match the advancing quality of observational data. As we project the impact of our calibrated model on cosmological constraints, we find that given the sensitivity of the
Euclid survey, a miscalibration of the HB could introduce biases in cluster cosmology analysis. Our work fills this critical gap, ensuring the HB calibration matches the expected precision of the
Euclid survey.
Key words: cosmological parameters / cosmology: theory / large-scale structure of Universe
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
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1 Introduction
The structure-formation process in the Universe is hierarchical, with smaller structures collapsing and merging to form larger ones. Galaxy clusters, the most massive virialized objects in the
Universe, lie at the apex of this hierarchy. They serve as valuable cosmological probes, offering insights into the growth of density perturbations and the geometry of the Universe (see, for
instance, Allen et al. 2011; Kravtsov & Borgani 2012, for reviews).
The cosmological exploitation of cluster surveys is primarily based on number count analysis. This method involves comparing the observed number of clusters in a survey, as a function of redshift and
of a given observable quantity, to the theoretical prediction of the halo mass function (HMF) within a cosmological model, thus enabling the derivation of constraints on cosmological parameters.
Numerous studies have been conducted in this area (e.g., Borgani et al. 2001; Holder et al. 2001; Rozo et al. 2010; Hasselfield et al. 2013; Planck Collaboration XX 2014; Bocquet et al. 2015; Mantz
et al. 2015; Planck Collaboration XXIV 2016; Bocquet et al. 2019; Abbott et al. 2020; Costanzi et al. 2021; Lesci et al. 2022a).
Complementing cluster count analysis is cluster clustering statistics, which examines the spatial distribution of clusters in the Universe (Mana et al. 2013; Castro et al. 2016; Baxter et al. 2016;
To et al. 2021; Lesci et al. 2022b; Sunayama et al. 2023; Romanello et al. 2024; Fumagalli et al. 2024). The halo bias (HB) is a fundamental concept in this analysis since it reflects the ratio
between the number overdensity of the cluster and of the matter distribution. This relationship is expected to bring cosmological information through the mass- and redshift-dependence of the HB and
to be linear on large scales (i.e., ≳ 100 Mpc) to guarantee the scale-independence of the HB.
The Euclid mission (Laureijs et al. 2011; Euclid Collaboration 2022, 2024c) is projected to provide significant advancements to cluster cosmology. Sartoris et al. (2016) have forecasted that the
combined cluster count and clustering analysis by the Euclid mission will provide constraints on the amplitude of the matter power spectrum and the mass density parameters independent and competitive
with other cosmological probes, underlining the potential of galaxy clusters as cosmological probes for ongoing and future missions.
At the heart of cluster cosmology are the theoretical models for the HMF and HB. Simplified models based on linear perturbation theory and spherical collapse have provided invaluable insights into
the potential of cluster counts and clustering as cosmological probes (see, e.g., Press & Schechter 1974; Bond et al. 1991). However, given the complexity and strongly non-linear nature of cluster
formation dynamics, a refinement of these models to the precision level required by available and forthcoming surveys has to rely on cosmological simulations as the primary method to capture such
complexity. Several studies have been dedicated to calibrating semi-analytical models for the HMF and HB, aiming to align these models’ predictions with the results from extensive sets of simulations
(see, for instance, Sheth & Tormen 1999; Sheth et al. 2001; Jenkins et al. 2001; Warren et al. 2006; Tinker et al. 2008, 2010; Bhattacharya et al. 2011; Watson et al. 2013; Despali et al. 2016;
Comparat et al. 2017; Euclid Collaboration 2023). These simulations not only accurately describe the gravitational interactions that predominantly drive structure formation, they but also attempt to
account for the effects of baryonic matter.
The influence of baryons, albeit a minor component in the Universe’s overall composition, plays a significant role in the formation of structures, particularly in the context of these simulations (
Cui et al. 2014; Velliscig et al. 2014; Bocquet et al. 2016; Castro et al. 2020). Given the sensitivity of baryon evolution to the inclusion and modeling of astrophysical processes occurring at
scales much smaller than those resolved in simulations, the modeling of baryonic feedback in hydrodynamical simulations remains a subject of active debate. At the scale of galaxy clusters, for
instance, baryonic feedback is known to reorganize the mass density profile of halos without disrupting the structures, thereby altering the mass enclosed within a given radius compared to
predictions from collisionless N-body simulations. Owing to the substantially greater computational demands of hydrodynamical simulations, it has become standard practice to derive the HMF from
gravitational N -body simulations, with subsequent post-processing to account for baryonic effects (see, e.g. Schneider & Teyssier 2015; Aricò et al. 2021). In this paper, we concentrate on the
initial step of calibrating the HB using collisionless simulations. This approach is intended to be a foundational phase, with the baryonic effects being integrated later, and we employed a
methodology akin to that used for the HMF (see, Castro et al. 2020; Euclid Collaboration 2024a). This strategy underscores our commitment to systematically exploring the cosmological parameter space,
acknowledging the importance of baryonic effects while methodically building toward their inclusion in our analysis.
Systematic errors in the calibration of the HMF and HB can significantly impact the final cosmological constraints. Studies such as those by Salvati et al. (2020), Artis et al. (2021), and Euclid
Collaboration (2023) have highlighted how inaccuracies in theoretical models can propagate biases into cosmological parameter inferences. In response to these challenges, Euclid Collaboration (2023)
presented a new, rigorously studied calibration of the HMF based on a suitably designed set of N -body simulations, offering the required accuracy to analyze Euclid cluster count data.
Semi-analytical modeling typically starts with a simplified physical model, such as the peak-background split (PBS; Mo & White 1996), which is then extended and refined by adding more degrees of
freedom. These additional degrees of freedom are subsequently fitted to simulations. Conceptually, PBS links the HB to the HMF by decomposing the density field into high- and low-frequency modes. The
high-frequency modes that cross the collapsing barrier describe the collapse of structures. In contrast, the low-frequency modes modulate the density field fluctuations, thereby enhancing the number
of peaks that cross the collapse threshold, therefore linking the clustering of collapsed objects with the local density field. Despite its qualitative consistency with simulations, the quantitative
precision of PBS must be enhanced, especially in the context of the Euclid mission’s requirements.
In this paper, we address the challenge of enhancing the accuracy of HB predictions to the level required to fully exploit the cosmological potential of the two-point clustering statistics from the
Euclid photometric cluster survey. Our approach involves calibrating a semi-analytical model to quantify the discrepancies between PBS predictions and simulation results. This calibration aims to
refine HB predictions, improving the reliability of cosmological parameter estimation derived from cluster counts and clustering.
This paper is organized as follows. We revisit the theoretical aspects used in this paper in Sect. 2. In Sect. 3, we describe the methodology used in our analysis. We present the HB model and its
calibration in Sect. 4 along with an assessment of our model’s impact in a forecast Euclid cluster cosmology analysis. Final remarks are made in Sect. 5. The implementation of our model is publicly
available at https://github.com/TiagoBsCastro/CCToolkit and presented in Sect. 5.
2 Theory
2.1 The halo mass function
The differential HMF is given by $dn dM dM=ρmMvf(v)d ln v,$(1)
where n is the comoving number density of halos with mass in the range [M, M + dM], v is the peak height, and the function vf (v) is known as the multiplicity function. The term pm is the comoving
cosmic mean matter density, $ρm=3H02Ωm,08πG,$(2)
where H[0] and Ω[m,0] are the current value of the Hubble parameter and the matter density parameter, and G is the gravitational constant. The peak height is defined as v = δ[c]/σ(M, z), where δ[c]
is the critical density for spherical collapse (Peebles 2020) and σ^2(M, z) is the filtered mass variance at redshift z so that it measures how rare a halo is. The mass variance is expressed in terms
of the linear matter power spectrum P[m](k, z) as $σ2(M,ɀ)=12π2∫0∞dk k2Pm(k,ɀ)W2(kR),$(3)
where R(M) = [3 M / (4 πρm)^1/3 is the Lagrangian radius of a sphere containing the mass M, and W(kR) is the Fourier transform of a top-hat filter of radius R.
The multiplicity function is considered universal if its cosmological dependence is solely through the peak height. However, numerous studies based on N-body simulations have challenged this
assumption. These analyses reveal that, while the initial approximation of HMF universality is generally valid, systematic deviations from this universality become evident at late times in the
universe’s evolution. This deviation has been demonstrated in various independent investigations, each indicating a nuanced understanding of the HMF’s behavior (e.g., Crocce et al. 2010; Courtin et
al. 2011; Watson et al. 2013; Diemer 2020; Ondaro-Mallea et al. 2021; Euclid Collaboration 2023).
The non-universality of the HMF is affected by both the halo definition and the residual dependence of Ac on cosmology. Various studies have shown this dependence, including Watson et al. (2013),
Despali et al. (2016), Diemer (2020), and Ondaro- Mallea et al. (2021) for the dependence on the halo definition, and Courtin et al. (2011) for the cosmology dependence of δc. In our study, we define
halos as spherical overdensities (SO) with an average enclosed mean density equal to Δ[vir](z) times the background density, where Δ[vir](z) is the non-linear density contrast of virialized
structures as predicted by spherical collapse (Eke et al. 1996; Bryan & Norman 1998). The multiplicity function for halo masses computed at the virial radius has been shown to preserve universality
better than other commonly assumed definition of halo radii (Despali et al. 2016; Diemer 2020; Ondaro-Mallea et al. 2021). As for δ[c], we use the fitting formula introduced by Kitayama & Suto (1996)
that ignores the effect of massive neutrinos; however, for the adopted values for total neutrino masses in this work, the fitting formula is still percent level accurate (LoVerde 2014).
In this paper, we use the HMF presented in Euclid Collaboration (2023): $v f(v)=A(p,q)2av2πe−av2/2(1+1(av2))(va)q−1,$(4)
where the parameters {a, p, q} depend on background evolution and power spectrum shape as $a=aRΩmaɀ(ɀ),$(5) $p=p1+p2(d ln σd ln R+0.5),$(6) $q=qRΩmqɀ(ɀ),$(7)
and where Ω[m] is the fractional density of matter in the Universe as a function of redshift, encompassing both baryonic and dark matter contributions $aR=a1+a2(d ln σd ln R+0.6125)2,$(8) $qR=q1+q2(d
ln σd ln R+0.5).$(9)
Lastly, the normalization parameter A is not a free parameter but a function of the other parameters: $A(p,q)={ 2−1/2−p+q/2π[ 2pΓ(q2)+Γ(−p+q2) ] }−1,$(10)
where Γ denotes the Gamma function. The adopted values for the HMF parameters are presented in Table 4 of Euclid Collaboration (2023) and depend on the halo-finder used. In this work, we mostly use
the ROCKSTAR calibration. The SUBFIND calibration is also used in Sect. 4.1.2 to assess the impact of the halo-finder in our model.
2.2 The linear halo bias
The overdensity of halos of mass M at the position r at redshift z, $δh(r,M,ɀ)=n(r,M,ɀ)/n¯(M,ɀ)−1,$(11)
is expressed in terms of the corresponding local number halo number density, n(r, M, z), and of the cosmic mean number density of such halos, $n¯(M,z)$. In linear theory, it is related to the matter
density contrast δm(r, z) as $δh(r,M,ɀ)=b(M,ɀ)δm(r,ɀ)+ϵ(r,M,ɀ),$(12)
where b(M, z) is the linear halo bias and ϵ is a stochastic term that in the following we assume to be associated with shot-noise.
It follows from Eq. (12) that the halo-halo, Ph , and halomatter power spectrum, Phm , are written as a function of the linear matter power spectrum, Pm , for sufficiently large scales as $Ph(k,M,ɀ)=
b2(M,ɀ)Pm(k,ɀ)+PSN,$(13) $Phm(k,M,ɀ)=b(M,ɀ)Pm(k,ɀ),$(14)
where P[SN] represents the shot-noise component. Under the assumption that halos offer a discrete Poisson sample of the underlying continuous matter density field, P[SN] denotes a shotnoise component
commonly assumed to be equivalent to the Poisson term, P[SN] = 1/N, where N is the mean number density of tracers. On the other hand, halos are known to correspond with high-density peaks of the
underlying matter distribution. Therefore, they are expected not to provide a purely Poissonian sampling of this continuous density field. In fact, Casas-Miranda et al. (2002) and Hamaus et al.
(2010) showed that positive and negative corrections to Poisson shot-noise are expected for low- and high-mass halos. In this paper, we parameterize P[SN] as $PSN=1−αn¯,$(15)
where α, a fitting parameter that we calibrate through simulations, controls the deviation from the assumption of Poisson noise.
Assuming the universality of the HMF, Mo & White (1996) derived the HB b(M, z) directly from the HMF by following the PBS framework. The PBS prediction for the HB as a function of the peak height ν
reads $bPBS(v)=1−1δc d ln v f(v)d ln v.$(16)
Although the PBS provides an estimate of the bias that presents the correct qualitative behavior, Tinker et al. (2010) claims a relatively poor performance of the PBS in reproducing results from N
-body simulations, with an accuracy of about 10–20%.
Given the correct qualitative behavior of the PBS prescription, in this paper, we aim to improve the prediction of the bias by calibrating a model for the bias as a function of b[PBS] – that is, we
assume Eq. (16) to be valid, with the HMF (4) – and model its difference with respect to the simulations.
Table 1
Cosmological parameters of the PICCOLO set of simulations.
3 Methodology
3.1 Simulations
3.1.1 N-body simulations
Table 1 presents the adopted values for the matter density parameter and baryonic density parameter at redshift 0 (Ω[m,0] and Ω[b,0]), the dimensionless Hubble parameter h, the spectral index of the
primordial power spectrum ns, and the amplitude of matter density fluctuations on scales of 8h^–1 Mpc σ[8] for the N-body simulations used in this work. We extended the set of PICCOLO simulations
introduced and used by Euclid Collaboration (2023) to calibrate the HMF model. We maintain the same technical configurations as the original PICCOLO simulations and refer to the above-mentioned HMF
paper for further details while summarizing the main aspects. The set comprises 69 cosmological boxes, each with a comoving size of 2000 h^–1 Mpc, and 4 × 1280^3 dark matter particles. The
simulations were conducted using OpenGADGET3, with initial conditions generated by monofonIC(Michaux et al. 2021), based on third-order Lagrangian Perturbation Theory (3LPT) at a redshift of z = 24.
The adopted gravitational softening is equivalent to one-fortieth of the mean inter-particle distance.
The original PICCOLO set of simulations included nine different choices for cosmological parameters, which were randomly chosen from the 95% confidence level hyper-volume of the joint SPT and DES
cluster abundance constraints (Costanzi et al. 2021). Those represent the cosmologies C0 to C8 in Table 1. To further stress our modeling and guarantee its robustness, we also add the cosmologies C9
and C10, which sample the (Ω[m,0], σ[8]) plane in the direction orthogonal to the degeneracy direction of the constraints from Costanzi et al. (2021), and in significant tension with such
constraints. For each cosmology, two white-noise realizations were created to generate initial conditions. For each noise realization, a pair was generated by fixing the amplitudes of the Fourier
modes of the density fluctuation field and pairing the phases (Angulo & Pontzen 2016), except for the reference C0 model, which had 20 realizations.
We further added three simulations with Einstein–de Sitter cosmology (EdS; i.e., Ω[m] = 1 and Ω[Λ] = 0) with powerlaw initial matter power spectrum $Pm(k)∝kns$ with n[s] ∈ {–1.5, –2.0, –2.5}. Those
simulations, which have the same box size and the same number of particles as the PICCOLO set, are only instrumental for the modeling and are not used for the calibration, as they are far from the
regime used to calibrate the model of Euclid Collaboration (2023).
Lastly, we carried out three pairs of simulations with massive neutrinos, again using the same box size as the PICCOLO set. Each pair has a total neutrino mass Σm[ν] ∈ {0.15, 0.30, 0.60} eV. The
simulation set-up for the neutrino simulations is the same used for the OpenGADGET3 simulations extensively validated in Adamek et al. (2023). The baseline cosmological parameters is the C0 set, with
Ω[ν,0] subtracted from Ω[m,0]. The simulations have the same primordial amplitude A[s], resulting in a lower σ[8] for increasing neutrino mass (see Table 1).
The initial conditions (ICs) for the neutrino simulations were generated using the FastDF (Elbers 2022) implementation to monofonIC (Michaux et al. 2021)^1. The forked repository with the FastDF
integration can be found here^2. We employed the same number of neutrino particles as the number of grid resolution elements used for the cold dark matter particles. The total neutrino mass specified
is attributed to a single massive neutrino species.
3.1.2 Approximate methods: PINOCCHIO
In this paper, we also analyzed 200 (100 pairs with each pair having fixed amplitudes and paired phases) halo catalogs simulated with the approximate LPT-based method implemented in the PINOCCHIO
code (Monaco et al. 2002, 2013; Munari et al. 2017). All these simulations have been carried out under the assumption of the C0 cosmological parameters. The rationale for this extra set of simulated
catalogs is to model the impact of fixing and pairing the Fourier mode amplitudes in the ICs on the cluster clustering.
3.2 Halo finders
Euclid Collaboration (2023) showed that the halo-finder adopted for the analysis of the N -body simulations can significantly alter the HMF. To understand if the halo-finder also impacts the HB, we
selected two algorithms to extract halo catalogs: ROCKSTAR (Behroozi et al. 2013a)^3 and SUBFIND (Springel et al. 2001; Dolag et al. 2009; Springel et al. 2021). Although all these algorithms rely on
the SO method to define halo boundaries, they differ in the method used to identify the center from which the spheres are grown and the criteria to classify between structures and sub-structures.
