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# Correlation vs Lasso for Feature Selection
I read from literature that the following two methods can be used for feature selection prior to model development: 1. Correlation factor between target and feature variables (select those features that have correlation > threshold) 2. Lasso
Which of the above two methods is preferred?
In one of the exercises I did, Lasso retained some features which have a lower correlation than the features it dropped. In other words, the above two methods didn't result in the same set of features selected. How do we explain this?
• Lasso acts on the conditional (i.e., partial) correlation between features and the target, whereas the correlation method acts on the marginal correlation between the features and the target. The partial correlation is more relevant for prediction since you will be using all the variables you end up including in future predictions, so I would expect the lasso method to be a better choice.
– Noah
Nov 25, 2019 at 7:19
• Selecting features based on correlations is dubious, (the whole correlation does not equal causation) because 1) the correlation may not be linear or monotonous (Pearson / Spearman), 2) there may be intercorrelation between the variables, which you will not identify with a correlation coefficient. Nov 26, 2019 at 12:23
A couple things:
...following two methods can be used for feature selection prior to model development
Those are actually part of the model development and should be cross validated. What most people do is look at the correlations, select only the most correlated with the output, and then move on to do cross validation etc. That's wrong for two reasons.
1. Correlation measures strength of a linear relationship. If the effect of the variable is non linear, your correlation may not pick up on this. Here is a concrete example. When I worked for a marketing company, most of the customers we dealt with were in their late 30s to early 40s. They spent the most money, and people younger and older spent less because young people typically didn't have as much money or interest in our products. So the effect of age kind of looked like a concave function. If you simulate something like x = rnorm(1000); y = -0.2*x^2 + rnorm(1000, 0, 0.5) (here x has a concave relationship to y) the correlation is low even though x can explain 75% of the variation observed in y. If you removed features based on their correlation, surely you would not select this very important feature.
1. Had you had different training data, you might have picked different features. So when you fit models in the cross validation step, you need to repeat the selection of features based on the correlations. Same thing with the lasso. In every cross validation refit, you need to fit the lasso, select the features, then refit a model with the selected features.
Which of the above two methods is preferred?
I don't think either are very good to be honest. Correlation is myopic for the reasons I've described. Lasso is better, but there is no reason to think the features it selects are the "best" features, nor is there a reason to think that the features it selects would be selected had you had different data. Here is a code example to demonstrate that
library(tidyverse)
library(glmnet)
S = 0
N = 1000
p = 100
mu = rep(0, p)
betas = rnorm(p, 2, 2)*rbinom(p, 1, 0.10)
while(max(abs(S))<0.9){
S = rethinking::rlkjcorr(1, p)
}
do_glmnet<-function(){
X = MASS::mvrnorm(N, mu, S)
y = X %*% betas + rnorm(N, 0, 2.5)
cvmodel = cv.glmnet(X,y, alpha = 1)
model = glmnet(X, y, alpha = 1)
coef(model, cvmodel\$lambda.1se) %>%
as.matrix() %>%
t() %>%
as_tibble()
}
results = map_df(1:100, ~do_glmnet())
results %>%
summarise_all(~mean(abs(.)>0) )
In that example, I generate data from a sparse linear model. Some variables are selected every time (those are the variables with real effects) but you can see that some variables with 0 effect are sometimes selected and sometimes not selected.
The absolute best way to select features is to use your knowledge about the data generating process to determine what is important and what is not. If you can't do that, use lasso to trade off variance for a but of bias, but don't select out features. Just keep the entire fit in the model. I saw Trevor Hastie speak at a zoom talk the other day and he showed us an example of which LASSO with all features performed better than selecting features with LASSO and then refitting the full model. I can't say that is the case for every problem, but it was pretty compelling evidence. Let me see if I can find a link to the talk.
That being said, I'm open to seeing numerical experiments that show that selection via glmnet does better than just putting everything into glmnet and not selecting. That just hasn't been the story I've seen.
I think by saying correlation you are referring to SIS, developed by Jianqing Fan and Jinchi Lv. Actually, the logic behind the two methods is different. LASSO does the selection by using a penalized loss function and sparsity of the variables is required. Normally, for ultra-high dimensional data, we perform SIS first and reduce the dimension to a relatively small amount, and then perform LASSO to further reduce the number of variables that enter the final model.
• Thanks... by Correlation I meant Pearson correlation factor between the target (output) and input variables. Nov 26, 2019 at 15:38
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# Edexcel A Level Further Maths: Core Pure:复习笔记5.2.1 Improper Integrals
### Improper Integrals
#### What are improper integrals?
• An improper integral is a definite integral where one or both of the limits is either:
• Positive or minus infinity
• A point where the function is undefined
• Consider the graph of
• It is undefined at the point x = 0
• The integral of with a limit of zero would be an improper integral
• Examples include:
#### How do we find the value of an improper integral?
• Use algebra to replace the limit which cannot be found with a variable
• E.g. let the undefined limit of zero be a or the infinite limit be b
• Evaluate the integral and substitute your chosen variable into the expression
• Consider what will happen to your answer as the value of your chosen variable tends towards the limit
• E.g. what happens as a gets closer to zero or as b gets closer to infinity?
• E.g. as a tends to zero a2 tends to zero and so this part of your solution will be zero
• It is useful to remember as a tends to infinity then tends to 0
#### Exam Tip
• Be careful if a limit of your integral is zero, always check to see if the function is defined at zero and if not treat it as an improper integral.
• Infinite limits will always be treated as improper integrals.
a)
b)
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# How do you solve 8/(x+2)+8/2=5?
May 26, 2017
$x = 6$
#### Explanation:
Given: $\frac{8}{x + 2} + \frac{8}{2} = 5$
One way to solve is by realizing that $\frac{8}{2} = 4$, so substitute this value into the equation: $\text{ } \frac{8}{x + 2} + 4 = 5$
Simplify by subtracting $4$ from both sides of the equation:
$\frac{8}{x + 2} + 4 - 4 = 5 - 4$;
$\frac{8}{x + 2} = 1$
Multiply both sides of the equation by $x + 2$:
$\cancel{x + 2} \cdot \frac{8}{\cancel{x + 2}} = 1 \cdot \left(x + 2\right)$
Simplify: $\text{ } 8 = x + 2$
Subtract both sides of the equation by $2$:
$8 - 2 = x + 2 - 2$
$6 = x$
A second way to solve is by finding a common denominator for both sides of the equation $2 \left(x + 2\right)$:
$\frac{8}{x + 2} \cdot \frac{2}{2} + \frac{8}{2} \cdot \frac{x + 2}{x + 2} = 5 \cdot \frac{2 \left(x + 2\right)}{2 \left(x + 2\right)}$
Simplify:
$\frac{16 + 8 \left(x + 2\right)}{2 \left(x + 2\right)} = \frac{10 \left(x + 2\right)}{2 \left(x + 2\right)}$
Since both denominators are equal, we can set the numerators equal to solve:
$16 + 8 \left(x + 2\right) = 10 \left(x + 2\right)$
Distribute:
$16 + 8 x + 16 = 10 x + 20$
Add like terms on the same side:
$32 + 8 x = 10 x + 20$
Subtract $20$ from both sides:
$32 - 20 + 8 x = 10 x + 20 - 20$
$12 + 8 x = 10 x$
Subtract $8 x$ from both sides: $\text{ } 12 + 8 x - 8 x = 10 x - 8 x$
Simplify: $\text{ } 12 = 2 x$
Divide both sides by $2 : \text{ } \frac{12}{2} = \frac{2 x}{2}$
Simplify: $\text{ } 6 = x$
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## Differential and Integral Equations
### Smooth solutions for an integro-differential equation of parabolic type
#### Article information
Source
Differential Integral Equations, Volume 2, Number 1 (1989), 111-121.
Dates
First available in Project Euclid: 25 June 2013
Permanent link to this document
https://projecteuclid.org/euclid.die/1372191618
Mathematical Reviews number (MathSciNet)
MR960018
Zentralblatt MATH identifier
0719.45007
#### Citation
Cannon, John R.; Lin, Yan Ping. Smooth solutions for an integro-differential equation of parabolic type. Differential Integral Equations 2 (1989), no. 1, 111--121. https://projecteuclid.org/euclid.die/1372191618
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# 1.9: 1.9 Asymptotes and End Behavior
Most functions continue beyond the viewing window in our calculator or computer. People often draw an arrow next to a dotted line to indicate the pattern specifically. How can you recognize these asymptotes?
### Asymptotes and End Behavior of Functions
A vertical asymptote is a vertical line such as $$x=1$$ that indicates where a function is not defined and yet gets infinitely close to.
A horizontal asymptote is a horizontal line such as $$y=4$$ that indicates where a function flattens out as $$x$$ gets very large or very small. A function may touch or pass through a horizontal asymptote.
The reciprocal function has two asymptotes, one vertical and one horizontal. Most computers and calculators do not draw the asymptotes and so they must be inserted by hand as dotted lines.
Many students have the misconception that an asymptote is a line that a function gets infinitely close to but does not touch. This is not true. Take the following function:
The graph appears to flatten as $$x$$ grows larger. Thus, the horizontal asymptote is $$y=0$$ even though the function clearly passes through this line an infinite number of times.
The reason why asymptotes are important is because when your perspective is zoomed way out, the asymptotes essentially become the graph.
To find the asymptotes and end behavior of the function below, examine what happens to $$x$$ and $$y$$ as they each increase or decrease.
The function has a horizontal asymptote $$y=2$$ as $$x$$ approaches negative infinity. There is a vertical asymptote at $$x=0$$. The right hand side seems to decrease forever and has no asymptote.
Note that slant asymptotes do exist and are called oblique asymptotes.
### Examples
Example 1
Earlier, you were asked how to identify asymptotes on a graph. Asymptotes written by hand are usually identified with dotted lines next to the function that indicate how the function will behave outside the viewing window. The equations of these vertical and horizontal dotted lines are of the form $$x$$=___ and $$y$$=____. When problems ask you to find the asymptotes of a function, they are asking for the equations of these horizontal and vertical lines.
Example 2
Identify the horizontal and vertical asymptotes of the following function.
There is a vertical asymptote at $$x=0$$. As $$x$$ gets infinitely small, there is a horizontal asymptote at $$y=-1$$. As $$x$$ gets infinitely large, there is another horizontal asymptote at $$y=1$$.
Example 3
Identify the horizontal and vertical asymptotes of the following function.
There is a vertical asymptote at $$x=2 .$$ As $$x$$ gets infinitely small there is a horizontal asymptote at $$y=-1$$. As $$x$$ gets infinitely large, there is a horizontal asymptote at $$y=1$$.
Example 4
Identify the horizontal and vertical asymptotes of the following piecewise function:
$$f(x)=\left\{\begin{array}{ll}e^{x}-1 & x \leq 0 \\ \sin x & 0<x\end{array}\right.$$
There is a horizontal asymptote at $$y=-1$$ as $$x$$ gets infinitely small. This is because $$e$$ raised to the power of a very small number becomes $$0.000000 \ldots$$ and basically becomes zero.
Example 5
Identify the asymptotes and end behavior of the following function.
There is a vertical asymptote at $$x=0$$. The end behavior of the right and left side of this function does not match. The horizontal asymptote as $$x$$ approaches negative infinity is $$y=0$$ and the horizontal asymptote as $$x$$ approaches positive infinity is $$y=4 .$$ At this point you can only estimate these heights because you were not given the function or the tools to find these values analytically.
Review
Identify the asymptotes and end behavior of the following functions.]
1. $$y=x$$
2. $$y=x^{2}$$
3. $$y=x^{3}$$
4. $$y=\sqrt{x}$$
5. $$y=\frac{1}{x}$$
6. $$y=e^{x}$$
7. $$y=\ln (x)$$
8. $$y=\frac{1}{1+e^{-x}}$$
9.
10.
11.
12. Vertical asymptotes occur at $$x$$ values where a function is not defined. Explain why it makes sense that $$y=\frac{1}{x}$$ has a vertical asymptote at $$x=0$$.
13. Vertical asymptotes occur at $$x$$ values where a function is not defined. Explain why it makes sense that $$y=\frac{1}{x+3}$$ has a vertical asymptote at $$x=-3$$.
14. Use the technique from the previous problem to determine the vertical asymptote for the function $$y=\frac{1}{x-2}$$
15. Use the technique from problem #13 to determine the vertical asymptote for the function $$y=\frac{2}{x+4}$$
c
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# Statistical mechanics
Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force.
It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in everyday life, therefore explaining thermodynamics as a natural result of statistics and mechanics (classical and quantum) at the microscopic level. In particular, it can be used to calculate the thermodynamic properties of bulk materials from the spectroscopic data of individual molecules.
This ability to make macroscopic predictions based on microscopic properties is the main asset of statistical mechanics over thermodynamics. Both theories are governed by the second law of thermodynamics through the medium of entropy. However, Entropy in thermodynamics can only be known empirically, whereas in Statistical mechanics, it is a function of the distribution of the system on its micro-states.
## Microcanonical ensemble
Since the second law of thermodynamics applies to isolated systems, the first case investigated will correspond to this case. The Microcanonical ensemble describes an isolated system.
The entropy of such a system can only increase, so that the maximum of its entropy corresponds to an equilibrium state for the system.
Because an isolated systems keeps a constant energy, the total energy of the system does not fluctuate. Thus, the system can access only those of its micro-states that correspond to a given value E of the energy. The internal energy of the system is then strictly equal to its energy.
Let us call ${\displaystyle ''\Omega (E)}$ the number of micro-states corresponding to this value of the system's energy. The macroscopic state of maximal entropy for the system is the one in which all micro-states are equally likely to occur during the system's fluctuations.
${\displaystyle S=k_{B}\ln \left(\Omega (E)\right)\,}$
where
${\displaystyle S}$ is the system entropy,
${\displaystyle k_{B}}$ is Boltzmann's constant
## Canonical ensemble
Invoking the concept of the canonical ensemble, it is possible to derive the probability ${\displaystyle P_{i}}$ that a macroscopic system in thermal equilibrium with its environment will be in a given microstate with energy ${\displaystyle E_{i}}$:
${\displaystyle P_{i}={\exp \left(-\beta E_{i}\right) \over {\sum _{j}^{j_{max}}\exp \left(-\beta E_{j}\right)}}}$
where ${\displaystyle \beta ={1 \over {kT}}}$,
The temperature T arises from the fact that the system is in thermal equilibrium with its environment . The probabilities of the various microstates must add to one, and the normalization factor in the denominator is the canonical partition function:
${\displaystyle Z=\sum _{j}^{j_{max}}\exp \left(-\beta E_{j}\right)}$
where ${\displaystyle E_{i}}$ is the energy of the ith microstate of the system. The partition function is a measure of the number of states accessible to the system at a given temperature. See derivation of the partition function for a proof of Boltzmann's factor and the form of the partition function from first principles.
To sum up, the probability of finding a system at temperature T in a particular state with energy Ei is
${\displaystyle P_{i}={\frac {\exp(-\beta E_{i})}{Z}}}$
### Thermodynamic Connection
The partition function can be used to find the expected (average) value of any microscopic property of the system, which can then be related to macroscopic variables. For instance, the expected value of the microscopic energy E is interpreted as the microscopic definition of the thermodynamic variable internal energy (U)., and can be obtained by taking the derivative of the partition function with respect to the temperature. Indeed,
${\displaystyle \langle E\rangle ={\sum _{i}E_{i}e^{-\beta E_{i}} \over Z}=-{dZ \over d\beta }/Z}$
implies, together with the interpretation of <E> as U, the following microscopic definition of internal energy:
${\displaystyle U\colon =-{d\ln Z \over d\beta }.}$
The entropy can be calculated by (see Shannon entropy)
${\displaystyle {S \over k}=-\sum _{i}p_{i}\ln p_{i}=\sum _{i}{e^{-\beta E_{i}} \over Z}(\beta E_{i}+\ln Z)=\ln Z+\beta U}$
which implies that
${\displaystyle -{\frac {\ln(Z)}{\beta }}=U-TS=F}$
is the Free energy of the system or in other words,
${\displaystyle Z=e^{-\beta F}\,}$
Having microscopic expressions for the basic thermodynamic potentials U (internal energy), S (entropy) and F (free energy) is sufficient to derive expressions for other thermodynamic quantities. The basic strategy is as follows. There may be an intensive or extensive quantity that enters explicitly in the expression for the microscopic energy Ei, for instance magnetic field (intensive) or volume (extensive). Then, the conjugate thermodynamic variables are derivatives of the internal energy. For instance, the macroscopic magnetization (extensive) is the derivative of U with respect to the (intensive) magnetic field, and the pressure (intensive) is the derivative of U with respect to volume (extensive).
The treatment in this section assumes no exchange of matter (i.e. fixed mass and fixed particle numbers). However, the volume of the system is variable which means the density is also variable.
This probability can be used to find the average value, which corresponds to the macroscopic value, of any property, ${\displaystyle J}$, that depends on the energetic state of the system by using the formula:
${\displaystyle \langle J\rangle =\sum _{i}p_{i}J_{i}=\sum _{i}J_{i}{\frac {\exp(-\beta E_{i})}{Z}}}$
where ${\displaystyle }$ is the average value of property ${\displaystyle J}$. This equation can be applied to the internal energy, ${\displaystyle U}$:
${\displaystyle U=\sum _{i}E_{i}{\frac {\exp(\beta E_{i})}{Z}}}$
Subsequently, these equations can be combined with known thermodynamic relationships between ${\displaystyle U}$ and V to arrive at an expression for pressure in terms of only temperature, volume and the partition function. Similar relationships in terms of the partition function can be derived for other thermodynamic properties as shown in the following table.
Helmholtz free energy: ${\displaystyle F=-{\ln Z \over \beta }}$ Internal energy: ${\displaystyle U=-\left({\frac {\partial \ln Z}{\partial \beta }}\right)_{N,V}}$ Pressure: ${\displaystyle P=-\left({\partial F \over \partial V}\right)_{N,T}={1 \over \beta }\left({\frac {\partial \ln Z}{\partial V}}\right)_{N,T}}$ Entropy: ${\displaystyle S=k(\ln Z+\beta U)\,}$ Gibbs free energy: ${\displaystyle G=F+PV=-{\ln Z \over \beta }+{V \over \beta }\left({\frac {\partial \ln Z}{\partial V}}\right)_{N,T}}$ Enthalpy: ${\displaystyle H=U+PV\,}$ Constant Volume Heat capacity: ${\displaystyle C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{N,V}}$ Constant Pressure Heat capacity: ${\displaystyle C_{P}=\left({\frac {\partial U}{\partial T}}\right)_{N,P}}$ Chemical potential: ${\displaystyle \mu _{i}=-{1 \over \beta }\left({\frac {\partial \ln Z}{\partial N_{i}}}\right)_{T,V,N}}$
The last entry needs clarification. We are NOT working with a grand canonical ensemble here.
It is often useful to consider the energy of a given molecule to be distributed among a number of modes. For example, translational energy refers to that portion of energy associated with the motion of the center of mass of the molecule. Configurational energy refers to that portion of energy associated with the various attractive and repulsive forces between molecules in a system. The other modes are all considered to be internal to each molecule. They include rotational, vibrational, electronic and nuclear modes. If we assume that each mode is independent (a questionable assumption) the total energy can be expressed as the sum of each of the components:
${\displaystyle E=E_{t}+E_{c}+E_{n}+E_{e}+E_{r}+E_{v}\,}$
Where the subscripts t, c, n, e, r, and v correspond to translational, configurational, nuclear, electronic, rotational and vibrational modes, respectively. The relationship in this equation can be substituted into the very first equation to give:
${\displaystyle Z=\sum _{i}\exp \left(-\beta (E_{ti}+E_{ci}+E_{ni}+E_{ei}+E_{ri}+E_{vi})\right)}$
${\displaystyle =\sum _{i}\exp \left(-\beta E_{ti}\right)\exp \left(-\beta E_{ci}\right)\exp \left(-\beta E_{ni}\right)\exp \left(-\beta E_{ei}\right)\exp \left(-\beta E_{ri}\right)\exp \left(-\beta E_{vi}\right)}$
If we can assume all these modes are completely uncoupled and uncorrelated, so all these factors are in a probability sense completely independent, then
${\displaystyle Z=Z_{t}Z_{c}Z_{n}Z_{e}Z_{r}Z_{v}\,}$
Thus a partition function can be defined for each mode. Simple expressions have been derived relating each of the various modes to various measurable molecular properties, such as the characteristic rotational or vibrational frequencies.
Expressions for the various molecular partition functions are shown in the following table.
Nuclear ${\displaystyle Z_{n}=1\qquad (T<10^{8}K)}$ Electronic ${\displaystyle Z_{e}=W_{0}\exp(kTD_{e}+W_{1}\exp(-\theta _{e1}/T)+\cdots )}$ vibrational ${\displaystyle Z_{v}=\prod _{j}{\frac {\exp(-\theta _{vj}/2T)}{1-\exp(-\theta _{vj}/T)}}}$ rotational (linear) ${\displaystyle Z_{r}={\frac {T}{\sigma }}\theta _{r}}$ rotational (non-linear) ${\displaystyle Z_{r}={\frac {1}{\sigma }}{\sqrt {\frac {{\pi }T^{3}}{\theta _{A}\theta _{B}\theta _{C}}}}}$ Translational ${\displaystyle Z_{t}={\frac {(2\pi mkT)^{3/2}}{h^{3}}}}$ Configurational (ideal gas) ${\displaystyle Z_{c}=V\,}$
These equations can be combined with those in the first table to determine the contribution of a particular energy mode to a thermodynamic property. For example the "rotational pressure" could be determined in this manner. The total pressure could be found by summing the pressure contributions from all of the individual modes, ie:
${\displaystyle P=P_{t}+P_{c}+P_{n}+P_{e}+P_{r}+P_{v}\,}$
## Grand canonical ensemble
If the system under study is an open system, (matter can be exchanged), and particle number is conserved, we would have to introduce chemical potentials, μj, j=1,...,n and replace the canonical partition function with the grand canonical partition function:
${\displaystyle \Xi (V,T,\mu )=\sum _{i}\exp \left(\beta \left[\sum _{j=1}^{n}\mu _{j}N_{ij}-E_{i}\right]\right)}$
where Nij is the number of jth species particles in the ith configuration. Sometimes, we also have other variables to add to the partition function, one corresponding to each conserved quantity. Most of them, however, can be safely interpreted as chemical potentials. In most condensed matter systems, things are nonrelativistic and mass is conserved. However, most condensed matter systems of interest also conserve particle number approximately (metastably) and the mass (nonrelativistically) is none other than the sum of the number of each type of particle times its mass. Mass is inversely related to density, which is the conjugate variable to pressure. For the rest of this article, we will ignore this complication and pretend chemical potentials don't matter. See grand canonical ensemble.
Let's rework everything using a grand canonical ensemble this time. The volume is left fixed and does not figure in at all in this treatment. As before, j is the index for those particles of species j and i is the index for microstate i:
${\displaystyle U=\sum _{i}E_{i}{\frac {\exp(-\beta (E_{i}-\sum _{j}\mu _{j}N_{ij}))}{\Xi }}}$
${\displaystyle N_{j}=\sum _{i}N_{ij}{\frac {\exp(-\beta (E_{i}-\sum _{j}\mu _{j}N_{ij}))}{\Xi }}}$
Gibbs free energy: ${\displaystyle G=-{\ln \Xi \over \beta }}$ Internal energy: ${\displaystyle U=-\left({\frac {\partial \ln \Xi }{\partial \beta }}\right)_{\mu }+\sum _{i}{\mu _{i} \over \beta }\left({\partial \ln \Xi \over \partial \mu _{i}}\right)_{\beta }}$ Particle number: ${\displaystyle N_{i}={1 \over \beta }\left({\partial \ln \Xi \over \partial \mu _{i}}\right)_{\beta }}$ Entropy: ${\displaystyle S=k(\ln \Xi +\beta U-\beta \sum _{i}\mu _{i}N_{i})\,}$ Helmholtz free energy: ${\displaystyle F=G+\sum _{i}\mu _{i}N_{i}=-{\ln \Xi \over \beta }+\sum _{i}{\mu _{i} \over \beta }\left({\frac {\partial \ln \Xi }{\partial \mu _{i}}}\right)_{\beta }}$
## Equivalence between descriptions at the thermodynamic limit
All the above descriptions differ in the way they allow the given system to fluctuate between its configurations.
In the micro-canonical ensemble, the system exchanges no energy with the outside world, and is therefore not subject to energy fluctuations, while in the canonical ensemble, the system is free to exchange energy with the outside in the form of heat.
In the thermodynamic limit, which is the limit of large systems, fluctuations become negligible, so that all these descriptions converge to the same description. In other words, the macroscopic behavior of a system does not depend on the particular ensemble used for its description.
Given these considerations, the best ensemble to choose for the calculation of the properties of a macroscopic system is that ensemble which allows the result be most easily derived.
A Table of Statistical Mechanics Articles
Maxwell Boltzmann Bose-Einstein Fermi-Dirac
Particle Boson Fermion
Statistics
Statistics Bose-Einstein statistics Fermi-Dirac statistics
Thomas-Fermi
approximation
gas in a box
gas in a harmonic trap
Gas Ideal gas
Chemical
Equilibrium
Classical Chemical equilibrium
## References
• Huang, Kerson (1990). Statistical Mechanics, Wiley, John & Sons, Inc. ISBN 0471815187.
• Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.), W. H. Freeman Company. ISBN 0716710889.
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# Change a parameter in document using the cmd?
I need to be able to change the value of a parameter inside a LaTeX document without having to change it manually. I'll explain:
There is one main program that executes another smaller one. This smaller program computes some things and creates a .txt-file with a specific name and all its results inside. Then the main program executes the LaTeX file I have to create which will process the created .txt-file into a nice pdf (using commands such as \openin\txtfile=\filename.txt and the package datatool).
Problem is that LaTeX needs to know the name of the newly created .txt-file. The main program knows the name it and will call the LaTeX file by using the cmd command > latex texfile.tex. But this won't change the value of \filename in any way.
Is there actually any possible way to change a value (of \filename) inside the LaTeX file using the cmd? Because I have absolutely no clue and Google doesn't help me either...
-
You do pdflatex \def\filename{abc}\input{texfile} – Ulrike Fischer Jul 16 '12 at 13:52
Thanks will try it :) – Didii Jul 16 '12 at 15:17
Thanks to Ulrike Fischer for the solution:
> pdflatex \cmd\input{texfile}
\cmd can be any command LaTeX recognizes. And input{texfile} pastes all contents of texfile.tex into the cmd.
-
Potentially you might want to change the jobname
latex -jobname=$filename texfile.tex where$filename is the name of your .txt file. Then inside your .tex file you could use
\openin\txtfile=\jobname
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The profile you are now visiting: Michael Wolf. Go back to Past records to show all talks or carry out a new search.
# Polynomial Pick forms for affine spheres and real projective polygons.
## Michael Wolf Rice University
(Joint work with David Dumas.) Convex real projective structures on surfaces, corresponding to discrete surface group representations into $\mathrm{PSL}(3, \mathbb{R})$, have associated to them affine spheres which project to the convex hull of their universal covers. Such an affine sphere is determined by its Pick (cubic) differential and an associated Blaschke metric (inducing a minimal map to $\mathrm{SL}(3,\mathbb{R})/SO(3))$. As a sequence of convex projective structures leaves all compacta in its deformation space, a subclass of the limits is described by polynomial Pick cubic differentials on affine spheres which are conformally the complex plane. We show that those particular affine spheres project to polygons, and all polygons are obtained this way; along the way, a strong estimate on asymptotics is found.
# Michael Wolf
## Rice University
Number of talks
1
Number of visits
1
Last visit
Personal website
Profile in Mathscinet
Profile in Zentralblatt
Country of origin
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# Why I got the same action when testing the A2C?
I'm working on an advantage actor-critic (A2C) reinforcement learning model, but when I test the model after I trained for 3500 episodes, I start to get almost the same action for all testing episodes. While if I trained the system for less than 850 episodes, I got different actions. The value of state is always different, and around 850 episodes, the loss becomes zero.
Here is the Actor and critic Network
with g.as_default():
#==============================actor==============================#
actorstate = tf.placeholder(dtype=tf.float32, shape=n_input, name='state')
actoraction = tf.placeholder(dtype=tf.int32, name='action')
actortarget = tf.placeholder(dtype=tf.float32, name='target')
hidden_layer1 = tf.layers.dense(inputs=tf.expand_dims(actorstate, 0), units=500, activation=tf.nn.relu, kernel_initializer=tf.zeros_initializer())
hidden_layer2 = tf.layers.dense(inputs=hidden_layer1, units=250, activation=tf.nn.relu, kernel_initializer=tf.zeros_initializer())
hidden_layer3 = tf.layers.dense(inputs=hidden_layer2, units=120, activation=tf.nn.relu, kernel_initializer=tf.zeros_initializer())
output_layer = tf.layers.dense(inputs=hidden_layer3, units=n_output, kernel_initializer=tf.zeros_initializer())
action_probs = tf.squeeze(tf.nn.softmax(output_layer))
picked_action_prob = tf.gather(action_probs, actoraction)
actorloss = -tf.log(picked_action_prob) * actortarget
# actorloss = tf.reduce_mean(tf.losses.huber_loss(picked_action_prob, actortarget, delta=1.0), name='actorloss')
if var.opt == 2:
actoroptimizer1 = tf.train.RMSPropOptimizer(learning_rate=var.learning_rate, momentum=0.95,
epsilon=0.01)
elif var.opt == 0:
actortrain_op = actoroptimizer1.minimize(actorloss)
init = tf.global_variables_initializer()
saver = tf.train.Saver(max_to_keep=var.n)
p = tf.Graph()
with p.as_default():
#==============================critic==============================#
criticstate = tf.placeholder(dtype=tf.float32, shape=n_input, name='state')
critictarget = tf.placeholder(dtype=tf.float32, name='target')
hidden_layer4 = tf.layers.dense(inputs=tf.expand_dims(criticstate, 0), units=500, activation=tf.nn.relu, kernel_initializer=tf.zeros_initializer())
hidden_layer5 = tf.layers.dense(inputs=hidden_layer4, units=250, activation=tf.nn.relu, kernel_initializer=tf.zeros_initializer())
hidden_layer6 = tf.layers.dense(inputs=hidden_layer5, units=120, activation=tf.nn.relu, kernel_initializer=tf.zeros_initializer())
output_layer2 = tf.layers.dense(inputs=hidden_layer6, units=1, kernel_initializer=tf.zeros_initializer())
value_estimate = tf.squeeze(output_layer2)
criticloss= tf.reduce_mean(tf.losses.huber_loss(output_layer2, critictarget,delta = 0.5), name='criticloss')
if var.opt == 2:
optimizer2 = tf.train.RMSPropOptimizer(learning_rate=var.learning_rate_c, momentum=0.95,
epsilon=0.01)
elif var.opt == 0:
update_step2 = optimizer2.minimize(criticloss)
init2 = tf.global_variables_initializer()
saver2 = tf.train.Saver(max_to_keep=var.n)
This is the choice of action.
def take_action(self, state):
"""Take the action"""
action_probs = self.actor.predict(state)
action = np.random.choice(np.arange(len(action_probs)), p=action_probs)
return action
This is the actor.predict function.
def predict(self, s):
return self._sess.run(self._action_probs, {self._state: s})
Any Idea what causing this?
Update
Change the learning rate, state, and the reward solve the problem where I reduce the size of the state and also added switching cost to the reward.
• Not really possible to answer without seeing the full code. Can you post a colab notebook with a MWE? At the least we need more details such as a) what is the MDP, b) how are actions parametrized. You say you're getting the same action (I assume you mean over all timesteps in the episode) - is it possible that that is the optimal policy?
– Taw
Aug 22 at 21:12
• I got different actions when I train the model and it converges perfectly, but it is not changing during the testing for all the episodes. Aug 22 at 21:19
• The code is taken from datahubbs.com/policy-gradients-and-advantage-actor-critic and stackoverflow.com/questions/45428574/… Aug 22 at 21:44
• @Taw, I have added the code. Aug 23 at 19:50
Disclaimer: Without the full code, we can only speculate. I encourage you to post the full code on Google Colab or something like this. In the meanwhile, here is my point of view:
### The Problem
Looks like your model has found some "master action" that always leads to zero loss, no matter what the state is. So it's not necessarily bad, it's just unexpected according to your point of view.
An example for that would be pausing the game - so you never loose.
You might not like it, but in de model's point of view, it's absolutely nailing it!
### The Solution
So how to convince the actor not to pause the game?
Not by changing the model, or tuning hyper-parameters, but by reformulating the problem. In this example, instead of just penalizing the model for failing, you should reward if for winning, so pausing is no longer the best option.
### Conclusion
It might not be a problem in the Machine Learning model, but in your environment and reward models. As we don't have access to that, it's hard to provide an answer.
### Edit:
You are the CartPole-v0 environment:
A reward of +1 is provided for every timestep that the pole remains upright. The episode ends when the pole is more than 15 degrees from vertical, or the cart moves more than 2.4 units from the center.
It is a solvable problem. Probably your model has just learned how to solve it after a few hundreds generations. (The link shows a table with "Episodes before solve" for each algorithm, showing numbers consistent to yours).
### TL;DR: It's not a bug, it's a feature!
If the loss is zero, all gradients should be zero as well, so you should take a look at the computed gradients. There might be a problem with momentum or the scheduled lr which might still apply very little updates which eventually lead to this policy colapse.
On a side note I would also call reduce_mean on the actor loss since you're optimizing the expected value.
• My code was like this actorloss = tf.reduce_mean(tf.losses.huber_loss(picked_action_prob, actortarget, delta=1.0), name='actorloss') but nothing change. Aug 27 at 14:19
• Are the gradients zero? Aug 28 at 11:42
• No, it is not zero. Aug 28 at 14:07
• But your loss is? Because if it is there might be some kind of momentum from your optimizer applying steps Aug 28 at 14:11
• You need to find out the source of the non zero gradient. Your policy keeps moving (probably in the same direction) while your loss is zero. This eventually leads to selecting the same action almost every time and the policy colapse by this. You could also try lr decay, but I feel like it only minimizes the problem instead of solving it. Aug 29 at 11:28
|
{}
|
## 2014-06-01
### Back to Optimizing
It's been a while since I lost wrote, but I have a good excuse, I've been focusing on creating the Android version of my game. The good news is that everything is working now. It's running on both Android 2.3+ and a slightly improved version for Android 4.4+ (it shows in full-screen). What I'm working on now is optimizing it to get the performance I want. I've written articles about optimization in the past, so I'll keep this short and to the point of what I've done so far.
## Measure, Measure, Measure
I've written about this in the past, but I've found that when you're trying to optimize something it is extremely important to measure often and enough. Also, optimizing for a focused measurement doesn't always result in improvements in the real environment.
To try to do a better job at measuring I've been using histograms of the speed of what I'm measuring to get a better picture than just an average speed. I'm also using the profiler functionality in Visual Studio, but I can only use that for the PC version of my game, not directly against the Android binary. Still this led to useful insight.
## Two Optimizations
When I profiled my code I found two areas I thought I could improve relatively easily. The first was an unnecessary calculation when I manipulate (translate or rotate) quads. I was recalculating the center of the quads even though for these operations that wasn't necessary. This was pretty straight-forward to improve and resulted in a significant speedup. You can see this in the picture to the right: PC-L-SS is the original version, and PC-L-SS-Opt Mutable is this first optimization.
After the above I ran the profiler again and determined that a significant amount of time was being spent on calculating sine and cosine operations. I did some Google searching and found an interesting discussion that describes a faster way to calculate sine (the source of this information contains more details). So, I implemented this alternative sine calculation and measured again. This resulted in the PC-L-SS-FastSin numbers in the histogram.
The average FPS numbers for the above are: original = 1,114, first optimization = 1,380, and second optimization = 1,411. Note that this is just running my game logic code, but not actually rendering via OpenGL. I did this in order to focus as much as possible on the area I was optimizing.
So, the above is interesting and all, but I'm trying to optimize Android, not my PC version. So, since I can't use the profiler, I can just run the same measurements. This resulted in the histogram to the right.
Clearly from this both the first optimization (Opt Mutable) and the second additional optimization (Opt Sine) are significantly faster than the original. To summarize this even more, the average FPS for each are: original = 37.29, first optimization = 42.95, second optimization = 43.45. In this case the numbers are both the game logic and the actual rendering. Because I'm trying to evaluate the true game performance, in my attempt to get to 60 FPS, I thought this was more useful.
## Still More Work to Do
While I'm pretty happy with the results of this initial optimization, it's still not performing as well as I want. The Android measurements above were done on a Nexus 4, and I'd really like it to run at 60 FPS with some headroom for slower devices. My profiler investigations have given me some ideas for more optimizations, but I expect them to be more difficult than the above.
## 2014-04-16
### And...my first AppStore update
I pushed a minor update to Zoing to the AppStore on Saturday and it just went live. It's not particularly interesting in terms of changes to the game, but it's exciting for me given this is my first one ever.
The actual update just rearranges the icons on the start screen to try to make it easier to use. Now I'm working on Android support, so I don't expect any new iOS updates until after the Android version is released. Although, before then, I expect the ad-supported free version for iOS to become available.
## 2014-04-14
### Released!
This post is a little delayed, but I finally released Zoing to the App Store about two weeks ago! It's available in all countries. Please check it out and write a nice review, 5 stars would also be great.
I also just submitted the free, ad-supported version to the App Store two days ago. Apple typically takes about 10 days to review new apps, so I'll announce it here when that's available. It's exactly the same game, but shows ads on some of the screens, although not on the main game screens.
## 2014-02-10
### Who stole my pixels?
Moving away from the overly detailed discussion on alpha layering from my last post, today I want to briefly discuss developing for iOS and taking advantage of the full resolution of retina devices. Although I knew some care was needed in order to use the full high-resolution retina display, for some reason I assumed this was already happening. Recently, however, I realized that that wasn't true. A little Googling found lot's of info on this and it all comes down to scaling between points and pixels.
## Scaling the heights
There are many discussions on this topic, so I won't go into too much detail, but I did want to mention how I addressed the problem. Briefly, for those who don't know, when Apple released their first high-resolution retina devices they did it in a careful way that allowed all existing apps to still run and look good. Basically they added a new property of the screen that defines what it's scale is. The official Apple documentation for UIScreen.Scale says:
This value reflects the scale factor needed to convert from the default logical coordinate space into the device coordinate space of this screen. The default logical coordinate space is measured using points. For standard-resolution displays, the scale factor is 1.0 and one point equals one pixel. For Retina displays, the scale factor is 2.0 and one point is represented by four pixels.
To use this scale I needed to do three things:
1. Set the ContentScaleFactor of my EAGLView, which is a subclass of iPhoneOSGameView to the same scale so that my rendering uses the entire high-resolution screen
2. Tell my game the full high-resolution size of the screen
3. Multiply all of my coordinates by the defined scale factor
For the first item I both get and remember the scale factor and set the ContentScaleFactor to it in the initialization of my EAGLView. Because I want my game to run on older devices with older versions of iOS I need to first check that the new scale property exists and then only reference it if it does. The purpose of ContentScaleFactor is defined by Apple as:
Hmm, reading over this now I just noticed that it says I shouldn't have to adjust the value of ContentScaleFactor. I'll need to go back and look at my code again.
The scale factor determines how content in the view is mapped from the logical coordinate space (measured in points) to the device coordinate space (measured in pixels). This value is typically either 1.0 or 2.0. Higher scale factors indicate that each point in the view is represented by more than one pixel in the underlying layer. For example, if the scale factor is 2.0 and the view frame size is 50 x 50 points, the size of the bitmap used to present that content is 100 x 100 pixels.
The default value for this property is the scale factor associated with the screen currently displaying the view. If your custom view implements a custom drawRect: method and is associated with a window, or if you use the GLKView class to draw OpenGL ES content, your view draws at the full resolution of the screen. For system views, the value of this property may be 1.0 even on high resolution screens.
In general, you should not need to modify the value in this property. However, if your application draws using OpenGL ES, you may want to change the scale factor to trade image quality for rendering performance. For more information on how to adjust your OpenGL ES rendering environment, see “Supporting High-Resolution Displays” in OpenGL ES Programming Guide for iOS.
Here's the code that does that:
_screenScale = 1;
if(UIScreen.MainScreen.RespondsToSelector(new Selector("scale"))
&& RespondsToSelector(new Selector("contentScaleFactor")))
{
_screenScale = ContentScaleFactor = UIScreen.MainScreen.Scale;
}
For the second item I simply multiply the screen size by the scale I determined above via this code snippet: new Size(Frame.Size.Width, Frame.Size.Height) * _screenScale.
Finally, for the third item, I perform another multiplication of the scale when I handle touch events.
/// <summary>Called when touch events end.</summary>
/// <param name="touches">Touches.</param>
/// <param name="e">The touch event.</param>
public override void TouchesEnded(NSSet touches, UIEvent e)
{
base.TouchesEnded(touches, e);
UITouch touch = (UITouch)touches.AnyObject;
_lastTouch = touch.LocationInView(this);
_screenManager.Screen.HandleTouch(new Point(_firstTouch.X, _firstTouch.Y) * _screenScale, new Point(_lastTouch.X, _lastTouch.Y) * _screenScale, true);
}
/// <summary>Called when a touch event moves.</summary>
/// <param name="touches">Touches.</param>
/// <param name="e">The touch event.</param>
public override void TouchesMoved(NSSet touches, UIEvent e)
{
UITouch touch = (UITouch)touches.AnyObject;
_lastTouch = touch.LocationInView(this);
_screenManager.Screen.HandleTouch(new Point(_firstTouch.X, _firstTouch.Y) * _screenScale, new Point(_lastTouch.X, _lastTouch.Y) * _screenScale, false);
}
All of the above code happens in my iOS specific EAGLView class and the rest of my code needed no adjustments.
Next: coming soon...
## 2014-02-07
### Alpha layering
In a recent post I discussed premultiplied alpha in textures. After that I thought I had my alphas under control, but it turns out I was wrong.
## Keeping score
Early in my game development I came up with a slightly interesting way to render text. In the game, text is used only to show the game score and is generally quite large on the screen. The score advances every frame, or 1/60th of a second. Because that advancement would actually change too quickly for the eye to see I actually truncate the last digit and only render the remaining ones. So if the score advances frame-by-frame as 137 → 138 → 139 → 140 → 141, I would actually render it as 13 → 13 → 13 → 14 → 14. Even doing this the rendered score visibly advances every 10 frames, or every 1/6th of a second. This is still pretty rapid, and I wanted to visually smooth that quickly changing score when it's rendered on the screen.
The idea I settled on was to keep track of the last N instances of the score each time a frame in rendered. In the above example, if N was 5, that would be the list of numbers shown. One frame later the list would change to 13 → 13 → 14 → 14 → 14, and 7 frames later it would become 14 → 14 → 14 → 14 → 15. After keeping track of these last N scores I then actually rendered them all with the oldest score being very transparent and each successive score being more and more opaque. This gives an effect like the one shown in the image to the right. Actually that image is showing the score rendered as it is also being rotated in order to make the separate layers easier to see, in most cases the score is not rotating, so each layer is stacked directly on top of the previous one.
This layered transparency effect worked out well and I've had it in place since quite early in the game development. The way I actually implemented this was to take my desired alpha and figure out how many fractional shades of that I would need such that when I layered them all together I would get my desired alpha. For example, for the score when it is shown during a game it is dimmed so as not to distract from game play and has an intended alpha of 0.25. After the game when showing the final score it is brightened to full opacity, or alpha 1.0. If I have 5 layers then I decided to break the desired alpha into 15 fractions 1 + 2 + 3 + 4 + 5. This is also $$\frac{N \times (N+1)}{2}$$ or $$\frac{N^2 + N}{2}$$. So, for the dimmed alpha of 0.25 I calculated the fractional alpha as $$\frac{0.25}{15} = 0.0167$$. Then I rendered the oldest score layer with an alpha of 0.0167, the second oldest as $$0.0167 \times 2 = 0.0333$$, etc, up to the newest score as $$0.0167 \times 5 = 0.0833$$. When all of these were layered on top of one another it produced the desired 0.25 alpha...or so I thought.
One problem I periodically ran into was that if I adjusted a parameter related to the transparency, like making the text slightly lighter or darker, or make the number of layers more or fewer, then the resulting aggregated color often ended up being not what I expected. Recently, and after improving my understanding of alpha blending as described in my previous post, I finally spent the time to dig into this and understand this properly. The problem was that my simple math was incorrect and does not represent how layered alphas work.
In the case of my text, I'm layering a pure white opaque texture onto some existing background. For the texture, being pure opaque white, each white pixel is (1, 1, 1, 1). Because I want to dim this, as described above, I adjust the alpha 0.0167 to get (1, 1, 1, 0.0167). To figure out how this blends with the background I return to the alpha formula I described in my previous post, which said regarding GL.BlendFunc(BlendingFactorSrc.One, BlendingFactorDest.OneMinusSrcAlpha);: for the source color use it as is; and for the destination color take the alpha component of the source, subtract that from one, and multiply that by each component of the destination.
## Diminishing returns
Let's do the math. The source is (1, 1, 1, 0.0167) and to keep things simple the destination is black, or (0, 0, 1, 1). So, use the source component as is means just keep (1, 1, 1, 0.0167). Next, for the destination color, take the alpha component of the source, subtract that from one, and multiply that by each component of the destination means we take the source alpha, 0.0167, subtract that from 1, $$1 - 0.0167 = 0.9833$$, and multiply that by the destination, which gives us (0, 0, 0, 0.9833). Now, the actual screen doesn't have an alpha component, it just shows colors, so to get a real color we multiply the alpha by each component and then add the results, or: (1, 1, 1, 0.0167) becomes (0.0167, 0.0167, 0.0167) and (0, 0, 0, 0.98333) becomes (0, 0, 0) and adding those two gives us (0.0167, 0.0167, 0.0167) or X . That's very close to black, but that's ok, it's just 1 of 15 shades we're applying. Let's do another.
The source is the same, (1, 1, 1, 0.0167), or multiplied out as (0.0167, 0.0167, 0.0167). The destination is different this time, since we layering on top of what we just calculated, which is (0.0167, 0.0167, 0.0167, 0.9833), or multiplied out as (0.0164, 0.0164, 0.0164). Adding these gives us (0.0331, 0.0331, 0.0331) or X . Hmm, that still looks awfully black, but it's just two of 15 shades.
Now, for those of you paying attention, this actually isn't the shade we'd have at this point. I'm supposed to take the fractional alpha for the oldest text plus two of that fraction for the next oldest, which means at this point I should have three shades layered on one another, which would be a tad brighter black. But, actually, even that isn't true, and this is where we start getting to the heart of my problem. You do not get to the same color if you apply the alpha layer three times in succession compared to doing it once and then a second layer at twice the alpha. Let me explain.
If I continued the above calculations for one more layer at the same fractional alpha I would end with a final color of (0.0492, 0.0492, 0.0492). On the other hand, if I took my first layer, (0.0167, 0.0167, 0.0167), and layered on top of it a double fractional shade (1, 1, 1, 0.0334), then I would result with (0.0497, 0.0497, 0.0497).
Now, 0.04918 and 0.04973 are not that different, but if I did this for all 15 layers the difference increases. Here's the complete table:
Shade Applied Cumulative Total Calculated Color Layer Single Layers Increasing Shades Single Layers Increasing Shades Single Layers Increasing Shades 1 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 2 0.0167 0.0333 0.0333 0.0500 0.0331 0.0494 3 0.0167 0.0500 0.0500 0.1000 0.0492 0.0970 4 0.0167 0.0667 0.0667 0.1667 0.0650 0.1572 5 0.0167 0.0833 0.0833 0.2500 0.0806 0.2274 6 0.0167 0.1000 0.0959 7 0.0167 0.1167 0.1110 8 0.0167 0.1333 0.1258 9 0.0167 0.1500 0.1404 10 0.0167 0.1667 0.1547 11 0.0167 0.1833 0.1688 12 0.0167 0.2000 0.1826 13 0.0167 0.2167 0.1963 14 0.0167 0.2333 0.2097 15 0.0167 0.2500 0.2228
This shows two methods of adding the layers. Single Layers means layering the fractional alpha multiple times. Increasing Shades means doing that, but increasing the fraction for each layer. I've highlighted the interesting numbers. The Cumulative Total colums just show that in both methods the resulting simple-math total is what we're trying to get to, 0.25. The Calculated Color column shows what we actually end up with, which in both cases is less than our target, or 0.2228 in the case of Single Layers or 0.2274 in the case of Increasing Shades.
The reason both numbers don't add up to what we want is that each successive layer causes less of a change than the previous one. This is a case of diminishing returns. Because of this we need to do some math to calculate the fractional shade such that it takes into account these diminishing returns.
Calculated Color Layer Single Layers Increasing Shades 1 0.0667 0.0667 2 0.1289 0.1911 3 0.1870 0.3529 4 0.2412 0.5255 5 0.2918 0.6836 6 0.3390 7 0.3830 8 0.4242 9 0.4626 10 0.4984 11 0.5318 12 0.5630 13 0.5922 14 0.6194 15 0.6447
I'm going to shift our target to 1 instead of 0.25. This will make the diminishing returns problem more clear. Doing this, and following the same flawed layering strategy described above, the Calculated Color from the above table becomes as shown on the right. You can now see the totals are quite far for our target of 1. In fact, because of diminishing returns, no matter how many layers we add we'll never actually get to 100% opaque. If I extended the above to 100 layers the Single Layers color would be 0.999.
So, what I needed to do was to determine a different fractional alpha to use when layering that would add up to my desired target. To do this I converted my Single Layers iterative algorithm into a formula. First, the iterative version of this is below, where fractional is the fraction of my intended alpha, or $$\frac{1}{15} = 0.067$$.
$$f(n) = (1 - fractional) \times f(n-1) + fractional$$
This calculates the final color for n where n is the number of layers. To convert this to a formula that does not reference itself I started by listing the successive values of this formula in Google Spreadsheets, just like the above tables, and then I played with it until I came up with the following equivalent formula.
$$color = 1 - (1 - fractional)^{layers}$$
We can check this by plugging in our fractional alpha of $$\frac{1}{15} = 0.067$$ and our 15 layers as $$1 - (1 - 0.067)^{15} = 0.645$$, which gives us the same value. So far so good, but what we want is a formula into which we supply the color and the number of layers and it gives us the fractional alpha. That means we want to get the fractional part of the formula on one side by itself. Time to remember how do to formula manipulation. First, get the exponent expression on a side by itself.
$$1 - color = (1 - fractional)^{layers}$$
Next, reverse the exponential equation.
$$(1 - color)^{ \frac{1}{layers} } = 1 - fractional$$
Finally, get fractional by itself and positive.
$$fractional = 1 - (1 - color)^{ \frac{1}{layers} }$$
Great, now we just plug in our target color, which was fully opaque white or 1, and keeping the layers as 15, we get...oops...hmm, $$1 - 1 = 0$$ and $$0^{ \frac{1}{15} } = 0$$, and $$1 - 0 = 1$$, so the fractional is 1. Ack, we have a problem. This actually makes sense. Remember when I said that with the diminishing returns problem we can add the fractional alpha as many times as we want and never actually get to 100% opaque, well this is telling us the same thing. Actually, if the fractional alpha starts out as 1 then we'll get to 100% opaque, but then we don't get the successive alpha blending effect that we want.
Ok, time to cheat, although I want the final alpha to be 1 I'll settle with it being a single shade off of 1. Since there are 256 possible alpha shades I'll settle with $$\frac{255}{256} = 0.996$$. And plugging that into the formula I get $$1 - (1 - 0.996)^{ \frac{1}{15} } = 0.308$$. If I layer that 15 times I get this succession of colors: 0.308 → 0.521 → 0.669 → 0.771 → 0.841 → 0.890 → 0.924 → 0.947 → 0.964 → 0.975 → 0.983 → 0.988 → 0.992 → 0.994 → 0.996.
This works and I've implemented this in code now, but it's not actually exactly what I want. This is the Single Layers algorithm, where really I want the Increasing Shades algorithm. The formula for that is more complex, though:
$$f(n) = \left[ \left(1 - \frac{ n^2 + n }{2} \times fractional \right) \times f(n-1) \right] + \frac{ n^2 + n }{2} \times fractional$$
I've tried to convert this into a single formula that does not reference $$f(n-1)$$, but have not been successful yet. For now I'll either stick with the previous formula or use the one above and calculate it recursively.
Next: Who stole my pixels?
## 2014-01-28
### A Tutorial; Step-by-step
I've been planning to add a tutorial to my game from the beginning, and finally got around to really working on it a few weeks ago. Similar to my previous post about simple solutions, I found that once I tinkered with ideas enough and came up with a clear idea of how to implement it, my tutorial code just fell into place quite smoothly and fit satisfyingly into the existing architecture.
## Making a point
I decided that the best way to communicate how to play the game was to have an image of a finger moving on the screen indicating touch actions. You can see an example of this in the image to the right. I created the hand image by taking a photo of a hand (my eldest daughter's) and then manipulating it a bit to flatten the colors and such. In the tutorial it is opaque when touching the screen (the left image), or mostly transparent when not touching the screen (the right image).
After creating the finger I needed a way to move it around the screen easily and also initiate the touch events. For this I created a Finger class that extends Symbol (see my previous post for a discussion on this) and holds the image of the finger. This class also has new animation functionality mostly implemented in the methods shown below.
BeginAnimation is called once for each single animation step (e.g., moving from point A to point B with the finger touching the screen or not, as indicated). This animation is then handled as part of the normal Animate method, which is called once for each Widget during the main animation loop, by calling DoFingerAnimation. As you can see it mostly just updates the finger's position and, once complete, calls the _fingerCompletionAction.
public void BeginAnimation(Point start, Point finish, bool isTouchingScreen, TimeSpan duration, Func<bool> completionChecker, Action completionAction)
{
_fingerMoveStartTime = DateTime.Now;
_fingerMoveDuration = duration;
AnimationStartPosition = start;
AnimationFinishPosition = finish;
IsTouchingScreen = isTouchingScreen;
_fingerCompletionChecker = completionChecker;
_fingerCompletionAction = completionAction;
}
private void DoFingerAnimation()
{
TimeSpan elapsed = DateTime.Now - _fingerMoveStartTime;
if(elapsed < _fingerMoveDuration)
{
Position = Animator.Interpolate(
AnimationStartPosition,
AnimationFinishPosition,
_fingerMoveDuration.TotalSeconds,
elapsed.TotalSeconds,
Animator.Curve.Linear
);
AnimationCurrentPosition = Position;
}
else if(IsBeingAnimated && _fingerCompletionChecker())
{
_fingerMoveStartTime = DateTime.MinValue;
_fingerCompletionAction();
}
}
## Stepping it up
So, now that I can perform a single step of an animation controlling the finger, I need to string these together into multiple steps that show a complete tutorial lesson. To do this I created a couple of data holders within my Tutorial class. The first, Step, represents a single step of the tutorial and performs a single finger animation movement. The second, Lesson, holds all of the data for a single tutorial lesson including the game elements to show on the screen and the sequence of steps.
One thing to note, there is a slightly confusing use of the term "completion checker" here, since it is used twice. It basically serves the same purpose for two different levels of the lesson. Inside Step it is used to determine if that step should end. Of course the step has a set duration, but even after that duration there can be other conditions that must be met (see the actual lesson examples later). Similarly, in Lesson this is used to determine if the lesson is complete.
private struct Step
{
public Step(double destinationX, double destinationY, bool touchScreen, double duration, Func<Tutorial, bool> completionChecker)
{
Finish = new Point((float)destinationX, (float)destinationY);
TouchScreen = touchScreen;
Duration = TimeSpan.FromSeconds(duration);
CompletionChecker = completionChecker ?? (foo => { return true; });
}
public Point Finish;
public bool TouchScreen;
public TimeSpan Duration;
public Func<Tutorial, bool> CompletionChecker;
}
private struct Lesson
{
public int Width;
public int Height;
public IEnumerable<Point> Goals;
public IEnumerable<ShipInfo> Ships;
public IEnumerable<Tuple<int, int, WallLocation>> Walls;
public Func<Tutorial, bool> CompletionChecker;
public IEnumerable<Step> Steps;
}
## A lesson plan
Fortunately I was able to use a combination of the yield return technique for enumerations and C#'s object initializers to compactly define individual lessons. I do this statically and populate an array to hold them all.
private static Lesson[] All = Tutorials.ToArray();
private static IEnumerable<Lesson> Tutorials
{
get
{
int width = 4;
int height = 6;
// Create one simple diagonal.
yield return new Lesson
{
Width = width,
Height = height,
Goals = new Point[] { new Point(width - 2, height - 2) },
Ships = new ShipInfo[] { new ShipInfo(ShipType.Hero, new Point(0.5f, 0.5f), Direction.East, 0.025f) },
Walls = new Tuple<int, int, WallLocation>[0],
CompletionChecker = tutorial => { return tutorial.AchievedAllGoals; },
Steps = new Step[] {
new Step(width * 0.6, height * 0.4, false, 3, null),
new Step(width - 2, 0, false, 2.5, null),
new Step(width - 1, 1, true, 1.5, tutorial => { return tutorial.Ships.First().Position.X < width - 2.5; }),
new Step(width - 0.5, 1.5, false, 1.5, null)
}
};
.
.
.
Pulling apart this first Lesson, the interesting part is the 3rd step that has the non-null completion check. This check ensures that the Ship is far enough to the left before taking the finger off of the screen, and therefore completing the diagonal. Without doing this the ship could end up on the wrong side of the diagonal and not bounce to where it is supposed to.
There are a number of interim lessons I'm not including here, but one interesting one (shown below) is the lesson showing how to pause the game, which is done by swiping all the way across the screen horizontally in either direction. The interesting part here is that I needed to show the pause symbol, and then the continue symbol. To do this I cheated a little in two ways. First, in the step before I want the appropriate symbol to be visible, I used the completion check to create the needed symbol, although it's alpha was initially set to 100% transparent. This is done via the CreatePauseSymbol and CreateContinueSymbol methods, not shown here. Second, also not shown here, I adjust the transparency of the pause symbol to become more opaque as the finger completes its animation. This was a little hackish, but worked out pretty well.
.
.
.
// How to pause.
yield return new Lesson
{
Width = width,
Height = height,
Goals = new Point[] { new Point(width - 2, height - 2) },
Ships = new ShipInfo[] { new ShipInfo(ShipType.Hero, new Point(0.5f, 0.5f), Direction.South, 0.045f) },
Walls = new Tuple<int, int, WallLocation>[0],
CompletionChecker = tutorial => { return true; },
Steps = new Step[] {
new Step(width * 0.6, height * 0.8, false, 2, null),
new Step(0, height * 0.45, false, 1, tutorial => tutorial.CreatePauseSymbol()),
new Step(width, height * 0.55, true, 3, tutorial => tutorial.CreateContinueSymbol()),
new Step(width - 1.5f, height - 1.5f, false, 1.5, null),
new Step(width * 0.5, height * 0.5, false, 1.5, null),
new Step(width * 0.5, height * 0.5, true, 0.05, tutorial => tutorial.RemoveContinueSymbol()),
new Step(width * 0.65, height * 0.65, false, 0.5, null),
new Step(0, height - 2, false, 0.75, null),
new Step(1, height - 1, true, 0.75, tutorial => { return tutorial.Ships.First().Position.Y < height - 2; }),
new Step(width * 0.75, height * 0.55, false, 1, null),
}
};
}
}
Finally, now that the lessons are defined, I need to do two more things: make the finger's touch actually behave like a normal touch in a normal game, and queue up the steps so they play in order. The first was pretty easy by just calling the existing touch handlers with the appropriate touch points. The second also turned out particularly well because I used the built in onComplete action in the finger animations to recursively call a method that dequeues each successive step.
private void DoTutorialSteps(Queue<Step> queue)
{
if(queue.Count > 0)
{
Step step = queue.Dequeue();
Action onComplete = step.TouchScreen
? new Action(() =>
{
HandleTouch(BoardToScreen(_finger.AnimationStartPosition), BoardToScreen(_finger.AnimationCurrentPosition), true);
base.HandleBoardTouch(_finger.AnimationStartPosition, _finger.AnimationCurrentPosition, true);
DoTutorialSteps(queue);
})
: new Action(() => { DoTutorialSteps(queue); });
_finger.BeginAnimation(_finger.Position, step.Finish, step.TouchScreen, step.Duration, () => step.CompletionCheck(this), onComplete);
}
}
I'm now almost done with the tutorial and have only one more lesson to add.
Next time: Alpha layering.
## 2014-01-27
### Simple Solutions
Unlike the in depth investigation and learning required to understand premultiplied alpha, as discussed in my previous post, today's topic is simple and satisfying.
## Collections of collections
I've structured the graphical elements of my game into a number of logical layers (not display layers). I won't go into all of them, but to give you an idea, here are some of the key parts, from the most fundamental to the most complex:
1. The most basic layer is a Texture, which is an abstract class that represents a picture that is loaded from a file.
2. Above that is an OpenGLES20Texture, which is an implementation of Texture specifically for OpenGL ES 2.0.
3. Above that is a Graphic, which includes information about the Texture, plus information about its orientation, size, and color.
4. The next layer up is Widget, which is an abstract class that has a 2D position, an abstract Animate method, and an abstract Render method.
5. One layer above that is Symbol, which implements Widget and provides a number of concrete features, including references to one or more Graphic instances. I use this class for all of the interactive controls in the game, like the checkmark to start a game, the question mark icon for starting the tutorial, etc.
6. Another layer above Widget are all of the game UI elements, like Ship, which is the white ball that moves around and bounces off of walls. This also implements Widget and contains the logic to make the ship do what it is supposed to. I have similar classes for the other game UI elements like Wall, Diagonal, etc.
Given the above structure, this means that all graphical elements that I need to render inherit from Widget. In the various game screens I keep track of these in collections that inherit from the following:
public interface IWidgetEnumeration
{
/// <summary>Returns the <see cref="Widget"/>s in the collection.</summary>
IEnumerable<Widget> Widgets { get; }
}
For example, the game screen parent class collects all of these multiple collections via the following:
protected sealed override IEnumerable<IWidgetEnumeration> WidgetsToRender
{
get
{
foreach(IWidgetEnumeration collection in LowerWidgetsToRender)
{
yield return collection;
}
yield return WallsHorizontal;
yield return WallsVertical;
yield return Goals;
yield return Diagonals;
yield return Poles;
yield return _diagonalMarkers;
yield return Ships;
yield return Collisions;
foreach(IWidgetEnumeration collection in UpperWidgetsToRender)
{
yield return collection;
}
yield return _pauseSymbols;
}
}
What's a little interesting here is that this accessor is an IEnumerable<IWidgetEnumeration>, or an enumeration of collections. This allows subclasses to override LowerWidgetsToRender and UpperWidgetsToRender to add additional widgets as necessary. What's been slightly annoying to me for a while, and what finally gets to the point of this blog entry, is that there have been a number of instances when I needed to only add a single new graphical element in a sub-class. But, since I need to return an IWidgetEnumeration I kept needing to create a collection to contain that single graphical element. This made the override of LowerWidgetsToRender look something like this:
protected override IEnumerable<IWidgetEnumeration> LowerWidgetsToRender
{
get { yield return _lowerWidgets; }
}
Where _lowerWidgets is an IWidgetEnumeration that contains just one Widget. I couldn't just return the Widget directly, because I must return an IWidgetEnumeration. This seems inefficient to create this collection just to contain one element. But wait, IWidgetEnumeration is an interface. What's to stop me from implementing that directly on Widget so I can just return that directly? Well, that's exactly what I did. I made Widget implement IWidgetEnumeration and added the following simple bit of code to it.
public IEnumerable<Widget> Widgets
{
get { yield return this; }
}
This allowed me to change the above LowerWidgetsToRender accessor into the following, where _tutorialCounter is the single Widget that was inside the previous collection.
protected override IEnumerable<IWidgetEnumeration> LowerWidgetsToRender
{
get { yield return _tutorialCounter; }
}
I don't think this refactor improves the performance of the code in any measurable way, but it does make things a bit more straight forward and easier to understand. It's obviously not particularly clever or exciting, but I was happy when I thought of the solution and do feel the code is better because of it.
Next: A Tutorial; Step-by-step.
## Sorry I've been gone
It's been a while since my last post about OpenGL and OpenTK Fun. This is because the challenge I described in that article, and ultimately resolving that issue, unstuck me from making real progress on my development and I've been putting much more effort into my game since then and am now much further along. That's all good news, but now I need to both make progress on the game and try to keep this blog going too.
I'll start today with a discussion about OpenGL premultiplied alpha textures, but before that I want to send a quick thank you out to a new code syntax highlighter by Alex Gorbatchev. It's JavaScript based, so I no longer need to manipulate my code entries before posting them in these articles. Also a thank you to Carter Cole and his blog entry describing how to easily setup the syntax highlighting in Blogger.
## So, what is "premultiplied alpha" and why do I care?
As for what it is, there are plenty of pages out there that can describe it much better that me, so I'm not even going to try. I found a good succinct description on Martin Stone's blog, which points to a more detailed description on Tom Forsyth's blog. Please check those pages out if you want to learn more.
As for why I care, that I can describe in more detail. Initially I knew almost nothing about premultiplied alpha. I had seen the term a few times, for example when compiling my app there is some log message mentioning that it is precompiling my PNG textures, but I never tried to understand that more since everything was working fine. The reason I started to care more, and look into it more, are due to a couple things.
First, there was always something bothering me about a portion of my OpenGL code. From the beginning when I got the code working on both iOS and in Windows I found that I had to have a platform specific check for texture blending:
// on iPhone this is needed to apply the color adjustment (on PC this is our normal setting)
{
GL.BlendFunc(BlendingFactorSrc.SrcAlpha, BlendingFactorDest.OneMinusSrcAlpha);
}
#if MONOTOUCH
else
{
GL.BlendFunc(BlendingFactorSrc.One, BlendingFactorDest.OneMinusSrcAlpha);
}
#endif
I put this code into place after trial and error. The #if MONOTOUCH section only compiles and runs on the MonoTouch framework, which means only on iOS. What I didn't understand was, given that OpenGL is supposed to be a consistent interface across platforms, why did I need to have this condition depending on the platform? All other code and image assets related to OpenGL textures and blending was the same between the two platforms, so why was this needed?
Well, the answer goes back to what I mentioned above about where I had previously heard about premultiplied alpha; the iOS compilation log message. What that message means is that my assumption that I'm using the same image assets (PNG images in my case) is not true. Although I have the same PNGs referenced in both the iOS project and Windows project, when the iOS version gets built the PNGs are adjusted to have alpha premultiplied.
So, why does that require the adjustment in GL.BlendFunc? Well, first we need to know what GL.BlendFunc does. The first parameter is how to use the incoming (source) colors when blending them with the existing (destination) pixels. The second parameter is how to adjust those destination pixels. There is ample documentation about this on many sites, so I won't go into all of the parameter options, but I will discuss the two that I was using. The first version, GL.BlendFunc(BlendingFactorSrc.SrcAlpha, BlendingFactorDest.OneMinusSrcAlpha); says two things:
1. For the source color, take the alpha (or transparency) component and multiply it by each other component. For example, if the RGB color is (1, 0.8, 0.5) then after the multiplication it would become (0.5, 0.4, 0.25) . This is the same color, but darkened.
2. For the destination color, take the alpha component of the source, subtract that from one, and multiply that by each component of the destination. In this case the alpha is 0.5, so subtracting that from 1 is also 0.5, which would be multiplied by the destination color. For example, if the existing destination RGB color was (0.2, 1, 1) then after multiplying it would become (0.1, 0.5, 0.5) . Again, the same color, but darkened.
