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# Formal Connection Between Symmetry and Gauss's Law

In the standard undergraduate treatment of E&M, Gauss's Law is loosely stated as "the electric flux through a closed surface is proportional to the enclosed charge". Equivalently, in differential form, and in terms of the potential (in the static case):

$$\nabla^2 \phi = -\frac{\rho}{\epsilon_0}$$

Now, when using the integral form, one typically uses the symmetries of a known charge distribution to deduce related symmetries in the electric field, allowing the magnitude of the field to be factored out of the integral. To do so, one usually relies on intuitive, heuristic arguments about how the field in question "ought to" behave$$^1$$.

I'm wondering how one goes about formalizing this notion in precise mathematical terms. In particular, it seems that there ought to be an equivalent statement for Gauss's law in differential form, along the lines of "symmetries in $$\rho$$ induce related symmetries in $$\phi$$". Is there a way to formally state this claim? In particular:

1. How would one formulate a proof of the conditions (necessary and sufficient) under which a symmetry of $$\rho$$ induces a symmetry in $$\phi$$?
2. When it exists, how does one explicitly state the induced symmetry in terms of the known symmetry?
3. Can such a result be generalized for arbitrary linear PDEs whose source terms exhibit some symmetry?

It seems to me like there must exist a concise, elegant, and general way to state and prove the above, but I can't quite seem to connect all the dots right now.

$$^1$$ See, for example, in Griffiths, Example 2.3, p. 72: "Suppose, say, that it points due east, at the 'equator.' But the orientation of the equator is perfectly arbitrary—nothing is spinning here, so there is no natural "north-south" axis—any argument purporting to show that $$\mathbf{E}$$ points east could just as well be used to show it points west, or north, or any other direction. The only unique direction on a sphere is radial."

• I have a follow-on question... Broken symmetry, where the symmetry of a solution is less than the symmetry of the equation, is common in quantum field theory. Why doesn’t it happen in, say, classical electrostatics? – G. Smith Nov 15 '18 at 1:36
• If you really want to say something simple in a more formal way, maybe think about symmetries as operators acting on some space of functions. If the operator maps a function to itself, it is symmetric. If they operator commutes with the Laplacian (or whatever your diff eq happens to be) then your equation is symmetric. Then it's easy to see that if a function on one side of a symmetric equation is symmetric, the other must be. – octonion Nov 15 '18 at 2:01

Let $$\mathcal{D}$$ be the operator corresponding to your equation ($$-\epsilon_0 \nabla^2$$ in this case). Let $$U$$ be some operator corresponding to the symmetry. It might be a rotation or parity transformation, etc.

If $$U f = f$$, we say the function $$f$$ is symmetric.

If $$U\mathcal D=\mathcal D U$$ as operators, then we say your equation is symmetric.

Let, $$\mathcal {D} f= g$$. If $$\mathcal{D}$$ is symmetric and $$g$$ is symmetric, then we can easily show $$\mathcal{D}f = \mathcal{D}\mathcal U f$$. If we can take an inverse of $$\mathcal{D}$$ we've proven $$f$$ is symmetric.

Taking an inverse of $$\mathcal{D}$$ is the the same thing as being able to solve the equation uniquely. In your particular case we can solve the equation uniquely if we restrict our function space to have some boundary conditions, say vanishing at infinity.

So this is the example you had in mind and this a formalization of the argument that $$\phi$$ must be symmetric.

Now if we can't solve the equation uniquely then there may be a loophole in the argument. A particular case I have in mind is a magnetic monopole which is rotationally symmetric, but the vector potential solution has a Dirac string and is not. But any two solutions $$f$$ and $$Uf$$ in this case are connected by a gauge transformation.

• That loophole is (I think, if I understand correctly) the point that I'm stuck on. It's perfectly clear to me that, if your equation has some symmetry, then acting on some particular solution $f$ with the relevant symmetry operator produces another particular solution $Uf$, but the stronger requirement that these two solutions are, in fact, equal doesn't seem to follow in a straightforward way, if at all. I guess what I'm looking for is a more detailed treatment of how/when former implies the latter? – TheMac Nov 15 '18 at 3:31
• I don't think you were touching on the loophole in your question. In your particular example the argument that $\phi$ is symmetric is valid. I've edited my answer to be clearer (this is why I prefer to answer in the comments btw) – octonion Nov 15 '18 at 4:35


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Documentation

# kfoldPredict

Predict labels for observations not used for training

## Description

example

Label = kfoldPredict(CVMdl) returns class labels predicted by the cross-validated ECOC model composed of linear classification models CVMdl. That is, for every fold, kfoldPredict predicts class labels for observations that it holds out when it trains using all other observations. kfoldPredict applies the same data used create CVMdl (see fitcecoc).

Also, Label contains class labels for each regularization strength in the linear classification models that compose CVMdl.

example

Label = kfoldPredict(CVMdl,Name,Value) returns predicted class labels with additional options specified by one or more Name,Value pair arguments. For example, specify the posterior probability estimation method, decoding scheme, or verbosity level.

example

[Label,NegLoss,PBScore] = kfoldPredict(___) additionally returns, for held-out observations and each regularization strength:

• Negated values of the average binary loss per class (NegLoss).

• Positive-class scores (PBScore) for each binary learner.

example

[Label,NegLoss,PBScore,Posterior] = kfoldPredict(___) additionally returns posterior class probability estimates for held-out observations and for each regularization strength. To return posterior probabilities, the linear classification model learners must be logistic regression models.

## Input Arguments

expand all

Cross-validated, ECOC model composed of linear classification models, specified as a ClassificationPartitionedLinearECOC model object. You can create a ClassificationPartitionedLinearECOC model using fitcecoc and by:

1. Specifying any one of the cross-validation, name-value pair arguments, for example, CrossVal

2. Setting the name-value pair argument Learners to 'linear' or a linear classification model template returned by templateLinear

To obtain estimates, kfoldPredict applies the same data used to cross-validate the ECOC model (X and Y).

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Binary learner loss function, specified as the comma-separated pair consisting of 'BinaryLoss' and a built-in, loss-function name or function handle.

• This table contains names and descriptions of the built-in functions, where yj is a class label for a particular binary learner (in the set {-1,1,0}), sj is the score for observation j, and g(yj,sj) is the binary loss formula.

ValueDescriptionScore Domaing(yj,sj)
'binodeviance'Binomial deviance(–∞,∞)log[1 + exp(–2yjsj)]/[2log(2)]
'exponential'Exponential(–∞,∞)exp(–yjsj)/2
'hamming'Hamming[0,1] or (–∞,∞)[1 – sign(yjsj)]/2
'hinge'Hinge(–∞,∞)max(0,1 – yjsj)/2
'linear'Linear(–∞,∞)(1 – yjsj)/2
'logit'Logistic(–∞,∞)log[1 + exp(–yjsj)]/[2log(2)]

The software normalizes the binary losses such that the loss is 0.5 when yj = 0. Also, the software calculates the mean binary loss for each class.

• For a custom binary loss function, e.g., customFunction, specify its function handle 'BinaryLoss',@customFunction.

customFunction should have this form

bLoss = customFunction(M,s)
where:

• M is the K-by-L coding matrix stored in Mdl.CodingMatrix.

• s is the 1-by-L row vector of classification scores.

• bLoss is the classification loss. This scalar aggregates the binary losses for every learner in a particular class. For example, you can use the mean binary loss to aggregate the loss over the learners for each class.

• K is the number of classes.

• L is the number of binary learners.

For an example of passing a custom binary loss function, see Predict Test-Sample Labels of ECOC Model Using Custom Binary Loss Function.

By default, if all binary learners are linear classification models using:

• SVM, then BinaryLoss is 'hinge'

• Logistic regression, then BinaryLoss is 'quadratic'

Example: 'BinaryLoss','binodeviance'

Data Types: char | string | function_handle

Decoding scheme that aggregates the binary losses, specified as the comma-separated pair consisting of 'Decoding' and 'lossweighted' or 'lossbased'. For more information, see Binary Loss.

Example: 'Decoding','lossbased'

Number of random initial values for fitting posterior probabilities by Kullback-Leibler divergence minimization, specified as the comma-separated pair consisting of 'NumKLInitializations' and a nonnegative integer.

To use this option, you must:

• Return the fourth output argument (Posterior).

• The linear classification models that compose the ECOC models must use logistic regression learners (that is, CVMdl.Trained{1}.BinaryLearners{1}.Learner must be 'logistic').

• PosteriorMethod must be 'kl'.

For more details, see Posterior Estimation Using Kullback-Leibler Divergence.

Example: 'NumKLInitializations',5

Data Types: single | double

Estimation options, specified as the comma-separated pair consisting of 'Options' and a structure array returned by statset.

To invoke parallel computing:

• You need a Parallel Computing Toolbox™ license.

• Specify 'Options',statset('UseParallel',true).

Posterior probability estimation method, specified as the comma-separated pair consisting of 'PosteriorMethod' and 'kl' or 'qp'.

• To use this option, you must return the fourth output argument (Posterior) and the linear classification models that compose the ECOC models must use logistic regression learners (that is, CVMdl.Trained{1}.BinaryLearners{1}.Learner must be 'logistic').

• If PosteriorMethod is 'kl', then the software estimates multiclass posterior probabilities by minimizing the Kullback-Leibler divergence between the predicted and expected posterior probabilities returned by binary learners. For details, see Posterior Estimation Using Kullback-Leibler Divergence.

• If PosteriorMethod is 'qp', then the software estimates multiclass posterior probabilities by solving a least-squares problem using quadratic programming. You need an Optimization Toolbox™ license to use this option. For details, see Posterior Estimation Using Quadratic Programming.

Example: 'PosteriorMethod','qp'

Verbosity level, specified as the comma-separated pair consisting of 'Verbose' and 0 or 1. Verbose controls the number of diagnostic messages that the software displays in the Command Window.

If Verbose is 0, then the software does not display diagnostic messages. Otherwise, the software displays diagnostic messages.

Example: 'Verbose',1

Data Types: single | double

## Output Arguments

expand all

Cross-validated, predicted class labels, returned as a categorical or character array, logical or numeric matrix, or cell array of character vectors.

In most cases, Label is an n-by-L array of the same data type as the observed class labels (Y) used to create CVMdl. (The software treats string arrays as cell arrays of character vectors.) n is the number of observations in the predictor data (X) and L is the number of regularization strengths in the linear classification models that compose the cross-validated ECOC model. That is, Label(i,j) is the predicted class label for observation i using the ECOC model of linear classification models that has regularization strength CVMdl.Trained{1}.BinaryLearners{1}.Lambda(j).

If Y is a character array and L > 1, then Label is a cell array of class labels.

The software assigns the predicted label corresponding to the class with the largest, negated, average binary loss (NegLoss), or, equivalently, the smallest average binary loss.