ROCKSTAR divides the simulation volume into 3D friend-of- friends (FOF; see, for instance, Davis et al. 1985) groups and runs a recursive 6D FOF algorithm on each group to create a hierarchy of FOF
subgroups. Halo centers are determined by averaging the positions of particles in the innermost subgroup. To improve consistency, we apply the CONSISTENT algorithm, which dynamically tracks halo
progenitors, to the extracted ROCKSTAR catalogs as demonstrated in Behroozi et al. (2013b). SUBFIND also determines halo centers using a parallel implementation of the 3D FOF algorithm but directly
assigns it to the particle with the lowest gravitational potential.
Among the halo-finders studied by Euclid Collaboration (2023), ROCKSTAR and SUBFIND are good representatives of the heterogeneity of possible results as they are close to the extremes, with SUBFIND
suppressing the abundance of objects more massive than 10^13 M[⊙] h^–1 by roughly 10%. See Knebe et al. (2011) for a more detailed comparison between the halo-finding algorithms.
3.3 Measuring the halo bias
To measure the HB, we bin the halo distribution in log[10](M[vir]/M[⊙] h^–1) with equispaced width intervals of 0.1 dex at each redshift. If the number of halos inside a bin was less than 10 000, we
merged it with its neighbor to avoid having bins where the power-spectrum measurements were primarily dominated by shot noise. We measured the cross-spectrum P[hm] between the halos in each mass bin
and the matter distribution traced by particles. The bias is then obtained as the ratio between this cross-spectrum and the matter power spectrum P[m](k). The matter density field was computed at the
initial conditions and rescaled according to the linear growth factor for the simulations without massive neutrinos. For the simulations with massive neutrinos, the matter density field was built
from the particle data from the same snapshot from which the halo catalog was extracted to consider the scale-dependent growth factor. We used the PYLIANS^4 Python libraries to construct the density
field and compute power spectra on a 1024^3 piecewise cubic spline mesh grid. PYLIANS averages the power spectra in k-space shells with the width given by the fundamental mode of the box, k[f] ≡ 2π/L
. Following Castro et al. (2020), we only considered modes with k values smaller than 0.05 h Mpc^–1 to measure the HB, to ensure the validity of the linear approximation. The maximum k used for the
measurements corresponds to the 16^th harmonic of the box and is much smaller than the Nyquist frequency of the grid used to compute the power spectrum.
To calculate the bias for each mass bin, we used the ratio of the halo-matter cross-spectrum and the matter power spectrum $bi,jsim=Phm(kj)i/Pm(kj),$(17)
where i and j are the mass bin and the k-shell indexes, respectively.
3.4 Halo bias calibration
We used a Bayesian approach with uninformative uniform priors on all parameters to fit our model for the linear HB parameters to simulation results. The best fits were obtained using the Dual
Annealing method to find the posterior maximum as implemented by Virtanen et al. (2020), and covariance between parameters was estimated using PyMC (Salvatier et al. 2016)^5. The No-U-Turn sampler
(NUTS) (Hoffman et al. 2014) was automatically assigned internally by PyMC to sample the likelihood.
We assumed a Gaussian likelihood for the bias, since the power spectrum estimation for a Gaussian field realization follows a χ^2-distribution when averaged over a shell. Since the number of modes N[
k] inside the shell increases rapidly with k, this distribution approaches a Gaussian distribution by the central limit theorem. However, the number of modes is small for the first bins, and
deviation from the Gaussian approximation is evident. To avoid this issue, we re-binned the measurement of the first 3 k-bins by merging them, ensuring that the bin with the fewest modes still
contains 117 modes to recover the Gaussian approximation’s validity.
We note that differently than Castro et al. (2020), we used ICs with fixed amplitudes. Therefore, the distribution of the simulated bias is not approximately a ratio of two Gaussian distributions but
approximately Gaussian itself since the denominator of Eq. (17) is not a random variable. The variance of the shell-average halo-matter cross-spectrum on simulations with random Gaussian initial
conditions is given by $(σPhmPm)2=1Nk(b2+1−αn¯ Pm),$(18)
where we assumed that the shot-noise contribution to the halo power spectrum follows Eq. (15) and a linear HB b. However, Zhang et al. (2022) showed that the predictions for random Gaussian initial
conditions overestimate the variance observed for biased tracers on simulations with fixed amplitudes. Therefore, we modified Eq. (18) as follows: $(σPhmPm)2≡σb2=1Nk(β b2+1−αn¯ Pm)+b2σsys2.$(19)
Here, β and σ[sys] are parameters we marginalize over, which control the variance suppression and the relative error due to the limited accuracy of the HMF used as the backbone for the PBS
prescription. We note that, based on the halo model (see, for instance Cooray & Sheth 2002), one should expect a value for β close to zero as in the limit where all halos are considered, the
shot-noise term on the right-hand side of Eq. (19) tends to zero so that one should recover the matter power spectrum that by construction has zero variance. Furthermore, the HMF presented in Euclid
Collaboration (2023) was shown to have percent-level accuracy; thus, σ[sys] is expected to assume similar values during the calibration, presuming the PBS framework is valid. However, it is crucial
to note that should σ[sys] significantly deviate from zero, such an occurrence could indicate a potential violation or limitation within the PBS framework, underscoring the necessity for careful
interpretation of these parameters.
We obtained the total log-likelihood by summing up all mass bins, modes, redshifts, and simulations. Following Euclid Collaboration (2023), we use the redshifts z ∊{2.00, 1.25, 0.90, 0.52, 0.29,
0.14, 0.0}, translating to a timespacing of about 1.7 Gyr. This spacing is larger than the characteristic dynamical time of galaxy clusters and effectively suppresses the correlation between the
results of different snapshots.
Similarly, we assume that the correlations between different mass bins and modes are negligible. This assumption is justified by linear theory, which posits that different modes evolve independently
in the linear regime, thus minimizing their mutual influence. In our analysis, we have three fitting parameters: α, β, and σ[sys] , in addition to the parameters of the bias model that are subject to
calibration (to be discussed in Sect. 4.1.3). This approach allows for a comprehensive calibration of the bias model, taking into account the shot-noise correction α, the suppression of variance β,
and the systematic uncertainties σ[sys] inherent to our method.
3.5 Forecasting Euclid’s cluster counts and cluster clustering observations
To understand the impact of the HB calibration on cosmological constraints, it is important to realistically forecast the cosmological information to be extracted from the Euclid photometric cluster
survey. For this purpose, we first quantify the impact of the HB on cluster counts and cluster clustering analyses. More precisely, the HB enters the modeling of the cluster counts covariance, which
we compute analytically following the model of Hu & Kravtsov (2003), as validated in Fumagalli et al. (2021). Regarding the cluster clustering, the HB enters both in the computation of the mean value
(power spectrum or two-point correlation function), and in the associated covariance matrix; in this work, we test the effect on the real-space two-point correlation function and its covariance,
following the model presented and validated in Euclid Collaboration (2024b).
After assessing the impact on the two statistics, we forecast how the accuracy of the HB calibration propagates on the cosmological constraints obtained by cluster counts and cluster clustering
experiments. We generate synthetic cluster abundance data as described in Section 2.5 of Euclid Collaboration (2023), assuming the HMF calibrated and the HB of this work as benchmarks. Through a
likelihood analysis, we constrain the cosmological parameters Ω[m,0] and σ[8], and the mass-observable relation (MOR) parameters A[λ], B[λ], C[λ], D[λ] (see Section 2.5 of Euclid Collaboration 2024a
), assuming flat priors for all the parameters. The MOR parameters describe the optical richness λ distribution as a function of the halo mass according to $〈 lnλ∣Mvir,ɀ 〉=lnAλ+Bλln
(Mvir3×1014h−1M⊙) +Cλln(H(ɀ)H(ɀ=0.6)),$(20)
where H(z) denotes the Hubble parameter at redshift z. The range for richness λ is considered to be between 20 and 2000, with the variation in logarithmic richness for a given virial mass and
redshift expressed as a log-normal scatter: $σlnλ∣Mvir,ɀ2=Dλ2.$(21)
The reference values for the parameters are A[λ] = 37.8, B[λ] = 1.16, C[λ] = 0.91, and D[λ] = 0.15. These values have been obtained refitting the parameters presented in Saro et al. (2015) to the
virial mass definition, with the assumption that halo profiles follow an NFW profile with a mass-concentration relationship as outlined in Diemer & Joyce (2019). The parameter values adopted in this
work are consistent with the model presented by Castignani & Benoist (2016) to assign cluster membership probabilities to galaxies from photometric surveys.
We perform the analysis on the synthetic catalogs, comparing them with the predictions made by using our bias and the one of Tinker et al. (2010), and compare the resulting posteriors with two
estimators: we quantify the broadening/tightening of the posterior’s amplitude by computing the difference of the figure of merit (∆FOM; see Huterer & Turner 2001; Albrecht et al. 2006) in the Ω[m,0]
− σ[8] plane, and the shift of the posterior’s position by computing the posterior agreement (Bocquet et al. 2019), which determines whether the difference between two posterior distributions is
consistent with zero.
Fig. 1
Constraints on the parameters in Eq. (19) fitted to the unbiased standard deviation of the 200 PINOCCHIO mocks with C0 cosmology. We fitted α and β considering k ≤ 0.05 h Mpc^−1, assuming a
Gaussian likelihood with error bar estimated from the measurements for 0.05 ≤ k/h Mpc^−1 ≤ 0.2 and assuming it to be constant in k and equivalent to the unbiased standard deviation.
4 Results
4.1 Calibration of the halo bias
4.1.1 Biased tracers statistics on fixed and paired simulations
Before modeling and calibrating the HB to the simulations, we investigate the impact of the variance suppression technique on the halo matter cross-spectrum.
In Fig.1, we present the constraints on the parameters in Eq. (19) fitted to the unbiased standard deviation of the 200 PINOCCHIO mocks with C0 cosmology. We fixed σ[sys] to zero, as this exercise
does not involve modeling errors on the bias. The parameters α and β were fitted using only modes with k ≤ 0.05 h Mpc^−1, under the assumption of a Gaussian likelihood. We estimated the error bars
from the measurements within the range 0.05 ≤ k/h Mpc^−1 ≤ 0.2, treating it as constant and equivalent to the unbiased standard deviation. This approach prevents the estimation of the expectation of
the mean and the error from using the same data. Figure 1 reveals a positive correlation between the parameters, with both assuming positive values. As expected, the posterior for the β parameter
peaks close to zero, validating the effectiveness of the fixed amplitudes technique in reducing the variance of biased tracer statistics. Notably, the small value of α found in our analysis suggests
that the shot noise is well modeled as Poissonian to within 1–3%.
In Fig. 2, we present the relative difference between Eq. (19) computed with the best-fit values of α and β, and the unbiased standard deviation of the measurements, for different mass bins and
redshift values. We present the results for three values z ∊ 0.0, 1.0, 2.0 spanning the redshift range of interest, while for the masses, we present the first, the intermediate, and the last occupied
bin for each redshift. We observe that the residuals of the fit always oscillate around zero, with no statistically significant mass, redshift, or k dependence.
In Fig. 3, we present the Pearson correlation coefficient ρ between the measurements of the halo-matter cross-power spectrum on a given simulation and its paired realization. The correlation
coefficient between two random variables X and Y is defined as $ρ(X,Y)=〈(X−〈X〉)(Y−〈Y〉)〉σXσY,$(22)
where σ[X] and σ[Y] are the standard deviation of the random variables X and Y. The cross-power spectrum was measured for k ≤ 0.05 h Mpc^−1, for different ranges of halo masses (as reported in each
panel) and redshift values (different columns). We also present the correlation coefficient between simulations that assumes uncorrelated white-noise realizations for comparison. We note that paired
simulations do not show a statistically significant difference in their correlation with respect to simulations that assume uncorrelated noise realizations. The same conclusion is obtained by running
a p-value test on all mass and redshift bins. This result justifies the assumption that different simulations are, in fact, independent, even if they have the same amplitudes and paired phases. This
conclusion aligns with the claims of Villaescusa-Navarro et al. (2018) that variance suppression techniques are unlikely to affect the halo abundance distribution. As the shot-noise term in Eq. (19)
dominates over the other terms for our sample selection, one could anticipate the independence of the bias results from fixed and paired simulations due to the independence of abundance fluctuations
and modes paring.
Lastly, we use the PINOCCHIO catalogs to assess the impact of neglecting the correlation between different mass bins and Fourier modes on the calibration likelihood presented in Sect. 3.4. We
measured the bias on the PINOCCHIO catalogs by applying the same mass and modes binning we used in our calibration and measured the correlation matrix between different simulations. In line with the
results presented in Euclid Collaboration (2024b) for the two-point correlation function, we explicitly verified in our analysis that the off-diagonal terms are sub-dominant and of the order of 10%,
validating our calibration likelihood.
4.1.2 Impact of the halo finder on the peak-background split performance
In Fig. 4, the impact of the halo finder on the PBS is examined through the bias ratio measured in halo catalogs generated from the same simulation using either ROCKSTAR or SUBFIND, alongside the
corresponding PBS prescription. For the ROCKSTAR catalogs, the standard error of the mean is further illustrated using an additional 19 realizations, with the assumed cosmology being C0. The analysis
spans different redshifts within the z ∊ [0, 2] range. It is observed that the PBS tends to underestimate the simulation-derived bias by approximately 10% at higher redshifts, though it shows
improved accuracy at z = 0. Given the minimal impact of the choice of halo finder on PBS’s overall accuracy, subsequent results focus exclusively on the ROCKSTAR halo catalogs. Despite PBS’s tendency
to underpredict the HB compared to simulations, it is noteworthy that the deviation remains consistently within 5–15% across all explored values of v and redshifts. Our future efforts will aim to
refine the PBS model by addressing these discrepancies, with the objective of achieving a simulation-calibrated HB model that is precise to within a few percent.
Fig. 2
Relative difference between Eq. (19) best fit and the unbiased standard deviation of the PINOCCHIO measurements. Different columns are for different redshifts, and the corresponding mass bins are
shown in each panel. The vertical dotted line demarcates two distinct sets of measurements: Those to the left of the line were utilized as data points in the parameter fitting process, while the
scatter of the points to the right was analyzed to estimate the variance. The gray regions highlight areas within a 5% deviation from the expected values.
4.1.3 Modeling the halo bias
In Fig. 5, we present the mean of the ratio of the measured bias with respect to the PBS prescription for different simulations. In the left panel, we use all the CO runs and show the ratio of the
bias as a function of v/(1 + z) for redshifts 0, 0.29, and 1.25. The factor (1 + z) is only used to scale the results from different red-shifts of the C0 model to the same range. In the middle panel,
we show the mean ratio as a function of v for the three EdS cosmologies. Lastly, in the right panel, we present the mean ratio as a function of the background evolution Ω[m](z) for the C0, C9, and C
10 cosmologies.
The left panel of Fig. 5 illustrates that the performance of PBS is influenced by the cosmological background evolution, encapsulated by Ω[m](z), yet appears unaffected by variations in v. This
contrasts with the observations in the central panel, where PBS performance varies with both v and changes in the power spectrum’s shape. This sensitivity to v is attributed to the limitations of the
HMF calibration by Euclid Collaboration (2023) when applied to an EdS cosmology that is far from its calibration regime, introducing artificial dependency.
However, it is important to note that, although these extrapolations to EdS scenarios are beyond the initial calibration range of the HMF model, the model’s accuracy is not disproportionately
affected across different values of n[s]. Indeed, in Euclid Collaboration (2023), it has been demonstrated that the HMF model retains a consistent level of precision across various EdS cosmologies
characterized by scale-free linear power spectra. Therefore, the dependence on the shape of the power spectrum is more likely related to the varying degree of accuracy of the PBS bias model as the
shape of the power spectrum changes.
We interpret the dependence of the PBS performance as a function of the shape of the power spectrum as follows. The extrapolation of the results on the central panel of Fig. 5 indicates that the PBS
performance on EdS cosmologies with a steeper power spectrum (n[s] < −2.5) degrades with decreasing n[s]. Within the PBS framework, the mass variance σ(R[L]) smoothed on the scale of the Lagrangian
patch R[L] is assumed to be dominated by the contribution of scales R[LSS] ≫ R[L] For a power-law power spectrum, it is $d ln σ d ln R=−(n+3)2.$(23)
Therefore, the ratio σ (R[L]) / σ(R[LSS]) tends to unity as n[s] tends to −3. On the one hand, this explains why n[s] = −2.5 presents better performance than the other cases as one of the PBS
assumptions is better suited. On the other hand, for n[s] ≡ −3, perturbations on all scales reach the collapse at the same time, and it is no longer possible to distinguish between peaks and a
large-scale modulation of a background perturbation, thus breaking PBS’s fundamental assumption.
The right panel of Fig. 5 shows that the residuals with respect to the PBS prediction increase linearly with the value of the density parameter Ω[m](z). While the slope of this linear dependence is
similar for the three cosmologies, the normalization is a decreasing function of the clustering amplitude S[8]. In fact, C9 is the simulation with the lowest $S8=(Ωm,0/0.3 σ8)=0.438$ while C10 has
the highest clustering amplitude S[8] = 1.07. The C1 to C8 simulations are not shown in this panel for better readability, but they cluster around C0 as they have similar S[8] values.
The better performance of the PBS in cosmologies with more clustering suggests that the difference between this model and the simulation results is related to the connection between Lagrangian
patches in the initial density field and the collapsed structures identified by the halo-finder. Collapsed structures stand out more clearly from the non-linear density field in more evolved and
clustered cosmologies. For less clustered models, large halos are still forming and overlapping due to ongoing mergers. This makes it more challenging to identify and link them to their corresponding
Lagrangian patches clearly. Not surprisingly, in the EdS cosmologies, the PBS best performance is for n[s] = −2.5, where the evolution of the power spectrum is the steepest.
The above line of reasoning suggests then that an SO algorithm could not be accurate in providing a one-to-one mapping between Lagrangian patches, destined to form virialized halos according to
spherical collapse, and for which the PBS method predicts the bias, and halos identified in the non-linearly evolved density field. In this vein, since both ROCKSTAR and SUBFIND are based on an SO
algorithm, it is not surprising that they predict similar deviations from PBS (see Fig. 4).