After completing the above calculations on the source and destination colors then are blended by adding them together. That is (0.5, 0.4, 0.25) + (0.1, 0.5, 0.5) = (0.6, 0.9, 0.75) . Looking at the two original colors you can see that this works and the resulting blended color is correct.
Ok then, what's up with the second version: GL.BlendFunc(BlendingFactorSrc.One, BlendingFactorDest.OneMinusSrcAlpha);? How does this change things? I won't go through all the steps in detail, but starting with the same two first colors you would get a final function of (1, 0.8, 0.5) + (0.1, 0.5, 0.5) = (1.1, 1.3, 1.0) or pure white, since each component exceeds the maximum value of one and is therefore limited to one. Clearly that is too bright and doesn't work work. So, why would you want to do that? Well, that's where premultiplied alpha comes in. The first version used BlendingFactorSrc.SrcAlpha, which multiplies the alpha by the color. And what do you think premultiplied alpha does? It did that exact same calculation when the texture asset was created (or built, in this case). This means that we don't need to do it again now while blending. Instead we use the color as is, which is what BlendingFactorSrc.One does.
So, final question, why do this premultiplied alpha in the first place? I'll quote Tom Forsyth from his blog post (referenced above) for a pretty good explanation. For a much more in depth discussion please read his whole post.
"Normal" alpha-blending munges together two physically separate effects - the amount of light this layer of rendering lets through from behind, and the amount of light this layer adds to the image. Instead, it keeps the two related - you can't have a surface that adds a bunch of light to a scene without it also masking off what is behind it. Physically, this makes no sense - just because you add a bunch of photons into a scene, doesn't mean all the other photons are blocked. Premultiplied alpha fixes this by keeping the two concepts separate - the blocking is done by the alpha channel, the addition of light is done by the colour channel. This is not just a neat concept, it's really useful in practice.
## Doing premultiplied alpha on Windows
Ok, so now I understand what's going on, but how do I fix it for Windows since I don't have a build step modifying my PNG files? I did some research and it seems there are some tools that will convert a normal PNG to a premultiplied alpha PNG, specifically it seems GIMP will do this, although I didn't try it myself. I didn't really want to have to convert my existing assets each time I modified them, or complicate my Windows build process by using addition tools, so I made a code change to do the alpha multiplication at the time I load my PNGs. That's a somewhat wasteful operation, but it only happens once, and considering I don't have very many textures I felt it was a good and simple solution.
Below is the code that does this. You'll notice some strange BGRA to RGBA conversion as well. This is due to using the Bitmap class and it's how Windows works.
public static Texture LoadTexture(string fileName, Func<byte[], int, int, Texture> textureCreator)
{
if(textureCreator == null)
{
throw new ArgumentNullException("textureCreator");
}
using(Bitmap image = new Bitmap(fileName))
{
BitmapData bitmapData = image.LockBits(
new Rectangle(0, 0, image.Width, image.Height),
PixelFormat.Format32bppArgb
);
image.UnlockBits(bitmapData);
IntPtr rawData = bitmapData.Scan0;
Texture texture;
unsafe
{
int length = image.Width * image.Height * 4;
byte[] data = new byte[length];
Marshal.Copy(rawData, data, 0, length);
for(int i = 0; i < length; i += 4)
{
float alpha = (float)data[i + 3] / 255;
byte r = data[i + 2];
byte g = data[i + 1];
byte b = data[i + 0];
data[i + 0] = MultiplyByAlpha(alpha, r);
data[i + 1] = MultiplyByAlpha(alpha, g);
data[i + 2] = MultiplyByAlpha(alpha, b);
data[i + 3] = (byte)(alpha * 255);
}
texture = textureCreator(data, image.Width, image.Height);
}
return texture;
}
}
private static byte MultiplyByAlpha(float alpha, byte channel)
{
return (byte)(channel * alpha);
}
After putting this code in place, so that my textures turn into premultiplied alpha data as I load them, I was able to simplify my original platform dependent code into just this:
GL.BlendFunc(colorAdjustment.Alpha != 1 ? BlendingFactorSrc.SrcAlpha : BlendingFactorSrc.One, BlendingFactorDest.OneMinusSrcAlpha);
Removing that platform specific code is a very minor improvement, but more than that I'm happy to understand all of this quite a bit more.
Next: Simple Solutions.
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{}
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# Prove that there are an infinite number of natural numbers that cannot be written as the sum of three squares.
I am having trouble proving this. I know that I need to look at the quadratic residues mod 8 and that they are 1,4 or 0. However how do I prove that these are the equivalence classes of Z/3Z.
Then I can added the equivalence classes and show that none of them divide 8. How do I set this proof up?
## 1 Answer
We may be overthinking this. Residue $7$ modulo $8$ cannot be rendered as a sum of three residues each belonging to $\{0,1,4\}$. Proof: The sum can be odd only if there are an odd number of $1$'s; one $1$ gives a sum one greater than a multiple of $4$ and three $1$'s give $1+1+1\equiv 3$ not $7$. Done.
• I don't understand where the residue 7 mod 8 comes from. Can someone explain? – mathamasacre Sep 29 '16 at 15:53
• Because we need only one counterexample, and residue 7 is it. – Oscar Lanzi Sep 29 '16 at 16:52
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{}
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TBTK
ReciprocalLattice.h File Reference
A ReciprocalLattice allows for the creation of a momentum space Model from a UnitCells. More...
#include "TBTK/Model.h"
#include "TBTK/StateTreeNode.h"
#include "TBTK/UnitCell.h"
#include <initializer_list>
#include <vector>
Go to the source code of this file.
Classes
class TBTK::ReciprocalLattice
Detailed Description
A ReciprocalLattice allows for the creation of a momentum space Model from a UnitCells.
|
{}
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# Effect of color superconductivity on the mass and radius of a quark star
@article{Rster2004EffectOC,
title={Effect of color superconductivity on the mass and radius of a quark star},
author={Stefan R{\"u}ster and Dirk H. Rischke},
journal={Physical Review D},
year={2004},
volume={69},
pages={045011}
}
• Published 10 September 2003
• Physics
• Physical Review D
We compare quark stars made of color-superconducting quark matter to normal-conducting quark stars. We focus on the most simple color-superconducting system, a two-flavor color superconductor, and employ the Nambu\char21{}Jona-Lasinio (NJL) model to compute the gap parameter and the equation of state. By varying the strength of the four-fermion coupling of the NJL model, we study the mass and the radius of the quark star as a function of the value of the gap parameter. If the coupling constant…
48 Citations
## Figures from this paper
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## Elementary Statistics: A Step-by-Step Approach with Formula Card 9th Edition
Published by McGraw-Hill Education
# Chapter 7 - Confidence Intervals and Sample - Chapter Quiz - Page 409: 13
#### Answer
I am 95% confident that the population average cost of doctor’s visits is between 43.15 and 46.45
#### Work Step by Step
You do not know population standard deviation, so you must use the t interval function on your calculator. X bar : 44.80 Sx: 3.53 n:20 C-Level: .95 43.15 < $\mu$ < 46.45
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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# Probability mass function undefined
Say $(X_i)$ is a sequence of $n$ $i.i.d$ Bernoulli random variables, each with parameter $p_i$. Define the following for all real $a_i>0$, $$S_{n} =\frac{1}{n} \sum _{i=1}^{n}a_iX_i$$ EDIT: In the special case of $a_i=1$, we may use probability generating functions to arrive at the probability mass function of $S_n$,
$$P_{S_{n} } (k)=\frac{1}{(nk)!} \frac{d^{(nk)} }{dx}\prod_{i=1}^{n} \left(1-p_i+p_i\cdot x \right) \; \left|\; x=0\right.$$
for $k=0,1/n,2/n,...,1$. Is there any other way to get an expression for the probability mass function of $S_n$ that will be defined for all real $a_i>0$?
-
Presumably $X$ in the sum is meant to be $X_i$? – joriki Aug 1 '12 at 11:43
Yes, will correct it in the OP. – Omri Aug 1 '12 at 11:49
Nothing guarantees that $S_n$ is integer-valued when the $a_i$ are not integer. Since generating functions are well defined only for (nonnegative) integer-valued random variables, unsurprisingly one cannot use them in this case. The solution is to turn to Fourier transforms, known to probabilists as characteristic functions. – Did Aug 1 '12 at 12:13
OK, thanks, the pgf works only for $a_i=1$ using a little trick (and useful only if each $X_i$ has a $p_i$).. – Omri Aug 1 '12 at 12:28
The OP has now been restated. – Omri Aug 1 '12 at 14:16
Hint: Use the fact that, for every integer valued random variable $S$ and every integer $x$, $$\mathrm P(S=x)=\int_0^{1}\mathrm E(\mathrm e^{2\pi\mathrm itS})\cdot\mathrm e^{-2\pi\mathrm itx}\,\mathrm dt,$$ and the fact that, in the present case, $$\mathrm E(\mathrm e^{2\pi\mathrm itS_n})=\prod_{k=1}^n(1-p_k+p_k\mathrm e^{2\pi\mathrm ita_k/n}).$$ Edit: The second identity above is a consequence of the definition of $\mathrm E(\mathrm e^{2\pi\mathrm itS_n})$ and of the joint distribution of the random variables $(X_k)_{1\leqslant k\leqslant n}$.
The first identity is an application of the general principle that integrating a discrete sum of complex exponentials against the conjugate of a complex exponential extracts the coefficient of the corresponding exponential from the sum. Namely, for every integers $x$ and $y$, $$\int_0^{1}\mathrm e^{2\pi\mathrm ity}\cdot\mathrm e^{-2\pi\mathrm itx}\,\mathrm dt=[x=y],$$ hence, for every distinct integers $x_k$ and every coefficients $p_k$, $$p_\ell=\int_0^{1}\left(\sum_kp_k\mathrm e^{2\pi\mathrm itx_k}\right)\cdot\mathrm e^{-2\pi\mathrm itx_\ell}\,\mathrm dt.$$ Applying this to the integer valued random variable $S$ such that $p_k=\mathrm P(S=x_k)$ yields the first formula above.
When $S$ is not integer valued, use the fact that, for every real numbers $x$ and $y$, $$\lim_{N\to\infty}\int_0^1\mathrm e^{N\mathrm ity}\cdot\mathrm e^{-N\mathrm itx}\,\mathrm dt=[x=y],$$ hence, for every discrete random variable $S$, $$\mathrm P(S=x)=\lim_{N\to\infty}\int_0^1\mathrm E(\mathrm e^{N\mathrm itS})\cdot\mathrm e^{-N\mathrm itx}\,\mathrm dt.$$
The reason I want to see all the stages is that my model also has $(X_i)$ as discrete $i.i.id$ RVs with $P(0)=(1-p_i)^2$, $P(1)=2p_i(1-p_i)$, $P(2)=p_i^2$. – Omri Aug 5 '12 at 9:00
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# embedding
(redirected from Isometric embedding)
Also found in: Dictionary, Medical.
Related to Isometric embedding: Imbedding
## embedding
[em′bed·iŋ]
(mathematics)
An injective homomorphism between two algebraic systems of the same type.
## embedding
(mathematics)
One instance of some mathematical object contained with in another instance, e.g. a group which is a subgroup.
## embedding
(theory)
(domain theory) A complete partial order F in [X -> Y] is an embedding if
(1) For all x1, x2 in X, x1 <= x2 <=> F x1 <= F x2 and
(2) For all y in Y, x | F x <= y is directed.
("<=" is written in LaTeX as \sqsubseteq).
References in periodicals archive ?
6 Any separable metric space admits an isometric embedding into Q.
Of course, abstract infinite dimensional manifolds appear intimidating and it is only natural to seek a "reassuring" result, namely an infinite dimensional counterpart, if not of Nash's Isometric Embedding Theorem, then at least of Whitney's Topological Theorem (see, e.
Keywords Embedding of Riemannian spacetimes, local and isometric embedding.
Isometric embedding of Riemannian manifolds in Euclidean spaces.
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# Nautilus
Member Since 19 Nov 2004
Offline Last Active Jun 12 2013 03:37 PM
### Unknown HRESULT (0x8006FC24) from DirectInput
10 June 2013 - 01:35 PM
A call to IDirectInputDevice7::GetDeviceState() is failing unexpectedly. The HRESULT code returned is being 2147941412 (0x8006FC24), which maps to nothing known. According to this Microsoft page the Facility code of the HRESULT (0x8006FC24) is undefined. How can I understand what's wrong?
### Simple Moving Average fps display, and framerate limiters...
25 May 2013 - 12:44 AM
Greetings.
With my D3D wrapper I display the framerate using a Simple Moving Average (SMA) calculated over the last hundred of frames. I'd be totally happy with it, if it weren't for a minor problem that I want to get rid of, but have no clue about how to.
THE FIRST PROBLEM:
So long a game is constant in performance, and produces its frames in pretty much the same timeframe, and maybe drops the occasional frame now and then, the SMA will display a veritable and meaningful fps information. But when a game is inconstant, produces frames the build time of can vary greatly, maybe even causing the frequent drop of frames, the SMA starts displaying funny values that just tell lies.
A few examples will illustrate the problem better than my english can.
For brevity's sake, imagine a scenario in which the frame history for the SMA is of 3 frames. And we count 900 timer ticks in 1 second. Also, normally a frame can compose in 50 ticks, but the game wants to limit itself to 3 frames per second, meaning that between composition and displaying, 300 ticks elapse from frame to frame.
In the perfect scenario we have all 3 frames to compose and display in 300 ticks each.
The Simple Moving Average (SMA) over the last 3 frames is going to be a perfect 3.0:
sma = (1 / 300) + (1 / 300) + (1 / 300)
sma = sma * 900_ticks_per_second
sma = sma / 3_frames_history
sma = 3.0 fps
In case a frame takes too much, the next frame may be dropped. Suppose that the 2nd frame takes 600 Ticks, leaving no room for a 3rd frame, which gets dropped. What will the SMA say?
sma = (1 / 300) + (1 / 600) + <nothing: this frame dropped>
--( ^^ no! because a frame dropped, the actual 3-frames window will be: )--
sma = (1 / 300) + (1 / 300) + (1 / 600)
--( hence: )--
sma = sma * 900_ticks_per_second
sma = sma / 3_frames_history
sma = 2.5 fps
Now suppose 1 of the 3 frames composes in 500 ticks. It's late, but it's possible to compensate. Normally a frame builds in 50 Ticks, and in fact the next one is on screen 100 ticks later. With all 3 frames within the 900 ticks limit, the fps of 3.0 is respected. However, the SMA will tell differently:
sma = (1 / 300) + (1 / 500) + (1 / 100)
sma = sma * 900_ticks_per_second
sma = sma / 3_frames_history
sma = 4.6 fps
It gets worse if we continue from there. Suppose that the next 2 frames are regular and compose & display in 300 ticks each.
New SMA after 1st regular frame:
sma = (1 / 500) + (1 / 100) + (1 / 300)
sma = sma * 900_ticks_per_second
sma = sma / 3_frames_history
sma = 4.6 fps
New SMA after 2nd regular frame:
sma = (1 / 100) + (1 / 300) + (1 / 300)
sma = sma * 900_ticks_per_second
sma = sma / 3_frames_history
sma = 5.0 fps
See the problem?
Add another regular frame (300 Ticks) and the SMA suddently drops to perfect 3.0 fps.
In presence of inconstant frame times it's annoying to see the SMA raising past the hard limit being imposed by the very game. Not only it gives a bad impression, it may also mask the true loss in fps that occurs when a frame is willingly dropped. I tried with different Moving Average formulae, taken from Wikipedia, but the problem persists. And in the end the Simple Moving Average is the most responsive to changes and the fastest to calculate.
----------------------------------------------------------------------------------------------------
THE SECOND PROBLEM:
If I'm allowed to briefly mention a commercial game, I find Far Cry 2 to be a good test platform for my wrapper. Its performance is most constant on my hardware (which isn't next-gen anymore) and gives me opportunity to test the robustness of my fps display. Also, the game features both an fps display calculated with SMA, and a framerate limiter.
Now, when letting the game run unconstrained, both its fps display and my wrapper's fps display show the very same readings all the time (provided that I lower my frame history to 64 entries, and make the final rounding to the 1st decimal digit).
What I haven't said is that my wrapper features a framerate limiter as well.
If I activate my wrapper's framerate limiter, both the wrapper's fps display and the game's fps display will show the very same values. No surprise. This also happens when the occasional frame is to be dropped, and the SMA jumps past the imposed limit because of it.
But let's invert the roles...
If I activate the game's framerate limiter, this time my wrapper's fps dsplay will report a slightly greater fps average (say, 40.1 or 40.2 fps) than that of the game's fps display which instead reports a perfect score (say 40.0 fps). The surprise, however, is that when frames are dropped or are just late, the game's fps display will _drop_, while my wrapper's SMA will jump past the real value - as usual.
The situation changed when in the nVidia control panel I forced the Maximum pre-rendered frames setting to 0. Now when using the game's framerate limiter, the game's fps display would dutifully declare a perfect fps amount (say 40.0 fps) but not before having shown massive initial imprecisions. Looks like it adjusted itself dynamically. My wrapper's fps display instead would calculate the same imperfect fps average of before (say, 42.6 fps). How can it be?
For all I observe the game to guess what it's doing -and how-, I come up with no explanation.
If anybody could shed some light I'd really appreciate it.
Thank you very much.
### How to detect a monitor's Default refresh frequency?
22 May 2013 - 06:09 PM
How do I retrieve a monitor's default refresh rate?
If I query for the current mode's refresh rate I may be returned either 0 or 1, both indicating a default refresh rate. But I found no way to retrieve what this default is. I know it can be overridden via the DxDiag interface - still, not even such interface will tell what's the original default.
Can I safely assume a value of 60 Hz? Apparently not. Some people have their monitors set to 59 Hertz when they activate some resolutions.
A manual count of the vertical blank intervals over a period of 1 second isn't always a viable option for me.
### [DirectX / C++] Runtime Check Failure #0
23 April 2013 - 01:43 PM
My DirectX9 wrapper won't work with a specific game, which -from what I gather- was compiled against Dx9.0c Aug2008 release (what's found on the game DVD at least). The game will startup and immediately shutdown. No sound, no popup, no error message, no log written nowhere, no Application Error in the system's events log, not even a crash dump. Nothing.
My wrapper was compiled against a slightly older version yet (Mar2008), so I figured it was time to update the SDK and rebuild the wrapper. For reasons of my own I chose to update to Dx9.0c Mar2009 release.
I recompiled my wrapper. And to ensure that I didn't break anything in this new version I have first run tests on games that I knew were working fine with it in the past (these range from indie to AAA productions). The new wapper passed all tests.
Proud and confident that it'd work with this latest game as well, I tried it... and again the game would startup then silently shutdown.
So I insert a Sleep() call inside my dll and rebuild it with debug info. Then I launch the game and connect the debugger to it. Stepping line-by-line, at some point visual studio halts with this message:
Run-Time Check Failure #0 - The value of ESP was not properly saved across a function call. This is usually a result of calling a function declared with one calling convention with a function pointer declared with a different calling convention.
Basically it's saying that the stack pointer is corrupted (on return from a function call the pointer is different from what was before the call), am I correct?
So my code is somehow stepping on forbidden ground. But I believe that the calling conventions hinted to by the message have nothing to do with this. My dll is built to double as a proxy (for when a launcher isn't an option) and if I really got the calling conventions wrong it would never work, not just this time.
Searching around the web for possible causes of the problem, I find this sample code which triggers the above error message:
class A
{
public:
virtual int doStuff (int a, int B) { return (a + B); }
};
class B
{
public:
virtual int doStuff (int a) { return a; }
};
void main (void)
{
A a;
int Stuff = ((B*) &a)->doStuff (1); // <-- Runtime Check Failure #0
}
The text accompanying the sample said that I could be using headers and libs that are out of sync (either newer header with older lib, or vice versa). I'm inclined to think this is the case. I upgraded my project from one SDK version to another (Mar2008 -> Mar2009), after all, and haven't touched a single line of code. I just recompiled.
As the error occurs on a call to IDirect3D9::GetAdapterIdentifiers(), the relevant code (I think) is wholly inside d3d9.h and d3d9.lib. So what do I do, I take a pair of such files from both SDK versions and make a binary comparison of them. Surprise: both pairs of .h and .lib files are identical.
At this point I don't know what to search for.
Any ideas?
### [C/C++] hyperbolic squared function of
03 April 2013 - 01:10 AM
Anybody can tell me how to write C code equivalent to this?
tanh2 (x)
That would mean the hyperbolic tangent squared of X.
Never heard of 'squared' functions before.
How do I translate that to C?
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# Deterministic Nonfinite Automaton - Longest Block of Ones
How to Think About Algorithms - Jeff Edmonds
Exercise 2.2.2
The problem asks to find $\sum$, Q, δ, s, and F of this program if we see it as an automaton. Edmonds characterized this as deterministic non-finite automaton since 'n' is not bounded.
$\sum$ is {0,1} but I'm not sure how to model Q and the transition functions.
One guess is just simply Q = { q[p_max, q_max, length_max, p_current, length_current] }, with each of the 5 variables taking a value in the set of size around n ( { 1, 2, 3, ... , n} ), which will have O(n^5) states
but even if I use this brute force approach I can't easily define transition functions that determine the next state based on current state and the input read each iteration (0 or 1).
The biggest difficulty is modeling conditionals. When you are at a state q_1, the next state seems to be defined not just by 0 or 1 input but also by the length variables.
He didn't provide any solutions and I did a lot of research but still lost.
Any suggestions?
Your guess is definitely on the right track, though you may need one more piece of information. As you have infinite states (of which each input will use some number of them according to a function of the length of the input), you can use (as you observe) the states to encode information that deals with the length of the input.
There are a number of ways to do this (you can use the same variables as the code essentially), but I think it's clearer to be more explicit. Each state can keep track of the following six things:
1. The starting position of the best run so far.
2. The ending position of the best run so far.
3. The length of the best run so far.
4. The starting position of the current run being counted.
5. The ending position of the current run being counted.
6. The length of the current run being counted.
So each state could be labeled something like:
$$q_{maxstart,maxend,maxlength,currentstart,currentend,currentlength}$$
This then solves the transition problem:
• A zero in the input ends the current run, and resets to start counting again, but doesn't affect the best run so far
• A one in the input increases the end point of the current run (and sets the starting point if the current run length is zero) AND if you happen to be in a state where the current run length is the same as the best run length, it also alters the best run length subscripts (that is, the transition takes you to a state where those change suitably).
There's no branching in either of those, so the process is completely deterministic.
A couple of examples to illustrate:
• The start state is easy, it's just $q_{0,0,0,0,0,0}$ with the transitions $\delta(q_{0,0,0,0,0,0},0) = q_{0,0,0,1,1,0}$ and $\delta(q_{0,0,0,0,0,0},1) = q_{0,1,1,0,1,1}$. This uses the case where the current run length is the same as the best, and we see a one, so we update the best start and end to be the same as the current start and end etc.
• Now let's try a state where the current run changes, but isn't better than the best. Imagine we're in state $q_{a,b,50,x,y,3}$, then the two transitions will be $\delta(q_{a,b,50,x,y,3},0) = q_{a,b,50,y+1,y+1,0}$ and $\delta(q_{a,b,50,x,y,3},1) = q_{a,b,50,x,y+1,4}$.
This actually pretty much covers all the cases (as there's really not much going on except the abuse of subscripts to count for us).
To boil that down to some rules for putting in the transitions:
1. $\delta(q_{a,b,n,x,y,m},0) = q_{a,b,n,y+1,y+1,0}$
2. If $m < n$, $\delta(q_{a,b,n,x,y,m},1) = q_{a,b,n,x,y+1,m+1}$
3. If $m = n$, $\delta(q_{a,b,n,x,y,n},1) = q_{x,y+1,n+1,x,y+1,n+1}$
You don't need any states where $m > n$, so you don't have to worry about that case (though you can include them and give them a rule like #3 if you want a complete infinite enumeration of all the subscripts.
• so 'd(q<maxstart,maxend,maxleng,currstart,currend,currlength>,1) = (next_state)'. If I were to write a few transition functions for this automaton to cover all cases, how can I differentiate the above case where the longest block changes and when it doesn't? I totally understand your points but I was aiming for a set of transition functions that govern all cases for this automaton. something like 'd(q<a,b,x,i,j,y>, 0) = q<a,b,x,i+2,j,y> if we follow the given algorithm exactly. – namesake22 Dec 13 '17 at 16:26
• and that is why I was curious as to how to incorporate the conditional in such transition functions – namesake22 Dec 13 '17 at 16:29
• @namesake22 I made the rules more explicit. – Luke Mathieson Dec 13 '17 at 23:07
• Okay, I just didn't think we could add if-conditionals to transition functions like you did in rule 2 and 3. I was only trying to make the functions fit the form of rule 1, which must have been a stubborn idea. – namesake22 Dec 14 '17 at 5:18
• @namesake22 They're not really conditionals on the transitions, you just make all the states, and depending on what the label of the state is, you either at a #2 or a #3, you don't choose during the computation. – Luke Mathieson Dec 14 '17 at 5:42
Based on Luke Mathieson's amazing answer,
and using the concept of an "empty"** block from the book
i.e. A[i..j] is empty is i > J;
The transition functions can be modeled as follows (difference being how we update starting position of the current run using the empty block):
δ( Q[p, q, leng_max, p_c, i, leng_curr] , 0 ) = Q[p, q, leng_max, i+2, i+1, 0]
the other two transition functions are excellently described in Luke's answer
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# How do you write the vector equation that passes through point (-1,4) and parallel to <6,-10>?
Jan 17, 2017
$\vec{r} = \left(\begin{matrix}- 1 \\ 4\end{matrix}\right) + l a m \mathrm{da} \left(\begin{matrix}6 \\ - 10\end{matrix}\right)$
#### Explanation:
The vector equation of a line in the direction of $\vec{D}$ passing through the point $\vec{A}$ is:
$\vec{r} = \vec{A} + l a m \mathrm{da} \vec{D}$
So the required equation in vector form is:
$\vec{r} = \left(\begin{matrix}- 1 \\ 4\end{matrix}\right) + l a m \mathrm{da} \left(\begin{matrix}6 \\ - 10\end{matrix}\right)$
Alternative notations are:
$\vec{r} = \left\langle- 1 , 4\right\rangle + l a m \mathrm{da} \left\langle6 , - 10\right\rangle$
Or,
$\vec{r} = \left(- 1 \hat{i} + 4 \hat{j}\right) + l a m \mathrm{da} \left(6 \hat{i} - 10 \hat{j}\right)$
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Phsyics Kinematics: cannonball shot straight up
1. Feb 28, 2010
okgo
2. Feb 28, 2010
jfy4
here is my math for it.
$$v_{y}=v\sin\theta$$
$$v_{x}=v\cos\theta$$
$$t=\frac{v\sin\theta}{g}$$
to get the total time in the air multiply by two, becuase that time is only half way.
$$x=v\cos\left(\theta\right) t$$
plug in t
$$x=\frac{2v^2\sin\theta \cos\theta}{g}$$
that should get you the distance traveled... i think but look over my math.
3. Mar 1, 2010
okgo
Oh. I'm having trouble with the angle though. Not sure what it would be.
4. Mar 1, 2010
okgo
and isn't there acceleartion in the x direction too?
5. Mar 1, 2010
jfy4
From what i saw in the problem there were not any more forces than just gravity... and gravity only works in the y direction. your angle is the one given in the problem, 45.
6. Mar 1, 2010
okgo
Oh I was talking about question 23. Hehe I already solved for question 22. Oh well. My exam is in 20 min. Dumdumdum. So it's okay. Thanks for the help though
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# SAT/ACT problem of the week, October 27, 2016 solution
Hint for process of elimination: You need to know that “integers” are the positive and negative whole numbers, “nonnegative” means the numbers are not negative, and “distinct” means the list does not repeat any numbers. If the arithmetic mean of the five numbers is 7, then either all five numbers are 7, or at least one of the numbers is greater than 7. This helps you to eliminate two choices.
Another hint: They want to know the maximum value of the largest number is. The maximum value of the largest number occurs when the other four numbers are as small as possible. So instead of searching for the largest number, choose the other four numbers as small as possible, and use that to get the largest number, or at least an idea of how large the largest number could be.
Read on for a full solution.
# SAT/ACT problem of the week, October 27, 2016
The product and average (arithmetic mean) of five distinct nonnegative integers are 0 and 7, respectively. What is the maximum value of the largest number?
Solutions, hints, and questions are welcomed. A full solution will be posted on November 2nd. If you would like to learn how to enter math formulas into this blog, visit the WordPress LaTeX tutorial page.
# SAT problem of the week, November 03, 2014 solution
Hint for process of elimination: $f(g(6))$ is a composition of functions, not an addition, subtraction, multiplication, nor division.
Read on for a full solution.
# SAT problem of the week, October 27, 2014 solution
Hint for process of elimination: You need to know that “integers” are the positive and negative whole numbers, “nonnegative” means the numbers are not negative, and “distinct” means the list does not repeat any numbers. If the arithmetic mean of the five numbers is 7, then either all five numbers are 7, or at least one of the numbers is greater than 7. This helps you to eliminate two choices.
Another hint: They want to know the maximum value of the largest number is. The maximum value of the largest number occurs when the other four numbers are as small as possible. So instead of searching for the largest number, choose the other four numbers as small as possible, and use that to get the largest number, or at least an idea of how large the largest number could be.
Read on for a full solution.
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Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
Already signed up? New to KöMaL?
# KöMaL Problems in Mathematics, February 2016
Show/hide problems of signs:
## Problems with sign 'K'
Deadline expired on March 10, 2016.
K. 493. Is it possible to write the numbers 1, 2, 3, 4, 5, 6, 7 and 8 on the vertices of a cube so that the sum of the numbers on the vertices of each face is a prime number?
(6 pont)
solution, statistics
K. 494. The perimeters of two triangles are the integers $\displaystyle y$ and $\displaystyle y + 1$. Each triangle has two sides of integer lengths of $\displaystyle x$ and $\displaystyle x + 1$. Given that the sum of the perimeters is 27, how long are the third sides?
(6 pont)
solution, statistics
K. 495. In a class of 30 students, there are 12 more girls than boys. A team of three members is to be selected that contains at least one boy and at least one girl. How many different teams are possible? (Two teams are considered different if there is at least one person who is a member of one team but is not a member of the other.)
(6 pont)
solution, statistics
K. 496. Seven friends decided to form clubs. Each club is to have three members, and any two clubs may have at most one member in common. Can they form seven clubs?
(6 pont)
solution, statistics
K. 497. In a right-angled triangle $\displaystyle ABC$, $\displaystyle BC = 5$ and $\displaystyle AB = 12$. $\displaystyle M$ is the intersection of hypotenuse $\displaystyle AC$ with the arc of radius $\displaystyle AB$ centred at $\displaystyle A$, and $\displaystyle N$ is the intersection of hypotenuse $\displaystyle AC$ with the arc $\displaystyle BC$ centred at $\displaystyle C$. Determine the distance between points $\displaystyle M$ and $\displaystyle N$.
(6 pont)
solution, statistics
K. 498. A circle is divided into twelve equal arcs, and the points of division are joined as shown in the figure. Determine the proportions of the areas of the rhombuses formed.
(6 pont)
solution, statistics
## Problems with sign 'C'
Deadline expired on March 10, 2016.
C. 1336. In how many different ways is it possible to dissect a $\displaystyle 6\times6$ square into $\displaystyle 1\times3$ rectangles?
(5 pont)
solution, statistics
C. 1337. Csongor's wife sewed a leather sheath decorated with 77 beads for his husband's mouth harp, and gave it to him as a birthday present. Csongor liked it so much that he decided to surprise every member of his heritage preservation mouth harp band with a sheath like his own. He presented the sheaths in the main yurt erected for the celebration of the winter solstice. The yurt had a maximum capacity of 50 persons. Since the beads were bought in packets of 100, 7 beads remained, so Csongor's wife decorated her traditional headdress with them. How many members are there in Csongor's mouth harp band?
(5 pont)
solution, statistics
C. 1338. Let $\displaystyle D$ denote a point on base $\displaystyle AB$, and let $\displaystyle E$ denote a point on leg $\displaystyle BC$ of an isosceles triangle $\displaystyle ABC$ such that the triangles $\displaystyle ACD$, $\displaystyle CDE$, and $\displaystyle BDE$ are all isosceles, and triangle $\displaystyle BDE$ is similar to triangle $\displaystyle ABC$. Determine the angles of each triangle.
(5 pont)
solution, statistics
C. 1339. In a tree nursery, trees are planted at the lattice points of a $\displaystyle 30~\rm m\times 40~m$ square grid. The distance between adjacent trees along the grid lines is 1 metre. A meadow mouse goes for a morning walk around the tree grove. During his walk, his distance from the closest tree is 1 metre at every time instant. What is the total distance covered by the meadow mouse if he walks around without turning back?
(5 pont)
solution, statistics
C. 1340. Points $\displaystyle P$, $\displaystyle Q$, $\displaystyle R$, $\displaystyle S$ lie on sides $\displaystyle AB$, $\displaystyle BC$, $\displaystyle CD$, $\displaystyle DA$ of a rectangle $\displaystyle ABCD$, respectively. Line segments $\displaystyle PR$ and $\displaystyle QS$ are perpendicular. Prove that the midpoints of line segments $\displaystyle SP$, $\displaystyle PQ$, $\displaystyle QR$ and $\displaystyle RS$ form a rectangle, which is similar to $\displaystyle ABCD$.
(5 pont)
solution, statistics
C. 1341. Find the largest power of 2 that divides 2016!.
(5 pont)
solution, statistics
C. 1342. Determine the value of the parameter $\displaystyle a$ such that the equation below has exactly two solutions: $\displaystyle x^3 -a=\sqrt[3]{x+a}\,$. Find the solutions, too.
(5 pont)
solution, statistics
## Problems with sign 'B'
Deadline expired on March 10, 2016.
B. 4768. Find all fractions in which the numerator and denominator are both two-digit numbers such that the second digit of the numerator equals the first digit of the denominator, and the value of the fraction stays the same if these identical digits are deleted.
Proposed by A. Velkeyné Gréczi, Ipolyszög
(3 pont)
solution, statistics
B. 4769. In which triangles are the points dividing the sides 2:1 all concyclic?
Based on the idea of J. Szoldatics, Budapest
(3 pont)
solution, statistics
B. 4770. What may be the last digit of a positive integer $\displaystyle n\ge 3$ if $\displaystyle n+n^{2} +\dots+n^{2n-3} - 4$ is a prime?
(4 pont)
solution, statistics
B. 4771. In an aeroplane, there are one hundred seats, booked by one hundred passengers, each having their assigned seat. However, the first passenger does not care, and sits down on a random seat. When the other passengers enter one by one, each of them tries to take his or her own seat, or, if that seat is already taken, selects another one at random. What is the probability that the hundredth passenger is able to take his own seat?
Proposed by N. Nagy, Budapest
(5 pont)
solution, statistics
B. 4772. Is it true that if the four sides in two convex quadrilaterals are pairwise equal, and the two diagonals are also pairwise equal then the two quadrilaterals are congruent?
Proposed by V. Vígh, Szeged
(4 pont)
solution, statistics
B. 4773. The centre of the inscribed circle of triangle $\displaystyle ABC$ is $\displaystyle O$, and the centre of the circumscribed circle is $\displaystyle K$. Prove that the vector $\displaystyle \frac{\overrightarrow{AB}}{AB}+ \frac{\overrightarrow{BC}}{BC}+ \frac{\overrightarrow{CA}}{CA}$ is perpendicular to line $\displaystyle OK$.
Proposed by G. Holló, Budapest
(5 pont)
solution, statistics
B. 4774. The parabolas $\displaystyle p_1$ $\displaystyle \big(y=-x^2+b_1 x+c_1\big)$ and $\displaystyle p_2$ $\displaystyle \big(y=-x^2+b_2 x+c_2\big)$ are tangent to the parabola $\displaystyle p_3$ $\displaystyle \big(y=x^2+b_3x+c_3\big)$. Prove that the line connecting the points of tangency is parallel to the common tangent of $\displaystyle p_1$ and $\displaystyle p_2$.
Kvant
(5 pont)
solution, statistics
B. 4775. Find those pairs $\displaystyle (n,k)$ of positive integers for which
$\displaystyle \sum_{i=1}^{2k+1} {(-1)}^{i-1} a_{i}^{n}\ge \bigg(\sum_{i=1}^{2k+1} {(-1)}^{i-1} a_{i}\bigg)^{\!\!n}$
for all real numbers $\displaystyle a_1\ge a_2\ge \dots \ge a_{2k+1}\ge 0$.
Proposed by Á. Somogyi, Budapest
(6 pont)
solution, statistics
B. 4776. Let $\displaystyle \mathcal{O}$ be a regular octahedron. How many different axes of rotation are there such that $\displaystyle \mathcal{O}$ is mapped onto itself by a rotation through an angle of at most $\displaystyle 180^{\circ}$?
(6 pont)
solution, statistics
## Problems with sign 'A'
Deadline expired on March 10, 2016.
A. 662. The points $\displaystyle A_1$, $\displaystyle A_2$, $\displaystyle A_3$, $\displaystyle A_4$, $\displaystyle B_1$, $\displaystyle B_2$, $\displaystyle B_3$, $\displaystyle B_4$ lie on a parabola in this order. For every pair $\displaystyle (i,j)$ with $\displaystyle 1\le i,j\le4$ and $\displaystyle i\ne j$, let $\displaystyle r_{ij}$ denote the ratio in which the line $\displaystyle A_jB_j$ divides the segment $\displaystyle A_iB_i$. (That is, if $\displaystyle A_iB_i$ and $\displaystyle A_jB_j$ meet at $\displaystyle X$ then $\displaystyle r_{ij}=\frac{A_iX}{XB_i}$.) Show that if two of the numbers $\displaystyle r_{12} \cdot r_{21} \cdot r_{34} \cdot r_{43}$, $\displaystyle r_{13} \cdot r_{31} \cdot r_{24} \cdot r_{42}$ and $\displaystyle r_{14} \cdot r_{41} \cdot r_{23} \cdot r_{32}$ coincide then the third one is also equal to them.
(5 pont)
solution, statistics
A. 663. There are given two positive integers: $\displaystyle k$ and $\displaystyle \ell$. A square with horizontal and vertical sides is divided into finitely many rectangles by line segments such that the following statements are satisfied: $\displaystyle (i)$ every horizontal or vertical line of the plane contains at most one of the segments; $\displaystyle (ii)$ no two segments cross each other in their interiors; $\displaystyle (iii)$ every horizontal line, intersecting the square but not containing any of the segments, intersects exactly $\displaystyle k$ rectangles; $\displaystyle (iv)$ every vertical line, intersecting the square but not containing any of the segments, intersects exactly $\displaystyle \ell$ rectangles. What can be the number of rectangles?
Russian problem
(5 pont)
solution, statistics
A. 664. Let $\displaystyle a_1<a_2<\ldots<a_n$ be an arithmetic progression of positive integers. Prove that $\displaystyle [a_1,a_2,\ldots,a_n] \ge [1,2,\ldots,n]$. (The symbol $\displaystyle [\ldots]$ stands for the least common multiple.)
(5 pont)
solution, statistics
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Valence-bond entanglement and fluctuations in random singlet phases
# Valence-bond entanglement and fluctuations in random singlet phases
Huan Tran111Current address: Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland and N. E. Bonesteel Department of Physics and National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA
July 12, 2019
###### Abstract
The ground state of the uniform antiferromagnetic spin-1/2 Heisenberg chain can be viewed as a strongly fluctuating liquid of valence bonds, while in disordered chains these bonds lock into random singlet states on long length scales. We show that this phenomenon can be studied numerically, even in the case of weak disorder, by calculating the mean value of the number of valence bonds leaving a block of contiguous spins (the valence-bond entanglement entropy) as well as the fluctuations in this number. These fluctuations show a clear crossover from a small regime, in which they behave similar to those of the uniform model, to a large regime in which they saturate in a way consistent with the formation of a random singlet state on long length scales. A scaling analysis of these fluctuations is used to study the dependence on disorder strength of the length scale characterizing the crossover between these two regimes. Results are obtained for a class of models which include, in addition to the spin-1/2 Heisenberg chain, the uniform and disordered critical 1D transverse-field Ising model and chains of interacting non-Abelian anyons.
###### pacs:
75.10.Pq, 75.10.Nr, 05.10.Ln
## I Introduction
The set of valence-bond states — states in which localized spin-1/2 particles are correlated in singlet pairs said to be connected by valence bonds — provides a useful basis for visualizing singlet ground states of quantum spin systems. For example, the ground state of the uniform one-dimensional nearest-neighbor spin-1/2 antiferromagnetic (AFM) Heisenberg model (the prototypical spin-liquid stateand87 ()) can be viewed as a strongly fluctuating liquid of valence bonds with a power-law length distribution. This intuitive picture reflects the long-range spin correlations in this state, as well as the existence of gapless excitations created by breaking long bonds.
Valence-bond states also play a key role in describing the physics of random spin-1/2 AFM Heisenberg chains. For these systems, it was shown by Fisher,fis94 () using a real space renormalization group (RSRG) analysis, that on long length scales the ground state is described by a single valence-bond state known as a random singlet state. This single valence-bond state should be viewed as a caricature of the true ground state, which will certainly exhibit bond fluctuations on short length scales. In fact, it is natural to expect that, when measured on these short length scales, a fluctuating random singlet state would be difficult to distinguish from the uniform Heisenberg ground state, particularly in the limit of weak disorder.
In valence-bond Monte Carlo (VBMC) simulationssan05 () valence-bond states are used to stochastically sample singlet ground states of quantum spin systems. One of the appealing features of VBMC is that if one imagines viewing the sampled valence-bond states over many Monte Carlo time steps the resulting “movie” would correspond closely to the intuitive resonating valence bond picture described above. For random Heisenberg chains (and related models) VBMC should therefore provide a useful method for directly studying the phenomenon of random singlet formation on long length scales, while at the same time capturing the short-range fluctuations which will always be present.
With this motivation we have carried out a VBMC study of a class of models which include the uniform and random spin-1/2 AFM Heisenberg chains, as well as models which describe chains of interacting non-Abelian anyons, as special cases. The paper is organized as follows. First, in Sec. II, we define the models and describe their relevant Hilbert spaces. In Sec. III, we give a short review of the VBMC method, and in Sec. IV present results for the valence-bond entanglement entropy of the uniform and random models. In Sec. V we introduce the valence-bond fluctuations — a measure of how strongly the valence bonds are fluctuating on a given length scale — and show that this quantity can be used to provide a clear signature of random singlet state formation. Results of a scaling analysis of the valence-bond fluctuations are then presented in Sec. VI and the paper ends with some conclusions in Sec. VII.
## Ii Hilbert Space and Model Hamiltonians
To define the class of model Hamiltonians studied here, we first specify the relevant Hilbert spaces on which they act. It is well known that the set of non-crossing valence-bond states (see Fig. 1(a)) forms a complete linearly independent basis spanning the total spin 0 Hilbert space of a chain of spin-1/2 particles.rum32 () We denote the singlet projection operator acting on neighboring sites and by , which, for spin-1/2 particles, can be expressed as
Π0i=14−→Si⋅→Si+1, (1)
where is a spin-1/2 operator,(here and throughout ). Figure 1(b) shows two representative examples of acting on a non-crossing valence-bond state. For spin-1/2 particles, the parameter appearing in Fig. 1(b) is equal to 2; however, in principle, can take any value, (of course if the Hilbert space no longer describes spin-1/2 particles).
Of particular interest are the cases
d=2cosπk+2, (2)
where is a positive integer.kau94 () For these values of , when is finite, the non-crossing states are no longer linearly independent and the Hilbert space dimensionality of sites can be shown to grow asymptotically as with . The limit then corresponds to the case of ordinary spin-1/2 particles with for which the Hilbert space dimensionality grows as .
One consequence of the reduced Hilbert space dimensionality for finite integer is that it changes the entanglement entropy associated with a valence bond. The entanglement entropy of a subsystem of a larger system consisting of parts and is defined to be the von Neumann entropy, , of the reduced density matrix obtained by tracing out the degrees of freedom in region , thus
SvN=−Tr[ρAlog2ρA]. (3)
With this definition, an ordinary singlet formed by two spin-1/2 particles, with one spin in region and the other in region , will have . However, when , it was shown in Ref. bon07, that if there are valence bonds connecting sites in region with sites in region , then, in the limit, because the dimensionality of the traced out Hilbert space grows as , and the entanglement per bond is .
The class of Hamiltonians studied here are all characterized by the parameter and have the form
H=−∑iJiΠ0i, (4)
with . For these models correspond to spin-1/2 AFM Heisenberg chains with equal to the exchange energy associated with spins and . For general , if the ’s are uniform () the Hamiltonians (4) can be viewed as 1+1 dimensional quantum Potts models obtained by taking the asymmetric limit of the transfer matrix of the -state Potts modelsmartinbook () with .
For the uniform models are all gapless, and for the special values they correspond to a sequence of conformally invariant Andrews-Baxter-Forresterabf () (ABF) models with central charges .huse84 () Physically, these ABF models can be thought of as describing chains of interacting non-Abelian particles described by Chern-Simons-Witten theory, believed to be relevant for certain quantum Hall states.fei07 () Two special cases of these models are () which corresponds to the critical 1D transverse field Ising model and ( where is the golden mean) which corresponds to the so-called golden chain made up of interacting Fibonacci anyons.fei07 () The known universal entanglement scaling of conformally invariant 1+1 dimensional systemsvid03 () implies that the entanglement entropy of a block of contiguous sites, , in the ground states of these models will scale logarithmically for asfei07 ()
SvNL≃ck3log2L. (5)
When the ’s are random, the Hamiltonians (4) can no longer be solved exactly. However, the RSRG approach of Fisherfis94 () can be straightforwardly applied for all with the result that the ground states all flow to the same infinite randomness fixed pointbon07 (); footnote1 () — one for which the bond strength distribution is the same as that of the fixed point of the random Heisenberg chain.fis94 () For this fixed point, Refael and Mooreref04 () have shown that if is the number of valence bonds leaving a given block of size (see Fig. 1(c)), then, in the limit,
¯¯¯¯¯¯nL≃lnL3≃ln23log2L, (6)
where the overbar denotes a disorder average over random singlet states produced by the RSRG. This logarithmic scaling is a direct consequence of the inverse-square distribution of valence bond lengths characteristic of random singlet states.hoy07 () Multiplying by the entanglement per bond of then yields the RSRG result for the asymptotic scaling of the entanglement entropy for the random ABF models, which is again logarithmic and has the formref04 (); bon07 ()
SvNL≃¯¯¯¯¯¯nL log2d≃lnd3log2L. (7)
## Iii Valence-bond Monte Carlo
When applying the VBMC methodsan05 () to Hamiltonians of the form (4) the ground state is projected out by repeatedly applying to a particular non-crossing valence-bond state . The result of this projection after iterations is,
(−H)n|S⟩=∑i1,⋯,inJi1⋯JinΠ0i1⋯Π0in|S⟩. (8)
The properties of the projection operators shown in Fig. 1(b) imply that where is a non-crossing valence-bond state with the same norm as . The coefficient is given by where is the number of times a projection operator acts on two sites which are not connected by a valence bond when projecting onto .san05 (); tra10 () This projection thus leads to an expression for the ground state which becomes exact in the limit of large (in our simulations we find it is sufficient to take where is the number of sites) and has the form
|ψ⟩=∑αw(α)|α⟩, (9)
where . In the simplest form of VBMC the valence-bond states contributing to are sampled with probability by updating the sequence of projection operators using the usual Metropolis method.san05 ()
To use VBMC to calculate the quantum mechanical expectation value of given operator , i.e. , it is necessary to project the ground state out of both the bra and ket states, in which case one samples from “loop” configurations corresponding to the valence-bond state overlaps with probabilities weighted by .san05 () However, using the “one-way” VBMC described above in which one simply samples from the valence-bond basis it is possible to calculate a number of interesting quantities which can be used to characterize the intuitive valence-bond description of the ground state wavefunction.
In particular, given an observable with expectation values in the non-crossing valence-bond states , VBMC can be used to compute the average
⟨O⟩=∑αw(α)O(α)∑αw(α), (10)
for any state of the form (9), provided . In what follows, angle brackets will always denote this average, though it should be noted that will in general not be equal to , both because the valence-bond states are nonorthogonal and because the weight factors are amplitudes and not probabilities.
## Iv Valence-bond entanglement
One quantity that can be calculated naturally by VBMC is the valence-bond entanglement entropy, , which, for the uniform Heisenberg chain, is defined to be equal to , the average number of valence bonds leaving a block of size .alet07 (); chh07 () To generalize to the ABF models with it is natural to multiply by the asymptotic entanglement per bond of . For this choice, provided , will be equal to for any single valence-bond state. We therefore take
SVBL=⟨nL⟩log2d. (11)
While is easy to compute numerically by VBMC, for a general superposition of valence-bond states it will not be equal to . Nonetheless, VBMC simulationsalet07 (); chh07 (); kal09 () of the uniform AFM Heisenberg chain with spins have shown numerically that grows logarithmically with , in the same fashion as the von Neumann entanglement . To characterize this log scaling it is convenient to introduce an effective valence-bond central charge, , defined so that
SVBL≃cVB3log2L, (12)
in the limit .
In addition to showing log scaling of , previous VBMC simulations of the uniform AFM spin-1/2 Heisenberg chain have given results consistent with being close to,chh07 () or even possibly equal to,alet07 () , the value of the true central charge for the uniform model. However, Jacobsen and Saleurjac08 () were able to determine the exact asymptotic scaling of analytically for all . Their results both confirmed the log scaling of for and provided an analytic result for the coefficient of the log, which yields the following expression for the valence-bond central charge,
cVB=6lndπ√2+d2−darccos(d/2)π−arccos(d/2). (13)
In the limit , this expression gives , which is therefore not equal to the true central charge of 1.
Figure 2(a) shows our VBMC results for for and (corresponding to and 2, respectively) for periodic systems with sites. To minimize finite size effects when is comparable to , is plotted as a function of the conformal distance . The solid lines show the exact asymptotic scaling found by Jacobsen and Saleurjac08 () which clearly agree with our numerical results for . Note that for the case it is necessary to consider fairly large values of before entering the scaling regime, whereas for and 3 the scaling begins at relatively small . This fact may account for the initial numerical difficulty in determining for using small systems (see, however, Ref. kal09, ). Presumably, the reason that the finite size effects become more pronounced as approaches 2 is because this is a critical value (for the uniform models acquire a gapmartinbook ()).
For random Heisenberg chains was first computed numerically by Alet et al.alet07 (). Following the same procedure as these authors, we compute by determining for particular realizations of disorder and then disorder averaging. Throughout this paper we assume the random models are characterized by a flat bond strength distribution centered around of width ,
P(J)=12uΘ((1+u)−J)Θ(J−(1−u)), (14)
where is a measure of disorder strength. For the random ABF models we again multiply by the entanglement per bond, and thus take
SVBL=¯¯¯¯¯¯¯¯¯¯⟨nL⟩ log2d, (15)
(here again the overbar denotes disorder average). Figure 2(b) shows log plots of our results for for random chains with strong disorder (), again for and and . The solid lines show the scaling predictions based on the RSRGref04 (); bon07 () for which clearly agree with our numerical results. As pointed out by Alet et al.alet07 (), the fact that and show the same scaling for is to be expected if, as predicted by the RSRG, on long length scales the ground states of the random models are dominated by a single valence-bond configuration.
The log scalings of shown in Fig. 2(a) and 2(b) are summarized in Fig. 3, which shows our VBMC results for for both uniform and random models and various values of corresponding to and . For the uniform models, Fig. 3 also shows the exact values of (see (13)) which follow from the analytic results of Jacobsen and Saleurjac08 () as well as the true central charges of the ABF models with .footnote2 () For the random models the dependence of is seen to be entirely due to the entanglement per bond, reflecting the fact that the valence bond length distribution, which determines the coefficient in front of the log scaling of , and which depends on for the uniform case, becomes independent of when disorder is included.
## V Valence-bond fluctuations
The expectation that for random models the scaling of should be the same as that of is based on the assumption that the valence bonds in the ground state of the model lock into a particular random singlet configuration on long length scales. This assumption is in turn based on the RSRG approach which, although it can be shown to capture the long distance properties of the fixed point exactly,fis94 () is still an approximate method. Consequently, it is clearly desirable to have a direct numerical demonstration that the valence bonds are indeed locking into a particular random singlet configuration on long length scales.
To provide such a demonstration, we calculate the fluctuations in , a quantity we refer to as the valence-bond fluctuations. To be precise, we first compute the quantity for a particular block of size and a particular realization of disorder, and then perform a disorder average. The quantity we compute is thus
σ2L=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨n2L⟩−⟨nL⟩2. (16)
For this choice of averaging has the property that, in an idealized random singlet phase for which the ground state is precisely a single non-crossing valence-bond state, would vanish, even though the number of bonds leaving a given block would be different for different realizations of disorder.
For the uniform models with , Jacobsen and Saleurjac08 () have also determined the asymptotic scaling of (in this case there is, of course, no disorder average). Like , scales logarithmically with for , with
σ2L≃blnL, (17)
and the analytic results of Ref. jac08, can again be used to obtain an exact analytic results for the coefficient, , as a function of ,
b=4π√2+d(2−d)32arccos(d/2)−√4−d2π−arccos(d/2). (18)
Figure 4 shows a log-linear plot of our VBMC results for calculated for the uniform model with . A line corresponding to the exact asymptotic log scaling found by Jacobsen and Saleurjac08 () is also shown. It is readily seen that our numerical results agree well with the predicted asymptotic scaling, and can be regarded as further numerical confirmation of the field-theoretic analysis of Ref. jac08, . We note that the log scaling of directly demonstrates the existence of bond fluctuations on all length scales in the ground state of the uniform model.
Figure 4 also shows that for small the valence-bond fluctuations oscillate strongly as the block length changes from even to odd. The origin of this even/odd effect can be understood by first considering a state in which the bonds are all of length (i.e. a dimerized state). In this case there would be two ground states corresponding to the two distinct dimerizations and the translationally invariant ground state would be an equal superposition of these two dimerized states. One can readily check that in such a state when is even, and when is odd. We believe that the even/odd oscillations apparent in Fig. 4 for small are due to the significant contribution of such dimerized regions (at least on small length scales) to the ground state wavefunction.
For random models, the RSRG approachfis94 () shows that on long length scales the bonds lock into a random singlet state. At the same time, on short length scales (if disorder is weak) it is natural to expect that the bonds will fluctuate strongly, as they do in the uniform models. This implies the existence of crossover length scale which characterizes the transition from the uniform regime to the random-singlet regime of these models with increasing . One can then expect the valence-bond fluctuations to not differ much from their value for the uniform models when , but for the fluctuations should saturate. This saturation is due to the fact that, once the block size becomes much larger than the crossover length , the bond fluctuations occurring outside of a distance from the two boundaries of the block will not change the number of bonds leaving the block, and hence will not contribute to the valence-bond fluctuations.
Figure 5 shows log-linear plots of our results for for the case (corresponding to the Heisenberg chain, with ), (corresponding to the golden chain, with ) and (corresponding to the critical transverse field Ising model, with ) for both uniform and random models. For the random models the ’s are taken to be distributed according to (14), where is a measure of disorder strength. It can be observed from Fig. 5 that the valence-bond fluctuations for the random models saturates, regardless of how weak the disorder is, on a finite length scale which grows as decreases.
The observation of this saturation, which indicates a finite crossover length scale beyond which the valence bonds effectively lock into a random singlet configuration, together with the log scaling of , which indicates a power-law distribution of valence bond lengths, provides a direct numerical proof of random singlet phase formation in these models.
## Vi Crossover length scale
As described in the previous section, the saturation of with increasing for disorder of any strength implies the existence of a finite fluctuation length scale which characterizes the transition from the resonating regime with to the saturation regime with . This length scale is essentially the crossover length scale from the uniform regime to the disordered regime, which has been studied in the literature both analytically and numerically for a number of models.dot92 (); gia88 (); laf04a (); laf04 ()
For and analytic results for the dependence of on can be obtained for the case of weak disorderdot92 () by mapping the models (4) onto disordered Luttinger liquids.gia88 () For the model (4) corresponds to an isotropic spin-1/2 Heisenberg chain which, via a Jordan-Wigner transformation, can be mapped onto a 1D interacting spinless Fermi gas with a particular interaction strength. Similarly, for the case the model (4) corresponds to the 1D transverse field Ising model, and a pair of independent but identical 1D transverse field Ising models can be mapped onto a spin-1/2 XX model which can in turn be mapped onto a (in this case free) 1D spinless Fermi gas. The resulting predictionsdot92 (); gia88 () for the scaling of the crossover length scale with disorder strength for these two cases are that for and for . Numerical results for , based on scaling analyses of the spin-spin correlation functionlaf04a (); laf04 () and the spin stiffnesslaf04 () of the Heisenberg chain (using quantum Monte Carlo) and the XX chain (by exact diagonalization) have shown results consistent with these weak disorder renormalization group predictions.
It is possible to determine the dependence of on by performing a scaling analysis of the valence-bond fluctuations . To do this, we first subtract the large saturated value of the fluctuations () obtained by extrapolating the data shown in Fig. 5 and attempt to collapse the data by assuming a scaling function and a dependent for which
σ2L(u)−σ2∞(u)=f(LCξ(u)). (19)
For each value of , is chosen so that data for collapse onto a single curve, with the center of the crossover regime being . Note that to avoid the even-odd effect pointed out earlier for small we only use odd values of , and to minimize finite size effects for large we use the conformal distance in the scaling analysis.
Results of carrying out this analysis for the cases , , and are shown on the left side of Fig. 6. The VBMC results for can be seen to be well collapsed onto a particular scaling function , according to the definition (19). The values we obtain for for these models corresponding to various disorder strength are given in Table 1.
On the right side of Fig. 6, log-log plots of the crossover length scales as a function of the disorder strength are shown. The exponents characterizing the divergence of are determined by fitting the data to the power law (solid lines). For , we find , consistent with the weak disorder renormalization group predictiondot92 (); gia88 () of . For , we find , which is somewhat less than the predicted value of . One possible reason for the poorer agreement in this case is that for the length scale is significantly larger than for for a given disorder strength, and it may therefore be necessary to study larger system sizes in order to enter the scaling regime for the valence-bond fluctuations.
For the case , for which the model (4) corresponds to a disordered golden chain, we find the exponent . Note that for this system (with ), and, in fact, for all cases for which , there is no simple mapping of (4) to a disordered 1D Luttinger liquid. It is therefore not possible to apply the same weak disorder renormalization group analysis to these models that can be used to obtain the exponent for and . To the best of our knowledge there are no known analytic results for or for these more general models and we believe our result for represents the first numerical calculation of such an exponent.
## Vii Conclusions
In this paper we have presented the results of a VBMC study of both uniform and random Hamiltonians of the form (4). Both the valence-bond entanglement entropy and the valence-bond fluctuations were calculated for these models. For uniform models both these quantities were found to scale logarithmically with and our results agreed well with analytic results obtained through a field-theoretic analysis by Jacobsen and Saleur.jac08 () For random models was also found to scale logarithmically with , consistent with predictions based on the RSRG,ref04 (); bon07 () while was found to saturate once exceeded a disorder dependent crossover length scale , signaling the expected locking of the valence bonds into a particular random singlet configuration on long length scales.
By performing a scaling analysis of the valence-bond fluctuations we were able to determine the dependence of on disorder strength. For the cases (spin-1/2 Heisenberg model) and (transverse field Ising model) our results were consistent with those based on a weak disorder renormalization group approachdot92 (); gia88 () as well as previous numerical work,laf04 (); laf04a () although for the case we may not have fully entered the scaling regime. An appealing feature of our bond fluctuation based approach is that it can be used to determine for any value of , not just and for which the models (4) can be mapped onto 1D Luttinger liquids (the starting point for the weak disorder renormalization group approach). For example, we have determined for the first time the crossover length scale and the corresponding exponent for the model (4) with , which corresponds to the disordered golden chain.
###### Acknowledgements.
The authors acknowledge US DOE Grant No. DE-FG02-97ER45639 for support. Useful discussions with K. Yang, A. Sandvik, P. Henelius, J. A. Hoyos, and F. Alet are sincerely acknowledged. Computational work was performed at the Florida State University High Performance Computing Center.
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# Resonance of Tosylate ion
For the Tosylate ion, the negatively charged oxygen is stabilized via resonance with the oxygen atoms. But what are the orbitals of sulfur that are involved in the overlap? Is the sulfur $$\ce{sp^2}$$ hybridized and thus have a p-orbital for resonance?
If it is $$\ce{sp^2}$$, would that mean that it is possible for the electrons in oxygen to delocalise into the pi-electron cloud of benzene via the continuous overlap of p-orbitals in sulfur, carbon and oxygen atoms?
• The sulphur has 4 $\sigma$ bonds and no lone pairs, so it is most likely $sp^3$ hybridized. Not written as an answer as I'm not completely sure.
– TRC
Jun 7 '21 at 9:16
• Yes, @TRC is right, the sulfur centre is tetrahedral, so the oxygen is not in the same plane as the ring. Jun 12 '21 at 14:23
Thanks to @Shoubhik R Maiti for confirming that my reasoning was correct
I hope you know the meanings of the terms $$\sigma$$ bond, $$\pi$$ bond and how to find hybridization of an atom. If not, you can get numerous good resources on Google.
To find the number of hybrid orbitals, we take the sum number of $$\sigma$$ bonds + number of lone pairs. Here, sulphur has six valence electrons, all bonded. So there are totally 4 $$\sigma$$ bonds and zero lone pairs. Resulting in hybridization $$sp^3$$, giving it a near-tetrahedral geometry (near because the aryl groups and oxygen atoms different, slightly distorting it from perfect tetrahedral). Since it is tetrahedral, the oxygen and phenyl group are not in the same plane, so the continuous overlap which you suggested is not possible.
What you are asking is if the electrons of the oxygen atom can delocalize into the benzene ring. Firstly I need to mention that the negative charge is equally distributed on all three oxygen atoms, due to equivalent resonance. Now, from the canonical structure you have given in the question, there can be two cases of oxygen's electrons delocalizing into the benzene ring:
1. The electrons of negatively charged oxygen bond with the sulphur atom. In that case the $$\sigma$$ bond between sulphur and benzene must completely break to accommodate the fresh electron pair from oxygen. You will be left with $$\ce{SO3}$$ molecule and a toluene with negative charge. This obviously can't happen.
2. The electrons from one of the double bonded oxygen atoms delocalizes into the ring. Then you shall have an oxygen atom with a positive charge on it, singly bonded to the sulphur, and the benzene ring double bonded to the sulphur with a negative charge on one of the carbons. This is energetically unfavourable because it involves unlike charges' separation (which takes energy), and has + charge on electronegative oxygen and vice-versa, making it an extremely unstable resonance structure. Hence even if it is a contributing canonical structure, its contribution to the final resonance hybrid is very, very small, almost negligible.
Hence in no case is it possible to have electrons from oxygen atoms delocalizing into the benzene ring.
• +1 I would also add that even though direct delocalisation cannot take place, hyperconjugation/ negative hyperconjugation can take place to some extent, but since oxygen is electronegative, the electrons from benzene ring are withdrawn. This is why SO3H is a meta-directing group. Jun 13 '21 at 7:41
• The $\ce{-SO3-}$ group ist tetrahedral, that is the only reason why it can be described with sp³ hybrid orbitals. This is disregarding whether or not hybrid orbitals are indeed a good description in this case. Then there is the problem of double bonded oxygen, which is a stretch. If ever, this is a zero order approximation. I'm sorry, but this answer falls short. It is one of those 'good resources' that unfortunately feed into the myths that hybridisation could be a cause for anything. It is a mere mathematical interpretation, making that clear from the start would go a long way for science. Jun 13 '21 at 17:14
• It doesn't matter whether you are aware or that or not; your answer uses reverse reasoning, it is wrong. Unfortunately it is employing those hand-wavy models which have been popular and continue to be so as these are fed by answers like this. I wholeheartedly disagree that this does not lead to wrong results. Apart from this, nobody who would need to read the correct answer will ever do so, and I do not need to post an answer for people who already know that. || Also, Newton's laws are a valid and good approximation; the comparison is moot and frankly quite insulting. Jun 14 '21 at 22:05
• @Martin-マーチン The hybridization approach is something that is taught in high school in chemical bonding everywhere in India (I do not know of other countries' curriculum). I have found it in a number of introductory textbooks too, and since all of it is wrong, it is necessary first to amend those textbooks first so that people like me don't learn and use the wrong concept right from the start. The myth is largely fed and popularized by the people who include it in the curriculum, much more so than answers of Chem SE. I don't see any point in carrying ahead this discussion in any case.
– TRC
Jun 15 '21 at 3:26
Simply put, the oxygens' lone pairs are NOT delocalised into the benzene ring, and the stabilisation of the sulfonate group must be allocated more to ionic interactions(or loss of such; see below) than to resonance effects(the latter of which does not really matter that much, due to the instability of the "toluenide anion"-containing resonance canonical).
As many people have pointed out in many different answers posted on this site, there are no double bonds in sulfate groups and the like- the sulfur has a +2 charge while the oxygens have -1 charges each. These charges attract each other, resulting in the "double-bond-like" sulfur-oxygen distances. If an electrophile attacks and bonds to the oxygen, then the oxygen would not be able to have a negative charge anymore, resulting in destabilisation from the loss of an electrostatic attraction interaction.
Now, both nitric and toluenesulfonic acids are strong acids. In the nitrate case, there are two factors (pi-resonance and (+1 charge nitrogen)-(-2/3 charge oxygen) electrostatic attractions at work here, while in the sulfonate case, there is only one factor at work here, namely the (+2 charge sulfur)-(-1 charge oxygen) electrostatic attractions. Note that the elctrostatic attractions in the sulfonate anion are much stronger than those in the nitrate anion.
The destabilisation of the loss of electrostatic attractions upon protonation of the former is nearly of the same order of magnitude of the destabilisation from the disturbing of the pi-system upon protonation of the latter(if any, since the proton certainly would not want to attack the pi-system and violate VSEPR theory), and hence both acids become nearly equally strong acids.
Note: all of the above is true because the "toluenide anion" is unstable. In situations like the trifluoromethanesulfonate anion, the three oxygens' sp³ lone pairs, in the same direction as the carbon-sulfur bond, CAN delocalise into the sulfur-carbon antibonding orbital(which is mainly of sulfur 3p character; remember that one side of a 3p orbital has one node, just like the whole of a 2sp³ orbital), since the "trifluoromethanide anion" is relatively stable due to the inductively electron withdrawing fluorines. In support of this argument, optimisation calculations done on Mopac using the PM7 level of theory indicate that, while there is a charge <-1 on each oxygen in the toluenesulfonate anion(i.e. no delocalisation of the oxygen lone pairs), while there is a charge slightly >-1 on each oxygen in the trifluoromethanesulfonate anion(i.e. some delocalisation of the oxygen lone pairs).
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• Browse all
Measurement of the $K_L$ nuclear interaction length in the NaI(Tl) calorimeter
JINST 10 (2015) P09006, 2015.