Cross-validated, negated, average binary losses, returned as an n-by-K-by-L numeric matrix or array. K is the number of distinct classes in the training data and columns correspond to the classes in CVMdl.ClassNames. For n and L, see Label. NegLoss(i,k,j) is the negated, average binary loss for classifying observation i into class k using the linear classification model that has regularization strength CVMdl.Trained{1}.BinaryLoss{1}.Lambda(j).

Cross-validated, positive-class scores, returned as an n-by-B-by-L numeric array. B is the number of binary learners in the cross-validated ECOC model and columns correspond to the binary learners in CVMdl.Trained{1}.BinaryLearners. For n and L, see Label. PBScore(i,b,j) is the positive-class score of binary learner b for classifying observation i into its positive class, using the linear classification model that has regularization strength CVMdl.Trained{1}.BinaryLearners{1}.Lambda(j).

If the coding matrix varies across folds (that is, if the coding scheme is sparserandom or denserandom), then PBScore is empty ([]).

Cross-validated posterior class probabilities, returned as an n-by-K-by-L numeric array. For dimension definitions, see NegLoss. Posterior(i,k,j) is the posterior probability for classifying observation i into class k using the linear classification model that has regularization strength CVMdl.Trained{1}.BinaryLearners{1}.Lambda(j).

To return posterior probabilities, CVMdl.Trained{1}.BinaryLearner{1}.Learner must be 'logistic'.

## Examples

expand all

X is a sparse matrix of predictor data, and Y is a categorical vector of class labels.

Cross-validate an ECOC model of linear classification models.

rng(1); % For reproducibility
CVMdl = fitcecoc(X,Y,'Learner','linear','CrossVal','on');

CVMdl is a ClassificationPartitionedLinearECOC model. By default, the software implements 10-fold cross validation.

Predict labels for the observations that fitcecoc did not use in training the folds.

label = kfoldPredict(CVMdl);

Because there is one regularization strength in CVMdl, label is a column vector of predictions containing as many rows as observations in X.

Construct a confusion matrix.

cm = confusionchart(Y,label);

Load the NLP data set. Transpose the predictor data.

X = X';

For simplicity, use the label 'others' for all observations in Y that are not 'simulink', 'dsp', or 'comm'.

Create a linear classification model template that specifies optimizing the objective function using SpaRSA.

t = templateLinear('Solver','sparsa');

Cross-validate an ECOC model of linear classification models using 5-fold cross-validation. Specify that the predictor observations correspond to columns.

rng(1); % For reproducibility
CVMdl = fitcecoc(X,Y,'Learners',t,'KFold',5,'ObservationsIn','columns');
CMdl1 = CVMdl.Trained{1}
CMdl1 =
classreg.learning.classif.CompactClassificationECOC
ResponseName: 'Y'
ScoreTransform: 'none'
BinaryLearners: {6x1 cell}
CodingMatrix: [4x6 double]

Properties, Methods

CVMdl is a ClassificationPartitionedLinearECOC model. It contains the property Trained, which is a 5-by-1 cell array holding a CompactClassificationECOC models that the software trained using the training set of each fold.

By default, the linear classification models that compose the ECOC models use SVMs. SVM scores are signed distances from the observation to the decision boundary. Therefore, the domain is $\left(-\infty ,\infty \right)$. Create a custom binary loss function that:

• Maps the coding design matrix (M) and positive-class classification scores (s) for each learner to the binary loss for each observation

• Uses linear loss

• Aggregates the binary learner loss using the median.

You can create a separate function for the binary loss function, and then save it on the MATLAB® path. Or, you can specify an anonymous binary loss function.

customBL = @(M,s)nanmedian(1 - bsxfun(@times,M,s),2)/2;

Predict cross-validation labels and estimate the median binary loss per class. Print the median negative binary losses per class for a random set of 10 out-of-fold observations.

[label,NegLoss] = kfoldPredict(CVMdl,'BinaryLoss',customBL);

idx = randsample(numel(label),10);
table(Y(idx),label(idx),NegLoss(idx,1),NegLoss(idx,2),NegLoss(idx,3),...
NegLoss(idx,4),'VariableNames',[{'True'};{'Predicted'};...
categories(CVMdl.ClassNames)])
ans=10×6 table
True      Predicted      comm         dsp       simulink    others
________    _________    _________    ________    ________    _______

others      others         -1.2319     -1.0488    0.048758     1.6175
dsp         dsp            -0.7387    -0.11534    -0.88466    -0.2613
others      others         -0.1251     -0.8749    -0.99766    0.14517
dsp         dsp             2.5867      6.4187     -3.5867    -4.4165
others      others       -0.025358     -1.2287    -0.97464    0.19747
others      others         -2.6725    -0.56708    -0.51092     2.7453
others      others         -1.1605    -0.88321    -0.11679    0.43504
others      others         -1.9511     -1.3175     0.24735    0.95111
simulink    others          -7.848     -5.8203      4.8203     6.8457

The software predicts the label based on the maximum negated loss.

ECOC models composed of linear classification models return posterior probabilities for logistic regression learners only. This example requires the Parallel Computing Toolbox™ and the Optimization Toolbox™

Load the NLP data set and preprocess the data as in Specify Custom Binary Loss.

X = X';

Create a set of 5 logarithmically-spaced regularization strengths from through .

Lambda = logspace(-6,-0.5,5);

Create a linear classification model template that specifies optimizing the objective function using SpaRSA and to use logistic regression learners.

t = templateLinear('Solver','sparsa','Learner','logistic','Lambda',Lambda);

Cross-validate an ECOC model of linear classification models using 5-fold cross-validation. Specify that the predictor observations correspond to columns, and to use parallel computing.

rng(1); % For reproducibility
Options = statset('UseParallel',true);
CVMdl = fitcecoc(X,Y,'Learners',t,'KFold',5,'ObservationsIn','columns',...
'Options',Options);
Starting parallel pool (parpool) using the 'local' profile ...
connected to 6 workers.

Predict the cross-validated posterior class probabilities. Specify to use parallel computing and to estimate posterior probabilities using quadratic programming.

[label,~,~,Posterior] = kfoldPredict(CVMdl,'Options',Options,...
'PosteriorMethod','qp');
size(label)
label(3,4)
size(Posterior)
Posterior(3,:,4)
ans =

31572           5

ans =

categorical

others

ans =

31572           4           5

ans =

0.0293    0.0373    0.1738    0.7596

Because there are five regularization strengths:

• label is a 31572-by-5 categorical array. label(3,4) is the predicted, cross-validated label for observation 3 using the model trained with regularization strength Lambda(4).

• Posterior is a 31572-by-4-by-5 matrix. Posterior(3,:,4) is the vector of all estimated, posterior class probabilities for observation 3 using the model trained with regularization strength Lambda(4). The order of the second dimension corresponds to CVMdl.ClassNames. Display a random set of 10 posterior class probabilities.

Display a random sample of cross-validated labels and posterior probabilities for the model trained using Lambda(4).

idx = randsample(size(label,1),10);
table(Y(idx),label(idx,4),Posterior(idx,1,4),Posterior(idx,2,4),...
Posterior(idx,3,4),Posterior(idx,4,4),...
'VariableNames',[{'True'};{'Predicted'};categories(CVMdl.ClassNames)])
ans =

10×6 table

True      Predicted       comm          dsp        simulink     others
________    _________    __________    __________    ________    _________

others      others         0.030309      0.022454     0.10401      0.84323
dsp         others          0.15837       0.25784     0.18567      0.39811
others      others         0.093212      0.063752     0.12927      0.71376
dsp         dsp           0.0057401       0.89678    0.014939     0.082538
others      others         0.085715      0.054451    0.083765      0.77607
others      others        0.0061121     0.0057884     0.02409      0.96401
others      others         0.066741      0.074103       0.168      0.69115
others      others          0.05236      0.025631     0.13245      0.78956

expand all

## Algorithms

expand all

The software can estimate class posterior probabilities by minimizing the Kullback-Leibler divergence or by using quadratic programming. For the following descriptions of the posterior estimation algorithms, assume that:

• mkj is the element (k,j) of the coding design matrix M.

• I is the indicator function.

• ${\stackrel{^}{p}}_{k}$ is the class posterior probability estimate for class k of an observation, k = 1,...,K.

• rj is the positive-class posterior probability for binary learner j. That is, rj is the probability that binary learner j classifies an observation into the positive class, given the training data.

## References

[1] Allwein, E., R. Schapire, and Y. Singer. “Reducing multiclass to binary: A unifying approach for margin classiﬁers.” Journal of Machine Learning Research. Vol. 1, 2000, pp. 113–141.

[2] Dietterich, T., and G. Bakiri. “Solving Multiclass Learning Problems Via Error-Correcting Output Codes.” Journal of Artificial Intelligence Research. Vol. 2, 1995, pp. 263–286.

[3] Escalera, S., O. Pujol, and P. Radeva. “On the decoding process in ternary error-correcting output codes.” IEEE Transactions on Pattern Analysis and Machine Intelligence. Vol. 32, Issue 7, 2010, pp. 120–134.

[4] Escalera, S., O. Pujol, and P. Radeva. “Separability of ternary codes for sparse designs of error-correcting output codes.” Pattern Recogn. Vol. 30, Issue 3, 2009, pp. 285–297.

[5] Hastie, T., and R. Tibshirani. “Classification by Pairwise Coupling.” Annals of Statistics. Vol. 26, Issue 2, 1998, pp. 451–471.

[6] Wu, T. F., C. J. Lin, and R. Weng. “Probability Estimates for Multi-Class Classification by Pairwise Coupling.” Journal of Machine Learning Research. Vol. 5, 2004, pp. 975–1005.

[7] Zadrozny, B. “Reducing Multiclass to Binary by Coupling Probability Estimates.” NIPS 2001: Proceedings of Advances in Neural Information Processing Systems 14, 2001, pp. 1041–1048.

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# Scalars and determining subspaces

1. Mar 7, 2013

### MoreDrinks

To determine if a subset of a vector space is a subspace, it must be closed under addition and scalar multiplication. As far as I can tell, this means adding two arbitrary vectors in the subset and having the sum be within the subset.

But...can the scalar be any number? Is there any limitation?

2. Mar 7, 2013

### micromass

If $W$ is a subspace of $V$, then $\alpha v\in W$ for any scalar $\alpha$ and for any $w\in W$.

3. Mar 7, 2013

### MoreDrinks

Thanks, my book is terrible.


text:
This repository is a digital service that collects, preserves, and distributes digital material. Repositories are important tools for preserving an organization's legacy; they facilitate digital preservation and scholarly communication.

Select a community to browse its collections.