On the other hand, we expect that collapsed structures have had more time to relax in cosmological models characterized by a higher value of S[8]. As a consequence, they are more likely to be
spherical. Again, this is in line with the better performance of the PBS on evolved cosmologies, as shown in the right panel of Fig. 5. Although suggestive, this interpretation of the deviations of
PBS predictions would require a dedicated analysis to track their origin in detail, which goes beyond the scope of the analysis presented here.
From the results shown in Fig. 5, it emerges that deviations from the PBS should depend on cosmic evolution, parameterized by Ω[m](z), on the slope of the linear power spectrum, and on the clustering
amplitude S[8]^6. To capture such dependencies, we adopted the following description of the correction to the PBS prediction for the linear HB: $bbPBS:=f(Ωm(ɀ),d ln σd ln R,S8) =A0f0(Ωm(ɀ))f1(d ln σd
ln R)f2(S8).$(24)
In the above expression, we assumed the following functional forms for the three above dependencies: $f0(x)=1+a1x,$(25) $f1(x)=1+b1x+b2x2,$(26) $f2(x)=1+c1x.$(27)
The parameters A[0], a[1], b[1], b[2], and c[1] are calibrated in the next section through a detailed comparison with simulations. A balance between simplicity and empirical accuracy drives the
parameterization chosen for these contributions. Specifically, we opted for a linear relationship for Ω[m](z) and S[8] while modeling the shape of the power spectrum using a quadratic function. In
order to assess the potential parameter redundancy in using an extra parameter for the shape of the power spectrum, we performed Watanabe-Akaike Information Criterion (WAIC) (Watanabe 2010) and
Pareto Smoothed Importance Sampling Leave-One-Out Cross-Validation (PSIS-LOO) (Vehtari et al. 2017) analyses comparing the model with b[2] free to vary with respect to b[2] fixed to 0. Both analyses
confirmed that the fewer degrees of freedom adopted by the surrogate model do not compensate for the decrease in the model’s predictability.
Lastly, we report that we do not observe any significant correlation between the model prediction residual and other cosmological parameters assumed in the simulations. Thus, we conclude that the
three f[i] components in Eq. (24) used in our analysis are sufficient to achieve our goal.
Fig. 3
Correlation coefficient ρ between the measurements of the halo-matter power spectrum, P[hm](k), in different simulations for k ≤ 0.05 h Mpc^−1 as a function of halo mass bin and redshift. The red
histograms show the distribution of the correlation coefficient between a simulation and its paired realization, while blue histograms are for the correlation coefficient between simulations with
uncorrelated white-noise realizations.
Fig. 4
Relative difference between the bias measured in halo catalogs and the bias predicted by the PBS model of Eq. (16) at different red-shifts in the range z ∊ [0, 2], Results refer to simulations
carried out for the C0 cosmology. In each panel, blue and red lines refer to the results obtained for the halo catalogs based on the application of ROCKSTAR and SUBFIND, respectively. For the
ROCKSTAR catalogs, we also show the standard error of the mean using other 19 realizations of the same cosmology.
4.2 Calibration of the halo bias correction
In Fig. 6, we present the marginalized two-dimensional and unidimensional constraints on the model parameters presented in Eqs. (24) to (27). The best-fit and 95% limits are reported in Table 2. We
calibrate our model using the subset of 60 PICCOLO simulations covering the cosmological parameters from C0 to C10.
In Fig. 7, we present the ratio between the mean of the observations on the 20 C0 simulations with respect to our model predictions. Different rows correspond to different redshifts, while each panel
corresponds to a different mass bin. The presented mass bins were selected as before to span from the least massive to the most massive occupied bin in that redshift. The shaded region in red
corresponds to the error on the mean, assuming that each measurement follows Eq. (19). The shaded regions in gray correspond to 2% and 4% regions. As can be seen, our model’s prediction presents a
performance below 2% for different masses and redshift regimes when not primarily dominated by the sample variance, as it happens, for instance, for k ≲ 10^−2 h Mpc^−1. This accuracy holds over the k
range up to which the onset of non-linearity occurs and the approximation of scale-independent bias breaks down, that is, k ≳ 0.05 h Mpc^−1 (marked by a vertical line).
We note that at large k values, non-linearity effects cause the bias measured from the simulations to take a scale dependence in exceeding the model prediction. As expected, this effect is smaller at
higher redshift, consistent with the effect of non-linearity shifting to larger k values.
Similarly to Fig. 7, in Fig. 8, we present the ratio between the mean of the observations on the C9 and C10 simulations with respect to our model predictions. C9 and C10 are the simulations with
lowest and highest S[8], respectively. Even for such extreme scenarios, our model performs well, thus confirming that our linear bias model, with the previously described calibration, can reproduce
results from simulations with an accuracy of a few percent for ΛCDM cosmologies.
Fig. 5
Mean of the ratio of the bias measured in simulations with respect to the PBS prescription. Left: ratio of the bias as a function of v/(1 + z) for different z, labeled by Ω[m](z), for all the CO
runs. Center, mean ratio as a function of v for the three EdS cosmologies of pure power-law shapes of the linear power spectrum at z = 0. We report the value of the three spectral indexes in the
inset. Right: mean ratio as a function of the background evolution Ω[m](z) for the C0, C9, and C10 cosmologies with varying S[8].
Fig. 6
Marginalized 68% and 95% confidence level contours on the model parameters presented in Eqs. (24) to (27). We calibrated our model using the subset of PICCOLO simulations C0–C10. (See Table 2 for
the best fit and confidence levels.)
Fig. 7
Ratio between the mean of the observations on the 20 C0 simulations with respect to our model predictions. Different rows correspond to different redshifts, while each panel corresponds to a
different mass bin. The shaded region in red corresponds to the error on the mean, assuming that each measurement follows Eq. (19). The shaded regions in gray correspond to 2% and 4% regions.
4.3 Cosmologies with massive neutrinos
We present our model’s performance when considering simulations with massive neutrinos in Fig. 9. This allowed us to assess the performance of our HB calibration for this minimal extension of ΛCDM.
In this case, the simulation’s bias has been computed by comparing it to the linear power spectrum of the corresponding cosmological model, which includes only the contributions from cold dark matter
and baryons. For consistency, the same choice of considering only CDM and baryon contribution is also made for the computation of the HMF entering in our model for the HB (see, for instance,
Castorina et al. 2014; Costanzi et al. 2013). From Fig. 9, it is evident that our model also precisely describes the bias in Λ(v)CDM models, despite the fact that such models were not used during the
HB calibration.
We note that, unlike for the pure ACDM models, in this case, the measured bias underpredicts the model bias at large k. We also note that the dependence of this effect on redshift, if any, goes in
the direction of being larger at higher z. Also, there is some hint for it to be slightly smaller for smaller values of m[v] and therefore of Ω[v]. These effects align with the expectation that such
deviations are not dominated by non-linear evolution but rather by the effect of neutrino-free streaming (Castorina et al. 2014).
Fig. 8
Similar to Fig. 7 but for the C9 and C10 cosmological parameters. Among the PICCOLO set, C9 and C10 correspond to the cosmologies with the lowest and the highest S[8], respectively.
Fig. 9
Similar to Fig. 7 but for simulations with massive neutrinos. For better plot readability, we only show the uncertainties (red shaded regions) for the simulation with total neutrino masses equal to
0.15 eV.
Fig. 10
Comparison between the HB predicted by our model with predictions from other models presented in the literature: Cole & Kaiser (1989). Sheth et al. (2001), Tinker et al. (2010), and Comparat et al.
(2017). We present both our benchmark model as well as the PBS predictions based on the HMF model of Euclid Collaboration (2023) used as a baseline of our model. Different columns correspond to
different redshifts. The relative difference with respect to our benchmark model is presented in the panels in the second row. We adopted a composite scale for the residual plot to show the dynamic
range of differences between the models: The scale is linear for values between [−10, 10] % and symmetric log outside. For reference, we show the zero line in black. The predictions of the models
from the literature have been computed using the COLOSSUS toolkit (Diemer 2018).
4.4 Comparison with previous models
In Fig. 10, we compare our model prediction with other models in the literature: Cole & Kaiser (1989), Sheth et al. (2001), Tinker et al. (2010), and Comparat et al. (2017). We present both our
benchmark model as well as the PBS predictions based on the HMF model of Euclid Collaboration (2023), which we use as a baseline of our model. Different columns correspond to different redshifts. The
relative difference to our benchmark model is presented in the panels in the second row. The predictions of the external models have been computed using the COLOSSUS toolkit (Diemer 2018)^7.
To ensure a fair comparison, we adopted the Planck-like CO cosmology, where the compared models have their peak performance among the PICCOLO cosmologies. All compared models have degraded
performance as we move from this benchmark cosmology while we have shown the robustness of our model in Figs. 7, 8, and 9. That is due to the models assuming either the universality of the bias
relation or a redshift-only dependence, while our method explicitly models the cosmology dependence of this relation. As such, comparisons between these models should be interpreted cautiously, as
the underlying cosmology influences the exact figures.
As for the comparison with the PBS prediction based on the HMF calibration by Euclid Collaboration (2023), the results shown here confirm those shown in Fig. 5: the PBS-based predictions
underestimate our simulations-based calibration by about 5–10%, almost independently of v. The models of Cole & Kaiser (1989) and Sheth et al. (2001) over- and under-estimate the bias by ~10%,
respectively. Their relatively poor performance is not surprising. The prediction by Cole & Kaiser (1989) corresponds to the PBS prediction when using the HMF from Press & Schechter (1974). As the
Press & Schechter (1974) HMF only qualitatively explains the abundance of halos, it is expected that the bias from the PBS will not perform much better. On the other hand, the Sheth et al. (2001)
model was calibrated on simulations. However, such simulations had a resolution that allowed those authors to cover a dynamic range significantly narrower than that accessible to our simulations. The
prediction by Cole & Kaiser (1989) differs from ours by an amount almost independent of y and redshift. On the other hand, the HB by Sheth et al. (2001) differs from ours in a v- and z-dependent way.
Notably, the model of Tinker et al. (2010) is only superior to the abovementioned models for low peak-height. The differences with respect to our model grow with redshift and peak-height. This could
be due to the heterogeneity of the simulations used to calibrate the model of Tinker et al. (2010) and the possible limitation of the model itself to capture the cosmological dependence of the HB. In
this paper, we calibrate to a set of simulations that have been run with the same code and setup. On the other hand, Tinker et al. (2010) based their analysis on a collection of simulations carried
out with different codes and configurations in terms of resolution and box sizes. Also, their model assumes a redshift dependency for the evolution, while from Fig. 6 we note that a parametrization
of this evolution through Ω[m](z) is more universal.
Lastly, the model by Comparat et al. (2017) shows a good agreement with our model. The most significant differences are at low-redshift where the model of Comparat et al. (2017) predicts a bias that
is smaller than ours by about 5%. This difference reduces to a sub-percent at high redshift. The agreement is unsurprising as the model of Comparat et al. (2017) was also calibrated on ROCKSTAR
Fig. 11
Percentage residuals of cluster counts (left panel) and cluster clustering covariance matrices (central and right panels), computed with the bias from Tinker et al. (2010) in comparison to the one
calculated using the bias calibrated in this study (Eq. (24)). We show the full covariance matrix for number counts in mass and redshift bins. For the two-point correlation function of galaxy
clusters, we show two blocks of the full covariance (low and high redshift bins) as a function of the radial separation.
4.5 Impact on cluster cosmology analysis
In this section, we forecast the impact of the HB calibration on cosmological analysis of cluster counts and cluster clustering from Euclid. We present the results for the bias model of Tinker et al.
(2010) and the model calibrated in this study. The rationale for assuming Tinker et al. (2010) is to use a widely used model in cluster cosmology representative of the difference in the bias models
presented in Fig. 10. Nonetheless, we do not expect the results to change significantly if we assumed the model of Comparat et al. (2017) that presents a better concordance with our model at high
redshift but worse at low redshift. Therefore, the overall impact on cosmological constraints will compensate partially as the clustering cosmological signal for Euclid peaks at lower redshifts.
As described in Sect. 3.5, we assess the effect of the HB calibration in a more realistic scenario, performing a likelihood analysis of both the individual analysis of number counts and cluster
clustering statistics and the combination of these probes. In all scenarios, the observable-mass relation (Eq. (20)) is calibrated by combining the probes with weak lensing (WL) mass estimates,
assuming a constant error of 1%. The mass calibration is the primary source of systematic uncertainty in cluster cosmology studies, and a 1% calibration represents the goal for stage IV surveys.
Therefore, the chosen setting offers a forecast of the utmost cosmological bias resulting from inaccurate modeling of the HB. Lastly, we assume three independent Gaussian likelihoods (Fumagalli et
al. 2024) for number counts, clustering, and WL masses.
In the left panel of Fig. 11, we start by presenting the percentage residuals of the number counts covariance. We show the full covariance matrix for the number counts analysis, with the mass
dependence within each redshift bin. Notably, the impact of the HB model is minimal at low redshift but becomes significant, reaching up to 20%, at higher redshifts, especially in the high-mass bins.
However, the impact of a different bias calibration is mitigated by the shot-noise contribution when the latter becomes dominant along the diagonal, as the HB only plays a role in the computation of
sample variance. To quantify the impact of such a discrepancy on parameter posteriors, we perform the likelihood analysis for a number counts experiment, as described in Sect. 3.5. From the
comparison of the two posteriors, we obtain ΔFOM = −0.67 and a perfect agreement between the positions of the two contours, meaning that the impact of the HB calibration is below other systematics.
As for the analysis of cluster counts, in the central and right panels of Fig. 11, we present the percent residuals for the clustering covariance as a function of the radial separation in both a
low-redshift (central panel) and high redshift (right panel) bin. Similar to our findings for the number counts, the most significant impact is observed on the off-diagonal elements, particularly at
high redshift. However, in contrast to number counts covariance, the inclusion of shot-noise in cluster clustering contributes to the off-diagonal elements, helping mitigate the effect of different
HB calibrations across all scales. This results in a difference that never exceeds 10%. Importantly, in the case of cluster clustering, the HB also affects the expected signal – either the two-point
correlation function or power spectrum – leading to a difference independent of the radial separation but increasing with redshift, reaching a 10% level in the high-redshift interval.
The cosmological forecasts from the clustering experiment show that the minimal variation in covariance terms produces a negligible difference in the posterior amplitude, equal to ΔFOM = 1.03.
However, comparing the two posteriors reveals an agreement at only 0.68 σ. This implies that the differences in the two-point correlation function translate into a sizeable shift in the cosmological
constraints. Notably, the difference in the posteriors induced by the different calibrations of the HB alone surpasses the 0.25 σ threshold commonly employed in other studies (Deshpande et al. 2024)
to flag systematic errors that, if exceeded, could accumulate and lead to a collectively significant difference.
The combined analysis (cluster counts + cluster clustering) posteriors are shown in Fig. 12. We notice that assuming the HB calibration by Tinker et al. (2010) still causes a shift in the posteriors
with respect to the HB calibration presented in this paper. This aligns with the forecast results for the cluster clustering analysis presented above. Although the difference is reduced to 0.39 σ,
the combination with number counts and weak lensing masses cannot compensate for the impact of the HB calibration that affects mostly the cluster clustering.
Fig. 12
Parameter posteriors at 68% and 95% confidence levels obtained by analyzing number counts, weak lensing masses, and cluster clustering computed with the HB calibrated in this work (cyan contours)
and the bias from Tinker et al. (2010) (orange contours). The error associated with weak lensing mass is set at 1% of the mass.
5 Conclusions
This paper presents a calibrated semi-analytical model for the HB in view of the joint cosmological exploitation of number counts and clustering of galaxy clusters from the Euclid survey. Our
approach began with the PBS model, based on the HMF of Euclid Collaboration (2023), and we extended it by introducing a novel parametric correction. This correction was designed to align the PBS
prediction with the results from an extended and homogenous set of N-body simulations that we carried out for vanilla ACDM models, varying cosmological parameters, and Λ(v)CDM models by varying the
sum of the neutrino masses. The simulations employed fixed and paired initial conditions (see, Angulo & Pontzen 2016), providing a robust, reduced variance framework for our analysis. We measured the
HB by examining the ratio of the matter-halo cross-spectrum. Additionally, we modeled the covariance of these measurements using 200 mock catalogs of the Euclid cluster survey, based on the
approximate LPT-based PINOCCHIO code. This ensured a thorough understanding of the uncertainties involved in our calibration of the HB. The key findings and implications of our study are summarized
in the following paragraphs.
The use of fixed and paired initial conditions for the simulations analyzed in our study proved highly advantageous for estimating the bias of tracers. By parametrizing the covariance of the bias
measurements with two parameters – one controlling the shot-noise contribution and the other for suppression due to fixing, respectively α and β in Eq. (19) – we observed significant effectiveness in
the variance suppression term. This was demonstrated in the constraints on the terms describing the variance in the halo-matter power spectrum, P[hm](k), shown in Fig. 1. Furthermore, our analysis of
the measurements of P[hm](k) between paired simulations, as illustrated in Fig. 3, revealed no significant correlation between them. This finding underscores the efficacy of the fixed and paired
simulation approach in providing reliable estimates of the bias factor characterizing the distribution of tracers (i.e., halos), which is free from the influences of inherent correlations that could
affect the results.
The impact of the choice of the halo finder used in the analysis of the N-body simulations on the performance of the PBS is shown in Fig. 4. While comparing the ROCKSTAR and SUBFIND halo finders, we
observed that the PBS generally underestimates the measured bias from simulations, particularly at higher red-shifts. However, the impact of the halo finder choice on the PBS prescription’s
performance is almost negligible, thus reinforcing the robustness of our approach in assessing the PBS performance across different redshift ranges.