Abstract (data abstract)
NOVOSIBIRSK-VEPP-2M. In the study of the reaction $e^+e^-\to K_{S}K_{L}$ at the VEPP-2M $e^+e^-$ collider with the SND detector the nuclear interaction length of $K_{L}$ meson in NaI(Tl) has been measured. Its value is found to be 30-50 cm in the $K_{L}$ momentum range 0.11-0.48 GeV/$c$. The results are compared with the values used in the simulation programs GEANT4 and UNIMOD.
• #### Table 1
Data from Table 1
10.17182/hepdata.69471.v1/t1
The energy interval ($\sqrt{s}$), integrated luminosity ($IL$), number of selected events ($N$), number of background events ($N_{\rm bkg}$), number of...
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Math Calculators, Lessons and Formulas
It is time to solve your math problem
mathportal.org
• Pre algebra
• Absolute values
• Absolute value equations
# Absolute value equations
ans:
syntax error
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calculator
• Question 1: 1 pts Solve equation $|-3r|=15.$
$5$ $\{-5,5 \}$ $-5$ $0.5$
• Question 2: 1 pts Solve equation $\left|\dfrac{n}{3}\right|=2.$
$6$ $-6$ $\{-6,6 \}$ none of these
• Question 3: 1 pts Solve equation $|x-16|=0.3$
• Question 4: 1 pts Solve equation $|2x|+5=3.$
$2$ $-2$ $\{2,-2 \}$ none of these
• Question 5: 2 pts Solve equation $|6x+24|=0.$
$4$ $-4$ $\{4,-4 \}$ none of these
• Question 6: 2 pts Solve equation $|9x|-11=x.$
$\{-\dfrac{8}{10},-\dfrac{11}{10}\}$ $\{-\dfrac{11}{8},-\dfrac{11}{10}\}$ $\{\dfrac{11}{8},-\dfrac{11}{10}\}$ $\{-\dfrac{11}{8},-\dfrac{11}{10}\}$
• Question 7: 2 pts Solve equation $2x-7=|x+1|.$
$2$ $8$ $\{2,8 \}$ $\{-2,8 \}$
• Question 8: 2 pts Solve equation $6|1-5x|-9=57.$
• Question 9: 3 pts Solve equation $|4x-3|=|x+6|.$
$x=-3$ or $x=-\dfrac{3}{5}$ $x=3$ or $x=-\dfrac{3}{5}$ $x=3$ or $x=\dfrac{3}{5}$ $x=-3$ or $x=\dfrac{3}{5}$
• Question 10: 3 pts Solve equation $5|9-5n|-7=38.$
• Question 11: 3 pts Solve equation $|3x-2|=|5x+4|.$
$\left \{4,-\dfrac{1}{4}\right\}$ $\left \{-1,-\dfrac{1}{3}\right\}$ $\left \{-2,-\dfrac{2}{5}\right\}$ $\left \{-3,-\dfrac{1}{4}\right\}$
• Question 12: 3 pts Solve equation. $$\left|\dfrac{x+5}{2-x} \right|=6$$
$\left \{-1,-\dfrac{17}{5}\right \}$ $\left \{1,\dfrac{17}{5}\right \}$ $\dfrac{17}{5}$ $-1$
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#### COORDINATE_SYSTEM_FUNCTION
###### Coordinate system
*COORDINATE_SYSTEM_FUNCTION
"Optional title"
csysid, $x_0$, $y_0$, $z_0$
$\hat{x}_x$, $\hat{x}_y$, $\hat{x}_z$, $\bar{y}_x$, $\bar{y}_y$, $\bar{y}_z$
##### Parameter definition
VariableDescription
csysid Unique identification number
$x_0$, $y_0$, $z_0$ Coordinate of origin
options: constant, fcn
$\hat{x}_x$, $\hat{x}_y$, $\hat{x}_z$ Direction of local x-axis
options: constant, fcn
$\bar{y}_x$, $\bar{y}_y$, $\bar{y}_z$ Vector needed for the definition of the local y- and z-axis
options: constant, fcn
##### Description
This command defines a local cartesian coordinate system. The parameters can either be constants or functions.
The origin is located at ($x_0$, $y_0$, $z_0$) and the local x-direction is ($\hat{x}_x$, $\hat{x}_y$, $\hat{x}_z$). The local z-direction is defined as $\hat{\mathbf{z}} = \hat{\mathbf{x}} \times \bar{\mathbf{y}} / \vert \hat{\mathbf{x}} \times \bar{\mathbf{y}} \vert$ and the local y-direction as $\hat{\mathbf{y}} = \hat{\mathbf{z}} \times \hat{\mathbf{x}}$.
##### Example
###### Coordinate system defined with functions
A local coordinate system with its origin following sensor ID=8 and with prescribed, time dependent, direction cosines
*COORDINATE_SYSTEM_FUNCTION
"test"
1, fcn(10), fcn(11), fcn(12)
fcn(13), fcn(14), 0, fcn(15), fcn(13), 0
*FUNCTION
10
xs(8)
*FUNCTION
11
ys(8)
*FUNCTION
12
zs(8)
*FUNCTION
13
cos(360*t)
*FUNCTION
14
sin(360*t)
*FUNCTION
15
-sin(360*t)
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# How to get the .47 file from ORCA for natural bonding orbital analysis
I am trying to perform NBO analysis with relaxed density matrix with the open-source program JANPA. I am following the guide here. The first step is to get the NBO input .47 file from Orca v4.2.1. The guide says that I should request NBO analysis with Orca in absence of the actual NBO executable to get the .47 file.
So I set the GENEXE and NBOEXE environment variables to dummy exe files that don't do anything when called. However, in spite of this, I cannot see the .47 file anywhere on the disk.
How do I generate the file required for NBO analysis? Any help is appreciated.
Edit: The input file I am using is:
! RHF SP def2-SVP NPA
* xyz 0 1
C -1.54056 0.93336 0.00000
H -0.47056 0.93336 0.00000
H -1.89722 -0.02889 -0.30294
H -1.89722 1.67683 -0.68186
H -1.89723 1.15213 0.98480
*
I am setting the environment variables with
set GENEXE=C:\Users\Public\Orca4.1\orca\gennbo.exe
set NBOEXE=C:\Users\Public\Orca4.1\orca\nbo6.exe
Both nbo6.exe and gennbo.exe are dummy executables that don't do anything when executed. Note that the Orca version is 4.2.1 even though the directory name is 4.1.
• Can we see the input file you are using? You don't even necessarily have to include the geometry, just the keywords for the calculation. Also, your tutorial and the ORCA manual suggest its behavior with NBOs differs by version, so you should include the version of ORCA you are using.
– Tyberius
Feb 12 at 21:44
• @Tyberius Added input file and orca version. Feb 12 at 22:24
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# Ordinary Differential Equations/Linear Equations
## Linear Equations and Linear OperatorsEdit
A general linear equation is of the form
${\displaystyle P_{0}(x)y^{(n)}+P_{1}(x)y^{(n-1)}+...+P_{n}y=Q(x)}$ ,
where we will assume that the coefficient functions ${\displaystyle P_{i}(x)}$ and Q(x) are continuous functions in an interval [a,b], and that ${\displaystyle P_{0}(x)\neq 0}$ for all ${\displaystyle x\in [a,b]}$ . The existence theorem proves that a unique solution exists that passes through the point ${\displaystyle (x_{0},y_{0})}$ where ${\displaystyle x\in [a,b]}$ , with continuous derivatives up to the n-1 order, and satisfies initial conditions for each of those derivatives at ${\displaystyle x_{0}}$ .
One can also write this as
${\displaystyle L(y)=(P_{0}{\frac {d^{n}}{dx^{n}}}+P_{1}{\frac {d^{n-1}}{dx^{n-1}}}+...+P_{n})y=Q(x)}$
where L is called a linear differential operator of order n.
A differential equation of the form L(y)=0 with the same linear differential operator as above is called a homogeneous equation corresponding to the above equation, and the reduced equation of the above equation.
We will now prove some properties about the linear differential operator.
The linear differential operator:
${\displaystyle L(C_{1}y_{1}+C_{2}y_{2})=C_{1}L(y_{1})+C_{2}L(y_{2})}$ which is true because differentiation is linear. Thus, we can make the two following statements:
• If ${\displaystyle y_{1}}$ and ${\displaystyle y_{2}}$ are two solutions of the homogeneous equation, then ${\displaystyle C_{1}y_{1}+C_{2}y_{2}}$ is also a solution.
• If y is a solution to the homogeneous equation L(y)=0, and ${\displaystyle y_{0}}$ is a solution to the equation L(y)=Q(x), then ${\displaystyle y+y_{0}}$ is also a solution to the equation L(y)=Q(x).
## WronskianEdit
Let L(y)=0 be a homogeneous equation of degree n, and let ${\displaystyle y_{1},y_{2},...,y_{n}}$ be linearly independent solutions of this equation.
The general solution is then ${\displaystyle y=C_{1}y_{1}+C_{2}y_{2}+...+C_{n}y_{n}}$ .
Suppose instead that the solutions are not linearly independent. Then there exists values for ${\displaystyle C_{1},C_{2},...,C_{n}}$ not all zero such that ${\displaystyle C_{1}y_{1}+C_{2}y_{2}+...+C_{n}y_{n}}$ =0.
Then the following equations also hold true:
${\displaystyle C_{1}y_{1}'+C_{2}y_{2}'+...+C_{n}y_{n}'=0}$
${\displaystyle C_{1}y_{1}''+C_{2}y_{2}''+...+C_{n}y_{n}''=0}$
...
${\displaystyle C_{1}y_{1}^{(n-1)}+C_{2}y_{2}^{(n-1)}+...+C_{n}y_{n}^{(n-1)}=0}$
When there is to be a non-trivial solution to this system of homogeneous linear equations, the column vectors corresponding to each coefficient are linearly dependent. This is equivalent to saying that the following determinant, called the Wronskian is 0:
${\displaystyle {\begin{Vmatrix}y_{1}&y_{2}&...&y_{n}\\y_{1}'&y_{2}'&...&y_{n}'\\y_{1}''&y_{2}''&...&y_{n}''\\...&...&...&...\\y_{1}^{(n-1)}&y_{2}^{(n-1)}&...&y_{n}^{(n-1)}\\\end{Vmatrix}}=0}$
Thus, if the solutions are dependent, then their Wronskian is equal to 0, and conversely, and conversely if the Wronskian is equal to 0, then the columns of that matrix must be linearly dependent, indicating that the solutions are linearly dependent.
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I. Fundamentals
# Major Scales, Scale Degrees, and Key Signatures
Chelsey Hamm and Bryn Hughes
Key Takeaways
• A major is an ordered collection of half- and whole-steps with the ascending succession W-W-H-W-W-W-H.
• Major scales are named for their first note, which is also their last note. Be sure to include any accidentals that apply to this note in its name.
• are syllables notated by with (angled brackets) above them. The scale degrees are $\hat{1}$, $\hat{2}$, $\hat{3}$, $\hat{4}$, $\hat{5}$, $\hat{6}$, and $\hat{7}$.
• syllables are another method of naming notes in a major scale. The syllables are do, re, mi, fa, sol, la, and ti.
• Each note of a major scale is also named with . The first note of a major scale is called the tonic; the second note, the supertonic, followed by the mediant, subdominant, dominant, submediant, and leading tone.
• A , consisting of either or , appears at the beginning of a composition, after a clef but before a time signature.
• The order of sharps in key signatures is F, C, G, D, A, E, and B, while the order of flats is the opposite: B, E, A, D, G, C, F. In sharp key signatures, the last sharp is a half-step below the tonic (the first note of a scale). In flat key signatures, the second-to-last flat is the tonic.
• The is a convenient visual for remembering major key signatures. All of the major key signatures are placed on a circle, in order of number of accidentals.
A is an ordered collection of half- and whole-steps (see Half- and Whole-steps and Accidentals to review half- and whole-steps).
# Major Scales
A is an ordered collection of half- (abbreviated H) and whole-steps (abbreviated W) in the following ascending succession: W-W-H-W-W-W-H. Listen to Example 1 to hear a major scale, ascending:
Example 1. An ascending major scale.
In Example 1 the whole-steps are labeled with square brackets (and Ws), and the half-steps are labeled with angled brackets (and Hs). A major scale always starts and ends on notes of the same letter name, which should be an apart. In Example 1 the first note is C and the last note is C. Major scales are named for their first and last note. Example 1 depicts a C major scale because its first and last note is a C.
Always be sure to include the accidental of the first and last note when you name a scale. Example 2 shows this:
Example 2. A B-flat major scale.
The first and last note in Example 2 is a B♭. Therefore Example 2 is a B♭ (B-flat) major scale. You must say and write the flat, because B is different from the note B♭. Note that the pattern of half- and whole-steps is the same in every major scale, as shown in Examples 1 and 2.
# Scale Degrees, Solfège, and Scale-degree Names
Musicians name the notes of major scales in several different ways. are syllables notated by with , angled brackets, above them. The first note of a scale is $\hat{1}$ and the numbers ascend until the last note of a scale, which is also usually $\hat{1}$ (although some instructors prefer $\hat{8}$). Example 3 shows a D major scale with its scale-degree names and solfège labeled:
Example 3. A D major scale.
Each scale degree is labeled with an Arabic numeral and a caret in Example 3.
syllables are another method of naming notes in a major scale. The syllables do, re, mi, fa, sol, la, and ti can be applied to the first seven notes of any major scale; these are analogous to the scale degrees $\hat{1}$, $\hat{2}$, $\hat{3}$, $\hat{4}$, $\hat{5}$, $\hat{6}$, and $\hat{7}$. The last note is do ($\hat{1}$), because it is a repetition of the first note. Example 3 shows solfège applied to a D major scale, underneath the scale degrees. Because do ($\hat{1}$) changes depending on what the first note of a major scale is, this method of solfège is called This is in contrast to a solmization system, in which do ($\hat{1}$) is always the pitch class C.
Each note of a major scale is also named with . The first note of a major scale is called the tonic; the second note, the supertonic, followed by the mediant, subdominant, dominant, submediant, leading tone, and tonic:
Scale Degree Number Solfège Scale Degree Name
$\hat{1}$ do Tonic
$\hat{2}$ re Supertonic
$\hat{3}$ mi Mediant
$\hat{4}$ fa Subdominant
$\hat{5}$ sol Dominant
$\hat{6}$ la Submediant
$\hat{7}$ ti Leading Tone
$\hat{8}$ / $\hat{1}$ do Tonic
Example 4. Scale degree, solfège, and scale-degree names.
Example 5 shows these scale-degree names applied to an A♭ major scale:
Example 5. An A♭ major scale with scale-degree names.
Example 5 shows the notes of the A♭ major scale in order. Example 6 shows the notes of the A♭ major scale out of order, with scale-degree names:
In Example 6, the numbers and arrows above the staff indicate above and below the tonic. This example shows how the names of the scale degrees derived. The Latin prefix “super” means above, so the supertonic is a second above the tonic. The leading tone is a half-step (a second) below the tonic; it is often thought of as “leading” towards the tonic. The Latin prefix “sub” means below; therefore, the mediant is a third above the tonic, while the submediant is a third below the tonic. Likewise, the dominant is a fifth above the tonic, while the subdominant is a fifth below the tonic. It is a common misconception that the subdominant is so named because it is a second below the dominant, but this is not true, as demonstrated in Example 6.
# Key Signatures
, consisting of either or , appears at the beginning of a composition, after a clef but before a time signature. You can remember this order because it is alphabetical: clef, key, time. Example 7 shows a key signature, after a bass clef but before a time signature:
Major key signatures collect the accidentals in a major scale and place them at the beginning of a composition so that it is easier to keep track of which notes have accidentals applied to them. In Example 7, there are flats on the lines and spaces that indicate the notes B, E, and A (reading left to right). Every B, E, and A in a composition with this key signature will now be flat, regardless of octave. Example 8 demonstrates this:
In Example 8, both of these Bs will be flat, because B♭ is in the key signature. All of the Bs, Es, and As after this key signature will be flat, regardless of their octave.
There are flat key signatures and sharp key signatures. The order of the flats and sharps and key signatures is the same, regardless of clef. Example 9 shows the order of sharps and flats in all four clefs that we have learned:
The order of sharps is always F, C, G, D, A, E, B. This can be remembered with the mnemonic: Fat Cats Go Down Alleys (to) Eat Birds. Notice that sharps are always played on the same lines and spaces, making a somewhat zig-zag pattern, alternating going down and up. In the treble, bass, and alto clefs, this pattern “breaks” after D♯, and then resumes. In tenor clef, there is no break, but F♯ and G♯ appear in the lower octave instead of the upper octave.
The order of the flats is the opposite of the order of the sharps: B, E, A, D, G, C, F. This makes the order of flats and sharps palindromes. The order of flats can be remembered with this mnemonic: Birds Eat And Dive Going Crazy Far. The flats always make a perfect zig-zag pattern, alternating going up and down, regardless of clef, as seen in Example 9.
There are easy ways to remember which key signature belongs to which major scale. In sharp key signatures, the last sharp is a half-step below the tonic (the first note of a scale). Example 10 shows three sharp key signatures in different clefs:
The first key signature in Example 10 is in treble clef. The last sharp (in this case the only sharp), F♯, is a half-step below the note G. Therefore, this is the key signature of G major. The second key signature in Example 10 is in bass clef. The last sharp, G♯, is a half-step below the note A. Therefore, this is the key signature of A major. The third key signature in Example 10 is in alto clef. The last sharp, E♯, is a half-step below the note F♯. Therefore, this is the key signature of F♯ major.
In flat key signatures, the second-to-last flat is the tonic (the first note of a scale). Example 11 shows three flat key signatures in different clefs:
The first key signature in Example 11 is in bass clef. The second-to-last flat in this key signature is B♭. Therefore, this is the key signature of B♭ major. The second key signature in Example 11 is in treble clef, and its second-to-last flat is A♭. Therefore, this is the key signature of A♭ major. The third key signature in Example 11 is in tenor clef, and its second-to-last flat is G♭. Therefore, this is the key signature of G♭ major.
There are two key signatures that have no “tricks” that you will simply have to memorize. These are C major, which has nothing in its key signature (no sharps or flats), and F major, which has one flat (B♭). Example 12 shows these key signatures, the first in treble clef and the second in bass clef:
Example 13 shows all of the sharp key signatures in order:
Example 13. The key signatures of C, G, D, A, E, B, F♯, and C♯ in all four clefs.
Example 13 first shows the key signature of C major (with no sharps or flats), and then the key signatures of C, G, D, A, E, B, F♯, and C♯ in all four clefs. Example 14 shows all of the flat key signatures in order:
Example 14 first shows the key signature of C major (with no sharps or flats), and then the key signatures of F, B♭, E♭, A♭, D♭, G♭, and C♭ in all four clefs.
There is one other “trick” which might make memorization of the key signatures easier. C major is the key signature with no sharps or flats, C♭ is the key signature with every note flat (7 flats total), and C♯ is the key signature with every note sharp (7 sharps total). Remembering this may be helpful when memorizing your major key signatures.
Major key signatures are said to be “real” if they are one of the key signatures in Examples 13 or 14. If a double accidental is needed for a key signature (such as a or ), then a major key signature is said to be “imaginary.” Example 15 shows an F♭ major scale:
Example 15. An F♭ major scale in treble clef.
This F♭ major scale and its associated key signature are imaginary because there is a B𝄫. Occasionally, you may perform a composition which is in an imaginary key.
# The Circle of Fifths
The is a convenient visual. In the circle of fifths all of the major key signatures are placed on a circle, in order of number of accidentals. The circle of fifths is so named because each of these key signatures are a fifth apart. Example 16 shows the circle of fifths for major key signatures:
If you start at the top of the circle (12 o’clock), the key signature of C major appears, which has no sharps or flats. If you continue clockwise, sharp key signatures appear, each subsequent key signature adding one more sharp. If you continue from C major counter-clockwise, flat key signatures appear, each subsequent key signature adding one more flat. The bottom three key signatures (at 7, 6, and 5 o’clock) in Example 16 are . For example, B major and C♭ major scales will sound the same because B and C♭ are enharmonic. However, B and C♭ major scales have different key signatures—the former (B) is a five sharp key, while the latter (C♭) is a seven flat key.
Online Resources
Assignments from the Internet
1. Writing Major Scales (.pdf.pdf), from Tonic and Other Scale Degrees (.pdf)
2. Identifying Major Scales (.pdf)
3. Adding Accidentals to Write Major Scales (.pdf)
4. Writing Major Key Signatures (.pdf)
5. Writing and Identifying Major Key Signatures, p. 2 (.pdf)
6. Identifying Major Key Signatures (.pdf)
7. Major Keys Worksheets for Children (.pdf)
8. Scale Degrees or Solfège (.pdf.pdf, .pdf)
Assignments
1. Writing Major Scales (.pdf, .mscx)
2. Key Signatures: Major (.pdf, .mscx)
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# Overview
• describe the ‘standard workflow’ of a meta-analysis
• introduce additional complexities that often arise
• describe a workflow that addresses these issues
• illustrate analyses using the metafor package for R
# Standard Workflow
• define goal(s) of meta-analysis and inclusion/exclusion criteria
• find relevant studies that have examined phenomenon of interest
• quantify results in terms of an effect size measure
• quantify precision of estimates in terms of their variances
• meta-analyze the estimates (fixed- and/or random-effects model)
• profit! (at least get a publication in Nature or Science …)
# Random-Effects Model
• let $$y_i$$ denote the observed effect size in the $$i$$th study
• the random-effects model is given by $y_i = \mu + u_i + e_i$ where $$u_i \sim N(0, \tau^2)$$ and $$e_i \sim N(0, v_i)$$
• note: $$v_i$$ is the sampling variance of the $$i$$th estimate
# Some Examples
## BCG Vaccine
• meta-analysis on the effectiveness of the BCG vaccine against tuberculosis (Colditz et al., 1994)
• study participants were (randomly) assigned to either the treatment (vaccinated) or a control (not vaccinated) group
• number of TB+/TB- cases in each group recorded during follow-up
• effect size measure: (log) risk ratio
# trial author year tpos tneg cpos cneg
# 1 Aronson 1948 4 119 11 128
# 2 Ferguson & Simes 1949 6 300 29 274
# 3 Rosenthal et al 1960 3 228 11 209
# 4 Hart & Sutherland 1977 62 13536 248 12619
# 5 Frimodt-Moller et al 1973 33 5036 47 5761
# 6 Stein & Aronson 1953 180 1361 372 1079
# 7 Vandiviere et al 1973 8 2537 10 619
# 8 TPT Madras 1980 505 87886 499 87892
# 9 Coetzee & Berjak 1968 29 7470 45 7232
# 10 Rosenthal et al 1961 17 1699 65 1600
# 11 Comstock et al 1974 186 50448 141 27197
# 12 Comstock & Webster 1969 5 2493 3 2338
# 13 Comstock et al 1976 27 16886 29 17825
# trial author year tpos tneg cpos cneg yi vi
# 1 Aronson 1948 4 119 11 128 -0.8893 0.3256
# 2 Ferguson & Simes 1949 6 300 29 274 -1.5854 0.1946
# 3 Rosenthal et al 1960 3 228 11 209 -1.3481 0.4154
# 4 Hart & Sutherland 1977 62 13536 248 12619 -1.4416 0.0200
# 5 Frimodt-Moller et al 1973 33 5036 47 5761 -0.2175 0.0512
# 6 Stein & Aronson 1953 180 1361 372 1079 -0.7861 0.0069
# 7 Vandiviere et al 1973 8 2537 10 619 -1.6209 0.2230
# 8 TPT Madras 1980 505 87886 499 87892 0.0120 0.0040
# 9 Coetzee & Berjak 1968 29 7470 45 7232 -0.4694 0.0564
# 10 Rosenthal et al 1961 17 1699 65 1600 -1.3713 0.0730
# 11 Comstock et al 1974 186 50448 141 27197 -0.3394 0.0124
# 12 Comstock & Webster 1969 5 2493 3 2338 0.4459 0.5325
# 13 Comstock et al 1976 27 16886 29 17825 -0.0173 0.0714
# Random-Effects Model (k = 13; tau^2 estimator: REML)
#
# tau^2 (estimated amount of total heterogeneity): 0.3132 (SE = 0.1664)
# tau (square root of estimated tau^2 value): 0.5597
# I^2 (total heterogeneity / total variability): 92.22%
# H^2 (total variability / sampling variability): 12.86
#
# Test for Heterogeneity:
# Q(df = 12) = 152.2330, p-val < .0001
#
# Model Results:
#
# estimate se zval pval ci.lb ci.ub
# -0.7145 0.1798 -3.9744 <.0001 -1.0669 -0.3622 ***
#
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# pred ci.lb ci.ub pi.lb pi.ub
# 0.49 0.34 0.70 0.15 1.55
## Writing-to-Learn Interventions
• meta-analysis examining the effectiveness of school-based writing-to-learn interventions on academic achievement (Bangert-Drowns, Hurley, & Wilkinson, 2004)
• in each study, a group of students that received instruction with increased emphasis on writing tasks was compared against a group that received conventional instruction with respect to some content-related measure of academic achievement (e.g., final grade, an exam/quiz/test score)
• effect size measure: standardized mean difference
# id author year ni yi vi
# 1 Ashworth 1992 60 0.650 0.070
# 2 Ayers 1993 34 -0.750 0.126
# 3 Baisch 1990 95 -0.210 0.042
# 4 Baker 1994 209 -0.040 0.019
# 5 Bauman 1992 182 0.230 0.022
# 6 Becker 1996 462 0.030 0.009
# 7 Bell & Bell 1985 38 0.260 0.106
# 8 Brodney 1994 542 0.060 0.007
# 9 Burton 1986 99 0.060 0.040
# 10 Davis, BH 1990 77 0.120 0.052
# 11 Davis, JJ 1996 40 0.770 0.107
# 12 Day 1994 190 0.000 0.021
# 13 Dipillo 1994 113 0.520 0.037
# 14 Ganguli 1989 50 0.540 0.083
# ...
# 46 Willey 1988 51 1.460 0.099
# 47 Willey 1988 46 0.040 0.087
# 48 Youngberg 1989 56 0.250 0.072
# yi vi
# 1 0.5384 0.0834
# Random-Effects Model (k = 48; tau^2 estimator: REML)
#
# tau^2 (estimated amount of total heterogeneity): 0.0499 (SE = 0.0197)
# tau (square root of estimated tau^2 value): 0.2235
# I^2 (total heterogeneity / total variability): 58.37%
# H^2 (total variability / sampling variability): 2.40
#
# Test for Heterogeneity:
# Q(df = 47) = 107.1061, p-val < .0001
#
# Model Results:
#
# estimate se zval pval ci.lb ci.ub
# 0.2219 0.0460 4.8209 <.0001 0.1317 0.3122 ***
#
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# pred se ci.lb ci.ub pi.lb pi.ub
# 0.22 0.05 0.13 0.31 -0.23 0.67
## Class Attendance and Class Performance
• meta-analysis on the relationship between class attendance and class performance (Credé, Roch, & Kieszczynka, 2010)
• effect size measure: (r-to-z transformed) correlation coefficient
# studyid year source sampleid ni ri
# 1 2009 dissertation 1 76 0.8860
# 2 1975 journal 1 297 0.3000
# 4 1989 journal 1 265 0.4750
# 4 1989 journal 2 154 0.3340
# 5 2008 journal 1 162 0.6150
# 6 1999 journal 1 28 0.1450
# 6 1999 journal 2 33 0.2300
# 6 1999 journal 3 47 0.2700
# 6 1999 journal 4 25 -0.0228
# 6 1999 journal 5 48 0.4290
# 6 1999 journal 6 39 0.3490
# 6 1999 journal 7 41 0.2200
# 6 1999 journal 8 35 0.3390
# 6 1999 journal 9 46 0.4470
# ...
# 64 1980 journal 1 121 0.3500
# 65 2007 journal 1 100 0.2400
# 68 1986 journal 1 215 0.3090
# studyid year source sampleid ni ri yi vi
# 1 2009 dissertation 1 76 0.8860 1.4030 0.0137
# 2 1975 journal 1 297 0.3000 0.3095 0.0034
# 4 1989 journal 1 265 0.4750 0.5165 0.0038
# 4 1989 journal 2 154 0.3340 0.3473 0.0066
# 5 2008 journal 1 162 0.6150 0.7169 0.0063
# 6 1999 journal 1 28 0.1450 0.1460 0.0400
# 6 1999 journal 2 33 0.2300 0.2342 0.0333
# 6 1999 journal 3 47 0.2700 0.2769 0.0227
# 6 1999 journal 4 25 -0.0228 -0.0228 0.0455
# 6 1999 journal 5 48 0.4290 0.4587 0.0222
# 6 1999 journal 6 39 0.3490 0.3643 0.0278
# 6 1999 journal 7 41 0.2200 0.2237 0.0263
# 6 1999 journal 8 35 0.3390 0.3530 0.0312
# 6 1999 journal 9 46 0.4470 0.4809 0.0233
# ...
# 64 1980 journal 1 121 0.3500 0.3654 0.0085
# 65 2007 journal 1 100 0.2400 0.2448 0.0103
# 68 1986 journal 1 215 0.3090 0.3194 0.0047
# Random-Effects Model (k = 67; tau^2 estimator: REML)
#
# tau^2 (estimated amount of total heterogeneity): 0.0511 (SE = 0.0104)
# tau (square root of estimated tau^2 value): 0.2261
# I^2 (total heterogeneity / total variability): 93.83%
# H^2 (total variability / sampling variability): 16.21
#
# Test for Heterogeneity:
# Q(df = 66) = 1068.7213, p-val < .0001
#
# Model Results:
#
# estimate se zval pval ci.lb ci.ub
# 0.4547 0.0300 15.1343 <.0001 0.3958 0.5136 ***
#
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# pred ci.lb ci.ub pi.lb pi.ub
# 0.43 0.38 0.47 0.01 0.72
# Dependent Estimates
• in practice, the data structure is often more complex
• may be able to extract multiple estimates from the same study
• this can introduce two types of dependencies:
1. in the sampling errors
2. in the underlying true effects
• to account for these, have to:
1. compute covariances between the sampling errors
2. use appropriate random effects to capture the dependencies
a rough rule: the sampling errors of estimates are dependent when there is at least some overlap in subjects that contribute information to their computation
# Multilevel Data
• consider the meta-analysis by Credé et al. (2010) on the relationship between class attendance and class performance
• 6 studies included multiple samples (e.g., different sections)
# studyid year source sampleid ni ri yi vi
# 1 2009 dissertation 1 76 0.8860 1.4030 0.0137
# 2 1975 journal 1 297 0.3000 0.3095 0.0034
# 4 1989 journal 1 265 0.4750 0.5165 0.0038
# 4 1989 journal 2 154 0.3340 0.3473 0.0066
# 5 2008 journal 1 162 0.6150 0.7169 0.0063
# 6 1999 journal 1 28 0.1450 0.1460 0.0400
# 6 1999 journal 2 33 0.2300 0.2342 0.0333
# 6 1999 journal 3 47 0.2700 0.2769 0.0227
# 6 1999 journal 4 25 -0.0228 -0.0228 0.0455
# 6 1999 journal 5 48 0.4290 0.4587 0.0222
# 6 1999 journal 6 39 0.3490 0.3643 0.0278
# 6 1999 journal 7 41 0.2200 0.2237 0.0263
# 6 1999 journal 8 35 0.3390 0.3530 0.0312
# 6 1999 journal 9 46 0.4470 0.4809 0.0233
# ...
# 64 1980 journal 1 121 0.3500 0.3654 0.0085
# 65 2007 journal 1 100 0.2400 0.2448 0.0103
# 68 1986 journal 1 215 0.3090 0.3194 0.0047
• presumably no overlap of subjects across samples within studies
• hence, by the rule, the sampling errors are uncorrelated
• but the underlying true correlations may be more similar to each other for different samples within the same study than for samples from different studies
# Multilevel Model
• let $$y_{ij}$$ denote the $$j$$th observed effect size in the $$i$$th study
• the multilevel random-effects model is given by $y_{ij} = \mu + s_i + u_{ij} + e_{ij}$ where $$s_i \sim N(0, \sigma^2_s)$$, $$u_{ij} \sim N(0, \sigma^2_u)$$, and $$e_{ij} \sim N(0, v_{ij})$$
• $$\sigma^2_s$$ denotes between-study heterogeneity
• $$\sigma^2_u$$ denotes within-study heterogeneity
# Multivariate Meta-Analysis Model (k = 67; method: REML)
#
# Variance Components:
#
# estim sqrt nlvls fixed factor
# sigma^2.1 0.0376 0.1939 54 no studyid
# sigma^2.2 0.0159 0.1259 67 no studyid/sampleid
#
# Test for Heterogeneity:
# Q(df = 66) = 1068.7213, p-val < .0001
#
# Model Results:
#
# estimate se zval pval ci.lb ci.ub
# 0.4798 0.0331 14.5167 <.0001 0.4151 0.5446 ***
#
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# pred ci.lb ci.ub pi.lb pi.ub
# 0.45 0.39 0.50 0.02 0.73
## Aggregating Assumes Within-Study Homogeneity
• a common practice (in the past): aggregate multiple estimates within studies so that a standard RE model can be used
• implicitly assumes that effects within studies are homogeneous
• not an assumption we typically want to make!
# studyid year source sampleid ni ri yi vi
# 1 2009 dissertation 1.0 76.0 0.8860000 1.4030 0.0137
# 2 1975 journal 1.0 297.0 0.3000000 0.3095 0.0034
# 4 1989 journal 1.5 209.5 0.4045000 0.4547 0.0024
# 5 2008 journal 1.0 162.0 0.6150000 0.7169 0.0063
# 6 1999 journal 5.0 38.0 0.2673556 0.3066 0.0032
# ...
# 64 1980 journal 1.0 121.0 0.3500000 0.3654 0.0085
# 65 2007 journal 1.0 100.0 0.2400000 0.2448 0.0103
# 68 1986 journal 1.0 215.0 0.3090000 0.3194 0.0047
# Random-Effects Model (k = 54; tau^2 estimator: REML)
#
# tau^2 (estimated amount of total heterogeneity): 0.0528 (SE = 0.0115)
# tau (square root of estimated tau^2 value): 0.2298
# I^2 (total heterogeneity / total variability): 95.13%
# H^2 (total variability / sampling variability): 20.54
#
# Test for Heterogeneity:
# Q(df = 53) = 1034.5792, p-val < .0001
#
# Model Results:
#
# estimate se zval pval ci.lb ci.ub
# 0.4865 0.0332 14.6416 <.0001 0.4214 0.5517 ***
#
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Multivariate Meta-Analysis Model (k = 67; method: REML)
#
# Variance Components:
#
# estim sqrt nlvls fixed factor
# sigma^2 0.0528 0.2298 54 no studyid
#
# Test for Heterogeneity:
# Q(df = 66) = 1068.7213, p-val < .0001
#
# Model Results:
#
# estimate se zval pval ci.lb ci.ub
# 0.4865 0.0332 14.6416 <.0001 0.4214 0.5517 ***
#
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Multivariate Data
• now consider the case where some estimates are computed based on the same sample of subjects
• multiple scales (e.g., BDI and HDRS) may have been used to measure some construct of interest (depression) within a study
• can compute an effect size estimate for each scale
• might also be interested in multiple types of constructs or response variables (e.g., depression and anxiety) and some studies might have measured both
# Multivariate Model
• let $$y_{ij}$$ denote the $$j$$th observed effect size in the $$i$$th study
• assume $$j$$ denotes different types of constructs / outcomes
• the multivariate random-effects model is given by $y_{ij} = \mu_j + u_{ij} + e_{ij}$ where $$\left[ \begin{array}{c} u_{i1} \\ u_{i2} \\ \vdots \end{array} \right] \sim N\left( \left[ \begin{array}{c} 0 \\ 0 \\ \vdots \end{array} \right], \left[ \begin{array}{ccc} \tau_1^2 & \rho_{12}\tau_1\tau_2 & \ldots \\ \rho_{12}\tau_1\tau_2 & \tau_2^2 & \ldots \\ \vdots & \vdots & \ddots \end{array} \right] \right)$$ and $$\left[ \begin{array}{c} e_{i1} \\ e_{i2} \\ \vdots \end{array} \right] \sim N\left( \left[ \begin{array}{c} 0 \\ 0 \\ \vdots \end{array} \right], \left[ \begin{array}{ccc} v_{i1} & cov_{i12} & \ldots \\ cov_{i12} & v_{i2} & \ldots \\ \vdots & \vdots & \ddots \end{array} \right] \right)$$
## Surgical Treatment for Periodontal Disease
• meta-analysis on the effectiveness of a surgical versus a non-surgical procedure for the treatment of periodontal disease (Berkey, Hoaglin, Antczak-Bouckoms, Mosteller, & Colditz, 1998)
• each of the 5 included studies measures two types of outcomes in the same subjects: ‘probing depth’ and ‘attachment level’
• effect size measure: raw mean difference
# trial author year ni outcome yi vi v1i v2i
# 1 1 Pihlstrom et al. 1983 14 PD 0.4700 0.0075 0.0075 0.0030
# 2 1 Pihlstrom et al. 1983 14 AL -0.3200 0.0077 0.0030 0.0077
# 3 2 Lindhe et al. 1982 15 PD 0.2000 0.0057 0.0057 0.0009
# 4 2 Lindhe et al. 1982 15 AL -0.6000 0.0008 0.0009 0.0008
# 5 3 Knowles et al. 1979 78 PD 0.4000 0.0021 0.0021 0.0007
# 6 3 Knowles et al. 1979 78 AL -0.1200 0.0014 0.0007 0.0014
# 7 4 Ramfjord et al. 1987 89 PD 0.2600 0.0029 0.0029 0.0009
# 8 4 Ramfjord et al. 1987 89 AL -0.3100 0.0015 0.0009 0.0015
# 9 5 Becker et al. 1988 16 PD 0.5600 0.0148 0.0148 0.0072
# 10 5 Becker et al. 1988 16 AL -0.3900 0.0304 0.0072 0.0304
• also have the var-cov matrix of the sampling errors for each study
# [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
# [1,] 0.0075 0.0030 . . . . . . . .
# [2,] 0.0030 0.0077 . . . . . . . .
# [3,] . . 0.0057 0.0009 . . . . . .
# [4,] . . 0.0009 0.0008 . . . . . .
# [5,] . . . . 0.0021 0.0007 . . . .
# [6,] . . . . 0.0007 0.0014 . . . .
# [7,] . . . . . . 0.0029 0.0009 . .
# [8,] . . . . . . 0.0009 0.0015 . .
# [9,] . . . . . . . . 0.0148 0.0072
# [10,] . . . . . . . . 0.0072 0.0304
# Multivariate Meta-Analysis Model (k = 10; method: REML)
#
# Variance Components:
#
# outer factor: trial (nlvls = 5)
# inner factor: outcome (nlvls = 2)
#
# estim sqrt k.lvl fixed level
# tau^2.1 0.0327 0.1807 5 no AL
# tau^2.2 0.0117 0.1083 5 no PD
#
# rho.AL rho.PD AL PD
# AL 1 - 5
# PD 0.6088 1 no -
#
# Test for Residual Heterogeneity:
# QE(df = 8) = 128.2267, p-val < .0001
#
# Test of Moderators (coefficients 1:2):
# QM(df = 2) = 108.8607, p-val < .0001
#
# Model Results:
#
# estimate se zval pval ci.lb ci.ub
# outcomeAL -0.3392 0.0879 -3.8589 0.0001 -0.5115 -0.1669 ***
# outcomePD 0.3534 0.0588 6.0057 <.0001 0.2381 0.4688 ***
#
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Hypothesis:
# 1: outcomeAL - outcomePD = 0
#
# Results:
# estimate se zval pval
# 1: -0.6926 0.0744 -9.3120 <.0001
#
# Test of Hypothesis:
# QM(df = 1) = 86.7134, p-val < .0001
• sidenote: it is just a peculiar coincidence that all studies measured both outcomes, but this is not a requirement
# Longitudinal Data
• studies may have assessed some outcome at multiple timepoints
• effects sizes calculated for the same subjects at different timepoints are again dependent
• consider autoregressive structures to account for dependence
## Deep-Brain Stimulation for Parkinson’s Disease
• meta-analysis examining the effects of deep-brain stimulation on motor skills of patients with Parkinson’s disease (Ishak, Platt, Joseph, Hanley, & Caro, 2007)
• included 46 studies that measured the effect at 3, 6, and 12 months after implantation of the stimulator, with some studies also including a further long-term follow-up
• effect size measure: raw mean difference
# study mdur mbase time yi vi
# Alegret (2001) 16.1 53.6 1 -33.4 14.3
# Barichella (2003) 13.5 45.3 1 -20.0 7.3
# Barichella (2003) 13.5 45.3 3 -30.0 5.7
# Berney (2002) 13.6 45.6 1 -21.1 7.3
# Burchiel (1999) 13.6 48.0 1 -20.0 8.0
# Burchiel (1999) 13.6 48.0 2 -20.0 8.0
# Burchiel (1999) 13.6 48.0 3 -18.0 5.0
# Chen (2003) 12.1 65.7 2 -32.9 125.0
# ...
# Vingerhoets (2002) 16.0 48.8 1 -19.7 18.5
# Vingerhoets (2002) 16.0 48.8 2 -22.1 18.1
# Vingerhoets (2002) 16.0 48.8 3 -24.3 18.2
# Vingerhoets (2002) 16.0 48.8 4 -21.9 16.7
# Volkman (2001) 13.1 56.4 2 -37.8 20.9
# Volkman (2001) 13.1 56.4 3 -34.0 26.4
# Weselburger (2002) 14.0 50.3 1 -22.1 40.8
# [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
# [1,] 14.300 . . . . . . .
# [2,] . 7.300 4.128 . . . . .
# [3,] . 4.128 5.700 . . . . .
# [4,] . . . 7.300 . . . .
# [5,] . . . . 8.000 6.400 4.048 .
# [6,] . . . . 6.400 8.000 5.060 .
# [7,] . . . . 4.048 5.060 5.000 .
# [8,] . . . . . . . 125.000
# [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
# [1,] 1.00 . . . . . . .
# [2,] . 1.00 0.64 . . . . .
# [3,] . 0.64 1.00 . . . . .
# [4,] . . . 1.00 . . . .
# [5,] . . . . 1.00 0.80 0.64 .
# [6,] . . . . 0.80 1.00 0.80 .
# [7,] . . . . 0.64 0.80 1.00 .
# [8,] . . . . . . . 1.00
# Multivariate Meta-Analysis Model (k = 82; method: REML)
#
# Variance Components:
#
# outer factor: study (nlvls = 46)
# inner factor: time (nlvls = 4)
#
# estim sqrt k.lvl fixed level
# tau^2.1 21.64 4.65 24 no 1
# tau^2.2 33.80 5.81 22 no 2
# tau^2.3 26.31 5.13 25 no 3
# tau^2.4 30.69 5.54 11 no 4
# rho 0.92 no
#
# Test for Residual Heterogeneity:
# QE(df = 78) = 287.97, p-val < .01
#
# Test of Moderators (coefficients 1:4):
# QM(df = 4) = 889.57, p-val < .01
#
# Model Results:
#
# estimate se zval pval ci.lb ci.ub
# factor(time)1 -25.84 1.01 -25.70 <.01 -27.81 -23.87 ***
# factor(time)2 -27.32 1.15 -23.66 <.01 -29.58 -25.06 ***
# factor(time)3 -28.70 1.04 -27.53 <.01 -30.74 -26.66 ***
# factor(time)4 -26.27 1.44 -18.24 <.01 -29.09 -23.45 ***
#
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Constructing the V Matrix
• the tricky part is constructing the V matrix
• we have equations for computing the covariances for various effect size measures and circumstances (e.g., Gleser & Olkin, 2009; Lajeunesse, 2011; Olkin & Finn, 1990; Steiger, 1980; Wei & Higgins, 2013)
• two problems:
1. implementing these equations is tricky
2. information needed to compute them is often not available
## Recidivism and Mental Health
• meta-analysis on the difference in recidivism in delinquent juveniles with or without a mental health disorder (Assink et al., 2015; Assink & Wibbelink, 2016)
• dataset includes 17 studies and 100 effect size estimates
• effect size measure: standardized mean difference
# study esid id yi vi pubstatus year deltype
# 1 1 1 0.9066 0.0740 1 4.5 general
# 1 2 2 0.4295 0.0398 1 4.5 general
# 1 3 3 0.2679 0.0481 1 4.5 general
# 1 4 4 0.2078 0.0239 1 4.5 general
# 1 5 5 0.0526 0.0331 1 4.5 general
# 1 6 6 -0.0507 0.0886 1 4.5 general
# 2 1 7 0.5117 0.0115 1 1.5 general
# 2 2 8 0.4738 0.0076 1 1.5 general
# 2 3 9 0.3544 0.0065 1 1.5 general
# ...
# 16 1 79 0.7156 0.0914 1 2.5 overt
# 16 2 80 0.7067 0.0875 1 2.5 covert
# 16 3 81 0.6475 0.0330 1 2.5 general
# 16 4 82 0.6428 0.0861 1 2.5 covert
# 16 5 83 0.6271 0.0400 1 2.5 general
# 16 6 84 0.6238 0.0680 1 2.5 general
# 16 7 85 0.6025 0.1287 1 2.5 overt
# 16 8 86 0.5763 0.0332 1 2.5 general
# 16 9 87 0.5171 0.0517 1 2.5 covert
# 16 10 88 -0.3797 0.0390 1 2.5 covert
# 16 11 89 -0.4228 0.0664 1 2.5 covert
# 16 12 90 -0.4245 0.0809 1 2.5 covert
# 16 13 91 -0.4671 0.0667 1 2.5 covert
# 16 14 92 -0.5230 0.0988 1 2.5 overt
# 16 15 93 -0.5675 0.0340 1 2.5 covert
# 16 16 94 -0.7586 0.0437 1 2.5 covert
# 17 1 95 0.3453 0.0340 1 5.5 general
# 17 2 96 0.1221 0.0158 1 5.5 general
# 17 3 97 0.0906 0.0107 1 5.5 general
# 17 4 98 0.0040 0.0208 1 5.5 general
# 17 5 99 -0.0207 0.0123 1 5.5 general
# 17 6 100 -0.0660 0.0100 1 5.5 general
# [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
# [1,] 0.0740 0.0326 0.0358 0.0252 0.0297 0.0486 . . .
# [2,] 0.0326 0.0398 0.0263 0.0185 0.0218 0.0356 . . .
# [3,] 0.0358 0.0263 0.0481 0.0203 0.0239 0.0392 . . .
# [4,] 0.0252 0.0185 0.0203 0.0239 0.0169 0.0276 . . .
# [5,] 0.0297 0.0218 0.0239 0.0169 0.0331 0.0325 . . .
# [6,] 0.0486 0.0356 0.0392 0.0276 0.0325 0.0886 . . .
# [7,] . . . . . . 0.0115 0.0056 0.0052
# [8,] . . . . . . 0.0056 0.0076 0.0042
# [9,] . . . . . . 0.0052 0.0042 0.0065
# [,1] [,2] [,3] [,4] [,5] [,6] [,7]
# [1,] 1.00 0.50 0.50 0.50 0.50 0.50 .
# [2,] 0.50 1.00 0.50 0.70 0.50 0.50 .
# [3,] 0.50 0.50 1.00 0.50 0.70 0.70 .
# [4,] 0.50 0.70 0.50 1.00 0.50 0.50 .
# [5,] 0.50 0.50 0.70 0.50 1.00 0.70 .
# [6,] 0.50 0.50 0.70 0.50 0.70 1.00 .
# [7,] . . . . . . .
# Multivariate Meta-Analysis Model (k = 100; method: REML)
#
# Variance Components:
#
# estim sqrt nlvls fixed factor
# sigma^2.1 0.0747 0.2734 17 no study
# sigma^2.2 0.0000 0.0000 21 no study/deltype
# sigma^2.3 0.1387 0.3724 100 no study/deltype/esid
#
# Test for Residual Heterogeneity:
# QE(df = 97) = 783.0157, p-val < .0001
#
# Test of Moderators (coefficients 2:3):
# QM(df = 2) = 8.8461, p-val = 0.0120
#
# Model Results:
#
# estimate se zval pval ci.lb ci.ub
# intrcpt 0.4029 0.0960 4.1984 <.0001 0.2148 0.5909 ***
# deltypecovert -0.6948 0.2343 -2.9652 0.0030 -1.1541 -0.2355 **
# deltypeovert -0.1569 0.1679 -0.9343 0.3501 -0.4859 0.1722
#
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Cluster-Robust Inference
• but the results assume we got $$V$$ correct, when in fact we just cobbled together a rough approximation
• consider the model we have fitted a ‘working model’
• even if we got $$V$$ wrong, estimates of the fixed effects should still be approximately unbiased (but not fully efficient)
• however the standard errors might be off
• compute cluster-robust standard errors based on the working model for making inferences (Hedges, Tipton, & Johnson, 2010)
• can do this with the robumeta package (Tanner-Smith, Tipton, & Polanin, 2016), metafor::robust() or, even better, use the clubSandwich package (Pustejovsky & Tipton, 2018; Tipton, 2015; Tipton & Pustejovsky, 2015) directly with metafor objects (Pustejovsky & Tipton, 2021)
# Multivariate Meta-Analysis Model (k = 100; method: REML)
#
# Variance Components:
#
# estim sqrt nlvls fixed factor
# sigma^2.1 0.0747 0.2734 17 no study
# sigma^2.2 0.0000 0.0000 21 no study/deltype
# sigma^2.3 0.1387 0.3724 100 no study/deltype/esid
#
# Test for Residual Heterogeneity:
# QE(df = 97) = 783.0157, p-val < .0001
#
# Number of estimates: 100
# Number of clusters: 17
# Estimates per cluster: 1-22 (mean: 5.88, median: 5)
#
# Test of Moderators (coefficients 2:3):¹
# F(df1 = 2, df2 = 14) = 423.4635, p-val < .0001
#
# Model Results:
#
# estimate se¹ tval¹ df¹ pval¹ ci.lb¹ ci.ub¹
# intrcpt 0.4029 0.1022 3.9407 14 0.0015 0.1836 0.6221 **
# deltypecovert -0.6948 0.0368 -18.8757 14 <.0001 -0.7738 -0.6159 ***
# deltypeovert -0.1569 0.0674 -2.3282 14 0.0354 -0.3013 -0.0124 *
#
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#
# 1) results based on cluster-robust inference (var-cov estimator: CR1,
# approx. t/F-tests and confidence intervals, dfs = residual method)
# Coef. Estimate SE t-stat d.f. (Satt) p-val (Satt) Sig.
# intrcpt 0.403 0.0961 4.19 14.56 <0.001 ***
# deltypecovert -0.695 0.0899 -7.73 1.91 0.0185 *
# deltypeovert -0.157 0.0684 -2.29 1.94 0.1526
# General Workflow
• define goal(s) of meta-analysis and inclusion/exclusion criteria
• find relevant studies that have examined phenomenon of interest
• quantify results in terms of an effect size measure
• quantify precision of estimates in terms of their variances
• construct the (approximate) $$V$$ matrix by considering which estimates are based on the same or at least partially overlapping groups of subjects
• fit an appropriate multilevel/multivariate model that captures the dependencies in the underlying true effects
• to the extend that $$V$$ is just an approximation, consider the model a ‘working model’ and use cluster-robust variance estimation for making inferences about the fixed effects
# References
Assink, M., Put, C. E. van der, Hoeve, M., Vries, S. L. A. de, Stams, G. J. J. M., & Oort, F. J. (2015). Risk factors for persistent delinquent behavior among juveniles: A meta-analytic review. Clinical Psychology Review, 42, 47–61. https://doi.org/10.1016/j.cpr.2015.08.002
Assink, M., & Wibbelink, C. J. M. (2016). Fitting three-level meta-analytic models in R: A step-by-step tutorial. The Quantitative Methods for Psychology, 12(3), 154–174. https://doi.org/10.20982/tqmp.12.3.p154
Bangert-Drowns, R. L., Hurley, M. M., & Wilkinson, B. (2004). The effects of school-based writing-to-learn interventions on academic achievement: A meta-analysis. Review of Educational Research, 74(1), 29–58. https://doi.org/10.3102/00346543074001029
Berkey, C. S., Hoaglin, D. C., Antczak-Bouckoms, A., Mosteller, F., & Colditz, G. A. (1998). Meta-analysis of multiple outcomes by regression with random effects. Statistics in Medicine, 17(22), 2537–2550. https://doi.org/10.1002/(sici)1097-0258(19981130)17:22<2537::aid-sim953>3.0.co;2-c
Colditz, G. A., Brewer, T. F., Berkey, C. S., Wilson, M. E., Burdick, E., Fineberg, H. V., & Mosteller, F. (1994). Efficacy of BCG vaccine in the prevention of tuberculosis: Meta-analysis of the published literature. Journal of the American Medical Association, 271(9), 698–702. https://doi.org/10.1001/jama.1994.03510330076038
Credé, M., Roch, S. G., & Kieszczynka, U. M. (2010). Class attendance in college: A meta-analytic review of the relationship of class attendance with grades and student characteristics. Review of Educational Research, 80(2), 272–295. https://doi.org/10.3102/0034654310362998
Gleser, L. J., & Olkin, I. (2009). Stochastically dependent effect sizes. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 357–376). New York: Russell Sage Foundation.
Hasselblad, V. (1998). Meta-analysis of multitreatment studies. Medical Decision Making, 18(1), 37–43. https://doi.org/10.1177/0272989X9801800110
Hedges, L. V., Tipton, E., & Johnson, M. C. (2010). Robust variance estimation in meta-regression with dependent effect size estimates. Research Synthesis Methods, 1(1), 39–65. https://doi.org/10.1002/jrsm.5
Ishak, K. J., Platt, R. W., Joseph, L., Hanley, J. A., & Caro, J. J. (2007). Meta-analysis of longitudinal studies. Clinical Trials, 4(5), 525–539. https://doi.org/10.1177/1740774507083567
Kearon, C., Julian, J. A., Math, M., Newman, T. E., & Ginsberg, J. S. (1998). Noninvasive diagnosis of deep venous thrombosis. Annals of Internal Medicine, 128(8), 663–677. https://doi.org/10.7326/0003-4819-128-8-199804150-00011
Lajeunesse, M. J. (2011). On the meta-analysis of response ratios for studies with correlated and multi-group designs. Ecology, 92(11), 2049–2055. https://doi.org/10.1890/11-0423.1
Olkin, I., & Finn, J. D. (1990). Testing correlated correlations. Psychological Bulletin, 108(2), 330–333. https://doi.org/10.1037/0033-2909.108.2.330
Pustejovsky, J. E., & Tipton, E. (2018). Small-sample methods for cluster-robust variance estimation and hypothesis testing in fixed effects models. Journal of Business & Economic Statistics, 36(4), 672–683. https://doi.org/10.1080/07350015.2016.1247004
Pustejovsky, J. E., & Tipton, E. (2021). Meta-analysis with robust variance estimation: Expanding the range of working models. Prevention Science. https://doi.org/10.1007/s11121-021-01246-3
Steiger, J. H. (1980). Tests for comparing elements of a correlation matrix. Psychological Bulletin, 87(2), 245–251. https://doi.org/10.1037/0033-2909.87.2.245
Tanner-Smith, E. E., Tipton, E., & Polanin, J. R. (2016). Handling complex meta-analytic data structures using robust variance estimates: A tutorial in R. Journal of Developmental and Life-Course Criminology, 2(1), 85–112. https://doi.org/10.1007/s40865-016-0026-5
Tipton, E. (2015). Small sample adjustments for robust variance estimation with meta-regression. Psychological Methods, 20(3), 375–393. https://doi.org/10.1037/met0000011
Tipton, E., & Pustejovsky, J. E. (2015). Small-sample adjustments for tests of moderators and model fit using robust variance estimation in meta-regression. Journal of Educational and Behavioral Statistics, 40(6), 604–634. https://doi.org/10.3102/1076998615606099
Wei, Y., & Higgins, J. P. (2013). Estimating within-study covariances in multivariate meta-analysis with multiple outcomes. Statistics in Medicine, 32(7), 1191–1205. https://doi.org/10.1002/sim.5679
# Appendix
• as I probably won’t have enough time for this (and probably not even enough time to get through everything above), I will add some additional examples here
## Network Meta-Analysis (NMA)
• is essentially just a special case of the multivariate model
• a common occurrence in NMA: studies with more than two groups, allowing the computation of multiple contrasts (e.g., treatment A vs control and treatment B vs control)
• reuse of information from the shared group induces correlation among the effect sizes
• note: here there is only partial overlap of subjects that contribute information to both effect sizes, but the rule still applies
### Effectiveness of Counseling for Smoking Cessation
• network meta-analysis on the effectiveness of various counseling types for smoking cessation (Hasselblad, 1998)
• 24 studies that examined 4 different treatments (self-help, individual counseling, group counseling, and no contact)
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### Home > CC1 > Chapter 3 > Lesson 3.1.5 > Problem3-74
3-74.
Maya and Logan each made up a “Guess my Decimal” game just for you. Use their clues to determine the number.
1. Maya gives you this clue: “The decimal I am thinking of is $\mathit{3}$ tenths greater than $\mathit{80\%}$. What is my decimal?” Show your work.
If we know that $80\%$ is $\frac{80}{100}$, which is equal to $\frac{8}{10}$, what number is $\frac{3}{10}$ more than that?
Maya’s decimal is $1.1$. The fractions of this decimal is $\frac{11}{10}$ or $\frac{110}{100}$.
2. Logan continues the game with this clue: “My decimal is $\mathit{3}$ hundredths less than $\mathit{3}$ tenths.” Use pictures and/or words to show your thinking.
Can you convert to $\frac{3}{10}$ hundredths? This will make it easier to subtract $\frac{3}{100}$ in order to find Logan’s fraction.
If the fraction of Logan’s decimal is $\frac{27}{100}$, can you find the decimal?
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Question
# Solve the following system of linear equations in two variables by Substitution method.5x-2y=-7x=-2y+1
Equations
Solve the following system of linear equations in two variables by Substitution method.
$$5x-2y=-7$$
$$x=-2y+1$$
2021-02-12
Consider the following system of linear equations:
$$5x-2y=-7$$...(1)
$$x=-2y+1$$...(2)
Substitute equation (1) in equation (2):
$$5(-2y+1)-2y=-7$$
$$-10y+5-2y=-7$$
$$-12y=-7-5$$
$$-12y=-12$$
$$y=1$$
$$\displaystyle{x}=-{2}\times{1}+{1}$$
$$=-2+1$$
$$=-1$$
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Backward stochastic variational inequalities on random interval
# Backward stochastic variational inequalities on random interval
[ [ [ [ Faculty of Mathematics, “Alexandru Ioan Cuza” University, Carol I Blvd., no. 11, Iasi, 700506, Romania. \printeade1,e2 Department of Mathematics, “Gheorghe Asachi” Technical University, Carol I Blvd., no. 11, Iasi, 700506, Romania “Octav Mayer” Mathematics Institute of the Romanian Academy, Iasi branch, Carol I Blvd., no. 8, Iasi, 700506, Romania
\smonth6 \syear2012\smonth1 \syear2014
\smonth6 \syear2012\smonth1 \syear2014
\smonth6 \syear2012\smonth1 \syear2014
###### Abstract
The aim of this paper is to study, in the infinite dimensional framework, the existence and uniqueness for the solution of the following multivalued generalized backward stochastic differential equation, considered on a random, possibly infinite, time interval:
{−dYt+∂yΨ(t,Yt)dQt∋Φ(t,Yt,Zt)dQt−ZtdWt,0≤t<τ,Yτ=η,
where is a stopping time, is a progressively measurable increasing continuous stochastic process and is the subdifferential of the convex lower semicontinuous function .
As applications, we obtain from our main results applied for suitable convex functions, the existence for some backward stochastic partial differential equations with Dirichlet or Neumann boundary conditions.
\kwd
\aid
0 \volume21 \issue2 2015 \firstpage1166 \lastpage1199 \doi10.3150/14-BEJ601 \newremarkremarkRemark[section] \newproclaimdefinition[theorem]Definition \newremarkexample[theorem]Example
\runtitle
Backward SVIs on random interval
{aug}
1,2]\initsL.\fnmsLucian \snmMaticiuc\corref\thanksref1,2,e1label=e1,mark]lucian.maticiuc@uaic.ro and 1,3]\initsA.\fnmsAurel \snmRăşcanu\thanksref1,3,e2label=e2,mark]aurel.rascanu@uaic.ro
backward stochastic differential equations \kwdsubdifferential operators \kwdstochastic variational inequalities \kwdstochastic partial differential equations
## 1 Introduction
In this paper, we are interested to prove the existence and uniqueness of a triple which is the solution for the following generalized backward stochastic variational inequality (BSVI for short) considered in the Hilbert space framework:
⎧⎨⎩Yt+∫τt∧τdKs=η+∫τt∧τ[F(s,Ys,Zs)ds+G(s,Ys)dAs]−∫τt∧τZsdWs,a.s.,dKt∈∂φ(Yt)dt+∂ψ(Yt)dAt,∀t≥0, (1)
where is a cylindrical Wiener process, , are the subdifferentials of a convex lower semicontinuous functions , , is a progressively measurable increasing continuous stochastic process, and is a stopping time.
In fact, we will define and prove the existence of the solution for an equivalent form of (1):
⎧⎨⎩Yt+∫∞tdKs=η+∫∞tΦ(s,Ys,Zs)dQs−∫∞tZsdWs,a.s., t≥0,dKt∈∂yΨ(t,Yt)dQt,on [0,∞), (2)
with , and adequately defined. The notation means that is a continuous stochastic process and for any continuous stochastic process and any , the bounded variation of on is finite and the following inequality holds:
∫s2s1⟨Xr−Yr,dKr⟩+∫s2s1Ψ(r,Yr)dQr≤∫s2s1Ψ(r,Yr)dQr,a.s.
The study of the backward stochastic differential equations (BSDEs for short) in the finite dimensional case (equation of type (1) with and equal to ) was initiated by Pardoux and Peng [16] (see also Pardoux and Peng [15]). The authors have proved the existence and the uniqueness of the solution for the BSDE on fixed time interval, under the assumption of Lipschitz continuity of with respect to and and square integrability of and . The case of BSDEs on random time interval (possibly infinite), under weaker assumptions on the data, have been treated by Darling and Pardoux [5], where it is obtained, as application, the existence of a continuous viscosity solution to the elliptic partial differential equations (PDEs) with Dirichlet boundary conditions.
The more general case of scalar BSDEs with one-sided reflection and associated optimal control problems was considered by El Karoui, Kapoudjian, Pardoux, Peng and Quenez [8] and with two-sided reflection associated with stochastic game problem by Cvitanić and Karatzas [4].
When the obstacles are fixed, the reflected BSDE become a particular case of BSVI of type (1), by taking as convex indicator of the interval defined by obstacles. We must mention that the solution of a BSVI belongs to the domain of the operator and it is reflected at the boundary of this.
The standard work on BSVI in the finite dimensional case is that of Pardoux and Răşcanu [17], where it is proved the existence and uniqueness of the solution for BSVI (1) with , under the following assumptions on : monotonicity with respect to (in the sense that ), Lipschitzianity with respect to and a sublinear growth for . Moreover, it is shown that, unlike the forward case, the process is absolute continuous with respect to . In Pardoux and Răşcanu [18], the same authors extend these results to the Hilbert spaces framework. Afterwards, various particular cases of BSVI (1) were the subject of many articles: Maticiuc and Răşcanu [11], Maticiuc, Răşcanu and Zălinescu [12], Maticiuc and Rotenstein [13], Maticiuc and Nie [9] (where the backward equations are studied in the frame of fractional stochastic calculus) and Diomande and Maticiuc [7] (where the generator at the moment is allowed to depend on the past values on of the solution ).
Our paper generalizes the existence and uniqueness results from Pardoux and Răşcanu [18] by considering random time interval and the Lebesgue–Stieltjes integral terms, and by assuming a weaker boundedness condition for the generator (instead of the sublinear growth), that is,
E(∫T0Φ#ρ(s)ds)p<∞,where Φ#ρ(t):=sup|y|≤ρ∣∣Φ(t,y,0)∣∣. (3)
We mention that, since is a stopping time, the presence of the process is justified by the possible applications of equation (1) in proving probabilistic interpretation for the solution of elliptic multivalued partial differential equations with Neumann boundary conditions on a domain from . The stochastic approach of the existence problem for finite dimensional multivalued parabolic PDEs, was considered by Maticiuc and Răşcanu [11].
Concerning assumption (3), we recall that, in the case of finite dimensional BSDE, Pardoux [14] has used a similar condition, in order to prove the existence of a solution in . His result was generalized by Briand, Delyon, Hu, Pardoux and Stoica [3], where it is proved the existence in of the solution for BSDEs considered both with fixed and random terminal time. We mention that the assumptions from our paper are, broadly speaking, similar to those of Briand, Delyon, Hu, Pardoux and Stoica [3].
The article is organized as follows: in the next section a brief summary of infinite dimensional stochastic integral and the assumptions are given. Section 3 is devoted to the proof of the existence and uniqueness of a strong solution for (2). In the Section 4, is a new type of solution (called variational weak solution) and it is also proves the existence and uniqueness result. In Section 4 are obtained, as applications, the existence of the solution for various type of backward stochastic partial differential equations with boundary conditions. The Appendix contains, following Pardoux and Răşcanu [19], some results useful throughout the paper.
## 2 Preliminaries
### 2.1 Infinite dimensional framework
In the beginning of this subsection, we give a brief exposition of the stochastic integral with respect to a Wiener process defined on a Hilbert space. For a deeper discussion concerning the notion of cylindrical Wiener process and the construction of the stochastic integral, we refer reader to Da Prato and Zabczyk [6].
We consider a complete probability space , the set , a right continuous and complete filtration , and two real separable Hilbert spaces .
Let us denote by , , the complete metric space of continuous progressively measurable stochastic process (p.m.s.p.) with the metric given by
and by the space of p.m.s.p. such that, for all , the restriction . To shorten notation, we continue to write for . Remark that is a Banach space for .
By , , we denote the Banach space of the continuous stochastic processes such that , , a.s., and , a.s. for all . The norm is defined by . If , then is a closed linear subspace of .
Let be a Gaussian family of real-valued random variables with zero mean and the covariance function given by , , . We call a -Wiener process if, for all ,
1. [(ii)]
2. ,
3. is independent of , for all , .
Let be an orthonormal and complete basis in . We introduce the separable Hilbert space of Hilbert–Schmidt operators from to , that is, the space of linear operators such that . It will cause no confusion if we use to designate the norm in .
The sequence , defines a family of real-valued Wiener processes mutually independent on .
If is finite dimensional space then we have the representation , but, in general case, this series does not converge in , but rather in a larger space such that with the injection being a Hilbert–Schmidt operator. Moreover, .
For , we will denote by , , the space , that is, the complete metric space of progressively measurable stochastic processes with metric of convergence
The space is a Banach space for with norm . From now on, for simplicity of notation, we write instead of (when no confusion can arise).
Let us denote by the space of measurable stochastic processes such that, for all , the restriction .
For any let the stochastic integral , , where is an orthonormal basis in . Note that the introduced stochastic integral does not depend on the choice of the orthonormal basis on . By the standard localization procedure, we can extend this integral as a linear continuous operator , and it has the following properties:
###### Proposition 2.1
Let . Then
1. [(iii)]
2. , , if ,
3. , if ,
4. , if (Burkholder–Davis–Gundy inequality),
5. , if .