• #### Bilinear Calderón--Zygmund theory on product spaces

(Journal des Math\'ematiques Pures et Appliqu\'ees, 2019-10)
We develop a wide general theory of bilinear bi-parameter singular integrals $T$. This includes general Calder\'on--Zygmund type principles in the bilinear bi-parameter setting: easier bounds, like estimates in the Banach ...
• #### Quantitative weighted estimates for Rubio de Francia's Littlewood--Paley square function

(Journal of Geometric Analysis, 2019-12)
We consider the Rubio de Francia's Littlewood--Paley square function associated with an arbitrary family of intervals in $\mathbb{R}$ with finite overlapping. Quantitative weighted estimates are obtained for this operator. ...
• #### An optimal scaling to computationally tractable dimensionless models: Study of latex particles morphology formation

(Computer Physics Communications, 2020-02)
In modelling of chemical, physical or biological systems it may occur that the coefficients, multiplying various terms in the equation of interest, differ greatly in magnitude, if a particular system of units is used. Such ...
• #### Some geometric properties of Riemann’s non-differentiable function

(Comptes Rendus Mathematique, 2019-11-06)
Riemann’s non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory ...
• #### Computing Multipersistence by Means of Spectral Systems

(Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation, 2019-07-08)
In their original setting, both spectral sequences and persistent homology are algebraic topology tools defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set Z of integer ...


text:
My Math Forum compute eigenvalue

 Applied Math Applied Math Forum

 February 12th, 2012, 01:17 PM #1 Newbie   Joined: Jan 2012 Posts: 5 Thanks: 0 compute eigenvalue hi, Can someone please help me find the eigenvalues for this: $\ dy/dx=y(y-1)(y-2)\$ I think I got the equilibrium steady states right: y =0, y=1, y=2 Not sure how to use those in order to find the eigenvalues. I know that the formula to find the eigenvalue is $\ Av=\lambda v\$ Any help will be greatly appreciated. Thks
 February 12th, 2012, 02:58 PM #2 Senior Member   Joined: Oct 2011 From: Belgium Posts: 522 Thanks: 0 Re: compute eigenvalue To what does A correspond in this differential equation?

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Features International Sugar Journal

# Vacuum [Full subscriber]

A definition of “cold” or “coldness” is “a relative absence of heat” (Chambers, 1941). Basically the word cold indicates a lack of heat energy; it does not represent an entity but a relative absence of something real, namely energy.  Similarly “vacuum” represents a relative absence of pressure, a real force. Since “vacuum” is not an entity, a vacuum pump cannot “pump” it. A vacuum pump is a device that removes air and/or gases from a vessel thus reducing the pressure inside it.  A vacuum can therefore be considered a as an air pressure lower than atmospheric pressure.  The term “vacuum…


text:
# spreg.TSLS¶

class spreg.TSLS(y, x, yend, q, w=None, robust=None, gwk=None, sig2n_k=False, spat_diag=False, vm=False, name_y=None, name_x=None, name_yend=None, name_q=None, name_w=None, name_gwk=None, name_ds=None)[source]

Two stage least squares with results and diagnostics.

Parameters
yarray

nx1 array for dependent variable

xarray

Two dimensional array with n rows and one column for each independent (exogenous) variable, excluding the constant

yendarray

Two dimensional array with n rows and one column for each endogenous variable

qarray

Two dimensional array with n rows and one column for each external exogenous variable to use as instruments (note: this should not contain any variables from x)

wpysal W object

Spatial weights object (required if running spatial diagnostics)

robuststring

If ‘white’, then a White consistent estimator of the variance-covariance matrix is given. If ‘hac’, then a HAC consistent estimator of the variance-covariance matrix is given. Default set to None.

gwkpysal W object

Kernel spatial weights needed for HAC estimation. Note: matrix must have ones along the main diagonal.

sig2n_kboolean

If True, then use n-k to estimate sigma^2. If False, use n.

spat_diagboolean

If True, then compute Anselin-Kelejian test (requires w)

vmboolean

If True, include variance-covariance matrix in summary results

name_ystring

Name of dependent variable for use in output

name_xlist of strings

Names of independent variables for use in output

name_yendlist of strings

Names of endogenous variables for use in output

name_qlist of strings

Names of instruments for use in output

name_wstring

Name of weights matrix for use in output

name_gwkstring

Name of kernel weights matrix for use in output

name_dsstring

Name of dataset for use in output

Examples

We first need to import the needed modules, namely numpy to convert the data we read into arrays that spreg understands and pysal to perform all the analysis.

>>> import numpy as np
>>> import libpysal

Open data on Columbus neighborhood crime (49 areas) using libpysal.io.open(). This is the DBF associated with the Columbus shapefile. Note that libpysal.io.open() also reads data in CSV format; since the actual class requires data to be passed in as numpy arrays, the user can read their data in using any method.

>>> db = libpysal.io.open(libpysal.examples.get_path("columbus.dbf"),'r')

Extract the CRIME column (crime rates) from the DBF file and make it the dependent variable for the regression. Note that PySAL requires this to be an numpy array of shape (n, 1) as opposed to the also common shape of (n, ) that other packages accept.

>>> y = np.array(db.by_col("CRIME"))
>>> y = np.reshape(y, (49,1))

Extract INC (income) vector from the DBF to be used as independent variables in the regression. Note that PySAL requires this to be an nxj numpy array, where j is the number of independent variables (not including a constant). By default this model adds a vector of ones to the independent variables passed in, but this can be overridden by passing constant=False.

>>> X = []
>>> X.append(db.by_col("INC"))
>>> X = np.array(X).T

In this case we consider HOVAL (home value) is an endogenous regressor. We tell the model that this is so by passing it in a different parameter from the exogenous variables (x).

>>> yd = []
>>> yd.append(db.by_col("HOVAL"))
>>> yd = np.array(yd).T

Because we have endogenous variables, to obtain a correct estimate of the model, we need to instrument for HOVAL. We use DISCBD (distance to the CBD) for this and hence put it in the instruments parameter, ‘q’.

>>> q = []
>>> q.append(db.by_col("DISCBD"))
>>> q = np.array(q).T

We are all set with the preliminars, we are good to run the model. In this case, we will need the variables (exogenous and endogenous) and the instruments. If we want to have the names of the variables printed in the output summary, we will have to pass them in as well, although this is optional.

>>> from spreg import TSLS
>>> reg = TSLS(y, X, yd, q, name_x=['inc'], name_y='crime', name_yend=['hoval'], name_q=['discbd'], name_ds='columbus')
>>> print(reg.betas)
[[88.46579584]
[ 0.5200379 ]
[-1.58216593]]

Attributes
summarystring

Summary of regression results and diagnostics (note: use in conjunction with the print command)

betasarray

kx1 array of estimated coefficients

uarray

nx1 array of residuals

predyarray

nx1 array of predicted y values

ninteger

Number of observations

kinteger

Number of variables for which coefficients are estimated (including the constant)

kstarinteger

Number of endogenous variables.

yarray

nx1 array for dependent variable

xarray

Two dimensional array with n rows and one column for each independent (exogenous) variable, including the constant

yendarray

Two dimensional array with n rows and one column for each endogenous variable

qarray

Two dimensional array with n rows and one column for each external exogenous variable used as instruments

zarray

nxk array of variables (combination of x and yend)

harray

nxl array of instruments (combination of x and q)

robuststring

mean_yfloat

Mean of dependent variable

std_yfloat

Standard deviation of dependent variable

vmarray

Variance covariance matrix (kxk)

pr2float

Pseudo R squared (squared correlation between y and ypred)

utufloat

Sum of squared residuals

sig2float

Sigma squared used in computations

std_errarray

1xk array of standard errors of the betas

z_statlist of tuples

z statistic; each tuple contains the pair (statistic, p-value), where each is a float

ak_testtuple

Anselin-Kelejian test; tuple contains the pair (statistic, p-value)

name_ystring

Name of dependent variable for use in output

name_xlist of strings

Names of independent variables for use in output

name_yendlist of strings

Names of endogenous variables for use in output

name_zlist of strings

Names of exogenous and endogenous variables for use in output

name_qlist of strings

Names of external instruments

name_hlist of strings

Names of all instruments used in ouput

name_wstring

Name of weights matrix for use in output

name_gwkstring

Name of kernel weights matrix for use in output

name_dsstring

Name of dataset for use in output

titlestring

Name of the regression method used

sig2nfloat

Sigma squared (computed with n in the denominator)

sig2n_kfloat

Sigma squared (computed with n-k in the denominator)

hthfloat

$$H'H$$

hthifloat

$$(H'H)^{-1}$$

varbarray

$$(Z'H (H'H)^{-1} H'Z)^{-1}$$

zthhthiarray

$$Z'H(H'H)^{-1}$$

pfora1a2array

$$n(zthhthi)'varb$$

__init__(y, x, yend, q, w=None, robust=None, gwk=None, sig2n_k=False, spat_diag=False, vm=False, name_y=None, name_x=None, name_yend=None, name_q=None, name_w=None, name_gwk=None, name_ds=None)[source]

Initialize self. See help(type(self)) for accurate signature.

Methods

 __init__(y, x, yend, q[, w, robust, gwk, …]) Initialize self.

Attributes

property mean_y
property pfora1a2
property sig2n
property sig2n_k
property std_y
property utu
property vm


text:
These are expanded notes prepared for a talk in a learning seminar on Caraiani-Scholze's paper On the generic part of the cohomology of compact unitary Shimura varieties, Spring 2016 at Columbia. We summarize the major ingredients of the proof, explain the preservation of perversity under the Hodge-Tate period map and deduce the main theorems: 1) the existence of Galois representations associated to the torsion classes in betti cohomology of certain compact unitary Shimura varieties; 2) after localized at a maximal ideal of the Hecke algebra satisfying a genericity assumption, the -cohomology is concentrated in the middle degree and torsion-free.

## Torsion Galois representations

Recall our set-up. Let be a PEL datum of type A:

• is a finite dimensional simple -algebra with center a CM field .
• is positive involution of second kind on (so is totally real).
• is a -module.
• is an alternating form that is -Hermitian, i.e., .

The PEL datum models the first cohomology of a polarized abelian variety of dimension with endomorphism by . Let be the group of the automorphisms of (as a -module) that preserves up to a similitude factor. Let be the associated unitary Shimura variety, which is a moduli space of such abelian varieties with -level structure.

Assume we are in one of the following two extremal cases:

• is a division algebra and as a -module. In this case is an anisotropic unitary group (in variables) and has no endoscopy. is an example of Kottwitz's simple Shimura varieties (including those considered by Harris-Taylor).
• , (equivalently , ). Assume to ensure is compact. Assume that is quasi-split at all finite places. In this case has most endoscopy and causes most difficulty in the stable trace formula. Wei may like this case more ( appears in his arithmetic fundamental lemma).

Assume , , are unramified outside a finite set of primes (and in the second case). Let be the unramified Hecke algebra. Let be a maximal ideal. The first main result constructs an associated torsion Galois representation.

Theorem 1 Assume for some . Thee there is a semisimple Galois representation unramified outside associated to .

The proof requires three major ingredients:

(1) Sug Woo Shin has constructed Galois representations attached to the system of (characteristic 0) Hecke eigenvalues appearing in the cohomology of the Shimura varieties and Igusa varieties (in the above two extremal cases), by stable trace formula. In particular, there exists a Galois representation associated to system of Hecke eigenvalues in .