Our modeling of the HB, as illustrated in Fig. 5, reveals significant insights into the cosmological dependency of the performance of the PBS. The background cosmological evolution influences the PBS
performance more than the peak height parameter v. This was particularly evident in different cosmologies, where PBS’s effectiveness varied with the shape of the power spectrum and the degree of
clustering evolution, as described by the S[8] parameter. Notably, in more clustered cosmologies, PBS improved its performance. This suggests a possible link between the ease of identifying collapsed
structures in cosmologies with more evolved clustering and their corresponding Lagrangian patches. This result led us to develop a refined model for the PBS correction, expressed in Eq. (24), which
incorporates terms depending on Ω[m](z), the local slope of the power spectrum, and S [8].
The calibration of our model parameters, with the best fit presented in Fig. 6 and Table 2, demonstrated its robust performance across a range of cosmological conditions. Figures 7 and 8 illustrate
our model prediction performance on the reference C0 simulations and on the C9 and C10 simulations. The accuracy of our model is particularly noteworthy, as it always remains below a 2% deviation for
different masses and redshift regimes, with the possible exception of unrealistic cases largely influenced by sample variance. Quite remarkably, this level of precision is maintained even in extreme
scenarios represented by the C9 and C10 simulations, which have the lowest and highest S[8] values, respectively.
The robustness of our HB calibration is further demonstrated in scenarios involving massive neutrinos, as showcased in Fig. 9. Despite not incorporating massive neutrino simulations during the
calibration phase, our model accurately predicts the HB in these cosmologies. Neutrinos are treated according to the model presented by Castorina et al. (2014, see also Costanzi et al. 2013) and the
measurements of the bias with respect to the matter power spectrum of cold dark matter and baryons, as was done for the HMF in simulations with massive neutrinos in Euclid Collaboration (2023). The
ability of our model to adapt and perform reliably in such scenarios without the need for recalibration highlights its robustness and versatility.
As for the comparison with HB models already introduced in the literature (see Figure 10), the models by Cole & Kaiser (1989) and Sheth et al. (2001) show significant deviations from our results,
likely due to their calibration in simulations covering narrower dynamic ranges and a variety of cosmological models. The HB model by Tinker et al. (2010) shows increasing discrepancies with our
model at higher redshifts and peak heights. Such differences could be attributed to its calibration on a heterogeneous set of simulations and inadequately accounting for the cosmological dependence
of the HB. In contrast, the model by Comparat et al. (2017) aligns more closely with our findings, particularly at higher redshifts. This agreement is expected, as their model was also calibrated
using ROCKSTAR catalogs.
As for the impact of changing the calibration of the HB on the derived cosmological posteriors, we showed in Fig. 11 the differences of the covariance matrices for a Euclid cluster count and cluster
clustering analysis using both our calibration and the one provided by Tinker et al. (2010). While the impact on number counts covariance is minimal at low redshifts, it becomes substantial, up to
20%, at higher redshifts. However, the presence of shot-noise in the analysis helps mitigate this effect. In cluster clustering, we observed that the HB calibration could lead to differences in the
two-point correlation function, particularly at high redshifts. This difference can potentially bias cosmological constraints beyond the 0.25 σ threshold commonly used to flag significant systematic
errors (Adamek et al. 2023). Moreover, the combined analysis of number counts, cluster clustering, and weak lensing masses demonstrates that even with these additional data, the calibration of HB
cannot be entirely compensated for. This highlights the importance of precise HB calibration in cluster cosmology, especially for a survey, such as the one being provided by Euclid, which will reach
an unprecedented sensitivity and level of statistics.
In summary, the analysis presented in this paper has systematically calibrated and tested the HB for a range of cosmological scenarios, demonstrating its critical impact on the precision of
cosmological analysis based on galaxy clusters for the Euclid mission. The resilience of our HB model against variations of cosmological models, including the presence of massive neutrinos and
different degrees of clustering amplitude, highlights its robustness and adaptability. Importantly, our model is robust against the halo finder definition, inheriting its dependence through the HMF
only. This is a remarkable feature, as the correspondence between halos in N-body simulations and real clusters in surveys remains a complex issue, with uncertainties in halo identification and
characterization potentially influencing the extraction of cosmological parameters. Future research should focus on understanding and quantifying these uncertainties, especially concerning
observational challenges, such as projection effects and the mass-observable relation. As we move forward, extending this precision to departures from the standard Λ(v)CDM framework will be crucial
to fully harnessing the capabilities of next-generation cosmological surveys.
Data availability
In Castro & Fumagalli (2024), we implement the model presented in this paper, together with the models for the HMF presented in Euclid Collaboration (2023) and for the impact of baryonic feedback on
cluster masses presented in Euclid Collaboration (2024a). The source code can be accessed in https://github.com/TiagoBsCastro/CCToolkit
It is a pleasure to thank Valerio Marra for constructive comments during the production of this work, Fabio Pitari and Caterina Caravita for support with the CINECA environment, Peter Berhoozi for
the support with ROCKSTAR, Oliver Hahn for the support with monofonIC, and Luca Di Mascolo for the support with PyMC. TC is supported by the Agenzia Spaziale Italiana (ASI) under - Euclid-FASE D
Attivita’ scientifica per la missione – Accordo attuativo ASI-INAF no. 2018-23-HH.0. SB, TC, PM, and AS are supported by the PRIN 2022 PNRR project “Space-based cosmology with Euclid: the role of
High-Performance Computing” (code no. P202259YAF), by the Italian Research Center on High-Performance Computing Big Data and Quantum Computing (ICSC), project funded by European Union –
NextGenerationEU – and National Recovery and Resilience Plan (NRRP) – Mission 4 Component 2, within the activities of Spoke 3, Astrophysics and Cosmos Observations, and by the INFN INDARK PD51 grant.
TC and AS are also supported by the FARE MIUR grant ‘ClustersXEuclid’ R165SBKTMA. AS is also supported by the ERC ‘ClustersXCosmo’ grant agreement 716762. MC and TC are supported by the PRIN 2022
project EMC2 – Euclid Mission Cluster Cosmology: unlock the full cosmological utility of the Euclid photometric cluster catalog (code no. J53D23001620006). KD acknowledges support by the DFG
(EXC-2094 – 390783311) as well as support through the COMPLEX project from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program grant agreement
ERC-2019-AdG 882679. We acknowledge the computing centre of CINECA and INAF, under the coordination of the “Accordo Quadro (MoU) per lo svolgimento di attività congiunta di ricerca Nuove frontiere in
Astrofisica: HPC e Data Exploration di nuova generazione”, for the availability of computing resources and support. We acknowledge the use of the HOTCAT computing infrastructure of the Astronomical
Observatory of Trieste – National Institute for Astrophysics (INAF, Italy) (see Bertocco et al. 2020; Taffoni et al. 2020). The Euclid Consortium acknowledges the European Space Agency and a number
of agencies and institutes that have supported the development of Euclid, in particular the Agenzia Spaziale Italiana, the Austrian Forschungsförderungsgesellschaft funded through BMK, the Belgian
Science Policy, the Canadian Euclid Consortium, the Deutsches Zentrum für Luft- und Raumfahrt, the DTU Space and the Niels Bohr Institute in Denmark, the French Centre National d’Etudes Spatiales,
the Fundação para a Ciência e a Tecnologia, the Hungarian Academy of Sciences, the Ministerio de Ciencia, Innovación y Universidades, the National Aeronautics and Space Administration, the National
Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Research Council of Finland, the Romanian Space Agency, the State Secretariat for
Education, Research, and Innovation (SERI) at the Swiss Space Office (SSO), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site (www.euclid-ec.org).
All Tables
Table 1
Cosmological parameters of the PICCOLO set of simulations.
All Figures
Fig. 1
Constraints on the parameters in Eq. (19) fitted to the unbiased standard deviation of the 200 PINOCCHIO mocks with C0 cosmology. We fitted α and β considering k ≤ 0.05 h Mpc^−1, assuming a
Gaussian likelihood with error bar estimated from the measurements for 0.05 ≤ k/h Mpc^−1 ≤ 0.2 and assuming it to be constant in k and equivalent to the unbiased standard deviation.
In the text
Fig. 2
Relative difference between Eq. (19) best fit and the unbiased standard deviation of the PINOCCHIO measurements. Different columns are for different redshifts, and the corresponding mass bins are
shown in each panel. The vertical dotted line demarcates two distinct sets of measurements: Those to the left of the line were utilized as data points in the parameter fitting process, while the
scatter of the points to the right was analyzed to estimate the variance. The gray regions highlight areas within a 5% deviation from the expected values.
In the text
Fig. 3
Correlation coefficient ρ between the measurements of the halo-matter power spectrum, P[hm](k), in different simulations for k ≤ 0.05 h Mpc^−1 as a function of halo mass bin and redshift. The red
histograms show the distribution of the correlation coefficient between a simulation and its paired realization, while blue histograms are for the correlation coefficient between simulations with
uncorrelated white-noise realizations.
In the text
Fig. 4
Relative difference between the bias measured in halo catalogs and the bias predicted by the PBS model of Eq. (16) at different red-shifts in the range z ∊ [0, 2], Results refer to simulations
carried out for the C0 cosmology. In each panel, blue and red lines refer to the results obtained for the halo catalogs based on the application of ROCKSTAR and SUBFIND, respectively. For the
ROCKSTAR catalogs, we also show the standard error of the mean using other 19 realizations of the same cosmology.
In the text
Fig. 5
Mean of the ratio of the bias measured in simulations with respect to the PBS prescription. Left: ratio of the bias as a function of v/(1 + z) for different z, labeled by Ω[m](z), for all the CO
runs. Center, mean ratio as a function of v for the three EdS cosmologies of pure power-law shapes of the linear power spectrum at z = 0. We report the value of the three spectral indexes in the
inset. Right: mean ratio as a function of the background evolution Ω[m](z) for the C0, C9, and C10 cosmologies with varying S[8].
In the text
Fig. 6
Marginalized 68% and 95% confidence level contours on the model parameters presented in Eqs. (24) to (27). We calibrated our model using the subset of PICCOLO simulations C0–C10. (See Table 2 for
the best fit and confidence levels.)
In the text
Fig. 7
Ratio between the mean of the observations on the 20 C0 simulations with respect to our model predictions. Different rows correspond to different redshifts, while each panel corresponds to a
different mass bin. The shaded region in red corresponds to the error on the mean, assuming that each measurement follows Eq. (19). The shaded regions in gray correspond to 2% and 4% regions.
In the text
Fig. 8
Similar to Fig. 7 but for the C9 and C10 cosmological parameters. Among the PICCOLO set, C9 and C10 correspond to the cosmologies with the lowest and the highest S[8], respectively.
In the text
Fig. 9
Similar to Fig. 7 but for simulations with massive neutrinos. For better plot readability, we only show the uncertainties (red shaded regions) for the simulation with total neutrino masses equal to
0.15 eV.
In the text
Fig. 10
Comparison between the HB predicted by our model with predictions from other models presented in the literature: Cole & Kaiser (1989). Sheth et al. (2001), Tinker et al. (2010), and Comparat et al.
(2017). We present both our benchmark model as well as the PBS predictions based on the HMF model of Euclid Collaboration (2023) used as a baseline of our model. Different columns correspond to
different redshifts. The relative difference with respect to our benchmark model is presented in the panels in the second row. We adopted a composite scale for the residual plot to show the dynamic
range of differences between the models: The scale is linear for values between [−10, 10] % and symmetric log outside. For reference, we show the zero line in black. The predictions of the models
from the literature have been computed using the COLOSSUS toolkit (Diemer 2018).
In the text
Fig. 11
Percentage residuals of cluster counts (left panel) and cluster clustering covariance matrices (central and right panels), computed with the bias from Tinker et al. (2010) in comparison to the one
calculated using the bias calibrated in this study (Eq. (24)). We show the full covariance matrix for number counts in mass and redshift bins. For the two-point correlation function of galaxy
clusters, we show two blocks of the full covariance (low and high redshift bins) as a function of the radial separation.
In the text
Fig. 12
Parameter posteriors at 68% and 95% confidence levels obtained by analyzing number counts, weak lensing masses, and cluster clustering computed with the HB calibrated in this work (cyan contours)
and the bias from Tinker et al. (2010) (orange contours). The error associated with weak lensing mass is set at 1% of the mass.
In the text
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Dec. 13 - Is time travel allowed?
Dear Friends,
Click the link if you don't receive the images/can't access the links.
Love and Light.
Is time travel allowed?
In our fifth online poll to find out what Plus readers would most like to know about the Universe you told us that you'd like to find out if time travel is allowed. We took the question to Kip Thorne
, Feynmann Professor of Theoretical Physics, Emeritus, at the California Institute of Technology, and here is his answer.
Kip Thorne
In brief: The laws of physics allow members of an exceedingly advanced civilisation to travel forward in time as fast as they might wish. Backward time travel is another matter; we do not know
whether it is allowed by the laws of physics, and the answer is likely controlled by a set of physical laws that we do not yet understand at all well: the laws of quantum gravity. In order for humans
to travel forward in time very rapidly, or backward (if allowed at all), we would need technology far far beyond anything we are capable of today.
Travelling forward in time rapidly
Albert Einstein's relativistic laws of physics tell us that time is "personal". If you and I move differently or are at different locations in a gravitational field, then the rate of flow of time
that you experience (the rate that governs the ticking of any very good clock you carry with you and that governs the aging of your body) is different from the rate of time flow that I experience.
(Einstein used the phrase "time is relative"; I prefer "time is personal".)
This personal character of time allows one person to travel forward in time much faster than another, a phenomenon embodied in the so-called twins paradox. One twin (call him Methuselah) stays at
home on Earth; the other (Florence) travels out into the Universe at high speed and then returns. When they meet at the end of the trip, Florence will have aged far less than Methuselah; for example,
Florence may have aged 30 years and Methuselah 4,500 years. (The twin that ages least is the one who undergoes huge accelerations, to get up to high speed, slow down, reverse direction, then
accelerate back and slow to a halt on Earth. The twin who leads the sedate life ages the most.)
A massive black hole is another vehicle for rapid forward time travel: If Methuselah remains in orbit high above the event horizon of a massive black hole (say, one whose gravitational pull is that
of a billion suns) and Florence travels down to near the event horizon and hovers just above it for, say, 30 years and then returns, Methuselah can have aged thousands or millions of years. This is
because time flows much more slowly near a black hole's event horizon (where the acceleration of gravity is huge) than far above it (where one can live sedately).
These time travel phenomena have been tested in the laboratory. Muons — short-lived elementary particles — travelling around and around in a storage ring at 0.9994 of the speed of light, at the
Brookhaven National Laboratory on Long Island, New York, have been seen to age 29 times more slowly than muons at rest in the laboratory. And atomic clocks on the surface of the Earth have been seen
to run more slowly than atomic clocks high above the Earth's surface — more slowly by about 4 parts in 10 billion.
Travelling backward in time: chronology protection
We physicists have been working hard since the late 1980s to understand whether the laws of physics allow backward time travel. We do not have a definitive answer yet, but the likely answer has been
summarised by Stephen Hawking, in his Chronology Protection Conjecture (see [1]): The laws of physics always conspire to prevent anything from travelling backward in time, thereby keeping the
Universe safe for historians.
We physicists have identified two mechanisms that might protect chronology: (1) The exotic material that is required in the manufacture of any time machine might be forbidden to exist, by the laws of
physics — forbidden to exist in the large amounts that time machines always require. (2) Time machines might always self-destruct, explosively, when one tries to activate them.
These mechanisms (1) and (2) are descriptive translations of mathematical results that we physicists have derived using the laws of physics expressed in their own natural language: mathematics. The
sentences (1) and (2) capture the essence of our calculations, but crucial details are lost in translation. For anyone who wishes to struggle to understand those details, good places to start are a
recent beautiful but highly technical review article by John Friedman (see [2]), and a much less technical but older and slightly outdated article by Matt Visser (see [3]).
I shall illustrate these chronology-protection mechanisms by an example of a time machine that my students Mike Morris and Ulvi Yurtsever and I invented and explored mathematically in 1989: a time
machine based on wormholes. (This is just one of many time-machine designs that have been studied. For others see Visser's review, [3])
Me crawling through a wormholeBlack Holes and Time Warps, [4], where you can find a more detailed description of this time machine.)
whose length is only a few centimetres but circumference is about that of my belly. (From my book
A wormhole-based time machine: A wormhole is a hypothetical tunnel through hyperspace that links one place in our Universe (e.g. my office at Caltech) to another place (e.g. the Caltech football
field). Each end of the wormhole (each mouth) looks like a crystal ball. Staring into it, one sees a distorted image of objects at the other end. Looking into the mouth in my office, I see the
football field, distorted; someone on the football field, looking into the mouth there, sees me and my office, distorted. The wormhole (tunnel) might be only 3 metres long, so if I enter the mouth in
my office and then travel just 3 metres through the tunnel, I emerge from the other mouth, onto the football field 300 metres from my office.
A wormhole as viewed from a higher-dimensional hyperspace. Our Universe is the two-dimensional sheet. The wormhole is a short cut through hyperspace from one location in our sheet (our Universe) to
Suppose, now, that a creature from an extremely advanced civilisation carries the football-field mouth out into the Universe on a "twins paradox" trip. When that mouth returns, it may have aged by
only one second while the mouth in my office aged by one day. The wormhole has become a time machine: If I enter one mouth and travel through it for only a few seconds, I emerge from the other mouth
one day in the future. Travelling through it in the other direction, I emerge one day in the past! (See [4].)
Exotic Matter and Vacuum Fluctuations: We do not know whether the laws of physics permit wormholes. We do know, however, that a wormhole will implode so quickly that nothing can traverse it, unless
it is held open by gravitationally repulsive forces that can only be produced by exotic matter. By the phrase "exotic matter" I mean matter that has negative energy and therefore anti-gravitates,
i.e. repels.