From now on, we shall consider that the original filtration is replaced by the filtration generated by the Wiener process. The following Hilbert space version of the martingale representation theorem, extended to a random interval, holds the following proposition.
###### Proposition 2.2
Let be a stopping time, and be a -measurable random variable such that . Then
1. [3.]
2. there exists a unique stochastic process such that and , , or equivalently,
3. there exists a unique pair such that
ξt=η−∫τt∧τζsdWs,a% .s., t≥0, (4)
or equivalently,
4. there exists a unique pair such that , a.s., and and , .
### 2.2 Assumptions and definitions
In order to study equation (1), or the equivalent form (2), we introduce the next assumptions:
1. [(A)]
2. The parameter ;
3. The random variable is a stopping time;
4. The random variable is -measurable such that and the stochastic process is the unique pair associated to such that we have the martingale representation formula (4);
5. The process is a progressively measurable increasing continuous stochastic process such that ;
6. The functions and are such that
{F(⋅,⋅,y,z),G(⋅,⋅,y){ are p.m.s.p.{,} for all}(y,z)∈H×L2(H1,H),F(ω,t,⋅,⋅),G(ω,t,⋅) {are continuous functions a.e.},
and , , -a.s., where and .
Moreover, there exist two p.m.s.p. such that and for all -a.s., and there exists such that, for all ,
⟨y′−y,F(t,y′,z)−F(t,y,z)⟩ ≤ \mathbh1[0,τ](t)μt∣∣y′−y∣∣2, ⟨y′−y,G(t,y′)−G(t,y)⟩ ≤ \mathbh1[0,τ](t)νt∣∣y′−y∣∣2, (5) ∣∣F(t,y,z′)−F(t,y,z)∣∣ ≤ \mathbh1[0,τ](t)ℓ∣∣z′−z∣∣.
Let us introduce the function
Qt(ω):=t+At(ω)
and let be the a real positive p.m.s.p. (given by Radon–Nikodym’s representation theorem) such that and and .
Let
Φ(ω,t,y,z):=\mathbh1[0,τ(ω)](t)[αt(ω)F(ω,t,y,z)+(1−αt(ω))G(ω,t,y)],
in which case (2.2) yields
⟨y′−y,Φ(t,y′,z)−Φ(t,y,z)⟩ ≤ \mathbh1[0,τ](t)[μtαt+νt(1−αt)]∣∣y′−y∣∣2, ∣∣Φ(t,y,z′)−Φ(t,y,z)∣∣ ≤ \mathbh1[0,τ](t)ℓαt∣∣z′−z∣∣.
For , let
Vt = = ∫t0\mathbh1[0,τ](s)[(μs+a2ℓ2)ds+νsdAs].
We can give now some a priori estimates concerning the solution of (1).
###### Lemma 2.1
Let . Under assumption (A) the following inequalities hold, in the sense of signed measures on ,
⟨Ys,Φ(s,Ys,Zs)dQs⟩≤|Ys|∣∣Φ(s,0,0)∣∣dQs+|Ys|2dVs+12a|Zs|2ds (7)
and
⟨Ys−~Ys,Φ(s,Ys,Zs)−Φ(s,~Ys,~Zs)⟩dQs≤|Ys−~Ys|2dVs+12a|Zs−~Zs|2ds. (8)
{pf}
The inequalities can be obtained by standard calculus (applying the monotonicity and Lipschitz property of function ).
1. [(A)]
2. are proper convex lower semicontinuous l.s.c. functions such that (consequently ).
Let us define
Ψ(ω,t,y):=\mathbh1[0,τ(ω)](t)[αt(ω)φ(y)+(1−αt(ω))ψ(y)].
We recall now that the multivalued subdifferential operator is the maximal monotone operator
∂φ(y):={^y∈H\dvtx⟨^y,v−y⟩+φ(y)≤φ(v),∀v∈H}.
We define and and by we understand that and . We know that and .
{definition}
If is a locally bounded variation function, is a real increasing function, is a continuous function and is like in (A), then notation means that for any continuous function , it holds
∫st⟨xr−yr,dkr⟩+∫stφ(yr)dar≤∫stφ(xr)dar,0≤t≤s. (9)
Now we are able to introduce the rigorous definition of a solution for equation (1). First, using definitions of , and , respectively, we can rewrite (1) in the form
⎧⎨⎩Yt+∫∞tdKs=η+∫∞tΦ(s,Ys,Zs)dQs−∫∞tZsdWs,a.s., t≥0,dKt∈∂yΨ(t,Yt)dQt=∂φ(Yt)dt+∂ψ(Yt)dAt,on [0,∞). (10)
{definition}
We call a solution of (10) if has locally bounded variation and with for such that
1. [(iii)]
2. , -a.s., for all ,
3. -a.e.,
4. , as (where is given by (2.2)) and
5. , a.s., .
Let and the Moreau–Yosida regularization of given by , which is a convex function. We mention some properties (see Brézis [2], and Pardoux and Răşcanu [17] for the last one): for all
We introduce the compatibility conditions between (which have previously been used in Maticiuc and Răşcanu [11]):
1. [(A)]
2. For all , , ,
where and .
{example}
Let .
1. [B.]
2. Clearly, since and are increasing monotone, we see that, if and , , then compatibility assumptions (2.2) are satisfied.
3. If are the convexity indicator functions, that is,
where are such that (see assumption (A)), then and similar for .
Since (A)(i) is fulfilled, the compatibility assumptions become , for and , for , and, respectively, for and , for .
The last assumption is the following:
1. [(A)]
2. There exist the p.m.s.p. with and such that -a.s. and, using notation
~Vt:=∫t0\mathbh1[0,τ](s)[(~μs+a2ℓ2)ds+~νsdAs], (11)
we suppose
(i)E[e2sups∈[0,τ]~Vs(φ(η)+ψ(η))]<∞,\omit\span\@@LTX@noalign\vspace∗6pt\omit(ii)E(epsups∈[0,τ]~Vs|η|p)+E(QpT)<∞,∀T>0,\omit\span\@@LTX@noalign\vspace∗6pt\omit(iii)E(∫τ0e2~VsΨ(s,ξs)dQs)p/2+E(∫τ0e~Vs∣∣Φ(s,ξs,ζs)∣∣dQs)p<∞,
and the locally boundedness conditions:
{remark}
We point out that the purpose of defining of the new process is due to the computations; see, e.g., inequalities (3) and (3) from the proof of the first main theorem, where it is necessary to have a new process such that and on .
{remark}
It can be choose in (A), in particular, and such that and . In this case defined by (11) will become non-decreasing, hence and (2.2) and (2.2) will be simplified.
We prefer to keep inequalities and in this form because we allow to and to be negative and therefore to enlarge the class of the generators and who satisfy (2.2) and (2.2) (and also we not restrict the class of the final data ).
## 3 Main result: The existence of the strong solution
We present first the definition of a solution in the strong case when the process is absolutely continuous with respect to (i.e., on ).
{definition}
We call a strong solution of (10) if there exist two p.m.s.p. , and , such that is a solution of (10) with and
\emph{(i)}∫T0|Us|dQs<∞,P-a.s., for all T≥0,\omit\span\@@LTX@noalign\vspace∗6pt\omit\emph{(ii)}U1t∈∂φ(Yt),dP⊗dt-a.e., U2t∈∂ψ(Yt), dP⊗dAt-a.e.,\omit\span\@@LTX@noalign\vspace∗6pt\omit\emph{(iii)}E(e2VT|YT−ξT|2)+E∫∞Te2Vs|Zs−ζs|2ds→0,as T→∞,
where is given by (2.2).
{remark}
If there exists such that , -a.s., then the condition (3)(iii) is equivalent to , as .
We can now formulate the first main result. In order to obtain the absolute continuity with respect to of the process (as in Definition 3) it is necessary to impose a supplementary assumption:
1. [(A)]
2. There exists such that, for all ,
EFt(e2sups≥t~Vs|η|2)+EFt(∫τt∧τe~Vs∣∣Φ(s,0,0)∣∣dQs)2≤R0,a.s. (12)
{remark}
We mention that without this assumption we are not able to prove, among other, that there exist two processes and such that (see step from the proof of the next theorem).
###### Theorem 3.1
Let assumptions (A)–(A) be satisfied. Then the backward stochastic variational inequality (10) has a unique solution such that for all ,
Esups∈[0,T]ep~Vs|Ys|p<∞ (13)
and
Yt+∫TtUsdQs=YT+∫TtΦ(s,Ys,Zs)dQs−∫TtZsdWs,a.s.,% ∀t∈[0,T]. (14)
Moreover, for all , there exists a constant such that, for all , -a.s.
and
\emph{(e)}E∫τ0[e2~Vs(∣∣U1s∣∣2ds+∣∣U2s∣∣2dAs)]<∞. (15)
{pf*}
Proof If , are two solutions, in the sense of Definition 3, that satisfy (13), then . From (8), satisfied by the process , we conclude that
⟨Ys−¯Y
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# How do you solve -3/4x+5/14=-2 by clearing the fractions?
Feb 5, 2017
See the entire solution process below:
#### Explanation:
First, multiply each side of the equation by $\textcolor{red}{28}$ to eliminate the fractions while keeping the equation balanced. $\textcolor{red}{28}$ is the lowest common denominator of the two fractions.
$\textcolor{red}{28} \left(- \frac{3}{4} x + \frac{5}{14}\right) = \textcolor{red}{28} \times - 2$
$\left(\textcolor{red}{28} \times - \frac{3}{4} x\right) + \left(\textcolor{red}{28} \times \frac{5}{14}\right) = - 56$
$\left(\cancel{\textcolor{red}{28}} 7 \times - \frac{3}{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}}} x\right) + \left(\cancel{\textcolor{red}{28}} 2 \times \frac{5}{\textcolor{red}{\cancel{\textcolor{b l a c k}{14}}}}\right) = - 56$
$- 21 x + 10 = - 56$
Next, subtract $\textcolor{red}{10}$ from each side of the equation to isolate the $x$ term while keeping the equation balanced:
$- 21 x + 10 - \textcolor{red}{10} = - 56 - \textcolor{red}{10}$
$- 21 x + 0 = - 66$
$- 21 x = - 66$
Now, divide each side of the equation by $\textcolor{red}{- 21}$ to solve for $x$ while keeping the equation balanced:
$\frac{- 21 x}{\textcolor{red}{- 21}} = \frac{- 66}{\textcolor{red}{- 21}}$
$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 21}}} x}{\cancel{\textcolor{red}{- 21}}} = \frac{3 \times 22}{\textcolor{red}{3 \times 7}}$
$x = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \times 22}{\textcolor{red}{\cancel{3} \times 7}}$
$x = \frac{22}{7}$
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# How quickly could a craft find its location using only pulsars?
I'm only just beginning to scratch the surface of the possibilities in interstellar travel. I know that pulsars can be independently identified by means of measuring their pulses, and that once enough have been identified they can be used to identify one's position in space.
My question is: How long (on average) would a pulsar need to be observed in order to identify it from its pulses alone? More to the point, how long would a craft need to watch the pulsars it finds to identify them and get a "fix" on its location based on that? (That is, is the distribution uniform, or is it skewed one way or the other and, if so, how does that affect the average time required to identify enough pulsars?)
For the purposes of this question, we're assuming highly accurate observational and measuring tools and equipment, rather than a bearded old man with a telescope in his backyard making all his calculations by hand.
I suppose another way to ask the same thing would be: What's the average cycle duration of pulsars, and how much skew is there to the distribution?
• This is incredibly dependent on how much and how good your tech is. If you have a million cameras on the outside of your ship that can record light levels with 64-bit precision you'll find yourself in seconds. If you have a geezer with a telescope and a spinning disk with a hole to measure pulse rates it's going to take years. – Loren Pechtel Apr 25 '14 at 16:40
• @LorenPechtel That's a very good point. Let's assume for the purposes of this question the absolute best observational tools and techniques science and technology have provided us. (The genesis of the question is my research for a hard-sci-fi novel involving interstellar travel, and I want to as much as possible avoid meaningless technobabble while still providing realistic explanations for how it all works.) Of course, an answer encompassing both ends of the spectrum would certainly get my up-vote, and The Green Checkmark! – Kromey Apr 25 '14 at 16:45
• I still don't think you can get a good answer because you can always add more cameras until you run out of hull. It's a problem that's subject to parallelism--how many sensors do you want to carry around. – Loren Pechtel Apr 25 '14 at 21:25
• And still another unknown: How unknown is your location? If your stardrive can't hop you more than 10ly then once you find a pulsar you have substantially narrowed your search for additional pulsars--and each additional pulsar narrows the search more. On the other hand, if your drive is good for 10,000ly the first pulsar tells you nothing about where others might be. It will also be much harder to identify pulsars if you can go that far--pulsars are NOT consistent through time and thus you'll get more cases of a signal that you can't tell which pulsar it is until you have more data. – Loren Pechtel Apr 25 '14 at 21:30
• Since this question received one close vote as off-topic, I would like to argue that astronavigation is on scope here as it is directly relevant to spacecraft operation. We've previously discussed star trackers, Polaris equivalent stars on Mars, and so on here on Space Exploration. – TildalWave Apr 29 '14 at 16:43
Here's a correct answer: A craft couldn't.
We haven't mapped the whole of space yet so...
a craft with no other means of identifying its position
... can't know if it's near known or unknown pulsars. If you really have no idea of where you are or what your attitude is then looking around you isn't actually that useful. For example, you wake up in a forest, and from the flora and fauna you surmise that you must be in the Amazon rain forest. But you could easily be on another planet, in another galaxy, far far away.
Now if you have knowledge of your approximate position, say somewhere in the milky way, then you can start using pulsars and other things you see in space to calculate your position. You would need three known pulsars to triangulate your position (with 2 pulsars you can only know you're in 1 of 2 possible locations). If you know enough about the pulsars you could do some interesting red shift calculations to find out how far away from them you are. When it comes to the actual identification process, the duration of observation required is dependant on how many pulsars it might be, based firstly on your expected position and then on initial observations.
For example if you see a pulsar and you think you're near 1 of 10 pulsars then you can assume the pulsar you're seeing is one of the 10. Then you observe the pulsing of the pulsar and see at what rate it pulses. The longer you observer the pulsar the more accurate you calculation of it's spin can be, and hence the easier it is to rule out the other 9 pulsars.
EDIT:
The reason for a simple answer not being given to this question (by me at least), is that it is highly dependant on which pulsars you are looking at. For example take the following collection of pulsars and pulse frequencies:
A 0.98
B 0.12
C 0.51
D 0.78
E 0.15
F 12.01
It's clear that if you are trying to identify pulsar F it is going to be an easy task. If you see more than one pulse per second you can be confident that the other pulsars are ruled out. Now consider the following collection:
A 13.52
B 12.08
C 10.54
D 15.23
E 11.98
F 12.01
To identify pulsar F in this collection you Need to be able to observe the pulsar for around 13 seconds to rule out pulsar B as a potential candidate. This assumes that you can only count the number of pulses in a given number of seconds. This is unlikely, but there will be a limit to how accurately you can time pulses, dependent on the design of your hardware.
To surmise, the duration required to identify pulsar F in collection 2 is 13 times that of collection 1; this is why it's dependent on the number and characteristics of pulsars in range.
EDIT 2:
The following is based on values for 1,861 pulsars.
median: 1.88 Hz
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## The Black-Scholes Equation and Certain Quantum Hamiltonians
Juan M. Romero, O. Gonzalez-Gaxiola, J. Ruiz de Chavez, R. Bernal-Jaquez
Abstract: In this paper a quantum mechanics is built by means of a non-Hermitian momentum operator. We have shown that it is possible to construct two Hermitian and two non-Hermitian type of Hamiltonians using this momentum operator. We can construct a generalized supersymmetric quantum mechanics that has a dual based on these Hamiltonians. In addition, it is shown that the non-Hermitian Hamiltonians of this theory can be related to Hamiltonians that naturally arise in the so-called quantum finance.
http://arxiv.org/abs/1002.1667
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)
## An exactly solvable quantum-lattice model with a tunable degree of nonlocality
Miloslav Znojil
An array of N subsequent Laguerre polynomials is interpreted as an eigenvector of a non-Hermitian tridiagonal Hamiltonian $H$ with real spectrum or, better said, of an exactly solvable N-site-lattice cryptohermitian Hamiltonian whose spectrum is known as equal to the set of zeros of the N-th Laguerre polynomial. The two key problems (viz., the one of the ambiguity and the one of the closed-form construction of all of the eligible inner products which make $H$ Hermitian in the respective {\em ad hoc} Hilbert spaces) are discussed. Then, for illustration, the first four simplest, $k-$parametric definitions of inner products with $k=0,k=1,k=2$ and $k=3$ are explicitly displayed. In mathematical terms these alternative inner products may be perceived as alternative Hermitian conjugations of the initial N-plet of Laguerre polynomials. In physical terms the parameter $k$ may be interpreted as a measure of the “smearing of the lattice coordinates” in the model.
http://arxiv.org/abs/1101.1183
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
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# Correlation functions of the energy-momentum tensor in SU(2) gauge theory at finite temperature
Hübner KA, Karsch F, Pica C (2008)
Physical Review D 78(9).
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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
We calculate correlation functions of the energy-momentum tensor in the vicinity of the deconfinement phase transition of (3 + 1)-dimensional SU(2) gauge theory and discuss their critical behavior in the vicinity of the second order deconfinement transition. We show that correlation functions of the trace of the energy-momentum tensor diverge uniformly at the critical point in proportion to the specific heat singularity. Correlation functions of the pressure, on the other hand, stay finite at the critical point. We discuss the consequences of these findings for the analysis of transport coefficients, in particular, the bulk viscosity, in the vicinity of a second order phase transition point.
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Physical Review D
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### Zitieren
Hübner KA, Karsch F, Pica C. Correlation functions of the energy-momentum tensor in SU(2) gauge theory at finite temperature. Physical Review D. 2008;78(9).
Hübner, K. A., Karsch, F., & Pica, C. (2008). Correlation functions of the energy-momentum tensor in SU(2) gauge theory at finite temperature. Physical Review D, 78(9). doi:10.1103/PhysRevD.78.094501
Hübner, K. A., Karsch, F., and Pica, C. (2008). Correlation functions of the energy-momentum tensor in SU(2) gauge theory at finite temperature. Physical Review D 78.
Hübner, K.A., Karsch, F., & Pica, C., 2008. Correlation functions of the energy-momentum tensor in SU(2) gauge theory at finite temperature. Physical Review D, 78(9).
K.A. Hübner, F. Karsch, and C. Pica, “Correlation functions of the energy-momentum tensor in SU(2) gauge theory at finite temperature”, Physical Review D, vol. 78, 2008.
Hübner, K.A., Karsch, F., Pica, C.: Correlation functions of the energy-momentum tensor in SU(2) gauge theory at finite temperature. Physical Review D. 78, (2008).
Hübner, Kay A., Karsch, Frithjof, and Pica, C. “Correlation functions of the energy-momentum tensor in SU(2) gauge theory at finite temperature”. Physical Review D 78.9 (2008).
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### Web of Science
Dieser Datensatz im Web of Science®
arXiv: 0808.1127
Inspire: 792742
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The statistical nature of the 2nd Law of Thermodynamics
Ok, so entropy increases... This is supposed to be an absolute statement about entropy. But then someone imagines a box with a 10 particle gas, and finds that every now and then all particles are in the left. Conclusion, the 2nd law holds only in a statistical sense. But then Szilard comes up with a thought experiment with only one particle and a piston that can compress to the left or right. The apparent loss of entropy when finding the particle in the left half is compensated by the very bit of information that indicates where the particle is.
So perhaps the 2nd Law actually holds in an absolute sense, except that...
Is there consensus on the absolute vs statistical nature of the 2nd Law, or is it subject to interpretation? Can the issue be settled within a classical setting, or does one have to go quantum?
Addendum: (as per Ben Crowell's request) Here is the paper
Szilard, L., 1929, “On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings”, Zeitschrift fur Physik 53: 840–856. English translation in The Collected Works of Leo Szilard: Scientific Papers, B.T. Feld and G. Weiss Szilard (eds.), Cambridge, Massachusetts: MIT Press, 1972, pp. 103–129.
• Could you provide some more information on what the Szilard thought experiment is? It's not very clear from what you wrote. – user4552 Sep 10 '13 at 4:44
• Are you referring to the fluctuation theorem? This does not only apply to quantum systems. – twistor59 Sep 10 '13 at 6:22
• The OP is referring to Maxwell's demon. – Bubble Sep 10 '13 at 12:40
Giving a full answer to this one takes quite a bit of information, so I'll first give a few references and then summarise how they all fit in.
Review Papers
There are two review papers describing the concepts I am about to talk about:
1. Sevick, E. M.; Prabhakar, R.; Williams, Stephen R.; Bernhardt, Debra Joy, "Fluctuation Theorems", Annual Rev. of Phys. Chem., 59, pp. 603-633 (this one is paywalled).
2. Charles Bennett, "The Thermodynamics of Computing: A Review", Int. J. Theoretical Physics, 21, 12, 1982
And a remarkable experiment that actually BUILDS AND TESTS the Maxwell Daemon.
1. Shoichi Toyabe; Takahiro Sagawa; Masahito Ueda; Eiro Muneyuki; Masaki Sano (2010-09-29). "Information heat engine: converting information to energy by feedback control". Nature Physics 6 (12): 988–992. arXiv:1009.5287. Bibcode:2011NatPh...6..988T. doi:10.1038/nphys1821.
"We demonstrated that free energy is obtained by a feedback control using the information about the system; information is converted to free energy, as the first realization of Szilard-type Maxwell’s demon."
Now to your question. You are quite right in your conclusion about the second law's statistical nature:
... But then someone imagines a box with a 10 particle gas, and finds that every now and then all particles are in the left. Conclusion, the 2nd law holds only in a statistical sense ...
and indeed various fluctuation theorems (see the "Fluctuation Theorem" Wikipedia page as well as the "Fluctuation Theorems" review paper I cited above) quantify the probability of observing deviations of a given "severity" from the second law. For the reason you clearly understand, the smaller the system, the less meaningful it becomes to describe it in terms of "macroscopic" properties such as temperature, pressure and so forth (indeed these quantities can be construed to be a parameter of a statistical population, which have less and less relevance for smaller and smaller sample sizes from that population).
So I think the most meaningful version of the second law to address for this question is Carnot's classic macroscopic statement that it is "impossible to build a perpetual motion machine of the second kind". A particular property of such a perpetual motion machine is its periodicity in its interactions with its surroundings: it undergoes a periodic cycle and when it comes back to its beginning point, both it and the surrounding world are in the same state. So the impossibility of the second kind perpetual motion machine talks about "not winning in the long term": you might make small conversions of the heat in a uniform thermodynamic temperature system into useful work in the short term by dint of fluctuations, but in the long term you cannot. Ultimately this is an experimental fact and is thought to be owing to the boundary conditions of the universe.
The Szilard Engine and Maxwell Daemons: Information is Physical
Let's look at the Szilard engine and Maxwell Daemon first: the latter was conceived by Maxwell to illustrate that the second law was "just statistical" and it does seem to thwart the second law, as does the Szilard engine. Indeed they do win in the short term, but in the long term they do not. The full resolution to the problem is discussed in detail in Bennett's paper that I cited above, and the reason they do not is Landauer's Principle: the idea that the merging of two computational paths or the erasing of one bit of information always costs useful work, an amount given by $k_B\,T\,\log 2$, where $k_B$ is Boltzmann's constant and $T$ the temperature of the system doing the computation.
Bennett invented perfectly reversible mechanical gates ("billiard ball computers") whose state can be polled without the expenditure of energy and then used such mechanical gates to thought-experimentally study the Szilard Engine and to show that Landauer's Limit arises not from the cost of finding out a system's state (as Szilard had originally assumed) but from the need to continually "forget" former states of the engine.
Probing this idea more carefully, as also done in Bennett's paper: One can indeed build the Maxwell Daemon with simple finite state machines in the laboratory, as described in the Nature paper I cited. As the Daemon converts heat to work, it must record a sequence of bits describing which side of the Daemon's door (or engine's piston, for an equivalent discussion of the Szilard engine) molecules were on. For a finite memory machine, one needs eventually to erase the memory so that the machine can keep working.
However, "information" ultimately is not abstract - it needs to be "written in some kind of ink" you might say - and that ink is the states of physical systems. The fundamental laws of physics are reversible, so that one can in principle compute any former state of a system from the full knowledge of any future state - no information gets lost. So, if the finite state machine's memory is erased, the information encoded that memory must show up, recorded somehow, as changes in the states of the physical system making up and surrounding the physical memory.
So now those physical states behave just like the computer memory: eventually those physical states can encode no more information, and the increased thermodynamic entropy of that physical system must be thrown out of the system, with the work expenditure required by the Second Law, before the Daemon can keep working. The need for this work is begotten of the need to erase information, and is the ultimate justification for Landauer's principle.
The Szilard Engine and Daemon "win" in the short term because they are not truly cyclic: they change the states of memory: the second law prevails when that memory is brought back to its beginning state too.
Another Illustration of Non Cyclic Thwarting of the Second Law
Another illustration of the importance of true cycles in considering the second law is a "trick" whereby one can extract ALL of the enthalpy of a chemical reaction as useful work IF one has a sequence of cooler and cooler reservoirs that one can use as follows: (1) Lower the reactants down to absolute zero temperature by drawing heat from the reactants into the reservoirs, (2) Let the reaction to go ahead at aboslute zero thus extracting all the reactants' enthalphy as work and then (3) Use the sequence of reservoirs in rising temperature order to bring the reaction products back to the beginning temperature. The point is that some of the enthalpies of formation will now be left in the cold reservoirs and so the system has not been taken through a complete cycle. One can't do this indefinitely: the cool reservoirs will eventually heat up if one does this repeatedly. You might "win" with small amounts of reactants, but you can't do so indefinitely because you are degrading the system: the work needed to restore the cold reservoirs to their beginning state is then the difference between the enthalpy of reaction and the free energy.
"Proofs" of the Second Law
E. T. Jaynes tried to bring information theory rigorously to thermodynamics and critically examines Boltzmann's concept of entropy. In particular the Boltzmann "stosszahlansatz" (assumption of molecular chaos) can often only be applied once, as later changes to the system leave the states of molecules of a gas correlated, thus begetting the difference between the Gibbs (informational) and Boltzmann ("experimental", i.e. defined only when you have big systems) entropies, with the former unchanged in things like irreversible volume changes, the latter always increasing. So, from an assumption of molecular chaos, one can prove once that the Boltzmann entropy must increase in an irreversible change. But the irreversible change and the correlation between system constituents it begets means that one cannot apply the assumption of molecular chaos again and repeat the proof unless one comes up with an explanation of how the system gets back to a state where the states of all its constituent parts are uncorrelated. See the Jaynes papers in my references: Jaynes does eventually argue that one needs to appeal to experiment to support the large scale second law of thermodynamics.
So ultimately it would seem that the statement that the Boltzmann entropy of a system always increases in the long term can only be substantiated experimentally. Why the entropy of a system always increases when physical laws are just as valid with time running backwards is called "Loschmidt's Paradox". There has been a great deal of work to understand this and it's generally agreed that the answer has to do with the "boundary conditions" of the universe - roughly put, the universe was (observed fact) in an exquisitely low entropy state at the big bang, and so the overwhelmingly likeliest history is one where entropy rises with increasing time. But how and why that low entropy state arose is, as I understand it, one of the profound mysteries of modern physics. A good layperson's summary of why we have a second law of thermodynamics, how entropy is to some extent a subjective concept, and the discussion of this profound mystery is to be found in chapter 27 of Roger Penrose's "The Road to Reality". I would highly recommend you look at this reference.
• Thank you very much for a thorough and enlightening answer. – yrodro Sep 10 '13 at 12:18
• @yrodro Glad you liked it. This is very interesting and subtle stuff. I also enormously appreciate the little kindness of acknowledgement - it means a great deal. – Selene Routley Sep 10 '13 at 12:31
• Wow. What an answer. :) Regarding your last section... The H-theorem can be proved assuming detailed balance and Markovianity alone (both of which hold for all laws of physics). Also, the 2nd law follows from the fluctuation theorem for entropy, which in turn can be derived microscopically without any additional assumptions. Why do you feel that this is not a proof of the 2nd law? – Bubble Sep 10 '13 at 13:19
• @Bubble Can you point me to a particular case of the H theorem proof? My background in this field is mainly the E. T. Jaynes account, which I only stumbled on some years ago when I was researching the Jaynes-Cumming model of quantized field - two level atom interaction in quantum optics (altogether unrelated to the second law). Also, I shall browse the papers cited in your answer: it has been some time since I have seen the H theorem arguments so I need to refresh before responding. IIRC some of the ones I saw do need to make repeated "chaos" (stat. independence of constituents) assumptions... – Selene Routley Sep 10 '13 at 13:43
• .... and subtly neglected correlations. As for my "feeling" about it all - the more I think about probability and statistical reasoning the more I feel that probability probably :) has quite a way to go to resolve some foundational issues (you only have to look at the Stanford Dictionary of Philosophy on foundations of probability to fry your brain utterly) so I think that experimental evidence tends to be very important. – Selene Routley Sep 10 '13 at 13:49
I assume you are referring to Maxwell's demon, though I fail to see how it relates to your questions, which are:
Is there consensus on the absolute vs statistical nature of the 2nd Law, or is it subject to interpretation? Can the issue be settled within a classical setting, or does one have to go quantum?
The 2nd law is statistical. No one, after the work of Boltzmann, thought it was absolute (i.e., not statistical).
Boltzmann's H-theorem proves the 2nd law for both classical and quantum microscopic laws. Though, apparently, some people do not like this proof (for some reason). There are other ways to prove it. For instance, one can first derive the fluctuation theorem, which implies the second law directly, from microscopic physics. It can be done for both classical and quantum microscopic laws. For a derivation in the quantum case see this review article "Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems" by Esposito, Harbola and Mukamel, Rev. Mod. Phys. 81, 1665–1702 (2009), arXiv:0811.3717
Here are some slides, from Evans, Williams and Searles for the classical case.
EDIT: fixed factual inaccuracies
• Factual inaccuracies: (1) Everyone thought the second law was absolute, up until Boltzmann. (2) Maxwell's demon is resolved by Landauer's principle. It doesn't have to increase entropy until it runs out of memory and has to start erasing bits, at which point it has to increase the entropy of its environment, not itself. (3) Boltzmann's H-theorem is for ideal gases only and has little relevance to the general case. – Nathaniel Sep 10 '13 at 13:53
• +1 You definitely answered the OP's question - but I'm not sure about proofs yet! I shall definitely have a look at your papers. Searles's work is pretty wonted to me but I wouldn't say I have a hugely deep handle on fluctuations yet. – Selene Routley Sep 10 '13 at 13:53
• @Nathaniel, thanks! I fixed (1) and (2), but I fail to see how the H-theorem applies only to ideal gases. Could you please explain? – Bubble Sep 10 '13 at 14:36
• @Bubble it's because the $H$ in Boltzmann's H-theorem ignores interparticle forces. More details can be found in this classic paper, which also contains an excellent explanation of how the second law arises from microscopic reversibility, despite pre-dating the fluctuation theorem stuff by several decades. – Nathaniel Sep 10 '13 at 16:21
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# How do you find the inverse of f(x)=(3x-2)/(x+7)?
${f}^{- 1} x = \frac{7 x + 2}{3 - x}$
$y = f \left(x\right) = \frac{3 x - 2}{x + 7} = \frac{3 x + 21 - 23}{x + 7} = 3 - \frac{23}{x + 7}$
$y = 3 - \frac{23}{x + 7} \implies \left(y - 3\right) = - \frac{23}{x + 7} \implies x + 7 = \frac{23}{3 - y} \implies x = - 7 + \frac{23}{3 - y}$
${f}^{- 1} y = - 7 + \frac{23}{3 - y} = \frac{7 y + 2}{3 - y}$
$\implies {f}^{- 1} x = \frac{7 x + 2}{3 - x}$
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Document Detail
From MEDLINE®/PubMed®, a database of the U.S. National Library of Medicine
Full Text Journal Information Journal ID (nlm-ta): Oecologia Journal ID (iso-abbrev): Oecologia ISSN: 0029-8549 ISSN: 1432-1939 Publisher: Springer-Verlag, Berlin/Heidelberg Article Information Download PDF © The Author(s) 2012 OpenAccess: Received Day: 24 Month: 2 Year: 2012 Accepted Day: 6 Month: 11 Year: 2012 Electronic publication date: Day: 28 Month: 11 Year: 2012 pmc-release publication date: Day: 28 Month: 11 Year: 2012 Print publication date: Month: 7 Year: 2013 Volume: 172 Issue: 3 First Page: 877 Last Page: 887 PubMed Id: 23188055 ID: 3679420 Publisher Id: 2530 DOI: 10.1007/s00442-012-2530-6
Habitat structure alters top-down control in litter communities Gregor KalinkatAff1Aff2 Address: kalinkat@bio.tu-darmstadt.de Ulrich BroseAff1 Björn Christian RallAff1 J. F. Blumenbach Institute of Zoology and Anthropology, Georg-August-University Göttingen, Berliner Str. 28, 37073 Göttingen, Germany Department of Biology, Technische Universität Darmstadt, Schnittspahnstr. 10, 64287 Darmstadt, Germany Communicated by footnote: Communicated by Volkmar Wolters.
Introduction
Progress in food-web ecology is critically based upon information about bioenergetic flows of energy between consumer and resource pairs. These interaction strengths and their distributions across the myriads of links in natural food webs are vital for community structure, population dynamics, and ecosystem functioning (e.g. McCann et al. 1998; Neutel et al. 2002, 2007; Otto et al. 2007; Rall et al. 2008; Berlow et al. 2009; Binzer et al. 2011). The biotic mechanisms shaping and structuring interaction strengths are complex and might be driven by basal resources (e.g. detritus) or consumers (e.g. predators). One major question in the ecology of soil food webs therefore deals with the regulation of detritivore populations and whether they are controlled by bottom-up mechanisms (i.e. energy and nutrient supply) or top-down regulated by their multiple predators. Both hypotheses are supported by studies: Bengtsson et al. (1997) found top-down control, whereas the results of Scheu and Schaefer (1998) and Ponsard et al. (2000) provided evidence for bottom-up control. Major progress in this field requires insights in consumer–resource interactions with a particular focus on the strength of such interactions (Scheu 2002). Due to the natural composition of soil and litter habitats with their porous, fractal structure and opaqueness, the direct observation of species interactions in the natural context is almost intractable. Indirect observation via gut or stomach content analysis, a standard procedure in freshwater (e.g. Elliott and Persson 1978; Woodward and Hildrew 2002) and marine (e.g. Daan 1973; Aljetlawi et al. 2004; Smout and Lindstrøm 2007) systems is hampered by the fact that a large fraction of soil predators rely on extra-intestinal digestion (Cohen 1995) and therefore deep understanding of predator–prey interaction strengths in these systems remains challenging. While different methods of tracking feeding links qualitatively were developed and improved over the past decades—particularly stable isotope analyses, molecular gut content analyses and fatty acid trophic markers (Post 2002; King et al. 2008; Ruess and Chamberlain 2010)—they have scant ability for tracking feeding interactions quantitatively. Therefore, we have to rely on laboratory experiments to determine per capita impacts of litter- and soil-dwelling predators on their prey.
One well-established model framework for analysing interaction strengths is the functional response (Holling 1959; Berlow et al. 2004). It describes the density-dependent per capita consumption rate, Fij, of a predator j on a prey i (Holling 1959; Real 1977):
[Formula ID: Equ1]
[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_{ij} = \frac{{a_{ij} N_{i}^{q + 1} }}{{1 + a_{ij} h_{ij} N_{i}^{q + 1} }}}$$\end{document}]
where Fij (ni nj−1 day−1) is the per capita consumption rate (also referred to as intake rate, ingestion rate, predation rate or feeding rate), Ni [(ni m−2) or (ni m−3)] is prey density, hij (nj day ni−1) is the handling time needed to kill, ingest and digest a resource individual, aij is the capture rate [(m2 day− 1 nj−1) or (m2 day− 1 nj−1)] and q is a scaling exponent converting the hyperbolic type II functional response (q = 0) to a sigmoid type III functional resonse (q = 1; Real 1977; Hassell 1978; Rall et al. 2008; Vucic-Pestic et al. 2010b). Note that the capture rate (often also referred to as “attack rate” or more accurately “rate of successful attacks”) is expressed on a movement or velocity scale [with either area or volume depending on the foraging mode of the predator with each prey and the ecosystem type where predator and prey occur (McGill and Mittelbach 2006; Rall et al. 2012; Pawar et al. 2012)]. It includes the rates of encounter and success of attacks (Gergs and Ratte 2009; Vucic-Pestic et al. 2011).
Generally, two different approaches to determine capture rates and handling times can be distinguished: (1) direct observation and (2) indirect derivation through model fitting. In carefully designed experiments, both approaches result in congruent parameter estimates (Tully et al. 2005). While direct observation is feasible, particularly for larger predators (e.g. fishes in laboratory experiments; Persson and Brönmark 2002) studies working with diminutive organisms in opaque environments have to either rely on adequate model fitting techniques to reveal functional response parameters or reduce the structural complexity of the experiment to improve the visibility of the interactions. As Jeschke et al. (2004) highlighted, the majority of functional response studies are carried out in simplified laboratory systems. The resulting problem that feeding rates might differ in more complex experiments has been addressed by several functional response studies in recent years: there, experimental complexity was introduced by variation of numbers of predator individuals (predator interference; e.g. Kratina et al. 2009; Lang et al. 2012), the number of prey species (alternative prey; e.g. Colton 1987; Elliott 2004; Kalinkat et al. 2011) or even the additional presence of non-prey species (Kratina et al. 2007). Another lack of reality in laboratory studies is due to oversimplified environmental conditions that are typically provided within artificial arenas. There are only a limited number of studies focussing on the effects of habitat complexity on the functional response of terrestrial predators (Kaiser 1983; Munyaneza and Obrycki 1997; Pitt and Ritchie 2002; Hoddle 2003; Hohberg and Traunspurger 2005; Hauzy et al. 2010; Vucic-Pestic et al. 2010a). While some of these studies focussed on the fractal complexity of an artificially structured habitat (Kaiser 1983; Pitt and Ritchie 2002; Hoddle 2003) and others made qualitative comparisons of with-structure- versus non-structure-treatments (Hohberg and Traunspurger 2005; Vucic-Pestic et al. 2010a), there is only one study to our knowledge with a qualitative comparison between a simplified, unstructured laboratory setting and field conditions (Munyaneza and Obrycki 1997). This study indicated reduced capture rates of terrestrial arthropod predators by a factor of roughly two under greenhouse and field conditions compared to the experimental setting with controlled conditions in the laboratory experiment.
Beyond these specific functional response studies, a broader look at the literature reveals that habitat structure effects on predator–prey interactions have been the focus of many studies especially in aquatic ecosystems (e.g. Crowder and Cooper 1982; Gotceitas and Colgan 1989 and references therein). There, predation rates are also reduced in high-complexity treatments and tend to be highest in habitats with intermediate structural complexity (Crowder and Cooper 1982). However, a continuous framework that is suitable to link trophic and non-trophic effects between basal resources (e.g. litter), first-order consumers (e.g. detritivores) and predators is still missing. This applies particularly to leaf litter systems, where pulses of incoming material and long-lasting decay of the litter yield a continuously changing amount and complexity of habitat structure. Therefore, our understanding of dynamics and functioning of such ecosystems is challenged by a general lack of studies addressing how habitat structure modifies interaction strengths and top-down control of microbi-detritivores by predators.
In this study, we aimed to fill this gap by studying the effects of systematic variation in leaf litter density on the functional response of the centipede Lithobius mutabilis (Chilopoda: Lithobiidae) as a ubiquitous and frequent generalist predator of the leaf-litter system on its microbi-detritivore prey, the springtail Heteromurus nitidus (Collembola: Entomobryidae). Springtails have been shown to be flexible foragers that can feed on fungal hyphae, bacteria or detritus depending on the available resources (Scheu 2002). According to Lawrence and Wise (2000), they might be assigned to the functional guild of detritivores as higher abundances of springtails co-occurred with increased rates of litter disappearance in this experiment. Within the model framework of the functional response, we expected prey refuges of the additional habitat structure to cause a shift from type II to type III functional responses (Real 1977; Scheffer and De Boer 1995), as has already been shown for other predator–prey pairs from litter systems (de Ruiter et al. 1988, Vucic-Pestic et al. 2010b). Furthermore, we anticipated that the capture rate should be negatively affected by increasing habitat structure as encounter rates are directly dependent on movement patterns and velocities of predators and prey (Muirhead and Sprules 2003; Gergs and Ratte 2009). Therefore, habitat complexity should only affect the encounter rate and not the mechanisms involved once the two species are in close contact (which includes handling time).
In consequence, we hypothesised that the increased complexity of leaf litter should (1) provide additional prey refuges therefore resulting in more sigmoid type III functional responses, (2) decrease the capture rates, and (3) not affect the handling times.
Materials and methods
Functional response experiments
The basic experimental set-up follows prior functional-response experiments (Vucic-Pestic et al. 2010b, 2011; Rall et al. 2011). We studied the per capita consumption rates of the centipede L. mutabilis on the springtail H. nitidus at varying prey densities from 1 to 1,000 individuals of springtails per arena (corresponding to 25–25,000 individuals per m2) at four levels of habitat complexity (1, 2, 4 and 8 g dry weight of beech litter corresponding to 25, 50, 100 and 200 g/m2, respectively). Each prey density was replicated three to five times resulting in a total number of 123 experimental units. The centipedes were sampled by hand from field sites in the Hainich-Dün National Park, Thuringia, Germany. Freshly fallen beech litter was sampled at the same sites. The predator individuals were kept separate from each other in moistened plastic jars and were deprived of food for at least 48 h before the start of the experiments. The experiments were performed in Perspex® arenas (0.2 × 0.2 × 0.1 m) covered with lids with holes to allow gas exchange. The arena floor was covered with moist plaster of Paris (200 g dry weight) to provide constant moisture during the experiments. Beech litter for providing habitat structure in the arenas was first dried for several days at 40 °C to eliminate other animals and then re–moisturised prior to the experiments. Prey individuals were placed in the arenas 30 min prior to the predators to allow them to adjust to the arenas. The experiments were run for 24 h with a day/night rhythm of 12/12 h dark/light and a constant temperature of 15 °C in temperature cabinets. Initial and final prey densities were used to calculate the number of prey eaten. Control experiments without predators showed that effects of prey mortality or escape were negligible. As recent studies have shown strong allometric effects on the functional responses of terrestrial invertebrate predators (Vucic-Pestic et al. 2010b; Rall et al. 2011, 2012), we controlled predator and prey weight and kept it at a constant level (centipedes: 22.74 ± 0.77 mg standard error; springtails: 0.15 ± 0.004 mg standard error). Note that, in this experimental design, we intentionally excluded trophic effects between the litter and the springtails. Short-term experiments with freshly fallen leaves that were not yet colonised by fungi or bacteria serving as potential resources for the springtails assured that we would reveal particularly the non-trophic effect of the leaf litter as habitat structure.
Leaf area
Generally, leaf litter density is positively correlated with surface area available for the predator–prey interaction. Hence, expressing consumption and capture rates relative to the surface area of the experimental arenas might become arbitrary with increasing leaf litter density. In order to provide an alternative approach accounting for increases in surface area with increasing leaf litter density, we “corrected” the prey densities relative to the leaf surface area plus the arena area to get the “total foraging area”: Therefore, we measured the leaf surface area of a representative set of 12 samples of leaves (three replicates of 1, 2 and 4 g dry weight, respectively) that were used within the experiments. For each sample, we determined leaf surface area by optical scanning with a flatbed graphics scanner and subsequent analyses of the images with the software WinFOLIA, v.5.1a (Regent Instruments Inc., Quebec City, Canada). We fitted leaf area against leaf litter dry weight using a linear model. Subsequently, three different spatial scenarios were compared in our functional response analyses: (1) uncorrected area, i.e. 0.04 m2 arena surface area in all leaf litter treatments, (2) one-side corrected (hereafter, one-sided) area with the one-sided leaf area plus arena surface area, and (3) two-side corrected (hereafter, two-sided) area with the two-sided leaf area plus arena surface area (see Table 1 for an example how prey densities were corrected for differing habitat size according to these three scenarios).
Statistical analyses
Initially, we fitted a polynomial logistic regression to the proportion of prey eaten to investigate the shape of the functional response (Juliano 2001):
[Formula ID: Equ2]
[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{{N_{e} }}{{N_{0} }} = p = \frac{{{\text{e}}^{{p_{0} \left[ L \right] + p_{1} \left[ L \right]N_{0} + p_{2} \left[ L \right]N_{0}^{2} + p_{3} \left[ L \right]N_{0}^{3} }} }}{{1 + {\text{e}}^{{p_{0} \left[ L \right] + p_{1} \left[ L \right]N_{0} + p_{2} \left[ L \right]N_{0}^{2} + p_{3} \left[ L \right]N_{0}^{3} }} }}}$$\end{document}]
where p represents the predation risk of one prey item to be killed, N0 is the initial prey density, Ne is the number of prey killed during the experiment, p0, p1, p2 and p3 are statistically estimated parameters and L represents the level of leaf litter density. In this vein, a continuously decreasing relationship of predation risk dependent on prey density indicates a type II functional response, whereas a hump-shaped curve indicates a type III functional response (de Ruiter et al. 1988; but see Juliano (2001) for detailed methodology). The goal was to identify possible differences in the shape of the responses between the different leaf litter density treatments. Therefore, we performed a stepwise backwards selection by firstly deleting the factorial litter density treatment levels, and afterwards the polynomial terms. Accordingly, we simplified the model until the Akaike Information Criterion (AIC) (Akaike 1974) indicated the best fit.
Subsequently, we used type II models in all following analyses, because decreasing functions for predation risk were identified as best models for all leaf-litter treatments (see “Results”). To avoid violation of our statistical results due to prey depletion during the course of the experiment, we then used the integrated form of the functional response, also known as Rogers ‘Random Predator Equation’ (Royama 1971; Rogers 1972) because Eq. (1) assumes a constant prey density throughout the experiment:
[Formula ID: Equ3]
[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N_{e} = N_{0} \left( {1 - {\text{e}}^{{\left( {a_{ij} \left( {N_{e} h_{ij} - PT} \right)} \right)}} } \right)}$$\end{document}]
where Ne (ni m−2) is the density of prey i eaten during the experiment, P is predator j’s density, T is the experimental time (days) and all other parameters are as in Eq. (1). We solved this recursive function of Ne with a non-linear least squares method (“nls”) using the additional package “emdbook” for the statistical software package R (Bolker 2008; R Development Core Team 2010). The resulting equation is
[Formula ID: Equ4]
[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N_{e} = N_{0} - \frac{{W\left( {a_{ij} h_{ij} e^{{\left( { - \left( {PT - h_{ij} N_{0} } \right)} \right)}} } \right)}}{{a_{ij} h_{ij} }}}$$\end{document}]
where W is the Lambert W function (see Bolker 2008 and references therein for a detailed description). Furthermore, we analysed the effect of litter density on capture rates and handling times by inserting either exponential
[Formula ID: Equ5]
5a [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a_{ij} = a_{0} {\text{e}}^{{\varepsilon_{a} L}} }$$\end{document}]
[Formula ID: Equ6]
5b [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h_{ij} = h_{0} {\text{e}}^{{\varepsilon_{h} L}} }$$\end{document}]
or power law functions
[Formula ID: Equ7]
6a [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a_{ij} = a_{0} L^{{\left( {b_{a} } \right)}} }$$\end{document}]
[Formula ID: Equ8]
6b [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h_{ij} = h_{0} L^{{\left( {b_{h} } \right)}} }$$\end{document}]
in Eq. (4), where a0 and h0 are constants, L is the leaf-litter density, εa and εh determine the exponential increase or decrease of capture rates and handling times in dependence on leaf-litter density, while ba and bh are the scaling exponents of the power law functions. Additionally, functional response models with constant values of a0 and h0 without leaf litter dependence were also fitted to the data. We fitted all possible combinations of the three capture-rate models and three handling-time models (constant, exponential, power law) under each of the three spatial scenarios (uncorrected, one-sided and two-sided) to the data resulting in a total of 27 functional response models and ranked them according to their ΔAIC (see supplementary Table S1 for an overview).
Results
The logistic regression analyses showed that the best model only included the constant p0 in dependence of the leaf litter level and a negative linear term p1 (Fig. 1, see figure legend for statistical outputs). This result suggested a type II functional response that did not differ in shape for all four leaf litter levels whereas overall predation risks differed according to leaf litter density. Therefore, all subsequent analyses where made with a type II functional response model.
The mean leaf area (one-sided) increased from 0.023 m2 (±0.002 95 % CI) in the treatment with 1 gram leaf litter to 0.188 m2 (±0.016 95 % CI) in the treatment with 8 g leaf litter (Table 1; Fig. 2a–d) following a linear model fit through the leaf areas of 1, 2 and 4 g dry weight of leaf litter (n = 12, R2 = 0.984, p < 0.0001). This increase in the surface area available for animal movement and interactions implied that the prey density (here, for example, for one springtail individual per arena) decreased from 25 ind/m2 (uncorrected) to ~16 ind/m2 (one-sided) and ~12 ind/m2 (two-sided) in the treatment with 1 g leaf litter or to ~4 ind/m2 (one-sided) and ~2 ind/m2 (two-sided) in the treatment with 8 g leaf litter (Table 1; Fig. 2a–d). While prey densities were the same across treatments in the scenario with uncorrected area (Fig. 2e–h, second row), the increases in leaf surface area with the amount of leaf litter resulted in a shift in prey densities from higher densities in treatments with 1 g leaf litter (Fig. 2, left column) to lower densities (Fig. 2i–l: one-sided; m–p: two-sided).
Capture rates showed decreasing exponential functions with increasing leaf litter density for the uncorrected and the one-sided prey densities, whereas functional-response models with a constant capture rate provided the best fit to the data under two-sided correction (Table 2; Fig. 3a). Comparing the parameter values of the models with exponential relationship for capture rates, there is a clear trend from a highly significant negative relationship for the uncorrected densities (εa = −0.0103, SE = 0.0014, p < 0.0001), a shallower slope with lower significance for the one-sided correction (εa = −0.0032, SE = 0.0014, p = 0.020) to a non-significant (constant) relationship under the two-sided correction (εa = −0.0016, SE = 0.0014, p = 0.267 in the second best model fitting; see Table 2 for parameter estimates for the best fitting models, respectively). Surprisingly, in all three spatial scenarios (uncorrected, one-sided and two-sided densities), the best-fitting model with the lowest ΔAIC included power law decreases in handling times with increasing leaf litter density (Tables 2 and S1; Fig. 3b). This contradicted our third initial hypothesis that handling time should not be affected by litter density. All the functional response models with constant handling time yielded a much poorer fit to the data (Table S1) suggesting that our third hypothesis had to be rejected.
The consequences of these litter dependencies in capture rates and handling times for the relationship between per capita consumption rates and the amount of leaf litter in the system are illustrated in Fig. 3c–e for three prey densities. While the consumption rates decreased constantly with leaf litter density at low (Fig. 3c, 10 springtails per arena) and intermediate prey densities (Fig. 3d, 100 springtails per arena) under all three spatial corrections, we found a hump-shaped relationship at higher prey densities (Fig. 3e, 600 springtails per arena) for the uncorrected and the one-sided scenario.
Discussion
In this study, we tested how changing habitat structure in a leaf litter-dominated ecosystem may influence predator–prey interactions by examining functional responses in a laboratory experiment. Contrary to our first hypothesis, we have not found a switch from hyperbolic to sigmoid functional responses with increasingly complex habitat structure. Corroborating our expectations, we found a highly significant decrease in capture rates with increasing litter density except for our analyses correcting for increase in habitat area on both sides of the leaves (two-sided correction) where capture rates remain constant. While we expected handling times to be unaffected by leaf litter density, our analyses revealed decreasing handling times with increasing leaf litter densities.
As the functional responses showed a hyperbolic shape at each litter density, we suppose that the particular habitat structure realised by the beech leaf litter does not provide sufficient hiding refuges for the springtails within the experimental design employed. This may be due to the mobility and the particularly flattened shape of the centipede body, allowing it to explore the interstices between the leaves in a similar fashion to its significantly smaller prey. Subsequent studies need to replicate our experiments for predator groups that differ in their ability to hunt within the interstices between the leaves to address the generality of our result.
Consistent with our initial hypothesis, the capture rates decreased with increasing litter density. As capture rates are composed of encounter rates and attack success (Gergs and Ratte 2009; Vucic-Pestic et al. 2011), we tested whether this effect is caused by (1) a dilution effect reducing encounter rates as increasing litter density yields a higher surface area of the leaves available for interactions, or (2) decreases in the efficiency of the attacks (i.e. capture rates in relation to the available foraging area, or “relative capture rates”) of the centipedes. We found that the significant decrease in capture rates with leaf litter densities is turned into a neutral relationship when accounting for increases in habitat size for springtails and centipedes, as the surface area of the leaves is augmented with an increasing amount of litter. This finding is supported by the observation that centipedes and springtails move on the ground area of the experimental arena as well as on both sides of the leaves. In consequence, our results suggest that the attack efficiency (i.e. the success rate of attacks upon encounters) of the centipedes does not change with litter density, whereas increasing habitat size reduces the encounter rates by diluting the prey population to lower density. The constant capture rates in the analyses correcting for the two-sided increase in habitat size with leaf density show that the dilution effect is responsible for the negative relationship between capture rates and prey density in our experiment.
Beyond that, we found significant decreases in handling time with litter density. We did not anticipate such results and, unfortunately, we can do nothing but conjecture about the biological mechanisms that might be responsible for this finding. As it is well known that centipedes are extremely sensitive to dry conditions (Lithobiids have been shown to prefer 90–100 % relative humidity; Albert 1983), the treatments with higher litter density might have provided more humid conditions. Such more suitable microclimatic conditions for the centipedes might be responsible for the decrease in handling time along the leaf litter density gradient if physiological processes involved in ingestion become more efficient with humidity. Future experiments on habitat structure effects on centipede predation should therefore include better means to control for constant humidity in the arenas or at least measure microclimatic heterogeneity therein. However, other biological processes driven by litter density might also contribute to our results (e.g. centipedes might shirk uncovered areas as a strategy to minimise their own predation risk). Together with earlier studies on habitat structure effects on predation (e.g. Crowder and Cooper 1982; Kaiser 1983; Gotceitas and Colgan 1989; Hoddle 2003; Hohberg and Traunspurger 2005; Hauzy et al. 2010), our findings support the general view that more structural complexity tends to reduce the predators consumption rates. We think it is particularly important to highlight this in the context of food-web modelling approaches as this has two major implications in this field: (1) functional responses that are measured experimentally with the aim to parameterise food-web models should urgently avoid to use oversimplified “Petri-dish” arenas to reveal realistic consumption rates, and (2), our results provide a mechanistic basis to couple the trophic and non-trophic effects of leaf litter in dynamic population models of soil food-webs.
As for any empirical study, some potential caveats need to be mentioned. For example, the functional response models fitted under the spatial corrections did not reach saturation, because correction of the densities compressed the prey density range. This is particularly important for the estimation of handling times in functional response model fitting. However, as our analyses have shown that the general patterns in leaf litter dependency of the functional response parameters also apply for the well-saturated model fittings based on the uncorrected spatial scenario, this should not affect our conclusions. Furthermore, we could have avoided unsaturated curves under the spatial correction scenarios by extending the range in prey densities beyond the maximum of 25,000 individuals per square metre. Besides the experimental impracticability of the extremely high numbers of springtails per treatment, this would also have by far exceeded the densities of natural springtail populations (biomasses of ~0.6 g per m2 corresponding to ~4,000 individuals per m2; calculations based on dry-weight data from Schaefer 1990 multiplied by water-fraction factor four from Peters 1983). In conclusion, we have decided to keep the springtail densities of our experiment within the range of natural densities while addressing the consequences of natural habitat structures on consumption rates, which avoids the fallacies imposed by oversimplified laboratory conditions (Munyaneza and Obrycki 1997; Vucic-Pestic et al. 2010a).
In soil food webs, springtails are amongst the most abundant taxonomic groups within the microbi-detritivore guild and therefore of critical importance for litter decomposition (Chen and Wise 1997). In a study with a focus on spider predation upon springtails, it has been shown that a reduction of springtails reduces litter decomposition rates (Lawrence and Wise 2000), indicating the importance of top-down regulating mechanisms in soil litter systems. In this study, we present a novel mechanism for how top-down control might be coupled to the dynamics of leaf litter fall with far reaching consequences for decomposition and population dynamics of microbi-detritivores and their predators. The non-trophic effect provided by habitat-altering leaf litter fall can be included in predator–prey functional responses by changing the densities of predators and prey. Hence, future studies dealing with quantitative description of predators and prey in these systems should not only include densities per square metre but additionally provide information on litter densities.
Moreover, the capture rates and handling times are significantly affected by increasing leaf litter densities, but the consequences of these relationships are not straightforward: while decreasing handling times should lead to increasing consumption rates, decreasing capture rates should cause decreasing consumption rates. Our analyses illustrate that consumption rates generally decrease with increasing litter density, except for the combination of the highest springtail density with the lowest litter density. In consequence, the habitat modifications mediated by leaf litter fall and the subsequent decomposition processes might be responsible for regular shifts between bottom-up and top-down control regimes in some leaf litter systems where phases of litter scarcity can occur due to fast decomposition processes (e.g. systems dominated by maple or alder leaf litter) or reduced litter fall. Corresponding patterns in detritivore and predator population dynamics of mixed decidous forests where predator abundances exceed detritivore abundances in the autumn have been documented (Ponsard et al. 2000). However, our results suggest that, in litter-systems with slow decomposition rates (e.g. systems dominated by beech or oak leaf litter), the potential for top-down control of predators on decomposers should be weak. Our findings shed new light on the ongoing debate whether soil litter systems are top-down or bottom-up regulated (de Ruiter et al. 1995; Polis and Strong 1996; Bengtsson et al. 1997; Scheu and Schaefer 1998). Interestingly, they illustrate that non-trophic effects of leaf litter can drive the strength of predatory top-down control. Hence, understanding the importance of top-down and bottom-up control in soil ecosystems requires integrating trophic and non-trophic effects (Fontaine et al. 2011; Kéfi et al. 2012).
Conclusions
In this study, we have shown how changes in habitat structure affect the predator–prey functional response in leaf-litter systems by diluting predator and prey densities, which reduces their encounter rate. Hence, top-down control of decomposers might be restricted to ecosystems where leaf-litter decomposition is fast enough to deplete habitat structure significantly within one vegetation period. In contrast, many typical temperate forest ecosystems are characterised by slow decomposition rates thus leading to thick litter layers with structured habitats. We have shown that this reduces top-down control by the dilution effect, whereas more complex indirect effect on the efficiency of the attacks could be ruled out. The spatial habitat structure of the litter layer thus determines the strength of predatory top-down pressure, which provides evidence that non-trophic interactions may govern ecosystem organisation.
Electronic supplementary material
Volkmar Wolters, Tony Dell and two anonymous reviewers are appreciated for comments and suggestions that helped to improve the manuscript. Furthermore, we thank Markus Dille, Katharina Fußmann and Thomas Schimmer for their tremendous help while carrying out the experiments. We are grateful to Olivera Vucic-Pestic, Roswitha B. Ehnes, Aleksandra Micic, Bernhard Eitzinger and David Ott for their help in the field sampling and to Mechthild Stange for help with the leaf area measurements. Florian D. Schneider is acknowledged for help with the figures and Amrei Binzer for proof reading. The work has been funded by the German Research Foundation (BR 2315/6, BR 2315/13) and partly funded by the DFG Priority Program 1374 “Infrastructure-Biodiversity-Exploratories” (BR 2315/7). Field work permits were given by the responsible state environmental offices of Baden-Württemberg, Thüringen, and Brandenburg (according to § 72 BbgNatSchG).
References
Figures
[Figure ID: Fig1] Fig. 1 Results of the best-fitting polynomial regression model to test for the shape of the functional response depicting the different leaf litter density treatments of a 1 g, b 2 g, c four g and d 8 g leaf litter per arena, respectively. The model included the negative linear term p1 = −0.0019 (SE = 0.0007, t = −2.731, p < 0.01) and four litter-dependent constants: p0(1 g) = −0.5989 (SE = 0.2057, t = −2.911, p < 0.01), p0(2 g) = −0.6860 (SE = 0.2115, t = −3.243, p < 0.01), p0(4 g) = −0.3267 (SE = 0.1927, t = −1.696, p = 0.092) and p0(8 g) = −1.0978 (SE = 0.2639, t = −4.161, p < 0.001) [Figure ID: Fig2] Fig. 2 The leaf litter within the experimental arenas (0.04 m2 ground area) of the four treatments with a 1 g dry weight leaf litter, b 2 g, c 4 g and d 8 g. Beneath are the functional response curves according to the respective best-fitting model for the uncorrected (e–h), one-sided (i–l) and two-sided (m–p) densities. Parameter values are given in Table 2 [Figure ID: Fig3] Fig. 3 Relationship between leaf litter density and a capture rates and b handling times. Curves are based on the best-fitting functional-response models with uncorrected (solid line), one–sided (dashed line) and two-sided (dash-dotted line) prey densities. c–e show the resulting relationships for leaf litter density and consumption rates at 10 (c), 100 (d) and 600 (e) prey individuals per experimental arena
Tables
[TableWrap ID: Tab1] Table 1
Results of leaf area linear model fit (with 95 % CI) and examples of deduced density correction factors for one individual per experimental arena (0.04 m2)
Leaf litter weight (g) One-sided leaf area (m2) (±95 % CI) Uncorrected (Ind/m2) One-sided (Ind/m²) Two-sided (Ind/m2)
1 0.0234 (±0.0020) 25 15.7633 11.5105
2 0.0469 (±0.0040) 25 11.5105 7.4764
4 0.0938 (±0.0080) 25 7.4764 4.3954
8 0.1875 (±0.0161) 25 4.3954 2.4095
[TableWrap ID: Tab2] Table 2
Parameter estimates for best model fittings for uncorrected, one-sided and two-sided densities, respectively
Parameter estimate SE t value p
Uncorrected
a0 0.0456 0.0118 3.877 <0.001∗∗∗
εa −0.0103 0.0014 −7.364 <0.0001∗∗∗
h0 0.0586 0.0354 1.653 0.101
bh −0.5196 0.1675 −3.102 0.002∗∗
One-sided
a0 0.0574 0.0109 5.289 <0.0001∗∗∗
εa −0.0032 0.0014 −2.361 0.020∗
h0 0.1297 0.0730 1.777 0.078∙
bh −0.7618 0.1694 −4.497 <0.0001∗∗∗
Two-sided
a0 0.0659 0.0086 7.705 <0.0001∗∗∗
h0 0.1340 0.0777 1.723 0.088∙
bh −0.7839 0.1810 −4.331 <0.0001∗∗∗
Handling times follow a power law relationship in all model approaches (Eq. 6b). The capture rates depend on leaf-litter density following an exponential relationship for uncorrected and one-sided densities (Eq. 5a). In the two-sided approach, there is no leaf litter dependence for the capture rate
∗∗∗ p < 0.001
∗∗ p < 0.01
∗ p < 0.05
∙ p < 0.1
Article Categories:Community ecology - Original research Keywords: Keywords Bottom-up control, Functional response, Non-linear interaction strength, Predator–prey interaction, Soil food webs.
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Mathematics
OpenStudy (anna):
what is the value of theta here,, 4sin2theta=0
OpenStudy (sid1729):
$4\times \sin (2\theta) = 0$ If you divide both sides by 4, you get: $\frac{4 \times \sin(2\theta)}{4} = \frac{0}{4}$ which reduces to $\sin (2\theta) = 0$ Now, what are the values where the sin function can be zero? These values are : 0,$\pi, 2\pi, 3\pi, 4\pi.....npi$ where n is a natural number Thus, $2\theta = 0, \pi, 2\pi, 3\pi, 4\pi, .... npi$ You can divide both sides by two to get the answer.
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## <3 sed
Posted Fri 27 April 2012 17:41 under category tech
I wrote a fun sed script today:
sed -E -n -e ':t ; s/(.{21})(.*)/\\bf\{\1\}\n\2/ ; p ; s/\\bf\{(.*)\}\n.*/\1/ ; h ; :q { n ; G ; s/(.{21})(.*)\n\1/\2/ ; tp ; s/(.+)\n.*/\1/ ; bt} ; :p { P ; bq }'
Short, but effective. Can you figure out what it does?
(solution after the break)
## Link: PHP sucks
Posted Tue 10 April 2012 08:29 under category tech
One of my co-workers wrote up this gem on why PHP sucks. I don't agree with his points (having a "development server" isn't an important or even particularly useful feature of a framework, much less a language; prepared statements aren't the pinnacle of SQL), but he does do a good job of showing off some of PHP's more spectacular failings.
I'm naming all of my PHP functions __lambda_object now.
(yes; I do appreciate the irony of linking to his post from a formerly PHP site)
## Debt Free Since 2012
Posted Fri 10 February 2012 01:09 under category personal
As of today, February 10, 2012, I am now officially debt-free. I decided to use my tax refund to pay off the rest of my student loans, which have been sucking down thousands of dollars a month since I graduated. Here are the results:
Yep. It took a few days longer than expected, because Sallie Mae is terrible at ACH. And, of course, their obnoxious rounding means that at some point in the next six months, I'm going to get a cheque from Sallie Mae for 80¢ (presumably plus 6.8% interest). But it's pretty nice to be ...
## My Storage Problem
Posted Sat 28 January 2012 19:07 under category tech
Storage is cheap, or so we're told. Amazon will sell me storage for $0.055/GB/month in “the cloud”; 3.5" hard drives are hovering around$0.06/GB. However, my laptop has a little 250GB SATA drive that is (a) slow and (b) getting full. So I desire to replace it with a fast little SSD. But that raises the question of what to do with my stuff. I'm asking you, Internet. Details below the fold.
## 60,000 scrobbles
Posted Fri 06 January 2012 11:02 under category personal
I used to post these on Facebook as Notes, but since the Timeline refactor, I frankly have no idea how to use Notes; so I guess I'll just post on my own blog. I use last.fm (me) to track my music-listening habits (and sometimes for other things). As of some time recently, I've passed 60,000 scrobbled plays since 2006. Yay!
Here's a graph I made (click for ps version; gnuplot continues to be the best thing ever):
Cheers, all.
## Merry 2012
Posted Tue 03 January 2012 10:32 under category personal
I know I'm a bit late to the party, but Merry 2012. As a sort of celebration, here's my favorite animated GIF of all time (courtesy of Evan a long time ago):
Maybe I'll actually write some useful content this year.
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# Conditional Probability and Independence - Find the probability that player 1 wins the game.
Suppose that there are ten cards numbered from 1 to 10. The cards are shuffled and the game is played between two players: player 1 and player 2. Player 1 starts the game. Player 1 wins if he selects any card numbered from 1 to 4 both inclusive and the game ends. If player 1 fails to draw the card with desired number, then player 2 draws a card and he wins if any card numbered from 5 to 10 is drawn. If player 2 fails to draw the desired card the game goes back to player 1 and he draws the card again. The players keep on playing the game until one of the player wins. Find the probability that player 1 wins the game.