(2) The construction of the (Hecke equivalent) Hodge-Tate period map from the infinite level Shimura variety to the flag variety (for any Shimura varieties of Hodge type), whose fibers are related to the Igusa varieties (for any Shimura varieties of PEL type). In particular,the main result of Chap. 4 shows that for any geometric point , the fiber above ,

Remark 1 Recall that the locally closed subspaces of the adic space form a stratification and becomes the Newton stratification on the Shimura variety via pullback along . To define , one identifies with the -affine Grassmannian (using is minuscule) and notices that a -point of gives a -bundle on the Fargues-Fontaine curve by modifying the trivial bundle along , hence corresponds to an element by classification of -bundles on the Fargues-Fontaine curve. Since a Newton stratum of the Shimura variety is a product of the corresponding Igusa variety and the Rapoport-Zink space, passing to the infinite level then can be realized as its local analogue for the infinite level Rapoport-Zink space (after Scholze-Weinstein), hence the fiber becomes the Igusa variety.

(3) The perversity of (for any compact Shimura variety of PEL type), which has the following consequence:

Theorem 2 (Perversity) Suppose is minimal (i.e., d= is minimal, i.e., is maximal) such that for some . Then is concentrated in degree .

Using these 3 ingredients, now we can finish the proof of Theorem 1.

Proof Let be any prime. We have a Hochschild-Serre spectral sequence which computes the cohomology of the classical Shimura variety using the cohomology of the perfectoid Shimura variety . It follows that for some . On the other hand, the Leray spectral sequence for the Hodge-Tate period map gives Therefore Since everything is compatible with the Hecke action away from , using the Ingredient (2) we know that there exists some , such that for some . Pick minimal such , by the perversity (Ingredient (3)), we know that is concentrated in one degree. Therefore -coefficient cohomology is also concentrated in one degree and torsion-free. By Poincare duality (applied to the dual system of Hecke eigenvalues), this is also true for . Hence shows up in , where the Galois representation lifting exists by Ingredient (1). ¡õ
Remark 2 As one can see from the proof, (as expected) the Galois representation is not found in , but rather goes through the cohomology of the perfectoid Shimura variety and Igusa tower, which secretly constructs congruences between automorphic forms of different weights and levels at .

## Vanishing of torsion

Now let us come to the second main result, which asserts the "generic part" of the cohomology of our compact unitary Shimura varieties vanishes outside the middle degree.

Theorem 3 Assume for some . Assume there is a prime which splits completely in and is unramified and decomposed generic at all places of above . Then is concentrated in the middle degree . In particular, is concentrated in the middle degree and torsion-free.
Remark 3 The existence of torsion Galois representations and vanishing results of similar sort (for locally symmetric spaces of ) are useful for proving modularity lifting results for over general number fields (after Calegari-Geraghty), where numerical coincidence in the usual Taylor-Wiles method fails.
Remark 4 Recall that a local Galois representation is decomposed generic means that its Frobenius eigenvalues are distinct and for any . Any characteristic 0 lift is then a direct sum of characters such is not a cyclotomic character (since there is no nontrivial extension between 's by the Euler characteristic formula). Such a local Galois representation corresponds to a generic principal series representation of under local Langlands (hence its name).

Generic principal series are mapped to zero under the Jacquet-Langlands correspondence to any group that is not quasi-split. As we saw last time, using this one deduces that only contributes to the "most ordinary part" of the Igusa variety:

Theorem 4 (Genericity) If is not -ordinary, then .
Remark 5 Since the shape of depends on the signature of , the assumption in Theorem 3 can be relaxed according the the signature of in order to ensure that is not quasi-split unless is -ordinary. For example, when the signature is for all but one infinite place, one needs to require that is unramified and decomposed generic at only one place of . In any case, when has sufficiently large image, the assumption is always satisfied by Chebotarev's density.

Theorem 3 now follows easily from the genericity and the three main ingredients.

Proof Since is compact, by Poincare duality, it suffices to show that By the same argument as in proof of Theorem 1, it suffices to show that for any , Again take any minimal as in the proof of Theorem 1. By genericity, we know must be -ordinary, hence is -dimensional (notice the order-reversing!) and . By perversity, vanishes for . ¡õ

## The Hodge-Tate period map preserves perversity

Finally, let us explain the proof of the perversity result. Let be any compact Shimura variety of PEL type with hyperspecial level at .

To motivate, recall two useful results for perverse sheaves in algebraic geometry:

• Any simple perverse sheaf on a scheme of finite type is of the form for some locally closed subscheme and a local system on . In particular, is concentrated in one degree.
• Any finite (more generally, any small) map preserves perversity: if is a perverse sheaf on , then is a perverse sheaf on .

By the minimality, we know that has support on the union So it has support in a closed subset of dimension equal to . The result then would follow if is "perverse". Why should it be? The intuition is that is an affine and partially proper (i.e., satisfies valuative criterion in the category of adic spaces). If we were working with schemes, this would mean is affine and proper, hence finite, and finite morphisms preserve perversity. Of course all the beauty of lies in its very non finite-type behavior, so we cannot literally say this. On the other hand, because of Ingredient (2), we only need to show the sheaf is perverse when restricted on an affinoid etale neighborhood of . Then we can pass to the special fiber (by the perfectoidness), where indeed becomes a finite map between affine schemes of finite type over the residual field.

The other issue is that admits the action of and is infinite dimensional. So it can only be perverse (or just constructible) after taking -invariants for open compact. So the idea for proving the perversity Theorem 2 is then to pass to finite levels and special fibers. Let us be more precise.

Proof Scholze showed that there exists a basis of affinoid etale neighborhoods of such that its pullback under (denoted by ) is affinoid perfectoid. For each such , we have a formal model and correspondingly a formal model for . For any such , for a sufficiently small pro- open compact subgroup of , we have a continuous action of on , which induces the trivial on the special fiber .

We choose such sufficiently small for each so that 's shrink to 1. Then we know that the fiber at can be computed using the cohomology of the special fiber using the right upper corner of the following diagram, When shrinking , we obtain Here is the specialization of . Now we can compute using the left lower corner as well, namely, first specialize, then apply (which now becomes a finite map between schemes of finite type). Because the general fact that specialization (aka, nearby cycle) preserves perversity, we know that each individual term in the direct limit is concentrated in degree . Hence the direct limit itself is also concentrated in degree . ¡õ


text:
# Question #f025e

May 20, 2017

$1$ millimolar = $1 \times {10}^{- 3}$ molars

#### Explanation:

One molar contains 1,000 millimolars. You can convert molar to millimolar by multiplying the figure by 1,000. In the metric system, the prefix "milli-" represents 10 to the power of -3, or one one-thousandth.

So 1,000 millimolars = 1 molar. In scientific notation,

$1 \times {10}^{3}$ millimolars = 1 molar

So, multiplying both sides by ${10}^{- 3}$ gives

$1$ millimolar = $1 \times {10}^{- 3}$ molars


text:
# Finding the limit of $(1-3 \cdot x)^{\frac{1}{x}}$

I am trying to find $\lim_{x \rightarrow0} (1-3 \cdot x)^{\frac{1}{x}}$

I thought about finding the limit of

$$(1-3 \cdot x)^{\frac{1}{x}}= e^{\frac{\ln(1-3 \cdot x)}{x}}$$

But that only works if $e^{\frac{\ln(1-3 \cdot x)}{x}}$ is continuous at $x=0$, which as far as I understand it is not. Am I right about that? And if I am, how do I find the limit?

Thanks!

-
Continuity at $0$ is irrelevant. To use many arguments, we do want the function to be continuous in a deleted neighbourhood of $0$. So there should exist an $\epsilon$ such that the function is continuous at all $x$ such that $0<|x|<\epsilon$. – André Nicolas May 21 '12 at 14:02

Instead of working with $e^{\frac1x\ln(1-3x)}$, work with $\frac{\ln(1-3x)}x$ directly: the hypotheses of l’Hospital’s rule are satisfied, so you can use it to find $$\lim_{x\to 0}\frac{\ln(1-3x)}x=\lim_{x\to 0}\ln (1-3x)^{1/x}=\ln\lim_{x\to 0}(1-3x)^{1/x}\;,$$ and then just exponentiate to get the desired limit.

-

$\lim_{x \rightarrow0} (1-3 \cdot x)^{\frac{1}{x}}$ = $\lim_{x \rightarrow0} ((1+(-3) \cdot x)^{\frac{1}{-3x}})^{{-3}}$ = $e^{{-3}}$

-
Wait but that is only true if $e^{\frac{\ln(1-3x)}{x}}$ is continuous at $x=0$ isn't it? – yotamoo May 21 '12 at 10:06

Hint 1: What happens if you substitute $x=\frac 1n$? You will get a well known series.

Hint 2: One of the definitions of $e^a$ is given by $$e^a=\lim_{n\to\infty}(1+\frac an)^n.$$

-
We haven't really dealt with series's, even though it seems like it converges to 1 as $x \rightarrow 0$ – yotamoo May 21 '12 at 9:55
I added a second hint. Does that help? – Simon Markett May 21 '12 at 10:24

Following the ideas of Simon and Prasad: if you know that $\,\displaystyle{\lim_{x\to\infty}\left(1+\frac{a}{f(x)}\right)^{f(x)}=e^a}\,$ for any function $\,f(x)\,$ s.t. $\,\displaystyle{\lim_{x\to\infty} f(x)=\infty}\,$ , then you're done as $$\lim_{x\to 0}(1-3x)^{1/x}=\lim_{x\to 0}\left(1-\frac{3}{1/x}\right)^{1/x}=e^{-3}$$

-

As above, the limit is $e^{-3}$. Your function doesn't have to be defined at $x = 0$; you're only interested in the limit to which it tends as $x \to 0$.

My workings would look like this:

$$\begin{eqnarray*} \lim_{x\to 0} (1-3x)^{1/x} &=& \lim_{x\to 0} e^{1/x \ln (1-3x)} \\ &=& e^{\lim_{x\to 0}1/x\ln (1-3x)} \\ &=& e^{\lim_{x\to 0} \frac{\frac{-3}{1-3x}}{1}} \text{ by l'Hospital} \\ &=& e^{-3} \end{eqnarray*}$$

I'm not sure if I'm abusing notation here or if I'm relying on implicit assumptions (maybe the continuity of $e^{x}$ when I take the limit into the exponent?). But this is just how I would work this out for myself.

Hope it helps :-)

-


text:
12:33 AM
@Skillmon @PauloCereda I am TeXing fanatic, eh fantastic chemical stuff right now....
See a very tiny excerpt:
...
.......
...........
Does one actually say in English:
"XYZ became superfluous"?
In the meaning: "XYZ is no longer important."

2 hours later…
2:33 AM
@DavidCarlisle Thanks! I asked an actual question and got a good answer and explanation. Bit more complicated than I had thought.