The quantum laws of physics do permit exotic matter to exist, and it has been created in the laboratory in very tiny amounts: in the so-called Casimir vacuum between two electrically conducting
plates, and in the so-called squeezed vacuum that is generated by optical physicists using nonlinear crystals.
The key to this negative energy is the fact that empty space (the vacuum) is filled with tiny fluctuations of all kinds of matter and fields that exist in the Universe. It is impossible to make these
fluctuations go away. They are a consequence of the quantum mechanical uncertainty principle: if, at one moment of time, there are no fluctuations at all of (for example) the electromagnetic field,
then the rate of change of the fluctuations must be infinitely large and a moment later the fluctuations will be enormous. The product of the strength of the fluctuations and the magnitude of their
rate of change is always bigger than a certain limit, given by the uncertainty principle. As a result, fluctuations are always present. We call them vacuum fluctuations because they are a property of
the vacuum, i.e. of otherwise empty space.
The Casimir Vacuum: When two electrically conducting plates are placed very close together, the vacuum fluctuational electric field parallel to the plates is strongly suppressed while that
perpendicular to the plates is little affected. The suppression reduces the fluctuational energy between the plates below what it would be in plate-free empty space, so the vacuum between the plates
(the Casimir vacuum) acquires negative gravitating energy. It has loaned some fluctuational energy to the electric fields inside the plates. The plates have catalysed this lending.
Wormhole held open by Casimir Vacuum: Two concentric spherical conducting plates are placed at the throat of a wormhole, with a tiny separation. (One dimension is suppressed here, so our Universe is
a 2-dimensional sheet and the plates look like circles rather than spheres.) If the plates' energies (including their mass's energy E=mc^2) are small enough, then the repulsive gravity due to the
Casimir vacuum between the plates can hold the wormhole open.
The laws of quantum physics say that vacuum fluctuations produce no gravity — or perhaps only an exceedingly tiny amount of gravity: the gravity that is accelerating the expansion of the Universe. In
other words vacuum fluctuations may be responsible for the so-called cosmological dark energy. But that dark energy is so tiny (10^-121 in dimensionless numbers) that it is irrelevant for my
discussion of time machines; so I shall say that the quantum fluctuations produce no gravity at all.
Or, rather, they produce no gravity under normal circumstances. One can devise ways, in fact, to make one region of empty space lend some of its vacuum fluctuations to an adjacent region. (This is
what experimental physicists do with the Casimir vacuum and with the squeezed vacuum.) When this happens, the lending region is left with a negative amount of gravitating energy, and the borrowing
region gets positive gravitating energy. The quantum laws place tight constraints on the amount of fluctuational energy that can be loaned. The larger the size of the lending region, the less energy
it can loan and therefore the less negative its energy can become. This is true in the Casimir vacuum, in the squeezed vacuum, and in all other variants of exotic matter. The laws of physics dictate
These constraints on the amount of negative gravitating energy might be severe enough to prevent one from ever accumulating enough of it to prevent a wormhole from imploding (see [5]). The reason is
that regions of space which do the borrowing and the devices which catalyse the borrowing might always have so much positive energy of their own, that their attractive gravity counteracts the
negative energy's repulsive gravity, and triggers all wormholes to implode. If that is the case, then wormhole-based time machines are forbidden: one can never travel through a wormhole before it
implodes. (John Friedman and colleagues have called this topological censorship, see [2].)
My personal guess is that these constraints on exotic matter do not prevent wormholes from being held open and thus do not protect chronology, but I could well turn out to be wrong. To learn the
truth, we physicists must develop a deeper understanding of quantum theory in warped spacetime than we now have — i.e. a deeper understanding of the combined laws of quantum theory and general
relativity, the laws of quantum gravity.
Time machine self destruction: If it turns out that wormholes can be held open, then doing so is not enough to guarantee that an ultra-advanced civilisation can convert a wormhole into a time machine
via a twins-paradox trip (carrying one mouth out into the Universe at high speed and then back). There is a second obstacle that must be surmounted — time machine self destruction:
Time machine self destruction: As the right wormhole flies back toward the left, at the end of its twins-paradox trip, vacuum fluctuations flow through the wormhole then out through the space between
them, returning to their starting point at the moment they left. Their gravitating energy grows extremely large, and perhaps destroys the wormhole at the moment it becomes a time machine. (Figure
adapted from my book Black Holes and Time Warps, [4].)
As the travelling mouth is returning to Earth, there comes a first moment when its wormhole can be used to travel backward in time. The first thing that can do so, and thereby meet itself before it
left, is an entity that enters one mouth, exits from the other before it entered, and then flies through the Universe back to its starting point at the highest possible speed, the speed of light —
arriving back at the first mouth at precisely the moment it started its trip. Even if no light or other light-speed radiation travels on this round-trip time-travel route, vacuum fluctuations will
always do so. They cannot be stopped. Upon arriving back at their starting point at the very moment when they left, the vacuum fluctuations will pile up on top of their younger selves. The result is
a duplicate of every fluctuation, and then, with another round trip, a quadrupling of every fluctuation, and so forth. The bottom line, according to a calculation that I did with my postdoc Sung-Won
Kim in 1990, is an explosive flow of gravitating fluctuational energy through the wormhole at precisely the moment when time travel is first possible — at the moment of time machine activation [4].
Will this explosive fluctuational energy destroy the wormhole and thence the time machine? At first Kim and I thought the wormhole could survive. However Stephen Hawking gave strong arguments to the
contrary, in his seminal 1991 research paper on chronology protection. The explosion is very likely to destroy the time machine when it is first activated, Hawking argued — and not just this time
machine, but any time machine that even the most advanced civilisation might conceive and build. Over the next few years many other physicists weighed in, with analyses of other time machine designs,
and it began to look like Hawking might be wrong: a sufficiently clever design might protect a time machine from self destruction. Then in 1996 Bernard Kay, Marek Radzikowski and Robert Wald
developed a powerful mathematical proof that the version of the laws of quantum physics which we were all using to analyse time machine self destruction are incapable of revealing the explosion's
outcome. The outcome is held tightly in the grip of the laws of quantum gravity, which we do not yet understand fully.
The fate of any time machine?
Hawking and I have a long history of bets with each other, about unsolved mysteries in physics. But we are not making a bet on this one, since for once we are on the same side. When we physicists
have mastered the laws of quantum gravity (Hawking and I agree), we will very likely discover that chronology is protected: the explosion always does destroy any time machine, when it is first
In June 2000, on the occasion of my 60th birthday, Hawking presented me with a tentative analysis of the explosion's outcome, using his own tentative version of the laws of quantum gravity. His
conclusion: if I try to use a very advanced civilisation's wormhole to travel backward in time, the quantum mechanical probability that I will succeed is one part in 10^60; see Hawking's article in
my birthday party book, [6]. That's an awfully small probability of surviving the explosion. Given the opportunity to try, I would not take the risk.
Other time machines: It is amazing what we can learn from the laws of physics, when we understand them well. One famous example is the laws' absolutely firm insistence that it is impossible to
construct a perpetual motion machine, even if one has all the tools of an exceedingly advanced civilisation. Another example is a proof by Hawking that to make a time machine, no matter how one goes
about it, one must use exotic matter — matter with negative energy — as an integral part of the device; wormholes illustrate this, but it is true in general. And a third example is the proof by Kay,
Radzikovsky and Wald that the laws of physics as we now know them will break down whenever a time machine is activated, no matter how one designs the machine. Again wormholes are just one example.
Hawking's theorem, and that of Kay, Radzikovsky and Wald, tell us that the fates of all time machines are held tightly in the grip of the laws of quantum gravity.
Progress in the quest to understand quantum gravity has been substantial over the past two decades. Complete success will come, I am convinced, within the next two decades or so — and it will bring
not only a clear understanding of whether backward time travel is possible, but also an understanding of many other mysteries, including how our Universe was born (see the Plus article What happened
before the Big Bang?).
No comments:
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13.6: Kepler's Laws of Planetary Motion
Last updated
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widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
• Describe the conic sections and how they relate to orbital motion
• Describe how orbital velocity is related to conservation of angular momentum
• Determine the period of an elliptical orbit from its major axis
Using the precise data collected by Tycho Brahe, Johannes Kepler carefully analyzed the positions in the sky of all the known planets and the Moon, plotting their positions at regular intervals of
time. From this analysis, he formulated three laws, which we address in this section.
Kepler’s First Law
The prevailing view during the time of Kepler was that all planetary orbits were circular. The data for Mars presented the greatest challenge to this view and that eventually encouraged Kepler to
give up the popular idea. Kepler’s first law states that every planet moves along an ellipse, with the Sun located at a focus of the ellipse. An ellipse is defined as the set of all points such that
the sum of the distance from each point to two foci is a constant. Figure \(\PageIndex{1}\) shows an ellipse and describes a simple way to create it.
Figure \(\PageIndex{1}\): (a) An ellipse is a curve in which the sum of the distances from a point on the curve to two foci (f[1] and f[2]) is a constant. From this definition, you can see that an
ellipse can be created in the following way. Place a pin at each focus, then place a loop of string around a pencil and the pins. Keeping the string taught, move the pencil around in a complete
circuit. If the two foci occupy the same place, the result is a circle—a special case of an ellipse. (b) For an elliptical orbit, if m << M , then m follows an elliptical path with M at one focus.
More exactly, both m and M move in their own ellipse about the common center of mass.
For elliptical orbits, the point of closest approach of a planet to the Sun is called the perihelion. It is labeled point A in Figure \(\PageIndex{1}\). The farthest point is the aphelion and is
labeled point B in the figure. For the Moon’s orbit about Earth, those points are called the perigee and apogee, respectively.
An ellipse has several mathematical forms, but all are a specific case of the more general equation for conic sections. There are four different conic sections, all given by the equation
\[\frac{\alpha}{r} = 1 + e \cos \theta \ldotp \label{13.10}\]
The variables \(r\) and \(\theta\) are shown in Figure \(\PageIndex{2}\) in the case of an ellipse. The constants α and e are determined by the total energy and angular momentum of the satellite at a
given point. The constant e is called the eccentricity. The values of \(\alpha\) and e determine which of the four conic sections represents the path of the satellite.
Figure \(\PageIndex{2}\): As before, the distance between the planet and the Sun is \(r\), and the angle measured from the x-axis, which is along the major axis of the ellipse, is \(\theta\).
One of the real triumphs of Newton’s law of universal gravitation, with the force proportional to the inverse of the distance squared, is that when it is combined with his second law, the solution
for the path of any satellite is a conic section. Every path taken by m is one of the four conic sections: a circle or an ellipse for bound or closed orbits, or a parabola or hyperbola for unbounded
or open orbits. These conic sections are shown in Figure \(\PageIndex{3}\).
Figure \(\PageIndex{3}\): All motion caused by an inverse square force is one of the four conic sections and is determined by the energy and direction of the moving body
If the total energy is negative, then 0 ≤ e < 1, and Equation \ref{13.10} represents a bound or closed orbit of either an ellipse or a circle, where e = 0. [You can see from Equation 13.10 that for e
= 0, r = \(\alpha\), and hence the radius is constant.] For ellipses, the eccentricity is related to how oblong the ellipse appears. A circle has zero eccentricity, whereas a very long, drawn-out
ellipse has an eccentricity near one.
If the total energy is exactly zero, then e = 1 and the path is a parabola. Recall that a satellite with zero total energy has exactly the escape velocity. (The parabola is formed only by slicing the
cone parallel to the tangent line along the surface.) Finally, if the total energy is positive, then e > 1 and the path is a hyperbola. These last two paths represent unbounded orbits, where m passes
by M once and only once. This situation has been observed for several comets that approach the Sun and then travel away, never to return.
We have confined ourselves to the case in which the smaller mass (planet) orbits a much larger, and hence stationary, mass (Sun), but Equation 13.10 also applies to any two gravitationally
interacting masses. Each mass traces out the exact sameshaped conic section as the other. That shape is determined by the total energy and angular momentum of the system, with the center of mass of
the system located at the focus. The ratio of the dimensions of the two paths is the inverse of the ratio of their masses.
You can see an animation of two interacting objects at the My Solar System page at PhET. Choose the Sun and Planet preset option. You can also view the more complicated multiple body problems as
well. You may find the actual path of the Moon quite surprising, yet is obeying Newton’s simple laws of motion.
Orbital Transfers
People have imagined traveling to the other planets of our solar system since they were discovered. But how can we best do this? The most efficient method was discovered in 1925 by Walter Hohmann,
inspired by a popular science fiction novel of that time. The method is now called a Hohmann transfer. For the case of traveling between two circular orbits, the transfer is along a “transfer”
ellipse that perfectly intercepts those orbits at the aphelion and perihelion of the ellipse. Figure \(\PageIndex{4}\) shows the case for a trip from Earth’s orbit to that of Mars. As before, the Sun
is at the focus of the ellipse.
For any ellipse, the semi-major axis is defined as one-half the sum of the perihelion and the aphelion. In Figure \(\PageIndex{4}\), the semi-major axis is the distance from the origin to either side
of the ellipse along the x-axis, or just one-half the longest axis (called the major axis). Hence, to travel from one circular orbit of radius r[1] to another circular orbit of radius r[2], the
aphelion of the transfer ellipse will be equal to the value of the larger orbit, while the perihelion will be the smaller orbit. The semi-major axis, denoted a, is therefore given by \(a = \frac{1}
{2} (r_{1} + r_{2})\).
Figure \(\PageIndex{4}\): The transfer ellipse has its perihelion at Earth’s orbit and aphelion at Mars’ orbit.
Let’s take the case of traveling from Earth to Mars. For the moment, we ignore the planets and assume we are alone in Earth’s orbit and wish to move to Mars’ orbit. From Equation 13.9, the expression
for total energy, we can see that the total energy for a spacecraft in the larger orbit (Mars) is greater (less negative) than that for the smaller orbit (Earth). To move onto the transfer ellipse
from Earth’s orbit, we will need to increase our kinetic energy, that is, we need a velocity boost. The most efficient method is a very quick acceleration along the circular orbital path, which is
also along the path of the ellipse at that point. (In fact, the acceleration should be instantaneous, such that the circular and elliptical orbits are congruent during the acceleration. In practice,
the finite acceleration is short enough that the difference is not a significant consideration.) Once you have arrived at Mars orbit, you will need another velocity boost to move into that orbit, or
you will stay on the elliptical orbit and simply fall back to perihelion where you started. For the return trip, you simply reverse the process with a retro-boost at each transfer point.
To make the move onto the transfer ellipse and then off again, we need to know each circular orbit velocity and the transfer orbit velocities at perihelion and aphelion. The velocity boost required
is simply the difference between the circular orbit velocity and the elliptical orbit velocity at each point. We can find the circular orbital velocities from Equation 13.7. To determine the
velocities for the ellipse, we state without proof (as it is beyond the scope of this course) that total energy for an elliptical orbit is
\[E = - \frac{GmM_{S}}{2a}\]
where M[S] is the mass of the Sun and a is the semi-major axis. Remarkably, this is the same as Equation 13.9 for circular orbits, but with the value of the semi-major axis replacing the orbital
radius. Since we know the potential energy from Equation 13.4, we can find the kinetic energy and hence the velocity needed for each point on the ellipse. We leave it as a challenge problem to find
those transfer velocities for an Earth-to-Mars trip.
We end this discussion by pointing out a few important details. First, we have not accounted for the gravitational potential energy due to Earth and Mars, or the mechanics of landing on Mars. In
practice, that must be part of the calculations. Second, timing is everything. You do not want to arrive at the orbit of Mars to find out it isn’t there. We must leave Earth at precisely the correct
time such that Mars will be at the aphelion of our transfer ellipse just as we arrive. That opportunity comes about every 2 years. And returning requires correct timing as well. The total trip would
take just under 3 years! There are other options that provide for a faster transit, including a gravity assist flyby of Venus. But these other options come with an additional cost in energy and
danger to the astronauts.
Kepler's Second Law
Kepler’s second law states that a planet sweeps out equal areas in equal times, that is, the area divided by time, called the areal velocity, is constant. Consider Figure \(\PageIndex{5}\). The time
it takes a planet to move from position A to B, sweeping out area A[1], is exactly the time taken to move from position C to D, sweeping area A[2], and to move from E to F, sweeping out area A[3].
These areas are the same: A[1] = A[2] = A[3].
Figure \(\PageIndex{5}\): The shaded regions shown have equal areas and represent the same time interval.
Comparing the areas in the figure and the distance traveled along the ellipse in each case, we can see that in order for the areas to be equal, the planet must speed up as it gets closer to the Sun
and slow down as it moves away. This behavior is completely consistent with our conservation equation, Equation \ref{13.5}. But we will show that Kepler’s second law is actually a consequence of the
conservation of angular momentum, which holds for any system with only radial forces.
Recall the definition of angular momentum from Angular Momentum, \(\vec{L} = \vec{r} \times \vec{p}\). For the case of orbiting motion, \(\vec{L}\) is the angular momentum of the planet about the
Sun, \(\vec{r}\) is the position vector of the planet measured from the Sun, and \(\vec{p}\) = m\(\vec{v}\) is the instantaneous linear momentum at any point in the orbit. Since the planet moves
along the ellipse, \(\vec{p}\) is always tangent to the ellipse.
We can resolve the linear momentum into two components: a radial component \(\vec{p}_{rad}\) along the line to the Sun, and a component \(\vec{p}_{perp}\) perpendicular to \(\vec{r}\). The cross
product for angular momentum can then be written as
\[\begin{align*} \vec{L} &= \vec{r} \times \vec{p} \\[4pt] &= \vec{r} \times (\vec{p}_{rad} + \vec{p}_{perp}) \\[4pt] &= \vec{r} \times \vec{p}_{rad} + \vec{r} \times \vec{p}_{perp} \ldotp \end
The first term on the right is zero because \(\vec{r}\) is parallel to \(\vec{p}_{rad}\), and in the second term \(\vec{r}\) is perpendicular to \(\vec{p}_{perp}\), so the magnitude of the cross
product reduces to
\[L = rp_{perp} = rmv_{perp}.\]
Note that the angular momentum does not depend upon \(p_{rad}\). Since the gravitational force is only in the radial direction, it can change only \(p_{rad}\) and not \(p_{perp}\); hence, the angular
momentum must remain constant.