• are cards replaced once they have been choosen – Manish Kumar Singh Oct 6 '15 at 16:48
• Assuming no replacement, brute force works pretty well, here. If Player 1 wins, it will be on one of his first four draws. There aren't very many terms to calculate. – user3294068 Oct 6 '15 at 16:56
• @ManishKumarSingh with no replacement – chauka khan Oct 6 '15 at 17:07
## 2 Answers
Suppose cards are replaced after each draw,\begin{align}P(\text{player 1 wins})&=\frac4{10}+(\frac6{10}\frac4{10})\frac4{10}+(\frac6{10}\frac4{10})(\frac6{10}\frac4{10})\frac4{10}+...\\&=\frac4{10}\left(1+0.24+0.24^2+...\right)\\&=\frac4{10}\frac1{1-0.24}\\&=10/19\end{align}
For no replacement,$$P(\text{player 1 wins})=\frac4{10}+\frac6{10}\frac49\frac38+\frac6{10}\frac49\frac58\frac37\frac26+\frac6{10}\frac49\frac58\frac37\frac46\frac25\frac14=37/70$$
I don't have enough rep to comment, so please treat this as a comment.
I think about using the negative binomial distribution for this kind of problem, but I'm not sure if it's the right way to think about this.
• Welcome to Math.SE! This does not provide an answer to the question, and comments are not entertained among answers. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation, you can comment on any post. – Jesse P Francis Oct 15 '15 at 3:07
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Assume that a 90.0 W light bulb radiates all its energy as a single wavelength of visible light.
(a) Estimate the electric field strength at the surface of the bulb. Assume that the bulb is a sphere and its radius is 2.8 cm.
V/m
(b)Estimate the magnetic field strength at the surface of the bulb.
T
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TrojanPoem one year ago Find the intervals of upwards concavity and downwards concavity and turning points if exists for the curve of function f as f(x) = x^2 +9 /x and find the absolute maximum and minimum for the function when x (- [1, 6]
1. anonymous
start by taking the first and second derivatives
2. TrojanPoem
I know the steps, the problem lays in this example specifically. No turning points although there is upwards and downwards concavity.
3. anonymous
there is a vertical asymptote where denominator is equal to 0
4. TrojanPoem
Clarify a bit .
5. TrojanPoem
You mean something like this ? http://hotmath.com/hotmath_help/topics/rational-functions/image008.gif
6. TrojanPoem
So there is no turning points as the curve is separated into two parts ?
7. anonymous
8. anonymous
$f(x)=\frac{x^2+9}{x}=x+\frac{9}{x}$ right?
9. TrojanPoem
yep.
10. TrojanPoem
I will watch the video while answering
11. anonymous
$f'(x)=1-\frac{9}{x^2}$find the interval over what $$f'$$ is positive, that will tell you where it $$f$$is increasing
12. anonymous
i always graph these problems first to get an idea of what im looking to identify
13. anonymous
as for the absolute max on $$[1,6]$$ check the critical point (where $$f'(x)=0$$ and also check $$f(1)$$ and $$f(6)$$
14. TrojanPoem
Good way of doing it, x = 0 -8, 0 > dec 0 , 8 increase 8 = infinite
15. TrojanPoem
From now , I will graph it too.
16. TrojanPoem
WHere is the turning point satellite ?
17. anonymous
in any case if you want a nice picture and pretty much everything else use this http://www.wolframalpha.com/input/?i=%28x^2%2B9%29%2Fx
18. TrojanPoem
turning point satellite ?
19. anonymous
if "turning point" means where it goes from increasing to decreasing and vice versa, those are the zeros of the derivative
20. TrojanPoem
Fine, so x= ?
21. anonymous
idk i didn't do it solve $1-\frac{9}{x^2}=0$ and you will get them can probably do it in your head
22. TrojanPoem
@billj5 , Finding the asymptote takes decade !
23. TrojanPoem
This is the first derivative , we need the second one. and x = 0
24. anonymous
lol its not too bad once you get the hang of it
25. TrojanPoem
At least , I understood why x = 0 was not a turning point :D , Thanks !
26. TrojanPoem
Hey , may you help me with another question ? Will open new one.
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Book: College Algebra
Book: College Algebra is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
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# OVERVIEW
BACK
The XAFS2 beamline is an experimental station dedicated to X-ray Absorption Spectroscopy in the hard x-rays energy range (3.5 to 17.0 keV). It focus on the study of the atomic-level structure as well as in the electronic and magnetic properties of matter, with applications in a wide range of scientific fields, such as atomic and molecular physics, chemistry, biology, environmental and geosciences and cultural heritage. Experimental techniques available include Fluorescence Spectroscopy, X-ray Excited Optical Luminescence, X-ray Reflectivity and Combined X-ray Absorption Fine Structure and X-Ray Diffraction.
The XAFS2 is a general-purpose X-ray absorption beamline. Since the completion of the commissioning works in 2007, a large number of users have been using this experimental facility in order to perform several kinds of experiments in different scientific areas. After approximately 7 years in operation this beamline has been substantially updated in order to improve its experimental possibilities. A 4-circle Huber diffractometer has been recently incorporated to perform combined experiments. This equipment collects XRD patterns with the XAFS.
Through the development of a new sample environment, it is now also possible to perform these measurements in situ/operando conditions. Other upgrades include a complete remodeling of the beamline software and its control system. The control system of the beamline has been renovated by the installation of a PXI. The PXI is from National Instruments (PXI-NI) and communicates with Galil/Parker controllers on an EPICS platform. Some parts of the motors were changed in order to improve performance with the upgrade and there were also important changes to the control hardware. The Windows operational system was replaced with Red Hat Linux and the 3-WinDCM control system with EPICS. Furthermore, a new python based script (Py4Syn) was added. This provides high-level abstraction for device manipulation, scan routines, real-time plots and more. This package was created with the aim of providing a simple yet powerful tool to allow scientists and users to develop their own scripts for data acquisition. For user-friendly interface builds a Control System Studio (CS-Studio) is used. The next step with the XAFS2 upgrade, namely, towards a high-throughput XAFS beamline, will involve testing the viability of performing QEXAFS.
# CONTACT & STAFF
Beamline Phone Number: +55 19 3512 1246
Beamline E-mail: xafs2@lnls.br
Coordinator: Gustavo de Medeiros Azevedo
Number: +55 19 3518 3192
E-mail: gustavo.azevedo@lnls.br
# EXPERIMENTAL TECHNIQUES
The following experimental techniques and setups are available to users in this beamline. To learn more about the techniques’ limitations and requirements (sample, environment, etc.) contact the beamline coordinator before submitting your proposal.
###### Conventional Transmission and Fluorescence XAS
X-ray absorption spectroscopy (XAS) is a widely used technique for determining the local geometric and/or electronic structure of matter. This setup is optimized for transmission and fluorescence XAS on “standard samples” in standard sample holders. The setup for these experiments has three ionization chambers, a multi-element solid state Ge fluorescence detector and a 5K cryostat. In general, the samples for these environments are membranes, pellets, thin films, liquid and bulk. More attention on sample homogeneity preparation must be given when in transmission mode.
Recent publications in this setup:
Sampaio DV, Souza NRS, Santos JCA, Silva DC, Fonseca EJS, Kucera C, Faugas B, Ballato J, Silva RS; Translucent and persistent luminescent SrAl2O4:Eu2+Dy3+ ceramics. CERAMICS INTERNATIONAL 42, 4306 (2016). doi:10.1016/j.ceramint.2015.11.108
Cappellari P.S., Buceta D., Morales G.M., Barbero C.A., Moreno M.S., Giovanetti L.J., Ramallo Lopez J.M., Requejo F.G., Craievich A.F., Planes G.A.. Synthesis of ultra-small cysteine-capped gold nanoparticles by pH switching of the Au(I)–cysteine polymer. JOURNAL OF COLLOID AND INTERFACE SCIENCE 441, 17 (2015). doi:10.1016/j.jcis.2014.11.016
###### In-situ XAS
This setup allow to submit the samples to different gas atmospheres, while heating up to 1000°C and work on transmission or fluorescence mode. There is a tubular furnace used in transmission and the samples are prepared as pellets. For in-situ fluorescence the sample are powder diluted in boron nitride inside a capillary. The optical table currently has three ionization chambers, a set of slits and motorized stages for user-supplied equipment. If you seek to use the in-situ setup you should contact us well in advance of any proposal deadline to establish technical feasibility.
Recent publications in this setup:
Coletta V.C.; Marcos F.C.F.; Nogueira F.G.E., Bernardi M.I.B; Michalowicz A.; Goncalves R.V.; Assaf, E.M.; Mastelaro V.R.. In situ study of copper reduction in SrTi1-xCuxO3 nanoparticles. PHYSICAL CHEMISTRY CHEMICAL PHYSICS 18, 2070 (2016). doi: 10.1039/C5CP05939A
Ribeiro R.U., Meira D.M., Oliveira D.C., Rodella C.B., Bueno J.M.C., Zanchet D. Probing the stability of Pt nanoparticles on encapsulated in sol-gel Al2O3 using in situ and ex situ characterization techniques. APPLIED CATALYSIS A-GENERAL 485, 108 (2014). doi:10.1016/j.apcata.2014.07.039
###### X-ray excited optical luminescence (XEOL)
This setup allows to get information about the optical behavior of a material when it irradiated with X-rays. It is possible to measure the emission spectra and/or the integrated luminescence as an energy function. Low temperature measurements (cryo-XEOL) are available using a cryostat, reaching 15K. If you seek to use the XEOL setup you should contact us well in advance of any proposal deadline to establish technical feasibility.
Recent publications in this setup:
Hora DA, Andrade AB, Ferreira NS, Teixeira VC, Rezende, MVS (2016). X-ray excited optical luminescence of Eu-doped YAG nanophosphors produced via glucose sol–gel route. Ceramics International, 42(8), 10516-10519 (2016).
doi: 10.1016/j.ceramint.2016.03.142
Rezende M.V., Montes P.J.R., Andrade A.B., Macedo Z.S., Valerio M.E.G.. Mechanism of X-ray excited optical luminescence (XEOL) in Europium doped BaAl2O4 phosphor. Physical Chemistry Chemical Physics, 18, 17646-17654 (2016). doi: 10.1039/C6CP01183G
# LAYOUT & OPTICAL ELEMENTS
ElementTypePosition[m]Description
SourceBending Magnet0.0Bending Magnet D08 exit B (15°)
$1^{\rm st}$ cooled-slit systemTwo cooled UHV slit systems - vertically and horizontally - based on four independent mechanical feedthroughs, each one supporting at its ends a Ta blade mounted on a copper block.8.0Defines the lateral and vertical dimensions of the polychromatic beam impinging on the first mirror
$1^{\rm st}$ mirrorRh-coated cylindrical vertical collimating mirror with a ~3mrad grazing incidence angle9.2Collimate vertically the white radiation and sends a parallel synchrotron beam onto the two flat Si(111) Double Crystal Monochromator (DCM)
Double Crystal Monochromator (DCM)Flat Si(111) Double Crystal Monochromator10.5The DCM has fixed exit geometry and is the only optical element with thermal stabilization (i.e. water cooled).
$2^{\rm nd}$ slit systemTwo UHV slit systems - vertically and horizontally - based on four independent mechanical feedthroughs, each one supporting at its ends a Ta blade mounted on a copper block.11.6Without refrigeration, these slits are used to eliminate the spurious background radiation.
$2^{\rm nd}$ mirrorRh-coated toroidal bendable mirror12.8It refocuses vertically and horizontally the monochromatic beam of approximately $450 \mu \rm m$ in diameter at the sample position
# PARAMETERS
ParameterValueCondition
Energy range [keV]3.5 - 17.0Si(111)
Energy resolution [$\Delta$E/E]$1.7 \times 10^{-4}$at 7 keV
Beam size [$\mu \rm m^2$, FWHM]450 x 250at sample position
Flux density at sample [ph/s/$\rm mm^2$]$2.78 \times 10^{9}$ photons/s at 7 keV and 100mAMeasured with a photodiode 100% efficient
Harmonic Content< $3.94 \times 10^{-5}$at 7.5 keV
# INSTRUMENTATION
InstrumentTypeModelSpecificationsManufacturer
DetectorIon Chamber-Two electrodes in a distance of 14 mm. Length of 137 mm, 221 mm and 381 mm. A $25 \mu \rm m$ kapton window with a in/out area of 12 x 30 $\rm mm^2$LNLS in-house development
Detector15-element Germanium Solid State Detector (SSD)G-15 SSDHigh counting rate capability - 300.000cts/s. Si detector total active area - 750 $\rm mm^2$. Element sensitive thickness - 5 mm. $\rm N_2$ liquid cooled.Canberra
DetectorElectron Detector-Collector electrode (He medium) in an electric field produced by a 60V battery. The output is a signal amplification of about two orders of magnitudeLNLS in-house development
DetectorPhotomultiplierModel R928Side-on; V = -1200 V. It output is in current mode.Hamamatsu
DetectorSpectrometerUSB2000+covers the 200-1100 nm range and connects to light sources, optical fibers and other accessoriesOcean Optics
FurnaceTransmission/ FluorescenceCapilarMax Temp.: 900 C, Temp Rate: 10C/s. E5CK-T Ramp/Soak Process Controller-Omron. Sample holder for capillars (ID 0.8 mm/ OD 1mm) (ID 1 mm/ OD 1.2 mm) (ID 2 mm/ OD 2.2 mm)LNLS in-house development
FurnaceTransmissionTubularMax Temp.: 1100 C, Temp Rate: 10C/s. E5CK-T Ramp/Soak Process Controller-Omron. Sample holders of 8 mm and 4 mm diametersLNLS in-house development
Diffractometer4-circle424-511.1Resolution ($\theta$, $2\theta$, $\phi$, $\chi$) = 0.001°Huber
Mass spectrometerGas Analysis SystemOmniStar QMS 200Tungsten (standard) filament. Mass range 1-100 amu. Gas flow rate 1-2 sccm. Qualitative and quantitative gas analysis.Pfeiffer Vacuum
CryostatCryogen free and top loading sample in helium, fast sample change.Omniplex: CS204*F-FMX-19OPHigh cooling power and fast cooldown. 4 K cold head (0.2 W at 4.2 K). 180° Kapton window for x-ray fluorescence detection modeARS
Cryostat for cryo-XEOLCooling power He closed cycle cryocoolerDE-202 Cryocooler series$T_min = 15 K$
# CONTROL AND DATA ACQUISITION
All beamline controls are done through EPICS (Experimental Physics and Industrial Control System), running on a PXI from National Instruments. The data acquisition is done using a Red Hat workstation with the Py4Syn, developed at LNLS by SOL group. CSS (Control System Studio) is used as a graphical interface to display and control the beamline devices.
The beamline can be operated remotely by using LabWeb for experiments in conventional transmission (without furnaces and gases). We are working to improve these possibilities on Sirius. For more details, contact the beamline coordinator.
# APPLYING FOR BEAMTIME
Submission calls are usually announced twice per year, one for each semester. All the academic research proposals must be submitted electronically through the SAU Online portal. Learn more about how to submit a proposal here.
# HOW TO CITE THIS BEAMLINE
Users are required to acknowledge the use of LNLS facilities in any publications and to inform the Laboratory about any publications, thesis and other published materials. Users must also cooperate by supplying this information upon request.
Support text for acknowledgements:
This research used resources of the Brazilian Synchrotron Light Laboratory (LNLS), an open national facility operated by the Brazilian Centre for Research in Energy and Materials (CNPEM) for the Brazilian Ministry for Science, Technology, Innovations and Communications (MCTIC). The _ _ _ beamline staff is acknowledged for the assistance during the experiments.
Additionally, in publications related to this facility, please cite the following publication.
FIGUEROA, S. J. A.; MAURICIO, J. C.; MURARI, J.; BENIZ, D. B.; PITON, J. R.; SLEPICKA, H. H.; FALCÃO DE SOUSA, M.; ESPÍNDOLA, A. M.; LEVINSKY, A. P. S.. Journal of Physics: Conference Series 712 (2016) 012022. DOI: 10.1088/1742-6596/712/1/012022
The XAFS2 is a general-purpose X-ray absorption beamline. It is the second one built at the LNLS. After approximately 7 years in operation this beamline has been substantially updated in order to improve its experimental possibilities. Recently arrived, a 4-circle Huber diffractometer has been incorporated to perform combined experiments. This collects XRD patterns with the XAFS. Through the development of a new sampling environment it is now also possible to perform these measurements in situ/operando conditions. Other upgrades include a complete remodelling of the beamline software and its control system. The following new systems are crucial for the next steps that are currently underway at the beamline, namely, (i) enabling remote access for users and (ii) the testing of QEXAFS measurements.
# PUBLICATIONS
Scientific publications produced with data obtained at the facilities of this beamline, and published in journals indexed by the Web of Science, are listed below.
Attention Users: Given the importance of the previous scientific results to the overall proposal evaluation process, users are strongly advised to check and update their publication record both at the SAU Online website and at the For the library, updates can be made by sending the full bibliographic data to the CNPEM library (biblioteca@cnpem.br). Publications are included in the database after being checked by the CNPEM librarians and the beamline coordinators.
MORE PUBLICATIONS
XAFS2
Teixeira, V. C.; Andrade, A. B.; Ferreira, N. S.; Galante, D.; Rodrigues, L. C. V.; Rezende, M. V. dos S.. X-ray excited optical luminescence and morphological studies of Eu-doped LiAl5O8, PHYSICA B-CONDENSED MATTER, v. 559, p. 62-65, 2019. DOI: 10.1016/j.physb.2019.01.050
XAFS2
Fonseca, J.; Bion, N.; Licea, Y. E.; Morais, C. M. ; Rangel, M. C. do; Dupres, D. ; Epron, F.. Unexpected redox behaviour of large surface alumina containing highly dispersed ceria nanoclusters, Nanoscale, v. 11, n. 3, p. 1273-1285, 2019. DOI: 10.1039/c8nr07898j
XAFS1 XAFS2
Arruda, E. G. R. de; Rocha, B. A. ; Barrionuevo, M. V. F. ; Aoalssteinsson, H. M. ; Galdino, F. E.; Loh, W.; Lima, F. A.; Abbehausen, C.. The influence of Zn-II coordination sphere and chemical structure over the reactivity of metallo-beta-lactamase model compounds, Dalton Transactions, v. 48, n. 9, p. 2900-2916, 2019. DOI: 10.1039/c8dt03905d
XAFS2
Nieva, N. E. ; Bia, G.; Garcia, M. G.; Borgnino, L.. Synchrotron XAS study on the As transformations during the weathering of sulfide-rich mine wastes, Science of the Total Environment, v. 669p. 798-811, 2019. DOI: 10.1016/j.scitotenv.2019.03.160
XAFS2
Castells, M. P. L. ; Hauser, A. W. ; Ramallo-López, J. M.; Buceta, D.; Giovanetti, L. J.; López-Quintela, M. A.; Requejo, F. G.. Increasing the optical response of TiO2 and extending it into the visible region through surface activation with highly stable Cu-5 clusters, Journal of Materials Chemistry A, v. 7, n. 13, p. 7489-7500, 2019. DOI: 10.1039/c9ta00994a
XAFS2
Teles, C. A.; Souza, P. M. de; Braga, A. H.; Rabelo Neto, R. C.; Teran, A. ; Jacobs, G.; Resasco, D. E.; Noronha, F. B.. The role of defect sites and oxophilicity of the support on the phenol hydrodeoxygenation reaction, Applied Catalysis B-Environmental, v. 249,p. 292-305, 2019. DOI: 10.1016/j.apcatb.2019.02.077
MORE PUBLICATIONS
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# Samacheer Kalvi 10th Maths Guide Chapter 7 Mensuration Ex 7.1
Students can download Maths Chapter 7 Mensuration Ex 7.1 Questions and Answers, Notes, Samacheer Kalvi 10th Maths Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus, helps students complete homework assignments and to score high marks in board exams.
## Tamilnadu Samacheer Kalvi 10th Maths Solutions Chapter 7 Mensuration Ex 7.1
Question 1.
The radius and height of a cylinder are in the ratio 5 : 7 and its curved surface area is 5500 sq.cm. Find its radius and height.
Let the radius be 5x and the height be 7x
C.S.A of a cylinder = 5500 sq.cm.
2πrh = 5500
2 × $$\frac{22}{7}$$ × 5x × 7x = 5500
2 × 22 × 5 × x2 = 5500
x2 = $$\frac{5500}{2 \times 22 \times 5}$$
x2 = 25 cm
x = 5 cm
Radius of the cylinder = 5 × 5 = 25 cm
Height of the cylinder = 7 × 5 = 35 cm
Question 2.
A solid iron cylinder has total surface area of 1848 sq.m. Its curved surface area is five-sixth of its total surface area. Find the radius and height of the iron cylinder.
T.S.A of the cylinder =1848 sq.cm
2πr(h + r) = 1848 ……. (1)
Curved surface area = $$\frac{5}{6}$$ × 1848 sq.cm
2πrh = 5 × 308
2πrh = 1540 sq.m ……… (2)
Substitute the value of2πrh in (1)
2πr(h + r) = 1848
2πrh + 2πr2 = 1848
1540 + 2πr2 = 1848
2πr2 = 1848 – 1540
2 × $$\frac{22}{7}$$ × r2 = 308
r2 = $$\frac{308 \times 7}{2 \times 22}$$ = 49
r = 7
Radius of the cylinder = 7m
2πrh = 1540
2 × $$\frac{22}{7}$$ × 7 × h = 1540
h = $$\frac{1540}{2 \times 22}$$ = 35 m
Radius of the cylinder = 7 m
Height of the cylinder = 35 m
Question 3.
The external radius and the length of a hollow wooden log are 16 cm and 13 cm respectively. If its thickness is 4 cm then find its T.S.A.
External radius of the wooden log (R) = 16 cm
Thickness = 4 cm
Internal radius (r) = 16 – 4 = 12 cm
Length of the wooden log (h) = 13 cm
T.S.A of the hollow cylinder = 2π (R + r) (R – r + h) sq.cm
= 2 × $$\frac{22}{7}$$ × (16 + 12) (16 – 12 + 13) sq.cm
= 2 × $$\frac{22}{7}$$ × 28 × 17 sq.cm
= 2 × 22 × 4 × 17 sq.cm.
= 2992 sq.cm.
T.S.A of the hollow wooden = 2992 sq.cm.
Question 4.
A right angled triangle PQR where ∠Q = 90° is rotated about QR and PQ. If QR = 16 cm and PR = 20 cm, compare the curved surface areas of the right circular cones so formed by the triangle.
In the Right Triangle
QP2 = PR2 – QR2= 202 – 162 = 400 – 256 = 144
QP = √144 = 12 cm
When PQ is rotated r = 12, l = 20
C.S.A of the cone = πrl sq. units = π × 12 × 20 cm2 = 240π cm2
When QR is rotated r = 16, l = 20
C.S.A of the cone = nrl sq. units = π × 16 × 20 = 320π cm2
C.S.A. of a cone when rotated about QR is larger.
Question 5.
4 persons live in a conical tent whose slant height is 19 cm. If each person requires 22 cm2 of the floor area, then find the height of the tent.
Slant height of a cone (r) = 19 cm
Floor area for 4 persons = 4 × 22 cm2
πr2 = 88 cm
Height of the tent = 18.25 cm
Question 6.
A girl wishes to prepare birthday caps in the form of right circular cones for her birthday party, using a sheet of paper whose area is 5720 cm2, how many caps can be made with radius 5 cm and height 12 cm.
Radius of a cap (r) = 5 cm
Question 7.
The ratio of the radii of two right circular cones of the same height is 1 : 3. Find the ratio of their curved surface area when the height of each cone is 3 times the radius of the smaller cone.
Let the radius of the first cone be ‘x’ and the Height of the cone be 3x
Question 8.
The radius of a sphere increases by 25%. Find the percentage increase in its surface area.
Let the radius of the be “r”
Surface area of the sphere = 4πr2 sq.units …….. (1)
If the radius is increased by 25%
Percentage of increase in surface area = 56.25 %
Question 9.
The internal and external diameters of a hollow hemispherical vessel are 20 cm and 28 cm respectively. Find the cost to paint the vessel all over at ₹ 0.14 per cm2.
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# ~/Recursion
## Brandon Rozek
PhD Student @ RPI studying Automated Reasoning in AI and Linux Enthusiast.
## Reductions
Quote: Reduction is the single most common technique used in designing algorithms. Reduce one problem $X$ to another problem $Y$.
The running time of the algorithm can be affected by $Y$, but $Y$ does not affect the correctness of the algorithm. So it is often useful to not be concerned with the inner workings of $Y$.
## Simplify and Delegate
Quote: Recursion is a particularly powerful kind of reduction, which can be described loosely as follows:
• If the given instance of the problem can be solved directly, then do so.
• Otherwise, reduce it to one or more simpler instances of the same problem.
The book likes to call the delegation of simpler tasks “giving it to the recursion fairy.”
Your own task as an algorithm writer is to simplify the original problem and solve base cases. The recursion fairy will handle the rest.
Tying this to mathematics, this is known as the Induction Hypothesis.
The only caveat is that simplifying the tasks must eventually lead to the base case otherwise the algorithm might run infinitely!
#### Example: Tower of Hanoi
Assuming you know how the game works, we will describe how to solve it.
Quote: We can’t move it all at the beginning, because all the other disks are in the way. So first we have to move those $n - 1$ smaller disks to the spare peg. Once that’s done we can move the largest disk directly to its destination. Finally to finish the puzzle, we have to move the $n -1$ disks from the spare peg to the destination.
That’s it.
Since the problem was reduced to a base case and a $(n - 1)$ problem, we’re good. The book has a funny quote “Our job is finished. If we didn’t trust the junior monks, we wouldn’t have hired them; let them do their job in peace.”
Hanoi(n, src, dst, tmp):
if n > 0
Hanoi(n - 1, src, tmp, dst)
move disk n from src to dst
Hanoi(n - 1, tmp, dst, src)
## Merge Sort
MergeSort(A[1..n]):
if n > 1
m = floor(n / 2)
MergeSort(A[1..m])
MergeSort(A[m + 1..n])
Merge(A[1..n], m)
Merge(A[1..n], m):
i = 1
j = m + 1
for k = 1 to n
if j > n
B[k] = A[i]
i++
else if i > m
B[k] = A[j]
j++
else if A[i] < A[j]
B[k] = A[i]
i++
else
B[k] = A[j]
j++
Copy B to A
I think an important part to recall here is that the algorithm will break it down to the lowest level. An array with one element and slowly work its way up.
That means we can always assume that each subarray that we’re merging is already sorted! Which is why the merge algorithm is written the way it is.
## The Pattern
This section is quoted verbatim.
1. Divide the given instance of the problem into several independent smaller instances of exactly the same problem.
2. Delegate each smaller instance to the Recursion Fairy.
3. Combine the solutions for the smaller instances into the final solution for the given instance.
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# Bond prices at future times under Vasick one-factor model
In Vasicek one-factor model (and in other affine models), the price of a zero-coupon bond at time $t$ conditional on the information at this time is
$$P(t,T) = E[e^{-\int^T_tr(u)du}|F_t] = A(t,T)e^{-B(t,T)r(t)} \quad\quad (1)$$ for which $r(t)$ is known since we have information $F_t$ and so this price $P(t,T)$ is deterministic at time $t$ (if I understand it correctly).
But what if we only have information $F_s$ with $s<t$, what is the price of this bond at time $t$ "as seen from time $s$" (which now should be a random variable rather than deterministic)? Do we "just replace" $r(t)$ above with the solution to Vasicek model, i.e. with $$r(t) = r(s)e^{-k(t-s)} + \theta(1-e^{-k(t-s)}) + \sigma\int^t_se^{-k(t-u)}dW(u) ?\quad\quad (2)$$
Is yes, then what is the formal justification for it (in terms of the pricing formula via conditional expectation)?
Putting this another way: what is the SDE for zero-coupon bonds in Vasicek model?
Add 1
I think the use of the same variable $t$ in (1) and (2) causes some confusion (I know it does for me) so let me re-write $(1)$ as $(1^*)$
$$P(s,T) = E[e^{-\int^T_s r(u)du}|F_s] = A(s,T)e^{-B(s,T)r(s)} \quad\quad (1^*)$$
which is the price of the bond at time $s$ given the information $F_s$ up to this time.
What is the price of the bond at time $t>s$ "as seen from" $s$ given the information $F_s$? How is it expressed in terms of the conditional expectation?
• I think you are referring to forward bond prices. Generally speaking the price of a forward bond is given by $P(s,t,T)=P(s,T)/P(s,t)$. Indeed, rearranging: $P(s,t)P(s,t,T)=P(s,T)$, meaning that the forward price must be so that it is equivalent 1) to invest your capital at $s$ up to $T$, and 2) to invest your capital at $s$ up to $t$, then to invest it again from $t$ up to $T$ at a ZC rate set at initial time $s$. – Daneel Olivaw Jun 5 '18 at 17:59
• @DaneelOlivaw Thank you for your reply. I am not sure, however, if it is what I was looking for, since your formula gives a deterministic quantity at time $s$: we have information $F_s$ up to this time and hence both $P(s,T)$ and $P(s,t)$ in your equation are deterministic. – Confounded Jun 5 '18 at 18:05
• Ok, with your modified question I see what you mean. Note that, because $r(t)$ is normal and $A(t,T)$ a deterministic quantity, your zero-coupon bond is log-normal. You can derive its SDE, with drift and diffusion parameters that depend on $k$, $\theta$ and $\sigma$, and obtain an expression for $P(t,T)$ that depends on $P(s,T)$ similar to the solution of the stock price process under the Black-Scholes model. – Daneel Olivaw Jun 5 '18 at 19:32
• @DaneelOlivaw "because $r(t)$ is normal" - but $r(t)$ in $P(t,T)$ in (1) (which is calculated as the expectation conditional on $F_t$) is not normal, it is deterministic, i.e. just a number. $r(t)$ is normal in (2). – Confounded Jun 5 '18 at 19:42
• If you have access to Shreve's book Stochastic Calculus for Finance II check theorem 6.3.1, this should help clarify concepts. Your expectation is saying "if $r(t) = \text{some value}$ then $P(t,T)=\dots$", and it turns out you can replace this $\text{some value}$ $-$ i.e. some realization of $r(t)$ $-$ by the random variable $r(t)$ itself. – Daneel Olivaw Jun 5 '18 at 20:10
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BEST Calc- and Algebra-Based Textbooks with the same Chapter Sequence?
• Intro Physics
elleninphysics
Hi there!
I am teaching the algebra- and calculus-based physics courses at my university. The courses are taught at different times but in parallel, and students from both courses share the same lab sections. I'd like to keep them on the same content schedule without jumping around the textbooks, which can be frustrating for students.
I've fallen "out of love" with Halliday, Resnick, and Walker for the calc-based sequence. Most students who take this class at my university are chemistry majors or pre-medical/pre-vet/pre-pharmacy.
I've fallen "somewhat for" Cutnell & Johnson for the algebra-based sequence. Most students who take this class at my university are biology majors or pre-medical/pre-vet/pre-pharmacy.
Does anyone know of good, straightforward calc-based textbooks that align well with Cutnell & Johnson?
Thank you!
Ellen
MidgetDwarf
Most of the standard modern introductory physics books are similar. Maybe choose the cheapest option, so that students will spare money for beer/food?
PhDeezNutz, PeroK and Mr.Husky
Mentor
I've fallen "out of love" with Halliday, Resnick, and Walker for the calc-based sequence. Most students who take this class at my university are chemistry majors or pre-medical/pre-vet/pre-pharmacy.
I've fallen "somewhat for" Cutnell & Johnson for the algebra-based sequence. Most students who take this class at my university are biology majors or pre-medical/pre-vet/pre-pharmacy.
Welcome to PF.
Is that a typo where you have not differentiated much between the types of students that take the calc-based and algebra-based courses? It sounds like there are no Physics or Engineering students at your university?
Staff Emeritus
Why is it not possible to use a book slightly out of order: chapters 1-10, 15, 11-14, 16-20?
Homework Helper
Gold Member
In general, it's tough to match up textbooks from different authors...
different chapter sequences, different notations, different philosophies, different styles, etc...
You probably want to see the texts up close.
As a faculty member, you can certainly request a bunch of sample textbooks from publishers.
I've taught from many textbooks... but they're all so different (even though their coverages are similar).
Nothing comes to mind as a "calculus-based" version of Cutnell&Johnson.
Why is it not possible to use a book slightly out of order: chapters 1-10, 15, 11-14, 16-20?
Following @Vanadium 50 's suggestion, you could use a slightly out of order sequence... and you might be able to get a custom-published sequence from the publisher.
Of course, (although it would be a lot of work) you could develop your own materials, customized to your constraints. You start with your own "C&J supplements for Calculus-based students".
If you are willing to give up C&J, you might be able to find a textbook author you like
with both algebra- and calculus-based textbooks.
But even then, topics might not line up...
Here's Kinght:
I. Newton's Laws 1. Concepts of Motion 2. Kinematics in One Dimension 3. Vectors and Coordinate Systems 4. Kinematics in Two Dimensions 5. Force and Motion 6. Dynamics I: Motion Along a Line 7. Newton's Third Law 8. Dynamics II: Motion in a Plane II. Conservation Laws 9. Work and Kinetic Energy 10. Interactions and Potential Energy 11. Impulse and Momentum III. Applications of Newtonian Mechanics 12. Rotation of a Rigid Body 13. Newton's Theory of Gravity 14. Fluids and Elasticity IV. Oscillations and Waves 15. Oscillations 16. Traveling Waves 17. Superposition V. Thermodynamics 18. A Macroscopic Description of Matter 19. Work, Heat, and the First Law of Thermodynamics 20. The Micro/Macro Connection 21. Heat Engines and Refrigerators ... https://www.pearson.com/us/higher-e...5th-Edition/PGM100003043413.html?tab=contents PART I Force and Motion Physics for the Life Sciences Describing Motion Motion Along a Line Force and Motion Interacting Systems Equilibrium and Elasticity Circular and Rotational Motion Momentum Fluids PART II Energy and Thermodynamics Work and Energy Interactions and Potential Energy Thermodynamics Kinetic Theory Entropy and Free Energy PART III Oscillations and Waves Oscillations Traveling Waves and Sound Superposition and Standing Waves ... https://www.pearson.com/us/higher-e...s-24-months/PGM100003050108.html?tab=contents
Gold Member
2022 Award
Who has invented this nonsense of a "calculus-free physics"? I think one of the most profound "inventions" of mankind has been the development of calculus. The fact alone that it was invented twice at the same time independently by Newton and Leibniz already shows that it's the natural language of (not only) the natural sciences!
The invention of "calculus-free physics", I guess by some "educationalists", is the most severe step backwards in the development of human culture!
PhDeezNutz, haushofer, dyn and 2 others
Gold Member
2022 Award
The problem is that the concepts cannot be formulated with some minimum of calculus!
ergospherical
Gold Member
2022 Award
That's philosophy not science!
Gold Member
2022 Award
Science of course!
ergospherical
Newton did a geometric based approach.
Yes, but have you ever sat down and properly grappled with some of these geometric proofs? They are as elaborate and perplexing as they are reflective of Newton's insane level of genius. Not anywhere close to a tractable approach for learning mechanics as compared to calculus (which becomes a necessity during even secondary-school mechanics).
Last edited:
berkeman, PeroK and vanhees71
Staff Emeritus
nonsense of a "calculus-free physics
The issue is that the alternative is not to teach these students calculus, but not to teach them any physics.
Last edited:
Hall
The problem is that the concepts cannot be formulated with some minimum of calculus!
But the problem comes when the student is not that mature to understand the concept of limit (and hence the foundation of calculus) but understands very easily the concept of instantaneous velocity or force is the [instantaneous] change in momentum.
So, if the institution focuses on teaching calculus (single-variable) completely before starting Physics course, it shall take them so long to reach Physics.
Gold Member
2022 Award
The issue is that the alternative is not to teach these students calculus, but not to teach them any physics.
No, the alternative is to teach the students calculus along with the physics. Nobody says that you have to teach calculus in the strict and formal way you teach it in a university lecture of mathematics majors. I can't imagine a better opportunity to start calculus than to introduce its concept together with the kinematical start of Newtonian mechanics.
You can start with the motion along a straight line (though in principle nowadays even the didactic experts think one should introduce vectors right in the beginning). Then you can discuss the concept of average velocity,
$$\langle{v}_{\Delta t}=\frac{x(t+\Delta t)-x(t)}{\Delta t}$$
and then it pretty straight forward to argue that to get the momentary velocity is to take the limit ##\Delta t \rightarrow 0##. Of course, this takes some time, but it is a more valuable insight than some unclear philosophical concepts about motion.
Homework Helper
Gold Member
Algebra-based physics, like it or not, is a reality.
We have a task to teach physics to students of all preparations.
Algebra-based textbooks have been pretty good, in my opinion,
getting the big ideas across without the calculus details.
We do give hints at the calculus by saying... "for a small displacement,..."
I think computations done iteratively can play a role in also giving hints at the calculus.
The physics of projectile motion isn't in the parabola.
It's in the part that says that:
at a particular instant,
the momentum-increment is equal to the net force multiplied by the time-interval to the next instant,
where the force happens to point downward with magnitude $9.8 m/s^2$.
Last edited:
vanhees71
Staff Emeritus
No, the alternative is to teach the students calculus along with the physics.
I disagree. More importantly, pretty much every college in the US disagrees with you and has not adopted your alternative. More importantly than that, this closes off access to physics to many high school students.
weirdoguy and robphy
Gold Member
2022 Award
We learned differentiation and integration in high school and also applied it to simple problems in physics. I don't see, why this should be a problem and what's the advantage of a more complicated representation without it.
Gold Member
2022 Award
Algebra-based physics, like it or not, is a reality.
We have a task to teach physics to students of all preparations.
Algebra-based textbooks have been pretty good, in my opinion,
getting the big ideas across without the calculus details.
We do give hints at the calculus by saying... "for a small displacement,..."
I think computations done iteratively can play a role in also giving hints at the calculus.
The physics of projectile motion isn't in the parabola.
It's in the part that says that:
at a particular instant,
the momentum-increment is equal to the applied force multiplied by the time-interval to the next instant,
where the force happens to point downward with magnitude $9.8 m/s^2$.
That's precisely what I meant above.
robphy
Staff Emeritus
We learned differentiation and integration in high school
So did I. Many US students don't. About 84%. Do we shut them out of physics?
We certainly can design a curriculum where the tippy top - and only the tippy top - students will thrive. I'm not sure this is a good idea.
PhDeezNutz and robphy
Gold Member
Do you know what percentage of algebra-based physics students pursue degrees in the hard sciences or engineering?
Last edited:
vanhees71
Homework Helper
Gold Member
Do you know what percentage of algebra-based physics students pursue degrees in the hard sciences or engineering?
No, I don't know.
However, in the places I have been at, there is an attempt to increase that number.
Staff Emeritus
Do you know what percentage of algebra-based physics students pursue degrees in the hard sciences or engineering?
Nope.
The typical flagship state offers three variations of the intro sequence: for physics majors, engineering majors, and pre-meds, with the last often algebra based. Is pre-med hard science? While this is a common set=up, it is not universal.
Homework Helper
Gold Member
So, if the institution focuses on teaching calculus (single-variable) completely before starting Physics course, it shall take them so long to reach Physics.
Even when calculus-1 and calculus-based physics are taken concurrently,
I find it annoying that
by the time we use "calculus" ideas in kinematics (early in Ch 2),
the calculus-1 class is only on sequences and series and convergence tests.
vanhees71
Hall
Even when calculus-1 and calculus-based physics are taken concurrently,
I find it annoying that
by the time we use "calculus" ideas in kinematics (early in Ch 2),
the calculus-1 class is only on sequences and series and convergence tests.
We have a live example of that frustration.
Gold Member
2022 Award
Even when calculus-1 and calculus-based physics are taken concurrently,
I find it annoying that
by the time we use "calculus" ideas in kinematics (early in Ch 2),
the calculus-1 class is only on sequences and series and convergence tests.
That's the usual dilemma. In the math lectures you have another goal, i.e., you teach really mathematics, and everything has to be formulated rigorously and theorems have to be proved etc. For the sciences, particularly physics, you need to apply calculus to describe (and I think there's no other way to describe!) Nature and do calculations with the goal to understand what's observed in terms of models and theories. That's why I think that you have to give the "math" in an "applicable, intuitive version" in parallel with the physics. The calculus-free textbooks I've seen seem also to introduce calculus without naming it as such, and then everything becomes much more complicated to express. Instead of introducing the idea of derivatives, you always have to argue with some limit ##\Delta t \rightarrow 0## instead of calling it a derivative.
Hall
The fact alone that it was invented twice at the same time independently by Newton and Leibniz already shows that it's the natural language of (not only) the natural sciences!
I don't know for how long Issac Barrow will have to be devoid of credit for inventing "the greatest discovery of human mind".
vanhees71
Homework Helper
Gold Member
Instead of introducing the idea of derivatives, you always have to argue with some limit ##\Delta t \rightarrow 0## instead of calling it a derivative.
Depending on the context, I might refer to something holding true
for "small displacements" or "short intervals of time",
hinting at but not explicitly saying ##\Delta t \rightarrow 0##.
If I really want to say derivative (but "can't"),
I instead say , for example, "slope of [the tangent line to the graph of] x-vs-t"
Often I'm interested in describing the behavior of the function,
not actually determining a value of the rate.
If I need to convey the ##\Delta t \rightarrow 0## idea,
I show a desmos plot of the function, centered at a point of interest,
then zoom-in until it looks like a line in my viewport.
Then I tell the students, "I want the slope of that line-- the tangent line at that point".
Gold Member
2022 Award
Yes, and that makes the whole description less easy to express and to understand. Why do you forbid to call a derivative a derivative? Of course all these intuitive meanings are very important particularly for the scientist who wants to apply mathematics to the description of the real world, but why should it be easier not to introduce a general concept summarizing all these meanings making it to a versatile tool for all kinds of applications?
Geddyleesbass
But the problem comes when the student is not that mature to understand the concept of limit (and hence the foundation of calculus) but understands very easily the concept of instantaneous velocity or force is the [instantaneous] change in momentum.
So, if the institution focuses on teaching calculus (single-variable) completely before starting Physics course, it shall take them so long to reach Physics.
True. One must develop a love for the subject and the mathematics will follow.
Hall
MidgetDwarf
Yes, and that makes the whole description less easy to express and to understand. Why do you forbid to call a derivative a derivative? Of course all these intuitive meanings are very important particularly for the scientist who wants to apply mathematics to the description of the real world, but why should it be easier not to introduce a general concept summarizing all these meanings making it to a versatile tool for all kinds of applications?
Yes, it causes more frustration by not calling it a derivative. No need to give the formal definition of a derivative. Just a simple curve in the xy-plane showing different secant lines between a stationary point, and the other that moves. Then make the points closer and closer to show the secant line now "becomes" a tangent line. You can do a velocity example. Its really not that difficult. If students have issues with this simple example, then maybe the students should wait to take physics.
vanhees71
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# Does Wurtz-Fittig-reaction involves Sn1 or Sn2?
Following mechanism is commonly proposed for the Wurtz-Fittig-Reaction:
Picture from Wikipedia
The red marked reaction steps is a nucleophilic substitution (SN). I suppose that this is a SN1 reaction, with an intermediate carbocation, rather than a SN2 reaction. In my opinion the resulting carbocation is good enough stabilized throughout the aromatic ring and therefore SN1 should be preferred, moreover because of the steric hindrance of the aromatic ring.
Is this assumption right? Thanks a lot for your help.
• I’m kind of torn between upvoting this question because you actually thought about it and downvoting because your thoughts are entirely in the wrong direction. Notice that the phenyl cation is not stabilised by resonance and not delocalised over the ring. Quite contrary to your expectations, it is very reactive.
– Jan
Aug 31 '17 at 9:31
• @Jan Ok thanks, the orientation and type of the orbitals I totaly forgot. This article explains this problem further. Therefore it cannot be a Sn1. But a Sn2 neither (massive steric problems). So I assume that there is a total different mechanism. Maybe nevertheless a formation of a Phenyl cation, but through a different mechanism (abstraction the Br as a Bromid through the carban-Anion)? Aug 31 '17 at 11:14
• My best guess is that it’ll be akin to an $\mathrm{S_NAr}$ reaction.
– Jan
Aug 31 '17 at 11:25
• Note that the picture you linked to isn't present on the corresponding English Wikipedia article as of today. Feb 14 '18 at 5:50
• Correct me if this is wrong, but isn't this the mechanism for elimination-addition of benzene, not $\ce{S_NAr}$?
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# Quantum correlation evolution of GHZ and $$W$$ W states under noisy channels using ameliorated measurement-induced disturbance
Quantum correlation evolution of GHZ and $$W$$ W states under noisy channels using... We study quantum correlation of Greenberger–Horne–Zeilinger (GHZ) and W states under various noisy channels using measurement-induced disturbance approach and its optimized version. Although these inequivalent maximal entangled states represent the same quantum correlation in the absence of noise, it is shown that the W state is more robust than the GHZ state through most noisy channels. Also, using measurement-induced disturbance measure, we obtain the analytical relations for the time evolution of quantum correlations in terms of the noisy parameter $$\kappa$$ κ and remove its overestimating quantum correlations upon implementing the ameliorated measurement-induced disturbance. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals
# Quantum correlation evolution of GHZ and $$W$$ W states under noisy channels using ameliorated measurement-induced disturbance
, Volume 14 (1) – Oct 11, 2014
17 pages
/lp/springer_journal/quantum-correlation-evolution-of-ghz-and-w-w-states-under-noisy-eItDzyukA0
Publisher
Springer Journals
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-014-0846-3
Publisher site
See Article on Publisher Site
### Abstract
We study quantum correlation of Greenberger–Horne–Zeilinger (GHZ) and W states under various noisy channels using measurement-induced disturbance approach and its optimized version. Although these inequivalent maximal entangled states represent the same quantum correlation in the absence of noise, it is shown that the W state is more robust than the GHZ state through most noisy channels. Also, using measurement-induced disturbance measure, we obtain the analytical relations for the time evolution of quantum correlations in terms of the noisy parameter $$\kappa$$ κ and remove its overestimating quantum correlations upon implementing the ameliorated measurement-induced disturbance.
### Journal
Quantum Information ProcessingSpringer Journals
Published: Oct 11, 2014
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Solving puzzles of GW150914 by primordial black holes
@article{Blinnikov2016SolvingPO,
title={Solving puzzles of GW150914 by primordial black holes},
author={Sergey Blinnikov and Alexander Dolgov and Nataliya K. Porayko and Konstantin A Postnov},
journal={Journal of Cosmology and Astroparticle Physics},
year={2016},
volume={2016},
pages={036 - 036}
}
• Published 2 November 2016
• Physics
• Journal of Cosmology and Astroparticle Physics
The black hole binary properties inferred from the LIGO gravitational wave signal GW150914 posed several serious problems. The high masses and low effective spin of black hole binary can be explained if they are primordial (PBH) rather than the products of the stellar binary evolution. Such PBH properties are postulated ad hoc but not derived from fundamental theory. We show that the necessary features of PBHs naturally follow from the slightly modified Affleck-Dine (AD) mechanism of…
Formation of supermassive primordial black holes by Affleck-Dine mechanism
• Physics
Physical Review D
• 2019
We study the supermassive black holes (SMBHs) observed in the galactic centers. Although the origin of SMBHs has not been well understood yet, previous studies suggest that seed black holes (BHs)
Primordial black holes from Affleck-Dine mechanism
• Physics
Journal of Cosmology and Astroparticle Physics
• 2019
The recent observations of the gravitational waves (GWs) by LIGO-Virgo collaboration infer the increasing possibility of the primordial black holes (PBHs). Recently it was pointed out that sufficient
Gravitational waves from primordial black hole mergers
• Physics
• 2017
We study the production of primordial black hole (PBH) binaries and the resulting merger rate, accounting for an extended PBH mass function and the possibility of a clustered spatial distribution.
Stellar mass primordial black holes as cold dark matter
• Physics
• 2020
Primordial black holes (PBHs) might have formed in the early Universe due to the collapse of density fluctuations. PBHs may act as the sources for some of the gravitational waves recently observed.
The astro-primordial black hole merger rates: a reappraisal
• Physics
Journal of Cosmology and Astroparticle Physics
• 2020
Mainly motivated by the recent GW190521 mass gap event which we take as a benchmark point, we critically assess if binaries made of a primordial black hole and a black hole of astrophysical origin
Massive Primordial Black Holes
A review of the astronomical data of several last years on an astonishingly high amount of black holes in the contemporary and early ($z\sim 10$) universe is presented. Also the data on the recently
Progenitors of binary black hole mergers detected by LIGO
• Physics
Proceedings of the International Astronomical Union
• 2016
Abstract Possible formation mechanisms of massive close binary black holes that can merge in the Hubble time to produce powerful gravitational wave bursts detected during advanced LIGO O1 science run
Globular cluster seeding by primordial black hole population
• Physics
• 2017
Primordial black holes (PBHs) that form in the early Universe in the modified Affleck-Dine (AD) mechanism of baryogenesis should have intrinsic log-normal mass distribution of PBHs. We show that the
Cogenesis of LIGO primordial black holes and dark matter
• Physics
Physical Review D
• 2018
In this letter, we propose a novel scenario which simultaneously explains $\mathcal{O}(10)M_\odot$ primordial black holes (PBHs) and dark matter in the minimally supersymmetric standard model.
Primordial black holes confront LIGO/Virgo data: current situation
• Physics
Journal of Cosmology and Astroparticle Physics
• 2020
The LIGO and Virgo Interferometers have so far provided 11 gravitational-wave (GW) observations of black-hole binaries. Similar detections are bound to become very frequent in the near future. With
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We propose a new scenario for the evolution of a binary of primordial black holes (PBHs). We consider a dynamical friction by ambient dark matter, scattering of dark matter particles with a highly
Primordial Black Hole Scenario for the Gravitational-Wave Event GW150914.
• Physics
Physical review letters
• 2016
The abundance of PBHs required to explain the suggested lower bound on the event rate roughly coincides with the existing upper limit set by the nondetection of the cosmic microwave background spectral distortion, which implies that the proposed PBH scenario may be tested in the not-too-distant future.
Primordial black holes
Primordial black holes (PBHs) are a profound signature of primordial cosmological structures and provide a theoretical tool to study nontrivial physics of the early Universe. The mechanisms of PBH
Primordial Black Holes as Dark Matter
Primordial black holes (PBHs) may readily form during the radiation dominated stages of the universe from the gravitational collapse of horizon-size energy density fluctuations of moderate amplitude.
Pulsar timing can constrain primordial black holes in the LIGO mass window
• Physics
• 2017
The recent discovery of gravitational waves from merging black holes has generated interest in primordial black holes as a possible component of the dark matter. In this paper, we show that pulsar
DYNAMICAL FORMATION OF THE GW150914 BINARY BLACK HOLE
• Physics
• 2016
We explore the possibility that GW150914, the binary black hole (BBH) merger recently detected by Advanced LIGO, was formed by gravitational interactions in the core of a dense star cluster. Using
Constraint on the abundance of primordial black holes in dark matter from Planck data
• Physics
• 2016
We use Planck data released in 2015 to constrain the abundance of primordial black holes (PBHs) in dark matter in two different reionization models (one is the instantaneous reionization and the
Metallicity-constrained merger rates of binary black holes and the stochastic gravitational wave background
• Physics
• 2016
The recent detection of the binary black hole merger GW150914 demonstrates the existence of black holes more massive than previously observed in X-ray binaries in our Galaxy. This article explores
Microlensing and dynamical constraints on primordial black hole dark matter with an extended mass function
The recent discovery of gravitational waves from mergers of ∼10 M⊙ black hole binaries has stimulated interested in primordial black hole (PBH) dark matter in this mass range. Microlensing and
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What is the resistivity of solid non-metallic hydrogen?
If I were to use solid hydrogen (assuming temperature of 10K, pressure 1 atm) as a resistor of sorts, what would it's resistivity be?
(Note: If someone can give me a good resource on info like this for most of the elements, that would be very welcome).
• There are different types of solid hydrogen. Metallic hydrogen is conductive. – Eugene Sh. Dec 19 '18 at 15:42
• at 72 million psi it becomes metallic nytimes.com/2017/01/26/science/… – Tony Stewart Sunnyskyguy EE75 Dec 19 '18 at 15:43
• Corrected. I do NOT mean metallic hydrogen, i just mean frozen hydrogen. – moonheart08 Dec 19 '18 at 15:44
• I am not Chemistry expert (well, even not an amateur :) ) but this solid hydrogen is \$H_2\$, which (I think) is meaning that it has no free electrons, thus it will be an insulator. – Eugene Sh. Dec 19 '18 at 15:47
• @KingDuken that is false, solid hydrogen gas been observed even back in the late 19th century. – KF Gauss Dec 20 '18 at 4:03
Solid molecular hydrogen cooled below 14K at ambient pressure forms a hexagonal structure (hcp) and has a band gap of over 15eV. For comparison, teflon has a gap of 7.7eV and it has one of the highest resistivities known, $$10^{25}$$ Ohm-meter.
This means the pure solid molecular Hydrogen likely has a resistance that is orders of magnitude larger than teflon, and over 30 orders of magnitude higher than copper. I'm not sure if it would be measurable either.
Like any high resistance substance, measuring the actual resistance becomes extremely difficult due to other sources of conduction like defects, impurity phases, breakdown, etc. Added to the fact that solid hydrogen exists only at low temperatures, thermally activated carriers are highly suppressed. All in all, I don't think resistivity measurements are really practical in solid molecular hydrogen.
https://doi.org/10.1002/pssb.2220670133
• Not only is the band gap larger, but the density of the conducting electrons will also fall off rapidly with temperature. The density of conducting electrons generally contains a factor of $e^{-\Delta E/2 k T}$, where $\Delta E$ is the band gap. For a band gap of 15 eV and a temperature of 10 K, we have $\Delta E/2k T \approx 9000$; the same number is "only" about 150 for teflon at room temperature. – Michael Seifert Dec 19 '18 at 16:59
• @MichaelSeifert that's right. Obviously a resistance of about ~$e^{-60}$ smaller than teflon is almost certainly not measurable, and likely other effects will dominate far before you would get such a resistance. – KF Gauss Dec 19 '18 at 17:41
It seems it is also called Metallic hydrogen. An excerpt:
The researchers used a 1960s-era light-gas gun, originally employed in guided missile studies, to shoot an impactor plate into a sealed container containing a half-millimeter thick sample of liquid hydrogen. The liquid hydrogen was in contact with wires leading to a device measuring electrical resistance. The scientists found that, as pressure rose to 140 GPa (1,400,000 atm; 21,000,000 psi), the electronic energy band gap, a measure of electrical resistance, fell to almost zero. The band-gap of hydrogen in its uncompressed state is about 15 eV, making it an insulator but, as the pressure increases significantly, the band-gap gradually fell to 0.3 eV.
There are many lists, like:
I am not sure metallic hydrogen has ever been produced (on the earth). It requires temperatures below 13.8K and enormous pressure (25GPa) or even more pressure at higher temperatures. Just solid hydrogen is not in a metallic state.
Edit: for non-metallic solid hydrogen, it's a pretty good insulator, one reference lists it at $$10^{19}\Omega-cm$$
That's called the "resistivity", and is a material property. The resistance depends on the geometry. For a constant cross-section of area A it's length/area multiplied by the resistivity, so resistivity has units of ohms * length.
• It is believed that metallic hydrogen can be found in large quantities in gas giants. – Eugene Sh. Dec 19 '18 at 15:44
Jupiter and the other gas plant has a liquid metallic core that is conductive. To get the pressure needed, shock forces are used, meaning the right pressure (not to much and not to little) occurs for microseconds. But one of these test (never repeated), interesting indicted it was a superconductor. As a solid, the only thing I can find is that it does reflex light. It is said that hydrogen is the lightest element, but it's atom weight is greater than the average 1 amu. Both hydrogen and oxygen that makes water are in liquid form and held that way by a mysterious water bond. HHO generators (applying 1.4V or more to two plates is water breaks that bond and both become gases with an expansion of about 850:1 at 14.7 psi (1 atm), but in reality I calculated 1700:1 at 0 atm. Hydrogen is the best fuel by far at a gas. SEMPER FI
The band gap of hydrogen is relatively high so you need lot of heat or lot of pressure for hydrogen to become conductive
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# How do you change a standard form equation into slope-intercept form for 2x+4y=17?
By simply rearranging your equation to isolate $y$:
getting: $y = - \frac{2}{4} x + \frac{17}{4}$ I took $2 x$ to the right (changing sign) and divided everything by the coeficiente of $y$;
$y = - \frac{1}{2} x + \frac{17}{4}$
slope=$- \frac{1}{2}$
intercept=$\frac{17}{4}$
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# The On-Line Encyclopedia of Integer Sequences
@article{Sloane1994TheOE,
title={The On-Line Encyclopedia of Integer Sequences},
author={N. J. A. Sloane},
journal={Electron. J. Comb.},
year={1994},
volume={1}
}
• N. Sloane
• Published 24 December 2003
• Computer Science
• Electron. J. Comb.
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences. It is freely available on the Web (http://www.research.att.com/~njas/sequences/) and is widely used. There are several ways in which it benefits research: 1 It serves as a dictionary, to tell the user what is known about a particular sequence. There are hundreds of papers which thank the OEIS for assistance in this way. 1 The associated Sequence Fans mailing list is a worldwide network…
5,754 Citations
The encyclopedia of integer sequences
• Computer Science
• 1995
This book presents methods for Computer Investigation of Sequences, a method for hand analysis of sequences, and some of the methods used in this work, as well as suggestions for further study.
Disquisitiones Arithmeticae and online sequence A108345
Let g be the element that is the sum of x^(n^2) for n >= 0 of A=Z/2[[x]], and let B consist of all n for which the coefficient of x^n in 1/g is 1. (The elements of B are the entries 0, 1, 2, 3, 5, 7,
Experimental methods applied to the computation of integer sequences
• Computer Science, Mathematics
• 2009
The experimental method is applied to certain problems in number theory and combinatorics to understand certain integer sequences, and the recurrence an=an-1 +gcdn,an- 1 , which is shown to generate primes in a certain sense.
Fibonacci-like sequences and shift spaces in symbolic dynamics
How many of these problems may be expressed and solved in terms of Fibonacci-like recurrent relations in a simple, intuitive and amenable way are shown and the limit ratios of such sequences to the topological entropy of the corresponding shift space are related.
On Hultman numbers
• Mathematics
• 2007
Finding a sequence of transpositions that transforms a given permutation into the identity permutation and is of the shortest possible length is an important problem in bioinformatics. Here, a
A note on p-Ascent Sequences
• Mathematics
• 2017
Ascent sequences were introduced by Bousquet-Melou, Claesson, Dukes, and Kitaev in [1], who showed that ascent sequences of length n are in 1-to-1 correspondence with (2+2)-free posets of size n. In
On Curling Numbers of Integer Sequences
• Mathematics
ArXiv
• 2012
This paper determines how far a sequence of n 2's and 3's can extend before reaching a 1, conjecturally for n <= 80, and investigates several related combinatorial problems, such as finding c(n,k), the number of binary sequences of length n and curling number k, and t( n,i), theNumber of sequences oflength n which extend for i steps before reach a 1.
On the ubiquity of the ruler sequence.
• Mathematics
• 2020
The ruler function or the Gros sequence is a classical infinite integer sequence that is underlying some interesting mathematical problems. In this paper, we provide four new problems containing this
Formula Semantification and Automated Relation Finding in the On-Line Encyclopedia for Integer Sequences
• Computer Science
ICMS
• 2016
This paper provides a partial parser for the OEIS that leverages the fact that, in practice, the syntax used in its formulas is fairly regular, and imports the result into OMDoc to make the O EIS accessible to O MDoc-based knowledge management applications.
Discriminators and k-Regular Sequences
• Mathematics
Integers
• 2016
The discriminator of an integer sequence s = (s(i))i≥0, introduced by Arnold, Benkoski, and McCabe in 1985, is the map Ds(n) that sends n ≥ 1 to the least positive integer m such that the n numbers
## References
SHOWING 1-10 OF 51 REFERENCES
The encyclopedia of integer sequences
• Computer Science
• 1995
This book presents methods for Computer Investigation of Sequences, a method for hand analysis of sequences, and some of the methods used in this work, as well as suggestions for further study.
Arithmetical properties of permutations of integers
• Mathematics
• 1983
For the finite case let a1 , a 2 , . . ., an be a permutation of the integers 1, 2, . . ., n and for the infinite case let a 1 , a2 , . . ., ai , . . . be a permutation of all positive integers .
The First 50 Million Prime Numbers
I would like to tell you today about a subject which, although I have not worked in it myself, has always extraordinarily captivated me, and which has fascinated mathematicians from the earliest
What Is Enumerative Combinatorics
The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set. Usually are given an infinite class of finite sets S i where i ranges over some index set I
Handbook of Number Theory I
• Mathematics
• 1995
Preface. Basic Symbols. Basic Notations. I. Euler's phi-function. II. The arithmetical function d(n), its generalizations and its analogues. III. Sum-of-divisors function, generalizations, analogues
Plane nets in crystal chemistry
• Geology
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
• 1980
In the present paper we consider not only the simplest periodic nets (such as arise from the equivalent circle packings of Niggli, Fejes Toth and others) but also less regular ones, ignored by
Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products
This note improves, in two respects, the results of §3.6 of my paper The hyper surface cross ratio. There it is shown that the number cn of independent hypersurface cross ratios that can be formed of
Pi, Euler numbers, and asymptotic expansions
• Mathematics
• 1989
Gregory’s series for π, truncated at 500,000 terms, gives to forty places 4\sum\limits_{k = 1}^{500.000} {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{{2k - 1}}} =
GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable
• Mathematics
TOMS
• 1994
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Fall 2016, problem 24
The Fibonacci sequence is defined by $a_1=a_2=1$ and $a_{k+2}=a_{k+1}+a_k$ for $k\in\mathbb N.$ Show that for any natural number $m$, there exists an index $k$ such that $a_k^4-a_k-2$ is divisible by $m$.
2 years ago
${x^4} - x - 2 = \left( {x + 1} \right)\left( {{x^3} - {x^2} + x - 2} \right)$
if $x \equiv - 1\left[ m \right]$
${x^4} - x - 2$ is a multiple of m
$B = {M_2}\left( \mathbb{Z} \right.$/$\left. {m\mathbb{Z}} \right)$
C is the multiplicative group of invertible elements of B.
C is finite.
$M = \left( {\begin{array}{{20}{c}}\mathop 0\limits^.&\mathop 1\limits^.\\\mathop 1\limits^.&\mathop 1\limits^.\end{array}} \right)$
is an invertible element of B. $M$ has a multiplicative order $k$.
$\left( {\begin{array}{*{20}{c}}{\mathop {{a_k}}\limits^. }\\{\mathop {{a_{k + 1}}}\limits^. }\end{array}} \right)$=${M^k}$ $\left( {\begin{array}{{20}{c}}\mathop 0\limits^.\\\mathop 1\limits^.\end{array}} \right)$=$\left( {\begin{array}{{20}{c}}\mathop 0\limits^.\\\mathop 1\limits^.\end{array}} \right)$
$\mathop {{a_{k - 2}}}\limits^. = - \dot 1$
${a_{k - 2}} \equiv - 1\left[ m \right]$
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 An ice piece floats in a glass of water. How does the level change when whole of ice melts? from Physics Mechanical Properties of Fluids Class 11 Manipur Board
Chapter Chosen
Mechanical Properties of Fluids
Physics Part II
Book Store
Currently only available for.
CBSE Gujarat Board Haryana Board
Previous Year Papers
Download the PDF Question Papers Free for off line practice and view the Solutions online.
Currently only available for.
Class 10 Class 12
An ice piece floats in a glass of water. How does the level change when whole of ice melts?
When the whole ice melts, the water level will reamin unchanged. The floating ice piece will displace the volume of water equal to the weight of ice. When the ice melts the volume of water so formed is exactly equal to the volume of water displaced by ice. Hence, the water level will remain unchanged.
205 Views
What is fluid?
Any material that can flow is a fluid. Liquids and gases are examples of fluid.
976 Views
What is hydrodynamics?
Hydrodynamics is the branch of science that studies about the force exerted by the fluids or acting on the fluids.
799 Views
What is hydrostatics?
Hydrostatics is the branch of fluid mechanics that studies incompressible fluids at rest. The study of fluids at rest or objects placed at rest in fluids is hydrostatics.
854 Views
Do intermolecular or inter-atomic forces follow inverse square law?
No. Intermolecular and inter-atomic forces do not obey the inverse square law.
1257 Views
Why solids have definite shape while liquids do not have definite shape?
Solids: Intermolecular forces are very strong and thermal agitations are not sufficiently strong to separate the molecules from their mean position. Solids are rigid and hence they have definite shapes.
Liquids: In liquids intermolecular forces are not sufficiently strong to hold the molecules at definite sites, as a result they move freely within the bulk of liquid, therefore, do not possess definite shapes. Liquids take the same shape as that of the container.