3 hours later…
5:12 AM
@cis I would not use superfluous to mean no longer important. I would read "XYZ became superfluous" as "XYZ is no longer necessary or needed". Although it might help to know what XYZ is.

2 hours later…
6:50 AM
@DavidPurton yes my guess above of two catcode 12 end of line wasn't quite right it was two catcode 13 you needed, but same idea:-)

7:16 AM
Re-watching Frank's talks from tex.stackexchange.com/questions/602379/…

2 hours later…
9:31 AM
@cis I would say "ignore XYZ, there are no elements there" ;-).
...I start suspecting that there is an inverse proportional law between how much work you throw at an answer and how much upvotes it gets... ;-P
2

@Rmano definitely. Look at my top scoring answers.

@DavidCarlisle @Rmano or my top scoring question

@CarLaTeX Your question was "how do you make a pineapple pizza?" I assume?

9:46 AM
@Rmano ... and exponential relation between votes and how many colours were used in the screenshots :)

@DavidCarlisle no roasted ducks?

3 hours later…
1:07 PM
I propose the following shortcuts in the style of i18n and a11y: y1s, n0o, and for @PauloCereda: o1h. And I almost forgot, for @DavidCarlisle: l3h
2

@JosephWright version 16 tomorrow?
2

@yo' the general idea of shortcuts is that they are shorter than the thing they are cutting

@DavidCarlisle said an Emacs person?

@UlrikeFischer Almost certainly

2:06 PM
@UlrikeFischer Off to CTAN

@yo' o1h
@yo' LOL <3

2:39 PM
@PauloCereda I saved this one for you to answer, as I know you would love to mention the editor with the original and greatest read-eval-print-loop tex.stackexchange.com/q/602487

@DavidCarlisle what is "repl"?

@DavidCarlisle 'Read Evaluate Print' - an interactive session such as pdftex \relax at the Command Prompt

@JosephWright yes I did wonder if that's what the OP meant, but I'm not sure

@DavidCarlisle It's what REPL means ...
@DavidCarlisle Indeed: want something in the PDF, no idea what it is

@JosephWright not an interactive lisp session built in to your editor?
;; This buffer is for text that is not saved, and for Lisp evaluation.
;; To create a file, visit it with C-x C-f and enter text in its buffer.

(+ 1 2 3)
6

(cos pi)
-1.0

2:54 PM
@MarcelKrüger Nice X-ray plan - I'd call it \shelxstream myself ;) (shelx.uni-goettingen.de, older than TeX)

@JosephWright The name shelx sounds like one of these modern sh replacements

@JosephWright should we have got this in to the last -dev release github.com/latex3/latex2e/pull/342/files
UnIcode Caligraphic and Script, Finally:
@MarcelKrüger ^ something for luamml to play with:-)

3:17 PM
@DavidCarlisle For non unicode-math fonts the design already takes this into account and just needs appropriate lookup tables. The same could be done for unicode-math, but first unicode-math has to decide what to do with these.
@DavidCarlisle Do you know of any test fonts which actually use these?

@MarcelKrüger is it expected that the fonts actually implement this internally or do you have to see teh variation selector and then dig out the alternative glyph?

ooh more siunitx

@MarcelKrüger No I wonder if Murray will get it in to Cambria Math, He was pushing for this...

@DavidCarlisle In general, OpenType contains special cmap tables for variation selectors and these should be used inside of the font. For TeX in particular, we still need to see the variation selector and dig out the alternative glyph based on this since the default support for variation selectors does not apply in math mode. I have some code for this somewhere (TM).

@DavidCarlisle Ideally yes, perhaps for -dev 2

3:29 PM
@MarcelKrüger ah OK thanks I think that sounds vaguely familiar so I may have known that once:-) So in theory you could support this in luatex even if the fonts don't change (if they have the two styles as stylistic variants for example so the glyphs are "somewhere") ?
@JosephWright unless we get braver and rip out the whole of color.sty and make it a wrapper for l3color

@DavidCarlisle Might be a bit much ;)

@PauloCereda Ping

@DavidCarlisle Right, but I would expect this to be mostly a output concern, I would expect I to be pretty unlikely that people start using 𝒜 and 𝒜 as LaTeX input.

@JosephWright Petra: oh no, the siunitx bloke again!
@AlanMunn pong :)

@PauloCereda Of course, I just fixed a bug that came in after the upload

3:36 PM
@JosephWright Can we blame @UlrikeFischer? :)

@MarcelKrüger sure but you may get that in conversions from Word to LaTex, and also if it worked then logically \mathscr{A} and \mathcal{A} would be defined to produce the variant selector pairs wouldn't they?
@MarcelKrüger remind me to come back and look at the transcript here in a years time and see if those A look different:-)

@DavidCarlisle Mapping \mathscr{A} to the glyph described by the variation selector will probably work by a different mechanic than mapping literal input of the variation selector. (Maybe a separate family for changing the default style? Or mathcodes representing internal glyph codes?) Something for Will to think about...

3:54 PM
@DavidCarlisle Seriously, we need to work out if the change is safe: I worry about odd colour models with xcolor

@JosephWright yes

@DavidCarlisle I think I got most of them, but we might need to add a way to silence the error for cases where I've missed something

@JosephWright skype ping

@JosephWright the patches I use in pdfmanagement for xcolor seems to work, until now I hadn't much problems. But probably we really need to get in contact with Uwe Kern to sort that out properly.

@UlrikeFischer Indeed

4:01 PM
@DavidCarlisle I just tried to convince Herbert to take over some of the seminar patches in hyperref, and the side-effect is that we discovered that seminar is broken in a new latex as it redefines \document ;-(.

@UlrikeFischer shame as it's so much better than beamer:-)
@UlrikeFischer no one reported that?

@UlrikeFischer Oopsie

@DavidCarlisle Joseph could take it over as a replacement ..
@DavidCarlisle not to us, and Herbert didn't seem to know it either.
@DavidCarlisle it was reported: tex.stackexchange.com/q/570913/2388 ;-) and I seem to have notified Herbert too.

@UlrikeFischer so you knew all that time

@DavidCarlisle my former me seems to have known it ;-). I wonder, does hyperref have to contain support for a class that is broken in a current latex? Perhaps I could move them into a "old-hyperref-patches" package and people could load or copy them if they need them.

4:20 PM
@UlrikeFischer Or just send them to the seminar mainatainer with a suggestion that while fixing \document he rolls in the hyperref patches?

@DavidCarlisle well the question is if he wants to fix. Last year when I notified him about the document problem, he wrote that he no longer wants to touch seminar. And I guess the guy in the question wrote him too and nothing happened.

@UlrikeFischer check them in to seminar-issues-for-a-future-maintainer
7

@DavidCarlisle Patch itslef is simple and I think safe, it's the knock-on

4:42 PM
Meanwhile, I'm working out if there are good ideas in breqn to extract, or if it's a tarpit

@DavidCarlisle Herbert just wrote that he will set it to unmaintained and move to obsolet.

@UlrikeFischer Oh wwow

@DavidCarlisle -- I can understand the "reserved" locations in the math alphanumerica (those letters are in the "letterlike symbols" block), but can't figure what the ones in this block are reserved for. (Unicode isn't always scrutable.)

@barbarabeeton ? same thing isn't it?

4:51 PM
@JosephWright do you think it will have a large impact? I never used it.

@UlrikeFischer No, but it's a long-standing presentation package

@DavidCarlisle -- Oh. wrong. I will bug Murray again.

5:08 PM
@DavidCarlisle -- If variation selectors are to be used for calligraphic letters, the least that should be done is to document it in TR#25. (Bug Murray.)

@barbarabeeton The two reserved ones in Extended-F are waiting on part (c) of the original proposal (unicode.org/L2/L2020/20252r-mod-ipa-a.pdf)

@MarcelKrüger -- Thanks. It's nice to have clear documentation for such things.

5:25 PM
@barbarabeeton oh the latin block I mentioned earlier? I thought you meant the math script letters

EL KABONG IS BACK

@DavidCarlisle -- Nope. But @MarcelKrüger provided the information about those gaps so I'm happy there.

@barbarabeeton so now just need stix and cambria math etc to add the tables for the variant selectors....
@barbarabeeton but typical unicode half job the variant selectors don't apply to lowercase and don't apply to bold script

@DavidCarlisle -- But the official Unicode variants aren't documented yet that I can find. So even if Cambria and stix provide the glyphs, there's no Unicode backing.

@barbarabeeton they are in unicode 14 which has been delayed tp September but the draft charts are up already as are the UCD data files.
@barbarabeeton this was from the unicode chart:
2 hours ago, by David Carlisle

5:37 PM
@JosephWright recently, I saw that \__quark_if_empty_if:o still uses \ifx internally, instead of \if (was among the bunch of minor speed ups in l3tl). Is there any particular reason why \@@_if_empty_if:o is needed twice (ones inside l3quark once inside l3tl)? Couldn't we just do a \cs_set_eq:NN in whichever module is loaded later, or just use the same internal macro (maybe moving it to the \__kernel name space).

@DavidCarlisle -- Yes, you already posted that. It only says "these are similar". The text at the top of that section is nice, but there need to be at least two distinct codings. Among the examples I provided, there were at least two that showed distinctly different math usages in the same document for script vs. calligraphic. That requirement hasn't yet been satisfied.

@barbarabeeton It has hasn't it? the unmodified U+1D49C can be an A in either style but there are now two variant selectors to force caligraphic/chancery or script/roundhand

@DavidCarlisle -- It really needs to be stated which variation selector is used for which shape.

@barbarabeeton that is exactly what the text in the image above does FE00 is \mathcal and FE01 is \mathscr
@barbarabeeton unless you are in \boldmath in which case you are still lost.

@DavidCarlisle -- Yes, you're correct. (I just had my annual eye exam, and my eyes are still fuzzy.) Okay, thanks. I will ask Murray, politely, to add this to TR#25. (And at the same time mention bold math.)

5:46 PM
@barbarabeeton I'm just wondering how best to get this in to w3.org/2003/entities/2007doc/script.html

@DavidCarlisle -- Bonne chance.

1 hour later…
6:50 PM
How much disk space does a full TeX Live installation take on a Windows machine?

@DavidCarlisle oh no
@DavidCarlisle ooh
@LaTeXereXeTaL My local install takes 8GB on my Linux box, I'd wager something similar on Windows.

7:05 PM
@PauloCereda Thank you.

@LaTeXereXeTaL Much as @PauloCereda says if you install everything

7:29 PM
@JosephWright what's the fun in not installing everything? :)

@DavidCarlisle -- Do you remember seeing any evidence that the two variation selectors were added to the script letters in block 2100 (letterlike symbols)?