Figure \(\PageIndex{6}\): The element of area \(\Delta\)A swept out in time \(\Delta\)t as the planet moves through angle \(\Delta \phi\). The angle between the radial direction and \(\vec{v}\) is \
Now consider Figure \(\PageIndex{6}\). A small triangular area \(\Delta A\) is swept out in time \(\Delta t\). The velocity is along the path and it makes an angle \(\theta\) with the radial
direction. Hence, the perpendicular velocity is given by \(v_{perp}= v\sin \theta\). The planet moves a distance \(\Delta\)s = v\(\Delta\)tsin\(\theta\) projected along the direction perpendicular to
\(r\). Since the area of a triangle is one-half the base (\(r\)) times the height (\(\Delta s\)), for a small displacement, the area is given by
\[\Delta A = \frac{1}{2} r \Delta s. \nonumber\]
Substituting for \(\Delta s\), multiplying by \(m\) in the numerator and denominator, and rearranging, we obtain
\[\Delta A = \frac{1}{2} r \Delta s = \frac{1}{2} r (v \Delta t \sin \theta) = \frac{1}{2m} r (mv \sin \theta \Delta t) = \frac{1}{2m} r (mv_{perp} \Delta t) = \frac{L}{2m} \Delta t \ldotp\]
The areal velocity is simply the rate of change of area with time, so we have
\[ \text{areal velocity} = \frac{\Delta A}{\Delta t} = \frac{L}{2m} \ldotp\]
Since the angular momentum is constant, the areal velocity must also be constant. This is exactly Kepler’s second law. As with Kepler’s first law, Newton showed it was a natural consequence of his
law of gravitation.
Kepler's Third Law
Kepler’s third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit. In Satellite Orbits and Energy, we derived Kepler’s third law for the special
case of a circular orbit. Equation \ref{13.8} gives us the period of a circular orbit of radius r about Earth:
\[T = 2 \pi \sqrt{\frac{r^{3}}{GM_{E}}} \ldotp \label{13.5.5}\]
For an ellipse, recall that the semi-major axis is one-half the sum of the perihelion and the aphelion. For a circular orbit, the semi-major axis (\(a\)) is the same as the radius for the orbit. In
fact, Equation \ref{13.5.5} gives us Kepler’s third law if we simply replace \(r\) with \(a\) and square both sides.
\[T^{2} = \frac{4 \pi^{2}}{GM} a^{3} \label{13.11}\]
We have changed the mass of Earth to the more general \(M\), since this equation applies to satellites orbiting any large mass.
Determine the semi-major axis of the orbit of Halley’s comet, given that it arrives at perihelion every 75.3 years. If the perihelion is 0.586 AU, what is the aphelion?
We are given the period, so we can rearrange Equation \ref{13.11}, solving for the semi-major axis. Since we know the value for the perihelion, we can use the definition of the semi-major axis, given
earlier in this section, to find the aphelion. We note that 1 Astronomical Unit (AU) is the average radius of Earth’s orbit and is defined to be 1 AU = 1.50 x 10^11 m.
Rearranging Equation \ref{13.11} and inserting the values of the period of Halley’s comet and the mass of the Sun, we have
\[\begin{split} a & = \left(\dfrac{GM}{4 \pi^{2}} T^{2}\right)^{1/3} \\ & = \left(\dfrac{(6.67 \times 10^{-11}\; N\; \cdotp m^{2}/kg^{2})(2.00 \times 10^{30}\; kg)}{4 \pi^{2}} (75.3\; yr \times 365\;
days/yr \times 24\; hr/day \times 3600\; s/hr)^{2}\right)^{1/3} \ldotp \end{split}\]
This yields a value of 2.67 x 10^12 m or 17.8 AU for the semi-major axis. The semi-major axis is one-half the sum of the aphelion and perihelion, so we have
\[\begin{split} a & = \frac{1}{2} (aphelion + perihelion) \\ aphelion & = 2a - perihelion \ldotp \end{split}\]
Substituting for the values, we found for the semi-major axis and the value given for the perihelion, we find the value of the aphelion to be 35.0 AU.
Edmond Halley, a contemporary of Newton, first suspected that three comets, reported in 1531, 1607, and 1682, were actually the same comet. Before Tycho Brahe made measurements of comets, it was
believed that they were one-time events, perhaps disturbances in the atmosphere, and that they were not affected by the Sun. Halley used Newton’s new mechanics to predict his namesake comet’s return
in 1758.
The nearly circular orbit of Saturn has an average radius of about 9.5 AU and has a period of 30 years, whereas Uranus averages about 19 AU and has a period of 84 years. Is this consistent with our
results for Halley’s comet?
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American Mathematical Society
Operator Theory and Analysis of Infinite Networks: Theory and Applications
Konrad Aguilar
Communicated by Notices Associate Editor Emily Olson
Operator Theory and Analysis of Infinite Networks: Theory and Applications
World Scientific Publishing Company, 2023, 448 pp.
By Palle E T Jorgensen and Erin P J Pearse
The quote that spoke to me the most (and the quote that I believe best summarizes the the content of this book) appears on page ix of the preface and states, “The literature on Hilbert space and
linear operators frequently breaks into a dichotomoy: axiomatic vs. applied. In this book, we aim at linking the two sides: After introducing a set of axioms and using them to prove some theorems, we
provide examples with explicit computations. For any application, there may be a host of messy choices for inner product, and often, only one of them is right (despite the presence of some axiomatic
isomorphism).” In particular, as a mathematician who works in a subfield of operator algebras that deals with many different metrics that induce the same topology, the last sentence of this quote is
something similar to what I tell my analysis/topology/functional analysis students all the time. And as an assistant professor at a liberal arts college who advises many undergraduate research
projects, the promise of explicit computations in the examples of this book opened up the possibility of finding new research projects for my students. I found the authors to have fulfilled their
promise as I read more and more of the text.
I have to admit that I was a little skeptical about the claim the authors had of “linking the two sides” of axiomatic and applied, and I wasn’t sure there was a place for this book in the wealth of
math textbooks we have. However, in the introduction the authors have sections devoted to “What This Book is About” (starting on page xlii) and “What This Book Isn’t About” (starting on page li) to
describe how this linking is accomplished. They also provide a detailed summary of each chapter. In particular, the section “What This Book Isn’t About” focuses on defending this textbook’s novel
approach to these subjects and how it approaches well-known topics in a new and enlightening way. For example, in regard to spectral theory on page lii, the authors clarify, “Our approach differs
from the extensive literature on spectral graph theory (see [Chu96] for an excellent introduction and an extensive list of further references) due to the fact that we eschew the basis for our
investigations. We primarily study as an operator on and with respect to the energy inner product. The corresponding spectral theory is radically different from the spectral theory of in .” And in
relation to operator algebras and an application of “infinite graphs to the study of quasi-periodicity in solid-state physics” on page liii, the authors state, “While periods and quasi-periods in
graphs play a role in our current results, they enter our picture in quite different ways, for example via spectra and metrics that we compute from energy forms and associated Laplace operators” and
that “There does not seem to be a direct comparison between our results and those of Guido et al.” In both of these cases and for the other topics they discuss in this section, they provide many
references to support their bold claims.
Of course, I shouldn’t have been skeptical of this book having a place in the literature since the authors bring their notable expertise to the text. I can confidently say that Palle E T Jorgensen is
known by everyone in the field of operator algebras and in many related fields due to his innumerous important contributions to these fields. The second author, Erin P J Pearse, has established
himself as an expert in fractal geometry and holds a patent related to his work in data science. I can’t think of a better pair of authors for a text that links the axiomatic and applied sides of
this particular subject.
Regarding the audience for this book, I believe it has something for everyone: from undergraduate to research mathematician (there are even conjectures scattered throughout the text, and I make some
comments about these below). And as I said in the intro, I, as a professor in a liberal arts college whose area is in analysis, will use it as a reference for finding connections between the analysis
presented in this text and my work to develop new undergraduate research projects for my students. While this text wouldn’t be sufficient for a course because it has no exercises, it could be used
for an independent study or a reading seminar.
I will now focus on some highlights from the text. Of course, there are more highlights than the following, but for the sake of brevity, I thought it best to focus on the following.
For the first highlight, I am going to take an unorthodox approach and begin by focusing on Chapter 16, near the end of the text, and the Appendices, before making my way to Chapter 1. As a
mathematician who works in operator algebras, when I first glanced at the text, a section entitled “The GNS Construction” caught my eye. However, there was something that initially confused me about
this title. This confusion stemmed from the fact that the GNS construction is a well-known construction that feels more foundational, so I would suspect something like this would find a home in the
Appendices. But as I read this section, I quickly realized why this content on the GNS appears in the text proper. The following statement from this chapter defends why this content is not reserved
for the Appendices (note that the Appendices are amazing, so much so that I spend time on them in this review next, but they do serve a different purpose than the text proper). “We provide the
following loose parallel [of the GNS construction with the Schoenberg-von Neumann theorem and also of Aronszajn’s theorem] for the interested reader to ponder further.” And the authors couldn’t be
more correct about my desire to ponder this parallel further. To make it even clearer how important these parallels are, “The statement of the GNS construction is more similar in flavor to that of
Aronszajn’s thoerem, but its proof is more similar to the Schoenberg-von Neumann theorem.” They then proceed to provide not only tables that display this important distinction, but also enlightening
sketches of the proofs of the GNS construction (Theorem 16.4) and the Schoenberg-von Neumann theorem (Theorem A.17). I provide the aforementioned tables here to give a glimpse into why I felt the
excitement that I felt.
Schoenberg-von Neumann:
GNS Construction:
The tables outline with incredible clarity the parallels between these constructions and I, for one, am grateful to the authors for providing them. A few pages before these tables, there is another
table that caught my eye as a mathematician who works in some of the “noncommutative areas” of mathematics such as: noncommutative geometry, noncommutative metric geometry, and noncommutative
topology. The next table in section 16.1 of Chapter 16 does an excellent job of displaying some noncommutative analogs to their commutative/classical counterparts, or in the authors’ terminology,
some quantum analogs to their probabilistic/classical counterparts.
Continuing in this strange route of presenting my highlights of this text, I move on to the Appendices. I believe that some of the best proofs or descriptions of classical results can be found in the
appendices of texts. One that particularly comes to mind is an appendix in John B. Conway’s A Course in Functional Analysis, which is Appendix C: The Dual of , which contains a nice presentation of
the Riesz-Markov-Kakutani representation theorem. In a similar way, the authors do not disappoint with their appendices. Appendix A mostly lists classical definitions in functional analysis, but
toward the end, gifts us with an illuminating sketch of the proof of the aforementioned Schoenberg-von Neuamann theorem (Theorem A.17) along with a proof of a powerful uniqueness corollary (Theorem
A.18). Appendix B provides some standard definitions as well as some detailed counterexamples that clarify some important concepts that appear in the text such as: a Hermitian operator that fails to
be essentially self-adjoint (Example B.14) and two self-adjoint operators whose product is not essentially self-adjoint (Example B.18). Moreover, they provide a generalization of the Krein
construction, while also offering a “more streamlined proof” a claim they make with which I fully agree. The diagram and table of Appendix C is also a great companion for the text proper since it,
for instance, provides a summary of some of the main properties of the Laplacian operator in the various contexts in which it appears in the text.
Next, we travel to the beginning of the text proper, where I would like to highlight some aspects of Chapters 1, 5, and 6. Indeed, Chapters 5 and 6 caught my eye since they focus on the Laplacian. I
find that the Laplacian is a difficult idea for students to grasp (I am including myself in the group of students that struggled with the Laplacian). I was pleasantly surprised with the presentation
of this operator. It first appears in Chapter 1, where the authors do an excellent job of setting the stage for the gentle pace of the rest of the text. They immediately opt to prefer explanation
over overwhelming detail but promise detail later and make good on that promise. This is made clear with statements like “We won’t worry about the domain of or T until Chapter 5.” I should note they
don’t exclude readers that are more familiar with the material since after this statement they address a familiar approach so as not to confuse mathematicians who know the subject material. Another
aspect I found impressive was the ability to utilize electrical resistance networks to motivate the definitions and results. The truly impressive part was the fact that while I have no knowledge
about resistance networks, through their explanations I have a firmer grasp on the mathematical definitions (as an added bonus, I may now be able to actually understand my brother, who is an
electrical engineer, when he talks about his job).
The amount of new research this text allows for is vast. The text not only contains a chapter devoted to Future Directions (Chapter 17), but also conjectures and open problems appear throughout the
text. My favorite one of these appears in Chapter 3 (Conjecture 3.48), where they provide a detailed description of the conjecture and its importance as well as produce a “Nonproof” “in the hope that
it will inspire the reader to find a correct proof.” I was very excited when I saw this and was delighted by the effort of the authors to provide their “Nonproof”!
Of course, the whole text is a pleasure to go through, but I believe the above highlights the novelty that this text furnishes to the mathematics community. In summary, this is a well done and
thought out example-driven text. It’s a great resource for students and professors alike, and I know that I will be using it as a resource for research with my students.
Photo of Konrad Aguilar is courtesy of Konrad Aguilar.
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The Conditional: ?: Operator in Ruby
The token or symbol used to define the operations performed on many operands is an operator. These operators may use one or more operands and perform different operations. There are sets of operators
defined for each programming language, and similarly, Ruby has its own too. The operands can be literal or expressions, and operators try to convert them into higher expressions.
This article will explore the Conditional: ?: Operator in Ruby. This operator is the only ternary operator, i.e.; it uses three operands for its operation. Let us know more about it.
The Conditional: ?: Operator
This operator is the only operator in Ruby that uses three operands and is the ternary operator. It has three parts, out of which the first part is the condition, and the other two are the possible
outcomes for the result of the condition.
We know that the condition may result in two outcomes which may be true or false. Hence, the other two parts have the code to be executed if the condition is correct. You will further understand this
part when we will discuss its syntax.
Condition ? True : False
In the syntax given above, we have three parts. Let’s discuss them one by one.
First Part (i.e., Condition in this case): In this part, the user enters the required condition he wants to test and upon which he wants the desired outcomes.
Second Part (i.e., True in this case): If the given condition is true, the written code should be executed.
Third Part (i.e., False in this case): Written code should be executed if the given condition is false.
In this syntax,? and : are part of the syntax and must be there.
Example 1:
#condition is true
x==3 ? ans=1 : ans=0
puts ans
Example 2:
#condition is false
x==3 ? ans=1 : ans=0
puts ans
This operator has relatively low precedence, and hence, it is not compulsory to put parentheses in the syntax. But, if the condition statements use the defined? Operator or the other outcomes use the
assignment operators; in that case, parentheses are necessary.
It is also necessary to put parentheses when methods used in the condition part end with an identifier. In this case, you can even put spaces between the syntax for proper explanation and correct
working of the operator. Look at the example below for an explanation:
a==2 ? b : c # This is legal
2==a ? b : c # Syntax error: a? is interpreted as a method name
(2==x) ? b : c # Okay: parentheses fixes the problem
2==a ? b : c # Spaces also eliminate the problem
This operator is right-associative, so the rightmost ones are grouped if these are used multiple times. For example:
v ? w : x ? y : z # The expression
v ? w : (x ? y : z) # is solved like this
This is how this operation works in Ruby.
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In the dynamic landscape of data analysis, understanding and interpreting relationships within datasets are critical for informed decision-making. Enter the Scatter Plot Maker, a user-friendly tool
designed to simplify the process of creating visually compelling scatter plots. This simple introduction aims to shed light on what a Scatter Plot Maker Calculator is and how it empowers individuals
and businesses to unlock meaningful insights from their data.
What is the Scatter Plot Maker?
Scatter Plot Maker are graphical representations that display individual data points on a two-dimensional graph. Each point represents the values of two variables, allowing for a visual examination
of their relationship. This type of visualization is particularly useful for identifying patterns, trends, and outliers within datasets.
Exploring the Foundation: The Scatter Plot Formula
At the core of any scatter plot lies a simple yet profound formula that enables the visualization of relationships between two variables. The scatter plot formula can be expressed as follows:
Y = f (x)
• y represents the dependent variable,
• x represents the independent variable,
• f(x) denotes the functional relationship between the variables.
The scatter plot graph calculator formula essentially allows us to graphically represent the relationship between two continuous variables, providing insights into correlations, clusters, and
outliers within the dataset.
Understanding Units and Measurements
To comprehend the scatter plot maker calculator formula fully, it's crucial to understand the units of the variables involved. The units of the dependent variable (y) and the independent variable (x)
must be compatible for meaningful interpretation.
For example,if x represents time in hours, and y represents distance in kilometers, the scatter plot will depict how distance changes concerning time. Ensuring consistency in units enhances the
accuracy of the scatter plot and facilitates a more insightful analysis.
Formula Table: Decoding the Essentials
Let's break down the scatter plot formula further and explore its components:
Term Description
y Dependent Variable (Vertical Axis)
x Independent Variable (Horizontal Axis)
f(x) Functional Relationship Between x andy
By understanding these components, users can grasp the essence of the scatter plot formula and apply it effectively to their datasets.
Practical Implementation: Scatter Plot Maker Calculator Tools
As the demand for data visualization escalates, various Scatter Plot Maker Calculator tools have emerged to simplify the process. These tools go beyond manual graph creation, offering features like
data import, customization, and real-time updates like our dilution ratio calculator. Some notable Scatter Plot Maker tools include:
Google Sheets Scatter Plot Maker:
Integrating seamlessly with Google Sheets, this tool allows users to create dynamic scatter plots with live data updates.