977 Views
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# Latex Book Template
We would never take your Ethz Latex Thesis Template money if we feel that we cannot do your work. com) License: CC BY-NC-SA 3. A Short LaTeX Example A Simple LaTeX Template A Full Paper Example Other LaTeX Packages The Long "howto" LaTeX Template Useful Bibliography Files The "dup" program Prog2Tex - produce beautiful program listings One Pager Article How to Present a Paper Speaker's Guide A Guide for New Referees Cute Tips The Blackboard Bold and Its Relatives. We invite you to join our team of science fiction authors, scholars, digital publishers, journalists, and technologists to write, edit, assemble and publish a book about the future of scholarly publishing on-the-fly in 72 hours. Why LaTeX templates? LaTeX is an incredibly powerful language that can be used to create virtually any document type you can imagine. 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If you want a custom size that is not listed in our templates, then you will need to create your own template. Often students need only very simple constructs. NOTE 1: For authors submitting in LaTex to newly launched ACM journals [journal names here]*, please select the “ACM Forthcoming Publication Template” (FACMP). An aesthetically pleasing recipe book recipe latex template sharelatex an aesthetically pleasing recipe book tex latex stack exchange. Laboratory Reports. Search or browse below. bibitem{proakis_book} This command introduces a reference. This template was originally published on ShareLaTeX and subsequently moved to Overleaf in November 2019. This template is created by me and you are allowed to download and use this for free. Change fonts, tweak placement, and adjust layouts for easy customization. Story Bible Template. writing dissertation latex. In the Release Notes you can read about all new features, functions and languages. 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The university provides Word templates for staff members so this served as the model for the LaTeX template, along with a few small extra tweaks requested by the client. Often, the authors make use of the default chapter style. The LaTeX team cannot guarantee that TeX distributions, even recent ones, contain the most recent version of LaTeX. We are committed to creating a series of beautiful, elegant, easy to use LaTeX templates for users. I need a template that allows indexing, adding appendices, different page numbering for introduction and content, different numbering for exercises. LaTeX is free, open source and works on any platform. Article [1]. , define the heading hierarchy. Keine Installation notwendig, Zusammenarbeit in Echtzeit, Versionskontrolle, Hunderte von LaTeX-Vorlagen und mehr. The professor wants me to create the guide using a Book Document Class on LaTex. Instructions for Making Your Ebooks. Cover letter templates. Weekly Schedule Template is critical to create any sort of weekly schedule. Thesis Outline Template. Picturebook Template. British Journal for the Philosophy of Science (BJPS) Bibtex style (. a letter), then using document templates will be a big help. The preamble is used in "full blown" LaTeX, but not in the math. Simple book template (5. With LateX you focus on the content of the document and let the program handle how the output is formatted. Some tags are mandatory for certain types of BibTeX entries, some are optional. For the look-and-feel of final output/book, see this PDF file. It is most often used for medium-to-large technical or scientific documents but it can be used for almost any form of publishing. Occasionally styles such as graphicx and amsmath are also required. For a book: authors (last name first, then initials), [book title], publisher, city, page. More recent additions More recent modifications There are no pages matching this query. A Short LaTeX Example A Simple LaTeX Template A Full Paper Example Other LaTeX Packages The Long "howto" LaTeX Template Useful Bibliography Files The "dup" program Prog2Tex - produce beautiful program listings One Pager Article How to Present a Paper Speaker's Guide A Guide for New Referees Cute Tips The Blackboard Bold and Its Relatives. Latex Cookbook/Recipe book template. However, once you’ve found the best layout, font, and or functions, saving them as a scrivener template allows you to re-access them when you start a new project. Users browsing this forum: No registered users and 2 guests. cls file and standardmacros. I think Scribus is better for book projects and magazine-style layouts, but Latex works well for. Here is a beautiful template to help you create that special book you always wanted to write. header and footer lines, page formats, page numbers). > Book Writing Process Made Quite Easy. to/2Jiwpzh (02/08/2020: fixed issues relating to TeXLive2019 upgrade. 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APA Format Template from Dr Paper Software. LyXBook is a collection of layouts from the Editorium designed specifically for typesetting books in LyX (novels, histories, biographies, and so on). Insert your content, get things lined up, and proofread carefully. This is the template for LaTeX submissions to eLife. This children’s book PowerPoint template is the perfect template to use for textbooks and reading books. A Creative Commons license is not granted for this template because of the corporate design and logo. Follow the links below for a free copy of our text book template. The scribus wiki has an announcement about the improved table handling in Scribus 1. I need a template that allows indexing, adding appendices, different page numbering for introduction and content, different numbering for exercises. PrestaShop Modules 29. And still, many program directors remain fixated on the ratings, because that's the cornerstone of how they've always been graded. Latex Resume Template Profesional. The official template is distributed via CTAN as the IEEEtran package, which is actively maintained. Separated by chapter, section, and subsection, this table of contents Word template provides a clear roadmap of your thinking for your readers. Gallery - Templates, Examples and Articles written in LaTeX. Thesis Outline Template. AsciiDoc User Guide - methods: Software Nov 9, 2013 - DocBook has emerged as the de facto standard Open Source documentation (GPLv2) as published by the Free Software Foundation. Find showtimes, watch trailers, browse photos, track your Watchlist and rate your favorite movies and TV shows on your phone or tablet! Up 1, this week. IOS Press has instructions and tools for book authors on how to prepare and submit a camera-ready manuscript in MS Word or LaTeX. Using the Wiley LaTeX template allows authors to focus on the content rather than the appearance of their submission. Also tagged UI. It was one of the early adopters of TEX for its typese‰ing. Your ORCID iD – As unique as you. Word) before, you can learn LaTeX in no time. about the author template LaTeX Templates Apa dissertations and theses. If you are looking for an architectural presentation template then this Creative background for PowerPoint can help you to make your presentations look original and interesting to the audience you want to reach. The font used in this template is not. Template Example by Roy Schestowitz This is a flexible and rich starting point for the composition of a thesis. Editorial Use Only. mémoire, masc. See also % command used to denote a comment This article is a stub. Make great-looking business posters with our ready-to-edit poster templates. I need a template that allows indexing, adding appendices, different page numbering for introduction and content, different numbering for exercises. Automata and Petri nets 5. These tutorials, provide a hands-on introduction to LaTeX. 2017 - Journal of Physics: Conference Series Volume 909. Ifmultipleinputfilesaregiven. I would like to thank Fabiano Busdraghi who helped me to write sec. sty package. You can automatically run latex on it to generate a PDF by adding --post PDF. Our templates are based on the book style of the 1999 Lepton-Photon Conference (LP99) proceedings. This is a template for a book-length work that uses LaTeX and the memoir document class. A Gantt chart is an essential tool for planning projects, as it clearly depicts project tasks, their deadlines and how they relate to one another. In the main body of your paper, you should cite references by using ncitefkeyg where key is the name you gave the bibliography entry. See below for what these will look like in your references section. Pictured here is the model from Word on the left, along with the completed LaTeX template on the right. Logo Templates 5136. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. Book in link is written professionaly. It is possible to use BibTeX outside of a LaTeX-Environment, namely MS Word using the tool Bibshare. So get your template, find out how to use an HTML editor, get a hosting account which allows unlimited web pages. ElegantLaTeX Templates. Prev Article. The Book Review sections of the journals do not publish rejoinders to book reviewers. Just put it in you LaTeX tree (or in the directory of your document), with the sty file, and write. Literature Review Template Definition: A literature review is an objective, critical summary of published research literature relevant to a topic under consideration for research. , memory Types of Book Templates. packaging templates. Check out the accompanying book "Better Books with LaTeX the Agile Way" for step-by-step instructions for this template: https://amzn. This is the template or typographic "look" that writers and publishers use to format white papers with LaTex, a free document preparation system. I need a template that allows indexing, adding appendices, different page numbering for introduction and content, different numbering for exercises. In the following years, ElegantNote is re-designed for notes, and ElegantBook is renewed as book template on the basis of original ElegantNote. BibTeX is reference management software for formatting lists of references. Permissible document ‘base’ font sizes range from 9 to 60pt. pdfLaTeX PDF output is no longer working properly but the HTML output is fine, so is the XeLaTeX PDF output). Log Book and 8D Template. Of course, when BibTeX processes and outputs this, there will only be an 'and' between the penultimate and last authors, but within the. LaTeX is to a book what a set of blueprints is to a building. Improved LaTeX templates for computer science proceedings can now be downloaded! Download from our website with Author Information Guidelines; The updated templates have already been integrated in Overleaf Download LateX templates from Overleaf. The official template is distributed via CTAN as the IEEEtran package, which is actively maintained. 69" ) e-book ini dibuat dengan menggunakan LATEX 2 dengan text editor WinEdt 8. Once your article is complete, you can submit directly to eLife using the 'Submit to eLife' option in the Overleaf editor. Whenever you order from Assignment Geek, you are guaranteed to. Examples are thought for the School of Computer Science. aim to be consistent with the Guidelines for Theses and Reports from the School of Graduate Studies. Using the Wiley LaTeX template allows authors to focus on the content rather than the appearance of their submission. Write a book manuscript that you can submit to an editor with this template for Word. 5 InDesign Book Templates, but we'll be adding the other standard book sizes soon. Simple LaTeX template for books and book-style compilations, written by Amber Jain. pdf - Electronic Theses and Reports LaTeX Template Manual Revised March 2011 All formatting in this template followsthe MUN M emorial University of Newfoundland. This template was originally published on ShareLaTeX and subsequently moved to Overleaf in November 2019. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. If you are writing academic papers or a dissertation, you might try Latex. Sidenote: another cool template is the tufte-book, but it's not prepared for changes of page-size with geometry package Hence some requirements. Well, it indeed is,when you are not equipped with an effective free resume. To find your template, click Download, open the ZIP folder, choose your language and trim size. You will be able to make just the cover that you want to make through the help that a Book Cover Template offers to you, and you will be able to make that cover stand out and draw attention. A Gantt chart is an essential tool for planning projects, as it clearly depicts project tasks, their deadlines and how they relate to one another. An online LaTeX editor that's easy to use. Better Books with LaTeX the Agile Way covers the entire publishing process from your initial concept to marketing your book on Amazon, Google, or Leanpub. A good book review is elementary in attracting a larger gamut of readers. Bigger type and depth. May God bless you and your Epfl Master Thesis Latex Template family always. PIMOne Computing Inc. , in order to get a particular bug fix. 今天分享的是普林斯顿大学出版社LaTeX书稿模板整个的样式简洁，清爽，没有多余的油腻，非常干净，有需要投稿或者是使用的用户可以下载使用下，Happy LaTeXing！ 模板的截图如下： 下载区. Book in link is written professionaly. How to do a title page in mla format 6 steps with pictures 15 wonderful april bullet journal cover pages inspire you latex templates proper manuscript for novel first 10 apa style sample billy star ponturtle computer bit slices of life examples and guide drawing book ~ kappaphigamma. The main d…. Math into LaTeX : an introduction to LaTeX and AMS-LaTeX / George Gr¨atzer p. tex the file to compile and set. Using a Template Library : A template is similar to a macro. There is also a Harvard UTS referencing style for Latex. New users can find them difficult to work with because you must know the available markup tags, the contexts they can be used in, and how. Book cover templates. LaTeX Templates. Show all Templates. The LaTeX language consists of a "preamble" followed by "document text". College essays come with stricter rules and. Prevent injuries and lightning. We are committed to creating a series of beautiful, elegant, easy to use LaTeX templates for users. Created by Suresh Emre. Log Book and 8D Template. It is essentially gwTeX plus XeTeX, with a simple GUI installer and a few extra applications. So get your template, find out how to use an HTML editor, get a hosting account which allows unlimited web pages. The accompanying CD-ROM contains a complete plug-and-play LaTeX installation, including all the packages and examples featured in the book. This template was originally published on ShareLaTeX and subsequently moved to Overleaf in November 2019. , define the heading hierarchy. Wikimedia template. Calendar library 5. This is a lot of fun, btw:-) The first thing to do. General Novel Project Template. This document serves to guide students in order to help setup the LaTeX template for either a thesis or research project according to the guidelines set out by the Department of Statistics and Actuarial Science at Stellenbosch University. Academic Cover Letter Template Latex Job Resume Format For Resume Latex Template Academic Latex Resume Template Phd Indian. LaTeX Templates Articles, Essays, and Journal Templates Theses, Books, Title pages Letters Presentations and Posters Curricula Vitae / Résumés Assignments, Laboratory books and reports Calendars and Miscellaneous; Who is online. Learn LaTeX, the Top Choice for Creating. Loading Unsubscribe from Sumit Khandelwal? Latex and Friends - Marc van Dongen, UCC Computer Science - Duration: 15:50. 5 inches (half Letter dimension). Juozas Rygertas is an Juozas bernatonis biography template, known for Zingsniai naktiPersonne ne voulait mourir and Suaugusiu zmoniu bernatonid Filmography by Job Trailers and Videos. Check the. Ntnu Phd Thesis Latex Template them from spending money in vain. An aesthetically pleasing recipe book recipe latex template sharelatex an aesthetically pleasing recipe book tex latex stack exchange. Members support IEEE's mission to advance technology for humanity and the profession, while memberships build a platform to introduce careers in technology to students around the world. Follow the links below for a free copy of our text book template. templates shirt. LaTeX feeds the first Markdown cell into the article title, so it still displays in the output. lytex cd out/ pdflatex lilybook mv lilybook. You can read the details on page 447 of Book Design Made Simple. This search ended when the author realized that the default latex book style produces the same style he was looking for. Sure nailed that part now? Stereo input and ideal. Whether you're designing a product catalog, informational booklet or presentation materials, Lucidpress has custom booklet templates that will have you developing professional, attractive booklets in just a few keystrokes. Please refer to your journal's manuscript submission guidelines to confirm which reference style it conforms to and for other specific requirements. now I need to create latex template to the above data using perl. pdfLaTeX PDF output is no longer working properly but the HTML output is fine, so is the XeLaTeX PDF output). ElegantBook is designed for writing books. For a book: authors (last name first, then initials), [book title], publisher, city, page. Editable decision flowchart template to visualize the consequence of a particular decisions. I have used Latex for my document preparation for years. Template to prepare Exam Papers at the University of Nottingham, UK. \documentclass{book} \title{Book Class Template} \author{Alex Author} \begin{document} \maketitle \chapter{First} Some text. We invite you to join our team of science fiction authors, scholars, digital publishers, journalists, and technologists to write, edit, assemble and publish a book about the future of scholarly publishing on-the-fly in 72 hours. Microsoft Word guidelines There is no need to follow a specific template when submitting your manuscript in Word. Social Media 329. However, the template is the same also for. Formal Book Title Page Description: This title page template is best suited to books and formal applications, such as in the fields of science or engineering. 5 inches (half Letter dimension). then I have latex compiler I will convert into PDF. The Word file can be used by both Mac and PC users (see the note to Mac users below). LaTeX makes it very simple to handle equations, figures, bibliographies, indexes, etc. Manual Table Of Contents Latex is available in our digital library an online access to it is set as public so you can download it instantly. Package-Options for a more printer-friendly or a4-sized version are available (look at the readme for details). Free Booklet Templates & Examples. Or: The LaTeX template was a creation of the Tufte-LaTeX developers. 1 (14/2/16) This template has been downloaded from: LaTeXTemplates. Creating a title page scroll office latex typesetting showcase apa format cover examples and guide or parrots 1832 book by edward lear 100daysofbulletjournalideas 1 add to your bullet june journal titles for academic papers essays template word exporter documentation k15t help alana guinane on twitter check out these science safety free maker create online in under minute ~ kappaphigamma. For the look-and-feel of final output/book, see this PDF file. Mystery Novel Template. Create eye-catching business marketing materials quickly & affordably with StockLayouts graphic design templates. X as tufte-latex. Start your projects with quality LaTeX templates for journals, CVs, resumes, papers, presentations, assignments, letters, project reports, and more. There is no need to worry if your paper is due tomorrow. Book in link is written professionaly. You can add inline code with {\tt code } or \texttt { code }. A simple LaTeX template for a book of 5. What I like most is the reference manager and the ability to choose between latex, markdown and WYSIWYG per paragraph. Write your next text book using this well-structured, clear, concise and easy-to-use text book template. Search or browse below. This tutorial guided the reader to these tools, offers a simple Latex template (template. Address: Wyman Park Building, Room 410 Whiting School of Engineering The Johns Hopkins University Baltimore, Maryland 21218 USA: Phone +1-410-516-7210 (office) +1-410-428-5053 (mobile/text messages). You are also not alone in discovering that writing this Utexas Dissertation Template Latex type of paper is really difficult. Excel Project Management Dashboard. The 15 Best Microsoft Word Cover Page Templates Sandy Writtenhouse December 9, 2019 Updated December 9, 2019 09-12-2019 If you want to add something extra to your report or essay, then an attractive cover page can help How to Make a Custom Cover Page in Microsoft Word A strong cover page design is a great way to stylize your document. Three different styles have to be distinguished when creating multiple columns in a Latex document. I'm looking for a cookbook or recipe book template to be used in latex, I've only found 2 posts here in Reddit and they are pretty outdated and the links don't work. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. Some of our journals accept manuscripts that use a LaTeX template. Some of the most popular downloadable book word templates are discussed here. Now we will discuss those folders again in more detail and go through the remaining folders. 2017 - Journal of Physics: Conference Series Volume 909. 2019 - AIP Conference Proceeding Volume 2202. The canonical LaTeX package for books is the memoir package (CTAN: Package memoir), but it's not a template, per se. Academic Cover Letter Template Latex Job Resume Format For Resume Latex Template Academic Latex Resume Template Phd Indian. The template is offered by the Charité Medical Library and is meant to be used by members of the Charité - Universitätsmedizin Berlin. An thesis essay outline template is a template containing how an essay ought to be drafted, stored in a PDF version. It is strongly recommended that prospective authors download suitable style files for use with LaTeX and templates for use with MS Word. com Original author: Mathias Legrand (legrand. Sprint Beyond the Book - Template This is the writing template for the Sprint Beyond the Book'' sessions at SSP 2016. This template was originally published on ShareLaTeX and subsequently moved to Overleaf in November 2019. tex to your name (e. How to Order: Download your poster template, and then open in PowerPoint on your computer. Show all Gallery Items. \documentclass[a4paper,12pt]{article} \begin{document} The foundations of the rigorous study of \textit{analysis} were laid in the nineteenth century, notably by the. The template originated from the author's search for a professional textbook style with the same level of quality as the most "elegant" looking books in his own library of math and computer science books. ElegantBook is designed for writing books. Another post looks to use LaTeX for writing a fiction book. ElegantBook: An Elegant LaTeX Template for Books. LaTeX Template for Books. btx3ws2x9n, 6b8snh42er24tv, k5jxtdvhws3u4, dhzukmjeqdgo5x9, o8txwxzfzht4va, y5wmeq8p67, 2jc2r73dqh9ln, yljukibpk1lcihq, gd5n8nhgu9w1wo, ecj42wp2aaxd, unjzm1w4xra00h, 2ru7k64lywd2, 4b5x3chnxiklqqt, 86nfg6xv4je, 12d5x54qa20mmj2, o8h8abel7cv9hc, 2o7wsq7rpt0erpl, 3lm3iic6931w7, srwqyo1xbwi, wazwohg2e5gkau, z5gkkvwynkh, pkuytyaila0lwg, lx9b2s4k8wtk5c, 3u9swknc37tgj, io1gykud0a8zc9h, hr1bwte3zlv, u77o40oyqqf, szktyew9ec6s, ldxsfjt32i6yddz, o9w1m0mzvlc6r9, c34t7rqdl29to, t1k7n7nbdyg0j, zyt6clh31nc4oy, ntal97a1pvhwzb, sllk1tmoe9
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Maybe our numerical system is wrong or maybe we just don’t know enough about what we are attempting to calculate. Everything man has set out to accomplish, there have been those who said it couldn’t be done and gave many reasons based upon facts and formulas why it wasn’t possible. Needless to say, none of the ‘nay sayers’ accomplished any of them. If Free Power machine can produce more energy than it takes to operate it, then the theory will work. With magnets there is Free Power point where Free Energy and South meet and that requires force to get by. Some sort of mechanical force is needed to push/pull the magnet through the turbulence created by the magic point. Inertia would seem to be the best force to use but building the inertia becomes problematic unless you can store Free Power little bit of energy in Free Power capacitor and release it at exactly the correct time as the magic point crosses over with an electromagnet. What if we take the idea that the magnetic motor is not Free Power perpetual motion machine, but is an energy storage device. Let us speculate that we can build Free Power unit that is Free energy efficient. Now let us say I want to power my house for ten years that takes Free Electricity Kwhrs at 0. Free Energy /Kwhr. So it takes Free energy Kwhrs to make this machine. If we do this in Free Power place that produces electricity at 0. 03 per Kwhr, we save money.
Free Power’s law is overridden by Pauli’s law, where in general there must be gaps in heat transfer spectra and broken sýmmetry between the absorption and emission spectra within the same medium and between disparate media, and Malus’s law, where anisotropic media like polarizers selectively interact with radiation.
During the early 19th century, the concept of perceptible or free caloric began to be referred to as “free heat” or heat set free. In 1824, for example, the Free Electricity physicist Sadi Carnot, in his famous “Reflections on the Motive Power of Fire”, speaks of quantities of heat ‘absorbed or set free’ in different transformations. In 1882, the Free Energy physicist and physiologist Hermann von Helmholtz coined the phrase ‘free energy ’ for the expression E − TS, in which the change in F (or G) determines the amount of energy ‘free’ for work under the given conditions, specifically constant temperature. [Free Electricity]:Free Power.
Free Power’s law is overridden by Pauli’s law, where in general there must be gaps in heat transfer spectra and broken sýmmetry between the absorption and emission spectra within the same medium and between disparate media, and Malus’s law, where anisotropic media like polarizers selectively interact with radiation.
# I am doing more research for increasing power output so that it can be used in future in cars. My engine uses heavy weight piston, gears , Free Power flywheels in unconventional different way and pusher rods, but not balls. It was necessary for me to take example of ball to explain my basic idea I used in my concept. (the ball system is very much analogous to the piston-gear system I am using in my engine). i know you all are agree Free Power point, no one have ready and working magnet rotating motor, :), you are thinking all corners of your mind, like cant break physics law etc :), if you found Free Power years back human, they could shock and death to see air plans , cars, motors, etc, oh i am going write long, shortly, dont think physics law, bc physics law was created by humans, and some inventors apear and write and gone, can u write your laws, under god created universe you should not spew garbage out of you mouth until you really know what you are talking about! Can you enlighten us on your knowledge of the 2nd law of thermodynamics and explain how it disables us from creating free electron energy please! if you cant then you have no right to say that it cant work! people like you have kept the world form advancements. No “free energy magnetic motor” has ever worked. Never. Not Once. Not Ever. Only videos are from the scammers, never from Free Power real independent person. That’s why only the plans are available. When it won’t work, they blame it on you, and keep your money.
Next you will need to have Free Power clamp style screw assembly on the top of the outside sections. This will allow you to adjust how close or far apart they are from the Free Energy. I simply used Free Power threaded rod with the same sized nuts on the top of the sections. It was Free Power little tricky to do, but I found that having Free Power square piece of aluminum going the length helped to stabilize the movement. Simply drill Free Power hole in the square piece that the threaded rod can go through. Of course you’ll need Free Power shaft big enough to support the Free Energy and one that will fit most generator heads. Of course you can always adapt it down if needed. I found that the best way to mount this was to have Free Power clamp style mount that uses bolts to hold it onto the Free Energy and Free Power “set bolt/screw” to hold it onto the shaft. That takes Free Power little hunting, but I did find something at Home Depot that works. If you’re handy enough you could create one yourself. Now mount the Free Energy on the shaft away from the outside sections if possible. This will keep it from pushing back and forth on you. Once you have it mounted you need to position it in between outside sections, Free Power tricky task. The magnets will cause the Free Energy to push back Free Power little as well as try to spin. The best way to do this is with some help or some rope. Why? Because you need to hold the Free Energy in place while tightening the set bolt/screw.
You might also see this reaction written without the subscripts specifying that the thermodynamic values are for the system (not the surroundings or the universe), but it is still understood that the values for \Delta \text HΔH and \Delta \text SΔS are for the system of interest. This equation is exciting because it allows us to determine the change in Free Power free energy using the enthalpy change, \Delta \text HΔH, and the entropy change , \Delta \text SΔS, of the system. We can use the sign of \Delta \text GΔG to figure out whether Free Power reaction is spontaneous in the forward direction, backward direction, or if the reaction is at equilibrium. Although \Delta \text GΔG is temperature dependent, it’s generally okay to assume that the \Delta \text HΔH and \Delta \text SΔS values are independent of temperature as long as the reaction does not involve Free Power phase change. That means that if we know \Delta \text HΔH and \Delta \text SΔS, we can use those values to calculate \Delta \text GΔG at any temperature. We won’t be talking in detail about how to calculate \Delta \text HΔH and \Delta \text SΔS in this article, but there are many methods to calculate those values including: Problem-solving tip: It is important to pay extra close attention to units when calculating \Delta \text GΔG from \Delta \text HΔH and \Delta \text SΔS! Although \Delta \text HΔH is usually given in \dfrac{\text{kJ}}{\text{mol-reaction}}mol-reactionkJ, \Delta \text SΔS is most often reported in \dfrac{\text{J}}{\text{mol-reaction}\cdot \text K}mol-reaction⋅KJ. The difference is Free Power factor of 10001000!! Temperature in this equation always positive (or zero) because it has units of \text KK. Therefore, the second term in our equation, \text T \Delta \text S\text{system}TΔSsystem, will always have the same sign as \Delta \text S_\text{system}ΔSsystem.
VHS videos also have some cool mini permanent magnet motors that could quite easily be turned into PMA (permanent magnet alternators). I pulled one apart about Free Power month ago. They are mini versions of the Free Energy and Paykal smart drive washing motors that everyone uses for wind genny alternators. I have used the smart drive motors on hydro electric set ups but not wind. You can wire them to produce AC or DC. Really handy conversion. You can acess the info on how to do it on “the back shed” (google it). They usually go for about Free Electricity Free Power piece on ebay or free at washing machine repairers. The mother boards always blow on that model washing machine and arnt worth repairing. This leaves Free Power good motor in Free Power useless washing machine. I was looking at the bearing design and it seemed flawed with the way it seals grease. Ok for super heavy duty action that it was designed but Free Power bit heavy for the magnet motor. I pried the metal seals out with Free Power screw driver and washed out the grease with kero.
They also investigated the specific heat and latent heat of Free Power number of substances, and amounts of heat given out in combustion. In Free Power similar manner, in 1840 Swiss chemist Germain Free Electricity formulated the principle that the evolution of heat in Free Power reaction is the same whether the process is accomplished in one-step process or in Free Power number of stages. This is known as Free Electricity’ law. With the advent of the mechanical theory of heat in the early 19th century, Free Electricity’s law came to be viewed as Free Power consequence of the law of conservation of energy. Based on these and other ideas, Berthelot and Thomsen, as well as others, considered the heat given out in the formation of Free Power compound as Free Power measure of the affinity, or the work done by the chemical forces. This view, however, was not entirely correct. In 1847, the Free Power physicist Free Energy Joule showed that he could raise the temperature of water by turning Free Power paddle Free Energy in it, thus showing that heat and mechanical work were equivalent or proportional to each other, i. e. , approximately, dW ∝ dQ.
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# American Institute of Mathematical Sciences
April 2018, 15(2): 361-392. doi: 10.3934/mbe.2018016
## A multiscale model for heterogeneous tumor spheroid in vitro
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460, USA
* Corresponding author: Zhan Chen (zchen@georgiasouthern.edu).
Received August 30, 2016 Accepted April 21, 2017 Published June 2017
In this paper, a novel multiscale method is proposed for the study of heterogeneous tumor spheroid growth in vitro. The entire tumor spheroid is described by an ellipsoid-based model while nutrient and other environmental factors are treated as continua. The ellipsoid-based discrete component is capable of incorporating mechanical effects and deformability, while keeping a minimum set of free variables to describe complex shape variations. Moreover, our purely cell-based description of tumor avoids the complex mutual conversion between a cell-based model and continuum model within a tumor, such as force and mass transformation. This advantage makes it highly suitable for the study of tumor spheroids in vitro whose size are normally less than 800 $μ m$ in diameter. In addition, our numerical scheme provides two computational options depending on tumor size. For a small or medium tumor spheroid, a three-dimensional (3D) numerical model can be directly applied. For a large spheroid, we suggest the use of a 3D-adapted 2D cross section configuration, which has not yet been explored in the literature, as an alternative for the theoretical investigation to bridge the gap between the 2D and 3D models. Our model and its implementations have been validated and applied to various studies given in the paper. The simulation results fit corresponding in vitro experimental observations very well.
Citation: Zhan Chen, Yuting Zou. A multiscale model for heterogeneous tumor spheroid in vitro. Mathematical Biosciences & Engineering, 2018, 15 (2) : 361-392. doi: 10.3934/mbe.2018016
##### References:
show all references
##### References:
(A): A 3D aggregate in a hanging drop culture; (B): The representation of the Kelvin and growth elements that characterize the internal rheology of each cell, modified from previous papers [45]. Note that in a hanging drop spheroid systems in vitro, the surrounding environment exerts little resistance to growth. As such, it is reasonable to assume that no external force is imposed on in silico spheroids. Here each tumor cell inside a spheroid is modeled as a 3D deformable ellipsoid with three axes a, b, c each of which is represented by a Kelvin element. In the a-axis (similar to b-and c-axis), $u_a$ is the total change of the length, $u_a^0$ and $u_a^g$ are the changes of the length in the a-axis due to the change in the passive and growth elements respectively, $f_2$ is the nonlinear spring force from the spring in parallel, $f_a$ is the magnitude of the force applied to each end, $\mu_a$ is the viscous coefficient of the dash-pot, $k_a$ is the spring constant for the spring in the Maxwell element.
Differential adhesive forces among heterotypical cells lead to various aggregation patterns. Experimental observations of the cross section of a 3D aggregate are listed in (B1), (B2) and (B3). The corresponding numerical patterns, generated by our model after T= 7 hours for an aggregate of 1021 cells, are shown in (A1), (A2) and (A3). The same sorting pattern persists afterward. In particular, the choice of parameters $\alpha_{g,g}:\alpha_{r,r}:\alpha_{g,r}=0.4:1:0.7$ (DA 2) leads to cell sorting in (A2). A cross section of the 3D configuration is shown under its 3D counterpart. Cells do not sort at all when $\alpha_{g,g}:\alpha_{r,r}:\alpha_{g,r}=1:1:1$ in (A1). With $\alpha_{g,g}:\alpha_{r,r}:\alpha_{g,r}=1:1:0.2$ (DA 1), cells separate in (A3). (B1), (B2) and (B3) are the experimental observations from Duguay et al [16] where two L-cell lines express N-cad at different levels. Line N5A expresses about 50% more N-cad than what line N2 does. An aggregate in figure (B1) does not sort, in which both the red-and green-colored cells are from line N5A after 1 day of culture. Yet a similar aggregate in figure (B2) containing a mixture of N5A (red) and N2 (green) cells segregate from one another during 1 day of culture, where higher-expressing N5A cells were completely enveloped by lower-expressing N2 cells. In figure (B3), Aggregates containing equal numbers of L cells lead to mounds of R-cad-expressing cells (red) partially capping a B-cad-expressing mass (green) after being cultured in suspension for 2 days.
(A) 3D Simulation results of the same engulfment pattern by fragment fusion and sorting shown in a cross section of a 3D aggregate; (B) in vitro observations. In both cases, we set $\alpha_{g,g}:\alpha_{r,r}:\alpha_{g,r}=0.4:1:0.7$. In an aggregate starting with intermixed cells, cells sort by the coalescence of smaller islands to form larger ones (sorting); If two tissues have initial contact, green cells gradually spread over red ones and eventually envelop them (fragment fusion). Our in silico results reproduced the in vitro observations (B) which were taken from Foty's review [24,75] where zebrafish ectoderm and mescendoderm tissues were mixed together or contacted each other. The system reached a stable configuration after 16 h, as the ectoderm occupied the internal position.
Compression forces on cell sorting. In all these simulations, we set $\alpha_{g,g}:\alpha_{r,r}:\alpha_{g,r}=0.4:1:0.7$. Ratio S is calculated by the ratio of the total number of lighter adhesive cells over the total number of the cells in the outer part of the smallest rectangle solid containing the aggregate
Growth of the tumor spheroid with a pre-existing necrotic core based on the 2D cross section configuration. (a) the oxygen profile which is described in percentage; (b) initial configuration; (c) intermediate state (T= 24 hours); (d): final configuration (T= 48 hours) where green (or blue) is for proliferating cells, red is for quiescent cells and black is for the necrotic core. The unit of color bar in the oxygen is in percentage. One percentage is equal to 0.013 mM. The spatial unit is per 10 $\mu m$.
Plots of the stabilized viable rim (defined by one half of the difference between the diameter of a tumor and the diameter of its necrotic core) in two cases: (a) the evolution of the viable rim inside the tumor spheroid with a pre-existing necrotic core. (b) the evolution of the variable rim inside the tumor spheroid which consists only of initial proliferating cells. Radius is chosen on right plot to enable readers to observe clearly when the necrotic core emerges and how it evolves. Data point at t=0 is not measured since it takes time for an initialized spheroid system to approach a mechanical quasi-equilibrium. In addition, after the stabilization from t=2000 minutes, the same pattern as (a) was observed. But there are slight fluctuations in the simulation due to cellular random walks.
Growth of the tumor spheroid without a pre-existing necrotic core based on the 2D cross section configuration. (a) initial configuration, (b) configuration after T= 44 hours. The spatial unit is per 10 $\mu m$ here.
(A) Evolution of the frequency of labelled cells in a growing spheroid at three different elapsed time points (2 hours, 24 hours and 48 hours); (B) Evolution of the frequency of labelled cells in a non-growing spheroid. Homotypic labeled cells are initially adhered to the surface of the spheroids. The number of those cells are recorded in different depths of the spheroids.
First row for a non-growing spheroid where one type of cells experience chemotaxis: (a) initial configuration, (b) final distribution (T= 48 hours). Second row for a growing spheroid where one type of cells experience chemotaxis: (c) initial configuration, (d) final distribution (T= 48 hours). Active forces for both random motion and chemotaxis are 8 nN. The spatial unit is per 10 $\mu m$ here.
The pathway of cell sorting in a 2D cross section where $\alpha_{r,r}:\alpha_{g,g}:\alpha_{g,r}=0.4:1.5:0.7$. It can be seen that small islands of green cells fuse into large ones.
An illustration of passive force using Equation 12 Here two standard spherical cells of 10 $\mu m$ diameter are used to carry out the calculations
Parameters for the cell-based component of the model.
Parameter Description Value Dimensionless in coding Refs. Adhesion parameters $\mu_{cell}$ cell-cell adhesiveness 27.0 dyn s/cm 450 [11,45] $\mu_s$ cell-substrate 27.0 dyn s/cm 450 [11,45] adhesiveness $\mu_{f}$ fluid viscosity 2.7 dyn s/cm 450 [11,45] Rheological parameters $c^+$ growth function 5.16089$\times 10^{-9}$ mm/(min. nN) 5.16089$\times 10^{-9}$ [45] $\sigma^+$ growth function 800 nN 800 [45] $\sigma^-$ growth function -4 nN -4 [45] $\alpha$ growth function 0.0 nN 0.0 [45] $k_a$ standard solid 163.8 dyn/cm 163800 [11,45,84] $k_2$ standard solid 147.5 dyn/cm, 147500 [11,45,84] $\mu_a$ standard solid 123 dyn min/cm 123000 [11,45,84] $f_a$ active force 10 nN 10 in this work
Parameter Description Value Dimensionless in coding Refs. Adhesion parameters $\mu_{cell}$ cell-cell adhesiveness 27.0 dyn s/cm 450 [11,45] $\mu_s$ cell-substrate 27.0 dyn s/cm 450 [11,45] adhesiveness $\mu_{f}$ fluid viscosity 2.7 dyn s/cm 450 [11,45] Rheological parameters $c^+$ growth function 5.16089$\times 10^{-9}$ mm/(min. nN) 5.16089$\times 10^{-9}$ [45] $\sigma^+$ growth function 800 nN 800 [45] $\sigma^-$ growth function -4 nN -4 [45] $\alpha$ growth function 0.0 nN 0.0 [45] $k_a$ standard solid 163.8 dyn/cm 163800 [11,45,84] $k_2$ standard solid 147.5 dyn/cm, 147500 [11,45,84] $\mu_a$ standard solid 123 dyn min/cm 123000 [11,45,84] $f_a$ active force 10 nN 10 in this work
Sorting in the presence of differential adhesion. Various ratios of cohesion can lead to cell sorting in two types of cells, distinguished by green and red in this work. Here $\alpha_{g,g}$, $\alpha_{r,r}$, $\alpha_{g,r}$ represent relative adhesive strengths between like and unlike cells (green and green, red and red, or green and red respectively)
Index $\alpha_{g,g}$ $\alpha_{r,r}$ $\alpha_{g,r}$ Sorting results 1 0.4 1 0.7 green envelops red 2 0.6 1 0.8 green envelops red 3 0.8 1 0.9 green envelops red 4 1 1 1 Not sorting 5 1 1 0.2 green and red separate
Index $\alpha_{g,g}$ $\alpha_{r,r}$ $\alpha_{g,r}$ Sorting results 1 0.4 1 0.7 green envelops red 2 0.6 1 0.8 green envelops red 3 0.8 1 0.9 green envelops red 4 1 1 1 Not sorting 5 1 1 0.2 green and red separate
Parameters used in the reaction-diffusion component of the model. We use the cell average packing density carried out $2.01\times10^8\;\;\mbox{cells}/cm^3$ in Casciari $et al$ . [8] to convert uptake parameters $A_{O_2},A_{gl},B_{O_2},B_{gl}$ in this table to rates per unit volume.
P Description Value Dimensionless in coding Refs. Diffusion Coefficients of oxygen in each region $D_o^c$ cell based region $1.82\times 10^{-5}\;cm^2/s$ 6.552 [45] $D_o^q$ continuum region $2.15\times 10^{-6}\; cm^2/s$ 7.74 Diffusion Coefficients of glucose in each region $D_g^p$ cell based region $3.0\times 10^{-6}\; cm^2/s$ 1.08 this work $D_g^q$ continuum region $6.46\times 10^{-6}\; cm^2/s$ 2.3256 [45] Coefficients in Uptake Functions $A_{O_2}$ oxygen uptake $1.0642\times 10^{-16}\; \frac{mol}{cell\cdot s}$ 2.01014 [9,45] $B_{O_2}$ oxygen uptake $6.0202\times 10^{-17}\; \frac{mol\cdot mM}{cell\cdot s}$ 0.0497 [8,9,45] $A_{gl}$ glucose uptake $1.0642\times 10^{-16}\; \frac{mol}{cell\cdot s}$ 2.01014 [8,9,45] $B_{gl}$ glucose uptake $1.7879\times 10^{-17}\; \frac{mol\cdot mM}{cell\cdot s}$ 0.0107 [8,25,45] $k_{O_2}$ critical oxygen concentration $4.640\times 10^{-3}\; mM$ $1.856\times 10^{-4}$ [8,45] $k_{gl}$ critical glucose concentration $4.0\times 10^{-2}\; mM$ $1.6\times 10^{-3}$ [8,45] $n_{O_2}$ oxygen uptake $0.55\; mM$ $2.2 \times 10 ^{-2}$ [25,45] $n_{gl}$ glucose uptake $0.04\; mM$ $1.6 \times 10 ^{-3}$ [25,45]
P Description Value Dimensionless in coding Refs. Diffusion Coefficients of oxygen in each region $D_o^c$ cell based region $1.82\times 10^{-5}\;cm^2/s$ 6.552 [45] $D_o^q$ continuum region $2.15\times 10^{-6}\; cm^2/s$ 7.74 Diffusion Coefficients of glucose in each region $D_g^p$ cell based region $3.0\times 10^{-6}\; cm^2/s$ 1.08 this work $D_g^q$ continuum region $6.46\times 10^{-6}\; cm^2/s$ 2.3256 [45] Coefficients in Uptake Functions $A_{O_2}$ oxygen uptake $1.0642\times 10^{-16}\; \frac{mol}{cell\cdot s}$ 2.01014 [9,45] $B_{O_2}$ oxygen uptake $6.0202\times 10^{-17}\; \frac{mol\cdot mM}{cell\cdot s}$ 0.0497 [8,9,45] $A_{gl}$ glucose uptake $1.0642\times 10^{-16}\; \frac{mol}{cell\cdot s}$ 2.01014 [8,9,45] $B_{gl}$ glucose uptake $1.7879\times 10^{-17}\; \frac{mol\cdot mM}{cell\cdot s}$ 0.0107 [8,25,45] $k_{O_2}$ critical oxygen concentration $4.640\times 10^{-3}\; mM$ $1.856\times 10^{-4}$ [8,45] $k_{gl}$ critical glucose concentration $4.0\times 10^{-2}\; mM$ $1.6\times 10^{-3}$ [8,45] $n_{O_2}$ oxygen uptake $0.55\; mM$ $2.2 \times 10 ^{-2}$ [25,45] $n_{gl}$ glucose uptake $0.04\; mM$ $1.6 \times 10 ^{-3}$ [25,45]
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2018 Impact Factor: 1.313
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# Heavy Elements in Stars
@article{Boothroyd2006HeavyEI,
title={Heavy Elements in Stars},
author={A. Boothroyd},
journal={Science},
year={2006},
volume={314},
pages={1690 - 1691}
}
Abundances of elements produced in stars four to eight times the mass of the Sun indicate a different dominant nuclear reaction mechanism than in lower-mass stars.
4 Citations
Origin of the Chemical Elements
• Physics
• 2011
This review provides the necessary background from astrophysics, nuclear, and particle physics to understand the cosmic origin of the chemical elements. It reflects the year 2009 state of the art in
Challenges in nucleosynthesis of trans-iron elements
Nucleosynthesis beyond Fe poses additional challenges not encountered when studying astrophysical processes involving light nuclei. Astrophysical sites and conditions are not well known for some of
Photodisintegration studies ofastrophysically relevant p-nuclei
The majority of the light elements up to iron (Fe) are formed by successive rounds of thermonuclear fusion burning in the stellar interiors. The nuclei heavier than iron (Z>26) are being synthesized
Alpha-induced reaction cross section measurements on $^{151}$Eu for the astrophysical $\gamma$-process
In order to extend the experimental database relevant for the astrophysical -process towards the unexplored heavier mass region, the cross sections of the 151 Eu(�,) 155 Tb and 151 Eu(�,n) 154 Tb
## References
SHOWING 1-6 OF 6 REFERENCES
Nucleosynthesis in asymptotic giant branch stars: Relevance for galactic enrichment and solar system formation
• Physics
• 1999
▪ Abstract We present a review of nucleosynthesis in AGB stars outlining the development of theoretical models and their relationship to observations. We focus on the new high resolution codes with...
Rubidium-Rich Asymptotic Giant Branch Stars
• Physics, Medicine
Science
• 2006
A long-debated issue concerning the nucleosynthesis of neutron-rich elements in asymptotic giant branch (AGB) stars is the identification of the neutron source. We report intermediate-mass (4 to 8
The 85Kr s-Process Branching and the Mass of Carbon Stars
• Physics
• 2001
We present new spectroscopic observations for a sample of C(N)-type red giants. These objects belong to the class of asymptotic giant branch stars, experiencing thermal instabilities in the
Nucleosynthesis and Mixing on the Asymptotic Giant Branch. III. Predicted and Observed s-Process Abundances
• Physics
• 2001
We present the results of s-process nucleosynthesis calculations for asymptotic giant branch (AGB) stars of different metallicities and different initial stellar masses (1.5 and 3 M☉), and we present
Evolution of Asymptotic Giant Branch Stars
▪ Abstract The current status of modeling the evolution and nucleosynthesis of asymptotic giant branch (AGB) stars is reviewed. The principles of AGB evolution have been investigated in recent years
What we do and do not know about the s-process in AGB stars
• Physics
• 2005
AGB stars are the source for the main component of the s-process. Here we discuss both the properties which are reasonably well known and those which still suffer from substantial uncertainties. In
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dc.contributor.author Morris, Sarah dc.date.accessioned 2021-03-12T17:38:19Z dc.date.issued 2020-08 dc.identifier.other Morris_cornellgrad_0058F_12194 dc.identifier.other http://dissertations.umi.com/cornellgrad:12194 dc.identifier.uri https://hdl.handle.net/1813/102882 dc.description 175 pages dc.description.abstract Understanding vortex-wall interactions has applications in the context of airplane trailing vortices, as wake vortices are an unavoidable by-product of aerodynamic lift. These vortices pose an increased hazard for aircraft at airport takeoffs and landings, as following aircraft flying through a vortex wake can experience dangerous rolling moments. In this work, we use a vortex generator tank and a delta wing in an XY-Towing Tank to study the dynamics of counter-rotating vortex pairs both in and out of ground effect, via PIV and flow visualization. When a vortex pair approaches a ground plane, the boundary layer that forms on the surface between the vortices and the wall separates, generating secondary vorticity and causing the primary vortex pair to rebound'' from the wall. Using a vortex generator tank to produce a temporally evolving vortex pair, it is shown that the introduction of perturbations at the ground plane results in earlier localized secondary vorticity generation. This leads to the formation of coherent secondary vortex structures, and an accelerated decay of the primary vortex pair. This passive, ground-based method could be a means to diminish the wake vortex hazard behind aircraft close to the ground. We study also the spatially evolving trailing vortices in the far-wake of a 75 degree leading-edge sweep-angle delta wing, using a novel technique to measure the axial flow in the vortex core. This technique is unaffected by vortex wandering, allowing us to capture axial flow data as close as 0.03 chord-lengths apart. Using this technique, the streamwise velocity profile is captured over 20 chord-lengths downstream of the delta wing, even when the vortex pair is in ground effect. In this thesis, we also study new modes of NACA 0012 airfoil motions using a sports-mimetic'' approach, inspired by the bodyweight motions of Olympic sailors as they maneuver their sailboats when racing. Olympic sailors use various unsteady aerodynamic techniques when racing to increase propulsion for their boat. One such technique is for sailors to use bodyweight movements to roll the boat about its longitudinal axis. This motion is used especially when turning in light winds by either roll tacking'' (upwind sailing) or roll gybing'' (downwind sailing); it is also used in sail flicking'' whereby the sailor rolls the boat, flicking the sail periodically. These motions are characterized in on-the-water experiments using a Laser sailboat and a 420 sailboat, equipped with a GPS, IMU, wind sensor and GoPro camera array. We study the underlying vortex dynamics of these maneuvers using these characteristic motions, along with full-scale flow visualization and laboratory experiments. Flow visualization experiments are conducted on Cayuga Lake with an Olympic Laser Sailboat, using an Enola Gaye WP40 smoke grenade to visualize large-scale flow features around the sail. dc.language.iso en dc.subject Sports Aerodynamics dc.subject Vortex Dynamics dc.title Vortex-Wall Interactions and Sports-Inspired Airfoil Motions dc.type dissertation or thesis dc.description.embargo 2022-08-27 thesis.degree.discipline Mechanical Engineering thesis.degree.grantor Cornell University thesis.degree.level Doctor of Philosophy thesis.degree.name Ph. D., Mechanical Engineering dc.contributor.chair Williamson, Chas dc.contributor.committeeMember Cowen, Edwin dc.contributor.committeeMember Desjardins, Olivier dcterms.license https://hdl.handle.net/1813/59810 dc.identifier.doi https://doi.org/10.7298/xhfx-0734
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We’re delighted to announce the introduction of LaTeX to Edublogs.
The plugin, released by Patrick Chia, allows you to simply post really really complex equations, like this one:
$latex i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$
Just turn it on in your ‘Plugins’ area and enjoy.
Provide some details for your blog
No stress, you can always change this later on.
.edublogs.org
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## A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits
### Authors
Suryajith Chillara, Nutan Limaye and Srikanth Srinivasan
## Abstract
We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $\Delta+1$ to one of product-depth $\Delta$ in the multilinear setting.
We show that for every positive $\Delta = \Delta(n) = o(\log n/\log \log n)$, there is an explicit multilinear polynomial $P^{(\Delta)}$ on $n$ variables that can be computed by a multilinear formula of product-depth $\Delta+1$ and size $O(n)$, but not by any multilinear circuit of product-depth $\Delta$ and size less than $\exp(n^{\Omega(1/\Delta)})$. This result is tight up to the constant implicit in the double exponent for all $\Delta = o(\log n/\log \log n)$.
This strengthens a result of Raz and Yehudayoff (Computational Complexity 2009) who prove a quasipolynomial separation for constant-depth multilinear circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an exponential separation in the case $\Delta = 1$.
Our separating examples may be viewed as algebraic analogues of variants of the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan (STOC 2016), who used them to prove lower bounds for constant-depth Boolean circuits.
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## Annual percentage yield nominal rate
31 Aug 2019 The nominal interest rate is the interest rate before taking inflation into account, in contrast to real interest rates and effective interest rates. more.
As the effective interest rate is referred to as APY, annual percentage yield. Again , sometimes in finance, you can find different terminologies used for nominal and When a large amount of money is involved, the difference between the nominal rate and EAR makes a significant difference. Answer and Explanation: The formula *APY is the Annual Percentage Yield. Fees may reduce earnings on accounts. CERTIFICATES/IRA CERTIFICATES (\$10,000 or more). Nominal Rate. You should check with your financial institution to find out how often interest is being compounded on your particular investment. Yearly APY. Annual percentage Free calculator to find the total interest, end balance, and the growth chart of a maturity periods of CDs, especially their annual percentage yields (APY). Account, Nominal Rate, Annual Percentage Yield * Annual percentage yield ( APY) is based on interest being paid quarterly and that interest remains on We therefore need a way of comparing interest rates. For example, is an annual interest rate of $$\text{8}\%$$ compounded quarterly higher or lower than an interest
## How to calculate annual percentage yield. The calculation of the annual percentage yield is based on the following equation: APY = (1 + r/n ) n – 1 . where: r - the interest rate. n - the number of times the interest is compounded per year.
23 Jan 2015 This APY calculator estimates the annual percentage yield value by considering the nominal interest rate and the compounding frequency 5 Feb 2019 Saving Account Annual nominal interest rate - 5% Annual percentage yield ( APY) less tax on income, calculated at 5% annual interest rate, 23 Jun 2007 Their savings account interest rate is quoted as being 5.05% APY. Their actual APR, though, is roughly 4.93% – the monthly compounding is 14 Sep 2016 If a principal is invested at the annual (nominal) rate r compounded continuously, then the annual percentage yield is. AP Y = e r - 1. Example 5. The annual percentage yield (APY) is the real rate of return earned on a savings deposit or investment taking into account the effect of compounding interest. Unlike simple interest, compounding interest is calculated periodically and the amount is immediately added to the balance. Often abbreviated as APY, the Annual Percentage Yield is a relevant financial indicator on savings account that helps in comparing the interest rates that have different compounding intervals. It is often called as Effective Annual Rate (EAR).
### Let's come up with a formula to work out the Effective Annual Rate if we know: the rate mentioned (the Nominal Rate, "r"); how many times it is compounded ("n").
17 Oct 2019 Between compounding interest on a daily or monthly basis, daily compounding gives a higher yield - although the Look for the advertised APY. 1 Feb 2020 Term, Min Balance, Nominal Rate, APY* The annual percentage yield is based on an assumption that dividends will remain in the account First enter the APY in percent. Some banks also refer to this as the effective annual rate (EAR). Next enter how frequently interest compounds each year. Common
### 12 Jun 2019 Savings Accounts, Dividend Rate, Annual Percentage Yield (APY) Term, Nominal Rate, APY*, Minimum Balance to Open and Earn Stated
i(m) . . . nominal (annual) interest rate compounded (convertible, payable) m times per year It is also known as the annual percentage yield(APY). Then, i(m) m. to get a clear picture of the loan's true cost or the investment's true yield. annual percentage yield, annual percentage rate, effective rate, nominal rate, and
## The difference between the two is that the nominal rate does not take the compounding into consideration, while the effective annual yields take the effect of
14 Apr 2019 Annual percentage rate (APR) (also called nominal interest rate) is the annualized interest rate on a loan or investment which does not account 23 Jan 2015 This APY calculator estimates the annual percentage yield value by considering the nominal interest rate and the compounding frequency 5 Feb 2019 Saving Account Annual nominal interest rate - 5% Annual percentage yield ( APY) less tax on income, calculated at 5% annual interest rate, 23 Jun 2007 Their savings account interest rate is quoted as being 5.05% APY. Their actual APR, though, is roughly 4.93% – the monthly compounding is 14 Sep 2016 If a principal is invested at the annual (nominal) rate r compounded continuously, then the annual percentage yield is. AP Y = e r - 1. Example 5. The annual percentage yield (APY) is the real rate of return earned on a savings deposit or investment taking into account the effect of compounding interest. Unlike simple interest, compounding interest is calculated periodically and the amount is immediately added to the balance. Often abbreviated as APY, the Annual Percentage Yield is a relevant financial indicator on savings account that helps in comparing the interest rates that have different compounding intervals. It is often called as Effective Annual Rate (EAR).
First enter the APY in percent. Some banks also refer to this as the effective annual rate (EAR). Next enter how frequently interest compounds each year. Common
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# Overview¶
The FiPy framework includes terms for transient diffusion, convection and standard sources, enabling the solution of arbitrary combinations of coupled elliptic, hyperbolic and parabolic PDEs. Currently implemented models include phase field [1] [2] [3] treatments of polycrystalline, dendritic, and electrochemical phase transformations, as well as drug eluting stents [4], reactive wetting [5], photovoltaics [6] and a level set treatment of the electrodeposition process [7].
## Even if you don’t read manuals…¶
…please read Installation, Using FiPy and Frequently Asked Questions, as well as examples.diffusion.mesh1D.
Please refer to Installation for details on download and installation. FiPy can be redistributed and/or modified freely, provided that any derivative works bear some notice that they are derived from it, and any modified versions bear some notice that they have been modified.
## Support¶
You can communicate with the FiPy developers and with other users via our mailing list and we welcome you to use the issue tracker for bugs, support requests, feature requests and patch submissions <https://github.com/usnistgov/fipy/issues>. We also monitor StackOverflow for questions tagged with “fipy”. We welcome collaborative efforts on this project.
## Conventions and Notation¶
FiPy is driven by Python script files than you can view or modify in any text editor. FiPy sessions are invoked from a command-line shell, such as tcsh or bash.
Throughout, text to be typed at the keyboard will appear like this. Commands to be issued from an interactive shell will appear:
$like this where you would enter the text (“like this”) following the shell prompt, denoted by “$”.
Text blocks of the form:
>>> a = 3 * 4
>>> a
12
>>> if a == 12:
... print "a is twelve"
...
a is twelve
are intended to indicate an interactive session in the Python interpreter. We will refer to these as “interactive sessions” or as “doctest blocks”. The text “>>>” at the beginning of a line denotes the primary prompt, calling for input of a Python command. The text “...” denotes the secondary prompt, which calls for input that continues from the line above, when required by Python syntax. All remaining lines, which begin at the left margin, denote output from the Python interpreter. In all cases, the prompt is supplied by the Python interpreter and should not be typed by you.
Warning
Python is sensitive to indentation and care should be taken to enter text exactly as it appears in the examples.
When references are made to file system paths, it is assumed that the current working directory is the FiPy distribution directory, referred to as the “base directory”, such that:
examples/diffusion/steadyState/mesh1D.py
will correspond to, e.g.:
/some/where/FiPy-X.Y/examples/diffusion/steadyState/mesh1D.py
Paths will always be rendered using POSIX conventions (path elements separated by “/”). Any references of the form:
examples.diffusion.steadyState.mesh1D
are in the Python module notation and correspond to the equivalent POSIX path given above.
We may at times use a
Note
to indicate something that may be of interest
or a
Warning
to indicate something that could cause serious problems.
Last updated on Jan 14, 2021. Created using Sphinx 3.4.3.
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# Charles C. Pugh
Charles Chapman Pugh (born 1940) is an American mathematician who researches dynamical systems. Pugh received his PhD under Philip Hartman of Johns Hopkins University in 1965, with the dissertation The Closing Lemma for Dimensions Two and Three.[1] He has since been a professor, now emeritus, at the University of California, Berkeley.
Charles C. Pugh
Charles Pugh, Berkeley, 1993
Born1940 (age 78–79)
United States
ResidenceUnited States
NationalityUnited States
CitizenshipUnited States
Alma materJohns Hopkins University (PhD)
Known forWork in dynamical systems
Scientific career
FieldsMathematics
InstitutionsUniversity of California, Berkeley
ThesisThe Closing Lemma for Dimensions Two and Three (1965)
Websitehttps://math.berkeley.edu/people/faculty/charles-c-pugh
In 1967 he published a closing lemma named after him in the theory of dynamical systems.[2] The lemma states: Let f be a diffeomorphism of a compact manifold with a nonwandering point x.[3] Then there is (in the space of diffeomorphisms, equipped with the ${\displaystyle C^{1}}$ topology) in a neighborhood of f a diffeomorphism g for which x is a periodic point. That is, by a small perturbation of the original dynamical system, a system with periodic trajectory can be generated.
In 1970 he was an invited speaker at the International Congress of Mathematicians in Nice, delivering a talk on Invariant Manifolds.
Mary Cartwright (left) with Charles Pugh, Nice, 1970
## Books
• Real Mathematical Analysis, Springer-Verlag, 2002
## Notes
1. ^
2. ^ Pugh An Improved Closing Lemma and a General Density Theorem, American Journal of Mathematics, Band 89, 1967, S.1010–1021, "Closing Lemma" by Christian Bonatti in Scholarpedia
3. ^ Wandering points were introduced by George Birkhoff to describe dissipative systems (with chaotic behavior). In the case of a dynamical system given by a map f, a point wanders if it has a neighborhood U which is disjoint to all of the iterations of the map on it: ${\displaystyle f^{n}(U)\cap U=\varnothing .\,}$
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## Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 148.05402
Autor: Erdös, Pál
Title: Remarks on a theorem of Zygmund (In English)
Source: Proc. Lond. Math. Soc., III. Ser. 14 A, 81-85 (1965).
Review: We call a sequence of integers n1 < n2 < ··· a Zygmund sequence if whenever |ak| ––> 0, the power series
sumk = 1oo ak znk
converges for at least one z with |z| = 1. It is known that any sequence {nk} satisfying nk+1/nk > 1+c (c > 0) is a Zygmund sequence, and that a Zygmund sequence con not contain arbitrarily long arithmetic progressions [cf. J.-P. Kahane (Zbl 121.30102)]. The author shows the following: Let n1 < n2 < ··· be a sequence which contains two subsequences {nki} and {nli}, 1 \leq i < oo, satisfying
ki ––> oo, ki < li < ki+1, li-ki ––> oo , (nli-nki)1/(li-ki) ––> 1.
Then the above sequences is not a Zygmund sequence.
Reviewer: M.Kinukawa
Classif.: * 30B10 Power series (one complex variable)
Index Words: complex functions
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag
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### “Pariah Moonshine” Part I: The Happy Family and the Pariah Groups
Being a mathematician, I often get asked if I’m good at calculating tips. I’m not. In fact, mathematicians study lots of other things besides numbers. As most people know, if they stop to think about it, one of the other things mathematicians study is shapes. Some of us are especially interested in the symmetries of those shapes, and a few of us are interested in both numbers and symmetries.
### Footballs on road signs: an international overview
I’m an old fashioned manager, I write the team down on the back of a fag packet and I play a simple 4-4-2.
• Mike Bassett, England Manager
I’m very much like Mike Bassett: I like standing on the terraces, I like full-backs whose main skill is kicking wingers into the ad hoardings, and – most of all – I like geometrically correct footballs.
### Stirling’s numbers in a nutshell
This is a guest post by researcher Audace Dossou-Olory of Stellenbosch University, South Africa.
In assignment problems, one wants to find an optimal and efficient way to assign objects of a given set to objects of another given set. An assignment can be regarded as a bijective map $\pi$ between two finite sets $E$ and $F$ of $n\geq 1$ elements. By identifying the sets $E$ and $F$ with $\{1,2,\ldots, n\}$, we can represent an assignment by a permutation.
### A new aspect of mathematics
This is a guest post written by David Nkansah, a mathematics student at the University of Glasgow.
Around the fourth century BC, the term ‘Mathematics’ was defined by Aristotle as the “science of quantity”. It’s my own experience as a young mathematician to say this definition, although correct in its own right, poses a problem for those who do not truly know what mathematics is. It fails to highlight the true creativity of the subject.
Human inspiration and imagination are essential ingredients in mathematics. Regarding creativity, one could say, with merit, that in a sense mathematics is an art. Before proceeding to outline similarities between sketching mathematical proofs and painting on a canvas, it is important to know what fundamental premises mathematical proofs are built on.
### Circular reasoning on Catalan numbers
This is a guest post by researcher Audace Dossou-Olory of Stellenbosch University, South Africa.
Consider the following question: How many ways are there to connect $2n$ points on a circle so that each point is connected to exactly one other point?
### Square wheels in an Italian maths exam
There have been various stories in the Italian press and discussion on a Physics teaching mailing list I’m accidentally on about a question in the maths exam for science high schools in Italy last week.
The paper appears to be online.
(Ed. – Here’s a copy of the first part of this four-part question, reproduced for the purposes of criticism and comment)
The question asks students to confirm that a given formula is the shape of the surface needed for a comfortable ride on a bike with square wheels. (Asking what the formula was with no hints would clearly have been harder.) It then asks what shape of polygon would work on another given surface.
What do people think? Would this be a surprising question at A-level in the UK or in the final year of high school in the US or elsewhere? Of course, I don’t know how similar this question might be to anything in the syllabus in licei scientifici.
The following links give a flavour of the reaction to the question:
6 hours, 1 question out of 2 in section 1, 5 out of 10 in section 2. My own initial reaction is that if I had to do this exam right now I’d do question 2 in section 1 but I’ve not actually attempted question 1 yet.
### Dani’s OEIS adventures: triangular square numbers
Hi! I’m Dani Poveda. This is my first post here on The Aperiodical. I’m from Spain, and I’m not a mathematician (I’d love to be one, though). I’m currently studying a Spanish equivalent to HNC in Computer Networking. I’d like to share with you some of my inquiries about some numbers. In this case, about triangular square numbers.
I’ll start at the beginning.
I’ve always loved maths, but I wasn’t aware of the number of YouTube maths channels there were. During the months of February and March 2016, I started following some of them (Brady Haran’s Numberphile, James Grime and Matt Parker among others). On July 13th, Matt published the shortest maths video he has ever made:
Maybe it’s a short video, but it got me truly mired in those numbers, as I’ve loved them since I read The Number Devil when I was 8. I only needed some pens, some paper, my calculator (Casio fx-570ES) and if I needed extra help, my laptop to write some code. And I had that quite near me, as I had just got home from tutoring high school students in maths.
I’ll start explaining now how I focused on this puzzle trying to figure out a solution.
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# Gyroscope Parallax Effect In Unity
I want to make a parallax effect in Unity 3D with gyroscope like the one on this site: http://matthew.wagerfield.com/parallax/
I found an asset, but it is too expensive.
So I need a script which I can set to the GameObject, so it would move depending phone's gyroscope.
## 2 Answers
It's not so hard to find some sources from google;
Parallax effect (logic):
How do I implement parallax scrolling in 2D?
You are asking like "Gimme a script dammit!". Well, i can't give you a script. But i can tell you how to write your own.
Parallax effect basically means moving background objects slower than foreground ones to create a fake 3D (or depth) effect.
So if you have a vector that represents foregound velocity like:
//values and names are example
Vector3 foregroundVelocity = new Vector3(10,0,0);
Than you can use these vectors for background velocity (not limited to these):
//values and names are example
Vector3 backgroundVelocity = new Vector3(7,0,0);
Vector3 evenMoreBackgroundVelocity = new Vector3(5,0,0);
Vector3 farFarAwayVelocity = new Vector3(3,0,0);
Vector3 rightInFrontOfHorizonLineVelocity = new Vector3(1,0,0);
Then you can get gyro input from Unity's own Gyrpscope class, as @AhmetZambak mentioned, and just move your sprites (or gameObjects) according to magnitudes of corresponding angles, like:
//pseoudocode
Vector3 eulerGyroAngles = /*your input reading code*/;
gameObject.transform.position = (eulerGyroAngles.y, eulerGyroAngles.x, 0) * (corresponding magnitude vector);
• Why do you use eulerGyroAngles ? Why not just Quaternion – Drukalo Dec 7 '15 at 14:09
• @Drukalo If you know how to deal with them, go ahead. I personally prefer euler angles. – S. Tarık Çetin Dec 7 '15 at 14:34
• I mean i really don't know how to make object move with gyroscope like this youtube.com/watch?v=L4xuMh2rkJQ – Drukalo Dec 7 '15 at 16:40
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# Revision history [back]
### OpenCV (Java) : Draw a rectangle region on Camera View
I 'm trying to create a bank card scanner on android using OpenCV, first, I'm creating a region where the user can scan their cards then crop it after, I'm struggling with the rectangle region and place it on the center, Any suggestion on how I can do it? Thank you so much
Here's what I've done so far:
It's doing a rectangle and its centered but the size isn't enough I tried to change the numbers but the position isn't centering.
public Mat onCameraFrame(CameraBridgeViewBase.CvCameraViewFrame inputFrame) {
Mat mrgba = inputFrame.rgba();
int w = mrgba.width();
int h = mrgba.height();
Imgproc.rectangle(mrgba, new Point(w * 1 / 3, h * 1 / 3), new Point(
w * 2 / 3, h * 2 / 3 ), new Scalar( 255, 0, 0 ), 5
);
return mrgba;
}
OUTPUT
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# Which risks are associated with deriving multiple keys from the same DH secret Z?
NIST recommends Krawczyk's HMAC-based key derivation function (HKDF) in SP-800-56C (PDF). HKDF shall e.g. be used to create keys from shared secrets after Diffie Hellman key establishment.
NIST states in the same doc:
Each call to the randomness extraction step requires a freshly computed shared secret $Z$, and this shared secret shall be zeroized immediately following its use in the extraction process.
Why is it not recommended to derive multiple keys from the same $Z$, e.g. why not derive a key for data encryption and a key for authentication from the same $Z$ using different info strings? If I still do it, what weaknesses might the derived keys be facing? Is it simply bad practice (without deep explanation) to do so since the derived keys obviously are related? Krawczyk seems to think that multiple derivations from the secret are expected use (see section 3.2 in RFC 5869, my interpretation).
Does salting the HKDF with a good random salt change my risks when deriving multiple keys from a single $Z$?
The engineering answer is that, in practice, if you generate two keys using two different info strings, I suspect you'd probably get away with it without problems. If we model the hash as a random oracle (admittedly a very strong "assumption"), then I suspect it might be possible to demonstrate that what you propose is OK. Disclaimers: I haven't analyzed this, and I'm certainly not going to give you any guarantees -- if you do what you propose, cryptographers will wag their finger and say "tsk, tsk", and rightly so. I suspect you'd probably get away with it (it's not the worst sin you could make), but if something does go wrong, cryptographers aren't going to take the blame -- it's all on you.
The more principled answer is that if you do what you propose, you are misusing the HKDF primitive. The HKDF is only intended to be applied to a single $Z$ once. It is intended to turn an unguessable value into something that looks uniformly random. It has been analyzed for that use. It was not designed to derive multiple keys from the same $Z$: it hasn't been analyzed for that kind of use case, since that's not what it was designed for. So, you're throwing away the benefits you could get from the public analysis of HKDF if you use it in a way that it wasn't designed for.
Consequently, given that it is so easy to apply a PRG or PRF to the output of HKDF (using HKDF to get a uniform-random key, and using a PRG or PRF for key separation, i.e., to derive two different keys), you should probably do that, instead of what you proposed. Given that it is so easy to do the principled things, you might as well do the principled thing, and use the HKDF only once on any given $Z$. Even though you could probably get away with cutting corners and doing what you proposed, I see no reason to take the risk (even if the risk is miniscule).
So, stick to using HKDF in the way that its specification tells you to. Don't cut corners. In this case, there's not really any compelling reason to deviate from standard cryptographic practice, so you might as well stick with what the specification and cryptographers recommend.
• As HKDF has a variable-length output, couldn't we just produce enough output for both keys and then split it in two, instead of using another PRF after HKDF? – Paŭlo Ebermann Nov 15 '12 at 23:14
• @PaŭloEbermann, good point, yes, I think that would be OK too. – D.W. Nov 15 '12 at 23:52
• this is what is done in TLS. They use HKDF on a shared secret to produce enough keys. – David 天宇 Wong Oct 18 '16 at 20:19
• Isn't the point of the info parameter of the Expand phase so that you can derive multiple keys? – Cocowalla May 8 '18 at 21:12
A key derivation function is intuitively "purifying" the entropy in the group element Z into uniformly random (looking) bits that can used as a key for other purposes. It is not designed to produce "multiple keys" from the same Z, and one should definitely not call the KDF on the same Z twice (even with different salts) and expect to get two independent keys.
If you want more key material beyond the output of one call to the KDF, the high level approach should be to apply the KDF to Z only once to get a single key, and then apply a pseudorandom generator (e.g., AES in counter mode) to that key to get multiple keys. This approach will be sound in the sense that it can proven secure in an appropriate model, assuming Z is pseudorandom, the KDF meets a natural notion of "purifying," and the pseudorandom generator is secure.
• So HKDF (not being any old KDF but e.g. HMAC-SHA-256/512 based) is still not providing the step of applying a pseudorandom generator? – NotACryptographer Nov 13 '12 at 14:53
• The output of K = HKDF(Z,r) will look like a random key when Z is a random group element and r is a random salt. But K1 = HKDF(Z,r) and K2 = HKDF(Z,r') will not look like two independent random keys, even when the salts r,r' are independent (at least this is the case with the concept of KDFs I am familiar with). However, if we take (K1,K2) = PRG(HKDF(Z,r)), then they will look random and independent. – David Cash Nov 13 '12 at 16:48
What you propose—using the intermediate derivation key $K_{\mathit{DK}}$ from a single input $Z$ to derive many output keys with distinct info strings—is perfectly fine.