8:12 PM
@barbarabeeton unicode.org/L2/L2021/21009.htm, B.1 — 25: "Consensus: The UTC accepts 52 variation sequences to distinguish roundhand and chancery style mathematical script alphabetic characters, for Unicode version 14.0. (Reference: L2/20-275R)"

8:49 PM
@MarcelKrüger With a postfix variation selector…
@MarcelKrüger I hope that HarfBuzz is able to cope with it

9:05 PM
@MarcelKrüger -- Yippee! Well, that takes care of the lightface. I should still check with Murray about the boldface and TR#25. Thanks. (Who's your mole on the UTC?)

9:36 PM
@barbarabeeton sorry I was out they are listed in the charts in the 2100 block as well as in the plane 1 math alphabet block.


text:
# Constructing Riemann surfaces using the covering spaces

In the paper "On the dynamics of polynomial-like mappings" of Adrien Douady and John Hamal Hubbard, there is a way of constructing Riemann surfaces. I recite it as follow:

A polynomail-like map of degree d is a triple $(U,U',f)$ where $U$ and $U'$ are open subsets of $\mathbb{C}$ isomorphic to discs, with $U'$ relatively compact in $U$, and $f: U'\rightarrow U$ a $\mathbb{C}$-analytic mapping, proper of degree $d$. Let $L \subset U'$be a compact connect subset containing $f^{-1}\left(\overline{U'}\right)$ and the critical points of $f$, and such that $X_0=U-L$ is connected. Let $X_n$ be a covering space of $X_0$ of degree $d^n$, $\rho_n:X_{n+1}\rightarrow X_n$ and $\pi_n:X_n\rightarrow X_0$ be the projections and let $X$ be the disjoint union of the $X_n$. For each $n$ choose a lifting $$\widetilde{f}_n\colon \pi_n^{-1}(U'-L)\rightarrow X_{n+1},$$ of $f$. Then $T$ is the quotient of $X$ by the equivalence relation identifying $x$ to $\widetilde{f}_n(x)$ for all $x\in \pi_n^{-1}(U'-L)$ and all $n=0,1,2,\ldots$. The open set $T'$ is the union of the images of the $X_n, n=1,2,\ldots$, and $F:T'\rightarrow T$ is induced by the $\rho_n$.

Why $T$ is a Riemann surface and isomorphic to an annulus of finit modulus? Is there anything special about the $\pi_n,\rho_n$? What kind of background do I need?

-

I will give an informal answer. Your covering space is just a collection of holed spaces right? The equivalence relation just projects the holed space down onto a space isomorphic to $U'-L$. It does this by pasting the image and the preimage of $f$ together. (In an informal sense with each iteration the covering space gets bigger. To see this consider the map $z \mapsto z^2$ on the punctured disk (disk without the origin) with radius 1 from the origin. With 1 iteration you cover the disk twice. Applying it another time on the double covering of the disk you cover it four times etc. Quotienting by all these iterations you get the punctured disk.) Hence you get something isomorphic to a space with an annulus of finite modulus. The latter space is just a classical (Parabolic) Riemann surface (Google it!).


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# Separating Positive and Negative Data Examples by Concepts and Formulas: The Case of Restricted Signatures

Jung, Jean Christoph, Lutz, Carsten, Pulcini, Hadrien and Wolter, Frank
(2020) Separating Positive and Negative Data Examples by Concepts and Formulas: The Case of Restricted Signatures. CEUR Workshop Proceedings, 2663.

We study the separation of positive and negative data examples in terms of description logic (DL) concepts and formulas of decidable FO fragments, in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols from the data and ontology that can be used for separation. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated. Our main results are that (a projective form of) the weak version is decidable in $\mathcal{ALCI}$ while it is undecidable in the guarded fragment GF, the guarded negation fragment GNF, and the DL $\mathcal{ALCFIO}$, and that strong separability is decidable in $\mathcal{ALCI}$, GF, and GNF. We also provide (mostly tight) complexity bounds.


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Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

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May 25, 2017

The "rungs" of the DNA strand are made up of nitrogenous bases.

#### Explanation:

The "rungs" of the DNA strand are made up of nitrogenous bases which include

• $\textcolor{b l u e}{\text{Guanine (G)}}$

• $\textcolor{g r e e n}{\text{Cytosine (C)}}$

• $\textcolor{\pi n k}{\text{Adenine (A),}} \mathmr{and}$

• $\textcolor{\mathmr{and} a n \ge}{\text{Thymine (T)}}$

$\textcolor{w h i t e}{- - - - - - - -} \textcolor{b l u e}{\text{G}}$ pairs with $\textcolor{g r e e n}{\text{C}}$
$\textcolor{w h i t e}{- - - - - - - -} \textcolor{\pi n k}{\text{A}}$ pairs with $\textcolor{\mathmr{and} a n \ge}{\text{T}}$

By pairs I mean forms hydrogen bonds, which are a type of intermolecular force of attraction and one of the strongest of them all.

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CryptoDB

Paper: Key Exchange Protocols: Security Definition, Proof Method and Applications

Authors: Anupam Datta Ante Derek John C. Mitchell Bogdan Warinschi URL: http://eprint.iacr.org/2006/056 Search ePrint Search Google We develop a compositional method for proving cryptographically sound security properties of key exchange protocols, based on a symbolic logic that is interpreted over conventional runs of a protocol against a probabilistic polynomial-time attacker. Since key indistinguishability and other previous specifications of secure key exchange suffer from specific compositionality problems, we develop a suitable specification of acceptable key generation. This definition is based on a simple game played by an adversary against a key exchange protocol and a conventional challenger characterizing secure encryption (or other primitives of interest). The method is illustrated using a sample protocol.
BibTeX
@misc{eprint-2006-21549,
title={Key Exchange Protocols: Security Definition, Proof Method and Applications},
booktitle={IACR Eprint archive},
keywords={cryptographic protocols /},
url={http://eprint.iacr.org/2006/056},


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# How to Use SFH235 IR Photodiode Correctly?

I have an SFH235 ir photo-diode (which datasheet can be found here) and only basic knowledge of electronics.

I want to connect the photodiode to an Arduino (the grand-plan is to add an ir diode and build a distance sensor).

I understand that photodiode produce current, and it produces more current the more intensive the light that falls on it.

I have seen schematics that connect photodiodes the same way that photoresistors are being connected. Others connect it to an "opamp"(why?) And that are also those articles that say I need to amplify the current. I have also seen capacitors added into the circuit.

Which is the correct way for my application? How can I connect the photodiode to the analogue input of Arduino (which measures voltage)?

The simplest way to connect a photodiode (even an LED can be used in this way as a photosensor) is as below:

simulate this circuit – Schematic created using CircuitLab

Note that the photodiode is reverse biased through the 1 Meg resistor R1. The photocurrent generated by the diode opposes the flow of leakage current through this reverse biased diode, said opposition increasing with more light on the diode junction. The voltage at the junction point thus rises in voltage with increase in light.

The schematic on the left will have fairly low voltage at the ADC pin. To increase this voltage, an op amp can be used as a non-inverting amplifier, as shown in the schematic on the right. In the example shown, this gain is Gain = 1 + (R3/R4) = 9.9182.

This voltage gain results in a greater reading at the ADC pin, potentially making use of a greater part of the ADC's input voltage range.

Other applications of op-amps for photodiodes include:

• as a transimpedance amplifier, to directly amplify the photocurrent, without use of a biasing resistor
• as a voltage follower (unity gain buffer) for the voltage divider formed between diode and biasing resistor

Note:
One must ensure, though, that the voltage after amplification does not exceed the 5 Volt supply. If the op-amp is powered from the same 5 Volt supply as the Arduino, this is automatically taken care of, as the op-amp's output gets clipped at its upper supply rail, or even lower if the op-amp is not a rail-to-rail type.

• If I use the simple way (left schematic) how can I calculate the input voltage on the ADC? How do I choose a value of R1? What confuses me is that when I measured voltage, current and resistance of the photodiode, all three changed with higher ir light intensity, I was thinking that only the current value is a parameter I can use to represent ir intensity. – Artium Jun 24 '13 at 19:02
• @Artium You would need to actually measure the voltage using a high impedance method (such as a 10x, or better, 100x probe on an oscilloscope ). Similarly, just start with R1=1M, then try 470k and 2.2M, to see which works best. The act of measuring the voltage or current would change the value, since the photocurrent magnitude is really very small, and the leakage through a typical inexpensive multimeter probe would drain away a lot of those femtoAmperes. – Anindo Ghosh Jun 24 '13 at 19:43
• So for the implicit input rails on the op-amp in the circuit at right, would the positive rail be connected to 5V and the negative rail to ground? – imallett Jan 12 '18 at 8:13
• @imallett Yes, the implicit supply rails would be connected to +5v and ground, as you say. – Anindo Ghosh Jan 13 '18 at 9:02
• "The voltage at the junction point thus rises in voltage with increase in light." Using the circuit at left (with battery instead of Arduino; measuring voltage at junction point relative to ground with multimeter), I get exactly the opposite. – imallett Feb 8 '18 at 5:14

As Members have said above, the Voltage & Current from Photo-diodes can be very small, especially so with IR. The OSRAM SFH203PFA shown for Instance can make 0.25-0.35 under Ambient or Mains lighting, but without Bias makes only a 0.003 3 Millivolt wave hit by IR Laser on my Phillips Oscilloscope. Amplification using an Op Amp works for me.


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# Tutorial: Fractional-order PID Controller Design for a System with Input-output Delay

In this tutorial, we shall illustrate the use of the FOMCON toolbox in connection with the problem of designing a controller for a system with a significant input-output delay. For this purpose, a fractional-order first-order plus dead time (FO FOPDT) system is considered. We treat the continuous-time case.

## 1  Control design goal

Consider a system, that exhibits fractional-order dynamics, as well as a comparatively significant input-output delay, and is accurately described by a fractional-order transfer function

$$G(s) = \frac{6.902}{2.686s^{0.73}+1}\mathrm{e}^{-2.9s}. \tag{1}$$

We assume, that the actuator of this system saturates at $u=\pm 1$. Suppose also that the "ideal" transfer function (1) is not known. However, experimental data from a step experiment is available:

In what follows, we use the tools available in the FOMCON toolbox to identify the system and design a suitable controller by means of constrained optimization.

## 2  System identification

Our first task is to identify the system from the experimental dataset shown in Fig. 1. To identify the system, we may use the fractional-order transfer function time-domain identification tool (fotfid).

Figure 2. Fractional-order transfer function time-domain identification tool graphical user interface

From observation of the registered transient response, we can conclude, that the system may be described by a first-order model with the static gain being close to $7$. Additionally, an input-output delay is observed and is close to $3$ seconds. Thus, our initial guess model is

\begin{equation*} G(s)=\frac{7}{s+1}\mathrm{e}^{-3s} \end{equation*}

and may be entered into the tool text fields as shown in Fig. 2. With this initial model, the system is identified in several iterations, and after a simple transformation1 using the normalize() command the system becomes

$$\tilde{G}(s)=\frac{6.914}{2.5027s^{0.71187}+1}\mathrm{e}^{-3s}. \tag{2}$$

In the next section we shall use this model to obtain a fractional-order PID controller for the plant in (1).