Microsoft Excel Scatter Plot Wizard:
Excel's Scatter Plot Wizard streamlines the process, guiding users through the selection of variables and customization options.
Online Scatter Plot Generators:
Platforms like Plotly, ChartGo, and Datawrapper provide web-based Scatter Plot Makers, enabling users to create interactive visualizations effortlessly.
Advantages of Scatter Plot Makers
Visual Clarity:
Scatter plots offer a clear visual representation of relationships within the data, making it easier to identify patterns and trends.
Data Correlation:
Scatter plots aid in discerning the correlation between variables, helping researchers make informed decisions based on the observed relationships.
Outlier Detection:
The visual nature of scatter plots makes it simple to identify outliers or anomalies in the dataset that may require further investigation.
What types of data can I visualize with a Scatter Plot Maker?
Scatter Plot Makers are versatile and can handle a wide range of data types. Whether you're dealing with numerical data sets, categorical variables, or even time-series data, these tools allow you to
visualize relationships between two variables effectively.
Is the Scatter Plot Maker suitable for large datasets?
Yes, the Scatter Plot Maker is designed to handle datasets of varying sizes. Its efficient algorithms and optimized performance ensure that you can create scatter plots even with large datasets,
allowing for comprehensive data analysis.
Can I import data from different file formats into the Scatter Plot Maker?
Absolutely. The Scatter Plot Maker supports the import of data from various file formats, including CSV, Excel, and other common formats. This feature enhances the tool's versatility, allowing you to
work with data from different sources seamlessly.
Can I collaborate with others using the Scatter Plot Maker?
Yes, many Scatter Plot Maker tools offer collaborative features, allowing multiple users to work on the same project simultaneously. This fosters teamwork and ensures that insights gained from
scatter plots can be shared and discussed among team members.
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On the same date Sapna retires from the business and the follow... | Filo
Question asked by Filo student
On the same date Sapna retires from the business and the following adjustments are to be made :
a. The firm's goodwill is to be revalued at .
b. The Assets and Liabilities are to be revalued as under - Stock ; Debtors ; Machiner; and Creditors ₹ 14,000 .
c. Satishan is to bring in and Kasturi is to bring in as additional capital.
d. Sapna is to be paid ₹ 16,200 in cash and balance on her capital is to be transferred to her loan account.
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Question Text On the same date Sapna retires from the business and the following adjustments are to be made :
Updated On Feb 13, 2024
Topic Calculus
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 134
Avg. Video Duration 2 min
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Derivative-Calculator.org – Derivative of atan(x) - Proof and Explanation
We want to find the derivative of $\mathrm{arctan}\left(x\right)$. Let $y=\mathrm{arctan}\left(x\right)$. Then, by definition, $x=\mathrm{tan}\left(y\right)$.
Taking the derivative of both sides with respect to $x$:
Now, we solve for $\frac{dy}{dx}$:
Using the identity ${\mathrm{sec}}^{2}\left(y\right)=1+{\mathrm{tan}}^{2}\left(y\right)$ and knowing $\mathrm{tan}\left(y\right)=x$:
Thus, the derivative of $\mathrm{arctan}\left(x\right)$ is:
To find the derivative of $\mathrm{arctan}\left(x\right)$, we start by letting $y=\mathrm{arctan}\left(x\right)$. This means that $x=\mathrm{tan}\left(y\right)$, representing the angle whose tangent
is $x$.
Next, we differentiate both sides of the equation $x=\mathrm{tan}\left(y\right)$ with respect to $x$. The derivative of $x$ with respect to $x$ is simply $1$.
On the right side, the derivative of $\mathrm{tan}\left(y\right)$ with respect to $y$ is ${\mathrm{sec}}^{2}\left(y\right)$, and by the chain rule, we multiply by $\frac{dy}{dx}$, which gives us ${\
Setting the derivatives equal, we get:
We then solve for $\frac{dy}{dx}$ by dividing both sides by ${\mathrm{sec}}^{2}\left(y\right)$:
We use the trigonometric identity ${\mathrm{sec}}^{2}\left(y\right)=1+{\mathrm{tan}}^{2}\left(y\right)$. Since $\mathrm{tan}\left(y\right)=x$ (from our earlier definition), we substitute $x$ for $\
Thus, the expression for the derivative simplifies to:
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Clear step by step for these 2 problems? (max & min)(concavity)
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Clear step by step for these 2 problems? (max & min)(concavity)
Hello, could anyone give clear step by steps to solve these problems? Thank you.
1) Let f(x)=4x?8x for x greater than 0. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima).
1. f is increasing on the intervals :______________
2. f is decreasing on the intervals:_____________
3. The relative maxima of f occur at x =_____________
4. The relative minima of f occur at x =______________
Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word "none". In the last two, your answer
should be a comma separated list of x values or the word "none".
2)Let f(x)=1/(5x^(2)+7). Find the open intervals on which f is concave up (down). Then determine the x-coordinates of all inflection points of f.
1. f is concave up on the intervals:____________
2. f is concave down on the intervals:____________
3. The inflection points occur at x =____________
Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word "none". In the last one, your answer
should be a comma separated list of x values or the word "none".
1 Reply
Look at the first derivative to see where the function is increasing or decreasing and look at the second derivative to see where the function is concave up or concave down.
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Learning over sets using kernel principal angles
We consider the problem of learning with instances defined over a space of sets of vectors. We derive a new positive definite kernel f(A, B) defined over pairs of matrices A, B based on the concept
of principal angles between two linear subspaces. We show that the principal angles can be recovered using only inner-products between pairs of column vectors of the input matrices thereby allowing
the original column vectors of A, B to be mapped onto arbitrarily high-dimensional feature spaces. We demonstrate the usage of the matrix-based kernel function f(A, B) with experiments on two visual
tasks. The first task is the discrimination of "irregular" motion trajectory of an individual or a group of individuals in a video sequence. We use the SVM approach using f(A, B) where an input
matrix represents the motion trajectory of a group of individuals over a certain (fixed) time frame. We show that the classification (irregular versus regular) greatly outperforms the conventional
representation where all the trajectories form a single vector. The second application is the visual recognition of faces from input video sequences representing head motion and facial expressions
where f(A, B) is used to compare two image sequences.
• Canonical Correlation Analysis
• Kernel Machines
• Large margin classifiers
Dive into the research topics of 'Learning over sets using kernel principal angles'. Together they form a unique fingerprint.
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How Do The Speeds V0, V1, And V2 (At Times T0, T1, And T2) Compare?
The comparison of speeds V0, V1, and V2, at each corresponding times T0, T1, and T2, can be done using numerous mathematical formulas and theories. Each formula is used based on the context of the
data given in the question. To further understand the comparison, let us break down the variables given in the question.
V0 and V1, are speeds at two different times. These are represented by T0 and T1. V2 is the speed at time T2. T0, T1, and T2, are the time points at which the speeds were recorded. The comparison of
the speeds V0, V1, and V2 can be done by calculating the ratio which is also known as the speed ratio.
The speed ratio is the ratio of the speed at two different time points. This ratio is calculated by dividing the speed at time T1 by the speed at time T0. This ratio helps to determine how consistent
the speed has been over the period of time.
Another way to compare the speeds V0, V1, and V2 is by calculating the change in speed over the period of time. The change in speed can be calculated by subtracting the speed at time T0 from the
speed at time T2. This calculation will give you the change in speed over the three time points.
The comparison of the speeds V0, V1, and V2 can also be done by calculating the average speed of the three time points. This is done by adding the speeds of the three points and then dividing the sum
by the three points. This will give you the average speed for the three time points.
In conclusion, the comparison of the speeds V0, V1, and V2 at times T0, T1, and T2, can be done by calculating the speed ratio, the change in speed over the entire time period, and the average speed
of the three time points. Each of the three methods will help to gain a better understanding of the comparison between the speeds.
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Florry the Lorry - Schengen Calculation Confusion
Published: Tue 19th April 2022
There has been a lot of confusion about How many days you can go away for in Europe and how people are showing a single day change giving them a lot more days. Take the following 2 examples, where
you originally get 41 days abroad, but by changing the start date by 1 day you suddenly get 65!
We use the example given from French Le Van who are amazing travellers who are doing the Schengen shuffle. Their travels are inspirational and photography is fantastic so well worth following.
I can see how this is confusing, by changing your date of entry/control by one day you got 24 more days! Let me explain why this works.
You're only allowed to be in Europe for 90 days out of 180, but it's rolling, which changes every day. In the first screenshot we can see that counting days backwards from 2nd April 2022 the 180 days
starts on 5th October 2021. In the second screenshot the date has changed to 3rd April 2022 the 180 days starts on 6th October 2021
So you can see by changing the date of Entry by 1 day, the sliding window changes by 1 day, this is important!
In the first screenshot we see 41 days so: 24+25+41 = 90, that's your limit so the calculator is right.
In the second screenshot we see 65 days. 24+25+65 = 110, so the calculator is wrong!
It's not, don't panic, the calculator is correct. Let me explain why.
We have to go back to the fact it's rolling. I said earlier that counting back 180 days from your Entry on 2nd April was 5th October 2021, but the next day it's changes
• 3rd April 2022 your 180 window starts on 6th October 2022.
• 4th April 2022 your window starts on the 7th October 2021.
• 5th April 2022 your window starts on the 8th October 2021.
• 6th April 2022 your window starts on the 9th October 2021.
Entering on the 2nd April 2022
If we keep listing all these days, you see that in 41 days from when you enter it will be:
13th May 2022 your window starts 14th November 2021, so let's calculate that:
Counting the days you were in Europe from 14th November we get 24 days from the first trip. 25 days from the second trip and 41 days from our current trip. so 24+25+41=90 youâ ve used it all up and
you have to back in the UK.
Entering on the 3rd April 2022
Now, if we change the entry date by 1 day and list the days again - they are the same:
• 3rd April 2022 your 180 window starts on 6th October 2022.
• 4th April 2022 your window starts on the 7th October 2021.
BUT now when we go 41 days forwards it's the 14th May 2022 and your window starts on the 15th November 2021 - This is the important but which people seem to miss.
Counting the days you were in Europe from 15th November we get 23 days from the first trip. 25 days from the second trip and 41 days from our current trip. so 23+25+41=89 So you can stay an extra
The next day the rolling 180 days changes again - so it starts on the 16th November 2021
Counting the days you were in Europe from 16th November we get 22 days from the first trip. 25 days from the second trip and 42 days from our current trip. so 22+25+42=89 So you can stay another day!
Keep this going, and you end up with being on 7th June 2022, the rolling 180 days starts on 9th December 2021
Counting your days now you get zero days from your first trip counted anymore, so it's just the second trip and the current trip: 0+25+65=90 and that is your last day in Europe.
I hope this clears up any confusion. If you have any questions let us know!
Related Posts
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Problem A
Everyone knows of the secret agent double-oh-seven, the popular Bond (James Bond). A lesser known fact is that he actually did not perform most of his missions by himself; they were instead done by
his cousins, Jimmy Bonds. Bond (James Bond) has grown weary of having to distribute assign missions to Jimmy Bonds every time he gets new missions so he has asked you to help him out.
Every month Bond (James Bond) receives a list of missions. Using his detailed intelligence from past missions, for every mission and for every Jimmy Bond he calculates the probability of that
particular mission being successfully completed by that particular Jimmy Bond. Your program should process that data and find the arrangement that will result in the greatest probability that all
missions are completed successfully. Note that the number of missions is equal to the number of Bonds, so that every Jimmy Bond needs to be assigned a single mission each.
Note: the probability of all missions being completed successfully is equal to the product of the probabilities of the single missions being completed successfully.
The first line will contain an integer $N$, the number of Jimmy Bonds and missions ($1 \le N \le 20$).
The following $N$ lines will contain $N$ integers between $0$ and $100$, inclusive. The $j$:th integer on the $i$:th line is the probability that Jimmy Bond $i$ would successfully complete mission
$j$, given as a percentage.
Output the maximum probability of Jimmy Bonds successfully completing all the missions, as a percentage.
Your answer should have an absolute error of at most $10^{-6}$.
Explanation of third sample
If Jimmy bond $1$ is assigned the $3$:rd mission, Jimmy Bond $2$ the $1$:st mission and Jimmy Bond $3$ the $2$:nd mission the probability is: $1.0 \cdot 0.13 \cdot 0.7 = 0.091 = 9.1\% $. All other
arrangements give a smaller probability of success.
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2
Sample Input 3 Sample Output 3
25 60 100 9.1
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A novel technique to secure the acute my
Acute myocardial infarcta, Privacy preserving data mining, Random projection, Data perturbation, Privacy level.
Privacy is an essential part in various applications of data mining, which mostly considers the areas of forensics, health, financial, behavioural, and other sort of confidential data. They may occur
due to the requirement to create user profiles, build models related to social network and to detect terrorism among others. For example, mining the data of a health care system may have the
necessity to dissect clinical records and other medical transactions. But the underlying problem is that the privacy laws may be broken when such different data sets of different users are combined.
It is unsafe to allow the health organizations to disclose the data even though the identifiers are deleted because the original information can be identified by the building of identification
attacks for connecting different data set [1]. Thus arises the need for better techniques which pay attention for securing private information. It also preserves the statistical behaviour and
characteristics which are necessary for data mining related applications. The approach, we discuss in this paper are defined in the following way: Assume there are N organizations A1; A2; …; AN,
where every organization, AI contains a transaction database DBi. It is quite common that some statistics related features of the union of the databases 2]. At this point, the original data are
generally perturbed and it is disclosed in its distorted form. Any user can access the released data. The work specifically takes into account a proposed technique to maintain privacy. This is
boosted by the result furnished by Kargupta et al. in their research, which pinpoints the drawbacks of additive data perturbation [3]. In particular, the research work well discovers the chances of
applying the technique of ‘project edge’ for building a modified form of data. The distorted data is revealed to the user who is mining the data. It can also be explained and proved that the
statistical properties are maintained well in the distorted form of data. The theorem of Johnson and Lindenstrauss [4] laid the foundation for this approach which proves that a collection of points
in a Euclidean space which is n-dimensional and can be mapped onto another subspace which is p-dimensional, where p=log n. Thus the pairwise length of the two points is secured by an atomic value.
Hence, it is understood that the original information is susceptible to change when the data is mapped onto a lower subspace, while preserving its statistical characteristics. It is assumed that the
confidential data is from the same domain and there is no sort of collusion between the parties.
The summarization of the work is listed as follows. In Section 2, the literature survey of the existing distortion techniques is elaborately presented. The challenges of the existing techniques and
the flaws of the other distortion techniques such as rule hiding, data swapping, k-Anonymity, random transformation, secure multiparty computation technique, random projection and morphological
operations are discussed. Section 3 presents a proposed ‘project edge’ technique which will further increase the privacy and accuracy level of images. Experimental works and results are also provided
to prove the efficiency of the proposed approach. Section 4 compares the hybrid ‘project edge’ technique with that of the existing distortion methods. The conclusion of the work is given in Section
Related work
This section offers a concise survey of the related papers in the area of data mining to maintain privacy.
Data distortion techniques
Data distortion or perturbation methods are mostly categorized into the probability distribution and the perturbation of the value. In the first method, the actual data are either substituted by
another data taken from the same set or substituted with the data set itself. The actual data are distorted by the addition or multiplication of noise, or by any other randomization algorithms in the
value distortion method [5]. The existing data perturbation techniques to distort the data are translation, rotation, scaling and hybrid perturbation techniques. This work, specifically concentrates
on the value distortion approach. It was the additive data perturbation technique that was proposed by Zhenmin et al. for the construction of decision tree classifiers [5]. It is referred to as a
translation based perturbation technique and it is easily susceptible to attacks. Randomization is carried out for every item of the actual information by the addition of noise generated randomly
which is selected from a distribution like a Gaussian. The original data is then reconstructed from its distorted form using algorithms like expectation maximization and then the categorization
prototypes are built. Kargupta et al. queried his points on the addition of noise. He addressed that adding noise might compromise the privacy because of the fact that the additive noise can be
withdrawn easily. The disadvantage of additive noise is overcome by the use of multiplicative noise for preserving the data privacy. There are two ways of introducing multiplicative noise [6,7]. One
way is multiplying the element by a number which is spawned randomly. The number spawned randomly owns a Gaussian distribution which is a truncated one with a mean equal to one and a low variance
level. The second one is choosing a log based conversion of the data and then to combine a Gaussian noise which is predefined as well as a multivariate one. Then the antilogarithm of the data is
found out. In general, the first one is beneficial when the data distributor only needs to cause small modifications to the inaugural one. The latter approach offers greater privacy level, but the
data utility is maintained in the logarithmic scale. The primary disadvantage of both the additive and multiplicative perturbations is that the pairwise similarity of data records is not preserved.
This report proposes an alternate plan of attack that tests to maintain the average of statistical features of the information. The perturbation of data that occurs by adding or multiplying noise
generally handles numerical data. In rotation based perturbation technique, each sub matrix is rotated independently and the properties of the data matrix were proved. The technique urges to generate
perfect centralized procedures for mining data while protecting privacy. The limitation is that it can be applied only to the data split column wise. It cannot be applied to row wise partitioned data
sets. The distortion of categorized data was studied first by Evfimievski et al. [8]. There was an evolution of a response method which is a randomized one to collect data via interviews. The
distortion of categorical data was again taken into account, specifically in association rule mining which was proposed by Evfimevski et al. [9]. The work was stretched forth by Agrawal et al. by
imparting in their framework, and a model for measuring the violation of privacy was brought into use [10]. The idea of γ amplification is used and is also applied in the model framework without the
presumption on the subject of distribution. The actual information is taken from the same distribution. This model was reconsidered by Dalenius et al. and they had explained for setting the
parameters efficiently for perturbation for reconstruction while preserving amplification [11].