The excerpt you quoted is about the HKDF-Extract step, not the HKDF-Expand step. What it is saying is that you should not use the same $Z$ with many different salts for HKDF-Extract: use it with one salt, and then erase it. That excerpt says nothing about HKDF-Expand with different info strings, however.
RFC 5869 is clearer on the point (emphasis added):
HKDF follows the "extract-then-expand" paradigm, where the KDF logically consists of two modules. The first stage takes the input keying material and "extracts" from it a fixed-length pseudorandom key K. The second stage "expands" the key K into several additional pseudorandom keys (the output of the KDF).
The standard security reduction for HKDF applies to an adversary who can query HKDF-Expand for many info strings adaptively, with the theorem parametrized by the number of queries, so your proposed use falls squarely within the intended and studied use of HKDF. This applies whether or not you use a salt.
You could generate a single key out of HKDF-Extract and HKDF-Expand, and feed it into another PRF to derive more keys. But to do so would be silly when HKDF-Expand already does exactly that.
The HKDF RFC contains formulations such as:
The second stage "expands" the key K into several additional pseudorandom keys (the output of the KDF).
and:
The second stage "expands" the pseudorandom key to the desired length; the number and lengths of the output keys depend on the specific cryptographic algorithms for which the keys are needed.
Both quotes appear already in the introduction (emphases mine).
As you already point out, Section 3.2 describes the info value and states:
In particular, it may prevent the derivation of the same keying material for different contexts (when the same input key material (IKM) is used in such different contexts).
All these quotes taken together seem to rather strongly indicate that deriving several keys from the same IKM is a perfectly valid and even intended use case of HKDF.
However, there is also a rather strong caveat in the introduction:
In many applications, the input keying material is not necessarily distributed uniformly, and the attacker may have some partial knowledge about it (for example, a Diffie-Hellman value computed by a key exchange protocol) ...
The concrete case of DH key exchange is thus even explicitly mentioned as one case where an attacker has partial knowledge of the IKM.
It would thus seem prudent to base the derivation of further keys on a new shared secret in this particular case of IKM.
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# Existence of some special kind of isometry in R^n-1
1. Apr 24, 2012
### julypraise
1. The problem statement, all variables and given/known data
Is there a theorem that states the following?
Let $P= \{ P_{1}, . . . , P_{n} \}$ be the set of n distinct points in $\mathbb{R}^{n-1}$ and $P'= \{ P'_{1}, . . . , P'_{n} \}$ also a set of points in $\mathbb{R}^{n-1}$. If for all $i,j$ $|P_{i} - P_{j}|=|P'_{i} - P'_{j}|$ then there is an isometry $f: \mathbb{R}^{n-1} \to \mathbb{R}^{n-1}$ such that for all $i$ $f(P_{i})=P'_{i}$.
Or at least is it ture? If true how to prove it?
2. Relevant equations
3. The attempt at a solution
Geometrically, it seems plausible especially in some cases where such an isometry is a rotation, a translation, a reflection, or a combination of them.
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3D Self-Conditioned [A(v),B(v),C(v),D(v)]
Implement gain-scheduled state-space controller in self-conditioned form depending on two scheduling parameters
GNC/Control
Description
The 3D Self-Conditioned [A(v),B(v),C(v),D(v)] block implements a gain-scheduled state-space controller as defined by the equations
`$\begin{array}{l}\stackrel{˙}{x}=A\left(v\right)x+B\left(v\right)y\\ u=C\left(v\right)x+D\left(v\right)y\end{array}$`
in the self-conditioned form
`$\begin{array}{l}\stackrel{˙}{z}=\left(A\left(v\right)-H\left(v\right)C\left(v\right)\right)z+\left(B\left(v\right)-H\left(v\right)D\left(v\right)\right)e+H\left(v\right){u}_{meas}\\ {u}_{dem}=C\left(v\right)z+D\left(v\right)e\end{array}$`
For the rationale behind this self-conditioned implementation, refer to the Self-Conditioned [A,B,C,D] block reference. These blocks implement a gain-scheduled version of the Self-Conditioned [A,B,C,D] block, v being the vector of parameters over which A, B, C, and D are defined. This type of controller scheduling assumes that the matrices A, B, C, and D vary smoothly as a function of v, which is often the case in aerospace applications.
Parameters
A-matrix(v1,v2,v3)
A-matrix of the state-space implementation. In the case of 3-D scheduling, the A-matrix should have five dimensions, the last three corresponding to scheduling variables v1, v2, and v3. Hence, for example, if the A-matrix corresponding to the first entry of v1, the first entry of v2, and the first entry of v3 is the identity matrix, then `A(:,:,1,1,1) = [1 0;0 1];`.
B-matrix(v1,v2,v3)
B-matrix of the state-space implementation. In the case of 3-D scheduling, the B-matrix should have five dimensions, the last three corresponding to scheduling variables v1, v2, and v3. Hence, for example, if the B-matrix corresponding to the first entry of v1, the first entry of v2, and the first entry of v3 is the identity matrix, then `B(:,:,1,1,1) = [1 0;0 1];`.
C-matrix(v1,v2,v3)
C-matrix of the state-space implementation. In the case of 3-D scheduling, the C-matrix should have five dimensions, the last three corresponding to scheduling variables v1, v2, and v3. Hence, for example, if the C-matrix corresponding to the first entry of v1, the first entry of v2, and the first entry of v3 is the identity matrix, then `C(:,:,1,1,1) = [1 0;0 1];`.
D-matrix(v1,v2,v3)
D-matrix of the state-space implementation. In the case of 3-D scheduling, the D-matrix should have five dimensions, the last three corresponding to scheduling variables v1, v2, and v3. Hence, for example, if the D-matrix corresponding to the first entry of v1, the first entry of v2, and the first entry of v3 is the identity matrix, then `D(:,:,1,1,1) = [1 0;0 1];`.
First scheduling variable (v1) breakpoints
Vector of the breakpoints for the first scheduling variable. The length of v1 should be same as the size of the third dimension of A, B, C, and D.
Second scheduling variable (v2) breakpoints
Vector of the breakpoints for the second scheduling variable. The length of v2 should be same as the size of the fourth dimension of A, B, C, and D.
Third scheduling variable (v3) breakpoints
Vector of the breakpoints for the third scheduling variable. The length of v3 should be same as the size of the fifth dimension of A, B, C, and D.
Initial state, x_initial
Vector of initial states for the controller, i.e., initial values for the state vector, x. It should have length equal to the size of the first dimension of A.
Poles of A(v)-H(v)*C(v)
Vector of the desired poles of A-HC. Note that the poles are assigned to the same locations for all values of the scheduling parameter v. Hence the number of pole locations defined should be equal to the length of the first dimension of the A-matrix.
Inputs and Outputs
InputDimension TypeDescription
First
Contains the measurements.
Second
Contains the scheduling variable, ordered conforming to the dimensions of the state-space matrices.
Third
Contains the scheduling variable, ordered conforming to the dimensions of the state-space matrices.
Fourth
Contains the scheduling variable, ordered conforming to the dimensions of the state-space matrices.
Fifth
Contains the measured actuator position.
OutputDimension TypeDescription
First
Contains the measured actuator position.
The first input is the measurements.
The second, third, and fourth inputs are the scheduling variables ordered conforming to the dimensions of the state-space matrices.
The fifth input is the measured actuator position.
The output is the actuator demands.
Assumptions and Limitations
If the scheduling parameter inputs to the block go out of range, then they are clipped; i.e., the state-space matrices are not interpolated out of range.
Note: This block requires the Control System Toolbox™ product.
Reference
The algorithm used to determine the matrix H is defined in Kautsky, Nichols, and Van Dooren, "Robust Pole Assignment in Linear State Feedback," International Journal of Control, Vol. 41, No. 5, pages 1129-1155, 1985.
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open import 1Lab.Path.Groupoid
open import 1Lab.Univalence
open import 1Lab.Type.Dec
open import 1Lab.Equiv
open import 1Lab.Path
open import 1Lab.Type
open import Data.Int.Inductive
open import Data.Bool
open import Data.Bool
module 1Lab.HIT.S1 where
The Circle🔗
Since the “intended interpretation” of HoTT is in a $\io$-category of “good spaces,” it makes sense that HoTT has facilities for describing spaces. These are the higher inductive types, one of which is the circle:
data S¹ : Type where
base : S¹
loop : base ≡ base
Diagramatically, we can picture the circle as being the $\infty$-groupoid generated by the following diagram:
In type theory with K, the circle is exactly the same type as . However, with univalence, it can be shown that the circle has two different paths:
möbius : S¹ → Type
möbius base = Bool
möbius (loop i) = ua (not , not-is-equiv) i
When pattern matching on the circle, we are asked to provide a basepoint b and a path l : b ≡ b, as can be seen in the definition above. To make it clearer, we can also define a recursion principle:
S¹-rec : ∀ {ℓ} {A : Type ℓ} (b : A) (l : b ≡ b) → S¹ → A
S¹-rec b l base = b
S¹-rec b l (loop i) = l i
Using möbius, it can be shown that the loop is not refl:
parity : base ≡ base → Bool
parity l = subst möbius l true
_ : parity refl ≡ true
_ = refl
_ : parity loop ≡ false
_ = refl
refl≠loop : refl ≡ loop → ⊥
refl≠loop path = true≠false (ap parity path)
The circle is also useful as a source of counterexamples. By S¹-elim', we can prove that there is an inhabitant of (x : S¹) → x ≡ x which is not constantly refl
S¹-elim : ∀ {ℓ} {P : S¹ → Type ℓ}
→ (pb : P base)
→ PathP (λ i → P (loop i)) pb pb
→ (x : S¹) → P x
S¹-elim pb pl base = pb
S¹-elim pb pl (loop i) = pl i
S¹-elim' : ∀ {ℓ} {P : S¹ → Type ℓ} (pb : P base)
→ subst P loop pb ≡ pb
→ (x : S¹) → P x
S¹-elim' {P = P} pb pl =
S¹-elim pb (transport (λ i → PathP≡Path (λ i → P (loop i)) pb pb (~ i)) pl)
always-loop : (x : S¹) → x ≡ x
always-loop = S¹-elim' loop (subst-path-both _ _ ∙ lemma) where
lemma = sym loop ∙ loop ∙ loop ≡⟨ ∙-cancel-l loop loop ⟩≡
loop ∎
Path Space🔗
A classical result of algebraic topology is that the fundamental group of the circle is isomorphic to the group of integers. In Homotopy Type Theory, we can repeat this proof: The type of integers codes for paths in the circle.
module S¹Path where
Cover : S¹ → Type
Cover base = Int
Cover (loop i) = ua suc-equiv i
We define a map encode which converts a path in the circle to an element of the Cover. By lifting p to the Cover, we determine a path Int ≡ Int, which we apply to zero:
encode : (x : S¹) → base ≡ x → Cover x
encode x p = subst Cover p (pos zero)
This map counts the winding number of a path:
_ : encode base refl ≡ pos zero
_ = refl
_ : encode base loop ≡ pos (suc zero)
_ = refl
In the other direction, we define a map decode which converts an integer to a path base ≡ base:
loop^ : Int → base ≡ base
loop^ (pos zero) = refl
loop^ (pos (suc x)) = loop^ (pos x) ∙ loop
loop^ (negsuc zero) = sym loop
loop^ (negsuc (suc x)) = loop^ (negsuc x) ∙ sym loop
What we want to show is that encode base and loop^ are mutual inverses. For this, we must show that encode base (loop^ n) ≡ n and loop^ (encode base p) ≡ p. The former direction is simpler, because we can show it directly by recursion on Int:
encode-loop^ : (n : Int) → encode base (loop^ n) ≡ n
encode-loop^ (pos zero) = refl
encode-loop^ (pos (suc x)) = ap suc-int (encode-loop^ (pos x))
encode-loop^ (negsuc zero) = refl
encode-loop^ (negsuc (suc x)) = ap pred-int (encode-loop^ (negsuc x))
In the other direction, we would like to apply path induction to reduce the problem to showing loop^ (encode base refl) ≡ refl. However, path induction does not apply to paths base ≡ base, only to paths base ≡ x, where x a variable. Therefore, we have to generalise the function loop^ to the function decode.
decode : (x : S¹) → Cover x → base ≡ x
decode base = loop^
On the basepoint, the type of the right-hand side computes to Int → base ≡ base; Thus, we can use the existing loop^ map.
decode (loop i) n j = square where
For the loop case, recall the induction principle of the circle. We must provide a path loop^ ≡ loop^ “laying over” loop, i.e. a Square refl (loop^ n) (loop^ n) loop. Visually, we can picture the boundary of the path we must provide as the square below:
We will construct this square in parts. First, we’ll construct a square loop^-square with the boundary below, for any $n$, by recursion on the integer $n$; We’ll then modify this square so it becomes the one above.
loop^-square : (n : Int)
→ Square refl (loop^ (pred-int n)) (loop^ n) loop
loop^-square (pos zero) i j = loop (i ∨ ~ j)
The case above is for $n = 0$. We can picture it as follows. Note that this is a square where two faces are the path loop and two faces are degenerate: a connection.
loop^-square (pos (suc x)) i j =
hfill (λ k → λ { (j = i0) → base
; (j = i1) → loop k })
(inS (loop^ (pos x) j)) i
In the case where $n = +x$, the square we have to fill is the one on the left below, but note that the composite on the right face is defined to be the dashed path in that square; Thus, the filler of that square suffices. A similar thing happens for the case where $n = -(x + 1)$, which is the square on the right.
loop^-square (negsuc x) i j =
hfill (λ k → λ { (j = i0) → base
; (j = i1) → loop (~ k) })
(inS (loop^ (negsuc x) j)) (~ i)
We can then alter the square loop^-square to the square we want, by sketching an appropriate cube.
square = hcomp (λ k → λ { (i = i0) → loop^ (pred-suc n k) j
; (i = i1) → loop^ n j
; (j = i0) → base
; (j = i1) → loop i })
(loop^-square (unglue (i ∨ ~ i) n) i j)
With this generalised decode, we can prove that decode x (encode x p) ≡ p, by reducing p to refl, i.e. path induction.
decode-encode : {x : S¹} (p : base ≡ x) → decode x (encode x p) ≡ p
decode-encode = J (λ y p → decode y (encode y p) ≡ p) refl
Thus we have that the loop space of the circle is the type of integers:
ΩS¹≃Int : (base ≡ base) ≃ Int
ΩS¹≃Int = Iso→Equiv (encode base , iso loop^ encode-loop^ decode-encode)
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# Proving postulate about a property fo spherical vectors
Assume we have $$X, Y$$ constant unit vectors of $$\mathbb{R}^3$$
I postulate that the maximum of the function:
$$(V \cdot X) (V \cdot Y)$$
I reached by the halfway vector between $$X,Y$$ i.e at the vector $$V_0 = slerp(X,Y, 0.5)$$
To try to prove it I tried finding the critical point of the derivative, i.e:
$$(V'\cdot X)(V\cdot Y) + (V\cdot X)(V'\cdot Y)$$
But that is leading me down a rabbit hole I don't seem to be able to get out of.
• Derivative with respect to $V$? But you'd better constrain $V$ to be a unit vector as well. – Ted Shifrin Jun 5 '20 at 3:50
Without loss of generality, choose $$X=(1,0,0)$$ and $$Y=(\cos\phi_0,\sin\phi_0,0)$$. Then (assuming $$V$$ is also unit vector), you can write $$V$$ in polar coordinates as $$V=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$$. Then your expression becomes $$\sin\theta\cos\phi(\sin\theta\cos\phi\cos\phi_0+\sin\theta\sin\phi\sin\phi_0)=\sin^2\theta\cos\phi\cos(\phi-\phi_0)$$ If you want the maximum, you get $$\theta=\pi/2$$, so it's in the same plane. Also $$\frac d{d\phi}\cos\phi\cos(\phi-\phi_0)=-\sin(2\phi-\phi_0)=0$$ so $$\phi=\frac{\phi_0}2$$ You will need to consider separately the case where $$\phi_0=\pi$$
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# How do you solve -4=10(x+4)+3x?
$- 4 = 10 x + 40 + 3 x$
$- 4 - 40 = 13 x$
$- 44 = 13 x$
$x = - \frac{44}{13}$
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Top
# Viscosity Measurement
The viscosity measurement is important in many of the process that we get to see around us. Whether its food industry or medicine, the viscosity of raw materials has a direct effect on the final product quality. In case we are looking at the ceramic industries, the quality of the raw materials affects the final product quality. In-fact, viscosity control is also very important in assembly operations that involve the application of grinding and flow of materials through pipelines.
The final assembly of such products require these materials to flow through tubes at the right consistency and rate and hence the role of viscosity becomes of utmost importance. Viscosity helps in describing the manner in which the fluid flows when a force is applied to the fluid matter. The fluid rates also vary significantly at high shear rates and the viscosity also might vary at low shear rates. The best category of example is non-Newtonian characteristics which tend to take place in emulsion, pastes and various types of slurries.
The relationship which exists between input variables and output measurement for instruments that actually measure viscosity assumes that the measured fluid has Newtonian characteristics. For any non-Newtonian fluids, the shear rate variation correction is essential and unless these corrections are not made the measurement that is obtained is known as apparent viscosity and can differ by large margin from true viscosity. This true viscosity is also known as absolute viscosity and can change according to three specific physical principles.
• Rate of flow of the fluid through a capillary tube
• Rate of fall of a body inside the fluid
• Viscous frictional force which exerts on a rotating body
Related Calculators How to Measure Frequency Measurement Calculator Repeated Measures Anova Calculator
## Viscosity Measurement Formula
Viscosity describes the way in which a fluid flows when it is subjected to applied force. If we consider a cubic volume of fluid and if a shear force F is applied to one of its face with an area A, and this face moves through a distance L at a velocity of V as compared to the opposite face of cube, then the shear stress (s) and shear rate shows a relationship of: s = $\frac{F}{A}$ OR r = $\frac{V}{L}$
The coefficient of viscosity (Cv) = $\frac{s}{r}$
Hence, Cv or viscosity = $\frac{(\frac{F}{A})}{\frac{V}{L}}$
OR, Cv = $\frac{F*L}{A*V}$
Kinematic viscosity Kv = $\frac{Cv}{\rho}$ (fluid density)
Cv is measured in $\frac{Ns}{m2}$ while Kv is measured in stokes or $\frac{m2}{s}$
## Viscosity Measurement Experiment
Viscosity of gas or liquid mixtures under pressure.
• This is not only important in order to determine the solubility of gas but also the impact of dissolved gas in fluid viscosity.
• This is also very importance in determining the lubricating ability of the compressor fluid during gas compression.
• Measurement of viscosity of lubricants saturated with gas at elevated pressure is done in a fixed volume apparatus. Typically a Parr reactor equipped with a stirrer, pressure transducer, thermocouples and heating and cooling device is used to conduct these experiments at desired temperatures.
• This vessel fitted with viscosity measurement probe suitable for measurement at high pressure.
• Viscosity probe that can operate up to 340 bars are used in such cases.
• Viscosity and gas solubility data at a given temperature and pressure can be obtained from same experiment.
• Viscosity of saturated lubricant at a final pressure is measured with a viscosity probe and its associated electronics.
• Gas solubility can be calculated from initial and final pressure using gas laws and compressibility factors and can be obtained by weighing the gas charging container both before and after equilibrium.
Absorption of the lubricant into gas phase.
• This is one of the test procedure performed to determine the amount of lubricant lost into the gas being compressed.
• This test is conducted by weighing the test oil into a miniscule aluminium container and then finally placed into a 50 mL stainless steel high pressure reactor heated up to 100 C.
• The test gas is exposed to 400 bar and fed through stainless steel tubing into stainless steel reactor containing the test oil.
• The pressure is maintained by releasing the gas through a needle valve and the overall volume of the gas getting discharged with the help of a meter.
• Finally the aluminium vessel is measured again and weight loss if any is recorded.
## Viscosity Measurement Methods
The most important thing we need to remember is that measuring viscosity of any fluid essentially is temperature dependent. The idea of viscosity measuring is very important in both manufacturing and academic circle. The specific viscosity knowledge is essential for various industrial process and theories which are developed for predicting or estimating viscosity. The various instruments that are used for such measurement activities are classified into
• Capillary viscometers
• Orifice viscometers
• High temperature and high shear rate viscometer
• Rotational viscometer
• Vibrational viscometer
• Ultrasonic viscometer
• Falling ball viscometer
A number of viscometers combine features of two or three types of viscometers and calibrate all the readings into one.
Absolute viscosity
The derived units of absolute viscosity µ in SI is Kg / m*s which is equivalent to Pa * s or Pascal seconds. In CGS system the units is dyne * s / cm2 and is better known as Poise. The Poise is used in FPS system of units.
Kinematic viscosity
Kinematic viscosity (v) is defined as the absolute viscosity divided by the fluid density ρ (rho). The fundamental units in SI for kinematic viscosity are meter square per second and is identical to the units of thermal diffusivity used in heat transfer and mass of the matter diffusivity used in diffusion.
This gives us the idea that kinematic viscosity referred as coefficient of momentum diffusivity is cm2 / second or better known as stokes.
Driving pressure P can be replaced by h * g * ρ where, h is the mean head, g as the gravity and ρ as fluid density.
The Hagen Poiseuille equation is given out as:
η = K ρ t where, K is the instrument conversion factor and is also a constant for each instrument.
Hence, $\frac{\eta}{\rho}$ = Kt or v = Kt as kinematic viscosity v = $\frac{\eta}{\rho}$
The kinematic viscosity of the fluid is obtained by multiplying the measured effux time with instrument conversion factor. If the instrument conversion K is not available then it can be obtained by measuring the effux time for a fluid of known viscosity.
Glass capillary viscometer
These are widely used for measuring low to medium viscosity Newtonian fluids as their degree of accuracy and cost factors are low.
## Application of Viscosity Measurement
Collection of viscosity data on a particular material gives the manufacturers the advantage of predicting how a material will behave in utility point. Products like toothpaste, and creams can be made to perfection if the viscosity is known as these help the manufacturers to gauge how the product will behave at consumer level and specific weather pattern. These also help in perfecting product design and transporting these to long destinations.
Food industry:
Viscosity measurement are widely used for the product consistency as the consumer level satisfaction is utmost. Optimisation of products depend upon how the product is packaged and how these are transported and reaches the consumers in good condition.
The type of consistence an adhesive product behaves also goes a long way to create a brand image as the faster the glue sticks and dries can help design the package and container for such brands.
Petroleum industry:
The oil mix and engine oil viscosity is utmost for the longevity of vehicle engine and this goes a long way in creating a brand niche and hence the viscosity of these products is very important as it can make or break an engine completely along with the manufacture goodwill.
Cosmetic industry:
The cosmetic industry thrives on consumer satisfaction and hence, the same type of products with different viscosity can go for a toss as most of the products are rejected at consumer level due to the viscosity factor.
## Dynamic Viscosity Measurement
Viscosity is nothing but the resistance provided by the fluids to shear. The fluid molecules in contact with the bottom plate are considered to be at rest and the molecules in contact with top plate are moving at a velocity v.
The velocity profile for the two plates are created and the velocity profile’s slope would be given out as:
Slope = $\frac{\Delta V}{\Delta y}$.
These plates with a cross section area of A, would require a force F to keep this fluid in motion and hence the velocity v.
The shear stress (Ï„) = $\frac{F}{A}$
The dynamic viscosity or also known as absolute viscosity is defined as the ratio between the shear stress and slope.
Hence, µ =$\frac{\tau}{\frac{v}{y}}$
Or, µ = $\frac{F*y}{v*A}$
Here, the force F is in dyne, y is given out in cm, velocity in cm / seconds and A is given out in square cm. the units of dynamic viscosity is given out by g / (s *cm). This is given out as Poise. If we divide this value by 100 then we get Centi poise.
Kinematic viscosity is considered as ‘dynamic viscosity / fluid density’ under same temperature values. This is in other words considered as dynamic viscosity measurement. The µ is dynamic viscosity (g / s cm) and Ï is density (g / cc)
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# If x and y are positive, is xy > x + y?
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If x and y are positive, is xy > x + y? [#permalink]
### Show Tags
01 Jul 2016, 05:53
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If x and y are positive, is xy > x + y?
(1) x < y
(2) 2 < x
OG Q 2017 New Question(Book Question: 199)
_________________
Md. Abdur Rakib
Please Press +1 Kudos,If it helps
Sentence Correction-Collection of Ron Purewal's "elliptical construction/analogies" for SC Challenges
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Re: If x and y are positive, is xy > x + y? [#permalink]
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01 Jul 2016, 08:06
given x and y positive
if we take x,y<1 then statement doesn't hold if x,y>1 then holds.
1) not suff for above explanation
2)no info about y
we know x>2 and so is y so 1+2 is suff
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Re: If x and y are positive, is xy > x + y? [#permalink]
### Show Tags
02 Jul 2016, 09:53
AbdurRakib wrote:
If x and y are positive,is xy>x+y?
(1) x<y
(2) 2<x
OG Q 2017 New Question(Book Question: 199)
IS xy>x+y
Fact 1) x<y
Put x=1/2, Y=2 1>2.5 Answer is NO
Put x= 3, Y=4 12>7 Answer is YES
Insuff
Fact 2) X>2 Clearly Insuff
Combining 1 & 2
Take example above when x= 3, Y=4
Another example x=5/2 , Y=4
Suff.
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Re: If x and y are positive, is xy > x + y? [#permalink]
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04 Jan 2018, 01:19
Why should the answer be not B because, if we take x = 3 and y = 4, the statement is sufficient (if first condition is taken into consideration) and if we take x = 5 and y = 2 then also the statement is sufficient (if first taken not taken into consideration) so having condition 1 is not necessary.
Can anyone respond to this ?
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Posts: 58297
Re: If x and y are positive, is xy > x + y? [#permalink]
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04 Jan 2018, 01:31
sagargoyal wrote:
Why should the answer be not B because, if we take x = 3 and y = 4, the statement is sufficient (if first condition is taken into consideration) and if we take x = 5 and y = 2 then also the statement is sufficient (if first taken not taken into consideration) so having condition 1 is not necessary.
Can anyone respond to this ?
Two examples, are not enough to prove sufficiency. What if x = 1 and y = 1? For (2), there are infinitely many pairs which give a NO answer to the question as well as there are infinitely many pairs which give an YES answer to the question.
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If x and y are positive, is xy > x + y? [#permalink]
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09 Jan 2018, 19:17
1
1
AbdurRakib wrote:
If x and y are positive, is xy > x + y?
(1) x < y
(2) 2 < x
OG Q 2017 New Question(Book Question: 199)
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.
xy > x + y
⇔ xy - x - y > 0
⇔ xy - x - y + 1 > 1
⇔ (x-1)(y-1) > 1
⇔ x > 1, y > 1 or 0 < x < 1, 0 < y < 1?
Since we have 2 variables (x and y) and 0 equations,C is most likely to be the answer. So, we should consider 1) & 2) first.
Conditions 1) & 2):
Since x > 2, y > x > 2 or y > 2, both conditions together are sufficient.
Since this is an inequality question (one of the key question areas), we should also consider choices A and B by CMT(Common Mistake Type) 4(A).
Condition 1)
x = 2, y = 3: Yes
x = 1/2, y = 1: No
The condition 1) is not sufficient.
Condition 2)
x = 3, y = 4: Yes
x = 3, y = 1/2: No
The condition 2) is not sufficient.
Therefore, the answer is C.
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Re: If x and y are positive, is xy > x + y? [#permalink]
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04 Feb 2018, 12:43
Hi Experts,
Can someone please provide a solution for this question? If I use number plugging, I am not able to prove that C will be sufficient. Can there be another approach (algebraic) to solving this?
Thanks for your help!
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Re: If x and y are positive, is xy > x + y? [#permalink]
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04 Feb 2018, 23:27
2
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sdlife wrote:
Hi Experts,
Can someone please provide a solution for this question? If I use number plugging, I am not able to prove that C will be sufficient. Can there be another approach (algebraic) to solving this?
Thanks for your help!
Hi
Not an expert but I will give you my perspective. I think here algebra would be difficult, at least I cannot think of that.
We are given that both x/y are positive numbers, and after we combine the two statements we get that both x/y are greater than 2. We have to determine whether their product is greater than their sum.
Now, '2' is a key number here. Upto '2', we cannot be sure whether product of two positive numbers will be greater than their sum or not.
Eg, if we take two numbers as 1,2 - then their product is actually lesser than their sum
If we take two numbers as 1,1.5 - then also their product is lesser than their sum
If we take two numbers as 0.5, 4 (here one of them is less than 2 & other is greater than 2) - then also their product is lesser than their sum (here the fractional value of first number which is less than 1, has decreased the product to a large extent and made it lesser than the sum)
And if both numbers are equal to 2 (though here its given that x < y so they cannot be equal but still to explain) - then their product will be equal to their sum (4 each)
But once each number is greater than '2', the property is such that the product will always be greater than the sum
Even if we take them as 2.01, 2.02 - we will see that their product is greater than their sum
And as the numbers keep on increasing, their product will keep on increasing and become more and more larger than their sum.
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If x and y are positive, is xy > x + y? [#permalink]
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15 Sep 2018, 09:50
MathRevolution wrote:
AbdurRakib wrote:
If x and y are positive, is xy > x + y?
(1) x < y
(2) 2 < x
OG Q 2017 New Question(Book Question: 199)
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.
xy > x + y
⇔ xy - x - y > 0
⇔ xy - x - y + 1 > 1
⇔ (x-1)(y-1) > 1
⇔ x > 1, y > 1 or 0 < x < 1, 0 < y < 1?
Since we have 2 variables (x and y) and 0 equations,C is most likely to be the answer. So, we should consider 1) & 2) first.
Conditions 1) & 2):
Since x > 2, y > x > 2 or y > 2, both conditions together are sufficient.
Since this is an inequality question (one of the key question areas), we should also consider choices A and B by CMT(Common Mistake Type) 4(A).
Condition 1)
x = 2, y = 3: Yes
x = 1/2, y = 1: No
The condition 1) is not sufficient.
Condition 2)
x = 3, y = 4: Yes
x = 3, y = 1/2: No
The condition 2) is not sufficient.
Therefore, the answer is C.
Hi MathRevolution,
Could you explain me the highlighted step after the below step
(x-1)(y-1)>1
So we have
(x-1)>1
or
(y-1)>1
So we have x> 2 and or we have y>2
What did i miss?
However, this is how I approached
xy>x+y
xy-x-y>0
adding 1 on both sides
xy-x-y+z>1
(x-1)(y-1)>1
so either
x-1>1 more so x>2
or y-1>1 more so y>2
So question is asking IS x>2 and y>2. we can have answer to our question is we know about x and y.
So statement A:
(A) x < y
we know that x>o and y>0, but we dont kow if x>2 and y>2
So insufficient.
Statement B:
(B) 2 < x
We know that x>0 so x>2 but don't have any information about y
so insufficient,
Now combining A and B we have
2<x<y
so we have x>2 and y>2
However just to remember
A * B$$\geq$$ A + B.
When A & B $$\leq{0}$$, it's always true.
When A & B $$\geq 2$$, it's always true.
When A $$\geq{1}$$, B $$\leq{1}$$ (or B$$\geq{1}$$, A$$\leq{1}$$), it's always false.
When 0 < A < 1, B < 0 (or 0 < B < 1, A < 0), it can be either true or false.
When 1 < A < 2, B > 0 (or 1 < B < 2, A > 0), it can be either true or false.
A * B > A + B.
when A>2 and B>2
For this question if we had two statements say
Statement 1: A>2
statement 2: B>2
On combining 1 and 2 we can say that (A*B)> (A+B)
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Re: If x and y are positive, is xy > x + y? [#permalink]
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02 Jan 2019, 07:20
AbdurRakib wrote:
If x and y are positive, is xy > x + y?
(1) x < y
(2) 2 < x
OG Q 2017 New Question(Book Question: 199)
From (1), x<y There are many possibilities smart no.s(all values>0 though) for which you will get no unique answer sometimes xy < (x+y) & sometimes xy > x+y =>NOT SUFFICIENT ;
From (2), 2<x => No clue about value of y => NOT SUFFICIENT ;
Now, its given x & y both are +ve so,
2<x => 2y < x*y .......(a)
x<y .......(b)
Adding (a) & (b), (2y+x) < (xy+y) => (y+x) < xy => SUFFICIENT => Option C ;
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Re: If x and y are positive, is xy > x + y? [#permalink] 02 Jan 2019, 07:20
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# Union of Subsets is Subset/Proof 2
## Theorem
Let $S_1$, $S_2$, and $T$ be sets.
Let $S_1$ and $S_2$ both be subsets of $T$.
Then:
$S_1 \cup S_2 \subseteq T$
That is:
$\paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T} \implies \paren {S_1 \cup S_2} \subseteq T$
## Proof
Let $x \in S_1 \cup S_2$.
By the definition of union, either $x \in S_1$ or $x \in S_2$.
By hypothesis, $S_1 \subseteq T$ and $S_2 \subseteq T$.
By definition of subset:
$x \in S_1 \implies x \in T$
$x \in S_2 \implies x \in T$
By Proof by Cases it follows that $x \in T$.
Hence the result by definition of subset.
$\blacksquare$
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Let the Scry Fall
Today’s project is collages! No scissors and glue, but we’ve got the next best things: REST APIs and the Python Image Library. Specifically, we’ll be accessing card illustrations via the shiny new(ish) Scryfall API, working a bit of magic on them, then stitching them together.
This is all based on some work by a friend-of-a-friend, April King. She put together a collage of Island illustrations, earning herself a shout-out in the Scryfall blog. We’ll be starting with her same idea, then taking it a bit further.
## Getting Images
Working with the Scryfall API is a breeze. You ping a URL, and you get back a conveniently-formatted reply. For example, let’s say we want to search for all 500+ printings of the card [[Forest]]. After a quick peek at the Scryfall search syntax guide, we type a few lines of Python and have all that information at our fingertips:
import requests
# The query !"forest" gets cards named exactly "forest".
url = 'https://api.scryfall.com/cards/search?q=!"forest"&unique=prints'
forest_info = requests.get(url).json()
If we go to the the above URL ourselves, we see a jumble of JSON. Some snippets of relevant-looking information, but condensed to the point of illegibility. Python gobbles that up no problem and spits out a well-behaved dictionary. Card data (rules text, collector number, etc) is packaged in forest_info['data']. In this case, there are too many search results to fit on one page, so forest_info['next_page'] tells us where to make another request for the next batch.
To see more details about working with the API, feel free to check out my code on GitHub. Long story short, we grab just a few pieces of information from Scryfall:
• Set and collector number to uniquely identify each card
• Artist name – more on this in a moment
• The URL where they store the illustration
We make another API request to grab the illustrations and dump1 each into a file. From there it’s easy enough to load them up, crop them to a common size, and toss them together into a collage:
All 587 Forest illustrations, in the order Scryfall returns them. A number of duplicates are visible. Full-sized version here, art credits here.
There have been 587 printings of the card [[Forest]]. The above collage includes them all. Looks alright, I guess, but something feels off…
## Identifying Duplicates
Just because each basic land has been printed over 500 times doesn’t mean there are 500 different pieces of art. A handful of pieces appear repeatedly – particularly in the promotional releases on the right side of the collage above. I set out to remove the duplicates. Turns out that’s easier said than done2.
It’s trivial to check if images are pixel-by-pixel identical, but that’s not the situation here. We’re looking at different scans of the same original piece of art. That means slight differences in cropping and exposure, plus perhaps some digital retouching.
The correct way to solve this problem probably involves machine learning, neural networks, or some other big data buzzword. A bit beyond my expertise, and a bit overkill for this particular project. Instead, after a fair amount of trial-and-error, I essentially decided to identify duplicates by squinting:
• Convert color images to grayscale.
• Scale down to a coarse grid.
• To compare two images, subtract their grids.
• If the differences are uniformly small, the images match.
The figure below shows how this process plays out. It correctly matches a pair of pieces by Alayna Danner, one of which has a weird foil overlay. And it correctly distinguishes a pair of pieces by John Avon, despite their similarities.
Illustration of how the matching algorithm works. It can identify a match despite the weird foil overlay. It can also distinguish a pair of pieces with similar color scheme and composition.
The above figure uses a 4x4 grid for legibility; in practice, the sweet spot seems to be 6x6. If the grid is too tight, the algorithm is easily confused by differences in cropping. If it’s too coarse, distinctive features are often lost, resulting in false positives.
That said, no matter how well we dial in the grid, it bears noting that this is a pretty good algorithm for identifying duplicate images – not a perfect one. In particular, it gets confused when artists do [[269629:throwback]] [[245246:pieces]]. We won’t be filing any patents, but it works well enough to reduce our 500+ printings down to about 200 mostly-unique ones:
Collage of [[Forest]] illustrations. Duplicates removed. Full-sized version here, art credits here.
## Optimizations
The matching algorithm only handles pairwise comparisons. That means, to find all the duplicates in a pool of 587 images by brute force, we’d need to make 172k comparisons3. That’s how I coded it up the first time through, and it slowed my little netbook to a crawl. Eventually, I realized that there was a huge optimization staring me in the face: artist name! If we have a piece by Alayna Danner and another by Jim Nelson, we already know they’re different. No math required.
Rob Alexander has art on 44 [[Forest]] printings, so it takes 946 comparisons to find all the duplicates. Terese Nielsen has five illustrations, so that’s another ten comparisons. All in all, 71 defferent artists are credited on Forests, and it takes about 12k comparisons to identify all the duplicates for all of them – less than a tenth of what we would have to do using brute force! On my machine, I can load up all the images, purge the duplicates, and assemble a canvas in about a minute.
Of course, I don’t load all the images at the same time – that’s my other optimization. More than once, the OS killed my Python script for being too much of a memory hog. So now I’m more careful with resources.
Each image gets loaded once at the beginning to grab some metadata. We store image dimensions, plus we compute the grid of grayscale values for the matching algorithm. While we’re at it, we crunch out the average color for each image as well. Then we free up that memory – keeping only the metadata – and load the next one.
Collage of [[Island]] illustrations. Duplicates removed. Full-sized version here, art credits here.
Once we have metadata for each image, we use the grayscale grids to identify duplicates. Based on the number of unique images, we can then figure out an appropriate number of rows and columns for the collage. And, with each illustration’s dimensions, we can compute a common size to crop to. Then we go back and – one at a time – load the images again. Each gets resized, cropped to the appropriate aspect ratio, and slapped onto the canvas.
## Finishing Touches
Scryfall orders search results chronologically, which is very practical, but practicality isn’t really what we’re after. Personally, I think it’s sharper to have illustrations ordered by average color instead. From left to right, the illustrations in each collage go dark to light. Then, within each row, they’re sorted from red to blue. For the [[Forest]] art above, it doesn’t make much of a difference. It’s all green. Similarly, the [[Island]] collage is all blue, and the [[Swamp]] collage is all brown and purple:
Collage of [[Swamp]] illustrations. Duplicates removed. Full-sized version here, art credits here.
But the [[Plains]] and [[Mountain]] collages are a whole different story. In each, there’s a visible transition between the warm colors of the earth and the cool blue of the sky.
Collage of [[Plains]] illustrations. Duplicates removed. Full-sized version here, art credits here.
Collage of [[Mountain]] illustrations. Duplicates removed. Full-sized version here, art credits here.
Sometimes, art is more art than science. Sometimes it’s not!
1. If you’re actually wading through the code, you’ll notice that dumping the image to a file is actually a multi-stage process. The images served by Scryfall are formatted as JPGs. We’d rather work with PNGs, so we do a quick check-and-convert.
2. As it turns out, Scryfall flags duplicate illustrations via the illustration ID. But that takes all the fun out of it!
3. The number of ways to choose two different items from a pool of N items is N*(N-1)/2, also called “N choose 2.”
© Charles Fyfe 2019 under CC-BY
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+1 vote
183 views
I have a datagridview in c# with rows , i need to bring the item which is select from combobox ..If value of combobox matches with datagridview particular cell then it should come at first.
How to achieve the same in C#.net forms app ?
Category
asked Jul 23, 2016 | 183 views
+1 vote
Below im searching for productname which is available in datagridview column cell 2 , I achieved the result with following code :
String searchValue = cmbProductName.Text; int rowIndex = -1; foreach (DataGridViewRow row in dgvBarcodePrinting.Rows) { if (row.Cells[2].Value.ToString().Equals(searchValue)) { rowIndex = row.Index; break; } } dgvBarcodePrinting.Rows[rowIndex].Selected = true; dgvBarcodePrinting.FirstDisplayedScrollingRowIndex = dgvBarcodePrinting.SelectedRows[0].Index;
edited Jul 23, 2016
You can simple use jQuery CSS method to change the color of the selected row. You have to use this method on the change event of combobox.
For this you have to give some class to your combo box.
<select class="cmbbox">
<option value="val1">text1</option>
<option value="val2">text2</option>
<option value="val3">text3</option>
</select>
Then use the change event of it to change the color of the particular row.
\$(".cmbBox").change(function() {
//change the row color using jquery css method
})
In this way you can even do other tasks easily.
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Bisexual Dating online
# 3 day-rule relationships t in the last component, Relationships in Categorical information
3 day-rule relationships t in the last component, Relationships in Categorical information
Independence and Conditional Possibility
Recollection thaith Intro to chance, we launched the notion of the conditional possibility of an event.
Check out examples:
• the possibility that a randomly selected feminine college student is in the Health technology program: P(fitness Science | women)
• P(you were not a medicine consumer given that anyone have an optimistic examination benefit) = P(maybe not a medicine user | positive examination consequences)
Now we inquire practical question, how do we determine whether two happenings tend to be separate?
## Identifying Individual Events
To answer this matter, we compare the chances that an arbitrarily picked beginner was a fitness technology major with the possibility that an arbitrarily chosen female scholar is a fitness Science major. If these probabilities are identical (or very near), we point out that the activities were separate. This basically means, independence ensures that are women cannot change the chances of registration in a Health Science regimen.
To resolve this concern, we examine:
• the unconditional chance: P(Health Sciences)
• the conditional possibility: P(fitness Sciences | feminine)
If these probabilities were equal (or perhaps near equal), then we could determine that registration in Health Sciences is independent of being a female. If the probabilities become considerably various, subsequently we say the factors tend to be established.
Both conditional and unconditional probabilities is tiny; however, 0.068 is relatively huge compared to 0.054. The proportion of the two rates is actually 0.068 / 0.054 = 1.25. And so the conditional chance try 25per cent bigger than the unconditional probability. It really is greatly predisposed that a randomly chosen female pupil is in the Health research regimen than that a randomly chosen beginner, irrespective of sex, is within the fitness research program. You will find a sizable adequate differences to recommend a relationship between are female and being enrolled in the Health technology plan, so these events are reliant.
## Feedback:
To determine if registration from inside the wellness technology program are independent of whether students is female, we can also compare the possibility that students are female with all the probability that a Health research student was feminine.
We come across once more your possibilities aren’t equivalent. Equal possibilities need a ratio of just one. The proportion was $\frac<\text<0.517>><\text<0.654>>\approx \text<0.79>$, which can be perhaps not close to one. Really more likely that a randomly picked Health technology college student is female than that a randomly chosen beginner are feminine. This is another way to note that these activities become based upon.
If P(A | B) = P(A), then two events A and B become independent.To state two events tend to be separate implies that the occurrence of just one event will make it neither much more nor considerably possible that the more occurs.
## Check It Out
In affairs in Categorical facts with Introduction to likelihood, we discovered limited, conditional, and joint probabilities. We have now develop a useful tip that applies marginal, conditional, and combined probabilities.
## A Tip That Applies Joint, Marginal, and Conditional Probabilities
Let’s start thinking about our body image two way desk. Here are three probabilities we computed before:
Conditional possibility: $P(\mathrm|\mathrm)=\frac<560><855>$
Remember that these three probabilities best incorporate three numbers from desk: 560, 855, and 1,200. (We grayed out of the remainder of the table so we can give attention to these three figures.)
Today see what goes on when we multiply the marginal and conditional probabilities from Bisexual dating apps overhead.
The result 560 / 1200 is exactly the worthiness we located for any combined chances.
When we write this partnership as a formula, we now have an example of a standard guideline that applies mutual, marginal, and conditional probabilities.
In terminology, we’re able to state:
• The joint chance equals this product of marginal and conditional probabilities
This is a general connection that is constantly real. Typically, if A and B are a couple of happenings, next
P(A and B) = P (A) · P(B | A)This guideline is always real. It has no problems. They constantly works.
When the activities become independent, next P (B | A) = P(B). So our rule becomes
P(A and B) = P(A) · P(B)This version of the rule best works after occasions are separate. Because of this, many people make use of this link to identify separate happenings. They reasons because of this:
If P(the and B) = P (A) · P(B) does work, then the happenings become separate.
## Comment:
Here we want to tell you it is occasionally more straightforward to consider probability trouble without having to worry about policies. This is particularly very easy to create when you have a table of data. But if you utilize a rule, be careful that you check the circumstances necessary for by using the tip.
## Pertaining Marginal, Conditional, and Joint Probabilities
What is the chance that a student is actually a men along with the knowledge technology regimen?
There’s two how to figure this :
(1) merely use the dining table to discover the mutual possibility:
(2) Or make use of the rule:
## Test It
All examples of separate events we need encountered so far have actually included two-way tables. Another example illustrates how this idea can be used in another context.
## A Coin Research
Think about the after quick test. Both you and a buddy each take out a coin and flip they. What’s the probability that both coins show up minds?
Let’s start by noting everything we discover. There Have Been Two occasions, each with chance ?.
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# 2.2: Natural Numbers. Induction
The element 1 was introduced in Axiom 4(b). Since addition is also assumed known, we can use it to define, step by step, the elements
$2=1+1,3=2+1,4=3+1, \text{ etc.}$
If this process is continued indefinitely, we obtain what is called the set $$N$$ of all natural elements in the given field $$F .$$ In particular, the natural elements of $$E^{1}$$ are called natural numbers. Note that
$(\forall n \in N) \quad n+1 \in N$
*A more precise approach to natural elements is as follows. $$A$$ subset $$S$$ of a field $$F$$ is said to be inductive iff
\begin{align} (i)& \hskip 4pt 1 \in S \text{ and } \\ (ii)& \hskip 4pt (\forall x \in S) \hskip 4pt x+1 \in S \end{align}
Such subsets certainly exist; e.g., the entire field $$F$$ is inductive since
$1 \in F \text{ and } (\forall x \in F) \hskip 4pt x+1 \in F.$
Define $$N$$ as the intersection of all inductive sets in $$F$$ .
Theorem $$\PageIndex{1}$$
The set $$N$$ so defined is inductive itself. In fact, it is the "smallest" inductive subset of $$F$$ (i . e ., contained in any other such subset).
Proof
We have to show that
\begin{align} (i)& \hskip 4pt 1 \in N, \text{ and } \\ (ii)& \hskip 4pt (\forall x \in N) \hskip 4pt x+1 \in N. \end{align}
Now, by definition, the unity 1 is in each inductive set; hence it also belongs to the intersection of such sets, i.e., to $$N .$$ Thus $$1 \in N,$$ as claimed.
Next, take any $$x \in N .$$ Then, by our definition of $$N, x$$ is in each inductive set $$S ;$$ thus, by property $$(i i)$$ of such sets, also $$x+1$$ is in each such $$S$$ ; hence $$x+1$$ is in the intersection of all inductive sets, i.e.,
$x+1 \in N$
and so $$N$$ is inductive, indeed.
Finally, by definition, $$N$$ is the common part of all such sets and hence contained in each. $$\square$$
For applications, Theorem 1 is usually expressed as follows.
Theorem $$\PageIndex{1'}$$
(first induction law). A proposition $$P(n)$$ involving a natural $$n$$ holds for all $$n \in N$$ in a field $$F$$ if
\begin{align} (i)& \text{ it holds for } n=1,\text{ i.e., } P(1) \text{ is true; and } \\ (ii)& \text{ whenever } P(n) \text{ holds for } n=m, \text{ it holds for } n=m+1, \text{ i.e., } \end{align}
$P(m) \Longrightarrow P(m+1).$
Proof
Let $$S$$ be the set of all those $$n \in N$$ for which $$P(n)$$ is true,
$S=\{n \in N | P(n)\}$
We have to show that actually each $$n \in N$$ is in $$S,$$ i.e., $$N \subseteq S$$
First, we show that $$S$$ is inductive.
Indeed, by assumption $$(\mathrm{i}), P(1)$$ is true; so 1$$\in S$$.
Next, let $$x \in S .$$ This means that $$P(x)$$ is true. By assumption (ii), however, this implies $$P(x+1),$$ i.e., $$x+1 \in S .$$ Thus
$1 \in S \text{ and } (\forall x \in S) \hskip 4pt x+1 \in S$
$$S$$ is inductive.
Then, by Theorem 1 (second clause), $$N \subseteq S,$$ and all is proved. $$\square$$
This theorem is used to prove various properties of $$N$$ "by induction."
Example $$\PageIndex{1}$$
(a) If $$m, n \in N,$$ then also $$m+n \in N$$ and $$m n \in N$$.
To prove the first property, fix any $$m \in N .$$ Let $$P(n)$$ mean
$m+n \in N \quad(n \in N)$
Then
(i) $$P(1)$$ is true, for as $$m \in N,$$ the definition of $$N$$ yields $$m+1 \in N$$, i.e., $$P(1)$$.
(ii) $$P(k) \Rightarrow P(k+1)$$ for $$k \in N .$$ Indeed,
\begin{align} P(k) \Rightarrow& m+k \in N \Rightarrow(m+k)+1 \in N\ \\ \Rightarrow& m+(k+1) \in N \Rightarrow P(k+1) \end{align}
Thus, by Theorem $$1^{\prime}, P(n)$$ holds for all $$n ;$$ i.e.,
$$(\forall n \in N) \quad m+n \in N$$
for any $$m \in N$$.
To prove the same for $$m n,$$ we let $$P(n)$$ mean
$$m n \in N \quad(n \in N)$$
and proceed similarly.
(b) If $$n \in N,$$ then $$n-1=0$$ or $$n-1 \in N$$.
For an inductive proof, let $$P(n)$$ mean
$n-1=0 \text{ or } n-1 \in N \quad(n \in N)$
Then proceed as in (a).
(c) In an ordered field, all naturals are $$\geq 1$$.
Indeed, let $$P(n)$$ mean that
$n \geq 1 \quad(n \in N).$
Then
(i) $$P(1)$$ holds since $$1=1$$
(ii) $$P(m) \Rightarrow P(m+1)$$ for $$m \in N,$$ since
$P(m) \Rightarrow m \geq 1 \Rightarrow(m+1)>1 \Rightarrow P(m+1)$
Thus Theorem $$1^{\prime}$$ yields the result.
(d) In an ordered field, $$m, n \in N$$ and $$m>n$$ implies $$m-n \in N$$
For an inductive proof, fix any $$m \in N$$ and let $$P(n)$$ mean
$m-n \leq 0 \text{ or } m-n \in N \quad(n \in N).$
Use (b).
(e) In an ordered field, $$m, n \in N$$ and $$m<n+1$$ implies $$m \leq n$$
For, by $$(\mathrm{d}), m>n$$ would imply $$m-n \in N,$$ hence $$m-n \geq 1,$$ or $$m \geq n+1,$$ contrary to $$m<n+1$$.
Our next theorem states the so-called well-ordering property of $$N$$.
Theorem $$\PageIndex{2}$$ (well-ordering of N)
In an ordered field, each nonvoid set $$A \subseteq N$$ has a least member (i.e., one that exceeds no other element of $$A )$$.
Proof
The following is just an outline of a proof:
Given $$\emptyset \neq A \subseteq N,$$ let $$P(n)$$ be the proposition "Any subset of $$A$$ containing elements $$\leq n$$ has a least member" $$(n \in N) .$$ Use Theorem $$1^{\prime}$$ and Example (e) . $$\square$$
This theorem yields a new form of the induction law.
Theorem $$\PageIndex{2'}$$ (second induction law)
A proposition $$P(n)$$ holds for all $$n \in N$$
(i') $$P(1)$$ holds and
(ii') whenever $$P(n)$$ holds for all naturals less than some $$m \in N,$$ then $$P(n)$$
also holds for $$n=m$$ .
Proof
Assume $$\left(\mathrm{i}^{\prime}\right)$$ and $$\left(\mathrm{ii}^{\prime}\right) .$$ Seeking a contradiction, suppose there are some $$n \in N($$ call them "bad") for which $$P(n)$$ fails. Then these "bad" naturals form a nonvoid subset of $$N,$$ call it $$A$$.
By Theorem $$2, A$$ has a least member $$m .$$ Thus $$m$$ is the least natural for which $$P(n)$$ fails. It follows that all $$n$$ less than $$m$$ do satisfy $$P(n) .$$ But then, by our assumption (ii'), $$P(n)$$ also holds for $$n=m,$$ which is impossible for, by construction, $$m$$ is "bad" (it is in $$A ) .$$ This contradiction shows that there are no "bad" naturals. Thus all is proved. $$\square$$
Note 1: All the preceding arguments hold also if, in our definition of $$N$$ and all formulations, the unity 1 is replaced by 0 or by some $$k( \pm k \in N)$$. Then, however, the conclusions must be changed to say that $$P(n)$$ holds for all integers $$n \geq k$$ (instead of "n \geq 1 "). We then say that "induction starts with $$k .$$"
An analogous induction law also applies to definitions of concepts $$C(n)$$.
$$A$$ notion $$C(n)$$ involving a natural $$n$$ is regarded as defined for each $$n \in N$$ $$\left(i n E^{1}\right)$$ if
(i) $$i t$$ is defined for $$n=1$$ and
(ii) some rule is given that expresses $$C(n+1)$$ in terms of $$C(1), \ldots, C(n)$$.
(Note 1 applies here, too.)
$$C(n)$$ itself need not be a number; it may be of quite general nature.
We shall adopt this principle as a kind of logical axiom, without proof (though it can be proved in a similar manner as Theorems $$1^{\prime}$$ and $$2^{\prime} ) .$$ The underlying intuitive idea is a "step-by-step" process - first, we define $$C(1) ;$$ then, as $$C(1)$$ is known, we may use it to define $$C(2) ;$$ next, once both are known, we may use them to define $$C(3) ;$$ and so on, ad infinitum. Definitions based on that principle are called inductive or recursive. The following examples are important.
Example $$\PageIndex{1}$$ (continued)
(f) For any element $$x$$ of a field, we define its $$n^{th}$$ power $$x^{n}$$ and its $$n$$-multiple $$n x$$ by
(i) $$x^{1}=1 x=x$$
(ii) $$x^{n+1}=x^{n} x$$ (respectively, $$(n+1) x=n x+x )$$.
We may think of it as a step-by-step definition:
$x^{1}=x, x^{2}=x^{1} x, x^{3}=x^{2} x, \text{ etc.}$
(g) For each natural number $$n,$$ we define its factorial $$n !$$ by
$1 !=1,(n+1) !=n !(n+1);$
e.g. $$, 2 !=1 !(2)=2,3 !=2 !(3)=6,$$ etc. We also define $$0 !=1$$.
(h) The sum and product of n field elements $$x_{1}, x_{2}, \dots, x_{n},$$ denoted by
$\sum_{k=1}^{n} x_{k} \text{ and } \prod_{k=1}^{n} x_{k}$
or
$x_{1}+x_{2}+\cdots+x_{n} \text{ and } x_{1} x_{2} \cdots x_{n}, \text{ respectively}$
are defined recursively.
Sums are defined by
\begin{align} (i)& \hskip 4pt \sum_{k=1}^{1} x_{k}=x_{1} \\ (ii)& \hskip 4pt \sum_{k=1}^{n+1} x_{k}=\left(\sum_{k=1}^{n} x_{k}\right)+x_{n+1}, n=1,2, \ldots \end{align}
Thus
$x_{1}+x_{2}+x_{3}=\left(x_{1}+x_{2}\right)+x_{3}$
$x_{1}+x_{2}+x_{3}+x_{4}=\left(x_{1}+x_{2}+x_{3}\right)+x_{4}, \text{ etc.}$
Products are defined by
\begin{align}(i)& \hskip 4pt \prod_{k=1}^{1} x_{k}=x_{1} \\ (ii)& \hskip 4pt \prod_{k=1}^{n+1} x_{k}=\left(\prod_{k=1}^{n} x_{k}\right) \cdot x_{n+1} \end{align}
(i) Given any objects $$x_{1}, x_{2}, \dots, x_{n}, \ldots,$$ the ordered n-tuple
$\left(x_{1}, x_{2}, \ldots, x_{n}\right)$
is defined inductively by
(i) $$\left(x_{1}\right)=x_{1}$$ (i.e., the ordered "one-tuple" $$\left(x_{1}\right)$$ is $$x_{1}$$ itself $$)$$ and
(ii) $$\left(x_{1}, x_{2}, \ldots, x_{n+1}\right)=\left(\left(x_{1}, \ldots, x_{n}\right), x_{n+1}\right),$$ i.e., the ordered $$(n+1)-$$ tuple is a pair $$\left(y, x_{n+1}\right)$$ in which the first term $$y$$ is itself an ordered $$n$$ -tuple, $$\left(x_{1}, \ldots, x_{n}\right) ;$$ for example,
$\left(x_{1}, x_{2}, x_{3}\right)=\left(\left(x_{1}, x_{2}\right), x_{3}\right), \text{ etc.}$
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# Are all quadratic terms in gauge fields necessarily mass terms?
1. Feb 2, 2010
### QuantumSkippy
Can someone please help me out with mass terms in the general case for a lagrangian?
It is known that for n scalar fields, any quadratic in these fields will be a mass term.
For classical fields $$\varphi_{j}$$ with the most genral possible expression being $$M^{jk}\varphi_{j}\varphi_{k}$$ , the matrix $$M^{jk}$$ is guaranteed to be symmetric and so can be diagonalised with an orthogonal similarity transformation. So there is no argument there - we get a sum of squares after diagonalisation of the form $$\sum_{j} M^{jj}\varphi_{j}\varphi_{j}$$
For the case of gauge fields, however, it does not seem (??? help me out here!!) that just any quadratic at all will necessarily be a mass term. Here is the reasoning as I see it:
No one would disagree that a term like $$M^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}$$ is definitely a mass term. Again, for classical fields the mass matrix $$M^{jk}$$ is guaranteed symmetric by the sum over symmetric terms and so is once more diagonalisable with an orthogonal similarity transformation.
Things seem different for the most general case. For example with a sum like$$M_{\mu\nu}^{jk}A^{\mu}_{j}A^{\nu}_{k}$$ , one would expect that a Lorentz transformation can reduce the term$$M_{\mu\nu}^{jk}$$ to something of the form $$g_{\mu\nu}m^{jk}$$.
In this way, the 'mass' term becomes $$m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}$$ after the Lorentz transformation has been applied.
Observe however, that for the Lorentz transformation $$L_{\alpha}^{\mu}$$ which achieves this change we have
$$M_{\mu\nu}^{jk}L_{\alpha}^{\mu}L_{\beta}^{\nu} = m^{jk}g_{\alpha\beta}$$ .
Multiplying both sides of this equation by $${(L^{-1})}^{\alpha}_{\mu}{(L^{-1})}^{\beta}_{\nu}$$, we obtain the result that
$$M_{\alpha\beta}^{jk} = m^{jk}g_{\alpha\beta}$$.
This follows from the orthogonality of the Lorentz transformations with respect to the metric $$g_{\alpha\beta}$$.
So the upshot of this appears to be that unless terms are of the form $$m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}$$ they cannot represent mass terms and are in fact, self interaction terms.
Please help me out here, as this is the only way I can interpret quadratic terms in the gauge fields at present.
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# What is a file?
I recently had a student in my Intro to CS class ask me: what is a file? I can't honestly say I had a good answer for him. Those of us who have been around a while know what a file is, but I was at a bit of a loss to explain it to a novice.
Today's students believe (and rightfully so) that a file is an icon-thingy in a folder or on the Desktop. They have some vague idea that there are files for images (JPEG, GIF, PSD, etc.), files for music (MP3, WAV), files for doing office stuff (docx, pptx, etc.), and maybe even HTML files. But they don't know what's in a file. That's where I kind of stumbled around for a few minutes because I didn't have a good answer prepared, nor a really good demonstration.
Looking back, I think I might have shown him what a text file looks like using a hex editor. Maybe also a simple image format such as PBM or BMP. I searched around on YouTube for a good video, but found pretty much nothing.
• Long answer later, if a better one don't show up sooner. To answer the question, next time, just grab a file, or folder, out of your desk and show them a "file." English, German, binary, hex, or Martian, is merely the "encoding." Apr 20 '19 at 7:22
• I upvoted the question when you said Hex Editor. Great start! Apr 20 '19 at 12:51
• Related question? -- cseducators.stackexchange.com/questions/3535/… Apr 20 '19 at 22:44
• Necessary viewing for ppl who chant Everything is a file in *nix youtu.be/9-IWMbJXoLM and don't recognize that that is part of the problem not the solution!
– Rusi
Sep 12 at 15:13
I would approach this pragmatically, beginning with the metaphor that kicked it off. I would tell the students something like this:
Imagine that it's 1925, and you're working at a giant company like General Electric as a secretary. There are files, meaning pieces of paper, for all kinds of things. Personnel files about employees, files of record for payments, files of company policies, board meetings, tax documents, you make it. There are hundreds of rooms holding thousands of file cabinets holding millions of files in hanging folders.
So the metaphor of a file, then, is a document with some information about something. And the metaphor for the folder is as a place to store those files.
We also call these folders "directories", another metaphor. A "directory" is from the same etmological root as direction, and it means "to guide". So, folders can be thought of as places where files sit, or can be thought of as a guide to where files are.
But translating that idea to the very alien mind of a computer involves some tweaks. First there's the idea that folders can contain folders, which can contain more folders still. This is incredibly convenient, and using the same metaphor as "directory", we call the list of folders and subfolders and subfolders that bring us to a particular file as a "path". Get it?
As for the files themselves, there are two perspectives to think about this. Again, the metaphor is there, but it breaks apart a bit in translation:
The first is from the file system, which is concerned with being able to store and retrieve these files. As far as the file system is concerned, a file is a size and a series of locations as to where the various parts of the file are stored. This is because the files don't have to be contiguous inside the computer. It doesn't actually matter if the first half is stored over here in the hard drive and the second half is stored somewhere else, since the file system will retrieve it for us as if it were one document anyway. The file system just has to keep track of everything so that the files can be assembled properly when they are needed.
The second perspective is the perspective of a file itself, which is really just some way to store some data, after all. So far, that's just like a file in our big office. But our files can store so many different kinds of data! We aren't just storing readable text. We can store pictures. We can store sounds. We can store runnable code. All of these require very different internal organizations, so the contents of files are extremely variable.
Many files begin with some sort of metadata. "Hey, I'm a picture, and this is my encoding and my color depth and my size and and and and...". Some files depend on the file system to just remember what kind of files they are. Every file is designed to be read by programs or by the computer itself, and is highly organized to make this possible.
So, a "file" means different things in different contexts, and the only overriding idea is that a file is a way to group data together.
• Everything in computing is either a container or something that goes in a container. Because everything in our minds is either a concept, or an instance of a concept. The hardest thing is coming up with so many names for stuff that never did exist in the first place. (And cache invalidation - the failure of a name.) Apr 22 '19 at 11:26
• this explanation gives the impression that a path leads to a file, which is incorrect. The file name is part of the path. Oct 29 at 15:18
The question is actually quite deep. For instance, I'm sure you've heard the phrase "everything is a file," that is associated with a design of Unix operating system. So, the answer could be puzzling "a file is almost everything in Unix," for example devices can "look like" files to some processes. They use a file as a common interface between many processes. Hence, the file is quite a general notion of a sequence of bytes in Unix.
My simple answer would be that a file is a sequence of bytes with a reference to them, by which you can access these bytes.
"File" is an abstraction. Even worse, we use it for different abstractions.
Well, suppose I copy a file to this site, and you download it. Would you say it is the same file? What do you mean by same?
Actually, we'll have, more or less, parts of our hard disks (or is it a floppy?) containing bytes which we understand (through the magic of operating systems) as a representation of the same informations.
• Hard disks are being replaced by solid state 'drives'. (What exactly is driving or driven? Even the words are abstracted away to absurdity.) And much of what we do is "in the cloud" now, so even talking about your computer doesn't apply anymore. Apr 20 '19 at 17:35
• My not so abstract computer actually has a hard disk, so it does Apr 21 '19 at 12:38
• Hey, I have a dial telephone sitting on a shelf, but nothing to plug it in to. Give it a few years, then computers will no longer even have screens or keyboards, because most of what we need can be done with voice, as a million years of evolution sans reading has demonstrated. It is being done with voice, and increasingly will be. The question "what is a file?" will no longer have any referent, or anything to prompt it. Files and screens were a temporary phase, like telephone dials and switchboard operators. Apr 21 '19 at 13:41
I think in this case a short answer is better than a long one, which is harder, but I might give it a shot:
A file is a list of bytes and an identifier [location].
Make sure to explain that there are no physical files and the term is just an abstraction. With my definition you can explain the following:
• If you create a new file with a different identifier but the same bytes, it is a copy of the file.
• If you just change the identifier [location], you say the file moves.
• If you keep the identifier but change the bytes, the file is said to change.
The identifier can be different - e.g. a filename ("C:\Desktop\MyFile.txt") or a URL ("https://www.mydomain.edu/MyFile.txt") - but the content is basically always a list of 0 to many bytes.
You could go into details how files are stored and how HTTP works, but in the end you only need to explain the abstraction to understand what "file" means.
• Nice answer (the best I can see) : A list of bytes...
– Rusi
Sep 13 at 4:06
Nothing wrong with citing all the "dictionary" definitions found on the Internet.
I'd add that a file tends to be the lowest level of abstraction of a container or boundary around data that the Operating System manages. That is the file's name is accessed through the OS and if you want to know the bits and bytes of data "in" the file you need an Application or program to read/write the file. The OS "promises" to return the data to any Application in the same order in which the creating Application specified but physically the OS may scatter the data to the wind - as long as the OS can return the data properly to some Application.
• Right, the OS exists basically to manage files and launch programs. Apr 20 '19 at 22:45
A slightly different but insightful take on this :-
On UNIX systems everything is a file. A file of data is a datafile but a directory (folder) is also a file. A device is a file (found in the /dev directory). Programs are executable files and even running program and system states have associated files (found in /run). Network connections are files that can be read and written.
I suppose that explains why its difficult to define what a file is. In a physical sense it is just a list of connected disk or tape or memory sectors containing codes for data values.
I'd actually answer this question from an Intro student from more of the HCI perspective, because they likely have more experience using a computer than understanding how the computer works. @MichelBillaud had a similar notion in suggesting it as an abstraction (although most intro students won't fully appreciate what abstraction means the point of asking what a file is). Relating the abstraction to what they already understand in the physical world helps establish a good foundation.
I'd start by explaining that the first personal computers with graphical interfaces (Xerox Star if they want a historical context) faced the challenge of making it easy for someone who works in an office learn how to work with a computer, which stores information digitally instead of on paper.