## 3  Controller design

The first step in controller design is choosing a suitable controller structure. In this case, we are concerned with the design of a fractional-order PID controller. However, the plant to be controlled exhibits a significant input-output delay. Therefore, we consider using a Smith predictor [1]. The basic structure of the predictor is given in the next figure.

Figure 3. Basic structure of a Smith predictor in a control loop

The design of the controller $C(s)$ is carried out in two steps. First we formulate an integer-order PID controller, and then optimize the parameters of the obtained controller, including the orders of the integrator and differentiator, finally arriving at a fractional-order PID controller working in a Smith predictor based control scheme. Note, that we use the parallel form of the PID controller.

### 3.1  Conventional PID controller tuning

We now describe the procedure for tuning a classical, integer-order PID controller for a fractional-order plant in FOMCON. We can use the iopid_tune graphical tool to first approximate the fractional-order model by a conventional FOPDT model, and then apply classical tuning formulae to get the PID controller parameters.

Suppose, that the identified plant has a workspace name of Gp. Then, we can use the integer-order PID tuning tool and fill the data field of the Fractional plant model box as shown in the following figure.

Figure 4. Integer-order PID tuning tool graphical interface

The fractional-order model is thence identified:

$$G(s)=\frac{6.70913}{6.53847s+1}\mathrm{e}^{-1.57628s}. \tag{3}$$

The result of the approximation is illustrated on the following figure.

Figure 5. Classical FOPDT model approximation of the fractional-order model

Now we use the Ziegler-Nichols tuning formula to obtain PID controller parameters:

$$K_p=0.741921,K_i=0.235339,K_d=0.584738. \tag{4}$$

We shall use these parameters as the point of departure for our fractional PID controller design.

### 3.2  Fractional-order PID controller optimization

The design is carried out using the fpid_optim tool. Since a custom controller structure is used, we shall use Simulink for evaluating the characteristics of the transient response of our system. We set the parameters of the optimization as shown in Fig. 6.

Figure 6. FPID optimization tool graphical interface

You may notice, that we use an LTI system Gp1 instead of the aforementioned Gp, which is exactly the same system with the delay term removed, since it is compensated for by the predictor.

Now we need to build the Simulink model. We start by creating a new model fpid_optimize_Gp.mdl, which is a copy of the default model, and then clicking the Edit button. The model should open in a Simulink editor window as shown in the next figure.

Our task now is to modify this model and to add the predictor in Fig. 3. The identified model is added as part of the predictor, while the original, "ideal" model is added as the plant replacing the block in the inital model. With this setup it is also possible to evaluate the impact of the difference of the reference and original model on the control system. The resulting model is shown in Fig. 8.

We can now proceed to optimize the controller. After 100 iterations, the following controller is obtained:

$$C(s) = 2.0717 + \frac{0.10262}{s^{0.94962}} + 0.47143s^{0.89562}. \tag{5}$$

The results2 of the optimization are presented in Fig. 9. A clear improvement of the control system performance can be observed.

Figure 9. Fractional-order PID controller optimization results

## Conclusions

In this tutorial, we have illustrated the application of the FOMCON toolbox to the problem of designing a fractional-order controller for a process, exhibiting both fractional-order dynamics and a significant delay. The obtained results are considered satisfactory. However, in case of a real system the described procedure may need corrections, depending on particular performance specifications.

## References

 [1] N. Abe and K. Yamanaka, “Smith predictor control and internal model control – a tutorial,” in SICE 2003 Annual Conference, vol. 2, 2003, pp. 1383–1387.

• 1 This transformation is not strictly necessary
• 2 Your results may be slightly different

1. Pritesh
Posted June 26, 2013 at 14:51 | Permalink

Hi ALEKSEI,

I am trying to repeat this exercise. Up to point 2, I don’t have any issue.

Could you please share figure 8 model with needed LTI model?

I want to learn FPID for Input/Output System.

Thanking You,

• Posted June 27, 2013 at 13:07 | Permalink

Hi Pritesh,

Sorry; the model supplied in the figure description was the one without the predictor implementation. I have corrected that now.

• Pritesh
Posted June 28, 2013 at 06:21 | Permalink

Hi ALEKSEI,
In model given by you, there is block called Fractional PID Controller.
Inside that block, it is FPID LTI. In that there is LTI function fpid_optimize_fpid.
How should I create the same? Is it Fractional Transfer function of FPID?
I have created ft type of Fractional Controller. Error: The “LTI system variable” parameter of the LTI block must be set to a valid LTI model
Once again Thanks

• Posted June 28, 2013 at 07:29 | Permalink

This model should be used with the FPID Optimization tool (Figure 6). You may set the model in the Simulation parameters options box. The model must be present in the current workspace. Otherwise, if you just want to simulate the transient response, you may replace the controller and the plant with those described in the tutorial using blocks from the FOMCON Simulink library.

• Pritesh Shah
Posted June 29, 2013 at 10:40 | Permalink

Hi ALEKSEI,
I agree with your comment. FPID Optimization tool will give me fotf type of object. But in simulation, Fractional PID Controller block needs LTI Block.
How should I get LTI block from FPID Optimization tool ?

• Posted June 29, 2013 at 23:20 | Permalink

You may use the fractional PID contorller block from the FOMCON Simulink library.

2. vamsi krishna
Posted February 19, 2014 at 21:01 | Permalink

sir,after several iteration for initial guess model we are doing the normalize for fractional order transfer function ,but here it is showing same initial model????????.can u help me please.,.,.,.,.

• Posted February 20, 2014 at 10:35 | Permalink

Please provide a screenshot depicting this situation.

3. anitha
Posted May 31, 2014 at 07:45 | Permalink

sir,
wht is the function to open fomcon Simulink library…

4. Sandip
Posted June 3, 2014 at 06:18 | Permalink

Hi Aleksei,
In this tutorial why we use saturation block ? what is function of this block in this model ?

Thanking You

• Posted June 3, 2014 at 07:41 | Permalink

The block models actuator saturation. This is a purely physical phenomenon caused by limitations of what the actuator can actually produce (for example, a pump with max. displacement capacity of 100L/min cannot pump 200L/min).

5. manohar
Posted June 6, 2014 at 05:35 | Permalink

sir,
Can this tool be used for an integer order plant transfer function..should the plant transfer function changed into an fractional order ..or else is there any way to change it from the toolbox.please suggest..
Thanking you

• Posted June 6, 2014 at 12:11 | Permalink

Yes, the tool supports conventional (i.e. integer-order systems) as well. You can also convert tf objects into fotf objects, since the former is technically a subclass of the latter.

6. manohar
Posted June 9, 2014 at 08:33 | Permalink

sir,
can you just give me an example of using this toolkit with an integer order plant transfer function…I will be very thankful as I tried a lot…
thanking you

Posted June 23, 2014 at 15:48 | Permalink

i dont have fopid matlab block.what can i do?

• Posted July 8, 2014 at 19:00 | Permalink

I do not understand what the problem is.

8. priyanka
Posted August 7, 2014 at 09:53 | Permalink

Sir,
how would i get the FOMCON simulink library? Is it present in MATLAB software or we have access it seperately? Please help me ?

• Posted August 8, 2014 at 12:15 | Permalink

It is included with the toolbox distribution which you can obtain from this website.

9. Priyanka
Posted August 13, 2014 at 10:15 | Permalink

I download the FOMCON toolbox and follow the path as you suggest on your site to add toolbox in MATLAB, but i could not open the toolbox in MATLAB. what should i do?

• Posted August 14, 2014 at 10:09 | Permalink

Could you elaborate? What exactly does “could not open the toolbox in MATLAB” mean?

10. suresh kumar
Posted August 19, 2014 at 16:34 | Permalink

how to apply astrom hagguland method tofind kd value at particular phasemargin where as kp,ki fromknown zigler nicholas method and how to tune fractional order pid fromthese two methods because i am doing a project on tuning and analysis of fractional order pid please reply as early as possible

11. Moupali Chakraborty
Posted August 20, 2014 at 19:55 | Permalink

Hi sir,
I have downloaded this toolbox.This is an awesome toolbox.It gives wonderful control response of the plant.Thank you very much.

12. Posted August 27, 2014 at 08:22 | Permalink

how to find lamda and mu

• Posted August 27, 2014 at 10:32 | Permalink

In this case they are obtained by means of optimization.


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Home / Information / Mathematics – 3

# Mathematics – 3

AIMS
• To make understand the basic concept and techniques of composition and resolution of vectors and computing the resultant of vectors.
• To enable to use the knowledge of gradient of a straight line in finding speed, acceleration etc.
• To enable to use the knowledge of conic in finding the girder of a railway bridge, cable of a suspension bridge and maximum height of an arch.
• To provide ability to apply the knowledge of differential calculus in solving problem like slope, gradient of a curve, velocity, acceleration, rate of flow of liquid etc.
• To enable to apply the process of integration in solving practical problems like calculation of area of a regular figure in two dimensions and volume of regular solids of different shapes.

SHORT DESCRIPTION
Vector : Addition and subtraction, dot and cross product.
Co-ordinate Geometry : Co-ordinates of a point, locus and its equation, straight lines, circles and conic.
Differential Calculus : Function and limit of a function, differentiation with the help of limit, differentiation of functions, geometrical interpretation of dydx , successive differentiation and Leibnitz theorem, partial differentiation.
Integral Calculus : Fundamental integrals, integration by substitutions, integration by parts, integration by partial fraction, definite integrals.
DETAIL DESCRIPTION
Vector
1 Apply the theorems of vector algebra.
1.1 Define scalar and vector.
1.2 Explain null vector, free vector, like vector, equal vector, collinear vector, unit vector, position vector, addition and subtraction of vectors, linear combination, direction cosines and direction ratios, dependent and independent vectors, scalar fields and vector field.
1.3 Prove the laws of vector algebra.
1.4 Resolve a vector in space along three mutually perpendicular directions
1.5 solve problems involving addition and subtraction of vectors.

2 Apply the concept of dot product and cross product of vectors.
2.1 Define dot product and cross product of vectors.
2.2 Interpret dot product and cross product of vector geometrically.
2.3 Deduce the condition of parallelism and perpendicularity of two vectors.
2.4 Prove the distributive law of dot product and cross product of vector.
2.5 Explain the scalar triple product and vector triple product.
2.6 Solve problems involving dot product and cross product.

CO-ORDINATE GEOMETRY

3 Apply the concept of co-ordinates to find lengths and areas.
3.1 Explain the co-ordinates of a point.
3.2 State different types of co-ordinates of a point.
3.3 Find the distance between two points (x1, y1) and (x2, y2 ).
3.4 Find the co-ordinates of a point which divides the straight line joining two points in certain ratio.
3.5 Find the area of a triangle whose vertices are given.
3.6 Solve problems related to co-ordinates of points and distance formula.