K-anonymity model
The difficulty that an owner of the data needs to portion out a quantity of identifiable data by not revealing one’s individuality is considered by the k-anonymity technique. Suppression and data
generalization are the techniques to overcome this problem. These techniques maintain the privacy related information. The best solution is to define all the quasiidentifiers which are used for
connecting to data from external sources. The information is released only when the person’s data which is on the waiver could not be keyed out from k-1 persons.
Data swapping technique
Fienberg et al. [11] had initially recommended the fundamental principle of data interchanging, which is a modified version of the technique proposed by Dalenius et al. This idea is implemented by
changing the data repository by exchanging some set of properties between the chosen set of tuples so that the data confidentiality is not disturbed. The marginal counts are also preserved. This
technique can be categorized under data perturbation. Many modifications and applications of the data swapping technique are quoted in their proposed technique.
Secure multiparty computation (SMC) technique
This technique Secure Multiparty Computation (SMC) takes into account the difficulty of accessing a subroutine of the confidential inputs from more than one party in such a way that only the output
of the function is revealed to the parties. The main building blocks of Secure Multiparty Computation (SMC) are the huge quantity of cryptographic protocols such as homomorphic and commutative
encryption, circuit evaluation protocol and oblivious transfer. A detailed idea about Secure Multiparty Computation (SMC) framework along with its applications to the field of data mining is reported
by Pinkas [12]. The work put forward by Goldreich offered a detailed introduction to Secure Multiparty Computation (SMC) [13]. It was explained clearly that any subroutine which is manifested by an
arithmetic circuit can be calculated by means of an arbitrarily circuit assessing protocol. But, this will make the approach impracticable for huge datasets. A set of Secure Multiparty Computation
(SMC) tools such as secure sum, inner product and set union beneficial for large-scale data privacy are briefed by Clifton et al. [14]. The techniques related to privacy preserving in data mining and
its state of the art is explained clearly by Agrawal et al. [15].
Distributed data mining approach (DDM)
This Distributed Data Mining (DDM) approach helps to compute the prototypes of data mining and to extract certain “patterns” at a given connection point by interchanging very few data between the
group of nodes that are taking part in [16,17]. Merugu et al. had proposed a paradigm for grouping distributed confidential data either in a semi supervised or in an unsupervised scenario [18]. An
algorithm proposed by Gowri et al. show a novel method for the process of clustering in which the accuracy of clustering the data is described very appropriately, this algorithm would help in the
process of pattern finding in the image. According to another algorithm, a model is built by each local site which transmits the model parameters to the global site. Here, it constructs a clustering
model. An algorithmic procedure to maintain privacy for a Bayesian network model is briefed by Meng et al. [19].
Rule hiding
The principal target of this technique is to translate the data repository in order to hide the sensitive rules and the complete fundamental patterns can still be considered. It was formally proven
by Atallah et al. that the best sanitization is an NP-hard problem to mask the confidential huge data sets in association rule mining [20]. Certain heuristic methods are used to overcome the
difficulties. For example, the perturbation-based association rule hiding technique is adopted out by changing a selected set of 1-values to 0-values or vice versa, hence that the frequent item sets
that generate the rule are handled or the relief of sensitive rules is taken down to a user-specified threshold [21]. Certain data attributes are replaced by a question mark in the blocking based
association rule hiding approach [22]. In this regard, the minimum support and assurance will be modified into a minimal interval. The data sensitivity is needed to be saved or maintained until the
support and/or the confidence of a confidential rule raises above the middle in the two areas.
Random orthogonal transformation
This part represents multiplicative distortion which uses orthogonal matrices generated randomly in the calculation of the inner product matrix. The deficiency of the applied method shall be analysed
and a general case is suggested that uses random projection matrices to protect data in a better way. The transformation of an orthogonal transformation is linear T: I^Tn → I^Tn, which maintains both
the size and the angles between the vectors [23].
Generally, orthogonal transformations can be represented by orthogonal matrices. Assume A and B are two data sets possessed by Vicky and Micky. Let A be an a[1] × b matrix, and B is an a[2] × b
matrix. The same attributes are observed in both the data sets. Assume R as a random orthogonal matrix of size n × n. Let us now take the linear conversion of the data sets:
X=AR; Y=BR; → (1)
Then, XX^T=AA^T; YY^T=BB^T → (2)
XY^T=ARR^TBT=AB^T → (3)
If the data owners Vicky and Micky outsource only the distorted version, it is possible to compute the pairwise angles and distances in between the row vectors. Hence, implementing a distance based
data mining application for a horizontally partitioned data becomes very easy. In the same way, if the data is transformed such that, X=AR; Y=BR; then we have, X^TY=A^TB and the pairwise similarities
are fully maintained in the disturbed data. Thus, a third person can examine the interconnection of the properties of column wise split data which is heterogeneously distributed without accessing the
sensitive data. The observer cannot guess the actual form of the original data because there are a number of inputs and a lot of transformation probabilities. Hence, the random orthogonal
transformation appears to protect data’s privacy in a more serious fashion, maintaining its utility. The transformation is called a ‘rotoinversion’ which is a rotation continued by a flip, when the
value of the determinant is -1. Hence, the original data can be identified by means of a proper revolution. Figures 1a and 1b display the working of a random transformation in 3D orthogonally. The
data are not perturbed well after random orthogonal transformation. Therefore, the technique of random rotation also does not secure the data up to the expected level [24].
Random projection
Basic mechanism: The technique of random projection projects an image of a higher dimensional plane to its subsequently lower-dimensional plane. The principle conception of random projection stems
from the Johnson’s theorem which is briefed below:
Lemma 1: (Johnson-Lindenstrauss Lemma).
For an integer p and for ε, where 0< ε <1, if m is non-negative, then. For a set P of p= |P| data points in IT^n, the mapping function is given by f: IT^n → IT^l, hence, for all a, b Є P, (1-ε) ||
a-b ||^2 ≤ || f (a) – f (b) ||^2 ≤ (1+ε) || a-b||^2
The lemma very clearly explains that a collection of p points in a k-dimensional expanse of Euclidean can be mapped onto a O (log p/ε^2) dimensional expanse and hence the pairwise similarities of any
two points are preserved by a small number. This wonderful characteristic infers that the data’s actual form can be changed by the reduction of its dimensionality yet still preserves its statistical
properties. Both the horizontal and vertical projection of the input data is carried out. The input image is illustrated in Figure 1a. The resultant images are displayed in Figures 2a and 2b. From
the result, it is seen clearly that the actual form of the data is disturbed and it is very hard to perceive. Other characteristics of the random projection technique and random matrices are
discussed in the following which helps in preserving the data utility.
The original data is given in Figure 1a.
Lemma 2: Assume M as a × b matrix. Every element m[ij] of M is independent and it is identically taken from a certain unknown distribution which has mean zero and variance as σ^2[m], then D [M^T M] =
a σ^2[m] I and D [MM^T] = b σ^2[m] I.
Proof: Let m[ij] and ε[ij] be the i^th and j^th entries of matrix M and M^T M
The random matrix entries are identically distributed and are independent and hence,
Also, it is noted that
Identically, we have
The observation was already made that vectors which have random directions are mostly perpendicular in an mdimensional space where m is greater than or equal to 1. The random projection technique
will be more powerful when it is combined with the other perturbation techniques.
(Projection by rows). Suppose Vicky and Micky are the owners of the data sets A and B respectively. Let A, B and C are the matrices where A=a × b1, B=a × b2 and C=k × a (k<a) be a random matrix.
Every element c[i, j] of C is unrelated and it is exactly taken from a certain distribution which has mean=0 and variance=σ^2[m]. Also, let
(Projection by columns). Suppose Vicky and Micky are the owners of the data sets A and B respectively. Let A, B and C are the matrices where A=a1 × b, A=a2 × b and C=b × k (k<b) be a random matrix.
Every matrix element C[i, j] of C is unrelated to each other and exactly taken from a certain distribution which has mean=0 and. Variance=σ^2[m] Also, let
From the above results, it is absolutely clear that the horizontal projection maintains the inner product of columns and that the vertical projection maintains that of rows. Thus it is shown that the
inner product is directly connected to many other distancerelated metrics. Some of them are:
The Euclidean distance of and b is given by the formula
The cosine angle of a and b, assuming that the data vectors are normalized to one.
Suppose the data vectors are normalized to one with mean=0, then the correlation coefficient of a and b is
When the data attributes are decreased by means of projection, the statistical relationships will be saved. Similarly, if the observations are compressed, then the kinship between the attributes will
really be saved. The mining procedures are applicable to the distorted data without disturbing the actual data. The drawback of random projection is it is highly unstable. Different projections
result in different clustering results. The technique can be more powerful when it is combined with other perturbation technique.
Morphological operations
The morphological operations such as dilation and erosion, which changes the structure and shape of an image, also suffer from certain limitations. Sagar concluded that the basic morphological
operations such as dilation and erosion are reactive to noise and obtrusions on the boundaries of a shape [25,26]. The operations will not produce better results if the objects are nearer with the
distance (2* size of structuring element). The drawbacks of the existing perturbation techniques are illustrated diagrammatically in Figure 3.
Proposed ‘Project Edge’ Technique
The algorithmic steps of the proposed technique are given as follows.
1). Input: An image A (a, b) and a matrix generated randomly C= (k<a) where k<a.
2). The input image A is projected into with and the perturbed form is
3). The boundary pixels of the image are detected using canny edge detection method.
4). Then the boundary pixel values of the perturbed image b[ab] (X) are further distorted to Y=d[ab] (X) in a nonlinear method.
5). The resultant image after doubly perturbed is Y.
6). The privacy level of the distorted image Y is calculated by the formula
Where V[ar] represents the variance, A represents the actual image and Y the output image after perturbation.
7). The root mean square error value is also computed by
The input image is perturbed twice. The size, dimensions and the original values of the input image are perturbed in step 2. Then the boundary pixels of the perturbed image are detected and again
perturbed. The perturbation technique applied to the boundary pixels follows a nonlinear method. Hence, it is impossible to estimate the original form of data A and B, if the distorted data only is
given. This is because the probability of finding out the solutions is infinite. Thus, it provides a very strong protection level of the image.
The shutter stock database is considered for the experimental work which is available in http://www.shutterstock.com/. Figure 4 shows a few sample medical images (AMI images) chosen from the
database. 750 medical images of various dimensions are extracted from the database. Some of the sample images are displayed in Figure 4. The resultant images after the application of proposed project
edge technique are shown in Figure 5. We also show in Table 1 that the proposed ‘Project edge’ technique increases the privacy level to a much greater level. The maximum privacy level hikes to 0.88.
The accuracy of the image is also increased. This is proven by the computation of root mean square error values in Table 2. The lower Root Mean Square Error (RMSE) values represent a higher accuracy
level. The average Root Mean Square Error (RMSE) value is 0.0024 which is very low and that accounts for higher accuracy.
^#Images ^#Dimensions Range of privacy level (0 to 1)
250 2D 0.80 - 0.86
200 3D 0.75 - 0.85
200 4D 0.76 - 0.87
100 5D 0.82 - 0.88
Table 1: Privacy level of perturbed images of different dimensions.
^#Images ^#Dimensions Range of RMSE
250 2D 0.0015 - 0.0023
200 3D 0.0013 - 0.0027
200 4D 0.0016 - 0.0025
100 5D 0.0020 - 0.0029
Root Mean Square Error (RMSE)
Table 2: Root Mean Square Error Value (RMSE) of perturbed images.
Comparison with Other Existing Techniques
The performance metrics of the proposed ‘project edge’ technique are compared with the other widely used techniques and the results are shown. Figures 6 and 7 present a comparison graph and chart of
the privacy level and the root mean square error value with the existing techniques and the proposed technique. The privacy level of the existing techniques and the proposed technique are shown
clearly in Figure 6 for an input image. The privacy level of the proposed technique is raised to “0.88” in the graph. The Root Mean Square Error (RMSE) value of the proposed technique is minimized to
0.0013. Thus the proposed technique possesses a maximum privacy level and a minimum Root Mean Square Error (RMSE) value compared to the existing perturbation techniques like translation, rotation,
multiplicative, dilation and erosion. The graph and the bar chart in the Figures 6 and 7 depict a higher privacy level and accuracy level for the proposed technique compared to the existing
techniques. A higher privacy level and a lower error value represent higher performance.
The proposed ‘project edge’ technique yields better results when compared with the other widely used existing techniques. This technique can be applied to images having dimension 2 and above. Higher
performance is achieved by the proposed technique since it perturbs twice the original image. The image is collapsed dimensionally and the original form of data is impossible to detect if only the
distorted data is given. The nonlinear technique is applied to modify the boundary pixel values after detecting the image boundaries. This in turn provides a stronger privacy level and the number of
solutions to guess and find out the original image is infinite.
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What is the P/E ratio *actually* saying (using real examples)
The P/E ratio is the most popular and the MOST abused and misunderstood valuation ratio. Let's discuss this.
Disclaimer: The companies used are only for illustrative purposes. NOT FINANCIAL ADVICE!
We keep hearing this ratio when people talk about buying stocks, especially old people. The problem is, most of these people have never really tried to dive into the P/E ratio. Just simple rules of
thumb that make no sense: "P/E of over 40 is expensive". Yeah, right. Look, on paper, it is a simple ratio: The market price of the stock divided by the earnings per share of the company. Well
atleast it seems simple. Price is easy to see. Earnings are subjective. What do we consider as earnings? Do we adjust for Other Income? Extraordinary items? Do we deep dive into notes to accounts to
see breakup of the numbers? It is a pain and I am so glad that I don't actually work as an equity research analyst. Can't imagine myself doing these adjustments regularly. Anyways, I am not diving
into all that. The problem is whenever we buy something in the market, our expectation is the company will continue doing well in the future, that is, generate lots of cash, so basically the stock
price will go up and some day down the line, we sell to someone else.
The company's market price is always going to reflect the expected future cash flows. People pay to buy Tesla shares not because it is making a lot of money today, but because they expect it to
generate a LOT of cash in the future and become the world's leader in the EV segment (I am not talking about stock traders here). That is why Tesla has a high P/E of something like 900. It is not a
profit making company, but the market expects it to have insane profits in the future. It may not turn out to be true, but then that is a risk we take on faith, research, belief etc. when investing.
But, the story does not end here. We're not just talking about future cash flows and profits here. We're also talking about reinvestment. Let me explain. Suppose you start selling ice cream in front
of a school. It takes you $1000 to setup everything like a counter, machines etc. (this is your capital). You sell ice cream worth $600 on first day and the cost of all that ice cream (ingredients)
to you was $500. You make a 20% profit margin [(600-500)/500], not bad for first day. Your return on capital is 10% (Profit/Capital. In this case, $100 profit divided by capital of $1000). The next
day, you want to sell more, but to sell more you need to increase your capacity. So out of the $100 you made on the first day, you put in $50 more in your capital. So your capital now is $1050.
Remember, the first day, your return on capital was 10%. If you want to maintain this, now you will have to earn a profit of $105 on second day because your capital now increased. So let's just
pretend you made a profit of $105 the next day. This is great. You maintained your return on capital. So, on the first day, you made $100, of which you put back in $50 to expand your capacity and
earn more. Next day, you earned $105. In total, the cash that you get to keep is $105 (second day earnings)+$50 (first day earnings)=$155.
But, what if, when you reinvested $50, you could not maintain 10% return? What if on the second day, you made only $100 again instead of $105? Your return on capital for second day went down to 9.5%
(100/1050). You did not invest the additional $50 efficiently. Or what if you earned more than 105. If you earn 110, your return is 10.47% (110/1050). You made an excellent reinvestment. Suppose you
have 2 businesses now instead of 1. The 1st business reinvested $50 inefficiently and earned 9.5%. The 2nd earned 10.47%. Which business is worth more? Which is a better business? The second
business. They generated more cash. They were more efficient capital allocators. They can earn more by investing less.
We went on a tangent here. But this is exactly what we should be using to judge P/E ratios. Are the businesses able to grow by making better returns on capital (earn $110 the next day instead of only
$100 after reinvesting $50)? Are they able to generate more cash with less reinvestment? Are they reinvesting too much for too little? Companies that are not going to grow very well in the future and
are wasteful with reinvestments will have a lower P/E. They are not cheap, they are just bad at reinvesting. Companies that may seem expensive may be having excellent returns on capital and big
expected growth in earnings. We need to look at P/E in this whole context. You can play with the spreadsheet for this here. It is not a well formatted spreadsheet, but you can get the point. You can
also read more by diving into Credit Suisse's note on this.
Okay, let's look at some real companies and see how their P/Es stack up.
I discussed this valuation earlier too. I projected out the company revenues, margins and reinvestments to arrive at the value of the company. Don't worry about how we made these projections as of
now. If you want to understand this process, I have written a guide on where you can go.
The ROIC (Return on invested capital) stays between 18-20% in my valuation. The implied P/E based on these assumptions is 21. I find the company fairly valued. You can explore my sheet here. The P/E
is pricing in moderate growth rates and assumes similar reinvestments going forward as they are now.
This is a valuation a friend of mine did. He projected out his own growth rates for the company. The company had an insane ROIC of 48% last year but my friend expects it to go down over time. The
company also has insane margins and growth expectations, which is why its price has recently rallied like anything.
The implied P/E pricing in these high growth rates comes out to be 28. It is all about reinvestments and growth. My friend does find the company overvalued as current P/E is 47. But that is not the
point. The point of this whole post has been to illustrate the P/E is about the expected growth rates, cash flows and capital efficiency and reinvestments. You can't just look at P/E as a single
standing ratio and make your judgements across companies or markets. There is more nuance to this. Investing is a simple but risky venture. I hope this post was helpful in helping to demystify the P/
E ratio. As always, you can play around with the spreadsheets that I have attached and do read the note by Credit Suisse that I have attached above.
Edit: I did not include the impact of interest/discount rates in the P/E ratio but that is definitely an important part of the equation. The excel sheets include that and you can play with them to
see how discount rates factor in.
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