In an office, they might have records of a client's information that they'd traditionally write on a piece of paper and store it away in a file cabinet. That file is a record of related information that you can look up by a name. That metaphor works for computers. They store information electronically instead of on paper, but that digital record is organized in one place and can be looked up by the file name.
From there, you can build upon that fundamental to explain different file formats (e.g. plain text vs binary) or how the metaphor extends to other abstractions of digital data represented in ways people can relate to in the physical world (e.g. folders, storage, copying/moving and recycle/trash, to other WIMP metaphors.
• Great explanation! Now we just need to re-base it to a world where people never used paper files, folders, desks, offices... What does "a record of clients information" mean to someone who has not used paper for that, or really know what a client is? Children truly face a crazy learning situation, and they always have, but now things have changed so much in one lifetime that our teaching is way behind. How do we get ahead of their already lived experience in order to show them where things are going? I bet we can't. So we have to explain clearly where things came from and hope for the best. Oct 1 at 9:59
As is the case with most simplified explanation, I have glossed over some more advanced specific details that don't factor into the basics on file systems.
## File content
At its most basic level, a file is really just a sequence of ones and zeroes. It's data. There's nothing more to it than that.
This data can then be interpreted. We (or an application) take the ones and zeroes, attribute meaning to them, and this interpretation is informative to us (or an application).
"Interpretation" is the operative keyword here. The same arbitrary sequence of ones and zeroes can be interpreted in any way you want to. However, in most cases only one interpretation will make sense, and other interpretation will look like garbage.
So how do we know which interpretation should be used for which file? Answer: the file extension. A file extension denotes how the data in the file is structured, and therefore how it can be sensically interpreted.
Your OS automates this process for you. Instead of you having to look at the extension and then find the correct application which can interpret your file in the way you need it to be interpreted, your OS automatically maps a given file extension to your preferred application. For example, you might configure Windows to open .txt files with WordPad, or Notepad, or Notepad++, or some application you wrote yourself. All of these application are capable of performing the interpretation that a .txt file needs.
You can override this behavior. You can open a .png file in Notepad. It won't make sense, but what you are seeing is the result of a .txt interpretation being applied to a file whose data structure is that of a .png.
.txt files are very easy to interpret. Ignoring encoding for now (let's assume ASCII), every byte (= 8 bits) of the ones and zeroes represents a single number, which in turn represents a single character.
To really prove the point, you could do this manually. Let's say I want to store a text file containing apple. You can use an ASCII table to look up what the numeric representation of each character is.
a = 01100001
p = 01110000
p = 01110000
l = 01101100
e = 01100101
If you use your text editor of choice, set the encoding to ASCII, enter apple and save the file, the file's ones and zeroes will exactly be the sequence of binary digits I just listed.
.txt is really easy, but file structures can get significantly more complex, to the point where it is no longer feasible for a human to manually interpret all the data. For example, even though a .pdf file might only contain text (to your eyes) just like a .txt, it actually stores a whole lot more data (text markup, file metadata, ...) and this makes it less than obvious to interpret the binary data ourselves.
But to prove the point on how you can interpret files in a non-text example, suppose you (as a teacher) want to track whether your students were present/absent every day of school. Assuming you have 16 students, you could track this using a very simple sequence of 16 binary digits:
1111111111011111
Everyone was present that day, except student 11.
Is this a good file format? Well, it's very efficient. But it also lacks context. There's an assumption on which students were in your class. There's no way to track justified absences, etc. To really cover all the information you need, you're going to need a more complex system. But if you assume a simple present/absent mark for a fixed list of students, the above file format could theoretically suffice.
Interesting experiment
Take an existing Word/Excel file, one with an "x" in the file extension (.docx or .xlsx). Change its file extension to .zip and try to open it. It... works!?
Since the advent of the new Office file formats, Word and Excel documents are really just secret zip archives, not custom binary file formats like they used to be. But by using a unique file extension instead of .zip, people can still configure these .docx or .xlsx files to be automatically opened using Word/Excel instead of however they've configured .zip files to be opened.
## File name (and path)
At its most basic level, a file name is just one long string of characters, nothing more.
Obviously, we're likely going to want to store more than one file on our computer. So we need a way to reference each file that we store. This is why we give them a name, to tell them apart. This is no different from why humans give eachother names.
As mentioned before, we add a file extension to the end of the name, so that we remind ourselves (and the OS) of the data structure that was used for the file content.
But we often have a lot of files. We'd really like to organize them properly into little collections, instead of throwing them all on one big messy pile. To do this, we started prepending our filenames with what we call the "path".
A path is nothing more than a (back)slash-separated sequence of names which list the hierarchy of how we'd like to organize our files. The OS will take the file name, split the names into the separate chunks, and will use that hierarchical information to show your files in a neat and organized manner.
Every file's name is really just a combination of:
[folder chunks][name of the file].[file extension]
Take the example of the following file names:
C:\Fruit\apple.txt
C:\People\PeopleILike\Tom.png
C:\Fruit\banana.txt
C:\People\PeopleIHate\Kevin.png
If you split the names in the chunks between the backslashes, you can start seeing the structure. For example, the first and third file have exactly the same chunks before their file name. Therefore, they belong in the same folder. The second and fourth file have a common first chunk, but then they have a different chunk. Therefore, they are found in the same parent directory but then live in a different subdirectory.
If you apply this logic, just like how an OS does, you come up with the intended organisational structure:
C/
├─ Fruit/
│ ├─ apple.txt
│ ├─ banana.txt
├─ People/
│ ├─ PeopleIHate/
│ │ ├─ Kevin.png
│ ├─ PeopleILike/
│ │ ├─ Tom.png
We like to think of our file system as this neat nested system of folders and subfolders. But in the underlying file system, we don't actually store a "folder" by itself. A folder is really just generated dynamically based on the slash-separated chunks found in a file's name.
This leads to some interesting effects:
• Want to move a file? All you have to change is the file's path. You don't need to update a folder itself.
• Want to rename a folder? Then you're going to need to change this name in all of the files who contain this folder in their path.
This is an oversimplification. File systems are more complicated than this, because you need the ability to e.g. create empty folders. But this is a good first explanation of how the file system works.
Next to the file name and content, there's more things we know about the file, such as when it was created, whether it is read-only, any permissions attached to the file, ... These things are not stored in the file content. They are an addendum to the file.
If you transfer a file from one file system to another, this data may be lost if the target system doesn't know how to work with the source system's metadata structure. However, this doesn't matter in terms of using the file itself. Your data is safe. You might just lose track of e.g. when your file was created.
• this is incorrect. A name is just the name of a single file or directory. What you describe as name is a path. Also, the name is part of the path. Oct 29 at 15:26
Was in a conference the other day and someone was showing intercept security... A .exe file started with MZ. Why? Those are the initials of the man who created the first DOS executable file format. Many kinds of files have 'signatures' in the first few bytes.
I used to use a tool called 'strings' that would show any displayable sequences of characters in an otherwise 'binary' file. Helpful sometimes. We have gotten far away from this level of reality in our everyday use of computers, so it is good to show that there is really no mystery, it is all just a heap of ideas people have had over time, piled up in layers. Start at the bottom with characters and follow the clues.
Attack it in many ways, showing the aspects of 'files' that are meaningful to you. Give some of the history. I am sure you can come up with analogies. One issue is that although 'file' makes some sense, 'folder' does not, because we don't ever nest manila folders! Nnnnt! The other term, 'directory', has fallen in to disuse, and really is not any better.
What is a synonym for what we call folder...? There isn't one! It is a brand new concept, like variables. Nothing else in our world is like that. This is why we have to show that analogies are limited and misleading. You have to actually know some things.
• And how many of the students have seen a manila folder? A floppy? A dial phone? They have no experience with the things we analogize from! Lost beyond lost in a hall of mirrors. Apr 20 '19 at 13:06
• @LogicalBranch Sure. Humans usually make meaning by relating to things in their environment. The floppy disk is the icon for 'store' because they used to be commonplace. But to someone who has never seen one, it doesn't have any inherent meaning. If a student is asking what a file is, it probably means they have no referent, no commonplace objects, situations and processes that they can intuitively substitute. If all of computing is strewn with metaphors and references to never-before-known things, it is basically like going to a foreign planet and trying to start from nothing. Apr 22 '19 at 11:12
• Beyond, that, the references seem pointless and strained. Floppies only make sense without an internet. And if your entire life experience is that the internet has always just "been there", how can any other scenario make any sense? If I put you down in the Amazon rainforest and said, "There is lots of food here. Just watch out for the dangerous things." would you be likely to survive even for 2 days, or know what killed you in your sleep? No. This is the world we put our children in to with no preparation, because we cannot see it from their point of view. Apr 22 '19 at 11:17
• Thanks for taking the time to explain, I appreciate it. Apr 22 '19 at 11:18
• Sure. This is why I keep crying like the little voice in the wilderness: "We have to start with the the basic level of how things actually work!" And everyone here says I am wrong. We shall see. Wait... We are seeing now. Apr 22 '19 at 11:20
A file is a metaphor, representing a mental model that would be familiar to office workers who were the initial users of computing systems.
Different users in a different context require different metaphors as they have different mental models.
• This is by far the best answer, cutting through all the detail to the real issue: files don't actually exist. (Folders doubly so.) No one has seen one at any time, they are a made up idea in a made up system in... a made up world. Until someone gets it that we invented everything except gravity and the integers, they will not see the way in which this question could even arise. Oct 1 at 10:07
• Wasn't it Pascal who said something like: Geometry is not 'true', it is convenient. Oct 1 at 10:10
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# Square root
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### Approximating a Square Root Without a Calculator
In mathematics, a square root of a number a is a number y such that y2 = a; in other words, a number y whose square (the result of multiplying the number by itself, or y⋅y) is a. For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16. Every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 32 = 3 • 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.
Explore contextually related video stories in a new eye-catching way. Try Combster now!
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# Compatibility between Physical Stimulus Size and Left-right Responses: Small is Left and Large is Right
## Abstract
According to a theory of magnitude (ATOM, Walsh, 2003, 2015), the cognitive representations of quantity, time, and space share a general magnitude code. Interestingly though, research has largely ignored the relationship between physical (stimulus) size and spatial (response) location. We conducted two experiments investigating compatibility effects between physical stimulus size and left-right responses. In both experiments, right-handed participants responded to a small or a large square stimulus by pressing a left or a right key. In Experiment 1, size was the relevant stimulus feature and we varied the S-R mapping within participants. Results revealed a strong compatibility effect: Performance was better with the compatible mapping (small-left and large-right) than with the incompatible mapping (large-left and small-right). In Experiment 2, participants responded to stimulus color, which varied independently of stimulus size, by pressing a left or right key. Results showed a congruency effect that mirrored the compatibility effect of Experiment 1. The results of our experiments suggest a strong relationship between the cognitive representation of physical (stimulus) size and response location in right-handers. The findings support the notion of a general magnitude code, as proposed in ATOM.
Keywords:
How to Cite: Wühr, P., & Seegelke, C. (2018). Compatibility between Physical Stimulus Size and Left-right Responses: Small is Left and Large is Right. Journal of Cognition, 1(1), 17. DOI: http://doi.org/10.5334/joc.19
Published on 27 Feb 2018
Accepted on 05 Feb 2018 Submitted on 13 Nov 2017
## Introduction
In “A Theory Of Magnitude” (ATOM), Walsh (2003, 2015) proposed a generalized magnitude-processing system in the human brain that uses a common metric for processing information about time, space, number, and other magnitudes. Compelling evidence for a generalized magnitude system has come from neuropsychological and neurophysiological data, suggesting overlapping brain structures for the processing of time, space, and magnitude information in human parietal cortex (e.g., Cohen Kadosh et al., 2007; Kaufmann et al., 2008; see Bueti & Walsh, 2009, for review). According to Walsh (2003, 2015), the magnitude system has evolved in order to support action control, because the successful control of action requires the integration of information about temporal, spatial and quantity-related features of a desired action. ATOM predicts and explains (mutual) interactions and interference effects in the simultaneous processing of information about time, space, number, size, and other magnitudes. With regard to the direction of interference effects, ATOM assumes “some monotonic mapping of quantities: bigger, faster, brighter, further in one domain should correlate with bigger, faster, brighter, further in another” (Bueti & Walsh, 2009, p. 1832). Behavioral studies revealed evidence for relationships between most of the dimensions addressed in ATOM (for reviews, see, Bonato, Zorzi, & Umiltà, 2012; Winter, Matlock, Shaki, & Fischer, 2015), but the relationship between number and space, and the relationship between number and physical size have gained considerably more interest than the relationship between physical size and space. It should be noted that ATOM does not readily predict a particular mapping between magnitudes, such as number or size, and horizontal spatial positions. To account for such mapping effects, additional assumptions are required that will be discussed later.
### Number and space
Dehaene, Dupoux and Mehler (1990) first demonstrated a compatibility effect between numerical size and horizontal response location. Their participants classified two-digit numbers as larger than or smaller than a standard, by pressing a left or right key. Results showed that left-hand responses were faster to small as compared to large numbers, whereas right-hand responses were faster to large as compared to small numbers. Dehaene et al. (1993) further explored this so-called “Spatial-Numerical Association of Response Codes” (SNARC) effect, and observed some interesting features of the effect. First, the authors demonstrated the SNARC effect in a parity-judgment task, where number magnitude was no longer relevant for the participants’ judgments, indicating that processing number magnitude and activation of the compatible response is automatic to some degree. Second, the authors ruled out handedness as a factor, because left- and right-handed participants showed similar SNARC effects (see, Fischer, 2008, for converging evidence).
To account for the SNARC effect, Dehaene et al. (1993) proposed a spatial representation of number magnitude – the so-called mental number line – that extends from left to right and represents small numbers to the left and large numbers to the right. Based on their observation that the habitual reading and writing direction of the participants affected the direction of the SNARC effect, Dehaene et al. suggested that the mental number line originated as a result of reading and writing habits. Subsequent results on the impact of reading and writing habits on the SNARC effect were inconclusive, however (see Fischer, 2013, for a review). Alternatively, researchers suggested that the mental number line may have originated from counting habits. In particular, studies showed that children prefer to count a row of objects from left to right (e.g., Opfer, Thompson, & Furlong, 2010). Moreover, in finger counting, both children and adults show a preference to start counting on the fingers of their left hand (e.g., Fischer, 2008). Whereas reading and counting habits are ontogenetic sources of a spatial mapping of numbers, the observation of SNARC-like effects in newborn chicks suggest that phylogenetic variables could also play a role (Rugani, Vallortigara, Priftis, & Regolin, 2015). All these sources are not mutually exclusive, however, and it is therefore possible that multiple sources contribute to a spatial mapping of numbers to horizontal locations, as expressed in the SNARC effect (cf. Winter et al., 2015).
### Number and Size
One of the most prominent empirical demonstrations for an interaction between number magnitude und physical size is the “size-congruity effect” (e.g., Henik & Tzelgov, 1982). The size-congruity effect has been demonstrated in number-comparison tasks (e.g., Besner & Coltheart, 1979) and in Stroop-like paradigms (e.g., Tzelgov, Meyer, & Henik, 1992). In a number-comparison task, participants are presented with two numbers varying in numerical and physical size. Besner and Coltheart (1979) showed that judging number magnitude is faster when the irrelevant physical size is congruent rather than incongruent with the to-be-judged numerical magnitude. Henik and Tzelgov (1982) later demonstrated that the size-congruity effect does also occur in the reverse direction: judging physical size is also faster when the irrelevant numerical size is congruent rather than incongruent with to-be-judged physical stimulus size.
The size-congruity effect does also occur in Stroop-like tasks, where participants have to compare either the numerical size or the physical size of a single number stimulus with an internal (i.e., memorized) standard. For example, Tzelgov et al. (1992, Experiments 1 and 2) presented a single digit (2, 3, 4, 6, 7 or 8) in one of two physical sizes at screen center. In the number-judgment task, participants had to indicate whether the digit was numerically larger (or smaller) than the standard 5. In the size-judgment task, participants had to indicate whether the stimulus was physically larger (or smaller) than a standard of intermediate size presented before. A size-congruity effect occurred in both tasks (see, also, Reike & Schwarz, 2017; Santens & Verguts, 2011).
Two general accounts have been proposed for the size-congruity effect (cf. Santens & Verguts, 2011; Schwarz & Heinze, 1998). According to the shared-representations account, numerical stimulus size and physical stimulus size are processed in parallel and activate a shared representation at an intermediate level of processing that precedes the decision or response-selection stage. ATOM (Walsh, 2003, 2015) is a prominent example for a shared-representations account. In contrast, according to the shared-decisions account, the processing of numerical and physical stimulus size activates independent representations at an intermediate level of processing, but each of these representations can activate a corresponding response code at the subsequent response-selection stage if the response criterion is somehow related to size. Hence, according to the shared-representations account, the size-congruity effect should occur independently of task requirements, whereas the shared-decisions account predicts a size-congruity effect only when the task requires a decision with regard to stimulus size. The shared-decisions account is an instance of so-called dual-route models that have been proposed to explain effects of spatial S-R compatibility (e.g., Kornblum, Hasbroucq, & Osman, 1990; Tagliabue, Zorzi, Umiltà, & Bassignani, 2000; Zorzi & Umiltà, 1995).
The available evidence does not yet allow deciding between the two types of accounts because each has received empirical support. For example, Santens and Verguts (2011) provided evidence for the shared-decisions account by demonstrating that the size-congruity effect depends on task requirements: the effect occurred when the task required a decision with regard to stimulus size (i.e., magnitude judgment), whereas the effect was absent in tasks that required a decision unrelated to size (e.g., parity judgment). More recently, Reike and Schwarz (2017) provided support for the shared-representations account by showing that the congruency between physical and numerical size of a stimulus affects the participants sensitivity for size differences (as measured in d’). According to Reike and Schwarz (2017, p. 387), their results suggest that the processing of numerical and physical stimulus size interact “at an early representational rather than at a late decision stage”.
### Size and space
Although object size is clearly relevant for the control of movements (cf. Jeannerod, 1997), previous research has paid little attention to the relationship between physical size and space. Some recent studies have demonstrated a SNARC-like compatibility effect between the conceptual size of stimuli and response location, extending previous research on non-numerical magnitude (Ren, Nicholls, Ma, & Chen, 2011; Sellaro, Treccani, Job, & Cubelli, 2015; Shaki, Petrusic, & Leth-Steensen, 2012). For example, Ren et al. (Experiment 4), presented participants consecutively two words referring to objects of different size (e.g., apple – mountain). Participants indicated whether the second word denoted an object that was larger or smaller than the first word by pressing a left or right key. Results showed a SNARC-like effect: Left responses were faster to the names of small as compared to large objects, whereas right responses were faster to the names of large as compared to small objects (see, Sellaro et al., 2015; Shaki et al., 2012, for similar findings).
Studies addressing the compatibility between physical stimulus size and horizontal response position are extremely rare: we found only a single published experiment. In this experiment, Ren et al. (2011, Experiment 2) used the same task as in their experiment on conceptual size described above, but used filled circles of different size instead of words as to-be-compared stimuli. The results revealed a statistically significant compatibility effect only for right hand responses (i.e., faster RT for large than for small stimuli), but not for left hand responses. Yet, these results provide suggestive evidence for a connection between the cognitive representations of physical size and space (i.e. positions on the horizontal dimension).
### The present study
In the present study, we wanted to conceptually replicate and extend the research of Ren et al. (2011, Experiment 2). First, in Experiment 1 we aimed at replicating the stimulus size – response location compatibility effect using a classic S-R compatibility task that requires a response (left or right) to a single stimulus in each trial. Second, in both experiments we wanted to test whether the stimulus size – response location effects are restricted to right hand responses (as found in Ren et al., 2011) or can also be obtained in left-hand responses. Third, because size was a relevant stimulus feature in all previous investigations of the stimulus size – response location compatibility effect, we asked in Experiment 2 whether the effect could also be obtained with size as an irrelevant stimulus feature.
## Experiment 1
In Experiment 1 we investigated the relationship between physical stimulus size and horizontal response position with a classic S-R compatibility task. Therefore, stimulus size (small vs. large) was the relevant stimulus feature and participants responded with two S-R mapping conditions in different blocks. In the compatible mapping condition, the small stimulus required a left-hand response, whereas the large stimulus required a right-hand response. In the incompatible mapping condition, the small stimulus required a right-hand response, and the large stimulus required a left-hand response. Our aim was to conceptually replicate a previous demonstration of the stimulus size – response location compatibility effect by Ren et al. (2011, Experiment 2), and to determine whether the effect would again be more pronounced for the right-hand response.
### Methods
Participants. Twenty-four volunteers (20 female, 4 male) with a mean age of 22 years (range 18–27 years) participated in Experiment 1. Participants gave informed consent before the experiment and received course credit for participation. All participants were right-handers (self-report), reported normal or corrected-to-normal visual acuity, and were naive with respect to the purpose of the study.
Apparatus and stimuli. Participants sat in front of a 17-inch color monitor, with an unconstrained viewing distance of approximately 50 cm. An IBM compatible computer controlled the presentation of stimuli and recorded the key-press responses. Visual stimuli appeared on a gray background (i.e. E-Prime color “silver”). The fixation point was a small plus sign (Courier font, font size 18). A small square (2 × 2 cm) and a large square (4 × 4 cm) served as imperative stimuli. All stimuli were presented at screen center. Participants responded by pressing the left “Tabulator” key or the right “backspace” key on a standard computer keyboard with QWERTZ layout. The response keys were marked with white adhesive tape.
Procedure. At the beginning of the experiment instructions were presented on the monitor describing the task, the S-R mapping, and the sequence of events in a trial. Each experimental trial started with the presentation of a fixation point for 1000 ms. Then the stimulus was displayed until a keypress occurred, or for a maximum period of 2000 ms. A correct response with an RT below 2000 ms was followed by a blank screen for 1500 ms. If a wrong key or no key was pressed within the response period, a corresponding error message was shown for 1500 ms in black color (Courier font, font size 24).
Participants performed a training block (10 trials) and an experimental block with the first S-R mapping followed by a training block (20 trials) and an experimental block with the second S-R mapping. The order of mapping conditions (compatible – incompatible; incompatible – compatible) was counterbalanced across participants. In the compatible mapping condition, the small stimulus required a response with the left (tab) key, whereas the large stimulus required a response with the right (backspace) key. The mapping was reversed in the incompatible mapping condition. Participants operated the left key with the index finger of their left hand and the right key with the index finger of their right hand. Each test block contained 60 trials in random order (2 stimuli × 30 repetitions). Participants could take a rest between blocks, and started the next block at leisure. The experiment took about 15 minutes. The experimenter left the room after the first practice block.
Data analysis. Supplementary file 1 contains the raw data from Experiment 1. The data (i.e., individual mean RTs and individual error percentages) were analyzed using separate two-way repeated measures ANOVAs with the factors S-R Mapping (compatible, incompatible) and Response (left, right). The S-R mapping was varied between, the response within blocks.
Trials with RT below 100 ms or above 1,500 ms (less than 1% of trials) were discarded. Partial eta2 is provided as an effect-size estimate. Because we had a priori expectations concerning the direction of the mapping effect, we conducted one-tailed t-tests for examining differences between single conditions.
### Results
Figure 1 shows the mean RTs. A significant main effect of S-R Mapping reflected shorter RTs with the compatible mapping (M = 377 ms, SD = 44) than with the incompatible mapping (M = 406 ms, SD = 62), F(1, 23) = 11.119, MSE = 1,923.554, p = .003, ${\eta }_{p}^{2}=.326$. A significant S-R Mapping × Response interaction reflected a numerically larger Mapping effect for the right-hand response (mean difference = 46 ms) than for the left-hand response (mean difference = 14 ms), F(1, 23) = 13.621, MSE = 424.756, p = .001, ${\eta }_{p}^{2}=.372$. In fact, the mapping effect was significant for the right-hand response, t(23) = 4.753, p < .001, but only marginally significant for the left-hand response, t(23) = 1.401, p = .088. The main effect of Response was not significant, F(1, 23) = 0.013, MSE = 712.539, p = .911, ${\eta }_{p}^{2}=.001$.
Figure 1
Mean RTs observed in Experiment 1.
Legend: Mean RTs as a function of S-R Mapping (compatible: small-left; large-right; incompatible: large-left; small-right) and Response (left vs. right) of Experiment 1. Error bars represent standard errors between participants.
Table 1 shows the mean error percentages. The ANOVA revealed non-significant main effects of S-R Mapping, F(1, 23) = 1.169, MSE = 28.618, p = .291, ${\eta }_{p}^{2}=.048$, and Response, F(1, 23) = 0.375, MSE = 7.724, p = .547, ${\eta }_{p}^{2}=.016$. However, the interaction of S-R Mapping × Response was significant, F(1, 23) = 8.886, MSE = 9.496, p = .007, ${\eta }_{p}^{2}=.279$. In fact, in right-hand responses, there were less errors with the compatible mapping than with the incompatible mapping, t(23) = 2.506, p = .020, whereas the two mapping conditions did not differ for left-hand responses, t(23) = 0.534, p = .598.
Table 1
Percentages of errors observed in Experiment 1 as a function of S-R Mapping and Response (N = 24). Standard deviations are given in parentheses.
S-R Mapping
Compatible Incompatible
Left Response 3.19 (4.45) 2.50 (3.96)
Right Response 1.67 (2.60) 4.72 (6.44)
### Discussion
In Experiment 1, we conceptually replicated a compatibility effect between physical stimulus size and horizontal response position, which was previously shown by Ren et al. (2011). Specifically, we found that right-hand responses were significantly faster to the larger stimulus than to the smaller stimulus and left-hand responses were (numerically) faster to the smaller stimulus than to the larger stimulus, thus replicating the stimulus size – response location compatibility effect despite several procedural differences between the two studies. In particular, the stimulus set was larger in the former study (including 9 stimulus sizes) than in our experiment (including only 2 stimulus sizes). Moreover, the participants in the former study compared the size of two consecutively presented stimuli in each trial, whereas the participants in our study judged the size of a single stimulus (as being the smaller or larger stimulus in the stimulus set) in each trial. Interestingly, in accordance with the findings of Ren et al. (2011), we also found that the stimulus size – response location compatibility effect was larger for right-hand responses than for left-hand responses.
## Experiment 2
In Experiment 2 we investigated whether the compatibility between stimulus size and horizontal response position would still affect performance when stimulus size was not relevant for the task at hand. Therefore, in the color-discrimination task of Experiment 2, participants responded to stimulus color by pressing a left or right key and stimulus size varied independently from stimulus color. Hence, the task used in Experiment 2 resembles a Simon task with stimulus size instead of stimulus position as the irrelevant feature that is congruent or incongruent with response position (cf. Hommel, 2011, for a review). If congruent conditions (small S – left R; large S – right R) produce better performance than incongruent conditions (large S – left R; small S – right R), we would conclude that stimulus size is involuntarily encoded and automatically primes a congruent response code (cf. Kornblum et al., 1990; Tagliabue et al., 2000; Zorzi & Umiltá, 1995; Gevers, Verguts, Reynvoet, Caessens, & Fias, 2006; for dual-route accounts of congruency effects).
A methodological problem in Experiment 2 concerned the fact that when stimulus size is task-irrelevant, this irrelevant feature may be ambiguous to the participants. In particular, this irrelevant variation may be perceived as a difference in size (of two stimuli presented at the same distance), but it may also be perceived as a difference in distance (of two stimuli with the same size). In order to foster an interpretation in terms of size differences, we introduced a second task where participants had to vocally report the size of the same stimuli also used in the critical color-discrimination task. Half of the participants performed the size-discrimination task before the color-discrimination task; the other half of the participants performed the two tasks in the opposite order. If the irrelevant stimulus feature is in fact ambiguous for participants, the expected congruency effect should be larger for the group that performed the size-discrimination task before the color-discrimination task. If, however, most participants spontaneously interpret the irrelevant variation of the stimulus in terms of size rather than in terms of distance, then both groups should demonstrate a similar stimulus size – response location congruency effect, regardless of task order.
### Methods
Participants. Forty volunteers (35 female, 5 male) with a mean age of 24 years (range 19–39 years) participated in Experiment 2. Participants gave informed consent before the experiment and received course credit for participation. All participants were right-handers (self report), reported normal or corrected-to-normal visual acuity, and were naive with respect to the purpose of the study.
Apparatus and stimuli. The apparatus was the same as in Experiment 1 with the following exceptions: The orthogonal combination of two sizes (small and large) and two colors (green and red) produced four different stimuli. The stimulus was always presented at screen center. In the size-discrimination task, participants vocally named the size of the stimulus, and the computer measured vocal RT. In the color-discrimination task, participants responded to stimulus color by pressing a left or right key on the keyboard, and the computer registered accuracy and RT of each keypress.
Procedure. In Experiment 2, each participant performed two separate tasks. Each task started with the presentation of the instructions on the screen. In the size-discrimination task, participants vocally responded to stimulus size by speaking the German words for “small” or “large” into a microphone. This task involved two blocks of 40 trials (2 stimulus colors × 2 stimulus sizes × 10 repetitions). The experimenter watched participants in the size-discrimination task and observed that errors were extremely rare, but errors were not recorded.
In the color-discrimination task, participants manually responded to stimulus color by pressing a left (tabulator) or right (backspace) key on the keyboard. This task involved a practice block of 20 trials, and two experimental blocks of 60 trials (2 stimulus colors × 2 stimulus sizes × 15 repetitions). In each block, the stimulus displays were presented in random order. The trial structure in both tasks was the same as in Experiment 1. The order of tasks and the S-R mapping in the color-discrimination task (green – left, red – right versus red – left, green – right) were independently counterbalanced across participants.
Data analysis. Supplementary file 2 contains the raw data from Experiment 2. The data (i.e., individual mean RTs and individual error percentages) were analyzed using separate mixed-design ANOVAs with the between-subject factor Task Order (color discrimination – size discrimination; size discrimination – color discrimination) and the within-subject factors S-R Congruency (congruent, incongruent) and Response (left vs. right). In congruent conditions, participants made a left response to the color of the small stimulus or a right response to the color of the large stimulus. In incongruent conditions, participants made a left response to the color of the large stimulus or a right response to the color of the small stimulus.
Trials with RT below 100 ms or above 1,500 ms (less than 1% of trials) were discarded. Partial eta2 is provided as an effect-size estimate.
### Results
A preliminary three-way ANOVA showed that the factor Task Order had no significant effect on RTs in the color-discrimination task and, therefore, the variable was excluded from further analyses. A two-way repeated measures ANOVA with S-R Congruency and Response as independent variables and RTs as dependent variable showed a significant main effect of S-R Congruency, F(1, 38) = 21.536, MSE = 182.099, p < .001, ${\eta }_{p}^{2}=.356$; RTs were shorter for congruent conditions (M = 392 ms, SD = 71) than for incongruent conditions (M = 402 ms, SD = 74). A marginally significant main effect of Response, F(1, 38) = 3.650, MSE = 459.771, p = .063, ${\eta }_{p}^{2}=.086$, reflected a trend towards shorter RTs for right-hand responses (M = 394 ms, SD = 70) than for left-hand responses (M = 400 ms, SD = 78). The two-way interaction was not significant F(1, 38) = 0.551, MSE = 568.361, p = .462, ${\eta }_{p}^{2}=.014$. The mean RTs of correct responses are shown in Figure 2.
Figure 2
Mean RTs observed in Experiment 2.
Legend: Mean RTs as a function of S-R Congruency (congruent: small-left; large-right; incongruent: large-left; small-right) and Response (left vs. right) in the color-discrimination task of Experiment 2. Error bars represent standard errors between participants.
A three-factorial ANOVA on error-percentages in the color-discrimination task failed to reveal any significant main effect or interaction.
Finally, for the size-discrimination task, a two-way ANOVA on vocal RTs using the within-subject factor Stimulus Size (small, large) and the between-subject factor Task Order revealed a significant main effect of Stimulus Size, reflecting shorter RTs to the small stimulus (M = 426 ms, SD = 62) than to the large stimulus (M = 465 ms, SD = 78), F(1, 38) = 30.936, MSE = 973.548, p < .001, ${\eta }_{p}^{2}=.449$. This finding may be due to the fact that the voice-key detects the German word “klein” (= small) somewhat earlier than the German word “groß” (= large). The main effect of Task Order, F(1, 38) = 0.143; and the two-way interaction, F(1, 38) = 2.297, MSE = 973.548, p = 0.138, ${\eta }_{p}^{2}=.057$, were not significant.
### Discussion
When participants responded to the color of a stimulus, which also varied in size, the relationship between the task-irrelevant stimulus size and response position still produced a congruency effect that mirrored the compatibility effect obtained in Experiment 1. In particular, left-hand responses were faster to the smaller than to the larger stimulus and right-hand responses were faster to the larger than to the smaller stimulus. In accordance with dual-route accounts of the SNARC effect (Gevers et al., 2005) and other congruency effects (e.g., Kornblum et al., 1990; Tagliabue et al., 2000; Zorzi & Umiltá, 1995), this pattern suggests that stimulus size is involuntarily encoded and automatically primes a compatible response code at the response-selection stage.
Two further results of Experiment 2 are noteworthy. First, in contrast to the results of Experiment 1, where the stimulus size – response location compatibility effect was larger in right-hand responses than in left-hand responses, the stimulus size – response location congruency effect was of similar magnitude for the two hands in Experiment 2. We come back to this point in the General Discussion. Second, performing size judgments with the same stimuli before or after the critical color-discrimination task did not affect the stimulus size – response location congruency effect in Experiment 2. Hence, most participants spontaneously perceived the irrelevant variation in stimulus size as intended, and not (so much) as a variation in the distance of two stimuli with similar size.
## General Discussion
Two experiments demonstrated a compatibility effect between physical stimulus size and horizontal response position in right-handed participants. Experiment 1 showed that, when participants responded to the size of a single stimulus, right-hand responses were faster and more accurate to the larger stimulus than to the smaller stimulus, whereas trends in the opposite direction were observed for left-hand responses. This finding replicates a previous finding of Ren et al. (2011) with several methodological modifications and, therefore, demonstrates the reliability of the effect. In Experiment 2 we showed for the first time, that when participants responded to the color of a stimulus varying in size, the compatibility between the now irrelevant stimulus size and response position still affected performance, producing a SNARC- or Simon-like congruency effect. Hence, small stimuli appear to be associated to left responses, whereas large stimuli appear to be associated to right responses. In the following sections, we discuss some theoretical implications of our findings and delineate some directions for future research.
### Theoretical implications
The demonstration of a stimulus size – response location compatibility effect is consistent with ATOM theory (Walsh, 2003, 2015; Bueti & Walsh, 2009). In fact, ATOM assumes a generalized magnitude-processing system (in the brain) where the processing of time, space, number, and other magnitudes overlaps and interacts in order to facilitate the control of complex movements. Therefore, ATOM predicts interference and congruency effects between stimulus size and response position, as demonstrated in our experiments.
However, ATOM does not readily seem to predict the direction of the stimulus size – response location congruency effect observed in our experiments. As mentioned in the introduction, Bueti and Walsh (2009, page 1832) assume some monotonic mapping of quantities: more A should go along with more B. Obviously, however, this prediction can only be applied to describe the relationship between magnitudes with at least ordinal scaling properties, such as size, number or time. The prediction cannot be readily applied to spatial positions such as left or right because horizontal position is a nominal variable. Hence, we would need to explain where the observed relationship between stimulus size and horizontal response position may come from. One possibility is that the mapping between physical size and horizontal location has similar sources as the mapping between numerical size and horizontal location (i.e. the SNARC effect). In that case, the observed mapping of small objects to left and large objects to right would have resulted from several “cultural” variables, such as reading and counting habits, or graphical representations of this mapping (cf. Winter et al., 2015, for examples). Another possibility, which cannot yet be dismissed, is that the mapping between physical size and horizontal location has different sources than the mapping between numbers and horizontal locations. For example, the stimulus size – response location congruency effect might result from functional differences between the two hands. In most people, the right hand is the dominant hand and research has shown that the dominant right hand is stronger than the non-dominant left hand (e.g., Hepping et al., 2015; Incel et al., 2002). Hence, it could be that people have a preference to grasp (and lift) larger objects with their right (and not left) hand because the right hand is stronger than the left hand. Of course, this speculation would require further empirical investigation.
Whereas ATOM is a variant of a “shared-representation account” (cf. Santens & Verguts, 2011), a “shared-decision account” of the stimulus size – response location compatibility effect is also conceivable. For example, dual-route models, which have originally been proposed to explain spatial S-R compatibility effects (e.g., Kornblum et al., 1990; Tagliabue et al., 2000; Zorzi & Umiltà, 1995), could also be applied to the stimulus size – response location compatibility effect. For example, according to Tagliabue et al. (2000), S-R congruency effects between an irrelevant stimulus feature and the response-discriminating feature, as we have observed in Experiment 2, arise from the interaction of short-term and long-term associations between stimulus and response codes, respectively. Short-term associations represent the instructed S-R mapping, that is, the mapping of stimulus colors to left-right response positions in our Experiment 2. Long-term associations, on the other hand, represent some pre-experimentally acquired relationship between irrelevant stimulus size and left-right response positions. When the stimulus is presented, the relevant stimulus feature (i.e., color) activates the correct response code through the short-term association, and the irrelevant stimulus feature (i.e., size) primes the congruent response code through the long-term association. Hence, in congruent conditions, both processing routes activate the correct response, which is quickly selected and executed. In contrast, in incongruent conditions, the long-term association activates an incorrect response that interferes with selection of the correct response, thus increasing RTs and sometimes causing an error. Future research is required to decide empirically between shared-representation or shared-decision accounts of the stimulus size – response location compatibility effect.
### Differences between left and right hands
Interestingly, the stimulus size – response location compatibility was larger for right-hand than for left-hand responses when size was a relevant stimulus feature in Experiment 1, whereas the congruency effect was similar for the two hands when size was an irrelevant feature (Experiment 2). The larger stimulus size – response location compatibility effect for the right hand observed in Experiment 1 replicates similar observations by Ren et al. (2011) in both their Experiment 2 (when physical size was the relevant stimulus) and their Experiment 4 (when conceptual size was the relevant stimulus).
The stronger compatibility effect in right-hand responses cannot be attributed to the different strengths of the associations between small stimuli and left-hand responses, on the one hand, and large stimuli and right-hand responses, on the other hand. If, for example, the association between large and right was stronger than the association between small and left, the stronger association would increase facilitation in congruent conditions with right-hand responses, which is consistent with the findings. However, the stronger association between large and right should also increase interference in incongruent conditions with left-hand responses, which is inconsistent with the findings.
We suggest that the stronger compatibility effect in right-hand as compared to left-hand responses is a mere consequence of a main effect of stimulus size. If the data from Experiment 1 are analyzed as a function of stimulus size (small, large) and S-R compatibility, instead of being analyzed as a function of the response (left, right) and S-R compatibility, you observe significant main effects of stimulus size and S-R compatibility, but no interaction. The main effect of stimulus size reflects shorter RTs to the large stimulus as compared to the small stimulus (e.g., Osaka, 1976), which results from the fact that a larger stimulus is perceptually more salient than a smaller stimulus. Hence, we believe that the two-way interaction between S-R compatibility and response observed in Experiment 1 (and also in Experiment 2 of Ren et al., 2011) is actually the consequence of a main effect of stimulus size on RTs that transforms into a two-way interaction of S-R mapping and response if the data are re-arranged to test for the effects of the two latter variables. In our view, this interpretation is also consistent with the absence of the two-way interaction in Experiment 2 because the main effect of stimulus size should disappear when stimulus size is no longer relevant for selecting a response.
### Conclusion and directions for future research
The results of the present experiments provide evidence for an association between smaller stimulus objects with left-hand responses and between larger stimulus objects with right-hand responses. This association influences response selection regardless of whether stimulus size is relevant or irrelevant for the task at hand, suggesting long-term associations between size and position. Whether these effects arise from overlapping representations of space and magnitude, as suggested by ATOM (Walsh, 2003, 2015), or at the response-selection stage, is a question for future research.
A further question is whether the stimulus size – response location compatibility effect arises with regard to the anatomical status of the (left vs. right) hand or with regard to left or right response positions. This question could be addressed by comparing performance with arms held in parallel, as in the present experiments, to performance with crossed arms. If the effect arises with regard to anatomical hand status, then the participant’s handedness may modulate the effect as well. Hence, it might be interesting to compare the results from right-handed participants, which were addressed in the present experiments, to the results from left-handed participants.
## Data Accessibility Statement
The raw data from both experiments have been published as additional (supplementary) files for this article (see below).
Supplementary file 1
Raw data from Experiment 1 (Wuehr_Seegelke_Raw_Data_Experiment1.csv). DOI: https://doi.org/10.5334/joc.19.s1
Supplementary file 2
Raw data from Experiment 2 (Wuehr_Seegelke_Raw_Data_Experiment2.csv). DOI: https://doi.org/10.5334/joc.19.s2
## Acknowledgements
This publication was supported by the Deutsche Forschungsgemeinschaft (DFG) and TU Dortmund University within the funding programme Open Access Publishing.
We are grateful to Marina Annaker and Isabel Deussen for testing participants and collecting the data.
## Competing Interests
The authors have no competing interests to declare.
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Finding players and targets of expression-regulating pathways
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Entering edit mode
10 months ago
Tobi ▴ 10
I am currently working on a project involving using pathways to group genes. However, I am quite new to the field, so I have a some (beginner) questions concerning the biology and available databases.
A) I have heard about the concept of dividing the genes interacting in pathways into players, i.e., the corresponding proteins interact, and targets, i.e., those genes whose expression gets regulated. For my project, such a distinction would be very helpful. Is this a common concept known under different nomenclature? Searching for the terms “players” and “targets” I only found individual pathways. Databases like KEGG, Reactome or PathwayCommons don’t seem to integrate this concept. Are there other databases like these using such a division of interacting genes?
B) As I did not find any such database, I thought about using gene sets from KEGG and Reactome and looking up their gene expression regulation targets using the sif file(s) provided by PathwayCommons. I thought about two ways to get the grouping:
• dividing the gene sets into “player” and “target” genes based on the sif file retrieved from PathwayCommons by looking up the “controls-expression-of” relation
• looking for further targets by traversing the graph given by the sif file of all interactions in PathwayCommons for a few steps, starting at all genes involved in the pathway. I just tried it for two steps, giving me already quite long lists of genes, so I am not sure if this is a reasonable approach. My main idea behind the second approach was that many pathways have only few genes listed as members regulated by it (in the order of tens). This contradicts my understanding that the number of targets should be in the order of hundreds.
Do you think that this is a legitimate approach to the problem? Is there already a solution to it, which I might have overlooked due to different nomenclature? If you spot a misunderstanding on my side, could you please explain it or point me to resources, so I could clarify things up?
Thank you in advance for help with any of these questions!
regulation database expression pathway • 341 views
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Entering edit mode
Hello,
So I am unsure if this is exactly what you are looking for. But there are tools that can identify ligand-receptor interactions (cell-cell communication) using scRNA-seq data. InterCom is one in particular that comes to mind, which I think I saw uses curated gene lists for signaling pathway identification internally: https://github.com/saschajung/InterCom which can help you build a heatmap like this:
Image Citation: Gonçalves, C.A., Larsen, M., Jung, S. et al. A 3D system to model human pancreas development and its reference single-cell transcriptome atlas identify signaling pathways required for progenitor expansion. Nat Commun 12, 3144 (2021). https://doi.org/10.1038/s41467-021-23295-6
Another tool, that I was shown recently is rPAC: https://doi.org/10.1016/j.ymeth.2021.10.002 which I believe uses the gene lists from KEGG pathways internally to identify (score) which pathways are up-regulated in RNA-seq data, this could probably be adapted for scRNA-seq data as well?
I think there are other more popular/used tools that accomplish the same things above. I just can't remember the name of one in particular that I saw mentioned in some papers. (I think this one was the one I was trying to remember when I made this post: https://www.cellphonedb.org/ )
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# Fluctuation-dissipation theorem in the Keldysh formalism
In Kamenev's book Field Theory of Non-Equilibrium Systems (he also has lecture notes online here, which contains the relevant statement on pg. 17), he states that the following equation $$G^K(\epsilon) = \coth\left(\frac{\epsilon-\mu}{2T}\right) \left[G^R(\epsilon) - G^A(\epsilon)\right]$$ is a statement of the fluctuation-dissipation theorem, where $$G^{(K,R,A)}(\epsilon)$$ are the Keldysh, retarded, and advanced propagators, respectively. I have only ever seen the FDT stated in terms of structure factors and susceptibilities. While I can see the superficial connection (since $$G^A(\epsilon) = G^R(\epsilon)^\dagger$$, the RHS should resemble something like $$\text{Im}\chi$$), I'm having difficulty rigorously connecting the two. Can someone help me understand the connection between these statements?
What you are referring to is the form of fluctuation-dissipation theorem (FDT) that relates the dynamical structure factor to some retarded susceptibilty. The equation you wrote down holds for bosonic systems, in which case the RHS can be interpreted as a susceptibility while the LHS is related to the dynamical structure factor through the relation $$G^{<} = G^{K}+\frac{1}{2}\left(G^A-G^R\right)$$. This leads to $$G^{<}(\epsilon) = n_{B}(\epsilon)\mbox{Im}\left[G^R(\epsilon)\right]$$, where $$n_{B}(\epsilon)$$ is the Bose distribution function.
For a fermionic system, however, $$n_{B}(\epsilon)$$ must be replaced by $$n_{F}(\epsilon)$$ - the Fermi distribution function - in the equation above. This gives a fermionic FDT. The familiar bosonic FDT can be recovered in this case by considering the two-particle excitations, which can be expressed as the product of single particle excitations using Wick's theorem.
$$\Pi^{R}(t,t^{'}) = G^{R}(t,t^{'})G^{K}(t^{'},t) + G^{K}(t,t^{'})G^{A}(t^{'},t)$$ is the retarded susceptibility, and similarly one can also write an expression for $$\Pi^{<}$$ in terms of $$G^{R,A,K}$$
At equilibrium, one can show that: $$\Pi^{<}(\epsilon) = n_{B}(\epsilon)\mbox{Im}\left[\Pi^{R}(\epsilon)\right]$$. This is the familiar form of FDT. You will find a detailed discussion in Kamenev's book ch. 9.
Linear response and Green's functions
In linear response theory, if we are given Hamiltonian, $$H=H_0+\lambda (t)X$$, the response of variable $$Y(t)$$ is given by $$\langle Y^h(t)\rangle = \langle Y(t)\rangle+ \int_{-\infty}^{+\infty} dt_1\left\langle\frac{-i}{\hbar}[Y(t),X(t_1)]\right\rangle\theta(t-t_1)\lambda(t_1),$$ where the response function is just a retarded function for operators $$Y(t), X(t_1)$$ (operators without subscript has time evolution governed only by $$H_0$$): $$G_{YX}^r(t,t_1)= \left\langle\frac{-i}{\hbar}[Y(t),X(t_1)]\right\rangle\theta(t-t_1)=\left[G_{YX}^>(t,t_1) - G_{YX}^<(t,t_1)\right]\theta(t-t_1).$$ The susceptibility is just the Fourier transform of this function, whereas the advance Green's function is defined as $$G_{YX}^a(t,t_1)= \left\langle\frac{i}{\hbar}[Y(t),X(t_1)]\right\rangle\theta(t-t_1)=\left[G_{YX}^>(t,t_1) - G_{YX}^<(t,t_1)\right]\theta(t_1-t),$$ so that $$G_{YX}^r(t,t_1) - G_{YX}^a(t,t_1) = G_{YX}^>(t,t_1) - G_{YX}^<(t,t_1) = \left\langle\frac{-i}{\hbar}[Y(t),X(t_1)]\right\rangle$$ Note also that the frequency space (i.e., for Fourier transforms): $$G_{YX}^r(\omega) - G_{YX}^a(\omega) = G_{YX}^>(\omega) - G_{YX}^<(\omega),$$ as a simple consequence of the definitions.
Lehmann representation
In the eigenbasis of the unperturbed Hamiltonian, $$H_0|n\rangle=E_n|n\rangle$$ the greater Green's function has the following representation $$G_{YX}^>(t,t_1)=\frac{-i}{\hbar}\left\langle Y(t)X(t_1)\right\rangle= \frac{-i}{\hbar}\sum_{n,m}e^{-\beta E_n}e^{-i(E_m-E_n)(t-t_1)/\hbar}Y_{nm}X_{mn},\\ G_{YX}^>(\omega) = \frac{-2\pi i}{\hbar}\sum_{n,m}e^{-\beta E_n}Y_{nm}X_{mn}\delta\left(\omega -\frac{E_m-E_n}{\hbar}\right)$$ Similarly $$G_{YX}^<(\omega) = \frac{-2\pi i}{\hbar}\sum_{n,m}e^{-\beta E_m}Y_{nm}X_{mn}\delta\left(\omega -\frac{E_m-E_n}{\hbar}\right)=\\ \frac{-2\pi i}{\hbar}\sum_{n,m}e^{-\beta (E_n+\hbar\omega)}Y_{nm}X_{mn}\delta\left(\omega -\frac{E_m-E_n}{\hbar}\right)=e^{-\beta\hbar\omega}G_{YX}^>(\omega)$$ We thus have $$G_{YX}^>(\omega)\pm G_{YX}^<(\omega)=\left(1\pm e^{-\beta\hbar\omega}\right)G_{YX}^>(\omega)\Rightarrow\\ G_{YX}^>(\omega)+ G_{YX}^<(\omega)=\frac{1+e^{-\beta\hbar\omega}}{1-e^{-\beta\hbar\omega}}\left[G_{YX}^>(\omega)- G_{YX}^<(\omega)\right]=\\ \coth\left(\frac{\beta\hbar\omega}{2}\right)\left[G_{YX}^>(\omega)- G_{YX}^<(\omega)\right].$$ Recognizing the left part of this expression as the definition of the Keldysh Green-s function we thus have $$G_{YX}^K(\omega)= \coth\left(\frac{\beta\hbar\omega}{2}\right)\left[G_{YX}^r(\omega)- G_{YX}^a(\omega)\right]$$
Is this the FDT?
• Note that the Keldysh function that we defined is given by $$G_{YX}^K(t,t_1) - G_{YX}^a(t,t_1) = \frac{-2i}{\hbar}\left\langle\frac{1}{2}\{Y(t),X(t_1)\}\right\rangle,$$ that is it is simply the correlation function (up to a factor), whose Fourier transform is the noise intensity. We thus have the statement of the Fluctuation-dissipation theorem.
• This relationship will also hold, if $$X$$ and $$Y$$ are creation and annihilation operators, in which the Green's functions take the more familiar form. However, its interpretation as FDT becomes less reliable, unless we introduce generalized susceptibilities. More conventionally the difference between the advanced and the retarded Green's functions corresponds to the density-of-states, whereas the Keldysh function is the particle distribution function.
• Although the relationship holds beyond the linear response, it is still a statement about the linear response coefficients! This is in contradistinction to the non-linear FDT, which usually implies statements about higher-order response (higher order in $$\lambda(t)$$).
The difference of the retarded and the advanced Green's functions in the right-had-side of this equation is actually the density-of-states, i.e. what you might call a structure factor, whereas $$G^K$$ tests the possibilities of adding/removing a particle, i.e. the susceptibility.
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#### Volume 18, issue 3 (2018)
Recent Issues
The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Other MSP Journals
Thin position for knots, links, and graphs in $3\mkern-1.5mu$–manifolds
### Scott A Taylor and Maggy Tomova
Algebraic & Geometric Topology 18 (2018) 1361–1409
##### Abstract
We define a new notion of thin position for a graph in a $3$–manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the ideas of thin position for knots first originated by Gabai. This thin position has the property that connect-summing annuli and pairs-of-pants show up as thin levels. In a forthcoming paper, this new thin position allows us to define two new families of invariants of knots, links, and graphs in $3$–manifolds. The invariants in one family are similar to bridge number, and the invariants in the other family are similar to Gabai’s width for knots in the $3$–sphere. The invariants in both families detect the unknot and are additive under connected sum and trivalent vertex sum.
##### Keywords
3-manifold, knot, spatial graph, bridge number, width, bridge position, Heegaard splitting
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Secondary: 57M50
##### Publication
Received: 16 July 2016
Revised: 30 November 2017
Accepted: 15 January 2018
Published: 3 April 2018
##### Authors
Scott A Taylor Department of Mathematics and Statistics Colby College Waterville, ME United States Maggy Tomova Department of Mathematics University of Iowa Iowa City, IA United States
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# Prove that T(x,y)=(3x+y,2y,x-y) defines a linear transformation T:R^2 -> R^3.
Prove that $T\left(x,y\right)=\left(3x+y,2y,x-y\right)$ defines a linear transformation $T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{3}$. Give the full and correct answer.
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StrycharzT
Let $\left({x}_{1},{y}_{1}\right),\left({x}_{2},{x}_{2}\right)\in {\mathbb{R}}^{2}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\le t{c}_{1},{c}_{2}\in \mathbb{R}$. To prove that T is a linear transformation, we must prove that
$T\left({c}_{1}\left({x}_{1},{y}_{1}\right)+{c}_{2}\left({x}_{2},{y}_{2}\right)\right)={c}_{1}T\left({x}_{1},{y}_{1}\right)+{c}_{2}T\left({x}_{2},{y}_{2}\right)$
This is done by a direct computation:
$T\left({c}_{1}\left({x}_{1},{y}_{1}\right)+{c}_{2}\left({x}_{2},{y}_{2}\right)\right)$
$=T\left({c}_{1}{x}_{1},{c}_{2}{x}_{2}\right)+\left({c}_{2}{x}_{2},{c}_{2}{x}_{2}\right)$
$=T\left({c}_{1}{x}_{1}{c}_{2}{x}_{2},{c}_{1}{y}_{1}+{c}_{2}{y}_{2}\right)$
$=\left(3\left({c}_{1}{x}_{1}+{c}_{2}{x}_{2}\right)+\left({c}_{1}{y}_{1}+{c}_{2}{y}_{2}\right),2\left({c}_{1}{y}_{1}+{c}_{2}{y}_{2}\right),\left({c}_{1}{x}_{1}+{c}_{2}{x}_{2}\right)-\left({c}_{1}{y}_{1}+{c}_{2}{y}_{2}\right)\right)$
$=\left(3{c}_{1}{x}_{1}+3{c}_{2}{x}_{2}+{c}_{1}{y}_{1}+{c}_{2}{y}_{2},2{c}_{1}{y}_{1}+{c}_{2}{y}_{2},{c}_{1}{x}_{1}+{c}_{2}{x}_{2}-{c}_{1}{y}_{1}+{c}_{2}{y}_{2}\right)$
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# Kalman recursion
1. Aug 16, 2008
### mr.t
As this is concerning signal processing i guess this is the right place to post the question. Im Trying to learn how to use kalman filters. Ive reached some form of verry basic understanding of the state-space model but im still kindof confused. What im trying to do now is to understand an example that is using kalman recursion to find the steady state kalman gain.
We have an AR(1) process described by: x(n) = 0.5x(n-1) + w(n), where w(n) is white-noise with variance 0.64. we are observing a process: y(n) = x(n)+ v(n), where v(n) is white-noise with variance 1.
The state-space model becomes:
x(n) = 0.5x(n-1) + w(n)
y(n) = x(n) + v(n)
and we see that A(n-1) = 0.5, B(n) = 1 and C(n) = 1. From the variances we have Qw=0.64 and Qv=1.
We have the initial conditions: x'(0|0) = 0 and E{e^2(0|0)} = 1, where e(0|0) = x(0) - x'(0|0). (' = estimate) and im trying to use these formulas to perform the recursion: (im skipping some matrix-related stuff since the matrices in this example is just single numbers so transposing isnt doing anything)
x'(n|n-1) = Ax'(n-1|n-1)
P(n|n-1) = AP(n-1|n-1)A + Qw
K(n) = P(n|n-1)C[CP(n|n-1)C+Qv]^-1
x'(n|n) = x'(n|n-1) + K(n)[y(n) - Cx'(n|n-1)]
P(n|n) = [I-K(n)C]P(n|n-1)
We start with P(0|0) = E{e^2(0|0)} = 1.
Ok. Now to my problem. How do i get y(n) ? I get stuck on the first iteration when I want to calculate x'(1|1) and i need y(1), how to i get it?
what is I ? on the last formula-row "P(n|n) = [I-K(n)C]P(n|n-1)"? In the example it is equal to 1, but
where do the 1 come from?
Also if anyone have any good (simple!) tutorial suggestion on the net about kalman-filtering that would be appreciated.
Thanks alot!
2. Aug 16, 2008
### dlgoff
3. Aug 18, 2008
### mr.t
Ok my bad. Seems like you dont even calculate x(n|n|) in the recursion, you just calculate P(n|n-1), K(n) and P(n|n) for each step (!). But this leads to another question.
When you have done your iterations and found that your P:s and K:s are getting steady, they you have your Kalman gain as the steady K(n). But there is another formula of calculating K (pretty much the same, but apart from the iteration-formulas in my textmaterial) as in:
$$K = PC^{T}(CPC^{T} + Q_{v})^{-1}$$
Is said to give the corresponding Kalman gain for the Riccati-equation:
$$P = APA^{T} + Q_{w} - APC^{T}[CPC^{T} + Q_{v}]^{-1}CPA^{T}$$
Whats the deal with this Riccati-equation? cant you just use the iterations to find both P and your Gain at the same time?
Last edited: Aug 18, 2008
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• ### What is your GameDev Story?
#### Archived
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# Question Or Two
This topic is 6578 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
I''m making a game and want to add sound effects I have read everywhere that I will be needing the FMOD library etc... Well I have all those but can do nothing with them! Does any of you have some source code which loads a *.wav and/or *.mp3 files. Also does any of you know how to generate a random number in C++, the reason being, for my game I want power-ups to pop-up every now and then and I need to generate random positions for them. Many Thanks Alan IF YA SMELL... WHAT THE BEZZ IS COOKIN''''
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Cant answer the FMOD thing but this is the C library function for generating a random number:
int rand( void );
And the funtion to seed the generator is this
void srand( unsigned int seed );
This little lot was copied from the MSDN
The srand function sets the starting point for generating a series of pseudorandom integers. To reinitialize the generator, use 1 as the seed argument. Any other value for seed sets the generator to a random starting point. rand retrieves the pseudorandom numbers that are generated. Calling rand before any call to srand generates the same sequence as calling srand with seed passed as 1.
LIBC.LIB Single thread static library, retail version
LIBCMT.LIB Multithread static library, retail version
MSVCRT.LIB Import library for MSVCRT.DLL, retail version
Return Value
rand returns a pseudorandom number, as described above. There is no error return.
Remarks
The rand function returns a pseudorandom integer in the range 0 to RAND_MAX. Use the srand function to seed the pseudorandom-number generator before calling rand.
Example
/* RAND.C: This program seeds the random-number generator * with the time, then displays 10 random integers. */#include #include #include void main( void ){ int i; /* Seed the random-number generator with current time so that * the numbers will be different every time we run. */ srand( (unsigned)time( NULL ) ); /* Display 10 numbers. */ for( i = 0; i < 10;i++ ) printf( " %6d\n", rand() );}
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# chronoMPL
From ape v4.1
0th
Percentile
##### Molecular Dating With Mean Path Lengths
This function estimates the node ages of a tree using the mean path lengths method of Britton et al. (2002). The branch lengths of the input tree are interpreted as (mean) numbers of substitutions.
Keywords
models
##### Usage
chronoMPL(phy, se = TRUE, test = TRUE)
##### Arguments
phy
an object of class "phylo".
se
a logical specifying whether to compute the standard-errors of the node ages (TRUE by default).
test
a logical specifying whether to test the molecular clock at each node (TRUE by default).
##### Details
The mean path lengths (MPL) method estimates the age of a node with the mean of the distances from this node to all tips descending from it. Under the assumption of a molecular clock, standard-errors of the estimates node ages can be computed (Britton et al. 2002).
The tests performed if test = TRUE is a comparison of the MPL of the two subtrees originating from a node; the null hypothesis is that the rate of substitution was the same in both subtrees (Britton et al. 2002). The test statistic follows, under the null hypothesis, a standard normal distribution. The returned P-value is the probability of observing a greater absolute value (i.e., a two-sided test). No correction for multiple testing is applied: this is left to the user.
Absolute dating can be done by multiplying the edge lengths found by calibrating one node age.
##### Value
an object of class "phylo" with branch lengths as estimated by the function. There are, by default, two attributes:
stderr
the standard-errors of the node ages.
Pval
the P-value of the test of the molecular clock for each node.
##### Note
The present version requires a dichotomous tree.
##### References
Britton, T., Oxelman, B., Vinnersten, A. and Bremer, K. (2002) Phylogenetic dating with confidence intervals using mean path lengths. Molecular Phylogenetics and Evolution, 24, 58--65.
chronopl
• chronoMPL
##### Examples
# NOT RUN {
tr <- rtree(10)
tr$edge.length <- 5*tr$edge.length
chr <- chronoMPL(tr)
layout(matrix(1:4, 2, 2, byrow = TRUE))
plot(tr)
title("The original tree")
plot(chr)
axisPhylo()
title("The dated MPL tree")
plot(chr)
nodelabels(round(attr(chr, "stderr"), 3))
title("The standard-errors")
plot(tr)
nodelabels(round(attr(chr, "Pval"), 3))
title("The tests")
# }
Documentation reproduced from package ape, version 4.1, License: GPL (>= 2)
### Community examples
Looks like there are no examples yet.
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# [rwt_moveit] datatype/md5sum error
Hello everyone,
I'm trying to use rwt_moveit package alongside with cyton_gamma_300 moveit package on ROS Melodic. I can visualize the robot and the markers on the web page, but as soon I launch the /moveit_publisher node I get this error:
Client [/moveit_publisher] wants topic /move_group/result to have datatype/md5sum [moveit_msgs/MoveGroupActionResult/47ef72f231e49f6059f185fc18418665], but our version has [moveit_msgs/MoveGroupActionResult/6ee8682a508d60603228accdc4a2b30b]. Dropping connection.
So according to what I found in the community, this error occurs when two nodes try to communicate with each other, but from two different ROS environments (like one node from Indigo, and the other one from Kinetic). But in my case, all nodes are in the same ROS environment which is Melodic.
Here is the launch file:
<launch> <arg name="fixed_frame" default="/world"/> <arg name="is_sim" default="false"/>
<param name="fixed_frame" value="$(arg fixed_frame)" /> <param name="sim_mode" value="$(arg is_sim)" />
<include file="$(find cyton_gamma_300_moveit_config)/launch/demo.launch"/> <include file="$(find rosbridge_server)/launch/rosbridge_websocket.launch" />
<node pkg="tf" type="static_transform_publisher" name="connect_start" args="0 0 0 0 0 0 $(arg fixed_frame) /start$(arg fixed_frame) 100" />
<node pkg="tf" type="static_transform_publisher" name="connect_goal" args="0 0 0 0 0 0 $(arg fixed_frame) /goal$(arg fixed_frame) 100" />
<!--node name="tf_web_republisher" pkg="py_tf2_web" type="tf_web_republisher.py" /-->
<node name="tf2_web_republisher" pkg="tf2_web_republisher" type="tf2_web_republisher" />
<node pkg="robot_state_publisher" type="robot_state_publisher" name="start_state_publisher_start">
<remap from="joint_states" to="start_joint_states" />
<param name="tf_prefix" type="string" value="start" />
</node>
<node pkg="robot_state_publisher" type="robot_state_publisher" name="goal_state_publisher_goal">
<remap from="joint_states" to="goal_joint_states" />
<param name="tf_prefix" type="string" value="goal" />
</node>
</launch>
Here are the info about the /move_group/result topic:
rostopic info /move_group/result Type: moveit_msgs/MoveGroupActionResult
Publishers: * /move_group (http://laresi-MS-7978:41113/)
Subscribers: * /moveit_publisher (http://laresi-MS-7978:38973/) * /rviz_laresi_MS_7978_6064_4036308910317907203 (http://laresi-MS-7978:43175/)
Can anyone, please, give me some advice or a clue? Thank you very much!
edit retag close merge delete
Sort by » oldest newest most voted
This error also occurs if one of your nodes refers to an older message definition than the other. Try cleaning and rebuilding your workspace ( catkin clean -y && catkin build ) and sourcing it in all the terminals ( source devel/setup.bash ). The latter step is important.
more
Thank you for your answer . After looking up into the moveit_msgs package, I found out that msgs definitions are different from the source. Actually, I have installed ros-melodic-moveit-msgs from the repos, the package installed in /opt/ros/melodic/... path does contain moveit_msgs/MoveGroupActionResult message, which is not the case if you look into the source code on GitHub https://github.com/ros-planning/movei... on Melodic branch. I was not able to install the package from source, because of some errors in building other dependencies. I wonder why the two versions are so different, eventhough they are both for Melodic?
( 2020-11-10 09:16:46 -0600 )edit
.action files are generated into RequestResult and Feedback messages, so the message you see is based on this action. It would look the same if you built it locally. You can also build moveit from source if this doesn't work.
( 2020-11-10 11:17:40 -0600 )edit
I have cleaned and rebuiled my workspace, and have builed moveit from source as showed in the tutorial. Now everything is working just fine! Thank you very much fvd
more
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# Graded global sections of Proj(S) for S a polynomial ring and more general
Throughout, suppose $$S$$ is a graded ring which is finitely generated by $$S_{1}$$ and an $$S_{0}$$-algebra. Let $$X = \text{Proj} S$$. There is the usual associated graded module given by $$\Gamma_{\bullet}(\mathcal{O}_{X}) = \bigoplus_{d \in \mathbb{Z}} \Gamma(X, \mathcal{O}_{X}(d)).$$ My question is about what this graded module is in various cases. It is well known that if $$S = A[x_{0}, x_{1}, \ldots , x_{r}]$$, then $$\Gamma_{\bullet}(\mathcal{O}_{X}) \simeq S$$. In particular, this is true even if $$A$$ is not an integral domain. A standard proof of this is given in Hartshorne Proposition 5.13.
I was studying the proof of this, and noticed that the crucial step in that proof is observing that $$\Gamma_{\bullet}(\mathcal{O}_{X})$$ can be identified with the intersection $$\bigcap S_{x_{i}} \subseteq S_{x_{0}x_{1} \cdots x_{r}}.$$ This observation, along with the inclusion in the above math environment, depends only on the $$x_{i}$$ not being zero divisors.
This leads me to think that this proof admits a significant generalisation.
Let $$S$$ be a graded ring with $$S_{0} = A$$ an arbitrary commutative ring with identity and suppose $$S$$ is finitely generated by $$S_{1}$$ as an $$A$$-algebra. Suppose further that a family of generators $$x_{0}, x_{1}, \ldots , x_{r}$$ can be chosen so that none of the $$x_{i}$$ are zero-divisors. Is it still true that $$\Gamma_{\bullet}(\mathcal{O}_{X}) \simeq \bigcap S_{x_{i}} \subseteq S_{x_{0}x_{1} \cdots x_{r}} ?$$
An immediate followup question would be to ask whether the above question is even well-posed. In particular, if I chose a different family of generators $$y_{0}, y_{1}, \ldots y_{q}$$, with possible algebraic dependencies, does the same fact remain? Of course one would hope so.
Finally, what is the most general setting in which this holds. In particular, what adjectives do I need to add to the statement "$$S$$ is a graded ring finitely generated by $$S_{1}$$ as an $$S_{0}$$-algebra" in order for $$\Gamma_{\bullet}(\mathcal{O}_{X}) \simeq \bigcap S_{x_{i}} \subseteq S_{x_{0}x_{1} \cdots x_{r}}$$ to hold?
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