4 Apply the concept of locus.
4.1 Define locus of a point.
4.2 Find the locus of a point.
4.3 Solve problems for finding locus of a point under certain conditions.

5 Apply the equation of straight lines in calculating various parameter.
5.1 Describe the equation x=a and y=b and slope of a straight line.
5.2 Find the slope of a straight line passing through two point (x1, y1,) and (x2, y2 ).
5.3 Find the equation of straight lines:
i) Point slope form.
ii) Slope intercept form.
iii) Two points form.
iv) Intercept form.
v) Perpendicular form.
5.4 Find the point of intersection of two given straight lines.
5.5 Find the angle between two given straight lines.
5.6 Find the condition of parallelism and perpendicularity of two given straight lines.
5.7 Find the distances of a point from a line.

6 Apply the equations of circle, tangent and normal in solving problems.
6.1 Define circle, center and radius .
6.2 Find the equation of a circle in the form:
i) x2 + y2 =a 2
ii) (x  h) 2 + (y  k) 2 =a 2
iii) x 2 + y 2 + 2gx + 2fy + c=0
6.3 Find the equation of a circle described on the line joining (x1, y1) and (x2, y2).
6.4 Define tangent and normal.
6.5 Find the condition that a straight line may touch a circle.
6.6 Find the equations of tangent and normal to a circle at any point.
6.7 Solve the problems related to equations of circle, tangent and normal.
7. Understand conic or conic sections.
7.1 Define conic, focus, directrix and eccentricity.
7.2 Find the equations of parabola, ellipse and hyperbola.
7.3 Solve problems related to parabola, ellipse and hyperbola.

DIFFERENTIAL CALCULUS
FUNCTION AND LIMIT
8. Understand the concept of functions and limits.
8.1 Define constant, variable, function, domain, range and continuity of a function.
8.2 Define limit of a function
8.3 Distinguish between f(x) and f(a).

8.4 Establish i) lim sinxx =1
x

. ii) lim tanxx =1.
x

9. Understand differential co-efficient and differentiation.
9.1 Define differential co-efficient in the form of
dydx = lim f(x+h)-f(x)h
h
9.2 Find the differential co-efficient of algebraic and trigonometrical functions from first principle.

10. Apply the concept of differentiation.
10.1 State the formulae for differentiation:
i) sum or difference
ii) product
iii) quotient
iv) function of function
v) logarithmic function
Find the differential co-efficient using the sum or difference
formula, product formula and quotient formula.
10.2 Find the differential co-efficient function of function and logarithmic function.

11. Apply the concept of geometrical meaning of dydx
11.1 Interpret dydx geometrically.
11.2 Explain dydx under different conditions
11.3 Solve the problems of the type:
A circular plate of metal expands by heat so that its radius increases at the rate of 0.01 cm per second. At what rate is the area increasing when the radius is 700 cm ?
12 Use Leibnitz’s theorem to solve the problems of successive differentiation.
12.1 Find 2nd, 3rd and 4th derivatives of a function and hence find n-th derivatives.
12.2 Express Leibnitz’s theorem
12.3 Solve the problems of successive differentiation and Leibnitz’s theorem.

13 Understand partial differentiation.
13.1 Define partial derivatives.
13.2 State formula for total differential.
13.3 State formulae for partial differentiation of implicit function and homogenous function.
13.4 State Euler’s theorem on homogeneous function.
13.5 Solve the problems of partial derivatives.

INTEGRAL CALCULUS
14 Apply fundamental indefinite integrals in solving problems.
14.1 Explain the concept of integration and constant of integration.
14.2 State fundamental and standard integrals.
14.3 Write down formulae for:
i) Integration of algebraic sum.
ii) Integration of the product of a constant and a function.
14.4 Integrate by method of substitution, integrate by parts and by partial fractions.
14.5 Solve problems of indefinite integration.

15 Apply the concept of definite integrals.
15.1 Explain definite integration.
15.2 Interpret geometrically the meaning of
15.3 Solve problems of the following types:
i) ii)

P* =Practical continuous assessment

## Robi 01886 Code Matching Number

Robi users or Non-Robi users can now get Robi Matching Number from any Retailer (All …


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## Here is why I don’t write about what I am actually thinking

The reason that I frequently don’t post is that I go through extensive periods studying for the actuarial examinations.  When I am intensely studying, much of everything else is ejected from my life.

I will write about what I study each day, but it may numb your brain.

I love actuarial mathematics, but the concepts, mathematics, and formulas are really ugly at first.  Normally, I appreciate nice little beautiful mathematical statements that express deep truths.  Many actuarial formulas, however, are daunting.  They might have a dozen variables, each with its own complex definition.  They relate to finance, which seems so far removed from the beauty of pure mathematics, and far from the math that relates to the natural world.

But, the concepts that I learned for my last exam are now my intimate friends.  What were once jumbles of symbols on the page are now complex expressions of important truths.

Take this ugly beast, for the price of a call option:

$\Delta =e^{-\delta h}\frac{C_u-C_d}{S(u-d)}$

$B = e^{-rh}\frac{uC_d-dC_u}{u-d}$

The price of the call option is $\Delta S +B$.

I started at the far end, the most complicated spot.  Now let me back up and explain the essentials.

When we trade on the stock market, we do not always trade the actual stocks, commodities, or currencies.  We may be trading derivatives of these assets.  That is, we may be trading financial instruments whose value is derived from these underlying assets.

In concrete terms, we might engage in a contract to buy an ounce of gold for $1000 in one year. This is a forward contract. We might buy an option to buy one share of Ankle Compuners in one year for$1100.  If Ankle is selling for say $1500 at that point, we exercise our option, buy the stock for$1100, then immediately sell it for $1500, instantly making$400.  If Ankle is selling for less than $1100, we do not exercise the option. We have lost whatever we paid for the option. This is known as a call option. We might buy an option to sell one share of Nash Motors in one year for$500.  If Nash is selling for say, $100 in one year, we exercise our option and sell the stock and have$500, instead of the $100 that Nash is worth. Why do we use derivatives, instead of the actual asset? To manage risk, like we did with our Nash stock. This is like insurance. To speculate on the market. We can buy an interest in the performance of Ankle, without having to spend$1000 for an entire share.  We might also be buying derivatives because of reduced transaction costs, or because they can help us to avoid the tax implications of owning the asset.

Here is our question.  Given the value of a stock, S, the risk-free bond rate r, the dividend rate, delta, and the volatility of the stock price, how do we find the price of the call option to buy Ankle Compuner in our first example?

We begin with the law of one price.  Two instruments with the same cash flows should have the same price.  We know how to calculate the price and payoff of stocks and bonds.  So if we can create a portfolio of stocks and bonds with the same payoff as the call option, they must have the same price.

In one year, if Ankle Compuner is selling for more than 1100 (we call this the exercise price, K), the payoff is the price at expiration (St)-K.  So if it is selling at 1300, the payoff is 1300-1100=200.

In one year, if Ankle Compuner is selling less than 1100, the payoff is 0.

The formulas above use binomial option pricing to find the price of this portfolio, and hence a theoretically correct price for this call option.  Theory is important here:  an incorrectly priced option will lead to an arbitrage opportunity.  In short, the options may be bought and sold instantly (perhaps within billionths of seconds …) for a profit.

I can’t explain any more here without getting in real deep.

While I am studying on this stuff, I am writing C++ and VBA programs, as a way to learn some computer languages that are in demand for actuaries.  My “native” language is LISP.  Other languages seem really ugly.  When you are first learning a language, the code ends up pretty ugly anyways.

So, here is my code:

#include <cstdlib>
#include <iostream>
#include <cmath>
#include <iomanip>

//Calculates price of call option, and portfolio required to duplicate option, using binomial tree

int main() {
using namespace std;
float S0, r,div, K, h, sigma, u, d, uS,dS,Cu,Cd,delta, bondValue, callPrice;
//get data
cout<<“Calculate the price of a call, using a one period binary tree”<<endl;
cout<<“Current Stock Price?”;
cin>>S0;
cout<<“Continuously Compounded Risk Free Rate?”;
cin>>r;
cout<<“Continuous Dividend Rate?”;
cin>>div;
cout<<“Strike Price?”;
cin>>K;
cout<<“Time to Expiration?”;
cin>>h;
cout<<“Volatility?”;
cin>>sigma;
//calculate likely upward and downward motions
u=exp((r-div)*h+sigma*sqrt(h));
d=exp((r-div)*h-sigma*sqrt(h));
uS=u*S0;
dS=d*S0;
cout<<“Upward Move “<<uS<<endl;
cout<<“Downward Move “<<dS<<endl;
//calculate payoff of call, given motions above
if (uS > K)
Cu = uS-K;
else Cu = 0;
if (dS > K)
Cd = dS-K;
else Cd = 0;
//calculate Delta, Beta, and call option price
delta=exp(-div * h)*(Cu-Cd)/(S0*(u-d));
bondValue=exp(-r * h)*((u*Cd)-(d*Cu))/(u-d);
callPrice=delta*S0+bondValue;
cout<<“To duplicate the call, buy “<<delta<<” shares of the stock for “<<delta*S0<<” dollars and borrow “<<-1 * bondValue<<” at the rate of “<<r<<” for a total price of “<<callPrice<<endl;
cout<<“Price of Call “<<callPrice<<endl;

//find the price of some other related calls
cout<<“Some other Call prices: “<<endl;
float step;
step = 0.05*K;
K= K-5*step;
for (int z=0;z<11;z++){
if (uS > K)
Cu = uS-K;
else Cu = 0;
if (dS > K)
Cd = dS-K;
else Cd = 0;

delta=exp(-div * h)*(Cu-Cd)/(S0*(u-d));
bondValue=exp(-r * h)*((u*Cd)-(d*Cu))/(u-d);
callPrice=delta*S0+bondValue;

cout<<setw(8)<<k<<” “<<callprice<<<span=”” class=”hiddenSpellError” pre=”” data-mce-bogus=”1″>endl;
K=K+step;
}
return 0;
}
//Wow what an ugly program

Here is a sample run:

If we buy 0.5483 shares of Ankle, and borrow 496.15 dollars at the continuously compounded risk-free rate of 10% (our current artificially induced nearly 0% interest rates are way uninteresting for examples), for a total cost of $52.18, we will duplicate the payoff for our call. So our call option should be$52.18.

Let’s check the math.

If we buy the one year $1100 call, and the stock price is 1221.40 in one year, our payoff will be 121.40. If the price is 1000, our payoff is 0. Suppose that we buy our$52.18 portfolio.

In one year, if the stock price is 1221.40, our 0.5483 share is worth $669.69. We borrowed 496.15 at 10% interest. In one year we must pay back $496.14e^{0.1} = \ 548.16.$ We sell the stock, and pay back the loan, for a payoff of$121.53.  Which is close enough due to all of the rounding that I did.

Now you see why I don’t write about what I think about all day.

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